Chapter 2 Full-Car Model
2.7 Conclusions
Then, taking equations (2.21), (2.22), and (2.23) as the assistance, the vehicle rotational motion can be clearly expressed.
2.6 Full-Car Model Validation
Now, we can not verify the validation of this full-car model, as result of the enormous experimental data. The production of the detail experimental data is needed time and cost.
However, we are unable to find that this detail data is available. Therefore, the validation of this full car model will be verified by experimental data in future, if this detail data is in hand.
2.7 Conclusions
The developed nonlinear vehicle model, contained 21 system orders, is addressed in this chapter. Using Euler transformation, the relationship between dynamic frames is obtained, and then, sprung mass system and unsprung mass system can be incorporated. Additionally, we bring the road condition into the full-car model to obtain the vehicle dynamics induced by the road conditions and the vehicle roll angle with respect to the road surface. In this way, the full-car model presents more accurate simulation with the real vehicle. Furthermore, the derivation of this full-car model does not aim at some special kind of the vehicle. Therefore,
this methodology can be widely applied to four-wheel vehicles with different dynamic characteristics.
Chapter 3
System Observability of Full-Car Model
The states observability matrix can reveal the connections between system states and system outputs. Therefore, it is possible to employ this technique to discover the connections between each vehicle dynamics. Owing to the conventional observability matrix is inadequate for the vehicle rollover dynamics, we proposed a novel nonlinear observability matrix and, based on this matrix, the feasibility of neglecting vehicle pitch motions is discussed.
The unsuitability of the conventional observability matrix is described briefly in section 3.1. Under our demands, a novel observability matrix is proposed in section 3.2. Based on the simulation results, as shown in section 3.3, the integrated yaw-roll model is developed in section 3.4.
3.1 Nonlinear Observability Matrix
Most of the conventional observability matrix techniques discuss the system observability around equilibrium points [30]. They need lots of math derivations but are only applicable to a small operation range. The rollover incidence encounters a large roll angle variation and 21 system states. Therefore, it is impractical to construct a nonlinear observability matrix based on conventional methods.
Hahn J. and Edgar T. [11] proposed a new covariance matrix for the observability of nonlinear systems. This approach computes the observability grammian around system equilibrium points by applying perturbations on states initial conditions. And next, the observability grammian matrix corresponding to each initial condition is summed up, which makes it a covariance matrix in essence. This approach employs the concept of covariance matrix for the system observability. Unfortunately, it is only suitable for the small operation range.
3.2 Novel Observability Matrix along a Trajectory
Extended from the covariance matrix discussed previously [11], we proposed a novel system observability matrix that is applicable to a nonlinear system with a large operation range. In this approach, the observability grammian in [11] was replaced with the states covariance matrix. Furthermore, the calculation of states covariance matrix is performed along a trajectory, instead of at equilibrium points.
The proposed states covariance matrix along a trajectory can be calculated by the following steps.
1. Choose a trajectory of interest and along this trajectory, many distinct operation points are specified. These operation points are treated as the states initial conditions for simulation later on.
2. The perturbations are applied to each operation point as the initial conditions for numerical simulations. States covariance matrix can be calculated for each operation point.
3. Repeat the Step 2 for each operation point along the trajectory.
4. State covariance matrix obtained from each operation point is summed up and normalized for the final outcome.
This approach takes advantage of computer computation power to replace intensive math derivation, which makes it particularly suitable for a complicated nonlinear system with a wide operation range. However, the development of this method is not fully completed yet.
More theoretical work is still on the way to ensure its feasibility.
3.3 Negligence of Pitch Motions
Table 3.1 shows the covariance matrix of the system along a stable trajectory. When comparing the numbers shown in “roll angle” column, we find relative small values in pitch motion and vertical motion, which imply that these two motions have less effect on the roll angle. The vertical motion can not be neglected in this case because it does play an important role of the CG lifting in a rollover incidence. The simulation fails to indicate its importance because the trajectory in this simulation is a stable one and does not involve much vertical motion. Therefore, if the order reduction must be made to the vehicle system, the vehicle pitch motion should be firstly considered.
～ ε ε θ θ φ φ ～ x 0.0716 0.0503 -0.1112 -0.0343 0.0658 -0.0084 x 0.0072 0.0647 -0.0596 -0.0517 0.3201 -0.0288 y -0.2036 0.1784 0.1091 -0.0396 0.2200 0.0146 y -0.3204 -0.0339 0.1580 -0.0385 0.1203 0.0598 z 0.0563 0.1083 -0.8329 0.0077 0.0138 -0.0010 z -0.0156 -0.0115 0.0070 -0.8555 0.0267 0.0756 ω2 -0.2453 -0.1590 0.2494 0.0332 0.1266 0.0465 ω 3 -0.1213 -0.0261 0.0383 0.0238 0.1578 0.0867 ω4 -0.3126 -0.2553 0.2106 0.0315 -0.1380 -0.0782
H1 0.1301 0.1468 0.1392 -0.1149 -0.2464 0.1154 H2 -0.1569 -0.1761 0.3201 -0.0898 0.2468 -0.0638 H3 0.4227 0.3330 -0.2791 -0.0034 0.4001 0.0164 H4 -0.4187 -0.3284 0.2091 0.0288 -0.4052 -0.0023
Table 3.1 States covariance matrix in partial part.
(subscript 1: front-left side, 2: front-right side, 3: rear-right side, and 4: rear-left side)
3.4 Integrated Yaw-Roll Model
To obtain a simplified vehicle model, the vehicle pitch motion is neglected from the full-car model. The vehicle pitch motion (ω ), as shown in equation (2.4), contains many y terms other than Euler pitch angle (θ). However, we chose to neglect the Euler angle (θ), instead of the vehicle pitch angular velocity (ω ), for simplicity for now. From equations y (2.8), rotational dynamics of the full-car model can be deduced to the following.
( )
The simplified vehicle model, with equations (3.1) for rotational dynamics, is named
“integrated yaw-roll model” in this thesis. Additionally, the integrated yaw-roll model is considered as the real vehicle in this thesis. In the following section, the construction of the vehicle rollover prediction system will be based on this integrated yaw-roll model.
Chapter 4
Vehicle Rollover Prediction System
The proposed prediction system manages to incorporate an observer-based estimator.
However, it is extremely difficult to construct a nonlinear observer for the full-car model, even for the integrated yaw-roll model. Owing to that, we proposed a novel separated yaw-roll model to reduce the intensive math derivation in the subsequent observer design.
Furthermore, two observers work with suitable sensor measurements and compute in the ADI-like scheme. With this separated yaw-roll model and this switching observer scheme, we are able to estimate every vehicle states accurately. Additionally, the stability and convergence analysis of this switching scheme is also represented in this thesis.
As discussed before, the response of a dynamic system can be well described by a precise system model and associated initial conditions. In this thesis, we took the integrated yaw-roll model to accompany with the real-time vehicle states, which are estimated by the switching observer scheme. Therefore, we are able to correctly predict vehicle states in the future time.
Additionally, this prediction method presents a strong evidence for a rollover occurrence.
Section 4.1 represents the separated yaw-roll model. The organization of the switching observer scheme is introduced in section 4.2, and the stability and convergence of that is presented in the next section. The suitable sensor and the suitable nonlinear observer are chosen in section 4.4 and 4.5. In the last section, the procedure of this whole system will be described briefly.
4.1 Separated Yaw-Roll Model
The novel separated yaw-roll model was obtained by breaking the integrated yaw-roll model into two subsystems, while preserving all the dynamics in the integrated yaw-roll model. As shown in the integrated yaw-roll model, Appendix A, the lateral force is present in both yaw dynamics and roll dynamics. To avoid the same states appearing in two sub-models, the lateral dynamics is ascribed to one model and the resulting lateral acceleration is
considered as the input to another model. With these arrangements, the integrated yaw-roll model is readily to be broken into two sub-models which are named “yaw model” and “roll model” in this thesis.
a) b)
(body frame)
(wheel frame)
B
ex B
ey
W
ex W
ey
(wheel frame) eWx
W
ey
(body frame)
(wheel frame)
B
ex B
ey
W
ex W
ey
Figure 4.1 Comparison of the full-car model and separated yaw-roll model. a) diagram of the full-car model (in the light gray) and the yaw model (in the dark gray) b) diagram of the full-car model (in the light gray) and the roll model
(in the dark gray)
4.1.1 Vehicle Yaw Model
The vehicle yaw model contains four degree-of-freedoms, which are longitudinal dynamics, lateral dynamics, yaw dynamics and wheel angular rate. The steering wheel angle, motor torque and brake torque, which originally considered as inputs to the integrated yaw-roll model, are treated as inputs to the yaw model. From some assumptions shown as followings, we can extract the vehicle yaw model from the integrated yaw-roll model.
z As discussed before, the Euler pitch angle (θ) and Euler pitch rate ( ) are neglected.
θ
z The vehicle states associated with the vehicle roll model are considered as inputs to the vehicle yaw model.
z With neglecting the vehicle pitch motion, we are able to consider only two tires (front and rear tires) in the integrated yaw-roll model for the vehicle yaw modeling.
z From the 3rd assumption, we consider to set two imaginary tires, which locate individually in the middle of axles, as shown in figure 4.1a.
z From the 4th assumption, variables of imaginary tires, such as the tire slip ratio, tire slip angle, tire steering angle, etc., are set the mean value of left and right tires, which originate from the full-car model.
z From the 4th assumption, the geometry parameter, the tread width of the vehicle, is set as zero. (sb1 = sb2 =0)
According to above-mentioned assumptions and equations (2.40) and (3.1), the dynamic equations of the yaw model are rearranged as follows:
( ) ( )
where the subscripts of tire forces ( ), “1” and “2”, represent the front and rear of two imaginary tires. Then, the derivation of the yaw model is competed.
Fi
4.1.2 Vehicle Roll Model
The roll model contains three degree-of-freedoms, which are vertical dynamics, roll dynamics and suspension dynamics. The lateral acceleration is treated as the input to the roll model.
z As discussed before, the Euler pitch angle (θ) and Euler pitch rate ( ) are neglected.
θ
z The vehicle states associated with the vehicle yaw model are considered as inputs to the vehicle roll model.
z With neglecting the vehicle pitch motion, we are able to consider only two tires (left and rear tires) in the integrated yaw-roll model for the vehicle roll modeling.
z From the 3rd assumption, we consider to set two imaginary tires, which locate in the middle of right end and in the middle of left end, as shown in figure 4.1b.
z From the 4th assumption, the nonlinear spring coefficient of the vehicle roll model is modified two times stiffer than that of the full-car model.
z From the 4th assumption, the geometry parameter, the wheelbase length of the vehicle, is set as zero. (l1 = l2 =0)
According to above-mentioned assumptions and equations (2.40) and (3.1), the dynamic equations of the roll model are rearranged as follows:
( )
∑
+=
− −
=
road z zi vehicle
x y z
x x
F F z
m
I I I I M
ε φ φ
φ 2sin cos
(4.2)
with 1
( ) (
1 2)
1 2
1 2 cos 2sin z 2 cos 2sin ( ) y y
z
x F F Z z F F
M sb h sb h ⎟+ − − +
⎠⎞
⎜⎝
⎛− +
⎟+
⎠⎞
⎜⎝
⎛ +
= φ φ φ φ
where the subscripts of tire forces ( ), “1” and “2”, represent the left and right of two imaginary tires. Then, the derivation of the roll model is competed.
Fi
Figure 4.2 Dynamic responses of two models (the integrated yaw-roll model and the separated yaw-roll model) in the general case
Figure 4.3 Dynamic responses of two models (the integrated yaw-roll model and the separated yaw-roll model) in the rollover case
4.1.3 Separated Yaw-Roll Model Validation
The validation of the separated yaw-roll model should be verified by the experimental data. As discussed before, we are unable to find the available data of the vehicle. Owing to that, we assume the integrated yaw-roll model has the accurate dynamic maneuver. Then, we design two simple driving maneuvers to compare with the dynamic response of the integrated yaw-roll model and the separated yaw-roll model. The comparison of simulation results will provide the evidence for the feasibility of the separated yaw-roll model.
In figure 4.2 and 4.3, the plot is arranged in the following order: longitudinal velocity in the upper left, lateral velocity in the upper middle, vertical velocity in the upper right, yaw rate in the lower left, roll rate in the lower middle, and roll angle in the lower right.
Furthermore, the integrated yaw-roll model outputs are drawn in solid-blue lines, and the separated yaw-roll model outputs are drawn in dash-green lines.
As shown in figure 4.2 and 4.3, two dynamic responses are going in the same maneuver.
Therefore, with these two simulation results, we can verify the validation of the separated yaw-roll model. Additionally, in the next section, we will use mathematical proof to show the stability of the separated yaw-roll model, and obtain the other evidence for the validation.
4.2 Switching Observer Scheme
After obtaining two subsystems for the separated yaw-roll model, the observer is constructed for each subsystem, respectively. These two observers work under a “switching scheme” in each simulation time step, which means, holding one subsystem when the other subsystem is doing the calculation. Although the separated yaw-roll model and integrated yaw-roll model possess the same system dynamics, the observer construction is indeed a lot simpler for the separated yaw-roll model accompanied with switching observer scheme.
4.2.1 Error Source
As discussed before, the proposed separated yaw-roll model is doing a switching scheme similar to conventional ADI methods. Therefore, same as the ADI method, the pertinent error can be attributed to two sources: switching time step error and round-off error [14].
The switching time step error mainly comes from the deviation generated by the high frequency content of the dynamic model, and this deviation propagates from one computation phase to another computation phase [14]. The round-off error is due to the discretization error and the digitization error [25]. The discretization error comes from the numerical approximation of a continuous-time function, which is needed for the computer numerical processing, and the digitization error comes from the finite length of digits. Therefore, the switching time step error largely depends on the switching time between two phases, and the round-off error largely depends on the computer hardware setup and the numerical method utilized to approximate a continuous-time differential equation.
In this thesis, we discussed the stability of this switching computation scheme for two cases. One employed “explicit Euler method” and the other one employed “Runge-Kutta method” as the accompanied numerical methods. At the end, the “Runge-Kutta method” is adopted for the vehicle rollover computer simulation in this thesis.
4.2.2 Preliminaries for the Stability Analysis of Switching Computation Scheme
Consider a nonlinear ordinary differential equation shown below.
(4.3)
(x(t)
F
x= )
where x∈ℜn, F:ℜn →ℜn, and 0≤t≤T . The above equation can stand for many nonlinear autonomous systems at the finite time. By separating the states into two groups, we can break down a complicated nonlinear differential equation into two sets of differential equations.
( )x(t) F1(x1(t),x2(t)) F2(x1(t),x2(t))
F = + (4.4)
where , , , and . Instead of using equation (4.3), the exact states values can be obtained from the following equation. time interval. One thing to be noted, up to this point, we have not made any approximation or simplification for the differential equation calculation.
In this thesis, we solve equation (4.5) by an ADI-like method in order to estimate the system (4.3). Following the ADI-like method, two simpler functions can be seen as two space dimensions, and operated in the switching scheme to simulate this system. Additionally, this method reveals that the computational process activates alternatively in one dimension, and is called the locally one-dimensional (LOD) method. The link between the ADI-like method and the LOD method can govern step-by-step convergence stability for this system.
4.2.3 Stability Analysis for “Explicit Euler Method” Approximation
In order for computers to calculate above differential equations, we need to approximate the continuous-time function and integral operation by suitable numerical methods. Here, we use a simple “explicit Euler method” to approximate equation (4.5) as to focus on the switching computation scheme for the stability analysis work. At the end of this section, we shall show that the local deviation between solutions, obtained from explicit Euler method accompanied with switching computation scheme and from explicit Euler method without switching scheme, decreases as time goes.
Proof:
By doing the Taylor series expansion on equation (4.5), we have the following equation.
(4.6)
Therefore, by neglecting the high order terms, we can have the following equation for the
“explicit Euler method” approximation.
( )
approximated by explicit Euler method. Then, the perturbed switching computational process for equation (4.7) can be written as follows as to obtain the solutions as close totn
where δ represents the perturbations during computation, and which mainly comes from i,n the round-off error at time t =tn, xˆi,n is the estimated value of xi,n at time t =tn by switching computation scheme. Subtracting (4.7) from (4.8), we have the following,
( ) ( )
where the error (εi) is defined as the difference between the estimated state ( ) and the state (
Therefore, by substituting (4.10) into (4.9), the error ε1,n+1 can be rewritten as follows,
Again, neglecting the high order terms of time step, we can rearrange (4.11) and (4.13) into a matrix form.
To find out the bound for the stability, the following matrix norm is introduced here.
n n
n
n A E D
E +1 = ⋅ +τ⋅ (4.15)
Therefore, from Cauchy-Schwarz inequality, the local error bound with perturbations can be derived from (4.15),
D
where D is the maximum bound of perturbations, and Dn ≤ . Because the perturbation D (δ ) has noting to do with the error ( ) propagation, we can find out a proper switching time step (
En
τ ) to facilitate An 2 <1. Therefore, the switching computation scheme can be stable.
The proof shown above only reveals the stability of this switching computation scheme, but not the convergence properties [14]. That is to say, the proof so far only guarantees that the error between switching computation value and discrete time system, shown in equation (4.6), does not grow. However, it does not say how accurate the estimated states xi,n
can be close to xi,n, let along , which is the exact states value of the differential equation. It is obvious that for
n
xi, n
xi,
to be close to , the solution obtained from numerical approximation method (
n
xi,
n
xi, ) must be close to . and as accurate as possible. For that reason, we use
“Runge-Kutta mthod” to approximate a continuous-time differential equation. The related stability issues are discussed in the following section.
n
xi,
4.2.4 Stability Analysis for “Runge-Kutta Method” Approximation
In this section, we provide the stability analysis of the switching scheme based on the Runge-Kutta method, which is adopted for the vehicle rollover simulations in this thesis. As shown in previous research that a high order Runge-Kutta method can guarantee the convergence of a nonlinear differential equation, we then assume no discretization error for the following proof work.
Proof:
By assuming the solution obtained by the Runge-Kutta method can be arbitrary close to the integral operation, the switching computational process can be written as follows,
(4.17) perturbations. In the above equation, we can see that when one state is active, the other state is inactive, and vice verse. Then, subtracting (4.5) from (4.17),
(4.18)
where the error (εi) is defined the difference between the estimated state ( ) and the exact state ( ).The last two terms of equation (4.18) can be processed as follow,
xˆi
where ε1(i ),n represents the intermediate states error during the switching scheme, for . From the dynamics viewpoint,
2
~
=1
i ε11,n represents the convergent rate for the states error, and ε represents for the computation accuracy or perturbations. Therefore, we can 12,n separate errors into two terms, which represent ε11,n and ε12,n in (4.19), and discuss, respectively.
Firstly, using the mean value theorem, we can obtain more information of the error Runge-Kutta method can obtain the numerical solution close to the exact solution of above equations, and linear respect to the time step (τ ), we can rewrite (4.20) as follows,
( )
where represents the exact solution of the integral operation shown above, which is
where represents the exact solution of the integral operation shown above, which is