Correct rotation with Quaternion

Question

I have some problems with a rotating marble.
I've tried it with Matrix.CreateFromYawPitchRoll and Matrix.CreateRotation but there were some problems, I think it's due to the Gimbal lock effect.

So, I've tried using quaternions instead, but nothing changed.
When moving on only an axis it works fine, but when the rotation occurs on two different axes the marble still rotates on wrong axes.

Here's my code:

// declarations
Vector3 Position = Vector3.Zero;
Vector3 Rotation = Vector3.Zero;
Quaternion qRotation = Quaternion.Identity;

AbsoluteBoneTransforms = new Matrix[Model.Bones.Count]; 
            Model.CopyAbsoluteBoneTransformsTo(AbsoluteBoneTransforms); 

In the Update method:

Position += speed;
Rotation = speed * MathHelper.ToRadians(-1.5f);

Quaternion rot = Quaternion.CreateFromAxisAngle(Vector3.Right, Rotation.Z) *
                Quaternion.CreateFromAxisAngle(Vector3.Backward, Rotation.X);

qRotation *= rot;

And in the Draw method:

effect.World = AbsoluteBoneTransforms[mesh.ParentBone.Index] * 
  Matrix.CreateFromQuaternion(qRotation) * Matrix.CreateTranslation(Position);

What's wrong?

EDIT

I've tried calculating directly the axis of rotation of my marble, instead of using combination of multiple axes:

angle += speed.Length() * angularVelocity;
qRotation = Quaternion.CreateFromAxisAngle(Vector3.Cross(speed, Vector3.Up), angle);
qRotation.Normalize();

angle is a float that keeps track of the current movement. This solution doesn't seem to create Gimbal lock, but marble rotations aren't correct, it seems that the rotating speed is not constant, but became faster and slower over time, I can't understand why.

Answer 1

Although I don't know any XNA, this might help you "cheat" your way into achieving that rolling effect of the marble:

Position += speed;

angle = Magnitude(speed) * MathHelper.ToRadians(-1.5f);

Quaternion rot = Quaternion.CreateFromAxisAngle(Vector3.Cross(speed, Vector3.UP), angle);

qRotation *= rot;

So, instead of multiplying quaternions and achieving a chaotic wobbly effect, you should only update the angular position and use that to achieve the rolling effect.

Later edit:

What happens when your sphere moves from position A to position B:

• the speed is used to translate it (you should consider adding a deltaTime to make it more FPS independent)
• you know how much the ball moves: |AB| = |speed| (for your particular case) so you can determine how much it has rolled/rotated around the cross(speed, up) vector/axis. This amount is the angle |AB|/(2*pi*radius) - you should incorporate the marble radius to achieve believable roll with no sliding/slippage
• lastly, qRotation is updated to reflect the sphere's current rotation w.r.t. to its starting pose

Of course, the correct way to achieve rolling behaviour from a physically accurate POV is to use the inertia tensor.

I implemented a short OpenGL demo using the following snippet to update the rolling sphere (and it worked):

  double radius = 0.1;
  static Quaternion qRot = identityq<double>();
  static double dTime = 0.002;

  Position += Velocity * dTime;
  double angle = length(Velocity) / radius * dTime;
  Vector3 axis = cross(Vector3(0,0,1), Velocity);
  normalize(axis);

  qRot = quat_from_axis_angle<double>(axis, angle) * qRot;