Keywords

14.1 Introduction

All-wheel-drive electric vehicles have been widely investigated and proved to be with better driving performance and handling performance than conventional vehicles [1, 2]. All-wheel-drive technology provides convenience to achieve advanced vehicle dynamics control, such as Traction Control (TC), Anti Brake Skid (ABS), Direct Yaw Control (DYC) or the integrated control systems [3, 4]. Moreover, all-wheel-drive technology makes the acting torque on each wheel available which boosts the investigations of tyre-road condition or vehicle behavior observation [5, 6]. Among the dynamics control technologies of all-wheel-drive electric vehicle, DYC is one of the most concerned topics [7, 8]. Y. Shibahata firstly proposed the basic principle of DYC system [9]. M. Nagai proposed a MMC controller to control the vehicle to follow the desired 2 DOF dynamic model with the feedback of yaw rate and side slip angle [10]. Y. Hori proposed a robustified MMC controller. He further claimed that, the research should focus on how to calculate the external yaw moment to make the vehicle follow the desired vehicle model [11]. Like Y. Hori stated, the following researchers mainly putted their focus on the control method to calculate the yaw moment. A. Goodarzi investigated the performance of optimal control on yaw moment generating [12]. Yu proposed an optimal controller with real-time online estimation of the tyre cornering stiffness [13]. Hedrick used sliding mode control during the process of the yaw moment generating [14]. To deal with the DYC control system in high nonlinear maneuver condition, Hori proposed a controller based on body slip angle fuzzy observer [15], and he further proposed a slip angle estimation block using the lateral tire force sensors [5]. Moreover, some researchers such as Wang have also made great achievements in AFS/DYC integrated control system [16, 17].

Above authors’ work have made significant contributions to the vehicle dynamics control. But in DYC’s basic principle, the desired vehicle lateral dynamics behavior is always calculated by 2 DOF dynamic model with inherent understeer characteristics [9]. As Wang stated in [17], the main problem of DYC mainly focuses on how to generate the desired moment which is indeed the most researchers focus on. In this chapter, based on our previous work of all-wheel-drive electric vehicle [18,19,20], a novel dynamic control principle-Relaxed Static Stability (RSS) will be proposed.

14.2 Pole Location Discussion of Vehicle Lateral Dynamic System

The state-space of 2 DOF dynamic model is expressed as:

$$ \dot{x} = Ax + B\delta $$
(14.1)
$$ B = \left[ {\begin{array}{*{20}c} { - \frac{{C_{f} }}{mu}} \\ { - \frac{{aC_{f} }}{{I_{z} }}} \\ \end{array} } \right] $$
(14.2)
$$ A = \left[ {\begin{array}{*{20}c} {\frac{{C_{f} + C_{r} }}{mu}} & {\frac{{aC_{f} - bC_{r} }}{{mu^{2} }} - 1} \\ {\frac{{aC_{f} - bC_{r} }}{{I_{z} }}} & {\frac{{a^{2} C_{f} + b^{2} C_{r} }}{{I_{z} u}}} \\ \end{array} } \right] $$
(14.3)

The pole location of the lateral dynamic system of several typical type vehicle will be discussed. Table 14.1 shows the specifications of a conventional front-drive passenger car. It is typical with ICE at front of the vehicle and a front C.G location. It can be easily concluded understeer characteristics.

Table 14.1 Conventional passenger car specifications
Full size table

Figure 14.1a shows the case when C.G location changes. The original pole location is conjugate complex values (−4.1, ± 4) which is noted by red points. With C.G location moves forward to 90%: 10% (red solid arrows), the distance from poles to the origin becomes larger. And they are still conjugate complex values which shows under damped characteristics. With C.G location moves backward to 40%: 60% (red dash arrows), the dynamics system becomes over damped, and the poles become double negative real poles. When the C.G location is 40%: 60%, it can be seen that a positive real pole occurs which indicates the system is unstable. With C.G location continues to move backward, the system will be unstable. Actually, in conventional vehicle overall configuration theory, forward C.G location has to be assured to obtain inherent understeer. For all-wheel-motor-drive vehicle, back C.G location is almost inevitable. It is easy to make the vehicle unstable with back C.G location. Taken the vehicle shown in Table 14.1 as example, the C.G location configuration arrange is constraint in the range from 100%: 0% to 40%: 60%. In Fig. 14.1c the pole locations with front tyre cornering stiffness changes is shown. The red solid arrow show the movement direction when the stiffness increases, and the dash arrow shows the stiffness decreases. When cornering stiffness of front tyres increases, the system goes over damped and double negative real poles occur. When the cornering stiffness of front tyres is higher than certain value (150% as origin), a positive real pole occurs which indicates an unstable characteristics. For conventional vehicle with front-drive, the traction force acting on front tyres makes front tyres’ stiffness lower than rear tyres. That makes a more stable tendency. But for all-wheel-drive vehicle, the traction force also acts on rear tyres which makes the stiffness much lower, so that it’s much easier to be unstable than conventional vehicle.

Fig. 14.1
figure 1

Pole locations distribution with vehicle specifications change (vehicle speed as 30 m/s)

Full size image

14.3 Basic Hierarchical Control Strategy of RSS

In this section, the basic overall control strategy of RSS will be proposed. With yaw moment M considered as control input, the dynamics model can be expressed as:

$$ \dot{x} = Ax + B\delta + CM $$
(14.4)

Consider M as feedback according to state vector x:

$$ M = Fx $$
(14.5)

Then Eq. (14.9) can be expressed as:

$$ \dot{x} = (A + CF)x + B\delta $$
(14.6)

The pole locations of the new dynamic system are determined by A + CF. Consider f1 and f2 as feedback coefficients. Through the manipulation with the characteristics matrix, the expression of the closed-looped pole location can be expressed as:

$$ p_{{1^{{\prime }} ,2^{{\prime }} }} = \frac{{I_{z} (C_{f} + C_{r} ) + m(a^{2} C_{f} + b^{2} C_{r} ) + f_{2} mu \pm \sqrt {\Delta_{2} } }}{{2I_{z} mu}} $$
(14.7)

where:

$$ \Delta_{2} = [I_{z} (C_{f} + C_{r} ) - m(a^{2} C_{f} + b^{2} C_{r} ) - f_{2} mu]^{2} - 4I_{z} m[mu^{2} - (aC_{f} - bC_{r} )][(aC_{f} - bC_{r} ) + f_{1} ] $$
(14.8)

It can be seen that the closed-looped pole locations of the vehicle is determined by both vehicle specifications m, L, Iz, Cf, Cr, a, b and parameters f1 and f2. That can be totally considered the CCV concept. The pole locations can be adjusted according to f1 and f2. According to desired target pole locations, feedback vector F can be solved, and the value of M can be determined. According to above basic discussion, the basic strategy of RSS can be expressed (Fig. 14.2).

Fig. 14.2
figure 2

Control schematic diagram of RSS

Full size image

In RSS, the desired pole locations should be determined first, which indicates the desired vehicle handling response. Figure 14.3 shows the desired pole locations in ‘Stable Mode’ and ‘Agile Mode’. In ‘Stable Mode’, the desired pole locations are indicated by red hollow circles. It can be seen that they are conjugate complex values arranged in straight lines which divide 2nd and 3rd quadrants. That shows an optimal damping ratio as 0.707 which is proved to be the optimal transient response. The moving direction of the pole locations when vehicle speed increases is shown by arrows. According to the passenger car case, it becomes under damped when vehicle speed exceeds 5 m/s. Moreover, the damping ratio becomes lower with vehicle speed increases. Based on that, the desired vehicle handling response is assumed to be with constant optimal damping ratio during the whole speed range, which could significantly increase the handling stability. Green shadow circles indicate the desired pole locations in ‘Agile Mode’. In each speed case, the desired pole locations are assumed to be one single negative real value which indicates critical damped. According to Ref. [21], neutral steer vehicle has greatest handling mobility, which is also the reason why Formula 14.1 race car is always designed to be neutral steer.

Fig. 14.3
figure 3

Desired pole locations

Full size image

14.4 Simulation Verification

A 7 DOF vehicle nonlinear dynamic model is used here (Fig. 14.4).

Fig. 14.4
figure 4

7 DOF vehicle dynamics model

Full size image

The dynamic equation of the vehicle body expressed in (o, x, y) coordinate frame can be written:

$$ \left\{ {\begin{array}{*{20}l} {m(\dot{u} - v\omega ) = (F_{x11} + F_{x12} )\cos \delta + F_{x21} + F_{x22} - (F_{y11} + F_{y12} )\sin \delta } \hfill \\ {m(\dot{v} + u\omega ) = (F_{y11} + F_{y12} )\cos \delta + F_{y21} + F_{y22} } \hfill \\ {I_{z} \dot{\omega } = a(F_{y11} + F_{y12} )\cos \delta - b(F_{y21} + F_{y22} ) + (F_{x22} + F_{x12} - F_{x21} - F_{x11} )B} \hfill \\ \end{array} } \right. $$
(14.9)

where v is the lateral speed of the vehicle. δ is the front wheel angle. Fxii, Fyii is the traction and lateral force of 4 tires as figure shown. Rotational dynamic equation of each wheel can be expressed:

$$ T_{i} - F_{xi} R_{t} = I_{w} \dot{\omega }_{i} $$
(14.10)

where Ti (i = 1–4) is the input torque of the motor acting on 4 wheels respectively. ωi is the rotation speed of the 4 wheels respectively. RSS concept provide the possibility for the vehicle to be designed as inherent unstable, and to be closed-looped stable with pole assignment. That has great significance for the future vehicle such as all-wheel-motor-drive vehicle with heavy battery package, or other configured vehicle which is inherent unstable according to traditional handling stability theory. In this section, this case will be discussed. If a vehicle is configured as inherent unstable, how RSS control improves the handling performance will be analyzed. The distance from C.G to front and rear axles (a, b) in Table 14.1 is changed to 1.6 and 0.94 m which is inherent unstable (oversteer) configured.

Figure 14.5 show the simulation results in ‘Stable Mode’ in a 100 km/h J-turn with inherent oversteer vehicle. Through Fig. 14.5, it can be seen that the vehicle without control loses stability and spins after 15 s when steer angle is inputted. The lateral acceleration of uncontrolled vehicle peaks about 0.6 g, and the yaw velocity peaks about 0.6 rad/s. The large side slip angle (−0.8 rad) indicates an extreme spin behavior of the uncontrolled vehicle. In comparison, the vehicle under RSS control is stable after the large steer angle is inputted. The lateral acceleration, yaw velocity and side slip angle remains a stable value after overshoot which indicates the vehicle behaves well under RSS control. Figure 14.5 shows the yaw moment demand for RSS control which peaks about −900 N m. Figure shows the comparison of real and imaginary part of the real-time pole of the vehicle. For uncontrolled vehicle, it can be seen that, in 15.5 s positive real part occurs which indicates the dynamic system of the vehicle becomes unstable. After 15.5 s, the poles of dynamic system of uncontrolled vehicle remains two real values with a positive pole. It indicate the vehicle is unstable during this time. The phenomenon matches the reality reflected by other figures. For the vehicle under RSS control, it can see that the poles remain two conjugate complex values which indicates the dynamic system is under damped, and it matches the control goal of ‘Stable Mode’.

Fig. 14.5
figure 5

Simulation in 100 km/h J-turn (Stable mode, inherent oversteer)

Full size image

14.5 Conclusion

In this chapter a novel concept-RSS is proposed which could be considered as CCV concept used in ground vehicle field. Different with existing DYC, RSS concept can be considered as a novel overall concept for ground vehicle which combines the mechanical and control systems. RSS control has two major advantages as following.

Firstly, RSS allows the vehicle to be designed as inherent static unstable which could significantly improves the overall configuration flexibility of vehicle. That has great significance for new-type configured vehicle, such as all-wheel-motor-drive vehicle with back heavy battery package and separate independent motors and controllers, as well as future advanced vehicle.

Secondly, based on pole assignment concept, RSS could significantly improve the handling performance of the vehicle. Moreover, it provides the possibility to easily define different handling performance demand according to the desired pole location determination. The ‘Stable Mode’ and ‘Agile Mode’ discussed in the paper give meaningful examples.