1. 4/15/2017
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. 4/15/2017
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
Combination of motion planning and physically realistic deformation simulation:
Generate a Probabilistic roadmap (PRM) for the rigid part of the environment
Search for a path using and trade off path- and deformation cost:

5. Planning Framework
Combination of motion planning and physically realistic deformation simulation:
Generate a Probabilistic roadmap (PRM) for the rigid part of the environment
Search for a path using and trade off path- and deformation cost:
Euclidean distance in configuration space

6. Planning Framework
Combination of motion planning and physically realistic deformation simulation:
Generate a Probabilistic roadmap (PRM) for the rigid part of the environment
Search for a path using and trade off path- and deformation cost:
Euclidean distance in configuration space
Deformation simulation

7. Dynamic Simulation of Deformable Objects
4/15/2017
Dynamic Simulation of Deformable Objects
Deformable modeling:
3D-tetrahedral model
Finite Element Method
Simulation framework:
Collision detection
Collision response
Simulation engine:
Quick introduction
Tetrahedral 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
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 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

10. Gaussian Processes (GPs)
4/15/2017
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 10 10

11. Gaussian Processes (GPs)
4/15/2017
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 11 11

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

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

17. Evaluation: Performance
Runtime requirements compared to a planner with integrated simulation:
Preprocessing simulations
Roadmap computation
Answering path queries
Planner with integrated simulation -
307min
(267min simulation)
10min
(9.7min simulation)
Planner with our GP-based estimation
~ 36h
42min
(2min GP- evaluation)
5.3s (1.8s GP- evaluation)
Long preprocessing, but only once per object
Independent of the environment
Speedup of 2 orders of magnitude during roadmap computation + query time

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!