Jacobians on Lie groups

Perturb with ⊕, measure with ⊖

For $f:\mathcal{M}\to\mathcal{N}$ the right Jacobian is the matrix that maps a small tangent step at $X$ to the resulting tangent step at $f(X)$:

If $\mathcal{N}$ is a vector space, $\ominus$ is ordinary subtraction. The simplest useful case is rotating a point, $f(R) = Rp$. Perturbing on the right, $R\operatorname{Exp}(\tau)p \approx R(I + [\tau]_\times)p = Rp - R[p]_\times\tau$, so

The demo perturbs a rotation by $\tau$ and compares where the points actually go (blue) with where the Jacobian predicts (hollow). The error shrinks quadratically in $\lVert\tau\rVert$, the signature of a correct first derivative.

The correction that makes Exp(a + b) behave

Adding a small $\delta$ to a rotation vector $\phi$ is not the same as composing a small rotation $\operatorname{Exp}(\delta)$. The two are related by the right Jacobian $J_r(\phi)$, the derivative of $\operatorname{Exp}$ in the sense of section 1:

The left Jacobian does the same job on the other side, $\operatorname{Exp}(\phi+\delta)\approx\operatorname{Exp}(J_l\delta)\operatorname{Exp}(\phi)$, and $J_l(\phi) = J_r(-\phi) = \operatorname{Exp}(\phi)J_r(\phi)$. The left Jacobian is exactly the $V$ matrix of Part 7. Both equal $I$ at $\phi = 0$ and drift away from it as the angle grows; $J_r^{-1}$ blows up as $\theta\to2\pi$. These are the closed-form sums of the BCH series from Part 4, for the case where one argument is small. The plot compares three ways to predict $\operatorname{Exp}(\phi+\delta)$ as $\lVert\phi\rVert$ grows.

Six building blocks for every residual

These are right Jacobians (right perturbations on inputs and outputs), following Solà et al. For $SO(3)$, $\operatorname{Ad}_R = R$. Each "check" button draws a random rotation, point and tangent vector, evaluates the analytic formula, estimates the same matrix by central differences in the tangent space, and reports the largest difference. Anything below about $10^{-7}$ is agreement.

For $SE(3)$ the same table holds with $6\times6$ blocks: $\operatorname{Ad}_T$ from Part 8, $J_r(\tau)$ with the $Q$ block of Barfoot, and the action Jacobian $\partial(Tp)/\partial\tau = [\,R\ \ -R[p]_\times\,]$. The last rows check two of these as well.

A pose-graph edge and a reprojection error

Jacobians compose like ordinary derivatives: $J_{f\circ g} = J_f\,J_g$. Two residuals cover most of SLAM and structure from motion.

Rotation pose-graph edge. A relative measurement $Z \approx R_1^\top R_2$ gives the residual $r = \operatorname{Log}(Z^\top R_1^\top R_2)$. Chaining the Log, compose and inverse rules gives

Reprojection (bundle adjustment). A camera with pose $T_{WC}$ sees a world point $p_W$ at pixel $z$. The camera-frame point is $p_C = T_{WC}^{-1}p_W$, and the residual is $r = \pi(p_C) - z$ with pinhole projection $\pi$. Perturbing the pose on the right:

Truncation error versus round-off

A central difference in the tangent space, $\bigl(f(X\oplus he_k)\ominus f(X\ominus\ldots)\bigr)/2h$, has truncation error proportional to $h^2$ and round-off error proportional to $\varepsilon_{\text{mach}}/h$. The total is smallest near $h\approx\varepsilon_{\text{mach}}^{1/3}\approx 10^{-5}$, where roughly 10 of the 16 digits survive. That is plenty to test an analytic Jacobian, but too slow and too noisy to use inside a large optimizer, and useless at a point where the function is not smooth. The standard practice is to write the analytic Jacobian, then keep a numerical check in the test suite, exactly like the buttons above. The plot shows the V-shaped error curve for $J_r$ itself.