Steering Research

Rack and Pinion 

Utilizes a rack and pinion gear system to turn rotational (steering wheel) motion into linear motion, which in turn moves the other linkages to make the wheels turn.

Manufacturing/Design of the Rack

Other forms of steering

introtosteeringlinkage.pdf

Other forms of steering are conventional steering or a bellcrank steering system. The conventional system will usually be heavier due to the gear box weight, whereas the bellcrank or crank and pinion is usually lighter. WIth the conventional steering system, different linkage geometries may be set up, with advantages and disadvantages to all the configurations as listed in the "intotosteeringlinkages" pdf. 

Steering Design

For basic steering to be achieved, there is a specific type of geometry wherein a turn, the inner wheel needs to turn less than the outer wheel for neither tire to slip in a turn. This geometry is called Ackermann geometry. 

Ackermann Steering Explained

How and why to use the Ackermann steering model

Determining Steering Geometry 

IRJET-V4I590.pdf

Mathematical-Model-to-Design-Rack-And-Pinion-Ackerman-Steering-Geomtery.pdf

The steering geometry (length of tie rods, steering arm, length of rack, etc) can be found by applying basic trigonometry principles as seen in the "mathematical-model-to-design-rack-and-pinion" pdf. The principles in this paper can be applied in python to iteratively find solutions by adding constraints (packaging, turn radius, max inner wheel angle, etc). The geometry can also be found through trial and error in Solidworks by creating and constraining the sketch correctly. 

Mathematical Model 

The idea of this model is to find four equations from the geometry of the system, these equations are from:

1. The relationship between the track, wheelbase and the center point of the back axle.
2. Geometry at zero toe condition (both tires facing straight).
3. The geometry of the inner wheel while turning.
4. The geometry of the outer wheel while turning.

After obtaining these equations using the Pythagorean theorem, there will be four unknowns and three equations, equation 1 was used to find the ideal value of outer wheel angle with a given inner wheel angle (knowing the track and wheelbase). The four unknowns will be x (steering arm), y(tie rod), q(rack travel) and d(distance from rack to front axle). When any one of them is fixed, the other three values can be solved, therefore, a python script is most helpful when used to fix one of the four variables and iteratively solve for the geometry. However this paper assumes that the rack will already have been found, so the rack length (p and r) will also have to be fixed and changed iteratively in order to solve the system. 

To validate the geometry, another equation (again found from geometry) can be used to find the actual outer wheel angle when given any inner wheel angle and known geometry values (x, y, q, d), so theoretically, the best geometry is the one that resembles the ideal Ackermann curve the most when plotting the actual vs ideal outer wheel angle (this can be found by finding the smallest avg delta between each point). For example:

Calculation Notes

Equation 1:   

cot(So)-cot(Si) = B/L

Equation 2:   

y^2 = ( (B-(p+2*r))/2)-x*sin(beta))^2+(d-x*cos(beta))^2

Equation 3:   

y^2 = ((B/2)-((p/2)+r-q)-x*sin(Si+beta))^2+(d-x*cos(Si+Beta))^2

Equation 4:   

y^2 = ((B/2)-((p/2)+r+q)+x*sin(So-beta))^2+(d-x*cos(So-beta))^2

To validate the values, two curves of the outer wheel angle can be plotted against a range of inner wheel angles, these two outer wheel angles are the actual and ideal Ackermann outer wheel angles. 

To find the actual outer wheel angle these equations are used: 

A = B/2 - (p/r +r)

C = ( y^2 - d^2 - x^2 - (A-q)^2)/(2*x)

K = C*(A-q) + (C^2*(A-q)^2-((A-q)^2+d^2)*(C^2-d^2))^0.5

So = arcsin(K) + Beta

After finding the actual outer wheel angle, two imaginary curves of outer wheel angle vs inner wheel angle can be plotted by doing a sweep of the values from 0 to 45 deg for inner wheel angle. After finding each outer wheel angle, the average delta can be found between the two. If the average delta is > 5 deg then the solution will be rejected due to too much deviation. 

CAD Model 

A 2D top-down view of the steering geometry was created to assess approximation of Ackermann when the wheels are turned. Model can be found in "MSXIV\Development\Dynamics\Steering\Ackermann 2D Model". 

Different values of the steering arm, tie rod, distance from the front axle, and the connection point on the suspension were changed. The geometry was validated through looking at how closely the dotted lines from the front wheels intersect the line from the back axle (remember this is the condition for Ackerman, where the wheels all turn around a central point). As these values were iterated through some patterns were found.

• The more negative the steering arm angle is, the closer the intersection 
• A longer steering arm gave better results
• Shorter distance from the steering arm to the wheel
• The tie rod length was not very relevant 
• Steering arm angle affected the turning radius of the wheels the most

A few geometries that resemble Ackermann was plotted into excel: https://docs.google.com/spreadsheets/d/1nAutwHmWkd5IAZSrZADMpOVEK2VDvbi-Nal5EFTHZkI/edit?ts=5cf2cd5b#gid=0.

Due to the negative steering arm angles, Ackermann vs anti-Ackermann geometries has to be compared. 

Designing Steering Email Chain

Original email: 

I am emailing you as a part of the Midnight Sun solar car design team that is responsible for steering. From a presentation slide on the ASC website about “suspension design” I saw a video demonstrating Ackermann steering where the steering arm angle was changed from the middle of the back axle to the middle of the car (and the second design achieved Ackermann more accurately). However in all literature we have come across, it recommends having the steering arm angles point to the back axle, could you clarify for us if it is safe to design our steering geometry so that the steering arm angle does not intersection on the back axle? Any response on this subject would be appreciated!

From Evan:

Hi Emily, most of the online videos that I’ve seen of Ackermann geometry are very simplified 2D representations of the vehicle. They generally assume 0 caster angle, 0 camber angle, no suspension travel, no tire slip angle, and that your steering arms are connected by a solid fixed length rod such as in the images below.

In this imaginary world you can build a 2D model that achieves perfect Ackermann steering geometry with the steering arms intersecting at the center of the rear axle. When in doubt, this may be an OK starting place but it may not be the most optimal scenario.

When I worked on suspension design for the PrISUm 11 solar car (Hyperion) I had a fully functional CAD model of the vehicle frame and suspension, such that I show a top down view and look at the wheel angles with the actual steering rack, tie rod, caster, and camber geometry. I think I convinced myself that obtaining perfect “Ackermann” steering through a wide range of steering motion with this more realistic vehicle representation is not possible. Granted, I did not have a vehicle dynamics or tire model that would allow me to take into account how much body roll (suspension travel) and tire slip angle would impact the actual tire turning circle centers. If you have the time/resources to further investigate the wheel turning circle centers on your solar car while taking into account more real world details as I’ve provided above then I’d highly recommend it and you may be able to come up with a more optimized steering arm angle than intersecting at the center of the rear axle. However, if you are running short on time to do this, if you set up your vehicle such that the steering arms intersect with the center of the rear axle that should be an OK starting point that will ensure you get some amount of Ackermann in your vehicle to reduce tire scrub (wear).

You mentioned safety in your question. In general I would say poor Ackermann geometry in a solar car should not present a significant safety risk to the driver of the vehicle, however it will result in increased tire wear around turns. For an ASC event, I personally would worry much more about lateral tire scrub and bump steer since most of the route will be traveling straight ahead. For FSGP your scrub radius and Ackermann geometry may have a more significant impact on tire wear than at ASC due to increased turning. At the end of the day, make sure you have plenty of extra tires for FSGP and even if your Ackermann geometry isn’t great you’ll still get through the event alright!

Hopefully this information is at least somewhat helpful to you as you work on setting up and tuning your vehicle’s suspension geometry.

From Bill:

Emily,

I agree with all of Evan's points about Ackerman being a tricky beast. If you do attempt to model it, keep in mind that a lot of modeling packages might assume a certain tire response. They might have a few tires to choose from, or even where you can put in your own data of plots of lateral force as a function of slip angle, normal load, and camber angle (maybe even tire pressure). I mention this because round cross-section tires generate more camber thrust than car tires. So it might be best to make your own models. Unfortunately this is a tricky beast that incorporates at least the following variables and a cascade through those variables that becomes more complicated as it goes, and then is iterative:

camber angle between tire and car

steering angle of steering wheel

effective steering angle of each steered wheel

roll angle of car

new steered angle of each wheel accounting for bump steer geometry

effective camber angle of tire on ground

any camber induced by the effect of caster plus steering angle (what I call steering-induced camber)

load on each wheel (from cornering force, which assumes a speed)

lateral thrust of each wheel taken from tire data by inputting slip angle, camber angle and normal load

The weight transfer and associated difference in lateral loads created by the two front tires at their respective steering angles is a reason that several F1 teams using anti-ackerman, which is where the inside wheel steers less since it has less normal load on it, and can't operate at as high a slip angle before it simply lets loose. Although an F1 car is optimizing to try to get as much lateral force from the inside tire as possible. Your goal is to minimize scrub. You are also working at lower cornering loads and with a higher Cg.

I think if it were me, I'd go about it by designing for slightly less than 'perfect ackerman' using the traditional method (so the inside wheel still turns more than the outside, but not as much as necessary to point at the point the car is turning around.), but also using an adjustable ackerman (realized by having an adjustable mount for the steering rack that allows longitudinal adjustment, which changes ackerman geometry) and just taking data for required energy to negotiate various radius corners at various speeds. Try that at a few settings of the rack, and then finding a setting that works well in as many likely scenarios as possible. As Evan said, it will still be a compromise. The reason I'd go for less than perfect is that a tire with less load can't handle as high a slip angle before letting loose and just dragging on the ground, and perfect ackerman assumes that both wheels will have the same slip angle.

New reply question: 

Thank you so much for your replies, I really appreciate the insight you gave us into Ackermann steering! One last question; we’ve found that a negative steering arm angle gives us a better approximation of Ackermann, however this is still not anti-Ackermann since the outer wheel does not turn more than the inner wheel. Would there be any concerns doing a model where the steering arms point towards the front axle?

Bill's reply:

I think you are well on the way to having a good idea of what it takes to design in proper Ackerman.

To answer your question, no there is no reason to be concerned about using angles going the opposite way of convention.

Your arms are pointing inward toward the front, but your rack also is aft of the outboard tie-rod ends. So the effective angle between the tie rod and the steering arm is near 90*. That 90* angle alone would give you zero Ackerman since a lot of what creates Ackerman is that on the inside wheel that angle is obtuse and getting more obtuse, and on the outside that angle is approaching 90*, then going acute. The point at which you are at a right angle is the point where you get the least movement and the farther you go from a right angle the more steering movement you get on that wheel. 

In your case, you start at about a right angle in both sides and one goes more acute at approximately the rate the other goes more obtuse. The rack being aft means that as the rack moves, the angle between the lateral axis of the car and the tie rod is going to be different on each side, which will give the different steering behaviors on each side, providing Ackerman. 

This is what I was alluding to with longitudinal adjustments of the rack to adjust Ackerman through testing. I've never seen a solar car team have an adjustable rack, but it would be possible as long as it is mounted securely.

Anti-Ackermann vs Ackermann 

https://www.f1technical.net/forum/viewtopic.php?t=8874

Slip Angle Wiki

The above forum contains comments that explain basic anti-Ackermann vs Ackermann quite well in layman's terms. 

Due to material properties within the tire, every type of tire will have a different slip angle (the difference in traveling direction and the direction the tire is pointed in) preference for maximum grip at different loading conditions. When cornering very fast and turning tightly, the vertical loads on the tires will change, and the tires will want to see a certain slip angle for maximum performance. Ackermann can be applied because pro-Ackermann will introduce more slip angle for the inner tire while anti-Ackermann will introduce less slip angle. Tires can prefer more or less slip angles in a turn. By knowing tire characteristics, the type of geometry can be chosen. Normally, when turning, the inner wheel will normally see reduced loads.