Adaptive Time Delay Impedance Control of Robot Manipulator via Voltage-Based Motor Control

Abstract

To accommodate the contact force between a robot and its environment, this paper presents an adaptive control framework for the impedance control of a manipulator with time delay estimation (TDE). To simplify the complex system model and yield adaptive feedback compensation, a voltage-based motor control approach was presented. Compared to the torque-based control model, the voltage-based control model is computationally more efficient and practically feasible. The proposed adaptive law was designed to compensate for the errors produced by the TDE. Through a stability analysis, the control framework was verified by semi-global uniform ultimate boundedness (SGUUB) stability. Experimental results are discussed, and the effectiveness of the adaptive control framework is demonstrated.

1. Introduction

One basic objective of a robot manipulator is to assist humans in performing a variety of tasks in complicated industrial processes, such as assembling and drilling [1,2], where a robot usually makes contact with the environment. Hence, the control of the contact force is important for the manipulator to realize efficient and rapid tracking performance [3,4]. However, owing to unknown external disturbances and modeling uncertainties, it was difficult to accurately follow the desired paths when the robot responded to the force applied from the environment.

To avoid inaccurate and unstable motions, various robust methods have been proposed for inaccurate and unstable motions [5,6]. Adaptive robust control methods were designed to compensate for environmental uncertainties [7,8]. In [9], the authors proposed a novel contact force/motion control strategy. The control tasks of the manipulator joints and valve-controlled cylinders were decoupled using a hierarchical decoupling controller. However, the establishment of joint models is complex. Hence, to maintain the stability of the contact force for the robot manipulator, an impedance control method was used to control the contact force of the manipulator [10]. In [11], the authors developed a robust adaptive model reference impedance controller for an n-link robotic manipulator. To reduce the calculation complexity, an adaptation law for robot parameter estimation was designed in the joint space. However, an accurate model of the contact force has not been proposed. In fact, it can usually be modeled as a mass-spring-damper system for when the robot makes contact with the environment [12].

To address the aforementioned challenges, some adaptive control approaches have been developed to perform the accurate impedance control of robots, such as sliding-mode control (SMC) [13,14], time delay control (TDC) [15,16], and neural network control [17,18]. SMC was found to be more robust to nonlinear uncertainties and unknown disturbances in systems [19,20]. When the switching gains of the SMC are beyond the upper limit of the nonlinear uncertainties, accurate motion control can be achieved. However, the upper bound was typically not known. The large difference between the switching gains and upper bound caused severe chattering [21,22]. The robot oscillates around the sliding manifold, leading to serious problems such as energy loss and imperfect tracking performance. The TDE was presented to compensate for the coupling dynamics and unknown disturbances [23,24]. The TDC was an easy control design that has been widely used in automatic pilot and robot control [25].

However, a TDE error still existed and led to the degradation of the control performance. To eliminate the TDE error, the proper control gains of TDC should be chosen [26,27]. Whether they are too large or too small, unsuitable control gains can weaken the controller’s performance. To decrease the efforts required to achieve suitable gains, researchers have employed neural networks to compensate for the uncertainty of dynamics [28]. Researchers have introduced a supervisory fuzzy sliding mode controller for surgical robots [29]. Supervisory fuzzy control was developed to dynamically adjust the output gains of the controller in real time, thereby effectively reducing the time required for gain tuning. The TDE error could be eliminated within a predefined control gain scope.

To simplify the control structure, voltage-based motor control was used to eliminate the complexities of the motor dynamics. Torque-based motor control is a common approach for robot control in most applications. However, most torque-based controllers usually omit the dynamics of the motor, which are important for high-speed tasks. Owing to the nonlinear uncertainties and unknown disturbances of the motor, it is more complex to develop a torque-based controller than a voltage-based one. Hence, a more direct method of using voltage as a signal to control motors has been employed. Compared to torque-based motor control, voltage-based motor control is more computationally simple and realistic. In [30], the authors investigated a voltage-based current estimation approach for speed control of DC (direct current) motor systems. This estimation approach was successfully used to achieve the precise speed control of DC motor systems. However, the proposed control method relied on establishing an accurate voltage-based current model. In [31], the authors proposed an adaptive voltage controller for three-wheeled omni-directional robots driven by DC motors. The proposed controller ensured accurate path tracking. However, the input voltage was dynamically limited based on real-time wheel and ground traction force estimates. In [32], the authors proposed a voltage-based impedance control method for mobile manipulators with a TDE. Compared to the usual current-based control method, the proposed method was computationally simpler, faster, and more robust, with negligible tracking error. However, the TDE error was not considered, which led to an inaccurate tracking performance.

The main aim of this study is to develop an adaptive control approach for the impedance control of a robot manipulator. The proposed method uses the TDE to construct an easy-to-control framework. Voltage-based motor control is computationally simpler than the conventional torque-based control strategy. Experiments have been performed on prototyped robots to analyze and demonstrate the effectiveness of the proposed scheme.

The major contributions of this study are as follows: (1) a voltage control model for the impedance control of the manipulator is presented; (2) an adaptive law is designed to eliminate TDE errors while conducting impedance control of the manipulator; and (3) experiments are designed to analyze and demonstrate the effectiveness of the proposed scheme, while maintaining the stability of the contact force between the robot and the environment.

2. Materials and Methods

2.1. Voltage-Based Motor Control

As shown in Figure 1, the dynamics of the robotic system can be described as:

$$\begin{array}{c}\hfill M\ddot{q}+C\dot{q}+G=T_r\end{array}$$

(1)

where $q={[q_1,q_2,\dots ,q_n]}^T\in {\mathbb{R}}^n$ represents the rotation angle of the robot’s joint, $M\in {\mathbb{R}}^{n\times n}$ represents the matrix of the robot’s inertia, $C\in {\mathbb{R}}^n$ represents the matrix of the centrifugal and Coriolis torques, $G\in {\mathbb{R}}^n$ represents the matrix for gravity, and $T_r\in {\mathbb{R}}^n$ represents the torque matrix.

The dynamics for the motor to provide the torque $T_r$ are as follows:

$$\begin{array}{c}\hfill J_m{\ddot{\phi }}_m+B+T_r={\tau }_m\end{array}$$

(2)

where ${\tau }_m\in {\mathbb{R}}^n$ denotes the motor torque, $J_m\in {\mathbb{R}}^{n\times n}$ denotes the matrix for the motor moment. $B\in {\mathbb{R}}^{n\times n}$ denotes the matrix for the motor coefficients. ${\phi }_m\in {\mathbb{R}}^n$ denotes the motor rotation angle.

The electrical equation to drive the DC motor can be described as:

$$\begin{array}{c}\hfill K_b{\dot{\phi }}_m+\varphi =V\end{array}$$

(3)

where $V\in {\mathbb{R}}^n$ denotes the matrix for the motor control voltages, $\varphi$ denotes the nonlinear model part for the motor control, and $K_b$ denotes the constant gain. Then, we have ${\dot{\phi }}_m=K_b^{-1}(V-\varphi )$. It can be obtained as:

$$\begin{array}{c}\hfill J_m{K}_b^{-1}(\dot{V}-\dot{\varphi })+B+T_r={\tau }_m\end{array}$$

(4)

Then, we have:

$$\begin{array}{cc}\hfill T_r& ={\tau }_m-J_m{K}_b^{-1}\dot{V}+J_m{K}_b^{-1}\dot{\varphi }-B\hfill \\ \hfill & =K\dot{V}+\mu \hfill \end{array}$$

(5)

where $K=-J_m{K}_b^{-1}$, $\mu =J_m{K}_b^{-1}\dot{\varphi }-B+{\tau }_m$. Integrating Equation (1), the dynamic model of the manipulator can be rewritten as:

$$\begin{array}{c}\hfill \int (M\ddot{q}+C\dot{q}+G-\mu )dt=\int \left(K\dot{V}\right)dt\end{array}$$

(6)

Then, we have:

$$\begin{array}{c}\hfill \int \left(M\ddot{q}\right)dt=\int \left(M\right)d\dot{q}=M\dot{q}-\int \left(\dot{q}\right)dM\end{array}$$

(7)

Furthermore,

$$\begin{array}{c}\hfill \int \left(C\dot{q}\right)dt=\int \left(C\right)dq=Cq-\int \left(q\right)DC\end{array}$$

(8)

Putting Equations (7) and (8) into Equation (6), we obtain:

$$\begin{array}{c}\hfill M\dot{q}+Cq+\int (G-\mu )dt-\int \left(\dot{q}\right)dM-\int \left(q\right)DC=KV\end{array}$$

(9)

Furthermore,

$$\begin{array}{c}\hfill M{K}^{-1}\dot{q}+C{K}^{-1}q+K^{-1}X=V\end{array}$$

(10)

where, $X=\int (G-\mu )dt-\int \left(\dot{q}\right)dM-\int \left(q\right)DC$. Y denotes the movement at the end of the manipulator, and $Y={(Y_x,Y_y,Y_z)}^T$. Hence, $\dot{Y}$ can be determined as follows.

$$\begin{array}{c}\hfill \dot{Y}=\mathrm{\Lambda }\dot{q}\end{array}$$

(11)

where $\mathrm{\Lambda }$ is Jacobian matrix of manipulator. Then, we have:

$$\begin{array}{c}\hfill \overline{M}\dot{Y}+\overline{C}q+\overline{X}=V\end{array}$$

(12)

where, $\overline{M}=M{K}^{-1}{\mathrm{\Lambda }}^{-1}$, $\overline{C}=C{K}^{-1}$, $\overline{X}=K^{-1}X$.

Assuming the constant matrix $M_d$, we have:

$$\begin{array}{c}\hfill M_d\dot{Y}+N=V\end{array}$$

(13)

where $N=(\overline{M}-M_d)\dot{Y}+\overline{C}q+\overline{X}$ and $M_d$ is diagonal.

The main mathematical notation and definitions are presented in

Table 1. Main mathematical notation and definitions.

Notation	Definition	Notation	Definition	Notation	Definition
M	inertia matrix	$q_1$	angle for joint 1	Y	movement of the end
C	centripetal–Coriolis matrix	$q_2$	angle for joint 2	$\mathrm{\Lambda }$	Jacobian matrix
G	gravitational torque	g	the gravity	$\overline{M}$	assistant variable
q	real angle of joints	$q_d$	desired trajectory	$\overline{C}$	assistant variable
$T_r$	torque for joint	$J_m$	motor moment	$\overline{X}$	assistant variable
$\overline{P}$	desired contact force	B	motor coefficient	N	assistant variable
$m_1$	mass of 1st limb	${\phi }_m$	motor rotation angle	$M_d$	constant matrix
$m_2$	mass of 2nd limb	${\omega }_j$	frequency	${\overline{P}}_R$	real contact force
$d_1$	distance from joint to center of 1st limb	${\tau }_m$	motor torque	$Y_d$	desired movement of the end
$d_2$	distance from joint to center of 2nd limb	V	control voltage	$M_R$	constant matrix
$l_1$	distance from joint 1 to joint 2	$\varphi$	nonlinear model part	$C_R$	constant matrix
$l_2$	distance from the end to joint 2	$K_b$	constant gain	$K_R$	constant matrix
$I_1$	moment of inertia of 1st limb	K	assistant variable	e	tracking error
$I_2$	moment of inertia of 2nd limb	$\mu$	assistant variable	$\widehat{N}$	estimation of N
X	assistant variable	$\overline{N}$	assistant variable	$t_d$	desired scale factor
$\mathrm{\Delta }u$	compensation for TDE error	$\sigma$	TDE error	$K_s$	control gain
K	output of control law	R	number of fuzzy rules	${\theta }_i$	constant factor
b	constant gain	c	constant gain	$V_1$	Lyapunov function
$V_2$	Lyapunov function	${\mu }_R$	assistant variable	$u_0$	assistant variable
t	scale factor	a	constant gain

.

2.2. Impedance Control Design

The principle of the proposed impedance control ATDIC (Adaptive Time Delay Impedance Control) is outlined in Figure 2. $q_d$ denotes the desired manipulator motion. $T_r$ can be calculated from the difference between $q_d$ and q, by using the proposed controller. q denotes the real motion of the manipulator.

An appropriate controller must be selected to adjust the contact force when the end of the manipulator is in contact with the environment. According to the mass-spring-damper system model, the trajectory of the manipulator is reshaped by the contact force. Hence, the impedance controller should guarantee a smooth force control with the desired tracking performance.

For the mass-spring-damper system, the real contact force, ${\overline{P}}_R$, between the end effector and environment can be written as:

$$\begin{array}{c}\hfill {\overline{P}}_R=M_R{\ddot{Y}}_d+C_R{\dot{Y}}_d+K_R{Y}_d\end{array}$$

(14)

where $Y_d$ denotes the desired movement at the manipulator end. $M_R$ denotes the inertia matrix between the robot and the environment. $C_R$ denotes the damping matrix between the robot and the environment. $K_R$ denotes the stiffness matrix between the robot and the environment. Then, we have:

$$\begin{array}{c}\hfill \overline{P}=M_R({\ddot{Y}}_d-\ddot{Y})+C_R({\dot{Y}}_d-\dot{Y})+K_R(Y_d-Y)\end{array}$$

(15)

where $\overline{P}$ denotes the desired contact force, which can be calculated using $K_R$ and the desired position $Y_d$. However, owing to noise from the inaccurate measurement of joint acceleration, $M_R$ is usually omitted. The robotic system is simplified as a spring-damper system by subtracting (15) from (14). Hence, we obtain (14) without considering the inertia matrix as follows:

$$\begin{array}{c}\hfill \dot{Y}=C_R^{-1}({\overline{P}}_R-\overline{P}-K_{R}Y)\end{array}$$

(16)

According to (13), (14), and (16), we have

$$\begin{array}{c}\hfill V=M_d({\dot{Y}}_d+K_R{C}_R^{-1}e-C_R^{-1}\overline{P})+N\end{array}$$

(17)

where $e=Y_d-Y$. From (13) and (17), the control system can be expressed as:

$$\begin{array}{c}\hfill \dot{e}+K_R{C}_R^{-1}e-C_R^{-1}\overline{P}=0\end{array}$$

(18)

When the manipulator contacts the environment, the real contact force ${\overline{P}}_R$ can be obtained directly from the force sensor. The controller should guarantee a smooth force control combined with the desired trajectory of the manipulator.

The manipulator’s control voltage, V, has been described as:

$$\begin{array}{c}\hfill V=M_{d}u+\widehat{N}\end{array}$$

(19)

where $\widehat{N}$ denotes the estimation of N and u denotes the designed control variable. According to the TDE, we have

$$\begin{array}{c}\hfill \widehat{N}=V_L-M_d{\dot{Y}}_L\end{array}$$

(20)

where L denotes the small delay time. From Equations (19) and (20), Equation (13) can be rewritten as follows:

$$\begin{array}{c}\hfill M_d\dot{Y}+\overline{N}=V_L\end{array}$$

(21)

where $\overline{N}=(M-M_d)\dot{Y}+M_d{\dot{Y}}_L-M_{d}u+N$. Therefore, the auxiliary control variable u is deigned as:

$$\begin{array}{c}\hfill u=u_0+\mathrm{\Delta }u\end{array}$$

(22)

where $u_0$ is used to achieve the desired dynamics of the system. $\mathrm{\Delta }u$ compensates for the TDE error $\sigma$. Hence, we have:

$$\begin{array}{c}\hfill u_0={\dot{Y}}_d+K_{s}e\end{array}$$

(23)

where $K_s\in {\mathbb{R}}^{n\times n}$ is the positive control gain.

To investigate the stability analysis, we chose the Lyapunov function candidate as

$$\begin{array}{c}\hfill V_1={\displaystyle \frac{1}{2}}{e}^{T}e\end{array}$$

(24)

Then, we have $\dot{V_1}$ as:

$$\begin{array}{cc}\hfill {\dot{V}}_1& =e^T\dot{e}\hfill \\ \hfill & =e^T({\dot{Y}}_d-\dot{Y})\hfill \\ \hfill & =e^T(M_d^{-1}N-M_d^{-1}V+{\dot{Y}}_d)\hfill \\ \hfill & =e^T(M_d^{-1}N-u-M_d^{-1}\widehat{N}+{\dot{Y}}_d)\hfill \\ \hfill & =e^T(M_d^{-1}N-M_d^{-1}\widehat{N}-{\dot{Y}}_d-K_{s}e-\mathrm{\Delta }u+{\dot{Y}}_d)\hfill \\ \hfill & =e^T(\sigma -K_{s}e-\mathrm{\Delta }u)\hfill \\ \hfill & =-K_s{e}^{T}e+e^T(\sigma -\mathrm{\Delta }u)\hfill \\ \hfill & \le -K_s{e}^{T}e+\sum _{i=1}^n|e_i|(|{\sigma }_i|-\mathrm{\Delta }u)\hfill \end{array}$$

(25)

From Equation (25), $\mathrm{\Delta }u$ should be sufficiently large to ensure $\left|\sigma \right|\le \mathrm{\Delta }u$. In fact, undesired chattering can be caused by larger $\mathrm{\Delta }u$. To eliminate the chattering problem, an adaptive control law is designed to regulate the value of $\mathrm{\Delta }u$ as:

$$\begin{array}{c}\hfill \mathrm{\Delta }u=tK\end{array}$$

(26)

where t denotes the scaled factor that limits the output range of the control law. K denotes the output of the control law. Then, we have:

$$\begin{array}{c}\hfill K={\displaystyle \frac{{\sum }_{r=1}^R{\theta }_i^r{\mu }_R\left(e\right)}{{\sum }_{r=1}^R{\mu }_R\left(e\right)}}\end{array}$$

(27)

where R denotes the number of fuzzy rules. ${\theta }_i={[{\theta }_i^1,{\theta }_i^2,\dots ,{\theta }_i^R]}^T$ denotes the constant factor of the membership function. ${\mu }_R$ denotes the membership function of FS.

$$\begin{array}{c}\hfill {\mu }_R\left(e\right)=max(min({\displaystyle \frac{e-a}{b-a}},{\displaystyle \frac{c-e}{c-b}}),0)\end{array}$$

(28)

where a, b, and c are constant gains. The relationship between e and K is encoded as a fuzzy rule. According to Equations (25) and (26), we obtain:

$$\begin{array}{c}\hfill {\dot{V}}_1=-K_s{e}^{T}e+e^T(\sigma -tK)\end{array}$$

(29)

To eliminate the TDE error, scaled factor t is varied according to $\sigma$. The optimal scaled factor is assumed to be $t_d$. To eliminate $\sigma$, the optimal output of the control law is $t_{d}K$ to eliminate $\sigma$. The error between t and $t_d$ is designed as:

$$\begin{array}{c}\hfill \tilde{t}=t-t_d\end{array}$$

(30)

and

$$\begin{array}{c}\hfill \sum _{i=1}^n{t}_i{K}_i=\sum _{i=1}^n{\tilde{t}}_i{K}_i+\sum _{i=1}^n{t}_{id}{K}_i\end{array}$$

(31)

2.3. Stability Analysis

To investigate the stability analysis, we reselect the Lyapunov function candidate as:

$$\begin{array}{c}\hfill V_2=V_1+{\displaystyle \frac{1}{2}}\sum _{i=1}^n{\tilde{t}}_i^2\end{array}$$

(32)

where ${\tilde{t}}_i^2>0$. Therefore, $V_2$ is positive definite. Then, we have:

$$\begin{array}{cc}\hfill {\dot{V}}_2& ={\dot{V}}_1+\sum _{i=1}^n{\tilde{t}}_i{\dot{\tilde{t}}}_i\hfill \\ \hfill & =-e^T{K}_{s}e+e^T(\sigma -tK)+\sum _{i=1}^n{\tilde{t}}_i{\dot{\tilde{t}}}_i\hfill \\ \hfill & =-e^T{K}_{s}e+\sum _{i=1}^n\left(e_i({\sigma }_i-t_i{K}_i)\right)+\sum _{i=1}^n{\tilde{t}}_i{\dot{\tilde{t}}}_i\hfill \end{array}$$

(33)

According to (31), (33) can be rewritten as:

$$\begin{array}{cc}\hfill {\dot{V}}_2& =-e^T{K}_{s}e+\sum _{i=1}^n(e_i({\sigma }_i-{\tilde{t}}_i{K}_i-t_{id}{K}_i))+\sum _{i=1}^n{\tilde{t}}_i{\dot{\tilde{t}}}_i\hfill \\ \hfill & =-e^T{K}_{s}e+\sum _{i=1}^n(e_i({\sigma }_i-t_{id}{K}_i))+\sum _{i=1}^n(-e_i{\tilde{t}}_i{K}_i+{\tilde{t}}_i{\dot{\tilde{t}}}_i)\hfill \end{array}$$

(34)

Hence, the adaptive law can be designed as:

$$\begin{array}{c}\hfill {\dot{\tilde{t}}}_i=e_i{K}_i\end{array}$$

(35)

Applying Equation (35) into Equation (34), we obtain:

$$\begin{array}{c}\hfill {\dot{V}}_2=-e^T{K}_{s}e+\sum _{i=1}^n\left(e_i[{\sigma }_i-t_{id}{K}_i]\right)\end{array}$$

(36)

Based on the universal approximation theorem, the optimal compensation error is limited as:

$$\begin{array}{c}\hfill |{\sigma }_i-t_{id}{K}_i|\le {\gamma }_i\left|e_i\right|\end{array}$$

(37)

where $\gamma$ denotes the positive constants, $0<{\gamma }_i<1$. Then, we have:

$$\begin{array}{cc}\hfill {\dot{V}}_2& \le -e^T{K}_{s}e+\sum _{i=1}^n{\gamma }_i{e}_i^2\hfill \\ \hfill & =\sum _{i=1}^n(-K_{si}{e}_i^2+{\gamma }_i{e}_i^2)\hfill \\ \hfill & =-e^T(K_s-\gamma )e\hfill \end{array}$$

(38)

If $K_{si}>{\gamma }_i$, so that $K_s-\gamma$ denotes the positive definite matrix, then:

$$\begin{array}{c}\hfill {\dot{V}}_2\le -e^T(K_s-\gamma )e\le 0\end{array}$$

(39)

which shows that ${\dot{V}}_2$ is negative semi-definite as long as $K_{si}>{\gamma }_i$. The tracking errors of the robot are eliminated within a finite amount of time by using the proposed method in combination with the adaptive law in Equation (35).

3. Experiments and Results

3.1. Experiment Setup

Three motion tasks are designed to test the effectiveness of the proposed controller. Figure 3 shows a schematic of the control system. Experimental results are shown in Figure 4, Figure 5, Figure 6, Figure 7, Figure 8, Figure 9 and Figure 10.

The robot manipulator was designed as a 4-DoF manipulator by SWUST Robotic Laboratory. To verify the effectiveness of the proposed control method, two joints of the robot manipulator were selected for the experiment. Hence, an experimental validation was performed on a 2-DoF prototype. The robot manipulator system contains three parts: mechanical structure, control system, and sensor system. The mechanical structure was constructed using an aluminum alloy and nylon fibers.

Table 2. Mass distribution of the robot manipulator.

Part	Mass (kg)	Motion Range (°)
Body Structure	2.06
Joint 1 Motor Unit	2.77
Joint 2 Motor Unit	2.59
Electronics Assembly	2.38
Motion Range of Joint 1		−90° to 90°
Motion Range of Joint 2		−150° to 150°
Total	10.8

lists the mass distribution of the robot manipulator. The total mass of the manipulator was 10.8 kg. Joint motion was driven using a Maxon DC motor (Maxon Group, Sachseln, Switzerland). The encoder of the DC motor collected the rotation angle data. The servo driver Elmo connected with the computer via a CAN bus to control the Maxon EC 45 Power Max brushless motors. The human–computer interface was designed on the computer with Visual Studio 2010 to monitor the change in the signals of the sensors mounted on the manipulator, sampled at 1 kH. Using this interface, information such as the position, velocity, and torque could be stored and analyzed.

The main mechanical parameters are presented in Figure 1. $q_1$ and $q_2$ represent the rotation angles of joints 1 and 2, respectively. $l_1$ and $l_2$ represent the distances between the different joints. $m_i$, $d_i$, $I_i$, and g represent the mass of the i-th robot link, distance from the joint to the center of the i-th link, moment of inertia of the i-th link, and gravity, respectively. In fact, $M\in {\mathbb{R}}^{2\times 2}$ denotes the real inertia matrix, $M=\left[\begin{array}{cc}{M}_{11}& M_{12}\\ M_{21}& M_{22}\end{array}\right]$, $M_{11}=m_1{d}_1^2+I_1+I_2+m_2(l_1^2+d_2^2+2l_1{l}_{2}cos{q}_2)$, $M_{12}=m_2{d}_2^2+I_2+m_2{l}_2{d}_{2}cos{q}_2$, $M_{21}=m_2{d}_2^2+I_2+m_2{l}_2{d}_{2}cos{q}_2$, $M_{22}=m_2{d}_2^2+I_2$; $C\in {\mathbb{R}}^{2\times 2}$ denotes the real centripetal–Coriolis matrix, $C=\left[\begin{array}{cc}{C}_{11}& C_{12}\\ C_{21}& C_{22}\end{array}\right]$, $C_{11}=-m_2{l}_1{d}_{2}sin{q}_2{\dot{q}}_2$, $C_{12}=-m_2{l}_1{d}_{2}sin{q}_2({\dot{q}}_1+{\dot{q}}_2)$, $C_{21}=m_2{l}_1{d}_{2}sin{q}_2{\dot{q}}_1$, $C_{22}=0$; $G\in {\mathbb{R}}^2$ is the real gravitational torque, $G=\left[\begin{array}{c}{m}_1{d}_{1}gcos{q}_1+m_{2}g(d_{2}cos(q_1+q_2)+l_{1}cos{q}_1)\\ m_2{d}_{2}gcos(q_1+q_2)\end{array}\right]$. The main mechanical parameters are designed as $m_1=3.5$ kg, $m_2=3.5$ kg, $d_1=0.25$ m, $d_2=0.25$ m, $I_1=1.3$$\mathrm{kg}·{\mathrm{m}}^2$, $I_2=1.9$$\mathrm{kg}·{\mathrm{m}}^2$, $l_1=0.52$ m, and $l_2=0.52$ m.

3.2. Case 1: Following a Circular Motion

3.2.1. Experimental Protocol

The motion task in Case 1 was designed to test the effectiveness of the proposed controller in the subsequent circular motion. As shown in Figure 4, the motion of the manipulator was performed by tracking the desired trajectory. The contact force was maintained at three levels: 140 N, 145 N, and 150 N. The initial values for $K_s$ and $\gamma$ were set as $K_s$ = 0.8, $\gamma$ = 0.6, respectively.

3.2.2. Results

As shown in Figure 7a,b, it is clear that the positions of the manipulator joints can accurately track the desired trajectory. The position tracking errors are shown in Figure 7c and are bounded. The control voltages of the joints are shown in Figure 7d. Obviously, the proposed control method could efficiently maintain the stability of the contact force. The contact force is shown in Figure 7e. The estimation of uncertainty $\widehat{N}$ is shown in Figure 7f, which shows the boundedness of the parameter estimation error.

The tracking performances of the different controllers are presented in

Table 3. Tracking performance of different controllers in Case 1.

Controllers	Tracking Error (°)		Contact Force (N)
	MEAN (°)	MSE (°)	MEAN (N)	MSE (N)
Impedance control (torque-based) [11]	4.4	3.2	3.3	2.5
Impedance control (voltage-based) [32]	3.5	2.4	3.2	2.6
ATDIC	2.8	1.4	2.4	1.7

. Compared with robust torque-based or voltage-based impedance controllers, the proposed adaptive impedance controller with time delay compensation yields a considerably improved response. The mean error (MEAN) and mean square error (MSE) were reduced to 2.8° and 1.4°, respectively. The chattering problem is strictly limited to the change in control signal. The compensation for TDE error $\mathrm{\Delta }u$ is shown in Figure 10a.

3.3. Case 2: Following a Triangle Circular Motion

3.3.1. Experimental Protocol

The motion task in Case 2 was designed to test the effectiveness of the proposed controller in triangular circular motion. As shown in Figure 5, the motion of the manipulator was performed by tracking the desired trajectory. The contact force was maintained at three levels: 140 N, 145 N, and 150 N. The initial values for $K_s$ and $\gamma$ were set as $K_s$ = 0.8, $\gamma$ = 0.6, respectively.

3.3.2. Results

As shown in Figure 8a,b, it is clear that the positions of the manipulator joints can accurately track the desired trajectory. The position tracking errors are shown in Figure 8c and are bounded. The control voltages of the joints are shown in Figure 8d. Obviously, the proposed control method could efficiently maintain the stability of the contact force. The value of the contact force is shown in Figure 8e. The estimation of uncertainty $\widehat{N}$ is shown in Figure 8f, which shows the boundedness of the parameter estimation error.

The tracking performances of the different controllers are listed in

Table 4. Tracking performance of different controllers in Case 2.

Controllers	Tracking Error (°)		Contact Force (N)
	MEAN (°)	MSE (°)	MEAN (N)	MSE (N)
Impedance control (torque-based) [11]	5.3	3.5	3.2	2.4
Impedance control (voltage-based) [32]	4.7	2.8	3.5	2.6
ATDIC	3.3	1.7	2.2	1.8

. Compared with the robust torque-based or voltage-based impedance controllers, the proposed adaptive impedance controller with time delay compensation yields a considerably improved response. The mean error (MEAN) and mean square error (MSE) were reduced to 3.3° and 1.7°, respectively. The chattering problem is strictly limited to the change in control signal. The compensation for TDE error $\mathrm{\Delta }u$ is shown in Figure 10b.

3.4. Case 3: Following a Sine Circular Motion

3.4.1. Experimental Protocol

The motion task in Case 3 was designed to test the effectiveness of the proposed controller following sinusoidal circular motion. As shown in Figure 6, the motion of the manipulator was performed by tracking the desired trajectory. The contact force was maintained at three levels: 140 N, 145 N, and 150 N. The initial values for $K_s$ and $\gamma$ were set as $K_s$ = 0.8, $\gamma$ = 0.6, respectively.

3.4.2. Results

As shown in Figure 9a,b, it is clear that the positions of the manipulator joints can accurately track the desired trajectory. The position tracking errors are shown in Figure 9c and are bounded. The control voltages of the joints are shown in Figure 9d. Obviously, the proposed control method could efficiently maintain the stability of the contact force. The value of the contact force is shown in Figure 9e. The estimation of uncertainty $\widehat{N}$ is shown in Figure 9f, which shows the boundedness of the parameter estimation error.

The tracking performances of the different controllers is listed in

Table 5. Tracking performance of different controllers in Case 3.

Controllers	Tracking Error (°)		Contact Force (N)
	MEAN (°)	MSE (°)	MEAN(N)	MSE (N)
Impedance control (torque-based) [11]	4.8	3.2	3.7	2.3
Impedance control (voltage-based) [32]	4.2	3.0	3.4	2.2
ATDIC	3.6	2.2	2.5	1.7

. Compared with robust torque-based or voltage-based impedance controllers, the proposed adaptive impedance controller with time delay compensation yields a considerably improved response. The mean error (MEAN) and mean square error (MSE) were reduced to 3.6° and 2.2°, respectively. The chattering problem is strictly limited to the change in control signal. The compensation for TDE error $\mathrm{\Delta }u$ is shown in Figure 10c.

4. Discussion

The robot manipulator is a 4-DoF manipulator. To verify the effectiveness of the proposed control method, two joints of the robot manipulator are selected for the experiment. Hence, an experimental validation is performed on a 2-DoF prototype. To simplify the analysis, actuator saturation and sensor noise are considered as model uncertainties. For future work, the proposed control method can also be applied to a 6-DoF robot manipulator. However, the configurations of the 6-DoF robotic manipulator should be changed from $q\in R^2$, $\tau \in R^2$, $M\in R^{2\times 2}$, $C\in R^{2\times 2}$, $G\in R^2$ to $q\in R^6$, $\tau \in R^6$, $M\in R^{6\times 6}$, $C\in R^{6\times 6}$, $G\in R^6$.

Compared to existing adaptive impedance control methods, the proposed adaptive impedance control provides a simpler design and better tracking performance. In Case 1, the contact force for mean error (MEAN) and mean square error (MSE) were reduced to 2.4 N and 1.7 N, respectively. In Case 2, the contact force for mean error (MEAN) and mean square error (MSE) were reduced to 2.2 N and 1.8 N, respectively. In Case 3, the contact force for mean error (MEAN) and mean square error (MSE) were reduced to 2.5 N and 1.7 N, respectively. With the voltage-based control model and time delay technique, fewer tracking errors and a more robust performance were achieved.

5. Conclusions

An adaptive control framework is proposed for impedance control of a manipulator with a TDE. A voltage control model for the impedance control of the manipulator is designed. Voltage-based motor control is computationally simpler than the conventional torque-based control strategy. The adaptive law is designed to eliminate TDE errors in order to conduct impedance control of the manipulator. The stability analysis is verified by semi-global uniform ultimate boundedness (SGUUB) stability. Experiments are designed to analyze and demonstrate the effectiveness of the proposed scheme. In Case 1, the mean error (MEAN) and mean square error (MSE) were reduced to 2.8° and 1.4°, respectively. In Case 2, the mean error (MEAN) and mean square error (MSE) were reduced to 3.3° and 1.7°, respectively. In Case 3, the mean error (MEAN) and mean square error (MSE) were reduced to 3.6° and 2.2°, respectively. The experimental validation is tested only on a 2-DoF prototype. For future work, the proposed control method can also be applied to a 6-DoF robot manipulator.