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.
Validity in Modal Logic Like in propositional logic, some formulas are always true. These formulas are valid. We can write this as: $\vDash \phi$ Formally we write: $\vDash \phi$ if and only if $M,w\vDash \phi$ for every pointed model $M,w$. Modal Logic Validity Example Consider the formula: \[\phi=(\square p\wedge\lozenge q)\implies\lozenge...
Typical Motion Models Typically there are two types of motion model: Odometry-based Used when systems are equipped with proprioceptive sensors such as wheel encoders. Velocity-based (Dead reckoning) Used based purely on velocities and time elapsed, using no sensors. Classification of Sensors Proprioceptive Sensors Measure values internally, measuring direct feedback on...
Language of Multi-Agent Modal Logic Currently we can say $\square\square\phi$ “I know that I know that $\phi$”. If we want to say “I know that you know that $\phi$”, with multiple agents, then we can annotate our boxes to define who knows what. To define this, we can use the...
Modelling Actions Actions are never carried out with absolute certainty. To incorporate the outcome of an action $u$ into the current belief, we use the conditional PDF $P(x\mid u,x’)$: This term specified the PDF that executing action $u$ changes the state from $x’$ to $x$. Generating a PDF for Actions...
State Estimation Consider that we want to determine whether a door is open or closed: A robot obtains measurement $z$ from its sensor We need to determine: \[P(\text{open}\mid z)\] Diagnostic vs. Causal $P(\text{open}\mid z)$ is diagnostic: What is the state of the door given the evidence? $P(z\mid\text{open})$ is causal: What...
We can complete image transformations using matrix operations. Arithmetic Operations These are pixel by pixel operations: Addition/Subtraction Multiplication/Division Logical Operations We should scale the final output so that we don’t overflow/underflow. In MatLab we can use imadd, imsubtract etc. Affine Transformations These are combinations of: Translation \[\begin{bmatrix}x'\\y'\end{bmatrix}=\begin{bmatrix}x\\y\end{bmatrix}+\begin{bmatrix}x_i\\y_i\end{bmatrix}\] Scaling \[\begin{bmatrix}x'\\y'\end{bmatrix} =...
Binary Images Simplest type of images Each pixel can either be black or white The size of the image is: \[M\times N\text{ bits}\] where $M$ and $N$ are the rows and columns. A histogram for a 1 bit image only has two peaks for dark and light. Grayscale Images Generally...
Random Variables Discrete Random Variables A set $X$ which contains a finite number of values ${x_1,x_2,\ldots,x_n}$ $P(X=x_i)$ (or $P(x_i)$) is the probability that the random variable $X$ takes on the value $x_i$. Continuous Random Variables $X$ takes on values along a bell curve. $p(X=x)$ (or $p(x)$) is a probability density...
A possible world is a description of a possible state of the universe, or at least those parts of the universe that we care about. Possible Worlds Simple Example Suppose we only care about whether it is currently raining in Liverpool. Then there are two possible worlds: $w_1$ it is...
Instantaneous Center of Rotation In a connected drive system (like a car), all wheels have to move along a circle. If travelling forward, the circle is infinely large. The centre of this turning circle is the convergence of all of the axels. This point is the ICR (Instantaneous Centre of...