Control of an Industrial PA10-7CE Robot Arm Based on Decentralized Neural Backstepping Approach

Abstract

This paper presents a discrete-time decentralized control strategy for trajectory tracking of a Mitsubishi PA10-7CE robot arm. A high order neural network is used to approximate a decentralized control law designed by the backstepping technique as applied to a block strict feedback form. The weights for each neural network are adapted online by extended Kalman filter training algorithm. The motion of each joint is controlled independently using only local angular position and velocity measurements. The stability analysis for closed-loop system via Lyapunov theory is included. Finally, the simulations results show the feasibility of the proposed scheme.

Appendices

Appendix

Stability Definitions

Let be a Multiple-Input Multiple-Output (MIMO) nonlinear system:

$$\begin{aligned} x(k+1)&= F(x(k), u(k)) \end{aligned}$$

(56)

$$\begin{aligned} y(k)&= h(x(k)) \end{aligned}$$

(57)

where $x \in \mathfrak R ^{n}$ is the system state, $u\in \mathfrak R ^{m}$ is the system input, $y \in \mathfrak R ^{p}$ is the system output, and $F\in \mathfrak R ^{n}\times \mathfrak R ^{m}\rightarrow \mathfrak R ^{n}$ is nonlinear function.

Definition 1

[4] System (56) is said to be forced, or to have input. In contrast the system described by an equation without explicit presence of an input $u$, that is

$$\begin{aligned} x(k+1)=F(x(k)) \end{aligned}$$

is said to be unforced. It can be obtained after selecting the input u as a feedback function of the state

$$\begin{aligned} u(k)=\vartheta (x(k)) \end{aligned}$$

(58)

Such substitution eliminates u:

$$\begin{aligned} x(k+1)=F(x(k), \vartheta (x(k))) \end{aligned}$$

(59)

and yields an unforced system (59) [12].

Definition 2

[4] The solution of (56)–(58) is semiglobally uniformly ultimately bounded (SGUUB), if for any $\Omega$, a compact subset of $\mathfrak R ^{n}$ and all $x(k_{0}) \in \Omega$, there exists an $\varepsilon > 0$ and a number $\mathbf N (\varepsilon ,x(k_{0}))$ such that $\Vert x(k) \Vert < \varepsilon \, \mathrm{for all}\, k \ge k_{0} + \mathbf N$.

Lemma 3

[20] Consider the linear time varying discrete-time system given by

$$\begin{aligned} x(k+1)&= A(k)x(k) + B(k)u(k) \nonumber \\ y(k)&= C(k)x(k) \end{aligned}$$

(60)

where $A(k), B(k)$ and $C(k)$ are appropriately dimensional matrices, $x \in \mathfrak R ^{n}$, $u \in \mathfrak R ^{m}$ and $y \in \mathfrak R ^{p}$. Let $\Phi (k(1), k(0))$ be the state transition matrix corresponding to $A(k)$ for system (60), $\Phi (k(1), k(0))=\prod _{k=k(0)}^{k=k(1)-1} A(k)$. If $\Phi (k(1), k(0)) < 1 \forall k(1) > k(0) > 0$, then system (60) is

1) globally exponentially stable for the unforced system ($u(k)=0$) and

2) Bounded Input Bounded Output (BIBO) stable [4], [20].

Corollary 1

[14] There is an exponential observer for a Lyapunov stable discrete-time nonlinear system (56)-(57) with $u=0$ if and only if the linear approximation

$$\begin{aligned} x(k+1)=Ax(k)+Bu(k)\nonumber \\ y(k)=Cx(k)\nonumber \\ A=\left. \frac{\partial F}{\partial x}\right|_{x=0}\text{,} \, B=\left. \frac{\partial F}{\partial u}\right|_{x=0}\text{,} \, C=\left. \frac{\partial h}{\partial x}\right|_{x=0} \nonumber \\ \end{aligned}$$

(61)

of the system (56)-(57) is detectable.

Fig. 11

Applied torque to joint 1

Fig. 12

Applied torque to joint 2

Fig. 13

Applied torque to joint 3

Fig. 14

Applied torque to joint 4

Fig. 15

Applied torque to joint 5

Fig. 16

Applied torque to joint 6

Fig. 17

Applied torque to joint 7

Applied Torques

The applied torques for each joint are illustrated in Figs. 11, 12, 13, 14, 15, 16, and 17, they are always inside of the prescribed limits given by the actuators manufacturer (see Table 2); that is, their absolute values are smaller than the bounds $\tau _{1}^{\text{ max}}$ to $\tau _{7}^{\text{ max}}$, respectively.

Table 2 Maximum torques