pose_guidance

POSE-Mode Guidance

Definition

POSE-mode guidance is the project’s own deterministic reference generator for the free-flying space manipulator: it produces the desired centre-of-mass orbit, the desired base attitude, and the desired end-effector (camera) pose and twist that the coordinated circumcentroidal controller then tracks. It is the guidance layer — distinct from the borrowed Giordano control law — and everything on this page is project-original (final.tex provenance, not Giordano). The controller it feeds is coordinated_circumcentroidal_control; its output is an analytic acceleration feedforward delivered into that stable inner loop.

The camera has two modes (terminology.md): POSE (the camera looks directly at the scheduled surface aim point — the default, achieving complete coverage) and ANCHOR (the scorer chooses the final aim point — retained for the risk-aware planning layer). This page covers the POSE-mode plan; the ANCHOR scoring metric is out of scope.

Regime — free-flying, project-original

The base is fully actuated (6-DOF), so guidance may command an arbitrary CoM orbit and base attitude independently — there is no momentum-conservation constraint coupling them. The constructions below (spherical-helix orbit, look-at standoff pose, startup ramp) are the project’s own; only the final change into circumcentroidal ($\oplus$) coordinates reuses Giordano’s kinematic maps (see circumcentroidal_motion).

Key Equations

Symbols per notation.md; guidance-construction quantities are defined inline at first use. Equation tags reference the project equation sheet current_sota §5 (a synthesis with final.tex as primary source).

CoM orbit reference and startup ramp

Continuous runtime guidance tracks a progress variable $s$ along the discrete helix $\Lambda$. A launch ramp scales the demanded CoM speed up from rest ($v_d(0)=0$, a gentle release), with startup time scale ${\tau }_{\mathrm{su}}$ and configured cruise speed $v_{\mathrm{ref}}$:

$$v_d(t)=[1-\exp (-t/{\tau }_{\mathrm{su}}\left)\right]v_{\mathrm{ref}}$$

(final.tex §2.1 / current_sota eq 5.1)

A second feedforward shaping warm-starts the acceleration. The blended feedforward acceleration ${\mathbf{a}}_{\mathrm{ff}}(t)$ begins at the cached seed value ${\mathbf{a}}_{cd}(0)$ (the centripetal demand sampled at release) and, across the blend window ${\tau }_{\mathrm{ff}}$, exponentially hands over to the live centripetal demand ${\mathbf{a}}_{cd}$. The blend weight is $0$ at $t=0$ and rises to $1$ as $t\to \infty$:

$${\mathbf{a}}_{\mathrm{ff}}(t)={\mathbf{a}}_{cd}(0)+[1-\exp (-t/{\tau }_{\mathrm{ff}}\left)\right]({\mathbf{a}}_{cd}-{\mathbf{a}}_{cd}(0))$$

(COMController.feedforward_accel; final.tex §2.1 / current_sota eq 5.2)

The shaping rises from the trusted seed to the live demand — it does not decay to zero. Physically the controller starts from the acceleration it already trusts at release and eases onto the current curved-path centripetal term.

Analytic spherical-helix tangent and curvature

The acceleration feedforward is the centripetal term needed to hold a curved path at speed:

$${\mathbf{a}}_{cd}=\Vert {\mathbf{v}}_{cd}{\Vert }^2\mathbf{\kappa }$$

(final.tex §2.2 / current_sota eq 5.3)

The curvature vector $\mathbf{\kappa }$ and unit tangent $\hat{\mathbf{t}}$ are evaluated analytically (replacing an older finite-difference form whose ripple polluted the feedforward). Map progress to the helix parameter $u\in [0,1]$ with $\theta =w_{\theta }u$, $\varphi ={\varphi }_0+w_{\varphi }u$, giving the spherical-helix position $\mathbf{p}(u)$:

$$\mathbf{p}(u)=R\left[\begin{array}{c}\cos \varphi \cos \theta \\ \cos \varphi \sin \theta \\ \sin \varphi \end{array}\right]\hat{\mathbf{t}}=\frac{{\mathbf{p}}^{\prime }}{\Vert {\mathbf{p}}^{\prime }\Vert }\mathbf{\kappa }=\frac{1}{\Vert {\mathbf{p}}^{\prime }{\Vert }^2}({\mathbf{p}}^{\prime \prime }-\frac{{\mathbf{p}}^{\prime }\cdot {\mathbf{p}}^{\prime \prime }}{\Vert {\mathbf{p}}^{\prime }{\Vert }^2}{\mathbf{p}}^{\prime })$$

(final.tex §2.2 / current_sota eqs 5.4, 5.5)

Because $\theta ,\varphi$ are linear in $u$, the derivatives ${\mathbf{p}}^{\prime },{\mathbf{p}}^{\prime \prime }$ follow in closed form, so $\hat{\mathbf{t}}$ and $\mathbf{\kappa }$ are exact.

Base attitude reference (offline build + runtime SLERP)

The reference-frame sequence is built once offline (each frame’s $z$-axis aligned to the next inward direction; the previous $x$-axis carried as a roll hint; sign flips removed). At runtime the desired base attitude ${\mathbf{R}}_{bd}(s)$ at continuous progress $s$ is the geodesic SLERP between the bracketing frames ${\mathbf{R}}_{i_0},{\mathbf{R}}_{i_1}$ (indices $i_0=\lfloor s\rfloor$, interpolation fraction $\alpha =s-i_0\in [0,1)$ — a local fraction, unrelated to the risk-layer $\alpha$):

$${\mathbf{R}}_{bd}(s)={\mathbf{R}}_{i_0}\exp (\alpha \log ({\mathbf{R}}_{i_0}^{\top }{\mathbf{R}}_{i_1}))$$

(final.tex §3 / current_sota eq 5.6)

This returns the stored frame exactly at samples; the base angular acceleration is taken negligible, ${\dot{\mathbf{\omega }}}_{bd}\approx 0$ (slow inward reference) — the assumption reused when the desired twist is differentiated below.

EE look-at pose and desired twist (POSE mode)

The desired EE pose is a look-at standoff construction from the surface aim point ${\mathbf{x}}_{\mathrm{surf}}$ and the CoM position ${\mathbf{p}}_c$: with standoff vector $\mathbf{r}={\mathbf{x}}_{\mathrm{surf}}-{\mathbf{p}}_c$, range $\rho =\Vert \mathbf{r}\Vert$, and aim direction $\hat{\mathbf{r}}=\mathbf{r}/\rho$, the look axis equals the aim direction, ${\mathbf{z}}_{ed}=\hat{\mathbf{r}}$, and the desired position is offset by the camera standoff $r_{\mathrm{cam}}$ (the pose construction eq 5.7 and the term standoff live on coordinated_circumcentroidal_control Claim 6 and terminology.md — not re-derived here). The whole desired pose is governed by the single direction $\hat{\mathbf{r}}$.

Reference rates follow at constant aim speed: $\dot{\mathbf{r}}={\dot{\mathbf{x}}}_{\mathrm{surf}}-{\mathbf{v}}_{cd}$, $\ddot{\mathbf{r}}={\ddot{\mathbf{x}}}_{\mathrm{surf}}-{\mathbf{a}}_{cd}$, $\dot{\rho }=\hat{\mathbf{r}}\cdot \dot{\mathbf{r}}$. The unit-vector rates (radial part removed by the projector $\mathbb{I}-\hat{\mathbf{r}}{\hat{\mathbf{r}}}^{\top }$):

$$\dot{\hat{\mathbf{r}}}=\frac{1}{\rho }(\mathbb{I}-\hat{\mathbf{r}}{\hat{\mathbf{r}}}^{\top })\dot{\mathbf{r}}\ddot{\hat{\mathbf{r}}}=\frac{1}{\rho }(\mathbb{I}-\hat{\mathbf{r}}{\hat{\mathbf{r}}}^{\top })\ddot{\mathbf{r}}-\frac{2\dot{\rho }}{\rho }\dot{\hat{\mathbf{r}}}-\Vert \dot{\hat{\mathbf{r}}}{\Vert }^2\hat{\mathbf{r}}$$

(final.tex §5.1.2 / current_sota eqs 5.8, 5.9)

The three $\ddot{\hat{\mathbf{r}}}$ terms are the projected aim acceleration, a Coriolis-like range term, and the centripetal term holding $\hat{\mathbf{r}}$ on the unit sphere. Desired EE linear velocity and acceleration then add the CoM motion and the standoff-scaled unit-vector rates:

$${\dot{\mathbf{p}}}_{ed}={\mathbf{v}}_{cd}+r_{\mathrm{cam}}\dot{\hat{\mathbf{r}}}{\ddot{\mathbf{p}}}_{ed}={\mathbf{a}}_{cd}+r_{\mathrm{cam}}\ddot{\hat{\mathbf{r}}}$$

(final.tex §5.1.2 / current_sota eqs 5.10, 5.11)

The desired angular velocity is the roll-free choice ${\mathbf{\omega }}_{ed}=\hat{\mathbf{r}}\times \dot{\hat{\mathbf{r}}}$ (eq 5.12, on coordinated_circumcentroidal_control Claim 6). Differentiating it gives the desired angular acceleration:

$${\dot{\mathbf{\omega }}}_{ed}=\hat{\mathbf{r}}\times \ddot{\hat{\mathbf{r}}}$$

(final.tex §5.1.3 / current_sota eq 5.13)

The choice is verified roll-free via ${\mathbf{\omega }}_{ed}\times \hat{\mathbf{r}}=\dot{\hat{\mathbf{r}}}$ and $\hat{\mathbf{r}}\cdot {\mathbf{\omega }}_{ed}=0$: one $(\dot{\hat{\mathbf{r}}},\ddot{\hat{\mathbf{r}}})$ computation feeds the entire 6-DOF desired twist. Stacking the linear and angular parts gives the world-frame desired twist:

$${\mathbf{\nu }}_{ed}=\left[\begin{array}{c}{\dot{\mathbf{p}}}_{ed}\\ {\mathbf{\omega }}_{ed}\end{array}\right]=\left[\begin{array}{c}{\mathbf{v}}_{cd}+r_{\mathrm{cam}}\dot{\hat{\mathbf{r}}}\\ \hat{\mathbf{r}}\times \dot{\hat{\mathbf{r}}}\end{array}\right]$$

(final.tex eq nu_e / current_sota eq 5.14)

Conversion to the local frame

Expressed in the rotating desired-EE frame (${\dot{\mathbf{R}}}_{ed}={\mathbf{\omega }}_{ed}^{\times }{\mathbf{R}}_{ed}$), the local twist and its rate are:

$${\mathbf{\nu }}_{ed}^{\mathrm{local}}=\left[\begin{array}{c}{\mathbf{R}}_{ed}^{\top }{\dot{\mathbf{p}}}_{ed}\\ {\mathbf{R}}_{ed}^{\top }{\mathbf{\omega }}_{ed}\end{array}\right]{\dot{\mathbf{\nu }}}_{ed}^{\mathrm{local}}=\left[\begin{array}{c}{\mathbf{R}}_{ed}^{\top }({\ddot{\mathbf{p}}}_{ed}-{\mathbf{\omega }}_{ed}\times {\dot{\mathbf{p}}}_{ed})\\ {\mathbf{R}}_{ed}^{\top }{\dot{\mathbf{\omega }}}_{ed}\end{array}\right]$$

(final.tex §5.1.4 / current_sota eqs 5.15, 5.16)

This local twist is then carried into circumcentroidal ($\oplus$) coordinates and differentiated (with ${\dot{\mathbf{\omega }}}_{bd}\approx 0$) to supply the analytic acceleration feedforward $\breve{\mathbf{M}}[0_3;{\dot{\mathbf{\nu }}}_{ed}^{\oplus }]$ into the controller. Those final two steps — ${\mathbf{\nu }}_{ed}^{\oplus }$ (eq 5.17) and ${\dot{\mathbf{\nu }}}_{ed}^{\oplus }$ (eq 5.18), built on Giordano’s maps ${\mathbf{G}}_{v_c},{\mathbf{G}}_{{\omega }_b}$ — live on coordinated_circumcentroidal_control Claim 6 and are not duplicated here.

Source / Provenance

• Ours (project-original). Eqs 5.1–5.16 — the startup speed ramp, the feedforward-blend warm-start, the analytic spherical-helix tangent and curvature, the SLERP base-attitude reference, the look-at standoff pose, and the desired-twist chain — are the project’s own deterministic POSE-mode guidance, tagged to final.tex (§2.1, §2.2, §3, §5.1). They are not stated in Giordano 2019, which supplies only the control law and the circumcentroidal coordinate machinery. Primary source: current_sota §5 (a synthesis with final.tex as primary).
• Borrowed piece. Only the final conversion into circumcentroidal coordinates (eqs 5.17–5.18) reuses Giordano’s kinematic maps ${\mathbf{G}}_{v_c},{\mathbf{G}}_{{\omega }_b}$; guidance introduces no new dynamic objects, reusing the machinery of circumcentroidal_motion. See giordano2019coordinated.
• Code. The blend warm-start (eq 5.2) is COMController.feedforward_accel; the guidance output feeds the working closed loop of coordinated_circumcentroidal_control (current_sota eq 4.12).

Related Topics

• coordinated_circumcentroidal_control — the controller this guidance feeds; holds the standoff pose (5.7), roll-free ${\mathbf{\omega }}_{ed}$ (5.12), and the $\oplus$ conversion (5.17, 5.18).
• circumcentroidal_motion — the $\oplus$ maps ${\mathbf{G}}_{v_c},{\mathbf{G}}_{{\omega }_b}$ reused to convert the desired twist.
• center_of_mass_regulation — the CoM PD loop that tracks the ramped orbit reference (5.1–5.3).
• trajectory_tracking — the tracking problem the analytic feedforward serves.
• cartesian_path_planning — Cartesian-space reference generation, the broader class this helix orbit sits in.

Open Questions

• The analytic feedforward (this page) is predicted to drop the operational EE-position floor $p_e$ toward the analytic cruise-lag limit; the gap closure vs the old finite-differenced demand is unverified pending the Phase-D deepening — see steady_state_error_floor.
• ANCHOR-mode guidance (scorer-chosen aim point) and its weighted-product scoring (current_sota §7.2) have no vault page; only the POSE-mode plan is covered here.
• The guidance-construction symbols ($\hat{\mathbf{r}},\rho ,{\mathbf{x}}_{\mathrm{surf}},{\mathbf{z}}_{ed},{\mathbf{p}}_{ed},{\mathbf{\nu }}_{ed},{\mathbf{a}}_{\mathrm{ff}},{\mathbf{R}}_{bd}$) are defined inline here but not yet in notation.md; central registration is a candidate for the next registry pass.