Nominal guidance & control of a free-flying space manipulator

Progress report · nominal phase. Where the end-effector error floor comes from — and why the nominal algorithms are locked down enough to enter the risk-aware phase.

Terminology

• Circumcentroidal coordinated control (the map) — a coordinate transformation that inertially decouples the centre-of-mass (CoM) translation from the attitude + end-effector dynamics about the CoM frame.
• Three subsystems — CoM: the decoupled loop , inertially free. Base + EE: the coupled circumcentroidal block .

Mission & system

Inspect the surface of a target satellite for damage. A camera replaces the traditional end-effector. The system CoM orbits the target on a smooth spherical helix; the base turns gradually to point its -axis radially inward.

I want to get an animated plot here since we have Plotly

Inspection mesh needs a new slide and explanation - this should be a .pcd if we can render/embed that in Plotly/Open3D somehow?

Coordinated control — a cascade on decoupled coordinates

The expanded form of Giordano eq-34b is the closed-loop control law:

• Inner loop — regulates CoM translation, rendered a decoupled system by the coordinate transform.
• Outer loop — regulates base attitude + EE pose, .
• A reduced-acceleration feedforward supplies the moving reference's inertial demand.

Source: singularities/final.tex §Control; Giordano, Ott, Albu-Schäffer, IEEE RA-L 2019.

What advanced (since last report)

• Deterministic POSE camera guidance — the camera looks directly at a scheduled surface point (smoothed projection of the orbit). Two modes instead of four; reaches complete coverage (0.9972) without the old reactive state machine's timing bugs.
• Analytic helix tangent — curvature feedforward in closed form on the smooth helix; the per-sample ripple from finite-differencing is eliminated.
• Launch ramp — : commanded speed ramps from rest so release is gentle, not impulsive.
• Orbit-synced mission end — ending on achieved orbit progress (not a fixed clock) cuts mission length −36% (1100 → 703 s) with coverage held.

★ The end-effector error floor — theory and measurement

Can tracking error be driven to zero, or is there a standing pose error the outer loop must hold? At steady state () the closed loop reduces to an algebraic balance.

Theoretical floor. What survives is a balance of the proportional term, the Coriolis force of the cruising arm, and the CoM coupling:

The idealized limit (perfect CoM tracking) is the Coriolis-only form : a floor set by the moving reference.

Measurement (current gold, 703 s).

FIGURE — to embed: CoM error (~1 mm) vs EE error (~0.104 m) time series, current gold. Live interactive plot already exists in the deployed report content/reports/checkin-jun25-26.html (the p_c & p_e panels). Embed a static export here once a browser is connected (or via an Orchestrator replot). The table above carries the decomposition quantitatively meanwhile.

So what. The CoM loop holds ~1 mm, but the operational EE floor is ~0.104 m — so the floor is not the CoM controller; it is the multibody coupling (the cruising arm's Coriolis force fed through the -reconstruction). The floor is architectural. The gap between the theoretical limit and the measured floor is precisely the research agenda below.

The system operates at its singularity floor

Over the real mission the conditioning median sits just below the 0.025 derate floor — the arm spends nearly half its working time near singularity. This is not academic; the handling earns its place.

The singular threshold from the velocity budget: . Four layers of defence:

• Tikhonov -regularization — , .
• Speed derating through impedance — gains and base wrench scaled by near singular.
• Damped inverse — three-tier: undamped → damped → hold-last.
• Hold-last freeze — below the inverses are frozen.

FIGURE — to embed: over the mission with the 0.025 derate band marked. Live plot in content/reports/checkin-jun25-26.html (the s_min_G panel). Current operational median = 0.0245.

★ A hypothesis we falsified — and kept anyway

We asked: does decoupling the CoM channel (a hierarchical inverse that keeps exact near an arm singularity) improve end-effector accuracy, camera pointing, or singularity clearance?

metric	old	new	Δ	read
CoMarm coupling defect	108×	—	fixed	exact to machine precision
(EE floor) p99	0.1544	0.1476	−4.4%	invariant
median	0.1095	0.1095	0.0%	floor unchanged
min	0.0190	0.0189	−0.1%	conditioning unchanged

Verdict — the answer is no, on all three. The end-effector floor already sits at its architectural limit; the pointing is excellent; the singularity risk lives in the arm, not the CoM. We keep the decouple because it makes the CoM channel mathematically exact — a correctness improvement, not a performance lever. A rigorously falsified hypothesis is a stronger result than a fragile positive one.

Open items — the bridge to risk-aware

The gap between the theoretical floor and the measured 0.104 m is a quantified agenda, not a loose end:

• Initial spikes — Tikhonov damping applied homogeneously to all coordinates spikes near singularity; a directional (filtered) damping should damp only the dying direction.
• Mismatched duration — characterise the residual phase effects over the full helix.
• True floor characterisation — close the theorymeasurement gap; attribute the operational floor to its terms.

Each becomes a proper investigation (lab-report method) before the floor numbers harden.

Nominal locked → ready for risk-aware planning

• Guidance is deterministic and reaches complete coverage; control is characterised against Giordano 2019.
• The error floor is understood — analytically (the balance) and empirically (~0.104 m), and shown to be architectural.
• The system operates at its singularity floor and is held there safely by four layers of handling.
• The ANCHOR mode + information-aware scorer are retained, inert, ready for risk-aware planning.

The thesis can begin: introduce uncertainty, and build risk-aware planning and control on a nominal baseline whose limits are now known and defensible.

Backup — provenance & open conventions

• Floor is config-dependent — operational median 0.104 m / p99 0.148 m; the 77 mm figure is the isolated Track-C config, not the full mission.
• Sign convention — slides use ; reconcile against current_sota eq (5.7).
• definitions — one velocity budget threaded through all three floors reproduces the code's 0.02 (cleanest to defend).
• 7-DOF redundancy — null-space (Khatib inertia-weighted) derived (redundant_jacobian.tex); future work.

Numbers from current gold (orbit-synced, 703 s) via the measurement facade; logs Inspection/logs/logs_Jun25_26. Brake (orbit-end) removed — not in current control.