Determining the torque needed to rotate an object to a given rotation

Fun question. Just like F = m a for linear forces, T = I alpha for angular forces. T is the torque, I is the inertia 3x3 tensor, and alpha is the angular acceleration. So basically your question amounts to finding an angular acceleration from a given change in rotation, and then multiplying that by I to get T.

Angular acceleration is a Vector3 whose direction is the axis of rotation and magnitude is rotational acc. in radians/sec^2. Since you already have two direction vectors (which need to be normalized), you can simply compute x = Vector3.Cross(oldPoint, newPoint) to get the required axis of rotation. This is the direction of alpha, but you still need the correct magnitude. We want radians/sec^2, so we need the angle between the two vectors. The magnitude of cross product is sin(theta) |v| |u|, since length of v and u are both 1, we just need Asin(x.magnitude).

Since you want to fully reach your newPoint in one frame, you can instead apply an impulse which is sort of like an instantaneous acceleration or change in velocity. So to summarize.

Vector3 x = Vector3.Cross(oldPoint.normalized, newPoint.normalized);
float theta = Mathf.Asin(x.magnitude);
Vector3 w = x.normalized * theta / Time.fixedDeltaTime;

This gives us the desired change in angular velocity (w). Now we just multiply by the inertia tensor. Unfortunately this is in some weird diagonal space. It is easiest to transform w into this space, compute T, then transform T back to global coords.

Quaternion q = transform.rotation * rigidbody.inertiaTensorRotation;
T = q * Vector3.Scale(rigidbody.inertiaTensor, (Quaternion.Inverse(q) * w));

Then just apply T to the rigidbody:

rigidbody.AddTorque(T, ForceMode.Impulse)

NOTE: PhysX seems to limit the speed of rotation, so this actually only works if the amount you are rotating by isn't too large.

Gentlemen, what is the sense of multiplying by 'inertiaTensorRotation' when it is always W=1 and XYZ=0, at least this is what Debug.Log shows me?

Taking hellcats's answer and fixing the "wobbliness" reported by aleiby, I came up with this :

 var x = Vector3.Cross(currentDir.normalized, newDir.normalized);
 float theta = Mathf.Asin(x.magnitude);
 var w = x.normalized * theta / Time.fixedDeltaTime;
 var q = transform.rotation * rigidbody.inertiaTensorRotation;
 var t = q * Vector3.Scale(rigidbody.inertiaTensor, Quaternion.Inverse(q) * w);
 rigidbody.AddTorque(t - rigidbody.angularVelocity, ForceMode.Impulse);

The fix is to substract the current angular velocity. I hope this will be useful for other people googling around for this.

If you modify the inertia tensor yourself, the ForceMode can sometimes be changed to VelocityChange.

  objectHeld.transform.localRotation = Quaternion.Euler(0.0f, rotationY, 0.0f);

I stumbled on this thread looking for the answer - so here it is for anybody else who's looking this example allows an object to rotate to face another object using torque only

change speed etc with a float rotationTorque and change the ridged body angular drag to suit

     Vector3 targetPosition = transform.InverseTransformPoint(new Vector3(target.transform.position.x,target.transform.position.y, target.transform.position.z));
         
     requiredTorqueX = (targetPosition.x / targetPosition.magnitude);    
     requiredTorqueY = (targetPosition.y / targetPosition.magnitude);
         
     rbody.AddRelativeTorque(((rotationTorque) *requiredTorqueY),((rotationTorque) * requiredTorqueX) * -1 , 0f, ForceMode.Force);

  

Hi hellcats, thanks for the detailed answer.

I am trying to implement your solution in my project and I am running into difficulty.

Here's my situation: http://imageshack.us/photo/my-images/706/explanationp.png

My object is travelling at a fixed speed on a heading in a 2-d coordinate space (z is always 0). On a mouse click I would like to apply a relative torque to turn it to a new heading. The rotational axis is always (0,0,1) - the z-axis.

The way I see it, there are two ways to rotate an object with a rigidbody: 1) switch on kinematics and do it manually or, 2) apply a rotational force. I'd like to use the second option.

My FixedUpdate function looks like this:

 void FixedUpdate () 
 {    
     rigidbody.AddRelativeTorque ( GetTorque(), ForceMode.VelocityChange);                                                                                  
     rigidbody.AddRelativeForce (forwardDirection * Time.deltaTime, ForceMode.VelocityChange); 
 }

The GetTorque function is:

 Vector3 GetTorque()
 {
     m_vHeading = rigidbody.velocity; 
     toTarget = (target.transform.position - gameObject.transform.position);
     
     x = Vector3.Cross(m_vHeading.normalized, toTarget.normalized);
     angle = Mathf.Asin(x.magnitude);
     Vector3 w = x.normalized * angle / Time.fixedDeltaTime;
 
     Quaternion q = gameObject.transform.rotation * rigidbody.inertiaTensorRotation;
     Vector3 T = q * Vector3.Scale(rigidbody.inertiaTensor, (Quaternion.Inverse(q) * w));
             
     return T;
 }
 

Where totarget is an empty gameobject controlled by mouse move. Can anybody tell me if I am on the right track? Am I over complicating something that should be simple. Any and all help is greatly appreciated.

Thanks, BinaryX