Determining the Number of Inverse Kinematic Solutions of a Constrained Parallel Mechanism Using a Homotopy Algorithm

This paper numerically determines the number of real-valued inverse kinematic solutions to a constrained parallel mechanism composed of three triangular platforms. The base and middle platforms are connected by three fixed-length legs, while the middle and distal platforms are connected by three variable length legs that extend out of the fixed-length legs in a collinear fashion. All legs are connected to the platforms via spherical joints at the corners. This mechanism is intended to replicate the motion of a human shoulder girdle. The constrained parallel mechanism has a multivalued solution to the inverse kinematics problem. A homotopy method was used to numerically compute the inverse kinematic solutions for over 100 cases. Each case was filtered for the number of real-valued solutions. The maximum number of real solutions was found to be 8, but in some cases there were fewer solutions.