Attitude measurement system based on micro-silicon accelerometer array

Transcription

1 Chaos, Solitons and Fractals 29 (2006) Attitude measurement system based on micro-silicon accelerometer array Li Qin *, Wendong Zhang, Huixin Zhang, Weixing Xu Key Laboratory of Instrumentation Science and Dynamic Measurement, North University of China, Ministry Education, Taiyuan, Shanxi , PR China Accepted 16 August 2005 Abstract This paper analyzes theoretically the strap-down inertia measurement system with nine micro-silicon accelerometers array and its attitude algorithm model. The error sources and characteristics in this measurement system are also analyzed. The algorithm model is validated by simulation experiment on the ground and the experimental results are presented. Finally, experimental errors are modified and compensated to improve algorithm precision based on micro-silicon accelerometer array. Ó 2005 Elsevier Ltd. All rights reserved. 1. Introduction The strap-down inertial measurement system with micro-silicon accelerometer array is mainly used to measure accurately attitude parameters of the flying object. Compared with the strap-down inertial measurement system composed with gyros and accelerometers, its main advantages are that it reduces cost and can resist high over-loading. The inertial measurement system is composed with nine micro-silicon accelerometers array. It will be installed to the flying object and flies with the flying object together. So this system can measure real-time acceleration values in their corresponding direction. According to these acceleration values, their corresponding attitudes can be obtained by attitude algorithm model. These data will provide necessary reference for inertial navigation. The paper mainly introduces the strap-down inertial measurement system and its attitude algorithm model based on the micro-silicon accelerometers array, and analyzes the error sources and characteristics in this measurement system. The authors present the results of simulating experiment on the ground. In the end in order to improve measurement precision, the researchers modify experimental errors through different methods. 2. Inertial measurement system and attitude algorithm model In theory six accelerometers are needed to describe the rigid bodyõs movement at least. Only when these six accelerometersõ positions and directions are arranged reasonably can the movement of the rigid movement be described * Corresponding author. address: qinli@nuc.edu.cn (L. Qin) /$ - see front matter Ó 2005 Elsevier Ltd. All rights reserved. doi: /j.chaos

2 142 L. Qin et al. / Chaos, Solitons and Fractals 29 (2006) accurately. However, in order to improve measurement precision it needs to compensate the effect of the intersection item in the angular velocity in algorithm of the attitude parameters [1]. The measurement system with nine micro-silicon accelerometers uses more accelerometers, making it possible to reduce the redundancy of the accelerometer signals caused by intersection item from the angle velocity. Therefore to a certain degree it improves measurement precision. The installation model is shown in Fig. 1. The inertia measurement system consists of nine micro-silicon accelerometers array, signal-conditioning device, signal acquisition device, signal memorizing device and central controlling device. When the flying object is high over-loading, it can still record and store real-time acceleration values during flying course. After the flying object is recovered, the attitude parameters can be solved ex post by computer. The measurement unit is shown as Fig. 2. The equation of the specific force measured with the accelerometer that is installed at any point of the flying object is as follows [2 4]: f b ¼ _V b en þ _x b nb Lb þ x b nb ðxb nb Lb Þþð2x b ie þ xb en ÞðV b en þ xb en Lb Þþg b þ r b ð1þ The flying parameters at any time may be gotten by specific force equation from the combination of accelerometers. Because this strap-down inertial measurement system replaces the function of gyros with the accelerometers, the calculating precision for the angular acceleration and velocity at any time is very important. The calculating formulas are showed as follows: 8 _x b z ¼ 1 ðf 2 b f d Þ= ðl b L d ÞþA b x =L f f f =L f þ 1 ½ e 2 YX e XY Š >< h i _x b x ¼ 1 f 2 e=l e f g =L g þ A b y =L g A b z =L e þ 1 ½ e 2 ZY e YZ Š ð2þ >: _x b y ¼ 1 ½ðf 2 h f i Þ= ðl h L i Þ ðf a f c 8 x b z1 ðkþ ¼xb z ðk 1ÞþR kt ðk 1ÞT _xb z >< ðtþdt x b x1 ðkþ ¼xb x ðk 1ÞþR kt ðk 1ÞT _xb x ðtþdt >: x b y1 ðkþ ¼xb y ðk 1ÞþR kt ðk 1ÞT _xb y ðtþdt Þ= ðl a L c ÞŠþ 1 ½ e 2 XZ e ZX Š 8 x b z xb x ¼ 1 ½ðf 2 h f i Þ=ðL h L i Þþðf a f c Þ=ðL a L c ÞŠ 1 ðe 2 ZX þ e XZ Þ >< x b z xb y ¼ 1 ½f 2 e=l e þ f g =L g A z =L e A y =L g Š 1 ðe 2 ZY þ e YZ Þ >: x b x xb y ¼ 1 ½ðf 2 b f d Þ=ðL b L d Þ A x =L f þ f f =L f Š 1 ðe 2 XY þ e YX Þ The rotation vector algorithm is used to calculate the attitude angle, and here we describe the tri-subsample and quadruple-subsample of the algorithm. The rotation vector can be denoted as U. The tri-subsample algorithm makes three ð3þ ð4þ Fig. 1. The nine-accelerometer array model.

3 L. Qin et al. / Chaos, Solitons and Fractals 29 (2006) Fig. 2. The gyro-free inertial measurement unit. equal interval sampling at the sampling interval h = t k t k 1, then the angle increment h 1, h 2 and h 3 can be obtained. We note the formula R 3 ih xðt 1 3 h k 1 þ sþds; i ¼ 1; 2; 3, then we can get: Uðt k 1 þ hþ ¼h þ X ðh 1 h 3 ÞþYh 2 ðh 1 h 3 Þ where h = h 1 + h 2 + h 3, X = 33/80, Y = 57/80. The quadruple-subsample algorithm makes four equal interval sampling at the sampling interval. We can get: Uðt k 1 þ hþ ¼h þ k 1 ðh 1 h 2 þ h 3 h 4 Þþk 2 ðh 1 h 3 þ h 2 h 4 Þþk 3 ðh 1 h 4 Þþk 4 ðh 2 h 3 Þ where h = h 1 + h 2 + h 3 + h 4, k 1 = 736/945, k 2 = 334/945, k 3 = 526/945, k 4 = 654/945. Set U ¼ bu x U y U z c h qðhþ ¼ cos U 0 sin U 0 U 2 sin 0 U 2 U 0 U 2 sin 0 i T x U 0 U 2 y U 0 U z ð5þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi U 0 ¼ U 2 x þ U2 y þ U2 z then the attitude matrix can be obtained by Q(T + h) = Q(T)q(h), and the attitude angles can also be calculated. 3. System simulation The simulation is made according to the bodyõs two dimensional rotation at fixed point. The origins of the inertial coordinate system and body coordinate system are coincident. The angular velocity along the OX axis of body coordinate system is x x = 0.4p rad/s, and the angular velocity along the OZ axis of inertial coordinate system is x z = 0.05p rad/s. Suppose the initial attitude angle are: pitch c =45, yaw u =45, roll h =0.Figs. 3 and 4 show the attitude angles calculated by the tri-subsample algorithm and the quadruple-subsample algorithm respectively, where de max is the absolute error of theoretic value to calculating value. As we can see from Figs. 3 and 4, the quadruple-subsample algorithm is superior to the tri-subsample algorithm. 4. Simulation experiment on the ground The gyro-free strap-down inertial measurement system is tested at a turning table with different rotate speeds. The axis of rotation is x which keeps horizontal. The body coordinate system and the inertial coordinate system are coincident at the initial time. The sampling frequency of the measurement system is 2 khz, and the recording time is 10 s. The rotate speed of the turning table, x x, is 300 circle/min. The noise and high frequency component can be got rid of by a digital low-pass filter. In this paper, a fourth order Butterworth low-pass filter with the edge frequency of 40 Hz is used. The angular velocity computed by using the

4 144 L. Qin et al. / Chaos, Solitons and Fractals 29 (2006) Fig. 3. Attitude angles by tri-subsample algorithm. Fig. 4. Attitude angles by quadruple-subsample algorithm. unprocessed data is shown in Fig. 5(a), and the angular velocity computed by using the processed data is shown in Fig. 5(b). As shown in Fig. 5, the angular velocity computed by using the unprocessed data is divergent, However after data processing, the precision of x x and x y have been increased one order of magnitude and x z has been increased two order of magnitude. 5. Error analysis In the gyro-free strap-down inertial-measurement system, the accelerometer is one unique measuring component, so all measurement errors result from the accelerometer, which is different from the system with gyro.

5 L. Qin et al. / Chaos, Solitons and Fractals 29 (2006) Fig. 5. The angular velocity. (a) Unprocessed, (b) processed. In this measurement system, the single axle accelerometer ADXL105 is used. There are different errors caused by accelerometer ADXL105 such as random error, cross-coupling error, etc. Random error can be compensated by the formula e m ðtþ ¼c 1 þ c 2 e k 2t, in which e m (t) is the error model of accelerometer ADXL105 at zero input [5]. Besides, the errors can be caused by: the approximation of mathematics model, inaccuracy of the original level and the original position and velocity, computing, the effect of sampling quantity and the finite word length effect of computer, alternation and inaccurate position of the measurement components. If the acquisition data are used to calculate attitude parameters of the flying object [6 8] without any processing, for example, the processing for noise disturbance, the measurement results will be not accurate. The errors of angular acceleration and angular velocity are mainly from constant error and random error, which make attitude error accumulate fast. If these errors can be estimated and be compensated [9], the precision of attitude algorithm will be improved greatly.

6 146 L. Qin et al. / Chaos, Solitons and Fractals 29 (2006) Fig. 6. Angular velocity error curve. (a) Uncompensated, (b) compensated. At present, there are two methods widely used in error compensation: (1) using error calculating to eliminate error accumulation of angular velocity caused by the accelerometer error, and (2) using random process theory to eliminate the error caused by random disturbance during the measurement of accelerometers. By establishing arithmetic matrix for different errors and doing simulating experiment by computer, the researchers can find the best attitude algorithm. It will be beneficial for navigation. In the following, the errors caused by sampling computation are compensated from two aspects. First, the angular velocity x b z, xb x and xb y of carrier can be obtained directly by the integration of the angular acceleration, so the error will be accumulated. Therefore, the above redundancies of the angular velocity can be calculated directly. Since the redundancies are solved from specific measured by accelerators, the algorithm error will not accumulate with time increasing. Suppose x^ is the angular velocity directly obtained from the integral of angular acceleration, set dx ¼ - x^, and Eq. (6) can be gotten: x x x y x x x z 5 ¼ 6 4 x y x z ðx^ x þ dx x Þðx^ ðx^ x þ dx x Þðx^ z þ dx z Þ ðx^ y þ dx y Þðx^ z þ dx z Þ 3 y þ dx y Þ 7 5 ð6þ With the ignorance of the quadratic small values it becomes: x y x x 0 dx x x x x y x x xy 6 ^ ^ ^ 4 x z 0 x 76 x 54 d x y 7 5 ¼ 6 x x x z x x xz ð7þ ^ 0 x z x y d x z x y x z x y xz The value of angular velocity solved by Eq. (7) can be compensated by the angular-integrated velocity, preventing the influence of integral iterative on calculating error. Second, when the accelerator is installed, its position error is inevitable, and this error will lead to Df error when the accelerator measures specific force. So it is necessary to compensate the value of acceleration to improve measurement precision before calculating flying parameters. Thus algorithm precision can be improved greatly (see Fig. 6, Df = 1 mm). 6. Conclusion The simulation model by computer and the simulation experiment on the ground proved that this attitude measurement system is suitable for the flying object like cannonball and calculating is approximately correct. But because many dynamic errors and iterative errors cannot avoidable in the process of measuring and calculating, the attitude parameters will produce excursion during flying. This measurement method has a great potentiality in attitude measurement with moderate precision. After some improvement such as by filtering and algorithm, algorithm precision can be improved further.

7 L. Qin et al. / Chaos, Solitons and Fractals 29 (2006) Acknowledgments A part of this work was performed at Key Laboratory of Instrumentation Science and Dynamic Measurement of Ministry Education. The authors would like to express sincere thanks to Mr. Wei Yang for his valuable discussions in the course of design and experiment. References [1] Chen JH, Lee SC, Debra DB. Gyroscope free strapdown inertial measurement unit by six linear accelerometers. J Guidance Control Dyn 1994;17(2): [2] Corey VB. Measuring angular acceleration with linear accelerometers. J Control Eng 1962(3): [3] Krishnan V. Measurement of angular velocity and linear acceleration using linear accelerometers. J Franklin Inst 1965; 280(4): [4] Padgaonkar AJ, Kriege KW, King AI. Measurement of angular acceleration of a rigid body using linear accelerometers. J Appl Mech 1975;42: [5] Ruiping N. One error correction method for accelerometer. J Transducer Technol 2002;21(4 6). [6] Miller Robin B. A new strapdown attitude algorithm. J Guidance Control Dyn 1983;6(4): [7] Savage Paul G. Strapdown inertial navigation integration algorithm design. Part 1: Attitude algorithms. J Guidance Control Dyn 1998;21(1): [8] Savage Paul G. Strapdown inertial navigation integration algorithm design. Part 2: Velocity and position algorithms. J Guidance Control Dyn 1998;21(2): [9] Ignagni MB. Efficient class of optimized coning compensation algorithms. J Guidance Control Dyn 1996;19(2):424 9.