ABSTRACT
In this chapter, we begin by examining the dynamics of rigid bodies that interact with other moving or stationary rigid bodies. All the bodies are components of a multibody system and are allowed to have a single point of interaction that can be realized through contact or some type of joint constraint. The kinematics for the case of point contact has been formulated in previous works [52-54]. On the other hand, the case of joint constraints can be easily handled because the type of joint clearly defines the degrees of freedom that are allowed for the rigid bodies that are connected through the joint. Then we will introduce a methodology for the description of the dynamics of a rigid body generally constrained by points of interaction. Our approach is to use the geometric properties of Newton’s equations and Euler’s equations to accomplish this objective. The methodology is developed in two parts, we first investigate the geometric properties of the basic equations of motion of a rigid body. Next we consider a multibody system that includes point interaction that can occur through contact or some type of joint constraint. Each body is considered initially as an independent unit and forces and torques are applied to the bodies through the interaction points. There is a classification of the applied forces with respect to their type
(e.g., constraint, friction, external). From the independent dynamic equations of motion for each body we can derive a reduced model of the overall system, exploiting the geometric properties of the physical laws. The framework that is developed can also be used to solve control problems in a variety of settings. For example, in addition to each point contact constraint, we can impose holonomic and/or nonholonomic constraints on individual bodies. Holonomic and nonholonomic constraints have been studied extensively in the areas of mechanics and dynamics, see, for example [4,37]. Holonomic control problems have been studied recently in the context of robotics and manipulation, for example [33,34]. On the other hand, nonholonomic control problems, which are more difficult to solve, have recently attracted the attention of a number of researchers and an extensive published literature is developing. Specific classes of problems have been studied, such as mechanical systems sliding or rolling in the plane, see, for example [9-11]. Another category of nonholonomic control problems deals with mobile robots and wheeled vehicles, for example [7,8,15,28-31,36,42-48,50,51]. Spacecrafts and space robots are a further class of mechanical systems with holonomic constraints. The reason for this characterization is the existence of certain model symmetries that correspond to conserved quantities. If these quantities are not integrable, then we have a nonholonomic problem. A number of works have been published in this area, see, for example [16,21,26,35,38,49,55], and the literature in this area is still developing. Important techniques based on the concept of geometric phases have also been developed for the solution of holonomic control problems [9,19,27,32,38-40].
