|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||In Press|
|Journal||Mechanism and Machine Theory|
|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|
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.