Haptic Transparency and Interaction Force Control for a Lower-Limb Exoskeleton

Accepted for publication in the IEEE T ransactions on Robotics (T -R O). Haptic T ransparenc y and Interaction F orce Control for a Lo wer -Limb Exoskeleton Emek Barıs ¸ K ¨ uc ¸ ¨ uktabak, Y ue W en, Sangjoon J. Kim, Matthew R. Short, Daniel Ludvig, Le vi Hargro ve, Eric J. Perreault, Ke vin M. L ynch, and Jose L. Pons Abstract —Controlling the interaction for ces between a human and an exoskeleton is crucial f or providing transpar ency or adjusting assistance or resistance levels. Howev er , it is an open problem to control the interaction forces of lower -limb exoskele- tons designed for unr estricted over ground walking . For these types of exoskeletons, it is challenging to implement for ce/torque sensors at every contact between the user and the exoskeleton for direct for ce measurement. Moreov er , it is important to compensate for the exoskeleton’ s whole-body gra vitational and dynamical for ces, especially for heavy lower -limb exoskeletons. Pre vious works either simpliﬁed the dynamic model by treating the legs as independent double pendulums, or they did not close the loop with interaction force feedback. The proposed whole-exoskeleton closed-loop compensation (WECC) method calculates the interaction torques during the complete gait cycle by using whole-body dynamics and joint torque measurements on a hip-knee exoskeleton. Furthermore, it uses a constrained optimization scheme to track desired interaction torques in a closed loop while considering physical and safety constraints. W e evaluated the haptic transparency and dynamic interaction torque tracking of WECC control on three subjects. W e also compar ed the performance of WECC with a controller based on a simpliﬁed dynamic model and a passiv e version of the exoskeleton. The WECC controller results in a consistently low absolute interaction torque err or during the whole gait cycle for both zero and nonzero desired interaction torques. In contrast, the simpliﬁed controller yields poor performance in tracking desir ed interaction torques during the stance phase. Index T erms —Physical human-robot interaction, exoskeletons, interaction force control, rehabilitation robots, assistive robots E. B. K ¨ uc ¸ ¨ uktabak is with the Center for Robotics and Biosystems, De- partment of Mechanical Engineering, Northwestern University , Evanston, IL, USA and Shirley Ryan AbilityLab, Chicago, IL USA. S. J. Kim and Y . W en are with the Shirle y Ryan AbilityLab and Department of Physical Medicine and Rehabilitation, Northwestern University , Chicago, IL, USA. M. R. Short is with the Shirley Ryan AbilityLab and Department of Biomedical Engineering, Northwestern University , Evanston, IL, USA D. Ludvig is with the Department of Biomedical Engineering, Northwestern Univ ersity . L. Hargrov e is with the Shirley Ryan AbilityLab, Center for Robotics and Biosystems, and Department of Physical Medicine and Rehabilitation, Northwestern Uni versity . E. J. Perreault is with the Department of Physical Medicine and Rehabili- tation, Department of Biomedical Engineering, Shirley Ryan AbilityLab and Center for Robotics and Biosystems. K. M. L ynch is with Center for Robotics and Biosystems and Department of Mechanical Engineering, Northwestern Univ ersity . J. Pons is with the Shirle y Ryan AbilityLab, Center for Robotics and Biosystems, Department of Mechanical Engineering, Department of Biomed- ical Engineering, and Department of Physical Medicine and Rehabilitation, Northwestern Uni versity . I . I N T R O DU C T I O N Lower -limb exosk eletons are used to assist impaired indi vid- uals in their daily life and as a tool for physical rehabilitation in clinical settings [1], [2]. They are also increasingly used to augment the abilities of healthy individuals [3]. In each of these applications, the exoskel eton must be capable of regulating the interaction forces and torques between the user and the exoskeleton, both for the safety of the user and the effecti v eness of the control mode. As an e xample, the transpar ent control mode attempts to zero the interaction forces between the user and the exoskele- ton. Transparenc y is useful for comparison when e valuating new rehabilitation control strategies [4] and for allowing users to move freely when wearing assistive de vices [5]. Other con- trol modes, used in rehabilitation, include “assist as needed” (AAN) and resistiv e control for robot-aided gait therapy [6]. W ith AAN, the robot provides only as much assistance force as needed to complete a gi ven task [7]. In resistive and error- augmentation methods, force is applied to make the task more challenging, which accelerates relearning of motor tasks [4], [8]. An important part of controlling interaction forces is com- pensating the weight of the lower -limb exoskeleton. How the weight of the exoskeleton is transferred to the ground or the user depends on its design. There are three main categories of design: a) Gr ounded trunk: The trunk of the exoskeleton is ﬁxed relativ e to a treadmill structure [9]–[11]. Due to the grounded trunk, only the masses of the individual leg links are felt by the user , and these masses can be compensated with acti ve control. b) Floating-base (trunk) without feet: These exoskele- tons are designed for unrestricted over ground walking and do not contact the ground [12]–[14]. Partial gra vity compensation can be achie ved for the swing leg, but it is not possible for the exoskeleton to completely carry its own weight since there is no contact with the ground. Such exoskeletons are designed to be as light as possible, which limits their maximum assis- tance/resistance and battery life. c) Floating-base with feet: These e xoskeletons are de- signed for unrestricted over ground walking and ha ve feet that contact the ground [15]–[21]. The weight of the exoskeleton can be transferred to the ground through the exoskeleton’ s foot contact. While this allows for stronger actuators and larger batteries, it comes at the cost of more complicated controllers for accurate interaction force control. 2 In this paper , we focus on ﬂoating-base lower -limb ex- oskeletons with feet, for which controlling interaction forces remains an open problem [17], [22]. Accurate interaction force control will enable natural implementations of AAN, resistiv e, and error-augmentation methods in o verground w alk- ing therapy using rehabilitation exoskeletons. Floating-base exoskeletons with feet also allow the possibility of larger assistiv e forces and longer battery life. Due to the high torque requirements of ﬂoating-base ex- oskeletons with feet, high reduction ratios are used, which re- sults in high friction and apparent inertia of motor rotors [23]. Also, these exoskeletons are heavy due to powerful actuators and large batteries. Therefore, it is essential to compensate for these factors to accurately control the interaction forces and torques between the user and the exoskeleton. Another challenge is measuring the interaction torques be- tween the human and the exoskeleton. Grounded exosk eletons typically have a small number of contacts with the user, and interaction forces can be directly measured by force sensors implemented at the contacts [10], [24], [25]. In contrast, ﬂoating-base exoskeletons have broadly distributed contacts with the user , including at the trunk and foot. This makes it challenging to implement force/torque sensors at all contacts. Therefore, it is required to indirectly estimate the interaction forces and torques through other sensory information. In this paper, we present a novel method to estimate and control user-exosk eleton interaction torques during the complete gait cycle of a ﬂoating-base hip-knee exoskeleton with feet. The controller , which is based on constrained optimization, outperforms the state of the art in achieving transparency (zero impedance) and nonzero impedances in the user-e xoskeleton interaction beha vior . The context of the contribution of this paper is established by the related work belo w . A. Related W ork Previous studies on controlling interaction forces with ﬂoating-base lower-limb exoskeletons with feet can be divided into three main categories. 1) Whole-body dynamics compensation with no interaction for ce feedback: A lo wer-limb exoskeleton can be modeled as a ﬂoating-base system considering all links and joints together , similar to a bipedal robot [26], [27]. V antilt et al. [16] implemented a whole-body ﬂoating-base model while considering external contact constraints such as ground contact or contact with a stool. They estimated the contact forces based on a complementary energy method [28]. Howe ver , this method does not allo w the calculation of interac- tion torques at the non-static contact points between the human and exoskeleton. Camardella et al. [15] also modeled the full-body dynamics based on the user-dependent continuous interpolation between left and right stance dynamics. These studies compensated for the gravitational and Coriolis torques calculated using the whole-body model. Ho wev er , modeling errors were not corrected with interaction force feedback, and desired interaction torques were used only as a feedforward term. 2) Simpliﬁed double pendulum model with interaction for ce feedback: Andrade et al. [17], [18], and Tue et al. [19] employ a simpler , non-whole-body model of the exoskeleton to help estimate interaction torques. In this simpliﬁed model, each leg is modeled as an independent double pendulum 2R robot. When the leg is in swing and stance, the hip and ankle joints are assumed to be pinned, respectively . This model is used to calculate the feedforward compensation torques, which are subtracted from the joint torque measurements to estimate the interaction torques between the human and exoskeleton. Interaction torque estimation is used as the input of a proportional-deriv ative (PD) or admittance controller . Since the backpack is not modeled and the legs are modeled independently , the dynamic compensation torques and inter- action torque estimates are not accurate for the stance leg. W ith this approach, the stance leg carries only its own weight instead of the weight of the entire exoskeleton. While this might be acceptable for light or grounded exoskeletons, it is not accurate for exoskeletons designed for over ground walking that provide a high level of support and long battery life. 3) User-dependent or user-instrumented appr oaches: Since physical parameters and walking gaits v ary from user to user , some interaction force controllers require additional sensors to be afﬁx ed to the user and/or custom tuning for each user . Shariﬁ et al. [20] implemented a neural network scheme to learn the whole-body dynamics of the exoskeleton together with the user . While this method does not require additional sensors to measure the joint torques, the main drawback is user-dependent learned dynamics, and a training phase is required for each user . Kazerooni et al. [21] implemented a hybrid controller where the exoskeleton’ s stance leg follows the joint positions of the user to provide haptic transparency . The joint positions of the user’ s leg are measured by incli- nometers attached to it. Moreov er , the proposed method is for haptic transparency only; if the desired interaction force is non-zero, a model of the user-dependent interaction properties are needed. 4) Summary: Selected features of interaction force/torque control studies on ﬂoating-base lower -limb exoskeletons that contact the ground are presented in T able I. These studies can be categorized into three main groups. First, there are studies where the whole-body dynamics of the exoskeleton are compensated with no interaction force feedback [15], [16], [20]. Second, some studies modeled the exoskeleton as two independent legs with no consideration of the back- pack [17]–[19]. These studies implemented PD or admittance controllers based on the interaction force/torque error . Third, there are methods that require additional instrumentation on the user [21] or user-dependent learning for model or gait state estimation [15], [20]. None of these proposed control strate gies implements optimization methods to enforce physical or safety limits to av oid actuator saturation or to constrain the motion to satisfy safety conditions. T o the best of our knowledge, there is no study that calculates interaction forces with full-body dynamics during the whole gait cycle and uses this estimation in a closed-loop control for a ﬂoating-base lo wer-limb exoskeleton that is in contact with the ground. 3 T ABLE I: Selected features of the interaction force/torque control studies. Note that this paper focuses on ﬂoating-base lower -limb exosk eletons with feet. Therefore, this table does not include studies with grounded exoskeletons and ﬂoating-base lower -limb exoskeletons without feet. Reference Model Interaction force feedback User dependent Instrumentation on user Physical/safety limit implementation Camardella et al., 2021 [15] Whole-body No Y es No No V antilt et al., 2019 [16] Whole-body No No No No Andrade et al., 2019-21 [17], [18] Independent double pendulums PD controller No No No T u et al., 2020 [19] Independent double pendulums Admittance controller No No No Shariﬁ et al., 2021 [20] Whole-body (learnt by NN) No Y es No No Kazerooni et al., 2006 [21] 3-link for swing no model for stance (position control) Sensitivity Ampliﬁcation No (transparency) Y es (non-zero reference) Y es No Our proposed method (WECC) Whole-body Admittance controller No No Y es B. Contributions In this paper , we propose a nov el subject-independent method to calculate interaction forces during both single and double stance conﬁgurations based on a whole-body dy- namic model. Moreov er , we propose constrained optimization combined with a virtual model controller to achie ve desired interaction torques while considering physical and safety constraints. W e call the proposed method whole-exoskeleton closed-loop compensation (WECC) because whole-body dy- namics of the exoskeleton is compensated and considered in the closed-loop controller . W e ev aluated the transparency and spring-damper haptic- rendering performance of our proposed WECC controller on treadmill walking. W e compared the performance of the WECC controller to a control approach based on a simpliﬁed dynamic model of the e xoskeleton (the “simpliﬁed” condition), where the legs are modeled as double pendulums independent of one another and the backpack is ignored [17]–[19], [23]. F or transparency ev aluations, we also tested the performance of a passiv e version of the exoskeleton with disassembled dri ves (the “no-driv e” condition). W e observed that the transparency of the swing leg was similar under the three conditions (WECC, simpliﬁed, and no-driv e), b ut our proposed WECC method outperformed the simpliﬁed controller and the no-drive condition for the stance leg. In nonzero impedance haptic rendering, the WECC and simpliﬁed controllers performed similarly for the swing leg, but our WECC controller tracked the desired interaction torques better for the stance leg. The core contributions of this article are as follows: 1) A continuous whole-body forward dynamical model for the whole gait cycle is developed based on the ratios of the vertical ground reaction forces at the feet. 2) A novel method to estimate the interaction forces during both single and double stance conﬁgurations is proposed. 3) A constrained-optimization-based virtual mass controller is implemented that considers the underactuation of the exoskeleton and its physical/safety constraints to control the interaction torques. 4) The transparency and haptic rendering performance of our proposed WECC controller is compared to a state- of-the-art controller based on a simpliﬁed dynamic model. I I . S Y S T E M D E S C R I P T I O N A N D M O D E L I N G A. Lower-limb Exoskeleton: ExoMotus-X2 A four-degrees-of-freedom (DoF) lo wer limb exoskeleton (ExoMotus-X2, Fourier Intelligence, Singapore), shown in Figure 1, was adapted and used as a platform to test the proposed interaction torque controller . This exoskeleton has four total activ e degrees of freedom at the hip and knee joints, and passi vely allows motion at the ankle joints. Each activ e joint is driv en by a Maxon EC60 100 W motor with a 1:122.5 reduction ratio obtained via harmonic gearing and a belt. Copley Accelnet A CK-055-06 motor driv ers are used to control the motors. A peak torque of 80 Nm and a peak velocity of 3.2 rad/s can be obtained at each joint. Due to the high friction caused by the large reduction ratio, it is not feasible to estimate the joint torques by only measuring the current through the motors. Therefore we modiﬁed the ExoMotus-X2 exoskeleton to add joint torque sensors at each driv en joint. Each torque sensor consists of a full Wheatstone bridge with four strain gauges on the limb just distal to the joint, similar to the method presented by Claros et al. [29]. These custom joint torque sensors provide measurements with a resolution of 0.3 Nm and 0.1 Nm at the hip and knee joints, respectiv ely . The strain gauges were calibrated by immobilizing the joints and hanging known weights at known locations while the exosk eleton was suspended by backpack attachment points. A close-up view of the strain gauge is shown in Figure 1 An inertial measurement unit (IMU) (T ech-IMU V4 by T echnaid S.L.) is mounted on the backpack to measure the orientation and angular velocity of that link. Using the encoder measurements from each motor and orientation estimation of the backpack, the orientation of each link in the sagittal plane with respect to the gravity vector is estimated. In addition, eight 3-DoF force plates (9047B, Kistler) under a custom 4 Fig. 1: ExoMotus-X2 lower -limb exoskeleton, featuring a close-up view of the strain gauge implementation (left) and a user wearing the exoskeleton (right). instrumented split-belt treadmill are used to measure ground reaction forces at each foot 1 . Furthermore, to have a tighter connection with the upper body , an additional trunk brace was adopted from another lower -limb exoskeleton (Exo-H3, T echnaid S.L.). Communication with motors and onboard sensors is estab- lished over a CAN b us using the CANOpen communication protocol. T o minimize communication delay and allow real- time visualization and tuning capabilities, an external PC is connected to the exoskeleton by a tether . Analog force plate readings are acquired with an Arduino Mega board connected to the same PC. The controllers and applications are imple- mented on an open-source software stack called CANOpen Robot Controller (CORC) [30] based on ROS and C++. High-lev el controller commands and sensor measurements are updated at 333 Hz, while the low-le vel current controller of the motors runs at 15 kHz. B. Dynamic Model Our proposed WECC control method is based on a whole- body model where we incorporate the saggital plane dynamics of ev ery link of the exoskeleton, including the backpack. 1 The external footplates provide ground truth, but the controller uses only relativ e ground reaction forces rather than absolute ground reaction forces, and estimates of these relative forces may be provided by sensors on the exoskeleton itself. This is discussed further in Sections III and VI. Left stance Flight Double stance Right stance F l < F lim F r ≥ F lim F l ≥ F lim F l , F r < F lim F r ≥ F lim F l ≥ F lim F l ≥ F lim F r < F lim F r ≥ F lim F l , F r ≥ F lim F r < F lim F l < F lim Fig. 2: State machine structure of the gait states. The variables F l and F r are the left foot and right foot v ertical force readings, respectiv ely . The threshold force that triggers state change is represented by F lim . This threshold can be an absolute force value or a ratio of the total vertical forces. When this state machine is used in real-time control, states are not allowed to change more than once ev ery 150 ms for robustness to noise. During walking, the focus of this paper , the ﬂight state never occurs. 1) Whole-body dynamics: The exoskeleton is modeled as a ﬂoating-base ﬁv e-link mechanism with four active joints. The dynamics of the human user are not modeled but are incorporated into the exoskeleton dynamics as a source of external torque at each joint. For each gait state, the constraints on the system and the equations of motion are modiﬁed accordingly . These gait states are no ground contact ( ﬂight ), contact by a single foot ( single stance ), and contact by two feet ( double stance ). Figure 2 presents the gait state machine. a) Single stance: In single stance, the ankle of the stance leg is assumed to be hinged to the ground. W ith this assumption, the e xoskeleton has ﬁ ve degrees of freedom. The generalized coordinates and equations of motion are M i ( q ) ¨ q + b i ( q , ˙ q ) + g i ( q ) = S ⊤ τ joint + τ int , i ∈ { ls , rs } , (1) where q = [ θ 0 , θ 1 , θ 2 , θ 3 , θ 4 ] ⊤ are the generalized coordinates corresponding to the backpack, hip, and knee angles, as shown in Figure 3a. The matrix and vectors M i ∈ R 5 × 5 , b i ∈ R 5 , g i ∈ R 5 , are the mass matrix, Coriolis and centrifugal torques, and gravitational torques respectiv ely during left stance ( i = ls) and right stance ( i = rs). The vector τ joint ∈ R 4 corresponds to joint torques, and S = [0 4 × 1 , I 4 × 4 ] is a selection matrix of actuated joints. The interaction torques applied to the exoskeleton by the user are given by the vector τ int ∈ R 5 . b) Double stance: This state can be expressed as an extension of the single stance model where one ankle is 5 assumed to be hinged on the ground and external forces are applied on the other ankle, M i ( q ) ¨ q + b i ( q , ˙ q ) + g i ( q ) = S ⊤ τ joint + τ int + J ⊤ j Γ j , ( i, j ) ∈ { ( ls , r ) , ( rs , l ) } , (2) where J j and Γ j corresponds to the contact Jacobian of the left ( j = l) or right ( j = r) ankle and the external forces at that point, respectively . Multiplying Equation (2) by the transpose of the constraint matrix H ∈ R 5 × 3 , where J H = 0 , results in the following equation of motion: H ⊤ ( M i ( q ) ¨ q + b i ( q , ˙ q ) + g i ( q ) − τ int ) = H ⊤ S ⊤ τ joint , i ∈ { ls , rs } . (3) The two position constraints on the other ankle represented in the constraint matrix H map the ﬁv e-dimensional Equation (2) onto the three-dimensional controllable space. An y three of the ﬁv e coordinates may be chosen as the generalized coordi- nates. 2 Since there are only three DoF , the system is ov eractu- ated during double stance; there exists a one-dimensional null space of internal “cocontraction” torques that ha ve no impact on motion. Overactuation in double stance complicates the estimation of interaction torques, as discussed in Section III, and may cause a discontinuity in the equations of motion and the dynamic compensation torques during transitions between single and double stance. T o sidestep these issues, we approximate double stance as a transition from left stance to right stance or vice versa. M ds ( q ) ¨ q + b ds ( q , ˙ q ) + g ds ( q ) = S ⊤ τ joint + τ int , (4) where M ds = α M ls + (1 − α ) M rs , (5) b ds = α b ls + (1 − α ) b rs , (6) g ds = α g ls + (1 − α ) g rs . (7) The variable α in Equations (5)–(7) corresponds to the in- terpolation factor from left-stance dynamics to right-stance dynamics. These interpolated dynamics are particular solutions to the equations of motion (3), as prov en in Appendix D. The interpolation factor α is calculated based on the ratio of the left vertical ground reaction force to the sum of both vertical ground reaction forces α = F l ,y F l ,y + F r ,y , (8) where F l ,y and F r ,y are left and right vertical ground reaction forces, respectiv ely . The rationale for deﬁning the α based on the ratios of the vertical ground reaction forces is presented in Appendix D. Interpolating the equation of motion from left-stance dynamics to right-stance dynamics allows smooth continuous modeling of the whole gait cycle that is consistent with the weight transition of the exoskeleton during walking. The difference of this method from the work of Camardella et al. [15] is that their kinematic-based gait se gmentation is user-dependent and requires a training procedure. 2 Interested readers may refer to [31] for the details of the constraint matrix H and Equations (2)–(3). (a) Single stance (b) Double stance (c) Flight (d) Simpliﬁed Fig. 3: Different gait states and representation of the gen- eralized coordinates. Whole-body model representations are shown in sub-ﬁgures (a), (b), and (c). Sub-ﬁgure (d) corre- sponds to the stance and swing phases of the simpliﬁed model. c) Flight: In the ﬂight state, the only external forces on the exoskeleton are due to interaction with the user . While this state does not occur during walking, we use this state’ s equation of motion for exoskeleton parameter estimation, as described in Section II-C. The equation of motion in general- ized coordinates is M ﬂy ( q ﬂy ) ¨ q ﬂy + b ﬂy ( q ﬂy , ˙ q ﬂy ) + g ﬂy ( q ﬂy ) = S ⊤ ﬂy τ joint + τ ﬂy int , (9) where q ﬂy = [ x 0 , y 0 , θ 0 , θ 1 , θ 2 , θ 3 , θ 4 ] is a vector of gener- alized coordinates corresponding to the linear and angular positions of the backpack and the hip and knee joint angles, as shown in Figure 3c. The variable M ﬂy ∈ R 7 × 7 is the mass matrix, b ﬂy ∈ R 7 is a vector of Coriolis and centrifugal forces, and g ﬂy ∈ R 7 a vector of gravitational forces. The v ariable S ﬂy = [0 4 × 3 , I 4 × 4 ] is a selection matrix of actuated joints, and τ ﬂy int ∈ R 7 is a vector of interaction forces and torques applied to the exoskeleton by the user . 2) Simpliﬁed dynamics: The dynamics described above are used in our proposed WECC controller . A simpler model, described here, considers each le g of the exoskeleton as an independent double pendulum and excludes the backpack [14], [19], [23]–[25], [32]. W e use this model for comparison to our approach. The hip joint is considered grounded for a swing leg, and the ankle joint is considered ﬁxed to the ground for a stance leg, as shown in Figure 3d. The equation of motion can be written M i ( q ) ¨ q + b i ( q , ˙ q ) + g i ( q ) = τ joint + τ int , i ∈ { st , sw } , (10) 6 T ABLE II: Estimated masses and inertias (about axes through the center of mass and perpendicular to the saggital plane) of links and motor rotors of the ExoMotus-X2 exoskeleton. Link m [kg] L [kgm 2 ] Backpack 10.3 0.201 Upper thigh 0.7 0.002197 Lower thigh 2.14 0.005901 Upper shank 0.64 0.002346 Lower shank 0.47 0.0007941 Foot 1.25 0.004441 Rotor (apparent) - 1 . 24610 − 4 × 122 . 5 2 = 1 . 87 where q = [ θ 1 , θ 2 ] ⊤ are the hip and knee joint positions, respectiv ely . The variables M i ∈ R 2 × 2 , b i ∈ R 2 , g i ∈ R 2 , are the mass matrix, Coriolis and centrifugal torques, and gravitational torques, respectiv ely , during stance ( i = st) and swing ( i = sw) of a single leg. C. P arameter Estimation Inertial parameters such as mass, mass moment of inertia, and center of mass (CoM) locations of each link were esti- mated from the CAD model of the ExoMotus-X2 exoskeleton. Dynamic terms such as M , b , and g are deﬁned as a function of these parameters, adjustable link lengths, and instantaneous joint states. Their values are calculated at ev ery time instant. Some of these parameters are presented in T able II. Note that the largest inertias come from the apparent rotor inertia which is proportional to the square of the reduction ratio (RR = 122.5). The high gear ratios also result in signiﬁcant friction at each joint. Friction at the joints is modeled as viscous friction combined with Coulomb friction as a function of joint speed, τ friction = c 0 ⊙ sign ˙ θ + c 1 ⊙ ˙ θ , (11) where ˙ θ is the v ector of joint velocities and c 0 and c 1 are the vectors of Coulomb and viscous coef ﬁcients, respectively . The symbol ⊙ indicates element-wise Hadamard product. T o conduct the parameter estimation experiments, the ex- oskeleton was suspended by backpack attachment points. Chirp motor torque commands with increasing frequency up to 3 Hz at different amplitudes were simultaneously giv en to the hip and knee joints of a single leg, τ motor = A i sin(6 π t 2 /T ) , (12) where A i ∈ { 7 . 5 , 10 , 12 . 5 , 15 } Nm and T = 60 s for each amplitude. Joint positions were measured during the experiments. The joint torques, τ joint , in Equations (1), (2) and (9) are the sum of motor torques, τ motor , and friction compensation torques, τ friction , τ joint = τ motor + τ friction . (13) As there was no contact with the ground at this conﬁgu- ration, the equation of motion of the ﬂight state was used to estimate the frictional parameters. Equation (9) is rewritten to include viscous and Coulomb friction constants by substituting τ joint in Equation (13) into Equation (9), T ABLE III: Estimated friction parameters of the ExoMotus- X2 exoskeleton and R 2 of the ﬁt. Joint c 0 [Nm] c 1 [Nms] R 2 Left hip 5.01 4.58 0.96 Right hip 3.26 4.67 0.94 Left knee 4.30 3.25 0.96 Right knee 4.45 5.16 0.96 M ﬂy ( q ﬂy ) ¨ q ﬂy + b ﬂy ( q ﬂy , ˙ q ﬂy ) + g ﬂy ( q ﬂy ) = S ⊤ ﬂy ( τ motor + c 0 ⊙ sign ˙ θ + c 1 ⊙ ˙ θ ) + τ int . (14) Recorded backpack and joint angular positions, joint angular velocities, joint angular accelerations, and CAD-based inertial parameters were used to calculate M ﬂy , b ﬂy , and g ﬂy at each time instant. Joint accelerations were calculated by taking the deriv ative of the joint velocities numerically by the nine-point difference method [33]. Then, in Matlab, accelerations were low-pass ﬁltered with a cutoff frequency of 10 Hz, while also incorporating delay compensation. Since the exoskeleton was hung from the backpack at a ﬁxed location, ¨ x and ¨ y were negligible. Also, as there was no user in the exoskeleton, rotational components of τ int were zero. The coef ﬁcients c 0 and c 1 were solved using the recorded data by least-square minimization on Equation (14). Estimated friction parameters for each joint and the R 2 value of the ﬁt are presented in T able III. I I I . I N T E R AC T I O N T O R Q U E C O N T RO L L E R A. Interaction T or que Estimation Estimating the interaction torques between the human and exoskeleton plays a signiﬁcant role in the transparenc y and haptic rendering capabilities of the robot. Placing force/- torque sensors at e very contact point between the human and exoskeleton would theoretically be ideal for measuring the interaction. Howe ver , due to the distributed contacts between the human and the exoskeleton and practical challenges of this approach, we chose to estimate the interaction torques through joint torque measurements. Joint torques were measured using our custom joint-torque sensors mentioned in Section II-A. Because the ExoMotus-X2 exoskeleton was designed for individuals with sensorimotor impairments and therefore lim- ited walking ability , we estimated the interaction torque during single and double stance only , excluding the ﬂight state. 1) Single stance interaction tor que estimation: W e estimate the interaction torque during single stance by subtracting the gravitational and Coriolis forces from the joint torque readings: τ int = − S ⊤ τ joint + b i ( q , ˙ q ) + g i ( q ) , i ∈ { ls , rs } . (15) As shown in T able II, the largest contributors to the inertial terms are the apparent rotor inertias due to the high reduction ratios. Howe ver , since the strain gauges are placed at the link (i.e., after the motor), the torque measurements are not af fected by the apparent rotor inertia. In addition, because the links hav e relativ ely low inertia and the walking speed during the 7 experiment is not high, the effects of inertial forces are sig- niﬁcantly smaller compared to gravitational forces. Therefore, acceleration-dependent inertial components are neglected in the interaction torque estimation. It is worth noting that τ int ∈ R 5 , which includes the interaction torque at the backpack joint in addition to the four leg joints. 2) Double stance interaction tor que estimation: Interaction torque estimation can be expressed by extending the left and right stance model with the additional external contact force at the right and left foot, respectiv ely , τ int = − S ⊤ τ joint + b ls ( q , ˙ q ) + g ls ( q ) − J ⊤ r F r (16) τ int = − S ⊤ τ joint + b rs ( q , ˙ q ) + g rs ( q ) − J ⊤ l F l , (17) where J i is the ankle Jacobian and F i is the external force applied on the left ( i = l) and right ( i = r) foot. Combining Equations 16 and 17 results in ten equations with rank seven. Nine unknowns in these equations are the ﬁv e-dimensional interaction torque vector , τ int , and the two-dimensional contact forces at the left ( F l ) and right ( F r ) feet. Since there are seven independent equations and nine unkno wns, it is not possible to uniquely solve for the interaction torque vector without additional information. Ratios of the vertical ground reaction forces (GRF) and the direction of the GRF in the sagittal plane are used to resolve the redundancy . In this study , we used external force plates to measure these terms, b ut they can also be obtained by wearable sensors, or sensors instrumented onto the exoskeleton, as explained in Section VI-C. The vertical ground reaction force ratio α is introduced in Equation (8) and is also used for the equations of motion during double stance. The direction of the force can be expressed as the ratio of the horizontal and vertical forces, γ = F l,x F l,y . (18) Equations (8) and (16)-(18) are con verted to a system of linear equations as K ξ = m , where K =     J ⊤ r 0 5 × 2 I 5 × 5 0 5 × 2 J ⊤ l I 5 × 5 [0 , 1] [0 , − α/ (1 − α )] 0 1 × 5 [0 , 0] [1 , − γ ] 0 1 × 5     , ξ =  F r ,x F r ,y F l,x F l,y τ ⊤ int  ⊤ , m =     − S ⊤ τ joint + b ls ( q , ˙ q ) + g ls ( q ) − S ⊤ τ joint + b rs ( q , ˙ q ) + g rs ( q ) 0 0     , (19) and solved for the interaction torques. 3) Simpliﬁed interaction tor que estimation: W ith the sim- pliﬁed double pendulum model, interaction torques at the hip and knee joints can be estimated by subtracting the dynamic effects from the joint torque measurements, τ int = τ joint + b i ( q , ˙ q ) + g i ( q ) , i ∈ { st , sw } . (20) It is important to note that with the simpliﬁed method, τ int ∈ R 2 does not include the interaction torque at the backpack. T ABLE IV: V irtual mass values for each link used during the experimentation. Link Gait State V irtual Mass [kgm 2 ] Backpack Single Stance 2 Backpack Double Stance 3.5 Thigh Single Stance 0.7 Thigh Double Stance 1.2 Shank Single Stance 0.5 Shank Double Stance 0.87 Moreov er , because the backpack link is not incorporated into the dynamic model, the backpack’ s weight is not subtracted, resulting in inaccurate interaction torque estimation for ex- oskeletons with hea vy backpacks. B. Contr ol 1) V irtual mass contr oller: T o control the interaction torques between the human and the e xoskeleton, we imple- mented a virtual mass contr oller [34], [35] to simulate a desired mass or inertia, with no damping, of each generalized coordinate of the exoskeleton. This can be achiev ed by setting the desired generalized acceleration to ¨ q ∗ = M − 1 virt ( τ int − τ ∗ int ) , (21) where τ ∗ int is the desired interaction torque, e.g., zero for transparency or nonzero for other desired impedances. Under the assumptions of perfect control of the generalized accelerations and accurate interaction torque estimation with little delay , this equation shows that the interaction torque error is proportional to the chosen virtual mass matrix M virt and joint accelerations. Therefore, lowering M virt results in better interaction torque tracking. Obviously , M virt cannot be decreased below a certain limit due to the actuation limits, modeling errors, and communication delays. Moreover , the limit on the virtual mass also depends on the interaction prop- erties between the human and exoskeleton. While the damping due to the soft tissues of the user makes the system more stable, a very stif f connection would increase the minimum allow able virtual mass for a stable response. W e used a diagonal M virt and tuned the virtual mass parameter for each generalized coordinate independently . The tuning process was done on a pilot user (a non-participant) by reducing each virtual mass as much as possible before compromising the stability of the control. The parameters were then veriﬁed with two other pilot users. This process resulted in different virtual mass parameters for single stance and double stance, as sho wn in T able IV. The virtual inertia of the backpack is relati vely lar ge to allo w better control performance at the leg segments. 2) Constrained optimization: While the desired accelera- tion ¨ q ∗ in Equation (21) is a ﬁve-dimensional vector , the exoskeleton has only four actuated joints. Therefore, it is not possible to perfectly track the desired generalized acceler- ations calculated by the virtual mass controller . Moreover , the maximum torque, power , and velocity of the motors bring additional limitations. T o track the desired generalized acceleration as accurately as possible under physical and safety 8 M − 1 virt - argmin x ∥ f ( x ) ∥ = ( ¨ q , τ motor ) s.t. Ax = b Dx ≤ f Constrained Optimization Coupled Human-Robot LS α = 1 DS α ∈ [0 1] RS α = 0 Interaction T orque Estimator ¨ q ∗ ( ¨ q , τ motor ) α α F grf α [ q , ˙ q , τ joint , F grf ] τ int τ err int τ * int Fig. 4: Schematic of the interaction force controller . constraints, we use real-time constrained optimization with OSQP [36] to solve for x = [ ¨ q ⊤ , τ ⊤ motor ] ⊤ ∈ R 9 , where τ motor is a vector of torque commands sent to the motor drivers: min x ∥ f ( x ) ∥ s.t. Ax = b Dx ≤ f . (22) The objectiv e function ∥ f ( x ) ∥ is the 2-norm of the dif ference between the actual and desired generalized accelerations, f ( x ) = [ I 5 × 5 , 0 5 × 4 ] x − ¨ q ∗ . (23) The equality constraint ensures that optimized v ariables satisfy the equation of motion of the corresponding gait state under physical limits, A = A EOM = [ M i , − S ⊤ ] , b = b EOM = [ − b i − g i + τ int + S ⊤ τ friction ] , (24) where i ∈ { ls , rs , ds } . Inequality constraints arise from physical and safety con- straints on motor torque, power , and v elocity . Motor torque and po wer constraints are presented in Equations (25) and (26), respectiv ely , D τ + max = [0 4 × 5 , I 4 × 4 ] , f τ + max = τ max D τ − max = [0 4 × 5 , − I 4 × 4 ] , f τ − max = τ max , (25) D P + max = [0 4 × 5 , I 4 × 4 ] , f P + max = P max | ˙ q | D P − max = [0 4 × 5 , − I 4 × 4 ] , f P − max = P max | ˙ q | , (26) where τ max = 80 Nm and P max = 100 W are the maximum allow able torque and power of the motors, respectiv ely , and the superscripts + and − are used to indicate the maximum and minimum limits, respectiv ely . The joint velocity constraint is con verted to an acceleration constraint such that the joint speed at the next time step will not exceed maximum limits, D ¨ q + = [0 4 × 1 , I 4 × 4 , 0 4 × 4 ] , f ¨ q + = ˙ q max − ˙ q ∆ t D ¨ q − = [0 4 × 1 , − I 4 × 4 , 0 4 × 4 ] , f ¨ q − = ˙ q max + ˙ q ∆ t , (27) where ˙ q max = 3 . 0 rad/s and ∆ t = 0 . 003 s is the control loop period. Individual D matrices and f vectors in Equations (25)– (27) are v ertically stacked to obtain the o verall D matrix and f vector in Equation (22). The controller is illustrated in Figure 4. Because we ha ve only a single objectiv e function for this task, a constrained optimization sufﬁces. In the case of multiple objecti ves with different priorities, a hierarchical optimization framew ork could be applied [37], [38]. This allows optimization of lo wer-priority tasks using freedoms in the design vector that do not af fect the objecti ve functions of higher-priority tasks. I V . E X P E R I M E N TA L V A L I DA T I O N Three healthy subjects (one male, two female, 71.6 ± 12.9 kg, 174 ± 1.7 cm, 29 ± 2.6 years) participated in this study to test the proposed interaction torque controller . The link lengths of the exoskeletons were adjusted for each participant so that they were comfortable and their joints were aligned with the exoskeleton’ s. The adjusted link lengths were also entered into software for automatic modiﬁcation of the dependent parameters such as M , b , and g . None of the subjects had used a lower -limb exosk eleton before this study . 9 At the beginning of the experiment, subjects were allowed about two minutes of walking with the exosk eleton on a treadmill for familiarization. Subjects were asked to always hold the hand-rails on the sides of the treadmill for safety and to balance in the frontal plane. W e tested the haptic rendering capabilities for zero desired interaction torque under three conditions and nonzero desired interaction torques under two different conditions. In addition to the proposed controller (WECC), the following two condi- tions were tested: a) No-drive: A passiv e exoskeleton with disassembled driv es was used. This allowed us to hav e a baseline condition where the main sources of friction and inertia (apparent rotor inertia) are eliminated. Subjects still felt the weight of the exoskeleton, howe ver . b) Simpliﬁed: The e xoskeleton legs are modeled and controlled as independent double pendulums. Equations (10) and (20) were used to estimate the interaction torques and follow desired accelerations, respectiv ely . The details of this controller are presented in Appendix A. The institutional revie w board (IRB) of Northwestern Uni- versity approved this study (STU00212684), and all proce- dures were in accordance with the Declaration of Helsinki. A. Evaluation of T ranspar ency Haptic transparency was tested for our proposed WECC method, the double-pendulum-based simpliﬁed controller , and the passi ve no-drive condition. For the WECC and simpliﬁed controller , zero desired interaction torque was commanded for each generalized coordinate. Each condition was tested for three trials of around 65 seconds walking on a treadmill with a speed of 1.1 km/h. The ﬁrst ﬁve seconds of data were discarded to focus on steady-state walking. A relati vely slow speed was selected due to the usual walking training speeds in physical rehabilitation settings for individuals with stroke or incomplete spinal cord injury and the joint velocity limits of 3.2 rad/s of the ExoMotus-X2 exoskeleton. Trials of the WECC and simpliﬁed controllers were performed in a randomized order for each subject. All trials of the passiv e no-driv e condition were performed consecutively for practical reasons, as this condition requires a different exosk eleton with disassembled dri ves. A metronome was played at 40 bpm, and subjects were asked to synchronize their heel strikes with the beats of the metronome to have similar step lengths between subjects and trials. Human-exoskeleton interaction torques and muscle acti vity were used to assess transparency performance. B. Evaluation of Haptic Rendering W e tested the interaction torque rendering capabilities of the WECC controller and the simpliﬁed controller when the desired interaction torques are generated by virtual springs and dampers at the joints, τ ∗ int = K ( q − q ∗ ) + C ˙ q , (28) where q ∗ is the neutral position vector of the virtual springs and K ∈ R 5 × 5 and C ∈ R 5 × 5 are diagonal stiffness and T ABLE V: V irtual stif fness and damping v alues used for haptic rendering ev aluation. Joint Leg Phase k i [Nm/rad] c i [Nms/rad] q i ∗ [degrees] Hip Stance 50 10 25 Hip Swing 30 6 25 Knee Stance 30 6 -45 Knee Swing 25 5 -45 damping matrices, respecti vely . A constant q ∗ i was used at each joint i , and the desired interaction torque at the backpack was set to zero by assigning K 0 and C 0 to zero. The constant neutral angles were chosen such that the desired interaction torques have a similar magnitude in both directions. Since the motion of a leg during stance is slower and has a smaller range, larger stif fness and damping constants were used for stance phases than for swing phases. The magnitudes of the stiffness and damping were chosen during pilot trials such that the rendered torque is signiﬁcant but does not prev ent the user from walking at the requested speed. The diagonal values of the stif fness ( k i ) and damping ( c i ) matrices, as well as q ∗ , are presented in T able V. T o a void discontinuities in the desired interaction torque during stance to swing transitions, the desired interaction torque was lo w-pass ﬁltered with a cutoff frequency of 5 Hz. C. Data Recording and Analysis Joint positions, velocities, and interaction torques were recorded during all trials. The exoskeleton without dri ves is instrumented with IMUs (T ech-IMU V4, T echnaid, Spain) at each link to measure joint motions, which are used in interaction torque calculation. Because IMUs were placed on the exoskeleton and not on the user , they were not af fected by possible human-robot joint misalignment. All exosk eleton data were collected at 333 Hz. Nine bipolar electromyography (EMG) electrodes (MA400 EMG Systems, Motion Lab Systems, USA) were placed on the belly of the follo wing muscles for the ev aluation of transparency experiments: rectus femoris (RF), vastus lateralis (VL), vastus medialis (VM), biceps femoris (BF), semitendi- nosus (ST), medial gastrocnemius (MG), lateral gastrocnemius (LG), and soleus (SOL). The electrodes were placed only on the left leg and not removed between trials. EMG data were recorded at 1500 Hz. The ra w EMG signals were ﬁrst bandpass ﬁltered between 20 and 500 Hz with a sixth-order Butterworth ﬁlter . Then, a notch ﬁlter between 59 and 61 Hz was applied to reduce noise due to po wer line interference. Finally , the signals were low-pass ﬁltered at 10 Hz after full-wave rectiﬁcation. Processed EMG signals were normalized with respect to the mean of the no-drive condition for each subject independently . For one subject, RF recordings were not successfully collected due to sensor failure. Exoskeleton-based measurements and processed EMG read- ings were windowed between the heel strikes of the corre- sponding leg. Windo wed data from the same joint on both legs and three subjects were lumped together for each condition. The mean absolute v alue of the whole cycle and only stance or swing phases for each gait cycle were used as sample points 10 and visualized in the corresponding box plots in the Results section. T o test the dif ference in means between each condition for each measure (i.e., interaction torque, muscle activity), we used two-way ANO V A analyses with condition and subject as ﬁxed ef fects, as well as the interaction between these variables. Post-hoc comparisons were made using Tuk ey’ s Honestly Signiﬁcant Difference (HSD) test for transparency experiments comparing the three conditions. A p -value of 0.05 was used to accept or reject null hypotheses. Though the effects of inter-subject variability were included in the ANO V A analyses, we only present the ANO V A results and post-hoc comparisons for the condition, as the focus of this paper is controller de velopment. T o ensure consistenc y with ANO V A analyses and pre vent the mixing of inter- and intra- subject variability , we removed the inter-subject variability from the data presented in the box plots and T ables in the following section. This is achiev ed by demeaning the data of each condition/subject pair and subsequently adding the ov erall mean of each condition, including three subjects. V . R E S U LT S A. T ranspar ency Participants were able to follow the metronome, and there was no signiﬁcant dif ference between the step time at different conditions (No-driv e: 1.49 ± 0.06 s, Simpliﬁed: 1.51 ± 0.06 s, WECC: 1.50 ± 0.06 s). Interaction torques during the trans- parency trials, normalized by the body mass of each subject (units of Nm/kg), are presented in Figure 5 and T able VI. Below we describe the interaction torques during a leg’ s swing phase, its stance phase (both single stance and double stance unless otherwise noted), and its entire cycle. The performance of the proposed WECC controller was consistent throughout the whole gait cycle with a mean absolute interaction torque around 0.07 ± 0.02 Nm/kg both at the hip and knee joints during both stance and swing phases. The simpliﬁed controller resulted in a similar mean absolute interaction torque during the swing phase (difference of mean absolute interaction torques ∆ ≤ 0.01 Nm/kg) but signiﬁcantly higher mean absolute interaction torque of 0.16 and 0.14 Nm/kg at the hip and knee joints, respecti vely , during the stance phase. During the stance phase, higher interaction torques were observed for the no-driv e condition than for WECC at both the hip ( ∆ ≥ 0.06 Nm/kg, p < 0 . 001 ) and knee ( ∆ ≥ 0.17 Nm/kg, p < 0 . 001 ) joints. During the swing phase, the no-drive condition resulted in signiﬁcantly higher interaction torques at the hip joint ( ∆ ≥ 0.05 Nm/kg, p < 0 . 001 ) and similar interaction torques at the knee joint ( ∆ ≈ 0.01 Nm/kg) compared to the proposed WECC method. A verage transparency performance during the whole cycle was best for the WECC controller , with mean absolute interaction torques of 0.076 ± 0.011 Nm/kg and 0.071 ± 0.011 Nm/kg at the hip and knee joints, respectiv ely . The simpliﬁed controller achiev ed better mean absolute interaction torque than the no- driv e condition at the knee joint ( ∆ ≈ 0.07 Nm/kg) a veraged ov er the whole gait cycle. At the hip joint, both performances were similar with a mean interaction torque difference around 0.01 Nm/kg. T ABLE VI: Mean and (standard deviation) of the interaction torques between the human and exosk eleton at the hip and knee joints during transparency experiments. All v alues are expressed in Nm/kg. p -values for the effect of condition in the two-way ANO V A and signiﬁcant post-hoc comparisons are presented in the right columns. Joint/Phase No-driv e ( c 1 ) Simpliﬁed ( c 2 ) WECC ( c 3 ) p T ukey’ s HSD Hip/Stance 0.13 (0.03) 0.16 (0.03) 0.08 (0.01) < 0.001 c 3 < c 1 < c 2 Hip/Swing 0.11 (0.02) 0.06 (0.01) 0.07 (0.01) < 0.001 c 2 < c 3 < c 1 Hip/WC 0.13 (0.02) 0.14 (0.02) 0.08 (0.01) < 0.001 c 3 < c 1 < c 2 Knee/Stance 0.25 (0.02) 0.14 (0.03) 0.07 (0.01) < 0.001 c 3 < c 2 < c 1 Knee/Swing 0.001 (0.01) 0.07 (0.01) 0.07 (0.01) < 0.001 c 1 < c 3 < c 2 Knee/WC 0.19 (0.02) 0.12 (0.02) 0.07 (0.01) < 0.001 c 3 < c 2 < c 1 T ABLE VII: Mean and (standard de viation) of muscle activi- ties during transparency experiments, normalized with respect to the mean of the no-driv e condition. p -v alues for the ef fect of condition in the two-way ANO V A and signiﬁcant post-hoc comparisons are presented in the right columns. Muscle No-driv e ( c 1 ) Simpliﬁed ( c 2 ) WECC ( c 3 ) p T ukey’ s HSD RF 1 (0.19) 0.83 (0.14) 0.77 (0.14) < 0.001 c 3 < c 2 < c 1 VL 1 (0.19) 0.64 (0.15) 0.59 (0.09) < 0.001 c 3 < c 2 < c 1 VM 1 (0.21) 0.59 (0.14) 0.56 (0.09) < 0.001 c 1 > c 2 , c 3 BF 1 (0.19) 1.32 (0.29) 1.94 (0.36) < 0.001 c 1 < c 2 < c 3 ST 1 (0.28) 1.57 (0.37) 2.51 (0.41) < 0.001 c 1 < c 2 < c 3 T A 1 (0.23) 0.97 (0.20) 1.06 (0.15) < 0.001 c 3 > c 1 , c 2 LG 1 (0.18) 1.56 (0.31) 1.91 (0.33) < 0.001 c 1 < c 2 < c 3 MG 1 (0.20) 1.58 (0.28) 1.95 (0.33) < 0.001 c 1 < c 2 < c 3 SOL 1 (0.17) 0.76 (0.15) 0.72 (0.09) < 0.001 c 3 < c 2 < c 1 Figure 6 and T able VII show the muscle activity of the various muscles during the transparency trials. It is seen that muscles responsible for hip ﬂexion (RF) and knee extension (RF , VL, VM) had the lo west activ ation with the WECC controller and highest with the passiv e no-drive condition. On the other hand, our controller resulted in the most activity , and the no-dri ve condition resulted in the least acti vity , for the muscles responsible for hip extension and knee ﬂexion (BF , ST). Furthermore, the SOL muscle, responsible for ankle plantar ﬂexion, was used the least with the WECC controller and the most under the no-drive condition. In contrast, LG and MG muscles, responsible for knee ﬂexion and ankle plantar ﬂexion, had the highest acti vation with the WECC controller and the lowest under the no-driv e condition. B. Haptic Rendering Properties of the rendered virtual spring-damper elements shown in T able V resulted in the desired interaction torques presented in Figure 7. The ﬁgure includes data from the joints of both legs and three subjects. Desired interaction torques vary in the range ± 15 Nm during the gait c ycle for all subjects. There was no signiﬁcant dif ference between the step times at different conditions (Simpliﬁed: 1.32 ± 0.13 s, WECC: 1.31 ± 0.13 s). The demonstration of the haptic rendering performance of the WECC control is also presented in this video 3 . 3 https://tinyurl.com/ExoHapticRendering 11 Fig. 5: Interaction torques between the human and exoskeleton during transparency trials. The ﬁgures on the top row sho w the interaction torques vs. normalized gait duration for a representative subject. The data includes e very step of both legs, and the shaded area represents ± standard de viation. On the bottom row , the box and whisker plots of the mean absolute interaction torques during the stance phase, swing phase, and the whole gait cycle are presented. Mean absolute values of the corresponding phase at each step of both legs of all three subjects are used as a data point after removing the inter-subject variability in box and whisker plots ( N ≈ 320 ). *** indicates p < 0 . 001 from T uke y’ s HSD test. T ABLE VIII: Mean and (standard deviation) of the interaction torque error between the human and e xoskeleton at hip and knee joints during haptic rendering experiments. All values are expressed in Nm/kg. p -v alues for the ef fect of condition in the two-way ANO V A are presented in the right column. Joint/Phase Simpliﬁed ( c 2 ) WECC ( c 3 ) p Hip/Stance 0.12 (0.02) 0.05 (0.01) < 0.001 Hip/Swing 0.04 (0.01) 0.04 (0.01) 0.49 Hip/WC 0.10 (0.02) 0.05 (0.01) < 0.001 Knee/Stance 0.21 (0.02) 0.05 (0.01) < 0.001 Knee/Swing 0.04 (0.01) 0.05(0.01) < 0.001 Knee/WC 0.17 (0.02) 0.05 (0.01) < 0.001 The rendering performance of our proposed WECC con- troller and the simpliﬁed controller are shown in Figure 8 and T able VIII. Similar to the performance of transparency trials, our controller’ s interaction torque tracking error w as consistent throughout the whole cycle. On the other hand, the simpliﬁed controller resulted in a large peak of interaction torque errors on both joints near the end of the single stance state. The WECC controller resulted in a normalized mean ab- solute interaction torque error of 0.049 ± 0.008 Nm/kg and 0.039 ± 0.008 Nm/kg at the hip joint during the stance and swing phases, respecti vely . The simpliﬁed controller resulted in a similar mean absolute error during the swing phase both at the hip and knee joints ( ∆ ≤ 0 . 01 Nm/kg). On the other hand, the av erage absolute interaction torque error was signiﬁcantly larger during the stance phase both at the hip ( ∆ ≥ 0 . 07 Nm/kg, p < 0 . 05 ) and knee ( ∆ ≥ 0 . 16 Nm/kg, p < 0 . 05 ) joints. A veraged over the whole cycle, the WECC controller resulted in an error of 0.050 ± 0.006 Nm/kg for both hip and knee joints. The errors of the simpliﬁed controller were signiﬁcantly larger both at the hip ( ∆ ≥ 0 . 05 Nm/kg, p < 0 . 05 ) and knee joints ( ∆ ≥ 0 . 12 Nm/kg, p < 0 . 05 ). V I . D I S C U S S I O N A. Interaction T or que T rac king The simpliﬁed controller and the proposed WECC controller performed similarly during the swing phase for both joints in the transparency and haptic rendering trials, as sho wn in Figures 5 and 8. Since the swing leg is not affected by 12 Fig. 6: Muscle acti vity during transparency experiments. Data are normalized with respect to the mean of the no-dri ve condition. Mean absolute values of the ﬁltered EMG data av eraged over the whole cycle at each step of all subjects are used after removing the inter-subject variability as a data point in bar plots ( N ≈ 160 ). * , ** , and *** indicate p < 0 . 05 , 0 . 01 , and 0 . 001 , respecti vely , from T ukey’ s HSD test. Fig. 7: Desired interaction torque due to the rendered spring damper elements. the weight of the backpack and the other leg, the two-link simpliﬁed model is suf ﬁcient and results in accurate interaction torque estimation and motor torque calculation for the swing leg. On the other hand, the simpliﬁed controller failed to render the desired interaction torques for the stance leg. This is because the model does not consider whole-body dynamics, resulting in incorrect interaction torque estimation for the stance leg. For the simpliﬁed controller , subjects feel addi- tional torques on their joints due to the uncompensated weight of the exoskeleton. The calculation of dynamical parameters and the differences between the two approaches are presented in Appendices B and C. As the proposed WECC controller uses whole-body dynamics and considers physical limitations, the interaction torque tracking performance was consistent throughout the entire gait cycle. The most transparent performance at the knee joint was obtained during the swing phase of the no-driv e condition. This is because the weight of the shank is insigniﬁcant, and there is low friction and no apparent rotor inertia. Because of the larger inertia, the no-driv e condition results in more interaction torque at the hip joint compared to the activ e controllers, where the weight of the leg is compensated. The highest amount of interaction torque was observed for the stance leg under the no-drive condition, as the stance leg of the subject needs to ov ercome the torques due to the weight of the exoskeleton’ s stance leg, swing leg, and backpack. The proposed method uses the exosk eleton model to es- timate interaction torque and control acceleration to track the desired interaction torque. Considering the model-based dynamics based on the gait state results in consistent per- formance throughout the whole gait cycle. This model-based approach depends on an accurate dynamic model. Thanks to CAD tools, this is not difﬁcult to obtain. Accurate friction modeling is more challenging. Performance can be improved by (1) using a more accurate friction model and (2) esti- mating joint accelerations in real time and implementing an inner acceleration control loop to compensate modeling errors. Estimating real-time accelerations also allows considering the neglected inertial terms in interaction torque estimation. In our post-experiment analyses, we observed that the mean absolute values of the neglected inertial torques were around 14% of the total dynamical forces. B. Muscle Activity The muscles responsible for hip ﬂexion (RF) and knee extension (RF , VL, VM) were activ ated the least with the proposed WECC controller and the most with the passiv e no- driv e condition during the transparency trials. This is mainly 13 Fig. 8: Interaction torque error during haptic rendering trials. The ﬁgures on the top ro w sho w the interaction torque error vs. normalized gait duration for a representative subject. The data includes every step of both legs, and the shaded area represents ± standard deviation. On the bottom ro w , the box and whisker plots of the mean absolute interaction torque error during the stance phase, swing phase, and the whole gait cycle are presented. Mean absolute values of the corresponding phase at each step of both legs of all three subjects are used after removing the inter-subject variability as a data point in box and whisker plots ( N ≈ 370 ). *** indicates p < 0 . 001 from ANO V A results. due to the accurate compensation of the exoskeleton dynamics for the WECC controller . The opposite trend was observ ed for the muscles responsible for hip extension and knee ﬂexion (BF , ST). These muscles were activ ated more with the WECC controller than for the no-driv e and simpliﬁed conditions. This is e xpected for BF and ST , remembering that the transparency goal for WECC is zero interaction torque, not assist. The uncompensated weight of the e xoskeleton in the no-dri ve and simpliﬁed conditions results in signiﬁcant hip-extension torque assist during early double stance, as sho wn in Figure 9 by the large positi ve po wer ﬂow from the exoskeleton to the user . The gastrocnemius muscles (LG, MG), which cause ankle plantar ﬂexion and knee ﬂexion, were activ ated most with the WECC controller and least with the no-drive condition. On the other hand, an opposite trend was observed for the SOL muscle, which causes only ankle plantar ﬂexion. High interaction torques in the knee ﬂexion direction (positi ve direction in Figure 5) for the simpliﬁed and no-dri ve conditions leads to less gastrocnemius activity in these conditions. This explains the different trend between gastrocnemius and SOL Fig. 9: Hip interaction power during transparenc y experiments. Positiv e and negati ve values indicate resistance and assistance to the human, respectiv ely . 14 Fig. 10: Mean interaction torque estimation of all subjects during the all WECC trials using the direction of the GRF ( γ = F x /F y ) and assuming horizontal GRF is negligible ( γ = 0 ). muscles. C. Overgr ound W alking The proposed method requires the ratios of the vertical GRF and the direction of GRF in the sagittal plane. While we used external force plates under the treadmill, these terms can also be obtained using wearable sensors or sensors instrumented onto the exosk eleton. For e xample, the ratio of the vertical forces can be measured with simple and cost-effecti ve FSR sensors or pressure pads. It is relati vely harder to measure the horizontal GRF because instrumented insoles usually provide only vertical forces. One can implement force sensors that provide two-directional force measurements [39] or indirectly estimate the anterior-posterior force assuming the foot does not slip on the ground [40] or using a model and estimated accelerations [41]. Because accelerations and decelerations are small for slow gait speeds, it is also possible to assume horizontal GRF to be near zero. The effect of this assumption on the interaction torque estimation was tested on the collected experimental data from all subjects during WECC conditions. Figure 10 shows that the interaction torque estimates are changed little by this assumption. The supplementary video includes preliminary over ground tests with FSR pads implemented at the bottom of the feet while assuming horizontal forces to be near zero. The user in the video climbs over a small step and walks forward and backward. W e also conducted a pilot test where a user walked a distance of 4.5 m four times with each controller (WECC and simpliﬁed) in haptic transparent mode, while holding onto parallel bars. W e observed similar trends between this ov erground walking test and the treadmill experiments. Mean absolute interaction torque results are presented in Figure 11. In the stance phase, the WECC controller resulted in a 35% and 65% decrease in mean absolute interaction torque at the hip and knee joints, respecti vely , compared to the simpliﬁed controller . The performance of the two controllers was similar in the swing phase (dif ference less than 1% and 5.5% for hip and knee, respectiv ely). Fig. 11: Mean absolute interaction torques during the pilot ov erground tests. V I I . C O N C L U S I O N W e have presented the whole-exoskeleton closed-loop com- pensation (WECC) controller to measure and control the interaction torques between a user and a ﬂoating base hip-knee exoskeleton that contacts the ground. The WECC controller calculates the interaction torques using joint torque measure- ments and gait state information. The desired joint interac- tion torques are achie ved by using a virtual mass controller together with a constrained optimization scheme that con- siders the whole-exosk eleton dynamics, physical limits, and safety constraints. The performance of the WECC controller was compared in terms of transparency and spring-damper haptic rendering with a commonly-used simpliﬁed double- pendulum model where the legs are modeled independent of one another . WECC control resulted in consistent interaction torque tracking during the whole gait cycle for both zero and nonzero desired interaction torques, while the simpliﬁed controller failed to track desired interaction torques during the stance phase. Future improvements to WECC control could include ex- plicit modeling of joint limits in the dynamic model, partic- ularly because the knee joint gets close to the physical limit during the gait cycle. This will improve the accuracy of the forward model and therefore the tracking of the optimized accelerations. Moreov er , an inner acceleration loop to com- pensate for the modeling inaccuracies, using additional IMUs or reinforcement learning -based strategies [42] for online optimization of system parameters, may be options to improve the current results. A limitation when comparing interaction torques between different controllers arises from the challenge of obtaining ground-truth data. Reﬂected interaction torques are inﬂuenced by forces at each contact point, necessitating the installation of F/T sensors at e very contact point for the measurement of these interaction torques. Because this is practically challenging, we used the whole-body dynamics of the exoskeleton to estimate the interaction torques as accurately as possible while comparing different controller performances. More accurate comparisons could be possible with an experimental setup where the contact points are kept at a minimum and all are sensorized. The WECC controller can potentially be used to adjust the interaction torques to assist a user in ov erground walking, 15 to resist for training in a clinical setting, or to provide haptic transparency on lower -limb exoskeletons with feet. Commercial versions of this type of exoskeleton are usually heavy , ha ve high friction, and do not hav e force-torque sensors. Consequently , they hav e been utilized primarily for position control rather than accurate interaction force control. The proposed WECC control overcomes these limitations and is especially beneﬁcial for heavy lower -limb exoskeletons where the whole-body dynamics need to be compensated. In future studies, this infrastructure will be used to in vestigate the ef fects of dyadic haptic interaction on performance and learning for complex multi-DOF lower-limb motions [43]–[45]. A P P E N D I X A. Simpliﬁed Contr oller For the simpliﬁed controller, the legs are modeled as two- link chains independent of each other . This model is used to estimate interaction torques as in Equation 20. Since an optimization scheme considering physical limits is not imple- mented in the related works in T able I, we do not use one for the simpliﬁed controller . Instead, Equation 10 is used as the forward model to track the desired acceleration commands sent by the virtual mass controller . The same controller gains for thigh and shank presented in T able IV are used. The simpliﬁed interaction torque estimation is used in the control loop; howe ver , the presented interaction torque results are calculated using the whole-body model to compare the results between conditions. The overall control structure is presented in Figure 12. The control structure runs independently for each leg. B. Calculation of the Dynamic P arameter s The mass matrix M , Coriolis v ector b , and gravitational vector g are calculated during single stance in both WECC and the simpliﬁed model as M i = X n ∈ N i J n S, i ⊤ m n J n S, i + J n R, i ⊤ Θ n J n R, i , (29) b i = X n ∈ N i J n S, i ⊤ m n ˙ J n S, i ˙ q + J n R, i ⊤ Θ n ˙ J n R, i ˙ q , (30) g i = − X n ∈ N i J n y , i ⊤ m n g , (31) where i ∈ { ls, rs, st, sw } . The subscripts i = ls and i = rs correspond to left stance and right stance for the whole-body model, and i = st, i = sw correspond to the state of a single leg being stance or swing for the simpliﬁed model as explained in Section II-B. The index n corresponds to the links of the exosk eleton in the following order: backpack ( n = 1 ), stance thigh ( n = 2 ), stance shank and foot ( n = 3 ), swing thigh ( n = 4 ), and swing shank and foot ( n = 5 ). For the whole-body model, all links are considered; therefore, N ls = N rs = { 1 , 2 , 3 , 4 , 5 } . On the other hand, the simpliﬁed model considers the legs separately; therefore, N st = { 2 , 3 } and N sw = { 4 , 5 } . The mass and mass moment of inertia at the center of mass of each link are represented by m n and Θ n , respectiv ely . The translational and rotational Jacobian of the center of mass of each link are shown by J n S, i and J n R, i , respecti vely . It is important to note that these Jacobians are calculated depending on the model and gait state (i.e., i ). For the whole-body model, J n S, ls-rs ∈ R 2 × 5 and J n R, ls-rs ∈ R 1 × 5 . For the simpliﬁed model, J n S, st-sw ∈ R 2 × 2 and J n R, st-sw ∈ R 1 × 2 . The v ariable J n y , i is the vertical component of the translational Jacobian. C. Differ ence Between Interaction T or que Estimation Using Simpliﬁed and WECC Models Due to the weight of the ExoMotus-X2 exosk eleton, gravita- tional torques g ha ve the largest contrib ution to the interaction torque. Therefore, the difference of g between the whole-body and simpliﬁed model is in vestigated in this subsection. The respective subsets of the whole-body Jacobians of the stance and swing center of masses are equal to the simpliﬁed stance and swing Jacobians, J n y , i [2 : 3] = J n y , st , ∀ n ∈ { 2 , 3 } , i ∈ { ls , rs } , J n y , i [4 : 5] = J n y , sw , ∀ n ∈ { 4 , 5 } , i ∈ { ls , rs } , (32) where the notation [a:b] denotes the subset of elements from index a (inclusive) to index b (inclusiv e). Using (31) and (32), the dif ference between the gra vitational torques on the swing leg in the whole-body model and the simpliﬁed swing model is g i [4 : 5] − g sw = X n ∈ 1 , 2 , 3 − J n y , i [4 : 5] m n g , i ∈ { ls , rs } . (33) The state of the swing leg or backpack does not affect the center of mass positions on the stance leg with respect to the stance ankle. Therefore, J n y , i [4 : 5] = 0 1 × 2 , ∀ n ∈ { 1 , 2 , 3 } , i ∈ { ls , rs } . (34) This leads to X n ∈ 1 , 2 , 3 − J n y , i [4 : 5] m n g = 0 , = ⇒ g i [4 : 5] = g sw , i ∈ { ls , rs } , (35) which shows the gravitational torques of the swing leg for the whole-body model are equi valent to the gravitational torques of the simpliﬁed swing model. On the other hand, using (31) and (32), the difference between the gravitational torques on the stance leg in the whole-body model and the simpliﬁed stance model is g i [2 : 3] − g st = X n ∈ 1 , 4 , 5 − J n y , i [2 : 3] ⊤ m n g i ∈ { ls , rs } , (36) which shows the simpliﬁed stance gravitational torque has an error proportional to the weight of the backpack and swing legs. For hea vy exoskeletons such as the ExoMotus-X2, this error results in signiﬁcant inaccuracy in interaction torque estimation. 16 M − 1 virt - 2-link feed-forward model (Eq. 10) Coupled Human-Robot LS α = 1 DS α ∈ [0 1] RS α = 0 Simpliﬁed Interaction T orque Estimator (Eq. 20) Whole-Body Interaction T orque Estimator V isualization ¨ q ∗ ( ¨ q , τ motor ) α α F grf α [ q , ˙ q , τ joint ] [ q , ˙ q , τ joint , F grf ] τ int τ err int τ * int α Fig. 12: Schematic of the interaction force controller used in the simpliﬁed condition. D. Double Stance Equation of Motion Equation (3) can be written as an extension of both left stance dyanmics ( i = ls) or right stance dynamics ( i = rs). Therefore, H ⊤ X ls = H ⊤ X rs , (37) where X ∈ { M ¨ q , b , g } . Multiplying Equations (5)-(7) by H ⊤ on the left, H ⊤ X ds = H ⊤ ( α X ls + (1 − α ) X rs ) = H ⊤ α X ls + H ⊤ X rs − H ⊤ α X rs = α ( H ⊤ X ls − H ⊤ X rs ) + H ⊤ X rs = H ⊤ X rs = H ⊤ X ls . (38) Therefore, the approximated dynamics of Equations (5)-(7) satisfy the equation of motion (3). The interpolation factor , α , is chosen based on the torques needed to carry the weight of the exosk eleton during the double stance. Let ˆ g ﬂy ≜ g ﬂy [3 : 7] and ˆ J i ,y ≜ J i ,y [3 : 7] . Index es from 3 to 7 correspond to the generalized coordinates during single and double stance (i.e., θ 0 − 4 ). The variable J i ,y is the vertical ( y ) component of the left ( i = l) or right ( i = r) ankle Jacobian in ﬂight state. The gravity v ector during left and right stance can be expressed by the sum of the gravity vector during the ﬂight state and the reﬂected joint torques due to the force on the stance foot, g ls = ˆ g ﬂy − ˆ J ⊤ l ,y m exo g g rs = ˆ g ﬂy − ˆ J ⊤ r ,y m exo g , (39) where m exo g is the total weight of the exoskeleton. Similarly , the gravity vector during double stance is equal to the sum of the ﬂight gravity vector and the reﬂected joint torques due to the vertical ground reaction forces on both legs, g ds = ˆ g ﬂy − ˆ J ⊤ l ,y F l ,y − ˆ J ⊤ r ,y F r ,y = α g ls + (1 − α ) g rs . (40) Substituting (39) into (40), ˆ g ﬂy − ˆ J ⊤ l ,y F l ,y − ˆ J ⊤ r ,y F r ,y = α ˆ g ﬂy − α ˆ J ⊤ l ,y m exo g + ˆ g ﬂy − ˆ J ⊤ r ,y m exo g − α ˆ g ﬂy + α ˆ J ⊤ r ,y m exo g . (41) Rearranging and simplifying (41) leads to αm exo g ( ˆ J ⊤ r ,y − ˆ J ⊤ l ,y ) = ˆ J ⊤ r ,y ( m exo g − F r ,y ) − ˆ J ⊤ l ,y F l ,y . (42) Assuming vertical acceleration is not signiﬁcant, i,e., m exo g = F l ,y + F r ,y results in α ( F l ,y + F r ,y )( ˆ J ⊤ r ,y − ˆ J ⊤ l ,y ) = F l ,y ( ˆ J ⊤ r ,y − ˆ J ⊤ l ,y ) = ⇒ α = F l ,y F l ,y + F r ,y . (43) 17 Fig. 13: Hip and knee joint positions for simpliﬁed and WECC controller during transparency (solid line) and haptic rendering experiments (dashed line) for a representative subject. E. Supplementary Kinematic Data Figure 13 shows the joint positions observed during the transparency and haptic rendering conditions. In both condi- tions, similar joint trajectories were observed. Ho wever , during the haptic rendering trials, a signiﬁcantly smaller range of motion was observed. This can be explained by the rendered torques due to the virtual spring and damper elements. A C K N O W L E D G M E N T This work was supported by the National Science Founda- tion / National Robotics Initiati ve (Grant No: 2024488), North- western Univ ersity and the T urkish Fulbright Commission. W e would like to thank T im Haswell for his technical support on the hardware improvements of the ExoMotus-X2 exoskeleton and Lorenzo V ianello for his help with the experimentation. Presented data will be av ailable upon request. R E F E R E N C E S [1] J. C. Moreno, J. Figueiredo, and J. L. Pons, Chapter 7 - Exoskeletons for lower-limb rehabilitation . Elsevier Ltd., 2018. [2] W . Huo, S. 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Kim, Y . W en, E. B. K ¨ uc ¸ ¨ uktabak, S. Zhan, K. L ynch, L. Hargro ve, E. J. Perreault, and J. L. Pons, “ A frame work for dyadic physical inter- action studies during ankle motor tasks, ” IEEE Robotics and Automation Letters , vol. 6, no. 4, pp. 6876–6883, 2021. [45] S. J. Kim, Y . W en, D. Ludvig, E. B. K ¨ uc ¸ ¨ uktabak, M. R. Short, K. Lynch, L. Hargrove, E. J. Perreault, and J. L. Pons, “Effect of dyadic haptic collaboration on ankle motor learning and task performance, ” IEEE T ransactions on Neural Systems and Rehabilitation Engineering , pp. 1–1, 2022. Emek Barıs ¸ K ¨ uc ¸ ¨ uktabak (S’21) received his B.S. degree in Mechanical Engineering from Middle East T echnical University , Ankara, T urkey in 2017 and M.S. degree in Mechanical Engineering from ETH Zurich, Switzerland in 2019. Currently , he is a Ph.D. student at the Center for Robotics and Biosystems at Northwestern University and the Legs+W alking Lab at the Shirley Ryan AbilityLab, Chicago, IL. His research interests include physical human-robot interaction, teleoperation, force control, and assistive robotics. Y ue W en recei ved the B.S. degree in Automation from W uhan University of T echnology , China, in 2011, the M.S. degree in Control Theory and En- gineering from Huazhong University of Science and T echnology , China, in 2014, and the Ph.D. degree in Biomedical Engineering from North Carolina State Univ ersity and the Univ ersity of North Carolina at Chapel Hill, Raleigh, NC, in 2019. He is currently an Assistant Professor in the Department of Mechanical and Aerospace Engineering at the Univ ersity of Central Florida, Orlando, FL. His research interests include the control and personalization of wearable robots, deep learning for neural machine interfaces, human-robot interaction, and gait rehabilitation. Sangjoon J. Kim earned his B.S. from the Depart- ment of Electrical and Computer Engineering at the Univ ersity of Wisconsin–Madison in 2012. Subse- quently , he attained both his M.S. and Ph.D. degrees in Mechanical Engineering from the Korea Ad- vanced Institute of Science and T echnology (KAIST) in 2014 and 2019, respectively . Currently , he serves as an Assistant Project Scientist at the University of California-Irvine, concurrently holding a position as a Research Engineer at Flint Rehabilitation. His primary focus lies in biomedical engineering and robotics. Matthew R. Short recei ved his B.S. degree in bio- engineering from T emple University , Philadelphia, P A, USA in 2018. He is currently pursuing a Ph.D. degree at Northwestern Univ ersity , Chicago, IL, USA, working in the Neurorehabilitation and Neural Engineering Lab at Shirley Ryan AbilityLab. His research interests include human-human interaction and ankle motor control in the context of post-stroke rehabilitation. Daniel Ludvig (S’06-M’11) receiv ed the B.Sc. de- gree in physiology and physics in 2003, and the M.Eng. and Ph.D. degrees in biomedical engineering in 2006 and 2010 respectively , all from McGill Univ ersity , Montreal, QC, Canada. He is currently a Research Assistant Professor in the Department of Biomedical Engineering at Northwestern Univ ersity and a Research Scientist at the Shirley Ryan Ability- Lab in Chicago, IL. Dr . Ludvig specializes in using quantitativ e and experimentally dri ven techniques to model human biomechanics and motor control. His current speciﬁc research interests lie in quantifying the role lower -limb neuromechanics play in healthy locomotion, and ho w alterations in these neuromechanics lead to injury and disease. 19 Levi Hargr ove receiv ed his MScE and PhD in Electrical Engineering from the University of New Brunswick (2005, 2008). He is currently the Director and Scientiﬁc Chair of Center for Bionic Medicine at the Shirley Ryan AbilityLab and an Associate Professor in the Departments of Physical Medicine & Rehabilitation and the McCormick School of En- gineering at Northwestern Uni versity . His research interests include signal processing, pattern recogni- tion, and control of bionic limbs. Eric J. Perreault is V ice President for Research and Professor of Biomedical Engineering and Phys- ical Medicine and Rehabilitation at Northwestern Univ ersity . His research addresses the neural and biomechanical factors in volv ed in the control of multi-joint movement and posture and how they are modiﬁed following neuromotor pathologies such as stroke and spinal cord injury . Eric is a fellow of the American Institute for Medical and Biological Engineering, and was recently chair of the National Advisory Board on Medical Rehabilitation Research, and president of the International Society for Electrophysiology and Kinesi- ology . Ke vin M. Lynch (S’90–M’96–SM’05–F’10) re- ceiv ed the B.S.E. degree in electrical engineering from Princeton University , Princeton, NJ, USA, and the Ph.D. degree in robotics from Carnegie Mellon Univ ersity , Pittsburgh, P A, USA. He is a professor of mechanical engineering at Northwestern Uni versity , where he directs the Center for Robotics and Biosystems and is a member of the Northwestern Institute on Complex Systems. He is a coauthor of the textbooks Principles of Robot Motion (Cambridge, MA, USA: MIT Press, 2005) and Modern Robotics: Mechanics, Planning, and Contr ol (Cambridge, U.K.: Cambridge Univ . Press, 2017) and the associated online courses and videos. His research interests include robot manipulation and locomotion, self-organizing multiagent systems, and physically collaborative human–robot systems. Jos ´ e L. Pons graduated in Mechanical Engineering (Univ ersidad de Navarra, Spain, 1992), and obtained a MSc in Information T echnologies (Universidad Polit ´ ecnica de Madrid, Spain, 1994), and a PhD in Physics (Universidad Complutense de Madrid, 1997), for his research work on dynamic optimiza- tion of robot mechanisms. Dr . Pons currently serves as the Scientiﬁc Chair of the Legs & W alking AbilityLab at Shirley Ryan AbilityLab . In addition to his research leadership role at SRAlab, he holds appointments as Professor at the Department of Physical Medicine & Rehabilitation, Feinberg School of Medicine and at the Departments of Biomedical Engineering and Mechanical Engineering, McCormick School of Engineering, Northwestern Uni versity .

Haptic Transparency and Interaction Force Control for a Lower-Limb Exoskeleton

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