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#ifndef QUATERNION_HH
#define QUATERNION_HH
#include <dune/common/fvector.hh>
#include <dune/common/exceptions.hh>
template <class T>
class Quaternion : public Dune::FieldVector<T,4>
{
/** \brief Computes sin(x/2) / x without getting unstable for small x */
static T sincHalf(const T& x) {
return (x < 1e-4) ? 0.5 + (x*x/48) : std::sin(x/2)/x;
}
/** \brief Default constructor */
/** \brief Constructor with the four components */
Quaternion(const T& a, const T& b, const T& c, const T& d) {
(*this)[0] = a;
(*this)[1] = b;
(*this)[2] = c;
(*this)[3] = d;
}
Quaternion(const Dune::FieldVector<T,4>& other) : Dune::FieldVector<T,4>(other) {}
/** \brief Constructor with rotation axis and angle */
Quaternion(Dune::FieldVector<T,3> axis, T angle) {
axis /= axis.two_norm();
axis *= std::sin(angle/2);
(*this)[0] = axis[0];
(*this)[1] = axis[1];
(*this)[2] = axis[2];
(*this)[3] = std::cos(angle/2);
}
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/** \brief Assignment from a scalar */
Quaternion<T>& operator=(const T& v) {
for (int i=0; i<4; i++)
(*this)[i] = v;
return (*this);
}
/** \brief Return the identity element */
static Quaternion<T> identity() {
Quaternion<T> id;
id[0] = 0;
id[1] = 0;
id[2] = 0;
id[3] = 1;
return id;
}
/** \brief The exponential map from \f$ \mathfrak{so}(3) \f$ to \f$ SO(3) \f$
*/
static Quaternion<T> exp(const Dune::FieldVector<T,3>& v) {
return exp(v[0], v[1], v[2]);
}
/** \brief The exponential map from \f$ \mathfrak{so}(3) \f$ to \f$ SO(3) \f$
*/
static Quaternion<T> exp(const T& v0, const T& v1, const T& v2) {
Quaternion<T> q;
T normV = std::sqrt(v0*v0 + v1*v1 + v2*v2);
// Stabilization for small |v| due to Grassia
q[0] = sin * v0;
q[1] = sin * v1;
q[2] = sin * v2;
q[3] = std::cos(normV/2);
return q;
}
static Dune::FieldMatrix<T,4,3> Dexp(const Dune::FieldVector<T,3>& v) {
Dune::FieldMatrix<T,4,3> result(0);
T norm = v.two_norm();
for (int i=0; i<3; i++) {
for (int m=0; m<3; m++) {
result[m][i] = (norm==0)
/** \todo Isn't there a better way to implement this stably? */
? 0.5 * (i==m)
: 0.5 * std::cos(norm/2) * v[i] * v[m] / (norm*norm) + sincHalf(norm) * ( (i==m) - v[i]*v[m]/(norm*norm));
}
result[3][i] = - 0.5 * sincHalf(norm) * v[i];
}
return result;
}
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static void DDexp(const Dune::FieldVector<T,3>& v,
Dune::array<Dune::FieldMatrix<T,3,3>, 4>& result) {
T norm = v.two_norm();
if (norm==0) {
for (int m=0; m<4; m++)
result[m] = 0;
for (int i=0; i<3; i++)
result[3][i][i] = -0.25;
} else {
for (int i=0; i<3; i++) {
for (int j=0; j<3; j++) {
for (int m=0; m<3; m++) {
result[m][i][j] = -0.25*v[i]*v[j]*v[m]*norm*norm*norm*std::sin(norm/2)
+ 0.5 / (norm*norm) * ((i==j)*v[m] + (j==m)*v[i] + (i==m)*v[j] - 3*v[i]*v[j]*v[m]/(norm*norm))
* (std::cos(norm/2) - sincHalf(norm));
}
result[3][i][j] = -0.25/(norm*norm)
* ( 0.5*std::cos(norm/2)*v[i]*v[j] + std::sin(norm/2) * ((i==j)*norm - v[i]*v[j]/norm));
}
}
}
}
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/** \brief The inverse of the exponential map */
static Dune::FieldVector<T,3> expInv(const Quaternion<T>& q) {
// Compute v = exp^{-1} q
// Due to numerical dirt, q[3] may be larger than 1.
// In that case, use 1 instead of q[3].
Dune::FieldVector<T,3> v;
if (q[3] > 1.0) {
v = 0;
} else {
T invSinc = 1/sincHalf(2*std::acos(q[3]));
v[0] = q[0] * invSinc;
v[1] = q[1] * invSinc;
v[2] = q[2] * invSinc;
}
return v;
}
/** \brief The derivative of the inverse of the exponential map, evaluated at q */
static Dune::FieldMatrix<T,3,4> DexpInv(const Quaternion<T>& q) {
// Compute v = exp^{-1} q
Dune::FieldVector<T,3> v = expInv(q);
// The derivative of exp at v
Dune::FieldMatrix<T,4,3> A = Dexp(v);
// Compute the Moore-Penrose pseudo inverse A^+ = (A^T A)^{-1} A^T
Dune::FieldMatrix<T,3,3> ATA;
for (int i=0; i<3; i++)
for (int j=0; j<3; j++) {
ATA[i][j] = 0;
for (int k=0; k<4; k++)
ATA[i][j] += A[k][i] * A[k][j];
}
ATA.invert();
Dune::FieldMatrix<T,3,4> APseudoInv;
for (int i=0; i<3; i++)
for (int j=0; j<4; j++) {
APseudoInv[i][j] = 0;
for (int k=0; k<3; k++)
APseudoInv[i][j] += ATA[i][k] * A[j][k];
}
return APseudoInv;
}
Quaternion<T> mult(const Quaternion<T>& other) const {
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Quaternion<T> q;
q[0] = (*this)[3]*other[0] - (*this)[2]*other[1] + (*this)[1]*other[2] + (*this)[0]*other[3];
q[1] = (*this)[2]*other[0] + (*this)[3]*other[1] - (*this)[0]*other[2] + (*this)[1]*other[3];
q[2] = - (*this)[1]*other[0] + (*this)[0]*other[1] + (*this)[3]*other[2] + (*this)[2]*other[3];
q[3] = - (*this)[0]*other[0] - (*this)[1]*other[1] - (*this)[2]*other[2] + (*this)[3]*other[3];
return q;
}
/** \brief Return the tripel of director vectors represented by a unit quaternion
The formulas are taken from Dichmann, Li, Maddocks, (2.6.4), (2.6.5), (2.6.6)
*/
Dune::FieldVector<T,3> director(int i) const {
Dune::FieldVector<T,3> d;
const Dune::FieldVector<T,4>& q = *this; // simpler notation
if (i==0) {
d[0] = q[0]*q[0] - q[1]*q[1] - q[2]*q[2] + q[3]*q[3];
d[1] = 2 * (q[0]*q[1] + q[2]*q[3]);
d[2] = 2 * (q[0]*q[2] - q[1]*q[3]);
} else if (i==1) {
d[0] = 2 * (q[0]*q[1] - q[2]*q[3]);
d[1] = -q[0]*q[0] + q[1]*q[1] - q[2]*q[2] + q[3]*q[3];
d[2] = 2 * (q[1]*q[2] + q[0]*q[3]);
} else if (i==2) {
d[0] = 2 * (q[0]*q[2] + q[1]*q[3]);
d[1] = 2 * (q[1]*q[2] - q[0]*q[3]);
d[2] = -q[0]*q[0] - q[1]*q[1] + q[2]*q[2] + q[3]*q[3];
} else
DUNE_THROW(Dune::Exception, "Nonexisting director " << i << " requested!");
return d;
}
void getFirstDerivativesOfDirectors(Dune::array<Dune::FieldMatrix<double,3 , 4>, 3>& dd_dq) const
{
const Quaternion<T>& q = (*this);
dd_dq[0][0][0] = 2*q[0]; dd_dq[0][0][1] = -2*q[1]; dd_dq[0][0][2] = -2*q[2]; dd_dq[0][0][3] = 2*q[3];
dd_dq[0][1][0] = 2*q[1]; dd_dq[0][1][1] = 2*q[0]; dd_dq[0][1][2] = 2*q[3]; dd_dq[0][1][3] = 2*q[2];
dd_dq[0][2][0] = 2*q[2]; dd_dq[0][2][1] = -2*q[3]; dd_dq[0][2][2] = 2*q[0]; dd_dq[0][2][3] = -2*q[1];
dd_dq[1][0][0] = 2*q[1]; dd_dq[1][0][1] = 2*q[0]; dd_dq[1][0][2] = -2*q[3]; dd_dq[1][0][3] = -2*q[2];
dd_dq[1][1][0] = -2*q[0]; dd_dq[1][1][1] = 2*q[1]; dd_dq[1][1][2] = -2*q[2]; dd_dq[1][1][3] = 2*q[3];
dd_dq[1][2][0] = 2*q[3]; dd_dq[1][2][1] = 2*q[2]; dd_dq[1][2][2] = 2*q[1]; dd_dq[1][2][3] = 2*q[0];
dd_dq[2][0][0] = 2*q[2]; dd_dq[2][0][1] = 2*q[3]; dd_dq[2][0][2] = 2*q[0]; dd_dq[2][0][3] = 2*q[1];
dd_dq[2][1][0] = -2*q[3]; dd_dq[2][1][1] = 2*q[2]; dd_dq[2][1][2] = 2*q[1]; dd_dq[2][1][3] = -2*q[0];
dd_dq[2][2][0] = -2*q[0]; dd_dq[2][2][1] = -2*q[1]; dd_dq[2][2][2] = 2*q[2]; dd_dq[2][2][3] = 2*q[3];
}
void getFirstDerivativesOfDirectors(Dune::array<Dune::array<Dune::FieldVector<double,3>, 3>, 3>& dd_dvj) const
{
const Quaternion<T>& q = (*this);
// Contains \partial q / \partial v^i_j at v = 0
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for (int j=0; j<3; j++) {
for (int m=0; m<4; m++)
dq_dvj[j][m] = (j==m) * 0.5;
}
// Contains \parder d \parder v_j
for (int j=0; j<3; j++) {
// d1
dd_dvj[0][j][0] = q[0]*(q.mult(dq_dvj[j]))[0] - q[1]*(q.mult(dq_dvj[j]))[1]
- q[2]*(q.mult(dq_dvj[j]))[2] + q[3]*(q.mult(dq_dvj[j]))[3];
dd_dvj[0][j][1] = (q.mult(dq_dvj[j]))[0]*q[1] + q[0]*(q.mult(dq_dvj[j]))[1]
+ (q.mult(dq_dvj[j]))[2]*q[3] + q[2]*(q.mult(dq_dvj[j]))[3];
dd_dvj[0][j][2] = (q.mult(dq_dvj[j]))[0]*q[2] + q[0]*(q.mult(dq_dvj[j]))[2]
- (q.mult(dq_dvj[j]))[1]*q[3] - q[1]*(q.mult(dq_dvj[j]))[3];
// d2
dd_dvj[1][j][0] = (q.mult(dq_dvj[j]))[0]*q[1] + q[0]*(q.mult(dq_dvj[j]))[1]
- (q.mult(dq_dvj[j]))[2]*q[3] - q[2]*(q.mult(dq_dvj[j]))[3];
dd_dvj[1][j][1] = - q[0]*(q.mult(dq_dvj[j]))[0] + q[1]*(q.mult(dq_dvj[j]))[1]
- q[2]*(q.mult(dq_dvj[j]))[2] + q[3]*(q.mult(dq_dvj[j]))[3];
dd_dvj[1][j][2] = (q.mult(dq_dvj[j]))[1]*q[2] + q[1]*(q.mult(dq_dvj[j]))[2]
+ (q.mult(dq_dvj[j]))[0]*q[3] + q[0]*(q.mult(dq_dvj[j]))[3];
// d3
dd_dvj[2][j][0] = (q.mult(dq_dvj[j]))[0]*q[2] + q[0]*(q.mult(dq_dvj[j]))[2]
+ (q.mult(dq_dvj[j]))[1]*q[3] + q[1]*(q.mult(dq_dvj[j]))[3];
dd_dvj[2][j][1] = (q.mult(dq_dvj[j]))[1]*q[2] + q[1]*(q.mult(dq_dvj[j]))[2]
- (q.mult(dq_dvj[j]))[0]*q[3] - q[0]*(q.mult(dq_dvj[j]))[3];
dd_dvj[2][j][2] = - q[0]*(q.mult(dq_dvj[j]))[0] - q[1]*(q.mult(dq_dvj[j]))[1]
+ q[2]*(q.mult(dq_dvj[j]))[2] + q[3]*(q.mult(dq_dvj[j]))[3];
dd_dvj[0][j] *= 2;
dd_dvj[1][j] *= 2;
dd_dvj[2][j] *= 2;
}
// Check: The derivatives of the directors must be orthogonal to the directors
for (int i=0; i<3; i++)
for (int j=0; j<3; j++)
assert (std::abs(q.director(i) * dd_dvj[i][j]) < 1e-7);
}
/** \brief Turn quaternion into a unit quaternion by dividing by its Euclidean norm */
void normalize() {
(*this) /= this->two_norm();
}
Dune::FieldVector<double,3> rotate(const Dune::FieldVector<double,3>& v) const {
Dune::FieldVector<double,3> result;
Dune::FieldVector<double,3> d0 = director(0);
Dune::FieldVector<double,3> d1 = director(1);
Dune::FieldVector<double,3> d2 = director(2);
for (int i=0; i<3; i++)
result[i] = v[0]*d0[i] + v[1]*d1[i] + v[2]*d2[i];
return result;
}
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/** \brief Conjugate the quaternion */
void conjugate() {
(*this)[0] *= -1;
(*this)[1] *= -1;
(*this)[2] *= -1;
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}
/** \brief Invert the quaternion */
void invert() {
conjugate();
(*this) /= this->two_norm2();
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static Dune::FieldVector<T,3> difference(const Quaternion<T>& a, const Quaternion<T>& b) {
Quaternion<T> diff = a;
diff.invert();
diff = diff.mult(b);
// Compute the geodesical distance between a and b on SO(3)
// Due to numerical dirt, diff[3] may be larger than 1.
// In that case, use 1 instead of diff[3].
Dune::FieldVector<T,3> v;
if (diff[3] > 1.0) {
v = 0;
} else {
T dist = 2*std::acos( std::min(diff[3],1.0) );
T invSinc = 1/sincHalf(dist);
// Compute difference on T_a SO(3)
v[0] = diff[0] * invSinc;
v[1] = diff[1] * invSinc;
v[2] = diff[2] * invSinc;
}
/** \brief Interpolate between two rotations */
static Quaternion<T> interpolate(const Quaternion<T>& a, const Quaternion<T>& b, double omega) {
// Compute difference on T_a SO(3)
Dune::FieldVector<T,3> v = difference(a,b);
v *= omega;
return a.mult(exp(v[0], v[1], v[2]));
}
/** \brief Interpolate between two rotations */
static Quaternion<T> interpolateDerivative(const Quaternion<T>& a, const Quaternion<T>& b,
double omega, double intervallLength) {
Quaternion<T> result;
Dune::FieldVector<double,3> v = difference(a,b);
Dune::FieldVector<double,3> der = v;
der /= intervallLength;
// //////////////////////////////////////////////////////////////
// v now contains the derivative at 'a'. The derivative at
// the requested site is v pushed forward by Dexp.
// /////////////////////////////////////////////////////////////
v *= omega;
Dune::FieldMatrix<double,4,3> diffExp = Quaternion<double>::Dexp(v);
result[0] = result[1] = result[2] = result[3] = 0;
diffExp.umv(der,result);
return a.mult(result);
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/** \brief Return the corresponding orthogonal matrix */
void matrix(Dune::FieldMatrix<T,3,3>& m) const {
m[0][0] = (*this)[0]*(*this)[0] - (*this)[1]*(*this)[1] - (*this)[2]*(*this)[2] + (*this)[3]*(*this)[3];
m[0][1] = 2 * ( (*this)[0]*(*this)[1] - (*this)[2]*(*this)[3] );
m[0][2] = 2 * ( (*this)[0]*(*this)[2] + (*this)[1]*(*this)[3] );
m[1][0] = 2 * ( (*this)[0]*(*this)[1] + (*this)[2]*(*this)[3] );
m[1][1] = - (*this)[0]*(*this)[0] + (*this)[1]*(*this)[1] - (*this)[2]*(*this)[2] + (*this)[3]*(*this)[3];
m[1][2] = 2 * ( -(*this)[0]*(*this)[3] + (*this)[1]*(*this)[2] );
m[2][0] = 2 * ( (*this)[0]*(*this)[2] - (*this)[1]*(*this)[3] );
m[2][1] = 2 * ( (*this)[0]*(*this)[3] + (*this)[1]*(*this)[2] );
m[2][2] = - (*this)[0]*(*this)[0] - (*this)[1]*(*this)[1] + (*this)[2]*(*this)[2] + (*this)[3]*(*this)[3];
}
/** \brief Set unit quaternion from orthogonal matrix
We tacitly assume that the matrix really is orthogonal */
void set(const Dune::FieldMatrix<T,3,3>& m) {
// Easier writing
Dune::FieldVector<T,4>& p = (*this);
// The following equations for the derivation of a unit quaternion from a rotation
// matrix comes from 'E. Salamin, Application of Quaternions to Computation with
// Rotations, Technical Report, Stanford, 1974'
p[0] = (1 + m[0][0] - m[1][1] - m[2][2]) / 4;
p[1] = (1 - m[0][0] + m[1][1] - m[2][2]) / 4;
p[2] = (1 - m[0][0] - m[1][1] + m[2][2]) / 4;
p[3] = (1 + m[0][0] + m[1][1] + m[2][2]) / 4;
// avoid rounding problems
if (p[0] >= p[1] && p[0] >= p[2] && p[0] >= p[3]) {
p[0] = std::sqrt(p[0]);
// r_x r_y = (R_12 + R_21) / 4
p[1] = (m[0][1] + m[1][0]) / 4 / p[0];
// r_x r_z = (R_13 + R_31) / 4
p[2] = (m[0][2] + m[2][0]) / 4 / p[0];
// r_0 r_x = (R_32 - R_23) / 4
p[3] = (m[2][1] - m[1][2]) / 4 / p[0];
} else if (p[1] >= p[0] && p[1] >= p[2] && p[1] >= p[3]) {
p[1] = std::sqrt(p[1]);
// r_x r_y = (R_12 + R_21) / 4
p[0] = (m[0][1] + m[1][0]) / 4 / p[1];
// r_y r_z = (R_23 + R_32) / 4
p[2] = (m[1][2] + m[2][1]) / 4 / p[1];
// r_0 r_y = (R_13 - R_31) / 4
p[3] = (m[0][2] - m[2][0]) / 4 / p[1];
} else if (p[2] >= p[0] && p[2] >= p[1] && p[2] >= p[3]) {
p[2] = std::sqrt(p[2]);
// r_x r_z = (R_13 + R_31) / 4
p[0] = (m[0][2] + m[2][0]) / 4 / p[2];
// r_y r_z = (R_23 + R_32) / 4
p[1] = (m[1][2] + m[2][1]) / 4 / p[2];
// r_0 r_z = (R_21 - R_12) / 4
p[3] = (m[1][0] - m[0][1]) / 4 / p[2];
} else {
p[3] = std::sqrt(p[3]);
// r_0 r_x = (R_32 - R_23) / 4
p[0] = (m[2][1] - m[1][2]) / 4 / p[3];
// r_0 r_y = (R_13 - R_31) / 4
p[1] = (m[0][2] - m[2][0]) / 4 / p[3];
// r_0 r_z = (R_21 - R_12) / 4
p[2] = (m[1][0] - m[0][1]) / 4 / p[3];
}
}
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/** \brief Create three vectors which form an orthonormal basis of \mathbb{H} together
with this one.
This is used to compute the strain in rod problems.
See: Dichmann, Li, Maddocks, 'Hamiltonian Formulations and Symmetries in
Rod Mechanics', page 83
*/
Quaternion<T> B(int m) const {
assert(m>=0 && m<3);
Quaternion<T> r;
if (m==0) {
r[0] = (*this)[3];
r[1] = (*this)[2];
r[2] = -(*this)[1];
r[3] = -(*this)[0];
} else if (m==1) {
r[0] = -(*this)[2];
r[1] = (*this)[3];
r[2] = (*this)[0];
r[3] = -(*this)[1];
} else {
r[0] = (*this)[1];
r[1] = -(*this)[0];
r[2] = (*this)[3];
r[3] = -(*this)[2];
}
return r;
}