[PPT] - Reactive Trajectory Deformation to Navigate Dynamic Environment PowerPoint Presentation

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Autonomous Motion in Dynamic Environment Our Approach : Trajectory Deformation Experimental Results Conclusion and Future Works

Reactive Trajectory Deformation to Navigate Dynamic Environment

Vivien Delsart and Thierry Fraichard

Inria Rhˆ

nes-Alpes, LIG-CNRS, Grenoble Universities (FR)

March 26, 2008

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Autonomous Motion in Dynamic Environment Our Approach : Trajectory Deformation Experimental Results Conclusion and Future Works

Autonomous Motion in Dynamic Environment

Where to move next? Dynamic environment Car-like model

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Autonomous Motion in Dynamic Environment Our Approach : Trajectory Deformation Experimental Results Conclusion and Future Works

Plan

Motion Autonomy : Previous Approaches Our Approach : Trajectory Deformation Experimental Results

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Autonomous Motion in Dynamic Environment Our Approach : Trajectory Deformation Experimental Results Conclusion and Future Works Reactive vs Deliberative Approaches Motion Deformation Path deformation

Motion determination - Deliberative vs Reactive Approaches

Two main approaches : Deliberative approaches

Solving of a motion planning problem Require a complete model of the environment High intrinsic complexity

Reactive approaches

Computation of the action to apply during the next time step Can operate on-line using local sensor information Convergence towards the goal not guaranteed

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Autonomous Motion in Dynamic Environment Our Approach : Trajectory Deformation Experimental Results Conclusion and Future Works Reactive vs Deliberative Approaches Motion Deformation Path deformation

Motion determination - Motion deformation

Principle : Modification of an initial motion in response to unexpected

bstacles

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Autonomous Motion in Dynamic Environment Our Approach : Trajectory Deformation Experimental Results Conclusion and Future Works Reactive vs Deliberative Approaches Motion Deformation Path deformation

Previous solutions : Path deformation

Deformation of a path, ie. a geometric curve Brock & Khatib [BK97] Lamiraux and al. [LBL02] Drawbacks :

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Autonomous Motion in Dynamic Environment Our Approach : Trajectory Deformation Experimental Results Conclusion and Future Works Trajectory Deformation Principle External forces Internal forces Total force

Our Solution : Trajectory Deformation (Teddy)

Trajectory ≡ Geometric path parametrized by time Features both spatial and temporal deformation Need to take in account a model of the future

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Autonomous Motion in Dynamic Environment Our Approach : Trajectory Deformation Experimental Results Conclusion and Future Works Trajectory Deformation Principle External forces Internal forces Total force

Principle of the approach

Robotic system : ˙ s = f (s, u), s ∈ S, u ∈ U (1) Trajectory ≡ [0; Tf ] → S Discrete trajectory : {n0, . . . , nN}, n ∈ S × T Trajectory deformation due to two forces : External forces due to obstacles Internal forces to maintain the connectivity

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Autonomous Motion in Dynamic Environment Our Approach : Trajectory Deformation Experimental Results Conclusion and Future Works Trajectory Deformation Principle External forces Internal forces Total force

External forces

Purpose : Fext(n) = Fext(s, t) exerted by the obstacles for collision avoidance In practice : Control points [BK97] cj = (pj, t) ∈ W × T Potential field : Vext(c) = kext(d0 − dwt(cj))2 if dwt(cj) < d0

therwise

(2) kext is the repulsion gain Distance dwt to the closest obstacle : dwt2 = ws2(x1 − x0)2 + ws2(y1 − y0)2 + wt2(t1 − t0)2 (3)

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Autonomous Motion in Dynamic Environment Our Approach : Trajectory Deformation Experimental Results Conclusion and Future Works Trajectory Deformation Principle External forces Internal forces Total force

External forces

Resulting force in W × T : Fwt

ext(cj, t) = −∇Vext(c) = kext(d0 − dwt(c)) d

||d|| (4) Forces applied in the configuration space : Fct

ext(q, t) = r

j=1

JT cj (q, t)Fwt

ext(cj)

(5) At last, Fext(s, t) derived from Fct

ext(q, t)

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Autonomous Motion in Dynamic Environment Our Approach : Trajectory Deformation Experimental Results Conclusion and Future Works Trajectory Deformation Principle External forces Internal forces Total force

Internal forces

Purpose : Maintain the connectivity of the trajectory R(n−) forward reachability set R−1(n+) backward reachability set 3 successive nodes n−, n, n+ Connectivity criterion : n must belong to R(n−) ∩ R−1(n+) In practice : Computation of R(n−) ∩ R−1(n+) if possible

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Autonomous Motion in Dynamic Environment Our Approach : Trajectory Deformation Experimental Results Conclusion and Future Works Trajectory Deformation Principle External forces Internal forces Total force

Internal forces

Case R(n−) ∩ R−1(n+) = Ø : Potential field defined between a node n and the centroid H Vint(n) = kintdst(n)2 kint attraction gain Resulting force : Fint(n) = −∇Vint(n) = kintdst(n) d ||d|| Case R(n−) ∩ R−1(n+) = Ø : Keep the connectivity with the past Moved to the closest node of R(n−)

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Autonomous Motion in Dynamic Environment Our Approach : Trajectory Deformation Experimental Results Conclusion and Future Works Trajectory Deformation Principle External forces Internal forces Total force

Total force applied

Total force applied on a node n : F(n) = Fext(n) + Fint(n) (6) More weight to kext move the trajectory away from obstacles faster More weight to kint increase the stiffness of the trajectory More weight to ws ensure to keep a great secure distance from obstacles More weight to wt increase the modifications applied on the speed on the system

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Autonomous Motion in Dynamic Environment Our Approach : Trajectory Deformation Experimental Results Conclusion and Future Works

Case Study (1) : Double Integrator System

State of the system A : (p, v) Dynamic of the system : ˙ p ˙ v

=

v a

(7)

Input control : u = a Constraints : v ≤ vmax a ≤ amax

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Autonomous Motion in Dynamic Environment Our Approach : Trajectory Deformation Experimental Results Conclusion and Future Works

Case Study (2) : Car-like system

State of the system A : (x, y, θ, φ, v) Dynamic of the system :       ˙ x ˙ y ˙ θ ˙ φ ˙ v       =       v cos(θ) v sin(θ) v tan(φ)/L ω a       (8) Input control : u = (a, ω) Constraints : v ∈ [0, vmax], |φ| ≤ φmax, |a| ≤ amax and |ω| ≤ ωmax

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Autonomous Motion in Dynamic Environment Our Approach : Trajectory Deformation Experimental Results Conclusion and Future Works

Experimental Results

Teddy in action

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Autonomous Motion in Dynamic Environment Our Approach : Trajectory Deformation Experimental Results Conclusion and Future Works

Conclusion and Future Works

Works done : Trajectory Deformation scheme Reactivity to information about obstacles behaviour Simulation over double integrator and car-like systems Future Works : Prediction validity Integration within a global navigation architecture Tests on an actual robotic system

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Autonomous Motion in Dynamic Environment Our Approach : Trajectory Deformation Experimental Results Conclusion and Future Works

Thank you for your attention! Questions ?

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