Inevitable Collision States: 1/15 Antoine Bautin, a Probabilistic - PowerPoint PPT Presentation

Probabilistic Inevitable Collision States Inevitable Collision States: 1/15 Antoine Bautin, a Probabilistic Perspective Luis Martinez, Thierry Fraichard Overview Antoine Bautin, Luis Martinez, Thierry Fraichard Context ICS Model of the future Probabilistic ICS e-Motion Team - LIG laboratory Probabilistic ICS Backward INRIA Rhones-Alpes ICS-Check Algorithm Forward ICS-Check Grenoble Universities Algorithm Results Conclusion

Probabilistic Context Inevitable Collision States Safe autonomous navigation of a robotic system in 2/15 open dynamic environments Antoine Bautin, Luis Martinez, Thierry Fraichard Overview Context ICS Model of the future Probabilistic ICS Probabilistic ICS Backward ICS-Check Algorithm Forward ICS-Check Algorithm Results Conclusion DARPA Urban challenge 2007: the technology is here but accidents took place ⇒ Motion safety remains an issue

Probabilistic Motion Safety Inevitable Collision States 3/15 Antoine Bautin, Motion safety requires to [Fraichard, 2007] Luis Martinez, Thierry Fraichard 1. reason about the future Overview 2. with an appropriate look ahead Context ICS Model of the future A concept that addresses these issues: Probabilistic ICS Probabilistic ICS Inevitable Collision States [Fraichard & Asama, 2004] Backward ICS-Check Algorithm Forward ICS-Check Related concepts: Algorithm Results ◮ Obstacle Shadow [Reif & sharir, 1985] Conclusion ◮ Region of Inevitable Collision [LaValle & Kuffner, 2001] ◮ Viability Kernel: Viability Theory [Aubin, 1991] ◮ Backward Reachable Set [Mitchell & Tomlin, 2003]

Probabilistic Inevitable Collision States Inevitable Collision States 4/15 Antoine Bautin, Luis Martinez, State in which whatever the control trajectory sequence Thierry Fraichard applied by the robotic system, a collision will eventually Overview occur Context ICS Model of the future Probabilistic ICS Probabilistic ICS Backward ICS-Check Algorithm Forward ICS-Check Algorithm Results Conclusion

Probabilistic Deterministic the model of the future Inevitable Collision States 5/15 Open environments are uncertain (prediction of the future Antoine Bautin, Luis Martinez, motion of obstacles) Thierry Fraichard → Safety requires to be conservative Overview Context Using a worst-case scenario e.g. : Growing discs ICS Model of the future [van den Berg & Overmars, 2007] Probabilistic ICS Probabilistic ICS Backward ICS-Check Algorithm Forward ICS-Check Algorithm Results Conclusion ⇒ every state is an ICS

Probabilistic Probabilistic model of the future Inevitable Collision States 6/15 Model for the future motion of Obstacles: Antoine Bautin, P occ [ B i , t ] ( x w , y w ) is available ∀ x w , y w , t , i Luis Martinez, Thierry Fraichard → Assumed available (can be built from various methods) Overview Context ICS Model of the future Probabilistic ICS Probabilistic ICS Backward ICS-Check Algorithm Forward ICS-Check Algorithm Results Conclusion Lookahead is set to the time when the distributions of the obstacles are uniform

Probabilistic Probabilistic ICS Inevitable Collision States 7/15 Antoine Bautin, Luis Martinez, Thierry Fraichard Contribution of this work : Overview Characterize ICS using a probabilistic model of the future Context Probabilistic ICS-checking algorithms ICS Model of the future Probabilistic ICS Probabilistic ICS Definition (New notion) Probabilistic ICS Backward ICS-Check Algorithm P ICS ( s ) = P ( s ∈ ICS( B )) = min ( P ICS [˜ u , B ] ( s )) Forward ICS-Check u ∈ ˜ Algorithm ∀ ˜ U Results Probabilistic ICS Checking Algorithm (New algorithm) Conclusion can be plugged into planning algorithm like Partial Motion Planning or RRT (future works)

Probabilistic Backward Probabilistic ICS-Check Algorithm Inevitable Collision States Direct adaptation of the Deterministic ICS Checker 8/15 [Martinez Gomez & Fraichard, 2008] Antoine Bautin, Luis Martinez, Key step 2 explained on next slide Thierry Fraichard 1. Select E with E ⊂ ˜ Overview U , a subset of the whole set of Context possible future trajectories (conservative ICS Model of the future approximation) Probabilistic ICS Probabilistic ICS 2. Compute P ICS [ B i , ˜ u j , t ] ( s ) for all t ,every B i and every Backward ICS-Check Algorithm u j ∈ E , s ∈ ˆ ˜ z c Forward ICS-Check Algorithm Results 3. Compute P ICS [ B i , ˜ u j ] ( s ) = � u j , t ] ( s ) for t 0 .. t la P ICS [ B i , ˜ Conclusion every B i and every ˜ u j ∈ E 4. Compute P ICS [˜ u j ] ( s ) = � i =1 ··· n b P ICS [ B i , ˜ u j ] ( s ) for every ˜ u j ∈ E 5. Compute P ICS ( s c ) = min( P ICS [˜ u j ] ( s c ))

Probabilistic Backward Probabilistic ICS-Check Algorithm Inevitable Collision States 9/15 Step 2: Compute P ICS [ B i , ˜ u j , t ] ( s ) Antoine Bautin, Luis Martinez, Thierry Fraichard z slice reasoning [Martinez Gomez & Fraichard, 2008] : ˆ Overview Planary System State: s = ( x , y , ˆ z ) Context ICS Model of the future Probabilistic ICS Probabilistic ICS Backward ICS-Check Algorithm Forward ICS-Check Algorithm Results Conclusion

Results for a ˆ z c slice Computing probabilistic ICS for : Point mass system with an initial state : ˙ x = 0 y = 10 ˙ 3 different control trajectories 1 obstacle moving down (probabilistically) Obstacle constant velocity : ˙ x = 0 y = − 10 ˙ Control trajectory : ¨ x = 0 ¨ y = − 1 ¨ x = +1 y = − 1 ¨ x = − 1 ¨ ¨ y = − 1

Probabilistic Results Inevitable Collision States The resulting probabilistic ICS set 10/15 Antoine Bautin, Luis Martinez, Thierry Fraichard Overview Context ICS Model of the future Probabilistic ICS Probabilistic ICS Backward ICS-Check Algorithm Forward ICS-Check Algorithm Results Conclusion

Probabilistic Results Inevitable Collision States 11/15 ICS set using 3 control trajectories and 3 obstacles Antoine Bautin, Luis Martinez, Thierry Fraichard Overview Context ICS Model of the future Probabilistic ICS Probabilistic ICS Backward ICS-Check Algorithm Forward ICS-Check Algorithm Results Conclusion

Probabilistic Complexity issue Inevitable Collision States 12/15 Antoine Bautin, Luis Martinez, Thierry Fraichard Starting from the obstacle trajectory, it is not know Overview beforehand which obstacle will influence the P ICS of the Context ICS state we want to check. Model of the future → compute P ICS for all the states that lead to a possible Probabilistic ICS Probabilistic ICS collision. Backward ICS-Check Algorithm Forward ICS-Check Algorithm → Find a more efficient algorithm Results Conclusion Start from the state to be checked : Evaluate a subset of forward reachable state

Probabilistic Forward Probabilistic ICS-Check Algorithm Inevitable Collision States 1. Select E 13/15 2. Compute P ICS [˜ u j , t ] ( s ) for all t and every ˜ u j ∈ E Antoine Bautin, Luis Martinez, 3. Compute P ICS [˜ u j ] ( s ) = � t 0 .. t la P ICS [˜ u j , t ] ( s ) for every Thierry Fraichard ˜ u j ∈ E Overview Context 4. Compute P ICS ( s c ) = min( P ICS [˜ u j ] ( s c )) ICS Model of the future Probabilistic ICS Probabilistic ICS Backward ICS-Check Algorithm Forward ICS-Check Algorithm Results Conclusion

Results Backward and Forward Pics-Check algorithms ICS-Check and ICS-Check overlay on Pics-Check

Probabilistic Conclusion Inevitable Collision States 15/15 Contribution: Antoine Bautin, Luis Martinez, ◮ Probabilistic ICS formulation of the ICS concept Thierry Fraichard ◮ Presentation of 2 Probabilistic ICS-Checkers algorithms Overview Context Backward Probabilistic ICS-Check Algorithm : ICS Model of the future ◮ Costly Probabilistic ICS Probabilistic ICS Forward Probabilistic ICS-Check Algorithm : Backward ICS-Check Algorithm ◮ Effective Forward ICS-Check Algorithm Results Future Works: Embedding of Pics-Check Algorithms in Conclusion navigation schemes 1. Reactive collision avoidance like ICS-Avoid [Martinez Gomez & Fraichard, 2009] 2. Global navigation scheme

Probabilistic Questions? Inevitable Collision States 15/15 Antoine Bautin, Luis Martinez, Thierry Fraichard Overview Context ICS Model of the future Probabilistic ICS Probabilistic ICS Backward ICS-Check Algorithm Forward ICS-Check Algorithm Thank you for your attention Results Conclusion