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Modulation Strategies for Dynamical Systems - Part 1
- 1. Modulation Strategies for Dynamical Systems Part 1 0 Organizers/Speakers: Nadia Figueroa, Seyed Sina Mirrazavi Salehian, Lukas Huber, Aude Billard June 29th, 2018
- 2. Modulation Strategies 1 Non-contact/Contact Transitions Obstacle avoidance Local refinement Start with an estimate of ሶ𝑥 = 𝑓 𝑥 from a set of demonstrations We can change the behavior of the system by: ሶ𝑥 = 𝑀 𝑥 𝑓 𝑥
- 3. Local refinement Legend 𝑥 Robot’s state R(x) Rotation Matrix [𝜅(𝑥)] Scaling factor We define the modulation as scaling and rotation: Define the parameter vector: For D>2, the parameter vector is expanded by parameters describing the rotation set. General formulation 2
- 4. Local refinement From Trajectory Data to Reshaping Parameters Trajectory data Reshaping parameters Legend 𝑥 Robot’s state R(x) Rotation Matrix [𝜅(𝑥)] Scaling factor 𝑥 𝑚 Orignal data-points 𝑥 𝑚 𝑜 New data-points 3
- 5. Local refinement Learning and Using the Reshaped Dynamics Reshaping parameters Local regression Construct modulation Original dynamics Reshaped dynamics Test input • Original dynamics must have the desired qualitative properties: • Point attractor • Limit cycle • Multiple attractors • The detail comes from local modulations. Robot 4
- 9. Local refinement Resulting reshaped SEDS models Reshaped models 8
- 10. Modulation Strategies 9 Non-contact/Contact Transitions Obstacle avoidance Local refinement Start with an estimate of ሶ𝑥 = 𝑓 𝑥 from a set of demonstrations We can change the behavior of the system by: ሶ𝑥 = 𝑀 𝑥 𝑓 𝑥
- 11. Obstacle avoidance 10 Intuition Method Experiments Task Modulate the initial DS such that: (I) The algorithm is in closed-form mode, computationally inexpensive. (II) No penetration/touching of the obstacle occurs. (III) Convergence regions are maintained. Task objectives ➢ Dynamic adaptation to disturbances ➢ Normal velocity on surface is zero ➢ Convergence to attractor (start to stop point)
- 12. Initial dynamics with single attractor 𝑥 𝑜 Obstacle avoidance Intuition Method Experiments Dynamic Modulation Matrix [1] Khansari-Zadeh, S. M., & Billard, A. (2012). A dynamical system approach to realtime obstacle avoidance. Autonomous Robots, 32(4), 433-454. [1] Legend 𝑥 ∈ ℝ 𝑑 Robot’s relative state (𝑥 = 𝑥 − 𝑥 𝑜 ) 𝐸 𝑥 Decomposition matrix 𝑛 𝑥 Normal to obstacle 𝑡𝑖 𝑥 Tangent to obstacle Γ 𝑥 Distance Function 𝐷 𝑥 Eigenvalue matrix 𝜆(Γ) Eigenvalues General formulation 𝑓 𝑥 𝑥 𝑀 𝑥 = 𝐸 𝑥 𝐷 𝑥 𝐸 𝑥 −1 11 𝑓 𝑥ሶ𝑥 = 𝑀 𝑥 𝑓 𝑥
- 13. 𝑥 𝑜 Obstacle avoidance Intuition Method Experiments 𝐸 𝑥 = 𝑛 𝑥 𝑡1 𝑥 ⋯ 𝑡 𝑑−1 𝑥 Dynamic Modulation Matrix [1] Khansari-Zadeh, S. M., & Billard, A. (2012). A dynamical system approach to realtime obstacle avoidance. Autonomous Robots, 32(4), 433-454. [1] Legend 𝑥 ∈ ℝ 𝑑 Robot’s relative state (𝑥 = 𝑥 − 𝑥 𝑜 ) 𝐸 𝑥 Decomposition matrix 𝑛 𝑥 Normal to obstacle 𝑡𝑖 𝑥 Tangent to obstacle Γ 𝑥 Distance Function 𝐷 𝑥 Eigenvalue matrix 𝜆(Γ) Eigenvalues General formulation 𝑛 𝑥 𝑡 𝑥 𝑓 𝑥 Free space 𝛤 𝑥 ≥ 0 Boundary region 𝛤 𝑥 = 0 𝑥 𝑀 𝑥 = 𝐸 𝑥 𝐷 𝑥 𝐸 𝑥 −1 12 𝑓 𝑥ሶ𝑥 = 𝑀 𝑥 𝑓 𝑥
- 14. 𝜆0 𝛤 ≤ 1, 𝜆𝑖 𝛤 ≥ 1, lim 𝛤→∞ 𝜆𝑖 𝛤 = 1 𝑥 𝑜 Obstacle avoidance Intuition Method Experiments Dynamic Modulation Matrix [1] Khansari-Zadeh, S. M., & Billard, A. (2012). A dynamical system approach to realtime obstacle avoidance. Autonomous Robots, 32(4), 433-454. [1] Legend 𝑥 ∈ ℝ 𝑑 Robot’s relative state (𝑥 = 𝑥 − 𝑥 𝑜 ) 𝐸 𝑥 Decomposition matrix 𝑛 𝑥 Normal to obstacle 𝑡𝑖 𝑥 Tangent to obstacle Γ 𝑥 Distance Function 𝐷 𝑥 Eigenvalue matrix 𝜆(Γ) Eigenvalues General formulation 𝑛 𝑥 𝑡 𝑥 𝑓 𝑥 Free space 𝛤 𝑥 ≥ 0 Boundary region 𝛤 𝑥 = 0 𝐷 𝑥 = 𝜆0 0 ⋱ 0 𝜆 𝑑−1 Conditions ➢ Compression in normal direction ➢ Stretching in tangential direction 𝑖 = 1 … 𝑑 − 1 ➢ No effect far away 𝑖 = 1 … 𝑑 − 1 Stretching/compression to guide flow 𝑥 𝜆0 0 = 0 arg max 𝛤 𝜆𝑖 = 0 𝑀 𝑥 = 𝐸 𝑥 𝐷 𝑥 𝐸 𝑥 −1 13 𝑓 𝑥ሶ𝑥 = 𝑀 𝑥 𝑓 𝑥
- 15. Obstacle avoidance Intuition Method Experiments 1 ≤ 𝑖 ≤ 𝑑 − 1 Dynamic Modulation Matrix 𝜆0 𝛤 ≤ 1, 𝜆 𝑡 𝛤 ≥ 1, lim 𝛤→∞ 𝜆𝑖 𝛤 = 1 𝜆0 0 = 0 arg max 𝛤 𝜆 𝑡 = 0 𝐸 𝑥 = 𝑛 𝑥 𝑡1 𝑥 ⋯ 𝑡 𝑑−1 𝑥 𝐷 𝑥 = 𝜆0 0 ⋱ 0 𝜆 𝑑−1 ሶ𝑥 = 𝑀 𝑥 𝑓 𝑥General formulation 𝑓 𝑥 𝑥 𝑀 𝑥 = 𝐸 𝑥 𝐷 𝑥 𝐸 𝑥 −1 14 Legend 𝑥 ∈ ℝ 𝑑 Robot’s relative state (𝑥 = 𝑥 − 𝑥 𝑜 ) 𝐸 𝑥 Decomposition matrix 𝑛 𝑥 Normal to obstacle 𝑡𝑖 𝑥 Tangent to obstacle 𝐷 𝑥 Eigenvalue matrix 𝜆(Γ) Eigenvalues Γ 𝑥 Distance Function
- 16. Obstacle avoidance Intuition Method Experiments 𝜆0 𝛤 = 1 − 1 𝛤 𝑥 + 1 𝜆𝑖 𝛤 = 1 + 1 𝛤 𝑥 + 1 1 ≤ 𝑖 ≤ 𝑑 − 1 > Inspired by Potential Flow in Fluid Dynamics Dynamic Modulation Matrix ሶ𝑥 𝜆0 𝛤 ≤ 1, 𝜆 𝑡 𝛤 ≥ 1, lim 𝛤→∞ 𝜆𝑖 𝛤 = 1 𝜆0 0 = 0 arg max 𝛤 𝜆 𝑡 = 0 𝐸 𝑥 = 𝑛 𝑥 𝑡1 𝑥 ⋯ 𝑡 𝑑−1 𝑥 𝐷 𝑥 = 𝜆0 0 ⋱ 0 𝜆 𝑑−1 ሶ𝑥 = 𝑀 𝑥 𝑓 𝑥General formulation 𝑓 𝑥 𝑥 𝑀 𝑥 = 𝐸 𝑥 𝐷 𝑥 𝐸 𝑥 −1 15 Legend 𝑥 ∈ ℝ 𝑑 Robot’s relative state (𝑥 = 𝑥 − 𝑥 𝑜 ) 𝐸 𝑥 Decomposition matrix 𝑛 𝑥 Normal to obstacle 𝑡𝑖 𝑥 Tangent to obstacle 𝐷 𝑥 Eigenvalue matrix 𝜆(Γ) Eigenvalues Γ 𝑥 Distance Function
- 17. Modulated dynamics Obstacle avoidance Intuition Method Experiments 𝜆0 𝛤 = 1 − 1 𝛤 𝑥 + 1 𝜆𝑖 𝛤 = 1 + 1 𝛤 𝑥 + 1 1 ≤ 𝑖 ≤ 𝑑 − 1 > Inspired by Potential Flow in Fluid Dynamics Dynamic Modulation Matrix ሶ𝑥 𝜆0 𝛤 ≤ 1, 𝜆 𝑡 𝛤 ≥ 1, lim 𝛤→∞ 𝜆𝑖 𝛤 = 1 𝜆0 0 = 0 arg max 𝛤 𝜆 𝑡 = 0 𝐸 𝑥 = 𝑛 𝑥 𝑡1 𝑥 ⋯ 𝑡 𝑑−1 𝑥 𝐷 𝑥 = 𝜆0 0 ⋱ 0 𝜆 𝑑−1 ሶ𝑥 = 𝑀 𝑥 𝑓 𝑥General formulation 𝑓 𝑥 𝑥 𝑀 𝑥 = 𝐸 𝑥 𝐷 𝑥 𝐸 𝑥 −1 16 Legend 𝑥 ∈ ℝ 𝑑 Robot’s relative state (𝑥 = 𝑥 − 𝑥 𝑜 ) 𝐸 𝑥 Decomposition matrix 𝑛 𝑥 Normal to obstacle 𝑡𝑖 𝑥 Tangent to obstacle 𝐷 𝑥 Eigenvalue matrix 𝜆(Γ) Eigenvalues Γ 𝑥 Distance Function
- 18. Obstacle avoidance Intuition Method Experiments Orthogonal Modulation Matrix (OMM) - Implementation [1] Khansari-Zadeh, S. M., & Billard, A. (2012). A dynamical system approach to realtime obstacle avoidance. Autonomous Robots, 32(4), 433-454. [1] 17
- 19. Obstacle avoidance Intuition Method Experiments Orthogonal Eigenvector Matrix (OEM) - Limitations OEM around a circular obstacle with convergence of all but one trajectories. OEM around an elliptical obstacle where a local minima occurs on the right side. [1] Khansari-Zadeh, S. M., & Billard, A. (2012). A dynamical system approach to realtime obstacle avoidance. Autonomous Robots, 32(4), 433-454. [1] 18
- 20. Obstacle avoidance Intuition Method Experiments Orthogonal Eigenvector Matrix (OEM) 𝑐 𝑥 = − 𝑥 𝑥 [1] Legend 𝑥 Robot’s relative state (𝑥 = 𝑥 − 𝑥 𝑜 ) 𝐸 𝑥 Decomposition matrix 𝑛 𝑥 Normal to obstacle 𝑡𝑖 𝑥 Tangent to obstacle 𝐷 𝑥 Eigenvalue matrix Γ 𝑥 Distance Function 𝜆(Γ) Eigenvalues 𝑀 𝑥 = 𝐸 𝑥 𝐷 𝑥 𝐸 𝑥 −1 𝐸 𝑥 = 𝑛 𝑥 𝑡1 𝑥 ⋯ 𝑡 𝑑−1 𝑥 𝐷 𝑥 = 𝜆0 0 ⋱ 0 𝜆 𝑑−1 ሶ𝑥 = 𝑀 𝑥 𝑓 𝑥General formulation Legend 𝑥 ∈ ℝ 𝑑 Robot’s relative state (𝑥 = 𝑥 − 𝑥 𝑜 ) 𝐸 𝑥 Decomposition matrix 𝑐 𝑥 Center line to obstacle 𝑡𝑖 𝑥 Tangent to obstacle 𝐷 𝑥 Eigenvalue matrix 𝜆(Γ) Eigenvalues Γ 𝑥 Distance Function 𝑓 𝑥 ሶ𝑥 𝑥 19 OEM around an elliptical obstacle where a local minima occurs on the right side.
- 21. OEM around a cylindrical obstacle where a local minima occurs on the right side. Obstacle avoidance Intuition Method Experiments Center Based Eigenvector Matrix (CEM) 𝑐 𝑥 = − 𝑥 𝑥 The new decomposition matrix 𝐸 𝑥 is not orthogonal anymore, but needs to have full rank. [2] Huber L., Billard A. & Slotine J.-J. (in preparation) Convergence ensured through contraction for dynamical system based obstacle avoidance in concave environments. [2] Legend 𝑥 Robot’s relative state (𝑥 = 𝑥 − 𝑥 𝑜 ) 𝐸 𝑥 Decomposition matrix 𝑛 𝑥 Normal to obstacle 𝑡𝑖 𝑥 Tangent to obstacle 𝐷 𝑥 Eigenvalue matrix Γ 𝑥 Distance Function 𝜆(Γ) Eigenvalues 𝑀 𝑥 = 𝐸 𝑥 𝐷 𝑥 𝐸 𝑥 −1 𝐷 𝑥 = 𝜆0 0 ⋱ 0 𝜆 𝑑−1 ሶ𝑥 = 𝑀 𝑥 𝑓 𝑥General formulation Legend 𝑥 ∈ ℝ 𝑑 Robot’s relative state (𝑥 = 𝑥 − 𝑥 𝑜 ) 𝐸 𝑥 Decomposition matrix 𝑐 𝑥 Center line to obstacle 𝑡𝑖 𝑥 Tangent to obstacle 𝐷 𝑥 Eigenvalue matrix 𝜆(Γ) Eigenvalues Γ 𝑥 Distance Function 𝑓 𝑥 ሶ𝑥 𝑥 20 CEM around an elliptical obstacle where a all but one trajectories converge to the attractor. 𝐸 𝑥 = 𝑐 𝑥 𝑡1 𝑥 ⋯ 𝑡 𝑑−1 𝑥
- 22. (a) Convex obstacles Obstacle avoidance Intuition Method Experiments Center Based Eigenvector Matrix (CEM) – Obstacle Shapes CBM can avoid obstacles as long as there exists a reference point from which there exists only one boundary point in each direction. (b) Star-shaped obstacles (c) Intersecting convex obstacles with common region (d) Intersecting obstacles without common region [2]
- 23. Obstacle avoidance Intuition Method Experiments Center Based Eigenvector Matrix (CEM) – Multiple Obstacles 22 > Weighted interpolation of corresponding modulated system of the orientation and magnitude separately 𝑥1 𝑜 ሶ𝑥1 𝜑 1 𝑓 𝑥 [2] Huber L., Billard A. & Slotine J.-J. (in preparation) Convergence ensured through contraction for dynamical system based obstacle avoidance in concave environments. [2]
- 24. Obstacle avoidance Intuition Method Experiments Center Based Eigenvector Matrix (CEM) – Multiple Obstacles 23 > Weighted interpolation of corresponding modulated system of the orientation and magnitude separately 𝜑 2 ሶ𝑥2 𝑓 𝑥 [2] Huber L., Billard A. & Slotine J.-J. (in preparation) Convergence ensured through contraction for dynamical system based obstacle avoidance in concave environments. [2]
- 25. Obstacle avoidance Intuition Method Experiments Center Based Eigenvector Matrix (CEM) – Multiple Obstacles 24 > Weighted interpolation of corresponding modulated system of the orientation and magnitude separately 𝑥1 𝑜 ሶ𝑥1 𝜑 1 𝜑 2 ሶ𝑥2 𝑓 𝑥 [2] Huber L., Billard A. & Slotine J.-J. (in preparation) Convergence ensured through contraction for dynamical system based obstacle avoidance in concave environments. [2]
- 26. Obstacle avoidance Intuition Method Experiments Center Based Eigenvector Matrix (CEM) – Multiple Obstacles 25 > Weighted interpolation of corresponding modulated system of the orientation and magnitude separately [2] Huber L., Billard A. & Slotine J.-J. (in preparation) Convergence ensured through contraction for dynamical system based obstacle avoidance in concave environments. [2] ሶҧ𝑥 𝑓 𝑥
- 27. Obstacle avoidance Intuition Method Experiments Center Based Eigenvector Matrix (CEM) – Moving Obstacles 26 > Modulation is applied partially in moving frame to avoid penetration of boundary, but also keeping attractor position close to the original one. [2] Huber L., Billard A. & Slotine J.-J. (in preparation) Convergence ensured through contraction for dynamical system based obstacle avoidance in concave environments. [2]
- 28. Obstacle avoidance Intuition Method Experiments Center Based Eigenvector Matrix (CEM) – Moving Obstacles 27 > Modulation is applied partially in moving frame to avoid penetration of boundary, but also keeping attractor position close to the original one. [2] Huber L., Billard A. & Slotine J.-J. (in preparation) Convergence ensured through contraction for dynamical system based obstacle avoidance in concave environments. [2]