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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>
{
public:
/** \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);
}
/** \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 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
T sin = (normV < 1e-4) ? 0.5 + (normV*normV/48) : std::sin(normV/2)/normV;
q[0] = sin * v0;
q[1] = sin * v1;
q[2] = sin * v2;
q[3] = std::cos(normV/2);
return q;
}
/** \brief Right quaternion multiplication */
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;
}
/** \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;
}
/** \brief Invert the quaternion */
void invert() {
(*this)[0] *= -1;
(*this)[1] *= -1;
(*this)[2] *= -1;
(*this) /= this->two_norm2();
}
/** \brief Interpolate between two rotations */
static Quaternion<T> interpolate(const Quaternion<T>& a, const Quaternion<T>& b, double omega) {
#if 0 // Interpolation in H plus normalization
Quaternion<T> result;
for (int i=0; i<4; i++)
result[i] = a[i]*(1-omega) + b[i]*omega;
result.normalize();
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return result;
#else // Interpolation on the geodesic in SO(3) from a to b
Quaternion<T> diff = a;
diff.invert();
diff.mult(b);
T dist = 2*std::acos(diff[3]);
T invSinc = (dist < 1e-4) ? 1/(0.5+(dist*dist/48)) : dist / std::sin(dist/2);
// Compute difference on T_a SO(3)
Dune::FieldVector<double,3> v;
v[0] = diff[0] * invSinc;
v[1] = diff[1] * invSinc;
v[2] = diff[2] * invSinc;
v *= omega;
return a.mult(exp(v[0], v[1], v[2]));
#endif
}
/** \brief Interpolate between two rotations */
static Quaternion<T> interpolateDerivative(const Quaternion<T>& a, const Quaternion<T>& b,
double omega, double intervallLength) {
Quaternion<T> result;
for (int i=0; i<4; i++)
result[i] = a[i] / (-intervallLength) + b[i] / intervallLength;
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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;
}