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Modular Inverse Dynamics and Wrench Recovery for Closed-Chain Mechanisms Using Path Assembly
Mohammad Dastranj, Jouni Mattila
TL;DR
Rigid closed chains can have indeterminate internal reactions despite fully specified motion, making it necessary to identify which actuator and wrench outputs are uniquely determined. The paper combines path-based motion reconstruction with nullspace-based recovery and evaluates it against Simscape Multibody, finding close agreement for passive positions, actuator efforts, and uniquely recoverable constraint-wrench components.
Problem
Redundant rigid constraints can leave reaction distributions nonunique even when motion is closure-consistent, so requested outputs must be invariant across admissible balance solutions.
Source: §I · p. 1Method
The workflow combines path-based motion reconstruction with balance-equation nullspace and dual criteria to recover uniquely determined actuator and joint-wrench outputs.
Source: §I · p. 1Results
The prescribed-motion comparison showed close agreement with the reference for passive positions, all seven actuator efforts, and valid constraint-wrench components.
Source: §V · p. 6Takeaways & Limitations
Balance, ideal-joint admissibility, and affine-family diagnostics support compatibility of the logged reactions with the numerical load family.
Source: B. Reconstruction and Load-Recovery… · p. 6
Abstract
from arXiv · showhide
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.
I. INTRODUCTION
Closed-chain motion can be fully specified while redundant constraints leave reaction distributions indeterminate, so recovery must distinguish invariant outputs from nonunique loads. The paper combines module-local path-based motion reconstruction with globally assembled body balances to recover uniquely determined actuator efforts and joint-wrench outputs.
I. INTRODUCTION
- Redundant rigid constraints can admit multiple reaction distributions even when motion is closure-consistent, so requested outputs must be tested for invariance across admissible balance solutions.Selected components may remain unique despite indeterminate internal reactions.Source: §I · p. 1
- The paper addresses the un demonstrated integration of reaction-uniqueness analysis into PACDM-based inverse dynamics for physical actuator and joint-wrench recovery.PACDM supplies closure resolution and local active-to-passive mappings from ordered transformation paths [11].Source: §I · p. 1
- Its workflow combines module-local closure reconstruction with globally assembled body-level balance equations and recovery of uniquely determined loads.Analytical second-order path differentiation supplies passive velocities and accelerations on a regular closure branch; nullspace and dual criteria identify determined outputs.Source: §I · p. 1
- The evaluation prescribes motion of a 7-degree-of-freedom heavy-duty hybrid serial–parallel manipulator with two closed-chain modules and compares results against Simscape Multibody.The common rigid-body, ideal-joint setting isolates reconstruction and load recovery from measurement noise and unmodeled physical disturbances.Source: §I · p. 1
A. Problem Formulation and Conventions
- The reconstruction problem gives the active trajectory and its first two derivatives while determining passive motion from closure on a selected regular branch.Module descriptions use ordered transformation paths sharing endpoints, and the reconstructed motion supplies kinematic inputs to physical body balances.Source: A. Problem Formulation and Conventions · p. 2
B. Analytical Motion Reconstruction
Analytical differentiation of assembled transformation paths extends closure-based motion reconstruction through second order. Module-local closure equations yield passive motion and its derivatives on a regular branch, with checks against full closure conditions.
B. Analytical Motion Reconstruction
- For each module, ordered paths sharing endpoints form logarithmic closure residuals whose coordinate Jacobian is reduced to selected passive equations.The module residuals enforce equality of path transformations, although their scalar components may be dependent.Source: B. Analytical Motion Reconstruction · p. 2
- Local reconstruction determines passive coordinates from active coordinates when the closure Jacobian has constant rank and the selected passive block is nonsingular.The resulting module maps are assembled in global coordinate order to map prescribed active rates to full coordinate rates.Source: B. Analytical Motion Reconstruction · p. 2
- Trajectory reconstruction uses an initialized row selection, prediction from the prior accepted configuration, Newton correction, and checks against full closure residuals.The resulting motion is checked through full velocity closure equations and acceleration mismatch matrices of the path pairs.Source: B. Analytical Motion Reconstruction · pp. 2–3
- Successive product-rule differentiation of path factors supplies first and second derivatives, with fixed factors contributing zero derivatives.Joint-factor derivatives use a constant generator, and the second derivative of an appended product includes the cross term 2 ˙T ˙F.Source: B. Analytical Motion Reconstruction · p. 2
- The closure acceleration bias comes from second derivatives of the logarithmic residual at closure and tangent velocity, where the relative transform derivative vanishes.Pairwise bias terms are assembled in residual order, then used with differentiated closure constraints to determine passive acceleration.Source: B. Analytical Motion Reconstruction · pp. 2–3
A. Physical Inverse-Dynamics Assembly
Physical inverse dynamics assembles spatial Newton–Euler balances over moving bodies and their incident joints. Ideal-joint reaction coordinates and active efforts are mapped into body balances and requested outputs with explicit wrench frames and physical meanings.
A. Physical Inverse-Dynamics Assembly
- For each moving body, spatial Newton–Euler balance computes the net wrench required from incident joints using body-frame motion, inertia, gravity, and known external loads.A balance is assembled for each physical moving body or rigid aggregate using its total spatial inertia.Source: A. Physical Inverse-Dynamics Assembly · p. 3
- Ideal revolute and prismatic joints represent their wrench as five reciprocal constraint-reaction coordinates plus an effort along the active joint motion screw.The reaction basis is orthogonal to the joint motion screw, reflecting the workless character of ideal constraints; passive joints have no actuation term.Source: A. Physical Inverse-Dynamics Assembly · p. 3
- The body-balance matrix places each joint wrench on its incident bodies with opposite action–reaction signs and transports it to the body reference frames.The joint follower frame and wrench reference are explicitly accounted for in the transport.Source: A. Physical Inverse-Dynamics Assembly · p. 3
- Requested outputs are selected by rows that preserve each actuator effort or wrench component’s joint, frame, component, and physical meaning.The body twist derivative is an ordinary derivative of body-frame components, so its linear part includes rotating-axis effects.Source: A. Physical Inverse-Dynamics Assembly · p. 3
B. Output Uniqueness and Recovery
Output uniqueness is determined by whether a requested linear output is invariant over the balance system’s nullspace. A dual formulation recovers determined outputs while rank, consistency, and residual checks govern numerical acceptance.
B. Output Uniqueness and Recovery
- For a consistent balance system, a requested output is unique exactly when its linear functional annihilates the balance matrix’s nullspace.A particular solution plus a nullspace basis describes the family of admissible load solutions.Source: B. Output Uniqueness and Recovery · p. 3
- A dual vector satisfying the transposed balance relation recovers a unique output directly from the body-balance requirements, even when the full load vector is nonunique.Thus, selected outputs can be determined without resolving every reaction coordinate.Source: B. Output Uniqueness and Recovery · p. 4
- Positive diagonal scales normalize balance equations, load coordinates, and outputs for dimensionless numerical tests and singular-value rank assessment.A full singular value decomposition provides the retained rank factors and complete right-nullspace basis.Source: B. Output Uniqueness and Recovery · p. 4
- Numerical output acceptance requires nullspace and dual residual tests, a sufficiently small normalized balance residual, and stable numerical rank under threshold variation.Outputs failing any condition are marked unavailable; undetermined load components of the representative solution are not reported as unique predictions.Source: B. Output Uniqueness and Recovery · p. 4
C. Diagnostics for Nonunique Reactions
After output recovery, the logged Simscape reactions are checked against body balance, ideal-joint admissibility, and the computed solution family. These diagnostics assess compatibility, not physical load sharing.
C. Diagnostics for Nonunique Reactions
- Logged Simscape reactions are checked after recovery, and these diagnostics do not modify the predicted outputs.Source: C. Diagnostics for Nonunique Reactions · p. 4
- Ideal-joint admissibility uses normalized excluded components and checks the signed free-axis torque for revolute joints or free-axis force for prismatic joints.Source: C. Diagnostics for Nonunique Reactions · p. 4
- Projected logged constraint-wrench coordinates and actuator efforts form the load vector for a balance residual; a second balance restores excluded wrench components with opposite signs on connected bodies.Source: C. Diagnostics for Nonunique Reactions · p. 4
- The normalized affine-family distance measures compatibility with the computed solution family, alongside raw balance and joint admissibility; it does not identify physical load sharing or model compliance.The nullspace coordinates are basis-dependent and serve only this diagnostic projection.Source: C. Diagnostics for Nonunique Reactions · p. 4
A. Manipulator and Simulation Setup
The evaluation uses a seven-degree-of-freedom heavy-duty hybrid serial–parallel manipulator with separately modeled tilt side links. Its prescribed-motion simulation and numerical settings define the validation setup.
A. Manipulator and Simulation Setup
- The manipulator model has 14 moving bodies and 18 coordinates—seven active and eleven passive—and its geometric and inertial parameters are protected by an NDA.Source: A. Manipulator and Simulation Setup · p. 4
- The lift and tilt modules use two and four paths, respectively, with Path 1 as reference; tilt paths 3 and 4 represent the original and additional side links.Source: A. Manipulator and Simulation Setup · p. 4
- Simscape runs a 30 s trajectory at 1 ms sampling with gravity and zero nongravitational external wrench; active motion is input and passive positions and joint loads validate reconstruction.Source: A. Manipulator and Simulation Setup · p. 4
- The assembly uses 84 balance equations, 97 load coordinates, and 223 requested scalar outputs, with scaled coordinates and a singular-value threshold of 10^-12 + 10^-10σ_max.Source: A. Manipulator and Simulation Setup · p. 4
B. Reconstruction and Load-Recovery Results
On the tested assembly branch, passive coordinates were reconstructed throughout the run, and all seven actuator efforts met recovery conditions. The uniquely recoverable constraint-wrench components agreed closely with Simscape, while diagnostics supported compatibility of logged reactions with the numerical load family.
B. Reconstruction and Load-Recovery Results
- All eleven passive coordinates were reconstructed throughout the run, with the largest absolute error of 5.09 × 10−9 rad at q8.Errors are raw and without angle wrapping; the comparison assesses reconstruction on the tested assembly branch.Source: B. Reconstruction and Load-Recovery… · p. 5, Table I · p. 5
- Balance and rank checks passed at every sample, and all seven actuator efforts satisfied recovery conditions; Figure 4 compares their histories and Table II quantifies agreement.The figure shows path-assembly and Simscape efforts, while Table II reports errors and full validity for all seven outputs.Source: B. Reconstruction and Load-Recovery… · p. 5, Figure 4 · p. 5, Table II · p. 6
- The largest force and torque RMSEs were 1.27 × 10−5 N and 6.32 × 10−6 N.m, respectively, for uniquely recoverable constraint-wrench components.37 of 54 force components and 28 of 54 moment components were valid throughout the run; validity counts describe recoverability, while error statistics measure agreement.Source: B. Reconstruction and Load-Recovery… · pp. 5–6, Table III · p. 6
- The ideal-joint admissibility residual was zero throughout, and balance and reaction-family diagnostics supported compatibility of logged reactions with the numerical load family.The temporal RMS values for the scaled raw body-balance residual and relative reaction-family distance were 6.41 × 10−7 and 6.96 × 10−8, respectively.Source: B. Reconstruction and Load-Recovery… · p. 6, Figure 5 · p. 6
V. CONCLUSION
The prescribed-motion comparison showed close agreement for passive positions, all seven actuator efforts, and valid constraint-wrench components. Reaction diagnostics supported compatibility with the admissible numerical load family without selecting a unique redundant reaction distribution; the findings are limited to the tested regular branch and rigid-body, ideal-joint model.
V. CONCLUSION
- The prescribed-motion comparison showed close agreement for passive positions, all seven actuator efforts, and valid constraint-wrench components.Source: §V · p. 6
- Reaction diagnostics supported compatibility with the admissible numerical load family without selecting a unique redundant reaction distribution.Source: §V · p. 6
- The findings apply to the tested regular branch and rigid-body, ideal-joint model; compliant load sharing remains outside the formulation.Source: §V · p. 6