# New Approach of the Variable Fractional-Order Model of a Robot Arm
## 3 The Robot Arm Description  
The presented solution is an element of building a comprehensive simulation environment for the robot's actuator in the form of an arm. The study of the system properties is primarily associated with the possibly precise reconstruction of the actual course of the arm's work. Mathematical models are used for this purpose. The model of the described arm does not consider the impact of the movement of the mobile platform, and it was assumed that all degrees of freedom would be taken away from it. The motion system of the presented arm is a simple open kinematic chain, in which the adjacent members relate to each other by means of a kinematic pair. The movement of individual links is independent, performed using dedicated harmonic gears with the ability to maintain a specific position. The mechanical structure of the robot arms allows for their further expansion. The modularity of the structure allows it to quickly change its configuration with no necessity to modify the operation of the entire system. The observation head can be replaced with a gripper or a specialized tool or supplemented with further links. The assumed possibilities of further expansion of the mechanical structure allow for its adaptation to the requirements of the future user, both in the defense and civil operations zone. The whole construction is controlled by a microprocessor system with implanted algorithms using original control procedures with controllers with variable non-integer orders of differentiation. They are a generalization of classical PD regulators. Regulators of this type can be included in the so-called adaptive regulators, where the variables are not so-called controller settings but differentiation and summation orders. Before commencing the work regulation process, appropriate executive systems had to be selected. During the numerical simulations, the following values of the manipulator parameter were assumed:  
* m1 = 1.0 [kg]--the weight of the drive systems;
* m2 = 7.0 [kg]--total weight of the manipulator;
* m = 0.5 [kg]--weights of the manipulator links;
* l = 0.5 [m]--the length of the manipulator links;* \(M_{0}(t)\), \(M_{1}(t)\)--driving torque. Selected ranges of displacements \(\varphi_{1}\), and \(\varphi_{2}\) of the manipulator links in the given time interval \(\Delta t\) were subjected to numerical analysis in the context of selecting the values and time courses of the drive torques required to complete the given trajectory of the manipulator links. Two basic cases of the initial positions of both links were considered. The calculations assumed the same time course, presented in Figure 1, for changes in the moments \(M_{0}(t)\), \(M_{1}(t)\), where:
* \(M_{0}\), \(M_{1}\)--fixed values of driving torque;
* \(\Delta t_{0}\)--start-up phase, an increase of drive torques to the assumed fixed values;
* \(\Delta t_{1}\)--phase of steady drive torques;
* \(\Delta t_{2}\)--phase of reduction of drive torques to zero;
* \(\Delta t_{3}\)--stop phase drive torques equal zero;
* \(\rm 1=0.5\) [m]--the length of the manipulator links. During the analysis, two prominent cases were studied (Figure 2a,b), mainly setting manipulators in the horizontal and vertical orientation. The first analyzed case is to set the manipulator in vertical position \(\varphi_{1}=\pi/2\), \(\varphi_{2}=\pi/2\) (Figure 2b) starting from horizontal position \(\varphi_{1}=0\), \(\varphi_{2}=\pi\) (Figure 2a) in time \(\Delta t=1\) [s]. Figure 1: The time course of the moment changes \(M_{0}(t)\), \(M_{1}(t)\). Figure 2: The extreme position of the manipulator’s arms: **(a)** folded; **(b)** maximum reach. From the application point of view, it is important to achieve the required position of the links with the minimum possible values of velocities and accelerations to stop the manipulator.