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 ${\ddot q}$, which is defined as $d{\dot q}/dt$. 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 ${\ddot q}$ in addition to velocity ${\dot q}$ and configuration $q$. Once again, both implicit and parametric models exist. For an implicit model, the constraints are expressed as

$$g_i({\ddot q},{\dot q},q) = 0 .$$ (13.27)

For a parametric model, they are expressed as

$${\ddot q}= f({\dot q},q,u) .$$ (13.28)