Overview

Four-wheel steering can help a vehicle maneuver by maximizing the available tractive force. This leads to a smaller turning radius at low-speeds and higher stability during high-speed lane changes. I will use control theory to ensure that the wheels of an FSAE vehicle are in an appropriate position. I will utilize the progress already created by T. Wisniewski, A. Sparrow, and P. Rowsell as a starting point and will update the circuitry and signal processing to create a vehicle that can utilize four-wheel steering.

Introduction

Background Information

Four-wheel steering is a growing trend on high-end production vehicles such as the Porsch 918 Spider and Acura RLX (1). It allows the vehicle to not only be more maneuverable at low speeds, but to grip the road better at high speeds. At low speeds, this is accomplished by turning the rear wheels in the opposite direction of the front wheels to create a smaller turning radius. While at high speeds, the rear wheels turn in the same direction as the front. If implemented correctly, the vehicle will be safer because of the improved stability and handling.

This system would give a Formula SAE team an advantage when competing in dynamic events due to the dynamic event types and layout. Formula SAE is an international, intercollegiate competition to design and build a small, Formula style vehicle. As of the 2016 Competition in Michigan, no other FSAE team has a four-wheel steering system. Teams are judged based on their performance in dynamics events, the quality and ingenuity of their designs, and the costs of their vehicles. Four-Wheel steering should reduce the track time for the skid-pad event by 1.35% per lap and reduce a lap time for autocross and endurance by 0.5% (4). A four-wheel steering set-up would help the design and dynamic events while playing a small tradeoff in cost.

BigPicture

Big Picture of the 2015 Frame with Four-Wheel Steering

During the fall of 2016, three mechanical engineering students from Washington University focused on creating a four-wheel steering prototype (2). They used the frame from the 2015 Wash U Racing race car and re-purposed the suspension to include two electronic linear actuators to rotate the rear-wheels independently from each other. The front wheels were mechanically connected to the steering wheel through a rack and pinion, as required by FSAE rules. Thus, through the control of an Arduino, the frame had four-wheel steering. The suspension geometry for the rear of the 2016 and 2017 vehicles are exactly the same; therefore, this design could be transferred to more recent versions for testing.

Linear Actuators without Tie-Rods

Schematic of the design for Four-Wheel Steering

Four-Wheel Steering Prototype on 2015 Frame

Outboard view of the set-up of the 2015 frame with actuators

Their set-up replaced a solid linkage from the upright to the frame that originally set the toe of the vehicle with an electrical, linear actuator. Four relays were used to switch the voltage on the actuators to change the direction of travel. In effect, this method controlled the speed of the motor by having the motor either running at full speed or not when the relays were switched. From the relay’s data specifications, the relays can respond on the order of 10 milliseconds. In order to better control the actuators, a new system will need to be implemented using a PWM signal. This will allow the motor’s speed to be controlled much more finely, will continue to run while the Arduino is calculating the next update, and will allow the user to shape the response to a more desirable one. Because this set-up was only attached to a bare frame, two potentiometers were used to simulate the speed of the vehicle and the angle of the steering wheel.

Control Panel on 2015 Frame

Original Control Panel on the 2015 vehicle with potentiometers to determine speed and steering angle

The Arduino, relays, and transistors were stored in a 3D-printed box. Due to time constraints, a breadboard was used to connect the various electrical components. This resulted in a tight fit in the electrical box and problems with keeping connections secured.

Electrical Box

Electrical box housing Arduino, relays, and other electronics for the original version

Finally, a control scheme will be implemented to ensure that the wheels are rotated appropriately to maximize available tractive force. Without this feature, the wheels may be hindering performance and endangering the driver. Implementing a four-wheel steering system for Wash U Racing will help the team better understand the vehicle’s dynamics, improve the performance of the vehicle, and open the door to more control systems.

Problem Statement

In order to establish an effective and reliable four-wheel steering system, the 2016 Wash U Racing race car frame will be retrofitted with a control system to independently control the angle of the rear wheels. The direction the wheels point in will be dependent on the angle of the steering wheel and the speed of the vehicle as well as other physical characteristics. With this information, the vehicle will know when to steer in concert (high-speeds) to reduce tire slip and when to steer in opposite directions (low-speeds) to reduce the radius of the turn. The control set-up should follow all of the rules dictated by FSAE.

Objectives

The objective of this project is to develop a stable four-wheel steering control scheme for an FSAE vehicle’s frame. It is much safer to conduct tests at low speeds to understand the dynamics and to visualize that the motors are working correctly. Future iterations will test the vehicle at speed to determine if the benefits are seen under race conditions. Various control schemes will be analyzed in Matlab to determine their effectiveness. Then a physical set-up will be built and tested. The set-up, at low speeds, should reduce the turning radius of the vehicle as well as to increase the maximum lateral acceleration of the vehicle. The minimum radius should decrease by at least 25% (3). Without four-wheel steering, the vehicle has a turning radius of 20 feet; therefore, the goal minimum radius is 15 feet. Additionally, the maximum acceleration should increase by 20% due to increased transaction during cornering. At high speeds, the tires should turn in concert to increase stability in lane changes. The control system needs to respond quickly enough to prevent any lag between driver input and motor output; otherwise, the vehicle will become unstable.

The desired steering angle was determined by being a fraction of the front toe angles. From before, the desired toe angles were determined using a quadratic fit function to the ideal Ackerman angles. This estimates the angle that the toe would be at given the wheel has rotated a certain percentage.

DesiredToeAngle

Matlab simulation of the desired toe angle

From here, we can work out to how far the actuator would have to travel to reach the desired toe angle. This is estimated as a simple ratio to convert from degrees to thousandths of an inch. However, due to the origin being non-differentiable, it was determined in previous work to have a transition region between the high and low speeds to smooth out the transition and prevent the controller from having huge swings in direction.

 

ActuatorPosition

Matlab simulation of the desired actuator position at low speeds

ActuatorPositionHighSpeed

Matlab simulation of the actuator position at high speeds

Method

Vehicle speed and the steering angle will determine what state the system is to determine the angle that the rear wheels should point and what motor voltage needs to be sent to achieve the desired input. This relies on reliable sensors that have been tested thoroughly.

Logic

Block diagram of the logical flow of the controller

Due to time constraints, I will use a PID controller to control the linear actuators. This will reduce the complexity of the controller and will allow time to test the hardware. In order to understand the whole system, I will understand how each components responds to build a comprehensive model. I will measure the response of the actuators to determine the gains associated with the controller. Additionally, I need to measure the response of the sensors before integrating them into the system. This will help control any error and will provide good ideas of where improvements can be made.

The code for this program has six main steps. It first needs to get information from the steering angle and the vehicle speed, convert that to usable information and determine the ideal turning radius of the rears. Then, it must measure the current angle of the rears, run it through the discrete-time PID controller, and finally send that information to the motors.

Pseudo Code

Pseudo Code of the controller

Data Collected

In order to choose the gains for the PID controller, it is vital to understand the plant model for the actuator. Using a DC voltage source and an NI Instrument board, I measured the step-response of the actuator. The Matlab simulation of the step-response mirrors the measured data. The motor was supplied 12V with a max of 2 amps to prevent any damage to the motor. The potentiometer mounted inside of the actuator housing was supplied five volts through the board, and, while the computer was logging data, the 12 volts was turned on and the motor extended. With the voltage turned off, the leads of the motor were quickly switched, and the motor retracted.

Testing Setup

Using the NI-Elvis Board and software to record the data from the H-Bridges and Actuator

StepResponse

Matlab simulation of a step response of the actuator

Using the plant model, I created a PID controller to control the distance the rear wheels travel. The gains for this model were chosen by increasing the proportional gain until the phase margin was below 50 degrees and then adding as much differential gain as possible while keeping the response speed below 0.6 inches per second, the physical limit of the actuators. With a little bit of integral control to remove steady-state errors, the model is able to respond quickly and accurately in simulation. This corresponded with a proportional gain of 50, an integral gain of 0.01, and a differential gain of 0.01. To model the speed of the switches and the code, a delay was added to the feedback loop to determine the effect of the delay. For the new system, the modeled delay was 10 microseconds.

SimulationBlockDiagram

Block diagram of the PID controller with a plant model and saturation and delay components

Next, I will measure the actual response with the h-bridges to determine how accurate the physical set-up matches the simulation result.
The previous method to reduce overshoot was to insert a variable delay in the code to get the actuators to settle closer to the intended target. The delay could be due to the relays being too slow to prevent the motors from overshooting.
Another issue is if the actuator’s potentiometer voltage drifts causing issues with the accuracy of the actuators position. More research will be done to see if the potentiometers do drift and if so whether a different circuit set-up will prevent this issue.

Results

First, I measured the response of the actuators to various PID gains. The first is the simple proportional integral (PI) controller. This is due to my implementation of a discrete time PID controller; however, the time that I integrate over means that the integral gain is extremely low. Thus, my PI controller is almost a proportional (P) controller. There is some readjustment in the right motor around the 2 second (2000 millisecond) mark, which could be due to the integral gain starting to make an adjustment and going too far.

PI

Proportional Integral (PI) response

Next, I used the gains that I estimated from my Matlab simulation using gains of K = 4.167, Ti = 100,000, and Td = 0.01*4.167 = 0.04167. This led to a very desirable response that seemed to track the input well. It is extremely similar to the PI controller with low integral gain but tends to be smoother and has fewer areas of readjustment.

EstimatedPID

Estimated PID gain experimental response

Finally, I decided to see what would happen if I could induce a underdamped response to the system. This is to see how close I am to a critically damped response. If my gains are close to creating an underdamped (oscillatory) response, then I know that I am close to a critically damped response.

Underdamped

Underdamped Response with gains K = 5.5 Ti = 50000 Td = 0.055 with percent overshoot of 3.2%

I would prefer to have an underdamped response rather than an underdamped response because I want the rear wheels to be moving laterally as little as possible to prevent the wheels from being over utilized. It is better to have the wheels be almost in the correct position than to be too active and causing the rear end to become unstable. Therefore, I plan on using the estimated PID gains due to them being close to a critical response while being on the safe side. I have finished wiring up the car but I made the mistake of only taking videos and taking no pictures. I will add pictures when I get back.
I measured the steering diameter with low-speed four-wheel steering and with only two-wheel steering by rolling the frame down a ramp and taking a sharp right at the end of the ramp. Not only do I measure the turning radius, but I can measure if there is a measurable delay in the response of the rear wheels. Two-wheel steering had a turning diameter of 25 feet; whereas, low-speed four-wheel steering had a measured turning diameter of 20 feet. This is a 20% decreases, which is close to my goal of 25%. I did not measure the non-delayed four-wheel steering turning diameter to compare.

Discussion

The four-wheel steering design was able to reduce the turning radius by 20% which is close to my goal of 25%. Through some minor adjustments to maximum toe angle and steering adjustments, I believe I could achieve that final 5\%. However, having a ten foot turning radius means that the car will not be struggling to make the turns in the dynamic events. Instead of creating a two-wheel steering design with extreme toe angles, a four-wheel steering design makes small turning radius’ achievable while keeping the toe angle reasonable. The tradeoff is complexity in design and steering design.

In theory, the car should be able to take the normal turns at a faster speed due to the slip of the wheel being minimized. I was not able to test this aspect of four-wheel steering out. When running the Matlab simulations and from the Bode plot, the phase margin occurs at 2.44 radians per second, but according to the Moose test conducted by Breuer, a driver’s max input speed occurs at 23.3 radians per second (1335 degrees per second); therefore, to prevent the system from lagging, the phase margin should occur at 23.3 radians per second or later (5). With the current actuator set-up, this is not possible. More research will have to be done to determine what characteristics the actuator needs.

In some ways, the system is hard to characterize. When steering the vehicle, it just works, but viewing the car from behind, the change is hardly noticeable. This bodes well for future work because the driver will only feel an improvement in performance without having to think about how the new system interacts with the car.

Future Work

In order for an Wash U to implement four-wheel steering into their vehicle, there are several steps to make the system more secure and stable. Faster actuators, a PCB for the H-bridges, and a slight redesign of the rear suspension will be necessary. Faster actuators will help the system respond quicker to increase the phase margin frequency. A PCB will allow the team to choose the H-Bridges they desire as well as to be able to have them securely connected with the Arduino. Redesigning the rear suspension will help the steering from influencing the dynamics of the suspension. These changes will allow car to have a successful four-wheel steering design that will help the team improve in a unique way.

References

[1]  M. Austin. The return of four-wheel steering. Popular Mechanics, Dec 2013. http://www.popularmechanics.com/cars/a9862/the-return-of-four-wheel-steering- 16311550/

[2]  T. Wisniewski, A. Sparrow, and P. Rowsell. Four-wheel steering. Technical report, Washington University in St. Louis, 2016.

[3]  M. Ariff, H. Zamzuri, M. Nordin, W. Yahya, S. Mazlan, and M. Rahman. Optimal control strategy for low speed and high speed four-wheel-active steering vehicle. Journal of Mechanical Engineering and Sciences, 8:1516–1528, Jun 2015.

[4]  J. Allwright. Four wheel steering (4ws) on a formula student racing car. Technical report, Swinburne University of Technology, 2015.

[5] J. J. Breuer. Analysis of driver-vehicle-interactions in an evasive manoueuvre – results of ,,moose analysis of driver-vehicle-interactions in an evasive maneuver – results of moose test studies. Technical report, Daimler-Benz AG, 1997.