The Inverse Kinematics of the Plane System 2-3 in a Mechatronic MP2R System, by a Trigonometric Method

Corresponding Author: Florian Ion Tiberiu Petrescu ARoTMM-IFToMM, Bucharest Polytechnic University, Bucharest, (CE), Romania E-mail: scipub02@gmail.com Abstract: Robots have penetrated today in almost all industrial fields, being much more precise than humans in the execution of operations, but also faster, more dynamic, more stable and more resilient, working 24 h of the 24 h possible, in any season, breaks, holidays, vacations and especially without getting sick. The most important fact is that they can also perform special operations that man can’t do. So, for example, they can work in toxic, contaminated, mined, dangerous, or airless environments (in the cosmos, under water or underground), they can also position the parts very precisely so they can be used for fine, special processing at high precision operations, excellent mechanical processing, exceptional medical operations and even in all medical operations today, helping the doctor manage the patient's surgery with amazing precision, especially when it comes to open heart, brain, kidneys, liver, etc. Robots should not be fed and yet they can have an extremely long life. They can help us conquer both the underwater spaces and the cosmic spaces. Robots have stunning work precision and a faster execution speed than a man. In addition, their very precise positioning makes robot operations a great advantage, which can no longer be neglected. In repetitive and tiring work they are irreplaceable. The most commonly used anthropomorphic robotic mechatronic systems, which are currently being used, have been studied by eliminating the heavy, matrix 3D spatial system, the study being simplified in a plan by considering the main work plan of the system and the plan, the rotation required to restore the spatial parameters of the anthropomorphic 3D system. In other words, we can greatly ease the work of the anthropomorphic robot engineer by moving from 3D systems to a 2D system. In this study we will study the inverse cinematics of the plan system, 2D, as the most important one. In general, end effector positions are known (required) and the positions of the 2D module elements and the necessary rotations of the actuators for their creation must be determined. In the kinematics we know the kinematic parameters xM and yM, which represent the scaled coordinates of the point M (endeffector M) and must be determined by analytical calculations the parameters φ20 and φ30.


Introduction
The most commonly used anthropomorphic robotic mechatronic systems, which are currently being used, have been studied by eliminating the heavy, matrix 3D spatial system, the study being simplified in a plan by considering the main work plan of the system and the plan, the rotation required to restore the spatial parameters of the anthropomorphic 3D system.
In other words, we can greatly ease the work of the anthropomorphic robot engineer by moving from 3D systems to a 2D system. In this study we will study the inverse cinematics of the plan system, 2D, as the most important one. In general, end effector positions are known (required) and the positions of the 2D module elements and the necessary rotations of the actuators for their creation must be determined. In the kinematics we know the kinematic parameters x M and y M , which represent the scaled coordinates of the point M (endefactor M) and must be determined by analytical calculations the parameters ϕ 20 and ϕ 30 . First we determine the intermediate parameters d and φ with already known (3) relations.
Robots have penetrated today in almost all industrial fields, being much more precise than humans in the execution of operations, but also faster, more dynamic, more stable and more resilient, working 24 h of the 24 h possible, in any season, breaks, holidays, vacations and especially without getting sick.
The most important fact is that they can also perform special operations that man can't do. So, for example, they can work in toxic, contaminated, mined, dangerous, or airless environments (in the cosmos, under water or underground), they can also position the parts very precisely so they can be used for fine, special processing at high precision operations, excellent mechanical processing, exceptional medical operations and even in all medical operations today, helping the doctor manage the patient's surgery with amazing precision, especially when it comes to open heart, brain, kidneys, liver, etc.
Robots should not be fed and yet they can have an extremely long life. They can help us conquer both the underwater spaces and the cosmic spaces.
Robots have stunning work precision and a faster execution speed than a man. In addition, their very precise positioning makes robot operations a great advantage, which can no longer be neglected. In repetitive and tiring work they are irreplaceable.
Today the moving mechanical systems are utilized in almost all vital sectors of humanity (Reddy et al., 2012). The robots are able to process integrated circuits (Aldana et al., 2013) sizes micro and nano, on which the man they can be seen only with electron microscopy (Lee, 2013). Dyeing parts in toxic environments, working in chemical and radioactive environments (Padula and Perdereau, 2013;Perumaal and Jawahar, 2013), or at depths and pressures at the deep bottom of huge oceans, or conquest of cosmic space and visiting some new exoplanets, are with robots systems possible (Dong et al., 2013) and were turned into from the dream in reality (Garcia et al., 2007), because of use of mechanical platforms sequential gearbox ).
The man will be able to carry out its mission supreme (Tang et al., 2013;Tong et al., 2013), conqueror of new galaxies (Flavio de Melo et al., 2012), because of mechanical systems sequential gear-box (robotics systems) (Garcia-Murillo et al., 2013).
Robots were developed and diversified (Lin et al., 2013), different aspects (He et al., 2013), but today, they start to be directed on two major categories: Systems serial (Liu et al., 2013;Petrescu and Petrescu, 2011b) and parallel systems . Parallel systems are more solid (Tabaković et al., 2013;Wang et al., 2013) but more difficult to designed and handled and for this reason, the serial systems were those which have developed the most. In medical operations or radioactive environments are preferred mobile systems parallel, because of their high accuracy positioning. Figure 1 shows the geometric-kinematic scheme of a base structure 3R.

Materials and Methods; the Trigonometric Method
From this platform you can study by adding any other modern n-R scheme. The platform (system) of Fig. 1 has three degrees of mobility, made by three actuators (electric motors) or actuators. The first electric motor trains the entire system in a rotation motion around a vertical axis O 0 z 0 . The motor (actuator) number 1 is mounted on the fixed member (bay, 0) and drives the mobile element 1 in a rotation motion around a vertical axis. On the mobile element 1, then all the other components (components) of the system are built.
There follows a planar (vertical) cinematic chain consisting of two movable elements and two kinematic motor couplings. It is the movable kinematic elements 2 and 3, the assembly 2,3 being moved by the second actuator mounted in the coupling A fixed on the element 1. Thus the second electric motor fixed by the element 1 will drive the element 2 in a relative rotation relative to element 1, but automatically it will move the entire kinematic chain 2-3.
The last actuator (electric motor) fixed by element 2 in B will rotate element 3 (relative to 2).
The rotation ϕ 10 made by the first actuator is also relative (between elements 1 and 0) and absolute (between elements 1 and 0).
The rotation ϕ 20 of the second actuator is also relative (between elements 2 and 1) and absolute (between elements 2 and 0) due to the positioning of the system.
The rotation θ = ϕ 32 of the third actuator is only relative (between elements 3 and 2), the corresponding absolute (between elements 3 and 0) being a function of θ = ϕ 32 and ϕ 20 .
The kinematic chain 2-3 (made up of moving kinematic elements 2 and 3) is a planar cinematic chain that falls into one plane or one or more parallel planes. It is a special cinematic system that will be studied separately. The kinematic coupler A (O 2 ) and B (O 3 ) become the first fixed coupler and the second movable coupler, both of which are C5 cinematic couplers, of rotation.
The kinematic chain 2-3 having the degree of mobility 2 must be driven by two motors.
It is preferred that the two actuators are two electric, DC, or alternating motors. The action can also be done with other engines. Hydraulic, pneumatic, sonic, etc.
The schematic diagram of the planar kinematic chain 2-3 ( Fig. 2) resembles its kinematic scheme.
The guide element 2 is connected to the fixed element 1 by the motor coupler O 2 and the drive element 3 is connected to the mobile element 2 by the motor coupler O 3 .
This results in a two-degree open cinematic chain made by the two actuators, ie the two electric motors mounted in the kinematic couplers A and B or O 2 or O 3 . Figure 3 shows the kinematic diagram of the open 2-3 chain (Petrescu, 2014).
In the triangle of some O 2 O 3 M we know the lengths of its three sides, d 2 , d 3 (constants) and d (variable), so that all the other elements of the triangle, namely its angles and the trigonometric functions of (we are particularly interested in sin and cos).
Different methods can be used to determine the angles ϕ 20 and ϕ 30 , of which two of them (as being most representative) will be presented: The trigonometric method and the geometric method.

Determination of Positions
Scalar position equations are written (2) The problem of this two scalar, trigonometric, two unknown equations (ϕ 20 and ϕ 30 ) is that they have been transcribed (they are trigonometric, transcendental equations, where the unknown does not appear directly ϕ 20 but in the form of cosϕ 20 and sinϕ 20 , so in reality in the two trigonometric equations we no longer have two unknown but four: cosϕ 20 , sinϕ 20 , cosϕ 30 and sinϕ 30 ).
To solve the system we need two more equations, so in the system (2) we have added two more trigonometric equations, namely the golden basic trigonometric equations as it is called for the angle ϕ 20 and separated for the angle ϕ 30 .
In order to solve the first two equations of the system (2) we write in the form (3) Each equation of the system (3) rises to square, then sums up both equations (raised to square) and obtains the equation of form (4) Now is the time to use the two trigonometric gold "equations of gold" written at the end of the system (2), with which Equation 4 gets simplified form (5) The terms of Equation 5 are arranged in the more convenient form (6) Insert (8) into (7) and amplify the right fraction with d so that the expression (7) takes the convenient form (9): Now is the moment of entering the cosine expression of the O 2 angle, depending on the sides of the triangle of some O 2 O 3 M (10): With relation (10) Equation 9 takes the simplified form (11): We want to eliminate sinφ 20 , which is why we have isolated the term in sin and square Equation 11, because by using the trigonometric gold equation of the angle φ 20 , we transform sin into cos, the equation becoming one of the second degree in cosφ 20 . After picking up the square (11) it takes shape (12): The gold formula is used so that the expression (12) takes the form (13) which is conveniently arranged by grouping the terms into the shape (14) The determinant of the second-degree Equation 14 just obtained, is calculated with the relation (15) The second order radical of the determinant is expressed as (16): The solutions of the second-degree equation in cosine (14) are written as (17) Next, in solutions (17) we replace the corresponding trigonometric angle of functions ϕ, the expressions (17) obtaining the form (18) The discriminative of the second-degree Equation 20 in cosine takes the form (21) The solutions of Equation 20 are written as (22) We obtained relations (23), from which the basic relationship is deduced (24) Repeat the procedure for determining the angle ϕ 30 , starting again from the system (2), in which the first two transcendental equations are rewritten in the form (25) Raises the two equations of the system (25) to the square and assembles, resulting in the equation of form (26), which is arranged in more convenient forms (27) and (28) We want to determine it first on the basket so that we will initially isolate the term in the sin, Equation 28 being in the form (29), which by square-up generates expression (30), the expression which is arranged in the form (31) We continue to write Equation 28 in the form (33), where the term is now isolated in the cosine for its elimination, in order to determine the sin: The relations (37) from which the expression (38) is deduced are also retained:

Velocities
From the system (2), only the relations (39) required in the study of velocities at the reverse kinematics are retained. It starts from the relationship that links the cosine of the angle to the sides of the triangle, a relationship that is time dependent and thus the simpler written value (relations 40) is obtained: The relationship (24) To determine ω 20 (relation 41) we need ϕ ɺ to be computed from (39) The relationship (38) is derived to obtain the angular velocity 30 30 ω ϕ ≡ ɺ , (expression 43). ϕ ɺ is calculated with the already known expression in the system (39) and M ɺ it is determined from the system (42) and with the help of the system (39) which determines also and d ɺ : Then the expression (41) is derived and the relation (46) generating the absolute angular acceleration 2 20 ε ε ≡ , which is calculated with ϕ ɺɺ the output from the system (44) and with 2 O ɺɺ extracted from the system (45) and for 2 O ɺɺ to be obtained it is also necessary d ɺɺ dermined also from (44): The relationship (43) is again obtained with time and the expression (48) of the absolute angular acceleration 3 30 ε ε ≡ which is determined with ϕ ɺɺ and M ɺɺ . ϕ ɺɺ it is determined from the system (44) and M ɺɺ withdrawn from the system (47) and needs and the d ɺɺ which is extracted from the system (44)

Discussion
Robots have penetrated today in almost all industrial fields, being much more precise than humans in the execution of operations, but also faster, more dynamic, more stable and more resilient, working 24 h of the 24 h possible, in any season, breaks, holidays, vacations and especially without getting sick. The most important fact is that they can also perform special operations that man can't do. So, for example, they can work in toxic, contaminated, mined, dangerous, or airless environments (in the cosmos, under water or underground), they can also position the parts very precisely so they can be used for fine, special processing at high precision operations, excellent mechanical processing, exceptional medical operations and even in all medical operations today, helping the doctor manage the patient's surgery with amazing precision, especially when it comes to open heart, brain, kidneys, liver, etc. Robots should not be fed and yet they can have an extremely long life. They can help us conquer both the underwater spaces and the cosmic spaces. Robots have stunning work precision and a faster execution speed than a man. In addition, their very precise positioning makes robot operations a great advantage, which can no longer be neglected. In repetitive and tiring work they are irreplaceable.
The most commonly used anthropomorphic robotic mechatronic systems, which are currently being used, have been studied by eliminating the heavy, matrix 3D spatial system, the study being simplified in a plan by considering the main work plan of the system and the plan, the rotation required to restore the spatial parameters of the anthropomorphic 3D system. In other words, we can greatly ease the work of the anthropomorphic robot engineer by moving from 3D systems to a 2D system. In this study we will study the inverse kinematics of the plan system, 2D, as the most important one. In general, end effector positions are known (required) and the positions of the 2D module elements and the necessary rotations of the actuators for their creation must be determined. In the kinematics we know the kinematic parameters x M and y M , which represent the scaled coordinates of the point M (endefactor M) and must be determined by analytical calculations the parameters ϕ 20 and ϕ 30 .

Conclusion
The most commonly used anthropomorphic robotic mechatronic systems, which are currently being used, have been studied by eliminating the heavy, matrix 3D spatial system, the study being simplified in a plan by considering the main work plan of the system and the plan, the rotation required to restore the spatial parameters of the anthropomorphic 3D system. In other words, we can greatly ease the work of the anthropomorphic robot engineer by moving from 3D systems to a 2D system. In this study we will study the inverse kinematics of the plan system, 2D, as the most important one. In general, end effector positions are known (required) and the positions of the 2D module elements and the necessary rotations of the actuators for their creation must be determined. In the kinematics we know the kinematic parameters x M and y M , which represent the scaled coordinates of the point M (endefactor M) and must be determined by analytical calculations the parameters ϕ 20 and ϕ 30 .

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.