information while underwater. (Washington Post, Tuesday, December 15, 2009, health and science Section pages E1 and E6.) As was mentioned, an important application of AUVs such as this is data collection of variables such as water temperature and salinity as a function of location, including depth.
Examples of mobile robots in manufacturing facilities include wheeled vehicles used for material transfer from one work station to another. Here a line painted on the floor may designate the path for the mobile robot to follow. Optical sensors sense the boundaries of the line and give commands to the steering system to cause the mobile robot to follow along the track. Schemes such as this can also be used for mobile robots whose assignment is to perform inventory checks or security checks in a large facility such as a warehouse. Here the path for the mobile robot is specified and the sensors acquire and store the required information as the robot makes its rounds.
There are two basic types of steering used by mobile robots operating on the ground. For both of these types of steering, the mobile robot may have one or two front wheels. One type is front‐wheel steering much like that of an automobile. This type of steering presents interesting challenges to the controller, because it yields a nonzero turning radius. This radius is limited by the length of the robot and the maximum steering angle.
The other type of steering involves independent wheel control for each side. By rotating the left and right wheels in opposite directions at the same speed, the robot can be made to turn while in place, i.e., at a zero turning radius. Tracked vehicles use this same type of differential‐drive steering strategy, there often referred to as skid steering.
Examples of mobile robots also include, as we mentioned earlier, AUVs such as underwater gliders, whose diverse applications range from oil/gas exploration and environmental monitoring to search and rescue and national harbor security. Due to the complex interaction between surrounding fluid and AUVs, hydrodynamics play an important role in determining vehicle dynamics which exhibits high nonlinearity. In addition, AUVs operate in open water environments typically in a truly three‐dimensional trajectory. Therefore, it is essential to establish the dynamic model of AUVs and further investigate how to control AUV’s dynamic motions given the unique propulsion and steering mechanisms such as buoyancy adjustment and control surfaces (e.g., a rudder or an elevator).
The objectives of this book are to serve as a textbook for a one‐semester graduate course on wheeled surface robots as well as AUVs and also to provide a useful reference for one interested in these fields. The book presumes knowledge of modern control and random processes. Exercises are included with each chapter. Prior facility with digital simulation of dynamic systems is very helpful but may be developed as one takes the course. The material lends itself well to the inclusion of a course project if one desires to do so.
1 Kinematic Models for Mobile Robots
1.1 Introduction
This chapter is devoted to the development of kinematic models for two types of wheeled robots. The kinematic equations are developed along with the basic geometrical properties of achievable motion. The two configurations considered here do not exhaust the myriad of possible configurations for wheeled robots; however, they serve as an adequate test bed for the development and discussion of the principals involved.
1.2 Vehicles with Front‐Wheel Steering
The first type of mobile robot to be considered is the one with front‐wheel steering. Here the vehicle is usually powered via the rear wheels, and the steering is achieved by way of an actuator for turning the front wheels.
In Figure 1.1, we have a diagram for a four‐wheel front‐wheel‐steered robot. The equations would also apply for the case of a single front wheel. The angle the front wheels make with respect to the longitudinal axis of the robot, yrobot, is defined as α, measured in the counter‐clockwise direction. The angle that the longitudinal axis, yrobot, makes with respect to the yground axis is defined as ψ, also measured in the counter‐clockwise direction. The instantaneous center about which the robot is turning is the point of intersection of the two lines passing through the wheel axes.
From geometry we have
which may be solved to yield the instantaneous radius of curvature for the path of the midpoint of the rear axle of the robot.
Figure 1.1 Schematic diagram of the front‐wheel steered robot.
From geometry we also have
or
which can be written as
(1.2)
If one held the steering angle α constant, the trajectory would result in a circle whose radius is dictated by the robot length and the actual steering angle used per equation (1.1).
Now the instantaneous curvature itself is defined as the ratio of change in angle divided by change in distance or change in angle per distance traveled. It is given by
which is the inverse of the instantaneous radius of curvature. Thus, the radius of curvature may be interpreted as
i.e., the change in distance traveled per radian change in heading angle.
The complete set of kinematic equations for the motion in robot coordinates are
(1.3a)
(1.3b)
(1.3c)
Converted to earth coordinates these become
(1.4a)
(1.4b)
(1.4c)
