Robust optimal constrained control of fully-constrained cable-driven parallel robots based on GSDRE

Abstract

This paper addresses constrained robust optimal control of Cable-Driven Parallel Robots (CDPRs) in detail. In the CDPRs, cables should remain in tension for all movements. Based on this fact, considering tension constraints, a robust optimal control scheme is presented for the cable robots in this study. To develop the idea, at first, the general nonlinear dynamic model of the CDPR is converted to a State-Dependent Coefficient (SDC) linear structure. In this model, uncertainties of the cable robot are included as a vector in the SDC linear structure. Next, a constrained robust optimal control algorithm is proposed based on Hamilton-Jacobi-Bellman (HJB) equations and Karush-Kuhn-Tucker (KKT) conditions using the state-dependent coefficient parameterization form. The proposed control scheme is formed based on the Generalized State-Dependent Riccati Equation (GSDRE), and a new equation is created to counteract with disturbances and uncertainty of the CDPR model. Finally, the stability of the closed-loop system is proved using Lyapunov’s second method. Simulation results show the effectiveness of the proposed control algorithm in terms of tracking and coping with the uncertainties and external disturbances.

Appendices

Appendix: Dynamics parameters

By choosing the parameters $2.25\left( \varvec{kg} \right) \le m \le 2.75\left( \varvec{kg} \right)$, $0.01\left( \varvec{kg} {\varvec{m}^2} \right) \le {I_z} \le 0.07\left( \varvec{kg} {\varvec{m}^2} \right)$, and $g = 9.806 \left( \varvec{m} {{\varvec{s}^2}} \right)$, the dynamic matrices of the 3 DoFs planar CDPR are expressed as follows [2]:

$$\begin{aligned} {M = \left[ {\begin{array}{*{20}{c}} m&{}0&{}0\\ 0&{}m&{}0\\ 0&{}0&{}{{I_z}} \end{array}} \right] ;\ C\left( {x,\dot{x}} \right) = 0; \ G = \left[ {\begin{array}{*{20}{c}} 0\\ {mg}\\ 0 \end{array}} \right] ;}\end{aligned}$$

The position of the cable attachment points is selected as [2].

Table 1 Root mean square and maximum of ${{\tilde{J}}}^*$

Appendix: Calculations

Consider the following defined remarks:

Remark 1

The symbol $\left[ \hspace{-1.49994pt}\left[ {.} \right] \hspace{-1.49994pt}\right]$ indicates that the inside parameter is a vector, e.g.,

$$\begin{aligned} \left[ \hspace{-1.49994pt}\left[ \theta \right] \hspace{-1.49994pt}\right] = {\left[ {\begin{array}{*{20}{c}} {\theta _1^T}&{\theta _2^T}&\ldots&{\theta _N^T} \end{array}} \right] ^T} \end{aligned}$$

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which N is the number of the vector elements.

Remark 2

The symbol $\left\{ . \right\}$ indicates the tensor space in which the inside parameter dimensions must be homogenized according to the problem.

Remark 3

${\theta _{i:}}$ and ${\theta _{:i}}$ represent the $i_{th}$ row and $i_{th}$ column of matrix ${\theta }$, respectively.

The derivatives of the matrices Q(z(t)), R(z(t)), and Y(z(t)) in terms of z(t) algebraically create more comprehensive dimensions than Q(z(t)), R(z(t)), and Y(z(t)). We use the following three tensor relations to perform accurate calculations.

$$\begin{aligned}&\left[ \hspace{-1.49994pt}\left[ {{e^T}\left( t \right) \cdot \frac{{\partial Q(z(t))}}{{\partial z\left( t \right) }} \cdot e\left( t \right) } \right] \hspace{-1.49994pt}\right] \nonumber \\ {}&\quad = \left[ {\begin{array}{*{20}{c}} {{e^T}\frac{{\partial {Q_{:1}}}}{{\partial {z_1}}}}&{} \ldots &{}{{e^T}\frac{{\partial {Q_{:n}}}}{{\partial {z_1}}}}\\ \vdots &{} \ddots &{} \vdots \\ {{e^T}\frac{{\partial {Q_{:1}}}}{{\partial {z_n}}}}&{} \cdots &{}{{e^T}\frac{{\partial {Q_{:n}}}}{{\partial {z_n}}}} \end{array}} \right] e\left( t \right) \end{aligned}$$

(72)

$$\begin{aligned}&\left[ \hspace{-1.49994pt}\left[ {{f^T}\left( t \right) \cdot \frac{{\partial R(z(t))}}{{\partial z\left( t \right) }} \cdot f\left( t \right) } \right] \hspace{-1.49994pt}\right] \nonumber \\ {}&\quad = \left[ {\begin{array}{*{20}{c}} {{f^T}\frac{{\partial {R_{:1}}}}{{\partial {z_1}}}}&{} \ldots &{}{{f^T}\frac{{\partial {R_{:n}}}}{{\partial {z_1}}}}\\ \vdots &{} \ddots &{} \vdots \\ {{f^T}\frac{{\partial {R_{:1}}}}{{\partial {z_n}}}}&{} \cdots &{}{{f^T}\frac{{\partial {R_{:n}}}}{{\partial {z_n}}}} \end{array}} \right] f\left( t \right) \end{aligned}$$

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$$\begin{aligned}&\left[ \hspace{-1.49994pt}\left[ {{f^T}\left( t \right) \cdot \frac{{\partial Y(z(t))}}{{\partial z\left( t \right) }} \cdot e\left( t \right) } \right] \hspace{-1.49994pt}\right] \nonumber \\ {}&\quad = \left[ {\begin{array}{*{20}{c}} {{f^T}\frac{{\partial {Y_{:1}}}}{{\partial {z_1}}}}&{} \ldots &{}{{f^T}\frac{{\partial {Y_{:n}}}}{{\partial {z_1}}}}\\ \vdots &{} \ddots &{} \vdots \\ {{f^T}\frac{{\partial {Y_{:1}}}}{{\partial {z_n}}}}&{} \cdots &{}{{f^T}\frac{{\partial {Y_{:n}}}}{{\partial {z_n}}}} \end{array}} \right] e\left( t \right) \end{aligned}$$

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Also, one can write,

$$\begin{aligned}&{\left\{ {\frac{{\partial A(z(t))}}{{\partial z(t)}} \cdot z(t)} \right\} = \left[ {\begin{array}{*{20}{c}} {\frac{{\partial {A_{1:}}}}{{\partial {z_1}}}z}&{} \ldots &{}{\frac{{\partial {A_{1:}}}}{{\partial {z_n}}}z}\\ \vdots &{} \ddots &{} \vdots \\ {\frac{{\partial {A_{n:}}}}{{\partial {z_1}}}z}&{} \cdots &{}{\frac{{\partial {A_{n:}}}}{{\partial {z_n}}}z} \end{array}} \right] } \end{aligned}$$

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$$\begin{aligned}&{\left\{ {\frac{{\partial B(z(t))}}{{\partial z(t)}} \cdot f(t)} \right\} = \left[ {\begin{array}{*{20}{c}} {\frac{{\partial {B_{1:}}}}{{\partial {z_1}}}f}&{} \ldots &{}{\frac{{\partial {B_{1:}}}}{{\partial {z_n}}}f}\\ \vdots &{} \ddots &{} \vdots \\ {\frac{{\partial {B_{n:}}}}{{\partial {z_1}}}f}&{} \cdots &{}{\frac{{\partial {B_{n:}}}}{{\partial {z_n}}}f} \end{array}} \right] } \end{aligned}$$

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