Redundant sensors can be used (Han, Chung, and Youm. 1995) or new parallel ...... Kim, Whee Kuk, Byun, Yong Kyu and Cho, Hyung Suk. 2000. Closed-form ...
The International Journal of Robotics Research http://ijr.sagepub.com/
Design of a Parallel-Type Gripper Mechanism Byung-Ju Yi, Heung Yeol Na, Jae Hoon Lee, Yeh-Sun Hong, Sang-Rok Oh, Il Hong Suh and Whee Kuk Kim The International Journal of Robotics Research 2002 21: 661 DOI: 10.1177/027836402322023240 The online version of this article can be found at: http://ijr.sagepub.com/content/21/7/661
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Byung-Ju Yi School of Electrical Engineering and Computer Science Hanyang University Korea
Heung Yeol Na
Design of a Parallel-Type Gripper Mechanism
MicroINSPECTION Inc. Korea
Jae Hoon Lee School of Electrical Engineering and Computer Science Hanyang University Korea
Yeh-Sun Hong School of Aeronautical and Mechanical Engineering Hankuk Aviation University Korea
Sang-Rok Oh Intelligent System Control Research Center, KIST Korea
Il Hong Suh Graduate School of Information and Communication Hanyang University Korea
Whee Kuk Kim Department of Control and Instrumentation Engineering Korea University Korea
Abstract A new parallel-type gripper mechanism is proposed in this work. This device has a parallelogramic platform that can be flexibly folded. Therefore, this mechanism not only can be used to grasp an object having irregular shape or large volume, but also can be utilized as a micro-positioning device after grasping objects. Forward position analysis and platform kinematics are investigated to deal with motion tracking and force control. Kinematic optimization is performed to design a parallel-type gripping mechanism so that it can reach the specified workspace, span the given range of the specified configuration parameters, and generate a desired force to grasp an object. A pneumatic rotator is employed for actuation and a miniaturized proportional 4/3-way directional valve is specially developed to deal The International Journal of Robotics Research Vol. 21, No. 7, July 2002, pp. 661-676, ©2002 Sage Publications
with feedback-based dynamic control. The proportional valve allows indirect force control by measuring the offset-load pressure raised by the contact between the grasped object and the parallel platform. In experimental work, the performance of the motion tracking and indirect force control has been shown to be successful.
KEY WORDS—parallel-type, gripper mechanism
1. Introduction Investigation on controlling the shape of robot configuration utilizing the kinematic redundancies has been called “configuration control.” However, studies on this subject have been confined traditionally to kinematically redundant serial manipulators (Colbaugh et al. 1989; Burdick 1989). On the other hand, studies on configuration control for 661
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THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / July 2002
parallel manipulators have been rare while quite a few studies on parallel manipulators have been reported (Erdman and Sandor 1991; Freeman and Tesar 1988; Hayward and Kurtz 1992; Kumar and Gardner 1990; Nahon and Angeles 1989; Ropponen and Nakamura 1990; Walker et al. 1989; Kang et al. 1990; Kim et al. 1996, 1997; Yi et al. 1997, 1999, 2000) In this study, we propose a parallel-type gripping manipulator that is configuration-controllable. Figure 1 shows a 3 DoF parallel manipulator that consists of several links and joints. Since the triangular-shaped platform always maintains its shape whether the platform is a rigid body or not, the shape of the platform system cannot be configured. In the literature, only a planar parallel mechanism having three chains has been considered. Now, consider the two parallel manipulators with four chains, given in Figure 2. Without loss of generality, the four-closed chain parallel mechanisms have better stiffness and load handling capacity via additional base-actuation, compared with the three-closed chain system. Note that the first parallel manipulator has a rigid platform, but that the second parallel manipulator has a non-rigid platform since the links on the platform are connected by joints. A conceptual schematic for this mechanism is shown in Figure 3. Note that two revolute joints exist at each vertex of the platform. Therefore, the shape of the platform can be changed. The motion degree of such parallel mechanism can be explained in terms of “Mobility.” Mobility is defined as the number of independent variables that must be specified in order to locate its elements relative to another. It is described by Erdman and Sandor (1989) M = N (L − 1) −
J
(N − Fi ),
Fig. 1. Parallel manipulator with 3 DoF.
rigid plate
(a)
(1)
i=1
where N , L, J , and Fi denote the dimension of the motion space allowed by the entire joints of the mechanism, the number of links, the number of joints, and the degree-of-freedom of the ith joint, respectively. Mobility also represents the minimum number of the system actuator. When M is greater than N , the system is called a kinematically redundant system. According to this, the mobility of the parallel manipulator shown in Figure 2(b) is 4, and thus it can control 3 degrees of motion in the operation space (i.e., the motion in the xand y-directions, and the rotation angle of the platform) along with one other, specific parameter. We consider this additional parameter as the shape of the platform. We expect that this parallel system can be utilized as a gripper mechanism by shaping the platform configuration. In general, conventional robot grippers have a simple geometry, which is often not appropriate to grasp an object having irregular shape or large volume. On the other hand, the proposed gripper mechanism is expected to be able to grasp any object having irregular shape or large volume and also it can be utilized as a micropositioning device after grasping.
(b) Fig. 2. Parallel manipulators with 4 chains.
Fig. 3. Conceptual schematic of the gripping mechanism.
Yi et al. / Parallel-Type Gripper Mechanism It can be observed that success of configuration controllability of the platform depends significantly on the decision as to where to place actuators. For instance, when the four base joints of Figure 4(a) are servo-controlled, the positions of every second joint are known. Then, the figure on the righthand side represents a simplified expression of the same system. We can notice that the resulting system does not have any additional mobility. In other words, those four inputs are appropriate to control the position and the shape of the platform. Now, consider Figure 4(b), in which a second joint of one of the activated serial chains and other three base joints are servo-controlled. The simplified expression of this system on the right-hand side does not have any additional mobility. Therefore, those four inputs are also appropriate to control the position and the shape of the platform. Finally, consider Figure 4(c) in which the base and the second joints of neighbouring two serial chains are activated. In this case, the resulting simplified expression shown on the right-hand side has one additional mobility, which might cause an arbitrary, free motion of the platform. Therefore, those four inputs are not appropriate to control the position and the shape of the platform. Beside the locations of the actuator and sensor, we have to consider several aspects in order to successfully execute shape control of the parallel-type gripping mechanism. They include the shape of the moving platform, the number of chains, and type of joint actuator (prismatic or revolute). Though Figure 2 has just one configuration parameter, the problem will become complicated as the number of the configuration parameters increases. In this paper, we investigate the complete kinematic analysis and design methodology for the planar gripper mechanism given in Figure 3.
(a) Mobility 0
(b) Mobility 0
(c) Additional Mobility 1 Fig. 4. Configuration controllability.
θ [X
2. Kinematics 2.1. Direct Kinematics In this section, we describe the direct (forward) kinematics of the parallel gripper model given in Figure 5. We assume that the given manipulator has a parallelogrammic platform. The center position (x, y), the orientation angle of the platform, and the angle α (configuration parameter) between link a1 and link a4 of the platform are to be controlled. It is well known that the direct kinematic solutions of general parallel manipulators are not unique. The proposed parallel manipulator is no exception. The position of the platform and the configuration angle may not be uniquely estimated by using sensor information as many as the number of the mobility (i.e., four). That is, there exist multiple solutions of direct kinematics. Several approaches have been reported to obtain closed-form direct kinematic solutions of parallel manipulators. Redundant sensors can be used (Han, Chung, and Youm 1995) or new parallel manipulators having a closed-form direct kinematic solution can be sought (Merlet 1996; Song and Kwon 2002; Kim, et al. 2000).
663
[X
l31 [Y
θ [Y
θ 32
θ [Z
l32
3 4
θ XZ
w O S P
α
θ 33 π −α
α Φ1
l 12
1
θ 12
l 11 θ 11
2
θ 23
l22 θ 2 2
l 21
Fig. 5. Parallel manipulator with mobility 4.
θ 21
θ 31
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In this section, we discuss the closed-form direct kinematic solution of the proposed system. Firstly, the closed-form direct kinematic solution of the system given in Figure 5 can be obtained by replacing the first and the second joints or the first and the third joints of each serial chain by prismatic joints. Merlet (1996) mentioned that PPR-type or PRP-type planar manipulators have a closed-form direct kinematic solution when the sensors are placed at each base joint. The only difference between Merlet’s model and the proposed parallel gripper is that the platform of our system is a parallelogram while Merlet considered a ternary platform. However, the way of obtaining the direct kinematic solution is identical. Secondly, consider the case where only revolute joints are employed. The closed-form direct kinematic solution for the RRR type planar parallel manipulator has been solved by Gosselin and Angeles (1988) and Shirkhodaie and Soni (1987). It was reported that this mechanism had a sixth-order polynomial. The proposed parallel gripper also possesses revolute joints. By applying the same methodology employed to analyze the direct kinematics of the RRR planar manipulator, the proposed parallel gripper possesses a 12th-order polynomial. Thus, it has at most 12 different configurations (see Appendix 1). Thus, this case always requires redundant sensors to estimate the forward position. In this paper, we choose the latter case employing additional sensors, since the linear joint is hard to implement. One additional sensor will be enough to estmate the forward position of the parallel gripper, since the resulting 4th-order polynomial is solvable. That is, we can select one configuration out of the four configurations. Since each base joint has its own position sensor, the locations of the second joints can be determined as shown in Figure 6. The simplified model consists of five four-bars with zero mobility in total. In the following, it is assumed that each chain has the same kinematic dimensions and the platform has parallelogrammic shape. Suppose that we attach one additional sensor at the second joint of the first subchain of the system. Then the position of A can be directly obtained. Figures 7 and 8 show the elbow-up and elbow-down configuration of the four-bar loop BADC, respectively. The parallel gripper of interest keeps the elbow-up configuration (0◦ < θ < 180◦ ) during operation. Thus, we consider the configuration given in Figure 7. ψ1 denotes the angle between the X axis of the reference coordinate and link AD and ψ2 is given as ψ2 = θ + ψ 1 ,
θ = cos
2 − c2 a12 + l22 2a1 l22
O [XSG [X P
n O [YSG [Y P
(3)
l32
a3 Φ
a4
m O ZYSG ZY P
w
α
h
a2
O XYSG XY P
k O YYSG YY P
a1
l 12
i
O XXSG XX P
l 22
θ
j O YXSG YX P
Fig. 6. A simplified version of Figure 3.
k O YY SYY P X ψX
YYG
θ
j O YXSYX P ψY
π −ψ Y
h O XY S XY P Fig. 7. Elbow-up configuration.
ψX
hO XY SXY P
X Yπ − ψ X
O ZXSG ZX P
l42
(2)
the angle between line AD and CD is given by the cosine law as −1
l
o
ψY
θ
k
YYG
Fig. 8. Elbow-down configuration.
j O YXSYX P ψY − π
Yi et al. / Parallel-Type Gripper Mechanism and the distance between the points A and C is denoted as (4) c = (X21 − X12 )2 + (Y21 − Y12 )2 .
cos(ψ1 + α) = (X12 − X41 )(l42 cos θ ∗ − a4 ) + (Y12 − Y42 )l42 sin θ ∗
(16)
AH 2
Then the location of the point D can be expressed in terms of the position information at A and C: X22 = X12 + a1 cos ψ1 = X21 + l22 cos ψ2
665
AGH = θ ∗ = cos−1
2 a42 + l42 − AH 2 , 2a4 l42
(17)
(5)
= X21 + l22 cos(θ + ψ1 ),
AH 2 = (X41 − X12 )2 − (Y41 − Y12 )2 .
Y22 = Y12 + a1 sin ψ1 = Y21 + l22 sin ψ2
(6)
= Y21 + l22 sin(θ + ψ1 ).
(18)
Then the center position of the parallogramic platform can be found as follows: x=
(X22 + X42 ) 2
(19)
y=
(Y22 + Y42 ) . 2
(20)
Rewriting eqs (5) and (6) gives (l22 cos θ − a1 ) cos ψ1 + (−l22 sin θ ) sin ψ1 = X12 − X21 (7) (l22 sin θ ) cos ψ1 + (l22 cos θ − a1 ) sin ψ1 = Y12 − Y21 (8) from which the unique solution for ψ1 can be obtained as ψ1 = arctan2(sin ψ1 , cos ψ1 )
(9)
2.2. First-Order Kinematics
with sin ψ1 =
cos ψ1 =
Conclusively, the forward position solution for this mechanism can be obtained by employing one additional sensor. In the inverse position analysis, all the joint angles of the parallel chain are easily obtained (Yi et al. 1998, 2000) for the given center position (x, y) of the platform, the orientation angle of the platform, and the configuration parameter α.
(l22 cos θ − a1 )(Y12 − Y21 ) − l22 sin θ (X12 − X21 ) 2 l22 + a12 − 2a1 l22 cos θ (10) (l22 cos θ − a1 )(X12 − X21 ) + l22 sin θ (Y12 − Y21 ) . 2 l22 + a12 − 2a1 l22 cos θ (11)
Then, the location of D can be obtained. A similar procedure can be applied to the four-bar loop BAGH. The location of G (X42 ,Y42 ) and the configuration parameter α are then obtained as X42 = X12 + a4 cos(ψ1 + α), Y42 = Y12 + a4 sin(ψ1 + α), α = −ψ1 + arctan 2(sin(ψ1 + α), cos(ψ1 + α))
(12) (13)
(21)
T u˙ ∗ = [x˙ y˙ ]T , 1φ˙ = θ˙11 θ˙12 θ˙13 α˙ .
(22)
where
(14)
(15)
Then, the velocity relation for the first chain is described as [1 Gu∗ φ ] ˙ u˙ = φ˙ = [1 Gua ]1 φ. (24) 0001 1
sin(ψ1 + α) = AH 2
˙ u˙ ∗ = [1 Gu∗ φ, φ ]1 φ
Now, augmenting the rate change of the configuration parameter, α, ˙ to the above velocity relation, a new output vector u˙ is defined as u˙∗ ˙u = . (23) α˙
where
(Y12 − Y41 )(l42 cos θ ∗ − a4 ) − (X12 − X42 )l42 sin θ ∗
The first-order kinematics relates the output velocity vector to the input joint velocity vector. In the following, G denotes the Jacobian and the left subscript of G denotes the number of serial subchains of the parallel manipulator, and the superscript and subscript on the right-hand side of G denote the dependent and independent parameters, respectively. Also, [Guφ ](∗,j ) and [Guφ ](i,∗) denote the j th column and the ith row of [Guφ ], respectively. [Guφ ](i,j ) denotes the (i, j ) element of [Guφ ]. Then, the velocity relation for the first serial subchain can be obtained by differentiating the position equations of the parallel chains (Appendix 2) with respect to time:
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THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / July 2002
wZ
The same relation can be obtained for the rest of the serial chains. Now, the inverse relation for each chain is denoted as i
˙ (i = 1∼ 4), φ˙ = [i Gua ]−1 u,
(25)
where we assume that there is no algorithmic singularity in the inversion of the Jacobian [i Gua ]. Considering the four base joints as independent joints where actuators and position sensors are placed, the velocity relation relating the output vector to the independent joint vector is constructed by selecting the first row of each inverse-Jacobian of the four serial sub-chains and forming a matrix relation as below: ˙ φ˙ a = [Gua ]u,
Z
(26)
w[
[
α
Z
[
wY
O S GP
X
Y
X
Y
wX
w XOXSG X P w YOYSG Y P
Φ
w ZOZSG Z P w [O[SG [ P
Fig. 9. Platform geometry.
where
[1 Gua ]−1 (1;∗)
[2 Gua ]−1 (1;∗) a [Gu ] = [ Gu ]−1 3 a (1;∗)
2.3. Platform Kinematics (27)
[4 Gua ]−1 (1;∗) and φ˙a = (θ˙11 θ˙21 θ˙31 θ˙41 )T . Now, inverting the relation of eq (26) yields a first-order forward kinematic relation of the system, given by u˙ = [Gau ]φ˙a ,
(28)
where
u −1 a (1;∗)
[1 G ]
[2 Gua ]−1 (1;∗) [G ] = [ Gu ]−1 3 a (1;∗) u a
.
Since there exists a duality between the velocity vector and static force vector, we have
Ta = [G ] Tu ,
(31)
y4 = y + b4 sin(π − ) − a4 sin(α − π + ).
[4 Gua ]−1 (1;∗)
u T a
x1 = x − b1 cos(α − π + ) + a1 cos(π − ), y1 = y − b1 sin(α − π + ) − a1 sin(π − ), x2 = x + b2 cos(π − ) + a2 cos(α − π + ), y2 = y − b2 sin(π − ) + a2 sin(α − π + ), x3 = x − a3 cos(π − ) + b3 cos(α − π + ), y3 = y + b3 sin(π − ) + a3 sin(α − π + ), x4 = x − b4 cos(π − ) − a4 cos(α − π + ),
−1
Tu = [Gua ]T Ta ,
Assume that the point Pi (i = 1∼ 4) of Figure 9 denotes the contact point of the grasped object and the parallel gripper. The number of the contact points will be possibly more than or less than four, but at least two. A force is transmitted through these points from the gripper to the grasped object. The locations of P1 through P4 are given as
By differentiating the above position equations, the velocity equation relating the velocity vector at the contact point to the velocity vector of the output position is obtained as
(29)
˙ P˙i = [i Gup ]u.
(30)
By using the principle of virtual work, we have a corresponding force relation, given by
where Tu and Ta denoting the output force vector and the input joint torque vector, respectively, consist of T
Ta = (τ11 τ21 τ31 τ41 ) , and Tu = (fx fy mz τα )T . Tu includes the dynamic load (TIu ) required for motion of the system as well as the effective external load (Teu ) applied to the parallel gripper.
Teu = [i Gup ]T FPi ,
(32)
(33)
where Teu , FPi denote the generalized operational load and the force vector transmitted through the Pi contact point, respectively. Finally, the total operational force applied to the parallel gripper is obtained by summing up the contribution from all contact forces: Tu =
4 i=1
[i Gup ]T Fpi .
(34)
Yi et al. / Parallel-Type Gripper Mechanism
667
Now, an offset torque vector to support the effective applied load vector is calculated from eqs (30) and (34):
force vector, the grasping force Tα of the platform is defined from
Teu = [Gup ]T Fp + [Gua ]T Ta = 0,
Ta = [Gau ]T(∗,4) Tα
(35)
where [Gau ]T(∗,4) denotes the 4th column of [Gau ]T . Now, the ratio of the 2-norm of the grasping force to that of the input load can be expressed as
where
Gup
T
= [1 Gup ]T [2 Gup ]T [3 Gup ]T [4 Gup ]T
Ta = Tα
and Fp = (FpT 1 FpT 2 FpT 3 FpT 4 )T .
p T u
p T a
Ta = −[G ] [G ] Fp = −[G ] Fp .
(36)
The general solution for eq (36) is given as Fp = −([Gap ]T )+ Ta + ([I] − ([Gap ]T )+ [Gap ]T )εε ,
TTα [Gau ](∗,4) [Gau ]T(∗,4) Tα TTα Tα
21 = p,
(39)
where the scalar p is given as
Equation (35) can be also expressed as u T a
(38)
21 p = [Gau ](∗,4) [Gau ]T(∗,4) .
(40)
Then, the 2-norm of the grasping force is defined by (37)
where the first term represents the particular solution resulting in minimum force norm, and the second terms denotes a homogeneous solution implying null space solutions. The first term of eq (37) can be employed for contact force estimation if the joint torque to support the externally applied load can be measured. For this, the following section introduces a servo-pneumatic actuating system equipped with pressure sensor that can effectively measure the joint torque.
3. Design of Parallel Gripper
Tα =
Ta . p
(41)
The global maximum grasping force is defined with respect to the entire workspace of the manipulator as Tα dW , (42) = w dW w F where the workspace of manipulators is denoted as W = dW.
(43)
w
The parallel gripper consists of a linkage part and a driving unit. In the previous design of parallel gripper (Yi et al. 1998), we employed DC motors to drive the system. However, real implementation of the previous prototype was impractical due to the size, volume, and heavy mass of the driving unit. Therefore, we considered using a miniaturized servopneumatic driving system that consists of a pneumatic rotator and a proportional directional valve. The objective of the kinematic optimization is to design a parallel-type gripping mechanism that can reach the specified workspace, span the given ranges of the specified configuration parameters, and generate a desired force to grasp an object.
Also, a manipulator should be designed so that it has wellconditioned workspace that allows its end-effector to move from one regular value to another without passing through a critical value (singularity). An isotropic index is a criterion to measure such phenomenon. In order to measure the motion isotropic characteristic of the system, the first-order kinematic property for the center position (x, y) of the platform and the orientation angle of the platform should be analyzed. u For this, a sub-Jacobian [Gu∗ a ] except the last row of [Ga ] is considered. Then, the isotropic index σI is defined as
3.1. Kinematic Design Indices
where σmin and σmax denote the minimum and maximum singular value of [Gua ], respectively. The global isotropic index is defined with respect to the entire workspace of the manipulator as σI dW -I = W . (45) W
3.1.1. Single Design Index We consider two kinematic indices; an isotropic index of the parallel manipulator and a grasping force of the parallel platform. In order to unify the dimension of Jacobian elements, we nondimensionalize the Jacobian elements with respect to the size of the moving platform. Based on the effective force relationship between the operational force vector and the input
σI =
σmin , σmax
(44)
The design of a manipulator system can be based on any particular criterion. However, the single criterion-based design
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does not provide sufficient control on the range of the design parameters involved. Therefore, multi-criteria based design has been proposed (Hayward and Kurtz 1992). However, the previous multi-criteria methods did not provide any systematic design procedure and flexibility in design. To consider these facts, a composite design index is proposed in the following section. 3.1.2. Composite Design Index In order to cope with multi-criteria based design, we employ the concept of a kinematic composite design index. The various design indices introduced above are usually incommensurate concepts due to differences in unit and physical meanings. Therefore, they should not be combined with normalization and weighting functions unless they are transferred into a common domain. As an initial step to this process, preferential information should be given to each design parameter and each design index. Then, each design index is transferred to a common preference design domain that ranges from zero to one. Here, the preference given to each design criterion is very subjective to the designer. Preference can be given to each criterion by weighting. This provides flexibility in design. For both Tα and σI , the best preference is given the maximum value, and the least preference is given the minimum value of the criterion. Then, the design index is transferred into the common preference design domain as below: I = -
-I − -I min , -I max − -I min
(46)
where “∼” implies that the index is transferred into the common preference design domain. Note that each composite design index is constructed such that a large value represents a I implies that the system possesses good better design. Large isotropic characteristics within the given workspace, and also large F implies that the system produces a large grasping force for an unit actuator load within the given workspace. A set of optimal design parameters is obtained based on the max-min principle. Initially the minimum values among the design indices for all sets of design parameters are obtained, and then a set of design parameters, which has the maximum of the minimum values, is chosen as the optimal set of design parameters. Based on this principle, the composite global design index (CGDI) is defined as the minimum value of the above mentioned design indices at a set of design parameters, and given as β ζ I , F . CGDI = min (47) The superscript Greek letters (β, ζ , etc) represent the degree of weighting, and usually a large value implies a large weighting. In general, the value of weighting is determined based on fuzzy measure such as normal, very, more or less, absolutely, and so on. Though those fuzzy measures can be defuzzified
as crisp values very subjectively, the following cases have been employed; normal is equivalent to 1, very is equivalent to 2, more or less is equivalent to 0.5, absolutely is equivalent to ∞, and so on (Terano, Asai, and Sugeno 1993). In order to evenly satisfy the several design objectives for all design indices, all of the weighting factors are set to 1.0. Now, a set of optimal design parameters is chosen as the set that has the maximum CGDI among all CGDI’s calculated for all sets of design parameters. 3.2. Optimization Methodology To deal with a nonlinear optimization with constrains, we employed a genetic optimization algorithm (Michalewics 1996) that ensures a global optimal solution. The link lengths and the base location of each serial subchain, and the dimensions of the platform of the parallel manipulator, can be cited as kinematic design parameters. Assume that the four serial subchains have a symmetric configuration. In other words, the first and second link lengths of the four serial chains are identical, respectively. Therefore, two design parameters exist for serial chains. The platform is an equilateral parallelogram. Therefore, it has one design parameter (a). The proposed gripper mechanism will be attached at the end of a macro-manipulator providing global motion. Therefore, we plan to place the four base locations of the four serial-chains around the end-position of a macro-manipulator symmetrically. For the base location of the parallel manipulator, only one design parameter (r: radius of the base platform) exists when the four base locations are placed symmetrically. In total, there exists four design parameters in the design of the proposed parallel-type gripping manipulator. Initially, we define the desired workspace and the range of the configuration parameter of the proposed parallel manipulator. Since this device is to be utilized as a micro-positioning device after grasping, the workspace will be smaller than that of the macro-manipulator. Here, the workspace will be given the inside area of a square centered at the middle of the workspace. The size of the workspace will be proportional to that of the platform. It is given as −0.25r ≤ 4x, 4y ≤ 0.25r,
(48)
where 4x, 4y denote the distances from the center of the workspace in the x- and y- directions, respectively. The range of orientation of the platform is given as −10◦ ≤ ≤ 10◦ ,
(49)
and the range of the configuration parameter will be given as 90◦ − η ≤ α ≤ 90◦ + η, where the angle η will be set as 30◦ .
(50)
Yi et al. / Parallel-Type Gripper Mechanism Now, kinematic constraints associated with the four design parameters are given as 0.01 m ≤ li1 , li2 , a ≤ 0.10 m, 0.01 m ≤ r ≤ 0.05 or 0.10 m .
(51)
We consider two different platform sizes since the platform size denoted by “a” is decided according to the size of object to be grasped. Kinematic optimization for the proposed parallel manipulator has been performed for the case of β = 1, ζ = 1, for the case of β = 3, ζ = 1 in which large weighting is given to the grasping force, and for the case of β = 1, ζ = 3, in which large weighting is given to the isotropic index. The optimal design parameters obtained from the optimization are given in Table 1, which denotes the case for rmax = 0.05 m, while Table 2 denotes the case for rmax = 0.10 m. In general, better kinematic isotropy (large ISO) is ensured for large base radius that, however, results in larger link size, compared with those of the smaller base platform. For rmax = 0.05 m, Figures 10 and 11 illustrate the optimized configurations for Case I and Case II, respectively. In general, the smaller the platform size is, the larger the base size is, and vice versa. To analyze the results, we plot the grasping force and the isotropic index for Case I with rmax = 0.05 m. We can observe from Figures 12 through 14 that overall trends of the two indices follow the given design objectives. Note that a trade-off exists between the isotropic characteristic and the grasping force. Based on the simulation result, we could conclude that the proposed parallelogramic gripping mechanism can be designed either for creating large grasping force or possessing good motion isotropy. The proposed parallel manipulator can also be employed as a good micro-positioning device, in that it has enough degree-
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of-freedoms (3), has light weight allowing high bandwidth, and possesses excellent positioning and sensing capabilities which are operational requirements for micro-positioning devices (Yamagata and Higuchi 1995). Figure 15 shows a folded configuration of the parallel gripper mechanism (α = 135◦ ). Figure 16 shows the parallel linkage detached from the assembly. Figure 17 represents a proportional 4/3-way directional valve developed for dynamic control of small-sized robotic systems such as human hands or parallel grippers (Ryu and Hong 1998). It consists of a DC-solenoid, a double flapper valve, and pressure sensors for measuring the load pressure. The schematics of the double flapper valve loaded with a pneumatic rotator is shown in Figure 18. The pressure signals are used to measure the load torque of the pneumatic rotator that drives the revolute joints of the parallel gripper. Figure 19 denotes the input signal-to-flapper displacement curve of the valve, which results from the force-todisplacement characteristics of the DC-solenoid and its return spring. The linearity is good enough to control the motion of the rotator with satisfactory accuracy. Furthermore, Figure 20 shows the input signal-to-output pressure curves of the valve, which are quite linear and symmetrical. The pressure difference of the two ports, A and B, measured by semiconductor-type sensors, can be employed for torque estimation. The step-input response of the pressure increase or decrease in the rotator chambers takes less than 0.35 seconds. The proposed pneumatic system has several merits; 1) it has high torque capability without reduction gears, 2) joint torque can be measured by cost and space saving pressure sensors, and 3) it has an inherent compliance due to the compressible characteristic of the air. As additional information, Figure 21 provides a plot relating the torque created at the pneumatic rotator to the pressure difference acting on it (Ryu and Hong 1998).
4. Experimental Results
Fig. 10. Optimized configuration for Case I.
Fig. 11. Optimized configuration for Case II.
Position tracking performance and indirect force control of the grasped object was performed in experiment. The plots in Figures 22(a) and (b) show the results of position tracking experiments for x- and y-directional trajectory, and Figures 22(c) and (d) correspond to those for rotational and folding motions, respectively. In the second experiment, the parallel gripper was used to grasp a cylinder with a symmetric shape. At the moment of grasping, the pressure difference at the proportional valve was measured by the pressure sensors integrated in the valves in order to estimate the reaction torque at each joint. Next, the grasping forces at all contact points could be easily obtained from the first term of eq (37). Figure 23 shows the finally estimated grasping forces at the four contact points. The force measurement is expected to be useful for the feedback control scheme.
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THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / July 2002 Table 1. Optimization Results for rmax = 0.05 m
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Table 2. Optimization Results for rmax = 0.10 m
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In real implementation, the employed pneumatic rotator exhibits a large frictional torque. Thus, in the initial startup of the mechanism, the motion of the gripper is not that smooth. To cope with this problem, the sensitivity of the pressure valve should be enhanced and the pneumatic rotator having less friction should also be developed.
5. Conclusions Conventional robot grippers have in general a simple geometry, which is often not appropriate to grasp an object having irregular shape or large volume. We propose a parallel mecha-
m
nism that is able to grasp any object having irregular shape or large volume by folding the parallelogramic platform, and also it can be utilized as a micro-positioning device after grasping. We employed small-size pneumatic actuators and also developed a miniaturized proportional valve to implement dynamic control. The smooth pressure sensitivity of the proportional valve allowed open-loop control of joint torque. By sensing the load pressure raised by the contact between the grasped object and the parallel platform the grasping force could be effectively estimated. Experimental results showed that the performances of the motion tracking and indirect force control were satisfactory.
Yi et al. / Parallel-Type Gripper Mechanism
(a) Isotropic index Fig. 12. Kinematic indices for Case I (ISO:GF=1:1).
(b) Grasping force
(a) Isotropic index Fig. 13. Kinematic indices for Case I (ISO:GF=1:3).
(b) Grasping force
(a) Isotropic index Fig. 14. Kinematic indices for Case I (ISO:GF=3:1).
(b) Grasping force
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THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / July 2002
Fig. 15. Configuration of parallel gripper.
(a)
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Fig. 16. Parallel linkage.
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Fig. 18. Schematic of double flapper valve loaded with a rotator.
Yi et al. / Parallel-Type Gripper Mechanism
673
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A1 B1 B4 A4 , A1 B1 B4 A4 ). For each configuration, the location of the floating link is decided: (B1 B4 ), (B1 B4 ), and (B1 B4 ). For A1 B1 B4 A4 , the circular contour of B3 B4 intersects with that of A3 B3 at B3 or B3 . Also, the three lines B1 B2 , B2 B3 , and A2 B2 may intersect at B2 or B2 . Thus, the number of possible combinations is twelve. However, since the parallelogramic closed-loop is kinematically independent of the four subchains, we can select the parallelogrammic structure B1 B2 B3 B4 rather than B1 B2 B3 B4 because B1 B2 B3 B4 forms a gripper that can grasp an object by folding the parallelogram. Then, the total number of configurations is reduced to 6. If we attach an additional sensor at B4 or A4 , the lines A1 B1 and B1 B4 intersect at two points (B1 and B1 ) since the location of B4 is known. Also the lines A3 B3 and B3 B4 have two intersecting points (B3 and B3 ). Therefore, the total number of possible configurations of the parallel gripper is counted as four. In real implementation, we select only one configuration among the four.
Appendix 1
Appendix 2
Gosselin and Angeles (1988) mentioned for the first time that the number of direct (or forward) kinematic solutions of the RRR-type 3 DoF planar manipulator was six. They employed a geometric approach. The contour of the couplar link of a four-bar, which is formed by two subchains and the ternary platform, intersects with the last link of the other chain at two points. This implies two configurations. Noting that a four-bar has three different configurations, there are six solutions in total. The same geometric approach will be applied to obtain the direct kinematic solution of the parallel gripper mechanism. Figure 24 represents a more detailed description of Figure 6. First of all, consider a four bar on the left-side. It is shown that it has three possible configurations (A1 B1 B4 A4 ,
The parallel gripper mechanism given in Figure 5 can be visualized as four serial-chains commonly satisfying the four output parameters. For the first chain, the center position (x, y) of the platform and the orientation angle for each serial chain are, respectively, given as 1 = θ11 + θ12 + θ13 + α = − (π − α), x = l11 c11 + l12 c11+12 +
a1 a4 c11+12+13 + c 1 , 2 2
y = l11 s11 + l12 s11+12 +
a1 a4 s11+12+13 + s 1 . 2 2
(52) (53)
(54)
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THE INTERNATIONAL JOURNAL OF ROBOTICS RESEARCH / July 2002 0.0 5
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For the second chain
For the third chain
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(55)
3 = θ31 + θ32 + θ33 + α = + α,
(58)
x = xo1 + l21 c21 + l22 c21+22 +
a2 a1 c21+22+23 + c 2 , 2 2
(56)
x = xo2 + l31 c31 + l32 c31+32 +
a3 a2 c31+32+33 + c 3 , 2 2
(59)
y = yo1 + l21 s21 + l22 s21+22 +
a2 a1 s21+22+23 + s 1 . 2 2
(57)
y = yo2 + l31 s31 + l32 s31+32 +
a3 a2 s31+32+33 + s 3 . 2 2
(60)
Yi et al. / Parallel-Type Gripper Mechanism 2.00
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Acknowledgments
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This work was supported by the Basic Research Program, KOSEF, under Grant R41-1998-000-00623-0.
0.50
References
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And finally for the fourth chain 4 = θ41 + θ42 + θ43 + π − α = + π,
(61)
x = xo3 + l41 c41 + l42 c41+42 +
a4 a3 c41+42+43 + c 4 , 2 2
(62)
y = yo3 + l41 s41 + l42 s41+42 +
a4 a3 s41+42+43 + s 4 , 2 2
(63)
where xoi and yoi (i = 1, 2, 3) denote the positions of the base of each serial chain with respect to the reference coordinate frame, and θij denotes the j th joint of the ith chain.
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