An Overview of Formulae for the Higher-Order Kinematics of Lower-Pair Chains with Applications in Robotics and Mechanism Theory

This paper presents a novel framework for Jacobian computation in motion optimization problems involving multi-link systems, where physical quantities are represented using higher-order time derivatives. In motion optimization of robots and humans, cost functions may incorporate higher-order time derivatives, such as jerk or the time variation of forces, to capture smoothness and perceptual characteristics, particularly in motion skill analysis and expressive behaviors, thereby necessitating Jacobian computations involving these quantities. However, such Jacobians are typically computed using numerical or automatic differentiation without explicitly exploiting the underlying multi-link structure, which can lead to increased computational cost and numerical instability. To address this limitation, we propose a structured Jacobian formulation for motion optimization, based on the comprehensive motion computation framework, in which physical quantities and their higher-order time derivatives are systematically represented along the multi-link structure. The proposed method systematically derives analytical expressions for Jacobians of kinematic and dynamic quantities, including momentum, forces, and joint torques, with respect to generalized coordinates and their higher-order derivatives. The resulting framework is applicable to both direct and inverse optimization. Through numerical experiments, we demonstrate that the proposed method improves computational efficiency compared to numerical and automatic differentiation, while achieving comparable accuracy. Furthermore, we demonstrate its effectiveness in inverse optimization by recovering cost function weights from motion data. Together, these results indicate that the proposed formulation provides a scalable and structured computational foundation for motion optimization involving higher-order time derivatives in multi-link systems.

This paper develops an arbitrary-order kinematic construction that links serial propagation, parallel-mechanism closure, and rigid-platform point fields within one dual screw framework. A cylindrical joint is retained as one native physical block, with revolute and prismatic joints obtained as special cases. For each fixed joint axis, ordinary Bell polynomials organize the derivatives of the exponential factor; across a chain, the noncommuting factors remain in their physical order. Initial-frame prefix and terminal-resolved covariant formulas then produce equivalent representations of the serial twist jet. For a parallel mechanism, repeated Leibniz differentiation, with joint-level derivatives organized by Bell polynomials, yields an arbitrary-order triangular active-passive closure recurrence: the same passive Jacobian is solved at every derivative order at a regular configuration, while the right-hand side contains only prescribed active data and lower-order jets. The resulting platform twist jet is mapped exactly to the point-independent affine invariants of the velocity, acceleration, jerk, and snap fields. The validation is deliberately complementary: a generic 3C chain with noncoplanar axes and nonzero rotational and translational cylindrical coordinates tests ordered serial propagation, an RR+RRR spherical wrist tests active-passive closure, and a Hunt-type 6-RUS mechanism with six active revolute joints tests an independently reconstructed platform jet and its affine fields. Independent differentiation of the rigid motion, evaluation of the affine fields, and the differentiated branch closures all agree through fourth order with residuals below $10^{-12}$ in the corresponding SI units. The formulation is purely kinematic and applies at configurations where the selected active-passive partition is regular.

The Lie group SE(3) of isometric orientation preserving transformation is used for modeling multibody systems, robots, and Cosserat continua. The use of these models in numerical simulation and optimization schemes necessitates the exponential map, its right-trivialized differential (often referred to as tangent operator), as well as higher derivatives in closed form. The $6\times 6$ matrix representation of the differential, $\mathbf{dexp}_{\mathbf{X}}:se\left( 3\right) \rightarrow se\left( 3\right)$ , and its first derivative were reported using a $3\times 3$ block partitioning. In this paper, the differential, its first and second derivative, as well as the Jacobian and Hessian of the evaluation maps, $\mathbf{dexp}_{\mathbf{X}}\mathbf{Z}$ and $\mathbf{dexp}_{\mathbf{X}}^{T}$, are reported avoiding the block partitioning. For all of them, higher-order approximations are derived. Besides the compactness, the advantage of the presented closed form relations is their numerical robustness when combined with the local approximation. The formulations are demonstrated for computation of the deformation field and the strain rates of an elastic Cosserat-Simo-Reissner rod.