Task Space Error Dynamics

Definition

Task space error dynamics describes how the outer-loop tracking errors of the
coordinated controller — base attitude ${\tilde{\mathbf{x}}}_b$, end-effector pose
${\tilde{\mathbf{x}}}_e$, and system-CoM position ${\tilde{\mathbf{x}}}_c$ — evolve
in time as functions of the measured twists. Because attitude is parameterized by the
unit quaternion (vector part $\mathbf{ϵ}$, scalar part $\eta$; see
notation.md), the rate of each error is not the raw twist but the twist
mapped through a configuration-dependent error-rate Jacobian ${\mathbf{J}}_{\tilde{x}}$
built from the current quaternion error. These maps are the kinematic bridge between the
reduced circumcentroidal velocity $\breve{\mathbf{v}}=[{\mathbf{\omega }}_b;{\mathbf{\nu }}_e^{\oplus }]$
(state of the coupled attitude+EE block, circumcentroidal_motion)
and the stacked outer-loop error $\tilde{\mathbf{x}}=[{\tilde{\mathbf{x}}}_b;{\tilde{\mathbf{x}}}_e]$
that the impedance law regulates (coordinated_control). They
underpin both the Lyapunov argument and the steady-state cruise-lag floor.

Key Equations

Error coordinates. With desired frames denoted by a ${}_d$ subscript and quaternion errors
$({\mathbf{ϵ}}_{e{e}_d},{\eta }_{e{e}_d})$ for the EE and $({\mathbf{ϵ}}_{b{b}_d},{\eta }_{b{b}_d})$
for the base:

$${\tilde{\mathbf{x}}}_c={\mathbf{R}}_{tc}^T({\mathbf{p}}_{tc}-{\mathbf{p}}_{t{c}_d}),{\tilde{\mathbf{x}}}_e=\left[\begin{array}{c}{\mathbf{p}}_{e{e}_d}\\ 2{\mathbf{ϵ}}_{e{e}_d}\end{array}\right],{\tilde{\mathbf{x}}}_b=2{\mathbf{ϵ}}_{b{b}_d},\tilde{\mathbf{x}}=\left[\begin{array}{c}{\tilde{\mathbf{x}}}_b\\ {\tilde{\mathbf{x}}}_e\end{array}\right]\in {\mathbb{R}}^9.$$

(Giordano §IV / current_sota §4.1) — attitude error enters only through the quaternion
vector part $\mathbf{ϵ}$, so no representation singularity arises.

Error-rate Jacobians (per task). The EE pose-error rate and the base attitude-error rate
map their respective twists through quaternion-built blocks:

$${\dot{\tilde{\mathbf{x}}}}_e={\mathbf{J}}_{{\tilde{x}}_e}{\mathbf{\nu }}_e,{\mathbf{J}}_{{\tilde{x}}_e}=\left[\begin{array}{cc}\mathbf{E}& 0\\ 0& -{\eta }_{e{e}_d}\mathbf{E}+[{\mathbf{ϵ}}_{e{e}_d}{]}^{\wedge }\end{array}\right],{\dot{\tilde{\mathbf{x}}}}_b={\mathbf{J}}_{{\tilde{x}}_b}{\mathbf{\omega }}_b,{\mathbf{J}}_{{\tilde{x}}_b}=-{\eta }_{b{b}_d}\mathbf{E}+[{\mathbf{ϵ}}_{b{b}_d}{]}^{\wedge }.$$

(Giordano eqs 24, 25 / current_sota eqs 4.1, 4.2) — the rotational block
$-\eta \mathbf{E}+[\mathbf{ϵ}{]}^{\wedge }$ is the standard quaternion-propagation
map; the EE translational block is the identity $\mathbf{E}$.

Compact stacked form. Block-diagonalizing the two Jacobians and folding in the CoM
contribution through ${\breve{\mathbf{G}}}_{v_c}=[0;{\mathbf{G}}_{v_c}]$:

$${\mathbf{J}}_{\tilde{x}}=\mathrm{blkdiag}({\mathbf{J}}_{{\tilde{x}}_b},{\mathbf{J}}_{{\tilde{x}}_e})\in {\mathbb{R}}^{9\times 9},\dot{\tilde{\mathbf{x}}}={\mathbf{J}}_{\tilde{x}}\breve{\mathbf{v}}+{\mathbf{J}}_{\tilde{x}}{\breve{\mathbf{G}}}_{v_c}{\dot{\tilde{\mathbf{x}}}}_c.$$

(Giordano eqs 26, 27 / current_sota eq 4.3) — the first term is the attitude+EE motion
about the CoM; the second injects the CoM-loop error rate ${\dot{\tilde{\mathbf{x}}}}_c$, which
is what couples the inner CoM loop into the outer task-error dynamics. The same
${\mathbf{J}}_{\tilde{x}}$ appears transposed in the impedance law
($-{\mathbf{J}}_{\tilde{x}}^T\breve{\mathbf{K}}\tilde{\mathbf{x}}$,
coordinated_control) and in the Lyapunov function
$V=\frac{1}{2}{\breve{\mathbf{v}}}^T\breve{\mathbf{M}}\breve{\mathbf{v}}+\frac{1}{2}{\tilde{\mathbf{x}}}^T\breve{\mathbf{K}}\tilde{\mathbf{x}}$.

Regime

Giordano 2019 (the master’s primary control source) is free-flying (fully-actuated
6-DOF base), matching this project. The quaternion error construction is shared with
Caccavale & Siciliano’s CLIK, which is derived for the free-floating regime — the
error-rate map is identical, but the underlying Jacobian relating $\dot{\mathbf{q}}$ to
the EE twist differs (generalized Jacobian vs. circumcentroidal Jacobian).

Source Support

• sukhanov2015dynamic — derives a free-flying planar
  model in which the end-point-to-target deviation coordinates appear explicitly as states,
  the precursor idea to carrying the task-space error directly in the dynamics.
• giordano2019coordinated — origin of the
  error-rate Jacobians (eqs 24-27) and the stacked error $\tilde{\mathbf{x}}$ used here;
  free-flying circumcentroidal coordinated control (the master’s primary control source).
• caccavale2001kinematic — quaternion-based CLIK error
  dynamics; uses the geometric Jacobian so no representation singularity enters the orientation
  error. Free-floating regime (generalized Jacobian), but the quaternion error-rate block
  is the same operator.

Related Topics

• unit_quaternion — supplies the error parameterization
  $(\mathbf{ϵ},\eta )$ from which every ${\mathbf{J}}_{\tilde{x}}$ block is built.
• coordinated_control — consumes $\tilde{\mathbf{x}}$ and
  ${\mathbf{J}}_{\tilde{x}}^T$ in the impedance law; this page supplies its outer-loop kinematics.
• closed_loop_inverse_kinematics — the CLIK family where
  the same quaternion error-rate dynamics first appears (Caccavale).
• circumcentroidal_motion — defines $\breve{\mathbf{v}}$ and
  ${\mathbf{\nu }}_e^{\oplus }$, the twists fed into ${\mathbf{J}}_{\tilde{x}}$.
• trajectory_tracking — the broader task these error dynamics serve;
  the steady-state floor steady_state_error_floor is
  obtained by setting $\dot{\tilde{\mathbf{x}}}=0$ in eq 4.3.

Open Questions

• Does ${\mathbf{J}}_{\tilde{x}}$ ever lose rank within the operating envelope? The block
  $-\eta \mathbf{E}+[\mathbf{ϵ}{]}^{\wedge }$ is singular only at error angle $\pi$
  ($\eta =0$, $\Vert \mathbf{ϵ}\Vert =1$); confirm tracking never approaches that.
• How is the CoM-coupling term ${\mathbf{J}}_{\tilde{x}}{\breve{\mathbf{G}}}_{v_c}{\dot{\tilde{\mathbf{x}}}}_c$
  bounded in the cascade stability argument relative to the inner CoM loop’s settling?