Real-time robust adaptive control of robots subjected to actuator voltage constraint

Abstract

This paper is concerned with the problem of design and implementation of a robust adaptive control strategy for electrically driven robots while considering to the constraints on the actuator voltage input. The proposed approach provides a flexible design framework and stable to deal with robustness compared with many other adaptive controllers, such as halting/slowing adaption techniques and adaptively adjusting command signal, which are proposed for robotic applications. The control design procedure is based on a new form of universal approximation theory and using Stone–Weierstrass theorem, to avoid saturation besides being robust against both structured and unstructured uncertainties associated with external disturbances and actuated manipulator dynamics. Moreover, the proposed approach eliminates problems arising from classic adaptive feedforward control scheme. The analytical studies as well as experimental results produced using MATLAB/SIMULINK external mode control on a two degree of freedom electrically driven robot demonstrate high performance of the proposed control schemes.

Appendices

Appendix 1

In order to develop the proposed control scheme, it is convenient to view each joint as a subsystem of the entire actuated manipulator system. Toward this end, let $V$ presented by

$$\begin{aligned} V=\left[ {{\begin{array}{llllll} {V_1 }&{} {V_2 }&{} \cdots &{} {V_j }&{} \cdots &{} {V_n } \\ \end{array}}}\right] ^{T}. \end{aligned}$$

(69)

By this definition, the dynamic model (10) can be expressed by a collection of $n$ second-order nonlinear dynamic which is described by following scalar differential equations:

$$\begin{aligned}&m_{ii} (q)\ddot{q}_i +n_i (q,\dot{q})\dot{q}_i +h_i (q,\dot{q})+\sum _{j=1,\, j\ne i}^n {m_{ij} (q)\ddot{q}_j } \nonumber \\&\quad =V_i \quad \left\{ {i = 1,2,\ldots ,n} \right\} , \end{aligned}$$

(70)

where the subscript “i” indicates the ith element, and $m_{ii} (q)$ is the varying effective inertia seen at the ith joint. Thus, Eq. (70) can be expressed as follows:

$$\begin{aligned} m_{ii} (q)\ddot{q}_i +d_i (q,\dot{q},\ddot{q})=V_{i} \qquad \left\{ {i = 1, 2,\ldots ,n} \right\} , \end{aligned}$$

(71)

where

$$\begin{aligned} d_i (q,\dot{q},\ddot{q})=n_i (q,\dot{q})\dot{q}_i +h_i (q,\dot{q})+\sum _{j = 1,\, j\ne i}^n {m_{ij} (q)\ddot{q}_j } \end{aligned}$$

(72)

is treated as disturbance. $d_i (q,\dot{q},\ddot{q})$ is imposed to the ith joint and contains the gravity, friction, coriolis, centrifugal torques for the ith joint, inertia coupling effects from the other joints as well. From the system point of view, (72) summarizes the coupling between the ith subsystem and the remaining subsystem. Plus and subtracting $\ddot{q}_i$ from (71) and multiplying both sides of it by $m_{ii}^{-1}$ yields available model, presented in Sect. 3.

Appendix 2

Proof of propositioin 1

Let $\mho$ to be a set of continuous function on $T$, and $T$ is Convex set in the form of (23). Now, suppose $\mathfrak {I}_1 (t)$ and $\mathfrak {I}_2 (t)$ are given as

$$\begin{aligned} \mathfrak {I}_1 (t)&= \sum _{i=1}^p {c_{i} \hbox {e}^{\lambda _i t}} \mathrm{cos}\left( {\omega _i t+\theta _i } \right) \nonumber \\ \mathfrak {I}_2 (t)&= \sum _{j=1}^p {\bar{{c}}_{j} \hbox {e}^{\bar{{\lambda }}_j t}} \mathrm{cos}\left( {\bar{{\omega }}_j t+\bar{{\theta }}_j } \right) \end{aligned}$$

(73)

we have

$$\begin{aligned}&\mathfrak {I}_1 (t)+\mathfrak {I}_2 (t)=\sum _{i = 1}^p {c_{i} \hbox {e}^{\lambda _i t}\mathrm{cos}\left( {\omega _i t+\theta _i } \right) }\nonumber \\&\qquad \qquad \qquad \qquad \quad + \sum _{j = 1}^p {\bar{{c}}_{j} \hbox {e}^{\bar{{\lambda }}_j t}\mathrm{cos}\left( {\bar{{\omega }}_j t+\bar{{\theta }}_j } \right) } \nonumber \\&\mathfrak {I}_1 (t).\mathfrak {I}_2 (t)=\frac{1}{2}\sum _{i=1}^p \sum _{j = 1}^{p} c_{i} \bar{{c}}_{j} \hbox {e}^{\left( {\lambda _i +\bar{{\lambda }}_j } \right) t}\mathrm{cos}\left( \left( {\omega _i +\bar{{\omega }}_j } \right) t\right. \nonumber \\&\qquad \qquad \qquad \quad \left. +\left( {\theta _i +\bar{{\theta }}_j } \right) \right) +\frac{1}{2}\sum _{i = 1}^p \sum _{j = 1}^p c_{i} \bar{{c}}_{j} \hbox {e}^{\left( {\lambda _i +\bar{{\lambda }}_j } \right) t}\nonumber \\&\qquad \qquad \qquad \qquad \mathrm{cos}\left( \left( {\omega _i -\bar{{\omega }}_j } \right) t+\left( {\theta _i -\bar{{\theta }}_j } \right) \right) \end{aligned}$$

(74)

Hence, $\mathfrak {I}_1 (t)+\mathfrak {I}_2 (t)$ and $\mathfrak {I}_1 (t).\mathfrak {I}_2 (t)\in \hbar$. Finally, for any arbitrary $\sigma \in \mathfrak {R}$ we can get

$$\begin{aligned} \sigma .\mathfrak {I}(t)=\sum _{i=1}^p {\sigma c_{i} \hbox {e}^{\lambda _i t}} \mathrm{cos}\left( {\omega _i t+\theta _i } \right) \end{aligned}$$

(75)

which is also in the form of (23). So, by considering (74) and (75) we can conclude that $\mho$ is algebra. Therefore, the first condition of Stone–Weierstrass Theorem is satisfied for $\mho$. We show that $\mho$ separates points on $T$. We choose the parameters of the $\mathfrak {I}(t)$ in the form of (23) as follows:

$$\begin{aligned} c_1 =1,\,\lambda _1 =-1,\,\omega _{1} =0,\,\theta _{1} =0. \end{aligned}$$

(76)

Since $t_1 \ne t_2$, then $\mathrm{e}^{-t_1 }\ne \mathrm{e}^{-t_2 }$ and therefore the second condition is also verified. To show that $\mho$ vanishes at no point of $T$, we simply observe that any system in the form of (23) with $\omega _{i} =0$, $\theta _{i} =0$, and $c_i >0$ has the property of

$$\begin{aligned} \forall t\in T,\, \mathfrak {I}(t)>0. \end{aligned}$$

(77)

Hence, $\mho$ vanishes at no point of $T$. Thus the three conditions of the Stone–Weierstrass theorem are satisfied. Therefore the result follows by Stone–Weierstrass Theorem. $\square$