How to avoid gimbal lock

Question

I am trying to write code with rotates an object.

I implemented it as:

Rotation about X-axis is given by the amount of change in y coordinates of a mouse and Rotation about Y-axis is given by the amount of change in x coordinates of a mouse.

This method is simple and work fine until on the the axis coincides with Z-axis, in short a gimble lock occurs.

How can I utilize the rotation arount Z-axis to avoid gimbal lock.

Answer 1

The simple solution is not to store the orientation of the object as angles around axes (X-, Y-, Z-axis), as for instance in euler angles.

Store the orientation of the object as a matrix or a quaternion.

This can cause gimbal lock, using euler angles:

class Object
{
    float m_angleAxisX;
    float m_angleAxisY;
    float m_angleAxisZ;
};

No gimbal lock:

class Object
{
    matrix m_orientation;   
};

No gimbal lock either:

class Object
{
    quaternion m_orientation;   
};

Now whenever the mouse is changed, multiply m_orientation with the orientation change coming from the mouse movement each frame.

Answer 2

This book (real-time rendering) helped me alot! See on page 66 and 70. It has very good graphics and explenation. Quaternions are on page 72 aswell! :)

Rotation about an Arbitrary Axis

This renders the camera with the rotation done by mouse input:

void Camera::getVectors(D3DXVECTOR3& up, D3DXVECTOR3& lookAt)
{
    float yaw, pitch, roll;
    D3DXMATRIX rotationMatrix;

    // Setup the vector that points upwards.
    up.x = 0.0f;
    up.y = 1.0f;
    up.z = 0.0f;

    // Setup where the camera is looking by default.
    lookAt.x = 0.0f;
    lookAt.y = 0.0f;
    lookAt.z = 1.0f;

    // Set the yaw (Y axis), pitch (X axis), and roll (Z axis) rotations in radians.
    pitch = m_rotation.x * 0.0174532925f;
    yaw   = m_rotation.y * 0.0174532925f;
    roll  = m_rotation.z * 0.0174532925f;

    // Create the rotation matrix from the yaw, pitch, and roll values.
    D3DXMatrixRotationYawPitchRoll(&rotationMatrix, yaw, pitch, roll);

    // Transform the lookAt and up vector by the rotation matrix so the view is correctly rotated at the origin.
    D3DXVec3TransformCoord(&lookAt, &lookAt, &rotationMatrix);
    D3DXVec3TransformCoord(&up, &up, &rotationMatrix);
}

// The Render function uses the position and rotation of the camera to build and update the view matrix
void Camera::render()
{
    D3DXVECTOR3 up, position, lookAt;

    // Setup the position of the camera in the world.
    position = (D3DXVECTOR3)m_position;

    getVectors(up, lookAt);

    // Translate the rotated camera position to the location of the viewer.
    lookAt = position + lookAt;

    // Finally create the view matrix from the three updated vectors.
    D3DXMatrixLookAtLH(&m_viewMatrix, &position, &lookAt, &up);

    return;
}

With the mouse input you modify yaw(head), pitch and roll.