Bharath Gopalakrishnan received his doctorate in Computer Science and Engineering (CSE). His research work was supervised by Prof. Madhava Krishna. Here’s a summary of  Bharath Gopalakrishnan’s thesis,  Motion Planning Under Uncertainty: A Chance Constrained Optimization Approach:

Motion planning is an important component of robot navigation and manipulation, and it is crucial in applications where the robot operates in full or partial autonomy. Some examples of these are multi-robot exploration, autonomous vehicles, and robot manipulation. In most of these applications, the robot’s state is not directly observable and can only be inferred from erroneous observations. Such situations require the planning algorithms to be robust to the uncertainties in actuation and perception. This thesis investigates the motion planning problem under uncertainty, with specific applications in navigation and path tracking. Applications of these kinds extensively use mathematical routines that solve constraints defined over random variables and are popularly known as chance constraints. Solving for chance constraints is extremely difficult and often intractable. This is primarily due to the fact that non-linear constraints defined over random variables result in a distribution that generally has no parametric form. In this thesis, we focus on constructing mathematical frameworks for solving such intractable optimization routines. This thesis’s key contribution is that we propose a series of formulations that can solve for chance constraints under parametric and non-parametric uncertainties. To achieve this, we construct the mathematical framework systematically over a sequence of stages, wherein each stage improvises over the limitations of the previous. For the first stage, we start with simple sampling-based approaches like the scenario approach. Though conceptually simple, this approach brings an added sample complexity in generating scenarios of the constraint functions. To overcome this limitation of the scenario approach, we proceed to the second stage, where we derive closed-form algebraic equations that would act as a surrogate to the otherwise intractable chance constraint. The novelty in this proposed approach stems from the nature of these surrogate functions, which are simple polynomial functions(mostly quadratic) of control variables and hence greatly simplify complex chance constrained optimization routines. While the proposed framework at this stage did enjoy closed-form algebraic solutions, it relied on the fact that the underlying random variables belonged to parametric forms of distributions (especially Gaussian). Hence we then proceed to the final stage where we solve for chance constraints under non-parametric uncertainties. We construct the proposed algorithm on the possibility of representing arbitrary distributions as functions in Reproducing Kernel Hilbert Space (RKHS). We systematically exploit the kernel trick to reformulate chance constrained optimization routines, such that it is tractable and computationally efficient. We detail out all the approaches proposed in these three stages over five chapters in this thesis. We primarily apply the proposed formulations on two challenging problems in motion planning:(i) reactive collision avoidance of mobile robots in uncertain dynamic environments and (ii) inverse dynamics based path tracking of manipulators under perception uncertainty. In all these applications, the underlying chance constraints are defined over highly nonlinear and non-convex functions of the uncertain parameters, making it highly intractable. We also extend one of the proposed formulations to a multi-robot navigation scenario, where each agent is responsible for computing safe maneuvers, considering perception and ego uncertainty.