Forward Kinematics and Compatibility Equations of a Joystick based on a 12–6 Stewart Redundant Parallel Mechanism

Remote control with force feedback is one of the key features of tele-robotics systems. In this paper we investigate about the mechanical properties of a joystick based on a 12-6 Stewart platform. In most of classical 6 DOF parallel mechanisms, the forward kinematics cannot be described with closed-form solutions, which prevents from real-time control and straightforward implementations. In this contribution we address a 12-6 Stewart redundant parallel mechanism. We present the forward kinematics of this mechanism and we analyze the corresponding compatibility equations.
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We introduce intermediate variables, which allow revealing scale relationships among feature points. Accordingly, we derive 15 quadratic isomorphic compatible equations, and then we convert the forward kinematics equations into 12 linear compatible equations. Based on matrix algebra, we determine the final and unique solution of the redundant system. Furthermore, we study the influence of the initial values and the structural parameters. We give simulation results showing both the efficiency and the precision of the proposed compared to classical approaches. Our method outperforms previous works, allowing a better design and a faster control for accurate and effective 6 DOF joysticks.