Dynamic Elements at MP3R

Corresponding Author: Florian Ion Tiberiu Petrescu ARoTMM-IFToMM, Bucharest Polytechnic University, Bucharest, (CE), Romania E-mail: scipub02@gmail.com Abstract: Mechatronic robotic systems are today widely used worldwide to ease human work, but especially where work is dangerous, in toxic, radioactive, chemical, explosive atmospheres, without air such as underwater or in the cosmos, or in places hard to reach the man. Robots can take the tedious repetitive work under any circumstances and they can perform a difficult operation for a long time, with no meal or rest breaks. Serial mobile mechanical systems are generally the most used mechatronic systems because they have good dynamics, high reliability and lower manufacturing cost with modest technologies. In general, anthropomorphic robotic structures are generally used in serial mechanical systems as they are more versatile, more economical, more reliable, more penetrating, faster and generally have a beautiful and innovative design. Anthropomorphic structures have been used for the first time in the automotive industry to facilitate human work, but also to replace it with repetitive, tiring, or toxic work. For this reason, the first anthropomorphic robots were manipulators and the following were dyeing robots in toxic environments, so that welding anthropomorphic, assemblies, those who checked technological lines and so on would still appear. Almost all operations in the automotive industry were automated based on anthropomorphic robots. For this reason, their study is today as necessary as ever for their continuous improvement. Anthropomorphic robots work at high speeds and therefore their dynamics is an extremely important issue. In this paper, we aim to present an original method of scientific, analytical study of the dynamics of the anthropomorphic mobile mechanical structures. Dynamics is the discipline that studies the real movement of a point, object, or a body, mechanical system ... The dynamic study attempts to capture the real movement of the studied object, as it is in reality. The movement of a body is derailed by the kinematic equations, the movement being generally studied by the kinematics, but when we are interested in the actual movements of a binding object, a dynamic motion study must be introduced. The dynamics besides the kinematics constrain the influence of the masses and forces on the movement of a body, as well as the elastic deformations, the inertial forces, or other external forces capable of influencing the movement of the body, including those caused by the bonds of the body, its object of that mechanism.


Introduction
Serial mobile mechanical systems are generally the most used mechatronic systems because they have good dynamics, high reliability and lower manufacturing cost with modest technologies. In general, anthropomorphic robotic structures are generally used in serial mechanical systems as they are more versatile, more economical, more reliable, more penetrating, faster and generally have a beautiful and innovative design.
Anthropomorphic structures have been used for the first time in the automotive industry to facilitate human work, but also to replace it with repetitive, tiring, or toxic work. For this reason, the first anthropomorphic robots were manipulators and the following were dyeing robots in toxic environments, so that welding anthropomorphic, assemblies, those who checked technological lines and so on would still appear. Almost all operations in the automotive industry were automated based on anthropomorphic robots. For this reason, their study is today as necessary as ever for their continuous improvement. Anthropomorphic robots work at high speeds and therefore their dynamics is an extremely important issue. In this paper, we aim to present an original method of scientific, analytical study of the dynamics of the anthropomorphic mobile mechanical structures.
Dynamics is the discipline that studies the real movement of a point, object, or body, mechanical system ... The dynamic study attempts to capture the real movement of the studied object, as it is in reality. The movement of a body is derailed by the kinematic equations, the movement being generally studied by the kinematics, but when we are interested in the actual movements of a binding object, a dynamic motion study must be introduced. The dynamics besides the kinematics constrain the influence of the masses and forces on the movement of a body, as well as the elastic deformations, the inertial forces, or other external forces capable of influencing the movement of the body, including those caused by the bonds of the body, its object of that mechanism (Fig. 1). Antonescu and Petrescu (1985;1989;Antonescu et al., 1985a-b;1986-19881994;1997;2000a-b;2001;Aversa et al., 2017a-e;2016a-o;Berto et al., 2016a-d;Cao et al., 2013;Dong et al., 2013;Comanescu et al., 2010;Franklin, 1930;He et al., 2013;Lee, 2013;Lin et al., 2013;Liu et al., 2013;Mirsayar et al., 2017;Padula and Perdereau, 2013;Perumaal and Jawahar, 2013;Petrescu, 2011;2015a-b;Petrescu and Petrescu, 1995a-b;1997a-c;2000a-b;2002a-b;2003;2005a-e;2011;2012a-b;2013a-b;2016a-c;Petrescu et al., 2009;2016;2017a-l).

Materials and Methods
In Fig. 1, the weight centers of the MP-3R system were represented. For each element, two elements were considered to be able to perform the calculations separately for the different directions of the parts of each element. Fig. 1: Geometry, kinematics and dynamics of an MP3R every elements' center of gravity Thus element 1 was separated into two parts O 0 O 1 with the center of gravity in G 1 and O 1 A with the center of gravity in G 1* . Element two was divided into two subelements: AO 2 with the center of gravity in G 2 and O 2 B with the center of gravity in G 2* . The last element (MP-3R's third element) was also reconsidered and divided into two sub-elements: BO 3 with the center of gravity in G 3 and O 3 M with the center of gravity in G 3* . All the centers of gravity positioned in the middle of the respective elements were considered to be the calculations, the elements being of the bar type (cylindrical or other shapes).
The dynamics of any system requires knowledge of the mechanical kinetic energy of the system. It is the starting point for the number one of determining dynamic calculations and relationships of any mechanical system. The problem with MP-3R systems is that they work spatially, so the kinetic energy of the system includes spatial elements (it can't fit only in a plan).
The Lagrange equation used has the known classical form (1): The most normal dynamic determination of a system is made using the Lagrange equations. From system (1) three different equations will be written. For this it is necessary to determine the kinetic energy equation of the considered system beforehand ( ) ( , ) k k q q ε ε = ɺ . In space, kinetic energy has six components (in the most general case) for each element: Three for linear velocities and three for angular velocities. In the case of linear velocities, rather than writing three kinetic energies (the same mass of the halved element and multiplied separately with the square of each scalar velocity component in the center of the mass), it is simpler to write only one resultant equation, i.e., to multiply half of the mass of the element respectively (in this case, each sub-item will be quoted as an element so that three elements will result in six) with the square of the absolute velocity of the considered element, determined (absolute velocity), in the center of the element. Thus, we will determine the absolute velocities in the mass centers of the elements and the squares of the absolute velocities, then together with the mechanical inertial moments and the squares of the angular velocities of the element determined on three movable axes (movable element) in rectangular shape (virtually a mobile, rectangular, solidarity coordinate system with each element is chosen). In the most general case for each of the six resulting elements, we will have maximum four expressions for the kinetic (mechanical) energy of the system. Next, the absolute velocities (and their squares) for each of the six system outputs (MP-3R) will be determined.
Further, the moments of mass (mechanical) inertia and the kinetic energy relations for each considered kinematic element (as already established there are six elements instead of three) will be determined.
For element 1, O 0 O 1 , determine the moment of mechanical inertia on the main axis, the only one that allows a rotation of the element (relation 11): The moment of mechanical inertia (mass) is denoted by J. It must be a special moment of geometric inertia, which is generally (correctly) noted with I. The moments of mass and geometric inertia always bind to each other through a physical-mathematical relationship. If the geometrical inertial moment is mainly used in the calculations of material resistance and machine tool design, mechanical mechanics, mechanics, mechanisms, robotics, motors, transmissions, (etc ...) the dynamic (physiological) study of the mechanisms and the components of the systems are made mandatory by the inertial masses in motion; the usual masses of the elements (denoted by m) are used in the translational movement and the inertial masses (denoted by J) have a determinant role in the rotation movement (of the system elements). There are mechanical (mass) inertial moments projected on a point, on an axis, or on a plane. The convention in mechanics and mechanisms is to generally use the moments of mass inertia projected at a point, usually the point being the center of gravity (mass or symmetry) of that element. For element 1, we use the center of gravity G 1 which, for the main z-axis of the element (which is the main axis of rotation) has the same (mechanical) inertial moment at any point of the axis (relation 11). For two rectangular axes x and y the inertial mass moment has the half-value (relation 12), for the most commonly used cases, when we have a cylindrical radial r 1 body. Another approximate relationship used for these inertial values when the body is long and very thin (when the radius is negligible in relation to the length) is the relation (13), where l 1 would be a1 if the radius r 1 would be negligible relative to the length a 1 . A more precise (general) relationship for this case would be (14): Next, we will only use the relation (12) because the systems studied have cylindrical elements with significant diameters (the approximate rays of the cylinders are large enough). If the shape of the element is not cylindrical, it can also be approximated by a cylinder.
For element 1, we have no rotation except for the z-axis. The kinetic energy of element one gets the form (15) (it is considered to be twice the kinetic energy): On element 1*, in the center of gravity G 1* the kinetic energy is written (16) On element 2 in center of gravity G 2 , the kinetic energy takes the form (17) On the element 2* in the center of gravity G 2* the kinetic energy takes the form (18 and 20): Intermediate relationships (19 and 21) are also used to determine the kinetic energies on the rotating element: Relationships (21) explain how to get expressions (19); Fig. 2, where you can see the two different rectangular triangles formed by the axes of point G 2* . The moments of mechanical inertia J 2* are known on the z 2* and x b axes, the inertial moment J 2 on the main axis of the element 2* and the inertial moment on the vertical axis y 2* but inclined towards the angle element (the element is located along the axis G 2*ya ): On element 3, in the center of gravity G 3 , the kinetic energy takes shape (22) where, the double of kinetic energy due to translation has the expression (23): The duplication of the kinetic energies due to the rotation of the element on the two axes is determined by the relations (24 and 25): On element 3*, in the center of gravity G 3* , the kinetic energy takes shape (27) and the final expression (31): where, double of the kinetic energy due to translation has the expression (28) The double of the kinetic energies due to the rotation of the element on the two axes is determined with the relations (29 and 30):

Results
The moments of the drive motors (variation of required moments of the three actuators) will be determined.
The kinetic energy of the entire system is written first, comprising the three elements each dissected in two (32). The kinetic energy relationship of the entire system (32) Lagrange equations of the second case have the known classical form (1): The expression (32) of the kinetic energy of the whole system is used. The independent parameters (the generalized coordinates) are written as (33). Q k represents the generalized forces (in us are even motor moments of actuators): ; ; The first derivative (relation 34) is the partial derivative of the total kinetic energy (of the whole system) to the independent parameter ω 10 (i.e., the partial kinetic energy of the system at the angular velocity of the first actuator is derived): ( ) The expression (34) There follows the partial derivative of the kinetic energy of the entire system with the independent parameter (36): The first Lagrange equation (of the three) can now be written as (37): By replacing the derivatives derived above in Equation (37) Next we repeat the previous procedure for the second element, partly deriving the total kinetic energy of the system in relation to the generalized coordinate (representing the angular velocity of the second actuator). The relationship is thus obtained (39) The resulting relationship (39) There follows a partial derivative of the kinetic energy of the system according to the angular displacement of the second actuator (41) Using the relations (40) and (41) The total kinetic energy of the system and the third element are now partially derived, partly deriving the total kinetic energy of the system from the generalized coordinate ω 30 (representing the angular velocity of the third actuator). The relationship is thus obtained (44) The absolute expression (44) The kinetic energy of the entire system is partly derived from the angular displacement of the third actuator and the expression (46) Using the relations (45) and (46) by introducing them into the Lagrange equation (47), the expression of the motor moment variation of the third actuator is obtained.

Discussion
By using the expressions (38), (43) and (48), the variations of the motor moments, actuator moments, can be determined for the entire operating range. It uses the angular displacements and angles determined in the first courses, values given in the form of functions (in direct kinematics), obtained from the studied relations (in the indirect kinematics), or determined by the conditions imposed on the end-effector to go through certain optimized trajectories preset), (review course 5). A dynamic synthesis can be made to optimize the choice of the three actuators.
So the motors are not dynamically influenced by the position of the first element, or more clearly, by the angle of rotation of the first element ( Fig. 1), the dynamic movement being influenced only by the positions of the second and third elements and by the angular velocities of the three actuators.

Conclusion
The work presents an analytical method for determination of dynamic parameters in a 3R robotics module.
The dynamics of any system requires knowledge of the mechanical kinetic energy of the system. It is the starting point for the number one of determining dynamic calculations and relationships of any mechanical system. The problem with MP-3R systems is that they work spatially, so the kinetic energy of the system includes spatial elements (it can't fit only in a plan).
The Lagrange equation used has the known classical form (1).

Acknowledgement
This text was acknowledged and appreciated by Dr. Veturia CHIROIU Honorific member of Technical Sciences Academy of Romania (ASTR) PhD supervisor in Mechanical Engineering.

Funding Information
Research contract: 1-Research contract: Contract number 36-5-4D/1986 from 24IV1985, beneficiary CNST RO (Romanian National Center for Science and Technology) Improving dynamic mechanisms.

Author's Contributions
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