Designing Variable Ackerman Steering Geometry for Formula Student Race Car

— This paper describes the different types of steering mechanisms and benchmarks a method for the selection of a steering system. It also explains the step-by-step method of designing a variable Ackerman steering geometry. The research uses the knowledge of Ackerman and trapezoidal systems, turning radius estimation, steering effort calculation, and the RMS error tool to design and justify the policies selected. The result comprises a detailed flow for designing a variable Ackerman steering geometry along with a set of Matlab codes that are required for the calculation of turning radius, space, angles on inner and outer wheels, and many other parameters. This paper offers essential knowledge for those who are new to the field and an overview for those interested in steering design.


I. INTRODUCTION
Formula SAE is an Engineering Design competition held by SAE International, which challenges students to build a formula-style race car evaluated in terms of engineering, performance, and efficiency. When it comes to high performance, vehicle dynamics plays a vital role in achieving the best possible outcomes. Along with mathematical modeling, good suspension and steering geometries are of utmost responsibility for the performance of the car. Since steering alone is responsible for maneuvering, so the selection and designing of a sound steering system become necessary for any car to be even drivable. Especially for land vehicles, the steering mechanism not only controls the maneuvering but also affects the suspension characteristics in motion. Thus, the selection of a correct mechanism is as important as designing and manufacturing vehicles [1]. In this paper, we will study the design of the steering mechanism. Firstly, to know the steering system is classified into three main types based on the mechanisms used. (As depicted in Figure 1) i. Ackerman or Pro Ackerman Steering mechanism ii. Parallel Steering mechanism iii. Anti-Ackerman or Reverse Ackerman Steering As we know that the car is a rigid body and for it to make a turn, all the tires should make the turn around the same center (As seen in Figure 1), or else the tires will either push or pull each other. This push or pull effect will force the tires to change their positions from the desired path by creating a scrub, and it will also result in momentum loss for the car.
So, to avoid this and help the car follow the pattern the above steering mechanisms are used based on different applications as each mechanism have their pros and cons.
Firstly, the Ackerman systems are relatively differentiated based on the amount of tire scrub (i.e., produced by each mechanism while the car is taking a turn).
There is no scrub in the Ackerman system; the scrub effect increases in parallel mechanisms and increases more in the anti-Ackerman systems. Also, the appropriate steering angle required for a turn is always a function of wheel load, road condition, turn speed (depends on turn radius), and tire characteristics. Since the car witnesses both high speed and low-speed corners, so there is no ideal steering mechanism possible unless controlling the steering angle for all the wheels is carried out independently.
The selection of Ackerman, Parallel, or Anti-Ackerman geometry depends on the vehicle's cornering velocity during the turn. The cornering speed decides the amount of load transfer, further affecting the significance of slip angles on the car. As the slip angles have a higher impact at higher loads, it leads to the creation of lateral force accordingly [2].
The cornering velocity is directly proportional to the load transfer. Also, a slight difference in slip angles doesn't create a significant impact on the performance (lateral force) of the car at similar loads. So, in low-speed corners, the Ackerman mechanism is preferred to prevent the momentum loss caused by the wheels' scrubbing. Whereas when the car is going in a high-speed turn, due to high load transfer, maintaining an adequate slip angle on the wheels becomes more important than preventing the tire scrub, and hence parallel or anti-Ackerman is preferred in this case. Thus the system decision is based on the vehicle capacity and cornering conditions of the race.
To study the steering geometry firstly in 2011 Lili and co-workers tried to produce an Ackerman geometry system, but this process was limited to the calculation of the length of parts like the tie rod and it did not compare the performance [3]. Later in 2016 Malu and co-workers directly elaborated the effect of the steering moment arm angle and also used an iterative method for its prediction [4].
Then, in a study from Malik and co-workers from 2017 a discussion was carried out on suspension parameters that affect the steering characteristics but the limitation of this study was not discussing the steering parameters [5]. Then in the same year Biswal and co-workers designed the suspension parameters and steering types still this study lacked in the determination of the steering parameters [6].
Again, in the same year, Raut and co-workers threw light on energy transfer systems and suspension parameters and have not discussed the steering parameters [7]. Afterward, in 2018 Gitay and co-workers discussed the value of the parameters again, stating the iterative method for its vale prediction [8].
Lastly, in late 2018 Naveen and co-workers elaborated on the types of the steering system and the basis of its selection, but again a method for steering parameters goes missing [9].
So, from this, it was clear that a precise flow with the correct methods to find the steering parameters have always been missing, which is one of the main objectives of this paper.
This research paper provides a step-by-step procedure on the decision and designing of steering parameters for a variable Ackerman geometry. It also includes a subtle overview of the selection of an appropriate steering mechanism.
Firstly, a discussion is carried out on the methodology adopted for the designing of the steering mechanism. This includes the calculations for determining the Ackermann steering condition, turning radius estimation, space requirement calculation, trapezoidal steering system, design of variable Ackerman system, cross-checking steering effort value, minimizing bump steer, and getting the tie rod inboard point. Then the manufacturing of the steering mechanism is elaborated also its incorporation in the car is shown. Later the results obtained are discussed. Also, a stepwise method for the designing of the steering mechanism which was adapted in the manufacturing of our formula student car is explained. Additionally, a comparison of theoretical and analytical values is done. Lastly, a conclusion is made based on the study.

II. METHODOLOGY
The whole process of designing the Ackerman geometry is broken down into the following nine simple steps. When Int. J. of Analytical, Experimental and FEA www.rame.org.in followed in order, these steps will result in a fully defined and functional variable Ackerman steering system. Lastly, the manufacturing of the steering mechanism is discussed.

A. Ackermann Steering Condition
The very slow movement of a vehicle results in a kinematic condition between the inner and outer wheels that allows a slip-free turn. This condition is termed as Ackerman condition and is usually expressed by Equation 1.

− =
Where is the steering angle of the inner wheel, and is the steering angle of the outer wheel.
The inner and outer wheels are defined based on the turning center O [10]. The Center of Mass of the steered vehicle will turn on a circle with a radius , which is determined using Equation 2.
Where is the cot-average of the inner and outer steer angles. The cot values are calculated by utilizing Equation The is termed as the turning radius of the vehicle [10].
All these parameters are shown in Figure 2.

B. Turning Radius Estimation
The Estimation of the turning radius range is another crucial yet less discussed step in designing the steering system. For a car to have the fastest corner and least lap time, there is a need to corner on the maximum turning radius possible for that turn. This is to provide maximum braking and acceleration capacity. So, all the corners on the track are studied, and the maximum possible turning radius for each corner is determined based on tire and engine specifications.
The smallest value from the list of those maximum possible radii is taken and further reduced to give the vehicle some additional cornering capability. It is denoted as the least turning radius required. This provides the turning radius range for the design of any geometry.
Matlab codes 2&3 (Annexure) were generated to determine the value of steered wheel angles and for the turning radius required to find the range of turning angle needed on the inner wheel.

C. Space Requirement Calculation
Calculation of the space required is necessary to ensure that the steering system opted fits the race track conditions before proceeding to further calculations. The racing track's width should always be more than the required space for the system to work as expected.
The space required can be calculated very easily by applying simple trigonometry given in Figure 3. = √( tan + 2 ) 2 + ( + ) 2 − tan (4) Where, ΔR is the space required for the vehicle to take a turn.
A Matlab code 4 (Annexure) was designed to determine the required turning space (Annexure) and compare it with all the competition's track sizes.

D. Trapezoidal Steering System
Achieving an Ackerman condition at all instances in a corner is not possible by a simple mechanical method so it requires all the wheels to be steered independently [11]. Therefore, a system is designed to perform the desired Ackerman system to the closest proximity possible.
The type of steering mechanism used to achieve the Ackerman geometry required by us was a symmetric fourbar linkage, called a trapezoidal steering mechanism as shown in Figure 4. The mechanism has two characteristic parameters: angle and offsets arm length . A steered position of the trapezoidal mechanism is also shown in Figure 5 to illustrate the inner and outer steer angles and . The relationship between the inner and outer steer angles of a trapezoidal steering mechanism is A Matlab code 5 (Annexure) was generated based on Equation 6 [10] to obtain the values of inner and outer wheel turning angles for the system designed.

Minimizing the RMS Error between Ackerman and Trapezoidal Systems
Since Ackerman's condition at all the instances of a turn can't be achieved simultaneously, so the steering for the whole range of turning radius is optimized [12]. This optimization using the root mean square (RMS) error gave us the least deviation possible from the Ackerman system across the entire range of turning radius. containing all these points is the solution for quickly varying the Ackerman geometry [11].

H. Minimizing Bump Steer
After finalizing and values, the last variable remaining in the coordinates of the tie rod outboard point is its height. This variable affects suspension characteristics more than it affects the steering characteristics. So the decision of this parameter is taken based on the suspension graphs obtained from the kinematic analysis of the system.
It is mainly varied to minimize the bump steer while examining the other graphs to avoid any unwanted variation of suspension parameters.
Other than suspension characteristics, it also affects the upright design and placing of the steering rack. Therefore sometimes, a little compromise is made on the performance in order to design and accommodate the systems adequately.
Adams Car was used to analyze and optimize the suspension parameters of the car [13].

I. Getting the Tie Rod Inboard Point
A straight line connecting the tie rod outboard point to the instantaneous center (IC) of the wheel is the domain for the inboard point. The height of the inboard point decides the height of the steering rack, so the decision made is based on the positioning of the steering rack [13,14].
After placing the steering rack, the intersection of the steering rack plane and the line connecting the outboard point to IC is the inboard point.

J. Material Selection and Manufacturing Process
As the design operates on fluctuating load cycles, it is essential to select material that has a high fatigue strength For assembly firstly the deep groove ball bearings were press-fitted in the casing at Satyamoorty industry, Vellore on a 20T hydraulic press. The Bevel gears were subsequently pressed in the respective casing using the same machine.
Gear meshing was done on the lathe machine in the same industry. The steering shaft and steering column were mated with bevel gear assembly at the workshop at Vellore institute of technology using a mallet. The complete assembly was secured with a help of a safety wire. The steering column was assembled with a coupler using a metal key, and the coupler was assembled with the steering column by aligning the internal splines of the coupler with splines on the steering rack. Figure 6 depicts the designed assembly of the entire steering geometry which was then used in the car.

B. Turning Radius Range
The turning radius range for our car turned out to be 3000 mm to ∞ (when the vehicle is in straight-line motion). The range was sufficient enough to perform up to the car's maximum capacity.

C. Wheel Turn Angle Range
For the corresponding minimum turning radius i.e. 3000 mm, the maximum inner wheel turning angle was obtained from the Matlab code and taken as 35° for further calculations. This gave us the turning angle range for the inner wheel from 0 to 35 degrees, corresponding to the required turning radius range. The range was properly functional, with all the clearances satisfied.    figure 9 are the same as in figure 7 for Ackerman geometry, thus verifying that the method and the calculations followed were correct.

D. RMS Minimization
The value for the first point of Ackerman geometry was obtained as = 30.14 at = 116 mm with the minimum RMS value of 2.622 * 10 −6 . This iteration was capable of running the car in all the events of the competition, though multiple outboard points were determined for event-wise better performance.
Matlab code 6 (Annexure) was used to obtain the required length of the steering moment arm to achieve the minimum RMS error which resulted in the graph shown in Figure 10.

E. Front View Suspension and Steering Geometry
The given sketch in Figure 11 represents the front view for the suspension and steering geometry designed on Solidworks. The geometry obtained from joining the outboard point of the tie rod to IC for finding the inboard point of the tie rod gave us the minimum possible bump steer.

G. Trapezoidal System
Matlab code 5 (Annexure) was used for finding the relative turning angles on the inner and outer wheels that are achieved from the designing of the trapezoidal system and the graph obtained is shown in Figure 13. The trapezoidal system designed was very close to the required Ackerman system, and the maximum possible deviation was observed to be 1.3° after comparing figure 7 and figure 13. The difference was small enough to run the car on this system.

IV. CONCLUSION
This paper helps to determine the right choice for the selection of the steering mechanism. In this paper, a vigorous study is carried out that resulted in many graphs which were