A geometric approach for forward kinematics analysis of a 3-SPS/S redundant motion manipulator with an extra sensor using conformal geometric algebra

Abstract

This paper proposes a geometric approach for forward kinematics analysis of a 3-SPS/S redundant motion mechanism using conformal geometric algebra (CGA). The forward kinematics of a parallel kinematic mechanism is generally very complicated and difficult to analyze. The proposed geometric method is useful because it provides simple, fast, and complete solutions to the problem and helps determine the relationship between the joints and end-effector without an iterative process. Thus, we introduce a geometric approach using CGA in an intuitive, stepwise fashion. We also use an extra sensor to provide more positional information, thereby allowing a unique solution to be selected geometrically from among the multiple solutions found using the geometric approach. In the mechanism considered herein, three identical legs linked by prismatic actuators are attached to a moving platform and to the base by two passive spherical joints. A passive leg is present at the center of the mechanism; it connects the center of the base to the platform, constraining the platform's movement. The added components constrain the movement of the platform, making it possible to analyze the mechanism using a geometric approach. Herein we present performance comparisons that validate use of the proposed approach in real-time applications and demonstrate its low computational load.