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.
Other forms of steering
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.
How and why to use the Ackermann steering model
Determining Steering Geometry
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:
- The relationship between the track, wheelbase and the center point of the back axle.
- Geometry at zero toe condition (both tires facing straight).
- The geometry of the inner wheel while turning.
- 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
| Variable | Meaning | Status |
|---|---|---|
So | Outer wheel angle | Find using eqn 1 using Si |
| Si | Inner wheel angle | Has to be set, arbitrary |
| W | Track width | Driven by areo |
| B | Distance kingpin to kingpin | Driven by areo |
| L | Wheelbase | Drive by areo |
| x | Steering arm length | Unknown |
| y | Tie-rod length | Unknown |
| p | Rack casing length | Has to be set |
| p+2r | Entire rack length | Has to be set |
| q | Rack travel | Unknown |
| d | Distance between rack and front axis | Unknown |
| Beta | Ackermann angle | Found from wheelbase and track |
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.