nonholonomic multibody mobile robots controllability and motion planning in the presence of obstacles

Nonholonomic Multibody Mobile Robots: Controllability and Motion Planning in the Presence of Obstacles Authors: Jerome Barraquand and Jean-Claude Latombe Published: 1991 Presented by: Jason Haas Main Contributions  Application of Controllability Rank Condition Theorem resulting in a general result on the controllability of nonholonomic robots  Application to multibody mobile robots – controllability results, even with inequality kinematic constraints  Implementation of planner for one- and two-body mobile robots Approach – Main Idea  Divide up the path into small steps  Small enough step size guarantees correctness  Pragmatic method for choosing granularity  Compute the control or next step repeatedly Controllability Generalization  Piecewise constant control inputs  Nonlinear control concepts  Accessibility  U-Accessibility  Weak U-Accessibility  Controllable  Locally  Weakly  Locally weakly  Lots of subtleties  System model – Controllability – U-Accessibility 0 q 1 q )( 0 qA U U ≡ subset of Controllability – Weak U- Accessibility Piece the accessible sets together 0 q 1 q )( 0 qA U 2 q )( 1 qA U )( 2 qA U )()()()( 2100 qAqAqAqWA UUUU ∪∪= Controllability – Controllable  A system is controllable if and only if ∀ q 0 ∈  Any state can reach any other state  System is locally controllable if and only if ∀ q ∈ is a neighborhood of q  Neighborhood is an open subset  Local controllability implies controllability via patching Controllability – Locally Controllable Controllability – Weak Controllability  A system is weakly controllable at q 0 if and only if Not a neighborhood, not an open subset  “A system is locally weakly controllable at q 0 if for every neighborhood U of q 0 , is also a neighborhood of q 0 ∀ q 0 ∈ .”  Weak controllability implies controllability via patching CqWA C =)( 0 )( 0 qWA U Controllability – Symmetry  Definition symmetric: accessibility relation (U- accessibility or weak U-accessibility) is symmetric (i.e. applies q 0 → q 1 and q 1 → q 0 ).  Local controllability implies controllability  Local weak controllability implies weak controllability if symmetric system [...]... above form nonintegrable /nonholonomic? (integrability) 2 Do constraints of the above form “restrict the set of configurations reachable from any given configuration?” (controllability) System Classification – Constraints  Set of k < n independent kinematic constraints    G ( q, q ) = ( G1 ( q, q ) ,  , G k ( q, q ) ) = ( 0,  ,0)  Definition – Gq = G (q,⋅)  − Subset of tangent space defined by Gq... a fine enough grained search Asymptotic optimality: if a solution path exists, the planner generates the solution with the minimal number of reversals (changes of sign of linear velocity) Practical only for 1-2 bodies (1991) Planner – System Model  No slipping Car / tractor –  Trailer –  Planner – Input      Start and goal configuration System model (equations of motion, constraints) Steering... (Frobenius)  > n-k ⇒ nonintegrable ⇒ nonholonomic  = n-k ⇒ integrable ⇒ holonomic Two propositions answer integrability question  1 Proplerly nonlinear kinematic constraints are nonholonomic  2 Holonomic ⇔ dim{ CLA( F )(q )} + dim{ G ( q, q, t )} = n    necessarily linear in q i.e can integrate ω (q) ⋅ q = 0  ⇒ LWC ⇒ controllable Planner – Claims     Applicable to multi-body mobile robots  Cars –... optimality  Any solution path can be tracked within ε  Search using Dijkstra’s algorithm will find minimum path  Need to keep 2x configuration space grid state Planner – Optimality (2)   Reapply proof for single-body Modify controls  Make steering angle granularity finer φ ∈ {φmin , φmin + δ iφ , , φmin + ( ri − 1)δ iφ } δ iφ = ( φmax − φmin ) ( ri − 1)  Why? Motion Planner – Complexity  Time  Overall... defined by Gq 1 (0, ,0)  Chart defined by Implicit Function Theorem u = ( uk +1 ,  , u n ) (independent) – mapping free from system model – System Classification – Equivalence  System equivalent to nonlinear control system  Kinematic inequalities on velocities map to inequalities on controls  Inequalities do not reduce dimension of control, only determine shape of control space System Classification... tangent to is integrable (tangent hyperplane integrated from subhyperplanes) Theorem: two conditions equivalent Controllability Rank Condition satisfied ⇔ (Chow, 1939)  CLA(F) = distribution  Vector ↔ basis, vector field ↔ distribution Locally weakly controllable (controllable) System Classification – Questions System Model   Constraints  G ( q, q, t ) = 0  ω (q) ⋅ q = 0 1 Are constraints of the above... cosθ dy/dt = v sinθ φ y θ θ dθ/dt = (v/L) tan φ φ L dx sinθ – dy cosθ = 0 |φ| < Φ x Configuration space is 3-dimensional: q = (x, y, θ) But control space is 2-dimensional: (v, φ) with |v| = sqrt[(dx/dt)2+(dy/dt)2] * Slide obtained from J.-C Latombe – Stanford CS 326 slides Lie Bracket* Maneuver made of 4 motions For example: -X X: Going straight X = (ν cos θ ,ν sin θ ,0 ) T -Y Y: Turning, angle φ T... Start and goal configurations live in same connected component of free configuration space Asymptotic completeness  Suppose solution τ of length N exists  Choose H≥N  Choose δt0 small enough 0 < d (τ , C − obstacles ) ≤ η 0 ≤ d (τ ,τ ε ) < ε < η  Bounds on R Planner – Optimality (1)   Assumption: some path exists with a finite number of reversals for a single-body mobile robot Asymptotic optimality... [X,Y] (δ t2 ) ν   Y = ν cos θ ,ν sin θ , tan φ  L   Lie bracket * Slide obtained from J.-C Latombe – Stanford CS 326 slides Y X (δt) Control Lie Algebra (2)  Recursively compute Lie brackets to find maximal distribution  Find hidden degrees of freedom  External product (e.g cross product)   Defines tangent space (where q lives) Frobenius Integrability Theorem      Condition 1 – distribution... allowed by constraints  Search node ties resolved with minimum length of curve of P1 Planner – Results (1)  Parallel parking  R = 8, φ ∈ [−30°,30°]  Run time = 20 s Planner – Results (2)  Cluttered workspace  R = 8, φ ∈ [−45°,45°]  Run time = 30 s Planner – Results (3)  Random obstacles  R = 9, φ ∈ [−45°,45°]  Run time = 120 s Planner – T-T Results (1)  Tractor-trailer parking  R = 7, φ ∈ . Nonholonomic Multibody Mobile Robots: Controllability and Motion Planning in the Presence of Obstacles Authors: Jerome Barraquand and Jean-Claude Latombe Published:. Haas Main Contributions  Application of Controllability Rank Condition Theorem resulting in a general result on the controllability of nonholonomic robots  Application to multibody mobile robots. constraints of the above form nonintegrable /nonholonomic? (integrability)  2. Do constraints of the above form “restrict the set of configurations reachable from any given configuration?” (controllability) (