Research ArticleForward Kinematics and Workspace Determination of a NovelRedundantly Actuated Parallel Manipulator
Haiqiang Zhang ,1,2 Hairong Fang ,1 Dan Zhang,2 Xueling Luo,2 and Qi Zou2
1School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, Beijing, China2Lassonde School of Engineering, York University, ON, Canada M3J 1P3
Correspondence should be addressed to Haiqiang Zhang; [email protected]
Received 8 March 2019; Accepted 21 April 2019; Published 12 May 2019
This paper presents a novel redundantly actuated 2RPU-2SPR parallel manipulator that can be employed to form a five-axis hybridkinematic machine tool for large heterogeneous complex structural component machining in aerospace field. On the contrary toseries manipulators, the parallel manipulator has the potential merits of high stiffness, high speed, excellent dynamicperformance, and complicated surface processing capability. First, by resorting to the screw theory, the degree of freedom of theproposed parallel manipulator is briefly addressed with general configuration and verified by Grübler-Kutzbach (G-K) criteria aswell. Next, the inverse kinematics solution for such manipulator is deduced in detail; however, the forward kinematics ismathematically intractable. To deal with such problem, the forward kinematics is solved by means of three back propagation(BP) neural network optimization strategies. The remarkable simulation results of the parallel manipulator demonstrate that theBP neural network with position compensation is an appropriate method for solving the forward kinematics because of itsvarious advantages, such as high efficiency and high convergence ratios. Simultaneously, workspaces, including joint space andworkspace of the proposed parallel manipulator, are graphically depicted based on the previous research, which illustrate thatthe proposed manipulator is a good candidate for engineering practical application.
1. Introduction
Various types of mechanisms have been applied in manyrobotic fields. Nowadays, some parallel manipulators haveattracted exhaustive attention from both academia andindustry. Compared to series manipulators, parallel manipu-lators are claimed to possess inherent advantages such ashigh stiffness, high loading capability, high precision, lowerror accumulation, fast response speed, and high orientationcapability. These characteristics allow the parallel manipula-tor to be widely used in various alternative fields, such ashigh-speed machine tools, high-speed pick and place applica-tions, surgical manipulators, and force/torque sensors [1–3].However, parallel manipulators suffer inherently from anunfavorable workspace. Therefore, it is of crucial significanceto make full use of the potential parallel manipulators to sat-isfy the current work requirements [4, 5].
Generally, some performance indices, such as dexterity,stiffness, accuracy, and workspace, will be involved in thedesign process of the parallel manipulator. Kinematics solu-tions, establishing the relationship between the joint spaceand operation workspace including position, velocity, andacceleration, are an extremely key issue. The inverse kine-matics solution is to find the joint angles or actuator lengthswhen the position and orientation of the moving platform aregiven [6], whereas the forward kinematics solution is to findthe terminal position and orientation in consideration of agiven set of joint angles or actuator lengths. Due to theinverse kinematics, a mathematical solution can be obtaineddirectly through the closed-loop vector of the parallel manip-ulator, but the forward kinematics solution is very complexand nonlinear, which presents a considerably difficult prob-lem. Therefore, computing the forward kinematics rapidlyis critically crucial in real-time control of the redundantly
HindawiInternational Journal of Aerospace EngineeringVolume 2019, Article ID 4769174, 14 pageshttps://doi.org/10.1155/2019/4769174
actuated 2RPU-2SPR parallel manipulator. In view of the lit-erature, there is an abundance of research on the forwardkinematics. Li and Xu [7] adopted the Newton iterativemethod to solve the forward position kinematics of the 3-PRS parallel manipulator. Fu et al. [8] studied the polynomialsolutions of the forward kinematics problem, converted itinto a first-degree polynomial equation and an eight-degreepolynomial equation, and verified the effectiveness and cor-rectness of the proposed algorithm. Joshi and Tsai [9] pro-posed a methodology for solving the forward kinematics ofa class of 3-DOF, 4-legged parallel manipulators in closedform and demonstrated the algorithm with the Triceptmanipulator that has at most twenty-four real solutions.Wu et al. [10] presented a simple numerical method for for-ward kinematics of a general 6-DOF parallel, which can gen-erate a unique actual solution. The method possesses moreadvantage than the conventional numerical iteration methoddoes because it does not rely on the initial value. Merlet [11]presented an efficient alternative algorithm, interval analysis,which can be numerically solved for the forward kinematics,that is, to determine all possible poses for given joint vari-ables. Tsai et al. [12] employed Bezou’s elimination methodand optimization techniques to solve the forward kinematicsproblem of the 3-PRS parallel mechanism, which makes itpossible for real-time control applications. Karimi andNategh [13] utilized a statistical approach including Batesand Watts to investigate the nonlinearity of the forwardkinematics, and the results demonstrate that the length ofthe region has a significant impact on the nonlinearity ofthe parallel manipulator. Li et al. [14] proposed a novel over-constrained three-degree-of-freedom spatial parallel manip-ulator and derived the analytical formulation of the forwardkinematics. Wang et al. [15] proposed a novel forwardkinematics algorithm, which is verity in the 3-RPS parallelmechanism, and the accuracy is, less than 10-6, sufficientlyhigh. However, it needs to select a reasonable initial value.
Recently, neural networks are employed to solve theforward kinematics of a parallel manipulator and draw par-ticular interests for numerous researchers, due to theirconsiderable ability to approximate nonlinear mapping func-tions. Parikh and Lam [16] proposed several neural networkstrategies for the Gough-Stewart parallel manipulator, andresults show that the hybrid strategy, combined with astandard Newton-Raphson numerical method, is the bestmethodology. Amir et al. [17] proposed a wavelet neural net-work, which can be employed as a powerful method for non-linear mapping equations, to approximate the desiredtrajectory of a hybrid robot manipulator. Sadjadian et al.[18] used neural network identification with different struc-tures to solve the forward kinematics considering highlycoupled and nonlinear characteristics of the mathematicalmodel which are almost impossible to solve analytically.Cheng et al. [19] generate the workspace of the 3-PRS par-allel robot by adopting the trained BP neural networkapproach. However, it is rare to use the neural networkfor solving the forward kinematics of the redundantlyactuated parallel manipulator.
The goal of this paper is to solve the forward kinematics,which promotes further practical applications of the
promising redundantly actuated 2RPU-2SPR parallel manip-ulator. The remainder of the paper is organized as follows. Theresearch background of the forward kinematics solution ispresented firstly. In Section 2, a novel redundantly actuated2RPU-2SPR parallel manipulator is proposed, which can beselected as the main body of the hybrid machine tool. In Sec-tion 3, the degree of freedom of the proposed parallel manipu-lator was briefly analyzed in a general configuration via thescrew theory, and the modified Grübler-Kutzbach (G-K)equation is utilized to verify the correctness for the degreeof freedom. Then, the inverse kinematics position is for-mulated. The BP neural network optimization strategiesproposed here are trained to solve the forward kinematicsbased on the inverse solution in Section 4. In Section 5,the joint space and workspace of the parallel manipulatorare performed, respectively, and the workspaces of theparallel manipulator are intuitively depicted by usingMATLAB programming. Finally, some concluding remarksare provided in the last section.
2. Manipulator Description
A five-axis hybrid kinematic machine tool is design to mill alarge complex structure component surface in the aerospacefield. As shown in Figure 1, it is composed of a novel three-degree-of-freedom parallel manipulator and two adjustableannular guides. The three-degree-of-freedom parallel manip-ulator is a novel redundantly actuated parallel manipulator,which consists of a moving platform connecting a fixed baseby four identical driving legs (or limbs) with a symmetricalstructure. Leg 1 and leg 3 connect by means of a fixed baseto the moving platform by a revolute joint (R), a prismaticjoint (P), and a universal joint (U) in sequence forming anRPU configuration kinematic chain, while leg 2 and leg 4connect by means of a fixed base to the moving platform bya spherical (S), prismatic (P), and revolute joint (R) insequence forming an SPR configuration kinematic chain,where the prismatic joint is activated independently by a lin-ear serve motor. The kinematic model of the parallel manip-ulator is shown in Figure 2. The moving platform and thebase are a square layout, and its circumcircle radii are a andb, respectively. The kinematic joint configuration arrange-ment satisfied certain geometric constraints, and the movingplatform will own three degrees of freedom, namely, onetranslation and two rotations [20, 21].
As shown in Figure 2, the Cartesian fixed coordinatereference frame B − xyz is attached to the centroid of thefixed base, whose coordinate origin B is a square centerof the circle, in which the x and y axes are the same asthe vector BB1 and BB4, and the z axis is determined bythe right hand rule. Analogously, the local coordinate ref-erence frame A − uvw is considered at the centroid of themoving platform. The coordinate origin A is the geometriccenter of the moving platform, the u and v axes are thesame as the vector AA1 and AA4, and the w axis is deter-mined by the right hand rule. The position vector of pointAi with regard to the Cartesian fixed coordinate referenceframe B − xyz can be indicated as (xi, yi, zi).
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3. Kinematics Analyses
In this section, the kinematics analysis, mobility, inversekinematics, and forward kinematics of the redundantly actu-ated 2RPU-2SPR parallel manipulator are mentioned.
3.1. Mobility Analysis. The mobility determination is the firstand foremost issue in designing the motion characteristic of aparallel manipulator. In order to determine the motion pat-tern of the 2RPU-2SPR parallel manipulator, mobility analy-sis including the number of degrees of freedom and motionclassification is indispensable. Furthermore, it is necessaryto analyze the constraint wrench screw provided by eachleg to the moving platform and comprehensively analyzethe constraint type of the moving platform, i.e., whether theyare a constraint force or constraint moment. Therefore, thedetailed analysis process is as follows [22].
In Figure 2, through the analysis of the screw theory, thetwist screw of the first RPU leg can be expressed as follows inthe fixed coordinate system
$11 = 1 0 0 ; 0 0 −b ,
$12 = 0 0 0 ; 0 m1 n1 ,
$13 = 1 0 0 ; 0 z1 −y1 ,
$14 = 0 e1 f1 ; f1y1 − e1z1 −f1x1 e1x1 ,
1
where m1 and n1 represent the direction cosine of the drivejoint, and e1 and f1 represent the direction cosine of the thirdrotation axis of the universal joint of the 1-st leg.
Then, considering the reciprocal product, the wrenchsystem of equation (1) can be easily obtained as
$c11 = e1 0 0 ; 0 e1z1 − f1y1 0 ,
$c12 = 0 0 0 ; 0 −f1 e1 ,2
where $c11 presents a constraint force that passes throughpointA1 and is parallel to $12, and $c12 represents a constraintmoment perpendicular to v and $13.
Analogously, the twist screw of the second SPR leg can beexpressed mathematically as
$21 = 0 0 1 ; 0 b 0 ,
$22 = 0 1 0 ; 0 0 −b ,
$23 = 1 0 0 ; 0 0 0 ,
$24 = 0 0 0 ; 0 −f1 e1 ,
$25 = 0 e1 f1 ; f1y2 − e1z2 −f1x2 e1x2
3
The wrench screw of equation (3) can be described as
$c21 = 0 e1 f1 ; 0 bf 1 −be1 , 4
where $c21 represents a constraint force that passes throughpoint B2 and is parallel to $25.
The twist screw of the third RPU leg can be expressednumerically as
$31 = 1 0 0 ; 0 0 b ,
$32 = 0 0 0 ; 0 m3 n3 ,
$33 = 1 0 0 ; 0 z3 −y3 ,
$34 = 0 e1 f1 ; f1y3 − e1z3 −f1x3 e1x3
5
The wrench screw of equation (5) can be obtained as
$c31 = e1 0 0 ; 0 e1z3 − f1y3 0 ,
$c32 = 0 0 0 ; 0 −f1 e1 ,6
where $c31 represents a constraint force that passes throughpointA3 and is parallel to $31, and $c32 represents a constraintcouple perpendicular to v and $33.
Figure 1: A five-axis hybrid kinematic machine tool.
TwistWrench
$c31
$25$c11
$13
$c12
$14
$12$44$24
$33 $45$34$c32
$32
$c21
$21
B2
$23
$22
$31 $c41 $43$42
$41 $11
Leg 1
Leg 4
Leg 2
Leg 3
disi
A uv
w
A1
A2
A3
A4
z
y
x
B
S B4
RPURPUSPR
S
SPR
R
B1
R
B3
U
Figure 2: Schematic diagram of the redundantly actuated parallelmanipulator.
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In a similar approach, the twist screw of the fourth SPRleg can be expressed mathematically as
$41 = 0 0 1 ; 0 −b 0 ,
$42 = 0 1 0 ; 0 0 b ,
$43 = 1 0 0 ; 0 0 0 ,
$44 = 0 0 0 ; 0 −f1 e1 ,
$45 = 0 e1 f1 ; f1y4 − e1z4 −f1x4 e1x4
7
The wrench screw of equation (7) can be obtained as
$c41 = 0 e1 f1 ; 0 −bf 1 be1 , 8
where $c41 represents a constraint force that passes throughpoint B4 and is parallel to $45.
So far, the constraint wrench screw of the redundantlyactuated parallel manipulator can be expressed as
$c11 = e1 0 0 ; 0 e1z1 − f1y1 0 ,
$c12 = 0 0 0 ; 0 −f1 e1 ,
$c21 = 0 e1 f1 ; 0 bf 1 −be1 ,
$c31 = e1 0 0 ; 0 e1z3 − f1y3 0 ,
$c32 = 0 0 0 ; 0 −f1 e1 ,
$c41 = 0 e1 f1 ; 0 −bf 1 be1
9
It is noteworthy that the direction vector A1A3 coincideswith the revolution joint axis of the moving platform, whoseinformation plays an important role in this additional poten-tial relation, which can be described as
z3 − z1y3 − y1
=f1e1, 10
By combining equation (10) and rearranging equation(9), here the twist screw of the moving platform can be for-mulated as follows:
$m1 = 1 0 0 ; 0 0 0 ,
$m2 = 0 0 0 ; 0 −f1 e1 ,
$m3 = 0 e1 f1 ; f1y3 − e1z3 0 0
11
In equation (11), $m1 represents one rotational degree offreedom around the x axis, $m2 represents a translationaldegree of freedom that is perpendicular to the moving plat-form, and $m3 represents one rotational degree of freedomof the moving platform around the v axis.
Generally, the number of degrees of freedom of the paral-lel manipulator can be calculated by utilizing the modifiedGrübler-Kutzbach (G-K) criterion, that is,
F = d n − g − 1 + 〠g
i=1f i + v − ς, 12
where F represents the degree of freedom of the mechanism,n represents the number of the components, g represents thenumber of the kinematic joints, d = 6 − λ represents theorder of the mechanism, f i represents the degree of freedomof the i-th kinematic joint, v represents the redundant con-straints of the mechanism, and ς represents the local degreeof freedom.
There is neither a constraint couple in the same directionnor a constraint force in collinearity among all the wrenchscrews of the parallel manipulator; therefore, there is no com-mon constraint, that is, λ = 0. Because there are only threelinearly independent variables in the six constraint screwsof the parallel manipulator, therefore, the parallel manipula-tor has three redundant constraints, that is, v = 3. As the par-allel manipulator has no local degree of freedom, so ς = 0. Wecan see from the schematic of the mechanism that the num-ber of the component is 10, the number of the kinematic jointis 12, and the relative freedom of all the kinematic joints inthe mechanism is 18.
Thus, the mobility number of the 2RPU-2SPR paral-lel manipulator can be determined easily by utilizingequation (12):
F = d n − g − 1 + 〠g
i=1f i + v − ς
= 6 × 10 − 12 − 1 + 18 + 3 − 0 = 313
In summary, the redundantly actuated 2RPU-2SPR paral-lel manipulator has three independent degrees of freedom,that is, two rotational degrees of freedom around the x axisand v axis and one translational degree of freedom along thez axis.
Compared with the existing 4-DOF parallel manipulatorpossessing four actuators [23], this novel redundantly actu-ated parallel manipulator has some key merits such as fouractuator joint variables with redundant actuation and threeredundant constraints, which make us draw more attentionto its advantages, i.e., increase the desirable workspace,improve the dexterity, and eliminate/decrease the singularity.Here, each universal joint is composed of two mutually per-pendicular rotational joints with “T” configuration type not“+” configuration type, and the “T” configuration type uni-versal joints designed in the mechanism can present a relativegreater rotation angle which can avoid singularity, increasethe workspace, and improve the kinematics and dynamicsperformance [24].
3.2. Inverse Kinematics. Figure 2 shows the structural dia-gram of the parallel manipulator, where the closed vectorconstraint equation can be written with regard to the fixedcoordinate system
bi + disi = p + ai 14
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In equation (14), bi is the position vector in the fixedcoordinate system, di represents the linear displacement ofthe i-th actuator, Aai is the position vector in the relativecoordinate system, and ai is the representation of vector Aaiwith respect to the fixed coordinate system, namely, ai = B
RA ⋅ Aai, where BRA is the rotation matrix of the parallelmanipulator. From the structure of the parallel manipulator,the position vector can be written as
b1 = 0 b 0 T ,
b2 = −b 0 0 T ,
b3 = 0 −b 0 T ,
b4 = b 0 0 T ,Aa1 = 0 a 0 T ,Aa2 = −a 0 0 T ,Aa3 = 0 −a 0 T ,Aa4 = a 0 0 T
15
As illustrated in Figure 2, the homogeneous coordinatetransformation matrix T of the redundantly actuated par-allel manipulator can be obtained through three times ofcomplex transformation, and the moving process is as fol-lows. First, angle α is rotated around the x axis. Second,distance h is translated along the new z axis. Finally, angleβ is rotated around the v axis of the moving platform. Itspose can be represented by position vector p x y zand rotation matrix BRA, and the transformation matrixcan be expressed as
T =BRA p
0 1= Rot x, α Trans z, h Rot v, β
=
cos β 0 sin β 0
sin α sin β cos α −sin α cos β −h sin α
−sin βcoα sin α cos α cos β h cos α
0 0 0 116
It can be seen from equation (16) that the positionvector of the moving platform p = x y z T is a functionof independent variables z, α, and β, namely,
x = 0,
y = −h sin α,
z = h cos α
17
So, the parasitic motion in the movement of the paral-lel manipulator is related to the workspace parameters,that is,
y = −z tan α 18
Equation (18) represents the coupling relationshipsbetween the position and orientation variables. The threeworkspace variables z, α, and β are independent of oneanother, so the inverse kinematics solution can be calcu-lated with the following analytical equation:
di = p + ai − bi , 19
where • represents the Euclidean norm of a vector, sothe unit vector si in the direction of activate leg can beexpressed as
si =p + ai − bi
di20
Taking the positive square root of the resulting equa-tion (19) and rearranging it lead to
d1 = z tan α − a ⋅ cos α + b 2 + z + a ⋅ sin α 2,
d2 = z + a ⋅ cos α cos β 2 + b − a ⋅ cos α 2 + z tan α + a ⋅ sin α sin β 2,
d3 = z tan α + a ⋅ cos α − b 2 + z − a ⋅ sin α 2,
d4 = z − a ⋅ cos α cos β 2 + b − a ⋅ cos α 2 + z tan α − a ⋅ sin α sin β 2
21
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If the structure parameters of the parallel manipulatorand pose of the moving platform are given, the displacementof the activated joint can be calculated by employing theinverse kinematics equation (21). The purpose of the inversekinematics solution is to determine the actuator lengthswhen the position and orientation (pose) of the moving plat-form are given. It is such length and pose data that areadopted as the training samples of BP neural networks inthe next section.
4. Forward Kinematics of the BPNeural Network
The forward kinematics solution is aimed at determiningthe position and orientation of the moving platform withrespect to the fixed reference frame once the linear dis-placement of the joint variables is given. In considerationof a rotation axis of the proposed parallel manipulatorlocated at the x axis around the fixed coordinate system,there is parasitic motion in the movement process, whichmakes the forward kinematics solution of the parallelmanipulator especially complex. Therefore, the forwardkinematics solution needs to be calculated by applying anartificial intelligence method named the BP neural networkin this section [25, 26].
4.1. BP Neural Network. The BP neural network is a dynamicsystem of the topological structure of the digraph, which
employs differentiable function as activation function topropagate forward. At the same time, the weight factor wij
(i = 1,2,3,4, and j presents the node from the input layer tothe hidden layer as well as from the hidden layer to the
wi5
wi6
wi7
wi8
wi4
wi3
wi2
wi1
wij𝛿
𝛿
Z
Zreal
𝛼
𝛽
f13(e)
f10(e)
f4(e)
f3(e)
f2(e)
f1(e)
f5(e)
f6(e)
f9(e)
f12(e)
f11(e)
+
−
Input layer (i)
Output layer (o)
Hidden layer (k)
d1
d2
d3
d4
Error back propagation
Figure 3: Topological structure of the BP neural network.
772 epochs0 100 200 300 400 500 600 700
Mea
n sq
uare
d er
ror (
mse
)
10-6
10-4
10-2
100
Best validation performance is 0.00027405 at epoch 766
TrainValidationTest
BestGoal
Figure 4: The training curve of the BP neural network.
6 International Journal of Aerospace Engineering
output layer) of network parameters is adjusted continuouslythrough error δ back propagation to minimize the meansquare errors of the network and realize the nonlinear map-ping from the actuator joint variable space to the operationvariable space without revealing the mathematical equationbeforehand. Its topological structure, as shown in Figure 3,contains the input layer, hidden layer, and output layer. Inthis case, the input layer consists of four nodes, i.e., four lin-ear input actuator displacements d = d1, d2, d3, d4 , and theoutput layer consists of three nodes, i.e., one position andtwo orientations parameter X = z, α, β [27, 28].
The number of the nodes (k) in the hidden layer is gener-ally determined by empirical formulas [29], that is,
k = m + o + l 22
Among them, m is the number of nodes in the inputlayer, o is the number in the output layer, and l is the constantwhose range is from one to ten. The number of nodes in thehidden layer is selected as six in this paper.
The differentiable and linear functions are usuallyemployed as alternative activation function of the BP neu-ral network. In this paper, the S-type logarithmic function,Tan-Sigmoid, is selected as the activation function of
output hidden layer nodes (neurons) in the predictionmodel, as follows:
f x =2
1 + e−x− 1 23
Back propagation requires that the activation functionused by the artificial nodes be differentiable, so the deriv-ative of the Tan-Sigmoid function can be expressed as
f ′ x =2e−x
1 + e−x 2 =1 − f x 2
224
Since the magnitude of different data is not the same,which may result in slower convergence ratios and longtraining time of the neural network, furthermore, therange of activation function in the output layer of the neu-ral network is limited to one interval, so it is necessary tonormalize the target data of the training network to therange [-1, 1]; the normalization criterion can be written as
y =2 x − xminxmax − xmin
− 1, 25
Table 1: Forward kinematics results of the parallel manipulator byutilizing the BP neutral network.
No. Values z (mm) α (rad) β (rad) Error/10-3
1Target 0.7 0.1 0
2.166008Calculated 0.701021 0.083644 -0.020175
2Target 0.7 0 0.1
2.264033Calculated 0.701452 -0.011170 0.124723
3Target 0.7 0.1 0.1
0.885016Calculated 0.701227 0.092165 0.107064
4Target 0.75 0.15 0
2.871267Calculated 0.751755 0.132945 -0.029886
5Target 0.75 0 0.15
2.332283Calculated 0.753355 -0.013697 0.174175
6Target 0.75 0.15 0.15
0.668443Calculated 0.752444 0.144920 0.155706
7Target 0.75 -0.1 0.15
2.768450Calculated 0.755368 -0.104542 0.182469
8Target 0.75 0.15 -0.1
5.261575Calculated 0.752359 0.114646 -0.152260
9Target 0.75 0 0.25
3.533075Calculated 0.755474 -0.039118 0.234595
10Target 0.75 0.25 0
3.746533Calculated 0.752127 0.254188 -0.044712
11Target 0.75 -0.25 0.25
9.037167Calculated 0.748228 -0.346240 0.200049
12Target 0.8 0.15 0.25
1.170758Calculated 0.800083 0.136980 0.244722
Table 2: Forward kinematics results of the parallel manipulator byutilizing the improved BP neutral network.
No. Values z (mm) α (rad) β (rad) Error/10-3
1Target 0.7 0.1 0
0.971667Calculated 0.689317 0.104087 -0.002263
2Target 0.7 0 0.1
0.293098Calculated 0.700993 -0.002258 0.102508
3Target 0.7 0.1 0.1
0.147458Calculated 0.700549 0.101409 0.099082
4Target 0.75 0.15 0
1.579175Calculated 0.732423 0.157081 -0.000093
5Target 0.75 0 0.15
0.496743Calculated 0.755302 -0.002327 0.151418
6Target 0.75 0.15 0.15
0.229083Calculated 0.749549 0.152573 0.149145
7Target 0.75 -0.1 0.15
0.917767Calculated 0.758538 -0.104923 0.154916
8Target 0.75 0.15 -0.1
3.714833Calculated 0.706730 0.159786 -0.104370
9Target 0.75 0 0.25
1.788858Calculated 0.732512 -0.004709 0.238476
10Target 0.75 0.25 0
2.240533Calculated 0.723416 0.253966 0.000679
11Target 0.75 -0.25 0.25
4.398025Calculated 0.697250 -0.249546 0.248398
12Target 0.8 0.15 0.25
0.471583Calculated 0.799934 0.152385 0.244868
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where xmin is the minimum value of x, xmax is the maxi-mum value of x, x is the input vector, and y is the outputvector of normalization.
The learning and training samples are set up by utiliz-ing the inverse solution equation (21) of the parallel manip-ulator. In this paper, we choose 5512 data values representingthe learning samples; the network training function istraingdx which updated weight and bias values according tothe gradient descent momentum and an adaptive learningrate. The network performance function is the meansquare error, which is the error between the network cal-culated outputs and the target outputs. The number ofiteration epochs is set as 4000, the expected error goal is1E − 7, the learning rate is 0.01, and the training networkis trained by the Levenberg-Marquardt (LM) optimizationalgorithm to improve the training speed and reduce theconvergence error.
After being successfully trained, once all the kinematicparameters are known, the forward kinematics can bedirectly computed, which is the main goal of this paper. Forthe sake of convenience, twelve groups of positions and pos-tures are taken as the test data, their inverse solutions arebrought into the trained network, taking the first group setof data as an example, and the iterative process is depictedin Figure 4. After 722 epochs, the optimum parameter values(z, α, andβ) can be obtained within the allowable error range,which implement the nonlinear mapping from the actuatorjoint variable space to the operation variable space [30, 31].
The forward kinematics solution of the parallel manip-ulator can be solved by minimizing the network perfor-mance function. The mean square error (MSE) of the BPneural network can be derived according to the error backpropagation equation
E = 112
〠3
i=1yi − yi
2, 26
where yi is the predicted value of the BP neural network,and yi is the desired target value; here yi representsparameters z, α, and β.
All of the selected parameters (z, α, and β) and the errorscalculated by the BP neural network are shown in Table 1.
From Table 1, we can see that the range of errors is(6 68 × 10−4, 9 03 × 10−3); the magnitude of most errors isalmost in the order of 10-3. However, the error of the thirdgroup and the sixth group is smaller than others, their mag-nitude of error is 10-4, and their orientation angles are thesame, which indicates that the mean square error is verysmall when the orientation angle is in the same value. Wealso find that in the eleventh group, the error is the biggestof all the results, and their orientation angles are in differentdirections, so the error is particularly large. By comparing thefifth group and the ninth group, it is illustrated that the meansquare error also becomes larger with the increase in the ori-entation angle. We hope that a smaller error is much better;therefore, much more efforts should be devoted to decreasethe mean square error.
4.2. Improved BP Neural Network. Since the position and ori-entation values of the parallel manipulator are not in an orderof magnitude, therefore, the improved BP neural network isused to train position z and orientation (α, β)separately,and the LM optimization algorithm is also used in the samemanner to solve the forward kinematics problem. The dataof the above twelve groups are predicted with the improvedBP neural network. The results of the parallel manipulatorfor calculation are shown in Table 2.
The results of the improved BP neural network are moreaccurate than those of the traditional BP neural network. Theorder of magnitude has basically reached 10-4, and there hasbeen a great improvement compared with the traditional BPneural network. However, the accuracy is still difficult to sat-isfy the high accuracy requirements of the parallel kinematicmachine tool. Therefore, error compensation is needed tofurther improve the calculation accuracy.
4.3. BP Neural Network with Position Compensation. Eventhough both BP neural network and improved BP neuralnetwork approaches yield good results for training andtesting for given desired data points, in order to obtain amore satisfactory accuracy, a BP neural network algorithmwith position error compensation is adopted; the calcula-tion process in Figure 5 represents the sequence followedfor minimizing the mean square error to obtain the opti-mum forward kinematics.
Step 1. Select the actuator displacement d0 = d1, d2, d3, d4 asthe initial input value of the loop program.
Step 2. Calculate the pose X = z, α, β of the moving plat-form with the improved BP neural network.
Step 3. Solve the actuator displacement d1 = d1, d2, d3, d4with the inverse kinematics solution.
Step 4. If the delta absolute value δ = d1 − d0 less than thepermission accuracy e, then export the pose parameter X =z, α, β ; otherwise, let dnew1 = d1 − δ as the new input of nextcycle repeat step 2 until we obtain the satisfactory values.
Analogously, the forward kinematics solution can bederived by employing the BP neural network with positioncompensation, and the approximation permission accuracye = 1 × 10−5; after calculation, the forward kinematics solu-tion results and mean square error can be obtained, as shownin Table 3.
From Table 3, we can see that the accuracy of the forwardkinematics solution is greatly improved by using the BP neu-ral network with the error position compensation method.The magnitude of the error has already reached 10-6, whichindicates that this optimization strategy has higher accuracythan the previous two methods described above. The calcula-tion results are shown in Table 3. It can be seen from the tablethat the maximum mean square error is 8 44 × 10−6, whilethe minimum one is 1 16 × 10−8. The experimental accuracyhas been significantly improved, which also illustrates theeffectiveness of the excellent algorithm for calculating theforward kinematics.
8 International Journal of Aerospace Engineering
Compared with three neural network strategies, asshown in Figure 6 (the 1st represents the general (tradi-tional) BP neural network, the 2nd represent the improved
BP neural network, and the 3rd represents the BP neuralnetwork with position compensation), we found that theBP neural network with position compensation has the besteffect for the forward kinematics solution of the parallelmanipulator. Comparing the 1st method with the 2ndmethod, the mean square error of the improved BP neuralnetwork method has much more improvement than that ofthe 1st method.
However, as depicted in Figure 7, the traditional BPneural network has a much lesser mean square error inthe z direction and a relatively bigger orientation errorincluding angles α and β, while the improved BP neuralnetwork method is just opposite to that of the 1st method.It is also shown that the 1st and 2nd are not good enough,and either method has shortcomings and inadequacies.Comparative analysis illustrated that the BP neural networkwith position compensation is fairly efficient and possesseshigh accuracy. It is also proved that the proposed algorithmcan be efficiently applied to forward kinematics of the2RPU-2SPR redundantly actuated parallel manipulator,which is critically important for the further researchincluding dynamic analysis, control simulation, and exper-iment prototype real-time control.
5. Workspace Analysis
The determination of joint and workspace is one of the mostimportant issues of the redundant actuation parallel manipu-lator, which directly reflects the comprehensive performanceof the parallel manipulator; its analysis results provide a the-oretical basis for the design and application of the 2RPU-2SPR parallel manipulator [32–34]. A real experiential proto-type (as depicted in Figure 8) is fabricated, whose workspaceis generated by adopting the main loop algorithm based onthe forward kinematics combining the structural parametersand the limitation conditions.
For convenience, we summarize the main loop algorithmprocedure as follows.
Mea
n sq
uare
erro
r
×10−3
×10−4
−1
0
1
2
3
4
5
6
3−4−2
0246
1 2 3
Figure 6: Comparison analysis of the mean square error.
Table 3: Forward kinematics results of the parallel manipulator byutilizing the BP neural network with position compensation.
No. Values z (mm) α (rad) β (rad) Error
1Target 0.7 0.1 0 1 4439E − 07
Calculated 0.7 0.099999 0
2Target 0.7 0 0.1 1 1688E − 08
Calculated 0.7 0 0.1
3Target 0.7 0.1 0.1 1 9760E − 07
Calculated 0.7 0.099999 0.099999
4Target 0.75 0.15 0 5 7654E − 07
Calculated 0.75 0.149999 0
5Target 0.75 0 0.15 5 0365E − 08
Calculated 0.75 0 0.15
6Target 0.75 0.15 0.15 8 9869E − 07
Calculated 0.75 0.149999 0.149999
7Target 0.75 -0.1 0.15 2 5847E − 07
Calculated 0.75 -0.099999 0.15
8Target 0.75 0.15 -0.1 6 8813E − 07
Calculated 0.75 0.149999 -0.099999
9Target 0.75 0 0.25 1 3156E − 06
Calculated 0.75 0 0.249999
10Target 0.75 0.25 0 8 4400E − 06
Calculated 0.749994 0.249994 0
11Target 0.75 -0.25 0.25 1 1985E − 06
Calculated 0.749995 -0.249994 0.249991
12Target 0.8 0.15 0.25 2 4444E − 06
Calculated 0.8 0.15 0.249998
Start
Initial input displacement d0
Calculate X with improved BPstrategy
Calculate d1 with inversekinematics
|𝛿| ≤ e
Export X
End
Y
N
dnew1=d1-𝛿
Figure 5: Flowchart of the position compensation.
9International Journal of Aerospace Engineering
In the actual movement process of the parallel manipula-tor, the following limitations should be considered:
(1) Limitation of the actuated joints di is defined as
dmin < di < dmax, 27
where dmin and dmax represent the minimum andmaximum actuator displacement, respectively.
(2) Joint angle constraints
Under consideration, the rotation angle Ri of the rev-olute joint R and the rotation angle Si of the sphericaljoint should satisfy the following inequality:
Rmin ≤ Ri = a cosei · diei di
≤ Rmax,
Smin ≤ Si = a cosei · diei ei
≤ Smax,28
where Rmin, Rmax, Smin, and Smax represent the mini-mum and maximum constraint angles of the revolutejoint R and the spherical joint, and ei represents thenormal vector of the fixed base.
(3) Singularity configuration constraint
At a singularity configuration, the parallel manipula-tor may lose one or more degrees of freedom, whichwill seriously affect the kinematic performance of theparallel manipulator, and then those configurationsshould be avoided. So the joints are kept away fromthe singularity configuration, and the rank of theconstraint screw system is limited to be three, that is,
Rank $c11, $c12, $c21, $c31, $c32, $c41 = 3 29
As has been described beforehand, the architecturalparameters and the maximum and minimum limit valuesof constraint conditions are shown in Table 4 as below,so the joint space and workspace can be depicted byMATLAB programming.
Mea
n sq
uare
erro
r
0
0.5
1
1.5
2
2.5
3Three BP neural network optimization strategies
z
𝛼
𝛽
3-2
0
2
1 2 3
×10−3
×10−4
Figure 7: Bar chart comparison of the three output parameters.
Figure 8: The experimental prototype of the parallel manipulator.
for d1 = d1 min to d1 max by step Δd1for d2 = d2 min to d2 max by step Δd2
for d3 = d3 min to d3 max by step Δd3for d4 = d4 min to d4 max by step Δd4
Table 4: Architecture parameters of the proposed mechanism.
Parameter Value Parameter Value
a/m 0.15 α/° [-60,60]
b/m 0.25 β/° [-60,60]
di/m [0.3,0.8] Ri/° [-45,45]
z/m [0.2,1.0] Si/° [-45,45]
10 International Journal of Aerospace Engineering
With consideration of four actuators, the joint workspacecan be combined by (d1, d2, d3), (d1, d2, d4), (d2, d3, d4),(d1, d3, d4), and so on; for the sake of convenience, we justprovide only one joint workspace (d1, d2, d3) to illustratethe forward kinematics solution, as depicted in Figures 9and 10, in which there is no empty and has a smooth surfaceand envelope boundary. Figure 9 indicates the joint space
that actuator joints can reach the points set. Figure 10 is theprojection in the d1 − d3 dimensional plane.
Simultaneously, the boundary of the workspace can beformulated by adopting the BP neural network with positioncompensation when given the feasible actuator joint variabledisplacement, which denotes the available utmost range ofthe end-effecter of the parallel manipulator. Figure 11 shows
0.3
0.3
0.4
0.4
0.5
0.5
0.6
0.6
0.70
0.70
0.800.80
0.80
0.700.6
0.50.4
0.3d2 (mm)
d 3 (m
m)
d1 (mm)
Figure 9: The three-dimensional view of the parallel manipulator.
0.80 0.70 0.6 0.5 0.4 0.3
0.3
0.4
0.5
0.6
0.70
0.80
d1 (mm)
d 3 (m
m)
Figure 10: The two-dimensional view of the joint space of the parallel manipulator.
11International Journal of Aerospace Engineering
the operation workspace of the redundantly actuated parallelmanipulator, and Figure 12 shows the projection of the two-dimensional view. From the two figures, we can see that thedistribution is uniform, the change is stable, and there is nosudden change, which illustrate that the proposed parallelmanipulator has a greater workspace and comparatively bet-ter orientation operation capability. Therefore, the parallelmanipulator can reach to a certain position with one or sev-eral different orientations with good dexterity, which is justrequired to further improve the flexibility in completing thecomplicated surface machining. What is more, due to thesymmetry of the parallel manipulator, the distribution of
the operation workspace about the coordinate axis also pre-sents certain symmetry.
6. Conclusions
In this paper, a novel 1T2R redundantly actuated 2RPU-2SPRparallel manipulator has been presented, which can beapplied to high-speedmachining in a serial and annular guiderail for milling some large heterogeneous complex freedomsurfaces in the aerospace field. According to the investigationof this paper, the following contributions can be drawn:
40200-20-40-60-40-2002040
0. 7
0. 8
0. 6
0. 5
0. 4
0. 3
0. 260
z (m
m)
𝛽 (°)𝛼 (°)
Figure 11: The operation workspace of the parallel manipulator.
0.2
0.3
0.4
0.5
0.6
0.7
0.8
-60 -40 40 60-20 0 20𝛽 (°)
z (m
m)
Figure 12: The two-dimensional view of the operation workspace of the parallel manipulator.
12 International Journal of Aerospace Engineering
(1) The mobility of the proposed redundantly actuated2RPU-2SPR parallel manipulator has been analyzedin detail based on the screw theory and Grübler-Kutzbach (G-K) criterion, and the correctness of theanalysis method is verified
(2) The inverse kinematic position of the redundantlyactuated parallel manipulator has been sequentiallyanalyzed, which lay the basis for the BP neural net-work training samples. The forward kinematics solu-tion problem of the redundantly actuated parallelmanipulator can be effectively solved by the BP neu-ral network with position compensation with highaccuracy, and the calculation results are in goodaccordance with the high precision requirement ofthe milling process, which lay a foundation for fur-ther dynamic modeling and real-time control of theexperimental prototype
(3) The joint space and workspace are depicted andconstructed. The analytic results illustrate that theproposed manipulator is a good candidate for engi-neering application. The proposed methodology torealize two rotations and one translation can alsoextended to other parallel manipulators includingredundantly actuated and nonredundantly actuatedparallel manipulators with a complicated forwardkinematics problem as well. In the future works, wewill focus on the dynamic characteristic analysis,error identification and compensation, sensitivityanalysis, and real-time control
Data Availability
The data used to support the findings of this study areavailable from the corresponding author upon request.
Conflicts of Interest
The authors declared no potential conflicts of interest withrespect to the research, authorship, and/or publication ofthis article.
Acknowledgments
The authors would like to acknowledge the financial sup-port of the Fundamental Research Funds for the CentralUniversities under Grant Nos. 2018JBZ007, 2018YJS136,and 2017YJS158; China Scholarship Council (CSC) underGrant No. 201807090079; the Natural Sciences and Engi-neering Research Council of Canada (NSERC); and theYork Research Chairs (YRC) program. Meanwhile authorHaiqiang Zhang is grateful to Advanced Robotics andMechatronics Laboratory and the science librarian JohnDupuis in York University.
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