Non-kinematic calibration of industrial robots using a rigid–flexible coupling error model and a full pose measurement method

Robotics and Computer Integrated Manufacturing 57 (2019) 46–58

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Non-kinematic calibration of industrial robots using a rigid–flexible coupling error model and a full pose measurement method

T

⁎

Chen Xiaoyana,b, , Zhang Qiujua,b, Sun Yilina,b a b

School of Mechanical Engineering, Jiangnan University,1800 Lihu Avenue, Wuxi 214122, China Jiangsu Key Laboratory of Advanced Food Manufacturing Equipment and Technology, 1800 Lihu Avenue, Wuxi 214122, China

A R T I C LE I N FO

A B S T R A C T

Keywords: Rigid-flexible coupling Non-kinematic calibration Robot accuracy Full pose measurement Dual-index

This study addresses the non-negligible flexibility problem of industrial robots and describes a rigid–flexible coupling error model for robot non-kinematic calibration. Simulation confirms that compliance errors are major sources of inaccuracies and affect robot accuracy with geometric errors. To reduce the impact of measurement noise and improve calibration performance, an enhanced approach is proposed for full pose measurement and identification optimisation. This approach is based on the self-adaption particle swarm optimisation (SAPSO) algorithm with a dual-index (i.e. observability index O1 and identification accuracy index Omin) and a modified Levenberg–Marquardt algorithm, which considers external constraints (i.e. structural interference and angular limitation). Simulation results illustrate that the 36 error parameters of the proposed error model can be identified with improved accuracy and stability using the proposed approach. Finally, experimental results obtained with a Staubli TX60L robot with a FARO laser tracker are introduced to demonstrate the practical effectiveness of our approach in robot accuracy improvement.

1. Introduction Industrial robots play a significant part in advanced automatic production lines designed to achieve the purpose of industry 4.0, with advantages of fast operation, high efficiency, modular structural design, flexible control system and high repeatability. The absolute positional accuracy of robots has become a prominent factor affecting robot accuracy with the widespread growth of industrial robot applications. Therefore, many researchers have focused on improving robot accuracy effectively by measuring and compensating absolute positional accuracy. The positioning errors of robots are typically divided into two types: geometric and non-geometric errors [1]. Robot accuracy is enhanced generally through robot calibration. Robot kinematic calibration has been studied extensively [2–9]. The classical kinematic calibration procedure comprises three basic steps [10]: (1) kinematic error modelling, (2) pose measurement and (3) kinematic parameter identification. Data of measurement poses required for kinematic parameter identification are obtained by a measurement system. Several commercial devices [11–15] and custom-designed tools [16–20] are adopted for robotic measurement systems. Laser tracker measurement systems were developed through high-performance laser technology, which has the advantages of a large measuring range, convenient adjustment and installation, good flexibility ⁎

and high precision and efficiency. Selecting the appropriate measurement system in practice is critical because it determines the performance of the experimental data and the efficiency of robot calibration. Laser tracker equipment is currently used as the optimum device that meets this target of industrial robot calibration. Data of measurement poses are important for achieving high accuracy and robust solutions of kinematic parameter identification. However, only a limited number of studies have directly addressed the problem of measurement and can thus significantly improve measurement efficiency. To estimate the effects of selected measurement poses on parameter identification, several quantitative performance measures [21] have been proposed and used as the objectives of measurement optimisation. Klimchik [22] used D-optimality criteria to select optimal measurement poses for planar anthropomorphic manipulators. Nearly all previous works [23–26] in the area of calibration optimisation provided users with iterative methods that minimise the objective function that depends on the singular values of the identification Jacobian (e.g. observability indexes). However, these approaches do not directly consider parameter estimation errors and may therefore lead to unexpected results (good observability indexes but low identification accuracy). Moreover, computation is time consuming due to poor convergence and high-dimensional data, and the effects of external constraints (e.g. measurement tools and measuring conditions) have

Corresponding author. E-mail address: [email protected] (X. Chen).

https://doi.org/10.1016/j.rcim.2018.07.002 Received 23 November 2017; Received in revised form 7 July 2018; Accepted 7 July 2018 0736-5845/ © 2018 Elsevier Ltd. All rights reserved.

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(Fig. 1a). The kinematic model of the Staubli TX60L robot is shown in Fig. 1b, which defines the seven link frames associated with the robot. This section aims to provide a solution for formulating a high-precision robot calibration model for industrial robots with non-negligible flexibility.

been ignored by these studies. Hence, a novel measurement optimisation method is required to provide an enhanced solution for robot calibration. As summarised in [27], robot kinematic calibration technology is nearly mature, in addition to certain optimisation problems. However, kinematic parameter identification may be influenced by non-geometric errors caused by several factors. Non-kinematic calibration is more difficult than kinematic calibration [28,29]. An efficient approach was proposed in [8]; this application considers the flexibility of the joints but not the geometric errors. To solve this problem, a non-kinematic error model was developed to consider not only geometric errors but also important non-geometric errors [30]. This sophisticated model, which features the flexibility of joints and links, can be developed using a CAD technique proposed in previous research [31]. However, from a mathematical point of view, this model is not suitable for calibration, which includes many redundant parameters. According to robotics literature, compliance errors caused by the flexibility of joints and links are the most critical non-geometric errors. Compliance errors must be highlighted under regular conditions. In addition to a complete kinematic error model, a sophisticated one that incorporates compliance errors is required. In this work, a Staubli TX60L robot is used as a prototype of an assisted tool in a theoretical analysis, experimental study and validation. The primary goal is to present a rigid–flexible coupling error model consisting of geometric and compliance errors for industrial robots with non-negligible flexibility. The enhancement of full pose measurement and optimisation method adopted for the external constraints is highlighted to efficiently obtain correct and stable values of geometric and compliance parameters. Furthermore, an optimised identification algorithm is utilised to obtain the comprehensive error parameters. The remainder of this paper is organised as follows. Section 2 presents the rigid–flexible coupling error model. Section 3 presents the main contribution, which is the proposed approach for full pose measurement optimisation. Section 4 describes a simulation study. Section 5 discusses the experimental results obtained for the non-kinematic calibration of a Staubli TX60L robot. Finally, Section 6 summarises the main results and contributions of this work.

2.1. Robot kinematic error model The robot kinematic model must satisfy certain requirements to be suitable for robot calibration. The model should be able to describe all geometric errors continuously. Accordingly, the proposed model modifies the D-H approach by utilising five parameters to describe the transformation. Then, the MDH model is used for the robot kinematic modelling in this study. Consequently, the kinematic homogeneous transformation between two consecutive link frames (Fi-1) and (Fi) can be written as follows:

Ai = Rot (z , θi ) Trans (0, 0, di ) Trans (ai , 0, 0) Rot (x i , αi ) Rot (yi , βi )

(1)

This paper uses Δθ, Δd, Δa, Δα and Δβ to represent joint angle deviation, link length deviation, link deviation, twist angle deviation and angle deviation of adjacent parallel axes, respectively. Differentiation method is used to establish the robot kinematic error model, as shown in Eq. (1).

dAi =

∂AiN ∂AiN ∂AiN ∂AiN ∂AiN Δθi + Δdi + Δai + Δαi + Δβi ∂θi ∂di ∂ai ∂αi ∂βi

(2)

The actual pose of the robot tool frame (Ftool) relative to the actual robot base frame (F0) is not equal to the nominal pose because of kinematic errors. The error equation is therefore given as follows: 0T R T

− 0TTN = dT

(3)

The actual robot base frame (F0) cannot be touched, and the measurement of the constructed robot base frame (Fb) has errors. Thus, the small transformation matrix BT0 is introduced to express the errors between the constructed robot base frame (Fb) and the actual robot base frame (F0), which can also be simplified by representing an error vector Δσ with six parameters. Then, the real error equation is updated as follows:

2. Robot rigid–flexible coupling error modelling

BT R

The Staubli TX60L robot is a 6-DOF serial robot with six revolute joints and can function as a typical representative of industrial robots

By substituting Eqs. (3) into (4), we can finally obtain the original robot kinematic error model as follows:

T

− 0TTN = dT R

(4)

Fig. 1. A 6-DOF serial robot: (a) The Staubli TX60L robot; (b) the kinematic model of the Staubli TX60L robot. (a) The Staubli TX60L robot is a 6-DOF serial robot with six revolute joints, and can be regard as a typical representative of industrial robots; (b) the kinematic model of the Staubli TX60L robot, which defines the seven link frames associated with the robot.

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dT R = dT + ( BT0 − E ) 0T TN

(5)

where E is a 4 × 4 identity matrix. With successive multiplication of the transformation matrices with the kinematic errors and a combination of Eqs. (1) and (2), the differential motion dT can be calculated as follows: M

dT =

∏ (Ai

+ dAi ) AT − 0TTN

(M = 6 in this work).

i=1

(6)

By substituting Eqs. (6) into (5), expanding the new equation and ignoring the high-order terms, we can obtain the final differential motion dTR. The position errors are directly extracted from dTR, and the conversion method between the direction cosine algorithm and Euler angle algorithm is used to acquire the orientation errors. Finally, the robot kinematic error model is established as follows:

P ΔXk = ⎡ Δk ⎤ = Hk *ηk ⎢ RΔk ⎥ ⎣ ⎦

(7) Fig. 2. Position errors caused by joint angle deviations and the link length deviations. The joint and link compliance is shown as the deviation of the joint angle and link length; the results of the position errors caused by the joint angle deviations and the link length deviations are presented with three sets of different deviations, and the 30 measurement poses are selected in the robot workspace randomly.

where

Hk = Mθ + Md + Ma + Mα + Mβ + Mσ

(8)

ηk = [Δθ Δd Δa Δα Δβ Δσ ]

(9)

where ΔXk, PΔk and RΔk denote the pose, position and orientation errors, respectively; Hk is the identification matrix; ηk represents the kinematics error parameters and Mθ,Md,Ma,Mα, Mβ and Mσ are the coefficient matrices corresponding to the error parameters.

and the external payload is deemed to cause the compliance errors. The centroid of the external payload does not coincide with the origin of the robot tool frame (Ftool). Thus, the vector 6Pm is represented between the centroid of the external payload with respect to the origin of the flange frame (F6), and the vector 6Ptool is represented between the origin of the robot tool frame (Ftool) and the flange frame (F6). Gravity demonstrates a downward vertical direction due to the external payload. The force vector can be given as follows:

2.2. Robot compliance error model Robot accuracy is also influenced by non-geometric errors, which are changeable during robot operation. Most non-geometric errors are sophisticated, such as friction coefficients and temperature variations. However, joint and link compliance can be predicted with the use of an appropriate model on the basis of the torque–torsion relation, hypothesis of Navier–Bernoulli and so on. Therefore, we focus on the effects of compliance errors caused by joint and link compliance on robot position accuracy in this section. Joint and link compliance can be shown as deviations of the joint angle and link length. The effects of link compliance can be divided into two parts: joint angle deviations and link length deviations. Joint compliance results in joint angle deviations. In this work, a simulation performed in the software MATLAB and is based on a Staubli TX60L robot is conducted to analyse the effects of joint angle and link length deviations on robot accuracy. Firstly, the nominal values of 30 measurement poses selected in the robot workspace are randomly obtained. The actual values can subsequently be computed by the robot kinematic model (Table 3) with the deviations of the original link length and joint angle. Link length and joint angle deviations are each magnified by 10 times the original values. Finally, the position error for each measurement pose is calculated, and then the three sets of position errors are determined (i.e. differences between actual and nominal position values). Relevant results of the position errors caused by joint angle and link length deviations are shown in Fig. 2. The effect of the variation in joint angle deviations on robot accuracy is more sensitive than that of link length deviations. Furthermore, structural stiffness is high for most industrial robots, thereby resulting in small compliance errors of the links. Therefore, superposition theory can be applied, the compliance errors can be aggregated to the joint angle deviations and the link length deviations caused by link compliance can be disregarded. On the basis of the hypothesis in a previous study [30], each robot joint is assumed to be a linear torsional spring with one stiffness coefficient in this work. Therefore, pose errors of the robot end-effector caused by compliance errors can be formulated by stiffness coefficients,

0F m

= [0 0

−G

(G = mg, g = 9.8 m/s2).

0 0 0]T

(10)

According to the definition of force Jacobian matrix, the force Jacobian of the external payload in the flange frame (F6) is as follows: T

6

m JF

E3 0⎤ (q) = ⎡ 0 6 ⎢ Ψ( 6 R Pm) E3 ⎥ ⎣ ⎦

(11)

where T ⎡ 0 − Pz Py ⎤ 0 − Px ⎥ Ψ(P ) = ⎢ Pz ⎢− P P 0 ⎥ y x ⎦ ⎣

(12)

Ѱ (P) is the antisymmetric matrix for a three-dimensional vector P. Then, the force vector in the flange frame (F6) is updated as follows: 0F 6

= 6mJF (q)* 0Fm

(13)

According to the correlation between the definitions of torque–torsion relation, kinematic Jacobian and force Jacobian, the pose errors of the origin of the flange frame (F6) can be calculated as follows:

ΔX60 = Jk (q)*K q−1 JF (q)* 0F6

(14)

where Jk(q) is the kinematic Jacobian, JF(q) is the force Jacobian and Kq represents the joint stiffness. On the basis of the kinematic Jacobian matrix, we can obtain the pose errors of the origin of the robot tool frame (Ftool) as follows: 0 ΔXtool =

tool 0 6 Jk (q)*ΔX6

(15)

where T

tool 6 Jk (q)

48

E − Ψ( 60R 6Ptool ) ⎤ =⎡ 3 ⎢0 ⎥ E3 ⎣ ⎦

(16)

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By substituting Eqs. (14) into (15), we can obtain the robot compliance error model and describe the effects of the compliance errors caused by the external payload as follows: 0 ΔPF = ΔXtool = RF *λ C

RF =

tool 6 Jk (q)*Jk (q)*diag

λ C−1 = K q = [λ1

λ2

λ3

Table 1 The assumed geometric errors and compliance errors of the Staubli TX60L robot (“–”does not exist).

(17)

{JF (q) 6mJF (q)* 0Fm} λ4

λ5

λ 6]T

(18) (19)

where λc is the compliance vector for robot joints and Kq is the vector of six stiffness coefficients corresponding to six joints of the Staubli TX60L robot.

ΔPg = Λ g K g

(20)

sin θ2 cos(θ2 + θ3) 0 ⎤ Λ g = Jk2,3 (q)*⎡ ⎢ 0 0 cos( θ 2 + θ3) ⎥ ⎦ ⎣

(21)

K g = [k1

(22)

RF

η ⎡ k⎤ Λ g] ⎢ λ C ⎥ = H *η ⎢ Kg ⎥ ⎣ ⎦

Δdi /(mm)

Δai /(mm)

Δαi /(°)

Δβi /(°)

−0.08 −0.4 0.09 0.03 – −0.04 – −0.05 0.02 0.03 0.05 0.2 −0.1 −0.05 – 0.03 −0.1 0.06 −0.008 – 0.06 0.1 −0.08 0.01 – −0.1 −0.3 0.03 0.07 – Xb = 0.3 mm, Yb = −0.2 mm, Zb = 0.1 mm, αb = 0.02°, βb = 0.01°, θb = 0.01° λ1 = 1.47E + 10, λ2 = 6.67E + 07, λ3 = 3.13E + 07, λ4 = 2.22E + 07, λ5 = 1.18E + 06, λ6 = 7.69E + 05 k1 = 0.0015, k2 = 0.00123, k3 = 0.00054

orientation errors. In this simulation, the least squares algorithm is applied to estimate the error parameters and is given as follows: k

Z = arg minf (η) = η

∑ (ΔX i − H iηi)2 = i=1

ΔX − Hη

2

(24)

According to our preliminary work [32], 14 is the minimum number of measurement poses under the condition of high identification accuracy. Therefore, with a random acquisition of the nominal pose values of 14 measurement poses throughout the entire robot workspace, the 14 actual pose values are calculated by the actual robot model with the assumed 39 error parameters (Table 1). Finally, on the basis of the least squares algorithm, the impact of measurement noise is disregarded. The 28 geometric parameters are identified using the four pose error models. The identification accuracy results obtained by the four pose error models are presented in Fig. 3. The identification accuracy rate of the 36-parameter error model is nearly 100%. When link self-gravity is

where Kg is a vector of the three link self-gravity coefficients. Finally, the rigid–flexible coupling error model for robot calibration can be expressed as follows:

ΔX = ΔXk + ΔPF + ΔPg = [Hk

1 2 3 4 5 6 Δσ

Kg

A robot compliance error model which presents the effects of the compliance errors caused by the external payload on robot accuracy is formulated in the previous section. Although only static calibration is considered in this work, the influence of the link self-gravity on robot accuracy remains non-negligible. Thus, link self-gravity and external payload are considered in developing the compliance errors. In our preliminary work [32], the additional compliance error model, which described the effects of the compliance errors caused by the link self-gravity, could be updated as follows:

k3]

Δθi /(°)

Kq[Nm/rad]

2.3. Calibration model of robot and simulation study

k2

i

(23)

The application of a non-redundant error model is the precondition of correct identification of the error parameters. Redundancy can be eliminated by applying numerical or analytical techniques [33]. The QR decomposition method is used to generate the desired error model in this study. In Section 2, we develop the complete kinematic error model, which comprises 30 geometric error parameters, of which six are error parameters of Δσ, which describes the small transformation between the actual robot base frame (F0) and the constructed robot base frame (Fb), and 24 are kinematic error parameters (DHM parameters for Links 1 to 6). Therefore, the rigid–flexible coupling error model considers 39 error parameters (i.e. 30 geometric parameters and 9 compliance error parameters). Using the QR decomposition method, we can obtain the non-redundant kinematic error model (the 28-parameter error model, which has 28 geometric error parameters) and the nonredundant rigid–flexible coupling error model (36-parameter error model, which has 28 geometric error parameters, 5 stiffness coefficients and 3 link self-gravity coefficients). In this section, a simulation study is conducted to further illustrate the effects of the compliance errors caused by the external payload and the link self-gravity on robot pose errors and the importance of these compliance errors in the correct identification of the actual kinematic parameters. For comparison, two other non-redundant models are used to identify the 28 geometric parameters. The first model is a 33-parameter error model, which is the 36-parameter error model without the three link self-gravity coefficients. The second non-redundant model is a 31-parameter error model, which is the 36-parameter error model without the five stiffness coefficients. All four error models are nonredundant pose error models (Section 3) that include position and

Fig. 3. Identification accuracy of the 28 parameters using the four pose error models. 36-parameter error model is the non-redundant rigid-flexible coupling error model, which takes into account not only the geometric errors, but also the compliance errors; 28-parameter error model is the non-redundant kinematic error model, 33-parameter error model is the 36-parameter one without the 3 link self-gravity coefficients, and 31-parameter error model is the 36parameter one without the 5 stiffness coefficients; based on the least square method, disregard the impact of the measurement noise, the 28 kinematic parameters are identified using the four pose error models respectively. 49

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that identification results using settled orientation are substantially volatile. Therefore, the pose error model is selected for use in our work.

disregarded, the 33-parameter error model incorrectly identifies several kinematic parameters. Therefore, the effects of the compliance errors caused by the link self-gravity on robot pose errors cannot be converted into the effects of the external payload. Therefore, the independent parameters of the link self-gravity must be considered for the compliance errors. As seen in Fig. 3, the effects of the compliance errors are associated with the impact of geometric errors on robot accuracy. They affect each other, so the coupling of the two kinds of errors is a major development of the robot calibration model. Consequently, the establishment of the rigid–flexible coupling error model is useful for robot calibration because the robot has non-negligible compliance errors.

3.2. Full pose measurement optimisation method The partial pose measurement method is usually adopted for robot calibration in industrial applications. This method only requires obtaining the end-effector position information without orientation information, and does not allow the user to identify certain robot parameters that must be estimated via end-effector orientation information. Meanwhile, the full pose measurement method requires full pose information, including end-effector position and orientation information. Conventional full pose measurement technique always uses a six-dimensional measurement system [5]. In this study, a three-dimensional measurement system is applied to achieve full pose measurement. An ‘X’-type measurement tool is set as the main device in this work (Fig. 5a) to obtain full pose information of the end-effector. The device can be mounted on four special magnetic nests which suit the 1.5-inch SMR of the Faro laser tracker (Fig. 5b). The main difficulty of this full pose measurement optimisation method is that the orientation information cannot be measured directly. To acquire this information, we can take advantage of the unique performance of this tool and propose an intermediate method where the orientation data are computed in the equations via a new Cartesian coordinate generated by several reference points. The ‘X’-type measurement tool has a mounting point set P′ (A′, B′, C′, ′ D ), and the four points of the point set are coplanar; the vector B′D′ perpendicularly intersects vector A′C′ at OT′ (Fig. 5a). Theoretically, the reference point set Pt (A, B, C, D) of the measurement tool attached to the robot mounting flange (Fig. 5b) shares the characteristics of the point set P′ (A′, B′, C′, D′). The four special magnetic nests have installation errors. Therefore, we use another mathematical method to establish the robot tool frame (Ftool). For example, a point set PB (A, B, C) is selected, and the point OT is obtained via the vertical vector BOT to the vector AC, which is the origin of the robot tool frame (Ftool). The directions X, Y, Z of the robot tool frame (Ftool) should be close to the robot flange frame (F6). Fig. 5b shows how the robot tool frame (Ftool) is established; the vector BOT is the direction X, and the vector CA is the direction Y. Finally, we can obtain the end-effector full pose information (i.e. OT in this study) via the new Cartesian coordinate (i.e. frame [F6]) generated by several reference points (i.e. point set PB (A, B, C) in this example). The measuring equipment, tools and methods in this measurement optimisation method, which are used to obtain the full pose information of measurement poses with high measurement precision, are thus confirmed. However, aside from measurement precision, characteristic information of measurement poses is a key factor affecting the performance of experimental data and the efficiency of robot calibration. Thus, the measurement optimisation method mainly aims to intelligently select the optimal measurement poses for robot calibration. The first step to achieving this goal is considering the external constraints to which intelligent selection is subjected. In this work, a laser tracker and an ‘X’-type measurement tool are used for robot calibration. One external constraint comes from the light emitted from the laser source, which should be directed to SMR without obstacles (Fig. 6a). This external constraint comprises two parts: (1) interference between robot and light and (2) angular limitation between the XY plane of the tool frame (Ftool) and the light emitted from the laser source. The other external constraint comes from the interference between the robot and the end-effector (i.e. ‘X’-type measurement tool with SMRs). These problems can be classified into two categories: structural interference and angular limitation. Firstly, we need to formulate a simplified physical model of the robot for interference detection. On the basis of the hierarchical bounding volumes method, a simplified physical model of the Staubli TX60L robot is formulated for the measurement optimisation method

3. Enhanced optimisation approach for robot calibration Although a robot is a highly nonlinear, high-coupling MIMO system, the success of parameter identification is influenced by measurement noise. In this section, a novel full pose measurement optimisation method is proposed to achieve the desired identification accuracy by obtaining the minimum number of optimal measurement poses. 3.1. Error model selection for parameter identification The position error model is a conventional calibration model established with position information, and the pose error model is established with full pose information. The rigid–flexible coupling pose error model is modelled as Eq. (23). Hence, the rigid–flexible coupling position error model can be updated as follows:

ΔP = [ E3 O3 ]*H *η

(25)

The QR decomposition method is used to estimate the redundancy parameters of this error model, and the non-redundant rigid–flexible coupling position error model (or position error model) is subsequently obtained. This error model considers 35 error parameters (i.e. 27 geometric error parameters, 5 stiffness coefficients and 3 link self-gravity coefficients). To compare this error model with the position model, the 36-parameter error model discussed in Section 2 is called the pose error model in this section. A simulation is performed using the same object (i.e. measurement poses) to compare the identification accuracy and breadth (i.e. number of identifiable parameters) of the pose error model and the position error model to select an improved error model for robot non-kinematic calibration. Two sets with 14 measurement poses are selected in the robot workspace randomly for comparison purposes; one set has varied orientations, and the other has the same orientation. Then, the 36 actual error parameter values (i.e. the 36 error parameters of the pose error model) are computed by the actual robot model with the 39 assumed error parameters (Table 1). On the basis of the least squares method, the parameters of the two error models are identified with the two sets of measurement poses. Meanwhile, the assumed measurement process is corrupted by measurement noise, which is simulated with the application of a normal distribution N1 (0, σ) (σ = 0.035[mm] and σ = 0.007[°]) and the resulting generation of normal random values with a standard deviation of position and a standard deviation of orientation. Results of the simulation experiment with measurement noise N1 indicate that 50 is the minimum number required to obtain reliable experimental results. When the number of repetitions exceeds 50, identification accuracy variations between simulation experiments are within 1%. Finally, each group is repeated 500 times by the normal random measurement noise to obtain more reliable statistics. Corresponding identification results are compared in Fig. 4. The average identification accuracy rate can reach 82.18%, and the identification breadth of the pose error model with varying orientations is 36. The pose error model has higher accuracy and wider breadth than does the position error model. However, neither model can provide an acceptable and correct identification result under the settled orientation. Furthermore, their standard identification accuracy rates indicate 50

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Fig. 4. Identification results using the position error model and the pose error model. The pose error model is the 36-parameter error model, the position error model is the 35-parameter error model without the orientation errors; based on the least square algorithm, the parameters of the two error models are identified with the two sets of measurement poses respectively; to obtained reliable statistics, each group is repeated 500 times by the normal random measurement noise, the two solid lines describe the identification accuracy of 36 parameters is obtained using the pose error model, the two bar charts describe the identification accuracy of 35 parameters is obtained using the position error model. Fig. 5. The experimental measurement tool: (a) “X”-type measurement tool; (b) the measurement tool with the SMRs. The“X”-type measurement tool can be mounted on four special magnetic nests which suit to 1.5-inch SMR of the Faro laser tracker, the measurement tool with the SMRs is used to obtain the end-effector full pose information included position and orientation information.

(Fig. 6b). This modelling method is also applicable to other industrial robots. The line set LR (Of1O1 = 375 mm, Of2Of3 = 400 mm, O3O4 = 450 mm) is represented for the main mechanical structure of the robot. Three sets of points are represented such that the distances between the points and these three lines, respectively, are less than the maximum radii r (r1 = 115 mm, r2 = 88 mm, r3 = 95 mm) of the joints. Values of radiuses are calculated by the robot structural parameters. Each set of points forms a cylinder with one straight line as the axis and two hemispheres with the two endpoints of the straight line as the centres (Fig. 6b). This geometry can describe the complex mechanical structure of the robot. Accordingly, when the distances between the closest point of the end-effector or the light and the line set LR are longer than the corresponding radius, the measurement system and the robot will not collide. The minimum distance algorithm is utilised to detect interference and can be presented as follows:

1, d > ri + s × rs f (d ) = ⎧ ⎨ ⎩ 0, d < =ri + s × rs

(26)

d = min (Pxi − Q yj )2 + (Pyi − Q yj )2 + (Pzi − Qzj )2

(27)

where P is the reference point set Pt (A, B, C, D); the points of the light are the closest points to the corresponding line set LR (Of1O1, Of2Of3, O3O4); Q is the point set of the line set LR, which is the closest point to the corresponding point of the point set Pt; rs is the radius of SMR, rs = 19.05 mm; s is the safety factor, we set s = 3 in this study. Then, we need to study the angular limitation between the XY plane of the tool frame (Ftool) and light emitted from the laser source. After a theoretical analysis and an experimental test, we obtain a useful con→ clusion. When the angle between the normal vector (e.g. NA ) of the XY → plane of the tool frame (Ftool) and incidence vector (Ns ) of the light emitted from the laser source is greater than 90°, the reflected light can be received by the Faro laser tracker, as shown in Fig. 6a. The angle 51

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Fig. 6. The foundation of measurement optimization method. (a) The Faro laser tracker and the measurement tool; (b) the simplified physical model for the Staubli → TX60L robot. (a) The laser tracker and “X”-type measurement tool are used for robot calibration, the angle between the normal vector (NA , for instance) of the XY → plane of the tool frame (Ftool) and incidence vector (NS ) of the light emitted from the laser source is one external constraint for measurement optimization; (b) the simplified physical model for the Staubli TX60L robot is used to solve the structural interference problem which is the other external constraint for measurement optimization.

which can balance the search capability between exploration and exploitation well. The SAPSO algorithm is applied to intelligent selection algorithm in this study because of these advantages. In this work, the optimisation problem for measurement pose intelligent selection should consider external constraints. These constraints are solved by interference detection and angle detection (i.e. the particles should satisfy two requirements: f(d) = 1 and f(θ) = 1). The robot joint limits must be considered also (Table 2). Therefore, the limitation of the space range for particles of the SAPSO algorithm can be determined by the aforementioned constraints. On the basis of this limitation, the Monte Carlo method is applied to generate the initial data (i.e., the number of particles is 500, and each particle contains a set of full pose information of the 14 measurement poses in this study) in the robot measurement space. The observability index O1 is utilised as an evaluation target for the SAPSO algorithm to reduce the effect of measurement noise. This index is used to measure the goodness of a set of measurement poses. The

detection algorithm can be expressed as follows:

1, π ≥ θ ≥ π /2 f (θ) = ⎧ ⎨ ⎩ 0, π /2 > θ ≥ 0

(28)

where

→ → (N *N ) θ = acr cos →A →S NA * NS

(29)

The next step in measurement pose intelligent selection is determining the intelligent selection algorithm. The number of measurement poses is less important than the optimised selection of robot configurations. According to previous work, 14 is the optimum and minimum number for robot calibration. The intelligent selection algorithm aims to choose the 14 optimal measurement poses in the robot measurement space intelligently, thereby potentially reducing the effect of measurement noise and improving identification performance. The main problem of this algorithm is measurement pose optimisation in the robot measurement space, and the dimension of the problem is 14. Particle swarm optimisation (PSO) is an evolutionary computation method based on swarm intelligence and is similar to genetic algorithms. The PSO algorithm has the advantages of fast searching speed, high dimension adaptability, high memory performance, simple structure and easy engineering realisation. The self-adaptive PSO (SAPSO) algorithm is the PSO algorithm with a self-adaptive inertia weight

Table 2 The joint limits of the Staubli TX60L robot.

qmin/° qmax/°

52

q1

q2

q3

q4

q5

q6

−180 180

−127.5 127.5

−142.5 142.5

−270 270

−122.5 133.5

−270 270

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Fig. 7. Flowchart of the proposed optimization approach for robot calibration. The proposed optimization approach of the full pose measurement and parameter identification for robot calibration: (1) the SAPSO algorithm with interference detection and angle detection is used to obtain optimal measurement poses intelligently; (2) the modified Levenberg-Marquardt is applied to estimate comprehensive error parameters (i.e. the geometric error parameters and compliance error parameters).

Omin = γ1*[Σ(Δθi − Δθi, real )2 + Σ(Δαi − Δαi, real )2 + Σ(Δβi − Δβi, real )2]

effects of the error parameters increase with the observability index. Through singular value decomposition, Eq. (23) is updated as follows:

ΔX = U ΣV ′η + ξ

+ γ2*[Σ(Δdi − Δdi, real )2 + Σ(Δai − Δai, real )2] + γ3*[Σ(Δλj − Δλj, real )2] + γ4 *[Σ(Δkm − Δkm, real )2]

(30)

(34)

⎡ σ1 ⎢⋮ ⎢0 Σ=⎢ ⎢0 ⎢⋮ ⎣0

0 ⋮ 0 0 ⋮ 0

⋯ ⋱ ⋯ ⋯ ⋱ ⋯

0⎤ ⋮⎥ σm ⎥ 0⎥ ⎥ ⋮⎥ 0⎦

where Omin is the identification accuracy index. The error parameters of Omin are calculated using Eq. (23). Parameters γ1, γ2, γ3 and γ4 are weight coefficients. The inertia weight w determines the capability of particles to inherit the current speed. A large value of w is beneficial to halting the exploitation and enhancing exploration capability. A small value is useful in revealing the exploitation capability and accelerating the convergence of the algorithm. Moreover, the inertia weight should be reduced to improve exploration capability and avoid premature convergence during exploitation for complex problems with highdimensional search spaces. Thus, the self-adaptive inertia weight w can adjust the inertia weight value by itself and make the SAPSO algorithm control the search capability between exploration and exploitation well. In this work, the self-adaptive inertia weight is given as follows:

(31)

The observability index O1 is the product of singular values and given as follows:

f (σ ) = 01 =

(σ1 σ2⋯σm)1/ m m

(32)

The identification accuracy index Omin is also used as an evaluation target for the SAPSO algorithm to improve identification performance. Therefore, the fitness function is given by the following:

min f = Omin *(f (σ ))−1

(33)

w=

53

(wmax − wmin )*(f − fmin ) ⎧w , f ≤ favg ⎪ min − (favg − fmin ) ⎨ , f > favg wmax ⎪ ⎩

(35)

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Fig. 8. Identification accuracy comparison of the proposed and conventional approach. The proposed approach employs the SAPSO algorithm with interference detection and angle detection to obtain the 14 optimal measurement poses, then 36 parameters are identified using the modified Levenberg–Marquardt algorithm; the conventional approach, which adopts the same error model, however, the 14 measurement poses are obtained by the DETMAX algorithm, and then the 36 parameters are identified using the least square algorithm; to obtained reliable statistics, each group is repeated 500 times by the normal random measurement noise with the normal distribution N1 (0, σ) (σ = 0.035[mm] and σ = 0.007[°]) and the normal distribution N2 (0, σ) (σ = 0.07[mm] and σ = 0.014[°]) respectively.

Fig. 9. Experimental platform with a Staubli TX60L robot for robot calibration. The experimental platform mainly includes a Staubli TX60L robot and a Faro laser tracker with the “X”-type measurement tool.

intelligently. In the proposed measurement optimisation strategy, the SAPSO algorithm considers not only the importance of the observability index O1 and the identification accuracy index Omin in the fitness function but also the information of external constraints (i.e. structural interference and angular limitation). Hence, the proposed measurement optimisation method has the advantages of reducing the effect of measurement noise and improving identification performance. These advantages are verified in the following section.

where wmax and wmin are the desirable maximum and minimum limits of w. On the basis of the range value setting methods introduced in [34, 35], we apply asimulation test with the research object and then adjust the two parameters slightly (wmax = 0.8, wmin = 0.5 in this study). f is the objective function value of the current particle; favg and fmin are the average and minimum objective function values, respectively, of all particles. The proposed full pose measurement optimisation method is based on the SAPSO algorithm, whose final purpose is to obtain a set of optimal measurement poses with the optimum minimum number 54

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sensitive to parameter ʋ (When the step length coefficient is satisfied 4 ≥ ʋ ≤ 100, results are acceptable.). Thus, we set ʋ = 10 in this study. A flowchart of the proposed optimisation approach for robot calibration is shown in Fig. 7. To confirm the advantages of the proposed approach, a simulation is designed with a contradistinctive calibration using the conventional approach, which adopts the same error model (i.e. the 36-parameter error model presented in Section 2). The 14 measurement poses are obtained by the DETMAX algorithm, which is a D-optimality method. This algorithm maximises the determinant of the identification matrix by adding or deleting measurement poses. The 36 parameters are then identified using the least squares algorithm. For our proposed approach, we use the SAPSO algorithm with interference and angle detections to obtain the 14 optimal measurement poses. Then, the 36 parameters are identified using the M–LM algorithm. The actual pose values are calculated by the actual robot model with the 39 assumed error parameters (Table 1). The two sets of measurement pose data are utilised to identify the parameters of the 36-parameter error model. The assumed measurement process is corrupted by measurement noise. Normal distribution N1 (0, σ) (σ = 0.035[mm] and σ = 0.007[°]) and normal distribution N2 (0, σ) (σ = 0.07[mm] and σ = 0.014[°]) are applied to generate normal random values with standard deviation to simulate measurement noise. According to the results of the simulation experiment with measurement noise N2, we can obtain 100 as the minimum number to obtain reliable experimental results. When the number of repetitions exceeds 100, the identification accuracy variations between simulation results are within 1%. Each group is repeated 500 times by the normal random measurement noise to obtain more reliable statistics. Results of identification accuracy using the two optimisation approaches are obtained. The two solid lines in Fig. 8 show the identification results of the 36 parameters using the proposed approach with two different parameters of measurement noise. The identification accuracy is high and stable, although the average identification accuracy rate is reduced from 95.8% to 92.7% when measurement noise is magnified by two times. The two dotted lines in Fig. 8 illustrate the identification results obtained by the conventional approach with the same two measurement noise parameters. The results show relatively low identification accuracy and poor stability with the larger measurements noise compared with the former. In summary, the proposed optimisation approach to full pose measurement and parameter identification for robot calibration has two primary advantages: (1) optimal measurement pose intelligent selection with external constraints and (2) high identification accuracy with high efficiency, good stability and comprehensive error parameters (i.e. geometric and compliance error parameters).

Table 3 Parameters identified of different calibration models (“−”does not exist, “!” cannot be identified). Parameters

Nominal values

Identified values (proposed error model)

Identified values (conventional kinematic error model)

Identified values (optimized position error model)

θ1[°] d1[mm] a1[mm] α1[°] θ2[°] d2[mm] a2[mm] α2[°] θ3[°] d3[mm] a3[mm] α3[°] θ4[°] d4[mm] a4[mm] α4[°] θ5[°] d5[mm] a5[mm] α5[°] θ6[°] d6[mm] a6[mm] α6[°] β2[°] Xb[mm] Yb[mm] Zb[mm] αb[°] βb[°] θb[°] λ2 [Nm/rad] λ3[Nm/rad] λ4 [Nm/rad] λ5 [Nm/rad] λ6 [Nm/rad] k1 k2 k3

0 0 0 −90 −90 0 400 0 90 20 0 90 0 450 0 −90 0 0 0 90 0 70 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

0.005 −0.132 0.139 −89.985 −90.007 – 400.346 0.073 90.107 19.867 −0.690 90.024 −0.020 449.672 −0.016 −89.989 −0.418 −0.080 1.195 89.971 2.139 69.155 −0.655 0.668 −0.014 0.698 0.185 ! −0.007 0.011 ! 2.43E + 06 −2.73E + 06 3.66E + 06 −1.38E + 06 −2.81E + 04 −4.46E-04 6.44E-04 −1.33E-03

−0.133 0.136 −0.752 −89.936 −89.865 – 400.158 0.166 89.907 20.468 −0.037 90.010 0.000 449.993 −0.249 −89.838 0.260 −0.203 ! ! ! 69.243 −0.663 ! −0.071 0.977 0.205 ! −0.011 −0.003 ! – – – – – – – –

−0.103 −0.978 −0.244 −89.946 −89.826 – 400.195 0.179 90.051 20.834 −2.222 90.103 0.048 449.232 −0.224 −89.975 0.286 −0.500 ! ! ! 69.593 −0.647 ! −0.042 0.412 0.187 ! −0.051 0.026 ! −9.27E + 07 −5.53E + 05 1.31E + 05 1.61E + 05 5.17 1.15E-04 4.35E-03 −5.82E-03

4. Comparison analysis of proposed and conventional approaches The rigid–flexible coupling error model is an approximate linear model in this study. Nevertheless, the real error model for the robot is nonlinear inevitably. The parameter identification process of the robot is a procedure of parameter estimation and optimisation. The Gauss–Newton method has fast convergence speed but insufficient iteration stability. By contrast, the steepest descent method has good stability but slow convergence. Thus, a modified Levenberg–Marquardt (M–LM) algorithm is adopted to estimate the error parameters. This modified algorithm can have both advantages. The M–LM algorithm is presented in the following equations:

η = (H T H + f (ΔX )*I )−1*H T *ΔX

5. Experimental results: calibration of Staubli TX60L robot To demonstrate the practical effectiveness and performance of the proposed calibration and compensation techniques and confirm the benefits of these methods from the perspective of industrial applications, this section addresses the geometric and compliance errors of a Staubli TX60L robot (Fig. 9), a representative industrial robot with nonnegligible flexibility. The advantages of the proposed error model (i.e. the 36-parameter error model presented in Section 2) are shown. The calibration and validation results of the proposed model are compared with those obtained by the conventional kinematic error model (i.e. the 24-parameter position error model with only non-redundant geometric error parameters) and the optimised position error model (Table 3, the 32-parameter position error model is optimised by the position error model presented in Section 3). The 14 measurement poses (Section 3.2) obtained are used for this experiment. The Faro laser tracker is utilised to measure the reference points of these measurement poses with the measurement tool (Fig. 5b). The actual pose values are calculated by the proposed method (Section 3.2). With the obtained experimental data, the parameter

(36)

i+1

⎧ f i (ΔX )*υ* ΔX , ΔX i + 1 < ΔX i ⎪ ΔX i i + 1 f (ΔX ) = ⎨ i 1 ΔX i + 1 , ΔX i + 1 ≥ ΔX i ⎪ f (ΔX )* * υ ΔX i ⎩

(37)

where ʋ is the step length coefficient. As f(ΔX) becomes increasingly small (f(ΔX)→0), the M–LM algorithm begins to resemble the Gauss–Newton method. When f(ΔX) is remarkably large, the M–LM algorithm becomes the steepest descent method [36]. According to the methods introduced in [37], this M–LM algorithm is not highly 55

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Fig. 10. Position errors before and after calibration using the three error models. The actual pose values of the 30 measurement poses of the Staubli TX60L robot are obtained by the Faro laser tracker, which are randomly selected in the entire robot workspace for validation, the differences between the three sets of calibration experiments is just the error models; the identification results using the three error models are used for robot error compensation respectively, and then the robot accuracy after calibration is obtained in this work.

calibration using the proposed calibration model are smaller than those after calibration using the kinematic error model or position error model in the same measurement environment. Average residual position error after calibration using the proposed error model is reduced significantly from 1.41 mm (before calibration) to 0.15 mm. To demonstrate the calibration methodology can be extended to other serial robots, a Staubli RX160L robot (Fig. 11) is used for this calibration experiment. Corresponding identification results are compared in Fig. 12. We can see that the advantages of the proposed calibration method are quite obvious. Average residual position error is reduced significantly from 3.72 mm (before calibration) to 0.23 mm using the proposed error model. Experimental procedures for robot calibration with the proposed error model should be called non-kinematic calibration, where compliance error parameters are also identified as geometric error parameters. Furthermore, non-kinematic calibration ensures an essential improvement of the robot position accuracy in the loaded mode. In this experiment, we regard the measurement tool (i.e., the measurement tool of the Staubli TX60L robot is 1.108 kg, and the measurement tool of the Staubli RX160L robot is 6.23 kg) as the external payload and consider the load factors in the process of robot calibration. As the mass of the external payload increases, the advantages of our non-kinematic calibration method become increasingly prominent. 6. Conclusions This paper presents a complete non-kinematic calibration framework with rigid–flexible coupling error model establishment, full pose measurement optimisation, full pose information acquisition and comprehensive parameter identification for serial industrial robots. In this work, we develop the rigid–flexible coupling error model, which considers not only kinematic errors but also non-negligible compliance errors (effects of external payload and link self-gravity) and the small transformation between the constructed robot base frame (Fb) and the actual robot base frame (F0). Contrary to the techniques in previous works, the proposed full pose measurement optimisation method adopts the SAPSO algorithm and considers external constraints (i.e. structural interference and angular limitation). This method evaluates the quality of measurement poses via the identification accuracy index Omin and the observability index O1 and considers the number of

Fig. 11. Experimental platform with a Staubli RX160L robot for robot calibration. The experimental platform mainly includes a Staubli RX160L robot and a Faro laser tracker with the “X”-type measurement tool.

identification results of the three error models are presented in Table 3. Finally, 30 measurement poses are randomly selected from the entire robot workspace for validation, and the enhancement results of absolute positional accuracy are shown in Fig. 10. Robot accuracy improvement due to calibration has been studied through residual error analysis before and after calibration. Comparison results are shown in Fig. 10. The residual position errors after 56

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Fig. 12. Position errors before and after calibration using the three error models. The actual pose values of the 30 measurement poses of the Staubli RX160L robot are obtained by the Faro laser tracker, which are randomly selected in the entire robot workspace for validation, the differences between the three sets of calibration experiments is just the error models; the identification results using the three error models are used for robot error compensation respectively, and then the robot accuracy after calibration is obtained in this work.

References

dimensions to be associated with the optimum minimum number from our previous work. Considering the difficulty in acquiring orientation information of the robot end-effector, a new Cartesian coordinate generated by position information of several reference points is presented to provide a good solution. The M–LM algorithm is applied to essentially increase the identification accuracy using the optimal full pose measurement method. The practical effectiveness and performance of the proposed approach are validated via an experimental study that covers the nonkinematic calibration of a Staubli TX60L robot with a FARO laser tracker. The geometric and compliance error parameters of the 36parameter error model are identified after robot calibration. The validation results demonstrate a significant enhancement in robot accuracy after the implementation of our approach. This finding allows us to achieve a robot absolute positional accuracy that equals 0.15 mm and meet most mission requirements. Furthermore, a developed experimental study via a Staubli RX160L robot is introduced to demonstrate the effectiveness of the proposed calibration methodology in robot accuracy improvement for other serial robots. This study mainly focuses on the non-kinematic calibration for the absolute positional accuracy of robots. This technique is based on an approximate linear error model. Satisfying the demand for the desired trajectory accuracy of industrial robots is difficult because of the increasing applications of such robots. Therefore, we will further explore this increase in applications in future studies. We need to focus on developing an error compensation technique that considers trajectory correction and smoothing methods and can be implemented off-line.

[1] J. Zhou, H.J. Kang, A hybrid least-squares genetic algorithm–based algorithm for simultaneous identification of geometric and compliance errors in industrial robots, Adv. Mech. Eng 7 (6) (2015) 1–12. [2] X. Zhang, Y. Song, et al., Stereo vision based autonomous robot calibration, Robot. Auton. Syst. 93 (2017) 43–51. [3] W. Wang, F. Liu, C. Yun, Calibration method of robot base frame using unit quaternion form, Precis. Eng 41 (2015) 47–54. [4] W. Zhu, B. Mei, Y. Ke, Kinematic modeling and parameter identification of a new circumferential drilling machine for aircraft assembly, Int. J. Adv. Manuf. Tech. 72 (5–8) (2014) 1143–1158. [5] A. Nubiola, I.A. Bonev, Absolute robot calibration with a single telescoping ballbar, Precis. Eng. 38 (3) (2014) 472–480. [6] W. Wang, G. Wang, C. Yun, A calibration method of kinematic parameters for serial industrial robots, Ind. Robot 41 (2) (2014). [7] H.N. Nguyen, J. Zhou, H.J. Kang, A calibration method for enhancing robot accuracy through integration of an extended Kalman filter algorithm and an artificial neural network, Neurocomputing 151 (Part 3) (2015) 996–1005. [8] E. Abele, M. Weigold, S. Rothenbucher, Modeling and identification of an industrial robot for machining applications, CIRP Annals 56 (1) (2007) 387–390. [9] Z.S. Roth, H. Zhuang, Y. Bai, Experiment study of PUMA robot calibration using a laser tracking system, Proceedings of the 2003 IEEE International Workshop on Soft Computing in Industrial Applications, NY, USA, Binghmton University, 2003, pp. 139–144. [10] Y. Guo, S. Yin, Y. Ren, et al., A multilevel calibration technique for an industrial robot with parallelogram mechanism, Precis. Eng 40 (2015) 261–272. [11] X. Xu, D. Zhu, et al., TCP-based calibration in robot-assisted belt grinding of aeroengine blades using scanner measurements, Int, J. Adv. Manuf. Tech. 90 (1–4) (2017) 635–647. [12] V.N. Borikov, O.V. Galtseva, et al., Method of noncontact calibration of the robotic ultrasonic tomograph, J. Phys. Conf. Ser. 671 (2016) 012014-1-012014-6. [13] M. Ulrich, C. Steger, Hand-eye calibration of SCARA robots using dual quaternions, Pattern Recogn. Image Anal. 26 (1) (2016) 231–239. [14] L. Wen, X. He, et al., Hand–eye calibration in visually-guided robot grinding, IEEE Trans. Cybern. 46 (11) (2016) 2634–2642. [15] M. Temesguen, O. Raúl, et al., Computationally efficient and robust kinematic calibration methodologies and their application to industrial robots, Robot. Comput. Integr. Manuf. 37 (2016) 33–48. [16] A. Filion, A. Joubair, et al., Robot calibration using a portable photogrammetry system, Robot. Comput. Integr. Manuf. 49 (2018) 77–87. [17] J.M. Ahola, T. Seppälä, et al., Calibration of the pose parameters between coupled 6-axis F/T sensors in robotics applications, Robot. Auton. Syst. 89 (2017) 1–8. [18] N. Mu, K. Wang, et al., Calibration of a flexible measurement system based on industrial articulated robot and structured light sensor, Opt. Eng. 56 (5) (2017) 054103. [19] A. Joubair, L. Zhao, et al., Use of a force-torque sensor for self-calibration of a 6DOF medical robot, Sens. Basel 16 (6) (2016) 798-1-798-19. [20] N. Erick, X. Ning, et al., Laser beam multi-position alignment approach for an

Acknowledgements This research is supported by National Natural Science Foundation of China (No. 51575236), and Postgraduate Research & Practice Innovation Program of Jiangsu Province (No. KYCX17_1462). Supplementary materials Supplementary material associated with this article can be found, in the online version, at doi:10.1016/j.rcim.2018.07.002. 57

Robotics and Computer Integrated Manufacturing 57 (2019) 46–58

X. Chen et al.

[21]

[22]

[23] [24] [25]

[26]

[27] [28]

[29] A. Klimchik, A. Pashkevich, et al., Compliance error compensation technique for parallel robots composed of non-perfect serial chains, Robot. Comput. Integr. Manuf. 29 (2) (2013) 385–393. [30] A. Joubair, I.A. Bonev, Non-kinematic calibration of a six-axis serial robot using planar constraints, Precis. Eng. 40 (2015) 325–333. [31] A. Klimchik, A. Pashkevich, et al., CAD-based approach for identification of elastostatic parameters of robotic manipulators, Finite Elem. Anal. Des. 75 (2013) 19–30. [32] X. Chen, Q. Zhang, Y. Sun, Intelligent selection and optimization of measurement poses for a comprehensive error model identification of 6-DOF serial robot, Proceedings of the 2016 23rd International Conference on, Mechatronics and Machine Vision in Practice (M2VIP), Nanjing, China, 2017 http://ieeexplore.ieee. org/document/7827279/. http://ieeexplore.ieee.org/document/7827279/ . [33] Y. Wu, A. Klimchik, et al., Geometric calibration of industrial robots using enhanced partial pose measurements and design of experiments, Robot. Comput. Integr. Manuf. 35 (2015) 151–168. [34] S. Zhai, T. Jiang, A new sense-through-foliage target recognition method based on hybrid differential evolution and self-adaptive particle swarm optimization-based support vector machine, Neurocomputing 149 (Part B) (2015) 573–584. [35] B. Tang, Z. Zhu, et al., A framework for multi-objective optimization based on a new self-adaptive particle swarm optimization algorithm, Inf. Sci. 420 (2017) 364–385. [36] J. Dong, K. Lu, et al., Accelerated nonrigid image registration using improved Levenberg-Marquardt method, Inf. Sci. 423 (2018) 66–79. [37] K. Amini, F. Rostami, A modified two steps Levenberg-Marquardt method for nonlinear equations, J. Comput. Appl. Math. 288 (2015) 341–350.

automated industrial robot calibration, Proceedings of the IEEE Annual International Conference on, Cyber Technology in Automation, Control, and Intelligent Systems (CYBER), Hong Kong, 2014, pp. 359–364. Yu Sun, J.M. Hollerbach, Observability index selection for robot calibration, Proceedings of the IEEE International Conference on, Pasadena, CA, USA, Robotics and Automation, 2008, pp. 831–836. A. Klimchik, Y. Wu, S. Caro, et al., Design of experiments for calibration of planar anthropomorphic manipulators, Proceedings of the IEEE/ASME International Conference on, Advanced Intelligent Mechatronics (AIM), 2011, pp. 576–581. H. Wang, T. Gao, et al., Finding measurement configurations for accurate robot calibration validation with a cable-driven robot, IEEE Trans. Robot. (2017) 1–14. A. Klimchik, S. Caro, et al., Optimal pose selection for calibration of planar anthropomorphic manipulators, Precis. Eng 40 (2015) 214–229. A. Klimchik, A. Pashkevich, et al., Optimization of measurement configurations for geometrical calibration of industrial robot, Int. J. Intell. Robot. Appl. (2012) 132–143. A. Klimchik, Y. Wu, et al., Optimal selection of measurement configurations for stiffness model calibration of anthropomorphic manipulators, Appl. Mech. Mater. 162 (2012) 161–170. G. Chen, T. Li, et al., Review on kinematics calibration technology of serial robots, Int. J. Precis. Eng. Manuf. 15 (8) (2014) 1759–1774. A. Klimchik, E. Magid, et al., Design of experiments for elastostatic calibration of heavy industrial robots with kinematic parallelogram and gravity compensator, IFAC Pap. Online 49 (12) (2016) 967–972.

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