jin2017reaction

Reaction Torque Control of Redundant Free-floating Space Robot

Authors: Jin, Zhou, Liu, Liu, Liu · Year: 2017 · Venue: International Journal of Automation and Computing (IJAC)
Raw: md

Summary

The paper derives an analytical expression for the reaction torque transmitted to the centroid of an uncontrolled (free-floating) satellite base by manipulator motion, and shows it takes the affine-in-acceleration form $T_0=M_R\ddot{\theta }+N_R$. Treating the “reaction-torque Jacobian” $M_R$ on equal footing with the generalized Jacobian $J_G$, the authors formulate (i) reactionless end-effector tracking when the arm is redundant ($n>m+l$) via an extended-Jacobian pseudoinverse, and (ii) base-disturbance minimization when redundancy is insufficient ($n<m+l$), solved either by a weighted least-squares (non-strict priority) or a null-space (strict priority) scheme. Singular value filtering (SVF) bounds the condition number of $J_G$ and $J_E$ to avoid dynamic singularity, and the methods are validated on a 7-DOF arm in a Linux/RTAI real-time controller-in-the-loop simulator.

Key Claims

• The reaction torque on the base centroid is affine in joint acceleration: $T_0=M_R\ddot{\theta }+N_R$ (Eq. 19), and two independent derivations — Newton-Euler symbolic inverse dynamics (Eq. 14) and the partitioned dynamics equation (Eqs. 16-18) — yield the same form; Example 1 shows the curves coincide.
• Because $T_0$ is linear in $\ddot{\theta }$, the inertia matrix $M_R$ can be treated as a Jacobian, enabling an extended Jacobian $J_E=[J_G;M_R]$ for simultaneous reactionless control plus end-effector tracking (Eq. 23-25).
• When $n<m+l$, zero reaction torque is unachievable; base disturbance is minimized as a constrained least-squares problem. The non-strict (weighted) solution (Eq. 28) gives smaller reaction torque but corrupts end-effector accuracy; the strict null-space solution (Eq. 33) preserves the primary tracking task at the cost of larger residual reaction torque.
• Quantitative result (Example 3, captured 100 kg load): base attitude disturbance drops from $(9.518,21.112,-9.006{)}^{\circ }$ without optimization to $(7.584,12.374,-6.745{)}^{\circ }$ with reaction-torque optimization.
• SVF replaces each singular value ${\sigma }_i$ with $f_{\upsilon ,{\sigma }_0}({\sigma }_i)=\frac{{\sigma }_i^3+\upsilon {\sigma }_i^2+2{\sigma }_i+2{\sigma }_{0\_J}}{{\sigma }_i^2+\upsilon {\sigma }_i+2}$, giving a full-rank pseudoinverse with bounded condition number through dynamic singularities (Eqs. 36-38).

Method

Regime: free-FLOATING. The base carries no external force/torque; only the joints are actuated, so linear and angular momentum are conserved and the base drifts in response to arm motion. This is explicitly NOT our free-flying (fully-actuated 6-DOF base) regime — here the base disturbance is the quantity to be suppressed by arm motion alone, not regulated by base thrusters/wheels.

Generalized Jacobian. From momentum conservation, the base twist is slaved to joint rates, $\left(\genfrac{}{}{0ex}{}{v_0}{w_0}\right)=J_{bm}\dot{\theta }$ (Eq. 6), with $w_b=-I_S^{-1}{I}_M\dot{\theta }$ (Eq. 8). Substituting into the end-effector kinematics $\dot{x}=J_S{w}_b+J_M\dot{\theta }$ (Eq. 7) gives the generalized Jacobian
$\dot{x}=(J_M-J_S{I}_S^{-1}{I}_M)\dot{\theta }=J_G\dot{\theta }(9).$

Reaction torque. Via Newton-Euler outward/inward recursion (Eqs. 11-13) the coupled wrench is $F_b=[F_0^T,T_0^T{]}^T=f_{ID}(\cdot )$. The base rotational equation $T_0=I_0{\dot{w}}_0+w_0\times I_0{w}_0$ (Eq. 15) combined with the partitioned dynamics
$\left[\begin{array}{cc}{\tilde{H}}_b& {\tilde{H}}_{bm}\\ {\tilde{H}}_{bm}^T& {\tilde{H}}_m\end{array}\right]\left[\begin{array}{c}{\dot{w}}_b\\ \ddot{\theta }\end{array}\right]+\left[\begin{array}{c}{\tilde{c}}_b\\ {\tilde{c}}_m\end{array}\right]=\left[\begin{array}{c}0\\ \tilde{\tau }\end{array}\right](16)$
yields ${\dot{w}}_b=-{\tilde{H}}_b({\tilde{H}}_{bm}\ddot{\theta }+{\tilde{c}}_b)$ (Eq. 18) and finally $T_0=M_R\ddot{\theta }+N_R$ with $M_R=-I_0{\tilde{H}}_b{\tilde{H}}_{bm}$ and $N_R=(I_S^{-1}{I}_M\dot{\theta })\times (I_0{I}_S^{-1}{I}_M\dot{\theta })-I_0{\tilde{H}}_b{\tilde{c}}_b$ (Eq. 19).

Control synthesis. Reactionless + tracking with redundancy: $\ddot{\theta }=J_E^{+}[\ddot{x}-{\dot{J}}_G\dot{\theta };-C_R]$ with $J_E=[J_G;M_R]$ (Eq. 23), optionally weighted by ${\lambda }_x,{\lambda }_b$ (Eq. 24). Disturbance minimization without sufficient redundancy: non-strict least squares (Eq. 28) vs. strict null-space projection $P=(I-J_G^{+}{J}_G)$, $\ddot{\zeta }=-[M_{R}P{]}^{+}[M_R{J}_G^{+}(\ddot{x}-{\dot{J}}_G\dot{\theta })+C_R]$ (Eq. 32), substituted into $\ddot{\theta }=J_G^{+}(\ddot{x}-{\dot{J}}_G\dot{\theta })+P\ddot{\zeta }$ (Eq. 33). Quaternion-based orientation feedback (Eqs. 41-44) closes the loop.

Notation flags. The OCR conflates several symbols: ${\dot{w}}_b$ vs. ${\ddot{x}}_b$ appear interchangeably (Eq. 18 uses ${\dot{w}}_b$ where Eq. 16 has the full base twist); Eq. 18 likely drops an inverse, i.e. should read ${\dot{w}}_b=-{\tilde{H}}_b^{-1}(\cdots )$. Eq. 20 writes $C_R$ where the derivation uses $N_R$. The pseudoinverse symbol alternates between $(\cdot {)}^{+}$ and $(\cdot {)}^{\#}$. $l$ is the (undefined-in-text) dimension of the base/reaction task. The SVF "${\sigma }_1$" in Eq. 38 is almost certainly ${\sigma }_l$.

Relevance to thesis

Although the paper is strictly free-floating, the reaction-torque-as-Jacobian idea is directly portable. For our free-flying base, the same $T_0=M_R\ddot{\theta }+N_R$ quantifies exactly the disturbance the base actuators must reject; minimizing $\Vert T_0\Vert$ via arm motion reduces the control effort/propellant the fully-actuated base spends holding attitude — a natural secondary objective and a candidate cost term in the risk-aware layer. The strict vs. non-strict priority dichotomy and the SVF singularity treatment are reusable in any redundancy-resolution stack, and the extended-Jacobian formulation is a clean precedent for stacking a base-effort task beneath end-effector tracking.

Connections

Topics: reaction_null_space · generalized_jacobian · dynamic_singularity · base_disturbance_minimization

Key Equations / Quotes

“the reaction torque has linear correlation with the angular acceleration, so the inertia matrix $M_R$ can be viewed as the Jacobian matrix, this is an important conclusion for the reaction torque control.” (Sec. 3.1)

$T_0=M_R\ddot{\theta }+N_R,M_R=-I_0{\tilde{H}}_b{\tilde{H}}_{bm}(19)$

$\ddot{\theta }={\left[\begin{array}{c}{J}_G\\ M_R\end{array}\right]}^{+}\left[\begin{array}{c}\ddot{x}-{\dot{J}}_G\dot{\theta }\\ -C_R\end{array}\right]=J_E^{+}\left[\begin{array}{c}\ddot{x}-{\dot{J}}_G\dot{\theta }\\ -C_R\end{array}\right](23)$

$\ddot{\theta }=J_G^{+}(\ddot{x}-{\dot{J}}_G\dot{\theta })-[M_{R}P{]}^{+}[M_R{J}_G^{+}(\ddot{x}-{\dot{J}}_G\dot{\theta })+C_R],P=(I-J_G^{+}{J}_G)(33)$

Open Questions

• The paper distinguishes algorithmic singularity (artificial, from null-space vector choice) from dynamic singularity but only treats the latter with SVF; how does SVF interact with the strict-priority $[M_{R}P{]}^{+}$ when $M_{R}P$ itself loses rank?
• No stability proof is given for the closed-loop quaternion feedback law combined with the SVF-modified pseudoinverse; bounded condition number does not by itself guarantee tracking convergence.
• How is the task-dimension parameter $l$ defined, and what determines the redundancy threshold $n\gtrless m+l$ for a 6-DOF base task?
• Validation is purely numerical (controller-in-the-loop RTAI simulator); no hardware results, so robustness to inertia-parameter error (critical after capturing an unknown load) is untested.