Mathematical Modeling of Wheeled Mobile Robots

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3. Kinematic Model

Creation of kinematic model was based on mobile robot GTR 2010, which was designed and constructed in department of applied mechanics, and also on experience from previous model GTR2006.

Chassis with four wheels is controlled differentially. Each one of the wheels is independently driven by separate speed servomechanism.

Mobile robot always moves in circular trajectory with diameter R (on trajectory projection of which on plane XY is circle). Direct motion occurs when trajectory radius approaches infinity. This radius changes in time depending on desired motion direction ^[7].

Axes of all wheels have to intersect in one point and that point is an immediate center of rotation ^[4].

The proposed four-wheel undercarriage will be steered differentially. Each of the wheels is driven individually by means of the servomechanism. This is a proposed design solution of the wheeled undercarriage ^[5]. In the middle-point of the undercarriage is situated a hinge, which enables tilting of both parts of the mobile robot and in this way it increases stability of the robot in terrain.

Figure 2. Mobile robot with sectional undercarriage

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Individual transformation matrixes among the neighbouring local coordinate systems will be multiplied symbolically using the software Mathcad in order to obtain the global transformation matrix, which describes the geometrical position and orientation of the centre of gravity in the global coordinate system, as well as there are obtained in this way the transformation matrixes describing the position and orientation of the individual undercarriage wheels.

Figure 3. Local coordinate systems

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The coordinate q1 represents a transformation of geometrical position of the gravity centre. This transformation corresponds to the rotation around the axis z. The coordinate q2 represents a transformation corresponding with translation toward the axis x.

According to the principle of the differential steering it is evident that if there is actual a requirement of a linear movement, so it is necessary that the turning radius is approximating to infinity. In this case this radius is given by the coordinate q1, which has to approximate to infinity. There was chosen also the third coordinate – q₃, which represents transformation of a translation along the y-axis. This coordinate ensures the linear movement of the undercarriage, as well as it simplifies the control of movement and calculations.

Based on constructed transformation matrices it is possible to construct simulation models for defining center of gravity geometric position of chassis and wheels in Matlab/Simulink environment.

Figure 4. Simulation model of position of center of gravity and wheels

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Simulation model of position of geometric center of gravity of chassis and individual wheels comprises of several subsystems that have common input. Every subsystem represents position of wheel, or position of center of gravity to beginning O (Instantaneous Center of Rotation).

Figure 5. Simulation model of velocity of chassis, center of gravity and wheels

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It is possible to use this simulation model to calculate estimated trajectory of center of gravity and wheels of chassis. It is also possible to examine behavior of model in any given combination of chassis dimensions. In connection to this there is also an option of chassis geometry optimization. Furthermore created model provides room for examining influence of terrain roughness on behavior of chassis of mechatronic system.

4. Dynamic Model

The aim of the system analysis from mechatronic point of view is to draw a dynamic model of system for control.

DC motor as a dynamic system is described by a system of differential equations that can be derived from the simplified diagram.

Figure 6. Simplified system diagram

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Subsequently it is possible to draw the motion equation:

(1)

Where M_E is moment torque, I_E, I_C and I_W is the sum of the moments of inertia of engine, clutch and wheel, where as following applies:

(2)

Losses occur when electrical energy is converted to mechanical energy (in the motor), or mechanical energy is converted to electrical energy. For the machine to be efficient, these losses must be kept to a minimum. Some losses are electrical, others are mechanical. Electrical losses are classified as copper losses and iron losses; mechanical losses occur in overcoming the friction of various parts of the machine. These losses represents the moment ML.

Dynamic model of a DC motor with separate constant excitation can be expressed as follows:

Ÿ Equation describing the electrical balance in the armature circuit

(3)

Equation (3) describes the sum of the voltages of individual members of the simplified electrical diagram, where R_E.i(t) is the voltage on winding resistance of engine, L_E di(t)/dt is the voltage on engine inductance and u_i(t) is the induced voltage, for which the following applies:

(4)

Where k_E is electric constant, ω_E(t) is angular velocity acting on the engine shaft.

Ÿ Equation of mechanical balance of moment on shaft:

(5)

(6)

(7)

Where I is moment of inertia of engine, M_L is moment, by which engine applies load (passive resistances.

Using Laplace transformation it is possible to derive from equations (1-7) the transfer function of DC motor:

(8)

Operator form of equation (8) of balance of moments is expressed by:

(9)

Block diagram of engine and wheels 1 and 2.

Figure 7 shows Dynamic model of DC motor with constant excitation.

Figure 7. Dynamic model of DC motor with constant excitation

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Figure 8 shows Dynamic model of DC motor with constant excitation.

Figure 8 and Figure 9 shows a model describing the dynamics of the engine and wheel of chassis. Since the chassis is controlled differentially, wheel velocities on right and left side must be the same. Therefore, for purpose of simplification, the right block represents engines and wheels 1 and 2, left block engines and wheels 3 and 4.

Figure 8. Block diagram of engine and wheels 1 and 2

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Figure 9. Block diagram of engine and wheels 3 and 4

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