Anthropomorphic Solid Structures n-r Kinematics

Transcription

1 American Journal of Engineering an Applie Sciences riginal Research Paper Anthropomorphic Soli Structures n-r Kinematics 1 Relly Victoria V. Petrescu, Raffaella Aversa, Bilal Akash, 4 Ronal B. Bucinell, 5 Juan. Corchao, 6 irila irsayar, Antonio Apicella an 1 Florian Ion T. Petrescu 1 ARoT-IFTo, Bucharest Polytechnic University, Bucharest, (CE Romania Avance aterial Lab, Department of Architecture an Inustrial Design, Secon University of Naples, 8101 Aversa (CE Italy Dean of School of Grauate Stuies an Research, American University of Ras Al Khaimah, UAE 4 Union College, USA 5 University of Salamanca, Spain 6 Department of Civil Engineering, Texas A& University, College Station, TX (Texas, USA Article history Receive: Revise: Accepte: Corresponing Author: Florian Ion T. Petrescu ARoT-IFTo, Bucharest Polytechnic University, Bucharest, (CE Romania scipub0@gmail.com Abstract: This paper presents an treats (in an original way the specific elements of the structures of robotic soli mobile anthropomorphic type. Are place on the wallpaper, the geometry an kinematics of the anthropomorphic robotic soli systems, in an original vision of the authors. ne presents the inverse kinematics of anthropomorphic systems, with mechanical elements an points: Geometry, cinematic, positions, isplacements, velocities an accelerations. They will be presente further two methos (as the most representatives: First one the metho trigonometric an secon one the geometric metho. Keywors: Anthropomorphic Robots, Direct Kinematic, Inverse Kinematic, R Systems, Velocities, Accelerations Introuction Toay, anthropomorphic structures are use more an more in almost all the fiels of inustrial. Robotic structures have emerge from the nee for automation an robotics of the inustrial processes. The first inustrial robots were calle upon by the heavy inustry an in particular by the automobiles inustry. The automotive inustry not only has requeste the appearance of the inustrial robots but even their subsequent evelopment (Angeles, 1989; Atkenson et al., 1986; Avallone an Baumeister, 1996; Baili, 00; Baron an Angeles, 1998; Borrel an Liegeois, 1986; Burick, The most use were an have remaine, the robots anthropomorphic, because they are more easily esigne, built, maintaine, are easily to hanle, more ynamics, robust, economic an in general they have a broaly working area. The structures of the soli anthropomorphic robots are mae up of elements an the couples of rotation, to which can a on an occasional basis an one or more couplers with translational moving. The couplers of rotation have been proven their effectiveness by moving them easier, more ynamic, step by step an especially being the most reliable. In general the couplers of rotation are moving more easily an more continuous, are actuate better an easier, control is less expensive an more reliable an programming the movements of rotation is also much simpler an more efficient (Ceccarelli, 1996; Choi et al., 004; Denavit, 1964; Di Gregorio an Parenti-Castelli, 00; Golsmith, 00; Grotjahn et al., 004; Guegan an Khalil, 00; Kim an Tsai, 00; Lee an Sanerson, 001; Liu an Kim, 00. All anthropomorphic structures are mae up of a basic structure R (Fig. 1. Starting from the basic structure R may buil then various robots 4R, 5R, 6R, 7R..., by the aition of moving elements an the couplers. Regarless of how many egrees of mobility has a soli structure final mobile, the basis for the thing is always represente by the soli structure R shown in Fig. 1. For this reason, the calculations presente in this stuy will be evelope for the basic structure R. The structure in Fig. 1 consists of three moving elements linke together by the couplers of rotation. It is a spatial structure, with a large area of movement (working. 017 Relly Victoria V. Petrescu, Raffaella Aversa, Bilal Akash, Ronal B. Bucinell, Juan. Corchao, irila irsayar, Antonio Apicella an Florian Ion T. Petrescu. This open access article is istribute uner a Creative Commons Attribution (CC-BY.0 license.

2 Relly Victoria V. Petrescu et al. / American Journal of Engineering an Applie Sciences 017, 10 (1:79.91 DI: /ajeassp Fig. 1. Schema geometry-cinematic of a structure R moern (anthropomorphic The platform (system as shown in Fig. 1 has three egree of mobility, carrie out by three actuators (electric motors. The first electric motor rives the entire system in a rotating movement aroun a vertical shaft 0 z 0. The engine (actuator number 1 is mounte on the fixe element (frame 0 an causes the mobile element 1 in a rotation movement aroun a vertical shaft 0 1. n the mobile element 1 are constructe then all the other elements (components of the system. It follows a kinematic chain plan (vertical, compose of two elements in movement an two couplers cinematic engines. It is about the kinematic elements in movement an, the assembly - being move by the actuator of the secon fitte in the coupler A, fixe on the element 1. Therefore the secon electric motor attache to the component 1 will rive the element in rotational motion relative to the item 1 an at the same time it will move the entire kinematic chain -. The last actuator (electric motor fixe by item, in B, will turn up the item (relative in relation to the. The rotation of the φ 10 carrie out by the first actuator, is an relative (between items 1 an 0 an absolute (between items 1 an 0. The rotation of the φ 0 carrie out by the secon actuator, is an relative rotation (between elements an 1 an absolute (between items an 0 ue to the arrangement of the system. The rotation of the θ φ carrie out by the thir actuator, is only a relative rotation (between elements an an the corresponing absolute rotation (between items an 0 is a function of the θ φ an φ 0 (Aversa et al., 016a; 016b; 016c; 016; 016e; 016f. The kinematic chain - (consisting of kinematic elements in movement an is a kinematic chain in plan, which fall within a single plan or in one compose of several plane parallel. It is a system kinematic special, which may be stuie separately. Then the element 1 (which it rives the kinematic chain - will be consiere as coupler, kinematic couplers engines A( an B( becoming, the first fixe coupler an the secon mobile coupler, both being kinematic couplers C 5, of rotation. For the etermination of the extent of mobility of the kinematic chain (in plane -, 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 (as we are talking about the two kinematic elements in movement note with an respectively an the C 5 represents the number kinematic couplers of the fifth-class, in this case C 5 (as we are talking about the couplers A an B or an : m C 6 4 (1 5 The kinematic chain - having to the egree of mobility, must be actuate by two motors. It is preferre that those two actuators to be two electric motors, irect current, or alternately. Alternatively the rive of motion may be one an with engines hyraulic, pneumatic, sonic, etc. Structural iagram of the plan kinematic chain - (Fig. resembles with its kinematic schema. The river element is linke to the element 1 consiere fixe, through the rive coupler an river element is linke to the mobile element through the rive coupler. 80

3 Relly Victoria V. Petrescu et al. / American Journal of Engineering an Applie Sciences 017, 10 (1:79.91 DI: /ajeassp Fig.. Structural iagram of the kinematic chain plan - linke to item 1 consiere fixe As a result there is the kinematic chain open with two egree of mobility, carrie out by the two actuators, i.e., the two electric motors, mounte on the motor kinematic couplers A an B or in question (Lorell et al., 00; erlet, 000; iller, 004; Petrescu et al., 009; Tsai, 000. Direct Kinematic of the Plan Chain - In Fig. can be tracke cinematic iagram of the chain plan - open. In irect cinematic are known cinematic parameters φ 0 an φ an must be etermine by the analytical calculation the parameters x an y, which represents the scalar coorinates of the point (eneffector. Are projecte the vectors an on the Cartesian axis system consiere fixe, xy ientical with the x y, to obtain the system of equations scaling (: x x x + x cosφ0 + cosφ cosφ y y y + y 0 + ( After shall be etermine in the form of Cartesian coorinates of the using the relations given by the system (, may be obtaine immeiately an the φ angle parameters using the relations estab-lishe in the framework of the system (: x + y x + y x x cosφ x + y y y x + y φ sign(sin φ arccos(cos φ ( Fig.. Kinematic iagram of the kinematic chain plan - linke to item 1 consiere fixe The system ( is written in a more concise manner in the form (4 which is erive with time to become the velocities system (5, which being erive with time generates in turn the system of accelerations (6: x cosφ0 + cosφ cosφ0 + cos( θ + φ0 π y sin( θ + φ0 π x v xɺ ω ω 0 ω0 sin φ ( θɺ + ω0 y v yɺ cosφ ω + cosφ ω cosφ0 ω0 + cos φ ( θɺ + ω x a ɺɺ x cosφ ω cosφ ω cosφ 0 ω 0 cos φ ( θɺ + ω0 y a ɺɺ y ω ω 0 ω0 sin φ ( θɺ + ω (4 (5 (6 Remark: The rotation spees of the actuators were consiere to be constant (relations 7: φɺ ω ct; θɺ ct si ω ct. 0 0 ε θɺɺ ε 0. 0 (7 The relations ( shall be erive an they an get the velocities (8 an the accelerations (9: 81

4 Relly Victoria V. Petrescu et al. / American Journal of Engineering an Applie Sciences 017, 10 (1:79.91 DI: /ajeassp x + y ɺ ɺ + y yɺ ɺ x ɺ + y yɺ x ɺ + y yɺ ɺ cosφ x y ɺ cosφ sin φ φɺ xɺ ( sin φ ɺ + cos φ φɺ yɺ (cos φ φɺ xɺ ( sin φ + yɺ (cosφ y cosφ x φɺ ɺ ɺ x + y y ɺ ɺ ɺ (8 x + y ɺ ɺ + y yɺ ɺ x ɺ + y yɺ ɺ + ɺɺ xɺ + x ɺɺ x + yɺ + y ɺɺ y xɺ + x ɺɺ x + yɺ + y ɺɺ y ɺ ɺɺ cosφ x y ɺ cosφ sin φ φɺ xɺ ( sin φ ɺ + cos φ φɺ yɺ (cos φ φɺ xɺ + yɺ cosφ ɺ φɺ + φɺɺ ɺɺ y cosφ yɺ φɺ x x cosφ φɺ ɺɺ ɺ φɺɺ ɺɺ y cosφ ɺɺ x yɺ φɺ xɺ cosφ φɺ ɺ φɺ x + x + y + y y ɺ ɺɺ ɺ ɺɺ ɺ ɺɺ (9 The following will etermine the positions, spees an accelerations, accoring to the positions of the scales point. It starts from the scalar coorinates of the point (the relationships 10: x y cosφ 0 0 (10 Is then etermine scaling spees an accelerations of the point, by ifferentiation of the system (10, in which shall be replace after erivation the proucts.cos or.sin with the respective positions, x or y, which become (in this way variables (see relations 11 an 1: xɺ ω y ω yɺ cosφ0 ω0 x ω ɺɺ x cosφ ω x ω ɺɺ y ω y ω (11 (1 Ha been put in evience in this moe the scalar spees an accelerations of point accoring to their original positions (scaling an absolute angular spee of the element. The angular spee was consiere to be constant. The technique of the etermination of the velocities an acceleration epening on the positions, is extremely useful in the stuy of the ynamics of the system, of the vibrations an noise cause by the respective system. This technique is common in the stuy of vibration system. Known the vibrations of scalar positions of point an then easily etermine the vibration velocities an accelerations that point an other points of the system all the functions of positions scaling known point. Also by this technique can calculate local noise levels at various points in the system an the overall level of noise generate, with a precision goo enough, compare with experimental measurements obtaine with aequate equipment. Absolute velocity of point (spee moule is given by relation (1 an the acceleration of can be written with expression (14: v xɺ + yɺ ω sin φ + ω cos φ ω0 ω a ɺɺ x + ɺɺ y ω cos φ + ω sin φ 4 ω0 ω (1 (14 In the following will etermine the cinematic scalar parameters of the point, eneffector, epening an on the parameters of the position of points an (the systems of relations 15-17: 8

5 Relly Victoria V. Petrescu et al. / American Journal of Engineering an Applie Sciences 017, 10 (1:79.91 DI: /ajeassp x x + cosφ y y + cosφ x x y y xɺ xɺ φɺ y ω 0 + ( y y ( ω0 y y ( ω0 ( y y y ω0 yɺ yɺ + cosφ φɺ x ω 0 + ( x x ( ω 0 x ( ω0 x ( x x θɺ + x ω0 yɺ yɺ cos φ ( ω0 xɺ xɺ sin φ ( ω0 ɺɺ x ( yɺ yɺ yɺ ω0 ɺɺ y ( xɺ xɺ θɺ + xɺ ω0 yɺ yɺ ( ( x x ω0 xɺ xɺ ( y ( y ω0 ɺɺ x ( x x ( ω0 θɺ + ( x x ω0 x ω0 ɺɺ y ( y y ( ω0 θɺ + ( y y ω0 y ω0 x ( x x ɺɺ ω0 + ( x x x ω0 ɺɺ y ( y y ω0 + ( y y y ω 0 ɺɺ x ( x x ( ω0 x ω0 ɺɺ y ( y y ( ω0 y ω0 ɺɺ x ( x x ( ω0 x ω 0 ɺɺ y ( y y ( ω0 y ω 0 Inverse Kinematic of the Plan Chain - (15 (16 (17 In Fig. can be tracke cinematic iagram of the chain plan - open. In inverse cinematic are known the parameters x an y (which represent the scalar coorinates of the point, eneffector, impose parameters an must be etermine by the analytical calculations the cinematic parameters φ 0 an φ. Determine first the intermeiate parameters an φ with relations (18: x + y ; x + y x x cos φ ; x + y y y x + y φ semn(sin φ arccos(cos φ (18 In triangle certain know the lengths of the three sies,, (constant an (variable so that it can be etermine epening on the lengths of the sies all other elements of the triangle, specifically its angles an trigonometric functions of their (particularly interest us sin an cos. You can use various methos an they will be presente further two (as the most representatives: etho trigonometric an geometric metho. Trigonometric etho, Determining Positions Writing of positions scaling Equations 19: cosφ0 + cosφ x 0 + y cos φ0 + sin φ0 1 cos φ + sin φ 1 (19 The problem with these two equations scaling, trigonometric, with two unknowns (φ 0 an φ is that they transcen (they are trigonometric equations, transcenental, where φ 0 unknown oes not appear irectly but in the form cosφ 0 an 0, so in reality the two trigonometric equations no longer have only two unknowns but four: cosφ 0, 0, cosφ an. To solve the system we nee more two equations, so that in the system (19 were also ae more two trigonometric equations, exactly the core golen as they are more say, for the angle φ 0 an separate for the φ angle. In orer to solve the first two equations of the system (19 are written in the form (0. cosφ x cosφ y 0 0 (0 Each equation of the system (0 are square, then ae together both equations (square to obtain the equation of the form (1: (cos φ0 + sin φ0 + x + y cosφ0 y sin φ0 (cos φ + sin φ (1 8

6 Relly Victoria V. Petrescu et al. / American Journal of Engineering an Applie Sciences 017, 10 (1:79.91 DI: /ajeassp Now is the time to use the two equations gol written at the en trigonometric system (, with which Equation 1 becomes simplifie expression (: Use golen formula an expression (9 becomes ( to be arrange conveniently by grouping terms of bringing to the form (1: + x + y cosφ y 0 0 ( x cos φ0 + cos x cos cosφ y y cos φ 0 0 ( Arrange the terms of the Equation in the most convenient form (. + ( y cos 0 ( x y cos φ0 x cos cosφ0 (1 + x + y ( x cosφ0 + y sin φ0 ( Discriminant of Equation 1 in the secon egree in cosine may be calculate by Equation : Divie the Equation with. an follows a new form (4: x + x + y cosφ0 + y 0 (4 cos + ( y cos ( x cos + y cos ( y cos y y (1 cos y sin ( bserving Fig., from system (18 may be explicit the relation (5: Raical secon orer of iscriminant is expresse as (: x + y (5 R y sin y sin ( Insert (5 to (4 an amplifie fraction the right to so that the expression (4 takes the form (6 a more useful form: + x cosφ0 + y 0 (6 Now, one can enter the expression cosine angle, accoring to some triangle (7: cos + (7 Using the expression (7 Equation 6 becomes simplifie (8: x cosφ cos y (8 0 0 Nees to remove 0, for which we isolate the term in sin an rose square Equation 8 for using the equation gol trigonometric angle φ 0 to transform sin in cosine, equation becoming one of the secon egree in cosφ 0. After squaring (8 becomes (9: x cos φ0 + cos x cos cosφ y sin φ 0 0 (9 Solutions to Equation 1 in the secon egree in cosine are written in the form (4: cos y sin cosφ x cos y sin x cos y sin 01, (4 Further, in solutions (4 are replace reports with corresponing trigonometric functions of angle φ, expressions (4 gaining form (5: x y cosφ0 cos 1, sin cos cos φ sin (5 cos( φ ± cosφ 0 cos( φ ± ne turns now to Equation 9 that it will be orer in form (6 with a view to resolving them in sin. Equation 6 rises to the square an using trigonometric equation golen of angle φ 0, is obtaine form (7: x cosφ cos y (

7 Relly Victoria V. Petrescu et al. / American Journal of Engineering an Applie Sciences 017, 10 (1:79.91 DI: /ajeassp x cos φ0 cos + y sin φ0 y cos 0 x x sin φ cos + y sin φ y cos 0 + φ φ ( x cos 0 sin φ 0 y cos 0 ( x cos ( x y sin 0 y cos sin 0 (7 Discriminant Equation 7 in the secon egree in cosine takes the form (8: y cos + ( x cos ( x cos + y x cos y cos ( x cos x sin Solutions to Equation 6 are written now as (9: y cos ± x sin 0 y cos ± x sin y cos x ± sin cos ± cosφ sin sin( φ ± 0 sin( φ ± (8 (9 From resulte relations (40 it can write the basic relationship (41: cosφ 0 cos( φ ± 0 sin( φ ± φ 0 (40 φ ± Ô (41 The proceure for etermining the angle φ, starting again from the system (19, where the first two transcenental equations are rewritten as (4 in orer to eliminate φ 0 angle this time: x + y + cosφ y + x cosφ + y (4 (44 x cosφ + y cos (45 Nee to etermine 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: x cosφ cos y (46 x cos φ + cos x cos cosφ y y cos φ ( y cos 0 cos φ x cos cosφ (47 (48 Equation 48 is a secon egree equation in cosine with solutions given by expression (49: cosφ cos cos ( cos ± + y ± y cos (1 cos cos sin ± y x cos y ± sin cosφ cos ± sin cos( φ cos φ cos( φ (49 Further write Equation 45 as (50, which is isolate this time cosine to eliminate it an then to etermine the term sin. The expression (5 is an equation of the secon egree in sin, amitting the solutions given by (5: x cosφ cos y (50 cosφ x cosφ y 0 0 (4 Equation 50 amounts to the square an obtain the equation of the form (51 suitable to be arrange in the form (5: Raises the two Equation 4 square an ae together, resulting in the equation of the form (4, which is arrange in the most convenient forms (44 an (45: x (1 sin φ cos y sin φ y cos + (51 85

8 Relly Victoria V. Petrescu et al. / American Journal of Engineering an Applie Sciences 017, 10 (1:79.91 DI: /ajeassp sin cos sin ( x cos 0 φ y φ y cos cos ( cos y + y cos (1 cos y cos sin y cos x sin cos cosφ sin sin( φ sin φ sin( φ Then, are qualifie only the relations (54 an (55: cosφ cos( φ sin( φ φ (5 (5 (54 φ (55 Trigonometric etho, Determining Velocities Starting from relations (56 require in the stuy of velocities in inverse kinematics: y cosφ x φɺ ɺ ɺ x + y y ɺ ɺ ɺ (56 It starts from the relationship linking the cosine of the angle Ô of the triangle sies, which erives in relation to time an thus obtains the value Ô ɺ written simply ɺ (relations 57: cos + ɺ cos sin ɺ ɺ cos ɺ ɺ ɺ sin 0 0 (57 It erives the Equation 4 to give the angular velocity ω φ ɺ (relation 58: ω φɺ φɺ ± ɺ ( To etermine ω 0 (relationship 58 we nee φ ɺ to be calculate from (56 an ɺ which is etermine from (57. In turn ɺ requires for its calculation ɺ that calculates all from system (56. Inlet velocities xɺ an yɺ are known, are require as input, or choose convenient, or can be calculate base on criteria impose. Similarly, one etermines an the angular velocity ω φ ɺ (relation 59. It erives the equation (55 to obtain the angular velocity ω φ ɺ (expression 60. Then, φ ɺ is calculate using the expression system known alreay (56 an ɺ is etermine from the system (59 an by the help of the system (56 which it causes an ɺ : cos + ɺ cos sin ɺ ɺ cos ɺ ɺ ɺ sin (59 ω φɺ φɺ ɺ (60 Trigonometric etho, Determining Accelerations It starts from the relationships (61 require in the stuy accelerations in inverse kinematics. The relation of the system (57 erive a secon time to time, gives the system (6: y cosφ x φɺɺ ɺɺ ɺɺ + yɺ φɺ xɺ cosφ φɺ ɺ φɺ x + x + y + y y ɺ ɺɺ ɺ ɺɺ ɺ ɺɺ cos + ɺ cos sin ɺ ɺ sin ɺ ɺ cos ɺ ɺɺ cos ɺɺ ɺ sin ɺ cos ɺ ɺ ɺɺ sin (61 (6 The following is erives the expression (58 to obtains expression (6, which generates angular acceleration absolute ε ε 0, that is calculate with φ ɺɺ out of the system (61 an with ɺɺ out of the system (45 an for the etermination of ɺɺ longer neee ɺɺ remove all from relations (61: 86

9 Relly Victoria V. Petrescu et al. / American Journal of Engineering an Applie Sciences 017, 10 (1:79.91 DI: /ajeassp ε ε ωɺ φɺɺ φɺɺ ± ɺɺ ( Now it erives a secon time (59 an obtains the system (64: cos x y y cos + x cos y y (69 cos + ɺ cos sin ɺ ɺ sin ɺ ɺ cos ɺ ɺɺ cos ɺɺ ɺ sin ɺ cos ɺ ɺ ɺɺ sin (64 It erives again the time relation (60 an we obtain the expression (65 of the angular acceleration absolute ε ε which is etermine with φ ɺɺ an ɺɺ : ε ε ωɺ φɺɺ φɺɺ ɺɺ (65 Geometric etho, Determining Positions It starts by writing positions equations, geometric (geometro-analytical (66. Scaling coorinates (x, y of point (eneffector are known an must be etermine an scaling coorinates of the point, which we enote by (x, y. The relationships of the system (66 are obtaine by writing the geometro-analytical equations of the circles, the rays an respectively: ( x x + ( y y x + y (66 Release binomials of the first equation of the system, enter the secon equation into the first, is also use an expression x + y, of amplify fraction with the factor convenient an obtaining the final expression of system (67, which is written together with the equation of the secon from system (66 in the new system (68 who nees to be aresse: + x + y y + x + y y x + y y cos x + y y cos x + y (67 (68 From the first equation of the system (68 explains the value of y, which rose an square (69: The phrase 'secon (69 is introuce into a secon relationship of (68 to give Equation 70, which are arrange conveniently in the form (71: y + cos + x cos y 0 cos x x x ( y cos 0 (70 (71 Equation 71 is an equation of the secon egree in x, which allows real solutions (7: x cos x x cos + ( y cos x cos y 1 cos x cos y sin x cos y sin x y cos sin ( cosφ cos sin cos( φ ± x cos( φ ± (7 It further etermines the unknown y by introucing value x obtaine from (7 in the first relationship of the system (69. ne obtains the expression (7: x y cos x cos sin y y + y x ± x y y y x cos ± sin ( cos ± cosφ sin sin( φ ± (( cos cos sin (7 87

10 Relly Victoria V. Petrescu et al. / American Journal of Engineering an Applie Sciences 017, 10 (1:79.91 DI: /ajeassp From (7 an (7 retain only the latest expression concentrate both in system (74: ( φ ( φ x cos ± y sin ± From Fig. may write Equation 75: (74 [( ɺ ( ɺ ] y x + y y y xɺ (8 x y y In step two we want to achieve yɺ for which multiplies the first relationship of the system (81 by (x an the secon with (-x, gather relations obtaine member with member an explains yɺ resulting relationship (8: x cosφ y 0 0 (75 [( ɺ ( ɺ ] x x + y y y yɺ (8 x y y Comparing systems (74 an (75 the resulting system (76, from that is eucting irect relation (77: ( φ ( φ cosφ cos ± sin ± φ (76 φ ± (77 Geometric etho, Determining Velocities Starting from the positions system (66, which is erive a function of time an obtains the velocities system (78. The system (78 is rewritten in simplifie form (79: ( x x ( xɺ xɺ + ( y y ( yɺ yɺ 0 ɺ + y yɺ 0 ( x x ɺ + ( y y yɺ ( x x ɺ + ( y y yɺ x ɺ + y yɺ 0 In (79 break out the brackets an get the system (80: x ɺ + y yɺ ( x ɺ + y yɺ ( x x ɺ + ( y y yɺ x ɺ + y yɺ 0 (78 (79 (80 The secon relationship of system (80 is inserte in the first, then the first expression is multiplie by (-1, so the system is simplifie, receiving the form (81: x ɺ + y yɺ ( x x ɺ + ( y y yɺ x ɺ + y yɺ 0 (81 The system (81 is solve in two steps. In the first step is multiplie the first relation of the system (81 by (y an the secon by (-y, after which the result expressions are gathere member by member to provie the following relation (8 in which it is explaine xɺ : Relationships (8 an (8 are written confine within the system (84: xɺ y h yɺ x h ( x x ɺ + ( y y yɺ h x y y Geometric etho, Determining Accelerations (84 Starting from the velocities system (84, which is erive a function of time an obtains acceleration system (85. The system (85 is rewritten as (86: ɺɺ x yɺ h + y hɺ x h + y hɺ ɺɺ y xɺ h x hɺ y h x hɺ h ( x y y ( x x ɺ + ( y y yɺ h ɺ ( x y y + h ( xɺ y + x yɺ yɺ y ɺ ( xɺ xɺ ɺ + ( x x ɺɺ x + ( yɺ yɺ yɺ + ( y y ɺɺ y ( xɺ xɺ ɺ + ( x x hɺ xɺ x y y ( yɺ yɺ yɺ + ( y y ɺɺ y + x y y xɺ y + x yɺ yɺ y ɺ h x y y ɺɺ x yɺ h + y hɺ x h + y hɺ ɺɺ y xɺ h x hɺ y h x hɺ ( x x y h + ( y y + x h y hɺ ɺ ɺ ɺ ɺ ɺ ɺ x y y ( x x ɺɺ x + ( y y ɺɺ y + y xɺ h x yɺ h + x y y (85 (86 88

11 Relly Victoria V. Petrescu et al. / American Journal of Engineering an Applie Sciences 017, 10 (1:79.91 DI: /ajeassp Geometric etho, Determining Angular Velocities an Accelerations nce etermine velocities an accelerations point, we can move on to etermining the absolute angular velocities an angular accelerations of the system. It starts from the system (75, which is erive a function of time an get the system (87: xɺ φɺ yɺ cosφ0 φɺ (87 For correct solution of the system (87 mounts the first relationship to the (-sinϕ 0 an the secon with (cosϕ 0, then collect both relationships obtaine (member by member an the explanation of the φ ɺ 0, obtaine searche expression (88: y cosφ x ω ω φ ɺ ɺ ɺ (88 The velocities system (87 is again erive the time an to obtain the absolute angular accelerations system (89: ɺɺ x cosφ φɺ φɺɺ ɺɺ y φɺ + cosφ φɺɺ (89 For correct solution of the system (89 amplify the first its relationship to the (-sinϕ 0 an the secon with (cosϕ 0 (see equations from system 90, then collect both relationships obtaine (member by member an the explanation of the φ ɺ 0, obtaine searche expression (91: ɺɺ x cosφ0 φɺ 0 sin φ0 φɺɺ 0 ( sin φ0 ɺɺ y 0 φɺ 0 + cos φ0 φɺɺ 0 (cos φ0 ε ε ω φɺɺ ɺɺ y cosφ ɺɺ x ( ɺ 0 0 (91 Remember the two relationships in the system (9: y cosφ0 x 0 ω ω0 φɺ ɺ ɺ 0 ɺɺ y cosφ ɺɺ x ε ε 0 ωɺ 0 φɺɺ Still express Equation 9, (Fig. : x x cosφ y y (9 (9 The relationships of the system (9 further are erive with the time an obtaine the velocities equations, system (94: xɺ xɺ φɺ yɺ yɺ cosφ φɺ (94 For correct solution of the system (94 amplify the first its relationship to the (-sinϕ an the secon with (cosϕ, then collect both relationships obtaine (member by member an the explanation of the φ ɺ, obtaine wante expression (95: ( y y cos φ ( x x ω ω φ ɺ ɺ ɺ ɺ ɺ (95 It erives the then with the time, the velocities system (94 an the absolute angular accelerations btain system (96: ɺɺ x ɺɺ x cosφ φɺ φɺɺ ɺɺ y ɺɺ y φɺ + cosφ φɺɺ (96 For correct solution of the system (96 amplify the first its relationship to the (-sinϕ an the secon with (cosϕ, then collect both relationships obtaine (member by member an by the explanation of the φ ɺɺ, obtaine wante expression (97: ε ε ω φɺɺ ( ɺɺ y ɺɺ y cos φ ( ɺɺ x ɺɺ x ɺ (97 Keep in the system (98 foun the two solutions an in system (99 centralize all four of them: ( y y cos φ ( x x ω ω φɺ ɺ ɺ ɺ ɺ ( ɺɺ y ɺɺ y cos φ ( ɺɺ x ɺɺ x ε ε ωɺ φɺɺ y cosφ0 x 0 ω ω0 φɺ ɺ ɺ 0 ( y y cos φ ( x x ω ω φɺ ɺ ɺ ɺ ɺ ɺɺ y cosφ0 ɺɺ x 0 ε ε 0 ωɺ 0 φɺɺ 0 ( ɺɺ y ɺɺ y cos φ ( ɺɺ x ɺɺ x ε ε ωɺ φɺɺ (98 (99 89

12 Relly Victoria V. Petrescu et al. / American Journal of Engineering an Applie Sciences 017, 10 (1:79.91 DI: /ajeassp Conclusion Toay, anthropomorphic structures are use more an more in almost all the fiels of inustrial. Robotic structures have emerge from the nee for automation an robotics of the inustrial processes. The first inustrial robots were calle upon by the heavy inustry an in particular by the automobiles inustry. The automotive inustry not only has requeste the appearance of the inustrial robots but even their subsequent evelopment. The most use were an have remaine, the robots anthropomorphic, because they are more easily esigne, built, maintaine, are easily to hanle, more ynamics, robust, economic an in general they have a broaly working area. The structures of the soli anthropomorphic robots are mae up of elements an the couples of rotation, to which can a on an occasional basis an one or more couplers with translational moving. The couplers of rotation have been proven their effectiveness by moving them easier, more ynamic, step by step an especially being the most reliable. In general the couplers of rotation are moving more easily an more continuous, are actuate better an easier, control is less expensive an more reliable an programming the movements of rotation is also much simpler an more efficient. This paper presents an treats (in an original way the specific elements of the structures of robotic soli mobile anthropomorphic type. Are place on the wallpaper, the geometry, kinematics an ynamics of the anthropomorphic robotic soli systems, in an original vision of the authors. Last part presents the inverse kinematics of anthropomorphic systems, with mechanical elements an points: Geometry, cinematic, positions, isplacements, velocities an accelerations, by two methos (as the most representatives: First one the metho trigonometric an secon one the geometric metho. Acknowlegement This text was acknowlege an appreciate by Assoc. Pro. Taher. Abu-Lebeh, North Carolina A an T State Univesity, Unite States, Samuel P. Kozaitis, Professor an Department Hea at Electrical an Computer Engineering, Floria Institute of Technology, Unite States, whom we thank an in this way. Funing Information Research contract: Contract number 6-5-4D/1986 from 4IV1985, beneficiary CNST R (Romanian National Center for Science an Technology Improving ynamic mechanisms. Contract research integration from ; Beneficiary: IS; TPIC: Research on esigning mechanisms with bars, cams an gears, with application in inustrial robots. Contract research. GR 69/ : NURC in 76; theme 8: Dynamic analysis of mechanisms an manipulators with bars an gears. Labor contract, no. 5/.01.01, the UPB, Stan for reaing performance parameters of kinematics an ynamic mechanisms, using inuctive an incremental encoers, to a itsubishi echatronic System PN-II- IN-CI All these matters are copyrighte! Copyrights: 94- qognhhtej, from :4:18; 46- vpstucgsiy, from :45:; 61- sqfsgqvutm, from :15:; 9- CrDztEfqow, from :7:5. Author s Contributions All the authors contribute equally to prepare, evelop an carry out this manuscript. Ethics This article is original. Authors eclare that are not ethical issues that may arise after the publication of this manuscript. References Angeles, J., An algorithm for the inverse ynamics of n-axis general manipulators using Kane's equations. Comput. ath. Applic., 17: DI: / ( Atkenson, C., H.A. Chae an J. Hollerbach, Estimation of Inertial Parameters of anipulator Loa an Links. 1st En., IT Press, Cambrige, assachuesetts. Avallone, E.A. an T. Baumeister, arks Stanar Hanbook for echanical Engineers. 10th En., cgraw-hill, New York, ISBN-10: , pp: Aversa, R., D. Parcesepe, R.V. Petrescu, G. Chen an F.I.T. Petrescu et al., 016a. Glassy amorphous metal injection mole inuce morphological efects. Am. J. Applie Sci., 1: DI: /ajassp Aversa, R., F.I.T. Petrescu, R.V. Petrescu an A. Apicella, 016b. Biomimetic finite element analysis bone moeling for customize hybri biological prostheses evelopment. Am. J. Applie Sci., 1: DI: /ajassp Aversa, R., R.V. Petrescu, F.I.T. Petrescu an A. Apicella, 016c. Smart-factory: ptimization an process control of composite centrifuge pipes. Am. J. Applie Sci., 1: DI: /ajassp

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