1. CS 326 A: Motion Planning http://robotics.stanford.edu/~latombe/cs326/2002 Coordination of Multiple Robots

2. Two Main Approaches Decoupled Planning: Plan for each robot independently of the others and coordinate them later  Several possible schemes Centralized Planning: Plan the motion of the robots in their “composite” configuration space

3. Planning with Moving Obstacles Dealing with moving obstacles require planning a trajectory , i.e., a path indexed by time  is conveniently represented in configuration x time space by a continuous curve whose tangent always projects positively along the time axis Obstacles map as forbidden regions in CT-space. Constraints on velocity constrain tangents to . Constraints on acceleration constrain curvature of 

4. Example t x y

5. Coordination of Multiple Robots  Does not require explicit introduction of time (except if there are moving obstacles)  Only requires using the same parameter to index the paths of the coordinated robots  Using the same indexing parameter corresponds to fixing the relative velocities of the robots

6. Decoupled Planning  Pure velocity tuning: (1) Separately plan a path of each robot to avoid collision with (static) obstacles (2) Compute the relative velocities of the robots to avoid inter-robot collision (e.g., coordination diagram)

7. Decoupled Planning  Pure velocity tuning:  Robot prioritization: - Plan path of a first robot in its C-space - Iterate: Plan trajectory of ith robot in its CT-space assuming that robots 1,…,i-1 are obstacles moving at some velocities

8. Centralized Planning  Plan collision-free path  in composite configuration C 1 x C 2 x…x C p space of the p robots  Forbidden regions in composite C-space are all configurations where either a robot collide with an obstacle or two robots collide with each other  The projection of  into C i is the path of the ith robot

9. Pros and Cons Assume p robots with n degrees of freedom each. Worst-case complexity of centralized planning is ~ e np Worst-case complexity of decoupled planning is ~ pe n << e np But decoupled planning is incomplete

10. Multi-Robot Example N robot = 6 x 5,000; N obst = 21,000 T av = 29 s Centralized planning 36 dofs

11. Centralized vs. Decoupled Planning Averages over 20 runs