void ExpMapQuaternion::pseudoLog_(RefVec out, const ConstRefVec& x, const ConstRefVec& y)
{
Eigen::Vector4d tmp;
toQuat q(tmp.data());
const toConstQuat xQ(x.data());
const toConstQuat yQ(y.data());
q = xQ.inverse()*yQ; //TODO double-check that formula
logarithm(out,tmp);
}
void ExpMapQuaternion::forceOnM_(RefVec out, const ConstRefVec& in)
{
toConstQuat inQuat(in.data());
toQuat outQuat(out.data());
outQuat = inQuat;
out.normalize();
}
Eigen::Matrix<double, 4, 3> ExpMapQuaternion::diffRetractation_(const ConstRefVec& x)
{
const Eigen::Map<const Eigen::Quaterniond> xQ(x.data());
Eigen::Matrix<double, 4, 3> J;
J << 0.5*xQ.w(), -0.5*xQ.z(), 0.5*xQ.y(),
0.5*xQ.z(), 0.5*xQ.w(), -0.5*xQ.x(),
-0.5*xQ.y(), 0.5*xQ.x(), 0.5*xQ.w(),
-0.5*xQ.x(), -0.5*xQ.y(), -0.5*xQ.z();
return J;
}
Eigen::Matrix<double, 3, 4> ExpMapQuaternion::diffPseudoLog0_(const ConstRefVec& v)
{
const toConstQuat vQ(v.data());
double n2 = vQ.vec().squaredNorm();
double n = sqrt(n2);
Eigen::Matrix<double, 3, 4> J;
if (n < prec && vQ.w()!=0 )
{
double a = 2/vQ.w();
double b = -2/(vQ.w()*vQ.w());
J << a, 0, 0, b*vQ.x(),
0, a, 0, b*vQ.y(),
0, 0, a, b*vQ.z();
}
else
{
// log(x,y,z,w) = f(x,y,z,w)*[x;y;z]
double f = atan2(2 * n * vQ.w(), vQ.w() * vQ.w() - n2) / n;
// df/dx = (x*atan((2*w*(x^2 + y^2 + z^2)^(1/2))/(- w^2 + x^2 + y^2 + z^2)))/(x^2 + y^2 + z^2)^(3/2) - ((2*w*x)/((x^2 + y^2 + z^2)^(1/2)*(- w^2 + x^2 + y^2 + z^2)) - (4*w*x*(x^2 + y^2 + z^2)^(1/2))/(- w^2 + x^2 + y^2 + z^2)^2)/(((4*w^2*(x^2 + y^2 + z^2))/(- w^2 + x^2 + y^2 + z^2)^2 + 1)*(x^2 + y^2 + z^2)^(1/2))
// df/dy = (y*atan((2*w*(x^2 + y^2 + z^2)^(1/2))/(- w^2 + x^2 + y^2 + z^2)))/(x^2 + y^2 + z^2)^(3/2) - ((2*w*y)/((x^2 + y^2 + z^2)^(1/2)*(- w^2 + x^2 + y^2 + z^2)) - (4*w*y*(x^2 + y^2 + z^2)^(1/2))/(- w^2 + x^2 + y^2 + z^2)^2)/(((4*w^2*(x^2 + y^2 + z^2))/(- w^2 + x^2 + y^2 + z^2)^2 + 1)*(x^2 + y^2 + z^2)^(1/2))
// df/dz = (z*atan((2*w*(x^2 + y^2 + z^2)^(1/2))/(- w^2 + x^2 + y^2 + z^2)))/(x^2 + y^2 + z^2)^(3/2) - ((2*w*z)/((x^2 + y^2 + z^2)^(1/2)*(- w^2 + x^2 + y^2 + z^2)) - (4*w*z*(x^2 + y^2 + z^2)^(1/2))/(- w^2 + x^2 + y^2 + z^2)^2)/(((4*w^2*(x^2 + y^2 + z^2))/(- w^2 + x^2 + y^2 + z^2)^2 + 1)*(x^2 + y^2 + z^2)^(1/2))
// g = (atan((2*w*n)/(- w^2 + n2)))/(n2)^(3/2) - ((2*w)/(n*(- w^2 + n2)) - (4*w*n)/(- w^2 + n2)^2)/(((4*w^2*n2)/(- w^2 + n2)^2 + 1)*n)
// g = (atan((2*w*n)/(- w^2 + n2)))/(n2*n) - ((2*w/n2)*(- w^2 + n2) - 4*w)/(4*w^2*n2+(- w^2 + n2)^2)
// df/dx = g*x = (x*atan((2*w*n)/(- w^2 + n2)))/(n2)^(3/2) - ((2*w*x)/(n*(- w^2 + n2)) - (4*w*x*n)/(- w^2 + n2)^2)/(((4*w^2*n2)/(- w^2 + n2)^2 + 1)*n)
// df/dy = g*y = (y*atan((2*w*n)/(- w^2 + n2)))/(n2)^(3/2) - ((2*w*y)/(n*(- w^2 + n2)) - (4*w*y*n)/(- w^2 + n2)^2)/(((4*w^2*n2)/(- w^2 + n2)^2 + 1)*n)
// df/dz = g*z = (z*atan((2*w*n)/(- w^2 + n2)))/(n2)^(3/2) - ((2*w*z)/(n*(- w^2 + n2)) - (4*w*z*n)/(- w^2 + n2)^2)/(((4*w^2*n2)/(- w^2 + n2)^2 + 1)*n)
// df/dw = -2/(w²+x²+y²+z²)
double g = (atan((2*vQ.w()*n)/(- vQ.w()*vQ.w() + n2)))/(n2*n) - ((2*vQ.w()/n2)*(- vQ.w()*vQ.w() + n2) - 4*vQ.w())/(4*vQ.w()*vQ.w()*n2+(- vQ.w()*vQ.w() + n2)*(- vQ.w()*vQ.w() + n2));
double dfdw = -2/(vQ.w()*vQ.w() + n2);
/*
* J = [ g.x²+f, g.y.x, g.z.x, df/dw.x]
* [ g.x.y, g.y²+f, g.z.y, df/dw.y]
* [ g.x.z, g.y.z, g.z²+f, df/dw.z]
*/
J << g*vQ.x()*vQ.x()+f, g*vQ.y()*vQ.x(), g*vQ.z()*vQ.x(), dfdw*vQ.x(),
g*vQ.x()*vQ.y(), g*vQ.y()*vQ.y()+f, g*vQ.z()*vQ.y(), dfdw*vQ.y(),
g*vQ.x()*vQ.z(), g*vQ.y()*vQ.z(), g*vQ.z()*vQ.z()+f, dfdw*vQ.z();
}
return J;
}
void ExpMapQuaternion::retractation_(RefVec out, const ConstRefVec& x, const ConstRefVec& v)
{
OutputType q;
exponential(q,v);
toQuat(out.data()) = (toConstQuat(x.data()))*(toConstQuat(q.data())); //out = x*exp(v)
}
