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.

中文翻译：

---

位移的伴随三角表示和通用3C链IKP的闭式解

在将刚体位移表示为伴随矩阵的基础上，本文介绍了位移的伴随三角表示（ATRD），作为旋转的三角表示的进一步概括。与最近报道的具有任意固定螺距的仿射螺杆位移的双重Rodrigues–Euler–Gauß–Gelman方程相比，ATRD建立在单位线和双重角度的乘积上，而不是在单位乘积上螺丝和真实角度。由于此概念上的差异，在对沿单位螺丝的位移进行参数化时，ATRD要求单位线的四个独立参数，而不是五个。作为计算运动学的结果，ATRD允许将解析解转移到3自由度反运动学问题（IKP），一般来说，