Exercises — Requirements — SMART (Specific, Measurable, Achievable, Relevant, Testable)

• Specific
• Measurable
• Achievable
• Relevant
• Testable

Memory hook: "Say My Awesome Rocket's Tested."

• (a) Measurable — "reasonably efficient" has no number, no units, no operator.
• (b) Achievable — the Tsiolkovsky Rocket Equation forbids $50\text{km/s}$ from chemical propulsion at that dry mass (parent note shows the mass ratio $\approx 83000$).
• (c) Testable (and Relevant) — "look nice" has no pass/fail criterion and traces to no mission objective.

• S ✅ — names the exact item ("bus structure").
• M ✅ — number (150), unit (kg), operator ($\le$).
• A — plausible, but you'd need a Mass Budget to confirm it. Cannot judge from the sentence alone.
• R — need the mission objective to confirm; not shown here.
• T ✅ — verifiable by inspection (weigh it).

So it clearly satisfies S, M, T; A and R need external context.

"The solar array shall generate $\ge 2.5\text{kW}\pm 5\%$ electrical power at end-of-life (15 years), when oriented normal to sunlight at $1\text{AU}$, across a temperature range of $-100^{\circ }\text{C}$ to $+120^{\circ }\text{C}$."

Checklist of what we added:

• S: "solar array … electrical power" (scope).
• M: $2.5\text{kW}$, $\pm 5\%$, operator $\ge$.
• A: leave to a power-budget analysis.
• R: implicitly traces to the mission's power need.
• T: verifiable by test under a solar simulator.

Use the Tsiolkovsky Rocket Equation: $\Delta v=I_{sp}{g}_0\ln \left(\frac{m_0}{m_f}\right)$

• WHAT/WHY the equation: it is the only relation linking exhaust speed and mass ratio to velocity change — the direct test of "can chemistry deliver this $\Delta v$?"
• Exhaust speed: $v_e=I_{sp}{g}_0=230\times 9.81=2256.3\text{m/s}$.
• Wet mass $m_0=50+10=60\text{kg}$; dry mass $m_f=10\text{kg}$; ratio $=6$.
• $\Delta v=2256.3\times \ln 6=2256.3\times 1.7918=4043\text{m/s}$.

$4043\text{m/s}\gg 500\text{m/s}$, so the requirement is Achievable with big margin. ✅

• (a) Inspection — measure with a coordinate-measuring machine.
• (b) Analysis — full-scale reentry is impractical to test, so use a thermal model.
• (c) Demonstration — run the end-to-end operational scenario.
• (d) Test — measure real output under a solar simulator.

See Verification and Validation for the four-method framework.

• S ✅ — item ("reaction wheel assembly"), axes ("each axis") and quantities are all named.
• M ✅ — torque $\ge 0.1\text{N⋅m}$, momentum $\ge 10\text{N⋅m⋅s}$, both with units and operators.
• A — plausible for flight-qualified wheels (check Technology Readiness Levels (TRL) $\ge 6$); not derivable from the sentence but not obviously violated.
• R — needs a trace to a pointing/slew objective. Not shown → flag for Requirements Traceability Matrix.
• T ✅ — "verified by test" plus measurable quantities gives a clear pass/fail.

Verdict: strong on S, M, T. Add a trace (R) and a TRL note (A) to make it airtight.

A valid chain (matching the parent's ice-mass example):

• Mission objective: "Observe Earth's ice-sheet mass balance to $\pm 10\text{Gt/year}$."
• ↓ System requirement: "Maintain radial altitude knowledge to $\pm 5\text{m}$ ($3\sigma$)."
• ↓ Subsystem requirement: the $\pm 3\text{m}$ GPS requirement above.

Why this proves Relevance: gravity-field (ice-mass) recovery needs precise orbit knowledge; orbit knowledge needs precise position fixes. Remove the GPS requirement and the chain snaps — the mission fails its objective. A requirement that traces to nothing can be cut, which is exactly the Relevance test.

Why it fails: "designed to" is a process requirement (it tells engineers what activity to perform), not a performance requirement (what the product must achieve). You cannot test whether someone "designed to" something — it fails Testable and is vague on Measurable.

Rewrite:

"The structure shall survive quasi-static launch loads of $\ge 8g$ axial and $\ge 4g$ lateral, verified by analysis and by qualification test to $1.25\times$ design limit load."

Now it constrains an outcome with numbers, a factor of safety, and a verification method — see Systems Engineering V-Model for where this sits in the flow.

Step 1 — invert Tsiolkovsky for the mass ratio (WHAT/WHY): we know $\Delta v$ and want propellant, so solve the rocket equation for $m_0/m_f$. $\frac{m_0}{m_f}=\exp \left(\frac{\Delta v}{I_{sp}{g}_0}\right)$

• $I_{sp}{g}_0=200\times 9.81=1962\text{m/s}$.
• Exponent $=120/1962=0.06116$.
• Ratio $=e^{0.06116}=1.06307$.

Step 2 — solve for propellant. With $m_f=8\text{kg}$: $m_0=1.06307\times 8=8.5046\text{kg},m_p=m_0-m_f=0.5046\text{kg}.$

Step 3 — write the SMART requirement (round up for margin):

"The propulsion subsystem shall provide $\Delta v\ge 120\text{m/s}$ using the cold-gas thruster ($I_{sp}\ge 200\text{s}$) with propellant mass $\ge 0.55\text{kg}$ and dry mass $\le 8\text{kg}$, verified by analysis (rocket-equation) and thruster-firing test."

Minimum propellant $\approx 0.50\text{kg}$; we spec $0.55\text{kg}$ to leave $\sim 10\%$ margin.

• Method: Test (with supporting Analysis for the momentum integral).
• Setup: Wheel mounted on a rigid stand with a calibrated torque sensor ($\pm 0.001\text{N⋅m}$) and an angular encoder; ambient temperature, vacuum recommended.
• Procedure: Command the wheel from $0$ to max RPM; log torque vs. time; compute stored momentum $L=I\omega$ from measured inertia $I$ and top speed $\omega$.
• Acceptance: measured torque $\ge 0.1\text{N⋅m}$ sustained $\ge 10\text{s}$ and $L\ge 10\text{N⋅m⋅s}$. Fold results into the Requirements Traceability Matrix and Verification and Validation records.

Angular momentum $L=I\omega$ (WHY: momentum storage is $I\omega$ — that's the quantity a wheel banks to later dump onto the spacecraft). $L=0.02\times 500=10.0\text{N⋅m⋅s}.$ It meets the requirement exactly ($L=10$). ⚠️ With zero margin, a real spec should demand $\ge 10$ with margin (e.g. design to $\omega =550\text{rad/s}$ giving $L=11$). Meeting a requirement "exactly" is a schedule risk — any manufacturing loss drops you below.

Mission objective: "Image coastal erosion at ground sample distance $\le 1\text{m}$ from $500\text{km}$ altitude."

↓ System requirement (points to the whole imaging chain — pointing stability so the ground pixel doesn't smear):

"The spacecraft shall maintain attitude pointing stability $\le 0.5\text{m}$ ground smear over one integration time (i.e. angular rate $\le 1\times 10^{-4}\text{rad/s}$), verified by analysis and hardware-in-loop test."

↓ Subsystem requirement (the actuator that delivers stability):

"The reaction wheel assembly shall provide control torque $\ge 0.1\text{N⋅m}$ about each axis with momentum storage $\ge 10\text{N⋅m⋅s}$, verified by test."

Interface: the camera-to-bus mechanical, electrical, and data boundary must be frozen in an Interface Control Document (ICD). Whole chain lives in the Requirements Traceability Matrix; each level is verified per the Systems Engineering V-Model.

WHY this geometry: a small pointing rate $\dot{\theta }$ sweeps the line-of-sight across the ground; the ground velocity of that sweep at range $h$ is $v_{\text{gnd}}=h\dot{\theta }$ (arc-length $=$ radius $\times$ angle). Smear is that velocity times how long the shutter is open.

• Ground sweep rate: $v_{\text{gnd}}=h\dot{\theta }=500000\text{m}\times 10^{-4}\text{rad/s}=50\text{m/s}$.
• Smear over integration: $s=v_{\text{gnd}}t=50\times 0.010=0.5\text{m}$.

Smear $=0.5\text{m}$, exactly the budget — so at this integration time the rate spec is the binding limit. (Same zero-margin warning as L4.3: in practice tighten $\dot{\theta }$ or shorten $t$.)

• Momentum: $L=I\omega =0.02\times 500=10.0\text{N⋅m⋅s}$ per wheel → meets $\ge 10$ (zero margin).
• Mass: $4\times 2.0\text{kg}=8.0\text{kg}>6\text{kg}$ budget → fails by $2\text{kg}$.

Conflict: the two requirements are inconsistent as written. Options, in order of preference:

1. Negotiate the Mass Budget up to $8\text{kg}$ if mission margins allow.
2. Drop to 3 wheels (no redundancy) — but Failure Modes and Effects Analysis (FMEA) will flag a single-point failure.
3. Select lighter wheels with higher spin rate (raise $\omega$ to keep $L$ while shrinking $I$ and mass).

Lesson: individual requirements can each be SMART yet collectively infeasible. Traceability and budgets exist to surface exactly this cross-coupling before hardware is cut.