Kinematic method links joint motion to platform behavior across robot mechanisms
Lie-Algebraic Bell Recurrences for Arbitrary-Order Twist Jets and Parallel-Mechanism Closure
Robotics
Summary
Robots and mechanical systems often have parts connected by joints that move in specific ways. This paper presents a mathematical method to describe motion changes of any order (speed, acceleration, jerk, and beyond) for complex assemblies that include serial chains and parallel mechanisms. The authors use special polynomials (called Bell polynomials) and algebraic tools to keep track of how motions combine, even when joints interact non-linearly. They validate their formulas by testing on different robotic arms and mechanisms, confirming accurate and consistent results.
What this means in practice
- •For robotic system designers: Calculate precise multi-order motion properties of robots with complex joint arrangements to optimize control and stability.
- •For mechanical engineers: Analyze and predict motion behavior in parallel mechanisms or linkages that include cylindrical joints using exact kinematic formulas.
Authors
Daniel Condurache
Abstract
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.