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.
Figures & tables
Fig. 1: left : Manipulator with a four-bar close-up. Yellow/gray markers denote active/passive joints. The close-up shows the identical parallel a/b side links; labels 10 and 12 represent 10a/10b and 12a/12b to avoid clutter. right : Lift and tilt closed-chain modules. Tilt paths 3 and 4 overlap in this side view.
Fig. 2: Lift and tilt path graphs. Suffixes B and F denote base and follower frames; a/b identify the parallel links.
Fig. 4: Actuator efforts from path assembly (solid) and Simscape (dashed): torques on the left and forces on the right. Each joint has an independent vertical scale, shared by the two methods; all efforts retain their physical units.
Joint
RMSE
Max. abs. error
q2
3.89×10−10
1.16×10−9
q4
5.97×10−10
3.01×10−9
q5
8.36×10−10
4.17×10−9
q6
2.73×10−10
8.46×10−10
q8
1.21×10−9
5.09×10−9
q9
1.21×10−9
4.90×10−9
TABLE I: Passive-coordinate errors (rad), without angle wrapping. All eleven coordinates have full coverage.
Joint
Unit
RMSE
Max. abs. error
q1
N.m
6.32×10−6
1.41×10−5
q3
N
1.27×10−5
7.05×10−5
q7
N
1.27×10−5
8.91×10−5
q13
N
2.24×10−6
1.16×10−5
q14
N.m
9.68×10−8
5.06×10−7
q15
N.m
9.96×10−8
5.04×10−7
TABLE II: Actuator-effort errors. All seven outputs are valid at every evaluated sample.
Group
Valid/ requested
Max. RMSE
Max. abs. error
Forces (N)
37/54
0.000407
0.00113
Moments (N.m)
28/54
0.00641
0.00704
TABLE III: Unique constraint-wrench recovery and errors relative to Simscape.
Fig. 5: Scaled raw body-balance residual (upper) and relative reaction-family distance (lower). Both diagnostics are dimensionless.
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.
Mohammad Dastranj, Jouni Mattila
Unit of Automation Technology and Mechanical Engineering, Faculty of Engineering and Natural Sciences, Tampere University, 33720 Tampere, Finland
This paper extends an existing algorithm for the first-order derivatives of rigid-body dynamics to the case of closed-chain kinematic systems modeled using constraint embed- ding. Many standard dynamics algorithms apply to both open- chain and constraint-embedded models, but existing efficient derivative methods assume joint velocity effects are locally config- uration invariant. We remove this assumption and derive adapted algorithms that extend dynamics derivatives to more general joint types, including those arising in constraint-embedded closed- chain models. Our results compare conventional pin-joint robot models with more complete actuation models that capture local closed chains. We show that the additional terms introduced by these generalizations have low computational impact when mod- eling actuation kinematics alone, but can incur higher cost when additional rigid bodies, such as motor rotors, are included in the actuation chain, or when considering non-local loops. Overall, these results enable more accurate dynamics computations for constraint-embedded actuation submechanisms to be adopted in model-predictive control and differentiable simulation.
Daniel J. Volpi, Patrick M. Wensing
Dept. of Aerospace & Mechanical Engineering, University of Notre Dame, IN-46556, USA
This paper discusses null-space wrench components in parallel manipulators. We examine the adaptation of the two most common characterizations of these components in grasp-like systems, namely, interaction forces and internal loads, to parallel manipulators with actuation redundancy. We identify critical oversights in the existing literature on the subject, resolve ambiguities related to the definitions of interaction forces and internal loads, and provide explicit methods for synthesizing equilibrating and manipulating joint torque vectors. A case study is also provided to justify the validity of our novel methods and correct erroneous results reported in the literature.