Modulation Strategies for Dynamical Systems-part 2
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In this presentation, we provide an introduction on modulation strategies for dynamical systems. This presentation is mainly converse the approach proposed for non-contact/contact transitions. You can find the paper here: lasa.epfl.ch/publications/uploadedFiles/Sina_Mirrazavi.pdf There is toy example which you can find here: https://github.com/sinamr66/CoDS_illustrative_example/tree/GUI_guid and C++ SDK: https://github.com/sinamr66/CoDs_SDK You can also find the main slides with animations and playable videos here: https://github.com/epfl-lasa/RSS2018Tutorial/blob/master/Presentations/Modulation%20-%20Part_2.pptx and an example of the implementation here: https://www.youtube.com/watch?v=fhfBBMH4XVg For more information contact Sina.Mirrazavi@epfl.ch
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Modulation Strategies for Dynamical Systems-part 2
- 1. Modulation Strategies for Dynamical Systems Part 2 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 Locally 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. Non-contact/Contact Transitions 2 Intuition Method Experiments Demonstration Learning Execution RealityExpected Mirrazavi Salehian, S. S. and Billard, A. (2018) A Dynamical System Based Approach for Controlling Robotic Manipulators During Non-contact/Contact Transitions. RA-L
- 4. Non-contact/Contact Transitions 3 Intuition Method Experiments Task: Endow the robotic system with a controller such that: (I) Ensuring stable contact, (II) At a desired location, (III) Leaving on the surface at a desired point Task objectives ➢The impact happens only once ➢The contact and departure points Mirrazavi Salehian, S. S. and Billard, A. (2018) A Dynamical System Based Approach for Controlling Robotic Manipulators During Non-contact/Contact Transitions. RA-L
- 5. Task objectives ➢The impact happens only once ➢The contact and departure points Task objectives ➢The impact happens only once: |𝑞1 𝑇 ሶ𝑥 𝑡∗ | ≤ 𝛿 Non-contact/Contact Transitions Intuition Method Experiments Legend 𝑥 Robot’s state Γ(x) The surface contact 𝑥 𝑐 Desired contact point 𝑥 𝑙 Desired departure point 𝑥 𝑐 𝑥 𝑙 Contact surface ➢ Non-penetrable, passive and planar. ➢ Γ 𝑥, 𝛼 Inelastic impact 4
- 6. Non-contact/Contact Transitions Intuition Method Experiments 𝑥 𝑐 𝑥 𝑙 Contact surface ➢ Non-penetrable, passive and planar. ➢ Γ 𝑥, 𝛼 0 < 𝛤 5 𝛤 = 0 𝛤 < 0 Free-motion region: 𝛤 𝑥, 𝛼 ≥ 𝜌 Transition region: 0 < 𝛤 𝑥, 𝛼 < 𝜌 Contact region: 𝛤 𝑥, 𝛼 = 0 0 < 𝛤 < 𝜌 𝜌 < 𝛤 𝜌 > 0 𝛤 𝑥 𝑐, 𝛼 = 0
- 7. Non-contact/Contact Transitions Intuition Method Experiments 𝑥 𝑐 𝑥 𝑙 Contact surface ➢ Non-penetrable, passive and planar. ➢ Γ 𝑥, 𝛼 6 𝛤 = 0 𝛤 < 0 𝜌 > 0 Free-motion region: 𝛤 𝑥, 𝛼 ≥ 𝜌 Transition region: 0 < 𝛤 𝑥, 𝛼 < 𝜌 Contact region: 𝛤 𝑥, 𝛼 = 0 0 < 𝛤 < 𝜌 𝜌 < 𝛤 𝛤 𝑥 𝑙, 𝛼 = 𝜌 0 < 𝛼 𝛤 𝑥 𝑐, 𝛼 = 0
- 8. Non-contact/Contact Transitions Intuition Method Experiments 𝑥 𝑐 7 𝑀 𝑥, ሶ𝑥 = 𝑄Λ𝑄−1 𝑄 = 𝑞1 ⋯ 𝑞 𝑑 𝑇 ሷ𝑥 = 𝑀 𝑥, ሶ𝑥 𝑓 𝑥, ሶ𝑥, 𝑡 Locally activeDirectional Modulation 𝑥 𝑙 𝑞1 𝑞2 𝑥 𝑐 Mirrazavi Salehian, S. S. and Billard, A. (2018) A Dynamical System Based Approach for Controlling Robotic Manipulators During Non-contact/Contact Transitions. RA-L Legend 𝑥 Robot’s state Γ(x) The surface contact 𝑥 𝑐 Desired contact point 𝑥 𝑙 Desired departure point
- 9. Legend Non-contact/Contact Transitions Intuition Method Experiments 𝑞1 8 𝑀 𝑥, ሶ𝑥 = 𝑄Λ𝑄−1 𝑄 = 𝑞1 ⋯ 𝑞 𝑑 𝑇 ሷ𝑥 = 𝑀 𝑥, ሶ𝑥 𝑓 𝑥, ሶ𝑥, 𝑡 Locally activeDirectional Modulation 𝑞2 𝑥 𝑐 𝑥 𝑙 Mirrazavi Salehian, S. S. and Billard, A. (2018) A Dynamical System Based Approach for Controlling Robotic Manipulators During Non-contact/Contact Transitions. RA-L 8 𝑥 Robot’s state Γ(x) The surface contact 𝑥 𝑐 Desired contact point 𝑥 𝑙 Desired departure point
- 10. Non-contact/Contact Transitions Intuition Method Experiments 9 𝑀 𝑥, ሶ𝑥 = 𝑄Λ𝑄−1 𝑄 = 𝑞1 ⋯ 𝑞 𝑑 𝑇 ሷ𝑥 = 𝑀 𝑥, ሶ𝑥 𝑓 𝑥, ሶ𝑥, 𝑡 Locally active 𝑥 𝑐 𝑥 𝑙 𝑞1 𝑞2 Free motion region: 𝜌 ≤ Γ 𝑥 Transition/contact regions: 0 ≤ Γ 𝑥 < 𝜌 Λ = I → M = 𝑄𝑄−1 ሷ𝑥 = 𝑓 𝑥, ሶ𝑥, 𝑡 Λ = 𝜆11 ⋯ 𝜆1𝑑 ⋮ ⋱ ⋮ 𝜆 𝑑1 ⋯ 𝜆 𝑑𝑑 Legend 𝑥 Robot’s state Γ(x) The surface contact 𝑥 𝑐 Desired contact point 𝑥 𝑙 Desired departure point
- 11. Non-contact/Contact Transitions Intuition Method Experiments 10 𝑀 𝑥, ሶ𝑥 = 𝑄Λ𝑄−1 𝑄 = 𝑞1 ⋯ 𝑞 𝑑 𝑇 ሷ𝑥 = 𝑀 𝑥, ሶ𝑥 𝑓 𝑥, ሶ𝑥, 𝑡 𝑥 𝑐 𝑥 𝑙 𝑞1 𝑞2 Λ = 𝜆11 ⋯ 𝜆1𝑑 ⋮ ⋱ ⋮ 𝜆 𝑑1 ⋯ 𝜆 𝑑𝑑 Controlling contact point Controlling impact velocity ➢ ∀𝑗 ∈ 1, … , 𝑑 𝜆1𝑗 = −2𝜔𝑞1 𝑇 ሶ𝑥 − 𝜔2 𝑞1 𝑇 𝑥 𝑓 𝑥, ሶ𝑥,𝑡 𝑞 𝑗 𝑓 𝑥, ሶ𝑥,𝑡 𝑇 𝑓 𝑥, ሶ𝑥,𝑡 ➢ ∀ 𝑖, 𝑗 ∈ 2,1 , 2,2 … , 𝑑, 𝑑 𝜆𝑖𝑗 = −2𝜔𝑞𝑖 𝑇 ሶ𝑥 − 𝜔2 𝑞𝑖 𝑇 𝑥 − 𝑥∗ ) 𝑓 𝑥, ሶ𝑥,𝑡 𝑞 𝑗 𝑓 𝑥, ሶ𝑥,𝑡 𝑇 𝑓 𝑥, ሶ𝑥,𝑡 Legend 𝑥 Robot’s state Γ(x) The surface contact 𝑥 𝑐 Desired contact point 𝑥 𝑙 Desired departure point
- 12. Legend 𝑥 Robot’s state Γ(x) The surface contact 𝑥 𝑐 Desired contact point 𝑥 𝑙 Desired departure point Non-contact/Contact Transitions Intuition Method Experiments 11 𝑀 𝑥, ሶ𝑥 = 𝑄Λ𝑄−1 𝑄 = 𝑞1 ⋯ 𝑞 𝑑 𝑇 𝑥 𝑐 𝑥 𝑙 𝑞1 𝑞2 Λ = 𝜆11 ⋯ 𝜆1𝑑 ⋮ ⋱ ⋮ 𝜆 𝑑1 ⋯ 𝜆 𝑑𝑑 q1 T ሷ𝑥 = −2𝜔𝑞1 𝑇 ሶ𝑥 − 𝜔2 𝑞1 𝑇 𝑥 q2 T ሷ𝑥 = −2𝜔𝑞2 𝑇 ሶ𝑥 − 𝜔2 𝑞2 𝑇 𝑥 − 𝑥∗ ⋮ qd T ሷ𝑥 = −2𝜔𝑞 𝑑 𝑇 ሶ𝑥 − 𝜔2 𝑞 𝑑 𝑇 𝑥 − 𝑥∗ lim 𝑡→∞ 𝑞𝑖 𝑇 𝑥 − 𝑞𝑖 𝑇 𝑥∗ = 0 ∀𝑖 ∈ 2 … 𝑑 lim 𝑡→∞ 𝑞1 𝑇 𝑥 = 0 𝑞1 𝑇 ሶ𝑥 𝑡∗ = 0 𝑥∗ = ൝ 𝑥 𝑐 𝑖𝑓 0 < 𝑞1 𝑇 𝑥 2𝑥 𝑙 − 𝑥 𝑐 𝑖𝑓0 = 𝑞1 𝑇 𝑥 lim 𝑡→∞ 𝑞𝑖 𝑇 𝑥 − 𝑞𝑖 𝑇 𝑥 𝑐 = 0 ∀𝑖 ∈ 2 … 𝑑lim 𝑡→∞ 𝑞𝑖 T 𝑥 − 𝑞𝑖 𝑇 (2xl − xc) = 0 ∀𝑖 ∈ 2 … 𝑑 Getting into contact with the surface with zero velocity Reaching 𝑥∗ while getting into contact with the surface
- 13. Legend 𝑥 Robot’s state Γ(x) The surface contact 𝑥 𝑐 Desired contact point 𝑥 𝑙 Desired departure point lim 𝑡→∞ 𝑞𝑖 T 𝑥 − 𝑞𝑖 𝑇 (2xl − xc) = 0 ∀𝑖 ∈ 2 … 𝑑 Non-contact/Contact Transitions Intuition Method Experiments 12 𝑀 𝑥, ሶ𝑥 = 𝑄Λ𝑄−1 𝑄 = 𝑞1 ⋯ 𝑞 𝑑 𝑇 𝑥 𝑐 𝑥 𝑙 Λ = 𝜆11 ⋯ 𝜆1𝑑 ⋮ ⋱ ⋮ 𝜆 𝑑1 ⋯ 𝜆 𝑑𝑑 q1 T ሷ𝑥 = −2𝜔𝑞1 𝑇 ሶ𝑥 − 𝜔2 𝑞1 𝑇 𝑥 q2 T ሷ𝑥 = −2𝜔𝑞2 𝑇 ሶ𝑥 − 𝜔2 𝑞2 𝑇 𝑥 − 𝑥∗ ⋮ qd T ሷ𝑥 = −2𝜔𝑞 𝑑 𝑇 ሶ𝑥 − 𝜔2 𝑞 𝑑 𝑇 𝑥 − 𝑥∗ lim 𝑡→∞ 𝑞𝑖 𝑇 𝑥 − 𝑞𝑖 𝑇 𝑥∗ = 0 ∀𝑖 ∈ 2 … 𝑑 lim 𝑡→∞ 𝑞1 𝑇 𝑥 = 0 𝑞1 𝑇 ሶ𝑥 𝑡∗ = 0 𝑥∗ = ൝ 𝑥 𝑐 𝑖𝑓 0 < 𝑞1 𝑇 𝑥 2𝑥 𝑙 − 𝑥 𝑐 𝑖𝑓0 = 𝑞1 𝑇 𝑥 2𝑥 𝑙 − 𝑥 𝑐
- 14. lim 𝑡→∞ 𝑞𝑖 T 𝑥 − 𝑞𝑖 𝑇 (2xl − xc) = 0 ∀𝑖 ∈ 2 … 𝑑 Non-contact/Contact Transitions Intuition Method Experiments 13 𝑀 𝑥, ሶ𝑥 = 𝑄Λ𝑄−1 𝑄 = 𝑞1 ⋯ 𝑞 𝑑 𝑇 𝑥 𝑐 𝑥 𝑙 Λ = 𝜆11 ⋯ 𝜆1𝑑 ⋮ ⋱ ⋮ 𝜆 𝑑1 ⋯ 𝜆 𝑑𝑑 q1 T ሷ𝑥 = −2𝜔𝑞1 𝑇 ሶ𝑥 − 𝜔2 𝑞1 𝑇 𝑥 q2 T ሷ𝑥 = −2𝜔𝑞2 𝑇 ሶ𝑥 − 𝜔2 𝑞2 𝑇 𝑥 − 𝑥∗ ⋮ qd T ሷ𝑥 = −2𝜔𝑞 𝑑 𝑇 ሶ𝑥 − 𝜔2 𝑞 𝑑 𝑇 𝑥 − 𝑥∗ lim 𝑡→∞ 𝑞𝑖 𝑇 𝑥 − 𝑞𝑖 𝑇 𝑥∗ = 0 ∀𝑖 ∈ 2 … 𝑑 lim 𝑡→∞ 𝑞1 𝑇 𝑥 = 0 𝑞1 𝑇 ሶ𝑥 𝑡∗ = 0 𝑥∗ = ൝ 𝑥 𝑐 𝑖𝑓 0 < 𝑞1 𝑇 𝑥 2𝑥 𝑙 − 𝑥 𝑐 𝑖𝑓0 = 𝑞1 𝑇 𝑥 2𝑥 𝑙 − 𝑥 𝑐 Legend 𝑥 Robot’s state Γ(x) The surface contact 𝑥 𝑐 Desired contact point 𝑥 𝑙 Desired departure point
- 15. lim 𝑡→∞ 𝑞𝑖 T 𝑥 − 𝑞𝑖 𝑇 (2xl − xc) = 0 ∀𝑖 ∈ 2 … 𝑑 Non-contact/Contact Transitions Intuition Method Experiments 14 𝑀 𝑥, ሶ𝑥 = 𝑄Λ𝑄−1 𝑄 = 𝑞1 ⋯ 𝑞 𝑑 𝑇 𝑥 𝑐 𝑥 𝑙 Λ = 𝜆11 ⋯ 𝜆1𝑑 ⋮ ⋱ ⋮ 𝜆 𝑑1 ⋯ 𝜆 𝑑𝑑 q1 T ሷ𝑥 = −2𝜔𝑞1 𝑇 ሶ𝑥 − 𝜔2 𝑞1 𝑇 𝑥 q2 T ሷ𝑥 = −2𝜔𝑞2 𝑇 ሶ𝑥 − 𝜔2 𝑞2 𝑇 𝑥 − 𝑥∗ ⋮ qd T ሷ𝑥 = −2𝜔𝑞 𝑑 𝑇 ሶ𝑥 − 𝜔2 𝑞 𝑑 𝑇 𝑥 − 𝑥∗ lim 𝑡→∞ 𝑞𝑖 𝑇 𝑥 − 𝑞𝑖 𝑇 𝑥∗ = 0 ∀𝑖 ∈ 2 … 𝑑 lim 𝑡→∞ 𝑞1 𝑇 𝑥 = 0 𝑞1 𝑇 ሶ𝑥 𝑡∗ = 0 𝑥∗ = ൝ 𝑥 𝑐 𝑖𝑓 0 < 𝑞1 𝑇 𝑥 2𝑥 𝑙 − 𝑥 𝑐 𝑖𝑓0 = 𝑞1 𝑇 𝑥 2𝑥 𝑙 − 𝑥 𝑐 Free motion region: 𝜌 ≤ Γ 𝑥 Λ = I M = I ሷ𝑥 = 𝑓 𝑥, ሶ𝑥, 𝑡 Legend 𝑥 Robot’s state Γ(x) The surface contact 𝑥 𝑐 Desired contact point 𝑥 𝑙 Desired departure point
- 16. lim 𝑡→∞ 𝑞𝑖 T 𝑥 − 𝑞𝑖 𝑇 (2xl − xc) = 0 ∀𝑖 ∈ 2 … 𝑑 Non-contact/Contact Transitions Intuition Method Experiments 15 𝑀 𝑥, ሶ𝑥 = 𝑄Λ𝑄−1 𝑄 = 𝑞1 ⋯ 𝑞 𝑑 𝑇 𝑥 𝑐 𝑥 𝑙 Λ = 𝜆11 ⋯ 𝜆1𝑑 ⋮ ⋱ ⋮ 𝜆 𝑑1 ⋯ 𝜆 𝑑𝑑 q1 T ሷ𝑥 = −2𝜔𝑞1 𝑇 ሶ𝑥 − 𝜔2 𝑞1 𝑇 𝑥 q2 T ሷ𝑥 = −2𝜔𝑞2 𝑇 ሶ𝑥 − 𝜔2 𝑞2 𝑇 𝑥 − 𝑥∗ ⋮ qd T ሷ𝑥 = −2𝜔𝑞 𝑑 𝑇 ሶ𝑥 − 𝜔2 𝑞 𝑑 𝑇 𝑥 − 𝑥∗ lim 𝑡→∞ 𝑞𝑖 𝑇 𝑥 − 𝑞𝑖 𝑇 𝑥∗ = 0 ∀𝑖 ∈ 2 … 𝑑 lim 𝑡→∞ 𝑞1 𝑇 𝑥 = 0 𝑞1 𝑇 ሶ𝑥 𝑡∗ = 0 𝑥∗ = ൝ 𝑥 𝑐 𝑖𝑓 0 < 𝑞1 𝑇 𝑥 2𝑥 𝑙 − 𝑥 𝑐 𝑖𝑓0 = 𝑞1 𝑇 𝑥 2𝑥 𝑙 − 𝑥 𝑐 Free motion region: 𝜌 ≤ Γ 𝑥 Λ = I M = I ሷ𝑥 = 𝑓 𝑥, ሶ𝑥, 𝑡 Legend 𝑥 Robot’s state Γ(x) The surface contact 𝑥 𝑐 Desired contact point 𝑥 𝑙 Desired departure point
- 17. Non-contact/Contact Transitions Intuition Method Experiments 16 Mirrazavi Salehian, S. S. and Billard, A. (2018) A Dynamical System Based Approach for Controlling Robotic Manipulators During Non-contact/Contact Transitions. RA-L
- 18. Non-contact/Contact Transitions Intuition Method Experiments 17 Mirrazavi Salehian, S. S. and Billard, A. (2018) A Dynamical System Based Approach for Controlling Robotic Manipulators During Non-contact/Contact Transitions. RA-L
- 19. Non-contact/Contact Transitions Intuition Method Experiments 18 Mirrazavi Salehian, S. S. and Billard, A. (2018) A Dynamical System Based Approach for Controlling Robotic Manipulators During Non-contact/Contact Transitions. RA-L