PhysibleMDM: Enforcing Inference-Time Physical Feasibility Constraints on Motion Diffusion Models via Noise Optimization

1 INTRODUCTION

• Infeasible generated motions can contain kinematic or dynamic artifacts that make them unreliable references for downstream tracking.Examples include joint-limit violations, self-collisions, ground penetration and slip, excessive torques, and inconsistent floating-base dynamics.Source: §1 · p. 1
• Training-time and pre-training physics approaches can be resource-intensive and do not guarantee that every final-model sample satisfies physical constraints.Retraining generators and tracking policies across refinement iterations can add cost.Source: §1 · p. 2
• Inference-time noise optimization evaluates objectives on fully denoised motions, but gradient-based methods require costly backpropagation and may be poorly informed by non-smooth contact-rich constraints.Efficient gradient-free optimization for robotic-motion feasibility while retaining characteristic style remains unexplored in the cited discussion.Source: §1 · p. 2
• PhysibleMDM jointly optimizes initial and posterior noise in a frozen MDM using Sep-CMA-ES, with noise standardization and autoregressive chaining.Its feasibility formulation combines kinematic constraints with constrained inverse-dynamics residuals, without a tracking controller or full trajectory simulation.Source: §1 · p. 2

2 RELATED WORK

• Sampling-based diffusion-noise updates support non-differentiable objectives without backpropagating through denoising, while PhysibleMDM jointly optimizes initial latent and posterior noise with Sep-CMA-ES.Sep-CMA-ES's diagonal covariance gives linear internal time and memory scaling with noise-variable count.Source: §2 · p. 3
• Physics has been incorporated through data refinement, simulated tracking, physics-informed learning, preference-based post-training, and generation-time planning or projection.Examples include PhysDiff's simulated-execution projection [Yuan et al., 2023] and CLoSD's alternating planning and execution [Tevet et al., 2025].Source: §2 · p. 3
• Training-time physics does not ensure feasibility of individual samples, especially under distribution shift, and predicted controls do not ensure executed states match predicted states.Simulated-tracking approaches also depend on a trained tracking controller.Source: §2 · p. 3

3 PRELIMINARIES: MOTION DIFFUSION MODELS

• The setup uses t for the current physical frame, F for predicted future frames, and P for past frames supplied as conditioning.Local future index f runs from 1 to F, with absolute frame ℓ = t + f.Source: §3 · p. 3
• A floating-base motion feature tuple contains root position and orientation, joint orientations, and link position and orientation features.The encoded vector uses six entries per position or orientation block.Source: §3 · p. 3
• A frozen DDPM iteratively denoises a noisy sequence conditioned on past motion, task commands, and style, with its denoiser predicting clean motion.Diffusion levels range from clean motion at d = 0 to initial noise at d = D.Source: §3 · p. 3
• Sampling starts from initial noise and injects posterior noise at reverse steps, with the noise tensors matching the future-motion shape and having independent standard-normal entries.Fixing all injected noise and conditioning makes the complete sampler deterministic.Source: §3 · pp. 3–4

4 OVERVIEW

• PhysibleMDM standardizes candidate noise, generates complete motions with a frozen MDM, and scores task objectives, kinematic violations, and constrained inverse-dynamics residuals.Batched Sep-CMA-ES updates the candidate population over a fixed iteration budget.Source: §4 · p. 4
• For long sequences, the method commits the first part of the best motion and repeats optimization conditioned on the preceding optimized motion.It encourages consistency between overlapping predictions.Source: §4 · p. 4
• The full-motion evaluations guide noise refinement without retraining the generator or relying on simulator rollouts.The passage specifies a frozen MDM and feasibility penalties evaluated on generated motions.Source: §4 · p. 4

5 DIFFUSION NOISE OPTIMIZATION

• For a frozen model and fixed conditioning, the generated sample is determined by the injected noise sequence, including initial and posterior-noise tensors.Posterior noise is injected only at steps with σdiff > 0; the complete reverse sampler maps noise and conditioning deterministically to a motion.Source: §5 · p. 4
• Each candidate’s initial and combined posterior noise is standardized separately using its empirical mean and sample standard deviation.Posterior statistics are shared across diffusion steps, while correlations and higher-order statistics may still change during optimization.Source: §5 · p. 4
• The optimization evaluates objectives and constraints on each fully denoised sample, requiring task objectives to be minimized subject to equality and inequality constraints.Positive inequality values indicate violations.Source: §5 · p. 4

6 PHYSICAL FEASIBILITY REFINEMENT OF GENERATED MOTION

• Under the assumed robot and contact model, feasibility requires valid geometry, consistent contact kinematics, and accelerations realizable by admissible torques and contact forces.An inverse-dynamics solve measures remaining mismatch without simulator rollouts or a tracking controller.Source: §6 · pp. 4–5

6.1 ROBOT CONFIGURATION AND DYNAMICS

• Each frame is represented by root position, root orientation, joint coordinates, generalized velocities, and generalized accelerations, with contact geometry obtained by forward kinematics.The configuration includes world-coordinate root positions and velocities; the floating base and joints make up the generalized velocity degrees of freedom.Source: §6.1 · p. 5

6.2 KINEMATIC FEASIBILITY CONSTRAINTS

• Kinematic constraints penalize ground penetration and active-contact slip, while contact-pattern preservation checks expected contact and flight behavior.The reference contact pattern is inferred from unoptimized samples and held fixed during each horizon; expected contact penalizes separation or excessive penetration, and expected flight penalizes insufficient clearance.Source: §6.2 · p. 5

6.3 DYNAMIC FEASIBILITY CONSTRAINTS

• Dynamic feasibility requires prescribed motion to satisfy floating-base equations of motion, actuator bounds, and contact-force limits, with no-slip kinematics checked separately.Admissible torques and contact forces alone do not establish that the prescribed acceleration is realizable.Source: §6.3 · p. 5

6.3.1 CONSTRAINED INVERSE DYNAMICS

• The constrained inverse-dynamics solve minimizes acceleration mismatch subject to actuator bounds and unilateral friction cones at active contacts.It accounts for the unactuated floating base through the equations of motion and returns torques and contact forces satisfying its constraints.Source: §6.3.1 · pp. 5–6, §6.3.2 · p. 6

6.3.2 ACCELERATION FEASIBILITY CONSTRAINT

• Acceleration feasibility sets the residual to zero and measures mismatch as the mean absolute residual across generalized acceleration components.The inner solve returns torques and contact forces satisfying its actuator and force-cone constraints.Source: §6.3.2 · p. 6

6.4 AUTO-REGRESSIVE REFINEMENT AND MOTION CONSISTENCY

• For long sequences, the method repeatedly optimizes future frames conditioned on past motion, commits the first Fkeep frames, and encourages consistency over the overlapping future frames.The best evaluated candidate is selected in each window before replanning.Source: §6.4 · p. 6

7 DIFFUSION NOISE OPTIMIZATION WITH SEP-CMA-ES

• Each raw noise candidate is standardized and evaluated through the complete sampler to produce a motion.The sampler uses the model and conditioning associated with that candidate.Source: §7 · p. 6
• The soft-constraint objective aggregates per-frame violations and applies weighted squared penalties to equality and inequality terms.The aggregation and fitness details are specified in Appendix E.1.Source: §7 · p. 6
• Sep-CMA-ES samples and ranks candidates, then adapts the search mean, step size, and diagonal covariance over a fixed iteration budget.Its diagonal covariance keeps complexity and memory linear in the number of optimization variables for a fixed population size, avoiding expensive gradients for non-smooth objectives.Source: §7 · p. 6

8.1 EXPERIMENTAL DETAILS

• The SMPL humanoid and custom biped use CAMDM-based generators trained on retargeted 100STYLE motions, conditioned on past motion, commanded root trajectory, and style.Experiments use eight denoising steps and autoregressive execution of optimized segments.Source: §8.1 · pp. 6–7

8.2 BASELINES

• The evaluation uses a frozen generator and does not require a pretrained tracking controller, comparing sampling, gradient, trajectory-refinement, and guided-diffusion baselines.Component ablations test posterior-noise optimization and noise standardization.Source: §8.2 · p. 7

8.3 PHYSICAL FEASIBILITY OPTIMIZATION SETTINGS

• Kinematic optimization penalizes contact-pattern deviations, slip, and self-collision; dynamic optimization instead penalizes acceleration-reconstruction error alongside slip and self-collision.Both settings include velocity tracking and autoregressive consistency; dynamic evaluation adds dynamics evaluation and inner inverse-dynamics solves.Source: §8.3 · p. 7

8.4 EVALUATION METRICS

• Evaluation measures feasibility, tracking, consistency, runtime, and smoothness; SMPL style accuracy and FID are also measured with a pretrained classifier.Higher Log Dimensionless Jerk indicates smoother motion.Source: §8.4 · p. 7

8.5 KINEMATIC FEASIBILITY OPTIMIZATION RESULTS

• 75%, 92%, and 74%: on SMPL, mean ground penetration, self-collision penetration, and slip decrease relative to unoptimized generation.Comparisons span 100STYLE and both embodiments; Table 1 covers SMPL kinematic feasibility and Table 2 covers the biped.Source: §8.5 · p. 7, Table 1 · p. 9, Table 2 · p. 9

8.6 DYNAMIC FEASIBILITY OPTIMIZATION RESULTS

• 89.5% and 79.5%: across 11 bipedal robot sequences, median reductions in mean acceleration-reconstruction error and mean ID residual relative to unoptimized motion.The sequences span motion styles, slopes, and friction settings; Table 3 covers dynamic feasibility during a two-foot jump.Source: §8.6 · p. 8, Table 3 · p. 9

8.7 TRAINING RL CONTROLLER ON OPTIMIZED VS. UNOPTIMIZED MOTION

• Training DeepMimic-style policies on the optimized two-foot jump reduces body-position, body-orientation, and joint-position tracking errors, but increases root-position and root-orientation errors.The reported reductions are 16.4%, 11.9%, and 7.4%; root errors increase by 39.6% and 41.3%, respectively.Source: §8.7 · p. 9

9 CONCLUSION

• On 11 bipedal robot sequences, median reductions were 89.5% in per-sequence mean acceleration-reconstruction error and 79.5% in mean inverse-dynamics residual.Across 100 SMPL motion styles, mean ground penetration, self-collision penetration, and slip fell by 75%, 92%, and 74%, respectively, relative to unoptimized generation.Source: §9 · p. 9
• Soft penalties do not guarantee strict physical feasibility, and optimized noise can drift from the learned distribution and degrade style.Evaluation is limited to bipedal embodiments with assumed dynamics and sticking contacts; repeated feasibility evaluations and inverse-dynamics solves can be costly for real-time use.Source: §9 · p. 10

A.1 MOTION DIFFUSION MODELS

• Motion diffusion models generate human-like motion under rich conditioning, but strict kinematic validity and contact-rich dynamical feasibility remain challenging.The generated motion commonly serves as a reference for downstream tracking controllers.Source: §A.1 · p. 18

A.2 INFERENCE-TIME OPTIMIZATION OF DIFFUSION MODELS

• Noise optimization evaluates objectives on generated outputs, while efficient gradient-free optimization of high-dimensional motion noise under non-smooth constraints remains underexplored.PhysibleMDM addresses this gap by jointly optimizing initial and posterior noise with Sep-CMA-ES.Source: §A.2 · p. 18

A.3 PHYSICAL FEASIBILITY OF GENERATED ROBOTIC CHARACTER MOTION

• Feasibility approaches based on physics-informed training or tracking-controller projection leave challenges in inference-time feasibility without retraining or a physics-based tracking controller.PhysibleMDM combines gradient-free noise optimization with kinematic penalties and constrained inverse-dynamics evaluations on fully generated motions.Source: §A.3 · p. 19

B.1 MOTION DATA AND PREPROCESSING

• 100STYLE locomotion clips spanning 100 styles are retargeted to SMPL and then to the bipedal robot; clips are resampled to 30 Hz and 50 Hz, respectively.Every contiguous P + F-frame window is used with stride one, omitting clips shorter than the window.Source: §B.1 · p. 20

B.2 MOTION REPRESENTATION, ARCHITECTURE, AND CONDITIONING

• The CAMDM-based transformer predicts 20 frames at 30 Hz for SMPL and 30 frames at 50 Hz for the bipedal robot, conditioned on past motion, commanded root trajectory, style, and attention biases.The architecture uses latent width 256, four layers, four heads, feed-forward size 1024, and dropout 0.2.Source: §B.2 · p. 20

B.3 TRAINING PROCEDURE AND HYPERPARAMETERS

• Training combines clean-motion reconstruction with losses on consecutive-frame feature differences, forward-kinematics body positions, and their consecutive-frame differences.A foot-contact term penalizes predicted foot displacement when reference displacement falls below a threshold.Source: §B.3 · p. 21

B.4 DIFFUSION SAMPLING AND AUTOREGRESSIVE MOTION GENERATION

• Autoregressive sampling commits 15 frames per SMPL call and 25 per robot call, retaining overlap for replanning; optimization methods select motions according to their specified procedures.The DDPM uses eight diffusion steps, and optimization methods use an inference batch size of 64.Source: §B.4 · p. 21

C.1 ROBOT CONFIGURATION AND DYNAMICS

• The decoder maps generated features to world-frame root and joint configurations, from which forward kinematics and floating-base dynamics define sticking contacts under friction cones.The configuration includes 23 spherical-joint rotations for SMPL or 20 revolute-joint angles for the bipedal robot.Source: §C.1 · pp. 22–23

C.2 MOTION SMOOTHING AND DERIVATIVE ESTIMATION

• Whittaker–Eilers filtering and finite differences estimate derivatives, but geometric constraints use decoded configurations and jerk diagnostics use unsmoothed finite differences.The differentiation stencils use Δt = 1/fs, with fs = 30 Hz for SMPL and 50 Hz for the bipedal robot.Source: §C.2 · p. 23

C.3 TASK OBJECTIVES

• The task objective combines command-derived planar root position and orientation tracking with overlap-feature consistency; consistency is zero on the first horizon or without overlap.Reference root trajectories integrate commanded planar velocity and yaw rate.Source: §C.3 · p. 24

C.4 KINEMATIC FEASIBILITY CONSTRAINTS

• Ground penetration and slip are averaged over contact points; majority-vote contact patterns are fixed per horizon and re-inferred autoregressively, while box-SAT overlap measures self-collision.The collision measure uses oriented-box approximations and is not mesh-intersection volume.Source: §C.4 · pp. 24–25

C.5.1 NOMINAL CONSTRAINED INVERSE DYNAMICS

• Nominal inverse dynamics fits torques and contact forces to generated motion via constrained least squares, respecting torque bounds and friction cones; its force residual is expressed as acceleration mismatch.The acceleration induced by the returned inverse-dynamics solution is compared with the generated motion's acceleration at the same state.Source: §C.5.1 · p. 25

C.5.2 RELAXED INVERSE DYNAMICS WITH A VIRTUAL WRENCH

• A relaxed inverse-dynamics formulation measures the smallest virtual root wrench compatible with the motion and constraints, but is used only for analysis, not physical-feasibility optimization.Source: §C.5.2 · p. 26

D.1 SOLVER FORMULATION AND NUMERICAL CONDITIONING; D.2 BATCHING AND COMPUTATIONAL EFFICIENCY; D.3 DIFFERENTIATION FOR GRADIENT-BASED BASELINES

• The inverse-dynamics inner loop is an SOCP solved with Clarabel; contact-tuple batching avoids recompilation, while gradient baselines differentiate selected contact problems without differentiating contact-set changes.Source: §D.1 · p. 27, §D.2 · p. 27, §D.3 · p. 27

E.1 CONSTRAINT SOFTENING AND CANDIDATE EVALUATION; E.2 NOISE PARAMETERIZATION AND OPTIMIZER INITIALIZATION; E.3 SEP-CMA-ES UPDATES

• Candidate fitness adds weighted squared feasibility violations to task loss; initial and posterior noise are separately standardized, and Sep-CMA-ES searches with diagonal covariance.Source: §E.1 · p. 27, §E.2 · p. 28, §E.3 · p. 28

E.3 SEP-CMA-ES UPDATES

• Sep-CMA-ES evaluates each population in parallel over K iterations, retains the best candidate’s noise and motion, commits Fkeep frames, and uses Fov frames as the next overlap target.The history is updated from the last P committed frames as the procedure advances.Source: §E.3 · p. 29

E.4 OPTIMIZATION PROCEDURE AND AUTOREGRESSIVE INTEGRATION

• Each candidate is scored by applying the generator to its noise and conditioning under the objectives and constraints; deterministic sampling lets the retained winning motion and noise avoid another sampling pass.The fixed K-iteration budget is repeated, and the best evaluated candidate supplies the committed frames and next history window.Source: §E.4 · p. 29

E.5 INFERENCE-TIME OPTIMIZATION BASELINES AND COMPARISON BUDGETS

• The comparison holds the MDM fixed and excludes retraining or finetuning methods (Xu et al., 2025; Watanabe et al., 2025) and pretrained tracking-controller rollouts (Yuan et al., 2023; Watanabe et al., 2025; Tevet et al., 2025).Within each experimental setting, optimization methods use the same task objectives and feasibility constraints.Source: §E.5 · p. 30

E.5.1 BASELINE METHODS

• Baselines span median-loss unoptimized samples, direct motion-space refinement, denoising guidance, independent noise sampling, and gradient-based noise optimization.Guided Diffusion applies guidance during denoising; Noise Best-of-N retains the best sampled noise.Source: §E.5.1 · p. 30
• Noise ZeNO optimizes initial noise, while its Physible adaptation extends optimization to initial and posterior noise under the shared loss.ZeNO uses reward evaluations on generated outputs and projects updates onto a sphere.Source: §E.5.1 · p. 30

E.5.2 ABLATIONS OF THE PROPOSED METHOD

• The ablations compare latent-noise-only search, which holds posterior noise fixed, with unstandardized search that passes raw noise directly to the sampler.The latent-noise-only variant optimizes latent noise; NNS omits centering and rescaling of both initial and posterior noise.Source: §E.5.2 · p. 31

E.5.3 COMPUTATIONAL BUDGETS

• Sampling-based noise methods and their ablations use K + 1 full reverse-sampling passes per horizon, whereas motion-space updates require no further denoiser calls.Each pass contains D batched denoiser evaluations.Source: §E.5.3 · p. 31, Table 5 · p. 32

F.1 ROBOT MODELS AND CONTACT SPECIFICATIONS

• The study specifies SMPL and bipedal robot embodiments, their actuator and ground assumptions, and embodiment-specific contact points and self-collision pairs.The collision checks include selected left–right body pairs, without within-limb or limb–torso pairs.Source: §F.1 · pp. 31–32

F.2 EXPERIMENTAL DETAILS FOR THE OPTIMIZATION SETTINGS; F.3 OPTIMIZATION HYPERPARAMETERS AND OBJECTIVE WEIGHTS

• Kinematic optimization uses 100 iterations per horizon, whereas dynamic optimization uses 2500; all methods use 64 samples and eight denoising steps.The experiments span 100STYLE motions and 11 bipedal sequences; the two-foot-jump sequence is used to compare optimization methods, with a fixed root-reference velocity command in dynamic experiments. Default settings use weighted-sum losses without a loss-weight curriculum.Source: §F.2 · p. 32, §F.3 · p. 33

F.4 EVALUATION METRICS

• Feasibility metrics average ground penetration and slip over candidate contact points and self-collision penetration over checked body pairs; acceleration reconstruction is mean absolute acceleration residual.The section also defines seed variability and motion repeatability measures, and reports LDJ as a smoothness diagnostic; higher LDJ indicates smoother motion.Source: §F.4 · pp. 34–35

F.5 STYLE CLASSIFIER AND FID EVALUATION

• Style accuracy and FID are evaluated using a classifier on overlapping 15-frame windows, with final scores computed over the full committed sequence.The classifier predicts 100 styles; its pre-head features provide FID embeddings compared with per-style reference statistics.Source: §F.5 · pp. 35–36

F.6 COMPUTATIONAL ENVIRONMENT AND RUNTIME MEASUREMENT; F.7 DOWNSTREAM RL TRACKING PROTOCOL

• Runtime measurements compare per-iteration and per-horizon optimization time, while downstream tracking trains separate policies for unoptimized and optimized two-foot-jump motions.Runtime tests use an i9-12900K CPU and RTX 4090 GPU; tracking uses Isaac Sim and evaluates both policies under identical flat-terrain conditions over the complete reference trajectory.Source: §F.6 · p. 36, §F.7 · pp. 36–37

G.1 QUANTITATIVE COMPARISON WITH BASELINES

• 0.676 vs. 0.939 mm ground penetration, 0.0152 vs. 0.0455 mm self-collision penetration, and 0.0484 vs. 0.0792 m s−1 ground slip: Physible outperforms Noise ZeNO on SMPL, while ZeNO has higher style accuracy (91.71% vs. 87.73%).On the bipedal robot, Physible reduces mean ground penetration, self-collision penetration, and slip by 40%, 85%, and 61% relative to Noise ZeNO. The broader comparison reports that Physible best addresses the multi-objective problem, while Sep-CMA-ES handles contact-pattern preservation more robustly than gradient-based baselines or Best-of-N sampling.Source: §G.1 · p. 38

G.2.1 OPTIMIZATION CONVERGENCE

• Noise GD reduces fitness fastest but does not reduce ground penetration as effectively, degrades style with more iterations, and direct motion optimization quickly increases jerk.Diag-CEM performs comparably to Sep-CMA-ES with slightly slower convergence; Physible shows strong convergence across objectives and constraints with only modestly increased jerk versus unoptimized motion.Source: §G.2.1 · p. 38, Figure 3 · p. 39

G.2.2 VARIABILITY ACROSS OPTIMIZATION SEEDS

• 0.85% of mean: the best weighted loss has this relative standard deviation across ten seeds, while generated motions have 0.0351 m root-trajectory RMSE and 0.0246 m body-position distance on average.The tested setting covers ten seeds, all 100 styles, and 150 frames per style; velocity tracking and motion consistency relative standard deviations are 0.90% and 1.84%, respectively.Source: §G.2.2 · p. 40, Table 8 · p. 40

G.3.1 INITIAL AND POSTERIOR NOISE OPTIMIZATION

• Latent-noise-only optimization yields modest gains over unoptimized motion and falls short of jointly optimizing initial and posterior noise.Noise standardization improves style accuracy from 76.01% to 87.73%, overall FID from 22.00 to 14.41, and mean per-style FID from 74.21 to 43.95; it also improves LDJ and motion-consistency error.Source: §G.3.1 · p. 41, §G.3.2 · p. 41

G.3.2 NOISE STANDARDIZATION

• Corrected NNS results show a feasibility–style tradeoff, with ground penetration 0.676 vs. 0.541 mm, self-collision penetration 0.0152 vs. 0.009 22 mm, slip 0.0484 vs. 0.0401 m s−1, and velocity-tracking error 0.0392 vs. 0.0334.Both methods retain residual violations.Source: §G.3.2 · p. 41

G.3.3 CONTACT PATTERN PRESERVATION

• Contact pattern preservation requires contact with limited penetration when contact is expected and separation otherwise, preventing the prolonged floating that ground-distance-only optimization can induce.With dynamic feasibility co-optimized, absent contacts implicitly discourage floating because contact forces cannot support the motion’s encoded accelerations; that safeguard is absent under purely kinematic optimization.Source: §G.3.3 · p. 41
• Over 240-frame neutral-walking clips, contact pattern preservation yields 1.166 mm mean penetration, 4.2% no-contact frames, and a 0.13 s longest contact loss, versus ground-distance loss at 0.002 mm, 91.7%, and 2.20 s.Some penetrations still exceed the 5 mm allowance; unoptimized motion has 3.3% no-contact frames and a 0.07 s longest contact loss.Source: §G.3.3 · p. 41, Figure 4 · p. 42

G.4.1 NEUTRAL WALKING: COMPARISON OF OPTIMIZATION METHODS

• Physible follows the integrated velocity command with negligible drift, largely preserves contact, and reduces deep ground penetration and slip in neutral walking.Other methods begin drifting around 1.5 s; slip remains slightly higher at some times for one contact point.Source: §G.4.1 · p. 45, Figure 7 · p. 46

G.4.2 CAT MOTION: GROUND-PENETRATION REDUCTION

• Physible reduces foot-ground penetration in the cat motion while retaining its characteristic poses.Matched snapshots and foot trajectories connect visible interactions with remaining penetration events.Source: §G.4.2 · p. 58, Figure 20 · p. 58

G.4.3 LAWN-MOWER MOTION: SELF-COLLISION AVOIDANCE

• In the evaluated lawn-mower sequence, Physible reduces arm interpenetration to negligible levels while retaining the characteristic arm-motion style.The targeted 150-frame experiment optimizes only the self-collision constraint for 100 iterations.Source: §G.4.3 · p. 59, Figure 21 · p. 59

G.4.4 STYLE PRESERVATION ACROSS SELECTED MOTIONS

• Cat and duck-foot retain 100% style-classification accuracy, while cross-over shows a temporary decrease after optimization.Classification uses 15-frame sliding windows averaged in 1 s bins over each 8 s clip.Source: Figure 24 · p. 62, §G.4.4 · p. 62

H.1.1 COMPARISON WITH BASELINES FOR TWO-FOOT-JUMP

• Physible reduces acceleration reconstruction from 0.454 to 0.081 and ID residual from 84.4 to 26.5 on the two-foot jump, with higher LDJ than other optimized methods.Noise GD attains lower errors (0.013 and 1.94) but takes longer and lowers LDJ; Physible trades larger residuals for smoother motion and lower computational cost.Source: §H.1.1 · p. 63

H.1.2 RESULTS ACROSS MOTION STYLES AND CONTACT CONDITIONS

• 89.5% and 79.5% are the median reductions in mean acceleration-reconstruction error and mean ID residual across 11 sequences, respectively.Both residual means and their 90th percentiles decrease in every evaluated sequence; mean LDJ is lower after optimization in every sequence.Source: §H.1.2 · p. 63, Table 9 · p. 64

H.2.1 VARIABILITY ACROSS OPTIMIZATION SEEDS

• Across ten two-foot-jump optimization seeds, acceleration reconstruction is 0.0877 ± 0.00979 and ID residual is 28.82 ± 3.38.Their relative standard deviations are approximately 11.2% and 11.7%; LDJ is −9.06 ± 0.12.Source: §H.2.1 · p. 65, Table 10 · p. 65

H.3.1 TWO-FOOT JUMP

• On the two-foot jump, Physible reduces vertical base acceleration error while retaining repeated ground contacts, the characteristic jump, and comparatively smooth trajectories.Noise GD has low acceleration error but floats throughout; Motion Sep-CMA-ES introduces substantial jitter, and Guided Diffusion increases errors and produces floating and penetration.Source: §H.3.1 · p. 66, Figure 25 · p. 67

H.3.2 LEFT HOP: UNOPTIMIZED VS. PHYSIBLE

• On the left hop, mean acceleration-reconstruction error falls from 0.511 to 0.009 and mean ID residual from 106.43 to 1.91, reductions of about 98% in both.The unoptimized vertical root-acceleration residual repeatedly reaches approximately 10 m s−2 or more; optimized planned and reconstructed root accelerations nearly coincide.Source: §H.3.2 · p. 80

H.3.3 UPHILL AND DOWNHILL WALKING: UNOPTIMIZED VS. PHYSIBLE

• Root linear-acceleration RMS mismatch falls 47% uphill and 84% downhill; angular mismatch falls 89% and 96%, respectively.Uphill motion retains appreciable linear mismatch, while optimization shifts root pitch positive uphill and negative downhill.Source: §H.3.3 · p. 92, Figure 51 · p. 93

H.4 DOWNSTREAM RL TRACKING

• For two-foot-jump tracking, optimized references reduce body-position, body-orientation, and joint-position errors by 16.4%, 11.9%, and 7.4%, but increase global root-position and orientation errors by 39.6% and 41.3%.The paper attributes the global drift to step-wise local acceleration corrections whose small residual inconsistencies accumulate over time.Source: §H.4 · p. 94

• The evaluation is limited to bipedal embodiments; future work should study manipulators, quadrupeds, and mobile manipulators.Source: I DETAILED LIMITATIONS AND FUTURE WORK · p. 95
• The method relies on accurate dynamics, inertial parameters, contact geometry, and contact characteristics, although these can be uncertain and time-varying.The authors propose incorporating uncertainty and evaluating performance under dynamics or contact mismatch.Source: I DETAILED LIMITATIONS AND FUTURE WORK · p. 95
• Assuming all ground contacts stick simplifies feasibility evaluation but rules out behaviors that deliberately use controlled slipping, such as moonwalk-like styles.The authors suggest jointly optimizing contact modes and motion.Source: I DETAILED LIMITATIONS AND FUTURE WORK · p. 95
• Majority-vote contact-pattern preservation can restrict substantial contact changes during optimization.Future alternatives should mitigate penetration and floating without overly constraining contact-pattern changes.Source: I DETAILED LIMITATIONS AND FUTURE WORK · p. 95
• Optimized noise is not guaranteed to remain within the training distribution, and strict constraint satisfaction can push it out of distribution and degrade style.The authors also identify difficulty choosing weights for competing objectives and constraints, and propose distribution-preserving mechanisms and earlier feasibility enforcement.Source: I DETAILED LIMITATIONS AND FUTURE WORK · p. 95
• Exactly zero ground penetration is generally infeasible to optimize, while setting the tolerated threshold too low can increase difficulty and slow convergence.The proposed contact-pattern constraint therefore permits limited penetration.Source: I DETAILED LIMITATIONS AND FUTURE WORK · p. 95
• Repeated horizon-wide feasibility evaluations and inverse-dynamics solves can be costly for long horizons or real-time use.Suggested directions include warm-starting, multi-resolution optimization, and amortized feasibility surrogates.Source: I DETAILED LIMITATIONS AND FUTURE WORK · p. 95