Heavy commercial vehicle yaw control simulation

3.1. Vehicle model

The simulated commercial vehicle is a two-axle off-road tipper with permanent rear-wheel drive. The chassis consists of rigid “backbone” tube with independent swinging half-axles. Front suspension is formed by two air-springs and two dampers, rear suspension is equipped with two air-springs with additional coil springs and antiroll bar. Frame with “backbone” tube, cabin, tipper body and accessories (bumper, fuel tanks etc.) is modeled as one rigid part to which the load representing part is fixed. Powertrain subsystem is represented by engine block part and driveline shafts. Between the engine block part and rear axle input shaft acted a torque whose value is calculated from throttle and clutch position, engine torque curve and actual gear ratio. Between output shafts of the classic differential additional torque is acting for simulation of the active right-left torque distribution. This torque represents the active differential which directly influences tire longitudinal force difference ($∆F_x$) and it is managed by control algorithm in Simulink. The rear drive axle is equipped by dual wheels. In the multibody simulation of the complete vehicle plays important role the appropriate tire model. So, in this case the PAC2002 tire model which is suitable for vehicle handling simulation on an even road is used. This tire model was developed on the base of Magic Formula 6.2 tire model, but it is adapted by MSC software company for ADAMS Car simulations. The tire model is simulated in mode including combined force-moment calculation with relaxation behavior.

For the handling simulations it is necessary to connect the dynamic model with so called vehicle test-rig. In this case ADAMS MDI-SDI Testrig which contains driver algorithm and predefined vehicle tests is used.

Fig. 2Graphical representation of the dynamic vehicle model

3.2. Control algorithm

The used control algorithm of the active yaw control system has hierarchical structure shown in Fig. 3. Sensor inputs for upper controller which ensures desired yaw torque calculation are wheel speeds, lateral acceleration, yaw rate and steering angle. Objective of the upper controller is to keep the yaw stability of the vehicle. The aim of lower controller is estimation of the right-left torque difference which produce the desired yaw torque so as to track the target yaw rate. The desired yaw rate for yaw torque estimation is calculated as follows [4]:

where:

${\dot{\mathrm{\Psi }}}_{des}$ – desired yaw rate,

$\dot{x}$ – vehicle longitudinal velocity,

$l_f$, $l_r$ – vehicle center of gravity distance from front and from rear axle,

$m$ – vehicle mass,

$C_{\alpha f}$, $C_{\alpha r}$ – cornering stiffness for each front and rear tire,

$L$ – wheelbase of the vehicle,

$\delta$ – steer angle.

Active yaw control strategies usually use a feedback controller to compute reference yaw moment from the difference of the measured yaw rate and desired yaw rate. This feedback controllers usually utilize sliding mode, linear quadratic regulation, model predictive control or robust control strategies [3]. In addition, the upper controller can also be feedforwarded by a map. Then the feedback controller compensates inaccuracies and disturbances or variation of the vehicle parameters. The presented control algorithm is based on feedback sliding mode controller without feedforward loop.

To sufficiently describe the vehicle dynamic state is except the vehicle yaw rate also a vehicle side slip angle estimation necessary. Otherwise, the vehicle drift with small yaw rate can occur without control algorithm determination of the vehicle instability. From this reason, active yaw control strategies with the vehicle side slip angle estimator to set the right controller boundaries are always used.

Fig. 3Structure of active yaw control system algorithm [4]

4.1. Vehicle stabilization

The main task of the proposed system is to stabilize the vehicle at the limit situations and keep it controllable for the driver. Due to the appropriate drive torque distribution, the limits of the total lateral tire forces acting on the drive axle can be enhanced which help to keep the vehicle handling stability [5]. For demonstration of the enhanced vehicle handling performance a single lane change maneuver on the road with lower friction coefficient was chosen. The friction coefficient was set to 0.5 which corresponds to drive on the wet asphalt or concrete road. The initial velocity was 50 km/h and steering wheel angle amplitude was set to 200 degrees which is front wheels steer angle about 8 degrees. Steering wheel angle during simulation is shown in Fig. 4. Simulated were both variants – with and without active differential.

Fig. 4Steering wheel angle during single lane change maneuver

4.2. Regulation of understeering

The second simulation was focused on potential vehicle understeering compensation through active differential. Heavy commercial vehicles are supposed to operate reliably in big range of loading. The transported cargo but change the understeering behavior in many cases. Therefore, it would be beneficial for vehicle handling to regulate the vehicle understeering or oversteering behavior through active differential. To demonstrate this effect on simulated vehicle the steady state constant radius cornering maneuver was chosen with radius 50 m. Vehicle accelerated during 20 secs from initial velocity 20 km/h.

4.3. Results summary

First maneuver represents transient behavior of the vehicle at the handling stability limits. Due to the active drive torque right-left distribution, the rear axle tire lateral force limit is enhanced, and the vehicle is during whole maneuver controllable. In Fig. 5 is the comparison of the resultant vehicle side slip angle during simulation. The red curve is side slip angle of the vehicle without active differential and the blue curve is vehicle with active drive torque distribution. It is obvious, that the vehicle without proposed system exceeded side slip angle limits and during 6th second of the simulation became unstable. Blue curve shows, that active differential is able to regulate effectively the side slip angle during transient dynamic state and keep the vehicle within the limits.

Fig. 5Vehicle side slip angle comparison during single lane change maneuver on wet road

Second maneuver represents steady state vehicle dynamic behavior and demonstrates the possibility of the vehicle understeering regulation. The comparison of the vehicle with and without active right-left torque distribution system is in Fig. 6. This graph shows dependence of the steer angle and vehicle lateral acceleration. Red curve is the steer angle of the vehicle without active yaw moment control and it is obvious, that with increasing lateral acceleration, the driver has to increase the steering wheel angle too, to compensates the understeering. The blue curve shows, that by the vehicle with active differential, the tire longitudinal force difference on the drive axle is able to keep the vehicle to follow constant radius trajectory also at higher lateral acceleration.

Fig. 6Vehicle understeering comparison during steady state cornering