This website houses notes from my studies at the University of Liverpool. If you see any errors or issues, please do open an issue on this site's GitHub.
The Meanings of $\square$ Consider the epistemic meaning of $\square$: I know that $\phi$ is true. and also the following model: stateDiagram-v2 direction LR w1 --> w2:a w2 --> w1 w2:w2<br>p From this graph we have: $M,w_1\vDash \square_ap$ $M,w_1\nvDash p$ This is bad as agent $a$ knows something that is...
Refer to the slides, from 81-94 for this lecture as there are a lot of diagrams. Consider we have a map with objects in it. We can make a likelihood field by applying a Gaussian distribution to the edge of each object in the map. Modelling a Robot with 6...
Formula games are formal games between two players: The “game board” is the formula. The Yes player tries to show that the formula is true. The No player tries to show that the formula is false. There is always a winning strategy for one of the players. Formula Games for...
Morphology is concerned with the shape of structures. Morphological Erosion 1 The final image is a subset ($\subseteq$) of the original image. Allows objects to shrink and separates objects & regions. It is a pixel-by-pixel operation, sliding the structuring element (SE) at every pixel: If the origin of the SE...
The aim of the sensor model is to determine: \[P(z\mid x)\] what is the probability of a measurement $z$, given that the robot is in state $x$. Beam-Based Sensor Model Consider each beam from the robot’s sensors independently. A scan $z$ is made up of $K$ measurements: \[z=\{z_1, \ldots,z_K\}\] Individual...
Rejection Sampling If we have a function $f$ we can sample random values from this function using the following method: Sample $x$ from a uniform distribution $[-b,b]$. Sample $y$ from $[0,\max f]$ If $f(x)>y$ then keep the sample. Otherwise reject. Sampling vs Odometry Sampling Determines a predicted new pose based...
The proof system $\mathbf K$ has the following axioms: $T$ - All the (substitution instances of) validities of propositional logic. $\mathbf K$ - $\square(\phi\implies\psi)\implies(\square\phi\implies\square\psi)$. and the following two rules: MP - If you have derived $\phi$ as well as $\phi\implies\psi$, then derive $\psi$. Necc - If you have derived $\phi$,...
Exteroceptive Sensors This is a class of sensors that measure information about a robot’s external environment. They are characterised by a number of different attributes: Field of View & Range Every sensor has a region of space that it can cover: The width of that region is the field of...
Feedback Control So far we have seen open loop control. This is where we make an observation, determine a plan to reach the goal and then execute that plan: This doesn’t account for errors in the action of moving to the goal, or changes in the environment. graph LR input...
Pixels, Neighbours & Connectivity $N_4$ Connectivity Each pixel has 4 pixels directly around it. $N_8$ Connectivity This includes diagonally adjacent pixels, so all pixels surrounding a pixel. Structuring Element This is a mask that can be used to define pixel neighbours: It is a small matrix of odd index. 1s...