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Advances in Mechanical Engineering

Volume 2013 (2013), Article ID 942580, 6 pages

http://dx.doi.org/10.1155/2013/942580

## 2-DOF Angle Measurement of Rocket Nozzle with Multivision

^{1}Postdoctoral Research Station of Instrument Science and Technology Discipline, Harbin Institute of Technology, P.O. Box 305, Harbin 150001, China^{2}Department of Automatic Measurement & Control, Harbin Institute of Technology, P.O. Box 305, Harbin 150001, China

Received 12 July 2013; Revised 29 October 2013; Accepted 31 October 2013

Academic Editor: Emanuele Zappa

Copyright © 2013 Yubo Guo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A real-time measurement method is presented for the 2-DOF swing angles of rocket nozzle by the use of multivision and rocket nozzle rotation axes. This method takes offline processing to measure the position of two nozzle rotation axes in image coordinate system by means of multi-vision and identify the rotation transformation relation between image coordinate system and fixed-nozzle coordinate system. During real-time measurement, the nozzle 2-DOF swing angles can be measured with transformation of marker coordinate from image coordinate system to fixed-nozzle coordinate system. This method can effectively resolve the problem of occlusion by markers in wide swing range of the nozzle. Experiments were conducted to validate its correctness and high measurement accuracy.

#### 1. Introduction

To ensure the flight control, range, and impact point of rocket nozzle, it is important to measure the 2-DOF (Degree of Freedom) swing angles of the rocket engine nozzle [1, 2]. At present, the two angles are obtained with geometric transformation and conversion based on the angular displacement of motion servo control rod measured with rotary potentiometer during the cold and hot test run of the nozzle. It is an indirect measurement method and is much deviated from the actual nozzle motions [3]. Reference [4] proposes to measure the geometric parameters of the rocket engine with laser tracker. Laser tracker can achieve high measurement accuracy. However, it is difficult to make adjustment, and it is usually used for static measurement. Reference [5] measures the pitching angle and yawing angle with electronic theodolite for investigation on the swing angle calibration of rocket engine nozzle. With electronic theodolite, the measurement accuracy of the swing angles can be ensured, but the efficiency is low due to point-by-point measurement and cannot meet real-time measurement requirements. Reference [6] puts forward a multivision measurement for nozzle motion parameters. It uses high speed cameras to achieve proper real-time measurement. For measuring the nozzle motion parameters, it adopts nozzle cone surface fitting method, which requires more markers and can only realize one-dimension swing angle measurement for the nozzle. What is more, the nozzle surface machining error and marker form error will certainly cause large measurement error.

Vision measurement shows broad application prospect in the field of measurement field [7–10] for its easy-to-move, noncontact, and real-time features. This paper presents a measurement method based on multivision and rocket nozzle rotation axes so as to improve the accuracy of vision-based measurement for rocket nozzle swing angles and to avoid occlusion problem during measurement.

#### 2. Multivision Measurement Model of Nozzle Motion

The coordinate systems for the multivision measurement of rocket nozzle swing angles are shown in Figure 1, including image coordinate systems (where is the number of cameras), world coordinate system , fixed-nozzle coordinate system , and dynamic nozzle coordinate system .

In image coordinate system, the origin is at the intersection point of optical axis and image plane, the coordinate axes are parallel to the lines and rows, respectively, and the markers are imaged at the two cameras. The image coordinate of marker center can be obtained by image processing. The coordinate of marker centers in world coordinate system and in image coordinate system conforms to vision image relation. With camera calibration, the coordinate in world system can be obtained from the image coordinate. The fixed-nozzle coordinate system is defined as a fixed coordinate system with nozzle at zero position, and the nozzle motion consists of translation in three directions relative to origin (rotation center) and swing angles around axes and . The dynamic nozzle coordinate system is fixed on the nozzle, and it coincides with while the nozzle is at zero position. The swing angles of the nozzle can be determined by the use of the transformation relations between coordinate systems and . The measurements of nozzle swing angles can be divided into two stages. The first stage is offline processing, including defining the transformation relation between fixed-nozzle coordinate system and world coordinate system and obtaining the coordinate of marker center in fixed-nozzle coordinate system while, nozzle is at zero. The second stage is real-time measurement, consisting of the following steps: firstly, take 3 or more marker centers image coordinates and employ vision image relation to complete marker measurement in world coordinate system; secondly, transform the marker coordinate from world coordinate system to fixed-nozzle coordinate system; finally, calculate nozzle swing angles according to marker coordinate in fixed-nozzle coordinate systems and values at zero position. The multivision coordinate measurement model, fixed-nozzle coordinate system model, and swing angle measurement model are established as below.

##### 2.1. Multivision Coordinate Measurement

Set the world coordinate of the center of marker (*⩽**n*) at , and the corresponding image point on camera (*⩽**m*) at ; then, at the pinhole model, it gives
where,

The viable is defined by camera calibration.

Apply the least square method to (1) to obtain the world coordinate of the center of marker as follows:

##### 2.2. Fixed-Nozzle Coordinate System Model

Rocket nozzle can rotate by certain angle around axes and , respectively. Therefore, the position of these axes in the world coordinate system can be defined with multivision, and the fixed-nozzle coordinate model can be accordingly established as follow.

Given axis across point , unit vector , the world coordinate of n markers on nozzle , after rotating around axis by angle , the world coordinate is , , . Then, where is the nozzle displacement along the axis during rotation. Equation (5) is denoted as

According to [11], and are calculated by

Given a point on the axis, taking , we can deduce that , world coordinate of n markers before and after rotation, displacement , rotation angle , and unit vector satisfy the following relations: where , , and are expressed as where is the element of matrix and is expressed as

Then, the least square solution of is

Thus, we can obtain unit vector of axis and any point on the axis by (7) and (12). Define similarly. Then, define with right hand rules. The intersection point of axes , , and is the rotation center of the nozzle, and the three unit vectors are row vectors of rotation transformation matrix of world coordinate system and fixed-nozzle coordinate system, while rotation center coordinate is translation vector.

##### 2.3. Swing Angle Measurement

Matrix and vector are used to denote the transformation relation between fixed coordinate system and world coordinate system. Suppose that the world coordinate of marker is when the nozzle is at zero position, and when moving to a certain position. Then, the corresponding coordinates and in fixed-nozzle coordinate system, respectively, satisfy

The swing angles and of the nozzle around axes and satisfy where is the translation in three directions of the rotation center.

Equation (14) is a nonlinear equation set for swing angle. When there are three or more markers and not all the markers are in the same line, the Gauss-Newton method can be used for iteration solution.

#### 3. Experimental Results

Figure 2 shows the multivision measurement system for swing angles of the nozzle. Four high speed cameras with infrared emitter and infrared filter are used as a multivision angle measurement system. Nozzle swing angles are simulated with dual axes rotating calibration equipment. The calibration equipment has high motion accuracy and can be used as motion reference. Infrared reflecting markers are arranged on the surface of the calibration equipment. The exposure and data acquisition of four cameras are the synchronized by synchronous control system, and the Ethernet is used to realize data transmission between the four cameras and the host computer.

Experiments are conducted by using ball markers with infrared reflecting surface. These markers will generate high-lighted spots at image plane. The marker image can be separated from the background by grey scale threshold setting. During the movement of the nozzle by large angle, the markers may be sheltered. The markers will be considered effective only when their image can be identified by two or more cameras. Otherwise, they will not be involved in the measuring process. The method in this paper can achieve 2-DOF angle measurement when 3 or more effective markers are identified. Figure 3 shows the marker image identification results at a given time. And in this case, all the cameras numbers 1, 2, 3, and 4 can identify four markers.

##### 3.1. Contrast Experiments for Geometrical Parameters of Cameras

The main parameters representing the geometrical configuration of cameras consist of camera focus, baseline length, and angle between optical axes of two cameras. Since the camera focus in the experiments is fixed as 24 mm, the effects of optical axis angle and baseline length are discussed as follows.

The cameras are arranged at circumference and in such a way that the baseline length and the optical axis angle of any adjacent cameras are the same (with optical axis oriented to the center of measurement field of view). Put 8 markers on the nozzle surface, let the nozzle make 2-DOF movement by ±20°, and take measurements at 1000 equally spaced positions. The maximum measurement errors are shown in Table 1.

The experiment results show that the system achieves the highest measurement accuracy at 90° optimal axis angle and 3 m baseline length.

##### 3.2. Contrast Experiments for Different Numbers of Markers

Different marker numbers will affect the measurement accuracy. When the nozzle makes 2-DOF movement by ±20°, measurements are taken at 1000 equally-spaced positions with the cameras arranged at 90° optimal axis angle and 3 m baseline length. The maximum measurement errors are shown in Table 2.

Above, Table 2 shows that the system measurement accuracy can be improved by increasing the number of markers.

Rotate the calibration equipment in 2-DOF by angle within ±20° with 90° optical axis angle and 3 m baseline length in case of 8 markers, and take real-time measurements at several positions with the method stated in this paper. Table 3 lists the absolute value of measurement errors of rotation angle, in which , denote the angle errors at given position in two directions, respectively. The Table shows that the maximum measurement error of swing angle is 0.18°. And the results confirm that this method can achieve high accuracy of 2-DOF angle measurement.

#### 4. Conclusion

This paper presents a real-time measurement method for swing angles of the rocket nozzle by means of multivision and rocket nozzle rotation axes. The experimental results show that this method can achieve a measurement accuracy that is better than 0.2° in ±20° angle range with 90°optical axis angle and 3 m baseline length in case of 8 markers. It is greatly significant for the actual measurement of swing angles of the rocket nozzle.

#### Acknowledgments

This work was financially supported by the Fundamental Research Funds for the Central Universities (Grant no. HIT.NSRIF.201138), the Research Fund for the Doctoral Program of Higher Education of China (Grant no. 20112302120029), the National Natural Science Foundation of China (Grant no. 51075095), and the National Natural Science Foundation of Heilongjiang province, China (E201045).

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