Vehicle roll stability control with active roll-resistant electro-hydraulic suspension

Abstract

This study examines roll stability control for vehicles with an active roll-resistant electro-hydraulic suspension (RREHS) subsystem under steering maneuvers. First, we derive a vehicle model with four degrees of freedom and incorporates yaw and roll motions. Second, an optimal linear quadratic regulator controller is obtained in consideration of dynamic vehicle performance. Third, an RREHS subsystem with an electric servo-valve actuator is proposed, and the corresponding dynamic equations are obtained. Fourth, field experiments are conducted to validate the performance of the vehicle model under sine-wave and double-lane-change steering maneuvers. Finally, the effectiveness of the active RREHS is determined by examining vehicle responses under sine-wave and double-lane-change maneuvers. The enhancement in vehicle roll stability through the RREHS subsystem is also verified.

Key words： electro-hydraulic suspension; roll stability; LQR; experiment

Introduction

According to statistics from the US National Highway Traffic Safety Administration, rollover crashes contribute 33% to the total number of fatal deaths in the USA, although they account for only 3% of all passenger car crashes [¹]. Vehicle rollover accidents are among the most common fatal causes of human death [²]. Consequently, vehicle roll motion control has become a prevalent research topic [³–⁶]. Vehicle suspension, which minimizes the vibrations of occupants caused by road disturbances, is the most important system for vehicle roll stability enhancement. Various strategies have been applied to vehicle suspension systems, and these can be generally categorized as passive suspension (optimization of the suspension system) [⁷–⁹], semi-active suspension [¹⁰–¹²], and active suspension [¹³–¹⁶]. Enhancing the roll motions of vehicles with active suspension systems is crucial.

Considering the importance of suspension systems in enhancing ride comfort and holding capacity, various suspension systems have been developed over the decades [¹⁷–²⁰]. Among them, the electro-hydraulic suspension has received extensive attention from industrial and academic circles. Du and Zhang [²¹] introduced an electrohydraulic suspension for vehicles with actuator constraints. Choi et al. [²²] presented a fuzzy controller for electro-hydraulic suspension systems with preview information on road disturbances. Sun et al. [²³] used electro-hydraulic suspension to improve vehicle ride comfort without sacrificing road holding capacity with limitations in vehicle parameter uncertainties. These studies have revealed that electro-hydraulic suspension is mainly used to enhance vehicle ride comfort. However, the application of this type of suspension in vehicle roll motion control has not been investigated.

For decades, researchers have exerted effort to enhance vehicle roll motion by using various control strategies. Kim [²⁴] developed an electrically actuated roll control system to enhance vehicle roll behavior under parameter perturbation. Yim [²⁵] used previewable information on steering input to design a static output feedback linear quadratic regulator (LQR) controller to avoid extreme roll motion situations. Huang et al. [²⁶] developed a state derivative-induced LQR controller to reduce the rollover risk of heavy articulated vehicles. Imine et al. [²⁷] used a twisting-sliding mode scheme to ensure roll stability via an active steering assistant system. Dal Poggetto and Serpa [²⁸] utilized a gain-scheduled LQR controller to reduce the possibility of vehicle rollover. The mechanisms through which vehicle velocity influences the controller gain were also investigated. Sun et al. [²⁹] developed an active stabilizer bar in consideration of fast-varying sprung mass. Pourasad et al. [³⁰] and Marzbanrad et al. [³¹] improved vehicle path following, roll, and handling performance simultaneously by using LQR controllers and fuzzy logic respectively. These studies have improved vehicle roll motion behavior with various control schemes. However, the application of electro-hydraulic suspension to the enhancement of vehicle roll motion still needs to be improved.

The integration of electro-hydraulic suspension and active controller can effectively enhance vehicle performance [³²–³⁴]. However, the majority of relevant studies mainly used one servo-valve to actuate one hydraulic suspension, and this usage consumes much energy. In this study, a cross-linked hydraulic suspension uses only one servo-valve to actuate two hydraulic suspensions; thus, less fuel is used. In addition, due to the natural merit of the cross-linked shape, the proposed electro-hydraulic suspension can effectively enhance vehicle roll stability without affecting other aspects, such as bounce and pitch motions. An active roll-resistant electro-hydraulic suspension (RREHS) subsystem controlled by the LQR scheme is also proposed to enhance vehicle performance. The primary contributions of this study are as follows: (1) An active RREHS subsystem is proposed, and (2) a vehicle model with four degrees of freedom (DOFs) and incorporates yaw and roll motions is developed and validated via field experiments.

The rest of the paper is organized as follows. In Section 2, the 4-DOF vehicle model and the LQR controller for generating roll-resistant moments are presented. In Section 3, the RREHS subsystem with an electro-servo valve actuator is introduced. In Section 4, vehicle model validation is conducted via field tests using a fully electric vehicle prototype. The effectiveness of the RREHS subsystem with an optimal control scheme is also investigated under sine-wave and double-lane-change maneuvers. Section 5 presents the main conclusions.

Vehicle model and LQR controller

A vehicle model that incorporates yaw and roll motions is formulated. Then, the LQR control scheme utilized to actuate the electro-hydraulic suspension is provided.

First, a 4-DOF vehicle model is utilized to characterize lateral and roll motions. The vehicle is assumed to run on an even road surface to simplify the modeling. Therefore, the wheel motions can be simplified as the average roll angle displacement of unsprung mass f_u, as shown in Fig. 1(b). The vehicle sideslip angle at the center of gravity (CG) b, yaw angle y, and roll angle f are the three other DOFs, as shown in Fig. 1. For brevity, the following assumptions are established: (i) roll angles f and f_u are small enough, and (ii) the vehicle’s longitudinal velocity v_x is considered to be constant.

Fig.1 4-DOF vehicle model. (a) Lateral representation; (b) vertical representation of half-vehicle.

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From Fig. 1, the dynamic equations on vehicle yaw motion can be derived as follows:

(1)\left[\begin{array}{ll}(m_{\mathrm{s}}+\mathrm{2}{m}_{\mathrm{uf}}+\mathrm{2}{m}_{\mathrm{ur}})v_x& \mathrm{2}(m_{\mathrm{uf}}{l}_{\mathrm{f}}-m_{\mathrm{ur}}{l}_{\mathrm{r}})\\ \mathrm{2}(m_{\mathrm{uf}}{l}_{\mathrm{f}}-m_{\mathrm{ur}}{l}_{\mathrm{r}})v_x& I_{zz}\end{array}\right]\left[\begin{array}{l}\dot{\beta }+\dot{\psi }\\ \ddot{\psi }\end{array}\right]=\left[\begin{array}{l}{\displaystyle \sum _{i=\mathrm{1}}^{\mathrm{4}}{F}_{y\mathrm{u}i}}+m_{\mathrm{s}}{h}_{\mathrm{o}\mathrm{s}}\ddot{\varphi }\\ M_{z\mathrm{u}}\end{array}\right],

where m_s is the sprung mass, m_uf and m_ur are the front and the rear unsprung masses, respectively, l_f and l_r are the lengths from CG to the front and rear axle, respectively, I_{zz}=I_{\mathrm{s}zz}+\mathrm{2}{m}_{\mathrm{uf}}(l_{\mathrm{f}}^{\mathrm{2}}+t_{\mathrm{f}}^{\mathrm{2}})+\mathrm{2}{m}_{\mathrm{ur}}(l_{\mathrm{r}}^{\mathrm{2}}+t_{\mathrm{r}}^{\mathrm{2}}) is the yaw inertia of the vehicle, h_os is the height from vehicle rolling center to CG of sprung mass, F_yui pertain to the lateral tire forces generated by the Dugoff tire model [³⁵], and M_zu=[l_f, l_f, –l_r, –l_r]F_yu is the yaw momentum with F_yu=[F_yu1, F_yu2, F_yu3, F_yu4]^T.

The suspension deflections and relative velocities can be obtained in consideration of vehicle geometry. The suspension forces (F_s) generated by the original mechanical suspension can be derived as follows:

(2){\mathit{F}}_{\mathrm{s}}={\mathit{C}}_{\mathrm{s}}\mathit{T}\left(\dot{\varphi }-{\dot{\varphi }}_{\mathrm{u}}\right)+{\mathit{K}}_{\mathrm{s}}\mathit{T}(\varphi -{\varphi }_{\mathrm{u}}),

where K_s=diag([k_sf, k_sf, k_sr, k_sr]), C_s=diag([c_sf, c_sf, c_sr, c_sr]), k_sf and k_sr are the stiffnesses of the front and the rear suspension, c_sf and c_sr are the dampers of the front and the rear suspension, respectively, and T=[–t_f, t_f, –t_r, t_r]^T is the geometry transformation matrix, and t_f and t_r are the half-track widths of the front and the rear axle.

Then, the moment balance of sprung mass along the longitudinal axis can be obtained as

(3)I_{\mathrm{s}}\ddot{\varphi }+m_{\mathrm{s}}g{h}_{\mathrm{os}}\sin \varphi -m_{\mathrm{s}}{a}_y{h}_{\mathrm{os}}\cos \varphi +{\mathit{T}}^{\mathrm{T}}{\mathit{F}}_{\mathrm{s}}+u(t)=\mathrm{0},

where I_{\mathrm{s}}is the roll inertia of sprung mass, a_y={\dot{v}}_y+v_x\dot{\psi } is the lateral acceleration, and u(t) is the active roll-resistant moment generated by the RREHS.

The force balance of the unsprung masses can also be obtained as

(4){\mathit{M}}_{\mathrm{u}}{\ddot{\mathit{Z}}}_{\mathrm{u}}+{\mathit{C}}_{\mathrm{s}}({\dot{\mathit{Z}}}_{\mathrm{u}}-\mathit{T}\dot{\varphi })+{\mathit{K}}_{\mathrm{s}}({\mathit{Z}}_{\mathrm{u}}-\mathit{T}\varphi )+{\mathit{C}}_{\mathrm{t}}({\dot{\mathit{Z}}}_{\mathrm{u}}-{\dot{\mathit{Z}}}_{\mathrm{g}})+K_{\mathrm{t}}({\mathit{Z}}_{\mathrm{u}}-{\mathit{Z}}_{\mathrm{g}})=\mathbf{0},

where M_u=diag([m_uf, m_uf, m_ur, m_ur]), C_t=diag([c_tf, c_tf, c_tr, c_tr]), K_t=diag([k_tf, k_tf, k_tr, k_tr]), c_tf and c_tr are the dampers of the front and the rear tyre, respectively, k_tf and k_tr are the vertical stiffnesses of the front and the rear tyre, respectively, Z_u=Tf_u is the vertical displacement of the unsprung masses, and Z_g=0 is the road input vector for wheels.

Equation (4) can be further rewritten by left-multiplying T^T as follows:

(5)I_{\mathrm{u}}{\ddot{\varphi }}_{\mathrm{u}}+c_{\mathrm{s}}\left({\dot{\varphi }}_{\mathrm{u}}-\dot{\varphi }\right)+k_{\mathrm{s}}({\varphi }_{\mathrm{u}}-\varphi )+c_{\mathrm{t}}{\dot{\varphi }}_{\mathrm{u}}+k_{\mathrm{t}}{\varphi }_{\mathrm{u}}-u(t)=\mathrm{0},

where I_{\mathrm{u}}=\mathrm{2}{m}_{\mathrm{uf}}{t}_{\mathrm{f}}^{\mathrm{2}}+\mathrm{2}{m}_{\mathrm{ur}}{t}_{\mathrm{r}}^{\mathrm{2}} is the equivalent roll inertia of unsprung mases, c_{\mathrm{s}}=\mathrm{2}{t}_{\mathrm{f}}^{\mathrm{2}}{c}_{\mathrm{sf}}+\mathrm{2}{t}_{\mathrm{r}}^{\mathrm{2}}{c}_{\mathrm{sr}}, k_{\mathrm{s}}=\mathrm{2}{t}_{\mathrm{f}}^{\mathrm{2}}{k}_{\mathrm{sf}}+\mathrm{2}{t}_{\mathrm{r}}^{\mathrm{2}}{k}_{\mathrm{sr}}, c_{\mathrm{t}}=\mathrm{2}{t}_{\mathrm{f}}^{\mathrm{2}}{c}_{\mathrm{tf}}+\mathrm{2}{t}_{\mathrm{r}}^{\mathrm{2}}{c}_{\mathrm{tr}}, and k_{\mathrm{t}}=\mathrm{2}{t}_{\mathrm{f}}^{\mathrm{2}}{k}_{\mathrm{tf}}+\mathrm{2}{t}_{\mathrm{r}}^{\mathrm{2}}{k}_{\mathrm{tr}}.

By defining \mathit{x}={\left[\varphi -{\varphi }_{\mathrm{u}},{\varphi }_{\mathrm{u}},\dot{\varphi },{\dot{\varphi }}_{\mathrm{u}}\right]}^{\mathrm{T}}, the state-space model for the subsystem in Eqs. (3) and (5) can be derived as follows:

(6)\dot{\mathit{x}}(t)=\mathit{A}\mathit{x}(t)+{\mathit{B}}_{1}w(t)+\mathit{B}u(t),

where w(t)=a_y–gf and

\mathit{A}=\left[\begin{array}{llll}\mathrm{0}& \mathrm{0}& \mathrm{1}& -\mathrm{1}\\ \mathrm{0}& \mathrm{0}& \mathrm{0}& \mathrm{1}\\ -{\displaystyle \frac{k_{\mathrm{s}}}{I_{\mathrm{s}}}}& \mathrm{0}& -{\displaystyle \frac{c_{\mathrm{s}}}{I_{\mathrm{s}}}}& {\displaystyle \frac{c_{\mathrm{s}}}{I_{\mathrm{s}}}}\\ {\displaystyle \frac{k_{\mathrm{s}}}{I_{\mathrm{u}}}}& -{\displaystyle \frac{k_{\mathrm{t}}}{I_{\mathrm{u}}}}& {\displaystyle \frac{c_{\mathrm{s}}}{I_{\mathrm{u}}}}& -{\displaystyle \frac{c_{\mathrm{s}}+c_{\mathrm{t}}}{I_{\mathrm{u}}}}\end{array}\right],\mathrm{\hspace{0.17em}}{\mathit{B}}_{\mathrm{1}}=\left[\begin{array}{l}\mathrm{0}\\ \mathrm{0}\\ {\displaystyle \frac{m_{\mathrm{s}}{h}_{\mathrm{os}}}{I_{\mathrm{s}}}}\\ \mathrm{0}\end{array}\right],\mathrm{\hspace{0.17em}}\mathit{B}=\left[\begin{array}{l}\mathrm{0}\\ \mathrm{0}\\ {\displaystyle \frac{\mathrm{1}}{I_{\mathrm{s}}}}\\ -{\displaystyle \frac{\mathrm{1}}{I_{\mathrm{u}}}}\end{array}\right].

The main objective of this work is to design a linear quadratic controller to improve vehicle dynamic performance [³⁶], which generally consists of the following aspects:

(i) Roll stability: Vehicle roll stability is generally characterized by the lateral load transfer ratio, which is affected by sprung mass roll angle and lateral acceleration. However, lateral acceleration cannot be minimized when the longitudinal velocity and path curvature are prescribed. We can minimize the roll angle of vehicle sprung mass f to help improve vehicle roll stability.

(ii) Suspension deflections: Excessive suspension deflections may deteriorate vehicle ride comfort and even cause structural damage. Considering that only the front axle is equipped with electro-hydraulic suspension, the suspension deflections at the front axle, which are proportional to (f–f_u) in this work, should be minimized to avoid vehicle performance deterioration.

(iii) Vehicle safety: A firm uninterrupted tire-terrain contact is a prerequisite for vehicle lateral stability, which is generally characterized by f_u in this work. Therefore, the tire-terrain angle difference should also be minimized.

The maximum value of the actuator output forces should be constrained so that the actuator input voltage is physically limited.

On the basis of these analyses, the quadratic cost function and the hard constraint on actuator force can be formulated as follows:

(7)J={\displaystyle {\int }_{\mathrm{0}}^t{\mathit{y}}^{\mathrm{T}}\mathit{Qy}\mathrm{d}t}+{\displaystyle {\int }_{\mathrm{0}}^t{\mathit{u}}^{\mathrm{T}}\mathit{R}\mathit{u}\mathrm{d}t},

(8)\left|u\right|\le u_{\max },\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\left|\varphi -{\varphi }_{\mathrm{u}}\right|\le {\varphi }_{\max },

where Q and R are the known positive-definite matrices with weight coefficients in y and u in their diagonals respectively, u_{\max } is the upper bound of control input, {\varphi }_{\max } is the maximum roll angle allowed, and y=Cx, where

\mathit{C}=\left[\begin{array}{llll}\mathrm{1}& \mathrm{1}& \mathrm{0}& \mathrm{0}\\ \mathrm{1}& \mathrm{0}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& \mathrm{1}& \mathrm{0}& \mathrm{0}\end{array}\right].

With these analyses, we define the controller gain as K=–R^–1B^TP, where P satisfies the Riccati function:

(9){\mathit{A}}^{\mathrm{T}}\mathit{P}+\mathit{PA}-\mathit{PB}{R}^{-\mathrm{1}}{\mathit{B}}^{\mathrm{T}}\mathit{P}+{\mathit{C}}^{\mathrm{T}}\mathit{QC}=\mathbf{0}.

Then, external force u(t)=Kx(t) is derived and generated with the electro-hydraulic actuator through Eq. (11), and the actuator outputs can be obtained with Eqs. (16) and (17). In consideration of vehicle performance, the weight matrices are set as

(10)\mathit{Q}=\left[\begin{array}{lll}{\mathrm{10}}^{\mathrm{5}}& \mathrm{0}& \mathrm{0}\\ \mathrm{0}& {\mathrm{10}}^{\mathrm{5}}& \mathrm{0}\\ \mathrm{0}& \mathrm{0}& \mathrm{1}\end{array}\right],\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathrm{ }\mathit{R}={\mathrm{10}}^{-\mathrm{4}}.

Electro-hydraulic suspension

Figure 2 shows a diagram of the proposed RREHS subsystem, which consists of a servo-valve (or proportional valve) hydraulic actuator, two cylinders, and several oil pipes.

The spool valve of the electronic servo-valve is controlled by electric current to produce displacement z_v. Oil supply high pressure p_s is generally stored outside the electronic servo-valve, and the moving spool valve distributes high-pressure oil to two oil circuits [³⁷].

The chamber section area differences in the two hydraulic cylinders are assumed to be negligible in this work. Thus, the vertical force generated by the RREHS subsystem acting on the vehicle system is equal to zero, and roll moment u(t) from the RREHS subsystem can be formulated as

(11)u(t)=(p_{\mathrm{1}}-p_{\mathrm{2}})A_{\mathrm{c}}{t}_{\mathrm{h}}-(p_{\mathrm{2}}-p_{\mathrm{1}})A_{\mathrm{c}}{t}_{\mathrm{h}}=\mathrm{2}\mathrm{\Delta }p{A}_{\mathrm{c}}{t}_{\mathrm{h}}.

where p_{\mathrm{1}} and p_{\mathrm{2}} are the pressures of pipelines, A_{\mathrm{c}} is the piston area, t_{\mathrm{h}} is the length of chambers, and \mathrm{\Delta }p is the pressure difference between top and bottom chambers.

The fluid quantities for each oil circuit can be derived as follows:

(12)\{\begin{array}{l}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}(V_{\mathrm{1t}}+V_{\mathrm{2b}})+{\displaystyle \frac{\mathrm{1}}{{\beta }_{\mathrm{e}}}}(V_{\mathrm{1t}}+V_{\mathrm{2b}}){\displaystyle \frac{\mathrm{d}{p}_{\mathrm{1}}}{\mathrm{d}t}}=Q_{\mathrm{1}}-C_{\mathrm{ip}}(p_{\mathrm{1}}-p_{\mathrm{2}})-C_{\mathrm{ep}}{p}_{\mathrm{1}},\\ {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}(V_{\mathrm{2t}}+V_{\mathrm{1b}})+{\displaystyle \frac{\mathrm{1}}{{\beta }_{\mathrm{e}}}}(V_{\mathrm{2t}}+V_{\mathrm{1b}}){\displaystyle \frac{\mathrm{d}{p}_{\mathrm{2}}}{\mathrm{d}t}}=Q_{\mathrm{2}}+C_{\mathrm{ip}}(p_{\mathrm{1}}-p_{\mathrm{2}})-C_{\mathrm{ep}}{p}_{\mathrm{2}},\end{array}

where b_e is the effective bulk modulus of the oil, C_ep and C_ip are the external and internal leakage coefficients of the cylinders, respectively, Q_{\mathrm{1}} and Q_{\mathrm{2}} represent the flow quantities of spool valve, V_{\mathrm{1t}} and V_{\mathrm{2t}} represent the volume of the top chamber of left and right cylinder, respectively, V_{\mathrm{1b}} and V_{\mathrm{2b}} represent the volume of bottom chamber of left and right cylinder, respectively.

Fig.2 Schematic of the electrohydraulic actuator.

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The volumes in each chamber can be determined with the relative displacements of the cylinders as follows:

(13)V_{\mathrm{1t}}=V_{\mathrm{10}}+A_{\mathrm{c}}(z_{\mathrm{s1}}-z_{\mathrm{u1}}),\mathrm{ }{V}_{\mathrm{1b}}=V_{\mathrm{20}}-A_{\mathrm{c}}(z_{\mathrm{s1}}-z_{\mathrm{u1}}),V_{\mathrm{2t}}=V_{\mathrm{10}}+A_{\mathrm{c}}(z_{\mathrm{s2}}-z_{\mathrm{u2}}),V_{\mathrm{2b}}=V_{\mathrm{20}}-A_{\mathrm{c}}(z_{\mathrm{s2}}-z_{\mathrm{u2}}),

where V₁₀ and V₂₀ are the initial volumes in the top and bottom chambers, respectively, z_{\mathrm{s1}} and z_{\mathrm{s2}} are the vertical displacements of the cylinders block, and z_{\mathrm{u1}} and z_{\mathrm{u2}} are the vertical displacements of the piston rod.

With Eqs. (12) and (13), the dynamic equations in the two oil circuits can be formulated as

(14)\{\begin{array}{l}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}(A_{\mathrm{c}}(z_{\mathrm{s1}}-z_{\mathrm{u1}})-A_{\mathrm{c}}(z_{\mathrm{s2}}-z_{\mathrm{u2}}))+{\displaystyle \frac{\mathrm{1}}{{\beta }_{\mathrm{e}}}}(V_{\mathrm{10}}+V_{\mathrm{20}}+A_{\mathrm{c}}(z_{\mathrm{s1}}-z_{\mathrm{u1}})-A_{\mathrm{c}}(z_{\mathrm{s2}}-z_{\mathrm{u2}})){\displaystyle \frac{\mathrm{d}{p}_{\mathrm{1}}}{\mathrm{d}t}}=Q_{\mathrm{1}}-C_{\mathrm{ip}}(p_{\mathrm{1}}-p_{\mathrm{2}})-C_{\mathrm{ep}}{p}_{\mathrm{1}},\\ {\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}(A_{\mathrm{c}}(z_{\mathrm{s2}}-z_{\mathrm{u2}})-A_{\mathrm{c}}(z_{\mathrm{s1}}-z_{\mathrm{u1}}))+{\displaystyle \frac{\mathrm{1}}{{\beta }_{\mathrm{e}}}}(V_{\mathrm{10}}+V_{\mathrm{20}}+A_{\mathrm{c}}(z_{\mathrm{s2}}-z_{\mathrm{u2}})-A_{\mathrm{c}}(z_{\mathrm{s1}}-z_{\mathrm{u1}})){\displaystyle \frac{\mathrm{d}{p}_{\mathrm{2}}}{\mathrm{d}t}}=Q_{\mathrm{2}}+C_{\mathrm{ip}}(p_{\mathrm{1}}-p_{\mathrm{2}})-C_{\mathrm{ep}}{p}_{\mathrm{2}},\end{array}

With Eq. (14), we derive

(15)\begin{array}{l}\mathrm{2}{A}_{\mathrm{c}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}((z_{\mathrm{s1}}-z_{\mathrm{u1}})-(z_{\mathrm{s2}}-z_{\mathrm{u2}}))+{\displaystyle \frac{\mathrm{1}}{{\beta }_{\mathrm{e}}}}(V_{\mathrm{10}}+V_{\mathrm{20}}){\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}(p_{\mathrm{1}}-p_{\mathrm{2}})\\ +{\displaystyle \frac{\mathrm{1}}{{\beta }_{\mathrm{e}}}}(A_{\mathrm{c}}(z_{\mathrm{s1}}-z_{\mathrm{u1}})-A_{\mathrm{c}}(z_{\mathrm{s2}}-z_{\mathrm{u2}})){\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}(p_{\mathrm{1}}+p_{\mathrm{2}})=Q_{\mathrm{1}}-Q_{\mathrm{2}}-(\mathrm{2}{C}_{\mathrm{ip}}+C_{\mathrm{ep}})(p_{\mathrm{1}}-p_{\mathrm{2}}).\end{array}

Considering that p₁+p₂=p_s, z_s1=–t_ff, z_s2=t_ff, z_u1=–t_ff_u, and z_u2=t_ff_u, we can further rewrite Eq. (15) as

(16)-\mathrm{4}{A}_{\mathrm{c}}{t}_{\mathrm{f}}{\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}(\varphi -{\varphi }_{\mathrm{u}})+{\displaystyle \frac{\mathrm{1}}{{\beta }_{\mathrm{e}}}}(V_{\mathrm{10}}+V_{\mathrm{20}}){\displaystyle \frac{\mathrm{d}}{\mathrm{d}t}}(p_{\mathrm{1}}-p_{\mathrm{2}})+(\mathrm{2}{C}_{\mathrm{ip}}+C_{\mathrm{ep}})(p_{\mathrm{1}}-p_{\mathrm{2}})=Q_{\mathrm{1}}-Q_{\mathrm{2}}.

For simplicity, we assume that the displacement of spool valve z_v is directly proportional to electric voltage u_v(t), namely, z_v=k_vu_v(t), where k_v is a known scalar.

The flow quantities in the two circuits can be presented as [³⁸]

(17)\{\begin{array}{l}{Q}_{\mathrm{1}}=k_{\mathrm{u}}{u}_{\mathrm{v}}(s(u_{\mathrm{v}})\sqrt{p_{\mathrm{s}}-p_{\mathrm{1}}}+s(-u_{\mathrm{v}})\sqrt{p_{\mathrm{1}}-p_{\mathrm{r}}}),\\ Q_{\mathrm{2}}=k_{\mathrm{u}}{u}_{\mathrm{v}}(s(-u_{\mathrm{v}})\sqrt{p_{\mathrm{s}}-p_{\mathrm{2}}}+s(u_{\mathrm{v}})\sqrt{p_{\mathrm{2}}-p_{\mathrm{r}}}),\end{array}

where

(18)

where C_d is the discharge coefficient, w is the spool valve area gradient, and p_r is the return pressure which equals to the atmospheric pressure generally.

Experiments and simulations

Model verification and numerical simulations

In this section, the proposed vehicle model is validated against experimental test results. A novel fully electric vehicle prototype is utilized to conduct various maneuvers, as shown in Fig. 3. The parameters of the vehicle model in the simulations, which will be illustrated in the next section, are identical to those of this vehicle, as listed in Table I in the Appendix. Table II shows the parameters of the servo-valve electro-hydraulic actuator.

Fig.3 Fully electric vehicle on a concrete pavement in the experimental tests.

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The experimental setup and corresponding devices are depicted in Fig. 4. A wheel torque meter is utilized to record the wheel steering angle with a steering ratio of 17.5. A velocity recorder, namely, RACELOGIC VBOX 3i, is used to obtain vehicle velocity v_x with the GPS navigation system. An inertial measurement unit (IMU) sensor is utilized to record yaw rate \dot{\psi }, roll angle f, and lateral acceleration a_y. These signals are then transferred to a laptop for further signal post-processing.

Fig.4 Experimental setup.

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The two different types of maneuver, namely, sine-wave and double lane change, are executed to validate the vehicle model by comparing the test and simulation results. The vehicle speeds are 40 and 80 km/h. The results are shown in Figs. 5 and 6.

Figure 5 shows the vehicle response comparisons at sine-wave maneuver. Considering that transient responses for the beginning and end of the sine-wave maneuver cannot be used to characterize vehicle performance, defining an effective time interval would be helpful. Here, the effective time interval starts at 3.9 s and ends at 11.1 s. Figure 5(a) shows the tire steering angle transferred from the steering wheel via the steering system that is utilized as the input of the subsequent simulations. Figure 5(b) shows that the yaw rate of the simulation fits that of the experimental result well, except that the peak values of the latter are smaller than the former. The reason for this result may be the filter embedded in the IMU sensor, which cuts off the signal peaks. Figure 5(c) shows that the lateral accelerations of the two signals are close to each other. Figure 5(d) indicates that the two roll angle responses fit well, but steady shift exists after the steering maneuver is finished.

Fig.5 Comparisons of simulation and experiment results at sine-wave maneuver: (a) Sine-wave steering angle, (b) yaw rate, (c) lateral acceleration, and (d) roll angle of sprung mass.

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Fig.6 Comparisons of simulation and experiment results at double-lane-change maneuver: (a) Sine-wave steering angle, (b) yaw rate, (c) lateral acceleration, and (d) roll angle of sprung mass.

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Figure 6 presents the vehicle responses of double-lane-change maneuver, which is used to compare the aperiodic dynamic characteristics with those of the sine-wave maneuver. Figure 6(a) shows the tire steering input for the experiment and simulation. Figure 6(b) shows the yaw rates for the two maneuvers. The figure reveals that although discrepancies exist in the transient behaviors of the two maneuvers, the general response tendencies are the same. The discrepancies may be attributed to the tire transient properties and several chassis apparatuses, such as rubber bushes, which are not formulated in the proposed vehicle model due to their complicated dynamic characteristics and minor importance in vehicle responses compared with suspension devices. Figure 6(c) shows that the simulation and experiment results have almost the same lateral accelerations. Figure 6(d) reveals that the roll angles in the simulation and experiment fit well. Thus, we conclude that the simulation results match the experimental results well, thereby validating the proposed vehicle model.

Validation of the control scheme

Simulations are conducted under the two steering maneuvers to validate the proposed control scheme and investigate the effectiveness of the LQR controller with electro-hydraulic suspension. Vehicle responses, including suspension deflection z_su at the left-front station, roll angle f, lateral load transfer rate (LTR) [³⁹], and tire-terrain contact force F_tz1 at the left-front station, are compared and analyzed. Meanwhile, the dynamic responses of the electro-hydraulic suspension, including roll-resistant moment u(t), electric voltage u_v(t), oil pressure p_i, and flow quantity Q_i at the two oil circuits, are also derived.

Figure 7 shows the vehicle dynamic responses under sine-wave maneuver for uncontrolled and controlled schemes. Figure 7(a) reveals that during the effective time interval, the suspension deflection is reduced by more than 30.7% with the RREHS subsystem. Figure 7(b) illustrates that the roll angle response is decreased by 29.2% or more with the proposed controller. Figure 7(c) indicates that LTR is decreased by over 10.7% during the effective time interval, which implies that vehicle roll stability is effectively enhanced. Figure 7(d) shows that tire force is reduced by 1.1% at the left-front station. Therefore, vehicle performance is improved by the electro-hydraulic suspension.

Fig.7 Vehicle responses under sine-wave steering maneuver at 40 km/h: (a) Suspension deflection z_su at the left-front station, (b) roll angle f; (c) lateral load transfer rate, and (d) tire-terrain contact force F_tz1.

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Figure 8 shows the dynamic responses of the electro-hydraulic suspension. Figure 8(a) indicates that a roll-resistant moment is imposed on the vehicle when the vehicle performs turning maneuver to enhance vehicle roll stability. Figure 8(b) shows the electric voltage required by the servo-valve to generate the desired roll-resistant moment. Figure 8(c) depicts the oil pressures of the two oil circuits in the RREHS subsystem, which can be derived with the desired moment u(t) and suspension geometry. Figure 8(d) shows the fluid flow quantities at the two circuits.

Fig.8 Dynamic responses of the electro-hydraulic suspension under sine-wave maneuver: (a) Roll-resistant moment u(t), (b) electric voltage of the servo-valve, (c) oil pressures at the two hydraulic circuits, and (d) flow quantities at the two circuits.

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Fig.9 Vehicle responses under double-lane-change steering maneuver at 80 km/h: (a) Suspension deflection z_su at the left-front station, (b) roll angle f; (c) lateral load transfer rate, and (d) tire-terrain contact force F_tz1.

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The vehicle and servo-valve responses under double-lane-change maneuver are also investigated, as shown in Figs. 9 and 10. In Fig. 9(a), the suspension deflection is decreased by 24.1% at t=3.01 s and by 46.7% at t=3.75 s. Figure 9(b) shows that roll angle f is decreased by more than 25.5% during the time interval [2.8, 4.0] s. In Fig. 9(c), we can observe that the LTR value is reduced by 28.9% at t=3.68 s, whereas it is only reduced by 0.6% at t=2.97 s. Figure 9(d) shows that the maximum tire force is slightly decreased.

Roll-resistant moment u(t) is derived in Fig. 10(a). The corresponding electric voltage u_v(t) required by the servo-valve is shown in Fig. 10(b). Figure 10(c) shows the obtained fluid pressures at the two fluid circuits of the RREHS subsystem. Figure 10(d) displays the fluid quantities Q_i at the two circuits.

Fig.10 Dynamic responses of the electro-hydraulic suspension under double-lane-change steering maneuver: (a) Roll resistant moment u(t), (b) electric voltage of the servo-valve, (c) oil pressures at the two hydraulic circuits, and (d) flow quantities at the two circuits.

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Conclusions

This study proposes a roll-resistant electro-hydraulic suspension to enhance vehicle roll stability when a vehicle is performing steering maneuvers. A 4-DOF vehicle model is proposed to investigate the integration of lateral and roll dynamics. Then, an RREHS subsystem is developed to enhance vehicle roll motion, and an LQR optimal control scheme is used to derive the roll-resistant moment. In addition, the vehicle model is validated against field experiment results. The effectiveness of the optimal control scheme is verified by implementing the scheme on the RREHS subsystem. The following conclusions are derived from the results. (1) A 4-DOF vehicle model that incorporates yaw and roll motions is derived and validated. (2) The optimal controlled RREHS subsystem can be utilized to enhance vehicle performance effectively, especially roll stability, under sine-wave and double-lane-change steering maneuvers.

Considering the differences in theoretical equations, numerical models, and engineering applications, validation of the equation through experiments is recommended. In our future work, the proposed controller will be experimented on by using a real vehicle.

Acknowledgement

This work was supported by the National Natural Science Foundation of China (Grant No. 5187051675) and the Natural Science Foundation of Hunan Provincial (Grant No. 2017JJ2031).

Open Access

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