Gaussian Processes as Continuous-time Trajectory Representations: Applications in SLAM and Motion Planning
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1 Gaussian Processes as Continuous-time Trajectory Representations: Applications in SLAM and Motion Planning Jing Dong License CC BY-NC-SA 3.0
2 Discrete time SLAM Downsides: Measurements distorted by motions Asynchronous measurements Not a compact representation Rolling shutter effect GPS: 5Hz Camera: 3.75Hz Asynchronous data C. H. Tong, P. Furgale, and T. D. Barfoot, Gaussian process gauss newton for non-parametric simultaneous localization and mapping, Intl. J. of Robotics Research, vol. 32, no. 5, pp , 2013.
3 Discrete time SLAM Downsides: Measurements distorted by motions Asynchronous measurements Not a compact representation Continuous-time representation Linear interpolation Splines Wavelets Gaussian process C. H. Tong, P. Furgale, and T. D. Barfoot, Gaussian process gauss newton for non-parametric simultaneous localization and mapping, Intl. J. of Robotics Research, vol. 32, no. 5, pp , 2013.
4 Outlines GP as continuous-time trajectory representation Extend sparse GP to Lie groups Use sparse GP in SLAM Use sparse GP in motion planning
5 Outlines GP as continuous-time trajectory representation Extend sparse GP to Lie groups Use sparse GP in SLAM Use sparse GP in motion planning
6 GP as 1D continuous-time trajectory with uncertainty X x=f(t) X t Rasmussen, Carl Edward. "Gaussian processes for machine learning." (2006).
7 GP with noise-free measurements X x=f(t) t=1s t*=1.5s? t=2s X t*? t Measurements Query Points Rasmussen, Carl Edward. "Gaussian processes for machine learning." (2006).
8 GP with noise-free measurements Rasmussen, Carl Edward. "Gaussian processes for machine learning." (2006).
9 GP with noisy measurements X x=f(t) t=1s t*=1.5s? t=2s X t*? t Measurement function: Measurements Query Points Rasmussen, Carl Edward. "Gaussian processes for machine learning." (2006).
10 GP with noisy measurements X x=f(t) t=1s? t=1.5s? t=2s? X Measurement function: t Measurements and Query points Measurement and Query share same time stamps: Rasmussen, Carl Edward. "Gaussian processes for machine learning." (2006).
11 Another perspective... Measurement function: Mean value: Maximum A Posteriori Estimation Sherman Morrison Woodbury formula Rasmussen, Carl Edward. "Gaussian processes for machine learning." (2006).
12 Nonlinear measurement function and high DOF... Problem definition: Mean value: MAP Estimation Gauss-Newton method T. Barfoot, C. H. Tong, and S. Sarkka, Batch continuous-time trajectory estimation as exactly sparse gaussian process regression, Robotics: Science and Systems (RSS), 2014.
13 Sparsity and Efficiency Gauss-Newton step Inverse Kernel (Prior) Measurements (Likelihood) Particular kernel is needed to make sparsity T. Barfoot, C. H. Tong, and S. Sarkka, Batch continuous-time trajectory estimation as exactly sparse gaussian process regression, Robotics: Science and Systems (RSS), 2014.
14 Sparse information matrix Information matrix: Generally the inverse kernel matrix is not sparse Trajectory Landmarks C. H. Tong, P. Furgale, and T. D. Barfoot, Gaussian process gauss newton for non-parametric simultaneous localization and mapping, Intl. J. of Robotics Research, vol. 32, no. 5, pp , 2013.
15 GPs generated by LTV-SDEs linear time-varying stochastic differential equations (LTV-SDEs) solution: Inverse kernel matrix is tridiagonal block-wise sparse if the GP is generated by a LTV-SDE T. Barfoot, C. H. Tong, and S. Sarkka, Batch continuous-time trajectory estimation as exactly sparse gaussian process regression, Robotics: Science and Systems (RSS), 2014.
16 Constant velocity LTV-SDEs V: change slowly X Inject white noise in acceleration Rewrite LTV-SDE T. Barfoot, C. H. Tong, and S. Sarkka, Batch continuous-time trajectory estimation as exactly sparse gaussian process regression, Robotics: Science and Systems (RSS), 2014.
17 Factor Graph view Trajectory Landmarks Tri-diagonal inverse kernel T. Barfoot, C. H. Tong, and S. Sarkka, Batch continuous-time trajectory estimation as exactly sparse gaussian process regression, Robotics: Science and Systems (RSS), Dong, Jing, Byron Boots, and Frank Dellaert. "Sparse Gaussian Processes for Continuous-Time Trajectory Estimation on Matrix Lie Groups." arxiv preprint arxiv: (2017).
18 Const-time Interpolation Any time could be interpolated by nearby two states Time complexity O(1) Measurements at any time fused in graph by interpolated factor T. Barfoot, C. H. Tong, and S. Sarkka, Batch continuous-time trajectory estimation as exactly sparse gaussian process regression, Robotics: Science and Systems (RSS), Dong, Jing, Byron Boots, and Frank Dellaert. "Sparse Gaussian Processes for Continuous-Time Trajectory Estimation on Matrix Lie Groups." arxiv preprint arxiv: (2017).
19 Results 2D SLAM cases,
20 Outlines GP as continuous-time trajectory representation Extend sparse GP to Lie groups Use sparse GP in SLAM Use sparse GP in motion planning
21 Example Lie Group: SO(2) θ S. Anderson and T. D. Barfoot, Full STEAM ahead: Exactly sparse gaussian process regression for batch continuous-time trajectory estimation on SE (3), (IROS), Dong, Jing, Byron Boots, and Frank Dellaert. "Sparse Gaussian Processes for Continuous-Time Trajectory Estimation on Matrix Lie Groups." arxiv preprint arxiv: (2017).
22 Gaussian Distribution on SO(2) Lie Group: SO(2) Tangent space: so(2) \in R^2 S. Anderson and T. D. Barfoot, Full STEAM ahead: Exactly sparse gaussian process regression for batch continuous-time trajectory estimation on SE (3), (IROS), Dong, Jing, Byron Boots, and Frank Dellaert. "Sparse Gaussian Processes for Continuous-Time Trajectory Estimation on Matrix Lie Groups." arxiv preprint arxiv: (2017).
23 Gaussian Distribution on SO(2) Expmap: so(2) -> SO(2) Logmap: SO(2) -> so(2) Lie Group: SO(2) Tangent space: so(2) \in R^2 S. Anderson and T. D. Barfoot, Full STEAM ahead: Exactly sparse gaussian process regression for batch continuous-time trajectory estimation on SE (3), (IROS), Dong, Jing, Byron Boots, and Frank Dellaert. "Sparse Gaussian Processes for Continuous-Time Trajectory Estimation on Matrix Lie Groups." arxiv preprint arxiv: (2017).
24 Gaussian Distribution on SO(2) Expmap: so(2) -> SO(2) Logmap: SO(2) -> so(2) Lie Group: SO(2) Tangent space: so(2) \in R^2 We define Gaussian on Lie group by defining Gaussian on tangent space, then map back by exponential map Note that tangent space is defined on a particular θ on Lie group S. Anderson and T. D. Barfoot, Full STEAM ahead: Exactly sparse gaussian process regression for batch continuous-time trajectory estimation on SE (3), (IROS), Dong, Jing, Byron Boots, and Frank Dellaert. "Sparse Gaussian Processes for Continuous-Time Trajectory Estimation on Matrix Lie Groups." arxiv preprint arxiv: (2017).
25 Local GP on SO(2) Ti Local GP range of Ti S. Anderson and T. D. Barfoot, Full STEAM ahead: Exactly sparse gaussian process regression for batch continuous-time trajectory estimation on SE (3), (IROS), Dong, Jing, Byron Boots, and Frank Dellaert. "Sparse Gaussian Processes for Continuous-Time Trajectory Estimation on Matrix Lie Groups." arxiv preprint arxiv: (2017).
26 Local GP on SO(2) Ti ti < t < ti+1 range of ξi(t) We define a local linear variable ξi(t) around Ti, which locally meets a zero-mean Gaussian process defined by kernel K(ti, t) S. Anderson and T. D. Barfoot, Full STEAM ahead: Exactly sparse gaussian process regression for batch continuous-time trajectory estimation on SE (3), (IROS), Dong, Jing, Byron Boots, and Frank Dellaert. "Sparse Gaussian Processes for Continuous-Time Trajectory Estimation on Matrix Lie Groups." arxiv preprint arxiv: (2017).
27 Constant Velocity LVT-SDE Const body frame velocity SDE: nonlinear Equivalence to LTV-SDE on loca variable Local constant velocity LTV-SDE S. Anderson and T. D. Barfoot, Full STEAM ahead: Exactly sparse gaussian process regression for batch continuous-time trajectory estimation on SE (3), (IROS), Dong, Jing, Byron Boots, and Frank Dellaert. "Sparse Gaussian Processes for Continuous-Time Trajectory Estimation on Matrix Lie Groups." arxiv preprint arxiv: (2017).
28 Outlines GP as continuous-time trajectory representation Extend sparse GP to Lie groups Use sparse GP in SLAM Use sparse GP in motion planning
29 Background Application: autonomous crop monitoring using UAV / UGV Get crop height / size / color / etc. Requirement: use low-cost sensors: camera, GPS, etc. Existing techniques: Structure from Motion (SfM): 3D reconstruction Difficulties: dynamic scene with crop growing: 4D reconstruction
30 Dataset Dataset: color images + GPS + IMU, on ground vehicle and UAV Ground dataset: Twice per week for 3 months: total 23 sessions With ground truth height / leaf chlorophyll at multiple sampling sites 50m Dong, Jing, et al. "4D Crop Monitoring: Spatio-Temporal Reconstruction for Agriculture." ICRA (2017).
31 Approach 3D Reconstruction GPS IMU 4D Reconstruction Point cloud analysis Dong, Jing, et al. "4D Crop Monitoring: Spatio-Temporal Reconstruction for Agriculture." ICRA (2017).
32 1. Multi-Sensor Fusion SLAM Multi-sensor (camera, IMU, GPS) fusion Using factor graph and maximum a posterior (MAP) estimation Optional dense reconstruction by PMVS 3D factor graph Dong, Jing, et al. "4D Crop Monitoring: Spatio-Temporal Reconstruction for Agriculture." ICRA (2017).
33 2. 4D Data Association and Reconstruction View-invariant feature matching by bounded search and homography warping Spatio-temporal (4D) factor graph optimization View-invariant robust feature matching Day i Day i+1 Spatio-temporal (4D) factor graph Dong, Jing, et al. "4D Crop Monitoring: Spatio-Temporal Reconstruction for Agriculture." ICRA (2017).
34 Results Dong, Jing, et al. "4D Crop Monitoring: Spatio-Temporal Reconstruction for Agriculture." ICRA (2017).
35 Outlines GP as continuous-time trajectory representation Extend sparse GP to Lie groups Use sparse GP in SLAM Use sparse GP in motion planning
36 Introduction Planning on high DOF system is difficult Sampling based approaches like RRT and RPM are slow due to high DOF State-of-the-art approaches are mostly optimization based Energy reduction (Zucker et al, 2013 IJRR, Ratliff et al, 2009 ICRA) Constrained optimization (Schluman et al, 2013 RSS) Factor Graph and message passing (Toussaint et al, 2010 Book) Main Contributions: Formulate motion planning as continuous trajectory estimation by GP Enable efficient motion planning by sparse GP and isam2
37 Motion Planning as Probabilistic inference Two requirements of motion planning: Start Feasibility: trajectory is collision-free Optimality: smooth output by reducing the energy used or trajectory length End Calculate the trajectory by maximize the probability Prior distribution: enforce smoothness Collision-free likelihood: enforce feasibility Dong, Jing, et al. "Motion Planning as Probabilistic Inference using Gaussian Processes and Factor Graphs." Robotics: Science and Systems (RSS) (2016).
38 Collision-free Likelihood Collision-free likelihood is calculated at each discretized points of trajectory Collision-free likelihood is defined by exponential family Start End Obstacle cost function is defined by hinge loss function on signed distance Signed distance field is pre-calculated Robot body use sphere model to approximate Dong, Jing, et al. "Motion Planning as Probabilistic Inference using Gaussian Processes and Factor Graphs." Robotics: Science and Systems (RSS) (2016).
39 GP prior and Factor graph Trajectory is represented by a GP Trajectory prior use constant velocity GP prior Additional prior on start/goal configuration (Since they are known) Inference problem represented by a factor graph Dong, Jing, et al. "Motion Planning as Probabilistic Inference using Gaussian Processes and Factor Graphs." Robotics: Science and Systems (RSS) (2016).
40 Duality between inference and energy minimization Trajectory Prior Collision-free Likelihood Smoothness Cost Collision Cost Dong, Jing, et al. "Motion Planning as Probabilistic Inference using Gaussian Processes and Factor Graphs." Robotics: Science and Systems (RSS) (2016).
41 Efficient planning: Use GP interpolation GP enable us to query any time of interest So we can check collision cost at more points with less states to optimize A single query state is interpolated by near by two states Interpolated collision cost factor is binary Dong, Jing, et al. "Motion Planning as Probabilistic Inference using Gaussian Processes and Factor Graphs." Robotics: Science and Systems (RSS) (2016).
42 Evaluation: Batch planning Dataset: 7-DOF WAM arm and 7-DOF PR2, total 114 problems Criteria: Success rate, average run-time, maximum run-time Dong, Jing, et al. "Motion Planning as Probabilistic Inference using Gaussian Processes and Factor Graphs." Robotics: Science and Systems (RSS) (2016).
43 Efficient planning: Replanning Example replanning problem: given current state and a new goal Naive approach: plan from scratch given current (as start) and goal states Start Current state New Goal Fact: planning problem changes little Original Goal Constraint on current state Goal changes Current state Dong, Jing, et al. "Motion Planning as Probabilistic Inference using Gaussian Processes and Factor Graphs." Robotics: Science and Systems (RSS) (2016). Mukadam, Mustafa, et al. "Simultaneous Trajectory Estimation and Planning via Probabilistic Inference." Robotics: Science and Systems (RSS) (2017).
44 Efficient replanning: Bayes tree and isam2 Original graph Goal changes Constraint on current state
45 Evaluation: Replanning Dataset: 7-DOF WAM arm and 7-DOF PR2, total 60 problems Dong, Jing, et al. "Motion Planning as Probabilistic Inference using Gaussian Processes and Factor Graphs." Robotics: Science and Systems (RSS) (2016).
46 Evaluation: Replanning Dataset: 7-DOF WAM arm and 7-DOF PR2, total 60 problems doc/pics/wam_replan_2.gif doc/pics/wam_replan_1.gif Dong, Jing, et al. "Motion Planning as Probabilistic Inference using Gaussian Processes and Factor Graphs." Robotics: Science and Systems (RSS) (2016).
47 Real-time Replanning on Robots Mukadam, Mustafa, et al. "Simultaneous Trajectory Estimation and Planning via Probabilistic Inference." Robotics: Science and Systems (RSS) (2017).
48 Thanks!
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