Cartesian Path Planning

Definition

Cartesian (task-space) path planning generates the joint and base motions required for the end-effector
to follow a prescribed inertial path ${\mathbf{v}}_E(t)$ — position and orientation, in prescribed time —
rather than planning in joint space. For a space manipulator the difficulty is that the end-effector
velocity depends on both arm and base motion through the system’s momentum balance, so a Cartesian path
that is feasible in isolation can still drive the system through a dynamic_singularity
where the generalized Jacobian loses rank. nanos2015avoiding studies this in the
free-floating regime (base actuators off; the base translates and rotates only in reaction to arm motion),
and finds initial system configurations / spacecraft attitudes such that a given Cartesian path avoids these
singularities, exploiting the entire reachable workspace including the path_dependent_workspace.
The method handles planar and spatial systems, zero or nonzero initial angular momentum, and does not require
arm redundancy. (Our free-flying base is fully actuated, so its momentum is not conserved — see Open Questions.)

Key Equations

Symbols per notation.md.

Momentum conservation plus the manipulator kinematics, stacked, relate the base angular velocity and joint
rates to the desired end-effector twist and the conserved CoM angular momentum $\mathbf{L}$
(source eq. 4; source writes ${\mathbf{h}}_{CM}$ for $\mathbf{L}$ and $\mathbf{A}$ for the stacked $9\times (N+3)$ map):

$$\mathbf{A}\left[\begin{array}{c}{\mathbf{\omega }}_b\\ \dot{\mathbf{q}}\end{array}\right]=\left[\begin{array}{cc}{\mathbf{R}}_0^{\top }& 0\\ 0& {\mathbf{C}}_T\end{array}\right]\left[\begin{array}{c}\mathbf{L}\\ {\mathbf{\nu }}_e\end{array}\right].$$

For a non-redundant spatial arm ($N=6$) this is solvable iff the generalized Jacobian is full rank.
The dynamic-singularity condition is the scalar (source eq. 6; source glyph ${\mathbf{S}}^{\ast }$, the free-floating
generalized Jacobian — here ${\mathbf{J}}_g$, GJM of Umetani–Yoshida):

$$S(\mathbf{q})=\det ({\mathbf{J}}_g(\mathbf{q}))=0.$$

The locus $S(\mathbf{q})=S_0$ defines singularity surfaces ($S_0=0$) and margin surfaces ($S_0\ne 0$) in joint
space; the planner keeps the path’s image clear of $S=0$ by choosing a feasible initial attitude.

Source Support

• nanos2015avoiding — primary; Cartesian (task-space) trajectory planning of a
  free-floating manipulator that follows an arbitrary end-effector path while avoiding dynamic singularities, by
  selecting the initial configuration/attitude (planar & spatial, with/without initial angular momentum, no redundancy).

Related Topics

• motion_planning — Cartesian path planning is the task-space specialization; this source contrasts it with the joint-space planning that prior work used to sidestep dynamic singularities.
• dynamic_coupling — the reason a Cartesian path is hard: end-effector motion is coupled to the (reactive) base through momentum conservation, making the singularities dynamic rather than kinematic.
• path_dependent_workspace — dynamic singularities partition the workspace into a path-independent region (PIW, always safe) and a path-dependent region (PDW); the method’s payoff is reaching the PDW along a singularity-free Cartesian path.
• trajectory_optimization — an alternative family that encodes singularity avoidance as a cost/constraint over the trajectory; here avoidance is instead achieved by a one-time choice of initial conditions for a prescribed path.
• cartesian_impedance_control — the execution-layer counterpart: planning yields a feasible Cartesian path; impedance/tracking control follows it, where Jacobian conditioning again governs safety near singularities.

Open Questions

• The source assumes a free-floating base (momentum conserved, motion is nonholonomic). On our free-flying
  base the momentum balance is replaced by actuated base dynamics — does the path-dependent-workspace partition and
  the $\det ({\mathbf{J}}_g)=0$ singularity locus even arise, or is it superseded by the circumcentroidal Jacobian
  conditioning ${\sigma }_{\min }({\mathbf{J}}_{{\nu }_e}^{\oplus })$ we track?
• The method selects an initial attitude for a given path with no online feedback; how does it degrade under the
  pose error and disturbances that a real tracking controller must absorb during execution?
• It is non-redundant by design ($N=6$ spatial). For our redundant arm, is initial-condition selection still the right
  lever, or does null-space redundancy_resolution make path-dependence avoidable directly?