Keywords

1 Introduction

The driving conditions of special vehicles are relatively complex, usually driving on emergency military roads, rural dirt roads, and roadless areas. In such off-road environments, the excitation generated by road roughness can seriously affect the smoothness, safety, and handling stability of special vehicles [1]. Active suspension can dynamically adaptively adjust the stiffness and damping characteristics of the suspension system based on the driving conditions of the vehicle, effectively reducing the impact and vibration transmitted from the road surface to the vehicle body. Therefore, research on suspension control strategies is of great significance for improving the overall performance of special vehicles.

In the research of active suspension control, vehicle state estimation is crucial and is also the foundation of suspension controller design. In most studies, acceleration sensors or displacement sensors are used to measure the acceleration of spring loaded/non spring loaded masses or the relative displacement of the suspension, and then these signals are processed in the back-end to obtain the required state variables. The Kalman filtering algorithm is a relatively classic algorithm in vehicle system state estimation. Reference [2] uses the Kalman filtering algorithm to estimate the vehicle’s center of mass sideslip angle in real-time and eliminate yaw rate errors; Reference [3] uses extended Kalman filtering to estimate vehicle state parameters such as yaw rate, center of mass sideslip angle, and longitudinal velocity, but there is currently little research applied to suspension state estimation.

Aiming at the problem of state variables required in the optimal control of active suspension of special vehicles, this paper establishes a vehicle vibration state estimation model based on Kalman filter algorithm, which estimates the state of the vertical displacement and vertical acceleration of the front and rear suspension and the body of the vehicle traveling on uneven roads. Based on the state estimation value, the linear quadratic form optimal control method is used to actively control the suspension, Provide a foundation for optimal control of active suspension of special vehicles.

2 Road Input Model and Vehicle Model

2.1 Pavement Input Model

The variation of the height q of the pavement relative to the datum plane along the length I of the road alignment is called the pavement longitudinal profile or pavement unevenness function q(I).

When measuring the unevenness of the pavement, the professional pavement meter or level can be used to measure on the actual pavement to get the unevenness value on the longitudinal section of the pavement, and the large amount of data obtained from the measurement can be imported into the computer for processing to get the statistical characteristic parameters such as the power spectral density function of the unevenness of the pavement, Gq(n), or the variance, σ2q, and so on.

As the vehicle vibration input of the road surface unevenness, the road surface power spectral density function is mainly used to describe its statistical characteristics, the road surface power spectral density function fitting expression is as follows:

$$ G_{q} \left( n \right) = G_{q} \left( {n_{0} } \right)\left( {\frac{n}{{n_{0} }}} \right)^{ - W} $$
(1)

where: Gq(n)-spatial power spectral density of pavement; Gq(n0)-pavement unevenness coefficient at reference spatial frequency; n-spatial frequency; n0-reference spatial frequency, taking the value of 0.1 m−1; W-frequency index, taking the value of 2.

The more common method to simulate the pavement unevenness excitation is to generate the pavement unevenness excitation by filtering white noise, which is suitable to meet the international and national standards of pavement unevenness excitation generation [4], and its time domain expression is as follows:

$$ \dot{q}\left( t \right) = - 2\pi n_{q} uq\left( t \right) + 2\pi n_{0} \sqrt {G_{q} \left( {n_{0} } \right)u} w\left( t \right) $$
(2)

where: u-vehicle speed; w(t)-standard Gaussian white noise with mean 0 and variance 1; q(t)-excitation of the road surface unevenness generated; nq-spatial lower cut-off frequency, which takes the value of 0.01/m. Considering that the driving environment of special vehicles is relatively harsh, the simulation is carried out by using the working condition of 30 km/h through the E-class road surface, in which Gq(n0) is 4096 × 10–6 m3.

2.2 4-Degree-of-Freedom Semi-Vehicle Model

A 4-degree-of-freedom 1/2 vehicle dynamics model is established, whose four degrees of freedom are the vertical motion of the body, the pitching motion, and the vertical motion of the front and rear wheels, respectively [5]. In the Fig. 1, ms is the mass of the half-vehicle model body, Jφ is the moment of inertia, zs is the displacement of the center of mass of the half-vehicle model body, φ is the pitch angle, l1 and l2 are the distances from the front and rear axles to the center of mass, zsf and zsr are the displacements of the concentrated mass on the front and rear axles, ksf and ksr are the stiffness coefficients of the front and rear suspension, csf and csr are the damping coefficients of the dampers of the front and rear suspensions, zuf and zur are the displacement of unsprung mass of the front and rear axle, ktf, ktr are the stiffness coefficients of the front and rear tires, and ztf, ztr are the road surface unevenness excitation of the front and rear wheels.

Fig. 1
A schematic of a vehicle suspension model with labeled components, including masses, springs, and dampers. At the top center, a horizontal bar labeled l, l 1, l 2, and Z s represents lengths and displacement. Two vertical lines extend downward from the ends of the horizontal bar, connecting to separate suspension systems on either side.

4-degree-of-freedom 1/2 vehicle suspension model

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According to D’Alembert’s principle, the differential equations of motion for the passive suspension dynamics of the semi-vehicle model can be listed, where Eqs. (3) and (4) are the differential equations of motion for the unsprung masses of the front and rear suspensions, Eq. (5) is the differential equation of motion for the center of mass of the vehicle body, and Eq. (6) is the differential equation of motion for the pitch angle.

$$ m_{uf} \ddot{z}_{uf} = k_{tf} \left( {z_{tf} - z_{uf} } \right) - c_{sf} \left( {\dot{z}_{uf} - \dot{z}_{sf} } \right) - k_{sf} \left( {z_{uf} - z_{sf} } \right) $$
(3)
$$ m_{ur} \ddot{z}_{ur} = k_{tr} \left( {z_{tr} - z_{ur} } \right) - c_{sr} \left( {\dot{z}_{ur} - \dot{z}_{sr} } \right) - k_{sr} \left( {z_{ur} - z_{sr} } \right) $$
(4)
$$ m_{s} \ddot{z}_{s} = c_{sf} \left( {\dot{z}_{uf} - \dot{z}_{sf} } \right) + k_{sf} \left( {z_{uf} - z_{sf} } \right) + c_{sr} \left( {\dot{z}_{ur} - \dot{z}_{sr} } \right) + k_{sr} \left( {z_{ur} - z_{sr} } \right) $$
(5)
$$\begin{aligned}J_{\varphi } \ddot{\varphi } &= - l_{f} \left[ {c_{sf} \left( {\dot{z}_{uf} - \dot{z}_{sf} } \right) + k_{sf} \left( {z_{uf} - z_{sf} } \right)} \right] \\&+ l_{r} \left[ {c_{sr} \left( {\dot{z}_{ur} - \dot{z}_{sr} } \right) + k_{sr} \left( {z_{ur} - z_{sr} } \right)} \right] \end{aligned} $$
(6)

Let the state variable of the system \(X = \left[ {\begin{array}{*{20}c} {z_{uf} } & {z_{ur} } & {z_{s} } & \varphi & {\dot{z}_{uf} } & {\dot{z}_{ur} } & {\dot{z}_{s} } & {\dot{\varphi }} \\ \end{array} } \right]^{T}\), The interference input is \(U = \left[ {\begin{array}{*{20}c} {z_{tf} } & {z_{tr} } \\ \end{array} } \right]^{T}\), The output is \(Y = \left[ {\begin{array}{*{20}c} {\ddot{z}_{uf} } & {\ddot{z}_{ur} } & {\ddot{z}_{s} } & {\ddot{\varphi }} & {z_{uf} - z_{tf} } & {z_{ur} - z_{tr} } & {z_{sf} - z_{uf} } & {z_{sr} - z_{ur} } \\ \end{array} } \right]^{T}\), Transform the above differential equations of motion into state space form:

$$ \begin{gathered} \dot{X} = AX + BU \hfill \\ Y = CX + DU \hfill \\ \end{gathered} $$
(7)

where A and B are the state and input matrices, respectively, and C and D are the output and direct transfer matrices, respectively. By building a four degree of freedom model of the semi vehicle in matlab/simulink, the vehicle parameters are shown in the table below (Table 1).

Table 1 Parameters required for vehicle simulation
Full size table

3 Kalman Filtering Algorithm

Kalman filtering consists of two main processes: the time update process (prediction) and the measurement update process (correction). The prediction process mainly uses the time updating equation to establish a priori estimates of the current state, project the values of the current state variables and the error covariance estimates forward in time, and construct a priori estimates for the next time state; the calibration process is responsible for the feedback, and uses the measurement updating equation to establish improved a posteriori estimates of the current state based on the a priori estimates from the prediction process as well as the current measurement variables [6].

Firstly, the state-space model of the above equation is discretized discrete by means of the Laplace transform.

$$ \begin{gathered} x\left( k \right) = Ax\left( {k - 1} \right) + Bw\left( k \right) \hfill \\ y\left( k \right) = Cx\left( k \right) + Dw\left( k \right) \hfill \\ \end{gathered} $$
(8)

where x(k) and x(k−1) are the state vectors at moments k and (k−1), respectively; and y(k) is the observation vector at moment k.

The specific algorithm flow is as follows [7]:

  1. a.

    Time updating process

State prediction equations:

$$ \hat{x}\left( {k|k - 1} \right) = A\hat{x}\left( {k - 1|k - 1} \right) + Bw\left( k \right) $$
(9)

Error covariance prediction:

$$ P\left( {k|k - 1} \right) = AP\left( {k - 1|k - 1} \right)AP^{T} + Q\left( k \right) $$
(10)

where Q(k) is the covariance of the system noise; P(k|k−1) is the propagation form of the covariance under the a priori state estimation, i.e. The time-updated expression for the covariance under the a priori state estimation.

  1. b.

    Measurement update process

Gain equations:

$$ K_{g} \left( k \right) = {{P\left( {k|k - 1} \right)C^{T} } \mathord{\left/ {\vphantom {{P\left( {k|k - 1} \right)C^{T} } {\left( {CP\left( {k|k - 1} \right)C^{T} + R} \right)}}} \right. \kern-0pt} {\left( {CP\left( {k|k - 1} \right)C^{T} + R} \right)}} $$
(11)

State update equations:

$$ x\left( {k|k} \right) = x\left( {k - 1|k - 1} \right) + K_{g} \left( k \right)\left[ {y\left( k \right) - Cx\left( {k|k - 1} \right)} \right] $$
(12)

Error covariance update expression:

$$ P\left( {k|k} \right) = \left[ {I - K_{g} \left( k \right)C} \right]P\left( {k|k - 1} \right) $$
(13)

4 LQR Control Based on State Estimation

Linear quadratic regulator (LQR) is one of the most fundamental optimal control problems for control problems and is commonly used to solve linear system control problems with Gaussian white noise inputs, i.e., to solve the optimal feedback control law problem that minimises a quadratic objective function [8]. It is mainly applicable to linear time-invariant systems as well as linear time-varying systems, and constitutes a linear dynamic feedback control law that is easy to compute and implement. The state space model of the passive suspension is transformed into a multivariate linear system state equation for the active suspension [9].

In the process of automobile suspension design, the performance index of the automobile usually refers to some parameters related to its safety and driving smoothness, which are mainly reflected in the tire dynamic displacement, body acceleration and suspension dynamic deflection. The linear-quadratic optimal control is used for the active suspension, in which the state variables of the suspension are all estimated using the Kalman algorithm, and the above parameters are selected as the performance evaluation indexes, and the integral values of their weighted squares can be obtained [10].

$$ \begin{aligned}J =& \mathop {\lim }\limits^{T \to \infty } \frac{1}{T}\int_0^T {\left[ {{q_1}{{\left( {{{\ddot z}_s}} \right)}^2} + {q_2}{{\left( {{z_{sf}} - {z_{uf}}} \right)}^2} + {q_3}{{\left( {{z_{sr}} - {z_{ur}}} \right)}^2}} \right.} \\&\left. { + {q_4}{{\left( {{z_{uf}} - {z_{tf}}} \right)}^2} + {q_5}{{\left( {{z_{ur}} - {z_{tr}}} \right)}^2}} \right]dt \end{aligned} $$
(14)

where q1, q2, q3, q4 and q5 are the weighting coefficients for body acceleration, front suspension dynamic deflection, rear suspension dynamic deflection, front tire dynamic displacement and rear tire dynamic displacement respectively.

Call the LQR toolbox in matlab, find the control rate [K] = lqr(A, B, Q, R, N), K is the optimal feedback control rate sought, change the value of q1, q2, q3, q4, q5, through repeated debugging until a good control effect occurs, to get q1 = 8000, q2 = 100, q3 = 100, q4 = 10, q5 = 10. The active suspension predictive control system was modeled and simulated, and the simulation results for each performance metric are shown in Fig. 2.

Fig. 2
Five-line graphs plot time on the horizontal axis labeled a to e with the following highest estimated values. Passive suspension (8.5, 8), L Q G based on Kalman filter (9.5, 67), L Q G based on Kalman filter (4.1, 40), L Q G based on Kalman filter (3.9, 39), and passive suspension (5.8, 29).

Comparison of simulation results for each performance index; a is the vertical acceleration of the car body; b is the dynamic deflection of the front suspension; c is the dynamic deflection of the rear suspension; d is dynamic displacement of front tires; e is dynamic displacement of rear tires

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According to the simulation verification results, compared with the previous passive suspension, the active suspension based on Kalman filter LQG control reduces the vertical acceleration of the vehicle by 35.36%, increases the dynamic deflection of the front and rear suspensions by 27.92% and 22.38% respectively, and increases the dynamic travel of the front and rear tires by 28.43 and 16.12%.

Although LQR control significantly reduces the vertical acceleration of the vehicle body, it reduces the bumpiness of the vehicle during driving and improves ride comfort. At the same time, the increase in suspension dynamic deflection also makes the suspension system more flexible during driving and better responsive to changes in the road surface. However, the optimization also sacrifices a portion of the tire travel, making the vehicle more bumpy and the riding experience worse when passing through off-road roads.

5 Conclusion

The Kalman filter algorithm is able to estimate the vibration state variables of special vehicles better, and also has good reliability under poor road conditions, which can be used for the active suspension control of special vehicles. In addition, based on the optimal control of the active suspension, the state variables estimated by Kalman filtering are used to replace the observed values of the sensors, which can satisfy the expected control effect within a certain permissible error range and reduce the installation of sensors while lowering the cost.