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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.
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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 • How can we model the deformation properties of objects? • How can the robot consider this information when planning its motions?
Planning with Deformation Cost • Estimating deformation is possible with finite element simulations • Manipulator planning: high-dimensional state space needs to be considered • Problem: too slow for online planning • Challenge: fast estimation of the deformation cost for manipulation robots • Our approach: • Define a subset of possible motions and simulate the deformations before planning (training data) • Estimate the cost of new motions by regression
Planning Framework • Generate a Probabilistic roadmap (PRM) for the rigid part of the environment • Search for a path using and trade off path- and deformation cost: Combination of motion planning and physically realistic deformation simulation:
Planning Framework • Generate a Probabilistic roadmap (PRM) for the rigid part of the environment • Search for a path using and trade off path- and deformation cost: Combination of motion planning and physically realistic deformation simulation: Euclidean distance inconfiguration space
Planning Framework • Generate a Probabilistic roadmap (PRM) for the rigid part of the environment • Search for a path using and trade off path- and deformation cost: Combination of motion planning and physically realistic deformation simulation: Euclidean distance inconfiguration space Deformation simulation
Dynamic Simulation of Deformable Objects Deformable modeling: • 3D-tetrahedral model • Finite Element Method Simulation framework: • Collision detection • Collision response Deformation simulations are costly and not suitable for online planning
Approximation & Assumptions • Our approach estimates the deformation cost based on training examples Assumptions • Obstacles are deformed but do not move • Ignore interactions between different objects • Consider only linear trajectories • Deformation cost depend only on the arm trajectory relative to an object and the material of the object
Deformation Cost Estimation • Given a set of sample trajectories and corresponding deformation cost values • Learn a predictive modelfor estimating the deformation costof 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 variance A covariance function models the influence of the data points on the query point variance mean training data
Gaussian Processes (GPs) Non-parametric model Covariance function: squared exponential … but the covariance function requires hyperparameters Learning the hyperparameters by maximizing the likelihood of the training data Popular: maximization via gradient methods Problem: significant cost of learning the GP from data
Problem Decomposition • We need many samples to accurately approximate the deformation cost • Problem: GP learning has cubic runtime complexity in the number of samples due to matrix inversion Approximation • Store all samples in a KD-tree for efficient organization and nearest neighbor queries • Select only trajectory samples that are “close” to build the GP
Nearest Neighbor Approximation • For 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 trajectory Trajectory distance function:
Considering the Kinematic Chain Simulation considers only the movement of the end-effector when generating samples • Consider the trajectories of different body parts (wrist, elbow, …) • Estimate the deformation cost of these trajectories using GP regression • Deformation cost of an edge in the roadmap: maximum of the individual trajectories Wrist trajectory End-effector trajectory
Evaluation: Prediction • Compare 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: Prediction • Compare 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: Performance • Long preprocessing, but only once per object • Independent of the environment • Speedup of 2 orders of magnitude during roadmap computation + query time Runtime requirements compared to a planner with integrated simulation:
Motion Planning Example Shortest path Trade-off between path cost and deformation cost
Motion Planning Example Shortest path Trade-off between path cost and deformation cost
Related Work • Planning 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]
Conclusion • Novel approach to manipulator motion planning considering deformable obstacles • Efficient estimation of the deformation cost along a trajectory using Gaussian process regression • GP training using a deformation simulation based on finite element method • Experiments illustrate an accurate cost estimation and online planning capabilities