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This paper presents an automatic procedure to enhance the accuracy of the numerical solution of an optimal control problem (OCP) discretized via direct collocation at Gauss–Legendre points. First, a numerical solution is obtained by solving a nonlinear program (NLP). Then, the method evaluates its accuracy and adaptively changes both the degree of the approximating polynomial within each mesh interval and the number of mesh intervals until a prescribed accuracy is met. The number of mesh intervals is increased for all state vector components alike, in a classical fashion. Instead, improving on state-of-the-art procedures, the degrees of the polynomials approximating the different components of the state vector are allowed to assume, in each finite element, distinct values. This explains the pnh definition, where n is the state dimension. With respect to the approaches found in the literature, where the degree is always raised to the highest order for all the state components, our methods allow a sensible reduction of the overall number of variables of the resulting NLP, with a corresponding reduction of the computational burden. Numerical tests on three OCP problems highlight that, under the same maximum allowable error, by independently selecting the degree of the polynomial for each state, our method effectively picks lower degrees for some of the states, thus reducing the overall number of variables in the NLP. Accordingly, various advantages are brought about, the most remarkable being: (i) an increased computational efficiency for the final enhanced mesh with solution accuracy still within the prescribed tolerance, (ii) a reduced risk of being trapped by local minima due to the reduced NLP size, and (iii) a gain of the robustness of the convergence process due to the better-behaved solution landscapes.

Minimum-Lap-Time Planning (MLTP) problems, which entail finding optimal trajectories for race cars on racetracks, have received significant attention in the recent literature. They are commonly addressed as optimal control problems (OCP) and are numerically discretized using direct collocation methods. Subsequently, they are solved as nonlinear programs (NLP). The conventional approach to solving MLTP problems is serial, whereby the resulting NLP is solved all at once. However, for problems characterized by a large number of variables, distributed optimization algorithms, such as the Alternating Direction Method of Multipliers (ADMM), may represent a viable option, especially when multi-core CPU architectures are available. This study presents a consensus-based ADMM approach tailored to solving MLTP problems through a distributed optimization algorithm. The algorithm partitions the problem into smaller sub-problems based on different sectors of a track, distributing them among multiple processors. ADMM is then used to ensure consensus among the distributed computational processes. The paper also outlines specific strategies leveraging domain knowledge to improve the convergence of the distributed algorithm.The ADMM approach is validated against the serial approach, and numerical results are presented for both single-lap and multi-lap scenarios. In both cases, the ADMM approach proves superior for problem dimensions of 70k+ variables compared to serial methods. In planning scenarios with complex vehicle models on long track horizons, i.e., for problems with 1M+ variables, the efficiency gain of the ADMM approach is substantial, and it becomes the only viable option to maintain computational times within acceptable limits.