Related
I am working on a simulation of plane movement. For now, I used Euler angles to transform "body frame" to "world frame" and it works fine.
Recently I learned about quaternions and their advantages over the rotation matrix (gimbal lock) and I tried to implement it using yaw/pitch/roll angles from the simulator.
Quaternion
If I understand correctly the quaternion represents two things. It has an x, y, and z component, which represents the axis about which a rotation will occur. It also has a w component, which represents the amount of rotation which will occur about this axis. In short, a vector, and a float. A quaternion can be represented as 4 element vector:
q=[w,x,y,z]
To calculate result (after full rotation) equation is using:
p'=qpq'
where:
p=[0,x,y,z]-direction vector
q=[w,x,y,z]-rotation
q'=[w,-x,-y,-z]
Algorithm
Create quaternion:
Using wikipedia I create quaternion by rotating around 3 axes (q):
Quaterniond toQuaternion(double yaw, double pitch, double roll) // yaw (Z), pitch (Y), roll (X)
{
//Degree to radius:
yaw = yaw * M_PI / 180;
pitch = pitch * M_PI / 180;
roll = roll * M_PI / 180;
// Abbreviations for the various angular functions
double cy = cos(yaw * 0.5);
double sy = sin(yaw * 0.5);
double cp = cos(pitch * 0.5);
double sp = sin(pitch * 0.5);
double cr = cos(roll * 0.5);
double sr = sin(roll * 0.5);
Quaterniond q;
q.w = cy * cp * cr + sy * sp * sr;
q.x = cy * cp * sr - sy * sp * cr;
q.y = sy * cp * sr + cy * sp * cr;
q.z = sy * cp * cr - cy * sp * sr;
return q;
}
Define plane direction (heading) vector:
p = [0,1,0,0]
Calculate Hamilton product:
p'=qpq'
q'= [w, -qx, -qy, -qz]
p' = (H(H(q, p), q')
Quaterniond HamiltonProduct(Quaterniond u, Quaterniond v)
{
Quaterniond result;
result.w = u.w*v.w - u.x*v.x - u.y*v.y - u.z*v.z;
result.x = u.w*v.x + u.x*v.w + u.y*v.z - u.z*v.y;
result.y = u.w*v.y - u.x*v.z + u.y*v.w + u.z*v.x;
result.z = u.w*v.z + u.x*v.y - u.y*v.x + u.z*v.w;
return result;
}
Result
My result will be a vector:
v=[p'x,p'y,p'z]
It works fine but the same as Euler angle rotation (gimbal lock). Is it because I use also euler angles here? I don't really see how it should work without rotation around 3 axes. Should I rotate around every axis separately?
I will be grateful for any advice and help with understanding this problem.
EDIT (how application works)
1. My application based on data streaming, it means that after every 1ms it checks if there are new data (new orientation of plane).
Example:
At the begging pitch/roll/yaw = 0, after 1ms yaw is changed by 10 degree so application reads pitch=0, roll=0, yaw = 10. After next 1ms yaw is changed again by 20 degrees. So input data will look like this: pitch=0, roll=0, yaw = 30.
2. Create direction quaternion - p
At the begging, I define that direction (head) of my plane is on X axis. So my local direction is
v=[1,0,0]
in quaternion (my p) is
p=[0,1,0,0]
Vector3 LocalDirection, GlobalDirection; //head
Quaterniond p,P,q, Q, pq; //P = p', Q=q'
LocalDirection.x = 1;
LocalDirection.y = 0;
LocalDirection.z = 0;
p.w = 0;
p.x = direction.x;
p.y = direction.y;
p.z = direction.z;
3. Create rotation
After every 1ms I check the rotation angles (Euler) from data streaming and calculate q using toQuaternion
q = toQuaternion(yaw, pitch, roll); // create quaternion after rotation
Q.w = q.w;
Q.x = -q.x;
Q.y = -q.y;
Q.z = -q.z;
4. Calculate "world direction"
Using Hamilton product I calculate quaternion after rotation which is my global direction:
pq = HamiltonProduct(q, p);
P = HamiltonProduct(pq, Q);
GlobalDirection.x = P.x;
GlobalDirection.y = P.y;
GlobalDirection.z = P.z;
5. Repeat 3-4 every 1ms
It seems that your simulation uses Euler angles for rotating objects each frame. You then convert those angles to quaternions afterwards. That won't solve gimbal lock.
Gimbal lock can happen anytime when you add Euler angles to Euler angles. It's not enough to solve this when going from local space to world space. You also need your simulation to use quaternions between frames.
Basically everytime your object is changing its rotation, convert the current rotation to a quaternion, multiply in the new rotation delta, and convert the result back to Euler angles or whatever you use to store rotations.
I'd recommend rewriting your application to use and story only quaternions. Whenever a user does input or some other logic of your game wants to rotate something, you immediately convert that input into a quaternion and feed it into the simulation.
With your toQuaternion and HamiltonProduct you have all the tools you need for that.
EDIT In response to your edit explaining how your application works.
At the begging pitch/roll/yaw = 0, after 1ms yaw is changed by 10 degree so application reads pitch=0, roll=0, yaw = 10. After next 1ms yaw is changed again by 20 degrees. So input data will look like this: pitch=0, roll=0, yaw = 30.
That is where gimbal lock happens. You convert to quaternions after you calculate the rotations. That is wrong. You need to use quaternions in this very first step. Don't do after 1ms yaw is changed by 10 degree so application reads pitch=0, roll=0, yaw = 10, do this:
Store the rotation as quaternion, not as euler angles;
Convert the 10 degree yaw turn into a quaternion;
Multiply the stored quaternion and the 10 degree yaw quaternion;
Store the result.
To clarify: Your steps 2, 3 and 4 are fine. The problem is in step 1.
On a side note, this:
It has an x, y, and z component, which represents the axis about which a rotation will occur. It also has a w component, which represents the amount of rotation which will occur about this axis
is not exactly correct. The components of a quaternion aren't directly axis and angle, they are sin(angle/2) * axis and cos(angle/2) (which is what your toQuaternion method produces). This is important as it gives you nice unit quaternions that form a 4D sphere where every point on the surface (surspace?) represents a rotation in 3D space, beautifully allowing for smooth interpolations between any two rotations.
This question already has answers here:
Quaternion rotation without Euler angles
(3 answers)
Closed 4 years ago.
So I'm writing a program where objects move around spacesim-style, in order to learn how to move things smoothly through 3D space. After messing around with Euler angles a bit, it seems they aren't really appropriate for free-form 3D movement in arbitrary directions, so I decided to move on to what seems to be best for the job - quaternions. I intend for the object to rotate around its local X-Y-Z axes at all times, never around the global X-Y-Z axes.
I've tried to implement a system of rotation using quaternions, but something isn't working. When rotating the object along a single axis, if no previous rotations were undertaken, the thing rotates fine along a given axis. However, when applying one rotation after another has been performed, the second rotation is not always along the local axis it's supposed to be rotating along - for instance, after a rotation of about 90° around the Z axis, a rotation around the Y axis still takes place around the global Y axis, rather than the new local Y axis which is aligned with the global X axis.
Huh. So let's go through this step by step. The mistake must be in here somewhere.
STEP 1 - Capture Input
I figured it would be best to use Euler angles (or a Pitch-Yaw-Roll scheme) for capturing player input. At the moment, arrow keys control Pitch and Yaw, whereas Q and E control Roll. I capture player input thus (I am using SFML 1.6):
///SPEEDS
float ForwardSpeed = 0.05;
float TurnSpeed = 0.5;
//Rotation
sf::Vector3<float> Rotation;
Rotation.x = 0;
Rotation.y = 0;
Rotation.z = 0;
//PITCH
if (m_pApp->GetInput().IsKeyDown(sf::Key::Up) == true)
{
Rotation.x -= TurnSpeed;
}
if (m_pApp->GetInput().IsKeyDown(sf::Key::Down) == true)
{
Rotation.x += TurnSpeed;
}
//YAW
if (m_pApp->GetInput().IsKeyDown(sf::Key::Left) == true)
{
Rotation.y -= TurnSpeed;
}
if (m_pApp->GetInput().IsKeyDown(sf::Key::Right) == true)
{
Rotation.y += TurnSpeed;
}
//ROLL
if (m_pApp->GetInput().IsKeyDown(sf::Key::Q) == true)
{
Rotation.z -= TurnSpeed;
}
if (m_pApp->GetInput().IsKeyDown(sf::Key::E) == true)
{
Rotation.z += TurnSpeed;
}
//Translation
sf::Vector3<float> Translation;
Translation.x = 0;
Translation.y = 0;
Translation.z = 0;
//Move the entity
if (Rotation.x != 0 ||
Rotation.y != 0 ||
Rotation.z != 0)
{
m_Entity->ApplyForce(Translation, Rotation);
}
m_Entity is the thing I'm trying to rotate. It also contains the quaternion and rotation matrices representing the object's rotation.
STEP 2 - Update quaternion
I'm not 100% sure this is the way it's supposed to be done, but this is what I tried doing in Entity::ApplyForce():
//Rotation
m_Rotation.x += Rotation.x;
m_Rotation.y += Rotation.y;
m_Rotation.z += Rotation.z;
//Multiply the new Quaternion by the current one.
m_qRotation = Quaternion(m_Rotation.x, m_Rotation.y, m_Rotation.z);// * m_qRotation;
m_qRotation.RotationMatrix(m_RotationMatrix);
As you can see, I'm not sure whether it's best to just build a new quaternion from updated Euler angles, or whether I'm supposed to multiply the quaternion representing the change with the quaternion representing the overall current rotation, which is the impression I got when reading this guide. If the latter, my code would look like this:
//Multiply the new Quaternion by the current one.
m_qRotation = Quaternion(Rotation.x, Rotation.y, Rotation.z) * m_qRotation;
m_Rotation is the object's current rotation stored in PYR format; Rotation is the change demanded by player input. Either way, though, the problem might be in my implementation of my Quaternion class. Here is the whole thing:
Quaternion::Quaternion(float Pitch, float Yaw, float Roll)
{
float Pi = 4 * atan(1);
//Set the values, which came in degrees, to radians for C++ trig functions
float rYaw = Yaw * Pi / 180;
float rPitch = Pitch * Pi / 180;
float rRoll = Roll * Pi / 180;
//Components
float C1 = cos(rYaw / 2);
float C2 = cos(rPitch / 2);
float C3 = cos(rRoll / 2);
float S1 = sin(rYaw / 2);
float S2 = sin(rPitch / 2);
float S3 = sin(rRoll / 2);
//Create the final values
a = ((C1 * C2 * C3) - (S1 * S2 * S3));
x = (S1 * S2 * C3) + (C1 * C2 * S3);
y = (S1 * C2 * C3) + (C1 * S2 * S3);
z = (C1 * S2 * C3) - (S1 * C2 * S3);
}
//Overload the multiplier operator
Quaternion Quaternion::operator* (Quaternion OtherQuat)
{
float A = (OtherQuat.a * a) - (OtherQuat.x * x) - (OtherQuat.y * y) - (OtherQuat.z * z);
float X = (OtherQuat.a * x) + (OtherQuat.x * a) + (OtherQuat.y * z) - (OtherQuat.z * y);
float Y = (OtherQuat.a * y) - (OtherQuat.x * z) - (OtherQuat.y * a) - (OtherQuat.z * x);
float Z = (OtherQuat.a * z) - (OtherQuat.x * y) - (OtherQuat.y * x) - (OtherQuat.z * a);
Quaternion NewQuat = Quaternion(0, 0, 0);
NewQuat.a = A;
NewQuat.x = X;
NewQuat.y = Y;
NewQuat.z = Z;
return NewQuat;
}
//Calculates a rotation matrix and fills Matrix with it
void Quaternion::RotationMatrix(GLfloat* Matrix)
{
//Column 1
Matrix[0] = (a*a) + (x*x) - (y*y) - (z*z);
Matrix[1] = (2*x*y) + (2*a*z);
Matrix[2] = (2*x*z) - (2*a*y);
Matrix[3] = 0;
//Column 2
Matrix[4] = (2*x*y) - (2*a*z);
Matrix[5] = (a*a) - (x*x) + (y*y) - (z*z);
Matrix[6] = (2*y*z) + (2*a*x);
Matrix[7] = 0;
//Column 3
Matrix[8] = (2*x*z) + (2*a*y);
Matrix[9] = (2*y*z) - (2*a*x);
Matrix[10] = (a*a) - (x*x) - (y*y) + (z*z);
Matrix[11] = 0;
//Column 4
Matrix[12] = 0;
Matrix[13] = 0;
Matrix[14] = 0;
Matrix[15] = 1;
}
There's probably something in there to make somebody wiser than me cringe, but I can't see it. For converting from Euler angles to a quaternion, I used the "first method" according to this source, which also seems to suggest that the equation automatically creates a unit quaternion ("clearly normalized"). For multiplying quaternions, I again drew on this C++ guide.
STEP 3 - Deriving a rotation matrix from the quaternion
Once that is done, as per R. Martinho Fernandes' answer to this question, I try to build a rotation matrix from the quaternion and use that to update my object's rotation, using the above Quaternion::RotationMatrix() code in the following line:
m_qRotation.RotationMatrix(m_RotationMatrix);
I should note that m_RotationMatrix is GLfloat m_RotationMatrix[16], as per the required parameters of glMultMatrix, which I believe I am supposed to use later on when displaying the object. It is initialized as:
m_RotationMatrix = {1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1};
Which I believe is the "neutral" OpenGL rotation matrix (every 4 values together represent a column, correct? Again, I get this from the glMultMatrix page).
STEP 4 - Display!
Finally, we get to the function run each cycle for the object that is supposed to display it.
glPushMatrix();
glTranslatef(m_Position.x, m_Position.y, m_Position.z);
glMultMatrixf(m_RotationMatrix);
//glRotatef(m_Rotation.y, 0.0, 1.0, 0.0);
//glRotatef(m_Rotation.z, 0.0, 0.0, 1.0);
//glRotatef(m_Rotation.x, 1.0, 0.0, 0.0);
//glRotatef(m_qRotation.a, m_qRotation.x, m_qRotation.y, m_qRotation.z);
//[...] various code displaying the object's VBO
glPopMatrix();
I have left my previous failed attempts there, commented out.
Conclusion - Sad panda
That is the conclusion of the life cycle of player input, from cradle to OpenGL-managed grave.
I've obviously not understood something, since the behavior I get isn't the behavior I want or expect. But I'm not particularly experienced with matrix math or quaternions, so I don't have the insight required to see the error in my ways.
Can somebody help me out here?
All you have done is effectively implement Euler angles with quaternions. That's not helping.
The problem with Euler angles is that, when you compute the matrices, each angle is relative to the rotation of the matrix that came before it. What you want is to take an object's current orientation, and apply a rotation along some axis, producing a new orientation.
You can't do that with Euler angles. You can with matrices, and you can with quaternions (as they're just the rotation part of a matrix). But you can't do it by pretending they are Euler angles.
This is done by not storing angles at all. Instead, you just have a quaternion which represents the current orientation of the object. When you decide to apply a rotation to it (of some angle by some axis), you construct a quaternion that represents that rotation by an angle around that axis. Then you right-multiply that quaternion with the current orientation quaternion, producing a new current orientation.
When you render the object, you use the current orientation as... the orientation.
Quaternions represent orientations around 3D compound axes.
But they can also represent 'delta-rotations'.
To 'rotate an orientation', we need an orientation (a quat), and a rotation (also a quat), and we multiply them together, resulting in (you guessed it) a quat.
You noticed they are not commutative, that means the order we multiply them in absolutely matters, just like for matrices.
The order tends to depend on the implementation of your math library, but really, there's only two possible ways to do it, so it shouldn't take you too long to figure out which one is the right one - if things are 'orbiting' instead of 'rotating', then you have them the wrong way around.
For your example of yaw and pitch, I would build my 'delta-rotation' quaternion from yaw, pitch and roll angles, with roll set to zero, and then apply that to my 'orientation' quaternion, rather than doing the rotations one axis at a time.
I have a weapon that bounces to the next enemy when it hits.
I first begin by calculating the delta and the getting the angle:
float deltaX = e->m_body->GetPosition().x - m_body->GetPosition().x;
float deltaY = e->m_body->GetPosition().y - m_body->GetPosition().y;
float angle = atan2((deltaY), deltaX) * 180 / M_PI;
Then I convert the angle to a vector and multiply it by 15 (the speed of the projectile):
b2Vec2 vec = b2Vec2(cos(angle*M_PI/180),sin(angle*M_PI/180));
vec *= 15.0f;
Finally, I apply the impulse to the body:
m_body->ApplyLinearImpulse(vec, m_body->GetPosition());
The problem is that the vector must be incorrect as the bullet does not go in the right direction. If I simply output the angle to the next enemy, it tends to output an angle that looks correct so the problem must be in the conversion to a vector.
I don't think you need to use any trigonometry functions here, because you already have the direction:
b2Vec2 direction = e->m_body->GetPosition() - m_body->GetPosition();
direction.Normalize(); // this vector now has length 1
float speed = ...;
m_body->ApplyLinearImpulse( speed * direction, m_body->GetWorldCenter() );
The problem is I have two points in 3D space where y+ is up, x+ is to the right, and z+ is towards you. I want to orientate a cylinder between them that is the length of of the distance between both points, so that both its center ends touch the two points. I got the cylinder to translate to the location at the center of the two points, and I need help coming up with a rotation matrix to apply to the cylinder, so that it is orientated the correct way. My transformation matrix for the entire thing looks like this:
translate(center point) * rotateX(some X degrees) * rotateZ(some Z degrees)
The translation is applied last, that way I can get it to the correct orientation before I translate it.
Here is what I have so far for this:
mat4 getTransformation(vec3 point, vec3 parent)
{
float deltaX = point.x - parent.x;
float deltaY = point.y - parent.y;
float deltaZ = point.z - parent.z;
float yRotation = atan2f(deltaZ, deltaX) * (180.0 / M_PI);
float xRotation = atan2f(deltaZ, deltaY) * (180.0 / M_PI);
float zRotation = atan2f(deltaX, deltaY) * (-180.0 / M_PI);
if(point.y < parent.y)
{
zRotation = atan2f(deltaX, deltaY) * (180.0 / M_PI);
}
vec3 center = vec3((point.x + parent.x)/2.0, (point.y + parent.y)/2.0, (point.z + parent.z)/2.0);
mat4 translation = Translate(center);
return translation * RotateX(xRotation) * RotateZ(zRotation) * Scale(radius, 1, radius) * Scale(0.1, 0.1, 0.1);
}
I tried a solution given down below, but it did not seem to work at all
mat4 getTransformation(vec3 parent, vec3 point)
{
// moves base of cylinder to origin and gives it unit scaling
mat4 scaleFactor = Translate(0, 0.5, 0) * Scale(radius/2.0, 1/2.0, radius/2.0) * cylinderModel;
float length = sqrtf(pow((point.x - parent.x), 2) + pow((point.y - parent.y), 2) + pow((point.z - parent.z), 2));
vec3 direction = normalize(point - parent);
float pitch = acos(direction.y);
float yaw = atan2(direction.z, direction.x);
return Translate(parent) * Scale(length, length, length) * RotateX(pitch) * RotateY(yaw) * scaleFactor;
}
After running the above code I get this:
Every black point is a point with its parent being the point that spawned it (the one before it) I want the branches to fit into the points. Basically I am trying to implement the space colonization algorithm for random tree generation. I got most of it, but I want to map the branches to it so it looks good. I can use GL_LINES just to make a generic connection, but if I get this working it will look so much prettier. The algorithm is explained here.
Here is an image of what I am trying to do (pardon my paint skills)
Well, there's an arbitrary number of rotation matrices satisfying your constraints. But any will do. Instead of trying to figure out a specific rotation, we're just going to write down the matrix directly. Say your cylinder, when no transformation is applied, has its axis along the Z axis. So you have to transform the local space Z axis toward the direction between those two points. I.e. z_t = normalize(p_1 - p_2), where normalize(a) = a / length(a).
Now we just need to make this a full 3 dimensional coordinate base. We start with an arbitrary vector that's not parallel to z_t. Say, one of (1,0,0) or (0,1,0) or (0,0,1); use the scalar product ·(also called inner, or dot product) with z_t and use the vector for which the absolute value is the smallest, let's call this vector u.
In pseudocode:
# Start with (1,0,0)
mindotabs = abs( z_t · (1,0,0) )
minvec = (1,0,0)
for u_ in (0,1,0), (0,0,1):
dotabs = z_t · u_
if dotabs < mindotabs:
mindotabs = dotabs
minvec = u_
u = minvec_
Then you orthogonalize that vector yielding a local y transformation y_t = normalize(u - z_t · u).
Finally create the x transformation by taking the cross product x_t = z_t × y_t
To move the cylinder into place you combine that with a matching translation matrix.
Transformation matrices are effectively just the axes of the space you're "coming from" written down as if seen from the other space. So the resulting matrix, which is the rotation matrix you're looking for is simply the vectors x_t, y_t and z_t side by side as a matrix. OpenGL uses so called homogenuous matrices, so you have to pad it to a 4×4 form using a 0,0,0,1 bottommost row and rightmost column.
That you can load then into OpenGL; if using fixed functio using glMultMatrix to apply the rotation, or if using shader to multiply onto the matrix you're eventually pass to glUniform.
Begin with a unit length cylinder which has one of its ends, which I call C1, at the origin (note that your image indicates that your cylinder has its center at the origin, but you can easily transform that to what I begin with). The other end, which I call C2, is then at (0,1,0).
I'd like to call your two points in world coordinates P1 and P2 and we want to locate C1 on P1 and C2 to P2.
Start with translating the cylinder by P1, which successfully locates C1 to P1.
Then scale the cylinder by distance(P1, P2), since it originally had length 1.
The remaining rotation can be computed using spherical coordinates. If you're not familiar with this type of coordinate system: it's like GPS coordinates: two angles; one around the pole axis (in your case the world's Y-axis) which we typically call yaw, the other one is a pitch angle (in your case the X axis in model space). These two angles can be computed by converting P2-P1 (i.e. the local offset of P2 with respect to P1) into spherical coordinates. First rotate the object with the pitch angle around X, then with yaw around Y.
Something like this will do it (pseudo-code):
Matrix getTransformation(Point P1, Point P2) {
float length = distance(P1, P2);
Point direction = normalize(P2 - P1);
float pitch = acos(direction.y);
float yaw = atan2(direction.z, direction.x);
return translate(P1) * scaleY(length) * rotateX(pitch) * rotateY(yaw);
}
Call the axis of the cylinder A. The second rotation (about X) can't change the angle between A and X, so we have to get that angle right with the first rotation (about Z).
Call the destination vector (the one between the two points) B. Take -acos(BX/BY), and that's the angle of the first rotation.
Take B again, ignore the X component, and look at its projection in the (Y, Z) plane. Take acos(BZ/BY), and that's the angle of the second rotation.
This question already has answers here:
Quaternion rotation without Euler angles
(3 answers)
Closed 4 years ago.
So I'm writing a program where objects move around spacesim-style, in order to learn how to move things smoothly through 3D space. After messing around with Euler angles a bit, it seems they aren't really appropriate for free-form 3D movement in arbitrary directions, so I decided to move on to what seems to be best for the job - quaternions. I intend for the object to rotate around its local X-Y-Z axes at all times, never around the global X-Y-Z axes.
I've tried to implement a system of rotation using quaternions, but something isn't working. When rotating the object along a single axis, if no previous rotations were undertaken, the thing rotates fine along a given axis. However, when applying one rotation after another has been performed, the second rotation is not always along the local axis it's supposed to be rotating along - for instance, after a rotation of about 90° around the Z axis, a rotation around the Y axis still takes place around the global Y axis, rather than the new local Y axis which is aligned with the global X axis.
Huh. So let's go through this step by step. The mistake must be in here somewhere.
STEP 1 - Capture Input
I figured it would be best to use Euler angles (or a Pitch-Yaw-Roll scheme) for capturing player input. At the moment, arrow keys control Pitch and Yaw, whereas Q and E control Roll. I capture player input thus (I am using SFML 1.6):
///SPEEDS
float ForwardSpeed = 0.05;
float TurnSpeed = 0.5;
//Rotation
sf::Vector3<float> Rotation;
Rotation.x = 0;
Rotation.y = 0;
Rotation.z = 0;
//PITCH
if (m_pApp->GetInput().IsKeyDown(sf::Key::Up) == true)
{
Rotation.x -= TurnSpeed;
}
if (m_pApp->GetInput().IsKeyDown(sf::Key::Down) == true)
{
Rotation.x += TurnSpeed;
}
//YAW
if (m_pApp->GetInput().IsKeyDown(sf::Key::Left) == true)
{
Rotation.y -= TurnSpeed;
}
if (m_pApp->GetInput().IsKeyDown(sf::Key::Right) == true)
{
Rotation.y += TurnSpeed;
}
//ROLL
if (m_pApp->GetInput().IsKeyDown(sf::Key::Q) == true)
{
Rotation.z -= TurnSpeed;
}
if (m_pApp->GetInput().IsKeyDown(sf::Key::E) == true)
{
Rotation.z += TurnSpeed;
}
//Translation
sf::Vector3<float> Translation;
Translation.x = 0;
Translation.y = 0;
Translation.z = 0;
//Move the entity
if (Rotation.x != 0 ||
Rotation.y != 0 ||
Rotation.z != 0)
{
m_Entity->ApplyForce(Translation, Rotation);
}
m_Entity is the thing I'm trying to rotate. It also contains the quaternion and rotation matrices representing the object's rotation.
STEP 2 - Update quaternion
I'm not 100% sure this is the way it's supposed to be done, but this is what I tried doing in Entity::ApplyForce():
//Rotation
m_Rotation.x += Rotation.x;
m_Rotation.y += Rotation.y;
m_Rotation.z += Rotation.z;
//Multiply the new Quaternion by the current one.
m_qRotation = Quaternion(m_Rotation.x, m_Rotation.y, m_Rotation.z);// * m_qRotation;
m_qRotation.RotationMatrix(m_RotationMatrix);
As you can see, I'm not sure whether it's best to just build a new quaternion from updated Euler angles, or whether I'm supposed to multiply the quaternion representing the change with the quaternion representing the overall current rotation, which is the impression I got when reading this guide. If the latter, my code would look like this:
//Multiply the new Quaternion by the current one.
m_qRotation = Quaternion(Rotation.x, Rotation.y, Rotation.z) * m_qRotation;
m_Rotation is the object's current rotation stored in PYR format; Rotation is the change demanded by player input. Either way, though, the problem might be in my implementation of my Quaternion class. Here is the whole thing:
Quaternion::Quaternion(float Pitch, float Yaw, float Roll)
{
float Pi = 4 * atan(1);
//Set the values, which came in degrees, to radians for C++ trig functions
float rYaw = Yaw * Pi / 180;
float rPitch = Pitch * Pi / 180;
float rRoll = Roll * Pi / 180;
//Components
float C1 = cos(rYaw / 2);
float C2 = cos(rPitch / 2);
float C3 = cos(rRoll / 2);
float S1 = sin(rYaw / 2);
float S2 = sin(rPitch / 2);
float S3 = sin(rRoll / 2);
//Create the final values
a = ((C1 * C2 * C3) - (S1 * S2 * S3));
x = (S1 * S2 * C3) + (C1 * C2 * S3);
y = (S1 * C2 * C3) + (C1 * S2 * S3);
z = (C1 * S2 * C3) - (S1 * C2 * S3);
}
//Overload the multiplier operator
Quaternion Quaternion::operator* (Quaternion OtherQuat)
{
float A = (OtherQuat.a * a) - (OtherQuat.x * x) - (OtherQuat.y * y) - (OtherQuat.z * z);
float X = (OtherQuat.a * x) + (OtherQuat.x * a) + (OtherQuat.y * z) - (OtherQuat.z * y);
float Y = (OtherQuat.a * y) - (OtherQuat.x * z) - (OtherQuat.y * a) - (OtherQuat.z * x);
float Z = (OtherQuat.a * z) - (OtherQuat.x * y) - (OtherQuat.y * x) - (OtherQuat.z * a);
Quaternion NewQuat = Quaternion(0, 0, 0);
NewQuat.a = A;
NewQuat.x = X;
NewQuat.y = Y;
NewQuat.z = Z;
return NewQuat;
}
//Calculates a rotation matrix and fills Matrix with it
void Quaternion::RotationMatrix(GLfloat* Matrix)
{
//Column 1
Matrix[0] = (a*a) + (x*x) - (y*y) - (z*z);
Matrix[1] = (2*x*y) + (2*a*z);
Matrix[2] = (2*x*z) - (2*a*y);
Matrix[3] = 0;
//Column 2
Matrix[4] = (2*x*y) - (2*a*z);
Matrix[5] = (a*a) - (x*x) + (y*y) - (z*z);
Matrix[6] = (2*y*z) + (2*a*x);
Matrix[7] = 0;
//Column 3
Matrix[8] = (2*x*z) + (2*a*y);
Matrix[9] = (2*y*z) - (2*a*x);
Matrix[10] = (a*a) - (x*x) - (y*y) + (z*z);
Matrix[11] = 0;
//Column 4
Matrix[12] = 0;
Matrix[13] = 0;
Matrix[14] = 0;
Matrix[15] = 1;
}
There's probably something in there to make somebody wiser than me cringe, but I can't see it. For converting from Euler angles to a quaternion, I used the "first method" according to this source, which also seems to suggest that the equation automatically creates a unit quaternion ("clearly normalized"). For multiplying quaternions, I again drew on this C++ guide.
STEP 3 - Deriving a rotation matrix from the quaternion
Once that is done, as per R. Martinho Fernandes' answer to this question, I try to build a rotation matrix from the quaternion and use that to update my object's rotation, using the above Quaternion::RotationMatrix() code in the following line:
m_qRotation.RotationMatrix(m_RotationMatrix);
I should note that m_RotationMatrix is GLfloat m_RotationMatrix[16], as per the required parameters of glMultMatrix, which I believe I am supposed to use later on when displaying the object. It is initialized as:
m_RotationMatrix = {1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1};
Which I believe is the "neutral" OpenGL rotation matrix (every 4 values together represent a column, correct? Again, I get this from the glMultMatrix page).
STEP 4 - Display!
Finally, we get to the function run each cycle for the object that is supposed to display it.
glPushMatrix();
glTranslatef(m_Position.x, m_Position.y, m_Position.z);
glMultMatrixf(m_RotationMatrix);
//glRotatef(m_Rotation.y, 0.0, 1.0, 0.0);
//glRotatef(m_Rotation.z, 0.0, 0.0, 1.0);
//glRotatef(m_Rotation.x, 1.0, 0.0, 0.0);
//glRotatef(m_qRotation.a, m_qRotation.x, m_qRotation.y, m_qRotation.z);
//[...] various code displaying the object's VBO
glPopMatrix();
I have left my previous failed attempts there, commented out.
Conclusion - Sad panda
That is the conclusion of the life cycle of player input, from cradle to OpenGL-managed grave.
I've obviously not understood something, since the behavior I get isn't the behavior I want or expect. But I'm not particularly experienced with matrix math or quaternions, so I don't have the insight required to see the error in my ways.
Can somebody help me out here?
All you have done is effectively implement Euler angles with quaternions. That's not helping.
The problem with Euler angles is that, when you compute the matrices, each angle is relative to the rotation of the matrix that came before it. What you want is to take an object's current orientation, and apply a rotation along some axis, producing a new orientation.
You can't do that with Euler angles. You can with matrices, and you can with quaternions (as they're just the rotation part of a matrix). But you can't do it by pretending they are Euler angles.
This is done by not storing angles at all. Instead, you just have a quaternion which represents the current orientation of the object. When you decide to apply a rotation to it (of some angle by some axis), you construct a quaternion that represents that rotation by an angle around that axis. Then you right-multiply that quaternion with the current orientation quaternion, producing a new current orientation.
When you render the object, you use the current orientation as... the orientation.
Quaternions represent orientations around 3D compound axes.
But they can also represent 'delta-rotations'.
To 'rotate an orientation', we need an orientation (a quat), and a rotation (also a quat), and we multiply them together, resulting in (you guessed it) a quat.
You noticed they are not commutative, that means the order we multiply them in absolutely matters, just like for matrices.
The order tends to depend on the implementation of your math library, but really, there's only two possible ways to do it, so it shouldn't take you too long to figure out which one is the right one - if things are 'orbiting' instead of 'rotating', then you have them the wrong way around.
For your example of yaw and pitch, I would build my 'delta-rotation' quaternion from yaw, pitch and roll angles, with roll set to zero, and then apply that to my 'orientation' quaternion, rather than doing the rotations one axis at a time.