Kinematic calibration of a 6-DOF parallel manipulator based on identifiable parameters separation (IPS)

Mechanism and Machine Theory 126 (2018) 61–78

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Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmachtheory

Research paper

Kinematic calibration of a 6-DOF parallel manipulator based on identifiable parameters separation (IPS) Yan Hu, Feng Gao∗, Xianchao Zhao, Baochen Wei, Donghua Zhao, Yinan Zhao State Key Lab of Mechanical System and Vibration, School of Mechanical Engineering, Shanghai Jiao Tong University, Shanghai 200240, PR China

a r t i c l e

i n f o

Article history: Received 19 December 2017 Revised 9 March 2018 Accepted 29 March 2018

a b s t r a c t This paper presents a new calibration method for a 6 degree-of-freedom (DOF) parallel manipulator based on identifiable parameters separation (IPS). This method begins with the parameters separation by the principle of IPS, and then identifies the subsets of the parameters sequentially and respectively. The method aims to identify as many parameters as possible just by simple and direct measuring procedures, which can lower the number of parameters in the complex identification model and thus reduce the coupling effect and uncertainty during identification. Moreover, the reduction of the parameters will allow fewer measurements and faster convergence for the identification algorithm. Firstly, the manipulator is briefly described and its geometric errors are studied. Secondly, the concept of IPS is introduced and the principle of IPS is summarized. Then, according to the proposed method, the corresponding calibration experiment is designed. Lastly, the experiment of this manipulator is conducted and the result shows the validity of the method for accuracy improvement. The rules of IPS and the simplification of complex identification enable the use of this method on other types of parallel mechanisms. © 2018 Elsevier Ltd. All rights reserved.

1. Introduction Parallel manipulator is a closed-loop kinematic chain mechanism whose end-effector is linked to the base by several independent kinematic chains [1]. Parallel manipulators are radically different, compared to traditional serial manipulators. Because the loads are distributed among the chains, parallel manipulators can allow high force or torque capacity. And they can also allow high structural rigidity and high dexterity due to their parallel coupling structure. Unlike conventional serial manipulators whose driving or transmission components are mostly mounted on the moving parts, parallel manipulators can achieve less body inertia which will allow high speed and high dynamic response. Moreover, unlike their serial counterparts whose positioning errors tend to propagate additively throughout the chain links, parallel manipulators can allow high accuracy due to noncumulative joint/link errors [2,3]. Since parallel manipulators present these good performances over their serial counterparts, they have been studied extensively for applications that require high speed, high stiffness and high precision since they were proposed [4]. However, the accuracy performance of parallel mechanism fulfills its requirement merely in theory. Previous research (from Wang and Masory) has shown that the error level of parallel mechanism end-effectors, mainly caused by the errors of manufacturing and assembling, is basically equivalent with that of serial

∗

Corresponding author. E-mail address: [email protected] (F. Gao).

https://doi.org/10.1016/j.mechmachtheory.2018.03.019 0094-114X/© 2018 Elsevier Ltd. All rights reserved.

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Y. Hu et al. / Mechanism and Machine Theory 126 (2018) 61–78

mechanisms based on the accuracy analysis of the Stewart Platform [5,6]. The subject of improving the positioning accuracy of parallel mechanisms has thus attracted researchers’ interests. Generally, the source of the manipulator inaccuracy includes geometric errors and non-geometric errors. It is widely recognized that the geometric errors of a manipulator come from the inevitable manufacturing imperfections and assembling errors as well as the natural wear and tear in service [7]. The non-geometric errors come from thermal distortions, compliance errors, clearance and friction [8,9]. As one of the two types of error sources that lead to manipulator inaccuracy, the geometric errors play a dominant role. The early work (from Becquet, Chen and Renders et al.) had proved that all non-geometric errors are responsible for only about 10% of the total positioning error [10–12]. Although some studies on the non-geometric errors can be found in the literature, most research is mainly focused on the geometric errors since then [13,14]. Considering the fact that the non-geometric errors vary from one robot to another and are also time-variant, we will restrict our analysis to the geometric errors in this paper, regardless of non-geometric errors and elastic deformations. The geometric errors of a robot are not affected by the poses of the end-effector, while the accuracy of the end-effector is influenced by the geometric errors. The kinematic calibration is recognized as a much more practical and effective way to improve the manipulator accuracy, compared to the method of imposing more strict manufacturing tolerances and more precise assembling procedures which are costly and time-consuming [6,13,15]. Kinematic calibration is defined as the process to identify the geometric errors between the actual and the nominal geometric parameters, and then to modify the kinematic model in the robot controller. Generally, the calibration can be divided into four steps: modeling, measurement, identification, and implementation [16]. There are lots of methods for error modeling and identification, and a good survey can be found in Sun’s work [17,18]. Examination of the literature shows that a variety of comprehensive studies and research work have been made in the area of kinematic calibration, and that a lot of methods have been proposed for kinematic calibration [19–23]. Typically, the methods can be classified into three major categories: the external calibration method, the self-calibration method, and the constrained calibration method. The characteristics and differences of the three types of method are particularly reported in reference [1]. Khalil et al. presented a comparison study on the methods of geometric parameter calibration by the Puma Stanford robots [24]. Zhuang studied the self-calibration for parallel mechanisms by Stewart platforms [20]. Early work from Gautier and Khalil studied the identifiable parameters, observation matrix and optimum configurations for robot calibration [25,26]. In recent years, plenty of creative approaches have been proposed, making a lot of supplements to the conventional calibration methods. Huang et al. developed a new approach using a minimum set of pose error measurements for six degree-of-freedom parallel mechanism machines [27]. Legnani et al. proposed a new measuring system based on wire-sensors for kinematic calibration [28]. Du and Zhang presented an original approach using a signal camera for automatically online calibration, and Renaud et al. proposed a new approach using legs observation for online calibration [29,30]. Wu et al. studied the selection of optimal measurement configurations and presented a new approach to find the calibration configurations which can make the positioning accuracy the best after conducting the calibration procedures [31]. So far, almost all of the existing kinematic calibration methods are following the conventional error modeling and identification procedure, with all the identifiable geometric parameters put together in one model and then identified together. In this paper, we are seeking the feasibility of reducing the number of the geometric parameters which need to be determined by the complex identification model. For this purpose, one way is to reduce the number of parameters in kinematics during the process of robot design [32], the other way is to improve the method of identification during the process of robot calibration. Our work is based on the study of the calibration of a 6 degree-of-freedom (DOF) parallel manipulator in our lab. As we know, parallel manipulators usually share the characteristic of strong coupling, which means that each output pose of the end-effector is related to all the actuated joint inputs. The identifiable parameters in the calibration model are thus coupled to the measured poses and joint displacements to some extent. Since the measuring process is prone to measurement noise or human error in practice, the identification results of these coupled parameters are easy to be affected by that noise or uncertainty especially when the number of the identifiable parameters is large. The identification results determine the accuracy of the kinematic model of a manipulator and thus determine the accuracy of the manipulator end-effector. From this point of view, we come up with the concept of the identifiable parameters separation (IPS) based on the analysis of the relationship of identifiable parameters. Rather than determine all the identifiable parameters together by one identification model, we identify the subsets of these parameters sequentially and respectively by taking advantage of the relationship of these parameters after the IPS. In this paper, a new calibration method for the 6-DOF parallel manipulator is presented based on our IPS, which aims to identify as many parameters as possible just by simple and direct measuring procedures and thus reduce the number of the remaining parameters that need to be identified by the subsequent identification model. Compared to the classical error parameters identification step, the proposed method can lower the number of the parameters for identification and thus reduce the coupling effect and uncertainty in the simplified identification model. Moreover, the reduction of the identifiable parameters will allow fewer measurements and faster convergence for identification algorithm. The objective of this work is to reduce the positioning error and achieve the desired positioning accuracy of the manipulator by the proposed calibration method. The required positioning accuracy of this manipulator is 1 mm and 0.1°. The noteworthy feature of this method is reducing the number of the geometric parameters that need to be determined by the complex identification model by sequentially and respectively identifying the subsets of the observable parameters. In the following sections, we will specifically explain how to use this method to conduct the kinematic calibration for the 6-DOF parallel manipulator. The remainder of the paper is organized as follows. Section 2 shows a brief description of the 6-DOF parallel manipulator, for which we will conduct the kinematic calibration later. Its kinematic model is presented and the geometric errors

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Fig. 1. The CAD model of the 6-DOF parallel manipulator.

are studied. Section 3 introduces the concept of IPS, on which the proposed method is based. The principle of the IPS is presented and the IPS-based error model is deduced. Then in Section 4, the experiment study is presented. According to the proposed method, the calibration experiment is designed and conducted. The calibration result is calculated and presented. 2. Description of the 6-DOF parallel manipulator Before calibration, it is necessary to have a good knowledge of the calibration target and to figure out which geometric parameter errors will contribute to the manipulator inaccuracy. Fig. 1 shows a computer aided design (CAD) model of this manipulator, which has the orthogonal structure consisting of 3-3-PUS chains. The manipulator has a moving platform (end-effector), six identical PUS kinematic chains and a fixed base. Each kinematic chain connects the fixed base to the moving platform by a prismatic joint, a universal joint, and a spherical joint in series. There are six linear actuators that actuate the six prismatic joints respectively thus making the moving platform move around with 6 degree of freedom. Based on the configurations of their prismatic joints, the six PUS chains can be divided into two groups. One group which has three prismatic joints whose direction vectors are coplanar in the horizontal plane, and the other group which has three prismatic joints whose direction vectors are parallel and in the vertical direction. The directions of the two axes of each universal joint are arbitrary. The center of each U joint is on the same line with the P joint from the same chain. The centers of the six spherical joints attached to the moving platform are theoretically coplanar and are distributed on a circle in a certain way. We define the theoretical home/zero position when the P joint, the center of U joint and the center of S joint from each chain are collinear. In this home position, the three horizontal chains are tangent to the circle on which the centers of the six S joints are distributed. Therefore, this manipulator has the symmetrical and orthogonal structure. Moreover the driving and transmission components of this manipulator are mounted on the ground. This will make the robot have large workspace and dexterity as well as good dynamic response and isotropy, which is suitable for high requirements, such as for high-precision 6-DOF motion simulation. 2.1. Inverse kinematics Since the kernel step of kinematic calibration is to find the actual geometric parameters for the kinematic model, and our error model is based on the inverse kinematics, we will present a brief description of the inverse kinematics of the 6-DOF parallel manipulator. As described above, the manipulator has six identical PUS kinematic chains, so we just choose one of them for clarity. Fig. 2 is the inverse kinematics diagram of the manipulator. Firstly, the moving platform frame {M} and the base reference frame {O} is defined as follows: As shown in Fig. 2(b), the moving platform frame is attached to the point M of the moving platform. The point M is the center point of the circle on which the centers of the six S joints are distributed. The radius of the circle is represent by rc . The x -axis of the moving platform frame is normal to this circle with its direction as shown in Fig. 2(a). The z -axis −−→ of the moving platform frame is in the direction of vector S1 M and then the y -axis can be determined by the right-hand rule. The base reference frame is a fixed frame. Its origin point O coincides with the point M when the manipulator is in its home position. As shown in Fig. 2(a), the x-axis of the base reference frame is vertically down. Its y-axis is parallel to the direction vector of the 1st P joint, with its direction as shown in Fig. 2(b) and then the z-axis can be determined by the right-hand rule.

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Fig. 2. Inverse kinematics diagram.

As seen in Fig. 2(a), point Si is the geometric center of the spherical joint (i = 1, 2, . . . , 6) and point Ui is the intersection point of the two axes of the universal joint (i = 1, 2, . . . , 6). Point Hi is the home position of point Ui (i = 1, 2, . . . , 6). The vector si represents the vector between M and Si with respect to the base reference frame {O}. The vector between Ui and Si is represented by li with respect to the base reference frame {O}. Its norm is represented by Li , which is actually the length of the rigid link. The vector ei (i = 1, 2, . . . , 6) is the unit direction vector of the prismatic joint, with respect to the base reference frame {O}. The scalar qi (i = 1, 2, . . . , 6) is the stroke of the ith linear actuator. hi is the position vector of the point Hi (i = 1, 2, . . . , 6) with respect to the base reference frame {O}. p is the position vector of the moving platform with respect to the base reference frame {O}. Let x = [ px , py , pz , α , β , γ ]T be the generalized coordinates of the moving platform, where px , py , pz are the coordinate components of the position vector p, and α , β , γ are the XYZ Euler angles which describe the orientation of the moving platform. The inverse kinematics is to find the solutions of the joint variables q1 , q2 , q3 , q4 , q5 , q6 corresponding to a given pose x of the moving platform. From the given pose x, we can construct the position vector p and the rotation matrix R of the moving platform, with respect to the base reference frame {O}. Then we can write the following equation according to the closed loop vector method:

p + si = hi + qi ei + li

, ( i = 1, 2, . . . , 6 )

(1)



Let si be the vector between M and Si , with respect to the moving platform frame {M}. Because point Si is attached to  the moving platform just like point M is, the vector si is known as a constant and the vector si can be computed by the following equation:

si = R si 

, ( i = 1, 2, . . . , 6 )

(2)

From Eq. (1), we can establish the following relation based on the norms of the vectors:

   p + R si  − hi − qi ei  = Li , ( i = 1 , 2 , . . . , 6 )

(3)

Squaring both side of Eq. (3), then we can deduce the solution of qi as follows where ς i stands for p + Rsi  − hi :

qi = ςi · ei −



(ςi · ei )2 − ςi · ςi + Li 2 , (i = 1, 2, . . . , 6)

(4)

2.2. Geometric errors Before the kinematic calibration, it is beneficial to examine which geometric errors can lead to manipulator inaccuracy. These geometric errors come from the deviations between the actual and the nominal geometric parameters. Eliminating ς i

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Table 1 The numbers of unknowns of the geometric errors. Geometric parameters

Geometric errors

Number of unknowns

si 

si 

3 × 3 × 3 × 1 × 60

hi ei Li Total number

in Eq. (4), we can obtain:





qi = p + Rsi  − hi · e i −



hi ei Li

6 = 18 6 = 18 6 = 18 6=6

 ( p + Rsi  − hi ) · ei )2 − ( p + Rsi  − hi )2 + Li 2 , (i = 1, 2, . . . , 6)

(5)

From the above equation, it is not difficult to find that si  , hi , ei and Li (i = 1, 2, . . . , 6) are the geometric parameters in the kinematics. These geometric parameters are the inherent kinematic parameters of the manipulator. Their actual values determine the accuracy of the kinematic model imbedded in the robot controller. These geometric parameters are chosen according to the sensitivity analysis of the manipulator accuracy. The way of sensitivity analysis can be found in most of literatures, which will not be discussed specifically in this paper. Because of the imperfect manufacturing of the S joint, the actual center of the sphere will deviate from the ideal center. In addition, the mounting positions of the six S joints on the moving platform may not completely coincide with the designed distributions. Then, there will be a deviation from si  , denoted by si  . Similarly, the actual home positions of the six U joints will deviate from their ideal positions, due to the manufacturing imperfections and assembling errors. For hi , these deviations are denoted by hi . Then for ei , which is the unit direction vector of the prismatic joint, there will also be some deviations due to the arrangement errors of the guide rail of each prismatic joint. These deviations are denoted by ei . Lastly, the actual length of each rigid link will also be not exactly equal to the designed dimension due to the manufacturing imperfections and assembling errors. The length deviation of each rigid link is accordingly denoted by Li . So, for the 6-DOF parallel manipulator, all the geometric errors are si  , hi , ei and Li (i = 1, 2, . . . , 6). According to the definition of geometric errors, we can get the following relations:

⎧    ⎪ ⎨si = si 0 + si h i = h i 0 + h i ⎪ ⎩ e i = e i 0 + e i L i = L i 0 + L i

, ( i = 1, 2, . . . , 6 )

(6)

where si 0 , hi 0 , ei 0 and Li 0 are the nominal/theoretical values of the geometric parameters si  , hi , ei and Li respectively. In these geometric errors, si  , hi and ei are three dimensional vectors while Li is one-dimensional scalar. For the ith PUS kinematic chain of the manipulator, si  , hi and ei have three unknowns respectively, and Li has one unknown. So the total number of unknowns of the geometric errors of the manipulator is sixty. The details are shown in Table 1. For the kinematic calibration, our task is to identify the total unknowns of the geometric errors, so as to have an accurate knowledge of the actual values of the geometric parameters of the kinematic model. Accordingly, the identifiability of the geometric errors becomes an issue in the first place. It is said that the error parameters are observable if the rank of the identification Jacobian is full [33]. For parallel mechanisms, it is hard to symbolically derive uniform rank conditions for the observability of geometric errors. But instead, the rank of the identification Jacobian can be computed numerically in the experimental study [21]. Here, we start with the assumption that all the geometric errors si  , hi , ei and Li are identifiable. Based on this assumption, we deduce the error model and then check the rank of the identification Jacobian numerically in the identification procedure. Later, the experiment results in Section 4 will verify the identifiability of these geometric errors. 3. IPS based error modeling This section describes the method of error modeling for the calibration of the 6-DOF parallel manipulator based on the identifiable parameters separation (IPS). Firstly, the concept of IPS is introduced and the principle for the IPS is presented. Then, based on the IPS, a simplified error model is deduced. The process of kinematic calibration is actually the process of the actual kinematic parameters acquisition. In fact, this acquisition is a kind of parameter identification. By measuring the actuator inputs and the end-effector outputs, the inherent kinematic parameters in the kinematic model can be identified through certain identification procedures. As mentioned in Section 1, parallel manipulators usually share the characteristic of strong coupling. The combined effect of the measurement noise, the artificial uncertainty and the large numbers of unknowns can be influential in the identification results of the kinematic parameters to some extent. Considering the fact that the noise and uncertainty are objective and unavoidable, it is easy to think of the improvement for the identification methods or models. Unlike conventional error parameters identification, which typically puts all the numerous geometric parameters together in one model and then identifies them through

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some algorithm, our method begins with the separation of these parameters and then identifies the subsets of these parameters sequentially and respectively. By the identifiable parameters separation, all the error parameters are divided into several groups. These groups are the subsets of the total error parameters and every two subsets have no intersection set. This can be explained by the mathematical set as follows:

H = G1 ∪ G2 ∪ · · · ∪ Gn Gi ⊂ H, Gi ∩ G j = ∅

∀i, j ∈ {1, 2, · · · , k} where the total parameters need to be identified are denoted by set H, and the divided groups are denoted by subset Gi . i is from 1 to k, if there are k groups. After the IPS, the subsets of the error parameters are sequentially identified. The purpose of the IPS is to pick out some of the error parameters which can be identified first just by simple and direct measuring procedures without a complex modeling process, and thus to reduce the number of the remaining error parameters that need to be identified by the subsequent identification model. This can lower the number of unknowns for the subsequent identification model and thus reduce the coupling effect and uncertainty in the model. Since the number of unknowns is reduced, the identification model can be simplified accordingly. Moreover, this will allow fewer measurements and faster convergence for the identification algorithm. That is the concept of the IPS. Our goal is to achieve the desired positioning accuracy of the 6-DOF parallel manipulator. Next, we will study the principle for the IPS based on this manipulator. As studied in Section 2, the geometric parameters that need to be identified are si  , hi , ei and Li . They are independent variables in the dimensional design of the manipulator. So logically, each of them can be identified independently and respectively in a variety of ways. This can eliminate the coupling effect between parameters. However, in practice, not all parameters can be measured directly, and there are always some parameters that need to be identified together indirectly through some complex model. The objective of the IPS is to find out the parameters that can be determined just by simple measuring procedures. Let us begin with si  . Although it is a constant vector with respect to the frame {M}, it is moveable relative to a fixed frame, such as the base frame {O}. When the pose of the manipulator end-effector changes, si  will change with respect to the ground. So we have to restrict the movement of the moving platform for the attempt to measure si  . Now assume that the moving platform is stationary. It is easy to construct a frame as the frame {M}. Then si  could be obtained based on the frame {M}. However, point Si is the equivalent spherical center of the ith S joint and is actually a virtual point inside the joint. That will make the direct measurement for si  very difficult and make other ways by indirect measurement very complicated. So it would not be appropriate for si  to be identified just by direct measurement. In other words, si  is inseparable. Accordingly, considering hi , although it is a constant vector with respect to the ground and point Hi is still, it is also hard to measure it. Since point Hi is also a virtual point inside the U joint, it not easy to be directly measured. If we want to obtain it by the definition of a universal joint, we may have to disassemble the kinematic chains. Therefore, hi is inseparable. Then think about ei . It is a non-dimensional direction vector of the P joint and is stationary with respect to the ground. Since it is easy to determine a line by several points along the line, the direct measurement for ei will be feasible. The P joint is a guide slider mechanism. When the slider moves along the guide rail, the position coordinates of any spot on the slider will make a line. The unit direction vector of that line is ei . The three dimensional (3D) coordinate measurement is easy to achieve and this will need no disassembly of the kinematic chains. So, ei is separable. Lastly, think about Li . Since the length of the rigid link is actually the distance between point Si and point Ui , the rigid link moves as the moving platform moves. Similar with si  , Li is moveable relative to the ground. In addition, point Si and Ui are virtual points, and are not easy to measure. So, Li is inseparable. Then all the geometric parameters of the manipulator can be divided into two groups. ei belongs to Group 1, while si  , ei and Li belong to Group 2. The parameters in Group 1 can be identified first independently without a complex modeling process. The remaining parameters in Group 2 are identified together subsequently. From the above analysis, the principle of the IPS can be summarized as a series of simple rules: 1. The parameters must be independent variables in the dimensional design of the manipulator. 2. The time derivative of the description of the parameters in a fixed frame must be zero: d (oPi )/dt = 0, where o Pi is the description of a parameters in the base frame {O}. 3. The parameters must have the characteristic of measurability. The direct measurement for the parameters should be really easy to achieve. 4. Mechanical integrity must be ensured. The way for direct measurement must keep the mechanical integrity with no chain disassembly. When a parameter meets all of the above conditions, the parameter can be picked out and then be identified firstly just by direct measuring procedures. Then we will present a way for the identification of the parameters in Group 1 and Group 2. Since there are many ways for error modeling, here we choose a simple and general approach for clarity. Group 1: Parameters in Group 1 are ei (i = 1, 2, . . . , 6). Getting a space straight line by linear fitting is an easy mathematical problem. The unit direction vector of this line is exactly ei . If there are n measured points for a line, the coordinates of the jth points is represented by (xj , yj , zj ). Then a least-square fitting method for the space straight line can be deduced as follows:

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Assuming that a space line goes through point (x0 , y0 , 0), the equation of the line is given by

( x − x0 ) f

( y − y0 )

=

g

=

z 1

Then the following matrix representation can be re-written:



f g

x0 y0

 



z x = 1 y

(7)

For the total n measured points, the following equation can be obtained:



f g

x0 y0



z1 1

... ...





zn x = 1 1 y1

... ...

xn yn



(8)

Then the least-squares solution is deduced:



f g

⎡ n



x0 ⎢ j=1 = ⎣ n y0

x jz j

n 

xj

j=1

y jz j

j=1

n 

yj

⎤⎡  n ⎥⎢ j=1 ⎦⎣  n

j=1

z j2

n 

⎤−1 zj

j=1

zj

n

⎥ ⎦

, ( j = 1, 2, . . . , n )

(9)

j=1

When f and g are solved by Eq. (9), the unit direction vector of the space line can be calculated. In practice, one can control the fit tolerance to ignore the points which are out of tolerance, and then to re-compute for the best fit line. Furthermore, lots of fitting algorithms are embedded in most data processing software and some measuring applications. That will bring much convenience for the identification. Compared to the traditional modeling and identification, the direct measuring and fitting approach is more concise and intuitive. Group 2: After the IPS, the rest of the parameters in Group 2 are si  , hi and Li . Since ei has been determined first, the number of unknowns here is reduced to 42. The next is the error modeling for the remaining parameters identification. Literature on robot calibration shows that the linear least-squares and nonlinear least-squares algorithms are commonly used as simple and practical ways for estimating the unknown parameters from measured data [16]. In this paper we choose the linear least-squares algorithms for clarity and without loss of generality. Considering that the inverse kinematic solution of parallel manipulators is much easier than the forward kinematic solution, our error model is based on the inverse kinematic solution by minimizing the inverse kinematic residuals [21]. According to the equations in Section 2, substituting Eq. (2) into Eq. (1) yields Eq. (10) as:

li = p + R si  − hi − qi ei ( i = 1 , 2 , . . . , 6 )

(10)

Then, squaring both sides of Eq. (10) yields:



Li 2 = p + Rsi  − hi − qi e i

T 



p + Rsi  − hi − qi e i ( i = 1 , 2 , . . . , 6 )

(11)

For a given pose of the manipulator’s end-effector, p and R are constants. Hence differentiating both sides of Eq. (11) yields:



2 Li d Li = 2 p + Rsi  − hi − qi e i

T 



Rdsi  − dhi − ei dqi (i = 1, 2, . . . , 6)

(12)

where d denotes a differential change of the entity. And Eq. (12) can be re-written as follows where ς i stands for p + Rsi  − hi :



( ς i − q i e i )T R d qi = (ςi − qi ei )T ei

dsi  

− ( ς i − q i e i )T

−Li

( ς i − q i e i )T e i

(ςi − qi ei )T ei

d hi d Li

, ( i = 1, 2, . . . , 6 )

(13)

In the above equation, the differentials dsi  , dhi and dLi are equal to the geometric errors si  , hi and Li respectively, assuming that the errors are tiny. Then dqi can be calculated as follows: m dqi = qi − qi

qi =

ςi · e i −



(ςi · ei )2 − ςi · ςi + Li 2



, ( i = 1, 2, . . . , 6 )

(14)

where m qi is the measured stroke of the ith linear actuator, and qi is the inverse kinematic solution for the ith linear actuator. Then the simplified error model for the remaining error parameters can be obtained:



m

(ςi − qi ei )T R qi − qi = ( ς i − q i e i )T e i

−(ςi − qi ei )T

(ςi − qi ei )T ei

si   h i , ( i = 1, 2, . . . , 6 ) ( ς i − q i e i ) T e i L i −Li

(15)

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A measured pose of the end-effector and the corresponding measured strokes of the six linear actuators compose one group of measurements. For each group of measurements, one can establish six independent equations of the six PUS kinematic chains according to Eq. (15). Each equation contains all the seven unknowns of the geometric errors of the corresponding kinematic chain. Since the error parameters of each kinematic chain can be solved independently, we need at least seven groups of measurements to solve for the 42 unknowns of the remaining geometric errors. However, in practice, we need a greater number of measurements and use the least squares estimation algorithm to overcome the effects of noise and uncertainty. Assuming that we have n groups of measurements, for the nth group of measurements we can get the following relation for the ith kinematic chain:



(ςi,n − qi,n ei )T Rn m qi,n − qi,n = (ςi,n − qi,n ei )T ei

−(ςi,n − qi,n ei )T

(ςi,n − qi,n ei )T ei

si   h i , ( n = 1, 2, . . . , 7, . . . , n ) (ςi,n − qi,n ei )T ei Li −Li

(16)

where m qi, n and qi, n are the measured and calculated stroke of the ith linear actuator according to the nth group of measurements. ς i, n stands for pn + Rn si  − hi . pn , Rn can be calculated from the nth measured pose of the end-effector. For all the n groups of measurements, we can get the following relation in the matrix form based on the ith kinematic chain:

⎡m

qi,1 − qi,1

⎡

⎤

⎢m qi,2 − qi,2 ⎥ ⎢ ⎥ .. ⎣ ⎦ . m

qi,n − qi,n

n×1

−(ςi,1 − qi,1 ei )T (ςi,1 − qi,1 ei )T R1 ⎢ (ς − q e )T e (ςi,1 − qi,1 ei )T ei i,1 i i ⎢ i,1 ⎢ (ςi,2 − qi,2 ei )T R2 −(ςi,2 − qi,2 ei )T =⎢ ⎢ (ς − q e )T e (ςi,2 − qi2 ei )T ei i,2 i i ⎢ i,2 ⎣ (ςi,n − qi,n ei )T Rn −(ςi,n − qi,n ei ) T (ςi,n − qi,n ei )T ei (ςi,n − qi,n ei )T ei   s i h i , ( n = 1, 2, . . . , 7, . . . , n ) Li 7×1

−Li

⎤

(ςi,1 − qi,1 ei )T ei ⎥ ⎥ ⎥ −Li ⎥ T (ςi,2 − qi,2 ei ) ei ⎥ ⎥ ⎦ −Li (ςi,n − qi,n ei )T ei

n×7

(17)

So the least-squares solution of the geometric errors of ith kinematic chain can be calculated by:

⎡m ⎤ qi,1 − qi,1   m s i  −1 ⎢ qi,2 − qi,2 ⎥ ⎥ h i =  T   T ⎢ .. ⎣ ⎦ . L i m

(18)

qi,n − qi,n

where  represents the n × 7 matrix in Eq. (17). Then the geometric parameters si  , hi , and Li can be updated by the solution through Eq. (6). In practice, the geometric errors are not so tiny to be treated as differential relations in Eq. (13), so the direct solution by Eq. (18) will not be accurate. We need to put iteration into this algorithm to solve for the accurate geometric parameters. As shown in Fig. 3, the iterative process begins with the nominal values of the geometric parameters in the kinematic model. By using the measured poses and strokes, the solution of the geometric errors can be obtained through the linear least-squares algorithms. Next, update the geometric parameters by the solved geometric errors and then put the updated geometric parameters into the kinematic model for the next circulation. When the solved geometric errors approach zero, the iteration process ends. The current values of the geometric parameters si  , hi , and Li are the identified values of them. For all the six kinematic chains, the iteration processes are the same. They can be running simultaneously thus to improve the numerical efficiency. Considering the fact that the demanded workspace of the manipulator is a six dimensional motion which contains three translations and three rotations in X, Y, Z dimensions, measuring configurations which go through all controllable DOFs are beneficial to increase the linear independence of the measured data. In this paper, we adopt full-pose measurement for the measuring configurations. Our measuring equipment can easily achieve the full-pose measurement. 4. Experiment study based on the IPS In this section, we will study how to design the calibration experiment according to the proposed method. Firstly, our measuring systems are introduced. The relevant calibration coordinate systems are built and their relations are analyzed. Then, the IPS-based calibration experiment is designed and the corresponding measuring steps are arranged. Based on this method, the calibration experiment for our 6-DOF parallel manipulator is conducted and the calibration result is presented. Lastly, a comparison experiment is presented to show the advantages of the IPS-based method over the conventional full parameter identification method. 4.1. Measuring systems According to the error model, the calibration experiment needs the spatial position measurement and the full-pose measurement. That can be easily satisfied by our laser tracking system. The measuring device is an API T3 laser tracker. Its

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Fig. 3. The iterative process for the solution of the error model.

measurement range is more than 60 m and 120 m diameter. Its absolute accuracy of a 3D coordinate is ±25 μm or 0.001 at the distance of 5 m. With the cooperation of the accessory SmartTRAK Sensor (STS), the six dimensional (6D) full-pose measurement which contains the position and the orientation can be obtained. The accuracy of the STS is 0.008 mm and 0.0035°. Meanwhile, the strokes of the six linear actuators are measured by the six linear encoders. For each linear actuator, there is a linear encoder installed side by side with its guide rail. When the moving part of one prismatic joint moves, the stroke can be measured by the relevant linear encoder. The accuracy of the linear encoders is 5 μm. For the kinematic calibration of the manipulator, the following coordinate systems are needed: The base reference frame {O}. As described in Section 2, it is a space-fixed frame in which the kinematics of the manipulator is described. The parameters identification and modification also have to be disposed in this frame. The moving platform frame {M}. As defined in Section 2, it is a body-fixed frame attached to the moving platform. The description of the position and orientation of the manipulator end-effector are based on this frame, and are determined by the transformation between the frame {M} and the frame {O}. The API laser tracker measuring frame {LM}. It is a virtual coordinate frame attached to the laser tracker. This frame is automatically generated by the laser tracker measuring system. All the original measured data are relative to this frame. The API STS frame {ST}. It is also a virtual coordinate frame attached to the API SmartTRAK Sensor. The full-pose measurement is actually the reflection of the homogeneous transformation of the frame {ST} with respect to the frame {LM}. Since the API STS is installed on the moving platform, the relationship between the frame {ST} and the frame {M} is constant. Then we can get the full-pose of the moving platform by measuring the STS. The fixed stone reference frame {FS}. It is an artificial coordinate frame for calibration and is fixed on the rack of the mechanism. The establishment of this frame is based on a marble block whose surfaces and edges have experienced precision finishing. This frame has the constant transformation relationships with the frame {O} and is easier to measure than the latter in practice. Since all the data measured by the laser tracker are expressed in the frame {LM}, we have to transform them to the frame {O} for post-processing. For this purpose, the frame {FS} is the connection between the frame {LM} and the frame {O}. In addition, each cycle of the manipulator calibration experiment, the placement of the API T3 laser tracker is random. The position and orientation of the frame {LM} is thus unknown, with respect to the frame {O}. However, if we can measure the frame {FS} in the first place, we can obtain the homogeneous transformation between the frame {FS} and the frame {LM}. Then we can obtain the exact position and orientation of the frame {LM} with respect to the frame {O}. Finally,

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Fig. 4. The coordinate systems for the kinematic calibration.

all the future measurements can be related with the frame {O}. That is the purpose of setting up the fixed stone reference frame {FS}. As shown in Fig. 4, all the relevant calibration coordinate systems are illustrated. The marble block is permanently fixed on the rack and stays still. The STS sensor and the spherical mounted reflector (SMR) target are instrument accessories. The STS sensor is used for 6D full-pose measurements while the SMR target for 3D coordinate measurements. When conducting calibration experiments, the STS sensor is installed on some spot of the moving platform, and the SMR target is installed on some spot of the slider. The API T3 laser tracker is located on the ground or on the rack to stay still. The placement of the laser track is random but able to cover all the measuring points. In this paper, we use the homogeneous transformations to describe the relationships between the coordinate systems. As mentioned above, the homogeneous transformation matrix of the frame {FS} relative to the frame {O} is constant and can be represented by the matrix O T . Accordingly, for each cycle of the calibration experiment, when the STS sensor and the FS S T are unchangeable. Here M T represents laser tracker are in position, the homogeneous transformation matrices M T and FLM ST ST S T represents the homogeneous the homogeneous transformation matrix of the frame {ST} relative to the frame {M}, and FLM transformation matrix of the frame {LM} relative to the frame {FS}. Since all the measured data by the laser trackers are relative to the frame {LM}, we can use the following derivation to transform them into the frame {O}: Let LM χ be the measured homogeneous coordinates of a point. Its homogeneous coordinates relative to the frame {O} represented by O χ can be obtained: O

χ = OF S T

FS LM LM T

χ

(19)

Let LM T be the measured homogeneous transformation matrix of the STS sensor. Then the transformation matrix of the ST n frame {M} relative to the frame {O} represented by O T can be obtained: M O M Tn

= OF S T

FS LM LM T ST Tn

M −1 ST T

( n = 1, 2, . . . , 7, . . . , n )

(20)

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Assuming that there are a total of n poses to be measured. 4.2. Design of the calibration experiment According to the result of the identifiable parameters separation for the 6-DOF parallel manipulator in Section 3, all the geometric parameters are divided into two groups. We first need to identify the parameters in Group 1, and then put them into the subsequent error model for the identification of the remaining parameters in Group 2. For Group 1, we can easily obtain ei by points measuring and linear fitting. For the identification of the parameters in Group 2, we need the full-pose measurements of the moving platform and the corresponding strokes of the six actuators. Those are the two main components of the calibration experiment. After knowing that, the next is the selection of measurement configurations. We can use the SMR target for the points measuring. For the identification of ei , the SMR target is attached on the slider of the ith actuator. The slider moves from one end of the guide rail to the other. For the purpose of clarity and without loss of generality, each time the slider moves forward five millimeters, one can take a measurement by the T3 laser tracker and record the coordinates of the SMR target. This will generate a series of relatively-uniform measurement points for each actuator. Similarly, for the full-pose measurements of the moving platform, one can randomly choose a series of relativelyuniform measurement poses inside the workspace of the manipulator. Furthermore, considering the isotropy, it is better to make the measurement poses symmetric about the symmetry axis of the workspace. In addition, the construction of the calibration coordinate systems is also a practical problem. Since the base reference frame {O} is defined based on the theoretical model of the manipulator, it is actually an invisible frame which is not easy to measure in practice. We need several steps to construct this frame in the calibration experiment. We use the laser tracker to scan the cylinders of the six sliders to obtain the axis of each cylinder. The axes of the three horizontally-arranged sliders can construct a space plane, while the axes of the three vertically-arranged sliders can construct a new cylinder geometrically. Then the origin of the frame {O} is determined by the intersection of the space plane and the axis of the new cylinder. The y-axis of the frame {O} is parallel to the direction vector of the 1st P joint. The cross product of the direction vectors of the 1st P joint and the 4th P joint will generate a new vector. The z-axis of the frame {O} is parallel to this new vector, and then the x-axis can be determined by the right-hand rule, with its direction as shown in Fig. 4. Then, according to the definition in Section 2, the moving platform frame {M} coincides with the frame {O} when the manipulator is in its home position. The construction of the frame {FS} is based on the marble block. There are many approaches to construct the frame {FS} by the surfaces and edges. However, once an approach is adopted, one must follow the same approach next time for the calibration experiments. Here we present one way to construct the frame {FS}. As shown in Fig. 4, use the laser tracker to scan the edge a and edge b to obtain the direction vector of each edge. Scan the surface p to obtain a space plane p. The cross product of the direction vectors of the two edges is defined as the z-axis of the frame {FS}. The intersection of the plane p and the z-axis is defined as the origin of the frame {FS}. The x-axis of the frame {FS} is parallel to the direction vector of edge b, and then the y-axis can be determined by the right-hand rule, with its direction as shown. For a parallel manipulator, the calibration is not once and for all. The manipulator must be calibrated termly to keep a good performance

Fig. 5. The prototype of the manipulator.

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Y. Hu et al. / Mechanism and Machine Theory 126 (2018) 61–78 Table 2 The nominal and the identified values of the geometric parameters. Geometric parameters

Nominal/theoretical values

Identified/actual values

si 

s1 0 = (0, 0, −450)T s2 0 = (0, −389.711432, 225)T s  = (0, 389.711432, 225)T

s1  = (0, 0, −450.193018)T s2  = (0, −388.719881, 227.091629)T s3  = (0, 391.033501, 223.084188)T s4  = (−18.642549, −227.090526, −388.885639)T s5  = (−19.074117, −223.527949, 391.718941)T s6  = (−19.125220, 450.638543, −2.221326)T h1 = (0.249816, 1199.734923, −449.609151)T h2 = (−3.055794, −989.263040, −813.029844)T h3 = (5.786219, −187.294704, 1277.176114)T h4 = (1181.927502, −225.729730, −393.529027)T h5 = (1180.680832, −225.600856, 387.249265)T h6 = (1181.457485, 449.225164, −5.383342)T e1 = (0, −1, 0)T e2 = (0.002269, 0.501817, 0.864971)T e3 = (−0.0 0 0401, 0.504294, −0.863532)T e4 = (−0.999991, 0.004265, 0)T e5 = (−0.999993, 0.0 0 0127, 0.0 03624)T e6 = (−0.999999, −0.0 0 0726, 0.0 0 0840)T L1 = 1199.735091 L2 = 1201.047086 L3 = 1202.333889 L4 = 1200.579801 L5 = 1199.765066 L6 = 1200.587701

30

s4 0 = (0, −225, −389.711432)T s5 0 = (0, −225, 389.711432)T s6 0 = (0, 450, 0)T h1 0 = (0, 1200, −450)T h2 0 = (0, −989.711432, −814.230485)T h3 0 = (0, −210.288568, 1264.230485)T h4 0 = (1200, −255, −389.7114312)T h5 0 = (1200, −225, 389.711432)T h6 0 = (1200, 450, 0)T e1 0 = (0, −1, 0)T e2 0 = (0, 0.5, 0.866602)T e3 0 = (0, 0.5, −0.866602)T e4 0 = (−1, 0, 0)T e5 0 = (−1, 0, 0)T e6 0 = (−1, 0, 0)T L1 0 = 1200 L2 0 = 1200 L3 0 = 1200 L4 0 = 1200 L5 0 = 1200 L6 0 = 1200

hi

ei

Li

Fig. 6. Photographs of the experiment.

of accuracy during service. The frame {O} should be constructed only the first time for the calibration experiment. The frame {FS} must be measured again once the location of the T3 laser tracker changes. Based on the above analysis, the corresponding measuring steps are arranged as follows: Firstly, ready the measuring instrument. Place the laser tracker in a suitable location and make sure it will not move. Install the STS sensor on the moving platform and paste the SMR target on the slider of one kinematic chain. Secondly, determine the position and orientation of the laser tracker. Since all the measured data are expressed in the frame {LM}, we should first know the relation between the laser tracker measuring frame and the fixed frame. Measure and construct the fixed stone reference frame {FS} to obtain the transformation matrix LM T. FS Thirdly, determine the base reference frame {O}. According to the previous instructions in this part, measure and construct the frame {O} to obtain the transformation matrix LM O T . Then the constant transformation matrix of the frame {FS} relative to the frame {O} can be calculated: O F ST

−1 = (LM O T)

LM FS T

(21)

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Fig. 7. The norms of the geometric errors during iterations.

Fourthly, determine the direction vectors of the six P joints. Measure and record the coordinates of the SMR target along the driving directions of each actuator. Then ei can be obtained by post-processing and linear fitting. Fifthly, determine the transformation matrix between the frame {ST} and the frame {M}. Take a full-pose measurement of the STS sensor when the manipulator is in home position. We can get the transformation matrix LM ST T0 . Then the transformation matrix of the frame {ST} relative to the frame {M} can be calculated: M ST T

=

LM −1 LM O

T

ST

T0

(22)

Sixthly, obtain the actual poses of the moving platform and the actual strokes of the six actuators under a series of given configurations. Input the nth pose to the manipulator controller and measure the STS sensor to get LM ST Tn . The strokes can be read from the linear encoders. The measured poses of the moving platform relative to the frame {O} can be obtained by Eq. (20). It should be noted that the transformation O T determined by the third step is permanent since it is the relation of FS two fixed frames. So in future experiments, this step can be skipped. When executing to the fourth step, the geometric parameters in Group 1 can be identified. Then combined with the measurements by the sixth step, the rest of the geometric parameters in Group 2 can be identified subsequently by the solution of the error model. 4.3. Experiment and the result The full-size prototype of the 6-DOF parallel manipulator is shown in Fig. 5, which is nearly 3.5 m high. The workspace of this manipulator is a cylinder with radius 300 mm and height 400 mm. The nominal length of the six rigid links is 1200 mm and the nominal radius rc is 450 mm. The acute central angle of the adjacent S joints is 30°. According to the kinematics in Section 2, the nominal values of the geometric parameters are listed in Table 2. In this table, the unit of si  , hi and Li is millimeter, while the unit of ei is 1. The calibration task is to obtain the actual values of the geometric parameters to modify the kinematics model in the controller. Some photographs of the calibration experiment are shown in Fig. 6. In the sub-graph on the upper-left corner of Fig. 6, the SMR target is attached on the slider of the actuator of one kinematic chain, for 3D coordinates measurements.

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Fig. 8. The pose errors before and after calibration.

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Table 3 The maximum pose errors during the verification experiment. Pose errors

px

py

pz

α

β

γ

Values

0.0743 mm

0.2858 mm

0.1885 mm

0.0122°

0.0118°

0.0159°

Fig. 9. The norms of the geometric errors during iterations of the comparison experiment.

The other three sub-graphs in Fig. 6 show some different poses of the moving platform during the 6D full-pose measurements. Considering the basic principles which are proposed in reference [34] for the selection of measuring configurations, we choose 66 relatively-uniform poses inside the workspace for the pose measurements. This number of poses can be beneficial to overcome the effects of noise and uncertainty. Among these poses, the angle of rotation about a fixed coordinate axis of the frame {O} ranges from −20 to −20°. According to the measuring steps illustrated previously, the required measurements are obtained. Through the linear fitting and the succeeding error model solving procedures, all the 60 unknowns of the geometric errors are identified. The identification results are listed in Table 2, which shows the actual values of the geometric parameters. The identification procedure of the rest parameters in Group 2 shows a good convergence. As shown in Fig. 7, the norms of the geometric errors of each kinematic chain quickly converge to zero within 5 iteration steps. Actually, the average computing time of the iterative routine by MATLAB is less than 1 s, which is quite satisfactory. Furthermore, the solution of the error model also indicates that the rank of the identification Jacobian is full in each iterative step, which numerically proves the identifiability of the geometric errors. According to Fig. 8, the pose errors of the 66 measured configurations of the moving platform are improved a lot after the calibration. Before calibration, the pose errors are almost up to 8 mm and 0.4° about the z-axis of the frame {O}. After the modification of the kinematics, the calibrated position and orientation errors of the measured poses are less than 0.2 mm and 0.040° about x-axis, 0.4 mm and 0.025° about y-axis, 0.4 mm and 0.035° about z-axis. In addition, another 13 poses are randomly chosen inside the demanded workspace of the manipulator to verify the accuracy of the calibrated manipulator. The poses are among the typical configurations of the workspace which covers the

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Fig. 10. The pose errors after calibration of the comparison experiment.

rotations from −15 to 15° and the shifts from −160 to 230 mm. During the verification experiment, each pose is measured five times. The maximum pose errors are listed in Table 3. Since the demanded workspace is smaller than the whole manipulator workspace, the results are better. 4.4. Comparison experiment In order to show the advantages of the IPS-based kinematic calibration method, a comparison experiment is presented between the IPS-based method and the direct full geometric parameter identification method. During the comparison experiment, the measuring configurations are the same with the 66 target poses of the IPS-based calibration experiment in previous paragraphs. The installation of the STS sensor and the location of the API laser tracker are also the same. The comparison experiment, following the IPS-based calibration experiment, is conducted in a relatively constant environment with the same temperature and humidity. Also, the error model with full geometric parameters is based on the inverse kinematic solution by minimizing the inverse kinematic residuals [21]. After the full-pose measurements and the iterative least-squares solving process, the result of the comparison experiment is obtained. As shown in Fig. 9, the norms of the geometric errors of each kinematic chain converge to zero within 6 iteration steps. The total number of iteration steps are 32. For better observation, the norms of ei have been expanded by a factor of 103 in Fig. 9. Compared to the iteration procedures with total 27 steps shown in Fig. 7, the convergence of the comparison experiment is 18% slower. That is to say, the IPS-based calibration method shows at least 15% faster convergence than the direct full geometric parameter identification method. Moreover, the average computing time of the full geometric parameter identification is 2 s, which is at least two times longer than the IPS-based method. Fig. 10 shows the calibration result of the full geometric parameter identification method. Pose errors of the 66 measured configuration are illustrated. This figure shows that the calibrated position and orientation errors of the measured poses are

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less than 0.6 mm and 0.10° about x-axis, 1.2 mm and 0.15° about y-axis, 1.0 mm and 0.15° about z-axis. The range of errors is relatively large. By contrast, the calibrated pose errors by our IPS-based method shown in Fig. 8 are much smaller. We can say that, for this 6-DOF parallel manipulator, the IPS-based calibration method brings an average of 50% better accuracy improvement than the direct full geometric parameter identification method. In our opinion, there may be two reasons for this good performance of the IPS-based method. One is that the first identification of ei can make the construction of base reference frame {O} more accurate. The other is that, the reduced number of parameters in the identification model can lower the coupling effect and uncertainty during identification thus to increase robustness. 5. Conclusions In this paper, a new kinematic calibration method for parallel mechanism is presented based on the concept of IPS. Compared to the existing calibration methods which are following the conventional error modeling and identification procedures, the proposed method begins with the parameters separation by the principle of the IPS, and then identifies the subsets of the parameters sequentially and respectively. This method aims to identify as many parameters as possible just by simple and direct measuring procedures. This can lower the number of the parameters in the complex identification model and thus reduce the coupling effect and uncertainty. The principle of the IPS is studied and summarized by simple rules, which are listed in Section 3. Based on this method, the corresponding calibration experiment is designed. The case study on our 6-DOF parallel manipulator is presented. The experiment results show great improvement of the accuracy of the manipulator by the proposed method. After calibration, the positioning accuracy of the manipulator is within 0.4 mm and 0.04°, which is better than the required positioning accuracy of 1 mm and 0.1°. The verification experiment shows good accuracy inside the demanded workspace after calibration. In the end, the comparison experiment shows the advantages of the IPS-based method over the direct full geometric parameter identification method, in terms of accuracy, speed of convergence and solving time. This method is developed from a particular type of parallel mechanism, but the idea of the IPS and the design of experiment might be used for other kinds of parallel mechanisms, such as the parallel mechanisms with linear actuators at the base introduced in reference [30]. Acknowledgments This work is partially supported by the National Natural Science Foundation of China (grant nos.51675328 and 51335007); Science and Technology Commission of Shanghai-based “Innovation Action Plan” Project (grant no.16DZ1201001). Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.mechmachtheory. 2018.03.019. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20]

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