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Modular Kinematic Reduction of Closed-Chain Mechanisms Using Path Assembly and Defect Homotopy

Mohammad Dastranj, Jouni Mattila

arXiv:2609.11338v1cs.RO

TL;DR

Closed kinematic chains make modular modeling difficult because nonlinear closure constraints couple active and passive coordinates, while existing approaches can require mechanism-specific treatment. The paper introduces PACDM, defect-homotopy branch acquisition, and local differential reduction from assembled path closures. On a 7-DoF heavy-duty manipulator, predictor-corrector continuation retained comparable accuracy while reducing runtime by approximately 45.8-fold.

  • Problem

    Closed-chain mechanisms complicate modular modeling and often require topology-specific treatment, motivating a path-based closure representation without mechanism-specific minimal scalar derivations.

  • Method

    PACDM assembles closure relations from ordered transformation paths, selects independent constraints by rank-revealing analysis, uses defect homotopy to recover physical branches, and derives local active-to-passive mappings.

  • Results

    45.8-fold runtime reduction was achieved by predictor-corrector continuation versus defect homotopy at every trajectory step, with trajectory RMS errors of 8.4702 × 10^-10 and 7.3249 × 10^-10 for the two modules.

  • Takeaways & Limitations

    The framework provides a module-level route from path geometry through closure construction and branch acquisition to reduced kinematic mappings for the tested manipulator configurations and trajectory.

  • Takeaways & Limitations

    The evaluation does not establish global convergence or statistical robustness across arbitrary mechanisms, and the method requires a regular selected passive branch within coordinate bounds.

Abstract

from arXiv · show

Closed kinematic chains complicate modular modeling by coupling active and passive coordinates through nonlinear closure constraints. This paper presents a Path-Assembled Closure Differential Mapping (PACDM) framework for modular closure resolution and kinematic reduction. Each closure element compares two ordered transformation paths with common endpoints, with their mismatch expressed through the logarithm on SE(3) and the corresponding Jacobian assembled from local transformation derivatives. Multi-path modules are constructed from a minimal set of pairwise closure elements, while rank-revealing analysis selects locally independent scalar constraints. A defect homotopy recovers closure-consistent passive coordinates from approximate estimates along a feasible and regular continuation path. At regular configurations, implicit differentiation yields the local active-to-passive differential mapping, which is subsequently used in a predictor-corrector continuation procedure for prescribed motion. The framework is evaluated on a seven-degree-of-freedom heavy-duty manipulator containing two-path and three-path closed-chain modules. Comparison with Simscape Multibody yields trajectory root-mean-square errors below 8.5 x 10^-10 rad, while predictor-corrector continuation is approximately 45.8 times faster than applying defect homotopy at every trajectory sample.

1. Introduction

Closed-chain mechanisms offer important robotic benefits but complicate motion equations, control, and reusable modular modeling. The paper addresses this by assembling path-based closure relations, recovering physical branches, and deriving local mappings for modular kinematic reduction.

  • Closed kinematic chains improve actuator economy, precision, load capacity, agility, and structural stability, but increase motion-equation complexity and control difficulty.
  • Existing modular formulations often require topology-specific treatment when extending serial or tree-based methods to closed-chain systems.
  • The paper targets a systematic closure representation built directly from ordered transformation paths without mechanism-specific derivation of minimal scalar closure equations.
  • PACDM compares two ordered paths with common endpoints, assembles multi-path modules from a graph-theoretically minimal set of pairwise closures, and uses rank-revealing analysis to select independent scalar constraints.
  • Defect homotopy acquires closure-consistent passive coordinates, after which local implicit differentiation supports active-to-passive mapping and predictor-corrector continuation within independent modules.

2. Path-Assembly Formulation of Closed-Chain Kinematic Constraints

The framework represents each module closure by comparing ordered transformation paths and differentiating their SE(3) mismatch. It assembles multi-path constraints, selects independent residual coordinates, and forms local differential mappings from path derivatives.

  • Each closed-chain module is processed from module-level coordinates and identified ordered paths, with directly coupled nominal modules treated as one module.
  • 2.1. Two-Path Closure Element: Two-path closure elements compare ordered transformation products sharing initial and terminal frames, imposing equality of their terminal-frame poses.
  • 2.1. Two-Path Closure Element: The closure mismatch is mapped into a six-dimensional local residual using the selected branch of the SE(3) logarithm and the angular-first vee convention.
  • 2.1. Two-Path Closure Element: The logarithmic residual is only locally valid within its selected branch, with the principal rotational representation becoming nonunique at relative rotation angle π.
  • 2.2. Closure Element Differentials: Module-level raw residuals and Jacobians are assembled before rank-revealing analysis removes redundant residual coordinates and retains independent constraints.
  • 2.2. Closure Element Differentials: The resulting PACDM Jacobian gives local sensitivity of the six-component residual to closure-element coordinates, with small-residual inverse-left-Jacobian evaluation stabilized by a Bernoulli-series switch.
  • 2.2. Closure Element Differentials: PACDM assembles path-wise transformation derivatives, transports them through adjoint prefixes, and applies the inverse left Jacobian to differentiate the logarithmic residual.

3. Passive-Coordinate Resolution via Defect Homotopy

Defect homotopy converts an approximate passive-coordinate estimate into a closure-consistent configuration by continuously deforming an artificial closure problem into the physical one. Corrector iterations and continuation checks maintain a selected regular branch while active coordinates remain fixed.

  • Approximate passive coordinates, combined with prescribed active coordinates and practical configuration information, provide the starting estimate for passive-coordinate resolution.
  • 3.1. Defect Homotopy: The defect transformation interpolates from the initial closure mismatch to identity, making the initial configuration exact at λ=0 and recovering the physical closure at λ=1.
  • 3.2. Passive-Coordinate Corrector and Continuation: Only passive coordinates are corrected while prescribed active coordinates remain fixed throughout the homotopy.
  • 3.2. Passive-Coordinate Corrector and Continuation: Independent residual rows are selected from the defected passive Jacobian at the exact artificial starting point and held fixed during continuation.
  • 3.2. Passive-Coordinate Corrector and Continuation: At each fixed homotopy value, damped Gauss–Newton corrections update passive coordinates until the selected defect residual converges, with scaling and admissible bounds optionally enforced.
  • 3.2. Passive-Coordinate Corrector and Continuation: Continuation reuses each converged passive solution as the next zero-order predictor and accepts a trial only after residual and reciprocal-condition checks pass.
  • 3.2. Passive-Coordinate Corrector and Continuation: Successful acquisition assumes a continuous intended branch within passive-coordinate bounds and the logarithm domain, with a nonsingular selected defected passive Jacobian.
  • 3.2. Passive-Coordinate Corrector and Continuation: Reducing the homotopy increment improves numerical resolution but does not guarantee continuation through loss of regularity or a branch boundary.

4. Kinematic Reduction and Active-to-Passive Mapping

The framework recovers closure-consistent passive coordinates, checks local regularity, and differentiates the closure equations to obtain an active-to-passive mapping for reduced kinematics and trajectory continuation.

  • Kinematic reduction: Closure-consistent configuration recovery enables reevaluation of the original residual and PACDM Jacobian, whose rank structure supports the subsequent kinematic reduction.The artificial defect is removed before assessing the physical closure conditions and constructing reduced relations.
  • Regularity conditions: The numerical closure rank counts locally independent scalar constraints, while full column rank of the passive Jacobian block is required for local passive-coordinate resolution.Rank loss in the passive block invalidates the intended active-to-passive parametrization; a simultaneous drop in overall rank indicates singularity.
  • Active-to-passive mapping: A nonsingular selected passive block permits the implicit function theorem to define passive coordinates locally as functions of active coordinates.Differentiating the selected closure condition yields the local differential mapping, which is combined with PACDM blocks to map active variations to complete module-coordinate variations.
  • Trajectory continuation: The defect-homotopy procedure acquires a desired physical branch, after which the local differential mapping predicts passive-coordinate changes for small active-coordinate increments.The predictor is formed from a closure-consistent configuration and supports continuation along prescribed trajectories.
  • Trajectory continuation: The first-order predictor is followed by direct correction that restores the nonlinear physical closure equations at the new trajectory point.The preceding row selector is retained during correction, then the physical Jacobian, ranks, and selector are reevaluated after convergence.
  • Trajectory continuation: Successful direct correction avoids complete defect-homotopy reacquisition at every nearby trajectory step, while failed checks trigger branch reacquisition.Acceptance requires convergence of the selected residual, satisfaction of the complete physical residual tolerance, and a suitable passive-block reciprocal condition.

5. Simulation and Results

The framework is evaluated on a 7-DoF manipulator with two- and three-path closed-chain modules using fixed-configuration recovery and continuous-trajectory continuation tests. It achieves near-reference passive-coordinate accuracy and substantially reduces continuation runtime.

  • 5. Simulation and Results: The study uses a 7-DoF manipulator containing one two-path and one three-path closed-chain module, with Simscape Multibody serving as the numerical reference model.The tests examine passive-coordinate acquisition at fixed configurations and continuation along a prescribed trajectory.
  • 5.1. Problem Formulation: Rank-revealing analysis selects independent residual rows from the passive-block Jacobian before fixed-configuration and trajectory computations.The implementation uses SVD for rank determination and CPQR of the transposed passive block to select independent rows.
  • 5.2. Fixed-Configuration Test: 1.150 × 10−10 is the largest reported fixed-configuration absolute error against Simscape Multibody across the recovered passive coordinates.The three-path second module maintains absolute errors below 7 × 10−11 across the tested configurations.
  • 5.3. Continuous-Trajectory Test: 8.4702 × 10−10 and 7.3249 × 10−10 are the trajectory RMSE values for Modules 1 and 2, respectively, with maximum absolute errors below 3.1 × 10−9.The local first-order prediction followed by direct closure correction maintains passive-coordinate solutions for both two-path and three-path modules.
  • 5.3. Continuous-Trajectory Test: 45.80-fold lower runtime is achieved by predictor–corrector continuation than by applying defect homotopy at every trajectory step, while retaining comparable numerical accuracy.Average runtime decreases from 980.542 s to 21.411 s for the tested trajectory.
  • 5. Simulation and Results: The numerical study is limited to the tested fixed configurations and prescribed trajectory rather than statistical robustness over arbitrary mechanisms, initial conditions, or operating configurations.The authors present the experiments as verification of the formulation and implementation for these cases.

6. Conclusion

The framework assembles closed-chain constraints from ordered transformation paths, resolves passive coordinates through defect homotopy, and derives local active-to-passive mappings for continuation. Tests on a seven-degree-of-freedom manipulator matched Simscape Multibody closely, while predictor–corrector continuation substantially reduced runtime.

  • The PACDM framework constructs closure relations from ordered transformation paths and uses rank-revealing analysis to identify locally independent scalar constraints.A graph-theoretically minimal set of two-path closure elements is sufficient for multi-path modules.
  • Defect homotopy recovers closure-consistent passive coordinates by removing an introduced defect along a feasible, regular continuation path.This procedure traces the connected solution branch from an approximate passive-coordinate estimate to the physical closure equations.
  • At regular configurations, partitioning the selected PACDM Jacobian yields a local active-to-passive differential mapping for subsequent modular modeling or control.The resulting intermediate Jacobian expresses variations of all module coordinates in terms of active-coordinate variations.
  • The numerical study used a seven-degree-of-freedom heavy-duty manipulator with two closed-chain modules containing two and three kinematic paths.All four fixed-configuration tests attained the required passive-coordinate Jacobian ranks and agreed closely with Simscape Multibody references.
  • 8.4702 × 10^-10 and 7.3249 × 10^-10 rad trajectory root-mean-square errors were obtained for the two modules, with runtime reduced approximately 45.8-fold.Average runtime was 21.411 s versus 980.542 s when defect homotopy was applied at every trajectory step.
  • The reported validation is limited to tested configurations and trajectory and does not establish global convergence or statistical robustness across arbitrary closed-chain mechanisms.The method also requires a regular passive-coordinate branch within prescribed coordinate bounds and away from branch boundaries.

A. Construction of Local Homogeneous Transformations

Local homogeneous transformations are composed along selected kinematic paths from fixed frame relations and joint-dependent exponential-map factors. Their coordinate derivatives are assembled in traversal order, with inverse transformations used for reversed traversal and separate contributions retained for repeated coordinates.

  • Each path transformation is formed by composing local homogeneous transformations between consecutive frames, representing fixed geometric relations or relative joint motions.The transformations map coordinates between adjacent frames and are composed successively along the selected path.
  • The frame convention defines each adjacent-frame transformation through a rotation in SO(3) and a translation in R^3.
  • A reversed traversal uses the inverse of the corresponding geometric transformation.
  • Fixed local transformations are coordinate-independent, so their coordinate derivatives vanish.They are determined directly by the constant relative rotation and translation between adjacent frames.
  • Joint-dependent factors use the exponential map of a local motion vector, with a direction sign accounting for traversal along the selected path.The exponential maps a six-dimensional motion vector into SE(3).
  • Joint frames may align the local z-axis with the physical joint axis without restricting arbitrary joint geometry.Surrounding fixed transformations encode the joint’s actual orientation and location.
  • Elementary joint transformations are written explicitly from the selected local motion primitives for revolute and prismatic joints.The local motion vectors and corresponding elementary transformations provide the joint-dependent factors used in path assembly.
  • The complete path derivative sums local factor contributions after fixed and joint-dependent factors are arranged in traversal order.A coordinate absent from a factor contributes zero, while repeated occurrences contribute separately through their transported terms.
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