Comparison of PD and LQR Methods for Spacecraft Attitude Control Using Star Trackers

Transcription

1 Comparison of PD and LQ Methods for Spaeraft Attitude Control Using Star raers Sott Beatt, Universit of New Meio, United States, ABSAC he wor ontained herein is a omparison of spaeraft attitude ontrol methodologies that use reation wheels for torque atuation and star traers to infer spaeraft orientation and angular rate. A Kalman filter was used to estimate Euler angles and angular rate using the measurements from three star traers. Using the linearied equations of motion for a rigid bod in spae, the linearied stabilit, effetiveness and robustness of a linear quadrati regulator (LQ ontrol design were ompared with that of a proportional-derivative (PD ontrol design. he goal of the stud was to determine the degree to whih the optimal gains alulated with the LQ ontrol law improve the performane of a spaeraft attitude ontrol sstem in omparison to the nonoptimal gains alulated with the PD ontrol law. NODUCON he orientation of an aerospae vehile fling in the Earth s atmosphere or orbiting the Earth at etreme altitudes is referred to as attitude. n the ase of a spaeraft; nowledge of its attitude is a ritial requirement when tring to ontrol the diretion that the spaeraft is pointing. A tpial spaeraft onfiguration will use rate gros to provide ontinuous real-time measurements of the spaeraft s attitude and star traers or a omparable instrument to provide aurate attitude updates. he use of star traers for attitude determination without rate gros is an interesting problem. A basi nertial Measurement Unit ontains orthogonall plaed grosopes for eah bod ais that spin at high PM to develop a grosopi stiffening effet. When an MU is mounted to a spaeraft bod the spaeraft will essentiall move around the gros. he MU will return measurements from sensors attahed to the gros whih determine the spaeraft s angle and spin rate about eah bod ais. he gros in an MU an provide ontinuous measurements, however gros have a tenden to drift and lose aura over time. A Star raer wors with a onsiderabl different prinipal. A Star raer uses a CCD imager looing awa from the Earth at a star field to tae an image of the star field; an onboard proessor will that ompare the images and determine the spaeraft s attitude based on the differenes between the two images. Star raers an provide attitude measurements that are muh more aurate then MU measurements, however a star traer annot provide measurements ontinuousl. A tpial implementation uses both an MU and one or more star traers; Star raer(s are used to determine the initial attitude and subsequent attitude updates for MU measurements. Simpl put, a reation wheel is a momentum ehange devie. he reation wheel spins at a moderate PM and as it aelerates to inrease or derease its angular veloit it gives momentum to or taes momentum from one bod ais of the spaeraft it is mounted to.

2 SPACE ENVONMEN & OBS A Molnia orbit was seleted for this simulation as it was for the previous wor. he tpial disturbanes at the simulation attitude inlude solar wind and gravit gradient effets. he orbit period is approimatel 1 hours and inlined at 63.5 degrees to redue the rate or perigee preession due to the J effet. Figures 1 and show diagrams of the Molnia Orbit and its period: Figure 1: Molnia Orbit Diagram Figure : Molnia Orbit Period he speified perigee and apogee radii of the orbit used in this wor are m and m respetivel. his gives approimate altitudes of 1m at perigee and 35785m at apogee. he eentriit for this orbit is.71. he perigee altitude is in the Low Earth Orbit (LEO region and the apogee altitude is in the Geosnhronous Earth Orbit (GEO region. Solar radiation pressure auses periodi variation in all of the orbital parameters of a spaeraft s presribed orbit. he distane between perigee and apogee are insignifiant ompared to the distane of the Earth from the Sun; therefore effet of the solar radiation flu is onstant at =135W/m. he fore imparted b solar radiation pressure is defined as: F F NCDEN EFLECED A = A = q (1 Where A is the eposed surfae area of the spaeraft and is the speed of light. he term q is the fration of radiation that is refleted. he total fore is then defined as: F A = F + F = ( 1+ q ( SP And, the torque on the spaeraft is: = F (3 SP CM CP SP

3 he term CMCP is the distane from the enter of mass of the spaeraft to the enter of pressure of the eposed surfae of the spaeraft. Gravit gradient effets on a spaeraft s attitude an beome signifiant if the moments of inertia for eah bod ais of the spaeraft are not equal. he gravitational effet on eah spaeraft bod ais is defined as: ( Y ( G X = 3Ω φ GY = 3Ω X θ G = μ where Ω = is the orbital angular veloit, 3 X inertia andφ, θ and,ψ are the spaeraft bod Euler angles. (4, Y and, are the spaeraft moments of SPACECAF AUDE EQUAONS OF MOON o auratel simulate and ontrol a spaeraft, rigorous nowledge of the spaeraft s attitude dnamis is required. hree referene frames pla an important role in this simulation: he bod frame has its origin at the enter of mass of the spaeraft and is fied to the spaeraft bod, the orbital referene frame is also loated at the enter of mass of the spaeraft, however, this frame maintains a nadir pointing orientation with the Ẑ O ais pointed at the Earth throughout the orbit, Xˆ O points along the orbital path and Yˆ O ompletes the right hand triad and points normal to the orbital plane. he Euler angles between the bod frame and the orbital frame represent the spaeraft pointing error. he final referene frame is the inertial frame whih is also loated at the spaeraft enter of mass. he inertial frame is defined as Xˆ points towards the Earth s Vernal Equino, Ẑ is perpendiular to Xˆ and normal to the Earth s orbit plane around the Sun and Yˆ ompletes the right handed triad. he Euler angles between the orbital frame and the inertial frame represent the nadir pointing requirement for the spaeraft. he A matri for the spaeraft is given in (5 and is derived from the Euler moment equations. (5 4 Ω ( + Ωh Ωh Ω& Ω( 1 h + h 3Ω ( h 1 h Ω& Ωh Ω ( + + Ωh Ω( + h 1 + h he states orrespond to roll, roll rate, pith, pith rate, aw and aw rate of the spaeraft bod with respet to the orbital frame sine the were obtained b taing the derivative of the total angular momentum with respet to the inertial frame.

4 CONOL SYSEM SPECFCAONS he simulated spaeraft has three star sensors aligned with the bod aes where one star sensor aptures and proesses an image ever 1 seonds. At eah time step where an image is taen b the star sensors and subsequentl an attitude measurement is taen, the dnamis of the spaeraft are froen in time to simplif the sstem suh that it is modeled as a linear time invariant sstem. he roll and pith inertias X and Y are 5, g-m and the aw inertia is 15, g-m so that the spaeraft is gravit gradient friendl. n order to auratel ompare the PD and LQ ontrol laws the same design riteria was hosen for the LQ as.1 degree. he PD ontroller is based on: & = A + Bu (6 where A = A( t = A and is the plant matri froen at time t. A state traing error is introdued whih represents the error between the desired attitude states and the atual attitude states. he state traing error is given as: ~ = (7 he ontrol input torques is determined from si ontrol gains and the si attitude states: h& h& h& X Y = = = VX VY V & φ + K Xφ & θ + K θ Y ψ& + K ψ (8 he final representation of equation (6 beomes: ~ & = A ~ + ( A & + Bu (9 Where is the urrent state vetor, A & represents the error between the desired & and atual &. he terms B and u are the ontrol and input matries, respetivel. he ontrol loop is losed b measuring the angular momentum of the reation wheels and the urrent attitude states in terms of the Euler angles and Euler angle rates. hen the augmented plant is determined b alulating Aaug = ( A BF Where F is the gain matri for the PD ontrol is the state referene vetor and the term ( he linear quadrati regulator (LQ alulates one step optimal gains for the ontrol of the dnami sstem at eah time step in the simulation. he linear quadrati regular is based on the matri iati equation [14] to minimie the ost funtion: V ~ Q ~ + u u (1 = Where ~ is the state traing error of the spaeraft, Q is the plant ovariane matri and is the sensor ovariane matri. For the one step optimiation the onst funtion beomes:

5 V ( Φ + Γ u Q ( Φ + Γ u + u u = (11 Φ + Γ u = and the optimal one step gain is defined as: where ( + 1 u 1 = ( + Γ QΓ Γ Q Φ (1 he gains are alulated for the single step optimiation using (1. he ovariane matri Q is hanging with time so it is epeted that the optimal gains derived b the LQ ontrol law will redue the magnitude of the torque ommands to the reation wheel resulting in a lower stored angular momentum in eah reation wheel. Given that the state measurements for both ontrol laws are disrete and that eah set of gains are alulated for a nominal plant matri at the time the measurement was taen; the gains must maintain the stabilit of the ontrolled sstem b eeping the eigenvalues of the augmented plant matri to the left of the j ω ais until the net state measurement is taen. he sstem with hanges in orbital position is represented b: ~ & = ( A + δ A & + Bu (13 whereδ A represents the hange in orbital position and attitude of the spaeraft between state measurements. herefore the orbital motion of the sstem between measurements must be bounded suh that the ontrolled sstem remains stable until the net state measurement is taen and a new set of gains are alulated. Figure 8 shows a graphial representation of this requirement. Figure 3: Disrete Measurements and Disturbane Bound Where t 1 and t are two disrete measurements where a nominal plant matri is alulated and the ontrol law gains are alulated. he termδa tn is the allowable bound for the spaeraft s orbital motion between state measurements. ESULS Several simulations were eeuted with varied moments of inertia and sample time periods. he orbit path for eah simulation was divided into 5 time steps where eah step equated to approimatel.7 degrees of true anomal. For the first set of simulations the moments of inertia were inreased for eah run b an order of magnitude and maintained their smmetr from X =5, Y =5 and =15 to X =5, Y =5 and =15 in g-m ; and for eah inrease in the moments of inertia the sample times were inreased in orders of magnitude from 1 seond to 1 seonds. For the seond set of simulations the ais and ais moments were inreased b perentages of +1.5%, +.5%, +5% and +1%. Furthermore, the ais was varied

6 b the same perentages on a third set of simulations. Varing the moments of inertia and the sample times provided a good robustness analsis. he LQ ontrol law produed a stable sstem with ever run. Eah of the simulations, however, had maimum sample times; one the maimum sample times were reahed the LQ ontrol law ould no longer reah a solution. he simulations also showed that the LQ ontrol law generated gains that provided overall improved ontrol of the spaeraft. hroughout all of the simulations the pith error from the LQ ontroller was muh the same as the error from the PD ontroller, however, the roll and aw errors were greatl redued. Furthermore, the ontrol torques to the reation wheels were also greatl redued whih lead to less angular momentum possessed b the reation wheels. Eah simulation has a randoml seleted point in the orbit where the plant and augmented plant models were subjeted to a step response analsis. n eah ase the LQ ontrolled step responses had less overshoot and faster settling times then the PD ontrolled step responses. CONCLUSON he wor herein implemented two ontrol laws for spaeraft attitude ontrol. A proportionalderivative ontrol law was shown to stabilie a linearied set of differential equations of motion whih formed a si degree of freedom simulation of a spaeraft orbiting the Earth in a Molnia orbit and a linear quadrati regulator was implemented to ontrol the same set of equations. Simulations were onduted using numerial analsis to simulate a spaeraft equipped with three star traers where onl one star traer was randoml seleted to provide a measurement for eah time step. A Kalman filter was implemented to estimate measurements of the other two star traers to determine the si states of the spaeraft s attitude. he analsis used the simulations to ompare the stabilit, effetiveness and robustness of the two ontrol laws. he simulations varied the moments of inertia of the spaeraft smmetriall b inreasing the moments but maintaining the shape of the spaeraft bod and non-smmetriall adjusting the moments to mae the shape of the spaeraft bod more retangular. he step response analses show that both ontrol laws were able to stabilie the plant and dampen osillations, however, the LQ ontrol law shifted the poles of the plant farther to the left of the j-w ais then the PD ontrol law. Effetiveness was analed b looing at step responses randoml eeuted during eah simulation and omparing the overshoot and settling times. Furthermore, a ost funtion was evaluated to measure the pointing errors of the spaeraft while it propagated through its orbit. his wor has shown that an optimal ontrol law an outperform a PD ontrol law using star traers as the onl measurement soure for attitude determination of a spaeraft. he simulations also showed that with optimal gains ame optimal torque ommands whih redued the total amount of angular momentum input into the spaeraft per time step. As a onsequene the reation wheels for the spaeraft ould be phsiall smaller if the LQ ontrol law was used instead of the PD ontrol law. Finall, the simulations suggested that a sample time range between 1 seonds and 3 seonds would be the most suitable for this spaeraft onfiguration sine sample times in that range had the best ombination of pointing errors, settling times and was reasonabl far from the maimum sample time of the LQ ontrol law. EFEENCES [1] Beatt, Sott M., Comparison of PD and LQ Methods for Spaeraft Attitude Control Using Star raers hesis, Universit of New Meio, 5