1. Multi-Step Motion Planning for Free-Climbing Robots Tim Bretl, Sanjay Lall, Jean-Claude Latombe, Stephen Rock Presenter: You-Wei Cheah

2. The Climbing Problem Goal: Enable multi-limbed robots to free-climb vertical rock Applications: – search and rescue – cave exploration – human assistance in rock and mountain climbing

3. Probabilities-Roadmap (PRM) motion planning Widely used for path planning through high- dimensional configuration spaces with multiple constraints Can construct feasible paths quickly Lacks a formal stopping criterion Question: How much time to spent on query?

4. LEMUR IIb Consists of 4 identical limbs attached to a circular chasis Total mass: 7kg Each limb has 3 revote joins DOF’s:  2 in-plane (yaw)  1 out-of plane (pitch)

5. Model Configurations are defined by 15 parameters: – the position/orientation (x p, y p, θ p ) of the pelvis – the joint angles (θ 1, θ 2, θ 3 ) of each limb. Holds lie on inclined plane are defined by – a 2-D point (x i, y i ) – a 3-D point (v i )

6. Model Friction modeled using Coulomb’s law LEMUR IIb maintains 3-hold and 4-hold stances while climbing Set of supporting holds is a stance, denoted σ Robot’s continuous motion with 4 supporting holds occurs on a 3-D manifold C σ4 3 supporting holds, motion occurs on a 6-D manifold C σ3 four additional constraints: quasi-static equilibrium, joint angle limits, joint torque limits, and (self-)collision

7. Climbing motion Switch between 3-hold and 4-hold stances σ 3 and σ 4 are adjacent if σ 4 = σ 3 ∪ {i} for some hold i Robot can only switched between adjacent stances σ and σ’ at points q t ∈ F σ ∩ F σ ′ If continuous path connecting q s to a transition point exists in that component, then reachable

9. One-step climbing algorithm Tries to build a path from some q s to a goal q g Sample uniformly at random the goal region for a goal configuration Build a PRM in the space of configurations that are collision free and satisfy the equilibrium test

10. Multi-step Planning Given a stance σ, a start configuration q s ∈ F σ, and a goal hold g: – construct a sequence of one-step motions that will bring the robot to a stance σ g that contains g. Graph search problem Nodes in the graph are components of feasible spaces and not particular configurations

11. Multi-step Planning

12. Performance Analysis One-step planning – Most one-step moves were planned quickly – Difficult moves took more time Multi-step planning – Total planning time approaches linear growth

13. Proposed modifications Given each one-step motion query, run for a short length of time Attempt to prove the motion is infeasible if a solution is not found If disconnection proof is not found, allow planner to run for an additional T max

14. Proving one-step disconnection Assumption: the feasible space F σ can be represented as a semialgebraic set Fnd a polynomial function g ∈ R[x 1, …, x n ] such that g(q s ) > 0, g(q g ) < 0, and P cut (g) is empty:

15. Future work Robot will have to plan based on locally sensed information that is incomplete or uncertain Advances in computational algebra might be able to produce practical algorithms for proving disconnections

16. Thank you