Barbara Frank, Cyrill Stachniss, Nichola Abdo, Wolfram Burgard University of Freiburg, Germany

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Using Gaussian Process Regression for Efficient Motion Planning in Environments with Deformable Objects. Barbara Frank, Cyrill Stachniss, Nichola Abdo, Wolfram Burgard University of Freiburg, Germany. Motivation. Enable a robot to consider deformable obstacles when planning its motions. - PowerPoint PPT Presentation

Transcript of Barbara Frank, Cyrill Stachniss, Nichola Abdo, Wolfram Burgard University of Freiburg, Germany

  • Barbara Frank, Cyrill Stachniss, Nichola Abdo, Wolfram Burgard

    University of Freiburg, GermanyUsing Gaussian Process Regression for Efficient Motion Planning in Environments with Deformable Objects

  • MotivationHow can we model the deformation properties of objects?How can the robot consider this information when planning its motions?Enable a robot to consider deformable obstacles when planning its motions

  • Planning with Deformation CostEstimating deformation is possible with finite element simulations Manipulator planning: high-dimensional state space needs to be consideredProblem: too slow for online planningChallenge: fast estimation of the deformation cost for manipulation robotsOur approach:Define a subset of possible motions and simulate the deformations before planning (training data)Estimate the cost of new motions by regression

  • Planning FrameworkGenerate a Probabilistic roadmap (PRM) for the rigid part of the environmentSearch for a path using and trade off path- and deformation cost:Combination of motion planning and physically realistic deformation simulation:

  • Planning FrameworkGenerate a Probabilistic roadmap (PRM) for the rigid part of the environmentSearch for a path using and trade off path- and deformation cost:Euclidean distance in configuration spaceCombination of motion planning and physically realistic deformation simulation:

  • Planning FrameworkGenerate a Probabilistic roadmap (PRM) for the rigid part of the environmentSearch for a path using and trade off path- and deformation cost:Euclidean distance in configuration spaceDeformation simulationCombination of motion planning and physically realistic deformation simulation:

  • Dynamic Simulation of Deformable ObjectsDeformable modeling: 3D-tetrahedral modelFinite Element Method

    Simulation framework:Collision detectionCollision response

    Deformation simulations are costly and not suitable for online planning

  • Approximation & AssumptionsOur approach estimates the deformation cost based on training examples

    AssumptionsObstacles are deformed but do not moveIgnore interactions between different objectsConsider only linear trajectoriesDeformation cost depend only on the arm trajectory relative to an object and the material of the object

  • Deformation Cost EstimationGiven a set of sample trajectories and corresponding deformation cost values

    Learn a predictive model for estimating the deformation cost of a new query trajectory

    Trajectory parametrization:Starting point on a sphere End point on a sphere Traveled distance

  • Gaussian Processes (GPs)GPs are a framework for non-parametric regression Model the data points (here deformation cost) as jointly Gaussian

    Predictive model for an input trajectory:

    Provides a mean and a predictive varianceA covariance function models the influence of the data points on the query point

  • Gaussian Processes (GPs)Non-parametric modelCovariance function: squared exponential

    but the covariance function requires hyperparameters

    Learning the hyperparameters by maximizing the likelihood of the training data

    Popular: maximization via gradient methodsProblem: significant cost of learning the GP from data

  • Problem DecompositionWe need many samples to accurately approximate the deformation costProblem: GP learning has cubic runtime complexity in the number of samples due to matrix inversion

    ApproximationStore all samples in a KD-tree for efficient organization and nearest neighbor queriesSelect only trajectory samples that are close to build the GP

  • Nearest Neighbor ApproximationFor each query trajectory, find the n closest neighbors from the training data (KD-tree)Train a local GP Similar to setting for training data far away from the query trajectoryTrajectory distance function:

  • Considering the Kinematic ChainSimulation considers only the movement of the end-effector when generating samplesConsider the trajectories of different body parts (wrist, elbow, )Estimate the deformation cost of these trajectories using GP regressionDeformation cost of an edge in the roadmap: maximum of the individual trajectoriesEnd-effector trajectoryWrist trajectory

  • Evaluation: PredictionCompare nearest-neighbor prediction (NN), GP with unit hyperparameters (GPStd), and GP with optimized hyperparameters (GPOpt)Leave-one-out cross validation:Predictive accuracy of deformation cost estimation:

  • Evaluation: PredictionCompare nearest-neighbor prediction (NN), GP with unit hyperparameters (GPStd), and GP with optimized hyperparameters (GPOpt)Cross validation D2 on D1:Predictive accuracy of deformation cost estimation:

  • Evaluation: PerformanceLong preprocessing, but only once per objectIndependent of the environmentSpeedup of 2 orders of magnitude during roadmap computation + query timeRuntime requirements compared to a planner with integrated simulation:

    Preprocessing simulationsRoadmap computationAnswering path queriesPlanner with integrated simulation -307min(267min simulation)10min(9.7min simulation)Planner with our GP-based estimation ~ 36h42min(2min GP- evaluation)5.3s (1.8s GP- evaluation)

  • Motion Planning ExampleTrade-off between path cost and deformation costShortest path

  • Motion Planning ExampleTrade-off between path cost and deformation costShortest path

  • Related WorkPlanning for deformable robots: [Kavraki et al. 98/00, Bayazit et al. 02, Gayle et al. 05]Planning in completely deformable environments: [Rodriguez et al. 06, Patil et al. 11]Application: medical simulation [Maris et al. 10, Alterovitz et al. 09]GP NN approximation for terrain modeling [Vasudevan et al. 09]

  • ConclusionNovel approach to manipulator motion planning considering deformable obstaclesEfficient estimation of the deformation cost along a trajectory using Gaussian process regressionGP training using a deformation simulation based on finite element methodExperiments illustrate an accurate cost estimation and online planning capabilities

  • Thanks for Your Attention!

    Simulation engine:Quick introductionTetrahedral model, linear elastic material, hookes law,Inner energy, that is computed, is a measure for how deformed an object is, and is our measure for deformation cost

    In action: video, demonstrates collision detection and collision response in each timestep for many tetrahedrons****