CS 326A: Motion Planning

1. CS 326A: Motion Planning Kynodynamic Planning + Dealing with Moving Obstacles+ Dealing with Uncertainty+ Dealing with Real-Time Issues

2. Kinodynamic Planning Problem Plan a robot’s trajectory that • satisfies a dynamic equation of motion of the form:s’ = f(s,u)where: • sis robot’s state configuration x velocity at time t • u is control input at time t • avoids collision with moving obstacles Outcome: (Piecewise-constant) control function u(t)

3. Problem Plan a robot’s trajectory that • satisfies a dynamic motion constraint of the form:s’ = f(s,u)where: • sis robot’s state configuration x velocity at time t • u is control input at time t • avoids collision with moving obstacles Outcome: (Piecewise-constant) control function u(t) Note similarity with non-holonomic constraint q’ = f(q,u)

4. Nonholonomic vs. Dynamic Constraints • Nonholonomic constraint: f(q,q’) = 0 • Dynamic constraints: f(q,q’,q’’) = 0 s = (q,q’)  g(s,s’) = 0 • Kinodynamic planning is done in state (configuration x velocity) space rather than in configuration space • The dimensionality of state space is twice that of configuration space

5.  Similarity of Method • Construction of a tree, by integrating equation of motion with selected control inputs • However, planning occurs in state space, instead of configuration space • Hence, PRM-like planner rather than deterministic one, to deal with greater dimensionality of the space

6. air thrusters gas tank air bearing Example: Space Robot Robot created to study issues in robot control and planning in no-gravity space environment

7. Navigation Among Moving Obstacles

8. y robot obstacles x Model, Sensing, and Control • The robot and the obstacles are represented as disks moving in the plane • The position and velocity of each disc are measured by an overhead camera every 1/30 sec

9. y f a x Model, Sensing, and Control • The robot and the obstacles are represented as disks moving in the plane • The position and velocity of each disc are measured by an overhead camera within 1/30 sec • The robot controls the magnitude f and the orientation a of the total pushing force exerted by the thrusters robot obstacles

10. Moving Obstacles in ConfigurationxTime Space y t x

11. Traditional PRM straight line segments

12. endgame region g s Kinodynamic Planning with PRM in State x Time Space t=3 t=4 t=5 t=4 t=2 t=3 t=1 t=2 t=1 t=2 t=3 Bi-directional search? The roadmap is a tree oriented along the time axis (because of the moving obstacles)

13. endgame region g s PRM in StatexTime Space Compare! The roadmap is a tree oriented along the time axis

14. 1 1 2 1 2 1 1 1 1 Sampling Strategy endgame region

15. Bi-Directional Search: Forward & Backward Integration Goal Region

16. t y x Example Run Obstacle map to cylinders in configurationtime space

17. Other Examples

18. But executing this trajectory is likely to fail ... • The measured velocities of the obstacles are inaccurate • Tiny particles of dust on the table affect trajectories and contribute further to deviation Obstacles are likely to deviate from their expected trajectories • Planning takes time, and during this time, obstacles keep moving The computed robot trajectory is not properly synchronized with those of the obstacles The robot may hit an obstacle before reaching its goal [Robot control is not perfect but “good” enough for the task]

19. Planning must take both uncertainty in world state and time constraints into account But executing this trajectory is likely to fail ... • The measured velocities of the obstacles are inaccurate • Tiny particles of dust on the table affect trajectories and contribute further to deviation Obstacles are likely to deviate from their expected trajectories • Planning takes time, and during this time, obstacles are moving The computed robot trajectory is not properly synchronized with those of the obstacles The robot may hit an obstacle before reaching its goal [Robot control is not perfect but “good” enough for the task]

20. The robot can handle uncertainty in an obstacle position by representing the set of all positions of the obstacle that the robot think possible at each time For example, this set can be a disc whose radius grows linearly with time Set of possible positions at time 2T Initial set ofpossible positions t = 0 t = T t = 2T Dealing with Uncertainty Set of possible positions at time T

21. The robot can handle uncertainty in an obstacle position by representing the set of all positions of the obstacle that the robot think possible at each time For example, this set can be a disc whose radius grows linearly with time Dealing with Uncertainty The robot must plan to be outside this disc at time t = T t = 0 t = T t = 2T

22. The robot can handle uncertainty in an obstacle position by representing the set of all positions of the obstacle that the robot think possible at each time (belief state) For example, this set can be a disc whose radius grows linearly with time The forbidden regions in configurationtime space are cones, instead of cylinders The trajectory planning method remains essentially unchanged Dealing with Uncertainty

23. Let t=0 the time when planning starts. A time limit d is given to the planner The planner computes the states that will be possible at t = d and use them as the possible initial states It returns a trajectory at some t < d, whose execution will start at t =d Since the PRM planner isn’t absolutely guaranteed to find a solution within d, it computes two trajectories using the same roadmap: one to the goal, the other to any position where the robot will be safe for at least an additional d. Since there are usually many such positions, the second trajectory is at least one order of magnitude faster to compute Dealing with Planning Time

24. Safe Region Planning an Escape Trajectory endgame region

25. The robot must monitor the execution of the planned trajectory and be prepared to re-plan a new trajectory Are we done? • Not quite ! • The uncertainty model may be itself incorrect, e.g.: • There may be more dust on the table than anticipated • Some obstacles have the ability to change trajectories • But if we are too careful, we will end up with forbidden regions so big that no solution trajectory will exist any more • So, it might be better to take some “risk”

26. If an obstacle has an unexpected position, the planner is called back to compute a new trajectory. Execution monitoring consists of using the camera (at 30Hz) to verify that all obstacles are at positions allowed by the robot’s uncertainty model Are we done? Note tradeoff between uncertaintymodel and number of re-plans The robot must monitor the execution of the planned trajectory and be prepared to re-plan a new trajectory

27. 1 2 3 5 4 6 7 Experimental Run with Re-Planning

28. 1 2 3 5 4 6 7 Experimental Run with Re-Planning

29. Another Experimental Run X Total duration : 40 sec

30. Another Experimental Run

31. 8 replanning operations Example in Simulation

32. Is this guaranteed to work? Of course not : • Thrusters might get clogged • The robot may run out of air or battery • The granite table may suddenly break into pieces • Etc ... [Unbounded uncertainty]