# New Approach of the Variable Fractional-Order Model of a Robot Arm
## 1 Introduction  
Robots are becoming more and more popular every day. It is hard to find an industry where robots are not used. The most popular form of commonly used robots are robotic manipulators, which are employed in the industry to perform repetitive tasks, search and rescue missions [1], and inspections [2]. In the case of robot arms, the most popular approaches are derived from Euler-Lagrange equations [3]. Their complexity, expressed by many relevant parameters, is often difficult to estimate, forcing a search for new solutions. Another new approach to modelling dynamics is based on fractional calculus.  
The concept of fractional calculus is not new and has been known since 1695, but it has recently gained popularity as a tool generalizing the commonly known calculus. It is becoming more and more popular in different research areas, ranging from electrical engineering, electromagnetism, electrochemistry, electronics, mechanics, and rheology to biophysics and economy [4; 5; 6]. Fractional calculus is also used to solve real-life problems, like control of autonomous vehicles [7], or to model and predict new cases of the COVID-19 [8]. In many instances fractional calculus allows modeling of the dynamics of more complex processes than their integer counterparts [9], for example, in the description of the transient behavior of a robot arm. The fractional system perspective in the study of the PRR (planar parallel robot) with three DOFs (degrees of freedom) was presented in [10]. Fractional calculus also has a generalization; when order varies with time, it is called variable fractional-order (VFO) calculus [11; 12].  
The concept of variable fractional-order integrals, and differentials was first proposed in 1993 [13]. Some complex physical phenomena show variable fractional order integrators, or differentiator properties [14]. Variable fractional-order calculus was implemented inmany different fields. The increasing attention on theoretical analysis and physical modeling using VFO can be noticed. The experimental study of variable fractional-order differentiators and integrators was examined in [14]. Additionally, the viscoelasticity behavior of rubbery state polymers was modeled using variable fractional-order calculus [15]. The comparison of the prediction of the proposed VFO model and those previously published in the literature demonstrated the higher accuracy of the former. VFO calculus is top-rated in control problems. A study was conducted comparing variable fractional-order PID (VFOPID) controllers for different types of unstable order derivatives [16]. The VFOPID controllers opened new possibilities for shaping its transient characteristics [17; 18; 19]. The simulation results of the VFOPID controller applied to plants with delay showing its advantage over both traditional and fractional-order PID controllers [20]. The VFOPID also provides additional flexibility during the designing phase. Another version of controllers which can be enhanced by VFO is fuzzy PID controllers [21]. The variable fractional-order fuzzy PID (VFOFPID) controller was applied along with an active tuned mass damper to improve building structures' performance when subjected to horizontal ground motions [22].  
In the paper, we propose a simple model of a robot arm using a variable fractional-order difference equation. In the previous studies, the variable fractional-order discrete-time PD controller for this arm was proposed [16; 23] along with the analysis of order function selection [24]. The variable fractional-order backward difference is introduced in Section 2. Next, the description of the robot arm is provided along with a discussion concerning its mathematical models. Then, some rules for the choice of the differentiation order functions are given. Finally, measured and simulated step responses of the robot arm are presented.