Base Disturbance Minimization

Definition

Base disturbance minimization is the use of a redundant manipulator’s spare degrees of freedom to
reduce (not necessarily null) the reaction wrench that arm motion transmits to the spacecraft base
while the end-effector still tracks its commanded path. In the free-floating regime assumed by the
primary source (jin2017reaction), the base is uncontrolled, so the
reaction torque $T_0$ on the base centroid directly disturbs the satellite attitude — undesirable for
solar-power generation and ground-station communication (the source’s stated motivations). Jin et al. treat the analytically-derived reaction torque as
a secondary task added to the end-effector tracking task: when the redundancy is sufficient
($n\ge m+l$) the reaction can be driven to zero (reactionless control, see
reactionless_motion); when it is not ($n<m+l$), the reaction can only be
minimized, which is the disturbance-minimization case proper.

Key Equations

Symbols per notation.md.

The base reaction torque is affine in the joint accelerations, with the reaction-torque inertia
matrix ${\mathbf{M}}_R=-{\mathbf{I}}_0{\tilde{\mathbf{H}}}_b{\tilde{\mathbf{H}}}_{bm}$ (Eq. 19) acting as
a “Jacobian” for the torque (source symbols: ${\mathbf{M}}_R,{\mathbf{N}}_R,{\mathbf{C}}_R$ are the
reaction-torque inertia / bias terms — the source writes Eq. 19 with ${\mathbf{N}}_R$ but the
optimization Eqs. 20–33 with ${\mathbf{C}}_R$ for the same bias; ${\mathbf{J}}_G$ is the source’s
generalized Jacobian; these are the paper’s notation, not yet in notation.md):

${\mathbf{T}}_0={\mathbf{M}}_R\ddot{\mathbf{\theta }}+{\mathbf{N}}_R.$

Disturbance minimization is then the constrained least-squares problem — track the end-effector while
shrinking $\Vert {\mathbf{T}}_0\Vert$ — posed for the redundancy-deficient case $n<m+l$:

\qquad\text{s.t.}\qquad \ddot{\boldsymbol x} - \dot{\boldsymbol J}_G\,\dot{\boldsymbol\theta} = \boldsymbol J_G\,\ddot{\boldsymbol\theta} .$$ The **strict task-priority** (null-space) solution keeps end-effector tracking inviolate and pushes the reaction minimization into the redundant subspace via the projector $\boldsymbol P=(\boldsymbol I-\boldsymbol J_G^{+}\boldsymbol J_G)$: $$\ddot{\boldsymbol\theta} = \boldsymbol J_G^{+}\!\left(\ddot{\boldsymbol x}-\dot{\boldsymbol J}_G\dot{\boldsymbol\theta}\right) - [\boldsymbol M_R\boldsymbol P]^{+}\!\left[\boldsymbol M_R\boldsymbol J_G^{+}\!\left(\ddot{\boldsymbol x}-\dot{\boldsymbol J}_G\dot{\boldsymbol\theta}\right)+\boldsymbol C_R\right].$$ The alternative **non-strict** form folds both tasks into one weighted pseudoinverse with weights $\lambda_x,\lambda_b$; there end-effector and reaction tasks trade off against each other, whereas the null-space form above protects the primary task at the cost of less reaction reduction. ## Source Support - [jin2017reaction](../sources/jin2017reaction.md) — primary. Derives the analytical reaction torque $\boldsymbol T_0=\boldsymbol M_R\ddot{\boldsymbol\theta}+\boldsymbol N_R$ for a redundant free-floating robot and gives both the weighted-least-squares (non-strict) and null-space (strict task-priority) solutions for base attitude-disturbance minimization; reports a drop from $\approx(9.5,21.1,-9.0)^\circ$ to $(7.6,12.4,-6.7)^\circ$ base disturbance with the optimization enabled, and avoids the dynamic singularity via singular-value filtering. ## Related Topics - [reaction_null_space](reaction_null_space.md) — supplies the joint motions that produce *zero* base reaction; disturbance minimization is the graceful-degradation case when the RNS is too small to null the reaction completely. - [reactionless_motion](reactionless_motion.md) — the $n\ge m+l$ limit of this page (reaction driven to exactly zero); minimization is what remains achievable below that DOF count. - [kinematic_redundancy_resolution](kinematic_redundancy_resolution.md) — the general machinery (null-space projection / weighted pseudoinverse) that here resolves redundancy toward low base reaction. - [generalized_jacobian](generalized_jacobian.md) — the source builds its tracking constraint on the generalized Jacobian $\boldsymbol J_G$, which already folds free-floating momentum conservation into the end-effector map. - [dynamic_coupling](dynamic_coupling.md) — the very effect being suppressed: arm motion couples into base attitude through the coupled inertia block $\tilde{\boldsymbol H}_{bm}$, which $\boldsymbol M_R$ folds in ($\boldsymbol M_R=-\boldsymbol I_0\tilde{\boldsymbol H}_b\tilde{\boldsymbol H}_{bm}$). See also [base_disturbance_rejection](base_disturbance_rejection.md) (the feedback-control counterpart), [task_priority_control](task_priority_control.md), and [momentum_conservation](momentum_conservation.md) (the conservation law that makes arm motion disturb an uncontrolled base in the first place). ## Open Questions - The source assumes a **free-floating** base (uncontrolled; momentum conserved, Eq. 16 has zero external base wrench). For our **free-flying** system the base is fully actuated, so attitude disturbance is a control-effort / fuel cost rather than an uncorrectable error — does minimizing $\|\boldsymbol T_0\|$ remain the right objective, or should it be recast as base-actuator-effort minimization? - The non-strict form lets the reaction task corrupt end-effector tracking via $\lambda_x,\lambda_b$; how should those weights be set when an actuated base can absorb residual reaction directly? - Jin et al. minimize reaction *torque* only (attitude), treating the reaction *force* $F_0$ separately; for a flying base that also translates, should force and torque disturbance be minimized jointly?