Robust strong duality for nonconvex optimization problem under data uncertainty in constraint

Yanfei Chai; Yanfei Chai

doi:10.3934/math.2021713

AIMS Mathematics

2021, Volume 6, Issue 11: 12321-12338. doi: 10.3934/math.2021713

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Research article

Robust strong duality for nonconvex optimization problem under data uncertainty in constraint

Yanfei Chai ^,

Department of Mathematics, Xi'an Polytechnic University, Xi'an 710048, China

Received: 08 February 2021 Accepted: 09 August 2021 Published: 26 August 2021
MSC : 90C46, 90C48

This paper deals with the robust strong duality for nonconvex optimization problem with the data uncertainty in constraint. A new weak conjugate function which is abstract convex, is introduced and three kinds of robust dual problems are constructed to the primal optimization problem by employing this weak conjugate function: the robust augmented Lagrange dual, the robust weak Fenchel dual and the robust weak Fenchel-Lagrange dual problem. Characterizations of inequality (1.1) according to robust abstract perturbation weak conjugate duality are established by using the abstract convexity. The results are used to obtain robust strong duality between noncovex uncertain optimization problem and its robust dual problems mentioned above, the optimality conditions for this noncovex uncertain optimization problem are also investigated.

Keywords:

weak conjugate function,
the inequality,
Robust abstract perturbational weak conjugate duality,
Robust strong duality

Citation: Yanfei Chai. Robust strong duality for nonconvex optimization problem under data uncertainty in constraint[J]. AIMS Mathematics, 2021, 6(11): 12321-12338. doi: 10.3934/math.2021713

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Abstract

1. Introduction

During the vehicle movement, the performance of the vehicle is affected by various vehicle structures and functions, such as power steering system, suspension system, braking system, etc. Moreover, in complex and high-speed environment, the vehicle's vertical, roll and pitch displacements contain a strong coupling relationship. Therefore, considering the motion and coupling characteristics of vehicle structures is meaningful to investigate vehicle dynamics. On the basis of geometric structure parameters of vehicle system and the nonlinear characteristics of shock absorber and leaf spring, the authors in ^[1] establish a nonlinear dynamic model for heavy vehicle. The correctness of the dynamic model is verified by testing the vertical acceleration data of the driver's seat, front wheel, middle wheel and rear wheel. To investigate the longitudinal driving behaviors of vehicle dynamics in the platoons, by taking the acceleration capability of heavy-duty vehicle into account, numerous heavy-duty vehicle platoon models are proposed ^{[2,3,4,5,6,7,8,9,10]}. Furthermore, by considering the lateral and longitudinal displacement characteristics of the vehicle, ^[11] represents a 2 DOF model of the vehicle and two diver cab models with time delays. In order to further describe the vehicle dynamics characteristics, in accordance with the two driver cab models in ^[11]. ^[15] and ^[16] further investigate the nonlinear lateral dynamics of a 2 DOF vehicle model. Based on longitudinal vehicle dynamics and by analyzing the dynamic of engine, torque converter, tire and capacitor pack, the authors of ^[17] present a dynamic model for a heavy-duty vehicle.

On the other side of research, the above mentioned vehicle models are mostly used to evaluate vehicle lateral and longitudinal dynamics characteristics, the influence of vehicle lateral and yaw dynamics characteristics are not considered enough. In practice, the tires not only provide horizontal and vertical forces to the vehicle, but also give vertical forces to the suspension system, especially in complex driving situations such as lane changes, cornering, or obstacle avoidance. In these cases, the vehicle's vertical, roll, and pitch dynamics are clearly coupled with lateral and yaw motion. Due to large inertia, high center of gravity and high roll center, heavy vehicles have poor stability when entering a turn or lane change, and the three-way coupling effect is large. Therefore, it is necessary to establish a three dimensional coupled vehicle model and study the influence of steering process on vehicle dynamics. More recently, more and more works focus on the coupling property of the vehicle. To reflect the steering influence on the overall response of the vehicle, ^[18] designs a novel 4 DOF hydraulic power steering (HPS) system. Simultaneously, ^[18] develops a 24 DOF model by taking the HPS system, the steering hand wheel angle, rack displacement, and hand wheel angles into account. According to the nonlinear characteristics of suspension damping and tire stiffness, ^[19] establishes a nonlinear three-way coupled lumped parameter model, and an improved nonlinear delay preview driver model was proposed based on ^[11], which was connected with the TCLP model to form a driver-vehicle closed-loop system. ^[20] establishes a complete vehicle model of a heavy truck, which not only investigates the nonlinear characteristics of suspension damping and tire stiffness, but also contains a modified preview driver model with nonlinear time delays to calculate the right front wheel steering angle for driving the vehicle along the desired route. In this paper, the kinematics and dynamics equations of cab and body are established by analyzing the three-way coupling effect of cab and body, as well as the dynamic characteristics of tire and suspension. Firstly, the dynamic relation of the tyre with deflection angle is introduced. Secondly, the coupling dynamics equation of cab was established by analyzing the three-way coupling effect of cab. Then, considering the dynamic characteristics of the vehicle suspension, the three-way coupling dynamic equation of the vehicle body is established. Finally, the kinematic and dynamic equations of cab and body are established based on the dynamic characteristics of tire and suspension and the Euler rotation theorem.

2. Nomenclature

Table 1. The symbols of the heavy-duty vehicle.

Definition	Symbol
forward traction (lateral traction) of the $i-th$ tire	$F_{xi}(i=1, \cdots, 6)(F_{yi})$
steering angle of the $i-th$ wheel	$\delta_i$
steering angular speed of the front axle tires	$\omega_t$
transverse (longitudinal) component of $i-th$ tire along the coordinate system $\{B\}$	$F_{Xi}(F_{Yi})$
suspension force, damping coefficient and spring constant of the $j-th$ spring between cab and body	$F_{cj}, H_{cj}$ and $K_{cj}$ $(j=1, 2, 3, 4)$
vertical displacement of the cab (body)	$z_c(z_b)$
pitching angle of the cab (body)	$\varphi_c(\varphi_b)$
roll displacements of the cab (body)	$\phi_c(\phi_b)$
longitudinal distance between origin of coordinates $\{C\}$ and cab rear (front) spring	$l_5(l_6)$
the distance between the origin of coordinates $\{C\}$ and $\{B\}$	$l_4$
the angle between the origin of coordinate $\{B\}$ and the sprung mass bar center of suspension	$\varphi_0$
transverse distance betweencab front spring and rear spring	$b_c$
resultant force of the cab inthe direction of axes $X_C$ , $Y_C$ and $Z_C$	$F_{xc}, F_{yc}$ and $F_{zc}$
resultant moment of the cab in the direction of axes $X_C$ , $Y_C$ and $Z_C$	$N_{xc}, N_{yc}$ and $N_{zc}$
total mass of the vehicle, cab and body	$m_s, m_t$ and $m_b$
velocity vectors of the cab in the coordinate system $\{C\}$ and $\{B\}$	$u_c, v_c$ and $w_c$ , $u_b, v_b$ and $w_b$
roll angle rate, pitch angle rate and yaw angle rate of the coordinate system $\{C\}$ and $\{B\}$	$p_c, q_c$ and $r_c$ , $p_b, q_b$ and $r_b$
vertical and transverse distance from the origin of $\{C\}$ and $\{B\}$ to the center of gravity of cab	$h_{os}$ and $e_{os}$ , $h_{ob}$ and $e_{ob}$
moment of inertia of a vehicle about axle $X_C, Y_C,$ $Z_C, X_B, Y_B$ and $Z_B$	$I_{xxc}, I_{yyc}$ and $I_{zzc},$ $I_{xxb}, I_{yyb}$ and $I_{zzb}$
moment of inertia of the cab about the axis $X_C$ and $Y_C$ ( $X_B$ and $Y_B$ )	$I_{xxsc}$ and $I_{yysc}$ $(I_{xxsb}$ and $I_{yysb})$
integral of the product of the $X_C$ and $Y_C$ ( $X_B$ and $Y_B$ ) deviation of an area element in a vehicle	$I_{xzc}(I_{xzb})$
compression displacement of the $j-th$ spring between cab and body	$x_{cj}$
cab center of gravity (body center of gravity) to cab front and rear spring transverse distance	$l_7$ and $l_8$ $(l_{15}$ and $l_{16})$
transverse (longitudinal) distance of cab center of gravity to set 1, set 2 and set 3 tires	$l_9, l_{10}$ and $l_{11}$ $(l_{12}, l_{13}$ and $l_{14})$
distance between the front axle and the rear axle in a suspension system	$l_3$
resultant force of the cab in the direction of axes $X_B, Y_B$ and $Z_B$	$F_{xb}, F_{yb}$ and $F_{zb}$
suspension force, damping coefficient and springconstant of the $j-th$ spring between the front axle or rear axle and the body of the suspension system	$F_{sj}, H_{sj}$ and $K_{sj}$
longitudinal transverse distance from the center of gravity of suspension system to the front axis of the suspension system	$l_1$
pitching angle of the left and right balance bars of the suspension system	$\varphi_{p1}$ and $\varphi_{p2}$
vertical displacement of the $j-th$ axle	$z_{uj}$
angle of inclination of the $j-th$ wheel shaft	$\phi_{uj}$
lateral distance between the left and right springs of the suspension system	$b_{s1}, b_{s2}$ and $b_{s3}$
damping forces of front suspension left and right springs	$F_{d1}$ and $F_{d2}$
resultant moment of the cab in the direction of axes $X_B, Y_B$ and $Z_B$	$N_{xb}, N_{yb}$ and $N_{zb}$
transverse (longitudinal) distance of cab center of gravity to set 1, set 2 and set 3 bearing spring	$l_{17}, l_{18}$ and $l_{19}$ $(l_{22}, l_{23}$ and $l_{24})$
transverse (longitudinal) distance of body center of gravity to set 1, set 2 and set 3 tires	$l_{25}, l_{26}$ and $l_{27}$ $(l_{28}, l_{29}$ and $l_{30})$
transverse distance (longitudinal distance) between the body center of gravity and the cab front (rear) spring	$l_{20}(l_{21})$
longitudinal distance from the center of gravity of the suspension system to the center of gravity of the rear axle of the suspension	$l_2$

| Show Table

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3. Model building

In this paper, the kinematic characteristics of a heavy-duty vehicle are considered to construct a 26 DOF vehicle body and a cab model. As shown in Figs. 1-4, the considered heavy vehicle has one front axle and two rear axles, which is called a three-axial vehicle. The degrees of freedom are vertical, roll and pitch displacements of the diver cab, vehicle body, the vertical and roll motion of three wheel axles, the pitch angles of the left and right balancing pole on rear suspension, and roll angle the of each tire. To further study the coupling property with each part, the vertical, roll and pitch motion of cab and body is modeled independently. Before introducing the related coordinate systems, the Euler's laws of motion is firstly given.

Figure 1. The established reference frame.

DownLoad: Full-Size Img PowerPoint

Figure 2. The top view of the three-axle heavy-duty vehicle.

DownLoad: Full-Size Img PowerPoint

Figure 3. The lateral view of the three-axle heavy-duty vehicle.

DownLoad: Full-Size Img PowerPoint

Figure 4. The top view of the three-axle heavy-duty vehicle.

DownLoad: Full-Size Img PowerPoint

Lemma 3.1. Observed from an inertial reference frame, the force applied to a rigid body is equal to the product of the mass of the rigid body and the acceleration of the center of mass, i.e.

$F^e = ma_c$

where $F^e$ is the resultant external force of the rigid body, $m$ is the rigid body mass, and $a_c$ is the acceleration of center of mass.

Lemma 3.2. . The fixed point $O$ (for example, the origin) of an inertial reference frame is set as the reference point. The net external moment applied to the rigid body is equal to the time rate of change of the angular momentum, i.e.

$M_O = \frac{dL_O}{dt}$

where $M_O$ is the is the external torque at point $O$ , $L_O$ is the angular momentum at point $O$ .

3.1. Establishing the related coordinate systems

To analyze the motion of heavy-duty vehicle, the corresponding coordinate frames are elaborated to describe the movement of the vehicle and indicated in . The moving coordinate frame $\{B\}$ is fixed to the vehicle's body and is called the body-fixed reference frame. The second coordinate frame $\{C\}$ is fixed to the cab and is called the cab-fixed reference frame. The third coordinate frame $\{E\}$ is fixed to the earth and is called the earth-fixed reference frame. The last coordinate frame $\{T_i\}$ is fixed to the $i-th$ , $(i = 1, 2, 3, 4, 5, 6)$ tire and is called the tire-fixed reference frame. In this paper, we assume that the body axes $X_B, Y_B, Z_B,$ the tire axes $X_{T_i}, Y_{T_i},$ and the cab axes $X_C, Y_C, Z_C,$ of heavy vehicle coincide with the principal axes of inertia, which are usually defined as:

$\bullet$ $X_B/X_{T_i}/X_C$ -longitudinal axis (directed from aft of the body/tire/cab to front).

$\bullet$ $Y_B/Y_{T_i}/Y_C$ -transverse axis (directed to right side of body/tire/cab).

$\bullet$ $Z_B/ Z_C$ -normal axis (directed from top to bottom).

3.2. Analysis of tire motion characteristics

To extract the kinetic model for the considered three-axial heavy-duty vehicle, the coordinate frame $T_{i}$ is designed for each tire, the corresponding schematic diagram is shown in Fig. 2. By taking the yaw angle into account, the forces produced by the engine are transformed into the forward traction and longitudinal traction on the suspension of heavy-duty vehicle. Based on the coordinate frame and Fig. 2, the forward and lateral traction of each tire can be expressed as

$\begin{equation} \begin{array}{lll} F_{Xi} = F_{xi}cos\delta_i-F_{yi}sin\delta_i,\\ F_{Yi} = F_{yi}sin\delta_i-F_{yi}cos\delta_i,\\ \dot{\delta_1} = \dot{\delta_2} = \omega_t,\delta_{3,4,5,6} = 0,\\ i = 1,\cdots,6. \end{array} \end{equation}$

(3.1)

3.3. Analysis of cab motion characteristics

In this subsection, the vertical, roll and pitch motion of the cab are considered to further accurately reflect the performance of spring suspension force between the cab and the body in the actual scenario. In accordance with the coordinate frame $\{C\}$ and Figs. 3-4, the spring force between the cab and body can be given as

$\begin{equation} \begin{array}{lll} F_{c1} = &K_{c1}(-z_c-\varphi_cl_5-z_b+(\varphi_b-\varphi_0)(l_4+l_5)-\frac{(\phi_b-\phi_c)b_c}{2})\\ &+H_{c1}(-\dot{z}_c-\dot{\varphi}_c l_5-\dot{z}_b+\dot{\varphi}_b(l_4+l_5)-\frac{(\dot{\phi}_b-\dot{\phi}_c)b_c}{2})\\ \end{array} \end{equation}$

(3.2)

$\begin{equation} \begin{array}{lll} F_{c2} = &K_{c2}(-z_c-\varphi_cl_5-z_b+(\varphi_b-\varphi_0)(l_4+l_5)+\frac{(\phi_b-\phi_c)b_c}{2})\\ &+H_{c2}(-\dot{z}_c-\dot{\varphi}_cl_5-\dot{z_b}+\dot{\varphi}_b(l_4+l_5)+\frac{(\dot{\phi}_b-\dot{\phi}_c)b_c}{2})\\ \end{array} \end{equation}$

(3.3)

$\begin{equation} \begin{array}{lll} F_{c3} = &K_{c3}(-z_c-\varphi_cl_5-z_b+(\varphi_b-\varphi_0)(l_4+l_5)-\frac{(\phi_b-\phi_c)b_c}{2})\\ &+H_{c3}(-\dot{z}_c-\dot{\varphi}_cl_5-\dot{z}_b+\dot{\varphi}_b(l_4+l_5)-\frac{(\dot{\phi}_b-\dot{\phi}_c)b_c}{2})\\ \end{array} \end{equation}$

(3.4)

$\begin{equation} \begin{array}{lll} F_{c4} = &K_{c4}(-z_c-\varphi_cl_5-z_b+(\varphi_b-\varphi_0)(l_4+l_5)+\frac{(\phi_b-\phi_c)b_c}{2})\\ &+H_{c4}(-\dot{z}_c-\dot{\varphi}_cl_5-\dot{z}_b+\dot{\varphi}_b(l_4+l_5)+\frac{(\dot{\phi}_b-\dot{\phi}_c)b_c}{2})\\ \end{array} \end{equation}$

(3.5)

Furthermore, by viewing the diver cab as a rigid body and employing the Lemmas 3.1 and 3.2, the kinematical equation of diver cab model is

$\begin{equation} \begin{array}{lll} F_{xc} = m_t(\dot{u}_c-v_cr_c)+m_s(w_cq_c-\dot{q}_ch_{os}-p_cr_ch_{os}-e_{os}q^2_c), \end{array} \end{equation}$

(3.6)

$\begin{equation} \begin{array}{lll} F_{yc} = m_t(\dot{v}_c-u_cr_c)+m_s(\dot{p}_ch_{os}-w_cp_c+q_c(p_ce_{os}-r_ch_{os})) \end{array} \end{equation}$

(3.7)

$\begin{equation} \begin{array}{lll} F_{zc} = m_s(\dot{w}_c+v_cp_c-u_cq_c+h_{os}(q^2_c+p^2_c)+e_{os}(p_cr_c-\dot{q}_c)), \end{array} \end{equation}$

(3.8)

$\begin{equation} \begin{array}{lll} N_{xc} = &I_{xxc}\dot{p}_c-I_{xzc}(\dot{r}_c+p_cq_c)+(I_{xxsc}-I_{yysc} -m_sh^2_{os})q_cr_c\\&+m_sh_{os}(\dot{v}_c+u_cr_c-w_cp_c), \end{array} \end{equation}$

(3.9)

$\begin{equation} \begin{array}{lll} N_{yc} = &I_{yyc}\dot{q}_c-I_{xzc}(p^2_c-r^2_c)+(I_{yysc}-I_{xxsc})p_cr_c-m_sh_{os}(\dot{u}_c\\ &-v_cr_c+w_cq_c)-m_se_{os}(\dot{w}_c+v_cp_c-u_cq_c), \end{array} \end{equation}$

(3.10)

$\begin{equation} \begin{array}{lll} N_{zc} = &I_{zzc}\dot{r}_c+I_{xzc}(q_cr_c-\dot{p}_c)+(I_{xxsc}-I_{yysc}+m_se^2_{os})p_cq_c\\ &-m_se_{os}w_cp_c, \end{array} \end{equation}$

(3.11)

When the vehicle is moving, the spring will produce spring force whether it is in a state of compression or tension. However, the direction of the force is opposite, so this paper considers the sign function and the direction of the spring displacement to determine the direction of the spring force. In order to obtain the dynamic force equation of the vehicle cab and body, we assume that the cab and body mass are evenly distributed, that is, the transverse distance between the cab and body center of gravity from the left and right tires is the same. By considering the definition of resultant force and resultant moment, the kinetic formula of longitudinal, transverse and vertical forces acting on the cab, as well as the yaw, pitch and roll moments is described as

$\begin{equation} \begin{array}{lll} F_{xc} = &c(\delta_1)c(\varphi_c)\sum^6_{i = 1}{F_{Xi}}+(s(\phi_c)s(\varphi_c)c(\delta_1)\\ &-s(\delta_1)c(\phi_c))\sum^6_{i = 1}{F_{Yi}}+(s(\delta_1)s(\phi_c)\\ &+c(\phi_c)s(\varphi_c)c(\delta_1))(F_{c1}+sign(x_{c2})F_{c2}\\ &+sign(x_{c3})F_{c3}+sign(x_{c4})F_{c4})\\ \end{array} \end{equation}$

(3.12)

$\begin{equation} \begin{array}{lll} F_{yc} = &s(\delta_1)c(\varphi_c)\sum^6_{i = 1}{F_{Xi}}+(s(\phi_c)s(\varphi_c)s(\delta_1)\\ &+c(\delta_1)c(\phi_c))\sum^6_{i = 1}{F_{Yi}}+(c(\phi_c)s(\phi_c)s(\delta_1)\\ &-c(\delta_1)s(\phi_c))(F_{c1}+sign(x_{c2})F_{c2}\\ &+sign(x_{c3})F_{c3}+sign(x_{c4})F_{c4})\\ \end{array} \end{equation}$

(3.13)

$\begin{equation} \begin{array}{lll} F_{zc} = &-s(\varphi_c)\sum^6_{i = 1}{F_{Xi}}+(s(\phi_c)c(\varphi_c))\sum^6_{i = 1}{F_{Yi}}\\ &+(c(\phi_c)c(\varphi_c)(F_{c1}+sign(x_{c2})F_{c2}) \end{array} \end{equation}$

(3.14)

$\begin{equation} \begin{array}{lll} N_{xc} = (F_{c1}+F_{c2})l_7-(F_{c3}+F_{c4})l_8, \end{array} \end{equation}$

(3.15)

$\begin{equation} \begin{array}{lll} N_{yc} = (F_{c1}-F_{c2})l_5+(F_{c3}-F_{c4})l_6, \end{array} \end{equation}$

(3.16)

$\begin{equation} \begin{array}{lll} N_{zc} = &(F_{X2}-F_{X1})l_9+(F_{X4}-F_{X3})l_{10}\\ &(F_{X5}-F_{X6})l_{11}+(F_{Y1}+F_{Y2})l_{12}\\ &-(F_{Y3}+F_{Y4})l_{13}-(F_{Y5}+F_{Y6})l_{14}, \end{array} \end{equation}$

(3.17)

where $sign(\cdot)$ denotes the symbolic function, $c(\cdot) = cos(\cdot)$ and $s(\cdot) = sin(\cdot)$ .

3.4. Analysis of suspension

For the considered heavy duty vehicle, two hydraulic dampers are fixed to the left and right front suspensions, and the balanced suspension does not have any shock absorbers. Thus, to represent the force situation of leaf spring in suspension system, the damping force of the two hydraulic dampers is considered for the front axle. The kinetic equation is given by

$\begin{equation} \begin{array}{lll} F_{s1} = K_{s1}(-z_b-(\varphi_b-\varphi_0)l_1-z_{u1}+\frac{(\phi_b-\phi_{u1})b_{s1}}{2})+F_{d1}, \end{array} \end{equation}$

(3.18)

$\begin{equation} \begin{array}{lll} F_{s2} = K_{s2}(-z_b-(\varphi_b-\varphi_0)l_1-z_{u1}-\frac{(\phi_b-\phi_{u1})b_{s1}}{2})+F_{d2}, \end{array} \end{equation}$

(3.19)

$\begin{equation} \begin{array}{lll} F_{s3} = &K_{s3}(-z_b+(\varphi_b-\varphi_0)l_2-\frac{\varphi_{p1}l_3}{2}-z_{u2}+\frac{(\phi_b-\phi_{u2})b_{s2}}{2})\\ &+H_{s3}(-\dot{z}_b+\dot{\varphi}_bl_2-\frac{\dot{\varphi}_{p1}l_3}{2}-\dot{z}_{u2}+\frac{(\dot{\phi}_b-\dot{\phi}_{u2})b_{s2}}{2}), \end{array} \end{equation}$

(3.20)

$\begin{equation} \begin{array}{lll} F_{s4} = &K_{s4}(-z_b+(\varphi_b-\varphi_0)l_2-\frac{\varphi_{p2}l_3}{2}-z_{u2}+\frac{(\phi_b-\phi_{u2})b_{s2}}{2})\\ &+H_{s4}(-\dot{z}_b+\dot{\varphi}_bl_2-\frac{\dot{\varphi}_{p2}l_3}{2}-\dot{z}_{u2}-\frac{(\dot{\phi}_b-\dot{\phi}_{u2})b_{s2}}{2}), \end{array} \end{equation}$

(3.21)

$\begin{equation} \begin{array}{lll} F_{s5} = &K_{s5}(-z_b+(\varphi_b-\varphi_0)l_2-\frac{\varphi_{p1}l_3}{2}-z_{u3}+\frac{(\phi_b-\phi_{u3})b_{s3}}{2})\\ &+H_{s5}(-\dot{z}_b+\dot{\varphi}_bl_2+\frac{\dot{\varphi}_{p1}l_3}{2}-\dot{z}_{u3}+\frac{(\dot{\phi}_b-\dot{\phi}_{u3})b_{s3}}{2}), \end{array} \end{equation}$

(3.22)

$\begin{equation} \begin{array}{lll} F_{s6} = &K_{s6}(-z_b+(\varphi_b-\varphi_0)l_2-\frac{\varphi_{p2}l_3}{2}-z_{u3}+\frac{(\phi_b-\phi_{u3})b_{s3}}{2})\\ &+H_{s6}(-\dot{z}_b+\dot{\varphi}_bl_2+\frac{\dot{\varphi}_{p2}l_3}{2}-\dot{z}_{u3}-\frac{(\dot{\phi}_b-\dot{\phi}_{u3})b_{s3}}{2}), \end{array} \end{equation}$

(3.23)

3.5. Analysis of body motion characteristics

This is analogous to the diver cab part, taking the body as a rigid body and according to the Lemmas 3.1 and 3.2, the force equation of body model can be expressed as

$\begin{equation} \begin{array}{lll} F_{xb} = m_t(\dot{u}_b-v_br_b)+m_b(w_bq_b-\dot{q}_bh_{ob}-p_br_bh_{ob}-e_{ob}q^2_b), \end{array} \end{equation}$

(3.24)

$\begin{equation} \begin{array}{lll} F_{yb} = m_t(\dot{v}_b-u_br_b)+m_b(\dot{p}_bh_{ob}-w_bp_b+q_b(p_be_{ob}-r_bh_{ob})), \end{array} \end{equation}$

(3.25)

$\begin{equation} \begin{array}{lll} F_{zb} = m_b(\dot{w}_b+v_bp_b-u_bq_b)+h_{ob}(q^2_b+p^2_b)+e_{ob}(p_br_b-\dot{q}_b)), \end{array} \end{equation}$

(3.26)

$\begin{equation} \begin{array}{lll} N_{xb} = &I_{xxb}\dot{p}_b-I_{xzb}(\dot{r}_b+p_bq_b)+(I_{xxsb}-I_{yysb}-m_bh^2_{ob})q_br_b\\ &+m_bh_{ob}(\dot{v}_b+u_br_b-w_bp_b), \end{array} \end{equation}$

(3.27)

$\begin{equation} \begin{array}{lll} N_{yb} = &I_{yyb}\dot{q}_b-I_{xzb}(p^2_b-r^2_b)+(I_{yysb}-I_{xxsb})p_br_b-m_bh_{ob}(\dot{u}_b\\ &-v_br_b+w_bq_b)-m_be_{ob}(\dot{w}_b+v_bp_b-u_bq_b), \end{array} \end{equation}$

(3.28)

$\begin{equation} \begin{array}{lll} N_{zb} = &I_{zzb}\dot{r}_b+I_{xzb}(q_br_b-\dot{p}_b)+(I_{yysb}-I_{xxsb}+m_be^2_{ob})p_bq_b\\ &-m_be_{ob}w_bp_b, \end{array} \end{equation}$

(3.29)

Recalling the problem of spring force direction and the assumption of uniform distribution of body mass, the longitudinal, transverse and vertical forces acting on the body are:

$\begin{equation} \begin{array}{lll} F_{xb} = &c(\delta_1)c(\varphi_b)\sum^6_{i = 1}F_{Xi}+(s(\phi_b)s(\varphi_b)c(\delta_1)\\ &-s(\delta_1)c(\phi_b))\sum^6_{i = 1}F_{Yi}+(s(\delta_1)s(\phi_b)\\ &+c(\phi_b)s(\varphi_b)c(\delta_1))(-F_{c1}-sign(x_{c2})F_{c2}\\ &-sign(x_{c3})F_{c3}-sign(x_{c4})F_{c4}+F_{s1}\\ &+sign(z_{u2})F_{s2}+sign(z_{u3})F_{s3}+sign(z_{u4})F_{s4}\\ &+sign(z_{u5})F_{s5}+sign(z_{u6})F_{s6}),\end{array} \end{equation}$

(3.30)

$\begin{equation} \begin{array}{lll} F_{yb} = &s(\delta_1)c(\varphi_b)\sum^6_{i = 1}F_{Xi}+(s(\phi_b)s(\varphi_b)s(\delta_1)\\ &+c(\delta_1)c(\phi_b))\sum^6_{i = 1}F_{Yi}+(c(\phi_b)s(\varphi_b)s(\delta_1)\\ &-c(\delta_1)s(\phi_b))(-F_{c1}-sign(x_{c2})F_{c2}\\ &-sign(x_{c3})F_{c3}-sign(x_{c4})F_{c4}+F_{s1}\\ &+sign(z_{u2})F_{s2}+sign(z_{u3})F_{s3}+sign(z_{u4})F_{s4}\\ &+sign(z_{u5})F_{s5}+sign(z_{u6})F_{s6}),\end{array} \end{equation}$

(3.31)

$\begin{equation} \begin{array}{lll} F_{zb} = &-s(\varphi_b)\sum^6_{i = 1}F_{Xi}+(s(\phi_b)c(\varphi_b)\sum^6_{i = 1}F_{Yi}\\ &+(c(\phi_b)c(\varphi_b)(-F_{c1}-sign(x_{c2})F_{c2}\\ &-sign(x_{c3})F_{c3}-sign(x_{c4})F_{c4}+F_{s1}\\ &+sign(z_{u2})F_{s2}+sign(z_{u3})F_{s3}+sign(z_{u4})F_{s4}\\ &+sign(z_{u5})F_{s5}+sign(z_{u6})F_{s6}),\end{array} \end{equation}$

(3.32)

$\begin{equation} \begin{array}{lll} N_{xb} = &-(F_{c1}+F_{c2})l_{15}+(F_{c3}+F_{c4})l_{16}\\ &+(F_{s1}+F_{s2})l_{17}+(F_{s3}+F_{s4})l_{18}\\ &-(F_{s5}+F_{s6})l_{19}, \end{array} \end{equation}$

(3.33)

$\begin{equation} \begin{array}{lll} N_{yb} = &(F_{c2}-F_{c1})l_{20}+(F_{c4}-F_{c3})l_{21}\\ &+(F_{s1}-F_{s2})l_{22}+(F_{s3}-F_{s4})l_{23}\\ &+(F_{s5}-F_{s6})l_{24},\end{array} \end{equation}$

(3.34)

$\begin{equation} \begin{array}{lll} N_{zb} = &(F_{X2}-F_{X1})l_{25}+(F_{X4}-F_{X3})l_{26}\\ &+(F_{X5}-F_{X6})l_{27}+(F_{Y1}+F_{Y2})l_{28}\\ &-(F_{Y3}+F_{Y4})l_{29}-(F_{Y5}+F_{Y6})l_{30}, \end{array} \end{equation}$

(3.35)

3.6. Diver cab and body nonlinear model

In this final subsection, a common type of diver cab and body model for heavy-duty vehicle is proposed.

In accordance with 3.1-3.17 and invoking the Euler rotation theorem, the dynamic and kinetic equation of driver cab are designed as

$\begin{equation} \begin{array}{lll} \dot{\eta}_c = J_c(\eta_c)\upsilon_c,\\ \dot{\upsilon}_c = G_c(\eta_c,\upsilon_c)[F^*_{Xi},F^*_{Yi}]^T+F_c(\upsilon)+u_c, \end{array} \end{equation}$

(3.36)

where $\eta_c = [x_c, y_c, z_c, \phi_c, \varphi_c, \delta_1]^T,$

$J_c(\eta_c) = \left[\begin{array}{ccc} J_1(\eta_c) & 0_{3\times3}\\ 0_{3\times3} & J_2(\eta_c)\\ \end{array}\right],$

$J_2(\eta_c) = \left[ \begin{array}{ccc} 1 & s(\phi_c)t(\varphi_c) & c(\phi_c)t(\varphi_c) \\ 0 & s(\phi_c) & -s(\phi_c)\\ 0 & s(\phi_c)/c(\varphi_c) & c(\phi_c)/c(\varphi_c) \end{array}\right],$

$G_c(\eta_c, \upsilon_c) = \left[\begin{array}{ccc} J_3(\eta_c)\\ 0_{3\times2}\\ \end{array}\right],$

$J_3(\eta_c) = \left[ \begin{array}{ccc} \frac{c(\delta_1)c(\varphi_c)}{m_t} & -s(\delta_1)c(\phi_c)+\frac{s(\phi_c)s(\varphi_c)c(\delta_1)}{m_t}\\ \frac{s(\delta_1)c(\varphi_c)}{m_t} & +c(\delta_1)c(\phi_c)+\frac{s(\phi_c)s(\varphi_c)s(\delta_1)}{m_t}\\ \frac{-s(\varphi_c)}{m_s} & \frac{s(\phi_c)c(\varphi_c)}{m_s} \end{array}\right],$

$F_c(\upsilon_c) = [F_{c1}(\upsilon_c), F_{c2}(\upsilon_c), F_{c3}(\upsilon_c), F_{c4}(\upsilon_c), F_{c5}(\upsilon_c), F_{c6}(\upsilon_c)]^T,$

$J_1(\eta_c) = \left[\begin{array}{ccc} c(\delta_1)c(\varphi_c) & -s(\delta_1)c(\phi_c)+s(\phi_c)s(\varphi_c)c(\delta_1)\\ s(\delta_1)c(\varphi_c) & c(\delta_1)c(\phi_c)+s(\phi_c)s(\varphi_c)s(\delta_1)\\ -s(\varphi_c) & s(\phi_c)c(\varphi_c) \end{array}\right.$

$\left.\begin{array}{ccc} s(\delta_1)s(\phi_c)+c(\phi_c)s(\varphi_c)c(\delta_1)\\ -c(\delta_1)s(\phi_c)+c(\phi_c)s(\varphi_c)s(\delta_1)\\ c(\phi_c)c(\varphi_c) \end{array}\right],$

$\upsilon_c = [u_c, v_c, w_c, p_c, q_c, r_c]^T$ , $F^*_{Xi} = \sum^6_{i = 1}F_{Xi}$ , $F^*_{Yi} = \sum^6_{i = 1}F_{Yi}$ ,

$u_c = \left[\begin{array}{cc} 0_{5\times 1} \\ \frac{((F_{X2}-F_{X1})l_9+(F_{X4}-F_{X3})l_{10}+(F_{X5}-F_{X6})l_{11}}{I_{zzc}}\\ \frac{+(F_{Y1}+F_{Y2})l_{12}-(F_{Y3}+F_{Y4})l_{13}-(F_{Y5}+F_{Y6})l_{14}))}{} \end{array} \right],$ $t(\cdot)$ represents the tangent function. The expansion equation of the matrix $F_c(\upsilon_c)$ is

$\begin{equation} \begin{array}{lll} F_{c1}(\upsilon_c) = &\frac{1}{m_t}[(s(\delta_1)s(\phi_c)+c(\phi_c)s(\varphi_c)c(\delta_1))*(F_{c1}\\ &+sign(x_{c2})F_{c2}+sign(x_{c3})F_{c3}+sign(x_{c4})F_{c4})\\ &-m_s(w_cq_c-\dot{q}_ch_{os}-p_cr_ch_{os}-e_{os}q^2_c)+m_tv_cr_c], \end{array} \end{equation}$

(3.37)

$\begin{equation} \begin{array}{lll} F_{c2}(\upsilon_c) = &\frac{1}{m_t}[(c(\phi_c)s(\varphi_c)s(\delta)-c(\delta)s(\phi_c))*(F_{c1}\\ &+sign(x_{c2})F_{c2}+sign(x_{c3})F_{c3}+sign(x_{c4})F_{c4})\\ &-m_s(\dot{p}_ch_{os}-w_cp_c+q_c(p_ce_{os}-r_ch_{os}))\\ &+m_tu_cr_c], \end{array} \end{equation}$

(3.38)

$\begin{equation} \begin{array}{lll} F_{c3}(\upsilon_c) = &\frac{1}{m_s}[c(\phi_c)c(\varphi_c)(F_{c1}+sign(x_{c2})F_{c2}+sign(x_{c3})F_{c3}\\ &+sign(x_{c4})F_{c4})]-(v_cp_c-u_cq_c+h_{os}(q^2_c+p^2_c)\\ &+e_{os}(p_cr_c-\dot{q}_c)), \end{array} \end{equation}$

(3.39)

$\begin{equation} \begin{array}{lll}F_{c4}(\upsilon_c) = &\frac{1}{I_{xxc}}(F_{c1}+F_{c2})l_7-(F_{c3}+F_{c4})l_8\\ &-m_sh_{os}(\dot{v}_c+u_cr_c-w_cp_c)+I_{xzc}(\dot{r}_c+p_cq_c)\\ &-(I_{xxsc}-I_{yysc}-m_sh^2_{os})q_cr_c,\end{array} \end{equation}$

(3.40)

$\begin{equation} \begin{array}{lll} F_{c5}(\upsilon_c) = &\frac{1}{I_{yyc}}(I_{xzc}(p^2_c-r^2_c)-(I_{yysc}-I_{xxsc})p_cr_c\\ &+m_sh_{os}(\dot{u}_c-v_cr_c+w_cq_c)\\ &+m_se_{os}(\dot{w}_c+v_cp_c-u_cq_c)\\ &+(F_{c1}-F_{c2})l_5+(F_{c3}-F_{c4})l_6), \end{array} \end{equation}$

(3.41)

$\begin{equation} \begin{array}{lll} F_{c6}(\upsilon_c) = &\frac{1}{I_{zzc}}[m_se_{os}w_cp_c-I_{xzc}(q_cr_c-\dot{p}_c)\\ &-(I_{yysc}-I_{xxsc}+m_se^2_{os})p_cq_c],\end{array} \end{equation}$

(3.42)

According to kinetic equations (18)-(42) and employing the Euler rotation theorem, the dynamic and kinetic equation of body are designed as

$\begin{equation} \begin{array}{lll} \dot{\eta}_b = J_b(\eta_b)\upsilon_b,\\ \dot{\upsilon}_b = G_b(\eta_b,\upsilon_b)[F^*_{Xi},F^*_{Yi}]^T+F_b(\upsilon)+u_b, \end{array} \end{equation}$

(3.43)

where $\eta_b = [x_b, y_b, z_, b\phi_b, \varphi_b, \delta_1]^T,$ $J_b(\eta_b) = \left[\begin{array}{ccc} J_1(\eta_b) & 0_{3\times3}\\ 0_{3\times3} & J_2(\eta_b)\\ \end{array}\right],$ $G_b(\eta_b, \upsilon_b) = \left[\begin{array}{ccc} J_3(\eta_b)\\ 0_{3\times2}\\ \end{array}\right],$ $\upsilon_b = [u_b, v_b, w_b, p_b, q_b, r_b]^T$ , $F^*_{Yi} = \sum^6_{i = 1}F_{Xi}$ , $F_b(\upsilon_b) = [F_{b1}(\upsilon_b), F_{b2}(\upsilon_b), F_{b3}(\upsilon_b), F_{b4}(\upsilon_b), F_{b5}(\upsilon_b), F_{b6}(\upsilon_b)]^T,$ $J_2(\eta_b) = \left[ \begin{array}{ccc} 1 & s(\phi_b)t(\varphi_b) & c(\phi_b)t(\varphi_b) \\ 0 & s(\phi_b) & -s(\phi_b)\\ 0 & s(\phi_b)/c(\varphi_b) & c(\phi_b)/c(\varphi_b) \end{array}\right],$ $F^*_{Xi} = \sum^6_{i = 1}F_{Xi}$ , $J_3(\eta_b) = \left[ \begin{array}{ccc} \frac{c(\delta_1)c(\varphi_b)}{m_t} & -s(\delta_1)c(\phi_b)+\frac{s(\phi_b)s(\varphi_b)c(\delta_1)}{m_t}\\ \frac{s(\delta_1)c(\varphi_b)}{m_t} & +c(\delta_1)c(\phi_b)+\frac{s(\phi_b)s(\varphi_b)s(\delta_1)}{m_t}\\ \frac{-s(\varphi_b)}{m_b} & \frac{s(\phi_b)c(\varphi_b)}{m_b} \end{array}\right],$ $u_b = \left[\begin{array}{cc} 0_{5\times 1} \\ \frac{((F_{X2}-F_{X1})l_{25}+(F_{X4}-F_{X3})l_{26}+(F_{X5}-F_{X6})l_{27}}{I_{zzb}}\\ \frac{+(F_{Y1}+F_{Y2})l_{28}-(F_{Y3}+F_{Y4})l_{29}-(F_{Y5}+F_{Y6})l_{30})}{} \end{array} \right],$ $J_1(\eta_b) = \left[\begin{array}{ccc} c(\delta_1)c(\varphi_b) & -s(\delta_1)c(\phi_cb)+s(\phi_b)s(\varphi_b)c(\delta_1)\\ s(\delta_1)c(\varphi_b) & c(\delta_1)c(\phi_b)+s(\phi_b)s(\varphi_b)s(\delta_1)\\ -s(\varphi_b) & s(\phi_b)c(\varphi_b) \end{array}\right.$

$\left.\begin{array}{ccc} s(\delta_1)s(\phi_b)+c(\phi_b)s(\varphi_b)c(\delta_1)\\ -c(\delta_1)s(\phi_b)+c(\phi_c)s(\varphi_b)s(\delta_1)\\ c(\phi_b)c(\varphi_b) \end{array}\right],$

The expansion equation of the matrix ${F_b(\upsilon_b)}$ is

$\begin{equation} \begin{array}{lll} F_{b1}(\upsilon_b) = &\frac{1}{m_t}[(s(\delta_1)s(\phi_b)+c(\phi_b)s(\varphi_b)c(\delta_1))*(-F_{c1}\\ &-sign(x_{c2})F_{c2}-sign(x_{c3})F_{c3}-sign(x_{c4})F_{c4})\\ &+F_{s1}+sign(z_{u2})F_{s2}+sign(z_{u3})F_{s3}\\ &+sign(z_{u4})F_{s4}+sign(z_{u5})F_{s5}+sign(z_{u6})F_{s6})\\ &-m_b(w_bq_b-\dot{q}_bh_{ob}-p_br_bh_{ob}-e_{ob}q^2_b)+m_tv_br_b], \end{array} \end{equation}$

(3.44)

$\begin{equation} \begin{array}{lll} F_{b2}(\upsilon_b) = &\frac{1}{m_t}[(c(\phi_b)s(\varphi_b)s(\delta)-c(\delta_1)s(\phi_b))*(-F_{c1}\\ &-sign(x_{c2})F_{c2}-sign(x_{c3})F_{c3}-sign(x_{c4})F_{c4})\\ &+F_{s1}+sign(z_{u2})F_{s2}+sign(z_{u3})F_{s3}\\ &+sign(z_{u4})F_{s4}+sign(z_{u5})F_{s5}+sign(z_{u6})F_{s6})\\ &-m_b(\dot{p}_bh_{ob}-w_bp_b+q_b(p_be_{ob}-r_bh_{ob}))\\ &+m_tu_br_b], \end{array} \end{equation}$

(3.45)

$\begin{equation} \begin{array}{lll} F_{b3}(\upsilon_b) = &\frac{1}{m_b}[c(\phi_b)c(\varphi_b)(-F_{c1}-sign(x_{c2})F_{c2}\\ &-sign(x_{c3})F_{c3}-sign(x_{c4})F_{c4})+F_{s1}\\ &+sign(z_{u2})F_{s2}+sign(z_{u3})F_{s3}+sign(z_{u4})F_{s4}\\ &+sign(z_{u5})F_{s5}+sign(z_{u6})F_{s6})\\ &+m_b(u_bq_b-v_bp_b-h_{ob}(q^2_b+p^2_b)\\ &-e_{ob}(p_br_b-\dot{q}_b)), \end{array} \end{equation}$

(3.46)

$\begin{equation} \begin{array}{lll}F_{b4}(\upsilon_b) = &\frac{1}{I_{xxb}}[(F_{c3}+F_{c4})l_{16}-(F_{c1}+F_{c2})l_{15}\\ &+(F_{s1}+F_{s2})l_{17}+(F_{s3}+F_{s4})l_{18}\\ &-(F_{s5}+F_{s6})l_{19}+I_{xzb}(\dot{r}_b+p_bq_b)\\ &-(I_{xxsb}-I_{yysb}-m_bh^2_{ob})q_br_b\\ &-m_bh_{ob}(\dot{v}_b+u_br_b-w_bp_b)],\end{array} \end{equation}$

(3.47)

$\begin{equation} \begin{array}{lll} F_{b5}(\upsilon_b) = &\frac{1}{I_{yyb}}[(F_{c2}-F_{c1})l_{20}+(F_{c4}-F_{c3})l_{21}\\ &+(F_{s1}-F_{s2})l_{22}+(F_{s3}-F_{s4})l_{23}\\ &-(F_{s5}-F_{s6})l_{24}+I_{xzb}(p^2_b-r^2_b)\\ &-(I_{yysb}-I_{xxsb})p_br_b\\ &+m_bh_{ob}(\dot{u}_b-v_br_b+w_bq_b)\\ &+m_be_{ob}(\dot{w}_b+v_bp_b-u_bq_b)], \end{array} \end{equation}$

(3.48)

$\begin{equation} \begin{array}{lll} F_{b6}(\upsilon_b) = &\frac{1}{I_{zzb}}[m_be_{ob}w_bp_b-I_{xzb}(q_br_b-\dot{p}_b)\\ &-(I_{yysb}-I_{xxsb}+m_be^2_{ob})p_bq_b], \end{array} \end{equation}$

(3.49)

4. Conclusions

In complex working conditions, there is a coupling relationship of the vertical, lateral and longitudinal dynamics of vehicles. By considering the kinetic character of the vertical, roll and pitch motion of the diver cab, vehicle body, the vertical and roll behavior of three wheel axles, the pitch angles of the left and right balancing pole on rear suspension, and roll angle the of each tire. In this paper, a common model of three-axles heavy-duty vehicle with 26 DOF have been proposed to extrude the kinetic characterization diver cab and vehicle body.

Acknowledgment

This work was supported by the National Natural Science Foundation of China under Grants U22A2043 and 62173172.

Conflict of interest

The author declares that there is no conflicts of interest in this paper.

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AIMS Mathematics

Robust strong duality for nonconvex optimization problem under data uncertainty in constraint

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Abstract

1. Introduction

2. Nomenclature

3. Model building

3.1. Establishing the related coordinate systems

3.2. Analysis of tire motion characteristics

3.3. Analysis of cab motion characteristics

3.4. Analysis of suspension

3.5. Analysis of body motion characteristics

3.6. Diver cab and body nonlinear model

4. Conclusions

Acknowledgment

Conflict of interest

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AIMS Mathematics

Robust strong duality for nonconvex optimization problem under data uncertainty in constraint

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Abstract

1. Introduction

2. Nomenclature

3. Model building

3.1. Establishing the related coordinate systems

3.2. Analysis of tire motion characteristics

3.3. Analysis of cab motion characteristics

3.4. Analysis of suspension

3.5. Analysis of body motion characteristics

3.6. Diver cab and body nonlinear model

4. Conclusions

Acknowledgment

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