In this paper, the kinematic structure of the geared robotic mechanism (GRM) is investigated

Transcription

1 KINEMATIC ANALYSIS OF GEARED ROBOTIC MECHANISM USING MATROID AND T-T GRAPH METHODS. Seyevahi Amirinezha Mustafa K. Uyguroğlu Department of Electrical an Enectronic Engineering Faculty of Engineering Eastern Meiterranean University Gazimağusa North Cyprus A. ABSTRACT In this paper, the kinematic structure of the geare robotic mechanism (GRM) is investigate with the ai of two ifferent methos which are base on irecte graphs an the methos are compare. One of the methos is Matroi Metho evelope by Talpasanu an the other metho is Tsai-Toka (T-T) Graph metho evelope by Uyguroglu an Demirel. It is shown that the kinematic structure of the geare robotic mechanism can be represente by irecte graphs an angular velocity equations of the mechanisms can be systematically obtaine from the graphs. The avantages an isavantages of both methos are emonstrate relative to each other. B. INTRODUCTION Kinematic an ynamic analysis of mechanical systems have been well establishe by using graph theory in recent years. Non-oriente an oriente graphs were use for this purpose. Nonoriente graph technique is mainly use for the kinematic analysis of robotic bevel-gear trains [1] [4]. The oriente graph technique has been use for electrical circuits an other types of

2 lumpe physical systems incluing mechanical systems in one-imensional motion since the early sixties [5] [8]. Chou et al. [9] use these techniques to three-imensional systems. Recently, Toka evelope a systematic approach, the so calle Network Moel Approach, for the formulation of three imensional mechanical systems [1], an Uyguroglu an Toka extene this approach for the kinematic an ynamic analysis of spatial robotic bevel-gear trains [11]. Most recently the oriente an non-oriente graph techniques were compare by Uyguroglu an Demirel [12], an the avantages of the oriente graph over the non-oriente graph were emonstrate using the kinematic analysis of bevel-gear trains. In orer to overcome the weaknesses of the methos evelope by Tsai an Toka, both metho were combine an the so calle T-T Graph metho was introuce []. On the other han, Talpasanu et al. [] evelope Matroi Metho for the kinematic analysis of geare mechanisms base on irecte graph as well. In this paper the kinematic structures of the GRM is investigate with the ai of Matroi Metho an T-T Graph metho. Since both methos are use irecte graph, the similarities an the ifferences are shown an the avantages of each system are inicate. C. Geare Robotic Mechanism GRMs are close-loop configurations which are use to reuce the mass an inertia of the actuators loas. Gear trains in GRMs are employe such that actuators can be place as closely as possible to the base. Figure 1 shows functional schematic of the GRM. It has 3 Degree of Freeom (i.e. it has 3 inputs) an the en-effector can have spatial motion (3 Dimensional) because this mechanism can generate two rotations about two intersecte axes an one rotation of en-effector about its axis. In this mechanism 4, 5, an 6 are sun gears (input links), 1 an 2

3 are carriers an 3, 7, a n 7 are planet gears. It is observe that links an joints (gear train) are use to transmit the rotation of the inputs to the en-effector. The motion of en-effector is prouce by links 4, 5, an 6 as inputs. The en-effector is attache to link 3 an carrie by link 2. M1, M2 an M3 are actuators. The rotation of link 3 is cause by M3 through 6 an 7 links an the rotation of links 1 an 2 is mae by M1 through link 4 an M2 through link 5 respectively Z14 7" Z18 A3 Z ' 17 Z17 A2 M Z11 Z Z Z8 M1 Z15 15 Z O Z1 Z9 Y M2 B2 B1 A1 Figure 1. The GRM mechanism. II. MATROID METHOD

4 In this section, Matroi metho [ ] is applie to the sample Geare Robotics Mechanism (GRM) to obtain the kinematic equations. First, its igraph is sketche an corresponing matrices are obtaine. Then, we calculate relative angular velocities by using these matrices an Screw theory. Associate igraph an corresponing matrices: The mechanism in Figure 1 consists of n 7 links, t 7 turning pairs an c 4 gear pairs. Note that k 11 is total number of joints (i.e. k t c an t n ). The following labeling, which is assigne to links an joints of sample mechanism, is use in Matroi metho [ ]: is assigne to groun link. 1, 2, 3, 4, 5, 6, an 7 are assigne to gears an carriers. 8, 9, 1, 11, 12, 13, an 14 are assigne to turning joints. 15, 16, 17, an 18 are assigne to meshing joints. Figure 2 shows associate igraph of the sample mechanism. In this igraph, noes present links as well as soli an ash arrows inicate turning an meshing joints respectively. Note that in each mechanism, corresponing to c gear pair there exist c funamental cycles hence in sample mechanism, there are 4 funamental cycles: C, C, C, a n C. In aition, Spanning Tree is efine such that there is not any cycle in igraph so in Figure 2 collection of soli arrows creates Spanning Tree.

5 Figure 2. Mechanism associate igraph. Here, one coul easily obtain the Incience Noe-Ege matrix from Figure 2: Γ (1) The entries of an Incient Noe-Ege Matrix are 1, -1 an. Each column represents an irecte ege which connects two noes an contains two nonzero entries. The arrow hea sie is 1 an the other sie is -1. Reuce Incient Noe-Ege matrix Γ is obtaine by eleting the first row where rows of this matrix are inepenent. Since k t c, Γ consists of two sub-matrices: G an * G which correspon to turning an gear pairs respectively.

6 * Г G G (2). Now, regaring to associate igraph in Figure 2, we can obtain Path matrix as follows: Ζ (3). Path matrix Z [ ], is a t n matrix an comes from the spanning tree. Here, z t, n (the entries of Path matrix) can be -1,, an +1. If ege t belongs to one of Spanning Tree s paths which are starte from noe n towar the groun link an its orientation is the same as path s irection, z t, n becomes +1. If it belongs to the path but the orientation is opposite, z t, n becomes -1. z if ege oes not belong to the relate path. t, n Spanning Tree Matrix is obtaine by using the formula: *T T T G Z (4). Then, Cycle-Basis matrix is obtaine as:

7 (5). С T U C C C C Actually, one can obtain Cycle-Basis matrix from Figure 2 irectly. 1) Inepenent equations for relative angular velocities: In this part, we obtain relative angular velocity equations of joints by means of Screw theory an then efine output relative velocities in terms of inputs. Let consier the ual vector (Screw) in Eq. (6): uˆ u r k c, k k k k c, k c, k c, k c, k L M N P Q R T (6). This 6 1 matrix can efine spatial geometry of z-axes of local frames attache to k joints. Note that each of these local frames has unit vector u 1 T with respect to itself. The first vector of Screw is unit vector of orientation of z axis: k z k axis with respect to the base frame an it presents the u D u (7) k, k where D is a pure rotational matrix about x axis:,k k 1 c o s s in s in c o s k k D (8)., k k k

8 So components of u k are: L ; M s in ; N c o s (9) k k k k k where k are offset angles between z-axis of base an z-axes of turning axes. So, in sample mechanism: a n 9 an unit vectors of revolute joints are: T T u u u u 1 a n u u u 1. The secon vector is the position vector of with respect to the reference frame: r I u (1) c, k c, k k T where I x y z is istance vector which orients from c to k an it has the form: c, k c, k c, k c, k T k c k c k c x x y y z z (11). Accoring to Table 1, components of position vector are: P z s in y c o s ; Q ; R (12). c, k c, k k c, k k c, k c, k Table 1. Coorinates of turning an gear pairs x k ( ) y B B B k z A A A A A A A A k So for each cycle, the P coefficients are calculate as follows: c, k Cycle C : P ; P ; P. 1 5,8 1 5,1 1 5,

9 Cycle C 16 ; ; ; : P P P P 1 6,9 1 6,1 1 6, ,1 6 Cycle C 17 ; ; ; : P P P P 1 7, , , ,1 7 Cycle C 18 ; ; : P P P 1 8, , ,1 8 Here, we efine twist matrix as a prouct between screw an relative velocity variables of pairs [ ]: sˆ uˆ θ (13) k c, k k where θ k T is partitione into two sub-matrices, θ t an T θ c relates to velocities of turning an gear pairs respectively. Now, we can obtain relative angular velocity equations by applying Haamar entry-wise prouct on Eq. (5) an Eq. (13) as follows: C sˆ (14) k c [ ] where sˆ sˆ s ˆ. In [ ], to two orthogonality conitions, one for relative velocities an k t c another for relative moments, are efine in orer to Eq. (14) hols true. Accoring to these two conitions, we can express equations of relative velocity variables of meshing an turning pairs in Eq. (15) an Eq. (16) respectively: θ T u θ (15) c t t

10 an because r c, c then T r θ (16) c, t t c Since Q R, we can simplify Eq. (16) in the following form:,, c t c t P θ (17) c, t t c where P c, t is coefficient matrix: P T P c, t c, t (18) In Eq. (17), P c, t are coefficients in terms of pitch iameter. These scalar coefficients are use to acquire inepenent equations of relative angular velocities. For sample mechanism, these inepenent equations are expresse in Eq. (19): 8 P P P P P 1 1 6,9 1 6, ,1 11 P P P 1 7, , , P P 1 5,8 1 5, , , (19) Moreover, we will later rewrite these coefficients in terms of gear ratios in Eq. (24): i ; i ; i ; i (2)

11 Up to here, we obtain a set of inepenent equations for relative angular velocities of turning pairs incluing input pairs an output ones. Now accoring to Kutzbach criterion [ ] in Eq. (21), we can calculate output relative velocities in terms of input relative velocities: E t r (21) where E is the number of inputs (Degree of Freeom), t is the number of turning pairs an r is the number of outputs (rank of Cycle-Basis matrix). So Eq. (17) is partitione in the following form: θ E P P r, E r, r θ r (22) Hence, solutions for output relative velocities θ r can be efine as functions of input relative velocities θ E : 1 r r E E θ P P θ (23) For sample mechanism in Figure 1, since E 3, t 7 a n r 4 so, a n are input velocities an,, a n are output velocities. Accoring to Eq. (2) an Eq. (22), we can rewrite Eq. (19) as follows: 8 9 i i i i i i 18 (24)

12 Note that in Eq. (24), we change the orer of thir an fourth columns in P matrix an thir an fourth rows in θ vector compare with Eq. (19). Accoring to Eq. (23), we can obtain inepenent equations for output relative velocities in terms of gear ratio for the sample mechanism: i i i i i i i i i i i i i i i i i i i (25) III. TSAI TOKAD (T T) GRAPH METHOD On the functional schematic of the GRM shown in Figure 3: is assigne to the groun. Links are numbere as 1, 2, 3, 4, 5, 6, an 7. Turning pairs axes are labele as a,b,c,, an e.

13 e c M1 a M2 b Figure 3: Functioal Schematic of the GRM. T-T Graph Representing of the GRM In the T-T graph metho, links are represente by noes an the oriente lines between these noes inicate the terminal pairs (ports), where a pair of meters, real or conceptual, are connecte to measure the complementary terminal variables which are necessary to escribe the physical behavior of the mechanism. The complementary terminal variables in mechanical systems are the terminal across (translational an rotational velocities) an the terminal through (forces an moments) variables.

14 The oriente graph representation of the turning-pair connection an the gear-pair connection are shown in Figure 4. 2 Carrier arm 3 Gear w 21 M 21 Gear 1 w 23 M 23 w 23 M 13 (a) 1 (b) 3 Transfer vertex Figure 4: (a) Turning-pair an graph representation, (b) Gear-Pair an graph representation The relation between the relative velocities an moments of the gears shown in Figure 4(b) is: w n M M n w (26) Where n N / N 1 / n an N 1 an N 2 are teeth numbers of gear 1 an gear 2, respectively. The sign of the gear ratio n is etermine base on the rotation irections of the gears. If both of the gears rotate in the same irection the sign is (+), otherwise the sign is (-). In orer to obtain the T-T graph representation of the GRM shown in Fig. 3, first, turning pairs are rawn by replacing them with their graph representations. The turning pairs axes labels are inserte into the graph as shown in Fig. 5.

15 4 a ω4 ω5 b 1 c 2 ω1 ω21 c e ω32 3 b ω61 ω Figure 5: T-T Graph representation of the turning pairs of the GRM. As it is seen from Fig.5, the turning pairs constitute the tree branches. Then, the graph representations of the gear pairs are rawn for completing the graph. At this point, it is require to etermine the transfer vertices representing the carrier arms for the gear pairs (4,1), (5,2), (6,7) an (7,3). In orer to etermine the transfer vertex, we will start from one of the noe representing the gear in meshes an go through the tree branches to reach the other noe representing the other gear. The vertex on this path, which has ifferent levels on opposite sies is the transfer vertex.

16 a b Path 1 ( 4 1) : vertex (pair axes a, b), b b c Path 2 (5 1 2 ) : vertex 1 (pair axes b, c), c c Path 3 ( ) : vertex 2 (pair axes c, ), e Path 4 ( ) : vertex 2 (pair axes, e). Therefore the sets of gear pair an corresponing carrier arm are (4,1)(), (5,2)(1), (6,7)(2) an (7,3)(2). Figure 6 shows the T-T graph representation of the GRM. The thin lines representing the gear meshes constitute the co-tree branches or links. ω5 ω4 b 4 a ω4 ω1 ω1 1 c 2 c ω21 ω21 e ω32 ω32 ω72 3 b ω51 ω61 ω62 ω72 ω Figure 6. T-T graph representation of the GRM. For the kinematic analysis of the GRM, we will consier only the angular velocities. Therefore the following terminal equations can be written by using Eq.(26) for the gear pairs: ( 4,1)( ) : w n w (27)

17 (5, 2 )(1) : w n w (28) ( 6, 7 )( 2 ) : w n w (29) ( 7, 3 )( 2 ) : w n w (3) From the graph shown in Fig. 5, the following funamental circuits (f-circuit) equations can be written easily. w w w (31) w w w (32) w w (33) 4 4 w w (34) 1 1 w w (35) w w (36) w w (37) w w (38)

18 Then, unknown angular velocities can be etermine in terms of input velocities by using the terminal an f-circuit equations. The angular velocities w, w, a n w are inputs an w, w, w, a n w are unknown angular velocities for the GRM. Using Eqs.(33), (33), an (27) w 1 can be obtaine as: w w 1/n w n w (39) Eqs.(35), (28), an (39) yiels: w w n w n w w n n w n w (4) For w 32, Eqs.(38), (3), (37), (36), (29), (32), an (4) are use. w w n w n w n n w n n w w w n n w n n w n n n n w n n n w n n w (41) Finally, w 72 can be obtaine from Eq.(41): w n w n n n w n n w n w (42) Or in more compact form: w 1 n 4 1 w 4 w n n n w 5 w n n n n n n n n n w 61 w n n n n n n (43) If we use the same notations for gear ratios as shown in Eq.(2) by using the following relations n i, n i, n i a n n i (44)

19 Then the following equation is obtaine which is the same as Eq.(25) w 1 i 1 5 w 4 w i i i w 5 w i i i i i i i i i w 61 w i i i i i i (45) Conclusion In this paper, the kinematic equations of the GRM are obtaine by using Matroi an T-T Graph Methos in sequence. Both methos use oriente graphs an represent the links with noes. In both methos, representations of turning pairs constitute the tree branches an gear pair representations constitute the links are co-tree branches. Matroi metho uses the oriente lines in orer to obtain the incient an path matrices an cycle-basis matrix is erive from these two matrices. Then using screw theory kinematic equations of the mechanism is obtaine. On the other han, T-T graph carries more information than Matroi. Since each line is a part of terminal graph of either turning of gear pair it carries complementary terminal variables. Therefore using terminal equations of the gear pairs, funamental circuit an funamental cut-set equations kinematic an static equations of the mechanism can be obtaine. This example shows that kinematic equations of the GRM can be obtaine more easily than Matroi Metho by using T-T Graph metho. The only avantage of the Matroi metho is the erivation of the relative velocities of the gears with respect to carrier arm. Since it uses the iameter of the gears an the iameters are written relative to the reference frame, the sign of the gear ratio is obtaine irectly without consiering the turning irection of the gears.

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