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REVIEW 4 major objections 4 minor 24 references

Towards Solutions of Manipulation Tasks via Optimal Control of Projected Dynamical Systems

T0 review · 4 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read The paper proposes solving manipulation planning as optimal control of a projected dynamical system, with signed distance functions as contact constraints and complementarity multipliers as forces.

desk verdict Credible proof-of-concept for PDS-based manipulation planning, but the printed ellipsoid signed-distance model is internally inconsistent and the non-convex extension is explicitly unproven. read the letter →

arxiv 2501.11946 v1 pith:D7TL5O3T submitted 2025-01-21 math.OC

Anton Pozharskiy , Armin Nurkanović , Moritz Diehl This is my paper · ORCID
classification math.OC MSC 49J4090C3393C85
keywords projecteddynamicalsystemsoptimalcontrolmanipulationplanningcomplementarityfiniteelementswithswitchdetectionsigneddistancefunctionsfrictionmathematicalprogramsconstraints
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

The paper proposes a single modeling route from contact-rich manipulation to a solved optimal control problem: write the dynamics as a projected dynamical system, translate it into an equivalent complementarity system using signed distance functions as constraints, discretize with finite elements with switch detection, and solve the resulting mathematical program. The central claim is that this route produces valid trajectories for planar pushing problems that involve multiple pushers, friction, and non-convex objects built as unions of convex ellipsoids, at computational costs on the order of one to three minutes on a desktop CPU. If this holds, manipulation planning no longer needs a separate discrete search over contact modes; the mode switches are found implicitly by the complementarity formulation and the switch-detecting discretization. The paper demonstrates the claim on two numerical examples and leaves the rigorous extension to non-convex unions of ellipsoids as an open question.

What carries the argument

The load-bearing object is the projected dynamical system and its equivalence to the differential complementarity system (DCS): with $C = \{x \mid c(x) \ge 0\}$, the projection $\dot{x} = P_{T_C(x)}(f(x))$ is replaced by $\dot{x} = f(x) + \nabla c(x)\lambda$ together with the complementarity conditions $0 \le c_i(x) \perp \lambda_i \ge 0$. For ellipsoidal objects, the signed distance functions are obtained as the optimal value of a convex program, and its KKT conditions plus the adjoint gradient $d_n(x) = \nabla_x(g(\alpha, p_d; x))\mu$ are embedded directly in the optimal control problem. The discretization that carries the argument is finite elements with switch detection (FESD), whose cross-complementarity constraints identify contact opening/closing and stick/slip switches; for friction, the paper multiplies those cross-constraints by $b_i = \min(\lambda_i + c(x_i))$ to relax them exactly when a contact is closed. The resulting mathematical program with complementarity constraints is solved by a Scholtes relaxation homotopy. Together these pieces turn a nonsmooth hybrid manipulation problem into one finite-dimensional nonlinear program.

What would settle it

Take the union-of-ellipsoids 'interior corner' configuration described in Section III-B, place a pusher into the corner, and compute the tangent cone of $C$ at the contact point; if that cone is not closed and convex, then Eq. (2) is not equivalent to Eq. (1), so the claimed trajectory for that non-convex object would not be a valid solution of the projected dynamical system.

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Extended reading notes

Core claim

The authors' central claim is that the nonsmooth, hybrid dynamics of planar manipulation can be modeled as a projected dynamical system, $\dot{x} = P_{T_C(x)}(f(x))$, and that this is computationally usable for optimal control once recast as the equivalent differential complementarity system $\dot{x} = f(x) + \nabla c(x)\lambda$, $0 \le c_i(x) \perp \lambda_i \ge 0$. Here $c_i$ are signed distance functions (for discs, explicit; for ellipsoids, defined as the solution of a convex program and embedded through its KKT conditions), and $\lambda_i$ are contact forces preventing overlap. The paper claims that direct optimal control of this system, discretized with finite elements with switch detection and a Scholtes relaxation homotopy, solves nontrivial planar manipulation tasks: a frictionless pusher rotating and returning a slider, and two disc pushers transporting an ellipsoidal slider under quasi-static Coulomb friction. For the first problem the full solve takes 53.58 seconds and for the second 161.598 seconds, which the authors present as evidence that the approach generates trajectories with reasonable computational effort. The paper also extends the switch-detection idea to friction by introducing a contact-closing indicator $b_i$ that relaxes cross-complementarity conditions when discontinuities appear.

Load-bearing premise

The equivalence between the projected dynamics and the complementarity system requires the constraint set to have a closed convex tangent cone, and for non-convex unions of ellipsoids the paper explicitly leaves this property unproven.

Editorial extensions

If this is right

  • Planar manipulation problems with sliding contacts can be posed and solved as single optimal control problems, with contact modes determined implicitly rather than enumerated.
  • Ellipsoidal objects, including non-convex unions of ellipsoids, can enter the same framework through implicitly defined signed distance functions embedded via KKT conditions.
  • Quasi-static friction can be incorporated by adding complementarity constraints on tangential velocity and a relaxed cross-complementarity condition for stick-slip switches.
  • The two demonstrated solves, 53.58 seconds for frictionless pushing and 161.598 seconds for two-pusher friction transport, indicate the approach is computationally practical at least at this scale.
  • If the framework extends as claimed, it gives a direct optimal control alternative to graph-based or sampling-based manipulation planners.

Reading between the lines

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

  • A natural next test is whether the same framework scales to three-dimensional ellipsoidal objects; the paper states the friction formulation in three dimensions as future work, and the convex-program signed distance machinery should carry over.
  • The open question about convexity of the tangent cone for unions of ellipsoids means the strongest advertised capability, non-convex objects, currently rests on numerical experience rather than proof; a failure there would not invalidate the convex single-ellipsoid results.
  • The $b_i$ relaxation idea could be reused in other hybrid optimal control problems where multipliers jump at events, since it only requires a computable indicator of active constraints.
  • The approach trades combinatorial mode selection for a larger nonlinear program, so its practical limit will likely be set by how many contacts and finite elements the solver can handle; benchmarking against mode-enumeration methods on identical tasks would be informative.
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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 / 4 minor

Summary. The paper proposes a modeling and numerical framework for planar manipulation tasks in which the contact dynamics are described as a projected dynamical system (PDS). The authors formulate an equivalent dynamical complementarity system using implicit signed distance functions, extend the model with a quasi-static friction complementarity formulation, discretize the resulting optimal control problem with finite elements with switch detection (FESD), and solve it using a Scholtes relaxation homotopy. Two numerical experiments are reported: a frictionless planar pushing task and a collaborative pushing task with friction, with reported solution times of 53.6 seconds and 161.6 seconds, respectively. The paper also claims a promising extension to non-convex objects modeled as unions of ellipsoids.

Significance. If the framework is correct and reproducible, it would be a useful direct optimal control route for nonsmooth manipulation problems, combining implicit signed-distance modeling with complementarity-based switch detection. The availability of source code in the nosnoc example set is a concrete strength, and the two example trajectories give plausible evidence for the proof-of-concept claim. However, the central ellipsoid signed-distance formulation is invalid as written due to a sign error that also makes the displayed KKT conditions inconsistent, and the claimed union-of-ellipsoids capability is not demonstrated by either experiment. These issues are load-bearing for the paper's main modeling contribution.

major comments (4)
  1. [Section III-B] The printed signed-distance optimization c(x) = min_{alpha,pd} alpha - 1 s.t. alpha <= (pd-pi)^T P(theta_i)(pd-pi), i=1,2, is unbounded below: for any pd, choosing alpha arbitrarily negative satisfies both constraints, so the minimum does not exist. The KKT conditions immediately below are internally inconsistent: the stationarity conditions 1 - mu1 - mu2 = 0 and 2*mu1*P(theta1)^T(pd-p1)+2*mu2*P(theta2)^T(pd-p2)=0 correspond to constraints alpha >= (pd-pi)^T P(theta_i)(pd-pi), while the displayed complementarity 0 <= mu_i perp (pd-pi)^T P(theta_i)(pd-pi) - alpha >= 0 has the sign reversed. With the corrected inequality direction alpha >= q_i(pd), the problem becomes bounded and the KKT conditions become consistent. As printed, however, the ellipsoid SDF, the gradient dn(x), and the equivalence between Equations (1) and (2) for ellipsoidal objects are not established. This is load-bearing because the collaborative pushing experiment in Section V-B relies on this SDF.
  2. [Section III-B, last paragraph] The claim that uniqueness of the contact point 'is sufficient to show that the tangent cone must be closed and convex at all points [17]' is not justified by the text; the tangent-cone property is a statement about the feasible set C, not about the SDF value function, and a unique gradient of c(x) does not by itself imply convexity of the tangent cone. The paper also explicitly leaves the union-of-ellipsoids case open ('It remains to be shown whether this formulation maintains the required properties of the set C'), yet the abstract asserts that the method handles 'non-convex objects modeled as unions of convex ellipsoids with reasonable computational effort.' No experiment in Section V uses a union of ellipsoids; Section V-B uses a single ellipsoid. The abstract claim is therefore not supported by the presented results.
  3. [Section V-B] The collaborative pushing experiment uses two disc pushers and one ellipsoidal slider, but the paper defines signed distance functions only for disc-disc contacts (Section III-A) and ellipse-ellipse contacts (Section III-B). The disc-ellipsoid signed distance function actually used in this experiment is not defined anywhere in the manuscript. The authors should specify how the disc-ellipsoid distance is obtained, for example as a degenerate-ellipse limit, and provide the corresponding KKT conditions, so that the numerical model can be reproduced from the paper alone.
  4. [Section IV-A] The index in the friction cross-complementarity relaxation using b_i is inconsistent: b_i is defined for i = 1, ..., Nfe - 1, but the constraints '0 <= b_n Gf_{n-1,ns} perp Hf_{n,j} >= 0' and '0 <= Gf_{n,i} perp b_n Hf_{n-1,ns} >= 0' are written with b_n for n = 2, ..., Nfe, which is out of range at n = Nfe. The intended quantity is presumably b_{n-1}. The authors should correct this and also clarify how the relaxation behaves when a contact is persistently active with lambda > 0 and c = 0, in which case b_i is not necessarily zero.
minor comments (4)
  1. [Section III-B] The phrase 'signed distance distance' should be corrected to 'signed distance'. The displayed complementarity condition contains a stray 'e' before '(pd - pi)^T' and should be cleaned up.
  2. [Section IV-A] The sentence 'We discretize the each of Ns control intervals' should read 'We discretize each of the Ns control intervals'.
  3. [Abstract] There is a missing space in 'switch detection[16]'; this is a minor formatting issue.
  4. [Footnote 1] The note that the source code is available 'On the branch pds_sdf' is useful, but the reproducibility statement would be improved by a commit hash or a tagged version.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the core PDS-to-complementarity derivation, FESD discretization, and SDF KKT embedding are self-contained or cite external foundational results; self-citations to prior method papers are not load-bearing predictions.

full rationale

The paper's central claim is a method demonstration: it formulates manipulation dynamics as a projected dynamical system (PDS), equivalently as a complementarity system (Eq. (1) to Eq. (2), citing the external equivalence result [2]), embeds a convex-program signed distance function from external work [19], and solves the resulting optimal control problem with FESD discretization and a Scholtes homotopy. The FESD method [16], NOSNOC toolkit [14], and MPCC solution approach [15] are self-citations, but they are code-supported prior method papers whose key discretization and relaxation equations are restated in Section IV rather than merely invoked as black-box predictions. No parameter is fitted to a subset of data and then announced as a prediction; the objective weights and discretization sizes are declared design choices. The openly stated limitation in Section III-B — "It remains to be shown whether this formulation maintains the required properties of the set C" — is an admitted gap for non-convex unions of ellipsoids, and it weakens the generality of the claim, but it is not a circular step. The same holds for the apparent sign error in the printed ellipsoid SDF optimization problem, which would make that program unbounded below as written; that is a correctness defect in the presented model, not an equivalence-by-construction between input and output. Overall, the derivation chain does not reduce to its own inputs, so the circularity score is low.

Assumptions & free parameters 2 free parameters · 6 assumptions · 0 invented entities

No new physical entities are introduced. The algebraic variables (lambda, friction multipliers, gamma, alpha, p_d, b_i) are standard complementarity variables or auxiliary algorithmic variables. The main external inputs are the OCP weights and discretization choices, which are declared design choices, and the quasi-static friction approximation.

free parameters (2)
  • OCP objective weights (Q_T, Q_u, q_theta) = Example 1: Q_T=diag(1e-3,1e-3,1e2,1e2,1e3), Q_u=1e-1 I; Example 2: Q_T=diag(1e-1,...,1e3), q_theta=1e2
    Hand-chosen weights define each planning task; they are explicit design choices, not fitted to data, but they shape the reported trajectories.
  • Discretization parameters (Nfe, ns, Ns, T) = Nfe=4, ns=4 or 2, Ns=30, T=20
    Fixed per example without a convergence study; results may change with refinement.
assumptions (6)
  • standard math PDS (1) is equivalent to the DCS (2) via Brogliato et al. [2].
    Section II relies on this equivalence to replace the projection operator with complementarity multipliers.
  • standard math The signed-distance program in Section III-B is convex and satisfies a constraint qualification, so KKT conditions are necessary and sufficient.
    Used to embed the ellipsoid distance and to extract dn(x)=grad_x g mu. Note that the displayed inequality direction is inconsistent as printed.
  • domain assumption LICQ: gradients of active signed-distance constraints are linearly independent.
    Section III-A states this guarantees existence and uniqueness for the frictionless PDS; it may fail at degenerate configurations.
  • domain assumption Quasi-static friction: friction force minimizes relative tangential velocity subject to a bound of mu_f lambda.
    Section III-C adopts this common modeling approximation; it is not full Coulomb dynamics and is not experimentally validated.
  • ad hoc to paper For unions of ellipsoids, the set C retains a closed and convex tangent cone, giving well-posed PDS solutions.
    Section III-B admits 'it remains to be shown' whether this holds, but the paper proceeds based on computational experience.
  • ad hoc to paper The b_i-based relaxation of friction cross-complementarity constraints preserves FESD switch-detection accuracy.
    Introduced without proof in Section IV-A for contacts that open or close.

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Cite this review

Pith. "Pith review of Towards Solutions of Manipulation Tasks via Optimal Control of Projected Dynamical Systems." pith.science (2026). https://pith.science/paper/D7TL5O3T

@misc{pith2026250111946,
  author       = {Pith},
  title        = {Pith review of: Towards Solutions of Manipulation Tasks via Optimal Control of Projected Dynamical Systems},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/D7TL5O3T}},
  note         = {Machine review of arXiv:2501.11946}
}
read the original abstract

We introduce a modeling framework for manipulation planning based on the formulation of the dynamics as a projected dynamical system. This method uses implicit signed distance functions and their gradients to formulate an equivalent gradient complementarity system. The optimal control problem is then solved via a direct method, discretized using finite-elements with switch detection. An extension to this approach is provided in the form of a friction formulation commonly used in quasi-static models. We show that this approach is able to generate trajectories for problems including multiple pushers, friction, and non-convex objects modeled as unions of convex ellipsoids with reasonable computational effort.

Figures

Figures reproduced from arXiv: 2501.11946 by the authors.

Figure 1
Figure 1. Several frames of the solution for the frictionless manipulation problem. [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Plots of solution to the frictionless pushing problem. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Several frames of the solution for the manipulation problem with [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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