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Mathematical model of a horse rider simulator and its program implementation in the “Wolfram Mathematica” computer mathematics system

Riding simulation machines allow training horse-back riders regardless of environmental and weather conditions, horse fatigue, stable remoteness and other factors. Riders can practice complex movements without fear of potential injury related to sometimes unpredictable behavior of the horse. This type of training machines requires special software for selecting the relevant action mode for the person during the rider’s practice. Thus, the selected behavior of the horse can be simulated and the injuries of horse-back rider can be avoided. For this purpose, the model of horse-back rider, taking into account its training level and the horse motion, is required. The model of “person – simulation machine” combination is proposed in the article. The proposed model allows developing operating modes taking into account the sportsman-training machine interaction. The mechanical model is an exosuit with a mobile pole, consisting of four links, the rider’s feet, shins, hips, and the corps are attached to. The pole corresponds to the mass center of the horse, the rider interacts with by standing on the stirrups. The pole motion is implemented by a telescopic link attached to immobile base. The relative rotations of the links are implemented by cylindrical hinges with negligible friction. The proposed model has been implemented as software in the “Wolfram Mathematica 11.3” environment. It has been designed for training machine dynamics simulation of the “horse ­– rider” system. The software includes several modules: 1) the module for specifying the structure of training machine mathematical model, generalized coordinates, and for auto-compilation of the corresponding system of differential equations; 2) the module for specifying the programmed model motion and calculation of the required control torques in the hinges; 3) the module for the Cauchy problem numerical solution; 4) the module for the animated visualizing of the model motion, and for exporting the obtained graphic results and numerical calculations. The developed software allows conducting dynamics analysis of mathematical model for the considered system based on the solution of both direct and inverse dynamics problems. Also it can be suggested for designing training machines with programmed operation mode. It has been shown that application of this software accelerates the development of training machines. Read more...

Modeling the dynamics of an exoskeleton link using the quaternion apparatus

The research is devoted to the development of an algorithm for applying the quaternion formalism to the modeling of the dynamics of exoskeletons and similar anthropomorphic mechanisms, such as robots, space suits, simulators, and related systems. The fundamental problem of developing exoskeletons, anthropomorphic mechanisms, and robotic systems is being investigated, which requires accelerating the process of analytical construction of mathematical models described by differential equations. To tackle this problem, the authors propose the use of hypercomplex number algebra, specifically quaternions. The application of quaternion algebra to the study of locomotion in anthropoid robotic devices such as exoskeletons with active propulsion systems that control the relative positions of the joints during movement should improve the construction of the mathematical model. These arguments determine the relevance of the research topic and the scientific novelty of the study. The development of high-speed methods for writing differential equations of motion based on quaternion algebra to describe the locomotion of spatial mechanical anthropoid systems determines the practical significance of the research results. The work presents a method for constructing an algorithm to model the dynamics of the shank of an exoskeleton, represented as a link connected by a spherical joint allowing rotation with respect to a fixed reference frame. The proposed mechanical model has been implemented as a program within the universal computer algebra system Wolfram Mathematica. The program is designed for simulating the dynamics of the exoskeleton link. Since the system does not provide built-in functions for working with analytically defined quaternions, the authors developed the required routines themselves. The program consists of several modules: a module for quaternion operations; a module for transformation matrices (used for validation and debugging of the quaternion module); a module for the automated formulation of the Lagrange equations of the second kind; a module for specifying the programmed motion of the model and computing the control torques in the joints; a module for numerically solving the Cauchy problem; and a module for animation and visualization of the model’s motion as well as for exporting the graphical results of the numerical simulations. The program’s results allow for the analysis of the dynamics of a mathematical model of a system based on the solution of the direct and inverse dynamics problem, and can be recommended for the design of exoskeletons, anthropomorphic robots, and manipulators with a programmable operating mode. Read more...