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Oliver Sander authoredQuaternion class to the Rotation class. Rotation axes and angles are a genuine rotation concept and don't really belong into a quaternion implementation. [[Imported from SVN: r8207]]
Oliver Sander authoredQuaternion class to the Rotation class. Rotation axes and angles are a genuine rotation concept and don't really belong into a quaternion implementation. [[Imported from SVN: r8207]]
rotation.hh 30.77 KiB
#ifndef ROTATION_HH
#define ROTATION_HH
/** \file
\brief Define rotations in Euclidean spaces
*/
#include <dune/common/fvector.hh>
#include <dune/common/array.hh>
#include <dune/common/fmatrix.hh>
#include <dune/common/exceptions.hh>
#include "quaternion.hh"
#include <dune/gfe/tensor3.hh>
#include <dune/gfe/unitvector.hh>
#include <dune/gfe/skewmatrix.hh>
template <class T, int dim>
class Rotation
{
};
/** \brief Specialization for dim==2
\tparam T The type used for coordinates
*/
template <class T>
class Rotation<T,2>
{
public:
/** \brief The type used for coordinates */
typedef T ctype;
/** \brief Dimension of the manifold formed by the 2d rotations */
static const int dim = 1;
/** \brief Coordinates are embedded in the euclidean space */
static const int embeddedDim = 1;
/** \brief Member of the corresponding Lie algebra. This really is a skew-symmetric matrix */
typedef Dune::FieldVector<T,1> TangentVector;
/** \brief Member of the corresponding Lie algebra. This really is a skew-symmetric matrix
This vector is not really embedded in anything. I have to make my notation more consistent! */
typedef Dune::FieldVector<T,1> EmbeddedTangentVector;
/** \brief Default constructor, create the identity rotation */
Rotation()
: angle_(0)
{}
Rotation(const T& angle)
: angle_(angle)
{}
/** \brief Return the identity element */
static Rotation<T,2> identity() {
// Default constructor creates an identity
Rotation<T,2> id;
return id;
}
static T distance(const Rotation<T,2>& a, const Rotation<T,2>& b) {
T dist = a.angle_ - b.angle_;
while (dist < 0)
dist += 2*M_PI;
while (dist > 2*M_PI)
dist -= 2*M_PI;
return (dist <= M_PI) ? dist : 2*M_PI - dist;
}
/** \brief The exponential map from a given point $p \in SO(3)$. */
static Rotation<T,2> exp(const Rotation<T,2>& p, const TangentVector& v) {
Rotation<T,2> result = p;
result.angle_ += v;
return result;
}
/** \brief The exponential map from \f$ \mathfrak{so}(2) \f$ to \f$ SO(2) \f$
*/
static Rotation<T,2> exp(const Dune::FieldVector<T,1>& v) {
Rotation<T,2> result;
result.angle_ = v[0];
return result;
}
static TangentVector derivativeOfDistanceSquaredWRTSecondArgument(const Rotation<T,2>& a,
const Rotation<T,2>& b) {
// This assertion is here to remind me of the following laziness:
// The difference has to be computed modulo 2\pi
assert( std::fabs(a.angle_ - b.angle_) <= M_PI );
return -2 * (a.angle_ - b.angle_);
}
static Dune::FieldMatrix<T,1,1> secondDerivativeOfDistanceSquaredWRTSecondArgument(const Rotation<T,2>& a,
const Rotation<T,2>& b) {
return 2;
}
/** \brief Right multiplication */
Rotation<T,2> mult(const Rotation<T,2>& other) const {
Rotation<T,2> q = *this;
q.angle_ += other.angle_;
return q;
}
/** \brief Compute an orthonormal basis of the tangent space of SO(3).
This basis is of course not globally continuous.
*/
Dune::FieldMatrix<T,1,1> orthonormalFrame() const {
return Dune::FieldMatrix<T,1,1>(1);
}
//private:
// We store the rotation as an angle
T angle_;
};
//! Send configuration to output stream
template <class T>
std::ostream& operator<< (std::ostream& s, const Rotation<T,2>& c)
{
return s << "[" << c.angle_ << " (" << std::sin(c.angle_) << " " << std::cos(c.angle_) << ") ]";
}
/** \brief Specialization for dim==3
Uses unit quaternion coordinates.
*/
template <class T>
class Rotation<T,3> : public Quaternion<T>
{
/** \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 Compute the derivative of arccos^2 without getting unstable for x close to 1 */
static T derivativeOfArcCosSquared(const T& x) {
const T eps = 1e-12;
if (x > 1-eps) { // regular expression is unstable, use the series expansion instead
return -2 + 2*(x-1)/3 - 4/15*(x-1)*(x-1) + 4/35*(x-1)*(x-1)*(x-1);
} else if (x < -1+eps) { // The function is not differentiable
DUNE_THROW(Dune::Exception, "arccos^2 is not differentiable at x==-1!");
} else
return -2*std::acos(x) / std::sqrt(1-x*x);
}
public:
/** \brief The type used for coordinates */
typedef T ctype;
/** \brief The type used for global coordinates */
typedef Dune::FieldVector<T,4> CoordinateType;
/** \brief Dimension of the manifold formed by the 3d rotations */
static const int dim = 3;
/** \brief Coordinates are embedded into a four-dimension Euclidean space */
static const int embeddedDim = 4;
/** \brief Member of the corresponding Lie algebra. This really is a skew-symmetric matrix */
typedef Dune::FieldVector<T,3> TangentVector;
/** \brief A tangent vector as a vector in the surrounding coordinate space */
typedef Quaternion<T> EmbeddedTangentVector;
/** \brief Default constructor creates the identity element */
Rotation()
: Quaternion<T>(0,0,0,1)
{}
Rotation<T,3>(const Dune::array<T,4>& c)
{
for (int i=0; i<4; i++)
(*this)[i] = c[i];
*this /= this->two_norm();
}
Rotation<T,3>(const Dune::FieldVector<T,4>& c)
: Quaternion<T>(c)
{
*this /= this->two_norm();
}
Rotation<T,3>(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 Rotation<T,3> identity() {
// Default constructor creates an identity
Rotation<T,3> id;
return id;
}
/** \brief Right multiplication */
Rotation<T,3> mult(const Rotation<T,3>& other) const {
Rotation<T,3> 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 The exponential map from \f$ \mathfrak{so}(3) \f$ to \f$ SO(3) \f$
*/
static Rotation<T,3> exp(const SkewMatrix<T,3>& v) {
Rotation<T,3> q;
Dune::FieldVector<T,3> vAxial = v.axial();
T normV = vAxial.two_norm();
// Stabilization for small |v| due to Grassia
T sin = sincHalf(normV);
// if normV == 0 then q = (0,0,0,1)
assert(!isnan(sin));
q[0] = sin * vAxial[0];
q[1] = sin * vAxial[1];
q[2] = sin * vAxial[2];
q[3] = std::cos(normV/2);
return q;
}
/** \brief The exponential map from a given point $p \in SO(3)$. */
static Rotation<T,3> exp(const Rotation<T,3>& p, const SkewMatrix<T,3>& v) {
Rotation<T,3> corr = exp(v);
return p.mult(corr);
}
/** \brief The exponential map from a given point $p \in SO(3)$.
There may be a more direct way to implement this
\param v A tangent vector in quaternion coordinates
*/
static Rotation<T,3> exp(const Rotation<T,3>& p, const EmbeddedTangentVector& v) {
assert( std::fabs(p*v) < 1e-8 );
// The vector v as a quaternion
Quaternion<T> vQuat(v);
// left multiplication by the inverse base point yields a tangent vector at the identity
Quaternion<T> vAtIdentity = p.inverse().mult(vQuat);
assert( std::fabs(vAtIdentity[3]) < 1e-8 );
// vAtIdentity as a skew matrix
SkewMatrix<T,3> vMatrix;
vMatrix.axial()[0] = 2*vAtIdentity[0];
vMatrix.axial()[1] = 2*vAtIdentity[1];
vMatrix.axial()[2] = 2*vAtIdentity[2];
// The actual exponential map
return exp(p, vMatrix);
}
/** \brief The exponential map from a given point $p \in SO(3)$.
\param v A tangent vector.
*/ static Rotation<T,3> exp(const Rotation<T,3>& p, const TangentVector& v) {
// embedded tangent vector
Dune::FieldMatrix<T,3,4> basis = p.orthonormalFrame();
Quaternion<T> embeddedTangent;
basis.mtv(v, embeddedTangent);
return exp(p,embeddedTangent);
}
/** \brief Compute tangent vector from given basepoint and skew symmetric matrix. */
static TangentVector skewToTangentVector(const Rotation<T,3>& p, const SkewMatrix<T,3>& v ) {
// embedded tangent vector at identity
Quaternion<T> vAtIdentity(0);
vAtIdentity[0] = 0.5*v.axial()[0];
vAtIdentity[1] = 0.5*v.axial()[1];
vAtIdentity[2] = 0.5*v.axial()[2];
// multiply with base point to get real embedded tangent vector
Quaternion<T> vQuat = ((Quaternion<T>) p).mult(vAtIdentity);
//get basis of the tangent space
Dune::FieldMatrix<T,3,4> basis = p.orthonormalFrame();
// transform coordinates
TangentVector tang;
basis.mv(vQuat,tang);
return tang;
}
/** \brief Compute skew matrix from given basepoint and tangent vector. */
static SkewMatrix<T,3> tangentToSkew(const Rotation<T,3>& p, const TangentVector& tangent) {
// embedded tangent vector
Dune::FieldMatrix<T,3,4> basis = p.orthonormalFrame();
Quaternion<T> embeddedTangent;
basis.mtv(tangent, embeddedTangent);
return tangentToSkew(p,embeddedTangent);
}
/** \brief Compute skew matrix from given basepoint and an embedded tangent vector. */
static SkewMatrix<T,3> tangentToSkew(const Rotation<T,3>& p, const EmbeddedTangentVector& q) {
// left multiplication by the inverse base point yields a tangent vector at the identity
Quaternion<T> vAtIdentity = p.inverse().mult(q);
assert( std::fabs(vAtIdentity[3]) < 1e-8 );
SkewMatrix<T,3> skew;
skew.axial()[0] = 2*vAtIdentity[0];
skew.axial()[1] = 2*vAtIdentity[1];
skew.axial()[2] = 2*vAtIdentity[2];
return skew;
}
static Rotation<T,3> exp(const Rotation<T,3>& p, const Dune::FieldVector<T,4>& v) {
assert( std::fabs(p*v) < 1e-8 );
// The vector v as a quaternion
Quaternion<T> vQuat(v);
// left multiplication by the inverse base point yields a tangent vector at the identity
Quaternion<T> vAtIdentity = p.inverse().mult(vQuat);
assert( std::fabs(vAtIdentity[3]) < 1e-8 );
// vAtIdentity as a skew matrix
SkewMatrix<T,3> vMatrix;
vMatrix.axial()[0] = 2*vAtIdentity[0];
vMatrix.axial()[1] = 2*vAtIdentity[1];
vMatrix.axial()[2] = 2*vAtIdentity[2];
// The actual exponential map
return exp(p, vMatrix);
}
static Dune::FieldMatrix<T,4,3> Dexp(const SkewMatrix<T,3>& v) {
Dune::FieldMatrix<T,4,3> result(0);
Dune::FieldVector<T,3> vAxial = v.axial();
T norm = vAxial.two_norm();
for (int i=0; i<3; i++) {
for (int m=0; m<3; m++) {
result[m][i] = (norm<1e-10)
/** \todo Isn't there a better way to implement this stably? */
? 0.5 * (i==m)
: 0.5 * std::cos(norm/2) * vAxial[i] * vAxial[m] / (norm*norm) + sincHalf(norm) * ( (i==m) - vAxial[i]*vAxial[m]/(norm*norm));
}
result[3][i] = - 0.5 * sincHalf(norm) * vAxial[i];
}
return result;
}
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<=1e-10) {
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*std::sin(norm/2)*v[i]*v[j]*v[m]/(norm*norm*norm)
+ ((i==j)*v[m] + (j==m)*v[i] + (i==m)*v[j] - 3*v[i]*v[j]*v[m]/(norm*norm))
* (0.5*std::cos(norm/2) - sincHalf(norm)) / (norm*norm);
}
result[3][i][j] = -0.5/(norm*norm)
* ( 0.5*std::cos(norm/2)*v[i]*v[j] + std::sin(norm/2) * ((i==j)*norm - v[i]*v[j]/norm));
}
}
}
}
/** \brief The inverse of the exponential map */
static Dune::FieldVector<T,3> expInv(const Rotation<T,3>& 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 Rotation<T,3>& 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;
}
/** \brief The cayley mapping from \f$ \mathfrak{so}(3) \f$ to \f$ SO(3) \f$.
*
* The formula is taken from 'J.C.Simo, N.Tarnom, M.Doblare - Non-linear dynamics of
* three-dimensional rods:Exact energy and momentum conserving algorithms'
* (but the corrected version with 0.25 instead of 0.5 in the denominator)
*/
static Rotation<T,3> cayley(const SkewMatrix<T,3>& s) {
Rotation<T,3> q;
Dune::FieldVector<T,3> vAxial = s.axial();
T norm = 0.25*vAxial.two_norm2() + 1;
Dune::FieldMatrix<T,3,3> mat = s.toMatrix();
mat *= 0.5; Dune::FieldMatrix<T,3,3> skewSquare = mat;
skewSquare.rightmultiply(mat);
mat += skewSquare;
mat *= 2/norm;
for (int i=0;i<3;i++)
mat[i][i] += 1;
q.set(mat);
return q;
}
/** \brief The inverse of the Cayley mapping.
*
* The formula is taken from J.M.Selig - Cayley Maps for SE(3).
*/
static SkewMatrix<T,3> cayleyInv(const Rotation<T,3> q) {
Dune::FieldMatrix<T,3,3> mat;
// compute the trace of the rotation matrix
T trace = -q[0]*q[0] -q[1]*q[1] -q[2]*q[2]+3*q[3]*q[3];
if ( (trace+1)>1e-6 || (trace+1)<-1e-6) { // if this term doesn't vanish we can use a direct formula
q.matrix(mat);
Dune::FieldMatrix<T,3,3> matT;
Rotation<T,3>(q.inverse()).matrix(matT);
mat -= matT;
mat *= 2/(1+trace);
}
else { // use the formula that involves the computation of an inverse
Dune::FieldMatrix<T,3,3> inv;
q.matrix(inv);
Dune::FieldMatrix<T,3,3> notInv = inv;
for (int i=0;i<3;i++) {
inv[i][i] +=1;
notInv[i][i] -=1;
}
inv.invert();
mat = notInv.leftmultiply(inv);
mat *= 2;
}
// result is a skew symmetric matrix
SkewMatrix<T,3> res;
res.axial()[0] = mat[2][1];
res.axial()[1] = mat[0][2];
res.axial()[2] = mat[1][0];
return res;
}
static T distance(const Rotation<T,3>& a, const Rotation<T,3>& b) {
Quaternion<T> diff = a;
diff.invert();
diff = diff.mult(b);
// Make sure we do the right thing if a and b are not in the same sheet
// of the double covering of the unit quaternions over SO(3)
T dist = 2*std::acos( std::min(diff[3],1.0) );
if (dist>=M_PI)
diff *= -1;
// 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].
return 2*std::acos( std::min(diff[3],1.0) );
}
/** \brief Compute the vector in T_aSO(3) that is mapped by the exponential map
to the geodesic from a to b
*/
static SkewMatrix<T,3> difference(const Rotation<T,3>& a, const Rotation<T,3>& 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( diff[3] );
// Make sure we do the right thing if a and b are not in the same sheet
// of the double covering of the unit quaternions over SO(3)
if (dist>=M_PI) {
dist -= M_PI;
diff *= -1;
}
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;
}
return SkewMatrix<T,3>(v);
}
/** \brief Compute the derivatives of the director vectors with respect to the quaternion coordinates
*
* Let \f$ d_k(q) = (d_{k,1}, d_{k,2}, d_{k,3})\f$ be the k-th director vector at \f$ q \f$.
* Then the return value of this method is
* \f[ A_{ijk} = \frac{\partial d_{i,j}}{\partial q_k} \f]
*/
void getFirstDerivativesOfDirectors(Tensor3<T,3, 3, 4>& 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];
}
static EmbeddedTangentVector derivativeOfDistanceSquaredWRTSecondArgument(const Rotation<T,3>& p,
const Rotation<T,3>& q) {
Rotation<T,3> pInv = p;
pInv.invert();
// the forth component of pInv times q
T pInvq_4 = - pInv[0]*q[0] - pInv[1]*q[1] - pInv[2]*q[2] + pInv[3]*q[3];
T arccosSquaredDer_pInvq_4 = derivativeOfArcCosSquared(pInvq_4);
EmbeddedTangentVector result;
result[0] = -4 * arccosSquaredDer_pInvq_4 * pInv[0];
result[1] = -4 * arccosSquaredDer_pInvq_4 * pInv[1];
result[2] = -4 * arccosSquaredDer_pInvq_4 * pInv[2];
result[3] = 4 * arccosSquaredDer_pInvq_4 * pInv[3];
// project onto the tangent space at q
EmbeddedTangentVector projectedResult = result;
projectedResult.axpy(-1*(q*result), q);
assert(std::fabs(projectedResult * q) < 1e-7);
return projectedResult;
}
/** \brief Compute the Hessian of the squared distance function keeping the first argument fixed
Unlike the distance itself the squared distance is differentiable at zero
*/
static Dune::FieldMatrix<T,4,4> secondDerivativeOfDistanceSquaredWRTSecondArgument(const Rotation<T,3>& p, const Rotation<T,3>& q) {
// use the functionality from the unitvector class
Dune::FieldMatrix<T,4,4> result = UnitVector<T,4>::secondDerivativeOfDistanceSquaredWRTSecondArgument(p.globalCoordinates(),
q.globalCoordinates());
// for some reason that I don't really understand, the distance we have defined for the rotations (== Unit quaternions)
// is twice the corresponding distance on the unit quaternions seen as a sphere. Hence the derivative of the
// squared distance needs to be multiplied by 4.
result *= 4;
return result;
}
/** \brief Compute the mixed second derivate \partial d^2 / \partial da db
Unlike the distance itself the squared distance is differentiable at zero
*/
static Dune::FieldMatrix<T,4,4> secondDerivativeOfDistanceSquaredWRTFirstAndSecondArgument(const Rotation<T,3>& p, const Rotation<T,3>& q) {
// use the functionality from the unitvector class
Dune::FieldMatrix<T,4,4> result = UnitVector<T,4>::secondDerivativeOfDistanceSquaredWRTFirstAndSecondArgument(p.globalCoordinates(),
q.globalCoordinates());
// for some reason that I don't really understand, the distance we have defined for the rotations (== Unit quaternions)
// is twice the corresponding distance on the unit quaternions seen as a sphere. Hence the derivative of the
// squared distance needs to be multiplied by 4.
result *= 4;
return result;
}
/** \brief Compute the third derivative \partial d^3 / \partial dq^3
Unlike the distance itself the squared distance is differentiable at zero
*/
static Tensor3<T,4,4,4> thirdDerivativeOfDistanceSquaredWRTSecondArgument(const Rotation<T,3>& p, const Rotation<T,3>& q) {
// use the functionality from the unitvector class
Tensor3<T,4,4,4> result = UnitVector<T,4>::thirdDerivativeOfDistanceSquaredWRTSecondArgument(p.globalCoordinates(),
q.globalCoordinates());
// for some reason that I don't really understand, the distance we have defined for the rotations (== Unit quaternions)
// is twice the corresponding distance on the unit quaternions seen as a sphere. Hence the derivative of the
// squared distance needs to be multiplied by 4.
result *= 4;
return result;
}
/** \brief Compute the mixed third derivative \partial d^3 / \partial da db^2
Unlike the distance itself the squared distance is differentiable at zero
*/
static Tensor3<T,4,4,4> thirdDerivativeOfDistanceSquaredWRTFirst1AndSecond2Argument(const Rotation<T,3>& p, const Rotation<T,3>& q) {
// use the functionality from the unitvector class
Tensor3<T,4,4,4> result = UnitVector<T,4>::thirdDerivativeOfDistanceSquaredWRTFirst1AndSecond2Argument(p.globalCoordinates(),
q.globalCoordinates());
// for some reason that I don't really understand, the distance we have defined for the rotations (== Unit quaternions)
// is twice the corresponding distance on the unit quaternions seen as a sphere. Hence the derivative of the
// squared distance needs to be multiplied by 4.
result *= 4;
return result;
}
/** \brief Interpolate between two rotations */
static Rotation<T,3> interpolate(const Rotation<T,3>& a, const Rotation<T,3>& b, T omega) {
// Compute difference on T_a SO(3)
SkewMatrix<T,3> v = difference(a,b);
v *= omega;
return a.mult(exp(v));
}
/** \brief Interpolate between two rotations
\param omega must be between 0 and 1
*/
static Quaternion<T> interpolateDerivative(const Rotation<T,3>& a, const Rotation<T,3>& b,
T omega) {
Quaternion<T> result(0);
// Compute difference on T_a SO(3)
SkewMatrix<T,3> xi = difference(a,b);
SkewMatrix<T,3> v = xi;
v *= omega;
// //////////////////////////////////////////////////////////////
// v now contains the derivative at 'a'. The derivative at
// the requested site is v pushed forward by Dexp.
// /////////////////////////////////////////////////////////////
Dune::FieldMatrix<T,4,3> diffExp = Dexp(v);
diffExp.umv(xi.axial(),result);
return a.Quaternion<T>::mult(result);
}
/** \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 rotation 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];
}
}
/** \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;
}
/** \brief Project tangent vector of R^n onto the tangent space */
EmbeddedTangentVector projectOntoTangentSpace(const EmbeddedTangentVector& v) const {
EmbeddedTangentVector result = v;
EmbeddedTangentVector data = *this;
result.axpy(-1*(data*result), data);
return result;
}
/** \brief The global coordinates, if you really want them */
const CoordinateType& globalCoordinates() const {
return *this;
}
/** \brief Compute an orthonormal basis of the tangent space of SO(3). */
Dune::FieldMatrix<T,3,4> orthonormalFrame() const {
Dune::FieldMatrix<T,3,4> result;
for (int i=0; i<3; i++)
result[i] = B(i);
return result;
}
};
#endif