The article considers the existing mathematical models of magneto-rheological substances and describes some of their properties. As a result of the open sources analysis, it was found that there are no exoskeleton models with variable-length links with adjustable stiffness, based on the application of magneto-rheological fluids. Therefore, the application of these fluids in other technical systems is considered. A mathematical model of an exoskeleton variable-length link with adjustable stiffness is proposed. This link can be used for supporting and strengthening the lower limbs of the human musculoskeletal system. The difference between the proposed mathematical model of the link and the existing ones lies in the fact that the section that changes its length is considered weighty. Therefore, the mathematical model of the link with a variable inertial characteristic, the moment of inertia relative to the axis perpendicular to the longitudinal axis of the link symmetry and passing through its beginning – the point where the link is fixed to the stationary mount with a cylindrical hinge, is considered. A method of motion control based on the assignment of differentiable functions is applied. The trajectory of the link movement is found, linear and angular velocities and accelerations are calculated. To showcase the link motion, the computer-animated visualization of the link motion control problem solution is presented. The control actions required for the implementation of the given motion have been calculated in the numerical experiment. The drag coefficient range of the magneto-rheological substance has been identified during the implementation of the proposed link motion. The software implementation of the proposed mathematical model of the exoskeleton variable-length link with adjustable stiffness has been done in the Wolfram Mathematica 11.3 universal computer math environment. The software package including the unit for deriving the differential equations of motion in analytical form, the kinematic trajectory synthesis unit, the computational experiment unit, and the unit for animated visualization of the model motion and its export in the wide-spread 'gif' video format has been developed.

Introduction. The article investigated one of the problems of creating exoskeletons — controlling the properties of magnetic rheological fluid in links of variable length with adjustable stiffness. Based on the research of domestic and foreign authors, the development and urgency of the topic was evaluated. The disadvantage of known exoskeleton models has been specified, i.e., the use of absolutely solid links, whose dynamics does not convey the dynamics of the human musculoskeletal system.   The scientific research aimed at the formation of a new direction in the development of exoskeletons that accurately simulate the biomechanics of movements.   Materials and Methods. Different states of structures of variable-length links with a magnetorheological fluid were studied. It has been noted that the links work on the principle of magnetic shock absorbers and consist of a piston rod, electromagnetic coils, and a housing filled with magnetorheological fluid. The ordering effect of an external magnetic field on the particles of a magnetorheological fluid was visualized and mathematically presented. The significance of such factors as time, charge density, magnetic field strength, as well as vectors of electric and magnetic induction, electric intensity and electric current density for this system was shown. The input parameter affecting the behavior of the magnetorheological fluid was determined. This was the magnetic field intensity. It was shown that the viscosity of the liquid varied depending on the shape of the magnetic particles (oblong or oblate ellipsoid).   Results. The dependences that were fundamental for solving the task were investigated and visualized. The magnetic field strength and the angle between the vector directed along a straight line connecting the centers of two micron particles, and the vector of the external magnetic field strength were taken as the basic parameters. It was shown how the magnetic moment, voltage and its antisymmetric part depended on them. It was established that to control the properties of a magnetorheological fluid, it was required to change: – the external magnetic field intensity; – the angle between the external magnetic field intensity and the orientation vector between the dipoles. Two values of force were compared: one – for a given link design, and the other — fixed when walking in the lower leg of a person. The consistency of these indicators was established.   Discussion and Conclusion. The scientific research results allowed us to present: – a method for controlling the properties of a magnetorheological fluid by an external magnetic field; – a variable-length link model with adjustable stiffness. The results obtained can be used in modeling multilink structures to create comfortable exoskeletons that interact synchronously with the human musculoskeletal system as a single human-machine system. The development is applicable to solving significant social and economic problems.

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A two-link model of exoskeleton with variable-length links for supporting the lower limbs of the human musculoskeletal system is proposed in the article. The researched model differs from the existing ones by the variable-length links, and by the angle calculation method. While in the existing models, the angles are calculated from the regular direction – from vertical, or from horizontal, – in the proposed research they are calculated between the links. As for practical exoskeleton implementation, the proposed method of angle calculation is appropriate to the actual working conditions of the electrical motors with the reduction gears installed in the hinges, which change the angles between the links. The construction of a variable-length exoskeleton link consists of two absolutely solid weighty sections located at both ends of the link and one weightless section between them in the center of the link. In the weightless section, there is a drive that creates a control longitudinal force, which realizes the increase or decrease in the length of the link in the required manner and provides the necessary maintenance of the length of the link when the person moves in the exoskeleton. The links are connected to each other using spherical hinges. Drives are installed in each hinge, creating control torques, which provide a relative rotational movement of the links. The jointly controlling longitudinal forces and moments realize the maintenance of the posture or the movement of the link in the required manner and, in relation to the exoskeleton, the repetition of the basic biomechanical properties of the human musculoskeletal system. The mathematical model in the form of the system of Lagrange differential equations of the second kind is obtained. The obtained mathematical model is examined for existence and uniqueness of the Cauchy solution. The kinematic trajectory of the link motion has been synthesized, which simulates the anthropomorphic movement of the supporting leg during the single-support phase of movement, and the control actions required for its implementation has been found. The significance of the results obtained in the process of modeling lies in the ability to create active exoskeletons, prostheses in medicine, anthropomorphic robots, and spacesuits that take into account the biomechanical features of the functioning of the human musculoskeletal system.

The objective of the study is the development of 3D variable-length link model with electric drives to be used in designing of next-generation comfortable exoskeletons. The developed link model has two inertial absolutely rigid sections on its ends and a variable- length section, considered weightless, in between. The mechanical part of the variable-length link model has been implemented in the universal computer math "Wolfram Mathematcia 11.3" environment by building the system of Lagrange – Maxwell differential equations. The electro-mechanical link model with electric drives has been implemented in the MatLab Simulink environment. The implemented model includes the following units: the trajectory synthesis unit per each degree of freedom, the unit for controlling torques calculation based on differential equations of motion, the unit for selecting electric motors with gears, the unit for calculating electric current per each motor and implementing the control system. The electric motors, reducers, rack and pinion gears implementing the specified and programmed link motion have been selected. The inertial and geometrical variable-length link parameters corresponding to the human tibia in the period of the single-support step phase have been selected. The drives implementing the link rotation are situated in the bottom link point in the combination of two orthogonal cylindrical hinges. One of these hinges is fixed to the supporting surface, the other one is fixed to the link end. This hinge combination simulates human ankle joint in the single-support step phase. The drive controlling the link length change is situated at the end of the bottom absolutely rigid weighty link section. The programmed trajectories for generalized coordinates are specified based on the simulation requirements of the anthropomorphic tibia motion. As a result, the electro-mechanical model of a variable- length link with parameters corresponding to the average man’s tibia has been developed. The drives and gears that allow implementing the motion close to anthropomorphic one have been selected. The implementation of this motion based on the developed software in the computer math "Wolfram Mathematica 11.3" environment and in the MatLab Simulink system has been demonstrated. The numerical calculations are presented.