Continuing the documentation of a self-study exercise of working through the Robot Mapping course offered by Autonomous Intelligent Systems lab at the University of Freiburg.

- Autonomous Intelligent Systems #1: Robot Mapping
- Autonomous Intelligent Systems #2: Robot Mapping
- Autonomous Intelligent Systems #3: Robot Mapping

**Note**: sheet 2 was skipped as it was just an example application of bayes theorem.

Before attempting the problem set (sheets) complete the slides + recording.

**Exercise 1.a**: Describe briefly the two main steps of the Bayes filter in your own words.

The 2 main steps consists of a predict step and an update step. The **predict step** propagates a belief of the state and error covariance through a system model to obtain an *a priori* estimate for the next time step. [1] The **update step** takes the *a priori* estimate and corrects/updates with a measurement, incorporating new information to obtain the *a posteriori* estimate.

**Exercise 1.b**: Describe briefly the meaning of the following probability density functions.

- \(p(x_t | u_t, x_{t-1})\) → distribution of the state at time,\(t\), given a control action applied from a state at time \(t-1\).
- \(p(z_t | x_t)\) → distribution of a given measurement at time \(t\), given the state at time \(t\).
- \(bel(x_t)\) →
*a priori*belief, and the current estimate of the distribution of the state at time \(t\).

**Exercise 1.c**: Specify the distributions that corresponds to the above mentioned 3 terms in the EKF.

- \(p(x_t | u_t, x_{t-1})\) → The mean is \(g(\mu_t, u_{t-1})\), while the covariance is the matrix
**Q**representing the process noise. - \(p(z_t | x_t)\) → The mean is \(h(\mu_t)\), while the covariance is the matrix
**R**representing the measurement noise. - \(bel(x_t)\) → \(\mu_t\) the state estimate at previous time step before putting through system model and covariance is \(\Sigma_t\).

**Exercise 1.d**: Explain in a few sentences all of the components of the EKF algorithm.

- \(\mu_t\) → state estimate at time \(t\), (n,1)
- \(\Sigma_t\) → covariance matrix at time \(t\), (n,n)
- \(\bar{\mu_t}\) → state estimate propagated through time using the process model., (n,1)
- \(\bar{\Sigma_t}\) → covariance matrix at time \(t\) updated with Jacobian,
**G**, and process noise,**Q**. (n,n) - \(g\) → function to update state space based on some motion/process/system model., (n,1)
- \(G_t^x \) → Jacobian of the motion matrix. (n,n)
- \(R_t\) → measurement noise, (m,m)
- \(h\) → measurement function, (m,1)
- \(H_t^x\) → Jacobian of h, (n,m)
- \(Q_t\) → process noise, (n,n)
- \(K_t\) → Kalman gain

**Exercise 2.a**: Derive the Jacobian matrix \(G_t^x \) of the noise-free motion function \(g\) with respect to the pose of the robot. Use the odometry motion model as in exercise sheet 1.

*Apologies for the sloppy hand writing. Will port to latex at some point*