CONSTRUCTION OF A MATHEMATICAL MODEL OF THE DYNAMICS OF AN AUTONOMOUS MOBILE ROBOT OF VARIABLE CONFIGURATION

to the axes of the base coordinate system. The construction of the mathematical model was carried out according to the Nyton-Euler method. The resulting mathematical model contains:


Introduction
Robots and manipulators (M) were designed to perform tasks that a person cannot cope with in some circumstances. For example, AMR of a variable configuration performs operations under dangerous or harmful conditions for humans. Increasingly, M is used for military purposes [1,2], in space [3,4], in nuclear energy [5], to overcome the consequences of man-made and natural disasters [6]. For work under extreme conditions, remotely controlled AMR with M are typically used [1][2][3][4][5][6]. To ensure effective remote control of AMR with M, it is necessary to design: -a kinematic scheme of AMR with M with actuators; -an information system of AMR with M; -a mathematical model taking into account the dynamic relationships of control channels; -control algorithms and software; -technology of transmission of commands from the operator.
To study the dynamics, such AMR can be considered as M on a mobile basis or a system of "carrier and transferred body" [1][2][3][4][5][6]. In [7], it is noted that in order to accurately model the dynamics of AMR with M, it is necessary to take into account the interaction between M and the platform, the interconnection of simultaneous rotation and sliding of rigid arms, as well as non-holonomic restrictions to prevent slipping and skidding of the platform.
A feature of the dynamics of AMR with a manipulator as a system of bodies is [8]: -changing the position of the center of mass of the system with the relative movement of the manipulator; -commensuration of non-diagonal and diagonal elements of the inertia tensor calculated relative to the axes of the base coordinate system associated with the center of mass of the AMR platform. Changing the position of the center of mass of the system causes the occurrence of disturbing momenta and causes a loss of orientation of the AMR in vertical planes. The loss of orientation or fall of AMR in the working area under extreme conditions not only leads to the impossibility of performing technological operations and causes material losses. When transporting radiation or explosive objects, a fall of AMR can have catastrophic consequences.
The non-diagonality of the tensor of the inertia of the body system relative to the basic coordinate system causes [8]: -the occurrence of inconsistency of the main central axes of inertia of the system of bodies with the axes connected to the center of mass of the AMR platform by the basic coordinate system; -inconsistency of control actions with the directions of the main central axes of inertia of the system; -interdependence of control channels.
When providing control commands to the drives of the AMR chassis, the non-diagonality of the inertia tensor causes the occurrence of angular velocities in the directions perpendicular to the direction of movement. The control system tries to work out perturbations and prevent the growth of angular velocities in the directions perpendicular to the direction of movement. Despite the operation of the control system, due to the non-diagonality and instability of the inertia tensor and cross-connection of the control channels, AMR with M loses orientation. Additional operation of the control system increases energy consumption, reduces battery life, and reduces the performance of AMR.
The movable base expands the service characteristics of M but imposes more stringent requirements on drives, devices of control, and surveillance systems.
The cycle of work of AMR with M can be divided into several stages [9]: -movement of the AMR platform from the starting point S to the finish point F with a fixed manipulator in compliance with the requirements of optimality (speed, performance, or efficiency, etc.); performing technological operations with a manipulator, attached or information equipment with a fixed or movable platform; -return of the AMR platform to the starting point S (or movement to a predetermined end point K) with a fixed manipulator.
At the first stage of operation, AMR is a trolley with an adaptive control system that moves under predetermined environmental conditions. To ensure the steady movement of the AMR from point S to point F, the control system must, based on the results of the exploration of the working area, draw up a dangerous route and generate control commands for each chassis drive. If in a mechanical system there is a relationship between dynamic characteristics and control channels, then the impact on the chassis drives causes disturbances in the drives of the manipulator. Such perturbations cause an increase in the generalized coordinates of M, as a result of which M begins to move relative to the platform and the system loses stability. Thus, in order to comply with the requirements of optimality, even for moving AMR with a fixed M, it is necessary to first establish the relationship of dynamic characteristics, to perform mathematical and simulation modeling of a mechanical system. Mathematical modeling of AMR dynamics of variable configuration without taking into account the relationship of the dynamic parameters of the mechanical system predetermines methodological and computational errors.
The relevance of this scientific topic is related to the need to build a mathematical model of AMR dynamics with a manipulator, taking into account the interrelationships of the dynamic parameters of a mechanical system. Such a mathematical model is the scientific basis for the development of adaptive control algorithms with a unit of current object self-identification. Synthesis of commands for actuators is carried out programmatically online after analyzing the dynamic parameters of a mechanical system, taking into account the interconnection of control channels. In practice, this will preserve the functionality and ensure the orientation of the AMR in vertical planes despite external disturbances and movements of the manipulator.

Literature review and problem statement
When working remotely under extreme a priori uncertain environmental conditions, various AMR designs of variable configuration are used. The use of AMR with manipulators is also appropriate in the implementation of technological operations with radiating [5] or explosive objects [1,2]. In the operation of such AMRs, the manipulator executes technological operations with the movement of the object of manipulation (cargo, tools, instrumentation) according to a given law of motion.
In different parts of the AMR trajectory of the changing configuration, the load parameters may change: geometric dimensions, shape, mass distribution. These features should be taken into account when building a mathematical model since the coefficients of differential equations of dynamics are functions of the generalized coordinates of the manipulator and depend on the law of motion and load parameters.
The presence of a movable base increases the maneuverability of manipulation AMR, expands the working space, and increases the number of permissible configurations of the manipulator, but reduces the accuracy of positioning the grip (attachments or tools).
Dynamic processes for M on a moving platform can be divided into three driving modes: -controlled movement of the platform along a given route with a "checked" M; -performance of technological operations M with the platform stopped; -M executes technological operations with controlled movement of the platform.
The mathematical model of the dynamics of the mechanical system "AMR with M" should be built and loaded into the microprocessor of the control system in a general case. The equations of the dynamics of a mechanical system for the listed modes of motion are special cases and are derived from a general mathematical model under certain conditions. In addition, the mathematical model should reflect the dynamic features of the studied design of AMR with M.
The controlled movement of the platform with the "checked" manipulator is well researched for various chassis design options. The "checked" manipulator changes the coordinates of the center of mass of the system but the elements of the inertia tensor are stationary [8]. When implementing this mode of movement, AMR is a controlled trolley that moves along the trajectory. The synthesis of the law of controlling the flat motion of a robot-trolley is a typical task: several methods of stabilizing movement along a given trajectory are known. In [10], the dynamics of a 4-wheelchair equipped with fixed devices were investigated. The mathematical model of the dynamics of the all-wheel drive and front-wheel drive trolley when cornering is given. In this case, the platform is considered a solid, the center of mass of the platform is the center of mass of the system. The linearized mathematical model presented in [10] does not reflect the relationship of dynamic parameters and control channels.
When moving the AMR with M along a given route, the control system must ensure minimal deviation of the center of mass of the platform from the specified trajectory. At the same time, the control moments are directed along the axes of the basic coordinate system associated with the center of mass of the AMR platform.
In [11], an unmanned vehicle is considered as a multistate hinge system. To describe the kinematics of the trolley, a modified Denavit-Hartenberg method was used. To build a dynamic model, the Newton-Euler algorithm is used. The proposed model takes into account the interrelationships of all parts of the vehicle, which makes it more representative for the development of adaptive control algorithms. Control is implemented using the Lyapunov functions. Taking into account the interrelationships of dynamic parameters ensures reliable tracking of the reference trajectory when kept in lanes, lane changes, avoiding obstacles, and during critical driving situations. It is noted that multi-body models provide more information, which is usually neglected when using a closed-form model. The model presented in [11] cannot be used to describe the dynamics of AMR with M but the expediency of using the Nyton-Euler method has been proven.
Paper [12] considers dynamic modeling of mobile multibody systems. This class of systems includes spatial wheeled vehicles subjected to constant rigid contacts. To build a nonlinear dynamic model, the Newton-Euler algorithm was used in combination with a projective approach based on an explicit contact model. The results of the implementation of the proposed method for modeling the dynamics of the bicycle are provided. Based on the mathematical model, an algorithm for controlling the movement along the trajectory has been developed. It has been proven that passive asymptotic stabilization of a bicycle can be ensured when it moves at sufficient initial speeds. The model presented in [12] does not take into account the peculiarities of the dynamics of AMR with M. At the same time, the effectiveness of the proposed control algorithms for cross-communication of channels is questionable.
Paper [13] describes dynamic models of tracked unmanned ground vehicles (UGV) in three-dimensional space instead of a simplified two-dimensional projection on a single plane in traditional vehicle theory. The determination of Euler angles is carried out relative to the coordinate system fixed on the main body. A dynamic model for rectilinear movement and steering modes is proposed. with the use of spatial forces and moments. Dynamic models are built according to the Newton-Euler method and the Lagrange method. With the help of the developed models, it is possible to establish rules for the distribution of vehicle load to ensure the stability of movement in three-dimensional space and implement the appropriate control algorithms. The model presented in [13] does not take into account the non-diagonality and nonstationarity of the tensor of inertia, especially the dynamics of AMR with M.
In [14], for the equations of controlled motion of the robot along the trajectory, a replacement of variables was found, which allows the linearization of the mathematical model on feedback. The synthesis of the law of control for an arbitrary "target path" was carried out taking into account the restrictions on phase coordinates and control actions. For AMR with M, linearization of the mathematical model is possible only for individual modes of motion. At the same time, the effectiveness of the proposed control algorithms for cross-linking channels is questionable.
Work [15] states that slipping the wheel can cause a significant deterioration in handling while driving a mobile robot. This paper proposes a method to prevent wheel slippage using a nonlinear predictive driving model. The limitations included in the optimization task change the force of interaction between each wheel and the ground. This approach has been tested under a dynamic modeling environment using the example of a wheeled mobile robot Pioneer 3-DX, which performs box pushing manipulations. For autonomous operation of AMR with M under extreme or predetermined conditions, the proposed method of preventing wheel slippage can be useful.
The trolley dynamics equations track properly the deviation of the trolley's center of mass from a given trajectory. But, in [12][13][14][15], issues related to the expediency and justification of the limits of application of the linearized mathematical model remained unresolved. In addition, the mathematical models proposed in [12][13][14][15] do not reflect the relationship of the dynamic parameters of the mechanical system. In [16], the design schemes for the formation of control actions on the chassis drives, taking into account the interconnection of the control channels, are given. The proposed schemes should be used in the development and design of driving chassis system for AMR with M; this will take into account and compensate for the non-diagonality and nonstationarity of the inertia tensor. But the mathematical models given in [16] cannot be used to describe the movement of M relative to the AMR platform.
M performs some technological operations with the platform stopped. Under this mode, the mathematical model of the system contains the equation of M dynamics relative to a non-moving base. Most often, the dynamics of manipulators are described using the Newton-Euler method and the Lagrange-Euler method [17].
When moving M relative to the AMR platform, the main central axes of inertia of the body system do not coincide with the axes of the base coordinate system [8]. As a result, the AMR inertia tensor in the coordinate system associated with the platform is non-diagonal and non-stationary. In [8], the results of the analysis of the elements of the inertia tensor of AMR are given. It is accepted that the mass M with a load of 10 % by weight of the AMR chassis. It is proved that the centrifugal and axial moments of inertia of AMR with a manipulator (relative to the axes of the base coordinate system) depend on the time and values of the generalized coordinates [8]. These features should be taken into account when building a mathematical model of movement M with a movable platform of AMR.
If M performs technological operations with controlled movement of the AMR platform, then the mathematical model should take into account the interrelationships of the dynamic parameters of the mechanical system. Under this mode of operation, it is advisable to consider the dynamics equation in relation to the coordinate system associated with the docking node of M and the platform. This approach is used in [18] where the mathematical model M of the spacecraft is presented, built according to the Newton-Euler method, taking into account the non-diagonality and nonstationarity of the inertia tensor. In [18], the mathematical model is constructed in the assumption that the mass of M is 1 % of the mass of the spacecraft's hull, and the center of mass of the spacecraft's hull is the center of mass of the mechanical system. Therefore, the mathematical model proposed in [18] does not take into account the change in the position of the center of mass of the system with the relative motion of the manipulator.
Paper [3] reports the results of the study of the dynamics and control of cosmic M with friction in the joints. To describe joint friction, the Coulomb, Stribek, and Lugre friction models are considered. The dynamic equation of the system is built by a recursive method based on the principle of variation in the speed of Jourdain. An active controller has been developed to track the trajectory of the system. The reliability of the proposed dynamic model is verified by comparing the results of numerical modeling and the results obtained from the ADAMS software. A link has been established between joint friction and low-speed movement in the process of tracking the trajectory. It is proved that friction in the joints of M changes the dynamic characteristics of the mechanical system and reduces the accuracy of control. The materials provided indicate that for the synthesis of effective control, it is necessary to apply mathematical models taking into account the relationship of the dynamic parameters of the mechanical system.
Paper [4] considered the technology of telerobotics for remote control of space M. Implementation of telerobotics makes it possible for M to copy the movements of the operator. Commands on M drive are formed simultaneously by the encoders of the control device. It is indicated that the optimality of the generated trajectory of M links depends on the professional training of the operator. However, no criteria have been established for assessing the compliance of movements of M and operator movements. It is also not determined whether the M mathematical model takes into account the relationship of the dynamic parameters of a mechanical system.
Paper [5] reports the design of a multifunctional service platform (MFMP). The MFMP model was built in accordance with the maintenance requirements of the Thermonuclear Test Reactor (CFETR). The mathematical model MFMP is constructed using the Newton-Euler method. The equivalent driving force of each drive is determined using an equivalent inverse Jacobi matrix. The accuracy of the model is tested by modeling in the ADAMS software environment. Based on the Newton-Euler method and the improved virtual principle of operation, a control system with stability control has been developed. The presented mathematical model is not applicable for AMR with M since the dynamic features of the mechanical system are not taken into account. But the presented methods of compiling and checking the accuracy of the mathematical model should be used.
Paper [6] reports a remote control system of M with five degrees of mobility. M is mounted on a movable chassis and is used as a robot -a savior. The operator remotely controls M using the computer interface, which displays the movement of the robot and records the current values of the characteristics of M. The FPGA controller controls M servo drives and exchanges data with the computer through a wireless module. To provide remote control over M in real time an interactive mixed programming interface VC and MATLAB has been built. The mathematical model of the robot-savior is constructed using software, so it is difficult to assess how much the relationship between dynamic parameters and control channels is taken into account. In addition, the developed system does not provide feedback from M to the operator, that is, it cannot be used to manipulate fragile or explosive objects.
Paper [7] reports a symbolic algorithm for obtaining the equation of motion of N-link M installed on a mobile platform. The kinematic scheme of M contains rotary-prismatic connections. To build the kinematic and dynamic equations of motion of the system, the Gibbs-Appel recursive formula is used. To increase the computational efficiency of the proposed algorithm, all mathematical operations are performed using matrices 3×3 and 3×1 only. It should be noted that the nonholonomic characteristics of a moving platform complicate the control equation. In addition, the proposed model does not make it possible to separate the effects of dynamic parameters on the interconnection of control channels.
In [19], a mobile wheeled platform with two manipulators is considered. The authors believe that the increase in the number of relationships between manipulators and the mobile platform, as well as the non-holonomic limitations of the mobile base, make manual derivation of motion equations almost impossible. To build a mathematical model of dynamics, an automatic system approach is proposed. To avoid calculating the "Lagrange multipliers" associated with nonholonomic constraints, motion equations are derived according to the Gibbs-Apelle recursive formula. To increase efficiency, all mathematical operations are implemented with matrices 3×3 and 3×1. The application of the proposed method for a wheeled platform with two manipulators demonstrates the possibility of building equations of motion but does not reflect the relationship of control channels.
In [20], pseudo symbolic dynamic modeling (PSDM) was presented to build simplified dynamic models of M, the design of which contains up to 7 degrees of mobility. The presented algorithm makes it possible to generate code in real time, simulate dynamics, and increase the efficiency of the model by eliminating minimally important elements. In addition, the authors of [20] developed an implementation of the algorithm in the MATLAB environment, which is publicly available. But in [20] the issues remained unresolved, related to the expediency and validity of the exclusion of minimally important elements.
In [21], a virtual model of kinematics and dynamics of M in MATLAB&SIMULINK was developed. The presented model uses a PID controller, and the dynamics equations are obtained by the Lagrange-Euler method. Work [22] considers the dynamic modeling of spatial three-link M using symbolic and numerical methods. To derive the equations of dynamics in the form of a space of states, an algorithm based on the Newton-Euler method is proposed. The algorithm is implemented in the Maple system, the simulation was carried out in MATLAB&SIMULINK. The mathematical models proposed in [20][21][22] do not reflect the relationship of the dynamic parameters of the mechanical system.
The above review of the literature proves that the application of the Newton-Euler method makes it possible to build a mathematical model to describe the three modes of motion of AMR with M. When overcoming the consequences of man-made and natural disasters, AMR with M performs technological operations in an a priori indefinite external environment. The working space of the AMR manipulator may be limited (for example, the width of the slot). The microprocessor of the control system should synthesize commands into executive drives in real time to take into account changes in workspace. This necessitates a change in the orientation of AMR in vertical planes, narrowing the set of values of the generalized coordinates of M, reducing the size of the working space of M, reducing the number of valid configurations of M. Mathematical modeling in real time will determine the optimal strategy for moving the manipulator with the controlled movement of the AMR platform. Taking into account the relationship of the dynamic parameters of the mechanical system could make it possible: -to preserve the functionality and ensure the orientation of AMR in vertical planes despite the movement of the manipulator; -to identify reserves to reduce energy consumption and increase the stability of AMR of a variable configuration when operating under a priori uncertain environmental conditions; -to improve the efficiency and survivability of AMR of a variable configuration in overcoming the consequences of man-made and natural disasters.

The aim and objectives of the study
The aim of this study is to build a mathematical model of the dynamics of AMR of variable configuration, taking into account the relationship of dynamic parameters of a mechanical system. This will make it possible to identify reserves to reduce energy consumption, increase stability, improve the efficiency and survivability of AMR of a variable configuration.
To accomplish the aim, the following tasks have been set: -to build the equation of motion of the center of mass of the AMR system of variable configuration along the trajectory, taking into account the relationship of the dynamic parameters of the mechanical system; -to establish the influence of the relationship of the dynamic parameters of the mechanical system on the equation of angular motion of AMR of the variable configuration; -to determine the relationship of the dynamic parameters of the mechanical system when the manipulator moves relative to the AMR platform.

1. Object and hypothesis of research
The object of this study is the dynamics of AMR with the manipulator. The main hypothesis is the relationship of the dynamic parameters of the mechanical system. The peculiarities of the dynamics of AMR with the manipulator are due to the change in the position of the center of mass of the system with the relative movement of the manipulator and the commensurate of the non-diagonal and diagonal elements of the inertia tensor calculated relative to the axes of the base coordinate system.
While building a mathematical model, the following assumptions were accepted: -the density of the AMR platform and the density of movable structural elements does not change over time; -all links of the manipulator are absolutely rigid; -the elements of kinematic pairs are absolutely rigid; -the drive elements in the hinges are absolutely rigid; -there is no elastic connection between the moving elements and the AMR platform.

2. The design of an autonomous mobile robot of variable configuration
The design scheme of AMR with the manipulator is shown in Fig. 1. The structure consists of an all-wheel drive AMR 4-wheel platform, and an anthropomorphic manipula-tor composed of a ring rotating around a vertical axis, and rod links -hands connected by rotary kinematic pairs of the fifth class.
In the assumption that all links are absolutely solid bodies, AMR with a manipulator is a dynamic system of non-deformed bodies with non-holonomic stationary connections.

3. Coordinate systems
To build a mathematical model, I shall introduce the right-hand Cartesian coordinate systems. In Fig. 1, the following notation is used: -АXYZ -inertial coordinate system; -СX С Y С Z С -movable base coordinate system. The reference point is connected to the point C -the center of mass of the platform. The axes are parallel to the main central axes of the inertia of the AMR platform. The СZ C axis is perpendicular to the plane of the platform movement, coincides with the local vertical, and is directed upwards to the manipulator ring, СX С located in the plane of movement of the platform and directed towards the movement, the СY С axis is located in the plane of movement of the platform and complements the coordinate system to the right-hand one; -ОX 0 Y 0 Z 0 -movable coordinate system. The reference point is connected to the point O -the center of mass of the system of bodies. The axes are parallel to the axes of the coordinate system СX С Y С Z С ; -О 1 X 1 Y 1 Z 1 -connected movable coordinate system. The reference point is connected to the point О 1 -the center of mass of the ring (docking node). The axes coincide with the main central axes of the inertia of the ring. In the initial position of the manipulator ring the axes of the coordinate systems О 1 X 1 Y 1 Z 1 are parallel to the axes of the coordinate system СX С Y С Z С ; - lator the axes of the coordinate systems О i X i Y i Z i are parallel to the axes of the coordinate system СX С Y С Z С ; -МX М Y М Z М -movable base coordinate system. The reference point is connected to the point M -the center of mass of the manipulator. In the initial position of the manipulator, the axes of the МX М Y М Z М coordinate systems are parallel to the axes of the coordinate system СX С Y С Z С .

Theoretical methods
The conclusions, scientific assumptions, and recommendations formulated in this paper are based on the fundamental positions of linear algebra, theoretical mechanics, and the theory of differential equations. The construction of a mathematical model was carried out according to the Nyton-Euler method.
The resulting mathematical model contains: -an equation of motion of the center of mass of the AMR system of variable configuration along the trajectory in the inertial coordinate system; -an equation of angular motion of AMR of variable configuration in the inertial coordinate system; -an equation of motion of the manipulator with respect to AMR.
To construct the equation of motion of the center of mass of AMR of a variable configuration along the trajectory in the inertial coordinate system, the theorem on the center of mass of a solid was applied [23,24].
To derive the equation of angular motion of AMR with a movable manipulator, a theorem on the change in the kinetic moment of a system of solids was employed [23,24].
To compile the equation of motion of the manipulator relative to the AMR platform, the theorem on the moments of the amount of motion was applied [23,24].
In the transformations of equations, the concept of a local derivative in the designated coordinate system is applied.

1. Equation of motion of the center of mass of an autonomous mobile robot of variable configuration along the trajectory
Applying the theorem on the motion of the center of mass of a solid [23,24], the following is recorded for the manipulator and platform in the inertial coordinate system   Suppose that the coordinate system CX C Y C Z C moves relative to the inertial coordinate system AXYZ at an angular velocity , Ω and the coordinate system О 1 X 1 Y 1 Z 1 moves relative to the coordinate system CX C Y C Z C with angular velocity 1 .
w Applying local derivatives, the following is received: is the operator of the local derivative in the coordinate system СX С Y С Z С ; , Ω -angular velocity of the coordinate system СX С Y С Z С relative to the inertial coordinate system AXYZ.
Using the rule of adding vectors and the concept of a local derivative, the following is recorded ( ) ; Ω --angular velocity of the coordinate system СX С Y С Z С relative to the inertial coordinate system AXYZ; 1 w -angular velocity of the coordinate system O 1 X 1 Y 1 Z 1 relative to the coordinate system СX С Y С Z С (or relative to the coordinate system ОX 0 Y 0 Z 0 ).
Substituting expressions (5) and (6) into formula (4): ( ) If one considers the intervals of movement of AMR with a fixed manipulator, then the relative speed and acceleration of the center of mass of the manipulator is zero, that is, Under these conditions, expression (7)  If one considers the intervals of movement of AMR at which the angular velocities are much smaller than linear velocities, in expression (7) one can neglect terms that contain the product of angular velocities as second-order small ones. In this case, expression (7) takes the form: -when moving AMR with the manipulator movable relative to the platform: So, from the equation of moving the center of mass of AMR with a fixed manipulator (10), after substituting formulas (13), the following is obtained: If AMR moves at limited angular velocities, then after substituting formulas (13), the following is obtained from equation (11): For the sections of the trajectory on which AMR moves with a movable manipulator, taking into account formulas (13), (12) is recorded in the form: Equation (16) is the equation of motion of the center of mass of AMR of a variable configuration along a trajectory in an inertial coordinate system in a general case. There are some cases of the equation of motion of the center of mass of AMR; (14) -if the manipulator does not move, (15) -if the system moves at a limited angular velocity.

2. The equation of angular motion of an autonomous mobile robot with a movable manipulator
Using the theorem on the change in the kinetic moment of the system of solids, the following is recorded for the manipulator: where m j is the mass of the j-th elementary part of the manipulator; j r -radius-vector of the j-th elementary part of the manipulator relative to the beginning of the inertial coordinate system (Fig. 2); j v -linear velocity of the j-th point of the manipulator relative to the beginning of the inertial coordinate system; M M -the main vector of the moment of external forces and forces in the hinges acting on the manipulator, relative to the beginning of the inertial coordinate system. Based on the theorem on the change in the amount of motion of the manipulator, the following is recorded: Formula (18)  Applying the theorem on the change of the kinetic moment, for the casing (platform) of the AMR, the following is recorded where m i is the mass of the i-th elementary part of AMR body; i r -radius-vector of the i-th elementary part of AMR body relative to the beginning of the inertial system; i v -linear velocity of the i-th point of AMR relative to the beginning of the inertial coordinate system; C M -the main vector of the moment of external forces and forces in the hinges acting on AMR hull relative to the beginning of the inertial coordinate system. Adding (17) and (18) (21) and (22) in (20), the following is obtained: Let's simplify equation (23) as a static moment of mass relative to the center of mass of the system; then equation (23) will take the following form: 2) If one considers the intervals of movement of AMR with a fixed manipulator, then the angular velocities in the kinematic pairs of the manipulator are zero 1   3) If one considers the intervals of movement of AMR at which the angular velocities are much smaller than linear velocities, in expression (24) one can neglect terms that contain the product of angular velocities as small second-order. After the mathematical transformations in equation (24), the following is obtained: (32) is the equation of motion of the manipulator relative to AMR housing in the general case.

Discussion of research results
Using the Newton-Euler method, a mathematical model of AMR dynamics of a variable configuration was built. A feature of the results obtained is to take into account the relationship of the dynamic parameters of the mechanical system.
As an example, the structure of AMR with a manipulator, which is designed to overcome the consequences of man-made and natural disasters, is considered. The features of the dynamics of AMR with the manipulator are due to the change in the position of the center of mass of the system with the relative motion of the manipulator and the commensuration of the non-diagonal and diagonal elements of the inertia tensor, calculated relative to the axes of the base coordinate system. The proposed mathematical model describes three modes: -controlled movement of the platform along a given route with fixed M; -movement of AMR, provided that the angular velocities are significantly less than linear velocities; -M performs technological operations with controlled movement of the platform.
The equations of the mathematical model are derived for the specified modes of motion of AMR with M. If one considers the intervals of movement of AMR at which the angular velocities are much smaller than linear velocities, then in the equations of dynamics one can neglect terms that contain the product of angular velocities, as small second-order. If one considers the intervals of movement of AMR with a fixed manipulator, then the angular velocities in the kinematic pairs of the manipulator are zero.
The mathematical model of the dynamics of AMR of changing configuration, taking into account the relationship of the dynamic parameters of the mechanical system, contains: -equation (16)  The AMR chassis control system provides movement in a limited space (work area) along a given trajectory using GPS navigation or orientation on a map or beacons. Within the working area, static and (or) moving obstacles are possible, which must be overcome with the slightest deviation from the specified route. The mathematical models of dynamics presented in [11][12][13][14][15] provide reliable tracking by the trolley of the reference trajectory but the considered design does not imply the presence of moving elements (M or attachments). In equations (14) to (16), the relationship of the dynamic parameters of the system from the direction and magnitude of the radius-vector .
OC p is established. The shift of the center of mass of the platform relative to the center of mass of the sys-tem of bodies by the OC p radius-vector is due to the presence of moving elements (M or attachments). The application of equations (14) to (16) provides tracking by the AMR platform of the reference trajectory in the presence of moving elements (M or attachments). This will prevent the occurrence of angular velocities in the directions perpendicular to the direction of movement of the platform, avoid slipping and skidding of the platform, and maintain the orientation of AMR with M.
Remote control of the manipulator provides high quality indicators of technological operations both with a non-moving and mobile chassis. The mathematical models proposed in [3][4][5][6][7][17][18][19] do not take into account the change in the position of the center of mass of the system with the relative motion of the manipulator. Models [19][20][21][22] do not make it possible to separate the influence of dynamic parameters on the interconnection of control channels.
The design of the grip and the kinematic scheme of the manipulator provides for the possibility of working with fragile or explosive objects. Equations (24) to (26) make it possible to establish the influence of dynamic parameters of the system on generalized speeds j w and accelerations j dw dt in the kinematic pairs of the manipulator. The movable base of M causes a change in the working area and an increase in the position error of the grip pole. The application of equations (24) to (26) provides software adjustment of the dynamic parameters of the manipulator when moving the AMR platform along the trajectory. Equation (32) enables control and software adjustment of the dynamic parameters of the manipulator and contact forces in the grip. The application of equation (32) makes it possible to fix the object and ensure its reliable retention when moving the AMR platform along the trajectory.
Adjusting the operation of the control system reduces energy consumption, prolong the time of autonomous work, and increases the productivity of AMR.
The limitations of this study are due to the consideration of only three modes of movement of AMR with M and the assumption that the elements of the kinematic scheme of M and the AMR platform are absolutely rigid bodies. For other modes of motion of AMR with M, it is necessary to refine the mathematical model based on reasonable assumptions and simplifications. The limitations of this study must be taken into account when trying to apply the developed mathematical model in practice, as well as in further theoretical studies. The disadvantage of the study is that the dynamics of kinematic pairs are not taken into account.
Advancing this study is the development of an adaptive control system with elements of artificial intelligence. As an object of control, AMR with a manipulator is a multi-connected system with a cross-internal connection of control channels, which is formed by the dynamic parameters of a mechanical system. According to the results of mathematical modeling using the proposed model, it is possible to develop algorithms for adaptive control using cross-connection of channels. This will reduce the number of responses of the control system, save energy, and prolong the time of autonomous work of AMR with M.

Conclusions
1. The equation of motion of the center of mass of the AMR system with a manipulator along the trajectory along the trajectory, taking into account the relationship of the dynamic parameters of the mechanical system, is built. At the same time, the influence of the dynamic parameters of the system on the direction and magnitude of the radius-vector OC p -the shift of the center of mass of the platform relative to the center of mass of the system of bodies -was established.
2. The influence of the relationship of dynamic parameters of a mechanical system on the equation of angular motion of AMR of changing configuration has been established. This takes into account the commensurate of the non-diagonal and diagonal elements of the inertia tensor, calculated relative to the axes of the base coordinate system.
3. The interrelation of dynamic parameters of the mechanical system during the movement of the manipulator relative to the AMR platform is determined. The influence of system parameters on generalized speeds j w and accelerations j dw dt in the kinematic pairs of the manipulator has been established. Thus, it is possible to control the contact forces in the grip, and if necessary, fix the object and ensure that it is held reliably.

Conflicts of interest
The author declares that she has no conflicts of interest in relation to the current study, including financial, personal, authorship, or any other, that could affect the study and the results reported in this paper.

Financing
The results reported in the paper were obtained by the author during research on the topic 0122U 001326 "Scientific and methodological support of applied research in the mechanics of mechatronic systems".

Data availability
All data are available in the main text of the manuscript.