Anthropomorphic Solid Structures nR Kinematics

Corresponding Author: Florian Ion T. Petrescu ARoTMM-IFToMM, Bucharest Polytechnic University, Bucharest, (CE) Romania Email: scipub02@gmail.com Abstract: This paper presents and treats (in an original way) the specific elements of the structures of robotic solid mobile anthropomorphic type. Are “placed on the wallpaper”, the geometry and kinematics of the anthropomorphic robotic solid systems, in an original vision of the authors. One presents the inverse kinematics of anthropomorphic systems, with mechanical elements and points: Geometry, cinematic, positions, displacements, velocities and accelerations. They will be presented further two methods (as the most representatives): First one the method trigonometric and second one the geometric method.


Introduction
Today, anthropomorphic structures are used more and more in almost all the fields of industrial. Robotic structures have emerged from the need for automation and robotics of the industrial processes. The first industrial robots were called upon by the heavy industry and in particular by the automobiles industry. The automotive industry not only has requested the appearance of the industrial robots but even their subsequent development (Angeles, 1989;Atkenson et al., 1986;Avallone and Baumeister, 1996;Baili, 2003;Baron and Angeles, 1998;Borrel and Liegeois, 1986;Burdick, 1988).
The most used were and have remained, the robots anthropomorphic, because they are more easily designed, built, maintained, are easily to handle, more dynamics, robust, economic and in general they have a broadly working area. The structures of the solid anthropomorphic robots are made up of elements and the couples of rotation, to which can add on an occasional basis and one or more couplers with translational moving. The couplers of rotation have been proven their effectiveness by moving them easier, more dynamic, step by step and especially being the most reliable. In general the couplers of rotation are moving more easily and more continuous, are actuated better and easier, control is less expensive and more reliable and programming the movements of rotation is also much simpler and more efficient (Ceccarelli, 1996;Choi et al., 2004;Denavit, 1964;Di Gregorio and Parenti-Castelli, 2002;Goldsmith, 2002;Grotjahn et al., 2004;Guegan and Khalil, 2002;Kim and Tsai, 2002;Lee and Sanderson, 2001;Liu and Kim, 2002).
All anthropomorphic structures are made up of a basic structure 3R (Fig. 1). Starting from the basic structure 3R may build then various robots 4R, 5R, 6R, 7R..., by the addition of moving elements and the couplers. Regardless of how many degrees of mobility has a solid structure final mobile, the basis for the thing is always represented by the solid structure 3R shown in Fig. 1. For this reason, the calculations presented in this study will be developed for the basic structure 3R. The structure in Fig. 1 consists of three moving elements linked together by the couplers of rotation. It is a spatial structure, with a large area of movement (working). The platform (system) as shown in Fig. 1 has three degree of mobility, carried out by three actuators (electric motors). The first electric motor drives the entire system in a rotating movement around a vertical shaft O 0 z 0 . The engine (actuator) number 1 is mounted on the fixed element (frame 0) and causes the mobile element 1 in a rotation movement around a vertical shaft O 0 O 1 . On the mobile element 1 are constructed then all the other elements (components) of the system.
It follows a kinematic chain plan (vertical), composed of two elements in movement and two couplers cinematic engines. It is about the kinematic elements in movement 2 and 3, the assembly 2-3 being moved by the actuator of the second fitted in the coupler A, fixed on the element 1. Therefore the second electric motor attached to the component 1 will drive the element 2 in rotational motion relative to the item 1 and at the same time it will move the entire kinematic chain 2-3. The last actuator (electric motor) fixed by item 2, in B, will turn up the item 3 (relative in relation to the 2).
The rotation of the φ 10 carried out by the first actuator, is and relative (between items 1 and 0) and absolute (between items 1 and 0).
The rotation of the φ 20 carried out by the second actuator, is and relative rotation (between elements 2 and 1) and absolute (between items 2 and 0) due to the arrangement of the system.
The kinematic chain 2-3 (consisting of kinematic elements in movement 2 and 3) is a kinematic chain in plan, which fall within a single plan or in one composed of several plane parallel. It is a system kinematic special, which may be studied separately. Then the element 1 (which it drives the kinematic chain 2-3) will be considered as coupler, kinematic couplers engines A(O 2 ) and B(O 3 ) becoming, the first fixed coupler and the second mobile coupler, both being kinematic couplers C 5 , of rotation.
For the determination of the extent of mobility of the kinematic chain (in plane) 2-3, shall apply to the structural formula given by the relationship (1), where m is the total number of moving parts of the kinematic chain plan; in our case m = 2 (as we are talking about the two kinematic elements in movement noted with 2 and 3 respectively) and the C 5 represents the number kinematic couplers of the fifth-class, in this case C 5 = 2 (as we are talking about the couplers A and B or O 2 and O 3 ): The kinematic chain 2-3 having to the degree of mobility 2, must be actuated by two motors. It is preferred that those two actuators to be two electric motors, direct current, or alternately.
Alternatively the drive of motion may be done and with engines hydraulic, pneumatic, sonic, etc. Structural diagram of the plan kinematic chain 2-3 ( Fig. 2) resembles with its kinematic schema.
The driver element 2 is linked to the element 1 considered fixed, through the drive coupler O 2 and driver element 3 is linked to the mobile element 2 through the drive coupler O 3 . As a result there is the kinematic chain open with two degree of mobility, carried out by the two actuators, i.e., the two electric motors, mounted on the motor kinematic couplers A and B or O 2 in question O 3 (Lorell et al., 2003;Merlet, 2000;Miller, 2004;Petrescu et al., 2009;Tsai, 2000).

Direct Kinematic of the Plan Chain 2-3
In Fig. 3 can be tracked cinematic diagram of the chain plan 2-3 open. In direct cinematic are known cinematic parameters φ 20 and φ 30 and must be determined by the analytical calculation the parameters x M and y M , which represents the scalar coordinates of the point M (endeffector M).
Are projected the vectors d 2 and d 3 on the Cartesian axis system considered fixed, xOy identical with the x 2 O 2 y 2 , to obtain the system of equations scaling (2): After shall be determined in the form of Cartesian coordinates of the M using the relations given by the system (2), may be obtained immediately and the φ angle parameters using the relations estab-lished in the framework of the system (3) The system (2) is written in a more concise manner in the form (4) which is derived with time to become the velocities system (5), which being derived with time generates in turn the system of accelerations (6) The relations (3) shall be derived and they and get the velocities (8) and the accelerations (9): The following will determine the positions, speeds and accelerations, according to the positions of the scales point Is then determined scaling speeds and accelerations of the point O 3 , by differentiation of the system (10), in which shall be replaced after derivation the products d.cos or d.sin with the respective positions, x O3 or y O3 , which become (in this way) variables (see relations 11 and 12): Had been put in evidence in this mode the scalar speeds and accelerations of point O 3 according to their original positions (scaling) and absolute angular speed of the element 2.
The angular speed was considered to be constant. The technique of the determination of the velocities and acceleration depending on the positions, is extremely useful in the study of the dynamics of the system, of the vibrations and noise caused by the respective system.
This technique is common in the study Also by this technique can calculate local noise levels at various points in the system and the overall level of noise generated, with a precision good enough, compared with experimental measurements obtained with adequate equipment.
Absolute velocity of O 3 point (speed module) is given by relation (13) In the following will determine the cinematic scalar parameters of the point M, endeffector, depending and on the parameters of the position of points O 3 and M (the systems of relations 15-17):

Inverse Kinematic of the Plan Chain 2-3
In In triangle certain O 2 O 3 M know the lengths of the three sides, d 2 , d 3 (constant) and d (variable) so that it can be determined depending on the lengths of the sides all other elements of the triangle, specifically its angles and trigonometric functions of their (particularly interest us sin and cos). You can use various methods and they will be presented further two (as the most representatives): Method trigonometric and geometric method.

Trigonometric Method, Determining Positions
The problem with these two equations scaling, trigonometric, with two unknowns (φ 20 and φ 30 ) is that they transcend (they are trigonometric equations, transcendental, where φ 20 unknown does not appear directly but in the form cosφ 20 and sinφ 20 , so in reality the two trigonometric equations no longer have only two unknowns but four: cosφ 20 , sinφ 20 , cosφ 30 and sinφ 30 ).
To solve the system we need more two equations, so that in the system (19) were also added more two trigonometric equations, exactly the core "golden" as they are more say, for the angle φ 20 and separate for the φ 30 angle. In order to solve the first two equations of the system (19) Now is the time to use the two "equations gold" written at the end trigonometric system (2) Insert (25) to (24) and amplified fraction the right to d so that the expression (24) takes the form (26) Needs to remove sinφ 20 , for which we isolated the term in sin and rose squared Equation 28 for using the equation gold trigonometric angle φ 20 to transform sin in cosine, equation becoming one of the second degree in cosφ 20 . After squaring (28) Radical second order of discriminant is expressed as (33): One turns now to Equation 29 that it will be order in form (36) The procedure for determining the angle φ 30 , starting again from the system (19) Raises the two Equation 42 squared and added together, resulting in the equation of the form (43), which is arranged in the most convenient forms (44) and (45) Need to determine first the cosine so that we isolate initially term in sin, Equation 45 with a special its form (46), which by squaring generates expression (47), an expression that arranges form (48) Further write Equation 45 as (50), which is isolated this time cosine to eliminate it and then to determine the term sin. The expression (52) is an equation of the second degree in sin, admitting the solutions given by (53) Then, are qualified only the relations (54) and (55):

Trigonometric Method, Determining Velocities
Starting from relations (56) required in the study of velocities in inverse kinematics: To determine ω 20 (relationship 58) we need φ ɺ to be calculated from (56)  ω φ ≡ ɺ (expression 60). Then, φ ɺ is calculated using the expression system known already (56) and M ɺ is determined from the system (59) and by the help of the system (56) which it causes and d ɺ : The following is derives the expression (58) to obtains expression (63), which generates angular acceleration absolute ε 2 ≡ ε 20 , that is calculated with φ ɺɺ out of the system (61) and with 2 O ɺɺ out of the system (45) and for the determination of 2 O ɺɺ longer needed d ɺɺ removed all from relations (61): It further determines the unknown y by introducing value x obtained from (72) in the first relationship of the system (69). One obtains the expression (73) From (72) and (73)

Geometric Method, Determining Velocities
Starting from the positions system (66), which is derived a function of time and obtains the velocities system (78). The system (78) is rewritten in simplified form (79): The second relationship of system (80) is inserted in the first, then the first expression is multiplied by (-1), so the system is simplified, receiving the form (81): The system (81) is solved in two steps.
In the first step is multiplied the first relation of the system (81) by (y) and the second by (-y M ), after which the result expressions are gathered member by member to provide the following relation (82) in which it is explained x ɺ : In step two we want to achieve y ɺ for which multiplies the first relationship of the system (81) by (x) and the second with (-x M ), gather relations obtained member with member and explains y ɺ resulting relationship (83): Relationships (82) and (83) are written confined within the system (84):

Geometric Method, Determining Accelerations
Starting from the velocities system (84), which is derived a function of time and obtains acceleration system (85). The system (85)

Geometric Method, Determining Angular Velocities and Accelerations
Once determined velocities and accelerations O 3 point, we can move on to determining the absolute angular velocities and angular accelerations of the system. It starts from the system (75), which is derived a function of time and get the system (87) The velocities system (87) is again derived the time and to obtain the absolute angular accelerations system (89) For correct solution of the system (89) amplify the first its relationship to the (-sinϕ 20 ) and the second with (cosϕ 20 ) (see equations from system 90), then collect both relationships obtained (member by member) and the explanation of the 20 φ ɺ , obtained searched expression (91) The relationships of the system (93) For correct solution of the system (94) amplify the first its relationship to the (-sinϕ 30 ) and the second with (cosϕ 30 ), then collect both relationships obtained (member by member) and the explanation of the 30 φ ɺ , obtained wanted expression (95) It derives the then with the time, the velocities system (94) and the absolute angular accelerations Obtain system (96) For correct solution of the system (96) amplify the first its relationship to the (-sinϕ 30 ) and the second with (cosϕ 30 ), then collect both relationships obtained (member by member) and by the explanation of the 30 φ ɺɺ , obtained wanted expression (97)

Conclusion
Today, anthropomorphic structures are used more and more in almost all the fields of industrial. Robotic structures have emerged from the need for automation and robotics of the industrial processes. The first industrial robots were called upon by the heavy industry and in particular by the automobiles industry. The automotive industry not only has requested the appearance of the industrial robots but even their subsequent development.
The most used were and have remained, the robots anthropomorphic, because they are more easily designed, built, maintained, are easily to handle, more dynamics, robust, economic and in general they have a broadly working area. The structures of the solid anthropomorphic robots are made up of elements and the couples of rotation, to which can add on an occasional basis and one or more couplers with translational moving. The couplers of rotation have been proven their effectiveness by moving them easier, more dynamic, step by step and especially being the most reliable. In general the couplers of rotation are moving more easily and more continuous, are actuated better and easier, control is less expensive and more reliable and programming the movements of rotation is also much simpler and more efficient.
This paper presents and treats (in an original way) the specific elements of the structures of robotic solid mobile anthropomorphic type. Are "placed on the wallpaper", the geometry, kinematics and dynamics of the anthropomorphic robotic solid systems, in an original vision of the authors. Last part presents the inverse kinematics of anthropomorphic systems, with mechanical elements and points: Geometry, cinematic, positions, displacements, velocities and accelerations, by two methods (as the most representatives): First one the method trigonometric and second one the geometric method.