Inertial Space Tracking

Definition

Inertial-space tracking is the control objective of driving a manipulator’s end-effector to a
desired position and orientation expressed in a fixed (inertial) reference frame, rather than in a
base-fixed frame, by commanding compensating joint motion. The distinction matters whenever the base
itself moves: in Holt & Desrochers (holt1992inertial) a 6-DOF PUMA
sits on a separately disturbed 3-DOF rotational platform whose motion is unknown to the
controller, so the arm must regulate the inertial-frame end-effector pose subject to that disturbance.
This is a terrestrial/lab testbed emulating a disturbed mount — not the free-flying regime (the
base is not commanded by the controller) and not the free-floating regime (the platform is an
externally driven mount, not a momentum-conserving reaction-coupled free body); for our free-flying
system the actuated base is a commanded degree of freedom rather than an unknown disturbance.

Key Equations

Symbols per notation.md.

The end-effector inertial-space pose error ${\tilde{\mathbf{x}}}_e={}^0{\mathbf{u}}_d-{}^0{\mathbf{u}}_k$
is mapped to a joint-displacement command through the (approximate-pseudoinverse) Jacobian and a base/EE
frame rotation $\mathbf{R}$, giving Holt’s kinematic control law (source eq. 19):

$\Delta {\mathbf{\theta }}_d={\mathbf{J}}^{‡}({\mathbf{\theta }}_k){}_0^3\mathbf{R}({\eta }_o)K_c({}^0{\mathbf{u}}_d-{}^0{\mathbf{u}}_k)$

with the velocity-level inverse-kinematics relation $\mathrm{d}\mathbf{u}=\mathbf{J}(\mathbf{q})\mathrm{d}\mathbf{q}$
(source eq. 1). Driving ${\tilde{\mathbf{x}}}_e\to 0$ as $k\to \infty$ is the tracking goal.

Notation flags. ${\mathbf{J}}^{‡}$ is Holt’s approximate pseudoinverse (block-partitioned
SVD inverse, source eq. 6) — a source-specific symbol absent from notation.md; it
reduces to ${\mathbf{J}}^{-1}$ when $\mathbf{J}$ is nonsingular but does not satisfy the
Moore–Penrose conditions at a singularity. The fixed-base manipulator Jacobian $\mathbf{J}$ here is
notation.md’s ${\mathbf{J}}_{{\nu }_e}$. $K_c=k_c\mathbf{I}$ ($0\le k_c\le 2$) is Holt’s scalar
tracking gain (a scalar times the identity) — not notation.md’s CoM-loop stiffness matrix
${\mathbf{K}}_c$ (${\mathbb{R}}^{3\times 3}$, SPD), and distinct from the EE stiffness ${\mathbf{K}}_e$;
${}^0\mathbf{u}$
is the source’s inertial EE pose (linear+angular), and ${\eta }_o$ here is the platform joint vector,
unrelated to the quaternion scalar part $\eta$ in notation.md.

Source Support

• holt1992inertial — formulates inertial-space end-effector tracking as
  disturbance rejection on a moving (disturbed) base; supplies the kinematic control law and shows that
  near kinematic singularities the control in certain (“forbidden”) directions weakens, leaving an
  unavoidable inertial-space tracking error.

Related Topics

• trajectory_tracking — inertial-space tracking is the frame choice for the
  tracked reference; tracking a time-varying ${}^0{\mathbf{u}}_d$ specializes it to a trajectory.
• base_disturbance_rejection — the same Holt problem, viewed as
  rejecting unknown base motion; inertial-space tracking is the objective, disturbance rejection the mechanism.
• singularity_robust_inverse — Holt’s approximate-pseudoinverse cap on
  ${\sigma }_{\min }$ is a singularity-robust inverse; both trade tracking accuracy for bounded joint rate near singularities.
• closed_loop_inverse_kinematics — Holt’s law is a discrete
  closed-loop inverse-kinematics scheme, feeding the inertial pose error back through ${\mathbf{J}}^{‡}$.
• resolved_motion_rate_control — the velocity-level
  $\mathrm{d}\mathbf{u}=\mathbf{J}\mathrm{d}\mathbf{q}$ mapping is the RMRC kernel inverted here.
• See also pseudoinverse_jacobian and kinematic_singularity,
  on which the approximate-pseudoinverse construction and the “forbidden directions” rest.

Open Questions

• Holt assumes the base disturbance is unknown and uncommanded; for our free-flying system the base is
  an actuated, commanded DOF — does inertial-space EE tracking still need arm-only compensation, or does it
  fold into coordinated base+arm control?
• The “forbidden-direction” error near singularities is intrinsic to the kinematic (${\mathbf{J}}^{‡}$)
  law; does a circumcentroidal / dynamics-level formulation change which directions become uncontrollable for
  a 6-DOF flying base?
• Holt’s arm is non-redundant (6-DOF on the 6-DOF task); does adding kinematic redundancy let inertial-space
  tracking preserve all task directions through a singularity?