This is a tutorial on how you can minimize an error with the help of a PID controller. This is a technique used in real-life self-driving cars and I learned about it in the free course Artificial Intelligence for Robotics - Programming a Robotic Car.

The error we are going to minimize here is the error a self-driving car has when it's following a series of waypoints. This error is called Cross Track Error, or simply CTE, and is the distance between the car's actual position and the position it should have, which is the position closest to the car on a line between the waypoints:

Why do you need to minimize this error? You could just determine if the car should steer left or right to reach the waypoint(as explained here: Turn left or right to reach a waypoint?). This is working, but the wheels will move really fast to the left or right when the car is driving straight towards the waypoint, which is not looking good. To minimize this behavior, a good way is to take the rolling average of the steering angles, but now the car will look like it's drunk. So we need a better way, which is a PID controller.

So what's a PID controller. A PID controller consists of three parts:

- P - Proportional
- I - Integral
- D - Derivative

...which can be summarized in a rather scary equation:

But this equation becomes just a few lines of code. The first part is the P, which is simply just a multiplication of the CTE with a constant you have to determine. If we just have the P part of the PID controller, the result is yet again the drunk behavior we wanted to avoid:

To avoid this drunk behavior, we need the D part of the equation. This D is simply the difference between the current CTE and the CTE last time step divided by the time step, and you multiply this with another constant of your choice. This will make the car smooth in to the correct direction:

The last part of the PID controller is the I. This is simply the sum of all CTE multiplied with a third constant of your choice. The idea behind the I part is that if the car has a built-in error in itself (real-life technology is not as accurate as in the computer) the I will minimize this error. So you can probably avoid the I part if you want. Anyway, if all this is abstract, I've made a youtube video that will show why we need the PID controller:

First of all you need a car, and you can use Unity's wheelcollider example WheelCollider Tutorial. Then you need a series of waypoints. Everything should look like this:

The first script you have to attach to the car. To this script you have to add all waypoints (in the correct order), and the wheels.

using UnityEngine; using System.Collections; using System.Collections.Generic; //Modified basic car controller from Unity //https://docs.unity3d.com/Manual/WheelColliderTutorial.html [System.Serializable] public class AxleInfo { public WheelCollider leftWheel; public WheelCollider rightWheel; public bool motor; public bool steering; } public class CarController : MonoBehaviour { public ListaxleInfos; //Car data float maxMotorTorque = 500f; float maxSteeringAngle = 40f; //To get a more realistic behavior public Vector3 centerOfMassChange; //The difference between the center of the car and the position where we steer public float centerSteerDifference; //The position where the car is steering private Vector3 steerPosition; //All waypoints public List allWaypoints; //The current index of the list with all waypoints private int currentWaypointIndex = 0; //The waypoint we are going towards and the waypoint we are going from private Vector3 currentWaypoint; private Vector3 previousWaypoint; //Average the steering angles to simulate the time it takes to turn the wheel float averageSteeringAngle = 0f; PIDController PIDControllerScript; void Start() { //Move the center of mass transform.GetComponent ().centerOfMass = transform.GetComponent ().centerOfMass + centerOfMassChange; //Init the waypoints currentWaypoint = allWaypoints[currentWaypointIndex].position; previousWaypoint = GetPreviousWaypoint(); PIDControllerScript = GetComponent (); } //Finds the corresponding visual wheel, correctly applies the transform void ApplyLocalPositionToVisuals(WheelCollider collider) { if (collider.transform.childCount == 0) { return; } Transform visualWheel = collider.transform.GetChild(0); Vector3 position; Quaternion rotation; collider.GetWorldPose(out position, out rotation); visualWheel.transform.position = position; visualWheel.transform.rotation = rotation; } void Update() { //So we can experiment with the position where the car is checking if it should steer left/right //doesn't have to be where the wheels are - especially if we are reversing steerPosition = transform.position + transform.forward * centerSteerDifference; //Check if we should change waypoint if (Math.HasPassedWaypoint(steerPosition, previousWaypoint, currentWaypoint)) { currentWaypointIndex += 1; if (currentWaypointIndex == allWaypoints.Count) { currentWaypointIndex = 0; } currentWaypoint = allWaypoints[currentWaypointIndex].position; previousWaypoint = GetPreviousWaypoint(); } } //Get the waypoint before the current waypoint we are driving towards Vector3 GetPreviousWaypoint() { previousWaypoint = Vector3.zero; if (currentWaypointIndex - 1 < 0) { previousWaypoint = allWaypoints[allWaypoints.Count - 1].position; } else { previousWaypoint = allWaypoints[currentWaypointIndex - 1].position; } return previousWaypoint; } void FixedUpdate() { float motor = maxMotorTorque; //Manual controls for debugging //float motor = maxMotorTorque * Input.GetAxis("Vertical"); //float steering = maxSteeringAngle * Input.GetAxis("Horizontal"); // //Calculate the steering angle // //The simple but less accurate way -> will produce drunk behavior //float steeringAngle = maxSteeringAngle * Math.SteerDirection(transform, steerPosition, currentWaypoint); //Get the cross track error, which is what we want to minimize with the pid controller float CTE = Math.GetCrossTrackError(steerPosition, previousWaypoint, currentWaypoint); //But we still need a direction to steer CTE *= Math.SteerDirection(transform, steerPosition, currentWaypoint) * -1f; float steeringAngle = maxSteeringAngle * PIDControllerScript.GetSteerFactorFromPIDController(CTE); //Limit the steering angle steeringAngle = Mathf.Clamp(steeringAngle, -maxSteeringAngle, maxSteeringAngle); //Average the steering angles to simulate the time it takes to turn the steering wheel float averageAmount = 30f; averageSteeringAngle = averageSteeringAngle + ((steeringAngle - averageSteeringAngle) / averageAmount); // //Apply everything to the car // foreach (AxleInfo axleInfo in axleInfos) { if (axleInfo.steering) { axleInfo.leftWheel.steerAngle = averageSteeringAngle; axleInfo.rightWheel.steerAngle = averageSteeringAngle; } if (axleInfo.motor) { axleInfo.leftWheel.motorTorque = motor; axleInfo.rightWheel.motorTorque = motor; } ApplyLocalPositionToVisuals(axleInfo.leftWheel); ApplyLocalPositionToVisuals(axleInfo.rightWheel); } } }

Next up is the PID controller, which you also have to attach to the car. You have to experiment to find the three constants tau_P, tau_I, and tau_D, but I'm using 40, 0.05, and 70. There's actually a way to generate these constants called Twiddle, but I've found it's easier to test different values.

using UnityEngine; using System.Collections; public class PIDController : MonoBehaviour { float CTE_old = 0f; float CTE_sum = 0f; //PID parameters public float tau_P = 0f; public float tau_I = 0f; public float tau_D = 0f; public float GetSteerFactorFromPIDController(float CTE) { //The steering factor float alpha = 0f; //P alpha = -tau_P * CTE; //I CTE_sum += CTE; //Sometimes better to just sum the last errors float averageAmount = 20f; CTE_sum = CTE_sum + ((CTE - CTE_sum) / averageAmount); alpha -= tau_I * CTE_sum; //D float d_dt_CTE = (CTE - CTE_old) / Time.fixedDeltaTime; alpha -= tau_D * d_dt_CTE; CTE_old = CTE; return alpha; } }

The last script is the funny math you need to make all this work.

using UnityEngine; using System.Collections; public static class Math { //Have we passed a waypoint? //From http://www.habrador.com/tutorials/linear-algebra/2-passed-waypoint/ public static bool HasPassedWaypoint(Vector3 carPos, Vector3 goingFromPos, Vector3 goingToPos) { bool hasPassedWaypoint = false; //The vector between the character and the waypoint we are going from Vector3 a = carPos - goingFromPos; //The vector between the waypoints Vector3 b = goingToPos - goingFromPos; //Vector projection from https://en.wikipedia.org/wiki/Vector_projection //To know if we have passed the upcoming waypoint we need to find out how much of b is a1 //a1 = (a.b / |b|^2) * b //a1 = progress * b -> progress = a1 / b -> progress = (a.b / |b|^2) float progress = (a.x * b.x + a.y * b.y + a.z * b.z) / (b.x * b.x + b.y * b.y + b.z * b.z); //If progress is above 1 we know we have passed the waypoint if (progress > 1.0f) { hasPassedWaypoint = true; } return hasPassedWaypoint; } //Should we turn left or right to reach the next waypoint? //From: http://www.habrador.com/tutorials/linear-algebra/3-turn-left-or-right/ public static float SteerDirection(Transform carTrans, Vector3 steerPosition, Vector3 waypointPos) { //The right direction of the direction you are facing Vector3 youDir = carTrans.right; //The direction from you to the waypoint Vector3 waypointDir = waypointPos - steerPosition; //The dot product between the vectors float dotProduct = Vector3.Dot(youDir, waypointDir); //Now we can decide if we should turn left or right float steerDirection = 0f; if (dotProduct > 0f) { steerDirection = 1f; } else { steerDirection = -1f; } return steerDirection; } //Get the distance between where the car is and where it should be public static float GetCrossTrackError(Vector3 carPos, Vector3 goingFromPos, Vector3 goingToPos) { //The first part is the same as when we check if we have passed a waypoint //The vector between the character and the waypoint we are going from Vector3 a = carPos - goingFromPos; //The vector between the waypoints Vector3 b = goingToPos - goingFromPos; //Vector projection from https://en.wikipedia.org/wiki/Vector_projection //To know if we have passed the upcoming waypoint we need to find out how much of b is a1 //a1 = (a.b / |b|^2) * b //a1 = progress * b -> progress = a1 / b -> progress = (a.b / |b|^2) float progress = (a.x * b.x + a.y * b.y + a.z * b.z) / (b.x * b.x + b.y * b.y + b.z * b.z); //The coordinate of the position where the car should be Vector3 errorPos = goingFromPos + progress * b; //The error between the position where the car should be and where it is float error = (errorPos - carPos).magnitude; return error; } }

And that's it, your passengers in your self-driving car don't need to get seasick anymore!