Dynamic prediction and compensation of aerocraft assembly variation based on state space model

<jats:sec><jats:title content-type="abstract-heading">Purpose</jats:title><jats:p>– This paper aims to clarify the predicting and compensating method of aeroplane assembly. It proposes modeling the process of assembly. The paper aims to solve the precision assembly of aeroplane, which includes predicting the assembly variation and compensating the assembly errors.</jats:p></jats:sec><jats:sec><jats:title content-type="abstract-heading">Design/methodology/approach</jats:title><jats:p>– The paper opted for an exploratory study using the state space theory and small displacement torsor theory. The assembly variation propagation model is established. The experiment data are obtained by a real small aeroplane assembly process.</jats:p></jats:sec><jats:sec><jats:title content-type="abstract-heading">Findings</jats:title><jats:p>– The paper provides the predicting and compensating method for aeroplane assembly accuracy.</jats:p></jats:sec><jats:sec><jats:title content-type="abstract-heading">Originality/value</jats:title><jats:p>– This paper fulfils an identified need to study how the assembly variation propagates in the assembly process.</jats:p></jats:sec>

section_content: The inertial navigation systems (INSs) are widely used as a sort of modernized navigation devices in aircrafts in the fields like civil applications and, especially, in military aviation. 

As the measuring elements of INSs are directly installed on the carriers, their installed position and assembly variation have a large impact on the navigation sensitivity of the aircrafts as well as the navigation accuracy of the INSs. 

Nowadays, the structure of aircrafts is more complex than before, and the number of cabins is increasingly large. As for the multi-cabin aircrafts, the accumulative effects of variation are more obvious between INSs and the power system. The purpose of decreasing assembly angular variation is to increase the consistency, stability and reliability of the aircrafts, and consequently improve their whole performance. With the development of computer-aided tolerancing (CAT), some researchers have proposed several mathematic methods of variation modeling (Anselmetti et al., 2010; Ameta et al., 2011). Assembly modeling and error controlling (Huang et al., 2007a; Mantripragada and Whitney, 1998) of multi-stage process is becoming an emerging field based on statistical and engineering research, and it has already made great development in the past two decades. In earlier studies, Mantriparagada et al., who followed the datum flow chain (DFC) theory (Mantripragada and Whitney, 1999), proposed that mechanical assembly process can be divided into two categories: 1 those whose assembly variation is completely determined by the machining errors of parts; and 2 those whose assembly variation is determined by the machining errors of parts as well as the fixtures. 

The position and orientation of the fixtures can be adjusted online. Then, according to the characteristic of these two categories, they described the size transformation of the parts in assembly process by the state space model. Ceglarek et al. from the University of Wisconsin-Madison proposed the A stream-of-variation analysis (SOVA) model based on key characteristics (KCs). In this model, all the constrains can be expressed by the unified virtual fixture model. It described the linear relation between key product characteristic (KPC) and key control characteristic (KCC), which laid the foundation of complicated multi-stage assembly process modeling. Based on this, Jin et al. (Jin and Shi, 1999; Shiu et al., 1996; Shi, 2006; Ding et al., 2002; Soman, 1999) took advantage of the variation vectors to describe the process parameters and product feature errors, so that the state space model for multi-stage assembly process is established. This paper focuses on the assembly process of aeroplanes which includes cabins and INSs, and establishes the assembly process state space model for their assembly accuracy research. The rest of the paper continues as follows: Section 2 analyzed and modeled the assembly mating features by using small displacement torsors and matrix transitions. Section 3 established the state space model of the assembly process. In Section 4, an aircraft assembly process is analyzed as an example. In the example, the assembly variation is predicted and compensated. Section 5 is the conclusion.
section_name: Introduction

section_content: Cabin assembly is very common in engineering. It can be divided into large-scale cabin-parts assembly and small-scale cabin-parts assembly according to assembly mode. The most common method of assembling large-scale cabin-parts is through riveting or welding, for example rockets. The method in small-scale cabin-parts assembly, however, is bolting, for example small-scale aeroplanes and spacecraft. One of the most significant indexes in cabin-parts assembly is the angular variation. 

This paper presents a study of the angular variation between different small-scale cabin-parts of a spacecraft assembly.
section_name: Assembly mating features analyzing and modeling

section_content: To describe the position and orientation of each part and feature, three frames are used, reference frame (O-X-Y-Z) R, part frame (O-X-Y-Z) P and feature frame (O-X-Y-Z) F (Figure 1 ). The reference frame is global frame, which is attached at the center of the bottom plane of part 1. The feature frame, which is used to describe the position and orientation of mating features, is attached at the center of mating feature planes. The part frame is set at the center of the top plane, sharing the same directions of the reference frame. 

Usually, the variation propagating relation between cabins is cascade propagation. There is a hypothesis that a part frame coincides with the feature frame between the part and the next one. This hypothesis is also suitable for cabin-INSs-assembly process. But the position and orientation of the next part frame can be obtained from the transformation of the last feature frame. 

Considering the part machining errors being far smaller than the part dimensions, the relationship between the actual frame and the nominal frame can be described by homogeneous transformation. In equation (1):

where,,, and a, b, c represent the relative position and orientation between the actual frame and the nominal frame, respectively.
section_name: Description of key mating features

section_content: In the process of cabin-cabin assembly and cabin-INSs assembly, mating features are all plane features. Hence, the plane features are set as an example to establish assembly variation model in this paper. 

In the standard design of productions, actual mating feature plane is an ideal feature plane which deviates from the nominal feature plane. The deviation can be represented by two sets of vectors: three translation vectors and three rotation vectors. Together, they form the small displacement torsors (Wu, 2010) :

By using the coordinate transition theory of robotic kinematics, equation ( 2 ) can be replaced by a 4 ϫ 4 homogeneous transition matrix, as the form of equation ( 1 ), the following matrix is obtained:

where, ␣, ␤ and ␥ are the projection on X, Y and Z axis of unit rotation vectors, respectively. And u, v and w are the projection on X, Y and Z axis of unit translation vectors, respectively. Set a two-cabin-aeroplane assembly process as an example in Figure 2. (O-X-Y-Z) R represents the reference frame, (O-X-Y-Z) p1 and (O-X-Y-Z) p2 are the part frame of cabin 1 and cabin 2, respectively, which includes machining errors. (O-X-Y-Z) p1= and (O-X-Y-Z) p2= are the nominal part frame of cabin 1 and cabin 2, respectively. The thin solid lines describe the nominal position of cabin 2, while the thick solid lines show the actual position of cabin 2. The dashed lines represent the actual feature plane with machining errors. Assume that the actual feature planes of cabins coincide with the mating plane, in which condition, (O-X-Y-Z) p1 in Figure 2 represents the actual plane of cabin 2 as well as the mating plane between cabin 1 and cabin 2. The actual feature plane with machining errors could be regarded as a coordinate transition of the nominal feature plane. H k-1, k and H k,k are coordinate transition matrices between the actual feature plane with machining errors and the nominal feature plane from the k-1 station to the k station:

The coordinate transition matrix between two actual mating feature planes of cabin k with machining errors can be obtained in Figure 2 :

where A k is the transition matrix between two nominal mating feature planes of cabin k, and A k c is the transition matrix between two actual mating feature planes of cabin k with machining errors. 

To establish the state space model of the aircraft assembly process, the state variables and other relative parameters of the assembly process and state space equation are filtered. Also, set the two cabins in Figure 2 as an example to analyze the variation propagation process. 

As for cabin 1, its part frame coincides with the feature frame between cabin 2 and itself. The position and orientation of the part frame relative to the reference frame can express the actual assembly variation after it is assembled. 

The part frame can be calculated by k coordinate transitions:

Equations ( 6 ) and ( 7 ) can be also written as:

where T k c and T k are the transition matrices for the actual positions with and without machining errors, respectively. dT k is the differential translation and differential rotation transition of the part frame relative to the reference frame, and it can be written as:

where ␦T can be defined as the variation infinitesimal matrix. For any neighboring space frame transition, equation ( 10 ) is satisfied:

P i in equation ( 10 ) can be replaced by A i (Whitney and Gilbert, 1994), then A i can be written as a simplified form:

where ␦R i is a 3 ϫ 3 variation rotation transition matrix, whose eigenvector is ␦ i. d i is a 3 ϫ 1 variation translation transition matrix. The differential translation component d and the differential rotation component ␦ after cabin k is assembled can be obtained by equations ( 6 )-( 11 ):

Equations ( 12 ) and ( 13 ) can be expressed in the form of:

where ⌬ yk ] T, ⌬ z ϭ [⌬ z1,. . ., ⌬ zk ] T, and W i are 3 ϫ k coefficient matrices. Equation ( 14 ) describes the total accumulated variation after cabin k is assembled. It obviously shows that the final position deviation relies on both translation and rotation variation, while the final orientation only depends on rotation variation. 

The summary legend of variable in this section is as follows:

,, and a, b, c ϭ the relative position and orientation between the actual frame and the nominal frame, respectively;
section_name: Modeling of variations

section_content: Suppose that there are k-1 cabins assembled in previous station, which form a subassembly s(k) in station k-1, and then cabin k is assembled to s(k) due to the assembly procedure. 

After the end of station k, there are k cabins on the new subassembly. As the variation of the actual assembly process comes not only from the variation state of the end cabin, but also from the relative motion of the neighboring cabins, the state variables get updated with the varying station: 15) where W is the coefficient matrix, 0 is the zero matrix. h k represents the current cabin position variation which is caused by the deformation of previous ones. This deformation is equivalent to a kind of feature frame transformation, which satisfies the superposition relationship of the variation propagation above. Hence, the form of h k is similar to equation (15). After above analysis, the state variables of the system are defined as X͑k͒, which represents the accumulated variation. ͑i͒ is the assembly variation which is contributed by cabin k, namely, it is the relative deviation between the actual part frame with machining errors and the nominal part frame. C ͑k͒ is the measuring matrix, which describes the assembly variation by measuring. v͑i͒ is the measuring error:

Based on equation ( 16 ) and Figure 3, the state space equation for the assembly and compensation process can be established:

The cabin assembly process can be regarded as a discreteevent-dynamic system, F͑k͒͑k͒ ϩ W͑k͒ is the system variation input. When the assembly process is determined, the variation input of each station can be estimated by parts variation and deformation. 

In the actual assembly process, compensation is after assembly. Therefore, the state space model can be improved as:

where H͑k͒ is the variation input after station k without compensation, who added A͑k͒X͑k͒ can be regarded as an intermediate variable. U͑k͒ is the differential vector which describes the feature variation in the compensation process. B͑k͒ is the transformation matrix of variation compensation vectors from the reference frame to the part frame. 

According to the state space model of the cabins assembly, measuring and compensation process, the system errors are the parts machining errors and the feature variation caused by parts deformation. The measuring result after assembly is the state observer value, which determines the compensation values. Eventually, the assembly variation state of a station is obtained, which is also the input state of the next station. 

In the actual assembly process, the final assembly variation can be predicted based on the parts machining errors and the feature variation caused by parts deformation. When the accuracy indexes are not satisfied, an error compensating is necessary. The compensation means the mating feature adjustment after the current station, which includes the adjustment of assembled parts and the parts to be assembled. The detail strategies include the adjustment of the next part relative assembly location and the repairing of assembled part mating feature such as scraping.
section_name: The state space model of assembly process

section_content: In this paper, an aircraft assembly process is analyzed as an example, which includes three cabins and an INS component. Figure 4 shows the components of the aircraft and corresponding frames. And the assembly sequence is cabin 3-cabin 2-the INS-cabin 1. 

The relative position of every nominal part frame is shown in Table I.
section_name: Example

section_content: In this example, assume that the assembly variation only comes from the machining errors and the deformation of parts. Through measuring the key mating feature plane of every component, the feature deviation caused by machining errors is shown in Table II. 

The deformation of parts can be calculated by FEA. 

In this example, cabin-cabin and cabin-INS assembly are all connected by screw bolts. The rotation deviation components of parts frame directly determined the navigation sensitivity of INSs. Hence, the differential rotation components are considered. 

In Figure 5, the assembly variation presents a trend of accumulation and propagation. In station 1, because of no deformation impact, the assembly variation is equal to the deviation of the part frame of cabin 1. However, in following stations, preload results in the parts deformation, which brings about corresponding deviation of every part frame. 

Calculating the deviation of part frames in assembly process based on the state space model, the final assembly variation is predicted. According to all the calculated state variables, the actual deviation of every part frame can be worked out to judge whether it meets the design requirements. If not, corresponding station need to be compensated.
section_name: Calculation of the assembly accumulated variation

section_content: Based on the actual situation, only latter stations can be compensated after obtaining the measuring result of current station. In this paper, next mating features of the assembled parts are compensated. For instance, after cabin 1 and cabin 2 are assembled, the mating feature of cabin 2 (which will be connected to cabin 3) is compensated. Based on the state space model and intermediate variables, which includes the measured parts error and the analyzed parts deformation, the final precision can be predicted. Thus, according to the predicted result, the compensation strategy can be determined. In this paper, because there is no rotation variation between nominal part frames, the nominal mating feature transition matrix A k is:

However, the actual mating feature transition matrix, which considered both deviation caused by machining errors and the deformation impact on assembly variation, can be obtained by: This variation matrix is calculated in extreme situations, so the mean variation matrix can be obtained by the Monte Carlo method: CM m ϭ ͫ 0. 00091 Ϫ0. 00088 0. 00132 0. 00086 Ϫ0. 00137 Ϫ0. 00065 Ϫ0. 00024 Ϫ0. 00106 0. 00732 0. 0116 0. 0135 0. 0128 ͬ

As in the paper, U͑k͒ is the differential vector which describes the feature variation in the compensation process. B͑k͒ is the transformation matrix of variation compensation vectors from the reference frame to the part frame. 

In this case, the reference frame coincides with the part frame in X, Y and Z axis, and we only care about the angular variation. So the matrix B in this example is a unit matrix. 

To make the assembly variation meet requirements, the compensation matrix is: D ϭ ͫ 0. 00091 Ϫ0. 00088 0. 00132 0. 00086 Ϫ0. 00137 Ϫ0. 00065 Ϫ0. 00024 Ϫ0. 00106 0. 00732 0. 0116 0. 0135 0. 0128 ͬ ϫ ΄ 1 1 1 1 0 1 1 1 0 0 1 1 0 0 0 1 ΅ Ϫ1 ϭ ͫ 0. 00091 Ϫ0. 00179 0. 0022 Ϫ0. 00046 Ϫ0. 00137 0. 00072 0. 00041 Ϫ0. 00082 0. 00732 0. 00432 0. 00182 Ϫ0. 00066 ͬ So the compensation differential vector is the vertical components of matrix D. So corresponding compensation is determined in Table IV. 

After the compensation of key mating features, the precision of the final assembly is improved by 73. 9, 50 and 4. 2 per cent about the X, Y and Z axis, respectively, which can be seen in Figure 6.
section_name: Assembly error compensating

section_content: In this paper, a dynamic prediction and compensation model of aircraft assembly is proposed. The small displacement torsor theory is used to describe the mating feature deviation, and then the variation propagating model in assembly process is established based on the state space model. Furthermore, the state space model includes the part deformation which is caused by assembly stress and error compensation. The deformation can be obtained by FEA, which is regarded as the error input of the model. Based on the model, the assembly precision can be predicted by calculation, and then, according to the variation minimum principle, the compensation matrix can be obtained. The real-time compensation in assembly process provides the quantitative guidance for part repairing and adjustment. It can eliminate the impacts of the machining error and part deformation on assembly precision to a great extent. In the case in this paper, the mating feature deviation and the part deformation have the same order. With the compensation of key mating features, the precision of the final assembly is improved by 73. 9, 50 and 4. 2 per cent about the X, Y and Z axis, respectively.
section_name: Summary

section_name: Corresponding author

section_content: Jianzhong Shang can be contacted at: jianzhongshang@ gmail. com For instructions on how to order reprints of this article, please visit our website: www. emeraldgrouppublishing. com/licensing/reprints. htm Or contact us for further details: permissions@emeraldinsight. com
section_name: Corresponding author

section_content: Yujun Cao, Xin Li, Zhixiong Zhang and Jianzhong Shang
section_name: Prediction and compensation of aerocraft assembly variation

section_content: Volume 35 • Number 2 • 2015 • 183-189
section_name: Assembly Automation