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Kinematics of an asymmetrical three-legged parallel manipulator by means of the screw theory
Mechanism and Machine Theory 45 (2010) 1013–1023
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Mechanism and Machine Theory j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / m e c h m t
Kinematics of an asymmetrical three-legged parallel manipulator by means of the screw theory Jaime Gallardo-Alvarado ⁎, Agustín Ramírez-Agundis, Héctor Rojas-Garduño, Benjamín Arroyo-Ramírez Department of Mechanical and Electronic Engineering, Instituto Tecnológico de Celaya, Av. Tecnológico y A. García Cubas, 38010 Celaya, Gto., Mexico
a r t i c l e
i n f o
Article history: Received 13 August 2008 Received in revised form 30 October 2009 Accepted 19 February 2010 Available online 8 April 2010 Keywords: Parallel manipulator Decoupled motion Screw theory Kinematics
a b s t r a c t In this work the kinematics of a three-legged parallel manipulator with asymmetrical limbs and decoupled motions over its moving platform is investigated by means of the theory of screws. The solution of the forward displacement analysis (FDA) is presented in echelon-form solution using a novel procedure based on simple geometric constraints, thus all the feasible locations of the moving platform, with respect to the fixed platform, are computed given a set of generalized coordinates. After, the velocity and acceleration analyses are carried out using the theory of screws. Finally, a numerical example is included and the results obtained via the screw theory are verified with the aid of special commercially available software like MSC.Adams©. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction It is well-known that the finite kinematic analysis of parallel manipulators with identical or symmetrical limbs usually is a hazardous task due to the coupled motions generated over the moving platform. Consider for instance that the forward position analysis of a Gough-Stewart platform, a hexapod introduced as a tire testing machine almost five decades ago [1,2] and reconsidered later as a flight simulator by Stewart [3], is a challenging intensive task called by Freudenstein as the “Everest of modern kinematics.” The main characteristic of this analysis is the inescapable generation of nonlinear kinematic constraint equations, which lead, unfortunately, to multiple solutions. In fact, the moving platform can reach forty different locations, with respect to the fixed platform, given a set of generalized coordinates, Raghavan [4]. Although effective methods had been reported in the literature to find all the feasible configurations of the Gough-Stewart platform, see for instance [5–8]; the solution presented as a univariate polynomial equation is not precisely a closed or echelon form solution and therefore the determination of the actual pose, position and orientation, of the moving platform with respect to the fixed platform, without using sensors as it is reported, among many others, by Han et al. [9], Parenti-Castelli and Di Gregorio [10], and Chiu and Perng [11]; is an open problem. Certainly, now we are able to calculate efficiently in a reasonable time all the solutions, but there is not, as far as the authors know, an algorithm capable of determining the solutions set the current pose of the moving platform. An option to avoid coupled motions over the moving platform, which evidently simplifies the forward displacement analysis, is the introduction of parallel manipulators with asymmetrical limbs. In this fashion, the introduction of spatial mechanisms with fewer than six degrees of freedom, known as limited-dof or defective parallel manipulators, deserves special attention. In Karouia and Hervé [12] a new class of asymmetrical spherical parallel manipulators is introduced and its structural synthesis is carried out by applying group theory. Gallardo-Alvarado [13] proposed a serial-parallel manipulator built with two asymmetrical limited-dof parallel manipulators assembled in series-connection. Refaat et al. [14] investigated the mobility analysis of four families of three degrees of freedom asymmetrical parallel manipulators using Lie group theory. In Gallardo⁎ Corresponding author. E-mail address: [email protected] (J. Gallardo-Alvarado). 0094-114X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2010.02.003
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Alvarado et al. [15] the kinematics of a 3-dof asymmetrical parallel manipulator with a passive central kinematic chain is carried using the theory of screws. In this contribution a three-legged parallel manipulator, known as a SP + SPR + SPS manipulator (SPR = Spherical, Prismatic and Revolute, respectively), is investigated by means of the theory of screws. The forward displacement analysis is carried out using a novel procedure which allows computation of all the feasible locations that the moving platform can reach given a set of generalized coordinates. On the other hand, the velocity and acceleration analyses are carried using the theory of screws. Simple and compact expressions to solve the forward velocity and acceleration analyses are derived here by applying the properties of reciprocal screws. Finally, the displacement, workspace, velocity and acceleration analyses of a numerical example is provided in order to prove the efficacy of the proposed method. Furthermore, the numerical results obtained via the screw theory are compared with numerical results obtained using special commercially available software like MSC.Adams©. 2. Description of the mechanism The parallel manipulator under study, see Fig. 1, is composed of a moving platform, body 1, and a fixed platform, body 0, connected to each other by means of three asymmetrical limbs, one of them is rigidly attached to the moving platform, of type SP, SPR and SPS. The prismatic pairs play the role of active joints providing, according to a revised version of the Kutzbach-Grübler formula, three degrees of freedom, two rotations and one translation, to the moving platform, thus this mechanism is of the type translational-rotational. An essential characteristic of the mechanism at hand, due to the revolute joint and the SP limb, is that the moving platform cannot rotate, with respect to the fixed platform, along an axis perpendicular to the plane formed by the points B1, B2 and B3. Hence, this spatial mechanism belongs to the class of mechanisms known as parallel manipulators with zero-torsion [16]. Furthermore, for simplicity the SP-type limb is normal to this plane. On the other hand a decoupled parallel manipulator can be understood in different ways: i) the translational and rotational displacements of the moving platform are achieved by means of two distinct groups of active joints, ii) not all the active limbs are connected directly to the moving platform, and iii) one degree of freedom of the moving platform is affected by only one of the active joints, such mechanisms are called fully decoupled parallel manipulators. The parallel manipulator at hand meets the first definition of decoupled parallel manipulator. In fact, please note that according to the reference frame xyzo the position of the fixed platform, with respect to the moving platform, can be determined by computing the coordinates of point S1, which is affected only by the generalized coordinate q1, whereas the orientation between both platforms is specified only by the generalized coordinates q2 and q3. It is worth mentioning that despite the loss of degrees of freedom, parallel manipulators with restricted rotations are interesting mechanisms with potential industrial applications that motivated an exhaustive research field. The tripod joint, a 3-SPS parallel manipulator designed for automotive transmissions, was investigated in detail by Durum [17] and was used as a robot finger by Tischler et al. [18]. Perhaps, the most studied type of zero-torsion mechanism is the 3-RPS parallel manipulator introduced by Hunt [19]. It is straightforward to show that the poor manipulability of the three degrees of freedom spatial mechanism can be ameliorated by assembling in series connections several 3-RPS parallel manipulators. The kinematics and dynamics of such spatial mechanisms, known as hyper redundant manipulators, were approached, respectively, by Lu and Leinonen [20] and Gallardo-Alvarado et al. [21]. The Clemens joint or reflected tripod, a 3-RSR parallel manipulator, is a constant-velocity coupling whose kinematics has been exhaustively studied, see for instance Peruzzini et al. [22], Canfield et al. [23], Dunlop and Jones [24]. Since its introduction, kinematicians found interesting applications of this singular mechanism such as haptic devices and robotic wrists. It seems to be an easy task, maybe in some particular topologies, but a problem of zero-torsion parallel manipulator is the correct geometric interpretation of the restricted rotations and their effects on the performance of the spatial mechanism at hand. The exact type of rotational degrees of freedom of some zero-torsion parallel mechanisms was approached successfully by Bonev [16] who proposed novel rotational angles in order to elucidate such problem.
Fig. 1. The parallel manipulator and its geometric scheme.
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3. Finite kinematics In this section the displacement analysis of the parallel manipulator is presented. 3.1. Inverse displacement analysis The inverse displacement analysis (IDA) of the parallel manipulator is stated as follows: Given the pose of the moving platform with respect to the fixed platform, compute the resulting length qi(i = 1, 2, 3) of each limb. Let XYZO and xyzo be reference frames attached, respectively, at the fixed platform and at the moving platform, see Fig. 1. It is well known that the homogeneous transformation matrix of body 1 with respect to body 0, 0T1, can be computed given the position, vector ro/O, and orientation, rotation matrix 0R1, of the moving platform with respect to the fixed platform as follows
0 1
T =
0 1
R
01×3
ro = O : 1
ð1Þ
Furthermore, the coordinates of the points Bi = (BXi, BYi, BZi)i = 1, 2, 3 attached at the moving platform, expressed in the reference frame XYZO, can be calculated using the transformation matrix 0T1 and the coordinates of the same points but expressed in the reference frame xyzo, bi = (bxi, byi, bzi)i = 1, 2, 3, as follows 2
3 2 3 BXi bxi 6 BYi 7 0 1 6 byi 7 6 7 6 7 4 BZi 5 = T 4 bz 5 i = 1; 2; 3 i 1 1
ð2Þ
Finally, according to Fig. 1 it is evident that the generalized qi coordinates result in 2
qi = ðBi −Si Þ • ðBi −Si Þ
i = 1; 2; 3
ð3Þ
where the dot • denotes the inner product of the usual three dimensional vectorial algebra. 3.2. Forward displacement analysis The forward displacement analysis of the parallel manipulator consists of finding the pose of the moving platform, with respect to the fixed platform, given the generalized coordinates qi(i = 1, 2, 3) of the spatial mechanism. Taking into account that 0 11 0
T
T = I4
ð4Þ
where 1T0 is the transformation matrix of body 0 with respect to body 1, and I4 is the identity matrix of order 4, a suitable strategy to solve the FDA is summarized in the following steps: (1) The reference frame XYZO is placed in such a way that the Y axis is perperpendicular to the plane formed by the centers of the spherical joints mounted on the fixed platform while the Z axis is along the points S1 and S2, with the origin O at point S1. Similarly, the reference frame xyzo is chosen in such a way that the y axis is along the SP-type limb while the z axis is along the points B1 and B2, with the origin o at point B1 (2) Expressed in the reference frame xyzo, compute, in this sequence, the coordinates of the points S1, S2 and S3 (3) Compute the transformation matrix 1T0 (4) From Eq. (4) compute the transformation matrix 0T1, which represents the pose of the moving platform with respect to the fixed platform. With these considerations in mind, the coordinates of the center of the spherical joint S1 are obtained directly, see Fig. 1, as follows S1 = ð0; −q1; 0Þ
ð5Þ
Eq. (5) depends only of q1 and it is a reminder that certainly the mechanism at hand can be classified as a decoupled parallel manipulator. On the other hand, the coordinates of the center of the spherical joint S2 = (X2, Y2, Z2) depend on the position of the point S1, but not of the position of point S3. This geometric condition simplifies considerably the computation of the coordinates of the point S2. In fact, a linear equation can be obtained from the constraint imposed by the revolute joint as follows ̂ ðS2 −B2 Þ• U−q 2 cosθ = 0
ð6Þ
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where Û = Uxî + Uyj ̂ + Uzk̂ is a unit vector along the axis of the revolute joint, and θ is the constant angle between the SPR limb and the axis of the revolute joint. Furthermore, two compatibility equations can be written as follows 2
ðS2 −B2 Þ•ðS2 −B2 Þ−q2 = 0
ð7Þ
and 2
ðS2 −S1 Þ•ðS2 −S1 Þ−S12 = 0
ð8Þ
Subtracting Eqs. (7) and (8), and solving simultaneously with Eq. (6) one can obtain X2 = C1 + C2 Z2
ð9Þ
Y2 = C3 + C4 Z2
ð10Þ
and
where, the coefficients are given by C1 = Uy q21 −B212 −S212 + q22 + 2UzB12 q1 + 2q2 cosθq1 = 2Uxq1 C2 = Uy B12 −Uzq1 = Uxq1 C3 = B212 −q21 + S212 −q22 = 2q1 C4 = −B12 = q1
g
:
Substituting Eqs. (9) and (10) into Eq. (8) it follows that 2 2 2 2 2 2 2 1 + C4 + C2 Z2 + 2ðC1 C2 + C3 C4 + q1 C4 ÞZ2 + C1 + C3 + 2q1 C3 + q1 −S12 = 0:
ð11Þ
Finally, the coordinates of the point S2 are computed upon Eqs. (9)–(11). These equations show that there are at most two possible positions for point S2. Similarly, in order to compute the coordinates of the point S3 = (X3, Y3, Z3), it is necessary to take into account the following three kinematic constraint equations 2
ð12Þ
2
ð13Þ
ðS3 −S1 Þ•ðS3 −S1 Þ−S13 = 0 ðS3 −S2 Þ•ðS3 −S2 Þ−S23 = 0 2
ðS3 −B3 Þ•ðS3 −B3 Þ−q3 = 0:
ð14Þ
Substracting Eq. (12) from Eq. (14), and Eq. (13) from Eq. (14) a linear system of two equations in the unknowns X3, Y3 and Z3 is generated. Solving these two linear equations, the variables X3 and Y3 are obtained in terms of the variable Z3 as follows X3 = C5 + C6 Z3
ð15Þ
Y3 = C7 + C8 Z3
ð16Þ
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where, the respective coefficients are given by 2
2
2
2
2 2 2 2 C5 = ðX2 q1 −S23 q1 + Y2 q1 −B13 sin βq1 + Z2 q1 −B13 cos βq1 +
2
2
2
2
2
2
2
2
q3 q1 −Y2 B13 sin β + Y2 q1 −Y2 S13 + Y2 q3 −Y2 B13 cos βÞ = 2ðX2 q1 −B13 sinβq1 −B13 sinβY2 Þ C6 = ð−Z2 q1 + B13 cosβq1 + Y2 B13 cosβÞ =
2
2
2
2
C7 = −ðB13 sinβX2 −B13 sinβS23 + B13 sinβY2 + B13 sinβZ2 − 2
2
2
2
2
2
B13 sin βX2 + S13 B13 sinβ−B13 cos βX2 + q1 X2 −q1 B13 sinβ− 2 S13 X2
+
g
:
ðX2 q1 −B13 sinβq1 −B13 sinβY2 Þ
2
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2 q3 X2 Þ = 2ðX2 q1 −B13 sinβq1 −B13 sinβY2 Þ
C8 = ðB13 sinβZ2 −B13 cosβX2 Þ = ðX2 q1 −B13 sinβq1 −B13 sinβY2 Þ
Substituting Eqs. (15) and (16) into Eq. (14), the following quadratic equation in the unknown Z3 is obtained 2 2 2 2 2 2 2 1 + C6 + C8 Z3 + 2ðC5 C6 −B13 cosβ−C6 B13 sinβ + C7 C8 ÞZ3 −2C5 B13 sinβ + C7 −q3 + C5 + B13 = 0:
ð17Þ
Finally, the coordinates of the point S3 are calculated upon the Eqs. (15)–(17). Clearly, Eq. (17) indicates that there are at most two possible positions available for the point S3. Once the coordinates of the points Si(i = 1, 2, 3), expressed in the reference frame xyzo, are computed, the corresponding rotation matrix 1R0 of body 0 with respect to body 1, see Gallardo-Alvarado et al. [21], results in 1 0
R = ½Rx Ry Rz
ð18Þ
where Ry = (S1 − B1)/||(S1 − B1)||, Rz = (S2 − S1)/||(S2 − S1)|| and Rx = Ry × Rz. Expression (18) confirms the decoupled motions of the mechanism under study because the orientation between the moving and fixed platforms, trough the computation of 1R0, depends only on the generalized coordinates q2 and q3. Certainly, the three limbs of the manipulator are actuated simultaneously but they affect in a different form the pose between the moving and fixed platforms. Furthermore, the homogeneous transformation matrix 1T0 results in 1 0
T =
1
R0 01×3
S1 : 1
ð19Þ
Therefore, according to Eq. (4), the transformation matrix 0T1 is obtained as the inverse of the matrix 1T0. It is straightforward to show that once the transformation matrix 0T1 is computed, the coordinates of the points Bi(i = 1, 2, 3), expressed in the reference frame XYZO, are easily determined, see subsection 3.1. Furthermore, the center of the moving platform, τ, is computed as follows τ = ðB1 + B2 + B3 Þ = 3
ð20Þ
According to Eqs. (11) and (17), for the parallel manipulator under study the moving platform can reach four different poses with respect to the fixed platform, given a set of generalized coordinates. Finally, the procedure introduced in this subsection is simple, systematic, can be easily translated into computer codes and, as far as the authors are aware, is a novel apportation of this work. 4. Velocity analysis The mathematical tool to approach the infinitesimal kinematics of the parallel manipulator under study is the theory of screws, and the modeling of the corresponding infinitesimal screws is depicted in Fig. 2. Let V = [P(V); D(V)]T = [ωT; vTO]T be the velocity state, or twist about a screw [25], of body 1 with respect to body 0; where ω is the angular velocity of the moving platform and vO is the linear velocity of a point fixed at the moving platform, which is
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Fig. 2. The infinitesimal screws of the parallel manipulator.
instantaneously coincident with the origin O of the global reference frame XYZO. The six-dimensional vector V can be written, see Sugimoto and Duffy [26], in screw form, the corresponding infinitesimal screws are depicted in Fig. 2, through any of the three connector limbs as a linear combination of the joint rate velocities ω as follows i 0 1 0 ω1 $i
i 1 2
i 2 3
i 3 4
i 4 5
i 5 6
þ1 ω2 $i þ2 ω3 $i þ3 ω4 $i þ4 ω5 $i þ5 ω6 $i = V i = 1; 2; 3
ð21Þ
or re arranging in a matrix–vector form Ji Ωi = V
ð22Þ
i = 1; 2; 3
where Ji = [0$1i 1$2i 2$3i 3$4i 4$5i 5$6i ] is the local Jacobian matrix of the i-th limb and Ωi = [0ωi1; 1ωi2; 2ωi3; 3ωi4; 4ωi5; 5ωi6]T is a matrix containing the joint rate velocities in the same limb. It is opportune to mention that, in order to satisfy the rank of the Jacobians of the SP and SPR limbs, the screws 4$51, 5$61 and 5$62 are included as fictitious or auxiliary elements. The inverse velocity analysis (IVA) is stated as follows: given a velocity state V which describes the instantaneous motion of the moving platform with respect to the fixed platform, compute the joint rate velocities, or in other words the i-th matrix Ωi, of the parallel manipulator. Clearly, the IVA is quickly determined by means of Eq. (22). In fact −1
Ωi = Ji V
ð23Þ
i = 1; 2; 3:
Naturally, an elementary condition to apply Eq. (23) is that the Jacobian matrix Ji must be invertible, otherwise, the i-th limb is at a singular configuration. On the other hand, the forward velocity analysis (FVA) is stated as follows: given the active joint rate velocities 3ωi4(i = 1, 2, 3), compute the velocity state V of the moving platform with respect to the fixed platform. To this aim, consider that the screws 4$51 and 5$61 are fictitious elements, then the application of the Klein form, {*; *}, of the Lie algebra se (3) of the screws associated to the spherical joint of the SP limb with both sides of Eq. (21) leads to the following three expressions n
0 1 $1 ; V
1 2 $3
o
= 0;
n
1 2 $1 ; V
o
1
= 3 ω4 ;
n
2 3 $1 ; V
o
ð24Þ
= 0:
Similarly, since the screw 5$62, of the SPR limb, is a fictitious element, the application of the Klein form of the screws 0$12, 1$22 and with both sides of Eq. (21) gives the following expressions n
0 1 $2 ; V
o
= 0;
n
1 2 $2 ; V
o
2
= 3 ω4 ;
n
1 2 $3 ; V
o
3
= 3 ω4 :
ð25Þ
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Casting in a matrix–vector form Eqs. (24)–(35) the input–output velocity equation results in T
1
2
3 T
J Δ V = ½0;3 ω4 ; 0; 0;3 ω4 ;3 ω4
ð26Þ I 0 3×3 3 where J = [0$11 1$21 2$31 0$12 1$22 1$23] is the active Jacobian of the parallel manipulator, and Δ = is an operator of polarity. I3 03×3 Once the velocity state V is computed, provided that the active Jacobian J is invertible, by means of Eq. (26), the angular velocity of the moving platform is obtained as the primal part of V, ω = P(V), while the linear velocity of the center of the moving platform, vC, is computed using the dual part of the velocity state, vO = D(V), as follows
vC = vO + ω × τ
ð27Þ
5. Acceleration analysis Let A = [P(A); D(A)T = [ω˙ T; (aO–ω × vO)T]T be the reduced acceleration state, or accelerator for brevity [27], of body 1 with respect to body 0; where ω˙ is the angular acceleration of the moving platform and aO is the linear acceleration of a point fixed at the moving platform, which is instantaneously coincident with the origin O of the global reference frame XYZO. The sixdimensional vector A can be written, for details the reader is referred to Rico and Duffy [28], in screw form through any of the three connector limbs as follows i 0 1 ˙ 1 $i 0ω
i 1 2
i 2 3
i 3 4
i 4 5
i 5 6
þ1 ω˙ 2 $i þ2 ω˙ 3 $i þ3 ω˙ 4 $i þ4 ω˙ 5 $i þ5 ω˙ 6 $i + $Lie i = A i = 1; 2; 3
ð28Þ
or re arranging in a matrix–vector form Ji Ω˙ i = A−$Lie i i = 1; 2; 3
ð29Þ
where Ω˙ i = [0ω˙ i1; 1ω˙ i2; 2ω˙ i3; 3ω˙ i4; 4ω˙ i5; 5ω˙ i6]T is a matrix containing the joint rate accelerations of the i-th limb and $Liei is the i-th Lie screw which is computed as follows $Lie i =
h
i 0 1 i 1 2 0 ω1 $i 1 ω2 $i
þ2 ωi3 2 $3i + … þ5 ωi6 5 $6i
i
h i + 1 ωi2 1 $2i 2 ωi3 2 $3i þ3 ωi4 3 $4i + … þ5 ωi6 5 $6i h i + 2 ωi3 2 $3i 3 ωi4 3 $4i þ4 ωi5 4 $5i þ5 ωi6 5 $6i h i h i + 3 ωi4 3 $4i 4 ωi5 4 $5i þ5 ωi6 5 $6i + 4 ωi5 4 $5i 5 ωi6 5 $6i
ð30Þ
where the brackets [* *] denote the Lie product of the Lie algebra se(3) of the Euclidean group SE(3). The inverse acceleration analysis (IAA) is stated as follows: given an accelerator A which describes the instantaneous motion of the moving platform with respect to the fixed platform, compute the joint rate accelerations, or in other words the i-th matrix Ω̇i, of the parallel manipulator. The IAA is quickly determined by means of Eq. (29). In fact −1 Ω˙ i = Ji ðA−$Lie iÞi = 1; 2; 3:
ð31Þ
On the other hand, the forward acceleration analysis (FAA) is stated as follows: given the active joint rate accelerations 2, 3), compute the resulting accelerator A of the moving platform with respect to the fixed platform. The application of the Klein form of the screws associated to the spherical joint of the SP limb with both sides of Eq. (28) brings the following three expressions
˙ i4(i = 1, 3ω
n
0 1 $1 ; A
o
=
n
0 1 $1 ; $Lie 1
o n o n o n o n 1 2 1 1 2 2 3 2 3 ; $1 ; A = 3 ω˙ 4 + $1 ; $Lie 1 ; $1 ; A = $1 ; Lie 1g:
ð32Þ
Similarly, since the screw 5$62, of the SPR limb, is a fictitious element, the application of the Klein form of the screws 0$12, 1$22 and 1$23 with both sides of Eq. (28) leads to n
o
n
o
0 1 $2 ; A
1 2 $2 ; A
n
=
n
0 1 $2 ; $Lie 2
2
= 3 ω˙ 4 +
1 2 D3 ; Ag= 3ω˙ 34
+
o
ð33Þ
n
o
n
o :
1 2 $2 ; $Lie 2
1 2 D3 ;DLie 3
ð34Þ ð35Þ
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Table 1 Data of the numerical example. B12 = B13 = 0.5 S12 = S13 = 0.75 β = γ = π/3 Ux = 0.866, Uy = 0, Uz = 0.5 q1 = 1.0 + 0.2 sin t q2 = 1.0 + 0.4 sin t cos t q3 = 1.0 − 0.2 sin t, 0 ≤ t ≤ 2π S1 = (0, 0, 0) S2 = (0, 0, 0.75) S3 = (0.6495190528, 0, 0.375)
Table 2 The four solutions of the FDA. Sol
B1
B2
B3
1 2 3 4
(0.290e−1, .998, 0.506e−1) (.793, −.606, 0.506e−1) (−.352, −.596, .721) (.528, .447, .721)
(.119, .970, .541) (.717, −.664, .541) (−0.800e−1, −.974, .541) (.394, .894, .541)
(.500, (.500, (.112, (.112,
.976, .217) −.976, .217]) −.669, .887) .669, .887)
Casting in a matrix–vector form Eqs. (32)–(35) the input–output acceleration equation results in 2
n
0 1 $1 ; $Lie 1
o
3
6 7 6 n o7 6 1 7 1 2 6 3 ω˙ 4 + $1 ; $Lie 1 7 6 7 6 7 n o 6 7 2 3 $1 ; $Lie 1 6 7 T 6 7 J ΔA = 6 7: n o 6 7 0 1 $ ; $ 2 6 7 2 Lie 6 7 6 n o7 6 2 7 1 2 6 3 ω˙ 4 + $2 ; $Lie 2 7 6 7 4 n o5 3 1 2 ˙ 4 + $3 ; $Lie 3 3ω
ð36Þ
Once the accelerator A is computed by means of Eq. (36), the angular acceleration of the moving platform is obtained as the primal part of the accelerator A, ω˙ = P(A). Furthermore, since the accelerator A satisfies the conditions of helicoidal vector fields
Fig. 3. Workspace of the center of the moving platform.
J. Gallardo-Alvarado et al. / Mechanism and Machine Theory 45 (2010) 1013–1023
Fig. 4. Forward velocity and acceleration analyses of the center of the moving platform.
1021
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[29], the linear acceleration of the center of the moving platform, aC, is computed using the dual part of the accelerator, D(A), as follows DðAC Þ = DðAÞ + ω˙ × τ
ð37Þ
where D(AC) = aC − ω × vC. Therefore aC = DðAÞ + ω × vC + ω˙ × τ
ð38Þ
Finally it is interesting to note that the computation of the accelerator A, by means of Eq. (36), does not require the computation of the passive joint rate accelerations of the parallel manipulator. 6. Numerical example In order to prove the efficacy of the method of analysis introduced in this contribution to solve the kinematics of the SP + SPR + SPS parallel manipulator, in this section a numerical example is presented. The data, using SI units, of the example are listed in Table 1. At time t = 0, the resulting four possible poses of the moving platform, with respect to the fixed platform, are listed in Table 2. Considering the intervals 0.7≤qi ≤1.2(i=1, 2, 3) for the generalized coordinates; the reachable workspace of the center of the moving platform is shown in Fig. 3. Naturally, only real solutions of the FDA were taken into account to produce the corresponding surfaces. Finally, the numerical results obtained for the forward velocity and acceleration analyses of the center of the moving platform using the theory of screws, taking solution 1 of Table 2 as the home position of the parallel manipulator, are reported in Fig. 4. Furthermore, in order to verify the results of the numerical example obtained using the screw theory, a faster and lower cost option than a physical prototype, is the use of special software like MSC.Adams© [30], which is the commercial denomination of a software package that realizes several kinds of analyses of mechanical devices. In particular Adams/View is a module of MSC. Adams©capable to make kinematic simulations of mechanisms by means of animations. The corresponding results of the simulation of the problem at hand using Adams/View are reported in Fig. 4. 7. Conclusions In this contribution the displacement, velocity and acceleration analyses of a SP + SPR + SPS parallel manipulator are carried out by means of the theory of screws. The forward displacement analysis is solved using a novel method based on simple geometric–kinematic constraints which allows computation of all the feasible poses of the moving platform, with respect to the fixed platform, given a set of generalized coordinates. Since only four solutions are available for the FDA, then the actual configuration of the parallel manipulator at hand can be easily determined without using sensors. Simple, elegant and compact expressions to solve the forward velocity and acceleration analyses are introduced here by taking advantage of the concept of reciprocal screws via the Klein form of the Lie algebra se(3). To this end, fictitious infinitesimal screws are considered in order to satisfy algebraic procedures. It is worth mentioning that with this method, the solution of the forward acceleration analysis is carried out without the computation of the passive joint rate accelerations of the parallel manipulator. Finally, the kinematics, including the determination of the workspace, of a numerical example is solved using the proposed method. Furthermore, the numerical results obtained via the screw theory for the forward velocity and acceleration analyses are compared with results generated with the help of special software like MSC.Adams©. Please note, according to Fig. 4, that the numerical results generated using the screw theory are in excellent agreement with those obtained with MSC.Adams©. References [1] V.E. Gough, Contribution to discussion of papers on research in automobile stability, control and in tyre performance, Proceedings Automation Division Institution of Mechanical Engineers (1956) 392–394. [2] V.E. Gough, S.G. Whitehall, Universal tyre test machine, Proceedings 9th International Technical Congress FISITA, 1962, pp. 117–135. [3] D. Stewart, A platform with six degrees of freedom, Proceedings Institution of Mechanical Engineers 180 (5) (1965) 371–378. [4] M. Raghavan, The Stewart platform of general geometry has 40 configurations, ASME Journal of Mechanical Design 115 (2) (1993) 277–282. [5] C. Innocenti, V. Parenti-Castelli, Direct position analysis of the Stewart platform mechanism, Mechanism and Machine Theory 25 (6) (1990) 611–621. [6] M.L. Husty, An algorithm for solving the direct kinematics of general Stewart-Gough platforms, Mechanism and Machine Theory 31 (4) (1996) 365–379. [7] C. Innocenti, Forward kinematics in polynomial form of the general Stewart platform, Proceedings ASME 1998 Design Engineering Technical Conference, 25th Biennial Mechanisms Conference, Atlanta, Georgia, CD-Rom format, 1998, Paper DETC98/MECH-5894. [8] L. Rolland, Certified solving of the forward kinematics problem with an exact algebraic method for the general parallel manipulator, Advanced Robotics 19 (9) (2005) 995–1025. [9] K. Han, W. Chun, Y. Youm, New resolution scheme of the forward kinematics of parallel manipulators using extra sensors, ASME Journal of Mechanical Design 118 (2) (1996) 214–219. [10] V. Parenti-Castelli, R. Di Gregorio, Determination of the actual configuration of the general Stewart platform using only one additional sensor, ASME Journal of Mechanical Design 121 (1) (1999) 21–25. [11] Y.J. Chiu, M.H. Perng, Forward kinematics of a general fully parallel manipulator with auxiliary sensors, The International Journal of Robotics Research 20 (5) (2001) 401–414. [12] M. Karouia, J.M. Hervé, Asymmetrical 3-dof spherical parallel mechanisms, European Journal of Mechanics - A/Solids 24 (1) (2004) 47–57. [13] J. Gallardo-Alvarado, Kinematics of a hybrid manipulator by means of screw theory, Multibody System Dynamics 14 (3) (2005) 345–366. [14] S. Refaat, J.M. Hervé, S. Nahavandi, H. Trinh, Asymmetrical three-DOFs rotational–translational parallel-kinematics mechanisms based on Lie group theory, European Journal of Mechanics - A/Solids 25 (3) (2006) 550–558.
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[15] J. Gallardo-Alvarado, H. Orozco-Mendoza, A. Maeda-Sánchez, Acceleration and singularity analyses of a parallel manipulator with a particular topology, Meccanica 42 (3) (2007) 223–238. [16] I.A. Bonev, Geometric Analysis of Parallel Mechanisms, thèse de doctorat, Université Laval, Canada, November 2002. [17] M.M. Durum, Kinematic properties of Tripode (Tri-pot) joints, ASME Journal of Engineering for Industry 978 (1975) 708–713. [18] C.R. Tischler, K.H. Hunt, A.E. Samuel, On optimizing the kinematic geometry of a dexterous robot finger, International Journal of Robotics Research 17 (10) (1998) 1055–1067. [19] K.H. Hunt, Structural kinematics of in-parallel actuated robot arms, ASME Journal of Mechanisms, Transmissions, and Automation in Design 105 (4) (1983) 705–712. [20] Y. Lu, T. Leinonen, Solution and simulation of position-orientation for multi-spatial 3-RPS parallel mechanisms in series connection, Multibody System Dynamics 14 (1) (2005) 47–60. [21] J. Gallardo-Alvarado, C.R. Aguilar-Nájera, L. Casique-Rosas, L. Pérez-González, J.M. Rico-Martínez, Solving the kinematics and dynamics of a modular spatial hyper-redundant manipulator by means of screw theory, Multibody System Dynamics 20 (4) (2008) 307–325. [22] W. Peruzzini, A. Oullet, R. Hui, Design and analysis of a three-dof parallel mechanism, Proceedings of the Ninth World Congress Theory Machines and Mechanisms IFToMM, Milan, Italy, 1995, pp. 2141–2145. [23] S.L. Canfield, A.J. Ganino, R.J. Salerno, C.F. Reinholtz, Singularity and dexterity analysis of the carpal wrist, Proceedings ASME 1996 Design Engineering Technical Conference, 24th Biennial Mechanisms Conference, Irvine, California, 1996, paper MECH-1156. [24] G.R. Dunlop, T.P. Jones, Position analysis of a 3-dof parallel manipulator, Mechanism and Machine Theory 32 (8) (1997) 903–920. [25] R.S. Ball, A Treatise on the Theory of Screws, Cambridge University Press, Cambridge, 1900 (Reprinted 1998). [26] K. Sugimoto, J. Duffy, Application of linear algebra to screw systems, Mechanism and Machine Theory 17 (1) (1982) 73–83. [27] K. Sugimoto, Kinematic and dynamic analysis of parallel manipulators by means of motor algebra, ASME Journal of Mechanisms Transmission and Automation in Design 109 (1) (1987) 3–7. [28] J.M. Rico, J. Duffy, An application of screw algebra to the acceleration analysis of serial chains, Mechanism and Machine Theory 31 (4) (1996) 445–457. [29] J. Gallardo-Alvarado, H. Orozco-Mendoza, R. Rodríguez-Castro, Finding the jerk properties of multi-body systems using helicoidal vector fields, Proceedings IMechE Part C: Journal of Mechanical Engineering Science 222 (11) (2008) 2217–2229. [30] MSC.Adams; available on line at http://www.mscsoftware.com.
Contents lists available at ScienceDirect
Mechanism and Machine Theory j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / m e c h m t
Kinematics of an asymmetrical three-legged parallel manipulator by means of the screw theory Jaime Gallardo-Alvarado ⁎, Agustín Ramírez-Agundis, Héctor Rojas-Garduño, Benjamín Arroyo-Ramírez Department of Mechanical and Electronic Engineering, Instituto Tecnológico de Celaya, Av. Tecnológico y A. García Cubas, 38010 Celaya, Gto., Mexico
a r t i c l e
i n f o
Article history: Received 13 August 2008 Received in revised form 30 October 2009 Accepted 19 February 2010 Available online 8 April 2010 Keywords: Parallel manipulator Decoupled motion Screw theory Kinematics
a b s t r a c t In this work the kinematics of a three-legged parallel manipulator with asymmetrical limbs and decoupled motions over its moving platform is investigated by means of the theory of screws. The solution of the forward displacement analysis (FDA) is presented in echelon-form solution using a novel procedure based on simple geometric constraints, thus all the feasible locations of the moving platform, with respect to the fixed platform, are computed given a set of generalized coordinates. After, the velocity and acceleration analyses are carried out using the theory of screws. Finally, a numerical example is included and the results obtained via the screw theory are verified with the aid of special commercially available software like MSC.Adams©. © 2010 Elsevier Ltd. All rights reserved.
1. Introduction It is well-known that the finite kinematic analysis of parallel manipulators with identical or symmetrical limbs usually is a hazardous task due to the coupled motions generated over the moving platform. Consider for instance that the forward position analysis of a Gough-Stewart platform, a hexapod introduced as a tire testing machine almost five decades ago [1,2] and reconsidered later as a flight simulator by Stewart [3], is a challenging intensive task called by Freudenstein as the “Everest of modern kinematics.” The main characteristic of this analysis is the inescapable generation of nonlinear kinematic constraint equations, which lead, unfortunately, to multiple solutions. In fact, the moving platform can reach forty different locations, with respect to the fixed platform, given a set of generalized coordinates, Raghavan [4]. Although effective methods had been reported in the literature to find all the feasible configurations of the Gough-Stewart platform, see for instance [5–8]; the solution presented as a univariate polynomial equation is not precisely a closed or echelon form solution and therefore the determination of the actual pose, position and orientation, of the moving platform with respect to the fixed platform, without using sensors as it is reported, among many others, by Han et al. [9], Parenti-Castelli and Di Gregorio [10], and Chiu and Perng [11]; is an open problem. Certainly, now we are able to calculate efficiently in a reasonable time all the solutions, but there is not, as far as the authors know, an algorithm capable of determining the solutions set the current pose of the moving platform. An option to avoid coupled motions over the moving platform, which evidently simplifies the forward displacement analysis, is the introduction of parallel manipulators with asymmetrical limbs. In this fashion, the introduction of spatial mechanisms with fewer than six degrees of freedom, known as limited-dof or defective parallel manipulators, deserves special attention. In Karouia and Hervé [12] a new class of asymmetrical spherical parallel manipulators is introduced and its structural synthesis is carried out by applying group theory. Gallardo-Alvarado [13] proposed a serial-parallel manipulator built with two asymmetrical limited-dof parallel manipulators assembled in series-connection. Refaat et al. [14] investigated the mobility analysis of four families of three degrees of freedom asymmetrical parallel manipulators using Lie group theory. In Gallardo⁎ Corresponding author. E-mail address: [email protected] (J. Gallardo-Alvarado). 0094-114X/$ – see front matter © 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2010.02.003
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Alvarado et al. [15] the kinematics of a 3-dof asymmetrical parallel manipulator with a passive central kinematic chain is carried using the theory of screws. In this contribution a three-legged parallel manipulator, known as a SP + SPR + SPS manipulator (SPR = Spherical, Prismatic and Revolute, respectively), is investigated by means of the theory of screws. The forward displacement analysis is carried out using a novel procedure which allows computation of all the feasible locations that the moving platform can reach given a set of generalized coordinates. On the other hand, the velocity and acceleration analyses are carried using the theory of screws. Simple and compact expressions to solve the forward velocity and acceleration analyses are derived here by applying the properties of reciprocal screws. Finally, the displacement, workspace, velocity and acceleration analyses of a numerical example is provided in order to prove the efficacy of the proposed method. Furthermore, the numerical results obtained via the screw theory are compared with numerical results obtained using special commercially available software like MSC.Adams©. 2. Description of the mechanism The parallel manipulator under study, see Fig. 1, is composed of a moving platform, body 1, and a fixed platform, body 0, connected to each other by means of three asymmetrical limbs, one of them is rigidly attached to the moving platform, of type SP, SPR and SPS. The prismatic pairs play the role of active joints providing, according to a revised version of the Kutzbach-Grübler formula, three degrees of freedom, two rotations and one translation, to the moving platform, thus this mechanism is of the type translational-rotational. An essential characteristic of the mechanism at hand, due to the revolute joint and the SP limb, is that the moving platform cannot rotate, with respect to the fixed platform, along an axis perpendicular to the plane formed by the points B1, B2 and B3. Hence, this spatial mechanism belongs to the class of mechanisms known as parallel manipulators with zero-torsion [16]. Furthermore, for simplicity the SP-type limb is normal to this plane. On the other hand a decoupled parallel manipulator can be understood in different ways: i) the translational and rotational displacements of the moving platform are achieved by means of two distinct groups of active joints, ii) not all the active limbs are connected directly to the moving platform, and iii) one degree of freedom of the moving platform is affected by only one of the active joints, such mechanisms are called fully decoupled parallel manipulators. The parallel manipulator at hand meets the first definition of decoupled parallel manipulator. In fact, please note that according to the reference frame xyzo the position of the fixed platform, with respect to the moving platform, can be determined by computing the coordinates of point S1, which is affected only by the generalized coordinate q1, whereas the orientation between both platforms is specified only by the generalized coordinates q2 and q3. It is worth mentioning that despite the loss of degrees of freedom, parallel manipulators with restricted rotations are interesting mechanisms with potential industrial applications that motivated an exhaustive research field. The tripod joint, a 3-SPS parallel manipulator designed for automotive transmissions, was investigated in detail by Durum [17] and was used as a robot finger by Tischler et al. [18]. Perhaps, the most studied type of zero-torsion mechanism is the 3-RPS parallel manipulator introduced by Hunt [19]. It is straightforward to show that the poor manipulability of the three degrees of freedom spatial mechanism can be ameliorated by assembling in series connections several 3-RPS parallel manipulators. The kinematics and dynamics of such spatial mechanisms, known as hyper redundant manipulators, were approached, respectively, by Lu and Leinonen [20] and Gallardo-Alvarado et al. [21]. The Clemens joint or reflected tripod, a 3-RSR parallel manipulator, is a constant-velocity coupling whose kinematics has been exhaustively studied, see for instance Peruzzini et al. [22], Canfield et al. [23], Dunlop and Jones [24]. Since its introduction, kinematicians found interesting applications of this singular mechanism such as haptic devices and robotic wrists. It seems to be an easy task, maybe in some particular topologies, but a problem of zero-torsion parallel manipulator is the correct geometric interpretation of the restricted rotations and their effects on the performance of the spatial mechanism at hand. The exact type of rotational degrees of freedom of some zero-torsion parallel mechanisms was approached successfully by Bonev [16] who proposed novel rotational angles in order to elucidate such problem.
Fig. 1. The parallel manipulator and its geometric scheme.
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3. Finite kinematics In this section the displacement analysis of the parallel manipulator is presented. 3.1. Inverse displacement analysis The inverse displacement analysis (IDA) of the parallel manipulator is stated as follows: Given the pose of the moving platform with respect to the fixed platform, compute the resulting length qi(i = 1, 2, 3) of each limb. Let XYZO and xyzo be reference frames attached, respectively, at the fixed platform and at the moving platform, see Fig. 1. It is well known that the homogeneous transformation matrix of body 1 with respect to body 0, 0T1, can be computed given the position, vector ro/O, and orientation, rotation matrix 0R1, of the moving platform with respect to the fixed platform as follows
0 1
T =
0 1
R
01×3
ro = O : 1
ð1Þ
Furthermore, the coordinates of the points Bi = (BXi, BYi, BZi)i = 1, 2, 3 attached at the moving platform, expressed in the reference frame XYZO, can be calculated using the transformation matrix 0T1 and the coordinates of the same points but expressed in the reference frame xyzo, bi = (bxi, byi, bzi)i = 1, 2, 3, as follows 2
3 2 3 BXi bxi 6 BYi 7 0 1 6 byi 7 6 7 6 7 4 BZi 5 = T 4 bz 5 i = 1; 2; 3 i 1 1
ð2Þ
Finally, according to Fig. 1 it is evident that the generalized qi coordinates result in 2
qi = ðBi −Si Þ • ðBi −Si Þ
i = 1; 2; 3
ð3Þ
where the dot • denotes the inner product of the usual three dimensional vectorial algebra. 3.2. Forward displacement analysis The forward displacement analysis of the parallel manipulator consists of finding the pose of the moving platform, with respect to the fixed platform, given the generalized coordinates qi(i = 1, 2, 3) of the spatial mechanism. Taking into account that 0 11 0
T
T = I4
ð4Þ
where 1T0 is the transformation matrix of body 0 with respect to body 1, and I4 is the identity matrix of order 4, a suitable strategy to solve the FDA is summarized in the following steps: (1) The reference frame XYZO is placed in such a way that the Y axis is perperpendicular to the plane formed by the centers of the spherical joints mounted on the fixed platform while the Z axis is along the points S1 and S2, with the origin O at point S1. Similarly, the reference frame xyzo is chosen in such a way that the y axis is along the SP-type limb while the z axis is along the points B1 and B2, with the origin o at point B1 (2) Expressed in the reference frame xyzo, compute, in this sequence, the coordinates of the points S1, S2 and S3 (3) Compute the transformation matrix 1T0 (4) From Eq. (4) compute the transformation matrix 0T1, which represents the pose of the moving platform with respect to the fixed platform. With these considerations in mind, the coordinates of the center of the spherical joint S1 are obtained directly, see Fig. 1, as follows S1 = ð0; −q1; 0Þ
ð5Þ
Eq. (5) depends only of q1 and it is a reminder that certainly the mechanism at hand can be classified as a decoupled parallel manipulator. On the other hand, the coordinates of the center of the spherical joint S2 = (X2, Y2, Z2) depend on the position of the point S1, but not of the position of point S3. This geometric condition simplifies considerably the computation of the coordinates of the point S2. In fact, a linear equation can be obtained from the constraint imposed by the revolute joint as follows ̂ ðS2 −B2 Þ• U−q 2 cosθ = 0
ð6Þ
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where Û = Uxî + Uyj ̂ + Uzk̂ is a unit vector along the axis of the revolute joint, and θ is the constant angle between the SPR limb and the axis of the revolute joint. Furthermore, two compatibility equations can be written as follows 2
ðS2 −B2 Þ•ðS2 −B2 Þ−q2 = 0
ð7Þ
and 2
ðS2 −S1 Þ•ðS2 −S1 Þ−S12 = 0
ð8Þ
Subtracting Eqs. (7) and (8), and solving simultaneously with Eq. (6) one can obtain X2 = C1 + C2 Z2
ð9Þ
Y2 = C3 + C4 Z2
ð10Þ
and
where, the coefficients are given by C1 = Uy q21 −B212 −S212 + q22 + 2UzB12 q1 + 2q2 cosθq1 = 2Uxq1 C2 = Uy B12 −Uzq1 = Uxq1 C3 = B212 −q21 + S212 −q22 = 2q1 C4 = −B12 = q1
g
:
Substituting Eqs. (9) and (10) into Eq. (8) it follows that 2 2 2 2 2 2 2 1 + C4 + C2 Z2 + 2ðC1 C2 + C3 C4 + q1 C4 ÞZ2 + C1 + C3 + 2q1 C3 + q1 −S12 = 0:
ð11Þ
Finally, the coordinates of the point S2 are computed upon Eqs. (9)–(11). These equations show that there are at most two possible positions for point S2. Similarly, in order to compute the coordinates of the point S3 = (X3, Y3, Z3), it is necessary to take into account the following three kinematic constraint equations 2
ð12Þ
2
ð13Þ
ðS3 −S1 Þ•ðS3 −S1 Þ−S13 = 0 ðS3 −S2 Þ•ðS3 −S2 Þ−S23 = 0 2
ðS3 −B3 Þ•ðS3 −B3 Þ−q3 = 0:
ð14Þ
Substracting Eq. (12) from Eq. (14), and Eq. (13) from Eq. (14) a linear system of two equations in the unknowns X3, Y3 and Z3 is generated. Solving these two linear equations, the variables X3 and Y3 are obtained in terms of the variable Z3 as follows X3 = C5 + C6 Z3
ð15Þ
Y3 = C7 + C8 Z3
ð16Þ
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where, the respective coefficients are given by 2
2
2
2
2 2 2 2 C5 = ðX2 q1 −S23 q1 + Y2 q1 −B13 sin βq1 + Z2 q1 −B13 cos βq1 +
2
2
2
2
2
2
2
2
q3 q1 −Y2 B13 sin β + Y2 q1 −Y2 S13 + Y2 q3 −Y2 B13 cos βÞ = 2ðX2 q1 −B13 sinβq1 −B13 sinβY2 Þ C6 = ð−Z2 q1 + B13 cosβq1 + Y2 B13 cosβÞ =
2
2
2
2
C7 = −ðB13 sinβX2 −B13 sinβS23 + B13 sinβY2 + B13 sinβZ2 − 2
2
2
2
2
2
B13 sin βX2 + S13 B13 sinβ−B13 cos βX2 + q1 X2 −q1 B13 sinβ− 2 S13 X2
+
g
:
ðX2 q1 −B13 sinβq1 −B13 sinβY2 Þ
2
1017
2 q3 X2 Þ = 2ðX2 q1 −B13 sinβq1 −B13 sinβY2 Þ
C8 = ðB13 sinβZ2 −B13 cosβX2 Þ = ðX2 q1 −B13 sinβq1 −B13 sinβY2 Þ
Substituting Eqs. (15) and (16) into Eq. (14), the following quadratic equation in the unknown Z3 is obtained 2 2 2 2 2 2 2 1 + C6 + C8 Z3 + 2ðC5 C6 −B13 cosβ−C6 B13 sinβ + C7 C8 ÞZ3 −2C5 B13 sinβ + C7 −q3 + C5 + B13 = 0:
ð17Þ
Finally, the coordinates of the point S3 are calculated upon the Eqs. (15)–(17). Clearly, Eq. (17) indicates that there are at most two possible positions available for the point S3. Once the coordinates of the points Si(i = 1, 2, 3), expressed in the reference frame xyzo, are computed, the corresponding rotation matrix 1R0 of body 0 with respect to body 1, see Gallardo-Alvarado et al. [21], results in 1 0
R = ½Rx Ry Rz
ð18Þ
where Ry = (S1 − B1)/||(S1 − B1)||, Rz = (S2 − S1)/||(S2 − S1)|| and Rx = Ry × Rz. Expression (18) confirms the decoupled motions of the mechanism under study because the orientation between the moving and fixed platforms, trough the computation of 1R0, depends only on the generalized coordinates q2 and q3. Certainly, the three limbs of the manipulator are actuated simultaneously but they affect in a different form the pose between the moving and fixed platforms. Furthermore, the homogeneous transformation matrix 1T0 results in 1 0
T =
1
R0 01×3
S1 : 1
ð19Þ
Therefore, according to Eq. (4), the transformation matrix 0T1 is obtained as the inverse of the matrix 1T0. It is straightforward to show that once the transformation matrix 0T1 is computed, the coordinates of the points Bi(i = 1, 2, 3), expressed in the reference frame XYZO, are easily determined, see subsection 3.1. Furthermore, the center of the moving platform, τ, is computed as follows τ = ðB1 + B2 + B3 Þ = 3
ð20Þ
According to Eqs. (11) and (17), for the parallel manipulator under study the moving platform can reach four different poses with respect to the fixed platform, given a set of generalized coordinates. Finally, the procedure introduced in this subsection is simple, systematic, can be easily translated into computer codes and, as far as the authors are aware, is a novel apportation of this work. 4. Velocity analysis The mathematical tool to approach the infinitesimal kinematics of the parallel manipulator under study is the theory of screws, and the modeling of the corresponding infinitesimal screws is depicted in Fig. 2. Let V = [P(V); D(V)]T = [ωT; vTO]T be the velocity state, or twist about a screw [25], of body 1 with respect to body 0; where ω is the angular velocity of the moving platform and vO is the linear velocity of a point fixed at the moving platform, which is
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Fig. 2. The infinitesimal screws of the parallel manipulator.
instantaneously coincident with the origin O of the global reference frame XYZO. The six-dimensional vector V can be written, see Sugimoto and Duffy [26], in screw form, the corresponding infinitesimal screws are depicted in Fig. 2, through any of the three connector limbs as a linear combination of the joint rate velocities ω as follows i 0 1 0 ω1 $i
i 1 2
i 2 3
i 3 4
i 4 5
i 5 6
þ1 ω2 $i þ2 ω3 $i þ3 ω4 $i þ4 ω5 $i þ5 ω6 $i = V i = 1; 2; 3
ð21Þ
or re arranging in a matrix–vector form Ji Ωi = V
ð22Þ
i = 1; 2; 3
where Ji = [0$1i 1$2i 2$3i 3$4i 4$5i 5$6i ] is the local Jacobian matrix of the i-th limb and Ωi = [0ωi1; 1ωi2; 2ωi3; 3ωi4; 4ωi5; 5ωi6]T is a matrix containing the joint rate velocities in the same limb. It is opportune to mention that, in order to satisfy the rank of the Jacobians of the SP and SPR limbs, the screws 4$51, 5$61 and 5$62 are included as fictitious or auxiliary elements. The inverse velocity analysis (IVA) is stated as follows: given a velocity state V which describes the instantaneous motion of the moving platform with respect to the fixed platform, compute the joint rate velocities, or in other words the i-th matrix Ωi, of the parallel manipulator. Clearly, the IVA is quickly determined by means of Eq. (22). In fact −1
Ωi = Ji V
ð23Þ
i = 1; 2; 3:
Naturally, an elementary condition to apply Eq. (23) is that the Jacobian matrix Ji must be invertible, otherwise, the i-th limb is at a singular configuration. On the other hand, the forward velocity analysis (FVA) is stated as follows: given the active joint rate velocities 3ωi4(i = 1, 2, 3), compute the velocity state V of the moving platform with respect to the fixed platform. To this aim, consider that the screws 4$51 and 5$61 are fictitious elements, then the application of the Klein form, {*; *}, of the Lie algebra se (3) of the screws associated to the spherical joint of the SP limb with both sides of Eq. (21) leads to the following three expressions n
0 1 $1 ; V
1 2 $3
o
= 0;
n
1 2 $1 ; V
o
1
= 3 ω4 ;
n
2 3 $1 ; V
o
ð24Þ
= 0:
Similarly, since the screw 5$62, of the SPR limb, is a fictitious element, the application of the Klein form of the screws 0$12, 1$22 and with both sides of Eq. (21) gives the following expressions n
0 1 $2 ; V
o
= 0;
n
1 2 $2 ; V
o
2
= 3 ω4 ;
n
1 2 $3 ; V
o
3
= 3 ω4 :
ð25Þ
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Casting in a matrix–vector form Eqs. (24)–(35) the input–output velocity equation results in T
1
2
3 T
J Δ V = ½0;3 ω4 ; 0; 0;3 ω4 ;3 ω4
ð26Þ I 0 3×3 3 where J = [0$11 1$21 2$31 0$12 1$22 1$23] is the active Jacobian of the parallel manipulator, and Δ = is an operator of polarity. I3 03×3 Once the velocity state V is computed, provided that the active Jacobian J is invertible, by means of Eq. (26), the angular velocity of the moving platform is obtained as the primal part of V, ω = P(V), while the linear velocity of the center of the moving platform, vC, is computed using the dual part of the velocity state, vO = D(V), as follows
vC = vO + ω × τ
ð27Þ
5. Acceleration analysis Let A = [P(A); D(A)T = [ω˙ T; (aO–ω × vO)T]T be the reduced acceleration state, or accelerator for brevity [27], of body 1 with respect to body 0; where ω˙ is the angular acceleration of the moving platform and aO is the linear acceleration of a point fixed at the moving platform, which is instantaneously coincident with the origin O of the global reference frame XYZO. The sixdimensional vector A can be written, for details the reader is referred to Rico and Duffy [28], in screw form through any of the three connector limbs as follows i 0 1 ˙ 1 $i 0ω
i 1 2
i 2 3
i 3 4
i 4 5
i 5 6
þ1 ω˙ 2 $i þ2 ω˙ 3 $i þ3 ω˙ 4 $i þ4 ω˙ 5 $i þ5 ω˙ 6 $i + $Lie i = A i = 1; 2; 3
ð28Þ
or re arranging in a matrix–vector form Ji Ω˙ i = A−$Lie i i = 1; 2; 3
ð29Þ
where Ω˙ i = [0ω˙ i1; 1ω˙ i2; 2ω˙ i3; 3ω˙ i4; 4ω˙ i5; 5ω˙ i6]T is a matrix containing the joint rate accelerations of the i-th limb and $Liei is the i-th Lie screw which is computed as follows $Lie i =
h
i 0 1 i 1 2 0 ω1 $i 1 ω2 $i
þ2 ωi3 2 $3i + … þ5 ωi6 5 $6i
i
h i + 1 ωi2 1 $2i 2 ωi3 2 $3i þ3 ωi4 3 $4i + … þ5 ωi6 5 $6i h i + 2 ωi3 2 $3i 3 ωi4 3 $4i þ4 ωi5 4 $5i þ5 ωi6 5 $6i h i h i + 3 ωi4 3 $4i 4 ωi5 4 $5i þ5 ωi6 5 $6i + 4 ωi5 4 $5i 5 ωi6 5 $6i
ð30Þ
where the brackets [* *] denote the Lie product of the Lie algebra se(3) of the Euclidean group SE(3). The inverse acceleration analysis (IAA) is stated as follows: given an accelerator A which describes the instantaneous motion of the moving platform with respect to the fixed platform, compute the joint rate accelerations, or in other words the i-th matrix Ω̇i, of the parallel manipulator. The IAA is quickly determined by means of Eq. (29). In fact −1 Ω˙ i = Ji ðA−$Lie iÞi = 1; 2; 3:
ð31Þ
On the other hand, the forward acceleration analysis (FAA) is stated as follows: given the active joint rate accelerations 2, 3), compute the resulting accelerator A of the moving platform with respect to the fixed platform. The application of the Klein form of the screws associated to the spherical joint of the SP limb with both sides of Eq. (28) brings the following three expressions
˙ i4(i = 1, 3ω
n
0 1 $1 ; A
o
=
n
0 1 $1 ; $Lie 1
o n o n o n o n 1 2 1 1 2 2 3 2 3 ; $1 ; A = 3 ω˙ 4 + $1 ; $Lie 1 ; $1 ; A = $1 ; Lie 1g:
ð32Þ
Similarly, since the screw 5$62, of the SPR limb, is a fictitious element, the application of the Klein form of the screws 0$12, 1$22 and 1$23 with both sides of Eq. (28) leads to n
o
n
o
0 1 $2 ; A
1 2 $2 ; A
n
=
n
0 1 $2 ; $Lie 2
2
= 3 ω˙ 4 +
1 2 D3 ; Ag= 3ω˙ 34
+
o
ð33Þ
n
o
n
o :
1 2 $2 ; $Lie 2
1 2 D3 ;DLie 3
ð34Þ ð35Þ
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Table 1 Data of the numerical example. B12 = B13 = 0.5 S12 = S13 = 0.75 β = γ = π/3 Ux = 0.866, Uy = 0, Uz = 0.5 q1 = 1.0 + 0.2 sin t q2 = 1.0 + 0.4 sin t cos t q3 = 1.0 − 0.2 sin t, 0 ≤ t ≤ 2π S1 = (0, 0, 0) S2 = (0, 0, 0.75) S3 = (0.6495190528, 0, 0.375)
Table 2 The four solutions of the FDA. Sol
B1
B2
B3
1 2 3 4
(0.290e−1, .998, 0.506e−1) (.793, −.606, 0.506e−1) (−.352, −.596, .721) (.528, .447, .721)
(.119, .970, .541) (.717, −.664, .541) (−0.800e−1, −.974, .541) (.394, .894, .541)
(.500, (.500, (.112, (.112,
.976, .217) −.976, .217]) −.669, .887) .669, .887)
Casting in a matrix–vector form Eqs. (32)–(35) the input–output acceleration equation results in 2
n
0 1 $1 ; $Lie 1
o
3
6 7 6 n o7 6 1 7 1 2 6 3 ω˙ 4 + $1 ; $Lie 1 7 6 7 6 7 n o 6 7 2 3 $1 ; $Lie 1 6 7 T 6 7 J ΔA = 6 7: n o 6 7 0 1 $ ; $ 2 6 7 2 Lie 6 7 6 n o7 6 2 7 1 2 6 3 ω˙ 4 + $2 ; $Lie 2 7 6 7 4 n o5 3 1 2 ˙ 4 + $3 ; $Lie 3 3ω
ð36Þ
Once the accelerator A is computed by means of Eq. (36), the angular acceleration of the moving platform is obtained as the primal part of the accelerator A, ω˙ = P(A). Furthermore, since the accelerator A satisfies the conditions of helicoidal vector fields
Fig. 3. Workspace of the center of the moving platform.
J. Gallardo-Alvarado et al. / Mechanism and Machine Theory 45 (2010) 1013–1023
Fig. 4. Forward velocity and acceleration analyses of the center of the moving platform.
1021
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J. Gallardo-Alvarado et al. / Mechanism and Machine Theory 45 (2010) 1013–1023
[29], the linear acceleration of the center of the moving platform, aC, is computed using the dual part of the accelerator, D(A), as follows DðAC Þ = DðAÞ + ω˙ × τ
ð37Þ
where D(AC) = aC − ω × vC. Therefore aC = DðAÞ + ω × vC + ω˙ × τ
ð38Þ
Finally it is interesting to note that the computation of the accelerator A, by means of Eq. (36), does not require the computation of the passive joint rate accelerations of the parallel manipulator. 6. Numerical example In order to prove the efficacy of the method of analysis introduced in this contribution to solve the kinematics of the SP + SPR + SPS parallel manipulator, in this section a numerical example is presented. The data, using SI units, of the example are listed in Table 1. At time t = 0, the resulting four possible poses of the moving platform, with respect to the fixed platform, are listed in Table 2. Considering the intervals 0.7≤qi ≤1.2(i=1, 2, 3) for the generalized coordinates; the reachable workspace of the center of the moving platform is shown in Fig. 3. Naturally, only real solutions of the FDA were taken into account to produce the corresponding surfaces. Finally, the numerical results obtained for the forward velocity and acceleration analyses of the center of the moving platform using the theory of screws, taking solution 1 of Table 2 as the home position of the parallel manipulator, are reported in Fig. 4. Furthermore, in order to verify the results of the numerical example obtained using the screw theory, a faster and lower cost option than a physical prototype, is the use of special software like MSC.Adams© [30], which is the commercial denomination of a software package that realizes several kinds of analyses of mechanical devices. In particular Adams/View is a module of MSC. Adams©capable to make kinematic simulations of mechanisms by means of animations. The corresponding results of the simulation of the problem at hand using Adams/View are reported in Fig. 4. 7. Conclusions In this contribution the displacement, velocity and acceleration analyses of a SP + SPR + SPS parallel manipulator are carried out by means of the theory of screws. The forward displacement analysis is solved using a novel method based on simple geometric–kinematic constraints which allows computation of all the feasible poses of the moving platform, with respect to the fixed platform, given a set of generalized coordinates. Since only four solutions are available for the FDA, then the actual configuration of the parallel manipulator at hand can be easily determined without using sensors. Simple, elegant and compact expressions to solve the forward velocity and acceleration analyses are introduced here by taking advantage of the concept of reciprocal screws via the Klein form of the Lie algebra se(3). To this end, fictitious infinitesimal screws are considered in order to satisfy algebraic procedures. It is worth mentioning that with this method, the solution of the forward acceleration analysis is carried out without the computation of the passive joint rate accelerations of the parallel manipulator. Finally, the kinematics, including the determination of the workspace, of a numerical example is solved using the proposed method. Furthermore, the numerical results obtained via the screw theory for the forward velocity and acceleration analyses are compared with results generated with the help of special software like MSC.Adams©. Please note, according to Fig. 4, that the numerical results generated using the screw theory are in excellent agreement with those obtained with MSC.Adams©. References [1] V.E. Gough, Contribution to discussion of papers on research in automobile stability, control and in tyre performance, Proceedings Automation Division Institution of Mechanical Engineers (1956) 392–394. [2] V.E. Gough, S.G. 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