This paper presents a generalized exponential formula for Forward and Differential Kinematics of open-chain multi-body systems with multi-degree-of-freedom, holonomic and nonholonomic joints. The notion of lower kinematic pair is revisited, and it is shown that the relative configuration manifolds of such joints are indeed Lie groups. Displacement subgroups, which correspond to different types of joints, are categorized accordingly, and it is proven that except for one class of displacement subgroups the exponential map is surjective. Screw joint parameters are defined to parameterize the relative configuration manifolds of displacement subgroups using the exponential map of Lie groups. For nonholonomic constraints the admissible screw joint speeds are introduced, and the Jacobian of the open-chain multi-body system is modified accordingly. Computational aspects of the developed formulation for Forward and Differential Kinematics of open-chain multi-body systems are explored by assigning coordinate frames to the initial configuration of the multi-body system, employing the matrix representation of SE(3) and choosing a basis for se(3). Finally, an example of a mobile manipulator mounted on a spacecraft, i.e., a six-degree-of-freedom moving base, elaborates the computational aspects.

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Mechanism and Machine Theory

Chhabra, R, & Emami, M.R. (M. Reza). (2014). A generalized exponential formula for forward and differential kinematics of open-chain multi-body systems. Mechanism and Machine Theory, 73, 61–75. doi:10.1016/j.mechmachtheory.2013.09.013