VIEWS: 13 PAGES: 6 CATEGORY: Research POSTED ON: 4/16/2012 Traditional Copyright
Initial Attitude Determination and C tation on the Basis o arameters K i d S. Mostov Andrey A. Soloviev T.-K. John Koo Department of Electrical Engineering and Computer Sciences University of California at Berkeley Abstract with an array of relatively inexpensive semiconductor The exclusion of gyroscopes from navigation system accelerometers. Here, the determination of the linear and architecture achieves an essential reduction in systems cost. angular parameters of motion is based on the weighted However, the absence of gyroscopes leads to the growth of averaging of data from the accelerometers. errors in the determination of navigation system angular However, an absence of gyroscopes leads to an orientation. This, in its turn, leads to the growth of errors in accelerated increase in attitude errors. This in turn causes a the determination of vehicle coordinates. Therefore, one of the rapid increase in errors of linear parameters of motion main problems in the construction of an integrated navigation (coordinates and velocity) between GPS fixes. Therefore, in system which is based on a gyro-free Inertial Navigation a gyro-free system, an abrupt degradation of navigation System (INS) and a single antenna Global Positioning System system accuracy can be observed. (GPS) is the determination of the initial angular orientation It is important to mention that a single antenna GPS and correction of gyro-free INS angular characteristics on the which is traditionally used in integrated navigation systems basis of linear navigation parameters (coordinates and velocity) that are obtained from GPS. This paper is devoted to regards the vehicle as a material point with no angular the description of the proposed solution method for this orientation. The use of a multiple-antenna GPS will allow problem. determination of the vehicle attitude, but will lead to an approximate twofold increase in the system cost. Thus, the Introduction critical point in the design of gyro-free navigation systems is the development of an algorithm for the initial attitude Currently, the rapid reduction of the cost of inertial determination and the correction of attitude errors on the sensors and GPS receivers can be observed. A persistent basis of GPS data. In this paper, the idea of a “virtual decrease in the price is predicted for the as well, due to the gyro”, consisting of distributed accelerometers combined continual improvements in semiconductor and with a single antenna GPS is presented. The paper has the micromachining technology. The drop of the cost of these following structure: sensors leads to the possibility of employing integrated Section two is devoted to the brief analysis of the navigation systems in a wide range of commercial existing approaches to the correction of angular parameters applications including the navigation of cars, yachts, used in gyroscopic navigation systems. Since the attitude agricultural vehicles, etc. The traditional approach to the correction algorithm has been developed for the gyro-free integrated navigation system design is based on the INS, section three describes briefly the error inducing combination of the determination of absolute navigation mechanism of the accelerometer based system. parameters (coordinates and velocity) which are obtained A formal problem statement and the basic idea of the from the GPS and the determination of the relative proposed approach are presented in section four. In section navigation parameters (relative changes in coordinates and five, the realization of the proposed approach is presented. velocity) which are obtained from the INS. Here the INS The validity of the proposed approach is verified by a provides the required update rate of the navigation simulation, the results of which are presented in section six. parameters between the GPS fixes and also during loss of a In conclusion, the plan for future research is discussed. GPS signal. However, the INS errors in determination of linear and angular parameters of motion grow continuously with time. 2. Related Work The correction of linear errors is based on the GPS data. A review of publications devoted to the problem of For the precise determination of angular parameters attitude correction has shown that currently there exist no accurate gyroscopes are used. The use of gyros in INS leads published research results addressing the problem of the to: correction of gyro-free INS angular parameters. An analysis a) high system costs ($2500 - $40000) and of the publications devoted to the attitude correction of b) the inclusion of an additional operation mode gyroscopic INS has shown that the proposed approaches are (gyroscope initialization) which takes additional time. based on specifications of gyro-augmented systems [2], [ 3 ] , Therefore, it is more economical to replace gyroscopes [41. 1034 0-7803-4269-0/97/S10.00 0 1998 l€EE For this reason, the direct application of traditional the errors (correction, it is necessary to perform joint methods to gyro-free navigation systems will lead to the correction of’deterministic and stochastic error components. increase in methodical errors. A comparison of the existing b) In c<asesof initial angular alignment (with initial approaches to the correction of angular orientation with the angular uncertainty & E ) , and after the loss of GPS signal proposed attitude correction approach for gyro-free (for example while in a tunnel or while moving in the navigation systems will be presented in detail in section 5. vicinity of skyscrapers, etc.), the components 2b) and 2c) grow rapidly with time and their magnitudes are essentially 3. Gyro-Free INS Error Analysis larger than magnitudes of components 1 and 2a). As mentioned above, an attitude correction algorithm Therefore, based on the conducted research and has been developed taking into consideration a gyro-free analysis, the development of attitude correction algorithm INS error analysis. This section presents the main results of will be performed in two successive stages: this analysis. Stage 1. The development of an algorithm for the An analysis of the growth of errors in the correction of the deterministic components of total error - determination of angular and linear parameters of motion as the basic part of the algorithm. has shown that: Stage 2. A combination of an algorithm of deterministic 1. The noise of accelerometers is the major source of error correclion with methods of stochastic error filtering errors in angular parameters (error component 1). The (particularly on the basis of adaptive Kalman filtering) - as errors show continuous and rather regular growth a modernization of the algorithm. characterized by a constant parameter. This paper is devoted to the description of the basic 2. The sources of errors in linear parameters (errors in part of the algorithm. the determination of coordinates and velocity) are the following: 4. Formall Problem Statement and the Basic a) accelerometer noise. The character of growth of this component (error component 2a) of error approximately Idea of the Proposed Solution Approach The formal statement of an angular characteristics coincides with the character of attitude error growth; correction problem is the basis for the proposed approach. b) a mismatch between angular characteristics of body A graphical interpretation used for the formal problem frame and an Earth-referenced frame which leads to an statement is presented in Fig. 1. For simplification, the incorrect determination of the vehicle’s direction of motion. graphical interpretation is initially presented for the two- This component (error component 2b) is characterized by dimensional case. continuous growth. Here, with the increase of mismatching angles’ magnitudes, an increase in the parameter which zEarth determines error growth rate can be observed. c) an incorrect compensation of gravity due to an zmov. angular mismatch between the body frame and the Earth- referenced frame. The growth of this error component Moving coordinate (error component 2c) is more rapid than that of the previous frame (related with axes component and has an abrupt character. This phenomenon can be explained by the fact that for the 2b) component, the accelerations have, in general, a sign-changing character. For the 2c) component the direction of gravity is always the same in the Earth-referenced frame. Conducted simulations and an analysis of simulation results have shown the possibility of classification of the error components. First, the errors can be categorized as either stochastic or deterministic: a) components 1 and 2a) have a stochastic nature; b) components 2b) and 2c) have a deterministic nature Figure 1. !$paceOrientation of the Body Frame, and are determined by the misalignment angles. On the other hand, errors can be sorted according to Earth-referenced Frame and Fictitious Frame their magnitude: As illustrated, there is an Earth-referenced coordinate a) In cases when the correction of angular parameters frame. Accelerations, measured by accelerometers, are is performed at the frequency equal to the frequency of determined in the moving frame or the body frame. Spacial updates of the linear parameters (or inversely proportional orientation of the body frame is determined by the spacial to the time between the two successive GPS fixes), orientation of INS accelerometers’ sensitive axes. Suppose components 1, 2a), 2b), 2c) are commensurable. Here, for that the anglle between the body frame axes and the Earth- 1035 referenced frame axes is equal to a . The determination of ( t G P S ) the change in vehicle coordinates is characterized vehicle coordinates is carried out by INS (taking into by the vector of relative displacement r. The coordinates of consideration the current value of a ) on the basis of the this vector are known in: relationships presented in [ 2 ] . However, as noted above, a) Earth-referenced frame, based on GPS data; accelerometer noise leads to the accumulation of error in b) fictitious frame, based on INS data. When the coordinates frame coincide with each other determination of a angle. Therefore, INS determines the (for example, after initial alignment) the components of the coordinates in some fictitious coordinate frame, rather than relative displacement vector will be equal in both in an Earth-referenced frame. The angle between the Earth- coordinate systems. However, in presence of angular referenced frame axes and the fictitious frame axes is equal mismatching the components of the relative displacement to a o . Thus, at the initial moment of time (after initial vector will be different in different coordinate systems. alignment) the fictitious frame and the Earth-referenced This can be explained by the rotation of relative frame coincide with each other. Thereafter, with displacement vector (due to the rotation of the fictitious accumulation of error in alpha determination, the fictitious frame) and incorrect compensation of gravity. Thus, it is frame rotates relative to the Earth-referenced frame and the necessary to rotate the fictitious frame for providing the angle between these coordinate frames becomes equal to coincidence of the relative displacement vector components in both coordinate systems. Obtained values of the rotation ao. Thus, the attitude correction algorithm must angles will determine angular mismatching, and thus, will periodically determine the values of angular mismatching determine the correction to the direct cosine matrix - Ccorr. (for the two dimension case - it is a o ;for the three Note that even in the absence of displacement (when the value of the displacement vector in the Earth-referenced dimensional case there are three mismatching angles: <Po, frame is equal to zero) the INS measures a non-zero vector of displacement which has the following components: e,, vo), provide the alignment of the fictitious frame and to make it collinear to the Earth-referenced frame. Taking into consideration a gyro-free INS structure, attitude correction algorithm must periodically define the correction Thus, this approach provides the possibility of to the direct cosine matrix Ccorr. which depends on realizing both the initial alignment (for example, when the misalignment of angles values. The multiplication of the vehicle is at rest) and the correction of attitude errors during current value of the direct cosine matrix by the Ccorr. must vehicle motion. be equivalent to the alignment of the fictitious frame with For the realization of the attitude correction algorithm the Earth-referenced frame. on the basis of the above-formulated idea, the following The formal problem statement and the basic idea of the steps must be taken: proposed algorithm are presented below. The graph of the 1. Prove the criteria of collinearity of the fictitious and basic idea is presented in Figure 2. the Earth-referenced frames: coincidence of relative displacement vector components in both coordinate systems. 2. Choose the trajectory of rotation of the fictitious coordinate system (the sequence of fictitious system rotations) that will provide the criteria fulfilment: coincidence of displacement vector components in both coordinate systems. 3. Determine the formal transformation of relative displacement vector components and the correction to the direct cosine matrix Ccorr that occur during the rotation of the fictitious coordinate system. 5 Realization of Attitude Correction Algorithm 5.1 Analysis of Existing Approaches Traditionally, the angular orientation correction has Figure 2. Graphical Interpretation of Proposed Basic been approached through the comparison of the relative Idea displacement vector components in an Earth-referenced and fictitious frames; a number of existing works discuss the During the time between two successive GPS fixes algorithms of attitude correction using such comparison [2]. These papers propose determining the correction to the direct cosine matrix on the basis of three vectors, obtained in the Earth-referenced and fictitious frames during three successive GPS fixes [2]. Three 3-dimensional vectors are necessary to determine the nine components of the direct cosine matrix. This method is based on the proposition that during three successive GPS fixes the angular mismatching between the coordinate systems remains constant. Thus, the nine components of the correction matrix also remain constant. However, conducted research has shown, that in the case of gyro-free navigation system this proposition leads to the essential error growth in determining misalignment angles. Therefore, this method is not applicable for a gyro-free system. In addition, there are cases for which the system of linear equations for determining the correction matrix components has multiple solutions. For example, this can be observed when the direction of vehicle motion remains constant during three successive GPS fixes. Therefore, for the realization of the attitude correction algorithm in gyro-free navigation systems it was proposed to match coordinate frames on the basis of a single GPS fix. Thus, the angular orientation correction will be performed on the basis of a single GPS fix rather than three GPS fixes. This will lead to a qualitative decrease in errors in attitude determination, since the frequency of attitude correction will be equal to the frequency of the linear parameters correction: the correction will be performed after each GPS fix. The obtained errors will be commensurable with the random error components presented above (components 1 and 2a). Random errors will be decreased using Kalman Figure 3. Proof of Alignment Criteria filtering during the second step of algorithm development. 5.2 Proof of Alignment Criteria Correctness 5.3 Fictitious Frame Rotation Trajectory Choice The next logical step is to prove the possibility of The next step in the attitude correction algorithm angular matching of the two coordinate frames on the basis development is to determine the trajectory of the fictitious of a single vector of relative displacement. frame rotation, which provides the fulfilment of the As outlined above, the criteria of collinearity between alignment criteria: coincidence of the relative displacement the fictitious and the Earth-referenced frames is the vector components in both coordinate frames. coincidence of the relative displacement vector components The traditional approach to solving this problem is in both coordinate frames. The formal proof of this criteria based on the analytical approach: definition of the relative is presented in Fig. 3. displacement vector components in a fictitious frame The proof is based on determining the values of the (obtained from INS) as a function of the relative mismatched angles; the components of the relative displacement vector components in the Earth-referenced displacement vector are the same in both coordinate frame (obtained from GPS or other external reference systems. The only possible value of mismatched angles is points) and angle misalignment. zero. Thus, the only incidence of orientation of the fictitious coordinate frame for which the component of the relative displacement vector in this frame coincides with the components of the relative displacement vector in the ‘Eurrh s i n g s i n 0 s i n p cosgsinp + s i n g s i n 0 c o s p -singcos8 cos 0cosp singcosrp + cosyrsinesinq s i n g s i n 9 - cosy sine cosp c o s g c o s e sine I Earth-referenced frame is the collinearity of the Earth- referenced and the fictitious coordinate frames. An exception is possible only with absolutely vertical movement of a vehicle. However, such a scenario is highly This approach leads to the necessity of solving a system of unlikely and can be virtually disregarded. nonlinear equations for determining the misalignment of angles. The traditional approach is the linearization of this 1037 nonlinear system and determination of misalignment angles second stage of the attitude correction algorithm as solutions of the linear system. However, the provided development - when the stochastic errors are taken into estimation demonstrates that this method remains valid consideration - current criteria is substituted by the least only for a small range of misalignment angles. In practice it mean squares criteria, realized through Kalman filtering. is possible for the misalignment angles to be out of this range - for example, after GPS signal loss or during the 5.3 Determining Formal Transformations for the initial alignment. In such cases the error of this method Fictitious Frame Rotations grows dramatically. In accordance with the chosen sequence of the In order to eliminate the outlined disadvantages it was fictitious frame rotations (steps 1 and 2 ) it is necessary proposed to include an intermediate criteria of alignment: determine the formal transformations of the relative with precise rotation around the vector of relative displacement vector components and the correction of the displacement. This allows for separation of the alignment direct cosine matrix which takes place during rotations. procedure into two independent steps, and significantly Formal transformations for the rotation 1 are as follows: simplifies the determination of misalignment angles for an Formal transformation of the correction to the direct arbitrary range of these angles. cosine matrix rotational axis: The successive alignment steps are as follows: SteD 1 (Fig. 4): p= RGPS RINS Rotation of the fictitious frame allows for the RGPSRINS coincidence of the angular characteristics of the relative rotational angle: displacement vector in both coordinate frames. After this r i procedure only the absolute values of vectors may differ. ZfiC (5.3) correction matrix transformation: 2 casa+p *(I-cosa) p p ( I - c o s a ) + p sina p p (1-eosa)-p sina X Y z x z Y b p (l-cosa)-pzsina X Y cosa+pZy(l -cosa) p p (I-cosa)+pxsina Y Z (5.4) 2 p X p Z ( 1 - c i ) s a ) + b sina p p (l-cosa)-pxsina casa+p y(l-cosa) 1 fict. Y Y z / Xfict. 'XEarth yr XEarth Before Step 1 After Step 1 Figure 4. First Step of Fictitious Frame Rotation Stea 2 (Fie. 5 ) : Rotation of the fictitious frame around the vector of L J L J relative displacement allows for equality of vector absolute It was proposed to perform step 2 as an iteration values in both coordinate systems. When vector procedure: where the fictitious coordinate frame is rotated components coincide, the coordinate frames are collinear. around the displacement vector in this frame. The coincidence of the displacement vector components in the fictitious frame and the Earth-referenced frame is the criteria for ending the iterations. The iterational procedure for step 2 is presented in Fig. 7. 6. Simulation Results The effectiveness of the proposed algorithm has been xfi confirmed by the simulations of the integrated navigation system performance. The characteristics of NovAtel RTK-2 were used as accuracy characteristics of a GPS receiver (the Before Step 2 After Step 2 frequency of the GPS is 2 Hz; the standard deviation of Figure 5. Second Step of Fictitious Frame Rotation error in the coordinates is 4 cm2). Accelerometers with It is important to mention that in this basic part of the standard deviation of noise of 25sm21s4 were used in the algorithm the vector component coincidence is considered simulations. Comparison of performance of an integrated as a criteria of the coordinate system matching. During the navigation system with and without the attitude correction 1038 algorithm is presented in Fig. 6. Here the character of error determination of the formal transformations of attitude correction parameters and the relative displacement Initial Conditions: { R fici Clcorr.} vector, which take place during fictitious frame rotations. v Correctness of the proposed method was confirmed by Zi+ 1 = [cosgcoscp cosysincpcose+ sinysine cosysincpsine- -sincp cOscpcase cos cp sin 0 sinycoscp sinysincpcose- cosysine sinysincpsine+ simulation results. Future work will be devoted to the combination of the deterministic error correction algorithm with filtering of random eirors. a) without anitude correction cosy coscp cosy sincp cos0 + sin y sine cosy sincpsine - s h y cos0 cos cp cos 0 coscp sine Jxci sinvcoscp sinvsincp cos 8 - cosy sin0 sin yr sincpsine + cosy cos0 I Verification of Criteria Fulfilment I t, seconds Figure 7. Iteration procedure of Second Step of b) with attitude correction Fictitious Framer Alignment Ay, meters in determination of y-coordinate ( A y ) for the gyro-free integrated navigation system is shown with and without attitude correction. The y-coordinate error increase with and without attitude correction is similar to the x- coordinate and z-coordinate error behavior. It is important to note that for the navigation system, which uses the attitude correction algorithm, the initial angular orientation was unknown. Thus, both initial alignment and attitude correction have been tested. t, seconds Simulation results have shown that using the attitude Figure 6. Comparison of coordinate error growth for correction algorithm allows to: a) define initial angular intlegrated gyro-free navigation system orientation of gyro-free INS; b) change qualitatively the character of errors in determining linear navigation References: parameters. Thus, a compensation of abrupt degradation of [l] J.H. Chen, S.C. Lee, D.B. DeBra “Gyroscope Free navigation system accuracy was achieved. Strapdown Inertial Measurement Unit by Six Linear Accelerometers”, Journal of Guidance, Control, and Conc1usion Dynamics, Vo1.17, No. 2. March-April 1994, p. 286-290. This paper presented the algorithm of gyro-free INS [2] I.Y. Bar-Itzhack, J. Reiner “Recursive Attitude angular parameters correction on the basis of single antenna Determination from Vector Observation: Direct Cosine Matrix ldenlification ”, Journal of Guidance, Control and GPS. The algorithm is based on the alignment of a fictitious coordinate frame with an Earth-referenced frame through Dynamics, vol. 7, no. 1, Jan-Feb, 1984, p.51-6 [ 3 ] R.E. Phillips, G.T. Schmidt “GPSANS Integration ”, fictitious frame rotation, with rotational axes and angles of rotation defining angular orientation correction parameters. AGARD 1996, p 9/1-18, vi+ 158 pp. The realization of attitude correction algorithm was [4] K.R. Britting “Inertial Navigation Systems Analysis ”, John Wiley, 1972 considered. This realization includes: choice of criteria of alignment of a fictitious frame with Earth-referenced frame and a proof of criteria correctness; choice of trajectory of fictitious frame rotation, which provides for the fulfilment of the alignment criteria; 1039