## 1 Introduction
One thus sets the weight of a Voronoi edge based on the travel time in an elevator and the traversal time of a passage. Using the weighted Voronoi graph, the robot finds the shortest path to the goal and follows the path. This paper considers a scenario where the robot detects and identifies an object utilizing on-board sensors. For instance, the robot can detect a nearby object utilizing its camera or Lidar. For object identification in the camera image, one can utilize real-time object identification methods, such as you only look once (YOLO) [21; 22; 23] or Fast R-CNN [24]. For real-time object identification using Lidar, one can apply classification approaches in [25; 26]. The classifier in [25; 26] classifies objects into car or pedestrian. How to identify objects using on-board sensors is not within the scope of our paper. If the robot detects an object (e.g., a human) while following a Voronoi edge, then it avoids the object utilizing reactive control laws. There are many papers on collision-avoidance control laws in 2D environments [27; 28; 29]. The authors of [27] addressed the dynamic window approach for collision avoidance. collision-avoidance controls in [27; 28; 29] did not consider the velocity of a moving object. In other words, collision-avoidance controls in [27; 28; 29] designed the robot's path assuming that all obstacles are static. Hence, collision-avoidance controls in [27; 28; 29] may not be suitable for avoiding a fast moving object. In our paper, we develop a reactive collision-avoidance control considering the velocity of a moving object explicitly. This distinguishes our paper from other papers which did not consider moving obstacles. For collision avoidance in the literature, obstacles were simplified as spheres or bounding boxes in [30; 31; 32; 33]. However, simplifying an object shape is not desirable, especially in the case where an object has arbitrary shapes. Our article develops reactive collision-avoidance controls which do not simplify an object shape. As far as we know, our paper is novel in addressing reactive collision-avoidance controls, which can avoid objects with arbitrary shapes. To the best of our knowledge, no paper in the literature handled how to make a robot follow the Voronoi graph while avoiding collision with other objects. Our paper integrates the proposed reactive collision-avoidance control with a path-tracking control, so that the robot can follow the Voronoi graph while avoiding collision with static map environments or (moving or static) objects with arbitrary shapes. To the best of our knowledge, the novelties of our paper can be summarized as follows:1. This paper addresses how to make the robot follow the building Voronoi graph, while avoiding collision with map environments as well as (moving or static) objects with arbitrary shapes. 2. This paper proposes a reactive collision-avoidance control, which achieves collision avoidance with map environments as well as (moving or static) objects with arbitrary shapes. We verify the effectiveness of the proposed approaches utilizing MATLAB simulations. This paper is organized as follows. Section 2 reviews the literature on robot path planning. Section 3 presents definitions and assumptions in this paper. Section 4 presents the path planner utilizing the building Voronoi graph. Section 5 introduces the reactive control for collision avoidance. Section 6 discusses how to make the robot follow the Voronoi path while avoiding collision. Section 7 provides MATLAB simulations to verify the effectiveness of the proposed approaches. Section 8 provides conclusions.