[Doc] Add remarkable identities for Jlog6

src/spatial/explog.hpp

@@ -461,6 +461,12 @@ namespace pinocchio
    *  \f]
    *  and
    *  \f[ \beta(x)=\left(\frac{1}{x^2} - \frac{\sin x}{2x(1-\cos x)}\right) \f]
+   *
+   *  \cheatsheet For \f$ M = log_6(a \times b) \f$,
+   *  \f$ \frac{\partial M}{\partial a} = Jlog_6(M) \times X_b^{-1} \f$
+   *  and \f$ \frac{\partial M}{\partial b} = Jlog_6(M) \f$. \n
+   *  For \f$ M = log_6(a^{-1}) \f$,
+   *  \f$ \frac{\partial M}{\partial a} = - Jlog_6(M) \times X_a \f$.
    */
   template<typename Scalar, int Options, typename Matrix6Like>
   void Jlog6(const SE3Tpl<Scalar, Options> & M,

unittest/explog.cpp

@@ -358,6 +358,32 @@ BOOST_AUTO_TEST_CASE(Jlog6_fd)
   BOOST_CHECK(Jfd.isApprox(Jlog, sqrt(step)));
 }
 
+BOOST_AUTO_TEST_CASE(Jlog6_of_product_fd)
+{
+  SE3 Ma(SE3::Random());
+  SE3 Mb(SE3::Random());
+
+  SE3::Matrix6 Jlog, Ja, Jb, Jfda, Jfdb;
+  Jlog6 (Ma.inverse() * Mb, Jlog);
+  Ja = - Jlog * (Ma.inverse() * Mb).toActionMatrixInverse();
+  Jb = Jlog;
+  Jfda.setZero();
+  Jfdb.setZero();
+
+  Motion dM; dM.setZero();
+  double step = 1e-8;
+  for (int i = 0; i < 6; ++i)
+  {
+    dM.toVector()[i] = step;
+    Jfda.col(i) = (log6((Ma*exp6(dM)).inverse()*Mb).toVector() - log6(Ma.inverse()*Mb).toVector()) / step;
+    Jfdb.col(i) = (log6(Ma.inverse()*Mb*exp6(dM)).toVector() - log6(Ma.inverse()*Mb).toVector()) / step;
+    dM.toVector()[i] = 0;
+  }
+
+  BOOST_CHECK(Jfda.isApprox(Ja, sqrt(step)));
+  BOOST_CHECK(Jfdb.isApprox(Jb, sqrt(step)));
+}
+
 BOOST_AUTO_TEST_CASE(Jexplog6)
 {
   Motion v(Motion::Random());