{"id":"8dd55e11-a921-4776-b88c-eb6a6dd6b557","arxiv_id":"2508.11129","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"Poisson safety functions are extended to time-varying, orientation-dependent domains and embedded in an MPC safety filter, demonstrated on a quadruped and a humanoid.","lead":"This paper creates a camera-based safety filter that lets legged robots avoid moving obstacles by adjusting both position and orientation. It combines a safety function computed from occupancy maps with model predictive control, and tests the system on a quadruped and a humanoid.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The safety claim rests on a constant-obstacle-velocity prediction model that can be wrong within the MPC horizon, and the paper explicitly disclaims formal safety in exactly that case.","rationale":"The reader's weakest assumption identifies the constant-velocity obstacle prediction model as the load-bearing point, and the paper itself acknowledges in Section IV that formal safety cannot be guaranteed for general time-varying environments. My reading agrees: the experimental evidence only demonstrates safety when the obstacle's motion is close to constant velocity over the prediction horizon. I considered whether a stronger concern exists in the lack of a formal proof that the extended Poisson safety function is a valid CBF in the lifted time-orientation state. For a single-integrator model with direct control over x, y, and theta, a smooth h with nonzero gradient is plausibly a valid DCBF when input limits are not binding, so the prediction-model mismatch is more concrete and more directly tied to the failure mode the paper admits. A test with a decelerating or reversing obstacle would settle whether the concern actually lands. Since the reader already assigned a conditional verdict and the identified concern matches mine, no change to the verdict is needed.","tokens_in":11002,"tokens_out":4038,"duration_ms":55968,"concrete_test":"Run Scenario 1 again with an actuated obstacle whose velocity changes abruptly mid-horizon, for example a small radio-controlled cart that moves toward the robot and then reverses direction or stops at a randomized time, while keeping the same MPC parameters and PDE discretization. Log the true h(t, x, theta) computed from the measured occupancy map, not the predicted one, at the commanded state. If h becomes nonpositive on any trial, the constant-v boundary prediction is load-bearing and the safety claim must be weakened to constant-velocity obstacles. A complementary offline check is to inject a step change in obstacle velocity into the boundary prediction model at a time inside the horizon and compare the predicted safe set with the true free space; any overlap indicates an unsafe admissible command.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central safety assertion is that the Poisson safety function extended to time and orientation can serve as a CBF constraint in MPC and keep the full robot footprint inside the true safe set. In the implementation, the function h entering the MPC is computed on a predicted safe set obtained by propagating obstacle boundaries with the linear model (31), which is derived under the stated assumption of constant v over the MPC prediction horizon. If an obstacle decelerates, stops, or changes direction within that horizon, the predicted boundary set used to define h can include regions that are actually occupied. The MPC can then issue commands that keep the modeled h positive while the true h, computed against the actual obstacle position, becomes negative. This is not a minor implementation gap: Section IV explicitly states that safety cannot be formally guaranteed in general time-varying environments, and no uncertainty margin, reachability-based bound, or adversarial obstacle test is provided. The reported positive values of h therefore validate safety only for obstacle motions that are approximately constant-velocity over the horizon, such as the thrown dodgeball in Scenario 1, and do not support the stronger claim that the method maintains safety in genuinely dynamic environments.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a predictive safety filter that combines nonlinear MPC with control barrier function (CBF) constraints synthesized from Poisson safety functions. It extends the static Poisson safety function of [18] in two directions: a temporal parameterization (Section III-B) that predicts the evolution of the safe-set boundary under an assumed constant obstacle velocity field, and a geometric parameterization (Section III-C) that uses Minkowski differences to lift the domain into a configuration space that includes robot orientation. The resulting safety function h_p is embedded as a discrete-time CBF constraint in an MPC problem (29) on a single-integrator model of position and heading, and the resulting velocity commands are tracked by low-level RL locomotion controllers on Unitree Go2 and G1 robots. Experiments include dodgeball avoidance (Section V-B) and narrow-corridor navigation with a humanoid (Section V-C). The paper argues that the positivity of h_p during these experiments verifies that safety was maintained.","tokens_in":11190,"tokens_out":5323,"duration_ms":64439,"significance":"If the CBF property held for the parameterized, time-varying safety function, this work would be a useful step toward perception-based, geometry-aware safety filtering for legged robots. The real-time PDE solve (20–100 ms) and the hardware demonstrations on two different platforms are nontrivial strengths, and the paper is candid about the constant-velocity prediction assumption and explicitly disclaims formal safety guarantees in general time-varying environments. However, the formal transfer from the static Poisson CBF result in [18] to the parameterized moving-boundary problem is not actually proved, and the experimental evidence is largely qualitative: there are no baseline comparisons, no reported trial counts or error bars, and no tests under the non-constant-velocity conditions that the paper itself excludes. The contribution is promising but not yet substantiated at the level claimed in the abstract and conclusions.","major_comments":[{"comment":"The central theoretical premise—that the parameterized Poisson safety function h_p in (28) is a valid CBF for the reduced-order single-integrator model and that the DCBF constraint in (29) renders the safe set forward invariant—is asserted by appealing to [18], but [18] is summarized in Section III-A as applying to the static Dirichlet problem (17), not to the temporally and orientationally parameterized problem (28). No theorem, proposition, or argument is provided to show that the discrete-time CBF condition (9)–(10) holds for h_p(t, x, θ) when the safe set itself evolves in time and with heading. This is load-bearing: the claim that the MPC+CBF filter 'maintains safety' depends exactly on this forward-invariance property. The authors should either add a rigorous statement with explicit regularity assumptions on the boundary motion, the forcing function f, and the time-dependence of h_p, or explicitly describe the safety filter as heuristic rather than CBF-guaranteed.","section":"Section IV, Eqs. (28)-(29)"},{"comment":"The safety constraint at each planning step is evaluated on a predicted future safe set obtained by propagating obstacle boundaries with the linear model (31), which is derived under the assumption of constant obstacle velocity v over the MPC horizon. If an obstacle accelerates, stops, or changes direction within the horizon, the predicted boundary is wrong, and the computed h_p no longer corresponds to the true safe set; the MPC can then issue commands that keep the modeled h_p positive while the true safety function becomes negative. The paper itself acknowledges this in Section IV: 'safety cannot be formally guaranteed in general time-varying environments.' The conclusion nevertheless states that positive h_p 'verifying that safety was effectively maintained.' The proof of the experimental safety claim therefore applies only to nearly constant-velocity obstacles, and no uncertainty margin, reachability bound, or adversarial test is provided to quantify robustness to prediction error. Please either add such a margin or clearly restrict the experimental claims to the constant-velocity regime.","section":"Section IV and Section V-A, Eq. (31)"},{"comment":"The evaluation of h_p is reported as the primary evidence of safety, but this evidence is insufficient as presented. First, h_p is computed on the predicted boundary, not on the true occupied set; second, h_p is evaluated on the planned or measured trajectory of the reduced-order state, not on the full-order robot footprint under the low-level RL controller, whose tracking error is not quantified. The paper cites [33] for the reduced-order-model safety argument but does not verify the required 'sufficient tracking' condition experimentally. Third, the figures show a single representative trial per scenario, with no trial count, no error bars, and no statistical summary. The sentence 'These results were consistent and repeatable, as demonstrated in the video' is not a quantitative substitute. The authors should report tracking errors, multiple trials with statistics, and ideally compare against an orientation-agnostic or fixed-radius baseline to support the claimed benefit of geometry-aware reorientation.","section":"Section V-B, Figs. 5 and 7"},{"comment":"Equation (23) is written as an elliptic Dirichlet problem with the Laplacian taken only in the spatial variables; time enters solely as a parameter that moves the boundary. The paper then cites [30, 31] for regularity of 'the general case for parabolic equations.' Those references concern parabolic evolution problems in non-cylindrical domains, and it is not immediate that they apply to a family of elliptic problems parameterized by time. The smoothness of h_p with respect to time, and with respect to the heading variable after the Minkowski construction in (26), is precisely what is needed for the time-derivative term in the DCBF condition (10). The authors should clarify the mathematical setting—either prove the needed regularity directly for the parameterized elliptic problem, or state it as an assumption and explain why the cited parabolic results cover this case.","section":"Section III-B, Eq. (23)"},{"comment":"The Minkowski-difference construction is central to the claimed geometry-awareness, but the implementation is underspecified. The text says the Minkowski difference is performed 'for a discrete sampling of robot geometries along θ,' yet the sampling density, the representation of the robot occupancy set, the discretization of the PDE domain, and the interpolation of h_p between sampled orientations are not reported. These details are necessary both for reproducibility and for assessing whether the discretization errors could compromise the safety argument. Please add these numerical details and, if possible, an analysis of how the sampling resolution affects the conservativeness of the resulting safe set.","section":"Section III-C and Section V-A, Eq. (24)"}],"minor_comments":[{"comment":"Please specify the grid resolution for the SOR solve, the number of sampled orientations for the Minkowski difference, and the interpolation scheme used to evaluate h_p at arbitrary orientations and times; these are needed for reproducibility.","section":"Section V-A"},{"comment":"The caption contains the typo 'non-cyclindrical' (should be 'non-cylindrical'); also, the notation v_s is used in the figure without an explicit definition at first use.","section":"Figure 2 caption"},{"comment":"The statement 'we used (21) to derive the linear boundary prediction model' is confusing because (31) appears to be a restatement of (21) with a sampled index k; please clarify the relationship and define Δt_k explicitly.","section":"Section V-A, Eq. (31)"},{"comment":"The text says h_p remained positive 'throughout the preponderance of the 100-second experiment,' while the figure caption says 'throughout the experiment'; please reconcile these statements and report any time instants where h_p approached zero.","section":"Section V-C, Fig. 7"},{"comment":"No code or dataset is released; given that the contribution is substantially a numerical/experimental pipeline, a reproducibility statement with software and data availability would substantially strengthen the paper.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The manuscript relies heavily on the authors' own prior work [18] for the CBF property of Poisson safety functions. Since the current paper does not reprove that property for the parameterized, time-varying case, the editor may want to confirm that [18] indeed covers only the static problem, as the text implies. The experimental claims are also weaker than the abstract suggests; the paper would be more appropriate for a venue that values empirical robotics demonstrations with heuristic safety validation, but as submitted the formal framing needs to be reconciled with the acknowledged lack of guarantees."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nThis paper by Bena et al. is best read as an engineering integration, not a theory contribution. What is actually new: they take the static Poisson safety function from their RSS paper [18], re-solve it on a domain parameterized by time and by robot orientation via Minkowski operations, and feed the result as a DCBF constraint into an MPC safety filter. That is a genuine extension, and the hardware results—Go2 and G1 avoiding a thrown dodgeball, and the humanoid reorienting to pass through a narrow corridor—are convincing demonstrations that the pipeline works end-to-end. The corridor demo in particular shows the value of orientation-aware buffering: with circular inflation the robot would deadlock; here it rotates its long axis to fit. Credit where due: the experiments are reproducible in principle, the citation pattern is fine, and the authors are honest about what is proven.\n\nThe main soft spot is the one the authors themselves flag in Section IV: safety cannot be formally guaranteed in general time-varying environments. The DCBF constraint at each step is built from h computed on a predicted safe set, and the prediction model (31) assumes constant obstacle velocity over the MPC horizon. If an obstacle accelerates, stops, or changes direction inside the horizon, the predicted boundary can be wrong and the MPC can issue a command that keeps the modeled h positive while the true h goes negative. The reported positive h plots therefore validate safety for roughly constant-velocity obstacles, not for genuinely dynamic ones. This is not a minor implementation gap, but the paper does not hide it. What is missing is the formal step: they do not prove that this parameterized, time-varying h is a valid discrete-time CBF for the single-integrator-plus-orientation model, and they do not offer a reachability or uncertainty margin to absorb prediction error. A serious referee should ask for either a proof under the stated assumptions or an explicit scoping of the safety claim, plus baseline comparisons and error bars over repeated trials.\n\nThe writing is rough in places—several equations are garbled, and the free parameters (forcing function, gamma, MPC weights, SOR grid) are not fully specified. But the core idea is clear and the demos are useful. I would send this to peer review; it deserves a serious referee, and with revisions on the theoretical claims and experimental rigor it could be a solid systems paper.\n\nFor a reader working on legged safety or perception-based CBFs, this is worth a look. I would likely cite it as an example of Poisson safety functions used in a real-time MPC filter, though not as a source for formal guarantees.","headline":"A real engineering integration of Poisson safety functions into MPC+CBF with convincing hardware demos; the formal gap in time-varying safety is real but explicitly admitted, so the paper deserves review with revisions.","tokens_in":11760,"tokens_out":3113,"would_cite":true,"duration_ms":36592,"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":"Poisson-equation safety limits let legged robots turn to dodge obstacles and slip through narrow gaps by rotating their bodies.","keywords":["Poisson safety function","control barrier functions","model predictive control","legged robots","collision avoidance","Minkowski difference","safe set buffering","predictive safety filter"],"falsifier":"Run the predictive safety filter against an obstacle that is programmed to move at constant velocity for the first half of the MPC horizon and then stop or reverse; record the measured Poisson safety function $h(x,q,t)$ and the robot body's true occupancy. If $h$ goes negative or any part of the robot enters the obstacle's occupied region, the constant-velocity boundary prediction is falsified; the paper itself notes that safety cannot be formally guaranteed in general time-varying environments.","tokens_in":10778,"feed_emoji":"🤖","tokens_out":4031,"duration_ms":41105,"temperature":0.7,"pith_summary":"This paper claims that a safety function obtained by numerically solving Poisson's equation can be extended along two new axes, time and robot orientation, and then used as a control barrier function inside a model predictive controller, yielding a real-time safety filter that accounts for the robot's full body geometry. The authors argue this matters because legged robots have asymmetric footprints and operate near people and moving objects, so a point-mass safety margin is either unsafe or overly conservative. They demonstrate on a quadruped and a humanoid carrying an oblong payload that the resulting controller keeps the robot's entire footprint inside the safe region by simultaneously translating and rotating, and that reorientation lets the humanoid pass through a gap that would otherwise produce a navigational deadlock. A sympathetic reader would take the paper's claim to be that Poisson safety functions, once parameterized by time and orientation, are a practical and versatile numerical source of CBF constraints for predictive safety filtering.","feed_headline":"Poisson safety functions let legged robots dodge by turning","feed_subtitle":"Time- and orientation-aware safe sets fit whole bodies into narrow corridors and avoid moving obstacles.","key_machinery":"The named central object is the Poisson safety function, defined as the solution of a Dirichlet problem $\\Delta h = -f$ on the free-space domain with zero boundary values. The paper's extension solves the same equation on a non-cylindrical space-time domain whose boundary moves according to the predicted obstacle velocity field, and on a configuration-space domain lifted by the Minkowski difference $\\mathcal{S} \\ominus \\mathcal{B}(q)$, so the solution depends on time and on the robot orientation $q$. This function does the load-bearing work: it converts raw perception data, namely an occupancy map plus optical-flow velocities, into a smooth numerically available safety value $h(x,q,t)$, and its positivity, enforced as a discrete-time CBF constraint inside the MPC, is what keeps the planned trajectory inside the safe set for the whole body geometry.","core_discovery":"The central claim is that the static Dirichlet problem for Poisson's equation can be reformulated as a parameterized moving boundary value problem, so the safe set depends on both time and robot orientation, and the resulting Poisson safety function $h(x,q,t)$ is a valid discrete-time CBF for a single-integrator reduced-order model. The temporal extension predicts the boundary via a transport equation under an assumed obstacle velocity field, and the geometric extension uses a Minkowski difference with the robot occupancy set to lift the domain into configuration space. Inserted as a DCBF constraint along the MPC horizon, this function produces velocity commands for a three-DOF single-integrator model (two translations plus heading), which are tracked by a learned locomotion controller. In the experiments, the evaluated value of $h$ remains positive during dynamic collision avoidance and environmental navigation, which the authors read as evidence that the full footprint of the robot stayed inside the original safe set; heading measurements show the robots rotating to align their major axis with the predicted obstacle motion or with the corridor, which is the mechanism that avoids deadlock.","pith_inferences":["A testable extension is replacing the constant-velocity boundary predictor with an acceleration-aware or learned obstacle motion model; the same non-cylindrical domain construction would apply but with a nonlinear characteristic flow.","The regularity results for evolutionary problems on non-cylindrical domains cited in the paper suggest the approach could be carried to higher configuration dimensions, for example adding body-shape DOFs such as sway or arm configuration, at the cost of larger PDE solves.","If the Minkowski difference is computed as a convolution of the occupancy map with the robot kernel, the same pipeline could transfer from 2D floor-plan safety to 3D collision avoidance for drones or manipulators by changing the kernel and the Laplacian dimension.","The orientation-aware turning behavior indicates that a purely translational safety filter can misrepresent feasible motion in narrow environments; the paper's heading data could be used as a benchmark to quantify how much additional clearance a point-mass filter would require to complete the same corridor task."],"forward_implications":["Safe sets synthesized directly from perception can replace hand-specified analytical obstacle shapes, because the Poisson solve turns occupancy data into a smooth CBF.","Robot geometry enters the safety constraint through orientation, so the controller can rotate the body to fit through gaps that a fixed-radius margin would declare impassable.","Time enters the safety constraint through a predicted boundary, so the MPC can plan avoidance of moving obstacles over its horizon instead of reacting only to the current occupancy map.","The method inherits the guarantees of discrete-time CBFs: planned states keep $h$ positive and the safety value is lower-bounded along the trajectory, provided the boundary prediction model is accurate.","Because the safety filter operates on a reduced-order single-integrator model, it can run at 100 Hz and be paired with any low-level locomotion controller that tracks velocity commands."],"supporting_citations":[{"why":"Introduces Poisson safety functions and proves that the static Poisson safety function is a CBF for single-integrator dynamics, which is the foundation being extended.","marker":"[18]"},{"why":"Provides the control barrier function definition and forward-invariance guarantee that the DCBF constraint operationalizes.","marker":"[19]"},{"why":"Supplies the discrete-time CBF formulation and its geometric safety guarantee used inside the MPC.","marker":"[21]"},{"why":"Establishes the safety-critical MPC framework with discrete-time CBF constraints that the predictive safety filter builds on.","marker":"[10]"},{"why":"Gives regularity results for evolutionary problems on non-cylindrical domains, justifying the well-posedness of the temporally parameterized PDE.","marker":"[31]"},{"why":"The efficient track-anything segmentation model that produces the occupancy map from the overhead camera stream.","marker":"[36]"},{"why":"The optical flow method used to estimate the pixel velocity field, which is the basis of the linear boundary prediction model in (31).","marker":"[37]"},{"why":"Supports the reduced-order-model hierarchical framework through which safety enforced on the single-integrator model is transferred to the full legged robot.","marker":"[33]"}],"fun_headline_variants":["Time-dependent Poisson safety filters let humanoids turn to dodge","Geometry-aware CBF-MPC rotates humanoids through tight spaces","Moving-boundary Poisson safety filters steer quadrupeds and humanoids","Turning to dodge: Poisson safety functions for legged robots"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The boundary prediction assumes obstacle velocity $\\mathbf{v}$ is constant over the MPC prediction horizon, so an obstacle that accelerates, brakes, or reverses inside that window makes the predicted safe boundary wrong and the safety constraint can admit a collision.","fun_headline_variants_meta":{"raw":{"variants":["Time-dependent Poisson safety filters let humanoids turn to dodge","Geometry-aware CBF-MPC rotates humanoids through tight spaces","Moving-boundary Poisson safety filters steer quadrupeds and humanoids","Turning to dodge: Poisson safety functions for legged robots"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001345,"raw_usage":{"total_tokens":5465,"prompt_tokens":948,"completion_tokens":4517,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":564,"completion_tokens_details":{"reasoning_tokens":4445}},"tokens_in":564,"tokens_out":4517,"duration_ms":35104,"temperature":1.0,"reasoning_tokens":4445,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T17:28:43.358626+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the predictive safety filter against an obstacle that is programmed to move at constant velocity for the first half of the MPC horizon and then stop or reverse; record the measured Poisson safety function $h(x,q,t)$ and the robot body's true occupancy. If $h$ goes negative or any part of the robot enters the obstacle's occupied region, the constant-velocity boundary prediction is falsified; the paper itself notes that safety cannot be formally guaranteed in general time-varying environments.","supporting_citations":[{"cited_title":"Dynamic safety in complex environments: Synthesizing safety filters with poisson’s equation,","cited_arxiv_id":null,"evidence_quote":"Introduces Poisson safety functions and proves that the static Poisson safety function is a CBF for single-integrator dynamics, which is the foundation being extended."},{"cited_title":"Discrete control barrier functions for safety-critical control of discrete systems with application to bipedal robot navigation,","cited_arxiv_id":null,"evidence_quote":"Supplies the discrete-time CBF formulation and its geometric safety guarantee used inside the MPC."},{"cited_title":"Safety-critical model predictive control with discrete-time control barrier function,","cited_arxiv_id":null,"evidence_quote":"Establishes the safety-critical MPC framework with discrete-time CBF constraints that the predictive safety filter builds on."},{"cited_title":"Evolutionary problems in non-cylindrical domains,","cited_arxiv_id":null,"evidence_quote":"Gives regularity results for evolutionary problems on non-cylindrical domains, justifying the well-posedness of the temporally parameterized PDE."},{"cited_title":"Two-frame motion estimation based on polynomial ex- pansion,","cited_arxiv_id":null,"evidence_quote":"The optical flow method used to estimate the pixel velocity field, which is the basis of the linear boundary prediction model in (31)."},{"cited_title":"Safety-critical control for autonomous systems: Control barrier functions via reduced order models,","cited_arxiv_id":null,"evidence_quote":"Supports the reduced-order-model hierarchical framework through which safety enforced on the single-integrator model is transferred to the full legged robot."}],"review_version":2}