1. CS 326 A: Motion Planning http://robotics.stanford.edu/~latombe/cs326/2004 Target Tracking and Virtual Cameras

2. Papers First paper: H.H. Gonzalez-Banos, C.Y. Lee, and J.C. Latombe. Real-Time Combinatorial Tracking of a Target Moving Unpredictably Among Obstacles. Proc. IEEE Int. Conf. on Robotics and Automation, 2002. Second paper: D. Nieuwenhuisen, M.H. Overmars. Motion planning for Camera Movements in Virtual Environments. 2003.

3. General Problem  Placement of camera / selection of viewpoint  Make “important” things visible, e.g, keep moving object (target) in view  Natural motion of camera

4. What is Known in Advance?  Environment?  Target trajectory? No distance constraint Constant distance target robot

5. What Goal is Possible?  Always keep the target in sight  Keep the target in sight as long as possible  Minimize time when target is not visible

6. Techniques  Known environment and target trajectory  Back-chaining of visibility regions (open-loop strategy)  Known environment, but unknown trajectory  Real-time dynamic programming, with horizon h

7. Target Tracking target robot

8. States Are Indexed by Time State = (robot-position, target-position, time) Action = (stop, up, down, right, left) Outcome of an action = 5 possible states, each with probability 0.2 ([i,j], [u,v], t) ([i+1,j], [u,v], t+1) ([i+1,j], [u-1,v], t+1) ([i+1,j], [u+1,v], t+1) ([i+1,j], [u,v-1], t+1) ([i+1,j], [u,v+1], t+1) right Each state has 25 successors

9. h-Step Planning Process Planning horizon h “Terminal” states: States where the target is not visible States at depth h Reward function: Target visible  +1 Target not visible  0 Maximizing the sum of rewards ~ maximizing escape time R (state) =  t where 0 <  < 1 discounting

10. h-Step Planning Process Planning horizon h The planner computes the optimal choice for the first step. This step is executed. And everything is repeated again … (sliding horizon)

11. h-Step Planning Process Planning horizon h h is chosen such that the computation over the tree can be done in one increment of time (real-time constraint)

12. Techniques  Known environment and target trajectory  Back-chaining of visibility regions (open-loop strategy)  Known environment, but unknown trajectory  Real-time dynamic programming, with horizon h One advantage: Flexibility Two difficulties: - Computation of visibility regions - Large branching factor

13. Techniques  Known environment and target trajectory  Back-chaining of visibility regions (open-loop strategy)  Known environment, but unknown trajectory  Real-time dynamic programming, with horizon h  Unknown environment and target trajectory  Risk-based approach (1 st paper)