Vol.37No.S
Transactions of Nanjing University of Aeronautics and Astronautics Nov.2020Isotropic Optimization of a Stewart‑Type Six‑Component
Force Sensor
ZHANG Tao ，XIAO Xinyi ，YANG Dehua *，WU Changcheng
College of Automation Engineering ，Nanjing University of Aeronautics and Astronautics ，Nanjing 211106，P.R.China
（Received 12May 2020；revid 23July 2020；accepted 30July 2020）
同胞
Abstract:This paper prents the isotropic optimization of a Stewart -type six -component force nsor.First ，the static model of the nsor is built by the screw theory and the forward isotropy indexes and the inver isotropy indexes are further prented.Second ，a comprehensive evaluation function is established to evaluate the isotropic performance of the nsor.By compromising all the isotropy indexes and solving the extreme value of the function ，the nsor optimization process is completed and an optimal solution of a t of nsor structure parameters is
obtained.Finally ，the design of the components and the asmbly of the prototype are established by 3D modeling software Pro -E.The verification of the isotropic performance of the nsor is conducted by the finite element analysis software ANSYS.The results obtained by our rearch can provide uful reference to the isotropic performance evaluation and structure design of the stewart -type six -component force nsor.Key words ：Stewart ；force nsor ；isotropy ；optimization ；finite element analysis CLC number ：TP212.1
Document code ：A性格英文
Article ID ：1005‑1120（2020）S‑0102‑07
0Introduction
With the ability of measuring three force com ‑
ponents and three torque components applied to it ，the six -component force/torque nsor is one of the most important and challenging nsors for re‑arch.It is widely ud in precision asmbly ，con‑tour tracking ，wind tunnel testing ，and force control of robots.The Stewart platform is a classic six -de‑gree -of -freedom （6-DOF ）parallel mechanism pro‑pod by Stewart ［1］.It has
been successfully applied to the field of six -dimensional force nsors ，thanks to its many advantages.First ，Stewart platform pos‑ss the distinguishing advantages of good stiff‑ness ，symmetric and compact design ［2］.Second ，Stewart platform has been studied for a long time and has become a mature theoretical system.Most importantly ，each elastic measurement limb of the Stewart platform -bad force nsor just sustains tensile strain or compressive strain along its axis.With the neglect of the gravity of the legs and the
frictional moment in the spherical pairs ，the nsor can realize the measurement of six -component force/torque without stress coupling in theory ［3］.
For optimal design ，structural design is particu‑larly important ，since many of the nsor ’s perfor‑mance are directly or indirectly related to the struc‑tural parameters.Isotropy is one of the most impor‑tant performance of force nsor ［4］.It leads to the minimum relative error in the force mapping ［2］.How to design a nsor with excellent performance has been extensively studied by many authors.Zhao ［5］propod a design method bad on the performance indices atlas.Yao et al.［6］listed the analytical rela‑tionship between the comprehensive performance in‑dex and the structural parameters ，which proved that the nsor can not satisfy both the forward iso‑tropic and the inver isotropic.Ref.［7］propod a design method bad on optimized triangular cones.
In this paper ，an optimization method is dis‑cusd.By establishing a comprehensive perfor‑mance evaluation function and solving the extremity
*Corresponding author ，E -mail address ：***************.
How to cite this article :ZHANG Tao ，XIAO Xinyi ，YANG Dehua ，et al.Isotropic optimization of a Stewart -type six -com ‑ponent force nsor ［J ］.Transactions of Nanjing University of Aeronautics and Astronautics ，2020,37(S)：102‑108.http ：//dx.doi/10.16356/j.1005‑1120.2020.S.013
No.S ZHANG Tao,et al.Isotropic Optimization of a Stewart -Type Six -Component Force Sensor of it ，the optimal t of structural parameters is ob‑tained when the comprehensive performance index is the best.The prototype is further modeled accord‑ing to the structural parameters.
1Statics Model of Stewart‑Type Force Sensor
As the foundation for the further rearch of
the nsor ，the static mathematical model of the six -component force nsor should be built first.As show in Fig.1，the classical six -component force/torque nsor bad on Stewart platform is com ‑pod of an upper platform ，a lower platform and six elastic legs connecting the tow platfor
ms with spherical joints.
The Cartesian coordinate OX B Y B Z B is t up with its origin located at the geometrical center of the upper platform.The X B axis is perpendicular to the line connecting B 1and B 6.Symbols are defined as follows ：B i （i =1，2，…，6）and A i （i =1，2，…，6）denote the position vectors of the center of the i -th spherical joint on the upper platform and the low ‑er platform with respect to the coordinate system ，respectively ；R B and R A denote the radius of the cir‑cles ，with which the centers of spherical joint locat‑ed ，on the upper platform and the lower platform ，respectively ；H denotes the distance between the upper and the lower platform ；αB denotes the twist angle between B 1and B 6；αA denotes the twist angle between A 1and A 6；In addition ，the twist angles be‑tween B 1and B 3，between B 3and B 5，between A 1and A 3，between A 3and A 5are
2π
3
.For the equilibrium of the upper platform ，the following equation can be obtained by screw theo‑ry ［8］
∑i =1
6
f i
$i =F
F
+F M (1)
where f i reprents the reaction produced on the i th elastic leg ；$i the unit line vector alone the i th leg ；F F and F M reprent the force vector and the mo‑ment vector applied on the center of the upper plat‑form ，respectively.
Eq.（1）can be rewritten in the form of matrix equation as
F =
G ·f
(2)
where F ={F F ，F M }T
={F x F y F z M x M y M z }T
is a six -dimensional vector compod of the external force and moment ；f ={f 1f 2f 3f 4f 5f 6}T
a vector
compod of the axial forces of the six legs ；G is the
force Jacobian matrix ［9］which is given by
G =
éëêùûú
S 1S O 1S 2S O 2S 3S O 3S 4S O 4S 5S O 5S 6S O 6ì
í
î
ïïïïS i =
B i -A i ||B i -A i S oi =A i B i ||B i -A i (3)
where
B i -A i =[R B cos φi -R A cos βi ,R B sin φi -R A sin βi H ]T
A i ×
B i =[R B H sin φi ,
-R B H cos φi R A R B sin (φi -βi )]T
|B
i
-A i |=
(R B cos φi -R A cos βi )2+(R B sin φi -R A sin βi )2+H 2
φi ={φ1,φ2,φ3,φ4,φ5,φ6}=
{αB /2,120-αΒ/2,120+αΒ/2,-120-αΒ/2,αΒ/2-120,-αΒ/2}βi ={β1,β2,β3,β4,β5,β6}=
{αΑ/2,120-αΑ/2,120+αΑ/2,-120-αΑ/2,αΑ/2-120,-αΑ/2
}
Fig.1
Statics model of Stewart -type force nsor
103
Vol.37
Transactions of Nanjing University of Aeronautics and Astronautics If G is not singular ，Eq.（2）can be expresd as
f =G -1·F =C ·F
(4)
where C is the inver mapping matrix.
Obviously ，the element of the matrix G is just determined by the five parameters of the Stewart platform ，including R B ，
R A ，αA ，αB and H .And the characteristics of matrix G will directly impact the performance of the nsor ，which is the basis of the force nsor design.Therefore ，structur‑al design is particularly important for optimal design.
2Isotropic Characterization of
Sensor
Different from common uniaxial force nsors ，the six -component force nsor bad on Stewart platform is designed to measuring all applying force components applied to the center of the upper plat‑form.So it is expected that the nsor have the same measurement nsitivity in six -force compo‑nent.In another word ，the nsor needs to be isotro‑pic.The isotropic nsor may be considered as the most slightly affected by the interferential noi ，the machining error ，and other error sources ［2］.From the perspective of mathematical analysis ，when the condition number ［5］of G matrix is 1，the nsor is isotropic.Generally speaking ，it is non -n in physics to measure isotropic performance between the force and the moment.Now ，it is usually to dis‑cuss the force isotropy and moment isotropy per‑ately.Bad on the discussion above ，matrixes G and C can be expresd as
G =[G F G M ]T (5)C =[C F C M ]
(6)
where
G F G T
F =éëêêêêù
ûúúúúψF 1000ψF 2000ψF 3G M G T
M =éëêêêêù
û
úú
úúψM 1000ψM 2000ψM 3C F C T F
=éëêêêêù
红楼梦对联
û
ú
ú
úúςF 1000ςF 2000ςF 3C M C T
M =éëêêêêùû
úú
úúςM 1000ςM 2000ςM 3L =
R 2A +R 2B -2R A R B cos [(αA -αB )/2]+H
2
m =R 2A +R 2B -2R A R B cos [(αA -αB )/2]
n =2R 2A sin 2
[(αA -αB )/2]
眼镜的品牌ψF 1=ψF 2=3m /L 2,ψF 3=6H 2/L 2
ψM 1=ψM 2=3H 2R 2B /L 2,ψM 3=3R 2B n /L
2
ςF 1=ςF 2=2L 2/3n ςF 3=L 2/6H 2
ςM 1=ςM 2=2mL 2/3H 2R 2B n
ςM 3=L 2/3R 2B n
where G F and C F are the first three rows of the G matrix and the first three columns of the C matrix ，respectively ，which are the force transfer factors.G M and C M are the last three rows of the G matrix and the last three columns of the C matrix ，respec‑tively ，which are the torque transfer factors.The four matrices G F ，G M ，C F ，and C M comprehensively reflect the isotropic performance of the nsor ，and are the basis for studying the isotropic performance and structural design of the nsor.
Many factors should be considered in the de‑sign of the six -component force nsor ，Including isotropy ，nsitivity ，stiffness ，magnification ，etc.Among them ，many indexes are mutually con‑strained.Not all nsors can achieve isotropic.So we need to define some indexes to measure the force isotropy performance and torque isotropy perfor‑mance of the nsor.Bad on the mathematical model established in the cond part and four matri‑ces ，the six -component force/torque nsor bad on Stewart platform isotropy indexes are defined as follows
（1）Forward force isotropy index I GF
I GF =1
cond (G F )=[λmin (G F G T
F )]1/2[λmax (
G F G T
F )]1/2
(7)
104
No.S ZHANG Tao,et al.Isotropic Optimization of a Stewart-Type Six-Component Force Sensor （2）Forward torque isotropy index I GM
I GM=1
cond(G M)=[λmin(G M G T M)]1/2
[λmax(G M G T M)]1/2
(8)
（3）Inver force isotropy index I CF
I CF=1
cond(C F)=[λmin(C F C T F)]1/2
[λmax(C F C T F)]1/2
(9)
（4）Inver torque isotropy index I CM
I CM=1
cond(C M)=[λmin(C M C T M)]1/2
[λmax(C M C T M)]1/2
(10)
whereλmin(·)andλmax(·)denotes the minimum and maximum eigenvalue of matrix，respectively.
Obviously，the larger the value of the four isot‑ropy indexes，the better the isotropic performance of the nsor.In particular，when the nsor satis‑fies both the forward force isotropy and the forward torque isotropy，it is said to satisfy the forward isot‑ropy；when the nsor satisfies both the rever force isotropy and the rever torque isotropy，it is said to satisfy the inver isotropy.When the nsor satisfies both the forward isotropy and the rever isotropy，it is said to satisfy the complete isotropy. 3Isotropic Optimization of Sensor For Stewart platform-bad force nsors，Ref.［5］has demonstrated that it is impossible to satisfy the forward isotropy and the rever isotropy simul‑taneously.In another word，it is impossible to achieve the complete isotropy.Therefore，
optimiz‑ing the design requires both consideration of design requirements and trade-offs of multiple performance indexes，in order to achieve the best overall isotropy performance.
As can be en from the definition of the above isotropy indexes，I GF，I GM，I CF and I CM all range from 0to1.The clor to1is，the better the isotropic performance of the nsor.In order to obtain the best comprehensive isotropy performance，multi-ob‑jective optimization bad on various performance indicators is required.The optimization problem is formulated as follows
Min(I=K1I GF+K2I GM+K3I CF+K4I CM)(11) where K1and K2denote the weight of the forward force isotropy index and the forward torque isotropy index，respectively；K3and K4the weight of the in‑ver force isotropy index and the inver torque isotropy index，respectively.Engineers can choo different weight ratios bad on concrete design goals.In this paper，all isotropy indexes here are considered together，the weights are t as
K1=K2=K3=K4=1(12) By solving the extreme of the function I，the best comprehensive isotropy index of Stewart plat‑form-bad force nsor is achieved.Meanwhile，one t of structural parameters with the best com‑prehensive isotropy index is obtained.
解忧杂货店读后感
The specific optimization process is reverd.
A cluster of feasible solutions within the design con‑straints is given.The values of the isotropy index and the optimization function I corresponding to each feasible solution are calculated，bad on the above discussion，the optimal solution is obtained when the optimization function gets the minimum. The design constraints are t as follows
ì
í
î
ïï
ïï
L=84mm
αB=20°
50mm<R B=R A<300mm
30°<αA<90°
(13)
As en from Fig.2，each point on the graph is a feasible design and the Y-coordinate gives corre‑sponding optimization function value of that design （point）.Obviously，the minimum value of the Y-coordinate is not4.This is becau that the classic Stewart structure six-component force nsor can‑not achieved the completely isotropy.In another word，I GF=I GM=I CF=I CM=1is impossible.So the minimum value of the optimization function always greater than4.Under this premi，it is known from the optimization results that the minimum value of the function is5.6569when I GF=I GM=I CF=I CM= 2/2and at this point the nsor’s comprehensive isotropy performance is optimal.
It is worth noting that from the engineering
105
Vol.37
Transactions of Nanjing University of Aeronautics and Astronautics point of view ，the solution clo
秋夜鲁迅>灵活的英文to the minimum value can also be regarded as the optimal solution.The solutions are listed in Table 1.
4Isotropic Verification by Finite Element Analysis
Finally ，in order to make the mechanical as‑
mbly more convenient ，we lect a t of structur‑al parameters in Table 1as the structural parameters of the prototype ：αA -αB =70°，R B =R A =100mm ，
L =84mm ，H =58.64mm.
The following points should be noted in the de‑sign and lection of components.
（1）The connection pair is chon to u flexi‑
ble hinge with three degrees of rotational freedom.The reason why the real ball joint is not ud is that it caus void and friction ，which affects the mea‑surement accuracy of the nsor.In addition to this ，the real ball hinge also limits the miniaturization of the nsor.
（2）Measurement of the reaction produced on the six elastic legs is obtained by six uniaxial force nsors.Uniaxial force nsors require further cali‑
bration to ensure measurement accuracy.
The parameters of the specific uniaxial force nsor are shown in Table 2.
The model is built through the 3D modeling
software Pro -E as shown in Fig.3.The detailed ex‑ploded view is shown in Fig.4.The 3D model was imported into the finite element analysis software ANSYS for further verification of the isotropic per‑
formance of the nsor.The better the isotropic per‑formance of the nsor ，the clor the deformation caud by the force/torque of different dimensions acts on the nsor.Since the nsor is symmetrical （the same performance in the X and Z directions ），only the following four cas need to be
simulated.
Fig.2Feasible solution t Table 1
Optimal solution t
Parameter t αA =50°
R B =228mm H=59.27mm αA =55°
R B =196mm H=59.16mm αA =60°
R B =172mm H=59.06mm αA =65°
R B =153mm H=59.09mm αA =70°
R B =139mm H=58.61mm αA =80°
R B =116mm H=58.74mm αA =90°
R B =100mm H=58.64mm αA =100°
R B =88mm H=58.59mm
Isotropic indexes I CF =0.7100I CM =0.7103I GF =0.7040I GM =0.7042I CF =0.7127I CM =0.7098I CF =0.7004I CM =0.7015I CF =0.7152I CM =0.7099I CF =0.7004I CM =0.6991I CF =0.7143I CM =0.7317I CF =0.7006I CM =0.7000I CF =0.7259I CM =0.7005I CF =0.7087I CM =0.6888I CF =0.7228I CM =0.7161I CF =0.6982I CM =0.6917I CF =0.7252I CM =0.7230I CF =0.6916I CM =0.6859I CF =0.7265I CM =0.7324I CF =0.6827I CM =0.6882
Value of function I
5.6569
5.6570
手脚心发热
5.6571
5.6571
5.6578
5.6578
5.6584
5.6596
Table 2Parameter of uniaxial force nsor
Parameter Sensitivity/(mV·V -1)
Nonlinearity/（%F.S ）Repeatability/（%F.S ）
Lag/（%F.S ）Zero output/（%F.S ）
Excitation voltage/V
Value 1.0±0.05±0.5±0.5±0.5±110—15
106
