# ALGORITHM OF AUTONOMOUS VEHICLE STEERING SYSTEM CONTROL LAW ESTIMATION WHILE THE DESIRED TRAJECTORY DRIVING

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2 OL. 11, NO. 15, AUGUST 016 ISSN CALCULATION OF STEERABLE HEELS TURN ANGLE Average wheel turn angle as a function of kinematic parameters of vehicle motion can be calculated, for instance, as follows [4]: l CoG lr lf mcog R R l K l K yr (1) = wheel turn angle; l = wheelbase; R = turning radius (respectively, 1 R is the trajectory curvature); CoG = velocity of vehicle center of gravity; l F = distance from vehicle center of gravity to front axle; l R = distance from vehicle center of gravity to rear axle; K = front wheels tire side slip constant; K yr = rear wheels tire side slip constant. In Bos handbook [5], while describing principles of operation of dynamic stability system, the equation is used whi can be applied for calculation of wheel turn angle: 1 l 1 l 1 = yaw rate; = longitudinal velocity of vehicle center of gravity; = aracteristic vehicle velocity, that is, the parameter whi generalizes geometrical and physical properties of vehicle. The performed tests demonstrated that Equation (1) provides acceptable results if, using the procedure described in article [6], the coefficients K and K yr are substituted with the functions of velocities (see Figure- ), and moreover, su calculations, as Eq. (), can describe only steady (quasi-stationary) driving modes. Comparisons of wheel turn angles according to the developed procedure with those according to Eq. () will be presented below, Equation () will be mentioned in the plots as the Bos equation. () Figure-. Tires side slip constants as a functions of vehicle velocity. It should be mentioned, that in order to reproduce the control law of steering wheel it is not sufficient to predefine the trajectory as a discrete set of Cartesian coordinates of vehicle center of gravity (x, y), it is also required to define the travelling velocity, for instance, as a parameter of time t in ea point of the trajectory (t, x, y), however, these initial parameters are also insufficient for calculations, since the vehicle center of gravity in general case can travel along one and the same trajectory, driving with various body sideslip angles β. Thus, let us consider the discrete data set (t, x, y, β) as initial data for calculations, whi is equal to the fact that in the tests the travelling trajectory of two different vehicle body points were recorded. 9313

3 OL. 11, NO. 15, AUGUST 016 ISSN The main difficulty in simulation of curvilinear vehicle driving is the calculation of lateral forces, acting in tires contact areas, since tire due to its elastic properties reacts to disturbances with some delay. Basic calculated equation of lateral force F S on the i th wheel is presented by empirical equation from [7] and [8]: F FSi (, FZ, k) 1 F arctank y ( t) (3) k y1 : k, y1 k y = dynamic tire side slip constants; = tire side slip angle; F Z = vertical wheel load. Analysis of Equation (3) revealed that it describes well the decrease of amplitudes of lateral force under dynamically varying external impacts (vertical force and side slip angle), but it does not describe in terms of physics the delay in increment of lateral force stipulated by tire elasticity. Thus, modifying Equation (3) with accounting for empirical dependence from [9] and [10], we obtain the final equation for calculation of lateral force: F Si t y0 L y0 F 1 F arctan k y1 = wheel velocity; (4) k y ( t) FSi L = relaxation length of pneumatic tire. Further on, the calculations were based on the vehicle bicycle model, the tire side slip angles and dynamic weight distribution over axles were determined by equations from [7]. The calculation procedure of steerable wheel turn angle while driving along desired trajectory is illustrated in Figure-3. Initial data [t,x,y,β] Calculation of vehicle CoG velocity [t, CoG ] elocity vector orientation angle [t,(ψ+β)] ehicle longitudinal axis orientation angle [t,ψ] Yaw rate [t,dψ/dt] Path curvature [t,r 1 ] Rear wheel tire side slip angle [t,α R ] CoG accelerations [t,a x,a y ] Dynamic weight distribution over axles [t,f ZF,F ZR ] Lateral forces on axles [t,f YF,F YR ] Front wheel tire side slip angle [t,α F ] Steerable wheels average turn angle [t,δ ] Figure-3. Calculation procedure of steerable wheels average turn angle upon driving along desired trajectory. Figure-4 and Figure-5 illustrate comparisons between acquired upon road tests steerable wheels average turn angles with calculations according to the developed procedure (Figure-3) and by Equation (). 9314

4 OL. 11, NO. 15, AUGUST 016 ISSN Figure-4. GAZ-313 minivan, Lane ange S = 0, velocity: 48 km/h. It can be seen, that the calculated wheel turn angle according to the proposed procedure is in good agreement with experimental data, coinciding both in phase and in amplitude of actions. Figure-5. GAZ-313 minivan, Slalom, 18 m; velocity: 37 km/h. The road test in Figure-5 was performed with lateral accelerations of about 5.5 m/s. Calculated and experimental lateral forces acting on vehicle are compared in Figure

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