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Nonovershooting Space Tether Deployment with Explicit Constraint of Positive Deployment Velocity Zheng H. Zhu* York University, Toronto, Ontario M3J 1P3, Canada Ming Liu†, Xingqun Zhan‡ Shanghai Jiao Tong University, Shanghai, 200240, China Baoyu Liu§ Shanghai Jiao Tong University, Shanghai, 200240, China

In this paper, we propose a control law for the passive tether deployment. First, a reference trajectory of nonovershoot tether length, non-negative deployment velocity and tether tension is designed with the consideration of the stability of internal dynamics of TSS. Second, a tension control law with pulse-width pulse-frequency modulation is developed to track the reference trajectory. The effectiveness of the proposed control law is demonstrated by simulations.

I. Introduction ETHERED spacecraft systems (TSS) have great potential in space explorations [1-5]. The critical issue to the success of a TSS mission is to successfully deploy the tether in a fast and stable manner [5] because the potential risk of tether jamming and wrapping around the main spacecraft. Many efforts have been devoted to the tether deployment control. For instance, Srinivas and Vadali [6] developed a asymptotically stable tension control law by Lyapunov’s second method. Pradeep [7] developed a simple linearized tension control law without Lyapunov function. However, the oversimplified dynamic model leads to an overshoot in tether length deployment if the tether is deployed quickly within two orbits. Sun and Zhu [8] developed a fractional-order control tension control law to improve the performance of tether deployment both in deployment speed and stability. Considering the tether cannot be compressed, Wen, Zhu et. al [9] developed a saturation function based control law to satisfy the non-negative tether tension constraint explicitly. It should be noted that these works assumed the tether is actively reeled in or out to achieve the desired performance. However, except for two missions - SEDS2 and YES2, all other tether missions relied on the gravity gradient to deploy and stabilize tethered satellites about the local vertical by simple and passive deployment mechanism. In such approach, the tether is monotonically deployed with a non-negative tether deployment velocity because there is no actively controlled mechanism to reel in and/or out tether [10]. Thus, there is

T

‡

Professor, Department of Mechanical Engineering, 4700 Keele Street; [email protected]. Associate Fellow and Lifetime Member AIAA (Corresponding Author) * PhD Student, School of Aeronautics & Astronautics, 800 Dongchuan Road; E-mail: [email protected]. Visiting PhD Student at Department of Earth and Space Science and Engineering, 4700 Keele Street. † Professor, School of Aeronautics & Astronautics, 800 Dongchuan Road; E-mail: [email protected] § PhD Student, School of Aeronautics & Astronautics, 800 Dongchuan Road, Shanghai; [email protected] 1 American Institute of Aeronautics and Astronautics

a need for a control law that is applicable for such simple passive tether deployment. II. Deployment Dynamics of Tethered Spacecraft Systems The current work is limited to the in-plane deployment of a TSS orbiting the Earth in a circular orbit. Assume the TSS is modeled as a dumbbell model consisting of a main satellite and a sub-satellite connected by an inextensible and massless tether with a maximum deployable tether length L. The in-plane libration angle of the TSS is denoted as the pitch angle  . In addition, it is assumed that the mass of the sub-satellite is negligible compared with the main satellite. Thus, the center of mass of the TSS resides in the main satellite. Based on these assumptions, the dimensionless differential equations of TSS deployment are given by [8]:



l  l  1 



2

 1  3cos 2   Tˆ 

l    2   1    3 cos  sin   0 l 





(1)

where the overhead dot denotes the time derivative, l   / L is the dimensionless tether length,  is the instantaneous tether length, Tˆ  T / (m 2 L) and T are the dimensionless and real tether tension, m is the mass of the sub-satellite,  is the orbital rate of TSS, and   t is the dimensionless time, respectively. The control task for the tether deployment is to ensure the system from an initial state [l l   ]T  [0 v0 0 0]T , where v0 is the initial deployment velocity, to a final state

[1 0 0 0]T . For the sake of convenience, the final equilibrium state is transferred to a zero state by introducing new variables [ x1 x2 x3 x4 ]T  [l  1 l   ]T . Then, Eq. (1) is transferred to a set of first-order differential equations,

x1  x2 x2  (1  x1 ) (1  x4 ) 2  1  3cos 2 x3   Tˆ x3  x4 x4  2

(2)

x2 (1  x4 )  3cos x3 sin x3 1  x1

It can be seen in Eq. (2) that x  0 is an equilibrium point. Accordingly, the initial state is transformed to x (0) = [1 v0 0 0]T , while the desired final state is x () = [0 0 0 0]T . The desired control law should deploy the tether to the final state subject to two constraints: (i) a nonovershooting tether length ( x1 ( )  0,   [ 0 , f ] ), (ii) positive deployment velocity ( x2 ( )  0,   [ 0 , f ] ), while keeping a positive tension ( Tˆ  0 ) in the deployment process. Here  0 and  f are the starting and ending time of the deployment. This work adopts the tension control law for the tether deployment control. Then, the tether tension can be expressed as the control input u  Tˆ  R , while defining the tether length as the output state, such as, y  x1  h( x )  R . Thus, the single-input-multiple-output system in Eq. (2) 2 American Institute of Aeronautics and Astronautics

can be transformed to the following single-input-single-output (SISO) system [11], such that,

 x  f ( x )  g ( x )u   y  h( x )

(3)

where  x2  0 (1  x )[(1  x ) 2  1  3cos 2 x ]   1 1 4 3     g ( x )  , f ( x )   x4  0      2 x2 (1  x4 )  3cos x3 sin x3  0  1  x1  Calculate the Lie derivatives of the output state h(x) and function g(x) as per [11],

h ( x ) h ( x )   L f h ( x )  x  f ( x )  x2 , Lg h( x )  x  g ( x )  0   L2 h ( x )  L ( L h ( x ))   ( L f h( x ))  f ( x )  (1  x )[(1  x ) 2  1  3cos 2 x ] f f 1 4 3  f x   Lg L0f h ( x )  Lg h ( x )  0   ( L1f h( x ))  1 1 L L h ( x )  L ( L h ( x ))   g ( x )  1  g f g f x 

(4)

Thus, the SISO system has the relative degrees of   2 . Accordingly, the dynamic system in Eq. (3) can be recast in terms of external and internal dynamics, y  1  y  L2f h( x )  Lg L f h( x )u or 1  2    Q (ξ , )  2  2 2 (1  2 )  3cos 1 sin 1 1  1

(5)

where η  [ x3 x4 ]T  R2 and ξ  [h( x ) L f h( x )]T =[ x1 x2 ]T  R 2 denote the internal state (libration angle and angular velocity) and external state (tether length and velocity) vectors of the SISO system. The stable output tracking control requires the zero dynamics   Q (0, ) stable or critically stable [11]. Substituting ξ  0 into the internal dynamics in Eq. (5) and linearizing the zero dynamics at zero equilibrium point yield, 1  2  2  3cos 1 sin 1  31

3 American Institute of Aeronautics and Astronautics

(6)

The eigenvalues of the linearized internal dynamics are  3i . Thus, the internal dynamics is critically stable and the stable output tracking control can be achieved. The tracking reference trajectory which could stabilize the internal dynamics is designed in Sec. III. III. Control Law Design The proposed new control law consists of two parts: the design of a nonovershooting reference trajectory for tether length and the nonovershooting trajectory tracking by tension control. A. Reference Trajectory Design Linearizing the internal dynamics in Eq. (5) in the neighborhood of   0 yields

1  2 2  31  2u

(7)

where u  2 / (1  1 )  l / l is a new control input. Equation (7) is stable if the close-loop state feedback control law u  k11  k22 is employed with the following control gains,

k1 = 0.5( p1 p2  3), k2  0.5( p1  p2 )

(8)

where p1 and p2 are the poles with negative real parts. Subsequently, the desired reference trajectory yr ( ) can be acquired by combing the equation u ( )  yr ( ) / [1  yr ( )] with an initial value yr (0) . Here, yr (0)  0.98 to avoid the singularity of internal dynamics at y (0)  1 . Thus, the desired reference trajectory is given by 

u (  ) d  yr ( )  [1  yr (0)]e 0 1

(9)

where   (0, ] . It is noted that Eq. (7) becomes a homogeneous equation with the linear feedback u  k11  k22 . Thus, the control input will always be u =0 with the zero initial

condition   0 , leading to a trivial trajectory yc ( )  yc (0) . To avoid this trivial trajectory, an exponential decay term k Z  e  is introduced to the feedback control law, such that, uˆ  k Z  e   k11  k2 2

(10)

where k Z is the positive gain whose value should (i) be sufficiently large to guarantee uˆ ( )  0 so that the requirement of non-negative deployment velocity ( yr ( )  0 ) is satisfied and (ii) not exceed an upper bond to ensure yr ()  0 so that the tether is fully deployed. B. Tension Control Law Design Assume the tether is deployed by a simple deployment system with a braking force to regulate the deployment velocity. In such a case, the braking force is applied by on and off control. Accordingly, the tension control input is defined in a discrete pulse form,


3 u 0

brake on brake off

(11) yc _ i 1 , yc _ i 1

Reference trajectory Real trajectory yc yc _ i , yc _ i yi , yi

a0 _ i n

aT _ i N n -th interval i

Fig. 1 The approximation with PWPF control To determine the pulse-width pulse-frequency of the applied control input, divide the total deployment time into J intervals. Each interval is further divided into N equal steps ( t ). Within i -th interval, assume the tether is freely deployed initially within the first n  0  n  N  steps and followed by applying braking force in the  N  n  rest steps, which is the pulse width of control input. This is to eliminate the tracking error at the end of interval, ei 1  yi 1  yc _ i 1  0 . Denote a0_ i and aT _ i as the deployment accelerations in the braking-off and braking-on periods respectively. Thus, they can be expressed in terms of state variables as per Eq. (2), such that,

a0 _ i  (1  1 ) (1  2 ) 2  1  3cos2 1  aT _ i  (1  1 ) (1  2 ) 2  1  3cos 2 1   3

(12)

If the interval is sufficient small, the a0 _ i and aT _ i can be approximated as constant in their respect periods. Based on the above assumptions, the tether length and deployment velocity at the end of i -th interval are deduced as

1 1 2 2 yi 1  yi  yi nt  a0 _ i  nt    yi  a0 _ i nt  ( N  n )t  aT _ i ( N  n )t  2 2 braking off

Denote yc   yi 1  yi  e

i 1  0

braking on

 yc _ i 1  yi . Equation (22) becomes


(13)

1 2 (a0 _ i  aT _ i )  nt   (a0 _ i  aT _ i ) Nnt 2  C 2

(14)

1 2 C  yi N t  aT _ i  N t   yc 2

(15)

with

and the pulse width in i-th interval

 3 0 C   ( N t ) 2  2  N  n  N C0   N   N  N 2  2C  others    3t 2   

(16)

where the symbol   denotes rounding down. The above pulse width  N  n  must be subjected to the constraint y  0 . Therefore, the brake will be forcibly turned off if the velocity becomes zero at the anytime within the interval. IV. Results and Discussion The effectiveness of the proposed tension controllers are demonstrated by simulation of a benchmark case with the same orbital and system parameters as used in Ref. [7] for comparison purposes. The TSS is assumed to orbit in a circular orbit with an altitude of 220km and the -3 orbital rate of 1.1804 10 rad / s . The maximum tether deployment length is 100km. All simulations were performed using Matlab 2016a with the numerical integrator ODE45 and default settings. A. Reference Trajectory Design The tether deployment should be in a fast and stable manner. Two competing scenarios are considered here in designing a reference trajectory for tether deployment. The first focuses on the control of maximum amplitude of in-plane angle, while the second pays more attention on rapid deployment. Therefore, two reference trajectories are designed with two sets of close-loop poles [ p1 p2 ]T  [0.5 0.5]T and [ p1 p2 ]T  [2 2]T with negative real parts. From Eq. (8), the associated control gains are calculated as [k1 k2 ]T  [1.375 0.5]T and k2 ]T  [0.5 2]T

Thus, the corresponding control laws are derived uˆ  0.2e  1.3751  0.52 and uˆ  5.23e  0.51  22 by tuning the control gain kZ to ensure (i) uˆ ( )  0 and (ii) no overshoot. Figure 2 shows the control input of internal dynamics of two reference trajectories while Fig. 3 shows the reference trajectories of tether deployment length and velocity generated by Eq. (9) in the time domain. It shows clearly the both reference trajectories reach the full length deployment while satisfy the constraints of positive deployment velocity and no overshoot in length deployment. [k1

.


Fig. 2 Time histories of uˆ  l / l with different poles.

Fig. 3 Reference trajectories with different poles. B. Tracking with PWPF modulation tension control law Assume initial tracking conditions for the PWPF control law for the two reference trajectories as [ y (0) y (0)]  [0.99 0.002] , [ y (0) y (0)]  [0.01 0.08] . The simulation period is set as 5 orbits and divided into J = 600 time intervals. Each interval is further divided into N  5 steps. The simulation results are shown in Figs. 4-8. First, the brake numbers, which is equivalent to control input in case of continuous tension control law, vs time interval is shown in Fig. 4. The total numbers of braking are 1187 and 340 in cases of p1,2  0.5 and p1,2  2 , respectively, where more brakes are applied in the case of slow deployment than in the case of fast deployment. The distribution of brake numbers over the time has a similar trend compared with the continuous tension control law demonstrated as shown in Fig. 5, as expected.


Fig. 4 Brake Numbers vs interval for different reference trajectories (a) p1,2  0.5 (b) p1,2  2 . Next, the time histories of tracking results of tether deployment length y and velocity y are shown in Fig. 5 for two different reference trajectories. The reference trajectories are successfully tracked by the PWPF modulation tension control law. It can be seen from Fig. 5(a) that the tether is deployed by 99.0% and 98.3% of full length in cases of p1,2  0.5 and p1,2  2 , respectively. As mentioned before, less brakes are applied in the case of fast

deployment ( p1,2  2 ), leading to a larger approximation error within each interval and consequently more accumulated error. However, the deployment velocities are always positive in both case, see Fig. 5(b), because there is no mechanism to real in tether in the current case. The frequent and sharp changes in velocities about the reference trajectories are attributed to the sudden application and release of brake force.


Fig. 5 Tracking results of tether deployment length y and velocity y with PWPF tension control law.

Fig. 6 Time histories of in-plane angle and angular velocity with PWPF tension control law. Figure 6 shows the time histories of in-plane angle and angular velocity realized by two different reference trajectories. It shows that the amplitude of in-plane angle approaches zero in almost the same way as the designed performance in the case of p1,2  0.5 while it reduces to


zero gradually into a small range 0.06 radian (3.43 deg) in the case of p1,2  2 . In low Earth orbit application, the small oscillation will decay over the time due to atmospheric drag.

Fig. 7 Time histories of control input u D (tether tension Tˆ ) with PWPF control law. Finally, the tether tension with the PWPF control law is shown in Figure 7. At the end of tether deployment, the brake is on and locked. The on-off braking actions are clearly depicted, where the zero and three represent brake on and off states respectively. The tether tension at both cases is eventually convergent around 3 within a small range at the end of deployment due to the 2 2 residual in-plane libration motion, such as, Tˆ  (1  2 )  1  3cos 1  3 1 0, 2 0 .

V. Conclusions This work proposes a new tension control law for the tether deployment of tethered spacecraft systems, subject to the constraints of nonovershoot of tether length, positive tether deployment velocity and positive tether tension. Simulation results show that the control law successfully track the reference trajectories. Acknowledgments This work is supported by the National Natural Science Foundation of China (No. 11372177) and the Discovery Grant of the Natural Sciences and Engineering Research Council of Canada.

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