Base Disturbance Rejection

Definition

Base disturbance rejection is the control objective of holding the end-effector on its
desired inertial-space trajectory despite unknown motion of the manipulator’s base, by
commanding compensating joint motion at the velocity (kinematic) level. In Holt & Desrochers
(holt1992inertial) the base (“platform”) disturbance enters as an
exogenous, unmeasured signal $\delta$ in the platform joints: the arm does not actuate or
stabilize the base — it treats the base motion as a disturbance to be cancelled at the end-effector
through resolved-motion-rate control. Note the regime: this is neither the free-floating case
(where arm motion reacts back onto an uncontrolled base via momentum coupling) nor our
free-flying case (where the base is fully actuated). It is a driven mobile platform — the base
is moved by an external agent and the arm rejects that motion kinematically. For our free-flying
system the analogous problem is rejecting residual base-attitude/translation error left by the base
controller; the same singularity caveats below apply.

Key Equations

Symbols per notation.md.

Holt’s source uses paper-local symbols ($J$, $du$, $\delta$, $\Delta {\theta }_d$, $K_c$, $J^{‡}$);
these are not in notation.md and are reproduced source-faithfully here, not
canonicalized (the wiki’s ${\mathbf{J}}_{{\nu }_e}$, $\dot{\mathbf{q}}$ are the corresponding
velocity-level objects).

The differential end-effector displacement splits into the measured arm contribution and the
unknown base-disturbance contribution $dv$ (Eq. 14):

$${}^{0}d{u}_{0,E}={}_3^0\mathbf{R}({\eta }_o+\delta ){}^3{J}_{3,E}(\theta )d\theta +dv.$$

The disturbance-rejection control law commands joint displacements that approximately cancel the
inertial-space error $({}^0{u}_d-{}^0{u}_k)$ via an inverse Jacobian (Eq. 19):

$$\Delta {\theta }_d={}^3{J}_{3,E}^{‡}({\theta }_k){}_0^3\mathbf{R}({\eta }_o)K_c({}^0{u}_d-{}^0{u}_k),K_c=k_c\mathbf{I},0\le k_c\le 2.$$

Here $J^{‡}$ is the approximate pseudoinverse (block-SVD of the $3\times 3$ sub-Jacobians,
thresholded at ${\sigma }_{\min }$) used so the law stays well-defined through singular regions — a
large disturbance can drive the arm into a singularity, so avoidance alone is insufficient.

Source Support

• holt1992inertial — primary. Experimentally rejects step,
  sinusoidal, and random disturbances in a 3-DOF platform’s rotational axis using a 6-DOF PUMA;
  introduces the approximate-pseudoinverse $J^{‡}$ control law and shows it behaves as a
  high-pass filter (low-frequency base disturbances attenuated most), with stability/performance set
  by disturbance amplitude and frequency.

Related Topics

• singularity_robust_inverse — Holt’s $J^{‡}$ is a
  singularity-robust inverse: it stays bounded through singular regions a disturbance may force the
  arm into, where the exact $J^{-1}$ blows up.
• inertial_space_tracking — the rejection objective is stated in
  inertial space (${}^0{u}_k\to {}^0{u}_d$); rejecting base motion is exactly holding the inertial-space
  track against an unknown base term.
• damped_least_squares — the canonical alternative to Holt’s thresholded
  block-SVD for keeping the inverse bounded near singularities (our notation.md
  ${\lambda }_J$, Chiaverini variable damping).
• trajectory_tracking — disturbance rejection is the
  trajectory-tracking problem with an additive exogenous base term $dv$ on the output.
• resolved_motion_rate_control — the rejection law is RMRC:
  inertial-space error is mapped to joint rates through the (approximate) inverse Jacobian.

Contrast with base_disturbance_minimization: minimization
(reaction-null-space, free-floating) keeps arm motion from exciting the base in the first place,
whereas rejection (here) lets the base move and actively cancels its effect at the end-effector.

Open Questions

• Holt treats the base disturbance $\delta$ as exogenous and unmeasured (driven platform). For our
  free-flying system the base is actuated and its state is estimated — does the kinematic-only
  $J^{‡}$ rejection still suffice, or must base and arm be co-designed (coordinated control)
  rather than the arm merely reacting?
• The high-pass-filter behavior attenuates low-frequency base disturbances least at DC. A standing
  base-pose offset (our cruise-lag floor, see trajectory_tracking) is
  effectively DC — how much steady-state EE error does pure kinematic rejection leave?
• Holt’s stability bound depends on knowing the maximum base deviation $\bar{\delta }$ a priori. What
  replaces that bound when the base disturbance is the (bounded but coupled) residual of a dynamic
  base controller rather than an independent platform?