Investigation of the Attitude Dynamics of the ISEE-3 Spacecraft

Investigation of the Attitude Dynamics of the ISEE-3 Spacecraft AOE5984: Advanced Spacecraft Dynamics and Control Kevin Langone December 12, 2003

Contents 1 Abstract 1 2 Introduction 1 3 Literature Review 2 4 Parameters 4 4.1 Reference Frames......................... 4 4.2 Units................................ 4 4.3 Spacecraft Dimensions...................... 5 4.4 Assumptions............................ 5 5 Mathematical Analysis 6 5.1 Orbit Stability........................... 6 5.2 3-axis Fixed Attitude Stability.................. 7 5.3 Stabilty of Axisymmetric Spinning Body............ 9 6 Simulation 10 6.1 Orbit................................ 11 6.2 Nominal Attitude......................... 11 6.3 3 Axis Stabilization........................ 17 6.4 Attached Spring-Mass-Damper.................. 20 7 Conclusion 22 i

1 Abstract This paper investigates the attitude propagation of the International Sun Earth Explorer-3 spacecraft during the halo orbit phase of its mission. The linearization technique is used to analyze the stability of both orbit and attitude for a spacecraft in the vicinity of L 1. Numerical simulation then models the nominal attitude propagation. Other simulations change ISEE-3 to a 3-axis stabilized spacecraft and a spinner with a spring-mass-damper installed. In the 3-axis case, the gains are varied to determine their effects. For the energy dissipator, the orientation is changed to examine the change in error-damping effectiveness. 2 Introduction The International Sun-Earth Explorer mission was a joint project between the National Aeronautics and Space Administration (NASA) and the European Space Agency (ESA) designed to study the interaction of the solar wind with the Earth s magnetosphere. The mission consisted of three spacecraft, the first two of which were launched in 1977 into highly elliptical Earth orbits. ISEE-3, launched in August of 1978 towards the L 1 point of the Sun-Earth system, became the first satellite to orbit a Lagrange point. The spacecraft was inserted into a halo orbit which kept it outside the solar exclusion zone (which would disrupt communications) but inside the antenna s field of view. Study of the spacecraft s orbit provided scientists and engineers with much new information on the mechanics of libration point orbits in the Sun-Earth system including trajectory correction and stationkeeping maneuver requirements. For over three years ISEE-3 was kept in its halo orbit about L 1 making upstream measurements of the solar wind. In June of 1982, the spacecraft performed the first of several maneuvers, sending it back to the vicinity of the Earth in order to make measurements of the Earth s geomagnetic tail. After several transits ISEE-3 left the Earth to intercept the comet Giacobini- Zinner. In September of 1985 ISEE-3 became the first spacecraft to make in-situ cometary measurements. It is currently in a heliocentric orbit and will return to Earth in 2014 barring further maneuvers. The purpose of this project is to simulate the dynamics of ISEE-3 during its halo orbit. Section 5 presents mathematical analysis to show the instabil- 1

ity of both the orbit and attitude in the absence of control. Section 6 then discusses the results of computer simulations which model the dynamics of ISEE-3. The orbit is considered independent of the attitude and computed first. The attitude simulation then models ISEE-3 in its nominal spinning attitude. Other models are included as well such as a three-axis stablization mode and the inclusion of an energy sink device to examine the effects that these changes have on the attitude propagation. 3 Literature Review Shortly after the development of space flight, the use of libration orbits for space missions became a topic of considerable interest. Robert Farquhar was one of the pioneers in the field; he completed thesis work on the subject and later worked as a mission designer for the ISEE-3 project. He, along with Muhonen and Richardson, discuss the proposed orbit for ISEE-3 while providing an introduction to libration point orbits along with a description of how the halo orbit is designed to fit the mission constraints [1]. In addition they provide appropriate equations of motion in the vicinity of L 1 and the required coupling between in-plane and out-of-plane frequencies needed to produce a halo orbit. Finally, these researchers conclude with a discussion about the transfer to the halo orbit and give a V budget associated with the midcourse corrections and stationkeeping maneuvers. In a later paper, Farquhar and Muhonen, along with Church, summarize ISEE-3 s maneuvers up to 1985 [2]. They discuss the successful maneuvering of the spacecraft that had only been predicted in the previous literature. Also, Farquhar and his colleagues present a summary of all the maneuvers including the launch and orbit maintenance for the halo orbit, transfer to the Earth s geomagnetic tail, and comet G-Z intercept trajectory via lunar swingby. Finally they report that the spacecraft initially carried 89 kg of hydrazine, enough for about 430 m/s of maneuvers. Stationkeeping for three years required only about 30 m/s, and attitude and spin control required about 32 m/s during that period. Thus maintenance of a halo orbit about L 1 was considerably less fuel costly than initially believed. Work continutes to further understand the dynamics of spacecraft in three body systems. For example, papers by Howell [3] and by Guzman, Cooley, Howell, and Folta [4] present more complete investigations of libration point orbits after ISEE-3 s launch. These investigations include computations of 2

orbit families and the combination of dynamical systems theory with computer simulation to study the nature of three body orbits. The study of orbits in the three body problem only makes up half of a spacecraft s dynamics; the attitude motion is certainly important as well. ISEE-3 was nominally spin-stabilized about its symmetry axis, which was to remain perpendicular to the ecliptic plane during its orbit. Spin-stabilization has been a concept of interest since the beginnings of space flight and considerable literature is available on the subject. Jones discusses a method for reorienting the spin axis of an axisymmetric spacecraft with minimum precession [5]. He concludes that a constant pitch rate is desirable during rotation of the spin axis. Later, Dvornychenko and Gerding examine the effects of non-circular orbits on spin stability [6]. They consider the effects of eccentricity and nodal regression and they establish a method for determining the maximum precession of the spin axis without solving the equations of motion. Steyn presents a method for spacecraft slew maneuvers using the shortest angular path [7]. A combination of passive gravity-gradient and active reaction wheel control is used to three-axis stabilize a nadir pointing spacecraft. Initially, the attitude stability of Earth-orbiting spacecraft was the primary focus of research efforts. Eventually though, the attitude motion of satellites in three-body orbits became a topic of interest as well. The first studies concerned the stability of a spacecraft located at a Lagrange point. Kane and Marsh [8] examined the stability of a spacecraft spinning with constant angular velocity at these points. At the triangular points, they conclude that the stability properties are nearly identical to the two-body case. At the colinear points however, the regions of instability are much larger than for the two-body case. Robinson [9] continues the work by considering a non-spinning body at the equilibrium points. He finds two regions of stability at the colinear points, similar to the Lagrange and Debra-Delp regions for the two-body case. Additionally, he finds two stable attitudes at the equilateral points. Later, Hitzl and Levinson analyze the attitude stability of an axisymmetric spinning satellite moving in a peridioc orbit of the restricted three body problem [10]. They employ Floquet theory to construct instability charts and present these for several orbits in the Earth-Moon system. Ashenberg later examines the pitch dynamics of a spacecraft moving in the elliptic restricted three-body problem in [11]. He shows that the pitch equation is of the same form for this problem as for the two-body case and that for stable pitch motion at a Lagrange point the satellite s minor axis is 3

tilted toward the larger primary. 4 Parameters 4.1 Reference Frames There are three primary frames of reference utilized in the following discussions. The first is the spacecraft body frame (b) which is fixed in the satellite. The 3 axis is the symmetry axis, and the 1 and 2 axes are mutually perpendicular to each other and the 3 axis and complete a right hand set. Since the spacecraft is considered axisymmetric the exact direction is somewhat arbitrary. An inertial reference frame (i) will also be used. For this case, the sun will be considered inertially fixed with the spacecraft and Earth moving around it. The inertial 3 axis is perpendicular to the ecliptic plane while the 1 and 2 axes remain fixed in the plane of rotation and complete a right hand set. The final reference frame of interest is the rotating libration point (rlp) frame. In this case the 3 axis is again perpendicular to the ecliptic plane. The 1 axis is along the line joining the Earth and Sun and points at the Earth at all times. The 2 axis is also in the ecliptic plane and completes the right hand set. 4.2 Units In the discussion of many three-body problems, canonical units are used to generalize the problem. A similar approach is taken here for the purpose of computations. Masses are normalized on the total mass of the primaries (Sun and Earth here) and distances on the mean distance between the primaries. Time is based on the revolutionary period of the smaller primary (in this case the Earth). The canonical units are denoted MU, DU, TU for the mass unit, distance unit, and time unit, respectively. The values for this problem are given below. 1MU = 2 10 31 kg 1DU = 1.5 10 11 m 1T U = 86400 365.25 2π 5.023 10 6 s 4

4.3 Spacecraft Dimensions The body of ISEE-3 is essentially a right circular cylinder of height 161 cm and diameter 173 cm with a mass of approximately 470 kg [1]. The moments of inertia for the body are then I 1 = I 2 = 1 4 mr2 + 1 12 ml2 = 189.44 kg m 2 I 3 = 1 2 mr2 = 175.83 kg m 2 Thus for the body the spin axis is the minor axis. However, ISEE-3 is equipped with a total of ten appendages that alter the inertia properties. These are listed in table 1; each is modeled as slender rod lying on the indicated axis (and thus it alters the moments of inertia about the other two axes by an amount equal to 1 3 ml2 ). All appendages are perpendicular to the symmetry axis with the exception of the axial antennas. With the inclusion of the appendages, the total moments of inertia are: I 1 = I 2 7000 kg m 2 I 3 13500 kg m 2 Thus the spin axis is the major axis of the spacecraft. ISEE-3 is included in Figure 1. A diagram of Table 1: Approximate size and mass of ISEE-3 appendages. Appendage Mass (kg) Length (m) Moment of Inertia (Each) Wire Antenna (4) 4 50 3333 Experiment Boom (2) 2 3 6 Inertia Boom (2) 2 3 6 Axial Antenna (2) 5 7 81 4.4 Assumptions In general, the following assumptions are implicit to all simulations unless otherwise noted: 1. ISEE-3 is a rigid body with constant moments of inertia as calculated above (in the body frame) 2. All assumptions associated with the circular restricted three-body problem 5

5 Mathematical Analysis 5.1 Orbit Stability The Lagrange points are those five points in the circular restricted three body problem at which the forces mutually cancel in the rotating reference frame. In this case, the stability of the L 1 point in the Sun-Earth system is of interest. The linearized equations of motion for a satellite near the Sun-Earth L 1 point are [1] ẍ 2ẏ (2B L1 + 1)x = 0 ÿ + 2ẋ + (B L1 1)y = 0 (1) z + B L1 z = 0 where B L1 = 4.06107. Using the substitutions x 1 = x x 2 = ẋ x 3 = y x 4 = ẏ x 5 = z x 6 = ż Figure 1: The ISEE-3 spacecraft. Reprinted from [1]. 6

the above equations can easily be written as the following first order system: 0 1 0 0 0 0 2B L1 + 1 0 0 2 0 0 x = A x A = 0 0 0 1 0 0 0 2 1 B L1 0 0 0 0 0 0 0 0 1 0 0 0 0 B L1 0 The eigenvalues of the matrix A are ±2.533 ± 2.087ı ± 2.015ı According to Hughes [12], the linear system given by equation 2 is unstable in the vicinity of the origin if any of the eigenvalues of A has a positive real part, which is true in this case. Furthermore, if the linear system is shown to be unstable, the nonlinear system is necessarily unstable as well. Thus motion in the vicinity of L 1 is an unstable motion and a spacecraft s trajectory will eventually diverge from that point in the absence of control. This fact is well known and has been proved in literature such as that by Szebehely [13]. 5.2 3-axis Fixed Attitude Stability The following discussion examines the attitude stability of a spacecraft at the L 1 point. The appropriate linearized attitude equations of motion applied to a non-spinning spacecraft subject to gravity gradient torque from both bodies are [9]: θ 1 kj 3 θ 1 = 0 (2) θ 2 J 1 θ 2 = (J 1 + 1) θ 3 (3) θ 3 + J 2 (1 + k)θ 3 = (J 2 1) θ 2 The θ i terms refer to Euler angles that relate the body and rotating libration point reference frames. The above equations apply to a body at the equilibrium point. This is not true for ISEE-3 but the distance away from L 1 7

is small compared to the distances from the primaries so this is not a bad approximation. The inertia coefficients and constant k are defined as: J 1 = I 2 I 3 J 2 = I 3 I 1 J 3 = I ( 1 I 2 m1 k = 3 I 1 I 2 I 3 r1 3 + m ) 2 r2 3 Examination of Equations 3 reveals that the first equation decouples from the other two. This equation is standard form second order and undamped; the corresponding stability condition is that the coefficient of the undifferentiated term by positive. Thus stability requires kj 3 < 0 (4) The other two equations can be written as a first order linear system in the same manner as in the previous section: 0 1 0 0 J x = A x A = 1 0 0 J 1 + 1 0 0 0 1 0 J 2 1 J 2 (1 + k) 0 The characteristic equation for the above system is (5) λ 4 + aλ 2 + b = 0 a = 1 + kj 2 J 1 J 2 b = J 1 J 2 (1 + k) (6) Routh s stability require the following to be true for system stability: a > 0 b > 0 a 2 > 4b (7) Using the above conditions along with Equation 4 yields the information depicted in Figure 2. There are two regions of stability, labeled I and II in the figure. Region I corresponds to I 3 > I 2 > I 1 while Region II indicates I 2 > I 1 > I 3. There are other restrictions on Region II though based on the constant k and the inertia coefficients. A graph similar to Figure 2 is available for Earth orbiting spacecraft subject to gravity gradient using Smelt parameters. In Section 6, ISEE-3 will be modeled in a 3-axis stabilized mode with its major axis nominally aligned with the ecliptic normal, which corresponds to a body on the border of Region I in Figure 2. 8

5.3 Stabilty of Axisymmetric Spinning Body The following analysis applies to the same situation as discussed in the previous section, except now the spacecraft is taken as an axisymmetric spinner. The linearized equations of motion for such a spacecraft at L 1 are [8] θ 1 [2 I(r + 1)] θ 2 [1 I(r + 1)]θ 1 = 0 (8) θ 2 + [2 I(r + 1)] θ 1 [1 I(r + 1)]θ 2 + k(i 1)θ 2 = 0 with the following quantities defined: I = I 1 I 3 r = η Ω τ = Ωt k = 3 [ ] 1 1 + m ρ + m 3 (1 ρ) 3 where η is the spin rate of the spacecraft, Ω is the spin rate of the rlp frame, m is mass ratio of the small primary to that of the large primary, ρ is the nondimensionalized distance from the large primary to L 1, and differentiation is with respect to τ. For the Sun-Earth system, ρ = 0.99 m = 3 10 6 k = 12.092 Figure 2: Attitude stability chart for non-spinning spacecraft at L 1. 9

The characteristic equation associated with Equation 8 is λ 4 + aλ 2 + b = 0 (9) a = I 2 (r + 1) 2 2I(r + 1) + 2 k(1 I) b = I 2 (r + 1) 2 I[2 + k(1 I)](r + 1) + 1 + k(1 I) The requirements for stability are again given by Equation 7. An instability plot is provided in Figure 3. Faster spins serve to stabilize the spacecraft, even if the major axis is not parallel to the ecliptic normal. For this system, spins in the opposite sense of the primary orbital motion are always unstable. ISEE-3 s spin rate was much faster than shown here and the spacecraft had I>1 so it would always be stable under the gravity-gradient torque if it were situated at L 1. Figure 3: Attitude stability chart for spinning spacecraft at L 1. 6 Simulation Simulation of ISEE-3 s dynamics are conducted with Matlab script files. For all simulations initial conditions for orbit and attitude are specified and 10

then differential equations of motion are numerically integrated using Matlab s ordinary differential equation solvers. 6.1 Orbit For the purpose of the orbit simulation the dimensions of the spacecraft are considered small compared to the dimensions of the orbit (i.e. a point mass) and thus the orbit is unaffected by the attitude and can be considered separately. The orbit propagation is modeled using Equations 1. Initial conditions are chosen such that the spacecraft begins on the RLP x-axis at the point of maximum z-amplitude. Initial velocities are selected so that the orbit makes two consecutive perpendicular crossings of the x-z plane, establishing the halo orbit. The time for one full revolution is approximately six months. The appropriate values are: x o = 216, 767km ẋ o = 0.00492km/s y o = 0 z o = 120, 000km ẏ o = 0.3km/s ż o = 0.0003km/s Orthographic views of the orbit (propagated for 170 days) are shown in Figure 4. The attitude simulations are conducted for much shorter times both for computational reasons and the fact that it should not take nearly as long to demonstrate stability/instability. 6.2 Nominal Attitude The first simulation models the actual attitude propagation of ISEE-3. The nominal attitude of the spacecraft kept the symmetry axis perpendicular to the ecliptic plane. ISEE-3 spun about this axis at a constant rate of approximately 20 revolutions per minute. The attitude of the spacecraft is characterized by a 3-1-3 sequence of Euler angle rotations (precession, nutation, and spin respectively) that relate the body frame to the rotating libration point frame. The corresponding rotation matrix is cθ 1 cθ 3 sθ 1 cθ 2 sθ 3 sθ 1 cθ 3 + cθ 1 cθ 2 sθ 3 sθ 2 sθ 3 R b rlp = cθ 1 sθ 3 sθ 1 cθ 2 cθ 3 sθ 1 sθ 3 + cθ 1 cθ 2 cθ 3 sθ 2 cθ 3 (10) sθ 1 sθ 2 cθ 1 sθ 2 cθ 2 11

Figure 4: ISEE-3 orbit simulation. 12

where c represents a cosine and s represents a sine. The appropriate kinetics equation is Euler s equation, expressed in the body frame [14]: I ω b i = ω b i Iω b i + g (11) where I is the principal moment of inertia matrix, ω is the angular velocity, and g is the torque vector. For many Earth-orbiting spacecraft, the gravity gradient torque is the dominant torque. Gravity gradient torque, expressed in the body frame, is computed by the expression [10] g gg = 3 GM r 3 â Iâ (12) where G is the gravitational constant, M is the central body mass, r is the distance to the central body s center of mass, and â is a unit vector from the spacecraft to the primary expressed in the body frame. In contrast to Earth satellites, the factor of r 3 in the denominator makes the gravity gradient torque small in magnitude for a spacecraft in the vicinity of L 1 (approximately 10 9 N m due to both the Sun and Earth). The torque caused by solar radiation pressure can indeed be considerably greater in magnitude. Exact computation depends heavily on the spacecraft geometry and surface reflectivity, but for this case the value can be approximated as 10 7 N m and considered constant [15]. For the purposes of this simulation, the solar radiation torque will be taken as acting about the rlp 2-axis. Thus, expressed in the body frame, 0 g slp = R b rlp 10 7 0 N m (13) When used in the numerical simulations, both the gravity gradient torque and the solar radiation torque will be included in Equation 11; however, the contribution of the gravity gradient torque is nearly negligible. In addition to the kinetics equation, a kinematics equation is necessary to determine the propagation of the attitude in time. As noted above, Euler angles are used to describe the attitude for visualization. For the purposes of numerical integration, however, Euler angles are generally a poor choice due to singularities and an abundance of trigonometric functions involved in the computations. Quaternions are used for this simulation because they avoid these problems. The initial conditions on Euler angles are first used 13

in Equation 10 to obtain a starting rotation matrix. The initial quaternions are then calculated from the expressions [14] q 4 = 1 2 1 + trace(r b rlp ) q = 1 4q 4 r 23 r 32 r 31 r 13 (14) r 12 r 21 where r ij is the entry from the ith row and j th column of R b rlp and the trace operation indicates the sum of the diagonal entries of the operand. The quaternions are then indicated with the appropriate kinematics equation [14]: q = Q( q)ω b rlp Q = 1 2 [ q ] + q 4 I d1 q T (15) where I d1 is the three-by-three identity matrix. Note that Equations 11 and 15 use two different angular velocities. The two vectors are related simply by ω b rlp = ω b i ˆr i3 2π (16) Here ˆr i3 indicates the third column of R b rlp and 2π is the angular velocity of the rlp frame with respect to the i frame. Once the integration is complete, the quaternions are converted back to Euler angles via the rotation matrix given in Equation 10 and the relation [14] R b rlp = (q 2 4 q T q)i 1 + 2qq T 2q 4 q (17) Note that once the rotation matrix is known, quadrant checks must be performed to discern the correct Euler angles, as the arc trigonometric functions may not return the expected values. With the above equations, the simulation has everything it needs to model ISEE-3 s dynamics. The first case is the nominal one, that is, all Euler angles are initially zero and the initial angular velocity is 20 rpm about the symmetry axis. The results are shown in Figure 5 for sixty seconds of propagation. As expected, the spacecraft exhibits a steady spin (rotating 20 times during the minute) and maintains its spin axis perpendicular to the ecliptic plane. Note that for the case of no nutation, the precession angle cannot be distinguished from the spin angle. The code is instructed to plot the sum of the two angles as only the spin angle (θ 3 ). Examination of longer duration simulations is instructive as well to see the behavior of the spin axis as it propagates through time. In the above case the 14

nutation stayed near enough to zero to be unnoticeable if not identically zero. Figure 6 displays the same simulation used above for a period of 150 minutes (the spin angle is not shown so as not to clutter the plot). The nutation angle begins to oscillate about zero with increasing amplitude, reaching a value of about 0.05 degrees in 9000 seconds. The case of the same initial angles (R b rlp =I 1 ) but with no spin is shown as well. For this case the angle between the ecliptic normal and the spin axis increases exponentially and reaches a value of approximately 0.33 degrees during the same span. Thus the spinning motion of the spacecraft serves to considerably slow the divergence of the spacecraft from its nominal attitude in the presence of disturbing torques. Figure 7 displays the data for a further investigation of the spin rate s effect on the attitude propagation. It shows the time for the spacecraft spin axis to deviate 0.01 degrees from the ecliptic normal for varying spin rates. Examination shows that the non spinning case actually diverges more slowly than slow spin rates but this trend quickly reverses past a spin rate of about 3 revolutions per minute. This is due to a slow (exponential) initial increase associated with the non-spinner but a relatively quick initial increase from the slow-spinning case which quickly tapers off. A larger choice for the angle of interest would likely yield a monotonic increase of the time to reach that angle with increasing spin rate. Referring again to Figure 6, even the spinning motion is insufficient to maintain the nominal pointing of the symmetry axis. Though it occurs slowly, Figure 5: Nominal attitude propagation. 15

Figure 6: Nutation and precession angles for spinning and non-spinning cases. Figure 7: Time for spin axis to deviate 0.01 degrees from ecliptic normal versus spin rate. 16

eventually the deviation between the ecliptic normal and the spin axis reaches an unacceptable level from a scientific perspective. Some additional method is needed to indefinitely maintain the attitude. ISEE-3 is equipped with a total of twelve thrusters: four radial, four spin-change, and four axial [2]. As noted in the Literature Review, during the halo orbit phase of the mission about 32 m/s of fuel was used for attitude maintenance and spin control. 6.3 3 Axis Stabilization Figure 6 shows that in the absence of any control or stabilizing mechanism the symmetry axis departs from its nominal direction (ecliptic normal). The next simulation uses ISEE-3 s thrusters in attempt to stabilize the spacecraft in a 3-axis fixed configuration. As noted in the previous section, ISEE-3 has a total of twelve thrusters onboard. It is assumed that the redundant configuration can generate torques about any axis. For the following simulation, a 2-1-3 rotation sequence is employed. The corresponding rotation matrix in terms of the Euler angles is cθ 1 cθ 3 + sθ 1 sθ 2 sθ 3 cθ 2 sθ 3 sθ 1 cθ 3 + cθ 1 sθ 2 sθ 3 R b rlp = cθ 1 sθ 3 + sθ 1 sθ 2 cθ 3 cθ 2 cθ 3 sθ 1 sθ 3 + cθ 1 sθ 2 cθ 3 (18) sθ 1 cθ 2 sθ 2 cθ 1 cθ 2 The Euler angles will now be integrated directly using the following relation: sθ 3 /cθ 2 cθ 3 /cθ 2 0 θ = S 1 (θ)ω b rlp S 1 (θ) = cθ 3 sθ 3 0 (19) sθ 3 sθ 2 /cθ 2 cθ 3 sθ 2 /cθ 2 1 Note that the singularity occurs at θ 2 = 90 whereas with the previous Euler angle sequence the singularity would occur at 0. Since the nominal attitude is all angles equal to zero, the singularity does not present a problem. Equation 11 is again used to model the kinetics. A simple linear PID controller is used to create the control torques. An initial error was created for the Euler angles, and the gains were varied to determine their effect. Several important results were noted: 1. Varying the proportional gain (k p ) has no effect on the time to damp out the disturbances. This is expected since the proportional time 17

multiplies the state, whereas the damping comes from the derivative terms. 2. Figure 8 displays the time for the controller to reduce the disturbance to 1/10th of the initial amplitude for different values of the derivative (k d ) and integral (k I ) gains. In general, reducing the integral gain and increasing the derivative gain reduce the time for the error to be damped out. 3. For a given value of the derivative gain, there is a maximum value of the integral gain beyond which the system response will diverge. This value of integral gain is plotted against derivative gain in Figure 9. Note that the relationship is exactly linear. 4. Figure 10 shows the total control effort required to correct the disturbance to 1/10th of its original value. Note that this figure corresponds almost identically with Figure 8. Thus the total control effort is directly related to the time required to reduce the disturbance. More total control effort corresponds directly to higher fuel consumption. 5. The total control effort required increases monotonically with increases in proportional gain. This result is expected since a higher proportional gain causes the motion to oscillate more. Thus a low proportional gain is desired for minimum fuel expenditure. Figure 8: Time to 1/10th amplitude using PID controller. 18

Figure 9: Corresponding integral and derivative gains for system divergence. Figure 10: Total control effort required to reduce disturbance by 90%. 19

6.4 Attached Spring-Mass-Damper The next simulation includes a spring-mass-damper fixed in the spacecraft. Figure 11 is a diagram of ISEE-3 with an included spring-mass-damper. The body frame is the same as previously established, but relocated to the center of mass with the inclusion of the damper. The principal moments of inertia are calculated with the damper at rest in the neutral position. s is a vector from the center of the b-frame to the neutral location of the damper; the r vector extends from the b-frame center to the current location of the damper. ξ is the displacement of the damper along its axis, whose unit normal vector is defined as n. Figure 11: ISEE-3 with a spring-mass-damper. In contrast to the previous two cases, this simulation does not establish the orbit separately from the attitude. The appropriate equations of motion are provided by Hughes [12]. The following symbols are defined as: c F irst moment of inertia J Second moment of inertia v Linear velocity of b frame m T otal spacecraft mass m d Mass of particle c visc V iscous damping constant k Spring constant The linear momentum of the spacecraft is p = mv c ω + m d ξn (20) 20

The angular momentum of the body frame is h = c v + Jω + m d ξs n (21) The linear momentum of the particle in the n-direction is also useful and defined as p n = m d (n T v n T s ω + ξ) (22) In order to use the numerical integrator, the time rates of change of these quantities must be defined. The time rate of change of linear momentum is ṗ = ω p + f (23) where f represents the external forces on the system. In this case the only external forces are the gravitational forces of the Sun and Earth, which are given by Newton s Law of Gravitation: f = GMm r 3 r (24) This force must be rotated into the body frame via the rotation matrix for use in Equation 23. The derivative of angular momentum is ḣ = ω h v p + g (25) Again the torque g is given by Equations 12 and 13. Finally, ṗ n = m d ω T n (v r ω) c visc ξ kξ (26) Note that the quantities v, ω, and ξ must be known to compute the right hand side of Equations 23, 25, and 26. These are computed by writing Equations 20, 21, and 22 as a linear system in terms of the above variables and inverting the coefficient matrix. This allows computation of v, ω, and ξ in terms of the current states. Finally, the necessary kinematics equations are Equation 15 and ẋ = v (27) The initial conditions on position and velocity are as indicated in Section 6.1. The damper is oriented at a 45 degree angle to both the body 1 and 2 axes. Other parameters are established as: 21

m d = 10 kg c visc = 150 N s m k = 1000 N m ξ o = 0.5 m ξo = 0 The initial attitude is specified as a 1 degree error on the nutation angle. Figure 12 shows the propagation of the nutation angle as the spacecraft spins. The damper does serve to eliminate the error in the state, though the time to damp in this case is on the order of days. Figure 12: Nutation damping using energy dissipation device. The next simulation used the same parameters but changes the orientation of the damper to determine the effect of this variable. The simulations were conducted for approximately three days; Figure 13 shows the final nutation angle after the given period for different damper orientations. The damper angle is measured relative to the spacecraft 2 axis. Results show that skewing the damper farther from the 2 axis improves the damping qualities, though the change is not large. 7 Conclusion This project examined the dynamics of the ISEE-3 spacecraft during the halo orbit phase of its mission. The linzearized equations of motion in the 22

vicinity of the solar L 1 point established the orbital motion. The first simulation modeled the nominal attitude propagation of ISEE-3, during which the spacecraft spun about its symmetry axis, which remained perpendicular to the ecliptic normal. Results showed that the spinning motion provided some stability to the motion and helped delay the deviation of the symmetry axis from its nominal direction. The next simulation modeled the spacecraft as 3-axis stabilized using the hydrazine thrusters to maintain the attitude. Higher derivative gains and lower integral gains reduce both the time and fuel required to damp out disturbances. The final simulation included an energy dissipation device in the spacecraft. This device served to damp out initial errors in the nutation angle, though the damping took place slowly. Figure 13: Final nutation angle after 250,000 seconds of damping. 23

References [1] R.W. Farquhar, D.P. Muhonen, and D.L. Richardson, Mission Design for a Halo Orbiter of the Earth, Journal of Spacecraft and Rockets, Vol. 14, No. 3, March 1979, pp. 170-177. [2] R.W. Farquhar, D.P. Muhonen, and L.C. Church, Trajectories and Orbital Maneuvers for the ISEE-3/ICE Comet Mission, Journal of the Astronautical Sciences, Vol. 33, No. 3, July-September 1985, pp. 235-254. [3] J.J. Guzman, D.S. Cooley, K.C. Howell, and D.C. Folta, Trajectory Design Strategies that Incorporate Invariant Manifolds and Swingby, Paper No. AAS 98-349, Spaceflight dynamics 1998; Proceedings of the AAS/GSFC International Symposium, Greenbelt, Maryland, May 1998. [4] K.C. Howell, Families of Orbits in the Vicinity of the Collinear Libration Points, Paper No. AAS 98-4465, AIAA/AAS Astrodynamics Specialist Conference and Exhibit, Boston, Massachusetts, August 1998. [5] A. Jones, Closed-Loop Pitch Control for Spin-Stabilized Spacecraft, Journal of Spacecraft and Rockets, Vol. 3, No. 4, April 1966, pp. 543-547. [6] V.N. Dvornychenko and R.B. Gerding, The Effect of Orbital Eccentricity and Nodal Regression on Spin-Stabilized Satellites, Celestial Mechanics, Vol. 16, November 1977, pp. 263-274. [7] W.H. Steyn, Near-Minimum-Time Eigenaxis Rotation Maneuvers Using Reaction Wheels, Journal of Guidance, Control, and Dynamics, Vol. 18, No. 5, September-October 1995, pp. 1184-1189. [8] T.R. Kane and E.L. Marsh, Attitude Stability of a Symmetric Satellite at the Equilibrium Points in the Restricted Three-Body Problem, Celestial Mechanics, Vol. 4, 1971, pp. 78-90. [9] W.J. Robinson, Attitude Stability of a Rigid Body Placed at an Equilibrium Point in the Restricted Problem of Three Bodies, Celestial Mechanics, Vol. 10, 1974, pp. 17-33. [10] D.L. Hitzl and D.A. Levinson, Attitude Stability of a Spinning Symmetric Satellite in a Planar Periodic Orbit, Celestial Mechanics, Vol. 20, August 1979, pp. 179-205. 24

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