---
abstract: 'Stochastic Model Predictive Control has proved to be an efficient method to plan trajectories in uncertain environments, e.g., for autonomous vehicles. Chance constraints ensure that the probability of collision is bounded by a predefined risk parameter. However, considering chance constraints in an optimization problem can be challenging and computationally demanding. In this paper, we present a grid-based Stochastic Model Predictive Control approach. This approach allows to determine a simple deterministic reformulation of the chance constraints and reduces the computational effort, while considering the stochastic nature of the environment. Within the proposed method, we first divide the environment into a grid and, for each predicted step, assign each cell a probability value, which represents the probability that this cell will be occupied by surrounding vehicles. Then, the probabilistic grid is transformed into a binary grid of admissible and inadmissible cells by applying a threshold, representing a risk parameter. Only cells with an occupancy probability lower than the threshold are admissible for the controlled vehicle. Given the admissible cells, a convex hull is generated, which can then be used for trajectory planning. Simulations of an autonomous driving highway scenario show the benefits of the proposed grid-based Stochastic Model Predictive Control method.'
author:
- 'Tim Brüdigam$^{1}$, Fulvio Di Luzio$^{2}$, Lucia Pallottino$^{2}$, Dirk Wollherr$^{1}$, and Marion Leibold$^{1}$[^1][^2][^3]'
bibliography:
- './references/refs\_fulvio.bib'
- './references/Dissertation\_bib.bib'
title: '**Grid-Based Stochastic Model Predictive Control for Trajectory Planning in Uncertain Environments**'
---

[^1]: \*The authors gratefully acknowledge the financial support by the BMW Group within the CAR@TUM project.

[^2]: $^{1}$T. Brüdigam, D. Wollherr, and M. Leibold are with the Chair of Automatic Control Engineering at the Technical University of Munich, Germany. [{tim.bruedigam; dw; marion.leibold}@tum.de]{}

[^3]: $^{2}$F. Di Luzio and L. Pallottino are with the Department of Information Engineering at the University of Pisa, Italy. L. Pallottino is also with the Research Center E. Piaggio of the University of Pisa. [f.diluzio@studenti.unipi.it; lucia.pallottino@unipi.it]{}
