Keep Out Zone

Definition

A keep-out zone (KOZ) is a forbidden region of configuration/position space that a maneuvering
spacecraft must not enter — in the capture context it is the volume occupied by the target resident
space object (RSO) and its appendages, enforced as a collision-avoidance constraint on the chaser’s
trajectory (virgili-llop2019convex). Geometrically the KOZ
constraint requires the chaser body set and the target set to be disjoint at all times; it is in
general non-convex (target shape, plus tumbling appendages make the zone time-varying). In the
cited work the chaser actuates both its base-spacecraft and its manipulator to execute a full
rototranslation around the zone — i.e. a free-flying maneuver, not a reactive free-floating one —
so the KOZ couples to the system center of mass and to the end-effector grapple geometry.

Key Equations

Symbols per notation.md. The obstacle/KOZ symbols below are reproduced
source-faithfully (sets $S_{(\cdot )}$, signed distance $d$, chaser CoM ${\mathbf{r}}_c$, node index $n$);
they are not in the canonical registry. Here ${\mathbf{r}}_c$ is the chaser CoM (source usage), not
the system CoM $\mathcal{C}$ of notation.md — flagged to avoid conflict.

Set-disjointness form of the KOZ constraint (non-convex):
$S_{\text{chaser}}\cap S_{\text{RSO}}=\varnothing .$

Signed-distance reformulation ($d>0$ separated, $d<0$ intersecting):
$d\ge 0.$

Linearization about a reference centroid trajectory ${\tilde{\mathbf{r}}}_c$ (the convex surrogate fed
to sequential convex programming, with ${\hat{\mathbf{n}}}_{\tilde{d}}=\tilde{\mathbf{d}}/\Vert \tilde{\mathbf{d}}\Vert$ the unit
separation direction):
${\tilde{d}}^{[n]}+{\hat{\mathbf{n}}}_{\tilde{d}}^{[n]\top }\left({\mathbf{r}}_c^{[n]}-{\tilde{\mathbf{r}}}_c^{[n]}\right)\ge 0.$

Source Support

• virgili-llop2019convex — primary: defines the KOZ as set
  non-intersection (Eq. 10), reformulates it via the signed distance $d\ge 0$ (Eq. 23, computed with
  GJK / expanding-polytope), and linearizes it (Eq. 36) so a non-convex KOZ can be handled inside a
  convergence-guaranteed sequential-convex-programming guidance scheme for chaser capture; also gives a
  conservative enclosing-sphere ($R_{KO}$) form and a convex-hull decomposition for non-convex targets.

Related Topics

• clearance_constraints — the KOZ is the inequality side of a clearance
  constraint: $d\ge 0$ is the zero-margin separating-distance requirement between chaser and target.
• motion_planning — KOZs are the obstacle/collision-avoidance constraints a
  planner must satisfy when routing the chaser to the grapple fixture.
• potential_field_planning — an alternative way to encode a KOZ:
  a repulsive field around the zone, contrasted with the constraint-based / penalty treatment used here.
• sequential_convex_programming — the method that makes the
  non-convex KOZ tractable: it is repeatedly linearized (Eq. 36) and solved as a convex sub-problem,
  with a convergence proof offered for the system-wide translation step.
• trajectory_optimization — the KOZ enters the optimal control problem
  as a path inequality constraint on the (transcribed) trajectory being optimized.

Open Questions

• The source’s KOZ is a translational obstacle constraint on the chaser CoM (with a separate attitude
  keep-out cone); for our redundant free-flying arm, does the relevant KOZ need to bound the full
  swept volume of the manipulator links, not just an enclosing sphere of the base?
• The enclosing-sphere $R_{KO}$ guarantees feasibility but is conservative and can render terminal
  grapple constraints infeasible when target protrusions penetrate it — what is the right
  non-conservative body model for our 6-DOF base plus arm during close approach?
• Linearizing $d\ge 0$ around ${\tilde{\mathbf{r}}}_c$ gives only a KKT (locally optimal) point of the
  non-convex problem; how sensitive is the converged trajectory to the seed ${\tilde{\mathbf{r}}}_c$ for
  a time-varying KOZ generated by an uncertain tumbling target?