1 Introduction

Assembly is the process of assembling parts (or components) into higher-level components or products. Large aerospace products are huge in size and complex in structure and have numerous components. Taking the production of aircraft as an example, the assembly workload usually accounts for more than half of the total production workload. The assembly of large components is one of the main tasks of aerospace assembly, and assembly accuracy plays a decisive role in the overall quality of aerospace products [1, 2].

With the increasing performance requirements of modern aerospace products, the fit tolerance between large components is becoming smaller and smaller. Due to machining error, tooling deformation, and measurement uncertainty [3, 4], the existing measurement-assisted assembly systems often lead to assembly difficulties [5]; that is, the large components may deviate from the ideal assembly trajectory, and the final pose error often exceeds the fit tolerance between the assembled components [6]. After forced assembly, there will be large assembly stresses, which will affect the overall performance and life of the final products [7].

With the development of automatic control technology, sensor technology, and computer technology, active compliance assembly has been applied to the precision assembly of components [8]. The active compliance assembly refers to the use of force sensors to measure the contact force between assembled components and generate control signals based on force control algorithms to realize the assembly of components [6]. Impedance control is the most commonly used active compliance control method. Unlike the direct control of force, impedance control adds a virtual mass-spring-damping (impedance) system between the terminal of the robot and the contact environment [9,10,11]. By adjusting the impedance of the terminal, the robot can output displacements and forces that satisfy the expected dynamic relationship. The interaction forces between the contact components in impedance control systems are usually measured by a six-dimensional force sensor (SDFS) [6, 12]. Due to the large mass and volume of large components, it is difficult to stably connect large components through a SDFS. To overcome this problem, in the previous research on the assembly of a helicopter rotor hub, we explored using distributed TDFS to sense the contact forces between assembled components [13]. Through experiments, it can be found that although the rotor hub and the principal axis can be assembled compliantly, there is still a problem of excessive contact force. The TDFS is the force measurement module for the active compliance assembly of the helicopter rotor hub, and its measurement accuracy directly affects the accuracy of the control system. Therefore, in order to reduce the contact force during the assembly process, it is necessary to improve the measurement accuracy of the force sensor.

There are two general methods to improve the measurement accuracy of TDFS. One is to filter the sensor to reduce signal noise. Xu et al. [14] simplified the model of a single-dimensional force sensor to a first-order inertial and a zero-order retainer to obtain the state and measurement equations of the single-dimensional force sensor and further derived a Kalman filtering algorithm for the multi-dimensional force sensor. Liu et al. [15] determined the initial filter value and filter parameters by deriving a Kalman filter algorithm based on the measurement model of a TDFS, and the filter algorithm was validated on a hardware platform consisting of a sensor, an A/D converter, and a DSP.

Another method to improve the measurement accuracy of TDFS is to decouple the sensor. Yu et al. [16] used finite element simulation to analyze the mechanical decoupling of sensors with complex elastic structures and calibrated the TDFS by the NI data acquisition system. In order to effectively reduce the impact of dimensional coupling on multi-dimensional force sensors, Zhao et al. [17] established a measurement model based on the structure of the TDFS and conducted decoupling and analysis on the model. Zhao et al. [18] also established a general calibration model for overconstrained parallel SDFS considering tensions and compressive stiffness of different branches and conducted calibration experiments. The experimental results show that the maximum type I error and type II error both decrease by more than 40%.

Through the above research, it can be found that filtering and decoupling the sensors can reduce the measurement error of the TDFS, and many scholars have made contributions. However, filtering and decoupling can only enhance the accuracy of the TDFS itself. In fact, the factors that cause the perception error of the contact force in the compliance assembly system also include the installation angle deviation of the force sensor, as shown in Fig. 1. Due to the lack of positioning reference for the installation of TDFS, if the TDFS coordinate system deviates from the base coordinate system (BCS) of the assembly system, there will be a deviation between the force component of F on each axis of the sensor coordinate system (the measured value of the TDFS) and the force component of F on each axis of the BCS. Since the calculation of contact force is conducted in the BCS, sensor installation angle errors will lead to contact force calculation errors, which will be analyzed in detail in Sect. 2.3.

Fig. 1
figure 1

Installation angle deviation of TDFS

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In order to improve the perception accuracy of the contact force in the active compliance assembly systems, this paper studies the calibration assembly of the TDFS installation angle. The essence of the calibration of the TDFS installation angle is to calculate the posture angle of the sensor coordinate system relative to BCS, so as to map the force measured by the TDFS to the BCS. Therefore, we can make a special fixture to ensure that the installed TDFS and the BCS are consistent. However, making specialized fixtures requires additional costs. We can also use a laser tracker to measure the posture parameters of the TDFS relative to the BCS [19] to calibrate the installation angle of the TDFS. However, the measurement uncertainty of the laser tracker itself can also affect the calibration accuracy of the TDFS [20]. The assembly mechanism for large components composed of multiple number-controlled positioners (NCPs) has redundant drives [21], so it is possible to calibrate the posture of TDFS according to the redundant force information inside the assembly mechanism. The high-precision automated assembly of the rotor hub and principal axis of the large helicopter is a typical active compliance assembly. This paper will take the compliance assembly system for helicopter rotor hub as an example to describe the calibration method of the TDFS installation angle in detail.

The primary goal of this paper is to improve the perception accuracy of contact force between assembled components in the active compliance assembly system by proposing a self-calibration method of the TDFS installation angle. The proposed calibration method does not require special fixtures or external measuring equipment, and the calibration accuracy will not be affected by the measuring equipment [22]. Therefore, this calibration method has cost advantages and higher accuracy. The rest of the paper is organized as follows. In Sect. 2, the influence of the installation angle deviation of the TDFS on the contact force calculation accuracy is analyzed based on the dynamic method of assembly mechanism. Section 3 proposes a self-calibration model of the TDFS installation angle. Section 4 analyzes the calibration error of the proposed method by Monte Carlo simulation and compares it with the error of traditional methods. Finally, the calibration experiment of the TDFS installation angle is conducted in the laboratory, and a series of pose adjustment and docking experiments of large components are conducted.

2 Calculation and error analysis of contact force of assembled components

2.1 Compliance assembly system of large gear components

The rotor hub and principal axis of the large helicopter are connected by precision gear components, which require high assembly precision, as shown in Fig. 2a. The radial positional deviation between the rotor hub and the principal axis is less than 0.05 mm, the radial angle deviation Δα is less than 3 × 10−4 rad, and the axial angle deviation Δβ is less than 2 × 10−4 rad. This is a typical minor clearance fitting mode, as shown in Fig. 2b. Therefore, the traditional measurement-aided assembly methods are challenging to meet the assembly accuracy of the gear components, and the compliance assembly must be adopted.

Fig. 2
figure 2

a, b Rotor hub and principal axis of large helicopter

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In the complaint assembly system shown in Fig. 3, the parallel assembly mechanism consists of three NCPs. The inter gear is connected to the terminal of the mechanism, and the outer gear is fixed on the pedestal. The TDFS is installed below the spherical hinge to measure the contact force of the spherical hinge. The NCP is essentially a three-axis robot that can move in the X-/Y-/Z-direction according to the programs [20, 23]. Let \({r}_i^t={\left[{x}_i^t{y}_i^t{z}_i^t\right]}^T\) represent the coordinates of the spherical hinge center (SHC) in the local coordinate system (LCS) Ol − xyzof the terminal, where i = {1, 2, 3}. The coordinate of the SHCi in the BCS Ob − xyz of the assembly mechanism can be expressed as \({r}_i^b={\left[{x}_i^b{y}_i^b{z}_i^b\right]}^T\). The SHC is the common point between the NCP and the terminal. According to the rigid body rotation principle [24], the coordinate of the SHCi in the BCS and the LCS satisfies Eq. 1.

Fig. 3
figure 3

Compliance assembly system for large gear components

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$${r}_i^b={R}_t^b{r}_i^t+{T}_t^b$$
(1)

where \({R}_t^b\) is the rotation matrix from the BCS to the LCS and \({T}_t^b\) is the position vector from the BCS to the LCS. Let \(\left({\alpha}_t^b,{\beta}_t^b,{\gamma}_t^b\right)\) and \(\left({x}_t^b,{y}_t^b,{z}_t^b\right)\) represent the posture angles and position coordinates of the terminal LCS relative to the BCS, respectively. Therefore, the following equations hold [24, 25].

$${R}_t^b=\left[\begin{array}{ccc}\cos {\alpha}_t^b\cos {\beta}_t^b& \cos {\alpha}_t^b\sin {\beta}_t^b\sin {\gamma}_t^b-\sin {\alpha}_t^b\cos {\gamma}_t^b& \cos {\alpha}_t^b\sin {\beta}_t^b\cos {\gamma}_t^b+\sin {\alpha}_t^b\sin {\gamma}_t^b\\ {}\sin {\alpha}_t^b\cos {\beta}_t^b& \sin {\alpha}_t^b\sin {\beta}_t^b\sin {\gamma}_t^b+\cos {\alpha}_t^b\cos {\gamma}_t^b& \sin {\alpha}_t^b\sin {\beta}_t^b\cos {\gamma}_t^b-\cos {\alpha}_t^b\sin {\gamma}_t^b\\ {}-\sin {\beta}_t^b& \cos {\beta}_t^b\sin {\gamma}_t^b& \cos {\beta}_t^b\cos {\gamma}_t^b\end{array}\right]$$
(2)
$${T}_t^b={\left[{x}_t^b{y}_t^b{z}_t^b\right]}^T$$
(3)

The spatial posture of the terminal can be represented by a six-dimensional vector\(w={\left[{\alpha}_t^b\kern0.5em {\beta}_t^b\kern0.5em {\gamma}_t^b\kern0.5em {x}_t^b\kern0.5em {y}_t^b\kern0.5em {z}_t^b\right]}^T\). It can be seen from Eq. 1 that the spatial posture and position of the gear components can be adjusted by changing the coordinate \({r}_i^b\) of the SHCi in the BCS through the cooperative motion of three NCPs [26].

2.2 Calculation model of assembly contact force

In order to achieve the compliance assembly of gear components, the contact force between the inner gear and the external environment (outer gear) needs to be measured accurately, and the impedance controller corrects the motion trajectory of the terminal based on the contact force information [27]. The force measurement module of the compliance assembly system is a TDFS. The force measured by the TDFS consists of three parts: the inertial force of the terminal (including the inner gear), the gravity of the terminal, and the contact force between the gear components. Let fe and Me represent the contact force and torque between the gear components, respectively. According to the Newton-Euler theorem, the dynamic model of the compliance assembly mechanism is as follows [28]:

$$\left\{\begin{array}{l}\sum\limits_{i=1}^3{F}_i^t+{\left({R}_t^b\right)}^T{G}_m+{f}_e={\textrm{ma}}^t\\ {}\sum\limits_{i=1}^3{\hat{r}}_i^t{F}_i^t+{\hat{r}}_m^t{\left({R}_t^b\right)}^T{G}_m+{M}_e={I}_t{\dot{\omega}}^t+{\omega}^t\times \left({I}_t{\omega}^t\right)\end{array}\right.$$
(4)

In Eq. 4, \({F}_i^t\) is the force applied to the spherical hinge in the LCS, which has the following relationship with the force \({F}_i^b\) exerted on the spherical hinge by NCPi in the BCS.

$${F}_i^t={\left({R}_t^b\right)}^T{F}_i^b$$
(5)

where Gm, m, and at represent the gravity matrix, mass, and acceleration of the terminal and \({\hat{r}}_i^t\) represents the anti-skew symmetric matrix of \({r}_i^t\), which can be expressed by Eq. 6. \({r}_m^t\) is the barycenter of the terminal. It and ωt represent the inertia tensor matrix and the angular velocity of the terminal, respectively.

$${G}_m={\left[0\ 0-\textrm{mg}\right]}^{\textrm{T}}$$
(6)
$${\hat{r}}_i^t=\left[\begin{array}{lll}0& -{z}_i^t& {y}_i^t\\ {}{z}_i^t& 0& -{x}_i^t\\ {}-{y}_i^t& {x}_i^t& 0\end{array}\right]$$
(7)

Theoretically, the force \({F}_i^b\) exerted by the NCP on the spherical hinge equals the spherical hinge contact force \({F}_i^s\) measured by the TDFS. However, due to the installation angle deviation of the TDFS, the following relationship exists between \({F}_i^b\) and \({F}_i^s\).

$$\left\{\begin{array}{l}{F}_i^b={R}_{s,i}^b{F}_i^s\\ {}{R}_{s,i}^b==\left[\begin{array}{ccc}\cos {\gamma}_{s,i}^b& -\sin {\gamma}_{s,i}^b& 0\\ {}\sin {\gamma}_{s,i}^b& \cos {\gamma}_{s,i}^b& 0\\ {}0& 0& 1\end{array}\right]\left[\begin{array}{ccc}\cos {\beta}_{s,i}^b& 0& \sin {\beta}_{s,i}^b\\ {}0& 1& 0\\ {}-\sin {\beta}_{s,i}^b& 0& \cos {\beta}_{s,i}^b\end{array}\right]\left[\begin{array}{ccc}1& 0& 0\\ {}0& \cos {\alpha}_{s,i}^b& -\sin {\alpha}_{s,i}^b\\ {}0& \sin {\alpha}_{s,i}^b& \cos {\alpha}_{s,i}^b\end{array}\right]\end{array}\right.$$
(8)

where \({\alpha}_{s,i}^b\), \({\beta}_{s,i}^b\), and \({\gamma}_{s,i}^b\) represent the posture angles of the TDFS relative to the BCS. Convert Eq. 4 into the matrix form.

$${F_e}^{\prime }={I}_N-{J}^{\prime }{F}^s-{R}_g{G}_m$$
(9)

where IN represent the inertia matrix of the terminal, J′ represent the Jacobian matrix of the assembly system considering the sensor posture deviation, and Fe′ represent the six-dimensional contact force of the assembled components. Their specific expressions are as follows:

$${I}_N=\left[\begin{array}{c}{\textrm{ma}}^t\\ {}{I}_t{\dot{\omega}}^t+{\omega}^t\times \left({I}_t{\omega}^t\right)\end{array}\right]$$
(10)
$${J}^{\prime }=\left[\begin{array}{ccc}{\left({R}_t^b\right)}^T{R}_{s,1}^b& {\left({R}_t^b\right)}^T{R}_{s,2}^b& {\left({R}_t^b\right)}^T{R}_{s,3}^b\\ {}{\left({R}_t^b\right)}^T{R}_{s,1}^b{\hat{r}}_1^t& {\left({R}_t^b\right)}^T{R}_{s,2}^b{\hat{r}}_2^t& {\left({R}_t^b\right)}^T{R}_{s,3}^b{\hat{r}}_3^t\end{array}\right]$$
(11)
$${F}^s={\left[{F}_1^s{F}_2^s{F}_3^s\right]}^T$$
(12)
$${R}_g=\left[\begin{array}{c}{\left({R}_t^b\right)}^T\\ {}{r}_m^t{\left({R}_t^b\right)}^T\end{array}\right]$$
(13)

According to Eq. 9, the control system can calculate the contact force Fe′ between assembled components in real time.

2.3 Error analysis of contact force

When the TDFS system coordinate is aligned with the BCS of the assembly system, the matrix \({R}_{s,i}^b\) in Eq. 8 is an identity matrix. In this case, the six-dimensional contact force Fe between the assembled components can be expressed by Eq. 14.

$${F}_e={I}_N-J{F}^s-{R}_g{G}_m$$
(14)

where

$$J=\left[\begin{array}{ccc}{\left({R}_t^b\right)}^T& {\left({R}_t^b\right)}^T& {\left({R}_t^b\right)}^T\\ {}{\left({R}_t^b\right)}^T{\hat{r}}_1^t& {\left({R}_t^b\right)}^T{\hat{r}}_2^t& {\left({R}_t^b\right)}^T{\hat{r}}_3^t\end{array}\right]$$
(15)

It can be calculated that the contact force error is

$$\triangle {F}_e={F_e}^{\prime }-{F}_e=\left(J-{J}^{\prime}\right){F}^s$$
(16)

It can be seen from Eq. 16 that the factors affecting contact force include the posture angle \(\left({\alpha}_{s,i}^b,{\beta}_{s,i}^b,{\gamma}_{s,i}^b\right)\) of the TDFS, the posture matrix \({R}_t^b\), and the spherical hinge contact force Fs. Due to the complexity of the contact force error model of the assembled components, it is challenging to analyze contact force errors by analytical methods. Therefore, the relationship between contact force error and posture angle error of TDFS will be studied through numerical simulation.

Since only the magnitude of the contact force error needs to be discussed, we can take a particular moment before assembly as an example for discussion. At this time, the contact force Fe′ between the assembled components is zero. In order to ensure the success rate of the assembly, the posture of the assembled components has been aligned before the assembly, so the posture matrix \({R}_t^b\) of the terminal can be set as an identity matrix. According to Eq. 16, the contact force error is independent of the inertial matrix IN, so the inertial matrix IN can be set as zero too. The following equation holds.

$${J}^{\prime }{F}^s=-{R}_g{G}_m$$
(17)

Equation 17 is an underconstrained equation. According to the principle of optimal driving force distribution, we can calculate the minimum two-norm solution of Eq. 17 [29].

$${\hat{F}}^s=-{\left(J^\prime J{^\prime}^T\right)}^{-1}{J}^{\prime }{R}_g{G}_m$$
(18)

Due to the posture angle deviation of the TDFS, the contact force error is

$$\varDelta {F}_e={J}^T{\left(J^\prime J{^\prime}^T\right)}^{-1}{J}^{\prime }{R}_g{G}_m-{R}_g{G}_m$$
(19)

In order to facilitate the analysis of the impact of posture angle on the contact force (torque) error ΔFe, let \(\varDelta f=\left|\varDelta {F}_e^x\kern0.5em \varDelta {F}_e^y\kern0.5em \varDelta {F}_e^z\right|\) and \(\varDelta M=\left|\varDelta {M}_e^x\kern0.5em \varDelta {M}_e^y\kern0.5em \varDelta {M}_e^z\right|\) represent the contact force error and the contact torque error where

$$\varDelta {F}_e={\left[\varDelta {F}_e^x\kern0.5em \varDelta {F}_e^y\kern0.5em \varDelta {F}_e^z\kern0.5em \varDelta {M}_e^x\kern0.5em \varDelta {M}_e^y\kern0.5em \varDelta {M}_e^z\right]}^T$$
(20)

since the posture angle of the TDFS after installation is very small, generally not exceeding 0.05 rad, and is symmetrically distributed. According to the parameters of the assembly mechanism shown in Table 1, the relationship curve between contact force (torque) error and installation angle error can be obtained. The calculation results are shown in Fig. 4.

Table 1 Parameters of the assembly mechanism
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Fig. 4
figure 4

a–f Relationship curve between contact force (torque) error and posture angle

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It can be observed from the calculation results that the variation trend of the contact force (torque) errors of the assembled components is roughly the same. As the installation angle error of the TDFS increases, the contact force (torque) errors also increase. When the installation angle error of a single TDFS reaches 5 × 10−2 rad, the contact force error even exceeds 80 N. Due to the inevitable installation angle errors of all TDFSs in the actual assembly systems, the factual contact force error should be greater. Therefore, it is essential to calibrate the installation angle of TDFS.

3 Calibration method for installation angle of TDFS

It can be seen from Eq. 9 that when the mechanism terminal is stationary and not in contact with other components, there is a coupling relationship between the contact forces of the spherical hinge [30]. Therefore, it is possible to establish a set of equations relating the force components of the TDFS and the terminal posture angles by adjusting the posture of the terminal. Another conventional method for calibrating the installation angle is to use a laser tracker to measure the measuring points (MPs) of the TDFS and then calculate the posture parameters. The following will describe the two methods for calibrating the posture angle parameters of TDFS in detail.

3.1 Self-calibration method for installation angle of TDFS

When the terminal of the assembly system is stationary and not in contact with other components, IN = 0, Fe′ = 0, and the following equation can be obtained.

$$\sum_{i=1}^3{\hat{r}}_i^t{F}_i^t+{\hat{r}}_m^t{\left({R}_t^b\right)}^T{G}_m=0$$
(21)

Expand Eq. 21.

$$\left[\begin{array}{ccc}{\hat{r}}_1^t{\left({R}_t^b\right)}^T& & \\ {}& {\hat{r}}_2^t{\left({R}_t^b\right)}^T& \\ {}& & {\hat{r}}_3^t{\left({R}_t^b\right)}^T\end{array}\right]\left[\begin{array}{c}{R}_{s,1}^b\\ {}{R}_{s,2}^b\\ {}{R}_{s,3}^b\end{array}\right]\left[\begin{array}{ccc}{F}_1^s& & \\ {}& {F}_2^s& \\ {}& & {F}_3^s\end{array}\right]=-{\hat{r}}_m^t{\left({R}_t^b\right)}^T\cdot {G}_m$$
(22)

Equation 22 is a nonlinear equation about posture angle \(\left({\alpha}_{s,i}^b{\beta}_{s,i}^b{\gamma}_{s,i}^b\right)\), and in order to simplify the solution, Eq. 22 can be seen as a linear equation about the matrix \({R}_{s,i}^b\).

Let \({r}_M=\left[\begin{array}{ccc}{\hat{r}}_1^t{\left({R}_t^b\right)}^T& & \\ {}& {\hat{r}}_2^t{\left({R}_t^b\right)}^T& \\ {}& & {\hat{r}}_3^t{\left({R}_t^b\right)}^T\end{array}\right]\) denote the spherical hinge matrix of the terminal, \(X=\left[\begin{array}{c}{R}_{s,1}^b\\ {}{R}_{s,2}^b\\ {}{R}_{s,3}^b\end{array}\right]\) denote the posture matrix of TDFSs, and \({F}_M=\left[\begin{array}{ccc}{F}_1^s& & \\ {}& {F}_2^s& \\ {}& & {F}_3^s\end{array}\right]\) denote the contact force matrix of the spherical hinges. Therefore, Eq. 22 can be written in the following form.

$${r}_M^iX={T}_M{F_M}^{-1}$$
(23)

Equation 23 contains 9 equations and 27 unknowns (matrix elements of \({R}_{s,i}^b\)). Therefore, at least three sets of posture matrices and contact forces measured by TDFSs are required to calculate X. Let \({r}_M^i\) and \({F}_M^i\) denote the spherical hinge matrix of the terminal and the contact force matrix of the spherical hinges in the i-th posture. Assuming that the terminal posture has been adjusted n times, Eq. 24 can be obtained.

$$\left[\begin{array}{c}{r}_M^1\\ {}{r}_M^2\\ {}\vdots \\ {}{r}_M^n\end{array}\right]X=\left[\begin{array}{c}{T}_M^1{\left[{F}_M^1\right]}^{-1}\\ {}{T}_M^2{\left[{F}_M^2\right]}^{-1}\\ {}\vdots \\ {}{T}_M^n{\left[{F}_M^n\right]}^{-1}\end{array}\right]$$
(24)

Let \(A={\left[\begin{array}{cccc}{r}_M^1& {r}_M^2& \cdots & {r}_M^n\end{array}\right]}^T\), \(B={\left[\begin{array}{cccc}{T}_M^1{\left[{F}_M^1\right]}^{-1}& {T}_M^2{\left[{F}_M^2\right]}^{-1}& \cdots & {T}_M^n{\left[{F}_M^n\right]}^{-1}\end{array}\right]}^T\), the least squares estimate of X can be calculated as \(\hat{X}={\left({A}^TA\right)}^{-1}{A}^TB\).

3.2 Measurement calibration method for installation angle of TDFS

Another conventional calibration method is to use measuring instruments (such as a laser tracker) to measure the relative posture relationship between the TDFS and the BSC. The main principle is to measure the MPs of the TDFS by the laser tracker [31] and then calculate the posture angle between the TDFS and the BCS of the assembly system. The specific method is as follows:

  1. (1)

    According to the technical parameters of the TDFS, the coordinates of the MP of the TDFS in the TDFS coordinate system can be obtained as \({P}_i^s={\left[{x}_i^s,{y}_i^s,{z}_i^s\right]}^T\) and i = {1, 2, 3, 4}, as shown in Fig. 5.

  2. (2)

    The coordinates of the MPs of the TDFS in the laser tracker measurement coordinate system (MCS) can be measured as \({P}_i^m={\left[{x}_i^m,{y}_i^m,{z}_i^m\right]}^T\) by the laser tracker.

  3. (3)

    According to equation \({P}_i^m={R}_s^m{P}_i^s+{T}_s^m\), the posture matrix of the TDFS relative to the laser tracker MCS can be calculated as \({R}_s^m\). The calculation methods include the least squares method, SVD method [32], and quaternion method [33].

Fig. 5
figure 5

Measurement of installation angle for TDFS

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  1. (4)

    Keeping the measurement station of the laser tracker fixed, drive the NCP to move in its travel space and use the laser tracker to measure the MP of the TDFS. It can be obtained that the coordinate of the MP in the MCS is \({P}_j^m={\left[{x}_j^m,{y}_j^m,{z}_j^m\right]}^T\), and at this time, the coordinate of the SHC in the BCS can be obtained as \({r}_j^b={\left[{x}_j^b,{y}_j^b,{z}_j^b\right]}^T\) according to the servo control system.

  2. (5)

    According to equation \({P}_j^m={R}_b^m{r}_j^b+{\overset{\check{} }{T}}_b^m\), the posture matrix of the BCS relative to the MCS can be calculated as \({R}_b^m\). It should be noted that due to the noncoincidence of the MP and SHC, \({\overset{\check{} }{T}}_b^m\) is not the position vector of the BCS relative to the laser tracker MCS.

  3. (6)

    According to the principle of coordinate system transfer, the posture matrix of the TDFS relative to the BCS can be calculated as \({R}_s^b={\left({R}_b^m\right)}^{-1}{R}_s^m\).

4 Accuracy analysis of installation angle calibration based on Monte Carlo method

Whether traditional measurement calibration methods or the proposed self-calibration method, the posture angle parameters of TDFS need to be solved through the least squares method, and the specific analytical expression for the angle parameters is difficult to be obtained. Therefore, the Monte Carlo MCS is used to analyze the calibration accuracy. The Monte Carlo method is a completely different calculation method from analytical methods. It uses random values for statistical experiments [34]. The Monte Carlo method can be used to evaluate the uncertainty of installation angle calibration.

4.1 Accuracy analysis of self-calibration of installation angle

Before conducting calibration accuracy analysis, it is necessary to identify the uncertainty sources that affect the self-calibration accuracy. The self-calibration method proposed in this article does not introduce an external measurement instrument, so the main factors affecting the calibration accuracy of the installation angle include the motion error of the NCP and the measurement error of the TDFS. Since NCP needs to be calibrated before being used for assembly, the main factor affecting the motion accuracy of NCP is repeated positioning error. The repetitive positioning error of NCP [35] is a random error with a normal distribution. The positioning uncertainty of the NCP used is σ = 10μm. Therefore, the motion error of each axis of the NCP follows an N(0, σ2) distribution. The accuracy analysis simulation process of the self-calibration of the TDFS installation angle is shown in Fig. 6.

Fig. 6
figure 6

Accuracy analysis simulation process of self-calibration

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The Monte Carlo simulation results are shown from Figs. 7, 8, 9.

Fig. 7
figure 7

Installation angle error distribution of TDFS1 using self-calibration

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Fig. 8
figure 8

Installation angle error distribution of TDFS2 using self-calibration

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Fig. 9
figure 9

Installation angle error distribution of TDFS3 using self-calibration

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4.2 Accuracy analysis of measurement calibration of installation angle

In the measurement calibration method, in addition to the motion error of NCP, factors that affect the calibration accuracy of the TDFS installation angle include the measurement error of the laser tracker. The laser tracker is a spherical coordinate measurement system, as shown in Fig. 10. The Cartesian coordinate P0 = (x, y, z) of the MP in the laser tracker MCS is a composite function (Eq. 25) of the parameter (r, θ, φ) of the laser tracker [36, 37], where r is the distance between the MP and the origin of the MCS and θ and φ are the horizontal angle and vertical angle of the MP relative to the MCS. Since both angle and distance measurement contain uncertainties, it ultimately leads to coordinate uncertainty of the MP. In Cartesian space, the uncertainty region can be represented by an uncertainty ellipsoid. The Leica AT901b laser tracker was used in the simulation. The uncertainty of the distance sensor is σ(r)==10 μ+5 μ/m and the uncertainty of the angle sensor is σ(θ) = σ(φ) = 2″. The accuracy analysis simulation process of the measurement calibration of the TDFS installation angle is shown in Fig. 11.

Fig. 10
figure 10

Measurement principle of laser tracker

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Fig. 11
figure 11

Accuracy analysis simulation process of measurement calibration

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$$\left\{\begin{array}{l}x=r\sin \varphi \cos \theta \\ {}y=r\sin \varphi \sin \theta \\ {}z=r\cos \varphi \end{array}\right.$$
(25)

The simulation results are shown in Figs. 12, 13, and 14.

Fig. 12
figure 12

Installation angle error distribution of TDFS1 using measurement calibration

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Fig. 13
figure 13

Installation angle error distribution of TDFS2 using measurement calibration

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Fig. 14
figure 14

Installation angle error distribution of TDFS3 using measurement calibration

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The standard deviation (σ) of the obtained TDFS installation angle is the uncertainty of the TDFS installation angle, and the uncertainties of the two calibration methods are shown in Table 2. It can be found that the average errors of the installation angle obtained by the two calibration methods are much smaller than its standard deviation and close to zero. Therefore, both the calibration method proposed in this article and the measurement calibration method can obtain an unbiased estimation of the installation angle of the TDFS, so that the installation angle of the TDFS can be calibrated effectively. It can also be found that the installation angle uncertainty of the proposed self-calibration method is lower than that of the measurement calibration method. Therefore, it can be believed that the self-calibration method can achieve more steady calibration results. Moreover, measurement calibration methods do not consider the measurement uncertainty of the TDFS itself. In practical applications, the proposed calibration method can achieve higher contact force calculation accuracy.

Table 2 Mean and uncertainty of TDFS installation angle (rad)
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5 Self-calibration and assembly experiment

To verify the proposed calibration method, an assembly system of gear components was built, and TDFS installation angle calibration and active compliance assembly experiments were conducted. The experimental assembly platform mainly consists of NCPs, TDFSs, terminal (bracket), gear components, servo system, and control software, as shown in Fig. 15. In the assembly experiment of gear components, the impedance control strategy was used to achieve compliant assembly of gear components. According to the principle of impedance control, the higher the perception accuracy of the contact force of the assembly system, the smaller the steady-state error of the assembly contact force.

Fig. 15
figure 15

Compliance assembly experiments platform for gear components

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The proposed calibration method and the traditional calibration method were used to calibrate the installation angle of the TDFS before conducting assembly experiments. The installation angles of the TFDS obtained using these two calibration methods are shown in Table 3.

Table 3 Calibrated installation angles of the TFDS (rad)
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It can be found from Table 3 that the difference between the TDFS installation angles obtained by these two calibration methods is less than 10−3 rad. Therefore, the consistency of these two calibration methods is good, which can confirm the effectiveness of these two calibration methods. It can also be found that the installation angle of the TDFS around the Z-direction is much greater than the other two directions, mainly due to the lack of positioning reference around the Z-direction between the TDFS and the NCP.

5.1 Impedance control experiment in free space

Before the inner gear contacts the outer gear, there is approximately 50 mm of free space along the assembly direction. In this space, the contact force between the assembled components is zero, and the inner gear should follow the theoretical trajectory. According to Eq. 16, if there is an error in the installation angle of the TDFS, it will lead to contact force errors, the impedance controller will mistakenly assume that the assembly components have already been contacted, and the impedance controller will correct the assembly trajectory, causing trajectory deviation. Free space impedance control experiments based on the installation angles obtained by two calibration methods were conducted. A total of 6 sets of experiments were conducted, and the initial and target posture and position of the terminal in free space of each experiment are shown in Table 4.

Table 4 Initial and target posture and position of the terminal
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In order to avoid the absolute positioning error of the assembly system affecting the results, the final position of the terminal is calculated based on the feedback value of the servo system. After these experiments were completed, calculate the deviations between the actual and theoretical position and posture of the terminal. The experimental results are shown in Fig. 16.

Fig. 16
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a–c Position error of the terminal. e–f Posture angle error of the terminal

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It can be observed that compared with the measurement calibration method, by using the self-calibration method, the average position deviation of the terminal in free space is reduced from 1.002 to 0.592 mm, and the average posture angle error is reduced from 7.088 × 10−3 rad reduced to 5.071 × 10−3 rad. According to the principle of impedance control, the proposed calibration method can achieve higher installation angle calibration accuracy and reduce contact force calculation errors more effectively. At the same time, the smaller the position and posture deviation when the assembled components come into contact, the greater the assembly success rate.

5.2 Compliance assembly experiments

Firstly, a laser tracker was used to measure the MPs of the outer gear and obtain the posture and position parameters of the outer gear relative to the BCS of the assembly system. Then, adjust the inner gear from the general posture to the assembly posture, ensuring that the inner gear can be inserted into the outer gear along the assembly direction. Finally, the assembly system makes the inner gear accelerate to 1.5 mm/s uniformly along the assembly direction and approach the outer gear component at a constant speed for assembly. The main control software records the contact force of the gear components during the assembly process in real time. When the contact force reaches the set threshold, gear components have completed assembly, or there is jamming between the gear components, resulting in assembly failure. The contact force thresholds in X-, Y-, and Z-directions set in this assembly experiment are (20, 20, 40) N, respectively. After calibrating the TDFS installation angle by the proposed calibration method, the contact force and torque of the gear components during the assembly process are shown in Fig. 17.

Fig. 17
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a–f Contact force (torque) during assembly of gear components

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It can be seen from Fig. 17 that the gear components come into contact at 45 s, and then, enter the compliance assembly stage. At 262 s, the contact force of the assembly components in the Z-direction reaches the threshold of 40 N, and the assembly is completed. The peak value of the contact force and torque of the assembly components occurs at the moment of contact. The maximum contact force is 12.1 N in the Z-direction, and the maximum contact torque is 3.48 N·m around the X-direction. During the whole assembly process, the contact force is within a safe range, and the overall trends of the contact force and torque in all directions are gradually decreasing and stabilizing. It can be considered that the proposed calibration method of the TDFS installation angle meets the requirements of active compliance assembly, and the feasibility of the relevant algorithms in practical applications has been preliminarily demonstrated.

In order to verify the proposed TDFS installation angle calibration method in reducing the assembly contact force in general, the assembly of the gear components according to the TDFS installation angle parameters obtained from two calibration methods was conducted and was compared with the assembly experiment without calibrating the TDFS installation angle. The maximum contact force and torque of the gear components during the assembly process were recorded, and a total of 10 sets of experiments were conducted. The experimental results are shown in Fig. 18.

Fig. 18
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a–c Maximum contact force of the gear components. e–f Maximum contact torque of the gear components

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It can be observed that compared to not calibrating the installation angle of TDFS, after calibrating the TDFS installation angle by using a laser tracker, the average value of the maximum contact force of the assembled components decreased from 8.41 to 6.58 N, and the average value of the maximum contact torque decreased from 5.22 to 3.89 N·m. After calibrating the TDFS installation angle using the proposed method, the average value of the maximum contact force was further reduced by 36.3%, and the average value of the maximum contact force was further reduced by 22.1%. Therefore, the proposed self-calibration method can significantly reduce the contact force of the assembled components in the impedance control system, which is very beneficial for improving the assembly quality.

6 Conclusion

In the compliance assembly system of large components based on distributed force sensors, the installation angle error of TDFS is an important factor affecting the assembly contact force. The traditional method for calibrating the installation angle of TDFS requires external measurement equipment such as a laser tracker, and the calibration accuracy is often unsatisfactory. In order to reduce the contact force of assembled components, this paper proposes a self-calibration method for the TDFS installation angle. Compared with the traditional calibration methods, the proposed method does not require external measuring equipment, and the control system can automatically adjust the terminal posture according to program settings and calculate the installation angle of the TDFS. Therefore, the proposed method has higher calibration efficiency. And since the proposed calibration method does not introduce external uncertainty, the calibration accuracy should also be higher than that of the traditional calibration method. The principal contributions of the paper include

  1. (1)

    Based on the dynamic model of the compliance assembly mechanism, the influence of the TDFS installation angle error on the perception accuracy of the contact force is analyzed. Theoretical calculations indicate that when the installation angle error of a single TDFS reaches 5 × 10−2 rad, the contact force error even exceeds 80 N. Therefore, it is very necessary to calibrate the TDFS installation angle.

  2. (2)

    The self-calibration model of the TDFS installation angle in the active compliance assembly system is established, majorly based on the relationship between the contact force of the spherical hinge and the posture of the assembly systems terminal.

  3. (3)

    In order to verify the effectiveness of the proposed calibration method, the Monte Carlo simulation is used to analyze the uncertainty of the calibration model. The results show that the installation angle of the TDFS obtained by the proposed calibration method is an unbiased estimation, so the self-calibration method can effectively calibrate the installation angle. Moreover, compared to the measurement calibration method, the installation angle uncertainty of the proposed calibration method has decreased by 30.1%. It can be believed that the self-calibration method can achieve more steady calibration results.

  4. (4)

    Taking the assembly of gear components as an example, a series of TDFS installation angle calibration experiments are conducted. The assembly experiment of the gear components shows that the proposed calibration method not only improves the alignment accuracy of the assembly feature, but also reduces the contact force of gear components during the assembly process compared to the traditional calibration methods. Therefore, the proposed calibration method can improve the success rate and quality of compliance assembly of large components.