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 ones-Alpes, LIG-CNRS, Grenoble Universities (FR) March 26, 2008 V.Delsart & T.Fraichard Reactive Trajectory Deformation 1/18
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 V.Delsart & T.Fraichard Reactive Trajectory Deformation 2/18
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 V.Delsart & T.Fraichard Reactive Trajectory Deformation 3/18
Autonomous Motion in Dynamic Environment Reactive vs Deliberative Approaches Our Approach : Trajectory Deformation Motion Deformation Experimental Results Path deformation Conclusion and Future Works 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 V.Delsart & T.Fraichard Reactive Trajectory Deformation 4/18
Autonomous Motion in Dynamic Environment Reactive vs Deliberative Approaches Our Approach : Trajectory Deformation Motion Deformation Experimental Results Path deformation Conclusion and Future Works Motion determination - Motion deformation Principle : Modification of an initial motion in response to unexpected obstacles V.Delsart & T.Fraichard Reactive Trajectory Deformation 5/18
Autonomous Motion in Dynamic Environment Reactive vs Deliberative Approaches Our Approach : Trajectory Deformation Motion Deformation Experimental Results Path deformation Conclusion and Future Works Previous solutions : Path deformation Deformation of a path, ie. a geometric curve Brock & Khatib [BK97] Lamiraux and al. [LBL02] Drawbacks : V.Delsart & T.Fraichard Reactive Trajectory Deformation 6/18
Autonomous Motion in Dynamic Environment Trajectory Deformation Principle Our Approach : Trajectory Deformation External forces Experimental Results Internal forces Conclusion and Future Works 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 V.Delsart & T.Fraichard Reactive Trajectory Deformation 7/18
Autonomous Motion in Dynamic Environment Trajectory Deformation Principle Our Approach : Trajectory Deformation External forces Experimental Results Internal forces Conclusion and Future Works Total force Principle of the approach Robotic system : s = f ( s , u ) , s ∈ S , u ∈ U ˙ (1) Trajectory ≡ [0; T f ] → S Discrete trajectory : { n 0 , . . . , n N } , n ∈ S × T Trajectory deformation due to two forces : External forces due to obstacles Internal forces to maintain the connectivity V.Delsart & T.Fraichard Reactive Trajectory Deformation 8/18
Autonomous Motion in Dynamic Environment Trajectory Deformation Principle Our Approach : Trajectory Deformation External forces Experimental Results Internal forces Conclusion and Future Works Total force External forces Purpose : F ext ( n ) = F ext ( s , t ) exerted by the obstacles for collision avoidance In practice : Control points [BK97] c j = ( p j , t ) ∈ W × T Potential field : � k ext ( d 0 − d wt ( c j )) 2 if d wt ( c j ) < d 0 V ext ( c ) = (2) 0 otherwise k ext is the repulsion gain Distance d wt to the closest obstacle : d wt 2 = w s 2 ( x 1 − x 0 ) 2 + w s 2 ( y 1 − y 0 ) 2 + w t 2 ( t 1 − t 0 ) 2 (3) V.Delsart & T.Fraichard Reactive Trajectory Deformation 9/18
Autonomous Motion in Dynamic Environment Trajectory Deformation Principle Our Approach : Trajectory Deformation External forces Experimental Results Internal forces Conclusion and Future Works Total force External forces Resulting force in W × T : ext ( c j , t ) = −∇ V ext ( c ) = k ext ( d 0 − d wt ( c )) d F wt (4) || d || Forces applied in the configuration space : r � F ct J T c j ( q , t ) F wt ext ( c j ) ext ( q , t ) = (5) j =1 At last, F ext ( s , t ) derived from F ct ext ( q , t ) V.Delsart & T.Fraichard Reactive Trajectory Deformation 10/18
Autonomous Motion in Dynamic Environment Trajectory Deformation Principle Our Approach : Trajectory Deformation External forces Experimental Results Internal forces Conclusion and Future Works 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 V.Delsart & T.Fraichard Reactive Trajectory Deformation 11/18
Autonomous Motion in Dynamic Environment Trajectory Deformation Principle Our Approach : Trajectory Deformation External forces Experimental Results Internal forces Conclusion and Future Works Total force Internal forces Case R ( n − ) ∩ R − 1 ( n + ) � = Ø : Potential field defined between a node n and the centroid H V int ( n ) = k int d st ( n ) 2 k int attraction gain Resulting force : F int ( n ) = −∇ V int ( n ) = k int d st ( n ) d || d || Case R ( n − ) ∩ R − 1 ( n + ) = Ø : Keep the connectivity with the past Moved to the closest node of R ( n − ) V.Delsart & T.Fraichard Reactive Trajectory Deformation 12/18
Autonomous Motion in Dynamic Environment Trajectory Deformation Principle Our Approach : Trajectory Deformation External forces Experimental Results Internal forces Conclusion and Future Works Total force Total force applied Total force applied on a node n : F ( n ) = F ext ( n ) + F int ( n ) (6) More weight to k ext move the trajectory away from obstacles faster More weight to k int increase the stiffness of the trajectory More weight to w s ensure to keep a great secure distance from obstacles More weight to w t increase the modifications applied on the speed on the system V.Delsart & T.Fraichard Reactive Trajectory Deformation 13/18
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 : � ˙ � v � � p = (7) ˙ v a Input control : u = a Constraints : � v ≤ v max a ≤ a max V.Delsart & T.Fraichard Reactive Trajectory Deformation 14/18
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 v cos( θ ) y ˙ v sin( θ ) ˙ θ = v tan( φ ) / L (8) ˙ φ ω a v ˙ Input control : u = ( a , ω ) Constraints : v ∈ [0 , v max ] , | φ | ≤ φ max , | a | ≤ a max and | ω | ≤ ω max V.Delsart & T.Fraichard Reactive Trajectory Deformation 15/18
Autonomous Motion in Dynamic Environment Our Approach : Trajectory Deformation Experimental Results Conclusion and Future Works Experimental Results Experimental Results Teddy in action V.Delsart & T.Fraichard Reactive Trajectory Deformation 16/18
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 V.Delsart & T.Fraichard Reactive Trajectory Deformation 17/18
Autonomous Motion in Dynamic Environment Our Approach : Trajectory Deformation Experimental Results Conclusion and Future Works Thank you for your attention! Questions ? V.Delsart & T.Fraichard Reactive Trajectory Deformation 18/18
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