Experimental Validation of a Torque-Controlled Variable Stiffness

IEEE/ASME TRANSACTIONS ON MECHATRONICS, VOL. 23, NO. 5, OCTOBER 2018

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Experimental Validation of a Torque-Controlled Variable Stiffness Actuator Tuned by Gain Scheduling Lin Liu

, Steffen Leonhardt

, Senior Member, IEEE, and Berno J. E. Misgeld

Abstract—A variable stiffness actuator (VSA) is an inherently parameter-dependent system due to the controllable stiffness element. Within the torque-controlled framework, the VSA is distinguished from the classical series elastic actuator (SEA). A frozen torque controller can directly determine the SEA performance with a fixed-stiffness spring selection. However, since the VSA operates at a set of operating points, the aim is to achieve an adaptive control approach. For this, we propose a gain-scheduled torque controller. The control performance is expected to recover robustness when stiffness values are varied from smaller to larger ones. Simultaneously the bandwidth is maximized, taking into account hardware limitations. In this way, a good tradeoff between stability and performance can be achieved. A key step in the gain-scheduled controller design is to implement the linear controllers, where the linear quadratic Gaussian (LQG) technique was applied to deal with the multiloop feedback (a cascade control scheme) in the VSA plant. A lever-arm based VSA [i.e., a mechanical-rotary variable impedance actuator (MeRIA)] was used to verify this new controller design approach by the simulations and experiments. The resulting gain-scheduled controller was also evaluated by taking into account the impedance control testing on a motion-supported platform for the knee joint. Index Terms—Gain-scheduling, impedance control, rehabilitation robotics, torque control, variable stiffness actuator (VSA).

I. INTRODUCTION MPLEMENTATION of passive-compliant elements is associated with safety performance in robotic devices. Against this background, the first series elastic actuator (SEA) equipped with a fixed compliance was developed [1]. Nevertheless, challenges of safety in the various human–robot contact tasks play an important role in the need to develop a novel SEA as a stiffness-

I

Manuscript received October 10, 2017; revised February 12, 2018 and June 28, 2018; accepted July 4, 2018. Date of publication July 9, 2018; date of current version October 15, 2018. Recommended by Technical Editor H. A. Varol. This work was supported by Chinese Scholarship Council under Grant 201307000008. (Corresponding author: Lin Liu.) The authors are with the Philips Chair for Medical Information Technology, Helmholtz-Institute for Biomedical Engineering, RWTH Aachen University, Aachen 52074, Germany (e-mail: [email protected]; [email protected]; [email protected]). This paper has supplementary downloadable multimedia material available at http://ieeexplore.ieee.org provided by the authors. Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TMECH.2018.2854416

, Member, IEEE

varying structure. This new type of actuator is generally referred to as a variable stiffness actuator (VSA). With increasing VSA research, discussion on the developing guidelines has clarified the design of adaptable compliance mechanisms, e.g., antagonistic or independent actuation setup [2], thereby determining the design requirements, e.g., output power, spring performance, sensors, and control [3], and fitting the application needs to the users, e.g., by the VSA datasheets [4]. In accordance with the overviews, the hardware components can be configured to each of the VSA designs from the literature. The vast majority of VSA consists of a coupled doubleactuation system that can export an adaptable passive compliance in operation. Then, in many applications, the focus is on link and stiffness positioning. A simple and effective way to achieve this is to design a decentralized controller, e.g., proportional-integral-derivative (PID) position control for the nearly decoupled lever-arm-based mechanism [5], [6]. Furthermore, model-based techniques have been proposed by associating them with different decoupling control approaches, e.g., feedback linearization [7], state feedback damping control using eigenmodel decoupling [8], and linear quadratic regulator (LQR)-based gain-scheduling using an approximation decoupling [9]. It is noteworthy that vibration suppression is the usual way to enhance the position tracking performance for a class of flexible joints [10], which is also discussed in the VSA literature, e.g., [8], [9]. In addition to the abovementioned applications, another important issue is how to perform “explosive movement tasks” in the VSA systems, e.g., throwing or kicking. To this end, optimal control was implemented to compute a control input trajectory. This trajectory was computed offline with the goal of exploiting the variable stiffness [11] and toward velocity maximization [12]–[14]. Another method to shape the mechanical impedance (both stiffness and damping) is the well-known impedance control approach [15] (i.e., active compliance). As proposed in [16], impedance control has been widely applied to the group of stiff robots and SEA, whereas one of the noticeable features is that the integrated force (or torque) sensing is used to construct an explicit inner loop that is cascaded with an outer position loop (i.e., impedance controller). The same approach has also been increasingly applied in the control of VSA-based robots, e.g., the hand-arm system [17] and the musculoskeletal system [18]. In these applications, some complementary strengths are taken

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into account, e.g., the VSA improves the capacities of the energy regulation and bandwidth range (which outperforms the SEA where the bandwidth is limited due to the fixed compliance), and impedance control increases the dynamic range and interaction stability of the global system. With corresponding advantages discussed above, the VSA research in the area of impedance control is also illustrated here. Specifically, the control demands are investigated in the VSAbased gait-assistance applications [19]–[21], and the target of further research is rehabilitation training. With these aims, adaptive impedance by control is necessary and appropriate to adjust the interaction between the patient and device in the different phases of rehabilitation (e.g., training models and control strategies by considering the human intention [22]). Especially for pursuing low-impedance actuation, the programmable controller is more convenient than improving the mechanical structure or spring. Corresponding further to the impedancecontrolled exoskeleton studies [23], [24], the actuator as a torque source is confined to the role of supporting the wearer’s movement by providing adequate speed, power, and bandwidth. Although the position control (see [5]–[9], [11]–[14]) plays an important role in the VSA research, the contribution of realizing torque control is still foremost in our specialized tasks. In the torque control loop of the VSA, the linear time-invariant (LTI) controller has been recognized, e.g., classical parameters setting [8], [17], [18], [20], [21], optimal control [25], and robust control used in our previous study [26]. However, the plant is nonlinear or time varying when the stiffness is changed in real time. In such cases, it will be difficult to always maintain the desired properties due to the tradeoff when using the LTI controllers. In other words, the LTI controller used on the torquecontrolled VSA is generally conservative. Based on the above-mentioned background, the main focus of this paper is to find a good tradeoff between torque tracking and stability in the torque-controlled VSA. Within the context of rehabilitation robotics, we pursue a high bandwidth actuator that can effectively increase the range of the wearer’s motion [20]. Meanwhile, maintaining stability makes an acceptable tradeoff for our system. For this purpose, a gain-scheduled torque controller based on a linear parameter-varying (LPV) plant is proposed. To predict the feasibility of this research, a VSA prototype, i.e., a mechanical-rotary variable impedance actuator (MeRIA) [26], was employed as the control objective. The starting point of the controller design is based on the LQR optimal theory. With the introduction of the integral action (i.e., LQRI), the synthesized regulator increases the steady-state tracking accuracy. Based on this, a desirable output impedance can also be ensured when implementing a cascade impedance control framework. Within our actuator torque control system, a cascaded torque velocity (i.e., motor velocity control) structure was used to ensure a robust output platform, that has been verified for the classical compliant actuators [16], [24], [26]–[29]. These previous studies employed the PID controller to control the motor velocity, which was also used in this study. For the developed multiloop feedback system mentioned above, a Kalman filter was designed for state estimation. Therefore, the resulting control method is attributed to the linear quadratic

Gaussian (LQG) technique [30]. Then, using the interpolating gain-scheduling control [31], the system recovers the stability margin in the entire range of stiffness with an effective vibration suppression. Simultaneously, a high performance actuator (i.e., maximization of the closed-loop torque bandwidth) is guaranteed throughout the design processes. The remainder of this paper is organized as follows. Section II presents the preliminaries related to the testing platform and control model derivation. Section III presents the theoretical background of the linear controller design and its implementation on minimal stiffness. Based on the synthesized LTI controller from the last section, Section IV focuses on the control issue of stiffness variation, which then motivated us to design the gain-scheduled controller. After discussing the torque control loop, Section V presents knee joint motion experiments on the impedance control loop. Finally, the conclusions are presented in Section VI. II. PRELIMINARIES This section begins with an introduction to the testing platform, in which all the experiments described here were implemented. This is followed by a description of the open-loop torque control model in a LPV form. A. System Setup Fig. 1 is a schematic diagram of the testing platform, showing the main system elements. The left panel shows the constructed rapid prototyping control system by dSPACE hardware (DS 1103, sampling frequency of 1 kHZ). Using the ControlDesk software the entire testing process can be controlled and monitored in real time, e.g., logical control and graphical data acquisition. The right panel shows the test bench equipped with the MeRIA that drives an orthosis to support the swing-motion testing. To ensure a safe testing environment, electric limit switches are fixed on the test bench. Moreover, the mechanical limit screw in the orthosis can be utilized to remain within an acceptable physiological range for the person involved. In this setup, the actuator output shaft (see Fig. 1, middle panel) is connected with a rotary adapter for the orthosis through coupling, the torque sensor is coupled between the actuator and orthosis, and the joint angle encoder is installed at the load side (i.e., the orthosis). With these two sensors, the feedback to close the cascade impedance control loop can be achieved. The middle panel shows the MeRIA design; its components include the following three parts. 1) Joint Actuation: The first speed-controlled brushless DC (BLDC) motor of 90 W (denoted as M1 ) with a harmonic drive is used as the actuation of the joint motion. 2) Stiffness Generation: The stiffness is generated by using two elastic bending bars that are installed on the output flange of the harmonic drive. The compliance is transmitted to the actuator output shaft via the cam followers. In the current prototype, the bending bar is fabricated using CK 85 (a hardened, tempered material); it has higher yield strength than DIN 1.4310; that is the material used

LIU et al.: EXPERIMENTAL VALIDATION OF A TORQUE-CONTROLLED VARIABLE STIFFNESS ACTUATOR

Fig. 1.

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Testing platform consisting of the control system, the MeRIA design, and the test bench.

Fig. 3. Fig. 2. Variable stiffness method used in the MeRIA. F denotes the external force, L e is the controllable effective length (by actuating the cam followers), and R e is the fixed-lever arm. The distance from the acting point of the external force (F ) to the pivot is the length of the lever arm (L e + R e ).

to fabricate the bending bars in our previous study [26]. With the enhanced property of the material, the improved safety (or carrying capacity of the spring) makes the mechanical drive more secure to perform the swing-motion testing: e.g., the experiments to apply the resisting torque on leg movement (see Section V). 3) Variable Stiffness System: The second positioningcontrolled BLDC motor of 23 W (denoted as M2 ) with a worm gear set is used as the actuation of the slider screw (with an 18 mm lead). The slider screw drives the cam followers for moving on the surface of the bending bar, which achieves the stiffness control (see below). The concept of stiffness control is shown in Fig. 2, from which the stiffness function (for two bending bars) can be calculated by using the stress analysis of a cantilever beam [32]: 6EI(Re + Le )2 /L3e , where E and I are the bending bar’s elastic modulus and area moments of inertia, respectively, both of which are constant after determining the material and dimensions. Accordingly, it is obvious that control of the effective length Le realizes a level-arm-based stiffness-varying principle. Note that the above-mentioned stiffness function is derived from a simplified model with nonlinear errors. Moreover, the assembly errors cannot be neglected when installing two symmetric bending bars. Therefore, we conducted an experiment to identify the stiffness (see the Appendix).

Fixed-load dynamic model of the MeRIA.

B. LPV Torque Control Model The compliant actuator with the fixed load side represents the worst-case scenario, regarding the stability of the torque control testing [1]. In this situation, the behavior of the actuator and load are completely decoupled. Starting from the above-mentioned points, a physical model of the fixed-load MeRIA is shown in Fig. 3. Assuming the actuator output torque Tj is fully acting on the spring, the following representation is derived using Hooke’s law: Tj = Kj (θ2 )θ1 /γ1

(1)

where the function Kj (θ2 ) [see (25)] is denoted as the static stiffness calculated by the motor angle position of M2 (denoted as θ2 ). Moreover, θ1 /γ1 represents the produced spring deflection that equals the motor angle position of M1 (denoted as θ1 ) after the Harmonic Drive (γ1 is the gear ratio). For the above equation, we write Tj = Tj (θ1 , θ2 ) as a function between θ1 and θ2 . Employing the MeRIA dynamics as proposed in [26], the model of the above-discussed system can then be expressed as Jm 1 θ¨1 + Bm 1 θ˙1 + Tj (θ1 , θ2 )/γ1 = Tm 1

(2)

Jm 2 θ¨2 + Bm 2 θ˙2 + Ts (θ1 , θ2 )/γ2 = Tm 2

(3)

with the armature current as input Tm 1 = km 1 im 1

(4)

Tm 2 = km 2 im 2

(5)

where Tm i , Bm i , Jm i , km i , and im i , respectively, are the electromagnetic torque, the damping, the moment of inertia

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TABLE I NOMINAL VALUES OF THE DYNAMIC MODEL

[33]. With the denoted reference θ˙1,ref , the velocity closed-loop transfer function is then written as Gcv (s, θ2 ) =

Gov (s)PI(s) θ˙1 (s) . = ˙θ1,ref (s) 1 + Gov (s)PI(s)

(7)

Combining (1) with (7), the torque open-loop transfer function from θ˙1,ref (s) to Tj (s) can be obtained Got (s, θ2 ) = = converted onto the motor shaft, torque constant, and armature current, for all of which the subscripts for i = 1, 2 correspond to the motor Mi . In (3), γ2 is the gear ratio of the worm gear. The function Ts (θ1 , θ2 ) is used to represent the resisting torque that is totally dependent on the degree of spring deflection during stiffness variation. For (2) to (5), the nominal values are listed in Table I, in which the moment of inertia, the gear ratio (excluding the energy losses), and torque constant were obtained from the datasheet. Moreover, the damping coefficient was estimated from the no-load testing of the motors. In this case, the motor torque is approximately equal to the damping torque, i.e., km im ≈ Bm θ˙m . With this, the damping value can be calculated with respect to the measured no-load speed and current. For such vertically arranged double-motor-based VSA, two methods are available to decompose the motor dynamics between (2) and (3). The first method is the implemented decentralized control approach without considering the coupling in our previous work [26] (see also [5], [6]), and the second method is attributed to the decoupling approximation method by assuming a small spring deflection [9] that is also appropriate to MeRIA whose maximal deflection angle is around 0.15 rad. Accordingly, stiffness variation is considered as a part of an exogenous disturbance in the actuator control. The experimental results in Section IV-C (i.e., the system output is slightly affected by the coupling) can then be used to verify the above conclusion. In practice, the controller of motor M2 is set to a positioning operating mode self-contained proportional-derivative (PD) controller. The positioning was calibrated using the manager software of the motor (FAULHABER Motion Manager). The torque control in the loop of motor M1 is discussed below. The torque control model is prepared after designing the cascaded inner motor velocity loop. Equation (2) can now be extracted from the dynamic equations of the MeRIA; then, the open-loop transfer function in terms of current input im 1 (s) and velocity output θ˙1 (s) can be acquired by substituting (1) and (4) into (2) Gov (s, θ2 ) =

km 1 γ12 s θ˙1 (s) = . (6) im 1 (s) Jm 1 γ12 s2 + Bm 1 γ12 s + Kj (θ2 )

A proportional-integral (PI) controller of the form PI(s) = KP v + KI v /s (where the proportional gain KP v = 1.9 A/rad/s and integral gain KI v = 570 A/rad) is then designed to shape the open-loop plant Gov (s, θ2 ) in (6). Here, the controller tuning rule is treated as a fast inner-loop design in cascade control

Tj (s) Tj (s) = Gcv (s, θ2 ) ˙θ1,ref (s) θ˙1 (s) b1 (θ2 )s + b0 (θ2 ) s3 + a2 s2 + a1 (θ2 )s

(8)

where a1 (θ2 ) =

KI v km 1 γ12 + Kj (θ2 ) , Jm 1 γ12

b0 (θ2 ) =

KI v km 1 Kj (θ2 ) , Jm 1 γ1

a2 = b1 (θ2 ) =

Bm 1 + KP v km 1 Jm 1 KP v km 1 Kj (θ2 ) . Jm 1 γ1

Equation (8) has been formulated in a LPV form with the scheduling variable Kj (θ2 ) as an exogenous signal. For the LPV-based controller design, a state-space representation is usually preferred. Therefore, the above transfer function needs to be converted without affecting the dynamics. For this, the observability canonical LPV state-space realization is employed x˙ = A(θ2 )x + B(θ2 )u (9) Got (θ2 ) : y = Cx + Du where x ∈ R3 is the state vector, u = θ˙1,ref is the input vector, and y = Tj is the output vector. The state-space realizations (A(θ2 ), B(θ2 ), C, D) are given by ⎡ ⎡ ⎤ ⎤ 0 0 0 b0 (θ2 ) ⎢ ⎢ ⎥ ⎥ 1 0 −a1 (θ2 ) ⎥, b (θ ) ⎥ B(θ2 ) = ⎢ A(θ2 ) = ⎢ ⎣ ⎣ 1 2 ⎦ ⎦ 0 1 −a2 0 C= 0

0

1 ,

D = 0.

Equation (9) is the final control model derived from the case of a fixed load. This model is then employed to design the torque controller discussed in the following section. III. LINEAR CONTROLLER DESIGN AND IMPLEMENTATION Section III-A presents the LQG control strategy that is identified by the separate process for optimal solution of the LQR and Kalman filter. Here, the LQR method generates a state feedback controller to close the control model with a commanded control signal. In this case, an exact model is required to obtain a correct steady-state value. However, this is difficult to achieve due to the uncertainties in the system. To prevent the steady-state error, an integral feedback added to LQR is used. With the introduced integral action, the resulting controller is denoted as LQGI [34]; the synthesized gain matrix in the LQGI controller can intuitively demonstrate the controller scheduling. Then, the control


experiments at minimum stiffness are discussed in Section III-B. Also, it should be noted that the case of a fixed load is applied to all the simulations and experiments on torque control.

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as stationary Gaussian white noise, and E{wd (t)} = 0,

E{wn (t)} = 0

E{wd (t)wdT (τ )} = Qw d δ(t − τ ) A. LQGI Control Strategy Taking a reference signal Tj,ref as the input of the torque loop, the control error dependent on the output feedback can be defined as e = Tj,ref − y.

(10)

The original plant [see (9)] in the LTI case can then be augmented with the new integral error state to prepare for the deterministic LQR control approach (noise-free) x˙ e

=

A

0

x

xe −C 0

x ˜ A˜

+

B

0

˜ B

u+

03×1 1

Tj,ref

(11)

xe (t) =

t

Kk = Pk C T Rw−1n

e(t) dt.

JLQR =

(˜ xT Qt x ˜ + Rt u2 )dt

(12)

0

with positive semidefinite matrix Qt ∈ R4×4 and positive weighting factor Rt ∈ R, by implementing an optimal state feedback control law ˜ T Pt x ˜ u = − Rt−1 B

Kt

(13)

where Pt = PtT 0 is the unique solution of the following algebraic Riccati equation (ARE): ˜ −1 B ˜ T Pt + Qt = 0. A˜T Pt + Pt A˜ − Pt BR t

(14)

The resulting optimal LQRI feedback gain Kt consists of both a state feedback gain Kx and an integral gain Ke . This can then be denoted as Kt = [Kx Ke ] (Kx ∈ R1×3 , Ke ∈ R). Next, we proceed to design the LQG estimation (without integral state) which is generally accompanied by the stochastic processes. To describe that, we reconstruct the original plant as Got :

x˙ = Ax + Bu + wd y = Cx + wn

(17)

where Pk = 0 is the unique solution by solving the dual representation of the ARE in (14) PkT

For (11), a conventional LQR optimization can be defined as to minimize the following cost function: ∞

(16)

where the optimal Kalman filter gain, Kk ∈ R3 , can be acquired by

0

where E{·} is the expected value operator, δ(t − τ ) is the delta function, and Qw d ∈ R3×3 and Rw n ∈ R are the covariance data. For the control task of the state estimation, a Kalman filter problem can be represented by minimizing the estimation ˆ}T {x − x ˆ}, where x ˆ is the error covariance, JLQG = E{x − x estimated state for x. Accordingly, the dynamic equation of the Kalman filter can be obtained by using the following general expression: ˆ) x ˆ˙ = Aˆ x + Bu + Kk (y − C x

where

E{wn (t)wnT (τ )} = Rw n δ(t − τ ) 1, for all t = τ δ(t − τ ) = 0, for all t = τ

(15)

where wd ∈ R3 and wn ∈ R are the system and sensor noise, respectively. This additional pair of exogenous inputs is modeled

APk + Pk AT − Pk C T Rw−1n CPk + Qw d = 0.

(18)

To calculate the controller, the weighting matrices are selected as proposed in [35]. Accordingly, Qt and Rt of the LQRI controller take the following form: Qt = diag(ρx C T C, qe ),

Rt = 1

(19)

where ρx is a weight for the original state vector and qe is a weight for the added integral error state. Qt is given as diagonal matrix; the advantage of this is that each state vector can be decoupled (with C T C that penalises only the plant outputs). Meanwhile, Qw d and Rw n for tuning the Kalman filter are given by Qw d = BB T ,

Rw n = ρk

(20)

where the determined Qw d in most cases is a suitable choice and ρk is the only value needed to iterate which reduces the workload. In this paper, based on the above points, we suggest the following iteration processes. 1) In LQRI design, the weights ρx and qe are determined by considering the tradeoff between stability and actuator performance. 2) In Kalman filter design, ρk ≥ 1 is constrained to maintain the stability margins. From the implemented closed-loop torque control structure in Fig. 4, the state-space form of the LQGI controller can be written as x˙ k = Ak xk + Bk vk (21) KLQGI : u = −Kt xk

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TABLE II CONTROLLER GAINS SYNTHESIZED BY THREE DIFFERENT INTEGRAL ERROR WEIGHTS

Fig. 4.

Block diagram of LQGI-based closed-loop torque control.

where

xk = Ak = Bk =

x ˆ

vk =

,

xe

Tj,ref

y

A − BKx − Kk C

−BKe

01×3

0

03×1

Kk

1

−1

.

The state-space form of the closed-loop system is then described by x˙ cl = Acl xcl + Bcl wcl Gct : (22) y = Ccl xcl + Dcl wcl where

⎡

⎤

x

⎡

Tj,ref

⎤

Fig. 5. Bandwidth experiments at minimum stiffness. (a) Small-torque bandwidth testing: the input chirp signal with peaks of ±2 N·m was varied up to 10 Hz. (b) Large-torque bandwidth testing: the input chirp signal with peaks of ±15 N·m was varied up to 5 Hz.

⎥ ⎥ ⎢ ⎢ ˆ ⎦, wcl = ⎣ wd ⎦ xcl = ⎣ x − x wn

xe ⎡ ⎢ Acl = ⎢ ⎣ ⎡

A − BKx

BKx

03×3

A − Kk C

−C

01×3

03×1

⎢ Bcl = ⎣ 03×1 1 Ccl = C

I3×3 I3×3 01×3

01×3

03×1

−BKe

⎤

⎥ 03×1 ⎥ ⎦ 0

⎤

⎥ −Kk ⎦ −1

0 , Dcl = 0

01×3

1 .

B. Controller Synthesis at Minimum Stiffness One of the control objectives is to maximize the bandwidth range of the MeRIA. Corresponding to the iteration processes (see Section III-A), ρx = 5 × 103 , qe = 1.2 × 107 , and ρk = 1 × 106 were selected to achieve the above purpose. The synthesized LQGI controller at minimal stiffness (201 N·m/rad) is denoted as KLQGI (θ21 ), and the associated LQRI gain Kt (θ21 ) and Kalman filter gain Kk (θ21 ) are listed in the first row of Table II. To clarify how the above controller is ultimately determined, the system’s responses under different controllers were

tested. In the experiments presented in Figs. 5 and 6, another two synthesized controllers were provided as a benchmark by fixing ρx and ρk then only tuning the integral error weights qe downward to qe,s = 3 × 106 and upward to qe,l = 3 × 107 , respectively. For these two controllers, the associated LQRI and Kalman filter gains are listed in the second and third rows of Table II, respectively. The first experiment (see Fig. 5) is designed to test the closedloop frequency response for the full torque range (i.e., small and large torque bandwidth [23]). Here, the transfer function estimation [28] is employed by denoting Gct (f ) = Pxy (f )/Pxx (f ), where f is the frequency in Hertz, Pxy (f ) is the cross-power spectral density of the input and output, and Pxx (f ) is the spectral power density of the input. In this experiment, a chirp signal was used as the reference. To depict the results, the frequency resolution was set to 0.2 Hz. Fig. 5(a) shows the frequency response with a small-torque input. The bandwidth is increased from 6.6 (by qe,s ) to 7.2 Hz (by qe ), which implies that the bandwidth is enlarged by increasing the integral error weight. However, when further increasing the weights (to qe,l ), the bandwidth is still restricted due to the limitations of the hardware (e.g., motor performance). Meanwhile, Fig. 5(b) shows the frequency response with a large-torque input. The bandwidth is saturated at 2.8 Hz and cannot be recovered


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Fig. 6. Step response at minimum stiffness. The amplitude of reference is created with peaks of ±15 N·m.

by tuning the controllers. In this case, the bandwidth is directly dependent on the mechanical stiffness (see Section IV-B). In the second experiment, the step response is shown in Fig. 6. By increasing the integral error weight, the system behavior is varied from critically damped (by qe,s ) to underdamped (by qe,l ). In other words, the time-domain performance is deteriorating. From the above discussion, the controller design was considered both in time and frequency domain. The KLQGI (θ21 )controlled system keeps a large-torque bandwidth of 2.8 Hz and a small-torque bandwidth of 7.2 Hz, which is the maximal bandwidth range that can be achieved at minimal stiffness. Simultaneously, the step response experiment in Fig. 6 shows its time-domain performance (blue solid line): the rise time is 0.16 s, settling time is 0.23 s, and overshoot is 8% (slightly underdamped). IV. GAIN-SCHEDULING APPROACH AND IMPLEMENTATION Section IV-A presents the controller resynthesis for the stiffness-varying cases. As an example of an experiment, the controller design at maximal stiffness of the MeRIA is discussed in Section IV-B. The studies on the gain-scheduling control are presented in Section IV-C. A. Controller Resynthesis for Stiffness Variation Fig. 7 shows the simulation results from two types of systems. The first type is controlled by the designed KLQGI (θ21 ) as presented in Section III-B, and the second is regulated by different controllers which are resynthesized at each simulated point (i.e., Kj = [300, 400, . . . , 1200] [N·m/rad]) by using the previously selected weights, i.e., ρx = 5 × 103 , qe = 1.2 × 107 , and ρk = 1 × 106 . To clearly reveal the VSA torque control problem, the upper limit of the stiffness range adopted in simulation (1200 N·m/rad) is larger than the upper limit (455 N·m/rad) that the MeRIA can achieve. Nevertheless, those stiffness values have been covered by the other prototypes (e.g., [5], [19], [32]), from which a classical torque control problem of VSA can be represented. Following the simulations, the fixed controller leads the tracking response, as shown in Fig. 7(a), to gradually be faster by increasing the stiffness, and the undesired oscillation occurs at the same time due to the tradeoff. The reason for that can also be explained by the corresponding open-loop margin analysis

Fig. 7. Simulations of the system responses by varying the stiffness from 300 (red solid line) to 1200 N·m/rad (red dashed line) with an interval of 100 N·m/rad. (a) and (b) depict the unit step response. (c) and (d) depict the Nichols diagram for the open-loop response in the frequency range from 0.1 to 1000 rad/s.

in Fig. 7(c), where the gain margin, GM = 19.2 dB, and phase margin, PM = 72.1◦ , at 300 N·m/rad are down to GM = 7.19 dB, and PM = 38.1◦ at 1200 N·m/rad. On the other hand, Fig. 7(b) shows that the oscillation can be totally suppressed by the resynthesized controller. In this case [as shown in Fig. 7(d)], the robustness is also recovered with the margins of GM = 22.4 dB and PM = 79.6◦ at 300 N·m/rad to GM = 25.5 dB and PM = 84.3◦ at 1200 N·m/rad. In conclusion, the simulation results show that the resynthesized controllers can increase the stability of the closed-loop system due to the effects of stiffness variation. Moreover, hardware limitation (see Section III-B) should also be considered in practice to verify the attainable response speed. Some experiments for the actuator are presented in the following section. B. Controller Synthesis at Maximum Stiffness At maximum stiffness (455 N·m/rad), the controller is also designed to meet the bandwidth maximization. However, because the weights used in the previous section cannot meet this requirement, we decreased the weights ρx to make a tradeoff. After testing iteratively, the weights ρx = 2.5 × 103 , qe = 1.2 × 107 , and ρk = 1 × 106 were selected. Here, the synthesized controller is denoted as KLQGI (θ22 ); the associated LQRI gain Kt (θ22 ) and Kalman filter gain Kk (θ22 ) are given as follows: Kt (θ22 ) = [3.12 × 10−4

0.071

Kk (θ22 ) = [534.1

0.0045]T .

1.78

2.81

− 3464]

Fig. 8 shows the bandwidth validation at maximum stiffness. In this experiment, the same chirp signal was used as proposed

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Fig. 8. Bandwidth experiments at maximum stiffness. In small-torque bandwidth testing, the input chirp signal was varied up to 15 Hz. In the large-torque bandwidth testing, the input chirp signal was varied up to 6 Hz.

Fig. 9. Step response at maximal stiffness. The amplitude of the reference is created with peaks of ±15 N·m.

in Section III-B. In the Kk (θ22 )-controlled system, the smalltorque bandwidth with input of ±2 N·m (blue solid line) is around 11.4 Hz. Furthermore, due to the enhanced mechanical compliance the large-torque bandwidth with input of ±15 N·m is around 4.4 Hz (green solid line), which is larger than in the case of minimum stiffness. The response for the KLQGI (θ21 )controlled system is also shown in Fig. 8, where only the smalltorque bandwidth is tested since the large-torque bandwidth is dependent only on the stiffness. The result shows that the bandwidth is also around 11.4 Hz (dashed red line) that inherits the maximization properties. Now we consider the time-domain performance for both controllers. As shown in Fig. 9, the step response for the KLQGI (θ22 )controlled system has a rising time of 0.11 s, settling time of 0.15 s, and an overshoot of 7%. However, the KLQGI (θ21 )controlled system is oscillating, where an overshoot of 14% is generated. To illustrate the tracking performance changing from above, the stability margin analysis was employed: the KLQGI (θ22 )-controlled system maintains better stability margins, GM = 20.2 dB and PM = 77.5◦ , than the KLQGI (θ21 )-controlled system, GM = 15.6 dB and PM = 64.4◦ . Based on the above discussion, the resynthesized controller maintains both bandwidth maximization and increased system performance. Notice that, when implementing KLQGI (θ22 ) on the smaller stiffness, the bandwidth cannot be maintained to meet our goal of a high bandwidth actuator. Therefore, controller resynthesis should always be considered in the VSA system for an acceptable tradeoff.

Fig. 10. Experiments using gain-scheduled controller K LQGI (θ2 ) versus frozen controller K LQGI (θ21 ). Real-time stiffness variation is given in (a). Square wave response is given in (b).

C. Gain-Scheduled LQGI Control Experiments The above-proposed KLQGI (θ21 ) (at 201 N·m/rad) and KLQGI (θ22 ) (at 455 N·m/rad) were designed at the endpoints of the stiffness range. For both controllers, the original plant is also stabilized for any fixed Kj ∈ [201, 455] N·m/rad by computing the eigenvalues of the closed-loop A-matrix [i.e., Acl in (22)]. From the above, both controllers have the overlapped stability region in the stiffness interval, such that the interpolated controller also stabilizes the closed-loop LPV system based on the theory of stability preserving [31]. Using the piecewise linear interpolation, the LQRI and Kalman filter gains are given Kt (θ2 ) =

455 − Kj (θ2 ) Kj (θ2 ) − 201 Kt (θ21 ) + Kt (θ22 ) 455 − 201 455 − 201 (23)

Kk (θ2 ) =

455 − Kj (θ2 ) Kj (θ2 ) − 201 Kk (θ21 ) + Kk (θ22 ) 455 − 201 455 − 201 (24)

to construct the gain-scheduled LQGI controller. Fig. 10 shows the square wave response for the gainscheduled and frozen controller with a real-time tuning stiffness. As shown in Fig. 10(a), the stiffness trajectory varies from 383 to 266 N·m/rad at an average speed of 29 N·m/rad/s, where the stiffness calculation associated with the measured effective length was done by the adaptive law of (25). For both control


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Fig. 12. Block diagram of the impedance-controlled human and orthosis system with the inner gain-scheduled torque controller. The symbols θj, ref and θj are the desired and measured joint angles, respectively. The impedance controller is in a proportional-derivative (PD) type that consists of a proportional gain K P s (virtual stiffness) and a derivative gain K D d (virtual damping).

Fig. 11. Experimental results of the closed-loop gain-scheduled system with a maximal rate of stiffness variation. (a) Square wave response for the stiffness control. The average velocity excited from low (204 N·m/rad) to high stiffness (422 N·m/rad) is +216 N·m/rad/s. In the varying direction from high to low stiffness, the average velocity is −238 N·m/rad/s. (b) System state with commanding a 15 N·m reference.

systems, the response speed is very close in the whole operating range, as shown in Fig. 10(b). In contrast, the response using gain-scheduled controller has less overshoot and is more stable at all excitation points (i.e., at 0.5 s, 1.5 s, 2.5 s, and 3.5 s, respectively). In addition, a comparison between the gain-scheduled LQGI controller and the H∞ loop-shaping controller used in our previous study [26] was also made using the experiment described in Fig. 10. Results of these experiments are presented in the Supplementary material. To distinguish differences between the gain-scheduled and frozen controller, the testing (shown in Fig. 10) is effective by considering only the changes in stiffness at a relatively slow speed and in one direction. However, the parameter variation rate (and direction that generally occurs in practice) affects the system output. To investigate this issue, the related experiments are presented in Fig. 11. As shown in Fig. 11(a), the stiffness was controlled with a maximal velocity that varied in both directions. To achieve that maximal velocity, the M2 (i.e., motor for stiffness variation) was running at a full speed (200 r/min after gear box). Fig. 11(b) shows that the system output is slightly affected when stiffness excitation starts from 0.5 to 1.5 s (disturbance on output 0.5 N·m) and from 2.5 to 3.5 s (disturbance on output −0.3 N·m). The above results are derived from the disturbance effects due to the dynamic coupling and the disturbance rejection property of the system. Moreover, during the entire process, the closedloop gain-scheduled system is stable with the maximal rate of stiffness variation, which is one of the main results in the controller design. In this section, the controllers designed at endpoints of stiffness were used to construct the gain-scheduling control scheme. This proposition was determined to produce a stable plant in the range of stiffness variation. In the interval between minimal and maximal stiffness, the design of the linear controllers for a

Fig. 13. Experimental setup. (a) EMG-sensor setup. (b) Photograph of the experiments in operation: the subject attaches the orthosis and maintains a sitting position.

higher number of operating points can also improve the accuracy of the gain-scheduling control scheme. V. KNEE JOINT MOTION EXPERIMENT So far, the torque control testing has been conducted in the fixed-load cases. However, testing of the free-output cases is important for real applications. For this, we employed the gainscheduled torque controller to implement an impedance control with a human test person.1 The diagram of the above control system is shown in Fig. 12. As mentioned previously, an adjustable impedance is desirable for the training purpose. Therefore, this section highlights the verification of the rendered stiffness (i.e., the relationship between output force and input position) through the testing of leg-swing motion. Fig. 13 shows the experimental setup; it consists of the following two components: 1) In Fig. 13(a), an electromyography (EMG) sensor was designed to measure the muscle activation [36], where the raw signal from the user’s leg is processed by the sensor circuit (comprising an amplifier circuit, full-wave rectifier circuit, and low-pass filter circuit), then transferred into the end system by an analog-to-digital-converter. 1 The experiment with the human subject was reviewed by the Ethics Committee at the RWTH Aachen Faculty of Medicine, Aachen, Germany (reference number: EK 145/18).

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with coupled stability, the stiffness constraint provided in [24] was considered, i.e., the mechanical stiffness always needs to bound the virtual stiffness. For example, in the above-mentioned impedance control experiments, the adjusted mechanical stiffness (Kj = 350 N·m/rad) and virtual stiffness (KP s = 40 and 120 N·m/rad) satisfy the condition of stiffness constraint.

VI. CONCLUSION

Fig. 14. Measured stiffness and EMG signal in both experiments. The torque, angle, and EMG data were obtained from a single extension.

The aim of this setup is to set a benchmark when applying different impedances on the leg. Here, the EMG data were recorded from the vastus lateralis muscle (knee extensors). The muscle location is pointed out by the electrodes located on the right leg. Before analyzing the EMG signal, the measurement was postprocessed by a secondorder Butterworth low-pass filter with a cutoff frequency of 2 Hz. However, because normalization of the EMG amplitude was not performed, this setup is not suitable for comparison of measurements between subjects [37]. 2) During testing, the subject was asked to extend the knee to overcome the resistance [see Fig. 13(b)]. Two experiments were conducted by adjusting the virtual stiffness (i.e., KP s ) to 40 and 120 N·m/rad, respectively. Meanwhile, the reference of the joint angle (i.e., θj,r ef ) was set to be a constant value, the virtual damping (i.e., KD d ) was set to zero, and the actuator stiffness was set to 350 N·m/rad. Note that these two experiments were performed in succession, without changing the EMG setup (e.g., the electrodes) and within the same test person. Therefore, a comparison of the EMG amplitude is acceptable, albeit without normalization [37]. Fig. 14 depicts the experimental results. The measured EMG signal shows that the test person needs to increase effort on moving the leg when virtual stiffness is enlarged from 40 (red dashed line) to 120 N·m/rad (blue dashed line). With the proposed impedance control system, the measured stiffness can satisfy a definite precision, i.e., 41 N·m/rad (red solid line) and 122 N·m/rad (blue solid line), respectively. Note that the stiffness measurement results mentioned above were obtained from the fitted curve (green solid line). The experimental results in Fig. 14 are based on the stable interaction between the impedance-controlled compliant actuator and the human test person. For this type of coupled system, some studies also mathematically represent the stability conditions based on passivity criterion [23], [29], in which the proposed schemes are conservative in the case of interaction with humans. When designing our impedance control system

In this paper, the torque control problem in the VSA application was presented and associated with the MeRIA prototype. Based on the designed gain-scheduled LQGI controller, both the robustness and tracking performance of the torque control loop are recovered in the entire stiffness range. At the same time, the bandwidth has been maximized. Based on these results, the target for maximal improvement of the actuator performance has been achieved. The stability of the closed-loop gain-scheduled system has also been verified at the full speed of stiffness variation. A general method for stiffness regulation in the VSA-based exoskeleton research has also been proposed [19], in which the actuator stiffness was varied from large to small in order to coordinate it with the stance and swing phase during human walking. Accordingly, the designed gain-scheduled torque controller might be a suitable choice for future research on the MeRIA-based exoskeleton. The method of actuator application also needs to be confirmed together with the specific stiffness regulation. The control framework applied in Section V can be classified into a so-called passive training model, i.e., the robot helps the test person to track a predefined trajectory without considering the human intention [22]. However, the effects of the devices and the human body should also be taken into account for the rehabilitation training purposes. The next stage of our research focuses on a VSA-based biofeedback control.

APPENDIX A. Stiffness Identification The experiments were still conducted on the test bench (see Fig. 1), where we started by fixing the bending bars (or the motor side) and then applied a force at the load side to generate a bending movement. With the above method, the stiffness can be calculated by the ratio of the applied torque (recorded by the torque sensor) to the deflection angle (recorded by the encoder). The tests shown in Fig. 15 were repeated with an interval of 0.01 m between the effective length (see Fig. 2) from 0.044 (corresponding to the maximal stiffness 455 N·m/rad) to 0.134 m (corresponding to the minimal stiffness 201 N·m/rad). Then, we have the following stiffness fitting equation: Kj (θ2 ) = 1.824 × 104 · Le (θ2 )2 − 6065 · Le (θ2 ) + 686.4 (25) where Le (θ2 ) ([m]) is denoted as the effective length associated with the M2 positioning θ2 .


Fig. 15.

Stiffness fitting result for minimal and maximal stiffness.

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[19] M. Cestari, D. Sanz-Merodio, J. C. Arevalo, and E. Garcia, “An adjustable compliant joint for lower-limb exoskeletons,” IEEE/ASME Trans. Mechatronics, vol. 20, no. 2, pp. 889–898, Apr. 2015. [20] P. Beyl, M. Van Damme, R. Van Ham, B. Vanderborght, and D. Lefeber, “Pleated pneumatic artificial muscle-based actuator system as a torque source for compliant lower limb exoskeletons,” IEEE/ASME Trans. Mechatronics, vol. 19, no. 3, pp. 1046–1056, Jun. 2014. [21] V. Grosu, C. Rodriguez-Guerrero, S. Grosu, B. Vanderborght, and D. Lefeber, “Design of smart modular variable stiffness actuators for roboticassistive devices,” IEEE/ASME Trans. Mechatronics, vol. 22, no. 4, pp. 1777–1785, Aug. 2017. [22] W. Meng, Q. Liu, Z. Zhou, Q. Ai, B. Sheng, and S. Xie, “Recent development of mechanisms and control strategies for robotassisted lower limb rehabilitation,” Mechatronics, vol. 31, pp. 132–145, 2015. [23] J. F. Veneman, R. Ekkelenkamp, R. Kruidhof, F. C. T. van der Helm, and H. van der Kooij, “A series elastic-and bowden-cable-based actuation system for use as torque actuator in exoskeleton-type robots,” Int. J. Robot. Res., vol. 25, no. 3, pp. 261–281, 2006. [24] H. Vallery, J. Veneman, E. van Asseldonk, R. Ekkelenkamp, M. Buss, and H. van Der Kooij, “Compliant actuation of rehabilitation robots,” IEEE Robot. Automat. Mag., vol. 15, no. 3, pp. 60–69, Sep. 2008. [25] B. J. E. Misgeld, M. Kramer, and S. Leonhardt, “Multivariable friction compensation control for a variable stiffness actuator,” Control Eng. Practice, vol. 58, pp. 298–306, 2017. [26] L. Liu, S. Leonhardt, and B. J. E. Misgeld, “Design and control of a mechanical rotary variable impedance actuator,” Mechatronics, vol. 39, pp. 226–236, 2016. [27] G. Wyeth, “Control issues for velocity sourced series elastic actuators,” in Proc. Austral. Conf. Robot. Automat., 2006, pp. 1–6. [28] F. Sergi, D. Accoto, G. Carpino, N. L. Tagliamonte, and E. Guglielmelli, “Design and characterization of a compact rotary series elastic actuator for knee assistance during overground walking,” in Proc. IEEE RAS EMBS Int. Conf. Biomed. Robot. Biomechatronics, 2012, pp. 1931–1936. [29] N. L. Tagliamonte and D. Accoto, “Passivity constraints for the impedance control of series elastic actuators,” Proc. Inst. Mech. Eng. I, J. Syst. Control Eng., vol. 228, no. 3, pp. 138–153, 2014. [30] M. Athans, “The role and use of the stochastic linear-quadratic-Gaussian problem in control system design,” IEEE Trans. Autom. Control, vol. 16, no. 6, pp. 529–552, Dec. 1971. [31] D. J. Stilwell and W. J. Rugh, “Stability preserving interpolation methods for the synthesis of gain scheduled controllers,” Automatica, vol. 36, no. 5, pp. 665–671, 2000. [32] J. Choi, S. Hong, W. Lee, S. Kang, and M. Kim, “A robot joint with variable stiffness using leaf springs,” IEEE Trans. Robot., vol. 27, no. 2, pp. 229–238, Apr. 2011. [33] M. A. Johnson and M. H. Moradi, “Some PID control fundamentals,” in PID Control: New Identification and Design Methods. London, U.K.: Springer-Verlag, 2005. [34] S. Skogestad and I. Postlethwaite, Multivariable Feedback Control: Analysis and Design, 2nd ed. Hoboken, NJ, USA: Wiley, 2005. [35] J. Doyle and G. Stein, “Multivariable feedback design: Concepts for a classical/modern synthesis,” IEEE Trans. Autom. Control, vol. 26, no. 1, pp. 4–16, Feb. 1981. [36] M. B. I. Raez, M. S. Hussain, and F. Mohd-Yasin, “Techniques of EMG signal analysis: Detection, processing, classification and applications,” Biol. Procedures Online, vol. 8, no. 1, pp. 11–35, 2006. [37] M. Halaki and K. Ginn, “Normalization of EMG signals: To normalize or not to normalize and what to normalize to?” in Computational Intelligence in Electromyography Analysis—A Perspective on Current Applications and Future Challenges. Rijeka, Croatia: InTech, 2012.

Lin Liu was born in Tianjin, China, in 1986. He received the Bachelor’s and Master’s degrees in mechanical engineering from the Southwest Jiaotong University, Chengdu, China, in 2009 and 2013, respectively. He is currently working toward the Ph.D. degree at the Philips Chair for Medical Information Technology, RWTH Aachen University, Aachen, Germany. His current research interests include the areas of mechatronic systems within the context of compliant actuators and rehabilitation robotics.

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Steffen Leonhardt (SM’06) was born in Frankfurt, Germany, in 1961. He received the M.S. degree in computer engineering from State University of New York System at Buffalo, NY, USA, the Dipl.-Ing. degree in electrical engineering and the Dr.-Ing. degree in control engineering from the Technische Universitat ¨ Darmstadt, Darmstadt, Germany, and the M.D. degree in medicine from J. W. Goethe University, Frankfurt, Germany. He became a Full Professor and the Head of the Philips endowed Chair of Medical Information Technology, RWTH Aachen University, Germany, in 2003. His research interests include physiological measurement techniques, personal health care systems, and feedback control systems in medicine. Dr. Leonhardt is an Associate Editor of the IEEE JOURNAL OF BIOMEDICAL AND HEALTH INFORMATICS, the IEEE TRANSACTIONS ON BIOMEDICAL CIRCUITS AND SYSTEMS, the Journal of Clinical Monitoring and Computing, and Biomedical Engineering/Biomedizinische Technik.

Berno J. E. Misgeld (M’12) was born in Euskirchen, Germany, in 1979. He received the Dipl.-Ing. (FH) degree in electrical and automation engineering from the University of Applied Sciences, Aachen, Germany, and the M.Sc. degree in informatics and control from Coventry University, Coventry, U.K., in 2003, respectively. He received the Dr.-Ing. degree in biomedical and control engineering from Ruhr-University Bochum, Bochum, Germany, in 2007. From 2006 to 2011, he was a Research and Development Engineer for guidance and flight control systems with Diehl-BGT-Defence, Ueberlingen, Germany. Since 2011, he has been a Senior Scientific Engineer in Biomechatronical Systems and Rehabilitation Robotics with the Chair of Medical Information Technology, RWTH Aachen University, Aachen, Germany. His research interests include feedback control and signal processing with application to biomedical systems, robotics and medicine.