Development of methods for modeling and solving problems of the control system dynamics with programmed constraints
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The project is aimed at developing methods for mathematical modeling of dynamics processes and solving problems of control systems for various purposes.Modification of methods for constructing and numerical solutions of controlled systems dynamics equations is provided,based on the methods of classical mechanics,applied mathematics,modern achievements of control theory and information technology.It is supposed to develop general approaches to solving the constructing problem of closed system dynamics equations corresponding to the set goals,and to develop computer modeling methods with the required accuracy.
To construct the dynamics equations,the methods proposed by the authors for constructing differential equations using known constraints equations with subsequent reduction to a given structure can be used.Modified Helmholtz conditions will allow us to obtain dynamic equations with dissipative forces that ensure the constraint stabilization.The research is aimed at creating methods,algorithms and software tools and artificial intelligence systems.
Based on the new approaches being developed,it is planned to solve the problems of celestial mechanics and cosmodynamics,determine the conditions for stabilizing the integrals of the equations of optimal motion in space ballistics,and the tasks of stabilizing the trajectory of a spacecraft with low thrust.In the mechanics of rigid body and systems of rigid bodies,the solution of the problems of manipulator control,robotic and anthropomorphic systems,protection of buildings and structures from external influences is provided.The tasks of modeling and controlling the dynamics of the horse-rider system and the development of recommendations for the design of simulators belong to the field of biomechanics.
The solution of modeling problems of the control systems under consideration should be accompanied by numerical calculations with the presentation of the results by graphical dependencies and other means of implementation.The list of tasks in the chosen research area corresponds to the current direction in each area and represents a new approach to solving the problems of controlling the purposeful movement of physical systems and their analogues.
Expected results:The work on the project is supposed to be carried out in accordance with the research directions of the participants.At the first stage,it is planned to develop methods for constructing dynamic equations describing processes corresponding to control purpose.For the formulation of control purpose,it is advisable to introduce the concept of program constraints that allow deviations from the constraint equations and control the behavior of solutions to dynamics equations in the neighborhood of undisturbed constraints.Determining the set of virtual displacements of the system allows you to modify the dynamic analogies of the system and obtain dynamic equations based on the principles of classical mechanics.During the realization of the project,it is planned to construction of dynamic equations based on the Hamilton-Ostrogradsky principle and the Gauss principle.A new approach to the problem of composing dynamic equations relates to inverse problems of dynamics as defined by A.S.Galiullin and allows us to construct control reactions that ensure the constraint stabilization.
For the construction of dynamic equations,the project also proposes a new approach that directly uses the constraint equations.According to the equations of differential constraints,corresponding to holonomic and nonholonomic constraints,the constraints perturbations equations are constructed,on the basis of which the set of systems of second-order differential equations can be constructed.Using modified Helmholtz conditions,a system in the form of Lagrange equations with dissipative forces providing constraints stabilization is distinguished from the resulting set.
New methods for constructing dynamic equations with constraints stabilization will be used to develop research on the Bertrand problem and to stabilize optimal trajectories of motion of a point of variable mass in rocket dynamics problems.In the works of Asimov D.M.the first integrals of the optimal motion equations for rocket of variable mass in the Lowden problem are determined.The project for the definition of stabilization conditions for the integrals corresponding to the specified initial conditions provides.Some problems of maneuvering spacecraft using low-thrust engines will be investigated.The solution of the stabilizing problem for the motion of the spacecraft along a given trajectory,ensuring stability with respect to external disturbances and the development of an appropriate numerical method for solving the equations of dynamics is provided.
By introducing the cyclic coordinates for reducing the dimension of a closed system dynamics equations a well-known solution of the nonholonomic constraints perturbation equations to be used.For the construction of dynamic equations with constraint stabilization in the form of Chaplygin equations and Voronets equations it is also provided.
The development of methods for the dynamic equations analytical construction should be accompanied by the development of appropriate numerical algorithms and programs.The construction of effective algorithms for solving systems of differential-algebraic equations using the first-order Euler difference scheme and Runge-Kutta methods is provided.The evaluation of the safe range of values of the coefficients of the linear system of constraint perturbation equations of perturbations using the Runge-Kutta method of the IV order will be carried out.Modification of the constraint stabilization methods will allow us to obtain solutions to some problems with singularities,in particular,the possibility of using the constraint stabilization method to bypass singular points is being considered.
It is planned to develop models of spatial models of anthropomorphic systems modeled by mechanical systems taking into account holonomic and nonholonomic constraints suitable for description biological prototypes.Mathematical modeling of the corresponding mechanisms will allow obtaining fundamental data for the design of robotic systems and the creation of algorithms for targeted control with constraint stabilization.It is planned to develop methods for modeling and solving of dynamics control problems for the horse-rider system based on the analysis of real movements.The work in this direction is the actual problems of designing simulators used in the training process and for physiotherapy procedures in medicine.
The solution of modeling problems and problems of control by specific systems is supposed to be accompanied by numerical calculations with the presentation of results by graphical dependencies and other means of implementation.
The project is aimed at developing methods for mathematical modeling of dynamics processes and solving problems of control systems for various purposes.Modification of methods for constructing and numerical solutions of controlled systems dynamics equations is provided,based on the methods of classical mechanics,applied mathematics,modern achievements of control theory and information technology.It is supposed to develop general approaches to solving the constructing problem of closed system dynamics equations corresponding to the set goals,and to develop computer modeling methods with the required accuracy.
To construct the dynamics equations,the methods proposed by the authors for constructing differential equations using known constraints equations with subsequent reduction to a given structure can be used.Modified Helmholtz conditions will allow us to obtain dynamic equations with dissipative forces that ensure the constraint stabilization.The research is aimed at creating methods,algorithms and software tools and artificial intelligence systems.
Based on the new approaches being developed,it is planned to solve the problems of celestial mechanics and cosmodynamics,determine the conditions for stabilizing the integrals of the equations of optimal motion in space ballistics,and the tasks of stabilizing the trajectory of a spacecraft with low thrust.In the mechanics of rigid body and systems of rigid bodies,the solution of the problems of manipulator control,robotic and anthropomorphic systems,protection of buildings and structures from external influences is provided.The tasks of modeling and controlling the dynamics of the horse-rider system and the development of recommendations for the design of simulators belong to the field of biomechanics.
The solution of modeling problems of the control systems under consideration should be accompanied by numerical calculations with the presentation of the results by graphical dependencies and other means of implementation.The list of tasks in the chosen research area corresponds to the current direction in each area and represents a new approach to solving the problems of controlling the purposeful movement of physical systems and their analogues.
Expected results:The work on the project is supposed to be carried out in accordance with the research directions of the participants.At the first stage,it is planned to develop methods for constructing dynamic equations describing processes corresponding to control purpose.For the formulation of control purpose,it is advisable to introduce the concept of program constraints that allow deviations from the constraint equations and control the behavior of solutions to dynamics equations in the neighborhood of undisturbed constraints.Determining the set of virtual displacements of the system allows you to modify the dynamic analogies of the system and obtain dynamic equations based on the principles of classical mechanics.During the realization of the project,it is planned to construction of dynamic equations based on the Hamilton-Ostrogradsky principle and the Gauss principle.A new approach to the problem of composing dynamic equations relates to inverse problems of dynamics as defined by A.S.Galiullin and allows us to construct control reactions that ensure the constraint stabilization.
For the construction of dynamic equations,the project also proposes a new approach that directly uses the constraint equations.According to the equations of differential constraints,corresponding to holonomic and nonholonomic constraints,the constraints perturbations equations are constructed,on the basis of which the set of systems of second-order differential equations can be constructed.Using modified Helmholtz conditions,a system in the form of Lagrange equations with dissipative forces providing constraints stabilization is distinguished from the resulting set.
New methods for constructing dynamic equations with constraints stabilization will be used to develop research on the Bertrand problem and to stabilize optimal trajectories of motion of a point of variable mass in rocket dynamics problems.In the works of Asimov D.M.the first integrals of the optimal motion equations for rocket of variable mass in the Lowden problem are determined.The project for the definition of stabilization conditions for the integrals corresponding to the specified initial conditions provides.Some problems of maneuvering spacecraft using low-thrust engines will be investigated.The solution of the stabilizing problem for the motion of the spacecraft along a given trajectory,ensuring stability with respect to external disturbances and the development of an appropriate numerical method for solving the equations of dynamics is provided.
By introducing the cyclic coordinates for reducing the dimension of a closed system dynamics equations a well-known solution of the nonholonomic constraints perturbation equations to be used.For the construction of dynamic equations with constraint stabilization in the form of Chaplygin equations and Voronets equations it is also provided.
The development of methods for the dynamic equations analytical construction should be accompanied by the development of appropriate numerical algorithms and programs.The construction of effective algorithms for solving systems of differential-algebraic equations using the first-order Euler difference scheme and Runge-Kutta methods is provided.The evaluation of the safe range of values of the coefficients of the linear system of constraint perturbation equations of perturbations using the Runge-Kutta method of the IV order will be carried out.Modification of the constraint stabilization methods will allow us to obtain solutions to some problems with singularities,in particular,the possibility of using the constraint stabilization method to bypass singular points is being considered.
It is planned to develop models of spatial models of anthropomorphic systems modeled by mechanical systems taking into account holonomic and nonholonomic constraints suitable for description biological prototypes.Mathematical modeling of the corresponding mechanisms will allow obtaining fundamental data for the design of robotic systems and the creation of algorithms for targeted control with constraint stabilization.It is planned to develop methods for modeling and solving of dynamics control problems for the horse-rider system based on the analysis of real movements.The work in this direction is the actual problems of designing simulators used in the training process and for physiotherapy procedures in medicine.
The solution of modeling problems and problems of control by specific systems is supposed to be accompanied by numerical calculations with the presentation of results by graphical dependencies and other means of implementation.