Dynamics of Mobile Systems with Controlled Configuration

This chapter deals with plane multilink snake-like mobile systems the links of which are connected by cylindrical (revolute) joints in such a way that the system moves due to bending changes in its configuration. The system moves along a horizontal plane; Coulomb's dry friction acts between the links of the system and the underlying plane. The dynamic mode of motion is considered. This mode implies the alternation of slow and fast phases. In the slow phases, part of the links are moving, while the other links are kept at rest by the friction forces. The joint torques (control torques) in these phases must be comparatively small to enable the friction forces to keep part of the links at rest. During the slow phases, the system's center of mass moves relative to the plane of motion. For the displacement of the center of mass to be considerable, the slow phases must have relatively long durations. In the fast modes, vice versa, the joint torques exceed significantly the friction torques, while the duration of these phases is small. The fast phases substantially change the configuration of the system but virtually do not displace the center of mass relative to the environment. Three- and two-link mechanisms are investigated in detail. It is proved that by combining slow and fast motions one can transfer these mechanisms from any initial position to any terminal position in the plane. Appropriate gaits and control modes are constructed. Optimal design and control parameters that maximize the average velocity of the motion of the mechanisms under consideration are found. Experiments that verify theoretical conclusions are described.

In this chapter, quasistatic motions of a multilink articulated snake-like mechanism along a rough horizontal plane are studied. The links of the mechanism are consecutively connected by revolute joints. Coulomb’s dry friction acts between the links and the underlying plane. The mechanism is actuated by the drives located at the joints. In the quasistatic motion mode, all links of the mechanism move with very small velocities and accelerations, which allows treating the entire motion as a continuous sequence of equilibria and use the static equations for calculations. The quasistatic motions are significantly inferior to the motions that alternate slow and fast phases in terms of the velocity, however, they impose minimal requirements on the force and power parameters of the drives. Various types of wave-like longitudinal motions of the mechanism are investigated. The control torques that provide these motions are calculated. Apart from a multilink snake-like system with revolute joints, a triangular-configuration system with prismatic (translational) joints is considered. This system moves along a rough horizontal plane due to the change in the lengths of the sides of the triangle.

Worm-like locomotion systems that consist of rigid bodies consecutively connected (by prismatic joints) into a chain are considered. The system moves along a straight line in a resistive environment. All bodies of the chain interact with the environment. The system is controlled by drives that generate the forces of interaction between the neighboring bodies. The control modes that provide a progressive motion of the system in a periodic manner in the environments with different resistance laws are designed. The design parameters and the control laws are optimized to maximize the average velocity of the motion of the chain or the energy consumption per unit path length.

The motions of multi-link systems in a liquid based on models with a finite number of degrees of freedom are investigated. It is assumed that the environment influences the elements of the bodies that move in it through the friction forces that are modeled by power-law functions of the velocities of these elements and are oriented opposite to the velocity vector. Under certain conditions, such laws, especially in the case of quadratic-law resistance, describe fairly well the forces of resistance to the motion of bodies in liquids possessing a viscosity. Equations of motion are derived for systems the links of which are connected by cylindrical or prismatic joints. The focus is put on the periodic motions at which the system configuration changes periodically and its center of mass moves by the same distance for each period along the given straight line. Parametric optimization and optimal control problems are solved with the aim to maximize the average velocity of the locomotion system movement in a periodic mode. The systems studied in this chapter model the swimming of fish and some animals as well as dynamics of artificial swimming vehicles. The results presented may be used to develop mobile robots moving in liquid.

In this chapter, we consider rectilinear motions of mobile systems that move in resistive media without external propelling devices (legs, wheels, caterpillars, fins, screw propellers, etc.) due to the motion of internal bodies. Such systems consist of a rigid housing and internal bodies that can move relative to the housing under the action of drives. The drives implement the interaction of the internal bodies with the housing of the locomotion system. The housing interacts also with the environment, whereas the internal bodies do not interact with the environment. A force applied by the drive to an internal body causes a reaction force applied to the housing, as a result of which the velocity of the housing relative to the environment changes. The change in the velocity of the housing leads to a change in the resistance (friction) force applied by the environment to the housing. The forces generated by the drives are internal forces for the mechanical system under consideration (the housing plus the internal bodies), while the environment resistance force is an external force. Thus, by controlling the motion of the internal bodies due to internal forces one can control the external force applied to the locomotion system and thereby the motion of the system as a whole. Mobile robots that move due to the motion of internal bodies are frequently called capsule robots (capsubots). The capsule robots have a number of advantages over mobile systems of other types. The capsule robots are simple in design, do not need complex mechanisms to transmit the motion from the drives to the propelling devices, are easy to miniaturize, and their housings can be made hermetic and geometrically smooth, without protruding pieces. The last circumstance enables the capsule robots to be used in vulnerable environments, in particular, in medicine for diagnosis inspections inside a human body or for precise delivery of a drug to an affected area. Capsule robots can be used also for motion inside thin tubes, for example, for their technical inspection, or in narrow slots. A number of control and optimization problems for capsule robots are solved.

This chapter addresses two-dimensional motions of capsule systems along a rigid horizontal plane in the presence of Coulomb's dry friction between the plane and the housing. It is required to transfer the system from a given initial position to a given final position with a desired orientation of the housing. For the case of a rectilinear motion, it is sufficient that the capsule robot have one internal body oscillating along a straight line parallel to the line of the motion of the robot. To perform a two-dimensional motion along a plane while changing the orientation of the housing a capsule robot must have a more complex structure containing several internal bodies or one internal body capable of moving along a curved trajectory the shape of which depends on the motion to be performed. Two types of mobile capsule system are considered in this chapter. One of the systems consists of a rigid housing and two internal bodies: a point mass that can move along a rectilinear guide relative to the housing and a rotor that can rotate about a vertical axis fixed in the housing. Controlling the motion of the internal bodies, one can control linear and angular motions of the housing. The other system contains only one internal body: a point mass that moves along a curvilinear trajectory relative to the housing. It is proved that one internal point mass suffices to control completely the position and orientation of the housing of the capsule system in the plane.

This chapter addresses the control of the motion of systems with movable internal point masses in the absence of external forces. In this case, it is impossible to control the motion of the center of mass, however, the orientation of the housing can be controlled. This possibility is of interest for the attitude control of spacecraft and other vehicles, as well as for the control of the orientation of a mobile capsule robot when performing rapid turns, in which case the influence of the external forces can be neglected as compared with the internal forces caused by the motion of the internal masses. The modes of the attitude control by means of one or several internal masses are constructed. The objective of the control is to provide a desired orientation for the housing. Three sections of this chapter are devoted to the plane motion of the system with one internal mass. In the plane motions, the housing performs parallel plane motions in the inertial space, while the internal point mass moves along a curve (not given in advance) in a plane that is parallel to the plane of the motion of the housing. Optimal control problems in which the housing is required to be rotated by a given angle in a minimal time are solved. It is assumed that the system is at rest in the initial and terminal states and that the initial position of the internal mass relative to the housing is given. The terminal position of the internal mass in the reference frame attached to the housing may be free, fixed partly (e.g., one coordinate is fixed, while the other coordinate is free) or fixed completely. The components of the relative velocity of the internal mass in the reference frame attached to the housing are used as the control variables. The absolute value of this velocity is subjected to a constraint. Two remaining sections deal with a spatial (triaxial) reorientation of the housing by using one internal point mass or several internal point masses. It is shown that using several (more than three) internal point masses allows substantially simplifying the structure of the control system.