###### Abstract  
Planning an achievable trajectory for a mobile robot usually consists of two steps: (i) finding a path in the form of a sequence of discrete waypoints and (ii) transforming this sequence into a continuous and smooth curve. To solve the second problem, this paper proposes algorithms for automatic dynamic smoothing of the primary path using a tracking differentiator with sigmoid corrective actions. Algorithms for setting the gains of the differentiator are developed, considering a set of design constraints on velocity, acceleration, and jerk for various mobile robots. When tracking a non-smooth primary path, the output variables of the differentiator generate smooth trajectories implemented by a mechanical plant. It is shown that the tracking differentiator with a different number of blocks also generates derivatives of the smoothed trajectory of any required order, taking into account the given constraints. Unlike standard analytical methods of polynomial smoothing, the proposed algorithm has a low computational load. It is easily implemented in real time on the on-board computer. In addition, simple methods for modeling a safety corridor are proposed, taking into account the dimensions of the vehicle when planning a polygon with stationary obstacles. Confirming results of numerical simulation of the developed algorithms are presented. autonomous wheeled vehicles; trajectory planning; path smoothing; tracking differentiator; design constraints; sigmoid function +
Footnote †: journal:  
0000-0001-5000-0000-0000-0000-0000-0000-0000-00000-00000-00000-00000-00000-00000-00000-00000-00000-00000-00000-00000-00000-000000-0  
reference trajectory has special points where the design constraints on vehicle velocity, acceleration, or jerk are violated, then spikes in control torques should be expected at these points. If such points occur regularly, this will lead to premature wear of the actuator. When tracking a non-smooth trajectory, the robot may stray from the route and have an accident. Thus, in trajectory control problems, the most important problem is to design smooth trajectories with continuous curvature that satisfy the design constraints of a particular wheeled robot and are achievable [5]. Planning an achievable trajectory for a mobile robot usually consists of two steps: (i) finding a path in the form of a sequence of discrete waypoints and (ii) converting this sequence into a continuous and smooth curve considering the movement time. Currently, many algorithms have been developed with different approaches that effectively solve the first problem depending on the reference conditions [6]. These include graph search algorithms (A* [7], etc. ), sampling-based algorithms (RRT [8], etc. ), reaction-based algorithms (DWA [9], etc. ), and their various modifications and combinations [10]. The subject of this research is the second problem. Interpolating curve algorithms are currently used to solve it [11]. They allow one to obtain an analytical description of the trajectory and, consequently, its derivatives of any required order. To obtain smooth curves, they use fragments of circles, polynomial splines [12], clothoids [13], generalized Cornu spirals [14], etc. A typical modern solution is B-splines [15]. The problem is reduced to the computation of polynomial coefficients of a stepped function of time (polynomial) that connects several neighboring points considering a specified time. For example, a smooth trajectory fragment can be obtained using a cubic B-spline and five reference points. To generate a trajectory subject to the design constraints of the mobile robot, additional algorithm or increasing the degree of the polynomial is required. For example, a polynomial curve of degree seven is used to constrain the first derivative in [16]. In addition, a smooth join of the separate fragments of the trajectory needs to be ensured. The implementation of analytical interpolation methods at the trajectory planning stage does not cause difficulties.