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Optimal design of a new spatial 3-DOF parallel robot with respect to a frame-free index WANG JinSong, LIU XinJun† & WU Chao Department of Precision Instruments, Tsinghua University, Beijing 100084, China
Optimal design is one of the most important issues in robots. Since the very beginning, the concepts of the Jacobian matrix, manipulability and condition number, which are used successfully in the field of serial robots, have been applied to parallel robots. Unlike serial robots, parallel robots are good for motion/force transmission. Their performance evaluation and design should be correspondingly different. This paper is an attempt to optimally design a novel spatial three-degree-of-freedom (3-DOF) parallel robot by using the concept of motion/force transmission. Accordingly, three indices are defined. The suggested indices are independent of any coordinate frame and could be applied to the analysis and design of a parallel robot whose singularities can be identified wholly by using the relative angle between the output and adjacent links, and by using the relative angle between the input and adjacent links. optimal design, parallel robots, index, transmission angle, motion/force transmission
1 Introduction Due to the fact that they have the advantages of high stiffness, high velocity, compactness, time-saving of machining, high load/weight ratio, and low moving inertia, parallel robots have been studied intensively for more than twenty years and are still receiving attention from universities and industry[1,2]. In this field, optimal design is one of the most important and challenging problems and is attracting more and more efforts[3, 4]. There are two issues involved: performance evaluation and dimension synthesis. Since the performance of a robot depends on several factors, it is usually difficult to say that a particular design is the only solution to a given problem, even for a mechanism with only one degree of freedom (DOF) and four links. When the number of links and number of DOF increase, the design becomes more complicated for the robot. Therefore, it is good to know how well a mechanism may run when it is still in the design stage. For parallel robots, several well-defined performance indices, which are popular in the field of serial robots,
have been developed extensively and applied to the design. However, a recent study[5] reviewed the most common indices that have been applied to the optimal design of parallel robots. They are the condition number[3] of the Jacobian matrix applied to increase its accuracy, and the global conditioning index[6] that is the computation over a kind of workspace of the robot. The conclusion of the paper is that these indices should not be used for parallel robots with mixed types of degrees of freedom (translations and rotations). As is well known, the four-bar mechanism has been studied for a very long time. The transmission angle is an important index for the design of such a mechanism as was pointed out by Alt[7], who defined the concept, using the forces tending to move the driven link and tending to apply pressure to the driven link bearings as a Received January 18, 2008; accepted August 25, 2008; published online January 12, 2009 doi: 10.1007s11431-008-0305-4 † Corresponding author (email:
[email protected]) Supported by the National Natural Science Foundation of China (Grant No. 50775118), High Technology Research and Development Program of China (863 Program) (Grant No. 2006AA04Z227), and National Basic Research Program of China (973 Program) (Grant No. 2007CB714000)
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simple index, to judge the force-transmission characteristics of a mechanism. By means of the index of transmission angle the quality of motion/force transmission in a mechanism can be judged in the design stage. Therefore, the transmission angle is an index evaluating the quality of motion/force transmission. It helps to decide the “best” among a family of possible mechanisms for the most effective force transmission[8]. Actually, a good transmission angle is the solution to most of the problems in planar mechanisms. For example, Alt[7] used the transmission angle to isolate better chains for various linkage applications. As was pointed out in ref. [9], though a good transmission angle is not a cure-all for every design problem, for many mechanical applications it can guarantee the performance of a linkage at high speed without unfavorable vibrations. The study in ref. [10] showed that when the transmission angle is equal to 90°, the most effective force transmission takes place and the output motion becomes less sensitive to the manufacturing tolerances on the link lengths, clearance between joints, and change of dimensions due to thermal expansion. Mechanisms having a transmission angle too far from 90° exhibit poor operational characteristics such as noise and jerk at high speeds[11,12]; and if it is 0, self-locking takes place. Thus the transmission angle of a mechanism provides a very good indication of the quality of its motion, the accuracy of its performance, its expected noise output, and its costs in general[13]. A large transmission angle usually leads to reasonable mechanical advantages and a high quality of motion transmission. The study of link mechanisms shows that transmission angle is significant not only as an indicator of good force and motion transmission but also as a prime factor in the linkage sensitivity to small design parameter errors. The smaller the transmission angle, the more sensitive the linkage will be[14]. Many studies have reached the conclusion that if the transmission angle becomes too small, the mechanical advantage becomes small, and even a very small amount of friction will cause the mechanism to jam. For the purpose of high speed, high accuracy, and high quality of motion transmission, the most widely accepted design limits for the transmission angle are (45°, 135°)[12] or (40°, 140°)[7]. Additionally, the transmission angle does not consider the dynamic forces due to velocity and accelerations. For this reason, it is widely used in the kinematic synthesis stage during which the lengths and mass properties of the links are
still unknown to the designer[15] that the kinematically determined transmission angle does not reflect the action of gravity or dynamic forces. It is obvious that the study of planar four-bar mechanisms has a very long history, longer than that of serial robots, and much longer than that of parallel robots. The Jacobian matrix, manipulability and condition number are concepts that have been proposed and used in the field of serial robots. The study results were, in general, directly transferred to parallel robots from the very beginning. A recent study[5] showed that the use of condition number of the Jacobian matrix of a mixed parallel robot is of dubious value when it is applied to the optimal design. However, without the Jacobian matrix the transmission angle can judge the quality of motion transmission and velocity and the performance sensitivity to the mechanical errors, tolerances and clearances. A planar four-bar mechanism is a single-closed-loop system. A parallel robot is a multi-closed-loop mechanism. Usually, a fully parallel robot has more or less the characteristic of a planar four-bar mechanism. We suggest that the design concept of the four-bar mechanisms should be used in the design of a parallel robot. In this paper, the local and global transmission indices (defined in Section 5) instead of the local and global conditioning indices will be proposed here as indices in the optimal design of a new spatial 3-DOF parallel robot, which can be kinematically considered as the combination of two planar mechanisms at any moment. The remainder of this paper is organized as follows. The next section describes the new spatial parallel robot. Section 3 investigates the traditional kinematic problems, e.g. inverse kinematics and Jacobian matrix. Section 4 recalls the classical concept of transmission angle, defines the forward and inverse transmission angles, and describes the relationship between the transmission angles and the singularities of mechanisms. Section 5 addresses the proposal and definition of some indices. Section 6 presents the optimal design of the new parallel robot using the recommended indices. Discussion and future work are presented in Section 7. Conclusions are given in the last section.
2 A new spatial 3-DOF parallel robot 2.1 Architecture description The new parallel robot, shown in Figure 1, contains a triangular plate referred to as the moving platform. The
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platform is an isosceles triangle, whose vertices are connected to the base through three legs. In this paper the three legs are referred to as the first (1), second (2) and third (3) legs, respectively. The first and second legs are in a same plane and have identical chains, each of which consists of a fixed length link, which is linked to an active slider through a passive revolute joint and is connected to the moving platform by a universal (or spherical) joint. In the case of universal joints in the two legs, the two revolute joints connected to the moving platform are collinear; the axes of the left four revolute joints are parallel to each other. The third leg is different from the former two legs; it consists of another fixed length link, which is linked to an active slider by a passive revolute joint and is connected to the moving platform through a passive cylinder joint. In this leg, axes of the revolute and cylinder joints are parallel to the x-axis. The three sliders are then attached to the base by three active prismatic joints. For the structure shown in Figure 1, drives for the three legs are all along the y-axis. This robot is actually a variation of HALF manipulator[16]. We here call the robot HALF*. It is undisputed that the moving platform of the robot has three spatial DOFs, which are two translations in the O-xy plane and one rotation about the x’-axis. It is not difficult to find out that during the movement the first and second legs act as a planar translational mechanism and the third leg serves as a mechanism providing a rotation. 2.2 Advantage One may see that the new parallel robot and the HALF manipulator have the same mobility, i.e. two translations and one rotation. However, there are some obvious differences between them. For example, for the HALF*, there is no parallelogram in the third leg. As is well known, a parallelogram increases the manufacturing difficulty and cost, and affects the accuracy of a device. From Figure 1, one may see that if the moving platform wants to translate along the x-axis with a constant orientation, the third leg can keep its configuration without any actuation. But, to implement the same function, the third leg of the HALF manipulator must be actuated. In other words, the new robot is x-translation-and-rotation decoupled. Therefore, compared with the HALF manipulator, the HALF* robot proposed here has the advantages in kinematics, architecture, manufacturing, energy cost, accuracy and assembling. Just like the HALF manipulator, the new robot is also 988
Figure 1 A new spatial 3-DOF parallel robot: (a) CAD model; (b) kinematical scheme.
good at the rotational capability. The tilting angle of the moving platform can be as high as ±50°. What is most important is that at any moment the new robot can be considered as the combination of two planar mechanisms. Especially, the third leg is actually a slider-crank mechanism. Planar mechanisms have been well studied. Their analysis and design are popular and are approbated by both academic community and industry. If a parallel robot is the combination of planar mechanisms, the analysis and design can be implemented leg by leg, using the classical methods used in the planar mechanisms. Its analysis and design will be simple and effective. In this sense, we think the proposal of such a kind of parallel robot is animate, and the application will be popular. Here, we should mention that when the new robot translates along the y-axis, the third leg will no longer keep its configuration if there is no actuation in the prismatic joint. That is the reason why we only indicate that the robot is x-translation-and-rotation decoupled. However, in the analysis and design of a robot actuated along the y-axis, we can only consider the performance along the x-axis[37]. Therefore, this drawback will not affect the analysis and design of the robot leg by leg. Section 6 will give a detailed description about this problem. 2.3 Disadvantage Although the new robot has some advantages compared with the HALF parallel manipulator, it still has its own disadvantages. For example, the adoption of cylinder joint may cause operational failure since it is a passive joint. To avoid this problem, we can apply the passive DOF to the robot. For example, in the HALF* robot, the joints connected to the moving platform in the first and second legs can be spherical joints, in each of which there is one passive DOF. On the other hand, to decrease
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the operational failure, the demand on the parallelism of axes for the related revolute joints in the robot should be very strict. This increases the manufacturing cost. Additionally, the positional workspace of the robot may be limited due to the cylinder joint. Therefore, the new robot can only be applied in the field where a relatively small workspace is required.
3 Traditional analysis of the new parallel robot 3.1
Inverse kinematic problem
Then, there are ( R − r + x) 2 + ( y1 − y )2 = R22 ,
(4)
( R − r − x)2 + ( y2 − y ) 2 = R22 , ( y3 − y − L1 sin φ ) + ( L3 − L1 cos φ ) = 2
2
(5) L22
.
O-xyz and can be written as T b1 = (− R y1 0)T , b2 = ( R y2 0)T , b3 = (0 y3 − L3 ) .
(1) In the reference frame O-xyz, position vectors pi (i=1, 2, 3) of points Pi can be written as p1 = ( x + r y 0)T and
p3 = (0 y + L1 sin φ − L1 cos φ ) , T
(2)
where (x, y, φ ) is the pose of the robot and φ is the rotating angle of the moving platform about the x′-axis. It is noteworthy that the x coordinate of point P3 at the
(6)
For a given pose (x, y, φ ), the inputs yi (i=1, 2, 3) can be obtained as y1 = ± R22 − ( R − r + x ) + y ,
(7)
y2 = ± R22 − ( R − r − x ) + y ,
(8)
y3 = ± L22 − ( L3 − L1 cos φ ) + y + L1 sin φ .
(9)
2
A kinematical scheme of the manipulator is developed as shown in Figure 1(b). Vertices of the output platform are denoted as platform joints Pi (i=1, 2, 3), and central points of the three revolute joints attached to the sliders are denoted as Bi (i=1, 2, 3). A fixed global reference frame O-xyz is located at the center point of line segment ab with the y-axis normal to the plane abc and the x-axis directed along ab. Another reference frame, called the moving frame: O′-x′y′z′, is located at the center of the side P1P2 . The y′-axis is perpendicular to the moving platform and x′-axis directed along P1P2. Since the first and second legs are the same in kinematic chains and the third leg is different, the geometric parameters for the first and second legs can be the same, but different from those of the third leg. Then, the geometric parameters will be O′P1=O′P2=r, B1P1=B2P2=R2, O′P3=L1, P3B3=L2, the normal distance L3 from the point O to the straightline path of the joint point B3, i.e. Oc=L3, and Oa= Ob=R. The inverse kinematic problem of this robot is somewhat similar to that of the HALF manipulator. It can be solved easily. Vectors bi (i=1, 2, 3) are defined as the position vectors of points Bi in the reference frame
p1 = ( x − r y 0)T ,
fixed length link of the third leg is always zero. The kinematic problem of the robot can be solved by writing bi − pi = Bi Pi . (3)
2
2
Therefore, there are eight inverse kinematic solutions for the robot. The configuration shown in Figure 1 corresponds to the solution when the “ ± ” signs in eqs. (7) —(9) are all “+”. In this paper, we are only concerned with this solution. Observing eqs. (6) and (9), one may see that the kinematics of the third leg is independent of the x coordinate. 3.2 Jocabian matrix Eqs. (4)—(6) can be differentiated with respect to time to obtain the velocity equations. This leads to an equation of the form A( y y (10) y )T = B ( x y φ)T , 1
2
3
where A and B are 3×3 matrices that can be expressed as 0 0 ⎡ y1 − y ⎤ ⎢ ⎥, A=⎢ 0 0 y2 − y ⎥ ⎢⎣ 0 0 y3 − y − L1 sin φ ⎥⎦ 0 ⎤ y1 − y ⎡r − R − x ⎢ B = ⎢R − r − x 0 ⎥⎥ , y2 − y ⎢⎣ 0 y3 − y − L1 sin φ J 33 ⎥⎦
(11)
in which J 33 = L1 [ ( y3 − y ) cos φ − L3 sin φ ] . If matrix A is nonsingular, the Jacobian matrix of the robot can be obtained as
J = A−1 B . (12) From eq. (11), one may see that elements in matrix B have no identical dimensions. This is a common problem for a parallel robot with mixed types of degrees of freedom. To deal with the non-homogeneity of the ma-
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trix, the mostly used method is dividing all elements in the third column by a link parameter. However, this solution is suspectable. For example, if the element J33 is divided by parameter L1 , then matrix B can be rewritten as ⎡r − R − x B* = ⎢⎢ R − r − x ⎢⎣ 0
y1 − y y2 − y y3 − y − L1 sin φ
0 ⎤ ⎥, 0 ⎥ ( y3 − y )cos φ − L3 sin φ ⎥⎦
(13) which indicates that L1=0 is not the condition of singularity. However, it is obvious that L1=0 leads to the singularity of the robot. In this sense, it is not difficult to see that dividing the related elements in the Jocabian matrix by a link parameter to solve the non-homogeneity problem is unreasonable. The condition number of a non-homogeneous Jocabian matrix will lose its physical meaning. Just as investigated in ref. [5], the conclusion is that the condition number should not be used in such a parallel robot. Instead of the indices related to the condition number, such as local conditioning index (LCI) and the global conditioning index (GCI), some indices based on the concept of transmission angle will be suggested for the optimal design of the new parallel robot studied here.
4 Transmission angle 4.1 Definition The transmission angle is something we are very familiar with without realizing it. In everyday life, we frequently try to move an object that is somehow constrained, which cannot move freely but is attached to something: the handle of a crank, a curtain on a rail, a sliding door. In all of these cases, the object may not be able to move even when we exert pressure against it. Let us take the case of the handle of a crank as an example. As shown in Figure 2, the crank is attached to the base with a constant counterclockwise moment M. To move it, we must apply a right-hand force F at the end of the crank. When the direction of the force is constant, depending on the position of the end point it will be more or less easy to start rotating. This is the reason why we sometimes feel comfortable and sometimes laborious when riding a bicycle. This is actually a matter of the transmission angle. Since the direction of motion of a crank is always perpendicular to the crank, the smaller angle between the force and crank is defined as the 990
transmission angle, denoted as μ. When the force is normal to the crank, i.e. identical to the direction of motion, force transmission is most effective; when the force is perpendicular to the direction of motion, force transmission is very inefficient.
Figure 2
The handle of a crank.
For the planar four-bar mechanism shown in Figure 3, if O1A is the input link, the force applied to the output link BO2 is transmitted through the coupler link AB. For sufficiently slow motions (negligible inertia forces), the force in the coupler link is pure tension or compression (negligible bending action) and is directed along AB. For a given force in the coupler link, the torque transmitted to the output bar (about point O2) is at a maximum when the angle μ between the coupler bar AB and the output bar BO2 is 90°. Therefore, angle ABO2 is called the transmission angle. In ref. [11], the transmission angle was defined as the smaller angle between the direction of the velocity difference vector of the driving link and the direction of the absolute velocity vector of the output link, both taken at the point of connection. Although there are other definitions (see refs. [7, 17], for instance), all these definitions are somehow related to a joint variable(s) of the mechanism.
Figure 3
Transmission angles.
When the transmission angle deviates significantly from 90°, the torque on the output bar decreases and may not be sufficient to overcome the friction in the system. For this reason, the deviation angle α =| 90° − μ | should not be too great.
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Meanwhile, at the moment that the angle γ between the input link O1A and the coupler link AB is 0 or 180°, as shown in Figure 3, the output point B will not move whatever the input is. Then, the motion of the input cannot be transmitted to the output effectively. This means the output point will lose a degree of freedom. Therefore, the deviation angle β =| 90° − γ | should not be too great either. In this paper, the angle μ is defined as the forward transmission angle, and the angle γ is referred to as the inverse transmission angle. If BO2 is an active link, the angle O1AB is the forward transmission angle and angle ABO2 the inverse transmission angle. Take the slider-crank mechanism shown in Figure 4 as an example, where angle O′P3B3 is the forward transmission angle and the angle between the coupler link and the normal to the straight-line path of the slider is the inverse transmission angle if the slider is actuated. For the 2-DOF 5R parallel robot shown in Figure 5, the forward transmission angle is defined as the angle between the two couplers AP and PC[18]; the inverse transmission angle is the angle between the input link
Figure 4
A slider-crank mechanism.
Figure 5
A 5R parallel robot.
and the coupler. For the 5R robot, there are two inverse transmission angles. By using the concept of transmission angle, as of 1961, the design of a 2-DOF five-bar mechanism was investigated[19]. Recently, a synthesis method for the 5R variable topology mechanism was proposed with transmission angle control[20]. The forward and inverse transmission angles of a PRRRP robot are defined as shown in Figure 6. The definitions follow those of the slider-crank mechanism and the 5R robot. 4.2 Relationship between transmission angle and singularity The transmission angle is an important index that can evaluate the quality of motion/force transmission. For this reason, the concept has been used in the design of almost all types of four-bar mechanisms, some other planar five-, six- and seven-bar mechanisms[21], the spatial four-link mechanisms[22] and even the 2-DOF spatial RSSRP mechanism[23]. Singularity is one of the most important problems in robots, especially in parallel robots. Since singularity leads to a loss of controllability and degradation of the natural stiffness, the analysis of parallel robots has attracted considerable attention[24−26]. There is a close relationship between the transmission angle and singularity. Take the four-bar mechanism shown in Figure 3 as an example. When μ=0 or 180° the mechanism is in the “dead point” which is the second kind of singularity according to the classification in ref. [24], while γ = 0 or 180° corresponds to the first kind of singularity where the mechanism has the self-jamming characteristic. For the slider-crank mechanism shown in Figure 4, the configuration O′P′3B′3 where μ=180° is the second kind of singularity, whereas the configuration O′P′3B′3 where γ = 0 is the first kind. The singularity of
Figure 6
A PRRRP parallel robot.
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the 5R parallel robot shown in Figure 5 has been listed systematically in ref. [27]. One may see that γ = 0 or 180° and μ = 0 or 180° represent the first and second kinds of singularities of the robot, respectively, and so are those of the PRRRP robot shown in Figure 6. Therefore, the forward and inverse transmission angles, μ and γ, are close relatives of the two kinds of singularities of the parallel robot. It is easy to conclude that if γ = 0 or 180° and μ = 0 or 180° occur at the same time, the robot will be in the third kind of singularity. In ref. [28], the authors also investigated the relationship between pressure angle, which is the complement of the transmission angle, and singularity, and reached the conclusion that in the singularity the pressure is at 90°. However, according to their definition of pressure angles, their discussion was actually only about the second kind of singularity. Unfortunately, based on the transmission angles, we cannot identify other singularities, such as architecture singularity[29] and some constraint singularities[30] of a parallel robot, especially a spatial parallel robot.
5 Definition of some indices 5.1 Local transmission index (LTI) The condition number of the Jacobian matrix is an index that has been used successfully in the design of serial robots. Although the condition number is depended on the coordinate frame, it has the main advantage of describing the kinematic behavior of a robot by means of a number. The index has also been applied to the analysis and design of parallel robots. It was used as an index to evaluate the accuracy/dexterity[31,32], and to describe the closeness of a pose to a singularity of a parallel robot[28,33]. In an optimal design, the condition number (or its reciprocal) is used to define a useful, good-condition or effective workspace (GCW or GEW) with a specified minimum LCI[27,34,35]. However, the minimum is still arbitrary or comparative since we cannot give it a definite value. Generally, it is not possible to define a mathematical distance to a singularity for a parallel robot[5]. Instead of LCI, another index will be defined here. Following the definition of transmission index proposed in ref. [36], an index based on the transmission angle is defined as χ = sin(TA) , (14) 992
where TA=μ or TA = γ . Then, there is 0 ≤ χ ≤1 .
(15)
A larger χ indicates a better motion/force transmission. Since at a different pose the transmission angle will be different, the index χ is referred to as the local transmission index (LTI) in this paper. The angle is defined as the figure formed by two lines diverging from a common point, or as that formed by two planes diverging from a common line. Thus, the angle is usually measured by the ratio of two linear parameters. Therefore, the LTI is definitely independent of any coordinate frame. This is one of its advantages and is the most important for the optimal design of mechanisms. For the purpose of high speed and high quality of motion/force transmission, the most widely accepted range for the transmission angle is (45°, 135°)[12] or (40°, 140°)[7]. Therefore, the LTI limit will correspondingly be
χ > sin(π 4) or χ > sin(2π 9) .
(16)
Then, unlike the LCI, the LTI has a significative limitation to its application. It is worth mentioning that in refs. [28,37], the authors defined a transmission index, i.e. cosα , by using the concept of the pressure angle α. Since the pressure angle is the complement of the transmission angle, the two definitions have the same meaning. However, the authors did not define indices to evaluate a workspace with good transmission and to judge the effectiveness of motion/force transmission of a robot over a whole workspace. The two kinds of indices will be very important in the design of a parallel robot. 5.2 Good-transmission workspace (GTW)
With the minimum of LTI, i.e. sin(π 4) or sin(2π 9) , we can identify a workspace for a mechanism. The corresponding workspace is referred to as a good-transmission workspace (GTW), which is defined as the set of poses where the transmission index for every transmission angle is greater than sin(π 4) or sin(2π 9) . In other words, within the GTW, at every pose the sine of every transmission angle is subject to the condition given by eq. (16). Let us see an example, i.e. the planar 5R symmetrical parallel robot shown in Figure 5. The kinematic problems of the robot were presented in ref. [27]. As analyzed in the paper, for a given position vector p = (x y)T of the output point P, the position vectors aℜ
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and cℜ , in the reference frame ℜ : O-xy, of points A and C, respectively, can be obtained. Then, there are ⎛ 2r 2 − a c 2 ℜ ℜ 2 ⎜ 2 r 2 ⎝
μ = cos −1 ⎜
γ i = cos
⎛ r2 −1 ⎜ 1 ⎜⎜ ⎝
2
⎞ ⎟, ⎟ ⎠
2 + r22 − oi ℜ p ⎞ ⎟ , i=1,2, ⎟⎟ 2r1r2 ⎠
(17)
(18)
where oi ℜ (i=1,2) are the position vectors of points Oi. Letting sin μ > sin(π 4) and sin γ i > sin(π 4) , we can obtain the GTW of the robot numerically. This means that for a given position ( x, y ) of point P to calculate the forward and inverse transmission angles μ and γi using eqs. (17) and (18), if they all are subject to the LTI constraint given in eq. (16), the point belongs to the GTW. For example, the GTW of the 5R parallel robot (see Figure 5) with parameters r1=0.85, r2=1.6 and r3=0.55 is shown as the region bounded by the loci of LTI=sin(π/4) in Figure 7. For comparison, in Figure 7, the good-condition workspace (GCW) specified by the loci of LCI=0.5 (LCI is defined as the reciprocal of the condition number of the Jacobian matrix) is also illustrated. One may see that the two kinds of workspaces defined by LCI and LTI are different from others. There are two points one should notice. Firstly, from a given LCI we don’t know how far the pose is from the singularity nor we can be sure whether the GCW (or the specified LCI value 0.5) is suitable or not. For example, observing Figure 7, one may see that at and near point a, the configuration when LCI=0.5 is still very near to the singularity. However, the transmission angle has a definite physical meaning. We can be sure that in the GTW defined by the LTI the motion/force transmission of the robot is effective. Secondly, for some configurations with a worse LCI value (for example less than 0.5 in the two examples) the robot is still effective in motion/force transmission; for some configurations with higher LCI the robot has worse motion/force transmission. Since the transmission angle has a specific meaning, we here prefer the design using the LTI.
Figure 7 Good-transmission workspace (GTW) and Good-condition workspace (GCW) of the 5R parallel robot with the parameters r1=0.85, r2=1.6 and r3=0.55. The regions bounded by the loci of LTI=sin(π/4) and LCI=0.5 are the GTW and GCW, respectively.
account the behavior within a considered workspace. In order to measure the global behavior of the motion/force transmission over the whole GTW, following the definition of GCI suggested in ref. [6], a global transmission index (GTI) is defined as n
Γ =
∫W ∑i χi
n dW
∫W dW
,
(19)
in which W denotes the good-transmission workspace, n the number of transmission angles and Γ min < Γ < 1 ( Γ min equals sin(π 4) or sin(2π 9) ). It is obvious that the GTI is also independent of any coordinate frame. Undoubtedly, for robots with different link lengths, their GTWs will be the same. If such a case occurs, we cannot judge which robot is better with respect to the GTW index itself. However, with the same GTWs, their GTIs may be different. The two indices, GTW and GTI, together will help us to optimally design a robot. Additionally, for a specified design problem, other performance indices such as stiffness and accuracy may be involved in identifying a better solution. This is not the content of the paper. To demonstrate the use of the proposed indices, i.e., LTI, GTI and GTW, the subsequent section will discuss the optimal design of the parallel robot proposed in Section 2.
5.3 Global transmission index (GTI)
6 Optimal design of the new parallel robot
The LTI χ can only judge the effectiveness of motion/force transmission of a robot at a single pose. Usually, a robot performs a task in a specified workspace but not at a pose. In a practical design, we should make a decision whether a robot is good or not by taking into
In this section, the optimal design of the robot will be implemented by means of the recommended indices, i.e. LTI, GTI and the GTW. Letting R-r in eqs. (4), (5), (7) and (8) be R1, i.e. R1=R-r, we can see that the kinematics of the first and second legs are actually those of the
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PRRRP symmetrical parallel robot shown in Figure 6. If the position of point O′ is specified, the kinematic equation of eq. (6) is actually that of a slider-crank mechanism shown in Figure 4. Therefore, the singular configuration of the robot is that when one of the PRRRP robot and the slider-crank mechanism is in singularity. The optimal design of the robot can be then divided into two parts: (a) design of the PRRRP parallel robot, and (b) design of the slider-crank mechanism. Since the transmission angle is the classical concept and tool in the analysis and design of planar mechanisms and the new decoupled parallel robot proposed in this paper is the combination of two planar mechanisms, the LTI, GTI and GTW can be applied to the optimal design of the new parallel robot. The optimal design will be implemented by using performance atlases. The advantages of this method were introduced in ref. [38].
r1max =
2cos(π 4) ≈ 0.8284 . 1 + cos(π 4)
(23)
Please note that some robots when r1 < r1min still have a GTW along the x-axis. But the workspace has two discontinuous regions, which are symmetric about the line x=0. For example, the GTW and some configurations of the robot with r1=0.54 and r1=1.46 are shown in Figure 8. One may see that when x=0, μ＜45°. There are two regions where LTI is subject to the constraint in eq. (16). Usually we do not prefer this kind of workspace in practice. In this paper, the GTW of such a robot is assumed to be zero.
6.1 Performance atlas
6.1.1 PRRRP parallel robot (the first and second legs). The kinematic analysis and optimal design of the PRRRP symmetrical parallel robots were studied in detail in ref. [39]. Normalization of the two-dimensional parameters R1 and R2 proposed in ref. [39] leads to two non-dimensional parameters r1 and r2; there are r1+r2=2 and 0 < r1 ≤ r2 ≤ 2 . Then, there is r1 ≤ 1 , which is actually the parameter design space of the PRRRP parallel robot. In the design process, only parameter r1 should be optimized. Using the kinematic analysis result of ref. [39], we can write −1 ⎛
2r 2 − r 2 ⎞ μ = cos ⎜⎜ 2 2 AB ⎟⎟ , ⎝ 2r2 ⎠
⎛ r12 + r22 − ri2y ⎞ ⎟ , i=1,2, ⎜ ⎟ 2r1r2 ⎝ ⎠
γ i = cos −1 ⎜
(20) (21)
where rAB = ry2 + 4r12 , ry = r22 − ( x − r1 )2 − r22 − ( x + r1 )2 and ri y = r22 + (−1)i 2 xr1 − r12 . Eqs. (20) and (21) imply that the forward and inverse transmission angles are independent of the y coordinate. Then, the LTI has no relation with the parameter. This means in the analysis and design we can ignore the workspace along the y-axis. This characteristic is identical with that about the LCI[39]. If the LTI is given by χ > sin(π/4) , the minimum and maximum values of r1 will be 2sin(π 8) r1min = ≈ 0.5535 , 1 + sin(π 8) 994
(22)
Figure 8 regions.
A PRRRP parallel robot with two discontinuous workspace
The relationship between the GTW length along the x-axis, denoted as Wx −GTW , and the normalized parameter r1 is shown in Figure 9, from which one may see that when r1=r1min the GTW is the largest; the length is inversely proportional to the parameter r1. Figure 10 illustrates the relationship between the GTI value and the normalized parameter r1. When r1=0.6813, the GTI reaches its maximum value. When r1 0.6813. 6.1.2 Slider-crank mechanism (the third leg). As described in Section 2, at any moment the third leg is actually the slider-crank mechanism. For example, if y=0, i.e. points O′ and O are coincident, then the kinematic equation (6) for the third leg can be rewritten as ( y3 − L1 sin φ )2 + ( L3 − L1 cos φ )2 = L22 .
(24)
This is the kinematic equation of the slider-crank mechanism shown in Figure 4. Using the normalization introduced in ref. [38], we will have three normalization
WANG JinSong et al. Sci China Ser E-Tech Sci | Apr. 2009 | vol. 52 | no. 4 | 986-999
parameters l1, l2 and l3 for parameters L1, L2 and L3. The parameter design space of the third leg, i.e. the slider-crank mechanism, is constrained by l1 + l2 + l3 = 3 , l1 ≤ l3 , and l1 + l2 > l3 ,
(25)
where li=Li/D, and D =(L1+L2+L3)/3 is the normalization factor. The parameter design space is shown in Figure 11.
As shown in Figure 4, since l1≤l3 the forward and inverse transmission angles of the normalized mechanism can be obtained as ⎛ l12 + l22 − cb3 ⎜ 2 l1 l 2 ⎝
μ = cos −1 ⎜
2
⎞ ⎟, ⎟ ⎠
⎛ l3 − l1 cosφ ⎞ ⎟, ⎟ l2 ⎠ ⎝
γ = cos −1 ⎜⎜
(26) (27)
where c is the position vector of the point O′ in the frame O-xyz, cb3 = y32 + l32 and y3 can be obtained
Figure 9 Relationship between the GTW length along the x-axis, i.e. Wx − GTW , and the normalized parameter r1 of the PRRRP parallel robot (the first and second legs of the new parallel robot).
from the kinematic equation (20) for a given orientation φ. One may see that the transmission angles are both relative to the orientation φ. Suppose that when φ∈(φmin, φmax) the two transmission angles μ and γ in eqs. (26) and (27) are all subject to the LTI constraint given by eq. (16). Here, φmin and φmax are the orientations where one of sinμ and sinγ is equal to sin(π/4) or sin(2π/9) and another one is subject to eq. (16). Then, the GTW for the orientation is defined as the relative angle between the two orientations φmin and φmax. If the orientational workspace is denoted as Wφ_GTW, we get Wφ _ GTW = φmax − φmin ,
Figure 10 Relationship between the GTI and the normalized parameter r1 of the PRRRP parallel robot (the first and second legs of the new parallel robot).
Figure 11 The Parameter design space of the slider-crank mechanism (the third leg of the new parallel robot).
(28)
which is the orientational capability of the mechanism. Figure 12 gives the relationship between Wφ_GTW and the normalized parameters l1, l2 and l3 when χ>sin(π/4). It shows that when l2 is near 1.5 and l3 is less than 1.0, the mechanisms have a larger GTW for the orientation (orientational capability). Figure 13 illustrates the relationship between the global transmission index (GTI) and the normalized parameters l1, l2 and l3 when χ>sin(π/4). It shows that the GTI is inversely proportional to parameter l3 and is proportional to parameter l1.
Figure 12 Atlas of the GTW for the orientation of the slider-crank mechanism (the third leg).
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Figure 13 Atlas of the global transmission index (GTI) of the slider-crank mechanism (the third leg).
6.2 Dimension optimization using the performance atlases
The optimal design using performance atlases was introduced in detail in ref. [38]. The design process for the studied robots is the same. Here, we also consider the design problem encountered in ref. [38], which is (a) desired positional workspace x × y = 10 mm × 10 mm in the O-xy plane, and (b) the desired rotational workspace as much as ±50° of the moving platform at every point in the workspace. The optimization process based on the atlases of Figures 9, 10, 12 and 13 can be summarized as follows. Step 1. Identification of an optimum region in the parameter design space. Please note that since all the design conditions cannot be the same, the identification of an optimum region is up to the designer. Here, we just give an example. For the first and second legs, an optimum region Ω 1− 2 can be identified using the atlases shown in Figures 9 and 10 when we use the performance constraints like Wx −GTW > 0.65 and Γ > 0.85 . From Figure 9,
Figure 14 An optimum region for the slider-crank mechanism (the third leg) when Wφ _ GTW ≥ 100° and Γ > 0.93 .
a non-dimensional robot in the parameter design space and all of its corresponding dimensional robots are similar in performance[38], the final design result based on the robot in the optimum region will be optimal. Therefore, the next step is to find an acceptable solution candidate in the optimum region. Step 2. Selection of a solution candidate from the optimum region. The obtained optimum regions Ω1−2 and Ω3 contain all possible solutions for the design. Since there does not exist the best but only a comparatively better solution for a design problem, one may pick up any one non-dimensional robot from the regions. For the first and second legs, without any reason r1=0.6 is selected here from the optimum region Ω1−2. Then, there is r2=2−r1=1.4. The GTW length along the x-axis and the GTI of the non-dimensional robot with r1= 0.6 and r2=1.4 are Wx −GTW = 0.7798 and Γ > 0.8517. For the third leg, we here pick up the non- dimensional robot with parameters l1 = 0.65, l2 = 1.5 and l3 = 0.85 from the optimum region Ω3. For the selected
one may see that when r1 < 0.6380 the length of GTW
robot, there are Wφ _ GTW = 102.7° and Γ > 0.9347.
is greater than 0.65. Figure 10 shows that if 0.5918 0.65 and Γ (r1 ) > 0.85] .
For the third leg, which is related to the orientational capability, an optimum region Ω3 should be determined with respect to the constraints of Wφ _ GTW ≥ 100° and better GTI, e.g. Γ > 0.93 . This region is shown as the hatched parts in the parameter design space in Figure 14. The identified optimum regions contain all possible non-dimensional solutions to the design problem. Since 996
Step 3. Determination of the dimensional parameters r, R, R2, L1, L2, and L3 with respect to the desired workspace. According to the normalization method introduced in refs. [38, 39], we should first determine the normalization factor, i.e. D. This factor can be obtained by comparing the desired workspace of a design problem and the workspace of the non-dimensional robot selected from the optimum region. For the first and second legs of the robot considered
WANG JinSong et al. Sci China Ser E-Tech Sci | Apr. 2009 | vol. 52 | no. 4 | 986-999
here, since the desired workspace along the x-axis is 10 mm, the factor D′ =10 mm/0.7798≈12.82 mm. Then, there are R2=D′r1=7.69 mm and R2=D′r2=17.95 mm. It is obvious that the GTIs for the non-dimensional and dimensional robots are the same. As an angle is only dependent on the ratio of related linear parameters but not on any one of the parameters itself, we cannot determine the normalization factor D of the third leg with respect to the orientational workspace ±50°. In this case, we should first determine parameter L1 of the moving platform according to the practical application and let the parameter be as small as possible for reducing the occupation space of the robot. Since parameters r and L1 are those of the moving platform, unless there is a special reason, we can let r=L1. Here, for the convenience of later comparison, we use the value of 9.77 mm optimized for the moving platform of the spatial 3-DOF parallel robot considered in ref. [38]. Then, we have L1=9.77 mm. Accordingly, we get D″=L1/l1=9.77 mm/0.65=15.03 mm, L2=D″l2=22.55 mm and L3=D″l3=12.78 mm. Since r=9.77 mm, we get R=R1+r=9.77+7.69=17.46 mm for the first and second legs. Step 4. Calculation of the input range. The input range to reach the desired workspace can be calculated by means of the inverse kinematic equations (7)—(9) of the spatial 3-DOF parallel robot. For the solution obtained in Step 3, y1= y2=[12.70 mm, 17.75 mm] and y3=[11.54 mm, 27.26 mm] if y=0. The final input range for each actuator should add the positional workspace along the y-axis. Then, there are y1= y2=[2.70 mm, 17.75mm] and y3=[1.54 mm, 27.26 mm] if the whole positional workspace is under the x-axis. Step 5. Checking the design result and adjusting, if necessary, the design solution. In this step, what should be checked depends on the application. For example, the designer may check whether the input range is suitable for his preferred commercial actuator. Whatever is checked, if the solution is not satisfactory, the designer can return to Step 2 and pick up another non-dimensional robot, or even return to Step 1 to identify another optimum region and adjust the design solution until the solution is satisfactory. This is actually the advantage of this design method. Please note that we get the orientational workspace Wφ_GTW about two orientations φmin and φmax as given in
eq. (28). In our design problem, or any others, the desired orientation range is ±50°. Since φmin is usually not equal to φmax, finally, we should adjust the original orientation φ0 of the moving platform to fit the requirement. For the example, as φmin= −69.5° and φmax=33.2°, the original orientation φ0 should be (φmax+φmin)/2, i.e. φ0=−18.15°. Ref. [38] discussed the optimal design problem of another variation of the HALF manipulator with respect to the LCI, GCI and orientational capability. With the same design problem, the optimized link lengths were 9.77 mm (for the moving platform), 21.82 mm (for the fixed length link) and 17.26 mm (for the base), which are very different from the results obtained in this paper. With these dimensions, the robot has, actually, no GTW along the x-axis for the first and second legs. There are Wφ _ GTW = 101.9° and Γ > 0.9033 for the third leg. The robot with the three dimensions is actually not a member of the optimum regions Ω1−2 and Ω3. Comparing the optimum region ΩMRC-GCI shown in Figure 6 in ref. [38] with Ω3, one may see that there is no common region between them. Therefore, the two designs are very different; and a robot with good GCI is likely to have bad motion/force transmission performance.
7 Discussion and future work As is shown in the paper, without the Jacobian matrix we can identify a workspace wherein the quality of the motion/force transmission of a parallel robot, which is kinematically the combination of planar mechanisms, is good by using the local transmission index. Many applications of planar four-bar mechanisms with one closed-loop chain indicate that the design of these mechanisms with respect to the transmission angle is successful. We also have no doubt about the condition number of the Jacobian matrix when it is applied to the design of a serial robot. The last two decades have witnessed huge growth in the design and application of parallel robots. Their capability in some application fields was once questioned and was even a source of disappointment. Being multiple closed-loop mechanisms, they have some of the characteristics of single closedloop mechanisms. It is worthwhile to think that why parallel robots did not inherit the design of four-bar mechanisms but that of serial robots. Clearly, it is time for all of us to ponder this question.
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As described in Section 4, by using the forward and inverse transmission angles, all singularities of a mechanism, i.e. the five-bar parallel robot and the slidercrank mechanism, can be identified. The forward and inverse transmission angles are the relative angles between the output and adjacent links and between the input and adjacent links, respectively. Therefore, if the singularities of a parallel robot can be identified wholly by using the two kinds of relative angles, the proposed indices, LTI, GTI and GTW, can be applied to its analysis and design. Meanwhile, we must recognize that the use of transmission angle and the proposed indices in parallel robots whose singularities cannot be found out by the said angles is challenging. The most important is how to define the inverse and forward transmission angles of these parallel robots. Whether the concept can be used in these robots needs further investigation. We have seen that, dramatically, without any advanced mathematic tool but only the relationship between points, the transmission angle, i.e. a joint variable of the mechanism, can describe the nature of a simple planar mechanism. But, the transmission angle is just a physical phenomenon. Although it implies something, the angle itself cannot reveal everything about robots, especially complicated robots. For instance, it cannot describe the architecture and constraint singularities of a parallel robot. However, the classical concept of motion/force transmission is noteworthy. After all, a parallel robot serves mostly as a system for motion/force transmission. The design for a parallel robot should fellow this rule.
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