Exponential Coordinates

Definition

Exponential coordinates parameterize a rotation $\mathbf{Q}\in \mathrm{SO}(3)$ by a 3-vector
$\mathbf{\psi }\in {\mathbb{R}}^3$ through the exponential map $\mathbf{Q}=\exp ({\mathbf{\psi }}^{\wedge })$,
where ${\mathbf{\psi }}^{\wedge }$ is the skew-symmetric (Lie-algebra $\mathfrak{so}(3)$) matrix of
$\mathbf{\psi }$; the inverse is the logarithm map $\mathbf{\psi }=\log (\mathbf{Q}{)}^{\vee }$, which is
unique while the principal rotation angle has magnitude $<\pi$. In misra2017optimal
this gives a minimal (3-parameter) attitude representation for the end-effector pose inside a
point-to-point trajectory-optimization problem, so the final-pose boundary condition reduces to a plain
${\mathbb{R}}^6$ vector constraint rather than a constraint on the $\mathrm{SE}(3)$ manifold. The concept is
regime-independent (it is a chart on $\mathrm{SO}(3)$); the cited use is a redundant space manipulator
whose base attitude is held fixed during the maneuver, so it is neither the fully-reactive free-floating
case nor a base actively thrusting throughout the maneuver.

Key Equations

Symbols per notation.md.

Notation flags. (i) The source’s exponential-coordinate vector is $\mathbf{\psi }$, not in
notation.md — used as in the source (axis-angle / rotation-vector coordinates of $\mathrm{SO}(3)$), not
coined. (ii) notation.md reserves $\eta$ for the unit-quaternion scalar part; the source reuses the
glyph $\eta$ for its stacked pose coordinate $[{\mathbf{r}}_e;\mathbf{\psi }]$ — to avoid conflict the
stacked vector is written $\mathbf{\chi }$ here. (iii) The source labels $\mathbf{\psi }$ the
“classical Rodrigues parameters”; the defining map $\mathbf{Q}=\exp ({\mathbf{\psi }}^{\wedge })$ is the
standard rotation-vector / axis-angle exponential map of $\mathrm{SO}(3)$, reported as written.

Exponential map and its skew argument (source Eq. preceding 11):

\exp(\cdot):\mathfrak{so}(3)\to\mathrm{SO}(3),\quad \log(\cdot):\mathrm{SO}(3)\to\mathfrak{so}(3).$$ Local-coordinate kinematics with stacked pose $\boldsymbol\chi=[\boldsymbol r_e;\boldsymbol\psi]\in\mathbb{R}^6$ (source Eqs. 11–12): $$\dot{\boldsymbol\chi}=\begin{bmatrix}\dot{\boldsymbol r}_e\\ \dot{\boldsymbol\psi}\end{bmatrix} =\boldsymbol G(\boldsymbol\chi)\,\boldsymbol J_a(\boldsymbol\theta)\,\dot{\boldsymbol\theta}, \qquad \boldsymbol G(\boldsymbol\chi)=\begin{bmatrix}\boldsymbol E&\boldsymbol 0\\ \boldsymbol 0&\boldsymbol A(\boldsymbol\psi)^{-T}\end{bmatrix}\in\mathbb{R}^{6\times6},$$ where $\boldsymbol A(\boldsymbol\psi)^{-T}$ maps end-effector angular velocity to $\dot{\boldsymbol\psi}$, and $\boldsymbol J_a(\boldsymbol\theta)$ is the source's analytic end-effector Jacobian. ## Source Support - [misra2017optimal](../sources/misra2017optimal.md) — defines $\boldsymbol\psi$ as the exponential coordinates of $\boldsymbol Q_e$ via $\boldsymbol Q_e=\exp(\boldsymbol\psi^\wedge)$, derives the $\boldsymbol G(\boldsymbol\chi)$ rate map (Eqs. 11–12), and uses it to express the final end-effector pose constraint (Eq. 13) as a 6-vector boundary condition in an SCP trajectory-optimization formulation. ## Related Topics - [unit_quaternion](unit_quaternion.md) — an alternative (4-parameter, singularity-free) attitude parameterization; the source notes quaternions or modified Rodrigues parameters could replace $\boldsymbol\psi$. Quaternions are the thesis convention ($\boldsymbol\epsilon,\eta$ in notation.md). - [optimal_control_bvp](optimal_control_bvp.md) — exponential coordinates supply the terminal-pose boundary condition (Eq. 13) for the optimal-control / boundary-value problem. - [trajectory_optimization](trajectory_optimization.md) — the parameterization is what lets the final $\mathrm{SE}(3)$ pose enter the SCP/trajectory-optimization formulation as a smooth vector constraint. - [motion_planning](motion_planning.md) — the broader point-to-point planning task this terminal representation serves. - [inertial_space_tracking](inertial_space_tracking.md) — both work in the inertial end-effector frame; exponential coordinates give a minimal chart for the inertial-frame attitude target. ## Open Questions - The source holds base attitude fixed ($\Omega_b=0$) and conserves linear momentum during the maneuver; does this minimal-attitude terminal constraint carry over unchanged when our 6-DOF actuated base is thrusting and re-orienting throughout the trajectory (full free-flying regime)? - Exponential coordinates lose uniqueness / chart-singularity at rotation angle $\pi$ (the log map is multivalued there); for large terminal re-orientations, does the planner need chart switching or a switch to [unit_quaternion](unit_quaternion.md) to stay well-posed? - The source equates $\boldsymbol\psi$ with the "classical Rodrigues parameters," yet the stated map is the rotation-vector exponential — is the singularity at $\pi$ (exp/log) the relevant one for our use, or the $\pm\pi$ blow-up of the Rodrigues (Gibbs) vector?