EE 451 - Path and Trajectory Planning H.I. Bozma November 12, 2014

Transcription

Outline
Path and Trajectory Planning
EE 451 - Path and Trajectory Planning
H.I. Bozma
Electric Electronic Engineering
Bogazici University
November 12, 2014
H.I. Bozma
Outline
Introduction
Path Planning Algorithms
Probabilistic Roadmap Methods (PRM)
Potential Fields
Trajectory Planning
H.I. Bozma
Outline
Introduction
Potential Fields
Trajectory Planning
Getting robot to move
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So far - Kinematics: The map btw configuration space DOF
and workspace position
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Velocity kinematics: The map btw configuration space
velocities and workspace velocities
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How to get the robot move and do what we want it to do?
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Assume: Initial and final configurations of the robot is given.
H.I. Bozma
Outline
Introduction
Potential Fields
Trajectory Planning
Getting robot to move - Motion planning
H.I. Bozma
Outline
Introduction
Potential Fields
Trajectory Planning
Path vs Trajectory Planning
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Path: A sequence of points (either in configuration or
workspace)
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Trajectory: A sequence of points with timing
H.I. Bozma
Outline
Introduction
Potential Fields
Trajectory Planning
Planning Considerations - Configuration (Joint) Space
H.I. Bozma
Outline
Introduction
Potential Fields
Trajectory Planning
Planning Considerations - Workspace
H.I. Bozma
Outline
Introduction
Potential Fields
Trajectory Planning
Obstacles
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No consideration of restrictions due to the workspace
(Obstacles!)
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Path and Trajectory Planning: Collision free paths or
trajectories
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Computationally complex!
H.I. Bozma
Outline
Introduction
Potential Fields
Trajectory Planning
Notation
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Configuration space Q
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Workspace W
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Obstacles in workspace Oi ⊂ W
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All obstacles O = ∪Oi
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Robot with q ∈ Q → A(q) ⊂ W
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Configuration space obstacles QO = {q ∈ Q | A(q) ∩ O =
6 ∅}
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Free configuration space Qfree = Q \ QO
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Initial configuration qs ∈ Qfree , Final configuration qf ∈ Qfree
H.I. Bozma
Outline
Introduction
Potential Fields
Trajectory Planning
Problem Statement:
Given qs , qf ∈ Qfree , find a collision-free path γ : [0, 1] → Qfree
such that γ(0) = qs and γ(1) = qf
H.I. Bozma
Outline
Introduction
Potential Fields
Trajectory Planning
Approaches
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Not feasible to generate Qfree completely!
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Search algorithm – Incremental generation of Qfree
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Roadmap methods
Potential fields
H.I. Bozma
Outline
Introduction
Potential Fields
Trajectory Planning
RM - General Approach
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Generate a representation for Qfree
Generate 1-D network of curves (time as varying parameter)
where each arc → Collision-free path btw two points
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Generate a path in Qfree from qs to qa
Generate a path in Qfree from qf to qb
Generate a path in Qfree from qa to qb
H.I. Bozma
Outline
Introduction
Potential Fields
Trajectory Planning
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Sampling the configuration space: A set of random
configurations are generated → Nodes
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Connect pairs of configurations: Plan trajectories between
nodes
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Enhancement: If trajectories are disconnected, generate new
paths to ensure connectivity
H.I. Bozma
Outline
Introduction
Potential Fields
Trajectory Planning
Example - PRM Approach
H.I. Bozma
Outline
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Advantages
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Introduction
Potential Fields
Trajectory Planning
No problems of local minima!
Problems
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Computational
Not reactive
H.I. Bozma
Outline
Introduction
Potential Fields
Trajectory Planning
Potential Field
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Imagine a surface U defined on workspace on which the robot
is visualized to move down the gradient
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Final configuration - Attractive (minimal point)
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Obstacle boundaries - Repelling
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Goal: One minimum (attractive) only with every other critical
point being nondegenerate!
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U(q) = Ua (q) + Ur (q) → F = Do U(q)
H.I. Bozma
Outline
Introduction
Potential Fields
Trajectory Planning
Attractive Field
Attractive force Ua,i =
1
T
2 ζi (oi (q) − oi (qf )) (oi (q) − oi (qf ))
1
1
T
2
2 ζi (oi (q) − oi (qf )) (oi (q) − oi (qf )) − 2 ζi d
H.I. Bozma
koi (q) − oi (qf )k ≤ d
koi (q) − oi (qf )k > d
Outline
Introduction
Potential Fields
Trajectory Planning
Example – RR Planar Robot
H.I. Bozma
Outline
Introduction
Potential Fields
Trajectory Planning
Repulsive Field
Ur ,i =
(
1
2 ηi
0
1
ρ(oi (q))
−
1
ρ0
2
H.I. Bozma
: ρ(oi (q)) ≤ ρ0
: ρ(oi (q)) > ρ0
Outline
Introduction
Potential Fields
Trajectory Planning
Example – RR Planar Robot, Repulsive Force
H.I. Bozma
Introduction
Potential Fields
Trajectory Planning
Outline
Repulsive Field
Ur ,2 =
1
2 η2
1
0.5
−1
2
1
0.52
0
−1
H.I. Bozma
Outline
Introduction
Potential Fields
Trajectory Planning
Mapping Workspace Forces to Joint Torques
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τ = JvT F
For each oi , construct the respective Jacobian Joi
P
P
Total torque τ = i JoTi Fai (q) + i JoTi Fri (q)
H.I. Bozma
Outline
Introduction
Potential Fields
Trajectory Planning
Gradient Descent Planning
1. q(0) = qs , k = 0
2. If kq(k) − qf k > (q(k))
q(k + 1) = q(k) + αi kττ (q(k))k
k =k +1
else return q
3. Go to (2)
H.I. Bozma
Outline
Introduction
Potential Fields
Trajectory Planning
General Performance
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Advantages
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Generate the torque trajectory to reach the goal
Reactive - As obstacles changes, so does the trajectory
Problems
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Stuck in local minima
May be difficult to compute attactive & repulsive forces
H.I. Bozma
Outline
Introduction
Potential Fields
Trajectory Planning
Revised Gradient Descent Planning
1. q(0) = qs , k = 0
2. If kq(k) − qf k > (q(k))
q(k + 1) = q(k) + αi kττ (q(k))k
k =k +1
else return q
3. If stuck in local minimum, move randomly to q 0
4. q(0) = q 0
5. Go to (2)
H.I. Bozma
Outline
Introduction
Potential Fields
Trajectory Planning
Why Trajectory Planning?
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In PRM’s, additional step of trajectory planning since we have
a sequence of points along the path!
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Discrete time computation:
γ : [0, 1] → Qfree s.t. γ(0) = qs and γ(1) = qf
Map this to time taken to do the task tf − t0
Convert to a sequence of joint configurations
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Inverse kinematics
Teach and playback
H.I. Bozma
Outline
Introduction
Potential Fields
Trajectory Planning
Planning a Trajectory
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Point to point – From q(0) → q(tf )
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Velocity constraints q(0)
˙
= κ0 q(t
˙ f ) = κtf (Usually both
κ0 = κtf = 0
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Acceleration constraints q¨(0) = α0 q¨(tf ) = αtf (Usually both
α0 = αt f = 0
H.I. Bozma
Outline
Introduction
Potential Fields
Trajectory Planning
Point to Point Trajectories
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P
Cubic polynomial trajectories q(t) = 4i=0 ai t i Discontinuties in jerk (Derivative of acceleration)
P
Quintic polynomial trajectories q(t) = 5i=0 ai t i
Linear segments with parabolic blends (LSPB)
Minimum time trajectory - A special case of LSPB
H.I. Bozma
Outline
Introduction
Potential Fields
Trajectory Planning
Cubic Polynomial Trajectory - Example
H.I. Bozma
Outline
Introduction
Potential Fields
Trajectory Planning
Quintic Polynomial Trajectory - Example
H.I. Bozma
Outline
Introduction
Potential Fields
Trajectory Planning
LSPB Trajectory - Example
H.I. Bozma
Outline
Introduction
Potential Fields
Trajectory Planning
Minimum Time Trajectory - Example
H.I. Bozma
Outline
Introduction
Potential Fields
Trajectory Planning
Multi-Point Trajectories
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A sequence of points – Endpoints are taken appropriately!
H.I. Bozma

EE 451 - Path and Trajectory Planning H.I. Bozma November 12, 2014

Transcription

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