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.

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