Virtual decomposition controller for flexible-joint robot manipulators with non-full-state feedback

Abstract

In this article, a tracking controller for flexible-joint robot manipulators with non-full-state feedback is investigated based on virtual decomposition control (VDC) theory. First, virtual decomposition principle is applied to decouple the entire system of flexible-joint robot manipulator into rigid-link subsystems and flexible-joint subsystems; then, subcontrollers are designed correspondingly for each link or joint subsystem. Importantly, to realize the closed-loop tracking control without state measurements on the link side, an extended Kalman filter is designed to achieve the proposed VDC approach via non-full-state feedback. In addition, strict stability analysis of the control system is given according to the theory of virtual stability and Lyapunov stability. Finally, validation example is provided to verify the proposed control approach.

Introduction

In general, due to the use of harmonic reducers, hydraulic, cable, or belts drives in the transmission, joint flexibility often occurs in the articulated robots. This feature reduces the frequency bandwidth of the system and makes the control problem much complicated. ¹

In recent years, there are extensive studies for flexible-joint robots. ^{2 –5} A simplified spring damping model of flexible joints and singular perturbation theory to decompose the flexible-joint robot into fast and slow subsystems was first proposed by Spong. ² Based on such simplified joint model, a simple proportional-differential (PD) controller for flexible-joint robot manipulator was proposed. ^3,4 The influence of the flexibility of joints on the dynamic performance of the robot is systematically analyzed. ⁵ However, aforementioned studies of flexible-joint robot manipulators are based on the complex traditional Lagrange integrated dynamics model, and thus make the control algorithms time-consuming, especially for the robot manipulators with high degree of freedoms. Meantime, the singular perturbation method used within the frame of traditional Lagrange integrated dynamic model is only applicable for the robot manipulator with weak joint flexibility.

It is worth noting that the virtual decomposition control (VDC) approach proposed by Zhu ⁶ provides possibilities to break through such restrictions. This approach has been used in the trajectory tracking control of modular and hydraulically powered robot manipulators, ^7,8 as well as impedance control for collision detection of robots. ⁹ It decomposes the overall robotic system into subsystems in respect to either single joint or single link. The subcontroller design for each subsystem is independent, while the coupling relationship between two adjacent subsystems is described by virtual power flow (VPF). Compared with the controller based on Lagrange integrated dynamic model, the calculations by this approach are only proportional to the quantity of subsystems.

Such VDC-theory-based controllers need full-state feedback from both motor side and link side in the flexible joints. However, in most engineering applications, the position and torque measurements on the link side are not always feasible due to the absence of sensors, in addition, angular speed and acceleration measurements of links are apt to be contaminated by external noise. To handle such problem, an extended Kalman filter (EKF) is designed in this article and is incorporated into the VDC approach to eliminate the state measurements of links for flexible-joint robot manipulators.

The main contribution of this article is that we develop a non-full-state feedback VDC approach for flexible-joint robot manipulators. First, different from the conventional Lagrange integrated treatment, the overall system model is decoupled into a certain number of subsystems in terms of either rigid link or flexible joint, which facilitates the dynamics modeling and controller design. Second, considering the state-unmeasurable problem on the link side, an efficient EKF with a new state variable definition is designed to observe the link states of position and torque as well as their differentials and realize the VDC approach with non-full-state feedback.

The article discusses in detail the VDC-based tracking control for flexible-joint robot manipulators via non-full-state feedback. In the second section, the dynamics of rigid-link subsystem and flexible-joint subsystem based on VDC approach and the forward dynamics of flexible-joint robot manipulator are presented. The third section deals with the design of control law for rigid-link subsystem and flexible-joint subsystem. In the fourth section, an EKF is proposed. In the fifth section, virtual stabilities of the subsystems and the whole system with EKF are analyzed by Lyapunov’s theorem. Numerical comparisons are made in the sixth section. Finally, some conclusions are given in the seventh section.

Dynamics modeling

The dynamics of an n degree-of-freedom flexible-joint manipulator can be expressed as link subsystems and flexible-joint subsystems using VDC approach, see Figure 1, where VPF (defined by definition 2.16 in the study by Zhu ⁶ ) is used to characterize the coupling dynamic interactions among subsystems.

Figure 1.

Schematic virtual decomposition of a serial robot manipulator.

Dynamics of rigid-link subsystem

The dynamics equation of the ith rigid-link subsystem expressed in its fixed frame can be written as

${\mathit{M}}_{{\mathit{B}}_i}\frac{\text{d}}{\text{dt}}\left({}^{\mathit{B}i}\mathit{V}\right)+{\mathit{C}}_{{\mathit{B}}_i}{}^{\mathit{B}i}\mathit{V}+{\mathit{G}}_{{\mathit{B}}_i}={}^{\mathit{B}i}\mathit{F}{}^{\mathbf{*}}$

1

where ${\mathit{M}}_{{\mathit{B}}_i}$ , ${\mathit{C}}_{{\mathit{B}}_i},\text{ and }{\mathit{G}}_{{\mathit{B}}_i}$ of equation (1) represents the inertial, centripetal/Coriolis, and gravitational terms, respectively. ${}^{\mathit{B}i}\mathit{V}\text{ and }{}^{\mathit{B}i}\mathit{F}{}^{\mathbf{*}}$ denote the generalized linear/angular velocity and net force/moment of frame $\left\{{\mathit{B}}_i\right\}$ . ⁶

Dynamics of flexible-joint subsystem

There have been extensive studies on modeling of flexible joints. The dynamics equation of the ith flexible-joint subsystem can be written as

$I_i{\ddot{\mathit{\phi }}}_i+f_i\dot{\mathit{\phi }}+k_i^{\mathrm{f}}·\left({\mathit{\phi }}_i-q_i\right)=u_i$

2

$k_i^{\mathrm{f}}·\left({\mathit{\phi }}_i-q_i\right)={\mathit{\tau }}_i$

3

where I_i denotes the rotor inertia of the motor multiplied by the gear ratio squared, f_i is the coefficient of friction, $k_i^{\mathrm{f}}$ is the stiffness of the flexspline, φ_i is the angle of the rotor divided by the gear ratio, q_i is the joint angle, τ_i is the physical torque passing through the flexspline, and u_i denotes the input torque.

Forward dynamics of flexible-joint robot manipulator

The dynamics of the flexible-joint robot is obtained by the linkage model of the link subsystem and the joint subsystem dynamics with generalized force and velocity mapping. It can be written as

${\mathit{T}}_{\mathrm{m}}^{\text{T}}{\mathit{M}}_{\mathrm{l}}{\mathit{T}}_{\mathrm{m}}\ddot{\mathit{q}}+\left({\mathit{T}}_{\mathrm{m}}^{\text{T}}{\mathit{M}}_{\mathrm{l}}{\dot{\mathit{T}}}_{\mathrm{m}}+{\mathit{T}}_{\mathrm{m}}^{\text{T}}{\mathit{C}}_{\mathrm{l}}{\mathit{T}}_{\mathrm{m}}\right)\dot{\mathit{q}}+{\mathit{T}}_{\mathrm{m}}^{\text{T}}{\mathit{G}}_{\mathrm{l}}\text{\hspace{0.17em}=\hspace{0.17em}}{\mathit{k}}^{\mathrm{f}}\left(\mathit{\varphi }-\mathit{q}\right)$

4

${\mathit{J}}_{\mathrm{m}}\ddot{\mathit{\phi }}+\mathit{f}\dot{\mathit{\phi }}+{\mathit{k}}^{\text{f}}\left(\mathit{\phi }-\mathit{q}\right)=\mathit{U}$

5

where ${\mathit{M}}_{\mathrm{l}}=\text{\hspace{0.17em}diag}\left[{\mathit{M}}_{{\mathit{B}}_{\text{1}}},{\mathit{M}}_{{\mathit{B}}_{\text{2}}},\dots ,{\mathit{M}}_{{\mathit{B}}_n}\right],{\mathit{C}}_{\mathrm{l}}=\text{\hspace{0.17em}diag}\left[{\mathit{C}}_{{\mathit{B}}_{\text{1}}},{\mathit{C}}_{{\mathit{B}}_{\text{2}}},\dots ,{\mathit{C}}_{{\mathit{B}}_{\mathit{n}}}\right],{\mathit{G}}_{\mathrm{l}}={\left[{\mathit{G}}_{{\mathit{B}}_{\text{1}}}^{\text{T}},{\mathit{G}}_{{\mathit{B}}_{\text{2}}}^{\text{T}},\dots ,{\mathit{G}}_{{\mathit{B}}_n}^{\text{T}}\right]}^{\text{T}},{\mathit{J}}_{\mathrm{m}}=\text{diag}\left[I_{m_{\text{1}}},I_{m_{\text{2}}},\dots ,I_{m_n}\right],{\mathit{k}}^{\mathrm{f}}=\text{\hspace{0.17em}diag}\left[k_1^f,k_2^f,\dots ,k_n^{\mathrm{f}}\right],\mathit{f}=\text{\hspace{0.17em}diag}\left[f_1,f_2,\dots ,f_n\right],\mathit{U}={\left[u_1,u_2,\dots ,u_n\right]}^{\text{T}};{\mathit{T}}_{\mathrm{m}}\in {ℝ}^{\text{6}n\times n}$ is the generalized force and velocity mapping matrix. ¹

Controller design

In this section, to deal with the model uncertainties for flexible-joint robot manipulator, adaptive strategy is adopted in the sub-controller design of rigid-link subsystems and flexible-joint subsystems.

Control objective

The control goal is to design a controller, which guarantees the actual link position; q ∈ ℝ ⁿ converges asymptotically to the desired q _d ∈ ℝ ⁿ , which is assumed to be third differentiable.

In brief, the control objective can be expressed as

$\forall \mathit{e}\left(0\right)\in {\text{R}}^n,\underset{t\to \infty }{\text{lim}}\mathit{e}\left(t\right)=0$

6

where the tracking error of the link position e (t) is

$\mathit{e}\left(t\right)={\mathit{q}}_{\mathrm{d}}\left(t\right)-\mathit{q}\left(t\right)$

7

Adaptive control law of rigid-link subsystem

Considering model uncertainties, the link subsystem can be expressed in a linear parameterization form as

${\mathit{Y}}_{{\mathit{B}}_i}{\mathit{\theta }}_{{\mathit{B}}_i}\stackrel{\text{def}}{=}{\mathit{M}}_{{\mathit{B}}_i}\frac{\text{d}}{\text{dt}}\left({}^{\mathit{B}i}\mathit{V}{}_{\mathrm{r}}\right)+{\mathit{C}}_{{\mathit{B}}_i}{}^{\mathit{B}i}\mathit{V}{}_{\mathrm{r}}+{\mathit{G}}_{{\mathit{B}}_i}$

8

where ${}^{\mathit{B}i}\mathit{V}{}_{\mathrm{r}}$ is the required generalize velocity in its fixed frame, ${\mathit{Y}}_{{\mathit{B}}_i}\in {ℝ}^{\text{6\hspace{0.17em}}\times \text{\hspace{0.17em}13}}$ is a regressor matrix, and ${\mathit{\theta }}_{{\mathit{B}}_i}\in {ℝ}^{\text{13}}$ is uncertainties parameter vector. ⁶

The control law of link subsystem is designed as

${}^{\mathit{B}i}\mathit{F}{{}_{\mathrm{r}}}^{\mathbf{*}}={\mathit{Y}}_{{\mathit{B}}_i}{\widehat{\mathit{\theta }}}_{{\mathit{B}}_i}+{\mathit{K}}_{{\mathit{B}}_i}\left({}^{\mathit{B}i}\mathit{V}{}_{\mathrm{r}}-{}^{\mathit{B}i}\mathit{V}\right)$

9

where ${}^{\mathit{B}i}\mathit{F}{{}_{\mathrm{r}}}^{\mathbf{*}}$ is the required net force/moment vector of the link, ${\widehat{\mathit{\theta }}}_{{\mathit{B}}_i}\in {ℝ}^{\text{13}}$ is the estimate of ${\mathit{\theta }}_{{\mathit{B}}_i}\in {ℝ}^{\text{13}},\text{ and }{\mathit{K}}_{{\mathit{B}}_i}\in {ℝ}^{\text{6\hspace{0.17em}}\times \text{\hspace{0.17em}6}}$ is a symmetric positive-definite matrix. ${\widehat{\mathit{\theta }}}_{{\mathit{B}}_i}$ can be updated by using the P function. ⁶

Control law of flexible-joint subsystem

The control law of the i th flexible-joint subsystem is

${\dot{q}}_{{\mathrm{r}}_i}={\dot{q}}_{{\mathrm{d}}_i}+{\mathit{\lambda }}_i\left(q_{{\mathrm{d}}_i}-q_i\right)$

10

${\mathit{\tau }}_{{\mathrm{r}}_i}=Z^{\text{T}}·{}^{\mathit{B}i}\mathit{F}{}_{\mathrm{r}}+k_{{\mathrm{p}}_i}\left({\dot{q}}_{{\mathrm{r}}_i}-{\dot{q}}_i\right)+t_{s_i}$

11

$t_{s_i}=P\left({\dot{q}}_{{\mathrm{r}}_i}-{\dot{q}}_i,k_{I_i},t_{s_i}^{-},t_{s_i}^{+}\right)$

12

${\mathit{\phi }}_{{\mathrm{d}}_i}=q_i+\frac{{\mathit{\tau }}_{{\mathrm{r}}_i}}{k_i^{\mathrm{f}}}$

13

${\dot{\mathit{\phi }}}_{{\mathrm{r}}_i}={\dot{\mathit{\phi }}}_{{\mathrm{d}}_i}+{\mathit{\lambda }}_{q_i}\left({\dot{q}}_{{\mathrm{r}}_i}-{\dot{q}}_i\right)+{\mathit{\lambda }}_{{\mathit{\phi }}_i}\left({\mathit{\phi }}_{{\mathrm{d}}_i}-{\mathit{\phi }}_i\right)+{\mathit{\lambda }}_{{\mathit{\tau }}_i}\left({\mathit{\tau }}_{{\mathrm{r}}_i}-{\mathit{\tau }}_i\right)$

14

$u=I_i{\ddot{\mathit{\phi }}}_{{\mathrm{r}}_i}+f_i{\dot{\mathit{\phi }}}_{{\mathrm{r}}_i}+k_{{\mathit{\phi }}_i}\left({\dot{\mathit{\phi }}}_{{\mathrm{r}}_i}-{\dot{\mathit{\phi }}}_i\right)+{\mathit{\tau }}_{{\mathrm{r}}_i}+k_{{\mathit{\tau }}_i}\left({\mathit{\tau }}_{{\mathrm{r}}_i}-{\mathit{\tau }}_i\right)$

15

where $q_{{\mathrm{d}}_i}\text{ and }{\dot{q}}_{{\mathrm{d}}_i}$ denote the desired position and angular velocity of the joint; λ_i, $k_{{\mathrm{p}}_i}$ , $k_{I_i}$ , ${\mathit{\lambda }}_{q_i}$ , ${\mathit{\lambda }}_{{\mathit{\phi }}_i}$ , ${\mathit{\lambda }}_{{\mathit{\tau }}_i}$ , $k_{{\mathit{\phi }}_i},\text{ and }{k}_{{\mathit{\tau }}_i}$ are all positive constants; $t_{s_i}^{+}>0\text{ and }{t}_{s_i}^{-}<0$ are positive constants as upper and lower bound of $t_{s_i}$ ; Z = [0, 0, 0, 0, 0, 1]^T for revolute-joint robot manipulator.

Extended Kalman filter

Considering the control law of rigid-link subsystem presented in equation (9) and the control law of flexible-joint subsystem presented in equations (10) –(15) needs full-state feedback including q_i, ${\dot{q}}_i$ , ${\ddot{q}}_i$ ,φ_i, ${\dot{\mathit{\phi }}}_i$ ,τ_i, and ${\dot{\mathit{\tau }}}_i$ . Motivated by Reif et al. ¹⁰ and Lightcap and Banks ¹¹ , an EKF with a new state variable definition is proposed in this article to observe the states q_i, ${\dot{q}}_i$ ,τ_i, and ${\dot{\mathit{\tau }}}_i$ on the link side, with only using the states φ_i and ${\dot{\mathit{\phi }}}_i$ on the motor side.

Different from the conventional treatment, ¹¹ the state variable x is defined here as

$\mathit{x}={\left[{\mathit{\tau }}^{\text{T}},{\dot{\mathit{\tau }}}^{\text{T}},{\mathit{\phi }}^{\text{T}},{\dot{\mathit{\phi }}}^{\text{T}}\right]}^{\text{T}}={\left[{\mathit{x}}_1^{\text{T}},{\mathit{x}}_2^{\text{T}},{\mathit{x}}_3^{\text{T}},{\mathit{x}}_4^{\text{T}}\right]}^{\text{T}}$

16

The time derivation of the state variable x is

$\mathit{f}\left(\mathit{x}\right)=\dot{\mathit{x}}=\left[\begin{array}{c}\dot{\mathbf{\tau }}\\ \begin{array}{c}{\mathit{k}}^{\mathrm{f}}[-\left({\left({\mathit{T}}_{\mathrm{m}}^{\text{T}}{\mathit{M}}_{\mathrm{l}}{\mathit{T}}_{\mathrm{m}}\right)}^{-1}+{\mathit{J}}^{-1}\right)\mathit{\tau }\text{+}\\ {\left({\mathit{T}}_{\mathrm{m}}^{\text{T}}{\mathit{M}}_{\mathrm{l}}{\mathit{T}}_{\mathrm{m}}\right)}^{-1}(\left({\mathit{T}}_{\mathrm{m}}^{\text{T}}{\mathit{M}}_{\mathrm{l}}{\dot{\mathit{T}}}_{\mathrm{m}}+{\mathit{T}}_{\mathrm{m}}^{\text{T}}{\mathit{C}}_{\mathrm{l}}{\mathit{T}}_{\mathrm{m}}\right)\dot{\mathit{q}}+\\ {\mathit{T}}_{\mathrm{m}}^{\text{T}}{\mathit{G}}_{\mathrm{l}})\text{\hspace{0.17em}\hspace{0.05em}+\hspace{0.17em}}{\mathit{J}}^{-1}\left(\mathit{U}-\mathit{f}\dot{\mathit{\phi }}\right)]\end{array}\\ \dot{\mathit{\phi }}\\ {\mathit{J}}^{-1}(\mathit{U}-\mathit{\tau }-\mathit{f}\dot{\mathit{\phi }})\end{array}\right]$

17

and the partial derivative of function f ( x ) with respect to the state variable x can be written as

$\frac{\mathit{\partial }\mathit{f}\left(\mathit{x}\right)}{\mathit{\partial }\mathit{x}}=\mathit{F}\left(t\right)=\left[\begin{array}{cccc}\mathbf{0}& \mathbf{\text{I}}& \mathbf{0}& \mathbf{0}\\ -{\mathit{k}}^{\text{f}}\left({\mathit{M}}^{-1}\text{\hspace{0.17em}+\hspace{0.17em}}{\mathit{J}}^{-1}\right)& \mathbf{0}& \mathbf{0}& -{\mathit{k}}^{\text{f}}{\mathit{J}}^{-1}\mathit{f}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{\text{I}}\\ {\mathit{J}}^{-1}& \mathbf{0}& \mathbf{0}& -{\mathit{J}}^{-1}\mathit{f}\end{array}\right]$

18

It is noteworthy that, in the EKF with conventional state variable definition, ¹¹ q and $\dot{\mathit{q}}$ are involved explicitly in the state variable x , which results in complicate calculations in equation (18) such as computing the partial derivatives of ${\left({\mathit{T}}_{\mathrm{m}}^{\text{T}}{\mathit{M}}_{\mathrm{l}}{\mathit{T}}_{\mathrm{m}}\right)}^{-1},\left({\mathit{T}}_{\mathrm{m}}^{\text{T}}{\mathit{M}}_{\mathrm{l}}{\dot{\mathit{T}}}_{\mathrm{m}}+{\mathit{T}}_{\mathrm{m}}^{\text{T}}{\mathit{C}}_{\mathrm{l}}{\mathit{T}}_{\mathrm{m}}\right),\text{ and }{\mathit{T}}_{\mathrm{m}}^{\text{T}}{\mathit{G}}_{\mathrm{l}}$ . While in this article, we redefined the state vector as $\mathit{x}={\left[{\mathit{\tau }}^{\text{T}},{\dot{\mathit{\tau }}}^{\text{T}},{\mathit{\phi }}^{\text{T}},{\dot{\mathit{\phi }}}^{\text{T}}\right]}^{\text{T}}$ , instead of q and $\dot{\mathit{q}}$ , τ and $\dot{\mathit{\tau }}$ are adopted as the states, then the corresponding partial derivatives of ${\left({\mathit{T}}_{\mathrm{m}}^{\text{T}}{\mathit{M}}_{\mathrm{l}}{\mathit{T}}_{\mathrm{m}}\right)}^{-1},\left({\mathit{T}}_{\mathrm{m}}^{\mathrm{\text{T}}}{\mathit{M}}_{\mathrm{l}}{\dot{\mathit{T}}}_{\mathrm{m}}+{\mathit{T}}_{\mathrm{m}}^{\text{T}}{\mathit{C}}_{\mathrm{l}}{\mathit{T}}_{\mathrm{m}}\right),\text{ and }{\mathit{T}}_{\mathrm{m}}^{\text{T}}{\mathit{G}}_{\mathrm{l}}$ are all zero, which improves the computing efficiency greatly. After τ and $\dot{\mathit{\tau }}$ are observed by the proposed EKF, the estimations of q and $\dot{\mathit{q}}$ can be derived from equation (3).

Observation vector h ( x ) is defined to denote the angular position and angular velocity of the joint motors

$\mathit{h}\left(\mathit{x}\right)=\left[\begin{array}{c}\mathit{\phi }\\ \dot{\mathit{\phi }}\end{array}\right]=\left[\begin{array}{c}{\mathit{x}}_{\text{3}}\\ {\mathit{x}}_{\text{4}}\end{array}\right]$

19

The partial differential equation of the observation vector with respect to the state is

$\frac{\mathit{\partial }\mathit{h}\left(\mathit{x}\right)}{\mathit{\partial }\mathit{x}}=\mathit{H}\left(t\right)=\left[\begin{array}{cccc}\mathbf{0}& \mathbf{0}& \mathbf{I}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}& \mathbf{0}& \mathbf{I}\end{array}\right]$

20

Substitute equations (16) to (20) into the EKF formula presented as equations (21) to (23), then

$\dot{\widehat{\mathit{x}}}=\mathit{f}\left(\widehat{\mathit{x}},\mathit{U}\right)+\mathit{K}\left(t\right)\left[\mathit{y}\left(t\right)-\mathit{h}\left(\widehat{\mathit{x}}\right)\right]$

21

$\dot{\mathit{P}}\left(t\right)=\mathit{F}\left(t\right)\mathit{P}\left(t\right)+\mathit{P}\left(t\right){\mathit{F}}^{\text{T}}\left(t\right)+\mathit{Q}\left(t\right)-\mathit{L}\left(t\right)\mathit{H}\left(t\right)\mathit{P}\left(t\right)$

22

$\mathit{L}\left(t\right)=\mathit{P}\left(t\right){\mathit{H}}^{\text{T}}\left(t\right){\mathit{R}}^{-1}\left(t\right)$

23

where $\widehat{\mathit{x}}$ is the estimation of x , U can be obtained through equation (15), $\mathit{y}\left(t\right)=\mathit{h}\left(\mathit{x}\right)$ is the output, and Q (t) and R (t) are the positive definite symmetric matrices that denote the covariances of the process noise and observation noise, respectively. L (t) is the Kalman gain and P (t) is a positive definite symmetric matrix that denotes the covariance under state x .

Stability analysis

Stability analysis of the overall system consists of the stability discussions of rigid-link subsystems, flexible-joint subsystems, and EKF.

Stablity analysis of the original system

According to definition 2.17 and theorem 2.1 in the study by Zhu ⁶ , if each subsystem and EKF is proved to be (virtually) stable, then the entire system is stable in the sense of Lyapunov.

From the control law (equation (9)) and the rigid-link subsystem dynamics (equation (1)), a nonnegative accompanying function for the link subsystem can be chosen as

$v_i^{\text{L}}=\frac{1}{2}[{\left({}^{\mathit{B}i}\mathit{V}{}_{\mathrm{r}}-{}^{\mathit{B}i}\mathit{V}\right)}^{\text{T}}{\mathit{M}}_{{\mathit{B}}_i}\left({}^{\mathit{B}i}\mathit{V}{}_{\mathrm{r}}-{}^{\mathit{B}i}\mathit{V}\right)+{\displaystyle \sum _{\mathit{\gamma }=1}^{13}\frac{{\left({\mathit{\theta }}_{\mathit{i}\mathit{\gamma }}-{\widehat{\mathit{\theta }}}_{\mathit{i}\mathit{\gamma }}\right)}^2}{{\rho }_{\mathit{i}\mathit{\gamma }}}}]$

24

From the control law (equations (10) to (15)) and the flexible-joint subsystem dynamics (equations (2) and (3)), a nonnegative accompanying function for the joint subsystem can be chosen as

$v_i^{\text{M}}=\frac{1}{2}[{\mathit{\text{I}}}_i{\left({\dot{\mathit{\phi }}}_{{\mathrm{r}}_i}-{\dot{\mathit{\phi }}}_i\right)}^2+k_i^{\mathrm{f}}\left(1+k_{\mathit{\tau }}\right){\left({\mathit{\phi }}_{{\mathrm{d}}_i}-{\mathit{\phi }}_i\right)}^2+\left(1+k_{\mathit{\tau }}\right){\mathit{\lambda }}_{q_i}\frac{t_s^2}{k_{I_i}}]$

25

Then, adding all the nonnegative accompanying function together, we get a nonnegative function of the entire system

$V\left(t\right)={\displaystyle \sum _{i=\text{\hspace{0.17em}1}}^n\left(v_i^{\text{L}}+v_i^{\text{M}}\right)}\ge 0$

26

The time derivation of equation (26) follows that

$\begin{array}{l}\dot{V}\left(t\right)\le -{\displaystyle \sum _{i=1}^n{\left({}^{\mathit{B}i}\mathit{V}{}_{\mathrm{r}}-{}^{\mathit{B}i}\mathit{V}\right)}^{\text{T}}{\mathit{K}}_{{\mathit{B}}_i}\left({}^{\mathit{B}i}\mathit{V}{}_{\mathrm{r}}-{}^{\mathit{B}i}\mathit{V}\right)+}\\ {\displaystyle \sum _{i=1}^n\{-\left(k_{{\mathit{\phi }}_i}+{\mathit{f}}_i\right){\left({\dot{\mathit{\phi }}}_{{\mathrm{r}}_i}-{\dot{\mathit{\phi }}}_i\right)}^2-\left(1+k_{{\mathit{\tau }}_i}\right)}\\ k_i^{\mathrm{f}}{\mathit{\lambda }}_{{\mathit{\phi }}_i}{\left({\mathit{\phi }}_{{\mathrm{d}}_i}-{\mathit{\phi }}_i\right)}^2-\left(1+k_{{\mathit{\tau }}_i}\right){\mathit{\lambda }}_{{\mathit{\tau }}_i}{\left({\mathit{\tau }}_{{\mathrm{r}}_i}-{\mathit{\tau }}_i\right)}^2-\\ \left(1+k_{{\mathit{\tau }}_i}\right){\mathit{\lambda }}_{q_i}{k}_{{\mathrm{p}}_i}{\left({\dot{q}}_{{\mathrm{r}}_i}-{\dot{q}}_i\right)}^2\}\end{array}$

27

According to theorem 2 in the study by Zhu, ¹³ if ${\dot{q}}_{\mathrm{d}}\in {\text{L}}_{\infty },{\ddot{q}}_{\mathrm{d}}\in {\text{L}}_{\infty },\text{ and }{q}_{\mathrm{d}}^{\left(3\right)}\in {\text{L}}_{\infty }$ , then leading to

${\dot{\mathit{q}}}_{\mathrm{r}}-\dot{\mathit{q}}\to 0$

${\dot{\mathit{q}}}_{\mathrm{d}}-\dot{\mathit{q}}\to 0$

${\mathit{q}}_{\mathrm{d}}-\mathit{q}\to 0$

Hence, the system is asymptotic stable in the sense of Lyapunov.

Stability analysis of EKF

We first expand f ( x ) and h ( x ) by

$\mathit{f}\left(\mathit{x}\right)-\mathit{f}\left(\widehat{\mathit{x}}\right)=\mathit{F}\left(t\right)\left(\mathit{x}-\widehat{\mathit{x}}\right)+\mathit{\varphi }$

28

$\mathit{h}\left(\mathit{x}\right)-\mathit{h}\left(\widehat{\mathit{x}}\right)=\mathit{H}\left(t\right)\left(\mathit{x}-\widehat{\mathit{x}}\right)+\mathbf{\chi }$

29

where ϕ and χ are the terms of the second and higher order in $\left(\mathit{x}-\widehat{\mathit{x}}\right)$ .

Define the observation error as

$\mathit{\zeta }\text{=\hspace{0.17em}}\mathit{x}-\widehat{\mathit{x}}$

30

then from equations (17), (21), (28), and (29), we get

$\dot{\mathit{\zeta }}\text{\hspace{0.17em}=\hspace{0.17em}}\left[\mathit{F}\left(t\right)-\mathit{L}\left(t\right)\mathit{H}\left(t\right)\right]\mathit{\zeta }+\mathit{\varphi }-\mathit{L}\left(t\right)\mathit{\chi }$

31

Define Y as

$\mathit{Y}={\mathit{P}}^{-1}\left(t\right)$

32

Consider a Lyapunov function as

$v_{\text{EKF}}\text{\hspace{0.17em}=\hspace{0.17em}}{\mathit{\zeta }}^{\text{T}}\mathit{Y}\mathit{\zeta }\ge \mathbf{0}$

33

The time derivation of equation (33) follows that

${\dot{v}}_{\text{EKF}}\left(t\right)\le \frac{\underset{¯}{q}}{2{\overline{\mathit{p}}}^2}{\Vert \mathit{\zeta }\left(t\right)\Vert }^2\le -\frac{\underset{¯}{q}\underset{¯}{p}}{2{\overline{p}}^2}{v}_{\text{EKF}}\left(t\right)$

34

where $\underset{¯}{q},\underset{¯}{p},\text{ and }\overline{p}\text{ satisfy }\underset{¯}{q}\mathbf{I}\le \mathit{Q}\text{ and }\underset{¯}{p}\mathbf{I}\le \mathit{P}\le \overline{p}\mathbf{I}$ , respectively. ¹⁰

Hence, the exponential stability of EKF is guaranteed.

Finally, the entire system with EKF is asymptotic stable. ¹¹

Numerical simulations

In order to verify the proposed controller (named as VDC + EKF), simulation tests on a two-link robot manipulator with revolute flexible joints (see Figure 2) are made.

Figure 2.

A two-link flexible-joint robot manipulator.

The parameters of the links and joints are listed in Tables 1 and 2.

Table 1.

The parameters of rigid links.

Links	l (m)	d (m)	m (kg)	Io (kg·m2)
1	0.5	0.25	10	1.0
2	0.5	0.25	4	0.4

Table 2.

The parameters of flexible joints.

Joints	I (kg·m2)	f (N·m·s)	kf (N·m/rad)
1	2.0	1.0	10,000
2	0.5	0.5	10,000

Set the process covariance matrix and observation covariance matrix as constant matrices, that is Q = 4I and R = 0.1I, where I is the identity matrix. The initial P is set as $\mathit{P}\left(0\right)=\mathbf{I}$

The desired angular position of links is given as

${\mathit{q}}_{\mathrm{d}}=\frac{1}{\text{4}\mathit{\pi }}\left[\begin{array}{l}\text{sin}(\text{4}\mathit{\pi }t)\text{\hspace{0.17em}+\hspace{0.17em}0.176}\\ \text{sin}(\text{4}\mathit{\pi }t)\text{\hspace{0.17em}+\hspace{0.17em}0.176}\end{array}\right]\text{rad}$

35

The control parameters of the links and joints are listed in Tables 3 and 4.

Table 3.

The control parameters of rigid links.

Links	1	2
KB	35	20
ρ 1	50	50
ρ 2	50	50
ρ 3	50	50

Table 4.

The control parameters of flexible joints.

Joints	1	2	Joints	1	2
λ	20	20	λq	0.7	0.6
k p	380	220	λφ	25	15
kI	2000	2000	λτ	0.005	0.005
$t_s^{-}$	−500	−500	k ϕ	210	110
$t_s^{+}$	500	500	k τ	0.5	0.5

Simulations comparisons are made between the proposed controller (VDC + EKF) and the composite controller that consists of a computed torque sub-controller for its slow subsystem, and a full-state feedback sub-controller for its fast subsystem (CT + FSFC). ¹²

To make overall quantitative evaluation of the two controllers, we adopt three criteria:

RMS—root mean square of the tracking error of the link computed by equations (36) and (37) on a trip of time T (T = 4 s)

$RMS[e_i(t)]=\sqrt{\frac{1}{T}{\displaystyle {\int }_0^T{\left|e_i(t)\right|}^2\text{dt}}}$

36

$RMS[\mathit{e}(t)]=\sqrt{\frac{1}{T}{\displaystyle {\int }_0^T{\Vert \mathit{e}(t)\Vert }^2\text{dt}}}$

37

Adjusting time—a period from the start to the moment trajectory tracking error of link falls into area of $\pm \text{1.3\hspace{0.17em}}\times {10}^{-\text{3}}$ rad.

Maximum torque—the largest torque input to the joint actuator.

From Figures 3 and 4 as well as Table 5, we can find that the proposed controller VDC + EKF has smaller trajectory tracking errors and shorter adjusting time than the controller CT + FSFC. Meanwhile, as shown in Figures 5 and 6 as well as Table 5, the maximum torque input to each joint of our controller (VDC + EKF) is smaller than that of the benchmark controller CT + FSFC.

Figure 3.

Tracking error of link 1.

Figure 4.

Tracking error of link 2.

Table 5.

Comparison of two controllers.

Parameters		VDC + EKF	CT + FSFC
RMS (rad)	e 1	0.0024	0.0029
	e 2	0.0023	0.0026
	|| e ||	0.0033	0.0039
Adjusting time (s)	joint 1	0.39	1.19
	joint 2	0.40	1.21
Maximum torque (Nm)	joint 1	92.20	844.51
	joint 2	28.86	301.34

VDC: virtual decomposition control; EKF: extended Kalman filter; RMS: root mean square; CT: computed torque; FSFC: full-state-feedback control.

Figure 5.

Torque input of joint 1.

Figure 6.

Torque input of joint 2.

Conclusion

In this article, the trajectory tracking control problem of flexible-joint robot manipulators is discussed based on VDC theory. Considering the uncertainty of the model parameters, the VDC law of the link subsystem and the joint subsystem are designed. Furthermore, an EKF observer is designed to achieve closed-loop tracking control with only position and velocity feedback measurement on the motor side, but without measuring the states of position, velocity, and acceleration, as well as torque and its derivative on the link side. Theoretical analysis and example simulation results verify the effectiveness and superiority of the proposed controller.