PaperlayerView exact revision on arXiv

Source-linked AI summary

Monte Carlo Motion Planning for Robot Trajectory Optimization Under Uncertainty

Lucas Janson, Edward Schmerling, Marco Pavone

arXiv:1504.08053v2cs.RO

TL;DR

Motion planning under uncertainty seeks low-cost paths that satisfy collision-probability constraints, while existing path-probability approximations can be inaccurate. MCMP combines variance-reduced Monte Carlo estimation with obstacle-inflation planning; simulations show computation can reach real-time speeds.

  • Problem

    Existing path collision-probability approximations can be quite inaccurate, while most motion-planning work assumes perfectly known robot states and deterministic actions.

    Source: §1 · pp. 2–3

  • Method

    MCMP uses variance-reduced Monte Carlo to estimate path collision probability and searches over obstacle inflation, solving a deterministic planning problem at each step.

    Source: §1 · p. 3

  • Results

    5–10 seconds total in 2–6-dimensional state spaces, excluding planning computation that may be cached offline; simulations also demonstrate computation at speeds amenable to real-time planning.

    Source: §7 · p. 14, §8 · p. 15

  • Takeaways & Limitations

    Monte Carlo estimates include a standard error, allowing users to assess estimate quality, unlike the described alternatives.

    Source: §7 · p. 14

    The paper's analysis is framed in a Gaussian setting.

    Source: §6 · p. 12

Abstract

from arXiv · show

This article presents a novel approach, named MCMP (Monte Carlo Motion Planning), to the problem of motion planning under uncertainty, i.e., to the problem of computing a low-cost path that fulfills probabilistic collision avoidance constraints. MCMP estimates the collision probability (CP) of a given path by sampling via Monte Carlo the execution of a reference tracking controller (in this paper we consider LQG). The key algorithmic contribution of this paper is the design of statistical variance-reduction techniques, namely control variates and importance sampling, to make such a sampling procedure amenable to real-time implementation. MCMP applies this CP estimation procedure to motion planning by iteratively (i) computing an (approximately) optimal path for the deterministic version of the problem (here, using the FMT* algorithm), (ii) computing the CP of this path, and (iii) inflating or deflating the obstacles by a common factor depending on whether the CP is higher or lower than a target value. The advantages of MCMP are threefold: (i) asymptotic correctness of CP estimation, as opposed to most current approximations, which, as shown in this paper, can be off by large multiples and hinder the computation of feasible plans; (ii) speed and parallelizability, and (iii) generality, i.e., the approach is applicable to virtually any planning problem provided that a path tracking controller and a notion of distance to obstacles in the configuration space are available. Numerical results illustrate the correctness (in terms of feasibility), efficiency (in terms of path cost), and computational speed of MCMP.

1 Introduction

The introduction motivates motion planning that accounts for uncertainty and presents MCMP as a way to estimate path collision probability and find low-cost paths subject to a safety tolerance. It frames variance-reduced Monte Carlo estimation and iterative obstacle inflation as the central contributions.

1 Introduction

  • Most motion-planning work assumes deterministic states and actions, whereas uncertainty can change path safety and quality in unstructured settings.The introduction identifies field robots and surgical robotic systems as examples.Source: §1 · p. 2
  • POMDP-based policy optimization can be computationally intensive online, so the paper instead plans open-loop trajectories and recomputes them in a receding-horizon fashion.The paper notes that the environment map may change after a short time horizon.Source: §1 · p. 2
  • Prior approaches either approximate path collision probability or require an explicit obstacle-space characterization, which can be unavailable.The paper argues that inaccurate approximations can mischaracterize the feasible domain of path optimization.Source: §1 · pp. 2–3
  • Its variance-reduced Monte Carlo estimator samples trajectories under a reference-tracking controller and is asymptotically exact, with an estimate of its own variance.The paper combines control variates and importance sampling, including for small collision probabilities.Source: §1 · p. 3
  • MCMP combines collision-probability estimation with deterministic planning and obstacle inflation or deflation to seek paths satisfying a target safety tolerance.The paper reports simulation results addressing feasibility, path cost, and computational speed.Source: §1 · p. 3

2 Background on Monte Carlo Variance Reduction

This section reviews Monte Carlo estimation and two variance-reduction techniques, then highlights the paper-specific challenge: choosing them so collision-probability estimates converge fast enough for real-time planning.

2 Background on Monte Carlo Variance Reduction

  • The section introduces Monte Carlo and variance reduction as background for the paper’s collision-probability estimation method.It points readers to a separate text for fuller coverage of Monte Carlo methods.Source: §2 · p. 4

2.1 Simple Monte Carlo

  • Simple Monte Carlo estimates an expectation by averaging function values over independent samples; here the function indicates whether a sampled trajectory collides, making its expectation the collision probability.The section also describes estimating uncertainty in the simple estimator from sample variance.Source: §2.1 · p. 4

2.2 Control Variates

  • Control variates reduce estimator variation by subtracting a scaled, correlated function whose expectation is known; asymptotically, the variance ratio is 1 − ρ^2.The scaling factor is estimated from simulated data, and its induced bias is asymptotically negligible relative to the variance.Source: §2.2 · pp. 4–5

2.3 Importance Sampling

  • For rare collision events, importance sampling draws from a distribution that makes collisions more common while retaining an unbiased estimate of the original probability.Simple Monte Carlo may require m ≫ 1/p samples for small relative uncertainty; a suitable sampling distribution can reduce variance substantially.Source: §2.3 · p. 5

2.4 Comments

  • Because there is no general rule for choosing a control variate or importance distribution, the paper’s contribution is to find choices for linear dynamics with Gaussian noise that support real-time estimation.Control variates and importance sampling can be combined in one estimator.Source: §2.4 · p. 6

3 Problem Statement

The problem statement formulates motion planning as path-cost optimization subject to a collision-probability safety constraint. It considers LQG tracking of nominal trajectories under linear dynamics, with discretized dynamics used for tractable probability computation.

3 Problem Statement

  • The paper considers linear dynamics with Gaussian process and measurement noise, and uses LQG-derived tracking policies for nominal trajectories.The nominal trajectories use continuous dynamics, while collision-probability computation uses discretized zero-order-hold approximate dynamics.Source: §3 · p. 6
  • The problem is to optimize a path cost while constraining the probability that the robot’s continuous random trajectory intersects the obstacle space.The formulation specifies initial and goal states in free space and discretizes the path when computing collision probability.Source: §3 · pp. 6–7
  • The formulation is a compromise between constrained POMDP optimization and open-loop planning; this paper restricts attention to LQG despite the stated potential for generalization.The authors justify the compromise by the greater complexity of constrained POMDPs and the need to replan when environment maps change.Source: §3 · p. 7
  • The subsequent estimation procedure aims to be asymptotically exact as the discretization step tends to zero, before MCMP generates approximate solutions to the discretized planning problem.Source: §3 · p. 7

4 The Problem of Computing Path CP

Common path-collision-probability approximations combine risks at waypoints, but they can miss collisions between waypoints, mishandle dependence, and approximate pointwise probabilities. Increasing waypoint count does not fix these problems: the additive estimate can diverge and the multiplicative estimate can approach 1 regardless of true path risk.

4 The Problem of Computing Path CP

  • Additive and multiplicative methods combine waypoint collision probabilities, but may miss between-waypoint collisions, mishandle dependence, and approximate individual waypoint probabilities.The additive method treats waypoint collisions as mutually exclusive; the multiplicative method treats them as independent, although neither assumption holds in general.Source: §4 · p. 7
  • As waypoint count grows, the additive estimate tends to infinity and the multiplicative estimate to 1, regardless of true path CP, potentially making every path infeasible under a fixed non-trivial constraint.Increasing waypoint resolution can improve between-waypoint coverage while worsening the dependence approximation.Source: §4 · pp. 7–8
  • Conditional multiplicative estimates can remain at least five times the true path CP, while standard additive and multiplicative estimates can be off by hundreds of percent.The reported discrepancies are especially relevant for small path CPs.Source: §4 · p. 8, Figure 2 · p. 9
  • For chance-constrained planning, CP estimates must be fast and accurate in absolute terms; additive estimates are conservative, while multiplicative estimates lack a finite-waypoint conservativeness guarantee.The paper notes that approximations may still measure relative CP between paths usefully in unconstrained problems.Source: §4 · p. 9
  • Monte Carlo estimates path CP by checking collisions along sampled tracked paths, accounting for waypoint dependence without relying on waypoint resolution or multivariate density integration.The paper says this addresses approximations (A)–(C).Source: §4 · pp. 9–10

5 Variance-Reduced Monte Carlo for Computing Pathwise CP

The estimator reduces Monte Carlo variance with a control variate based on fast CP approximations and importance sampling that steers samples toward collisions. The combined procedure is presented as Algorithm 1.

5.1 Control Variates

  • 5.1 Control Variates: Fast probability approximations serve as control variates because they can be computed quickly and act as correlated proxies for path CP.For a useful control variate, its expectation should be known or rapidly computable and it should be highly correlated with the target; tested correlations were regularly around 0.8.Source: §5.1 · p. 10

5.2 Importance Sampling

  • 5.2 Importance Sampling: A mixture of distributions shifts noise toward close obstacles at selected waypoints, making collisions more likely in samples while keeping likelihood ratios tractable.The shifts are distributed according to Mahalanobis distance to reduce likelihood-ratio variance.Source: §5.2 · pp. 10–11

5.3 Combining the Two Variance-Reduction Techniques

  • 5.3 Combining the Two Variance-Reduction Techniques: Algorithm 1 combines the control variate and importance distribution by sampling from Q and computing likelihood ratios P/Q.The passage introducing the algorithm says its full details are omitted.Source: §5.3 · p. 11

6 MCMP Algorithm

MCMP repeatedly solves a deterministic planning problem with inflated obstacles, estimates the resulting path’s collision probability, and adjusts the inflation by bisection toward a target safety level. The paper cautions that inaccurate CP approximations can cause unnecessary cost or even apparent infeasibility.

6 MCMP Algorithm

  • MCMP uses bisection to adjust obstacle inflation, solving the deterministic planning problem and estimating each candidate path’s CP against a target constraint.When estimated CP exceeds the target, the algorithm raises the lower inflation bound; otherwise, it lowers the upper bound.Source: §6 · pp. 11–12
  • Overestimating CP can produce costlier paths or inflate obstacles enough to close all routes to the goal, even when Monte Carlo estimation finds a feasible path.Underestimation by multiplicative approximations can instead violate the safety constraint.Source: §6 · pp. 11–12
  • In the paper’s Gaussian setting, a small-CP path may become much safer with only a slight increase in obstacle clearance, although extra inflation can close a homotopy class or make the problem infeasible.The paper notes that Gaussian tails fall off rapidly.Source: §6 · p. 12

7 Numerical Experiments

The experiments evaluate MCMP across linear dynamical systems and obstacle sets, with planning based on differential FMT* and cached data reducing online replanning time. They show runtime near real time, while highlighting how trajectory smoothing and homotopy changes affect the bisection procedure.

7 Numerical Experiments

  • The deterministic planning step used differential FMT* [23] on fixed nodes; Table 1 reports performance statistics averaged over 400 MCMP runs—20 runs each for 20 SBMP sample sets.The study implemented variance-reduced path CP estimation and MCMP in Julia [3].Source: §7 · p. 12, Table 1 · p. 14
  • Within a homotopy class, smoothing makes path CP vary continuously with inflation factor, but it may jump between classes.When CP decreases at a class transition, bisection can stall; block and backtrack blocks the riskier class and resets the lower bound, increasing computation time but returning a path at the goal CP.Source: §7 · p. 13, Figure 3 · p. 13
  • For double-integrator systems, unsmoothed trajectories make path CP piecewise constant with small discontinuities as the inflation factor changes.When bisection ends around the target CP, the selected path satisfies the safety constraint, explaining mean true CP below the goal in Table 1 rows 4 and 5.Source: §7 · p. 13
  • 5–10 seconds total: MCMP approached real-time runtimes in 2–6-dimensional state spaces, excluding planning computation that could be cached offline.Planning and probability-estimation times were similar; the few thousand simulations used would not let simple Monte Carlo certify even one path CP within (0.9%, 1.1%).Source: §7 · p. 14
  • The conditional multiplicative approximation stayed within a factor of 2 of true CP at the tested discretization, but could overestimate or underestimate it.Unlike the approximations, Monte Carlo provides a standard error that can indicate estimate accuracy.Source: §7 · p. 14

8 Conclusion

The paper presents variance-reduced Monte Carlo for pathwise collision-probability estimation and uses it in a chance-constrained planning meta-algorithm. Simulations report computation speeds amenable to real-time planning, while broader controllers, uncertainty models, and planning improvements remain future work.

8 Conclusion

  • The method combines a planning-specific control variate and importance distribution with Monte Carlo to estimate pathwise collision probability quickly and provably accurately.The estimation procedure is used in a simple meta-algorithm to generate low-cost paths that are not conservative with respect to a nominal path CP constraint.Source: §8 · p. 15
  • Simulation results confirm the theory and demonstrate computation at speeds amenable to real-time planning.Source: §8 · p. 15
  • Further work includes parallelization, broader controllers and uncertainty models, improved meta-algorithms, and applying Monte Carlo to other planning objectives.For heavier-tailed distributions, suitable control variates or importance distributions remain to be developed.Source: §8 · p. 15

A Problem Formulation

The formulation seeks a minimum-cost nominal path whose stochastic execution under a reference-tracking LQG controller avoids obstacles with high probability. It represents the stochastic trajectory in continuous and discrete time to support collision-probability evaluation.

A.1 Continuous-time formulation

  • The model assumes linear stochastic robot dynamics with Gaussian process and measurement noise, and defines free space as the complement of obstacle space.The initial state and goal region lie in free space.Source: §A.1 · p. 17
  • A nominal solution follows the deterministic system, and deviation variables describe differences between stochastic and nominal state, control, and output.Source: §A.1 · p. 17
  • The LQG controller minimizes a quadratic cost by combining LQR state feedback with a Kalman-filter estimate of state deviation.Source: §A.1 · p. 17

A Problem Formulation

  • The planning objective is a minimum-cost nominal path subject to a bound on the probability that its stochastic dynamics enter obstacle space.Source: §A.1 · p. 17

A.2 Discrete Approximation

  • The problem evaluates collision probability along the continuous curve traced by the random trajectory connecting the discrete states.Source: §A.2 · p. 18
  • The discrete approximation uses fixed-timestep linear dynamics with Gaussian process and measurement noise.The discrete LQG controller is specified through its feedback and filtering equations.Source: §A.2 · p. 18
  • The discrete formulation uses recursion to compute waypoint marginal distributions along the full LQG-controlled trajectory, starting from the initial mean and covariance.Source: §A.2 · p. 18
Loading 1504.08053v2…