1 American Institute of Aeronautics and Astronautics

Simulation of the Dynamics of a Short Tethered Satellite System

H.J. Pernicka*, M. Dancer†, A. Abrudan‡

University of Missouri-Rolla Rolla, Missouri, 65409-0050

and

J. Harrington§

Georgia Institute of Technology Atlanta, Georgia, 30319

This study investigates the dynamics of a short tethered satellite system composed of two

satellites joined by a ten-meter tether. Despite the many studies in the literature regarding tethered satellites and their dynamics, none that specifically concern “short” tethers have been located to date. Equations of motion are derived and numerically integrated to produce the tether tension, system attitude, orbit, and libration characteristics. Techniques have been implemented which add corrections to the integrated states and help reduce long-term simulation errors. Also considered are the effects of perturbations due to the Earth’s geopotential, solar and lunar gravity, aerodynamic drag and solar radiation pressure.

Nomenclature α = in-plane tether libration angle β = out-of-plane tether libration angle f, g = dynamic and constraint vectors F1, F2 = force acting on satellites H = angular momentum H = Hamiltonian function I = identity matrix K, K = operators L = tether length λ = Lagrange multiplier m = total system mass m1, m2 = satellite masses M = applied external moment µ = satellite mass ratio ωo = orbit mean motion r1, r2 = satellite position vector s = system state vector t = time T = tether tension force T = tether tension magnitude

* Associate Professor, Mechanical and Aerospace Engineering, 1870 Miner Circle, Rolla, MO 65409-0050, Senior Member AIAA. † Undergraduate Student, Mechanical and Aerospace Engineering, 1870 Miner Circle, Rolla, MO 65409-0050, Student Member AIAA. ‡ Undergraduate Student, Mechanical and Aerospace Engineering, 1870 Miner Circle, Rolla, MO 65409-0050, Student Member AIAA. § Graduate Research Assistant, School of Aerospace Engineering, Georgia Institute of Technology, Student Member AIAA.

AIAA/AAS Astrodynamics Specialist Conference and Exhibit16 - 19 August 2004, Providence, Rhode Island

AIAA 2004-5311

Copyright © 2004 by Henry Pernicka. Published by the American Institute of Aeronautics and Astronautics, Inc., with permission.

2 American Institute of Aeronautics and Astronautics

V = cost function v1, v2 = satellite velocity x, y, z = Cartesian coordinates of satellite

I. Introduction The concept of using Tethered Satellite Systems (TSS) for space endeavors has been envisioned for several

decades. A variety of uses including satellite power generation, formation flying, and passive attitude control have been suggested for such systems. To date only a handful of tether experiments have been conducted in space. Early experiments tested tether deployment techniques and tether survivability while more recent experiments have investigated the orbital dynamics of such systems. Of the tether experiments that have flown, only the Tether Physics Survivability (TiPS, 1996) experiment has been able to provide a long-term history of tether motion. However, experiments and studies such as the TiPS project1 have focused on satellite pairs connected by a long tether, on the order of kilometers. The study of short-tethered satellite systems, on the order of meters, has not yet received much attention.

The recent NASA Distributed Space Systems (DSS) initiative includes the use of tightly controlled satellite formations. Such formations may be flown using "free flying" configuration control schemes, or alternatively by using tethers to constrain the formation geometry. The University of Missouri-Rolla is in the process of developing a microsatellite in support of future DSS missions. The Missouri-Rolla Satellite (MR SAT) project will help develop this technology using two small spacecraft connected by a short tether of ten meters length. It is imperative to the success of the mission to develop a comprehensive dynamics model to simulate the dual-satellite formation so that accurate estimates of tether tension can be made. Another key area of concern is ensuring that the tether not go slack, which requires realistic simulations using models of high fidelity. Lastly, simulating the deployment process is critical as well in creating a reliable design for the hardware used to separate the spacecraft.

II. Previous Contributions Despite the many studies in the literature regarding tethered satellites and their dynamics, none that specifically

concern “short” tethers have been located to date. The Tethers in Space Handbook2 provides a good overview of TSS fundamentals. Misra and Modi’s3 useful survey paper contains a comprehensive list of tether-related studies.

More specifically, an energy method was used in Ref. 4 to study the dynamics of a tethered satellite system under the influence of an oblate gravitational field. The purpose was to investigate the potential of pumping the tether between a satellite pair in order to add energy and boost its orbit around an oblate planet. The tether was assumed massless and very long, on the order of 50 km - 200 km. One conclusion related to this study was that tether tension is maximized when the tether is aligned with the local vertical and minimized when horizontal.

One comprehensive study on tethered satellite system dynamics is the analysis by the Naval Research Laboratory of their TiPS project1. The purpose of the TiPS project was to provide information about the dynamics and survivability of tethered space systems. Their experimental system consisted of two bodies with a total system mass of 53.5 kg connected by a 4 km long tether on an initial circular orbit of approximately 900 km altitude. An accurate prediction of the system dynamics was necessary for the study because the long tether length and associated oscillatory motions initiated ground tracking difficulties. Ref. 1 provides an outline of the methods used to predict the tethered orbit. Their system was modeled as two end masses connected by a massless, extensible tether with longitudinal damping. Their approach involved supplementing traditional equations for the center of mass of a system with specific equations for the end bodies. Orbital and attitude determination was acquired by numerically integrating the system of equations. From this study it was concluded that the orbital determination of the center of mass of the system cannot be decoupled from the attitude determination in the presence of a tether because neither of the bodies being tracked can be considered the center of mass of the system.

In Ref. 5, two different approaches were used to approximate the dynamics of a tethered satellite system. The first used the position and velocity of one of the end masses to formulate equations of motion for the system with respect to that end mass. The motivation was to provide a model that could provide information regarding the entire system based on the tracking information of a single satellite in the system using a perturbed two-body model6. The second approach focused on referencing the center of mass to derive equations of motion for the system. Their analysis provided results for a massless and inextensible tether of extensive lengths on the order of 10 km to 100 km. It was concluded that the adjusted center of mass approach provided more accurate results for short observational periods. This was credited to the simplicity of such a model.

3 American Institute of Aeronautics and Astronautics

III. Problem Definition The main objective of the MR SAT project is to develop a low-cost satellite pair whose purpose is to test the

ability of satellites to stay in formation either by use of tethers or by active control schemes using low thrust and intersatellite communications. In order to achieve this, the mission will consist of two distinct mission phases following the orbital insertion of the system. The first phase will involve the separation of the satellite pair as the ten meter long tether is unwound from its reeling mechanism. The tethered satellite pair will then remain in a passively controlled orbit as data related to the system dynamics is collected. The second phase will involve the separation of the satellite pair by disconnecting the tether. It is at this point of the mission when the ability of the satellites to remain in “free flying” formation will be tested.

Developing a model that can accurately simulate all phases of this mission is important for several reasons. Accurate knowledge of tether tension is needed to make design choices such as tether material, spool motor, sensors, and other structural components. Understanding uncommon dynamics scenarios such as tether deployment and tether libration are beneficial to communications, tracking, and attitude determination and control methods needed for the spacecraft.

Reference 2 provides an analytic estimate of tether tension based on a simple dynamic model. This result was used for preliminary calculations early in the satellite design process, as well as for a validation of the numerical results presented in this paper. From Ref. 2, the tether tension can be approximated as

( )2

1 23 oT m L m mω= (1)

where T is the tether tension, oω is the orbit mean motion, L is the tether length (10 meters in this study), m1 and m2

are the masses of the two tethered spacecraft, and 1 2m m m+ . Estimating m1= 15 kg and m2= 5 kg (consistent

with the MR SAT project) and assuming an orbital altitude of 400 km (with oω = 0.001131 rad/sec), equation (1) gives the tension in the tether as 0.1439 millinewtons.

With an estimate of the tether tension available, basic spacecraft bus design could commence. However, a more detailed analysis using a model of higher fidelity was necessary to ensure that the tether dynamics would not prevent the proper operation of the other spacecraft subsystems. Of particular concern was the low value of the tension and the possibility that the tether might go slack due to perturbations from drag, solar radiation pressure, third-body (lunar and solar) effects, and Earth’s geopotential effects. Also of concern were the effects of initial misalignment errors and rates that might occur as the tether is deployed. These concerns motivated the current study of tether dynamics with a high-fidelity model.

V. Dynamic Model A. Equations of Motion

Developing a tethered satellite system simulation begins with deriving the equations of motion (EOMs). Due to the presence of the tether tension the motion of the individual masses cannot be assumed Keplerian. This presents a challenge in developing methods for estimating the unique motion of a TSS. There exist many approaches to this problem, and several have been published as indicated previously. Here the EOMs were derived using Cartesian coordinates to locate each mass, thus facilitating the use of standard numerical integration algorithms.

The coordinate frame and position vectors used to describe the system are depicted in Fig. 1. The system consists of a single inertial frame with the origin located at the center of the Earth. The axes are aligned such that z is defined as pointing north, x points in the direction of the Vernal Equinox, and y completes the right-hand coordinate frame.

4 American Institute of Aeronautics and Astronautics

Figure 1 Coordinate frame.

Mass m1 is located with coordinates x1, y1, z1 and mass m2 with coordinates x2, y2, z2 measured from the origin of the inertial frame at the center of the Earth. Summing the forces on each mass produces the scalar EOMs as

1 1 1

1 1 1

1 1 1

2 2 2

2 2 2

2 2 2

x

y

z

x

y

z

T xm x F

LT y

m y FL

T zm z F

LT x

m x FL

T ym y F

LT z

m z FL

∆= +

∆= +

∆= +

∆= − +

∆= − +

∆= − +

∑

∑

∑

∑

∑

∑

(2-7)

where L is the tether length (the tether is assumed inextensible due to its short length), T is the tension in the tether,

1 , ,x y zF∑ and

, ,2x y zF∑ represent the component sum of all other forces acting on m1 and m2, respectively,

2 1 2 1 2 1, , andx x x y y y z z z∆ − ∆ − ∆ − .

In this study, 1 , ,x y zF∑ and

, ,2x y zF∑ were defined to include Earth’s geopotential (the 12 by 12 Grace Gravity

Model01 was used), lunar and solar third-body perturbations (JPL’s DE405 ephemeris model was used to locate the Sun and Moon), solar radiation pressure (SRP), and aerodynamic drag. Representative values for the MR SAT spacecraft were used in estimating the SRP and drag terms.

In order to numerically integrate equations (2)-(7), the unknown tension T must first be eliminated. This can be done fairly straightforwardly by applying Newton’s second law to the system of masses so that the internal tension force does not appear in the resulting equations of motion and by writing cm cmd dt =H M in scalar form for the spacecraft pair as a system. Unfortunately, expressing the EOMs in this manner produces a singularity. (The location of the singularity depends upon which scalar EOM is eliminated by the tether constraint,

[ ]2 2 2

2 1 2 1 2 1

1 2( ) ( ) ( )L x x y y z z= − + − + − = 10 meters, and occurs at either 1 2x x= , 1 2y y= , or 1 2z z= .) An alternate

formulation was thus sought to eliminate this singularity.

5 American Institute of Aeronautics and Astronautics

B. Singularity Removal

In the derivation of the EOMs, a constraint was defined to maintain the constant tether length. This constraint was applied to the equations of motion in order to eliminate the unknown tension force. As mentioned, performing the analysis in this manner introduces a singularity. An alternate approach is to substitute the EOMs into the tether length constraint equation and, instead of canceling the unknown constraint forces, calculate the value of the constraint forces. To illustrate this approach, begin with the tether constraint in the form

22 2 2 2

2

10

2

dL x y z x x y y z z

dt= ∆ + ∆ + ∆ + ∆ ∆ + ∆ ∆ + ∆ ∆ =

(8)

Substitute the equations of motion (2)-(7) into equation (8) to obtain

2 2 2 0x y zx y z xK yK zK∆ + ∆ + ∆ + ∆ + ∆ + ∆ = (9) where the operator ( )K ⋅ is defined as

( )( ) ( ) ( ) ( )2 1

2 2 1 1

F FT TK

m L m m L m⋅ ⋅

⋅

∆ ⋅ ∆ ⋅− + − −∑ ∑ (10)

Rearranging equation (9) results in

2 2 2

2 1

1 1x y zTL x y z xK yK zK

m m+ = ∆ + ∆ + ∆ + ∆ + ∆ + ∆

(11)

where a new operator, ( )K ⋅

, is defined to be

( )( ) ( )2 1

2 1

F FK

m m⋅ ⋅

⋅ −∑ ∑ (12)

To simplify equation (11), define the total mass and mass ratio as

1 2

2

1 2

m m m

m

m mµ

+

+

(13)

and equation (11) then becomes

( )2 2 2

1 x y z

TLx y z xK yK zK

mµ µ= ∆ + ∆ + ∆ + ∆ + ∆ + ∆

− (14)

and the tether tension is then found to be

( ) ( )2 2 21x y z

mT x y z xK yK zK

L

µ µ−= ∆ + ∆ + ∆ + ∆ + ∆ + ∆ (15)

6 American Institute of Aeronautics and Astronautics

Now that the tether tension is known, equations (2-7) can be numerically integrated to determine the state history of each satellite. By inspection of equation (15), as long as all forces on the two satellites are well defined and the tether length is nonzero, the tether tension can be found without the singularity difficulties encountered by other methods.

Equation (8) constrains the tether length as constant, but the length can be allowed to vary for study of the tether deployment phase. Following the same general approach from above, let the tether constraint be instead

22 2 2 2 2

2

1

2

dL x y z x x y y z z L LL

dt= ∆ + ∆ + ∆ + ∆ ∆ + ∆ ∆ + ∆ ∆ = +

(16)

Using the operator defined in equation (10) produces

2 2 2 2

x y zx y z xK yK zK L LL∆ + ∆ + ∆ + ∆ + ∆ + ∆ = + (17)

Rearranging equation (17) results in

2 2 2 2

2 1

1 1x y zTL x y z xK yK zK L LL

m m+ = ∆ + ∆ + ∆ + ∆ + ∆ + ∆ − −

(18)

Solving for the tether tension yields

( ) ( )2 2 2 21x y z

mT x y z xK yK zK L LL

L

µ µ−= ∆ + ∆ + ∆ + ∆ + ∆ + ∆ − − (19)

If the tether deployment profile is defined (i.e. ( )L t , ( )L t , and ( )L t are known), the tether deployment phase

can then be simulated (provided 0L ≠ ). Such a profile would typically be available when using an active control system to deploy the tether. Note that equation (19) reduces to equation (15) when 0L L= = , which corresponds to a tether of constant length.

C. System State Correction

As is well known, caution must be used when EOMs are numerically integrated as the numerical error can be significant sometimes making assessment of results difficult. Many studies have been conducted on developing methods to preserve certain constraints during numerical integration, with two cited here7,8. A new approach, termed “state correction,” is used here in an effort to reduce the numerical integration error. Equation (8) only constrains L to zero. In general this constraint would allow L to vary linearly with time, and choosing the initial conditions properly would cause L to remain constant. Unfortunately, round-off error in determining the tether tension (equation (19)) and truncation error induced through integration will cause failure of the system to follow the equation (8) constraint exactly and hence lead to slight variations in the tether length profile. In an attempt to remedy this induced error, a technique has been included in the numerical simulation that modifies the system state at each integration step to “correct” the induced error. The basic plan of the error correction method is to modify the system state as little as possible so that the new state satisfies the tether profile constraints

2 2 2 2L x y z= ∆ + ∆ + ∆ (20)

21

2

dL x x y y z z LL

dt= ∆ ∆ + ∆ ∆ + ∆ ∆ =

(21)

The next step is to determine what is meant by “modify the system state as little as possible.” The difference

between the modified state and the integrated state minimizes the function

7 American Institute of Aeronautics and Astronautics

1

2TV Ws s∆ ∆ (22)

where [ ]1 1 1 1 1 1 2 2 2 2 2 2Tx y z x y z x y z x y z∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆ ∆∆s is the change in the system state. W is a

weighting matrix that determines which states are adjusted the least, and is defined as

2 2 2 2

2 2 2 2

1 1 2 2

r v r v

r v r vCM CM CM CMW diag I I I I

(23)

With W defined as above, the change in the system state will minimize the percent change in the position and

velocity of the two satellites. The error correction problem is now a parameter optimization problem with constraints. Adjoining the constraints to equation (22) gives

1

2T TH W= s s f∆ ∆ + λ (24)

where

2 2 2 2x y z L

x x y y z z LL

∆ + ∆ + ∆ −

∆ ∆ + ∆ ∆ + ∆ ∆ −

f (25)

is evaluated at the corrected state. The change in the state is then found from

0TH

W∂ ∂

= =∂ ∂

fs

s s∆

∆ ∆+ λ (26)

0=f (27)

To solve equations (26) and (27), apply Newton’s method which yields

1

1

n

n n

−

+

∂= −

∂

gx x g

x x

(28)

where

sx

∆

λ (29)

T

W∂

+∂

fs

g sf

∆∆

λ (30)

22

21

0

fg s

f

s

T

ii

i

fW λ

=

∂ ∂+

∂ ∂ ∂=∂ ∂

∂

∑s

x∆ ∆

∆

(31)

8 American Institute of Aeronautics and Astronautics

Once equation (28) converges to a solution, the corrected state *s is obtained from

* = +s s s∆ (32)

Equation (32) is applied at the beginning of each integration step in an attempt to reduce the truncation error produced during the previous integration step and to enforce the tether constraint equation by allowing more accurate calculation of the tether tension. The results presented in the next section include examples with and without state correction (equation (32)) applied.

VI. Results A variety of simulations were conducted of interest to the MR SAT project. Since the launch vehicle and thus the

orbit that MR SAT will be injected into are not yet known, it was necessary to investigate a range of the applicable parameter space. While all of the simulations cannot be fully described here due to space constraints, a sample set summarizing the results is given. In this manner the tether behavior over a range of parameter space in LEO is provided that can assist future planning of tethered missions.

The initial conditions of the TSS were specified using two orientation angles α and β as shown in Fig. 2 that quantify the tether libration within the orbit plane (α) and out-of the orbit plane (β). In Fig. 2, r represents the radial direction from the center of the Earth, h is the angular momentum vector, and ө is directed along the local horizon. The EOMs were numerically integrated using MATLAB® with the ode113.m routine.

Figure 2 Definition of tether orientation angles.

9 American Institute of Aeronautics and Astronautics

The first simulation was a “baseline” case in which the model only incorporated a geopotential field corresponding to a spherically shaped Earth. No perturbations were included. The orbit of the center of mass is circular at 400 km altitude and 51 deg inclination. The resulting tension and tether orientation angles are shown in Fig. 3. Note the tension value of 0.144 millinewtons confirms the value found by equation (1) of 0.1439 millinewtons.

Figure 3 Baseline case at 400 km altitude: Initial α = 0, β = 0, α = 0, β = 0 , no perturbations.

Figure 4 shows the tether orientation for cases where an initial error of 20 degrees in α or β is present. As has been established by previous studies, the out-of-plane libration induces an in-plane libration. In both cases the tension never goes below 0.1 millinewtons, but does vary up to 0.2 millinewtons.

Figure 4 Initialα = 20/0 deg, β = 0/20 deg, α = 0, β = 0 , no perturbations.

10 American Institute of Aeronautics and Astronautics

Figure 5 shows a simulation with an initial β = 20 deg and 0.005α = rad/sec. A plot is included in this figure that shows β versus α. An obvious near-periodic motion has been established between the in-plane and out-of-plane motions.

Figure 5 Initialα = 0, β = 20 deg, α = 0.005 rad/sec, β = 0 , no perturbations.

Figure 6 shows the tension history for two cases with initial 0.05α = rad/sec or 0.05β = rad/sec. In both cases the tension fluctuates significantly but the tether remains taut.

Figure 6 Initialα = 0, β = 0 deg ; left plot α = 0.05 rad/sec, β = 0 , right plot α = 0, β = 0.05 rad/sec , no

perturbations.

11 American Institute of Aeronautics and Astronautics

Figure 7 shows a simulation using the baseline parameters ( 0, 0, 0, 0α β α β= = = = ) but now with all perturbations active (solar/lunar third-body gravity, SRP, drag, Earth 12x12 geopotential). The tension in the tether is maintained even in the presence of these primary perturbations. Note the duration of the propagation has been increased to twelve hours in this simulation.

Figure 7 Perturbed baseline case: initial α = 0, β = 0, α = 0, β = 0 , all perturbations active.

Figure 8 shows the case shown in Fig. 5 with all perturbations acting. The nearly periodic behavior between the in-plane and out-of-plane motion is still present, although now not as well defined. Other simulations conducted (but not shown here) with an eccentricity of 0.025 for the center of mass show further deterioration in the periodic behavior. However, tether tension is still maintained.

Figure 8 Initialα = 0, β = 20 deg, α = 0.005 rad/sec, β = 0 , all perturbations active.

12 American Institute of Aeronautics and Astronautics

Figure 9 shows the baseline case simulated at a higher orbital altitude of 900 km with no perturbations active. The tether tension is slightly reduced as expected. While not shown, with perturbations active the tether tension remains above 0.1 millinewtons.

Figure 9 Baseline case at 900 km altitude: initial α = 0, β = 0, α = 0, β = 0 , no perturbations.

Figure 10 shows a pair of identical simulations except that state correction was used for the case shown in the right hand plots. The plots show the desired (10 meters) and integrated tether length and the corresponding percent error. When state correction was used, an order of magnitude improvement was realized. The use of state correction is an area of continuing study at this time.

Figure 10 Initialα = 0, β = 20 deg, α = 0.005 rad/sec, β = 0 : State correction in right hand plots.

VII. Conclusion

A simulation routine was developed to study the motion of a short, inextensible tether in low Earth orbit. All perturbations expected to have a significant effect on the tether motion were included in the model, including Earth’s geopotential terms through degree and order twelve, lunar and solar gravity, solar radiation pressure, and aerodynamic drag. In the analysis performed for a tether of ten meters length, all data obtained indicates that the tether will not go slack once it is successfully deployed. Deployment errors will lead to libration by the tether, but tension will still be maintained (unless the deployment errors are excessive). Future efforts will examine the deployment aspects in more detail.

13 American Institute of Aeronautics and Astronautics

Acknowledgements Dr. Brad Miller from the University of Missouri-Rolla is gratefully acknowledged for many helpful discussions. The financial support of this work from the Mechanical and Aerospace Engineering Department at the University of Missouri-Rolla is also appreciated.

References 1Alfriend, K. T., Barnds, W. J., Coffey, S. L., and Stuhrenberg, L. M., “Attitude and Orbit Determination of a

Tethered Satellite System,” AAS 95-351, AAS/AIAA Astrodynamics Specialists Conference, Halifax, Nova Scotia, Canada, August 14-17, 1995.

2Cosmo, M.L. and Lorenzini, E.C., “Tethers in Space Handbook,” prepared for the NASA Marshall Space Flight

Center, Third Edition, December 1997. 3Misra, A.K. and Modi, V.J., “A Survey on the Dynamics and Control of Tethered Satellite Systems,” Advances

in the Astronautical Sciences, Vol. 62, Univelt, 1987, pp. 667-719 (also AAS paper 86-230). 4Breakwell, John V., and Gearhart, James W., “Pumping a Tethered Configuration to Boost its Orbit Around an

Oblate Planet,” The Journal of Astronautical Sciences, Vol. 35, No. 1, January-March 1987, pp. 19-39. 5Cochran, J. E, Jr., Cho, S., Lovell, T. A., and Cicci, D. A., “Modeling Tethered Satellite Dynamics for

Identification and Orbit Determination,” The Journal of the Astronautical Sciences, Vol. 48, No. 1, January-March 2000, pp. 89-108.

6Cho, S., Cochran, J. E., Jr., and Cicci, D. A., “Identification and Orbit Determination of Tethered Satellite

System,” AAS 98-101, AAS/AIAA Astrodynamics Specialists Conference, Boston, MA, August 10-12, 1998. 7Gonzalez, O., “Mechanical Systems Subject to Holonomic Constraints: Differential-Algebraic Formulations and

Conservative Integration”, Physica D, 1997. 8Leimkuhler, B. and Skeel, R. D., “Symplectic Numerical Integrators in Constrained Hamiltonian Systems”,

Journal of Computational Physics, 1994, pp. 117-125.