- Email: [email protected]
Climber motion optimization for the tethered space elevator
ARTICLE IN PRESS Acta Astronautica 66 (2010) 1458–1467
Contents lists available at ScienceDirect
Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro
Climber motion optimization for the tethered space elevator Paul Williams , Wubbo Ockels Faculty of Aerospace Engineering, Delft University of Technology, P.O. Box Postbus 5058, 2600 GB Delft, The Netherlands
a r t i c l e i n f o
abstract
Article history: Received 23 May 2009 Received in revised form 30 September 2009 Accepted 2 November 2009 Available online 30 November 2009
The tethered space elevator could provide a revolutionary means for enabling cheap transportation to geostationary altitude and beyond. Assuming that such a system can be built, one of the dynamic design problems is determining a means of moving the elevator along the tether so as to minimize the residual in-plane librational motion of the elevator ribbon and counterweight. In particular, this paper studies the problem from the point of view of dynamic optimization. A simplified dynamic model of the elevator system is derived that accounts for the fundamental libration modes and the motion of the elevator. The model is used to solve an optimal control problem that results in zero final in-plane librational motion of the ribbon. The results show that it is possible to eliminate the in-plane librational oscillations by reversing the direction of the elevator for a short time. It is also shown that it is possible to remove excess librational energy by controlled movement of the elevator on the ribbon. The effects of out-of-plane librations are also considered, but the climber motion does not have enough control authority to damp the out-of-plane motion over short time scales. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Space elevator Climber Numerical optimization Optimal control
1. Introduction The tethered space elevator is a radical concept that has existed in the literature even prior to the modern spaceflight era [1]. The basic concept requires a tether or ribbon extending from geostationary altitude to the Earth’s surface. However, in order to make the system feasible, the ribbon is required to extend past geostationary altitude. The system center of orbit is designed to remain at geostationary altitude so that the system spins in orbit at the same rate as the Earth rotates. This means that the tip of the ribbon is always located at the same location on the Earth’s surface. Although Tsiolkovsky [1] is often credited with its invention, it was not until Pearson [2] that the concept was cemented in engineering terms. At the time of publication, no engineering material existed
Corresponding author. Tel.: + 61 3 9859 6877; fax: + 61 3 9925 6108.
E-mail addresses: [email protected] (P. Williams), [email protected] (W. Ockels). 0094-5765/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2009.11.003
that could come close to the requirements needed for an Earth orbiting elevator. Recent developments in carbon nanotubes [3] offer one potential means for enabling the development of a full scale elevator system, albeit with many technical challenges to overcome. Several previous works have dealt with various aspects of the dynamics of the elevator system. Cohen and Misra [4] derived a continuum model representation of the elevator ribbon to calculate the frequencies and mode shapes. Less rigorous estimation of frequencies and mode shapes has been presented elsewhere [5]. Williams [6] used a lumped mass model of the ribbon to calculate frequencies and mode shapes, and also simulated the effects of the passage of the elevator up to geostationary altitude. However, the elevator motion was prescribed according to an acceleration-constant speed-deceleration schedule. Cohen and Misra [7] used a simplified model of the elevator system to produce an analytic approximation for the libration response of the ribbon due to an elevator’s motion under the assumption of constant velocity motion. They also showed that the use of
ARTICLE IN PRESS P. Williams, W. Ockels / Acta Astronautica 66 (2010) 1458–1467
multiple ascending or descending elevators, when spaced appropriately, can be used to produce a zero net libration angle for the elevator ribbon. Studies of the dynamics of other elevator systems have also been undertaken, using models of varying fidelity [8–11]. One of the major challenges faced by the space elevator is control of the ribbon oscillations and counterweight librations. In particular, the out-of-plane librations are driven by resonance caused by lunar perturbations [12]. When an elevator climbs the ribbon, its motion induces a perturbation to the in-plane librations. If the elevator motion is not controlled, there is the potential for growth of the in-plane librations. If the elevator system is ultimately produced, it will be necessary to consider active control of the elevator during ascent, so that severe lateral oscillations are not induced. Furthermore, it is likely that motion planning will be required so as to reduce the resulting libration amplitude of the ribbon following passage of the elevator to the desired altitude. Although it is not feasible to control out-of-plane counterweight oscillations by movement of the elevator, it is possible to eliminate in-plane oscillations. Present technology allows very rapid computation of optimal trajectories, which would enable planning of trajectories just prior to beginning ascent, or even real-time feedback control. This paper presents a numerical analysis of the optimal trajectories for the ascent of a climber to geostationary altitude. A simplified model of the system is used to produce the optimal trajectories.
2. Dynamic elevator model 2.1. Simplified elevator model To study the dynamics of the climber on the elevator system, detailed models of the system can be derived. However, to study the optimal means for controlling the climber to minimize the librations of the ribbon and counterweight, a simplified model of the system is used. The model treats only the rigid body motion of the ribbon. It considers the ribbon to be completely rigid, and the climber and counterweight to be point masses. The model derivation is a simplified version of the one presented in [7], with extension to consider out-of-plane motion. The reference coordinate system for the model is shown in Fig. 1. The origin of the rotating coordinate system is located at the Earth’s surface at the anchor point, which is located on the Equator. The counterweight, mc , is located at a distance of l from the anchor point, equal to the length of the ribbon. The elevator, me , is at a distance d along the ribbon. The deviation of the ribbon from the vertical is measured by the in-plane libration angle y, and the out-of-plane libration angle f. The radius of the Earth is denoted by RE, and its angular velocity is given by o. The plane of motion is the orbital plane of the elevator, i.e., the Equatorial plane, and the out-ofplane angle is measured positively towards the North pole. Let the position of the elevator, counterweight, and an arbitrary point along the ribbon at a distance s
1459
mc y
me d
w
x In-Plane Projection z x Out-of-Plane Projection
Fig. 1. Simplified representation of the tethered elevator system anchored to the Earth.
from the anchor be written in the rotating coordinate frame as r e ¼ ðRE þ d cos y cos fÞiþ ðd sin y cos fÞj þ ðd sin fÞk
ð1Þ
r c ¼ ðRE þl cos y cos fÞiþ ðl sin y cos fÞj þ ðl sin fÞk
ð2Þ
rðsÞ ¼ ðRE þs cos y cos fÞi þ ðs sin y cos fÞj þðs sin fÞk
ð3Þ
where i, j, and k are unit vectors along the x-, y- and z-axes, respectively. The inertial velocity of the elevator in the rotating frame is given by r_ e ¼ ½d_ cos y cos f dðy_ þ oÞsin y cos f _ cos y sin fi þ½d_ sin y cos f þ dy_ cos y cos f þ oðR df E _ sin y sin fj þ ½d_ sin f þ df _ cos fk þ d cos y cosfÞ df ð4Þ with similar expressions for the counterweight and arbitrary point along the ribbon. Note, however, that the radial velocity terms _l and s_ do not appear in these equations because the ribbon length is assumed to be constant. It may be possible to control the ribbon oscillations by reeling the main elevator ribbon, but this possibility is not considered here. The total kinetic energy of the system is calculated by T¼
1 1 1 mc r_ c r_ c þ me r_ e r_ e þ 2 2 2
Z
l
_ rðsÞ _ ds rðsÞrðsÞ
ð5Þ
0
where r(s) is the line density of the ribbon given by " !# r g0 R2E 3 RG ðsþ RE Þ2 rðsÞ ¼ r Am exp sRG 2 s þRE 2R2G
ð6Þ
where r is the mass density of the ribbon material, Am is the maximum cross-sectional area of the ribbon at geostationary orbit radius (RG), g0 is the value of gravity at sealevel, and s is the ribbon design stress. The potential energy due to the Newtonian gravitational field of the
ARTICLE IN PRESS 1460
P. Williams, W. Ockels / Acta Astronautica 66 (2010) 1458–1467
Earth is calculated as
mmc V ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðRE þ l cos y cos fÞ þ l2 sin2 y cos2 f þ l2 sin2 f mme qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðRE þd cos y cos fÞ þd2 sin2 y cos2 f þd2 sin2 f Z l mrðsÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds 2 0 ðRE þ s cos y cos fÞ þs2 sin2 y cos2 f þ s2 sin2 f ð7Þ where m is the Earth’s gravitational parameter. After application of Lagrange’s equations, the equation of motion for the libration of the ribbon are given by _ y_ þ oÞ M m ddð sin y 2 y€ ¼ 2 e o2 RE M1 cos f M1 " mRE sin y me d þ M1 cos f ðR2E þ 2RE d cos y cos f þ d2 Þ3=2 þ þ
0
ðR2E þ 2RE scosycosf þ s2 Þ3=2
_ tan f ds þ 2ðy_ þ oÞf
M2 2 m dd_ o RE cos y sin f ðo þ y_ Þ2 cos f sin f 2 e f_ M1 M1 " mRE cos y sin f mc l þ M1 ðR2 þ 2RE l cos y cos f þ l2 Þ3=2 E
me d ðR2E þ2RE d cos y cos f þ d2 Þ3=2 Z l rðsÞs 0
ðR2E þ 2RE s cos y cos f þ s2 Þ3=2
# ds
ð9Þ
where M1 ¼ me d2 þmc l2 þ
Z
l
rðsÞs2 ds
ð10Þ
0
M2 ¼ me d þ mc l þ
Z
l
rðsÞs ds
_ ¼x x_ 3 ¼ f 4 € x_ 4 ¼ f x_ 5 ¼ d_ ¼ x6 ð12Þ
2.2. The need for elevator speed optimization #
f€ ¼
þ
x_ 2 ¼ y€
_ ; x ¼ d; x ¼ d; _ u ¼ d. € where x1 ¼ y; x2 ¼ y_ ; x3 ¼ f; x4 ¼ f 5 6
ð8Þ
þ
x_ 1 ¼ y_ ¼ x2
x_ 6 ¼ d€ ¼ u
mc l ðR2E þ2RE l cos y cos f þl2 Þ3=2 Z l rðsÞs
examines methods of eliminating the residual libration after climbing the ribbon. For numerical simulation purposes, the libration equation and elevator motion are described by a set of four first-order differential equations defined as follows:
ð11Þ
0
The equations of motion for the elevator represent a damped nonlinear oscillator. The damping (positive and negative) is provided by movement of the elevator along _ The in-plane libration equation shows that the ribbon, d. there is a non-zero forcing contribution of the elevator as _ y_ þ oÞ=M . it moves along the ribbon, given by 2me ddð 1 Note that the presence of the orbital angular velocity acts to create a Coriolis force as the elevator moves along the ribbon. Thus, even for zero initial librations, an in-plane motion is always created due to elevator movement. The same is not true of the out-of-plane motion. In the absence of other external perturbations, the out-of-plane motion is not excited by the in-plane motion. However, if the out-of-plane motion is excited then the two modes become coupled. Librations due to elevator motion are undesirable as they increase the tension in the ribbon and couple to lateral oscillations [6]. The next section
The fundamental modes of vibration of the space elevator ribbon have been discussed in detail in [4,6,12]. The librational frequencies can be obtained from Eqs. (8) and (9) by linearization around the equilibrium position. The fundamental frequency is a pendulum motion (libration) in the Equatorial plane with a period on the order of 5 days. The second frequency is an out-of-plane libration with a period of approximately 1 day. The higher frequencies [6] consist of lateral oscillations of the ribbon itself, displaced from the line connecting the anchor point and the counterweight. The lateral oscillations resemble string oscillations. The elevator motion induces both librational and string oscillations, with the magnitude of the disturbance proportional to the mass of the elevator. As an example, Fig. 2 shows the results of a simulation of a 1000 kg elevator climbing a 70,000 km long flexible ribbon to geostationary altitude [6]. The elevator moves at a constant speed of 50 m/s, with acceleration and decelerations of 4 m/s2 at the beginning and end of the journey. Note that the unstrained ribbon length is 70,000 km, but due to elasticity the counterweight extends beyond an altitude of 70,000 km. The major effect of the motion on the dynamics is the initialization of the first libration mode, shown in Fig. 2a. There is almost no damping in the system and hence the libration continues well after the elevator ceases climbing. Fig. 2b shows the lateral response of the ribbon, generated by projecting the displacement of the tether along the vector between the anchor point and the counterweight. It can be noted from the numerical simulation that the lateral oscillations are small compared to the overall libration, and hence initial studies of minimizing the librational oscillations are warranted. Rather than climbing at a constant velocity, a better strategy is to control the elevator’s ascent in such a way that the librational oscillations are minimized. However, it is not an intuitive design task—we utilize optimal control to obtain a rapid solution to this problem.
ARTICLE IN PRESS P. Williams, W. Ockels / Acta Astronautica 66 (2010) 1458–1467
1461
3.1. Numerical optimization of elevator ascent 3.1.1. Basic ascent problem Consider the problem of finding the optimal climber € to minimize the cost function acceleration uðÞ ¼ dðÞ Z tf ðu2 þc1 x26 Þ dt ð13Þ J1 ¼
10 -50 5 -100 0
2
4 6 Time (days)
x 10 4
0
t0
subject to the equations of motion given by Eq. (12), the initial conditions
x (k m)
y (km)
50
fx1 ; x2 ; x3 ; x4 ; x5 ; x6 gt0 ¼ f0; 0; 0; 0; 0; 0g
10 0
8
and the terminal constraints
Displacement Relative to Ribbon Ends (km)
fx1 ; x2 ; x3 ; x4 ; x5 ; x6 gtf ¼ f0; 0; 0; 0; hG ; 0g
2 1 0 -1 -2
-3 x1 8 04
10 8 6 Dis 4 ta Rib nce A bon lon (km g )
6 4 2
2 0 0
ð14Þ
ys)
da e(
Tim
Fig. 2. Response of system to elevator climbing to geostationary altitude at 50 m/s (&=elevator position): (a) libration response and (b) ribbon deflection along anchor-counterweight line.
3. Optimization of climber motion In [7], Cohen and Misra analyzed the equations of motion for the ribbon libration under elevator forcing effects and proposed synchronizing the ascent or descent of N climbers to null the effects of the residual ribbon libration at the end of the climb to geostationary altitude. The method works as long as the climbers undergo the same motion profile and are specifically spaced along the tether. Any faults in the elevator’s motors or disruption in power supply could result in failure of the method to eliminate the residual oscillations. It is of interest to be able to plan the ascent of a single climber to geostationary altitude such that the final ribbon libration is nullified. This is not possible by maintaining a single climber at constant speed. The advantage of optimizing the motion of a single climber is that it is possible to re-plan trajectories in the case of motor failures or other unscheduled events. If solutions can be generated quickly, then real-time optimal feedback can be used to adjust the elevator’s ascent profile to mitigate librational disturbances. Furthermore, coordinating multiple climbers could be a logistical issue depending on the goals of a given space elevator mission.
ð15Þ
Note that the elevator motion is prescribed kinematically and that its motion couples to the librational dynamics (x1 and x2) by means of Coriolis forces associated with its linear velocity in the rotating frame. This particular problem is restricted to the in-plane motion of the system, and hence the out-of-plane states could be omitted. If a solution to the problem defined by Eqs. (13)–(15) exists, then it is possible to nullify the libration of the ribbon through proper scheduling of the elevator’s speed on the ribbon. Note that the cost function minimizes a weighted functional of the climber acceleration and climber speed, where c1 is a weighing parameter. 3.1.2. Time optimal ascent An important class of maneuvers that can be used to establish lower bounds on the elevator’s ascent is the minimum time maneuver. In such a maneuver, both the elevator acceleration and velocity are bounded to maximum permissible levels, and the cost function is selected as the final time, J2 ¼ tf
ð16Þ
The cost function is subject to the constraints _ d_ max d_ min rdr
ð17Þ
€ d€ max d€ min rdr
ð18Þ
as well as the equations of motion defined by Eqs. (12). In addition, the maneuver is subject to the same initial and terminal constraints given in Eqs. (14) and (15). 3.1.3. Librational stabilization problem In addition to the ascent problems, the ability to damp librational oscillations induced from other sources is also of interest. This has practical use for such tasks as direct removal of excess librational energy, or manipulation of the ribbon position for avoiding debris. A simplified stabilization problem can be posed as, minimize: Z tf u2 dt ð19Þ J3 ¼ 0
subject to the equations of motion given by Eq. (12), the initial conditions fx1 ; x2 ; x3 ; x4 ; x5 ; x6 gt0 ¼ fy0 ; 0; f0 ; 0; hG ; 0g
ð20Þ
ARTICLE IN PRESS 1462
P. Williams, W. Ockels / Acta Astronautica 66 (2010) 1458–1467
and the terminal constraints
3.2. Numerical solution method An efficient solution to the nonlinear optimal control problem described in Section 3.1 can be obtained by application of direct transcription techniques combined with nonlinear programming. In this paper, the continuous optimal control problem is transcribed into a discrete parameter optimization problem by way of the Legendre pseudospectral method [13,14]. This process is automated in the software DIRECT [15]. Solutions are obtained in a MATLAB environment. It should be noted that although we do not consider the possibility directly in this work, solutions can be obtained in real-time for the problems posed in this paper. This is possible due to the long time scales involved in the dynamics, and the speed of the DIRECT software. This means that it is possible to obtain closed-loop solutions employing sampled-data feedback. This possibility should be explored in future work. 4. Numerical results 4.1. Time optimal ascent As noted in the previous section, it is important to establish a lower bound to the feasible maneuver time for realistic constraints on the elevator’s maximum speed and acceleration. The time optimal ascent problem is solved for the case where the elevator is 1000 kg, the ribbon is 100,000 km long with a maximum cross-sectional area of 10 mm2, and the material strength is 35 GPa with density 1300 kg/m3. These values have been taken from Ref. [7]. The optimal control problem is solved using the Legendre pseudospectral method with the time domain discretized into 200 nodes. The elevator’s maximum speed is limited to 50 m/s, with a maximum acceleration of 4 m/s2. The optimal solution is presented in Figs. 3–5. Fig. 3 shows that the libration angle of the ribbon peaks at approximately 0.0121 after approximately 8 days. The optimal time of ascent to nullify the ribbon libration is 9.383 days. Figs. 4 and 5 show that the elevator is able to maintain a constant speed throughout the vast majority of the ascent. Although hard to discern on the time scale, the initial acceleration and deceleration phases occur at the maximum limit. Fig. 5 shows that in order to bring the final libration angle and libration rate of the ribbon to zero at the final time, the elevator’s speed must be reversed for just over half a day after about 8 days. At the higher altitude, the forcing effect is larger and is therefore able to force the ribbon swing back earlier than would have otherwise occurred. Once the swing back has commenced, the elevator is able to resume its course to geostationary, where it comes to a stop. Hence, the swing back is much faster than the initial swing away from the local vertical.
-0.002 -0.004 -0.006 -0.008 -0.01 -0.012 0
2
4
6
8
10
Time (days) Fig. 3. Ribbon libration angle for time optimal elevator ascent to geostationary.
Elevator Distance along Ribbon (L/HG)
where y0 and f0 are the initial in-plane and out-of-plane angles to be specified.
Ribbon Libration Angle (deg)
ð21Þ
1
0.8
0.6
0.4
0.2
0 0
2
4 6 Time (days)
8
10
Fig. 4. Time optimal elevator ascent to geostationary.
60
Elevator Speed (m/s)
fx1 ; x2 ; x3 ; x4 ; x5 ; x6 gtf ¼ f0; 0; free; free; hG ; 0g
0
30
0
-30
-60 0
2
4 6 Time (days)
8
10
Fig. 5. Elevator speed schedule for time optimal ascent to geostationary altitude.
ARTICLE IN PRESS P. Williams, W. Ockels / Acta Astronautica 66 (2010) 1458–1467
4.2. Effect of time of ascent
1463
Young’s modulus is 35 GPa, the ribbon density is 1300 kg/ m3, the maximum cross-sectional area is 10 mm2, and the ribbon length is 100,000 km. The results of numerical optimizations are shown in Figs. 6–8. Fig. 6 shows the librational response of the ribbon to the elevator’s motion for the optimized trajectories as a function of the time of travel. For clarity, the time scale is normalized by time of travel, and the time of travel is shown on a third axis. For the fastest travel time, the libration angle of the ribbon reaches a peak just after the midway point. As the travel time increases, the ribbon undergoes multiple oscillation
In this section, the acceleration and speed constraints on the elevator are lifted, and smooth unconstrained optimal solutions are sought as a function of different system parameters. This means that we are solving the optimal control problem defined by Eqs. (13)–(15), without enforcing Eqs. (17) and (18). In this particular subsection, the time of ascent is varied and the elevator’s motion is optimized to eliminate the final ribbon libration. The time of ascent is varied from 4 to 30 days in increments of 1 day. The elevator mass is 1000 kg, the
Libration Response (deg)
0.005 0 -0.005 -0.01 -0.015 -0.02 -0.025 30
-0.03 20
-0.035 1
0.8
0.6 0.4 Normalized Ti me
0.2
0
0
p Tri
)
ys
da e(
10
Tim
Fig. 6. Ribbon libration angle as a function of time of travel.
Elevator Velocity (m/s)
300
200
100
0
-100 0 ip Tr
10 e
m Ti s)
ay
(d
20 30
0
0.2
0.6 e alized Tim
0.4 Norm
Fig. 7. Optimal elevator speed as a function of time of ascent.
0.8
1
ARTICLE IN PRESS 1464
P. Williams, W. Ockels / Acta Astronautica 66 (2010) 1458–1467
x 104 4
Elevator Position (km)
3.5 3 2.5 2 1.5 1 0.5 0 1
30 0.8
0.6 Normalized
0.4 Time
10 0.2
Time Trip
0
0
20 s) (day
cycles, but the libration angle always remains negative. The optimal solution for eliminating the residual libration does not involve simply climbing at a constant speed. Fig. 7 shows the time history of the elevator speed. For the fastest trip time, the speed peaks at approximately 15% of the trip time, then slowly decreases to zero. The elevator reverses direction after approximately 45% of the trip time, which causes the libration to reach its peak quickly. The elevator then accelerates back towards geostationary, reaching a second peak of 200 m/s, after which the elevator is slowed back to a stop. Increasing the transit time decreases the velocity peaks at the beginning and end of the transfer, and reduces the need for the elevator to travel back down the ribbon. Increasing the time of travel also leads to a decrease in the peak libration amplitude. In addition, the libration peak shifts towards the end of the ascent. For very long travel times, the ribbon undergoes a series of low amplitude oscillations about a non-zero mean, culminating in a slightly larger peak just prior to returning to the local vertical. Fig. 8 shows the elevator position on the ribbon as a function of transit time. This clearly shows that fast ascent times can only be achieved by reversing the elevator motion. 4.3. Effect of elevator mass The Coriolis forces exerted on the ribbon as a result of the elevator’s motion along it are directly proportional to the elevator mass. Hence, the torque acting on the ribbon to disturb it away from the vertical position depends directly on the elevator mass. In this section, the same basic system is used to simulate the effects of elevator mass on the optimal climber trajectories. The ribbon is 100,000 km long with Young’s modulus 35 GPa, the maximum cross-sectional area is 10 mm2, and mass density of 1300 kg/m3. The travel time is fixed at 10 days,
Libration Response (deg)
Fig. 8. Normalized distance of elevator along ribbon for optimal ascent as a function of time of ascent.
0.1 0 -0.1 -0.2 -0.3 0 Ele
x 104
va
tor
10
1 Ma
ss
2
(kg
)
0
5 ) (days e im T
Fig. 9. Ribbon libration angle as a function of elevator mass.
and the elevator mass is varied between 1000 and 20,000 kg. Optimal elevator trajectories are shown in Figs. 9–11. Fig. 9 shows that the peak ribbon angle increases in direct proportion to the elevator mass. This shape of the variation in ribbon angle is essentially the same, irrespective on the elevator mass. Initially the ribbon remains in vertical equilibrium and only starts to display noticeable deviations after approximately 1 day of travel. The libration peaks after approximately 7 days, after which the ribbon rapidly converges back to the stable equilibrium position. Figs. 10 and 11 show the variation of the elevator position along the ribbon and the elevator’s speed, respectively. These figures indicate that the optimal control inputs are relatively insensitive to elevator mass. This is an important result because it means that the elevator load can be changed substantially
ARTICLE IN PRESS P. Williams, W. Ockels / Acta Astronautica 66 (2010) 1458–1467
1465
Elevator Position (km)
x 104 4 3 2 1 0 10
2 Tim e
5 (da ys)
1 0 0
or
vat
Ele
ss Ma
) (kg
4
0
x1
40 20 0
1
10
as
s(
kg
2
)
0
30 20 10
5 ys) Time (da
4.4. Effect of tether length The final parameter that is examined in this work is the ribbon length. Using the same parameters as in the previous example, but with an elevator mass of 1000 kg, the ribbon length is varied between 70,000 and 160,000 km. Results of the optimal trajectories are shown in Figs. 13–15. Increasing the tether length increases the ribbon mass and the rotational inertia of the system. The maximum cross-sectional area is always located at geostationary altitude, and hence increasing the ribbon length requires mass to be distributed between geostationary altitude and the counterweight. The
5000
7500 10000 Mass (kg)
12500
15000
0 -0.005 -0.01 -0.015 10 Tim
2 1.5
5
e(
Fig. 11. Optimal elevator speed as a function of elevator mass.
1
da
0 0.5
ys)
on
Ribb
th Leng
5
x 10 (km)
Fig. 13. Ribbon libration angle as a function of tether length.
x 104 Elevator Position (km)
without impacting the open-loop control. Note, however, that this conclusion is only valid within the assumption that the ribbon remains straight. Increasing the elevator mass will create larger Coriolis forces and hence induce larger lateral ribbon vibrations. This was established through simulations in [6] and the main result is summarized in Fig. 12, which shows that the maximum lateral deviation of the tether from the line connecting the anchor and the counterweight is linearly proportional to the elevator mass.
2500
Fig. 12. Maximum lateral deflection of a flexible elevator ribbon during transit to geostationary altitude at a constant speed of 450 m/s.
Libration Response (deg)
60
rM
40
0
80
-20 0 x 104 E lev ato
50
0
Fig. 10. Normalized distance of elevator along ribbon for optimal ascent as a function of elevator mass.
Elevator Velocity (m/s)
Max. Lateral Oscillation (km)
60
4 3 2 1 0 10 Tim
2
5
e(
da
ys
)
0
1
1.5 km) ngth (
0.5
5
x 10
n Le
Ribbo
Fig. 14. Normalized distance of elevator along ribbon for optimal ascent as a function of tether length.
additional mass at higher altitudes directly increases the rotational inertia. Fig. 13 shows the effect of ribbon length on the libration response for the optimal ascent maneuver. This shows that variations in ribbon length have only a minor effect on the optimal trajectories. As
ARTICLE IN PRESS P. Williams, W. Ockels / Acta Astronautica 66 (2010) 1458–1467
100
50
0 0.5
x 105R
10
1
ibb
on
1.5 ng th (km )
Le
2 0
Libration Response (deg)
Elevator Velocity (m/s)
1466
4 2 0 -2 2 In 1.5 itia lA 1 ng le 0.5 (d eg )
5 ys) (da e m Ti
0 10
0
Time
15
5 a (d ys)
Fig. 15. Optimal elevator speed as a function of tether length.
the ribbon length increases, the peak libration angle decreases. This is caused by the increased system inertia, requiring larger disturbing forces to displace the counterweight. In reality, the ribbon length will be virtually constant during operations. Hence, even changes on the order of a few hundred kilometers would have negligible impact on the control of the elevator. Fig. 14 shows the elevator position on the ribbon, and Fig. 15 shows the elevator speed as a function of ribbon length. These results highlight the result that the elevator control is largely unaffected by changes in ribbon length.
Elevator Velocity (m/s)
Fig. 16. Effect of initial in-plane libration angle on optimal libration stabilization using elevator.
4.5. Libration stabilization
100 0 -100 2 In
1.5 lA 1 ng le 0.5 (d eg )
itia
0 5 10 0
e Tim
15
s)
(day
Fig. 17. Effect of initial in-plane libration angle on elevator velocity for libration stabilization.
Libration Response (deg)
The final case considered in this paper is utilizing the elevator for direct control of the in-plane libration of the ribbon. The stabilization problem is solved by varying the initial in-plane angle from 01 to 21, and fixing the maneuver time at 15 days. The ribbon is 100,000 km long with Young’s modulus 35 GPa, the maximum crosssectional area is 10 mm2, and mass density of 1300 kg/ m3. The elevator mass is 10,000 kg. The in-plane libration response is shown in Fig. 16 for the case where the out-ofplane angle is zero. The results show that it is possible to remove excess librational energy by moving the elevator on the ribbon in a judicious manner. Fig. 17 shows the variation in elevator velocity as a function of the initial libration angle. The maximum speed of the elevator increases as a function of the initial libration angle, peaking at approximately 100 m/s for initial angles of approximately 21. The same sequence of problems was resolved in the presence of out-of-plane oscillations. The initial out-of-plane angle is fixed at 0.51. Fig. 18 shows the difference in the in-plane libration angle for the optimal response compared to the case with no out-of-plane librations. Although the elevator cannot directly control the out-of-plane angle, it is still able to remove the excess in-plane librational energy. This control scheme can therefore be effective for controlling the long-term evolution of the in-plane ribbon dynamics. It might be possible to utilize the technique for applications such as
200
0.3 0.2 0.1 0 -0.1 -0.2 2 Ini tia
1
lA
ng
le
(de
g)
0
15
10
0
5
)
ays Time (d
Fig. 18. Difference in in-plane libration response caused by 0.51 out-ofplane initial libration angle, shown as a function of initial in-plane libration angle.
ARTICLE IN PRESS P. Williams, W. Ockels / Acta Astronautica 66 (2010) 1458–1467
debris mitigation by moving the ribbon out of the path of certain debris. This possibility should be considered in future work. 5. Conclusions The control of the ascent of a space elevator to geostationary altitude has been considered. The fundamental mode of vibration of an anchored ribbon is a rigid body oscillation about the anchor point on the Earth’s surface. This mode can lead to oscillations of the counterweight, which can be located up to 100,000 km above the anchor point. As the elevator climbs the ribbon, Coriolis forces create a torque that initiates the librational mode. The lack of damping in space causes this mode to persist. Using a simplified model that only accounts for the two fundamental libration modes, it is possible to solve the ascent problem using optimal control. The results show that it is possible to eliminate the librational oscillations by reversing the direction of the elevator for a short time. For a maximum speed of 50 m/s, with accelerations limited to 4 m/s2, the optimal trip time is on the order of 9.38 days. By varying the elevator mass and ribbon length, results show that the optimal ascent trajectories are relatively insensitive to such parameters. It is also possible to use the elevator as a means for controlling the librational response of the ribbon, and to remove excess energy that can arise from other sources such as perturbations. Results show that the technique is useful for removing in-plane oscillation energy, but that out-ofplane librations require another means of control.
1467
References [1] K.E. Tsiolkovsky, Speculations between earth and sky, Isd-vo ANSSSR, Moscow, 1895, p. 35 (In Russian, reprinted in 1959). [2] J. Pearson, The orbital tower: a spacecraft launcher using the Earth’s rotational energy, Acta Astronautica 2 (1975) 785–799. [3] S. Iijima, Helical Microtubules of Graphitic Carbon, Nature 354 (1991) 56–58. [4] S.S. Cohen, A.K. Misra, Elastic oscillations of the space elevator ribbon, Journal of Guidance Control, and Dynamics 30 (6) (2007) 1711–1717. [5] B.C. Edwards, E.A. Westling, The Space Elevator: A Revolutionary Earth-to-Space Transportation System, 2003. [6] P. Williams, Dynamic multibody modeling for tethered space elevators, Acta Astronautica 65 (3–4) (2009) 399–422. [7] S.S. Cohen, A.K. Misra, The effect of climber transit on the space elevator dynamics, Acta Astronautica 64 (2009) 538–553. [8] P. Williams, C. Blanksby, Study on prolonged rendezvous using tethers, AAS/AIAA Astrodynamics Specialist Conference, August 7–11, 2005, Embassy Suites Hotel, Lake Tahoe Resort, Paper AAS 05-321. [9] P. Williams, C. Blanksby, Prolonged payload rendezvous using a tether actuator mass, Journal of Spacecraft and Rockets 41 (5) (2004) 889–892. [10] C.R. McInnes, Dynamics of a particle moving along an orbital tower, Journal of Guidance, Control and Dynamics 28 (2) (2005) 380–382. [11] H.A. Fujii, T. Watanabe, T. Kusagaya, D. Sato, M. Ohta, Dynamics of a flexible space tether equipped with a crawler mass, Journal of Guidance, Control, and Dynamics 31 (2) (2008) 436–440. [12] L. Perek, Space elevator: stability, Acta Astronautica 62 (8–9) (2008) 514–520. [13] J. Elnagar, M.A. Kazemi, M. Razzaghi, The pseudospectral legendre method for discretizing optimal control problems, IEEE Transactions on Automatic Control 40 (10) (1995) 1793–1796. [14] I.M. Ross, F. Fahroo, Legendre pseudospectral approximations of optimal control problems, Lecture Notes in Control and Information Sciences, vol. 295, Springer, New York, pp. 327–342. [15] P. Williams, User’s guide to DIRECT version 1.17, Technical Report, Melbourne, Australia, March 2005.
Contents lists available at ScienceDirect
Acta Astronautica journal homepage: www.elsevier.com/locate/actaastro
Climber motion optimization for the tethered space elevator Paul Williams , Wubbo Ockels Faculty of Aerospace Engineering, Delft University of Technology, P.O. Box Postbus 5058, 2600 GB Delft, The Netherlands
a r t i c l e i n f o
abstract
Article history: Received 23 May 2009 Received in revised form 30 September 2009 Accepted 2 November 2009 Available online 30 November 2009
The tethered space elevator could provide a revolutionary means for enabling cheap transportation to geostationary altitude and beyond. Assuming that such a system can be built, one of the dynamic design problems is determining a means of moving the elevator along the tether so as to minimize the residual in-plane librational motion of the elevator ribbon and counterweight. In particular, this paper studies the problem from the point of view of dynamic optimization. A simplified dynamic model of the elevator system is derived that accounts for the fundamental libration modes and the motion of the elevator. The model is used to solve an optimal control problem that results in zero final in-plane librational motion of the ribbon. The results show that it is possible to eliminate the in-plane librational oscillations by reversing the direction of the elevator for a short time. It is also shown that it is possible to remove excess librational energy by controlled movement of the elevator on the ribbon. The effects of out-of-plane librations are also considered, but the climber motion does not have enough control authority to damp the out-of-plane motion over short time scales. & 2009 Elsevier Ltd. All rights reserved.
Keywords: Space elevator Climber Numerical optimization Optimal control
1. Introduction The tethered space elevator is a radical concept that has existed in the literature even prior to the modern spaceflight era [1]. The basic concept requires a tether or ribbon extending from geostationary altitude to the Earth’s surface. However, in order to make the system feasible, the ribbon is required to extend past geostationary altitude. The system center of orbit is designed to remain at geostationary altitude so that the system spins in orbit at the same rate as the Earth rotates. This means that the tip of the ribbon is always located at the same location on the Earth’s surface. Although Tsiolkovsky [1] is often credited with its invention, it was not until Pearson [2] that the concept was cemented in engineering terms. At the time of publication, no engineering material existed
Corresponding author. Tel.: + 61 3 9859 6877; fax: + 61 3 9925 6108.
E-mail addresses: [email protected] (P. Williams), [email protected] (W. Ockels). 0094-5765/$ - see front matter & 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.actaastro.2009.11.003
that could come close to the requirements needed for an Earth orbiting elevator. Recent developments in carbon nanotubes [3] offer one potential means for enabling the development of a full scale elevator system, albeit with many technical challenges to overcome. Several previous works have dealt with various aspects of the dynamics of the elevator system. Cohen and Misra [4] derived a continuum model representation of the elevator ribbon to calculate the frequencies and mode shapes. Less rigorous estimation of frequencies and mode shapes has been presented elsewhere [5]. Williams [6] used a lumped mass model of the ribbon to calculate frequencies and mode shapes, and also simulated the effects of the passage of the elevator up to geostationary altitude. However, the elevator motion was prescribed according to an acceleration-constant speed-deceleration schedule. Cohen and Misra [7] used a simplified model of the elevator system to produce an analytic approximation for the libration response of the ribbon due to an elevator’s motion under the assumption of constant velocity motion. They also showed that the use of
ARTICLE IN PRESS P. Williams, W. Ockels / Acta Astronautica 66 (2010) 1458–1467
multiple ascending or descending elevators, when spaced appropriately, can be used to produce a zero net libration angle for the elevator ribbon. Studies of the dynamics of other elevator systems have also been undertaken, using models of varying fidelity [8–11]. One of the major challenges faced by the space elevator is control of the ribbon oscillations and counterweight librations. In particular, the out-of-plane librations are driven by resonance caused by lunar perturbations [12]. When an elevator climbs the ribbon, its motion induces a perturbation to the in-plane librations. If the elevator motion is not controlled, there is the potential for growth of the in-plane librations. If the elevator system is ultimately produced, it will be necessary to consider active control of the elevator during ascent, so that severe lateral oscillations are not induced. Furthermore, it is likely that motion planning will be required so as to reduce the resulting libration amplitude of the ribbon following passage of the elevator to the desired altitude. Although it is not feasible to control out-of-plane counterweight oscillations by movement of the elevator, it is possible to eliminate in-plane oscillations. Present technology allows very rapid computation of optimal trajectories, which would enable planning of trajectories just prior to beginning ascent, or even real-time feedback control. This paper presents a numerical analysis of the optimal trajectories for the ascent of a climber to geostationary altitude. A simplified model of the system is used to produce the optimal trajectories.
2. Dynamic elevator model 2.1. Simplified elevator model To study the dynamics of the climber on the elevator system, detailed models of the system can be derived. However, to study the optimal means for controlling the climber to minimize the librations of the ribbon and counterweight, a simplified model of the system is used. The model treats only the rigid body motion of the ribbon. It considers the ribbon to be completely rigid, and the climber and counterweight to be point masses. The model derivation is a simplified version of the one presented in [7], with extension to consider out-of-plane motion. The reference coordinate system for the model is shown in Fig. 1. The origin of the rotating coordinate system is located at the Earth’s surface at the anchor point, which is located on the Equator. The counterweight, mc , is located at a distance of l from the anchor point, equal to the length of the ribbon. The elevator, me , is at a distance d along the ribbon. The deviation of the ribbon from the vertical is measured by the in-plane libration angle y, and the out-of-plane libration angle f. The radius of the Earth is denoted by RE, and its angular velocity is given by o. The plane of motion is the orbital plane of the elevator, i.e., the Equatorial plane, and the out-ofplane angle is measured positively towards the North pole. Let the position of the elevator, counterweight, and an arbitrary point along the ribbon at a distance s
1459
mc y
me d
w
x In-Plane Projection z x Out-of-Plane Projection
Fig. 1. Simplified representation of the tethered elevator system anchored to the Earth.
from the anchor be written in the rotating coordinate frame as r e ¼ ðRE þ d cos y cos fÞiþ ðd sin y cos fÞj þ ðd sin fÞk
ð1Þ
r c ¼ ðRE þl cos y cos fÞiþ ðl sin y cos fÞj þ ðl sin fÞk
ð2Þ
rðsÞ ¼ ðRE þs cos y cos fÞi þ ðs sin y cos fÞj þðs sin fÞk
ð3Þ
where i, j, and k are unit vectors along the x-, y- and z-axes, respectively. The inertial velocity of the elevator in the rotating frame is given by r_ e ¼ ½d_ cos y cos f dðy_ þ oÞsin y cos f _ cos y sin fi þ½d_ sin y cos f þ dy_ cos y cos f þ oðR df E _ sin y sin fj þ ½d_ sin f þ df _ cos fk þ d cos y cosfÞ df ð4Þ with similar expressions for the counterweight and arbitrary point along the ribbon. Note, however, that the radial velocity terms _l and s_ do not appear in these equations because the ribbon length is assumed to be constant. It may be possible to control the ribbon oscillations by reeling the main elevator ribbon, but this possibility is not considered here. The total kinetic energy of the system is calculated by T¼
1 1 1 mc r_ c r_ c þ me r_ e r_ e þ 2 2 2
Z
l
_ rðsÞ _ ds rðsÞrðsÞ
ð5Þ
0
where r(s) is the line density of the ribbon given by " !# r g0 R2E 3 RG ðsþ RE Þ2 rðsÞ ¼ r Am exp sRG 2 s þRE 2R2G
ð6Þ
where r is the mass density of the ribbon material, Am is the maximum cross-sectional area of the ribbon at geostationary orbit radius (RG), g0 is the value of gravity at sealevel, and s is the ribbon design stress. The potential energy due to the Newtonian gravitational field of the
ARTICLE IN PRESS 1460
P. Williams, W. Ockels / Acta Astronautica 66 (2010) 1458–1467
Earth is calculated as
mmc V ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðRE þ l cos y cos fÞ þ l2 sin2 y cos2 f þ l2 sin2 f mme qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðRE þd cos y cos fÞ þd2 sin2 y cos2 f þd2 sin2 f Z l mrðsÞ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ds 2 0 ðRE þ s cos y cos fÞ þs2 sin2 y cos2 f þ s2 sin2 f ð7Þ where m is the Earth’s gravitational parameter. After application of Lagrange’s equations, the equation of motion for the libration of the ribbon are given by _ y_ þ oÞ M m ddð sin y 2 y€ ¼ 2 e o2 RE M1 cos f M1 " mRE sin y me d þ M1 cos f ðR2E þ 2RE d cos y cos f þ d2 Þ3=2 þ þ
0
ðR2E þ 2RE scosycosf þ s2 Þ3=2
_ tan f ds þ 2ðy_ þ oÞf
M2 2 m dd_ o RE cos y sin f ðo þ y_ Þ2 cos f sin f 2 e f_ M1 M1 " mRE cos y sin f mc l þ M1 ðR2 þ 2RE l cos y cos f þ l2 Þ3=2 E
me d ðR2E þ2RE d cos y cos f þ d2 Þ3=2 Z l rðsÞs 0
ðR2E þ 2RE s cos y cos f þ s2 Þ3=2
# ds
ð9Þ
where M1 ¼ me d2 þmc l2 þ
Z
l
rðsÞs2 ds
ð10Þ
0
M2 ¼ me d þ mc l þ
Z
l
rðsÞs ds
_ ¼x x_ 3 ¼ f 4 € x_ 4 ¼ f x_ 5 ¼ d_ ¼ x6 ð12Þ
2.2. The need for elevator speed optimization #
f€ ¼
þ
x_ 2 ¼ y€
_ ; x ¼ d; x ¼ d; _ u ¼ d. € where x1 ¼ y; x2 ¼ y_ ; x3 ¼ f; x4 ¼ f 5 6
ð8Þ
þ
x_ 1 ¼ y_ ¼ x2
x_ 6 ¼ d€ ¼ u
mc l ðR2E þ2RE l cos y cos f þl2 Þ3=2 Z l rðsÞs
examines methods of eliminating the residual libration after climbing the ribbon. For numerical simulation purposes, the libration equation and elevator motion are described by a set of four first-order differential equations defined as follows:
ð11Þ
0
The equations of motion for the elevator represent a damped nonlinear oscillator. The damping (positive and negative) is provided by movement of the elevator along _ The in-plane libration equation shows that the ribbon, d. there is a non-zero forcing contribution of the elevator as _ y_ þ oÞ=M . it moves along the ribbon, given by 2me ddð 1 Note that the presence of the orbital angular velocity acts to create a Coriolis force as the elevator moves along the ribbon. Thus, even for zero initial librations, an in-plane motion is always created due to elevator movement. The same is not true of the out-of-plane motion. In the absence of other external perturbations, the out-of-plane motion is not excited by the in-plane motion. However, if the out-of-plane motion is excited then the two modes become coupled. Librations due to elevator motion are undesirable as they increase the tension in the ribbon and couple to lateral oscillations [6]. The next section
The fundamental modes of vibration of the space elevator ribbon have been discussed in detail in [4,6,12]. The librational frequencies can be obtained from Eqs. (8) and (9) by linearization around the equilibrium position. The fundamental frequency is a pendulum motion (libration) in the Equatorial plane with a period on the order of 5 days. The second frequency is an out-of-plane libration with a period of approximately 1 day. The higher frequencies [6] consist of lateral oscillations of the ribbon itself, displaced from the line connecting the anchor point and the counterweight. The lateral oscillations resemble string oscillations. The elevator motion induces both librational and string oscillations, with the magnitude of the disturbance proportional to the mass of the elevator. As an example, Fig. 2 shows the results of a simulation of a 1000 kg elevator climbing a 70,000 km long flexible ribbon to geostationary altitude [6]. The elevator moves at a constant speed of 50 m/s, with acceleration and decelerations of 4 m/s2 at the beginning and end of the journey. Note that the unstrained ribbon length is 70,000 km, but due to elasticity the counterweight extends beyond an altitude of 70,000 km. The major effect of the motion on the dynamics is the initialization of the first libration mode, shown in Fig. 2a. There is almost no damping in the system and hence the libration continues well after the elevator ceases climbing. Fig. 2b shows the lateral response of the ribbon, generated by projecting the displacement of the tether along the vector between the anchor point and the counterweight. It can be noted from the numerical simulation that the lateral oscillations are small compared to the overall libration, and hence initial studies of minimizing the librational oscillations are warranted. Rather than climbing at a constant velocity, a better strategy is to control the elevator’s ascent in such a way that the librational oscillations are minimized. However, it is not an intuitive design task—we utilize optimal control to obtain a rapid solution to this problem.
ARTICLE IN PRESS P. Williams, W. Ockels / Acta Astronautica 66 (2010) 1458–1467
1461
3.1. Numerical optimization of elevator ascent 3.1.1. Basic ascent problem Consider the problem of finding the optimal climber € to minimize the cost function acceleration uðÞ ¼ dðÞ Z tf ðu2 þc1 x26 Þ dt ð13Þ J1 ¼
10 -50 5 -100 0
2
4 6 Time (days)
x 10 4
0
t0
subject to the equations of motion given by Eq. (12), the initial conditions
x (k m)
y (km)
50
fx1 ; x2 ; x3 ; x4 ; x5 ; x6 gt0 ¼ f0; 0; 0; 0; 0; 0g
10 0
8
and the terminal constraints
Displacement Relative to Ribbon Ends (km)
fx1 ; x2 ; x3 ; x4 ; x5 ; x6 gtf ¼ f0; 0; 0; 0; hG ; 0g
2 1 0 -1 -2
-3 x1 8 04
10 8 6 Dis 4 ta Rib nce A bon lon (km g )
6 4 2
2 0 0
ð14Þ
ys)
da e(
Tim
Fig. 2. Response of system to elevator climbing to geostationary altitude at 50 m/s (&=elevator position): (a) libration response and (b) ribbon deflection along anchor-counterweight line.
3. Optimization of climber motion In [7], Cohen and Misra analyzed the equations of motion for the ribbon libration under elevator forcing effects and proposed synchronizing the ascent or descent of N climbers to null the effects of the residual ribbon libration at the end of the climb to geostationary altitude. The method works as long as the climbers undergo the same motion profile and are specifically spaced along the tether. Any faults in the elevator’s motors or disruption in power supply could result in failure of the method to eliminate the residual oscillations. It is of interest to be able to plan the ascent of a single climber to geostationary altitude such that the final ribbon libration is nullified. This is not possible by maintaining a single climber at constant speed. The advantage of optimizing the motion of a single climber is that it is possible to re-plan trajectories in the case of motor failures or other unscheduled events. If solutions can be generated quickly, then real-time optimal feedback can be used to adjust the elevator’s ascent profile to mitigate librational disturbances. Furthermore, coordinating multiple climbers could be a logistical issue depending on the goals of a given space elevator mission.
ð15Þ
Note that the elevator motion is prescribed kinematically and that its motion couples to the librational dynamics (x1 and x2) by means of Coriolis forces associated with its linear velocity in the rotating frame. This particular problem is restricted to the in-plane motion of the system, and hence the out-of-plane states could be omitted. If a solution to the problem defined by Eqs. (13)–(15) exists, then it is possible to nullify the libration of the ribbon through proper scheduling of the elevator’s speed on the ribbon. Note that the cost function minimizes a weighted functional of the climber acceleration and climber speed, where c1 is a weighing parameter. 3.1.2. Time optimal ascent An important class of maneuvers that can be used to establish lower bounds on the elevator’s ascent is the minimum time maneuver. In such a maneuver, both the elevator acceleration and velocity are bounded to maximum permissible levels, and the cost function is selected as the final time, J2 ¼ tf
ð16Þ
The cost function is subject to the constraints _ d_ max d_ min rdr
ð17Þ
€ d€ max d€ min rdr
ð18Þ
as well as the equations of motion defined by Eqs. (12). In addition, the maneuver is subject to the same initial and terminal constraints given in Eqs. (14) and (15). 3.1.3. Librational stabilization problem In addition to the ascent problems, the ability to damp librational oscillations induced from other sources is also of interest. This has practical use for such tasks as direct removal of excess librational energy, or manipulation of the ribbon position for avoiding debris. A simplified stabilization problem can be posed as, minimize: Z tf u2 dt ð19Þ J3 ¼ 0
subject to the equations of motion given by Eq. (12), the initial conditions fx1 ; x2 ; x3 ; x4 ; x5 ; x6 gt0 ¼ fy0 ; 0; f0 ; 0; hG ; 0g
ð20Þ
ARTICLE IN PRESS 1462
P. Williams, W. Ockels / Acta Astronautica 66 (2010) 1458–1467
and the terminal constraints
3.2. Numerical solution method An efficient solution to the nonlinear optimal control problem described in Section 3.1 can be obtained by application of direct transcription techniques combined with nonlinear programming. In this paper, the continuous optimal control problem is transcribed into a discrete parameter optimization problem by way of the Legendre pseudospectral method [13,14]. This process is automated in the software DIRECT [15]. Solutions are obtained in a MATLAB environment. It should be noted that although we do not consider the possibility directly in this work, solutions can be obtained in real-time for the problems posed in this paper. This is possible due to the long time scales involved in the dynamics, and the speed of the DIRECT software. This means that it is possible to obtain closed-loop solutions employing sampled-data feedback. This possibility should be explored in future work. 4. Numerical results 4.1. Time optimal ascent As noted in the previous section, it is important to establish a lower bound to the feasible maneuver time for realistic constraints on the elevator’s maximum speed and acceleration. The time optimal ascent problem is solved for the case where the elevator is 1000 kg, the ribbon is 100,000 km long with a maximum cross-sectional area of 10 mm2, and the material strength is 35 GPa with density 1300 kg/m3. These values have been taken from Ref. [7]. The optimal control problem is solved using the Legendre pseudospectral method with the time domain discretized into 200 nodes. The elevator’s maximum speed is limited to 50 m/s, with a maximum acceleration of 4 m/s2. The optimal solution is presented in Figs. 3–5. Fig. 3 shows that the libration angle of the ribbon peaks at approximately 0.0121 after approximately 8 days. The optimal time of ascent to nullify the ribbon libration is 9.383 days. Figs. 4 and 5 show that the elevator is able to maintain a constant speed throughout the vast majority of the ascent. Although hard to discern on the time scale, the initial acceleration and deceleration phases occur at the maximum limit. Fig. 5 shows that in order to bring the final libration angle and libration rate of the ribbon to zero at the final time, the elevator’s speed must be reversed for just over half a day after about 8 days. At the higher altitude, the forcing effect is larger and is therefore able to force the ribbon swing back earlier than would have otherwise occurred. Once the swing back has commenced, the elevator is able to resume its course to geostationary, where it comes to a stop. Hence, the swing back is much faster than the initial swing away from the local vertical.
-0.002 -0.004 -0.006 -0.008 -0.01 -0.012 0
2
4
6
8
10
Time (days) Fig. 3. Ribbon libration angle for time optimal elevator ascent to geostationary.
Elevator Distance along Ribbon (L/HG)
where y0 and f0 are the initial in-plane and out-of-plane angles to be specified.
Ribbon Libration Angle (deg)
ð21Þ
1
0.8
0.6
0.4
0.2
0 0
2
4 6 Time (days)
8
10
Fig. 4. Time optimal elevator ascent to geostationary.
60
Elevator Speed (m/s)
fx1 ; x2 ; x3 ; x4 ; x5 ; x6 gtf ¼ f0; 0; free; free; hG ; 0g
0
30
0
-30
-60 0
2
4 6 Time (days)
8
10
Fig. 5. Elevator speed schedule for time optimal ascent to geostationary altitude.
ARTICLE IN PRESS P. Williams, W. Ockels / Acta Astronautica 66 (2010) 1458–1467
4.2. Effect of time of ascent
1463
Young’s modulus is 35 GPa, the ribbon density is 1300 kg/ m3, the maximum cross-sectional area is 10 mm2, and the ribbon length is 100,000 km. The results of numerical optimizations are shown in Figs. 6–8. Fig. 6 shows the librational response of the ribbon to the elevator’s motion for the optimized trajectories as a function of the time of travel. For clarity, the time scale is normalized by time of travel, and the time of travel is shown on a third axis. For the fastest travel time, the libration angle of the ribbon reaches a peak just after the midway point. As the travel time increases, the ribbon undergoes multiple oscillation
In this section, the acceleration and speed constraints on the elevator are lifted, and smooth unconstrained optimal solutions are sought as a function of different system parameters. This means that we are solving the optimal control problem defined by Eqs. (13)–(15), without enforcing Eqs. (17) and (18). In this particular subsection, the time of ascent is varied and the elevator’s motion is optimized to eliminate the final ribbon libration. The time of ascent is varied from 4 to 30 days in increments of 1 day. The elevator mass is 1000 kg, the
Libration Response (deg)
0.005 0 -0.005 -0.01 -0.015 -0.02 -0.025 30
-0.03 20
-0.035 1
0.8
0.6 0.4 Normalized Ti me
0.2
0
0
p Tri
)
ys
da e(
10
Tim
Fig. 6. Ribbon libration angle as a function of time of travel.
Elevator Velocity (m/s)
300
200
100
0
-100 0 ip Tr
10 e
m Ti s)
ay
(d
20 30
0
0.2
0.6 e alized Tim
0.4 Norm
Fig. 7. Optimal elevator speed as a function of time of ascent.
0.8
1
ARTICLE IN PRESS 1464
P. Williams, W. Ockels / Acta Astronautica 66 (2010) 1458–1467
x 104 4
Elevator Position (km)
3.5 3 2.5 2 1.5 1 0.5 0 1
30 0.8
0.6 Normalized
0.4 Time
10 0.2
Time Trip
0
0
20 s) (day
cycles, but the libration angle always remains negative. The optimal solution for eliminating the residual libration does not involve simply climbing at a constant speed. Fig. 7 shows the time history of the elevator speed. For the fastest trip time, the speed peaks at approximately 15% of the trip time, then slowly decreases to zero. The elevator reverses direction after approximately 45% of the trip time, which causes the libration to reach its peak quickly. The elevator then accelerates back towards geostationary, reaching a second peak of 200 m/s, after which the elevator is slowed back to a stop. Increasing the transit time decreases the velocity peaks at the beginning and end of the transfer, and reduces the need for the elevator to travel back down the ribbon. Increasing the time of travel also leads to a decrease in the peak libration amplitude. In addition, the libration peak shifts towards the end of the ascent. For very long travel times, the ribbon undergoes a series of low amplitude oscillations about a non-zero mean, culminating in a slightly larger peak just prior to returning to the local vertical. Fig. 8 shows the elevator position on the ribbon as a function of transit time. This clearly shows that fast ascent times can only be achieved by reversing the elevator motion. 4.3. Effect of elevator mass The Coriolis forces exerted on the ribbon as a result of the elevator’s motion along it are directly proportional to the elevator mass. Hence, the torque acting on the ribbon to disturb it away from the vertical position depends directly on the elevator mass. In this section, the same basic system is used to simulate the effects of elevator mass on the optimal climber trajectories. The ribbon is 100,000 km long with Young’s modulus 35 GPa, the maximum cross-sectional area is 10 mm2, and mass density of 1300 kg/m3. The travel time is fixed at 10 days,
Libration Response (deg)
Fig. 8. Normalized distance of elevator along ribbon for optimal ascent as a function of time of ascent.
0.1 0 -0.1 -0.2 -0.3 0 Ele
x 104
va
tor
10
1 Ma
ss
2
(kg
)
0
5 ) (days e im T
Fig. 9. Ribbon libration angle as a function of elevator mass.
and the elevator mass is varied between 1000 and 20,000 kg. Optimal elevator trajectories are shown in Figs. 9–11. Fig. 9 shows that the peak ribbon angle increases in direct proportion to the elevator mass. This shape of the variation in ribbon angle is essentially the same, irrespective on the elevator mass. Initially the ribbon remains in vertical equilibrium and only starts to display noticeable deviations after approximately 1 day of travel. The libration peaks after approximately 7 days, after which the ribbon rapidly converges back to the stable equilibrium position. Figs. 10 and 11 show the variation of the elevator position along the ribbon and the elevator’s speed, respectively. These figures indicate that the optimal control inputs are relatively insensitive to elevator mass. This is an important result because it means that the elevator load can be changed substantially
ARTICLE IN PRESS P. Williams, W. Ockels / Acta Astronautica 66 (2010) 1458–1467
1465
Elevator Position (km)
x 104 4 3 2 1 0 10
2 Tim e
5 (da ys)
1 0 0
or
vat
Ele
ss Ma
) (kg
4
0
x1
40 20 0
1
10
as
s(
kg
2
)
0
30 20 10
5 ys) Time (da
4.4. Effect of tether length The final parameter that is examined in this work is the ribbon length. Using the same parameters as in the previous example, but with an elevator mass of 1000 kg, the ribbon length is varied between 70,000 and 160,000 km. Results of the optimal trajectories are shown in Figs. 13–15. Increasing the tether length increases the ribbon mass and the rotational inertia of the system. The maximum cross-sectional area is always located at geostationary altitude, and hence increasing the ribbon length requires mass to be distributed between geostationary altitude and the counterweight. The
5000
7500 10000 Mass (kg)
12500
15000
0 -0.005 -0.01 -0.015 10 Tim
2 1.5
5
e(
Fig. 11. Optimal elevator speed as a function of elevator mass.
1
da
0 0.5
ys)
on
Ribb
th Leng
5
x 10 (km)
Fig. 13. Ribbon libration angle as a function of tether length.
x 104 Elevator Position (km)
without impacting the open-loop control. Note, however, that this conclusion is only valid within the assumption that the ribbon remains straight. Increasing the elevator mass will create larger Coriolis forces and hence induce larger lateral ribbon vibrations. This was established through simulations in [6] and the main result is summarized in Fig. 12, which shows that the maximum lateral deviation of the tether from the line connecting the anchor and the counterweight is linearly proportional to the elevator mass.
2500
Fig. 12. Maximum lateral deflection of a flexible elevator ribbon during transit to geostationary altitude at a constant speed of 450 m/s.
Libration Response (deg)
60
rM
40
0
80
-20 0 x 104 E lev ato
50
0
Fig. 10. Normalized distance of elevator along ribbon for optimal ascent as a function of elevator mass.
Elevator Velocity (m/s)
Max. Lateral Oscillation (km)
60
4 3 2 1 0 10 Tim
2
5
e(
da
ys
)
0
1
1.5 km) ngth (
0.5
5
x 10
n Le
Ribbo
Fig. 14. Normalized distance of elevator along ribbon for optimal ascent as a function of tether length.
additional mass at higher altitudes directly increases the rotational inertia. Fig. 13 shows the effect of ribbon length on the libration response for the optimal ascent maneuver. This shows that variations in ribbon length have only a minor effect on the optimal trajectories. As
ARTICLE IN PRESS P. Williams, W. Ockels / Acta Astronautica 66 (2010) 1458–1467
100
50
0 0.5
x 105R
10
1
ibb
on
1.5 ng th (km )
Le
2 0
Libration Response (deg)
Elevator Velocity (m/s)
1466
4 2 0 -2 2 In 1.5 itia lA 1 ng le 0.5 (d eg )
5 ys) (da e m Ti
0 10
0
Time
15
5 a (d ys)
Fig. 15. Optimal elevator speed as a function of tether length.
the ribbon length increases, the peak libration angle decreases. This is caused by the increased system inertia, requiring larger disturbing forces to displace the counterweight. In reality, the ribbon length will be virtually constant during operations. Hence, even changes on the order of a few hundred kilometers would have negligible impact on the control of the elevator. Fig. 14 shows the elevator position on the ribbon, and Fig. 15 shows the elevator speed as a function of ribbon length. These results highlight the result that the elevator control is largely unaffected by changes in ribbon length.
Elevator Velocity (m/s)
Fig. 16. Effect of initial in-plane libration angle on optimal libration stabilization using elevator.
4.5. Libration stabilization
100 0 -100 2 In
1.5 lA 1 ng le 0.5 (d eg )
itia
0 5 10 0
e Tim
15
s)
(day
Fig. 17. Effect of initial in-plane libration angle on elevator velocity for libration stabilization.
Libration Response (deg)
The final case considered in this paper is utilizing the elevator for direct control of the in-plane libration of the ribbon. The stabilization problem is solved by varying the initial in-plane angle from 01 to 21, and fixing the maneuver time at 15 days. The ribbon is 100,000 km long with Young’s modulus 35 GPa, the maximum crosssectional area is 10 mm2, and mass density of 1300 kg/ m3. The elevator mass is 10,000 kg. The in-plane libration response is shown in Fig. 16 for the case where the out-ofplane angle is zero. The results show that it is possible to remove excess librational energy by moving the elevator on the ribbon in a judicious manner. Fig. 17 shows the variation in elevator velocity as a function of the initial libration angle. The maximum speed of the elevator increases as a function of the initial libration angle, peaking at approximately 100 m/s for initial angles of approximately 21. The same sequence of problems was resolved in the presence of out-of-plane oscillations. The initial out-of-plane angle is fixed at 0.51. Fig. 18 shows the difference in the in-plane libration angle for the optimal response compared to the case with no out-of-plane librations. Although the elevator cannot directly control the out-of-plane angle, it is still able to remove the excess in-plane librational energy. This control scheme can therefore be effective for controlling the long-term evolution of the in-plane ribbon dynamics. It might be possible to utilize the technique for applications such as
200
0.3 0.2 0.1 0 -0.1 -0.2 2 Ini tia
1
lA
ng
le
(de
g)
0
15
10
0
5
)
ays Time (d
Fig. 18. Difference in in-plane libration response caused by 0.51 out-ofplane initial libration angle, shown as a function of initial in-plane libration angle.
ARTICLE IN PRESS P. Williams, W. Ockels / Acta Astronautica 66 (2010) 1458–1467
debris mitigation by moving the ribbon out of the path of certain debris. This possibility should be considered in future work. 5. Conclusions The control of the ascent of a space elevator to geostationary altitude has been considered. The fundamental mode of vibration of an anchored ribbon is a rigid body oscillation about the anchor point on the Earth’s surface. This mode can lead to oscillations of the counterweight, which can be located up to 100,000 km above the anchor point. As the elevator climbs the ribbon, Coriolis forces create a torque that initiates the librational mode. The lack of damping in space causes this mode to persist. Using a simplified model that only accounts for the two fundamental libration modes, it is possible to solve the ascent problem using optimal control. The results show that it is possible to eliminate the librational oscillations by reversing the direction of the elevator for a short time. For a maximum speed of 50 m/s, with accelerations limited to 4 m/s2, the optimal trip time is on the order of 9.38 days. By varying the elevator mass and ribbon length, results show that the optimal ascent trajectories are relatively insensitive to such parameters. It is also possible to use the elevator as a means for controlling the librational response of the ribbon, and to remove excess energy that can arise from other sources such as perturbations. Results show that the technique is useful for removing in-plane oscillation energy, but that out-ofplane librations require another means of control.
1467
References [1] K.E. Tsiolkovsky, Speculations between earth and sky, Isd-vo ANSSSR, Moscow, 1895, p. 35 (In Russian, reprinted in 1959). [2] J. Pearson, The orbital tower: a spacecraft launcher using the Earth’s rotational energy, Acta Astronautica 2 (1975) 785–799. [3] S. Iijima, Helical Microtubules of Graphitic Carbon, Nature 354 (1991) 56–58. [4] S.S. Cohen, A.K. Misra, Elastic oscillations of the space elevator ribbon, Journal of Guidance Control, and Dynamics 30 (6) (2007) 1711–1717. [5] B.C. Edwards, E.A. Westling, The Space Elevator: A Revolutionary Earth-to-Space Transportation System, 2003. [6] P. Williams, Dynamic multibody modeling for tethered space elevators, Acta Astronautica 65 (3–4) (2009) 399–422. [7] S.S. Cohen, A.K. Misra, The effect of climber transit on the space elevator dynamics, Acta Astronautica 64 (2009) 538–553. [8] P. Williams, C. Blanksby, Study on prolonged rendezvous using tethers, AAS/AIAA Astrodynamics Specialist Conference, August 7–11, 2005, Embassy Suites Hotel, Lake Tahoe Resort, Paper AAS 05-321. [9] P. Williams, C. Blanksby, Prolonged payload rendezvous using a tether actuator mass, Journal of Spacecraft and Rockets 41 (5) (2004) 889–892. [10] C.R. McInnes, Dynamics of a particle moving along an orbital tower, Journal of Guidance, Control and Dynamics 28 (2) (2005) 380–382. [11] H.A. Fujii, T. Watanabe, T. Kusagaya, D. Sato, M. Ohta, Dynamics of a flexible space tether equipped with a crawler mass, Journal of Guidance, Control, and Dynamics 31 (2) (2008) 436–440. [12] L. Perek, Space elevator: stability, Acta Astronautica 62 (8–9) (2008) 514–520. [13] J. Elnagar, M.A. Kazemi, M. Razzaghi, The pseudospectral legendre method for discretizing optimal control problems, IEEE Transactions on Automatic Control 40 (10) (1995) 1793–1796. [14] I.M. Ross, F. Fahroo, Legendre pseudospectral approximations of optimal control problems, Lecture Notes in Control and Information Sciences, vol. 295, Springer, New York, pp. 327–342. [15] P. Williams, User’s guide to DIRECT version 1.17, Technical Report, Melbourne, Australia, March 2005.