Motion Planning and ControlUsing RRTs

[Selected Slides/Movies]

Michael M. Curtiss (MS Studennt)

Michael S. Branicky (Advisor)Electrical Engineering & Computer Science

Case Western Reserve University

May 2002

New Applications of RRTs

We have applied/extended them to:

• Nonlinear planning– pendulum swing-up, acrobot

• Prioritized multi-agent planning– air traffic control

Pendulum Swing-Up• A pendulum of mass m and

length l• Motor at joint can apply

discrete torques• Initial configuration: =0 (down), dot=0• Goal configuration: = (up), dot=0

Pendulum Swing-Up

• A pendulum of mass m and length l

• Motor at joint can apply torques {-1,0,1}

dual tree, 5600 nodes

single tree, 3300 nodes

Pendulum Swing-Up (Cont.)

Torques = {-4, -2, -1, 0, 1, 2, 4}, 4000 iterations

Acrobot Swing-Up

[adapted from Sutton & Barto]

Acrobot: The Movie

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• Planning for simplified air traffic control– Airplanes take off from one of six airports and fly to a

destination airport– Airplanes cannot occupy the same cell at the same time,

except adjacent to airports– Airplanes cannot fly directly in front of or behind other

airplanes (preventing swapping)

• Prioritized Planning– A path is planned for each agent in turn– Paths are treated as obstacles in space-time for all future

agents, and are immutable once planned

Multi-Agent Planning

Multi-Agent Planning: The Movie

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Nonholonomic Airplanes• Six airports• W-Space = [-1,1] x [-1,1]• Safety-radius of 0.03• Rate-constrained turning• Unicycle equations of motion:

• Prioritized Planning:

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Dynamic Safety Envelopes

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• Lregion=(v2/2)Amax

• Can always stop without

hitting other agents

Hybrid Trajectories

• Hybrid problems require finding valid trajectories from sinit to sgoal

• Trajectory is defined as a sequence of states s, where s=(x,q)

Hybrid RRT Algorithm

BUILD_RRT(sinit)

1 T.init(sinit); 2 for k=1 to K do

3 srand RANDOM_STATE();

4 EXTEND(T, srand); 5 return T

EXTEND(T, s)

1 snear NEAREST_METRIC_NEIGHBOR(s, T)

2 if (NEW_STATE(s, snear, snew, unew) then

3 T.add_vertex(snew)

4 T.add_edge(snear, snew, unew)

Hybrid RRT Changes

• Include discrete state in state space (S=XQ)• Redefine distance metric

– Non-trivial systems must account for discrete state changes in the distance metric

– Stair climbing example, (S=[-20,20]2{1,2,3,4}):

• Introduce switching as an operator– Unrestricted (switching control) or restricted (stair climber)– Autonomous (pogo stick) or controlled (gear shifting)

(c1,c2) = x1-x22 + 20 q1-q2

Stair-Climber Example

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