Nonholonomic Motion Planning: Steering Using Sinusoids

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Nonholonomic Motion Planning Steering Using
Sinusoids
R. M. Murray and S. S. Sastry
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Motion Planning without Constraints

• Obstacle positions are known and dynamic
  constrains on robot are not considered.

From Planning, geometry, and complexity of robot
motion By Jacob T. Schwartz, John E. Hopcroft
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Problem with Planning without Constraints
Paths may not be physically realizable
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Mathematical Background

• Nonlinear Control System

• Distribution

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Lie Bracket

• The Lie bracket is defined to be

• The Lie bracket has the properties

1.)
(Jacobi identity)
2.)
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Physical Interpretation of the Lie Bracket
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Controllability

• A system is controllable if for any

• Chows Theorem

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Classification of a Lie Algebra

• Construction of a Filtration

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Classification of a Lie Algebra

• Regular

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Classification of a Lie Algebra

• Degree of Nonholonomy

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Classification of a Lie Algebra

• Maximally Nonholonomic

• Growth Vector

• Relative Growth Vector

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Nonholonomic Systems

• Example 1

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Nonholonomic Systems

• Example 2

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Phillip Hall Basis
The Phillip Hall basis is a clever way of
imposing the skew-symmetry of Jacobi identity
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Phillip Hall Basis

• Example 1

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Phillip Hall Basis

• A Lie algebra being nilpotent is mentioned
• A nilpotent Lie algebra means that all Lie
  brackets higher than a certain order are zero
• A lie algebra being nilpotent provides a
  convenient way in which to determine when to
  terminate construction of the Lie algebra
• Nilpotentcy is not a necessary condition

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Steering Controllable Systems Using Sinusoids
First-Order Systems

• Contract structures are first-order systems with
  growth vector

• Contact structures have a constraint which can be
  written

• Written in control system form

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Steering Controllable Systems Using Sinusoids
First-Order Systems
More general version
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Derive the Optimal Control First-Order Systems

• To find the optimal control, define the Lagrangian

• Solve the Euler-Lagrange equations

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Derive the Optimal Control First-Order Systems
Example
Lagrangian
Euler-Lagrange equations
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Derive the Optimal Control First-Order Systems

• Optimal control has the form

where is skew symmetric

• Which suggests that that the inputs are sinusoid
  at various frequencies

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Steering Controllable Systems Using Sinusoids
First-Order Systems Algorithm
yields
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Hopping Robot (First Order)

• Kinematic Equations
• Taylor series expansion at l0
• Change of coordinates

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Hopping Robot (First Order)

• Applying algorithm 1
• a. Steer l and ? to desired values by
• b. Integrating over one period

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Hopping Robot (First Order)

• Nonholonomic motion for a hopping robot

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Steering Controllable Systems Using Sinusoids
Second-Order Systems
Canonical form
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Front Wheel Drive Car (Second Order)

• Kinematic Equations
• Change of coordinates

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Front Wheel Drive Car (Second Order)

• Sample trajectories for the car applying
  algorithm 2

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Maximal Growth System

• Want vectorfields for which the P. Hall basis is
  linearly independent

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Maximal Growth Systems
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Chained Systems
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Possible Extensions
Canonical form associated with maximal growth 2
input systems look similar to a reconstruction
equation
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Possible Extensions

• Pull a Hattonplot vector fields and use the body
  velocity integral as a height function

• The body velocity integral provides a decent
  approximation of the systems macroscopic motion

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Plot Vector Fields