# Three-Dimensional Path-Following with Articulated 6 DoF Robot and ToF Sensors
## 1 Introduction  
One of the basic tasks in robotics is to lead the robot arm (for example, the arm of an industrial robot) or the robot itself (for example, in mobile robotics) from the start point to the target point without collisions. This involves making strategic decisions and defining a sequence of valid robot configurations (including position, orientation, and speed) that constitute the obstacle-free path leading to the desired destination and taking environmental constraints into account [1]. Planning of the route can be global or local. In global path planning, all spaces where obstacles exist are predetermined so the obstacle-free route can be calculated in advance [2, 3]. When the environment is unknown a priori, or it is not static, local path planning is needed to dynamically respond to the environment changes. Both approaches may also be combined, for example, by providing the reference path, calculated by the path planner first, and then, as the robot moves, by correcting it using local planning as the response to changes in the environment detected by the robot's sensors [4]. Recent studies in path planning, advanced inverse kinematics (IK), and sensor fusion present significant progress, addressing challenges in efficiency, adaptability, and precision [3]. Efficient and precise navigation often needs advanced sensors like IMUs (Inertial Measurement Units), LiDARs, radars, mono and stereo vision systems, etc. Data from various sensors are usually combined (which is called sensor fusion) to provide complex information about the state of the robot and the environment [3]. Then, advanced algorithms for calculating inverse kinematics task, path planning, obstacle detection, and avoidance are applied to allow the robot to reach the desired target position. When the robot moves, it usually follows the path calculated by some path planning algorithm; however, this path usually does not exist physically in the environment. It is \"virtual\",and various control techniques are applied to minimise the tracking error and, of course, to perform the task collision-free [5]. In particular, Artificial Intelligence (AI) methods are currently widely explored and applied with numerous successes for data processing and path planning [3; 6]. However, despite undoubtedly advantages and often excellent results, there are also drawbacks and limitations. For example, they usually require preparing and tunning the AI model, which is not an easy task, and gathering a large amount of training data. Thus, the training process, particularly of deep learning algorithms, is often time-consuming, requires a high level of AI expert knowledge, demands high computational power, and consumes a lot of energy [3; 7]. The additional drawback of some advanced methods are also the costs of some types of sensors, for example, 3D cameras and long-range LiDARs. This leaves a space for exploring simpler or cheaper solutions [8], especially when movement accuracy is not of the first importance. One of the simplest variants for performing the task of moving a robot from the start point to the target position is to move it along a line that is physically defined in the environment. In such a case, and if there is only one possible path to the target (i.e., the path does not split), only local planning is needed, and it is sufficient to just move the robot along the path. This approach is commonly known as \"line-following\" and is usually applied in mobile robotics. Typically, the line is drawn on the surface where the robot moves. Thanks to sensors, the robot controller detects if the robot pulls off the line to the left or to the right and corrects the robot movement by turning the robot to the direction of the sensor that detected the line, to keep the robot back over the line (Figure 1). Such an approach focuses on correcting the local robot position error relative to the path. In this scenario, although the robot moves in a 2-dimensional (2D) space, the problem is 1-dimenional.