Denoc Formulation

Definition

The Decoupled Natural Orthogonal Complement (DeNOC) is a velocity-transformation method for
deriving the equations of motion of a serial (or branched) multibody chain from the Newton–Euler
equations, due to Saha and extending the Natural Orthogonal Complement of Angeles & Lee. A
velocity-transformation matrix ${\mathbf{N}}_{\mathrm{voc}}$ maps the generalized joint rates
$\dot{\mathbf{q}}$ to the stacked link twists $\mathbf{t}$, and is factored into a lower
block-triangular factor ${\mathbf{N}}_l$ (twist propagation along the chain) and a block-diagonal
factor ${\mathbf{N}}_d$ (per-joint motion subspaces). Premultiplying the stacked Newton–Euler
equations by ${\mathbf{N}}_{\mathrm{voc}}^{\top }$ annihilates the non-contributing (constraint)
wrenches and collapses the system to the minimal generalized equations of motion, yielding a
recursive $\mathcal{O}(n)$ algorithm for the generalized inertia and convective-inertia matrices.
In virgili-llop2016spacecraft the DeNOC is applied to a
spacecraft-plus-arm with the base appended as link $0$, so the same formulation serves both a
free-flying and a free-floating base — the regime is set by how ${\dot{\mathbf{q}}}_0$ (base
twist) is driven or left free, not by the DeNOC machinery itself.

Key Equations

Symbols per notation.md.

Velocity transformation and its decoupling (Eqs. 16–17; the source writes the DeNOC matrix $N$ —
renamed here ${\mathbf{N}}_{\mathrm{voc}}$ to avoid clashing with the null-space projector
$\mathbf{N}$ in notation.md):

\boldsymbol N_{\mathrm{voc}} = \boldsymbol N_l\,\boldsymbol N_d .$$ Projection of the stacked Newton–Euler equations $\boldsymbol M_{\mathrm{NE}}\dot{\boldsymbol t}+\dot{\boldsymbol M}_{\mathrm{NE}}\boldsymbol t=\boldsymbol w$ onto the joint space removes the non-contributing wrenches ($\boldsymbol N_{\mathrm{voc}}^\top\boldsymbol w^{n}=0$), giving the generalized equations of motion (Eqs. 38–40): $$\boldsymbol M\,\ddot{\boldsymbol q} + \boldsymbol C\,\dot{\boldsymbol q} = \boldsymbol\tau,\qquad \boldsymbol M = \boldsymbol N_{\mathrm{voc}}^\top \boldsymbol M_{\mathrm{NE}}\,\boldsymbol N_{\mathrm{voc}},\qquad \boldsymbol C = \boldsymbol N_{\mathrm{voc}}^\top\!\left(\boldsymbol M_{\mathrm{NE}}\dot{\boldsymbol N}_{\mathrm{voc}} + \dot{\boldsymbol M}_{\mathrm{NE}}\boldsymbol N_{\mathrm{voc}}\right).$$ The block-stacked link-mass operator $\boldsymbol M_{\mathrm{NE}}=\mathrm{blkdiag}(\boldsymbol M_0,\dots,\boldsymbol M_n)$ is the source's $M$; the resulting **generalized inertia is the canonical $\boldsymbol M$ of [notation.md](../notation.md)** (the source labels it $H$). The base–arm coupling appears as the off-diagonal block (Eq. 44): $$\boldsymbol M = \begin{bmatrix}\boldsymbol M_0 & \boldsymbol M_{0m}\\ \boldsymbol M_{0m}^\top & \boldsymbol M_m\end{bmatrix}.$$ ## Source Support - [virgili-llop2016spacecraft](../sources/virgili-llop2016spacecraft.md) — derives the DeNOC twist propagation, the $\boldsymbol N_l\boldsymbol N_d$ factorization, and the $\mathcal{O}(n)$ recursive forms of $\boldsymbol M$ and $\boldsymbol C$ for a spacecraft-with-arm; states explicitly that the same SPART formulation handles a *flying* or a *floating* base. ## Related Topics - [ffsm_dynamics](ffsm_dynamics.md) — the DeNOC is one route to assemble the FFSM equations of motion that this page's $\boldsymbol M\ddot{\boldsymbol q}+\boldsymbol C\dot{\boldsymbol q}=\boldsymbol\tau$ form expresses. - [free_floating_dynamics](free_floating_dynamics.md) — recovered from the *same* DeNOC model by leaving the base twist $\dot{\boldsymbol q}_0$ unactuated (zero base wrench), contrasting our free-flying use where $\dot{\boldsymbol q}_0$ is commanded. - [generalized_inertia_matrix](generalized_inertia_matrix.md) — the DeNOC produces $\boldsymbol M=\boldsymbol N_{\mathrm{voc}}^\top\boldsymbol M_{\mathrm{NE}}\boldsymbol N_{\mathrm{voc}}$ recursively in $\mathcal{O}(n)$. - [generalized_jacobian](generalized_jacobian.md) — the per-link horizontal slices $\boldsymbol J_i$ of $\boldsymbol N_{\mathrm{voc}}$ are the velocity Jacobians; the free-floating GJM is a further reduction folding momentum conservation. - [dynamic_coupling](dynamic_coupling.md) — the base–arm coupling block $\boldsymbol M_{0m}$ (source $H_{0m}$) is exactly the inertial coupling between arm motion and base reaction. ## Open Questions - The source presents one formulation for both regimes but its worked control example is the (free-floating) Desired/Zero Reaction Maneuver. What changes in the DeNOC-derived $\boldsymbol M,\boldsymbol C$ partition when the base is *actuated* (free-flying) — does the coupling block $\boldsymbol M_{0m}$ stop being a constraint and become a control channel? - The DeNOC gives the full coupled $\boldsymbol M$; how does this relate to Giordano's circumcentroidal split $\boldsymbol\Gamma$ that decouples the CoM from the attitude+EE block — is one a coordinate transform of the other?