1. Introduction

Space tether system is a promising technology over decades. It has wide potential applications in the space debris mitigation & removal, space detection, power delivery, cargo transfer and other newly science & technic missions. Recently, there is continuous interest in the space tether systems, in leading space agencies such as, NASA's US National Aeronautics and Space Administration, ESA's European Space Agency, and JAXA's Japan Aerospace Exploration Agency [1]. Their interest technologies include the electrodynamic tether (EDT) propulsion

© 2016 The Author(s). Licensee InTech. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and eproduction in any medium, provided the original work is properly cited. © 2018 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

technology, retrieval of tethered satellite system, multibody tethered system and space elevator system. Compared with existing technologies adopted by large spacecraft such as the rocket or thruster, the space tether technology has the advantages of fuel-efficiency (little or no propellant required), compact size, low mass, and ease-of-use [2]. These advantages make it reasonable to apply the space tethered system for deorbiting the fast-growing low-cost micro/nanosatellites and no-fuel cargo transfer. The difficulty associated with space tether system is to control & suppress its attitudes during a mission process for the technology to be functional and practical. Many works have been devoted to solving this problem, and one effort is to use the optimal control due to its good performances in the complex and unstable nonlinear dynamic systems. In this chapter, a new piecewise onboard parallel optimal control algorithm is proposed to control and suppress the attitudes of the space tether system. To test its validity, two classical space tether systems, the electrodynamic tether system (EDT) and partial space elevator (PSE) system are considered and tested.

system. Up to data, limited devotions have been done on the libration suppression of a PSE system using tension control only. Williams used optimal control to design the climber's speed function of a climber for a full space elevator [10]. Modeled by simplified dynamic equations, an optimal control problem is solved, and the solution results in zero in-plane libration motion of the ribbon in the ending phase of climber motion. The study shows that to eliminate the inplane oscillations by reversing the direction of the elevator is possible. Kojima et al. [11] extended the mission function control method to eliminate the libration motion of a three body tethered system. The proposed method is effective when the total tether length is fixed and the maximum speed of the climber no more than 10 m/s. Although these efforts are useful to suppress the libration motion of the PSE system, it still difficult to control the attitudes of such

Piecewise Parallel Optimal Algorithm http://dx.doi.org/10.5772/intechopen.76625 95

To overcome the challenges in aforementioned works, we propose a parallel onboard optimal algorithm contains two phases. Phase 1 concerns the reference state trajectory optimization within a given time interval, where an optimal control model is formulated based on the timescale separation concept [3, 12] to simplify the dynamic calculations of the EDT & PSE system. An open-loop optimal state trajectory is then obtained by minimizing a cost function subject to given constraints. The state trajectory of paired state and control input variables is solved approximately by the direct collocation method [13] that is based on the Hermite-Simpson method [14]. In this phase, the simplified dynamic model is by used. Phase 2 concerns the tracking of the open-loop optimal state trajectory within the same interval. A closed-loop optimal control problem is formulated in a quadrature form to track the optimal state trajectory obtained in phase 1. Unlike phase 1, all the major perturbative forces are included, and more realistic geomagnetic and gravitational field models are considered. While the system is running the process in phase 2 with one CPU, the next phase 1 calculation is running in another CPU with data modification based on the errors obtained in the last calculation program. The simulation results demonstrate the effectiveness of the approach in fast satellite deorbit by EDTs in equatorial orbit. Furthermore, for fast transfer period of the partial space elevator, the propose method also shows good effect on suppression the libration angles of the

Assume two CUPs are used to process the calculation. CUP-1 is used to determine the openloop optimal control trajectory of dynamic states employing the simple dynamic equations. The obtained optimal state trajectory will be tracked by CPU-2 using closed-loop RHC. While the system is tracking the i-th interval, the (i + 1)-th optimal trajectory is being calculated in CPU-1. Once the tracking for the i-th interval is finished (implemented by CPU-2), the real finial state Si will be stored in the memory and the (i + 1)-th optimal trajectory can be tracked. By repeating the above process, the optimal suppression control problem is solved in a parallel

a system in the transfer period.

climber and the end body with tension control only.

piecewise manner until the transfer period is over.

2. Optimal control algorithm

2.1. Control scheme

An EDT system with constant tether length is underactuated. The electric current is the only control input if there are no other active forces such as propulsion acting on the ends of an EDT. The commonly adopted control strategy in the literature is the current regulation using energybased feedback in this underactuated control problem. Furthermore, many efforts have been done to solve this problem with optimal control. Stevens and Baker [3] studied the optimal control problem of the EDT libration control and orbital maneuverer efficiency by separating the fast and slow motions using an averaged libration state dynamics as constraints instead of instantaneous dynamic constraints in the optimal control algorithm. The instantaneous states are propagated from the initial conditions using the optimal control law in a piecewise fashion. Williams [4] treated the slow orbital and fast libration motions separately with two different discretization schemes in the optimal control of an EDT orbit transfer. The differential state equations of the libration motion are enforced at densely allocated nodes, while the orbital motion variables are discretized by a quadrature approach at sparsely allocated nodes. The two discretization schemes are unified by a specially designed node mapping method to reflect the coupling nature of orbital and libration motions. The control reference, however, is assumed known in advance.

A PSE system is consisted with one main satellite and two subsatellites (climber & end body) connected to each other by tether(s). The difficulty associated to such a system is to suppress the libration motion of the climber and the end body. This libration is produced by the moving climber due to the Coriolis force, which will lead the system unstable. While the climber is fast moving along the tether, the Coriolis force will lead to the tumbling of the PSE system. Thus, the stability control for suppressing such a system is critical for a successful climber transfer mission. To limit the fuel consuming, tension control is widely used to stable the libration motion of the space tethered system due to it can be realized by consuming electric energy only [5]. Many efforts have been devoted to suppressing the libration motion of space tethered system such as, Wen et al. [6] stabled the libration of the tethered system by an analytical feedback control law that accounts explicitly for the tension constraint. The study shows good computational effect, and the proposed method requires small data storage ability. Ma et al. [7] used adaptive saturated sliding mode control to suppress the attitude angle in the deployment period of the space tethered system. Optimal control [8, 9] is also proved as a way to overcome the libration issue. The above tension control schemes are helpful for both two-body and three-body tethered system. Up to data, limited devotions have been done on the libration suppression of a PSE system using tension control only. Williams used optimal control to design the climber's speed function of a climber for a full space elevator [10]. Modeled by simplified dynamic equations, an optimal control problem is solved, and the solution results in zero in-plane libration motion of the ribbon in the ending phase of climber motion. The study shows that to eliminate the inplane oscillations by reversing the direction of the elevator is possible. Kojima et al. [11] extended the mission function control method to eliminate the libration motion of a three body tethered system. The proposed method is effective when the total tether length is fixed and the maximum speed of the climber no more than 10 m/s. Although these efforts are useful to suppress the libration motion of the PSE system, it still difficult to control the attitudes of such a system in the transfer period.

To overcome the challenges in aforementioned works, we propose a parallel onboard optimal algorithm contains two phases. Phase 1 concerns the reference state trajectory optimization within a given time interval, where an optimal control model is formulated based on the timescale separation concept [3, 12] to simplify the dynamic calculations of the EDT & PSE system. An open-loop optimal state trajectory is then obtained by minimizing a cost function subject to given constraints. The state trajectory of paired state and control input variables is solved approximately by the direct collocation method [13] that is based on the Hermite-Simpson method [14]. In this phase, the simplified dynamic model is by used. Phase 2 concerns the tracking of the open-loop optimal state trajectory within the same interval. A closed-loop optimal control problem is formulated in a quadrature form to track the optimal state trajectory obtained in phase 1. Unlike phase 1, all the major perturbative forces are included, and more realistic geomagnetic and gravitational field models are considered. While the system is running the process in phase 2 with one CPU, the next phase 1 calculation is running in another CPU with data modification based on the errors obtained in the last calculation program. The simulation results demonstrate the effectiveness of the approach in fast satellite deorbit by EDTs in equatorial orbit. Furthermore, for fast transfer period of the partial space elevator, the propose method also shows good effect on suppression the libration angles of the climber and the end body with tension control only.
