[PPT] - L ECTURE 14: P OTENTIAL F IELDS FOR L OCAL P LANNING , N AVIGATION I PowerPoint Presentation

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16-311-Q INTRODUCTION TO ROBOTICS

LECTURE 14:

POTENTIAL FIELDS

FOR LOCAL PLANNING, NAVIGATION

INSTRUCTOR: GIANNI A. DI CARO

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RECAP: BEHAVIOR ARBITRATION

Environment

Sense Act

Behavior 3 Behavior n Behavior 1 Behavior 2

A r b i t r a t i

n

Conflict resolution module

It might be needed also to access / adjust sensory systems

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RECAP: BEHAVIOR ARBITRATION STRATEGIES

Averaging / Composition B1 ⊕ B2 Voting {R1, R2, R3}: X {R4}: Y

⇒ X

Least Commitment {R1}: DON’T X Fixed priority B1(t) ≻ B2(t), ∀t Alternate B2([t1,t2]), B1([t2,t3]) Variable priority B1(t1) ≻ B2(t1), B2(t2) ≻ B1(t2), Subsumption Suppression: BNew ≻ BOld Inhibition: BNew ⋀ BOld ⇒ ∅

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N AV I G AT I O N , L O C A L P L A N N I N G , G L O B A L P L A N N I N G

Robot has a global, detailed workspace map and has a model of its kino-

dynamics: it can plan the path or the trajectory to qGoal from the current configuration q (deliberative paradigm) General navigation task: define motion controls for (effectively) reaching a desired qGoal configuration while avoiding obstacles and being compliant with kinematic (and dynamic) constraints

Robot has a local, detailed workspace map and has a model of its kino-

dynamics: it can plan (iteratively) the local path or the trajectory towards qGoal from the current configuration q

Robot has partial, local, workspace information, collected during motion via

sensors (e.g., camera, range finder):

it can plan online the local path or the trajectory towards qGoal from the

current configuration q (deliberative approach), using kino-dyno knowledge

it can use a reactive approach, using local workspace information as a

(mostly memory-less) input to perform local navigation aiming to qGoal

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USE THE COMPOSITION APPROACH: POTENTIAL FIELDS

Motor schemas / Potential field methods for navigation tasks The robot is represented in configuration space as a particle under the influence of an artificial potential field U(q) which superimposes:

1. Repulsive forces from obstacles
2. Attractive force from goal(s)

U(q) = Uatt(q) + Urep(q) F ⃗(q) = −∇ ⃗ U(q)

Different behaviors feels different fields, and the arbiter combines their proposed motion vectors

Following a gradient descent moves the robot towards the minima (goal = global minimum)

R. Arkin, Behavior-Based Robotics, MIT Press, 1998

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POTENTIAL FIELDS AT WORK

Shape and mathematical properties of the potential functions matter …

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Oriolo: Autonomous and Mobile Robotics - Motion Planning 3

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objective: to guide the robot to the goal q g

attractive potential

two possibilities; e.g., in C = R2

paraboloidal conical

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AT T R A C T I V E P O T E N T I A L

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AT T R A C T I V E P O T E N T I A L

Oriolo: Autonomous and Mobile Robotics - Motion Planning 3

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paraboloidal: let e = q g — q and choose ka > 0
the resulting attractive force is linear in e
conical:
the resulting attractive force is constant

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AT T R A C T I V E P O T E N T I A L

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fa1 behaves better than fa2 in the vicinity of q g but

increases indefinitely with e continuity of fa at the transition requires

a convenient solution is to combine the two profiles:

conical away from q g and paraboloidal close to q g i.e., kb = ½ ka

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objective: keep the robot away from CO

repulsive potential

assume that CO has been partitioned in advance in

convex components COi

for each COi define a repulsive field

where kr,i > 0; ° = 2,3,...; ´ 0,i is the range of influence

f COi and

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R E P U L S I V E P O T E N T I A L

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equipotential contours the higher °, the steepest the slope Ur,i goes to 1 at the boundary of COi

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R E P U L S I V E P O T E N T I A L

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fr,i is orthogonal to the equipotential contour passing

through q and points away from the obstacle

the resulting repulsive force is
fr,i is continuous everywhere thanks to the convex

decomposition of CO

aggregate repulsive potential of CO

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R E P U L S I V E P O T E N T I A L

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total potential

force field:
superposition:

local minimum global minimum

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T O TA L P O T E N T I A L

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planning techniques

three techniques for planning on the basis of ft
1. consider ft as generalized forces:

the effect on the robot is filtered by its dynamics (generalized accelerations are scaled)

2. consider ft as generalized accelerations:

the effect on the robot is independent on its dynamics (generalized forces are scaled)

3. consider ft as generalized velocities:

the effect on the robot is independent on its dynamics (generalized forces are scaled)

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P L A N N I N G / N AV I G AT I O N U S I N G P O T E N T I A L F I E L D

Robot has mass (inertia)! Kinematics

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technique 1 generates smoother movements, while

technique 3 is quicker (irrespective of robot dynamics) to realize motion corrections; technique 2 gives intermediate results

strictly speaking, only technique 3 guarantees (in the

absence of local minima) asymptotic stability of q g; velocity damping is necessary to achieve the same with techniques 1 and 2

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P L A N N I N G / N AV I G AT I O N U S I N G P O T E N T I A L F I E L D

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on-line planning (is actually feedback!)

technique I directly provides control inputs, technique 2 too (via inverse dynamics), technique 3 provides reference velocities for low-level control loops the most popular choice is 3

off-line planning

paths in C are generated by numerical integration of the dynamic model (if technique 1), of (if technique 2), of (if technique 3) the most popular choice is 3 and in particular i.e., the algorithm of steepest descent

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P L A N N I N G / N AV I G AT I O N U S I N G P O T E N T I A L F I E L D

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local minima: a complication

if a planned path enters the basin of attraction of a

local minimum q m of Ut, it will reach q m and stop there, because ft (q m ) = —rUt(qm) = 0; whereas saddle points are not an issue

repulsive fields generally create local minima, hence

motion planning based on artificial potential fields is not complete (the path may not reach q g even if a solution exists)

workarounds exist but consider that artificial potential

fields are mainly used for on-line motion planning, where completeness may not be required

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L O C A L M I N I M A : A C O M P L I C AT I O N

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workaround no. 1: best-first algorithm

build a discretized representation (by defect) of Cfree

using a regular grid, and associate to each free cell of the grid the value of Ut at its centroid

planning stops when q g is reached (success) or no

further cells can be added to T (failure)

build a tree T rooted at q s: at each iteration, select the

leaf of T with the minimum value of Ut and add as children its adjacent free cells that are not in T

if success, build a solution path by tracing back the

arcs from q g to q s

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W O R K A R O U N D 1 : B E S T- F I R S T A L G O R I T H M

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best-first evolves as a grid-discretized version of

steepest descent until a local minimum is met

the best-first algorithm is resolution complete
at a local minimum, best-first will “fill” its basin of

attraction until it finds a way out

its complexity is exponential in the dimension of C,

hence it is only applicable in low-dimensional spaces

efficiency improves if random walks are alternated

with basin-filling iterations (randomized best-first)

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W O R K A R O U N D 1 : B E S T- F I R S T A L G O R I T H M

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workaround no. 2: navigation functions

path generated by the best-first algorithm are not

efficient (local minima are not avoided)

if the C-obstacles are star-shaped, one can map CO to

a collection of spheres via a diffeomorphism, build a potential in transformed space and map it back to C

a different approach: build navigation functions, i.e.,

potentials without local minima

another possibility is to define the potential as an

harmonic function (solution of Laplace’s equation)

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W O R K A R O U N D 2 : N AV I G AT I O N F U N C T I O N S

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an efficient alternative: numerical navigation functions
with Cfree represented as a gridmap, assign 0 to start

cell, 1 to cells adjacent to the 0-cell, 2 to unvisited cells adjacent to 1-cells, ... (wavefront expansion)

!! !" " ! ! ! ! # # # # $ $ $ $ % % % & & & & ' ' ' ' ' ' ' ( ( ( ( ( ( ( ( ( ) ) ) ) ) ) ) * * * * !" !" !" !! !! !# !# !# !# !$ !$ !$ !% !& !% !& !' !( !) !*

start goal solution path: steepest descent from the goal

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16-311-Q INTRODUCTION TO ROBOTICS

LECTURE 14:

POTENTIAL FIELDS

FOR LOCAL PLANNING, NAVIGATION

INSTRUCTOR: GIANNI A. DI CARO

RECAP: BEHAVIOR ARBITRATION

RECAP: BEHAVIOR ARBITRATION STRATEGIES

N AV I G AT I O N , L O C A L P L A N N I N G , G L O B A L P L A N N I N G

USE THE COMPOSITION APPROACH: POTENTIAL FIELDS

POTENTIAL FIELDS AT WORK

attractive potential

paraboloidal conical

AT T R A C T I V E P O T E N T I A L

AT T R A C T I V E P O T E N T I A L

AT T R A C T I V E P O T E N T I A L

increases indefinitely with e continuity of fa at the transition requires

conical away from q g and paraboloidal close to q g i.e., kb = ½ ka

repulsive potential

convex components COi

where kr,i > 0; ° = 2,3,...; ´ 0,i is the range of influence

R E P U L S I V E P O T E N T I A L

equipotential contours the higher °, the steepest the slope Ur,i goes to 1 at the boundary of COi

R E P U L S I V E P O T E N T I A L

through q and points away from the obstacle

decomposition of CO

R E P U L S I V E P O T E N T I A L

total potential

local minimum global minimum

T O TA L P O T E N T I A L

planning techniques

the effect on the robot is filtered by its dynamics (generalized accelerations are scaled)

the effect on the robot is independent on its dynamics (generalized forces are scaled)

the effect on the robot is independent on its dynamics (generalized forces are scaled)

P L A N N I N G / N AV I G AT I O N U S I N G P O T E N T I A L F I E L D

technique 3 is quicker (irrespective of robot dynamics) to realize motion corrections; technique 2 gives intermediate results

absence of local minima) asymptotic stability of q g; velocity damping is necessary to achieve the same with techniques 1 and 2

P L A N N I N G / N AV I G AT I O N U S I N G P O T E N T I A L F I E L D

technique I directly provides control inputs, technique 2 too (via inverse dynamics), technique 3 provides reference velocities for low-level control loops the most popular choice is 3

paths in C are generated by numerical integration of the dynamic model (if technique 1), of (if technique 2), of (if technique 3) the most popular choice is 3 and in particular i.e., the algorithm of steepest descent

P L A N N I N G / N AV I G AT I O N U S I N G P O T E N T I A L F I E L D

local minima: a complication

local minimum q m of Ut, it will reach q m and stop there, because ft (q m ) = —rUt(qm) = 0; whereas saddle points are not an issue

motion planning based on artificial potential fields is not complete (the path may not reach q g even if a solution exists)

fields are mainly used for on-line motion planning, where completeness may not be required

L O C A L M I N I M A : A C O M P L I C AT I O N

workaround no. 1: best-first algorithm

using a regular grid, and associate to each free cell of the grid the value of Ut at its centroid

further cells can be added to T (failure)

leaf of T with the minimum value of Ut and add as children its adjacent free cells that are not in T

arcs from q g to q s

W O R K A R O U N D 1 : B E S T- F I R S T A L G O R I T H M

steepest descent until a local minimum is met

attraction until it finds a way out

hence it is only applicable in low-dimensional spaces

with basin-filling iterations (randomized best-first)

W O R K A R O U N D 1 : B E S T- F I R S T A L G O R I T H M

workaround no. 2: navigation functions

efficient (local minima are not avoided)

a collection of spheres via a diffeomorphism, build a potential in transformed space and map it back to C

potentials without local minima

harmonic function (solution of Laplace’s equation)

W O R K A R O U N D 2 : N AV I G AT I O N F U N C T I O N S

cell, 1 to cells adjacent to the 0-cell, 2 to unvisited cells adjacent to 1-cells, ... (wavefront expansion)

start goal solution path: steepest descent from the goal

W O R K A R O U N D 3 : WAV E F R O N T E X PA N S I O N