The adjoint trigonometric representation of displacements and a closed‐form solution to the IKP of general 3C chains

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.

On Orientation, Position, and Attitude Singularities of General 3R Chains

“…Given a third spear Λ Λ Λ N , orthogonal to Λ Λ Λ A and Λ Λ Λ B , the directed dual angle between Λ Λ Λ A and Λ Λ Λ B with respect to the normal Λ Λ Λ N is inferred by means of a suitable orientation signum s ∈ +1, −1 as ϑ AB ∼ = s •|ϑ AB | in formal accordance with the case of direction vectors [3,5]. For comparison purposes, a certificate ' θ 0' is introduced for dual angles as…”

“…The special case of intersecting spears with θAB = 0 defines a 'rotor' and the special case of parallel spears with ϑ AB = 0 defines a 'translator' [25]. 2 The pendant for the spherical case is introduced in [3]; a dual-angle concept permitting cylindrical motions along arbitrary reference spears is introduced in [5].…”

“…12) translates to the three cases rank N N) decreases, the chain features a singular configuration of 'orientation type' [21]. 5 Algebraically, this is expressed as…”

Connections between matrix products for 3-vectors and geometric algebra

“…Line geometry is "concerned with the set of lines of three-dimensional space" and has "relations to mechanics and spatial kinematics" and "therefore applications [..] in mechanism design and robotics" [26]. In this context, the present article has been motivated by an analysis [4] in which the kinematics of spatial chains with three cylindrical joints are solved analytically by employing the algebra and geometry of oriented lines in space. In the scope of that analysis, the symmetric matrix a b ∈ sym(3) has been defined for two linear independent vectors a, b ∈ R 3 as…”

“….= {AA}0 by means of the reversion A of Equation (C.2) in the appendix. 4 A geometric algebra Cp,q,r ∼…”