M1-T1 — Three Questions: Where Am I, Where Should I Go, How Do I Get There

Learning objectives

After this lesson, you should be able to:

• State the question that navigation, guidance and control each answer.
• Name the signals that pass between the blocks, with their symbols and units.
• Explain why the word navigation means different things in aerospace and in ground robotics, and use the aerospace meaning.
• Show that a final miss is the sum of one guidance term, one control term and one navigation term.

The intuition

Meet Hopper, the vehicle that carries this course. It is a small rocket lander on a test range: one gimballed engine, an inertial measurement unit (accelerometers and gyroscopes), an altimeter and radio range beacons around the range. Its job is a hop: lift off from the launch pad, fly 100 m downrange, and land on a target pad. Nothing about this job is special to rockets. A delivery drone, a ship docking, an aircraft on approach and a satellite pointing at a ground station all do the same three things.

1. Navigation answers where am I, and how am I moving? It turns raw sensor readings into an estimate of the vehicle's state: position, velocity, attitude and their rates.
2. Guidance answers where should I go, and what should I ask for next? It turns the estimated state and the target into a command: a point to reach, a velocity to hold, a direction to point.
3. Control answers how do I make the vehicle do that? It turns the command and the estimate into actuator signals: a gimbal angle, a throttle setting, a wheel torque.

The vehicle itself closes the loop. Its actuators move it, its sensors observe the result, and navigation turns the observations into the next estimate.

Look at which arrows carry estimates and which carry truth. Guidance and control never see the true state x. They see x̂, the navigation block's best guess. Everything that follows in this course is, in one way or another, a consequence of that drawing.

Three blocks, three kinds of mathematics

The split is not only tidy, it matches how the work is done. Navigation is estimation: it combines noisy measurements with a model of motion, and its mathematics is probability and filtering. Control is feedback: it makes a dynamic system follow a command despite disturbances, and its mathematics is stability and the frequency response. Guidance is geometry and optimization: it decides what to ask for given where the vehicle is and what it wants, and its mathematics is relative motion and optimal control. Different engineers usually own the three, and each can be tested alone: navigation against recorded sensor data, control against a simulated vehicle, guidance against a simulated target.

The blocks are not always separate programs. A predictive controller can compute a trajectory and the actuator signals in one optimization, and a missile autopilot is often designed together with its guidance law. The separation is a way of thinking that survives such mergers: even when one program does two jobs, you can still ask which job went wrong.

Two meanings of the word navigation

Ground robotics and aerospace use the word differently, and a reader of both will meet the clash. On this site's Navigation track, navigation means getting a mobile robot from A to B: maps, planners, obstacle avoidance. In aerospace, and in this course, it means only the estimation question: where am I? What the ground-robot community calls navigation is roughly this course's navigation plus its guidance plus some route planning. Neither use is wrong. When a page says navigation, check which one it means.

Planning also needs a place in this vocabulary. Choosing a route through a building, ordering ten delivery stops or finding a path around obstacles are decisions over long horizons and discrete choices. They sit above guidance, which works continuously in real time on the path or target it is handed. The Navigation track teaches that planning, and this course starts where it ends.

Navigation says where the vehicle is. Guidance says where it should be next. Control makes it so. The first and the third work from the estimate, never from the truth.

Definitions

State $x$

The true condition of the vehicle: position (m), velocity (m/s), attitude (rad) and angular rates (rad/s). Never observed directly.

Control input $u$

What the actuators are told to do: gimbal angle (rad), throttle (0 to 1), wheel torque (N m).

Disturbance $d$

Forces the designer does not command: wind, payload change, model error. Unknown in flight.

Measurement $z$

What the sensors report, with noise $v$. A function of the state, not the state.

Estimate $\hat{x}$

Navigation's best guess of the state. Its error is $\varepsilon = \hat{x} - x$, the same sign convention as in Probability for Robotics.

Command $r$

What guidance asks the controller to achieve, in the same units as the state it refers to (a position, a velocity, an attitude).

Target $p_T$

Where the mission wants the vehicle to end: a pad position, a waypoint, an orbit.

With these symbols each block is one equation. The vehicle and its sensors are the physical part:

The three blocks you design are:

Notice the arguments. Neither $G$ nor $C$ contains $x$. This is the most important fact of the loop, and the next section shows what it costs.

Derivation: where a miss comes from

Suppose Hopper is ordered to land on a pad at position $p_T$, and ends its flight at $x(T)$ at touchdown time $T$. The miss is $x(T) - p_T$. Which block caused it? Work along one axis. Three small, honest facts about the blocks are enough.

1. $$\hat{x} = x + \varepsilon$$

   This is the definition of the estimation error. The controller sees the left side, the world is the right side without the error.

2. $$\hat{x}(T) = r(T) + c$$

   A controller can only compare the estimate with the command. The most it achieves is to bring them together, up to a leftover $c$ (a lag, a saturated actuator, a gain that was too low). Call $c$ the control error. It is visible to the controller.

3. $$r(T) = p_T + g$$

   Guidance ends its command at the target plus an error $g$ of its own: a stale target, a wrong coordinate frame, a terminal condition that was never reachable. Call $g$ the guidance error.

4. $$x(T) - p_T = \hat{x}(T) - \varepsilon - p_T$$

   Rearrange the first line to get the truth, $x = \hat{x} - \varepsilon$, and subtract the target.

5. $$x(T) - p_T = (r + c) - \varepsilon - p_T = g + c - \varepsilon$$

   Substitute the second line for $\hat{x}(T)$, and then the third for $r(T) - p_T$. The miss is one term per block.

The last line is the only equation of this lesson that you must remember. It has three consequences.

• Control cannot remove the navigation error. Even with perfect guidance ($g = 0$) and a perfect controller ($c = 0$) the vehicle misses by $-\varepsilon$. It lands exactly where it believes the pad is, and the belief is wrong.
• The three errors can be separated only with the truth. The controller sees $c$ and guidance sees nothing, but $\varepsilon$ is the difference from a quantity that no on-board block has. Test ranges therefore record the truth separately: a surveyed marker, a motion-capture system, a simulator that owns the true state.
• The same miss has different cures. A miss of 4 m can come from any of the three terms, and the fix for each lives in a different block and in a different team.

The estimate, not the truth, closes the loop. A vehicle can only be as accurate as its navigation, and no amount of control quality can buy that back.

Lesson M1-T4: From Mission Requirement to GNC Budget turns this identity into a budget: it gives each term a size and combines them.

A worked example

Hopper flew three test flights, each ending about 4 m from a pad at 100.0 m along the range. The test range logs, for each flight, the final command $r$, the final estimate $\hat{x}$ and the true final position $x$ from a surveyed marker. Which block do you send each flight to?

1. Record three numbers per flight. Flight A: $r = 100.0$, $\hat{x} = 100.2$, $x = 96.2$ (metres). Flight B: 104.0, 104.1, 104.0. Flight C: 100.0, 96.1, 96.0.
2. Compute the four quantities. The miss is $x - p_T$. The three terms are $g = r - p_T$, $c = \hat{x} - r$ and $\varepsilon = \hat{x} - x$.
3. Check the identity. For flight A the miss is $96.2 - 100.0 = -3.8$ and $g + c - \varepsilon = 0.0 + 0.2 - 4.0 = -3.8$. The program below asserts that this holds for every flight.
4. Read the dominant term. Flight A: the estimate is 4.0 m ahead of the truth. Hopper flew to where it thought the pad was, and the fault is in navigation (a beacon coordinate error, a bad altimeter bias). Flight B: the command itself ended 4.0 m from the pad and everything downstream was accurate, so the fault is in guidance (a stale target, a coordinate frame mix-up). Flight C: the command and the estimate were right, but the vehicle was left 3.9 m short of the command, so the fault is in control (a saturated gimbal, too little gain).
5. Note what the control engineer would have seen. Looking only at on-board data, flights A and B look like good flights: the estimate ended on or next to the command. Only flight C looks bad from inside. That is why $\varepsilon$ and $g$ hide from the controller and need the truth to find.

The same bookkeeping as a program. It uses only the Python standard library, and the site's tests run it and compare its output with the output below.

PAD = 100.0   # position of the pad along the range (m)

flights = {
    # name: (final command r, final estimate x_hat, true final position x), all in metres
    'A': (100.0, 100.2, 96.2),
    'B': (104.0, 104.1, 104.0),
    'C': (100.0, 96.1, 96.0),
}

for name, (r, x_hat, x) in flights.items():
    miss = x - PAD       # what happened
    g = r - PAD          # guidance: where the command ended, relative to the pad
    c = x_hat - r        # control: how far the estimate was left from the command
    e = x_hat - x        # navigation: estimate minus truth
    assert abs(miss - (g + c - e)) < 1e-9
    terms = {'guidance': g, 'control': c, 'navigation': e}
    culprit = max(terms, key=lambda k: abs(terms[k]))
    print(f'flight {name}: miss {miss:+.1f} m = guidance {g:+.1f} + control {c:+.1f} - (navigation {e:+.1f})  -> {culprit}')

flight A: miss -3.8 m = guidance +0.0 + control +0.2 - (navigation +4.0)  -> navigation
flight B: miss +4.0 m = guidance +4.0 + control +0.1 - (navigation +0.1)  -> guidance
flight C: miss -4.0 m = guidance +0.0 + control -3.9 - (navigation +0.1)  -> control

Lab M1-L1: Who Missed the Pad? gives you the same three logs for a longer flight and asks you to name the block before you look at any plot.

In robotics

The loop is easy to find in real systems once you know what to look for. The names change, the questions do not.

Two practical habits follow from this lesson.

• Log the interface signals. The command $r$, the estimate $\hat{x}$ and, wherever you can get it, the truth $x$, are the three signals that let you split a miss into its three terms. Autopilot logs usually carry the setpoints and the estimator outputs; see UAV M6-T13: Flight logs and debugging for how to read them. A project that logs only the actuator outputs cannot tell navigation from guidance.
• Separate the blocks for testing. Run control on a simulated vehicle with the true state as input to see what the loop could do with perfect navigation. If the miss disappears, navigation is the problem. The same trick, turned the other way, isolates guidance.

The UAV track has already shown these blocks for a multirotor: UAV M5-T15: Guidance and control integration looks at the hand-overs between them, and UAV M6-T5: PX4 state estimation and EKF2 at the navigation block of PX4. The Navigation track's The Navigation Stack at a Glance draws the same chain for a ground robot, with planning added on top. This course goes under all three of those pages.

Common misconceptions

Key takeaways

1. Navigation estimates the state, guidance decides what to ask for, and control makes the vehicle do it. The vehicle and its sensors close the loop.
2. Guidance and control work from the estimate $\hat{x}$, never from the true state $x$.
3. The miss at the end of a flight is $g + c - \varepsilon$: one guidance term, one control term and one navigation term.
4. Control cannot repair a wrong estimate. A vehicle lands where it believes the target is.
5. Splitting a miss into its three terms needs the command, the estimate and the truth to be logged, so a test range records all three.
6. Aerospace navigation means estimation only. Ground robotics often uses the word for the whole job of getting from A to B.

References and further reading

Each source is marked free or paid. Free to read does not mean free to reuse: the course resources page lists every source with its licence.

• D. Ghose, NPTEL “Navigation, Guidance and Control” (Indian Institute of Science) , the first module. The closest single free course to this one in scope. Free to read.
• R. W. Beard and T. W. McLain, Small Unmanned Aircraft: Theory and Practice, Princeton University Press, 2012, the introductory chapter, which sets out the architecture of an unmanned aircraft system. Paid book; the authors' slides and code are free.
• K. J. Åström and R. M. Murray, Feedback Systems, 2nd edition, Princeton University Press, 2021 , chapter 1 (Introduction), for feedback in general. Free to read as the authors' PDF; the book is copyrighted.
• C. J. Dennehy, Best Practices for the Design, Development, and Operation of Robust and Reliable Space Vehicle GN&C Systems, NASA/TP-20230005922, 2023 . Lessons collected from real crewed and robotic programmes. Free to read; a US Government work.
• P. D. Groves, Principles of GNSS, Inertial, and Multisensor Integrated Navigation Systems, 2nd edition, Artech House, 2013, the introduction. The reference for the navigation block. Paid book.

Knowledge check

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What’s next

Related content

Nothing below is required to finish this lesson.

• Lab
  M1-L1: Who Missed the Pad?

  Hopper lands 4 m from its pad in three different flights. From the logged truth, estimate and command signals, decide in each flight whether navigation, guidance or control is responsible.

Other courses on this site treat neighbouring topics from their own angle:

• UAV M5-T15: Guidance and control integration
• UAV M1-T12: Complete UAV system architecture and data flow
• Navigation M1-T2: The Navigation Stack at a Glance
• Physical AI: The Physical AI stack

This course leaves the following to the pages that already teach them:

• The ground-robot navigation stack (costmaps, global and local planners)
• UAV hardware and companion-computer architecture