holonomic constraints
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This paper presents an investigation of modeling and solving of differential equations in the study of mechanical systems with holonomic constraints. The 2D and 3D mathematical models of constrained motion are made. The structure of the models consists of nonlinear first or second order differential equations. Cases of free movement and movement with resistance are investigated. Solutions of the Cauchy problem of obtained differential equations were obtained by Runge–Kutta method.

Symmetry preserving difference schemes approximating equations of Hamiltonian systems are presented in this paper. For holonomic systems in the Hamiltonian framework, the symmetrical operators are obtained by solving the determining equations of Lie symmetry with the Maple procedure. The difference type of symmetry preserving invariants are constructed based on the three points of the lattice and the characteristic equations. The difference scheme is constructed by using these discrete invariants. An example is presented to illustrate the applications of the results. The solutions of the invariant numerical schemes are compared to the noninvariant ones, the standard and the exact solutions.

Based on Lagrangian mechanics, use of velocity constraints as a special set of quasi-velocities helps derive explicit equations of motion. The equations are applicable to holonomic and nonholonomic constrained multibody systems. It is proved that in proposed quasi-spaces, the Lagrange multipliers are eliminated from equations of motion; however, it is possible to compute these multipliers once the equations of motion have been solved. The novelty of this research is employing block matrix inversion to find the analytical relations between the parameters of quasi-velocities and equations of motion. In other words, this research identifies arbitrary submatrices and their effects on equations of motion. Also, the present study aimed to provide appropriate criteria to select arbitrary parameters to avoid singularity, reduce constraints violations, and improve computational efficiency. In order to illustrate the advantage of this approach, the simulation results of a 3-link snake-like robot with nonholonomic constraints and a four-bar mechanism with holonomic constraints are presented. The effectiveness of the proposed approach is demonstrated by comparing the constraints violation at the position and velocity levels, conservation of the total energy, and computational efficiency with those obtained via the traditional methods.

Mobile robots are endeavoring toward full autonomy. To that end, wheeled mobile robots have to function under non-holonomic constraints and uncertainty derived by feedback sensors and/or internal dynamics. Speed control is one of the main and challenging objectives in the endeavor for efficient autonomous collision-free navigation. This paper proposes an intelligent technique for speed control of a wheeled mobile robot using a combination of fuzzy logic and supervised machine learning (SML). The technique is appropriate for flexible leader-follower formation control on straight paths where a follower robot maintains a safely varying distance from a leader robot. A fuzzy controller specifies the ultimate distance of the follower to the leader using the measurements obtained from two ultrasonic sensors. An SML algorithm estimates a proper speed for the follower based on the ultimate distance. Simulations demonstrated that the proposed technique appropriately adjusts the follower robot’s speed to maintain a flexible formation with the leader robot.

Abstract Multi-joint snake robot is a vital reconnaissance, surveillance and attack weapon in national defence and military in the future. To study the trajectory tracking problem of a multi-joint snake robot with high redundancy and multi-degree of freedom in the plane, an adaptive trajectory tracking controller of a multi-joint snake robot considering non-holonomic constraints is proposed in this paper. The adaptive trajectory tracking controller replaces unknown parameters in the environment wi t h estimated values, which effectively solves the negative effects caused by uncertain and time-varying environmental parameters in the process of the robot movement and realizes the stability of the controller. Firstly, a new dynamical model of a multi-joint snake robot is established through coordinate transformation. Secondly, the control objective of the controller of the multi-joint snake robot is established. Thirdly, the proposed controller of the multi-joint snake robot is designed by the Backsteppi n g method to realize the control of the joint angle tracking error, link angle tracking error, actuator torque error and motion speed error of the robot. Then, a suitable Lyapunov function is found to verify the stability of the controller. Finally, through the MATLAB simulation and prototype experiment, the motion process of the multi-joint snake robot is observed, the trajectory tracking performance of the robot is analyzed, and the effectiveness of the adaptive trajectory tracking controller is verified.

Hybrid A* algorithm has been widely used in mobile robots to obtain paths that are collision-free and drivable. However, the outputs of hybrid A* algorithm always contain unnecessary steering actions and are close to the obstacles. In this paper, the artificial potential field (APF) concept is applied to optimize the paths generated by the hybrid A* algorithm. The generated path not only satisfies the non-holonomic constraints of the vehicle, but also is smooth and keeps a comfortable distance to the obstacle at the same time. Through the robot operating system (ROS) platform, the path planning experiments are carried out based on the hybrid A* algorithm and the improved hybrid A* algorithm, respectively. In the experiments, the results show that the improved hybrid A* algorithm greatly reduces the number of steering actions and the maximum curvature of the paths in many different common scenarios. The paths generated by the improved algorithm nearly do not have unnecessary steering or sharp turning before the obstacles, which are safer and smoother than the paths generated by the hybrid A* algorithm for the autonomous ground vehicle.

The research results on the derivation of equations of motion (a mathematical model) of a mechanical system constrained by holonomic constraints of the first and second kind, which contains only tangential components of the constraint of the second kind, are presented in the article. These tangential components are taken as control parameters. Besides, the controllability of the plate, considered in the Appel problem, is investigated.