Generation and Forward Displacement Analysis of Two New Classes of Analytic 6-SPS Parallel Manipulators • • • • • • • • • • • • • • • • • • •
• • • • • • • • • • • • • • • •
Xianwen Kong, Clément M. Gosselin∗ Département de Génie Mécanique Université Laval, Québec Québec, G1K 7P4 Canada email:
[email protected]
Received 11 May 2000; accepted 6 January 2001
Analytic manipulators are manipulators with a characteristic polynomial of fourth degree or lower. Using the component approach to generate analytic 6-SPS parallel manipulators (PMs), the generation process is reduced to the generation of analytic components for 6-SPS PMs. Two new classes of analytic components for 6-SPS PMs are generated at first. Then, two new classes, IX and X, of analytic 6-SPS PMs are generated. The forward displacement analysis (FDA) of the new analytic 6-SPS PMs is also performed. The FDA of the 6-SPS PMs of class IX is reduced to the solution of one univariate cubic equation and two univariate quadratic equations, in sequence, while that of the 6-SPS PMs of class X is reduced to the solution of three univariate quadratic equations in sequence. Both of the new analytic 6-SPS PMs have at most eight assembly modes. © 2001 John Wiley & Sons, Inc.
1. INTRODUCTION Analytic manipulators are manipulators with a characteristic polynomial of fourth degree or lower. The forward displacement analysis (FDA) of analytic parallel manipulators (PMs) and the inverse displacement analysis of analytic serial manipulators can be performed analytically. Analytic manipulators have attracted considerable attention from researchers recently as these manipulators are suitable for practical manipulator design from the
* To whom all correspondence should be addressed.
Journal of Robotic Systems 18(6), 295–304 (2001) © 2001 by John Wiley & Sons, Inc.
kinematic point of view. The generation of analytic spatial serial manipulators,12 analytic planar PMs34 analytic spherical PMs,56 and analytic 6-SPS (Gough-Stewart platform) PMs7–11 has been dealt with in the literature. The 6-SPS PM is one of the most typical PMs proposed. Here S denotes a spherical joint which is passive while P denotes an actuated prismatic joint. It is constructed by connecting a moving platform and a base with six SPS legs (Fig. 1). In Figure 1 and throughout, each leg is represented by a dashdot line segment between the centers of its two S joints, each platform is represented by a transparent polygon in thick line whose vertices are the centers
296 Journal of Robotic Systems—2001
Figure 1.
The 6-SPS parallel manipulator.
of the S joints on the platform. A dotted line within the thick-line polygon is used to represent a nonplanar platform. The dotted-line separates the vertices of the polygon into two groups. The vertices in the group with three or more vertices are located on one plane, while the other vertices are not located on the plane. For a planar platform, there exists no dotted line. Up to now, eight classes of analytic 6-SPS PMs have been found.12 As the 6-SPS PM with similar planar platforms is a special case of the 6-SPS PM with linearly-related planar platforms proposed in refs. 10 and 11, class VIII analytic 6-SPS PM introduced in ref. 12 should be referred to as the 6-SPS PM with linearly-related planar platforms instead of the 6-SPS PM with similar planar platforms. Generating new analytic 6-SPS PMs is still an open issue. Using the component approach to generate analytic 6-SPS PMs,9 the generation of analytic 6-SPS PMs is reduced to the generation of analytic components for 6-SPS PMs. Two new classes of analytic components for 6-SPS PMs are generated at first. Then, two new classes of analytic 6-SPS PMs are generated. At last, the FDA of the new analytic 6-SPS PMs is performed.
2. THE COMPONENT APPROACH TO THE GENERATION OF ANALYTIC 6-SPS PMs A component is a part of the 6-SPS PM for which the number of inputs is equal to the number of
Figure 2. The PL (a) and the general LB (b) components for 6-SPS parallel manipulators.
degrees of freedom. Here, points, (straight) line segments, and rigid bodies should be regarded as elements of mechanisms. In the description of the components, the letters P L B and b, respectively, stand for point, line segment, rigid body, and planar rigid body. There are many classes of components for 6-SPS PMs.9 The ones related to this article are the PL and LB components. The PL component [Fig. 2(a)] is composed of a point B1 B2 and a line segment A1 A2 connected by two SPS legs Ai Bi , while the LB component [Fig. 2(b)] is composed of a line segment B1 B2 B3 B4 B5 and a rigid body A1 A2 A3 A4 A5 connected by five SPS legs Ai Bi . A 6-SPS PM can be decomposed into one or more components. A 6-SPS PM is analytic if all of its components are analytic. Thus, the generation of 6-SPS PMs is reduced to the generation of analytic components for 6-SPS PMs.
Kong and Gosselin: Generation and Forward Displacement Analysis 297
3. GENERATION OF NEW CLASSES OF ANALYTIC LB COMPONENTS In this section, we focus on the generation of analytic LB components. Up to now, only one class of analytic LB component has been proposed. That is the LB component with a planar base proposed by Zhang and Song.7 Here and throughout, it is denoted by the Lb component. The configuration analysis of the Lb component can be performed by solving a univariate quartic equation and a univariate quadratic equation in sequence. There are at most eight assembly modes for this class of analytic LB component.
3.1. Basic Idea for the Generation of New Analytic LB Components To facilitate the generation of new analytic LB components, the geometric conditions, revealed in refs. 3 and 13, to reduce the degree of characteristic polynomial for PMs are reviewed. It is revealed in ref. 13 that the maximum number of solutions to the forward displacement analysis of the true Stewart platform (i.e., a specific 6-SPS PM with three PL components) is 12. The number is smaller than 16 in the case of a general 6-SPS PM with three PL components. The reason for this is that the forward displacement analysis of the true Stewart platform can be reduced to that of a planar parallel structure. The geometric characteristic of the true Stewart platform is that all the straight line segments in its PL components are parallel. It was shown in ref. 3 that the collinearity of both platforms in a 3-RPR planar PM will lead to a decrease in the degree of the characteristic polynomial and the number of real assembly modes of the planar PM. The above results lead to a natural way of generating possible analytic LB components. A possible analytic LB component can be obtained by imposing the following geometric constraints, which are similar to those revealed in refs. 3 and 13 for PMs, to the LB component 1. There should exist two PL components in the LB component and the lines in the PL components are parallel to each other. These conditions ensure that the FDA of the LB component can be reduced to that of a planar parallel structure.
Figure 3.
The Lb PL//PL component.
2. The rigid body in the LB component should be coplanar. This condition guarantees that the platforms of the planar parallel structure are collinear. For convenience, the LB component satisfying the above two conditions is denoted by Lb PL//PL (Fig. 3). The degree of the characteristic polynomial of the Lb PL//PL component should be lower than that of the general LB component. The question now addressed is whether the Lb PL//PL component is analytic. In order to answer this question, let us perform the configuration analysis of the Lb PL//PL component.
3.2. Configuration Analysis of the LbPL//PL Component For purposes of simplification and without loss of generality, the coordinate system O–XY Z [Fig. 4(a)] is attached to the base with O being coincident with A1 , the Z-axis coinciding with A1 A2 and the X-axis intersecting A4 A5 . The XB -axis is attached along the moving line with OB being coincident with B1 B2 . The geometric parameters for the moving line B B and the base are* zA2 xA3 zA3 xA4 zA4 zA5 xB3 xB4 . Here, ai = xAi yAi zAi T and bi = xBi yBi zBi T denote the position vector of Ai and Bi in the coordinate B system O–XY Z, while xBi denotes the coordinate of Bi along the axis XB . * yA3 = 0 as the base of the Lb component is planar and located on the O–XZ plane.
298 Journal of Robotic Systems—2001 moving line with respect to the coordinate system O–XY Z. The configuration analysis can be performed by solving the following equations B
B
oB + xBi xi − ai T oB + xBi xi − ai = L2i i = 1 2 5
(1)
When the Lb PL//PL component is projected onto the O–XY coordinate plane [Fig. 4(a)], an imaginary 3-RR planar parallel structure (C1 C2 C3 –D1 D2 D3 ) with collinear platforms is obtained. Here, C1 , C2 , C3 , D1 , D2 and D3 are, respectively, the projection of A1 A2 , A4 A5 , A3 , B1 B2 , B4 B5 and B3 . The imaginary 3-RR planar parallel structure with collinear platforms is called the equivalent 3-RR planar parallel structure for the Lb PL//PL component [Fig. 4(a)]. C1 C2 C3 and D1 D2 D3 are called the base and moving platform, respectively, while Ci Di is called the leg of the equivalent 3-RR planar PM. The configuration analysis of the Lb PL//PL component can be performed using the following steps. Step 1. Calculate z and iz . Step 2. Compute the geometric parameters for the equivalent 3-RR planar parallel structure. Step 3. Perform the configuration analysis of the equivalent 3-RR planar parallel structure to calculate x, y, ix and iy .
3.2.1. Computation of z and iz Figure 4. Configuration analysis of the Lb PL//PL components: (a) reduction of the Lb PL//PL component to its equivalent 3-RR planar parallel structure with collinear platforms; (b) equivalent 3-RR planar parallel structure with collinear platforms.
The configuration analysis of the Lb PL//PL component can be stated as follows: for a given set of inputs Li = Ai Bi i = 1 2 5, find the pose (position and orientation) of the moving line. Here, the position of the moving line is denoted by the position vector of OB , oB = x y zT , in the coordinate system O–XY Z, and the orientation of the moving line by the unit vector, xi = ix iy iz T along the
The solution of the first two equations in Eq. (1) yields z = L21 − L22 + z2A2 /2zA2
(2)
The solution of the fourth and fifth equations in Eq. (1) gives the coordinate of B4 along the Z-axis zB4 = L24 − L25 + z2A5 − z2A4 /2zA5 − zA4
(3)
iz can then be obtained as B
iz = zB4 − z/xB4
(4)
Kong and Gosselin: Generation and Forward Displacement Analysis 299
3.2.2. Computation of the Geometric Parameters for the Equivalent 3-RR Planar Parallel Structure The dimensions of the base and the moving platform of the equivalent 3-RR planar parallel structure are denoted by a1 = C1 C2 , a2 = C1 C3 , b1 = D1 D2 and b2 = D1 D3 . The leg lengths of the 3-RR planar parallel structure are denoted by l1 = C1 D1 , l2 = C2 D2 and l3 = C3 D3 . Solving Eq. (2) and the first equation in Eq. (1), we have l1 = L21 − z2 1/2
a1 = xA4
(7)
a2 = xA3 1/2 B2 b1 = xB4 − zB4 − z2
(8)
b2 =
(9) (10)
The coordinate of B3 along the z-axis is zB3 =
B B z + xB3 zB4 − z/xB4
(11)
Thus, l2 can be calculated as l2 = L23 − zB3 − zA3 2 1/2
(12)
It can be seen that once the inputs Li of the Lb PL//PL component are given, the geometric parameters, ai , bi , and li , for the equivalent 3-RR planar parallel structure can be uniquely determined.
3.2.3. Configuration Analysis of the Equivalent 3-RR Planar Parallel Structure For clarity, the equivalent 3-RR planar parallel structure C1 C2 C3 –D1 D2 D3 is shown separately in Figure 4(b). Two coordinate systems are established in order to perform the configuration analysis of the 3-RR planar parallel structure. The coordinate system O–XY is attached to the base with O being coincident with C1 and the X-axis passing through C2 and C3 . The coordinate system OD –XD YD is attached to the moving platform with OD being coincident with D1 and the XD -axis passing through D2 and D3 . The configuration analysis of the 3-RR planar parallel structure can be stated as follows: for a
(13)
iy = sin
The closure equations of loops C1 D1 D2 C2 C1 and C1 D1 D3 C3 C1 in the complex form are l1 ei + b1 ei − a1 = l2 ei
(6)
It is obvious that
B B b1 xB3 /xB4
ix = cos
(5)
Solving Eq. (3) and the fourth equation in Eq. (1), we have l3 = L24 − zB4 − zA4 2 1/2
given set of leg lengths li = Ci Di i = 1 2 3, find the pose of the moving platform. Here, the position vector of OD x yT , in the coordinate system O–XY is used to denote the position of the moving platform, and the unit vector ix iy T along the Xi axis with respect to the coordinate system O–XY is used to denote the orientation of the moving platform. To facilitate the derivation, is used to denote the angle between the axes X and XD . It is clear that
l1 ei + b2 ei − a2 = l3 ei Multiplying the two members of each of the above equations with their complex conjugates respectively, we have l1 b1 cos − − l1 a1 cos − a1 b1 cos = d1 l1 b2 cos − − l1 a2 cos − a2 b2 cos = d2
(14)
where 1 2 − l12 − a2i − bi2 di = li+1 2
i = 1 2
From Figure 4, it can be seen that x = l1 cos y = l1 sin
(15)
The configuration analysis of the 3-RR planar parallel structure with collinear platforms can be performed by solving Eqs. (14) and (15) in conjunction with the corresponding condition given below. As the complexity of the configuration analysis of different classes of 3-RR planar parallel structures with collinear platforms varies to some extent, 3-RR planar parallel structures can be classified into the following two classes, namely: 1. The 3-RR planar parallel structure with nonsimilar collinear platforms, i.e., a 3-RR planar parallel structure which satisfies b1 /a1 = b2 /a2
(16)
2. The 3-RR planar parallel structure with similar collinear platforms, i.e., a 3-RR planar parallel structure which satisfies b1 /a1 = b2 /a2 = k = 0
(17)
300 Journal of Robotic Systems—2001
The 3-RR planar parallel structure with nonsimilar collinear platforms. The configuration analy-
sis of the planar 3-RR planar parallel structure with non-similar collinear platforms can be performed as follows. Multiplying the first equation in Eq. (14) by a2 b2 and then subtracting from it the second equation in Eq. (14) multiplied by a1 b1 , we have l1 b1 b2 a2 − a1 cos − − l1 a1 a2 b2 − b1 cos = a2 b2 d1 − a1 b1 d2
(18)
Multiplying the second equation in Eq. (14) by b1 and then subtracting from it the first equation in Eq. (14) multiplied by b2 , we get −b1 b2 a2 − a1 cos + l1 a1 b2 − a2 b1 cos = b1 d2 − b2 d1
c0 = −2z1 z3 c1 = l12 z23 − z24 + z22 − 2z3 z5 + z4 + z1 2 c2 = 2−l12 z3 z4 + z5 z4 + z1 c3 = l12 z24 + z25 − l12 z22 For each value of x obtained by solving Eq. (24), y can be calculated as y = ±l12 − x2 1/2
(25)
For a given set of x and y, ix and iy can be derived from Eqs. (21), (22), and (13) as ix = cos = z3 x − z4 /z2
(19)
The substitution of Eq. (15) into Eqs. (18) and (19) then yields z2 x cos + z2 y sin − z1 x = z5
where
(20)
iy = sin = z5 + z1 x − xz3 x − z4 /z2 y (26) if y = 0 ix = cos = z3 x − z4 /z2 iy = sin = ±1 − ix2 1/2
(27)
if y = 0
where z1 = a1 a2 b2 − b1 z2 = b1 b2 a2 − a1 z5 = a2 b2 d1 − a1 b1 d2 and z2 cos = z3 x − z4
(21)
where z3 = a1 b2 − a2 b1
The 3-RR planar parallel structure with similar collinear platforms. The configuration analysis
z 4 = b1 d 2 − b 2 d 1 From the substitution of Eq. (21) in Eq. (20), we get z2 y sin = z5 + z1 x − xz3 x − z4
(22)
of the 3-RR planar parallel structure with similar collinear platforms can be performed as follows. Substituting Eq. (17) into Eq. (14), we have kl1 a1 cos − − l1 a1 cos − ka21 cos = d1
From Eq. (15), we have x2 + y 2 = l12
The configuration analysis of the 3-RR planar parallel structure with non-similar collinear platforms is reduced to the solution of one univariate cubic equation and one univariate quadratic equation in sequence. Similar to the process proposed in ref. 3, it can be proved that there are at most two solutions to Eq. (24) in the interval −l1 l1 . Thus, there are at most four solutions to the configuration analysis of the 3-RR planar parallel structure with nonsimilar collinear platforms.
(23)
kl1 a2 cos − − l1 a2 cos − ka22 cos = d2
(28)
Substituting Eqs. (21)–(23) into the identity sin2 + cos2 = 1 multiplied by z22 y 2 , we get
Multiplying the second equation in Eq. (28) by a1 and then subtracting from it the first equation multiplied by a2 , we get
c0 x3 + c1 x2 + c2 x + c3 = 0
−kt1 cos = t2
(24)
(29)
Kong and Gosselin: Generation and Forward Displacement Analysis 301
where
Then, from Eq. (34), we get t1 = a1 a2 a2 − a1
y = t3 + t1 + t2 x/kt1 iy
t2 = a1 d2 − a2 d1 From Eqs. (29) and (13), we obtain the value for ix ix = cos = −t2 /kt1
(30)
For each solution of ix , iy can be calculated as iy = sin = ±1 − ix2 1/2
(31)
Thus, two solutions for ix iy T can be obtained using Eqs. (31) and (30). Multiplying the first equation in Eq. (28) by a22 and then subtracting from it the second equation in Eq. (28) multiplied by a21 , we have kl1 a1 a2 a2 − a1 cos − − l1 a1 a2 a2 − a1 cos =
a22 d1 − a21 d2
(32)
The substitution of Eq. (15) into Eq. (32) yields kt1 x cos + kt1 y sin − t1 x = t3
(33)
where t3 = a22 d1 − a21 d2 For a given value of ix and iy obtained above, the corresponding values of x and y can be obtained as follows. As singularities occur when iy = 0, we assume that iy = 0. Substituting Eq. (30) into Eq. (33), we have kt1 yiy = t3 + t1 + t2 x
(34)
The substitution of Eqs. (34), (31), and (30) into Eq. (23) multiplied by k2 t12 sin2 gives a quadratic in x, namely f0 x2 + f1 x + f2 = 0
(35)
where
It can be easily seen from Eqs. (31), (30), (36), and (37) that the configuration analysis of the 3-RR planar parallel structure with similar collinear platforms is reduced to the solution of two univariate quadratic equations in sequence. The maximum number of solutions to the configuration analysis of the 3-RR planar parallel structure with similar collinear platforms is 4.
3.3. Two New Classes of Analytic LB Components According to the complexity of the configuration analysis of the Lb PL//PL component, two new classes of analytic LB components can be generated. They are: 1. Class II analytic LB component. Class II analytic LB component is a Lb PL//PL component containing an equivalent 3-RR planar parallel structure with non-similar collinear platforms. 2. Class III analytic LB component. Class III analytic LB component is a Lb PL//PL component containing an equivalent 3-RR planar parallel structure with similar collinear platforms.
4. GENERATION OF TWO NEW CLASSES OF ANALYTIC 6-SPS PMs Corresponding to the two new classes, classes II and III, of analytic LB components, two new classes of analytic 6-SPS PMs can be generated (Fig. 5), namely 1. Class IX analytic 6-SPS PM. The class IX analytic 6-SPS PM is the one containing the class II analytic LB component. 2. Class X analytic 6-SPS PM. The class X analytic 6-SPS PM is the one containing the class III analytic LB component.
f0 = k2 + 1t12 + 2t1 t2 f1 = 2t3 t1 + t2 f2 = t32 − l12 k2 t12 − t22 The solution of Eq. (35) gives x = −f1 ± f12 − 4f0 f2 1/2 /2f0
(37)
(36)
where the class numbers of the analytic 6-SPS PMs refer to the classification scheme presented in [12].
302 Journal of Robotic Systems—2001 of the coordinate system of OB –XB YB ZB with respect to the coordinate system O–XY Z is used to denote the orientation of the moving platform. The set of equations for the FDA of the 6-SPS PM is B
B
oB + Rbi − ai T oB + Rbi − ai = L2i i = 1 2 6
(39)
The FDA of the two new classes of analytic 6-SPS PMs follows the same general steps presented below. Step 1. Perform the configuration analysis of the LB component to obtain x y z ix iy and iz . Figure 5. lators.
New classes of analytic 6-SPS parallel manipu-
5. THE FDA OF NEW CLASSES OF ANALYTIC 6-SPS PMs For purposes of simplification and without loss of generality, two coordinate systems are established. The coordinate system O–XY Z is attached to the base with O being coincident with A1 , the Z-axis coinciding with A1 A2 and the X-axis intersecting A4 A5 . The coordinate system OB –XB YB ZB is attached to the moving platform with OB being coincident with B1 B2 and the XB -axis passing through B3 and B4 B5 . YB is chosen to keep B6 located on the coordinate plane OB –XB YB . The geometric parameters for the platforms of the 6-SPS PM are zA2 , xA3 , zA3 , xA4 , B B B B zA4 , zA5 , xA6 , yA6 , zA6 , xB3 , xB4 , xB6 and yB6 . Here, ai = xAi yAi zAi T and bi = xBi yBi zBi T denote the position vector of Ai and Bi in the coordinate system B B B B O–XY Z, while bi = xBi yBi zBi T denotes the position vector of Bi in the coordinate system O–XB YB ZB . The FDA of the 6-SPS PM can be stated as follows: for a given set of inputs Li = Ai Bi i = 1 2 6, find the pose (position and orientation) of the moving platform. Here, the position vector of OB , oB = x y zT , in the coordinate system O–XY Z is used to denote the position of the moving platform, the direction cosine matrix ix jx kx jy ky R = (38) iy iz jz kz
The configuration analysis of the LB component is to solve the first five equations in Eq. (39) which is actually Eq. (1). It can be performed following the procedure presented in the Section 3.2. Step 2. Calculate jx jy jz kx ky and kz . These can be obtained following the procedure below. It is evident that ix jx + iy jy + iz jz = 0
(40)
jx2 + jy2 + jz2 = 1
(41)
The sixth equation of Eq. (39) is B
B
xB6 ix + yB6 jx + x − xA6 2 B
B
B
B
+ xB6 iy + yB6 jy + y − yA6 2 + xB6 iz + yB6 jz + z − zA6 2 = L26
(42)
Manipulation of Eqs. (40)–(42) gives jz = −F6 ± F62 − E6 Q6 1/2 /E6 jx = G5 jz + H5 jy = G6 jz + H6 where E6 = G25 + G26 + 1 F6 = G5 H5 + G 6 H6 Q6 = H52 + H62 − 1 G5 = B6 iz − C6 iy /)6 H5 = −D6 iy /)6
(43)
Kong and Gosselin: Generation and Forward Displacement Analysis 303 Table I. No 1 2 3 4
Real solutions for Example 1. ix
iy
iz
jx
jy
jz
kx
ky
kz
−01530 −09882 −00160 09777 −01490 −01477 01436 −00382 09890 −01530 −09882 −00160 01489 −00071 −09888 09771 −01537 01483 08393 05437 −00160 05432 −08362 00752 00275 −00718 −09972 08393 05437 −00160 −05377 08247 −01754 −00822 01558 09845
G6 = C6 ix − A6 iz /)6
y
z
01472 01472 46118 46118
92313 92313 79982 79982
79750 79750 79750 79750
are (see Fig. 2): zA2 = 5, xA3 = 7, zA3 = 4, xA4 = B 10 zA4 = 1, zA5 = 7, xA6 = 6, yA6 = 8, zA6 = 2, xB3 = 3, B B B xB4 = 6, xB6 = 3, yB6 = 3. The inputs of the manipulator are L1 = 122, L2 = 97, L3 = 104, L4 = 132, L5 = 113, L6 = 68. Of the eight sets of solutions we obtained, four sets are real (Table I).
H6 = D6 ix /)6 )6 = A6 iy − B6 ix B
A6 = 2yB6 x − xA6 B
B6 = 2yB6 y − yA6
Example 2. The dimensions of the base and the moving platform of the class X analytic 6-SPS PM are (see Fig. 2): zA2 = 5, xA3 = 5, zA3 = 4, xA4 = B 10 zA4 = 1, zA5 = 7, xA6 = 6, yA6 = 8, zA6 = 2, xB3 = 3, B B B xB4 = 6, xB6 = 3, yB6 = 3. The inputs of the manipulator are l1 = 15, l2 = 2, l3 = 2, l4 = 15, l5 = 2, l6 = 2. Of the eight sets of solutions we obtained, two sets are real (Table II).
B
C6 = 2yB6 z − zA6 B2
x
B2
D6 = xB6 + yB6 + x − xA6 2 + y − yA6 2 B
+ z − zA6 2 − L26 + 2xB6
× ix x − xA6 + iy y − yA6 + iz z − zA6 For any set of ix iy iz jx jy and jz kx ky and kz can be obtained as kx = iy jz − jy iz ky = iz jx − jz ix
7. CONCLUSIONS
(44)
kz = ix jy − jx iy
Two new classes, IX and X, of analytic 6-SPS PMs have been proposed. Both of these classes of 6-SPS PMs have at most eight sets of solutions to their FDA problem. The FDA was reduced to the solution of one univariate cubic equation and two univariate quadratic equations in sequence for class IX analytic 6-SPS PM and to the solution of three univariate quadratic equations in sequence for class X analytic 6-SPS PM. The characteristic of new analytic 6-SPS PMs is that they contain an analytic LB component with two PL components, while the characteristic of the analytic LB components with two PL components is that the body is coplanar and the lines in the PL components are parallel to each other. The reason why the new analytic 6-SPS PMs are analytic is their FDA can be reduced to the configuration analysis of analytic planar parallel structures.
The analysis presented above shows that the FDA was reduced to the solution of one univariate cubic equation and two univariate quadratic equations in sequence for class IX and to the solution of three univariate quadratic equations in sequence for class X. Both of these classes of 6-SPS PMs have at most 8 sets of solutions to their FDA problem.
6. EXAMPLES Numerical examples are now presented to illustrate the application of the method proposed in the previous sections. Example 1. The dimensions of the base and the moving platform of the class IX analytic 6-SPS PM Table II. No 1 2
Real solutions for Example 2.
ix 06887 06887
iy
iz
jx
jy
jz
kx
07251 −00160 −07243 06887 00324 00345 07251 −00160 07006 −06708 −02434 −01872
ky
kz
x
y
z
−00107 −01564
09995 09699
61196 61196
69131 69131
79750 79750
304 Journal of Robotic Systems—2001 It is noted that the analytic 6-SPS PM of class X will undergo finite motion once it reaches a configuration in which its equivalent 3-RR planar parallel structure has a finite degree of freedom. Also, the generation of all possible analytic 6-SPS PMs is still an open problem. The work presented in this article is useful in the context of the development of fast 6-SPS PMs.
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