Research and experiment of repairable space telescope interface system

Space technology is the core competitiveness of a powerful country. Space maintenance technology has just started in our country. The biggest innovation of this paper is to design a set of interface system, which is suitable for on-orbit maintenance of space science instruments. It represents that this technology from scratch in our country. A new measuring method is proposed. According to the homogeneous coordinate transformation, the mathematical models of the relationships between fit clearances of interface system and the six-degree of freedom change of the maintenance module are derived. According to the index requirements, the corresponding fit clearance and repeat positioning accuracy are determined. Set up an experimental platform, and use autocollimator and laser tracker to measure the repeated positioning accuracy of the module. Based on the theory of small displacement, the least square method is used to deduce the rotation angle and translation amount of the module through the change of the characteristic point coordinates. The experiment proves that the interface meets the requirement of on-orbit maintainable operation, and the experimental data proves that the theoretical equation is correct. Providing engineering reference for other space maintenance scientific instruments.


I. INTRODUCTION
In a complex space environment, all space science instruments may at any time affect their value due to failure, degradation or obsolescence. And most of the spacecraft on-orbit fault is usually only a problem with individual component. These problems will cause the space science instrument function decline, or even lead to the loss of the entire spacecraft function, forming a huge waste. Re-launching spacecraft to replace them would only lead to increased costs, but space maintenance and service technologies could effectively address this problem. On-orbit maintenance can quickly restore its functions, which has good timeliness and economy. Secondly, the reliability of the whole machine can be effectively improved by replacing the vulnerable and consumable parts [1].
NASA astronauts repaired the skylab solar windsurfing board on May 26, 1975, which was the first repair operation carried out by mankind in space [2]. The on-orbit capture of the SMM probe in April 1984 was the first time in human history to repair a satellite in space orbit [3]. The MIR space station launched by the Soviet Union in 1986 was the first modular space station assembled on-orbit in human history [4]. Typical on-orbit maintenance tasks include Japan ' s ETS-VII ( Engineering Test Satellite-VII ) ; America's Orbital Express; German Robot Technology Experiment [5]- [7]. And the most representative one is the HST launched in 1990. During the more than ten years from 1993 to 2009, the Hubble Telescope accepted five on-orbit service tasks. Including repairs, replacement of faulty parts, equipment upgrades and installation of new measuring elements. Hubble's detection capability has been increased by 70 times and its life span has been extended through on-orbit maintenance, contributing endless power to scientific research [8]- [12].
Before and after the replacement of the back-end module of the space telescope, the stability of the optical focal plane directly determines the imaging quality, and the stability of the focal plane is closely related to the spatial layout of the back-end module interface. Chen Xiqu studied the effect of substrate temperature on the focal plane array [13]; Zhang Yong studied the adjustable focal length and controllable focal plane of a liquid crystal lens driven by a double electrode [14]; Li Jin studied the focal plane of the collimator for geometric calibration of optical cameras [15]. The clearance of the interface mechanism determines the repeated positioning accuracy of the back-end module, which must be reasonably allocated. A large number of scholars have studied the clearance: Ravindra Arora studied the influence of electrical clearance in air for overhead traction system [16]; Xu Minmin studied the influence of rolling bearing radial clearance on the state of wind turbine [17]; Gu Yingkui studied the effect of bearing clearance on dynamic characteristics of a spur gear system [18]; Teng Fei studied the effect of variation mechanism of three-dimensional blade tip clearance on aero-engine [19].
This paper takes the on-orbit maintenance of the space telescope as the research object, and a set of interface mechanism is designed in detail to meet the requirements of on orbit thermal stress release according to the kinematics principle. According to the homogeneous coordinate transformation, the theoretical equations of rotation angle and fit clearance are deduced, and the best fit clearance satisfying the repeated positioning accuracy is given. The space microgravity environment is simulated, and the measured rotation angle and translation by the autocollimator and laser tracker. The experimental data shows that the theoretical equation is correct, the clearance is reasonable, and the index requirements are met.
The paper is organized as follows: Section 2 introduces the positioning principle and spatial layout of the interface system. According to the principle of focal plane stability, the best spatial distribution of the interface is given. Section 3 introduces the mathematical model of repeated positioning accuracy and distributes the interface clearance reasonably according to the requirement of the index. Section 4 introduces the internal composition and detailed design of the interface system. Section 5 introduces the experimental method of repeatable positioning accuracy of the back-end module and analyzes the experimental data. Section 6 draws conclusion.

A. Positioning principle of interface
The two mechanical modules that need to be connected to the maintainable interface have separable ability. The separated two mechanical modules have good reset ability through the connectable interface and can maintain good position stability. The interface structure cannot have problems such as over-constrained or under-constrained. After the space telescope is put into orbit, the deformation between the optical system and the supporting structure is greater than the allowable deformation of the optical system due to gravity unloading, stress release, thermal deformation and other factors. In order to ensure the stability of the focal plane, a method must be adopted to isolate the optical system to avoid the influence of unknown stress. The idea of kinematic connection comes from multi-degree-of-freedom robot, which can release the thermal stress well and maintain high stiffness and pointing accuracy by removing the degree of freedom constraint of deformation direction. The calculation equation of spatial degree of freedom is (1).
In (1), f is the degree of freedom of mechanism, n is the number of components, j is the number of kinematic pairs, i p is the degree of freedom of the i kinematic pair. As shown in Fig. 1, the interface mechanism adopts three points support. Interface A restricts the three translation of the module in space and releases three rotational degrees of freedom in space. Interface B restricts the translation of the module along the Y and Z directions, releasing the other four degrees of freedom. Support point A and B together restrict the rotation of the module around the X and Z directions. Interface C restricts the translation of the module along the Z direction and releases the other 5 degrees of freedom. The three points ABC together restrict the rotation of the module around the Y direction. In this system, the components include optical instrument and installation platform. The number of components is 2 and the number of kinematic pairs is 3, that is, n is 2 and j is 3. The degrees of freedom of each kinematic pair are 3, 4 and 5 respectively. And (1) is used to solve for f as 0. The three support points of ABC jointly realize the statically determinate support of the module [20]. The principle of releasing thermal stress on-orbit：Since the degrees of freedom of the three translation directions of point A are completely limited, when the space temperature changes, the module will deform uniformly along the XY directions. Point B moves along the axial direction and point C moves on the XY plane. The entire module will only produce rotational deformation around point A on the XY plane. The thermal stress and thermal deformation caused by the temperature are released through the translational movement of the module. The interface structure is designed using kinematics principles, which is suitable for installation of space science instruments with drastic temperature changes, ensuring the long-term normal operation of the whole machine on-orbit.

B. Space layout of installation interface
The plane of three points ABC is parallel to the focal plane, and the center of the three points coincides with the projection of the center of the focal plane in the direction of the optical axis. Point C is located at the front of the module, two points AB are located on either side of the module. The projection of the three points ABC on the XY plane is shown in Fig. 3   As shown in the Fig. 3, the coordinate system is established with the origin of point A, the Z axis is perpendicular to the paper and outward. Only knows that AB is located on both sides of the module and C is located inside the module. Table 1 below is the spatial coordinates of the three points ABC.
Among them, the unknown variable is Y X Y B ,C ,C , their variation is 0 L  . The center of the triangle formed by ABC three points should coincide with the center of the square whose side length is L .Therefore, we have the following (2).
In the Fig. 3, O is the centroid of the triangle formed by three points, O1 is the centroid of the three points after the temperature change, where the change of the centroid is the pointing accuracy of the focal plane of the back-end module, which is (4).
In order to obtain the highest pointing accuracy under the same temperature change, (4) is the smallest, we obtain:

A. Pose transformation matrix
The translation and rotation angle of the back-end module after the replacement are ( , , , , , ) The rigid body fixed coordinate system and the reference coordinate system coincide before the change. It is assumed that the position change sequence of the rigid body is rotated around the X-axis, Y-axis, and Z-axis of the reference coordinate system in turn, and finally it is translated along the three axes. The fixed coordinate system of the rigid body becomes blue after rotating around the X-axis, becomes green after rotating around the Y-axis, and becomes pink after rotating around the Z-axis. The color of the fixed coordinate system is still pink after the translation change. The change process is shown in Fig. 4. The point P on the rigid body becomes P3 after three rotation transformations and then P4 after translation. The change from point P to P4 is realized through the change matrix F.
Solve the (6): cos cos cos sin sin sin cos cos sin cos sin sin sin cos sin sin sin cos cos sin sin cos sin cos sin sin cos cos cos When the rotation angle of the rigid body around the three axes are less than 30″, (6) can be simplified as (7).
At this time, the magnitude of the maximum difference between (6) and (7) is 10 -8 , which can be considered negligible. Proving 1 FF  . Therefore, the position transformation matrix of the rigid body is expressed as 1 F . Similarly, when the rotation angle of the module around each axis is less than 30″, the rotation order or translation order around XYZ axes can be changed at will. There are 24 different combinations of the form of the module change matrix F ,all of which can be approximated with 1 F Means.

B. Angle error analysis
According to formula F 1 , we believe that the theoretical rotation angle of the rigid body are ,, X Y Z R R R , but the real rotation angle of the rigid body after spatial change is expressed as The real rotation angle of the rigid body around each axis can be expressed as the spatial coordinate change of the other axis unit vectors. ,, i j k are the unit vector of the three axes in the reference coordinate system, and (9) represents ,, The three unit vectors change to ,, i j k    after spatial variation.
   (10) cos cos cos sin sin sin cos cos sin cos sin sin sin cos sin sin sin cos cos sin sin cos sin cos sin sin cos cos cos The direction of space vector remains unchanged after translation. In order to simplify the calculation, the translation of the rigid body is omitted. After the rigid body changes, the real rotation angle of X-direction can be expressed as the angle between YZ j of the projection of j onto the YZ plane and j ,or, the angle between YZ k of the projection of k onto the YZ plane and k . The real rotation angle of Y-direction can be expressed as the angle between XZ i of the projection of i onto the XZ plane and i ,or, the angle between XZ k of the projection of k onto the XZ plane and k .The real rotation angle of Z-direction can be expressed as the angle between XY i of the projection of i onto the XY plane and i ,or, the angle between XY j of the projection of j onto the XY plane and j .
When j is used to represent 1 sin sin sin cos cos , (sin cos ) (sin sin sin cos cos ) When the rotation angles of rigid body around three axes are close to zero, the above (11) can be simplified as 1 2 2 cos cos , (sin ) (cos cos ) When k is used to represent 1 cos cos , (cos cos ) (sin sin cos sin cos ) When the rotation angles of rigid body around three axes are close to zero, the above (13) can be simplified as 1 2 2 cos cos , (cos cos ) (sin ) When the rotation angles of rigid body around three axes are close to zero, (12) and (14) have the same expression. Therefore, the value of the rotation angle in the X-direction is unique.
When k is used to represent When the rotation angles of rigid body around three axes are close to zero, the above (17) can be simplified as When the rotation angles of rigid body around three axes are close to zero, (16) and (18) have the same expression. Therefore, the value of the rotation angle in the Y-direction is unique.
When i is used to represent 1 When the rotation angles of rigid body around three axes are close to zero, the above (20) can be simplified as 1 2 2 cos cos , (cos cos ) (sin ) When j is used to represent 1 sin sin sin cos cos , (sin sin sin cos cos ) (cos sin sin sin cos ) When the rotation angles of rigid body around three axes are close to zero, the above (22) can be simplified as ' 1 2 2 cos cos , (cos cos ) (sin ) When the rotation angles of rigid body around three axes are close to zero, (21) and (23) have the same expression. Therefore, the value of the rotation angle in the Z-direction is unique.
When the rotation angles of rigid body around three axes are close to zero, the rotation angle of a rigid body around each axis has the same form. Taking the X-direction as a reference, the difference between the real and theoretical rotation angles is expressed as We can see from (12) and (13) that there are only two variables, so X R  can be simplified to a function with only two variables. Suppose that two different variables have the same variation range, draw the following Fig. 5. When the rotation angle of the rigid body around each axis is less than30″, it can be seen that the maximum value of X R  is 0.00003″. Similarly, the rotation angle around YZ axis has the same conclusion. Therefore, the theoretical rotation angle of rigid body can nearly equal to the real rotation angle of rigid body. Similarly, when the rotation angle of the module around each axis is less than 30″, the rotation order or translation order around XYZ axes can be changed at will. The theoretical rotation angle of rigid body can replace the real rotation angle of rigid body.

C. Theoretical repeated positioning accuracy analysis
The spatial coordinates of each feature point on the rigid body in reference coordinate system are ( , , ) After the change, the spatial coordinates of the feature point in the reference coordinate system are ( , , ) According to the homogeneous coordinate transformation, (26) can be obtained.
The coordinate variation of the characteristic points on the rigid body is ( , , )

IV. Interface system design
The A-point mechanism adopts a typical ball head-ball socket structure, which is characterized by limiting the three-dimensional translational freedom of the ball in the socket and releasing the rotational freedom in all directions. And integrate the electric drive locking mechanism to form the A-point positioning locking point, where the locking is to deal with the huge impact of the rocket when it is launched. It is released automatically by the electric drive after on-orbit.
A-Point mechanism consists of positioning ball, ball and socket assembly, operating rod, locking mechanism, L plate, etc. The specific model is shown in Fig. 7-a: The connection between the L plate and the module are called the moving part, the connection between the ball-socket assembly and the module frame are called the fixed part. As shown in Fig. 7-a, the connection of A-point can be tightened and separated by the rotation of the screw thread. The ball-socket assembly is composed of the ball-socket base and the ball. Meanwhile, the ball can only rotate freely in the socket and cannot be detached. In order to enable the screw to be accurately imported into the ball socket, a polished rod is set between the ball head and the thread at the front end of the screw for easy import. When the two parts of the ball head and the ball socket are firmly connected, the ball can be closely attached to the ball socket by the screw to form a closed ball head-ball socket structure. In this way, the three-dimensional translational freedom of the ball in the sphere nest is limited and the freedom of rotation in each direction is released. The B-point mechanism is a typical shaft-hole structure, which can realize the restriction of two-dimensional translational freedom and releases the translation of the other dimension and three-dimensional rotational freedom. The shaft is inserted and pulled out with a rack and pinion structure to realize the lateral movement of the shaft, and finally realize the lateral positioning function when the module is fixed. The The C-point mechanism adopts a typical key-slot structure, which is used to limit the degree of freedom in one direction. It also integrates an electric drive locking mechanism. The C-point mechanism is composed of positioning shaft, positioning hole, eliminate gap components, locking mechanism, etc. As shown in Fig. 7-c： The ball head and the positioning hole are fitted with clearance to facilitate the shaft insertion and extraction. In addition, the clearance elimination and positioning function is realized by elastic steel ball assembly. The two sets of elastic steel ball components are distributed symmetrically, acting as a balanced force. The elastic steel ball is pre-tensioned by a compression spring with an elastic coefficient of 11N / mm . When the spring is compressed, two sets of pre-tension forces will axially restrain the positioning shaft.

A. The experimental principle
The plane formed by the three points ABC should be parallel to the direction of disassembly and assembly of the module. Point C is located at the front end of the module, and the first contact is achieved when the module is pushed to the positioning position. The positioning shaft installed on the module is the ball head, and the corresponding rectangular hole on the module frame. The axis of the positioning shaft is consistent with the direction of disassembly and assembly. Therefore, the positioning and disengagement of point C is directly realized by disassembling the module without additional operations. This is the main difference from points A and B. Point A is located on the side of the module, and cooperate after contact at point C. On the module is a ball and socket structure, and on the module frame is a combination of a positioning ball with a screw hole and a ball-socket seat. The two work together to make the module as a whole tighten. Point B is located on the other side of the module, the locating shaft of the ball head is installed on the module, and the circular hole is installed on the module frame. Since the axis of the B-point positioning shaft is perpendicular to the installation direction, the positioning shaft of B-point on the module is last pushed into the corresponding positioning hole. And at the same time, when the module exits, the positioning shaft of B-point must first exit along the axis direction.
The micro-gravity simulation ground test includes the falling tower method, which is implemented by building high towers on the ground or digging deep wells; Weightless airplane method, the plane flies in a parabolic attitude and uses acceleration to offset gravity; Liquid buoyancy balance gravity method, so that the test piece is immersed in water; Non-contact suspension method, which includes a variety of electromagnetic suspension, air suspension, acoustic suspension, electrostatic suspension, light suspension, particle suspension and so on. The hanging wire system is a test system that connects the experimental target and the gravity compensation device through the pulley block and the hanging wire to simulate the ground micro-gravity environment. Compared with other systems, this system is simple in structure and easy to implement. The experiment time can be adjusted freely to ensure sufficient and reliable data.
In order to verify the correctness of the theoretical repeat positioning accuracy, this article uses the following verification method. The coordinate system is established with the center of the interface A as the origin of coordinates, and the directions of the XYZ three coordinate axes are shown in Fig. 8. The models of the two autocollimators are Collapex AC300, the measurement accuracy is 0.015″. The layout of the autocollimator is shown in 2 and 3 in Fig. 8, 2 and 3 are matched with plane mirrors 4 and 5 on the module respectively. Autocollimator 2 measures the rotation angle of the module in the Y and Z directions, and autocollimator 3 measures the rotation angle of the module in the X direction. The model of laser tracker 1 is Leica AT960, the measurement accuracy is 2μm. The laser tracker measures the four target balls (6789) which on the same plane of the module, and calculates the six degrees of freedom of the module according to the coordinate changes of these four points measured by the laser tracker. At the same time, it is compared with the measurement results of the autocollimator to verify the measurement angle of the autocollimator.

FIGURE 8. Experimental schematic diagram
In order to obtain the repeated positioning accuracy of the measured objects, the least square method is used to fit the data measured by the laser tracker. The data processing of the laser tracker is as follows: Definition Equation (30) Find the partial derivatives of the rotation angle of objective function E and the partial derivatives of the translation of objective function E in turn, and make the partial derivatives equal to zero. We can get the system of linear equations with six variables. By solving the equations, we can get the rotation angle and translation of the rigid body.
Before the experiment, the coordinate system of the laser tracker is transformed. The global coordinate system of the laser tracker is converted according to the coordinates of known points on the workpiece, and the coordinates of the target ball to be measured by the back-end module are compensated at the same time. The compensation consists of two parts: In the Z direction, one part is the radius of the target ball and the other part is half of the thickness of the experimental piece, and the XY direction is only the radius of the target ball. Verify the facula stability of the two instruments in a laboratory environment. The laser tracker continuously records the spatial coordinates of four target balls for 30 minutes, and calculates the spatial coordinate changes of the four target balls. The maximum standard deviations of the coordinate changes of the four target balls in the XYZ directions are 0.38μm，0.43μm and 0.40μm. The two autocollimations were also continuously record for 30 minutes, and the rotation angular standard deviations of the module in the XYZ directions were calculated as 0.0051″, 0.0063″ and 0.0054″. The test results show that the facula of the two measuring instruments have high stability within 30 minutes, meeting the requirements of the precision measurement of the system, and the effect of facula drift is negligible. This experiment uses a hanging wire system to unload the gravity of the module. The hanging wire system includes fixed pulley, hanging wire, hanging frame, and counterweight. The fixed pulley is installed on the hanging frame, the hanging wire passes through the fixed pulley, one end is connected the module, and the other end is connected the counterweight with the same weight as the module. By adjusting the mass of the counterweight, the whole experimental device can finally realize microgravity simulation of the order of magnitude 0.001g. On-orbit maintenance feasibility test and repeated positioning measurement device are shown in Fig. 9.  Table 2 is the record result of 10 times plugging and unplugging the module measured by the autocollimator, and Table 3 is the record result of the four target balls measured by the laser tracker. Process the data recorded by the tracker according to (32). As shown in Fig. 9: The rotation angles of the module in XYZ directions measured by two kinds of instruments.  According to the Grubbs criterion, there is no gross error in the measured data. Based on the analysis of the experimental data, the rotation angle measured by the autocollimator is taken as the reference. The maximum error of the angle measured by the laser tracker relative to the data of the autocollimator in the three directions of XYZ is less than 11%(as shown in Fig. 11), which proves that the rotation angle measured by the two instruments is true and reliable.  Fig. 12-a). Both measurement results are better than the index .The repeat positioning accuracy of translation in the XYZ directions measured by the laser tracker is 9.3  μm, 9  μm and 10  μm (as shown in Fig. 12-b). During the experiment of the two instruments, the measured repeat positioning accuracy are slightly lower than the theoretical value in the XY directions, and slightly higher than the theoretical value in the Z direction. Analyzing the experimental data and taking the theoretical repeat positioning accuracy as the benchmark, it can be seen that the max error of the module repeat positioning accuracy measured by the two instruments relative to the theoretical repeat positioning accuracy is less than 12% (as shown in Fig. 12-c). However, there are some acceptable errors between experimental results and theoretical derivation due to errors in machining, installation, and personnel etc. Which is basically consistent with the theoretical results, to prove that the experimental data is true and reliable.

Ⅵ. Conclusion
In this paper, taking the on-orbit maintenance of the space telescope as the research object, referring to the kinematics principle, a detailed design of a set of interface mechanisms that can meet its on-orbit rapid disassembly and installation. The repeated positioning accuracy of the module is calculated according to the homogeneous coordinate transformation, a simplified change matrix is obtained. The relationship between the theoretical rotation angle and the fitting clearance is derived, and the fitting clearance of the mechanism is designed with the rotation angle index as a constraint. A method combining autocollimator and laser tracker is proposed to measure the repeat positioning accuracy of the module. The error between the angle repetition positioning accuracy of the module in XYZ directions measured by the autocollimator and theoretical value is 7.9%, 11.9%, 2.1%. The error between the angle repetition positioning accuracy of the module in XYZ directions measured by the laser tracker and theoretical value is 5.3%, 4.6%, 1.4%. The error between the translation repetition positioning accuracy of the module in XYZ directions measured by the laser tracker and theoretical value is 7%, 10%, 10%. The experimental results are basically consistent with the theoretical results, that is, the correctness of the theoretical results is verified. It is also verified that the laser tracker can measure the six degrees of freedom of a rigid body when the accuracy is allowed. It provides reference for other on-orbit repairable interface system.