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REVIEW 4 major objections 5 minor 1 cited by

On Solving the Dynamics of Constrained Rigid Multi-Body Systems with Kinematic Loops

T0 review · 4 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Proximal ADMM is the most robust dual solver for constrained rigid-body simulation with kinematic loops.

desk verdict A substantial, honest benchmark with a suggestive mass-ratio finding, but the ADMM-vs-splitting comparison has methodological loose ends and no released artifacts. read the letter →

arxiv 2504.19771 v1 pith:YR4PI65E submitted 2025-04-28 cs.RO

Vassilios Tsounis , Ruben Grandia , Moritz Bächer This is my paper · ORCID
classification cs.RO MSC 70E5570F4090C33
keywords constrainedrigid-bodydynamicskinematicloopsnonlinearcomplementarityproblemsecond-orderconeprogrammingalternatingdirectionmethodofmultipliersprojectedGauss-Seidelfrictionalcontactsimulationill-conditionedforward
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper claims that for simulating constrained rigid-body systems with kinematic loops, large mass ratios, and frictional contacts, an ADMM-based solver for the dual forward-dynamics problem is substantially more robust and accurate than the projective splitting solvers used in most physics engines. The authors formulate the time-stepping problem in maximal coordinates as a nonlinear complementarity problem, add constraint stabilization and softening, and compare eight solver variants on a benchmark suite ranging from toy problems to full-scale closed-loop mechanisms. In the Boxes-Fixed experiment with a mass ratio of $10^4$, ADMM-NCP was the best solver on 100% of the samples across all metrics and was the only solver able to converge on some samples, while the splitting solvers underestimated joint reactions and let constraints drift. If the claim is correct, physics engines for closed-loop mechanisms and high inertial disparity should move to ADMM-style dual solvers rather than projected Gauss-Seidel variants.

What carries the argument

The load-bearing object is the dual forward-dynamics NCP: find $\lambda$ in the composite cone $K$ such that the augmented constraint velocity $\hat v(\lambda)$ lies in $K^{*}$ and is complementary to $\lambda$. The problem data is the Delassus matrix $D = J^{T} M^{-1} J$ and the free velocity $v_f$; the nonlinearity comes from the De Saxcé correction $\Gamma(v^+(\lambda))$, which enforces maximal dissipation and is fixed iteratively to obtain a convex second-order cone program. ADMM-NCP solves that convexified problem with proximal ADMM, whose key device is an extra proximal term $\eta \|x - x^{-}\|^2$ added to the Delassus quadratic, making the regularized system strictly convex without biasing the solution; it then projects globally onto $K$ and updates the dual variable. The splitting solvers instead apply block-wise projected SOR with local projectors, either simple Euclidean cone projections or per-contact QCP solvers.

What would settle it

Re-run the identical curated Boxes-Fixed sample sequence using the actual production implementations of the splitting-based solvers (their native projectors, relaxation schedules, and termination criteria) instead of the in-house proxies; if any of them converges to ADMM-NCP-level residuals on a substantial fraction of the high-mass-ratio samples, the paper's central claim is falsified.

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Extended reading notes

Core claim

On the paper's own terms, the central discovery is empirical: a proximal ADMM applied to the NSOCP transcription of the dual forward dynamics is the most robust and accurate solver among those compared, especially when the Delassus matrix is ill-conditioned by hyperstatic kinematic loops and inertial disparity. The evidence is the Boxes-Fixed mass-ratio grid, where splitting-based methods degrade once the mass ratio exceeds roughly 1000, while ADMM-NCP keeps converging except at very high absolute mass scales tied to the spectral radius of $D$. ADMM's global simultaneous projection avoids the ordering bias of per-contact iteration, yields minimum-norm contact reactions with negligible internal coupling, and can still render usable forces even when it does not converge within the iteration budget.

Load-bearing premise

The comparison assumes the re-implemented splitting solvers faithfully represent the production physics-engine solvers they stand in for; if their local contact projections, relaxation schedules, or termination checks are unrepresentative, the measured superiority of ADMM-NCP could be an artifact of the proxies rather than a property of the methods.

Editorial extensions

If this is right

  • Closed-loop mechanisms with passive joints and mass ratios above roughly 1000 should be simulated with an ADMM-based dual solver rather than projected Gauss-Seidel to avoid constraint drift and force underestimation.
  • ADMM-NCP produces minimum-norm contact reactions with virtually no internal coupling, so contact force distributions on resting and sliding bodies are physically plausible without per-contact ordering heuristics.
  • Constraint stabilization and softening improve constraint satisfaction for all solvers but do not overturn the relative ranking; the NCP's complementarity structure can be preserved under both augmentations.
  • ADMM-NCP convergence depends more on the spectral radius of the Delassus matrix than on its condition number, so preconditioning the Delassus system is the natural next lever for making the method faster.
  • Early stopping based on objective-function improvement drastically cuts iteration counts but costs multiple orders of magnitude in accuracy, so it is not a free performance win.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • Editorial inference: if the empirical ranking holds up, the engineering bottleneck shifts from crafting per-contact local projectors to building fast linear solvers and preconditioners for the regularized Delassus matrix, because ADMM turns each iteration into one dense linear solve plus a global projection.
  • Editorial inference: the same dual ADMM should extend naturally to compliant, deformable, or viscoelastic contact models and to batched GPU simulation, where the global Delassus solve can be amortized across many problem instances.
  • Editorial inference: the paper's mass-ratio grid suggests a minimal robustness benchmark — a two-body fixed-joint drag scenario sweeping the mass ratio from 10 to $10^6$ — that could be adopted as a standardized stress test for any new forward-dynamics solver.
  • Editorial inference: a natural testable follow-up is whether the ADMM advantage persists at larger time-steps or with higher-order integrators, since the current study fixes semi-implicit Euler and the accuracy gap may narrow or widen with integration order.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

4 major / 5 minor

Summary. The paper develops and evaluates dual solvers for the time-stepping forward dynamics of constrained rigid multi-body systems with kinematic loops, formulated as Nonlinear Complementarity Problems (NCPs) and their convex relaxations. The authors derive the dual NCP/NSOCP from an augmented-Lagrangian/proximal perspective, describe a set of projective splitting solvers (PGS variants, NBGS, RaiSim variants) and two ADMM-based solvers (ADMM-CCP and ADMM-NCP), and benchmark them on a suite of twelve problems ranging from toy examples to robotic and Audio-Animatronics systems. The central empirical claim is that ADMM-NCP is the most robust and accurate solver for ill-conditioned systems, with the Boxes-Fixed experiment showing that splitting-based solvers fail to propagate constraint reactions at mass ratios above roughly 1000, while ADMM-NCP remains accurate and converges on samples where no other solver converges.

Significance. If the comparison is accepted, the paper makes a practically important point: the splitting-based first-order solvers used in the majority of physics engines are not reliable for closed-loop mechanisms with large mass ratios, whereas a globally coupled ADMM with spectral penalty adaptation can be. The derivation of the dual problem and the proximal framework is standard and clearly presented, and the extensive benchmark suite, performance profiles, and explicit statement of solver configurations are genuine strengths. The paper also usefully documents artifacts such as non-minimum-norm internal forces and the effects of constraint softening and stabilization. The headline claim, however, rests on the fairness of the solver comparison, and on that point the manuscript currently leaves load-bearing gaps: termination criteria differ between solver classes, SOR parameters are not reported, and equal iteration counts are not shown to be equal computational budgets.

major comments (4)
  1. [Sec. VI-E and Sec. VIII-A] The comparison uses equal Nmax = 10^4 for all solvers, but the per-iteration cost is not equal: ADMM-NCP (Alg. 6, line 6) solves a dense regularized system (D + (eta + rho)I) each iteration, while the splitting solvers (Alg. 5) perform block projections. The reported 'Mean Iteration Time' metric (Sec. VII-D) is collected but is not used as the budget for the performance profiles; a solver that does more work per iteration is favored by an iteration-count comparison. Please report wall-clock time (or flop-equivalent cost) for the headline experiments and show that the conclusions are unchanged under equal-time budgets.
  2. [Sec. VI-F, Eqs. (170a-c)] ADMM-NCP terminates on proxy residuals defined on the ADMM iterates (r_p = x_i - y_i, r_d = eta(x_i - x_{i-1}) + rho(y_i - y_{i-1}), r_cp = [x_i^T z_i]), whereas the splitting solvers terminate on the actual NCP residuals (156)-(158). If the proxy residuals are systematically looser than the true residuals, the robustness advantage of ADMM-NCP in Figs. 15-19 could partly reflect a more permissive stopping rule. Please report, for the same samples, the true NCP residuals (156)-(158) achieved by each solver at termination, or otherwise justify that the proxy residuals are equivalent for these problems.
  3. [Sec. VI-E and Table I] The SOR parameters omega0, omega_min, and gamma used in Alg. 5 are not reported, and the PROX/SORProx baseline is omitted 'purely due to time considerations' (Sec. VI-E). Since the SOR relaxation factor strongly affects convergence of PGS-type methods, the comparison may not reflect the best achievable behavior of the splitting class. Please provide the parameter values used for each solver, and ideally a sensitivity study over the relaxation parameters, before drawing the broad conclusion that splitting-based solvers are unsuitable above mass ratio 1000.
  4. [Sec. VIII-C, Boxes-Fixed and Fig. 18] The mass-ratio conclusion is drawn from a single problem family in which the two bodies are connected by a fixed joint and only ADMM-NCP converges. The paper itself notes that ADMM-NCP is 'more sensitive to the absolute scale of the heaviest body' and that its convergence depends on the spectral radius rho(D) rather than only the condition number. This is a load-bearing qualification: the claim that splitting solvers fail above mass ratio ~1000 needs to be demonstrated across several problem classes (including the Fourbar mass-ratio run, Fig. 47) with reporting of both the mass ratio and the condition/spectral data, and with the solver-class comparison made at equivalent effort and equivalent termination residuals. As written, the evidence supports the weaker statement that, on these particular problems and with these particular configurations, the splitting solvers tested did not converge while ADMM-NCP did.
minor comments (5)
  1. [Abstract] The abstract contains 'particularly interest' and 'aggravatingly over the complete set'; both should be corrected (e.g., 'particularly interested' and 'aggregately').
  2. [Fig. 13 and Fig. 14 captions] Several figure labels contain typos: 'Boxes-on-Plane' in Fig. 13 should be 'Box-on-Plane', and 'PGS-CPP' in Fig. 14 should be 'PGS-CCP'.
  3. [Sec. VIII-C, Fourbar] The sentence 'An final curated run' should read 'A final curated run'.
  4. [Sec. V-C] References to 'Baumgarter' should be 'Baumgarte' to match the standard spelling in the literature.
  5. [Sec. VIII-A] The paper does not state whether the implementation, benchmark definitions, or raw data will be released; given that the experimental comparison is the core contribution, a code/data availability statement would substantially improve reproducibility.

Circularity Check

0 steps flagged · score 1.0 of 10

No derivation-level circularity; the curated benchmark protocol is the only self-referential element and it is not load-bearing.

full rationale

The paper's analytical derivation chain is standard and externally grounded: the dual NCP (15) is built from the Delassus matrix D=J^T M^{-1}J (11), the free velocity vf (12)/(80), and the De Saxcé-corrected cone complementarity (52)-(55), with the NSOCP equivalence attributed to Cadoux [51], De Saxcé [30], and Acary et al [31]. No step defines D, vf, or K in terms of ADMM-NCP's output, and no fitted parameter is renamed as a prediction; the solver comparison is an empirical benchmark. The only self-referential element is the curated-run protocol in Sec. VIII-C: 'we use ADMM-NCP as a reference solver to generate a sequence of problems that are then re-solved by all others.' This is a benchmark-design choice rather than a circular derivation, because the resulting samples are static dual problems re-solved by every solver under identical tolerances, and the central Boxes-Fixed claim is corroborated by independent-run time-series (Figs. 15-17) and the mass-ratio grid (Fig. 18). Sec. VI-E does state a coverage limitation: 'purely due to time considerations, the authors regretfully have not been able to include the PROX algorithm in this evaluation' — this weakens the generality of the splitter comparison but does not make the ADMM-NCP result equivalent to its inputs. Similarly, the exclusion of ADMM-CCP from Boxes-Fixed after divergence in initial testing is a methodological choice, not a circular reduction. Overall the derivation is self-contained and the empirical claims are falsifiable against external solvers, so circularity is minimal.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claims are empirical, so free parameters are the hand-chosen solver and softening parameters that affect the comparisons. The mathematical axioms are standard in the field; the only ad hoc one is the CRBD-to-CABD transfer assertion. No invented entities are introduced.

free parameters (3)
  • Constraint softening parameter set = {d0=0.9, dw=0.95, w=0.001, m=0.5, p=2.0, T=0.02, beta=1.0}
    Chosen by hand for each constraint; the paper states there is no automatic procedure and that poor choices cause oscillations (Sec. V-D, Sec. VIII-C). Results of softening runs depend on these values.
  • ADMM spectral adaptation parameters = tau0=0.2, tau=0.05, alpha=10.0, eta=1e-6
    Used in the ADMM-NCP solvers (Eq. 172). The convergence and solution quality of ADMM-NCP depend on these hand-set values (Sec. VI-F).
  • Solver budgets and tolerances = Nmax=1e4, eps_abs=1e-12 (high-precision); Nmax=1e3, eps_abs=1e-6 (high-throughput)
    Maximum iteration counts and convergence tolerances determine which solvers are judged to 'converge'; these choices directly shape the performance profiles (Sec. VIII-C).
assumptions (4)
  • domain assumption The dual NCP is equivalent to the NSOCP (Eq. 16) as established by a 'coarse proof' in Carpentier et al. [32].
    The paper relies on this equivalence to justify solving the NCP via ADMM and to define residuals; a coarse proof is accepted from the cited literature.
  • domain assumption The De Saxce operator is invariant with respect to bias velocities v*, allowing restitution and stabilization terms to be absorbed into vf (Sec. V-B, V-C).
    The authors state this is proven in Appendix A-B; the proof is in the omitted appendix, so we take it as an assumption here.
  • ad hoc to paper If a method works in maximal-coordinate CRBD, it will also work in constrained articulated-body dynamics (CABD) because CABD reduces dimensionality.
    Stated without proof in Sec. I; it is used to justify the choice of CRBD as the benchmark formulation.
  • domain assumption Hard-contact rigid-body model with Coulomb friction and Newton restitution is a valid test model for physical plausibility.
    Standard assumption in non-smooth mechanics (Sec. II); the benchmark's meaning depends on it.

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Pith. "Pith review of On Solving the Dynamics of Constrained Rigid Multi-Body Systems with Kinematic Loops." pith.science (2026). https://pith.science/paper/YR4PI65E

@misc{pith2026250419771,
  author       = {Pith},
  title        = {Pith review of: On Solving the Dynamics of Constrained Rigid Multi-Body Systems with Kinematic Loops},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YR4PI65E}},
  note         = {Machine review of arXiv:2504.19771}
}
read the original abstract

This technical report provides an in-depth evaluation of both established and state-of-the-art methods for simulating constrained rigid multi-body systems with hard-contact dynamics, using formulations of Nonlinear Complementarity Problems (NCPs). We are particularly interest in examining the simulation of highly coupled mechanical systems with multitudes of closed-loop bilateral kinematic joint constraints in the presence of additional unilateral constraints such as joint limits and frictional contacts with restitutive impacts. This work thus presents an up-to-date literature survey of the relevant fields, as well as an in-depth description of the approaches used for the formulation and solving of the numerical time-integration problem in a maximal coordinate setting. More specifically, our focus lies on a version of the overall problem that decomposes it into the forward dynamics problem followed by a time-integration using the states of the bodies and the constraint reactions rendered by the former. We then proceed to elaborate on the formulations used to model frictional contact dynamics and define a set of solvers that are representative of those currently employed in the majority of the established physics engines. A key aspect of this work is the definition of a benchmarking framework that we propose as a means to both qualitatively and quantitatively evaluate the performance envelopes of the set of solvers on a diverse set of challenging simulation scenarios. We thus present an extensive set of experiments that aim at highlighting the absolute and relative performance of all solvers on particular problems of interest as well as aggravatingly over the complete set defined in the suite.

Figures

Figures reproduced from arXiv: 2504.19771 by the authors.

Figure 1
Figure 1. A visual depiction of the non-smooth initial-value problem (8) using [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Free-body diagram of a simple constrained system with: (1) [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Binary collisions between bodies Ak and Bk for arbitrary contact with index k. In all configurations, the gap-function (33) can be used to compute the signed distance dc,k nc,k (black arrows) of Bk w.r.t. Ak , where nc,k is the normal vector (blue arrows) w.r.t the plane that is tangent to the two contacting points of the bodies. Choosing Ak as the reference body is a merely a matter of convention. When dc,k > 0 (ca… view at source ↗
Figures from the paper (47 more)
Figure 4
Figure 4. Figure 4: The set-valued force laws along the normal and tangent directions. The [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: The Coulomb friction cone Kµ (a) and the relationships between the former to its dual cone K∗ µ and polar cone Ko µ (b). As conjugates of Kµ, the dual and polar cones are those whose elements make obtuse angles with each λ ∈ Kµ, i.e. x ∈ K∗ µ , x T λ ≥ 0, and x ∈ Ko µ …
Figure 6
Figure 6. Figure 6: Examples of impedance shaping using the sigmoidal shaping functions [PITH_FULL_IMAGE:figures/full_fig_p016_6.png]
Figure 7
Figure 7. Figure 7: A depiction of the geometric interpretation of the natural-map function. [PITH_FULL_IMAGE:figures/full_fig_p022_7.png]
Figure 8
Figure 8. Figure 8: Benchmark Suite: a set of twelve constrained rigid multi-body systems spanning three categories. The top, middle and bottom rows respectively depict the Primitive, Robotics, and Audio-Animatronics® problem categories, and the system complexity is increasing from left t…
Figure 9
Figure 9. Figure 9: The different force distributions exhibited by each solver on the hard [PITH_FULL_IMAGE:figures/full_fig_p030_9.png]
Figure 10
Figure 10. Figure 10: Box-on-Plane: Time-series plots of the first run on the hard-contact problem. The PGS-CCP, PGS-NCP, RAISIM-DS and ADMM-NCP solvers yield similar motions, while the others exhibit significant translational and rotational deviations along the Y and Z-axes. When sliding,…
Figure 11
Figure 11. Figure 11: Box-on-Plane: Time-series plots of the second run with MuJoCo-like constraint softening enabled. All solvers render minimum-norm contact forces and the motions exhibit significantly less deviations due to the improved symmetry of the former. NBGS and RAISIM still over…
Figure 11
Figure 11. Figure 11: While the box is unforced, the softening leads to [PITH_FULL_IMAGE:figures/full_fig_p032_11.png]
Figure 12
Figure 12. Figure 12: Box-on-Plane: A comparison focusing on the NCP-type solvers. While the overall motions, forces and optimization objectives generated by all solvers can seem very similar, a detailed view of the NCP dual, NCP complementarity and NCP natural-map residuals can exemplify …
Figure 14
Figure 14. Figure 14: Boxes-Fixed: Examples of the constraint violation accumulated by each solver as well as the forces rendered. The translational deviation of the box causes a shifting of the overall CoM of the system that leads to the rear contacts opening while sliding. with αj , γk =…
Figure 15
Figure 15. Figure 15: Boxes-Fixed: Time-series plots of the motions, forces and performance metrics for the dragging run. The top box is of mass mt = 103 kg while the bottom box mb = 0.1 kg with a system mass-ratio of rm = 104 . Despite not converging, ADMM-NCP was still able to realize th…
Figure 16
Figure 16. Figure 16: Boxes-Fixed: Time-series plots of the dropping run showing the effects of the impact on the constraint violation (center-bottom) and overall motion (top row). These effects are also apparent in the NCP complementarity residual (bottom-right) were the solvers with larg…
Figure 17
Figure 17. Figure 17: Boxes-Fixed: Time-series plots of the combined drop and dragging run that also enables constraint stabilization. The augmentation to the problem leads to significant improvement w.r.t constraint violation (center-bottom) for all solvers. In the case of the NBGS (green…
Figure 19
Figure 19. Figure 19: Boxes-Fixed: performance profiles showing the relative performance between solvers. In this scenario ADMM-NCP is the outright winner, as indicated by being the best solver on 100% of the samples across all metrics. It was the only solver to ever be able to converge on…
Figure 20
Figure 20. Figure 20: Nunchaku: A rendering that showcases the motions of the system in the corresponding set of experiments. The system is initialized to drop from a fixed initial height. After impacting the plane, all four corners of the bottom box are in contact, and the system is able …
Figure 21
Figure 21. Figure 21: Nunchaku: Summary of the first run, solving the hard-contact problem without augmentations. All solvers exhibit non-zero constraint penetration, i.e. configuration-level constraint residuals. Considering the absence of constraint stabilization, their performance is ra…
Figure 22
Figure 22. Figure 22: Nunchaku: Performance profiles showing the relative performance between solvers on this problem. While PGS-CCP is the least performing in terms of number of iterations (top left), it demonstrates accuracy comparable to ADMM-NCP. RAISIM and RAISIM-DS also seem to perfo…
Figure 23
Figure 23. Figure 23: Fourbar: Renderings of the motion of the system and active contact forces along the different phases of the experiment. The four-bar is dropped from an initial height from which it begins to shift onto one side due to gradually accumulating numerical drift due to the …
Figure 25
Figure 25. Figure 25: Fourbar: Performance profiles of the full set of solvers. In contrast to the other primitive problems where the differences between solvers were less pronounced, in this case ADMM-NCP proves to be the best solver overall by a wide margin. Even w.r.t the number of iter…
Figure 24
Figure 24. Figure 24: Fourbar: Convergence profiles of the most prominent solvers, with both X and Y axes are rendered in log10 scale. The NCP and CCP objectives (top row) are juxtaposed with the accuracy performance metrics to highlight the crucial differences between them. RAISIM-DS-ES r…
Figure 26
Figure 26. Figure 26: Fourbar: Summary of the first run on the un-augmented hard-contact problem. The top row depicts the absolute position of the designated base link, i.e. the lowest of the two horizontal bodies. The second row depicts the net contact forces acting on the base along the …
Figure 27
Figure 27. Figure 27: Walker: The forces distributed over the non-convex collision geometry of the system’s planar feet rendered by the ADMM solver. Black and white spheres indicated contacts that are sticking and opening, respectively. The collection of contacts on the heel of the left fo…
Figure 29
Figure 29. Figure 29: Walker: Time-series plots of the third run where stabilization is applied to all constraint sets. ADMM-NCP exhibits the best overall performance on this problem with significantly better performance w.r.t the accuracy metrics of the NCP dual, complementarity and natur…
Figure 28
Figure 28. Figure 28: From these we can conclude that although ADMM [PITH_FULL_IMAGE:figures/full_fig_p041_28.png]
Figure 30
Figure 30. Figure 30: Walker: Summary of the third run that combines constraint stabilization with constraint softening for only the contact constraints. The softening significantly improves the convergence rate of all splitting-based solvers, while the constraint stabilization ensures tha…
Figure 31
Figure 31. Figure 31: IronMan © MARVEL: the Audio-Animatronics® figure, featured in real-life at Disneyland Paris in France, and shown here with (left) and without (right) the external shell geometries. The internal mechanical assembly reveals an exceptionally complex humanoid system. thro…
Figure 34
Figure 34. Figure 34: IronMan: performance profiles TABLE III Benchmarks: Data contents of each dual problem sample. Type Entry Description Problem Definition D Delassus matrix vf free-velocity vector µ friction coefficients dt time-step Problem Dimensions nb no. of bodies nj no. of joints…
Figure 33
Figure 33. Figure 33: IronMan: Summary of the second run in which the system is actuated to track a reference trajectories using a naive joint-space PID controller. to beat RAISIM-DS-ES and RAISIM-DS in terms of the number of iterations and NCP dual residual. The degree of this disparity b…
Figure 35
Figure 35. Figure 35: Benchmarks: performance profiles generated over all problems in the suite. of fixed-size corresponding to each of the categories de￾fined in Sec. VII-C. This ensures that the sampling process collects a diverse set of samples spanning all cases of ill￾conditioning. Al…
Figure 36
Figure 36. Figure 36: A visualization of the geometry implied by the constraints of [PITH_FULL_IMAGE:figures/full_fig_p051_36.png]
Figure 37
Figure 37. Figure 37: Conic-section types: (top left) a circle when [PITH_FULL_IMAGE:figures/full_fig_p052_37.png]
Figure 38
Figure 38. Figure 38: The bisection-method applied to a sample conic-section (purple) [PITH_FULL_IMAGE:figures/full_fig_p054_38.png]
Figure 39
Figure 39. Figure 39: Boxes-Fixed: A heatmap-like rendering of solver iterations istop and the complementarity residual rncp as functions of the bottom and top bodies with respective masses mb and mt. The data is generated while the bottom box is at rest on the plane [PITH_FULL_IMAGE:figu…
Figure 40
Figure 40. Figure 40 [PITH_FULL_IMAGE:figures/full_fig_p057_40.png]
Figure 41
Figure 41. Figure 41: Nunchaku: Summary of the second run that enables constraint stabilization. PGS-CCP PGS-NCP NBGS RAISIM RAISIM-DS RAISIM-DS-ES ADMM-NCP 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 1 2 3 4 5 6 rb,z (m) Time (s) Body Position Z 0 2 4 6 8 10 12 0 1 2 3 4 5 6 nc Time (s) Nubmer of Conta…
Figure 42
Figure 42. Figure 42: Nunchaku: Summary of the third run that enables constraint softening. PGS-CCP PGS-NCP NBGS RAISIM RAISIM-DS RAISIM-DS-ES ADMM-NCP 0 0.2 0.4 0.6 0.8 1 1.2 1.4 0 1 2 3 4 5 6 rb,z (m) Time (s) Base Position Z 100 101 102 103 104 0 1 2 3 4 5 6 istop Time (s) Number of Ite…
Figure 43
Figure 43. Figure 43: Nunchaku: Summary of the fourth run that enables both constraint stabilization and softening [PITH_FULL_IMAGE:figures/full_fig_p058_43.png]
Figure 44
Figure 44. Figure 44: Fourbar: Summary of the second run that enables constraint stabilization. PGS-CCP PGS-NCP NBGS RAISIM RAISIM-DS RAISIM-DS-ES ADMM-NCP -0.05 -0.04 -0.03 -0.02 -0.01 0 0.01 0.02 0.03 0.04 0.05 0 2 4 6 8 10 12 14 16 rb,y (m) Time (s) Base Position Y 0 0.01 0.02 0.03 0.04…
Figure 45
Figure 45. Figure 45: Fourbar: Summary of the third run that enables constraint softening [PITH_FULL_IMAGE:figures/full_fig_p059_45.png]
Figure 46
Figure 46. Figure 46: Fourbar: Summary of the fourth run that enables both constraint stabilization and softening. PGS-CCP PGS-NCP NBGS RAISIM RAISIM-DS RAISIM-DS-ES ADMM-NCP -0.025 -0.02 -0.015 -0.01 -0.005 0 0.005 0.01 0.015 0.02 0.025 0 2 4 6 8 10 12 14 16 rb,x (m) Time (s) Base Positio…
Figure 47
Figure 47. Figure 47: Fourbar: Summary of solving the un-augmented problem in the presence of a large mass-ratio [PITH_FULL_IMAGE:figures/full_fig_p060_47.png]
Figure 48
Figure 48. Figure 48: Walker: Time-series plots of solving the un-augmented problem with environment collisions and gravity disabled. PGS-CCP PGS-NCP NBGS RAISIM RAISIM-DS RAISIM-DS-ES ADMM-NCP -0.1 -0.05 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0 1 2 3 4 5 6 7 8 9 10 rb,z (m) Time (s) Base P…
Figure 49
Figure 49. Figure 49: Walker: Time-series plots of solving the un-augmented problem with external collisions and gravity enabled [PITH_FULL_IMAGE:figures/full_fig_p061_49.png]
Figure 50
Figure 50. Figure 50: Walker: Time-series plots of solving the stabilized problem using high-throughput solver settings. Primitive 0 0.2 0.4 0.6 0.8 1 100 101 102 103 Proportion of problems Within factor of the best performing solver Number of Iterations 0 0.2 0.4 0.6 0.8 1 100 102 104 106…
Figure 51
Figure 51. Figure 51: Performance profiles w.r.t each benchmark system category. [PITH_FULL_IMAGE:figures/full_fig_p062_51.png]
Figure 52
Figure 52. Figure 52: Performance profiles w.r.t each benchmark sample category. [PITH_FULL_IMAGE:figures/full_fig_p063_52.png]

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Pith tools

Reviewed August 16, 2026 · model on record in the stance chip above.