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REVIEW 3 major objections 5 minor 25 references

Joint parameter and state estimation for regularized time-discrete multibody dynamics

T0 review · 3 major / 5 minor · reviewed 2026-08-08 · deepseek-v4-flash

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Pith's one-line read This paper claims that offline identification of regularized, frictional multibody dynamics can be solved as a joint state-and-parameter nonlinear least squares problem, and demonstrates it by identifying 13 parameters of a real Furuta…

desk verdict Useful and honest engineering method, but the central parameter-recovery claim is not actually tested because the only synthetic experiment uses a different discrete model than the data generator. read the letter →

arxiv 2502.06599 v1 pith:56VQVZRR submitted 2025-02-10 math.OC cs.SYeess.SY

Hannes Marklund , Martin Servin , Mats G Larson This is my paper · ORCID
classification math.OCcs.SYeess.SY MSC 70E5565K1090C30
keywords multibodydynamicsparameterestimationstateinversedifferentiablephysicsregularizedconstraintsfrictioncomplementarityconditions
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 tries to establish that offline parameter identification for regularized, frictional multibody dynamics can be recast as a single nonlinear least squares problem in which the hidden states and the physical parameters are optimized together. The key move is to measure mismatch with inverse-dynamics impulse residuals rather than forward simulation error: each residual asks what external impulse would be needed to make the model follow the observed state-to-state transitions, so unobserved constraint deviations appear as forces instead of requiring full state observability. The authors demonstrate the claim on a real Furuta pendulum, identifying 13 inertial, frictional, tilt, and motor parameters in roughly six seconds and a handful of Levenberg-Marquardt iterations, with cross-validation costs close to training costs for well-excited trajectories. They also show the method's limits: very stiff constraints make the explicit inverse-dynamics derivatives misdirect the optimizer, so the approach needs sufficiently soft constraints or an implicit variant.

What carries the argument

The central object is the inverse-dynamics impulse residual: for each state transition, the difference between the external impulse the discrete equations of motion require and the impulse the model actually supplies, computed by explicitly solving for the constraint multipliers from the regularized time-stepper. Concatenating these residuals with observation residuals forms a sparse nonlinear least squares objective over the full state trajectory and the parameters. The machinery that makes it work includes Levenberg-Marquardt on the quaternion manifold with box constraints, forward-mode automatic differentiation with custom differentiation rules for the linear complementarity problems introduced by dry friction and for zero rotation vectors, and an exploitation of the block-bidiagonal structure of the Jacobian that keeps the computation fast. The residuals do the conceptual work: they convert a trajectory-matching problem into a force-balance problem, which is what makes the unobserved constraint deviations estimable rather than a barrier.

What would settle it

Take the recorded swing-up trajectory used in the paper, set the hinge compliance to $10^{-7}$ m/N while keeping $h=0.01$ s and $\kappa=100$, and run the optimization under the 20-iteration cap. The paper's explanation predicts the explicit inverse-dynamics derivatives will misdirect the optimizer and stall; if it instead converges to the same parameters and cross-validation costs as the $10^{-4}$ case, the claimed stiffness limit is falsified.

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Extended reading notes

Core claim

The central claim, stated on the paper's own terms, is that joint state and parameter estimation for regularized time-discrete multibody dynamics with dry friction is a tractable nonlinear least squares problem once inverse dynamics replaces forward simulation. States are free optimization variables, so the optimizer can place the unobserved constraint deviations wherever they are most consistent with the recorded encoder angles and control signal, while the parameters are shared across the whole trajectory. The paper reports that on a real Furuta pendulum this identifies a 13-dimensional parameter space, including moments of inertia, friction coefficients, gravity direction, and motor gain, in about 6.6 seconds and six Levenberg-Marquardt iterations, with cross-validation scores of order $10^{-5}$ for release and swing-up trajectories. It also reports a boundary: when kinematic constraints are too stiff, the derivatives from the explicit inverse-dynamics solve provide misdirection and the optimization stalls or drifts.

Load-bearing premise

The method's usefulness rests on the assumption that the explicit inverse-dynamics solve yields derivative directions that push the optimizer toward a good minimum at the constraint stiffnesses one would actually use; the paper itself shows this assumption fails when constraints are very stiff.

Editorial extensions

If this is right

  • Simulation-error rollouts are not needed for offline identification: the equation-error formulation with estimated states converges quickly enough for interactive use, at least on systems like the Furuta pendulum tested here.
  • The method reduces tuning to three quantities, time step, constraint compliance and damping, and a single state-error weight, and the experiments show broad plateaus over which identified parameters and costs change little.
  • Identified parameters should be treated as shadow parameters that absorb discretization effects, not necessarily as direct physical measurements; the paper demonstrates this by showing estimated parameters depend on time-step even on synthetic data.
  • The stiff-constraint failure means practical deployments must keep joint and contact compliance above a stiffness threshold, or use a different implicit inverse-dynamics derivative.
  • Cross-validation across similarly excited trajectories is a viable check: models tuned on swing-up and release data transfer to each other with costs within a small factor of training cost.

Reading between the lines

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

  • The paper's hypothesis that instabilities during optimization cause the derivatives to provide misdirection is testable by replacing the explicit inverse-dynamics solve with an implicit one; if that variant removes the stiff-constraint stall, the explanation is confirmed and the method's scope extends to stiffer joints and contacts.
  • The reported sparsity of the Jacobian suggests the per-iteration work scales with the number of time-steps, so the same approach could plausibly handle much longer trajectories or larger mechanisms, though the paper only demonstrates one two-body system.
  • The inconsistency of the linearized uncertainty estimates across time-series implies that any error bars on identified parameters should be treated with caution until a nonlinear or sampling-based uncertainty treatment is applied, which the paper explicitly leaves open.
  • Because identified parameters compensate for discretization errors, comparing optimizations at several time-steps could give a practical sense of which parameters are physical and which are tuning artifacts, an idea the paper does not develop.
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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

3 major / 5 minor

Summary. The paper develops an offline joint state and parameter estimation method for regularized time-discrete multibody dynamics with frictional constraints. States and parameters are estimated jointly by minimizing a weighted nonlinear least squares objective that combines inverse-dynamics impulse residuals (Section 3.2, Eqs. (3.7)--(3.8)) with observation residuals. The optimization is performed with a Levenberg--Marquardt algorithm on manifolds with box constraints, using forward-mode automatic differentiation and custom differentiation rules for linear complementarity problems and quaternion rotations (Section 2.7, Appendix D). The method is evaluated on synthetic data and on real data from a pendulum and a Furuta pendulum, including a 13-dimensional parameter identification problem (inertia, friction, gravity tilt, and motor gain). The paper reports fast convergence (on the order of seconds), good cross-validation scores, robustness of results across method parameters, and identifies stiff constraints as a failure mode.

Significance. The paper addresses a practically important problem: identifying parameters of fixed-time-step multibody simulators from partially observed trajectories, where constrained degrees of freedom are not directly measured. The inverse-dynamics formulation is a sensible alternative to simulation-error optimization and is supported by several technically strong components: custom JVPs through the LCP solver and quaternion tangent-space handling, exploitation of the block-bidiagonal sparsity of the residual Jacobian, an on-manifold Levenberg--Marquardt implementation, and evaluation on real hardware with cross-validation across multiple recorded trajectories. The main contribution would be a fast offline identification pipeline for a class of regularized discrete multibody models. However, as detailed below, the central identification claim is not fully validated: the synthetic experiments never run the estimator in the same discretization that generated the data, and the real-data results show that several of the 13 parameters are not identifiable. The paper is worth publishing after these gaps are addressed.

major comments (3)
  1. [Section 4.7, Figures 6--7] The synthetic parameter-recovery study does not provide a matched-discretization validation of the identification claim. Observation data are generated at a time step of h = 10^-4 s with compliance epsilon = 10^-6 m/N (Appendix G.3, Table G.5), while the estimator is run at h = 10^-2 s for the parameter sweeps in Figure 6 and Figure 7. The text explicitly states that 'the estimated parameters do not approach the true values that were used to generate the observations' and attributes this to the time-step mismatch. As a result, the paper contains no experiment in which the data-generating model and the estimator share the same discrete dynamics, regularization, and time step. Without such an ideal-case recovery test, the central claim that the optimization identifies a known parameter vector with 'impressive precision' is unsupported: the real-data results have no ground truth, and the synthetic sweeps confound model mismatch with estimation error. Please add a matched-discretization experiment (same h, same compliance, same model class) with known parameters and noisy observations, and report recovery errors for all identified parameters.
  2. [Conclusions, Section 4.6, Table 4] The conclusion that the method identifies parameters in a 13-dimensional space 'with impressive precision' is contradicted by the paper's own Table 4. For JAxx and JAyy, the linearized standard-deviation estimates (200, 2000, 200, 50, 3000, 80 kg m^2 in the three scenarios) are larger than the estimated values, which are reported as zero, and the text states that 'we lack information to specify JAxx and JAyy.' Moreover, parameters such as JAzz, JBzz, and b1 vary across the three scenarios by amounts far exceeding the reported linearized standard deviations. This shows that only a subset of the 13 parameters is identifiable from these data, and that the phrase 'impressive precision' overstates the evidence. Please either restrict the claim to the identifiable subset, report an identifiability/conditioning analysis, or substantially temper the conclusion.
  3. [Section 4.9, Figure 6] The stiff-constraint failure is left as an unexplained breakdown, and the suggested cause is speculative. The text says 'we suspect that instabilities during optimization cause the derivatives to provide misdirection,' but the experiments do not distinguish derivative misdirection from a more basic loss of identifiability, poor conditioning of the inverse-dynamics residual, or an optimizer failure at small compliances. Because the method's central robustness claim depends on the inverse-dynamics residuals and their derivatives being informative, please provide a concrete diagnostic (e.g., eigenvalue or condition-number analysis of the approximated Hessian, or a comparison between the explicit inverse-dynamics derivative and a finite-difference check across the compliance range) or explicitly state, with supporting evidence, that the failure is a derivative issue rather than an identification issue.
minor comments (5)
  1. [Section 3.2, Eq. (3.11)] The right-hand side of Eq. (3.11) is written as Delta p_{u,k}, but the residual defined in Eq. (3.7) is Delta p_{u,k+1}; the index should be made consistent.
  2. [Table 4] The column headers '0 1 2' in Table 4 are not defined in the table or its caption; please state that these correspond to the release, pulse, and swing-up scenarios (or otherwise define them).
  3. [Figure 5] The axis labels 'JAzz (kg/s)' and 'JByy (kg/s)' in the right column of Figure 5 are dimensionally wrong for moments of inertia; these should be kg m^2.
  4. [Section 4.7, Figure 6] The statement that the identified parameters 'weakly depend on the choice of time-step' is hard to reconcile with the large relative change in b1 between h = 10^-2 s and h = 10^-3 s shown in the left column of Figure 6; please clarify whether this variation is considered weak relative to the parameter's scale or the achievable accuracy.
  5. [Throughout] There are several typographical errors that should be corrected, including 'notataion' (Section 2.3), 'funcionality' (Section 2.7), 'Comined errors' (Table 3 caption), 'a prioi' and 'denomenator' (Appendix E), and 'Paramer values' (Figure 7 caption).

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: inverse-dynamics residuals are independent equation-error terms, and cross-validation/synthetic studies provide external checks.

full rationale

The paper's derivation chain is self-contained. The objective (3.15) minimizes inverse-dynamics residuals (3.7)-(3.8) plus observation residuals (3.6); these residuals are defined directly from the regularized discrete Newton-Euler/Spook equations (2.26)-(2.31), not from the parameters being estimated or from the observed output being reinserted as a target. The states x0:n-1 are free optimization variables, and the impulse residuals measure the inconsistency between a proposed state trajectory and the discrete dynamics, which is an equation-error formulation rather than a circular definition. Cross-validation in Sections 4.5-4.6 holds parameters fixed and evaluates against held-out time-series, so the reported CV scores are genuine out-of-sample predictions. The synthetic study in Section 4.7 uses externally generated data with known parameter values and explicitly reports that the estimates do not approach the true values at a mismatched time-step, an honest limitation discussed in Section 4.9, not a fitted input disguised as a prediction. The only load-bearing external input is the Spook time-stepper from the prior literature [12], and no uniqueness theorem or ansatz is imported through self-citation. The acknowledged unidentifiability of JAxx and JAyy and the stiff-constraint breakdown are stated limitations, not hidden circular steps.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central method rests on several hand-chosen regularization and weighting parameters (kappa, epsilon, tau, h) and on an arbitrary normalization of friction radii. No new physical entities are introduced. The 13 estimated parameters are the goal of the method, not extra assumptions. The 'shadow parameter' interpretation in Section 4.9 honestly acknowledges that estimated parameters are discrete-model-specific.

free parameters (5)
  • state error weight kappa = kappa = 100 used for real Furuta; swept 0.001 to 100000 in experiments
    Controls relative weighting of inverse-dynamics impulse residuals and observation residuals. The paper treats it as a method parameter and shows results depend on it strongly: small kappa overfits, large kappa leads to infeasible optimization.
  • hinge compliance epsilon = epsilon = 1e-4 m/N (real experiments); 1e-6 m/N in synthetic generation
    Regularization coefficients are method parameters, not estimated; too stiff or too soft constraints degrade or break convergence (Figures 4 and 6).
  • constraint damping tau1, tau2 = 0.02 s
    Fixed regularization damping for the frictional hinges; not estimated or swept.
  • internal friction radii r1, r2 = 1 m (arbitrary)
    Section 4.3 states the reason for r1 = r2 = 1 m is not physical but an arbitrary choice to unify force and torque multipliers. Only r*mu is identifiable, so this normalization determines the reported numerical values of mu1 and mu2.
  • time-step h = h = 0.01 s for real Furuta; swept 1e-3 to 1e-1 s in synthetic study
    Discretization step is a method parameter; parameter estimates depend weakly on h except for large h, where errors increase (Figure 6).
assumptions (6)
  • domain assumption The Spook regularized time-discrete integrator with compliance and damping is an adequate model of the physical pendulum dynamics.
    Section 2.3 builds the method on Spook; if the discrete model cannot represent the real system, the fitted parameters are shadow parameters rather than physical ones (Section 4.9).
  • domain assumption Coulomb dry friction can be represented as a mixed linear complementarity problem with linearized friction cones.
    Section 2.4 and Appendix F; the estimator's custom differentiation rules differentiate through this LCP solver.
  • domain assumption Observation and state noises are zero-mean Gaussian and homoscedastic, and state errors can be modeled as impulse errors with covariance Omega.
    Section 3.2 and Appendix E; this justifies the nonlinear least squares objective and the inverse-dynamics residual weighting.
  • domain assumption The motor torque is proportional to the control signal with no delay: tau_motor = K u.
    Section 4.3; motor gain K is estimated and is unidentifiable in the release scenario.
  • domain assumption The observations are joint angles from encoders at 1000 Hz with only additive noise; no encoder delay or quantization is modeled.
    Section 4.1 and Section 3.1; delays would bias the inverse dynamics residuals.
  • standard math Lemke's algorithm and block-LDLT factorization solve the LCPs exactly enough that AD through them yields correct derivatives.
    Appendix A and Appendix D.1; the custom JVP rule assumes the active set is correctly identified by the solver.

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Cite this review

Pith. "Pith review of Joint parameter and state estimation for regularized time-discrete multibody dynamics." pith.science (2026). https://pith.science/paper/56VQVZRR

@misc{pith2026250206599,
  author       = {Pith},
  title        = {Pith review of: Joint parameter and state estimation for regularized time-discrete multibody dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/56VQVZRR}},
  note         = {Machine review of arXiv:2502.06599}
}
read the original abstract

We develop a method for offline parameter estimation of discrete multibody dynamics with regularized and frictional kinematic constraints. This setting leads to unobserved degrees of freedom, which we handle using joint state and parameter estimation. Our method finds the states and parameters as the solution to a nonlinear least squares optimization problem based on the inverse dynamics and the observation error. The solution is found using a Levenberg-Marquardt algorithm with derivatives from automatic differentiation and custom differentiation rules for the complementary conditions that appear due to dry frictional constraints. We reduce the number of method parameters to the choice of the time-step, regularization coefficients, and a parameter that controls the relative weighting of inverse dynamics and observation errors. We evaluate the method using synthetic and real measured data, focusing on performance and sensitivity to method parameters. In particular, we optimize over a 13-dimensional parameter space, including inertial, frictional, tilt, and motor parameters, using data from a real Furuta pendulum. Results show fast convergence, in the order of seconds, and good agreement for different time-series of recorded data over multiple method parameter choices. However, very stiff constraints may cause difficulties in solving the optimization problem. We conclude that our method can be very fast and has method parameters that are robust and easy to set in the tested scenarios.

Figures

Figures reproduced from arXiv: 2502.06599 by the authors.

Figure 1
Figure 1. Abstract representation of the impulse residuals [PITH_FULL_IMAGE:figures/full_fig_p016_1.png] view at source ↗
Figure 2
Figure 2. Images of the pendulum and the Furuta pendulum. [PITH_FULL_IMAGE:figures/full_fig_p022_2.png] view at source ↗
Figure 3
Figure 3. A render of the Furuta pendulum with annotated joint angles and lengths. [PITH_FULL_IMAGE:figures/full_fig_p024_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: The two upper plots show three of the estimated parameters, tuned to the [PITH_FULL_IMAGE:figures/full_fig_p029_4.png]
Figure 5
Figure 5. Figure 5: The left column of plots shows simulated observation trajectories using models [PITH_FULL_IMAGE:figures/full_fig_p030_5.png]
Figure 6
Figure 6. Figure 6: The identified parameters are found to weakly depend on the choice [PITH_FULL_IMAGE:figures/full_fig_p031_6.png]
Figure 6
Figure 6. Figure 6: The left plot column shows the resulting parameters and errors after optimization [PITH_FULL_IMAGE:figures/full_fig_p032_6.png]
Figure 7
Figure 7. Figure 7: Paramer values versus and errors versus the state error weight in the simulated [PITH_FULL_IMAGE:figures/full_fig_p033_7.png]

Discussion (0). Continue with ORCID to comment.

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