# The Adjoint Trigonometric Representation of Displacements and a Closed-Form Solution to the IKP of General 3C Chains

 Title The Adjoint Trigonometric Representation of Displacements and a Closed-Form Solution to the IKP of General 3C Chains Publication Type Journal Article Year of Publication 2020 Authors Bongardt B Journal Journal of Applied Mathematics and Mechanics Date Published 05/2020 Abstract Based on the representation of rigid body displacements as adjoint matrices, the article introduces the adjoint trigonometric representation of displacements (ATRD) as a further generalization of the trigonometric representation of rotations. In comparison to the dual Rodrigues–Euler–Gauß–Gelman equation, recently reported for affine screw displacements with arbitrary, fixed pitches, the ATRD is built upon a product of a unit line and a dual angle, instead of upon a product of a unit screw and a real angle. Due to this conceptual difference, the ATRD requires four independent parameters of a unit line instead of five when parametrizing a displacement along a unit screw. As a consequence for computational kinematics, the ATRD permits transferring the analytic solution to the inverse kinematics problem (IKP) of 3-DOF, general, spherical 3R-chains into a closed-form solution to the IKP of 6-DOF, general, affine 3C-chains. URL https://onlinelibrary.wiley.com/doi/epdf/10.1002/zamm.201900214 DOI 10.1002/zamm.201900214