Constrained Optimal Control of Cable-Driven Parallel Robots Based on SDRE

Abstract

This paper addresses constrained optimal control of Cable-Driven Parallel Robots (CDPRs) in details. As it is known, in a CDPR mechanism, cables should remain in tension at all movement maneuvers in contrast to conventional robots. In this paper, considering this constraint, an optimal control scheme is presented for this class of robots. To this end, first, the general dynamic model of the CDPR is converted to 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 optimal control algorithm based on the Hamiltonian and Karush-Kuhn-Tucker (KKT) conditions is proposed using the state-dependent coefficient parameterization form. The proposed control scheme is formed based on the state-dependent Riccati equation (SDRE), and the stability of the closed-loop system is proved using Lyapunov second method. Simulation results show the effectiveness of the proposed control algorithm in terms of tracking and coping with the uncertainties and external disturbances.

Supported by Electrical Engineering Department of Tehran Polytechnic.

Appendices

A Appendix

The dynamic matrices of the 3 DOF planar CDPR are presented as follows [9]:

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

where parameters $m = 2.5\left( \boldsymbol{kg} \right)$, ${I_z} = 0.03\left( \boldsymbol{kg\,{m^2}} \right)$, and are considered. The position of the cable attachment points is considered as Article [9].

B Appendix

Consider the following notations:

Notation B.1: The symbol $\left[ \left[ {.} \right] \right]$ indicates that the inside parameter is a vector, e.g.,

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

(64)

which N is the number of the vector elements.

Notation B.2: The symbol $\left\{ . \right\}$ indicates the tensor space in which the inside parameter dimensions must be homogenized according to the problem.

Notation B.3: ${\theta _{i:}}$ and ${\theta _{:i}}$ represent the $i_{th}$ row and $i_{th}$ column of matrix ${\theta }$, respectively.

Since algebraically, the derivatives of the matrices R(z(t)) and Q(z(t)) in terms of z(t) create more extensive dimensions than R(z(t)) and Q(z(t)), the vectors ${\left[ \left[ {{e^T}\left( t \right) \cdot \frac{{\partial Q(z(t))}}{{\partial z\left( t \right) }} \cdot e\left( t \right) } \right] \right] }$ and ${{\left[ \left[ {{f^T}\left( t \right) \cdot \frac{{\partial R(z(t))}}{{\partial z\left( t \right) }} \cdot f\left( t \right) } \right] \right] }}$ are used. With regards to Notation B.1, after multiplying their internal parameters, an answer vector is formed for each of them. Hence, the dimensions of the problem are not affected. According to the discussed topics and taking advantage of Notation B.2, the tensor space ${\left\{ {{e^T}\left( t \right) \cdot \frac{{\partial Q(z(t))}}{{\partial z\left( t \right) }} \cdot e\left( t \right) } \right\} }$ and ${\left\{ {{f^T}\left( t \right) \cdot \frac{{\partial R(z(t))}}{{\partial z\left( t \right) }} \cdot f\left( t \right) } \right\} }$ can be calculated. The exact amount of vectors ${\left[ \left[ {{e^T}\left( t \right) \cdot \frac{{\partial Q(z(t))}}{{\partial z\left( t \right) }} \cdot e\left( t \right) } \right] \right] }$ and ${\left[ \left[ {{f^T}\left( t \right) \cdot \frac{{\partial R(z(t))}}{{\partial z\left( t \right) }} \cdot f\left( t \right) } \right] \right] }$ are obtained as Eqs. (65) and (66).

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

(65)

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

(66)

According to Notation B.3, the mentioned tensor spaces are computed.

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

(67)

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

(68)