sensors Article

Mathematical Model and Calibration Experiment of a Large Measurement Range Flexible Joints 6-UPUR Six-Axis Force Sensor Yanzhi Zhao 1,2, *, Caifeng Zhang 1,2 , Dan Zhang 3 , Zhongpan Shi 1,2 and Tieshi Zhao 1,2 1

2 3

*

Key Laboratory of Parallel Robot and Mechatronic System of Hebei Province, Yanshan University, Qinhuangdao 066004, China; [email protected] (C.Z.); [email protected] (Z.S.); [email protected] (T.Z.) Key Laboratory of Advanced Forging & Stamping Technology and Science of Ministry of Education of China, Yanshan University, Qinhuangdao 066004, China Department of Mechanical Engineering, Lassonde School of Engineering, York University, 4700 Keele Street, Toronto, ON M3J1P3, Canada; [email protected] Correspondence: [email protected]; Tel./Fax: +86-335-805-7031

Academic Editor: Vittorio M. N. Passaro Received: 16 June 2016; Accepted: 5 August 2016; Published: 11 August 2016

Abstract: Nowadays improving the accuracy and enlarging the measuring range of six-axis force sensors for wider applications in aircraft landing, rocket thrust, and spacecraft docking testing experiments has become an urgent objective. However, it is still difficult to achieve high accuracy and large measuring range with traditional parallel six-axis force sensors due to the influence of the gap and friction of the joints. Therefore, to overcome the mentioned limitations, this paper proposed a 6-Universal-Prismatic-Universal-Revolute (UPUR) joints parallel mechanism with flexible joints to develop a large measurement range six-axis force sensor. The structural characteristics of the sensor are analyzed in comparison with traditional parallel sensor based on the Stewart platform. The force transfer relation of the sensor is deduced, and the force Jacobian matrix is obtained using screw theory in two cases of the ideal state and the state of flexibility of each flexible joint is considered. The prototype and loading calibration system are designed and developed. The K value method and least squares method are used to process experimental data, and in errors of kind I and kind II linearity are obtained. The experimental results show that the calibration error of the K value method is more than 13.4%, and the calibration error of the least squares method is 2.67%. The experimental results prove the feasibility of the sensor and the correctness of the theoretical analysis which are expected to be adopted in practical applications. Keywords: six-axis force sensor; parallel mechanism; flexible joints; calibration experiment

1. Introduction With the ability of measuring three force components and three torque components, the six-axis force sensor is one kind of the most important and challenging sensors, used widely in many research areas such as wind tunnel balances, thrust stand testing of rocket engines, and in robotics, the automobile industry, aeronautics, etc. [1]. Compared with single-axis sensors, not only the volumes and prices of multi-axis force sensors are considered, but also their structures, in order to achieve a balance between the isotropy of force/torque and that of sensitivity [2]. Recently, researchers all over the world have done a lot of work on six-axis force sensors. Since the Stewart platform was applied to the measurement of space six-axis forces by measuring the forces in the six legs with convection elements in 1983 [3], parallel structure have been widely used in six-axis force sensors [4], stimulated by the advantages of good stiffness, symmetric and compact Sensors 2016, 16, 1271; doi:10.3390/s16081271

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structure, and straightforward mapping expression between the wrenches applied on the platform and the measured leg forces. Nguyen et al. [5] developed a Stewart platform-based sensor with LVDT’s mounted along the legs for force/torque measurement in the presence of passive compliance. Durand changed the traditional Stewart structure and put piezoelectric quartz inside the Stewart structure and pre-tightened it [6], therefore, the sensor was more compact and could be used to measure tensions. Dwarakanath et al. [7,8] introduced the usage of ring-shaped sensing element in the Stewart platform sensor, and presented a simply supported, ‘joint less’ six-axis parallel force sensor. Kim et al. [9] put forward a six-axis wrist force sensor using FEM for an intelligent robot. Although research on the traditional Stewart platform-based six-axis force sensor is quite mature, the sensor with common joints has a lower precision, and the performance of each direction is different. In comparison, the sensor with flexible joints has features of compact structure, fast response, small accumulated error, no mechanical friction, and high measurement accuracy because of the use of integrated design, so it has broad application prospects [10–14]. Kerr [15] suggested that the Stewart platform with instrumented elastic legs can be used as a six-axis force sensor. Jin et al. [16] proposed a novel isotropic six-axis force sensor based on a variation of Stewart platform, whose three pairs of elastic legs are perpendicular to the three orthogonal surfaces of the basic cube. Gao et al. [17] developed a six-axis controller based on Stewart platform-based force sensor, and introduced the elastic joints to replace the real spherical joints which made the miniaturization possible. Liang et al. [18] designed and developed a new six-axis sensor system with a compact monolithic elastic element (EE), which detected the tangential cutting forces Fx , Fy and Fz (i.e., forces along x-, y-, and z-axis) as well as the cutting moments Mx , My and Mz (i.e., moments about x-, y-, and z-axis) simultaneously. Unfortunately, most of the sensors mentioned above are used in a small range of applications. Nowadays, the large measurement range six-axis force sensor is more and more widely needed in applications such as aircraft landing gear momentum tests, rocket thrust tests, spacecraft docking and wind tunnel tests, whose precision requirements are also getting increasingly higher. However, when the parallel six-axis force sensor is used in large measurement range occasions, the traditional joints are difficult to be processed into flexible joints, and the improvement of measurement accuracy will be affected, so parallel six-axis force sensors with large measurement range and high accuracy are urgently required. When the force transmission relation of the sensor is established, the stiffness of each flexible joint and the whole stiffness which has a direct impact on the measuring accuracy, bearing capacity and dynamic performance, and is one of the important indexes to measure the performance of sensors must be considered. Research on stiffness of the sensors based on parallel mechanism began in the 1990s [19], which gave Gosselin’s minimalist stiffness mapping model of parallel mechanism under no loading. Then, Griffis and Duffy [20] proposed a kind of parallel mechanism with branches which are assumed to be wire springs. Under the premise of joints are equivalent, they considered the effect of differential motion, and established the complete stiffness model by the analytical method. Chakarov [21] analyzed the effect of external load on the overall stiffness matrix in force redundant parallel mechanism. Pashkevich et al. [22] proposed a kind stiffness modeling method of the over-constrained parallel mechanism with flexible branches and flexible drive joints. Zhang and Wei [23] derived the stiffness model of the mechanism and evaluated the global stiffness using the sum of the diagonal elements of the stiffness matrix. However, existing institutions to complete stiffness modeling methods are very complex, and there is little research combining together stiffness and force Jacobian matrix. In this paper, a kind of six-axis force sensor based on 6-UPUR parallel mechanism with flexible joints, which has large measurement range and high accuracy is proposed. Meanwhile, the complete mathematical model considering the flexibility of the joints is established, and the calibration experiment is completed. The structure of this paper is as follows: after the introduction, Section 2 concerns the structural analysis of the sensors, including the intrinsic disadvantages of the traditional parallel sensors based on the Stewart platform and the structure features of the large measurement range six-axis force sensor of 6-UPUR parallel mechanism with flexible joints. The measuring principle, mathematical model of

6, summarizing the work that has been done. 2. Sensor Structure Figure 1a illustrates the traditional parallel sensor diagram, which is based on the Stewart platform Sensors 2016, 16, and 1271 composed of a measuring platform, a lower fixed platform and six elastic legs 3 of 20 connecting the two platforms with traditional spherical joints. Considering the traditional spherical joints are difficult to processed into flexible joints in a large the structure which included the ideal state and the state of flexibility ofaffected, each flexible is considered measurement range situation, and the measurement accuracy will be whichjoint restricts its application in six-axis force sensors based on flexible parallelon mechanism, this paper proposes in Section 3. Section 4 introduces the experimental research static calibration of the sensora kind prototype. of structure model of six-axis force sensor based on a parallel 6-UPUR mechanism. As shown in the Section presents Sensors52016, 16, 1271the results of the experiment. The paper is concluded in Section 6, summarizing 3 of 20 Figure 1b, it consists of a measuring platform, a fixed platform and six measuring legs divided into work that has been done. groupsinofSection legs and two legs4 in each group located in aresearch verticalon plane. measuring leg isthree considered 3. Section introduces the are experimental staticEach calibration of the contains a single-axis force sensor and connects the fixed platform with flexible universal joint, and 2. sensor Sensorprototype. StructureSection 5 presents the results of the experiment. The paper is concluded in Section the measuring platform withdone. combined spherical joint. Through the improvement of the 6,connects summarizing the work that has been Figure 1a illustrates the traditional parallel diagram, which is based onwhich the Stewart platform traditional Stewart platform mechanism, andsensor the introduction of flexible joints have the and2.peculiarities composed of a measuring platform, a lower fixed platform and six elastic legs connecting the two of non-clearance, friction-less and high sensitivity to replace the traditional spherical Sensor Structure joints, it is possible to develop a sensor with a large measurement range. platforms with traditional spherical joints. Figure 1a illustrates the traditional parallel sensor diagram, which is based on the Stewart platform and composed of a measuring platform, a lower fixed platform and six elastic legs connecting the two platforms with traditional spherical joints. Considering the traditional spherical joints are difficult to processed into flexible joints in a large measurement range situation, and the measurement accuracy will be affected, which restricts its application in six-axis force sensors based on flexible parallel mechanism, this paper proposes a kind of structure model of six-axis force sensor based on a parallel 6-UPUR mechanism. As shown in Figure 1b, it consists of a measuring platform, a fixed platform and six measuring legs divided into three groups of legs and two legs in each group are located in a vertical plane. Each measuring leg contains a single-axis force sensor and connects the fixed platform with flexible universal joint, and connects the measuring platform with combined spherical joint. Through the improvement of the traditional Stewart platform mechanism, and the introduction of flexible joints which have the (b) traditional spherical (a) peculiarities of non-clearance, friction-less and high sensitivity to replace the joints, it is possible to develop a sensor with a large measurement range. Figure 1. (a) The force sensor structure based on the traditional Stewart platform; (b) the force sensor Figure 1. (a) The force sensor structure based on the traditional Stewart platform; (b) the force sensor structure based on 6-UPUR parallel mechanism. structure based on 6-UPUR parallel mechanism. The characteristic parameters of the parallel 6-UPUR six-axis force sensor include radius R of Considering traditional joints platform, are difficult processedangle into flexible in a large the measuring the platform, radiusspherical r of the lower the to positioning A, B, C, joints the center distance l1 ofrange the U situation, joint and location center distance l2 ofaccuracy the R joint,will as shown in Figurewhich 2. measurement and the measurement be affected, restricts its

application in six-axis force sensors based on flexible parallel mechanism, this paper proposes a kind of structure model of six-axis force sensor based on a parallel 6-UPUR mechanism. As shown in Figure 1b, it consists of a measuring platform, a fixed platform and six measuring legs divided into three groups of legs and two legs in each group are located in a vertical plane. Each measuring leg contains a single-axis force sensor and connects the fixed platform with flexible universal joint, and connects the measuring platform with combined spherical joint. Through the improvement of the traditional (b) (a) Stewart platform mechanism, and the introduction of flexible joints which have the peculiarities of Figure 1. (a) The force sensor based on the Stewart platform;spherical (b) the force sensorit is possible non-clearance, friction-less and structure high sensitivity to traditional replace the traditional joints, structure based on 6-UPUR parallel mechanism. to develop a sensor with a large measurement range. The characteristic parameters of the parallel 6-UPUR six-axis force sensor include radius R of the The characteristic parameters of the parallel 6-UPUR six-axis force sensor include radius R of measuring platform, radiusradius r of the platform, the positioning angle A, B,A,C,B,the distance l1 the measuring platform, r oflower the lower platform, the positioning angle C, center the center of the U joint and location center distance l of the R joint, as shown in Figure 2. Figure 2. Schematic diagram of characteristic 2 distance l1 of the U joint and location center distance l2 of the R joint,parameters. as shown in Figure 2.

Figure 2. of characteristic parameters. Figure 2. Schematic Schematicdiagram diagram of characteristic parameters.

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3. Mathematical Model 3.1. Ideal Mathematical Model 3.1.1. Force Analysis There are reacting forces on measuring legs when an external force loaded on the measuring platform of the parallel sensor whose surface is assumed to be friction-less and continuous, and the reacting force is considered along the axis of each measuring leg. According to the space static balance Sensors 2016, 16, 1271 4 of 20 conditions of the measuring platform, the following equation can be obtained on screw theory [24]: 3. Mathematical Model 6 ÿ

3.1. Ideal Mathematical Model

fi Si “ F` P M

(1)

i “1

3.1.1. Force Analysis

where fi represents the reacting force on the measuring Si represents theonunit vector along the There are reacting forces on measuring legs when legs; an external force loaded the line measuring ith measuring leg; F and M, respectively, represent the ith loaded force vector and toque vector on the platform of the parallel sensor whose surface is assumed to be friction-less and continuous, and the reacting force is considered along the axis of each measuring leg. According to the space static balance center of the measuring platform; P is the dual sign. conditions of the measuring platform, the following equation can be obtained on screw theory [24]: Equation (1) can be rewritten in the form of matrix expression as: 6

fS i

i

 F  M

Fω “ GFf f

(1)

i=1

(2)

where fi represents the reacting force on the measuring legs; Si represents the unit line vector along

“ ‰ the“ith F measuring leg; F and M,Trespectively, represent the ith loaded forceforce vectorapplied and toque vector where Fω is the vector of six-axis external on the measuring x Fy Fz Mx My Mz on the center of the measuring T platform;  is the dual sign. platform; f “ r f 1 f 2 f 3 f 4 f 5 f 6 s is the vector composed of the reacting forces of the six measuring Equation (1) can be rewritten in the form of matrix expression as: legs; GFf is the first-order static influence coefficient matrix. If GFf is non-singular matrix, the inverse F Fω  G f f (2) mapping between Fω and f is: f “ six-axis GF Fw external force applied on the measuring (3) = [ ] is the vectorf of where ] = [ is the vector composed of the reacting forces of the six measuring platform; ´ ¯ ´1 f F the first-order static influence coefficient matrix. If is is non-singular matrix, the the inverse where, Glegs; is the force Jacobi matrix, and it is the mapping matrix from external force F “ Gf mapping between Fω and f is:

loaded on the measuring platform of the sensor to the force produced on the six measuring leg. GFf is f  G Ff F w

(3)

closely related with the geometric structure of the sensor, and the characteristics of GFf determine force Jacobiother matrix,features and it is the mapping matrix from the external force ( ) is theand stiffness,where, isotropy,=sensitivity many of the sensor. 3.1.2.

loaded on the measuring platform of the sensor to the force produced on the six measuring leg. determine is closely related withModel the geometric structure of the sensor, and the characteristics of Ideal Mathematical stiffness, isotropy, sensitivity and many other features of the sensor.

Each leg of the parallel 6-UPUR mechanism can be seen as a series open-chain mechanism which 3.1.2. Ideal Mathematical Model is composed of six links and six joints, and the U joint is equivalent to two joints whose axes are vertical of the can included be seen as ain series whichis connected and intersect.Each Thelegbase isparallel called6-UPUR link 0, mechanism which is not theopen-chain six links.mechanism The first link is composed of six links and six joints, and the U joint is equivalent to two joints whose axes are to the base by the first rotational joint; the second link is connected to the first link by the second vertical and intersect. The base is called link 0, which is not included in the six links. The first link is rotationalconnected joint, and sobase on. by the first rotational joint; the second link is connected to the first link by the to the rotational joint, and so on. to the base fixedly is referred to as {0}; the coordinate system on The second coordinate system attached The coordinate system attached base fixedly is referred to as {0}; the coordinate system on the reference point of the end of the legtoisthereferred to as {P}; coordinate system attached to the ith link the reference point of the end of the leg is referred to as {P}; coordinate system attached to the ith link fixedly is referred to as{i}, as shown in Figure 3. fixedly is referred to as{i}, as shown in Figure 3.

Figure Schematic diagram of the linkslinks and joints. Figure 3. 3. Schematic diagram of the and joints.

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The differential motion vector of central reference point P which is on the measuring platform is: ´ DP “

dP

δh

¯T

” ı . “ GφP φ

(4)

” ı ´ ¯ where: δh “ δhx δhy δhz is the angle variable, d P “ d Px d Py d Pz is the displacement variable, GφP is ”. . . ıT the first-order static influence coefficient matrix, φ “ φ1 . . . φ6 is the differential motion of the six legs: « ” ı GφP “

S1 S2 0 S1 ˆ pP ´ R1 q S2 ˆ pP ´ R2 q S3

S4 S5 S6 S4 ˆ pP ´ R4 q S5 ˆ pP ´ R5 q S6 ˆ pP ´ R6 q

ff (5)

S1 . . . S6 are the directions of joints’ axes of each leg, and they are: ! S1 “

0

0

1

)T

” S j “ T j ´1

0

cosαp j´1q j

´sinαp j´1q j

ıT

(6)

Tj is rotational transformation matrix: ” Tj “

ı a jk

S j ˆ a jk

pj “ 1, 2, ¨ ¨ ¨ , 6q

Sj

(7)

a jp j`1q is the direction of the common normal line between adjacent axes: ” a12 “

cosθ1

sinθ1

0

ıT

” a j p j `1 q “ T j ´1

cosθ j

P ´ Rn “

cosαp j´1q j sinθ j 6 ´ ÿ

sinαp j´1q j sinθ j

ıT

(8) pj “ 2, 3, ¨ ¨ ¨ , 6q

¯ apr´1qr apr´1qr ` dr Sr ` T j Pp6q

(9)

r “ n `1

where apr´1qr is length of the common normal line between adjacent axes, dr is offset along the axis of rotation. Pp6q is the representation of the point P in coordinate at the end of the leg, and the left leg and the right leg are separately expressed as: p6q

Pl

” “

b ´

where:

R2

´

l22 4

ı l2 C 2 cos 2

´l2 sin C2

p6q

Pr

” “

b ´

R2

´

l22 4

ı ´ l22 cos C2

(10)

´l2 sin C2

˘2 ff ` ? ? R 1 ´ cosA ´ r 1 ´ cosB C “ arccos 1 ´ l22 ” ı ” ı According to the above equations, GφP and GφP can be obtained, separately: «

l

» — — — ” ı — GφP “ — — l — – » — — — ” ı — P Gφ “ — — r — –

0 0 0 1 1 0 a5 ` y p xp a1 ` a4 ` d3 ` d6 ` z p 0 0 ´pa4 ` d3 ` d6 ` z p q 0 0 0 ´1 1 0 y1p ´ a5 ´x1p a1 ` a4 ` d3 ` d6 ` z1p 0 a4 ` d3 ` d6 ` z1p 0

(11)

r

0 0 0 1 0 0 0 0 0 1 0 0

0 1 0 xp 0 ´pa4 ` d6 ` z p q 0 ´1 0 ´x1p 0 a4 ` d6 ` z1p

0 0 1 a5 ` y p d6 ` z p 0 0 0 1 y1p ´ a5 d6 ` z1p 0

1 0 0 0 ´x p ´y p fi

1 0 0 0 ´x1p ´y1p

ffi ffi ffi ffi ffi ffi ffi fl

fi ffi ffi ffi ffi ffi ffi ffi fl

(12)

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Because the z axis of the fixed coordinate system O-xyz which is on the center of the lower platform is the direction of S1 , and selection of coordinate system has a direct impact on the first-order static influence coefficient matrix, so it is necessary to take the symmetry of the load platform structure into account, thus: « ff OR ” ı ” ı 0 Oi O GP `O ˘O GφP pi “ 1, 3, 5q φ i “ O r S P Oi O O R O R i i « ff (13) OR ” ı ” ı 0 Oi P O GP ` ˘O Gφ pj “ 2, 4, 6q φ j “ l S O P Oi O O R O OR i

i

When movement of the upper platform is known, movement of the joints in each leg can be written: ” ı´1prq . pr q (14) φ “ GφP DP pr “ 1, 2, ¨ ¨ ¨ , 6q When six active members of the mechanism are determined, equations of the six active movements from the above equations are taken out and expressed as: ” ı´1paq . p aq φα “ GφP DP α: ” ı´ 1p b q . pbq φ β “ GφP DP

(15)

β:

.. . . pfq

φζ

” ı´1p f q “ GφP DP ζ:

. p aq

where φα is the differential motion of the six legs, its subscripts are leg’s number and joint’s number ” ı´1paq ” ı´1paq respectively, GφP represents the α-th line of the inverse matrix GφP . α: ” ı . q Combining the above six equations to constitute a matrix expression, like this: q “ GP DP . ” ı . ” ı ” ı´1 q The inverse solution is DP “ GqP q, GqP “ GP . Furthermore, the first-order static influence coefficient matrix can be obtained: GFf “

ˆ”

O

” ı 1 GφP ´ 1 p3, :q

O

” ı 1 GφP ´ 2 p3, :q

O

” ı 1 GφP ´ 3 p3, :q

O

” ı 1 GφP ´ 4 p3, :q

O

” ı 1 GφP ´ 5 p3, :q

O

ı T ˙ ´1 ” ı 1 GφP ´ 6 p3, :q

(16)

” ı´1 ” ı´1 p3, :q pi “ 1, . . . , 6q represents the third row vectors of the matrix O GφP pi “ 1, . . . , 6q. where O GφP i i ” ı´1 f Thus, the force Jacobi matrix is GF “ GFf , and that is the mapping matrix from the external force applied on the measuring platform of the sensor to the force produced on the six elastic legs in the ideal condition. 3.2. Mathematical Model of the Sensor with Flexible Joints The six-axis force sensor adopts the structure that all joints are flexible joints with single degree of freedom, and its 3D model is shown in Figure 4. Each leg is a split structure. The lower part of the leg is composed of a flexible universal joint with an integral structure and a lower positioning block. Each elastic leg is connected to the measuring and fixed platforms through the upper and lower positioning blocks by bolts. Thus, it’s realized that the decomposition of the six-axis external force to the six legs.

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Each elastic leg is connected to the measuring and fixed platforms through the upper and lower Each elastic leg is connected to theit’s measuring andthe fixed platforms through the upper and force lower positioning blocks by bolts. Thus, realized that decomposition of the six-axis external Each elastic leg is connected to the measuring and fixed platforms through the upper and lowerto positioning by bolts. Thus, it’s realized that the decomposition of the six-axis external force to the six legs.blocks positioning blocks by bolts. Thus, it’s realized that the decomposition of the six-axis external force to the six legs. Sensors 7 of 20 the six2016, legs.16, 1271 Force-measuring Force-measuring platform Force-measuring platform platform

Upper positioning block Upper positioning block Upper positioning block Flexible universal joint Flexible universal joint Flexible universal Single-axis force joint sensor Single-axis force sensor Single-axis force sensor

Measuring leg Measuring leg Measuring leg

Lower positioning block Lower positioning block Lower positioning block

Fixed platform Fixed platform Fixed platform

Figure4.4.Diagram Diagramofofthe thelarge largemeasurement measurementrange rangeforce forcesensor sensorofof6-UPUR 6-UPURparallel parallelmechanism mechanismwith with Figure Figure 4. Diagram of the large measurement range force sensor of 6-UPUR parallel mechanism with flexible joints. flexible Figure 4.joints. Diagram of the large measurement range force sensor of 6-UPUR parallel mechanism with flexible joints. flexible joints.

The upper positioning block is composed of two flexible rotation joint, which form a flexible The upper upper positioning positioning block block is is composed composed of of two two flexible flexible rotation rotation joint, joint, which which form form aa flexible The flexible spherical joint with composite flexible universal joint by assembling relationship. Itsform frontaview and The upper positioning block is composed of twoby flexible rotation joint, which flexible spherical joint with composite flexible universal joint assembling relationship. Its front view view and spherical joint with composite flexible universal joint by assembling relationship. Its front and graphic model are composite shown in Figure 5.universal By using joint the knowledge of material mechanics, theview rotational spherical joint with flexible by assembling relationship. Its front and graphic model model are shown shown in in Figure Figure 5. By using using the the knowledge knowledge of material material mechanics, the rotational rotational graphic 5. obtained: By of mechanics, the stiffnessmodel of theare flexible rotary joint is k1 = knowledge 8.592 × 1044 (N·m/rad). graphic are shown in Figure 5. By using the of material mechanics, the rotational stiffness of of the the flexible flexible rotary 8.592 ׈10 104 (N·m/rad). (N¨m/rad). stiffness rotary joint joint is is obtained: obtained: kk11 == 8.592 stiffness of the flexible rotary joint is obtained: k1 = 8.592 × 104 (N·m/rad).

(a) (b) (a) (b) (a) (b) Figure 5. (a) The front view of flexible rotation joint; (b) the graphic model of flexible rotation joint. Figure5.5.(a) (a)The Thefront frontview view of of flexible Figure flexible rotation rotation joint; joint; (b) (b) the the graphic graphic model model of offlexible flexiblerotation rotationjoint. joint. Figure 5. (a) The front view of flexible rotation joint; (b) the graphic model of flexible rotation joint. The lower positioning block is an integral structure, which is composed of two symmetrical positioning integral structure, symmetrical The universal lower positioning block is an structure which is composed flexible joints, with the same as the flexible universal joint inof thetwo upper part of the The lower positioning block is an integral structure, which is composed of two symmetrical joint flexible universal joints, with the same structure as the flexible universal in the upper part of the leg. Its front view and graphic model of a single joint are shown in Figure 6. By using the knowledge flexible universal joints, with the same structure as the flexible universal joint in the upper part of the leg. Its front view and graphic model of a single joint are shown in Figure 6. By using the knowledge Its front mechanics, the rotational stiffness of a single joint is obtained: k2 = 4.0852 × 104 (N·m/rad). of material leg. Its front view and graphic model of a single joint are shown in Figure 6. By using the4 knowledge (N¨m/rad). of material mechanics, mechanics, the the rotational rotationalstiffness stiffnessofofaasingle singlejoint jointisisobtained: obtained:k k2==4.0852 4.0852ˆ× 10 104 (N·m/rad). of material mechanics, the rotational stiffness of a single joint is obtained: 2k2 = 4.0852 × 104 (N·m/rad).

(b) (a) (b) (a) (b) (a) Figure 6. (a) The front view of a single joint; (b) the graphic model of a single joint. Figure 6. (a) The front view a single (b) the the graphic model model of aasingle single joint. Figure The front viewof single joint joint;;; (b) Figure 6. 6. (a)(a) The front view ofofaasingle joint (b) the graphic graphic model of of a singlejoint. joint. f

Considering the effect of elastic deformation of joints on GF , so the following will re-establish the complete force Jacobian matrix.

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Considering the effect of elastic deformation of joints on , so the following will re-establish the complete force Jacobian matrix. Figure 7 is schematic diagram flexible element coordinate systems’ establishment a leg. Figure 7 is thethe schematic diagram of of flexible element coordinate systems’ establishment onon a leg. where, O-xyz is the fixed coordinate system, O -x y z , i = 1,2,3, . . . ,6 is the local coordinate system where, O-xyz is the fixed coordinate system, i1i1-xi1i1yi1i1 zi1,i1i = 1,2,3,…,6 is the local coordinate system whichis isestablished establishedononthe thereference referencepoint pointofofthe thelower lowerpositioning positioningblock, block,OOi2i2-x-xi2yi2i2yzi2i2zis the local which the local i2 is coordinate system which established reference point standard one-axis force sensor, coordinate system which is is established onon thethe reference point of of thethe standard one-axis force sensor, zi3the is the local coordinate system which is established on reference the reference point of flexible the flexible Oi3O-xi3i3-x yi3i3zyi3i3is local coordinate system which is established on the point of the U U joint, the local coordinate system which is established reference point the joint, Oi4-xOi4yi4i4-x zi4i4 yisi4 zthe coordinate system which is established on on thethe reference point of of the i4 islocal integrated positioning block, O -x y z is the reference coordinate system which is established integrated positioning block, Oip-xipipyipipzip ipis the ip reference coordinate system which is established on theon the reference theofend of a flexible reference point point of theof end a flexible seriesseries leg. leg.

Figure 7. 7. Schematic diagram of of thethe coordinate systems forfor thethe flexible joints. Figure Schematic diagram coordinate systems flexible joints.

When there is the six-axis external force loaded on the end of the flexible series leg, by the When there is the six-axis external force loaded on the end of the flexible series leg, by the principle principle of virtual work, the following equation can be obtained: of virtual work, the following equation can be obtained:  ΔX ı  #j  J j X j   Δx j ,”Δy j , Δz j , Δαxj , Δαy j , Δαz j  (17) ∆X j “ J j X j “ ∆x j , ∆y j , ∆z j , ∆α x j , ∆αy j , ∆α jz j 1, 2, 3, 4   pj “ 1, 2, 3, 4q (17) Fj  J Fj F  F “J F j

Fj

where ΔXj is deformation vector of the end of leg’s reference point, Xj is the elastic deformation where ∆Xj is deformation vector of the end of leg’s reference point, Xj is the elastic deformation vector vector of the end of the j-th basic flexible unit, Jj is the pose transformation matrix, Fj = [fx, fy, fz, mx, my, T of the end of the j-th basic flexible unit, Jj is the pose transformation matrix, F j = [fx , fy , fz , mx , my , mz ] = , , ıT] is the reaction force on the end mz]T is the six-axis non-coplanar force, ” [ , , , is the six-axis non-coplanar force, F “ f , f , f , m , m , m is the reaction force on the end of xj yj zj x j matrix. yj zj j transformation of the j-th basic flexible unit, JFj is the force theAccording j-th basic flexible unit, JFj is the force transformation to the deformation superposition principlematrix. of series leg, the total deformation vector According to the deformation superposition principle of series leg, total deformation of the end of flexible series leg’s reference point caused by movement andthe rotation deformationvector vectorof endbasic of flexible series reference point caused by movement and rotation deformation vector of ofthe every flexible unitleg’s can be obtained: every basic flexible unit can be obtained: 4 4 X   X j   J j X j  J 1 X 1  J 2 X 2  J 3 X 3  J 4 X 4 (18) 4 4 ÿ j 1 ÿ j 1 X“ ∆X j “ J j X j “ J1 X1 ` J2 X2 ` J3 X3 ` J4 X4 (18) It can be known by the definition ofj“ the j “1 1 stiffness matrix, the relationship between the six-axis external force at the end of the leg reference point and the end deformation of the leg is: It can be known by the definition of the stiffness matrix, the relationship between the six-axis (19) F point  KX and the end deformation of the leg is: external force at the end of the leg reference where K is the stiffness matrix of the end of flexible series leg. F “ KX(21), the following equation can be obtained: (19) Substituting Equations (17) and (20) into Equation 4 where K is the stiffness matrix1 of the4 end of flexible series 4leg. X  K F   J j X j   J j K j1 Fj   J j K j1 J Fj F (20) Substituting Equations (17) and j 1 (20) into j 1Equation (21), j 1 the following equation can be obtained:

4 of the jth 4 basic flexible 4 element. The end reference point where Kj (j = 1,2,…,4) is the stiffness matrix ÿ ÿ ÿ ´1 ´1 1 X“ K´flexible F “ series J j X j “leg are J j Kestablished Jas j Kshown is and the coordinate system of each j Fj “ j J Fj F in Figure 8. Op-xpypzp (20) j “1 j “1 j“1 the reference coordinate system established on the center point of the upper platform. Oip-xipyipzip, i =

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where Kj (j = 1,2, . . . ,4) is the stiffness matrix of the jth basic flexible element. The end reference point and the coordinate system of each flexible series leg are established as shown in Figure 8. Op -x9pof yp20 zp is Sensors 2016, 16, 1271 the reference coordinate system established on the center point of the upper platform. Oip -xip yip zip , i = 1,2, . , 6reference is the reference coordinate is established the reference theofend 1,2,…, 6 is. .the coordinate systemsystem which which is established on theon reference point point of theofend of the flexible series leg. the flexible series leg.

(a)

(b)

Figure 8. Schematic diagram of flexible element coordinate systems’ establishment: (a) (a) TheThe structure Figure 8. Schematic diagram of flexible element coordinate systems’ establishment: structure diagram; (b) Top view. diagram; (b) Top view.

On the assumption that the stiffness of the upper platform and all of branches is infinite, On the assumption that the stiffness of the upper platform and all of branches is infinite, neglecting neglecting the minor deformation caused by force and the thickness of upper platform, when there the minor deformation caused by force and the thickness of upper platform, when there is a six-axis is a six-axis external force, the geometric compatibility conditions between the end reference point external force, the geometric compatibility conditions between the end reference point and the reference and the reference point of the upper platform of the ith flexible series leg: point of the upper platform of the ith flexible series leg:  Δxi  »  Δx  fi » fi  Δy   Δy  ∆xi ∆x i   —   ffi Op T — ffi ∆y ∆yi  O »R O p  OOp R T SOp ri    Δzi fi — — ffi  Δz  ffi T T ip ip — ffi — ffi q R ´ R S pr     i ∆z   ffi J X    O O —ΔX ∆z — ffi (21)(21) i ip ip i i – fl O Δα T p ∆X “ — “ Δα ffi — ffi “ Ji ∆Xi  x  ffi  033 R O p  T xi  — — ffi O ∆α ∆α ip x x 03ˆ3  i  — — ffi Oip R  Δα  ffi  – fl ∆α ∆αyi  Δαyi  –  y y  fl     ∆αzi ∆α  Δα z z  Δαzi  where ΔX∆X is displacement vector at the reference point of of thethe upper platform. where is displacement vector at the reference point upper platform. According to to thethe synthesis principle of of thethe space force system, thethe relationship between thethe According synthesis principle space force system, relationship between external force vector at the reference point of the upper platform andand thethe reaction force vector on on thethe external force vector at the reference point of the upper platform reaction force vector sixsix flexible series branches is established byby thethe following expression: flexible series branches is established following expression: fi˛  ¨ f fi  f xi   » f x  x  f xi     — ffi‹  ˚ f yi f y  f y ffi  f yi  fi — — ffi‹ ffi ˚»Op O p — 6 6 — ffi ˚ ffi‹ ÿ —         R 0 6 6 ÿ R 0 3fˆ 3 Oip 33 f z f z ffi f O — ˚ — ffi‹ zi zi ip – fl         F“— “ “ JFi Fi ffi ˚ ffi‹ — J Fi Fi F — ffi‹ ˚ r SOpprRi q O pOpRR  Omp xiR  — m x  mx ffi  m xi   i  1 i  1 S   i “ 1 i “ 1 — ffi ffi‹ OO i Oip ipip  Oip   — ˚ – ˝ – fl‚ my  m fl myi   y  myi   mz   mzi  m    mz   zi    According to the definition of stiffness matrix of flexible parallel mechanism, there is: According to the definition of stiffness matrix of flexible parallel mechanism, there is: »

6

6

(22)(22)

6

6 6 1 ÿ ÿ F  KΔX   J Fiÿ F6  J Fi K i ΔX i  J Fi K i J i ΔX ´1 F “ K∆Xi 1“ i JF F “ J K ∆X “ JFi Ki Ji ∆X ii1 i Fi i i 1 i

(23) (23)

i “1 i “1 i“1be obtained by the above equation: The stiffness matrix of the upper platform reference point can 6

K =  J Fi K i J i-1

(24)

i =1

The relationship between the end force of each flexible element and the end force of the leg is: Fij  J Fij Fi

(25)

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The stiffness matrix of the upper platform reference point can be obtained by the above equation: K“

6 ÿ

1 JFi Ki J´ i

(24)

i “1

The relationship between the end force of each flexible element and the end force of the leg is: Fij “ JFij Fi

(25)

The relationship between the end force of each flexible element and the six-axis external force acting on the sensor’s upper platform is: 1 ´1 Fij “ JFij Ki J´ i K F

(26)

The relationship between the six forces expressed in the local coordinate system and the six external forces acting on the sensor platform is: Oip

Fij

1 ´1 “ JOip JFij Ki J´ i K F

(27)

For Equations (21)–(27), there are: i = 1,2, . . . ,6; j = 1, . . . ,4. By extracting equations when j = 2, it can be obtained: » — — — — — –

O1p f 12x O2p f 22x

.. .

O6p

f 62x f

fi

» ´

1 ´1 JO1p J12 K1 J´ 1 K

¯

´ ¯1: ffi — ´1 ´1 ffi — J J K J K O2p 22 2 2 ffi — 1: ffi “ — .. ffi — — . fl – ´ ¯ 1 ´1 JO6p J62 K6 J´ K 6

1:

fi » ffi — ffi — ffi — ffi — ffi — ffi — fl — –

fx fy fz mx my mz

fi ffi ffi ffi ffi ffi ffi ffi fl

(28)

f

That is f = [GF s1 F, [GF s1 is the mapping matrix from the external force/toque applied on the measuring platform of the sensor to the force produced on the six elastic legs considering elastic deformation of the flexible joints. 3.3. Comparative Analysis of Numerical Simulation and Mathematical Models In the above two sections, the mathematical models of the two cases are obtained. By using the ANSYS finite element simulation software platform, the external force F 1 = [5000 0 0 0 0 0 ]T , F 2 = [0 5000 0 0 0 0]T , F 3 = [0 0 5000 0 0 0]T , F 4 = [0 0 0 5000 0 0]T , F 5 = [0 0 0 0 5000 0]T , F 6 = [0 0 0 0 0 5000]T are respectively loaded in the geometry center of the measuring platform in the three-dimensional model of the sensor, and the size of the force produced on the six elastic legs are obtained by the simulation calculation. The accuracy comparison of the theoretical values and simulation values is shown in Figure 9 and Table 1. In the ideal case, the theoretical value of the mathematical model is recorded as the first theoretical value, and the theoretical value considering elastic deformation of flexible joints is recorded as the second theoretical values.

5000 0 0 0 0] , F3 = [0 0 5000 0 0 0] , F4 = [0 0 0 5000 0 0] , F5 = [0 0 0 0 5000 0] , F6 = [0 0 0 0 0 5000] are respectively loaded in the geometry center of the measuring platform in the three-dimensional model of the sensor, and the size of the force produced on the six elastic legs are obtained by the simulation calculation. The accuracy comparison of the theoretical values and simulation values is shown in Figure 9 and Table 1. In the ideal case, the theoretical value of the mathematical model is recorded as the 2016, first theoretical value, and the theoretical value considering elastic deformation of flexible joints 11 of 20 Sensors 16, 1271 is recorded as the second theoretical values. 2 0 00 1 5 00 1 0 00 500 0 - 5 00 - 1 000 - 1 500 - 2 000

Size of Force(N)

Size of Force(N)

2000 1500 1000 500 0 -500 -1000 -1500 -2000

f1

f2

f3

f4

f5

f6

f1

f2

f3

f4

f5

f6

The first theoretical v 1083. -610.3 -1734. 1769. 624.6 -1154. alues

The frist theoretical -1210 1484.4 274.4 -274.4 -1484.4 1210 values

The second theoretica 1015. -540.4 -1555. 1555. 540.3 -1015. l values

The second theoretical values

-1337.6 1629.8 312.4 -317.7 -1704.8 1300.6

Simulation values

Simulation values

-1181.6 1400.2

963.3 -527.9 -1517. 1516. 527.5 -959.9

Sensors 2016, 16, 1271 800 700 600 500 400 300 200 100 0

f1

-264.1 -1403.1 1118.3

11 of 20

(b) Size of Torque(N·m)

Size of Force(N)

(a)

265

3 0 00 2 0 00 1 0 00 0 - 1 000 - 2 000 - 3 000 - 4 000 - 5 000

f2

f3

f4

f5

f6

The first theoretical 717.8 values

699.6

669.3

683.7

676.6

661.9

The first theoretical -4423.4 2274 2311.9 2330.9 2165.3 -4313.5 values

The second theoretical values

605.1

605.1

605.1

605.1

605.1

605.1

The second theoretical values

-4033.6 2016.8 2016.8 2016.8 2016.8 -4033.6

Simulation values

577.5

577.7

578.5

578.4

577.3

577.1

Simulation values

-3774.2 1917.4 1918.6 1921.6 1923 -3800.4

f1

5 0 00 4 0 00 3 0 00 2 0 00 1 0 00 0 - 1 000 - 2 000 - 3 000 - 4 000 - 5 000

f1

f2

f3

f4

f5

f6

(d) Size of Torque(N·m)

Size of Torque(N·m)

(c)

f2

f3

f4

f5

f6

4 0 00 3 0 00 2 0 00 1 0 00 0 - 1 000 - 2 000 - 3 000 - 4 000

f1

f2

f3

f4

f5

f6

The first theoretical -1785.8 3820.7 3830.7 -3938.7 -3963.6 -2105.4 values

The first theoretical 3451.2 -3467.8 3533.5 -3485.6 3327.9 -3438.4 values

The second theoretical values

-1629.6 3493.6 3493.6 -3493.6 -3493.6 -1938.7

The second theoretical values

3067.3 -3067.2 3066.9 -3066.9 3067.2 -3067.3

Simulation values

-1528.9 3237.9 3276.9 -3268.6 -3267.6 -1819.7

Simulation values

2902.6 -2906.8 2893.9 -2890.2 2906.5 -2909

(e)

(f)

Figure 9. (a) Comparison of theoretical values and simulation values loading force along X-axis; (b)

Figure 9. (a) Comparison of theoretical values and simulation values loading force along X-axis; Comparison of theoretical values and simulation values loading force along Y-axis; (c) Comparison (b) Comparison of theoretical values and simulation values loading force along Y-axis; (c) Comparison of theoretical values and simulation values loading force along Z-axis; (d) Comparison of theoretical of theoretical values and simulation values loading force along Z-axis; (d) Comparison of theoretical values and simulation values loading torque along X-axis; (e) Comparison of theoretical values and values and simulation valuestorque loading torque along X-axis; (e)of Comparison of theoretical values and simulation values loading along Y-axis; (f) Comparison theoretical values and simulation simulation valuestorque loading torque along Y-axis; (f) Comparison of theoretical values and simulation values loading along Z-axis. values loading torque along Z-axis. Table 1. Error comparison between numerical simulation and mathematical models.

Table 1. Error comparison between numerical simulation and mathematical models. F2

F1

Fx

IS1(%)

12.5

DS2(%)

5.69 IS1 (%) 2 DS (%) 13.2 IS(%) 2.35 DS(%) 24.3 IS(%) DS(%) 4.55 IS(%) 17.2 DS(%) IS(%) 6.43 DS(%) 16.8 IS(%) 6.18 DS(%)

Fx

Fy

Fz

Mx

IS(%)

Fy DS(%) IS(%)

Fz

DS(%)

Mx IS(%) DS(%) M

y

My IS1

IS(%)

Mz DS(%)

F3

F4

F5

F6

F1

15.6F 2

F14.3 3

F4

16.7F 5

F18.4 6

20.3

12.5 5.69 13.2 2.35 24.3 4.55 17.2 6.43 16.8 6.18 18.9 5.37

2.3015.6

2.46 14.3 2.46 17.9 17.9 3.42 3.42 15.7 15.7 4.39 4.39 20.5 20.5 4.87 16.9 4.87 6.20 16.9 22.1 6.20 5.64

16.7 2.54 20.3 3.75 18.2 4.41 21.3 4.72 20.5 6.44 20.6 5.76

2.5418.4

2.37 20.3 5.47 21.5 16.3 5.48 2.27 17.2 14.7 4.62 4.58 13.5 12.6 5.78 15.7 4.65 6.14 21.3 18.2 6.47 5.16

5.47

16.42.30

16.4 5.67 5.67 21.121.1 4.524.52 18.6 18.64.93 4.9318.0 7.32 18.0 19.3 7.325.23

20.32.37

21.5 3.75 5.48 18.217.2 4.414.58 12.6 21.34.65 4.7221.3 6.47 20.5 14.5 6.445.24

16.3 2.27 14.7 4.62 13.5 5.78 15.7 6.14

state;14.5 DS2

IS(%)between the 18.9 19.3the simulation 22.1 values in 20.6 18.2 (%) is the error theoretical and the ideal (%) is the error Mz between the theoretical and the simulation values in the case of considering the deformation stiffness of DS(%) 5.37 5.23 5.64 5.76 5.24 5.16 flexure joints. IS1 (%) is the error between the theoretical and the simulation values in the ideal state; DS2 (%) is the error between the theoretical and the simulation values in the case of considering the deformation stiffness of flexure joints.

As shown in Table 1, the measurement errors of the six elastic legs are reduced to 10% after considering the deformation stiffness error of flexure joints, which proves that the mathematical model is effective.

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As shown in Table 1, the measurement errors of the six elastic legs are reduced to 10% after considering the deformation stiffness error of flexure joints, which proves that the mathematical12model Sensors 2016, 16, 1271 of 20 is effective.

4. Calibration Experiment 4. Calibration Experiment 4.1. Prototype 4.1. Prototype In order to prove the superiority of the proposed sensor structure and prove the correctness of In order to prove the superiority of the proposed sensor structure and prove the correctness of the theoretical analysis, a prototype of the large measurement range six-axis force sensor of 6-UPUR the theoretical analysis, a prototype of the large measurement range six-axis force sensor of 6-UPUR parallel mechanism with flexible joints was manufactured, as shown in Figure 10. Considering the parallel mechanism with flexible joints was manufactured, as shown in Figure 10. Considering the manufacturing process and economic cost, the material property of each component was selected as manufacturing process and economic cost, the material property of each component was selected as shown in Table 2. The main structure parameters and the measuring range of the sensor are shown shown in Table 2. The main structure parameters and the measuring range of the sensor are shown in in Tables 3 and 4. Tables 3 and 4.

Figure10.10.Prototype Prototype large measurement range six-axisforce forcesensor sensorofof6-UPUR 6-UPURparallel parallel Figure of of thethe large measurement range six-axis mechanism with flexible joints. mechanism with flexible joints. Table Material properties thecomponents. components. Table 2.2. Material properties ofof the Components Components

Materials Materials

Modulus Poisson Poisson Ratio ElasticElastic Modulus Ratio

Force-measuring platform Hard aluminum alloy Force-measuring Flexible joints Hard aluminum 40CrNiMoA alloy platform Fixed platform Q235

Flexible joints

A (˝ ) 65.6

A (°)

B (°)

65.6Fx (N)

F40.0 y (N) ˘10,000

B (˝ )

Fy (N)

Q235 C (˝ )

R (mm)

210 Gpa R (mm)

l1 (mm)

40.0 3. Structure 40.0 466.7 Table parameters of 300.0 six-axis force300.0 sensor.

2700 kg/m3 3 7830 2700kg/m kg/m3 7850 kg/m3

7830 kg/m3

0.25 l2 (mm)

7850 kg/m3

163.6

(°) R (mm) R force (mm)sensor. l1 (mm) Table 4.CMeasuring range of the six-axis Fz (N) 40.0

Mx (N¨m)466.7My (N¨m) 300.0 Mz (N¨m)

˘10,000 ˘5000 ˘5000 ˘5000 Table 4. Measuring range of the six-axis force sensor.

4.2. Calibration Experiment Fx (N)

0.30 0.30 0.30 0.25

206 Gpa 40CrNiMoA 0.30 Table 3. Structure parameters of six-axis force sensor.

Fixed platform

˘10,000

70 Gpa 206 70 Gpa Gpa 210 Gpa

Density Density

Fz (N)

Mx (N·m)

My (N·m)

Mz (N·m)

l2 (mm)

300.0 163.6 Overload Capacity 120%

Overload Capacity

To measure the loaded external force accurately, the sensor should be calibrated by using a special ±10,000 ±10,000which±10,000 ±5000 in directions ±5000 of x, y, z separately. 120% calibration system, can generate±5000 forces and torques In this

4.2. Calibration Experiment To measure the loaded external force accurately, the sensor should be calibrated by using a special calibration system, which can generate forces and torques in directions of x, y, z separately. In this paper the calibration system, which mainly included a hydraulic loading system, calibration

platform, parallel 6-UPUR six-axis force sensor with flexible joints, signal processing device, data platform, parallel six-axiscalibration force sensor with flexible joints, processing device, acquisition device, 6-UPUR data processor, software system and so signal on, as shown in Figures 11data and acquisition device, data processor, calibration software system and so on, as shown in Figures 11 and 12, was designed and manufactured. 12, was designed and Considering that manufactured. the structure of the six-axis force sensor of 6-UPUR parallel mechanism with Considering that the structure of the six-axis sensorerror of 6-UPUR parallel mechanism flexible joints is very complex and the presence offorce measuring can’t be ignored because 13 ofwith the Sensors 2016, 16, 1271 of 20 flexible joints is very complex and the presence of measuring error can’t be ignored because of the influence of these factors such as the design principles, manufacturing and processing errors, so influencepoint of these factors such as therange design manufacturing andlinear processing errors, so multiple loading in the sensor andprinciples, the method of least squares fit are needed in paper the calibration system, which mainly included a hydraulic loading system, calibration platform, multiple point loading in the sensor range and the method of least squares linear fit are needed in calibration experiments. Thus, the linear relationship between inputs and outputs of the large parallel 6-UPUR six-axis force sensor with flexible joints, signal processing device, data acquisition calibration experiments. Thus, the linear relationship between inputs and outputs of the large measurement range 6-axis force sensor of 6-UPUR parallel mechanism with flexible joints can be device, data processor, calibration software system and so on, as shownwith in Figures 11joints and 12, measurement range 6-axis force sensor of 6-UPUR parallel mechanism flexible canwas be calculated. designed and manufactured. calculated.

Figure Figure 11. 11. Structure Structure of of the the calibration calibration experiment experiment system. system. Figure 11. Structure of the calibration experiment system.

Figure 12. Diagram of the calibration experiment system. Figure 12. Diagram of the calibration experiment system. Figure 12. Diagram of the calibration experiment system.

Considering that the structure of the six-axis force sensor of 6-UPUR parallel mechanism with

Detailed experimental steps are described are as follows: flexible joints is very complex andare thedescribed presence of measuring error can’t be ignored because of the Detailed experimental steps are as follows: (1) Each axis force/torque within is divided into 10 load points in two positive influence of these factors such as the thesensor designrange principles, manufacturing and processing errors,and so (1) Eachpoint axisdirections, force/torque within the sensor range is divided 10 load points in two and negative asthe shown in Table multiple loading in sensor range5;and the method into of least squares linear fitpositive are needed negative directions, as shown 5; relationship between inputs and outputs of the large in calibration experiments. Thus, in theTable linear measurement range 6-axis force sensor of 6-UPUR parallel mechanism with flexible joints can be calculated. Detailed experimental steps are described are as follows:

position, and connect the calibration hardware system, as shown in Figure 13; (3) At each loaded point, conduct loading and unloading experiments in turn, and record the output voltage of each measuring leg corresponding to each load point; (4) Repeat Steps (2) and (3), conduct loading and unloading experiments in the opposite direction and record the experimental data; Sensors 2016, 16, 1271 14 of 20 (5) Repeat Steps (2)–(4), the calibration experiments of the six axis force sensor are carried out and all the experimental data are obtained; (6) the data. The the calibration matrixisofdivided the sensor andin thetwo linearity of (1) Check Each and axis process force/torque within sensor range intois10obtained, load points positive the sensor is determined. and negative directions, as shown in Table 5; (2)

Select the force/torque component of the load. The load unit is installed in the corresponding Table 5 Loading points of calibration force/torque. position, and connect the calibration hardware system, as shown in Figure 13; (3) At each loaded point, conduct loading and unloading experiments in turn, and record the output Loading Points 1 2 3 4 5 8 9 10 voltage of each measuring leg corresponding to each load6 point; 7 (4) Repeat Steps (2) and (3), conduct loading and unloading experiments in the opposite direction Positive 1000 2000 3000 4000 5000 6000 7000 8000 9000 10,000 Force and record the experimental data; (5)(N) Repeat Steps (2)–(4), calibration of the six and Negative 10,000 the 9000 8000 experiments 7000 6000 5000axis force 4000 sensor 3000are carried 2000 out 1000 all the experimental data are obtained; Positive 500 the 1000 1500 2000 matrix 2500of the3000 4000and the 4500linearity 5000of Torque (6) Check and process data. The calibration sensor 3500 is obtained, the sensor is determined. (N·m)

Negative

5000

4500

4000

3500

3000

2500

2000

1500

1000

(a)

(b)

(c)

(d)

(e)

(f)

500

Figure 13. (a) Force loading along x-axis; (b) force loading along y-axis; (c) force loading along z-axis; Figure 13. (a) Force loading along x-axis; (b) force loading along y-axis; (c) force loading along z-axis; (d) torque torque loading loading around around x-axis; x-axis; (e) (e) torque torque loading loading around around y-axis; y-axis; (f) (f) torque torqueloading loadingaround aroundz-axis. z-axis. (d)

4.3. Experimental Method

Table 5. Loading points of calibration force/torque.

In the analysis of experimental data, the forces that are loaded on the prototype during the Loading Points 1 linearly 2 independent 3 4 vectors, 5 6 7as: 8 9 10 experiment can be described as six denoted Force (N) Torque (N¨m)

Positive Negative Positive Negative

1000 10,000 500 5000

2000 9000 1000 4500

3000 8000 1500 4000

4000 7000 2000 3500

5000 6000 2500 3000

6000 5000 3000 2500

7000 4000 3500 2000

8000 3000 4000 1500

9000 2000 4500 1000

10,000 1000 5000 500

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4.3. Experimental Method In the analysis of experimental data, the forces that are loaded on the prototype during the experiment can be described as six linearly independent vectors, denoted as: » — — — — FS “ — — — –

Fx 0 0 0 0 0

0 Fy 0 0 0 0

0 0 Fz 0 0 0

0 0 0 Mx 0 0

0 0 0 0 My 0

0 0 0 0 0 Mz

fi ffi ffi ffi ffi ffi ffi ffi fl

(29)

When the external forces are loaded on the measuring platform of the sensor, the six single-axis force sensors will sense the force produced on the measuring legs, and then the change of output voltage of the Wheatstone bridges can be measured out. We can get a set of relationships: Fs “ GV

(30)

where G is the calibration matrix between the loaded forces and the output voltages; V is the output voltage matrix. Thus the calibration matrix between the loaded forces and the output voltages of the six measuring legs can be calculated: G “ F s V ´1 (31) Besides, the error matrix evaluating the accuracy of the six-axis force sensor is defined as follows: Err “

FS ´ F6ˆ6 FFS

(32)

where F FS is the full range of the sensor, F 6 ˆ6 can be calculated based on the mapping matrix between the loaded force and the output voltage by Equation (32); the error matrix Err is a comprehensive evaluation. The diagonal components of the error matrix separately represent the measurement errors of the six different directions of the loaded external force, and other components represent the interference errors between different directions. 5. Experiment Results 5.1. Calibration Results and Analysis Through the calibration system described above, the calibration experiments based on the sensor prototype are carried out, and the data of the six measuring legs are obtained. Taking a complete static calibration experimental data and using the K value method and least square method to substitute the loading forces and the voltages data which are collected from six measuring legs into the corresponding equation, the static calibration matrix GK , G2C and the error matrix ErrK , Err2C can be obtained by the analysis and process of the experimental data, therefore, the performance of force-measuring of the sensor is obtained: » fi ´0.0694 ´0.2151 0.2443 ´0.4298 ´0.5077 0.2468 — ´0.5148 0.4170 0.2045 ´0.2858 0.2491 ´0.1535 ffi — ffi — 0.2528 0.2198 0.2476 0.2298 0.2288 0.2653 ffi — ffi GK “ — (33) ffi — ´0.1336 0.1129 0.1250 0.0064 ´0.0033 ´0.1244 ffi — ffi – ´0.0807 ´0.0332 ´0.0452 0.1647 0.1822 ´0.0460 fl ´0.1247

0.0999

´0.0879

0.0881

´0.1108

0.0831

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»

ErrK

— — — — “— — — –

0.0431 0.0279 0.0607 0.0939 0.0956 0.0519

0.0240 0.0264 0.0216 0.0291 0.1340 0.0174

0.0249 0.0118 0.0333 0.0270 0.0295 0.0696

0.0054 0.0083 0.0141 0.0208 0.0252 0.0216

0.0117 0.0082 0.0038 0.0230 0.0163 0.0219

0.0093 0.0556 0.0062 0.0135 0.0036 0.0076

fi ffi ffi ffi ffi ffi ffi ffi fl

(34)

From Equation (34), it can be noted that when the K value method is used to decouple, the calibration error of each axis is: F x (4.31%), F y (2.64%), F z (3.33%), M x (2.08%), M y (1.63%), M z (0.76%). The maximum error of the I kind is 4.31%, which appears in F x ; the maximum error of the II kind is 13.40%, which is appears in F y , when loaded in M y . So the calibration error obtained by this method is 13.40%. » fi ´0.0929 ´0.2095 0.2377 ´0.4204 ´0.5071 0.2437 — ´0.4518 0.4031 0.1981 ´0.2675 0.2288 ´0.1496 ffi — ffi — 0.2546 0.2748 0.2717 0.2628 0.2603 0.2678 ffi — ffi G2C “ — (35) ffi — ´0.1202 0.1127 0.1251 0.0148 ´0.0137 0.1157 ffi — ffi – 0.0680 ´0.0508 ´0.0380 0.1533 0.1703 ´0.0373 fl ´0.0877 0.0757 ´0.1058 » 0.0122 0.0068 0.0124 — 0.0144 0.0066 0.0111 — — 0.0072 0.0062 0.0089 — Err2C “ — — 0.0169 0.0088 0.0203 — – 0.0167 0.0268 0.0074 0.0070 0.0061 0.0112

0.1067 0.0026 0.0022 0.0059 0.0039 0.0049 0.0047

0.0917 fi 0.0071 0.0061 ffi ffi 0.0039 ffi ffi ffi 0.0063 ffi ffi 0.0038 fl

´0.1174 0.0070 0.0038 0.0028 0.0068 0.0059 0.0020

(36)

0.0042

From Equation (36), it can be noted that when the least squares method is used to decouple, the calibration error of each axis is: F x (1.22%), F y (0.66%), F z (0.89%), M x (0.39%), M y (0.59%), M z (0.42%).The maximum error of the I kind is 1.22%, which appears in F x ; the maximum error of the II kind is 2.68%, which appears in F y , when loaded in M y , so the calibration error obtained by this method is 2.68%. From the results of the calibration experiment, the calibration error comparison of the two kinds of decoupling methods is as shown in Table 6. The results show that the accuracy of least squares method is much better than that of the K value calibration method. Table 6. The calibration error comparison of the two kinds of decoupling methods. Decoupling Method K value method Least squares method

Type I Error (%)

Type II Error (%)

Fx

Fy

Fz

Mx

My

Mz

Maximum Value

Maximum Value

4.31

2.64

3.33

2.08

1.63

0.76

4.31

13.40

1.22

0.66

0.89

0.39

0.59

0.42

1.22

2.68

5.2. Linearity Analysis The input signal and the output signal of the sensor are not completely linear, and there is always an error. The ratio of the error to the measurement range is called the linearity of the sensor. When a certain force/torque is loaded to one direction of the sensor, the output voltage of the six legs will change with the change of the loading force/torque. Figure 11 shows the changing curves between the sensor’s measuring force and the standard loading force when the force or torque is loaded in one direction. Figures 14 and 15 are the variations of the voltage of each leg with the loading force/torque changing, when the force/torque is loaded in the three directions of X, Y and Z.

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(a)

(b)

(c) Figure14. 14.(a) (a)Voltage Voltagecurves curves legs loading force along X-axis; (b) Voltage curves legs loading Figure of of legs loading force along X-axis; (b) Voltage curves of legsofloading force force along Y-axis; (c) Voltage curves of legs loading force along Z-axis. along Y-axis; (c) Voltage curves of legs loading force along Z-axis.

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(a)

(b)

(c) Figure15. 15.(a) (a)Voltage Voltagecurves curvesofoflegs legsloading loadingtorque torquealong alongX-axis; X-axis;(b) (b)Voltage Voltagecurves curvesofoflegs legsloading loading Figure torque along Y-axis; (c) Voltage curves of legs loading torque along Z-axis. torque along Y-axis; (c) Voltage curves of legs loading torque along Z-axis.

Table 7 is the linearity of each leg obtained by least square method. As shown in Table 6, the Table 7 is the linearity of each leg obtained by least square method. As shown in Table 6, the linearity of each leg of the sensor is less than 1%, which shows that the sensor has a good linearity. linearity of each leg of the sensor is less than 1%, which shows that the sensor has a good linearity.

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Table 7. Linearity of six-axis force sensor. Linearity (%) Items Fx Fy Fz Mx My Mz

Leg 1

Leg 2

Leg 3

Leg 4

Leg 5

Leg 6

0.22 0.20 0.14 0.21 1.00 0.23

0.18 0.14 0.18 0.19 0.75 0.39

0.23 0.29 0.71 0.82 0.51 0.25

0.24 0.22 0.64 0.32 0.24 0.15

0.08 0.13 0.57 0.39 0.4 0.15

0.15 0.21 0.97 0.92 0.4 0.39

6. Conclusions In this paper, to overcome the influence of the gap and friction of the traditional joints on the parallel six-axis sensors’ precision and stability, we have successfully demonstrated a kind of six-axis force sensor based on 6-UPUR parallel mechanism with flexible joints, which has large measurement range and high accuracy. The force mathematical model of the sensor is established on the screw theory; according to the relations of the stiffness and deformation compatibility condition, the stiffness matrix considering flexibility of each flexible joint is built up; then the complete mathematical model is established. The sensor prototype and the calibration system are manufactured and static calibration experiments were carried out on the sensor. The results show that the measurement error is less than 2.68%, which shows that the sensor has high measuring accuracy and good linearity. The experimental results prove the feasibility of the large measurement range six-axis force sensor of 6-UPUR parallel mechanism with flexible joints. The design and loaded research of the parallel six-axis force sensor has reference significance. Acknowledgments: The authors would like to acknowledge the project supported by the National Natural Science Foundation of PR China (NSFC) (Grant No. 51105322), the Natural Science Foundation of Hebei Province (Grant No. E2014203176), the Natural Science Research Foundation of Higher Education of Hebei Province (Grant No. QN2015040) and China’s Post-doctoral Science Fund (No. 2016M590212). Author Contributions: Yanzhi Zhao and Tieshi Zhao conceived and designed the sensor; Zhongpan Shi performed the experiments, Yanzhi Zhao and Caifeng Zhang analyzed the data and wrote the paper; Dan Zhang offered help in the process of writing. All authors reviewed and approved the manuscript. Conflicts of Interest: The authors declare no conflict of interest.

Abbreviations 6-UPUR A kind of mechanism configuration. U-universal pair, P-prismatic pair, R-revolute pair. LVDT linear variable differential transformer.

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