M1-T1 — Why Robots Need Probability

Learning objectives

After this lesson, you should be able to:

• Name the five sources of uncertainty in a robot and give an example of each.
• Explain what a probability means in two ways: as a long-run frequency and as a degree of belief.
• Distinguish a state, an estimate, an error and an uncertainty.
• Show with an expected-cost calculation why the most likely state can lead to the wrong action.

The intuition

Rover, the small two-wheeled robot of this course, is told to drive 2.00 m along a corridor. You mark where it stops and repeat the run ten times. The marks are spread over several centimetres. The command was the same every time, so the spread comes from things the command does not control.

None of these can be removed completely. A better sensor makes the spread smaller, not zero. So the honest description of what Rover knows after its drive is not “I am at 2.00 m” but “I am probably within a few centimetres of 2.00 m, and almost certainly within ten”. Probability is the language that makes that sentence precise.

What a probability means

A probability is a number between 0 and 1 attached to a statement. There are two ways to read it, and a robotics engineer uses both.

• Long-run frequency. “The probability that Rover docks is 0.9” means: repeat the drive many times under the same conditions and about 90% of the runs end on the dock. This reading describes the robot and its world. You can measure it with an experiment.
• Degree of belief. “The probability that Rover is in the kitchen is 0.9” is about one situation, now. Rover either is in the kitchen or is not; nothing is repeated. The number describes what the robot knows: how strongly the evidence it has supports the statement. New evidence changes the number even though the robot has not moved.

The two readings obey exactly the same rules (the next lesson states them), which is why the same mathematics serves for testing a robot over a thousand runs and for running a filter inside it.

Best guess and belief

Suppose Rover's range sensor says the door ahead is 0.60 m away. A program that stores the single number 0.60 has thrown away something it knew: that the sensor is good to about 10 cm, so the door might be at 0.50 m or 0.70 m. A program that stores a belief keeps the whole picture: every possible distance, with a weight that says how plausible it is.

A belief is not vagueness. It is a precise, numeric statement of what the robot does and does not know, and it is the input every decision in this course will use.

Definitions

State $x$

The true value of the quantities you care about: here the distance to the door; later a pose $(x, y, \theta)$, a velocity, a map. The robot never observes it directly.

Measurement $z$

What a sensor reports. It depends on the state, but it is not the state.

Estimate $\hat{x}$

A single number the robot computes as its best guess of the state.

Error $e = \hat{x} - x$

The difference between the estimate and the truth. It is one unknown number: if the robot knew it, it would subtract it.

Uncertainty

A description of the errors that are possible and how likely each is. It is known even though the error is not. A standard deviation of 0.10 m is an uncertainty.

Probability $P(A)$

A number in $[0, 1]$ attached to a statement $A$: 0 means “cannot be true”, 1 means “certainly true”.

Belief $\mathrm{bel}(x)$

The robot's probability for each possible state, given everything it has measured and commanded so far.

Written with the notation used in the rest of the course, the belief at time $t$ is a conditional probability: the probability of the state given the data.

Two kinds of uncertainty behave differently, and it is useful to keep them apart:

• Noise (also called aleatoric uncertainty) is the scatter that remains however much you know: the next range reading will differ from this one. Averaging reduces its effect; nothing removes it.
• Ignorance (epistemic uncertainty) is what you could know but do not: the wheel radius, where the robot was switched on, whether the door is open. More data shrinks it.

Derivation: when the best guess gives the wrong action

Why carry a belief at all? Because the action that is best for the most likely state is not always the best action. The argument takes four lines.

Rover approaches the door and must choose between two actions: continue at speed, or slow down. There are two possibilities for the world: the door is closer than the stopping distance (call this $H$, for hazard), or it is not. Rover's belief gives $P(H) = p$. Each action has a cost in each case:

1. $$\mathbb{E}[\text{cost} \mid \text{continue}] = p\,C_c + (1-p)\cdot 0 = p\,C_c$$

   The expected cost of an action is each cost multiplied by its probability, added up. It is the average cost per approach if the same situation were met many times.

2. $$\mathbb{E}[\text{cost} \mid \text{slow down}] = p\,C_s + (1-p)\,C_s = C_s$$

   Slowing down costs the same lost time whether or not the hazard was real.

3. $$\text{slow down when}\quad p\,C_c > C_s$$

   Choose the action with the smaller expected cost.

4. $$p > p^{*} = \frac{C_s}{C_c}$$

   Divide both sides by $C_c$. The threshold depends only on the ratio of the two costs.

Now compare with a robot that uses only its best guess. If $p < 0.5$ the most likely state is “no hazard”, and for that state the best action is to continue. So the best-guess robot continues whenever $p < 0.5$. The belief robot slows down whenever $p > C_s / C_c$. If a collision is fifty times worse than a lost second, that threshold is 0.02. For every belief between 2% and 50% the two robots disagree, and the best-guess robot is the one that eventually hits the door.

The best guess answers “what is most likely?”. A decision needs “what could happen, how likely is it, and what would it cost?”. Only a belief can answer that.

A worked example

Put numbers on the figure above. The range sensor reads $z = 0.60$ m and its noise has a standard deviation of $\sigma = 0.10$ m. At the current speed Rover needs 0.45 m to stop. A collision costs $C_c = 50$ and slowing down costs $C_s = 1$ (in any unit, for example seconds of delay).

1. Best guess. The estimate is 0.60 m and the stopping distance is 0.45 m, so there is a margin of 0.15 m. The best-guess robot continues.
2. Belief. Treat the belief about the true distance $d$ as a bell-shaped (Gaussian) curve centred on 0.60 m with standard deviation 0.10 m. Module 2 studies this curve; here we only need the area of its left tail.
3. Probability of the hazard. The hazard is $d < 0.45$. That is $(0.45 - 0.60)/0.10 = -1.5$ standard deviations below the centre. The area of a Gaussian below −1.5 standard deviations is 0.0668, so $p = 0.067$: about one approach in fifteen.
4. Threshold. $p^{*} = C_s / C_c = 1/50 = 0.02$.
5. Decision. $0.067 > 0.02$: slow down. In expected cost, continuing costs $0.067 \times 50 = 3.34$ per approach and slowing down costs 1.
6. How close may Rover get at this speed? The hazard probability falls to 0.02 when the reading is $0.45 + 2.054 \times 0.10 = 0.655$ m. Below a reading of 0.655 m the robot should already be slowing: a margin of about two standard deviations, not zero.

The same calculation with the course library. Every number above is printed by this program, and the site's tests run it.

from robotprob import gaussian_cdf, gaussian_ppf

z, sigma = 0.60, 0.10          # reading and its standard deviation (m)
stop = 0.45                    # stopping distance (m)
C_c, C_s = 50.0, 1.0           # cost of a collision, cost of slowing down

p = gaussian_cdf(stop, z, sigma)               # P(true distance < stopping distance)
p_star = C_s / C_c
print('P(hazard)          =', round(p, 4))
print('threshold          =', round(p_star, 4))
print('cost if continue   =', round(p * C_c, 2))
print('cost if slow down  =', round(C_s, 2))
print('decision           =', 'slow down' if p > p_star else 'continue')
print('slow below reading =', round(stop - gaussian_ppf(p_star) * sigma, 3), 'm')

P(hazard)          = 0.0668
threshold          = 0.02
cost if continue   = 3.34
cost if slow down  = 1.0
decision           = slow down
slow below reading = 0.655 m

In robotics

Every part of a robot's software that touches the real world carries uncertainty, and the rest of this course is about the tools each part uses.

Three concrete places you will meet it in practice:

• Messages carry it. In ROS 2 the pose and velocity messages used by localization (geometry_msgs/PoseWithCovariance, inside nav_msgs/Odometry) include a 6 × 6 covariance matrix next to the estimate. A consumer that ignores it is the best-guess robot of this lesson. Module 4 explains how to read that matrix.
• Flight controllers gate on it. A drone's state estimator compares each new measurement with what it expected and rejects one that is improbably far off, for example a GPS jump. “Improbably far” is defined by the estimator's own uncertainty. The UAV course describes this for PX4 and EKF2; modules 7 and 8 build the mathematics.
• Humanoids and manipulators need it at contact. Whether a foot is firmly on the ground or a part is seated in a fixture is never observed directly; it is inferred from noisy forces and joint positions, as a probability.

If you have followed the Physical AI lessons, State, observation, belief introduces the same three words without mathematics. This course supplies the mathematics.

Common misconceptions

Key takeaways

1. Uncertainty enters a robot through actuation, sensing, the environment, the model and the computation. It can be reduced but not removed.
2. A probability can be read as a long-run frequency (a property of the robot and its world) or as a degree of belief (a property of what the robot knows). Both follow the same rules.
3. The state is the truth, the estimate is a best guess, the error is their unknown difference, and the uncertainty describes which errors are plausible.
4. A belief assigns a probability to every possible state, given all the data so far. It is the quantity every filter in this course computes.
5. Good decisions weigh each possible outcome by its probability and its cost. The most likely state alone can recommend an action that is, on average, much worse.

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.

• S. Thrun, W. Burgard and D. Fox, Probabilistic Robotics, MIT Press, 2005, chapter 1 (Introduction) and section 2.1. The standard textbook for this course. Paid book.
• C. Stachniss, “Probability Primer for Probabilistic Robotics” (video lecture) . Free to watch.
• T. D. Barfoot, State Estimation for Robotics, 2nd edition, Cambridge University Press, 2024 , chapter 1. Free to read as the author's PDF; the book is copyrighted.
• J. Tsitsiklis and P. Jaillet, MIT RES.6-012 Introduction to Probability , lecture 1. Free videos and notes, licence CC BY-NC-SA 4.0.
• ROS 2 geometry_msgs/PoseWithCovariance message definition . Free documentation.

Knowledge check

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What’s next

You now have the vocabulary. The next lesson gives the three rules that every probability, frequency or belief, must obey.

Related content

Nothing below is required to finish this lesson.

• Lab
  M1-L1: Noisy Robot Motion Simulator

  The same command gives a different end position every run. Simulate noisy motion, measure the spread, and predict how often the robot stops inside the docking zone.

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

• Physical AI: State, observation, belief
• SLAM M2-T4: Probability and Uncertainty