Geometry and Inverse Kinematic at the MP3R Mobile Systems

Corresponding Author: Florian Ion Tiberiu Petrescu ARoTMM-IFToMM, Bucharest Polytechnic University, Bucharest, (CE), Romania E-mail: scipub02@gmail.com Abstract: The development and diversification of machines and mechanisms with applications in all fields require new scientific researches for the systematization and improvement of existing mechanical systems by creating new mechanisms adapted to modern requirements, which involve increasingly complex topological structures. The modern industry, the practice of designing and building machinery is increasingly based on the results of scientific and applied research. Each industrial achievement has backed theoretical and experimental computer-assisted research, which solves increasingly complex problems with advanced computing programs using an increasingly specialized software. The robotization of technological processes determines and influences the emergence of new industries, applications under special environmental conditions, the approach of new types of technological operations, manipulation of objects in the alien space, teleoperators in the top disciplines like medicine, robots covering a whole field greater service provision in our modern, computerized society. Movable, robotic, mechatronic mechanical systems have entered nearly all industrial spheres. Today, we can no longer conceive of industrial production without these extremely useful systems. They are still said to steal from people's jobs. Even so, it should be made clear that these systems create value, work in difficult, repetitive, non-pausing, high-quality work, without getting tired, without getting sick, without salary, and producing value who are paid and people left without jobs, so that they can work elsewhere in more pleasant, more advantageous conditions, with the necessary breaks. In other words, robots do not destroy people but help them in the process of work. Let us not remember the fact that in some environments people could not even work. In fact, the robot's profitability for work without stopping, repetitive, and qualitative, is no longer in question. In addition, there are many heavy operations that are absolutely necessary for the presence of robots. You can not create microchips with people directly without interposing the robot. Man can not directly work with objects of such small size. Neither difficult medical operations can be designed without robotic mechatronic systems. The most used robotic mechanical mechanical systems are the anthropomorphic ones in the class of serial systems. To this we have studied the direct kinematics in previous castings, and in this paper we are going to study the inverse kinematics.


Introduction
The development and diversification of machines and mechanisms with applications in all fields require new scientific researches for the systematization and improvement of existing mechanical systems by creating new mechanisms adapted to modern requirements, which involve increasingly complex topological structures.
The modern industry, the practice of designing and building machinery is increasingly based on the results of scientific and applied research. Each industrial achievement has backed theoretical and experimental computer-assisted research, which solves increasingly complex problems with advanced computing programs using an increasingly specialized software.
The robotization of technological processes determines and influences the emergence of new industries, applications under special environmental conditions, the approach of new types of technological operations, manipulation of objects in the alien space, teleoperators in the top disciplines like medicine, robots covering a whole field greater service provision in our modern, computerized society.
Movable, robotic, mechatronic mechanical systems have entered nearly all industrial spheres.
Today, we can no longer conceive of industrial production without these extremely useful systems. They are still said to steal from people's jobs. Even so, it should be made clear that these systems create value, work in difficult, repetitive, non-pausing, high-quality work, without getting tired, without getting sick, without salary, and producing value who are paid and people left without jobs, so that they can work elsewhere in more pleasant, more advantageous conditions, with the necessary breaks. In other words, robots do not destroy people but help them in the process of work.
Let us not remember the fact that in some environments people could not even work. In fact, the robot's profitability for work without stopping, repetitive, and qualitative, is no longer in question. In addition, there are many heavy operations that are absolutely necessary for the presence of robots. You can't create microchips with people directly without interposing the robot. Man can not directly work with objects of such small size. Neither difficult medical operations can be designed without robotic mechatronic systems.
The most used robotic mechanical mechanical systems are the anthropomorphic ones in the class of serial systems. To this we have studied the direct kinematics in previous castings, and in this paper we are going to study the inverse kinematics.

Materials and Methods
Inverse kinematic manipulators and serial robots will be exemplified for the 3R cinematic model (Fig. 1). In the inverse kinematics, we already know the direct relation relations (1) and we have to determine the inverse relations, ie to determine the independent rotation 10 20 30 , , φ φ φ of the three movable elements, depending on the kinematic parameters imposed to the enforcer x M , y M , z M , known (given, imposed). With the determined independent angles, then the relative rotations corresponding to the movements of the three drive motors in the rotation couplers (drives of the actuators) will be located: The fixed coordinate system was denoted by x 0 O 0 y 0 z 0 . The mobile systems (rigidized) of the three mobile elements (1, 2, 3) have indices 1, 2 and 3. Their orientation has been chosen conveniently.
The system (1) It is desirable to solve the system (1') directly by obtaining independent exact solutions.
The first step is the multiplication of the equation (1.1) with 10 sinφ − and of the relation (1.2) with cosφ 10 , after which the two resulting expressions are obtained by obtaining the trigonometric equation (2) which is solved with the solutions (3), i.e. for the first independent parameter φ 10 trigonometric functions of cosine and sinus functions: When we want to get the value of an angle directly when we know sin and cos functions, we use the expression (4): 10 10 10 semn(sin ) arccos(cos ) The angle is given directly by the arctic function, and its sinus sign, which can be +1 or -1, sends the angle in its quadrant, in the top or bottom half circle.
At the next step we multiply equation (1.1) with 10 cosφ and relation (1.2) with 10 sinφ , we add the obtained expressions and obtain the trigonometric equation (5) With the notations (7) we obtain for the equation system (6) the direct and exact solutions (8); equations (6) take shape (6 '): The system (6 ') is written in the form (6' '): Equations (6'') rise to squares each and then add, yielding (6''') expression: sin φ and thus generating the solutions for the sin function.
With the two sin and cos phrases it is possible to calculate exactly the value of the angle, which will be given by the arctic, and will take over the upper semicircle for a positive sinus, and the inferior half circle for a negative sinus sign.
The algorithm can be resumed for the angle To make sure that all solutions satisfy the system simultaneously, trigonometric 30 φ angle values are extracted directly from the system (6 ''). Their expression depends directly on the value of the angle calculated at the previous step ( 20 φ ), but all values surely satisfy the system from which they were deducted:

Determination of Actuator Angular Speeds
We start from equation 2: We derive the equation (2) and obtain the relation (9): The angular velocity of the first actuator has the expression (11): From the derived (6") system we obtain the angular speeds of the other two actuators. Draw (6") and result the system (12) We multiply the first relationship of the system (12) The relationship (13) is written in the form (14) The relationship (14) is in the form (15) From (15) we explain the angular velocity of the second actuator, and we get the relation (16) Then we multiply the first relation of the system (12) The relationship (17) is written in the form (18) From (18) we explain the angular velocity of the last actuator, and we get the relation (19):   1  20  2  20  30  3  20  30 cos sin sin( ) The angular speeds of the three actuators will be further explained in the system (20) Several parameters must be calculated for their determination.

Discussion
The development and diversification of machines and mechanisms with applications in all fields require new scientific researches for the systematization and improvement of existing mechanical systems by creating new mechanisms adapted to modern requirements, which involve increasingly complex topological structures.
The modern industry, the practice of designing and building machinery is increasingly based on the results of scientific and applied research.
Each industrial achievement has backed theoretical and experimental computer-assisted research, which solves increasingly complex problems with advanced computing programs using an increasingly specialized software.
The robotization of technological processes determines and influences the emergence of new industries, applications under special environmental conditions, the approach of new types of technological operations, manipulation of objects in the alien space, teleoperators in the top disciplines like medicine, robots covering a whole field greater service provision in our modern, computerized society.
Movable, robotic, mechatronic mechanical systems have entered nearly all industrial spheres.
Today, we can no longer conceive of industrial production without these extremely useful systems. They are still said to steal from people's jobs. Even so, it should be made clear that these systems create value, work in difficult, repetitive, non-pausing, high-quality work, without getting tired, without getting sick, without salary, and producing value who are paid and people left without jobs, so that they can work elsewhere in more pleasant, more advantageous conditions, with the necessary breaks. In other words, robots do not destroy people but help them in the process of work.
Let us not remember the fact that in some environments people could not even work. In fact, the robot's profitability for work without stopping, repetitive, and qualitative, is no longer in question. In addition, there are many heavy operations that are absolutely necessary for the presence of robots. You can not create microchips with people directly without interposing the robot. Man can not directly work with objects of such small size. Neither difficult medical operations can be designed without robotic mechatronic systems.
The most used robotic mechanical mechanical systems are the anthropomorphic ones in the class of serial systems. To this we have studied the direct kinematics in previous castings, and in this paper we are going to study the inverse kinematics.

Conclusion
The inverse kinematics is the one that corresponds to the daily reality in which the robots are programmed to work in order to perform certain operations, to observe some imposed trajectories so that they move precisely to achieve and achieve the desired trajectory and all necessary kinematic parameters.

Acknowledgment
This text was acknowledged and appreciated by Dr. Veturia CHIROIU Honorific member of Technical Sciences Academy of Romania (ASTR) PhD supervisor in Mechanical Engineering and by Prof. BERTHOLD GRUNWALD, Past Director Mercedes Benz Daimler AG, Germany and Past Head Department of Automotive Engineering from Bucharest Polytechnic University, whom we thank and in this way.

Funding Information
Research contract: Contract number 27-7-7/1987, beneficiary Central Institute of Machine Construction from Romania (and Romanian National Center for Science and Technology).
All these matters are copyrighted.

Author's Contributions
All the authors contributed equally to prepare, develop and carry out this manuscript.

Ethics
This article is original. Authors declare that are not ethical issues that may arise after the publication of this manuscript.