© H. Zhang et al., Published by EDP Sciences, 2025

1 Introduction

Industrial manipulator has been widely used in industry sectors like machining, welding, assembling, carrying, etc. Ultrasonic testing is an important method in non-destructive testing (NDT) to detect performance, quality and flaw for parts without destruct [1]. It is promising to introduce manipulator into the NDT process, which will take full advantages of high precision and efficiency of the manipulator. At the same time, missing and failure detection induced by man-made factors can be relieved obviously. Mineo et al. [2] collaborating with TWI Technology Center designed a prototype robotic ultrasonic testing system which can dramatically increase work efficiency when testing complex geometry components. It has been proved that propagation law of ultrasonic wave is similar to ray, they both follow Snell's Law [3]. Echo has highest intensity when sound axis is perpendicular to reflective surface. In addition, propagation distance can be calculated accurately according to travel time. Based on the above principle, we proposed a kind of calibration method which is more convenient for manipulator in ultrasonic testing system to calibrate the initial position of the manipulator.

According to robotic theory, manipulator's absolute position and posture is determined by initial position and structure parameters. They are not absolutely accurate in practice. Errors introduced by manufacturing tolerance, set-up error, wear and tear generally exist in manipulators. It is possible to lose initial position after repairing or transporting. Positional accuracy will also be influenced after collision. It is necessary to calibrate initial position at that time [4]. In literature [5], calibration is divided into four processes, i.e. modeling, measurement, identification and correction. A laser-tracker is used to calibrate precision of the manipulator. In early stage, physical contact methods are mainly used. For example, Gill [6] used ball-bar to measure accuracy for robot. Swayze [7] used wire potentiometer to measure posture, then structure parameters are calibrated according to such data. To meet automatic demand, non-contact methods are developed. Ha [8] used laser sensor to measure Z-axis position of end-effector and used grid plate to measure XY-axis position of end-effector. In literature [9], Ulrich et al. mounted camera on end-effector to construct a hand-eye serial robot system, which can calibrate robot kinematics, the hand-eye transformations, and, for camera-guided robots, the interior orientation of camera simultaneously. Selami et al. [10] attached a 3-D positioning and posture (3DPP) sensor onto the industrial robot, which can acquire orientation data in real time. Zhuang et al. [11] used theodolite to measure poses of platform and transform the pose to basic coordinate system to determine transformation relationship between the theodolite and the base coordinate system. Wang et al. [12] used laser interferometry based tracker to verify efficiency of his new approach on improving accuracy.

On the other hand, some mathematical methods for solving large scale non-linear problems have been widely used in robot calibration. For example, Won et al. [13] used the least squares method to estimate the pose transformation relationship between the robot's base coordinate system and the laser tracker measurement coordinate system. An improved robot observability metric combined with the Binary Simulated Annealing Algorithm (BSAA) is introduced to optimize the selection of calibration sampling data. Bastl et al. [14] introduced a newly developed method based on multi-objective deep learning evolutionary algorithm to find optimal estimates of the robot parameters. Zhang et al. [15] proposed a method to quantify uncertainty in robot pose errors. In the proposed method, a distribution-free joint prediction model is designed to realize the simultaneous prediction of points and uncertainty intervals. Ma et al. [16] provided a method that utilizes the Levy flight and Sparrow Search Algorithm (LSSA) to optimize the measurement pose of the robot, reducing the number of measurement poses and improving calibration accuracy. In the aspect of dual-robot operation for ultrasonic testing, the calibration process to obtain spatial relationships among different frames is an essential step to obtain accurate testing results. Zheng et al. [17] formulate an AXP = YCQ equation to transform the calibration problem into the equation solving, the Kronecker-product-based closed-form method and the Gauss-Newton iterative method are proposed to calculate the results. Guo et al. [18] proposed a four-posture calibration method to calibrate the transformation relationship of the irregular-shaped tool frame relative to the robot flange frame. Though many effective algorithms have been proposed to improve accuracy, the mathematical modes of these algorithms are complex, and the combination of the mathematical mode with specific calibration method is a complex process.

In this research, we use ultrasonic testing system as measuring device. The initial position is calculated according to deviation relationships between the end-effector and the joints based on D-H convention. Simulation and experiment results demonstrate that the ultrasonic transducer can be used to calibrate the initial position of the manipulator without increasing other devices. This method makes manipulator calibration in the ultrasonic testing system easy and low cost, especially convenient to be used in factory.

2 Configuration of ultrasonic testing system

In literature [19], Ma et al. designed an ultrasonic test system for complex surface. Scanning machine of the system adopts Cartesian form. Though it can test complex surface, spatial complex structure can not be scanned by this system. On account that Cartesian scanning machine is limited by its own structure, it is difficult for such machine to scan spatial complex structure. A six degree of freedom (6-DOF) articulated manipulator is introduced as scanning machine to fill gaps. Compared with traditional Cartesian scanning machine, the manipulator featured with flexible motion space is more suitable to scan spatial structure. Structure diagram of ultrasonic testing manipulator system is shown in Figure 1. The system includes manipulator motion module, ultrasonic testing module and auxiliary module. The 6-DOF manipulator is a scanning machine carrying ultrasonic transducer to scan parts. Its corresponding servo controller is composed of a motion control part and a servo motor driver. The ultrasonic testing module is made up of a pulse transmitting-receiving device and an ultrasonic transducer. The auxiliary module mainly consists of water pump, tank, jet and some other supports. All the above modules are controlled by an industrial computer to realize coordination controlling, data processing and result analyses.

It is suitable for adopting pulse-echo method to test complex parts like rails and ship plates [20]. Principle of the test method is similar to radar or some other acoustic detection devices which can detect defects by launching ultrasonic pulse and receiving reflected pulse. While the difference is that ultrasonic transducer is based on piezoelectric effect generally existing in crystal [21]. The crystal produce vibration after motivated by periodic electric pulse, then the vibration transforms into periodic elastic waves. The elastic waves propagate forward in acoustic form and enter into part coupling with liquid. Reflected waves back into crystal and transform into tiny electrical signal in polarization direction. Based on the above principle, a pulse transmitting-receiving device is introduced to motivate transducer and receive the electrical signal. In receiving state, the transducer transform reflected elastic waves into the electrical signal. The received signal is processed by amplifier and filter to improve signal quality. Then, the processed signal is collected by A/D part and converted into digital signal for computer. Finally, the received signal is displayed in time-amplitude form, called A-Scan. In addition, it is further processed by program and displayed as B-Scan or C-Scan image. The B-Scan image presents a cutaway view parallel to the sound beam. The C-Scan view presents a cutaway view perpendicular to the sound beam.

Fig. 1 Configuration of ultrasonic testing system with manipulator.

3 Pulse-echo based measuring method

The ultrasonic testing system shown in Figure 1 is used to test complex parts. In addition, it can be used to calibrate initial position of the manipulator without introducing other devices. The presented method is based on pulse-echo principle.

Newton had calculated sound velocity in air according to Boyle's Law in 1687. Laplace amended Newton's computing process later in 1816. He took adiabatic process into consideration, and introduced equation of state, which made theoretical value more accurate [22]. Based on above background, a semi-empirical formula is proposed to calculate sound velocity:

$$v_L=\sqrt{\frac{\gamma {\text{K}}_T}{{\rho }_L}}$$(1)

where γ is a ratio between specific heat at constant pressure and volume, K_T is isothermal bulk modulus, and ρ_L is liquid density.

Transmitted pulse and series of echo pulse in time domain with different propagation distances are shown in Figure 2. According to equation (1), sound velocity keeps consistent when propagating in homogeneous medium with constant temperature and pressure. Distance between transducer and reflective surface can be calculated according to lag between transmitted pulse and surface echo pulse. Thus, relative position between end-effector and reflector can be measured.

On the other hand, relative posture between end-effector and reflector can also be measured based on ultrasonic theory. When ultrasonic wave propagate from liquid into solid, we can acquire according to Newton's laws (F=ma) that:

$${\text{P}}_{\text{L}}-P_{\text{S}}=m\frac{\left(v_L-v_S\right)}{\mathrm{d}t}.$$(2)

Among which, P_L , P_S and v_L , v_S is normal pressure and normal velocity respectively on the boundary. Considering that two mediums are fitted closely, hence two normal velocities are equal (i.e. v_L=v_S). It can be further inferred that P_L=P_S. Two pressures can be decomposed into static pressure and dynamic pressure. The difference between static and dynamic is time-invariant and time-variant:

$$\begin{array}{l}{\text{P}}_{\text{L}}=P_{\text{L}}^{\text{S}}+P_{\text{L}}^{\text{D}}\\ P_{\text{S}}=P_S^{\text{S}}+P_S^{\text{D}}\end{array}.$$(3)

Since static pressures keep consistent on the boundary, i.e. P_L^S= P_S^S, dynamic pressures are also equal, i.e. P_L^D = P_S^D. In this research, dynamic pressures are mainly meant sound pressure, therefore P_L^D and P_S^D can be marked as p_L and p_S. In addition, liquid only can transmit tensile and pressure stress, which is different from solid (solid can transmit shear stress simultaneously). Therefore, shear stress is zero in liquid, i.e. τ_L = 0.

Ultrasonic wave satisfies three boundary conditions based on the above analysis, i.e. v_L=v_S, p_L=p_S and τ_L=0. Corresponding mode conversion will be caused on the boundary due to such three conditions as shown in Figure 3.

A comparison between Figures 2 and 3 shows that pulse amplitude in Figure 3 is lower than that in Figure 2. On account of mode conversion which is caused by oblique incidence, orientation of reflected wave deflects a certain angle. In consequence, receiving energy declines obviously. This phenomenon, which is similar with ray, is called Snell's Law. Such principle can be used to measure whether end-effector is perpendicular with reflector. Posture of manipulator can be determined accordingly.

We can infer from the above analysis that lag between transmitted pulse and echo pulse can be used to calculate relative position, and echo pulse intensity can be used to measure posture. Experiment demonstrates that such principle can be introduced to calibrate the initial position.

Fig. 2 Transmitted pulse and echo pulse in time domain.

Fig. 3 Mode conversion in fluid-solid interface.

4 Calibration model of manipulator

Many mathematical methods such as double quaternions [23], geometrical method [24], moth-flame optimization algorithm [25], neural-network [26], etc. have been proposed to solve kinematic problems. Among these methods, D-H convention is a common model to describe joint relationships in space. In this research, kinematic model is constructed based on D-H convention. We devote to explore deviation relationships between the joints and the end-effector. During process of construct kinematic mode, we find that the manipulator can be broken up into two parts, i.e. “arm” and “wrist” [27]. The initial position of “arm”, which includes first three joints, can be calibrated according to position deviation. The initial position of “wrist”, which includes later three joints, can be calibrated according to posture deviation.

4.1 Kinematic model

Staubli-TX90L is introduced in this research as an example, and its structure diagram is shown in Figure 4. Coordinates are established on each joint based on D-H convention [28] to describe spatial relationship between each joint. Pose in Figure 4 is specified as the initial position.

Relationship between two adjacent coordinates is expressed by rotation-transformation matrix Tⁱ_i-1:

$$\begin{array}{l}{T}_{i-1}^i=Rot\left(Z_{i-1},{\theta }_i\right)Trans\left(\mathrm{0,0},d_i\right)Trans\left(a_i,\mathrm{0,0}\right)Rot\left(X_{i\text{-1}},{\alpha }_i\right)\\ =\left[\begin{array}{cccc}\hfill \cos {\theta }_i\hfill & \hfill -\sin {\theta }_i\cos {\alpha }_i\hfill & \hfill \sin {\theta }_i\sin {\alpha }_i\hfill & \hfill a_i\cos {\theta }_i\hfill \\ \hfill \sin {\theta }_i\hfill & \hfill \cos {\theta }_i\cos {\alpha }_i\hfill & \hfill -\cos {\theta }_i\sin {\alpha }_i\hfill & \hfill a_i\sin {\theta }_i\hfill \\ \hfill 0\hfill & \hfill \sin {\alpha }_i\hfill & \hfill \cos {\alpha }_i\hfill & \hfill d_i\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\end{array}$$(4)

Among which θ_i ,d_i,ɑ_i and α_i is D-H parameter:

• θ_i—joint angle from X_i-1 axis to X_i axis around Z_i-1 axis,

• d_i—joint distance form X_i-1 axis to X_i axis along Z_i-1 axis,

• α_i—offset angle from Z_i-1 axis to Z_i axis around X_i axis,

• ɑ_i—offset distance from Z_i-1 axis to Z_i axis along X_i axis,

All the D-H parameters in Staubli TX90L are listed in Table 1.

Taking the D-H parameters into equation (5), then specific relationship between each joint can be expressed as:

$$\begin{array}{c}{\text{T}}_0^1=\left[\begin{array}{cccc}\hfill c_1\hfill & \hfill 0\hfill & \hfill \text{-}{s}_1\hfill & \hfill a_1{c}_1\hfill \\ \hfill s_1\hfill & \hfill 0\hfill & \hfill c_1\hfill & \hfill a_1{s}_1\hfill \\ \hfill 0\hfill & \hfill \text{-}1\hfill & \hfill 0\hfill & \hfill {\text{d}}_1\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]{\text{T}}_1^2=\left[\begin{array}{cccc}\hfill c_2\hfill & \hfill {\text{-s}}_2\hfill & \hfill 0\hfill & \hfill a_2{c}_2\hfill \\ \hfill s_2\hfill & \hfill {\text{c}}_2\hfill & \hfill 0\hfill & \hfill a_2{s}_2\hfill \\ \hfill 0\hfill & \hfill \text{0}\hfill & \hfill 1\hfill & \hfill {\text{d}}_2\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\\ {\text{T}}_2^3=\left[\begin{array}{cccc}\hfill c_3\hfill & \hfill 0\hfill & \hfill s_3\hfill & \hfill \text{0}\hfill \\ \hfill s_3\hfill & \hfill 0\hfill & \hfill \text{-}{c}_3\hfill & \hfill \text{0}\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill \text{0}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]{\text{T}}_3^4=\left[\begin{array}{cccc}\hfill c_4\hfill & \hfill 0\hfill & \hfill s_4\hfill & \hfill \text{0}\hfill \\ \hfill s_4\hfill & \hfill 0\hfill & \hfill \text{-}{c}_4\hfill & \hfill \text{0}\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill {\text{d}}_4\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\\ {\text{T}}_4^5=\left[\begin{array}{cccc}\hfill c_5\hfill & \hfill 0\hfill & \hfill s_5\hfill & \hfill \text{0}\hfill \\ \hfill s_5\hfill & \hfill 0\hfill & \hfill \text{−}{c}_5\hfill & \hfill \text{0}\hfill \\ \hfill 0\hfill & \hfill 1\hfill & \hfill 0\hfill & \hfill \text{0}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill \text{1}\hfill \end{array}\right]{\text{T}}_5^6=\left[\begin{array}{cccc}\hfill c_6\hfill & \hfill {\text{-s}}_6\hfill & \hfill 0\hfill & \hfill \text{0}\hfill \\ \hfill s_6\hfill & \hfill {\text{c}}_6\hfill & \hfill 0\hfill & \hfill \text{0}\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill & \hfill {\text{d}}_6\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right]\end{array}$$(5)

where c_i and s_i is simplification of cosθ_i and sinθ_i.

The relationship between two joints is described by matrix multiplication. For example, relationship between end-effector (X₆, Y₆, Z₆) and basic coordinates (X₀, Y₀, Z₀) is described as:

$$T_0^6=T_0^1{T}_1^2{T}_2^3{T}_3^4{T}_4^5{T}_5^6=\left[\begin{array}{cccc}\hfill n_x^6\hfill & \hfill o_x^6\hfill & \hfill a_x^6\hfill & \hfill p_x^6\hfill \\ \hfill n_y^6\hfill & \hfill o_y^6\hfill & \hfill a_y^6\hfill & \hfill p_y^6\hfill \\ \hfill n_z^6\hfill & \hfill o_z^6\hfill & \hfill a_z^6\hfill & \hfill p_z^6\hfill \\ \hfill 0\hfill & \hfill 0\hfill & \hfill 0\hfill & \hfill 1\hfill \end{array}\right].$$(6)

The equation (6) can be simplified as:

$$T_0^6=\left[\begin{array}{cc}\hfill R_0^6\hfill & \hfill P_0^6\hfill \\ \hfill 0\hfill & \hfill 1\hfill \end{array}\right],$$(7)

where P⁶₀ represents positional component and R⁶₀ represents directional component. The other coordinates (X_i, Y_i, Z_i) can also be expressed as Pⁱ₀ and Rⁱ₀ relative to basic coordinate system (X₀, Y₀, Z₀).

Fig. 4 Structure Diagram of Staubli-TX90L.

4.2 Deviation model

4.2.1 Deviation model in arm

Though absolute joint angle can not be determined before initial position has been calibrated, relative joint angle can be measured by encoder accurately. When sending rotational command to joint i, corresponding angle change is expressed as δθ_i. The position and posture relationship between joint i relative to joint i−1 is expressed as Tⁱ_i-1, and its corresponding differential transform δTⁱ_i-1 is estimated by Taylor series as:

$$\delta T_{i-1}^i=\frac{\partial T_{i-1}^i}{\partial {\theta }_i}\delta {\theta }_i+\frac{1}{2!}\frac{{\partial }^2{T}_{i-1}^i}{\partial {\theta }_i^2}\delta {\theta }_i^2+\cdots +\frac{1}{n!}\frac{{\partial }^n{T}_{i-1}^i}{\partial {\theta }_i^n}\delta {\theta }_i^n.$$(8)

The differential transform is estimated as linear relationship, then higher order items in equation (8) are neglected. So the equation (8) is simplified as:

$$\delta T_{i-1}^i=\frac{\partial T_{i-1}^i}{\partial {\theta }_i}\delta {\theta }_i.$$(9)

Based on the above estimation, changes of coordinates (X₄,Y₄,Z₄) in point “W” relative to basic coordinates as shown in Figure 4 are estimated as:

$$\delta T_0^4=\frac{\partial T_0^4}{\partial {\theta }_1}\delta {\theta }_1\text{+}\frac{\partial T_0^4}{\partial {\theta }_2}\delta {\theta }_2\text{+}\frac{\partial T_0^4}{\partial {\theta }_3}\delta {\theta }_3\text{+}\frac{\partial T_0^4}{\partial {\theta }_4}\delta {\theta }_4.$$(10)

P⁴₀=[p⁴_x, p⁴_y, p⁴_z]^T is contained within the T⁴₀ expressing position relationship of point “W” relative to basic coordinates, which is similar to the P⁶₀ in equation (7). Therefore position relationship in δT⁴₀ can be extracted as:

$$\left[\begin{array}{c}\hfill \delta p_x^4\hfill \\ \hfill \delta p_y^4\hfill \\ \hfill \delta p_z^4\hfill \end{array}\right]=[\begin{array}{c}\hfill \text{-}{s}_1{s}_{23}{d}_4\text{-}{s}_1{a}_2{c}_2\text{-}{c}_1{d}_2\text{-}{a}_1{s}_1\hfill \\ \hfill c_1{s}_{23}{d}_4\text{+}{c}_1{a}_2{c}_2\text{-}{s}_1{d}_2+a_1{c}_1\hfill \\ \hfill 0\hfill \end{array}\begin{array}{l}\begin{array}{cc}\hfill {\text{−c}}_1{\text{c}}_{23}{\text{d}}_4{\text{-c}}_1{\text{a}}_2{\text{s}}_2\hfill & \hfill {\text{c}}_1{\text{c}}_{23}{\text{d}}_4\hfill \end{array}\\ \begin{array}{cc}\hfill \text{−}{s}_1{c}_{23}{d}_4\text{-}{s}_1{a}_2{s}_2\hfill & \hfill s_1{c}_{23}{d}_4\hfill \end{array}\\ \begin{array}{cc}\hfill \text{−}{s}_{23}{d}_4\text{-}{a}_2{c}_2\hfill & \hfill {\text{−s}}_{23}{d}_4\hfill \end{array}\end{array}]\left[\begin{array}{c}\hfill \delta {\theta }_1\hfill \\ \hfill \delta {\theta }_2\hfill \\ \hfill \delta {\theta }_3\hfill \end{array}\right].$$(11)

Deviation relationship between first three joints and the “W” point is expressed by equation (11), where c₂₃ and s₂₃ is simplification of cos(θ₂+θ₃) and sin(θ₂+θ₃) respectively. Therefore, actual joint angles can be calculated according to joint deviation and position deviation.

Nominal and actual joint angles are expressed as θ_i^N and θ_I^A. The relationship is θ_i^A = θ_i^N + Δθ_i, where Δθ_i is offset exiting in joints. Calibration method of the first three joints is summarized as follows:

(1) Calibrate joint 1

We make joint 3 changing a small angle δθ₃, and keep the other joints not changed, i.e. δθ₁=0, δθ₂=0, δθ₄=0, δθ₅=0, δθ₆=0. Positional deviation of point “W” can be calculated according to equation (11):

$$\{\begin{array}{c}\hfill \delta p_{x1}^4=c_1{c}_{23}{d}_4\delta {\theta }_3\hfill \\ \hfill \delta p_{y1}^4=s_1{c}_{23}{d}_4\delta {\theta }_3\hfill \\ \hfill \delta p_{z1}^4=-s_{23}{d}_4\delta {\theta }_3\hfill \end{array}.$$(12)

Actual angle of joint 1 relative to the initial position is:

$${\theta }_1^{\text{A}}=\text{arctan}\left(\frac{\delta p_{y1}^4}{\delta p_{x1}^4}\right).$$(13)

(2) Calibrate joint 2

Then we make joint 3 turning bake to previous angle and make joint 2 changing a small angle δθ₂. Positional deviation of point “W” can be calculated similarly from equation (11):

$$\delta p_{z2}^4=\left(-s_{23}{d}_4-a_2{c}_2\right)\delta {\theta }_2$$(14)

where −s₂₃d₄ can be calculated from equation (12). Actual angle of joint 2 relative to the initial position is:

$${\theta }_2^{\text{A}}=\text{arccos}\left(\frac{\delta p_{z1}^4}{\delta {\theta }_3{a}_2}\text{-}\frac{\delta p_{z2}^4}{\delta {\theta }_2{a}_2}\right).$$(15)

(3) Calibrate joint 3

Sum angle of joint 2 and joint 3 can be calculated from equation (12):

$${\theta }_2^{\text{A}}\text{+}{\theta }_3^{\text{A}}=\arcsin \left(\text{-}\frac{\delta p_{z1}^4}{d_4\delta {\theta }_3}\right).$$(16)

Actual angle of joint 3 relative to the initial position can be acquired combine with equation (15):

$${\theta }_3^{\text{A}}=\arcsin \left(\text{-}\frac{\delta p_z^4}{d_4\delta {\theta }_3}\right)\text{-arccos}\left(\frac{\delta p_{z1}^4}{\delta {\theta }_3{a}_2}\text{-}\frac{\delta p_{z2}^4}{\delta {\theta }_2{a}_2}\right).$$(17)

The offsets exist in such three joints can be calculated according to actual angles, i.e. Δθ_i=θ^A_i − θ^N_I (i=1,2,3). It is obviously in equations (13), (15), and (17) that actual joint angles can be determined by measuring relative positions δp⁴_x, δp⁴_y and δp⁴ _z.

4.2.2 Deviation model in wrist

The later three joints in wrist are calculated according to posture relationship. The posture relationship between end-effector (X₆, Y₆, Z₆) and coordinates (X₃, Y₃, Z₃) is expressed as:

$$\begin{array}{l}{R}_3^6=R_3^4{R}_4^5{R}_5^6\\ =\left[\begin{array}{ccc}\hfill c_4{c}_5{c}_6+s_4{s}_6\hfill & \hfill -c_4{c}_5{s}_6+s_4{c}_6\hfill & \hfill c_4{s}_5\hfill \\ \hfill s_4{c}_5{c}_6-c_4{s}_6\hfill & \hfill -s_4{c}_5{s}_6-c_4{c}_6\hfill & \hfill s_4{s}_5\hfill \\ \hfill s_5{c}_6\hfill & \hfill -s_5{s}_6\hfill & \hfill -c_5\hfill \end{array}\right]\end{array}$$(18)

Since R⁶₀= R³₀R⁶₃, we can infer that R⁶₃= (R³₀)⁻¹ R⁶₀. Among which, R³₀ can be determined accurately after the first joints have been calibrated. If R⁶₀ can be measure by ultrasonic method, the later three joints can be calibrated according to equation (18). Assuming R⁶₃ has been determined by R⁶₃= (R³₀)⁻¹ R⁶₀, and its result is:

$$R_3^6=\left[\begin{array}{ccc}\hfill n_x^s\hfill & \hfill o_x^s\hfill & \hfill a_x^s\hfill \\ \hfill n_y^s\hfill & \hfill o_y^s\hfill & \hfill a_y^s\hfill \\ \hfill n_z^s\hfill & \hfill o_z^s\hfill & \hfill a_z^s\hfill \end{array}\right].$$(19)

Actual angle of the later three joints is θ^R₄,θ^R₅ and θ^R₆. Among which θ^R₅ is:

$${\theta }_5^{\text{R}}=A\tan 2\left(\pm \sqrt{{(a_x^{\text{s}})}^2+{(a_y^s)}^2},a_z^s\right).$$(20)

Axis of joint 4 and joint 6 will be collinear when joint 5 turn to a specific angle θ^R₅=0. Manipulator will stay at singular situation which need to avoid. At that time, manipulator loses one degree of freedom. We need to avoid singular situation in calibration process.

θ^R₄ and θ^R₆ can be calculated from equation (18):

$${\theta }_4^{\text{R}}=A\tan 2\left(\frac{a_y^{\text{s}}}{s_5},\frac{a_x^{\text{s}}}{s_5}\right),$$(21)

$${\theta }_6^R=A\tan 2\left(\frac{-o_z^{\text{s}}}{s_5},\frac{n_z^s}{s_5}\right).$$(22)

Finally, the offsets exist in later three joints is Δθ_i=θ^R_i− θ^N_i (i=4,5,6).

4.3 Measuring deviation via ultrasound

According to the Section 4.2, all six joints can be calibrated in condition that relative position $\delta p_x^4,\delta p_y^4,\delta p_z^4$ and posture R⁶₀ can be measured accurately. A calibration sample as shown in Figure 5 is placed in work space of the manipulator. Three reflective surfaces of the sample must keep perpendicular to three coordinate axes. At the beginning of calibration process, the manipulator needs to be driven to calibration pose. Selection standard of such pose is keeping ultrasonic transducer receiving reflective wave in whole calibration process.

Posture of the end-effector R⁶₀ can be determined by searching strongest reflective signal from calibration sample whose posture has been confirmed definitely. Relative position $\delta p_x^4,\delta p_y^4,\delta p_z^4$ can also be determined by calibration sample. In Figure 4, origin of coordinates (X₄, Y₄, Z₄) and coordinates (X₅, Y₅, Z₅) coincides in point “W”. The relationship between end-effector (X₆, Y₆, Z₆) and coordinates (X₅, Y₅, Z₅) can be expressed as:

$$T_0^6=T_0^5{T}_5^6.$$(23)

Position coordinates of point “W” can be acquired by calculating positional components P⁵₀ in T⁵₀:

$$\{\begin{array}{c}\hfill p_x^w=p_x^5=p_x^6-a_x^5{d}_6\hfill \\ \hfill p_y^w=p_y^5=p_y^6-a_y^5{d}_6\hfill \\ \hfill p_z^w=p_z^5=p_z^6-a_z^5{d}_6\hfill \end{array}.$$(24)

Relative position of coordinates (X₆,Y₆,Z₆) is equal to relative position of point “W” in condition that posture of end-effector not changed. It is also the relative position of coordinates (X₄,Y₄,Z₄), i.e. $\delta p_x^4,\delta p_y^4,\delta p_z^4$. Finally, the relative position and posture is acquired through such calibration sample.

Fig. 5 Pose relationship between manipulator and sample.

5 Simulation and experiment of the calibration model

Deviation between the end-effector and the joints has been analyzed before. It can be used to calculate offsets. In this research a Staubli TX90L manipulator, an ultrasonic testing device and an industrial computer (32G RAM) are introduced to verify the calibration model.

5.1 Simulation result

Staubli Robot Suite (SRS, v7.3) is a kinematic software coupled with Staubli TX90L manipulator. External influences can be eliminated by using simulative method to verify calibration model. Manipulator needs to be driven to calibration pose (θ₁=30.53°,θ₂=31.15°, θ₃=109.49°,θ₄=0°,θ₅=39.36°,θ₆=30.54° in this research) referring to Figure 5. Six joint angles are read from software, and this group of angles is regarded as “nominal” angle. Identical offsets are introduced into six joints, and corresponding angles are regarded as “actual” angle. Δθ^I in Table 2 are ten groups of introduced offsets increased from −5° to 5°. Δθ^C_i (i=1,2,3,4,5,6) in Table 2 are calculated offsets of six joints based on calibration model. Calibration accuracy can be evaluated by comparing deviation between introduced offsets and calculated offsets.

Ultimate goal of calibration is to improve positional accuracy. Therefore, positional deviation of the end-effector can reflect calibration effect more directly. Table 3 lists positional deviation before and after calibration according to simulation data in Table 2. ΔX, ΔY and ΔZ express deviation between “actual” and “nominal” position of the end-effector.

The results in Table 3 indicate that small offsets in joints will induce obvious deviation into end-effector. Calibration process can decrease deviation dramatically. Posture data in simulation process can be read from software, while posture data in actual calibration is acquired based on pulse-echo principle. Deviation may exist between measured and actual data. Therefore, it is necessary to verify effectiveness of calibration model by actual experiment.

5.2 Experiment result

To further verify effectiveness of the calibration model, we construct the ultrasonic testing system as shown in Figure 1 to calibrate the initial position for the manipulator in the system. A constrained plan is built in sample; this plan is parallel to X-Y plan in coordinate system and distance is 30 mm. There are 20 “nominal” points uniformly distributing on the plan. Positional deviation of such points before and after calibration is shown is Table 4.

Referring to literature [29], we use Matlab to draw 3D diagrams as shown in Figure 6 to express data in Table 4 more intuitively. Coordinate system in Figure 6 identifies with workspace coordinate system of the sample in Figure 5. Solid points represent “nominal” value which are distributing uniformly on constraint plan. Triangular points represent actual value after calibration, and circular points represent actual value before calibration. Comparing the following points, it is obvious that 20 points is closer to plan after calibration.

Furthermore, we used statistical approach to compare positional deviation before and after calibration as shown in Table 5. It is obvious that average and maximum value decreases dramatically after calibration comparing with the value before calibration. However, standard deviations do not present obvious change comparing with such two items. We can draw a conclusion from statistics that calibration process can decrease positional deviation relative to “nominal” point, but such calibration process can not improve repeated positioning accuracy. About 3 mm deviations still exist after calibration. On the one hand, it is due to approximation error existing in calibration model. On the other hand, it is influenced by external factors like manufacturing error, gear and tear, etc. Though position accuracy can be improved by calibrating the initial position, other parameters need to be considered simultaneously to improve positional accuracy.

Verify experiment implemented on 20 points indicate that this calibration method can decrease position error caused by offsets.

Fig. 6 Distribution of marks before and after calibration.

6 Conclusion

This paper presents a hands-on method to calibrate the initial position of the manipulator based on D-H convention. It is a low-cost method, which is highly suitable for calibrating the manipulators in ultrasonic testing systems. The experiment results demonstrate that the positioning accuracy can be improved through the presented method. Key findings and evidences are as follows:

• Ultrasonic wave is mainly applied in nondestructive test. It is concluded that ultrasonic wave can also measure relative position and posture. Measurement accuracy can satisfy demands on calibrating offsets for manipulator.

• Whole manipulator is break up into two parts (i.e. “arm” and “wrist”). Then, position and posture can be considered individually. “Arm” part decides position, and “wrist” part decides posture. Matrix relationship between end-effector and joints can be simplified as trigonometric functions based on such separation method. It can simplify computing process and avoid computing error caused by iteration.

• A calibration sample is designed to measuring relative position and posture. Design principle of the sample is based on pulse-echo method and “arm-wrist separation” method. The offsets can be calculated according to relative position and posture.

• In simulation, ten group of offsets from −5° to 5° are introduced into the nominal joint angles. After calibration, the positing accuracy in X, Y and Z direction are increased by 96%, 88% and 94% respectively.

• In experiment, the positioning accuracy in X, Y and Z direction are increased by 85.8%, 69.2% and 81.4% respectively indicating that the offsets can be calibrated by such method. All the experimental results verify the effectiveness of the proposed calibration method used in offsets identification and compensation.

Considering that the algorithm accuracy and the measurement precision are two important elements to influence the calibration result of the joint offsets. The higher-order Taylor series needs to be retained and the high-precision ultrasonic transducer can be used to further improve calibration accuracy, which are our important work in the future.

Funding

This research was funded by the Research Start-Up Project of NCUT grant number [No. 11005136025XN076-0192025], the Youth Research Special Project of NCUT grant number [No. 2025NCUTYRSP006].

Conflicts of interest

The authors declare no conflict of interests.

Data availability statement

The datasets used or analyzed during the current study are available from the corresponding author on reasonable request.

Author contribution statement

Conceptualization: H.Z.; Methodology: H.Z.; Resources: Y.L.; Data Curation: Y.C.; Writing − Original Draft Preparation: H.Z.; Writing − Review & Editing: X.S.

References

All Tables

All Figures

	Fig. 1 Configuration of ultrasonic testing system with manipulator.
In the text

	Fig. 2 Transmitted pulse and echo pulse in time domain.
In the text

	Fig. 3 Mode conversion in fluid-solid interface.
In the text

	Fig. 4 Structure Diagram of Staubli-TX90L.
In the text

	Fig. 5 Pose relationship between manipulator and sample.
In the text

	Fig. 6 Distribution of marks before and after calibration.
In the text