Regularity Properties and Deformation of Wheeled Robots Trajectories - - PowerPoint PPT Presentation

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

• f 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?

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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 = (β1 . . . βh) 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)

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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]

(δm, δs) δe = δm + δs δm: mobility δs: steerability δe: maneuvrability

C: Caster F: Fixed S: Steering

Type Figure Examples Kinematic equations (3,0) Omni-directional robots ˙ x = η1 cos θ − η2 sin θ ˙ y = η1 sin θ + η2 cos θ ˙ θ = η3 (2,0) Differential drive ˙ x = −η1 sin θ ˙ y = η1 cos θ ˙ θ = η2 (2,1) Unicycle ˙ x = −η1 sin(θ + β) ˙ y = η1 cos(θ + β) ˙ θ = η2 ˙ β = ζ1 (1,1) Bicycle, car-like robots ˙ x = −η1L sin θ sin β ˙ y = η1L cos θ sin β ˙ θ = η1 cos β ˙ β = ζ1 (1,2) Kludge ˙ x = −η1(2L cos θ sin β1 sin β2 +L sin θ sin(β1 + β2)) ˙ y = −η1(2L sin θ sin β1 sin β2 −L cos θ sin(β1 + β2)) ˙ θ = η1 sin(β2 − β1) ˙ β1 = ζ1 ˙ β2 = ζ2 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

◮ and, in addition, the function arctan(˙

x/˙ y) is C 1 and piecewise C 2 (the angle of the tangent vector) (i.e. the trajectory is curvature continuous, because κ = ˙ θ/v)

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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)

˙ x = −η1 sin θ (1) ˙ y = η1 cos θ (2) ˙ θ = η2 (3)

◮ The admissible non-halting trajectories are exactly (x, y) where

x, y are C 1 and piecewise C 2 and arctan(˙ x/˙ y) is C 1 and piecewise C 2

◮ (⇒) Suppose η1 and η2 satisfy the assumptions (η1, η2 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

◮ one can compute η1 =

• ˙

x2 + ˙ y 2, C 0 and piecewise C 1

◮ one can compute θ = arctan(˙

x/˙ y), C 0 and piecewise C 1

◮ but to have η2 = ˙

θ C 0 and piecewise C 1, one need to assume, as a supplementary condition, that θ = arctan(˙ x/˙ y) is C 1 and piecewise C 2

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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

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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) ∀t ≥ τ C′(t) = F(C(t))

◮ Affine transformation: F(x) = u + Mx where u

is a translation and M is a matrix. Remark: the set

• f all affine transformations of the plane forms a group of

dimension 6 (2 for the translation and 4 for the matrix multiplication)

◮ Not all affine transformations deform C into an

admissible C′

◮ How to characterize the set of admissible affine

transformations?

Admissible

τ

Non-admissible Initial trajectory

[Pham, RSS 2011]

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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 arctan(˙ ˙x/˙ y) is C 1 and piecewise C 2 ⇒ the admissible affine deformations form a subgroup

• f dimension 1 of the general affine group

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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

• rientation corrections

−10 10 20 30 10 20 30 40 50

Feedback control

−10 10 20 30 5 10 15 20 25 30 35 40 45 50

Gap-filling for probabilistic planners

5 10 15 20 25 30 35 −10 −5 5 10 15 20 25 30 35

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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

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