Motion Planning

Definition

Motion planning is the generation of a feasible, collision-free trajectory that carries a robot
from an initial to a goal configuration while satisfying its kinematic, dynamic and task
constraints; for a space manipulator it is the prerequisite step whose output is then executed by
trajectory tracking control (dai2022review).
dai2022review organizes the space-arm literature into two streams —
obstacle-avoidance-driven planning (pseudo-inverse / null-space, artificial potential field,
configuration-space, $A^{\ast }$, and RRT methods) and motion-requirement-driven planning (joint-space
vs. Cartesian-space trajectories parameterized by polynomials, splines, Bézier or NURBS curves and
refined by intelligent optimization). Regime caveat: that review assumes the free-floating
regime — base position uncontrolled, linear and angular momentum conserved, with a nonholonomic
constraint and kinematic/dynamic coupling between arm and base; our free-flying system has a
fully-actuated 6-DOF base, so its momentum is not conserved and the planner is not bound by that
constraint. zelinsky1993planning is a terrestrial
ground-mobile-robot treatment (complete-coverage path planning via the distance/path transform),
included only as a representative grid-search / numeric-potential-field planner — not a space source.

Key Equations

Symbols per notation.md.

There is no single governing equation; the defining object is the configuration-space (C-space)
obstacle map that a collision-free planner must avoid. For an obstacle $B$,
(dai2022review, §2):

$$C{O}_{\text{robot}}(B)=\{\mathbf{q}\in \mathcal{Q}:\text{robot}(\mathbf{q})\cap B\ne \varnothing \},$$

so a free path is a curve $\mathbf{q}(s)$, $s\in [0,1]$, joining start to goal with
$\mathbf{q}(s)\notin C{O}_{\text{robot}}(B)$ for all obstacles. Here $\mathbf{q}$ is the
generalized configuration and $\mathcal{Q}$ its configuration space ($\mathcal{C}$ in
notation.md is reserved for the system-CoM frame, not the C-space); $s$ is reused
for the runtime progress variable in our guidance notation (see notation.md). For a
free-floating arm the relevant Jacobian is the dynamic one ${\mathbf{J}}^{\ast }$ (Papadopoulos; per
notation.md, distinct from the generalized Jacobian matrix ${\mathbf{J}}_g$), whose
singular set differs from the fixed-base kinematic singular set — a planner must avoid the
dynamically singular configurations ($\det {\mathbf{J}}^{\ast }=0$), not just the fixed-base kinematic
ones ($\det {\mathbf{J}}_m=0$). The review states this qualitatively for the Rybus et al. Bézier-curve
method (dai2022review, §3.1: “these configurations differ from the
kinematically singular configurations obtained for a fixed-base manipulator”); the determinant
conditions are our notation-faithful restatement.

Source Support

• dai2022review — primary. Review of space-robotic-arm trajectory
  planning; gives the two-stream taxonomy, the planner families, the C-space obstacle map, and the
  free-floating dynamic-singularity caveat. Assumes the free-floating regime throughout.
• zelinsky1993planning — supporting. Terrestrial ground-mobile
  robot; introduces the path transform (a numeric potential field over a grid) for complete-coverage
  paths, illustrating the grid-search / potential-field planner family on a wheeled platform.

Related Topics

• sampling_based_motion_planning — the RRT / random-sampling
  family that dai2022review lists among obstacle-avoidance planners; a
  concrete instance of motion planning.
• trajectory_optimization — the optimization-based refinement
  (polynomial/Bézier parameterization, QP, intelligent algorithms) that turns a feasible plan into an
  optimal one; the §3.2 stream of the review.
• trajectory_tracking — the downstream control step that executes the
  planned trajectory; the review names tracking as the consumer of planning output.
• potential_field_planning — the artificial / numeric potential-field
  planner family used by both sources (Khatib-style APF in the review; the path transform in
  zelinsky1993planning).
• next_best_view — supplies the inspection viewpoints/goals that a coverage or
  point-to-point planner must reach for our inspection mission.

Open Questions

• dai2022review assumes a free-floating base, so its planners
  fold momentum conservation and the dynamic singularity $\det {\mathbf{J}}^{\ast }=0$ into the plan. For
  our free-flying fully-actuated base, momentum is not conserved — which of these planner families
  remain necessary, and which constraints drop out entirely?
• The review concentrates kinematic/dynamic coupling into the planning layer. With an actuated
  base we can instead exploit coupling at the control layer (e.g. circumcentroidal decoupling) — does
  the planning problem then reduce to ordinary Cartesian/joint-space trajectory generation?
• zelinsky1993planning’s coverage path is 2-D grid-based on a
  ground robot; what is the right lift of a “complete-coverage” notion to a 3-D inspection trajectory
  for a 6-DOF flying base plus redundant arm?