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2 International Journal of Applied Mathematics and Computer Science

2 2025

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Probabilistic lane segmentation using a low-dimensional linear parametrization

Acuña Carlos, Arechavaleta Gustavo, Castelán Mario

International Journal of Applied Mathematics and Computer Science

2025

Vol. 35, no. 1

179--189

Lane detection is an important module for active safety systems since it increases safety and reduces traffic accidents caused by driver inattention. Illumination changes or occlusions make lane detection a challenging task, especially if the detection is performed from a single image. Consequently, this paper presents a probabilistic approach based on the Kalman filter, which uses information from previous image frames to estimate the lane that could not be detected in the current image frame, considering uncertainty in the prediction as well as in the detection. To this end, a principal component analysis of the segmented curvature is introduced with the purpose of dimensionality reduction, moving from a large dimensional pixel representation to a considerably reduced space representation. Furthermore, the proposed approach is compared with a fully connected pretrained CNN model for lane detection, demonstrating that the proposed method has a lower computational cost in addition to a smoother transition between lane estimates.

DMOC-based robot trajectory optimization with analytical first-order information

Villanueva-Piñon Carla, Arechavaleta Gustavo

International Journal of Applied Mathematics and Computer Science

2025

Vol. 35, no. 1

83--96

Discrete mechanics and optimal control (DMOC) is a numerical optimal control framework capable of solving robot trajectory optimization problems. This framework has advantages over other direct collocation and multiple-shooting schemes. In particular, it works with a reduced number of decision variables due to the use of the forced discrete Euler-Lagrange (DEL) equation. Also, the transcription mechanism inherits the numerical benefits of variational integrators (i.e., momentum and energy conservation over a long time horizon with large time steps). We extend the benefits of DMOC to solve trajectory optimization problems for highly articulated robotic systems. We provide analytical evaluations of the forced DEL equation and its partial differentiation with respect to decision variables. The Lie group formulation of rigid-body motion and the use of multilinear algebra allow us to efficiently handle sparse tensor computations. The arithmetic complexity of the proposed strategy is analyzed, and it is validated by solving humanoid motion problems.