Suddhu | June 16th 2020, RPL reading group | paper link

Overview

• Captures complex belief distributions and noisy dynamics

• System where multimodality and complex behaviors cannot be neglected, at the cost of larger computational effort

• GP-SUM comprises of:
  • Dynamic models expressed as a GP
  • Complex state represented as weighted sum of Gaussians

• Can be viewed as both a sampling technique and a parametric filter

• Applications: (i) synthetic benchmark task, (ii) planar pushing

1. Introduction

• Cases where state belief cannot be approximated as a unimodal Gaussian distribution

• Example: push grasping (Dogar and Srinivasa)

• Theoretical contribution: Bayes updates of belief without linearization, or unimodal Gaussian approximation

2. Related work

• Gaussian processes have been applied to learn dynamics models, planning/control, system identification, and filtering.

• GP-Bayes filters learn dynamics and measurement models:
  • GP-EKF: linearizes GP model to guarantee final Gaussian (Ko and Fox)
  • GP-UKF: similar to the unscented Kalman filter (Ko and Fox)
  • GP-ADF: computes first two moments and returns Gaussian (Deisenroth and Hanebeck)
  • GP-PF: no closed form distributions, particle depletion (Ko and Fox)

• Task-specific:
  • MHT: multi-hypothesis tracking, joint distribution as a Gaussian mixture (Blackman)
  • MPF: manifold particle filter, sampling-based algorithm to decompose state space using contacts (Koval)

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3. Background on GP filtering

3.1 Bayes filters

• Track the state of system x_t probabilistically:
  • action u_(t-1) evolves state from x_(t-1) to x_t
  • Observation z_t occurs
  • Belief:

    

• Two steps: prediction update and measurement update

3.1.1 Prediction update

• given system dynamics and previous belief, we compute prediction belief:

• Integral cannot be solved analytically: linearize dynamics and assume Gaussian distribution (EKF)

3.1.2 Measurement update

• Given prediction belief and new measurement z_t we recover belief:

  

• For closed form: linearize observation model and assume Gaussian

3.1.3 Recursive belief update

• Combine Equation 1 and Equation 2 to express belief with respect to previous belief, dynamic model and observation model:

3.2 Gaussian processes

• Real systems may have unknown or inaccurate models, can be learnt via GPs

• Advantages:
  • Learns high-fidelity model from noisy data
  • Estimates uncertainty of predictions from data (ie: quality of regression)

• Output and latent function:

  $y(x) = f(x) + \epsilon \quad \text{where} \quad \epsilon \sim N(0, \ \sigma^2)$

• We need data and kernel function:

• Mean and variance equations (kernel: square exponential):

4. GP-SUM Bayes filter

• Dynamics model p(x_t | x_(t-1), u_(t-1)) and measurement model p(z_t | x_t) are represented by GP

• Considers the weak assumption that belief is approximated by sum of Gaussians
  • at infinite samples this converges to the real distribution

4.1 Updating the prediction belief

• Belief from standard Bayes filter:

• Prediction belief (combining Equation 1 and Equation 5):
  • updated with previous prediction belief, dynamics model, and observation model

• Prediction belief at (t-1) approximated as sum of Gaussians
  • M_(t - 1) # of components and w_(t - 1), i weight of Gaussian mixture

• Sampling from this distribution, we can simplify Equation 7 as:

• Dynamics model can be expressed as Gaussian (GP):

• Weight depends on the observations:

• Final prediction belief:

• Other GP-Bayes filters approximate the prediction belief as a single Gaussian, and previous approximation accumulate over time

4.2 Recovering belief from prediction belief

• From Equation 5 and Equation 10, the belief is a sum of Gaussians:

• p(z_t | x_t) N(x_t | mu_t, j, Sigma_t, j) won't be proportional to a Gaussian, so they recover belief through an approximation (Deisenroth and Hanebeck)
  • won't go into detail but preserves the first two moments of p(z_t | x_t) N(x_t | mu_t, j, Sigma_t, j)
  • only needed to recover belief, does not accumulate as iterative filtering error

4.3 Computational complexity

• GP-SUM complexity increases linearly with # of Gaussians sampled M

• Also depends on size of GP model n

• cost of propagating prediction belief:

5. Results

5.1 1D non-linear synthetic model

Dynamics model (especially sensitive around 0):

Measurement model:

5.2 Real task: uncertainty in pushing

• Planar pushing yields stochastic behavior due to friction variability and uncertain interactions

• Heteroscedastic: noise in pushing dynamics depends on the action

• Use GP-SUM to propagate uncertainty in planar pushing by learning the dynamics of system with a heteroscedastic GP (Bauza and Rodriguez)
  • train GP:
    • input = contact_points, pusher_velocity, pusher_direction
    • output = object_displacement

• Notable difference, there is no measurement model. State distribution can only expand over time, and weights are equal.

• Propagating uncertainty of pushing can:
  • inform strategies that reduce uncertainty (active)
  • better perform pose estimation

    

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  6. Discussion and future work

  • Situations where Gaussian beliefs are unreasonable
  • Future work:
    • # of samples: adjusting over time (particle filter methods)
    • computational cost: scale linearly with # of samples.
    • GP extensions: can use sparse GPs (Pan and Boots)