PDE-based control synthesis for a planar cable-driven continuum arm

Classical wave and beam models cannot describe the arc-like planar configuration of a continuum arm. Based on the Cosserat rod theory, a strain vector and the rotation matrix are introduced into this work. Two groups of nonlinear partial differential equations (PDEs) are obtained by analyzing kinetic and potential energy densities and virtual power done by the cable tension, which are explicit both in the spatial coordinates and in the physical parameters. Although involving algebraic–trigonometric descriptions, the obtained dynamical model is the most concise formulation but difficult to use for control design. We develop a feedforward plus proportional–derivative feedback boundary control scheme for the nonlinear PDE system to regulate the planar shape of the continuum arm and then linearize the dynamical equations about the equilibrium to assess the control performance. Some interesting results are the closed-loop stability and the well-posedness analysis of the linearized model, which have not previously been noted. Finally, numerical results validate the dynamical equations and the performance of the controller.