- Email: [email protected]
R-CUBE, a decoupled parallel manipulator only with revolute joints
Mechanism and Machine Theory
Mechanism and Machine Theory 40 (2005) 467–473
www.elsevier.com/locate/mechmt
R-CUBE, a decoupled parallel manipulator only with revolute joints Weimin Li a b
a,* ,
Feng Gao b, Jianjun Zhang
a
School of Mechanical Engineering, Hebei University of Technology, Tianjin 300130, PR China School of Mechanical Engineering, Shanghai Jiaotong University, Shanghai 200030, PR China
Received 20 March 2004; received in revised form 6 September 2004; accepted 7 September 2004 Available online 27 October 2004
Abstract A new 3-DOF translational parallel manipulator, named R-CUBE, is presented in this paper. This manipulator has decoupled motion in x, y, and z axes, and employs only revolute joints. The structure and kinematics of the manipulator are studied, which shows there exist rotational redundant constraints to the moving platform. A modified structure is also introduced to avoid the redundant constraints. The mechanism singularity is examined. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: R-CUBE; Decoupled motion; Redundant constraints; Singularity
1. Introduction Since Gough firstly introduced a 6-DOF parallel mechanism—the variable-length-strut octahedral hexapod—for tire detecting device [1] and afterwards Stewart used the same mechanism for flight simulator [2], parallel mechanisms have been used in many practical applications such as heavy load platforms, good kinematic and dynamic performance machines, fine motion and precise positioning manipulators etc. *
Corresponding author. Tel./fax: +86 22 26564224. E-mail address: [email protected] (W. Li).
0094-114X/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2004.09.001
468
W. Li et al. / Mechanism and Machine Theory 40 (2005) 467–473
In recent years, theoretical researches of parallel mechanisms have been increased rapidly in a variety of areas, especially in mechanism design methodology. Hunt [3] studied the structural kinematics of parallel manipulators on the basis of screw theory and enumerated promising kinematic structures. Pernette et al. [4] presented the method for design of a 3-DOF parallel translating manipulator with U–P–U joints kinematic chains. The planar 2-DOF parallel manipulators are 5-bar linkages with 5 revolute joints [5] or 3 revolute joints and 2 prismatic pairs. The planar 3-DOF parallel manipulators [6,7] are 8-bar linkages with 2 ternary links connected through three in-parallel legs, each leg consisting of two links. The Delta robot [8] is with 3 translational degrees of freedom. Herve´ and Sparacino [9] enumerated a numerous structural types of limbs for Delta-like robots. Appleberry [10] described a truly new translational parallel manipulator with 3-UPU limbs. More recently, several types of composite pairs and new kinds of subchains with specific degrees of freedom are proposed by Gao et al. [11] and several new type of 2-, 3-, 4- and 5-DOF parallel robotic mechanisms are obtained. However, most of these manipulators have coupled motion between the position and orientation of the end-effector. Recent research on 3-DOF parallel manipulators has been leaning toward the decoupling of the position and orientation of the end-effector and the elimination of complicated multi-DOF joints. Tsai presented the design of a 3-DOF translational platform that employs only revolute joints and planar parallelograms [12] and a spatial 3-UPU (universal– prismatic–universal) parallel manipulator [13]. Zhao and Huang [14] studied the kinematics of an overconstrained 3-RRC (revolute–revolute–cylindrical) translational manipulator. Baron and Bernier [15] studied the design of a 3-DOF Star manipulator which was firstly proposed by Herve´ and Sparacino [16]. Carricato and Parenti-Castelli [17] presented a family of 3-DOF parallel manipulators with pure translational motion. Kong and Gosselin [18] studied the kinematics and singularity of a 3-CRR translational parallel manipulator, and before long, Kim and Tsai [19] discussed the optimal design about the same manipulator. In this paper, the ‘‘R-CUBE’’, a decoupled parallel manipulator that employs only revolute joints to achieve translational motion of the moving platform is presented. The kinematic structure of the manipulator is described. The kinematics of the manipulator is studied and a modified structure is given. The mechanism singularity is examined.
2. Kinematic structure The kinematic structure and geometry of the R-CUBE parallel manipulator is shown in Fig. 1, where a moving platform is connected to a fixed frame by three limbs. Each limb includes a planar parallelogram and an arm that is consisted of two links and three joints. The structure size of the jth limb is represented by lj1 ðl0j1 Þ, lj2, lj3. The joints Aj ðA0j Þ for j = 1, 2, 3 are connected to the fixed frame, while the joints Bj for j = 1, 2, 3 are connected to the moving platform. Furthermore, the three axes of the revolute joints Bj for j = 1, 2, 3 are parallel to the x, y, and z axes, respectively. The rotational actuator can be applied to Aj or A0j for j = 1, 2, 3. If the lengths of lj1 for j = 1, 2, 3 are equal, then the workspace of the manipulator is a cube. This is why the manipulator is so called. The position of the platform is defined by hj for j = 1, 2, 3. When hj = 0, the moving platform is just at the center of the cube.
W. Li et al. / Mechanism and Machine Theory 40 (2005) 467–473
469
Fig. 1. R-CUBE parallel manipulator with 3-RRRRRRR joints: (a) structure and (b) geometry.
One advantage of this manipulator is economic in production for only revolute joints are applied. Another advantage is that the moving platform can easily get high speed as a stroke of the end-effector only needs the actuator rotates within 180°. As a general knowledge, the planar parallelogram has only one translational DOF (degree of freedom). Hence each planar parallelogram can be considered as a 1-DOF translational pair with two links connected to each side of it, and one of the links is just the fixed frame. So equivalently, each limb consists of three joints, one pair and three links. Then the DOFs of the mechanism can be calculated as [20] F ¼ kðl m 1Þ þ
m X
fi ¼ 6ð11 12 1Þ þ 3ð1 þ 1 þ 1 þ 1Þ ¼ 0
ð1Þ
i¼1
where F denotes the DOFs of the mechanism, l and m represent the number of links and number of joints (or pairs), respectively, fi denotes DOFs associated with the ith joint (or pair), and k is a constant which value is 6 for spatial mechanism or 3 for planar (or spherical) mechanism. From the result of Eq. (1), the mechanism has no freedom. However this is not the real case, so we should examine the mechanism in a new way. For convenience, we let $ be the special Plu¨cker coordinates for describing the displacement of the output link of a limb in a parallel mechanism, which is $j ¼ ð mxj
myj
mzj
xaj
xbj
xcj Þ
ð2Þ
where mj ð mxj myj mzj Þ expresses the translation of the output link of the limb j in three directions, and xj ð xaj xbj xcj Þ represents the rotation of the output link of the limb j with respect to three EularÕs angles, a, b and c. The special Plu¨cker coordinates mxj, myj, mzj, xaj, xbj, xcj can be taken as 1 or 0. When taking 1, it means that the limb j has that DOF, when taking 0, it means that the limb j has no that DOF. Then the new way to determine the DOFs of the parallel mechanism can be expressed as the following Freedom criterion
470
W. Li et al. / Mechanism and Machine Theory 40 (2005) 467–473
The special Plu¨cker coordinates of the end-effector equal to the intersection of the special Plu¨cker coordinates of all the limbs connecting from the fixed frame to the end-effector. This criterion can be described in the following general Eq. (3), which is very useful not only for mechanism analysis but also for mechanism design $ ¼ $1 \ $2 \ $n
ð3Þ
In the present mechanism, n = 3, and as the planar parallelogram is considered as a 1-DOF planar translational pair, it holds that $1 ¼ ð 1 1
1
1
0
0Þ
ð4Þ
$2 ¼ ð 1 1
1
0
1
0Þ
ð5Þ
$3 ¼ ð 1 1
1
0
0
1Þ
ð6Þ
Then the DOFs of the output platform will be determined from Eq. (3), that is $ ¼ ð1
1
1
0 0
0Þ
ð7Þ
It is pointed out that the mechanism has three translational DOFs. If we examine Eqs. (4)–(6), we can find out that each limb provides two rotational constraints to the moving platform, or in another words, each rotational parameter of the platform is constrained by two limbs. As a result, there are three redundant constraints on rotation. This is why the result of Eq. (1) is zero.
3. Discussion about redundant constraints The manipulator has three redundant revolute constraints, which may open some trouble in engineering. In production, two constraints to one parameter may be not exactly the same, as the manufacture error exists. Then we had to making clearance to fit this occasion or the mechanism may not move. To overcome this shortage, we should throw off the redundant revolute constraints. The simple way to do this is to rewrite Eqs. (4)–(6) as follows: $1 ¼ ð 1 1
1
1
1
0Þ
ð8Þ
$2 ¼ ð 1 1
1
0
1
1Þ
ð9Þ
$3 ¼ ð 1 1
1
1
0
1Þ
ð10Þ
Then we rearrange the kinematic structure of the three limbs in a new way, which satisfies Eqs. (8)–(10) respectively. The new kinematic structure is described in Fig. 2, in which each limb provides only one rotational constraint to the moving platform and no redundant constraints exist. Using the general Eq. (3), we can get the special Plilcker coordinates of the moving platform, which still satisfies Eq. (7), so the rearranged kinematic structure should be correct.
W. Li et al. / Mechanism and Machine Theory 40 (2005) 467–473
471
Fig. 2. Another structure of the R-CUBE with 3-RRRRR joints.
4. Kinematics The forward and inverse kinematic analyses are trivial since the decoupled characteristic exists. Referring to Fig. 1(b), each limb constrains point P to lie on a plane which passes through points Mj2, Mj3, and Bj, and is perpendicular to the axis of x, y, and z, respectively. The position of jth plane is determined only by hj whenever the length lj1 is given. Consequently, the position of P is determined by the intersection of three planes, i.e., the intersection of hj for j = 1, 2, 3. If the distance from Mj1 to Mj2 is m0j, then a simple kinematic relation can be written as 2
px
3
2
a01 þ m01 þ l11 sin h1
3
7 6 7 6 4 py 5 ¼ 4 a02 þ m02 þ l21 sin h2 5
ð11Þ
a03 þ m03 þ l31 sin h3
pz
where p ¼ ½ px py pz T denotes the position vector of the end-effector. Taking the time derivative of Eq. (11) and applying the principle of virtual work yields 2 3 2 3 p_ x h_ 1 6_ 7 7 1 6 ð12Þ 4 h2 5 ¼ J 4 p_ y 5 _h3 p_ z and 2
T1
3
2
fx
3
6 7 6 7 4 T 2 5 ¼ J T 4 fy 5 T3 fz
ð13Þ
472
W. Li et al. / Mechanism and Machine Theory 40 (2005) 467–473
where f ¼ ½ fx fy fz T is the vector of the end-effector output force, Tj for j = 1, 2, 3, denotes the jth rotational actuator torque, and J is a diagonal matrix that holds 2 3 l11 cos h1 6 7 ð14Þ l21 cos h2 J ¼4 5 l31 cos h3
5. Singularity discussion From Eq. (12) we can find out that the rotational actuator speed is nonlinear to the velocity of the end-effector. Moreover, if hj = ±90°, then detjJj = 0, for any expected velocity of the end-effector, the rotational speed of the actuator will be infinite. When hj is not equal but close to ±90°, then detjJj ! 0, the required rotational speed of the actuator may be still too high to reach. So the value of the hj must be designed in an appropriate range whenever the speed limit of the end-effector is given. Suppose the desired velocity of the end-effector is ve, and the permissible rotational speed of the actuator is ne, then the absolute maximum value of the hj for j = 1, 2, 3 can be obtained from Eq. (12) me ð15Þ cos j hj j¼ lj1 ne Let hjm ¼ arccos
me lj1 ne
ð16Þ
Then hj should satisfy hjm 6 hj 6 hjm
ð17Þ
Whenever the mechanism design satisfies Eq. (17), no singularity will exist.
6. Conclusions A new 3-DOF translational parallel manipulator, named R-CUBE, is conceived. A modified structure is introduced to avoid the rotational redundant constraints. The forward and inverse kinematic analyses are carried through. This manipulator has some outstanding advantages. Firstly, the x, y, and z motions of the manipulator are decoupled. Secondly, the manipulator is economic in production and simple in calibration. Thirdly, the end-effector of the manipulator can easily get high speed. Finally, singularities exist only at the edge of the manipulatorÕs workspace and can be easily avoided by proper design. The advantages show that this manipulator has extensive application potential in industry.
W. Li et al. / Mechanism and Machine Theory 40 (2005) 467–473
473
Acknowledgments This research is supported by National Natural Science Foundation of China (No. 50475055) and 863 Project Office of China.
References [1] V.E. Gough, S.G. Whitehall, Universal tyre test machine, in: Proceedings of the FISITA Ninth International Technical Congress, May, 1962, pp. 117–137. [2] D. Stewart, A platform with six degrees of freedom, Proceedings of the Institution of Mechanical Engineers, London, UK 180 (15) (1965) 371–386. [3] K.H. Hunt, Kinematic Geometry of Mechanisms, Clarendon Press, Oxford, 1978. [4] E. Pernette, S. Henein, I. Magnani, R. Clavel, Design of a 3-DOF parallel translating manipulator with U–P–U joints kinematic chains, in: Proceedings of the IROSÕ97, 1997, pp. 1637–1642. [5] F. Gao, X.-J. Liu, W.A. Gruver, Performance evaluation of two degrees of freedom planar parallel robots, Mechanism and Machine Theory 33 (2) (1998) 661–668. [6] C.M. Gosselin, E. Lavoie, On the kinematic design of spherical three-degrees-of-freedom parallel manipulators, International Journal of Robotic Research 12 (4) (1993) 394–402. [7] C.M. Gosselin, B. Angeles, The optimum kinematic design of a planar three-degrees-of-freedom parallel manipulator, ASME Journal of Mechanisms Transmissions and Automation in Design 110 (1) (1998) 35–41. [8] R. Clavel, DELTA, a fast robot with parallel geometry, in: Proceedings of the International Symposium on Industrial Robotics, Switzerland, 1988, pp. 91–100. [9] J.M. Herve´, F. Sparacino, Structural synthesis of parallel robots generating spatial translation, in: 5th IEEE International Conference on Advanced Robotics, 1991, pp. 808–813. [10] W.T. Appleberry, Antirotation positioning device, US patent 5,156,062, 1992. [11] F. Gao, W.M. Li, X.C. Zhao, Z.L. Jin, H. Zhao, New kinematic structures for 2-, 3-, 4- and 5-DOF parallel manipulator designs, Mechanism and Machine Theory 37 (11) (2002) 1395–1411. [12] L.W. Tsai, G.C. Walsh, R. Stamper, Kinematics of a novel three DOF translational platform, in: Proceedings of the 1996 IEEE International Conference on Robotics and Automation, Minneapolis, MM, 1996, pp. 3446–3451. [13] L.W. Tsai, Kinematics of a three-DOF platform manipulator with three extensible limbs, in: J. Lenarcic, V. Parenti-Castelli (Eds.), Recent Advances in Robot Kinematics, Kluwer Academic Publishers, London, 1996, pp. 401–410. [14] T.S. Zhao, Z. Huang, A novel three-DOF translational platform mechanism and its kinematics, in: Proceedings of the 2000 ASME Design Engineering Technical Conferences, Baltimore, MD, MECH-14101, 2000. [15] L. Baron, G. Bernier, The design of parallel manipulators of star topology under isotropic constraint, in: Proceedings of the 2001 ASME Design Engineering Technical Conferences, Pittsburgh, PA, DAC-21025, 2001. [16] J.M. Herve´, F. Sparacino, STAR: a new concept in robotics, in: International Conference 3K-ARK, 1992, pp. 176– 183. [17] M. Carricato, V. Parenti-Castelli, A family of 3-DOF translational parallel manipulators, in: Proceedings of the 2001 ASME Design Engineering Technical Conferences, Pittsburgh, PA, DAC-21035, 2001. [18] X.W. Kong, C. Gosselin, Kinematics and singularity analysis of a novel type of 3-CRR 3-DOF translational parallel manipulator, International Journal of Robotic Research 21 (9) (2002) 791–798. [19] H.S. Kim, L.W. Tsai, Design optimization of a Cartesian parallel manipulator, Journal of Mechanical Design 125 (1) (2003) 43–51. [20] L.W. Tsai, Mechanism Design: Enumeration of Kinematic Structures According to Function, CRC Press, Boca Raton, FL, 2000.
Mechanism and Machine Theory 40 (2005) 467–473
www.elsevier.com/locate/mechmt
R-CUBE, a decoupled parallel manipulator only with revolute joints Weimin Li a b
a,* ,
Feng Gao b, Jianjun Zhang
a
School of Mechanical Engineering, Hebei University of Technology, Tianjin 300130, PR China School of Mechanical Engineering, Shanghai Jiaotong University, Shanghai 200030, PR China
Received 20 March 2004; received in revised form 6 September 2004; accepted 7 September 2004 Available online 27 October 2004
Abstract A new 3-DOF translational parallel manipulator, named R-CUBE, is presented in this paper. This manipulator has decoupled motion in x, y, and z axes, and employs only revolute joints. The structure and kinematics of the manipulator are studied, which shows there exist rotational redundant constraints to the moving platform. A modified structure is also introduced to avoid the redundant constraints. The mechanism singularity is examined. Ó 2004 Elsevier Ltd. All rights reserved. Keywords: R-CUBE; Decoupled motion; Redundant constraints; Singularity
1. Introduction Since Gough firstly introduced a 6-DOF parallel mechanism—the variable-length-strut octahedral hexapod—for tire detecting device [1] and afterwards Stewart used the same mechanism for flight simulator [2], parallel mechanisms have been used in many practical applications such as heavy load platforms, good kinematic and dynamic performance machines, fine motion and precise positioning manipulators etc. *
Corresponding author. Tel./fax: +86 22 26564224. E-mail address: [email protected] (W. Li).
0094-114X/$ - see front matter Ó 2004 Elsevier Ltd. All rights reserved. doi:10.1016/j.mechmachtheory.2004.09.001
468
W. Li et al. / Mechanism and Machine Theory 40 (2005) 467–473
In recent years, theoretical researches of parallel mechanisms have been increased rapidly in a variety of areas, especially in mechanism design methodology. Hunt [3] studied the structural kinematics of parallel manipulators on the basis of screw theory and enumerated promising kinematic structures. Pernette et al. [4] presented the method for design of a 3-DOF parallel translating manipulator with U–P–U joints kinematic chains. The planar 2-DOF parallel manipulators are 5-bar linkages with 5 revolute joints [5] or 3 revolute joints and 2 prismatic pairs. The planar 3-DOF parallel manipulators [6,7] are 8-bar linkages with 2 ternary links connected through three in-parallel legs, each leg consisting of two links. The Delta robot [8] is with 3 translational degrees of freedom. Herve´ and Sparacino [9] enumerated a numerous structural types of limbs for Delta-like robots. Appleberry [10] described a truly new translational parallel manipulator with 3-UPU limbs. More recently, several types of composite pairs and new kinds of subchains with specific degrees of freedom are proposed by Gao et al. [11] and several new type of 2-, 3-, 4- and 5-DOF parallel robotic mechanisms are obtained. However, most of these manipulators have coupled motion between the position and orientation of the end-effector. Recent research on 3-DOF parallel manipulators has been leaning toward the decoupling of the position and orientation of the end-effector and the elimination of complicated multi-DOF joints. Tsai presented the design of a 3-DOF translational platform that employs only revolute joints and planar parallelograms [12] and a spatial 3-UPU (universal– prismatic–universal) parallel manipulator [13]. Zhao and Huang [14] studied the kinematics of an overconstrained 3-RRC (revolute–revolute–cylindrical) translational manipulator. Baron and Bernier [15] studied the design of a 3-DOF Star manipulator which was firstly proposed by Herve´ and Sparacino [16]. Carricato and Parenti-Castelli [17] presented a family of 3-DOF parallel manipulators with pure translational motion. Kong and Gosselin [18] studied the kinematics and singularity of a 3-CRR translational parallel manipulator, and before long, Kim and Tsai [19] discussed the optimal design about the same manipulator. In this paper, the ‘‘R-CUBE’’, a decoupled parallel manipulator that employs only revolute joints to achieve translational motion of the moving platform is presented. The kinematic structure of the manipulator is described. The kinematics of the manipulator is studied and a modified structure is given. The mechanism singularity is examined.
2. Kinematic structure The kinematic structure and geometry of the R-CUBE parallel manipulator is shown in Fig. 1, where a moving platform is connected to a fixed frame by three limbs. Each limb includes a planar parallelogram and an arm that is consisted of two links and three joints. The structure size of the jth limb is represented by lj1 ðl0j1 Þ, lj2, lj3. The joints Aj ðA0j Þ for j = 1, 2, 3 are connected to the fixed frame, while the joints Bj for j = 1, 2, 3 are connected to the moving platform. Furthermore, the three axes of the revolute joints Bj for j = 1, 2, 3 are parallel to the x, y, and z axes, respectively. The rotational actuator can be applied to Aj or A0j for j = 1, 2, 3. If the lengths of lj1 for j = 1, 2, 3 are equal, then the workspace of the manipulator is a cube. This is why the manipulator is so called. The position of the platform is defined by hj for j = 1, 2, 3. When hj = 0, the moving platform is just at the center of the cube.
W. Li et al. / Mechanism and Machine Theory 40 (2005) 467–473
469
Fig. 1. R-CUBE parallel manipulator with 3-RRRRRRR joints: (a) structure and (b) geometry.
One advantage of this manipulator is economic in production for only revolute joints are applied. Another advantage is that the moving platform can easily get high speed as a stroke of the end-effector only needs the actuator rotates within 180°. As a general knowledge, the planar parallelogram has only one translational DOF (degree of freedom). Hence each planar parallelogram can be considered as a 1-DOF translational pair with two links connected to each side of it, and one of the links is just the fixed frame. So equivalently, each limb consists of three joints, one pair and three links. Then the DOFs of the mechanism can be calculated as [20] F ¼ kðl m 1Þ þ
m X
fi ¼ 6ð11 12 1Þ þ 3ð1 þ 1 þ 1 þ 1Þ ¼ 0
ð1Þ
i¼1
where F denotes the DOFs of the mechanism, l and m represent the number of links and number of joints (or pairs), respectively, fi denotes DOFs associated with the ith joint (or pair), and k is a constant which value is 6 for spatial mechanism or 3 for planar (or spherical) mechanism. From the result of Eq. (1), the mechanism has no freedom. However this is not the real case, so we should examine the mechanism in a new way. For convenience, we let $ be the special Plu¨cker coordinates for describing the displacement of the output link of a limb in a parallel mechanism, which is $j ¼ ð mxj
myj
mzj
xaj
xbj
xcj Þ
ð2Þ
where mj ð mxj myj mzj Þ expresses the translation of the output link of the limb j in three directions, and xj ð xaj xbj xcj Þ represents the rotation of the output link of the limb j with respect to three EularÕs angles, a, b and c. The special Plu¨cker coordinates mxj, myj, mzj, xaj, xbj, xcj can be taken as 1 or 0. When taking 1, it means that the limb j has that DOF, when taking 0, it means that the limb j has no that DOF. Then the new way to determine the DOFs of the parallel mechanism can be expressed as the following Freedom criterion
470
W. Li et al. / Mechanism and Machine Theory 40 (2005) 467–473
The special Plu¨cker coordinates of the end-effector equal to the intersection of the special Plu¨cker coordinates of all the limbs connecting from the fixed frame to the end-effector. This criterion can be described in the following general Eq. (3), which is very useful not only for mechanism analysis but also for mechanism design $ ¼ $1 \ $2 \ $n
ð3Þ
In the present mechanism, n = 3, and as the planar parallelogram is considered as a 1-DOF planar translational pair, it holds that $1 ¼ ð 1 1
1
1
0
0Þ
ð4Þ
$2 ¼ ð 1 1
1
0
1
0Þ
ð5Þ
$3 ¼ ð 1 1
1
0
0
1Þ
ð6Þ
Then the DOFs of the output platform will be determined from Eq. (3), that is $ ¼ ð1
1
1
0 0
0Þ
ð7Þ
It is pointed out that the mechanism has three translational DOFs. If we examine Eqs. (4)–(6), we can find out that each limb provides two rotational constraints to the moving platform, or in another words, each rotational parameter of the platform is constrained by two limbs. As a result, there are three redundant constraints on rotation. This is why the result of Eq. (1) is zero.
3. Discussion about redundant constraints The manipulator has three redundant revolute constraints, which may open some trouble in engineering. In production, two constraints to one parameter may be not exactly the same, as the manufacture error exists. Then we had to making clearance to fit this occasion or the mechanism may not move. To overcome this shortage, we should throw off the redundant revolute constraints. The simple way to do this is to rewrite Eqs. (4)–(6) as follows: $1 ¼ ð 1 1
1
1
1
0Þ
ð8Þ
$2 ¼ ð 1 1
1
0
1
1Þ
ð9Þ
$3 ¼ ð 1 1
1
1
0
1Þ
ð10Þ
Then we rearrange the kinematic structure of the three limbs in a new way, which satisfies Eqs. (8)–(10) respectively. The new kinematic structure is described in Fig. 2, in which each limb provides only one rotational constraint to the moving platform and no redundant constraints exist. Using the general Eq. (3), we can get the special Plilcker coordinates of the moving platform, which still satisfies Eq. (7), so the rearranged kinematic structure should be correct.
W. Li et al. / Mechanism and Machine Theory 40 (2005) 467–473
471
Fig. 2. Another structure of the R-CUBE with 3-RRRRR joints.
4. Kinematics The forward and inverse kinematic analyses are trivial since the decoupled characteristic exists. Referring to Fig. 1(b), each limb constrains point P to lie on a plane which passes through points Mj2, Mj3, and Bj, and is perpendicular to the axis of x, y, and z, respectively. The position of jth plane is determined only by hj whenever the length lj1 is given. Consequently, the position of P is determined by the intersection of three planes, i.e., the intersection of hj for j = 1, 2, 3. If the distance from Mj1 to Mj2 is m0j, then a simple kinematic relation can be written as 2
px
3
2
a01 þ m01 þ l11 sin h1
3
7 6 7 6 4 py 5 ¼ 4 a02 þ m02 þ l21 sin h2 5
ð11Þ
a03 þ m03 þ l31 sin h3
pz
where p ¼ ½ px py pz T denotes the position vector of the end-effector. Taking the time derivative of Eq. (11) and applying the principle of virtual work yields 2 3 2 3 p_ x h_ 1 6_ 7 7 1 6 ð12Þ 4 h2 5 ¼ J 4 p_ y 5 _h3 p_ z and 2
T1
3
2
fx
3
6 7 6 7 4 T 2 5 ¼ J T 4 fy 5 T3 fz
ð13Þ
472
W. Li et al. / Mechanism and Machine Theory 40 (2005) 467–473
where f ¼ ½ fx fy fz T is the vector of the end-effector output force, Tj for j = 1, 2, 3, denotes the jth rotational actuator torque, and J is a diagonal matrix that holds 2 3 l11 cos h1 6 7 ð14Þ l21 cos h2 J ¼4 5 l31 cos h3
5. Singularity discussion From Eq. (12) we can find out that the rotational actuator speed is nonlinear to the velocity of the end-effector. Moreover, if hj = ±90°, then detjJj = 0, for any expected velocity of the end-effector, the rotational speed of the actuator will be infinite. When hj is not equal but close to ±90°, then detjJj ! 0, the required rotational speed of the actuator may be still too high to reach. So the value of the hj must be designed in an appropriate range whenever the speed limit of the end-effector is given. Suppose the desired velocity of the end-effector is ve, and the permissible rotational speed of the actuator is ne, then the absolute maximum value of the hj for j = 1, 2, 3 can be obtained from Eq. (12) me ð15Þ cos j hj j¼ lj1 ne Let hjm ¼ arccos
me lj1 ne
ð16Þ
Then hj should satisfy hjm 6 hj 6 hjm
ð17Þ
Whenever the mechanism design satisfies Eq. (17), no singularity will exist.
6. Conclusions A new 3-DOF translational parallel manipulator, named R-CUBE, is conceived. A modified structure is introduced to avoid the rotational redundant constraints. The forward and inverse kinematic analyses are carried through. This manipulator has some outstanding advantages. Firstly, the x, y, and z motions of the manipulator are decoupled. Secondly, the manipulator is economic in production and simple in calibration. Thirdly, the end-effector of the manipulator can easily get high speed. Finally, singularities exist only at the edge of the manipulatorÕs workspace and can be easily avoided by proper design. The advantages show that this manipulator has extensive application potential in industry.
W. Li et al. / Mechanism and Machine Theory 40 (2005) 467–473
473
Acknowledgments This research is supported by National Natural Science Foundation of China (No. 50475055) and 863 Project Office of China.
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