###### Abstract
However, they are not suitable for smoothing complex routes with a large number of reference points and fragments of different shapes in real time due to the large computational load and computation time [17]. In addition, the complex analytical trajectory can take up much memory when stored on the on-board computer. To solve this problem, we developed a universal dynamic smoothing method that automatically generates a smooth curve that satisfies the design constraints of a particular robot for any number of reference points. At the same time, complicating the shape of the reference curve does not lead to a more complex algorithm and increased computational operations. Only nominal values of vehicle velocity and acceleration, etc., are required for its presetting. The algorithm can be implemented both offline at the motion planning stage and online on an on-board computer with any primitive software. The main difference from the above-mentioned interpolating curve algorithms is that within the framework of the dynamic smoothing method, we obtain a smooth trajectory and its derivatives of any required order in the form of continuous vector signals without analytical description. It is possible to store sequences of waypoints in the on-board computer memory (it takes very little space) for different operational scenarios and activate their processing in real time. The complexity of the above advantages is provided via a dynamic tracking differentiator. In traditional applications, this computational algorithm is used to process the sensor signal of a state variable of the control plant and recover its first derivative. This allows no installation of velocity or acceleration sensors (they usually produce noisy signals [18]). The tracking differentiator is an autonomous system of second-order differential equations having a canonical input-output form [19; 20; 21; 22; 23]. The input corrective influence is formed as a nonlinear function of the differentiator variables and the external sensor signal to stabilize the tracking error (the uncertainty between the differentiator output variable and the sensor signal) and to filter out noise disturbances if they are present in the measurements. As a result, the state variables of the differentiator reproduce the useful measurement signal and its derivative, which are used in the feedback loop. In the deterministic case, discontinuous correction [21; 22] is used, and in the general case of noisy measurements, continuous analogs of the sign function (e.g., hyperbolic tangent [20]), more complex fuzzy logic methods [23], and others are used. In this paper, we propose to apply a tracking differentiator with continuous correction to track the initial non-smooth trajectory of a wheeled robot. First, we need to smooth the external signal subject to design constraints. Second, we recover its derivatives of any desired order. Increasing the dynamic order of the differentiator solves the second problem. To solve the first problem, we developed a special algorithm for the design of corrective influences with the generation of S-shaped sigmoidal local links. Sigma functions (modifications of the hyperbolic tangent) are smooth and bounded [24]. The gains that depend on the design constraints of the robot define the domain of their values. In a closed-loop system, the output variable of the differentiator tracks the external signal with some accuracy, which is a measure of the smoothing of the external signal at angular points. The other variables of the differentiator (derivatives of the output variable) \"track\" the introduced sigmoid functions and, therefore, do not move out of the specified ranges. Thus, the tracking differentiator automatically generates a smooth trajectory and its derivatives in real time with specified constraints. These signals are used in the tracking system of the control plant as reference achievable influences.