Motorsports Physics — Modeling and Simulation of the Counter-steering
Not to be confused with the same term that was used in motorcycles, the counter-steer, opposite lock, power slide, or Scandinavian flick is one of the most famous signatures of motorsports. Start by reviewing some clips that utilize such a technic, the underlying mechanism will be revealed in this article based on a general single-track model. Furthermore, a sliding-mode-based controller was also proposed to demonstrate the possibilities to stabilize the counter-steering operation. Enjoy!
Some understanding of the terms over/understeer will be beneficial while reading this article, readers may refer to the previous posts (link).
What is Counter-steering?
To grab the basic ideas of what counter-steering is, let us start with three clips that each utilizes such a technique in different scenarios.
Demonstration
In this first clip from an actual F1 race track, the vehicle unintentionally exhibits oversteer behavior at the exit of the corner. The driver, Fernando Alonso Díaz, successfully recovered the diverging yaw motion by aggressively steering in the opposite direction of the corner.
But what if the counter-steering is itself the goal? This second video demonstrated a series of clips where the driver purposely induced an oversteer condition through power slipping the rear axle, then performing the counter-steering. Notice that the front wheel roughly coincides with the vehicle’s groundspeed heading.
Alternatively, the vehicle in this last case enters the oversteer condition by locking the rear axle instead of overly powering it.
What could we learn?
Counter-steering is a methodology that steers the front wheel to a smaller slip angle to stabilize the yaw rate even under an oversteer situation. The observation that the front wheel points roughly to the ground speed during counter-steering also suggested this argument.
Additionally, the oversteer could be induced through a large longitudinal slip of the rear axle(powering or locking), weight transferring to the front axle(braking/deceleration), or simply caused by a large body slip-angle.
Modeling of the Vehicle Dynamics
To study the mechanism behind the counter-steering technique, a single-track model with a nonlinear quasi-steady-state tire model will be utilized. Readers may refer to these two articles(link-1, link-2) for more detail about the terms and the assumptions. Important variables were listed here for convenience: the front and rear wheelbases “l1, l2”, vehicle heading “ψ”, yaw-rate “ω”, vehicle speed “v”, slip angles of front-wheel/rear-wheel/body “α1, α2, β”, front steering angle “δ”, and the lateral/normal forces “Fy_i/Fz_i” resulted/applied on the tires.
The universality of the tire modeling
Although a specific tire model was used to model the counter-steering in this article, the discussion could easily be generalized to different scenarios as long as two of the following characteristics match. Namely, the lateral force of the rear tire saturates or degrades at a larger slip angle, and the vehicle has a steerable front wheel.
Equilibrium states (Handling diagram)
Before digging into the full dynamics, we could gain some insight by first inspecting the Handling diagram(link, or [1]). Focusing on the unstable solution(orange hollow circle) at the top right with a negative steering angle “δ₁<0”. The fact that it occurs on the branch that has a sliding rear wheel and also has a positive yaw rate makes it a candidate for the counter-steering operating state.
Phase (State-space) trajectory analysis
Compare to the small-signal linearization conducted in the previous article(link), analyzing the full phase trajectory captures the global behavior instead of just local characteristics. In the following figure, the trajectories of the state “[β,ω]” under a fixed steering angle “δ=0.1” was calculated given the initial conditions(black dots) and shown as blue colored traces.
The green dots were the stable(solid) and unstable(hollow) equilibrium points determined from the flow of the trajectories, and the Region of Attraction(ROA) of the stable equilibrium point was painted in orange. Furthermore, the corresponding front/rear wheel slip angles “αᵢ” were overlayed as the additional coordinate in the figure. The observation that the ROA extends in the direction of a sliding front wheel(more understeer) agrees with the prediction made in the previous article (link).
The stable equilibrium point (solid green dot) represents the steady state of a typical cornering, in which both the steering angle “δ>0” and the yaw rate “ω>0” share the same sign. On the other hand, despite the instability, the equilibrium state at the bottom right provides a cornering direction “ω<0” opposite to the steering angle “δ>0”, hence the name Counter-steering.
Phase portrait v.s. Handling diagram
The equilibrium points observed in the phase portrait can be matched with the ones solved through the Handling diagram. Such relation is depicted in the following figure where 5 phase portraits were each generated under a fixed steering angle.
Additionally, you may notice that the stable equilibrium point and its ROA once vanishes and then appear again as the steering angle increase. This unique equilibrium point(solid yellow dot) claims its stability by an overly steered front wheel, effectively slipping the front wheel and creating an understeer condition. Readers may refer to the simulation provided here(video link).
The counter-steering
Oversteer recovery
We are now ready to model and simulate the counter-steering with all the ingredients ready. Take the scenario from the first clip(link), assuming that the steering angle is large enough, or simply due to a power slipped rear axle, the vehicle exhibits oversteer while leaving the corner. This leads to an excessive yaw rate “ω, y-axis” and builds the body slip angle “β, x-axis” to a dangerous level, the corresponding state trajectory was visualized as “Stage A” in the following phase portrait.
Consequently, if the driver was able to switch the steering angle to an opposite direction, performing a counter-steering, the state will then follow the new portrait and avoids further skidding shown as “Stage B”.
The animation of this scenario was provided in the following video. Notice that the front wheel points roughly to the ground speed during the counter-steering stage, this matches the observation in the actual clips.
Maintaining the counter-steering operation
Since the counter-steering state is itself an equilibrium point, it is possible to keep the counter-steering operation for a long period as demonstrated in the second clip(link). Referencing to the discussion in the previous post(link), the single-track model with front-wheel steering was controllable at the equilibrium states and implies the existence of the stabilizing controller.
In the following figure, phase portraits with two steering angles were overlayed and the separatrixes were given as dotted lines. If the steering angle was switched between these two conditions whenever the separatrix was reached, the state will be limited in a specific region as shown by the purple trajectories.
Again, the animation of this scenario was provided in the following video.
Conclusion and takeaways
This article reveals the underlying mechanism of the counter-steering technique which is to maintain the traction of the front wheel through the steering input. In addition, model-based phase portrait analyses were conducted with numerical simulations to provide some intuition. At last, as an equilibrium state, a toy controller was designed to regulate the state around the counter-steering operation point.
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.