# 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 2018 Authors Bongardt B Type of Article - PREPRINT - Keywords adjoint matrices, adjoint trigonometric representation of displacements, dual numbers, Euler–Rodrigues rotation formula, inverse kinematics, line geometry, Plücker vectors, principle of transference, screw theory, trigonometric representation of rotations 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 Euler-Rodrigues 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 parameterizing 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. Full Text PREPRINT / see researchgate entry