holt1994inertial

Inertial-Space Disturbance Rejection for Space-Based Manipulators

Authors: Holt, Desrochers · Year: 1994 · Venue: NASA Conference Publication (N94-26281); work from Rensselaer Polytechnic Institute / Lockheed Engineering & Sciences
Raw: md

Summary

A kinematic control law is implemented that drives a 6-DOF PUMA manipulator (mounted on a 3-DOF platform) to track desired end-effector position and attitude in inertial space, while rejecting unknown disturbances in the platform axes. The key technical device is an approximate pseudoinverse Jacobian $J^{‡}$ that exploits the PUMA Jacobian’s block structure so it remains well-defined through kinematic singularities at roughly one-quarter the cost of the full pseudoinverse. The method is validated experimentally with step, sinusoidal, and random platform disturbances, including operation near singular configurations.

Key Claims

• Under the realistic case where only the nominal platform location and an upper bound on the disturbance are known (not the exact disturbance signal), direct end-point sensing is required to recover the end-effector location; assuming the disturbance is exactly known is unrealistic under measurement delay, poorly-known platform kinematics, or a non-rigid platform.
• The approximate pseudoinverse $J^{‡}$ is defined for all configurations and equals $J^{-1}$ when $J$ is nonsingular, but does not satisfy the Moore-Penrose conditions when $J$ is singular.
• $J^{‡}$ costs about $4\times$ less than the full pseudoinverse: two $3\times 3$ SVDs are an $O(2(N/2{)}^3)$ operation versus $O(N^3)$ for the $6\times 6$ SVD. Measured computation times confirm this (the approximate pseudoinverse far faster than the full pseudoinverse per Table 1).
• The approximation error $\Vert J{J}^{‡}J-J{\Vert }_2$ is bounded: $\le \Vert J{\Vert }_2$ when both $B,E$ singular; $\le \Vert E{\Vert }_2$ when only $B$ nonsingular; $\le \Vert B{\Vert }_2$ when only $E$ nonsingular; and exactly $0$ when both are nonsingular.
• The closed-loop system is linear with time-varying coefficients $M_{k,k-1}$; with $K_c=k_{c}I$, the spectral radius of $M_k$ is controlled by $k_c\in [0,2]$ (eigenvalues lie on a circle of radius $k_c$, or at zero if $J$ singular).
• Empirically the $J^{‡}$ controller behaves like a high-pass filter: low-frequency disturbance components are attenuated most; relative stability depends on disturbance amplitude and relative performance on disturbance frequency.
• Near singularities the control becomes very weak in certain (“forbidden”) directions, implying unavoidable tracking error there; setting ${\sigma }_{min}$ appropriately keeps the arm from crossing the singularity (avoiding flex/noflex “chattering”).

Method

The differential kinematics use $du=J(q)dq$ (Eq. 1), with $J\in {\mathbb{R}}^{6\times 6}$ for the PUMA. Expressed in frame 6 the Jacobian has block lower-triangular form (Eq. 3):
${}^6{J}_{3,9}=\left[\begin{array}{cc}B& 0\\ D& E\end{array}\right],$
so $J^{-1}=\left[\begin{array}{cc}{B}^{-1}& 0\\ -E^{-1}D{B}^{-1}& E^{-1}\end{array}\right]$ (Eq. 4). The approximate pseudoinverse replaces the inverses of the blocks with their pseudoinverses (Eq. 6):
$J^{‡}\triangleq \left[\begin{array}{cc}{B}^{†}& 0\\ -E^{†}D{B}^{†}& E^{†}\end{array}\right].$
The SVD is taken only on the $3\times 3$ submatrices $B$ (linear/position block) and $E$ (angular block), with a minimum-singular-value threshold ${\sigma }_{min}$ that gates when a singular value is treated as zero. Smaller ${\sigma }_{min}$ shrinks the “well” around the singular point (extending the region where $J^{‡}=J^{-1}$) at the cost of large, discontinuous norms; larger ${\sigma }_{min}$ widens the singular region and smooths the norm.

The disturbance-rejection control isolates the platform dependence via a coordinate transform (Eq. 14):
${}^{0}d{u}_{0,E}={}_3^0\mathbf{R}({\eta }_o+\delta ){}^3{J}_{3,E}(\theta )d\theta +dv,$
where ${\eta }_o$ is the nominal platform joint vector, $\delta$ the disturbance, and $dv$ the end-effector motion caused by the (unknown) platform displacement. The discrete control law (the ”$J^{‡}$ control law”, Eq. 19) is
$\Delta {\theta }_d={}^3{J}_{3,E}^{‡}({\theta }_k){}_0^3\mathbf{R}({\eta }_o)K_c({}^0{u}_d-{}^0{u}_k),$
using the nominal rotation ${}_0^3\mathbf{R}({\eta }_o)$ (the disturbance $\delta$ is not known to the controller). With one-period actuation delay the closed loop becomes (Eq. 23) ${}^0{u}_k=(I-M_{k,k-1}){}^0{u}_{k-1}+M_{k,k-1}{}^0{u}_d+\Delta v_k$. A stable gain follows from $k_c=2/\sqrt{{\tan }^2\alpha +1}$ (Eq. 26), where $\alpha$ approximates the worst-case argument of the spectrum of ${}_3^0\mathbf{R}({\eta }_o+\bar{\delta }){}_0^3\mathbf{R}({\eta }_o)$.

Regime note. This is not a free-floating model: the base is a 3-DOF actuated/disturbed platform and the controller compensates platform motion using the arm. There is no momentum-conservation / reaction coupling treatment of an uncontrolled base; the platform motion enters purely as a kinematic disturbance $\delta$ on the arm Jacobian. It is closer in spirit to a free-flying / fixed-base manipulator with base disturbance than to the free-floating space-manipulator literature it cites (Papadopoulos & Dubowsky). The contribution is kinematic (Jacobian-level), not dynamic.

Relevance to thesis

For a free-flying space manipulator, base motion (commanded or perturbed) couples directly into the end-effector. This paper’s framing — track the end-effector in inertial space while the base moves — is exactly the coordinated-control objective for a fully-actuated 6-DOF base, with the platform disturbance $\delta$ playing the role of base pose uncertainty. The block-structured singularity-robust inverse and its certified error bound $\Vert J{J}^{‡}J-J{\Vert }_2\le \Vert B{\Vert }_2$ (or $\Vert E{\Vert }_2$) give a cheap, analyzable way to pass through kinematic singularities, relevant to the nominal guidance layer before adding risk. The bound on $\delta$ (rather than exact knowledge) and the need for end-point sensing prefigure the risk-aware/uncertainty layer.

Connections

Topics: generalized_jacobian · dynamic_singularity · pseudoinverse_jacobian · coordinated_control

Key Equations / Quotes

“The usual method of dealing with singularities of the Jacobian is to avoid them. This approach is not applicable to the disturbance rejection problem since a sufficiently large disturbance could force the manipulator into a singular configuration.” (Sec. 2)

“the manipulator must avoid not just singular points, but singular regions, since the norm of $J^{-1}$ becomes very large in the neighborhood of a singularity.” (Sec. 2)

Approximation error bounds (Eqs. 10-13):
$\Vert J{J}^{‡}J-J{\Vert }_2\le \Vert J{\Vert }_2(B,E\text{ sing.}),\le \Vert E{\Vert }_2(B\text{ nonsing.}),\le \Vert B{\Vert }_2(E\text{ nonsing.}),=0(B,E\text{ nonsing.}).$

“the $J^{‡}$ controller is like a high-pass filter; the lowest frequency components of the disturbance signal are attenuated the most.” (Sec. 9)

Open Questions

• The analysis is purely kinematic; how would arm-base dynamic coupling (reaction torques on a real free-flying base) change the disturbance-rejection guarantees?
• The error bound $\Vert J{J}^{‡}J-J{\Vert }_2$ characterizes deviation from a true generalized inverse, but the paper does not translate this into a tracking-error bound on ${}^0{u}_d-{}^0{u}_k$ near singularities — what is the achievable tracking accuracy in the “forbidden” directions?
• ${\sigma }_{min}$ trades off chattering vs. tracking; no systematic rule is given for choosing it from disturbance bound $\delta$.
• Only step-disturbance results are presented in detail; sinusoidal/random results are summarized qualitatively (high-pass behavior) but not quantified here.