Multi-Step Motion Planning for Free-Climbing Robots

Tim Bretl, Sanjay Lall, Jean-Claude Latombe, Stephen Rock

Presenter: You-Wei Cheah

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

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?

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)

Model

• Configurations are defined by 15 parameters:– the position/orientation (xp, yp, θp) of the pelvis

– the joint angles (θ1, θ2, θ3) of each limb.

• Holds lie on inclined plane are defined by– a 2-D point (xi, yi)

– a 3-D point (vi)

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 Cσ4

• 3 supporting holds, motion occurs on a 6-D manifold Cσ3

• four additional constraints: quasi-static equilibrium, joint angle limits, joint torque limits, and (self-)collision

Climbing motion

• Switch between 3-hold and 4-hold stances• σ3 and σ4 are adjacent if σ4 = σ3 {i} for some ∪

hold i• Robot can only switched between adjacent

stances σ and σ’ at points qt F∈ σ ∩ Fσ′

• If continuous path connecting qs to a transition point exists in that component, then reachable

One-step climbing algorithm

• Tries to build a path from some qs to a goal qg

• 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

Multi-step Planning

• Given a stance σ, a start configuration qs F∈ σ, and a goal hold g:– construct a sequence of one-step motions that

will bring the robot to a stance σg that contains g.

• Graph search problem• Nodes in the graph are components of feasible

spaces and not particular configurations

Multi-step Planning

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

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 Tmax

Proving one-step disconnection

• Assumption: the feasible space Fσ can be represented as a semialgebraic set

• Fnd a polynomial function g R[x∈ 1, … , xn] such that g(qs) > 0, g(qg) < 0, and Pcut(g) is empty:

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

Thank you