Keywords

1 Introduction and Motivation

The development of vehicular technology has led to a surge of innovations in chassis control. Still, such advancement is restricted by the very basic mechanics mechanism of the tire-ground contacts. To maintain steerability for drivers, classical active systems (e.g., ESC) commonly keep the vehicle within the linear SS boundaries that in return, give up the potential in the nonlinear regions [1, 2]. In advanced chassis with independent torque control, more possibilities are shown with the utilization of large combined slips, such as drifting control in [3] and obstacle avoidance in [4]. Indeed, such large-side slip motion is stabilizable while not elevated since the maximal yawing is yet within the SS boundary in [1, 5].

1.1 Problem Formulation

The motivation of the paper comes from the demand to stabilize the oversteer vehicle after global instability is induced by Saddle-Node bifurcation [1]. Instead of moving towards driting where a large body side-slip is required, we aim to achieve high cornering speed while retaining a small body side-slip. The key finding traces that the maximal achievable yaw rate \(\omega _{z,\textrm{max,c}}\) in SS is extended than the well-known open-loop one \(\omega _{z,\textrm{max,ol}}\):

$$\begin{aligned} \omega _{z,\textrm{max,c}} \left( \approx \frac{\mu _\textrm{f} +\mu _\textrm{r}}{2} \frac{1}{v_x g} \right) > \omega _{z,\textrm{max,ol}} \left( = \frac{\textrm{min}\{\mu _{\textrm{f}},\mu _\textrm{r}\}}{v_x g} \right) , \end{aligned}$$
(1)

where \(\mu \) is the contact coefficients and g is the gravitational constant (see [5] for detailed parameter definitions).

1.2 Key Contributions

  1. I)

    we propose a new mechanics-based approach to manipulate the vehicle dynamics in critical maneuvers, while maintaining predictable body motions in either yaw or lateral side-slip.

  2. II)

    a desired SS equilibrium can be created even for oversteered vehicles in globally unstable domains, where an elevated performance of SS cornering can be achieved;

  3. III)

    methodology for the control systems development based on the proposed strategy is demonstrated, and the simulated results verify its feasibility and functionality.

Fig. 1.
figure 1

(a) Modification of SS steering chara (-OL: open-loop; -C: with proposed control; US: understeer; OS: oversteer; NS: neutral); (b) Visualised control intervention.

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2 Methodology

The proposed method is applied to a 3-DOF (degree of freedom) planar vehicle model where control system development is also introduced.

2.1 Vehicle and Tire Model in Combined Slips

A planar 3-DOF vehicle model assumes the form [1] (check [6] for parameters):

$$\begin{aligned} \begin{aligned} m \left( \dot{v_x} - v_y \omega _z \right) &= \left( F_{x1} +F_{x2} \right) \sin \delta + \left( F_{y1} +F_{y2} \right) \cos \delta +\left( F_{x3} +F_{x4} \right) , \\ m \left( \dot{v_y} + v_x \omega _z \right) &= \left( F_{y1} +F_{y2} \right) \cos \delta +\left( F_{y3} +F_{y4} \right) +\left( F_{x1} +F_{x2} \right) \sin \delta , \\ I_z \dot{\omega }_{z} &= \left( F_{y1} +F_{y2} \right) l_{\textrm{f}} \cos \delta - \left( F_{y3} +F_{y4} \right) l_{\textrm{r}} + F_{x2} d_{2} -F_{x1} d_{1}, \end{aligned} \end{aligned}$$
(2)

where \(F_{\ldots }\) are the tire forces (1, 2, 3, 4 refer to four wheels, see Fig. 1) and \(v_{x},v_{y},\omega _z\) are the motion states. The tire force at the combined slip is described by adopting the combination of the steady-state Magic formula and non-steady-state TMeasy model, which finally gives the force calculation in the form of

$$\begin{aligned} F_{i} = \mu _{i} F_{zi} \sin \bigl (C \left( B s_{i} - E B s_{i} + E \arctan B s_{i} \right) \bigr ) , i \in [1,2,3,4]. \end{aligned}$$
(3)

The longitudinal and lateral forces are derived based on (3), respectively:

$$\begin{aligned} F_{xi} = \dfrac{s_{xi}}{s_{i}} F_{i} , F_{yi} = \dfrac{s_{yi}}{s_{i}} F_{i} . \end{aligned}$$
(4)

where \(s_{xi}, s_{yi}\) are the slip ratios defined in two dimensions [5] and \(s_i= \sqrt{s_{xi}^2+s_{yi}^2}\).

2.2 Preliminaries of Extending Steady-State Cornering

Using OS vehicles as a demonstration, the contact force in front wheels cannot be fully used due to the imbalance of yawing moment \(\sum M_{z} \ne 0\), thus, we apply certain opposite longitudinal forces in the side wheels to create extra yaw torque \(M_{z,x} = F_{x2}d_2-F_{x1}d_1\) in (2) while generating no sum in longitudinal acceleration \(\sum F_{x} = 0\). This \(M_{z,x}\) contributes in a similar way to ESC, but not for stabilization, to directly modify the combined tire characteristics in the axle which consequently, leads to the enlarged lateral force generation \(\varDelta F_{y}\) and additional yaw moment \(M_{z,y}\). To reach SS motions, balances in three planar dimensions \(\textbf{x} = [v_x, v_y, \omega _z]^T\) in (2) have to be satisfied:

$$\begin{aligned} \sum F_{x,\ldots }(\overline{\textbf{x}}; \textbf{u}) = -\overline{v}_y \overline{\omega }_{z,\textrm{ss}} , \sum F_{y,\ldots }(\overline{\textbf{x}}; \textbf{u}) = \overline{v}_x \overline{\omega }_{z,\textrm{ss}} , \sum M_{z,\ldots }(\overline{\textbf{x}}; \textbf{u}) = 0 , \end{aligned}$$
(5)

where \(\overline{\textbf{x}}\) denote the equilibrium of state variables and \(\textbf{u}\) is the controllable inputs that are the longitudinal slip ratios \([s_{x1},s_{x2}]^T\).

Analytically, in maximal conditions, full contact forces both in the front and rear axle are generated and balanced assuming an SS condition:

$$\begin{aligned} \sum _{i=1}^{4} F_{y,i} + \varDelta F_{y} = \overline{v}_x \overline{\omega }_{z,\textrm{ss}} , \sum M_{z,\ldots } = \varDelta F_{y} l_{\textrm{f}} - \left( F_{x1}d_1-F_{x2}d_2\right) = 0, \end{aligned}$$
(6)

where in the case of the open-loop system, \(\sum _{i=1}^{4} F_{y,i} = v_x \omega _{z,\textrm{max,ol}}\). Thus, the increment of the SS yaw rate can be derived \(\varDelta \omega _{z,\textrm{max}} = \varDelta F_{y}/v_x\), and the unbalanced moment is compensated through longitudinal tire forces.

Fig. 2.
figure 2

Extension and modification of SS steering chara (-OL: open-loop; -CL: with proposed control; US: understeer; OS: oversteer), where the red dot-dash line refers to the OL boundary in (1).

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The solution of Eq. (5) satisfying (6) can be numerically found by inversing the nonlinear systems as shown in Fig. 2, where the external input parameters are \([\omega _{z,\textrm{des}},v_{x,\textrm{des}},\delta ]\), and the solution set is \([\overline{v}_{y},\overline{s}_{x1},\overline{s}_{x2}]\). The black line refers to the SS cornering achieved by the proposed methodology, where the green region indicates the absolute elevated performance w.r.t classical boundary (check (1)).

Stability around the new elevated equilibrium \(\overline{\textbf{x}}\) in Fig. 1 is checked through phase planes in Fig. 3. In this maneuver, driver is not able of stabilizing the motions (left of Fig. 3), and even the drifting would be restricted in the OL boundaries (red dotted line). While in the right-side panel, one can achieve an out-of-limit yawing speed in a more stable and simple manner. The quick demonstration in Figs. 2 and 3 points out the capability of the proposed methodology in critical conditions.

Fig. 3.
figure 3

Phase planes of (a) an uncontrolled open-loop OS vehicle system and of (b) a same OS vehicle but with feedforward control torque.

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2.3 Control System Development

Based on the created equilibrium \(\overline{\textbf{x}}\) and the calculated longitudinally controlled slips \(\overline{s}_{x1},\overline{s}_{x2}\), the control law \(\textbf{T}_{\textrm{c}}\) in the side wheels is structured as

$$\begin{aligned} \textbf{T}_{\textrm{c}} = \textbf{T}_{\textrm{c,ff}}(\overline{s}_{x1},\overline{s}_{x2}) + \textbf{T}_{\textrm{c,fb}}(\textbf{x},\overline{\textbf{x}}) , \end{aligned}$$
(7)

where the feedforward term \(\textbf{T}_{\textrm{c,ff}}\) calculated from (6) is applied for creating the SS equilibrium and the feedback term \(\textbf{T}_{\textrm{c,fb}}\) is to accomplish better local stability against deviations. By substituting \(\overline{s}_{x1},\overline{s}_{x2}\) into the tire formula (3, 4), one can derive that

$$\begin{aligned} \textbf{T}_{\textrm{c,ff}} = [F_{x1}(\overline{s}_{x1}) R_w, F_{x2}(\overline{s}_{x2}) R_w]^T . \end{aligned}$$
(8)

Consequently, the feedback controller can be assumed in an elegant way that

$$\begin{aligned} \textbf{T}_{\textrm{c,fb}} = [\dfrac{d_2}{d_1+d_2} M_{\textrm{pd}}, \dfrac{d_1}{d_1+d_2} M_{\textrm{pd}}]^T , M_{\textrm{pd}}= \textbf{K} \left( \textbf{x}(t) - \overline{\textbf{x}} \right) , \end{aligned}$$
(9)

where \(\textbf{K}\) is the coefficient matrix for control gains of motion states. This torque \(\textbf{T}_{\textrm{c}}\) is further applied to the rotational dynamic model of wheels, which for simplicity here, is not listed. The matrix of control gains is

$$\begin{aligned} \textbf{K} = \dfrac{1}{I_{z}}\begin{bmatrix} 0 & P_{y} & P_{\omega } \end{bmatrix} \end{aligned}$$
(10)

where \(P_{y} = -0.0772 \, \mathrm {N^2 m s^3}\) and \(P_{\omega } = 1.2375 \, \mathrm {N^2 m^2 s^3}\) are the optimal gains derived at the new equilibrium (check [6] for methods).

3 Simulated Result and Discussion

Simulations are carried out in Matlab/Simulink with the Vehicle Dynamics Toolbox, and a path with a radius of 48 m is designed to test the cornering capability. The longitudinal speed is set at 20 m/s, which means that to stably exit the corner, the minimum yaw rate is \(20/48 = 0.417\) rad/s larger than the \(\omega _{z,\textrm{max,ol}} = 0.4\) rad/s of the oversteered vehicle in Fig. 2. Thus, without control, theoretically the corner cannot be passed at this speed.

Fig. 4.
figure 4

Three types of external control intervention are compared: without control, with only feedforward torque \(\textbf{T}_{\textrm{c,ff}}\), with both \(\textbf{T}_{\textrm{c}}\), where the (a) road map, (b1-b2) motion state variables are checked, respectively.

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Figure 4 presents the kinematics and dynamics data in passing the corner. Compared to the case without control, the proposed controller helps the vehicle to get through the corner successfully, and in road maps, little difference can be spotted. While in the dynamic responses, further incorporation of feedback portion improve the settling processes by faster decaying, which finally gets the vehicle stabilized at \(\overline{\omega }_{z} = 0.43\) rad/s exceeding the road requirement.

4 Conclusion

In sum, the proposed method demonstrates the effectiveness and advantages of motion stabilization and performance elevation for OS vehicles compared to naturally unstable drifting. A more promising application is for vehicles at different steering characteristics, where on the basis of advanced chassis, one can create desired motions way beyond the classical handling boundary.