Posture adjustment method of large aircraft components based on multiple numerical control positioners

In the digital assembly system of large aircraft components (LAC), multiple numerical control positioners (NCPs) are usually used as actuators to adjust the position and posture of LAC to realize the docking of LAC. The posture adjustment mechanism (PAM) composed of multiple NCPs is a redundant actuated parallel mechanism (RAPM). The traditional full position control (FPC) method may lead to interference between NCPs, resulting in the deformation of NCPs and bracket, affecting the service life of the equipment. This paper proposes a posture adjustment method based on hybrid control, which divides the motion axes of the NCPs into the position control axis and the force control axis to avoid the internal force of posture adjustment caused by the cooperative motion error of the NCPs. The driving force of the axis of the RAPM under the same posture adjustment trajectory is uncertain. To further reduce the internal force of posture adjustment, a driving force distribution method based on the dynamic model of PAM is proposed. Then, the principle of selecting the position control axis of the PAM is analyzed, and the optimization strategy of the position control axis based on the condition number of the Jacobian matrix is studied to improve the motion performance of the PAM. Finally, the posture adjustment experiment of LAC is carried out. The results show that the hybrid control method based on the optimum contact force can significantly reduce the interaction force between NCPs. The experimental results of posture adjustment accuracy verification show that the optimization strategy of position control axis makes the accuracy of single posture adjustment meet the requirements of LAC docking, which can effectively improve the docking efficiency of LAC.


Introduction
The aircraft manufacturing process can be divided into three stages: part manufacturing, assembly, and test, where assembly is combining parts (or components) into higherlevel components or products according to the design requirements. Large aircraft is huge in size, complex in structure, and has a large number of components, and the assembly workload usually accounts for more than 50% of the total workload of aircraft manufacturing [1]. The docking of LAC is one of the main tasks of aircraft assembly, and the docking accuracy is a crucial factor affecting the final quality of aircraft products [2]. The traditional docking of LAC relies on a large number of special toolings and manual operations, which is labor-intensive, inefficient, and difficult to improve the assembly accuracy, and the design and manufacture of a large number of special toolings increase the cost and cycle of aircraft manufacturing [3]. With the rapid development of digital measurement technology [4], automatic control technology [5], and sensor technology [6], automatic docking system has been gradually adopted in the assembly of LAC [7,8]. In the automatic docking system, the position and posture of LAC are measured by digital measuring equipment (such as laser tracker), and the accurate positioning and posture adjustment of the LAC are realized by digital posture adjustment system. Compared with the manual docking, the automatic docking has obvious advantages [9]. It can better and faster complete the docking task [10], improve the quality of assembly, shorten the design and manufacturing cycle of tooling, and reduce the cost of product manufacturing.
In the digital posture adjustment system, the NCP is usually used as the actuator to adjust the position and posture of LAC [11]. The NCP is an electromechanical automatic device that can move along the X\Y\Z direction according to the predetermined program [12]. Multiple NCPs are connected with the bracket that fixes the LAC, constituting a redundant actuated parallel PAM. Compared with the ordinary parallel mechanism, the RAPM has higher stiffness [13,14], greater bearing capacity [15], and other characteristics. Nahon [16], Kumar [17], Kokkinis [18], and other scholars pointed out that the RAPM can reduce the internal force of the mechanism, improve the force transmission capacity, and optimize the energy consumption by adjusting the driving force of the joint. Choudhury and Ghosal pointed out that the redundant drive can improve the stability of the mechanism [19] and improve the force transmission capability and the motion accuracy of the platform. Therefore, adjusting the posture of LAC by RAPM can improve the accuracy and stability of posture adjustment of LAC. Therefore, the PAM for posture adjustment of LAC is mainly RAPM.
Currently, there are two major problems during the control of redundant actuated parallel PAM: the distribution of the driving force of RAPM and the control strategy of RAPM. The traditional control method of redundant actuated parallel PAM based on multi NCPs is the FPC method. The FPC method is to plan the posture adjustment trajectory of LAC according to the posture adjustment task and calculate the displacement curves of each axis of the NCP according to inverse kinematics. The obtained displacement curves of each axis are used as the input of the position controller to drive the motor to realize the coordinated movement of multiple NCPs [21]. However, the motion error of each axis always exists in the actual posture adjustment process, which leads to the position error of the ball joint center (BJC) [22]. If active position control is applied to all axes of the NCPs, it may cause interaction between the NCPs [23]. Defined the maximum interaction force between the NCPs as the internal force of posture adjustment. The larger internal force of posture adjustment will deform and even damage the bracket and the NCPs. Therefore, Zhu et al. proposed to cancel the external drive of some axes of the NCP and take them as the follower axes to avoid the internal force of posture adjustment [24]. However, this posture adjustment control method will make the PAM become an ordinary parallel mechanism and reduce the stiffness of the PAM. And the bracket will be used to transfer the force that the passive axis needs, which will still cause the bracket to be subjected to additional tension or compression.
Aiming at the problem that the drive axes of the RAPM interfere with each other, Tao and Luh controlled the position and the force of the redundant robot simultaneously [25]. They achieved high tracking performance of the position and the force when the kinematics model was uncertain. Kim added multiple active joints to the non-redundant parallel robot to construct a redundant parallel robot and used the independent control signals to control the redundant branches of the mechanism, which not only retains the good kinematic and static properties of the mechanism but also increases the effective workspace of the mechanism [26]. Tang et al. proposed a hybrid control strategy for 5-DOF redundant drive parallel machine tools and used force servo control for redundant branches, which improved the stability and reliability of the machine tool motion [27]. Procházka et al. used force control for the redundant axis of the RAPM with kinematic parameter error and used position control for the other axes, which improved the performance of the mechanism without mutual interference of the drive axes [28]. These studies show that adopting control methods different from other branches for the redundant branches of the parallel mechanism cannot only avoid the mutual constraints between drive axes but also improve the kinematic or dynamic performance of the mechanism.
The number of drive axes of the redundant actuated PAM is more than the degree of freedom (DOF) of the mechanism. Hence, the driving force distribution of the mechanism under the same posture adjustment trajectory is uncertain [29]. Since the LAC are connected with the NCP through the ball joint, the contact force of the ball joint is equal to the driving force of the NCP on the LAC. Chu and Huang used a PID controller to control the contact force of the ball joint in the hybrid control of posture adjustment system for LAC [30]. The experimental results show that this control method can effectively reduce the internal force of posture adjustment. However, the kinematics model of the PAM has not been constructed in this control method, and the optimal contact force of the ball joint is constantly changing. Controlling the contact force of the ball joint to zero may lead to a large interaction force between the NCPs at a specific time during the posture adjustment process. Zhang et al. obtained the minimum 2-norm solution of the driving force of the NCP by constructing the generalized inverse matrix [31]. The smaller driving force can improve the control accuracy and motion stability of the system. However, under this driving force distribution mode, there may still be a large interaction between NCPs [32]. Therefore, this paper uses the Newton-Euler method to build the kinematics model of the PAM, and solves the optimal driving force of the posture adjustment of LAC with the constraint conditions of the interaction between NCPs, and then controls the contact force of the ball joint in real-time by force servo control, so that the posture adjustment system has the minimum posture adjustment internal force during the whole posture adjustment process.
The rest of the paper is organized as follows. "Section 2" introduces the basic principle of posture adjustment of LAC. "Section 3" proposes a driving force distribution method of the redundant actuated parallel PAM without internal force by establishing the kinematics model of the PAM. "Section 4" analyzes the coupling relationship between the drive axes of the PAM and points out that the critical reason for the internal force of the posture adjustment is that the number of drive axes of the PAM exceeds the DOF of the mechanism, and then studies the optimization strategy of the position control axis based on the condition number of the Jacobian matrix of the PAM to improve the kinematic performance. "Section 5" carries out the posture adjustment experiments of LAC to study the interaction force between NCPs and the posture adjustment accuracy during the actual posture adjustment process.

Posture adjustment principle of LAC
In the wing-fuselage docking system shown in Fig. 1, the fuselage has been fixed at the predetermined position, and the wing is connected with three NCPs through a ball joint composed of ball head and ball socket [33]. The three-dimensional force sensor (TDFS) is installed below the ball joint to measure the contact force of the ball joint and serves as the connecting part between the ball joint and the NCP body [6]. In the figure, the coordinate system O w -xyz is the world coordinate system (WCS) of the whole posture adjustment system, and other coordinate systems are expressed in this coordinate system. The local coordinate system (LCS) O f -xyz is the consolidated coordinate system of the LAC (wing), which is determined by the theoretical design model of the aircraft.
It is assumed that the LCS O l -xyz and the WCS O w -xyz are coincident initially. The LAC coordinate system O l -xyz first rotates γ angle around the Z axis of the coordinate system O g -xyz, and then rotates β angle around the Y axis of the coordinate system O w -xyz, and then rotates α angle around the X axis of the coordinate system O w -xyz, and finally translates the distance (x, y, z) along the XYZ axis of the coordinate system O w -xyz to obtain the LCS O l -xyz. According to the principle of rigid body rotation [34], the rotation matrix from the coordinate system O w -xyz to the LAC coordinate system O l -xyz can be expressed as:

Kinematic model of PAM
The distribution of the driving force of the RAPM is uncertain. The unreasonable driving force distribution method may cause the NCP to produce a large driving force, which may exceed the motor load and affect the stability of the system and may also cause a large interaction between the NCPs. Therefore, it is necessary to build the kinematic modeling of the PAM and study the relationship between the driving force of the NCP and the motion trajectory of the PAM. According to Newton's law, for the PAM composed of three NCPs, the motion equation of the LAC in the WCS is: where w i is the force exerted by the NCP on the LAC under the WCS, m = [00 − G] T is the gravity matrix of the LAC (including brackets), and w l is the acceleration of the LAC. According to the Euler equation, the dynamic equation of the LAC in the LCS is: i is the force on ball head in the LSC, l m is the barycenter position of the LAC, l is the inertia tensor matrix of the LAC, l = ̇ is the angular velocity of the LAC relative to its own coordinate system, and ̇ l =̇ ̇ + ̈ is the angular acceleration [36].

Modeling and solution of NCP driving force
The driving force of the NCP is not only used to drive the motion of the LAC but also used to overcome its friction and inertia force. According to Newton's second law, the motion equation of the NCPi is: In Eq. (6), p i is the driving force of the NCPi, i is the friction coefficient matrix, N,i is the load pressure matrix, i is the inertia coefficient matrix, and is the acceleration matrix. Their specific expressions are as follows: Substitute Eq. (6) into Eqs. (3) and (4), and the following equations can be obtained.
Transform Eq. (12) into the form of matrix multiplication. where: In Eq. (14), 3×3 is the third-order identity matrix and ̂ l i is the anti-skew symmetric matrix of l i , which can be written as follows: Equation (13) is an indefinite equation with countless solutions. That is, the driving force solution of the PAM is uncertain. The general form of the solution of Eq. (13) is: ized inverse matrix of W and ζ is any 9-order column vector. When = 0 , the minimum 2-norm solution of the driving force can be obtained. However, there is still an interaction force between NCPs under this condition. Therefore, the additional constraint equation of driving force is constructed according to the interaction between NCPs. Let F ij represent the interaction force between NCPi and NCPj, and the following equation can be obtained [37].
Let F ij = 0 and expand it.
Simultaneous Eqs. (13) and (20), and the driving force solution without internal force can be solved.

Analysis of coupling relationship between drive axes
It can be seen from Eq. (2) that when the posture and position of LAC change, the displacement variation T of NCPi can be expressed as: where Δ w l is the posture matrix variation of LAC, Δ l i is the position variation of BJCi in the LCS, also known as the deformation of BJCi, and Δ w l = ΔxΔyΔz T is the position variation of LAC. Let Δ = [Δ Δ Δ ] T represent the posture angle variation of the LAC around its own coordinate system, w l ′ represent the posture matrix after rotation, and w l represent the posture matrix of the LAC relative to its own coordinate system. Therefore, the posture matrix variation of LAC can also be expressed as: When Δ approaches to infinitesimal, Eq. (23) can be obtained: Substitute Eq. (23) into Eq. (21), and the following equation can be obtained.
of each axis of the NCP and the posture and position variation of the LAC can be established.
, and J is the Jacobian matrix of the PAM, which can be expressed as: If the bracket is not deformed, the deformation variations of the BJCs are zero, and the following equation can be obtained.
Since the dimension of vector Δ is greater than 6, Eq. (27) is over-determined. The necessary and sufficient condition for Eq. (27) to have a unique solution is R( ) = R( , Δ ) . At this time, the displacement Δ of the NCPs is linearly related to the Jacobian matrix J. Therefore, in order to prevent the deformation of the bracket, the displacement variation Δ w i of each NCP needs to meet certain constraints. The deviation between the actual displacement of the NCP and the theoretical displacement can be greatly reduced by calibrating the PAM. However, under the existing technical conditions, it is difficult to eliminate the deviation. Therefore, the traditional FPC method leads the bracket to deform easily, resulting in a large interaction between the NCPs, affecting the assembly quality of LAC.
To sum up, the root cause that generates posture adjustment internal force is that the number of drive axes of the PAM exceeds the degree of freedom of the PAM terminal. If only parts of the axes of the PAM are position controlled to adjust the posture and position of LAC, and the other axes are controlled by force servo to release excessive constraints. This hybrid control method can reduce the posture adjustment internal force caused by the uncoordinated movement during the posture adjustment process.

Construction and simulation of NCP control model
When the number of the position control axes exceeds the DOF of the PAM terminal, the driving axes will inevitably interfere with each other, causing deformation or damage to the mechanism [38]. When the number of position control axes of the PAM is less than its DOF, the mechanism's motion is uncertain. Therefore, the basic principle for selecting the position control axis of the PAM is that the number of selected position control axes should be equal to the DOF of the mechanism. For the PAM composed of three NCPs, in order to realize the complete control of the six DOFs of the terminal, six axes should be selected as the position control axes, and three axes should be selected as the force control axes. The structural block diagram of the hybrid control system of the PAM is shown in Fig. 2.
The position control axis is the active axis, which is controlled according to the pre-planned posture adjustment trajectory to realize the posture adjustment and positioning of the LAC. The force control axis is the redundant axis, which is controlled by the force servo to avoid mutual interference between the drive axes. The principle of force servo control is to compensate the displacement output by the NCP according to the ball joint contact force measured by the TDFS, so that the contact force can follow the expected value [39]. Take the X axis as an example, the control block diagram of the force control axis is shown in Fig. 3, where F x c represents the ball joint contact force in the direction of the force control axis measured by the TDFS, X d represents the theoretical trajectory of the force control axis planned by the posture adjustment system, X e represents the trajectory of the ball head, X o represents the trajectory output by the NCP, and F x r represents the optimum contact force of the ball joint. In order to analyze the ability of the force control axis to follow the optimal contact force during the posture adjustment process, it is necessary to build a model of the force control axis and conduct simulation research. The NCP has three DOFs in X\Y\Z directions and can realize linkage control in three directions. The NCP mainly consists of a pedestal, an X direction slider, a Y direction slider, a telescopic column, a TDFS, and a ball socket, as shown in Fig. 4.
The transmission principle of the X-direction slider is shown in Fig. 5, mainly composed of a servo motor, a ball screw nut pair, a linear guide rail, and a sliding platform, where x represents the angular displacement of servo motor, K x 1 represents the torsional stiffness of the lead screw, B x 1 represents the rotational damping, L x represents the lead of the screw, K x 2 represents the linear stiffness of the linear guide rail, B x 2 represents the sliding damping, and M x represents the load inertia. The transmission principle of the Y-direction slider is the same as that of the X-direction slider. The parameters of the X-direction slider of the NCP are shown in Table 1.
In order to obtain the contact force between the ball head and the ball socket in the simulation, it is necessary to simplify the contact force model of the ball joint (X-direction force control axis), as shown in Fig. 6. When X e = X o , the output position of the NCP coincides with the position of the ball head center, and the contact force is zero. When X e ≠ X o , the contact force is not zero, and the NCP is deformed. Let K x e represent the stiffness of NCP in the X-direction, and the X-direction contact force of the ball joint is:  For the simulation of the force control axis, the response speed and steady-state error to the optimum contact force are investigated first. Establish the force control axis simulation model as shown in Fig. 7 in Simulink. Input the unit step signal at the optimum contact force port and use PID tuner to adjust the PID parameters, so as to increase the response speed of the system and reduce the overshoot and steadystate error of the system. The unit step signal response result of the optimum contact force after PID parameter tuning is shown in Fig. 8.
The simulation results show that the overshoot is 6.8%, the adjustment time is about 0.13 s, and the final steady-state error is 1. Therefore, the designed force control system is stable, without vibration, and has a fast response speed.
Then, the tracking effect of the force control axis to position and the contact force error are studied by simulation. Since the position of the ball joint and the optimum contact force are constantly changing during the posture adjustment process. Let X e = 50sin(0.01 × t − ∕2) + 50 represent the trajectory signal and F r x = 20sin(0.1 × t) represent the optimum contact force signal, and the simulation time is 100 s. The displacement curve and displacement error curve of the force control axis are shown in Fig. 9. The output displacement of the force control axis shows a good tracking effect, and the maximum displacement error is only 0.0072 mm. The accuracy of the output position of the force control axis meets the requirements and can accurately follow the displacement of the ball head.
The contact force curve and contact force error curve of the force control axis are shown in Fig. 10. The contact force error changes smoothly, and the maximum contact force error is less than 6 N, which can meet the requirements of posture adjustment. Step response of force control axle

Selection principle of position control axis
According to the basic principles of mechanism, if the number of the largest linear independent groups of the Jacobian matrix of the spatial parallel mechanisms is less than the number of position control axes, there will be interference between the position control axes, and the selection of position control axes is unreasonable. According to Eq. (24), the Jacobian vectors corresponding to the X\Y\Z axis of NCPi are i,x = 1000z w i − y w i , i,y = 000z w i 0x w i , and i,z = 001y w i − x w i 0 , respectively. Since the Z-direction of the NCP needs to bear the gravity of LAC, the Z axis of each NCP is selected as the position control axis. The Jacobian matrix corresponding to the Z axes of the three NCPs is: The rank of 3 z is 3, and the 6 DOFs of the PAM terminal cannot be constrained, so the other three axes of the PAM need to be selected as the position control axes. When the X axes of three NCPs are selected as the position control axes, the Jacobian matrix of the PAM is: The rank of 3,3 z,x is less than or equal to 5, which cannot limit the 6 DOFs of the terminal. Similarly, three Y axes cannot be selected as the position control axes. Therefore, two X axes and one Y axis, or two Y axes and one X axis  should be selected as the position control axes. If two X axes (e.g., X axis of NCP 1 and X axis of NCP 2) and one Y axis (e.g., Y axis of NCP 1) are selected as the position control axes, the Jacobian matrix of the PAM is: The necessary condition for the reasonable selection of the position control axes is that the rank of the Jacobian matrix of the PAM is 6, so the determinant of the Jacobian matrix is not equal to zero.
The solution is y w 1 − y w 2 ≠ 0 . If two Y axes (e.g., Y axis of NCP 1 and Y axis of NCP 2) and one X axis (e.g., X axis of NCP 1) are selected as the position control axes, the Jacobian matrix of the PAM is:  To sum up, it can be seen that the principle of selecting the position control axis of the PAM is that the three Z axes of NCPs should be selected as the position control axes, but three X axes or three Y axes cannot be selected as the position control axes simultaneously. If two X axes are selected as position control axes, the Y coordinate values of the selected NCP cannot be equal. Similarly, if two Y axes are selected as position control axes, the X coordinate values of the selected NCP cannot be equal.

Optimization strategy of position control axis
According to the position control axis selection theory, three axes from the six horizontal axes of the NCPs should be selected as the position control axes. For the PAM shown in Fig. 1, there are 16 reasonable combinations of position control axis, as shown in Table 2, where 1X1Y2X means that the X axis of NCP 1, Y axis of NCP 1, and X axis of NCP 2 are selected as the position control axes, and other combinations are similar. Since the force control axis is mainly used to track the contact force, the kinematic performance of the PAM is mainly determined by the position control axis.
The kinematic performance of the mechanism can be studied by the condition number of the matrix. For the linear equations = , the condition number of coefficient matrix A is expressed by cond( ) , which reflects the sensitivity of solution vector x to the error of constant vector b. The greater the condition number, the stronger the sensitivity. For a general matrix A, its condition number is equal to the product of the 2 norm of A and the 2 norm of the inverse of A.
where max ( ) is the maximum singular value of matrix A and min ( ) is the minimum singular value of matrix A, therefore cond( ) ≥ 1.
Let Δ denote the theoretical input of the PAM, denote the input error, Δ denote the output of the PAM terminal, and denote the output error. According to Eq. (27), the following equation can be obtained.
Since Δ = Δ , = , according to the properties of the norm, the following formula can be obtained.
Since the Jacobian matrix J is reversible.  Similarly, the following formula can be obtained according to the properties of the norm.
where ‖ ‖ ‖Δ ‖ is the input error rate of the PAM, and ‖ ‖ ‖Δ ‖ is the output error rate of the PAM. The smaller the Jacobian matrix condition number cond( ) , the larger the lower bound and the smaller the upper bound of the output error of the mechanism, the more stable the system is, and the PAM will have good kinematic performance. The larger the Jacobian matrix condition number cond( ) , the smaller the lower bound and the larger the upper bound of the output error of the mechanism, the more uncontrollable the system is, and the PAM will not have good kinematic performance. When the condition number cond( ) tends to infinity, the mechanism will be in a singular configuration. If the position control axis is not selected according to the position control selection principle described in "Sect. 4.3," the Jacobian matrix J of the mechanism may be singular, and the PAM will not be controllable. Therefore, when selecting the position axis of the PAM, the condition number of the Jacobian matrix of the mechanism should be as small as possible.
The first three columns of Jacobian matrix J of the PAM are the velocity Jacobian matrix v , and the last three columns are the angular velocity Jacobian matrix , so Jacobian matrix J can be written as follows: For the combinations of position control axis in Table 2, the velocity Jacobian matrices v are the same, and the condition numbers are the same too, so only the condition numbers of angular velocity Jacobian matrix need to be considered. Since NCP 1 and NCP 2 are symmetrically distributed, 8 groups of combinations (from No. 1 to No. 8) are selected for condition number simulation calculation. The simulation parameter values are shown in Table 3, and the simulation results are shown in Fig. 11.
It can be found from Fig. 11 that the position control axis combination with the minimum condition number of Jacobian matrix is constantly changing during the process of posture adjustment. Therefore, the position control axis combination should be corrected in time according to the Jacobian matrix conditions, so that the PAM will always have the best kinematic performance and adjustment accuracy.

Posture adjustment experiment of LAC
To further evaluate the feasibility of the posture adjustment method of LAC proposed in this paper for reducing the internal force of posture adjustment and improving the accuracy of posture adjustment, a posture adjustment system of simulated LAC is built in the laboratory, and a series of the posture adjustment experiments are carried out. The experimental posture adjustment system mainly includes three NCPs, simulated LAC (wing and fuselage), servo control system, laser tracker, etc., as shown in Fig. 12.
The stroke of the NCP is (300, 200, 150) mm, the absolute positioning accuracy is (8,8,8) μm, and the maximum load of single NCP is 300 N. The laser tracker used is Leica AT901, and the measurement uncertainty of the angle sensor is ( ) = ( ) = 2 ε , and the measurement uncertainty of the distance sensor is (r) = 10μ + 5μ/m. The simulated wing is equipped with 6 measuring points, and the theoretical coordinates of these MPs in the LCS are shown in Table 4. The posture and position of the simulated wing can be calculated by measuring the coordinates of these MPs with the laser tracker. The servo control system is composed of B&R industrial computer, servo driver, servo motor, analog-to-digital (AD) module, TDFS, cable, etc. The topology of the servo control system is shown in Fig. 13. The B&R industrial computer is used to process the instructions of the upper computer and carry out the bottom control of the servo system. The servo driver is used for the high-precision closed-loop control of the motor. The servo motor is used to drive the NCP to move according to the preset command. The TDFS is used to measure the contact force of ball joint, the effective range of the force sensor is 500 N, the sensitivity is 0.001, and the accuracy is 0.3%. The AD module is used to convert the analog signal measured by the TDFS into the digital signal that can be recognized by the industrial computer. The conversion accuracy of the AD module is 16 bit, and the conversion time is 40 μs. The cable is used for communication and power transmission between modules.

Interaction force between NCPs
Firstly, the interaction force between NCPs when using the optimal contact force control (OFC) method is investigated through experiments and compared with the FPC method, the zero contact force control (ZFC) method, and the fol-  Table 5.
After the posture adjustment is completed, the interaction force between NCPs can be calculated by Eq. (19) according to the contact force datum of ball joints. The maximum interaction force between NCPs under different posture adjustment control methods is shown in Fig. 14.
It can be found from Fig. 14 that due to the motion error of the PAM, the FPC method will lead to a large interaction force between the NCPs, and the maximum interaction force can reach 93.6 N. The larger internal force of posture adjustment may cause the deformation of NCP and bracket, and affect the service life of the equipment. The FAC method takes some axes as the follower axes to avoid mutual interference between the drive axes. Compared with the FPC method, the interaction force between the NCPs can be significantly reduced, and the average interaction force is reduced from 73.2 to 34.6 N. However, the interaction force between NCPs is still large, which is mainly due to the friction of the follower axis, and the bracket needs to transmit the driving force that the follower axis needs. The ZFC method avoids transmitting the driving force that the follower axis needs through the bracket, and the interaction force between the NCPs is further reduced. The OFC method can theoretically convert the force that NCPs output into the driving force for LAC movement completely. However, due to the kinematic model error of the PAM and the force servo control error, the interaction force between NCPs still exists. However, compared with other control methods, the interaction force of this control method is the smallest, and the average value is only 10.2 N. The OFC method can ensure that the internal force of posture adjustment will not lead to the deformation of NCP and bracket.

Posture adjustment accuracy of LAC
Next, the influence of different combinations of position control axis on posture adjustment accuracy will be verified by experiments. In these experiments, three position control axis combinations are adopted. The first combination is the position control axes with the minimum Jacobian condition number of the PAM when the LAC is in the theoretical target posture and position. In fact, since the position control axis of the FAC method cannot be switched during the posture adjustment process, this position control axis selection method is usually adopted in the FAC method. The second combination is the position control axes with the minimum average Jacobian condition number A cn . The calculation method of the average Jacobian condition number is shown in Eq. (41). The third combination is that the control system automatically adjusts the position control axes according to the position control axes optimization strategy proposed in "Section 4.4," so that the PAM always has the minimum Jacobian condition number.
The posture adjustment accuracy is mainly reflected by the position deviation Δr = √ Δx 2 + Δy 2 + Δz 2 of the measuring point (MP), where Δx , Δy , and Δz denote the position deviation of the MP in the X\Y\Z directions, respectively. According to the principle of coordinate system transformation and the target posture and position of the LAC in Table 5, the theoretical target coordinates of the MPs on the LAC can be calculated, as shown in Table 6.
After completing the posture adjustment, the MPs are measured by the laser tracker, and the position deviations of the MPs under different position control axis combinations are shown in Fig. 15.
It can be found from Fig. 15 that when the first position control axis combination is adopted for posture adjustment, the position deviations of the MPs are large, and the average value is 2.37 mm. Generally, the position deviation of the MP after the posture adjustment is required to be less than 1 mm, so this position control axis selection method cannot meet the accuracy requirements of the docking of the LAC. In the FAC method with this position control axis combination, the large internal force of posture adjustment will also cause the deformation of the bracket or NCPs, so the posture adjustment  Fig. 15 Position deviations of the MPs accuracy of the FAC method usually cannot meet the requirements of LAC docking. When the position control axis combination with the minimum average condition number A cn is used for posture adjustment, the position deviation of the MP is reduced to 1.60 mm, which still cannot meet the accuracy requirements of LAC docking. Therefore, the posture adjustment accuracy of the above two fixed position control axis combinations cannot meet the requirements of LAC docking. When these fixed position control axis combinations are used for posture adjustment, in order to make the posture adjustment accuracy meet the requirements, it is often necessary to repeatedly adjust the position and posture of the LAC, which will undoubtedly reduce the efficiency of aircraft large parts docking. When the position control axis combination is adjusted dynamically according to the Jacobian condition number of the PAM, the average position deviation of the MPs has greatly reduced to 0.59 mm, and a single posture adjustment can meet the accuracy requirements of the docking.

Conclusion
In the digital docking system of LAC, the PAM composed of multiple NCPs is RAPM. Due to the motion error of the NCP, the traditional FPC method may lead to interference between the NCPs, resulting in the deformation of the NCP and the bracket, affecting the service life of the equipment. Aiming at these problems, this paper divides the motion axes of the NCPs into the position control axis and the force control axis and proposes a posture adjustment method based on hybrid control to avoid the posture adjustment internal force caused by the coordinated motion error of the NCP. Since the driving force distribution of RAPM is uncertain, a driving force distribution method based on the kinematic model of PAM is proposed, which further reduces the posture adjustment internal force. Then, the root causes of the internal force of posture adjustment are analyzed, and the general principle of position control axis selection is put forward. Finally, an optimization strategy of the position control axis based on the condition number of the Jacobian matrix is studied to improve the kinematic performance and accuracy of PAM.
To verify the effectiveness of the posture adjustment control method proposed in this paper for reducing the internal force of posture adjustment and improving the accuracy of posture adjustment, a simulated LAC docking experimental platform is built to carry out the posture adjustment experiment. The experimental results show that compared with the FPC method and the FAC method, the hybrid control method based on optimal contact can significantly reduce the interaction force between NCP, and the maximum interaction force in the experiment is not more than 14.3 N. Compared with ZFC method, the interaction force between NCPs of the OFC method is also lower, which further proves the rationality of the hybrid control method based on the optimal contact force. By measuring the MPs on the LAC, it can be found that the position control axis optimization strategy based on the condition number of the Jacobian matrix of the PAM can make the PAM always have high motion performance, which can ensure that a single posture adjustment can meet the accuracy requirements of the LAC docking.
Author contribution Wenmin Chu: methodology, simulation, data curation, analysis, experiment approach, writing; Gen Li: validation, supervision, review and editing; Shuanggao Li: experiment platform, editing the manuscript; Xiang Huang: providing the problem and the industrial requirements, project administration.

Declarations
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Conflict of interest
The authors declare no competing interests.