  Convex optimization is crucial in controlling legged robots, where stability
and optimal control are vital. Many control problems can be formulated as
convex optimization problems, with a convex cost function and constraints
capturing system dynamics. Our review focuses on active balancing problems and
presents a general framework for formulating them as second-order cone
programming (SOCP) for robustness and efficiency with existing interior point
algorithms. We then discuss some prior work around the Zero Moment Point
stability criterion, Linear Quadratic Regulator Control, and then the feedback
model predictive control (MPC) approach to improve prediction accuracy and
reduce computational costs. Finally, these techniques are applied to stabilize
the robot for jumping and landing tasks. Further research in convex
optimization of legged robots can have a significant societal impact. It can
lead to improved gait planning and active balancing which enhances their
ability to navigate complex environments, assist in search and rescue
operations and perform tasks in hazardous environments. These advancements have
the potential to revolutionize industries and help humans in daily life.
