Introduction. In the modern world, the demand for automatic mobile wheeled vehicles is growing, and the possibilities of coordinating their operation are being actively explored. Issues of autonomy and controllability of mobile robots are of considerable scientific and applied interest. In particular, approaches to the calculations of multiphysical processes, simulating the movement of specific types of wheels under different conditions are considered. Some studies focus on the dynamics of deformable contacting bodies during sliding, rolling and rotation. The results of these studies are used in robotics [1-3]. The control of mobile wheeled vehicles is considered on the example of electric scooters “Segway” [4].

Mobile mechanisms with simple and efficient wheel motion control systems are needed primarily for operating in narrow spaces [5], for conducting research under difficult industrial conditions [6].

In [7-11], the possibilities of accurate modeling of the dynamics of wheeled vehicles are shown, taking into account the sliding, spinning, and rolling of the wheels.

At the same time, the phenomena arising from the combination of friction effects are not described in the literature. The presented work aims at investigating the effect of multicomponent friction on the dynamics of a robot with a differential drive. Tasks involve building a mathematical model of robot movement with multicomponent friction and simulating program movements.

Materials and Methods. The scheme of a two-wheeled differential drive robot is considered (Fig. 1). The continuous movement of wheels of radius R along the support surface is assumed.

The gravity center of platform C is shifted relative to the geometric center of platform A along X axis by distance along Y — axis — by distance . Points and — centers of mass of the wheels (index 1 corresponds to the left wheel, index 2 —to the right).

The following designations are introduced:

— mass of the robot platform;

— moment of inertia of the platform relative to the center of mass C;

— fixed coordinate system; determine the position of point A in

y — angle of the course of the platform;

— wheel steering angles (index 1 corresponds to the left wheel, index 2 — to the right);

— wheels width;

— engine torques applied to the wheels.

It is assumed that only the normal bearing reaction acts on the support wheel 4 (Fig. 1). On the move, the support wheel is located on X axis at a distance from the center of the platform

Research Results

Mathematical model of robot dynamics. When describing the movement of a differential drive robot in the framework of nonholonomic mechanics, expressions are used that take into account the limitations associated with zero wheel slip relative to the surface:

(1)

where — projections on Х and Y axes of velocities of points , which are centers of the contact areas of the wheels and the supporting surface.

Note that for a nonholonomic model, a point contact of the wheels and the surface is assumed, i.e., the contact area of each wheel degenerates into a single point of contact.

Consider the expressions:

(2)

With their help, equality for the angular velocities of the wheels can be obtained:

(3)

In the general case, when the presence of a contact platform is assumed, we obtain the equations of dynamics of a differential drive robot:

(4)

Here,

— projections of the friction force on the directions of the slip speeds of the centers of the wheel contact patches;

— projections of the friction force in directions perpendicular to the speeds of the centers of the wheel contact patches;

— summands of the friction torque under spinning for two wheels;

— angular acceleration of platform spinning;

— angular accelerations of wheels projected onto their axis of rotation;

— weight of each wheel;

and — inertia torques of the wheel relative to the corresponding axes.

To form the control actions, a model of the robot dynamics described in the framework of nonholonomic mechanics is used [12]:

(5)

It can also be represented as:

(6)

where and — left sides of the equation (5).

Model of engines on driving wheels. Let us describe a model of engines capable of generating control torque on driving wheels

The equations of motion of wheeled DC motors can be expressed by the formulas [13]:

(7)

Here,

— inductance of electric motor conductors;

R — resistance to electric current in the motor circuit;

— power supply for the engine k;

— current passing through the motor armature circuit;

— structural constants of engines.

It is assumed that the drive wheel motors have the same characteristic. From (7), it can be seen that

(8)

Given (8), in the first equation from system (7), we obtain a differential equation with respect to moment

(9)

Equation (9) is divided by multiplier , and we get:

(10)

Let us introduce the notation:

. (11)

In this case, the control moments can be described:

(12)

where

— constants;

— parameter that determines the speed of transients in the electrical part of engines.

When ignoring the switching time in the electromechanical system of the engine:

(13)

Авторы [14] получили похожие формулы.

The authors [14] obtained similar formulas.

Model under study. Let us consider models describing the dynamics of a differential drive robot.

Model 1 is described by equations (6) and (12), when the duration of transients is assumed to be vanishingly small

(14)

Here, are calculated by formulas (3).

Equations (4), (13), as well as the Coulomb friction model, are used to describe Model 2:

(15)

Model 3 is represented by equations (4), (12). In addition, multicomponent friction presented in [12] with the help of Pade decompositions is taken into account:

, (16)

(17)

(18)

(19)

Here,

constants that are found from the formulas in [12];

α — angle characterizing the direction of linear sliding speeds relative to the contact points of wheels

— coefficient whose sign depends on the rolling direction.

To determine the program control, the motion laws of the robot should be specified . By (5), it is possible to express values depending on time. We use known values to calculate control voltages and , supplied to the wheel drives:

(20)

The motion of the system with the given initial conditions was modeled in the Wolfram Mathematica package.

Consider To integrate the system, the integration step was automatically selected and the built-in numerical methods of Wolfram Mathematica were used with an accuracy of the results of at least 10–6.

Figures 2 and 4 show that the control guarantees faithful realization of the program laws of motion after the completion of transients in models 1 (nonholonomic) and 2 (with Coulomb friction). In Model 3 (multicomponent sliding, rolling and twisting friction), after the completion of transients due to software control, a constant error occurs in the angular velocity of platform rotation. In all the models considered, the sliding speed of the wheels in the transverse direction quickly decreases to zero. The cases of matching dependences obtained for different models are shown in Figures 2 and 3.

To check the fulfillment of the conditions of continuous motion of the supporting wheel, we write down the theorem on the variation of the kinetic moment of the system in projections on Y axis:

According to (4):

From the last equation, we obtain the condition of continuous motion of the supporting wheel:

Considering (5):

The robot's accelerations did not exceed 1 m/s2; therefore, this inequality is fulfilled at every moment of time.

Discussion and Conclusion. The simulation results allowed for drawing a number of conclusions. Software control based on nonholonomic model 1 is acceptable for robots whose design does not take into account the contact friction of the wheels that occurs under a combination of sliding, spinning and rolling. However, when forming control actions, to achieve more accurate adherence to the program motion laws, it is required to take into account the model of multicomponent friction. The resulting solution can be used to build a control system with a model predictive control. Such a development will be the subject of further research.