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  • Robots

    To investigate human and animal locomotion, a number of legged robots were developed in our group since 2004. Read about the bipedal robot BioBiped or the research in the Locomorph project focusing on morphology and morphosis strategies in locomotion.

  • Prostheses

    To investigate models of the muscle-tendon dynamics on humans we developed the research platform PAKO. Using our insights on gait biomechanics, walking and running could be realized with the robotic Walk-Run Ankle prosthesis.

  • Facilities

    Several indoor and outdoor facilities with state-of-the-art measurement equipment helps us to perform experiments on humans, animals and robots. Details can be found here: Facilities.

  • Experiments

    Experimental studies are performed in both research projects and in teaching courses at the Sports Science Institute at TU Darmstadt. Outcomes from student research and educational projects on biomechanics can be found in the awarded Teaching Wiki of our institute.

  • Models

    Models help us to study the fundamental principles of human and animal locomotion. The derived biomechanical concepts can be applied to bipedal robots, exoskeletons or prosthesis. In the European project Balance, we are working on an active orthosis.


  • The German Research Foundation (DFG) has approved the new graduate school “LokoAssist - Seamless Integration of Assistance Systems for Natural Movement of Humans” at TU Darmstadt. LokoAssist is carried out together with partners from the University of Heidelberg and the Central Institute for Mental Health in Mannheim. The graduate school will be funded for four and a half years from 2022 with an estimated 6 million euros. More information
  • LEARN&ACT DAY of the Movement Academy Motions and Emotions will be postponed due to the current situation

Latest Publications

Model-based Control for Gait Assistance in the Frontal Plane

Abstract: There is a growing interest in developing devices for human gait assistance. Most research focuses on the sagittal plane's assistance, assuming that walking is predominantly a sagittal plane motion. However, movements in the frontal plane have particular importance for balance control as well as load carrying. This paper studies hip abduction assistance and proposes using a simple human locomotion model as a guideline for designing and controlling assistive devices. We introduce the force modulated compliant hip (FMCH) model in the frontal plane, which applies the ground reaction force to tune the hip abductor/adductor stiffness. The effects of this model-based control approach on walking gait are investigated through leg muscles' activation, metabolic costs, and adaptability to a new condition (carrying 38 kg extra load) for seven healthy young subjects using an experiment-based simulation in OpenSim. Our results show 6.72 ± 0.6% and 11 ± 0.3% reduction in metabolic cost for walking at a freely selected speed and while load carrying, respectively. Also, compensating for increased muscle activation while carrying extra load due to the hip compliance adjustment by leg force feedback approves the adaptability of the proposed control approach. The publication can be found at BioRob2022.

Exploring surface electromyography (EMG) as a feedback variable for the human-in-the-loop optimization of lower limb wearable robotics

Abstract: Human-in-the-loop (HITL) optimization with metabolic cost feedback has been proposed to reduce walking effort with wearable robotics. This study investigates if lower limb surface electromyography (EMG) could be an alternative feedback variable to overcome time-intensive metabolic cost based exploration. For application, it should be possible to distinguish conditions with different walking efforts based on the EMG. To obtain such EMG data, a laboratory experiment was designed to elicit changes in the effort by loading and unloading pairs of weights (in total 2, 4, and 8 kg) in three randomized weight sessions for 13 subjects during treadmill walking. EMG of seven lower limb muscles was recorded for both limbs. Mean absolute values of each stride prior to and following weight loading and unloading were used to determine the detection rate (100% if every loading and unloading is detected accordingly) for changing between loaded and unloaded conditions. We assessed the use of multiple consecutive strides and the combination of muscles to improve the detection rate and estimated the related acquisition times of diminishing returns. To conclude on possible limitations of EMG for HITL optimization, EMG drift was evaluated during the Warmup and the experiment. Detection rates highly increased for the combination of multiple consecutive strides and the combination of multiple muscles. EMG drift was largest during Warmup and at the beginning of each weight session. The results suggest using EMG feedback of multiple involved muscles and from at least 10 consecutive strides (5.5 s) to benefit from the increases in detection rate in HITL optimization. In combination with up to 20 excluded acclimatization strides, after changing the assistance condition, we advise exploring about 16.5 s of walking to obtain reliable EMG-based feedback. To minimize the negative impact of EMG drift on the detection rate, at least 6 min of Warmup should be performed and breaks during the optimization should be avoided. Future studies should investigate additional feedback variables based on EMG, methods to reduce their variability and drift, and should apply the outcomes in HITL optimization with lower limb wearable robots. The publication can be found at Frontiers in Neurorobotics.

For further projects and publications of M. Grimmer and G. Zhao please check ResearchGate, Google Scholar, ORCID or LOOP and ResearchGate.

Unified GRF-based control for adjusting hopping frequency with various robot configurations

In our newly published work in the EPA (electric-Pneumatic Actuator) project, we promoted the concept of Force Modulated Compliance (FMC) control in generating various hoppings (at different frequencies) with various robot configurations (in presence/absence of PAMs).

Abstract: The highly dynamic hybrid nature of legged locomotion makes it a very challenging task to control. A proper control strategy, besides the ability to generate stable motions, should also possess generalization capabilities and adjustability to different conditions. In this regard, this work takes a step forward in promoting the concept of Force Modulated Compliance (FMC) control as a bioinspired reflex-based control approach and assesses its potential in generating various hopping motions. The FMC is an easy-to-tune and simplified representation of the neuromuscular control, which functions as an adjustable spring modulated by the ground reaction force. We implemented the FMC controller on EPA-Hopper-II which is an extension of our previous electric-pneumatic actuation (EPA) designs. By optimizing control parameters for hopping frequencies of 1.5−3.5Hz and different PAM configurations in the robot structure, we show that FMC can generate stable hopping motions and adapt to different configurations. The optimization results also show that leg stiffness has a high correlation with normalized energy consumption. Increases in leg stiffness decrease the normalized energy consumption and vice versa. The outcomes of this study, particularly the highlighted generalization of the FMC control, open new doors for efficient control of legged robots and assistive devices.

For further publications of the author please check: Google Scholar, ResearchGate

Bioinspired Legged Robot Design via Blended Physical and Virtual Impedance Control

In our recent publication of the EPA (electric-Pneumatic Actuator) project, we took leverage of the EPA design by simultaneous physical and virtual impedance control of a hopper robot.

Abstract: In order to approach the performance of biological locomotion in legged robots, better integration between body design and control is required. In that respect, understanding the mechanics and control of human locomotion will help us build legged robots with comparable efficient performance. From another perspective, developing bioinspired robots can also improve our understanding of human locomotion. In this work, we create a bioinspired robot with a blended physical and virtual impedance control to configure the robot’s mechatronic setup. We consider human neural control and musculoskeletal system a blueprint for a hopping robot. The hybrid electric-pneumatic actuator (EPA) presents an artificial copy of this biological system to implement the blended control. By defining efficacy as a metric that encompasses both performance and efficiency, we demonstrate that incorporating a simple force-based control besides constant pressure pneumatic artificial muscles (PAM) alone can increase the efficiency up to 21% in simulations and 7% in experiments with the 2-segmented EPA-hopper robot. Also, we show that with proper adjustment of the force-based controller and the PAMs, efficacy can be further increased to 41%. Finally, experimental results with the 3-segmented EPA-hopper robot and comparisons with human hopping confirm the extendability of the proposed methods to more complex robots.

For further publications of the author please check: Google Scholar, ResearchGate

Adjustable Compliance and Force Feedback as Key Elements for Stable and Efficient Hopping

In this is recent publication of the EPA (electric-Pneumatic Actuator) project we demonstrated the role of the mechanical and controlled compliance in improving efficiency and performance in the MARCO-EPA-robot hopping. Compliation of PAM for tuning mechanical compliance with virtual complaince control through electric motor shows the advantages of the EPA design.

Abstract: Achieving efficient and human-like hopping motions with consistent consecutive patterns requires proper mechanical design and control. In this regard, we introduce a new bioinspired design and control approach comprised of a hybrid actuation system and force-based compliance control. Combining an electric motor with a pneumatic artificial muscle makes up the so-called Electric-Pneumatic Actuation (EPA) system which enables presetting the leg compliance. Using the ground reaction force for online adjustment of EPA allows controlling the leg stiffness. This approach combines a simplified version of reflex-based control and physical impedance adjustment, that resembles the human neuromuscular system in a parsimonious way. By optimizing PAM pressures and control gains for different targeted hopping heights using Bayesian optimization, we show experimentally that the proposed approach generates various stable hopping motions and yields efficient performance. Moreover, our method is shown to be capable of creating more human-like hopping patterns compared to the previous studies on the same robot.

For further publications of the author please check: ResearchGate, ORCID or LOOP

The mechanisms and mechanical energy of human gait initiation from the lower-limb joint level perspective

Guoping Zhao and Martin Grimmer published a fundamental biomechanical study on human gait initiation. If you like to know how to do it, the publication can be found at Scientific Reports.

Abstract: This study aims to improve our understanding of gait initiation mechanisms and the lower-limb joint mechanical energy contributions. Healthy subjects were instructed to initiate gait on an instrumented track to reach three self-selected target velocities: slow, normal and fast. Lower-limb joint kinematics and kinetics of the first five strides were analyzed. The results show that the initial lateral weight shift is achieved by hip abduction torque on the lifting leg (leading limb). Before the take-off of the leading limb, the forward body movement is initiated by decreasing ankle plantarflexion torque, which results in an inverted pendulum-like passive forward fall. The hip flexion/extension joint has the greatest positive mechanical energy output in the first stride of the leading limb, while the ankle joint contributes the most positive mechanical energy in the first stride of the trailing limb (stance leg). Our results indicate a strong correlation between control of the frontal plane and the sagittal plane joints during gait initiation. The identified mechanisms and the related data can be used as a guideline for improving gait initiation with wearable robots such as exoskeletons and prostheses.

For further projects and publications of G. Zhao and M. Grimmer please check ResearchGate and ResearchGate, Google Scholar, ORCID or LOOP.

Soft pneumatic elbow exoskeleton reduces the muscle activity, metabolic cost and fatigue during holding and carrying of loads

In 2019 Martin Grimmer got in contact with John Nassour (Technische Universität Chemnitz, now Technische Universität München) who was working on pneumatic actuators. Together they designed a human-machine interface to create the elbow exoskeleton “Carry” to assist both elbows during holding and carrying of loads. One pneumatic actuator is used for each elbow. Together with Guoping Zhao they evaluated the functionality of Carry at a moderate assistance level (7,2Nm elbow torque in total of both arms) and found reductions of up to 50% for the muscle activity, up to 61% for the net metabolic rate, and up to 99% for fatigue in a group study of 12 individuals. A tethered setup was used for the test, which could be used at stationary work places. However, the authors were further interested if an autonomous setup would be possible. Based on theoretical estimations they found that system weights of about 3 to 4kg are required to support about 100 repetitions. A mobile version of Carry would be great to support load handling tasks e.g. of construction or relocation workers. The significant benefits of Carry indicate this device could prevent systemic, aerobic, and/or possibly local muscle fatigue that may increase the risk of joint degeneration and pain due to lifting, holding, or carrying. The publication can be found at Scientific Reports.

For further projects and publications of M. Grimmer and G. Zhao please check ResearchGate, Google Scholar, ORCID or LOOP and ResearchGate.

Parallel Compliance Design for Increasing Robustness and Efficiency in Legged Locomotion—theoretical background and applications

This is a recent publication in the EPA (electric-Pneumatic Actuator) project in which the theoretical background on the advantages of adding parallel compliance in increasing the robustness against environment uncertainties or perturbations are presented. This is the second part of the study titled Parallel compliance design for increasing robustness and efficiency in legged locomotion – proof of concept. published in IEEE Transactions on Mechatronics. This follow-up paper shows some results of robustness improvement with the EPA-Hopper using parallel PAMs. This article is published in IEEE/ASME Transactions on Mechatronics.


—Bipedal locomotion in uncertain environments is a challenging control problem. In order to reduce the effect of imprecise and noisy measurements, performance enhancement, and energy consumption reduction, many researchers employ compliant elements in the robot structure, parallel to the control system. However, there is no systematic methodology for concurrent design of compliant components and the controller. In a primary article, we introduced a method for the simultaneous design of the controller and compliant elements to increase the walking robustness against uncertainties, based on hybrid zero dynamics (HZD) analysis. The overall controller comprising the HZD controller and parallel compliance (PC) is called the HPC controller. In this article, we present two levels of extension: 1) extended HPC (EHPC): an extended HPC with fewer constraining assumptions; 2) concurrent control and parallel compliance design (CPC): a generalized version of concurrent control and PC design, which is applicable for any gait control approach, and is not limited to HZD. In this article, apply the Lyapunov, boundedness, and input to state stability concepts to analyze the EHPC’s walking robustness. Detailed step-by-step design of this controller for a compass gait model and the MATLAB codes are also provided. An experimental study on a hopper robot supports the generalization in the CPC method to apply to other controllers while pneumatic artificial muscles are utilized as tunable PCs.

For further publications of the author please check: ResearchGate, ORCID or LOOP

Hopping frequency influences elastic energy reuse with joint series elastic actuators

Aida is investigating methods to improve the locomotion efficiency of bipedal robots. As a template, the hopping task was selected to identify optimal spring (tendon) stiffness values to support monoarticular serial elastic actuators within hopping at different frequencies. When comparing the results to previous works on walking and running, it turns out that the optimal stiffness values identified for the preferred walking and moderate running speeds are in a similar range as the values for the range of investigated hopping frequencies. In conclusion we assume that first, the biological tendon properties are in relation to those determined by our approach and second, that the properties were optimized in humans to also achieve an efficient locomotion. Results were recently published in the Journal of Biomechanics.

Robotic limb design struggles to combine energy efficiency with human-like levels of movement versatility. High efficiency and a range of angles and torques are characteristics of human hopping at different frequencies. Humans use muscles in combination with tendons to achieve the required joint actuation. Therefore, we consider whether appropriately tuned series elastic actuators (SEAs) placed at the leg joints can be used to reduce the functional gap between robots and humans. Human hip, knee, and ankle biomechanics were recorded over a range of hopping frequencies to extract joint angles and torques, which were used as an input to a mechanical simulation SEA model. This model was used to optimize the SEA stiffness of each joint to either minimize peak power or energy requirements. This work investigated the relationship between hopping frequency and SEA stiffness, the utility of using SEAs at each joint, and the reasons behind humans’ preferred hopping frequency. Although the constant stiffness values across different hopping frequencies are suitable for the knee and the ankle, a variable serial elastic actuator stiffness could still further reduce energy requirements. Optimal SEA stiffness was found to reduce peak power requirements by up to 73% at the ankle and up to 66% at the knee, with greatest benefits found around the preferred frequency. However, no SEA benefits were found for the hip and above the preferred hopping frequency for the knee. These insights could be used to aid in the design of robotic and assistive devices to achieve versatile and energy efficient human-like movements.

For further projects and publications of A. Mohammadi Nejad Rashty and M. Grimmer please check ResearchGate, Google Scholar and ResearchGate, Google Scholar, ORCID or LOOP.

Neuromechanical force-based control of a powered prosthetic foot

Previously Hartmut Geyer, a former member of the lab, developed a neuromechanical template model, which uses force, length and velocity feedback in combination with muscle modeling to enable walking in simulation and with a powered prosthetic foot (BiOM). Maziar Sharbafi had the idea of the FMCA based control (force modulated compliant ankle), which simplifies this concept and directly translates the feedback signals to an appropriate joint torque. Through efforts from Amirreza Naseri and Martin Grimmer, we were able to test FMCA with the Walk-Run-Ankle. Results were recently published in Wearable Technologies.

Abstract: This article presents a novel neuromechanical force-based control strategy called FMCA (force modulated compliant ankle), to control a powered prosthetic foot. FMCA modulates the torque, based on sensory feedback, similar to neuromuscular control approaches. Instead of using a muscle reflex-based approach, FMCA directly exploits the vertical ground reaction force as sensory feedback to modulate the ankle joint impedance. For evaluation, we first demonstrated how FMCA can predict human-like ankle torque for different walking speeds. Second, we implemented the FMCA in a neuromuscular transtibial amputee walking simulation model to validate if the approach can be used to achieve stable walking and to compare the performance to a neuromuscular reflex-based controller that is already used in a powered ankle. Compared to the neuromuscular model-based approach, the FMCA is a simple solution with a sufficient push-off that can provide stable walking. Third, to assess the ability of the FMCA to generate human-like ankle biomechanics during walking at the preferred speed, we implemented this strategy in a powered prosthetic foot and performed experiments with a non-amputee subject. The results confirm that, for this subject, FMCA can be used to mimic the non-amputee reference ankle torque and the reference ankle angle. The findings of this study support the applicability and advantages of a new bioinspired control approach for assisting amputees. Future experiments should investigate the applicability to other walking speeds and the applicability to the target population.

For further projects and publications of M. Grimmer and M. Sharbafi please check ResearchGate, Google Scholar, ORCID or LOOP and ResearchGate, Google Scholar, ORCID or LOOP.

Lower limb joint biomechanics-based identification of gait transitions in between level walking and stair ambulation

Gait transitions in between level walking and stair ambulation were investigated in one of our projects involving Martin Grimmer and the Department of Electrical Engineering and Information Technology of Ulrich Konigorski. A team of Postdocs, Phd candidates and students performed the one of the largest human gait studies ever, involving an instrumented staircase, in the Locomotion Laboratory. The work was recently published in the PLOS ONE.

Abstract: Lower limb exoskeletons and lower limb prostheses have the potential to reduce gait limitations during stair ambulation. To develop robotic assistance devices, the biomechanics of stair ambulation and the required transitions to level walking have to be understood. This study aimed to identify the timing of these transitions, to determine if transition phases exist and how long they last, and to investigate if there exists a joint-related order and timing for the start and end of the transitions. Therefore, this study analyzed the kinematics and kinetics of both transitions between level walking and stair ascent, and between level walking and stair descent (12 subjects, 25.4 yrs, 74.6 kg). We found that transitions primarily start within the stance phase and end within the swing phase. Transition phases exist for each limb, all joints (hip, knee, ankle), and types of transitions. They have a mean duration of half of one stride and they do not last longer than one stride. The duration of the transition phase for all joints of a single limb in aggregate is less than 35% of one stride in all but one case. The distal joints initialize stair ascent, while the proximal joints primarily initialize the stair descent transitions. In general, the distal joints complete the transitions first. We believe that energy- and balance-related processes are responsible for the joint-specific transition timing. Regarding the existence of a transition phase for all joints and transitions, we believe that lower limb exoskeleton or prosthetic control concepts should account for these transitions in order to improve the smoothness of the transition and to thus increase the user comfort, safety, and user experience. Our gait data and the identified transition timings can provide a reference for the design and the performance of stair ambulation- related control concepts.

For further projects and publications of M. Grimmer please check: ResearchGate, Google Scholar, ORCID or LOOP

Doppler Radar for the Extraction of Biomechanical Parameters in Gait Analysis

Doppler Radar for the extraction of biomechanical parameters during walking was investigated in one of our latest studies in cooperation with the Signal Processing Group of Prof. Zoubir. Ann-Kathrin Seifert and Martin Grimmer performed a series of experiments in the Locomotion Laboratory. The work was recently published in the IEEE Journal of Biomedical and Health Informatics.

Abstract: The applicability of Doppler radar for gait analysis is investigated by quantitatively comparing the measured biomechanical parameters to those obtained using motion capturing and ground reaction forces. Nineteen individuals walked on a treadmill at two different speeds, where a radar system was positioned in front of or behind the subject. The right knee angle was confined by an adjustable orthosis in five different degrees. Eleven gait parameters are extracted from radar micro-Doppler signatures. Here, new methods for obtaining the velocities of individual lower limb joints are proposed. Further, a new method to extract individual leg flight times from radar data is introduced. Based on radar data, five spatiotemporal parameters related to rhythm and pace could reliably be extracted. Further, for most of the considered conditions, three kinematic parameters could accurately be measured. The radar-based stance and flight time measurements rely on the correct detection of the time instant of maximal knee velocity during the gait cycle. This time instant is reliably detected when the radar has a back view, but is underestimated when the radar is positioned in front of the subject. The results validate the applicability of Doppler radar to accurately measure a variety of medically relevant gait parameters. Radar has the potential to unobtrusively diagnose changes in gait, e.g., to design training in prevention and rehabilitation. As contact-less and privacy-preserving sensor, radar presents aviable technology to supplement existing gait analysis tools for long-term in-home examinations.

For further projects and publications of M. Grimmer please check: ResearchGate, Google Scholar, ORCID or LOOP

A biarticular passive exosuit to support balance control can reduce metabolic cost of walking

In this research the advantages of a biarticular thigh exosuit in supporting human walking is demonstrated. Find the details in our recently published paper (open access) by Barazesh, H and Sharbafi, M. A. in Bioinspiration & Biomimetics.


Nowadays, the focus on the development of assistive devices just for people with mobility disorders has shifted towards enhancing physical abilities of able-bodied humans. As a result, the interest in the design of cheap and soft wearable exoskeletons (called exosuits) is distinctly growing. In this paper, a passive lower limb exosuit with two biarticular variable stiffness elements is introduced. These elements are in parallel to the hamstring muscles of the leg and controlled based on a new version of the FMCH (force modulated compliant hip) control framework in which the force feedback is replaced by the length feedback (called LMCH). The main insight to employ leg length feedback is to develop a passive exosuit. Fortunately, similar to FMCH, the LMCH method also predicts human-like balance control behaviours, such as the VPP (virtual pivot point) phenomenon, observed in human walking. Our simulation results, using a neuromuscular model of human walking, demonstrate that this method could reduce the metabolic cost of human walking by 10%. Furthermore, to validate the design and simulation results, a preliminary version of this exosuit comprised of springs with constant stiffness was built. An experiment with eight healthy subjects was performed. We made a comparison between the walking experiments while the exosuit is worn but the springs were slack and those when the appropriate springs were contributing. It shows that passive biarticular elasticity can result in a metabolic reduction of 14.7±4.27%. More importantly, compared to unassisted walking (when exosuit is not worn), such a passive device can reduce walking metabolic cost by 4.68±4.24%.

For further publications of the author please check: ResearchGate, ORCID or LOOP

Bio-inspired neuromuscular reflex based hopping controller for a segmented robotic leg

The bio-inspired neuromuscular reflex based controller can generate stable hopping motion in a real robot. Check out our recently published paper (open access) by Zhao et al. in Bioinspiration & Biomimetics for more details.


It has been shown that human-like hopping can be achieved by muscle reflex control in neuromechanical simulations. However, it is unclear if this concept is applicable and feasible for controlling a real robot. This paper presents a low-cost two-segmented robotic leg design and demonstrates the feasibility and the benefits of the bio-inspired neuromuscular reflex based control for hopping. Simulation models were developed to describe the dynamics of the real robot. Different neuromuscular reflex pathways were investigated with the simulation models. We found that stable hopping can be achieved with both positive muscle force and length feedback, and the hopping height can be controlled by modulating the muscle force feedback gains with the return maps. The force feedback neuromuscular reflex based controller is robust against body mass and ground impedance changes. Finally, we implemented the controller on the real robot to prove the feasibility of the proposed neuromuscular reflex based control idea. This paper demonstrates the neuromuscular reflex based control approach is feasible to implement and capable of achieving stable and robust hopping in a real robot. It provides a promising direction of controlling the legged robot to achieve robust dynamic motion in the future.

For further publications of the author please check: ResearchGate, ORCID or LOOP

Biarticular muscles in light of template models, experiments and robotics: a review

Read our recent review paper about biarticular muscles to learn about the scientific discoveries from simulation models, evidence from human experiments and beneficial design principles in robotic applications. Link to the published paper (open access): Link to Royal Society Interface.

[[|Link to Royal Society Interface]]

Abstract: Leg morphology is an important outcome of evolution. A remarkable morphological leg feature is the existence of biarticular muscles that span adjacent joints. Diverse studies from different fields of research suggest a less coherent understanding of the muscles’ functionality in cyclic, sagittal plane locomotion. We structured this review of biarticular muscle function by reflecting biomechanical template models, human experiments and robotic system designs. Within these approaches, we surveyed the contribution of biarticular muscles to the locomotor subfunctions (stance, balance and swing). While mono- and biarticular muscles do not show physiological differences, the reviewed studies provide evidence for complementary and locomotor subfunction-specific contributions of mono- and biarticular muscles. In stance, biarticular muscles coordinate joint movements, improve economy (e.g. by transferring energy) and secure the zig-zag configuration of the leg against joint overextension. These commonly known functions are extended by an explicit role of biarticular muscles in controlling the angular momentum for balance and swing. Human-like leg arrangement and intrinsic (compliant) properties of biarticular structures improve the controllability and energy efficiency of legged robots and assistive devices. Future interdisciplinary research on biarticular muscles should address their role for sensing and control as well as non-cyclic and/or non-sagittal motions, and non-static moment arms.

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