Estimation of Electric Drive Vehicle Sideslip Angle Based on EKF

Abstract

The sideslip angle is the most crucial state variable in the stability control system for vehicles, particularly in the electric drive vehicle field. But recently, there is no other way that could obtain the sideslip angle directly with low cost. Due to the characteristic of strongly nonlinear vehicle system, this paper discusses how to evaluate the sideslip angle with the extended Kalman filter (EKF) algorithm and build both double-track kinematics model and tire model, and then, we proposed the conception of nonlinear state space description and analyzed the result with a simulation method ultimately.

67.1 Introduction

Recently, most of researchers focus on the study of vehicle state and parameter estimation, and clearly, the accuracy of state and parameter estimation affects the control performance of vehicle control system.

The sideslip angle is the most crucial state variable in the stability control system for vehicles. At present, there are two kinds of method for achieving the sideslip angle [1–4]: The first one depends on the GPS sensor signal for detection, and the other one employs estimated method which adopts the principle of kinematics and the kinetic. The former is not suitable for general usage because of high cost; therefore, the latter becomes the majority of the recent studies.

Taking eight-wheel independent drive vehicle as research object, this paper develops a double-track kinematics model, makes real-time estimation for sideslip angle by using EKF algorithm, and then verifies the estimated result with real-time simulation method.

67.2 Modeling of Dynamics

67.2.1 Vehicle Dynamics Model

Assuming that the longitudinal speed makes no changes on the moment of vehicle turning, without taking into consideration longitudinal acceleration and deceleration, we can develop a double-track kinematics model, which contains transverse movement and yawing motion, as shown in Fig. 67.1.

Fig. 67.1

Double-track kinematics model of vehicle with 2 DOF

Double-track kinematics model of vehicle with 2 DOF is expressed as follows:

$$ \begin{aligned} a_{y} & = \frac{1}{m}(F_{y11} \cos \delta_{11} + F_{x11} \sin \delta_{11} + F_{y12} \cos \delta_{12} + F_{x12} \sin \delta_{12} + F_{y21} \cos \delta_{21} \\ & \quad + F_{x21} \sin \delta_{21} + F_{y22} \cos \delta_{22} + F_{x22} \sin \delta_{22} + F_{y31} + F_{y32} + F_{y41} + F_{y42} ) \\ \end{aligned} $$

(67.1)

$$ \begin{aligned} a_{x} & = \frac{1}{m}( - F_{y11} \sin \delta_{11} + F_{x11} \cos \delta_{11} - F_{y12} \sin \delta_{2} + F_{x12} \cos \delta_{12} - F_{y21} \sin \delta_{21} \\ & \quad + F_{x21} \cos \delta_{21} - F_{y22} \sin \delta_{22} + F_{x22} \sin \delta_{22} + F_{x31} + F_{x32} + F_{x41} + F_{x42} ) \\ \end{aligned} $$

(67.2)

$$ \begin{aligned} \dot{\gamma } & = \frac{1}{{I_{z} }}[a(F_{y11} \cos \delta_{11} + F_{x11} \sin \delta_{11} + F_{y12} \cos \delta_{12} + F_{x12} \sin \delta_{12} ) + b(F_{y21} \cos \delta_{21} \\ & \quad + F_{x21} \sin \delta_{21} + F_{y22} \cos \delta_{22} + F_{x22} \sin \delta_{22} ) - c(F_{y31} + F_{y32} ) - d(F_{y41} + F_{y42} ) \\ & \quad + \frac{l}{2}(F_{x12} \cos \delta_{12} + F_{x22} \cos \delta_{22} + F_{x32} + F_{x42} - F_{x11} \cos \delta_{11} - F_{x21} \cos \delta_{21} \\ & \quad - F_{x31} - F_{x41} )] \\ \end{aligned} $$

(67.3)

$$ \begin{aligned} \dot{\beta } & = \frac{1}{mV}[F_{y11} \cos (\beta - \delta_{11} ) + F_{y12} \cos (\beta - \delta_{12} ) + F_{y21} \cos (\beta - \delta_{21} ) \\ & \quad + F_{y22} \cos (\beta - \delta_{22} ) - F_{x11} \sin (\beta - \delta_{11} ) - F_{x12} \sin (\beta - \delta_{12} ) \\ & \quad - F_{x21} \sin (\beta - \delta_{21} ) - F_{x22} \sin (\beta - \delta_{22} ) + (F_{y31} + F_{y32} + F_{y41} + F_{y42} ) \\ & \quad \cos \beta - (F_{x31} + F_{x32} + F_{x41} + F_{x42} )\sin \beta ] - \gamma \\ \end{aligned} $$

(67.4)

$$ \begin{aligned} \dot{V} & = \frac{1}{m}[F_{y11} \sin (\beta - \delta_{11} ) + F_{y12} \sin (\beta - \delta_{12} ) + F_{y21} \sin (\beta - \delta_{21} ) \\ & \quad + F_{y22} \sin (\beta - \delta_{22} ) + F_{x11} \cos (\beta - \delta_{11} ) + F_{x12} \cos (\beta - \delta_{12} ) \\ & \quad + F_{x21} \cos (\beta - \delta_{21} ) + F_{x22} \cos (\beta - \delta_{22} ) + (F_{y31} + F_{y32} + F_{y41} + F_{y42} ) \\ & \quad \sin \beta + (F_{x31} + F_{x32} + F_{x41} + F_{x42} )\cos \beta ] \\ \end{aligned} $$

(67.5)

where $F_{xij}$ is the tire longitudinal forces, $F_{yij}$ is the lateral tire forces, m is the vehicle mass, and $I_{z}$ is the yaw moment of inertia. The forward velocity V, steering angle $\delta_{ij}$, yaw rate γ, and the vehicle slip angle β are then used to calculate the tire slip angles $\alpha_{ij}$, where

$$ \left\{ {\begin{array}{*{20}l} {\alpha_{1} = \delta_{1} - (\beta + a\gamma /V_{x} )} \hfill \\ {\alpha_{2} = \delta_{2} - (\beta + a\gamma /V_{x} )} \hfill \\ {\alpha_{3} = \delta_{3} - (\beta + b\gamma /V_{x} )} \hfill \\ {\alpha_{4} = \delta_{4} - (\beta + b\gamma /V_{x} )} \hfill \\ {\alpha_{5} = \alpha_{6} = - \beta + c\gamma /V_{x} } \hfill \\ {\alpha_{7} = \alpha_{8} = - \beta + d\gamma /V_{x} } \hfill \\ \end{array} } \right. $$

(67.6)

67.2.2 Tire Model

The Dugoff tire model is chosen for our study. It is expressed as follows:

$$ \left\{ {\begin{array}{*{20}l} {F_{yij} = - C_{\alpha ij} \tan \alpha_{ij} f(\lambda )} \hfill \\ {f(\lambda ) = \left\{ {\begin{array}{*{20}l} {(2 - \lambda )\lambda ,} \hfill & {{\text{if}}\,\lambda < 1} \hfill \\ {1,} \hfill & {{\text{if}}\,\lambda \ge 1} \hfill \\ \end{array} } \right.} \hfill \\ {\lambda = \frac{{\mu_{\hbox{max} } F_{zij} }}{{2C_{\alpha ij} \left| {\tan \alpha_{ij} } \right|}}} \hfill \\ \end{array} } \right. $$

(67.7)

The model describes a tire model with the characteristic of nonlinear lateral force [5] and applies less parameter, where $C_{\alpha ij}$ is the lateral stiffness and $F_{zij}$ is the normal load on the tire.

When the vehicle sideslip angle changes, a lateral tire force is created with a time lag. This transient behavior of tires can be formulated using a relaxation length δ [6], and the dynamic lateral forces can be written as follows:

$$ \dot{F}_{yij} = \frac{V}{{\delta_{ij} }}( - F_{yij} + \bar{F}_{yij} ) $$

(67.8)

where $\bar{F}_{yij}$ is calculated from the quasi-static Dugoff tire model.

67.3 Stochastic State Space Representation

The nonlinear stochastic state space representation of the system described in the previous section is given as follows:

$$ \left\{ {\begin{array}{*{20}l} {\dot{X}(t) = f(X(t),U(t)) + w(t)} \hfill \\ {Y(t) = h(X(t),U(t)) + v(t)} \hfill \\ \end{array} } \right. $$

(67.9)

The input vector U comprises the steering angle and the normal forces

$$ \begin{aligned} U & = [\delta_{11} ,\delta_{12} ,\delta_{21} ,\delta_{22} ,F_{z11} ,F_{z12} , \ldots ,F_{z42} ]^{T} \\ & = [u_{1} ,u_{2} ,u_{3} ,u_{4} ,u_{5} ,u_{6} , \ldots ,u_{12} ]^{T} \\ \end{aligned} $$

(67.10)

The measure vector Y comprises yaw rate, vehicle velocity, and longitudinal and lateral accelerations

$$ Y = [\gamma ,V,a_{x} ,a_{y} ]^{T} = [y_{1} ,y_{2} ,y_{3} ,y_{4} ]^{T} $$

(67.11)

The state vector X comprises yaw rate, vehicle velocity, sideslip angle at the COG, lateral forces, and longitudinal tire forces.

$$ \begin{aligned} X & = [\gamma ,V,\beta ,F_{y11} ,F_{y12} , \ldots ,F_{y42} ,F_{x11} ,F_{x12} , \ldots ,F_{x42} ]^{T} \\ & = [x_{1} ,x_{2} ,x_{3} ,x_{4} ,x_{5} , \ldots ,x_{12} ,x_{13} , \ldots ,x_{19} ]^{T} \\ \end{aligned} $$

(67.12)

The process and measurement noise vectors ω and v, respectively, are assumed to be white, zero mean, and uncorrelated.

The particular nonlinear function $f(.)$ of the state equations is given by

$$ \left\{ {\begin{array}{*{20}l} {f_{1} = \frac{1}{{I_{z} }}[a(x_{4} \cos \mu_{1} + x_{12} \sin \mu_{1} + x_{5} \cos \delta_{12} \mu_{2} + x_{13} \sin \mu_{2} )} \hfill \\ \qquad{ + b(x_{6} \cos \mu_{3} + x_{14} \sin \mu_{3} + x_{7} \cos \mu_{4} + x_{15} \sin \mu_{4} )} \hfill \\ \qquad{ -\, c(x_{8} + x_{9} ) - d(x_{10} + x_{11} ) + \frac{l}{2}(x_{13} \cos \mu_{2} + x_{15} \cos \mu_{4} } \hfill \\ \qquad{ + x_{17} + x_{19} - x_{12} \cos \mu_{1} - x_{14} \cos \mu_{3} - x_{16} - x_{18} )];} \hfill \\ {f_{2} = \frac{1}{m}[x_{4} \sin (x_{3} - \mu_{1} ) + x_{5} \sin (x_{3} - \mu_{2} ) + x_{6} \sin (x_{3} - \mu_{3} )} \hfill \\ \qquad {+ x_{7} \sin (x_{3} - \mu_{4} ) + x_{12} \cos (x_{3} - \mu_{1} ) + x_{13} \cos (x_{3} - \mu_{2} )} \hfill \\\qquad { + x_{14} \cos (x_{3} - \mu_{3} ) + x_{15} \cos (x_{3} - \mu_{4} ) + (x_{8} + x_{9} + x_{10} + x_{11} )} \hfill \\ \qquad {\sin x_{3} + (x_{16} + x_{17} + x_{18} + x_{19} )\cos x_{3} ];} \hfill \\ {f_{3} = \frac{1}{{mx_{2} }}[x_{4} \cos (x_{3} - \mu_{1} ) + x_{5} \cos (x_{3} - \mu_{2} ) + x_{6} \cos (x_{3} - \mu_{3} )} \hfill \\\qquad { + x_{7} \cos (x_{3} - \mu_{4} ) + x_{12} \sin (x_{3} - \mu_{1} ) + x_{13} \sin (x_{3} - \mu_{2} )} \hfill \\ \qquad {+ x_{14} \sin (x_{3} - \mu_{3} ) + x_{15} \sin (x_{3} - \mu_{4} ) + (x_{8} + x_{9} + x_{10} + x_{11} )} \hfill \\ \qquad {\cos x_{3} + (x_{16} + x_{17} + x_{18} + x_{19} )\sin x_{3} ] - x_{1} ;} \hfill \\ {f_{4} = \frac{{x_{2} }}{{\sigma_{11} }}( - x_{4} + \bar{F}_{y11} (\alpha_{11} ,\mu_{5} ));} \hfill \\ {f_{5} = \frac{{x_{2} }}{{\sigma_{12} }}( - x_{5} + \bar{F}_{y12} (\alpha_{12} ,\mu_{6} ));} \hfill \\ \qquad \ldots \hfill \\ {f_{11} = \frac{{x_{2} }}{{\sigma_{42} }}( - x_{11} + \bar{F}_{y42} (\alpha_{42} ,\mu_{12} ));} \hfill \\ {f_{12} = 0;} \hfill \\ \qquad \ldots \hfill \\ f_{19} = 0 \hfill \\ \end{array} } \right. $$

(67.13)

The observation function $h(.)$ is given by

$$ \left\{ {\begin{array}{*{20}l} {h_{1} = x_{1} ;} \hfill \\ {h_{2} = x_{2} ;} \hfill \\ {h_{3} = \frac{1}{m}( -\, x_{4} \sin u_{1} + x_{12} \cos u_{1} - x_{5} \sin u_{2} + x_{13} \cos u_{2} } \hfill \\ \qquad{ - x_{6} \sin u_{3} + x_{14} \cos u_{3} - x_{7} \sin u_{4} + x_{15} \sin u_{4} } \hfill \\ \qquad { + x_{16} + x_{17} + x_{18} + x_{19} )} \hfill \\ {h_{4} = \frac{1}{m}(x_{4} \cos u_{1} + x_{12} \sin u_{1} + x_{5} \cos u_{2} + x_{13} \sin u_{2} } \hfill \\ \qquad{+ x_{6} \cos u_{3} + x_{14} \sin u_{3} + x_{7} \cos u_{4} + x_{15} \sin u_{4} } \hfill \\ \qquad {+ x_{8} + x_{9} + x_{10} + x_{11} )} \hfill \\ \end{array} } \right. $$

(67.14)

67.4 EKF Algorithm

The first-order EKF is presented as follows.

• (a) Initialization:

  The initial state and the initial covariance are determined by

  $$ \begin{aligned} \bar{X}_{0} & = E[X_{0} ], \\ p_{0} & = E[(X_{0} - \bar{X}_{0} )(X_{0} - \bar{X}_{0} )^{T} ] \\ \end{aligned} $$

  (67.15)

• (b) Time Update:

  The prediction of the state is given by

  $$ \bar{X}_{{k\left| {k - 1} \right.}} = f(\bar{X}_{{k - 1\left| {k - 1} \right.}} ,U_{k} ) $$

  (67.16)

  The predicted covariance is computed as

  $$ P_{{k\left| {k - 1} \right.}} = AP_{{k - 1\left| {k - 1} \right.}} A^{T} + Q $$

  (67.17)

• (c) Measurement Update:

  The filter gain is calculated by

  $$ K_{k} = P_{{k\left| {k - 1} \right.}} H^{T} [HP_{{k\left| {k - 1} \right.}} H^{T} + R]^{ - 1} $$

  (67.18)

  The state estimation is determined by

  $$ \bar{X}_{k\left| k \right.} = \bar{X}_{{k\left| {k - 1} \right.}} + K_{k} [Y_{k} - h(\bar{X}_{{k\left| {k - 1} \right.}} )] $$

  (67.19)

  The estimated covariance is

  $$ P_{k\left| k \right.} = [I - K_{k} H]P_{{k\left| {k - 1} \right.}} $$

  (67.20)

  $A_{k}$ and $H_{k}$ are the process and measurement Jacobians at step k of the nonlinear equations around the estimated states.

  $$ \left\{ {\begin{array}{*{20}l} {A_{k} = \frac{{\partial f(\bar{X}_{{k - 1\left| {k - 1} \right.}} ,U_{k} ,0)}}{\partial X}} \hfill \\ {H_{k} = \frac{{\partial h(\bar{X}_{{k\left| {k - 1} \right.}} ,0)}}{\partial X}} \hfill \\ \end{array} } \right. $$

  (67.21)

67.5 Simulation

• (a) High-speed hunting driving in good road condition

  Fig. 67.2

  

  Sideslip angle

• (b) Low-speed hunting driving in good road condition

  Fig. 67.3

  

  Sideslip angle

• (c) Hunting driving in low-friction road condition

  Fig. 67.4

  

  Sideslip angle

In Figs. 67.2, 67.3 and 67.4, they show that the estimated value of sideslip angle is basically in accordance with that of simulation under the three typical motions.

67.6 Conclusion

This paper applies EKF filter algorithm for evaluating the sideslip angle of vehicles, chooses Dugoff tire model which is based on the double-track kinematics model, and then extends a nonlinear state space for the filter estimation. The simulation result shows that the filter method using EFK would have better performance for vehicle sideslip angle.