13. Differential Models

This chapter provides a continuous-time counterpart to the state
transition equation,
, which was crucial in
Chapter 2. On a continuous state space, (assumed
to be a smooth manifold), it will be defined as
,
which intentionally looks similar to the discrete version. It will
still be referred to as a state transition equation. It will also be
called a *system* (short for *control system*), which is a
term used in control theory. There are no obstacle regions in this
chapter. Obstacles will appear again when planning algorithms are
covered in Chapter 14. In continuous time, the state
transition function yields a velocity as opposed to the
next state. Since the transitions are no longer discrete, it does not
make sense to talk about a ``next'' state. Future states that satisfy
the differential constraints are obtained by integration of the
velocity. Therefore, it is natural to specify only velocities. This
relies on the notions of tangent spaces and vector fields, as covered
in Section 8.3.

This chapter presents many example models that can be used in the
planning algorithms of Chapter 14. Section
13.1 develops differential constraints for the case in
which is the C-space of one or more bodies. These constraints
commonly occur for wheeled vehicles (e.g., a car cannot move
sideways). To represent dynamics, constraints on acceleration are
needed. Section 13.2 therefore introduces the *phase
space*, which enables any problem with dynamics to be expressed as
velocity constraints on an enlarged state space. This collapses the
higher order derivatives down to being only first-order, but it comes
at the cost of increasing the dimension of the state space. Section
13.3 introduces the basics of Newton-Euler mechanics
and concludes with expressing the dynamics of a free-floating rigid
body. Section 13.4 introduces some concepts from
advanced mechanics, including the Lagrangian and Hamiltonian. It also
provides a model of the dynamics of a kinematic chain of bodies, which
applies to typical robot manipulators. Section 13.5
introduces differential models that have more than one decision maker.

- 13.1 Velocity Constraints on the Configuration Space
- 13.1.1 Implicit vs. Parametric Representations
- 13.1.2 Kinematics for Wheeled Systems
- 13.1.3 Other Examples of Velocity Constraints

- 13.2 Phase Space Representation of Dynamical Systems
- 13.2.1 Reducing Degree by Increasing Dimension
- 13.2.2 Linear Systems
- 13.2.3 Nonlinear Systems
- 13.2.4 Extending Models by Adding Integrators

- 13.3 Basic Newton-Euler Mechanics

- 13.4 Advanced Mechanics Concepts
- 13.4.1 Lagrangian Mechanics
- 13.4.2 General Lagrangian Expressions
- 13.4.3 Extensions of the Euler-Lagrange Equations
- 13.4.4 Hamiltonian Mechanics

- 13.5 Multiple Decision Makers