{"id":"73a8e902-01e6-44b5-a369-bc9c00c685d7","arxiv_id":"2602.06864","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Branch-and-rejoin trajectory optimization improves robot robustness to uncertain contact timing, with 21.6- and 40-percentage-point success gains in cart-pole and egg-catching tests.","lead":"SURE is a trajectory-optimization method that plans multiple possible contact-time branches and merges them into one common final path, so robot plans stay valid when a bump or catch happens earlier or later than expected. In two testbeds it raised success rates from 44.8% to 66.4% (cart-pole) and from 45% to 85% (egg catching) versus nominal planning.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"SURE's robustness is only guaranteed for contact times exactly at branching nodes; off-grid contacts leave the scheduler using a branch with mismatched post-impact state, and the paper provides no bound on this error.","rationale":"The reader's identified weakest assumption about time alignment at the rejoin node is not, in my view, the most load-bearing issue: a time base can be reset when switching to the common final trajectory, since only state continuity at the merge is needed for the concatenated trajectory to be executable. The deeper weakness is the mismatch between discrete planning nodes and continuous actual contact times. The SURE optimization guarantees that each branch, if tracked from its exact post-impact initial state R(x_i), rejoins and reaches the terminal state. But the scheduler cannot magically obtain that exact initial state when contact occurs off-grid. The paper provides no analysis of this error, and the reported aggregate success rates could mask a strong dependence on branch density. This is a genuine gap in the argument, not merely a missing implementation detail. That said, the paper's empirical results are internally consistent and the branch-and-rejoin idea is sound for exactly-on-grid contact times. The concern is addressable with a branch-count sweep and per-distance-to-branch analysis, so the appropriate verdict remains CONDITIONAL, as the reader concluded.","tokens_in":15156,"tokens_out":16403,"duration_ms":174597,"concrete_test":"In the cart-pole simulation, run the SURE plans with a fine sweep of wall positions (e.g., 1000 samples), and compute success rate as a function of the normalized distance between the actual contact time and the nearest branching node's nominal contact time. Then repeat the SURE optimization with |K|=3, 5, 10, and 20. If success dips near the midpoint between branches and/or degrades as |K| decreases, the discrete-branch approximation is responsible for the claimed robustness; if not, the mismatch is benign.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that SURE robustly handles contact-timing uncertainty. In Sec. IV-B, branches are generated only for the finite set of nodes K (5 branches in cart-pole, 10 in egg catch). Constraint (2e) initializes branch i at R(x_i), the post-impact state for a contact exactly at node i. In execution, the scheduler (Sec. IV-C1) selects the 'nearest' branch when contact is detected, but actual contact times are continuous (wall position and release height vary continuously). For a contact occurring between nodes, the true post-impact state differs from every R(x_i); the rejoin guarantee (2f) applies only to trajectories starting exactly at R(x_i). The paper does not analyze the resulting initial-state error, does not specify the nearest-branch metric, and reports only aggregate success rates. Consequently, the demonstrated robustness may depend on the chosen branch density and on the tracking controller's ability to absorb these mismatches, rather than being a property of the branch-and-rejoin formulation. The abstraction of a continuous uncertainty interval into |K| discrete pre-impact states is the unexamined load-bearing approximation.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes SURE, a trajectory optimization framework for contact-rich tasks under uncertain contact timing. It augments a nominal multiple-shooting OCP with a branching phase: from each of |K| discrete pre-impact nodes, a short branch is optimized from the corresponding post-impact state, and all branches are constrained to rejoin a common final trajectory at a shared state. Two control uses are proposed: contact-triggered trajectory scheduling and a single 'robust nominal' middle branch. The method is evaluated in simulation on a cart-pole wall-stabilization task and on hardware for egg catching, reporting higher success rates than nominal baselines and lower computation time than a full tree OCP baseline. The central claim is that branching plus rejoining gives robustness to contact-timing uncertainty at modest optimality loss.","tokens_in":15457,"tokens_out":10010,"duration_ms":106077,"significance":"The branching-and-rejoining idea is a genuinely useful middle ground between a single nominal trajectory and a full tree OCP, and the two case studies are evaluated independently of the optimization constraints through closed-loop stochastic simulation and hardware trials, which is a real strength. The robust nominal trajectory also outperforms the nominal plan without a scheduler, suggesting the improvement is not only an artifact of an oracle switch. However, the finite-branch approximation and the unspecified timing of the scheduler are currently unexamined, so the paper's central robustness claim is not yet fully supported. If the authors close that gap, this could be a practical contribution to contact-rich trajectory optimization.","major_comments":[{"comment":"The formulation builds branches only at the finite set of nodes K. In execution, the contact time is continuous within [K0, Ke], and for a contact that occurs between two branching nodes the actual post-impact state is not equal to any R(x_i) used in (2e). The rejoining guarantee (2f) then does not apply to the selected branch. The paper does not define the 'nearest' branch metric, does not quantify the initial-state mismatch, and does not study sensitivity to branch density. Since the abstract claims robustness to contact-timing uncertainty rather than only to the discrete node set, this is a load-bearing approximation. Please add an error bound (e.g., via Lipschitz continuity of the reset map and dynamics) or a systematic sensitivity study (e.g., success rate vs |K| and vs contacts forced at midpoints between nodes).","section":"Sec. IV-B, Eqs. (2e)-(2f), and Sec. IV-C1"},{"comment":"State equality (2f) is the only rejoining condition. The trajectory-scheduling description does not state how time bases are aligned when a branch is activated at an arbitrary detection time and then joins the common trajectory. If the branch has a different duration from the corresponding common-trajectory segment, the reference time at the switch is undefined. Please specify the exact scheduling rule (branch selection metric, time reparameterization, and the switch condition at the merge state), and confirm that the closed-loop evaluator implements that rule.","section":"Sec. IV-C1, Eq. (2f)"},{"comment":"The claim that the robust nominal trajectory 'keeps the relative velocity bounded below 2.67 m/s across all tested initial heights' is partly a check of constraints (19c) at the branching nodes, because v_lim is minimized subject to those constraints. For release heights that do not coincide with branch nodes, no bound is guaranteed by the optimization. To support the robustness claim, either evaluate at off-grid heights (e.g., midpoints between branches) or prove an interpolation bound from the finite branch constraints.","section":"Sec. V-B3, Eq. (19c), Fig. 9"},{"comment":"The guard constraints are not stated consistently. In Eq. (1), g(x_i)>0 in (1e) is immediately followed by g(x_i)=0 at i=c in (1g), so the domain of (1e) must exclude the contact node. In Eq. (2), (2l) bounds g only for i<K0; nothing in the general formulation prevents a guard crossing after K_e. The case-specific text in Sec. V-A2 adds constraint (11), but the general SURE problem (2) should include g(x_i)>d for all i not in K. Also, if d is a decision variable in (2), the objective does not contain d, so d is not determined unless fixed as in the case studies. These points need to be cleaned up for the formulation to be well-posed.","section":"Secs. III-IV, Eqs. (1e), (1g), (2l)"},{"comment":"Success rates are reported as point estimates with no confidence intervals, and Table I does not state the number of simulation trials per initial condition. In Table II, the 'Success Rate' column appears to count a 0.5 score as half-success, but this scoring rule is not defined in the text. Because the central evidence is empirical, please report exact n, confidence intervals or a significance test, and the scoring rule explicitly.","section":"Tabs. I-II and Sec. V"}],"minor_comments":[{"comment":"The phrases 'improvement of 21.6%' and 'by 40%' mix percentage points and relative improvements; please state which is meant (e.g., 21.6 percentage points versus a 48% relative improvement for the cart-pole total).","section":"Abstract and Sec. V"},{"comment":"The justification for choosing the middle branch as 'the most representative trajectory' is informal. Please label this as a heuristic or provide a formal argument for why the midpoint of the branching interval is the best single nominal trajectory.","section":"Sec. IV-C2"},{"comment":"In the egg-catching SURE formulation, state continuity at contact is assumed (x_{i,0}=x_i), whereas the general formulation (2e) uses a reset map R(x_i). Since the egg mass is small this is a reasonable approximation, but it should be explicitly flagged as a case-specific simplification.","section":"Sec. V-B2"},{"comment":"The evaluation ranges for the wall position and restitution coefficient are wider than the d=0.05 design range. This is a strength, but the caption and text should state clearly that the closed-loop evaluation is over a broader uncertainty set than the one used in optimization.","section":"Sec. V-A3, Fig. 5"}],"recommendation":"major_revision","confidential_remarks":"I think the paper is promising and the empirical results are worth preserving, but the central robustness claim is currently supported only for the discrete branch nodes; the continuous contact-time case is the main unexamined approximation. I would recommend requiring the authors to add either a formal error bound or a sensitivity study with off-grid contacts and varying branch counts, and to specify the scheduling/time-alignment mechanism. If those additions are made, the paper would likely be suitable for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the branch-and-rejoin piece is genuinely new, the experiments are honestly executed, and the robust-nominal-without-scheduler comparison is a nice control. The soft spot is that the paper does not actually analyze what happens when contact occurs between branches, even though its central claim is robustness to continuous timing uncertainty. That does not sink the paper, but it needs to be acknowledged and bounded before I take the headline claim at face value.\n\nWhat's new: prior work [39] terminates at the latest contact and replans; tree OCP [11] is exponential. SURE forces all post-impact branches to rejoin a common final trajectory, which is a clean way to keep the problem size linear in the number of branches. The two case studies are appropriate: cart-pole with wall, and egg catching on a real Z1. The simulations use a PGS contact solver independent of the optimization model, and the hardware trials are real. The result that the robust nominal trajectory (middle branch) beats the nominal trajectory even without a scheduler is the most convincing piece — it shows the optimizer is genuinely reshaping the motion, not just relying on the scheduler to fix things.\n\nSoft spots, in order of severity:\n\n1. Off-grid contact times. Branches are generated only at a finite set of nodes. In execution, contact can happen at any time. The scheduler picks the 'nearest' branch, but the paper never defines that metric and never bounds the mismatch between the true post-impact state and the chosen branch's initial state. The stress-test note is right: the robustness guarantee only holds exactly at branch nodes. Empirically it works, but that is because the branch density and the feedback controller are absorbing the error. The paper should at least discuss this approximation and show sensitivity to branch count.\n\n2. The abstract says 'improves the success rate by 40%' — that is a 40 percentage-point increase (from 45% to 85%), not a 40% relative improvement. Same for the 21.6% in cart-pole. That is an overstatement that will mislead casual readers.\n\n3. No code or data release, and no confidence intervals on the success rates. For a methods paper that depends on empirical demonstration, this makes the results hard to verify. The hardware trial count (20 per trajectory) is small but not unusual.\n\nThe rejoin constraint only enforces state equality, not time alignment, and the paper does not discuss how the scheduler maps the detected contact time to a time index on the branch. That is a real implementation detail worth clarifying, though the experiments suggest it is handled in practice.\n\nBottom line: this paper deserves a serious referee. The gap between the discrete branch formulation and the continuous uncertainty claim is addressable in revision, and the empirical evidence, while imperfect, is consistent. I would cite it if I were working on contact-rich trajectory optimization.","headline":"SURE is a solid incremental advance with real empirical gains; the main weakness is that its robustness claim outruns the discrete-branch formulation, and the abstract overstates the improvement.","tokens_in":15879,"tokens_out":2658,"would_cite":true,"duration_ms":28403,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A branch-and-rejoin trajectory optimizer explicitly plans for uncertain contact timing, shifting motions from brittle single-path plans to families of safe responses.","keywords":["trajectory optimization","contact timing uncertainty","hybrid systems","branching and rejoining","robust control","trajectory scheduling","cart-pole","robot manipulation"],"falsifier":"In the cart-pole simulation, deliberately shift the detected contact time by a fraction of the branch's time step so that the scheduler switches to a branch at a node whose nominal time does not match the actual switch time, and check whether the closed-loop success rate drops toward the nominal level. If the state-only rejoin were sufficient, this mismatch should not matter; if tracking fails, the timing alignment assumption is falsified.","tokens_in":15042,"feed_emoji":"🤖","tokens_out":4542,"duration_ms":42342,"temperature":0.7,"pith_summary":"This paper introduces SURE, a trajectory optimization formulation for robotic tasks in which the exact time of a contact is not known in advance. Instead of planning one deterministic trajectory or a full tree of possibilities, SURE lets a pre-impact trajectory branch at several candidate contact times, runs a short post-impact branch from each, and forces all branches to rejoin a common final trajectory. In simulation of a cart-pole balancing against a wall of uncertain position, the scheduled branches raise success from 44.8% to 66.4%; in a physical egg-catching experiment, the middle 'robust nominal' branch raises success from 45% to 85%. The rejoining trick costs only about 5% in optimality while cutting computation by more than half compared to a full tree formulation.","feed_headline":"Branch-and-rejoin plans beat uncertain contact times","feed_subtitle":"One optimization pass creates branches for each possible impact moment, then merges them into a shared finish — lifting egg-catch success fr","key_machinery":"The branching-and-rejoining structure: from each node in the branching phase K, a short branch of Nr+1 nodes evolves under free dynamics and is constrained to end exactly at the first common post-impact node x_{K_e+1}. Guard constraints g(x)=d at K0 and g(x)=-d at Ke define the uncertainty interval. This lets the optimizer plan the entire family of contact responses in one nonlinear program, with the number of decision variables growing only with the number of branches and branch length, not with full paths to the terminal state.","core_discovery":"The central claim is that robustness to contact-timing uncertainty can be obtained by explicitly optimizing a bundle of trajectories: one common pre-contact trajectory, a branch from each candidate contact state, and a single shared post-contact trajectory that all branches must join at a fixed node. The uncertainty interval is encoded by guard constraints that fix the contact-triggering function g to +d at the earliest contact node and to -d at the latest. This structure yields a trajectory that is safe across the whole interval, and when a contact sensor is available the controller can switch to the branch whose pre-impact state matches the detected contact time. The paper shows that even","pith_inferences":["Because the rejoin constraint only matches states, not arrival times, the scheduler's reference-time alignment at the switch is left unspecified; a mismatch between branch time grid and common-trajectory time grid could degrade tracking and deserves a dedicated test.","The same branch-and-rejoin structure could be applied to other uncertain parameters — for example, the restitution coefficient or object mass — by treating them as the branching variable instead of contact time.","The robust nominal trajectory's benefit comes from globally reshaping the motion (e.g., building a 'buffering distance' before impact), which suggests that learned policies conditioned on the uncertainty range could inherit this robustness without online replanning.","One testable extension is to relax the state-equality rejoin to a terminal cost on the mismatch, which would trade a little more optimality for even better conditioning of the optimization."],"forward_implications":["Explicitly modeling contact-timing uncertainty in the optimizer yields safer motions than nominal plans, even when the controller only tracks a single 'robust nominal' trajectory.","When a contact sensor exists, scheduling among the precomputed branches provides a further gain in success rate (e.g., 55.3% to 66.4% in the cart-pole study).","The rejoining constraint makes the approach computationally scalable: at 7 nodes per branch the added cost over the nominal trajectory is 4.87% while computation drops 55.85% relative to a tree OCP.","The same formulation extends to multiple contact transitions by repeating the branching phase for each transition.","The optimizer can treat the uncertainty width d as a decision variable, letting the solver find the largest uncertainty region over which the task remains feasible."],"fun_headline_variants":["Branch-and-rejoin plans boost contact success by 40%","One trajectory bundle handles uncertain contact times","Robust robot plans: branch, then merge for contact safety","SURE path planning: prepare for any impact moment","Uncertain contact? Optimize all branches at once"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The whole approach assumes that enforcing state equality at the rejoin node is sufficient — that a branch arriving at the shared trajectory with the same state but potentially a different time can be followed without defining how the reference time base aligns at the switch.","fun_headline_variants_meta":{"raw":{"variants":["Branch-and-rejoin plans boost contact success by 40%","One trajectory bundle handles uncertain contact times","Robust robot plans: branch, then merge for contact safety","SURE path planning: prepare for any impact moment","Uncertain contact? Optimize all branches at once"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000137,"raw_usage":{"total_tokens":947,"prompt_tokens":666,"completion_tokens":281,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":410,"completion_tokens_details":{"reasoning_tokens":204}},"tokens_in":410,"tokens_out":281,"duration_ms":3601,"temperature":1.0,"reasoning_tokens":204,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T03:46:29.858343+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"In the cart-pole simulation, deliberately shift the detected contact time by a fraction of the branch's time step so that the scheduler switches to a branch at a node whose nominal time does not match the actual switch time, and check whether the closed-loop success rate drops toward the nominal level. If the state-only rejoin were sufficient, this mismatch should not matter; if tracking fails, the timing alignment assumption is falsified.","supporting_citations":[],"review_version":1}