Rational Motion Interpolation Under Kinematic Constraints of Spherical 6R Closed Chains

The work reported in this paper brings together the kinematics of spherical closed chains and the recently developed free-form rational motions to study the problem of synthesizing rational interpolating motions under the kinematic constraints of spherical $6R$ closed chains. The results presented in this paper are an extension of our previous work on the synthesis of piecewise rational spherical motions for spherical open chains. The kinematic constraints under consideration are workspace related constraints that limit the position of the links of spherical closed chains in the Cartesian space. Quaternions are used to represent spherical displacements. The problem of synthesizing smooth piecewise rational motions is converted into that of designing smooth piecewise rational curves in the space of quaternions. The kinematic constraints are transformed into geometric constraints for the design of quaternion curves. An iterative algorithm for constrained motion interpolation is presented. It detects the violation of the kinematic constraints by searching for those extreme points of the quaternion curve that do not satisfy the constraints. Such extreme points are modified so that the constraints are satisfied, and the resulting new points are added to the ordered set of the initial positions to be interpolated. An example is presented to show how this algorithm produces smooth spherical rational spline motions that satisfy the kinematic constraints of a spherical $6R$ closed chain. The algorithm can also be used for the synthesis of rational interpolating motions that approximate the kinematic constraints of spherical $5R$ and $4R$ closed chains within a user-defined tolerance.