Abstract
In this paper, a novel relative position and orientation (R-P&O) measurement method for large-volume components is proposed. Based on the method, the parallel distances between the cooperative point pairs (CPPs) are collected by multiple pairs of wireless ranging sensors which are installed on respective components and finally turned into the R-P&O. Accordingly, a measurement model is built and an algorithm is designed to solve the model, in which the radial basis function neural network (RBFNN) produces a preliminary solution by offline training and the differential evolution (DE) strategy finds the accurate solution by online heuristic searching. Furthermore, the crucial parameters and the performance of the algorithm are analyzed through simulating a virtual alignment process which proves that the RBFNN-DE algorithm can quickly and accurately find the global optimal solution in the whole effective workspace. Besides the theory study, a ranging device based on ultrasound has been developed along with a calibration method. Depending on the device, an experiment of actual alignment is implemented to verify the algorithm. Experimental results indicate that the error of R-P&O is no more than 4.1 mm and 0.32° when the ranging error is 0.1 mm, compared with the measurement result of indoor GPS (iGPS).
1. Introduction
In aviation, aerospace, and ship manufacturing, the product assembly is characterized by large volume, high accuracy, and complex processes. As the key production procedure, component alignment greatly affects the quality of manufacturing, the priority target of which is measuring R-P&O that means the position and orientation of the moving component coordinate system relative to the immovable component coordinate system. The traditional method is generally to figure out the geometric size of the component by standard gauges with analog transfer, which has been unable to meet the increasing requirements of alignment. Hence, it is of great theoretical and practical value to research for a high-accuracy digital measurement system for the alignment process of a large-volume component [1, 2].
The common digital measuring instrument includes a laser tracker [3, 4], theodolite [5], iGPS [6, 7], industrial camera [8, 9], coordinate measuring machine (CMM) [10, 11], and so on, which depends on transforming the relevant points into the reference coordinate system of a third party to fit the R-P&O of components [12]. Because of the indirect acquisition of R-P&O, this method involves a complex coordinate system transformation and unnecessary error [13]. Considering that the component to be measured has a large volume, it is difficult to cover all the target points with a single instrument. Alternatively combining multiple instruments, the error caused by a transfer station needs to be taken into account [14]. Besides, for all of the above instruments the price is expensive, several of the instruments (theodolite and CMM) are no longer able to work when the R-P&O is changing.
Inspired by the forward kinematics problem (FKP) in the field of robotics, the R-P&O can be calculated based on the distances of CPPs. A pair of corresponding points from two individual components with known coordinates in a local coordinate system is called a CPP. FKP refers to using the kinematic equations of a robot to compute the R-P&O of the end effector through the joint parameters [15]. The difference is that for each measurement task of alignment, the position of CPPs can be different, and they don’t have to be regularly distributed in two planes such as the Stewart platform [16]. In order to solve the complex nonlinear equations contained in the FKP, there are two kinds of methods that are proposed, namely the numerical method [17] and analytical method [18]. The mathematical model of the numerical method is simple, but it is quite dependent on the initial value and cannot deduce all solutions. On the contrary, the elimination process of the analytical method is very complex which may introduce an imaginary root. In addition, a lot of intelligent algorithms were proposed recently, such as particle swarm optimization [19], hybrid immune algorithm [20], and neural network algorithm [21].
The traditional methods for FKP can lead to multisolutions. However, the measurement needs to be synchronized with the alignment process, and there is no condition for judgment manually after each calculation. On the other hand, it is difficult to make a general rule to pick the correct solution for various alignment components, so we turn to intelligent algorithms. Compared with other algorithms, RBFNN has the characteristics of approaching any nonlinear function with arbitrary-precision and having a good global approximation ability. It is also suitable for the solution of multidimensional parameters and can be easily adjusted by changing the number and width of network nodes. Accordingly, it has a natural advantage to solve the FKP. However, this depends on the quality of training, and the result cannot be satisfactory with poor training quality. DE has few controlled parameters and can search for solutions by using individual differences. Its fast convergence capability can solve the real-time problem robustly. It also applies to the solution of multidimensional parameters. Adversely, it easily falls into a local optimal solution and relies on a good initial value. Therefore, considering the complementary advantages of the two algorithms, the preliminary result of RBFNN is used as the initial value of DE, and then a quite accurate result can be solved.
In this paper, a measurement model is built to calculate the R-P&O of large-volume components by parallel ranging, and an intelligent algorithm combining RBFNN [22] and DE [23] is proposed to solve the model in Section 2. For the purpose of analyzing the performance and crucial parameters of the algorithm, a virtual alignment is simulated through a hypothetical trajectory in Section 3. Furthermore, a kind of ranging device is developed based on an ultrasonic signal, and the calibration method of the device is introduced. Finally, the novel method is verified by the actual alignment experiment in Section 4.
2. The Method for R-P&O Measuring
2.1. The Measurement Model Based on Parallel Ranging
In the alignment process, the R-P&O measurement based on parallel ranging involves the immovable and moving component. Meanwhile, a number of wireless sensors are installed on CPPs of the components to measure the distances, which provide support for calculating the R-P&O. The schematic diagram of the measurement model is shown in Figure 1.
The symbols are defined as follows: (a) is the base coordinate system (BCS), which is fixed to the immovable component(b) is the moving coordinate system (MCS), which is fixed to the moving component(c) is the coordinate of CPPs in BCS, which can be obtained based on a 3D model of the design or calibration in advance(d) is the coordinate of CPPs in MCS, which can be obtained based on a 3D model of the design or calibration in advance(e) is the distance between a CPP, which is obtained by a ranging device(f) is the expression of R-P&O in matrix form [24], which is the target to be solved(g) is a rotation matrix, which is uniquely determined by the Euler angle (h) is a translation matrix, which is uniquely determined by displacement
Our aim is to determine the R-P&O ; that is, by . It can be expressed as (1):
According to the Euclidean distance between CPPs, we can set up equations. Because we need to solve 6 independent parameters, there must be . Then, it can be transformed into an optimization problem:
Equation (2) with a complex structure contains a large number of trigonometric functions. In the process of iterative optimization, it is easy to fall into the local minimum and get the solution which does not conform to the actual situation. At the same time, the speed of convergence and the quality of the solution depend on the selection of initial value. In view of this, a solution method based on RBFNN-DE is proposed, which can quickly and accurately calculate the R-P&O.
The algorithm flow of RBFNN-DE is shown in Figure 2. Firstly, according to the specific alignment components, the possible variation space of the R-P&O is designated, that is, the workspace. Then, we start to generate a number of R-P&O in the workspace randomly, and use the inverse function of (1) to find the theoretical distance of CPPs corresponding to each R-P&O. A large number of data samples consisting of R-P&O and distance are randomly assigned to the test set and the training set. The training set is responsible for calculating the parameters of the network, and the test set is responsible for evaluating the error of the network. At this point, if the actual measured distance of CPPs at a certain time in the alignment process is input to the network, the preliminary R-P&O calculation result can be obtained.
Combining and , we can construct a solution space that may contain the accurate R-P&O and randomly generate multiple values in this space to initialize population . Using the difference between individuals in , the mutant population was created. Then, we cross with the original population to get the population . Finally, we compare the individuals in with those in and select the relatively better individuals to form the new population . The process is iterated until the population satisfies the terminating condition, and at that time the optimal value is picked from the as the final result of the RBFNN-DE algorithm.
2.2. Preliminary Estimation Based on RBFNN
As shown in (3), the radial basis function is radial symmetry, defined as a monotone function of the Euclidean distance from any point to the center point , and represents the width of the function:
RBFNN with a three-layer structure is constructed as Figure 3.
Through RBFs, the input data ( dimension) can be projected nonlinearly to the high dimensional space ( dimension). Furthermore, RBFs are combined linearly to obtain the output ( dimensions) which approximates the original function as shown by (4). represents the weight of the hidden layer to the output layer:
The parallel distances can be determined uniquely by according to the inverse kinematics problem (IKP). Thus, we can generate the set containing large amounts of data for training the neural network and the set for testing the performance of RBFNN.
In order to simplify the training process, the center of RBF is specified at each training sample, namely , and all the width of RBF is set to constant . By matrix expression, we can get (5):
Using the least squares method (LSM) and singular value decomposition (SVD) [25], can be calculated, which means RBFNN for specific workspace is completely constructed. Given an input , there is .
Since the center of RBF is consistent with the training sample, the network may be overfitting with a large error in the set . Thus, the network can only estimate the R-P&O preliminarily, and it is necessary to get more accurate results by processing further.
2.3. Accurate Computation Based on DE Algorithm
DE is a heuristic random search algorithm based on group differences, including mutation, crossover, and selection process. It is characterized by a differential strategy for individual mutation, global optimization, and good stability. The following are the main steps of DE algorithm.
Step 1. Population Initialization. According to the error displayed on the set and the preliminary result calculated by RBFNN, the upper bound and lower bound of the search space are determined. In the search space, individuals are randomly generated to form the generation population according to (6), and is used to generate uniform distributed random numbers from 0 to 1:
Step 2. Mutation. The most obvious feature of the DE algorithm is to utilize the difference vector between individuals to generate a new individual. The method we used is revealed in (7): are unequal integers in the range , while is the variation parameter controlling the influence degree of the difference vector on the new individual. If there is , the will be assigned again according to (7).
Step 3. Crossover. The purpose of crossover is to further improve the diversity of the population. Equation (8) describes the method of generating a new crossover individual by randomly selecting from the mutation or ordinary individual at every dimension, where is called crossover parameter:
Step 4. Selection. Equation (9) describes how to select the better individual from ordinary or crossover individuals according to the greedy strategy, which eventually becomes a member of the next generation:
Step 5. Iteration. The above steps will be iterated until reaching maximum iterations or the accuracy restriction . The optimal individual is an accurate estimation of R-P&O.
Combining RBFNN and DE, the whole flow of the algorithm in the form of pseudocode is presented in Algorithm 1.
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3. Algorithm Simulation and Result Analysis
3.1. The Generation of Simulation Data
According to the actual working conditions, a trajectory of the alignment process is simulated. The trajectory is a set of six dimensional vectors, which represent the R-P&O of the dynamic coordinate system observed in the static coordinate system, and each dimension in the vector is continuous on the trajectory. In order to analyze the performance of the algorithm, a number of discrete key frames on the trajectory are picked as representatives. Since the distance of every CPP is known at each frame of the trajectory, the R-P&O can be calculated based on the RBFNN-DE algorithm. Finally, the performance of the algorithm is analyzed by comparing the calculated and simulated trajectory. The main steps of simulation data generation are as follows.
Step 1. Delineating the Scope of Workspace The virtual alignment conditions can be outlined as follows: the size of the junction surface is , the original distance of components is , and the angle of relative orientation is not greater than 25° during the alignment process. These conditions can be expressed by (10):
Step 2. Defining the Coordinates of CPPs . In the case of using 8 CPPs, the coordinate of each point in the local coordinate system is randomly generated according to (11). It delineates the possible distribution of CPPs in the cube range:
Step 3. Creating the Trajectory of Simulated Movement. The quadratic function is used to create the simulation trajectory of moving coordinate system origin as shown in (12). Each ranging 10 to 90 is corresponding to a frame of the alignment process:
At each frame, the orientation of the moving coordinate system satisfies the assumption of 0° to 25°. The orientation should be smaller and smaller, and finally approaching 0°. Therefore, we use to create random values ranging from 0° to 25°, and then arrange them in descending order as shown in (13). The final result will correspond to every frame on the trajectory.
After the above settings, the simulation data about alignment trajectory is generated. As Figure 4 shows, the moving part will move from top to bottom and rotate, approaching the stationary part. The boxes of the solid line border represent the space in which and are located, respectively, and the boxes with the dotted border represent the key frames that the box will pass through.
3.2. Solving Trajectory Based on RBFNN-DE Algorithm
Let the number of the RBF center and randomly generate 1000 samples as the training set and another 1000 samples as the test set in . Through setting different width and analyzing the normalized percentage error of R-P&O, the most suitable can be selected in Figure 5. The larger means the narrower radius of RBF, a smaller error in the training set but more likely overfitting. To enhance the generalization ability, the fitting function can be smoothed by reducing . Therefore, the that has the best performance in the test set is selected.
The final generated RBFNN is applied to the simulation alignment in Section 2.1, and the calculation error of results is presented in Figure 6, in which the uncertainty of the R-P&O is . Accordingly, can be set to for the subsequent DE algorithm.
The average number of iteration and the average time consumption of the DE algorithm are most affected by . For analyzing their relationship, let , and , respectively. The relationship is illustrated in Figure 7.
When , the average number of iteration is 168, and the average time consumption is 0.3 s. The final calculation error is presented in Figure 8.
The final uncertainty of R-P&O is 0.002 mm, 0.003 mm, and 0.002 mm and 0.0002°, 0.0001°, and 0.0001° in Figure 8, and the trajectory solved by the RBFNN-DE algorithm is quite close to the theoretical trajectory, which means that the algorithm is effective and accurate.
3.3. The Influence of Other Conditions on Calculation Results
The structure of the measurement model can affect the measurement error, in which the most significant factor is the number of CPPs. Keeping the number of CPPs fixed, the mean value of R-P&O uncertainty is calculated by different CPPs generated randomly. The relationship between the number of CPPs and R-P&O uncertainty is illustrated in Figure 9, which reveals that the more number of CPPs, the higher accuracy of results, because of more constraints.
Besides, the noise in the distance measurement can also affect the calculation result in the actual alignment process. To illustrate this, the Gauss noise with a standard deviation of is added to the theoretical distances to simulate actual measured values; and to analyze the relationship between and the calculated uncertainty of R-P&O in the above measurement model, the R-P&O of is chosen. The results in Figure 10 reveals that with the increase of the uncertainty increases.
According to the accuracy requirements of the R-P&O measurement, the ranging device with corresponding accuracy can be selected from various sensors. For example, when using ultrasonic wave as a ranging signal, the ranging accuracy can reach 0.1 mm, which means the corresponding uncertainty of R-P&O is 0.24 mm and 0.032° in the above measurement model according to Figure 10.
4. The Measurement System of R-P&O
4.1. Composition of the System
The method proposed in this paper requires the support of a wide-area distance measurement sensor. The sensor should contain two parts: the transmitter and the receiver, and the measuring signal should be sprinkled to cover a certain area of space. As long as the receiver is in the signal coverage area and the signal is received correctly, the measured distance can be calculated according to the time of flight (TOF). When the component is moving, the measurement system is still able to collect the distances between CPPs continuously. The principle of the ranging sensor is presented in Figure 11.
Based on the principle of ranging, a separate type and high-accuracy ultrasonic ranging device (such as Figure 12) has been developed.
The size of the device is . There are 4 locate holes at the bottom of the shell used to connect with the measured component. The GPS is used as the time synchronization signal, and the ranging accuracy can reach . The sound absorption sponge is used to limit the divergence of the sound wave and suppress multiple bounces of sound waves between the surfaces of the devices. After the data is collected, the result is transmitted back to the host computer by radio frequency (RF). The device is powered by a lithium battery and does not require an external power at work. When multiple devices work simultaneously, the carrier signal ensures that the ultrasonic signal will not be confused and the time-sharing sending ensures that the RF signal will not interfere.
4.2. Calibration of the System
Since the method requires knowing the coordinates of CPPs in the coordinates of components, the relative position of CPPs and the locate holes should be calibrated before the device used. Assuming that a CPP is at the center of the ultrasonic sensor surface, the device is calibrated by CMM (such as that shown in Figure 13), and the calibration data is shown in Table 1 taking a device as an example. When the devices are installed on the components, the coordinates of CPPs can be converted to the coordinate system of the components by 4 locate holes, to avoid measuring CPPs before every time the device used.
Because the real cooperative point of an ultrasonic sensor is not on the surface, the measuring distance is not equal to the distance of CPPs. Thus, there is a systematic error that requires compensation [26]. The following experiment is designed to find the value of compensation for each pair of devices. Since the GPS signal needs to be received outdoors, the repeater of the GPS signal is used to introduce the signal into the room, and the ranging device can work normally in the room (such as Figure 14).
The receiver is fixed as the center of the circle (such as Figure 15), and the transmitter is placed at different positions on the different radii within 30° (the maximum receiving angle of the receiver), when the theoretical distance is measured by CMM as a comparison. The measurement result of one pair of the devices is shown in Table 2, which indicates that the compensation of the device is .
4.3. Verification Experiment of the System
In order to verify the method in actual alignment, a verification experiment of the system, such as Figure 16, is constructed. Six pairs of calibrated ultrasonic ranging devices are installed on two -size metal plates laid face to face with the known installation position of the devices. One piece of plate with receivers is immovable and the other with transmitters moves closer to the immovable one in a different orientation. For investigating the accuracy of the R-P&O measurement, 7 measurement results of the moving plate at different locations are picked out for analysis compared with the measurement result of iGPS.
In the experiment, the original input data contains the coordinates of CPPs and the measured distances (as shown in Table 3), and the final results are presented in Table 4. The experiment indicates that the maximum value of the R-P&O deviation in each dimension is no more than , namely 4.1 mm and 0.32° in the form of a synthetic value, while the ranging accuracy of the sensor is 0.1 mm. The accuracy level is lower than the simulation result of 0.24 mm and 0.032° in Section 2.3 because the metal plate is deformed and the structural parameters of the measurement model are different. However, it is still suitable for some large-volume component alignment processes, such as the assembly of a ship section.
From the above deviation values, it can be seen that the system has a higher position accuracy and lower orientation accuracy in the connection direction of CPPs. The reason is that this direction is the main direction of the distance measurement and the measured distances are insensitive to the rotation angle in this direction.
5. Conclusion
In this paper, a novel method for R-P&O measurement of large-volume components is proposed. Firstly, the model of R-P&O measurement by parallel ranging is established, and the solution method based on the RBFNN-DE algorithm is proposed, where RBFNN is to get the preliminary solution by a trained network and DE is to obtain the accuracy solution by heuristic random searching.
Then, taking the simulation trajectory as a sample, the basis of the critical parameters in the algorithm is given. The calculation results of the simulation experiment prove that the accuracy of the measured R-P&O is superior to and . Furthermore, the influence of the number of CPPs and ranging noise on the algorithm individually is given. In the condition of simulated ranging accuracy reaching 0.1 mm, the calculated accuracy of R-P&O is 0.24 mm and 0.032°.
Finally, the self-developed ranging device according to the requirements of measuring R-P&O is introduced. Meanwhile, the calibration experiment of the device is described, in which the installation parameters and compensation of the device are obtained. Afterwards, using six pairs of ranging devices and two pieces of metal plate, an actual alignment verification experiment is built. Compared with the measurement data of iGPS, the maximum value of R-P&O deviation is no more than 4.1 mm and 0.32° in the form of a synthetic value, while the ranging accuracy of the sensor is 0.1 mm. The experiment verifies that the novel R-P&O measurement method is effective and accurate comparing to the large volume.
The measurement accuracy of R-P&O can be improved by reducing ranging error and increasing the number of CPPs. Thus, a ranging device with higher accuracy that supports multiple sensors to work simultaneously will be the focus of future research. Meanwhile, the dynamic performance of the system in tracking the R-P&O based on this method will be studied.
Data Availability
All data are provided in full in the Algorithm Simulation and Result Analysis section of this paper.
Conflicts of Interest
The authors declare that there is no conflict of interest regarding the publication of this paper.
Acknowledgments
This study was cosupported by the National Defense Basic Scientific Research (no. JCKY2013206C003) and the Open Fund of the State Key Laboratory of Precision Measurement Technology and Instruments (no. PIL1404).