Modular Inverse Dynamics and Wrench Recovery for Closed-Chain Mechanisms Using Path Assembly

The gist: This paper develops a modular inverse-dynamics workflow that combines path-based motion reconstruction with recovery of uniquely determined actuator and joint-wrench outputs for rigid closed chains using path assembly

How it works

The core of the method involves a modular inverse-dynamics workflow that combines path-based motion reconstruction with recovery of uniquely determined actuator and joint-wrench outputs. This workflow extends the inherited kinematics through analytical second-order path differentiation to reconstruct passive velocities and accelerations on a regular closure branch The reconstruction problem concerns the prescribed motion of a rigid closed-chain manipulator with ideal revolute and prismatic joints, where the active trajectory and its first two time derivatives are given.

The analytical motion reconstruction relies on the closure representation and local differential mapping provided by path-assembly [11]. For each independently resolved closed-chain module, the active and passive coordinates form the vector qbm = col(qbm,a, qbm,p). Closure is expressed within the selected local logarithm chart as ρl = h Log (T⟨−⟩l) −1T⟨+⟩li∨ = 06. The resulting module-coordinate Jacobian Rm = ∂ρm/∂qbm is used to determine the passive coordinates from the active coordinates under the assumption that Rm has constant rank nm,p in a neighborhood of the considered configuration and that the selected passive block Rm,p is nonsingular.

Inverse Dynamics and Wrench Recovery

Following motion reconstruction, spatial Newton–Euler balances are assembled over an explicitly specified physical body-and-joint graph. For each physical moving body or rigid aggregate, a balance is assembled using its total spatial inertia. Joint wrenches are expressed as wj = Cjλj + Ujuj, where λj contains the reaction coordinates and active efforts. The load vector χ contains the reaction coordinates and active efforts in joint order. The assembly over the moving bodies yields Aχ = b, y = Lχ, where b = col(b1,..., bnb) contains the wrench requirements in (12).

Output Uniqueness and Recovery

Output uniqueness follows the established nullspace criterion for reaction components and their linear combinations. For a consistent balance system, a particular solution χ0 and a basis Nχ of ker(A) describe all solutions as χ = χ0 + Nχη, where η is arbitrary. The first condition removes the term lT k Nχη, and the remaining equivalences follow from ker(A)⊥ = range(AT). Consequently, any such dual vector dk ∈ R6nb recovers the output as yk = dTk b. The numerical calculation uses positive diagonal scales Se, Sx and Sy for the balance equations, load coordinates and outputs, respectively. A full singular value decomposition An = UΣVT [13] provides the retained factors Ur, Σr and V r for numerical rank r.

Diagnostics for Nonunique Reactions

Nonunique reactions are assessed through the compatibility of the logged Simscape loads with the rigid-body balance equations and ideal-joint conditions. Joint admissibility is assessed from ej, with moments and forces normalized by their respective scales, and the signed quantity STj wSSc,j. A small family distance supports compatibility of the projected loads with the computed solution family and is interpreted together with raw balance and joint admissibility.

Simulation and Results

The formulation is evaluated against Simscape Multibody under prescribed motion of a 7-degree-of-freedom heavy-duty hybrid serial–parallel manipulator with two closed-chain modules. The results show close agreement with the reference for passive positions, actuator efforts and uniquely recoverable constraint-wrench components. Table II reports that all seven actuator efforts are valid at every evaluated sample. Table III confirms that 37 of 54 force components and 28 of 54 moment components are valid throughout the run. The diagnostics support compatibility of the logged reactions with the admissible numerical load family without selecting a unique redundant reaction distribution.

Conclusion

This work connected analytical path-based motion reconstruction to physical inverse dynamics and recovery of uniquely determined loads. The prescribed-motion comparison showed close agreement for passive positions, all seven actuator efforts and the valid constraint-wrench components. The reaction diagnostics supported compatibility with the admissible numerical load family without selecting a unique redundant reaction distribution. These findings apply to the tested regular branch and rigid-body, ideal-joint model; compliant load sharing remains outside the formulation.

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The largest absolute error is 5.09 × 10−9 rad at q8

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The maximum absolute scaled residual across the 84 raw body-balance equations, and drel is the relative form of rfam in that section The temporal RMS values are 6.41 × 10−7 and 6.96 × 10−8, with maxima 7.04 × 10−7 and 9.05 × 10−8, respectively The ideal-joint admissibility residual is zero throughout.

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All eleven coordinates have full coverage

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The largest force and torque RMSEs are 1.27 × 10−5 N and 6.32 × 10−6 N.m, respectively All eleven coordinates have full coverage

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Table III reports 37 of 54 force components and 28 of 54 moment components as valid throughout the run Valid: uniquely recoverable components satisfying all recovery checks throughout the run

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The largest individual component RMSE, and all error statistics use valid outputs The valid counts include structurally zero ideal-joint components

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The requested set comprises all three components at each of the 18 joints; validity identifies uniquely recoverable outputs satisfying the checks in Section III-B, independently of their agreement with Simscape The two maxima may occur in different components

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The largest force and torque RMSEs are 1.27 × 10−5 N and 6.32 × 10−6 N.m, respectively

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Valid: uniquely recoverable components satisfying all recovery checks throughout the run

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The largest individual component RMSE, and all error statistics use valid outputs

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The largest force and torque RMSEs are 1.27 × 10−5 N and 6.

1. Bold header: Modular Inverse Dynamics for Closed-Chain Systems

This improvement enables AI systems to perform motion reconstruction by combining path-based motion reconstruction with recovery of uniquely determined actuator and joint-wrench outputs, addressing the problem where Rigid closed chains can admit indeterminate internal reactions even when their motion is fully specified.

2. Bold header: Unique Output Recovery via Nullspace Criterion

The system can reliably determine specific required outputs by utilizing Established nullspace and dual criteria identify and recover unique outputs, ensuring that selected outputs can be determined even when the complete load vector is nonunique.

3. Bold header: Compatibility Diagnostics for Load Families

The AI system can assess if physical loads are valid using checks like body balance, idealjoint admissibility and affine-family diagnostics, which helps in assessing compatibility with the admissible numerical load family without needing to select a specific unique reaction distribution.

Rigid closed chains can admit indeterminate internal reactions even when their motion is fully specified. This paper develops a modular inverse-dynamics workflow that combines path-based motion reconstruction with recovery of uniquely determined actuator and joint-wrench outputs. Analytical second-order differentiation of assembled transformation paths supplies passive velocities and accelerations on a regular closure branch. Spatial Newton--Euler balances are then assembled over the physical body-and-joint graph. Established nullspace and dual criteria identify and recover unique outputs, while balance, ideal-joint admissibility and affine-family diagnostics assess nonunique reference reactions. Evaluation against Simscape Multibody uses prescribed motion of a seven-degree-of-freedom heavy-duty hybrid serial--parallel manipulator under common rigid-body, ideal-joint assumptions. The results show close agreement with the reference for passive positions, actuator efforts and uniquely recoverable constraint-wrench components. Reaction diagnostics support compatibility with the admissible numerical load family.