1. Motion Planning and Control Using RRTs [Selected Slides/Movies] Michael M. Curtiss (MS Studennt) Michael S. Branicky (Advisor) Electrical Engineering & Computer Science Case Western Reserve University May 2002

2. New Applications of RRTs We have applied/extended them to: Nonlinear planning –pendulum swing-up, acrobot Prioritized multi-agent planning –air traffic control

3. Pendulum Swing-Up A pendulum of mass m and length l Motor at joint can apply discrete torques Initial configuration:  =0 (down),  dot=0 Goal configuration:  =  (up),  dot=0

4. Pendulum Swing-Up A pendulum of mass m and length l Motor at joint can apply torques {-1,0,1} dual tree, 5600 nodes single tree, 3300 nodes

5. Pendulum Swing-Up (Cont.) Torques = {-4, -2, -1, 0, 1, 2, 4},4000 iterations

6. Acrobot Swing-Up [adapted from Sutton & Barto]

7. Acrobot: The Movie

8. Planning for simplified air traffic control –Airplanes take off from one of six airports and fly to a destination airport –Airplanes cannot occupy the same cell at the same time, except adjacent to airports –Airplanes cannot fly directly in front of or behind other airplanes (preventing swapping) Prioritized Planning –A path is planned for each agent in turn –Paths are treated as obstacles in space-time for all future agents, and are immutable once planned Multi-Agent Planning

9. Multi-Agent Planning: The Movie

10. Nonholonomic Airplanes Six airports W-Space = [-1,1] x [-1,1] Safety-radius of 0.03 Rate-constrained turning Unicycle equations of motion: Prioritized Planning:

11. Dynamic Safety Envelopes L region =(v 2 /2)A max Can always stop without hitting other agents

12. Hybrid Trajectories Hybrid problems require finding valid trajectories from s init to s goal Trajectory is defined as a sequence of states s, where s=(x,q)

13. Hybrid RRT Algorithm BUILD_RRT(s init ) 1 T.init(s init ); 2 for k=1 to K do 3s rand  RANDOM_STATE(); 4EXTEND(T, s rand ); 5 return T EXTEND(T, s) 1 s near  NEAREST_METRIC_NEIGHBOR(s, T) 2 if (NEW_STATE(s, s near, s new, u new ) then 3T.add_vertex(s new ) 4T.add_edge(s near, s new, u new )

14. Hybrid RRT Changes Include discrete state in state space (S=X  Q) Redefine distance metric –Non-trivial systems must account for discrete state changes in the distance metric –Stair climbing example, (S=[-20,20] 2  {1,2,3,4}): Introduce switching as an operator –Unrestricted (switching control) or restricted (stair climber) –Autonomous (pogo stick) or controlled (gear shifting)  (c 1,c 2 ) =  x 1 -x 2  2 + 20  q 1 -q 2 

15. Stair-Climber Example