Closed Form Expressions for Higher Derivatives of Screw Systems

Abstract

A central element in the kinematic analysis is the determination of partial derivatives of the twist of a member in a serial kinematic chain with respect to the joint parameters (angles, translations). This requires partial derivatives of the screw system, generated by a given ordered set of joint screws. While the closed form expression of first and second order derivatives are widely known in terms of screw products, and even the derivatives up to fifth order have been reported, a general closed form expression for derivatives of arbitrary order remained an open issue. Such a closed form expression of partial derivatives of the joint screws for any order is reported in this paper. The final result for the n-th partial derivative involves n-fold nested Lie bracket (screw products) of the joint screws. The crucial observation that gives rise to the closed form expression is that the derivative is given as a sum of terms with complementary joint index ranges.

Derivatives of Screw Systems in Body-Fixed Representation

“…It is straightforward to derive explicit expressions for higher order derivatives noting the bilinearity of the Lie product and inserting (6). This has been pursued in [2,6,7,10,11], and the closed form for arbitrary orders was presented recently [9]. It remains to derive closed form relations for the body-fixed twists.…”

“…Moreover, comparing (9) with (6) shows that the derivative of the spatial twist is identically zero, exactly when the partial derivative of the body-fixed twist is not. The second partial derivative follows immediately from as (9) and the bilinearity of the Lie bracket as…”

“…It is important to notice the index range in (9). As it is clear from the kinematics that the contribution of the ith joint twist to the body twist only depends on the relative configuration of body i and body r , and thus on the joint variables q i+1 , .…”

“…However, explicit closed form expressions have not yet been reported. In this chapter such closed form expressions are presented complementing these for the known spatial representation [9].…”

“…It is well-know that for a kinematic chain the partial derivatives of the spatial screw representation is given in terms of Lie brackets, and explicit expressions have been reported [6,7,9,11]. Partial derivatives of the body-fixed representation are also of great importance for MBS kinematics and dynamics, as well sensitivity analysis and optimization [5,13].…”