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Task-oriented design method and research on force compliant experiment of six-axis wrist force sensor Jiantao Yao a,b,∗, Dajun Cai a, Hongyu Zhang a, Hang Wang a, Dezu Wu a, Yongsheng Zhao a,b a b

Parallel Robot and Mechatronic System Laboratory of Hebei Province, Yanshan University, Qinhuangdao 066004, China Key Laboratory of Advanced Forging & Stamping Technology and Science, Ministry of Education of China, Yanshan University, Qinhuangdao 066004, China

a r t i c l e

i n f o

Article history: Received 19 December 2014 Revised 15 December 2015 Accepted 28 January 2016 Available online 26 February 2016 Keywords: Wrist force sensor Parallel structure Task Peg-in-hole assembly

a b s t r a c t This paper analyzes the task-oriented design method of six-axis force sensor and proposes the task model of the sensor. The task mathematical model of the sensor is established based on the idea of task ellipsoid. The models of force ellipsoid and moment ellipsoid are also established. The relational expression between the task model and ellipsoid model of sensor is obtained. Then, a fully pre-stressed dual-layer parallel six-axis wrist force sensor is proposed, whose static mathematical model is also established. The sensor task model for assembly work is proposed and the analytical expression between the sensor structure parameters and task model is deduced. According to the assembly work, the sensor structure is designed speciﬁcally, and the speciﬁc structure sizes of the sensor are obtained. Then the new sensor prototype manufactured for peg-in-hole assembly is processed. The calibration experiment and peg-inhole assembly experiment on the prototype are completed and each performance index is well examined by the experiment results. The experiment results also lay the foundation for the practical application of six-axis force sensor. © 2016 Elsevier Ltd. All rights reserved.

1. Introduction With the ability of measuring three-component force and threecomponent torque, six-axis wrist force sensor has been one of the most important sensors in robot force control and force position control. Parallel mechanisms possess the distinguishing advantages of stability, high rigidity, carrying capacity, no error accumulation, accuracy and inverse solution easily [1], and the parallel mechanism ideas are introduced into the design of the force sensitivity element structure of six-axis force sensor by many scholars. Gaillet and Reboulet [2] proposed the six-axis force sensor based on Stewart platform in 1983. Chen [3], Kerr [4], Nguyen et al. [5], Ferraresi et al. [6], Xiong [7] had researched the design problems of Stewart platform-based force sensor. Meanwhile, Romiti [8], Sorli [9], Kerr [10], Kang [11] developed the Stewart platform-based force sensor prototype. Gao et al. [12] designed a miniature six-axis force sensor which is used for the robot wrist and ﬁnger based on the design idea of elastic hinge instead of spherical pair. Dwarrakanath [13] designed a Stewart platform based force-torque sensor with a circular sensitive element, and then a Stewart platform-based force ∗ Corresponding author at: Parallel Robot and Mechatronic System Laboratory of Hebei Province, Yanshan University, Qinhuangdao 066004, China. Tel./fax: +86 335 807 4581. E-mail addresses: [email protected] (J. Yao), [email protected] (Y. Zhao).

http://dx.doi.org/10.1016/j.mechatronics.2016.01.007 0957-4158/© 2016 Elsevier Ltd. All rights reserved.

sensor [14] was designed based on one-way constraint and point contact. Ranganath et al. [15] designed a force-torque sensor based on a Stewart platform in a near-singular conﬁguration from the angle of the sensor sensitivity. Pacchierotti et al. [16] presented a particular peg-in-hole application that a complex robotic hand using with sensor is considered to return the force of the object to the haptic master. Jia et al. [17] designed a new six-axis heavy force sensor based on the Stewart platform structure. It’s a very complicated problem to do the structure optimization for the parallel structure six-axis force sensor, and many factors need to be considered for its design and applied research. Therefore, a reasonable optimization principle is the basis of the optimization design and performance evaluation. Unfortunately, the optimization principle is still diﬃcult due to the relative complexity of the six-axis force sensor, although the deﬁnition of performance index of conventional one-dimensional force sensor is fairly standard. In the past, the researches mainly focused on the sensor itself, rather than associated with the tasks. Actually, the structure, size and performance of sensor should be suitable for the execution of the prescriptive tasks. Currently, most scholars conducts structural optimization design mainly basing on the isotropic principle, and the isotropic is regarded as an important performance index of six-axis force sensor structural evaluation [7,18–21], such as force isotropy, torque isotropy, sensitivity isotropy, fully isotropic, etc. These indices are

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obtained from the sensor itself without considering the speciﬁc and actual situation. Sensor is used to perform a certain task, such as peg-in-hole assembly, planar contour tracking, bolt into the hole and grasping, where sensor does not necessarily need isotropic structure. Actually, those task models are not spherical but ellipsoidal, while the isotropic is not optimal [22]. Therefore, according to the applications of six-axis wrist force sensor, the task model of sensor is established. The sensor elastomeric structure based on the performance evaluation criterion of the task model is also designed. The conﬁguration, structure, size and performance suitable for the speciﬁed task are obtained, which have important significance for the design and application of the six-axis wrist force sensor. In this paper, the mathematical model and task model of sixaxis force sensor are established, and the analytical expression between the task ellipsoid and the force ellipsoid is deduced. Besides, the sensor’s whole structure is designed based on the task model of the peg-in-hole assembly. Then the structure parameters of sensor are calculated. Furthermore, the sensor prototype is manufactured and the experiment of the peg-in-hole assembly is carried out.

Sensor task model can be regarded as a set of generalized force which meets the task requirement. Sensor task model is divided into two cases: the known task and unknown task. The known task model is corresponding to the case that the requirement of measuring task is clear, and it can be described particularly with a set of force wrench. Whereas, the unknown task model is corresponding to the case that the requirement of measurement task is unknown and it can’t be described with the set of force wrench. 2.1.1. The establishment of the known task model For grasping planning, Li and Sastry [22] used task ellipsoid to model the known grasp task and the similar way also can be used to describe mathematical model of the known task [7]. Task model built in the reference [7] is a hyper-ellipsoid of combining force with moment. However, in this paper, the task model is divided into task force ellipsoid model and task moment ellipsoid model, respectively. The general expression of the task force ellipsoid in sensor task coordinate system can be obtained as

+

+

2 a3 FGtz

+ a4 FGtx FGty + a5 FGty FGtz + a6 FGtz FGtx + a7 FGtx

+ a8 FGty + a9 FGtz + a10 ≤ 0

(1)

where FGtx , FGty and FGtz represent three orthogonal force components, and ai (i = 1, 2,…,10) represents real coeﬃcient. Eq. (1) can be rewritten as

a1 (FGtx − uF Gt )2 + a2 (FGty − vF Gt )2 + a3 (FGtz − wF Gt )2 + a4 (FGtx − uF Gt )(FGty − vF Gt ) + a5 (FGty − vF Gt )(FGtz − wF Gt ) + a6 (FGtz − wF Gt )(FGtx − uF Gt ) + a11 ≤ 0

(2)

where (uFGt , vFGt , wFGt ) represents the center of task force ellipsoid, uF Gt = − 2a2 a6 a9 − 4a2 a7 a3 − a6 a5 a8 − a4 a5 a9 + 2a4 a8 a3 + a7 a25 (2λF Gt ),

vF Gt = (−2a1 a5 a9 + 4a1 a8 a3 + a6 a4 a9 + a6 a5 a7 − a26 a8 − 2a7 a4 a3 )/(2λF Gt ), 2 2 2 a11 =

2 2 2 b1 MGtx + b2 MGty + b3 MGtz + b4 MGtx MGty + b5 MGty MGtz

+ b6 MGtz MGtx + b7 MGtx + b8 MGty + b9 MGtz + b10 ≤ 0

(3)

where MGtx , MGty and MGtz represent the three orthogonal moment components, and bi (i = 1, 2,…,10) represents real coeﬃcient. Eq. (3) can be rewritten as:

b1 (MGtx − uMGt )2 + b2 (MGty − vMGt )2 + b3 (MGtz − wMGt )2 + b4 (MGtx − uMGt )(MGty − vMGt ) + b5 (MGty − vMGt )(MGtz − wMGt ) + b6 (MGtz − wMGt )(MGtx − uMGt ) + b11 ≤ 0

(4)

where (uMGt , vMGt , wMGt ) represents the center of task moment ellipsoid; uMGt = − 2b2 b6 b9 − 4b2 b7 b3 − b6 b5 b8 − b4 b5 b9 + 2b4 b8 b3 + b7 b25 (2λMGt ),

vMGt = (−2b1 b5 b9 + 4b1 b8 b3 + b6 b4 b9 + b6 b5 b7 − b26 b8 − 2b7 b4 b3 )/(2λMGt ), wMGt = −2b6 b2 b7 − b24 b9 + b4 b5 b7 + b4 b6 b8 + 4b2 b1 b9 − 2b5 b1 b8 (2λMGt ), 4b2 b10 b2 − b2 b2 − 4b6 b9 b7 b2 + 2b6 b5 b7 b8 − 4b6 b10 b4 b5 6

6 8

+2b6 b4 b9 b8 − 16b2 b10 b1 b3 + 4b2 b3 b27 + 4b2 b1 b29 − b24 b29 + 4b1 b28 b3 −4b1 b9 b5 b8 + 4b10 b25 b1 − 4b8 b7 b4 b3 + 2b4 b9 b5 b7 + 4b10 b24 b3 − b25 b27

(4λMGt ), λMGt = b24 b3 − b4 b2 b1 b3 + b26 b2 − b4 b5 b6 + b25 b1 .

2.1. The establishment of the task model

2 a2 FGty

Similarly, the general expression of the task moment ellipsoid in sensor task coordinate system can be obtained as

b11 =

2. The method of establishing the sensor’s task model

2 a1 FGtx

λF Gt = a24 a3 − 4a2 a1 a3 + a26 a2 − a4 a5 a6 + a25 a1 .

4a6 a10 a2 − a6 a8 − 4a6 a9 a7 a2 + 2a6 a5 a7 a8 − 4a6 a10 a4 a5 +2a6 a4 a9 a8 − 16a2 a10 a1 a3 + 4a2 a3 a27 + 4a2 a1 a29 − a24 a29 + 4a1 a28 a3 −4a1 a9 a5 a8 + 4a10 a25 a1 − 4a8 a7 a4 a3 + 2a4 a9 a5 a7 + 4a10 a24 a3 − a25 a27

(4λF Gt ), wF Gt = (−2a6 a2 a7 − a24 a9 + a4 a5 a7 + a4 a6 a8 + 4a2 a1 a9 − 2a5 a1 a8 )/(2λF Gt ),

Eqs. (2) and (4) can be written in matrix form as:

FTGt CTF Gt CF Gt FGt

≤1

(5)

MTGt CTMGt CMGt MGt ≤ 1

(6)

where

FGt = [FGtx − uF GtFGty − vF Gt FGtz − wF Gt ]T , a1 a4 /2 a6 /2 a2 a5 /2 , CTF Gt CF Gt = − a111 a4 /2 a6 /2 a5 /2 a3

MGt = MGtx − uMGt MGty − vMGt MGtz − wMGt b1 b4 /2 b6 /2 1 T b2 b5 /2 . CMGt CMGt = − b b4 /2 11 b6 /2 b5 /2 b3

T

,

As matrix CTF Gt CF Gt is a real symmetric matrix, we have (1)–( 7) with the orthogonal diagonal factorization of the matrix.

CTF Gt CF Gt = QF Gt diag(λF Gtx , λF Gty , λF Gtz )QTF Gt = [QF Gtx QF Gty QF Gtz ]diag(λF Gtx , λF Gty , λF Gtz )[QF Gtx QF Gty QF Gtz ]T (7) where λFGtx , λFGty and λFGtz represent the eigenvalues of matrix CTF Gt CF Gt , and the square roots of their reciprocal are half the length of the sensor’s task force ellipsoid spindles, respectively. The three column vectors of matrix QFGt constitute a complete orthonormal feature vector system of matrix CTF Gt CF Gt , which represent the direction of the task force ellipsoid’s three spindles, respectively. Similarly, we have Eq. (8) with the orthogonal diagonal factorization of the matrix:

CTMGt CMGt = QMGt diag(λMGtx , λMGty , λMGtz )QTMGt = [QMGtx QMGty QMGtz ]diag(λMGtx , λMGty , λMGtz )[QMGtx QMGty QMGtz ]T (8) where λMGtx , λMGty and λMGtz represent the eigenvalues of matrix CTMGt CMGt , and the square roots of their reciprocal are half-length of the sensor’s task moment ellipsoid spindles, respectively. The three column vectors of matrix QMGt constitute a complete orthonormal feature vector system of matrix CTMGt CMGt , which represent the direction of the task moment ellipsoid’s three spindles, respectively.

J. Yao et al. / Mechatronics 35 (2016) 109–121

Fig. 1. Sketch map of given task mathematic model.

Fig. 2. Sketch map of unknown task mathematic model.

Eqs. (5)–(8) indicate that task force model and task moment model of the sensor are two ellipsoids formed by a set of contact forces and moments on the tested parts, respectively. As shown in Fig. 1, Ot –Xt Yt Zt is the task coordinate system, ellipsoid OFGt is the sensor’s force ellipsoid, OFGt (uFGt , vFGt , wFGt ) is the center of force ellipsoid, the speciﬁc location of OFGt is determined by the contact forces on the tested parts when the task is not executed, namely the values of uFGt , vFGt and wFGt refer to the contact forces along three coordinates axis, respectively, for example the gravity and so on. The half lengths three spindles of the force ellipsoid’s can be given by aF Gt = 1/ λF Gtx , bF Gt = 1/ λF Gty and cF Gt = 1/ λF Gtz , which are equal to the square roots of the eigenvalues reciprocal of matrix CTF Gt CF Gt and whose speciﬁc values are the maximum contact forces along the three coordinates axis in the process of the task execution. Ellipsoid OMGt is the sensor’s moment ellipsoid and OMGt (uMGt , vMGt , wMGt ) is the center of moment ellipsoid. The values of uMG t, vMG t and wMG t refer to the contact moments along three coordinates axis when the task is not executed. The half lengths of the moment ellipsoid’s three spindles can be given by aMGt = 1/ λMGtx , bMGt = 1/ λMGty and cMGt = 1/ λMGtz , which are the square roots of the eigenvalues’ reciprocal of matrix CTMGt CMGt , and their speciﬁc values are the maximum moments along three coordinates axis in the process of the task execution. 2.1.2. The establishment of the unknown task model For unknown task, which cannot be described speciﬁcally javascript:void(0); like the previous section because of the lack of information about the sensor’s measuring task. Therefore, provided that the capacity of bearing all forces and moments in each direction is required to be the same, the requirement can be described by the task model of the sphere. The equation of task force sphere in the sensor’s task coordinate system can be obtained as

(FUtx − uF Ut )2 + (FUty − vF Ut )2 + (FUtz − wF Ut )2 ≤ rF2Ut

(9)

where Futx, Futy and Futz represent three orthogonal force components; (uFGt , vFGt , wFGt ) represents the center of task force sphere. rFUt represents the radius of task force sphere. Similarly, the expression of task force moment sphere in sensor’s task coordinate system can be obtained as 2 (MUtx − uMUt )2 + (MUty − vMUt )2 + (MUtz − wMUt )2 ≤ rMUt

111

(10)

where MUtx , MUTy and MUTz represent three orthogonal moment components; (uMUt , vMUt , wMUt ) represents the center of the task moment sphere. Eqs. (9) and (10) represent the force and moment mathematical models of unknown task whose radii are rFUt and rMUt , respectively, and spheres center are OFUt (uFUt , vFUt , wFUt ) and OMUt (uMUt , vMUt , wMUt ) respectively, as shown in Fig. 2. The physical meaning about the radius and the center of the two task spheres is the same as the half lengths of the force and moment ellipsoid’s three spindles and the center of known task mathematical models, respectively.

Eqs. (9) and (10) can be expressed in matrix form as T FUt CTF Ut CF Ut FUt ≤ 1

(11)

T MUt CTMUt CMUt MUt ≤ 1

(12)

where

FUt = [FUtx − uF Ut FUty − vF Ut FUtz − wF Ut ]T , CTF Ut CF Ut = I3 /rF2Ut , MUt = [MUtx − uMUt MUty − vMUt MUtz − wMUt ]T , 2 CTMUt CMUt = I3 /rMUt .

2.2. The bearing ellipsoid analysis of parallel six-axis force sensor Parallel six-axis force sensor mainly consists of measuring platform, base platform, and measuring legs. The measured force is applied on the measuring platform, whose surface is assumed to be continuous and frictionless. There exists reacting force on measuring legs if six-dimension external force is applied on the measuring platform. If other applying forces on the legs are ignored, the reacting force is along the axis of each leg. For the equilibrium of the measuring platform, the mathematical model of sensor can be obtained based on screw theory, as m

f i $i = Fs + ∈ Ms

(13)

i=1

where fi represents the axial force produced on ith leg; $i represents the unit line vector along the axis of the ith leg; where Fs and Ms represent the applying force vector and torque vector acted on the coordinate center of the measuring platform. Eq. (13) can be rewritten in the form of matrix equation as

F = Gs f

(14) ]T

where F = [Fx Fy Fz Mx My Mz is the vector of six-dimension external force applied on the measuring platform; F can be rewritten as

F F= s Ms

(15)

f = [f1 f2 , …, fm ] (m ≥ 6) is the vector composed of the reacting forces of the m legs; Gs is the force Jacobian matrix [23], which can be also known as measuring mapping matrix of the sensor. It is given by

S1 Gs = S01

S2 S02

··· ···

Sm S0m

(16)

When n ≥ 6, Gs is not the square matrix. It is easy to ﬁnd that the ﬁrst three rows of matrix Gs have nothing to do with the dimension while the last three rows of matrix Gs have the dimension of length. In order to make the physical

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J. Yao et al. / Mechatronics 35 (2016) 109–121 s tR

concept clearer, the expression can be written as

Gs =

GF s GMs

represents the rotation matrix that coordinate system{t} relative 0

(17)

to coordinate system{s}; s PtO = [ pz

−pz py 0 −px ]. px 0

−py

Substituting Eqs. (25) and (26) into Eq. (27)

where GF = [G]1,2,3 and GMF = [G]4,5,6 are the positive mapping matrix of force vector and moment vector, respectively. Generally, the stiffness of each elastic leg is the same, and when n ≥ 6, the reacting forces of the legs can be calculated with the Moore–Penrose generalized inverse of matrix G [24] as

f = G+ s F = Cs F

(18)

where G+ s is the Moore–Penrose generalized inverse of matrix Gs , and we have Cs = G+ s . Considering the different dimensions of force and moment, the matrix Cs can be expressed as

Cs = [CF s CMs ]

(19)

where GFS = [G]1,2,3 and GMS = [G]4,5,6 are the positive mapping matrix of force vector and moment vector, respectively. Substituting Eqs. (15) and (19) into (18), the matrix f can be expressed as

f = CF s FS +CMs MS

(20)

Gs = JGt

(28)

If both sides of Eq. (28) are inversed, the equation can be obtained as

Cs

= Ct J−1

If the force Jacobian matrix is converted from the reference coordinate system of sensor to the coordinate system of task, the equation can be obtained as

Ct = Cs J

(30)

Substituting Eq. (19) into Eq. (30), the equation can be obtained as

[CF t CMt ] = [CF s CMs ] J

[CF t CMt ] = [CF s CMs ]

⇒

(21)

fm = CMs MS

(22)

(31)

Expanding Eq. (31), then we have

Let

fF = CF s FS

(29)

s tR

S(s Pto )ts R

0 s tR

CF t = CF s ts R + CMs S(s Pto )ts R CMt = CMs ts R

(32)

The descriptions of force ellipsoid and moment ellipsoid are According to (21) and (22), two spheres are deﬁned in the reacting force space of legs by ||fF || = 1 and ||fM || = 1. Furthermore, mapping the two spheres into the space of applied force, the force ellipsoid and the moment ellipsoid can be obtained and expressed as

FTs CTF s CF s Fs = 1

(23)

MTs CTMs CMs Ms = 1

(24)

2.3. The coordinate transform between the force ellipsoid and the moment ellipsoid The sensor bearing ellipsoids are described in the reference coordinate system of sensor, while the sensor task models are described in the reference coordinate system of task; therefore, both are needed to be transformed to the same coordinate system. Since the purpose of this paper is task-oriented design of the sensor, convert the sensor ellipsoids from coordinate system of sensor to the coordinate system of task. According to Eq. (14), sensor’s static mathematical models can be described in the reference coordinate system of sensor as follows: s

F = Gs f

(25)

Eq. (25) can be described in the task reference coordinate system as t

F = Gt f

(26)

Furthermore, the relationship between the generalized force in the basic coordinate system and the task coordinate system can be expressed as s

F = Jt F

where J = [

(27) s 0 tR ]; s PtO S (s Pto )ts R ts R

represents the matrix that the original

point of coordinate system{t} relative to the coordinate system{s}.

FtT CTF t CF t Ft

=1

(33)

MtT CTMt CMt Mt = 1

(34)

where

CF t = CF s ts R + CMs S(s Pto )ts R,

CMt = CMs ts R.

Eqs. (33) and (34) indicate that the shape of the sensor’s moment ellipsoid remains constant when the coordinate system is not rotated. While the shape of the sensor’s force ellipsoid is changed as long as the coordinate system is changed. Obviously, the sensor’s force ellipsoid has no invariance after coordinate transition and coordinate rotation; however, the sensor’s moment ellipsoid has invariance after coordinate transition but has no invariance after coordinate rotation. For the practical application, the users hope the equation is as simple as possible. Therefore, it’s better to establish the basic coordinate system and task coordinate system in the same direction, namely, the coordinate transformation can occur without coordinate rotation. 2.4. The sensor design based on the known task model According to the ellipsoid theory, the force ellipsoid and the moment ellipsoid of the sensor should contain the force ellipsoid and the moment ellipsoid of task, respectively. So the length constraints of the ellipsoids’ spindles can be given by

λF sx = kF x λF Gtx λF sy = kF y λF Gty λF sz = kF z λF Gtz

(35)

where λFsx , λFsy and λFsz represent the eigenvalues of matrix CFs T CFs , and the square roots of their reciprocal are half length of the sensor’s force ellipsoid spindles, respectively; kfx , kfy and kfz are positive real numbers, which are not more than 1. So the force ellipsoid of sensor meeting the task requirement can be expressed in the task coordinate system, as t T Fs QF s diag

(λF sx , λF sy , λF sz )QTF s Fs = 1 t

(36)

J. Yao et al. / Mechatronics 35 (2016) 109–121

113

Fig. 3. Prototype of the pre-stressed six-axis force sensor: 1 – measuring platform; 2 – adjust shim; 3 – sleeve; 4 – measuring leg; 5 – pre-stressing platform; 6 – pre-stressing bolt; 7 – pre-stressing nut; 8 – middle platform; and 9 – ﬁxed base.

where the three column vectors of tions of the sensor’s force ellipsoid Similarly, the moment ellipsoid quirement can be expressed in the t

matrix QFs represent the direcspindles. of sensor meeting the task retask coordinate system, as

MTs QMs diag(λMsx , λMsy , λMsz )QTMs Ms = 1 t

(37)

where the three column vectors of matrix QMs represent the directions of the sensor’s moment ellipsoid spindles; λMsx , λMsy and λMsz are the eigenvalues of matrix CMs T CMs , and the square roots of their reciprocal are half length of the sensor’s moment ellipsoid spindles, respectively, which satisfy the constraints as

λMsx = kMx λMGtx λMsy = kMy λMGty λMsz = kMz λMGtz

(38)

where kMx , kMy and kMz are positive real numbers which are not more than 1. Making Eqs. (36) and (37) equal to Eqs. (33) and (34), respectively, we can have

QF s diag(λF sx , λF sy , λF sz )QTF s = CTF t CF t

QMs diag(λMsx , λMsy , λMsz )QTMs = CTMt CMt

(39)

We can obtain the structural parameters satisfying the task requirements through solving Eq. (39). 3. Task-oriented model of the parallel force sensor prototype 3.1. The structure model of the fully pre-stressed wrist force sensor This paper proposes a fully pre-stressed dual layer parallel sixaxis force sensor. Fig. 3 illustrates the structure of dual-layer fully pre-stressed sensor. The external force is applied on the measuring platform, the middle platform is connected to the measuring platform by the ﬂange, and the seven measuring legs are connected to the middle platform, ﬁxed base and pre-stressing platform by the cone-shaped spherical pairs with unilateral constraint. The seven measuring legs can be divided into two groups, which is, three elastic legs above the measuring platform and the other four under the measuring platform, respectively. The six-dimension external force applied on the measuring platform is delivered to the seven measuring legs through the middle platform in order to realize the measurement of six-dimension force. The structure design of the pre-stressed branches is an important item in the whole design. The pre-stressed branches are composed of the ﬁxed base, sleeves, gasket, and pre-stressing platform. The ﬁxed base is connected to the sleeve by the screws, prestressing platform is connected to sleeve by the bolts, and the gasket is made of the low stiffness and high plasticity aluminum alloy

Fig. 4. Structure sketch of fully pre-stressed parallel six-axis force sensor.

material. When tightening the pre-stressed bolts, the gasket will have a certain amount of deformation, and the pre-stressing platform will have a certain press displacement in the direction of the ﬁxed base. In this way, measuring legs can bear an anticipant prestressing force. In order to change the pre-stressing force, the relative displacement between ﬁxed base and pre-stressed platform can be adjusted by changing the gasket thickness. In order to minimize the error during the mechanical machining and assembly, the measuring legs are designed to be an integral structure whose material is 40CrNi2Mo. Both ends of measuring legs are cone-shaped. Resistance strain gauges are attached to the inside of cylindrical bore, making up measuring bridge to measure the applied force along axes of measuring legs. There exist four openings around each measuring leg to reduce the impact of partial load besides the applied force along axes of measuring legs. Based on the fully pre-stressed sensor structure, the structural model can be established as shown in Fig. 4. The sensor reference coordinate system {} is established in the geometric center of the measuring platform’s lower surface. The axis XS and the projection on the measuring platform’s lower surface of the line from the coordinate system origin to the ball joint 1 are overlap. The axis ZS is vertical upward. bi and Bi (i = 1, 2,…, 7) denote the position vectors of the measuring platform’s upper surface and the center of ith spherical joint on the pre-stressing platform’s lower surface with respect to the basic reference coordinate system, respectively, and bj and Bj (j = 4, 5, 6, 7) denote the position vectors of the measuring platform lower surface and the center of jth spherical joint on the ﬁxed base upper surface with respect to the basic reference coordinate system, respectively. The top three measuring legs are evenly distributed, and the substratum four measuring legs are symmetrical distribution. The measuring legs are connected to corresponding platforms by the cone-shaped spherical pairs with unilateral constraint and the seven measuring legs are equal in length denoted by l. R denotes the circle radius where ball joint point on the prestressing platform and ﬁxed base located, r denotes the circle radius where the ball joint point on measuring platform located. Hu denotes the distance between the pre-stressing platform’s lower surface and measuring platform’s upper surface. Hd denotes the distance between the ﬁxed base’s upper surface and measuring platform’s lower surface. h denotes the thickness of measuring platform. α , β and γ denote the directional angles of ball joint B1 of pre-stressing platform, ball joint b1 of ﬁxed base, ball joint point b4 on measuring platform’s lower surface with respect to X

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axis of the frame {}, respectively. Furthermore, α is equal to zero. The force Jacobian matrix can be obtained as

⎡

⎤

b7 − B7 b1 − B1 b2 − B2 ··· ⎢ |b1 − B1 | |b2 − B2 | |b7 − B7 | ⎥ Gs = ⎣ B7 × b7 ⎦ B1 × b1 B2 × b2 ··· |b1 − B1 | |b2 − B2 | |b7 − B7 |

(40)

Substituting the sensor’s structure parameters into Eq. (30), the speciﬁc expressions of the force Jacobian matrix can be gained.

bi − Bi = [r cos γi − R cos εi r sin γi − R sin εi −Hu ]T , b j − B j = [r cos γ j − R cos ε j r sin γ j − R sin ε j Hd ]T ,

|bi − Bi | = l1 =

|b j − B j | = l2 =

R2 + r 2 − 2Rr cos(γi − εi ) + Hu 2 ,

R2 + r 2 − 2Rr cos(ε j − γ j ) + Hd 2 ,

Bi × bi = [Rh sin εi − r (Hu + h ) sin γi r (Hu + h ) cos γi

Fig. 5. The diagram of relation between task coordinate and sensor coordinate in assembly work.

− Rh cos εi rR sin(γi − εi )] , T

B j × b j = [rH sin γ j −rH cos γ j −Rr sin(ε j − γ j )]T

(i = 1, 2, 3; j = 4, 5, 6, 7), 2π 2π r = [γ1 γ2 γ3 γ4 γ5 γ6 γ7 ] = 0 − γ −γ + πγ + π −γ , 3 3 ε = [ε1 ε2 ε3 ε4 ε5 ε6 ε7 ] = αα + 2π α − 2π β −β + πβ + π −β . 3

3

What can be gained from Eq. (40) is

G = GS GTS

(41)

Substituting Eq. (40) into Eq. (41), the equation can be obtained as

⎡

11

⎢ 0 ⎢ 0 G=⎢ ⎢ 1 ⎣ 2 0

0

22

0 0

0

33

0

0 0 −2 1

3 1

1 3

2 1

0

0 0

44

55

0 0

0

⎤

0 0 ⎥ −2 1 ⎥ ⎥ 0 ⎥ ⎦ 0

66

where

1 (3r2 − 6rR cos α + 3R2 + 8r2 cos2 γ 2l 2 − 16rR cos β cos γ + 8R2 cos2 β ), 1 2 = 2 (3r 2 − 6rR cos α + 3R2 + 8r 2 sin γ 2l

3.2. The structure parameters design of the wrist sensor for the assembly task The sensor prototype is designed for peg-in-hole assembly, and connected to the shaft with the measuring platform by the ﬂange. As Fig. 5 shows, the coordinate system Os –Xs Ys Zs is the reference coordinates of the sensor. The origin Os of the coordinate system is on the centroid of the lower surface of sensor’s measuring platform. Besides, Ot –Xt Yt Zt is the task coordinate system. The two coordinate axes are parallel, respectively. The coordinate of the origin Os in the centroid of shaft is (0, 0, Zt 0 ). Assuming the length of the shaft is 180 mm and the diameter is 12 mm. According to Eqs. (5) and (6), the task force ellipsoid and the task moment force ellipsoid can be expressed as

Ft T CtF T CtF Ft ≤ 1

(42)

Mt T CtM T CtM Mt ≤ 1

(43)

According to the principle of the equivalent stress in peg-inhole assembly, and combine the models of the force ellipsoid and moment ellipsoid, we can gain the following parameters:

11 = 22

− 16rR sin β sin γ + 8R2 sin β ), 1 = 2 (3Hu2 + 4Hd2 ), l 1 2 = 2 [3r 2 (Hu + h ) − 6rR cos α (Hu + h )h 2l 2

33 44

2

+ 3R2 h2 + 8r 2 sin

γ Hd2 ],

1 2 2 [3r 2 R2 sin α + 4r 2 R2 sin (γ − β )], l2 3 1 = − 2 rR sin α Hu , 2l 1 2 = 2 [3r2 (Hu + h ) − 3rR cos α (Hu + 2h) 2l + 3R2 h − 8r 2 cos2 γ Hd + 8rR cos β cos γ Hd ], 1 3 = − 2 [3r2 (Hu + h ) − 3rR cos α (Hu + 2h) 2l

66 =

2

+ 3R2 h − 8r 2 sin

γ Hd + 8rR sin β sin γ Hd ].

T CtF CtF

16.5 0 0

=

0 16.5 0

T CtM CtM

=

0.0 0 098 0 0

0 0 0.29

0 0.0 0 098 0

0 0 0.0039

The sensor ellipsoid can be gained according to Eqs. (41), (33), and (34)

FTs CF T CF Fs = 1

(44)

Ms T CM T CM Ms = 1

(45)

where

CTF CF =

0 0

CTM CM =

XF =

XF

XM 0 0

0

Y F 0 0

Y M 0

0 0

ZF 0 0

,

ZM

,

1 2 2 3h R − 6r Hu hR − 6r h2 R + 3r 2 Hu2 12λ1

J. Yao et al. / Mechatronics 35 (2016) 109–121

115

Table 1 The key structure parameters of sensor prototype.

α (deg.)

β (deg.)

γ (deg.)

R (mm)

r (mm)

l (mm)

Hu (mm)

Hd (mm)

h (mm)

0

30

45

40

25

31

27.1293

25.8428

9.1794

+ 6r 2 Hu h + 3r 2 h2 l22 + 8Hd2 r 2 l12 cos2 γ ,

1 2 2 3Hu r + 6r 2 Hu h − 6rhHu R + 3r 2 h2 12λ2

Y F =

2

− 6rRh2 + 3R2 h2 l22 + 8Hd2 r 2 l12 sin

ZF = l12 l22

γ ,

3Hu2 l22 + 4Hd2 l12 ,

1

XM = (3R2 − 6rR + 3r2 )l22 12λ2 2 + 8 r 2 sin γ − 2rR sin β sin γ + R2 sin β l12 , 1

Y M = (3R2 − 6rR + 3r2 )l22 12λ1 + 8 r 2 cos2 γ − 2rR cos β cos γ + R2 cos β l12 , ZM = l22 [4r2 R2 sin2 (γ − β )], l1 = l2 =

Fig. 6. Calibration system of the full pre-stressed six-axis wrist force sensor.

r 2 + R2 − 2rR + Hu2 ,

⎧ 2 l 2 2 ⎪ ⎪ ⎪ ⎨12λ {3[(R − r )(h + Zt0 ) − rHu ] + 8[(Zt0 + Hd )r cos γ − Zto R cos β ] } = 16.5kF x

r 2 + R2 − 2rR cos(γ − β ) + Hd2 ,

1

l2 {3[R(h − Zt0 ) + r (Hu + Zt0 − h )]2 − 8[(Zt0 + Hd )r sin γ − Zto R sin β ]2 } = 16.5kF y ⎪ ⎪ ⎪12λ2 ⎩ l 2 /(3Hu2 + 4Hd2 ) = 0.29kF z

λ1 = (r2 cos γ (Hu + Hd + h ) − Rr cos β (Hu + h ) − rR cos γ (h + Hd ) + R2 h cos β )2 , λ2 = (r2 sin γ (Hd + Hu + h ) − rR sin γ (h + Hd ) + R2 h sin β − Rr sin β (Hu + h ))2 .

(48)

In the above formula, l1 and l2 denote the measuring legs’ length of the upper and lower platform, respectively, generally l1 = l2 = l. Converting the sensor ellipsoids from the reference coordinate system of sensor to the coordinate system of task, substituting Eqs. (44) and (45) into Eqs. (43) and (42), respectively, we can have

FtT CTF t CF t Ft = 1

(46)

MtT CTMt CMt Mt = 1

(47)

where

CTF t CF t =

XF t

CTMt CMt =

XF t =

0 0

XMt 0 0

0

Y F t 0 0

Y Mt 0

0 0

ZF t 0 0

,

l2 {3[(h − Zt )R − (h + H1 − Zt )r]2 12λ1

+ 8[(Zt + H2 )r cos γ − Zt R cos β ] }, 2

Y F t =

2

l {3[(h − Zt )R − (h + H1 − Zt )r]2 12λ2

− 8[(Zt + H2 )r sin γ − Zt R sin β ] },

ZF t = l

2

2

l2

2

2 [3(R − r ) + 8(r 2 cos2 γ − 2rR cos β cos γ + R2 cos β )] = 0.0 0 098kMy ⎪ ⎪ λ1 ⎪ ⎩12 2 2 2 2 l /[4r R sin (γ − β )] = 0.0039kMz

(49) where kfx = kfy = kfz = kmx = kmy = kmz = 0.9, making Eqs. (48) and (49) simultaneous to solve the equation set. The key structure parameters of the sensor prototype can be determined, as shown in Table 1.

4. Calibration experiments of the task sensor 4.1. Calibration system of the six-axis force sensor

,

ZMt

⎧ 2 l ⎪ 2 2 2 2 ⎪ ⎪ ⎨12λ [3(R − r ) + 8(r sin γ − 2rR sin β sin γ + R sin β )] = 0.0 0 098kMx

2

3H12 + 4H2 ,

XMt =

l2 2 2 [3(R − r ) + 8(r 2 sin γ − 2rR sin β sin γ + R2 sin β )], 12λ2

Y Mt =

l2 2 [3(R − r ) + 8(r 2 cos2 γ − 2rR cos β cos γ + R2 cos β )], 12λ1

ZMt = l 2 /[4r2 R2 sin2 (γ − β )]. According to Eqs. (35)–(38), the equations satisﬁed the peg-inhole assembly task can be obtained as

The calibration system of the full pre-stressed six-axis wrist force sensor is shown in Fig. 6, which mainly includes loading platform, six-axis force sensor, AD data acquisition device, computer, data acquisition software and so on. Since six-axis force sensor structure is very complex and the presence of measuring error is inevitable because of affection by the design principles, manufacturing and processing errors and other factors. It’s an effective mean to improve the measurement accuracy by using static calibration experiments to compensate measuring errors above. The calibration matrix of six-axis force sensor expresses the linear relationship between input and output. If the sensor is a linear system, the calibration matrix can be obtained by loading six groups of linearly independent loading force/torque and collecting the output voltage of each measuring leg. In general, the sensor is a nonlinear system, so multi-point loading within the sensor range and the method of least squares linear ﬁt are needed to calibration experiments. Thus, the liner relationship between inputs and outputs of 6-axis force sensor can be obtained. The detailed experimental steps are listed as follows. (1) Record the initial voltage before loading and reset the initial voltage.

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Fig. 7. The output value of each dimension when loading x-direction force.

Fig. 9. The output value of each dimension when loading z-direction force.

Fig. 8. The output value of each dimension when loading y-direction force.

Fig. 10. The output value of each dimension when loading x-direction torque.

(2) Each dimension force/torque of the sensor within the range is divided into a number of load points every 1 kgf/0.1 kgf m. (3) For each load point, conduct loading and unloading experiments one by one in order, and record the output voltage of each measuring leg corresponding to each load point. (4) According to step (3), conduct loading and unloading experiments in the opposite direction of corresponding dimension and record the experimental data. (5) According to steps (3) and (4), the calibration experiments of the six dimensional force/torque component of the sensor are carried out one by one and record the experiment data. (6) Repeat steps (3)–(5) three times, complete static calibration experiments are carried out three times and experimental data are recorded. (7) Check and process the data, solve the sensor calibration matrix and determine the sensor’s linearity, repeatability, hysteresis error and other static performance index.

Fig. 11. The output value of each dimension when loading y-direction torque.

4.2. Results With the calibration system above, the static calibration experiment is carried out for the developed six-axis force sensor, the force-measuring data on six dimensions is obtained. Using the least square method for calibration experiment, the error matrix EL and the repetition error ER can be obtained by the analysis and process of the experimental data, therefore, the performance of force-measuring is obtained. Taking a complete static calibration experimental data, substituting the standard loading force and the voltage data collected from seven measuring legs into the related formula, the static calibration matrix Gc can be obtained. Substituting the calibration matrix into the related formula, we can gain the loading force/torque measuring results after the calibration. Figs. 7–12 show the changing curves between the sensor’s measuring force and the standard loading force when the force or torque is loaded in one direction. As can be seen from the ﬁgures, the sensor outputs have a high linearity and the dimensional coupling value is small in different

Fig. 12. The output value of each dimension when loading z-direction torque.

direction.

⎡

0.021 −0.010 −0.009 ⎢ 0.001 0.019 −0.015 ⎢ 0.037 0.038 0.036 Gc = ⎢ ⎢ 0.0 0 0 0.005 −0.004 ⎣ −0.005 0.003 0.002 −0.0 0 0 0.0 0 0 0.0 0 0

0.023 −0.022 0.003 0.004 −0.033 −0.033 −0.001 −0.001 −0.003 0.003 −0.001 0.001

⎤

−0.027 0.025 −0.003 −0.003⎥ −0.034 −0.036⎥ ⎥ 0.001 0.001 ⎥ ⎦ 0.003 −0.003 −0.001 0.001

(50)

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117

Fig. 13. The solid model of measuring leg. Fig. 14. The prototype of six-axis force sensor.

⎡

0.0058 ⎢ 0.0010 ⎢0.0039 EL = ⎢ ⎢0.0027 ⎣ 0.0065 0.0158

0.0031 0.0058 0.0044 0.0182 0.0068 0.0019

0.0107 0.0027 0.0050 0.0104 0.0075 0.0066

0.0030 0.0026 0.0046 0.0081 0.0084 0.0046

0.0129 0.0075 0.0059 0.0123 0.0078 0.0035

⎤

0.0039 0.0055⎥ 0.0109 ⎥ ⎥ (51) 0.0061⎥ ⎦ 0.0056 0.0117

Substituting the data obtained from three times loading into the formula, the sensor’s repeatability error matrix can be obtained as follows:

⎡

0.0066 ⎢0.0045 ⎢0.0086 ER = ⎢ ⎢0.0070 ⎣ 0.0080 0.0050

0.0068 0.0024 0.0088 0.0080 0.0100 0.0040

0.0068 0.0024 0.0088 0.0080 0.0100 0.0040

0.0075 0.0038 0.0070 0.0070 0.0060 0.0040

0.0081 0.0030 0.0092 0.0050 0.0013 0.0040

⎤

0.0030 0.0057⎥ 0.0030⎥ ⎥ (52) 0.0090⎥ ⎦ 0.0090 0.0056

From the static calibration experiment results, the maximum non-linearity error of developed fully pre-stressed dual-layer parallel six-axis force sensor is 1.17% of full range. The maximal dimensional coupled nonlinear error is 1.82% of full range. Maximum repeatability error among each direction is 0.88% of full range. The maximal dimensional coupled repeatability error is 1.00% of full range. Maximum return error among each direction is 0.44% of full range.

6. Experiment study on force compliant control of the peg-in-hole assembly task The peg-in-hole assembly task is very common in the industrial production. The peg-in-hole assembly process can be divided into three stages, approaching stage, contacting stage and getting into hole stage. In the peg-in-hole assembly process, if the robot’s position precision can’t be ensure or the relative position of the shaft and the hole are unknown to the robot, stuck and wedged will be occurred easily during the contact of the shaft and the hole, once the above situation appear, the assembly will be failure. Therefore, we installed the six-axis force sensor on the end effecter of the parallel robot, converting the detected signal into the force position adjustment amount. Furthermore, the robot can do the right amount of translation and rotation through the control of the robot industrial computer, so that make the clamping workpiece in the best assembly position and ensure the successful completion of the assembly work. This paper mainly discusses the non-chamfered clearance peg-in-hole assembly strategy. In the assembly process, ﬁrstly a coarse positioning is carried out to the shaft and the hole platform, then the robot’s end effecter clamping the shaft move to the position of the assemble hole, and at last the peg-in-hole assembly task is completed based on the detected information from the sensor and the correlation force control algorithm. 6.1. The compliant control algorithm

5. Development of sensor prototype The fully pre-stressed parallel six-axis force sensor prototype is developed based on the designed drawings of the parts. Fig. 13 illustrates the measuring leg of one-dimensional force sensor, which is processed once according to the size requirement and can achieve the measurement of one-dimensional force in the axial direction. Fig. 14 illustrates the six-axis force sensor prototype with the loading cap, and the calibration force is applied on the sensor through the threaded hole in the positive and negative x-direction, y-direction and z-direction of the loading cap. This sensor is cylindrical in shape whose max outside diameter is 170 mm, height is 90 mm, mass is about 1.5 kg, measurement range is ±100 N in force and ±100 N m in moment, and resolution ratio is 0.1%F.S., the errors of the I kind and the II kind for the developed fully pre-stressed six-axis force sensor through the calibration are not more than 0.41%F.S. and 0.64%F.S., respectively. The sensor can be installed easily in the robotic wrist and also can be applied to industrial production occasions that require six-dimensional force measurement.

In this paper, the control method is based on the position control, and the force feedback servo loop is added to the parallel robot, then the compliant selection matrix is also added to the parallel robot to choose the direction of the force that needs to be controlled. The whole compliant control system block diagram is shown in Fig. 15, force signals are obtained from the force sensor, the signal processing is conducted through the USB data isolation and low-pass ﬁltering in order to make the force signal accurate, and then the force signals in work coordinate system are obtained after the force coordinate conversion. Compared with the given force vector, the product of force error Fe and compliant control matrix Kp can be compensated as the position error. The determinant S and determinant Kp are used in compliant control. S is the choice determinant of the compliant control, it is a 6 × 6 diagonal matrix whose diagonal elements are 0 and 1 which are used to select the degree of freedom that we need to position control and force control. Kp is a 6 × 6 force feedback matrix, which is obtained based on the mechanical structure stiffness and the position system stiffness and integrated the desired closed

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Fig. 15. The whole compliance control system block diagram.

Fig. 17. The whole compliant control experimental platform.

Fig. 16. Schematic diagram of force sense perception system.

loop stiffness of the robot. The force and position expression of the entire force feedback system can be obtained

x(n + 1 ) = x(n ) + S xd + S x f

(53)

x(n + 1) in Eq. (53) is the target position that the robot reaches. x(n) is the starting position of the robot. xd is the step size that the robot moves. xf is the position correction of several directions that need to be controlled by the force. It can be obtained as

x f = k p (F − Fr )

(54)

Consequently, only choosing the appropriate force feedback matrix and the degree of freedom that need to be controlled, the compliant assembly can be perfectly accomplished. 6.2. Experimental platform The compliant assembly control experiment is made up of three parts: the fully pre-stressed parallel six-axis force sensor, data acquisition system, 6-UPS parallel robot. The robot uses the vc++ to write the whole control system interface to make the manipulation more humanization. The force sensing system diagram of the compliant assembly system is shown in Fig. 16. The whole compliant control experimental platform is shown in Fig. 17. The six-axis force sensor is shown in Fig. 14, which has the characteristics of high measuring accuracy, small coupling, good stability and so on. The robot in the compliant assembly is 6-UPS parallel robot which has the characteristics of the large stiffness weight ratio, simple structure, low movement mass, good dynamic performance, larger working space and so on.

Fig. 18. Peg-in-hole assembly ﬂow.

6.3. Experimental procedure Before starting the compliant assembly experiment, the compliant assembly control system software is needed to start ﬁrstly and the processing tasks are completed in the control interface. The peg-in-hole assembly ﬂow is shown in Fig. 18. After starting the 6-axis force sensor, the touching force of the shaft and the hole is real-time monitored by the system. When the measuring force

J. Yao et al. / Mechatronics 35 (2016) 109–121

119

Fig. 19. The results of the peg-in-hole assembly.

is more than the threshold set by the system, the shaft and the hole platform contact with each other and the parallel robot begins to conduct the compliant control according to the touching force information combined with force control algorithm. After the parallel robot making an adjustment, it continues to measure the force applied on its end effecter, and cycle the above processes until the shaft inserted into the hole smoothly. At the last stage of the program, the detection is needed that whether the shaft inserted into the bottom of hole. The threshold is set to be 100 N, and the length of the shaft is 45 mm. Namely, fZ > 100 N and z > 45 mm are regarded as the detection condition of the hole bottom. If the measuring information meets the condition, the program stops running and peg-in-hole assembly is completed. The assembly gap between the shaft and the hole is 0.1 mm, and the peg-in-hole assembly experiment can be completed smoothly.

6.4. Results In general, the process of parallel robot peg-in-hole assembly is divided into two stages: non-contact stage and contact stage. The non-contact stage is completed by the position control of the parallel robot, making the shaft which installed on its end effecter close to the hole platform gradually; the contact stage is completed by the force control of the parallel robot and the purpose is to adjust the robot pose deviation to make the shaft insert the hole smoothly. The peg-in-hole assembly pose status is shown in Fig. 19. At the initial stage of the assembly, the shaft spins a small angel to make the adjustment more obvious during the contacting.

The contrast analysis diagram between the inserted depth of the shaft and the assembly force during the assembly process can be shown in Figs. 20 and 21. According to the theoretical derivation, in the initial stage of assembly, the force acting on the shaft along X-axis and Y-axis is relatively large and the ﬂuctuation is also obvious. With the increasing of inserted depth of the shaft, the center line of the shaft and the hole tend to overlap basically, and the forces acting on the shaft and hole along X-axis and Y-axis become small until to meet the deﬁned limits. Thus, the peg-in-hole assembly is completed successfully and the entire process takes 190 s which satisﬁes the requirement of the experiment. According to experimental data and the experimental phenomena, six-axis force sensor can capture the experimental data accurately when the shaft and the hole contacting each other. According to the principle of compliant control, the shaft position posture can be adjusted automatically, the center line of the shaft and the hole overlap gradually and ﬁnally the assembly task is completed. The assembly process of adjustment error is less than 0.1 mm, which satisﬁes the requirement of the experiment. 7. Conclusion In order to meet the requirement in assembly task, a taskoriented full pre-stressed six-axis wrist force sensor is introduced in this paper. According to the speciﬁc tasks, the task models based on force and moment ellipsoids are proposed, then a structural model of the sensor is determined, and the new sensor’s structure sizes are obtained by the speciﬁc theoretical calculation. Furthermore, the full pre-stressed wrist force sensor prototype is

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Fig. 20. The curve about Fx changing with the increase of depth.

Fig. 21. The curve about Fy changing with the increase of depth.

designed and manufactured. Aimed at the performance characteristics of the sensor, calibration experiment is conducted, and then the calibration matrix of the sensor is obtained. In order to verify each performance index of the sensor, the sensor experiment platform based on the peg-in-hole assembly is built and the assembly experiment is carried out. As shown in the experimental results, the sensor achieves a good function of force feedback in the process of assembly and possesses high sensitivity and small error. Besides, the experimental data is agreed with the expected data, the whole experiment achieves the desired results and veriﬁes each performance of the sensor. Acknowledgments This research is sponsored by the ﬁnancial support of the Major State Basic Research Development Program of China (973 Program) (Grant no. 2013CB7330 0 0), the NSFC (Grant nos. 51305383, 51275439), and Specialized Research Fund for the Doctoral Program of Higher Education (SRFDP) (20131333120 0 07), Hebei Province Natural Science Research Project for Distinguished Young Scholars in Higher Education Institutions (No. YQ2014040). The reviewers are also acknowledged for their critical comments.

Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.mechatronics.2016.01. 007.

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