Augmented Virtual Stiffness Rendering of a Cable-driven SEA for Human-Robot Interaction

Ⅰ. INTRODUCTION

Human-centered robotic systems have been rapidly advanced in recent years and widely applied in various applications and human-robot interaction is of fundamental importance [1]. These systems are usually designed to operate in direct contact with humans or to perform human-like tasks by interacting with unstructured environments. For example, human-centered robots have attracted huge research attention in rehabilitation robotics, such as the MIT-MANUS [2], the ARMin [3], the RiceWrist [4], etc. For these applications to interact with the human upper limb, safety is critically important and accurate impedance rendering is also beneficial.

The elastic component was introduced into actuation by Pratt and Williamson twenty years ago [5]. Afterwards, series elastic actuators came to be popular. They offer a number of advantages, including greater shock tolerance, more accurate and stable force control, the capacity for energy storage, lower reflected inertia, and less inadvertent damage to the human or environment. Variously adapted series elastic actuator (SEAs) have appeared for different applications, e.g., a non-back drivable SEA for prosthesis [6], a compact rotary SEA for knee assistance [7], an exotendon-driven rotary SEA for human-robot interaction [8], etc.

There exists a number of cable-driven devices for human-robot interaction, for example, the STRING-MAN [9], the lower extremity powered exoskeleton (LOPES) [10], the universal haptic drive (UHD) [11], etc. The force is generated by cable tension and transmitted to another joint or target place. A cable can intentionally and easily change the torque direction, which allows detachment of the motor from the robot location and enables power transmission to remote distance. A lightweight cable for transmission means low inertia, less energy loss and space occupation [12], [13]. Besides, due to its properties of unidirectional force endurance and breaking apart when the tension exceeding the threshold, cable actuation can be made as a safe solution for robots interacting with human.

In order to achieve compliant interaction between human and robot, impedance control, which was first described by Hogan in [14], is the fundamental approach to shape a given system's behavior to match a predefined dynamic model. Many impedance control approaches have been proposed and the key differences lie on the inner torque controllers. For example, in [15]$-$[19], pure PID (proportional-integral-derivative) controllers have been used for torque control. In [20]$-$[24], disturbance observer (DOB) based control methods have been employed to improve robustness and impedance rendering accuracy. In [25]$-$[29], adaptive controllers based on neural network have been designed to provide good dynamic and robustness performance. Cascaded impedance control structures wrapping torque control around an inner velocity or position loop [15]$-$[17], have also achieved good performance.

However, stable torque control does not suffice for human robot interaction, and the system must guarantee passivity in the presence of uncertain contact dynamics. By detailed analysis of dynamically interacting systems, Colgate introduced passivity as the interaction stability constraint and derived the necessary and sufficient conditions for passive interaction with any external environment [30].

Both low stiffness for good transparency and high stiffness bigger than the physical spring constant are required for high-fidelity human-robot interactive tasks. However, this is challenging under the passivity constraint [17], which leads to a conservative impedance controller. For that, the passivity constraints can be relaxed by putting a phase lead-lag compensator into the stiffness control structure to achieve coupled stability [17], [31].

In this paper, we address the stiffness rendering problem for a cable-driven SEA system to interact with the human arm. The aims are to achieve low stiffness for good transparency as well as high stiffness bigger than the physical spring constant, to render the virtual stiffness with good accuracy and to assess the rendering accuracy with quantified metrics. We have employed the 2-DOF control method together with a compensator to handle the competing requirements on tracking performance, noise and disturbance rejection, and energy optimization for the cable-driven SEA system. By adding a stiffness compensator into the impedance control structure, the virtual stiffness has been rendered in the extended range of 0.1 to 2.0 times of the physical spring constant with guaranteed relaxed passivity. Quantified metrics in both the time and frequency domain have verified good rendering accuracy.

The paper is organized as follows. Section Ⅱ gives the problem formulation. Section Ⅲ presents the comprehensive methods on controller design and quantified metrics for rendering assessment. Extensive simulation and experimental results are shown in Section Ⅳ. Finally, Section V concludes the paper.

Ⅱ. PROBLEM FORMULATION

The concept of a cable-driven series elastic actuator interacting with human is illustrated in Fig. 1. The DC motor is characterized by inertia $J_A$ and damper $b_f$, and the spring constant is $K_s$. The motor produces the torque $\tau_A$, and the output torque is $\tau_o$. The motor velocity is $\omega_A$ and it leads to the cable displacement $\varphi_A$. When interacting with the human hand, the resulted motion is denoted as $\varphi_L$ and the interaction torque is represented as $\tau_L$.

Figure  1.  Human-robot interaction with a cable-driven series elastic actuator.

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We assume that the inertia of the cable and spring can be ignored, and thus $\tau_o = \tau_L$. Without taking into account the nonlinear factors, the system dynamics can be characterized by the following equations:

For an interaction system with the human, it is critically important to regulate the dynamic relationship between the motion $\varphi_L$ and interaction torque $\tau_L$ in a desired way. Define

and

The problem is to design the control strategy such that the rendered virtual stiffness $K(s)$: 1) covers a wide range, with lower stiffness to achieve good transparency and high stiffness that can be bigger than the physical spring constant $K_s$, 2) is accurate, and 3) satisfies the passivity requirement for stable human-robot interaction.

Ⅲ. METHODS

A cascaded impedance, torque and velocity control structure is illustrated in Fig. 2. This hierarchical control strategy provides a purposeful and convenient way for analysis of human-robot interaction, and has been used in many previous works with PID subcontrollers [15]$-$[17].

Figure  2.  Diagram of a cascaded impedance, torque and velocity control approach. $\varphi_{L, d}$ is reference position. $C_I$ is the impedance controller that generates the desired torque $\tau_d$. $C_\tau$ denotes the torque controller that generates the desired velocity $\omega_d$. $C_v$ represents the velocity controller that generates the motor torque $\tau_A$. $G_M$ is the motor model from the torque $\tau_A$ to motor velocity $\omega_A$.

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We approach the control problem on the basis of this structure. Instead of PID controllers, we use the 2-DOF control method and a compensator for the inner torque control, and put a phase-lead compensator into the impedance controller to augment the stiffness rendering capability of the system. In our previous works [32], [33], we have demonstrated advantages of the 2-DOF method over the PI method for the torque control of a cable-driven SEA. The 2-DOF torque control method performed better than the PI method in the presence of noise, disturbance and motor saturation.

A The Cable-Driven SEA With Velocity-sourced Motor

A velocity controlled motor model has been taken for the cable-driven SEA system as shown in Fig. 3. The inner velocity loop helps to track the reference velocity signal accurately and quickly, as well as to overcome possible undesirable and unpredictable system nonlinearities.

Figure  3.  Diagram of the cable-driven SEA with velocity-sourced motor.

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The dynamics from the desired velocity $\omega_d$ and motion $\varphi_L$ to the output torque $\tau_L$ can be derived as

where

and

B Parameterization of All Stabilizing 2-DOF Controllers

A closed loop system using the 2-DOF configuration to stabilize the plant $P(s)$ is shown in Fig. 4 [34]. It has two inputs $r$ and $y$ instead of one input in the 1-DOF control structure and $C(s)$ can be written as $C(s) = \left[{\begin{array}{*{20}{c}} {{C_1}(s)}&{{C_2}(s)} \end{array}} \right]$. It can independently deal with the tracking and disturbance or noise rejection requirements, and achieve a zero overshoot step response for $P(0)\ne0$.

Figure  4.  The 2-DOF control configuration.

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Let

where $a(s)$ and $b(s)$ are coprime and $a_0\ne0$. Let

where $p(s)$ and $q(s)$ are coprime and $p_0\ne0$, be any stabilizing controller. Thus, $c(s) = a(s)p(s) + b(s)q(s)$ is stable. Factorize $c(s)$ as $c(s)=f(s)h(s)$ such that deg $f(s)=n$ and deg $h(s)=m$. Let

Then, $M(s), N(s), X(s)$ and $Y(s)$ are all stable transfer functions satisfying

Theorem 1: The set of all stabilizing 2-DOF controllers that give stable closed-loop systems can be given by the Youla-Kucera parameterization

where $Q_1(s)$ and $Q_2(s)$ are two arbitrary stable functions [35].

Thus, if one stabilizing controller was found, then all other stabilizing controllers can be obtained.

Systematic design procedures for stabilizing 2-DOF controller are given as the following steps [36]:

Step 1: Denote $P(s)$ as

where $a(s)$ and $b(s)$ are coprime and $a_0\ne0$.

Step 2: Find a stable polynomial $d_\rho(s)$ such that

where $\rho$ is a positive number used to give a relative weight to $u(t)$ and tracking error $e(t)$.

Step 3: Find a stable polynomial $d_{\lambda, k}(s)$ such that

where $\lambda$ is a positive number used to give a relative weight to $d(t)$ and $r(t)$, and $k$ is a positive number used to give a relative weight to $n(t)$ and $r(t)$.

Step 4: The feedback part of the optimal controller ${C_2}(s) = {q(s)}/{p(s)}$ with the same order as the plant is the unique strictly proper and type 0 pole-placement controller such that

Step 5: The feedforward part of the optimal controller is

Step 6: The optimal performance index is given by

If $\rho<1$, the controller will put more emphasis on $e(t)$ than $u(t)$, which means that the system will achieve better tracking performance with more energy, and vice versa. If $\lambda > k$, the controller will put more emphasis on $d(t)$ than $n(t)$, which means that the system will achieve better disturbance rejection, otherwise, it achieves better noise rejection. It provides us a way to adjust competing performance requirements between reference tracking, disturbance rejection and energy consumption minimization by the parameters $\rho$, $\lambda$ and $k$ to get a desired performance.

This control strategy provides a feasible and convenient way to independently deal with competing requirements of reference tracking, disturbance or noise rejection for the cable-driven series elastic actuator. It also can obtain an intrinsic stable system since only the products and sums of stable transfer functions are involved, which benefits the stability of stiffness control.

C Torque Control With the 2-DOF Method

Series elastic actuators are originally designed to pursue accurate and stable torque or force control. For this reason, it is natural to utilize a torque or force control loop that facilitates the stiffness control performance.

To enhance stability, robustness and control accuracy, a stabilizing 2-DOF controller is adopted for torque regulation, as shown in Fig. 5. To realize accurate torque control under motion $\varphi_L(s)$, there are two steps to follow.

Figure  5.  Torque control with the 2-DOF method. $C_\tau$ denotes the 2-DOF controller with subcontrollers $C_1(s)$ and $C_2(s)$. Disturbance $d$, noise $n$ and motor saturation can be taken into consideration. (a) The torque controller is designed under the condition of $\varphi_L(s)=0$. (b) The compensator $C_L(s)$ is added to minimize the effects caused by the interactive motion.

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Firstly, we let $\varphi_L(s)=0$ and follow the design procedure to obtain the 2-DOF controller $C_{\tau}(s)$ as shown in Fig. 5(a). This gives

where

Secondly, we need to design a compensator $C_L(s)$ to compensate for the effects caused by interactive motion $\varphi_L(s)$ as shown in Fig. 5(b). Let:

Then, there is

For an ideal torque controller $C_\tau(s)$, there is $G_1(s)\simeq 1$, and (20) can be simplified as ${\tau _L}(s) \simeq {\tau _d}(s)$. Thus, exact torque control can be achieved with a 2-DOF controller $C_\tau(s)$ and a feedforward compensator $C_L(s)$. Also, the disturbance $d$ and noise $n$ can be rejected by the 2-DOF controller $C_\tau(s)$, and the system remains stable.

D Stiffness Control Under Relaxed Passivity Constraints

The initial stiffness rendering structure based on the 2-DOF torque control is shown in Fig. 6. The desired virtual stiffness $K_d(s)$ is proportional to the physical spring stiffness $K_s$, i.e., $K_d(s)=I_d K_s$. Without loss of generality, assume that the reference position $\varphi_{L, d}=0$, and there is

Figure  6.  Initial architecture of stiffness control for the cable-driven SEA. $K_d$ is the desired virtual stiffness.

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For a system interacting with human or unknown environment, there are two necessary and sufficient conditions for $Z(s)$ as defined in (4), which should be satisfied to guarantee passivity [30]:

1) $Z(s)$ must be stable such that $Z(s)$ has no poles in the right-half of the complex $s$-plane.

2) Re$(Z(j\omega))\geq0$ for all $\omega$ for which $j\omega$ is not the pole of $Z(s)$.

The first constraint has been satisfied intrinsically by the 2-DOF method based stiffness control structure. For the second constraint, it is also equivalent to bounding the phase of $Z(s)$ to be in the range of $[-90^\circ, \, 90^\circ]$. However, this constraint is very conservative and excessively restrains the performance.

As indicated in [17], the system is not passive if it is required to display a virtual pure stiffness higher than the physical spring constant only through cascaded torque-velocity control. This unexpected phenomenon is also true in the 2-DOF method based impedance control structure when examining the bode plot as shown in Fig. 7.

Figure  7.  A case of rendering a pure stiffness $2K_s$ without stiffness compensation. The phase has values below $-90^\circ$ when frequency changes from 1 rad/s to 600 rad/s, thus the system is not passive. The max phase lag is $32^\circ$ at the frequency $75$ rad/s (about 12 Hz).

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To realize a virtual pure stiffness higher than its physical spring stiffness, a phase lead compensator $C_z$ is added into the stiffness control structure as presented in Fig. 8. It provides a phase lead $\phi_d$ dragging the phase of $Z(s)$ to the bounds of $[-90^\circ, +90^\circ]$ at the specific frequency. A pure derivative regulator can also bring a lead effect, but it may cause system instability.

Figure  8.  High stiffness rendering structure with a compensator $C_Z(s)$.

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Let

The phase of $C_z(j\omega)$ with regard to frequency $\omega$ is

This implies that $p>z$ must be satisfied to have a phase lead effect. The magnitude of $C_z(j\omega)$ with regard to frequency $\omega$ is

For human upper limb motion, we can limit the movement frequency to be below 5 Hz [23], [37] and focus more on performance. In order to choose a suitable compensator $C_z(s)$, $A(\omega)$ and $\phi(\omega)$ at $\omega=20\pi$ rad/s for each group of $p$ and $z$ can be plotted, as shown in Fig. 9.

Figure  9.  (a) and (b) are the plots of magnitude $A(j\omega)$ and phase $\phi(j\omega)$ regarding $p$ and $z$ for $\omega=20\pi$ rad/s.

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E Metrics for Performance Assessment

Quantified metrics are needed to assess the rendering quality of the control strategy. Let the control error be

where $K_a$ is the system actual rendered stiffness.

In the time domain, the correlation coefficient, maximum tracking error $T_m$ and mean squared torque tracking error $T_2$ can be examined to quantify the accuracy of tracking an input signal. But this quantitative method cannot provide a perspective to assess the rendering accuracy in frequency domain. Additionally, if the actual stiffness $K_a$ is directly calculated according to $\tau_L/(-\varphi_L)$ in the time domain, the denominator may be zero. For these two reasons, two other metrics based on $H_\infty$ and $H_2$ norm in the frequency domain have been taken to assess the accuracy and to avoid zero denominator [38]. The error frequency response is

where the actual stiffness response $K_a(s)$ from signal $-\varphi_L$ to signal $\tau_L$ can be estimated by MATLAB function "spafdr".

Then, the $H_\infty$ metric with respect to the system frequency response is defined as

The $H_2$ metric with respect to the system frequency response is defined as

The $M_\infty$ metric offers the worst matching difference between the system desired and actual frequency responses, while the $M_2$ metric provides a global matching error across all frequencies. The worst matching frequency is denoted as $\omega_\infty$. It indicates a better rendering accuracy if smaller $M_\infty$ and $M_2$ are achieved. Actually, we only need to calculate the two metrics in the frequency range of interest $\Omega=\{1, 2, \ldots, 100\}$ rad/s. The two metrics can be replaced by the following:

Ⅳ. EXPERIMENTS AND RESULTS

A The Experimental Platform

The concept of our cable-driven series elastic actuation system is presented in Fig. 10. It is a 1-DOF platform designed for physical human-robot interaction [39]. The realized prototype is shown in Fig. 11.

Figure  10.  The structure of our 1-DOF cable-driven SEA platform for physical human-robot interaction: ① cable for transmission; ② DC motor, including a reduction gear head and a rotary encoder; ③ linear springs as the elastic components; ④ handle for human-robot interaction; ⑤ sliding guide; ⑥ magnetic linear encoder to measure the displacement of the slider, including a reading head and a magnetic tape attached parallel to the sliding guide; ⑦ cable winch; ⑧ pulleys for cable transmission and redirection.

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Figure  11.  The prototype of the 1-DOF cable-driven SEA platform: ① servo controller for the DC motor; ② data acquisition board; ③ Simulink Real-Time target computer; ④ scopes to display the data in real-time; ⑤ host computer.

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A brush rotary DC motor (Maxon RE30 60 W 24 V) is used as the velocity source. The motor has a MR-228452 500CPT incremental encoder mounted and a GP32A 14:1 planetary gear head installed on its shaft. The motor velocity is managed by a servo controller (Escon 50/5 409510). Both sides of the handle are connected to the motor in a cable-spring series structure. The handle is mounted at a slider and can move along the guide by cable transmission or human arm motion. The two cables are redirected by the pulleys and wrapped around the winch. The two springs have an initial length of 87.7 mm, and can be extended to the maximal length of 224.7 mm. They are pretensioned to half of their maximum displacement range. There is also a slider at each conjunction of the springs and the cables. The displacement of the handle and the deformations of the two springs are measured by three magnetic linear encoders (MLS105) mounted at each slider, with the precision of 5 $\mu$m. The length of the sliding guide is 60 cm, which is sufficient for the linear movement range of the upper limb.

Control of the actuator and data acquisition of the sensors are realized by a MATLAB/Simulink real-time target system running on a standard computer. A data acquisition board (HUMUSOFT MF634 PCI-express multifunction I/O card) is inserted into the PCI-Express x1 slot of the target computer to send the velocity command as analog signals to the motor servo controller and receive the encoder readings. The control algorithm is implemented with MATLAB/Simulink, compiled in the host computer, downloaded to the target computer and is run in real-time.

To build a precise motor model for the simulation, the parameters $J_A$ and $b_f$ were identified using MATLAB system identification toolbox. Since $\tau_A$, $\tau_L$, $\omega_A$ can be obtained, the signals $\tau_A-\tau_L$ and $\omega_A$ can be used to estimate these parameters.

All parameters of the cable-driven series elastic actuator are shown in Table Ⅰ. To achieve a better velocity tracking performance, the velocity controller parameters $K_{pv}, K_{iv}$ were tuned as 0.0457 Nm/(rad/s) and 1.3455 Nm/(rad/s). Taking those parameters into (6), there is

Table  Ⅰ.  THE CABLE-DRIVEN SEA PARAMETERS

Description	Value
Reflected total inertia $J_A$	$6.90\times {10^{ - 4}}\, {\rm{kg}} \cdot {{\rm{m}}^{\rm{2}}}$
Coefficient of viscous friction $b_f$	${\rm{0}}{\rm{.0059\, Nm/(rad/s)}}$
Stiffness of double spring $K_s$	$2 \times 0.0242\, {\rm{Nm/rad}}$
Radius of cable winch $r$	$7.25\, {\rm{mm}}$
Ratio of gear head $K_g$	$14:1$

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Following the design procedure of the stabilizing 2-DOF controller, the parameters $\rho, \lambda, k$ are 0.0005, 1, and 1. Then

By referring to Fig. 9, $p=100, z=50$ with a max phase lead of $18^\circ$ and a magnitude of 3 dB at the frequency of 11 Hz as presented in Fig. 12 were chosen for $C_z(s)$.

Figure  12.  The bode plot of $C_z(s)$ for $p=100, z=50$.

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B Simulations and Results

To validate the phase lead effect of the stiffness compensator $C_z(s)$, a group of simulations with desired stiffness $K_d$ varying from $0.2K_s$ to $1.8K_s$ were conducted and compared between the compensated and the uncompensated stiffness rendering method for the cable-driven SEA system. When $I_d$ are 1.1 and 1.4, the bode plots of simulation results are shown in Fig. 13. It is obvious that the compensated systems are all stable and act closely to the desired pure spring at low frequencies.

Figure  13.  Comparison results of the system's actual rendered stiffness $Z(s)$ between the compensated method and the uncompensated method. The phase of each case has been improved to over $-90^\circ$ noticeably and the magnitude has changed less than 1 dB at low frequencies.

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During the next simulation, the proposed compensated 2-DOF stiffness rendering method is applied to control the cable-driven SEA system with respect to nonlinear factors of disturbance, noise and motor saturation.

The responses to the same sinusoidal reference signal with a frequency of 2 Hz for different desired stiffness $0.5K_s, 1.2K_s$ and $1.8K_s$ are shown in Fig. 14. They all tracked the reference in a satisfactory way and obtained a slight tracking error. With the increase of the desired stiffness, the actual torque comes closer to the desired torque.

Figure  14.  Responses to sinusoidal signal when rendering different desired stiffness with our method. The actual output torque is $\tau_L$, and the desired torque $\tau_d$ is computed by $-{\varphi_L}{K_d}$. A closer match result of the actual and the desired signal means a more accurate stiffness rendering.

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Furthermore, five quantified metrics including the correlation coefficient, maximum torque tracking error $T_m$, mean squared torque tracking error $T_2$, $M_\infty$ and $M_2$ are all calculated and compared with a classic control approach, which used PID controller. The quantified results are listed in Table Ⅱ. For all five metrics, the 2-DOF method performed better for different stiffness rendering.

Table  Ⅱ.  SIMULATED RESULTS OF QUANTIFIED METRICS TO EVALUATE STIFFNESS RENDERING ACCURACY FOR THE COMPENSATED 2-DOF METHOD AND COMPENSATED PI NETHOD

	$0.5K_s$		$1.0K_s$		$1.5K_s$		$2.0K_s$
	2-DOF	PI	2-DOF	PI	2-DOF	PI	2-DOF	PI
Correlation coefficient	0.9662	0.9461	0.9976	0.9857	0.9998	0.9685	0.9991	0.9502
$T_m$(Nm)	0.0606	0.0819	0.0365	0.0888	0.0208	0.1480	0.0422	0.2236
$T_2$(Nm)	0.0338	0.0371	0.0178	0.0319	0.0096	0.0788	0.0216	0.1306
$M_\infty$ (Nm/rad)	0.0172	0.0477	0.0093	0.0743	0.0030	0.0871	0.0097	0.0459
$M_2$(Nm/rad)	0.1301	0.2391	0.0688	0.2794	0.0204	0.3894	0.0741	0.3831
$\omega_\infty$(Hz)	5.0	6.6	5.0	7.5	6.6	10.0	5.7	6.6

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It should be noted that the noise signal $n$ and disturbance signal $d$ were both white Gaussian noise with variances of 1 and 0.1, respectively, which were more challenging than in the practical system. The frequencies at which the maximal error magnitude reached were all above 5 Hz. It is clear that the 2-DOF method based compensated stiffness rendering approach shows superior performances of tracking and robustness.

C Experiments and Results

To further verify the performances of the compensated 2-DOF method, several experiments have been conducted on the test platform shown in Fig. 11. When the handle was driven by human hand along the guide, the results of the motion $\varphi_L$, the actual output torque $\tau_L$ and the desired output torque $\tau_d$ for different desired stiffness are shown in Fig. 15. The bode plots of the actual stiffness response analyzed by MATLAB function "spafdr" and the desired stiffness response are presented in Fig. 16. The five quantified metrics in these experiments are listed in Table Ⅲ. Different virtual impedances varying from $0.1K_s$ to $2.0K_s$ are all verified.

Figure  15.  Experimental results of tracking the desired torque generated by human hand motion for different desired stiffness rendering.

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Figure  16.  Bode plots of system stiffness $K(s)$. $K_d(s)$ is the desired virtual stiffness. $K_a(s)$ is the system actual stiffness response from $-\varphi_L$ to $\tau_L$ in experiment, which is analyzed by MATLAB function "spafdr". Although there are some discrepancies at high frequency, but we are only interested in the response at low frequency.

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Table  Ⅲ.  EXPERIMENTAL RESULTS OF QUANTIFIED METRICS TO EVALUATE STIFFNESS RENDERING ACCURACY FOR THE COMPENSATED 2-DOF METHOD FOR DIFFERENT DESIRED STIFFNESS RENDERING (FROM $0.1K_s$ TO $2.0K_s$)

	$0.1K_s$	$0.2K_s$	$0.3K_s$	$0.5K_s$	$0.7K_s$	$0.9K_s$	$1.1K_s$	$1.3K_s$	$1.5K_s$	$1.7K_s$	$1.9K_s$	$2.0K_s$
Correlation Coefficient	0.7046	0.9124	0.9455	0.9926	0.9950	0.9973	0.9997	0.9997	0.9994	0.9986	0.9981	0.9979
$T_m$ (Nm)	0.0153	0.0202	0.0164	0.0173	0.0130	0.0115	0.0087	0.0087	0.0074	0.0081	0.0103	0.0078
$T_2$(Nm)	0.0075	0.0084	0.0076	0.0075	0.0065	0.0060	0.0050	0.0047	0.0041	0.0036	0.0037	0.0036
$M_\infty$ (Nm/rad)	0.0188	0.0147	0.0159	0.0117	0.0163	0.0206	0.0083	0.0103	0.0128	0.0145	0.0110	0.0394
$M_2$ (Nm/rad)	0.0658	0.0569	0.0539	0.0492	0.0874	0.0701	0.0407	0.0434	0.0517	0.0603	0.0659	0.1209
$\omega_{\infty}$ (Hz)	5.8	3.9	8.6	4.4	6.4	7.0	6.5	6.4	7.1	11.5	6.4	10.0

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The output torque tracked the desired torque with little error for different virtual stiffness. The correlation coefficients were all above $0.9$ except for the worst case $0.7046$ for $K_d=0.1K_s$, and the $M_2$ and $M_\infty$ were congruously small for all $K_d$, which indicates good tracking performance. The discrepancies between the actual and the desired stiffness response were all small at low frequencies. Passive interactions have been achieved for $K_d\leq2K_s$ at low frequency, and the system did not show any unstable oscillations when interacting with human hand.

Ⅴ. CONCLUSION

In this work, we addressed the stiffness rendering problem for a cable-driven SEA system to interact with the human. A cascaded velocity-torque-impedance control structure was established by taking the motor as a velocity-source. The 2-DOF control method together with a compensator were implemented to handle the competing requirements on tracking performance, noise and disturbance rejection, and energy optimization to achieve high fidelity torque control in cable-driven SEA system. By adding a phase-lead compensator into the impedance controller, the stiffness rendering capability has been augmented with guaranteed relaxed passivity, and thus broke the conservative limitations by the conventional passivity requirement. In this way, the rendered stiffness can go higher than the physical spring constant or go low to achieve good transparency. Extensive simulations and experiments were performed, and the virtual stiffness has been rendered in the extended range of 0.1 to 2.0 times of the physical spring constant with guaranteed relaxed passivity for physical human-robot interaction below 5 Hz. The rendering accuracy was verified by quantified metrics.