{"id":"dac817bd-3b2b-466b-88d8-82c255e29e7e","arxiv_id":"2501.11946","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"A manipulation-planning framework that solves an optimal control problem over projected, complementarity-based contact dynamics, demonstrated on planar pushing with friction and ellipsoidal objects.","lead":"Robots that push objects must cope with contacts switching between touching and sliding, which makes planning hard. The authors write pushing as a projected dynamical system with signed-distance contact constraints, then solve the resulting optimal control problem to plan trajectories for two planar pushing tasks.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section III-B's printed ellipsoid signed-distance program is unbounded below: the inequality direction contradicts the KKT conditions, so the ellipsoid and union-of-ellipsoid model is invalid as written.","rationale":"I read the paper as a proof-of-concept for planning manipulation via optimal control of projected dynamical systems, with the strongest evidence being two solved planar pushing examples. The central claim requires the implicit signed-distance formulation for ellipsoids to be correct, because it is the basis for the collaborative pushing experiment and for the claimed extension to unions of ellipsoids. The printed optimization problem in Section III-B is not merely missing a proof; it is internally inconsistent. The objective is unbounded below under the stated constraints, and the KKT equations correspond to the opposite inequality. This is a direct mathematical error, not a disagreement with outside consensus. The reader's weakest_assumption focused on the well-posedness of non-convex unions of ellipsoids, which the paper explicitly leaves open; I agree that is a real gap. However, I judge the sign error in the ellipsoid SDF to be the single most load-bearing concern, because it invalidates the written model even in the convex ellipsoid case used in the experiment. The concern is fixable: replacing alpha <= q_i with q_i <= alpha yields the intended convex program and the KKT conditions as mostly printed (up to the complementarity sign), and the code is nominally available in NOSNOC. For this reason I do not recommend rejecting the paper outright; the appropriate outcome remains the reader's CONDITIONAL verdict, contingent on correcting the equations, re-deriving the gradient, and verifying the collaborative pushing result with the corrected SDF. I mark agreement as partial because the reader identified the same sign inconsistency in the rationale but did not make it the headline weakest assumption.","tokens_in":7581,"tokens_out":9097,"duration_ms":100090,"concrete_test":"Take two unit circles with centers at (0,0) and (1.5,0), so P1=P2=I. Solve the printed optimization problem from Section III-B: min alpha - 1 s.t. alpha <= ||pd-p1||^2 and alpha <= ||pd-p2||^2. A solver will report unbounded below, whereas the corrected formulation min alpha - 1 s.t. ||pd-pi||^2 <= alpha returns alpha = 0.75^2 = 0.5625 and c = -0.4375, correctly indicating overlap. Re-running the collaborative pushing example with this corrected SDF would confirm whether the published 161.6 s trajectory is reproducible.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central modeling contribution for ellipsoids is the implicit signed-distance function c(x) in Section III-B: c(x) = min_{alpha,pd} alpha - 1 subject to alpha <= (pd-pi)^T P(theta_i)(pd-pi), i=1,2. As printed, this problem is unbounded below: for any pd, taking alpha to -infinity satisfies both constraints, so the minimum does not exist. The KKT conditions displayed immediately afterward are also inconsistent with the stated inequality. The stationarity condition 1 - mu1 - mu2 = 0 together with the gradient stationarity 2*mu1*P1^T(pd-p1)+2*mu2*P2^T(pd-p2)=0 is the KKT system for constraints q_i(pd) <= alpha (i.e., alpha - q_i(pd) >= 0), not for alpha <= q_i(pd). The complementarity line 0 <= mu_i perp q_i(pd) - alpha >= 0 has the sign reversed relative to those stationarity conditions. Therefore the extraction of the signed-distance gradient dn(x)=nabla_x(g(alpha,pd;x))mu is not justified by the written equations, and the equivalence between the PDS in Eq. (1) and the complementarity system in Eq. (2) fails at the level of the ellipsoid model. This is more load-bearing than the separately acknowledged open question about unions of ellipsoids, because even a single pair of ellipsoids has no well-defined c(x) as written. The collaborative pushing experiment in Section V-B uses this SDF, so its reported trajectory cannot be checked against the stated model until the inequality direction is corrected and the corrected KKT system is provided.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":7916,"tokens_out":7244,"duration_ms":74606,"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":[{"comment":"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.","section":"Section III-B"},{"comment":"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.","section":"Section III-B, last paragraph"},{"comment":"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.","section":"Section V-B"},{"comment":"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.","section":"Section IV-A"}],"minor_comments":[{"comment":"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.","section":"Section III-B"},{"comment":"The sentence 'We discretize the each of Ns control intervals' should read 'We discretize each of the Ns control intervals'.","section":"Section IV-A"},{"comment":"There is a missing space in 'switch detection[16]'; this is a minor formatting issue.","section":"Abstract"},{"comment":"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.","section":"Footnote 1"}],"recommendation":"major_revision","confidential_remarks":"The central idea is promising and the numerical evidence is suggestive, but the ellipsoid SDF formulation contains a sign error that invalidates the printed model, and the abstract overstates the union-of-ellipsoids results. These are fixable within the manuscript's scope, so I recommend major revision rather than rejection. The paper fits the journal's scope, and the reliance on the authors' prior FESD work is disclosed appropriately."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The core idea is worth taking seriously: formulate planar manipulation as optimal control of a projected dynamical system, convert it to a complementarity system, discretize with FESD, and solve as an MPCC. The two examples are plausible evidence for the proof-of-concept claim—a frictionless planar push and a collaborative push with friction, solved in 53.6 and 161.6 seconds—and the code is available in NOSNOC. That part is credible and reproducible.\n\nWhat is actually new is the combination: the manipulation-specific complementarity model, the implicit ellipsoid signed-distance function embedded via KKT conditions, and the quasi-static friction complementarity relaxation. The spherical case is well-founded and the citation pattern is honest.\n\nThe soft spots are significant. First and most load-bearing, the ellipsoid SDF in Section III-B is invalid as written. The optimization problem is unbounded below: for any contact point, alpha can go to negative infinity. The displayed KKT conditions do not match the inequality direction—stationarity is for constraints alpha >= q_i, while the complementarity is for q_i - alpha. The correct version is a sign flip, but the collaborative pushing experiment uses this model, so its trajectory cannot be checked against the printed equations. This is more than a cosmetic typo because it sits at the center of the claimed contribution.\n\nSecond, the authors themselves state that it remains to be shown whether unions of ellipsoids preserve the required properties of the set C. That is an acknowledged open question, and the PDS-DCS equivalence depends on it. Third, the friction cross-complementarity relaxation is heuristic; there is no proof or numerical convergence study supporting it.\n\nWho is this for? Researchers in nonsmooth optimal control and contact-rich manipulation. It deserves a serious referee, but not acceptance as-is. I would send it to peer review with the expectation of major revision: correct the ellipsoid formulation, provide the corrected KKT system, and address the union-of-ellipsoids well-posedness with either a proof or a systematic numerical study.","headline":"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.","tokens_in":8460,"tokens_out":3336,"would_cite":false,"duration_ms":35657,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["49J40","90C33","93C85"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["projected dynamical systems","optimal control","manipulation planning","complementarity systems","finite elements with switch detection","signed distance functions","friction","mathematical programs with complementarity constraints"],"falsifier":"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.","tokens_in":7311,"feed_emoji":"🤖","tokens_out":6514,"duration_ms":64563,"temperature":0.7,"pith_summary":"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.","feed_headline":"Two pushing tasks solved in minutes by one model","feed_subtitle":"A projected-dynamics formulation handles friction, multiple pushers, and ellipsoidal sliders without preselecting contact modes.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the equivalence between complementarity systems, projected systems, and differential inclusions on which the PDS-to-DCS reformulation rests.","marker":"[2]"},{"why":"Provides the existence and uniqueness theory for projected dynamical systems used to justify well-posedness under mild assumptions.","marker":"[13]"},{"why":"Introduces finite elements with switch detection for projected dynamical systems, which is the numerical discretization at the core of the method.","marker":"[16]"},{"why":"Supplies the relaxation homotopy method used to solve the resulting mathematical programs with complementarity constraints.","marker":"[15]"},{"why":"Source of the convex-program formulation for signed distance between convex primitives, embedded via KKT conditions in the ellipsoidal model.","marker":"[19]"},{"why":"Supplies the adjoint-based gradient extraction $d_n(x) = \\nabla_x(g(\\alpha,p_d;x))\\mu$ used to differentiate the implicit signed distance function.","marker":"[6]"},{"why":"Gives the variational-analysis results used to argue that the tangent cone is closed and convex for ellipsoidal constraint sets.","marker":"[17]"},{"why":"Basis of the quasi-static friction model that the paper extends with complementarity conditions on tangential velocities.","marker":"[10]"}],"fun_headline_variants":["Projected dynamics solve planar pushing tasks","Optimal control of projected dynamics handles multiple pushers","Model turns manipulation into complementarity problem","Friction, multi-pusher tasks solved via projected dynamics","Complementarity formulation solves manipulation tasks"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Projected dynamics solve planar pushing tasks","Optimal control of projected dynamics handles multiple pushers","Model turns manipulation into complementarity problem","Friction, multi-pusher tasks solved via projected dynamics","Complementarity formulation solves manipulation tasks"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000305,"raw_usage":{"total_tokens":1734,"prompt_tokens":911,"completion_tokens":823,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":527,"completion_tokens_details":{"reasoning_tokens":755}},"tokens_in":527,"tokens_out":823,"duration_ms":6570,"temperature":1.0,"reasoning_tokens":755,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T17:41:16.912492+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"On the equivalence between com- plementarity systems, projected systems and differential inclusions","cited_arxiv_id":null,"evidence_quote":"Supplies the equivalence between complementarity systems, projected systems, and differential inclusions on which the PDS-to-DCS reformulation rests."},{"cited_title":"Projected dynamical systems and variational inequalities with applications , volume 2","cited_arxiv_id":null,"evidence_quote":"Provides the existence and uniqueness theory for projected dynamical systems used to justify well-posedness under mild assumptions."},{"cited_title":"Finite Elements with Switch Detection for Numerical Optimal Control of Projected Dynamical Systems","cited_arxiv_id":"2404.05367","evidence_quote":"Introduces finite elements with switch detection for projected dynamical systems, which is the numerical discretization at the core of the method."},{"cited_title":"Solving mathematical programs with complementarity constraints arising in nonsmooth optimal control","cited_arxiv_id":null,"evidence_quote":"Supplies the relaxation homotopy method used to solve the resulting mathematical programs with complementarity constraints."},{"cited_title":"Differentiable collision detection for a set of convex primitives","cited_arxiv_id":null,"evidence_quote":"Source of the convex-program formulation for signed distance between convex primitives, embedded via KKT conditions in the ellipsoidal model."},{"cited_title":"High Accuracy Numerical Optimal Control for Rigid Bodies with Patch Contacts through Equivalent Contact Points -- Extended Version","cited_arxiv_id":"2403.13931","evidence_quote":"Supplies the adjoint-based gradient extraction $d_n(x) = \\nabla_x(g(\\alpha,p_d;x))\\mu$ used to differentiate the implicit signed distance function."},{"cited_title":"Rockafellar and R","cited_arxiv_id":null,"evidence_quote":"Gives the variational-analysis results used to argue that the tangent cone is closed and convex for ellipsoidal constraint sets."},{"cited_title":"A quasi-static model and simulation approach for pushing, grasping, and jam- ming","cited_arxiv_id":null,"evidence_quote":"Basis of the quasi-static friction model that the paper extends with complementarity conditions on tangential velocities."}],"review_version":1}