Reaction wheels — momentum management, zero-crossing

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WHY do reaction wheels exist?

The catch: the internal trick cannot cancel an external torque forever. External torques inject real angular momentum into the whole system. The wheel absorbs it by spinning faster and faster — until it hits redline. That's why momentum management exists.

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WHAT is conserved — the core law

HOW: derive the body–wheel coupling from scratch

Consider a single wheel with spin axis fixed in the body, spin inertia $I_w$, wheel speed $\Omega_w$. Let the body have inertia $I_b$ about that axis and rate $\omega_b$. Along the spin axis:

$H_{\text{tot}} = I_b\,\omega_b + I_w\,\Omega_w$

Why this form? Angular momentum is (inertia)×(angular rate), and along a common axis these add as scalars.

Differentiate. With no external torque $\dot H_{\text{tot}}=0$:

\quad\Rightarrow\quad \boxed{\;\dot\omega_b = -\frac{I_w}{I_b}\,\dot\Omega_w\;}$$ > [!intuition] Read the minus sign > Spin the wheel up ($\dot\Omega_w>0$) and the body accelerates the *opposite* way ($\dot\omega_b<0$). That IS how you turn: the wheel's motor torque $\tau_m = I_w\dot\Omega_w$ reacts against the body — hence "**reaction** wheel." Now add an external torque $\tau_{\text{ext}}$ along the axis: $$\frac{d}{dt}\big(I_b\omega_b + I_w\Omega_w\big) = \tau_{\text{ext}}$$ If the controller holds the body still ($\omega_b\approx 0$, $\dot\omega_b\approx0$), then: $$I_w\,\dot\Omega_w = \tau_{\text{ext}} \quad\Rightarrow\quad \Omega_w(t) = \Omega_w(0) + \frac{1}{I_w}\int_0^t \tau_{\text{ext}}\,dt'$$ > [!formula] Wheel momentum accumulation > The wheel's stored momentum equals the **time-integral of the external torque**: > $$H_w(t) = H_w(0) + \int_0^t \tau_{\text{ext}}\,dt'$$ > A tiny but *constant-sign* torque (like a fixed solar-pressure offset) grows $H_w$ without bound → **saturation**. --- ## Momentum management (desaturation / "momentum dumping") > [!definition] Momentum management > The process of removing accumulated wheel momentum using an **external** actuator (thrusters or magnetic torquers), so the wheel returns toward a safe speed while the body stays pointed. > [!intuition] Why you NEED something external > Wheels only *shuffle* momentum between body and wheel — the sum is fixed. To change the **total**, you must apply a real external torque. Magnetorquers push against Earth's field $\vec B$; thrusters throw mass out. Either changes $\vec H_{\text{tot}}$, letting the controller slow the wheel back down. Magnetorquer dump law (the workhorse in LEO): $$\vec\tau_{\text{mag}} = \vec m \times \vec B, \qquad \vec m = -\,k\,(\vec H_w - \vec H_{\text{target}})\times \vec B$$ This produces a torque that drives $\vec H_w$ toward target (a cross-product-only control — it can't touch the component along $\vec B$, hence you rely on orbital variation of $\vec B$). --- ## Zero-crossing — the subtle enemy > [!definition] Zero-crossing > The instant a wheel's speed passes through $\Omega_w = 0$ (changing sign) as its momentum is regulated near zero. > [!intuition] Why zero is a bad place to sit > Near $\Omega_w=0$ two nasty things happen: > 1. **Bearing friction reverses sign** and is highly nonlinear (stiction). Around zero the friction torque flips direction, so the motor's needed torque is discontinuous → the wheel "hangs," then jerks. This injects a **jitter** disturbance into the spacecraft — deadly for a telescope. > 2. **Motor torque ripple / cogging** is worst near zero rate where there's no averaging over rotation. > > Result: a wheel dawdling at zero speed produces micro-vibrations exactly when you want the finest pointing. > [!intuition] Why we deliberately keep wheels spinning (bias momentum) > To *avoid* zero-crossing we run each wheel at a **non-zero bias speed** $\Omega_{\text{bias}}$. The controller regulates $H_w$ toward a bias value, not toward zero, so friction never reverses. In a 4-wheel pyramid you can even set biases so no wheel ever crosses zero during a maneuver. > [!mistake] "Just command the wheel through zero, it's only friction." > **Why it feels right:** friction seems small compared to motor torque, so who cares. **Why it's wrong:** at zero speed, static (stick) friction can *exceed* the tiny torque command needed for fine pointing; the wheel sticks and the pointing error grows until the motor overcomes stiction, then it snaps — an impulsive disturbance. **Fix:** bias the wheels away from zero, or route the required momentum through other wheels in a redundant array so no single wheel crosses zero. > [!mistake] "Reaction wheels store momentum, so they solve secular torques forever." > **Why it feels right:** you *can* absorb any torque for a while. **Why it's wrong:** a constant-sign external torque integrates without bound; the wheel saturates in finite time. **Fix:** periodic momentum dumping with an external actuator is mandatory — wheels are a *buffer*, not a *sink*. ![[3.5.48-Reaction-wheels-—-momentum-management,-zero-crossing.png]] --- ## Worked examples > [!example] 1 — Slew a satellite with one wheel > $I_b = 500\ \text{kg·m}^2$, wheel $I_w = 0.05\ \text{kg·m}^2$. We spin the wheel from $0$ to $\Omega_w = 300\ \text{rad/s}$ in $10\ \text{s}$. What body rate results (no external torque, starts at rest)? > > **Step — conservation:** $I_b\omega_b + I_w\Omega_w = 0$. > *Why?* No external torque → total $H$ stays at its initial value, $0$. > $$\omega_b = -\frac{I_w\Omega_w}{I_b} = -\frac{0.05\times 300}{500} = -0.03\ \text{rad/s}$$ > The body slowly rotates opposite to the wheel. **Why the minus sign matters:** to point *toward* a target you spin the wheel *away* from it. > > **Motor torque used:** $\tau_m = I_w\dot\Omega_w = 0.05 \times (300/10) = 1.5\ \text{N·m}$. > [!example] 2 — Time to saturation from solar pressure > A constant disturbance $\tau_{\text{ext}} = 2\times10^{-5}\ \text{N·m}$. Wheel max momentum $H_{\max} = 0.05\ \text{kg·m}^2 \times 600\ \text{rad/s} = 30\ \text{N·m·s}$. Starting from bias $H_w(0)=0$. > > **Step — integrate:** $H_w(t) = \tau_{\text{ext}}\,t$. > *Why?* Constant torque → linear accumulation ($\int \text{const}\,dt = \text{const}\cdot t$). > $$t_{\text{sat}} = \frac{H_{\max}}{\tau_{\text{ext}}} = \frac{30}{2\times10^{-5}} = 1.5\times10^{6}\ \text{s} \approx 17.4\ \text{days}$$ > **Why it matters:** you must schedule a dump well before ~17 days, else the wheel maxes out and *loses attitude control*. > [!example] 3 — Magnetorquer dump feasibility > Earth field $B = 3\times10^{-5}\ \text{T}$, magnetorquer dipole $m = 30\ \text{A·m}^2$ perpendicular to $B$. > > **Step — max torque:** $\tau = mB\sin\theta$, max at $\theta=90^\circ$. > *Why?* $\vec\tau=\vec m\times\vec B$; cross product peaks when perpendicular. > $$\tau_{\max} = 30\times 3\times10^{-5} = 9\times10^{-4}\ \text{N·m}$$ > To dump $30\ \text{N·m·s}$: $t = 30 / 9\times10^{-4} \approx 3.3\times10^{4}\ \text{s} \approx 9.3\ \text{h}$. > **Why it matters:** dumping is slow and can only act perpendicular to $\vec B$, so you spread it across the orbit as $\vec B$ rotates. --- > [!recall]- Feynman: explain to a 12-year-old > Imagine you're sitting on a spinny office chair holding a bicycle wheel. If you spin the wheel one way, YOU spin the other way — that's how a satellite turns itself without pushing on anything outside. But the wheel can only spin so fast. If a tiny steady wind (space has gentle "winds" from sunlight and gravity) keeps nudging you the same way, the wheel spins faster and faster to fight it, until it can't go any faster — it's "full." Then you need to gently push on something *outside* (Earth's magnet or a puff of gas) to empty the wheel back down. Also, if the wheel ever slows to a stop and tries to spin backward, it gets a little "stuck-then-jerk" (like a squeaky door hinge). That jerk shakes your telescope. So we keep the wheels always spinning a bit, never letting them freeze at zero. > [!mnemonic] BUFFER + BIAS > **B**uffer, not sink (wheels fill up). **U**se external torque to dump. **F**riction flips at zero. **F**inite time to saturate. **E**mpty before redline. **R**un at **BIAS** speed to dodge zero-crossing. --- ## #flashcards/physics Why does spinning a reaction wheel one way rotate the spacecraft the other way? ::: Conservation of total angular momentum: $I_b\omega_b + I_w\Omega_w = \text{const}$, so $\dot\omega_b = -(I_w/I_b)\dot\Omega_w$. Can reaction wheels cancel an external torque indefinitely? ::: No — wheel momentum is the time-integral of external torque; a constant-sign torque saturates the wheel in finite time. What is momentum management (desaturation)? ::: Using an external actuator (thrusters/magnetorquers) to remove accumulated wheel momentum so the wheel returns to a safe speed. Why can't reaction wheels themselves change total angular momentum? ::: They only redistribute momentum internally between body and wheel; changing the total requires an external torque. What is zero-crossing and why is it harmful? ::: A wheel's speed passing through zero; bearing stiction reverses sign nonlinearly, causing a stick–slip jitter that corrupts fine pointing. How do we avoid zero-crossing? ::: Run wheels at a non-zero bias momentum (or route momentum through a redundant wheel array) so no wheel's speed reverses. Magnetorquer torque law? ::: $\vec\tau = \vec m \times \vec B$; only acts perpendicular to $\vec B$, so dumping relies on the field varying over the orbit. Time to saturation for constant torque $\tau$? ::: $t_{\text{sat}} = H_{\max}/\tau$ where $H_{\max}=I_w\Omega_{\max}$. Motor torque needed to accelerate a wheel? ::: $\tau_m = I_w\dot\Omega_w$ (reacts against the body). --- ## Connections - [[Attitude Determination and Control System (ADCS)]] - [[Conservation of Angular Momentum]] - [[Control Moment Gyroscopes (CMG)]] — related, uses gimbaled spinning wheels - [[Magnetorquers and Earth's Magnetic Field]] - [[Gravity Gradient Torque]] · [[Solar Radiation Pressure]] · [[Aerodynamic Drag Torque]] - [[PID and Feedback Control]] — the loop that commands wheel torque - [[Bearing Friction and Stiction]] ## 🖼️ Concept Map ```mermaid flowchart TD RW[Reaction wheel flywheel] AM[Total angular momentum conserved] POINT[Precise pointing no fuel] EXT[External torques drag solar gravity] ACC[Wheel momentum accumulation] SAT[Wheel saturation redline] MM[Momentum management] DUMP[Dump via thrusters or torquers] MINUS[Body spins opposite to wheel] RW -->|obeys| AM AM -->|enables| POINT RW -->|spin up gives| MINUS EXT -->|inject real| AM EXT -->|integrated over time| ACC ACC -->|grows toward| SAT SAT -->|requires| MM MM -->|performs| DUMP DUMP -->|removes| ACC POINT -->|limited by| SAT ``` ## 🔊 Hinglish (regional understanding) > [!intuition]- Hinglish mein samjho > Dekho, reaction wheel ek chhoti si flywheel hoti hai jo satellite ke andar motor se ghumti hai. Physics ka core rule hai: total angular momentum conserve hota hai. Matlab agar tum wheel ko ek direction me ghumaoge, to satellite ka body ulti direction me ghum jaayega — bina kisi fuel ke, sirf electric power se. Isiliye telescopes aur imaging satellites years tak precise pointing kar paate hain, kyunki har baar thruster fire nahi karna padta. > > Lekin ek problem hai. Space me chhote-chhote external torques hote hain — solar radiation pressure, gravity gradient, thodi si atmospheric drag. Ye lagataar ek hi taraf push karte rehte hain. Wheel in torques ko absorb karti hai apni speed badha ke, aur kyunki $H_w = \int \tau_{ext}\,dt$, ye momentum bina ruke badhta jaata hai. Ek din wheel apni **max speed (saturation)** pe pahunch jaati hai — phir wo aur help nahi kar sakti. Isliye humein **momentum dumping** karna padta hai: magnetorquer (Earth ke magnetic field ke against push) ya thruster se external torque de ke wheel ko wapas slow karte hain. Yaad rakho — wheel ek *buffer* hai, *sink* nahi. > > Ab **zero-crossing** ka funda. Jab wheel ki speed zero ke aaspaas hoti hai aur sign change karti hai, to bearing ki friction (stiction) achanak ulti ho jaati hai — wheel thodi der "stuck" ho jaati hai phir jhatke se chalti hai. Ye jhatka satellite me micro-vibration daalta hai, jo fine pointing ko barbaad kar deta hai. Isse bachne ke liye hum wheels ko hamesha ek **bias speed** pe chalate hain, kabhi zero pe rukne nahi dete. Isse friction kabhi reverse nahi hoti aur pointing smooth rehti hai. ![[audio/3.5.48-Reaction-wheels-—-momentum-management,-zero-crossing.mp3]]