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While parallel programming, particularly on graphics processing units (GPUs), and numerical optimization hold immense potential to tackle real-world computational challenges across disciplines, their inherent complexity and technical demands often act as daunting barriers to entry. This, unfortunately, limits accessibility and diversity within these crucial areas of computer science. To combat this challenge and ignite excitement among undergraduate learners, we developed an application-driven course, harnessing robotics as a lens to demystify the intricacies of these topics making them tangible and engaging. Our course's prerequisites are limited to the required undergraduate introductory core curriculum, opening doors for a wider range of students. Our course also features a large final-project component to connect theoretical learning to applied practice. In our first offering of the course we attracted 27 students without prior experience in these topics and found that an overwhelming majority of the students felt that they learned both technical and soft skills such that they felt prepared for future study in these fields. 

Nonlinear Model Predictive Control (NMPC) is a state-of-the-art approach for locomotion and manipulation which leverages trajectory optimization at each control step. While the performance of this approach is computationally bounded, implementations of direct trajectory optimization that use iterative methods to solve the underlying moderately-large and sparse linear systems, are a natural fit for parallel hardware acceleration. In this work, we introduce MPCGPU, a GPU-accelerated, real-time NMPC solver that leverages an accelerated preconditioned conjugate gradient (PCG) linear system solver at its core. We show that MPCGPU increases the scalability and real-time performance of NMPC, solving larger problems, at faster rates. In particular, for tracking tasks using the Kuka IIWA manipulator, MPCGPU is able to scale to kilohertz control rates with trajectories as long as 512 knot points. This is driven by a custom PCG solver which outperforms state-of-the-art, CPU-based, linear system solvers by at least 10x for a majority of solves and 3.6x on average. 

There has been a growing interest in parallel strategies for solving trajectory optimization problems. One key step in many algorithmic approaches to trajectory optimization is the solution of moderately-large and sparse linear systems. Iterative methods are particularly well-suited for parallel solves of such systems. However, fast and stable convergence of iterative methods is reliant on the application of a high-quality preconditioner that reduces the spread and increase the clustering of the eigenvalues of the target matrix. To improve the performance of these approaches, we present a new parallel-friendly symmetric stair preconditioner. We prove that our preconditioner has advantageous theoretical properties when used in conjunction with iterative methods for trajectory optimization such as a more clustered eigenvalue spectrum. Numerical experiments with typical trajectory optimization problems reveal that as compared to the best alternative parallel preconditioner from the literature, our symmetric stair preconditioner provides up to a 34% reduction in condition number and up to a 25% reduction in the number of resulting linear system solver iterations.