Configuration Dependent Stability Domain

Definition

The configuration-dependent stability domain is the subset of manipulator
configurations $\mathbf{q}$ for which the closed-loop tracking error dynamics of a space
manipulator are asymptotically stable. Because the dynamic-coupling coefficients that enter
the error dynamics are functions of the joint angles (and of the base attitude), some
configurations satisfy the stability conditions and others do not; the admissible set therefore
has a configuration-dependent boundary, not a fixed one. In rutkovskii2023control
the domain is derived for a free-FLYING robot (fully-actuated, gas-jet base plus electromechanical
arm) installing a payload: it is the region in the joint-angle plane $({\alpha }_1,{\alpha }_2)$ where the
end-point regulation loop stays stable, used to pick the initial configuration and bound the
admissible joint excursion during the operation. It generalizes the notion of a singularity-free
region $\Omega$ (notation.md) from a purely kinematic rank condition to a dynamic, closed-loop
stability condition.

Key Equations

Symbols per notation.md. The joint angles ${\alpha }_1,{\alpha }_2$, the
configuration-dependent coupling coefficients $d_{ij}(\mathbf{q})$, and their determinant
$\Delta d$ are local to rutkovskii2023control and are not
in notation.md (do not confuse $\Delta d$ with any canonical symbol).

Linearized end-point error dynamics under PD control (12)–(13) yield the characteristic
polynomial ${\sum }_{j=0}^4{c}_j{\lambda }^j=0$, whose Routh-Hurwitz coefficients depend on the
coupling determinant

$$\Delta d=k_{0x}{k}_{0y}(d_{11}{d}_{22}-d_{12}{d}_{21}),d_{ij}=d_{ij}(\mathbf{q}).$$

The configuration-dependent stability domain is then the joint-angle region satisfying the
sign condition (necessary for $c_j>0\forall j$)

$${\Omega }_{\Delta d}=\{({\alpha }_1,{\alpha }_2):\Delta d(\mathbf{q})>0|\vartheta ,{\alpha }_3\text{fixed}\},$$

where $\vartheta$ is the base attitude and ${\alpha }_3$ the fixed end-link angle. Where
$d_{ij}(\mathbf{q})\le 0$ the regulated coordinates $X_{\epsilon },Y_{\epsilon }$ become
unstable, so the domain boundary is set by the configuration-dependent sign of the coupling, not
by actuator limits.

Source Support

• rutkovskii2023control — primary and sole source: derives
  the stability domain $\Delta d>0$ in joint-angle coordinates $({\alpha }_1,{\alpha }_2)$ from a
  Routh-Hurwitz analysis of the linearized end-point error dynamics, and shows its topology depends
  on base attitude $\vartheta$ (more) and the gripper angle ${\alpha }_3$ (less), Fig. 2. Free-FLYING
  regime (actuated base); contrasts explicitly with the paper’s free-FLOATING mode (attitude
  control disabled), where the working area narrows and dynamic singularities appear.

Related Topics

• lyapunov_stability — the domain is the set where the closed-loop error
  has an asymptotically stable equilibrium; here certified via Routh-Hurwitz on the linearized
  characteristic polynomial rather than a Lyapunov function.
• dynamic_coupling — the coefficients $d_{ij}(\mathbf{q})$ encode
  base$\leftrightarrow$arm coupling; their configuration dependence is precisely what makes the
  stable set configuration-dependent.
• coordinated_control — the domain bounds where coordinated base-plus-arm
  regulation of the end point stays stable; it informs initial-configuration selection and admissible
  joint excursion.
• ffsm_dynamics — the coupling coefficients descend from the FFSM equations of
  motion; the linearization (small joint rates) is what reduces them to the $d_{ij}$ form.
• technical_controllability — the source’s prior admissibility
  condition; the stability domain is a stronger, closed-loop requirement layered on top.

Open Questions

• The domain is derived for a planar, linearized (small-rate), self-braking-actuator case; does
  the $\Delta d>0$ characterization survive in 3-D and at higher joint rates, where the neglected
  $f_K,f_{\alpha }$ velocity-product terms re-enter?
• The source bounds stability via a static sign condition on $d_{ij}(\mathbf{q})$ along a
  quasi-static install; how does this relate to the kinematic singularity-free region $\Omega$ and
  to the dynamic ${\sigma }_G$ thresholds used elsewhere in this wiki for the fully-coupled FFSM?