How to design a motor-assisted steering system based on Simulink?

1. Working Principle

The following figure is the structural schematic diagram of the electric steering system:

The following figure is the schematic diagram of electric power steering:

The forward input system block diagram of the EPS system is:

The reverse input system block diagram of the EPS system is:

The overall function transfer block diagram based on the Simulink model is:

2. Mathematical model and parameters of electric power steering system

2.1 Dynamic Analysis of the System

Establish a dynamic model with the steering wheel as the research object:

The dynamic equation is established with the pinion as the research object:

2.2 Motor model analysis

This system uses a DC motor as the drive motor, with a rated voltage of U, an inductance of L, an armature resistance of R, a back electromotive force constant of Kb, a motor torque coefficient of Ka, and a speed of N. The following relationship holds true:

When the motor tends to a stable state, the inductance does not work, and the current is stable, the above formula can be simplified to:

The motor output torque is:

In the formula,

G1 is the transmission ratio of the motor reduction mechanism, and the assist torque of the motor acting on the steering system column is:

Establish a dynamic model with the motor as the research object:

Where δ is the front wheel steering angle.

There are two control strategies for DC motors, proportional control and proportional plus derivative control (PD).

If the system uses a PD controller to control the motor, the motor drive voltage U is:

2.3 Mathematical model of electric power steering system

Where T0 is the steering resistance torque, Mr is the steering resistance torque mainly in situ, Mz is the return torque, wheel steering angle α, sideslip angle φ, vehicle speed v, road surface, tire type and air pressure, vehicle weight, front axle load and friction of each transmission system and other related nonlinear functional relationships. To simplify the analysis, the following formula is used to express the resistance torque:

The above equation is a two-variable differential equation. To analyze its characteristics, a matrix equation is established. Let:

According to the above, we can get the matrix equation:

The input to the system is:

The state equation of the system is:

The output equation of the system is:

The linear two-degree-of-freedom car model equation is as follows:

The recursive process is simplified here, and finally we get:

Compared with cars with ordinary steering systems, the EPS system has a smaller steady-state gain of yaw rate than the ordinary steering system. At the same time, the gain at different vehicle speeds can be obtained by modifying the parameter values ​​in the M file.

2.4. Automobile EPS structural parameters

%Transfer function parameters of vehicle center of mass sideslip angle & yaw angle to front wheel steering angle%

%Parameter settings%

k1=-62715;%Front axle lateral stiffness%

k2=-110128;%Rear axle lateral stiffness%

m=1956;% vehicle mass%

a=1.654;%Distance from front axle to center of mass%

b=1.752;%Distance from rear axle to center of mass%

Iz=3984;%Moment of inertia of the vehicle around the Z axis%

L=3.406;% wheelbase%

u=20;% vehicle speed%

d=0.1;% front wheel trail%

R=0.1;% armature resistance%

Wn=((k1*k2*L*L)/(m*Iz*u*u)+((a*k1-b*k2)/Iz))^(1/2);% Natural frequency of vehicle steering response%

K=m/(L*L)*(a/k2-b/k1);%Stability factor%

Kv=250;

eta=(-m*(a*a*k1+b*k2)+Iz*(k1+k2))/(2*m*Iz*L*((1+K))^0.5);% Resistance ratio of vehicle steering response%

Gbeta0=((1+m*a*u*u/(L*b*k2))/(1+K*u*u))*(b/L);% Steady-state side slip angle gain%

Gomiga0=(1/(1+K*u*u))*(u/L);% Steady-state yaw rate gain%

Tbeta=-((Iz*u)/(L*b*k2))*(1/(1+m*a*u*u/(L*b*k2)));%

Tomiga=-m*a*u/(L*b*k2);

Ka=0.02;%Motor torque coefficient%

Kb=0.02;%Back electromotive force constant%

Ks=1.5'%Torque sensor torsional stiffness%

Kp=1;%PD control coefficient%

Kd=0.15'%PD control coefficient%

Jep=0.06;% pinion equivalent total inertia moment%

Ir=0.08;%Front wheel equivalent moment of inertia%

Jm=0.005;%Motor moment of inertia%

G1=20;%Motor reduction gear ratio%

G2=20;% Steering wheel to front wheel transmission ratio%

Br=0.3;%Front wheel equivalent friction coefficient%

Bm=0.01;%Motor damping coefficient%

Bsw=0.1;% Steering wheel equivalent damping coefficient%

Bep=0.3;% pinion equivalent damping coefficient%

lamude=1+(Ka*Kp)/R*G1;%Assistance coefficient%

K1=(G1*Ka*Kp/R+Ks)/Jep%Matrix equation coefficient%

K2=-(G1*Ka/R*(Kp-Ks+Kv))/Jep%Matrix equation coefficient%

K3=-(G1*Ka/R*Kd+Bsw)/Jep

K4=-(G1*Ka*(Kp+Kb*G1)/R+Bep-Bm)/Jep

mel=G1*Ka*Kd*Ks/R;

Wp=(lamude*Ks/(Ir+Jm*G1^2))^0.5;

Tpd=mel/(lamude*Ks);

setapd=(Br+Bm*G1*G1+Ka*Kb*G1*G1/R+mel)/(2*(lamude*Ks*(Ir+Jm*G1*G1))^0.5);%PD controller EPS system damping ratio%

2.5 Simulation Results

The figure above shows the system response of the pinion angle relative to the steering wheel torque input under the PD control state when V=10m/s. Since this simulation is only for the characteristic analysis of the EPS system, the input torque signal is set to constant torque. Under the condition of motor assistance, the instantaneous response of the system has been significantly improved. As the steering resistance decreases due to the increase in vehicle speed, the pinion rotation resistance decreases, and the time for the system to adjust to the steady state is shortened. As can be seen from Figure 5, when the system has electric power steering, the system responds very quickly, and the system response tends to be stable in about 0.1 seconds, indicating the good response characteristics of the system.

The electric power steering adopts the PD control strategy. Compared with the ordinary power steering, it has oscillation instability according to relevant literature, while the yaw angle output response of the electric power steering system quickly stabilizes.

The EPS system control method has a significant impact on the transient response of the vehicle. The EPS system with PD control method suppresses the irregular fluctuation of the yaw angular velocity and makes it quickly tend to a steady-state value, which is beneficial to improving the transient response quality of the vehicle, but the reaction time of the system is slightly prolonged.

It can be seen from the above figure that when there is electric power steering, the pinion angle response of the car's steering system is very stable, and the car's yaw angle output response tends to be more stable than the car's yaw angle output response without electric power steering.

The natural frequency of the steering system of the EPS system is smaller than that of the ordinary steering system and is close to the natural frequency of the vehicle's yaw angle. Therefore, a large resonance peak wave appears at about 0.05 seconds. When the driver's operating frequency is close to this frequency range, the vehicle's yaw angular velocity is very sensitive to the steering wheel angle, and the vehicle is easy to lose control. Increasing the natural frequency of EPS is conducive to improving the vehicle's handling stability.