Abstract
A companion study introduced joint durability into the dimensional design of the Theo Jansen walking linkage and found its classical "holy numbers" Pareto-dominated, but it modelled the revolute joints as ideal, clearance-free pins, so its wear figures were relative rankings, not a prediction of in-service degradation. Here we relax that idealization. We build a forward-dynamic model of the Jansen leg in which a revolute joint becomes a clearance joint with a continuous normal contact law (Lankarani-Flores, hysteresis-damped) and Ambrosio friction, integrated as a constraint-stabilized differential-algebraic system, and couple it to the Archard law in a wear->clearance->impact feedback loop. Three findings emerge. First, neglecting clearance underestimates the peak joint load: the clearance model gives a peak contact force of ~104 N at the load-bearing pin against ~48 N for the ideal joint (~2x amplification), rising to ~426 N when two joints carry clearance at once. Second, the coupling is strongly impact-sensitive--single trajectories are non-monotonic and can reverse the design ranking, a chaos consistent with the literature--so designs must be compared statistically; over an ensemble of 16 randomized phases the optimized joint is robustly more durable, with per-cycle wear ~9-7x lower (peak force ~4x lower) at one clearance joint and still ~1.7x lower on both with two (p<0.01 throughout). Third, the wear is strongly non-uniform--it concentrates on a ~10 deg load arc--so assuming uniform clearance growth underestimates local clearance growth by ~36x. The clearance-free durability advantage thus survives the chaotic, multi-joint, non-uniformly-worn coupling in the ensemble mean. We deliver the first clearance-coupled forward-dynamic model of the Jansen leg and specify a falsifiable protocol to test each prediction.
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Jun 20, 2026cs.RO
The Jansen linkage is a single-degree-of-freedom planar leg mechanism whose eleven "holy numbers" were evolved by Theo Jansen to optimize the foot-path gait alone, with no regard for the wear of its revolute joints. This paper introduces a durability objective into the design of the Jansen leg. A parametric forward-kinematic model (two-circle-intersection solver), an inverse-dynamic model (constraint-Jacobian / Lagrange-multiplier formulation of a seven-body, ten-joint system, independently cross-verified by a reduced-DOF energy method), and an Archard wear model are coupled to evaluate, for any set of link lengths, both gait quality and the per-cycle sliding wear at every pin. Because the wear is computed on ideal, clearance-free revolute joints, the resulting wear figures are a relative comparative ranking rather than an absolute life prediction. A bi-objective problem -- composite gait error versus total joint wear, subject to step-length, ground-clearance, duty-factor and assembly constraints -- is solved with NSGA-II. Under the adopted gait metric the classical Jansen design is Pareto-dominated: for a representative design, link-length adjustments within +/-29% simultaneously flatten the stance (-28%), smooth the stance velocity (-58%) and reduce total joint wear by ~56%. A sensitivity study shows the wear advantage is robust across a crank-speed x payload envelope (48%-56%) and identifies the link lengths that most strongly govern wear. A variance-based global (Sobol) analysis confirms that two link lengths dominate the wear variance, and a Monte-Carlo manufacturing-tolerance study shows the wear advantage degrades gracefully under realistic fabrication error. The framework provides a practical route to longer-lived walking linkages and a baseline for future wear-clearance-impact coupled studies.
Jichao Wang
Date pendingcs.RO
Precision reducers couple contact geometry, bearing support, structural deformation, and loading history. This paper presents a dynamic toolkit connecting distributed local contacts to the complete mechanical reaction path. Work-conjugate maps transfer displacement, reaction, and tangent contributions between contacts and rigid or reduced coordinates, with explicit allocation of contact and body elasticity. An implicit generalized-alpha solution distinguishes trial evaluations from accepted history. Contact records then support performance protocols and configured geometric or constitutive feedback. The numerical studies focus on mechanical coupling and pressure recovery. An idealized annular housing retains physical interfaces while its structural coordinates are reduced. Two reductions with similar static errors have cross-port response errors of 24.975 and 0.312 percent over the same frequency band relative to a common parent model. In a shared-pin example, a 20 micrometer radial displacement of one wheel changes the load on a second, fixed wheel by approximately 75 N. Removing cross-station compliance removes this incremental transfer on the tested sleeve-seating branch. A double-wheel cycloidal assembly relates torsional branch response to aggregate contact-load variation and normalized pressure fields. The pressure maxima remain sensitive to resolution despite small discrete force residuals. Further formulations specify how motion and contact records support precision, vibration, heat, wear, and durability models with their required inputs. The framework separates model representation and numerical resolution while retaining common definitions of motion, force, and observation.
Jiacheng Miao, Chao Liu, Qiliang Wang +2
May 7, 2026cs.RO
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