This project is part of Udacity’s Self-Driving-Car Nanodegree. The project description and build instructions can be found here, the required simulator here.
The goal of this project is the implementation and tuning of a PID controller that is used to control the steering angle of a vehicle so that it can follow its assigned trajectory on the lake track.
PID control overview
The PID controller is one of the most frequently used control loop feedback mechanisms in practice. The control loop implemented for this project is shown below.
The plant in a control loop is the system that we want to control, the system whose behaviour we want to affect. In this project the system is the vehicle in the simulator, and we want it to drive in the center of the lane. The input into the system is the actuating signal, in our case a specific position of the steering wheel. The vehicle will take the provided steering command and change its lateral position according to its specific system dynamics. The lateral position of the vehicle is the output of the system, the controlled variable. The basic idea of a control system is to figure out how to generate the approriate actuating signal, so that the system produces the desired output. In our case this means, how to move the steering wheel, so that the vehicle drives in the center of the lane. The system output is fed back and compared to the reference variable: the center of the lane. This shows us how far off the vehicle is from where it should actually be. The difference is measured by the error term, which in our case is the cross track error, the distance of the vehicle from the centerline. If the vehicle’s position is near the lane center, the cross track error is close to zero, and that is exactly what we want, a zero error. And here is where the contoller comes into play. The controller’s task is to convert the error term into a suitable actuator command: an appropriate position of the steering wheel. This shall be done in a way, that over time the error converges to zero. So how does the PID controller accomplish this task?
Let’s consider that the vehicle is too far to the left. Then it needs to turn to the right and vice versa. But how much?
Proportional controller term
One way to set the steering wheel angle is to use the so-called proportional control. It determines the steering command by multipliying the present cross track error e(t) with a scaling factor called the proportional gain KP. The proportional controller steers all the more, the further away the vehicle is from the centerline, or in other words the higher the cross track error e(t) is. When the vehicle approaches the center of the lane, the steering angle gets smaller and smaller, being zero at the moment the vehicle reaches the centerline. However, the vehicle may still overshoot, since its orientation typically is not aligned with the centerline. If the proportional gain is set too high, the system can become unstable with increasing oscillation magnitudes (left video below). In contrast, if the gain is set to low, the steering commands will not be large enough to keep the vehicle inside the lane (center video). The gain must have a certain magnitude to provide steering commands that are large enough to navigate the vehicle around the curve. However, the proportional controller alone is not sufficient to do this safely, as the abruptly increasing cross track error in the curves leads to overshooting tendencies resulting in strong oscillations (right video). We need to consider an additional control term.
| high KP (unstable) | low KP (unresponsive) | medium KP (still oscillating) |
|---|---|---|
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Derivative controller term
A good candidate for an extra measurement is the cross track error rate, the time derivative of the error. By multiplying it with its own gain KD we get the derivative controller term, which measures how fast the error evolves over time. In our case this means, how fast the vehicle is moving to or away from the center of the lane. The derivative gain KD needs to be tuned simultaneously to the proportional gain KP. When the vehicle approaches the centerline, the cross track error e(t) and its derivative de/dt have opposite signs and therefore act in opposite directions.
Conceptually we can think that increasing the proportional gain KP will increase the pull that the vehicle feels towards the center of the lane, and increasing the derivative gain KD increases the resistance the car will feel against moving too quickly towards it.
The derivative term can be interpreted as a prediction of the future error, and it is used to reduce overshooting and oscillation tendencies. If it is too low (for a given proportional gain), the system is called under damped, and it will still overshoot and oscillate. If it is too high, the system will be over damped, and it will take a long time to settle. Properly choosing the derivative gain allows the car to approach the desired trajectory quickly with a cross track error rate close to zero. This is being called critically damped. By adding the derivative control term, the intense oscillations of the plain P Controller could be reduced, and the vehicle now is able to take the curves.
Although a plain PD controller is sufficient to safely navigate the vehicle on this track, for reasons of completeness, let’s take a look on the last term of the PID controller.
Integral controller term
Environmental factors or mechanical defects can change the vehicle’s nominal behavior and thus the performance of the controller. For example if there’s a heavy crosswind the vehicle may drift sidewards unless the driver counteracts the wind force with a corrective steering command. The vehicle may experience a residual lane offset, called the steady-state error. One way to address this problem is to add yet another term: the integral term. This third measurement sums up the cross track error over the time, by taking the integral ∫e(t)dt. It gives an indication of whether the vehicle spends more time on one side of the trajectory than on the other. The integral term is multiplied by its own gain KI. If the gain is too large, normal controller fluctuations could get exaggerated and the controller can become unstable. However, if the gain is too low, the response to the dynamic changes could take to long. If the gain is just right, the controller can quickly reduce the steady-state error and return to its nominal performance.
PID control tuning
The combination of these three terms results in the PID control: a versatile controller that uses the present, the past, and a prediction of the future error to calculate the appropriate actuator commands. Each of the errors contributes some amount to the overall output of the controller, whereby the contributions are weighted by the corresponding PID gains. By properly tuning each of the gains, different characteristics of the dynamic system can be addressed, as shown in the subsequent table. However, note that the PID gains are dependent of each other, and changing one of them can effect the other two. The table shall only give a reference about the general effects of each of the gains on the system’s behaviour.
Project result
The PID gains were tuned manually, by observing their effects on the driving behaviour in the simulator and considering their influence on the system’s behaviour, as shown in the table above. The finally chosen gains are
Kp = 0.12
Ki = 0.00
Kd = 3.50
Since there was no systematic bias in the simulator causing a steady-state error, the integral
term was neglected, and a plain PD controller was used to complete this project. The
implementation of the controller can be found in the src folder.
