## 1 Introduction  
At the Department of Production systems and Robotics, Faculty of Mechanical Engineering, Technical University of Kosice [1], we are developing our own concept of a robotic arm with a modular structure. The arm features variable DOF (degrees of freedom), but our priority is to fine-tune the practical 6 DOF. The kinematic arrangement of the robotic arm is serial, with multiangle curvatures between individual autonomous modules. The robot is referred to by its acronym of URM (unlimited rotation module). In contrast to common designs of serial kinematics in angular arrangement, the design of our modular robot features several specificities and peculiarities [2]. A few solutions similar to ours can even be found on the market. However, none of them contains as many innovations as the solution proposed by us. The advantages of the URM's modular setup include the full autonomy of each module (in terms of its mechanics, data, power), wireless power transmission between individual modules (achieved through wireless power modules similar to those used in wireless charging of modern mobile phones), that the entire setup does not include a cable harness, and that therefore each module can rotate indefinitely, without any angular constraint. Passive curvatures between the individual modules may have various angular values, and thus have a direct effect on the size and shape of the robot's workspace. Each module is energetically independently (each module contains powerful accumulators enabling the robot to function for a certain time when the feed from the mains is interrupted). Some of the main disadvantages of our modular design are the reduced load-bearing capacity of the robot's end effector, reduced power efficiency compared to the standard cable harness arrangement, more complicated steering, and the occurrence of singular states. It was exactly the last of the items mentioned that necessitated further investigation and finding a solution. It goes without saying that one of the main goals is to maximally reduce the costs of maintenance of such a modular system, which is in line with the solutions promoted under 14.0. This closely related issue has been dealt with by the authors in the paper [3]. The steering mechanism, that is to say, the inverse kinematics for our modular system, is described in our paper [4]. A partial goal for our team is to also attain sufficient repeatability of the position accuracy of the robotic modular arm. There are several scientific papers using the ISO 9283 standard [5]. One of the main research goals is to avoid singularities [6; 7] in the workspace in the process of selecting the end effector trajectories. The key role here is played by the Jacobian, both in its analytical and its geometrical form [8]. Evaluation of the Jacobian and determination of articulated dependencies are done in terms of vector quantities rather than through the joint angles approach. Various approaches and modified iteration methods are used here [9; 10], Quadratic [11]. Inverse solution of such sets of equations is a demanding processes. That is why it is an advantage to have the mechanism's workspace displayed also when the end-effector positions acquire singular states. We used the Monte Carlo method for mapping (visualizing) the workspace, especially due to its simplicity and frequent deployment in solving similar problems [12; 13; 14]. When this method is used, the computation time increases with the number of samples. The main problem is decreasing precision when the number of samples decreases. In some cases, the methods are given more accurate instructions by dividing the workspace into specific areas and those are computed separately [15; 16]. Alternately, improved algorithms yield more precise results where workspace boundaries are concerned [17]. Thus, some works using the Monte Carlo simulation focus on a rough estimate of the workspace.