subsystem_decomposition

Subsystem Decomposition: COM · Base · End-Effector

The wiki’s subsystem axis. The free-flying coordinated controller acts on three distinct
subsystems; this page names them, defines the axis that tags them, and tabulates which failure
mechanism owns which subsystem — the attribution discipline that
error_floor.md §4 demands.

Statement

The circumcentroidal coordinate transform $\mathbf{\Gamma }$ (notation.md) sends
the full coupled dynamics into transformed coordinates
$\mathbf{z}=[{\mathbf{v}}_c;{\mathbf{\omega }}_b;{\mathbf{\nu }}_e^{\oplus }]$ with a block-diagonal
inertia $\hat{\mathbf{M}}=\mathrm{blkdiag}(m\mathbf{E},\breve{\mathbf{M}})$
(circumcentroidal_decoupling). That block structure is the
taxonomy — the three subsystems the thesis tracks (errors ${\tilde{\mathbf{x}}}_c,{\tilde{\mathbf{x}}}_b,{\tilde{\mathbf{x}}}_e$;
inputs ${\mathbf{f}}_c,{\mathbf{\tau }}_b^{\oplus },{\mathbf{w}}_e^{\oplus }$):

• COM (com) — the decoupled $3\times 3$ loop $m{\dot{\mathbf{v}}}_c={\mathbf{f}}_c$, closed by
  the SPD impedance $m{\ddot{\tilde{\mathbf{x}}}}_c+{\mathbf{D}}_c{\dot{\tilde{\mathbf{x}}}}_c+{\mathbf{K}}_c{\tilde{\mathbf{x}}}_c=0$.
  Inertially free (${\mathbf{M}}_c=m\mathbf{E}$); sees no Coriolis coupling.
• Base attitude (base) — ${\mathbf{\omega }}_b$, the upper block of the coupled $9\times 9$
  circumcentroidal subsystem $\breve{\mathbf{M}}$.
• End-effector (ee) — ${\mathbf{\nu }}_e^{\oplus }$, the lower block of that same $9\times 9$
  $\breve{\mathbf{M}}$.

base and ee are not independent: they share the reduced inertia $\breve{\mathbf{M}}$ and the
CoM-coupling Coriolis ${\mathbf{C}}_c{\mathbf{v}}_c$, forming the coupled attitude+EE state
$\breve{\mathbf{v}}=[{\mathbf{\omega }}_b;{\mathbf{\nu }}_e^{\oplus }]$. Only com truly decouples.
Frames $\mathcal{C}/\mathcal{B}/\mathcal{E}$ are in notation.md; the frame schematic is
Giordano Fig. 1 (library/coordinated_control/coordinated_control.md).

The subsystem axis

This decomposition is the wiki’s subsystem axis — a list-valued typed field (closed vocab
com | base | ee; omit ⇒ subsystem-agnostic). A page carries every subsystem its claim touches:
a CoM-loop result is [com]; a circumcentroidal-block mechanism is [base, ee]; a whole-controller
overview is [com, base, ee]. Ask “what touches the COM?” with Dataview
WHERE contains(subsystem, "com"). The axis is the join key across the three layers: the concept
wiki (here), the code wiki (sims_implementation_map), and
the logged evidence (evidence_map).

Failure modes are subsystem-specific (the attribution discipline)

Decompose before diagnosing: each subsystem has its own singular set and its own failure
mechanism, and conflating them mis-assigns cause. Distilled from
error_floor.md §4:

Subsystem	Block	Trigger / singular set	Characteristic failure	Mechanism
com	$3\times 3$ decoupled	none intrinsic (${\mathbf{M}}_c=m\mathbf{E}$)	near-singular spikes in ${\tilde{\mathbf{x}}}_c,{\dot{\tilde{\mathbf{x}}}}_c$	open — likely forward-Euler if ${\mathbf{M}}_c^{-1}{\mathbf{D}}_c$ develops a fast mode; not Tikhonov
base, ee	$9\times 9$ $\breve{\mathbf{M}}$	${\sigma }_G={\sigma }_{\min }(\mathbf{\Gamma })\to 0$ ($\equiv$ ${\mathbf{J}}_{{\nu }_e}^{\oplus }$ rank loss)	velocity-reconstruction blow-up; ${\mathbf{\omega }}_b$ period-2 ripple	forward-Euler on ${\breve{\mathbf{M}}}^{-1}\breve{\mathbf{D}}$ (discretization_stability_omega_b); Tikhonov / derate act on this block (singularity_threshold_cascade)
control map	${\mathbf{J}}_{\tilde{x}}$	quaternion singular set (distinct from ${\sigma }_G$)	error-floor premise ${\mathbf{J}}_{\tilde{x}}^{\top }\breve{\mathbf{K}}$ loses invertibility	steady_state_error_floor

The discipline. Name the subsystem and its block before attributing a fix. Tikhonov
regularization acts on the base+ee $9\times 9$ $\mathbf{\Gamma }$-solve — it does not act on the
decoupled $3\times 3$ CoM loop. Whether near-singular CoM spikes are the same forward-Euler artifact
or a genuinely separate CoM-loop effect is an open question, decided by the $dt$-independent
lag-1 autocorrelation of the logged ${\dot{\tilde{\mathbf{x}}}}_c$ ($\approx -1$ at $dt=0.03$
flipping to $\approx +1$ at $dt=0.01$ ⇒ forward-Euler). Evidence:
evidence_map. Do not write “Tikhonov causes the COM spikes” until
that test is run — this is exactly the mis-attribution
error_floor.md §4 flags.

Assumptions

• Nonsingular ${\mathbf{J}}_{{\nu }_e}^{\oplus }\Rightarrow \mathbf{\Gamma }$ invertible: the decoupling
  holds on the singularity-free region $\Omega$ (singularity_threshold_cascade).
• $n=6$ (nonredundant) for the clean $3\times 3$ / $9\times 9$ split. The redundant $n=7$ case adds a
  self-motion / null-space channel (frozen below $\sim 10^{-3}$) — a redundant DOF outside this
  three-subsystem taxonomy.
• Hat-vs-breve is load-bearing: $\hat{\mathbf{M}}$ full $12\times 12$, $\breve{\mathbf{M}}$ reduced
  $9\times 9$ (terminology.md, circumcentroidal_decoupling).

Proof sketch

The block-diagonal $\hat{\mathbf{M}}=\mathrm{blkdiag}(m\mathbf{E},\breve{\mathbf{M}})$ that
licenses the split is derived in
dynamics_modifications_7dof.md and
circumcentroidal_decoupling; the subsystem-specific failure-mode
separation (CoM forward-Euler vs base+EE ${\sigma }_G$ blow-up) is established in
error_floor.md §1–4 and
omega_b_forward_euler_instability.md.
This page is the taxonomic synthesis, not a new theorem: it names the three subsystems the proven
decomposition produces and records which mechanism owns which block.

Source / provenance

• Literature: giordano2019coordinated — the $\mathbf{\Gamma }$
  split (eq 21) producing the decoupled CoM + coupled attitude+EE blocks; Fig. 1 frames.
• Ours: the subsystem axis and the failure-mode-by-subsystem attribution table — the diagnostic
  discipline that the CoM-spike attribution question motivated.

Caveats

• base and ee are a coupled pair, not independent subsystems; [base, ee] reflects the shared
  $9\times 9$ block. Tagging a circumcentroidal-block claim with only one of them understates the
  coupling.
• The CoM-spike attribution is unresolved; this page records the open question and the deciding
  test, it does not assert the answer.
• Distinct singular sets: ${\sigma }_G$ (circumcentroidal, base+ee) $\ne$ the quaternion singular set
  of ${\mathbf{J}}_{\tilde{x}}$ (the error-floor domain). Do not merge them.

Related

• Math: circumcentroidal_decoupling (the decoupling derivation) ·
  steady_state_error_floor ·
  discretization_stability_omega_b.
• Loops: center_of_mass_regulation (com) ·
  momentum_dumping (base) ·
  circumcentroidal_motion (ee) ·
  coordinated_control.
• Singularity: singularity_threshold_cascade ·
  damped_least_squares.
• Registries: notation.md (frames $\mathcal{C}/\mathcal{B}/\mathcal{E}$; the
  ${\mathbf{v}}_c/{\mathbf{\omega }}_b/{\mathbf{\nu }}_e^{\oplus }$ blocks) ·
  terminology.md (circumcentroidal, hat/breve).
• Bridge: sims_implementation_map (concept↔code) ·
  evidence_map (logs).

Implementation (sims wiki)

External — into the code wiki via the sims_wiki/ symlink (resolves in Obsidian, not GitHub).

• com_controller — the decoupled $3\times 3$ CoM loop (com).
• base_controller / base_guidance — base-attitude channel (base).
• ee_guidance — end-effector channel (ee).
• breve_controller — the coupled circumcentroidal [base, ee] law + singularity stack.