Switching from Flat to Spatial Motion to 3R Mechatronic Systems

Corresponding Author: Florian Ion Tiberiu Petrescu ARoTMM-IFToMM, Bucharest Polytechnic University, Bucharest, (CE), Romania E-mail: scipub02@gmail.com Abstract: The anthropomorphic robots are part of the classical series of mechatronic systems, being in the form of arms and having at least three space rotation, to which other components may eventually be added, thus lengthening the entire kinematic chain. You can also add all the planetary or spatial rotating arms or others that are translating. At the end we always have the end effector element that can be a manipulator, that is, a hand to grasp the objects, in which case one can speak of a prehensive device, that is a gripping device that today imitates very well a human hand even if it is one mechanical, may also be a painting, cutting or welding device, or one for machining. The base support and schematics of all anthropomorphic robots remain the 3R space system. It has been presented in other works and studied matrix spatially, or more simply in a plan, but in this case, it is necessary to move from the working plane to the real space, or vice versa, passage that we will present in this study. Projections of point M on planar axes will be marked with the higher P (Plan) index to distinguish them from the corresponding space axes. Due to the fact that the vertical projection plane is removed from the Oρ axis with a constant distance a2 + a3, (the vertical working plane does not project directly on the Oρ axis, but on an axis parallel to it distal to the length a2 + a3) the projection of the M point on the horizontal plane of the space will not fall in M 'but at the point M''. Therefore, the projections of M on the axes Ox and Oy will not be those of point M' but those of point M'', according to the relations given by the system (2). We want to remove the angle of 90°C from the relations (2), which had an important explanatory role in the understanding of the phenomenon, to see how the equation of transition from plane to spatial axes is written, here (in the horizontal plane of space) about a rotation, whose relations should not be automatically detained, but deduced logically, which is why we will immediately move from the logically determined system (2) to the convenient system (3), which will now be obtained from (2) the angle of 90°C from the trigonometric relations. Perhaps the method used may seem rather difficult, but compared to spatial matrix methods, it is extremely straightforward and direct, contributing to transforming the space movement into a flat, much easier to understand and studied movement. In the system (4) we centralize all the transition relations from the plane to the spatial movement.


Introduction
shows the kinematic diagram of the planar chain and Fig. 2 shows the kinematic scheme of the space chain.
The transition from the plane to the space movement will then be continued.
The x 2 Oy 2 plane dimensions will be projected onto the zOρ axes. Thus, the length on the horizontal vertical axis Oy will be projected onto the spatial vertical axis Oz by adding the constant a 1 ϕ and the length of the horizontal plan axis Ox will be projected on the horizontal spatial axis Oρ by adding the constant d 1 , according to the relations given by the system (1) (1) Projections of point M on planar axes will be marked with the higher P (Plan) index to distinguish them from the corresponding space axes.
Due to the fact that the vertical projection plane is removed from the Oρ axis with a constant distance a 2 + a 3 , (the vertical working plane does not project directly on the Oρ axis, but on an axis parallel to it distal to the length a 2 + a 3 ) the projection of the M point on the horizontal plane of the space will not fall in M 'but at the point M'' (Fig. 2).
Therefore, the projections of M on the axes Ox and Oy will not be those of point M' but those of point M'', according to the relations given by the system (2) We want to remove the angle of 90°C from the relations (2), which had an important explanatory role in the understanding of the phenomenon, to see how the equation of transition from plane to spatial axes is written, here (in the horizontal plane of space) about a rotation, whose relations should not be automatically detained, but deduced logically, which is why we will immediately move from the logically determined system (2) to the convenient system (3), which will now be obtained from (2) the angle of 90°C from the trigonometric relations:  Perhaps the method used may seem rather difficult, but compared to spatial matrix methods, it is extremely straightforward and direct, contributing to transforming the space movement into a flat, much easier to understand and studied movement.
In the system (4) we centralize all the transition relations from the plane to the spatial movement: For simpler determination of speeds and accelerations in the system (4) from which it departs, it is denoted a 2 + a 3 by a, so that (4) acquires the simplified aspect (6): The spatial positioning system (6) is derived from time and the spatial velocity system (7) is obtained: The space velocity system (7) derives from time and the spatial acceleration system (8) is obtained, which is restricted to the shape (9) The space velocity system (7) is restricted to the shape (10), which by using the notations u and v is rewritten in the simplified form (11) and the acceleration system (9) can be restricted to the shape (12), with the notations w, t: Next, we will present the positions, velocities and spatial accelerations, all written down within the system (13)

Results
The spatial position vector module of the end effector point M in the fixed Cartesian space system is given by the relation (14) The modulus of the absolute speed vector of point M is obtained with the relation (15): The M-point absolute acceleration vector module is obtained with relation (16)

Discussion
Simple transition from plan to spatial computing can help us modify our work so that instead of performing all spatial matrices, let's study the planar system, then add the equation of transition from plane to spatial mode and so the same results will be obtained as if we had all the difficult spatial calculations done, practically just in plan, simplified. Man is accustomed to seeing the plan better than space, but especially to judge and reason more easily the plane phenomena than the spatial phenomena.

Conclusion
The anthropomorphic robots are part of the classical series of mechatronic systems, being in the form of arms and having at least three space rotation, to which other components may eventually be added, thus lengthening the entire kinematic chain. You can also add all the planetary or spatial rotating arms or others that are translating.
At the end we always have the end effector element that can be a manipulator, that is, a hand to grasp the objects, in which case one can speak of a prehensive device, that is a gripping device that today imitates very well a human hand even if it is one mechanical, may also be a painting, cutting or welding device, or one for machining.
The base support and schematics of all anthropomorphic robots remain the 3R space system. It has been presented in other works and studied matrix spatially, or more simply in a plan, but in this case, it is necessary to move from the working plane to the real space, or vice versa, passage that we will present in this study.
Projections of point M on planar axes will be marked with the higher P (Plan) index to distinguish them from the corresponding space axes.
Due to the fact that the vertical projection plane is removed from the Oρ axis with a constant distance a 2 + a 3 , (the vertical working plane does not project directly on the Oρ axis, but on an axis parallel to it distal to the length a 2 + a 3 ) the projection of the M point on the horizontal plane of the space will not fall in M 'but at the point M''.
Therefore, the projections of M on the axes Ox and Oy will not be those of point M' but those of point M'', according to the relations given by the system (2).
We want to remove the angle of 90°C from the relations (2), which had an important explanatory role in the understanding of the phenomenon, to see how the equation of transition from plane to spatial axes is written, here (in the horizontal plane of space) about a rotation, whose relations should not be automatically detained, but deduced logically, which is why we will immediately move from the logically determined system (2) to the convenient system (3), which will now be obtained from (2) the angle of 90°C from the trigonometric relations.
Perhaps the method used may seem rather difficult, but compared to spatial matrix methods, it is extremely straightforward and direct, contributing to transforming the space movement into a flat, much easier to understand and studied movement. In the system (4) we centralize all the transition relations from the plane to the spatial movement.

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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