Control moment gyroscopes (CMG) — high torque, singularity

3.5.49 · Physics › Guidance, Navigation & Control (GNC)

CMGs ki zaroorat HAI KYO?

Spacecraft ko fast reorient (slew) karna padta hai — imaging satellites, ISS, telescopes. Fuel exhale kiye bina internal torque banane ke do tarike:

• Reaction wheel: wheel ki speed change karo $\Rightarrow$ torque $=I\dot{\omega }$. Bada torque paane ke liye badi angular acceleration chahiye, jo bahut motor power leta hai aur jaldi saturate ho jaata hai.
• CMG: wheel speed fixed rakho, apni axis rotate karo $\Rightarrow$ torque $=\Omega \times h$, jahan $h$ already bada hai. Chhoti gimbal rate $\Omega$, bahut bada torque. Yeh hai torque amplification.

Ek single CMG torque KAISE banata hai — derive karo

First principles. Rotation ke liye Newton–Euler kehta hai torque = angular momentum ka rate of change: $\vec{\tau }=\frac{d\vec{h}}{dt}.$

Kyunki $h=|\vec{h}|$ constant hai, sirf $\hat{s}$ rotate karta hai. Agar gimbal $\hat{s}$ ko gimbal axis $\hat{g}$ ke around rate $\Omega =\dot{\delta }$ par rotate karta hai (gimbal angle $\delta$), toh ek rotating unit vector ka rule yeh hai: $\frac{d\hat{s}}{dt}=\vec{\Omega }\times \hat{s},\vec{\Omega }=\dot{\delta }\hat{g}.$

Yeh step kyun? Fixed length ka ek vector jo angular velocity $\vec{\Omega }$ ke through rotate ho raha hai woh hamesha $\vec{\Omega }\times (\text{vector})$ ke roop mein change hota hai — yeh rotation ki definition hai. $h$ se multiply karo:

Multiple CMGs: Jacobian aur total momentum

Ek CMG sirf ek plane mein torque deta hai. Teeno axes control karne ke liye ek cluster use karte hain (typically 4, pyramid mein). Maano har CMG $i$ ka gimbal angle ${\delta }_i$ hai; total stored momentum hai: $\vec{H}(\vec{\delta })={\sum }_i{\vec{h}}_i({\delta }_i).$

Body par output torque hai: $\vec{\tau }=-\dot{\vec{H}}=-{\sum }_i\frac{\partial {\vec{h}}_i}{\partial {\delta }_i}{\dot{\delta }}_i=-A(\vec{\delta })\dot{\vec{\delta }},$ jahan Jacobian $A(\vec{\delta })=\left[\frac{\partial {\vec{h}}_1}{\partial {\delta }_1}|\cdots |\frac{\partial {\vec{h}}_n}{\partial {\delta }_n}\right]\in {\mathbb{R}}^{3\times n}.$

Jacobian kyun? Yeh sirf chain rule hai: har column batata hai ki woh CMG abhi kis direction mein push kar sakta hai, apne current gimbal angle ke hisaab se. Ek commanded torque ${\vec{\tau }}_{cmd}$ produce karne ke liye gimbal rates solve karo: $\dot{\vec{\delta }}=-A^{+}{\vec{\tau }}_{cmd},A^{+}=A^T(A{A}^T{)}^{-1}\text{(pseudo-inverse).}$

Singularity KIYA hoti hai?

Ek useful singularity measure: $m=\sqrt{\det (A{A}^T)}\stackrel{\text{singular}}{\to }0.$

Escapable vs. trapped

• Hyperbolic (escapable): nearby gimbal motions exist karte hain jo momentum ko wall se door le jaate hain — uske around steer karo.
• Elliptic / internal (trapped): har escape ke liye momentarily $\hat{u}$ ki taraf momentum build karna padta hai; standard steering atak jaata hai. Null-motion aur singularity-robust (SR) inverse se handle karo: $\dot{\vec{\delta }}=-A^T(A{A}^T+\lambda I{)}^{-1}{\vec{\tau }}_{cmd}+(I-A^{+}A)\vec{d}.$

Worked examples

Active recall

Connections

• Reaction wheels — momentum storage & saturation
• Gyroscopic precession & rigid-body Euler equations
• Jacobian & pseudo-inverse (Moore–Penrose)
• Spacecraft attitude control & slew maneuvers
• Null space & rank of a matrix
• Singularity-robust inverse & damped least squares
• Momentum management & desaturation (magnetic torquers, thrusters)

Concept Map

Deep Dive

• D1 · Foundations — har symbol zero se
• D2 · Visual walkthrough — derivation pictures mein
• D3 · Worked examples — har scenario
• D4 · Exercises — graded, full solutions
• D5 · Question bank — concept traps