Shape Control of Manipulators with Hyper Degrees of Freedom

This paper provides a theoretical framework for controlling a manipulator with hyper degrees of freedom (HDOF). An HDOF manipulator has the capability to achieve various kinds of tasks. To make full use of its capability, shape control is proposed here; that is, not only the tip of a manipulator, but also its whole body is controlled. To formulate control objectives for shape control, we define a shape correspondence between an HDOF manipulator and a spatial curve that prescribes a desired shape. The shape correspondence is defined by using solutions of a nonlinear optimization problem termed the shape-inverse problem. We give theorems on the existence of the solutions, and on an existence region that allows us to convert shape-control problems into more tractable ones. A shape-regulation control problem is considered first to bring an HDOF manipulator onto a given time-invariant curve. The idea of estimating the desired curve parameters is the crucial key to solving the problem by Lyapunov design. The derived shape-regulation law includes the estimator, which infers the desired curve parameters corresponding to the desired joint positions on the curve. The idea of the desired curve-parameter estimation is also effective for shape tracking where a time-varying curve is used for prescribing a moving desired shape. Considering an estimator with second-order dynamics enables us to find two shape-tracking control laws by utilizing conventional tracking methods in manipulator control. We show the simulation results of applying the derived shape-tracking control laws to a 20-DOF manipulator. KEY WORDS—manipulator, shape control, dynamics, kinematics, kinematic degrees of freedom, geometry of curves The International Journal of Robotics Research Vol. 18, No. 6, June 1999, pp. 584-600, ©1999 Sage Publications, Inc.