In practice, navigation of mobile robots in confined environments is often done using a spatially discrete cost-map to represent obstacles. Path following is a typical use case for model predictive control (MPC), but formulating constraints for obstacle avoidance is challenging in this case. Typically the cost and constraints of an MPC problem are defined as closed-form functions and typical solvers work best with continuously differentiable functions. This is contrary to spatially discrete occupancy grid maps, in which a grid's value defines the cost associated with occupancy. This paper presents a way to overcome this compatibility issue by re-formulating occupancy grid maps to continuously differentiable functions to be embedded into the MPC scheme as constraints. Each obstacle is defined as a polygon -- an intersection of half-spaces. Any half-space is a linear inequality representing one edge of a polygon. Using AND and OR operators, the combined set of all obstacles and therefore the obstacle avoidance constraints can be described. The key contribution of this paper is the use of fuzzy logic to re-formulate such constraints that include logical operators as inequality constraints which are compatible with standard MPC formulation. The resulting MPC-based trajectory planner is successfully tested in simulation. This concept is also applicable outside of navigation tasks to implement logical or verbal constraints in MPC.

翻译：在实际应用中，移动机器人在受限环境中的导航通常采用空间离散的代价地图来表示障碍物。路径跟踪是模型预测控制（MPC）的典型应用场景，但在此情况下，为避障问题构建约束条件具有挑战性。通常，MPC问题的代价函数与约束条件被定义为闭式函数，且常规求解器对连续可微函数处理效果最佳。这与空间离散的占据栅格地图相矛盾，因为栅格地图中每个栅格的值定义了占据状态所对应的代价。本文提出一种方法，通过将占据栅格地图重构为连续可微函数，并将其作为约束条件嵌入MPC框架，从而克服这一兼容性问题。每个障碍物被定义为多边形——即半空间的交集。每个半空间代表多边形的一条边，可用线性不等式表示。通过使用AND与OR逻辑运算符，可以描述所有障碍物的组合集合，进而构建避障约束。本文的核心贡献在于利用模糊逻辑，将包含逻辑运算符的约束重构为与标准MPC框架兼容的不等式约束。基于此方法构建的MPC轨迹规划器在仿真测试中取得了成功。该概念亦可推广至导航任务之外，用于在MPC中实现逻辑或语言描述的约束条件。