Redundant Manipulator Kinematics and Dynamics on Differentiable Manifolds

Edward J. Haug and Adrian Peidro, "Redundant Manipulator Kinematics and Dynamics on Differentiable Manifolds," ASME. J. Comput. Nonlinear Dynam. November 2022; 17(11): 111008. https://doi.org/10.1115/1.4055313

​Redundant manipulators provide versatility and enhance performance potential by using a greater number of inputs than outputs to be controlled. This flexibility enables obstacle avoidance and performance optimization, in addition to achieving specified outputs. However, this requires new analytical and computational tools that enable control strategies that select from an infinite number of admissible inputs that yield the specified output, while realizing enhanced manipulator performance. Topological attributes of Euclidean space, in which manipulators function, are employed to create a differentiable manifold structure that provides explicit parameterization of the infinite number of inputs associated with a desired output. This representation is employed to demonstrate achievement of output specifications, while avoiding obstacles in the manipulator’s workspace by switching between nominal input trajectories far from obstacles to self-motions that prevent collisions otherwise. It is also applied for mapping the self-motion manifold of a seven degree of freedom robot arm that is impossible to analyze using existing methods. A time domain implementation of the parameterization is presented that provides velocity and acceleration information required for control of manipulator dynamics. Computational methods are presented that enable real-time implementation of results derived on modern high-speed microprocessors, for use in computer-based manipulator control systems.