2013 American Control Conference (ACC) Washington, DC, USA, June 17-19, 2013
Orbit and attitude control for gravimetry drag-free satellites: when disturbance rejection becomes mandatory * Enrico Canuto, Member ISA, Senior Member IEEE
Abstract— The paper outlines orbit and attitude control problems of a long-distance two-satellite drag-free formation for the Earth gravity monitoring. Modeling and control design follows the Embedded Model Control and shows how disturbance dynamics and rejection are mandatory. Orbit and attitude control can be treated separately except in the thrust dispatching law of the all-propulsion actuation. Orbit and attitude control split into three sub-problems to be designed in a hierarchical way. In both cases the inner loop is a wide-band drag-free control aiming to zero the linear non-gravitational accelerations of the orbit control and the total angular acceleration in the attitude case. Drag-free demands for disturbance measurement and rejection by means of a specific disturbance dynamics and observer. The orbit outer loops are the altitude and formation distance controllers. The attitude outer loops are in charge of rejecting the residual drag-free bias and drift, which demands a narrow-band control suitable for star tracker measurements, and of aligning the optical axes of each satellite, which demands accurate sensor and wide bandwidth. Simulated and experimental results are provided.
I. INTRODUCTION Post GOCE (Gravity field and steady state Ocean Circulation Explorer) [1] and post GRACE (GRAvity recovery and Climate Experiment) [2] space missions will rely on a formation of free-falling ‘proof masses’ and on the measurement of their distance variations for revealing anomalies of the Earth gravity. Spacecrafts hosting freefalling masses are referred to as drag-free satellites. Two alternatives are possible. 1) Free-falling mass solution: the satellite chases the proof mass to keep it centered in the cage through a control system fed by mass position sensors and actuated by thrusters mounted on the spacecraft. 2) Accelerometer solution: the proof mass is endowed with an active suspension system keeping the mass centered in the cage. Since the known suspension force equals the nongravitational forces acting on the satellite, the latter can be directly cancelled by the on-board thrusters driven by a dragfree control. The paper is concerned with this solution. Gravimetry is improved by a long-distance formation as in the GRACE mission, by making each satellite drag-free as in the GOCE mission and by disposing of a high-accuracy long-distance measurement like that provided by laser * Part of this work has been done under a contract with Thales Alenia Space, Torino, Italy, within a study granted by the European Space Agency. E. Canuto is with Politecnico di Torino, Dipartimento di Automatica e Informatica, Corso Duca degli Abruzzi 24, 10129 Torino, Italy (e-mail:
[email protected]).
978-1-4799-0176-0/$31.00 ©2013 AACC
interferometers. Formation distance requirements are dictated by the distance variation range of interferometers, which is in the order of several hundreds of meters. During normal operations, attitude control must be rather accurate to allow the laser beam pointing to the receiving optics. We assume that each satellite launches its own laser beam to be aligned with the incoming beam of the companion satellite. Each satellite must dispose of sensors capable of accurately measuring axis misalignment. Laser optics is designed such as laser pointing is achieved by a two degrees-of-freedom attitude control (pitch and yaw). The third degree (roll) is used for servoing solar panels to sun direction. Attitude, during laser pointing acquisition, and roll, during normal operations, are measured by star trackers. Formation control need to measure the relative satellite position which is done by differential satellite-to-satellite navigation instruments like GPS receivers. Drag-free control requires accelerometers providing linear and angular accelerations. Drag-free, formation and attitude control are actuated by a propulsion assembly as in [3] and [4], which in this case consists of eight small proportional thrusters capable of a few millinewton thrust. The control problem is tackled within the Embedded Model Control [5], which calls for a control unit around the real-time embedded model of the controllable dynamics to be complemented by the dynamics of the disturbance to be rejected. Orbit and drag-free control can be considered as subsystems of the formation control. Laser pointing and roll control take advantage of an angular drag-free control, which is demanded to zero angular accelerations. Alternative design of drag-free satellites has been mostly developed in the frequency domain as in [6], [7], [8]. Section II is devoted to orbit dynamics and control. A perturbed dynamics around a common circular reference orbit is formulated, which leads to Hill’s equations. Only orbit altitude and drag-free control are treated showing that a ‘stochastic’ disturbance dynamics is mandatory to estimate and reject unknown disturbance. Formation distance control is only mentioned as an objective. Orbit control is arranged in a hierarchical way where drag-free control is the inner loop. Section III treats attitude and pointing control: they are designed in a hierarchical way as the orbit control. Angular drag-free control becomes the inner loop and repeats the CoM drag-free equations. Attitude control employing star trackers becomes the outer loop in charge of rejecting the residual drag-free bias and drift, and of forcing body axes to track a common orbital frame. When laser pointing control
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becomes operational, star tracker attitude restricts to bias rejection. It complements drag-free control at lower frequencies and leaves pointing control to align satellite axes. Experimental and simulated results are shown. II. ORBIT DYNAMICS AND CONTROL A. Reference frames Three main frames are necessary. The inertial frame I = {O, i1 , i2 , i3 } is the Earth-centered equatorial frame frozen at some date. The formation frame L = C , l1 , l2 , l3 is the common local orbital frame, defined by the mean and the mean velocity position = r ( r1 + r2 ) / 2 = v ( v1 + v2 ) / 2 through the following relations r / r×v / v r , , v= l1 = l2 × l3 , l2 = l v,r= r . (1) = 3 r / r×v / v r
{
}
Formation to inertial transformation Rli derives from (1). The third frame is the body frame of each satellite Ck = Ck , bk1 , bk 2 , bk 3 centered in the satellite CoM Ck , k = 1, 2 , where one refers to the ahead satellite along l1 and two to the following. The first axis bk1 is assumed to be aligned with the outcoming laser beam; the third axis bk 3 is aligned with the direction normal to the solar panel plane. The body to inertial transformation is Rki . Given a coordinate vector, say r , measurement, reference, prediction, tracking error and orbit average are denoted by r , r , rˆ , r= r − r and
{
}
rave ( t ) = ∫
t
t−P
r (τ ) dτ / P ,
(2)
where P = 2π / ω is the orbital period of a reference orbit. The body to formation transformation holds Rki = Rli Rkl B. Orbit and formation dynamics Orbit and formation control aims to make the relative position ∆r= r1 − r2 drag-free, to keep constant the average length d of ∆r and the orbit average altitude h . Orbit and formation dynamics is obtained from perturbation equations (Clohessy-Wiltshire/Hill’s equations [10]) with respect to a reference circular orbit of radius r , velocity v and angular rate ω ≅ 1.17 mrad/s f ≅ 0.18 mHz . The reference frame L = C , l1 , l2 , l3 is defined as in (1). The perturbed position and rate coordinates (longitudinal, radial and cross-track) of each satellite k in the frame L are
{
∆rk
}
(
)
T
= ∆rkx ∆rkz ∆rky , ∆ v k ∆vkx
∆vkz
T
∆vky / ω . (3)
Velocity coordinates ∆vkj are the local time derivatives ∆rkj and the longitudinal rate is defined as in [3] by (4) ∆= vkx ∆rkx + 2ω∆rkz . Confining the effects of J2 (Earth flatness) and of a small eccentricity ( e < 0.005 ) into periodic terms, the modified Hill’s equation [3] of the perturbations (3) becomes 0 ∆rk −2 F I ω ∆rk ∆ v = −Ω 2 F T ∆ v + R l u + d / ω + ( ) k k k k k , (5) 0 + ω η g 2 ( 2θ ) r + ε G2 ( 2θ ) δ rk
Matrices and parameters in (5) are as follows: = Ω diag {0, ω , ω} , ε < 0.01,η < 0.002 0 2ω 0 . (6) F= 0 0 0 , θ= θ 0 + ωt 0 0 0 In (5), G2 ( 2θ ) is aperiodic matrix and g 2 ( 2θ ) (common mode) is a periodic vector, whose expressions can be found in [3]. Both terms depend on J2 and e , and their components satisfy G2ij ≤ 1 and g 2i ≤ 1 . u k and d k are the command vector and the non-gravitational acceleration to be rejected by drag-free control, respectively. The sum a= u k + d k is k the total non-gravitational acceleration. u k and d k are expressed in body coordinates and are converted into reference orbit coordinates by Rkl ≅ Rkl , where equality assumes that the attitude control is active. The following transformation (7) ∆r = ∆r1 − ∆r2 , ∆ v =− ∆ v1 ∆ v 2 allows to convert (5) into differential perturbation equations. They have the same form of (5), except for their command and drag, which become (8) u= u1 − u 2 , ∆d = d1 − d 2 . ∆ The common mode g 2 drops from differential equations. Two kinds of measurements are available: (i) the body accelerometer measurements a k of a= u k + d k , (ii) the k satellite-to-satellite measurements rk and v k of rk and velocity v k . Accelerometers (10 Hz sampling) are affected by drift d ak and by zero-mean noise a k at discrete time ti = iT , T = 0.1s . Their equation is (9) a k ( i ) =a k ( i ) + d ak ( i ) + a k ( i ) . GPS-like data (1 Hz sampling) are affected by zero mean noise rk and v k . The measurements of the mean position and rate in the local frame, at the discrete time j ( i ) = floor ( i / N ) , N = 10 , can be written as T = r ( j ) Rli ( r1 ( j ) + r2 ( j ) ) / 2 , (10) i T = v ( j ) Rl ( v1 ( j ) + v 2 ( j ) ) / 2 − ωl × r ( j ) upon knowledge of the frame angular rate ωl . Similar expressions hold for differential position and rate in (7).
( ) ( )
C. Control objectives and hierarchy The ideal control objectives are the following. Drag-free: the non-gravitational acceleration of each satellite must be zero, which is demanded by the accelerometer solution addressed in this paper, i.e. (10) ak = 0 . 2. Formation: the average distance must be constant, i.e. (11) d ave = d . 3. Orbit: the average altitude must be constant, i.e. (12) rave = r . Drag-free requirements are expressed by a bowl-shaped upper limit Sa ( f ) of the spectral density of a k . The density profile reaches a minimum in the frequency band
1.
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= B
{1 mHz ≅ 5f ≤ f ≤ 0.1 Hz} ,
(13)
and diverges outside B (see Fig. 1). At low Earth orbits (300 to 400 km of altitude) suitable for gravity monitoring, drag rejection up to f d ≅ 0.5 Hz has been shown necessary to cope with (13) [4]. That is made possible by an accelerometer bandwidth wider than the Nyquist frequency of 5 Hz. Orbit control must be multi-rate and hierarchical. The inner loop, based on accelerometer measurements, makes each satellite drag-free as in (10) for 0 ≤ f ≤ f d . The outer loop keeps constant the average formation distance and the average orbit height. Hierarchy must be completed with a frequency coordination since the outer loop command becomes the reference of the inner loop and must not contrast drag-free control. In practice the frequency content of the reference signals are restricted to stay within the low frequency band where Sa ( f ) diverges. Requirements in terms of average quantities is mandatory for making formation and orbit commands to be harmonic-free for f ≥ f , such as to respect the bowl-shaped limit Sa ( f ) . The outer loop employs the average measurements rave (n) , in (10), that are sampled at … of r( j) n ( j ) = round ( j /= M ) with M round ( P / ( NT ) ) ≅ 5400 . Here only orbit and drag-free control are treated. The total orbit command decomposes as uk (i ) = u dk ( i ) + u ok ( n ( i ) ) + u fk ( n ( i ) ) , (14) n ( i ) = round ( i / ( NM ) ) where d , o and f denote drag-free, orbit and formation. D. Orbit altitude control
disturbance state xd to be cancelled: δ m 1 1 1/ 2 δ m = δ r ( n + 1) 0 1 1 δ r ( n ) + xd 0 0 1 xd 1/ 2 0 0 w . (17) P +b 1 u x ( n ) + 1 0 r , b = w 2 ω 0 0 1 d ∆rave ( n ) = δ m ( n ) , δ r ( n ) = ∆r ( n ) − ∆r ( n − 1)
bd= xd ( n ) + wr ( n ) x (n)
The rate ω y in (15) has been replaced in (17) by the average ω . The last two terms in (15) have been cancelled because they are zero-mean The noise signals wr and wd have been added for driving the disturbance d x , which is mainly caused by the accelerometer drift d ak in (9) [9]. The order of magnitude d a = d ak for a GOCE class accelerometer satisfies d a < 0.5 μm/s 2 . The following lemma is straightforward. Lemma 1. Equation (17) is controllable by u x and observable from ∆rave .□ The altitude requirement (12) can be rewritten in terms of δ m , as the zero average perturbation δ m ( n ) = 0 . The following theorem states the conditions for the expected value of δ m to tend asymptotically to zero. Theorem 2. If wr and wd are zero mean and bounded, and the feedback gains km and kr stabilize the controllable part of (17), the control law ux ( n ) = ( −kmδ m ( n ) − kr δ r ( n ) − xd ( n ) ) / b (18)
In the GOCE mission [11] the average orbit height has been regulated from the ground station to stay within ensures the following limit is attained, i.e. fluctuations of about 10 m. The uplinked command was lim n →∞ E {δ m ( n )} = 0 . (19) impulsive and about one pulse per week was sufficient. Disposing of more accurate thrusters, a continuous control is Proof. The proof follows by replacing (18) in (17), which possible. Altitude dynamics follows by restricting (5) to the yields the closed-loop equation perturbed dynamics of the orbit radius rk = rlk 3 , where k km 1− 1 − r δ m δ m 0 lk 3 = rk / rk is defined as in (1). Dropping k , the altitude = 2 2 ( n ) + wr ( n ) . (20) δ r ( n + 1) δr perturbation to be zeroed is defined as δ r= r − r , r being 1 −km 1 − kr the reference. The following results descends by applying two asymptotically stable eigenvalues (5) to δ r , and confirms the well-known fact that altitude can Fixing 0 < γ =− 1 λ ≤ 1 , , the gains in (20) become j = 1, 2 j j be actuated by a longitudinal thrust u x . (21) km= 2 ( γ 1 + γ 2 − γ 1γ 2 ) , kr= γ 1γ 2 . Theorem 1. The altitude error ∆r along r satisfies ω y 1 A noise estimator [9] estimates the noise components wr ∆r = r − ω y2η r sin ( 2θ ) , (15) ( ux + d x ) − (t ) , and updates as in the next Section E. and w x 2ω y 2ω y d d
where ω y and ω y are the local vertical local horizontal angular rate and acceleration. ω y enters (15) unlike (5), since r does not track a circular orbit because of J2 and eccentricity.□ The average ∆rave ( t ) of ∆r sampled at t = nP can be written as the difference ∆rave ( n= ) m ( n ) − m ( n − 1=) δ m ( n )
= m (n)
1 nP 2π = r (τ )dτ , P ∫ 0 P ωy
.
(16)
The following average and sampled equation descends from (15) and (16), and includes the first-order dynamics of the
E. Drag-free control Consider equation (5) which applies to each satellite CoM Ck and to formation center C ; the latter coordinates are in (7). The only way to make C drag-free is to make Ck dragfree as expressed by (10). Accordingly, the control law becomes the disturbance rejection (22) u dk ( i ) = −d k ( i ) . The problem to be solved is the implementation of (22), taking into account that a k in (9) is the error signal (22) a= uk + dk . k The issue corresponds to the classical feedback disturbance
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Secondly, a noise estimator, that correlates w with the innovation expressed by the model error between plant and model output, is designed [9]. For the model error be available, the controllable dynamics from command to measurements must become part of the control unit together with the disturbance dynamics (embedded model). The closed-loop system made by embedded model and noise estimator constitutes a state predictor. In this case, the disturbance dynamics has been designed from experimental data of the thermosphere density and of the thruster noise [4]. A second order stochastic dynamics is the result. Restricting to a generic coordinate and satellite, the embedded model reads as 1) Ax ( i ) + w ( i ) + Bu ( i ) , a (= i ) Cx ( i ) + ea ( i ) x (i +=
1 − β β 0 β 1 = 1 , B = 0 , w = A 0 0 0 0 1 C = [1 0 0]
wa w= d , x ws
xa x d xs
. (23)
Equation (23) must be complemented with the equation (24) d ( i ) + d a ( i ) = xd ( i ) + wa ( i )
The variables xd , sd are the disturbance states, and the drag-free control law (22) converts into (25) ud ( i ) = − xd ( i ) .
The noise wa cannot be rejected being simultaneous to u ( i ) [5]. The next theorem states how to measure xd . Theorem 4. The only way to measure xd in real-time is to correlate w to the model error ea= a − xa . In view of the linear and time invariant (LTI) equation (23), a necessary and sufficient condition, under bounded ea , to guarantee bounded measurement error is the static correlation (26) w ( i ) = Lea ( i ) , that ensures that A − LC is asymptotically stable. Proof. The proof follows from Lemma 1 and from L being the gain vector of a LTI state observer. The goal is to guarantee that ea is bounded. The latter encodes the effects of the neglected dynamics [9], which fact demands to restrict the state observer bandwidth to below 0.5 Hz . The latter value is sufficient to low-Earth-orbit drag-free control.□ 1
Acceleration [µm/s2]
rejection that requires integrative action, and more generically to the internal model principle. To this end d k must be split into predictable and not predictable components x dk and w ak , i.e. (22) d= x dk + w ak . k Theorem 3. Assuming full uncertainty, i.e. that no relation of d k is known with any variable except a k in (9), the control law (22) can be implemented if and only if d k is the output of the dynamic model x dk (= i + 1) Ad x dk ( i ) + Fd ( a k ) , x= x dk 0 dk ( 0 ) (22) = d k ( i ) Cd x dk ( i ) + Fw ( a k ) driven by a k . The feedback functions Fd and Fw must be designed to guarantee closed-loop stability. Proof. Assume to know exactly d k ( i ) at time ti = iT . If d k ( i ) = constant , the value should be conserved in (22) and Fd = 0 , Fw = 0 (the reset signal in classical control). Otherwise d k ( i ) must be conserved in (22), but corrected by the measurement a k through Fd and Fw . The latter operators must stabilize the closed loop made by (22), (22), (22) and (9). □ Embedded Model Control [5] splits the design of (22) in two steps. Firstly the disturbance dynamics, driven by the noise vector w is designed, which amounts to fix x dk , A and C in (22), and to replace the feedback operators with (22) = Fd ( ⋅) Gd w= , Fw ( ⋅) H d w .
0.1
0.01
0.001
0.0001 0.001
0.01
1
0.1
10
Frequency [Hz]
Fig. 1. Spectral density of the longitudinal residual acceleration of the GOCE satellite (courtesy of Thales Alenia Space Italia).
Fig. 1 shows the experimental drag-free residuals of the GOCE satellite along the axis aligned with the velocity vector. The straight line segment above the spectral profiles is the upper limit Sa ( f ) , which is equal to 0.025 μm/s 2 / Hz in the range B of (13). Residuals vary with the epoch because of the solar activity. Simulated profiles, not reported here, show the same shape. III. ATTITUDE DYNAMICS AND CONTROL A. Attitude dynamics The attitude dynamics of each satellite k with respect to the formation frame L in (1) can be written in the 321 Euler angles {ψ k , θ k , φk } assuming small yaw and pitch. Roll is the sum φ= ϕk + ϕk of a reference roll ϕk and of the error k ϕk . The reference roll ϕk makes the solar panel pointing toward the sun. The body angular rate coordinates ω can be written in the non-rotating body axes as follows
( ( ) )
T
ω y ω z X ϕ ωb − [ 0 ω 0] , (27) ωT ω x = showing that xd must account for the accelerometer bias = and drift d a in (9). In fact the latter cannot be separated where ω denotes rotating body coordinates, and X ⋅ () b from d in order to make the following Lemma to hold. denotes a Euler rotation. Gravity gradient, gyro and Lemma 2. Equation (23) is observable from a . □ magnetic accelerations are included in the unknown drag The state variable xa accounts for the thruster-to- acceleration d as in the orbit formulation, since it is k accelerometer dynamics, that for GOCE-class measured by accelerometers and rejected by the wide band accelerometers may be simplified to be a delay, i.e. β = 1 . angular drag-free control. All the torques must be rejected As a consequence = a xa + ea , ea being the model error [5]. 4322
including gravity gradient. The state equations holds 0 q k Q I q k = + ω u k + d k k k 0 0 ω k a=
(36) = v k ω k NT + Qq k where the derivative of Qq k ≅ Qω k is included in the unknown disturbance d k to be rejected by drag-free control. Dropping coordinate and satellite subscripts, the following embedded model is adopted [4], [9] x ( j += 1) Ax ( j ) + Bu ( j ) + Gw ( j ) , (37) = q ( j ) Cx ( j ) + eq ( j )
, (28) 0 0 −ω 0 0 0 ω 0 0 where now a k is an angular acceleration vector. with the following matrices Measurements are provided by two star trackers through the q 1 1 0 0 0 0 rotation equation T T = = A 0 1 1 1 , x v= , G 1 0 . (38) , B = (29) = q k × Rli Rski Rskbk X ( −ϕk ) − I xd 0 0 1 0 0 1 where Rskbk is the instrument matrix and Rli is the measure of wq wd , C [1 0 0] wT = the orbital transformation from GPS measurements. = Combining star trackers and orbital frame errors, (29) can be The state equation is observable from q . In (37) q is the rewritten at the star tracker sampling time as follows attitude, = v ω NT ± ω qh is the attitude increment obtained q k ( j ) =q k ( j ) + d qk ( j ) + q k ( j ) , (30) from (36) and the interconnected angle qh (it may be zero), xd is the stochastic state accounting for accelerometer bias where d qk denotes bias and q k zero-mean random errors. and drift, and eq is the model error. When optical sensors are available for measuring the Theorem 5. Under bounded w , the control law reciprocal misalignment of the optical axes bk1 , k = 1, 2 , uq ( j ) = − xd ( j ) − kq q ( j ) − kω ( v ( j ) ± ω qh ) , (39) their measurements apply to the differential dynamics 0 ∆q Q I ∆q satisfies (33) less a zero-mean bounded error, if and only if = ∆ω 0 0 ω + ∆= . the gains kq and kω stabilize the closed-loop matrix (31) ∆ ∆ a u + d
ϕk = q k = θ k , Q ψ k
( )
( )
1 Ac = − kq
∆q = q1 − q 2 , ∆ω = ω1 − ω 2 ,... The measurements, restricted to pitch and yaw read as ∆θ k =(1 + εθ k ) (θ k − θ j ) + ∆bθ k + ∆θ k , (32) ∆ψ k =(1 + εψ k ) (ψ k −ψ j ) + ∆bψ k + ∆ψ k where j ≠ k , εθ k and εψ k are scale errors, ∆bθ k and ∆bψ k are biases. B. Control requirements and hierarchy Ideal control requirements split as follows. Residual acceleration must be zeroed as in (10). Attitude and rate must be zeroed: (33) = ω k 0,= qk 0 . 3. Laser pointing requires to zero the optical axis misalignment, i.e. (34) = ∆θ 0,= ∆ψ 0 . Zero attitude applies to roll, to pitch and yaw when optical sensors are not available. Since attitude refers to the reference frame L , zero attitude is equivalent to pointing requirement, except that the latter asks for more accurate sensors. A similar hierarchy as in Section II is adopted, the drag-free control being the inner loop, and the attitude control the outer loop. The total control u k is the sum (35) uk (i ) = u dk ( i ) + u qk ( j ( i ) ) + u pk ( j ( i ) ) , 1. 2.
Where j ( i ) = round ( i / N ) , q refers to attitude and p to pointing control. The drag-free control u dk is the same as the CoM drag-free in Section II.E and is not repeated here. C. Attitude control using star tracker The embedded model is obtained upon the following definition of the angular increment
1 . 1 − kω
(40)
Proof. The theorem follows from the closed-loop state equation which consists of (37) and (39), and the asymptotic closed-loop equations under w = 0 , namely
( k x I − kω Q ) q k
=0 ⇒ q k =0
0 ⇒ ωk = 0 vk = ω k NT + Qq k =
.
(41)
The gains kq and kω are designed to achieve a BW f q close to 0.005 Hz which is well less than the drag-free BW to attenuate the effect of the star tracker errors.□ The problem is now to measure xd and ∆q . It is solved as in Section II.E by correlating noise to model error eq in (37). Unlike (26) correlation cannot be static because dim w < dim x [9]. The noise estimator is = w ( j ) Lea ( j ) + Mp ( j )
p ( j + 1) = (1 − δ ) p ( j ) + ea ( j )
.
(42)
The gains L , M and δ must be designed to stabilize the state observer and to guarantee the target estimation error. D. Pointing control Assume the attitude control provides the cancellation of the accelerometer bias and drift, i.e. (43) uq ( j ) = − xd ( j ) . Then either pointing control is zeroing the differential rate and attitude in (31) through the generic time-continuous law (44) u pk ( t ) = α k ( −k∆q ∆qˆk ( t ) − k∆ω ∆ωˆ k ( t ) ) , where α1 + α 2 = 1 . Because of scale factor error and bias in (32), the state measurements in (44) are corrupted as
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∆qˆk ≅ (1 − ε k ) ( qk − q j ) + bk
∆ωˆ k ≅ (1 − ηk ) (ωk − ω j ) + ck
.
(45)
progressive deployment of different candidate optical sensors that possess different FOV and accuracy. The higher and bias error on the left side is due to star trackers.
The following theorem estimates the misalignment bias caused by scale errors and bias in (45). Theorem 6. The asymptotic misalignment ∆q= q1 − q2 guaranteed by the control law (44) in (31), when affected by the errors in (45), holds −1
Attitude [microrad]
limt →∞ ∆q =b (1 − ε ) + k∆ω ck∆−q1 (1 − ε ) ≅ b −1
ε= α1ε1 + α 2ε 2 , b = −α1b1 + α 2 b2 , c = −α1c1 + α 2 c2
. (46)
Proof. The proof is straightforward. □
40 60 Time- [ks]
80
V. CONCLUSIONS
Fig. 2 and Fig. 3 show the simulated result from a simplified orbit subject to small eccentricity and J2. In Fig. 2, the initial mean altitude is about 700 m lower than the reference, which asks for altitude acquisition while respecting thrust bound. Thrust saturation occurs until 1000 ks in Fig. 3. The orbit period is about 5.4 ks. After altitude acquisition, command becomes equal and opposite to estimated bias and drift. Without control, altitude would drift to about -1200 m in 2500 ks mainly because of accelerometer bias. The target bound of the mean altitude error is 10 m.
The paper outlines a mandatory design in terms of disturbance dynamics, measurement and rejection for the orbit and attitude control of a two-satellite formation at low Earth orbit. A hierarchical control design has been adopted, taking advantage of the wide-band acceleration measurements capable of cancelling non-gravitational CoM and angular accelerations. Narrower outer loops can be designed to zero attitude and angular rate (attitude control) and to keep constant orbit altitude and formation distance (orbit/formation control). Simulated and experimental results are provided.
Mean Altitude Error
200
REFERENCES
0
[1]
-200
Altitude [m]
-50
Fig. 4. Time profile of pointing errors for a sequence of sensors
A. Orbit control
-400
Altitude control No altitude control
-600 -800 -1000 -1200 -1400 0
500
1000
1500 Time [ks]
2000
2500
3000
Fig. 2. Average and sampled altitude error with and without control. Command and Disturbance
-7
x 10
Disturbance Command
15 2
Acceleration [m/s ]
0
20
IV. SIMULATED AND EXPERIMENTAL RESULTS
20
Pitch θ Yaw ψ
50
10 5 0 -5 500
1000
1500 Time [ks]
2000
2500
Fig. 3. Longitudinal disturbance and command.
Command quantization is of the order of micronewton, which is sufficiently small for command values to change each orbit in the average. Micronewton quantization can be achieved by electrical micro-propulsion. Thrust-stands are available to measure compatible noise levels [12]. B. Attitude control Preliminary simulated results are shown in Fig. 4, and refer to pitch and yaw pointing errors. Fig. 4 shows a
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