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3Russian Journal of Nonlinear Dynamics, 2022, vol. 18, no. 5, pp. 771-785. Full-texts are available at http://nd.ics.org.ru DOI: 10.20537/nd221208
NONLINEAR ENGINEERING AND ROBOTICS
MSC 2010: 70C20
Stiffness Modeling of a Double Pantograph Transmission System: Comparison of VJM and MSA
Approaches
W. K. Shaker, A. Klimchik
This paper deals with the stiffness modeling of the double pantograph transmission system. The main focus is on the comparison analysis of different stiffness modeling approaches: virtual joint modeling (VJM) and matrix structural analysis (MSA). The aim of this work is to investigate the limitations of the considered approaches. To address this issue, corresponding MSA-based and VJM-based stiffness models were derived. To evaluate the deflections of the end effector, the external loads were applied in different directions at multiple points in the robot workspace. The computational cost and the difference in end-effector deflections were studied and compared. MSA was found to be 2 times faster than VJM for this structure. The results obtained showed that the MSA approach is more appropriate for the double pantograph mechanism.
Keywords: stiffness modeling, parallel robot, double pantograph, virtual joint modeling, matrix structural analysis
1. Introduction
Parallel manipulators have been widely used in modern applications as a machine tool. As a result, considerable attention has been paid to the tool accuracy for more precise industrial operations. In addition to that, parallel structures have some advantages over serial manipulators
Received September 08, 2022 Accepted November 18, 2022
This work was supported by the Russian Science Foundation (project number 22-41-02006).
Walid K. Shaker
Robotics and Computer Vision program, Innopolis University
ul. Universitetskaya 1, Innopolis, Republic of Tatarstan, 420500 Russia
Alexandr Klimchik [email protected]
School of Computer Science, University of Lincoln
Brayford Way, Brayford Pool, Lincoln LN6 7TS, United Kingdom
such as higher payload and higher rigidity. Since the external load is distributed among several legs, these structures have higher stiffness and higher accuracy as the error is not accumulated at the end-effector (EE) as in the case of serial robots [1].
The double pantograph, often referred to as the scissor lifting mechanism, is a 1-DOF transitional mechanism consisting of two serial kinematic legs that move simultaneously and are attached to the EE platform, as shown in Fig. 1. It has one horizontal prismatic joint and it can be actuated using an electric motor or a hydraulic system. The main application of scissor lifts is the vertical load transmission with or without human intervention. They are commonly used in the assembly and disassembly of parts, construction maintenance, and other industrial tasks such as landing access [2].
To address the accuracy problem, stiffness analysis can be considered as a best practice because it allows us to calculate the deflections at the EE under an applied external load and thus to improve the overall accuracy by calculating the difference between the desired pose and the actual pose of the robot [3]. In the literature, three main approaches to stiffness modeling are determined. FEA (finite element analysis) [4, 5] is the most accurate approach suitable for nonlinear analysis; however, it is computationally expensive as it decomposes the structure's links into a large number of finite small elements. The matrix structural analysis (MSA) [6, 7] applies the same idea as FEA, but handles the computational cost as the structure elements are significantly simplified. It is more suitable for parallel robots. However, it can also be utilized to model serial manipulators [8]. Finally, the virtual joint modeling (VJM) [9-11], which is the simplest technique as it assumes that all links are rigidly expanded with virtual joints that represent the elasticity of links and joints. In this work, MSA and VJM are implemented and a comparison between the two methods is made.
Fig. 1. Scissor lifting mechanism
2. Kinematics
In order to implement the stiffness modeling of the double pantograph robot, forward and inverse kinematics have to be derived for each kinematic chain. The kinematics scheme is illustrated in Fig. 2.
kz
__
~x X j
LX
z z \
/1 S/Ai /V\<13 1
X X
dx
Fig. 2. Kinematic scheme of the double pantograph system 2.1. Forward kinematics
As shown, the local coordinate frames of both serial chains coincide, and they are located in the middle of the horizontal prismatic joint, £-axis. Since we only have one prismatic joint, it will drive both legs with half of its stroke. For example, when the active prismatic joint is advanced with dx, each chain will advance with a horizontal distance It is also noticeable that, each leg contains three revolute joints and that its rotation axis, the y-axis, is perpendicular to the prismatic joint axis. These 2 legs also coincide at the upper local coordinate frame, which is the EE frame. As the lower and upper local frames are shifted only in the direction of the z-axis, this means that, if one leg is translated in the +£-axis first, it will translate in the — £-axis later and vice versa. Since the robot structure consists of 2 similar chains, the forward kinematics of each chain i can be written as a product of the following homogeneous transformations:
T TbaseTx
{di,i)Ry (q
passive i,l )Tx (l)Ry (q passive i,2)Tx (l)Ry (qpassive i,
3)Tx(di,2)Ttool, (2.1)
where l is the link length. For the ith leg, dj,j is half of displacement of the jth prismatic joint, and qpassive ij is a revolute joint variable for the jth passive joint. The transformation between the global coordinate frame and the local coordinate frame is described by the matrix Tbase, while the transformation between the ends of the legs is given by the matrix Ttool. Table 1 presents the translational and rotational parameters for each leg.
ql = atan2 (y/l2 -dx2, dx^j, (2.2)
q3 = 180° — qi, (2.3)
q2 = q3— qi. (2.4)
Table 1. Kinematics Parameters
Parameter i = l i = 2
di, l di, 2 dx 2 dx 2 dx 2 dx 2
^passive i, 1 ~(h
^passive i,2 (h ~(h
^passive i,3 h %
Thus, given dx from the prismatic joint, the height of the pantograph dz can be calculated as T(3, 4).
2.2. Inverse kinematics
The inverse kinematics aims to calculate the active and passive joints of the robot at each height dz. It is simple and straightforward, as it can be written as
• Active prismatic joint: dx = \Jl2 — .
• Passive revolute joints q1, q2, q3 can be obtained from Eqs. (2.2)-(2.4).
2.3. Structure limits
This type of structure should involve some constraints on motion to avoid a failure of the structure. Hence, the structure limits should be analyzed as follows.
Since I2 - ^ 0, dz < 21.
Similarly, z = 2\ZP — dx2, so l2 — dx2 ^ 0, thus, dx ^ I. Assuming that l = 0.5 m, dx ^ 0.5, dz ^ 1,
• when the structure is totally expanded, dz = 1, dx = 0, q1 = q3 = 90°, and q2 = 0;
• when it is totally compressed, dx = 0.5, dz = 0, q1 = 0, and q2 = q3 = 180°.
In both cases, the structure will break because it is not allowed to drive the robot at these limits. Thus, 0.1 m and 0.9 m are assigned as the minimum and maximum heights that the robot can reach safely, as shown in Fig. 3.
3. Stiffness of the links
We considered the links as cylindrical beams made of aluminum. Generally, 3D beam elements could have a stiffness matrix K in the following form:
K =
Kn K12 K21 K22
(3.1)
where K11 represents the force / torque reaction due to the deflection of the left end at the left end point of the beam, K12 represents the force / torque reaction due to the deflection of the right end at the left end point of the beam, K21 represents the force / torque reaction due to the deflection of the left end at the right end point of the beam, and K22 represents the force / torque reaction due to the deflection of the right end at the right end point of the beam.
x-axis
Fig. 3. Structure limits of the double pantograph system. It is assumed to be driven within 0.1-0.9 m
For a regular beam:
Kn —
Ki2 =
¥ o
n 12EIz
0 0 0 0
K22 —
L3
0 0 0
¥ 0 0
0
0
6 EIZ
0 0
0
6 EI, IT
0 0
12EI, 0
6EI
3L
0
6EJ.
L2
GI p I
u. 0
~Tr~
4EI„
EA I
0 0 0 0
K
0
r2EIz L3
0 0 0
L2 0
0 0
12£/y
0
6£/y
0
= K T 21 — K12 )
L 0
4EIz
0 0 0
GIp I
0 0
n 6EIz U--j^r
6EIy
IT 0
2EI_y_ I
0
0 0
0
2EI,
0 0
V2EIy
0
6EI«
L2
0 0
0
GIp I
0
0
0
6 EIU
0
L
U--JJ-
0 0
0
6 EI,, L2
0 0
0
4EIz
(3.2)
(3.3)
(3.4)
(3.5)
0
0
L
L
where L is the length of the beam, Iy is the principle moment of inertia around the y axis, Iz is the principle moment of inertia around the z axis, Ip is the torsional moment of inertia, E is Young's modules of the aluminum beam, and G is Coulomb's modules of the aluminum beam.
4. Virtual joint modeling
The VJM model for the double pantograph shown in Fig. 4 can be modeled as 2 separate chains.
^^ Passive Joints Virtual Spring
Fig. 4. VJM model
The kinematic model for each leg can be represented as depicted in Fig. 5, where Ac and Ps refer to the active joint and the prismatic joint, respectively.
Base
Ac I Ps
c >
Link Ps ) Link
V rv—y J
Fig. 5. Kinematic model representation for a single chain
In VJM, an active prismatic joint could be presented by adding a 1DOF virtual spring, while elastic links can be expressed as rigid links followed by a 6DOF virtual spring, as illustrated in Fig. 6.
Fig. 6. Extended kinematic model (VJM) model for a single chain
The corresponding transformation can be obtained by extending Eq. (2.1) as follows:
(4.1)
T = TbaseTx(di,1)Tx(^i,1)Ry(^passive i,1)Tx(l)T3D,2-7)Ry(^passive i,2)Tx(l)
T3D (®i,8- 13)Ry (qpassive i,3)Tx (di,2)Ttool ,
where Tbase and Ttool are identity matrices in our case. For the ith leg, 9ij is the jth virtual
joint. The T3D \ 6ijis the 6-DOF virtual spring and can be described as the following transformations:
T3D i^i,j-(j+5}) = Tx (@i,j )Ty (^i,j+1)Tz (@i ,j+2) Rx (®i, j+3)Ry (^i,j+4)Rz ^ij+5^ (4.2)
In [9], the classical Cartesian stiffness matrix for each chain can be calculated as
K0 i = Je, iK-}jl)-1 (4.3)
where the Je is the numerical Jacobian matrix calculated with respect to the virtual joint variables, and Ke is the aggregated spring stiffness matrix. It is a 13 x 13 diagonal matrix containing three major diagonal components which are the stiffness parameters of the flexible elements. The first component is a scalar value representing the active joint stiffness and it is assumed as 106 The second and third components are 6 x 6 matrices representing the stiffness properties of the two links, respectively:
K =
Kactive 0 0
0 K22 , 6x6 0
0 0 K22,6X6
(4.4)
13x13
K22 is used because virtual joints are attached to the right-end of the links. Using the classical Cartesian stiffness matrix K0 obtained in Eq. (4.3), the Cartesian stiffness matrix of the chain i can be found:
Kc, i = Kc, i — Kc, iJq, iKCq, i, (4.5)
where
Kcq,i = JTi (K0,i) Jqj-1 jTi (K0,i) . (4.6)
In Eq. (4.6) Jq is the numerical Jacobian matrix calculated with respect to the passive joint variables. It is a 6 x 3 matrix since the single chain consists of three passive joints.
Therefore, the Cartesian stiffness matrix Kc of the whole robot can be computed as the summation of the Cartesian stiffness matrices of the two chains [9]:
Kc = £ Kcl. (4.7)
i=1
Consequently, the EE deflection At can be obtained from Hook's law, as shown below:
Ai6xl = K-lW6xl. (4.8)
This At represents the 3D deflection. As we are only interested in the positional displacement, the final deflection value due to an applied load can be obtained from the following equation:
t = \J Ai(l)2 + Ai(2)2 + Ai(3)2. (4.9)
5. Matrix Structural Analysis
The MSA model for the double pantograph [12] is depicted in Fig. 7 where the structure is represented by 21 nodes. These nodes are connected to each other based on the structure connection type. The following table explains the nodes connections and connection types.
Table 2. Nodes Connections and Connection Types
Flexible Link Passive Joints Rigid Joints Elastic Joints Rigid Support
(3, 5} (0, 3) (0, i) (1,2) (0)
(4, 6) (2, 4) <5, 8)
(7, 9) (5, 6) (6, 7)
(8, 10) (9, 11) (13, 16)
(11, 13) (10, 12) (14, 15)
(12, 14) (13, 14)
(15, 17) (17, 19)
(16, 18) (18, e)
(19, e)
The rigid support constraint at node 0 can be expressed as follows:
°6x6 I6x6
W0 Atn
J6x1 ■
(5.1)
The deflection and loading constraints for flexible links could be described as follows:
I6x6 °6x6
66
K11 K12'
i,1 i,1
t K 21 K 22 T6x6 Ki,1
W i
Wj 06x1
Ati _°6x1_
Al
(5.2)
where i and j are the link nodes. It should be noted that the stiffness matrices used in flexible links constraints should be global matrices; thus, a transformation from local stiffness matrices to the global coordinate system is applied as described below:
K11 i,1 K12! i,i
K21 . i,1 K22 i,1.
QK1QT QKj QT
QK2W QK22QT
(5.3)
Rigid Support
Fig. 7. MSA model
Q ^ Passive Joint Rigid Joint Elastic Joint
where Q is a 6 x 6 matrix consisting of two main diagonal rotation matrices R that describe the rotation of the end of the link:
Q =
R3x3 03x3 °3x3 R3x3
The passive joints allow rotations around the y-axis; then the constraints are
where
05x6 05x6 \r,y \j ÏW-1 i 05x 1
\r,y \j 05x 6 05x6 W- J 05x 1
\p,y \j 01x6 01x6 01x6 Ati 0ix1
01x6 \p,y 01x6 01x6 .01x1.
yï =
\j =
0 0 0 0 1 0
1 0 0 0 0 0' 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1
(5.4)
(5.5)
(5.6)
(5.7)
while rigid joints constraints are as follows:
06x6 06x6 ^6x6 ^6x6 ^6x6 ^6x6 06x6 °6x6
W
W 06xl
Ati _°6xl_
Aj
(5.8)
Since the active elastic joint 1-2 is translated along the x-axis, the constraints can be described by the following equation:
05x6 05x6
^6x6 ^6x6
Ae,x 0
Al,2 0lx6
Al,2
06x6
Al,2
66
W1
W2 At1 AÎ2
5xi
06xl
0^1
(5.9)
where
\e,x _ Al,2 =
1 0 0 0 0 0
010000' 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 1
The external force is applied to the node e and is donated by the following equation:
\T,X _ Al,2 =
(5.10)
(5.11)
^6x6 06x6
We Ate
= We
ext •
(5.12)
We can aggregate all of the system linear equations in matrix form using the above-mentioned Eqs. (5.1)—(5.12). Since we have 21 nodes, and each node is represented by wrench and deflection, the aggregated matrix could be described as shown below:
^282x246 B282x6
c
6x246
D
6x 6
Wa
agl26xl
At
agl20xl
Ate
J282xl
We
ext
(5.13)
The above equation can be written in the following form:
Mx = v,
(5.14)
where the main matrix M is 288 x 252. The system is overconstrained, since the M matrix is not invertible. To solve such a system, we need to find the smallest 2-norm x that at the same time provides the least residual e:
e = Mx - v. (5.15)
As the minimum of \\e\\2 coincides with the minimum of (Mx— v)T(Mx — v), the solution to this least squares problem is given by a pseudoinverse similar to finding the extremum as follows:
2MT (Mx — v) = 0,
MT Mx = MT v, x = (MT M )-1 MTv,
then x can be calculated as
x = M+v. (5.16)
So, the EE deflection At is the last 6 elements of the vector x:
At
6x1
06x246 ^6x6
x252x1 (5-17)
and it is obtained due to the applied wrench Wext. The implementation can be accessed from this repository [13].
6. Results
VJM and MSA are implemented to calculate deflection due to applied loads in different directions. The robot EE deflection is found at multiple points in the workspace. Since the structure is translated along the z-axis, deflection calculations are performed at 30 different points from the minimum to maximum height limits introduced in Section 2.3.
2.4 2.2 S 2.0
SO
<| 1.8-I L6"
I 1-4-
a>
a 1.21.0-
le-4
• • •
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Fig. 8. Deflection using MSA due to wrench [100, 0, 0, 0, 0, 0]N
Figure 8 shows the deflections, using the MSA approach, due to the 100^ load applied in the direction of the x-axis. Figures 9 and 10 present the deflections due to 100^ load applied in the direction of the y-axis, while the deflections due to 100^ load in the z-axis using MSA are shown in Fig. 11.
According to the deflections scatter plot analysis, it is notable that the deflections computed by the VJM approach due to an external force applied in the x-axis and the z-axis are not reasonable as compared to the results obtained from the MSA method. This is because of some limitations of using VJM to model this type of structures. The idea is that, when implementing VJM, we extended the kinematic model for each chain with virtual joints, but there is no way to introduce passive joints constraints which connect the two legs.
le—3
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
z
Fig. 9. Deflection using MSA due to wrench [0, 100, 0, 0, 0, 0]N
le—3
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
z
Fig. 10. Deflection using VJM due to wrench [0, 100, 0, 0, 0, 0]N
le—4
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
z
Fig. 11. Deflection using MSA due to wrench [0, 0, 100, 0, 0, 0]N
Since VJM limitations do not allow us to compare the deflections due to forces in the x and z directions with the corresponding deflections of MSA, a comparison between MSA and VJM is made in terms of deflection deference due to an applied load in the y direction and
computation time. Consequently, we can only judge the performance of the models based on applying loads in the direction of the axis of rotation, the y-axis, which is not affected too much by the absence of connecting passive joints.
As shown in Figs. 9 and 10, the maximum deflection is found at the maximum height that the EE can reach. The maximum deflection of the VJM approach is shown to be 5.2 mm, while only 1.3 mm is the maximum deflection of the MSA. The deflection deference percentage between MSA and VJM results at the highest points is around 76 % due to the applied load in the y-axis. This deference is due to passive constraints between the two legs which are not applied in the VJM implementation.
Computationally, it is found that the total time elapsed to execute the MSA approach for deflection calculations of 30 points is 2.77 seconds, while 5.77 was recorded as an execution time for the same points using VJM approach.
7. Discussion
In fact, the classical structure for the VJM approach is a parallel connection of strictly serial chains. But this is not the case in the double pantograph structure as the two chains are connected through the passive joints (5, 6) and (13, 14) described in the MSA model (Fig. 7). Hence, we cannot add the Cartesian stiffness matrices for the two legs by applying Eq. (4.7) directly. This is the main reason for such a difference between the MSA and VJM results, especially when the robot EE is subjected to loads in the x-axis and the z-axis as the VJM model in this case represents a loose structure in the x and z directions.
The VJM results can be enhanced by deriving the Cartesian stiffness matrix for the whole system at once. As explained in [14], the potential energy of the system can be expressed as
i 2
E(e1,e2) = -1£efK0iei. (7.1)
i=1
At the equilibrium, this energy must be minimized subject to geometrical constraints. The physical interpretation of these constraints is described by the passive joints between the two chains. Mathematically, it can be presented as
t = g(di, qj, i = 1, 2. (7.2)
It can be introduced in the Lagrangian function as follows:
1 2 2
m, K qi, q2) = -Y,diK0i°i + E- Si(0i> (7-3)
i=1 i=1
Following Eg. (7.3), we can obtain the Cartesian stiffness matrix for the system including the constraints. This will be implemented in future work as an approach to improve the VJM results. In addition, we can think to model the VJM as a series of stacked parallel chains instead of left and right serial chains, which may allow introducing the central passive joints and performing the coupling through kinematics.
However, VJM results in sensible deflection values when applying force in the direction of the axis of rotation, the y-axis, and we can visually notice that both MSA and VJM result in approximately similar deflection values in this case. Therefore, a comparison is made in the following section to analyze the results of both MSA and VJM due to an applied force in the y-axis.
8. Conclusion
This paper presents the stiffness analysis of the double pantograph transmission system. Stiffness modeling is implemented using VJM and MSA techniques to find the EE deflection at different points in the workspace. After deflection calculations, deflection scatter plots are built to analyze the maximum deflection due to 100N force along x, y, z directions, respectively. It was found that the VJM method has some limitations to model this structure, which prevent evaluating deflection with an external load in the x, z directions. A comparative analysis is performed to compare both MSA and VJM computation complexity and deflection deference. The program was run to compute the execution time for both MSA and VJM. MSA reported 2.77 seconds, while VJM is 5.77 seconds. Thus, MSA is about 2 times faster than VJM for this structure. Eventually, MSA has less computational cost and no limitations to model the double pantograph structure compared to the VJM approach.
Conflict of interest
The authors declare that they have no conflicts of interest.
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