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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 has the properties The Lie bracket is defined to be 1.) 2.) (Jacobi identity)

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Physical Interpretation of the Lie Bracket

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Controllability Chow’s Theorem A system is controllable if for any

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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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Optimal control has the form Derive the Optimal Control: First-Order Systems Which suggests that that the inputs are sinusoid at various frequencies where is skew symmetric

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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 l=0 Change of coordinates

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Applying algorithm 1 a. Steer l and ψ to desired values by b. Integrating over one period Hopping Robot (First Order)

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Nonholonomic motion for a hopping robot Hopping Robot (First Order)

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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 Hatton…plot vector fields and use the body velocity integral as a height function The body velocity integral provides a decent approximation of the system’s macroscopic motion

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Plot Vector Fields

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