Forward kinematics of a cable-driven parallel robot with pose estimation error covariance bounds

This paper presents four forward kinematics algorithms for an over-constrained six degree-of-freedom (DOF) cable-driven parallel robot (CDPR) that in addition to computing the end-effector's pose, also provide covariance bounds on the pose estimation error. The first two proposed methods are based on cable-length and cable-length-squared loop-closure equations and the use of unconstrained attitude parameterizations to describe the orientation of the CDPR end-effector. The second pair of methods involve constrained attitude parameterizations and are also based on cable-length and cable-length-squared loop-closure equations. Nonlinear least-squares optimization is used in each of these methods to iteratively compute the forward kinematics solution and determine covariance bounds on the pose estimation error. Attitude identities are used to obtain analytic expressions for the computations whenever possible. The forward kinematics algorithms are validated through Monte-Carlo simulations, where Euler-angle-sequence, quaternion, and DCM parameterizations of the end-effector attitude are implemented and the accuracy of the covariance bounds is demonstrated. It is also shown that the method based on the cable-length-squared loop-closure equations yields improved convergence properties compared to the cable-length loop-closure equations.