1. Nonholonomic Multibody Mobile Robots: Controllability and Motion Planning in the Presence of Obstacles By Jerome Barraquand and Jean-Claude Latombe Presenter: Yubin Zou

2. What is the problem? The motion of Multi-body mobile robot ▫See it as Multi-body mobile vehicle ▫tractors towing several trailers sequentially hooked The controllability of its motion ▫How to control multi-body vehicle for avoiding the obstacles and reaching the specific position

3. Outline Configuration state and Configuration space The constraint among bodies ▫Constraint equations The structure of planner ▫Tree Searching ▫Three key parameters Experiment Result

4. Configuration State and configuration space The simplest example: one-body vehicle In this case, the configuration state is (x, y,θ), so the configuration space is 3-dimension.

5. Extend to two-body vehicle Two-body vehicle The configuration state is (x, y,θ1,θ2), in other word, the configuration space is 4-dimension.

6. Extend to n-body vehicle N-body vehicle The configuration state is (x, y,θ1,θ2,…, θn), so the configuration space of n-body vehicle is n+2 dimension. Therefore, velocity state is (x’, y’,θ1’,θ2’,…, θn’), they are the velocity component following different dimensions. what is the relationship among (x, y,θ1,θ2,…, θn) and (x’, y’,θ1’,θ2’,…, θn’)?  constraint among bodies ▫How other bodies move when the tractor is moving?

7. The constraint among bodies The simple example: one-body vehicle Let [x’, y’, θ1’] represent the decomposition of velocity vector V, and x’, y’, θ1’ is the velocity component following different dimensions

8. Extend to n-body vehicle Here is the distance from kth body to (k+1)th body. n

9. Extend to n-body vehicle For a configuration state, (x, y,θ1,θ2,…, θn) is known, then (x’, y’,θ1’,θ2’,…, θn’) can be obtained through three equations only when are known.

10. Planner So far, we have already constructed the configuration space and known the constraint among parameters. Tree Searching ▫The discretization of the continuous configuration space (Decompose configuration space into a multiple of cells)

11. Cell The number of cells: ▫2^R(n+2) where n +2 is the number of dimension, and R is the resolution of decomposition, the bigger R make each cell smaller. Each cell has equal size ▫(∆x, ∆y, ∆θ1, ∆θ2,…, ∆θn) One cell is explored ▫One cell is said to be explored when it contains a configuration state which has already been expanded.

12. Tree Searching The initial State : P (x, y,θ1,θ2,…, θn), The goal State : P’ The successor function : Using three functions:

13. ▫After getting (x’, y’,θ1’,θ2’,…, θn’), integrate them with the constant internal time ∆t. ▫We can obtain the metric distance L between the expanded state and the successor state, then adding L to the expanded state can obtain the successor state (x, y,θ1,θ2,…, θn) (x’, y’,θ1’,θ2’,…, θn’) If x’, y’,θ1’,θ2’,…, θn’ are constant, then the successor state is (x + x’∆t, y + y’∆t, θ1+ θ1’∆t, θ2 + θ2’∆t,…, θn + θn’∆t) Action Expanded state Tree Searching(continued) input parameters output

14. Tree Searching(continued) ▫After getting (x’, y’,θ1’,θ2’,…, θn’), integrate them with the constant internal time ∆t. ▫We can obtain the metric distance L between the expanded state and the successor state, then adding L to the expanded state can obtain the successor state Actions: ▫The number of actions |V|* depends on r Because, r is the discretization parameters given to searching. are selected from [-90,90].

15. Tree Searching and planner Rules End Searching Condition ▫The current configuration state is in the same cell with the goal configuration state (Finish Task!) Searching Depth ▫It cuts searching at the depth H. Three key parameters of this planner ▫R: The resolution of configuration decomposition  Decide the search space ▫H: Tree Searching Depth ▫∆t: the interval time between current control and next control ▫These three parameters depend on the robot precision

16. Experiment Results R = 8

17. Experiment Results R = 8,

18. Experiment Results R = 9,

19. Thank you!