Often an autonomous mobile vehicle can be characterized as nonholonomic, or is said to have nonholonomic constraints. Such constraints can have several effects on autonomous vehicles.
1) A vehicle is limited in how it can move. It has fewer local degrees of freedom than it has globally. In other words, while a position and orientation (i.e., a posture) may be achievable, a vehicle's current configuration often limits its ability to reach the posture without careful maneuvering or judicious use of its local degrees of freedom.
2) Variables describing vehicle position and orientation are dependent on each other. For example, an automobile (a tricycle-like vehicle) cannot alter its orientation without changing its position. It cannot spin "on a dime." An aspect that must be considered for such a vehicle is its minimum turning radius described later in this chapter.
3) Variables describing vehicle position and orientation depend on a history of previous values.1 A primary difficulty in developing a path generator for a nonholonomic vehicle lies in giving a path to the vehicle that fits within its local degrees of freedom. Several examples of nonholonomic vehicles and their equations of motion are now given.
Examples of nonholonomic vehicles used throughout this thesis are the dual wheel and tricycle. The wheel, as a basic nonholonomic vehicle, is included here as a reference.
The wheel as considered here is constrained to roll without slipping. It is constrained to move in the direction that the wheel is pointing; that is, it can not move sideways. This is the nonholonomic constraint.
This vehicle type is shown in Figure 1. x and y locate the wheel center point in two dimensions. Theta is the angle subtended by the x-axis and the wheel direction, and v is the wheel velocity.
Dual wheel can be thought of as similar to a military tank; a vehicle driven by two independent, parallel motorized wheels. As with the wheel, it is also constrained to move in the direction that it is pointing. This is the nonholonomic constraint.
This vehicle type is shown in Figure 2. The midpoint of the axis connecting the wheel centers is at (x, y). Vehicle orientation with respect to the x-axis is given by (, and vehicle velocity is denoted by ).
The tricycle is subject to two nonholonomic constraints. The first is that the velocity of the front wheel is in the direction that it is pointing. The second is that the rear wheels can not move sideways. T
The midpoint of the axis connecting the rear wheel centers is denoted by (x, y). The distance between the rear wheel axis and the front wheel center is l. The orientation of the tricycle body with respect to the x-axis is (, and the steering wheel angle with respect to the tricycle body is (. The tricycle body velocity is (. C is the intersection point of the rear wheel axis and the steering wheel axis. If the front wheel angle were fixed, C would be the center of the circle circumscribed by the tricycle. s is the arc length of the path traversed by the rear axis midpoint. The distance between C and the rear axis midpoint is r(s).
Tricycle differs from the Wheel and Dual Wheel in that the speed, (, and the steering angle, (, determine the body turning rate, .