Synthesis of Spatial Two-link Coupled Serial Chains

We address the synthesis of serial chain spatial mechanisms with revolute joints in which the rotations about the joints are coupled via cables and pulleys. Such coupled serial chain mechanisms o er a middle ground between the more versatile and compact serial chains and the simpler closed chains by combining some of the advantages of both types of systems. In particular, we focus on the synthesis of single degree-of-freedom, coupled serial chains with two revolute joints. We derive precision point synthesis equations for two precision points by combining the loop closure equations with the necessary geometric constraints in terms of the unknown mechanism parameters. This system of equations can now be solved linearly for the link vectors after a suitable selection of free choices. We optimize over the free choices to generate an end e ector trajectory that closely approximates a desired end e ector trajectory for motion generation and path following applications. NOMENCLATURE ~ A1 Vector denoting position of the base joint. ~ B1 Vector representing link 1 in the rst position. ~ C1 Vector representing link 2 in the rst position. E Combined error measure for positions and orientations. ~ Pi Vector representing the end e ector location in the i position. Qi A quaternion. Q[ŝa; =2] A unit quaternion corresponding to a rotation about axis ŝa. R[ŝa; ] The rotation matrix denoting a rotation about an axis ŝa. ~ R[ŝa; ] The quaternion rotation operator. R1 Coupling ratio between joints 1 and 2. ŝa The axis of rotation. û The axis of rotation of joint 1. v̂ The axis of rotation of joint 2. ŵ The composite axis of rotation of end e ector. Angle of rotation about axis û of joint 1. Angle of rotation about axis v̂ of joint 2. Composite angle of rotation about axis ŵ. Angle of rotation Quaternion multiplication.