Closed-Loop Kinematic and Indirect Force Control of a Cable-Driven Knee Exoskeleton: A Lyapunov-Based Switched Systems Approach

Lower-limb exoskeletons can aid restoring mobility in people with movement disorders. Cable-driven exoskeletons can offload their actuators away from the human body to reduce the weight imposed on the user and enable precise control of joints. However, ensuring limb coordination through bidirectional motion control of joints using cables raise the technical challenge of preventing the occurrence of undesired cable slackness or counteracting forces between cables. Thus, motivation exists to develop a control design framework that integrates both a joint control loop to ensure suitable limb tracking and a cable control loop to maintain cable tension properly. In this article, a two-layer control structure consisting of high and low-level controllers are developed to ensure a knee-joint exoskeleton system follows the desired joint trajectories and adjusts the cable tension, respectively. A repetitive learning controller is designed for the high-level knee joint tracking objective motivated by the periodic nature of the desired leg swings (i.e., to achieve knee flexion and extension). Low-level robust controllers are developed for a pair of cables, each actuated by an electric motor, to track target motor trajectories composed of motor kinematics and offset angles to mitigate cable slackness. The offset angles are computed using admittance models that exploit measurements of the cable tensions as inputs. Each electric motor switches its role between tracking the knee joint trajectory (i.e., the motor acts as the leader motor to achieve flexion or extension) and implementing the low-level controller (i.e., the motor acts as the follower motor to reduce slackness). Hence, at any time, one motor is the leader and the other is the follower. A Lyapunov-based stability analysis is developed for the high-level joint controller to ensure global asymptotic tracking and the low-level follower controller to guarantee global exponential tracking. The designed controllers are implemented during leg swing experiments in six able-bodied individuals while wearing the knee joint cable-driven exoskeleton. A comparison of the results obtained in two trials with and without using the admittance model (i.e., exploiting cable tension measurements) is presented. The experimental results indicate improved knee joint tracking performance, smaller control input magnitudes, and reduced cable slackness in the trial that leveraged cable tension feedback compared to the trial that did not exploit tension feedback.

reducing the energetic costs of walking, and improving endurance [2]. Exoskeleton devices are equipped with actuators to provide limb assistance such as electrical motors [3], [4], [5], hydraulic [6] and pneumatic [7], [8] actuators. The selection of actuators is important since they increase the inertia of the exoskeleton potentially increasing the metabolic costs and inducing misalignment, which may lead to discomfort, pain, and even injury [9]. Cable-driven exoskeletons have been developed to reduce the weight and inertia of traditional rigid exoskeletons and provide precise joint control for limb assistance.
Cable-driven exoskeletons can offload the actuators away from the human body and provide torque at the joints using Bowden cables (that are connected to a separate actuation unit). Decoupling the actuators from the user can enable light-weight exoskeletons [10] for lower-limb [11], [12] and upper-limb [10], [13], [14] assistance. Cables cannot provide compressive forces; therefore, at least two cables are needed to achieve bi-directional motion control of one joint (agonist-antagonist movement) [15]. Thus, a constructive control design is motivated to ensure suitable coordination between cables to elicit smooth limb motion and avoid two major potential issues [16]: (1) cable slackness that can degrade the system's response time and control performance; and (2) counteracting cable forces being applied concurrently at the joint, which can lead to an unstable joint kinematic response and thus yield unnatural joint motion. This article develops a systematic layered control framework to mitigate the aforementioned issues and ensure knee joint tracking.
Solutions to prevent cable slackness are to pre-load and maintain cable tension using mechanical units such as compression springs in series with the Bowden cable sheath [17], pulling springs in series with the cable [18], and helical torsion springs [19]. Although these mechanical approaches can adjust cable tension, the system's response time increases (due to the spring's loading time) and its tension cannot be automatically controlled. Alternatively, designing control strategies can digitally adjust the cable tension by controlling the actuators directly (e.g., prescribe desired cable stiffness) to customize the interaction for different users. Force control has been widely implemented in upper-limb exoskeletons [20], [21], lower-limb exoskeletons [22], [23], [24], ankle actuators [25], exosuits [26], cycling [27], [28], and surgical manipulators [29]. Indirect force controllers exploit impedance or admittance-based models to generate desired trajectories based on kinematic and force inputs, respectively. This article exploits admittance models for the electric motors to adjust the tension of cables and thus mitigate slackness and counteracting forces in the knee exoskeleton.
Learning-based controllers are motivated for joint tracking since limb movements such as leg swings are repetitive or periodic. Repetitive control (RC) and iterative learning control (ILC) are two primary learning control methods for periodic tasks illustrated in applications such as upper-limb rehabilitation robots [30], lower-limb exoskeletons [31], functional electrical stimulation (FES)-cycling [27], and industrial robots [32], [33]. Learning control approaches seek to improve tracking performance by leveraging learned inputs from previous trials. ILC is usually implemented for discontinuous tasks in which the initial conditions are reset to the same values at the beginning of each trial [34]. Furthermore, ILC has been combined with model predictive control (MPC) [35] and reference-free learning MPC [36] to improve tracking performance. RC is implemented for continuous operation and the initial conditions are not reset [37]. Differently from our previous work in [38] that designs a robust controller to track the knee joint trajectory without experimental implementation, a repetitive learning controller is developed in this article leveraging the design in [37] to reduce the reliance on high-frequency and high-gain control.
In this article, a control framework segregated into a highlevel joint controller and a low-level motor controller are designed, analyzed, and experimentally implemented using a cable-driven knee-joint exoskeleton. A pair of electric motors provide bi-directional motion about the knee joint. One motor is called the leader motor that implements the high-level input to track the desired joint kinematics (magnitude and direction). The complementary motor is called the follower motor that implements a low-level input to prevent undesired cable slackness and counteracting forces. When the desired direction of joint motion flips, the roles of the motors switch (i.e., the leader motor becomes the follower motor and vice versa). Compared to our previous work in [38], the main contributions in this article are as follows: 1) Low-level control inputs are designed using a robust control approach for the electric motors when each of them acts as the follower motor. The low-level controllers track the leader motor's angle and a kinematic offset term is generated using an admittance model, which exploits feedback of the tension of the cable (indirect force control) connected to the follower motor. 2) A repetitive learning controller is designed as a representative high-level controller to track periodic kneejoint trajectories and achieve leg swings while reducing the need for high gain and high-frequency feedback. 3) A Lyapunov-based stability analysis segregated into two theorems is developed to demonstrate that the novel designs of the high-level and low-level closed-loop controllers with input switching ensure global asymptotic and exponential tracking, respectively. The high-level and low-level control inputs, which leverage kinematic and cable force feedback, are shown to be bounded. 4) Experiments are conducted in six able-bodied individuals to demonstrate the performance of the low-level and high-level controllers. The results of two trials with (Controller A) and without (Controller B) using the admittance model (i.e., exploiting cable tension measurements as indirect force control) are discussed and compared. The article is organized as follows. The dynamic models are introduced in Section II. In Section III, the repetitive learning controller is designed for the high-level control layer to FIGURE 1. Schematic depicts the two-layer control system illustrated at the moment of achieving knee flexion as an example (see direction of the arrow in exoskeleton). The high-level knee joint controller exploits joint position q and velocityq to compute the control input u that is forwarded to the leader motor (e.g., in this example to the flexion motor). The admittance models with state variables δ n ,δ n ,δ n corresponding to extension and flexion motors, n = {ex, fl} respectively, exploit cable tension feedback T ex , T fl to generate the target kinematic offset to compute the input into the follower motor. Thus, the high-level and low-level control inputs are allocated to the leader and follower motors, respectively, to achieve bi-directional motion (i.e., extension and flexion). The low-level control inputs for the extension and flexion motors are denoted as u ex , u fl , respectively. In this schematic, the high-level knee joint controller computes the knee flexion input u; thus, the flexion motor serves as the leader motor and the extension motor serves as the follower motor. To achieve knee extension, the roles of the leader and follower motors switch. track a periodic knee joint trajectory. The designed low-level controllers for the electric motors and the admittance models are introduced in Section IV. The Lyapunov-based stability analyses for the high-level and low-level control loops are developed in Section V. Experimental results for controllers with and without the indirect force control are presented in Section VI, a discussion is presented in Section VII and the conclusion is presented in Section VIII.

II. DYNAMIC MODELS A. CABLE-DRIVEN ACTUATOR SYSTEM
The lower-limb exoskeleton applies torque about the knee joint and is actuated by electric motors using customized cable-driven mechanisms (i.e., forces are transmitted via Bowden cables). A pair of electric motors is used to achieve knee joint extension (ex) and flexion ( f l). To ensure smooth coordination between the pair of motors, the following roles are defined for each motor at any given time 1) The leader motor is the driving motor controlling the knee joint kinematics to achieve the desired knee joint angle and direction (i.e., flexion or extension). 2) The follower motor is the complementary motor seeking to mitigate cable slackness and prevent undesired counteracting forces with the leader motor. A control framework inspired in our previous theoretical work in [38] is developed to design a high-level controller to track the desired knee joint trajectory and low-level controllers to mitigate cable slackness. The relationships between the designed high-level and low-level controllers with the roles of the motors at any given time are described as follows: 1) The high-level controller generates the control input to track the knee joint trajectory that is forwarded to the leader motor. 2) The low-level controllers generate control inputs for the pair of motors to avoid cable slackness and undesired counteracting forces when the role of each motor is assigned as the follower motor.
3) The control input for each motor switches between the high-level controller and its low-level controller depending if the motor is the leader or follower, respectively. Thus, at any given time, one motor is the leader and the second motor is the follower. 4) The low-level controller for a given motor is called active if the control input is fed into the motor (i.e., follower role). The low-level controller is called inactive if the control input is not being fed into the motor (i.e., motor has the leader role). A schematic of the two-layer control system is presented in Fig. 1 , where the knee joint control input is denoted by u (high-level control input), and the follower motor control inputs for the extension and flexion motors are denoted by u ex , u f l (low-level control inputs), respectively.

B. KNEE-SHANK AND POWERED EXOSKELETON DYNAMIC MODEL
The knee-joint shank dynamics and the powered exoskeleton can be modeled as [39] M(q)q + G(q) + P(q,q) = τ e (t ), where q : R ≥t 0 → R,q : R ≥t 0 → R, andq : R ≥t 0 → R are the measurable knee joint angle, measurable joint angular velocity, and unmeasurable angular acceleration, t 0 ∈ R ≥0 denotes the initial time. The inertia of the combined knee-shank and exoskeleton is denoted as M : R → R >0 ; G : R → R denotes the gravitational effects; P : R × R → R denotes the passive viscoelastic and damping effects. The torque applied by the leader motor is denoted as τ e : R ≥t 0 → R, and can be defined as where B e : R ≥t 0 → R >0 is the control effectiveness of the leader motor at time t, and is defined as where B f l , B ex ∈ R >0 are the control effectiveness values of the flexion and extension motor, respectively. The signum function sgn(·) = 1 for (·) > 0, sgn(·) = 0 for (·) = 0, and sgn(·) = −1 for (·) < 0. The high-level knee joint control input u : R ≥t 0 → R is designed in Section III.

C. FOLLOWER MOTOR DYNAMIC MODEL
The follower motor system dynamics can be modeled as follows where n = {ex, f l} is the index corresponding to the extension and flexion motor, θ n ,θ n ,θ n : R ≥t 0 → R denote the motor's measurable angular position, velocity, and unmeasurable acceleration. The inertia of the motor is denoted by J n ∈ R >0 , the damping coefficient is denoted by D n ∈ R, an additive disturbance is denoted by d n : R ≥t 0 → R, and the input effectiveness for each motor is denoted as B n ∈ R >0 , ∀n. The assigned low-level follower control input is denoted as u n : R ≥t 0 → R, ∀n and designed in Section IV.
The following properties and assumptions for (1) and (4) are exploited in the subsequent control design and stability analysis.
Property 5: The control effectiveness of the motors are bounded as B ≤ B n ≤ B, ∀n, where B, B ∈ R >0 are known constants [44].

III. HIGH-LEVEL JOINT CONTROL DEVELOPMENT
The objective of the high-level control design is to track desired, periodic knee kinematic trajectories to move the shank in the extension and flexion directions. Thus, a repetitive learning controller inspired by [27] is developed for the highlevel knee joint control objective.
To quantify the joint tracking objective, the position tracking and filtered tracking errors denoted by ξ, η : where q d ,q d ,q d : are known constants, and γ ∈ R >0 is a selectable control gain. The desired knee-joint trajectories are periodic in the sense that The open-loop error system can be obtained by taking the time derivative of (6), pre-multiplying by M, substituting for (1), (2) and (5), and performing some algebraic manipulation to yield 1 where the auxiliary signals χ, By using Properties 1, 3 and 4, the mean value theorem and the explicit boundedness of the periodic desired trajectories, the auxiliary signals χ, W d , N d can be upper bounded as where ρ, β η , ∈ R >0 are known positive constants, and the composite error vector z : Given the open-loop error system in (7), the high-level control input u : R ≥t 0 → R can be designed as where ψ 1 , ψ 2 , ψ 3 , ψ 4 ∈ R >0 are selectable positive control gains. The designed control input u in (15) is composed of the repetitive learning termŴ d that acts as a feedforward term exploiting the periodic nature of the desired knee trajectory, the feedback term ψ 1 η that exploits the instantaneous joint kinematic tracking error, and the sliding mode terms that compensate for the disturbances, the uncertain terms upper bounded by a constant in (13), the state-dependent uncertainty in (11), and the learning cross term that influence the highlevel closed-loop error system. The repetitive learning control where μ ∈ R >0 is a control gain and T is the known time period of the desired leg trajectories. The saturation function sat w 1 ,w 2 (·) is defined as where w 1 > w 2 ∈ R are upper and lower bounds. The closed-loop error system can be expressed by substituting (15) into (7) and performing some algebraic

manipulation as
where the unknown learning estimation errorW d ∈ R is defined asW d W d −Ŵ d . Due to the periodicity and boundedness of W d (t ), the following relationship is derived (19) and the learning estimation errorW d can be expressed as using the repetitive learning update law in (16).

IV. LOW-LEVEL FOLLOWER MOTOR CONTROL DEVELOPMENT
The objective of the low-level follower motor control design is to mitigate cable slackness and prevent counteracting forces with the leader motor. To quantify the performance of the follower motor, the measurable error for each motor e n : R ≥t 0 → R, ∀n, can be defined as where α ∈ R >0 is a selectable constant control gain, and the subscriptn denotes the complementary motor for motor n (e.g., if n = ex, thenn = f l, and vice versa). Remark 1: As depicted in Fig. 2 , when a motor is serving as a leader motor, the state-dependent and time-invariant mapping between the knee joint angle q and the corresponding motor's angular position θn denoted by φn : R ≥t 0 → R exists, and can be defined as where vn, wn ∈ R are known constants based on the geometry of the exoskeleton design. The target offset angles, velocities, and accelerations for both motors can be expressed as δ n ,δ n ,δ n : R ≥t 0 → R, ∀n, which are generated from the following admittance models: The desired virtual mass, damper, and stiffness parameters are denoted as M a , D a , K a ∈ R >0 . The parameters in (23) and (24) are selected such that the transfer functions are passive [47,Lemma 6.4]. The measurable cable tension signals corresponding to the cables connected to the extension and flexion motors are denoted as T ex , T f l : R ≥t 0 → R, which are bounded with bounded u and u n , ∀n. The desired cable tension is denoted as T d ∈ R >0 , and the selectable parameters in the admittance model are denoted as a 1 , a 2 , a 3 , a 4 , k T ∈ R. The saturation functions sat a 1 ,a 2 (·) and sat a 3 ,a 4 (·) are defined similarly to (17) using their corresponding upper and lower bounds.
The filtered tracking error for each motor r n : R ≥t 0 → R, ∀n, can be defined as where β ∈ R >0 is a selectable constant control gain. Taking the time derivative of (25), pre-multiplying by J n , substituting for (4) and (21), and performing some algebraic manipulation yields Taking the time derivative of (22) twice and substituting into (1), and then substituting it into the previous equation yields where the auxiliary signal χ n : R ≥t 0 → R, ∀n, is defined as Using Properties 1-4, Properties 6-7, and Assumption 2, the auxiliary signal can be bounded as |χ n | ≤ c n,1 + c n,2 l n , ∀n, where c n,1 , c n,2 ∈ R ≥0 , ∀n, are known constants, and l n : The low-level follower motor control input can be designed as u n = k n,1 r n + k n,2 + k n,3 l n + k n,4 |u| + k n,5 |δ n | sgn(r n ), (31) where k n,1 , k n,2 , k n,3 , k n,4 , k n,5 ∈ R >0 , ∀n, are positive selectable control gains. The designed control input u n in (31) is composed of a feedback term k n,1 r n leveraging the instantaneous tracking error and the high-frequency control terms that are multiplied by the signum function to compensate for disturbances and the cross-terms due to the composite error l n , the high-level control input u and the indirect force termδ n that influence the low-level closed-loop error system.
Remark 2: Since the high-level knee joint control input u can be implemented separately in the high-level control loop using the knee joint kinematics, then the low-level control input u n is implementable.
The closed-loop error system can be obtained by substituting (31) into (27) as J nṙn = χ n − e n − B n k n,1 r n + k n,2 + k n,3 l n + k n,4 |u|+k n,5 |δ n | sgn(r n ) + J n B e vn M −1 u+αδ n .

V. STABILITY ANALYSIS
The stability of the high-level knee joint kinematic controller and low-level follower motor controller can be examined independently through the following two theorems. Theorem 1 shows that the closed-loop knee kinematic controller with repetitive learning achieves asymptotic tracking. Theorem 2 shows that the low-level robust follower motor controllers achieve exponential tracking. Lemma 1 is used to prove that the inactive low-level following motor controller is uniformly bounded. Theorem 1: Given the closed-loop error system in (18), the high-level controller in (15) and repetitive learning control law in (16) ensures global asymptotic tracking in the sense that provided the following sufficient gain conditions are satisfied Proof: Let V h : R 3 × R ≥t 0 → R be a positive-definite, continuously differentiable Lyapunov function candidate defined as which satisfies the following inequalities where γ 1 min 1 2 , 1 2 c m , 1 2μ , γ 2 max 1 2 , 1 2 c M , 1 2μ , and (37) Let z h (t ) be a Filippov solution to the differential inclu- is an upper semi-continuous, nonempty, compact and convex valued map on Q ⊂ R, where Q is an open and connected set, as defined in [48], [49]. The vector h 1 is defined by using (5) and (6) Due to the sliding mode terms in (15), which induce a discontinuity at any point in Q, then a generalized solution using Filippov solutions [48] at a certain point can be found by examining the behavior of its derivative at nearby points utilizing differential inclusions (i.e., since a solution to the system may not exist in the classical sense [49]). Hence, since the control input in (15)  ∈V h (z h (t ), t ), whereV h is the generalized time derivative of (35) along the Filippov trajectories ofż h = h 1 (z h ) and is defined as in [49] The inequality in (40) Provided the sufficient gain conditions in (34) (16),Ŵ d ∈ L ∞ , and along with the fact that W d ∈ L ∞ due to the periodicity and boundedness as shown in (12),W d ∈ L ∞ . Since ξ,ξ, η ∈ L ∞ , then q,q ∈ L ∞ since q d ,q d ∈ L ∞ , and thus, u ∈ L ∞ . Then, q ∈ L ∞ in (1).

Lemma 1: The low-level motor follower controller in (31) is uniformly bounded when inactive since
Proof: When motorn is the leader motor, its low-level motor follower controller is inactive (meaning, motorn is using the high-level control input u) to achieve the joint tracking objective. According to Theorem 1, the high-level control input u ∈ L ∞ , and thus the high-level error signals ξ, η, q ∈ L ∞ . The motor n is the follower motor and ensures θn, θ n , e n , r n ∈ L ∞ by Theorem 2. Hence, the inactive low-level follower controller of motorn will remain bounded χn, Vn ∈ L ∞ , ∀n.

A. EXPERIMENTAL SETUP AND PROTOCOL
Experiments in six able-bodied individuals (five males aged 22-30 years and one female aged 22 years) were conducted to illustrate the performance of the developed controllers in (15) and (31), and repetitive learning control law in (16). Written informed consent was obtained prior to participation as approved by the Institutional Review Board at Syracuse University. Participants wore a cable-driven leg exoskeleton in a standing position, while the electric motors applied torque at the knee joint to achieve knee extension and flexion. Fig. 3 illustrates the exoskeleton testbed with 2 brushless 24 VDC electric motors (EC60 flat, Maxon) mounted on an external actuation unit used to actuate a cable-driven mechanism and apply torque about the knee joint. The angles of each motor θ n , ∀n, and knee joint q were measured using optical encoders (H1, US Digital). Load-cells (OMEGA) were installed in series with the cables to measure cable tension T n , ∀n. The controllers were implemented on a desktop computer (Windows 10 OS) running a real-time target (QUARC 2.6, Quanser) via MATLAB/Simulink 2018a (MathWorks Inc) with a sample rate of 1 kHz. A Quanser QPIDe and a Q-8 DAQ boards were used to read the encoders and control the motor drivers (Maxon) operating in current-controlled mode.
Two trials were implemented for each individual with two different low-level controllers (Controller A and B) to compare the tracking performance with and without using the cable tension feedback to update the desired low-level motor follower trajectories, respectively. Controller A is the lowlevel controller designed in (31) with error signals e n , r n in (21) and (25), respectively, that leverage cable tension feedback through the use of the admittance models (23) and (24). Controller B is the low-level controller that does not use cable tension feedback developed in [38], in which the error signals are implemented as and thus, the low-level motor follower control inputs (Controller B) is designed as u n = k n,1 r n + k n,2 + k n,3 l n + k n,4 |u| sgn(r n ), ∀n, (53) respectively. It is noticeable that Controller B tracks the leader motor's kinematics and does not exploit cable tension feedback, i.e., δ n is not used to compute e n in (51), which is equivalent to setting α = 0 in (21) and k n,5 = 0 in (31). Hence, the admittance models in (23) and (24) are not leveraged in the implementation of Controller B.
Leveraging the knee-joint range of motion obtained in previous work for treadmill walking [51], the range of the desired knee angle trajectory was defined between 10 to 50 degrees with a period of T = 3 seconds as depicted in Fig. 4 (top row  plots). The sinusoid desired trajectory was implemented to test the repeatability of the developed control methods in multiple participants, prior to the implementation of the controllers in complex gait experiments. This trajectory also facilitates a standardized and consistent protocol to draw comparisons between the Controller A and Controller B across participants. The experiments are implemented with a duration of 60 seconds (20 leg swings in total). Table 1 includes the selection of the control gains for the high-level knee joint kinematic controller in (6), (15) and (16), the gains for the low-level follower control inputs (for both Controller A and Controller B) in (21), (25), (31), (52), and (53), and the parameters of the admittance models in (23) and (24). Table 2 summarizes the knee joint tracking error, follower motors tracking errors, and the cable tension performance for each participant with Controller A and B. The knee joint tracking performance and the high-level control input u for Subject 1 are presented in Fig. 4, where the desired knee joint trajectory q d is depicted in blue, the actual knee joint angle q is depicted in red, and the knee joint tracking error ξ is depicted  in gray. Fig. 5 shows the high-level control input u computed by Controllers A and B, where the red curves indicate the mean input u across the six participants and the shaded areas depict the standard deviation. The repetitive learning control lawŴ d defined in (16) is presented in Fig. 6 for Subject 1. The low-level follower motor tracking performance for Subject 1 is illustrated in Fig. 7, where the desired trajectory is depicted in red and the actual motor angle is depicted in blue. Fig. 8 illustrates the mean of the measured cable tension T n , ∀n across the six participants, and the shaded areas depict the standard deviation.

VII. DISCUSSION
The experimental results demonstrate the feasibility of the developed low-level follower controllers designed in (31) to mitigate cable slackness and counteracting forces while the high-level controller in (15) to achieve satisfactory knee joint tracking performance. In the previous theoretical work [38], the objective of the low-level follower controller is to activate the complementary motor to closely follow the kinematics of the leader motor. This controller ensures the motors are tracking close to each other; however, the lack of feedback of the actual cable tension limits the capacity to cope with  slackness and thus, adjust the cable tension correspondingly. The low-level follower controllers in this article directly exploit cable tension feedback to mitigate cable slackness and counteracting forces. The desired trajectory for the follower motor n includes not only the complementary motorn kinematics, but also the kinematic offset motor angle δ n , which is generated from the admittance models in (23) and (24) that leverage the cable tension measurements to adjust the cable tension indirectly (i.e., through indirect force control). To demonstrate the improvement in tracking performance, low-level Controllers A and B were implemented in each participant with the same high-level controller u in (15).

A. HIGH-LEVEL KNEE JOINT TRACKING PERFORMANCE
The mean knee joint position error is 1.44 deg and 0.42 deg across participants for Controllers A and B, respectively. The average knee joint tracking error is not significantly different between the two controllers, but Controller A has less average standard deviation compared to Controller B. The knee joint tracking performance for Subject 1 is presented in Fig. 4 to compare the two controllers. The high-level joint controller in (15) smoothly brought the knee joint angle to track the desired steady-state within 4 cycles. The repetitive learning control inputŴ d shown in Fig. 6 illustrates the learning of the periodic dynamics during the leg swings. The repetitive learning control term adjusts the control input due to the periodicity of the leg motion and exploits the inputs recorded in the previous iteration.
With the same high-level controller u, the joint tracking error (difference between q d and q) obtained with the low-level Controller B [ Fig. 4(b)] is bigger and the joint angle has a larger variation than the joint performance obtained with the low-level Controller A [ Fig. 4(a)], despite the fact that the mean tracking errors for both low-level controllers are similar. In Fig. 4, the high-level control input u has a larger magnitude range with low-level Controller B [ Fig. 4   curves indicate the measured motor angles, and the dotted curves indicate when the low-level follower controller is inactive (e.g., such motor is the leader motor and receives u as its input). The average tracking errors for the Controller A corresponding to the extension and flexion motors are −0.03, −3.34 rad, respectively, while the average tracking errors for the Controller B are 1.47, −2.34 rad, respectively. Since the error signals e n , r n used in Controllers A and B are different (i.e., Controller A leverages (21) and (25), while Controller B uses (51) and (52)), it is not feasible to directly compare the low-level tracking performance between them since they fundamentally implement a different approach (e.g., kinematic plus indirect force control versus only kinematic control). Controller B is purely tracking the leader motor, while Controller A further includes the target offset angle δ n computed from the admittance models. Overall, Controller A provides smaller control input magnitudes into the system and minimizes the standard deviation of the tracking performance compared to Controller B.
It is worth noting that the average tracking error is larger when the flexion motor is serving as a follower motor (i.e., when the knee is doing extension movement) for both controllers. During experiments, the participants were standing upright and thus, the high-level controller lifted the shank to counteract the effects of gravity during the flexion movement. During the extension movement, the effect of gravity naturally aids the extension of the shank to reach the standing position, resulting typically in faster movements (compared to flexion motion). The high-level controller acts as a brake during knee extension, mitigating deviations from the desired trajectory due to gravity (and other disturbances). Consequently, the faster movement makes the flexion follower motor apply a larger input to follow the extension (leader) motor during knee extension. These observations are supported by the computed control inputs and tracking performance, where the average high-level control input u across all experiments are 2.54 A and −0.86 A in knee flexion and extension movement, respectively, and the larger tracking errors in Fig.  7(c) and (d).

C. CABLE TENSION AND SLACKNESS
Controllers A and B can also be examined based on the applied tension to achieve the tracking objective (e.g., in addition to the joint kinematic tracking performance). The measured cable tensions for both low-level controllers are illustrated in Fig. 8, where the percentage of the total time that each cable is experiencing slackness are 27.29(ex), 1.86( f l ) and 39.98(ex), 0.21( f l ) (%) for Controllers A and B, respectively. These percentages are computed as the summation of the time the cable is experiencing slackness (T n < T d ) divided by the total time (see the percentages for all participants reported in Table 2). The target offset motor angle δ n , ∀n, generated by the admittance model seeks to adjust the motor angle to maintain the cable tension close to the desired tension. The cable tensions are different between the two controllers in Fig. 8, where the average peak tension for the extension cable (red) is 42 lb for Controller A and 62 lb for Controller B. This additional cable tension demand computed by Controller B is partially due to cable slackness, which is happening in each swing period (when the cable tension reaches 0). When the cable slackness happens, the pulley system takes additional time to tension the cable while the controller increases the control input to minimize the error, and this results in higher pulling forces once the cable is finally tensioned (i.e., resulting in increased cable tension). As reported in Table 2, the difference between the mean extension cable slackness percentage of Controller A and Controller B is 12.69% (i.e., Controller A experiences slackness on average 12.69% less time than Controller B across all participants). For both controllers, the flexion cable experience slackness less than 2% of the total time (see the mean values across all participants in Table 2). Overall, Controller A successfully reduces the cable slackness and peak cable tension compared to Controller B. In addition, Controller A required less control effort than Controller B, which can aid mitigating unsafe actuator control forces applied to the lower-limb exoskeleton worn by participants.

D. SUMMARY
This article describes the design, analysis, implementation, and preliminary comparison of two low-level controllers for a cable-driven knee exoskeleton. The comparative analysis shows that Controller A exhibits several advantages over Controller B. First, Controller A demonstrates reduced variability in knee tracking error and requires less high-level control command, resulting in better tracking performance compared to Controller B. Second, Controller A shows a notable reduction in cable slackness compared to Controller B, and the peak cable tension is also lower than Controller B. Overall, the study demonstrates the benefits in performance of the new control design (Controller A) over the previously developed controller (Controller B) in [38]. Nevertheless, the selection of the control approach (e.g., a kinematic approach versus a kinematic plus an indirect force control approach) to be implemented for a given exoskeletal application is dependent on the availability of actuators, sensors, and computational resources to achieve a desired tracking objective. Ultimately, human perception of the control algorithm (e.g., tolerance of the applied forces by the device) is another important consideration for determining a suitable control approach using exoskeletons.

VIII. CONCLUSION
A two-layer control structure was developed for a lowerlimb cable-driven exoskeleton to activate a pair of motors to achieve desired knee joint tracking. A repetitive learning controller was developed for the high-level controller to track periodic desired knee joint trajectories. The lowlevel controller was developed to mitigate the potential cable slackness and counteracting forces. The target trajectory for each motor is updated exploiting measurements of the cable tension in real-time. A Lyapunov-based stability analysis is developed to ensure global asymptotic tracking performance for the knee joint tracking objective. The low-level control subsystems achieve exponential tracking as demonstrated in the Lyapunov-based stability analysis, while a bounded and stable tracking performance is guaranteed under switching between both low-level control subsystems. The control algorithms are successfully implemented in real-time experiments in six able-bodied individuals comparing two different lowlevel controllers in separate trials (i.e., Controllers A and B). The experimental results illustrate the potential of the developed controllers to improve low-level and knee-joint tracking performance, while preventing potential cable slackness and reducing the peak cable tensions to achieve the tracking objectives. Future work includes leveraging the developed controllers for treadmill walking experiments by combining the hip and ankle joints together with the knee joint. In addition, a gait-phase detection algorithm is currently being developed to ensure smooth multi-joint coordination for walking. Future efforts also include developing an adaptive-based control extension to enhance the performance of the low-level controllers during gait experiments. An adaptive admittance controller is also motivated for customizing the virtual mass, damper, and spring in the admittance model in real-time.