Chapter 27 — Key Takeaways (Guidance, Navigation, and Control)

A one-page reference. Reread this before an exam, or before you write your mission's GN&C approach.

The loop, in one line

Navigation (where am I, how sure?) → Guidance (where do I go, by what path?) → Control (what do I command the actuators?) → the vehicle moves → sensors read → back to navigation. Round and round, tens to thousands of times a second. Closed-loop = measure the result and correct the error; open-loop = fire a plan and hope. The flight lives in the correcting, not the plan.

The three boxes and their tools

Box Question Core tool Owner chapter
Navigation where am I & how sure? Kalman / extended Kalman filter (state estimation) Ch. 13 (filter); Ch. 14 (attitude)
Guidance where do I go & how? powered explicit guidance (PEG); velocity-to-be-gained; Lambert targeting this ch.; Ch. 13 (Lambert)
Control what do I command now? PID feedback law this chapter

Key equations (with symbols and units)

Equation Meaning Notes
$u = K_p e + K_i\!\int e\,dt + K_d\,\dfrac{de}{dt}$ PID control law ($u$ = command, $e$ = error) P = present, I = past, D = future
$K = \dfrac{\sigma_{\text{pred}}^2}{\sigma_{\text{pred}}^2+\sigma_{\text{meas}}^2}$ Kalman gain (0–1) defined in Ch. 13 — used, not re-derived
$\hat{x}^+ = \hat{x}^- + K(z-\hat{x}^-);\ (\sigma^+)^2 = (1-K)\sigma_{\text{pred}}^2$ Kalman update fused $\sigma$ beats both inputs
$v_{\text{go}} = v_{\text{target}} - v_{\text{now}}$ velocity-to-be-gained guidance nulls it; cutoff when zero
$M_c = F\sin\delta\,L$ control torque from a gimbal ($\delta$ deflection, $L$ arm) in N·m
$t = D/c,\quad c \approx 3.0\times10^{5}\ \text{km/s}$ light-time (distance / speed of light) sets the autonomy requirement

The PID terms — what each fixes

Term Responds to Cures Costs
P ($K_p$) present error the bulk of the correction steady-state droop; overshoot if too high
I ($K_i$) accumulated past error steady-state droop (drives error to 0) lag; integral windup when saturated
D ($K_d$) rate of change of error overshoot / ringing (adds damping) amplifies sensor noise

Physical picture: P = spring, D = damper → P+D makes the error a damped oscillator; tune toward critical damping. Droop cure = integral, not more gain — high $K_p$ destabilizes (here, stable only for $0 < K_p < 2$).

Decision aid — "which idea do I reach for?"

Situation Reach for
Estimate the state from noisy sensors + a model Kalman / EKF (predict grows σ, update shrinks σ)
A drifting IMU pair it with an absolute sensor (star tracker, GNSS, radar) via the filter
Steer an ascent to an exact orbit despite dispersions closed-loop PEG (recompute steering + cutoff)
Hit a point at a time (rendezvous, injection) Lambert / velocity-to-be-gained targeting
Hold an attitude or rate against disturbances PID control loop
Persistent offset under a constant disturbance add the integral term
Overshoot / ringing add (or raise) the derivative term
Round-trip light-time > reaction time make the loop autonomous (onboard)

Sensors → estimation → actuation (the integrated loop)

  • Sensors: IMU (accel + gyro, drifts), star tracker (arcsec attitude), sun/Earth sensors, GNSS (LEO), DSN ranging/Doppler (deep space), landing radar / terrain-relative nav. Each is a shadow of the state.
  • Estimator: the EKF fuses them into state + covariance.
  • Actuators: thrust-vector control (gimbaled engine), RCS thrusters, reaction wheels/CMGs (Ch. 14), magnetorquers (LEO).
  • Architecture: nested loops — fast rate loop ⊂ attitude loop ⊂ slow guidance loop; gains scheduled as the vehicle changes.

Autonomy and "seven minutes of terror"

  • Autonomy level is set by light-time. Moon: ~1.3 s one-way (ground can supervise the slow loop). Mars: ~4–24 min one-way (≈28 min round trip) ≫ ~7 min EDL → fully autonomous, no help possible.
  • Autonomous loops demand radiation-hardened (Ch. 26), highly reliable/redundant (Ch. 32) computers, and graceful degradation (Apollo's 1202: shed low-priority tasks, keep guidance alive).

Numbers worth memorizing

  • Speed of light $c \approx 3.0\times10^{5}\ \text{km/s}$; Mars one-way light-time $\approx 4$–$24\ \text{min}$.
  • Kalman gain lives in $[0,1]$; fused variance uses inverse-variance addition ($1/\sigma_+^2 = 1/\sigma_{\text{pred}}^2 + 1/\sigma_{\text{meas}}^2$).
  • Discrete proportional loop $\theta_{n+1}=(1-K_p)\theta_n+\text{const}$ stable for $0
  • Loop rates: booster attitude tens of Hz; agile inner/rate loops hundreds–thousands of Hz.

Themes surfaced

  • Space is an unforgiving environment (theme 2): you survive by correcting relentlessly, and — past a few light-seconds — alone; the autonomous loop is the difference between a mission and a crater.
  • Orbital mechanics is beautiful (theme 3): the same $F=ma$ that describes motion, wrapped in feedback, lets you synthesize the physics you want (a virtual spring–damper); a thermostat and a Mars landing are the same loop.

Mission / astrotools additions this chapter

  • MDR: added the GN&C approach note (navigation / guidance / control / autonomy, per track).
  • Code: an illustrative PID controller helper — a teaching aid, not a canonical astrotools module (like Ch. 13's kalman_update); together they sketch a full navigate-then-control loop.