In this lecture we will discuss the dynamical system and measurement models that underlie the common filter. The mathematical lenses of dynamical systems and probabilities help to model motions and noise. To motivate us, we will use a position tracking example. In order to track a moving object, the robot must model the dynamical system of motion. The dynamical system describes how the state of the object changes in time, as well as how the robot measures the state. In a simple example, the state, x, will be indexed by time steps, t. The state will be comprised of position v, in meters, and velocity dvdt measured in meters per second. Due to dynamics, the state changes in each time step. Going from the current time step t to the next time step t+1. This change is captured by A, the state transition matrix. Sometimes notated as pi. The state transition matrix combines state information to describe the state at the next step in time. The transition simplifies the current state to depend only on the previous state making our mathematical lives easier. With the state being in position in velocity, we know that the position must change in time based on the velocity. The state transition matrix captures this with the given formulation. The robot will not directly measure X unfortunately, but the robot may observe portions of x through it's sensors. This portion is labeled z, where the relationship between the state and measurement is given by the mixing matrix, c. For completeness, the term u is included. Which represents any external input not dependent on the state, x. We will not explore this extra term in this module, instead, it is set to zero. Creditly both X and Z contain noise even in this model. State X is noisy because the linear model does not capture all physical interactions. Observation Z are noisy because sensors contain noise in their measurements. Based on the dynamical model, we can construct a graph of the information that we receive in time. At any given time, we know the two pieces of information. The previous state, x of t-1, and the current measurement, z of t. With this information we want to compute the current state, x of t. Remember, we never trust a single value to represent our state in our measurement due to noise. To provide an estimate that captures this uncertainty, we will transform this dynamical model into a set of probability expressions. Given the state dynamics, the probability of the current state is conditioned on the probability of the previous state. Essentially, this means that our current state cannot move too far from our previously known state. A large movement would probabilistically be very unlikely. One tricky thing is to relate the sensor reading z to the true state x. Our sensors only give us a single measurement but we want a probability distribution. To do this, we estimate the probability of drawing the observation conditioned on where we are now, state x of t. For instance, seeing a ball right in front of us provides a very certain estimate with very little variance. Conversely, an observation made at a distance will have a wide variance since we are not sure exactly how far away it is. We will continue to use the Gaussian distribution as a nice mathematical model. For the common filter that means choosing a gas unit to model the state with mean and covariance notated by n. Now given both the dynamical system model and the probabilistic model, we can combine the two ideas. First, the linear dynamical system means that we can utilize the state transition matrix A to model the probability of the next state based on the current state. Additionally the probability of an observation z of t can be modeled with the relationship to the matrix c of the dynamical system model. We then append the notion of noise to the system in both measurements and observations through new m and new o. Applying the Gaussian probability distribution model, both the state and the noise are represented with means and variances. Before combining the knowledge of state evolution with the measurements made, we can consolidate the expressions based on certain properties of the Gaussian distribution. First we can apply the linear transformation through the matrixes A and C. Applying a linear transformation on a Gaussian distribution yields another Gaussian distribution with modified means and variances. Similarly, we can add two Gaussian distributions to form yet another Gaussian where the new mean and variance are the summation of the original two. In the next section, we will show how these two probability distributions can be fused to provide a better estimate of the true state.