Lecture 9 | MIT 6.832 (Underactuated Robotics), Spring 2019

The lecture provides a rigorous and insightful introduction to computational Lyapunov function synthesis using sums of squares (SOS) optimization. The instructor builds upon previous material, clearly motivating the need for a scalable method to verify positivity of Lyapunov functions for all states. The transition from linear systems to nonlinear systems is well-structured, starting with the quadratic case and then generalizing to polynomial functions. The explanation of SOS decomposition is particularly clear, with a simple example illustrating how a polynomial can be rewritten as a sum of squares, thereby making its positivity evident. The instructor also appropriately notes the limitations of SOS, such as the existence of positive polynomials that are not SOS (e.g., Motzkin polynomial), which adds nuance to the presentation. The use of semidefinite programming as the underlying optimization tool is well-justified, and the mention of practical implementation in Drake is useful for students. However, the lecture assumes prior knowledge of Lyapunov theory and optimization, which may be challenging for beginners. The presentation is primarily theoretical, with limited discussion of real-world applications or numerical examples, which could enhance understanding. The sources are not explicitly cited within the lecture, but the course website is provided for further reference. Overall, the lecture is of high quality, offering a solid foundation for researchers and advanced students in control theory. The adéquation between title and content is excellent, as the lecture directly addresses computational methods for underactuated robotics. The main weakness is the lack of concrete examples or case studies, which could help illustrate the practical utility of the methods. Nevertheless, the lecture is a valuable resource for those interested in advanced control techniques.

The title accurately describes the content: a lecture on underactuated robotics, specifically focusing on computational methods for Lyapunov functions.

8/10

Lecture from MIT OpenCourseWare, presented by an expert in the field, with clear mathematical derivations and references to established methods (sums of squares, semidefinite programming). The content is rigorous and well-structured, though it is a lecture and not peer-reviewed.

Markers derived by PSI from the transcript: the creator did not define chapters.

• 00:00Introduction and recap of previous lecture on computational Lyapunov functions.
• 02:00Review of sampling-based linear programming approach for Lyapunov function search.
• 04:00Discussion of quadratic Lyapunov functions for linear systems and semidefinite programming.
• 06:00Introduction of the idea to extend to nonlinear systems using polynomial Lyapunov functions.
• 08:00Explanation of sums of squares (SOS) decomposition and its role in certifying positivity.
• 10:00Example of SOS decomposition for a simple polynomial and linear constraints involved.
• 12:00Discussion of limitations: not all positive polynomials are SOS, mention of Motzkin polynomial.
• 14:00Formulation of sums of squares optimization as a convex problem and implementation in Drake.

• Underactuated Robotics Course Website — Official course website providing lecture notes, assignments, and additional resources.

• Underactuated Robotics Course Website — Course materials align with the lecture content.