Constrained Control of Fully-Constrained Cable-Driven Parallel Robots With Elastic Cables: An Optimal Robust Approach

Abstract

The paper discusses the robust optimal control of <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Cable-Driven Parallel Robots</i> (CDPRs) with elastic cables, addressing the limitations of treating cables as rigid bodies. This simplification is inadequate for applications with large workspaces, where cable vibrations significantly affect performance. The proposed control strategy ensures that cables remain under tension by applying positive inputs, thus preventing compressive forces. The initial step involves transforming the nonlinear dynamics of the CDPR system, which includes both the elastic cables and motor dynamics, into a linear framework using <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">State-Dependent Coefficient</i> (SDC) parameterization. This allows for the retention of nonlinearities within the system and input matrices while encapsulating dynamic uncertainties and external disturbances. Subsequently, the constrained control is robustly designed utilizing optimization techniques rooted in <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Hamilton-Jacobi-Bellman</i> (HJB) equations and <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Karush-Kuhn-Tucker</i> (KKT) conditions. The robust optimal constrained controller is formulated using the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Generalized State-Dependent Riccati Equation</i> (GSDRE), followed by stability analysis conducted through Lyapunov's second method. Finally, the effectiveness of the proposed method is validated through simulations, demonstrating its capability to maintain cable tension and damp vibrations, which are critical for the operational efficiency of CDPRs in practical applications.