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A probabilistic data-driven model for planar pushing

Maria Bauza, Alberto Rodriguez

arXiv:1704.03033v2cs.ROcs.LGstat.ML

TL;DR

Planar pushes can produce variable outcomes, while prior data-driven models had not captured both expected behavior and experimental variability. This paper uses variational heteroscedastic Gaussian processes to predict push outcomes and their input-dependent variance. The model outperforms a standard analytical model with an order of 100 samples, and its accuracy tails off at about 103 training samples.

  • Problem

    Planar pushing can produce convergent, divergent, or multi-modal outcomes, and prior data-driven planar-pushing work had not modeled both expected behavior and experimental variability.

    Source: §I · p. 1

  • Method

    The paper uses variational heteroscedastic Gaussian processes to model the mean and input-dependent variance of planar-push outcomes.

    Source: §VIII · p. 7

  • Results

    An order of 100 samples is sufficient to outperform a standard analytical model, and VHGP accuracy tails off at about 103 training samples.

    Source: §VIII · p. 7

  • Takeaways & Limitations

    Input-dependent-noise models can represent pushes with different levels of stability, and the learned model's accuracy tails off after a reduced amount of training data.

    Source: §IV · p. 3, A. Evaluation on Different Objects and… · p. 5

    The instantaneous dynamics retain a Gaussianity and unimodality assumption, although input-dependent noise can express multi-modal behavior when integrated over time.

    Source: §VIII · p. 8

Abstract

from arXiv · show

This paper presents a data-driven approach to model planar pushing interaction to predict both the most likely outcome of a push and its expected variability. The learned models rely on a variation of Gaussian processes with input-dependent noise called Variational Heteroscedastic Gaussian processes (VHGP) that capture the mean and variance of a stochastic function. We show that we can learn accurate models that outperform analytical models after less than 100 samples and saturate in performance with less than 1000 samples. We validate the results against a collected dataset of repeated trajectories, and use the learned models to study questions such as the nature of the variability in pushing, and the validity of the quasi-static assumption.

I. INTRODUCTION

Planar pushing can produce different outcome distributions even for pushes indistinguishable at sensor and actuator resolution, motivating models that capture both expected behavior and variability. The paper uses input-dependent-noise Gaussian processes to model mean and variance, building on prior data-driven work that had not modeled both expected behavior and experimental variability.

I. INTRODUCTION

  • Indistinguishable pushes can yield divergent, convergent, or multi-modal outcomes, so some pushes are more precise than others.Pushing is a primitive manipulation task and can also occur in more complex behaviors such as grasping.Source: §I · p. 1
  • The paper learns a compact data-driven model of the mean and variance of a pushed object's expected behavior.The motivation is that physical interaction laws compound microscopic variation, so their accuracy may be statistical rather than exact.Source: §I · p. 1
  • The model uses Heteroscedastic Gaussian processes, including their variational implementation, to represent noise that depends on the pushing action.The paper uses the dataset by Yu et al. and adds repeated pushes across a grid of locations and directions to validate predicted variances.Source: §I · p. 1
  • Evaluation compares predictions across four objects and four materials against normal Gaussian processes and a common analytical model, then validates distributions against repeated pushes.Prediction accuracy uses NMSE and NLPD; distribution validation uses KL divergence.Source: §I · p. 2

II. RELATED WORK

  • Prior data-driven planar-pushing work studied control, but, to the authors' knowledge, had not modeled both expected behavior and experimental variability.The related work identifies Zhou et al. as a data-efficient, physics-inspired friction model.Source: §II · p. 2
  • The work builds on empirical pushing data and variational heteroscedastic Gaussian processes, which explicitly model transition-noise dependence on the input action.The cited variational approach addresses noise underestimation associated with a simplified maximum-likelihood algorithm.Source: §II · p. 2

III. PUSHER-SLIDER DATA

The paper represents planar pushing as an input-output relationship between a pusher's action and the object's displacement. It records pusher-object interaction with motion tracking and force sensing.

III. PUSHER-SLIDER DATA

  • The pusher-slider interaction is recorded using a Vicon tracking system and a force-torque sensor.The setup is designed to provide a clean interface between pusher and slider and is described in Yu et al. [2].Source: Figure 2 · p. 2
  • The input space specifies pusher velocity, contact point, and push angle, while the output is object displacement in a push-aligned frame.For objects without symmetries, contact points are evenly spaced around the perimeter.Source: Figure 3 · p. 2
  • The study uses a large general-purpose pushing dataset for learning and a separate repeated-push dataset to validate predicted outcome variance.The modeled outputs are center-of-mass x and y displacement and orientation change after a push lasting ∆t seconds.Source: §III · p. 2

A. Learning Data

The learning data come from a prior robotic planar-pushing dataset spanning objects and sliding surfaces.

A. Learning Data

  • The prior dataset records pusher and object trajectories and interaction forces for stainless-steel objects of different shapes sliding on different materials.The setup uses a high-precision industrial robot with a cylindrical pusher.Source: A. Learning Data · p. 3

B. Validation Data

The model predicts an input-dependent expected outcome and variance, and repeated pushes provide empirical distributions for validating variance predictions. The added dataset also supports study of pushing variability, including multi-modality.

B. Validation Data

  • The model predicts expected motion and variance as functions of the action u = (vp, c, β).The input-dependent variance is the key complexity motivating heteroscedastic rather than standard Gaussian processes.Source: B. Validation Data · p. 3
  • A validation dataset collected 100 repetitions per push, approximating object-motion distributions across initial contact angles from −1.5 to +1.5 radians in 0.1-radian steps and 11 evenly spaced initial contact points.The pusher follows a straight 1 cm trajectory at 20 mm/s; this grid allows extraction of expected mean and variance of object motion.Source: B. Validation Data · p. 3
  • The repeated-push dataset provides empirical evidence for studying pushing variability, including multi-modal effects.The stated role of the dataset is to validate the model's predicted outcome variance.Source: B. Validation Data · p. 3

IV. HETEROSCEDASTIC GAUSSIAN PROCESSES

Classical Gaussian processes assume constant Gaussian observation noise, while planar pushing exhibits input-dependent variability. Heteroscedastic Gaussian processes model both the process mean and variance, and VHGP uses a variational implementation for this purpose.

IV. HETEROSCEDASTIC GAUSSIAN PROCESSES

  • A conventional GP’s predictive variance combines constant process noise with input-dependent regression error, which reflects confidence in the predicted mean.The regression-error term depends strongly on the density of training points near the input.Source: §IV · p. 3
  • HGPs model input-dependent noise and can improve mean estimation by giving less noisy observations more relevance.This addresses pushing dynamics in which some input regions are more stable than others.Source: §IV · p. 3
  • VHGP uses Bayesian variational theory to derive closed-form process mean and variance estimates, with separate kernels for the mean and log-variance.The mean and log-variance use ARD-SE kernels.Source: §IV · p. 3
  • GP and VHGP inference scale alike with training data, while VHGP replaces constant noise with the expected input-dependent process noise.The predictive mean equation is essentially unchanged from the GP case.Source: §IV · p. 4

V. THE LEARNED MODEL

The learned model uses three independent VHGPs for the displacement and rotation outputs. Its predicted variability changes with push location and direction, and the modeled standard deviations are compared with repeated-push measurements.

V. THE LEARNED MODEL

  • Three independent VHGPs predict Δx, Δy, and Δθ; this ignores output correlations but performs sufficiently when enough training data are used.Combining multi-task GPs with heteroscedastic GPs is proposed as future work.Source: §V · p. 4
  • For a square on plywood at 20 mm/s, Δx is greatest for a centered, perpendicular push, while rotation peaks at specified off-center locations and angles.Maximum rotation occurs at c = 0.75, β = +30°, or c = 0.25, β = −30°.Source: §V · p. 4
  • 10%–40%: modeled push standard deviations are this fraction of expected output magnitude, with variability mapped over contact location and direction.Figure 6 compares modeled and experimental standard deviations.Source: §V · p. 4
  • Higher-noise regions are well defined, predictions match validation data to a degree, and measurement precision suggests noise principally comes from pushing rather than sensing.For a given object, the regressed noise shape remains more or less constant across surfaces.Source: §V · p. 4
  • Uncertainty estimates can distinguish more- from less-stable pushes and support multi-step prediction by propagating uncertainty.The model captures interaction uncertainty beyond deterministic predictions of standard analytical models.Source: §V · p. 4

VI. EVALUATION OF THE LEARNED MODEL

The evaluation uses normalized mean square error (NMSE) and normalized log probability density (NLPD) to assess different aspects of prediction. Their definitions make them suitable for different comparisons.

VI. EVALUATION OF THE LEARNED MODEL

  • NMSE measures normalized squared prediction error relative to observation variance and sums the individual output NMSEs.It assesses expected values rather than motion variability, making it useful for comparison with deterministic pushing models.Source: §VI · p. 4
  • NLPD evaluates how likely observations are under a probabilistic model, combining the individual output probabilities.Lower NLPD indicates higher likelihood; the metric penalizes underconfident and overconfident models.Source: §VI · p. 4
  • Deterministic models cannot be fairly evaluated with NLPD because that metric assesses predictive uncertainty as well as observed outcomes.NMSE instead compares expected predictions with observations.Source: §VI · p. 4

A. Evaluation on Different Objects and Materials

VHGP performance saturates with fewer than 1,000 samples across evaluations on multiple object shapes and materials. Its expected-displacement predictions also resemble the validation data, while performance saturation suggests planar pushing can be learned with reduced exploration.

A. Evaluation on Different Objects and Materials

  • Fewer than 1,000 samples suffice to saturate VHGP performance across four surfaces and four object shapes.At a 0.2-second push duration, this corresponds to about five minutes of experiments in the reported setup.Source: A. Evaluation on Different Objects and… · p. 4
  • For a square on plywood at 20 mm/s, predicted expected displacements across contact locations and push angles resemble the validation data.The plotted symmetries reflect the square's symmetry.Source: Figure 5 · p. 5
  • Performance remaining roughly constant after some data collection suggests pushing can be learned through reduced environmental exploration.Source: A. Evaluation on Different Objects and… · p. 5

B. Comparison with Other Models

The data-driven models outperform the analytical model after approximately 100 samples, while VHGP's input-dependent noise captures stochastic behavior that standard GP does not. The analytical model is less reliable at high velocities, where its quasi-static assumption omits inertia.

B. Comparison with Other Models

  • GP and VHGP have similar NMSE, but their NLPD difference supports modeling input-dependent noise in pushing.NMSE evaluates the most likely outcome; the models differ in how they account for stochastic behavior.Source: B. Comparison with Other Models · p. 5
  • The analytical comparison assumes quasi-static interaction, uniform pressure and Coulomb friction, and an ellipsoidal limit surface.Source: B. Comparison with Other Models · p. 5
  • The proposed data-driven model outputs a distribution of motion outcomes that encloses unexpected behaviors, and both GPs and VHGPs outperform the analytical model after approximately 100 samples.Source: B. Comparison with Other Models · p. 5
  • The analytical model is less reliable at high velocities, consistent with its quasi-static assumption and omission of inertia.The data-driven model includes pusher velocity as an input and represents dynamic effects through distributional uncertainty.Source: B. Comparison with Other Models · p. 5
  • Predicted standard deviations are compared with observed variability, and some high-variance regions appear independent of surface type.The authors suggest that some pushes are more stable than others regardless of surface.Source: Figure 6 · p. 6

VII. VALIDATION

Because the original training and testing data lacked enough repeated pushes to estimate ground-truth variance, the authors collected repeated trajectories for validation. Average and median KL divergences indicate that the model predicts both expected motion and its distribution reasonably well.

VII. VALIDATION

  • The original dataset lacked enough repeated pushes to estimate ground-truth variance, so a repeated-trajectory benchmark was collected for validation.The benchmark estimates each repeated push's expected motion and variance.Source: §VII · p. 6
  • Average KL divergence over pushes evaluates model predictions against the benchmark distributions, assuming each real push distribution comprises three independent Gaussians.Source: §VII · p. 6
  • Average and median KL divergences indicate that the model predicts expected motion and its distribution reasonably well.The table reports KL divergence between validation data and the models.Source: §VII · p. 6, Table II · p. 6

VIII. DISCUSSION

The discussion presents VHGPs as data-driven probabilistic models of planar pushing that capture action-dependent variability. It reports improved prediction and distribution validation, while examining where the model’s assumptions and scope constrain its use.

Surfaces and shapes

  • Across the tested surfaces, NMSE and NLPD errors are reasonably similar, while the circle is easier to learn than the other tested shapes.The authors suggest that equivalent circle contact points make the contact-point dimension redundant; higher NLPD on Delrin and plywood may reflect greater degradation over time.Source: Figure 7 · p. 7, Figure 8 · p. 7

Results

  • After about 100 samples, GPs and VHGPs outperform the analytical model; VHGPs outperform GPs on NLPD, and accuracy tails off at about 10^3 training samples.The reported training data can be captured in about five minutes in the authors’ setup.Source: Figure 9 · p. 7, §VIII · p. 7

Data-driven modeling

  • VHGP models capture the mean and action-dependent variance of planar pushes, addressing variability that can change by up to an order of magnitude across actions.The model can also help identify actions with unpredictable behavior.Source: §VIII · p. 7

Scope

  • The learned models are specific to each object and material, so generalization across shapes and materials remains for future work.The authors identify less computationally expensive training as a requirement for that future work.Source: §VIII · p. 7

Distribution validation

  • On repeated pushes across more than 300 actions, KL-divergence evaluation shows a clear improvement over a normal GP.The validation dataset contains 100 identical pushes for each action.Source: §VIII · p. 7

Gaussianity and multimodality

  • Gaussianity and unimodality remain key assumptions, although some repeated-push output distributions are not clearly Gaussian and VHGPs still approximate them well.The authors note that input-dependent noise can express multimodal behavior when integrated over time.Source: §VIII · p. 8, Figure 10 · p. 8

Velocity dependence

  • When data from different velocities are time-scaled and combined, performance peaks around 50–80 mm/s and degrades at higher velocities, indicating that the quasi-static assumption does not hold beyond that range.The authors therefore include pusher velocity as an explicit model input.Source: §VIII · p. 8
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