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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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