Optimal control of a two-body limbless crawler along a rough horizontal straight line
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ORIGINAL PAPER
Optimal control of a two-body limbless crawler along a rough horizontal straight line Nikolay Bolotnik
· Tatiana Figurina
Received: 24 April 2020 / Accepted: 3 October 2020 © Springer Nature B.V. 2020
Abstract An optimal control problem is solved for a two-body limbless locomotor crawling along a straight line on a horizontal rough plane. Coulomb’s dry friction acts between the locomotor’s bodies and the underlying plane. The control is performed by the force of interaction between the bodies. The system should be moved from the state of rest by a given distance in a minimal time, provided that the relative positions of the bodies in the initial and terminal states coincide and the velocities of the bodies at the terminal instant are equal to zero. A particular attention is given to the case where the bodies are prohibited to change the direction of their motion. Keywords Limbless locomotion · Coulomb’s friction · Optimal control
1 Introduction This paper is related to the dynamics and control of limbless locomotion systems. Such systems can move in nonlinear resistive environments without special propelling devices (wheels, legs, caterpillars, fins, etc.) N. Bolotnik (B) · T. Figurina Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, 101-1 Vernadsky Ave, Moscow, Russia 119526 e-mail: [email protected] T. Figurina e-mail: [email protected]
due to the change in their configurations. These systems consist of a number of bodies (links) connected by cylindrical (revolute) or prismatic (translational) joints. The bodies interact with one another and with the environment. The systems are controlled by the forces of interaction between their bodies; these forces are internal forces with respect to the locomotor. The interaction between the bodies changes the velocities of these bodies relative to the environment, which leads to the change in the resistance forces applied by the environment to the locomotor components. The forces of interaction with the environment are external forces for the locomotor. Therefore, by changing the internal interaction forces, one can control the external forces, controlling thereby the motion of the entire system. This principle of motion underlies locomotion of some limbless animals, in particular snakes and worms, and can be utilized in artificial locomotors (mobile robots). The motion of systems with revolute joints along a horizontal rough plane is studied in [1–4]. It is assumed that all links of the system have contact with the plane and the Coulomb’s dry friction acts between the bodies and the underlying plane. Dynamic and quasi-static modes of motion are considered. In the dynamic mode, slow and fast motions alternate [1]. In the slow phases, part of the links move slowly, while the remaining links are kept fixed due to friction. During the slow phases, the center of mass of the system is moving relative to the plane. In the fast phases, the system quickly changes its configuration, while the center of mass virtually
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