The Study of the Dynamics of a Cebâşev Manipulator

: The paper presents a new method for determining the dynamic parameters of a Cebâşev conveyor mechanism, solvable in two steps. In the first step, the classic method of preserving the kinetic energy of the entire mechanism is used to determine the rotating mass of the Cebâşev mechanism denoted by J*, this being the dynamic rotating mass of the entire mechanism reduced to the driving element 1 (of the crank type; classical rotating mass is called the mechanical or mass moment of inertia). In the second step (with the help of a new method), the dynamic angular speed of the Cebâşev mechanism's crank is directly determined, by using the mechanism's kinetic energy conservation a second time. This eliminates the need to use a more laborious classical method in the second step, such as differential Newton, Lagrange of type 1, Laplace transform, or Fourier, a finite difference method.


Introduction
A definition of dynamical physics or mechanics says that it is a branch of mechanics (see mechanical meaning 1) that deals with forces and their relationship primarily to motion, but sometimes to the equilibrium of bodies (Merriam-Webster, 2022). Dynamics is the discipline that deals with actual motion in any field, being the most important motion, which is why the study of dynamics in mechanics, as well as in other fields, is very important. Today, however, dynamics cover a multitude of aspects, starting from forces in systems and going to new technologies and technological processes. A methodology for the flexible implementation of collaborative robots in intelligent manufacturing systems is presented in the paper (Giberti et al., 2022). A robot arm design optimization method using a kinematic redundancy resolution technique is presented in (Maaroof et al., 2021). Trajectory control of industrial robots using multilayer neural networks driven by iterative learning control can be found in the paper (Chen and Wen, 2021). Dynamic and friction parameters of an industrial robot with repeatability identification, comparison and analysis are other important aspects of dynamic and robotic processes in the industry (Hao et al., 2021). The impact of gravity compensation on reinforcement learning in goalsetting tasks for robotic manipulators is a relatively new problem in dynamic disciplines (Fugal et al., 2021). Another dynamic new aspect is the mechatronic redesign of a manual assembly workstation in collaboration with wiring assemblies (Palomba et al., 2021), which can be directly associated with new technological processes.
Another aspect of the dynamic process appears in (Yamakawa et al., 2021) through the development of a high-speed, low-latency, remote-controlled robotic manual system. Accessible educational resources for teaching and learning robotics (Pozzi et al., 2021) is also a dynamic aspect, but different from the physicalmechanical one that is of particular interest to us in this study. Dynamic identification of the parameters of a pointing mechanism taking into account joint play (Sun et al., 2021) is a basic dynamic process. The impact of cycle time and payload of an industrial robot on resource efficiency (Stuhlenmiller et al., 2021) is also an important aspect of dynamic processes. Today, adaptive position (or force) control of a robot manipulator (Gierlak, 2021;Geng et al., 2021), as well as trajectory control (Colan et al., 2021;Liu et al., 2021;Engelbrecht et al., 2021;Alizade et al., 2021;Scalera et al., 2021), are important dynamic processes.
There is no need to discuss the importance of the articulated robots studied in this study because today they represent 90% of the industrial robots used almost everywhere, these articulated robots have the task of making practically all the components of a machine, moving and manipulating, or others having the role of processing. Industrial robots are automatic, programmable machines 240 with multiple axes of motion that can move to perform a task. A motion axis is a joint in the robot body where a segment can move. For example, a three-axis robot can rotate at its base, move its arm up and down and rotate the handle at the end of the arm. They impress with their versatility, which allows them to be used in new types of application areas. Whether on the floor, ceiling, or wall-thanks to the integrated power supply and compact control system-such a wellestablished system offers maximum precision in the smallest spaces. The Safe-Robot functionality enables innovative automation concepts. Whether it is suitable for controlled atmosphere rooms, hazardous areas, hygienic or splash-proof design, it is always accurate and fast in every pattern and movement. Whether in dusty, wet, or sterile environments, such a robot achieves top performance in any production environment. Robots have penetrated everywhere, including the medical area and operating rooms . A special problem in the dynamic processes of robots (Petrescu and Petrescu, 2015a-b, 2016 is the study of their kinematics, closely related to dynamics. Another important dynamic aspect (Essomba, 2021;Miguel-Tomé, 2021) is the balancing of technological processes. The influence of forces (Petrescu, 2022) is actually what determines the dynamic, real operation of all mechanical processes, including robots. In this way, it is possible to control the trajectory of robots and/or spacecraft (or drones) (Alpers, 2021;Caruso et al., 2021;Ebel et al., 2021;Thompson et al., 2021;Vatsal and Hoffman, 2021;Al Younes and Barczyk, 2021;Pacheco-Gutierrez et al., 2021;Stodola et al., 2021;Raviola et al., 2021;Medina and Hacohen, 2021;Malik et al., 2021).
The Cebasev manipulator robot ( Fig. 1) described in the current work is a simple mechanism with planar movement, which can draw an almost straight line (Fig. 2) for a certain period of time with only one rotary motor that actuates the mobile element 1. AB, BT and DC are the length of the mobile elements 1, 2 and 3 and 1 (or 1), 2, (or 2) and 3 (or 3), are the angles that position the mobile elements in relation to the Ox axis.
This study will briefly present a new method for determining the dynamic parameters of the Cebâşev robot mechanism. The manipulator robot works in one plane ( Fig. 1), but has a support (pivotal column) that supports it and can rotate it 360 degrees.

Methods
The centers of symmetry (of mass) of the three mobile elements are considered to be positioned according to Fig. 3, where element 2, the connecting rod, has its center of symmetry right in coupling C due to the constructive dimensions indicated by Cebâşev. S1, S2 (or C) and S3 represent the centers of symmetry of the three mobile elements belonging to the Cebâşev robot. m 1 , m 2 and m 3 represent the masses of the three mobile elements each concentrated in the corresponding center of symmetry. JS1, JS2 (or JC) and JS3 represent the rotating masses of the three mobile elements considered each around the axis of rotation perpendicular to the plane of the robot in the corresponding center of symmetry. The initial relations of kinematic links between the kinematic elements of the mechanism, positions and speeds are established, to determine the speeds of mobile elements 2 and 3 depending on the known kinematic parameters of the first element, 1, driver (1-6): Practically, system (6) solves the connection between the speeds of the three moving elements of the Cebâşev mechanism.
Next, the kinetic energy conservation Eq. (7) is written on the entire mechanism for all mobile elements: One then expresses twice the kinetic energy of the entire mechanism, which is also a constant (8): The equation of conservation of double the kinetic energy of the entire mechanism is ordered (9): Then the previous equation is reduced to element 1 of the mechanism which is mobile and active (control and driving), by dividing the equation of twice the kinetic energy of the entire mechanism by the square of the angular velocity of crank 1 (10) After replacing the speeds written in the first equations, the double of the kinetic energy divided by the square of the angular speed of crank 1 takes the form (11) This equation is reduced to the simplified form (12) In dynamic models it is known that the entire mechanism can be replaced with a single crank 1 whose rotating mass will take the form (13): or with a disc (or crack) with the center of rotation in coupling A, whose rotating mass will have the form (14): 242 To avoid the difficulties of using in step 2 the Lagrange equation of type 1 or 2 (as the case may be), the Newton equation in differential form, or the use of the Laplace or Fourier integral, or the use of other differential equations to linearize the system, will be used in following a simple, original, direct method, which also expresses the conservation of kinetic energy the second time and in this way it will be possible to directly determine the dynamic angular velocity of crank 1 and the corresponding dynamic angular acceleration, directly by deriving the already obtained dynamic angular velocity (15), (Petrescu, 2022): The values of the angular velocities of mobile elements 2 and 3 (16) can now be found immediately: To determine the reduced (dynamic) angular accelerations of mobile elements 2 and 3, the values of the kinematic angular velocities are replaced in the respective kinematic equations with the dynamic ones already obtained. Then the dynamic kinematics of the Cebâşev mechanism is determined, by using the old kinematic equations, in which both the angular velocities and the angular accelerations will be replaced with their dynamic values. The calculation program and the simulations performed in Mathcad can be found in the appendix.

Results
The mechanical (mass) moment of inertia of the entire mechanism reduced to crank 1 in coupling A can be traced in the diagram in Fig. 4. The dynamic angular velocities and dynamic angular accelerations of the three mobile elements can be traced in the diagrams of Fig. 5-10. For a comparative study between the kinematic and dynamic values, the diagrams in Fig. 11-16 were drawn.
All calculations and drawn diagrams can be found in the appendix. As you can see, the dynamics of the mechanism determined in two simple steps allows the calculation and plotting of the dynamic parameters very simply, in just two steps, the first being a classic one obtained by conservation of kinetic energy and the second step being a modern one where kinetic energy conservation is also used a second time (Petrescu, 2022), thus avoiding the need to use classical laborious methods, such as Newton's differential equations, the Lagrange equation of 1st order (for this mechanism), the Laplace transform or the Fourier integral, or various finite difference methods.
The method is simple to implement on any type of controller, a PID controller being simply designed and adapted thanks to this new method proposed in the current work, a method that can be generalized (Petrescu, 2022).

Discussion
Even if the presented dynamic method is the most general, the simple stand the most efficient, but also the fastest, it can be improved, because its dynamic effect is reduced to the inertial masses of the elements (both translation and rotation), introduced off the complete kinematic energies, during rotation and translation for each mobile element of the robot machine, these replacing the effect of the inertial forces in the mechanism on the real dynamic movement, without taking into account also the dynamic effects due to the links between the elements, i.e., the effects imposed by the kinematic couplings. For the introduction of these effects, it is necessary to correct the known, input angular speed,, with a dynamic factor D, by amplifying the rotating input speed with the dynamic factor D due to kinematic couples. In this type of mechanism (modular group), which contains two moving elements linked together by a kinematic coupling C5 (of the fifth class) of rotation, the dynamic coefficient D imposed by the mechanism has the value given by formula 17 (Petrescu, 2012).
The dynamic diagrams (noted with DD) after the correction applied by taking into account the dynamic coefficient D, which also represents the influence of the couples, have the forms (17-28). For a comparative study between the kinematic and dynamic values, the diagrams in Fig. 23-28 were drawn.
It is easy to see that within the robot car, the couplings have a calming effect on the dynamic operation, reducing both the speeds and the angular accelerations of the mobile kinematic elements they connect.
All numerical simulations were performed with the help of the Mathcad program.
A professional method of determining the dynamic operation of a mechanism is a particular and laborious one, which determines the dynamics of a robot or a machine directly on the respective mechanism using its force diagram. Such a method, however, could only be presented separately, in another work.

Conclusion
The robot mechanism presented in the work is a pivoting column that supports and rotates at 360 degrees a Cebâşev planar robot.
The method is simple to implement on any type of controller, a PID controller being simply designed and adapted thanks to this new method proposed in the current work, a method that can be generalized.
As you can see, the dynamics of the mechanism determined in two simple steps allows the calculation and plotting of the dynamic parameters very simply, in just two steps, the first being a classic one obtained by conservation of kinetic energy and the second step being a modern one where kinetic energy conservation is also used a second time, thus avoiding the need to use classical laborious methods, such as Newton's differential equations, the Lagrange equation of 1st order (for this mechanism), the Laplace transform or the Fourier integral, or various finite difference methods.

247
The dynamic angular velocities and dynamic angular accelerations of the three mobile elements can be traced in the diagrams of Fig. 5-10.
For a comparative study between the kinematic and dynamic values, the diagrams in Fig. 11-16 were drawn.