Path Dependent Workspace

Definition

For a free-floating space manipulator (uncontrolled base; arm motion reacts on the spacecraft),
Papadopoulos & Dubowsky partition the reachable workspace into two regions. The Path Independent
Workspace (PIW) is the set of inertial locations that can never be reached in a dynamically
singular configuration — reachable by any path lying within it. The Path Dependent Workspace
(PDW) is the complement: locations that may be reached in a dynamically singular configuration
depending on the path the end-effector takes to get there, so they are reachable by some paths but
not others. The phenomenon exists because the spacecraft attitude is not a function of the joint
angles alone but of the history (path) of the motion — a consequence of the non-integrability of
angular momentum — so a given workspace point maps to infinitely many configurations $\mathbf{q}$,
some of which are singular. The PDW/PIW split is a free-floating artifact; it presumes an
uncontrolled base.

Key Equations

Symbols per notation.md.

A dynamic singularity is a rank loss of the free-floating end-effector Jacobian ${\mathbf{J}}^{\ast }$
(here at fixed base, ${}^0{\mathbf{J}}^{\ast }(\mathbf{q})$; the base-attitude transform $\mathrm{diag}({\mathbf{T}}_0,{\mathbf{T}}_0)$ is always invertible):

$\det [{}^0{\mathbf{J}}^{\ast }(\mathbf{q})]=0.$

The set of singular configurations forms hypersurfaces ${\mathbf{Q}}_{s,i}$ in joint space. Each maps,
via the end-effector’s barycentric distance $R=\Vert {\mathbf{r}}_E-{\mathbf{r}}_{\mathrm{CM}}\Vert$
from the system centre of mass, to a spherical shell in inertial space; each hypersurface sweeps a
shell-bounded volume $(R_{\min ,i},R_{\max ,i})$. The PDW is the union of these volumes and the PIW is
its complement within the reachable workspace ${\mathcal{W}}_R$:

\qquad \text{PIW}=\mathcal W_R \setminus \text{PDW}.$$ (Symbols $R,\,R_{\min,i},\,R_{\max,i},\,\boldsymbol Q_{s,i}$ and the reachable workspace $\mathcal W_R$ are not in notation.md; introduced source-faithfully here. $\boldsymbol J^{*}$ is the registered Papadopoulos Jacobian — distinct from $\boldsymbol J_g$ and $\boldsymbol J_{\nu_e}^{\oplus}$.) ## Source Support - [papadopoulos1993dynamic](../sources/papadopoulos1993dynamic.md) — primary and sole source: defines the PDW/PIW partition, proves the singularities of $\boldsymbol J^{*}$ are path-dependent via the non-integrability of angular momentum, and gives the planar two-link example (reaching point B from A fails on the straight path AB but succeeds on the detour ACB). Also notes design levers that shrink the PDW: large/infinite base inertia, or mounting the arm at the base CoM (planar case eliminates the PDW). ## Related Topics - [dynamic_singularity](dynamic_singularity.md) — the rank-deficiency event ($\det{}^{0}\boldsymbol J^{*}=0$) whose path-dependent placement in inertial space *creates* the PDW. - [free_floating_dynamics](free_floating_dynamics.md) — the regime that gives rise to the phenomenon; an actuated base would let attitude track a command rather than drift with the path. - [angular_momentum_conservation](angular_momentum_conservation.md) — its *non-integrability* is the root cause: attitude depends on path history, so a point maps to many (some singular) configurations. - [generalized_jacobian](generalized_jacobian.md) — the Umetani GJM ($\boldsymbol J_g$) is the companion free-floating map; $\boldsymbol J^{*}$ here plays the same role and shares the dynamic-singularity property. - [dynamic_coupling](dynamic_coupling.md) — base reaction to arm motion is the coupling that makes attitude (hence reachability) path-dependent. - [kinematic_singularity](kinematic_singularity.md) — fixed-base singularities are path-independent (solve $\det\boldsymbol J(\boldsymbol q)=0$); they remain a subset of the dynamic singularities here and bound the reachable workspace. ## Open Questions - The PDW/PIW partition assumes an **uncontrolled (free-floating) base**. For our **free-flying** system the base attitude is a commanded state, not a path integral of arm motion — does the path-dependence collapse, leaving only the (path-independent) configuration-dependent singularity set, or does a residual coupling-driven analogue survive? - Papadopoulos shows large base inertia shrinks the PDW; the free-flying analogue would be base actuation authority. Is there a quantitative trade between control bandwidth/torque margin and an effective PIW for an actuated base? - "The efficient construction of paths to reach points in the PDW is still an open area of research" (source). For an actuated base, does this reduce to standard singularity-avoidance planning, or does coupled base-arm guidance change the path-construction problem?