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

1. 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

2. 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?

3. 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

4. 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:

5. 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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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:

14. 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

15. 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:

16. 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:

17. 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:

18. Motion Planning Example Shortest path Trade-off between path cost and deformation cost

19. Motion Planning Example Shortest path Trade-off between path cost and deformation cost

20. 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]

21. 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

22. Thanks for Your Attention!