barbara frank, cyrill stachniss, nichola abdo, wolfram burgard university of freiburg, germany
DESCRIPTION
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 PresentationTRANSCRIPT
Barbara Frank, Cyrill Stachniss,
Nichola Abdo, Wolfram Burgard
University of Freiburg, Germany
Using Gaussian Process Regression for Efficient Motion Planning
in Environments with Deformable Objects
Motivation
How 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 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:
Euclidean distance inconfiguration space
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:
Euclidean distance inconfiguration space
Deformation simulation
Combination of motion planning and physically realistic 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 model
for 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
meantrainingdata
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
End-effector trajectory
Wrist 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
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
Runtime requirements compared to a planner with integrated simulation:
Motion Planning Example
Trade-off between path cost and deformation cost
Shortest path
Motion Planning Example
Trade-off between path cost and deformation cost
Shortest path
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
Thanks for Your Attention!