Lecture 13: Dynamical Systems based Representation. Contents:
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1 Lecture 13: Dynamical Systems based Representation Contents: Differential Equation Force Fields, Velocity Fields Dynamical systems for Trajectory Plans Generating plans dynamically Fitting (or modifying) plans Imitation based learning Thanks to my collaborator Auke Ijspeert (EPFL) for many of the contents on the slides for this lecture. Lecture 13: RLSC - Prof. Sethu Vijayakumar 1
2 Movement policies as Dynamical Systems des f (,, goal) des Discreet & Rhythmic Movement Primitives Represent complex movements in globally stable attractor landscapes of nonlinear autonomous differential equations Choose kinematic representation for easy reuse in different workspace location Ensure easy temporal and spatial scaling (topological equivalence) Use local learning to modify the attractors according to demonstration of teacher and self-learning Lecture 13: RLSC - Prof. Sethu Vijayakumar 2
3 Discreet & Rhythmic Movement superposition Open loop with oscillators Closed loop control in horizontal plane Open (vertical) + Closed (horizontal) loop control Lecture 13: RLSC - Prof. Sethu Vijayakumar 3
4 What is a Differential Equation? Differential equation: an equation that describes how state variables evolve over time, for instance: y ( c y) Lecture 13: RLSC - Prof. Sethu Vijayakumar 4
5 Some Definitions Ordinary differential equation: differential equation that involves only ordinary derivatives (as opposed to partial derivatives) Autonomous equation: differential equation that does not (explicitly) depend on time Linear differential equation: differential equation in which the state variables only appear in linear combinations Nonlinear differential equation: differential equation in which some state variables appear in nonlinear combinations (e.g. products, cosine, ) Fixed point: point at which all derivatives are zero (can be an attractor, a repeller, or a saddle point, cf later) Limit cycle: periodic isolated closed trajectory (can only occur in nonlinear systems) Lecture 13: RLSC - Prof. Sethu Vijayakumar 5
6 Interesting Regimes of Differential Eqs. From Strogatz 1994 Unstable node saddles Attractor Attractors Limit cycles Chaos Lecture 13: RLSC - Prof. Sethu Vijayakumar 6
7 First Order Linear Systems First order linear system: y ( c y) How to solve this equation, for a given y(t=0), c, and? Two methods: analytical solution or numerical integration Analytical solution: y( t) ( y c)exp( 0 t) Numerical integration: Euler method, Runge-Kutta, c Lecture 13: RLSC - Prof. Sethu Vijayakumar 7
8 First Order Linear Systems y ( c y) y( t) ( y c)exp( 0 t) c Lecture 13: RLSC - Prof. Sethu Vijayakumar 8
9 Second Order Linear Systems y ( ( c x) y) x y Lecture 13: RLSC - Prof. Sethu Vijayakumar 9
10 Second Order Linear Systems y ( ( c x) y) x y Lecture 13: RLSC - Prof. Sethu Vijayakumar 10
11 Second Order Non-linear System y x 2cos 2cos y cos x x cos y Repeller saddles Attractor Lecture 13: RLSC - Prof. Sethu Vijayakumar 11
12 Learning a movement by demonstration Movement recognition Joint angles Trajectory formation system Movement execution (Inv. Dyn.) Task of the trajectory formation system: To encode demonstrated trajectories with high accuracy, To be able to modulate the learned trajectory when: Perceptual variables are varied (e.g. timing, amplitude) Perturbations occur Lecture 13: RLSC - Prof. Sethu Vijayakumar 12
13 Encoding a trajectory Traditionally, the problem of replaying a trajectory has been decomposed into two different issues: One of encoding the trajectory, and One of modifying the trajectory, for instance, in case the movement is perturbed, or when it requires to be modulated. Our approach: combine both abilities in a nonlinear dynamical system Aim: to encode the trajectory in a nonlinear dynamical system with well defined attractor landscape Lecture 13: RLSC - Prof. Sethu Vijayakumar 13
14 Nonlinear Dynamical Systems Approach Discrete movements dy/dt Rhythmic movements dy/dt y y Single point attractor Limit cycle attractor Lecture 13: RLSC - Prof. Sethu Vijayakumar 14
15 Two types of movement recordings Sensuit «Kinesthetic» demonstration Lecture 13: RLSC - Prof. Sethu Vijayakumar 15
16 Pointing Demonstration position velocity Demo for One DOF Lecture 13: RLSC - Prof. Sethu Vijayakumar 16
17 Shaping Attractor Landscapes Goal: g z g y z y z z z Can one create more complex dynamics by nonlinearly modifying the dynamic system: z z? z g y z y f z Lecture 13: RLSC - Prof. Sethu Vijayakumar 17
18 Discreet Control Policy Output System Goal: g Amplitude and phase system Lecture 13: RLSC - Prof. Sethu Vijayakumar 18
19 Shaping Attractor Landscapes A globally stable learnable nonlinear point attractor: Trajectory Plan Dynamics Canonical Dynamics Local Linear Model Approx. z z g y z y f x, v z y where v g x v x f v v x x, v z v k i1 k i1 i i w b v w i 1 wi exp di x ci and x 2 x x g x Lecture 13: RLSC - Prof. Sethu Vijayakumar 19
20 Learning the Attractor Given a demonstrated trajectory y(t) demo and a goal g Extract movement duration Adjust time constants of canonical dynamics to movement duration Use LWL to learn supervised problem ydemo ytarget z f x, v y Also extended to rhythmic primitives : [ Stefan Schaal, Sethu Vijayakumar et al, Proc. of Intl. Symp. Rob. Res.(ISRR) (2001) ] Lecture 13: RLSC - Prof. Sethu Vijayakumar 20
21 Trajectory following & Generalization Backhand Demonstration Backhand Reproduction Lecture 13: RLSC - Prof. Sethu Vijayakumar 21
22 Lecture 13: RLSC - Prof. Sethu Vijayakumar 22 Basic oscillator: Output signal: 1 ( 0) r r r N i i N i T i i m I w z y z y y z 1 1 ) ) ( ( ],, [ v x T v x I ], [ Rhythmic Control Policies
23 Modulation of Goal: Anchor Lecture 13: RLSC - Prof. Sethu Vijayakumar 23
24 Drumming: Modulating Frequency Drumming: Kinesthetic Demo Imitation and Modulation Lecture 13: RLSC - Prof. Sethu Vijayakumar 24
25 Biped Locomotion using DMPs 5-link biped robot [Nakanishi, Morimoto, Endo, Cheng, Schaal, and Kawato, 2004] Height 40cm Mass hip 2.0kg knee 0.64kg foot 0.06kg Approach Movement primitives as a central pattern generator (CPG) Synchronization between CPG and robot - phase resetting and frequency adaptation - phase regulation between coupled oscillators Lecture 13: RLSC - Prof. Sethu Vijayakumar 25
26 Control Architecture Rhythmic Movement primitives periodic pattern generation learning from demonstrated trajectories Phase resetting [Nakanishi, Morimoto, Endo, Cheng, Schaal, and Kawato, 2004] CPG reset the phase of the oscillator at heel contact oscillator output with local models Dynamical primitives (index: i=1~# of oscillators) Frequency adaptation Phase regulation among oscillators coupling among oscillators Robot and Environment phase resetting frequency update Lecture 13: RLSC - Prof. Sethu Vijayakumar 26
27 Experiments: Bipedal Locomotion Lecture 13: RLSC - Prof. Sethu Vijayakumar 27
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