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A new tension control law for deployment of tethered satellites
Mechanics Research Communications,Vol. 24, No. 3, pp. 247-254, 1997 Copyright ~ 1997 Elsevier Sdenec Lid Printed in the USA. All fights r~ervcd 0093-6413/97 $17.00 + .00
Pergamon
F I l S0093-6413(97)00021-9
A NEW
TENSION CONTROL
DEPLOYMENT
LAW FOR
OF TETHERED
SATELLITES
S. P r a d e e p D e p a r t m e n t of Aerospace Engineering Indian I n s t i t u t e of Science Bangalore 560012 India
(Received 3 May 1996; accepted for print 19 February 1997)
1
Introduction
Tethered satelliteshave been proposed for a variety of uses such as creation of artificialgravity, gravity gradient stabilization,minimization of impact loads during spacecraft docking, electrodynamic drives for spacecraft, power generators for spacecraft, exploration of the upper atmosphere, orbital transfer, aerodynamic deceleration of satellitesto prepare for reentry, and as part of space escalators. The issue of deployment and retrievalof tethers from orbiting spacecraft (the Space Shuttle is the currently used the parent spacecraft) is critical to the success of the tethered satellite mission. The problem of deployment and retrieval of tethers from the Space Shuttle is extremely complicated because it leads to nonlinear time varying differentialequations. One of the methods proposed for deployment and retrievalof tethered satellitesis the tension control law [1]. In this method, the tension in the tether is varied in accordance with s predetermined control law to extend or retrieve the tether to the required length, while minimizing the oscillationsof the tether. The governing equations being nonlinear, Liapunov's second method may be used to determine the tension control law. This approach has two drawbacks: firstly,Liapunov functions can be found only by trial and error, and consequently, the experience of the analyst plays a major role in the selection of the control law. The relaxation of any one of the assumptions made in the derivation of the equations would lead to a new set of governing equations, which in turn implies a new problem as far as the determination of the Liapunov function is concerned. Secondly, the procedure leads to a nonlinear control law. Despite these limitations, Vadali [2] and Fujii and Ishijima [3] have successfullyemployed this technique. In this paper, a new method to derive tension control laws is proposed, which overcomes the aforementioned limitations of the Liapunov approach. The governing nonlinear differentialequations are linearized about the equilibrium point, and assuming a linear tension control law, theorems from analytical mechanics are used to determine the tension in the tether so that the linear equations become asymptotically stable. This is then translated to the stabilityof the equilibrium point of the nonlinear system by using the theorem of stability in the firstapproximation. It cannot be concluded that the stability is global, because only a linearized equation is investigated. It is assumed that the domain of attraction is sufficientlylarge for the problem on hand. The advantages of the method put forward herein are: • It provides a procedure for the determination of the tension control law, unlike previous appxoaches. 247
248
S. PRADEEP
• Since linear equations are used for the determination of the tension control law, the control law that results is linear, unlike the existing results which lead to nonlinear control laws.
2
Theoretical Background
The theoretical background required for the analysis carried out in this paper is briefly reviewed in this section.
2.1
Stability of Nonlinear Equations
Consider a physical system governed by the following first order vector differential equation: = g(t; y)
(1)
where y and g are (n x 1) vectors. Let y = W(t) be a particular solution of equation(I), i.e., ~ = g(t; 7)- It is desired to study the stability of the solution y = ~ (t). Define the new variable x(t) = y(t) - ~(t)
(2)
x(t) = 0 corresponds to the particular solution y = ~?(t). This is called the equilibrium solution or the unperturbed state. Differentiating equation (2) leads to
x(t)
= #(t)-iT(t) =
g(t; y] . . . . . Yn) - g(t; rh . . . . . r],)
= =
g(t;xa +qa . . . . , x ~ + , ~ ) - g ( t ; q ~ . . . . ,~]~) f(t, x) [ since n (t) is a known function of t ]
[ using equation (1) ] (3)
Equation (3) is the equation of perturbed motion. It is as formidable as (1) and therefore, no simplification has been achieved in the transition from the original equation to the perturbed equation. However, in many cases, it is possible to simplify the perturbed nonlinear equation by expanding the nonlinear function in a power series in a neighbourhood of the origin, truncating the series to include only the linear terms, and investigating the resulting equation for stability [4]. If equation (3) takes the form x = A(t)x + h(t; x) after the series expansion of f, then ~ = A(t)x is called the equation in the first approximation. The theorem of stability in the first approximation, cited below, greatly simplifies the analysis of the stability of equation (3).
Theorem of Stability in the First Approximation ~>
If A i.s a c o n s t a n t m a t r / z ~ h o s e eigcn~atues
of t h e equation :~ = A x + h(t; x) with It x(0) ][ sufJ:tcient[~ s~atL ezists, a n d t h e sotut~on x = 0 is as~raptoticatt~ stabte. II Thus, all one needs to do to investigate the stability of the nonlinear equation (3) is to examine the eigenvaiues of the matrix A. If they are all negative, then the zero solution of equation (3) is asymptotically stable. This is a great simplification, because the determination of eigenvalues of a constant ~oefficient matrix is a trivial affair.
2.2
Stability of Linear Second Order Equations
In dynamics, most often one comes across linear multidimensional second order constant coefficient ordinary differential equations of the form
Mit+ (D+G) 4+ (K + S) q=O
(4)
where q is a vector whose components are the generalized coordinates, M is the symmetric, positive definite inertia matrix, D is a symmetric positive definite matrix arising from damping terms, G is a skew-symmetric
CONTROL OF TETHERED SATELLITES
249
matrix arising from gyroscopic forces, K is the symmetric stiffness matrix, and S is a skew-symmetric matrix, called the circulatory matrix. Many elegant results regarding the stability of this equation have been proven. Two theorems are of interest in this paper. One of them is the KTC (Kelvin-Talt-Chetayev) theorem and the other is Zajac's extension of the KTC theorem. Both are stated below ([5, 6]). The KTC Theorem ~> If the zero solution oJ the equation M~ solution of the equation M ~ + G ~ + K q = 0 is also stable. I
+
K q = 0 is stable, t h ~ t h e zero
Zajac's Extension of the KTC Theorem v lJ the zero solution oJ the equation M{~ + Kq = 0 is .stable, thev~ t h e zero sotut/o~t oj M~I+D~I+Kq = 0 is asljmptoticathj stable iJ t h e ~ a t r / z D is posit/be
semi- d,efinite.
3 3.1
I
Analysis Governing
Equations
Consider the deployment of a tethered satellite from the Space Shuttle. Assume that the tether is rigid and of negligible mass, the tether remains straight during deployment, there are no aerodynamic effects, and that the motion is planar. The equations of motion of the tether and the attached satellite during deployment are [2, 3]
(s) where ! is the instantaneous length of the tether, 9 is the pitch angle, f~ is the orbital rate, T is the tension in the tether, and m is the mass of the subsatellite. It is more convenient to work with nondimensional equations, and therefore, time, length of the tether and tether tension are nondimensionalized as follows: T = A =
~t t / l,.a=
where tm~ is the maximum tether length. The nondimensional equations corresponding to (5) are A"-A(I+0')2+A-3Acos20 0,,+(~_~)(l+0,)+3cos0sin0
=
-~
=
0
(6)
where the prime indicates differentiation w.r.t, r, the nondimensional time. The problem is to take the system from the initial state A = 0, A' = c, 0 -- 0 and O~ =- 0, (where c is a constant whose value is to be decided by the designer), to the final state A = 1, A' = 0, O = 0, and 0~ --- 0 by controlling the tension in the tether, and ensuring that the tension is always positive. 3.2
An
Outline
of the
Proposed
Method
By means of the tension control law which will soon be specified, it is desired that the system finally attains the state (A A' 0 ~)T = (I 0 0 0)T from the initial state (A A' 0 0')T ----(0 c 0 0)T. Rephrase this problem of dynamical evolution as follows[7]: pretend that the system, to start with, was in equilibrium in the state
(;~;~'O ¢)r =
(I 0 0 0)r
(7)
250
S. PRADEEP
and that it was disturbed by an astronaut to the state (~ A' O 0') T = (0 c 0 0) r
(8)
If the equilibrium at state (7) were asymptotically stable, then, without any action on the part of the astronaut, the system would, by itself, return to equilibrium. In saying this, the problem of dynamical evolution of the system from the initial state (8) to the final state (7), has been transformed into the problem of asymptotic stability of the state (7). Nothing has changed, except the viewpoint. At this juncture, objections may be raised at the claim that, "the system would, by itself, return to the equilibrium." Bear in mind that this is merely the changed viewpoint. In reality, the tension in the tether would cause the tether to extend from ,k = 0 to A = 1. Nevertheless, we feign that the tether extends by itself, and from this, deduce the tension required for this. A serious objection might stem in the mind of the reader whether the perturbation to state (8) from the equilibrium state (7) is indeed small. Is it justified to say that such a large deployment of the tether is small? We are agin brought back to the question of "the domain of attraction" alluded to in the introduction. The analysis is carried out assuming that the perturbations are small. The truth of this assumption is confirmed in the numerical simulation. Proceeding further, assuming that perturbations about the equilibrium are small, the equations are linearized about the equilibrium point. The tension is assumed to be a linear function of the state variables. Prom the theorem of stability in the first approximation, if the zero solution of the linearized system were asymptotically stable, then the zero solution of the nonlinear system is also asymptotically stable if the nonlinearities were sufficiently small in magnitude. Assuming that the nonlinearities are small, one can concentrate on the linearized equations. The gains in the tension control law are selected so that all eigenvaiues of the system matrix have negative real parts. T h e gains are, however, not chosen this way. Instead, the set of first order linear equations is transformed into a set of second order equations M q " + (D + G) q ' + (K + S) q = 0. Select the control gains such that S = 0, thus eliminating circulatory forces. Consider the system with purely potential forces: M q " + K q = O. Select the gains to make K positive definite. This will make the system stable. According to the K T C theorem, the gyroscopic forces do not disturb the stability of equilibrium. Therefore, one gyroscopic matrix is as good as another. Those gains in the control law which appear in the gyroscopic matrix may be set to zero, so that the tension control law is simplified. Finally, select gains in the control law to make D positive semi-definite. With these choices, the zero solution of the linear system M q " + (D + G) q ' + ( K + S) q = 0 will be asymptotically stable b~¢ virtue of Zajac's extension of the K T C theorem, and by the theorem of stability in the first approximation, the zero solution of the corresponding nonlinear system will also be asymptotically stable. Translating this back to reality, by the selection of the control gains as described above, the tether will extend from g = 0 and finally settle at g = e,~o~ with ~ = 0 = b = 0. The advantages of this method are: 1. it leads to a linear control law unlike the Liapunov approach, which most often leads to a nonlinear control law, and 2. the control law is derived using the physics of the problem, and not by trial and error, as in the Liapunov approach. The detailed formulation is now presented.
3.3
Formulation
Equations (6) are two simultaneous second order ordinary differential equations. They are first transformed into a set of four first order equations.
CONTROL OF TETHERED SATELLITES
251
D e f i n e I/1 = A, 1/2 = At, 113 = 0, Y4 = 0 t a n d e q u a t i o n s (6) b e c o m e Yl ' y2 t Ya t
-~ ..~ =
Y2 yl(l+ya)2-yl+3ylcos2ya-T Y4
(9)
A s s u m e t h a t t h e s y s t e m is in e q u i l i b r i u m in t h e s t a t e Yl = 1, Y2 = 0, Ya = 0, Y4 = 0. T r a n s f o r m t h e c o o r d i n a t e s so t h a t t h e o r i g i n c o r r e s p o n d s to t h e e q u i l i b r i u m s t a t e . Define n e w v a r i a b l e s Xl = yl - 1, x2 = ~ - 0 = ~ , x3 = Y3 - 0 = y~, x4 = y4 - 0 = y4. E q u a t i o n s (9) b e c o m e Xl t
~
x2'
---- (1 + xl) (1
X2
X3 t
~
+
X4) 2
(1 + Xl) + 3 (1 + x l ) cos':' x3 --
--
(10)
X4
E q u a t i o n s (10), w h e n l i n e a r i z e d , is Xl t X2 P X3 X4 t
=
x2
_--
3xl +2x4+3--
~
(11)
X4
=
- - 2 x 2 -- 3 x a
The second and fourth of equations (II) axe now e x p r e s s e d in t h e following s e c o n d o r d e r f o r m :
(12) x3 "
2
0
C h o o s e T t o b e a l i n e a r f u n c t i o n of t h e s t a t e variables:
(13)
T~- klXl + k 2 x 2 + k a x a + k a x a + k s Combining equations (12) and (13),
i)x3 ( +('22+,)(xi)0
-3 + kl
+
0 Let ks = 3. This makes equation (14) homogeneous: - 3 + kl +
0 F r o m e q u a t i o n (15), t h e d a m p i n g m a t r i x is
(k,)
(3_,)o ,)(xi)3(0)0
(14)
(is)
k2 ~-
D=
k4
(16)
a n d t h e g y r o s c o p i c m a t r i x is
G=
0
k4--42 /
4 -- k4 2
0
(17)
252
S. PRADEEP
T h e s y s t e m is now of t h e form
Mq"
+ (D + G) q' + ( K + S) q = 0 where q = (xl xa) T and
K + S= ( -3 +
ka )3
Select k3 = 0. T h i s will eliminate the circulatory forces. T h e s y s t e m under purely potential forces now becomes
+ 2
3
=
It
0
3
X3
(is) 0
which is of the form Mq" + Kq = O. Equation (18) is stable if kl :> 3. According to the K T C theorem, if a gyroscopic m a t r i x is added so as to alter the s y s t e m equation to Mq" + Gq' + Kq = 0, the zero solution of the new equation will also be stable, provided the zero solution of the original equation was stable. One gain in t h e tension control law, namely, k4, appears in the gyroscopic matrix as m a y be seen from (17). From what was j u s t stated, no m a t t e r what the value for k4 is, it will not disturb the property of stability. It is a d v a n t a g e o u s to choose k4 to be zero, so that the tension control law will be simplified. T h e d a m p i n g matrix, with the choice for k4 incorporated, is
D=(
k20 0)0
T h i s m a t r i x is positive semi-definite if k2 > 0. W i t h t h e aforestated choices, the tension is
T= klxl + k2x2 + k.~ with
3.4
(19)
k1 > 3 k2 > 0 k~ = 3 Numerical
Example
Consider deployment of a tethered satellite from the Space Shuttle in a circular orbit of radius 220 km, with an orbital rate of 1.1804 × 10 -3 r a d / s . T h e tether is a s s u m e d to be 100 k m long. T h e initial conditions are xl(0) = - 1 , x2(0) = 0.5 (assumed), x3(0) = 0, x4(0) = 0. T h e initial condition x2(0) = 0.5 implies dX/dT = 0.5, or, ~ = 5 9 m / s . T h e initial condition xl(0) = - 1 creates a singularity in equation (10). To circumvent this [2], select xl(0) to be - 0 . 9 9 . T h e initial conditions now become xl(0) = - 0 . 9 9 , x2(0) = 0.5, x3(0) = 0, x,(0) = 0. T h e desired final conditions are xl(0 ) = 0, x2(0) = 0, xa(0) = 0, x4(0) = 0. T h e gains are chosen to be kl -- 4, T = 4xl + 2x~ + 3.
k2 =
2 and k.~ = 3. From equation (19), t h e tension control law is
T h e nonlinear equations are: 21 t
=
22
22'
= =
(1 + 21) (1 + x4) 2 - (1 + 21) + 3 (1 + 21) cos~ 2~ x4
x3'
(20)
where T - - 4xl + 2 x 2 4-3 T h e nonlinear e q u a t i o n s were solved numerically for two orbits. T h e plots of g, 9 and the tension versus time are s h o w n in Fig.(1). It is seen t h a t the length settles down to g = 100 k m (corresponding to x 1 = 0 ) in two orbits. 8 settles down to zero in roughly one orbit. T h e behaviour of t a n d 0 show t h a t t h e p e r t u r b a t i o n is small indeed a n d lies within the d o m a i n of attraction. T h e tension is always positive, a n d settles down to the equilibrium value of 3.
CONTROL OF TETHERED SATELLITES
4
253
Conclusion
M a n y space missions involving tethers are being planned. Tethered satellitesare proposed for a variety of uses. One of the primary issues in tether utilizationis fast deployment of payloads. Equations governing deployment of tethers from orbiting spacecraft are time varying and nonlinear. The dynamics and control of tethers extending from an orbiting spacecraft is studied in this paper. One of the methods proposed for deployment of tethers is the tension control law. In this method, the length of the tether is controlled by varying the tension in the tether. The tension in the tether is determined using the Liapunov approach. It carrieswith it the inherent drawback of the Liapunov method, viz., the tension can be determined only by trialand error. In addition, the tension control law that results is nonlinear. In this paper, a new method to determine the tension control law is proposed. It is based on linearizing the equations of motion, and hence the tension control law that results is linear. The linear equations are converted to second order, and the tension control law is determined by using theorems in analytical mechanics. It is shown that in this way the tension control law can be determined without resorting to trial and error.
Acknowledgement The author would like to thank the Space Technology Cell, Indian Institute of Science, Bangalore for financial support.
References [1] Rupp, C., "A Tether Tension Control Law for Tethered Satellites Deployed along Local Vertical," NASA TMX-64963, (1975). [2] Vadali, S.R., "Feedback Tether Deployment and Retrieval," Journal of Guidance, Control and Dynamics, Vol.14, No.2, p.469, (1991) [3] Fujii, H, and Ishijima, S., "Mission Function Control for Deployment and Retrieval of a Subsatellite," Journal of Guidance, Control and Dynamics, Vol.12, No.2, p.243, (1989). [4] Cesari, L., Asymptotic Behaviour and Stability Problems in Ordinary Differential Equations, Academic Press, New York, (1963). [5] Zajac, E.E. '~rhe Kelvin-Tait-Chetayev theorem and extensions," Journal of the Astronautical Sciences, 11, p.46 (1964). [6] Zajac, E.E. "Comments on 'Stability of damped mechanical system,' and a further extension," AIAA Journal, 3, p.1749 (1965). [7] Pradeep, S., "Dynamics and Control of Space Structures During Operations Involving Time Varying Phenomena," ISTC/AE/SP/051, (1994), Space Technology Cell, Indian Institute of Science, Bangalore.
254
S. PRADEEP
1.0
k 0
0 G (rad)
I
__
I
I
x,/--
-I'C
I
I
I
1
1
4 A
T
2 0 0
1-0 1.5 0.5 time (number of orbits)
Figure 1: Deployment Dynamics
2.0
Pergamon
F I l S0093-6413(97)00021-9
A NEW
TENSION CONTROL
DEPLOYMENT
LAW FOR
OF TETHERED
SATELLITES
S. P r a d e e p D e p a r t m e n t of Aerospace Engineering Indian I n s t i t u t e of Science Bangalore 560012 India
(Received 3 May 1996; accepted for print 19 February 1997)
1
Introduction
Tethered satelliteshave been proposed for a variety of uses such as creation of artificialgravity, gravity gradient stabilization,minimization of impact loads during spacecraft docking, electrodynamic drives for spacecraft, power generators for spacecraft, exploration of the upper atmosphere, orbital transfer, aerodynamic deceleration of satellitesto prepare for reentry, and as part of space escalators. The issue of deployment and retrievalof tethers from orbiting spacecraft (the Space Shuttle is the currently used the parent spacecraft) is critical to the success of the tethered satellite mission. The problem of deployment and retrieval of tethers from the Space Shuttle is extremely complicated because it leads to nonlinear time varying differentialequations. One of the methods proposed for deployment and retrievalof tethered satellitesis the tension control law [1]. In this method, the tension in the tether is varied in accordance with s predetermined control law to extend or retrieve the tether to the required length, while minimizing the oscillationsof the tether. The governing equations being nonlinear, Liapunov's second method may be used to determine the tension control law. This approach has two drawbacks: firstly,Liapunov functions can be found only by trial and error, and consequently, the experience of the analyst plays a major role in the selection of the control law. The relaxation of any one of the assumptions made in the derivation of the equations would lead to a new set of governing equations, which in turn implies a new problem as far as the determination of the Liapunov function is concerned. Secondly, the procedure leads to a nonlinear control law. Despite these limitations, Vadali [2] and Fujii and Ishijima [3] have successfullyemployed this technique. In this paper, a new method to derive tension control laws is proposed, which overcomes the aforementioned limitations of the Liapunov approach. The governing nonlinear differentialequations are linearized about the equilibrium point, and assuming a linear tension control law, theorems from analytical mechanics are used to determine the tension in the tether so that the linear equations become asymptotically stable. This is then translated to the stabilityof the equilibrium point of the nonlinear system by using the theorem of stability in the firstapproximation. It cannot be concluded that the stability is global, because only a linearized equation is investigated. It is assumed that the domain of attraction is sufficientlylarge for the problem on hand. The advantages of the method put forward herein are: • It provides a procedure for the determination of the tension control law, unlike previous appxoaches. 247
248
S. PRADEEP
• Since linear equations are used for the determination of the tension control law, the control law that results is linear, unlike the existing results which lead to nonlinear control laws.
2
Theoretical Background
The theoretical background required for the analysis carried out in this paper is briefly reviewed in this section.
2.1
Stability of Nonlinear Equations
Consider a physical system governed by the following first order vector differential equation: = g(t; y)
(1)
where y and g are (n x 1) vectors. Let y = W(t) be a particular solution of equation(I), i.e., ~ = g(t; 7)- It is desired to study the stability of the solution y = ~ (t). Define the new variable x(t) = y(t) - ~(t)
(2)
x(t) = 0 corresponds to the particular solution y = ~?(t). This is called the equilibrium solution or the unperturbed state. Differentiating equation (2) leads to
x(t)
= #(t)-iT(t) =
g(t; y] . . . . . Yn) - g(t; rh . . . . . r],)
= =
g(t;xa +qa . . . . , x ~ + , ~ ) - g ( t ; q ~ . . . . ,~]~) f(t, x) [ since n (t) is a known function of t ]
[ using equation (1) ] (3)
Equation (3) is the equation of perturbed motion. It is as formidable as (1) and therefore, no simplification has been achieved in the transition from the original equation to the perturbed equation. However, in many cases, it is possible to simplify the perturbed nonlinear equation by expanding the nonlinear function in a power series in a neighbourhood of the origin, truncating the series to include only the linear terms, and investigating the resulting equation for stability [4]. If equation (3) takes the form x = A(t)x + h(t; x) after the series expansion of f, then ~ = A(t)x is called the equation in the first approximation. The theorem of stability in the first approximation, cited below, greatly simplifies the analysis of the stability of equation (3).
Theorem of Stability in the First Approximation ~>
If A i.s a c o n s t a n t m a t r / z ~ h o s e eigcn~atues
of t h e equation :~ = A x + h(t; x) with It x(0) ][ sufJ:tcient[~ s~atL ezists, a n d t h e sotut~on x = 0 is as~raptoticatt~ stabte. II Thus, all one needs to do to investigate the stability of the nonlinear equation (3) is to examine the eigenvaiues of the matrix A. If they are all negative, then the zero solution of equation (3) is asymptotically stable. This is a great simplification, because the determination of eigenvalues of a constant ~oefficient matrix is a trivial affair.
2.2
Stability of Linear Second Order Equations
In dynamics, most often one comes across linear multidimensional second order constant coefficient ordinary differential equations of the form
Mit+ (D+G) 4+ (K + S) q=O
(4)
where q is a vector whose components are the generalized coordinates, M is the symmetric, positive definite inertia matrix, D is a symmetric positive definite matrix arising from damping terms, G is a skew-symmetric
CONTROL OF TETHERED SATELLITES
249
matrix arising from gyroscopic forces, K is the symmetric stiffness matrix, and S is a skew-symmetric matrix, called the circulatory matrix. Many elegant results regarding the stability of this equation have been proven. Two theorems are of interest in this paper. One of them is the KTC (Kelvin-Talt-Chetayev) theorem and the other is Zajac's extension of the KTC theorem. Both are stated below ([5, 6]). The KTC Theorem ~> If the zero solution oJ the equation M~ solution of the equation M ~ + G ~ + K q = 0 is also stable. I
+
K q = 0 is stable, t h ~ t h e zero
Zajac's Extension of the KTC Theorem v lJ the zero solution oJ the equation M{~ + Kq = 0 is .stable, thev~ t h e zero sotut/o~t oj M~I+D~I+Kq = 0 is asljmptoticathj stable iJ t h e ~ a t r / z D is posit/be
semi- d,efinite.
3 3.1
I
Analysis Governing
Equations
Consider the deployment of a tethered satellite from the Space Shuttle. Assume that the tether is rigid and of negligible mass, the tether remains straight during deployment, there are no aerodynamic effects, and that the motion is planar. The equations of motion of the tether and the attached satellite during deployment are [2, 3]
(s) where ! is the instantaneous length of the tether, 9 is the pitch angle, f~ is the orbital rate, T is the tension in the tether, and m is the mass of the subsatellite. It is more convenient to work with nondimensional equations, and therefore, time, length of the tether and tether tension are nondimensionalized as follows: T = A =
~t t / l,.a=
where tm~ is the maximum tether length. The nondimensional equations corresponding to (5) are A"-A(I+0')2+A-3Acos20 0,,+(~_~)(l+0,)+3cos0sin0
=
-~
=
0
(6)
where the prime indicates differentiation w.r.t, r, the nondimensional time. The problem is to take the system from the initial state A = 0, A' = c, 0 -- 0 and O~ =- 0, (where c is a constant whose value is to be decided by the designer), to the final state A = 1, A' = 0, O = 0, and 0~ --- 0 by controlling the tension in the tether, and ensuring that the tension is always positive. 3.2
An
Outline
of the
Proposed
Method
By means of the tension control law which will soon be specified, it is desired that the system finally attains the state (A A' 0 ~)T = (I 0 0 0)T from the initial state (A A' 0 0')T ----(0 c 0 0)T. Rephrase this problem of dynamical evolution as follows[7]: pretend that the system, to start with, was in equilibrium in the state
(;~;~'O ¢)r =
(I 0 0 0)r
(7)
250
S. PRADEEP
and that it was disturbed by an astronaut to the state (~ A' O 0') T = (0 c 0 0) r
(8)
If the equilibrium at state (7) were asymptotically stable, then, without any action on the part of the astronaut, the system would, by itself, return to equilibrium. In saying this, the problem of dynamical evolution of the system from the initial state (8) to the final state (7), has been transformed into the problem of asymptotic stability of the state (7). Nothing has changed, except the viewpoint. At this juncture, objections may be raised at the claim that, "the system would, by itself, return to the equilibrium." Bear in mind that this is merely the changed viewpoint. In reality, the tension in the tether would cause the tether to extend from ,k = 0 to A = 1. Nevertheless, we feign that the tether extends by itself, and from this, deduce the tension required for this. A serious objection might stem in the mind of the reader whether the perturbation to state (8) from the equilibrium state (7) is indeed small. Is it justified to say that such a large deployment of the tether is small? We are agin brought back to the question of "the domain of attraction" alluded to in the introduction. The analysis is carried out assuming that the perturbations are small. The truth of this assumption is confirmed in the numerical simulation. Proceeding further, assuming that perturbations about the equilibrium are small, the equations are linearized about the equilibrium point. The tension is assumed to be a linear function of the state variables. Prom the theorem of stability in the first approximation, if the zero solution of the linearized system were asymptotically stable, then the zero solution of the nonlinear system is also asymptotically stable if the nonlinearities were sufficiently small in magnitude. Assuming that the nonlinearities are small, one can concentrate on the linearized equations. The gains in the tension control law are selected so that all eigenvaiues of the system matrix have negative real parts. T h e gains are, however, not chosen this way. Instead, the set of first order linear equations is transformed into a set of second order equations M q " + (D + G) q ' + (K + S) q = 0. Select the control gains such that S = 0, thus eliminating circulatory forces. Consider the system with purely potential forces: M q " + K q = O. Select the gains to make K positive definite. This will make the system stable. According to the K T C theorem, the gyroscopic forces do not disturb the stability of equilibrium. Therefore, one gyroscopic matrix is as good as another. Those gains in the control law which appear in the gyroscopic matrix may be set to zero, so that the tension control law is simplified. Finally, select gains in the control law to make D positive semi-definite. With these choices, the zero solution of the linear system M q " + (D + G) q ' + ( K + S) q = 0 will be asymptotically stable b~¢ virtue of Zajac's extension of the K T C theorem, and by the theorem of stability in the first approximation, the zero solution of the corresponding nonlinear system will also be asymptotically stable. Translating this back to reality, by the selection of the control gains as described above, the tether will extend from g = 0 and finally settle at g = e,~o~ with ~ = 0 = b = 0. The advantages of this method are: 1. it leads to a linear control law unlike the Liapunov approach, which most often leads to a nonlinear control law, and 2. the control law is derived using the physics of the problem, and not by trial and error, as in the Liapunov approach. The detailed formulation is now presented.
3.3
Formulation
Equations (6) are two simultaneous second order ordinary differential equations. They are first transformed into a set of four first order equations.
CONTROL OF TETHERED SATELLITES
251
D e f i n e I/1 = A, 1/2 = At, 113 = 0, Y4 = 0 t a n d e q u a t i o n s (6) b e c o m e Yl ' y2 t Ya t
-~ ..~ =
Y2 yl(l+ya)2-yl+3ylcos2ya-T Y4
(9)
A s s u m e t h a t t h e s y s t e m is in e q u i l i b r i u m in t h e s t a t e Yl = 1, Y2 = 0, Ya = 0, Y4 = 0. T r a n s f o r m t h e c o o r d i n a t e s so t h a t t h e o r i g i n c o r r e s p o n d s to t h e e q u i l i b r i u m s t a t e . Define n e w v a r i a b l e s Xl = yl - 1, x2 = ~ - 0 = ~ , x3 = Y3 - 0 = y~, x4 = y4 - 0 = y4. E q u a t i o n s (9) b e c o m e Xl t
~
x2'
---- (1 + xl) (1
X2
X3 t
~
+
X4) 2
(1 + Xl) + 3 (1 + x l ) cos':' x3 --
--
(10)
X4
E q u a t i o n s (10), w h e n l i n e a r i z e d , is Xl t X2 P X3 X4 t
=
x2
_--
3xl +2x4+3--
~
(11)
X4
=
- - 2 x 2 -- 3 x a
The second and fourth of equations (II) axe now e x p r e s s e d in t h e following s e c o n d o r d e r f o r m :
(12) x3 "
2
0
C h o o s e T t o b e a l i n e a r f u n c t i o n of t h e s t a t e variables:
(13)
T~- klXl + k 2 x 2 + k a x a + k a x a + k s Combining equations (12) and (13),
i)x3 ( +('22+,)(xi)0
-3 + kl
+
0 Let ks = 3. This makes equation (14) homogeneous: - 3 + kl +
0 F r o m e q u a t i o n (15), t h e d a m p i n g m a t r i x is
(k,)
(3_,)o ,)(xi)3(0)0
(14)
(is)
k2 ~-
D=
k4
(16)
a n d t h e g y r o s c o p i c m a t r i x is
G=
0
k4--42 /
4 -- k4 2
0
(17)
252
S. PRADEEP
T h e s y s t e m is now of t h e form
Mq"
+ (D + G) q' + ( K + S) q = 0 where q = (xl xa) T and
K + S= ( -3 +
ka )3
Select k3 = 0. T h i s will eliminate the circulatory forces. T h e s y s t e m under purely potential forces now becomes
+ 2
3
=
It
0
3
X3
(is) 0
which is of the form Mq" + Kq = O. Equation (18) is stable if kl :> 3. According to the K T C theorem, if a gyroscopic m a t r i x is added so as to alter the s y s t e m equation to Mq" + Gq' + Kq = 0, the zero solution of the new equation will also be stable, provided the zero solution of the original equation was stable. One gain in t h e tension control law, namely, k4, appears in the gyroscopic matrix as m a y be seen from (17). From what was j u s t stated, no m a t t e r what the value for k4 is, it will not disturb the property of stability. It is a d v a n t a g e o u s to choose k4 to be zero, so that the tension control law will be simplified. T h e d a m p i n g matrix, with the choice for k4 incorporated, is
D=(
k20 0)0
T h i s m a t r i x is positive semi-definite if k2 > 0. W i t h t h e aforestated choices, the tension is
T= klxl + k2x2 + k.~ with
3.4
(19)
k1 > 3 k2 > 0 k~ = 3 Numerical
Example
Consider deployment of a tethered satellite from the Space Shuttle in a circular orbit of radius 220 km, with an orbital rate of 1.1804 × 10 -3 r a d / s . T h e tether is a s s u m e d to be 100 k m long. T h e initial conditions are xl(0) = - 1 , x2(0) = 0.5 (assumed), x3(0) = 0, x4(0) = 0. T h e initial condition x2(0) = 0.5 implies dX/dT = 0.5, or, ~ = 5 9 m / s . T h e initial condition xl(0) = - 1 creates a singularity in equation (10). To circumvent this [2], select xl(0) to be - 0 . 9 9 . T h e initial conditions now become xl(0) = - 0 . 9 9 , x2(0) = 0.5, x3(0) = 0, x,(0) = 0. T h e desired final conditions are xl(0 ) = 0, x2(0) = 0, xa(0) = 0, x4(0) = 0. T h e gains are chosen to be kl -- 4, T = 4xl + 2x~ + 3.
k2 =
2 and k.~ = 3. From equation (19), t h e tension control law is
T h e nonlinear equations are: 21 t
=
22
22'
= =
(1 + 21) (1 + x4) 2 - (1 + 21) + 3 (1 + 21) cos~ 2~ x4
x3'
(20)
where T - - 4xl + 2 x 2 4-3 T h e nonlinear e q u a t i o n s were solved numerically for two orbits. T h e plots of g, 9 and the tension versus time are s h o w n in Fig.(1). It is seen t h a t the length settles down to g = 100 k m (corresponding to x 1 = 0 ) in two orbits. 8 settles down to zero in roughly one orbit. T h e behaviour of t a n d 0 show t h a t t h e p e r t u r b a t i o n is small indeed a n d lies within the d o m a i n of attraction. T h e tension is always positive, a n d settles down to the equilibrium value of 3.
CONTROL OF TETHERED SATELLITES
4
253
Conclusion
M a n y space missions involving tethers are being planned. Tethered satellitesare proposed for a variety of uses. One of the primary issues in tether utilizationis fast deployment of payloads. Equations governing deployment of tethers from orbiting spacecraft are time varying and nonlinear. The dynamics and control of tethers extending from an orbiting spacecraft is studied in this paper. One of the methods proposed for deployment of tethers is the tension control law. In this method, the length of the tether is controlled by varying the tension in the tether. The tension in the tether is determined using the Liapunov approach. It carrieswith it the inherent drawback of the Liapunov method, viz., the tension can be determined only by trialand error. In addition, the tension control law that results is nonlinear. In this paper, a new method to determine the tension control law is proposed. It is based on linearizing the equations of motion, and hence the tension control law that results is linear. The linear equations are converted to second order, and the tension control law is determined by using theorems in analytical mechanics. It is shown that in this way the tension control law can be determined without resorting to trial and error.
Acknowledgement The author would like to thank the Space Technology Cell, Indian Institute of Science, Bangalore for financial support.
References [1] Rupp, C., "A Tether Tension Control Law for Tethered Satellites Deployed along Local Vertical," NASA TMX-64963, (1975). [2] Vadali, S.R., "Feedback Tether Deployment and Retrieval," Journal of Guidance, Control and Dynamics, Vol.14, No.2, p.469, (1991) [3] Fujii, H, and Ishijima, S., "Mission Function Control for Deployment and Retrieval of a Subsatellite," Journal of Guidance, Control and Dynamics, Vol.12, No.2, p.243, (1989). [4] Cesari, L., Asymptotic Behaviour and Stability Problems in Ordinary Differential Equations, Academic Press, New York, (1963). [5] Zajac, E.E. '~rhe Kelvin-Tait-Chetayev theorem and extensions," Journal of the Astronautical Sciences, 11, p.46 (1964). [6] Zajac, E.E. "Comments on 'Stability of damped mechanical system,' and a further extension," AIAA Journal, 3, p.1749 (1965). [7] Pradeep, S., "Dynamics and Control of Space Structures During Operations Involving Time Varying Phenomena," ISTC/AE/SP/051, (1994), Space Technology Cell, Indian Institute of Science, Bangalore.
254
S. PRADEEP
1.0
k 0
0 G (rad)
I
__
I
I
x,/--
-I'C
I
I
I
1
1
4 A
T
2 0 0
1-0 1.5 0.5 time (number of orbits)
Figure 1: Deployment Dynamics
2.0