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.