gait analysis research paper

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• Published: 02 March 2006

Gait analysis methods in rehabilitation

• Richard Baker 1 , 2 , 3 , 4  

Journal of NeuroEngineering and Rehabilitation volume  3 , Article number:  4 ( 2006 ) Cite this article

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Introduction

Brand's four reasons for clinical tests and his analysis of the characteristics of valid biomechanical tests for use in orthopaedics are taken as a basis for determining what methodologies are required for gait analysis in a clinical rehabilitation context.

Measurement methods in clinical gait analysis

The state of the art of optical systems capable of measuring the positions of retro-reflective markers placed on the skin is sufficiently advanced that they are probably no longer a significant source of error in clinical gait analysis. Determining the anthropometry of the subject and compensating for soft tissue movement in relation to the under-lying bones are now the principal problems. Techniques for using functional tests to determine joint centres and axes of rotation are starting to be used successfully. Probably the last great challenge for optical systems is in using computational techniques to compensate for soft tissue measurements. In the long term future it is possible that direct imaging of bones and joints in three dimensions (using MRI or fluoroscopy) may replace marker based systems.

Methods for interpreting gait analysis data

There is still not an accepted general theory of why we walk the way we do. In the absence of this, many explanations of walking address the mechanisms by which specific movements are achieved by particular muscles. A whole new methodology is developing to determine the functions of individual muscles. This needs further development and validation. A particular requirement is for subject specific models incorporating 3-dimensional imaging data of the musculo-skeletal anatomy with kinematic and kinetic data.

Methods for understanding the effects of intervention

Clinical gait analysis is extremely limited if it does not allow clinicians to choose between alternative possible interventions or to predict outcomes. This can be achieved either by rigorously planned clinical trials or using theoretical models. The evidence base is generally poor partly because of the limited number of prospective clinical trials that have been completed and more such studies are essential. Very recent work has started to show the potential of using models of the mechanisms by which people with pathology walk in order to simulate different potential interventions. The development of these models offers considerable promise for new clinical applications of gait analysis.

For the purposes of this paper gait analysis will be assumed to refer to the instrumented measurement of the movement patterns that make up walking and the associated interpretation of these. The core of most contemporary gait analysis is the measurement of joint kinematics and kinetics. Other measurements regularly made are electromyography (EMG), oxygen consumption and foot pressures. A systematic physical examination of the patient is usually conducted as part of a gait analysis.

Rehabilitation is a clinical discipline and this paper will thus concentrate on clinical gait analysis. Richard Brand [ 1 , 2 ] proposed four reasons for performing any clinical test (see Table 1 ). The third of these might actually be taken as a definition of the word clinical i.e. a clinical test is one conducted in order to select from among different management options for a patient (including the possibility of not intervening).

Much contemporary gait analysis is done for the purpose of clinical research . This differs from clinical testing in that the reason is not to make clinical decisions for the individual patient, but to learn about a condition affecting a group of patients or the effect of an intervention. It is important to remember that the criteria for valid clinical research may not be the same as those for valid clinical testing. For example if a measurement made on a patient cannot be relied upon because of random errors then that measurement will not be useful for clinical purposes. By increasing the number of patients in a sample however, even measurements with quite large random errors can result in meaningful conclusions in clinical research. This paper will focus on gait analysis for clinical use. It will also focus on methodology rather than areas of clinical application.

Brand's [ 1 , 2 ] other three possible reasons for performing any clinical test are to distinguish between disease entities (diagnosis), to determine the severity, extent or nature of a disease or injury (assessment), and to predict outcomes of intervention (or the absence of intervention). The monitoring of the progress of a patient's condition either following intervention or in its absence might be regarded as an additional reason. This modification of Brand's approach is summarised in Table 2 .

Brand went on to propose a number of criteria for assessing the usefulness of biomechanical measurements in general which, with some modification, can be used as criteria for the usefulness of all clinical gait analysis. These are listed in Table 3 . The first requirement of any clinical measurement is that it should characterise the patient, that is if the patient attends on two separate occasions, between which his or her condition might be considered as stable, the measurements taken should be similar. This requires that the measurement technique itself is repeatable but also that the quantity being measured is stable and independent of factors such as mood, motivation or pain. Measurements can be repeatable and stable without necessarily being accurate (representative of a specific physical quantity). Such tests can be clinically useful but will be much easier to interpret if they are also accurate. In an era of evidence based clinical practice it is essential that any measurement techniques are appropriately validated which must include assessments of both their repeatability and accuracy.

In order to perform a diagnostic function it is necessary for measurements to be able to distinguish normal from abnormal patterns of movement and also between the characteristics of one disease entity and another. There are two aspects to this. The first is having measurement systems capable of working to adequate precision. The second is a knowledge of what characterises normal walking or a particular disease entity.

The requirement for patient assessment pre-supposes that a diagnosis does not give sufficient information to determine the most appropriate management for a patient and that measuring the precise characteristics of a patient's condition are essential for this. Measurements thus have to be sufficiently precise to reveal clinically important differences between patients with the same diagnosis. For monitoring purposes measurements need to be sufficiently precise to be able to determine whether a patient's condition is stable, improving or deteriorating.

Brand suggested that the measurement technique should not affect the function it is measuring. The walking performed in a gait analysis laboratory however, with the patient concentrating on what they are doing in an idealised environment, is not necessarily representative of their normal walking. At the very least this must be taken into account when interpreting results.

Gait analysis should reveal information that is useful to the clinician and this will generally require that results are reported in terms analogous to accepted clinical concepts. It must be cost-effective, that is the benefit of performing the test must be worth the cost. This balance need not necessarily be determined in purely financial terms but the financial cost of gait analysis is a significant factor. Finally there is no point doing any clinical test if the results could be obtained sufficiently well by simply observing the patient

The information obtained by assessing the patient is that used for selecting management options. This process does not, therefore, make further demands on the measurement systems but does require an understanding of how the patient's condition is likely to be affected by an intervention (or none) to a level sufficient to determine which options are preferable. Prediction of outcomes takes this one stage further to being able to determine not only which management option is best but also how the patient will be after that intervention.

This sequential analysis of the four potential purposes of clinical tests reveals a progression from just requiring reliable and precise measurements to the additional requirement of having an understanding of how such information is incorporated into clinical practice. The state of the art is that the measurement component of gait analysis can reasonably be described as an objective process whereas the interpretation component is predominantly subjective.

Making the interpretive component more objective can be achieved in two ways. The first is to develop a general theory of how people walk whether they have recognised pathology or not. As long ago as 1982 Cappozzo lamented, "The approaches to clinical gait analysis and evaluation are not supported by general theories" [ 3 ] and despite over 20 years of intense activity this is still a reasonable summary of the state of the art. The second approach, which must operate in the absence of the former, is to conduct clinical research to ascertain the outcome of particular interventions on groups of patients characterised by certain measurements. Most of the knowledge base used in the interpretive component of gait analysis comes from such studies. It is because there are relatively few studies available to base such interpretations on that the subjective element of interpretation is necessary in contemporary clinical gait analysis.

Modern clinical gait analysis traces its origins back to the early 1980s with the opening of the laboratory developed by the United Technologies Corporation at Newington, Connecticut and those provided with equipment by Oxford Dynamics (later to become Oxford Metrics) in Boston, Glasgow and Dundee. Retro-reflective markers were placed on the skin in relation to bony landmarks. These were illuminated stroboscopically and detected by modified video cameras. If two or more cameras detect a marker and the position and orientation of these cameras are known then it is possible to detect the three-dimensional position of that marker [ 4 ].

Whilst the basic principles remain the same as the earliest systems, the speed, accuracy and reliability has advanced beyond all recognition. It is not uncommon now to find clinical systems using 8, 10 or more cameras functioning at over 100 Hz and capable of detecting reliably the presence of many tens of markers of between 9 and 25 mm diameter. Calibration of the systems (the determination of the position, orientation and optical and electronic characteristics of the cameras) can generally be accomplished in less than a minute. Marker positions from clinical trials can be reconstructed and markers labelled automatically in real time (although this feature is often not essential for clinical studies). The determination of the accuracy of such systems is now generally limited by the accuracy of any alternative means to determine marker position and can be taken to be of the order of 1 mm. This is probably an order of magnitude smaller than other sources of error in determining joint kinematics and kinetics. This particular measurement technology has thus reached a mature state of development that, whilst advances will almost certainly continue, already probably delivers all that is required by conventional gait analysis [ 5 ].

The same cannot be said of the computer models used to derive joint kinematics and kinetics from the marker position data supplied by the measurement hardware. Almost all commercially available clinical systems use some variant of the Conventional Gait Model [ 6 ] which has been referred to as the Newington, Gage, Davis [ 7 ], Helen Hayes, Kadaba [ 8 , 9 ] or Vicon Clinical Manager (VCM) model. This was developed using the minimum number of markers possible to determine 3-dimensional kinematics and kinetics [ 10 , 11 ] of the lower limb at a time when measurement systems were only capable of detecting a handful of markers. It assumes three degree of freedom joints for the hip and knee and a two degree of freedom joint at the ankle. The model is hierarchical requiring the proximal segments to have been detected in order that distal segments can be defined and incorporates regression equations to determine the position of the hip joint centre with respect to pelvic markers. Kinetics are determined using an inverse dynamics approach which generally requires considerable filtering to give any useful signals. An alternative system the Cleveland Clinic Model based around a cluster of markers on a rigid base attached to each segment is the only other widely used model. Unfortunately documentation of this model in the scientific literature is very poor.

The problem of limited repeatability

The primary problem of current measurement technology is that of reliability in routine clinical use. Several studies have now been reported in which a single subject has been analysed in a number of different laboratories [ 12 – 14 ]. These have shown a degree of variability between sites that would appear to be sufficient to undermine clinical applications. In retrospect, the original studies of the reliability are flawed. There was no such study of the Davis implementation of the model and the statistics used by Kadaba et al [ 8 , 9 ] to report reliability of their implementation probably acted to mask deficiencies. In particular, use of relative measures of reliability such as the coefficient of multiple correlation (CMC) makes interpretation of findings difficult. Almost all reliability studies have been done on subjects without pathology where marker placement is reasonably straightforward. Reliability for clinical populations is rarely reported in the literature and is almost certainly inferior.

At least one recent study has shown that it is possible to get levels of reliability sufficient to justify the continued clinical use of gait analysis within a single centre [ 15 ]. Too few centres however are providing evidence to establish that this is the rule rather than the exception.

Whilst not the most exciting field of research, a very real need of clinical gait analysis is for the development of techniques for establishing the reliability of measurement techniques and of methods of quality assurance that will ensure that the very highest standards of reliability are achieved in routine clinical practice .

Source of error: Model calibration

There are two principal sources of error. The first is the difficulty determining the anthropometry of the individual subject (known as model calibration ). This has two aspects, placing markers accurately with respect to specific anatomical landmarks and determining the location of the joint centres (and other anatomical features) in relation to these markers. Failure to place markers accurately is probably the single greatest contributor to measurement variability in contemporary clinical gait analysis. This is partly a matter of appropriate staff training and quality assurance but at least as important, and more fundamental, is the problem that many of the landmarks used to guide marker placement are not themselves particularly well defined in patients with certain conditions [ 16 ]. Even when bony landmarks are sharply defined an increasing number of patients have a considerable thickness of subcutaneous fat that makes palpation difficult.

The Conventional Gait Model uses regression equations to determine the position of the hip joint centre in relation to the pelvis. Both Bell's [ 17 – 19 ] and Davis' [ 7 ] equations are commonly used and there is now good evidence that neither is satisfactory in healthy adults [ 20 ]. There have still been no published studies of whether either is valid for healthy children. Children with orthopaedic conditions including cerebral palsy may often have dysplasia of the hip or deformity of the pelvis, and it is exceedingly unlikely that any form of regression equation could be used in these patients to determine hip joint position.

Methods for moving away from anatomical landmarks and regressions equations for determining joint centres have been around for nearly a decade, the process being known as anatomical calibration [ 21 ]. They rely on calibration movements to be performed before capturing walking data and some form of fitting of the measured marker positions to an underlying model of how the body moves. The simplest example is probably the determination of the hip joint centre. It is assumed that the hip joint moves as a ball and socket joint about centre of rotation fixed in the pelvis. Any marker on the femur would thus be expected to describe a path on the surface of a sphere centred on the hip joint centre when the hip joint is moving. A least squares fit of the measured data to such a sphere allows the location of that joint centre to be determined [ 20 , 22 ]. Similar approaches are applicable to determine that axis of the knee joint which for this purpose has often been assumed to be a simple hinge joint.

Various approaches to fitting data to an underlying model have been attempted and many seem to give reasonable results [ 20 , 22 – 28 ]. Such techniques have not so far been widely accepted into clinical practice probably because there is a perception that such calibration trials are too difficult for patients to execute. At least one clinical lab however has now committed itself to implementing such techniques into routine practice and has reported failure to perform test adequately in only one of over 700 patients tested so far.

Further studies are needed to confirm these studies and to identify which of the range of available optimisation techniques is the best suited to clinical applications. Comprehensive reliability studies are again needed to demonstrate the advantages of using such models over the conventional model .

Sources of error: Soft tissue artefact

The second source of error is the degree of movement of the skin, muscle and other soft tissues in relation to the bones that occurs during walking. This is perhaps most marked in relation to the rotational profile of the hip. Lamoreux [ 29 ], as far back as 1991, reported that with optimal placement of thigh wands only 65% of transverse plane hip joint rotation was detected and that with poor placement this could be as little as 35%.

The problem of skin and other soft tissue movement is more problematic than that of model calibration. Lu and O'Connor where the first to propose fitting a model of how the body is expected to move to marker co-ordinate data [ 30 ] using an optimisation approach. This model uses a least squares fit, similar to some of the techniques described above for model calibration, and thus makes no assumptions about the nature of the soft tissue movement. Other similar models have now been made commercially available [ 26 ]. More recent studies have started to try to map out the movement of markers with respect to the underlying bones [ 31 , 32 ]. If such movement can be characterised as a function of joint angle then, in principle, this knowledge could be built into a model to allow such movements to be compensated for. Such mapping is only likely to be useful if it can be shown that soft tissue movement is consistent across a range of subjects and activities. It is not clear at present whether these conditions are satisfied. A particular problem in regard to mapping soft tissue movement is that of defining what the "true" movement of the bones is. In the absence of any gold standard a variety of assumptions are being used most of which have serious limitations.

Significant work is needed in this area. A gold standard method for determining joint movement is required.

Maps of soft-tissue movement as a function of joint angle are required and work done to establish how these vary from individual to individual and from task to task.

Marker sets need to be defined based on the optimum placement of markers given knowledge of the soft-tissue displacements.

Finally it is possible that knowledge of likely soft-tissue displacement could be built into the optimisation algorithms allowing for better estimates of the movements of the underlying skeleton.

The development of a gold standard method for determining joint movement will probably require a move away from skin-mounted markers (or other sensors). Once such technology is available however it is quite possible that this will supersede the presently available systems. The cost of any such new systems however is likely to prohibit ready clinical availability in the foreseeable future.

There has been some work done on markerless optical methods. By placing a number of video cameras around a subject and tracing the silhouette of the walking subject on each it is possible to generate a 3-dimentional silhouette of that subject. This has already been achieved but the next step of using such a silhouette to determine the co-ordinate systems associated with the moving body segments has not yet been satisfactorily achieved.

It is possible that the problem of skin movement can only be satisfactorily addressed by making direct measurements of bone position. It is now possible to take 3-dimensional images of bones (and muscles) using MRI but only within a very restricted capture volume [ 33 – 35 ]. The image processing problem of automatically determining a bone embedded axis system from such images has yet to be solved satisfactorily. Similarly both uniplanar and biplanar cine fluoroscopy [ 36 – 40 ] has been used to detect the 3-dimensional movement of the internal knee prostheses during a variety of movements. This is possible because a knowledge of the exact size and shape of the prosthetic components and their opacity to x-rays greatly simplifies the image processing problem. Using similar techniques to determine the movement of joints has also been reported [ 41 – 43 ]

Using 3-dimensional imaging techniques to directly determine bone movements during walking either as a technology with potential clinical applicability or for use as a gold reference standard from which to improve the implementation of conventional marker based technologies is one of the greatest challenges in this area.

Methods for interpreting clinical gait analysis data

The second element of clinical gait analysis is the interpretation of data. Conventions for describing 3-dimensional joint kinematics and kinetics are well formulated. Many laboratories are augmenting conventional kinematics and kinetics with muscle length and, less commonly, moment arm graphs. Normal patterns of movement as represented by these data are now generally fairly well understood by clinical specialists although there is actually very little normative data published in the peer-reviewed literature. Similarly, many abnormal patterns of movement are quite widely recognised by clinicians but there few published attempts at formal classification of these [ 44 – 46 ]. Many clinicians have learnt to associate particular abnormal patterns in particular patient groups with particular impairments of body structure and function. Intervention based on such an understanding often leads to a normalisation of gait patterns at subsequent assessments (e.g. [ 47 – 55 ]). It is on this basis that clinical gait analysis operates at present.

Despite the widespread acceptance of many of these conventions there are still problems. Baker [ 56 ] demonstrated that the Euler sequence used to calculate pelvic angles gives rise to data that can be mis-leading to clinicians and proposed an alternative to correct this which is yet to be adopted widely within clinical analysis. Methods for interpreting angles in three dimensions, either in terms of Euler/Cardan rotations or the Grood and Suntay convention [ 57 , 58 ] are not well understood either by clinicians or many bioengineers. A recent attempt to standardise the reporting of joint angles [ 59 ] proposed a different convention to that of the Conventional Gait Model and the continuing debate as to which is preferable illustrates this confusion [ 60 , 61 ]. Joint moments are generally reported with reference to orthogonal axis systems fixed in the distal (Conventional Gait Model) or proximal segments (or occasionally the laboratory axis system). These differ significantly depending on the axis system chosen [ 6 , 62 ] yet there has been no debate about which if any is preferable. Reporting moments about orthogonal axis systems and joint rotations about non-orthogonal ones leads to difficulties in relating the moments to the changes in joint angles to which they are related. The use of muscle moment arms will be discussed further below but it is interesting that there is no straightforward definition of the meaning of the term moment arm in three dimensions [ 63 ] and it is often not clear how such data should be interpreted.

A consistent, comprehensive and clear method for describing joint kinematics and kinetics in three dimensions would be of immense benefit for the clinical gait analysis community.

Perhaps the most important limitation of our present understanding of human walking, however, is that it is primarily descriptive. We know what happens rather than why it happens. Many in the clinical gait analysis community regard kinematics as descriptive but contend that kinetics explain movement patterns. This is almost certainly misguided. Kinetics are simply another set of measurements and can thus only be descriptive.

There have been various attempts at establishing a theory of walking but none is particularly convincing. Saunders, Inman and Eberhart's determinants of normal walking [ 64 ] are perhaps the best known of these. Recent publications however have questioned how the detail of these reflects experimental data [ 65 – 70 ]. Gage [ 71 , 72 ] based his pre-requisites of gait on earlier work by Perry [ 73 ] but these are best regarded as pointers to where particular patients are deficient rather than explanations of how they are achieving walking with or without pathology.

Perhaps the closest we have come so far to understanding why we walk the way we do has come from the work of Pandy and Anderson [ 74 , 75 ]. They have shown that it is possible to construct a mathematical simulation of muscle function during normal walking based on the assumption that the total consumption of energy per unit distance walked is minimised. The authors, however, commented that the model seems more dependent on the boundary conditions imposed than on the nature of the optimisation function. Further, because of the complex nature of the optimisation process driving the model it is still difficult to explain how the precise characteristics of any particular feature of the walking pattern affect the overall calculation of energy expenditure. So far such a model has only been constructed for normal walking.

An obvious challenge in the emerging field of computational biomechanics is to apply similar techniques to model walking with particular forms of pathology.

Conceptually, modifying such models to incorporate a specific abnormality of the musculo-skeletal anatomy such as a leg length discrepancy or contracture of a particular muscle is reasonably straightforward. It is much less certain whether such techniques can be applied at all to patients with neuromuscular pathology who are most frequently seen by clinical gait analysis services. Optimisation techniques assume that movements are controlled in such a way that a specific control function is minimised. In many neuromuscular conditions (Cerebral Palsy, Parkinson's disease, adult hemiplegia) the problem is one of a loss of central control and this would appear to invalidate any techniques modelling human movement as an optimised process.

If such models are developed it will be interesting to see whether they give any insights into the clinical management of patients. Further it will be interesting to see whether their use leads to an understanding of why we walk the way we do which can be formulated as theories that are applicable without the use of such complex models.

Perhaps the greatest challenge in clinical gait analysis is still to answer the question. "Why do we walk the way we do and why don't our patients?".

Whilst the answer to this question still seems as far away as ever, significant advances have been made over recent years in understanding the mechanisms by which we walk particularly in the way that muscles act. For many years it was assumed that a muscle's anatomical position determines how it acts. It was assumed for example that the action of the hamstrings, passing behind the knee, was always to flex the knee. It is only comparatively recently that biomechanists have come to appreciate that any individual muscle has an effect on all the segments of the body and that in some circumstances this may result in a muscle having an action different to its anatomical function [ 76 – 81 ]. It is now fairly well accepted, for example, that the hamstrings functions as a knee extensor during early stance in normal walking because its effect in extending the hip has a secondary tendency to extend the knee which is greater than its direct effect as an anatomical knee flexor [ 82 ].

Such work depends on knowing the joint kinematics and kinetics and inertial properties of the body segments. These can be used to estimate the forces in individual muscles [ 81 – 83 ]. This is an indeterminate problem so is dependent on an optimisation approach (and the validity of this in neuromuscular pathology is questioned in the same way as that of the simulations described above). Once the muscle forces are known forward modelling can be used to determine the effect that a given muscle is having on any segment (or joint) of the body. Until very recently the first part of this problem, the estimation of muscle forces had not been achieved which limited the application of the second part, the forward modelling to data obtained from the simulations described above [ 74 , 75 ]. Recently methods have been develop to estimate the muscle forces required to generate measured joint kinematics and ground reaction forces and have been used both to understand the function of individual muscles during pathological gait and predict the effect of interventions [ 84 , 85 ]. These have been based on scaled models of the adult musculo-skeletal anatomy.

A further area of challenge is in using 3-dimensional imaging techniques to model musculo-skeletal deformities to allow the generation of patient-specific models of walking.

There is also considerable debate at present about the validity of these techniques (the simulations, the estimations of muscle forces and the forward modelling). Whilst the general principles are sound the techniques are known to be extremely sensitive to certain aspects of their implementation (and may be sensitive to many more). For example the forward modelling in particular is sensitive to how the interaction between the foot and the floor is modelled with there being no clear consensus as to the most appropriate method for this [ 74 , 77 , 81 ].

Implementation of these models must be based on robust techniques being developed to validate models, the first step of this is in rigorous analysis of the sensitivity of models to the assumptions on which they are based.

Methods for understanding the effect of intervention

Understanding how to interpret clinical gait analysis data is not itself sufficient to allow selection from amongst treatment options (Table 1 ). For this it is also necessary to know what effect the available interventions are likely to have on someone's walking pattern. If we had a general theory of walking then it might be possible to develop a theoretical basis for considering the effect of any intervention. For patients whose walking could be modelled using a simulation based on specific musculo-skeletal abnormalities it might be possible to use similar simulations to model what might happen if partial correction of those abnormalities were attempted (obviously full correction would restore normal walking!). The author is unaware of any published work at this level at present.

There are then two methods for understanding the effects of intervention in these patients; clinical research to establish what the actual effect of a given intervention is or using knowledge of the mechanisms of walking to predict the effect of modifying the characteristics of the musculo-skeletal anatomy.

By far the most common approach to date has been the use of clinical research – the comparison of gait patterns before and after a particular intervention [ 47 – 55 , 86 – 88 ]. Even so there have been comparatively few studies that have given conclusive findings. Many studies which claim to have done so have quite serious methodological flaws. This is particularly true of research into orthopaedic surgery for children with CP where researchers have used retrospective audits of clinical practice to try and answer specific questions. Many of these studies attempt to make inferences about individual procedures which have only ever been performed as part of a multi-level surgical package [ 47 , 49 – 51 , 54 , 55 ]. It is impossible to tell from these studies which effects are due to the particular procedure being considered and which are due to the overall package. Several studies have attempted to separate out those effects by dividing patients into those who have and those who have not had a particular procedure as part of the overall package of surgery and use methods to compare groups similar to those that would be used for a randomised clinical trial [ 47 , 54 ]. The validity of this approach is questionable, however, because generally the two groups of patients were not similar to start with. Those that had the procedure had it because it was considered that the patient needed it and vice versa. Comparison of the two groups to give insight into the effect of the procedure is thus invalid.

Perhaps the most challenging field of research for clinical gait analysis is in the design and conduct of prospective clinical trials to ascertain the effects of specific treatments on specific patient groups.

An alternative to the use of clinical trials is to use knowledge of the mechanisms of walking as a basis for modelling the effect of changing that mechanism. Reports of such studies are now starting to emerge. For example Arnold et al. [ 84 ] have reported a subject specific model of a cerebral palsy patient with a stiff knee gait and used it to predict the effect of three different potential interventions. These indicated a preferable intervention and the post-intervention gait data showed at least qualitative agreement with the theoretical predictions.

Application of such techniques to a wider range of clinical problems represents another exciting sphere of research in clinical gait analysis. It may well be that such techniques are limited to the fairly narrow range of interventions that are based on correction of the mechanisms for very specific aspects of walking but identifying the range of potential applications will be an important part of this process .

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Richard Baker

Gait CCRE, Murdoch Children's Research Institute, Parkville, Victoria, Australia

Department of Mechanical and Manufacturing Engineering, University of Melbourne, Parkville, Australia

Musculoskeletal Research Centre, La Trobe University, Bundoora, Victoria, Australia

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Baker, R. Gait analysis methods in rehabilitation. J NeuroEngineering Rehabil 3 , 4 (2006). https://doi.org/10.1186/1743-0003-3-4

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Adaptive detection in real-time gait analysis through the dynamic gait event identifier.

Graphical Abstract

1. Introduction

• High-precision multi-event real-time gait detection: We present a novel gait detection method suitable for complex settings, characterized by high precision and the ability to handle multiple events in real time. This method not only elevates the accuracy in identifying crucial gait events but also showcases its applicability and adaptability across dynamic and varying scenarios, being responsive to diverse gait events.
• Innovative weighted sleep time approach: We introduce an innovative weighted sleep time approach, which, by dynamically modulating the algorithm’s sensitivity and dormancy period, significantly enhances the accuracy and adaptability in detecting gait events.
• Adaptive threshold decision-making: We have developed an adaptive threshold decision-making rule aimed at real-time adjustment of detection thresholds for gait events. This rule is particularly effective in adapting to changes in the amplitude of gait curves across various scenarios, thus substantially improving the overall performance and adaptability in gait event detection.

2.1. Dynamic Gait Event Identifier

• Δ y k = y k − y k − 1 represents the change in Euler angles between consecutive gait data points, signifying the instantaneous gait dynamics.
• Δ t k denotes the time interval between these data points, reflecting the temporal aspect of gait changes.
• α and β are weighting coefficients designed to balance the immediate gait changes against their rate over time, thereby accommodating the diverse dynamics of gait patterns. These coefficients are defined as follows: α = σ Δ y σ Δ y + μ Δ v (4) β = μ Δ v σ Δ y + μ Δ v (5)
• σ Δ y is the standard deviation of the immediate changes in Euler angles ( Δ y k ), a statistical measure capturing the variability within the gait data.
• μ Δ v represents the mean rate of change in the Euler angles, encapsulating the average velocity of gait alterations across the dataset.
• B a r sets the threshold, distinguishing between positive and negative gait events in the context of the algorithm’s classification process.

2.2. Weighted Sleep Time Method

2.3. adaptive threshold decision rules, 3. experiment, 3.1. data gathering, 3.2. evaluation indicator, 3.3. optimization methodology, 3.3.1. optimization of sleeptime.

• Sensitivity analysis: The sensitivity metric, which indicates the true positive rate of detecting gait events, showed peak values for both toe-off (TO) and heel strike (HS) events in the range of 60 to 63 s. This suggests an optimal balance between event detection capability and the algorithm’s responsiveness within this sleeptime interval.
• Average difference optimization: The average difference, reflecting the precision in localizing detected events, reached its optimum at a sleeptime of 60 s. This optimal point signifies the highest alignment between detected events and their actual occurrences, thereby minimizing localization error.
• MCC performance: The Matthews correlation coefficient, a comprehensive measure of classification accuracy, exhibited optimal performance within the 55 to 60 s range. Given the MCC’s value in assessing the balance between various aspects of binary classification performance, this finding underscores the efficacy of the DGEI methodology within the specified sleeptime range.

3.3.2. Optimization of Bar

• MCC considerations: The MCC metric, which offers a balanced evaluation of the algorithm’s classification capabilities, identified the 5 to 10 range as optimal. This interval demonstrates a robust performance across detecting true positives and negatives while minimizing errors.
• Sensitivity insights: For the sensitivity metric, values above 8 consistently achieved a performance exceeding 95%. This high sensitivity indicates the algorithm’s effective detection of gait events at higher bar settings.
• Average difference analysis: The average difference across varying bar values showed minimal variation, suggesting that this metric was less sensitive to changes in the bar parameter. This stability implies that the bar setting’s impact on event localization precision is comparatively uniform.

3.3.3. Coupling Analysis of Hyperparameters

3.4. peak detection, 3.5. error analysis, 4. discussion, 5. conclusions, author contributions, institutional review board statement, informed consent statement, data availability statement, conflicts of interest.

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Liu, Y.; Liu, X.; Zhu, Q.; Chen, Y.; Yang, Y.; Xie, H.; Wang, Y.; Wang, X. Adaptive Detection in Real-Time Gait Analysis through the Dynamic Gait Event Identifier. Bioengineering 2024 , 11 , 806. https://doi.org/10.3390/bioengineering11080806

Liu Y, Liu X, Zhu Q, Chen Y, Yang Y, Xie H, Wang Y, Wang X. Adaptive Detection in Real-Time Gait Analysis through the Dynamic Gait Event Identifier. Bioengineering . 2024; 11(8):806. https://doi.org/10.3390/bioengineering11080806

Liu, Yifan, Xing Liu, Qianhui Zhu, Yuan Chen, Yifei Yang, Haoyu Xie, Yichen Wang, and Xingjun Wang. 2024. "Adaptive Detection in Real-Time Gait Analysis through the Dynamic Gait Event Identifier" Bioengineering 11, no. 8: 806. https://doi.org/10.3390/bioengineering11080806

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A comprehensive survey on gait analysis: History, parameters, approaches, pose estimation, and future work

Affiliations.

• 1 Information Technology, Indira Gandhi Delhi Technical University for Women, New Delhi 110006, New Delhi, India. Electronic address: [email protected].
• 2 Information Technology, Indira Gandhi Delhi Technical University for Women, New Delhi 110006, New Delhi, India.
• 3 Computer Science and Engineering, National Institute of Technology, New Delhi 110040, New Delhi, India.
• PMID: 35659390
• DOI: 10.1016/j.artmed.2022.102314

Human gait is a periodic motion of body segments-the analysis of motion and related studies is termed gait analysis. Gait Analysis has gained much popularity because of its applications in clinical diagnosis, rehabilitation methods, gait biometrics, robotics, sports, and biomechanics. Traditionally, subjective assessment of the gait was conducted by health experts; however, with the advancement in technology, gait analysis can now be performed objectively and empirically for better and more reliable assessment. State-of-the-art semi-subjective and objective techniques for gait analysis have limitations that can be mitigated using advanced machine learning-based approaches. This paper aims to provide a narrative and a comprehensive analysis of cutting-edge gait analysis techniques and insight into clinical gait analysis. The literature of the previous surveys during the last decade is discussed. This paper presents an elaborated schema, including gait analysis history, parameters, machine learning approaches for marker-based and marker-less analysis, applications, and performance measures. This paper also explores the pose estimation techniques for clinical gait analysis that open future research directions in this area.

Keywords: Clinical gait analysis; Human gait analysis; Objective analysis review; Pose estimation; Semi-subjective analysis.

Copyright © 2022 Elsevier B.V. All rights reserved.

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BRIEF RESEARCH REPORT article

Inertial measurement unit-based real-time feedback gait immediately changes gait parameters in older inpatients: a pilot study.

• 1 Department of Orthopedic Surgery, Graduate School of Medical and Dental Sciences, Kagoshima University, Kagoshima, Japan
• 2 Department of Physical Therapy, School of Health Sciences, Faculty of Medicine, Kagoshima University, Kagoshima, Japan
• 3 Sports Science Area, Department of Mechanical Systems Engineering, Daiichi Institute of Technology, Kagoshima, Japan
• 4 Course of Health Sciences, Graduate School of Health Sciences, Kagoshima University, Kagoshima, Japan
• 5 Department of Rehabilitation, Tarumizu Municipal Medical Center, Tarumizu Central Hospital, Kagoshima, Japan

The effect of gait feedback training for older people remains unclear, and such training methods have not been adapted in clinical settings. This study aimed to examine whether inertial measurement unit (IMU)-based real-time feedback gait for older inpatients immediately changes gait parameters. Seven older inpatients (mean age: 76.0 years) performed three types of 60-s gait trials with real-time feedback in each of the following categories: walking spontaneously (no feedback trial); focused on increasing the ankle plantarflexion angle during late stance (ankle trial); and focused on increasing the leg extension angle, which is defined by the location of the ankle joint relative to the hip joint in the sagittal plane, during late stance (leg trial). Tilt angles and accelerations of the pelvis and lower limb segments were measured using seven IMUs in pre- and post-feedback trials. To examine the immediate effects of IMU-based real-time feedback gait, multiple comparisons of the change in gait parameters were conducted. Real-time feedback increased gait speed, but it did not significantly differ in the control ( p = 0.176), ankle ( p = 0.237), and leg trials ( p = 0.398). Step length was significantly increased after the ankle trial ( p = 0.043, r = 0.77: large effect size). Regarding changes in gait kinematics, the leg trial increased leg extension angle compared to the no feedback trial ( p = 0.048, r = 0.77: large effect size). IMU-based real-time feedback gait changed gait kinematics immediately, and this suggests the feasibility of a clinical application for overground gait training in older people.

1 Introduction

Decreased gait speed, propulsion, and range of motion of the lower extremities have been reported as typical changes in gait mechanics due to aging and various motor dysfunctions ( Boyer et al., 2017 ). Such changes can lead to decreasing mobility and quality of life and an increased risk of adverse events ( Abellan van Kan et al., 2009 ). Thus, it is important for older people to maintain gait speed as one of the determiners of gait ability.

Previous studies showed that gait speed did not increase by functional training such as resistance training alone ( Kim et al., 2001 ; Ouellette et al., 2004 ). In order to improve gait ability, it is necessary to establish effective gait training procedures guided by individual gait characteristics (such as gait feedback training) ( Franz et al., 2014 ; Schenck and Kesar, 2017 ; Genthe et al., 2018 ; Browne and Franz, 2019 ; Liu et al., 2020 ; Liu et al., 2021 ). Over the recent years, gait practice by using wearable sensors has been reported in clinical applications ( Gordt et al., 2018 ; Hinton et al., 2023 ; Silva-Batista et al., 2023 ; Hinton et al., 2024 ), but its effect on increasing gait speed remains unclear. More specifically, previous studies had conducted gait feedback training using foot motion or foot pressure measured by a sensor attached to the dorsal foot and insole ( Sungkarat et al., 2011 ; Byl et al., 2015 ). These reports have limited the target of gait feedback training, and it is necessary to establish methods that can be adapted to individual gait abnormalities in order to promote effective walking.

In order to apply an effective gait practice for older people in clinical settings, a system of gait feedback using multiple parameters was considered necessary. In clinical practice and cohort fields, we have analyzed human movement such as gait using inertial measurement units (IMUs) ( Miyazaki et al., 2021a ; Miyazaki et al., 2021b ; Matsuzawa et al., 2021 ; Araki et al., 2023 ), and they are utilized in gait practice. However, there are few reports on the effect of gait feedback training performed in clinical practice ( Hinton et al., 2023 ; Hinton et al., 2024 ), and more clinical data on this subject are needed. We have previously reported an IMU-based gait feedback system with real-time feedback of joint angles during overground gait, which showed increasing gait speed immediately with joint angle changes in young healthy adults ( Miyazaki et al., 2023 ). There are various types of feedback ( Sigrist et al., 2013 ), and our feedback system consists of extrinsic feedback, in which the knowledge of the result is provided by auditory stimulation. In addition, auditory stimulation is reported to be more effective than visual stimulation for dynamic postural control ( Hasegawa et al., 2020 ). Our system uses auditory stimulation, can be implemented with a PC and IMU, making it easy to use in a clinical setting. Therefore, this system may be applicable to overground gait training for older people in clinical settings.

The purpose of this study was to examine whether IMU-based real-time feedback gait for older inpatients immediately changes gait parameters. The findings of this study offer fundamental data regarding effective gait practice for older inpatients in clinical settings. We hypothesized that IMU-based real-time feedback gait would lead to increased gait speed immediately, and specific changes in gait kinematics for each feedback target would also be observed.

2 Materials and methods

2.1 participants.

Seven older inpatients (mean age, 76.0 ± 7.1 years; including three women, four patients with orthopedic conditions, two patients post-stroke, and one patient with metabolic disease) who could walk several minutes without walking aids participated in this study ( Table 1 ). The exclusion criteria were as follows: (1) lower-limb impairments such as pain that affected the measurement of gait and physical performance, (2) severe dementia, and (3) not consenting to participate in this study. Basic information, including disease, age, sex, height, and body mass index, was recorded. In addition, the five-times-sit-to-stand test (FTSS) was used as an indicator of physical performance. The FTSS involved standing up and sitting down five times from a sitting position, as quickly as possible, without pushing off ( Mong et al., 2010 ). In the FTSS, well-trained assessors recorded the time taken to perform five consecutive chair-stands (timed to 0.1 s) from a seated position on a 45-cm-tall chair, with arms folded across the chest.

Table 1 . Participants’ demographics.

The study was approved by the Ethics Committee on Epidemiological Studies of Tarumizu Central Hospital (approval number: 20-8), and all participants provided written informed consent before participating in the study.

2.2 Feedback trials

As in previous studies ( Miyazaki et al., 2023 ), gait parameters measured before and after the gait trials were compared to examine the immediate effects. During feedback trials, participants were instructed to modify their lower limb motion during gait under three types of feedback, and they walked on a 30-m walkway for 60 s in each trial ( Liu et al., 2021 ; Miyazaki et al., 2023 ). Three feedback trials were performed ( Figure 1 ): (i) a feedback trial without feedback (no feedback trial) and two feedback trials with real-time feedback during overground gait to (ii) increase the ankle plantarflexion angle during the late stance (ankle trial) and (iii) increase the leg extension angle, which is defined by the location of the ankle joint relative to the hip joint in the sagittal plane ( Miyazaki et al., 2019 ), during the late stance (leg trial). Gait kinematics used as feedback targets were the ankle plantarflexion angle and leg extension angle at late stance. These parameters have been related to propulsion during gait ( Hsiao et al., 2015a ; Hsiao et al., 2015b ; Browne and Franz, 2017 ; Browne and Franz, 2019 ), and they could be a feasible target for gait feedback training ( Browne and Franz, 2019 ; Liu et al., 2020 ; Liu et al., 2021 ). Before each feedback trial, participants were explained the gait modification during each feedback trial by using verbal instructions and pictures. The details of the explanation were as follows: no feedback trial, “walk at your usual pace during this trial”; ankle trial, “push back the ground harder before you swing your leg so that it makes a beep sound during this trial”; and leg trial, “extend your leg farther backward before you swing your leg so that it makes a beep sound during this trial” ( Miyazaki et al., 2023 ).

Figure 1 . Experimental protocol of the IMU-based real-time feedback gait. At pre-gait (spontaneous gait) and post-gait (replicate gait without feedback) measurements, gait parameters were measured using IMUs. Pre-gait measurements also determined the threshold of feedback. During the feedback trials, participants modified gait in response to the beep sound when the participant’s current joint angle (solid line) reached the threshold angle (dot line). The threshold was set at a 20% increase in the peak values of each joint angle during spontaneous gait. IMUs: inertial measurement units.

Before and after each feedback trial, participants walked along the 14-m walkway twice to measure gait parameters using IMUs ( Figure 1 ). Spontaneous and replicate gait were measured pre- and post-feedback trials, and post-gait measurements were made without feedback. Each gait feedback trial consisted of one feedback trial and two gait measurement pre- and post-feedback trials ( Figure 1 ), and they were randomly performed according to the Microsoft Excel Rand function. In addition, an approximate 2-min standing break interval was provided between each trial ( Miyazaki et al., 2023 ). We measured the length of each patient’s right thigh and shank by using a measuring tape before the pre-gait measurement.

2.3 Methodology of the IMU-based real-time feedback gait

IMU-based real-time feedback gait was performed using a mobile PC (One-Mix3Pro, Tech-One Co. Ltd, Tokyo, Japan), and the joint angles calculated by IMUs were displayed on a PC ( Figure 1 ) ( Miyazaki et al., 2023 ). Gait parameters were measured using seven IMUs (MTw Awinda, Xsens, Enschede, NL), and the IMUs consisted of a 3D gyroscope, 3D accelerometer, and 3D magnetometer. The sampling frequency was 100 Hz. The 3-axis acceleration and tilt angles in a global coordinate system were obtained from the magnetic and inertial data using a Kalman filter on MT Manager software (4.7.2, Xsens, the Netherlands). The reliability of IMUs has been reported previously ( Ferrari et al., 2010 ). Before gait measurements, IMUs were attached by elastic belts to the posterior sacrum, bilateral anterior thighs, shanks, and dorsal feet. For the dorsal feet, IMUs were fixed on their shoes ( Figure 1 ). IMUs were also attached frontally and vertically against the frontal plane where possible, and they were calibrated so that the vertical direction of the coordinate system followed the direction of gravity during static standing ( Miyazaki et al., 2019 ). The timing of the maximal posterior tilt angle of the sensor attached to each shank was used to determine the timing of initial contact ( Revi et al., 2020 ). The PC screen displayed the joint angles calculated by the IMUs in real-time, and the threshold of the feedback was set at a 20% increase in the peak values of each joint angle during a spontaneous gait during the pre-feedback trial ( Miyazaki et al., 2023 ). Participants were provided continuous real-time auditory feedback, and beep sounds were emitted when the participant’s current joint angle reached the threshold, during each feedback trial ( Figure 2 ).

Figure 2 . Comparisons of the changes in gait parameters. (A) Gait speed, (B) cadence, (C) step length, (D) maximum leg extension angle at late stance, (E) maximum ankle plantarflexion angle at late stance, and (F) increment of velocity at late stance. *: p < 0.05.

2.4 Data analysis

Low-pass filtering was performed on the joint angle, and acceleration data were measured using IMUs with a 10 Hz and 20 Hz cutoff frequency ( Arumukhom Revi et al., 2021 ; Araki et al., 2024 ). For spatiotemporal parameters, cadence was calculated by identifying heel contact during the maximum posterior tilt angle of the sensor on the shank ( Revi et al., 2020 ). Stride length and gait speed were also calculated based on the walking time measured by IMUs. The joint angles including the hip, knee, and ankle were calculated as relative Euler angles measured from IMUs fixed on the pelvis, thigh, shank, and foot segments ( Araki et al., 2023 ; Miyazaki et al., 2023 ). In addition, the leg extension angle was determined based on the location of the ankle joint relative to the hip joint in the sagittal plane, estimated from the tilt angle matrix measured by IMUs and the vector of the thigh and shank segment coordinated by segment length ( Miyazaki et al., 2019 ). Previous studies have confirmed the validity of using IMUs to determine these gait parameters ( Miyazaki et al., 2019 ), and maximum ankle plantarflexion angle and leg extension angle during the late stance were calculated ( Miyazaki et al., 2023 ). The increment of velocity was calculated using the anterior acceleration measured with the IMU fixed on the sacrum during late stance, which has also been reported as the association to the impulse of the anterior ground reaction force such as an indicator commonly used as propulsion force ( Miyazaki et al., 2019 ). Thus, data processing was performed using the mathematical software MATLAB R2020a (Mathworks Inc., MA, United States).

2.5 Statistical analysis

The mean values of the variables determined for the bilateral lower extremities during 10 strides (five from the two gait measurements pre- and post-feedback trial, respectively) were used as the representative values. To confirm the normal distribution of the data, the Shapiro–Wilk test was conducted. To examine the immediate effects of the feedback trials (no feedback, ankle, and leg) on each gait parameter, the t-test and Mann–Whitney U-test were conducted. Then, to compare the change in gait parameters before and after the feedback trial, Friedman analysis was performed, and the Bonferroni method or Shaffer method was used to perform the multiple comparisons test. Calculations of r were performed to estimate the effect size of the group comparison. The effect size was classified into small (r = 0.10), medium (r = 0.30), and large (r > 0.50) effect sizes, as described previously ( Cohen, 2013 ). All statistical analyses were performed using the software Statistical Package for the Social Sciences (SPSS 25, IBM, NY, United States), and the significance level was set at p = 0.05.

3.1 Spatiotemporal gait parameters

In comparisons of pre- and post-feedback trials ( Table 2 ), the step length was found to be significantly increased after the ankle trial ( p = 0.043) and showed a tendency to increase after the leg trial ( p = 0.063). Gait speed did not change after the control ( p = 0.176), ankle ( p = 0.237), and leg trials ( p = 0.398). Cadence also did not significantly change after the control ( p = 0.237), ankle ( p = 0.176), and leg trials ( p = 0.237).

Table 2 . Individual changes in spatiotemporal and kinematic gait parameters after feedback trials.

On comparison of the changes in spatiotemporal gait parameters, Gait speed and stride length did not differ between each feedback trial ( Figures 2A, B ). Cadence was found to differ significantly between each feedback trial, and it was decreased during the leg trial compared with the no feedback trial ( p = 0.023, r = 0.54: large, Figure 2C ).

3.2 Kinematic gait parameters

In comparisons of pre- and post-feedback trials ( Table 2 ), ankle plantarflexion angle was found to be significantly increased after the ankle ( p = 0.018) and leg trials ( p = 0.028). The leg extension angle was significantly increased after the leg trial ( p = 0.028). There was a significant increment of velocity after the ankle ( p = 0.018) and leg trials ( p = 0.018).

On comparison of the changes in kinematic gait parameters, leg extension angle was found to differ significantly between each feedback trial ( p < 0.050), and it increased during the leg trial compared with the no feedback trial ( p = 0.048, r = −0.49: medium, Figure 2D ). The ankle plantarflexion angle and increment of velocity differed between each feedback trial ( p = 0.066). The ankle plantarflexion angle showed a tendency to increase after the ankle trial compared with the no feedback trial ( p = 0.098, r = −0.432: medium, Figure 2E ), and there was a higher increment of velocity after the leg trial compared with the no feedback trial ( p = 0.098, r = −0.432: medium, Figure 2F ).

4 Discussion

In this study, we examined the immediate effects of IMU-based real-time feedback gait, focused on either the ankle or leg motion, during overground gait on gait kinematics in older inpatients. IMU-based real-time feedback gait in 60 s immediately changed spatiotemporal and kinematic gait parameters according to the feedback targets. Therefore, this study demonstrated the immediate effect of IMU-based real-time feedback gait focused on the motion of each joint, and it suggests the feasibility of its clinical application for overground gait training in older people.

IMU-based real-time feedback increased gait speed and showed a moderate effect size, but it was not significantly different for each feedback trial. In the ankle and leg trials, gait speed changed by a mean of 0.02–0.03 m/s, and a minimal detectable change in gait speed in community-dwelling older people (0.04–0.06 m/s) has not been observed ( Perera et al., 2006 ). A previous report of older adults shows a similar trend, with no immediate increase in gait speed (mean change 0.08 m/s) after gait training in patients following a stroke ( Hinton et al., 2023 ). In healthy participants using this gait training system, gait speed increased immediately after feedback trials (mean change 0.15–0.19 m/s) ( Miyazaki et al., 2023 ), and these increases were close to or larger than 0.17 m/s, which is reported as the minimal detectable change in healthy participants ( Meldrum et al., 2014 ). Of other spatio-temporal gait parameters, step length was increased after the ankle trial, showed a tendency to increase after the leg trial, and showed a large effect size. In addition, change in cadence was smaller in the leg trial than in the no feedback trial. Participants have experienced increased gait speed by changing gait strategies that alter either cadence or stride length or both ( Howard et al., 2013 ; Baudendistel et al., 2021 ; Tateuchi et al., 2021 ). An immediate effect was observed in healthy adults, and a moderate to large effect size was shown for older inpatients. Thus, for older people in clinical settings, this gait training system may be effective in increasing gait speed through changing their gait strategy by considering intervention time, fatigue, and other factors.

In gait kinematic parameters, the leg extension angle was also significantly increased after the leg trial (mean change 3.2°), and change in the leg extension angle was larger in the leg trial than in the no feedback trial. The ankle plantarflexion angle was significantly increased after the ankle (mean change 6.4°) and leg trials (mean change 3.3°), and higher increment of velocity was observed after both trials. In addition, the ankle plantarflexion angle was significantly increased in the ankle trial compared to the no feedback trial; meanwhile, there was a higher increment of velocity in the leg trial compared to the no feedback trial. These parameters also showed a moderate or greater effect size. Sufficient forward movement of the center of gravity ensured an increase in leg extension angle ( Bowden et al., 2006 ; Balasubramanian et al., 2007 ; Turns et al., 2007 ), which also leads to an increase in the propulsion force ( Hsiao et al., 2015a ; b ; Hsiao et al., 2016 ; Browne and Franz, 2017 ). Similar to the leg extension angle, the increment of velocity during the late stance is an indicator of the propulsion force ( Miyazaki et al., 2019 ), and ankle plantarflexion angle also contributes to increase in step length and propulsion force during gait ( Hsiao et al., 2015a ; Hsiao et al., 2015b ; Zelik and Adamczyk, 2016 ; Browne and Franz, 2017 ). In addition, the measurement error of the leg extension angle is reported as 1.4°–1.9° ( Miyazaki et al., 2019 ), and the minimal detectable change is also reported as 3.8° for the leg extension angle ( Kesar et al., 2011 ) and 2.6° for the ankle plantarflexion angle ( Molina-Rueda et al., 2021 ); changes in these parameters in the leg and ankle trials of the current study were close to or larger than these figures. These gait kinematics during the late stance would be akin to an increase in push-off power, and we facilitated their immediate change using real-time feedback. Therefore, this IMU-based real-time feedback gait is capable of immediately changing gait parameters related to forward propulsion, giving it the potential to improve walking efficiency in older people in clinical settings.

4.1 Potential implications for effective gait practice for older inpatients in clinical settings

Although the sample was small, we were able to implement the protocol for older inpatients. This study did not show a similar immediate effect to that reported for healthy young participants ( Miyazaki et al., 2023 ), but we believe that the current IMU-based real-time feedback gait system has potential for clinical application. The strength of this system is that multiple parameters can be selected, so it is necessary to consider which parameters are most informative, and further study is needed to realize gait practice using the most appropriate feedback target for individuals. Decreasing ankle push-off and propulsion force at the late stance have been reported as gait parameters that change with aging ( Boyer et al., 2017 ) along with dependence on the proximal joint compared with healthy young adults ( DeVita and Hortobagyi, 2000 ; Hortobagyi et al., 2016 ; Kuhman et al., 2018 ; Conway and Franz, 2020 ). In older adults at risk for mobility, disability showed a faster preferred gait speed and physical function in the group with increased stride length compared with the group with increased cadence ( Baudendistel et al., 2021 ). Conversely, another report demonstrates the relationship between ankle power and forward shift of the center of gravity during gait in older people ( Sloot et al., 2021 ). In this study, the mean values of participants were 1.02 m/s for gait speed and 10.56 s for FTSS. In addition, this study especially showed immediate changes in gait kinematics during ankle trials. Therefore, the ankle motion might be a suitable target of IMU-based real-time feedback gait for efficiently increasing gait speed in older people without physical function decline, who did not meet the criteria for physical function decline in sarcopenia (<1.0 m/s for gait speed and/or 12.0 s for FTSS) ( Chen et al., 2020 ).

4.2 Limitations

Our study had several limitations. First, this study examined only the immediate effect and was not able to examine long-term intervention effects. Second, this system used only auditory feedback, making it difficult to set the threshold between the lower and upper limits. In previous studies, gait feedback training using audio and visual feedback was performed using treadmills and monitors that fitted within the optimal range of thresholds ( Schenck and Kesar, 2017 ; Liu et al., 2020 ; Liu et al., 2021 ). Comparisons with other feedback methods such as auditory and vibratory stimulations are also needed. Third, fatigue after each trial was not assessed, and it is unclear whether the intensity of gait feedback was appropriate. Fourth, the small sample size may have increased the variability of outcome measures. Finally, physical function was measured only by FTSS. In this study, participants did not meet the criteria for physical function decline in sarcopenia ( Chen et al., 2020 ). In clinical settings, it is anticipated that this gait training will be implemented for inpatients with poorer physical function. More detailed and varied measurements of physical function, such as individual muscle strength and balance ability, are needed. It is necessary to accumulate several cases to verify the effectiveness of gait training under controlled conditions of disease and physical function. Since the latter systems are not feasible in a clinical setting, we believe that our gait feedback system is more likely to be used in clinical settings. Despite these limitations, this study showed that an immediate change in gait kinematics was observed, and it provides evidence of effective overground gait training for older people in clinical settings.

5 Conclusion

In this study, IMU-based real-time feedback gait immediately changed gait parameters according to the types of each joint motion at late stance during overground gait in older inpatients. This IMU-based real-time feedback gait system also allows multiple gait parameters to be selected, which could lead to effective overground gait training using feedback targets appropriate for each inpatient in clinical settings. To achieve effective gait practice in clinical settings for older inpatients, further study is needed to clarify the long-term effects of IMU-based real-time feedback gait on gait parameters and the appropriate target of gait practice for each individual.

Data availability statement

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation.

Ethics statement

The studies involving humans were approved by the Declaration of Helsinki and approved by the Ethics Committee on Epidemiological Studies of Tarumizu Central Hospital (approval number: 20-8). The studies were conducted in accordance with the local legislation and institutional requirements. The participants provided their written informed consent to participate in this study.

Author contributions

TM: writing–original draft, methodology, funding acquisition, formal analysis, and data curation. RK: writing–review and editing, supervision, and methodology. YT: writing–review and editing, investigation, data curation, and conceptualization. DS: writing–review and editing, project administration, investigation, and data curation. SA: writing–review and editing, formal analysis, and conceptualization. HM: writing–review and editing. YU: writing–review and editing. SN: writing–review and editing and investigation. YN: writing–review and editing, formal analysis, and conceptualization. MK: writing–review and editing, supervision, project administration, and methodology.

The author(s) declare that financial support was received for the research, authorship, and/or publication of this article. This work was partly supported by a research grant from the Mikiya Science and Technology Foundation.

Acknowledgments

The authors wish to thank Tomokazu Kaji and Toshihiro Takenaka for coordinating this study and Yuki Turuda for building the gait feedback system.

Conflict of interest

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

Publisher’s note

All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors, and the reviewers. Any product that may be evaluated in this article, or claim that may be made by its manufacturer, is not guaranteed or endorsed by the publisher.

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Keywords: clinical application, gait training, wearable sensor, gait analysis, propulsion

Citation: Miyazaki T, Kiyama R, Takeshita Y, Shimose D, Araki S, Matsuura H, Uto Y, Nakashima S, Nakai Y and Kawada M (2024) Inertial measurement unit-based real-time feedback gait immediately changes gait parameters in older inpatients: a pilot study. Front. Physiol. 15:1384313. doi: 10.3389/fphys.2024.1384313

Received: 09 February 2024; Accepted: 24 July 2024; Published: 06 August 2024.

Reviewed by:

Copyright © 2024 Miyazaki, Kiyama, Takeshita, Shimose, Araki, Matsuura, Uto, Nakashima, Nakai and Kawada. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

*Correspondence: Takasuke Miyazaki, [email protected]

Disclaimer: All claims expressed in this article are solely those of the authors and do not necessarily represent those of their affiliated organizations, or those of the publisher, the editors and the reviewers. Any product that may be evaluated in this article or claim that may be made by its manufacturer is not guaranteed or endorsed by the publisher.

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Gait analysis: Approaches and applications

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Gait analysis methods in rehabilitation

Richard baker.

1 Hugh Williamson Gait Analysis Service, Royal Children's Hospital, Parkville, Victoria, Australia

2 Gait CCRE, Murdoch Children's Research Institute, Parkville, Victoria, Australia

3 Department of Mechanical and Manufacturing Engineering, University of Melbourne, Parkville, Australia

4 Musculoskeletal Research Centre, La Trobe University, Bundoora, Victoria, Australia

This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( http://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Introduction

Brand's four reasons for clinical tests and his analysis of the characteristics of valid biomechanical tests for use in orthopaedics are taken as a basis for determining what methodologies are required for gait analysis in a clinical rehabilitation context.

Measurement methods in clinical gait analysis

The state of the art of optical systems capable of measuring the positions of retro-reflective markers placed on the skin is sufficiently advanced that they are probably no longer a significant source of error in clinical gait analysis. Determining the anthropometry of the subject and compensating for soft tissue movement in relation to the under-lying bones are now the principal problems. Techniques for using functional tests to determine joint centres and axes of rotation are starting to be used successfully. Probably the last great challenge for optical systems is in using computational techniques to compensate for soft tissue measurements. In the long term future it is possible that direct imaging of bones and joints in three dimensions (using MRI or fluoroscopy) may replace marker based systems.

Methods for interpreting gait analysis data

There is still not an accepted general theory of why we walk the way we do. In the absence of this, many explanations of walking address the mechanisms by which specific movements are achieved by particular muscles. A whole new methodology is developing to determine the functions of individual muscles. This needs further development and validation. A particular requirement is for subject specific models incorporating 3-dimensional imaging data of the musculo-skeletal anatomy with kinematic and kinetic data.

Methods for understanding the effects of intervention

Clinical gait analysis is extremely limited if it does not allow clinicians to choose between alternative possible interventions or to predict outcomes. This can be achieved either by rigorously planned clinical trials or using theoretical models. The evidence base is generally poor partly because of the limited number of prospective clinical trials that have been completed and more such studies are essential. Very recent work has started to show the potential of using models of the mechanisms by which people with pathology walk in order to simulate different potential interventions. The development of these models offers considerable promise for new clinical applications of gait analysis.

For the purposes of this paper gait analysis will be assumed to refer to the instrumented measurement of the movement patterns that make up walking and the associated interpretation of these. The core of most contemporary gait analysis is the measurement of joint kinematics and kinetics. Other measurements regularly made are electromyography (EMG), oxygen consumption and foot pressures. A systematic physical examination of the patient is usually conducted as part of a gait analysis.

Rehabilitation is a clinical discipline and this paper will thus concentrate on clinical gait analysis. Richard Brand [ 1 , 2 ] proposed four reasons for performing any clinical test (see Table ​ Table1). 1 ). The third of these might actually be taken as a definition of the word clinical i.e. a clinical test is one conducted in order to select from among different management options for a patient (including the possibility of not intervening).

Reasons performing clinical tests as stated by Brand [1, 2])

Much contemporary gait analysis is done for the purpose of clinical research . This differs from clinical testing in that the reason is not to make clinical decisions for the individual patient, but to learn about a condition affecting a group of patients or the effect of an intervention. It is important to remember that the criteria for valid clinical research may not be the same as those for valid clinical testing. For example if a measurement made on a patient cannot be relied upon because of random errors then that measurement will not be useful for clinical purposes. By increasing the number of patients in a sample however, even measurements with quite large random errors can result in meaningful conclusions in clinical research. This paper will focus on gait analysis for clinical use. It will also focus on methodology rather than areas of clinical application.

Brand's [ 1 , 2 ] other three possible reasons for performing any clinical test are to distinguish between disease entities (diagnosis), to determine the severity, extent or nature of a disease or injury (assessment), and to predict outcomes of intervention (or the absence of intervention). The monitoring of the progress of a patient's condition either following intervention or in its absence might be regarded as an additional reason. This modification of Brand's approach is summarised in Table ​ Table2 2 .

Reasons performing clinical gait analysis (modified from Brand [1, 2])

Brand went on to propose a number of criteria for assessing the usefulness of biomechanical measurements in general which, with some modification, can be used as criteria for the usefulness of all clinical gait analysis. These are listed in Table ​ Table3. 3 . The first requirement of any clinical measurement is that it should characterise the patient, that is if the patient attends on two separate occasions, between which his or her condition might be considered as stable, the measurements taken should be similar. This requires that the measurement technique itself is repeatable but also that the quantity being measured is stable and independent of factors such as mood, motivation or pain. Measurements can be repeatable and stable without necessarily being accurate (representative of a specific physical quantity). Such tests can be clinically useful but will be much easier to interpret if they are also accurate. In an era of evidence based clinical practice it is essential that any measurement techniques are appropriately validated which must include assessments of both their repeatability and accuracy.

Criteria for biomechanical measures (extracted from text of Brand [1])

In order to perform a diagnostic function it is necessary for measurements to be able to distinguish normal from abnormal patterns of movement and also between the characteristics of one disease entity and another. There are two aspects to this. The first is having measurement systems capable of working to adequate precision. The second is a knowledge of what characterises normal walking or a particular disease entity.

The requirement for patient assessment pre-supposes that a diagnosis does not give sufficient information to determine the most appropriate management for a patient and that measuring the precise characteristics of a patient's condition are essential for this. Measurements thus have to be sufficiently precise to reveal clinically important differences between patients with the same diagnosis. For monitoring purposes measurements need to be sufficiently precise to be able to determine whether a patient's condition is stable, improving or deteriorating.

Brand suggested that the measurement technique should not affect the function it is measuring. The walking performed in a gait analysis laboratory however, with the patient concentrating on what they are doing in an idealised environment, is not necessarily representative of their normal walking. At the very least this must be taken into account when interpreting results.

Gait analysis should reveal information that is useful to the clinician and this will generally require that results are reported in terms analogous to accepted clinical concepts. It must be cost-effective, that is the benefit of performing the test must be worth the cost. This balance need not necessarily be determined in purely financial terms but the financial cost of gait analysis is a significant factor. Finally there is no point doing any clinical test if the results could be obtained sufficiently well by simply observing the patient

The information obtained by assessing the patient is that used for selecting management options. This process does not, therefore, make further demands on the measurement systems but does require an understanding of how the patient's condition is likely to be affected by an intervention (or none) to a level sufficient to determine which options are preferable. Prediction of outcomes takes this one stage further to being able to determine not only which management option is best but also how the patient will be after that intervention.

This sequential analysis of the four potential purposes of clinical tests reveals a progression from just requiring reliable and precise measurements to the additional requirement of having an understanding of how such information is incorporated into clinical practice. The state of the art is that the measurement component of gait analysis can reasonably be described as an objective process whereas the interpretation component is predominantly subjective.

Making the interpretive component more objective can be achieved in two ways. The first is to develop a general theory of how people walk whether they have recognised pathology or not. As long ago as 1982 Cappozzo lamented, "The approaches to clinical gait analysis and evaluation are not supported by general theories" [ 3 ] and despite over 20 years of intense activity this is still a reasonable summary of the state of the art. The second approach, which must operate in the absence of the former, is to conduct clinical research to ascertain the outcome of particular interventions on groups of patients characterised by certain measurements. Most of the knowledge base used in the interpretive component of gait analysis comes from such studies. It is because there are relatively few studies available to base such interpretations on that the subjective element of interpretation is necessary in contemporary clinical gait analysis.

Modern clinical gait analysis traces its origins back to the early 1980s with the opening of the laboratory developed by the United Technologies Corporation at Newington, Connecticut and those provided with equipment by Oxford Dynamics (later to become Oxford Metrics) in Boston, Glasgow and Dundee. Retro-reflective markers were placed on the skin in relation to bony landmarks. These were illuminated stroboscopically and detected by modified video cameras. If two or more cameras detect a marker and the position and orientation of these cameras are known then it is possible to detect the three-dimensional position of that marker [ 4 ].

Whilst the basic principles remain the same as the earliest systems, the speed, accuracy and reliability has advanced beyond all recognition. It is not uncommon now to find clinical systems using 8, 10 or more cameras functioning at over 100 Hz and capable of detecting reliably the presence of many tens of markers of between 9 and 25 mm diameter. Calibration of the systems (the determination of the position, orientation and optical and electronic characteristics of the cameras) can generally be accomplished in less than a minute. Marker positions from clinical trials can be reconstructed and markers labelled automatically in real time (although this feature is often not essential for clinical studies). The determination of the accuracy of such systems is now generally limited by the accuracy of any alternative means to determine marker position and can be taken to be of the order of 1 mm. This is probably an order of magnitude smaller than other sources of error in determining joint kinematics and kinetics. This particular measurement technology has thus reached a mature state of development that, whilst advances will almost certainly continue, already probably delivers all that is required by conventional gait analysis [ 5 ].

The same cannot be said of the computer models used to derive joint kinematics and kinetics from the marker position data supplied by the measurement hardware. Almost all commercially available clinical systems use some variant of the Conventional Gait Model [ 6 ] which has been referred to as the Newington, Gage, Davis [ 7 ], Helen Hayes, Kadaba [ 8 , 9 ] or Vicon Clinical Manager (VCM) model. This was developed using the minimum number of markers possible to determine 3-dimensional kinematics and kinetics [ 10 , 11 ] of the lower limb at a time when measurement systems were only capable of detecting a handful of markers. It assumes three degree of freedom joints for the hip and knee and a two degree of freedom joint at the ankle. The model is hierarchical requiring the proximal segments to have been detected in order that distal segments can be defined and incorporates regression equations to determine the position of the hip joint centre with respect to pelvic markers. Kinetics are determined using an inverse dynamics approach which generally requires considerable filtering to give any useful signals. An alternative system the Cleveland Clinic Model based around a cluster of markers on a rigid base attached to each segment is the only other widely used model. Unfortunately documentation of this model in the scientific literature is very poor.

The problem of limited repeatability

The primary problem of current measurement technology is that of reliability in routine clinical use. Several studies have now been reported in which a single subject has been analysed in a number of different laboratories [ 12 - 14 ]. These have shown a degree of variability between sites that would appear to be sufficient to undermine clinical applications. In retrospect, the original studies of the reliability are flawed. There was no such study of the Davis implementation of the model and the statistics used by Kadaba et al [ 8 , 9 ] to report reliability of their implementation probably acted to mask deficiencies. In particular, use of relative measures of reliability such as the coefficient of multiple correlation (CMC) makes interpretation of findings difficult. Almost all reliability studies have been done on subjects without pathology where marker placement is reasonably straightforward. Reliability for clinical populations is rarely reported in the literature and is almost certainly inferior.

At least one recent study has shown that it is possible to get levels of reliability sufficient to justify the continued clinical use of gait analysis within a single centre [ 15 ]. Too few centres however are providing evidence to establish that this is the rule rather than the exception.

Source of error: Model calibration

There are two principal sources of error. The first is the difficulty determining the anthropometry of the individual subject (known as model calibration ). This has two aspects, placing markers accurately with respect to specific anatomical landmarks and determining the location of the joint centres (and other anatomical features) in relation to these markers. Failure to place markers accurately is probably the single greatest contributor to measurement variability in contemporary clinical gait analysis. This is partly a matter of appropriate staff training and quality assurance but at least as important, and more fundamental, is the problem that many of the landmarks used to guide marker placement are not themselves particularly well defined in patients with certain conditions [ 16 ]. Even when bony landmarks are sharply defined an increasing number of patients have a considerable thickness of subcutaneous fat that makes palpation difficult.

The Conventional Gait Model uses regression equations to determine the position of the hip joint centre in relation to the pelvis. Both Bell's [ 17 - 19 ] and Davis' [ 7 ] equations are commonly used and there is now good evidence that neither is satisfactory in healthy adults [ 20 ]. There have still been no published studies of whether either is valid for healthy children. Children with orthopaedic conditions including cerebral palsy may often have dysplasia of the hip or deformity of the pelvis, and it is exceedingly unlikely that any form of regression equation could be used in these patients to determine hip joint position.

Methods for moving away from anatomical landmarks and regressions equations for determining joint centres have been around for nearly a decade, the process being known as anatomical calibration [ 21 ]. They rely on calibration movements to be performed before capturing walking data and some form of fitting of the measured marker positions to an underlying model of how the body moves. The simplest example is probably the determination of the hip joint centre. It is assumed that the hip joint moves as a ball and socket joint about centre of rotation fixed in the pelvis. Any marker on the femur would thus be expected to describe a path on the surface of a sphere centred on the hip joint centre when the hip joint is moving. A least squares fit of the measured data to such a sphere allows the location of that joint centre to be determined [ 20 , 22 ]. Similar approaches are applicable to determine that axis of the knee joint which for this purpose has often been assumed to be a simple hinge joint.

Various approaches to fitting data to an underlying model have been attempted and many seem to give reasonable results [ 20 , 22 - 28 ]. Such techniques have not so far been widely accepted into clinical practice probably because there is a perception that such calibration trials are too difficult for patients to execute. At least one clinical lab however has now committed itself to implementing such techniques into routine practice and has reported failure to perform test adequately in only one of over 700 patients tested so far.

Sources of error: Soft tissue artefact

The second source of error is the degree of movement of the skin, muscle and other soft tissues in relation to the bones that occurs during walking. This is perhaps most marked in relation to the rotational profile of the hip. Lamoreux [ 29 ], as far back as 1991, reported that with optimal placement of thigh wands only 65% of transverse plane hip joint rotation was detected and that with poor placement this could be as little as 35%.

The problem of skin and other soft tissue movement is more problematic than that of model calibration. Lu and O'Connor where the first to propose fitting a model of how the body is expected to move to marker co-ordinate data [ 30 ] using an optimisation approach. This model uses a least squares fit, similar to some of the techniques described above for model calibration, and thus makes no assumptions about the nature of the soft tissue movement. Other similar models have now been made commercially available [ 26 ]. More recent studies have started to try to map out the movement of markers with respect to the underlying bones [ 31 , 32 ]. If such movement can be characterised as a function of joint angle then, in principle, this knowledge could be built into a model to allow such movements to be compensated for. Such mapping is only likely to be useful if it can be shown that soft tissue movement is consistent across a range of subjects and activities. It is not clear at present whether these conditions are satisfied. A particular problem in regard to mapping soft tissue movement is that of defining what the "true" movement of the bones is. In the absence of any gold standard a variety of assumptions are being used most of which have serious limitations.

The development of a gold standard method for determining joint movement will probably require a move away from skin-mounted markers (or other sensors). Once such technology is available however it is quite possible that this will supersede the presently available systems. The cost of any such new systems however is likely to prohibit ready clinical availability in the foreseeable future.

There has been some work done on markerless optical methods. By placing a number of video cameras around a subject and tracing the silhouette of the walking subject on each it is possible to generate a 3-dimentional silhouette of that subject. This has already been achieved but the next step of using such a silhouette to determine the co-ordinate systems associated with the moving body segments has not yet been satisfactorily achieved.

It is possible that the problem of skin movement can only be satisfactorily addressed by making direct measurements of bone position. It is now possible to take 3-dimensional images of bones (and muscles) using MRI but only within a very restricted capture volume [ 33 - 35 ]. The image processing problem of automatically determining a bone embedded axis system from such images has yet to be solved satisfactorily. Similarly both uniplanar and biplanar cine fluoroscopy [ 36 - 40 ] has been used to detect the 3-dimensional movement of the internal knee prostheses during a variety of movements. This is possible because a knowledge of the exact size and shape of the prosthetic components and their opacity to x-rays greatly simplifies the image processing problem. Using similar techniques to determine the movement of joints has also been reported [ 41 - 43 ]

Methods for interpreting clinical gait analysis data

The second element of clinical gait analysis is the interpretation of data. Conventions for describing 3-dimensional joint kinematics and kinetics are well formulated. Many laboratories are augmenting conventional kinematics and kinetics with muscle length and, less commonly, moment arm graphs. Normal patterns of movement as represented by these data are now generally fairly well understood by clinical specialists although there is actually very little normative data published in the peer-reviewed literature. Similarly, many abnormal patterns of movement are quite widely recognised by clinicians but there few published attempts at formal classification of these [ 44 - 46 ]. Many clinicians have learnt to associate particular abnormal patterns in particular patient groups with particular impairments of body structure and function. Intervention based on such an understanding often leads to a normalisation of gait patterns at subsequent assessments (e.g. [ 47 - 55 ]). It is on this basis that clinical gait analysis operates at present.

Despite the widespread acceptance of many of these conventions there are still problems. Baker [ 56 ] demonstrated that the Euler sequence used to calculate pelvic angles gives rise to data that can be mis-leading to clinicians and proposed an alternative to correct this which is yet to be adopted widely within clinical analysis. Methods for interpreting angles in three dimensions, either in terms of Euler/Cardan rotations or the Grood and Suntay convention [ 57 , 58 ] are not well understood either by clinicians or many bioengineers. A recent attempt to standardise the reporting of joint angles [ 59 ] proposed a different convention to that of the Conventional Gait Model and the continuing debate as to which is preferable illustrates this confusion [ 60 , 61 ]. Joint moments are generally reported with reference to orthogonal axis systems fixed in the distal (Conventional Gait Model) or proximal segments (or occasionally the laboratory axis system). These differ significantly depending on the axis system chosen [ 6 , 62 ] yet there has been no debate about which if any is preferable. Reporting moments about orthogonal axis systems and joint rotations about non-orthogonal ones leads to difficulties in relating the moments to the changes in joint angles to which they are related. The use of muscle moment arms will be discussed further below but it is interesting that there is no straightforward definition of the meaning of the term moment arm in three dimensions [ 63 ] and it is often not clear how such data should be interpreted.

Perhaps the most important limitation of our present understanding of human walking, however, is that it is primarily descriptive. We know what happens rather than why it happens. Many in the clinical gait analysis community regard kinematics as descriptive but contend that kinetics explain movement patterns. This is almost certainly misguided. Kinetics are simply another set of measurements and can thus only be descriptive.

There have been various attempts at establishing a theory of walking but none is particularly convincing. Saunders, Inman and Eberhart's determinants of normal walking [ 64 ] are perhaps the best known of these. Recent publications however have questioned how the detail of these reflects experimental data [ 65 - 70 ]. Gage [ 71 , 72 ] based his pre-requisites of gait on earlier work by Perry [ 73 ] but these are best regarded as pointers to where particular patients are deficient rather than explanations of how they are achieving walking with or without pathology.

Perhaps the closest we have come so far to understanding why we walk the way we do has come from the work of Pandy and Anderson [ 74 , 75 ]. They have shown that it is possible to construct a mathematical simulation of muscle function during normal walking based on the assumption that the total consumption of energy per unit distance walked is minimised. The authors, however, commented that the model seems more dependent on the boundary conditions imposed than on the nature of the optimisation function. Further, because of the complex nature of the optimisation process driving the model it is still difficult to explain how the precise characteristics of any particular feature of the walking pattern affect the overall calculation of energy expenditure. So far such a model has only been constructed for normal walking.

Conceptually, modifying such models to incorporate a specific abnormality of the musculo-skeletal anatomy such as a leg length discrepancy or contracture of a particular muscle is reasonably straightforward. It is much less certain whether such techniques can be applied at all to patients with neuromuscular pathology who are most frequently seen by clinical gait analysis services. Optimisation techniques assume that movements are controlled in such a way that a specific control function is minimised. In many neuromuscular conditions (Cerebral Palsy, Parkinson's disease, adult hemiplegia) the problem is one of a loss of central control and this would appear to invalidate any techniques modelling human movement as an optimised process.

If such models are developed it will be interesting to see whether they give any insights into the clinical management of patients. Further it will be interesting to see whether their use leads to an understanding of why we walk the way we do which can be formulated as theories that are applicable without the use of such complex models.

Whilst the answer to this question still seems as far away as ever, significant advances have been made over recent years in understanding the mechanisms by which we walk particularly in the way that muscles act. For many years it was assumed that a muscle's anatomical position determines how it acts. It was assumed for example that the action of the hamstrings, passing behind the knee, was always to flex the knee. It is only comparatively recently that biomechanists have come to appreciate that any individual muscle has an effect on all the segments of the body and that in some circumstances this may result in a muscle having an action different to its anatomical function [ 76 - 81 ]. It is now fairly well accepted, for example, that the hamstrings functions as a knee extensor during early stance in normal walking because its effect in extending the hip has a secondary tendency to extend the knee which is greater than its direct effect as an anatomical knee flexor [ 82 ].

Such work depends on knowing the joint kinematics and kinetics and inertial properties of the body segments. These can be used to estimate the forces in individual muscles [ 81 - 83 ]. This is an indeterminate problem so is dependent on an optimisation approach (and the validity of this in neuromuscular pathology is questioned in the same way as that of the simulations described above). Once the muscle forces are known forward modelling can be used to determine the effect that a given muscle is having on any segment (or joint) of the body. Until very recently the first part of this problem, the estimation of muscle forces had not been achieved which limited the application of the second part, the forward modelling to data obtained from the simulations described above [ 74 , 75 ]. Recently methods have been develop to estimate the muscle forces required to generate measured joint kinematics and ground reaction forces and have been used both to understand the function of individual muscles during pathological gait and predict the effect of interventions [ 84 , 85 ]. These have been based on scaled models of the adult musculo-skeletal anatomy.

There is also considerable debate at present about the validity of these techniques (the simulations, the estimations of muscle forces and the forward modelling). Whilst the general principles are sound the techniques are known to be extremely sensitive to certain aspects of their implementation (and may be sensitive to many more). For example the forward modelling in particular is sensitive to how the interaction between the foot and the floor is modelled with there being no clear consensus as to the most appropriate method for this [ 74 , 77 , 81 ].

Methods for understanding the effect of intervention

Understanding how to interpret clinical gait analysis data is not itself sufficient to allow selection from amongst treatment options (Table ​ (Table1). 1 ). For this it is also necessary to know what effect the available interventions are likely to have on someone's walking pattern. If we had a general theory of walking then it might be possible to develop a theoretical basis for considering the effect of any intervention. For patients whose walking could be modelled using a simulation based on specific musculo-skeletal abnormalities it might be possible to use similar simulations to model what might happen if partial correction of those abnormalities were attempted (obviously full correction would restore normal walking!). The author is unaware of any published work at this level at present.

There are then two methods for understanding the effects of intervention in these patients; clinical research to establish what the actual effect of a given intervention is or using knowledge of the mechanisms of walking to predict the effect of modifying the characteristics of the musculo-skeletal anatomy.

By far the most common approach to date has been the use of clinical research – the comparison of gait patterns before and after a particular intervention [ 47 - 55 , 86 - 88 ]. Even so there have been comparatively few studies that have given conclusive findings. Many studies which claim to have done so have quite serious methodological flaws. This is particularly true of research into orthopaedic surgery for children with CP where researchers have used retrospective audits of clinical practice to try and answer specific questions. Many of these studies attempt to make inferences about individual procedures which have only ever been performed as part of a multi-level surgical package [ 47 , 49 - 51 , 54 , 55 ]. It is impossible to tell from these studies which effects are due to the particular procedure being considered and which are due to the overall package. Several studies have attempted to separate out those effects by dividing patients into those who have and those who have not had a particular procedure as part of the overall package of surgery and use methods to compare groups similar to those that would be used for a randomised clinical trial [ 47 , 54 ]. The validity of this approach is questionable, however, because generally the two groups of patients were not similar to start with. Those that had the procedure had it because it was considered that the patient needed it and vice versa. Comparison of the two groups to give insight into the effect of the procedure is thus invalid.

An alternative to the use of clinical trials is to use knowledge of the mechanisms of walking as a basis for modelling the effect of changing that mechanism. Reports of such studies are now starting to emerge. For example Arnold et al. [ 84 ] have reported a subject specific model of a cerebral palsy patient with a stiff knee gait and used it to predict the effect of three different potential interventions. These indicated a preferable intervention and the post-intervention gait data showed at least qualitative agreement with the theoretical predictions.

Competing interests

The author has received research funding from Oxford Metrics Plc (Oxford, UK)

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• Published: 26 July 2024

Adaptive hierarchical origami-based metastructures

• Yanbin Li   ORCID: orcid.org/0000-0003-0870-4507 1   na1 ,
• Antonio Di Lallo 1   na1 ,
• Junxi Zhu 1 ,
• Yinding Chi 1 ,
• Hao Su   ORCID: orcid.org/0000-0003-3299-7418 1 , 2 , 3 &
• Jie Yin   ORCID: orcid.org/0000-0002-6297-1262 1  

Nature Communications volume  15 , Article number:  6247 ( 2024 ) Cite this article

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• Applied mathematics
• Mechanical engineering

Shape-morphing capabilities are crucial for enabling multifunctionality in both biological and artificial systems. Various strategies for shape morphing have been proposed for applications in metamaterials and robotics. However, few of these approaches have achieved the ability to seamlessly transform into a multitude of volumetric shapes post-fabrication using a relatively simple actuation and control mechanism. Taking inspiration from thick origami and hierarchies in nature, we present a hierarchical construction method based on polyhedrons to create an extensive library of compact origami metastructures. We show that a single hierarchical origami structure can autonomously adapt to over 10 3 versatile architectural configurations, achieved with the utilization of fewer than 3 actuation degrees of freedom and employing simple transition kinematics. We uncover the fundamental principles governing theses shape transformation through theoretical models. Furthermore, we also demonstrate the wide-ranging potential applications of these transformable hierarchical structures. These include their uses as untethered and autonomous robotic transformers capable of various gait-shifting and multidirectional locomotion, as well as rapidly self-deployable and self-reconfigurable architecture, exemplifying its scalability up to the meter scale. Lastly, we introduce the concept of multitask reconfigurable and deployable space robots and habitats, showcasing the adaptability and versatility of these metastructures.

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Introduction.

Versatile shape-morphing capability is crucial for enabling multifunctionality in both biological and artificial systems, allowing them to adapt to diverse environments and applications 1 , 2 , 3 . For example, the mimic octopus can rapidly transform into up to 13 distinct volumetric shapes, mimicking various marine species 1 . In the realm of artificial systems, there has been a range of strategies proposed to create shape-morphing structures, including continuous forms of beams, plates, and shells 4 , 5 , 6 , bar-linkage networks or mechanical kinematic mechanisms 7 , 8 , 9 , 10 , 11 , 12 , 13 , folding or cutting-based origami/kirigami structures 12 , 14 , 15 , 16 , 17 , 18 , 19 , 20 , 21 , 22 , and reconfigurable robotic structures composed of assembled magnetic or jointed modules 23 , 24 , 25 , 26 , 27 , 28 . These structures have found broad applications in transformable architecture 21 , 29 , reconfigurable robotics 25 , 30 , biomedical devices 8 , 31 , flexible spacecraft 32 , 33 , multifunctional architected materials 20 , 34 , reprogrammable shape-morphing matter 6 , 35 , 36 , as well as deployable structures that can undergo dramatic volume change for convenient storage and transport 15 , 29 , 32 , 33 , 37 , 38 , 39 .

However, despite these advancements, artificial shape-morphing structures have yet to rival their biological counterparts in terms of the diversity of attainable volumetric shapes, as well as the efficiency and autonomy with which such versatile shape morphing can be achieved through simple actuation and control 6 , 23 , 24 , 26 , 27 , 35 , 36 . One of the primary challenges resides in the tradeoff between theoretically allowable versatility of shape-morphing, which encompasses the quantity and diversity/type of reconfigured shapes, and practical controllability in terms of actuation. For instance, while previously reported structures 11 , 23 , 24 , 26 , 27 , 28 , 36 have demonstrated the ability to change into a vast number of distinct shapes, they often require exceedingly complex actuation and control systems. This complexity can render the shape morphing process tedious, time-consuming, and energy-inefficient. On the other hand, certain structures may exhibit simpler reconfiguration kinematics 3 , 5 , 6 , 7 , 15 , 31 , 40 , 41 , 42 , enabling them to feasibly attain desired shapes. However, their specified structural forms may largely limit the achievable reconfigured shapes within few specific categories. These challenges, along with others such as complex reconfiguration kinematics, poor re-programmability, lack of inverse design capability, and limited functionality of the reconfigured shapes, as summarized in Supplementary Table  1 , could considerably impede the broad applications of shape-morphing structures in areas such as reconfigurable architecture, metamaterials, and robotics (see more details in Supplementary Note  1 ). The versatility of shape morphing is intricately linked to a structure’s mobility, i.e., the number of degrees of freedom (DOF). Theoretically, structures with a higher number of DOFs tend to exhibit greater versatility in shape morphing 11 , 23 , 24 , 25 , 26 , 27 , 35 , 36 . However, this very versatility in theory often makes it exceedingly difficult to actuate structures with higher DOFs, considering the potential need for distributed actuation of each DOF 25 .

Conventional rigid mechanism-based origami structures, constrained by their folding interconnections, are limited to morphing between their original and compact states due to one single DOF. This limitation simplifies actuation and deployment but sacrifices the potential for achieving a variety of shapes 12 , 13 , 16 , 18 , 22 , 38 , 40 , 43 . To address this limitation, recent advances have introduced modular origami metastructures composed of assembled polyhedron-shaped modules 26 , 36 , 39 , such as cubes and tetrahedrons, etc. These structures offer more than four mobilities. For example, recent studies demonstrated that a single unit cell consisting of six extruded cubes could transform into four different configurations using four distributed pneumatic actuators to control folding angles 39 . However, when scaling up to a 4 × 4 × 4 periodic meta-structures to achieve similar transformations, it requires a staggering 96 distributed actuators for each DOF 39 , resulting in low actuation efficiency. More recently, we proposed shape-morphing planar kinematic origami/kirigami modules composed of a closed-loop connection of eight cubes 36 . These modules can be manually transformed into over five different configurations via kinematic bifurcation. When assembled into a 5 × 5 array, they theoretically offer over 10,000 mobilities through bifurcation 36 . However, practically, they pose grand challenges in terms of actuation and control. Similarly, discrete kinematic cube-based modules are often assembled into lattice, chain, or hybrid architectures and used in robotic structures with higher DOFs for multifunctional modular reconfigurable robots 25 . Although these modular origami and robotic structures offer enhanced shape-morphing capabilities, they typically require control and actuation systems for each module. This complexity results in lengthy and intricate reconfiguration steps, as well as complex and time-consuming actuation, morphing kinematics, and reconfiguration paths, primarily due to their redundant DOFs 11 , 25 , 26 , 27 , 35 , 36 (Supplementary Table  1 and related discussions in Supplementary Note  1 ).

Drawing inspiration from planar thick-panel origami 12 , 18 , 22 , 36 and hierarchical materials/structures 44 , 45 , 46 , 47 in nature and engineering, here, we propose leveraging hierarchical architecture of spatial closed-loop mechanisms interconnected both within (locally) and across (globally) each hierarchical level to address the versatility-actuation tradeoff in an example system of highly reconfigurable hierarchical origami metastructures. As illustrated in Fig.  1 a, a base or level-1 structure is a spatial closed-loop mechanism consisting of n rigid linkages and n rotational hinges, an n R looped mechanism. Simply replacing each rigid linkage in a k R looped mechanism with the level-1 structure creates a level-2 “ k R” spatial looped flexible mechanism (Fig.  1b ), since each linkage becomes an n R looped mechanism, with k being the number of rotational hinges at level 2 (note that k is not necessarily equal to n ). The rotary hinges can employ origami line folds and the rigid links can take variously shaped structural elements, such as thick plates and polyhedrons (e.g., cubes, triangular or hexagonal prisms) (Fig.  1c ). The polyhedrons can be combinatorically connected at their edges using rotary hinges at each hierarchical level, offering extensive design space for diverse reconfigurable hierarchical metastructures (Fig.  1d – f and Supplementary Note  2 ).

Schematic illustrations of a level-1 metastructure composed of an n R spatial looped mechanism with n rotary hinges and n rigid linkages ( a ) and a level-2 metastructure composed of a “ k R” spatial looped mechanism at level 2 and n R looped mechanisms at level 1 ( b ). c The designs of rotary hinges and rigid linkages in the forms of respective origami line fold and different polyhedrons. d Illustration of two types of reconfigurable metastructures using planar and spatial tessellation of thin plates and prims, respectively. Examples of 3D-printed prototypes of self-reconfigurable level-1 ( e ) and level-2 ( f ) origami-based robotic metastructures actuated by electrical servomotors. Scale bar: 3 cm. The level-1 and level-2 metastructures are composed of closed-loop connections of 8 and 32 cubes, respectively. g Demonstration of the advantages of hierarchical looped mechanism in creating self-reconfigurable metastructures with versatile shape morphing under fewer reconfiguration DOFs (actuated servomotors) than 3.

We demonstrate the unprecedented properties of the metastructures arising from their hierarchical architecture of spatial closed-loop mechanisms. We find that hierarchical closed-loop mechanisms naturally introduce intricate geometric constraints that dramatically reduce the number of active DOFs required for shape morphing, even when involving a large number of structural elements (Fig.  1f, g ). Benefiting from this hierarchical coupling of closed-loop mechanisms, we show that these hierarchical origami metastructures can be efficiently actuated and controlled while achieving a wealth of versatile morphed shapes (over 10 3 ) through simple reconfiguration kinematics with low actuation DOF (≤3) (Fig.  1g ). The proposed construction strategy unlocks a vast design space by orchestrating combinatorial folding both within and across each hierarchical level, relying on spatial closed-loop bar-linkage mechanisms. It effectively overcomes the intrinsic limitations in our previous ad-hoc shape-morphing designs with similar structural elements 36 , including geometric frustrations, large number of DOF, and a lack of generalizability due to the use of units with specific shapes 13 (see Supplementary Note  1.2 for detailed comparison). Compared to the state-of-the-art shape-morphing systems 7 , 8 , 9 , 10 , 11 , 14 , 16 , 18 , 22 , 24 , 25 , 26 , 27 , 28 , 29 , 30 , 32 , 36 , 37 , 39 , 40 , 43 , 48 , our combinatorial and hierarchical origami-inspired design shows superior multi-capabilities, including high reconfiguration and actuation efficiency (requiring less time and fewer transition steps and actuations), simple kinematics and control, high (re)-programmability, a large number of achievable shapes, and potential multi-functionalities (see Supplementary Note  1.1 and Supplementary Table  1 for detailed comparison). We explore the underlying science of versatile shape morphing and actuation in the hierarchical origami metastructures, as well as their applications in self-reconfigurable robotics, rapidly self-deployable and transformable buildings, and multi-task reconfigurable space robots and infrastructure.

Hierarchical origami-based shape-morphing structures with combinatorial design capability

Figure  2a–c and Supplementary Figs.  1 – 3 illustrate the hierarchical approach employed to construct a category of planar thick-panel origami-based shape-morphing structures. In Fig.  2a , the level-1 structure represents an over-constrained rigid spatial bar-linkage looped mechanism, characterized by the number of linkages being equal to or greater than the connected bars. This structure consists of n (where n  = 4, 6, 8) rigid cubes (Fig.  2a , i) serving as linkages interconnected by n hinge joints (i.e., line folds) at cube edges functioning as rotatable bars (see details in Fig.  2a , ii) 36 . These hinges are highlighted by yellow lines in Fig.  2a , iii. An example of a level-1 structure with n  = 8 is shown in Fig.  2a , ii and iii, while additional examples with n  = 4 and 6 are depicted in Supplementary Fig.  1a–c .

a–c Schematics of constructing level-1 ( a ), level-2 ( b ), and level-3 ( c ) reconfigurable and deployable structures using hierarchical closed-loop rigid bar (line hinges)-linkage (cubes) mechanisms (column ii) as different-leveled structural motifs (column iii). The representative morphed architectures with internal structural loops (ISLs) are shown in column iv. d Schematics of selected combinatorial designs by either combinatorically hinging two adjacent cubes at one of the four cube edge pairs at level 1 (i) and level 2 (ii) or flipping any level-1 structure with asymmetric hinge locations on top and bottom surfaces (ii) or combined. e Comparison of the maximum initial structural DOFs of different hierarchical structures composed of 4, 6, and 8 cubes at level 1. f Comparison of the combinatorically designed four categories of level-2 structures in ( b ) (insets and Supplementary Fig.  6 ) on the number of combinatorial level-2 hinge connections, reconfiguration modes, and morphed configurations with ISLs.

The connectivity between the cubes, namely the placement of the joints, dictates the spatial folding patterns of the structure (Supplementary Figs.  1 and 2 ). Broadly, the deployment follows four fundamental structural motifs, defined here as the mechanism-based connecting systems used to construct each leveled structure: one 2R chain-like mechanism and three 4R, 6R 10 , 18 , 22 , or 8R closed-loop mechanisms 41 , where n R denotes mechanisms with n rotational links and n rotatable (R) joints (see Supplementary Fig.  3 , and detailed definitions in Supplementary Note  3 ). For two adjacent cube faces, four potential edge locations exist to accommodate hinge joints (Fig.  2d , i). Consequently, a structure with n cubes theoretically allows for 4 n combinatorial sets of connections, offering an extensive design space for level-1 structures (Supplementary Fig.  4 ). Specially, we define this multiple design possibility by the placement of hinge joints in all leveled structures as their combinatorial design capability. As illustrated later, the combinatorial design capability of our proposed systems can be considerably expanded given the structural asymmetries and the multiple choices of structural motifs. Depending on the chosen connectivity, level-1 structures composed of n cubes exhibit an initial maximum number of 2 ( n  = 4), 3 ( n  = 6) and 5 ( n  = 8) DOFs (Fig.  2e ), which can be utilized for morphing into a diverse array of distinct 3D architected structures (as exemplified in Fig.  2a , iv, and further elaborated in Supplementary Fig.  1 and Supplementary Movie  1 ).

By substituting the higher-level linkages with the lower-level basic or hierarchical structures (e.g., Fig.  2a–c , i–iii and Supplementary Fig.  3a, b ) in the four fundamental structural motifs (2R, 4R, 6R, and 8R), we can create a class of flexible spatial hierarchical mechanism-based origami structures by combinatorically choosing any type of the n R linkages as different-level structural motifs (Supplementary Fig.  3c ). Notably, the term “flexible spatial mechanism” refers to mechanical mechanisms with bars and linkages arranged in 3D space, where the length of linkages is not fixed and varies during reconfiguration. For example, Fig.  2c , ii illustrates a level-3 structure comprising 8R linkages at level 1, 4R linkages at level 2, and 2R linkages at level 3, denoted as <8R, 4R, 2R>. The sequence from left to right corresponds to the structural motifs used from lower-level structure to higher-level structure.

The associated level-2 structure is depicted in Fig.  2b and denoted as <8R, 4R>. Additional examples of hierarchical origami structures with varying numbers of cubes at level 1 are presented in Supplementary Figs.  2 , 5 and 6 . Upon deployment, these structures can continuously transform into a multitude of intricate architected forms featuring internal structural loops (ISLs): internal voids within reconfigured architected structures enclosed by boundary structural components (as illustrated in Fig.  2a–c , iv, Supplementary Figs.  5c and  6 ). These ISLs efficiently facilitate different-level kinematic bifurcations, where a singular configuration state triggers a sudden increase in structural DOFs, leading to additional subsequent reconfiguration branches. This is in sharp contrast to the counterparts composed of four cubes at level 1, which are primarily limited to simple chain-like configurations (Supplementary Fig.  2a ) despite having a greater number of initial DOFs in the hierarchical structures of <4R, 4R> and <4R, 4R, 4R> (Fig.  2e ).

Moreover, the design space of hierarchical structures can be considerably expanded by combinatorically (1) adjusting the connectivity at higher-level bars (Fig.  2d , ii) and (2) manipulating structural asymmetries at the lower-level linkages given the asymmetric patterned joints on the top and bottom surfaces across the thickness, e.g., simple upside-down flipping (see Fig.  2d , ii for an example of level 2 structure). As an illustration, the insets in Fig.  2f and Supplementary Fig.  5 show four selected categories of combinatorial <8R, 4R> level-2 structures created by flipping the level-1 8R linkages and modifying the connections at the level-2 joints. By employing combinatorial design strategies involving mechanism hierarchy, spatial fold patterning across multiple levels of bars, and folding asymmetries in the linkages, we can generate an extraordinary vast design space encompassing millions of configurations, even within a simple level-2 structure (see analysis in Supplementary Note  2 ).

Compared to state-of-the-art 2D 14 , 16 , 40 , 48 and 3D origami designs 12 , 18 , 22 , 36 , 42 including our previous ad-hoc design of specific tessellated closed-loop mechanism of cubes 36 , this hierarchical approach offers several advantages: Firstly, it largely broadens the range of designs by allowing combinatorial connections within and across each hierarchical mechanism, which are either disabled or severely limited in previous studies 12 , 18 , 22 , 36 , 42 . Secondly, it effectively avoids geometric frustration in our previous ad-hoc designs 36 , which refers to structural constraints arising from deformation incompatibility during deployment 44 , 45 . This avoidance is made possible by the compatible reconfigurations of differently leveled spatially looped mechanisms (Fig.  2a–c , ii). Thirdly, this fundamental design principle establishes a versatile structural platform that can be applied to various shaped building blocks, overcoming the limitations in our previous ad-hoc designs 36 and other studies associated with specific structural elements 22 , 26 , 36 , 38 , 39 , 42 . Fourthly, it possesses the intrinsic benefit of structural hierarchy 46 , 47 , 49 , favoring higher-level structures with greater diversity and quantity of actuated reconfigured shapes under simple control and actuation.

Within this extensive design space, designs of particular interest are those that exhibit high reconfiguration capabilities via collision-free kinematic paths involving only a few active structural DOF during shape-changing processes. Such designs enable rich shape-morphing capability with simple and reliable control. After comparison (Supplementary Note  2 ), we identified an optimal category composed of four identical <8R> type of level-1 structures (see Category 1 in Fig.  2f and Supplementary Fig.  5b , with detailed definitions provided in Supplementary Notes  2 and 3 ) to showcase their extensive shape-morphing behavior under few active DOF. These designs boast the highest structural symmetries and the largest number of ISLs, facilitating bifurcation and shape diversity (Fig.  2f ).

Continuously evolving versatile shape morphing

Figure  3a provides a comprehensive view of the shape-morphing configurations diagram of one exemplary optimal <8R, 4R> level-2 structure selected from Category 1 in Fig.  2b (see Fig.  3a , i for its hierarchical design details). These structures were fabricated by assembling the 3D-printed rigid square facets (in white) into hollow cubes via interlocking mechanisms and flexible printed line hinges made of rubber-like materials (in black) (Supplementary Fig.  7a , see “Methods” and Supplementary Movie  2 for details). This design not only facilitates straightforward assembly but also allows for easy disassembly and reassembly of facets into hierarchical structure (Supplementary Fig.  7b–d ). For clarity, configurations with folding angles that are multiples of 90° are displayed since these angles correspond to kinematic bifurcations, as discussed later.

a Shape-morphing configurations diagram in the 3D-printed prototype exhibiting hierarchical transition tree-like features. The branches in the transition tree of represent the bifurcated configurations. Scale bar: 3 cm. b The variation of flexible level-2 link length with the opening angle of hinges during the shape transition from node M D to M E , and node M E to M F in reconfiguration loop 1 in ( a ). Inset shows the eigenvalues v kk as a function of the rotating angle in both level-1 and level-2 structures. c The relationship between the number of reconfiguration paths and the number of kinematic bifurcation configuration states for the combinatorically designed category I–III level-2 systems. d One selected combinatorial design of the shape-morphing level-2 structures by rearranging the level-1 hinges (i), and some of its representative reconfigured shapes (ii–v). Scale bar: 3 cm. e Comparison among the total number of hinges, the number of rotated joints, and the number of reconfiguration DOFs during the reconfiguration loop from node M A to M F and back to M A in ( a ).

With the inherent capacity for versatile shape changes provided by the level-1 linkage structure (Supplementary Fig.  7b ), the level-2 structure can continuously evolve, adopting various representative complex architectures along multiple reconfiguration paths (indicated by different colored lines in Fig.  3a ). Notably, these shapes bear a striking resemblance to trucks, trophies, tunnels, shelters, and various architectural structures (see more details in Supplementary Fig.  8 and representative reconfiguration processes in Supplementary Movie  3 ).

To systematically represent all reconfigured shapes and their corresponding shape transitions in Fig.  3a (ii), we employ a data-tree-like diagram (Supplementary Fig.  9 ), inspired by graph theory used in computer science to elucidate logical relationships among adjacent data nodes 50 (Supplementary Note 4 ). In this diagram, both nodes and line branches are assigned specific physical meanings, signifying individual reconfigured shapes and the relative shape-morphing kinematic pathways connecting them. As shown in Fig.  3a and Supplementary Fig.  9 , starting from a compact state (node M A ), the analyzed level-2 structure can traverse a closed-loop shape-morphing path (termed reconfiguration loop 1, RL-1, or a parent loop). Along this path, it transitions from simple chain-like structures (e.g., node M A  → M B  → M C  → M D ) to intricate architectures featuring ISLs (e.g., node M D  → M E  → M F ). Subsequently, starting from node M E with ISLs, it can further transform into nodes M F , M 5 , M 6 , or return to M D ). Theoretically, this continuous evolution in shape arise from the varying link lengths of the flexible level-2 linkage as line folds exhibit changing folding angles (Fig.  3b and Supplementary Fig.  11 , see the analysis in Supplementary Notes  5 – 7.1 , which examines length variations in level-2 links during two representative shape-morphing processes from node M D to M E and from node M E to M F ).

Benefitting from both chain-like and closed-loop mechanisms embedded in the morphed structural configurations, the parent loop gives rise to several subtrees (e.g., at node M A , M B or M 2 , M E , and M F ). These subtrees, in turn, branch into more paths through kinematic bifurcations (e.g., at node M 11 , M 15 , and M 17 ), as depicted in the inset of Fig.  3b . These bifurcations can be accurately predicted based on the number of null eigenvalues v kk in the kinematics model (see Supplementary Note  7.2 for detailed theoretical analysis). Importantly, node M 6 and node M 10 , located in different subtrees, are interconnected to form another reconfiguration loop (i.e., RL-2). This allows for direct transformation between two configurations or nodes that traverse different subtrees efficiently, without the need to return to the initial configuration and repeat redundant transforming steps, as required in previous reconfigurable structures 11 , 14 , 16 , 23 , 24 , 25 , 26 , 28 , 36 . Comparable hierarchical transition tree structures featuring bifurcated branches and interconnected nodes are observed in most of the four categories of other combinatorial <8R, 4R> level-2 structures (Supplementary Fig.  11 ). These structures are obtained by rearranging multilevel joint locations on top or bottom surfaces or by flipping the level-1 linkage (as seen in the level-2 representative in Fig.  3d and Supplementary Fig.  7d , e ). Consequently, a multitude of versatile and distinct morphed configurations are generated (Supplementary Figs.  12 and 13 ) based on differing hinge connectivity.

Additionally, for all combinatorial designs (Supplementary Fig.  5b ), we observed that the number of reconfiguration paths increases approximately linearly with the number of bifurcated nodes or configurations (Fig.  3c ). Notably, starting from a defined fold pattern, the same level-2 structure can generate nearly 10 3 reconfiguration paths with approximately 100 bifurcation nodes, thereby bestowing extensive shape-morphing capabilities (see analysis in Supplementary Note  8 ). In comparison to previous designs 12 , 16 , 18 , 22 , 26 , 29 , 31 , 38 , 39 , 40 , 43 , 48 that offer only a few shape-morphing paths from a defined fold pattern, our hierarchical design strategy enables a high number ( N  ~ 10–10 3 ) of kinematic transitions, demonstrating substantial versatility in generating numerous shapes and architectures.

Given that each reconfigured shape in Fig.  3a is defined by internal fold rotation angles that are multiples of 90°, we can accurately represent each shape by collecting spatial vectors v of the body center coordinates of all structural elements into a shape matrix M (see “Methods” for details). This matrix takes the explicit form M  = ( v 1 , v 2 , v 3 , …, v n ) (with n  = 32 for the level-2 structures shown in Fig.  3 and Supplementary Fig.  5 , see “Methods” and Supplementary Note  4 for details). Consequently, we can systematically annotate all reconfigured shapes in Fig.  3a using their corresponding shape matrices M k (with k as the shape index, see inset in Fig.  3a and Supplementary Fig.  9 ). Once the initial shape matrix M A is known, we can theoretically determine all the reconfigured shapes of the level-2 structure in Fig.  3a accordingly (see “Methods” for details). Importantly, this annotation approach is generalizable and can be applied to all other hierarchical origami metastructures presented in this work.

Remarkably, despite the level-2 structure’s total of 36 joints, only a small number of them are needed to drive the shape-morphing process, referred to as active reconfiguration DOF (Fig.  3e ). For example, when considering the multistep shape-morphing process from node M D to node M A , i.e., M D  → M E  → M F  → M A in Fig.  3a , it exhibits only 2, 2 and 1 DOF, respectively, even though it involves the rotation of 16, 8, and 24 joints (Fig.  3e and more details in Supplementary Figs.  14 and 15 ). This is in contrast to our previous ad-hoc design of cube-based reconfigurable metastructures 36 . Despite the presence of multiple closed-chain loops, they often function as independent units that barely couple with each other during shape morphing due to the specific architecture design of these metastructures, which results in high mobilities over 10,000 36 , making it impossible for control and actuation. In contrast, the reduction in active joints in this work is due to the specific interconnectivity of the looped level-1 and level-2 structures as geometric constraints, which dramatically reduces the number of active joints required while enabling high reconfigurability. Additionally, the multilevel closed-loop interconnectivity simplifies the control of shape-morphing paths in terms of simple transition kinematics, as demonstrated below.

Simple transition kinematics during shape morphing

The transition kinematics describes the quantitative relationship among the folding angles during the shape morphing of hierarchical structures. In Fig.  4a , we utilize the transformation matrix T (d, γ) to describe the relative spatial relationship of the four links, where d is the shortest distance between adjacent joints, and γ is the opening angle between adjacent cube-based links, as shown in Fig.  4b, c and Supplementary Fig.  16 . For a looped mechanism, it holds that \({\sum }_{i=1}^{m}{{{{\bf{T}}}}}_{i}={{{\bf{I}}}}\) , where m  = 8 and m  = 4 for the level-1 and level-2 links, respectively, and I is the identity matrix (see Supplementary Note  6 for details). With such simple equations, we can readily derive the relationship among the joint angles for all the transition paths using the local Cartesian coordinate systems presented in Fig.  4c (see Supplementary Note  7.1 for details).

a Schematics of level-2 structures with labeled hinge connections on top and bottom surface. b Schematics of the opening angles γ kj ( k , j are integers with 1 ≤  k  ≤ 4 and 1  ≤  j  ≤ 8 denote the link and hinges opening angles, respectively) between adjacent cubes in level-1 structure. c Construction of eight local coordinate systems for the 8 hinges of level-1 structure. d The reconfiguration kinematics from node M 7 (i) to node M 13 (iii) in Fig.  3a, b : the involved shape-changing details of level-1 link #1 and #3 (ii) and variations of the rotating angles for all folds (iv). Scale bar: 3 cm. e The reconfiguration kinematics from node M 15 (i) to node M 21 (v) by bypassing node M 25 (iii) in Fig.  3a, b : the involved shape-changing links #1 and #2 for the process from node M 15 to M 25 (ii) and links #2 and #4 for the process from node M 25 to M 21 (iv) and variations of all folds during these two processes (vi). Scale bar: 3 cm. f Low reconfiguration DOFs for the reconfiguration process in ( d ) (1 DOF) and ( e ) (1 or 2 DOF(s)).

To illustrate the simplicity of transition kinematics, we select two representative reconfiguration paths (node M 7  → node M 13 and node M 15  → node M 21 in Fig.  3a ) that transform from simple chain-like structures to complex architectures with ISLs (“Methods”). Figure  4d, e shows their detailed transition kinematics for these paths. It is observed that both shape-morphing paths involve only local and stepwise transition kinematics. For example, when transitioning from node M 7 to M 13 (Fig.  4d , i and iii), only the joints in link #2 and #4 (Fig.  4d , i) are engaged in sequential rotations (Fig.  4d , ii), while the remaining joints in link #1, link #3, and level-2 joints remain stationary (Fig.  4d , iv) (see “Methods” for details). Similarly, the reconfiguration kinematics from node M 15 to M 21 , bypassing node M 25 , follows a straightforward linear angle relationship, as shown in Fig.  4e , i–vi. Despite these two reconfiguration processes representing the most complex shape morphing (see more details in Supplementary Figs.  17 and 18 ), they can be achieved using simple kinematics-based control. Moreover, Fig.  4f shows that the number of active DOFs for each step remains below 3 during these two reconfiguration processes, thanks to the specific looped interconnectivity of hierarchical structures. This is superior to previous designs, which either featured condensed 11 , 12 , 14 , 16 , 18 , 22 , 26 or completely discrete internal connections 25 , 27 .

Given the unveiled simple transition kinematics of hierarchical structure and the low number of active DOFs during shape morphing, next, we explore and demonstrate their potential applications such as autonomous robotic transformers with adaptive locomotion, rapidly deployable self-reconfigurable architectures, and multifunctional space robots.

Autonomous multigait robotic transformer

To achieve autonomous shape morphing in the hierarchical origami structure, we utilize servomotors to actuate the active joints, while passive joints are secured using metal pins (Fig.  5a , i). These servomotors are powered by onboard rechargeable batteries and controlled through a customized circuit board equipped with a Bluetooth signal receiver (Fig.  5a , ii, see more details in “Methods” and Supplementary Note  9 ). This setup enables untethered shape morphing via a developed remote control system (Fig.  5a , iii, see more details in Supplementary Figs.  19 – 21 and Supplementary Movies  4 and  8 ).

a Schematics of untethered actuation design details for the level 1 eight-cube-based structure: 5 electrically powered servomotors for active hinge rotation (i), onboard power system and Bluetooth wireless receiver to conduct reconfiguration order (ii) from customized remote control software (iii). b Demonstrated untethered shape morphing in the level-1 structure through looped mechanisms. c – e Shape transformation in level-1 structure for multigait locomotion. Scale bar: 3 cm. c Forward (i) and sideway locomotion (ii). d Locomotion gait switch from reconfiguration to legged walking; e Legged walking with carried payload on flat surface (i) and 10°-sloped surface (ii). f Demonstration of the specific positions of 22 active servomotors and the rolling locomotion of level-2 structure. Scale bar: 3 cm. g Locomotion speeds of both level-1 and level-2 structures in ( c – f ).

Thanks to the specific kinematics, even though there are a total number of 8 joints in a level-1 structure and 32 joints in a level-2 structure, only 5 (Fig.  5a ) and 22 (Fig.  5f ) servomotors are needed to accomplish all the reconfiguration paths in these structures (Fig.  5b–e in level 1, and Figs.  5 f and 6a–c in level 2, respectively, see details in Supplementary Fig.  22 ). Importantly, the number of active servomotors involved in the reconfiguration paths does not exceed 3 (Fig.  6d ). For the level-1 structure, it can rapidly and continuously transform from the compact planar state to 6R and 8R-looped linkage configurations via looped mechanisms within a few seconds (Fig.  5b and Supplementary Movie  4 ). Additionally, it can assume simple 2R chain-like configurations via chain-like mechanisms (Fig.  5c ).

Self-deployment into bridge and/or shelter-frame-like ( a ) and fully open 4-story building-like structures ( b ) with high loading capacity of over 10 kg ( c ). Scale bar: 3 cm. d Comparison among the total number of hinges, the total number of servomotors, the rotated hinges, and the actively actuated servomotors during the shape transformation shown in ( b ).

Next, we delve into harnessing active shape morphing for autonomous robotic multigait (Fig.  5c–e ) and rolling (Fig.  5f ) locomotion. By following the chain-like reconfiguration loop path (Supplementary Fig.  7b ), the level-1 structure can repeatedly transform its body shape to achieve impressive multigait robotic locomotion. For instance, it can perform forward or backward locomotion (one cycle is shown in Fig.  5c , i) at a rapid speed of approximately 1000 mm/min (3.07 body length/min) (Fig.  5g ). Alternatively, it can change its movement direction from forward motion to sideway motion (Fig.  5c , ii) or switch its reconfiguration locomotion mode to a bipedal crawling mode (Fig.  5d and see more details in Supplementary Fig.  19c ). Moreover, it is capable of carrying some payload (around 1 kg, equivalent to its self-weight) and climbing sloped surfaces (10°, Fig.  5e ) at reduced speeds of approximately 225 mm/min and 190 mm/min (Fig.  5g ), respectively. Furthermore, a similar chain-like reconfiguration allows us to demonstrate rolling-based mobility in the level-2 structure (Fig.  5f and Supplementary Movie  5 ) at a speed of about 600 mm/min (Fig.  5g ).

Rapidly deployable and scalable self-reconfigurable architectures

Moreover, the compact level-2 structure can effectively self-transform and rapidly deploy into architectural forms resembling bridges, tunnels, and shelters (Fig.  6a, b and Supplementary Movie  6 ), both with and without internal looped structures. This transformation occurs within 2 min, a notable advance compared to previous studies that required several hours and complex algorithms 11 , 23 , 24 , 25 , 26 . Additionally, it can rapidly self-deploy into a fully open multi-story building-like structure, expanding its occupied volume fourfold (Fig.  6b , v). It can also quickly revert to a compact large cube (Fig.  6b , iv and Supplementary Movie  6 ). Due to its specific structural features (Supplementary Fig.  23 , see more details in Supplementary Note  10 ), the reconfigured level-2 structure can bear substantial loads without collapsing, such as approximately 13 kg (over 3.5 times its self-weight) for the bridge- or tunnel-like structures and about 10 kg (over 2.5 times its self-weight) for the multi-story structure (Fig.  6c ).

Notably, during the self-deployment from a compact planar structure to a complex multi-story open structure in Fig.  6b , the number of active motors remains low, never exceeding 3, despite the total number of 36 joints and 22 motors (Fig.  6d ). For example, during the reconfiguration from the compact cube to the fully open structure (Fig.  6b , iv–v), only 2 active servomotors drive the rotation of 16 joints (Fig.  6d ), demonstrating high reconfiguration efficiency.

As proof of concept, we demonstrate that these spatial hierarchical mechanism designs can be up-scaled to meter-sized buildings by assembling heavy-duty cardboard packing boxes (box side length 0.6 m). Starting from flat-packed cardboards with minimal space requirements, they can be rapidly assembled for easy deployment and reconfiguration into various structurally stable meter-scale tunnels, shelters, and multi-story open structures (Fig.  7a and Supplementary Movie  7 ). Remarkably, the total volume occupied by the deployed multi-story open architecture is 200 times larger than the initial volume of the flat-packed cardboards (Supplementary Fig.  24 ). Collectively, these properties make the proposed design promising for potential applications as temporary emergency shelters and other autonomously rapidly deployable and reconfigurable temporary buildings.

a Meter-scale demonstration of deployable, shape-morphing architectures using cubic packaging boxes (side length of 60 cm). Scale bar: 30 cm. b Schematics of potential conceptual applications in versatile reconfigurable space robots and habitats.

The hierarchical and combinatorial designs in both the links and joints at multiple levels of hierarchical structures provide an extensive design space for creating various spatial looped folding patterns and architected origami-inspired structures capable of shape morphing. It creates hierarchical origami-based metamaterials with (1) fewer active reconfiguration mobilities, (2) simple reconfiguration kinematics to facilitate practical control and actuation, and (3) rich shape-morphing capability adaptable to various applications. The hierarchical architecture couples the closed-loop mechanisms within and across each hierarchical level. Despite the large number of joints involved, the hierarchical looped mechanisms inherently impose geometric constraints that dramatically reduce the number of active DOFs required for shape morphing. This reduction greatly simplifies both actuation and control without sacrificing rich shape-morphing capability, which previously required the actuation of each DOF individually in reconfigurable origami metamaterials and robots. It also enables the feasibility of inverse designs, allowing for imitating target shapes and structures (Supplementary Figs.  20 ,  25 and  26 , see theoretical details in Supplementary Note  11 ).

Our design strategy combines structural hierarchy with over-constrained looped kinematic mechanism without considering elastic deformation in the hinges and cubes. Practically, the elastic deformation or slack, especially in the hinges, could cause the system to be floppy or potentially deviate from the desired non-bifurcated and/or bifurcated kinematic paths. As demonstrated in the multimaterial 3D-printed level-2 structure in Fig.  3a , the soft hinges are printed thin with little stiffness to ensure almost free rotation. Thus, in addition to bending for rotation motion, the hinges also undergo certain twisting deformation, potentially causing the structure to deviate from their ideal kinematic paths. However, deviations occur only during the complex reconfiguration processes, e.g., from configuration M 7 to M 13 in Fig.  3a . Such deviations are suppressed when the reconfiguring structure exhibits structural symmetries, e.g., from configuration M D to Configuration M E in Fig.  3a preserving x - y and z - y plane symmetries. The slack can be avoided by fabricating hinges with a low ratio of bending stiffness to twisting stiffness. This will help to suppress its twisting deformation to follow the kinematic paths without making the structure overly floppy. For systems fabricated with high-precision rigid links and hinges, slack or elastic deformation can be minimized or eliminated, as demonstrated in the prototype of both level-1 and level-2 structures with 3D-printed rigid cubes and rigidly rotatable hinges in Fig.  5a . Similar to studied 2D rigidly foldable origami structures, the reconfiguration kinematics of the system becomes energy scale independent. Thus, the system can rigorously follow its bifurcated reconfiguration kinematic path via fewer number of actuation hinges to smoothly reconfigure into all desired configurations without any locking issues as demonstrated in Figs.  5 and 6 .

We note that there are several limitations of this work. First, the load bearing capacity of some reconfigured 3D architectures is still limited, which could hinder their practical engineering and structural applications, especially at meter scales. The load bearing capacity is dependent of not only the transformed architectures (see the free body diagrams of force analysis for example in Supplementary Fig.  23 ), but also the bending stiffness of both cubes and hinges and the structural designs of the hinges. The hinges are imitated with 3D-printed soft rubber-like materials or tapes with low bending rigidity that facilitate the bending and rotation motion but sacrifice the load-carrying capabilities. The load bearing capacity could be improved by using stronger materials with high bending rigidity or locking hinges or devices at either 90° or 180° folded angles. Second, the shape-morphing capability for robotic applications is limited to multi-gait motion demonstrated in this work. How to leverage the rich shape-morphing capability for diverse and adaptive robotic locomotion in unstructured environments remains to be uncovered. Third, the demonstration of self-deployment and self-reconfiguration is limited to centimeter-scale prototypes while the meter-scale demo is done manually due to the limitation of both power and servomotors. At large scales, the heavier self-weight of cubes cannot be neglected, which requires high-torque servomotors and high-power batteries to generate sufficient torque output to counter the gravity and drive the folding.

Moving forward, these limitations also open new opportunities for future researches in morphing matter. First, this work explores only a small region of the tremendous design space in morphing matter to showcase its potential. The vast combinatorial folding patterns arise from the combinatorial connections in the base units, as well as within and across each hierarchical mechanism (Supplementary Fig.  3 ). These combinatorial hierarchical mechanisms are generalizable and can be applied to construct similar reconfigurable hierarchical metastructures composed of any shape-morphing spatial closed-loop mechanism for easy actuation and control yet rich shape morphing. For example, the cube units can be replaced by other composed geometrical shapes, such as thick plates with substantially reduced thickness dimension, tetrahedrons, and triangular-shaped prisms, or extended to genuine volumetric 3D structures (examples are provided in Supplementary Figs.  27 and 28 , with more details in Supplementary Note  12 ).

Second, this work focuses on exploring the reconfiguration kinematics of the hierarchical origami systems by modeling the system as idealized hierarchical rigid mechanisms and neglecting the deformation in both the cubes and hinges. However, in scenarios when such elastic deformation are non-negligible, similar to the non-rigidly deformable origami metamaterials in origami engineering, the over-constrained looped kinematic mechanisms become energy scale dependent, considering the potentially involved complex deformation in the cubes, hinges, and architectures during reconfiguration such as bending, stretching, twisting, and shearing or combined. Consequently, it will transform the rigid mechanisms into both reconfigurable and deformable architected materials and structures, which couples kinematics with mechanics. Such coupling will enrich new kinematics, mechanics, transformed configurations, reconfiguration paths, and reprogrammable mechanical behaviors such as multistability and stiffness anisotropy. Specially, how the energy scale affects the kinematic bifurcated paths and how the coupled kinematic bifurcation and elasticity change both the reconfigurations and mechanical responses of bifurcated mechanical metamaterials remain to be uncovered. We envision such studies could also find broad applications in reprogrammable mechanical computing, mechanical memory, and mechanical metamaterials.

Third, considering these multi-capabilities in conjunction with scalability, modularity, and disassemblability, we envision diverse applications in robotics, architecture, and even in space. Figure  7b conceptually illustrates potential applications in multitask adaptive shape-morphing space robots and habitat (Supplementary Movie  8 ). The hierarchical origami architectures could be deployed with largely increased exposed surface areas for enhanced solar energy harvesting, and reconfigured to avoid debris collision or accommodate more docking stations. It could also serve as reconfigurable space habitat or be des-assembled into modular robots for multitask exploration. For large-sized structures, the feasibility of actuation in a space environment is considerably higher, primarily due to the absence of gravity and the absence of ground-based collisions that can impede complex shape-morphing processes on Earth.

Sample fabrication of cube-based origami structures

To demonstrate the shape morphing in cube-based origami structures, we used two ways to fabricate and assemble the hollow cubes. One is for quick shape-morphing demonstration by directly 3D printing individual cubes with cube size of 2 cm (Stratasys Connex Objet-260 with stiff materials of Vero PureWhite) and connecting them with adhesive plastic tapes (Scotch Magic Tape, 6122) as free-rotation hinges (Supplementary Figs.  5 ,  8 and  11 – 13 ). The other is for easy assembly and disassembly demonstration by 3D printing Lego-like pieces of thin rigid plates (Fig.  3 ). Two types of thin plates were printed (Supplementary Fig.  7a ): one is a thin rigid plate with interlocking teeth (Vero PureWhite) for assembling into a hollow cube, the other is a connection piece composed of two connected thin rigid plates with soft hinges made of rubber-like materials (Agilus-black) through 3D multimaterial printing. The connection piece is used to connect two neighboring cubes at any selected hinge locations with the soft hinges facilitating the free rotation of cubes. The cube size is 3 cm.

Fabrication of autonomous robotic transformers

The cubes were 3D-printed with ABS printing materials (QIDI Tech X-Max 3D printer) with cube size of 81.5 mm and mass of 40 g. To ensure the compact contacts between the 3D-printed cube components, we created open areas at the joints positions and use the U-shaped bracket to hold electronic elements (Fig.  5a , i). Each motor (DSservo RDS3225) was powered by a 3.7 V LiPo battery and controlled via its specific control board (Adafruit ItsyBitsy nRF52840 Express). Additional chips were incorporated for accommodating the JST connector for the battery (Adafruit Pro Trinket LiIon/LiPoly Backpack Add-On) and for adaption of the supply voltage (SparkFun Logic Level Converter—Bi-Directional). The control boards were identified by a numeric ID and communicated with each other via Bluetooth by following a serial framework, where each controller receives the information from the previous one and sends them to the next one. More details can be found in Section S10 of Supplementary Information.

Fabrication of meter-scale samples

The cubes used in the meter-scale shape-morphing architectures in Fig.  7a were heavy-duty cardboard packing boxes (Recycled Shipping Box, Kraft) with dimensions of 0.6 m × 0.6 m × 0.6 m. Boxes were connected using the fiber-reinforced ultra-adhesive tape (BOMEI PACK Transparent Bi-Directional Filament Strapping Tape).

Fundamental principles governing the shape transformation

Given the mechanical kinematic mechanism’s structural features, the core of the shape transformations in these structures involves changes in the spatial positions of specific structural elements resulting from the directional rotations of internal hinges and their interconnections. Technically 10 , 22 , 51 , this operation can be mathematically modeled using a rotation matrix t (see details in Supplementary Note  4.2 ). Thus, the shape transformations of any leveled structures can be denoted as:

where M ′ represents the transformed shape from shape M , with both M and M ′ reflected in Fig.  3a and Supplementary Fig.  9 as M k , and t represents the mathematical operations between them. In our analysis, we initially build a fixed global Cartesian coordinate system at the bottom center of the original shape (see the inset at the initial shape in Fig.  3a , ii). Subsequently, we construct a local coordinate system at each fold to derive the body center coordinates of the rotated cube structural components (Supplementary Fig.  10a, b ) in each shape-morphing process. Mathematically, we can thus determine the new positions of the rotated cubes as follows:

where v n_local and v n_new represent the body center vectors of cube # n before and after shape morphing, respectively, in the local and fixed global coordinate systems. t n is a general functional form including all directional rotations of cube # n (Supplementary Fig.  10b–e ), see the systematic analytical details in Supplementary Note  4.2 . d n is the translational vector between the fixed global coordinate system and the local coordinate system of cube # n . Note that all shape matrices of the initial and reconfigured shapes are described in the fixed global coordinate systems.

We validate the theoretical framework by modeling the shape-morphing process of reconfiguration loop 1, i.e., from the initial shape M A to shape M F , passing through shapes M B , M C , M D and M E . The shape matrix of the initial shape is determined first in the fixed global coordinate system. Based on Eqs. ( 1 ) and ( 2 ), we rationally derive the new positions of the rotated cubes in new shapes accordingly. Specifically, we analyze the process from morphing from shape M D to shape M E , where a total of 24 cubes are involved. To provide a representative example, we select cube #20 and theoretically derive its new spatial positions. Subsequently, we compare these derived solutions with experimental results to validate our proposed theoretical framework.

Starting from the initial shape M A , we derive the shape matrix of M D , as expressed in Eq. ( 3 ). Consequently, in the fixed global coordinate system, we obtain the explicit spatial vector for cube #20 \({{{{\boldsymbol{v}}}}}_{20}^{{{{{\bf{M}}}}}_{D}}\) with \({{{{\boldsymbol{v}}}}}_{20}^{{{{{\bf{M}}}}}_{D}}={(1,3,1)}^{T}\) . During the shape-morphing process, cube #20 undergoes rotation along the x- axis within the locally built coordinate systems (Supplementary Fig.  10a , iv). To derive its new positions, we first calculate its spatial vector in the local coordinate systems, represented by \({{{{\boldsymbol{v}}}}}_{20\_local}^{{{{{\bf{M}}}}}_{D}}={(1,1,1)}^{T}\) . Utilizing a 90° x -directional rotation, we then derive its new coordinates in the global coordinate systems using Eq. ( 2 ) with explicit derivation details as:

Within the built fixed global coordinate systems, we extract the experimental result pertaining to the spatial position of cube #20 in the global coordinate system, denoted as \({{{{\boldsymbol{v}}}}}_{30}^{{{{{\bf{M}}}}}_{E}}={(1,3,-1)}^{T}\) . The theoretical model is in excellent agreement with the experimental result. In order to derive the shape matrices of shapes M D and M E in reconfiguration loop 1, we need firstly determine the shape matrix of the initial shape M A , which is presented with explicit components as:

Then, combining Eqs. ( 1 )–( 4 ), we can finally obtain the shape M D and shape M E as:

Reconfiguration kinematics in Fig.  4

The following gives the reconfiguration kinematic details for the morphing process shown in Fig.  4 d, e . Specially, we label the opening angles of four level-1 link structure as γ km ( k and m are integers with 1 ≤  k  ≤ 4 as the k th link while 1 ≤  m  ≤ 8 as the m th rotating folds between two adjacent cubes m and m  + 1 ( m  + 1 → 1 when m  = 8), see details in Fig.  4a, b ), and the level-2 folds angles separately as γ B11 , γ B11’ , γ T21 and γ Τ21’ (Fig.  4a, b and B and T represent the bottom and top surfaces, respectively).

The selected two reconfiguration processes exhibit only local and stepwise transition kinematics. From shape M 7 to M 13 (Fig.  4d , i and iii), only the folds of links #2 and #4 (Fig.  4d , i) are involved with sequential rotations (Fig.  4d , ii) and the remaining folds of link #1, link #3 and the level-2 keep unchanged (Fig.  4d , iv). For kinematic details of the reconfigured links #2 and #4 shown in Fig.  4d , iv, during the initial process ①  →  ② , we only need to linearly change the folds angle γ m4,6 ( m  = 2 or 4) from 180° to γ 0 (Here we set γ 0 as 150° while it ranges from 90° to 180°; see more details in Supplementary Fig.  17 ) and meanwhile linearly increase γ m2,8 from 0° to γ 0 . Then, in the following process ② → ③ , we can maintain folds angles γ m2,4,6,8 as γ 0 while both linearly decreasing γ m1,5 from 180° to sin -1 [(sin γ 0 ) 2 /(1 + (cos γ 0 ) 2 ] (≈ 109.5° for γ 0  = 150°) and augmenting γ m3,7 from 0° to 180°−sin −1 [(sin γ 0 ) 2 /(1 + (cos γ 0 ) 2 ] (≈ 70.5° for γ 0  = 150°). Lastly, for ③ → ④ → ⑤ , we can simultaneously transfigure links #2 and #4 as 8R-looped rigid linkage with kinematics as γ m1,5  = sin −1 [(sin γ m2 ) 2 /(1 + (cos γ m2 ) 2 ], γ m3,7  = 180°−sin −1 [(sin γ m2 ) 2 /(1 + (cos γ m2 ) 2 ] (γ m2 reducing from γ 0 to 90°) while γ m2  = γ m4  = γ m6  = γ m8 (see Supplementary Note  6 ) to reach shape M 13 . Moreover, as illustrated in Fig.  4e , vi that displays sequential and local kinematic features, we note that the reconfiguration kinematics from shape M 15 to M 21 by bypassing shape M 25 (Fig.  4e , i–v) are much simpler with only linear angle relationships.

Inverse design to imitate target shapes

Inverse design to imitate target shapes for special application scenarios can also be accessible for our hierarchical structures. However, the imitating process of our inverse design is different from previous designs by presetting material/structural patterns to purposely retain the target shapes. Our inverse design method is based on the selection algorithm from the reconfigured shape library by following several steps.

First is to build a database for the configuration library. Each cube can be treated as a spatial voxelated pixel with its geometrical center represented by a vector. Then, we can use a matrix to characterize a morphed shape, where the spatial positions of composed cubes are described by their corresponding vectors. For example, for all the combinatorically designed level-2 structures shown in Supplementary Fig.  5 , for one special design k , all its reconfigured shapes can be summarized into:

where M kn represent the mathematically expressed forms of the n th reconfigured shapes in the transition tree for the k th combinatorically designed level-2 structures.

Second is to compose all the combinatorically designed level-2 structures into the database matrix D in the form of:

where z stands for the maximum number of reconfigured shapes by the k th level-2 structure.

Third is to discretize the target shape into cube-shaped voxelated pixels and mathematically convert it into a mathematical matrix T .

Last is to find the shapes in the database that match for the target shape by comparing the matrix T with the components of database matrix, i.e., D ij . There are two criterions to find out the optimal imitated shape: (1) find the smallest value of the error function Errf defined as:

wherein || || represent the mode of matrix and usually \({\Vert {{{\bf{T}}}}-{{{{\bf{D}}}}}_{ij}\Vert }_{\max }\) is determined as \(\Vert {{{\bf{T}}}}\Vert+{\Vert {{{{\bf{D}}}}}_{ij}\Vert }_{\max }\) for simplicity. (2) The conditions that guarantee the imitated shapes whose cube pixels are with approximately the same absolute spatial positions with the target shape, i.e.:

Finally, we can obtain the most approximately imitated shape M km from the database. The inverse design method is briefly summarized in Supplementary Fig.  25 .

Simulation by customized software

A model has been developed for simulation in ROS-Gazebo (Supplementary Figs.  20 and 21 and Supplementary Movies  5 and  8 ). For simplicity, a single design composed of the four lateral faces of a cube is used to model every module of the robot. The connections between the modules are modeled as revolute joints (either passive or actuated). Additional blocks are used to replicate the positions and masses of the motors in the real system. Kinematic constraints are implemented to model the robot as a closed kinematic chain.

Data availability

The authors declare that the data supporting the findings of this study are available within the article and its Supplementary Information files.  Source data are provided with this paper.

Code availability

The code used for the analyses is deposited via Zenodo at https://doi.org/10.5281/zenodo.12690922 .

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Acknowledgements

J.Y. acknowledges the funding support from NSF (CMMI-2005374 and CMMI-2126072). H.S. acknowledges the funding support from NSF 2231419. The authors acknowledge the helpful discussions with Dr. K. Bertoldi and Dr. M. Yim.

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These authors contributed equally: Yanbin Li, Antonio Di Lallo.

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Department of Mechanical and Aerospace Engineering, North Carolina State University, Raleigh, NC, 27606, USA

Yanbin Li, Antonio Di Lallo, Junxi Zhu, Yinding Chi, Hao Su & Jie Yin

Lab of Biomechatronics and Intelligent Robotics, Joint NCSU/UNC Department of Biomedical Engineering, North Carolina State University, Raleigh, NC, USA

University of North Carolina at Chapel Hill, Chapel Hill, NC, USA

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Contributions

Y.L. and J.Y. proposed the idea. Y.L. conducted theoretical and numerical calculations. Y.L. and Y.C. designed and performed experiments on shape-morphing prototypes. A.D. and J.Z. designed and performed experiments on untethered actuation of shape-morphing prototypes. Y.L., A.D., H.S. and J.Y. wrote the paper. H.S. and J.Y. supervised the research. All the authors contributed to the discussion, data analysis, and editing of the manuscript.

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Correspondence to Yanbin Li , Hao Su or Jie Yin .

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Li, Y., Di Lallo, A., Zhu, J. et al. Adaptive hierarchical origami-based metastructures. Nat Commun 15 , 6247 (2024). https://doi.org/10.1038/s41467-024-50497-5

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Received : 28 November 2023

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DOI : https://doi.org/10.1038/s41467-024-50497-5

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