1. Search-Based Footstep Planning
Armin Hornung, Daniel Maier, Maren Bennewitz
Presentation by Dominique Gordon

2. Introduction

3. Humanoid Robots vs. Wheeled Robots
Step over obstacles
Many degrees of freedom
Not yet feasible to plan whole body motions in real world

4. Overview A* search algorithm wA* search algorithm
ARA* search algorithm
R* search algorithm
Extensions to 3D
Adaptive-Level-of-Detail Planning

5. Today!
First evaluation of anytime search-based footstep planning over long distances!

6. Search-Based Footstep Planning

7. Robot’s State State s = (x, y, 𝛳) of stance foot
Footstep action a = (∆x, ∆y, ∆𝛳) relative to stance foot
Expanding a state s - determine all successor states s’
s’s checked for collisions and discarded if invalid

8. A* Search Expands states according to f(s) = g(s) + h(s)
g(s) = actual cost of best path to current state
h(s) = heuristic that estimates cost to final goal
Heuristics must be ADMISSIBLE
h(s) ≤ the actual cost

9. A* Search with SLD as Heuristic
f(x) = = 449
f(x) = = 393
f(x) = = 447

10. A* Search Results Randomized Method
Finds optimal path
Performance dependent on quality of the heuristic
Slow
Randomized Method
Fast
No guarantees on solution quality

11. Anytime Search-Based Footstep Planning

12. Weighted A* Search (wA*)
Inflates h with a factor w≥1
w=1 is standard A*
Finds paths faster, expands fewer states
Bias towards states that are closer to goal
Trades quality of solution for efficiency

13. wA* Search with SLD as Heuristic, w=10
f(x) = (374) = 3815
f(x) = (253) = 2670
f(x) = (329) = 3408

14. Anytime Repairing A* (ARA*)
Series of wA* searches while reusing previous information
Initially large w
As time allows, search runs with lower values of w
With enough time, ARA* reaches w = 1
Cost of best solution is guaranteed no worse than w times optimal solution
speedup is achieved by not re-computing the state values that have been correctly computed in the previous iterations

15. Anytime Repairing A* (ARA*) explained
A* searches with decreasing w.
w = 2.5 w = 1.5 w =1
The corresponding ARA* search iterations.
w = 2.5 w = 1.5 w =1

16. ARA* Results Still depends on quality of heuristic
Heuristic of euclidean distance create local minima around obstacles
Heuristic of 2D Dijkstra path potentially inadmissible because no stepover

17. Randomized A* Search (R*)
Depends less on quality of heuristic
Series of short-range wA* searches
Generates k successor states at distance ∆ in random direction
Goal states within ∆ is added to successors as well
Collision checked
Finds “easy” local path - least number of expansions
If requires too many expansions, marked AVOID
Iteratively lowers w as time permits
In our experiments, we used a threshold of 100 expansions to detect that a path can not be found easily

18. R* explained by me K = 4, ∆ = 1, w = 1 (for demo purposes)
Heuristic is SLD
∆=1
g(S2)=5
∆=1
g(S1)=7 S2
h(S2)=4 S1 S3
h(S3)=3
h(S1)=5
Goal
∆=1
g(S3)=5
Start
h(S4)=3.5
∆=1
g(S4)=3 S4
f(S1) = = 12
f(S2) = 5 +4 = 9
f(S3) = 5 +3 = 8
f(S4) = = 6.5
At every iteration, R* selects the next state s to expand from
While normal A* expands s by generating all the immediate successors of state s, R* expands s by generating K states
Only looks at 4 states -- so fast!

19. R* with Euclidean distance heuristic

20. ARA* vs. R* Experiments
ARA* with 2D Dijkstra heuristic expands non-optimal path
Inadmissible
ARA* with Euclidean distance heuristic fails to find a plan in time
R* with Euclidean finds good initial solutions quickly

21. Extensions to 3D 3D sensing builds elevation map
Footstep state now also has height
Body of robot collision-checked against environment to execute step
Non-traversable if obstacle exceeds the max step height
Extended Dijkstra heuristic

23. Adaptive-Level-of-Detail Planning

24. Large open spaces with no obstacles Close to obstacles
2D path easy to compute
Easy for humanoid to follow with corresponding walking controller
Close to obstacles
Footstep planning allows robots to step close to or over obstacles
More efficient than 2D path

25. fast 2D planning in open spaces
Adaptive-Level-of-Detail planning combines
fast 2D planning in open spaces
with
footstep planning in areas with obstacles

26. Preprocessing: Classify the environment
open area vs. close to obstacles
traversable vs. non traversable obstacles

27. Execution of Adaptive-Level-of-Detail Planning
A walking controller to execute planned footsteps
A velocity controller to follow the 2D path

28. Experiments with Adaptive-Level-of-Detail Planning
2D-only path = fastest computation, not optimal
Global footstep plan = longest to compute, efficient path
Adaptive-Level-of-Detail planning combines benefits
Footstep planning only invoked where necessary
Path costs 2% higher than optimal, 51% lower than 2D

29. Summary Anytime algorithms ARA* R*
3D sensing and collision checking extends footstep planning to 3D
Adaptive level-of-detail planning

30. Personal Opinions Impressed by adaptive-level-of-detail planning
Great application of AI search algorithms
Didn’t explain ARA* and R* algorithms well
Lack of resources for algorithms

31. Questions?

32. Works Cited
Search-Based Footstep Planning by Armin Hornung, Daniel Maier, Maren Bennewitz
Anytime Search-Based Footstep Planning with Suboptimality Bounds by Armin Hornung, Andrew Dornbush, Maxim Likhachev, Maren Bennewitz
ARA*: Anytime A* with Provable Bounds on Sub-Optimality by Maxim Likhachev, Geogg Gorgon, Sebastian Thrun
R* Search by Maxim Likhachev and Anthony Stentz