Chapter 4 : Instantaneous Kinematic Analysis

The instantaneous kinematic analysis of a manipulator defines the direction and magnitude (translational and rotational) of impending motion at a specific point in time. This analysis is position dependent, and assumes apriori knowledge of the position information. From the position kinematic analysis, a kinematic transformation between joint space and tool space can be represented as: ( ) x = f θ (4.1) Instantaneous or velocity analysis follows directly from the position analysis. Here, the input velocity vector, ω is mapped into the output space velocity vector, v, by the matrix, J called the Jacobian of the manipulator: v J = ω . (4.2) This matrix equation demonstrates that the Jacobian is necessarily a linear relationship between the input and output velocities. Obtaining the Jacobian, however, may involve a complex process. When the vector function f from the position analysis is available explicitly, velocity analysis (the Jacobian) results from direct differentiation. However, in most parallel manipulators the kinematic equations are implicit, making differentiation non-trivial. A number of alternative methods are available for Jacobian determination. For example, the Jacobian can be created from a set of linear velocity equations based on the known manipulator position information. Screw theory provides another well-known approach to developing a system Jacobian. However, these approaches result in several complications when applied to parallel manipulators. In this analysis, although the kinematic position equations are implicit, direct differentiation is used to obtain the Jacobian matrix.