Passive stability of nonlinear vehicle dynamics
Analysis and simulations
In nonlinear vehicle dynamics, the equilibrium operation points could be solved by the handling diagram with proper assumptions, but whether or not these points were stable requires further analysis. In this article, the passive (open-loop) stability of the equilibrium points in vehicle dynamics will be discussed and the corresponding criteria will be introduced. Additionally, simulations and animated dynamics were also provided to give some intuition about the mechanism.
The handling diagram and the dynamic modeling
A single-track model with a nonlinear tire lateral force will be utilized throughout this article with the assumption of a constant speed and small slip angles. With the block diagram shown below, it is clear that the tire model is the major source of the nonlinearity if both of the assumptions were satisfied.
The equilibrium operation points
In reference to this previous article, the equilibrium solutions can be graphically searched using the handling diagram. Since the dynamic characteristics will be discussed under a fixed speed, it is logical to seek solutions under some given speed. The process of finding the solution(s) could be demonstrated as follows, with a steering angle specified as “δ”, the solution(s) was the intersection between the handling curves, and the shifted constant speed contour depicts as the family of the solid blue lines.
The resulting equilibrium points can be interpreted from two perspectives. The first is the difference between the front and the rear slip angles “α₁-α₂” which can be directly read out from the intersection point (hollowed circles) with the handling curves.
The second is the curvature “1/R” which can be inferred from finding the intersection (solid circles) of the original constant speed coutour and the horizontal line passing through the solution. Although the latter representative was not unique to the operating points, it will play its role in the upcoming sections.
Dynamics of small variation
An equilibrium itself was not sufficient to claim stability without further analyzing the dynamics. In this section, the dynamic equation of the small variation of the states will be derived under a constant speed, in other words, only the lateral dynamics were captured.
The detail will be hidden in the manipulation shown as follows and please feel safe to jump straight to the resulting second-order linear system. Beware that the assumption of a constant speed holds well for the scenario of a high-speed cornering but might not be true in a sharp turn where the lateral and the longitudinal dynamics were highly coupled, i.e. brake while turning.
The passive stability
From the second-order characteristic equation derived in the last section, the passive stability could be ensured if the characteristic equation shown below has each eigenvalue with a negative real part. Equivalently, stable if and only if a positive stiffness(colored in blue) and positive damping (colored in orange) [2].
Before we proceed, the normalized cornering stiffness “Φ” may be defined as an alternative to the original one “C” as shown.
Positive damping
The positivity of the damping could be maintained by satisfying the following inequality. Interestingly, even with one of the tires slipping, for example, Φ₁<0, the damping still has a chance to be positive. In contrast, if both tires were slipping, there is no hope.
Positive stiffness
On the other hand, the condition for a positive stiffness could be deduced with the process shown as follows. The relation between the steering angle, the slip angles, and the curvature “δ=(α₁-α₂)+L/R” was also incorporated to reach the result.
The stability criteria
With all the ingredients prepared, the final criteria that ensure the stability were summarized below as two inequalities. If we considered a scenario where both tires were gripping “Φ₁,₂>0” in which the positive damping was already satisfied, the only possible to reach an instability condition is to violate the positivity of the stiffness through a negative ratio of steering to the normalized lateral acceleration “∂δ/∂η”.
A method provided in [1] helps determine the stability by finding the maximum possible “η” for each different speed. This was depicted in the following figure where the maximum stable “η” for multiple constant speed curves were searched by finding their tangent point to the handling diagram. The corresponding boundary was then portraited as the orange line in the plane spanned by the normalized lateral acceleration “η” and the curvature “L/R”. Obveously, instability only occurs if the operation point was in the generalized over-steer condition.
Non-trivial stable operation point
There is a chance that, with one of the tires slipping, both the damping and the stiffness were still positive. For example, from the figure in the section where we solve for equilibrium points, with a large steering angle “δ₂”, there is a stable equilibrium point with a slipping front tire “Φ₁<0” and a negative steering slope “∂δ/∂η”.
The analogy of this condition is when a driver overly steered the front tire which stabilizes the vehicle counterintuitively. You may already predict such an operation has smaller damping compared to the normal “both gripping” driving, which is true and can be observed in the simulation.
Numerical examples and simulations
Because the handling diagram is realized under a semi-static condition, the concentrated information is hard to interpret. This is where the dynamic simulations could help and fill in the transient responses that the semi-static analysis struggles to provide.
Numerical model
Trajectories top view
Let us first examine the following figure that depicts the top-viewed trajectories with various fixed speeds and step-injected steering angles. Starting from a relatively lower speed “v₀=10,15 m/s” and a small steering angle “δ=0.17 rad”, the vehicle exhibited an under-steer characteristic as predicted by the handling diagram, in other words, a larger radius with a higher speed. With a further increased speed “δ=0.17 rad, v₀=20 m/s” the vehicle will eventually become oversteered and experienced a diverged body-slip angle with the trajectory colored in orange.
Another interesting observation is that such an unstable condition can be avoided by increasing the steering angle “δ=0.34 rad, v₀=20 m/s”. This is the non-trivial stable operation point we discussed earlier.
State responses
Considering that the body slips are small and hard to observe on a large scale, it often helps to examine the detail of the state responses as time plots. In the responses shown below, some conditions will eventually diverge and therefore cropped reasonably.
If we focus on the three experiments “A, B, and C” that share an identical steering angle of “δ=0.31 rad”. An interesting pattern appears as an increasing speed also helps stabilizes the vehicle “B to C”. This is again another example that reaches the non-trivial stable operation condition, the slightly under-damped result also occurs as predicted from the analysis.
To fully understand the mechanism that governs these results, the corresponding handling diagram for cases “A, B, and C” were also illustrated as follows. For case “A” the operation point was stable and belongs to the case where both front and rear tires were gripping “Φ₁,₂>0”. In cases “B and C”, things start to get a bit interesting since the intersection only occurs when the front tire was slipping and causes the simple criteria provided in [1] unapplicable.
Recalling the full criteria of the stability, the positivity of the stiffness was only satisfied with a speed higher than a certain value which “B and C” both satisfied. Furthermore, the positivity of the damping was only possible when at least one tire is gripping, in which case “B” was marginally satisfying and may easily diverge due to a larger state variation which is the case in the simulation.
Dynamics from the driver’s perspective
The dynamics viewed from a vehicle(driver) perspective often provide a more instinctive interpretation, which is provided in the following video for all three of the cases. For more scenarios, you may refer to the playlist through: https://youtube.com/playlist?list=PLcq0_jtfHo6foUDNBy-EvI3Op1hhoa9-k.
Case ”A”, under-steer, stable
Case ”B”, over-steer, unstable
Case ”C”, over-steer, stable
Conclusion and takeaways
As a complement to the previous article which introduces the handling diagram that solves for equilibrium points. This article focuses on the justification of whether these equilibrium points were stable without actively controlling the steering angle, that is, the passive stability. Consequently, a stable criterion was derived for the small variation model and a simplified version was possible if only ordinary operation modes were considered.
What hasn’t been covered?
First, the model used for the stability analysis so far does not handle large variation which is nature in nonlinear systems. Second, the possibility of stabilization through an actively controlled steering angle (or any other input) was neglected. Last but not least, the assumption of a semi-static condition for longitudinal motion limits the applicability to those phenomena that exhibit coupled longitudinal and lateral motion such as the brake-induced over-steer.
Other possibilities?
There are indeed other interesting operation points that were not covered in this article, not to mention that stability could be enforced by actively changing the steering angle input say from a stunt driver. For instance, refer to the following video that shows the so-called counter steering technic.
References
[1] H. B. Pacejka, Tire and Vehicle Dynamics. Butterworth-Heinemann, 2012.
[2] S. H. Żak en S. H. Żak, Systems and Control. Oxford University Press, 2003.