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Overview of Part IV:
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Planning Algorithms
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Exercises
IV
. Planning Under Differential Constraints
Subsections
Overview of Part IV:
Planning Under Differential Constraints
13
. Differential Models
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Velocity Constraints on the Configuration Space
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Implicit vs. Parametric Representations
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Implicit representation
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Parametric constraints
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Conversion from implicit to parametric form
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Kinematics for Wheeled Systems
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A simple car
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A differential drive
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A simple unicycle
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A car pulling trailers
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Other Examples of Velocity Constraints
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Pushing a box
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Flying an airplane
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Rolling a ball
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Trapped on a surface
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Phase Space Representation of Dynamical Systems
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Reducing Degree by Increasing Dimension
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The scalar case
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The vector case
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Higher order differential constraints
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Linear Systems
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Nonlinear Systems
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Extending Models by Adding Integrators
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Better unicycle models
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A continuous-steering car
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Smooth differential drive
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Basic Newton-Euler Mechanics
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The Newtonian Model
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3
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Motions of Particles
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Motion of a single particle
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Motion of a set of particles
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Motion of a Rigid Body
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Advanced Mechanics Concepts
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Lagrangian Mechanics
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Calculus of variations
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Hamilton's principle of least action
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Applying actions
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Procedure for deriving the state transition equation
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2
General Lagrangian Expressions
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Conversion to a phase transition equation
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4
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Extensions of the Euler-Lagrange Equations
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Incorporating velocity constraints
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Nonconservative forces
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Hamiltonian Mechanics
13
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Multiple Decision Makers
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Differential Decision Making
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Differential Game Theory
Further Reading
Exercises
14
. Sampling-Based Planning Under Differential Constraints
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Introduction
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Problem Formulation
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Different Kinds of Planning Problems
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Terms from planning literature
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Terms from control theory
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Obstacles in the Phase Space
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Additional constraints on phase variables
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The region of inevitable collision
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Reachability and Completeness
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Reachable Sets
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Reachable set
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Time-limited reachable set
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Backward reachable sets
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2
The Discrete-Time Model
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Reachability graph
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Resolution completeness for
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Resolution completeness for
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Motion Primitives
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Sampling-Based Motion Planning Revisited
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3
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1
Basic Components
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Distance and volume in
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Sampling theory
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Collision detection
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System Simulator
14
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Local Planning
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4
General Framework Under Differential Constraints
14
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4
Incremental Sampling and Searching Methods
14
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Searching on a Lattice
14
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A double-integrator lattice
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Extensions and other considerations
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Incorporating State Space Discretization
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RDT-Based Methods
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Other Methods
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Feedback Planning Under Differential Constraints
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Problem Definition
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5
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Dynamic Programming with Interpolation
14
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Decoupled Planning Approaches
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Different Ways to Decouple the Big Problem
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2
Plan and Transform
14
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6
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3
Path-Constrained Trajectory Planning
14
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Expressing systems in terms of
,
, and
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3
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Determining the allowable accelerations
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3
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The path-constrained phase space
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4
Computing optimal solutions via dynamic programming
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A bang-bang approach for time optimality
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Gradient-Based Trajectory Optimization
Further Reading
Exercises
15
. System Theory and Analytical Techniques
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1
Basic System Properties
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Stability
15
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Lyapunov Functions
15
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3
Controllability
15
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Continuous-Time Dynamic Programming
15
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Hamilton-Jacobi-Bellman Equation
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The discrete case
15
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2
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The continuous case
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Variants of the HJB equation
15
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2
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2
Linear-Quadratic Problems
15
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2
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3
Pontryagin's Minimum Principle
15
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3
Optimal Paths for Some Wheeled Vehicles
15
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3
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1
Dubins Curves
15
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3
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2
Reeds-Shepp Curves
15
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3
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3
Balkcom-Mason Curves
15
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4
Nonholonomic System Theory
15
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4
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1
Control-Affine Systems
15
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4
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2
Determining Whether a System Is Nonholonomic
15
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4
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Completely integrable or nonholonomic?
15
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4
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2
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2
Distributions
15
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Lie brackets
15
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4
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2
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The Frobenius Theorem
15
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3
Determining Controllability
15
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4
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3
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The Lie algebra
15
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4
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3
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2
Lie algebra of the system distribution
15
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4
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3
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3
Philip Hall basis of a Lie algebra
15
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4
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3
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4
Controllability of driftless systems
15
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4
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3
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5
Handling Control-Affine Systems with Drift
15
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5
Steering Methods for Nonholonomic Systems
15
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5
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1
Using the P. Hall Basis
15
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5
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2
Using Sinusoidal Action Trajectories
15
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5
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2
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1
Steering the nonholonomic integrator
15
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5
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2
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2
First-order controllable systems
15
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5
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2
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3
Chained-form systems
15
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5
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3
Other Steering Methods
Further Reading
Exercises
Steven M LaValle 2009-09-20
