Unity aircraft physics

Question

I want to make a simple aircraft controller, what looks like little realistic in unity. I watch some video from airplane physics. and make a simple script in unity, but if I start, my plane cant move or if I change drag to zero, it cant lift. I tried to use real data and get it from wiki(F22 Raptor). To my game object, I gave the rigidbody component mass = 19670 kg. Engine thrust = 2 * 116000.0f Newton.

    private void calculateEnginePower()
    {
        EnginePower = engineThrust * ThrottleInput;
    }

    private void calculateForces()
    {
        angleOfAttack = Vector3.Angle(Vector3.forward, rb.velocity);
        angleOfAttack = Mathf.Clamp(angleOfAttack, 0, 90);

        coefficient = Mathf.Pow(1225.04f * rb.velocity.magnitude, 2) - 1; //M^2-2 where: M is mach.         

        if (coefficient > 0.0f)
            coefficientLift = (4 * angleOfAttack) / Mathf.Sqrt(coefficient);
        lift = 1.2754f * 0.5f * Mathf.Pow(rb.velocity.magnitude, 2) * coefficientLift * 78.04f; // densy1.2754 kg/m3, speed m/s , (F22)Wing area: 840 ft² (78.04 m²)

        coefficientDrag = 0.021f;
        rb.drag = coefficientDrag * 0.5f * Mathf.Pow(rb.velocity.magnitude,2) * 1.2754f * 78.04f;

        rb.AddForce(transform.up * lift);
        rb.AddForce(transform.forward * EnginePower);
    }

used these formulas:

for Lift force: Lift formula for Lift coefficient: Cl formula for Drag: Drag formula and for Drag coefficient: I used data from wiki too (0.021f).

Answer 1

So there are a number of issues with your code. I've outlined them below;

Calculate Forces

---

Issue: angleOfAttack = Vector3.Angle(Vector3.forward, rb.velocity);

• Vector3.forward and rb.velocity are both in world-space. AoA is the angle between the local chord-line of your wing and the aircraft's velocity.
• Vector3.Angle will return an unsigned angle. AoA must work in both positive and negative directions otherwise negative pitch and inverted flight would not be possible.

Solution: Move rb.velocity to local-space and solve for AoA with trigonometry.

// *flip sign(s) if necessary*
var localVelocity = transform.InverseTransformDirection(rb.velocity);
var angleOfAttack = Mathf.Atan2(-localVelocity.y, localVelocity.z);

---

Issue: coefficient = Mathf.Pow(1225.04f * rb.velocity.magnitude, 2) - 1;

• 4α/sqrt(M^2−1) is a supersonic wave coefficient for M > 1. At zero velocity this equation will reduce to sqrt(-1) which is an imaginary number that will produce NaN. Mach is expressed as M=V/C where V=velocity and C=the speed of sound. Your 1225.04f constant must be C in units of km/h and not m/s as required. You're also multiplying and not dividing as given in the equation.

Solution: Simplify your equations with Lifting Line Theory.

var aspectRatio = (wingSpan * wingSpan) / wingArea; 
var inducedLift = angleOfAttack * (aspectRatio / (aspectRatio + 2f)) * 2f * Mathf.PI;
var inducedDrag = (inducedLift * inducedLift) / (aspectRatio * Mathf.PI);

Source: Aerospaceweb.org

---

Issue: rb.drag = coefficientDrag * 0.5f * Pow(rb.velocity.mag,2) * 1.2754f * 78.04f;

• rb.drag is not required since we're calculating and applying drag manually.

Solution: Set the rb.drag property to the smallest possible value.

rb.drag = Mathf.Epsilon; // set in Awake

---

Issue: rb.AddForce(transform.up * lift);

• transform.up is not correct for lift. Lift acts perpendicular to velocity while drag acts parallel.

Solution: Compute the lift direction by crossing the normalized velocity vector with the aircraft's lateral direction and apply drag opposite to velocity.

// *flip sign(s) if necessary*
var dragDirection = -rb.velocity.normalized;
var liftDirection = Vector3.Cross(dragDirection, transform.right);
rb.AddForce(liftDirection * lift + dragDirection * drag);

---

Your lift equation looks OK, so putting it all together would look something like this; (Untested)

public float wingSpan = 13.56f;
public float wingArea = 78.04f;

private float aspectRatio;

private void Awake ()
{
    rb.drag = Mathf.Epsilon;
    aspectRatio = (wingSpan * wingSpan) / wingArea;
}

private void calculateForces ()
{
    // *flip sign(s) if necessary*
    var localVelocity = transform.InverseTransformDirection(rb.velocity);
    var angleOfAttack = Mathf.Atan2(-localVelocity.y, localVelocity.z);

    // α * 2 * PI * (AR / AR + 2)
    var inducedLift = angleOfAttack * (aspectRatio / (aspectRatio + 2f)) * 2f * Mathf.PI;

    // CL ^ 2 / (AR * PI)
    var inducedDrag = (inducedLift * inducedLift) / (aspectRatio * Mathf.PI);

    // V ^ 2 * R * 0.5 * A
    var pressure = rb.velocity.sqrMagnitude * 1.2754f * 0.5f * wingArea;

    var lift = inducedLift * pressure;
    var drag = (0.021f + inducedDrag) * pressure;

    // *flip sign(s) if necessary*
    var dragDirection = rb.velocity.normalized;
    var liftDirection = Vector3.Cross(dragDirection, transform.right);

    // Lift + Drag = Total Force
    rb.AddForce(liftDirection * lift - dragDirection * drag);
    rb.AddForce(transform.forward * EnginePower);
}