Handling Diagram in Nonlinear Vehicle Dynamics
The generalized notion of over and understeer
Despite a simple mechanism, the dynamics of a vehicle are highly nonlinear and “entertaining”. Unlike the non-holonomic constraints commonly applied in the robotic community, the tire of road vehicles was assumed to always have some non-zero slip angle to create lateral force, which also opens the box of Pandora. In this article, the dynamics will be simplified as a single-track vehicle and the possible quasi-steady-state operation point will be exploited with the introduced Handling diagram. In short, this article intends to answer the following question:
Under some constant speed, what is the minimum cornering radius?
Nonlinear Lateral Vehicle Dynamics
The single-track model
Also referred to as a bicycle model, the single-track model is a common simplification of the vehicle that neglects the wheel track yet captures most of the characteristics that will be discussed in this article. The dimensions and the physical variables were depicted in the following figure including the front and rear wheelbases “l1, l2”, vehicle heading “ψ”, yaw-rate “ω”, vehicle speed “v”, slip angles “α1, α2, β”, front steering angle “δ”, and the lateral forces “Fy_1, Fy_2” generated by the tire.
Quasi-steady-state tire model
Realistically, the side forces of the tire originate from the deformation of its structure, in other words, the forces caused by a changed geometry such as the slip angle require some “relaxation length” to be built up. However, with the assumption of a certain vehicle speed, the settling time is negligible and the tire model will be replaced by a static mapping throughout this article. For one’s reference, the most well-known empirical mapping under such an assumption was the “Magic Formula” [1].
Full picture
With all the ingredients prepared, the lateral dynamics of the vehicle can be realized in the block diagram shown below with the steering angle treated as an input. The kinematic relation captures the geometric, the static function represents the tire model, and finally, the dynamics are governed by the forces and inertia. Readers should be aware that the vehicle speed “v” was assumed to be near-constant compared to other variables and the slip angles were small (<15deg) in this scenario.
The equilibrium solution
The dynamic model alone was sufficient to describe the behavior of a vehicle, but for most applications, the equilibrium states were also interested and require further manipulation of the equations. Unlike linear systems which have trivial solutions [2], solving the equilibrium states of a nonlinear system often leads to ad-hoc methods. In this section, the so-called “Handling diagram” will be introduced to solve the equilibrium state of a nonlinear vehicle dynamic.
Variables and degree of freedom
Under a steady-state operation, without those degenerated cases, the degree of freedom from the dynamics will be eliminated while the steering angle “δ” and speed “v” were treated as input constants. In total, four variables were fully determined as “ω_ss, β_ss, δ_const, v_const” with the subscript omitted for simplicity. To further provide an instinctive point of view, the resultant steady-state yaw-rate “ω” will be rephrased by the cornering radius “R=v/ω” in some of the following discussions.
First attempt
Before diving into the method of the Handling diagram, let us tackle this problem with a more primitive approach. Observe that the dynamic equation will return a zero vector at equilibrium states, from the two dynamic equations (colored in yellow) shown in the last section, the following relations could be claimed. Symbol “η” was introduced as the identical ratio between lateral and normal force of each tire “Fy_i/Fz_i” and found to be also equal to the ratio of lateral acceleration to the gravity “ay/g”.
This result suggests that the ratio “η” could be used as an entry. As shown in the figure below, with each lateral force “F_yi” normalized to the normal force “F_zi”, a proper steady-state solution must occur on a common horizontal line “η” that has intersections with both of the tire model curves somewhere. The corresponding cornering radius “R=v²/ηg” and the yaw-rate “ω=ηg/v” are determined by the vehicle speed “v” while the slip angles “α1, α2”, and the steering angle input “δ” are also straightforward to derive following the kinematic relations.
For the example shown in the figure above, the maximum lateral acceleration “ay=η_max*g” or the minimum cornering radius “R=v²/(η_max*g)” was limited by the normalized curve of the rear tire. Additionally, with a constant vehicle speed “v”, the trend of the slip angles and the steering angle with a varying lateral acceleration (or the yaw-rate equivalently) was also captured nicely in the figure at right.
The Handling diagram
The aforementioned method works well to find the steering angle given other conditions but suffers if asked to solve for the others given some steering angle. This can be overcome by utilizing the geometric relation between slip angles and the steering angle “δ=(α1-α2)+Lω/v” where “L=l1+l2”. With that in mind, it is reasonable to subtract the normalized tire characteristic curve horizontally and creates the so-called Handling curve as a function of the ratio “η”. The actual operation is a bit confusing and may refer to the following figure as an example, noticeably, the maximum ratio “η” remains the same as before.
The power of the Handling diagram will be demonstrated through the following examples with one of the variables from {ω, δ, v} treated as the independent variable while another as constant. In total there were 1 independent, 1 fixed, and 2 steady-state constraints which fully determined the solution if exist. Although the body slip angle “β” is missing in the Handling diagram, it could always be calculated after knowing the tire slip angle and other variables.
Fixed “v”, changing “ω” or equivalently the ratio “η”
Under some constant speed “v”, the term “Lω/v” from the steering angle relation “δ=(α1-α2)+Lω/v” is proportional to the yaw-rate “ω” or the ratio “η”, such a flat contour was depicted on the first quadrature of the diagram shown as follows. The resultant steering angle was then simply the distance between the intersection of both curves. Suppose the ratio “η1” was selected, then the corresponding steering angle of one of the valid operation conditions is the sum of the green and blue colored arrows.
Fixed “v”, changing “δ”
This is an extension of the previous one, the idea is to simply seek for any intersection of the curves with the constant speed “v” contour shifted with the selected steering angle “δ” as shown below. In this case, the resulting yaw-rate “ω” or the lateral acceleration ratio “η” is the dependent output. Without a surprise, following the ordinary solutions close the main branch which both front and rear tires were gripping, as the steering angle increases, the lateral acceleration increases while the cornering radius decreases.
Fixed “δ”, changing “v”
As a complement to the last scenario, the trend with changing speed was also possible to be visualized using the Handling diagram.
Fixed “R”, changing “v”, and the generalized notion of oversteer
With a modification of the “load line” from constant speed to a fixed radius “R=v/ω” as a vertical line shown in the figure, the same trick works similarly. Furthermore, the idea of oversteer and understeer for nonlinear tires could be formalized with this utilization of the Handling diagram. For one’s reference, an oversteered vehicle will require (quasi-statically) a smaller steering angle as the speed increases while circling a fixed radius. For the specific example shown below, the vehicle will exhibit an oversteer characteristic above speed “v1” as the steering angle decreases along the increasing speed, and vice versa.
Conclusion and caviats
As demonstrated in this article, the handling diagram provides a powerful method to predict the behavior of a nonlinear vehicle with a nonlinear tire model. Practically, since it can be regressed from experimental data by running a circular track, it also served as an indication of the performance and potentially helps the design or tunning.
Beware that the solution given by the Handling diagram does not imply stability, which is the topic to be covered in the future.
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.