A Human-Robot Cooperative and Personalized Compliant Joint Controller for Upper-Limb Rehabilitation Robots: The Elbow Joint Validation

Background: Appropriate training modalities for post-stroke upper-limb rehabil6 itation are key features for effective recovery after the acute event. This work presents a novel 7 human-robot cooperative control framework that promotes compliant motion and renders differ8 ent high-level human-robot interaction rehabilitation modalities under a unified low-level control 9 scheme. 10 Methods: The presented control law is based on a loadcell-based impedance controller provided 11 with positive-feedback compensation terms for disturbances rejection and dynamics compensation. 12 We developed an elbow flexion-extension experimental setup, and we conducted experiments to 13 evaluate the controller performances. Seven high-level modalities, characterized by different levels of 14 (i) impedance-based corrective assistance, (ii) weight counterbalance assistance, and (iii) resistance, 15 have been defined and tested with 14 healthy volunteers. 16 Results: The unified controller demonstrated suitability to promote good transparency and 17 render compliant and high-impedance behavior at the joint. Superficial electromyography results 18 showed different muscular activation patterns according to the rehabilitation modalities. Results 19 suggested to avoid weight counterbalance assistance, since it could induce different motor relearning 20 with respect to purely impedance-based corrective strategies. 21 Conclusion: We proved that the proposed control framework could implement different phys22 ical human-robot interaction modalities and promote the assist-as-needed paradigm, helping the 23 user to accomplish the task, while maintaining physiological muscular activation patterns. Future 24 insights involve the extension to multiple degrees of freedom robots and the investigation of an 25 adaptation control law that makes the controller learn and adapt in a therapist-like manner. 26

From a low-level point of view, achieving compliant motion is a fundamental, yet challenging, 120 task in rehabilitation robotics. In fact, if achieving rigid behavior of the robot can be considered a 121 trivial task, obtaining its opposite can be challenging since the low-level controller should reject the 122 dissipative effects introduced by the robot hardware. At the same time, one of the key characteristics 123 of the motor recovery process is not to limit, in any way, any intention of movement coming from the 124 user and, possibly, of guiding the subject's voluntary movements towards the correct task execution. 125 Compliant motion in rehabilitation robotics can thus be addressed as a compromise between good 126 trajectory tracking and minimization of interaction forces. 127 Usually, rehabilitation robots and exoskeletons are provided with high-ratio transmission gear-128 boxes that are kinematically inefficient, and that can introduce static and viscous friction. In this 129 scenario, the perceived compliance cannot be guaranteed by the back-drivability of the motor itself. 130 Still, it can be implemented by adding an elastic element in series with the actuation unit, i.e., 131 series elastic actuators (SEA) [9,11,15,71], or with compliant controllers that add virtual springs 132 and dampers to shape the virtual mechanical impedance at the joint. 133 In the literature, several low-level controllers have been proposed to achieve compliant motion, 134 and in turn, to implement the previously described training modalities. Among all, impedance 135 control is one of the most common approaches, and it has been demonstrated to be a very efficient 136 solution for neurorehabilitation [48]. The impedance control belongs to those control schemes that 137 permit a compliant physical human-robot interaction. It implements dynamic control that relates 138 force/torque and position: a torque/force output is generated from a position input. In particular, 139 impedance control is characterized by a nested loop architecture. An inner torque-feedback loop 140 implements the transparent behavior and promotes the mechanical compliance (i.e., it "softens" the 141 control). An outer position-feedback loop corrects for trajectory tracking errors by applying forces 142 or torques aimed at the completion of the task (i.e., it "stiffens" the control). Furthermore, two 143 different variants of the impedance control can be identified. When the actuation unit is inherently 144 back-drivable, the torque control can be implemented through an open-loop current control loop 145 (i.e., implicit impedance). In the other cases, a load-cell or an elastic element is exploited in series 146 as a feedback signal for the closed-loop torque control loop (i.e., explicit impedance) [8,63]. 147 Regarding the rehabilitation domain, both impedance controllers in joint-space [34,39,56] and stroke rehabilitation, is based on a SEA-based joint-space impedance control that promotes the 156 coordinated motion of the shoulder, through the assistance of the scapulohumeral rhythm [37].
implementation, the literature is still swampy and fragmented. Firstly, most literature reviews 183 focused on the desired rehabilitation behavior and did not investigate the implementation on the 184 robots' hardware [45]. Secondly, the literature proposes several custom solutions that strongly de-185 pend on the kinematics, mechanics, and electronics of the developed robots. Finally, some research 186 groups already proposed that a mixture of assistance, correction and resistance with impedance-187 control laws could be used to gradually increase the amount of expected voluntary muscle activ-188 ity [16]. However, a generalization and validation of these approaches are still lacking. 189 With this work, we identified a cooperative control framework that implements multiple high-190 level human-robot interaction modalities with a unified low-level explicit impedance control law. 191 We will now describe and demonstrate its ability to promote different features, such as favoring 192 good transparency of the joint, compensating for the weight of the robot and of the impaired limb, 193 assisting the motion along the desired trajectory, recovering from task deviations, or challenging 194 the user by applying resistance and increased gravity to the motion. 195 3 Unified compliant control framework 196 The unified controller relies on the concepts of compliant control and, in particular, impedance 197 control. The overall scheme of the proposed controller is presented in Fig. 1. The virtual mechanical 198 impedance is implemented in the outer position-loop I(s), which is in charge of correcting for 199 deviations from the desired angular position. Namely, it is driven by the difference between the 200 commanded reference (i.e., θ r ) and the measured angle (i.e., θ l ). The inner torque-loop F (s) is in 201 charge of controlling the torque output at the load axis. Since its dynamics should not influence the 202 outer loop, the inner loop is usually implemented at a higher control frequency. Thus, it is supposed 203 to be fast enough so that its dynamics can be neglected with respect to the outer impedance loop.

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In this work, we consider an exemplary single-degree-of-freedom joint, shown in Fig. 2, as a 205 platform to validate the controller and its functionalities. The actuation chain is composed of an 206 electric motor coupled with a high-ratio transmission gearbox. The unit is also provided with an 207 incremental encoder that measures the joint angle, and a reaction torsional load-cell provides torque 208 feedback at the output load axis. The dynamics of the one degree-of-freedom actuation system is 209 Figure 1: Block diagram of the unified controller scheme based on explicit impedance control law. Inner torque control F (s) is in red and outer impedance control I(s) is in blue. Dotted lines represent positive-feedback compensations.
as follows: where θ m is the motor displacement, while τ m and τ l respectively represent the motor torque and   Figure 2: The actuation chain consists of an electric motor provided with an angular encoder, a transmission gearbox, a torsional torque sensor and an generalized aluminum bar load. The motor driver acquires input signals from the actuation chain and commands torque set-points to the electric motor.
target torque of the actuator (τ m ) through a Proportional-Integrative-Derivative (PID) controller, 232 with feedback from the torsional load-cell (τ l ), that in the Laplace form is (2): To compensate for static and viscous friction introduced by high-ratio gearboxes, an additional where τ c is the Coulomb friction torque,θ is the measured joint velocity,θ c is the Coulomb joint 238 velocity threshold, and f v is the viscous friction coefficient. The hyperbolic tangent function ensures 239 the Coulomb term to be continuous and smooth acrossθ = 0 in order to avoid undesired oscillations.

240
The τ fg term is then combined with the PID torque output estimated τ m term as input to the 241 actuator. The actual torque actuated at the load axis is then measured by the load-cell (τ l ) and 242 fed back to the PID controller to track the reference torque (τ r ). The impedance control can be regarded as an outer position loop that takes a reference in terms of 245 angular position (i.e., θ r ) and, by means of a virtual mechanical impedance, produces a reference 246 torque (i.e., τ r ) that in turn is fed to the inner control loop. The total reference torque can be seen Instead, the measured torque at the load axis consists of the actual torque generated by the 252 robotic system and can be broken down into four main components, as shown in (5): where τ comp and τ imp represent the actuation torques commanded to the motor as in Eq. 4, τ ext is 254 composed of the external torque that the user exerts to the motor, and τ r es represents the residual 255 disturbance torque that the inner torque controller can not reject. To derive the feedback impedance-based term (i.e., τ imp ), considering a first order impedance, the 258 transfer function I(s) between the reference target (θ s ) and the impedance-based torque term (τ imp ) 259 is characterized by two parameters: virtual spring (K s ) and virtual damper (K d ), and it can be 260 implemented in the well-know form (6): that in the time domain becomes (7): where τ imp is the desired impedance control torque that is used as a set-point by the inner torque 263 loop, while θ d and θ are, respectively, the desired and measured joint angle positions. The virtual stiffness, by means of the virtual spring constant K s , pulls the joint link towards 265 its desired configuration (i.e., the spring corrects for deviations from its equilibrium point, which is 266 continuously adapted to follow the desired angular trajectory). At the same time, the virtual damper 267 (K d ) dissipates the spring energy and damps oscillations. Overall, the role of these parameters is 268 to render, as shown in Figure 3 for the elbow joint, a second-order system by virtualizing a spring-

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For the sake of simplicity, in this section, we consider the general single-degree-of-freedom joint 279 shown in Fig. 3, which can be reduced to a rigid pendulum system. The torque acting at the load 280 axis can be described with the dynamic equation of the system, which includes both the robot and 281 the human, as follows (8): where J l is the inertia moment, f l is the viscous friction at the load axis, and G represents gravita- in our work, we only compensate for gravitational and viscous frictional torques. 291 We therefore introduce the simplified compensation term: wheref l is the estimated viscous friction coefficient,Ĝ link represents the weight compensa-293 tion term for the robot components, andĜ w c represents the weight compensation of the human 294 component. The weight compensation term for the robot can be modeled as in (10): where m is the robot link mass, l its center-of-mass distance, and g is the gravitational acceleration.

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As for the gravitational compensation term of the human (Ĝ wc ), we need to make explicit 297 reference to the single-degree-of-freedom joint used as a demonstrative example (Fig. 3). Of course, regulated by means of a weighting factor (ranging from 0% to 100%) that accounts for misalignment 302 and uncertainties in the anthropometric data as in (11): where W is the weighting factor, m f and m h are the masses of forearm and hand, while l f 304 and l h are their centers of mass. With this dynamic compensation, only inertial, centrifugal, and 305 residual frictional torques are to be overcome if the user wants to perform a voluntary movement 306 (i.e., they are not included in the compensation term).

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The feedforward compensation torque formulation can be obviously generalized if a n-degree-308 of-freedom robot is concerned.

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In this case, the dynamics compensation terms can also include Coriolis and Centrifugal torques.  The literature proposed several approaches and control modalities for robot-mediated therapy to-315 wards the goal of personalized therapy. In this work, we included seven high-level human-robot 316 interaction modalities, ranging from those that assist the movement the most to challenging strate-317 gies. In this section, we first describe the desired behavior for each of the proposed modalities.

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Then, we propose a match between the high-level behavior and a set of parameters for the pre-319 sented low-level unified controller that can render the desired behavior. is not aware of the predefined exercise task a priori. At low-level, the virtual stiffness is removed, 354 and low damping is kept to avoid undesired oscillations and dampen the motion.

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Active Transparent mode (T) In Active Transparent mode, the user performs the task, and 356 the robot follows the movement without assisting (nor resisting) the movement. In other words, 357 this modality enables the robot to be dynamically transparent to users' voluntary movements, by 358 compensating the exoskeleton weight at each configuration along the task.

359
Regarding its implementation, the low impedance behavior is achieved by means of a zero-  a trade-off in the impedance parameters is needed to induce a physiological muscular activation 382 aimed at completing the task in an assisted-as-needed fashion.

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To define the quantitative values of stiffness, damping and weight assistance for each modality, 384 we separately ran some preliminary tests on two healthy subjects, which were not recruited for the 385 rehabilitation modalities assessment to avoid bias. The parameters of the controller were empirically 386 determined according to the perceived behavior. The damping K d is kept proportional to the 387 stiffness K s to avoid undesired oscillations and jerky movements as: . 389 where the ratio factor α = 5. To guarantee safety in human-robot interaction, the coupled stability of the human-robot system 392 is a fundamental requirement [10]. In particular, given two separately stable systems, the coupled 393 stability is always guaranteed. However, an unstable robotic system can become stable after cou- parameters [1]. Also in this case, it is suggested that "there is a trade-off between achievable 417 stiffness and low undesired interaction torques" [66] and that "a desired stiffness higher than the 418 physical spring stiffness is not allowed for passivity" [9]. Given the above mentioned considerations, 419 it is clear that, even for simple controllers, the passivity-based control design problem is non-trivial.

420
High control bandwidth for the torque control loop (i.e., having a high-fidelity torque tracking) might not be beneficial for achieving stability and passivity. For these reasons, in this work we 422 did not explicitly investigated the coupled stability, nor we analytically derived its passivity, but 423 we followed qualitative guidelines to empirically tune the control parameters and display a stable 424 behavior of the coupled system.

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The experimental set-up and its connection are described in Fig. 4b, while its final realization is shown in Fig. 4a. The main features of the presented experimental set-up are reported in Table   459 1.      The desired movement speed was kept the same across all modalities. θ r (t) = P 0 + P 1 (t − P 2 ) P3 (P 4 − t) P5 , P 2 ≤ t ≤ P 4 (13) where the P n parameters are used to configure the desired trajectory. P 0 represents the initial 504 position offset, P 2 and P 4 are the start and the stop time, P 3 and P 5 are the interpolators' orders 505 for the raising and falling phases, and P 1 is related to movement amplitude A 0 by means of Eq. 14.    To validate the implemented control strategies and to investigate how they affect the user's 518 behavior, we also registered the muscular activity. In particular, we recorded the biceps and triceps 519 (long head) muscles, as shown in Fig. 4b

537
As for the capability of the system to promote physical human-robot transparency, results demon-538 strated that the torque controller accurately followed the commanded torque (i.e., the anti-gravity 539 torque) in both dynamic conditions. The maximum residual resistive torque during back-driving 540 movements was about ± 0.3 Nm, which was perceived as negligible by the user that was performing    As for the performances of the impedance controller, Fig. 9 shows the relationship between

556
We recruited 14 voluntary healthy volunteers, with median age of 25 years [24 -27]. Table 2 shows 557 the empirically obtained parameters that we used for the human-robot interaction assessment, as 558 described in Section 3.3. using an impedance control logic, which doesn't guarantee an accurate position tracking, and since 566 no effort was required from the user, in Passive mode we can notice higher errors, but the trajectory 567 variability is minimal. Finally, in W mode, by which the controller does not correct for trajectory 568 deviation, the tracking RMSE was slightly higher than the other modes.

570
In Figure 11, we present the average envelope profiles of muscular contraction (biceps and triceps 571 brachii ), and the torque output for each of the presented modality.   Furthermore, the integrated EMG (iEMG) results are reported in Fig. 12 for each modality.

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The Friedman test revealed significant differences among training modalities for the iEMG index for Passive mode (P) In Passive mode, the robot entirely performs the movement and the subjects 579 were asked to simulate the "passive" behavior by relaxing their muscles along the movement, and 580 by not counteracting to residual trajectory errors. As expected, the normalized activation of biceps 581 and triceps was minimal, which confirmed the user's "passive" behavior ( Figure 11. Considering 582 the biceps activation during the flexion phase we found a significant difference (i.e., p-values < 0.05) 583 for all modalities except from W mode. Instead, triceps contraction during the extension phase 584 resulted in significant difference with C+W, W, R, and Ch modes (Table 3).

585
Active Corrective mode with Weight counterbalance (C+W) Concerning the C+W 586 mode, we can appreciate a distinct activation of the biceps (agonist muscles) during the elbow 587 flexion phase. The biceps activation was not different from W and T modes (p-values > 0.05),

588
while it was significantly different from P, where the subjects were almost relaxed, C, where the 589 controller did not compensate for gravity, and R and Ch modalities, where the subject were making 590 23 more effort (p-values < 0.05). In contrast, the triceps (antagonist muscles) contracted during elbow 591 extension. In fact, since the controller was counterbalancing for the arm weight, the users could 592 not exploit the gravity force to extend the elbow during the second phase of the task. Indeed, the 593 iEMG during the triceps extension was significantly higher than the one detected in the P mode 594 where the muscle was relaxed (p-value < 0.001) and, at the same time, we did not detect differences 595 from the W mode (p-value = 0.431).

596
Active Corrective mode without Weight counterbalance (C) When in C mode, the acti- phases. The biceps contracted during the lifting phase, and the triceps during the descending phase.

606
In this mode, since the controller did not correct for trajectory deviation, the trajectory RMSE was 607 higher than the previous modes ( Figure 10).    Table 4: P-values results of the post-hoc analysis comparing integrated EMG index among training modalities during elbow extension movement.

628
Regarding the torque output results presented in Fig. 11, the right plots show the torque output 629 generated by the elbow-joint system to the users' arm interface. In P mode, the measured torque 630 consisted of the torque generated by the motor to complete the task. Such torque is equal to 631 the inverse-dynamic torque needed to passively move the human-robot system along the desired terbalance. We tested the unified controller performances with an elbow flexion-extension test-bed.

665
The experimental results showed that the developed set-up, combined with the proposed low-level 666 controller, exhibited very low impedance at the joint level, imposing negligible resistive torques (less 667 than 0.3 Nm) on the user's free-motion movements. Notably, since the impedance-based corrective 668 term of the unified controller is superimposed to the Transparent control mode, achieving a baseline dynamic transparent behavior was a fundamental step to implement compliant rehabilitation The Transparent (T) mode was considered the baseline reference, since it describes the behavior 713 by which neither assistance nor resistance is provided to the user during the task. In fact, the 714 muscular effort registered in this modality corresponds to the natural free task execution. During 715 elbow-flexion we observed a medium biceps contraction, while the triceps was characterized by a 716 slight co-contraction. During the extension phase, instead, a modulated contraction of the biceps 717 is used to control the downward motion provided by gravity, while the triceps were again not 718 significantly activated, given that the movement was performed in favour of gravity. 719 We also observed that Assistive (C, W and C+W) modes promoted similar biceps contractions 720 that are significantly higher with respect to Passive mode. However, when the weight counterbal-721 ance was active (i.e, C+W and W modalities), the triceps experienced greater contraction with 722 respect to the other training modes. Therefore, these results indicate that such modalities in-723 duced the physiological contraction of biceps muscles, and that the controller was inducing slightly 724 greater motor antagonistic activation when additional weight counterbalance assistance was present.

725
Comparing results obtained in T mode with the C mode, we could interestingly observe that the 726 activation profiles in the two modalities were comparable, despite the C mode allows a reduced 727 effort and avoid any fail in task execution, providing assistance whether the user is not capable 728 of completing the task or is too slow. We can also observe that, given that the participants were 729 performing controlled movements (i.e., healthy subjects followed a trajectory pre-defined in position 730 and velocity) with comparable performances, the controller was able to induce muscular patterns 731 in the A, A+W and W modes that are not significantly different from the baseline T mode. We 732 can also verify that the torque output in this modalities roughly followed the robot weight coun-733 terbalance term, and that the residual dynamic torque to complete the tasks was generated by 734 users' voluntary contraction. Therefore, we can derive that the proposed control system is able to 735 correctly implement the assist-as-needed paradigm, helping the user to accomplish the task while 736 inducing the physiological muscular activation pattern.

737
Instead, in R and Ch modes, the statistical analysis confirmed that, for both biceps and triceps, 738 significant greater muscular contraction levels were reached with respect to other modalities. In The results demonstrated that the proposed unified controller was able to provide low-impedance 745 and high-impedance correction, low-resistance and high-resistance behavior, rendering different 746 perceived human-robot interaction modalities. The developed controller, thanks to its inner explicit 747 torque feedback control, could reject most of the disturbance torques introduced by the high-ratio 748 gearbox, without the need for an accurate model-based compensation.

749
From the rehabilitation point of view, the goal is to achieve efficient motor control that should 750 be as similar as possible to the free task scenario, i.e., the Transparent mode. However, we noticed 751 that W and C+W solutions, which both involved anti-gravity compensation, imply an agonist-752 antagonist coordination that is completely different from the natural one, and therefore they could 753 induce unnatural muscular synergies. Instead, purely corrective strategies (such as C mode), around 754 the desired trajectory, modulate the assistance without impacting the muscle recruitment strategy, Instead, the proposed R and Ch methods were able to motivate and induce challenging exercises 760 to the subject, training both agonist and antagonist muscles. For this reason the presented approach 761 could also be applied to the recovery from sports and non-sports injuries. In fact, the controller 762 might assist the motion during early stages of the physiotherapy, then, by switching modality, it 763 might improve the muscle mass recovery.

764
Overall, the controller and the developed hardware confirmed suitability to implement the train- In this paper, we presented and tested a human-robot cooperative controller for upper-limb robot-775 mediated rehabilitation. The design of the control framework took inspiration from motor learning 776 and neurophysiological aspects, which suggest that good collaboration between the impaired subject 777 and the therapeutic device is needed to induce an effective motor recovery. In this sense, we 778 found strong evidence that the proposed controller guaranteed dynamic transparency -to promote 779 users' voluntary movements -and produced variable assistance and resistance levels -to tune the 780 rehabilitation treatment according to subject's performance and involvement. 781 We demonstrated that a proper combination of stiffness, damping, and weight assistance of the 782 presented unified controller can render different physical human-robot interaction and, consequently, 783 promote different human-robot interaction rehabilitation modalities. We also proved that assistance 784 based on anti-gravity weight counterbalance (i.e., W and C+W modes) changes the muscular 785 effort with respect to purely corrective assistance (i.e. C mode). Thus it does not train the 786 same muscular synergic coordination of natural free task movements. We believe that, since a 787 collaborative controller should provide the minimal amount of assistance to complete the tasks, 788 the presented high-level modalities can be considered as different points of a continuum, and we 789 posit that they can be potentially selectable according to the stage of motor recovery, involving the 790 subject in the completion of the rehabilitation treatment. Our results suggest that the presented 791 collaborative framework is suitable for these purposes. Future works will extend this approach to 792 multiple degrees of freedom robots and investigate the optimal adaptation control law that makes 793 the controller learn and adapt to the subject's performances in a therapist-like manner.