Baker–Campbell–Hausdorff formula: in non-commutative Lie groups, when composing exp(Z) = exp(X)exp(Y) it is not true that Z = X+Y. Instead, Z can be calculated using the BCH formula:
Z = X + Y + [X,Y]/2 + [X-Y,[X,Y]]/12 - [Y,[X,[X,Y]]]/24 + ... See http://en.wikipedia.org/wiki/Baker–Campbell–Hausdorff_formularelease/4.3a0
Frank Dellaert
parent
67b4834bdb
commit
5e00c58ea7
15
cpp/Lie.h
15
cpp/Lie.h
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@ -104,4 +104,19 @@ namespace gtsam {
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inline Vector logmap(const Vector& p) { return p;}
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inline Vector logmap(const Vector& p1,const Vector& p2) { return p2-p1;}
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/**
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* Three term approximation of the BakerÐCampbellÐHausdorff formula
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* In non-commutative Lie groups, when composing exp(Z) = exp(X)exp(Y)
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* it is not true that Z = X+Y. Instead, Z can be calculated using the BCH
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* formula: Z = X + Y + [X,Y]/2 + [X-Y,[X,Y]]/12 - [Y,[X,[X,Y]]]/24
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* http://en.wikipedia.org/wiki/BakerÐCampbellÐHausdorff_formula
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*/
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template<class T>
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T BCH(const T& X, const T& Y) {
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static const double _2 = 1. / 2., _12 = 1. / 12., _24 = 1. / 24.;
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T X_Y = bracket(X, Y);
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return X + Y + _2 * X_Y + _12 * bracket(X - Y, X_Y) - _24 * bracket(Y,
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bracket(X, X_Y));
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}
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} // namespace gtsam
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117
cpp/testRot3.cpp
117
cpp/testRot3.cpp
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@ -17,45 +17,51 @@ Point3 P(0.2,0.7,-2.0);
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double error = 1e-9, epsilon = 0.001;
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/* ************************************************************************* */
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TEST( Rot3, constructor) {
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TEST( Rot3, constructor)
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{
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Rot3 expected(eye(3, 3));
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Vector r1(3), r2(3), r3(3);
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r1(0)=1;r1(1)=0;r1(2)=0;
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r2(0)=0;r2(1)=1;r2(2)=0;
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r3(0)=0;r3(1)=0;r3(2)=1;
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r1(0) = 1;
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r1(1) = 0;
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r1(2) = 0;
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r2(0) = 0;
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r2(1) = 1;
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r2(2) = 0;
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r3(0) = 0;
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r3(1) = 0;
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r3(2) = 1;
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Rot3 actual(r1, r2, r3);
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CHECK(assert_equal(actual,expected));
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}
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/* ************************************************************************* */
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TEST( Rot3, constructor2) {
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Matrix R = Matrix_(3,3,
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11.,12.,13.,
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21.,22.,23.,
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31.,32.,33.);
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TEST( Rot3, constructor2)
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{
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Matrix R = Matrix_(3, 3, 11., 12., 13., 21., 22., 23., 31., 32., 33.);
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Rot3 actual(R);
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Rot3 expected(11,12,13,
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21,22,23,
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31,32,33);
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Rot3 expected(11, 12, 13, 21, 22, 23, 31, 32, 33);
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CHECK(assert_equal(actual,expected));
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}
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/* ************************************************************************* */
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TEST( Rot3, constructor3) {
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TEST( Rot3, constructor3)
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{
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Rot3 expected(1, 2, 3, 4, 5, 6, 7, 8, 9);
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Point3 r1(1, 4, 7), r2(2, 5, 8), r3(3, 6, 9);
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CHECK(assert_equal(Rot3(r1,r2,r3),expected));
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}
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/* ************************************************************************* */
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TEST( Rot3, transpose) {
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TEST( Rot3, transpose)
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{
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Rot3 R(1, 2, 3, 4, 5, 6, 7, 8, 9);
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Point3 r1(1, 2, 3), r2(4, 5, 6), r3(7, 8, 9);
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CHECK(assert_equal(inverse(R),Rot3(r1,r2,r3)));
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}
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/* ************************************************************************* */
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TEST( Rot3, equals) {
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TEST( Rot3, equals)
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{
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CHECK(R.equals(R));
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Rot3 zero;
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CHECK(!R.equals(zero));
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@ -71,7 +77,8 @@ Rot3 slow_but_correct_rodriguez(const Vector& w) {
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}
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/* ************************************************************************* */
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TEST( Rot3, rodriguez) {
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TEST( Rot3, rodriguez)
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{
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Rot3 R1 = rodriguez(epsilon, 0, 0);
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Vector w = Vector_(3, epsilon, 0., 0.);
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Rot3 R2 = slow_but_correct_rodriguez(w);
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@ -79,17 +86,20 @@ TEST( Rot3, rodriguez) {
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}
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/* ************************************************************************* */
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TEST( Rot3, rodriguez2) {
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Vector v(3); v(0) = 0; v(1) = 1; v(2) = 0;
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TEST( Rot3, rodriguez2)
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{
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Vector v(3);
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v(0) = 0;
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v(1) = 1;
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v(2) = 0;
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Rot3 R1 = rodriguez(v, 3.14 / 4.0);
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Rot3 R2(0.707388,0,0.706825,
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0,1,0,
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-0.706825,0,0.707388);
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Rot3 R2(0.707388, 0, 0.706825, 0, 1, 0, -0.706825, 0, 0.707388);
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CHECK(assert_equal(R1,R2,1e-5));
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}
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/* ************************************************************************* */
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TEST( Rot3, rodriguez3) {
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TEST( Rot3, rodriguez3)
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{
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Vector w = Vector_(3, 0.1, 0.2, 0.3);
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Rot3 R1 = rodriguez(w / norm_2(w), norm_2(w));
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Rot3 R2 = slow_but_correct_rodriguez(w);
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@ -170,6 +180,34 @@ TEST(Rot3, log)
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CHECK(assert_equal(R5,R3*R2));
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}
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/* ************************************************************************* */
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class AngularVelocity: public Point3 {
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public:
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AngularVelocity(const Point3& p) :
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Point3(p) {
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}
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AngularVelocity(double wx, double wy, double wz) :
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Point3(wx, wy, wz) {
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}
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};
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AngularVelocity bracket(const AngularVelocity& X, const AngularVelocity& Y) {
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return cross(X, Y);
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}
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/* ************************************************************************* */
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TEST(Rot3, BCH)
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{
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// Approximate exmap by BCH formula
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AngularVelocity w1(0.2, -0.1, 0.1);
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AngularVelocity w2(0.01, 0.02, -0.03);
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Rot3 R1 = expmap<Rot3> (w1.vector()), R2 = expmap<Rot3> (w2.vector());
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Rot3 R3 = R1 * R2;
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Vector expected = logmap(R3);
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Vector actual = BCH(w1, w2).vector();
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CHECK(assert_equal(expected, actual,1e-5));
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}
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/* ************************************************************************* */
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// rotate derivatives
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@ -228,12 +266,14 @@ TEST( Rot3, compose )
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Rot3 actual = compose(R1, R2);
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CHECK(assert_equal(expected,actual));
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Matrix numericalH1 = numericalDerivative21<Rot3,Rot3,Rot3>(compose, R1, R2, 1e-5);
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Matrix numericalH1 = numericalDerivative21<Rot3, Rot3, Rot3> (compose, R1,
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R2, 1e-5);
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Matrix actualH1 = Dcompose1(R1, R2);
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CHECK(assert_equal(numericalH1,actualH1));
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Matrix actualH2 = Dcompose2(R1, R2);
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Matrix numericalH2 = numericalDerivative22<Rot3,Rot3,Rot3>(compose, R1, R2, 1e-5);
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Matrix numericalH2 = numericalDerivative22<Rot3, Rot3, Rot3> (compose, R1,
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R2, 1e-5);
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CHECK(assert_equal(numericalH2,actualH2));
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}
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@ -287,28 +327,19 @@ TEST( Rot3, xyz )
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// z
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// | * Y=(ct,st)
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// x----y
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Rot3 expected1(
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1, 0, 0,
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0, ct,-st,
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0, st, ct);
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Rot3 expected1(1, 0, 0, 0, ct, -st, 0, st, ct);
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CHECK(assert_equal(expected1,Rot3::Rx(t)));
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// x
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// | * Z=(ct,st)
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// y----z
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Rot3 expected2(
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ct, 0, st,
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0, 1, 0,
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-st, 0, ct);
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Rot3 expected2(ct, 0, st, 0, 1, 0, -st, 0, ct);
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CHECK(assert_equal(expected2,Rot3::Ry(t)));
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// y
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// | X=* (ct,st)
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// z----x
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Rot3 expected3(
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ct, -st, 0,
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st, ct, 0,
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0, 0, 1);
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Rot3 expected3(ct, -st, 0, st, ct, 0, 0, 0, 1);
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CHECK(assert_equal(expected3,Rot3::Rz(t)));
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// Check compound rotation
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@ -339,7 +370,8 @@ TEST( Rot3, yaw_pitch_roll )
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TEST( Rot3, RQ)
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{
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// Try RQ on a pure rotation
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Matrix actualK; Vector actual;
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Matrix actualK;
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Vector actual;
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boost::tie(actualK, actual) = RQ(R.matrix());
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Vector expected = Vector_(3, 0.14715, 0.385821, 0.231671);
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CHECK(assert_equal(eye(3),actualK));
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@ -358,11 +390,7 @@ TEST( Rot3, RQ)
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CHECK(assert_equal(Vector_(3,0.0,0.0,0.1),Rot3::roll (0.1).ypr()));
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// Try RQ to recover calibration from 3*3 sub-block of projection matrix
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Matrix K = Matrix_(3,3,
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500.0, 0.0, 320.0,
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0.0, 500.0, 240.0,
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0.0, 0.0, 1.0
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);
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Matrix K = Matrix_(3, 3, 500.0, 0.0, 320.0, 0.0, 500.0, 240.0, 0.0, 0.0, 1.0);
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Matrix A = K * R.matrix();
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boost::tie(actualK, actual) = RQ(A);
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CHECK(assert_equal(K,actualK));
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@ -370,6 +398,9 @@ TEST( Rot3, RQ)
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}
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/* ************************************************************************* */
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int main(){ TestResult tr; return TestRegistry::runAllTests(tr); }
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int main() {
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TestResult tr;
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return TestRegistry::runAllTests(tr);
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}
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/* ************************************************************************* */
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