Satellite Attitude Determination with AttitudeSensors and Gyros using Steady-state Kalman Filter
Vaibhav V. Unhelkar , Hari B. Hablani Student, email: firstname.lastname@example.org. Professor, email: email@example.com
Department of Aerospace Engineering, Indian Institute of Technology Bombay, Mumbai - 400076
AbstractAttitude determination, along with attitude control,is critical to functioning of every space mission. Here, weinvestigate the application of a steady-state Kalman Filter forattitude determination using attitude sensors (sun sensor andhorizon sensors) and rate integrating gyros. The three-axissteady-state Kalman Filter formulation, adopted from literature,offers obvious computational advantages for on-board implemen-tation on spacecrafts. Thus, closed loop simulation, involvingthe attitude determination formulation, true attitude dynamicsand attitude control, for a satellite are developed in MATLABr-Simulink. Based on the simulation, performance of the filter foran Earth-pointing Low Earth Orbit satellite is presented, in orderto examine the applicability of the steady-state Kalman Filter forsatellite attitude determination.
Maintaining a desired orientation in space, with a spec-ified level of accuracy, is a mission requirement for everyspacecraft. Attitude determination along with attitude controlis responsible for satisfying this requirement. The level ofaccuracy required varies, depending on the function of thespacecraft, which in turn effects the selection of sensors andcomplexity of algorithm.
During the past four decades, a lot of research has beendone in the area of spacecraft attitude determination. Variousalgorithms exist in the literature, with varied level of com-plexity and applicability. The choice of algorithm for a missiondepends on pointing accuracy requirements, the type of sensorsavailable and capability of the on-board computer.
Following  and , which provide a comprehensivesurvey of existing attitude determination approaches, attitudedetermination methods can be broadly categorized as
Single Frame Methods Sequential Methods
Single frame methods are essentially those methods whichdo not require any a priori information/estimate and giveinformation of attitude based only on current measurements.These methods require a minimum of two independent vec-tor measurements, so as to determine the complete attitudeinformation. Sensors measuring reference vectors, such as,Sun sensors, Horizon sensors and Star trackers, are used todetermine attitude using single frame methods. Methods alsoexist which optimally utilize information from more than tworeference vectors to provide the attitude information. Someexamples of single frame methods include, TRIAD, QuEst(Quaternion Estimator) and Davenports q-method.
Sequential methods are those which utilize past informationand provide the solution of the attitude using both thepast estimate and current measurement. This mixing of pastestimate with the current measurement is usually achieved byemploying filters (such as, Kalman Filter). These methods aregenerally more accurate and robust than single frame methodsand can also provide reliable attitude information when somemeasurements are missing. Furthermore, sequential methodsbeing recursive have minimum memory requirements. Correc-tion or updates to the estimate can be achieved through anyof the reference sensors used for single frame methods.
Here, we present simulation for a sequential attitudedetermination algorithm using attitude measurements (fromSun sensors and Horizon sensors) and gyros. Applicationof Kalman Filter for these set of sensors has been exploredin detail . However, calculation of Kalman Gains iscomputationally burdensome. Hence, the performance of asteady-state Kalman Filter is examined for the above setof sensors. The three-axis Kalman Filter formulation hasbeen adopted from , which develops upon the single-axissteady-state analysis of . Corrections obtained from attitudemeasurements are assumed to be obtained from a single-framemethod, and are modeled as true attitude with white noise.The rate integrating gyro output provides the incrementalattitude information corrupted by varying drift-rate bias,white noise and quantization noise.
II. SENSOR MODELSIn this section, we describe the sensor models being used
in our attitude determination algorithm.A. Rate Integrating Gyros
Gyros are inertial sensors which measure the informationrelated to angular velocity in the body frame. Gyros are ofvarious types - such as, mechanical gyros, ring laser gyros(RLG), fiber optic gyros (FOG) - and can be classified basedon their accuracy, mechanisms and form of output. Rate gyrosmeasure angular rate directly, while the rate-integrating gyrosmeasure integrated angular rate. Rate integrating gyros aretypically more accurate . For our analysis rate integratinggyros are used; also, the mathematical parameters of the gyroerrors are taken corresponding to that of Fiber Optic Gyros.
A generic model of the rate integrating gyro is used (),allowing for testing the estimation algorithms for differenttypes of gyros. The rate integrating gyros measure the incre-mental attitude information along with some noise. The overall
measurement equation along with the errors for the gyro isgiven as
= k + Tgyrobk + k + q,k (1)
in = Amisalign (3)
The term k expresses the true change in the spacecraftattitude, while the
kdenotes the rate integrating gyro
output. The gyro drift rate bias is denoted by bk, while zero-mean noise due to continuous time random-walk rate vectornv(t) and drift acceleration nu(t) is expressed by k. Thevariance of
kis a 33 diagonal matrix for which the diagonal
element is 2 = 2Tgyro +
2/sand 2u (rad
2/s3) are power spectral densities of the scalarelements of nv and nu, respectively. The gyro drift-rate biaschanges and this change is represented as
bk = bk1 + k (4)
where k is a zero-mean discrete random-rate noise vector,with variance of each element being 2 =
2uTgyro (). The
term q,k represents the quantization error of the gyro. Lastly,scale and misalignment errors occur due to the mechanicalmisalignments in the system, or intrinsic sensor errors. Theseerrors are quantified using the misalignment matrix, whichdescribes the transformation between the expected and actualsensor axes,
Sx xy xzyx Sy yzzx zy Sz
(5)For simulations presented later, we have used the parameters
of a rate integrating type Fiber Optic Gyro; the same are listedin Table I. No misalignment or scale errors are considered.The sampling rate of the gyros for the simulation is taken as100Hz.
TABLE IGYRO PARAMETERS
Parameter Value Unitsv 7.27 rad/
u 3 104 rad/s3/2e 15 rad
Tgyro 0.01 s
B. Attitude Sensors
Attitude sensors provide information of total attitude, asopposed to gyros which provide information about the incre-mental attitude. These sensors measure a reference vector andby comparing it with its modeled vector provide the requisiteattitude information. Determination of attitude from vectormeasurements is accomplished by the use of single framemethods. In our analysis, we assume the vector measurementshave been processed, and the output of attitude sensors isdescribed directly as the true attitude along with white noise.
Sun sensor and horizon sensors are used as the two attitudesensors. Sun sensors determine the direction of the sun (sun
vector) with respect to the satellite. A reference value of thesun vector can be determined using the position of the satellite,which can be compared against the measurement to determinethe yaw attitude. However, one disadvantage of using sunsensor is that no information is available during the eclipsephase of the satellite, i.e. when the satellite is in the shadowof the earth. This disadvantage is more prevalent in the LowEarth Orbits. Horizon sensors, also known as earth sensors,sense the earths radiation in certain frequency bands, namely,narrow infrared wavelength band corresponding to emissionline of CO2 molecule. Typically, heading and attitude are usedseparately, where sun sensors are used to measure heading andhorizon sensors are used to measure pitch and roll. Thus alongwith the sun sensor, the horizon sensors provide completeattitude information.
As mentioned earlier, a simple model considering onlywhite noise as the attitude error in the sensor output is used forthe two attitude sensors. The standard deviation of the discretewhite noise (n) is different for sun sensor (yaw) and horizonsensors (roll and pitch). For simulation, the standard deviationin the discrete time white noise along the three axis is taken as,The attitude sensors provide output at a lower sampling rate as
TABLE IIATTITUDE SENSOR PARAMETERS
Parameter Value Unitsn,roll 0.0667 degn,pitch 0.1000 degn,yaw 0.0333 degTatt 1 sec
compared to the gyros. For our simulation, they are assumed tobe working at 1 Hz. Apart from the above mentioned sensors,on board hardware is required to measure the position of thespacecraft. For our analysis, we assume the position of thesatellite is known perfectly, and no error models for positionerrors and orbit propagation are considered.
III. ATTITUDE DETERMINATION FORMULATION
A steady-state three-axis Kalman Filter adopted from (and ) is used for attitude determination. The steady-state filter is computationally better than the recursive gainKalman filter as the Kalman Gains are constant and can becomputed beforehand. This eliminates the need of on-boardmatrix computations; since, calculation of Kalman g