Post on 03-Apr-2018
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13.2 Phase Space Representation of Dynamical
Systems
The differential constraints defined in Section 13.1 are often called kinematic because they can be expressed in
terms of velocities on the C-space. This formulation is useful for many problems, such as modeling the possible
directions of motions for a wheeled mobile robot. It does not, however, enable dynamics to be expressed. For
example, suppose that the simple car is traveling quickly. Taking dynamics into account, it should not be able to
instantaneously start and stop. For example, if it is heading straight for a wall at full speed, any reasonable model
should not allow it to apply its brakes from only one millimeter away and expect it to avoid collision. Due to
momentum, the required stopping distance depends on the speed. You may have learned this from a drivers
education course.
To account for momentum and other aspects of dynamics, higher order differential equations are needed. There
are usually constraints on acceleration , which is defined as . For example, the car may only be able to
decelerate at some maximum rate without skidding the wheels (or tumbling the vehicle). Most often, the actions
are even expressed in terms of higher order derivatives. For example, the floor pedal of a car may directly set the
acceleration. It may be reasonable to consider the amount that the pedal is pressed as an action variable. In this
case, the configuration must be obtained by two integrations. The first yields the velocity, and the second yields
the configuration.
The models for dynamics therefore involve acceleration in addition to velocity and configuration . Once
again, both implicit and parametric models exist. For an implicit model, the constraints are expressed as
(13.27)
For a parametric model, they are expressed as
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(13.28)
Subsections
13.2.1 Reducing Degree by Increasing Dimension
13.2.1.1 The scalar case
13.2.1.2 The vector case
13.2.1.3 Higher order differential constraints
13.2.2 Linear Systems
13.2.3 Nonlinear Systems
13.2.4 Extending Models by Adding Integrators
13.2.4.1 Better unicycle models
13.2.4.2 A continuous-steering car
13.2.4.3 Smooth differential drive
Next: 13.2.1 Reducing Degree by Up: 13. Differential Models Previous: 13.1.3.4 Trapped on a
Steven M Lavalle 2010-04-24
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