Kinematic Synthesis of Spatial R-r Dyads for Path following Revisited Using the Rotation Matrix Approach

We revisit the dimensional synthesis of a spatial two-link, two revolute-jointed serial chain for path following applications, focussing on the systematic development of the design equations and their analytic solution for the three precision point synthesis problem. The kinematic design equations are obtained from the equations of loop-closure for end-eeector position in rotation-matrix/vector form at the three precision points. These design equations form a rank-deecient linear system in the link-vector components. The nullspace of the rank deecient linear system is then deduced analytically and interpreted geometrically. Tools from linear algebra are applied to systematically create the auxiliary conditions required for synthesis and to verify consistency. An analytic procedure for obtaining the link-vector components is then developed after a suitable selection of free choices. Optimization over the free choices is possible to permit the matching of additional criteria and explored further. Examples of the design of optimal two-link coupled spatial R-R dyads are presented where the end-eeector interpolates three positions exactly and closely approximates an entire desired path. NOMENCLATURE ~ A 1 Vector denoting position of the base joint. ~ B 1 Vector representing link 1 in the rst position. ~ C 1 Vector representing link 2 in the rst position. E Combined error measure for positions and ori-entations. ~ P i Vector representing the end eeector location in the i th position. R^ s a ; ] The rotation matrix denoting a rotation about an axis ^ s a. R 1 Coupling ratio between joints 1 and 2. ^ s a The axis of rotation. ^ u The axis of rotation of joint 1. ^ v The axis of rotation of joint 2. ^ w The equivalent axis of rotation of end eeector. ~ X Right nullspace vector. ~ Y Left nullspace vector. Angle of rotation about axis ^ u of joint 1. Angle of rotation about axis ^ v of joint 2. Composite angle of rotation about axis ^ w. Angle of rotation