3. APPROXIMATION OF THE LATERAL STABILITY REGIONS AND ITS APPLICATION METHODS FOR VEHICLE CONTROLAND ITS APPLICATION METHODS FOR VEHICLE CONTROL
3.1 Approximation of lateral stability regions for passenger cars
3. APPROXIMATION OF THE LATERAL STABILITY REGIONS
nonlinear eects of the vehicle, the denition of stability can be dicult. Unstable motion of the vehicle by various factors can be caused, e.g., sudden change in the adhesion coecient, insucient lateral force at large lateral slip. These factors must be taken into account during the control design in order to guarantee the motion stability of the vehicle even in risky trac situations. From control engineering aspect, the criterion of stable vehicle motion is in relation with the performance of the vehicle on the motion, as it is illustrated by the following examples.
• It can dicult to determine the unstable motion of the vehicle. Due to nonlin-earities in the lateral dynamics, the stability regions are generally computed on constant longitudinal velocities, see e.g. [67, 68]. Nevertheless, sometimes the unstable motion of the vehicle leads to a signicant reduction in the ve-locity, at which the stability of the vehicle is restored. Thus, if the stability of the vehicle is evaluated depending on the velocity, the scenario is only locally unstable, but globally stable.
• Moreover, the increasing error in the path tracking of the autonomous vehicle is caused by deciencies in the lateral control. In this case the error can be handled as a performance problem. Performance level depend also on the designed controller, not only on the vehicle dynamics itself.
These examples illustrate that it may be dicult to nd an appropriate criterion to evaluate stability in the context of vehicle dynamics with nonlinearities. The stabil-ity of motion is described by the original work of Lyapunov, see [69]. It motivates that stability is recommended to interpret as a character of the motion. Thus, in this section the motion of the vehicle based on collected data is analyzed.
The viewpoint of analyzing vehicle motion with data-driven tools yields a de-cision tree based on collected data. The data is provided by the sensors of the autonomous vehicles, such as inertial and gyro sensors, GPS velocity measurement and wheel speed sensors. Moreover, the data-driven analysis contains scenarios which are considered to be acceptable or unacceptable from the viewpoint of the path tracking of the autonomous vehicle. These scenarios are called good or bad instances in the dataset. The purpose of the decision tree generation is to nd the set of relations with which the a current scenario can be classied as acceptable or unacceptable. Each scenario requires a denition of a criterion to be acceptable.
The approach of this thesis is based on the idea that the motion of the vehicle is generally acceptable for the human passengers in the linear region of the tyre force characteristics. In this case the side-slip angle of the axles can provide information about the characteristics of the motion. Thus, the dened criterion expresses the similarity between the current side-slip of the front axle(1 +α1) and the expected side-slip based on the linear formulation of the vehicle [FNG18a]. The selection criterion is similar to that, which has been used to road surface estimation (2.49).
This criterion scales the nonlinear behavior of vehicle by comparing the current
3.1. Approximation of lateral stability regions for passenger cars 51
measurement to a linear model:
−ε1 < |1 +αf|
|1 +δ−β−lfvψ˙
x |
−1≤ε1, (3.1)
where ε1 is an experimentally dened parameter, l1 is the distance between the center of gravity of the vehicle and the front axle. ψ˙ denotes the yaw rate, β is the vehicle side-slip vx is longitudinal velocity and δ is the steering angle. This condition expresses that an instance is said to be 'acceptable', if (3.1) is fullled. It approximates the stable states of the vehicle. Otherwise, the instance is classied as 'unacceptable', which is related to the unstable state of the vehicle. Thus, condition (3.1) results in the datasetS1 with the classication of 'acceptable' or 'unacceptable' for the stability set approximation.
In the followings, a machine learning-based algorithm is proposed for approxi-mating the lateral stability regions of the vehicle. This method is a pure data-driven approach, which means it uses only the available measured signals from the onboard system. The main advantage of this approach is that it is more suitable for the nonlinear dynamics of the vehicle, which appears at high velocities.
In the following a brief description of the results of the stability set approximation is presented using the previously introduced C4.5 method, see Subsection 2.3.3. The approximation is based on the assumption that criterion (3.1) is able to distinguish the 'acceptable' and 'unacceptable' states, which provides information about the stability of the vehicle. However, the dened condition is not identical with a vehicle stability criterion, as presented above. Although the class of 'acceptable' instances contains only stable states, it can provide a conservative inner approximation of the stability set. The details of the set computation are found in [FNG18a].
Illustration of the analysis results
The purpose of this section is to demonstrate the reachability sets of the lateral vehicle model, which are computed through the machine learning algorithm. In the analysis the data-mining WEKA software is used, in which C4.5 algorithm has been implemented [70].
The attributes of the instances are αf, αr slip at the front and rear wheels, β side slip of the vehicle, ψ˙ yaw-rate, vx longitudinal velocity, µ adhesion coecient and C class. The C class has two values, i.e., acceptable and unacceptable, and the instances are classied by the algorithm. During the analysis the training set contains approximately 1.2 million instances, while the test set for the validation has 2million members. In the example a mid-size passenger car is used.
The generated trees are evaluated by the cross-validation technique, the results can be found in Table 3.1. The rst column in Table 3.1 shows the minimum number of instances which are contained in a leaf. The second column illustrates the percentage of the correctly classied instances. The sizes of the produced trees are in the last column. Note that the increasing number of the minimum objects
decreases both the percentage of the correctly classied instances and the sizes of the trees.
Tab. 3.1: Relationship between the tree size and the object number Min. Objects Correctly Classied Inst. Size of Tree
2 99.7343% 2431
10 99.6426% 1339
100 99.2948% 315
500 98.9136% 97
1000 98.7037% 61
5000 98.1892% 17
In the following, the classication with minimum 500 objects is used, because this object number has a reasonable percentage of correctly classied instances.
Moreover, the generated tree is suciently small to be used for further analyses. In Figure 3.1 the instances in the test set which are classied as 'good' are illustrated.
Note that the results of the decision tree appropriately cover the test set.
-8 -6 -4 -2 0 2 4 6
f (deg) -4
-3 -2 -1 0 1 2 3 4
r (deg)
Test set C4.5 algorithm
Fig. 3.1: Results of the decision tree
Figure 3.2 shows the results of the decision tree and the classied test sets in the plane ofαf and αr at dierent velocities. The two slip attributes have high impacts on the resulting decision tree, which shows that the calculated sets t well. Note that the sizes of the sets become larger with increasing velocity. It means that the vehicle can reach larger regions of slips at high velocities. This tendency is conrmed by the experience in vehicle dynamics.
Figure 3.3 illustrates the results of the classication and the test sets in the plane of yaw rate and side slip at dierent velocities, in which the sets also t well. These attributes have lower impacts on the logic relations in the decision tree. The regions
3.1. Approximation of lateral stability regions for passenger cars 53
20 40 60
-5 80 100
vx (km/h) 120
f (deg)
0 5
r (deg) 5 -5 0
Test set C4.5 algorithm
Fig. 3.2: αf and αr sets depending on velocityvx
of reachable β and ψ˙ increase depending on the longitudinal velocity, similarly to the tendency at the lateral slipsαf, αr, see Figure 3.2.
30 40 50 60
5 70 80 90
v x (km/h)
100 110
(deg)
0 50
-5 -50 0
Test set C4.5 algorithm
Fig. 3.3: ψ˙ and β sets depending on velocity vx
In the following, the impact of the adhesion coecient on the reachability regions of the vehicle model is illustrated. Figure 3.4 shows the regions of the slips at dierent adhesion coecients, in which the velocity of the vehicle is xed atvx = 90 km/h. Note that the illustrated sets become smaller at high adhesion coecients.
The reason for this behavior is that the adhesion coecient highly inuences the lateral forces of the wheels. At high µ the small slip angle generates high lateral force, while at smallµ the higher slip angle induces high lateral force.
Figure 3.5 shows the sets ofψ˙ andβ at dierent adhesion coecients and at the
10
1 (deg) .3 0
.4
10 .5 .6
5
(-)
2 (deg) .7
0 .8
.9
-5 -10 -10
Test set C4.5 algorithm
Fig. 3.4: αf and αr sets depending on the adhesion coecient
xed velocity of 90km/h. The tendency of the set sizes is similar to the previous case. The size of the sets becomes smaller at high adhesion coecients and larger at low adhesion coecients. The calculated sets t to the test sets well.
50
.3 0
.4 .5
10 (-).6
.7
5 .8
(deg) .9
0 -5 -10 -50
Test set C4.5 algorithm
Fig. 3.5: ψ˙ and β sets depending on the adhesion coecient