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Motoneuron-Inspired Sampling for Model Predictive Path Integral Control

Alexis Poignant, Jan Babič

arXiv:2609.28325v1cs.RO

TL;DR

MPPI’s performance under finite rollout budgets depends on the structure of its proposal distribution. This paper introduces a motoneuron-inspired proposal and evaluates it against Gaussian and spectrum-matched controls. Structured sampling improves executed-control smoothness, while task effects vary with conditions and actuation.

  • Problem

    MPPI performance under finite rollout budgets depends strongly on proposal structure, and articulated-system sampling scales with both control horizon and actuator count.

    Source: §I · p. 1

  • Method

    The paper introduces Spike-MPPI, a motoneuron-inspired proposal with shared drive and motor-pool recruitment, discharge, and twitch dynamics, evaluated on torque-driven and antagonistic Ant models.

    Source: §I · p. 2, G. Robot Models · p. 7

  • Results

    Across flat experiments, Poisson reduces control-increment magnitude by approximately 43–53% and Spike by approximately 51–61% relative to Standard; spectrum matching reproduces substantial task-level behavior, while Spike retains additional smoothness and coordination effects.

    Source: §IV · pp. 12–13

  • Takeaways & Limitations

    The results support treating proposal design as a combination of second-order spectral structure and higher-order statistical organization.

    Source: §V · p. 13

    Experiments are simulation-only, and the simplified antagonistic actuator and motoneuron models are not quantitatively validated muscle–tendon models.

    Source: §IV · p. 13

Abstract

from arXiv · show

Model Predictive Path Integral (MPPI) control relies on stochastic trajectory sampling, and its performance under limited rollout budgets depends strongly on the structure of the proposal distribution. Standard implementations commonly perturb control sequences with Gaussian noise, despite growing evidence that temporally correlated and structured sampling can improve finite-budget control. We introduce Spike-MPPI, a motoneuron-inspired proposal that generates temporally structured perturbations through a simplified model of motoneuron dynamics. The proposal is evaluated within a common MPPI framework on torque-actuated and antagonistically actuated MuJoCo Ant models against standard Gaussian sampling and spectrum-matched Gaussian controls. Results show that structured sampling substantially improves executed-control smoothness, while its effect on task performance depends on rollout condition and robot actuation. Spectrum matching reproduces a substantial part of the observed behavior, while the full Spike proposal retains additional effects beyond second-order spectral structure. These results support treating proposal design as a combination of second-order spectral structure and higher-order statistical organization.

I. INTRODUCTION

MPPI’s finite-budget performance depends strongly on its proposal distribution, motivating temporally structured sampling. The paper introduces Poisson and motoneuron-inspired Spike proposals and evaluates them in a shared MPPI setting.

I. INTRODUCTION

  • Finite rollout budgets make MPPI performance strongly dependent on the proposal distribution, while articulated-system sampling also grows with horizon and actuator count.Structured temporal parameterizations are motivated as a way to concentrate samples on trajectories rather than treating every control coordinate independently.Source: §I · p. 1
  • Prior work motivates structured proposals through temporally correlated noise, reduced temporal parameterization, and event-based sampling –.These approaches include sparse knot points, spline interpolation, colored noise, sample reuse, and compound-Poisson events.Source: §I · p. 1
  • Biological motor pools provide a model for coordinated, temporally extended activation through recruitment, discharge dynamics, and twitch responses –.These mechanisms motivate structured perturbations without prescribing a gait or learned policy.Source: §I · pp. 1–2
  • The paper introduces sparse marked Poisson proposals and Spike-MPPI, which combines shared drive, ordered recruitment, hysteresis, rate coding, renewal discharge, and heterogeneous twitches.The samplers are evaluated on torque-driven and antagonistic MuJoCo Ant models against Standard and spectrum-matched Gaussian proposals.Source: §I · pp. 2–3

A. Problem Formulation

  • The problem formulation considers online receding-horizon control of an articulated system, with nominal control sequences initialized at zero and subsequently warm-started by shifting the prior solution.The predictive dynamics include rigid-body dynamics, contacts, actuator limits, and task-dependent environmental interactions.Source: A. Problem Formulation · p. 3

B. Model Predictive Path Integral Control

The paper holds the MPPI framework constant while changing how perturbation sequences are generated. It describes Gaussian sampling as a baseline and Poisson sampling as a marked-event process decoded through twitch kernels.

B. Model Predictive Path Integral Control

  • MPPI generates candidate control sequences by adding sampled perturbations to the nominal sequence, evaluates their costs, and updates the sequence using normalized importance weights.Across sampling methods, the nominal sequence, predictive dynamics, objective, actuator limits, temperature adaptation, and update are shared; only perturbation generation differs.Source: B. Model Predictive Path Integral… · p. 3

C. Gaussian Sampling

  • Standard Gaussian sampling perturbs native actuator coordinates with Gaussian noise temporally correlated by a stationary first-order filter.The experiments use correlation coefficient ρs = 0.25 and exploration standard deviation equal to 30% of each actuator’s control scale.Source: C. Gaussian Sampling · p. 3

D. Poisson Sampling with Twitch kernels

  • Poisson sampling replaces Gaussian perturbations with sparse signed events and decodes them through a causal twitch kernel.Events have separate sign and recruitment-depth marks; positive and negative pool outputs route to agonist and antagonist channels for the antagonistic robot. Pre-clipping RMS is matched to Standard, although realized post-clipping variance may differ near control bounds.Source: D. Poisson Sampling with Twitch kernels · pp. 3–4

E. Spike Sampling

Spike replaces fixed Poisson event generation with a reduced-order stochastic motor-pool model. Its proposal combines shared drive, recruitment and discharge dynamics, and heterogeneous twitch responses.

E. Spike Sampling

  • Spike is a reduced-order motor-pool proposal that adds common drive, ordered recruitment, hysteresis, rate coding, refractory renewal discharge, and heterogeneous twitches.The stated aim is to introduce structured priors into MPPI, not to reproduce complete spinal electrophysiology.Source: E. Spike Sampling · pp. 4–5
  • A shared stochastic drive feeds the agonist and antagonist pools, while lower derecruitment thresholds let recruited units remain active as drive falls.For the antagonistic Ant, background drive permits simultaneous agonist–antagonist recruitment and physical co-contraction.Source: E. Spike Sampling · pp. 4–5
  • Active units increase firing rate with suprathreshold drive and discharge through a refractory shifted-Gamma renewal process; later-recruited units are stronger and faster.Motor-unit-specific twitches convert sparse discharges into temporally extended perturbations; Spike’s pre-clipping RMS is scaled to match Standard.Source: E. Spike Sampling · p. 5

F. Spectrum-Matched Gaussian Sampling

Spectrum-matched Gaussian proposals preserve the structured proposals’ finite-horizon second-order spectra while removing their event-based and motor-pool statistics. This permits comparison of spectral structure with the additional organization in the full proposals.

F. Spectrum-Matched Gaussian Sampling

  • Gaussian-Poisson and Gaussian-Spike preserve their target proposals’ temporal spectra and autocorrelation but remove event-based and motor-pool statistics.The comparison tests whether behavior associated with a structured proposal is reproduced by its finite-horizon second-order spectrum.Source: F. Spectrum-Matched Gaussian Sampling · pp. 5–6
  • Calibration perturbations estimate each target’s finite-horizon spectral matrix, scalar for torque-driven Ant and 2 × 2 for antagonistic Ant.The matrix can represent second-order agonist–antagonist coupling; its spectral factor is obtained by eigendecomposition.Source: F. Spectrum-Matched Gaussian Sampling · pp. 5–6
  • In a representative torque-driven Ant trace, structured proposals show visibly smaller command increments than Standard after MPPI averaging.The figure compares the hip command increment and executed command for leg 1 over the same time window.Source: Figure 3 · p. 6
  • Each spectrum-matched sample transforms white Gaussian noise in the frequency domain, with conjugate symmetry and real DC and Nyquist coefficients yielding real perturbations.The resulting Gaussian proposal reproduces the target proposal’s finite-horizon power spectrum in expectation.Source: F. Spectrum-Matched Gaussian Sampling · p. 6

G. Robot Models

The two Ant models share morphology, environment, task objective, and planning dynamics, differing in joint actuation. The antagonistic model adds impedance effects from simultaneous agonist–antagonist activation.

G. Robot Models

  • The models share the same rigid-body morphology, joint topology, environment, task objective, and planning dynamics, differing only in joint actuation.The torque-driven Ant has eight bidirectional motors, while the antagonistic Ant uses sixteen nonnegative actuator commands.Source: G. Robot Models · p. 6
  • The torque-driven Ant uses eight bidirectional joint motors; Spike subtracts agonist and antagonist pool outputs to form its eight-dimensional command.Standard samples eight Gaussian actuator coordinates, while Poisson generates eight signed twitch-decoded channels.Source: G. Robot Models · p. 6
  • The plant adds no activation filter, isolating proposal timing: Standard has no twitch decoder, Poisson uses a common kernel, and Spike a heterogeneous twitch bank.The antagonistic pair uses nonnegative agonist and antagonist commands.Source: G. Robot Models · pp. 6–7
  • In the antagonistic model, differential activation controls directional torque while simultaneous activation changes joint stiffness and damping.Equal activation cancels the directional torque term but retains the impedance terms; Spike generates both motor-pool outputs directly.Source: G. Robot Models · p. 7

H. Tasks

Tasks share a closed stadium track, with progress defined by the robot or manipulated object depending on the task. The rollout objective combines progress with stability and control regularization, and transfer studies vary terrain, morphology, task, and planner dynamics.

H. Tasks

  • Running tracks robot-torso progress on flat terrain for the main rollout-budget comparison and reuses the task for terrain, morphology, and planner-mismatch studies.The robot is rewarded for forward progress while upright and on the road.Source: H. Tasks · p. 7
  • Box pushing tracks box progress and shapes reward for coupled robot–box progress and reduced distance before contact.Candidate trajectories are rejected if the box is lifted above its limit or tips below the specified up-axis threshold.Source: H. Tasks · p. 7
  • Sled towing tracks sled progress; because the rope already couples robot and sled, the task omits approach-to-object shaping.The task applies lift and up-axis limits to the sled.Source: H. Tasks · p. 7
  • The rollout cost rewards primary-body progress and, for object tasks, coupled robot progress and approach improvement; it penalizes loss of upright posture and control deviation.Leaving the road corridor or falling assigns infinite cost.Source: H. Tasks · p. 7
  • Generalization studies vary terrain, leg morphology, task, and planner dynamics without changing proposal parameters.They include rocky and mixed tracks, altered leg lengths, pushing and towing, and planner-model mismatch.Source: H. Tasks · p. 7

I. Experimental Protocol

The protocol fixes controller and proposal settings, compares rollout-budget and screening conditions, and evaluates matched-seed runs using task, optimization, and executed-control metrics.

I. Experimental Protocol

  • The controller uses a 20 ms step and 75-step horizon; Poisson and Spike use six recruitment levels and an 8 Hz base rate.Exploration scale is 30% of each actuator’s control scale; Poisson and Spike share the stated base twitch constants.Source: §I · p. 8
  • Flat running compares 32 and 256 full-fidelity candidates with Screen 256, which previews 256 and fully evaluates 32 candidates.Screening selects 24 by preview cost and samples the remaining 8 uniformly; transfer studies use unscreened N = 32.Source: §I · p. 8
  • Each flat-study configuration uses 10 matched seeds, each with one discarded warm-up lap and 10 retained laps; seeds, not laps, are statistical replicates.All seeds complete retained evaluation in the main flat-running experiment.Source: §I · pp. 8–9
  • Reported metrics include main-body velocity, MPPI-weighted rollout cost, executed-command smoothness, and computation time per control step.Smoothness is measured from successive executed commands; computation time includes candidate generation, MPPI computation, plant integration, and progress projection.Source: §I · p. 9
  • For Ant-2, co-contraction and differential action quantify simultaneous antagonist activation and directional actuator use, while phase-locking measures relative command phase consistency.These actuation measures are not task-performance objectives; coordination is summarized within and between legs.Source: §I · p. 9

2) Spectral shaping:

The section frames three hypothesis-driven contrasts to separate temporal spectral structure from the full Spike proposal’s additional organization. An auxiliary spectrum-matched Poisson control is excluded from the inferential analysis.

2) Spectral shaping:

  • The first contrast asks whether spectrum-matched Gaussian-Spike differs from Standard Gaussian sampling.This isolates the behavior associated with matching Spike’s finite-horizon spectrum against the standard proposal.Source: 2) Spectral shaping · p. 9
  • The second contrast asks whether the full Spike proposal differs from its spectrum-matched Gaussian counterpart.This tests for effects beyond spectral matching.Source: 2) Spectral shaping · p. 9
  • The third contrast asks whether motoneuron-inspired Spike differs from the simpler Poisson event-based proposal.Gaussian-Poisson is an auxiliary descriptive control and is not included in the hypothesis-driven inferential analysis.Source: 2) Spectral shaping · p. 9
  • Contrasts use two-sided exact seed-blocked permutation tests, with random seed as the independent experimental unit and pairing by seed and retained-lap index.Holm correction is applied separately by outcome metric and hypothesis family; significance is assessed at α = 0.05.Source: 2) Spectral shaping · p. 9

III. RESULTS

On flat running, structured proposals consistently smooth executed controls, while velocity differences vary with rollout budget and robot actuation. Spike also changes Ant-2 actuator use and command coordination, with some coordination gains reproduced by spectrum-matched Gaussian sampling.

A. Flat Racing Comparison

  • Spike reduces mean control-increment norm by approximately 51–61% versus Standard across both robots and all rollout conditions, exceeding Poisson’s 43–53% reduction.Spike has smaller control increments than Poisson in every condition, even when task velocity is not higher than Standard.Source: Figure 6 · p. 10, A. Flat Racing Comparison · p. 11
  • At N = 32, Ant-2 velocity rises from 4.77 m/s with Standard to 5.14 m/s with Spike, a 7.8% increase; Spike also has the lowest weighted rollout cost.The respective weighted rollout costs are −2.74 for Standard, −2.85 for Poisson, and −3.01 for Spike.Source: Figure 6 · p. 10, A. Flat Racing Comparison · p. 10
  • At N = 32, Ant velocity is 5.09 m/s with Standard, 5.25 m/s with Poisson, and 5.30 m/s with Spike; at N = 256, Standard leads Ant at 6.21 m/s.On Ant-2 at N = 256, Spike reaches 5.93 m/s versus 5.78 m/s for Standard, while Poisson reaches 5.57 m/s.Source: Figure 6 · p. 10, A. Flat Racing Comparison · pp. 10–11
  • With rollout screening, Ant velocities are 6.03, 5.97, and 5.93 m/s for Standard, Poisson, and Spike; Spike reaches 5.83 m/s on Ant-2.On Ant-2, Poisson falls from 5.69 to 5.39 m/s under screening; Spike is numerically faster than Standard across all three rollout conditions.Source: Figure 6 · p. 10, A. Flat Racing Comparison · p. 11
  • On Ant-2, co-contraction falls from 0.73 with Standard to 0.47 with Poisson and 0.50 with Spike at N = 32; Spike has the largest differential action.At N = 32, differential action is 0.25 for Standard and Poisson and 0.34 for Spike; these measures are not monotonic performance objectives.Source: Figure 7 · p. 10, A. Flat Racing Comparison · p. 11
  • At N = 32, Spike raises within-leg coordination over Standard from 0.068 to 0.138 on Ant and from 0.070 to 0.150 on Ant-2.Gaussian-Spike reproduces much of the increase, though Spike remains numerically more coordinated in the reported comparisons.Source: Figure 8 · p. 10, A. Flat Racing Comparison · p. 11
  • At N = 32, mean computation time is 14.2 ms/step for Standard and 16.4 ms/step for Spike on Ant; evaluating 256 full rollouts raises computation to approximately 97–117 ms/step.Screening reduces that cost to approximately 29–44 ms/step.Source: A. Flat Racing Comparison · p. 11

B. Generalization and Robustness

Across terrain, tasks, altered morphologies, and planner–plant integrator mismatches, Spike generally retains smoother control, but velocity and completion outcomes depend on the scenario. The mismatch experiments expose a completion-reliability trade-off despite smoothness and, in some conditions, higher velocity.

B. Generalization and Robustness

  • The transfer studies are descriptive and are not included in the flat-running experiment’s hypothesis-driven inferential analysis.Velocity and smoothness are conditioned on completed laps and should be interpreted with completion rate, especially when early termination is frequent.Source: B. Generalization and Robustness · p. 11
  • On mixed terrain with Ant-2, Spike completes 8.0 laps/seed versus 1.6 for Standard and raises mean velocity from 3.83 to 4.11 m/s.Across all four terrain–morphology combinations, Spike has the highest mean velocity and lowest control increments.Source: B. Generalization and Robustness · p. 11, Table III · p. 12
  • Across object-manipulation tasks, Spike produces the smoothest control in every robot–task combination, but Standard remains fastest for Ant pushing.Spike reaches the highest mean velocity on Ant-2 for both pushing and towing; Ant towing is essentially tied at 2.650 m/s for Standard and 2.647 m/s for Spike.Source: B. Generalization and Robustness · pp. 11–12, Table IV · p. 12
  • Under altered leg lengths, Spike has the lowest control increments in every condition, while velocity leaders vary by robot and configuration.For same-side Ant-2, Spike raises mean velocity from 3.95 to 4.41 m/s and reduces control-increment magnitude from 1.06 to 0.48.Source: Table V · p. 12, B. Generalization and Robustness · p. 12
  • Under integrator mismatch, Spike remains smoothest in all four conditions and is fastest for both robots under fast RK4, but Ant-2 completion is lower than Standard’s.Ant-2 completes 5.0 and 3.9 laps/seed with Spike under the two mismatches, versus 7.8 and 8.8 with Standard.Source: Table VI · p. 12, B. Generalization and Robustness · p. 12

IV. DISCUSSION

Across tested conditions, structured proposals produced smoother executed controls, while task-level effects varied with rollout budget and actuation. Spectrum matching explained part, but not all, of Spike’s effects.

IV. DISCUSSION

  • Poisson reduced control-increment magnitude by approximately 43–53% and Spike by approximately 51–61% relative to Standard across flat experiments.The same ordering persisted across terrain, task, and morphology variations.Source: §IV · p. 12
  • Spectrum-matched Gaussian reproduced substantial Spike behavior, with no detected additional flat-running velocity or weighted-rollout-cost effect, while Spike retained differences in smoothness, co-contraction, and coordination.The experiments do not isolate which higher-order temporal or amplitude mechanisms account for the residual differences.Source: §IV · pp. 12–13
  • At N = 32, temporally coherent proposals may increase the chance of useful rollouts; at larger budgets, Standard has more candidates and weakly correlated components may partially cancel.Source: §IV · p. 13
  • Spike differed from Poisson consistently in smoothness and coordination, while task-performance differences appeared primarily on antagonistic Ant-2 rather than torque-driven Ant.The additional motor-pool structure primarily affected temporal organization and use of commands, with task-level benefits depending on actuation model.Source: §IV · p. 13
  • Across terrain, manipulation, and morphology variations, Spike retained the lowest executed-control increments, but substantial planner-model mismatch could coincide with reduced completion.Spike could remain the smoothest proposal and retain high velocity under mismatch.Source: §IV · p. 13
  • Evaluation was simulation-only, and the antagonistic actuator and motoneuron models were simplified abstractions rather than validated muscle–tendon models.Structured samplers add modest computational overhead; under substantial model inaccuracy, smoother control can coexist with reduced completion.Source: §IV · p. 13
  • Future work could optimize sparse event sequences, adapt proposal components from control performance, or use MPPI as a teacher for closed-loop policy learning.Source: §IV · p. 13

V. CONCLUSION

  • The conclusion frames MPPI proposal design as combining second-order spectral structure with higher-order statistical organization.Source: §V · p. 13

APPENDIX

The appendix tables provide Holm-adjusted and raw permutation p-values for four hypothesis-driven contrasts.

APPENDIX

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