p fields for local planning, navigationgdicaro/16311-fall17/slides/... · 2017. 9. 30. · oriolo:...

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 o 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)

F⃗(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 …

Oriolo: Autonomous and Mobile Robotics - Motion Planning 3 4

• 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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• 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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• 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


• 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 of COi and

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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 1at the boundary of COi

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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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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


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


• 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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• 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 planningpaths 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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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


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


• 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


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


• 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)

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start goal

solution path: steepest descent from the goal

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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

p fields for local planning, navigationgdicaro/16311-fall17/slides/... · 2017. 9. 30. · oriolo:...

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