Vehicle cornering behavior and Ackermann geometry

[AbsoluteUnit]

TL;DR Summary

The Ackermann steering geometry is such that the lateral slip, tan(a) (where 'a' is the sideslip angle) tends to zero during stead-state low velocity turns. Is it reasonable to expect this vehicle behavior during simulation (see attached)?

Dear all,

I am writing a vehicle dynamics simulation for my thesis topic. However, I came into a conundrum when testing the cornering behavior of my vehicle. The problem is inherently complex due to its many subsystems, but I'll try to give as much detail without bogging the thread down.

Problem description
I am using a proprietary vehicle dynamics model to benchmark my own model. The issue is that I am finding discrepancies in the cornering behavior of my model and the proprietary model for the small turn angle, low-velocity case, but I am not entirely sure why. Essentially, my model exhibits a lower lateral displacement than predicted by the proprietary model and lower lateral acceleration, but the only reason I can think of is that the lateral slip of my tires aren't behaving similarly (after re-checking my equations multiple times) due to something else that I am missing.

My model is a front-steer vehicle that uses the Ackermann steering geometry to dictate the steer angle of the inner and outer tires (relative to the turn center). My understanding is that the Ackermann steer geometry is such that the vehicle can circumnavigate a curve at low (or zero) speeds without inducing a sideslip at the road-tire interface.

Since I haven't had much experience with vehicle modeling prior to this, I am looking for a second opinion on whether the following vehicle behavior looks "typical", or am I overlooking something very crucial? Should I just document my model's performance in the thesis and not worry so much about it?

Thank you for your insights.

(I did not attach the proprietary model output since I am just interested in whether the model behavior seems reasonable by intuition/prior experience. If it seems "off", that would at least give me an idea of whether I should be worried or not.)

---------------------------------
Scenario:
Initial longitudinal speed, Vx0 = 5 m/s
Initial lateral speed, Vy0 = 0 m/s
Initial angular speed, Psi0 = 0 rad/s

Input steer = 5 degrees (direct steer input, no steering ratio)
Wheel steers are recomputed for each wheel using the Ackermann steer ratio.

Vehicle properties
mass, m = 1778kg
yaw MoI, Jzz = 3800 kg-m^2
static weight distribution(front track/rear track) = approx. (9600N/7800N)

half-track width, w = 1m
front-track-to-centroid length, l1 = 1.33m
rear-track-to-centroid length, l2 = 1.62m
Centroid height = 0.53m

Load transfer is computed assuming a rigid vehicle model using the superposition of quasi-static longitudinal and lateral load shifts, i.e. without considering a suspension system.

Tire properties
Longitudinal slip stiffness, Cs = 89000 N/slip
Lateral slip stiffness, Ca = 45000 N/slip
static radius, Rg = 0.3m
Rolling MoI, Jyw = 2.00 kg-m^2

pavement friction coef., mu = 0.87

• The Dugoff tire model was used to compute tire forces.
• The force curves produced were qualitatively similar to the force curves produced by the proprietary model.
• The slip stiffness are assumed to be static (i.e. not load-sensitive).

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Two scenarios are attached:
(1) Turning without maintaining longitudinal velocity (no input torque)
(2) Turning with constant velocity

(1) Turning without maintaining longitudinal velocity (no input torque)
(left) Vehicle displacements (right) Vehicle-body velocities

Vehicle trajectory

(2) Turning with constant longitudinal velocity
(left) Vehicle displacements (right) Vehicle-body velocities

Vehicle trajectory

 

[Baluncore]

The Ackermann three link steering is an approximation that works in the real world.
You will need to study the alignment error, as a function of turn radius before simulating a turn with an unknown (toe-in or toe-out) alignment error.

 

[jack action]

Are both models using the same Akermann geometry, i.e. with or without considering slip angles?

 

[AbsoluteUnit]

The Ackermann three link steering is an approximation that works in the real world.
You will need to study the alignment error, as a function of turn radius before simulating a turn with an unknown (toe-in or toe-out) alignment error.

Thanks! Adding complexity to the model is something I plan to do after I submit my thesis (and if my advisor is still keen on letting me work on the problem), for now I have to contend with naive application of Ackermann steer. But this is really good information, since I have been looking for something more realistic than the Ackermann steer.

Are both models using the same Akermann geometry, i.e. with or without considering slip angles?

Unfortunately this isn't something I can probe directly, since the documentation of the proprietary model is a bit vague :( A bit of a crap shoot on my part really, but I don't have access to other more well-documented models like CarSim so I have to hope the proprietary model is well-tested.

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Either way, I believe I found the reason of the difference between the proprietary model and my model by accident when tinkering around with the model:
For the wheel spin equation:

Jyw*(dω/dt) = Td - Tb*sign(ωR_e) - F_xwR_e - RR​

Where

Re := tire effective radius,​

Fwx := tire longitudinal force,​

ω := tire angular velocity,​

Td := driving torque, Tb := braking torque.​

​

Turns out that the proprietary model does not include rolling resistance, RR and longitudinal resistance(?), M_xw = F_xw*R_e in their model (i.e. I had to "emulate" these indirectly), which explains the discrepancy (besides the question of steering geometry, which is still a good question).

Exhibit one [Exh. 1] shows the propriety model's lateral displacement and velocity without "emulated" tire RR and M_xw. [Exh. 2] shows the same model with an (uncalibrated) emulated RR and M_xw. One can observe how the lateral velocity in [Exh. 2] degrades over time when the latter effects are considered.

It's kind of interesting how RR and M_xw are typically small compared to applied driving (T_d) and braking torques (T_b), but consequences are significant when not included in the model.

Well, that's one mystery off my table (at least, I hope this is the correct diagnosis), thank you for your time!

(The following are vehicles NOT under constant velocity, so the scenario is equivalent to the first case in my original post)
[Exh. 1] Proprietary model 5 deg steer without RR and Mxw

[Exh. 2] Proprietary model 5 deg steer with "emulated" RR and Mxw
(I haven't yet calibrated the model inputs to produce roughly similar longitudinal slips to my model to "mimic" RR and Mxw properly, so this is for illustration only)