I would like to understand how to properly solve an ODE involving Lie group/algebra elements.

As is commonly seen in robotics and engineering, consider the kinematic equation that describes the orientation of a planar object using an element of the matrix Lie group, $\text{SO}(2)$: $$ \dot{\Phi} = \Phi \omega^\wedge, $$ where $\Phi \in \text{SO}(2)$, $\omega \in \mathbb{R}$, and the "hat-map" is defined as $$ \begin{align*} ^\wedge : \mathbb{R} &\to \mathfrak{so}(2) \\ a &\mapsto \begin{bmatrix} 0 & -a \\ a & 0 \end{bmatrix}. \end{align*} $$

The objective is to solve the first-order ODE using valid group operations that preserve the orthonormal constraints of matrices in $\text{SO}(2)$ (i.e., using only the group operator, which is defined to be matrix multiplication).

Note that the kinematic equation is of the form $$ \frac{d\Phi}{dt} + \Phi(t) p(t) = q(t) $$ with $p(t)=-\omega^\wedge$ and $q(t)=0$. Assuming $\omega$ is constant, we can use $\text{exp}(-\omega^\wedge t)$ as the integrating factor, we have $$ \begin{align} \underbrace{(\dot{\Phi} - \Phi\omega^\wedge)}_1e^{-\omega^\wedge t} &= \underbrace{0}_2 \\ \frac{d}{dt}[\Phi e^{-\omega^\wedge t}] &= 0 \\ \int_0^t \frac{d}{d\tau}[\Phi e^{-\omega^\wedge \tau}]d\tau &= 0 \end{align} $$

Using the Fundamental Theorem of Calculus, we find the solution $$ \begin{align} \underbrace{\Phi(t) e^{-\omega^\wedge t} - \Phi_0}_3 &= 0 \\ \Phi(t) &= \Phi_0 e^{\omega^\wedge t}. \end{align} $$

The solution seems correct and this approach gives me intuition into Lie integrator techniques, including how to digitally implement this technique by discretizing. However, the use of the Fundamental Theorem of Calculus seems incorrect as it causes us to subtract two group elements, which is not defined.

Stepping through the use of operators and symbols:

  1. This subtraction is okay because $\dot{\Phi}$ and $\Phi\omega^\wedge$ are in the tangent space, which is a vector space.

  2. $0$ is not defined in the manifold $\text{SO}(2)$.

  3. The subtraction of two group elements is not defined.

What is the correct approach here? Do I need to redefine the integral to use the exponential map as is done for various algorithms in [1]? What background theory am I misunderstanding? How do I correctly deal with items 2 and 3?

Coming from engineering, my differential geometry background is weak.

[1] C. Hertzberg, R. Wagner, U. Frese, and L. Schröder, “Integrating generic sensor fusion algorithms with sound state representations through encapsulation of manifolds,” Inf. Fusion, vol. 14, no. 1, pp. 57–77, 2013.

  • $\begingroup$ You may try to diagonalize over $\mathbb{C}$ which then reduces to solving two linear ODE with complex coefficients $\endgroup$ – daruma Jun 19 '18 at 23:40
  • $\begingroup$ Your concern with subtraction of group elements should start earlier, what exactly is $\frac{dΦ}{dt}$? Analytically it is the limit of a difference quotient, involving a difference of group elements. You might want to consider $Φ^{-1}\frac{dΦ}{dt}$ as element of the Lie algebra. Or change to the language of vector fields as sections in the tangent bundle of a manifold. $\endgroup$ – LutzL Jun 20 '18 at 10:55

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