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Singularity analysis and detection of 6-UCU parallel manipulator
Robotics and Computer-Integrated Manufacturing 30 (2014) 172–179
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Robotics and Computer-Integrated Manufacturing journal homepage: www.elsevier.com/locate/rcim
Singularity analysis and detection of 6-UCU parallel manipulator Guojun Liu a,n, Zhiyong Qu a, Xiaochu Liu b, Junwei Han a a b
Institute of Electro-Hydraulic Servo Simulation & Test System (IEST), Harbin Institute of Technology, Harbin, China College of Mechanical and Power Engineering, Harbin University of Science and Technology, Harbin, China
art ic l e i nf o
a b s t r a c t
Article history: Received 26 July 2012 Received in revised form 5 September 2013 Accepted 14 September 2013 Available online 18 October 2013
6-UCU kind Gough–Stewart platform (GSP) has been used extensively in practice. The singularity of GSP has been studied by many scholars, but their works mainly focused on finding the methods to divide the cases of singularity or searching the solutions with Jacobian matrices. On the other hand, this paper studies the singularities of 6-UCU parallel manipulator caused by not only the active joints but also passive universal joints. Two types of singularity are derived based on a degree of freedom method by using screw theory. Singularity detection is essential to certify the absence of singularity within a prescribed workspace or a reachable workspace for a practical use of the 6-UCU parallel manipulator. Algorithms are proposed by using evolutionary strategy to detect the singularity in the desired or reachable workspace of the 6-UCU parallel manipulator. Case studies are presented to demonstrate the effectiveness of the proposed singularity detection methods. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Singularity analysis Singularity detection Gough–Stewart platform 6-UCU parallel manipulator
1. Introduction Compared with the spherical joint, the universal joint can bear more tension and rotate in larger angular range, and then 6-UCU kind GSP has been extensively used in practice, such as the tyre test machine designed by Gough in 1955 [1–3] (In the reviewer communication part of Stewart's paper [2], Gough pointed out “In point of fact, the universal joint systems attaching the jacks to the platform are identical to those attaching the jacks to the foundation”.), the first motion simulator patented by Cappel in 1967 [4], the commercial flight simulators [5], the docking test system in China and other over 30 motion simulators produced by our lab, and Moog FCS 5000E motion base [6]. Singularities are critical configurations in which the kinetostatic behavior of a mechanism suddenly changes with respect to a full-cycle condition [7]. The singularity of GSP has been studied by many researchers. Merlet [8] used Grassman geometry to study the singularity of the 3/6 GSP systematically. Gosselin and Angeles [9] studied singularities of GSP on the input–output velocity map. Ma and Angeles [10] studied architecture singularities of GSP. St.-Onge and Gosselin [11] presented a method to determine analytically and represent graphically the singularity loci of the general GSP. Huang et al. [12] analyzed the singularities of 3/6 and 6/6 GSP based on the kinematic status of the machinery and the linear-complex. All the references mentioned above only considered the singularity caused by the Jacobian matrix derived from
n
Corresponding author. Tel.: þ 86 15046108395. E-mail address: [email protected] (G. Liu).
0736-5845/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.rcim.2013.09.010
the input–output velocity map, but they did not considered the singularity caused by the passive joints. The singularity analysis approaches based solely on input–output equations may fail to detect certain singularities in the general closed-loop case as shown in [13–15]. In order to study singularity in a more general framework, Zlatanov et al. [16] proposed a method for finding and classifying all the singularities of arbitrary non-redundant mechanism based on the velocity-equation, but their method may be nonimmediate or ambiguous in some cases due to the deliberate choice of looking at singularities from the standpoint of the device's end user [7]. Zhao and Zhou [17] presented a methodology to analyze the singularity of spatial parallel manipulator with the theory of reciprocal screws, but they did not consider the singularity caused by the actuated joints. Zhao et al. [18] addressed the singularity of spatial parallel manipulator with terminal constraints, but they also did not consider the singularity caused by the actuated joints. In order to identify and interpret the causes of singular events clearly, Conconi and Carricato [7] presented a singularity analysis approach of general parallel kinematic chains on the basis of the physical causes that originate the phenomena. In order to analyze simply and interpret the causes of singular events clearly, the singularity analysis of 6-UCU kind GSP is studied based on a degree of freedom theory by using screw theory in this paper with considering the passive joints and the actuated joints. In practice, it is crucial during the design phase of a robot to determine whether there are singularities within a given workspace or trajectory or not [19]. A fast straight yes–no answer of the singularity detection is important [19]. Su et al. [20] set the objective function as the square of det (J) (J is the Jacobian matrix),
G. Liu et al. / Robotics and Computer-Integrated Manufacturing 30 (2014) 172–179
and used real-coded genetic algorithm to search the minimum value to detect the singularity of GSP for the given workspace, but it is not considered the constraints of leg length limits, and also the six-dimensional workspace is very difficult to obtain. Merlet [21] used interval analysis to detect the singularity of GSP by searching det (J), but the main drawbacks of this method is that some expertise in interval analysis is needed to get an efficient implementation [19]. Blaise et al. [6] proposed an algorithm using Matlab to look at the sign of the determinant of the Jacobian matrix for a wide range of constant-orientation workspaces to detect singularity of GSP. Another method is using symbolic language program for the singularity detection: first obtain the analytic expression of det (J) ¼0, and then fixed three mobility variables to draw the singular loci surface and workspace to determine that whether there is or no singularity in the workspace [22–24], but the singularity is only determined at some special postures and not the whole reachable 6D workspaces. In our best knowledge, all the singularity detection methods of the 6-UCU parallel manipulator are not considered the singularity caused by the passive universal joints. The singularity detection algorithms of the 6-UCU parallel manipulator with considering both the passive and active joints are proposed in this paper. In this paper, the singularities of the 6-UCU parallel manipulator are analyzed in Section 2. The singularity detection algorithms are proposed in Section 3. Some cases are studied in Section 4 to illustrate the use of the singularity detection algorithms proposed in this paper. The conclusions are given in Section 5.
From the causes of physical phenomena, the singularities of 6-UCU parallel manipulator can be divided into two types: leg singularity and actuator singularity. If the connective of the moving platform is changed to less than 6 at an instantaneous pose, this type singularity is defined as leg singularity. If the actuated joints cannot drive the whole 6-UCU parallel manipulator effectively at an instantaneous pose, this type singularity is defined as actuator singularity. In order to analyze the singularities of the 6-UCU parallel manipulator completely, we define an instantaneous reference frame with its origin located at point O1 and the axes u, v, and w parallel to the axes x, y, and z of frame O xyz. Then we express all the joint screws with respect to this instantaneous reference frame. Fig. 2 depicts the equivalent kinematic chain of a UCU leg, where the lower universal joint is replaced by two intersecting i i unit screws, $^ 01 (subscript 0 stands for zero pitch) and $^ 02 , and the upper universal joint is replaced by two intersecting unit screws, i i $^ 05 and $^ 06 . The cylinder joint is replaced by two colinear unit i i screws, $^ 13 (subscript 1 stands for infinite pitch) and $^ 04 . There are six joint screws associated with each leg. The third joint is the only actuated joint, and the remaining five are passive. The actuated joint screw is an infinite pitch screw, and the remaining five are zero pitch screws. Let sj;i be a unit vector along the jth joint axis of the ith leg. Then the six unit joint screws of a leg can be written as " # s1;i ^$ i ¼ 01 ðpi li ni Þ s1;i
2. Singularity analysis of the 6-UCU parallel manipulator The 6-UCU parallel manipulator consists of a moving platform, a fixed base and six identical legs. Each leg consists of two universal joint at both ends of the leg and a cylindrical joint in the middle. The piston of the ith (i¼1–6) leg is attached to the platform with a universal joint at point P i , and the cylinder is attached to the base with a universal joint at point Bi . In practice, the piston and piston rod of the cylinder not only retract (or extend) along the axis of the cylinder actively, but also rotate along the axis passively. As shown in Fig. 1, the world frame O xyz is fixed on the base at origin O, and the body frame O x1y1z1 is fixed on the moving platform at origin O1.
" i $^ 02 ¼
s2;i
#
ðpi li ni Þ s2;i "
i $^ 13 ¼
" i $^ 04 ¼
0
#
s3;i #
s4;i pi s4;i
" i $^ 05 ¼
Fig. 1. Coordinate systems of 6-UCU parallel manipulator.
173
#
s5;i pi s5;i
Fig. 2. Equivalent kinematic structure of a UCU leg.
174
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" i $^ 06 ¼
s6;i
#
pi s6;i
where pi ¼ O1 Pi , bi ¼ OBi , li ni ¼ Bi Pi , ni ¼ s4;i ¼ s3;i , and all these vectors are expressed in frame O xyz.0 is a 3 1 zero column vector. 2.1. Leg singularity We now consider each leg as an open-loop chain and express the instantaneous twist of the moving platform in terms of the joint screws i i i i i i i i i i i i $p ¼ θ_ 1 $^ 01 þ θ_ 2 $^ 02 þ d_ 3 $^ 13 þ θ_ 4 $^ 04 þ θ_ 5 $^ 05 þ θ_ 6 $^ 06
ð1Þ
i i i i i i where θ_ 1 , θ_ 2 , d_ 3 , θ_ 4 , θ_ 5 , θ_ 6 are the joint-velocities of the ith leg. $p is the twist of the moving platform. " # w $p ¼ vp
Fig. 3. Lower universal joint in leg singularity.
where w is the angularity velocity vector of the moving platform in frame O xyz. vp is the translatory velocity vector of the point O1 fixed in moving platform in frame O xyz. The vector subspace spanned by the ith leg screws, Si , is i i i i i i ð2Þ Si ¼ span $^ 01 ; $^ 02 ; $^ 13 ; $^ 04 ; $^ 05 ; $^ 06 ; r i ¼ dim Si According to reciprocal screw theory [25], we can get all the inverse screws $irj (j ¼1,...,ci ) of the ith leg. where $irj is the jth inverse screw of the ith leg, and ci is the order of the inverse screw system of leg i. Define the vector subspace spanned by the ith leg inverse screws to be C i as C i ¼ span ð$ir1 ; …; $irci Þ;
ci ¼ dim ðC i Þ
ð3Þ
Then we have i
r þci ¼ 6
ð4Þ
Define the vector subspace spanned by all the leg inverse screws to be C pb as 1r 6r 6r C pb ¼ span ð$1r 1 ; …; $c1 ; …; $1 ; …; $c6 Þ;
cpb ¼ dim ðC pb Þ
ð5Þ
pb
where c is the dimension of the wrench systems of the moving platform relative to the fixed base. Then the connectivity of moving platform relative to fixed base, r pb , is [25] r pb ¼ 6 cpb
ð6Þ
It is can be deduced from Eqs. (2) to (6) that: for the 6-UCU parallel manipulator, if there is a inverse screw of leg i, then r pb is less than 6 and the 6-UCU parallel manipulator is in a singular pose. Then consider the singularity caused by the passive joints of the 6UCU parallel manipulator. When the first rotational joint axis (s1;i ) of the lower universal joint and the axial direction of the same leg (s3;i ) are collinear, then the 6-UCU parallel manipulator is in a leg i i singularity. As shown in Fig. 3, if the screws $^ 01 and $^ 13 of the ith leg are in collinear, then it can be deduced that (from the reciprocal screw theory [25]) there is a wrench $ir01 constrained the movement
of the moving platform. $ir01 is pass through point P i , and parallel to i $^ 02 . It is can be deduced from Eqs. (2) to (6) that the connectivity of the moving platform relative to the fixed base is less than 6, and then the 6-UCU parallel manipulator is in a singular pose. When the second rotational joint axis (s6;i ) of the upper universal joint and the axial direction of the same leg (s3;i ) are collinear, then the 6-UCU parallel manipulator is in leg singularity.
Fig. 4. Upper universal joint in leg singularity. i i As shown in Fig. 4, if the screws $^ 06 and $^ 13 of the ith leg are in collinear, then it can be deduced that (from the reciprocal screw theory [25]) there is a wrench $ir01 constrained the movement of
the moving platform. $ir01 is pass through point Bi , and parallel to i $^ 05 . It is can be deduced from Eqs. (2) to (6) that the connectivity of the moving platform relative to the fixed base is less than 6, and then the 6-UCU parallel manipulator is in a singular pose.
2.2. Actuator singularity The validity condition of actuated joints in parallel mechanisms is [25] “the selection of actuated joints should ensure that, in a general configuration, the DOF of the mechanism with the F actuated joints blocked will be zero”. When there is no leg singularity, and then if the actuated joint can drive the 6-UCU parallel manipulator effectively, cpb must be 6. As shown in Fig. 5, block the actuated joint of the ith leg, there is only one inverse screw $ir01 of the ith leg. The pitch of $ir01 is zero, and the unit vector along $ir01 is s3;i . $ir01 is expressed as " # s3;i $ir01 ¼ pi s3;i
ð7Þ
G. Liu et al. / Robotics and Computer-Integrated Manufacturing 30 (2014) 172–179
175
operators with the simulated binary crossover (SBX) operator [27] and polynomial mutation operator [28]. The methods to evaluate a solution that violates some constraints are frequently using the penalty functions, but the coefficients of the penalty functions are very difficult to determine with no systematical method [26]. This paper will use the constraint handling method proposed by Deb [29] with no need to determine any coefficients. The flowchart of the evolutionary strategy used in this paper is shown in Fig. 6, and all the parameters are set as in Table1.
3.2. Leg singularity detection algorithms As pointed out above, when the rotational joint axis of the universal joint, which is fixed to the base or the moving platform, and the axial direction of the same leg are collinear, then the 6-UCU parallel manipulator is in a leg singularity.
Fig. 5. Screws of a UCU leg (when the actuated joint is blocked).
Then the vector subspace spanned by all the leg inverse screws, C pb , is 2r 3r 4r 5r 6r C pb ¼ span ð$1r 01 ; $01 ; $01 ; $01 ; $01 ; $01 Þ
Define the matrix J1 as J1 ¼ $1r ; $2r ; $3r ; $4r ; $5r ; $6r " 01 01 01 01 01 01 s3;2 s3;3 s3;1 ¼ p1 s3;1 p2 s3;2 p3 s3;3 " ¼
ð8Þ
s3;4 p4 s3;4
s3;5 p5 s3;5
s3;6 p6 s3;6
n1
n2
n3
n4
n5
n6
p 1 n1
p2 n2
p3 n3
p 4 n4
p 5 n5
p6 n6
#
#
ð9Þ
We can deduce from Eq. (9) that J1 is the transpose of the Jacobian matrix J, and then the rank of J is equivalent to cpb . Then if det (J) ¼0, the actuators cannot drive the 6-UCU parallel manipulator effectively, and it is in a singular pose.
3. Singularity detection algorithms of 6-UCU parallel manipulator In practice, it is crucial during the design phase of a robot to determine whether there are singularities within a given workspace or trajectory or not [19]. A fast straight yes–no answer of the singularity detection is important [19]. Many authors have studied the singularity detection of the GSP with only considering the actuator singularity mentioned in Section 1, but there is no one who considered the leg singularity. The singularity detection algorithms of the 6-UCU parallel manipulator with considering of both the passive and active joints are proposed in this section. Fig. 6. Flowchart of evolutionary strategy.
3.1. Evolutionary strategy Evolutionary algorithms, such as genetic algorithms, evolution strategies, particle swarm optimization algorithms are widely used to optimize [26]. Because evolutionary strategy is generally a real number encoding algorithm and with elitism preserved, then this paper uses the (μþλ) evolutionary strategy in the singularity detection procedures. For the given workspace, the search variables are limited by the lower and upper bounds of the Cartesian coordinate variables, then it is not a constrained problem. For the reachable workspace, the search variables are limited by the leg length ranges, then it is a constrained problem. To deal with all the two cases described above, we apply the crossover and mutation
Table1 Parameters of evolutionary strategy. Parameters name
Parameter values
Population size N Number of iteration μ λ Tournament selection size Distribution index for SBX Distribution index for polynomial mutation
50 2000 50 50 2 20 20
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G. Liu et al. / Robotics and Computer-Integrated Manufacturing 30 (2014) 172–179
For the ith leg, define angles ϕi1 and ϕi2 as ϕi1
¼ cos
1
ϕi2
ðs1;i U ni Þ;
¼ cos
1
ðs6;i U ni Þ
ð10Þ
Define ϕ1 min , ϕ1 max , ϕ2 min and ϕ2 max as ϕ1 min ¼ min ð ϕ11
ϕ21
ϕ31
ϕ41
ϕ51
ϕ61 Þ
ð ϕ11
ϕ21
ϕ31
ϕ41
ϕ51
ϕ61 Þ
ϕ2 min ¼ min ð ϕ12
ϕ22
ϕ32
ϕ42
ϕ52
ϕ62 Þ
ϕ2 max ¼ maxð ϕ12
ϕ22
ϕ32
ϕ42
ϕ52
ϕ62 Þ
ϕ1 max ¼ max
ð11Þ
where, min and max stand for minimize and maximize, respectively. Define ϕmin as ϕmin ¼ min ð ϕ1 min
ð180 3 ϕ1 max Þ
ϕ2 min
ð180 3 ϕ2 max Þ Þ ð12Þ
Leg singularity detection algorithms are propose as follow.
3.2.1. Leg singularity detection algorithm for the given workspace or trajectory Step 1. Create a subprogram to calculate the value of ϕmin using Eq. (12) at a pose. Step 2. Set the objective to minimize f 1 ¼ ϕmin . Step 3. Use the evolutionary strategy algorithms (as shown in Fig. 6) to search f 1 min , where f 1 min is the minimum value of f 1 in the given workspace or trajectory. Step 4. If f 1 min is equivalent to zero, there is a leg singularity in the given workspace or trajectory. If f 1 min is not equivalent to zero, there is no leg singularity in the given workspace or trajectory.
3.2.2. Leg singularity detection algorithm for the reachable workspace Step 1. Create a subprogram to calculate the value of ϕmin using Eq. (12) at a pose. Step 2. Set the objective to minimize f 1 ¼ ϕmin . Step 3. Use the constraint handling method proposed by Deb [29] to deal with the link length limits and the evolutionary strategy algorithms (as shown in Fig. 6) to search f 1 min , where f 1 min is the minimum value of f 1 in the reachable workspace. Step 4. If f 1 min is equivalent to zero, there is a leg singularity in the reachable workspace. If f 1 min is not equivalent to zero, there is no leg singularity in the reachable workspace.
Step 6. Use the evolutionary strategy algorithms (as shown in Fig. 6) to search f 2 min , where f 2 min is the minimum value of f 2 in the given workspace or trajectory. Step 7. Use the principle of continuity to determine the actuator singularity, namely, if 0 is included in the ranges of det (J), then there is actuator singularity in the given workspace or trajectory; if no 0 is included in the ranges of det (J), then there is no actuator singularity in the given workspace or trajectory.
3.3.2. Actuator singularity detection algorithm for the reachable workspace Step 1. Create a subprogram to calculate det (J) at a pose. Step 2. Set the objective to minimize f1 ¼det (J). Step 3. Use the constraint handling method proposed by Deb [29] to deal with the link length limits and the evolutionary strategy algorithms (as shown in Fig. 6) to search f 1 min , where f 1 min is the minimum value of f 1 in the reachable workspace. Step 4. Create another subprogram to calculate det (J) at a pose. Step 5. Set the objective to minimize f2 ¼ det (J). Step 6. Use the constraint handling method proposed by Deb [29] to deal with the link length limits and the evolutionary strategy algorithms (as shown in Fig. 6) to search f 2 min , where f 2 min is the minimum value of f 2 in the reachable workspace. Step 7. Use the principle of continuity to determine the actuator singularity, namely, if 0 is included in the value ranges of det (J), then there is actuator singularity in the reachable workspace; if no 0 is included in the value ranges of det (J), then there is no actuator singularity in the reachable workspace. 4. Case study In order to illustrate the singularity detection methods proposed in this paper, some cases are studied in this section. 4.1. Case study of leg singularity detection algorithms An electric driven 6-UCU parallel manipulator shown in Fig. 7 is studied in this section, whose upper joints and lower joints are both universal joints with large-scale rotation capacity.
3.3. Actuator singularity detection algorithms If there is no leg singularity in the given workspace (or trajectory), and then the actuator singularity detection algorithms are propose as follow.
3.3.1. Actuator singularity detection algorithm for the given workspace or trajectory Step 1. Create a subprogram to calculate det (J) at a pose. Step 2. Set the objective to minimize f1 ¼ det (J). Step 3. Use the evolutionary strategy algorithms (as shown in Fig. 6) to search f 1 min , where f 1 min is the minimum value of f 1 in the given workspace or trajectory. Step 4. Create another subprogram to calculate det (J) at a pose. Step 5. Set the objective to minimize f2 ¼ det (J).
Fig. 7. An electric driven 6-UCU parallel manipulator produced by our lab.
G. Liu et al. / Robotics and Computer-Integrated Manufacturing 30 (2014) 172–179
The parameters of the electric driven GSP are shown as 2 3 0:6748 0:8687 2:2336 6 0:4150 1:0187 2:2336 7 6 7 6 7 6 1:0897 0:1500 2:2336 7 T 6 7; B ¼6 0:1500 2:2336 7 6 1:0897 7 6 7 4 0:4150 1:0187 2:2336 5 2
0:6748 0:8139
0:8687 2:2336 3 0:1000 0:6677 0:7548 0:6677 7 7 7 0:6548 0:6677 7 7; 0:6548 0:6677 7 7 7 0:7548 0:6677 5
6 0:3203 6 6 6 0:4935 T P ¼6 6 0:4935 6 6 4 0:3203 0:8139 lmin ¼ 1:4453;
0:1000
0:6677
lmax ¼ 2:0690
where the ith (i¼1,…,6) column of B is the position vector of point Bi in frame O xyz. The ith column of P is the position vector of point P i in frame O1 x1y1z1. lmin and lmax are the minimum and maximum lengths of the legs. At home position, frames O xyz and O1 x1y1z1 are coincident. All the units of lengths and angles are expressed in meter and degree, respectively. T is the matrix transpose. In the process of design, the mechanical interferences of the joints and interferences between links are taken full account to ensure that the interferences do not occur at all in the course of the movement, and then they are not considered in singularity detection. As shown in Fig. 7, all the axes of the rotational joints fixed to the base of the electric driven 6-UCU parallel manipulator are in the plane constructed by the attachment points of the lower universal joints, and are perpendicular to the corresponding shorter edges of hexagon constructed by the attachment points. All the axes of the rotational joints fixed to the moving platform of the electric driven 6-UCU parallel manipulator are perpendicular to the corresponding shorter edges of hexagon constructed by the attachment points of the upper universal joints, and are form 451 angles from the plane constructed by the attachment points of the upper universal joints. 4.1.1. Leg singularity detection case study for the given workspace or trajectory The workspace are given as x A ½ 0:500 m; 0:500 m, y A ½ 0:500 m; 0:500 m, z A ½ 0:350 m; 0:350 m, ϕ3 ¼ ϕ2 ¼ ϕ1 ¼ 0 3 , where x, y and z are the position variables of point O1 along X-, Y-, and Z-axes of frame O xyz. ϕ1 , ϕ2 , and ϕ3 are the roll, pitch and yaw angles of the moving platform with relative to the frame O xyz. For the electric driven 6-UCU parallel manipulator, use the algorithms proposed in Section 3.2.1 to test the leg singularity in the given workspace. The presented algorithm is
177
programmed using matlab2011a in a Windows XP environment. The CPU of the used computer is an Intel(R) Xeon(TM) with 2.66 GHz. The computation time is 218.7 s. After 2000 iterations, the search result of ϕmin is 20.41, and the search results are shown in Fig. 8, where deg stands for degree. Because the minimum value of ϕmin is not zero, then there is no leg singularity in the given workspace of the electric 6-UCU parallel manipulator. 4.1.2. Leg singularity detection case study for the reachable workspace For the electric driven 6-UCU parallel manipulator, use the algorithms proposed in Section 3.2.2 to test the leg singularity in the reachable workspace. The computation time is 215.2 s. After 2000 iterations, the search result of ϕmin is 11.01, and the search results are also shown in Fig. 8. Because the minimum value of ϕmin is not zero, then there is no leg singularity in the reachable workspace of the electric 6-UCU parallel manipulator. 4.2. Case study of actuator singularity detection algorithms In order to illustrate the actuator singularity detection methods proposed in Section 3.3, some practical 6-UCU parallel manipulators are studied in this section. 4.2.1. Actuator singularity detection case study for the given workspace or trajectory Use the algorithms proposed in Section 3.3.1 to test the actuator singularity in the given workspace of the electric driven 6-UCU parallel manipulator as shown in Section 4.1. The computation time for search of minimum det (J) is 139.1 s. After 2000
Fig. 9. A hydraulic driven 6-UCU parallel manipulator in a singular pose. 3
25
det(J)(case4.2.1) −det(J)(case4.2.1) det(J)(case4.2.2.1) −det(J)(case4.2.2.1) det(J)(case4.2.2.2) −det(J)(case4.2.2.2)
2
case4.1.1 case4.1.2
Objective function value
Minimum angle[deg]
30
20 15
1 0 −1 −2 −3
10
0
500
1000 1500 Iteration number
2000
Fig. 8. Minimum angles of the electric driven 6-UCU parallel manipulator corresponding with iteration.
−4
0
500
1000
1500
2000
Iteration number
Fig. 10. Minimum det (J) and det (J) of the 6-UCU parallel manipulators corresponding with iteration.
178
G. Liu et al. / Robotics and Computer-Integrated Manufacturing 30 (2014) 172–179
iterations, the search result of minimum det (J) is 2.8, and the search results are shown in Fig. 10. The computation time for search of minimum det (J) is 149.4 s. After 2000 iterations, the search result of minimum det (J) is 0.9, and the search results are also shown in Fig. 10. The minimum and maximum values of det (J) are 2.8 and 0.9 respectively. Use the principle of continuity to determine the actuator singularity, namely, 0 is not included in the value ranges of det (J), then there is no actuator singularity of the electric 6-UCU parallel manipulator in the given workspace. It is consist with the singularity detection result of Ma et al. [24] by using an algebraic method. 4.2.2. Actuator singularity detection case study for the reachable workspace In order to illustrate the actuator singularity detection methods proposed in Section 3.3.2, two practical 6-UCU parallel manipulators, a singular hydraulic and a nonsingular electric driven 6-UCU parallel manipulators are studied in this section.
4.2.2.2. A nonsingular electric driven 6-UCU parallel manipulator. The electric driven 6-UCU parallel manipulator is a 5000E motion base (from Moog FCS) [6], whose upper joints and lower joints are both universal joints [6]. The parameters of this 6-UCU parallel manipulator are shown as follows [6]: 2 3 1:0668 0:127 1:1769 6 0:423418 0:987298 1:1769 7 6 7 6 7 6 7 0:643382 0:860298 1:1769 7 BT ¼ 6 6 0:643382 0:860298 1:1769 7; 6 7 6 7 4 0:423418 0:987298 1:1769 5 2
1:0668 0:516382
6 0:296418 6 6 6 0:8128 T P ¼6 6 0:8128 6 6 4 0:296418 0:516382
4.2.2.1. A singular hydraulic driven 6-UCU parallel manipulator. The hydraulic driven 6-UCU parallel manipulator is shown is Fig. 9, whose upper joints and lower joints are both universal joints with large-scale rotation capacity. As Huang et al. [30] pointed out that “We find that, for this class of 6/6-Gough–Stewart manipulators, there are also some special singularity cases where six lines associated with the six extensible links of the manipulator can intersect one common line and the unwanted motion of the manipulator is a pure rotational motion.”, and then the hydraulic driven 6-UCU parallel manipulator is in a singular pose as shown in Fig. 9. The parameters of the hydraulic driven 6-UCU parallel manipulator are shown as follows: 2
1:1787
6 0:3945 6 6 6 0:7842 T B ¼6 6 0:7842 6 6 4 0:3945 2
1:1787 0:3849
6 0:1598 6 6 6 0:5447 PT ¼ 6 6 0:5447 6 6 4 0:1598
0:2250 1:1333 0:9083 0:9083 1:1333 0:2250 0:4067 0:5367 0:1300 0:1300 0:5367
1:9226
3
1:9226 7 7 7 1:9226 7 7; 1:9226 7 7 7 1:9226 5 1:9226 3 0:2838 0:2838 7 7 7 0:2838 7 7; 0:2838 7 7 7 0:2838 5
0:3849 0:4067 0:2838 lmin ¼ 1:4400; lmax ¼ 2:2200 where the ith (i¼1,…,6) column of B is the position vector of point Bi in frame O xyz. The ith column of P is the position vector of point P i in frame O1 x1y1z1. lmin and lmax are the minimum and maximum lengths of the legs. At home position, frames O xyz and O1 x1y1z1 are coincident. All the units of lengths and angles are expressed in meter and degree, respectively. T is the matrix transpose. For the hydraulic driven 6-UCU parallel manipulator, use the algorithms proposed in Section 3.3.2 to search the minimum of det (J) and det (J) in the reachable workspace, the results are also shown in Fig. 10. The computation time for search of minimum det (J) is 173.1 s. The computation time for search of minimum det (J) is 149.8 s. The minimum values of det (J) and det (J) are 0.3 and 0.6 respectively, namely, the minimum and maximum values of det (J) are 0.3 and 0.6 respectively. Because the extreme values are in opposite signs, then according to the principle of continuity, there is a pose with det (J) ¼0, namely, there is the actuator singularity in the reachable workspace. This result is consistent with the fact as shown in Fig. 9.
lmin ¼ 1:143;
0:127
1:1769 3 0 0:767334 07 7 7 0:127 07 7; 0:127 07 7 7 0:767334 0 5 0:640334
0:640334
0
lmax ¼ 1:651
where the ith (i¼1,…,6) column of B is the position vector of point Bi in frame O xyz. The ith column of P is the position vector of point P i in frame O1 x1y1z1. lmin and lmax are the minimum and maximum lengths of the legs. At home position, frames O xyz and O1 x1y1z1 are coincident. All the units of lengths and angles are expressed in meter and degree, respectively. T is the matrix transpose. For the electric driven 5000E motion base, use the algorithms proposed in Section 3.3.2 to search the minimum of det (J) and det (J) in the reachable workspace, the results are also shown in Fig. 10. The computation time for search of minimum det (J) is 143.4 s. The computation time for search of minimum det (J) is 163.1 s. The minimum values of det (J) and det (J) are -3.5 and 2.0 respectively, namely, the minimum and maximum values of det (J) are 3.5 and 2.0 respectively. Because the extreme values are in the same sign, then according to the principle of continuity, there is no pose with det (J)¼ 0, namely, there is no actuator singularity in the reachable workspace. It is consist with the singularity detection result of Blaise et al. [6] by using a numerical method.
5. Conclusion The 6-UCU kind GSP has been widely used in practice. Based on a degree of freedom theory by using screw theory, the singularities of the 6-UCU parallel manipulator caused by the legs and actuators are analyzed in this paper, and two singular types are defined as leg singularity and actuator singularity. Two cases of leg singularities of the 6-UCU parallel manipulator are found in this paper. The singularity test problems of 6-UCU parallel manipulator are classified into different types from the application viewpoints. In order to detect all the singularities of the 6-UCU parallel manipulator, singularity detection procedures are proposed correspondingly, which are using evolutionary strategy algorithm to search the extreme values of the different objective functions. The effectivenesses of the proposed singularity detection procedures are confirmed by studying cases. As may be seen from the results, the computation time is in general quite acceptable.
Acknowledgment This paper is financially supported by the National Natural Science Foundation of China under Grant no. 51105096 and the 921 Manned Space Project of China under Grant no. HgdJG00401D04.
G. Liu et al. / Robotics and Computer-Integrated Manufacturing 30 (2014) 172–179
References [1] Gough VE, Whitehall SG. Universal tyre testing machine. In: Eley G, editor. Proceedings of the 9th international automobile technical congress, discussion pp. 250ff. Federation Internationale des Societes d'Ingenieurs des Techniques de l'Automobile (FISITA), IMechE 1, London, UK; 1962: p. 117–37. [2] Stewart D. A platform with six degrees of freedom. In: Proceedings of the Institution of Mechanical Engineers, 1965–1966; vol. 180, Pt 1 no. 15: p. 371– 86 (including communications and author's reply). [3] Fichter EF, Kerr DR, Rees-Jones J. The Gough–Stewart platform parallel manipulator: a retrospective appreciation. Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical Engineering Science 2009;223(1):243–81. [4] Cappel KL. Motion simulator. US Patent 3295224; 1967. [5] Ma O. Mechanical analysis of parallel manipulators with simulation, design and control applications (PhD dissertation). McGill University, Montreal, Canada; 1991: p. 122–3. [6] Blaise J, Bonev I, Monsarrat B, Briot S, Lambert JM, Perron C. Kinematic characterisation of hexapods for industry. Industrial Robot: An International Journal 2010;37(1):79–88. [7] Conconi M, Carricato M. A new assessment of singularities of parallel kinematic chains. IEEE Transactions on Robotics 2009;25(4):757–70. [8] Merlet J-P. Singular configurations of parallel manipulators and Grassmann geometry. International Journal of Robotics Research 1989;8(5):45–56. [9] Gosselin C, Angeles J. Singularity analysis of closed-loop kinematic chains. IEEE Transactions on Robotics and Automation 1990;6(3):281–90. [10] Ma O, Angeles J. Architecture singularity of platform manipulators. In: Proceedings of the IEEE international conference on robotics and automation, Sacramento, CA; 1991: p. 1542–7. [11] St.-Onge BM, Gosselin CM. Singularity analysis and representation of the general Gough–Stewart platform. International Journal of Robotics Research 2000;19(3):271–88. [12] Huang Z, Li Q, Ding H. Theory of parallel mechanisms. Netherlands: Springer; 2012; 217–88. [13] Zlatanov D, Fenton RG, Benhabib B. Singularity analysis of mechanisms and robots via a velocity-equation model of the instantaneous kinematics. In: Proceedings of the 1994 IEEE international conference on robotics and automation, San Diego, California, USA; 1994: p. 986–91. [14] Zlatanov D, Fenton RG, Benhabib B. Classification and interpretation of the singularities of redundant mechanisms. In: Proceedings of the 1998 ASME design engineering technical conferences, Atlanta, Georgia, USA; 1998: p. 1–11.
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[15] Zlatanov D, Bonev IA, Gosselin CM. Constraint singularities of parallel mechanisms. In: Proceedings of the 2002 IEEE international conference on robotics and automation, Washington, DC., USA; 2002: p. 496–502. [16] Zlatanov D, Fenton RG, Benhabib B. Identification and classification of the singular configurations of mechanisms. Mechanism and Machine Theory 1998;33(6):743–60. [17] Zhao JS, Zhou K. A novel methodology to study the singularity of spatial parallel mechanisms. International Journal of Advanced Manufacturing Technology 2004;23(9–10):750–4. [18] Zhao JS, Feng ZJ, Zhou K, Dong JX. Analysis of the singularity of spatial parallel manipulator with terminal constraints. Mechanism and Machine Theory 2005;40(3):275–84. [19] Merlet J-P. Parallel robots. 2nd Ed.. Netherlands: Springer; 2006; 206–7. [20] Su YX, Duan BY, Peng B, Nan RD. Singularity analysis of fine-tuning Stewart platform for large radio telescope using genetic algorithm. Mechatronics 2003;13(5):413–25. [21] Merlet J-P. A formal-numerical approach for robust in-workspace singularity detection. IEEE Transactions on Robotics 2007;23(3):393–402. [22] Li HD, Gosselin CM, Richard MJ. Determination of the maximal singularity-free zones in the six-dimensional workspace of the general Gough–Stewart platform. Mechanism and Machine Theory 2007;42(4):497–511. [23] Jiang QM, Gosselin CM. Singularity equations of Gough–Stewart platforms using a minimal set of geometric parameter. Journal of Mechanical Design 2008;130(11):1123031–8. [24] Ma J, Huang Q, Xiong H, Han J. Analysis and application of the singularity locus of the Stewart platform. Chinese Journal of Mechanical Engineering 2011;24 (1):133–40. [25] Kong X, Gosselin C. Type synthesis of parallel mechanisms. Berlin, Heidelberg: Springer; 2007; 19–42. [26] Yu XJ, Gen M. Introduction to evolutionary algorithms. London: SpringerVerlag; 2010; 193–258. [27] Deb K, Agrawal RB. Simulated binary crossover for continuous search space. Complex Systems 1995;9:115–48. [28] Deb K, Goyal M. A combined genetic adaptive search (GeneAS) for engineering design. Computer Science and Informatics 1996;26(4):30–45. [29] Deb K. An efficient constraint handling method for genetic algorithms. Computer Methods in Applied Mechanics and Engineering 2000;186:311–38. [30] Huang Z, Cao Y. Property identification of the singularity loci of a class of Gough–Stewart manipulators. International Journal of Robotics Research 2005;24(8):675–85.
Contents lists available at ScienceDirect
Robotics and Computer-Integrated Manufacturing journal homepage: www.elsevier.com/locate/rcim
Singularity analysis and detection of 6-UCU parallel manipulator Guojun Liu a,n, Zhiyong Qu a, Xiaochu Liu b, Junwei Han a a b
Institute of Electro-Hydraulic Servo Simulation & Test System (IEST), Harbin Institute of Technology, Harbin, China College of Mechanical and Power Engineering, Harbin University of Science and Technology, Harbin, China
art ic l e i nf o
a b s t r a c t
Article history: Received 26 July 2012 Received in revised form 5 September 2013 Accepted 14 September 2013 Available online 18 October 2013
6-UCU kind Gough–Stewart platform (GSP) has been used extensively in practice. The singularity of GSP has been studied by many scholars, but their works mainly focused on finding the methods to divide the cases of singularity or searching the solutions with Jacobian matrices. On the other hand, this paper studies the singularities of 6-UCU parallel manipulator caused by not only the active joints but also passive universal joints. Two types of singularity are derived based on a degree of freedom method by using screw theory. Singularity detection is essential to certify the absence of singularity within a prescribed workspace or a reachable workspace for a practical use of the 6-UCU parallel manipulator. Algorithms are proposed by using evolutionary strategy to detect the singularity in the desired or reachable workspace of the 6-UCU parallel manipulator. Case studies are presented to demonstrate the effectiveness of the proposed singularity detection methods. & 2013 Elsevier Ltd. All rights reserved.
Keywords: Singularity analysis Singularity detection Gough–Stewart platform 6-UCU parallel manipulator
1. Introduction Compared with the spherical joint, the universal joint can bear more tension and rotate in larger angular range, and then 6-UCU kind GSP has been extensively used in practice, such as the tyre test machine designed by Gough in 1955 [1–3] (In the reviewer communication part of Stewart's paper [2], Gough pointed out “In point of fact, the universal joint systems attaching the jacks to the platform are identical to those attaching the jacks to the foundation”.), the first motion simulator patented by Cappel in 1967 [4], the commercial flight simulators [5], the docking test system in China and other over 30 motion simulators produced by our lab, and Moog FCS 5000E motion base [6]. Singularities are critical configurations in which the kinetostatic behavior of a mechanism suddenly changes with respect to a full-cycle condition [7]. The singularity of GSP has been studied by many researchers. Merlet [8] used Grassman geometry to study the singularity of the 3/6 GSP systematically. Gosselin and Angeles [9] studied singularities of GSP on the input–output velocity map. Ma and Angeles [10] studied architecture singularities of GSP. St.-Onge and Gosselin [11] presented a method to determine analytically and represent graphically the singularity loci of the general GSP. Huang et al. [12] analyzed the singularities of 3/6 and 6/6 GSP based on the kinematic status of the machinery and the linear-complex. All the references mentioned above only considered the singularity caused by the Jacobian matrix derived from
n
Corresponding author. Tel.: þ 86 15046108395. E-mail address: [email protected] (G. Liu).
0736-5845/$ - see front matter & 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.rcim.2013.09.010
the input–output velocity map, but they did not considered the singularity caused by the passive joints. The singularity analysis approaches based solely on input–output equations may fail to detect certain singularities in the general closed-loop case as shown in [13–15]. In order to study singularity in a more general framework, Zlatanov et al. [16] proposed a method for finding and classifying all the singularities of arbitrary non-redundant mechanism based on the velocity-equation, but their method may be nonimmediate or ambiguous in some cases due to the deliberate choice of looking at singularities from the standpoint of the device's end user [7]. Zhao and Zhou [17] presented a methodology to analyze the singularity of spatial parallel manipulator with the theory of reciprocal screws, but they did not consider the singularity caused by the actuated joints. Zhao et al. [18] addressed the singularity of spatial parallel manipulator with terminal constraints, but they also did not consider the singularity caused by the actuated joints. In order to identify and interpret the causes of singular events clearly, Conconi and Carricato [7] presented a singularity analysis approach of general parallel kinematic chains on the basis of the physical causes that originate the phenomena. In order to analyze simply and interpret the causes of singular events clearly, the singularity analysis of 6-UCU kind GSP is studied based on a degree of freedom theory by using screw theory in this paper with considering the passive joints and the actuated joints. In practice, it is crucial during the design phase of a robot to determine whether there are singularities within a given workspace or trajectory or not [19]. A fast straight yes–no answer of the singularity detection is important [19]. Su et al. [20] set the objective function as the square of det (J) (J is the Jacobian matrix),
G. Liu et al. / Robotics and Computer-Integrated Manufacturing 30 (2014) 172–179
and used real-coded genetic algorithm to search the minimum value to detect the singularity of GSP for the given workspace, but it is not considered the constraints of leg length limits, and also the six-dimensional workspace is very difficult to obtain. Merlet [21] used interval analysis to detect the singularity of GSP by searching det (J), but the main drawbacks of this method is that some expertise in interval analysis is needed to get an efficient implementation [19]. Blaise et al. [6] proposed an algorithm using Matlab to look at the sign of the determinant of the Jacobian matrix for a wide range of constant-orientation workspaces to detect singularity of GSP. Another method is using symbolic language program for the singularity detection: first obtain the analytic expression of det (J) ¼0, and then fixed three mobility variables to draw the singular loci surface and workspace to determine that whether there is or no singularity in the workspace [22–24], but the singularity is only determined at some special postures and not the whole reachable 6D workspaces. In our best knowledge, all the singularity detection methods of the 6-UCU parallel manipulator are not considered the singularity caused by the passive universal joints. The singularity detection algorithms of the 6-UCU parallel manipulator with considering both the passive and active joints are proposed in this paper. In this paper, the singularities of the 6-UCU parallel manipulator are analyzed in Section 2. The singularity detection algorithms are proposed in Section 3. Some cases are studied in Section 4 to illustrate the use of the singularity detection algorithms proposed in this paper. The conclusions are given in Section 5.
From the causes of physical phenomena, the singularities of 6-UCU parallel manipulator can be divided into two types: leg singularity and actuator singularity. If the connective of the moving platform is changed to less than 6 at an instantaneous pose, this type singularity is defined as leg singularity. If the actuated joints cannot drive the whole 6-UCU parallel manipulator effectively at an instantaneous pose, this type singularity is defined as actuator singularity. In order to analyze the singularities of the 6-UCU parallel manipulator completely, we define an instantaneous reference frame with its origin located at point O1 and the axes u, v, and w parallel to the axes x, y, and z of frame O xyz. Then we express all the joint screws with respect to this instantaneous reference frame. Fig. 2 depicts the equivalent kinematic chain of a UCU leg, where the lower universal joint is replaced by two intersecting i i unit screws, $^ 01 (subscript 0 stands for zero pitch) and $^ 02 , and the upper universal joint is replaced by two intersecting unit screws, i i $^ 05 and $^ 06 . The cylinder joint is replaced by two colinear unit i i screws, $^ 13 (subscript 1 stands for infinite pitch) and $^ 04 . There are six joint screws associated with each leg. The third joint is the only actuated joint, and the remaining five are passive. The actuated joint screw is an infinite pitch screw, and the remaining five are zero pitch screws. Let sj;i be a unit vector along the jth joint axis of the ith leg. Then the six unit joint screws of a leg can be written as " # s1;i ^$ i ¼ 01 ðpi li ni Þ s1;i
2. Singularity analysis of the 6-UCU parallel manipulator The 6-UCU parallel manipulator consists of a moving platform, a fixed base and six identical legs. Each leg consists of two universal joint at both ends of the leg and a cylindrical joint in the middle. The piston of the ith (i¼1–6) leg is attached to the platform with a universal joint at point P i , and the cylinder is attached to the base with a universal joint at point Bi . In practice, the piston and piston rod of the cylinder not only retract (or extend) along the axis of the cylinder actively, but also rotate along the axis passively. As shown in Fig. 1, the world frame O xyz is fixed on the base at origin O, and the body frame O x1y1z1 is fixed on the moving platform at origin O1.
" i $^ 02 ¼
s2;i
#
ðpi li ni Þ s2;i "
i $^ 13 ¼
" i $^ 04 ¼
0
#
s3;i #
s4;i pi s4;i
" i $^ 05 ¼
Fig. 1. Coordinate systems of 6-UCU parallel manipulator.
173
#
s5;i pi s5;i
Fig. 2. Equivalent kinematic structure of a UCU leg.
174
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" i $^ 06 ¼
s6;i
#
pi s6;i
where pi ¼ O1 Pi , bi ¼ OBi , li ni ¼ Bi Pi , ni ¼ s4;i ¼ s3;i , and all these vectors are expressed in frame O xyz.0 is a 3 1 zero column vector. 2.1. Leg singularity We now consider each leg as an open-loop chain and express the instantaneous twist of the moving platform in terms of the joint screws i i i i i i i i i i i i $p ¼ θ_ 1 $^ 01 þ θ_ 2 $^ 02 þ d_ 3 $^ 13 þ θ_ 4 $^ 04 þ θ_ 5 $^ 05 þ θ_ 6 $^ 06
ð1Þ
i i i i i i where θ_ 1 , θ_ 2 , d_ 3 , θ_ 4 , θ_ 5 , θ_ 6 are the joint-velocities of the ith leg. $p is the twist of the moving platform. " # w $p ¼ vp
Fig. 3. Lower universal joint in leg singularity.
where w is the angularity velocity vector of the moving platform in frame O xyz. vp is the translatory velocity vector of the point O1 fixed in moving platform in frame O xyz. The vector subspace spanned by the ith leg screws, Si , is i i i i i i ð2Þ Si ¼ span $^ 01 ; $^ 02 ; $^ 13 ; $^ 04 ; $^ 05 ; $^ 06 ; r i ¼ dim Si According to reciprocal screw theory [25], we can get all the inverse screws $irj (j ¼1,...,ci ) of the ith leg. where $irj is the jth inverse screw of the ith leg, and ci is the order of the inverse screw system of leg i. Define the vector subspace spanned by the ith leg inverse screws to be C i as C i ¼ span ð$ir1 ; …; $irci Þ;
ci ¼ dim ðC i Þ
ð3Þ
Then we have i
r þci ¼ 6
ð4Þ
Define the vector subspace spanned by all the leg inverse screws to be C pb as 1r 6r 6r C pb ¼ span ð$1r 1 ; …; $c1 ; …; $1 ; …; $c6 Þ;
cpb ¼ dim ðC pb Þ
ð5Þ
pb
where c is the dimension of the wrench systems of the moving platform relative to the fixed base. Then the connectivity of moving platform relative to fixed base, r pb , is [25] r pb ¼ 6 cpb
ð6Þ
It is can be deduced from Eqs. (2) to (6) that: for the 6-UCU parallel manipulator, if there is a inverse screw of leg i, then r pb is less than 6 and the 6-UCU parallel manipulator is in a singular pose. Then consider the singularity caused by the passive joints of the 6UCU parallel manipulator. When the first rotational joint axis (s1;i ) of the lower universal joint and the axial direction of the same leg (s3;i ) are collinear, then the 6-UCU parallel manipulator is in a leg i i singularity. As shown in Fig. 3, if the screws $^ 01 and $^ 13 of the ith leg are in collinear, then it can be deduced that (from the reciprocal screw theory [25]) there is a wrench $ir01 constrained the movement
of the moving platform. $ir01 is pass through point P i , and parallel to i $^ 02 . It is can be deduced from Eqs. (2) to (6) that the connectivity of the moving platform relative to the fixed base is less than 6, and then the 6-UCU parallel manipulator is in a singular pose. When the second rotational joint axis (s6;i ) of the upper universal joint and the axial direction of the same leg (s3;i ) are collinear, then the 6-UCU parallel manipulator is in leg singularity.
Fig. 4. Upper universal joint in leg singularity. i i As shown in Fig. 4, if the screws $^ 06 and $^ 13 of the ith leg are in collinear, then it can be deduced that (from the reciprocal screw theory [25]) there is a wrench $ir01 constrained the movement of
the moving platform. $ir01 is pass through point Bi , and parallel to i $^ 05 . It is can be deduced from Eqs. (2) to (6) that the connectivity of the moving platform relative to the fixed base is less than 6, and then the 6-UCU parallel manipulator is in a singular pose.
2.2. Actuator singularity The validity condition of actuated joints in parallel mechanisms is [25] “the selection of actuated joints should ensure that, in a general configuration, the DOF of the mechanism with the F actuated joints blocked will be zero”. When there is no leg singularity, and then if the actuated joint can drive the 6-UCU parallel manipulator effectively, cpb must be 6. As shown in Fig. 5, block the actuated joint of the ith leg, there is only one inverse screw $ir01 of the ith leg. The pitch of $ir01 is zero, and the unit vector along $ir01 is s3;i . $ir01 is expressed as " # s3;i $ir01 ¼ pi s3;i
ð7Þ
G. Liu et al. / Robotics and Computer-Integrated Manufacturing 30 (2014) 172–179
175
operators with the simulated binary crossover (SBX) operator [27] and polynomial mutation operator [28]. The methods to evaluate a solution that violates some constraints are frequently using the penalty functions, but the coefficients of the penalty functions are very difficult to determine with no systematical method [26]. This paper will use the constraint handling method proposed by Deb [29] with no need to determine any coefficients. The flowchart of the evolutionary strategy used in this paper is shown in Fig. 6, and all the parameters are set as in Table1.
3.2. Leg singularity detection algorithms As pointed out above, when the rotational joint axis of the universal joint, which is fixed to the base or the moving platform, and the axial direction of the same leg are collinear, then the 6-UCU parallel manipulator is in a leg singularity.
Fig. 5. Screws of a UCU leg (when the actuated joint is blocked).
Then the vector subspace spanned by all the leg inverse screws, C pb , is 2r 3r 4r 5r 6r C pb ¼ span ð$1r 01 ; $01 ; $01 ; $01 ; $01 ; $01 Þ
Define the matrix J1 as J1 ¼ $1r ; $2r ; $3r ; $4r ; $5r ; $6r " 01 01 01 01 01 01 s3;2 s3;3 s3;1 ¼ p1 s3;1 p2 s3;2 p3 s3;3 " ¼
ð8Þ
s3;4 p4 s3;4
s3;5 p5 s3;5
s3;6 p6 s3;6
n1
n2
n3
n4
n5
n6
p 1 n1
p2 n2
p3 n3
p 4 n4
p 5 n5
p6 n6
#
#
ð9Þ
We can deduce from Eq. (9) that J1 is the transpose of the Jacobian matrix J, and then the rank of J is equivalent to cpb . Then if det (J) ¼0, the actuators cannot drive the 6-UCU parallel manipulator effectively, and it is in a singular pose.
3. Singularity detection algorithms of 6-UCU parallel manipulator In practice, it is crucial during the design phase of a robot to determine whether there are singularities within a given workspace or trajectory or not [19]. A fast straight yes–no answer of the singularity detection is important [19]. Many authors have studied the singularity detection of the GSP with only considering the actuator singularity mentioned in Section 1, but there is no one who considered the leg singularity. The singularity detection algorithms of the 6-UCU parallel manipulator with considering of both the passive and active joints are proposed in this section. Fig. 6. Flowchart of evolutionary strategy.
3.1. Evolutionary strategy Evolutionary algorithms, such as genetic algorithms, evolution strategies, particle swarm optimization algorithms are widely used to optimize [26]. Because evolutionary strategy is generally a real number encoding algorithm and with elitism preserved, then this paper uses the (μþλ) evolutionary strategy in the singularity detection procedures. For the given workspace, the search variables are limited by the lower and upper bounds of the Cartesian coordinate variables, then it is not a constrained problem. For the reachable workspace, the search variables are limited by the leg length ranges, then it is a constrained problem. To deal with all the two cases described above, we apply the crossover and mutation
Table1 Parameters of evolutionary strategy. Parameters name
Parameter values
Population size N Number of iteration μ λ Tournament selection size Distribution index for SBX Distribution index for polynomial mutation
50 2000 50 50 2 20 20
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G. Liu et al. / Robotics and Computer-Integrated Manufacturing 30 (2014) 172–179
For the ith leg, define angles ϕi1 and ϕi2 as ϕi1
¼ cos
1
ϕi2
ðs1;i U ni Þ;
¼ cos
1
ðs6;i U ni Þ
ð10Þ
Define ϕ1 min , ϕ1 max , ϕ2 min and ϕ2 max as ϕ1 min ¼ min ð ϕ11
ϕ21
ϕ31
ϕ41
ϕ51
ϕ61 Þ
ð ϕ11
ϕ21
ϕ31
ϕ41
ϕ51
ϕ61 Þ
ϕ2 min ¼ min ð ϕ12
ϕ22
ϕ32
ϕ42
ϕ52
ϕ62 Þ
ϕ2 max ¼ maxð ϕ12
ϕ22
ϕ32
ϕ42
ϕ52
ϕ62 Þ
ϕ1 max ¼ max
ð11Þ
where, min and max stand for minimize and maximize, respectively. Define ϕmin as ϕmin ¼ min ð ϕ1 min
ð180 3 ϕ1 max Þ
ϕ2 min
ð180 3 ϕ2 max Þ Þ ð12Þ
Leg singularity detection algorithms are propose as follow.
3.2.1. Leg singularity detection algorithm for the given workspace or trajectory Step 1. Create a subprogram to calculate the value of ϕmin using Eq. (12) at a pose. Step 2. Set the objective to minimize f 1 ¼ ϕmin . Step 3. Use the evolutionary strategy algorithms (as shown in Fig. 6) to search f 1 min , where f 1 min is the minimum value of f 1 in the given workspace or trajectory. Step 4. If f 1 min is equivalent to zero, there is a leg singularity in the given workspace or trajectory. If f 1 min is not equivalent to zero, there is no leg singularity in the given workspace or trajectory.
3.2.2. Leg singularity detection algorithm for the reachable workspace Step 1. Create a subprogram to calculate the value of ϕmin using Eq. (12) at a pose. Step 2. Set the objective to minimize f 1 ¼ ϕmin . Step 3. Use the constraint handling method proposed by Deb [29] to deal with the link length limits and the evolutionary strategy algorithms (as shown in Fig. 6) to search f 1 min , where f 1 min is the minimum value of f 1 in the reachable workspace. Step 4. If f 1 min is equivalent to zero, there is a leg singularity in the reachable workspace. If f 1 min is not equivalent to zero, there is no leg singularity in the reachable workspace.
Step 6. Use the evolutionary strategy algorithms (as shown in Fig. 6) to search f 2 min , where f 2 min is the minimum value of f 2 in the given workspace or trajectory. Step 7. Use the principle of continuity to determine the actuator singularity, namely, if 0 is included in the ranges of det (J), then there is actuator singularity in the given workspace or trajectory; if no 0 is included in the ranges of det (J), then there is no actuator singularity in the given workspace or trajectory.
3.3.2. Actuator singularity detection algorithm for the reachable workspace Step 1. Create a subprogram to calculate det (J) at a pose. Step 2. Set the objective to minimize f1 ¼det (J). Step 3. Use the constraint handling method proposed by Deb [29] to deal with the link length limits and the evolutionary strategy algorithms (as shown in Fig. 6) to search f 1 min , where f 1 min is the minimum value of f 1 in the reachable workspace. Step 4. Create another subprogram to calculate det (J) at a pose. Step 5. Set the objective to minimize f2 ¼ det (J). Step 6. Use the constraint handling method proposed by Deb [29] to deal with the link length limits and the evolutionary strategy algorithms (as shown in Fig. 6) to search f 2 min , where f 2 min is the minimum value of f 2 in the reachable workspace. Step 7. Use the principle of continuity to determine the actuator singularity, namely, if 0 is included in the value ranges of det (J), then there is actuator singularity in the reachable workspace; if no 0 is included in the value ranges of det (J), then there is no actuator singularity in the reachable workspace. 4. Case study In order to illustrate the singularity detection methods proposed in this paper, some cases are studied in this section. 4.1. Case study of leg singularity detection algorithms An electric driven 6-UCU parallel manipulator shown in Fig. 7 is studied in this section, whose upper joints and lower joints are both universal joints with large-scale rotation capacity.
3.3. Actuator singularity detection algorithms If there is no leg singularity in the given workspace (or trajectory), and then the actuator singularity detection algorithms are propose as follow.
3.3.1. Actuator singularity detection algorithm for the given workspace or trajectory Step 1. Create a subprogram to calculate det (J) at a pose. Step 2. Set the objective to minimize f1 ¼ det (J). Step 3. Use the evolutionary strategy algorithms (as shown in Fig. 6) to search f 1 min , where f 1 min is the minimum value of f 1 in the given workspace or trajectory. Step 4. Create another subprogram to calculate det (J) at a pose. Step 5. Set the objective to minimize f2 ¼ det (J).
Fig. 7. An electric driven 6-UCU parallel manipulator produced by our lab.
G. Liu et al. / Robotics and Computer-Integrated Manufacturing 30 (2014) 172–179
The parameters of the electric driven GSP are shown as 2 3 0:6748 0:8687 2:2336 6 0:4150 1:0187 2:2336 7 6 7 6 7 6 1:0897 0:1500 2:2336 7 T 6 7; B ¼6 0:1500 2:2336 7 6 1:0897 7 6 7 4 0:4150 1:0187 2:2336 5 2
0:6748 0:8139
0:8687 2:2336 3 0:1000 0:6677 0:7548 0:6677 7 7 7 0:6548 0:6677 7 7; 0:6548 0:6677 7 7 7 0:7548 0:6677 5
6 0:3203 6 6 6 0:4935 T P ¼6 6 0:4935 6 6 4 0:3203 0:8139 lmin ¼ 1:4453;
0:1000
0:6677
lmax ¼ 2:0690
where the ith (i¼1,…,6) column of B is the position vector of point Bi in frame O xyz. The ith column of P is the position vector of point P i in frame O1 x1y1z1. lmin and lmax are the minimum and maximum lengths of the legs. At home position, frames O xyz and O1 x1y1z1 are coincident. All the units of lengths and angles are expressed in meter and degree, respectively. T is the matrix transpose. In the process of design, the mechanical interferences of the joints and interferences between links are taken full account to ensure that the interferences do not occur at all in the course of the movement, and then they are not considered in singularity detection. As shown in Fig. 7, all the axes of the rotational joints fixed to the base of the electric driven 6-UCU parallel manipulator are in the plane constructed by the attachment points of the lower universal joints, and are perpendicular to the corresponding shorter edges of hexagon constructed by the attachment points. All the axes of the rotational joints fixed to the moving platform of the electric driven 6-UCU parallel manipulator are perpendicular to the corresponding shorter edges of hexagon constructed by the attachment points of the upper universal joints, and are form 451 angles from the plane constructed by the attachment points of the upper universal joints. 4.1.1. Leg singularity detection case study for the given workspace or trajectory The workspace are given as x A ½ 0:500 m; 0:500 m, y A ½ 0:500 m; 0:500 m, z A ½ 0:350 m; 0:350 m, ϕ3 ¼ ϕ2 ¼ ϕ1 ¼ 0 3 , where x, y and z are the position variables of point O1 along X-, Y-, and Z-axes of frame O xyz. ϕ1 , ϕ2 , and ϕ3 are the roll, pitch and yaw angles of the moving platform with relative to the frame O xyz. For the electric driven 6-UCU parallel manipulator, use the algorithms proposed in Section 3.2.1 to test the leg singularity in the given workspace. The presented algorithm is
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programmed using matlab2011a in a Windows XP environment. The CPU of the used computer is an Intel(R) Xeon(TM) with 2.66 GHz. The computation time is 218.7 s. After 2000 iterations, the search result of ϕmin is 20.41, and the search results are shown in Fig. 8, where deg stands for degree. Because the minimum value of ϕmin is not zero, then there is no leg singularity in the given workspace of the electric 6-UCU parallel manipulator. 4.1.2. Leg singularity detection case study for the reachable workspace For the electric driven 6-UCU parallel manipulator, use the algorithms proposed in Section 3.2.2 to test the leg singularity in the reachable workspace. The computation time is 215.2 s. After 2000 iterations, the search result of ϕmin is 11.01, and the search results are also shown in Fig. 8. Because the minimum value of ϕmin is not zero, then there is no leg singularity in the reachable workspace of the electric 6-UCU parallel manipulator. 4.2. Case study of actuator singularity detection algorithms In order to illustrate the actuator singularity detection methods proposed in Section 3.3, some practical 6-UCU parallel manipulators are studied in this section. 4.2.1. Actuator singularity detection case study for the given workspace or trajectory Use the algorithms proposed in Section 3.3.1 to test the actuator singularity in the given workspace of the electric driven 6-UCU parallel manipulator as shown in Section 4.1. The computation time for search of minimum det (J) is 139.1 s. After 2000
Fig. 9. A hydraulic driven 6-UCU parallel manipulator in a singular pose. 3
25
det(J)(case4.2.1) −det(J)(case4.2.1) det(J)(case4.2.2.1) −det(J)(case4.2.2.1) det(J)(case4.2.2.2) −det(J)(case4.2.2.2)
2
case4.1.1 case4.1.2
Objective function value
Minimum angle[deg]
30
20 15
1 0 −1 −2 −3
10
0
500
1000 1500 Iteration number
2000
Fig. 8. Minimum angles of the electric driven 6-UCU parallel manipulator corresponding with iteration.
−4
0
500
1000
1500
2000
Iteration number
Fig. 10. Minimum det (J) and det (J) of the 6-UCU parallel manipulators corresponding with iteration.
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iterations, the search result of minimum det (J) is 2.8, and the search results are shown in Fig. 10. The computation time for search of minimum det (J) is 149.4 s. After 2000 iterations, the search result of minimum det (J) is 0.9, and the search results are also shown in Fig. 10. The minimum and maximum values of det (J) are 2.8 and 0.9 respectively. Use the principle of continuity to determine the actuator singularity, namely, 0 is not included in the value ranges of det (J), then there is no actuator singularity of the electric 6-UCU parallel manipulator in the given workspace. It is consist with the singularity detection result of Ma et al. [24] by using an algebraic method. 4.2.2. Actuator singularity detection case study for the reachable workspace In order to illustrate the actuator singularity detection methods proposed in Section 3.3.2, two practical 6-UCU parallel manipulators, a singular hydraulic and a nonsingular electric driven 6-UCU parallel manipulators are studied in this section.
4.2.2.2. A nonsingular electric driven 6-UCU parallel manipulator. The electric driven 6-UCU parallel manipulator is a 5000E motion base (from Moog FCS) [6], whose upper joints and lower joints are both universal joints [6]. The parameters of this 6-UCU parallel manipulator are shown as follows [6]: 2 3 1:0668 0:127 1:1769 6 0:423418 0:987298 1:1769 7 6 7 6 7 6 7 0:643382 0:860298 1:1769 7 BT ¼ 6 6 0:643382 0:860298 1:1769 7; 6 7 6 7 4 0:423418 0:987298 1:1769 5 2
1:0668 0:516382
6 0:296418 6 6 6 0:8128 T P ¼6 6 0:8128 6 6 4 0:296418 0:516382
4.2.2.1. A singular hydraulic driven 6-UCU parallel manipulator. The hydraulic driven 6-UCU parallel manipulator is shown is Fig. 9, whose upper joints and lower joints are both universal joints with large-scale rotation capacity. As Huang et al. [30] pointed out that “We find that, for this class of 6/6-Gough–Stewart manipulators, there are also some special singularity cases where six lines associated with the six extensible links of the manipulator can intersect one common line and the unwanted motion of the manipulator is a pure rotational motion.”, and then the hydraulic driven 6-UCU parallel manipulator is in a singular pose as shown in Fig. 9. The parameters of the hydraulic driven 6-UCU parallel manipulator are shown as follows: 2
1:1787
6 0:3945 6 6 6 0:7842 T B ¼6 6 0:7842 6 6 4 0:3945 2
1:1787 0:3849
6 0:1598 6 6 6 0:5447 PT ¼ 6 6 0:5447 6 6 4 0:1598
0:2250 1:1333 0:9083 0:9083 1:1333 0:2250 0:4067 0:5367 0:1300 0:1300 0:5367
1:9226
3
1:9226 7 7 7 1:9226 7 7; 1:9226 7 7 7 1:9226 5 1:9226 3 0:2838 0:2838 7 7 7 0:2838 7 7; 0:2838 7 7 7 0:2838 5
0:3849 0:4067 0:2838 lmin ¼ 1:4400; lmax ¼ 2:2200 where the ith (i¼1,…,6) column of B is the position vector of point Bi in frame O xyz. The ith column of P is the position vector of point P i in frame O1 x1y1z1. lmin and lmax are the minimum and maximum lengths of the legs. At home position, frames O xyz and O1 x1y1z1 are coincident. All the units of lengths and angles are expressed in meter and degree, respectively. T is the matrix transpose. For the hydraulic driven 6-UCU parallel manipulator, use the algorithms proposed in Section 3.3.2 to search the minimum of det (J) and det (J) in the reachable workspace, the results are also shown in Fig. 10. The computation time for search of minimum det (J) is 173.1 s. The computation time for search of minimum det (J) is 149.8 s. The minimum values of det (J) and det (J) are 0.3 and 0.6 respectively, namely, the minimum and maximum values of det (J) are 0.3 and 0.6 respectively. Because the extreme values are in opposite signs, then according to the principle of continuity, there is a pose with det (J) ¼0, namely, there is the actuator singularity in the reachable workspace. This result is consistent with the fact as shown in Fig. 9.
lmin ¼ 1:143;
0:127
1:1769 3 0 0:767334 07 7 7 0:127 07 7; 0:127 07 7 7 0:767334 0 5 0:640334
0:640334
0
lmax ¼ 1:651
where the ith (i¼1,…,6) column of B is the position vector of point Bi in frame O xyz. The ith column of P is the position vector of point P i in frame O1 x1y1z1. lmin and lmax are the minimum and maximum lengths of the legs. At home position, frames O xyz and O1 x1y1z1 are coincident. All the units of lengths and angles are expressed in meter and degree, respectively. T is the matrix transpose. For the electric driven 5000E motion base, use the algorithms proposed in Section 3.3.2 to search the minimum of det (J) and det (J) in the reachable workspace, the results are also shown in Fig. 10. The computation time for search of minimum det (J) is 143.4 s. The computation time for search of minimum det (J) is 163.1 s. The minimum values of det (J) and det (J) are -3.5 and 2.0 respectively, namely, the minimum and maximum values of det (J) are 3.5 and 2.0 respectively. Because the extreme values are in the same sign, then according to the principle of continuity, there is no pose with det (J)¼ 0, namely, there is no actuator singularity in the reachable workspace. It is consist with the singularity detection result of Blaise et al. [6] by using a numerical method.
5. Conclusion The 6-UCU kind GSP has been widely used in practice. Based on a degree of freedom theory by using screw theory, the singularities of the 6-UCU parallel manipulator caused by the legs and actuators are analyzed in this paper, and two singular types are defined as leg singularity and actuator singularity. Two cases of leg singularities of the 6-UCU parallel manipulator are found in this paper. The singularity test problems of 6-UCU parallel manipulator are classified into different types from the application viewpoints. In order to detect all the singularities of the 6-UCU parallel manipulator, singularity detection procedures are proposed correspondingly, which are using evolutionary strategy algorithm to search the extreme values of the different objective functions. The effectivenesses of the proposed singularity detection procedures are confirmed by studying cases. As may be seen from the results, the computation time is in general quite acceptable.
Acknowledgment This paper is financially supported by the National Natural Science Foundation of China under Grant no. 51105096 and the 921 Manned Space Project of China under Grant no. HgdJG00401D04.
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