Inspired from the inchworm’s locomotion, self-propelled robots driven by autogenous internal force and environmental resistance have attracted great attention from applied mathematicians, experimentalists to engineers because of the underlying applications in various real-world problems and health care technologies, as well as the challenges associated to the construction of suitable models, often involving dynamical systems including non-smooth phenomena and delay effects.

This special issue aims to bring together researchers working in self-propelled locomotive mechanisms and non-smooth dynamical systems to present new theoretical developments and to show how to make the best use of them in real applications. The contributions considered in this issue focus on self-propulsion robots involving non-smooth phenomena, such as impact and friction, as well as time-delay effects across different scales, potentially from the meso- to the micro-scale. The special issue contains thirteen articles, seven on robot modelling, four on robot optimisation and two on robot experiments. Following is a brief summary of each article to provide the readers with a quick guide to the content.

Shmatkov considered the problem of changing the spatial orientation of a rigid body by using a movable mass interacting with the body in the presence of external forces. Equations of motion of the system were obtained, and an example of spatial reorientation of a mechanical system was presented. The proposed concept of this work can be used to control robotic systems and spacecrafts.

Chernousko studied the three-dimensional motion of a rigid body controlled by several auxiliary internal masses under the assumption that external forces acting upon the system are negligible. The required body reorientation was achieved by means of circular movements of auxiliary point masses within certain bounded domains inside the body. Two versions of possible motions of the point masses that ensured the desired reorientation were presented. This new concept is valid for fast turns of the body about its centre of mass and can be used potentially for micro-robots, spacecraft, and other vehicles.

Bolotnik and Figurina considered a two-body model of a limbless crawler moving on an inclined rough plane, and the motion of the crawler was controlled by the interactive forces between these two bodies. The authors theoretically proved that if no constraint was imposed on the control force, the crawler can be driven from any initial state of rest into an arbitrarily small neighbourhood of any prescribed terminal state of rest.

Tkachenko et al. proposed a mathematical model for the motion of a mobile robot that can be controlled using an alternating magnetic field. The robot, composed of two spherical bodies with a magnetisable material and a non-magnetic elastic coupling, can move along the bottom of a vessel filled with a liquid. Influences of control and environmental parameters, such as frequency of the magnetic field and viscosity and density of surrounding liquid, on the motion of the robot were studied.

Impact plays a two-fold role in self-propelled robots. On one hand, undesired robot-obstacle or robot-robot impacts that cannot be avoided may cause instability in robots. On the other hand, impacts can be utilised for robot propulsion and directional control. Tan et al. studied the near-grazing bifurcations (a transition from non-impact to impact) of a single-degree-of-freedom impact system with two-side elastic constraints and stabilised the near-grazing dynamics by using a deep reinforcement learning control method. The proposed control method allows the continuous transition from non-impact to impact periodic motion through grazing.

Yan et al. investigated the dynamics of a vibro-impact capsule moving on an intestinal substrate with the consideration of a circular fold through mathematical modelling and numerical simulation. Their bifurcation analyses suggested that the capsule can always perform period-1 motion when the driving force is small, and fold crossing requires a large excitation amplitude, especially when the duty cycle ratio of the driving excitation is small. The findings of this work can provide fundamental insight into the optimisation and control of capsule’s locomotion in the small intestine.

Figurina and Knyazkov studied the motion of a capsule system along a line on a rough horizontal plane. The capsule consists of a hull and an internal mass moving periodically inside the hull parallelly to the line of motion of the system. A periodic regime of motion was defined which was proved to be unique and stable with respect to its initial conditions.

Zhu et al. optimised a self-propelled capsule moving in the digestive tract in terms of its moving speed, impact force and energy consumption. NSGA-II, Monte Carlo, and Six-Sigma algorithms were combined to conduct a multi-objective optimisation of capsule’s parameters based on a reliability analysis. The proposed optimisation method and the optimisation results are expected to be used for improving the self-propelled capsule design and its application in wireless controllable endoscopy.

Xue et al. studied a worm-like robot driven by internal vibrations and optimised its locomotion velocity and energy consumption. Pareto front method was employed and extended to reach a coordinated optimisation for different friction environments. Optimal amplitude and frequency of the vibrations were determined by combining the identified Pareto region with the stick–slip locomotion regime to achieve this bi-objective optimisation target.

Zhao et al. investigated the coordination of the actuation phases in a multi-module vibration-driven robot with linear or nonlinear connections. A particle swarm optimisation algorithm was employed to optimise the phase-difference coordination pattern corresponding to the maximal average steady-state velocity of the robot. The findings of this work may provide useful guidelines for the design and control of earthworm-like vibration-driven robots.

Zarychta et al. analysed the construction of a neural-network based, closed-loop control of a discontinuous capsule drive. Neural network was used to determine the dependence between an open-loop controller’s output and system’s state. Robustness of the neural controller with respect to the variation of parameters of the controlled system was analysed and compared with the original, optimised open-loop control. It is expected that the proposed method can facilitate the construction of closed-loop controllers for systems involving non-smooth phenomena, such as impact and friction.

Kamamichi and Furuta studied a self-propelled board driven by the inclined rectilinear motion of an internal mass and demonstrated the concept experimentally. A mathematical model development and numerical simulations were presented to study the relations of the frequency and the inclined angle of reciprocal motion of the internal mass to its locomotion velocity and periodic motions.

Duong et al. studied the dynamic response of a vibro-impact capsule moving up on an inclined track with stochastic slopes. A mathematical model of the system was developed and verified experimentally at three inclined angles under different control parameters. The effect of random perturbation of the inclined angle on the capsule was also examined. The results of this work are expected to be used for the design and control of the vibro-impact self-propelled robots in various stochastic conditions.

In summary, all papers presented in this special issue are at the vanguard of the modelling, optimisation and experimental study of self-propelled robots, with special focus on engineering and medical applications. The authors have significantly enriched this exciting and relatively new branch with their contributions and have paved the road for future research directions and open questions to be tackled by both young and experienced scientists in the area and beyond. Along these lines, we would like to close this issue by cordially thanking the authors for their efforts in preparing these high-quality works, the reviewers for their valuable time and peer-review work and the Editor-in-Chief, Professor Anna Pandolfi, for her friendly support and cooperation during the realisation of this special issue.