Cable-driven robotic interface for lower limb neuromechanics identification

This paper presents a versatile cable-driven robotic interface to investigate the single-joint joint neuromechanics of the hip, knee and ankle. This endpoint-based interface offers highly dynamic interaction and accurate position control, as is typically required for neuromechanics identification. It can be used with the subject upright, corresponding to natural posture during walking or standing, and does not impose kinematic constraints on a joint, in contrast to existing interfaces. Mechanical evaluations demonstrated that the interface yields a rigidity above 500N/m with low viscosity. Tests with a rigid dummy leg and linear springs show that it can identify the mechanical impedance of a limb accurately. A smooth perturbation is developed and tested with a human subject, which can be used to estimate the hip neuromechanics.


I. INTRODUCTION
An accurate characterisation of lower limb neuromechanics is required to understand lower limb neurophysiology and to design appropriate control for a robotic walking aid. To identify the limbs neuromechanics, a rigid robotic interface equipped with powerful actuators is typically required to apply force/position disturbances while sensors record the resulting modification of position/force. Available robotic interfaces to identify the lower limb neuromechanics include motor-driven dynamometers [1], [2], [3], [4], [5], [6] and gait rehabilitation exoskeleton devices [7], [8], whose characteristics are compared in Table I.

A. Existing neuromechanics estimation devices
Motor-driven dynamometers are robotic interfaces that perform single joint rotations while the two limbs are fixed to the interface. It can be used for single joint identification and physical therapy, providing isotonic, isometric and isokinetic experimental conditions. Such robotic interfaces have been used to estimate the torque-angle relation of the ankle joint [2], injury or disease induced increase in ankle joint stiffness [9], passive resistance torque increased Email: {h.huang14, e.burdet}@imperial.ac.uk. All authors but Bouri are or were with the Department of Bioengineering, Imperial College of Science, Technology and Medicine, UK (https://www.imperial.ac.uk/humanrobotics). Farkhatdinov is with the School of Electronics Engineering and Computer Science, Queen Mary University of London, UK. Arami is with the Department of Mechanical and Mechatronics Engineering, University of Waterloo, Canada. Bouri is with the Biorobotics Laboratory,École Polytechnique Fédérale de Lausanne. We thank Jonathan Eden for editing the text. This work was funded in part by the EU-FP7 grants ICT-601003 BALANCE, ICT-611626 SYMBITRON. at the knee [10] or at all joints [11] due to spinal cord injury (i.e. spasticity), and the difference in ankle range of motion (ROM) in individuals with cerebral palsy [12], [13]. These interfaces however impose movement constraints on a joint, which may result in unnatural motions and accounts for variability in torque-angle identification results using single-joint motor-driven dynamometer relative to estimations from multi-joint inverse dynamics [2]. Furthermore, motor-driven dynamometers are not equipped with body weight support for hip joint measurements. Therefore, hip joint neuromechanics investigations with those interfaces could only involve healthy participants [14], or be used with impaired individuals only in postures not requiring weight bearing.
Gait rehabilitation robotic exoskeletons are interfaces affixed to the body. They can provide controlled gait assistance and be used to analyse neuromechanical factors such as spasticity and voluntary muscle force level [7], or joint impedance during leg swinging [8]. However, exoskeletons constrain the joints movement, and may induce nonnegligible vibrations due to the difficulty to design a rigid mechanical structure. Furthermore, they may not be suitable to develop specific experiment protocols required to measure the lower limb neuromechanics. For example, the stiffness estimation method of [15], [16] relies on perturbation pulse trains of 1.72 • amplitude and 150ms pulse width, which is challenging to implement on existing gait rehabilitation exoskeletons due to their limited rigidity and torque capabilities.
Until present, most existing devices focused on ankle or knee joint measurements, while devices targeting the hip joint could usually be used only for isokinetic motion [14], [17] or for multi-joint torque perturbations [8] that may limit the measurements accuracy. In view of these functional limitations, we developed a robotic interface that can be used to systematically investigate the single-joint neuromechanics of the hip, knee and ankle joint in a natural upright position, without constraining the targeted joint motion, and with negligible structural vibrations even during highly dynamic perturbations.

B. Functional requirements
We want to develop a versatile device with a powerful actuator, a rigid mechanical structure and suitable control, arXiv:1908.02689v1 [cs.RO] 7 Aug 2019  [8], [19] that can be used to implement various protocols, including isometric and isokinetic conditions as well as brief mechanical perturbations, in order to characterise the lower body joints neuromechanics. To implement isometric conditions, the device should be able to provide a strong standstill torque to resist the subject's maximum voluntary contractions. The joint torque measurements obtained during a stair climbing experiment [20] were used as a reference for our device's actuator's standstill torque. To perform isokinetic experiments and quantify velocity-dependent muscle or joint behaviour, our device should also be able to implement a full range of movements with constant velocity range from 0 to 250 • /s [21]. Finally, the device should be able to realise the highly dynamic movements required for estimating mechanical joint impedance. For instance, evaluating ankle reflexes as in [15], [16] requires a fast ankle angle perturbation with 1.72 • amplitude and 150ms pulse width. In order to perform similar controlled position perturbations on the hip joint, considering that a human leg contributes to 15-20% of the body weight [22], a torque up to 100Nm would be required for a 90kg subject.
Large forces and accelerations required to characterise the lower limb neuromechanics demand a powerful and thus heavy actuator. Therefore, this actuator should be rigidly fixed away from the moving limb and reliable motion transmission should be used. A pneumatic cylinder connection [23] or a two-bar linkage transmission [24] could be used in this purpose, which would however result in a limited workspace. A pretensioned cable transmission can provide actuation with low inertia and without backlash [25], and is therefore selected for our interface.
The structure of the interface should be rigid, to minimise undesired vibrations and deformations. An additional structure is required to support the subject in an upright posture such as for walking. This structure should be mechanically independent from the robotic interface in order to avoid the transmission of vibrations from the motor. Finally, to avoid constraining the hip and knee joints motion such as with the Lokomat [7] and LOPES [8] exoskeletons, we developed an end-point based interface interacting with the extremity of the examined limb while the rest of the body can move freely.
This paper presents and validates the resulting Neuromechanics Evaluation Device (NED). Section II explains its design concept, its components and control. Section III analyses the resulting kinematics and parameters sensitivity, with mechanical characterisations. Typical applications are illustrated in section IV.

A. General description
Considering all the design factors presented in the previous section, our solution included a cable driven device with a large actuator placed outside of the workspace transmitting power to the limb, while the subject is supported in a natural upright posture by an independent structure. The developed Neuromechanics Evaluation Device (NED) is illustrated in Fig. 1.
As shown in Fig.1a, the subject is half seated on a rigid chair (of length 0.55m, width 0.7m and height 1.5m) with one leg suspended in the workspace and attached to the system via a foot fixture. The leg is moved by the motor (AM8061, Beckhoff, Germany) located at the bottom of the workspace, via a steel cable (7x7 galvanised steel with PVC coating). Two load-cells (TAS510, HT sensors, China) are placed between the extremities of the foot fixture and the cable to measure the respective interaction forces. The front and rear pulleys can be locked at different positions along the rail (3m in the horizontal direction and 1.5m vertically) in order to keep the cable perpendicular to the leg (as shown in Fig.1a) and tensed. The cable tension is adjusted by the turnbuckles placed in series with the cable. All structures are made with aluminum strut profiles (40x40L, Bosch Rexroth, Germany) and bolted to the cement floor.
The open seat enables the experimenter to perform hip experiments at different knee angles as shown in Fig.1b. In this case the knee joint can be kept at a specific joint angle using a knee brace (T-scope, Breg), which enables us to study the influence of the knee angle. By adjusting the seat, it becomes also possible to study the knee neuromechanics as shown in Fig.1c.
In order to select an actuator based upon our design criteria, motors produced by Beckhoff, ETEL and Infranor were evaluated depending on their technical specifications including motor peak torque, standstill torque, moment of inertia and possible gearbox reduction ratio. The actuator selection process is detailed in Appendix A. Appendix D lists NED's components and their characteristics, including the selected actuator.

B. Cable transmission
The cable transmission, using the Capstan effect, requires a sufficiently large contact force between the cable and the pulley to prevent slippage. The Capstan equation defines the force relation between both the cable and the contact area of a cylinder while pulling a leg forward (Fig.1d), where T load is the front cable tension which bares larger loading, T hold the force required to hold the loading on pulley, µ the friction coefficient between the cable and pulley, and φ the contact angle. The minimum contact angle φ to prevent slippage is calculated from the friction coefficient and the expected cable force on both sides of the pulley. Assuming a joint torque of 150Nm [20] and a plastic-metal friction coefficient of 0.1-0.3, the minimum number of cable turns must be four. Conservatively, the cable was wounded five times around the pulley.

C. Control system
The control architecture of NED is depicted in Fig. 2. The supervisor computer provides the position command to the control computer (CX5130, Beckhoff), which performs the real-time control and monitors the motor controller (AX5112, Beckhoff) and motor (AM8061, Beckhoff). The cable system transmits the motion. In the example of a position disturbance, the angular displacement of motor shaft ∆θ m is monitored by the software limits to avoid overstretching the leg. The two load-cells at the extremities of the ankle fixture record the interaction forces F 1 , F 2 , which are fed back via a resister bridge unit. Muscle activity of the subject are recorded with surface electromyography (EMG) electrodes and filtered by an amplifier. Safety relays, which are controlled by a laser safety system and emergency buttons, are connected to the power supply of the motor controller in case of need. Both the control computer and motor controller are powered separately by a power supply (PRO ECO 120W, Weidmuller, Finland). The operating software environment TwinCAT plans and implements the fastest point-to-point motion with given speed, acceleration and jerk limits.
In addition to considering the feedback error, the motor controller uses three sensor channels. Two resister bridges (EL3351, Beckhoff) are used to measure the interaction force between the device and the subject's leg, and an analog channel (EL3255, Beckhoff) can be used e.g. for measuring the leg motion.

D. Safety measure and ergonomics
Safety is a critical factor for robotic interfaces which are in contact with the human body. Therefore, we implemented redundant hardware and software measures to ensure safety throughout our experiments. A safety system was developed to define the allowable range of motion as shown in Fig.3. The laser box emits a signal to the photodiode in the receptor box, which controls two safety relays. If the laser beam is blocked by any obstacle, e.g. leg moving beyond the expected range, safety relays will shut down the motor controller. Second, software safety measures implemented in the motor controller shut down the power when a position, speed, acceleration or power/torque limit is reached. Specific software limits define the workspace in which the leg should move depending on the targeted experiment. Finally, the power supply of both the motor and motor controller will shut down if any of the three emergency buttons is pressed. These buttons, which are available to the subject and experimenters during the experiment, are connected to another two safety relays and a master switch. In total, four safety relays (PSR-MS35, Phoenix Contact, Finland) control the power supply of the controller. If any safety threshold is reached the power motor is set off.
Beside the safety measures described above, various factors are included to provide a comfortable environment for different subjects. The dimensions of NED, that includes rail lengths and chair size, are designed based on subjects of height between 1.5-1.8m. The rigid chair is covered with Off-plane motion induced error. Panel (a) illustrates the measurement error resulting from side-way motions. θ1 and θ2 are the misalignments between the load-cell and the desired leg motion, L1 and L2 the distances between the foot and both pulleys. The maximum permissible side-way motion xu can be calculated by trigonometry. Panel (b) depicts the error resulting from a large leg rotation. Since both pulleys are fixed for each experiment, large leg motions will cause an angle between the load-cell and line of motion, which is described as angles θ3 and θ4. L3 and L4 are the distances between the foot and the pulleys. On the other hand, L 3 and L 4 are the distances between the foot and the ideal pulley location. x d and y d are the distances between the ideal pulley location and actual pulley location. memory-foamed cushions to increase experiment contentment, and handrail location is adjustable to optimised body weight support.

III. SYSTEM CHARACTERISATION
This section first analyses the kinematics of the developed system. It then examines sensor measurement errors and solutions to ensure an accurate recording. Lastly, a series of system identification tests are performed to characterise the system in different dynamical conditions.

A. Kinematics and the sensitivity analysis
NED is designed to measure the lower limb joints' flexionextension biomechanics assuming that the cable and leg motions are restricted to the sagittal plane (with high enough cable pretension). For small angular displacements with the knee stretched and locked to maintain the leg straight, the kinematics is: where ρ m is the motor pulley diameter,θ m is the motor speed,ẋ is the cable linear motion speed, L the leg length andθ the hip joint rotation speed.
To ensure that the aforementioned planar movement assumption is valid we investigated how lateral leg movements can potentially influence the accuracy of the biomechanical measurements. We assume that during an experiment the leg-cable attachment point can displace sideways by an undesired distance x u measured from the normal plane of movement as shown in Fig. 4a. Then, the cable force measurements are affected by the off-plane configuration described by the angles θ 1 = arctan(x u /L 1 ) and θ 2 = arctan(x u /L 2 ) as shown in Fig. 4a. By considering different leg lengths {80-95cm}, hip angles {5 • -60 • } and different pulley locations {80-165cm horizontal and 45-110cm vertical}, it is shown that the side-way motion should be limited within 14.3cm to result in a force measurement error below 5%. A side-way motion test (with 200N cable pretension) shows that a 14cm side-way motion requires an external force of 225N. Experiments should therefore be limited within such force limitations.
As large angular displacement cannot be considered as linear motion, we further investigated the influence of limb rotation upon measurement accuracy. For each experiment, the pulley locations are relocated and fixed to yield a perpendicular cable connection minimising the measurement errors (shown as the gray dashed line in Fig. 4b). Moving far away from the initial position will cause measurement error due to angles θ 3 = (L 2 3 + L 2 3 − Y 2 d )/(2 * L 3 * L 3 ) and θ 4 = (L 2 4 + L 2 4 − X 2 d )/(2 * L 4 * L 4 ) (the black lines). By considering different leg lengths {80-95cm}, different experiment hip angles {5 • -60 • } and different sizes of leg motions, the largest acceptable leg motion before reaching an 5% error in measurement is 21cm. This can be considered as maximum acceptable position displacement to design experiments.

B. Cable's tension spatial and temporal dependency
The mass of the load-cells, connecting the elements and the cable itself (0.5kg) will slightly bend the cable and create cable sagging as shown in the left panel of Fig. 5. As the cable is not perfectly straight, we observe that the tension does not change monotonically (as shown in Fig. 5b with cable tension measured from a single load-cell during a back and forth motion) and this discontinuity in force measurements is caused by a misalignment between the load-cells' axis and the cable's motion. Trigonometric calculations showed that a pretension of 200N limits the relative error between the measured and actual tensions As in most cable-based systems, NED has mechanical characteristics that can slightly change over time. During the validation tests, it was observed that the measured cable tension will drop for 1N every 33.6s (during a cyclic movement test of speed 750mm/s with a pretension values of 200N while holding a 18kg dummy leg, which will be described in Section IV-A). This tension drop is negligible since most movements for neuromechanics evaluation require short perturbations with duration <1s (as will be developed in Section IV-B). Mathematical description of the cable temporal dependency is described in Appendex B.

C. Cable system modelling
We performed system identification tests to demonstrate that the behaviour of the designed interface can be characterised with the second order linear dynamical system under different speed. Hence, the transfer function describing the cable's dynamics with cable tension, ∆F (s), as input, and cable displacement, ∆X(s), as output can be expressed as: with M x the mass of the moving components on the cable, B x and K x the cable viscosity and stiffness, respectively. For system identification tests, we pre-programmed NED's controller to perform 10 saw-shape displacement patterns with ±60mm amplitude and different speeds of 20-750mm/s as shown in Fig. 6a. The force acting on the cable was measured with a load-cell and recorded at 1kHz. All ten trials at a given speed conditions were used to estimate  The parameters identified at different speeds are shown in Fig. 6b, demonstrating that our interface is characterised as high stiffness and low viscosity (which reduces with the speed). This indicates that (despite the inherent cable compliance) NED is a rigid device that can be used to identify the lower limb mechanics. The performance of the fitting were confirmed by normalised root mean square error value (NRMSE) with values higher than 70%. Due to both low variance in estimated impedance and relatively high NRMSE value, the mechanical characteristic of NED can be described by the average transfer function parameter's values, with the averaged Bode plot shown in Fig. 6c which has two poles, at 1.6 and 17.2Hz respectively. Further identification of the cable's nonlinearities is described in Appendix C.

IV. VALIDATION
In this section, we demonstrate how NED can be used for lower limb neuromechanics characterisation. First, two experiments were conducted to identify the dynamic parameters of a dummy leg and a pair of springs seperately, and then the experiment identification results were compared to know mechanic properties of the components. We then determined an optimal position perturbation for estimating hip joint stiffness of healthy subjects.

A. Dummy leg mechanics
To validate the functionality of the developed interface, experiments are carried to identify the mechanical properties dummy leg and compare with the values obtained from CAD calculations (Fig. 7a). The design parameters of the leg were: mass 18kg (resembles the leg mass of a 90kg subject), length of 70cm and moment of inertia of 1.84kg·m 2 with respect to the hip joint. This mechanical dummy leg was fixed in NED for neuromechanics experiments. During the experiment, the leg was rotated about the hip joint (flexion/extension sequences) with the amplitude of 5 • (6cm endpoint displacement) and with the speed range of 20-750mm/s as shown in Fig.7d. The prior mentioned flexion/extension movements were tested at five different hip angles between 15 • and 55 • to test the the influence of gravity to the identification results. In total, 20 repetitions were performed at each combined condition of speed {20-750mm/s} and hip angle {15 • -55 • }.
As described in Section III-C, the developed robot exhibits high stiffness and low viscosity which resembles a rigid device. Therefore, the cable dynamics in series with the leg could be neglected, and the recorded displacements and cable-leg interaction forces were used to estimate a linear second-order model of the mechanical dummy leg (with the nonlinear least square method tfest of Mathworks Matlab): where ∆τ is the change of interactive torque, I the leg inertia, B the viscous parameter of the joint, K the hip joint stiffness, F 1 and F 2 are the two load-cells' signals.
It is important to note that the change in interaction torque (∆(F 1 − F 2 )L) was used here rather than change in one load-cell measurement (∆F 1 ), as the purpose of this section was to estimate dummy leg impedance through position displacement and resulting force changes. The estimated impedance values are shown in Fig. 7b with the NRMSE value depicted in Fig. 7c which suggests that the dummy leg's dynamic parameters were successfully identified for velocities larger than 40mm/s (with NRMSE > 80%). The inertia estimated in this dynamic identification is close to the value predicted from the CAD parameters (1.84kg·m 2 ) while the viscosity and stiffness values are both low. These results indicate that NED can be used to identify the hip mechanical impedance.

B. Stiffness estimation
To evaluate whether NED can be used to identify stiffness, we used two parallel springs attached to the cable as shown in Fig.8a. The stiffness of these two springs was then identified using a smooth position displacement as described in [26]. In this method, stiffness K can be simply computed from: on the constant position plateau where inertia and viscosity have little influence, as ∆θ = ∆θ = 0, see Fig.8b. Twenty tests were carried out for each perturbation with the amplitude of 2-8mm and duration 50-150ms. Stiffness was then evaluated from (5) with mean displacement ∆θ and mean measured torque ∆τ during the plateau region. The results of Fig. 8d demonstrate that this method can identify stiffness accurately, with estimations improved with a larger perturbation amplitude and no observable difference in different duration or perturbation direction. Since the load-cells' measurement suffers from noise with standard deviation of 0.29N and maximum 1N, a large amplitude with stronger spring force will increase the signal to noise ratio and improve the estimation. In the mean time, the angle measurement of the motor shaft has a resolution of 0.019 o corresponding to a 0.35mm cable displacement and angle of 0.46 • for a 90cm long leg. This also implies a small perturbation amplitude (e.g. 2mm amplitude) will suffer from measurement errors.

C. Optimal position perturbation to identify stiffness
A pilot study with one healthy subject (male, weight: 54kg, height: 172cm) was carried out to evaluate the feasibility of using NED to identify hip stiffness with the technique described in the previous section. The experimental protocol was approved by the Imperial College Research Ethics Committee, and all procedures were performed according to the principles described in the Declaration of Helsinki. The subject was informed on the device and experiment, and signed an informed consent form prior to the experiment. The participant's weight and leg length (from the anterior superior iliac spine to the lateral malleolus) were measured to estimate the leg inertia. A lockable knee brace was used to fix the knee joint at 0 • angle. The subject was half seated on the chair with one leg suspended, was asked to support his body weight on the handle and relax the lower limb. A harness was attached to the ankle of the tested leg which was connected to the cable and the motor (Fig.1). The participant was given an emergency stop and can stop the experiment whenever needed. The laser safety system was initiated and adjusted to define the range of motion of the tested leg at 15 • . The motor performed a slow motion to move the participant's leg to define the comfortable range of motion for additional system adjustments. A single pulse perturbation was also given to provide the participant with an experience of a perturbation and adjust the system's safety.
As described in Section II-C, the motor controller implements the fastest displacement given the safety limits in speed, acceleration and jerk. The position perturbation with a constant displacement plateau was thus determined by these limits as well as by the plateau duration and displacement amplitude. Our goal was to use a perturbation of minimum amplitude and duration, which would disturb the subject minimally and thus avoid any voluntary reaction. However, this comes in trade-off with the perturbation amplitude, which should be large enough to maximise the signal to noise ratio (SNR) of the stiffness estimation. On the other hand, very large accelerations to yield a fast perturbation could cause cable oscillations which would disturb the subject and affect the stiffness estimation quality. To achieve fast displacement with minimal cable oscillations and stable force measurements, we increased the perturbation amplitude iteratively (starting from 6cm) and reduced the controller's dynamic limits (speed, acceleration and jerk) while recording the magnitudes of cable oscillation. In total, 50 combinations of the speed, acceleration and jerk limits were tested for position perturbation command with the hip flexed at 15 • .
The resulting optimal perturbation is shown in Fig. 9a. The amplitude of the perturbation is largely above the position resolution (0.35mm cable displacement) and from the test with the spring of Fig. 8, should have a large signal to noise ratio. Considering the large motor variability in human movements, we selected a larger displacement than required to maintain a high signal to noise ratio. The collected kinematics and interaction forces were again least square fitted to estimate the hip joint impedance using (4). As shown in Fig. 9a, the optimised perturbation resulted in consistent and reproducible motions with negligible force oscillations. Fig.9b shows the estimated joint impedance (I, B, K) of three different perturbation amplitudes {15,17.5,20}mm with a 150ms long plateau. The green line shown in Fig. 9b is the inertia calculated from the anatomical model [22] using the subject's weight and leg length. We can see that the variance of estimation is small (11% for inertia, 10% for stiffness and 19% for viscosity). The inertia estimate is close to the anatomical model, and the values of viscoelasticity are in the same order of magnitude as reported in [8].

V. DISCUSSION
Investigating the lower-limb neuromechanics is critical to understanding the control of standing and walking in healthy and neurologically affected individuals, as well as to efficiently control assistive and rehabilitation devices for performance augmentations. However, so far only few studies could use a single device to investigate lower-limb neuromechanics of different joints and specific to the hip. Importantly, hip joint viscoelasticity investigation was only performed in multi-joint torque perturbation [8], which is usually limited in accuracy, and never with precise single joint position displacement. In this context, we have developed and validated a novel robotic interface named NED (Neuromechanics Evaluation Device) to investigate the lower-limb neuromechanics.
NED can apply a large range of dynamic interactions to a subject's leg at static posture or during movement. This enables the neuromechanics identification of hip and knee joints in flexion/extension. Importantly, NED allows the experimenter to estimate a subject lower limb neuromechanics in a natural upright posture under controlled environment, which also makes the device well suited for carrying out investigations on patients' neuromechanics. The device can be quickly adapted to a subject's specific anatomy and to carry out various measurements. The use of a closed mechanical cable loop with powerful actuator fixed outside the rigid supporting structure enables us to implement highly dynamic environments with little vibrations.
In this paper, NED's mechanics was characterised, and its performance to estimate a lower limb neuromechanics was demonstrated through the identification of a dummy leg and a spring's mechanical impedance. As a result of a powerful actuator and stiff mechanical frame of NED, it was possible to achieve accurate and repeatable position perturbation which enabled more efficient dynamics identification of individual leg joint compare to the mechanisms with rigid links [2], [24]. The good match of the identified and measured parameters as well as the range of protocols that can be implemented on NED makes it an effective tool to identify the hip, knee and ankle joint biomechanics. The techniques developed in this paper could be used to systematically investigate hip joint viscoelasticity as described in [27].

A. Motor selection
The motor was selected in four steps to identify an actuator fulfilling the design criteria of Section I-B. Actuators from three manufacturers were considered: ETEL, Beckhoff and Infranor, their specifications were used in simple simulations to determine the most appropriate actuator.
First, actuators with unsuitable features (such as extreme power rated torque <1Nm or >200Nm or excessive dimensions such as length >1m) were removed from the list. Then simulation of impedance estimation was carried out to define the minimum motor peak torque required for our robot. A typical perturbation profile for impedance identification [26] with 100ms rise time and 0.03 • amplitude was selected to move a simulated leg. The simulated model represented a 90kg subject's leg (with inertia calculated as in [22]). Motors with too low dynamics were removed from the list. Different gearbox ratio and pulley radius were considered in the same time to achieve an optimal solution for each motor.
Significant standstill torque is required for isometric experiments when the subject is performing motions involving maximal voluntary isometric contractions (MVIC). Therefore, a third selection was conducted by comparing the MVIC data available in literature [20], [28] with the selected motors capabilities, and motors with insufficient standstill torque were excluded.
In general, human movements are slower than a typical actuator's speed. Therefore, reducing the actuator's speed enables us to increase the output torque and thus select smaller motors. In our setup the motor pulley amplifies the motor torque by a factor of L/ρ, where L is the leg length and ρ the pulley diameter. While a pulley of small size would provide a large torque amplification and thus enable us selecting a smaller motor, a small pulley will be fragile. To have a sufficiently rigid pulley, we impose a radius larger at least 1cm larger than the motor shaft. Considering all these factors, the motor Beckhoff AM8061 with a gearbox of reduction ratio 5:1 was selected, with a pulley of diameter of 5.5cm.

B. Cable temporal dependency
To characterise the drop in cable tension over time we measured the tension force while during slow periodic cable displacements of ± 6 cm with the average cable speed was 0.2 mm/s. The initial pretension was 200 N. In total five cyclic motions were performed for different initial positions of the cable and the results are shown in Fig. 10). A relatively small decrease in cable tension, f (t), was observed which could be modelled with an exponential function: where t is the time in seconds with compensated result shown in Fig.10 right panel. As described in Section III-B, this tension drop is negligible for fast perturbations. However, it might be required to consider cable tension drop for timely tests, depending on the experimental tasks.

C. Cable Nonlinearities
As described in the main text, a system identification showed that NED's mechanical characteristics can be de-scribed as a linear second order system with high stiffness and low viscosity. To further assess the nonlinear characteristics (such as Coulomb friction) within the system, an additional identification was carried out by adding a piecewise linear Hammerstein-Wiener (HW) model as shown in Fig.11a. The average value identified in Section III-C was used as the linear system described in the centre of Fig.11a. The light blue circles and black triangles in Fig.11b shows how using the HW model slightly improves the identification, which is probably due to consideration of motor Coulomb friction.