## 2 Related Work  
In this section, we provide a brief overview of previous works on person-following robots and target person trajectory prediction. One of the earliest person-following robot techniques reported in 1998 integrated a color-based method and a sophisticated contour-based method to track a person using a camera [8]. Subsequent works studied robots that follow the target from behind using a visual sensor [9] or 2D laser scanners [10], a robot that accompanies the person side-by-side using a LiDAR sensor [11], and a robot that acts as a leader in airports to guide passengers from their arrival gate to passport control [12]. Robotics and computer vision are fields in which technological improvements emerge every day. Several tracking methods for mobile robots based on various sensing sensors for people and obstacle detection have been reported. These methods include the use of 2D laser scanners in indoor environments [13] and in both indoor and outdoor environments [10]. Range finders provide a wide FoV for human detection and tracking [11]. However, such range sensors provide poor information for people detection compared to visual sensors. RGB-D cameras such as Orbbec Astra [7] and Kinect [14] are visual sensors that have been adopted in recent years and are currently available on the market. These cameras are suitable for indoor environments, easy to use on robots, and provide synchronized depth and color images in real time. In this study, we used an Orbbec Astra RGB-d camera to track the people and the LiDAR sensor to detect obstacles and navigate when the target is lost. Several human-following robot systems have been proposed in different outdoor or indoor environments in recent years. However, there have been a few attempts to address the target recovery from a failure situation using the online trajectory prediction. [PERSON] et al. [15] used a pre-trained appearance model to detect and track a person. If the person is lost, the robot continues to navigate to the last observed position (LOP) of the target person and then rotates 360\({}^{\circ}\) until it finds the person again. However, the recovery task fails if the robot cannot find the person after reaching the LOP and completing the 360\({}^{\circ}\) rotation. [PERSON] and [PERSON] [16] proposed a specific person detection method for mobile robots in indoor and outdoor environments in which the robot moves toward the LOP continuously if it loses the target. Our previous work [7] adopted a deep learning technique for tracking persons, used a color feature to identify the target person, and navigation to the LOP, then a random searching to recover the person if he/she is completely lost. [PERSON] and [PERSON] [17] used two 3D LiDAR sensors together with AdaBoost and a Kalman filter to generate points could for people detection in outdoor environments. To recover the target, they used two electronically steerable passive array radiator (ESPAR) antennas as a receiver and a transmitter to estimate the position of the target, then started the searching operations. However, this method has poor accuracy for human detection. [PERSON] et al. [18] proposed an approach based on an online convolutional neural network to detect and track the target person. They used trajectory replication-based techniques to recover missing targets. However, the aforementioned systems used simple methods to recover the target person, such as navigation to the LOP, random searching, or rotation. These methods usually fail if the target is not nearby the LOP or take a long time to recover the target compared to the methods, which predict a target's trajectory. [PERSON] et al. [19] developed a recovery function based on a logarithmic function. However, this method is not compatible with environments involving multiple people. [PERSON] et al. [20] adopted Euclidean clustering and histogram of oriented gradient (HOG) features with a support vector machine (SVM) classifier to detect the legs of persons.