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On the feasibility of utilising an array of planar parallel robots to service adjoining workspaces
Mechanism and Machine Theory 128 (2018) 382–394
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Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmachtheory
Research paper
On the feasibility of utilising an array of planar parallel robots to service adjoining workspaces Mats Isaksson∗, Jason Miller, Mostafa Nikzad School of Engineering, Swinburne University of Technology, Hawthorn, VIC 3122, Australia
a r t i c l e
i n f o
Article history: Received 22 September 2017 Revised 1 June 2018 Accepted 2 June 2018
Keywords: Parallel kinematics Screw theory Additive manufacturing Subtractive manufacturing 3D printing
a b s t r a c t Parallel robots feature several favourable properties, including the potential for high acceleration and accuracy. However, in order to minimise cycle times and optimise the usage of factory space, industrial applications commonly include multiple closely mounted robots, working on the same or closely located objects. As parallel robots generally suffer from a limited workspace-to-footprint ratio, attempting to utilise parallel robots for such applications typically leads to issues with mechanical interference. This paper investigates the feasibility of dividing a planar workspace into several smaller areas, each of them serviced by a dedicated planar parallel mechanism. Applications where this arrangement could be beneficial include 2D manufacturing operations, such as water jet cutting and laser cutting. Additionally, the proposed arrangement could be extended by actuating either the work object fixture or the entire array of mechanisms in one additional degree of freedom. The resulting architecture could be beneficial for 3D manufacturing operations, such as additive or subtractive manufacturing. The presented analysis studies the limitations of such architectures in terms of mechanical interference and achievable kinematic performance. Solutions to reduce mechanical interference are introduced and screw-theory-based indices are employed to evaluate the achievable kinematic performance. © 2018 Elsevier Ltd. All rights reserved.
1. Introduction In order to optimise the usage of factory space, industrial robots are commonly mounted closely together. Cycle time and production cost can often be reduced if several robots perform different tasks on the same object. A recent trend is the utilisation of coordinated collaborating robots [1]. Typical application for such robots include arc welding, where one robot manipulates the work object while other robots perform the welding [1–3]. Such an approach can save floor space and improve accessibility of the work object [3]. Coordination may be achieved by robot-to-robot communication or by utilising a single controller for all participating robots. The latter option saves additional floor space and provides several other benefits, including the possibility for improved collision avoidance [1]. Parallel robots [4] exhibit several benefits compared to similar-sized serial robots [5], including the potential for high stiffness and low moving mass, which enables higher acceleration and reduced cycle times. As the position errors of the actuators are not cumulative, the position accuracy is typically high. Additionally, the actuators are commonly mounted on the fixed base, leading to straightforward cabling solutions. However, parallel robots also suffer from several limitations,
∗
Corresponding author. E-mail address: [email protected] (M. Isaksson).
https://doi.org/10.1016/j.mechmachtheory.2018.06.001 0094-114X/© 2018 Elsevier Ltd. All rights reserved.
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Fig. 1. Parameters of the studied 2-DOF planar parallel mechanism. The view is from the intended work object. The manipulated platform can pass below the guide-ways. The single link and the parallelogram operate in different vertical planes. The only potential mechanical interferences are between the link L1 and the cart C2 and between the links L2a and L2b ; however, such configurations are outside the intended workspace of the mechanism.
typically including a limited workspace-to-footprint ratio [6]. Due to mechanical interference, workspace sharing is difficult, and even attempting to arrange parallel robots to service separate but adjoining workspaces is challenging. Herein, we make a distinction between the reachable workspace, which is the entire workspace that the tool can service without mechanical limitations or singularities, and the utilised workspace, which is a subsection of the reachable workspace where the kinematic performance is considered acceptable. Typically, the utilised workspace excludes the border regions of the reachable workspace. Potential interference between the tools of two neighbouring mechanisms at the border of each mechanism’s utilised workspace is unavoidable; however, such interference is straightforward to handle with software algorithms, leading to only minor constraints on the independence of each mechanism. The challenge for a parallel mechanism is that its arm system is often closer to a neighbouring mechanism than its tool, leading to potential interference also between the arm systems of the mechanisms. Modelling all such interferences is complex and introduces additional constraints on how independently the two mechanisms can service their respective workspace. In terms of collision avoidance, it is preferable that when the tool is in its border regions, it is the section of the mechanism that is closest to the neighbouring mechanism. This is only possible for some parallel mechanisms and typically means utilising the outer regions of the reachable workspace, where the kinematic performance may not be adequate. Research and development of parallel robots that can be mounted closely with minimal mechanical interference is of significant industrial interest [7]. In this paper, we focus on applications requiring two or three degrees of freedom. We investigate the feasibility of dividing a planar workspace into several smaller areas, each of them serviced by a dedicated two-degree-of-freedom (2-DOF) planar parallel mechanism. Applications where such an arrangement could be beneficial include water jet cutting and laser cutting. Additionally, this arrangement could be extended by actuating either the work object fixture or the entire array of mechanisms in one additional degree of freedom. The resulting architecture could be beneficial for additive manufacturing, as the utilisation of multiple print heads could dramatically reduce the build times. As a 3D printing platform with multiple print heads does not require independent 3-DOF mechanisms to manipulate each print head, the proposed architecture could be a cost-efficient solution. In addition to reduced number of actuators, such an architecture has several other potential benefits, including improved accuracy, space savings, and low-cost components. This paper provides an evaluation of the described architecture for a particular choice of the individual 2-DOF mechanisms. It is demonstrated how the objectives of optimal kinematic performance and low risk of mechanical interference are inversely correlated. By utilising intuitive screw-theory-based performance indices, the potential and limitations of the proposed architecture are fully quantified. The remainder of this paper is organised as follows. Section 2 introduces the individual 2-DOF parallel mechanisms, including their kinematic parameters. Section 3 reviews the screw-theory-based performance indices employed for the subsequent kinematic analysis. Section 4 begins with a kinematic analysis of a single mechanism and continues to study the feasibility of utilising multiple mechanisms of the same type, first arranged in a one-dimensional array and then in a two-dimensional array. Finally, Section 5 provides conclusions and ideas for future work. 2. Individual planar mechanism The individual unit of the proposed architectures is the mechanism illustrated in Fig. 1. It is a planar parallel mechanism with two positional DOFs and constant orientation of the manipulated platform. It includes two actuated carts on parallel linear guide-ways. Possible actuation schemes include screws, belt or rack-and-pinion. One actuated cart C1 includes a revolute joint Rc1 , while the second actuated cart C2 includes two revolute joints Rc2a and Rc2b . The manipulated plat-
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form includes three revolute joints Rp1 , Rp2a and Rp2b . Three links L1 , L2a and L2b of equal length connect the joints on the actuated carts and the manipulated platform, where L2a and L2b are arranged to always be parallel. The links could be counterbalanced to avoid bending forces on the joints Rc1 , Rc2a and Rc2b . By arranging the single link and the links in the parallelogram to operate in different planes below the guide-ways, mechanical interference between the single link and the links in the parallelogram and between the links and the guide-ways is not an issue. 2.1. Kinematic parameters Fig. 1 includes the kinematic parameters of the mechanism. The tool centre point (TCP) is located on the axis of joint Rp1 . This joint axis is in the centre of a perpendicular line between the joint axes of Rp2a and Rp2b . The utilised workspace of the TCP is outlined in a dotted line. This workspace is a subsection of the entire reachable workspace, where certain kinematic constraints are fulfilled. These constraints will be introduced in Section 3. A coordinate system has been introduced with its x-axis and y-axis coinciding with two of the workspace borders. The width of the workspace in the x-direction is denoted by wx and its width in the y-direction by wy . The width wx is not relevant to the kinematic optimisation but necessary to calculate the required length of the guide-ways.
T
The planar position of the TCP is given by x = x y . The distance between Rp2a and Rp2b equals 2p and the angle between a line parallel to the y-axis and a line perpendicular to the axes of Rc2a and Rc2b is denoted by α . The x- and y-coordinates of the joint axis of Rc1 are denoted by q1 and d1 , respectively. The x- and y-coordinates of an axis in the middle between Rc2a and Rc2b are denoted by q2 and d2 , respectively. The kinematic lengths of L1 , L2a and L2b are equal and denoted by l. The kinematic length is the perpendicular distance between the joint axes of the joints at the ends of the link. The illustrated angles μ1 , μ2 , μp and μr are employed in the kinematic analysis and will be discussed in Section 3. The guide-ways are of equal length. The stroke length of the prismatic actuators is denoted by gl , while their smallest and largest x-coordinate is denoted by gmin and gmax , respectively. 2.2. Inverse kinematics and length of the guide-ways The kinematics of the mechanism in Fig. 1 is identical to that of the over-constrained mechanisms described by Liu et al. [8]. Pythagoras’ Theorem provides the equations
l 2 − ( x − qi )2 − ( y − di )2 = 0,
i = 1, 2.
(1)
The inverse kinematics of the mechanism is derived by selecting the smallest of the two qi values fulfilling (1) according to
qi (x, y ) = x −
l 2 − ( y − di )2 ,
i = 1, 2.
(2)
As can be seen from (2), physical solutions require that l ≥ |y − di | for 0 ≤ y ≤ wy . The length of the guide-ways are selected to be equal. The required stroke length gl and the positions gmin and gmax are calculated by utilising (2) according to
q1min =
q1 (0, d1 ), if d1 > 0, q1 (0, 0 ), if d1 ≤ 0,
q2min =
q2 ( 0, d2 ), q2 ( 0, wy ),
if d2 < wy , if d2 ≥ wy ,
(3)
(4)
q1max = q1 (wx , wy ),
(5)
q2max = q2 (wx , 0 ),
(6)
gmin = min(q1min , q2min ),
(7)
gmax = max(q1max , q2max ),
(8)
gl = gmax − gmin .
(9)
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3. Transmission indices The kinematic performance evaluation of the studied mechanisms is performed by utilising screw-theory-based indices. Screw theory was introduced by Ball [9,10] and revived by Hunt [11] significantly later. Refined by the work of several successive researchers, screw theory has gradually become the state-of-the-art method for kinematic analysis of parallel mechanisms. The utilised indices characterise the transmission efficiency between each actuated cart and its corresponding link system, the transmission efficiency between each link system and the manipulated platform, and how efficiently the parallelogram constrains the platform’s rotation. ˆ for unit wrenches and Tˆ for unit twists. The input twist TˆIi of a kinematic chain i is the We employ the notation W twist of the actuated joint of that chain, while the output twist TˆOi of the same chain is the free motion of the manipulated ˆ Ci j of chain i are the genplatform when all actuated joints except the one of chain i are locked. The constraint wrenches W eralised forces that can be transmitted from the manipulated platform to the fixed base through chain i when its actuated joint is unlocked. The same chain i can provide several constraint wrenches denoted by different values of j. The actuation ˆ Ai of chain i is the generalised force, in addition to the constraint wrenches, that can be can be transmitted when wrench W the actuated joint of chain i is locked. Based on the work in [12,13], Wang et al. [14] defined an input transmission index (ITI) and an output transmission index (OTI) for a kinematic chain i according to
|Wˆ Ai ◦ TˆIi | , ˆ |WAi ◦ TˆIi |max
ITIi =
OTIi =
(10)
|Wˆ Ai ◦ TˆOi | . |Wˆ Ai ◦ TˆOi |max
(11)
The value of ITIi is the normalised instantaneous work between the input twist and actuation wrench of a chain, measuring the transmission efficiency between the actuator and the passive links in the chain. The value of OTIi is the normalised instantaneous work performed between the chain’s actuation wrench and its output twist, measuring the transmission efficiency between the passive links and the manipulated platform. In the special case that the denominator in (10) or (11) is equal to zero, the corresponding value of the index is zero by definition. As pointed out by Liu et al. [15], a zero value of ITIi corresponds to a type one singularity [16] of the mechanism, while for a non-redundant mechanism, a zero value of OTIi corresponds to a type two singularity [16]. The transmission indices for the entire mechanism are found by taking the minimum over the n chains according to
ITI = min(ITIi ),
i = 1 . . . n,
OTI = min(OTIi ),
i = 1 . . . n.
(12)
(13)
Marlow et al. [17] introduced the intrachain constraint index (ICCI) to provide a measure of the transmission ability of ˆ ICi on their local output a closed-loop subchain. The ICCI measures the work performed by the local constraint wrenches W twists TˆICi according to
ICCIi =
|Wˆ ICi ◦ TˆICi | , |Wˆ ICi ◦ TˆICi |max
(14)
and
ICCI = min(ICCIi ),
i = 1 . . . k,
(15)
ˆ ICi are the local constraints where k = 2 for the studied four-bar closed-loop subchain. The local constraint wrenches W wrenches in the closed loop subchain. The local output twists TˆICi are the additional twists gained in a chain’s local twist ˆ ICi is suppressed. system when a local constraint wrench W The indices ITI, OTI and ICCI are dimensionless and frame invariant with a range between zero and unity. They may be applied to parallel mechanisms with arbitrary DOFs and have been utilised for analysis of a large range of parallel mechanisms [15,17–25]. We continue by calculating OTI, ITI1 , ITI2 and ICCI for the mechanism in Fig. 1. Consider an orthogonal coordinate system with its z-axis coinciding with the joint axis of Rp1 and its origin at an arbitrary point on this axis. Note that this coordinate system is different from the one illustrated in Fig. 1. In such a coordinate system, the closest point of the joint axes of Rp1 , Rp2a , Rp2b , Rc1 , Rc2a , Rc2b are given by rp1 , rp2a , rp2b , rc1 , rc2a , rc2b , respectively. The choice of coordinate system means that rp1 = 03×1 . Additionally, the closest point of an axis parallel to Rc2a and located between Rc2a and Rc2b is given by
rc2 =
rc2a + rc2b 2
(16)
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The actuation wrench of chain one is a zero pitch screw given by
ˆ A1 = W
1 1 −rc1 rp1 − rc1 = . l rp1 × (rp1 − rc1 ) l 03×1
(17)
ˆ A2a and W ˆ A2b , respectively, and expressed The local wrenches in the parallelogram formed by L2a and L2b are denoted by W as
ˆ A2s W
rp2s − rc2s 1 = , l rp2s × rp2s − rc2s
s = a, b.
(18)
ˆ A2 and W ˆ C2 , As long as L2a and L2b are not collinear, the wrenches (18) can be substituted by two equivalent wrenches W ˆ A2 is a zero pitch actuation wrench and W ˆ C2 an infinite pitch constraint wrench [26–28], expressed as where W
ˆ A2 = W
rp1 − rc2 1 1 −rc2 = , l rp1 × rp1 − rc2 l 03×1
(19)
ˆ C2 = 03×1 . W zˆ
(20)
The two output twists are zero pitch screws expressed as
TˆO1 =
TˆO2 =
zˆ , rc2 × zˆ
(21)
zˆ , rc1 × zˆ
(22)
while the two input twists are identical infinite pitch screws expressed as
03×1 TˆIi = , xˆ
i = 1, 2,
(23)
ˆ A1 and TˆI1 is calculated by where xˆ is a unit vector in the direction of the x-axis in Fig. 1. The reciprocal product between W utilising (17) and (23) according to
ˆ A1 ◦ TˆI1 = W
l cos π2 − μ1 −rc1 · xˆ = = sin μ1 . l l
(24)
The value of ITI1 is calculated by inserting (24) into (10) according to
ITI1 =
|Wˆ A1 ◦ TˆI1 | | sin μ1 | = = |sin μ1 |. |Wˆ A1 ◦ TˆI1 |max | sin μ1 |max
(25)
An equivalent calculation gives
ITI2 = |sin μ2 |.
(26)
ˆ A1 and TˆO1 is calculated from (17) and (21) according to The reciprocal product between W
ˆ A1 ◦ TˆO1 = W
−rc1 · (rc2 × zˆ ) zˆ · (rc2 × rc1 ) l 2 sin μr = = = l sin μr . l l l
(27)
Inserting (27) into (11) provides the OTI for chain one according to
OTI1 =
|Wˆ A1 ◦ TˆO1 | l | sin μr | = = |sin μr |. |Wˆ A1 ◦ TˆO1 |max l | sin μr |max
(28)
The corresponding calculation for OTI2 provides
OTI2 = |sin μr |.
(29)
Inserting (28) and (29) into (13) gives
OTI = min(OTI1 , OTI2 ) = |sin μr |.
(30)
We continue by calculating the ICCI for the studied mechanism. For a planar closed-loop subchain composed of a general ˆ ICi and TˆICi , producing two ICCI results [17]. The wrenches of the closed-loop four-bar linkage, there are two pairs of W
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ˆ IC1 = W ˆ A2a and W ˆ IC2 = W ˆ A2b , where W ˆ A2a and W ˆ A2b are subchain before its treatment as a generalised kinematic pair are W ˆ IC1 is suppressed is a rotation around Rp2b , expressed as given by (18). The local output twist TˆIC1 possible if the wrench W
TˆIC1 =
zˆ , rp2b × zˆ
(31)
ˆ IC2 is suppressed is a rotation around Rp2a , given by while the local output twist TˆIC2 possible if the wrench W
TˆIC2 =
zˆ . rp2a × zˆ
(32)
Employing (18) and (31) gives
ˆ IC1 ◦ TˆIC1 = W
−rc2a zˆ zˆ · (rp2b × zˆ ) = · (rp2b × rc2a ) = · (rp2b × (rc2 + rp2a ) = l l l zˆ zˆ · (rp2b × rc2 ) = · ( pl sin(π − μp )zˆ ) = p sin μp . l l
(33)
Hence, according to (14),
ICCI1 =
|Wˆ IC1 ◦ TˆIC1 | = |sin μp |. |Wˆ IC1 ◦ TˆIC1 |max
(34)
An equivalent calculation gives
ICCI2 =
|Wˆ IC2 ◦ TˆIC2 | = |sin μp |. |Wˆ IC2 ◦ TˆIC2 |max
(35)
Hence, the ICCI for the parallelogram is given by
ICCI = min(ICCI1 , ICCI2 ) = |sin μp |.
(36)
It should be noted that for a general four bar linkage ICCI1 = ICCI2 . Utilising basic trigonometry in Fig. 1, the angles μ1 , μ2 , μr and μp may be expressed as functions of y according to
μ1 ( y ) = μ2 ( y ) =
π 2
π 2
− arcsin
− arcsin
y − d1 l
=
π 2
+ arcsin
d1 − y , l
(37)
d2 − y , l
(38)
μr (y ) = π − μ1 (y ) − μ2 (y ) = − arcsin μp ( y ) = π − α − μ2 ( y ) =
π 2
d1 − y l
− α + arcsin
+ arcsin
d2 − y , l
(39)
d2 − y . l
(40)
4. Dimensional synthesis This section demonstrates how to utilise the indices from Section 3 to derive kinematically optimal architectures of the type described in Section 2 and quantify their performance. We begin by analysing a single mechanism, for which mechanical interference is not an issue, and continue to consider the complications arising when combining several such mecha-
nisms, first in a row and then in a two-dimensional array. As the performance in x
in x +
y
T
y
T
is identical to the performance
for all , we only consider the worst case performance when y varies between 0 and wy .
4.1. Single mechanism The values of OTI, ITI1 and ITI2 are closely correlated. Referring to Fig. 1, it is clear that longer link lengths l leads to angles μ1 and μ2 closer to 90 degrees and therefore ITI1 and ITI2 closer to the optimal value of one. However, longer link lengths also means that μr and therefore OTI tends to zero. By studying Fig. 1, it can be observed that in the range 0 ≤ y ≤ wy , the minimum value of ITI1 occur for y = wy , while the minimum value of ITI2 occur for y = 0. By differentiating (30) with respect to y, it can be shown that the minimum value of OTI occurs when the TCP is in the middle between the guide-ways, that is for
y=
d1 + d2 . 2
(41)
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Utilising (25), (26) and (30), the minimum values ITI1 , ITI2 and OTI may be expressed as
min(ITI1 (y )) = ITI1 (wy ) = sin
min(ITI2 (y )) = ITI2 (0 ) = sin
min(OTI(y )) = OTI
d1 + d2 2
π 2
π
− arcsin
2
+ arcsin
d1 − wy l
d2 l
= sin 2 arcsin
,
(42)
,
d2 − d1 2l
(43)
.
(44)
In order to maximise min(OTI, ITI1 , ITI2 ) in the entire utilised workspace, we employ the condition min(ITI1 ) = min(ITI2 ) = min(OTI ) = sin μmin , where μmin is our selected performance criterion. Using (42)–(44), this provides
d1 = wy − l cos μmin ,
(45)
d2 = l cos μmin ,
(46)
d2 − d1 = 2l sin
μ
min
2
.
(47)
Inserting (45) and (46) into (47) gives
wy = 2 cos μmin − sin l
μ
min
2
Solving (48) for a positive μmin gives
μmin = 2 arcsin
9−4
wy l
−1
4
.
(48)
,
0≤
wy ≤ 2, l
(49)
For a single mechanism or a row of mechanisms aligned along the x-axis, the lowest value of ICCI can be maximised independently of OTI, ITI1 and ITI2 by employing the condition ICCI(0 ) = ICCI(wy ). Utilising (15) and (40), this condition provides
|sin (α + μ2 (0 ) )| = |sin (α + μ2 (wy ) )|.
(50)
The optimal value of α may be expressed as
α=
π − μ2 ( 0 ) − μ2 ( w y ) 2
.
(51)
Utilising (38), (46) and (48), α can be expressed in μmin according to
α=
μmin π −μ min + arcsin 2 sin 2 2
− cos (μmin )
2
.
Hence, utilising (49), an optimal value of α can be calculated based on The achievable μmin depends on the ratio
wy l
(52) wy . l
according to (49). This equation provides a compact characterisation of
the potential of the studied mechanism. Fig. 2 shows how μmin varies for 0 ≤ wy l
wy l
≤ 2. As can be seen, the largest μmin
occurs when the ratio is zero, while μmin is zero when the ratio equals two. In order to achieve high values of μmin , the ratio has to be small, which is either achieved by a large l or a small wy . As the length of the links (l) strongly affect both the manipulator footprint and its dynamic properties, a practically useful mechanism with a high value of μmin is only possible if wy is relatively small. The literature [29–32] suggests that values of μmin between 35–45 and 135–145 degrees w are acceptable. It can be seen in Fig. 2 that μmin ≥ 40 degrees is achieved for ly ≤ 0.85. By instead utilising perpendicular guide-ways, The maximum achievable μmin equals 90 degrees. This is an interesting concept that merits further evaluation. However, utilising parallel guide-ways means that the workspace may be extended indefinitely in one direction and less issues with interference for an architecture including a two-dimensional array of mechanisms. Hence, this study is limited to a parallel arrangement of the guide-ways. Fig. 3(a)–(c) exemplify the resulting mechanisms and their utilised workspace for l = 1 m, p = 0.05 m and three different values of μmin . The workspace width wx was selected arbitrarily to 1 m, while the workspace width wy was calculated from (48). The dotted rectangle illustrates the workspace in which min(OTI, ITI1 , ITI2 , ICCI) ≥ sin μmin . The length of the guideways was calculated according to (9). In order to eliminate the potential of mechanical interference involving the guideways, the carts, linkages and the manipulated platform are designed to operate below the guide-ways. Fig. 3(d)–(f) show the values of the transmission indices for the three variants.
M. Isaksson et al. / Mechanism and Machine Theory 128 (2018) 382–394
Fig. 2. Illustration of (49). The maximum achievable μmin is plotted for 0 ≤
389
wy l
≤ 2.
Fig. 3. Examples of architectures and indices for l = 1 m and three different values of μmin . The selected wx is 1 m for all variants, while the resulting wy are 1.21 m, 0.85 m and 0.44 m, respectively, and the required stroke lengths of the prismatic actuators are 1.50 m, 1.36 m and 1.21 m, respectively.
4.2. One row of mechanisms In this section, we consider the feasibility of arranging m of the mechanisms from Section 2 in a row to obtain a combined utilised workspace of mwx × wy . Each mechanism services an area wx × wy . Potential mechanical interference in the border region between the utilised workspace of each mechanism is unavoidable. The width of this region depends on the size of the tool. The path planning software is tasked with controlling interference in this zone, for example by preventing the tool of one mechanism to enter its border zone while the neighbouring mechanism is in its corresponding zone.
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Fig. 4. (a) Bottom view of four mechanisms arranged in a row. (b) The two rightmost mechanisms in (a) viewed parallel to the x-axis. (c) The two rightmost mechanisms in (a) viewed parallel to the y-axis. (d) The transmission indices for the mechanisms.
If the row only includes two mechanisms, additional interference can be avoided by arranging the mechanisms to work opposite each other; however, if the row includes more than two mechanisms, a special arrangement is required. The proposed solution is to let the additional mechanisms operate in different planes, as illustrated in Fig. 4. Fig. 4(a) shows a bottom view of the complete architecture including four mechanisms, Fig. 4(b) and (c) show different side views of the two rightmost mechanisms in Fig. 4(a), while Fig. 4(d) shows the transmission indices. The kinematic parameters were selected by first choosing μmin = 40 degrees, p = 0.05 m, wy = 0.5 m and thereafter employing (48) to calculate the corresponding optimal value of l = 0.59 m. The width wx of each workspace was arbitrarily selected to wx = 0.5 and the calculated stroke length of the prismatic actuators equals 0.71 m. The combined utilised workspace of the four mechanisms is 1 m2 . As can be seen in Fig. 4(b) and (c), an architecture including four mechanisms per row requires two vertical levels of guide-ways. These guide-ways would be supported by a framework of beams parallel to the y-axis, connected at the endpoints of each guide-way. The required number of guide-way levels is given by ceil( m 2 ), where m is the number of mechanisms per row and the ceil function provides the least integer that is greater than or equal to its input. The number of guide-way levels is the limiting factor of the proposed design, as more guide-way levels requires additional height of some of the manipulated platforms, which at some point becomes unfeasible.
4.3. Multiple rows of mechanisms This section analyses the feasibility of arranging m × n of the mechanisms from Section 2 in an array with a combined utilised workspace of mwx × nwy . Three major issues must be considered; the entire combined utilised workspace must be reachable, potential mechanical interference must be limited and the kinematic properties in entire workspace must be acceptable. As can be seen from Fig. 3, in order for the entire combined utilised workspace to be reachable, the workspace width wy of a single mechanism must be larger than the distance between the guide-ways, as in Fig. 3(a) and (b). This clearly limits the achievable kinematic performance.
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Fig. 5. The constraints required to eliminate mechanical interference between mechanisms in different rows.
In order to analyse mechanical interference, we begin by calculating a different expression for the angle α in Fig. 1. In the case of multiple rows, the value given by (52) is no longer the optimal choice. The reason for this is that the y-dimension of the actuated carts affects potential mechanical interference, which in turn affects the achievable value of OTI, ITI1 and ITI2 . A value of α = 90 degrees would minimise the required y-dimension of the actuated cart C2 and be optimal to maximise OTI, ITI1 and ITI2 ; however, it would also reduce the achievable ICCI. In order to find the optimal compromise, we employ the condition
min(ICCI ) = min(OTI ) = min(ITI1 ) = min(ITI2 ) = sin μmin ,
(53)
min(ICCI ) = ICCI(wy ).
(54)
where Employing (15) and (40), the equation ICCI(wy ) = | sin μmin | can be rewritten as
l cos μmin cos α − l sin μmin sin α + d2 − wy = 0.
(55)
Substituting d2 by employing (46) gives
cos μmin cos α − sin μmin sin α = Utilising (48) to eliminate
wy l
wy − cos μmin . l
(56)
and solving for α gives
α = −μmin + arccos cos μmin − 2 sin
μ
min
2
.
(57)
The first constraint is that the actuated carts on different guide-ways cannot collide. The only possible collisions is between the cart C1 of a mechanism in one row and the cart C2 of a mechanism in a neighbouring row. As illustrated in Fig. 5(a), such collisions can be avoided by introducing the condition
wy + d1 −
c1y − (d2 + p cos α + rj ) > c, 2
(58)
where c1y is the y-dimension of the cart C1 , rj is the radius of the revolute joints and c is a clearance margin between carts. Eliminating d1 , d2 and l utilising (45), (46) and (48) gives
k1 (μmin )wy + k2 (μmin ) > 0, where
k1 (μmin ) =
cos μmin − 2 sin μ2min
, cos μmin − sin μ2min
k2 (μmin ) = −p cos(α ) −
c1y − rj − c, 2
(59)
(60)
(61)
and α is a function of μmin according to (57). The function k1 (μmin ) is positive for μmin < 43 degrees while k2 (μmin ) is typically negative and we may therefore plot the smallest wy that fulfils (59) as a function of μmin . The three lowest curves, denoted by c-c, in Fig 6 are examples of such curves. The curves are for three different values of p and constant values of rj = 0.01 m, c1y = 0.03 m and c = 0.01 m. As can be seen, in order to achieve μmin = 40 degrees with p = 0.05, the value of wy must be at least 0.36 m. Additionally, in order to simplify the task for the path planning software, it is much preferable if also potential collisions between carts and the manipulated platform are eliminated when the manipulated platform is within its dedicated workspace. This requires a stronger condition on wy . As can be seen from Fig. 5(b), all potential collisions between the cart C1 of a mechanism in one row and the manipulated platform of a mechanism in neighbouring row are eliminated by the condition
d1 −
c1y − ( p cos α + rj ) > c. 2
(62)
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Fig. 6. The minimum wy required in order to eliminate the described mechanical interference as a function of μmin for c1y = 0.03, rj = 0.01 m, c = 0.01 m and three different values of p. The curves marked c-c are for the condition (59) while the curves c-p refer to the stricter condition (66).
Fig. 7. (a) Example of an architecture with 16 mechanisms. As long as the TCP of each mechanism remains in its dedicated workspace, mechanical interference may only occur at the borders of this workspace. (b) The lowest value of each performance index equals sin μmin , where μmin = 36.7 degrees.
Similarly, as shown in Fig. 5(c), all potential collisions between the cart C2 of a mechanism in one row and the manipulated platform of a mechanism in neighbouring row are eliminated by the condition
wy − ( p cos α + rj ) − (d2 + p cos α + rj ) > c.
(63)
Eliminating d1 from (62) by utilising (45) and d2 from (63) by employing (46) and rearranging the terms gives
wy − l cos μmin − p cos α − rj − c −
c1y > 0, 2
wy − l cos μmin − p cos α − rj − c − p cos α − rj > 0.
(64)
(65)
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Eliminating l from (64) and (65) by utilising (48), these conditions can be summarised as
k1 (μmin ) wy + k3 (μmin ) > 0, 2 where
(66)
k3 (μmin ) = −p cos(α ) − rj − c − max p cos α + rj ,
c1y 2
(67)
and α is given by (57) and k1 is given by (60). The three upper curves in Fig. 6 shows the minimum wy required for a given μmin in order to fulfil (66). The curves are for three different values of p and constant values of rj = 0.01 m, c1y = 0.03 m and c = 0.01 m. The conclusion is that this arrangement is only feasible for relatively large values of wy . In order to achieve smaller wy , compromises on the parallelogram width (2p) or μmin are required. The disadvantages of a low value of p includes increased link forces and increased sensitivity to manufacturing errors in the link lengths. Fig. 7 exemplifies an architecture that fulfils both (59) and (66). As long as each mechanism remains in its dedicated workspace, the only potential mechanical interference is between manipulated platforms and these are only possible in the border zones of each mechanism’s workspace. The selected parameter values are μmin = 36.7 degrees, wy = 0.5 m, p = 0.04 m, rj = 0.01 m, c1y = 0.03 m and c = 0.01 m. The calculated lengths of the links and the guide-ways are l = 0.51 m and gl = 0.71 m, respectively. 5. Conclusion and future work This paper analysed the feasibility of a mechanical architecture including an array of 2-DOF parallel mechanisms of a specific type. Such architectures could be useful for two dimensional manufacturing operations, such as laser cutting or water jet cutting. Additionally, by including an actuation scheme for the work object in one DOF, it could be utilised for three dimensional manufacturing operations, such as additive manufacturing. Screw-theory-based indices characterising the kinematic performance of the investigated architecture were derived. The kinematic performance was measured by the worst case of the transmission angles in the utilised workspace. For an architecture including a single 2-DOF mechanism or a row of such mechanisms, the kinematic performance is a decreasing function of wy /l, where wy is the width of the utilised workspace in the direction perpendicular to the guide-ways and l is the length of the links. When the ratio wy /l approaches zero, the worst case transmission angle tends to its maximum value of 60 degrees, while the worst case transmission angle tends to zero when the ratio approaches two. It was observed that although it is theoretically possible to use a large number of mechanisms per row, the practicality of this solution deteriorates when the number of mechanisms per row is increased. For architectures including a two-dimensional array of 2-DOF mechanisms, it was shown that practical limitations on component sizes and the potential of mechanical interference put additional restrictions on the achievable kinematic performance and limits how closely the rows can be arranged. Planned future work involves building a prototype of a 3D printer utilising a two-dimensional array of mechanisms and an actuated print bed. There are several issues that require further study, including cabling solutions and the development of a suitable path planning algorithm. Additionally, it would be of interest to perform a similar feasibility study for an arrangement utilising perpendicular guide-ways, as such solutions enables an increase of the worst-case transmission angles. References [1] T. Brogårdh, Robot control overview: an industrial perspective, Model. Identif.Control 30 (3) (2009) 167–180. [2] G. Michalos, S. Makris, N. Papakostas, D. Mourtzis, G. Chryssolouris, Automotive assembly technologies review: challenges and outlook for a flexible and adaptive approach, CIRP J. Manuf. Sci. Technol. 2 (2) (2010) 81–91. [3] P.G. Ranky, Collaborative, synchronous robots serving machines and cells, Ind. Robot 30 (3) (2003) 213–217. [4] J.-P. 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Hunt, Kinematic Geometry of Mechanisms, Oxford University Press, Oxford, UK, 1978. [12] G. Sutherland, B. Roth, A transmission index for spatial mechanisms, J. Eng. Ind. 95 (2) (1973) 589–597. [13] C. Chen, J. Angeles, Generalized transmission index and transmission quality for spatial linkages, Mech. Mach. Theory 42 (9) (2007) 1225–1237. [14] J. Wang, C. Wu, X.J. Liu, Performance evaluation of parallel manipulators: motion/force transmissibility and its index, Mech. Mach. Theory 45 (10) (2010) 1462–1476. [15] X. Liu, C. Wu, J. Wang, A new approach for singularity analysis and closeness measurement to singularities of parallel manipulators, J. Mech. Rob. 4 (4) (2012) 041001-01–041001-10. [16] C. Gosselin, J. Angeles, Singularity analysis of closed-loop kinematic chains, IEEE Trans. Rob. Autom. 6 (3) (1990) 281–290. [17] K. Marlow, M. Isaksson, S. Dai, S. Nahavandi, Motion/force transmission analysis of parallel mechanisms with planar closed-loop subchains, J. Mech. 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Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmachtheory
Research paper
On the feasibility of utilising an array of planar parallel robots to service adjoining workspaces Mats Isaksson∗, Jason Miller, Mostafa Nikzad School of Engineering, Swinburne University of Technology, Hawthorn, VIC 3122, Australia
a r t i c l e
i n f o
Article history: Received 22 September 2017 Revised 1 June 2018 Accepted 2 June 2018
Keywords: Parallel kinematics Screw theory Additive manufacturing Subtractive manufacturing 3D printing
a b s t r a c t Parallel robots feature several favourable properties, including the potential for high acceleration and accuracy. However, in order to minimise cycle times and optimise the usage of factory space, industrial applications commonly include multiple closely mounted robots, working on the same or closely located objects. As parallel robots generally suffer from a limited workspace-to-footprint ratio, attempting to utilise parallel robots for such applications typically leads to issues with mechanical interference. This paper investigates the feasibility of dividing a planar workspace into several smaller areas, each of them serviced by a dedicated planar parallel mechanism. Applications where this arrangement could be beneficial include 2D manufacturing operations, such as water jet cutting and laser cutting. Additionally, the proposed arrangement could be extended by actuating either the work object fixture or the entire array of mechanisms in one additional degree of freedom. The resulting architecture could be beneficial for 3D manufacturing operations, such as additive or subtractive manufacturing. The presented analysis studies the limitations of such architectures in terms of mechanical interference and achievable kinematic performance. Solutions to reduce mechanical interference are introduced and screw-theory-based indices are employed to evaluate the achievable kinematic performance. © 2018 Elsevier Ltd. All rights reserved.
1. Introduction In order to optimise the usage of factory space, industrial robots are commonly mounted closely together. Cycle time and production cost can often be reduced if several robots perform different tasks on the same object. A recent trend is the utilisation of coordinated collaborating robots [1]. Typical application for such robots include arc welding, where one robot manipulates the work object while other robots perform the welding [1–3]. Such an approach can save floor space and improve accessibility of the work object [3]. Coordination may be achieved by robot-to-robot communication or by utilising a single controller for all participating robots. The latter option saves additional floor space and provides several other benefits, including the possibility for improved collision avoidance [1]. Parallel robots [4] exhibit several benefits compared to similar-sized serial robots [5], including the potential for high stiffness and low moving mass, which enables higher acceleration and reduced cycle times. As the position errors of the actuators are not cumulative, the position accuracy is typically high. Additionally, the actuators are commonly mounted on the fixed base, leading to straightforward cabling solutions. However, parallel robots also suffer from several limitations,
∗
Corresponding author. E-mail address: [email protected] (M. Isaksson).
https://doi.org/10.1016/j.mechmachtheory.2018.06.001 0094-114X/© 2018 Elsevier Ltd. All rights reserved.
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Fig. 1. Parameters of the studied 2-DOF planar parallel mechanism. The view is from the intended work object. The manipulated platform can pass below the guide-ways. The single link and the parallelogram operate in different vertical planes. The only potential mechanical interferences are between the link L1 and the cart C2 and between the links L2a and L2b ; however, such configurations are outside the intended workspace of the mechanism.
typically including a limited workspace-to-footprint ratio [6]. Due to mechanical interference, workspace sharing is difficult, and even attempting to arrange parallel robots to service separate but adjoining workspaces is challenging. Herein, we make a distinction between the reachable workspace, which is the entire workspace that the tool can service without mechanical limitations or singularities, and the utilised workspace, which is a subsection of the reachable workspace where the kinematic performance is considered acceptable. Typically, the utilised workspace excludes the border regions of the reachable workspace. Potential interference between the tools of two neighbouring mechanisms at the border of each mechanism’s utilised workspace is unavoidable; however, such interference is straightforward to handle with software algorithms, leading to only minor constraints on the independence of each mechanism. The challenge for a parallel mechanism is that its arm system is often closer to a neighbouring mechanism than its tool, leading to potential interference also between the arm systems of the mechanisms. Modelling all such interferences is complex and introduces additional constraints on how independently the two mechanisms can service their respective workspace. In terms of collision avoidance, it is preferable that when the tool is in its border regions, it is the section of the mechanism that is closest to the neighbouring mechanism. This is only possible for some parallel mechanisms and typically means utilising the outer regions of the reachable workspace, where the kinematic performance may not be adequate. Research and development of parallel robots that can be mounted closely with minimal mechanical interference is of significant industrial interest [7]. In this paper, we focus on applications requiring two or three degrees of freedom. We investigate the feasibility of dividing a planar workspace into several smaller areas, each of them serviced by a dedicated two-degree-of-freedom (2-DOF) planar parallel mechanism. Applications where such an arrangement could be beneficial include water jet cutting and laser cutting. Additionally, this arrangement could be extended by actuating either the work object fixture or the entire array of mechanisms in one additional degree of freedom. The resulting architecture could be beneficial for additive manufacturing, as the utilisation of multiple print heads could dramatically reduce the build times. As a 3D printing platform with multiple print heads does not require independent 3-DOF mechanisms to manipulate each print head, the proposed architecture could be a cost-efficient solution. In addition to reduced number of actuators, such an architecture has several other potential benefits, including improved accuracy, space savings, and low-cost components. This paper provides an evaluation of the described architecture for a particular choice of the individual 2-DOF mechanisms. It is demonstrated how the objectives of optimal kinematic performance and low risk of mechanical interference are inversely correlated. By utilising intuitive screw-theory-based performance indices, the potential and limitations of the proposed architecture are fully quantified. The remainder of this paper is organised as follows. Section 2 introduces the individual 2-DOF parallel mechanisms, including their kinematic parameters. Section 3 reviews the screw-theory-based performance indices employed for the subsequent kinematic analysis. Section 4 begins with a kinematic analysis of a single mechanism and continues to study the feasibility of utilising multiple mechanisms of the same type, first arranged in a one-dimensional array and then in a two-dimensional array. Finally, Section 5 provides conclusions and ideas for future work. 2. Individual planar mechanism The individual unit of the proposed architectures is the mechanism illustrated in Fig. 1. It is a planar parallel mechanism with two positional DOFs and constant orientation of the manipulated platform. It includes two actuated carts on parallel linear guide-ways. Possible actuation schemes include screws, belt or rack-and-pinion. One actuated cart C1 includes a revolute joint Rc1 , while the second actuated cart C2 includes two revolute joints Rc2a and Rc2b . The manipulated plat-
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form includes three revolute joints Rp1 , Rp2a and Rp2b . Three links L1 , L2a and L2b of equal length connect the joints on the actuated carts and the manipulated platform, where L2a and L2b are arranged to always be parallel. The links could be counterbalanced to avoid bending forces on the joints Rc1 , Rc2a and Rc2b . By arranging the single link and the links in the parallelogram to operate in different planes below the guide-ways, mechanical interference between the single link and the links in the parallelogram and between the links and the guide-ways is not an issue. 2.1. Kinematic parameters Fig. 1 includes the kinematic parameters of the mechanism. The tool centre point (TCP) is located on the axis of joint Rp1 . This joint axis is in the centre of a perpendicular line between the joint axes of Rp2a and Rp2b . The utilised workspace of the TCP is outlined in a dotted line. This workspace is a subsection of the entire reachable workspace, where certain kinematic constraints are fulfilled. These constraints will be introduced in Section 3. A coordinate system has been introduced with its x-axis and y-axis coinciding with two of the workspace borders. The width of the workspace in the x-direction is denoted by wx and its width in the y-direction by wy . The width wx is not relevant to the kinematic optimisation but necessary to calculate the required length of the guide-ways.
T
The planar position of the TCP is given by x = x y . The distance between Rp2a and Rp2b equals 2p and the angle between a line parallel to the y-axis and a line perpendicular to the axes of Rc2a and Rc2b is denoted by α . The x- and y-coordinates of the joint axis of Rc1 are denoted by q1 and d1 , respectively. The x- and y-coordinates of an axis in the middle between Rc2a and Rc2b are denoted by q2 and d2 , respectively. The kinematic lengths of L1 , L2a and L2b are equal and denoted by l. The kinematic length is the perpendicular distance between the joint axes of the joints at the ends of the link. The illustrated angles μ1 , μ2 , μp and μr are employed in the kinematic analysis and will be discussed in Section 3. The guide-ways are of equal length. The stroke length of the prismatic actuators is denoted by gl , while their smallest and largest x-coordinate is denoted by gmin and gmax , respectively. 2.2. Inverse kinematics and length of the guide-ways The kinematics of the mechanism in Fig. 1 is identical to that of the over-constrained mechanisms described by Liu et al. [8]. Pythagoras’ Theorem provides the equations
l 2 − ( x − qi )2 − ( y − di )2 = 0,
i = 1, 2.
(1)
The inverse kinematics of the mechanism is derived by selecting the smallest of the two qi values fulfilling (1) according to
qi (x, y ) = x −
l 2 − ( y − di )2 ,
i = 1, 2.
(2)
As can be seen from (2), physical solutions require that l ≥ |y − di | for 0 ≤ y ≤ wy . The length of the guide-ways are selected to be equal. The required stroke length gl and the positions gmin and gmax are calculated by utilising (2) according to
q1min =
q1 (0, d1 ), if d1 > 0, q1 (0, 0 ), if d1 ≤ 0,
q2min =
q2 ( 0, d2 ), q2 ( 0, wy ),
if d2 < wy , if d2 ≥ wy ,
(3)
(4)
q1max = q1 (wx , wy ),
(5)
q2max = q2 (wx , 0 ),
(6)
gmin = min(q1min , q2min ),
(7)
gmax = max(q1max , q2max ),
(8)
gl = gmax − gmin .
(9)
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3. Transmission indices The kinematic performance evaluation of the studied mechanisms is performed by utilising screw-theory-based indices. Screw theory was introduced by Ball [9,10] and revived by Hunt [11] significantly later. Refined by the work of several successive researchers, screw theory has gradually become the state-of-the-art method for kinematic analysis of parallel mechanisms. The utilised indices characterise the transmission efficiency between each actuated cart and its corresponding link system, the transmission efficiency between each link system and the manipulated platform, and how efficiently the parallelogram constrains the platform’s rotation. ˆ for unit wrenches and Tˆ for unit twists. The input twist TˆIi of a kinematic chain i is the We employ the notation W twist of the actuated joint of that chain, while the output twist TˆOi of the same chain is the free motion of the manipulated ˆ Ci j of chain i are the genplatform when all actuated joints except the one of chain i are locked. The constraint wrenches W eralised forces that can be transmitted from the manipulated platform to the fixed base through chain i when its actuated joint is unlocked. The same chain i can provide several constraint wrenches denoted by different values of j. The actuation ˆ Ai of chain i is the generalised force, in addition to the constraint wrenches, that can be can be transmitted when wrench W the actuated joint of chain i is locked. Based on the work in [12,13], Wang et al. [14] defined an input transmission index (ITI) and an output transmission index (OTI) for a kinematic chain i according to
|Wˆ Ai ◦ TˆIi | , ˆ |WAi ◦ TˆIi |max
ITIi =
OTIi =
(10)
|Wˆ Ai ◦ TˆOi | . |Wˆ Ai ◦ TˆOi |max
(11)
The value of ITIi is the normalised instantaneous work between the input twist and actuation wrench of a chain, measuring the transmission efficiency between the actuator and the passive links in the chain. The value of OTIi is the normalised instantaneous work performed between the chain’s actuation wrench and its output twist, measuring the transmission efficiency between the passive links and the manipulated platform. In the special case that the denominator in (10) or (11) is equal to zero, the corresponding value of the index is zero by definition. As pointed out by Liu et al. [15], a zero value of ITIi corresponds to a type one singularity [16] of the mechanism, while for a non-redundant mechanism, a zero value of OTIi corresponds to a type two singularity [16]. The transmission indices for the entire mechanism are found by taking the minimum over the n chains according to
ITI = min(ITIi ),
i = 1 . . . n,
OTI = min(OTIi ),
i = 1 . . . n.
(12)
(13)
Marlow et al. [17] introduced the intrachain constraint index (ICCI) to provide a measure of the transmission ability of ˆ ICi on their local output a closed-loop subchain. The ICCI measures the work performed by the local constraint wrenches W twists TˆICi according to
ICCIi =
|Wˆ ICi ◦ TˆICi | , |Wˆ ICi ◦ TˆICi |max
(14)
and
ICCI = min(ICCIi ),
i = 1 . . . k,
(15)
ˆ ICi are the local constraints where k = 2 for the studied four-bar closed-loop subchain. The local constraint wrenches W wrenches in the closed loop subchain. The local output twists TˆICi are the additional twists gained in a chain’s local twist ˆ ICi is suppressed. system when a local constraint wrench W The indices ITI, OTI and ICCI are dimensionless and frame invariant with a range between zero and unity. They may be applied to parallel mechanisms with arbitrary DOFs and have been utilised for analysis of a large range of parallel mechanisms [15,17–25]. We continue by calculating OTI, ITI1 , ITI2 and ICCI for the mechanism in Fig. 1. Consider an orthogonal coordinate system with its z-axis coinciding with the joint axis of Rp1 and its origin at an arbitrary point on this axis. Note that this coordinate system is different from the one illustrated in Fig. 1. In such a coordinate system, the closest point of the joint axes of Rp1 , Rp2a , Rp2b , Rc1 , Rc2a , Rc2b are given by rp1 , rp2a , rp2b , rc1 , rc2a , rc2b , respectively. The choice of coordinate system means that rp1 = 03×1 . Additionally, the closest point of an axis parallel to Rc2a and located between Rc2a and Rc2b is given by
rc2 =
rc2a + rc2b 2
(16)
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The actuation wrench of chain one is a zero pitch screw given by
ˆ A1 = W
1 1 −rc1 rp1 − rc1 = . l rp1 × (rp1 − rc1 ) l 03×1
(17)
ˆ A2a and W ˆ A2b , respectively, and expressed The local wrenches in the parallelogram formed by L2a and L2b are denoted by W as
ˆ A2s W
rp2s − rc2s 1 = , l rp2s × rp2s − rc2s
s = a, b.
(18)
ˆ A2 and W ˆ C2 , As long as L2a and L2b are not collinear, the wrenches (18) can be substituted by two equivalent wrenches W ˆ A2 is a zero pitch actuation wrench and W ˆ C2 an infinite pitch constraint wrench [26–28], expressed as where W
ˆ A2 = W
rp1 − rc2 1 1 −rc2 = , l rp1 × rp1 − rc2 l 03×1
(19)
ˆ C2 = 03×1 . W zˆ
(20)
The two output twists are zero pitch screws expressed as
TˆO1 =
TˆO2 =
zˆ , rc2 × zˆ
(21)
zˆ , rc1 × zˆ
(22)
while the two input twists are identical infinite pitch screws expressed as
03×1 TˆIi = , xˆ
i = 1, 2,
(23)
ˆ A1 and TˆI1 is calculated by where xˆ is a unit vector in the direction of the x-axis in Fig. 1. The reciprocal product between W utilising (17) and (23) according to
ˆ A1 ◦ TˆI1 = W
l cos π2 − μ1 −rc1 · xˆ = = sin μ1 . l l
(24)
The value of ITI1 is calculated by inserting (24) into (10) according to
ITI1 =
|Wˆ A1 ◦ TˆI1 | | sin μ1 | = = |sin μ1 |. |Wˆ A1 ◦ TˆI1 |max | sin μ1 |max
(25)
An equivalent calculation gives
ITI2 = |sin μ2 |.
(26)
ˆ A1 and TˆO1 is calculated from (17) and (21) according to The reciprocal product between W
ˆ A1 ◦ TˆO1 = W
−rc1 · (rc2 × zˆ ) zˆ · (rc2 × rc1 ) l 2 sin μr = = = l sin μr . l l l
(27)
Inserting (27) into (11) provides the OTI for chain one according to
OTI1 =
|Wˆ A1 ◦ TˆO1 | l | sin μr | = = |sin μr |. |Wˆ A1 ◦ TˆO1 |max l | sin μr |max
(28)
The corresponding calculation for OTI2 provides
OTI2 = |sin μr |.
(29)
Inserting (28) and (29) into (13) gives
OTI = min(OTI1 , OTI2 ) = |sin μr |.
(30)
We continue by calculating the ICCI for the studied mechanism. For a planar closed-loop subchain composed of a general ˆ ICi and TˆICi , producing two ICCI results [17]. The wrenches of the closed-loop four-bar linkage, there are two pairs of W
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ˆ IC1 = W ˆ A2a and W ˆ IC2 = W ˆ A2b , where W ˆ A2a and W ˆ A2b are subchain before its treatment as a generalised kinematic pair are W ˆ IC1 is suppressed is a rotation around Rp2b , expressed as given by (18). The local output twist TˆIC1 possible if the wrench W
TˆIC1 =
zˆ , rp2b × zˆ
(31)
ˆ IC2 is suppressed is a rotation around Rp2a , given by while the local output twist TˆIC2 possible if the wrench W
TˆIC2 =
zˆ . rp2a × zˆ
(32)
Employing (18) and (31) gives
ˆ IC1 ◦ TˆIC1 = W
−rc2a zˆ zˆ · (rp2b × zˆ ) = · (rp2b × rc2a ) = · (rp2b × (rc2 + rp2a ) = l l l zˆ zˆ · (rp2b × rc2 ) = · ( pl sin(π − μp )zˆ ) = p sin μp . l l
(33)
Hence, according to (14),
ICCI1 =
|Wˆ IC1 ◦ TˆIC1 | = |sin μp |. |Wˆ IC1 ◦ TˆIC1 |max
(34)
An equivalent calculation gives
ICCI2 =
|Wˆ IC2 ◦ TˆIC2 | = |sin μp |. |Wˆ IC2 ◦ TˆIC2 |max
(35)
Hence, the ICCI for the parallelogram is given by
ICCI = min(ICCI1 , ICCI2 ) = |sin μp |.
(36)
It should be noted that for a general four bar linkage ICCI1 = ICCI2 . Utilising basic trigonometry in Fig. 1, the angles μ1 , μ2 , μr and μp may be expressed as functions of y according to
μ1 ( y ) = μ2 ( y ) =
π 2
π 2
− arcsin
− arcsin
y − d1 l
=
π 2
+ arcsin
d1 − y , l
(37)
d2 − y , l
(38)
μr (y ) = π − μ1 (y ) − μ2 (y ) = − arcsin μp ( y ) = π − α − μ2 ( y ) =
π 2
d1 − y l
− α + arcsin
+ arcsin
d2 − y , l
(39)
d2 − y . l
(40)
4. Dimensional synthesis This section demonstrates how to utilise the indices from Section 3 to derive kinematically optimal architectures of the type described in Section 2 and quantify their performance. We begin by analysing a single mechanism, for which mechanical interference is not an issue, and continue to consider the complications arising when combining several such mecha-
nisms, first in a row and then in a two-dimensional array. As the performance in x
in x +
y
T
y
T
is identical to the performance
for all , we only consider the worst case performance when y varies between 0 and wy .
4.1. Single mechanism The values of OTI, ITI1 and ITI2 are closely correlated. Referring to Fig. 1, it is clear that longer link lengths l leads to angles μ1 and μ2 closer to 90 degrees and therefore ITI1 and ITI2 closer to the optimal value of one. However, longer link lengths also means that μr and therefore OTI tends to zero. By studying Fig. 1, it can be observed that in the range 0 ≤ y ≤ wy , the minimum value of ITI1 occur for y = wy , while the minimum value of ITI2 occur for y = 0. By differentiating (30) with respect to y, it can be shown that the minimum value of OTI occurs when the TCP is in the middle between the guide-ways, that is for
y=
d1 + d2 . 2
(41)
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Utilising (25), (26) and (30), the minimum values ITI1 , ITI2 and OTI may be expressed as
min(ITI1 (y )) = ITI1 (wy ) = sin
min(ITI2 (y )) = ITI2 (0 ) = sin
min(OTI(y )) = OTI
d1 + d2 2
π 2
π
− arcsin
2
+ arcsin
d1 − wy l
d2 l
= sin 2 arcsin
,
(42)
,
d2 − d1 2l
(43)
.
(44)
In order to maximise min(OTI, ITI1 , ITI2 ) in the entire utilised workspace, we employ the condition min(ITI1 ) = min(ITI2 ) = min(OTI ) = sin μmin , where μmin is our selected performance criterion. Using (42)–(44), this provides
d1 = wy − l cos μmin ,
(45)
d2 = l cos μmin ,
(46)
d2 − d1 = 2l sin
μ
min
2
.
(47)
Inserting (45) and (46) into (47) gives
wy = 2 cos μmin − sin l
μ
min
2
Solving (48) for a positive μmin gives
μmin = 2 arcsin
9−4
wy l
−1
4
.
(48)
,
0≤
wy ≤ 2, l
(49)
For a single mechanism or a row of mechanisms aligned along the x-axis, the lowest value of ICCI can be maximised independently of OTI, ITI1 and ITI2 by employing the condition ICCI(0 ) = ICCI(wy ). Utilising (15) and (40), this condition provides
|sin (α + μ2 (0 ) )| = |sin (α + μ2 (wy ) )|.
(50)
The optimal value of α may be expressed as
α=
π − μ2 ( 0 ) − μ2 ( w y ) 2
.
(51)
Utilising (38), (46) and (48), α can be expressed in μmin according to
α=
μmin π −μ min + arcsin 2 sin 2 2
− cos (μmin )
2
.
Hence, utilising (49), an optimal value of α can be calculated based on The achievable μmin depends on the ratio
wy l
(52) wy . l
according to (49). This equation provides a compact characterisation of
the potential of the studied mechanism. Fig. 2 shows how μmin varies for 0 ≤ wy l
wy l
≤ 2. As can be seen, the largest μmin
occurs when the ratio is zero, while μmin is zero when the ratio equals two. In order to achieve high values of μmin , the ratio has to be small, which is either achieved by a large l or a small wy . As the length of the links (l) strongly affect both the manipulator footprint and its dynamic properties, a practically useful mechanism with a high value of μmin is only possible if wy is relatively small. The literature [29–32] suggests that values of μmin between 35–45 and 135–145 degrees w are acceptable. It can be seen in Fig. 2 that μmin ≥ 40 degrees is achieved for ly ≤ 0.85. By instead utilising perpendicular guide-ways, The maximum achievable μmin equals 90 degrees. This is an interesting concept that merits further evaluation. However, utilising parallel guide-ways means that the workspace may be extended indefinitely in one direction and less issues with interference for an architecture including a two-dimensional array of mechanisms. Hence, this study is limited to a parallel arrangement of the guide-ways. Fig. 3(a)–(c) exemplify the resulting mechanisms and their utilised workspace for l = 1 m, p = 0.05 m and three different values of μmin . The workspace width wx was selected arbitrarily to 1 m, while the workspace width wy was calculated from (48). The dotted rectangle illustrates the workspace in which min(OTI, ITI1 , ITI2 , ICCI) ≥ sin μmin . The length of the guideways was calculated according to (9). In order to eliminate the potential of mechanical interference involving the guideways, the carts, linkages and the manipulated platform are designed to operate below the guide-ways. Fig. 3(d)–(f) show the values of the transmission indices for the three variants.
M. Isaksson et al. / Mechanism and Machine Theory 128 (2018) 382–394
Fig. 2. Illustration of (49). The maximum achievable μmin is plotted for 0 ≤
389
wy l
≤ 2.
Fig. 3. Examples of architectures and indices for l = 1 m and three different values of μmin . The selected wx is 1 m for all variants, while the resulting wy are 1.21 m, 0.85 m and 0.44 m, respectively, and the required stroke lengths of the prismatic actuators are 1.50 m, 1.36 m and 1.21 m, respectively.
4.2. One row of mechanisms In this section, we consider the feasibility of arranging m of the mechanisms from Section 2 in a row to obtain a combined utilised workspace of mwx × wy . Each mechanism services an area wx × wy . Potential mechanical interference in the border region between the utilised workspace of each mechanism is unavoidable. The width of this region depends on the size of the tool. The path planning software is tasked with controlling interference in this zone, for example by preventing the tool of one mechanism to enter its border zone while the neighbouring mechanism is in its corresponding zone.
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Fig. 4. (a) Bottom view of four mechanisms arranged in a row. (b) The two rightmost mechanisms in (a) viewed parallel to the x-axis. (c) The two rightmost mechanisms in (a) viewed parallel to the y-axis. (d) The transmission indices for the mechanisms.
If the row only includes two mechanisms, additional interference can be avoided by arranging the mechanisms to work opposite each other; however, if the row includes more than two mechanisms, a special arrangement is required. The proposed solution is to let the additional mechanisms operate in different planes, as illustrated in Fig. 4. Fig. 4(a) shows a bottom view of the complete architecture including four mechanisms, Fig. 4(b) and (c) show different side views of the two rightmost mechanisms in Fig. 4(a), while Fig. 4(d) shows the transmission indices. The kinematic parameters were selected by first choosing μmin = 40 degrees, p = 0.05 m, wy = 0.5 m and thereafter employing (48) to calculate the corresponding optimal value of l = 0.59 m. The width wx of each workspace was arbitrarily selected to wx = 0.5 and the calculated stroke length of the prismatic actuators equals 0.71 m. The combined utilised workspace of the four mechanisms is 1 m2 . As can be seen in Fig. 4(b) and (c), an architecture including four mechanisms per row requires two vertical levels of guide-ways. These guide-ways would be supported by a framework of beams parallel to the y-axis, connected at the endpoints of each guide-way. The required number of guide-way levels is given by ceil( m 2 ), where m is the number of mechanisms per row and the ceil function provides the least integer that is greater than or equal to its input. The number of guide-way levels is the limiting factor of the proposed design, as more guide-way levels requires additional height of some of the manipulated platforms, which at some point becomes unfeasible.
4.3. Multiple rows of mechanisms This section analyses the feasibility of arranging m × n of the mechanisms from Section 2 in an array with a combined utilised workspace of mwx × nwy . Three major issues must be considered; the entire combined utilised workspace must be reachable, potential mechanical interference must be limited and the kinematic properties in entire workspace must be acceptable. As can be seen from Fig. 3, in order for the entire combined utilised workspace to be reachable, the workspace width wy of a single mechanism must be larger than the distance between the guide-ways, as in Fig. 3(a) and (b). This clearly limits the achievable kinematic performance.
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391
Fig. 5. The constraints required to eliminate mechanical interference between mechanisms in different rows.
In order to analyse mechanical interference, we begin by calculating a different expression for the angle α in Fig. 1. In the case of multiple rows, the value given by (52) is no longer the optimal choice. The reason for this is that the y-dimension of the actuated carts affects potential mechanical interference, which in turn affects the achievable value of OTI, ITI1 and ITI2 . A value of α = 90 degrees would minimise the required y-dimension of the actuated cart C2 and be optimal to maximise OTI, ITI1 and ITI2 ; however, it would also reduce the achievable ICCI. In order to find the optimal compromise, we employ the condition
min(ICCI ) = min(OTI ) = min(ITI1 ) = min(ITI2 ) = sin μmin ,
(53)
min(ICCI ) = ICCI(wy ).
(54)
where Employing (15) and (40), the equation ICCI(wy ) = | sin μmin | can be rewritten as
l cos μmin cos α − l sin μmin sin α + d2 − wy = 0.
(55)
Substituting d2 by employing (46) gives
cos μmin cos α − sin μmin sin α = Utilising (48) to eliminate
wy l
wy − cos μmin . l
(56)
and solving for α gives
α = −μmin + arccos cos μmin − 2 sin
μ
min
2
.
(57)
The first constraint is that the actuated carts on different guide-ways cannot collide. The only possible collisions is between the cart C1 of a mechanism in one row and the cart C2 of a mechanism in a neighbouring row. As illustrated in Fig. 5(a), such collisions can be avoided by introducing the condition
wy + d1 −
c1y − (d2 + p cos α + rj ) > c, 2
(58)
where c1y is the y-dimension of the cart C1 , rj is the radius of the revolute joints and c is a clearance margin between carts. Eliminating d1 , d2 and l utilising (45), (46) and (48) gives
k1 (μmin )wy + k2 (μmin ) > 0, where
k1 (μmin ) =
cos μmin − 2 sin μ2min
, cos μmin − sin μ2min
k2 (μmin ) = −p cos(α ) −
c1y − rj − c, 2
(59)
(60)
(61)
and α is a function of μmin according to (57). The function k1 (μmin ) is positive for μmin < 43 degrees while k2 (μmin ) is typically negative and we may therefore plot the smallest wy that fulfils (59) as a function of μmin . The three lowest curves, denoted by c-c, in Fig 6 are examples of such curves. The curves are for three different values of p and constant values of rj = 0.01 m, c1y = 0.03 m and c = 0.01 m. As can be seen, in order to achieve μmin = 40 degrees with p = 0.05, the value of wy must be at least 0.36 m. Additionally, in order to simplify the task for the path planning software, it is much preferable if also potential collisions between carts and the manipulated platform are eliminated when the manipulated platform is within its dedicated workspace. This requires a stronger condition on wy . As can be seen from Fig. 5(b), all potential collisions between the cart C1 of a mechanism in one row and the manipulated platform of a mechanism in neighbouring row are eliminated by the condition
d1 −
c1y − ( p cos α + rj ) > c. 2
(62)
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Fig. 6. The minimum wy required in order to eliminate the described mechanical interference as a function of μmin for c1y = 0.03, rj = 0.01 m, c = 0.01 m and three different values of p. The curves marked c-c are for the condition (59) while the curves c-p refer to the stricter condition (66).
Fig. 7. (a) Example of an architecture with 16 mechanisms. As long as the TCP of each mechanism remains in its dedicated workspace, mechanical interference may only occur at the borders of this workspace. (b) The lowest value of each performance index equals sin μmin , where μmin = 36.7 degrees.
Similarly, as shown in Fig. 5(c), all potential collisions between the cart C2 of a mechanism in one row and the manipulated platform of a mechanism in neighbouring row are eliminated by the condition
wy − ( p cos α + rj ) − (d2 + p cos α + rj ) > c.
(63)
Eliminating d1 from (62) by utilising (45) and d2 from (63) by employing (46) and rearranging the terms gives
wy − l cos μmin − p cos α − rj − c −
c1y > 0, 2
wy − l cos μmin − p cos α − rj − c − p cos α − rj > 0.
(64)
(65)
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Eliminating l from (64) and (65) by utilising (48), these conditions can be summarised as
k1 (μmin ) wy + k3 (μmin ) > 0, 2 where
(66)
k3 (μmin ) = −p cos(α ) − rj − c − max p cos α + rj ,
c1y 2
(67)
and α is given by (57) and k1 is given by (60). The three upper curves in Fig. 6 shows the minimum wy required for a given μmin in order to fulfil (66). The curves are for three different values of p and constant values of rj = 0.01 m, c1y = 0.03 m and c = 0.01 m. The conclusion is that this arrangement is only feasible for relatively large values of wy . In order to achieve smaller wy , compromises on the parallelogram width (2p) or μmin are required. The disadvantages of a low value of p includes increased link forces and increased sensitivity to manufacturing errors in the link lengths. Fig. 7 exemplifies an architecture that fulfils both (59) and (66). As long as each mechanism remains in its dedicated workspace, the only potential mechanical interference is between manipulated platforms and these are only possible in the border zones of each mechanism’s workspace. The selected parameter values are μmin = 36.7 degrees, wy = 0.5 m, p = 0.04 m, rj = 0.01 m, c1y = 0.03 m and c = 0.01 m. The calculated lengths of the links and the guide-ways are l = 0.51 m and gl = 0.71 m, respectively. 5. Conclusion and future work This paper analysed the feasibility of a mechanical architecture including an array of 2-DOF parallel mechanisms of a specific type. Such architectures could be useful for two dimensional manufacturing operations, such as laser cutting or water jet cutting. Additionally, by including an actuation scheme for the work object in one DOF, it could be utilised for three dimensional manufacturing operations, such as additive manufacturing. Screw-theory-based indices characterising the kinematic performance of the investigated architecture were derived. The kinematic performance was measured by the worst case of the transmission angles in the utilised workspace. For an architecture including a single 2-DOF mechanism or a row of such mechanisms, the kinematic performance is a decreasing function of wy /l, where wy is the width of the utilised workspace in the direction perpendicular to the guide-ways and l is the length of the links. When the ratio wy /l approaches zero, the worst case transmission angle tends to its maximum value of 60 degrees, while the worst case transmission angle tends to zero when the ratio approaches two. It was observed that although it is theoretically possible to use a large number of mechanisms per row, the practicality of this solution deteriorates when the number of mechanisms per row is increased. For architectures including a two-dimensional array of 2-DOF mechanisms, it was shown that practical limitations on component sizes and the potential of mechanical interference put additional restrictions on the achievable kinematic performance and limits how closely the rows can be arranged. 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