Regularity Properties and Deformation of Wheeled Robots Trajectories Quang-Cuong Pham and Yoshihiko Nakamura Nakamura-Takano Laboratory Department of Mechano-Informatics University of Tokyo
Outline I: Regularity properties of wheeled robots trajectories II: Affine deformation of wheeled robots trajectories
Outline I: Regularity properties of wheeled robots trajectories II: Affine deformation of wheeled robots trajectories
I: Regularity properties of wheeled robots trajectories II: Affine deformation of wheeled robots trajectories Introduction: planar trajectories of wheeled robots ◮ Consider a planar path ( x , y ) of a wheeled robot ( x , y , θ ) ◮ Examples: ◮ Any wheeled robot must stop (halt) at the red point to avoid discontinuity of the velocity vector ◮ A car-like robot must halt at the red point to re-orient its front wheels (lack of curvature-continuity) [Fraichard and Scheuer 2004] ◮ Any wheeled robots can execute this path without halting ◮ Question: for a given wheeled robot, what are the paths that can be executed without halting? 1 / 11
I: Regularity properties of wheeled robots trajectories II: Affine deformation of wheeled robots trajectories General kinematic equations of wheeled robots ◮ General kinematic equations [Campion et al. 1996] � ˙ ξ = B ( ξ , β ) η ˙ = β ζ where ◮ ξ = ( x , y , θ ) ◮ β = ( β 1 . . . β h ) contains the steering angles of the steering wheels ( h = 0 if there is no such wheel) ◮ η : basically, the rotation velocities of the wheels ( = ˙ φ ) ◮ ζ : steering velocities of the wheels 2 / 11
I: Regularity properties of wheeled robots trajectories II: Affine deformation of wheeled robots trajectories Assumptions Assumptions on the inputs: ◮ ζ (the wheel steering velocities) is piecewise C 0 , but not necessarily C 0 ◮ This assumption is implicitly made when authors permit curvatures with discontinuous derivatives [Fraichard and Scheuer 2004] ◮ η (basically, the wheel rotation velocities) is C 0 and piecewise C 1 ◮ This assumption is implicitly made when authors require the car speed to be continuous Reformulation of the question: characterize the non-halting (i. e. v ( t ) > 0 for all t ) trajectories that can be generated by these inputs (admissible non-halting trajectories) 3 / 11
I: Regularity properties of wheeled robots trajectories II: Affine deformation of wheeled robots trajectories Classification of wheeled robots ◮ Wheeled robots can be classified in 5 types [Campion et al. 1996] Type Figure Examples Kinematic equations x ˙ = η 1 cos θ − η 2 sin θ Omni-directional (3,0) y ˙ = η 1 sin θ + η 2 cos θ robots ˙ = θ η 3 ( δ m , δ s ) ˙ = − η 1 sin θ x Differential ˙ = η 1 cos θ (2,0) y drive δ e = δ m + δ s ˙ θ = η 2 δ m : mobility x ˙ = − η 1 sin( θ + β ) y ˙ = η 1 cos( θ + β ) δ s : steerability (2,1) Unicycle ˙ θ = η 2 ˙ = β ζ 1 δ e : maneuvrability C: Caster x ˙ = − η 1 L sin θ sin β y ˙ = η 1 L cos θ sin β Bicycle, (1,1) ˙ F: Fixed car-like robots θ = η 1 cos β ˙ β = ζ 1 S: Steering x ˙ = − η 1 (2 L cos θ sin β 1 sin β 2 + L sin θ sin( β 1 + β 2 )) y ˙ = − η 1 (2 L sin θ sin β 1 sin β 2 − L cos θ sin( β 1 + β 2 )) (1,2) Kludge ˙ θ = η 1 sin( β 2 − β 1 ) ˙ = β 1 ζ 1 ˙ β 2 = ζ 2 4 / 11
I: Regularity properties of wheeled robots trajectories II: Affine deformation of wheeled robots trajectories Result ◮ Class I: For robots of degree of maneuvrability 3 (i.e. types (3,0), (2,1), (1,2)), a non-halting trajectory ( x , y ) is admissible if and only if x and y are C 1 and piecewise C 2 ◮ Class II: For robots of degree of maneuvrability 2 (types (2,0) and (1,1)) a non-halting trajectory ( x , y ) is admissible if and only if x and y are C 1 and piecewise C 2 ◮ y ) is C 1 and piecewise C 2 (the ◮ and, in addition, the function arctan (˙ x / ˙ angle of the tangent vector) (i.e. the trajectory is curvature continuous, because κ = ˙ θ/ v ) 5 / 11
I: Regularity properties of wheeled robots trajectories II: Affine deformation of wheeled robots trajectories Example of proof for a class II robot ◮ Consider e. g. a two-wheel differential drive: type (2,0) ˙ = − η 1 sin θ (1) x y ˙ = η 1 cos θ (2) ˙ θ = η 2 (3) ◮ The admissible non-halting trajectories are exactly ( x , y ) where x , y are C 1 and piecewise C 2 and arctan (˙ y ) is C 1 and piecewise C 2 x / ˙ ◮ ( ⇒ ) Suppose η 1 and η 2 satisfy the assumptions ( η 1 , η 2 are C 0 and pw C 1 ) ◮ From (3), one has that θ is C 1 and piecewise C 2 ; and arctan (˙ x / ˙ y ) = θ ◮ From (1) and (2), one has that x and y are C 1 and piecewise C 2 ◮ ( ⇒ ) Suppose x and y are C 1 and piecewise C 2 x 2 + ˙ y 2 , C 0 and piecewise C 1 ◮ one can compute η 1 = � ˙ y ) , C 0 and piecewise C 1 ◮ one can compute θ = arctan (˙ x / ˙ θ C 0 and piecewise C 1 , one need to assume, as a ◮ but to have η 2 = ˙ y ) is C 1 and piecewise C 2 supplementary condition, that θ = arctan (˙ x / ˙ 6 / 11
Outline I: Regularity properties of wheeled robots trajectories II: Affine deformation of wheeled robots trajectories
I: Regularity properties of wheeled robots trajectories II: Affine deformation of wheeled robots trajectories Deformation of trajectories ◮ Planning trajectories for nonholonomic robots (e.g. cars, submarines, quadrotors, satellites,...) is difficult and time-consuming ◮ When facing unexpected events (obstacle, change of goal position, state perturbations, etc.), it may therefore be more advantageous to deform a previously planned trajectory than re-plan anew 7 / 11
I: Regularity properties of wheeled robots trajectories II: Affine deformation of wheeled robots trajectories Affine deformations of trajectories ◮ A transformation F deforms a trajectory C = ( x ( t ) , y ( t )) t ∈ [0 , T ] into C ′ at a time instant τ by ∀ t < τ C ′ ( t ) = C ( t ) Initial trajectory τ C ′ ( t ) = F ( C ( t )) ∀ t ≥ τ ◮ Affine transformation: F ( x ) = u + Mx where u is a translation and M is a matrix. Remark: the set Admissible Non-admissible of all affine transformations of the plane forms a group of dimension 6 (2 for the translation and 4 for the matrix multiplication) [Pham, RSS 2011] ◮ Not all affine transformations deform C into an admissible C ′ ◮ How to characterize the set of admissible affine transformations? 8 / 11
I: Regularity properties of wheeled robots trajectories II: Affine deformation of wheeled robots trajectories Affine deformations of wheeled robots trajectories ◮ Assume the admissibility conditions discussed previously for wheeled robots trajectories ◮ Class I robots: admissible trajectories ( x , y ) are C 1 and piecewise C 2 ⇒ the admissible affine deformations form a subgroup of dimension 2 of the general affine group (which is of dimension 6) ◮ Class II robots: admissible trajectories ( x , y ) are C 1 and piecewise C 2 and y ) is C 1 and piecewise C 2 ⇒ the arctan (˙ ˙ x / ˙ admissible affine deformations form a subgroup of dimension 1 of the general affine group 9 / 11
I: Regularity properties of wheeled robots trajectories II: Affine deformation of wheeled robots trajectories Advantages of affine deformations ◮ Advantages of affine deformations ◮ single step (as opposed to iterative approximations) ◮ no trajectory re-integration ◮ exact, algebraic, corrections ◮ Examples Position and Feedback control Gap-filling for orientation corrections probabilistic planners 50 50 45 35 40 30 35 40 25 30 20 25 30 15 20 10 20 15 5 10 0 10 5 −5 0 −10 0 5 10 15 20 25 30 35 0 −10 0 10 20 30 −10 0 10 20 30 10 / 11
I: Regularity properties of wheeled robots trajectories II: Affine deformation of wheeled robots trajectories Conclusion ◮ We have classified wheeled robots in two classes in function of their degrees of maneuvrability and characterize the non-halting trajectories for each class ◮ Based on this characterization, we have identified the admissible affine deformations for each class of wheeled robots 11 / 11
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