16-311-Q I NTRODUCTION TO R OBOTICS L ECTURE 14: P OTENTIAL F IELDS FOR L OCAL P LANNING , N AVIGATION I NSTRUCTOR : G IANNI A. D I C ARO
RECAP: BEHAVIOR ARBITRATION Conflict resolution It might be needed also to module access / adjust sensory systems A r Behavior 1 b i Behavior 2 t r a Behavior 3 t i Behavior n o n Sense Environment Act 2
RECAP: BEHAVIOR ARBITRATION STRATEGIES Voting Fixed priority {R1, R2, R3}: X B1(t) ≻ B2(t), ∀ t {R4}: Y Alternate ⇒ X B2([t 1 ,t 2 ]), B1([t 2 ,t 3 ]) Variable priority Averaging / B1(t 1 ) ≻ B2(t 1 ), Composition B2(t 2 ) ≻ B1(t 2 ), B1 ⊕ B2 Subsumption Suppression: B New ≻ B Old Least Commitment Inhibition: B New ⋀ B Old ⇒ ∅ {R1}: DON’T X 3
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 General navigation task: define motion controls for (e ff ectively) reaching a desired q Goal configuration while avoiding obstacles and being compliant with kinematic (and dynamic) constraints • Robot has a global, detailed workspace map and has a model of its kino- dynamics: it can plan the path or the trajectory to q Goal from the current configuration q ( deliberative paradigm) • 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 q Goal 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 q Goal 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 q Goal 4
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: U(q) = U att (q) + U rep (q) ⃗ (q) = − ∇ ⃗ U(q) F 1. Repulsive forces from obstacles 2. Attractive force from goal(s) 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 5
POTENTIAL FIELDS AT WORK Shape and mathematical properties of the potential functions matter … 6
attractive potential AT T R A C T I V E P O T E N T I A L • objective: to guide the robot to the goal q g • two possibilities; e.g., in C = R 2 paraboloidal conical Oriolo: Autonomous and Mobile Robotics - Motion Planning 3 4 7
AT T R A C T I V E P O T E N T I A L • paraboloidal: let e = q g — q and choose k a > 0 • the resulting attractive force is linear in e • conical: • the resulting attractive force is constant Oriolo: Autonomous and Mobile Robotics - Motion Planning 3 8 5
AT T R A C T I V E P O T E N T I A L • f a 1 behaves better than f a 2 in the vicinity of q g but increases indefinitely with e • a convenient solution is to combine the two profiles: conical away from q g and paraboloidal close to q g continuity of f a at the transition requires i.e., k b = ½ k a Oriolo: Autonomous and Mobile Robotics - Motion Planning 3 9 6
repulsive potential R E P U L S I V E P O T E N T I A L • objective: keep the robot away from CO • assume that CO has been partitioned in advance in convex components CO i • for each CO i define a repulsive field where k r , i > 0 ; ° = 2,3,... ; ´ 0, i is the range of influence of CO i and Oriolo: Autonomous and Mobile Robotics - Motion Planning 3 7 10
R E P U L S I V E P O T E N T I A L the higher ° , equipotential the steepest the slope contours U r , i goes to 1 at the boundary of CO i Oriolo: Autonomous and Mobile Robotics - Motion Planning 3 11 8
R E P U L S I V E P O T E N T I A L • the resulting repulsive force is • f r , i is orthogonal to the equipotential contour passing through q and points away from the obstacle • f r , i is continuous everywhere thanks to the convex decomposition of CO • aggregate repulsive potential of CO Oriolo: Autonomous and Mobile Robotics - Motion Planning 3 12 9
total potential T O TA L P O T E N T I A L • superposition: • force field: global minimum local minimum Oriolo: Autonomous and Mobile Robotics - Motion Planning 3 10 13
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 planning techniques • three techniques for planning on the basis of f t 1. consider f t as generalized forces: the effect on the robot is filtered by its dynamics (generalized accelerations are scaled) Robot has mass (inertia) ! 2. consider f t as generalized accelerations: the effect on the robot is independent on its dynamics (generalized forces are scaled) Kinematics 3. consider f t as generalized velocities: the effect on the robot is independent on its dynamics (generalized forces are scaled) Oriolo: Autonomous and Mobile Robotics - Motion Planning 3 11 14
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 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 Oriolo: Autonomous and Mobile Robotics - Motion Planning 3 12 15
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 • 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 • 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 Oriolo: Autonomous and Mobile Robotics - Motion Planning 3 16 13
L O C A L M I N I M A : A C O M P L I C AT I O N local minima: a complication • if a planned path enters the basin of attraction of a local minimum q m of U t , it will reach q m and stop there, because f t ( q m ) = — r U t ( q m ) = 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 Oriolo: Autonomous and Mobile Robotics - Motion Planning 3 14 17
workaround no. 1: best-first algorithm 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 • build a discretized representation (by defect) of C free using a regular grid, and associate to each free cell of the grid the value of U t at its centroid • build a tree T rooted at q s : at each iteration, select the leaf of T with the minimum value of U t and add as children its adjacent free cells that are not in T • planning stops when q g is reached (success) or no further cells can be added to T (failure) • if success, build a solution path by tracing back the arcs from q g to q s Oriolo: Autonomous and Mobile Robotics - Motion Planning 3 15 18
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 • at a local minimum, best-first will “fill” its basin of attraction until it finds a way out • the best-first algorithm is resolution complete • 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) Oriolo: Autonomous and Mobile Robotics - Motion Planning 3 19 16
workaround no. 2: navigation functions 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 • path generated by the best-first algorithm are not efficient (local minima are not avoided) • a different approach: build navigation functions, i.e., potentials without local minima • 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 • another possibility is to define the potential as an harmonic function (solution of Laplace’s equation) Oriolo: Autonomous and Mobile Robotics - Motion Planning 3 17 20
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