# Inertial Forces and Friction in Propulsion of a Rigid Body
## 1 Introduction  
The propulsion mechanism of mobile robots, designed as closed capsules without external moving parts or openings, has attracted considerable attention in recent research. Such robots can be conceptualized as enclosed structures containing one or more internal masses capable of oscillatory motion. By controlling the dynamics of these internal movements--either linear or rotational--the entire system can achieve directed motion. The underlying principle involves the generation of controlled inertial forces, which modulate both the normal force exerted by the robot on the surface and the resulting propulsion force. Commonly referred to as vibration-driven or inertia-driven robots, these systems offer significant advantages over traditional locomotion methods by eliminating the need for wheels, legs, or tracks. Their sealed design enhances robustness and facilitates operation in challenging environments, including solid and fluid media. The theoretical foundations of such systems were extensively studied by [PERSON], who analyzed the dynamics and optimal control of rigid bodies equipped with internal moving masses. In paper [1], a simplified mechanical model was proposed, consisting of a rigid body capable of horizontal motion with an internally oscillating mass. Parameters for maximizing the system's average velocity were derived. Subsequent work [2] investigated controlled horizontal motion under dry friction forces, where the movement of a bounded internal mass provided propulsion. In paper [3], the periodic translational motion of atwo-mass system--comprising a container interacting with its environment and an internal moving mass--was examined. The study demonstrated how simple oscillatory motions of the internal mass induced displacement of the entire system. Experimental validation of this principle was achieved using pendulum systems and rotating masses in mobile robots designed for tubular environments. Further advancements were made by [PERSON] et al. [4; 5; 6], who analyzed the dynamics of systems with two internal movable masses. Their work addressed optimal control problems for motion on rough surfaces, where horizontal and vertical movements of internal masses were used to regulate frictional forces and optimize velocity. These studies considered various friction laws, including piecewise linear, quadratic, and Coulomb friction. The results demonstrate that by modulating the internal masses' dynamics, the external reaction forces could be controlled, enabling directed motion and velocity regulation. The simplicity and versatility of vibration-driven robots make them promising candidates for a range of applications, including locomotion on surfaces, in dense media (e.g., soil), and within pipes. Potential biomedical applications include microrobots for drug delivery or diagnostic purposes. For instance, [PERSON] et al. [7] presented optimal control strategies for two-body systems in nonlinear viscous media, while [PERSON] and [PERSON] [8; 9] explored similar setups in resistive environments, including interconnected modular systems. Optimization challenges for these systems have been addressed in various contexts. [PERSON] [10] studied linear motion on inclined surfaces, while [PERSON] and [PERSON] [11] conducted experimental research on systems with unbalanced rotors. [PERSON] [12] examined systems comprising rigid bodies and internal movable masses on rough surfaces, emphasizing rotational dynamics and friction modelling. [PERSON] and [PERSON] [13; 14] provided a comprehensive analysis of periodic motion induced by rotating unbalanced masses, identifying stable motion modes. [PERSON] and [PERSON] [15] investigated systems with eccentrics (unbalanced rotating wheels), achieving incremental motion on rough surfaces. The idea of an inertial propulsion system can be used in various mixed mobile solutions. For example, recently, [PERSON] et al. [16] have investigated an inertial drive of a wheeled platform based on two revolving eccentric bodies.