### Basic Information

**Course Code**: SC651**Course Name**: Estimation on Lie Groups**Course Offered In**: Spring 2022**Instructors**: Prof. Ravi Banavar**Prerequisites**: MA 108 or SC 639. An understanding of Lie groups is not necessary as they are covered in the content.**Difficulty (on a scale of 5)**: 4

### Course Content

For the uninitiated, estimation refers to process of improving the knowledge of state of a system from the noisy sensor measurements (which need not be the complete state) and Lie groups can be seen as generalizations of the Euclidean spaces. In the course, the focus is to develop and understand state estimation algorithms for state spaces which are Lie groups. One example would be to estimate the orientation of a spacecraft (where the state space is the set of all possible rotation matrices) from unit vector measurements of fixed stars. From the website: Kalman filtering: Studying R. E. Kalman’s first two papers on linear filtering and prediction problems. Manifolds, Lie groups and bundles: Introduction to the machinery of manifolds and Lie groups, group actions, the adjoint and co-adjoint actions, the infinitesimal generator, fiber bundles Structure of estimators on Lie groups - Setting the framework for attitude estimation in satellites and spacecraft, Visual Simultaneous and Localization (VSLAM) in mobile robotics, working out the structure and design of observers in both cases. Extended Kalman Filters (EKF) on Lie groups - Understanding and developing the notion of linearization on a Lie group and the equivalent EKF.

### Feedback on Lectures

A mix of recorded lectures and live lectures were given. Several tutorial sessions were also conducted to help develop a better understanding of group theory and about Lie groups. The lectures were very interactive in nature. Professor would often schedule TA sessions on topics as needed and decided by the audience.

### Feedback on Evaluations

The evaluation was based on three assignments given. In the first one, a linear Kalman Filter was to be designed for different systems and present an exhaustive analysis of them. In the second one, we were asked to implement a gradient-based lie group estimator on SO(3) for the case when a spacecraft can measure two stars (Reference: https://arxiv.org/abs/1907.09234). In the third one, we had to present our understandings of a recent paper on VSLAM which employs an equivariant estimator (Reference: https://doi.org/10.1016/j.automatica.2021.109803). The assignments are very open ended and the problem statements are not very precisely given, however, contacting the professor and TAs clarified any doubts upon contacting. The grading was fairly lenient as some people who missed an assignment were not given below AB.

### Study Material and Resources

Several papers on the recent advances were discussed in the class.

- Stochastic Processes and Filtering Theory - A. H. Jazwinski, Elsevier 1970.
- Kalman Filtering Theory - A. V. Balakrishnan, Optimization Software Inc, 1970.
- Optimal Filtering - B. D. O. Anderson and J. B. Moore, Prentice Hall, 1979.
- Introduction to Mechanics and Symmetry - J. Marsden and T. Ratiu, SpringerVerlag, 1994.
### Follow-up Courses

None

### Final Takeaway

This is among the few courses which are related to differential geometry and control theory (sensor estimation). The prerequisites are fairly minimum and hence enable anyone to develop intuition for the advanced topics like lie groups.