GNC/base_guidance.py — Base Guidance

Middle layer of the guidance tower. Extends [[com_guidance]] with desired base attitude $\bm R_{bd}(s)$, $\bm\omega_{bd}$, and the impedance-derate ramp $\gamma(\sigma_G)$.

Offline Frame Sequence Build

A roll-continuous frame field is constructed once at init along the orbit:

• $z$-axis: inward $\hat{\bm z}_i = -\hat{\bm p}_i$
• $x$-axis: parallel-transported from the previous frame to remove sign flips
• enforce_frame_sequence_continuity removes any residual discontinuities

$$\bm R_i = \text{frame}(\hat{\bm z}_i,\ \bm x_{\mathrm{hint}})$$

The sequence is a stored field $\{\bm R_i\}$; no online optimisation.

Runtime SLERP

At progress $s$, bracket the stored frames at $i_0=\lfloor s\rfloor$, $i_1=i_0+1$, $\alpha=s-i_0$:

$$\bm R_{bd}(s) = \bm R_{i_0}\exp\!\big(\alpha \log(\bm R_{i_0}^\top\bm R_{i_1})\big)$$

(eq 5.6, final.tex §3)

Returns the stored frame exactly at integer samples. The geodesic interpolant guarantees $\bm R_{bd}(s)\in\mathrm{SO}(3)$ at every fractional progress.

Negligible Base Angular Acceleration

Desired base angular velocity $\bm\omega_{bd}$ comes from a central finite difference of the frame field:

$$\bm\omega_{bd} \approx \frac{\log(\bm R_{-}^\top\bm R_{+})}{\Delta t},\qquad \Delta t = \frac{\Delta u}{\dot s}$$

(body-frame increment — omega_body_frame = True flag)

Because the inward reference rotates slowly along the orbit, the angular acceleration is negligible:

$$\dot{\bm\omega}_{bd} \approx \bm 0$$

(final.tex §3)

Impedance Derate Ramp $\gamma(\sigma_G)$

$\sigma_G\equiv\sigma_{\min}(\bm\Gamma)$ measures proximity to singularity. The ramp:

$$\gamma(\sigma_G) = \begin{cases} \sigma_{\mathrm{crit},1} & \sigma_G \le \sigma_{\mathrm{crit},2}\\[4pt] 1 & \sigma_G \ge \sigma_{\mathrm{crit},1}\\[4pt] \sigma_{\mathrm{crit},1} + (1-\sigma_{\mathrm{crit},1})\dfrac{\sigma_G - \sigma_{\mathrm{crit},2}}{\sigma_{\mathrm{crit},1}-\sigma_{\mathrm{crit},2}} & \text{otherwise} \end{cases}$$

(eq 6.4, final.tex §7) $\sigma_{\mathrm{crit},1}=v_{\mathrm{ref}}/x_{\max}\approx 0.143$; $\sigma_{\mathrm{crit},2}=\sigma_{\mathrm{crit},1}^2$.

Derate Applied to the Demand

derate_desired_by_gamma scales the CoM desired by $\gamma$ before smoothing:

$$\bm v_c \leftarrow \gamma \bm v_c,\qquad \bm a_c \leftarrow \gamma^2\bm a_c,\qquad \bm\omega_b \leftarrow \gamma \bm\omega_b$$

$\bm a_c\propto\lVert\bm v_c\rVert^2$ so the quadratic scaling keeps centripetal demand consistent. The slowdown is emergent: softening $\breve{\bm K}$ relocates the cruise-lag equilibrium to $\tilde{\bm x}_{ss}\to\tfrac{1}{\gamma}\tilde{\bm x}_{ss}$ (eq 4.14) rather than throttling the command.