PID: ACC99-IEEE0173
Finite Thrust Control for Satellite Formation Flying with State Constraints. Randal W. Beard
1
Fred Y. Hadaegh
2
1 Department of Electrical & Computer Engineering, Brigham Young University 2 Jet Propulsion Laboratory, California Institute of Technology
Abstract
distance constraint is rigidly enforced. Using Euclidean geometry, and Newtonian mechanics, the controls are constructed such that the relative distance constraints remain satis ed throughout the maneuver. Wang & Hadaegh pioneered the development of formation ying strategies for tightly controlled satellite formations. In [1] they developed nearest neighbor tracking laws to maintain relative position and attitude between spacecraft. Their approach is extended in [2] to the problem of continuous rotational slews. In [3], adaptive control laws are designed to reject common space disturbances. The application of space-based formation ying to interferometry is discussed in [4]. DeCou [5] studies passive formation control for geocentric orbits in the context of interferometry. McInnes [6] uses Lyapunov control functions to maintain a constellation of satellites in a ring formation. Ulybyshev [7] uses an LQ regulator approach for relative formation keeping. Formation initialization has been studied in [8]. An approach related to [1] is reported in [9, 10]. The basic idea is to treat the satellite formation as a rigid body, and then control the formation as a whole. This approach has been used to study the problem of rotating a constellation of satellites, such that the fuel across the formation is minimized and equalized [11]. A similar approach is followed in this paper.
This paper considers the problem of rotating a constellation of spacecraft using on/o thrusters such that the relative distance between the spacecraft is maintained to within a speci ed tolerance. The basic idea is to resolve all of the constraints into a sphere around the desired trajectory of each spacecraft. As the constellation rotates, the sphere sweeps out a tube that constrains the motion of the spacecraft. The control is activated when the spacecraft encounters the constraint sphere, the direction and magnitude of the thrust are calculated to maximize the travel of the spacecraft before the next required thrust, thus minimizing thrust.
1 Introduction Multiple spacecraft formation ying is emerging as an enabling technology for a number of planned NASA missions. Of particular interest to this paper is separated spacecraft interferometry. Some interferometry missions impose the constraint that the constellation be rotated, or retargeted, while maintaining the relative distance between the spacecraft in the constellation. The objective of this paper is to derive control laws for rotating a constellation of spacecraft using on/o thrusters such that the relative
Corresponding Author: email
[email protected]
1
2 De nitions and Assumptions
Cl-1
The motivation for this work is the proposed NASA space-based interferometry missions. To construct images using interferometry requires a large number of scienti c observations in the plane perpendicular to the direction of the star. In addition, the formation will be required to reorient from one star to another. The spacecraft will have a high precision metrology sensor that is used to measure relative distances to the accuracy of fractions of a wavelength of light. The hardware associated with the metrology system requires that the relative position and attitude of the spacecraft be constrained to maintain \metrology lock." If these constraints are violated, then the metrology system must be reinitialized. This paper considers the problem of maintaining relative position constraints. Therefore, attitude dynamics will be ignored. Attitude constraints will be considered in another paper. The translational motion for each spacecraft is assumed to be given by
z rd l
CR Cl+1 ω
Figure 1: The desired rotation of the constellation.
d , i.e., ij
i
d i
j
Let
d j
ij :
ij 4 = min ij 2;
Then the constraints are guaranteed to be satis ed if
`
v = F0 f
M` _
(
r ? r ) ? (r ? r ) <
r_ = ( v `
Cl
` `
r (t) ? r (t) <
where M` is the mass and F` is the maximum thrust of the `th spacecraft, and f` is a unit vector. The force exerted on the `th spacecraft is assumed to have nite or zero magnitude, but the thrust direction is assumed to be unconstrained. In this paper, we will consider the problem of rotating the constellation about a unit vector z, centered at a coordinate frame CR in space, as shown in gure 1, where C` is the coordinate frame of the `th spacecraft, and ! is the angular rate of rotation of the entire formation. Let rd`(t) be the desired position of the `thspacecraft in CR . Let dij be the desired distance between Ci and Cj , and suppose that the relative distance between Ci and Cj is constrained to be within ij of
d `
`
(1)
for all ` = 1; : : : ; N and t > 0, since
( i
i
r ? r ) ? (r ? r ) r ? r + r ? r j
d i
0, given that the constraint is satis ed at time t = 0. Note that
it is not necessarily d
important that r` ? r` ! 0; simply that the constraint remains satis ed for all time. The idea of the control derived in this paper is to re the thrusters at the boundaries of the constraint sphere, which minimizes the number of times the thrusters must be red, thus minimizing fuel expended. For the convenience of the reader, the major assumptions made throughout the paper are summarized below.
A1. The constellation is in free space. A2. The magnitude of the thrusters is nite but can re in any direction.
A3. Each spacecraft is a rigid body with mass that is time-invariant.
A4. The relative position of each spacecraft can
be determined with respect to the inertial coordinate frame CO .
v+ = v+ ; x x + v+ ; y y + v+ ; z z : `
3
`
`
`
`
`
`
`
`
`
vl+( t )
vO+(
r dl ( t )
t)
θ
rO( t + ∆ )
-
v- ( t ) X z O l
ε vl- ( t )
θ
ε
+
ε
d
R v- ( t )
rl ( t )
O
rO( t )
d
R + x
Figure 3: The desired velocity change assuming impulsive thrust.
2
r dO( t )
x
z
l
ε 2
l
yl
Considering the geometry of the problem, it is clear that the x` and y` components of the velocFigure 4: Geometry in the x` ? y` Plane ity determine how far along the tube the spacecraft will travel. To maximize the distance trav- which implies that eled, the x` and y` components will be chosen ! such that the spacecraft skims the inside edge of d R ? = 2 ` + ? 1 the constraint tube, as shown in Figure 3. The
;
= sin (4)
rO (t^) z` component is chosen such that the spacecraft drifts across the constraint tube, avoiding the walls of the tube as shown in Figure 5. where it is assumed that 0 + 2 . + ? + Let vO and vO be the projection of v` (t^) and From Figure 4, vO+(t^) can be resolved in terms ? ^ v` (t) on the x` ? y` plane, respectively. Then of vO?(t^) and vO?(t^) z` as
v+ = v+; x x + v+; y y v? = v? ; x x + v? ; y y : O
`
`
O
`
`
`
`
`
`
`
`
`
`
"
v?(t^) v+(t^)
+ = cos( + ? ? ) ?
v (t^)
v (t^) # ? (t^) z v
+ sin(+ ? ? )
? ^ v (t) z ;
(2)
O
O
O
O
Let rO (t^) and rdO (t^) be the projection of r`(t^) ` O and rd`(t^) on the x` ? y` plane, and let + and ` O ? be de ned as in Figure 4.
The de nition of inner product gives where vO+(t^) will be determined later. Substi
? vO (t^); rO (t^) = vO?(t^) rO (t^) cos( ? ?); tuting from Equation (2) gives 1 v+(t^) = q which implies that
O +
? 2 ? 2
v (t^) ! O
? v` ; x ` + v` ; y ` ^ ^ v
O (t); rO (t) ? ? 1
?
; (3) = ? cos cos(+ ? ?) v`?; x` + sin(+ ? ?) v`?; y` x`
v (t^) rO (t^)
O + cos(+ ? ?) v`?; y` ? sin(+ ? ? ) v`?; x` y` : (5) where it is assumed that 0 ? . From Figure 4 it can be seen that Let be the amount of time required for the d R ? = 2 spacecraft to reach the boundary of the tube afsin(+) =
` ^
; ter thrust. From the geometry of Figure 4 it can rO (t) 4
The objective is to pick the velocity v`+(t^) such that r` and rd` have the same angular position at time t^ + , as shown in the Figure 5. De ne d and h to be the lengths shown in Figure 5. Since the center of the constraint sphere is de ned by rd`(t^), we have that * + +(t^) v
d = r` (t^) ? rd` (t^); O (7)
v+ (t^) O
h = r` (t^) ? rd` (t^); z` : (8) To determine the distance from the center of the projected constraint sphere to rd`(t^+), we must rst determine the magnitude of the average velocity vd of v`d in the velocity plane. The projection of v`d onto the velocity plane is R`d ! cos(!t). Therefore the average velocity over a time period is Z Rd 1 d R`d ! cos(!t)dt = ` sin(! ): v = 0 The required distance is therefore R`d sin(!). From the geometry of Figure 5, it is clear that
be seen that
r (t^ +q) ? r (t^) = q
r (t^) 2 ? (R ? =2)2 + 2R :
O
O
d `
O
d `
By the fundamental theorem of calculus we also know that
r (t^ + ) ? r O
^
O (t)
Z
+ (t^)d
=
O
0+
= O (t^)
= +(t^) :
v
v v
O
Therefore q
q
rO (t^) 2 ? (R`d ? =2)2 + 2R`d =
+
v (t^) : (6) O
De ne the \velocity" plane at time t^ to be the plane that passes through r`(t^) and that is parallel to the plane spanned by z` and vO+(t^). The origin of the plane is de ned to be the point where r`(t^) intersects the plane. The orthogonal v+ ^ ^ vectors z` and kv+k form a basis for the plane. r`(t + ) ? r`(t) = 2 ? h z` + Note that the desired position vector rd`(t) in d vO
: (9) + R sin( ! ) ? d ` tersects the plane in, at least, one location. Let
v+ O t^ + be the time of intersection. The geometry Since the actual position of the spacecraft will of the velocity plane is shown in Figure 5. be in the velocity plane during the interval t 2 d (t^; t^ + ), we have that rl ( t ) - r ( t ) rl ( t + ∆ ) l O
O
r (t^ + ) ? r (t^) = `
d ϕ
h
vT- (
ε
zl
`
From Figure 5, it is clear that
v+(t^) = v T
vT+( t )
t)
Z
+ (t^)
T
"
0
v+d : T
+^ v O (t)
cos(')z` + sin(') + ^
v (t)
# :
O
Integrating we obtain
vO+( t )
r (t^ + ) ?" r (t^) = v+(t^) # +(t^) v cos(')z + sin(')
+ ^
: (10) v (t)
vO+( t )
`
`
T
`
Figure 5: Geometry of the Tangent Plane
O
O
5
we see from Figure 5 that
Equating Equations (9) and (10) gives
v`+; z` = vT+(t^) cos('): (18) (11) T
d + vT+ (t^) sin(') = R`d sin(! ): (12) The velocity v`+(t^) can therefore be found using the following algorithm.
+
v (t^) and Finally, the relationship between O
+
v (t^) can be seen from Figure 5 to be T Algorithm 3.1
h+
v+(t^) cos(') = 2
v+(t^) = v+(t^) sin('):
(13) Input. , R`d , !, v`?(t^), r`(t^), rd`(t^). Equations (6), (11), (12), and (13) represent Compute (in order).
+
v (t^) , four equations in the four unknowns O
+ 1. rO (t^) = hr`; x` i x` + hr` ; y` i y`,
v (t^) , and '. Plugging Equation (13) into T 2. vO?(t^) from Equation (2), Equation (6) gives 3. ? from Equation (3),
+
v (t^) sin(') T q 4. + from Equation (4), q 2 d d + = krO (t)k ? (R` ? =2)2 + 2R` : (14) 5. kvv+ ((tt^^))k from Equation (5), Using Equation (14) in Equation (12) and solv6. d from Equation (7), ing for gives 7. h from Equation (8), q p 2 d + krO (t)k ? (R`d ? =2)2 + 2R`d 8. from Equation (15), =
d R` 9. vT+(t^) from Equation (16), 10. ' from Equation (17), (15) = 1 sin?1 ( ): !
11. vO+(t^) from Equation (13), From Equations (11) and (12) we get
+ 12. v ( t^); x` from Equation (5), `
+ 2 2
v (t^) = (=2 ? h)2 + (Rd sin(! ) ? d)2 ; ` T 13. v`+ (t^); y` from Equation (5),
which implies that 14. v`+ (t^); z` from Equation (18). q
+
v (t^) = 1 (=2 ? h)2 + (Rd sin(! ) ? d)2 : Output. T `
(16) v`+ = v`+; x` x` + v`+; y` y` From Equation (11) we obtain + v`+; z` z` O
T
O
O
"
=2
? h
#
The velocity vector v`+(t^) was derived assumvT+(t^) ing impulsive thrust. It is necessary to determine how to transition between v`?(t^) and v`+(t^)
+ Finally vO (t^) is obtained from Equation (13) using nite thrusters. Finite thrusters will also Noting that by construction constrain the angular rate of the constellation !. + * These issues will now be addressed. + +
v v A trajectory that transitions between v`+(t^) vT+ = v`+; z` z` + v`+;
vO+
vO+
; and v`?(t^) is shown in Figure 6. O O '
= cos?1
:
(17)
6
incurred while calculating the velocity in Algorithm 3.1: let c represent the computational delay. Second, the look ahead must be long enough to
guarantee that
while trusting, the convO+(t) d
straint r`(t) ? r` (t) < , will never be viovO(t) lated. Let wc be the worst case trust time. As a worst case assume that we are required to reverse the direction of v`?, i.e., v`+ = ?v`?. This is Figure 6: Reference trajectory transitioning be- clearly a conservative worst-case scenario. From tween v`? and v`+. the fundamental theorem of calculus we obtain ?2v`? = F`fM` wc ; Let ` be the time required to transition be` tween v`?(t^) and v`+(t^). Then by the fundamenwhich implies that tal theorem of calculus
? Z
v M` 2 ` : wc = F` f` d : M` v`+ (t^) = M` v`? (t^) + F Actual Trajectory
`
0
`
Assuming that the thrust direction is constant The look-ahead duration should therefore be wc + c. throughout the transition, we have The feedback control can be described as fol? + ? ^ ^ lows: M` v` (t) ? v` (t) = F` f` ` : This implies that the direction of the thrust must Algorithm 3.2 When be ? r ` (t) + (wc + c )v` 2 S 2 (t + wc + c ); +^ ?^ (19) f` =
vv`+((tt^)) ?? vv`?((tt^))
; Set t~ = t, t^ = t + wc + c, ` ` Compute v`+(t^) using Algorithm 3.1. and that the duration of the thrust must be
+ Compute f` and ` from equations (19)
v (t^) ? v? (t^) M` ` ` ` = : (20) and (20). F` Fire the thrusters in direction f` at time t = The diculty is that the `th spacecraft must t^ ? ` =2 for duration ` . start thrusting before it reaches the boundary of the constraint, but v`+(t^) is computed using data Set v`? = v`+(t^). calculated at the constraint boundary. To alleviate this problem we create a nite look-ahead. If ^ is the desired look ahead duration, and if the spacecraft were to continue at a constant ve^ then In this paper we have derived a closed-loop conlocity of v`?(t^) over the period (t; t + ), trol law for controlling a constellation of space? ^ ^ craft, such that the formation rotates at a conr`(t + ) = r`(t) + v` ; stant rate in free space, and such that the relative distances between the spacecraft remain which is the desired look-ahead operator. The duration of the look-ahead must account within speci ed constraints. The spacecraft are for two things. First, the computational delay assumed to have on/o thrusters that can re
4 Conclusions
7
in any arbitrary direction. Simulation results have not been included because of space limitations. Future e orts consist of including rotational dynamics, formation spin-up and spindown modes, and the ability to track arbitrary trajectories.
[7] Y. Ulybyshev, \Long-term formation keeping of satellite constellation using linear-quadratic controller," Journal of Guidance, Control and Dynamics, vol. 21, pp. 109{115, JanuaryFebruary 1998. [8] R. Beard, R. Frost, and W. Stirling, \A hierarchical coordination scheme for satellite formation initialization," in AIAA Guidance, Navigation and Control Conference, (Boston, MA), 1998. [9] R. Beard, \Architecture and algorithms for constellation control," Technical Report, Jet Propulsion Laboratory, California Institute of Technology, 4800 Oak Grove Dr., Pasadena, CA 91109, March 1998. [10] R. W. Beard and F. Y. Hadaegh, \Constellation templates: An approach to autonomous formation ying," in World Automation Congress, (Anchorage, Alaska), May 1998. [11] R. W. Beard, T. W. McLain, and F. Y. Hadaegh, \Fuel equalized retargeting for separated spacecraft interferometry," in American Control Conference, (Philidelphia, PA), 1998.
Acknowledgment
Portions of this work were performed while the rst author was an ASEE summer faculty fellow at the Jet Propulsion Laboratory, California Institute of Technology.
References [1] P. K. C. Wang and F. Y. Hadaegh, \Coordination and control of multiple microspacecraft moving in formation," The Journal of the Astronautical Sciences, vol. 44, no. 3, pp. 315{355, 1996. [2] P. Wang, F. Hadaegh, and K. Lau, \Synchronized formation rotation and attitude control of multiple free- ying spacecraft," in AIAA Guidance, Navigation and Control Conference, 1997. [3] F. Y. Hadaegh, W.-M. Lu, and P. K. C. Wang, \Adaptive control of formation ying spacecraft for interferometry," in IFAC, IFAC, 1998. [4] K. Lau, S. Lichten, L. Young, and B. Haines, \An innovative deep space application of gps technology for formation ying spacecraft," in American Institute of Aeronautics and Astronautics, Guidance, Navigation and Control Conference, pp. 96{381, July 1996. [5] A. B. DeCou, \Orbital station-keeping for multiple spacecraft interferometry," The Journal of the Astronautical Sciences, vol. 39, pp. 283{ 297, July-Sept. 1991. [6] C. R. McInnes, \Autonomous ring formation for a planar constellation of satellites," Journal of Guidance, Control and Dynamics, vol. 18, no. 5, pp. 1215{1217, 1995.
8
