CSCI 7000 ADVANCED ROBOTICS

~VISHAL VERMA

Rapidly Exploring Random Trees

Agenda

Intro to Motion PlanningProblem FormulationIntro to RRTsRRT Algorithm RRT AnalysisImplementing RRT PlannersExamples

Intro to Motion Planning

Used in: Robotics (:D) Spacecraft Computer Graphics/Animations Computational Biology Virtual Prototyping Vehicle safety

Problem Formulation

State Space ‘X’

init X and Xgoal XCollision Detector D : X {true,false}Inputs – Set ‘U’ of actionsIncremental simulator: calculate x(t+t) due to

u(t)Metric ‘ρ’: for distance Search for continuous path

Intro to RRTs

Search high dimensional spacesConsider algebraic constraints

ObstaclesConsider local constraints

Differential constraints of Motion Non Holonomic constraints

Differential Constraints

Constraints from mechanical constructionExamples:

0 0 0 + 0

Non Holonomic Constraints

Controllable DOF < Total DOFExample – Car:

Total DOF – 3 [x,y,θ] Controllable DOF – 2 [x, θ]

Constraints introduced: Cannot make sharp turns

Concept of RRTs

Intuitively: Monte-Carlo Search Biased to favor largest Voronoi regions

Binary Tree: Searched Systematically NP-Hard

RRT: Searched (pseudo)randomly Pull tree toward unexplored portions

RRT - Justification

Other similar options: Randomized Potential Field Method:

Depends on a good heuristic potential function Difficult to find with obstacles/Differential Constraints

Probabilistic Roadmap approach Generates many random configurations Connects with local planner Good for Holonomic Local planner too complicated for non holonomic Needs non-linear control system

RRT Algorithm

RRT Analysis

Probabilistically Complete

Probability of success increases exponentially with vertices

RRT as Disc:

Implementing RRT Planners

RRTs as building blocksSingle v/s Double RRT PlannersNo biasing -> slow convergenceBiasing:

RRT-GoalBias RANDOM_STATE() modeled as coin toss with small bias

to goal RRT-GoalZoom

RANDOM_STATE() modeled with bias to area around goal

Size of area closest RRT Vertex to goal

The RRT CONNECT() Routine

Replaces EXTEND()Multiple calls to EXTEND()

Better for holonomic planning

EXTEND() still better for non-holonomic

Lack of good metric

Bidirectional RRT

Further Thoughts

More than 2 RRTs? Computation time divided

Construct RRTs/Explore state space Interconnect RRTs

Probabilistic Roadmap: Limiting/Extreme version of this Max separate RRTs merged

Example: Growing RRT

Example: Holonomic Planning

Example: Holonomic Planning

Example: Holonomic Planning

Example: Non - Holonomic Planning

Example: Non - Holonomic Planning

Example: Non - Holonomic Planning

Example: Kinodynamic Planning

Questions?