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Loci of singular configurations of a 3-DOF spherical parallel manipulator
Robotics and Autonomous Systems 48 (2004) 77–91
Loci of singular configurations of a 3-DOF spherical parallel manipulator G¨ursel Alıcıa,∗ , Bijan Shirinzadehb b
a School of Mechanical, Materials and Mechatronics Engineering, University of Wollongong, 2522 NSW, Australia Robotics & Mechatronics Research Laboratory, Department of Mechanical Engineering, Monash University, 3800 Vic., Australia
Received 25 February 2003; received in revised form 29 May 2004; accepted 8 July 2004
Abstract In order to evaluate the dexterity and kinematic accuracy of a robot manipulator during the design, trajectory planning and control stages, it is of prime importance to know its singular configurations. In this paper, a systematic method to obtain singularity loci of parallel manipulators exemplified by a 3 DOF spherical parallel manipulator is addressed. Firstly, the inverse position equations are formed, and the velocity equation containing the actuator and the end effector (platform) velocities, and Jacobian matrices is derived. Secondly, the determinants of the manipulator Jacobian matrices are evaluated for a specified set of geometric parameters. Finally, the contours of the determinants at 0.0 plane, which are the singularity loci of the manipulator in Cartesian space, are generated. Two different orientation matrices are chosen to demonstrate that the representation of singularity loci is formulation dependent. The main difference of our work from previously published work is that there is no need of expressing singularity loci mathematically. This saves a considerable amount of computational time. The proposed method allows the user to generate the loci of singular configurations of any kinematics chain whose Jacobian matrices are expressed analytically. The method is a quick tool for the designers to decide whether the singularities are acceptable or not. Further, as the singularity loci indicate the acceptable boundaries of singularities, the designers can make use of it to accurately quantify the manipulator dexterity and kinematic accuracy. © 2004 Elsevier B.V. All rights reserved. Keywords: Robot manipulator; Degree of freedom
1. Introduction Many robotic applications are based on manipulating a tool or an object and keeping it steady relative to a user-defined point. Anthropomorphic, open-chain serially connected robot manipulators have traditionally been ∗ Corresponding author. Tel.: +61 2 4221 4145; fax: +61 2 4221 3101. E-mail address: [email protected] (G. Alıcı).
0921-8890/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.robot.2004.07.009
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employed to satisfy such a requirement. However, for applications such as robotics surgery, micromanipulation, and micromachining, etc. needing high precision positioning and high stiffness, parallel manipulators that consist of one or more closed kinematic chains can serve better than serially connected manipulators. One of the disadvantages of parallel manipulators is the difficulty of trajectory planning mainly due to singular configurations which are sometimes referred to as “special” or “critical” configurations [1,2]. The determination of these configurations is a challenging problem and has motivated many researchers to study various techniques in this area [3–13]. This paper focuses on the determination of singular configurations of a 3 DOF spherical parallel manipulator without deriving any analytical expression for the singularity contours. Singular configurations of a mechanism are undesirable due to the fact that the degrees of freedom of the mechanism change instantaneously. While the singular positions of serially connected manipulators result in the loss of one or more degrees of freedom [4,5,6], those of parallel manipulators result in either a gain or a loss of one or more degrees of freedom [3,7–11]. At singular configurations, the determinant of the manipulator Jacobian becomes zero. The singularities of several mechanisms have been obtained. However, nobody has used a technique not based on deriving analytical expressions for the variables making the determinant of the manipulator Jacobian matrices zero. Usually, the determinants are large in size and are difficult to handle mathematically. Gosselin and Angeles [7] introduced the concept of two Jacobian matrices, which relate input velocities to output velocities, for parallel manipulator having an equal number of inputs and outputs. The singularity of each matrix corresponds to a loss or a gain of a degree of freedom and singularity of both occurs only when the mechanism is architecturally singular. Later, Daniali et al. [9] have claimed that the singularity of both matrices is not necessarily architecture-dependent; depending on the Jacobian formulation of any of the three types of singularities described in [7] can occur. Sefrioui and Gosselin [8] have reported on the quadratic nature of the singularity curves of planar three-degree-of-freedom parallel manipulators. The roots of the determinants of the manipulator’s Jacobian are used to obtain a graphical representation of these curves in the manipulator’s Cartesian workspace. Merlet [12] has studied the singular configurations of a six-degree of freedom parallel manipulator using a geometric method rather than finding the roots of the determinants of the manipulator Jacobian matrix. Zlatanov et al. [13] have presented a generalised approach to determine the singular configurations of any mechanism with arbitrary chains and an equal number of inputs and outputs. They used a velocity equation including the velocities of active and passive joints in order to determine singular configurations. Basu and Ghosal [10] have proposed two methods (algebraic and geometric) to determine singular configurations of platform-type multi-loop spatial mechanisms containing spherical joints on the platform. Forward position analysis of parallel manipulators is a difficult task due to nonlinear equations giving multiple solutions [14–20]. This follows that a spherical manipulator with prismatic actuators has multiple configurations for a unique set of actuator displacements. On the other hand, inverse position analysis of parallel manipulators is straightforward and a set of platform orientation angles gives a unique actuator length. Further, it has been suggested in this study that it is not necessary to solve forward position equations and present singularity contours in terms of actuator inputs. The determinants are calculated for a range of output angles. After obtaining singularity contours in the operational space of the manipulator, actuator lengths corresponding to those orientation angles are calculated by employing inverse position formulation. The method proposed in this study consists of: (a) forming the inverse position equations for the manipulator, (b) deriving the velocity equation containing the actuator and the end effector (platform) velocities and Jacobian matrices, (c) calculating the determinants of the manipulator Jacobian matrices for a specified set of geometric parameters, and (d) generating the contours of the determinants at 0.0 plane which are the singularity loci of the manipulator in terms of platform orientation angles. This approach is simple, yet systematic and does not necessitate deriving the explicit expressions for the roots of the determinants in order to plot the singularity contours. This is the main difference between previously published work [16,17] and this study. Two separate orientation matrices are chosen to demonstrate that the singularity contours presented are formulation dependent. The singularity contours disclose the acceptable boundaries of singularities, and thus provide designers with a more accurate representation
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of dexterity and kinematic accuracy. It is concluded that the proposed method is useful in trajectory planning and in the design of the spherical parallel manipulator and other general parallel manipulators.
2. Inverse position formulation The manipulator considered in this study is depicted in Fig. 1, where two pyramid-like rigid bodies are connected together by a spherical pair at the common apex O. The coordinate system A is fixed to the base and the coordinate system B is attached to the moving platform which is able to rotate about three perpendicular axes. The coordinates of the connection points in the base and the platform are given with respect to the frame A and frame B, respectively. Note that both coordinate systems are located at the center of the spherical pair at point O. Three prismatic actuators connect the platform and the base to each other. Each prismatic actuator is connected to the base and the platform via an unactuated universal pair, and an unactuated spherical pair, respectively. i and B i For the provided platform orientation angles, θ 1 , θ 2 and θ 3 , and the specified coordinates of the vectors A (i = 1, 2, 3), the analytical expressions for the actuator lengths Li can be derived. Referring to Fig. 1, the following relationship can be written; i i = [R]B i − A L
(1)
i is the vector for the length of each prismatic actuator and [R] is the 3 × 3 rotation matrix defining the where L orientation of the coordinate frame B with respect to the coordinate frame A. Eq. (1) can be rewritten as follows; i )2 i − A L2i = ([R]B
(2)
Eq. (2) may also be expressed as; Ti [R]B i L2i = A2i + Bi2 − 2A
(3)
Fig. 1. Spherical parallel manipulator made up of two pyramid-like base and platform. “S”, “P” and “U” denote a spherical pair, a prismatic pair and a universal pair, respectively.
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Euler angles can be employed to describe the orientation of a rigid body relative to a fixed reference frame. The following XYZ Euler angle representation [21] is chosen for [R] in Eq. (3); cθ2 cθ3 −cθ2 sθ3 sθ2 [R] = Rx (θ1 )Ry (θ2 )Rz (θ3 ) = sθ1 sθ2 cθ3 + cθ1 sθ3 −sθ1 sθ2 sθ3 + cθ1 cθ3 −sθ1 cθ2
r11 = r21 r31
r12 r22
r13 r23
r32
r33
−cθ1 sθ2 cθ3 + sθ1 sθ3
cθ1 sθ2 sθ3 + sθ1 cθ3
cθ1 cθ2 (4)
where cθi ≡ cos θi , sθi ≡ sin θi and Rx , Ry and Rz are the basic rotation matrices about the three coordinate axes, respectively. The rotation matrix [R] is formed when the coordinate system B is rotated by θ 1 about its x-axis, then rotated by θ 2 about its modified y-axis and finally rotated by θ 3 about its modified z-axis. By substituting the position vectors Ai = (Aix , Aiy , Aiz ) and Bi = (Bix , Biy , Biz ) together with the rotation matrix into Eq. (3), the inverse position equations of the manipulator can be obtained. Each actuator length can be calculated from the following expression; L2i = A2i + Bi2 − 2[Aix (r11 Bix + r12 Biy + r13 Biz ) + Aiy (r21 Bix + r22 Biy + r23 Biz ) +Aiz (r31 Bix + r32 Biy + r33 Biz )]
(5)
It must be noted that for each value of θ 1 , θ 2 and θ 3 , the coordinates of the connection points Bi with respect to the stationary frame A can be determined from i )A = [R]B i (B
(i = 1, 2, 3)
(6)
3. Singularity analysis The nonlinear kinematic equations describing the relationship between input and output position vectors are expressed as; F (L, Θ) = 0
(7)
where F is three-dimensional for the problem at hand and is a function of the input vector L, which represents the set of the position of actuated joints, and the output vector Θ, which represents the coordinates of the output point. Taking the first time derivative of Eq. (7) leads to the relationship between the input and the output velocity vectors as follows; ∂F ˙ ∂F ˙ ˙ =0 ˙ + DΘ L+ Θ = 0 ⇒ CL ∂L ∂Θ
(8)
˙ represent the actuator (joint) velocity ˙ and Θ where C and D are configuration-dependent 3 × 3 Jacobian matrices, L vector and the end effector (platform) velocity vector, respectively. The singular positions are the singular values of the C and D matrices. Based on the singularities of the C and D matrices, there exist three types of singularities for closed-loop kinematic chains [7]. The first type of singularity is faced when det[C] = 0 and det[D] = 0. This ˙ 1, L ˙ 2 and implies that the platform loses one or more degrees of freedom. Nonzero values of the actuator velocities L ˙ 3 can generate zero values for the platform velocities θ˙ 1 , θ˙ 2 and θ˙ 3 . This type of singularity is trivial for parallel L manipulators with prismatic actuators. The second type happens when det[C] = 0 and det[D] = 0. This implies that the output point (the platform) gains one or more degrees of freedom; the output is movable even when all
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the actuated joints are locked. Another consequence of this type singularity is that the effective stiffness of the manipulator seen at the output point does not exist in some directions. The third type is encountered when det[C] = 0 and det[D] = 0, simultaneously. The third type of singularity corresponds to configurations where the manipulator can experience finite motions when its actuators are locked or where a finite input does not produce output motion. So, the singularities occur whenever C, D or both become singular. This type of singularity is avoided by a proper choice of the kinematic parameters during the design phase.
4. Singularity contours The inverse position equation for each actuator is rewritten in the form of Eq. (5), their first time derivative is taken, and the resulting equations are expressed in the form of Eq. (8), as given below; ˙ 1 + c12 L ˙ 2 + c13 L ˙ 3 + d11 θ˙ 1 + d12 θ˙ 2 + d13 θ˙ 3 = 0 c11 L ˙ =0 ˙ 1 + c22 L ˙ 2 + c23 L ˙ 3 + d21 θ˙ 1 + d22 θ˙ 2 + d23 θ˙ 3 = 0 ⇒ CL ˙ + DΘ c21 L ˙ ˙ ˙ ˙ ˙ ˙ c31 L1 + c32 L2 + c33 L3 + d31 θ1 + d32 θ2 + d33 θ3 = 0 where c11 = 2L1 , c21 = 0, c31 = 0,
c12 = 0,
c13 = 0
c22 = 2L2 , c23 = 0 c32 = 0, c33 = 2L3
Fig. 2. Variation of det[D] with θ 2 and θ 3 and at θ 1 = 60◦ plane (top plot), singularity contours for det[D] at 0.0 plane (bottom plot).
(9)
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Fig. 3. Variation of det[D] with θ 1 and θ 3 and at θ 2 = 120◦ plane (top plot), singularity contours for det[D] at 0.0 plane (bottom plot).
Fig. 4. Variation of det[D] with θ 1 and θ 2 and at θ 3 = 180◦ plane (top plot), singularity contours for det[D] at 0.0 plane (bottom plot).
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Fig. 5. Variation of actuator lengths with θ 2 and θ 3 at θ 1 = 30◦ plane.
Fig. 6. Variation of actuator lengths with θ 1 and θ 3 at θ 2 = 130◦ plane.
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Fig. 7. Variation of actuator lengths with θ 1 and θ 3 at θ 3 = 230◦ plane.
Fig. 8. Variation of the loci of the actuator lengths at the planes of L1 = 400.1622 mm (top), L2 = 233.9549 mm (middle), L3 = 677.0713 mm (bottom) with θ 2 and θ 3 at θ 1 = 30◦ plane.
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and for i = 1, 2, 3 di1 = −2Aiy Bix Q1 − 2Aiy Biy Q2 − 2Aiy Biz Q3 − 2Aiz Bix Q4 − 2Aiz Biy Q5 − 2Aiz Biz Q6 di2 = −2Aix Bix Q7 − 2Aix Biy Q8 − 2Aix Biz Q9 − 2Aiy Bix Q10 − 2Aiy Biy Q11 − 2Aiy Biz Q12 −2Aiz Bix Q13 − 2Aiz Biy Q14 − 2Aiz Biz Q15 di3 = −2Aix Bix Q16 − 2Aix Biy Q17 − 2Aiy Bix Q18 − 2Aiy Biy Q19 − 2Aiz Bix Q20 − 2Aiz Biy Q21 Q1 = cos θ1 sin θ2 cos θ3 − sin θ1 sin θ3 Q2 = −cos θ1 sin θ2 sin θ3 − sin θ1 cos θ3 Q3 = −cos θ1 cos θ2 Q4 = sin θ1 sin θ2 cos θ3 + cos θ1 sin θ3 Q5 = −sin θ1 sin θ2 sin θ3 + cos θ1 cos θ3 Q6 = −sin θ1 cos θ2
Fig. 9. Variation of the loci of the actuator lengths at the planes of L1 = 400.1622 mm (top), L2 = 233.9549 mm (middle), L3 = 677.0713 mm (bottom) with θ 1 and θ 3 at θ 2 = 130◦ plane.
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Q7 = −sin θ2 cos θ3 Q8 = sin θ2 sin θ3 Q9 = cos θ2 Q10 = sin θ1 cos θ2 cos θ3 Q11 = −sin θ1 cos θ2 sin θ3 Q12 = sin θ1 sin θ2 Q13 = −cos θ1 cos θ2 cos θ3 Q14 = cos θ1 cos θ2 sin θ3 Q15 = −cos θ1 sin θ2 Q16 = −cos θ2 sin θ3
Fig. 10. Variation of the loci of the actuator lengths at the planes of L1 = 400.1622 mm (top), L2 = 233.9549 mm (middle), L3 = 677.0713 mm (bottom) with θ 1 and θ 2 at θ 3 = 230◦ .
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Q17 = −cos θ2 cos θ3 Q18 = −sin θ1 sin θ2 sin θ3 + cos θ1 cos θ3 Q19 = −sin θ1 sin θ2 cos θ3 − cos θ1 sin θ3 Q20 = cos θ1 sin θ2 sin θ3 + sin θ1 cos θ3 Q21 = cos θ1 sin θ2 cos θ3 − sin θ1 sin θ3 The expressions for the determinants of the C and D Jacobian matrices in Eq. (9) are used to generate singularity contours for a given set of geometric parameters of the manipulator, and for a range of orientation angles.
5. Numerical analysis and discussion The coordinates of the base connection points in the reference frame A are chosen as A1 = (−30, −240, −360) mm, A2 = (60, 210, −330) mm, A3 = (−210, 30, 240) mm and those of the platform in the frame B are chosen as B1 = (180, 30, 300) mm, B2 = (−240, −30, 240) mm, B3 = (30, −210, 360) mm. It is assumed that there are no physical limitations on the operation ranges of the platform; 0 ≤ θ1 , θ2 , θ3 ≤ 360◦ . Note from Eq. (9) that det[D] is large in size and difficult to handle analytically while det[C] = 8L1 L2 L3 . Mathematically, either L1 or L2 or L3 must be zero in order to make det[C] zero. But, in reality the actuator inputs
Fig. 11. Variation of det[D] with θ 2 and θ 3 and at θ 1 = 60◦ plane (top plot), singularity contours for det[D] at 0.0 plane (bottom plot).
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cannot be zero. They can only reach their limit positions. Therefore, this type of singularity occurs when one of the three actuators reaches either its maximum limit or minimum limit. det[D] might become zero for certain orientation angles. This follows that an infinitesimal motion of the end effector is possible even if the actuators are at halt. This implies that the end effector (platform) gains one or more degree of freedom and the stiffness of the manipulator in some directions disappears. For the chosen numerical values, the singularity loci at different planes of θ 1 , θ 2 and θ 3 are shown in Figs. 2–4. Please note that it is not possible to depict the variation of det[D] as a function of the three platform orientation angles. However, for a fixed value of any of the platform orientation angles, say θ 3 the variation of det[D] with the other two platform orientation angles, θ 1 , θ 2 can be presented in a 3D plot, and the corresponding singularity loci at the 0.0 plane of det[D] can be generated. Therefore, the depicted numerical results are illustrative to demonstrate the methodology rather than to give the whole of singular configurations. Assume that the manipulator has a singular configuration of (θ 1 , θ 2 , θ 3 ). Like any given (θ 1 , θ 2 , θ 3 ) configuration of the manipulator, there will exist a unique set of actuator displacements (L1 , L2 , L3 ), as described by Eq. (5). The numerical results given in Figs. 5–7 demonstrate this typical characteristic of parallel manipulators. On the other hand, for a given set of actuator displacements (L1 , L2 , L3 ), there might be as many as eight sets of (θ 1 , θ 2 , θ 3 ) solutions [20]. This distinct characteristic of parallel manipulators is demonstrated in the numerical results provided in Figs. 8–10, which are the loci of the actuator lengths at the planes of L1 = 400.1622 mm, L2 = 233.9549 mm, L3 = 667.0713 mm. These unique set of actuator lengths are calculated for a manipulator configuration of (30◦ , 130◦ , 230◦ ). The implications of Figs. 8–10 are that there are a multiple number of (θ 1 , θ 2 , θ 3 ) configurations resulting in the same actuator lengths. Some of them might be singular configurations, which need to be omitted from the planned trajectory. These singular configurations are determined from the singularity contours shown in Figs. 2–4. Hence, by making use of both Figs. 2–4 and Figs. 8–10, a manipulator trajectory free from singular configurations can be generated.
Fig. 12. Variation of det[D] with θ 1 and θ 3 and for θ 2 = 120◦ (top plot), singularity contours for det[D] at 0.0 plane (bottom plot).
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The authors believe that a note on how Figs. 5–10 are generated will be useful for the sake of clarity. Obviously, it is not possible to depict the variation of each actuator length Li = (i = 1, 2, 3) as a function of the three platform orientation angles (θ 1 , θ 2 , θ 3 ) in the same plot. However, for a fixed value of any of the platform orientation angles, e.g. θ 3 , the variation of each actuator length Li = (i = 1, 2, 3) with the other two remaining platform orientation angles, (θ 1 , θ 2 ), can be presented in a 3D plot (as shown in Figs. 5–7), and the loci of the actuator lengths at the planes of L1 = 400.1622 mm, L2 = 233.9549 mm, L3 = 667.0713 mm (as shown in Figs. 8–10), can be generated. As the manipulator is actuated in joint space, it might be required to determine the actuator inputs bringing the platform to singular configurations. If the singular configurations are known in the task (Cartesian) space of the platform, it is straightforward to calculate the corresponding actuator inputs and remove these inputs from the joint (actuation) space trajectory. Also, the trajectory planned either in joint space or task space can be projected onto the singularity loci in the same space to determine how close the trajectory is from the singularities. The numerical results presented in this section demonstrate that the proposed method can provide designers with a quick representation of singularity boundaries and a corresponding distribution of actuator displacements. In order to show that the singularity representation is formulation dependent, and further show that the method is practical, a different orientation matrix R is chosen to determine the singularity contours. To this end, we have chosen roll, pitch and yaw angles for representing the orientation of the platform, and the resulting rotation matrix is;
cθ2 cθ3
[R] = Rz (θ3 )Ry (θ2 )Rx (θ1 ) = cθ2 sθ3 −sθ2
sθ1 sθ2 cθ3 − cθ1 sθ3 sθ1 sθ2 sθ3 + cθ1 cθ3 sθ1 cθ2
cθ1 sθ2 cθ3 + sθ1 sθ3
r11
cθ1 sθ2 sθ3 − sθ1 cθ3 = r21 cθ1 cθ2
r31
r12
r13
r22
r23
r32
r33
Fig. 13. Variation of det[D] with θ 1 and θ 2 and for θ 3 = 180◦ (top plot), singularity contours for det[D] at 0.0 plane (bottom plot).
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The procedure presented above is followed; the numerical results corresponding to Figs. 2–4 are shown in Figs. 11–13, respectively. Note that the representation of the singularity contours has changed considerably. The singularity loci are described with respect to the coordinate system fixed to the platform, whose orientation with respect to the fixed base is described by the rotation matrix R. Therefore, a different orientation matrix gives a different representation of singularity loci. It must be recalled that the singularities of a specific manipulator cannot change with the formulation, but their representation depends on the formulation.
6. Conclusions We have presented a method based on the determinants of the Jacobian matrices to generate the singularity contours of a 3 DOF spherical parallel manipulator in terms of output orientation angles. This approach does not require deriving analytical expressions for the singularity contours. We have not been involved in solving forward position equations of the manipulator which are difficult to solve and result in multiple solutions. Instead, the determinants are calculated for a range of output angular positions which are then substituted into the inverse position equations in order to determine actuator inputs (lengths) needed to attain the given output positions. The proposed method which is quick, simple and systematic allows the user to generate the loci of singular configurations of any manipulator and thus, the user will be able to decide whether the singularities are acceptable or not. The method can be extended to any kinematic chain whose Jacobian matrices are described analytically. Two different orientation representation matrices are chosen to demonstrate that the representation of singularity loci is formulation dependent. The singularity loci disclose the acceptable boundaries of singularities, and will provide designers with a more accurate quantification of dexterity and kinematic accuracy. We conclude that the proposed method is useful in trajectory planning and the design of parallel manipulators. It is believed that this method contributes to previously published work from the point of view of being simple and systematic, and requiring small computational time.
Acknowledgements The authors thank anonymous reviewers for their constructive criticisms and suggestions.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
K. Sugimoto, J. Duffy, K.H. Hunt, Special configurations of spatial mechanisms and robot arms, Mech. Mach. Theor. 17 (2) (1982) 119–132. H.S. Yen, L.I. Wu, The stationary configurations of planar six-bar kinematic chains, Mech. Mach. Theor. 23 (4) (1988) 287–293. G. Alıcı, Determination of singularity contours for five-bar planar parallel manipulators, Robotica 18 (6) (2000) 569–575. S.L. Wang, K.J. Waldron, A study of singular configurations of serial manipulators, ASME J. Mech. Transm. Autom. Des. 109 (1) (1987) 14–20. Z.C. Lai, D.C.H. Yang, A new method for the singularity analysis of simple six-link manipulators, Int. J. Robot. Res. 5 (2) (1987) 66–74. G.L. Long, R.P. Paul, Singularity avoidance and the control of an eight-revolute-joint robot manipulator, Int. J. Robot. Res. 11 (6) (1992) 503–515. C.M. Gosselin, J. Angles, Singularity analysis of closed-loop kinematic chains, IEEE Trans. Robot. Autom. 6 (3) (1990) 281–290. J. Sefrioui, C.M. Gosselin, On the quadratic nature of the singularity curves of planar three-degree-of-freedom parallel manipulators, Mech. Mach. Theor. 30 (4) (1995) 533–551. H.R.M. Daniali, P.J. Zsombor-Murray, J. Angeles, Singularity analysis of planar parallel manipulators, Mech. Mach. Theor. 30 (5) (1995) 665–678. D. Basu, A. Ghosal, Singularity analysis of platform-type multi-loop spatial mechanisms, Mech. Mach. Theor. 32 (3) (1997) 375–389. C.L. Collins, G.L. Long, The singularity analysis of an in-parallel hand controller for force-reflected teleoperation, IEEE Trans. Robot. Autom. 11 (5) (1995) 661–669. J.P. Merlet, Singular configurations of parallel manipulators and Grassman geometry, Int. J. Robot. Res. 8 (5) (1989) 45–56.
G. Alıcı, B. Shirinzadeh / Robotics and Autonomous Systems 48 (2004) 77–91
91
[13] D. Zlatanov, R.G. Fenton, B. Benhabib, A unifying framework for classification and interpretation of mechanism singularities, ASME J. Mech. Des. 117 (1995) 566–572. [14] C. Innocenti, V. Parenti-Castelli, Echelon form solution of direct kinematics for the general fully-parallel spherical wrist, Mech. Mach. Theor. 28 (4) (1993) 553–561. [15] C.M. Gosselin, J. Sefrioui, M.J. Richard, On the direct kinematics of spherical three-degree-of-freedom parallel manipulators of general architecture, ASME J. Mech. Des. 116 (1994) 508–594. [16] C.M. Gosselin, L. Perreault, C. Vaillancourt, Simulation and computer-aided kinematic design of three-degree-of-freedom spherical parallel manipulators, J. Robot. Syst. 12 (12) (1995) 857–869. [17] J. Sefrioui, C.M. Gosselin, Etude et representation des lieux de singularite des manipulateurs paralleles spheriques a trios degrees de liberte avec actionneurs prismatiques, Mech. Mach. Theor. 29 (4) (1994) 559–579. [18] Z. Huang, Y.L. Yao, A new closed-form kinematics of the generalized 3-DOF spherical parallel manipulators, Robotica 17 (6) (1999) 475–485. [19] Y. Zhang, C.D. Crane III, J. Duffy, Determination of the unique orientation of two bodies connected by a ball-and-socket joint from four measured displacements, J. Robot. Syst. 15 (5) (1998) 299–308. [20] P. Ji, H. Wu, Algebraic solution to forward kinematics of a 3-DOF spherical parallel manipulator, J. Robot. Syst. 18 (5) (2001) 251–257. [21] J.J. Craig, Introduction to Robotics: Mechanics and Control, Addison and Wesley, 1989.
¨ Gursel Alici received the BSc degree with high honours from Middle East Technical University, Gaziantep, Turkey, in 1988 and the MSc degree from Gaziantep University in 1990 (both degrees in mechanical engineering); and the PhD degree in Robotics from Oxford University, UK, in 1993. He is currently a Senior Lecturer at the School of Mechanical, Materials and Mechatronics Engineering in the University of Wollongong, Australia. His research interests are in the areas of mechanics, design, control and calibration of mechanisms/robot manipulators/parallel manipulators, motion design, robotic tooling, robotics surgery and organ modelling, and micro/nano robotic systems.
Dr Bijan Shirinzadeh received his BE and MSE in Mechanical and Aerospace Engineering from the University of Michigan, USA, and PhD in Mechanical Engineering from University of Western Australia. From 1990 through 1994, he was a senior research scientist in Commonwealth Scientific Industrial Research Organization (CSIRO), Australia. Dr. Bijan Shirinzadeh is currently an Associate Professor, and the Director of Robotics & Mechatronics Research Laboratory (RMRL) which he established in 1994 in the Department of Mechanical Engineering at Monash University, Australia. His current research interests include autonomous systems, sensory-based control, laser-interferometry-based tracking and guidance, micro-nano manipulation systems, virtual reality and haptics, systems kinematics and dynamics, and automated fabrication and manufacturing.
Loci of singular configurations of a 3-DOF spherical parallel manipulator G¨ursel Alıcıa,∗ , Bijan Shirinzadehb b
a School of Mechanical, Materials and Mechatronics Engineering, University of Wollongong, 2522 NSW, Australia Robotics & Mechatronics Research Laboratory, Department of Mechanical Engineering, Monash University, 3800 Vic., Australia
Received 25 February 2003; received in revised form 29 May 2004; accepted 8 July 2004
Abstract In order to evaluate the dexterity and kinematic accuracy of a robot manipulator during the design, trajectory planning and control stages, it is of prime importance to know its singular configurations. In this paper, a systematic method to obtain singularity loci of parallel manipulators exemplified by a 3 DOF spherical parallel manipulator is addressed. Firstly, the inverse position equations are formed, and the velocity equation containing the actuator and the end effector (platform) velocities, and Jacobian matrices is derived. Secondly, the determinants of the manipulator Jacobian matrices are evaluated for a specified set of geometric parameters. Finally, the contours of the determinants at 0.0 plane, which are the singularity loci of the manipulator in Cartesian space, are generated. Two different orientation matrices are chosen to demonstrate that the representation of singularity loci is formulation dependent. The main difference of our work from previously published work is that there is no need of expressing singularity loci mathematically. This saves a considerable amount of computational time. The proposed method allows the user to generate the loci of singular configurations of any kinematics chain whose Jacobian matrices are expressed analytically. The method is a quick tool for the designers to decide whether the singularities are acceptable or not. Further, as the singularity loci indicate the acceptable boundaries of singularities, the designers can make use of it to accurately quantify the manipulator dexterity and kinematic accuracy. © 2004 Elsevier B.V. All rights reserved. Keywords: Robot manipulator; Degree of freedom
1. Introduction Many robotic applications are based on manipulating a tool or an object and keeping it steady relative to a user-defined point. Anthropomorphic, open-chain serially connected robot manipulators have traditionally been ∗ Corresponding author. Tel.: +61 2 4221 4145; fax: +61 2 4221 3101. E-mail address: [email protected] (G. Alıcı).
0921-8890/$ – see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.robot.2004.07.009
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employed to satisfy such a requirement. However, for applications such as robotics surgery, micromanipulation, and micromachining, etc. needing high precision positioning and high stiffness, parallel manipulators that consist of one or more closed kinematic chains can serve better than serially connected manipulators. One of the disadvantages of parallel manipulators is the difficulty of trajectory planning mainly due to singular configurations which are sometimes referred to as “special” or “critical” configurations [1,2]. The determination of these configurations is a challenging problem and has motivated many researchers to study various techniques in this area [3–13]. This paper focuses on the determination of singular configurations of a 3 DOF spherical parallel manipulator without deriving any analytical expression for the singularity contours. Singular configurations of a mechanism are undesirable due to the fact that the degrees of freedom of the mechanism change instantaneously. While the singular positions of serially connected manipulators result in the loss of one or more degrees of freedom [4,5,6], those of parallel manipulators result in either a gain or a loss of one or more degrees of freedom [3,7–11]. At singular configurations, the determinant of the manipulator Jacobian becomes zero. The singularities of several mechanisms have been obtained. However, nobody has used a technique not based on deriving analytical expressions for the variables making the determinant of the manipulator Jacobian matrices zero. Usually, the determinants are large in size and are difficult to handle mathematically. Gosselin and Angeles [7] introduced the concept of two Jacobian matrices, which relate input velocities to output velocities, for parallel manipulator having an equal number of inputs and outputs. The singularity of each matrix corresponds to a loss or a gain of a degree of freedom and singularity of both occurs only when the mechanism is architecturally singular. Later, Daniali et al. [9] have claimed that the singularity of both matrices is not necessarily architecture-dependent; depending on the Jacobian formulation of any of the three types of singularities described in [7] can occur. Sefrioui and Gosselin [8] have reported on the quadratic nature of the singularity curves of planar three-degree-of-freedom parallel manipulators. The roots of the determinants of the manipulator’s Jacobian are used to obtain a graphical representation of these curves in the manipulator’s Cartesian workspace. Merlet [12] has studied the singular configurations of a six-degree of freedom parallel manipulator using a geometric method rather than finding the roots of the determinants of the manipulator Jacobian matrix. Zlatanov et al. [13] have presented a generalised approach to determine the singular configurations of any mechanism with arbitrary chains and an equal number of inputs and outputs. They used a velocity equation including the velocities of active and passive joints in order to determine singular configurations. Basu and Ghosal [10] have proposed two methods (algebraic and geometric) to determine singular configurations of platform-type multi-loop spatial mechanisms containing spherical joints on the platform. Forward position analysis of parallel manipulators is a difficult task due to nonlinear equations giving multiple solutions [14–20]. This follows that a spherical manipulator with prismatic actuators has multiple configurations for a unique set of actuator displacements. On the other hand, inverse position analysis of parallel manipulators is straightforward and a set of platform orientation angles gives a unique actuator length. Further, it has been suggested in this study that it is not necessary to solve forward position equations and present singularity contours in terms of actuator inputs. The determinants are calculated for a range of output angles. After obtaining singularity contours in the operational space of the manipulator, actuator lengths corresponding to those orientation angles are calculated by employing inverse position formulation. The method proposed in this study consists of: (a) forming the inverse position equations for the manipulator, (b) deriving the velocity equation containing the actuator and the end effector (platform) velocities and Jacobian matrices, (c) calculating the determinants of the manipulator Jacobian matrices for a specified set of geometric parameters, and (d) generating the contours of the determinants at 0.0 plane which are the singularity loci of the manipulator in terms of platform orientation angles. This approach is simple, yet systematic and does not necessitate deriving the explicit expressions for the roots of the determinants in order to plot the singularity contours. This is the main difference between previously published work [16,17] and this study. Two separate orientation matrices are chosen to demonstrate that the singularity contours presented are formulation dependent. The singularity contours disclose the acceptable boundaries of singularities, and thus provide designers with a more accurate representation
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of dexterity and kinematic accuracy. It is concluded that the proposed method is useful in trajectory planning and in the design of the spherical parallel manipulator and other general parallel manipulators.
2. Inverse position formulation The manipulator considered in this study is depicted in Fig. 1, where two pyramid-like rigid bodies are connected together by a spherical pair at the common apex O. The coordinate system A is fixed to the base and the coordinate system B is attached to the moving platform which is able to rotate about three perpendicular axes. The coordinates of the connection points in the base and the platform are given with respect to the frame A and frame B, respectively. Note that both coordinate systems are located at the center of the spherical pair at point O. Three prismatic actuators connect the platform and the base to each other. Each prismatic actuator is connected to the base and the platform via an unactuated universal pair, and an unactuated spherical pair, respectively. i and B i For the provided platform orientation angles, θ 1 , θ 2 and θ 3 , and the specified coordinates of the vectors A (i = 1, 2, 3), the analytical expressions for the actuator lengths Li can be derived. Referring to Fig. 1, the following relationship can be written; i i = [R]B i − A L
(1)
i is the vector for the length of each prismatic actuator and [R] is the 3 × 3 rotation matrix defining the where L orientation of the coordinate frame B with respect to the coordinate frame A. Eq. (1) can be rewritten as follows; i )2 i − A L2i = ([R]B
(2)
Eq. (2) may also be expressed as; Ti [R]B i L2i = A2i + Bi2 − 2A
(3)
Fig. 1. Spherical parallel manipulator made up of two pyramid-like base and platform. “S”, “P” and “U” denote a spherical pair, a prismatic pair and a universal pair, respectively.
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Euler angles can be employed to describe the orientation of a rigid body relative to a fixed reference frame. The following XYZ Euler angle representation [21] is chosen for [R] in Eq. (3); cθ2 cθ3 −cθ2 sθ3 sθ2 [R] = Rx (θ1 )Ry (θ2 )Rz (θ3 ) = sθ1 sθ2 cθ3 + cθ1 sθ3 −sθ1 sθ2 sθ3 + cθ1 cθ3 −sθ1 cθ2
r11 = r21 r31
r12 r22
r13 r23
r32
r33
−cθ1 sθ2 cθ3 + sθ1 sθ3
cθ1 sθ2 sθ3 + sθ1 cθ3
cθ1 cθ2 (4)
where cθi ≡ cos θi , sθi ≡ sin θi and Rx , Ry and Rz are the basic rotation matrices about the three coordinate axes, respectively. The rotation matrix [R] is formed when the coordinate system B is rotated by θ 1 about its x-axis, then rotated by θ 2 about its modified y-axis and finally rotated by θ 3 about its modified z-axis. By substituting the position vectors Ai = (Aix , Aiy , Aiz ) and Bi = (Bix , Biy , Biz ) together with the rotation matrix into Eq. (3), the inverse position equations of the manipulator can be obtained. Each actuator length can be calculated from the following expression; L2i = A2i + Bi2 − 2[Aix (r11 Bix + r12 Biy + r13 Biz ) + Aiy (r21 Bix + r22 Biy + r23 Biz ) +Aiz (r31 Bix + r32 Biy + r33 Biz )]
(5)
It must be noted that for each value of θ 1 , θ 2 and θ 3 , the coordinates of the connection points Bi with respect to the stationary frame A can be determined from i )A = [R]B i (B
(i = 1, 2, 3)
(6)
3. Singularity analysis The nonlinear kinematic equations describing the relationship between input and output position vectors are expressed as; F (L, Θ) = 0
(7)
where F is three-dimensional for the problem at hand and is a function of the input vector L, which represents the set of the position of actuated joints, and the output vector Θ, which represents the coordinates of the output point. Taking the first time derivative of Eq. (7) leads to the relationship between the input and the output velocity vectors as follows; ∂F ˙ ∂F ˙ ˙ =0 ˙ + DΘ L+ Θ = 0 ⇒ CL ∂L ∂Θ
(8)
˙ represent the actuator (joint) velocity ˙ and Θ where C and D are configuration-dependent 3 × 3 Jacobian matrices, L vector and the end effector (platform) velocity vector, respectively. The singular positions are the singular values of the C and D matrices. Based on the singularities of the C and D matrices, there exist three types of singularities for closed-loop kinematic chains [7]. The first type of singularity is faced when det[C] = 0 and det[D] = 0. This ˙ 1, L ˙ 2 and implies that the platform loses one or more degrees of freedom. Nonzero values of the actuator velocities L ˙ 3 can generate zero values for the platform velocities θ˙ 1 , θ˙ 2 and θ˙ 3 . This type of singularity is trivial for parallel L manipulators with prismatic actuators. The second type happens when det[C] = 0 and det[D] = 0. This implies that the output point (the platform) gains one or more degrees of freedom; the output is movable even when all
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the actuated joints are locked. Another consequence of this type singularity is that the effective stiffness of the manipulator seen at the output point does not exist in some directions. The third type is encountered when det[C] = 0 and det[D] = 0, simultaneously. The third type of singularity corresponds to configurations where the manipulator can experience finite motions when its actuators are locked or where a finite input does not produce output motion. So, the singularities occur whenever C, D or both become singular. This type of singularity is avoided by a proper choice of the kinematic parameters during the design phase.
4. Singularity contours The inverse position equation for each actuator is rewritten in the form of Eq. (5), their first time derivative is taken, and the resulting equations are expressed in the form of Eq. (8), as given below; ˙ 1 + c12 L ˙ 2 + c13 L ˙ 3 + d11 θ˙ 1 + d12 θ˙ 2 + d13 θ˙ 3 = 0 c11 L ˙ =0 ˙ 1 + c22 L ˙ 2 + c23 L ˙ 3 + d21 θ˙ 1 + d22 θ˙ 2 + d23 θ˙ 3 = 0 ⇒ CL ˙ + DΘ c21 L ˙ ˙ ˙ ˙ ˙ ˙ c31 L1 + c32 L2 + c33 L3 + d31 θ1 + d32 θ2 + d33 θ3 = 0 where c11 = 2L1 , c21 = 0, c31 = 0,
c12 = 0,
c13 = 0
c22 = 2L2 , c23 = 0 c32 = 0, c33 = 2L3
Fig. 2. Variation of det[D] with θ 2 and θ 3 and at θ 1 = 60◦ plane (top plot), singularity contours for det[D] at 0.0 plane (bottom plot).
(9)
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Fig. 3. Variation of det[D] with θ 1 and θ 3 and at θ 2 = 120◦ plane (top plot), singularity contours for det[D] at 0.0 plane (bottom plot).
Fig. 4. Variation of det[D] with θ 1 and θ 2 and at θ 3 = 180◦ plane (top plot), singularity contours for det[D] at 0.0 plane (bottom plot).
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Fig. 5. Variation of actuator lengths with θ 2 and θ 3 at θ 1 = 30◦ plane.
Fig. 6. Variation of actuator lengths with θ 1 and θ 3 at θ 2 = 130◦ plane.
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Fig. 7. Variation of actuator lengths with θ 1 and θ 3 at θ 3 = 230◦ plane.
Fig. 8. Variation of the loci of the actuator lengths at the planes of L1 = 400.1622 mm (top), L2 = 233.9549 mm (middle), L3 = 677.0713 mm (bottom) with θ 2 and θ 3 at θ 1 = 30◦ plane.
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and for i = 1, 2, 3 di1 = −2Aiy Bix Q1 − 2Aiy Biy Q2 − 2Aiy Biz Q3 − 2Aiz Bix Q4 − 2Aiz Biy Q5 − 2Aiz Biz Q6 di2 = −2Aix Bix Q7 − 2Aix Biy Q8 − 2Aix Biz Q9 − 2Aiy Bix Q10 − 2Aiy Biy Q11 − 2Aiy Biz Q12 −2Aiz Bix Q13 − 2Aiz Biy Q14 − 2Aiz Biz Q15 di3 = −2Aix Bix Q16 − 2Aix Biy Q17 − 2Aiy Bix Q18 − 2Aiy Biy Q19 − 2Aiz Bix Q20 − 2Aiz Biy Q21 Q1 = cos θ1 sin θ2 cos θ3 − sin θ1 sin θ3 Q2 = −cos θ1 sin θ2 sin θ3 − sin θ1 cos θ3 Q3 = −cos θ1 cos θ2 Q4 = sin θ1 sin θ2 cos θ3 + cos θ1 sin θ3 Q5 = −sin θ1 sin θ2 sin θ3 + cos θ1 cos θ3 Q6 = −sin θ1 cos θ2
Fig. 9. Variation of the loci of the actuator lengths at the planes of L1 = 400.1622 mm (top), L2 = 233.9549 mm (middle), L3 = 677.0713 mm (bottom) with θ 1 and θ 3 at θ 2 = 130◦ plane.
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Q7 = −sin θ2 cos θ3 Q8 = sin θ2 sin θ3 Q9 = cos θ2 Q10 = sin θ1 cos θ2 cos θ3 Q11 = −sin θ1 cos θ2 sin θ3 Q12 = sin θ1 sin θ2 Q13 = −cos θ1 cos θ2 cos θ3 Q14 = cos θ1 cos θ2 sin θ3 Q15 = −cos θ1 sin θ2 Q16 = −cos θ2 sin θ3
Fig. 10. Variation of the loci of the actuator lengths at the planes of L1 = 400.1622 mm (top), L2 = 233.9549 mm (middle), L3 = 677.0713 mm (bottom) with θ 1 and θ 2 at θ 3 = 230◦ .
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Q17 = −cos θ2 cos θ3 Q18 = −sin θ1 sin θ2 sin θ3 + cos θ1 cos θ3 Q19 = −sin θ1 sin θ2 cos θ3 − cos θ1 sin θ3 Q20 = cos θ1 sin θ2 sin θ3 + sin θ1 cos θ3 Q21 = cos θ1 sin θ2 cos θ3 − sin θ1 sin θ3 The expressions for the determinants of the C and D Jacobian matrices in Eq. (9) are used to generate singularity contours for a given set of geometric parameters of the manipulator, and for a range of orientation angles.
5. Numerical analysis and discussion The coordinates of the base connection points in the reference frame A are chosen as A1 = (−30, −240, −360) mm, A2 = (60, 210, −330) mm, A3 = (−210, 30, 240) mm and those of the platform in the frame B are chosen as B1 = (180, 30, 300) mm, B2 = (−240, −30, 240) mm, B3 = (30, −210, 360) mm. It is assumed that there are no physical limitations on the operation ranges of the platform; 0 ≤ θ1 , θ2 , θ3 ≤ 360◦ . Note from Eq. (9) that det[D] is large in size and difficult to handle analytically while det[C] = 8L1 L2 L3 . Mathematically, either L1 or L2 or L3 must be zero in order to make det[C] zero. But, in reality the actuator inputs
Fig. 11. Variation of det[D] with θ 2 and θ 3 and at θ 1 = 60◦ plane (top plot), singularity contours for det[D] at 0.0 plane (bottom plot).
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cannot be zero. They can only reach their limit positions. Therefore, this type of singularity occurs when one of the three actuators reaches either its maximum limit or minimum limit. det[D] might become zero for certain orientation angles. This follows that an infinitesimal motion of the end effector is possible even if the actuators are at halt. This implies that the end effector (platform) gains one or more degree of freedom and the stiffness of the manipulator in some directions disappears. For the chosen numerical values, the singularity loci at different planes of θ 1 , θ 2 and θ 3 are shown in Figs. 2–4. Please note that it is not possible to depict the variation of det[D] as a function of the three platform orientation angles. However, for a fixed value of any of the platform orientation angles, say θ 3 the variation of det[D] with the other two platform orientation angles, θ 1 , θ 2 can be presented in a 3D plot, and the corresponding singularity loci at the 0.0 plane of det[D] can be generated. Therefore, the depicted numerical results are illustrative to demonstrate the methodology rather than to give the whole of singular configurations. Assume that the manipulator has a singular configuration of (θ 1 , θ 2 , θ 3 ). Like any given (θ 1 , θ 2 , θ 3 ) configuration of the manipulator, there will exist a unique set of actuator displacements (L1 , L2 , L3 ), as described by Eq. (5). The numerical results given in Figs. 5–7 demonstrate this typical characteristic of parallel manipulators. On the other hand, for a given set of actuator displacements (L1 , L2 , L3 ), there might be as many as eight sets of (θ 1 , θ 2 , θ 3 ) solutions [20]. This distinct characteristic of parallel manipulators is demonstrated in the numerical results provided in Figs. 8–10, which are the loci of the actuator lengths at the planes of L1 = 400.1622 mm, L2 = 233.9549 mm, L3 = 667.0713 mm. These unique set of actuator lengths are calculated for a manipulator configuration of (30◦ , 130◦ , 230◦ ). The implications of Figs. 8–10 are that there are a multiple number of (θ 1 , θ 2 , θ 3 ) configurations resulting in the same actuator lengths. Some of them might be singular configurations, which need to be omitted from the planned trajectory. These singular configurations are determined from the singularity contours shown in Figs. 2–4. Hence, by making use of both Figs. 2–4 and Figs. 8–10, a manipulator trajectory free from singular configurations can be generated.
Fig. 12. Variation of det[D] with θ 1 and θ 3 and for θ 2 = 120◦ (top plot), singularity contours for det[D] at 0.0 plane (bottom plot).
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The authors believe that a note on how Figs. 5–10 are generated will be useful for the sake of clarity. Obviously, it is not possible to depict the variation of each actuator length Li = (i = 1, 2, 3) as a function of the three platform orientation angles (θ 1 , θ 2 , θ 3 ) in the same plot. However, for a fixed value of any of the platform orientation angles, e.g. θ 3 , the variation of each actuator length Li = (i = 1, 2, 3) with the other two remaining platform orientation angles, (θ 1 , θ 2 ), can be presented in a 3D plot (as shown in Figs. 5–7), and the loci of the actuator lengths at the planes of L1 = 400.1622 mm, L2 = 233.9549 mm, L3 = 667.0713 mm (as shown in Figs. 8–10), can be generated. As the manipulator is actuated in joint space, it might be required to determine the actuator inputs bringing the platform to singular configurations. If the singular configurations are known in the task (Cartesian) space of the platform, it is straightforward to calculate the corresponding actuator inputs and remove these inputs from the joint (actuation) space trajectory. Also, the trajectory planned either in joint space or task space can be projected onto the singularity loci in the same space to determine how close the trajectory is from the singularities. The numerical results presented in this section demonstrate that the proposed method can provide designers with a quick representation of singularity boundaries and a corresponding distribution of actuator displacements. In order to show that the singularity representation is formulation dependent, and further show that the method is practical, a different orientation matrix R is chosen to determine the singularity contours. To this end, we have chosen roll, pitch and yaw angles for representing the orientation of the platform, and the resulting rotation matrix is;
cθ2 cθ3
[R] = Rz (θ3 )Ry (θ2 )Rx (θ1 ) = cθ2 sθ3 −sθ2
sθ1 sθ2 cθ3 − cθ1 sθ3 sθ1 sθ2 sθ3 + cθ1 cθ3 sθ1 cθ2
cθ1 sθ2 cθ3 + sθ1 sθ3
r11
cθ1 sθ2 sθ3 − sθ1 cθ3 = r21 cθ1 cθ2
r31
r12
r13
r22
r23
r32
r33
Fig. 13. Variation of det[D] with θ 1 and θ 2 and for θ 3 = 180◦ (top plot), singularity contours for det[D] at 0.0 plane (bottom plot).
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The procedure presented above is followed; the numerical results corresponding to Figs. 2–4 are shown in Figs. 11–13, respectively. Note that the representation of the singularity contours has changed considerably. The singularity loci are described with respect to the coordinate system fixed to the platform, whose orientation with respect to the fixed base is described by the rotation matrix R. Therefore, a different orientation matrix gives a different representation of singularity loci. It must be recalled that the singularities of a specific manipulator cannot change with the formulation, but their representation depends on the formulation.
6. Conclusions We have presented a method based on the determinants of the Jacobian matrices to generate the singularity contours of a 3 DOF spherical parallel manipulator in terms of output orientation angles. This approach does not require deriving analytical expressions for the singularity contours. We have not been involved in solving forward position equations of the manipulator which are difficult to solve and result in multiple solutions. Instead, the determinants are calculated for a range of output angular positions which are then substituted into the inverse position equations in order to determine actuator inputs (lengths) needed to attain the given output positions. The proposed method which is quick, simple and systematic allows the user to generate the loci of singular configurations of any manipulator and thus, the user will be able to decide whether the singularities are acceptable or not. The method can be extended to any kinematic chain whose Jacobian matrices are described analytically. Two different orientation representation matrices are chosen to demonstrate that the representation of singularity loci is formulation dependent. The singularity loci disclose the acceptable boundaries of singularities, and will provide designers with a more accurate quantification of dexterity and kinematic accuracy. We conclude that the proposed method is useful in trajectory planning and the design of parallel manipulators. It is believed that this method contributes to previously published work from the point of view of being simple and systematic, and requiring small computational time.
Acknowledgements The authors thank anonymous reviewers for their constructive criticisms and suggestions.
References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12]
K. Sugimoto, J. Duffy, K.H. Hunt, Special configurations of spatial mechanisms and robot arms, Mech. Mach. Theor. 17 (2) (1982) 119–132. H.S. Yen, L.I. Wu, The stationary configurations of planar six-bar kinematic chains, Mech. Mach. Theor. 23 (4) (1988) 287–293. G. Alıcı, Determination of singularity contours for five-bar planar parallel manipulators, Robotica 18 (6) (2000) 569–575. S.L. Wang, K.J. Waldron, A study of singular configurations of serial manipulators, ASME J. Mech. Transm. Autom. Des. 109 (1) (1987) 14–20. Z.C. Lai, D.C.H. Yang, A new method for the singularity analysis of simple six-link manipulators, Int. J. Robot. Res. 5 (2) (1987) 66–74. G.L. Long, R.P. Paul, Singularity avoidance and the control of an eight-revolute-joint robot manipulator, Int. J. Robot. Res. 11 (6) (1992) 503–515. C.M. Gosselin, J. Angles, Singularity analysis of closed-loop kinematic chains, IEEE Trans. Robot. Autom. 6 (3) (1990) 281–290. J. Sefrioui, C.M. Gosselin, On the quadratic nature of the singularity curves of planar three-degree-of-freedom parallel manipulators, Mech. Mach. Theor. 30 (4) (1995) 533–551. H.R.M. Daniali, P.J. Zsombor-Murray, J. Angeles, Singularity analysis of planar parallel manipulators, Mech. Mach. Theor. 30 (5) (1995) 665–678. D. Basu, A. Ghosal, Singularity analysis of platform-type multi-loop spatial mechanisms, Mech. Mach. Theor. 32 (3) (1997) 375–389. C.L. Collins, G.L. Long, The singularity analysis of an in-parallel hand controller for force-reflected teleoperation, IEEE Trans. Robot. Autom. 11 (5) (1995) 661–669. J.P. Merlet, Singular configurations of parallel manipulators and Grassman geometry, Int. J. Robot. Res. 8 (5) (1989) 45–56.
G. Alıcı, B. Shirinzadeh / Robotics and Autonomous Systems 48 (2004) 77–91
91
[13] D. Zlatanov, R.G. Fenton, B. Benhabib, A unifying framework for classification and interpretation of mechanism singularities, ASME J. Mech. Des. 117 (1995) 566–572. [14] C. Innocenti, V. Parenti-Castelli, Echelon form solution of direct kinematics for the general fully-parallel spherical wrist, Mech. Mach. Theor. 28 (4) (1993) 553–561. [15] C.M. Gosselin, J. Sefrioui, M.J. Richard, On the direct kinematics of spherical three-degree-of-freedom parallel manipulators of general architecture, ASME J. Mech. Des. 116 (1994) 508–594. [16] C.M. Gosselin, L. Perreault, C. Vaillancourt, Simulation and computer-aided kinematic design of three-degree-of-freedom spherical parallel manipulators, J. Robot. Syst. 12 (12) (1995) 857–869. [17] J. Sefrioui, C.M. Gosselin, Etude et representation des lieux de singularite des manipulateurs paralleles spheriques a trios degrees de liberte avec actionneurs prismatiques, Mech. Mach. Theor. 29 (4) (1994) 559–579. [18] Z. Huang, Y.L. Yao, A new closed-form kinematics of the generalized 3-DOF spherical parallel manipulators, Robotica 17 (6) (1999) 475–485. [19] Y. Zhang, C.D. Crane III, J. Duffy, Determination of the unique orientation of two bodies connected by a ball-and-socket joint from four measured displacements, J. Robot. Syst. 15 (5) (1998) 299–308. [20] P. Ji, H. Wu, Algebraic solution to forward kinematics of a 3-DOF spherical parallel manipulator, J. Robot. Syst. 18 (5) (2001) 251–257. [21] J.J. Craig, Introduction to Robotics: Mechanics and Control, Addison and Wesley, 1989.
¨ Gursel Alici received the BSc degree with high honours from Middle East Technical University, Gaziantep, Turkey, in 1988 and the MSc degree from Gaziantep University in 1990 (both degrees in mechanical engineering); and the PhD degree in Robotics from Oxford University, UK, in 1993. He is currently a Senior Lecturer at the School of Mechanical, Materials and Mechatronics Engineering in the University of Wollongong, Australia. His research interests are in the areas of mechanics, design, control and calibration of mechanisms/robot manipulators/parallel manipulators, motion design, robotic tooling, robotics surgery and organ modelling, and micro/nano robotic systems.
Dr Bijan Shirinzadeh received his BE and MSE in Mechanical and Aerospace Engineering from the University of Michigan, USA, and PhD in Mechanical Engineering from University of Western Australia. From 1990 through 1994, he was a senior research scientist in Commonwealth Scientific Industrial Research Organization (CSIRO), Australia. Dr. Bijan Shirinzadeh is currently an Associate Professor, and the Director of Robotics & Mechatronics Research Laboratory (RMRL) which he established in 1994 in the Department of Mechanical Engineering at Monash University, Australia. His current research interests include autonomous systems, sensory-based control, laser-interferometry-based tracking and guidance, micro-nano manipulation systems, virtual reality and haptics, systems kinematics and dynamics, and automated fabrication and manufacturing.