Control parameters design of spacecraft formation flying via modified biogeography-based optimization
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Abstract
For spacecraft formation flying (SFF) missions, effective control of relative motion is a critical issue. This paper investigates the problem of feedback parameters design in the trajectory tracking controller of SFF. To overcome the difficulty in manual parameters adjustment, a modified biogeography-based optimization (M-BBO) algorithm is employed by transforming the parameters tuning into an optimization problem. In the developed M-BBO, the new component is a hybrid operator, where the search mechanism of grasshopper optimization algorithm is integrated into the migration operation of biogeography-based optimization (BBO). It helps M-BBO to achieve a better balance between exploitation and exploration abilities, thereby facilitating the generation of promising candidate solutions. During the optimization process, the performance indicator is a linear weighted function that considers the tracking error and fuel consumption of the SFF controller. Simulation results show that the parameters obtained via M-BBO ensure accurate control at low cost, and comparative experiments with other versions of BBO are conducted to prove M-BBO’s merit in terms of convergence performance.
Keywords
Spacecraft formation flying Control parameters optimization Biogeography-based optimization1 Introduction
Spacecraft formation flying (SFF) is a significant technology for certain space missions [1]. With notable benefits such as low cost, high efficiency and great flexibility, SFF expands the functions of a traditional single spacecraft [2, 3]. To guarantee the practical implementation of SFF, it is required to perform valid control of relative position between the spacecraft.
One major goal of SFF control is that the spacecraft can track a desired relative trajectory, and extensive researches have focused on the design of SFF controller. Some popular techniques of SFF control included linear quadratic control, model predictive control and slide mode control [4, 5, 6]. To obtain the expected control performance, the controller parameters commonly need to be identified through the designers’ experience, which is a challenging task in the control system [7].
Recently, owing to automating the parameter adjustment, the application of bio-inspired optimization algorithms to control parameters design has attracted much attention. Steinberg and Page [8] developed a backstepping control scheme with parameter optimization done by a genetic algorithm (GA), which was favorable for the multi-axis control of a high-performance aircraft. In [9], the GA was also employed to optimally search the control gains of pitch autopilot for aircraft landing. Lu et al. [10] suggested a control strategy of turbine engine based on particle swarm optimization (PSO), where proportional–integral–differential parameters are tuned via PSO. Deng and Duan [11] optimized the selection of control parameters in the carrier landing system by pigeon-inspired optimization (PIO). Aiming at satellite formation keeping, Soyinka and Duan [12] regarded mean orbit elements as feedback and proposed the chaotic artificial bee colony (ABC) algorithm to estimate the feedback gain. It can be seen that combining the bio-inspired approaches with the controller design works well in a number of studies.
When it comes to the emerging bio-inspired methods for global optimization, biogeography-based optimization (BBO) cannot be underestimated, because it has demonstrated remarkable success in diverse engineering cases [13, 14]. BBO is modeled after the distribution mechanism of biological species among habitats. Generally, BBO has an excellent exploitation ability, which triggers the hybridization of BBO with other evolutionary algorithms that are good at exploration. Here, exploitation refers to the tendency for the search agents to change on a relatively small scale and move locally, while they are encouraged to be highly randomized and search abruptly during exploration. For instance, the mutation operator of differential evolution (DE) was merged with BBO to increase the population diversity while preserving the exploitation [15]. Such hybrid heuristics algorithms established on the basis of BBO were also reported in [16, 17]. In this study, a modified BBO (M-BBO) method is presented to enhance the balance in BBO between exploitation and exploration. The strategy of updating the solutions in grasshopper optimization algorithm (GOA) [18] is introduced into the standard BBO migration operation. According to [18], GOA is able to explore the search space well, and this is the motivation of redesigning the migration operator.
The novelty of this study is twofold. First, the task of acquiring the controller parameters of SFF along elliptical orbits is converted to an optimization problem, and BBO is considered as a solver to find the parameters. Second, M-BBO that absorbs the advantage of GOA is proposed, where a new migration operator integrates the exploitation of BBO and the exploration of GOA.
The rest of this paper is as follows. The relative translational dynamics and controller design of SFF are described in Sect. 2. The basic BBO and its modified version are contained in Sect. 3. The detailed implementation of M-BBO for control parameters optimization is presented in Sect. 4, followed by numerical simulations and experimental results in Sect. 5. Finally, Sect. 6 concludes the paper.
2 System model of SFF
2.1 Relative motion dynamics
2.2 Lyapunov-based control
From the stability theorem in [21], Eq. (10) can indicate that the closed loop system with the control law \( {\mathbf{u}} \) is asymptotically stable.
3 M-BBO algorithm
3.1 Principles of the basic BBO
In natural biogeography, the extent to which a habitat is suitable for living is judged by the habitat suitability index (HSI), and the numerous factors that can exert influence on the HSI are collectively called the suitability index variables (SIVs). As a population-based algorithm, BBO treats each candidate solution as a habitat. The goodness of one individual is measured by HSI, and each component of the individual is analogous to a SIV.
In BBO, the solutions change probabilistically by the migration operation, which distinguishes BBO from reproductive manners like GA. Besides, for each solution, the migration rate is governed by its quality. Notice that the good solutions are more likely to pass on their own features to others, while the poor ones are inclined to accept new information from others. It is these features that make BBO become a powerful optimization tool [13].
3.2 New migration operator of M-BBO
As mentioned above, in the migration, the solution features of good individuals might substitute certain features of poor individuals, which means that BBO exploits the information of the current population finely. Nevertheless, the existence of these features in both good and poor individuals results in the lack of population diversity and exploration capacity. To make BBO possess better optimization performance, exploitation and exploration are abilities that should be well counterpoised. Moreover, the no free lunch theorem [24] has proven that there are no techniques for all the optimization problems, which is also the reason that M-BBO is needed.
4 Control parameters design based on M-BBO
4.1 Problem formulation
This study looks on the parameters identification of SFF controller as a continuous-domain optimization problem. It can partly alleviate the workload of designers because the conventional cut-and-try method usually calls for plenty of experience and tests. In addition, by tuning the control parameters via optimization algorithms, the control characteristic can satisfy different design requirements.
4.2 Implementation procedure of M-BBO for parameters optimization
- Step 1
Input the orbital elements of the leader, the initial position \( {\varvec{\uprho}}_{0} \) and desired trajectory \( {\varvec{\uprho}}_{d} \) of the follower in the LVLH frame, the total control time \( t_{f} \), the weights \( \omega_{1} \) and \( \omega_{2} \), the search ranges of decision variables \( \left( {X_{j}^{l} ,X_{j}^{u} } \right) \) and the parameters in M-BBO (including population size \( N_{\text{P}} \), maximum optimization generation \( N_{\text{G}} \), maximum migration rates \( I \) and \( E \), maximum mutation rate \( \sigma_{\text{max} } \))
- Step 2
Generate an initial population and initialize generation number \( n_{\text{G}} = 0 \)
- Step 3
Evaluate the weighted optimization objective value in Eq. (20) for each individual in the population
- Step 4
Apply elitism mechanism: updating the best solution \( H_{\text{best}} \), which will replace the worst solution of the current population if \( H_{\text{best}} \) does not exist in the present generation
- Step 5
If the maximum generation number is reached (\( n_{\text{G}} \ge N_{\text{G}} \)), output the optimal control parameters. Otherwise, go to step 6
- Step 6
Compute the immigration rates, emigration rates and mutation rates according to the ranking result of the fitness
- Step 7
Perform the new migration operator as discussed in Sect. 3.2
- Step 8
Conduct the mutation operator as discussed in Sect. 3.1
- Step 9
Update the generation number \( n_{\text{G}} = n_{\text{G}} + 1 \) and go to step 3 for the next iteration
5 Simulation
To validate the effectiveness of the developed strategy of using M-BBO to optimize the control parameters in SFF, an example of two-spacecraft formation is demonstrated. The leader is assumed to be perfectly controlled in an elliptical orbit with semi-major axis 6878.137 km and eccentricity 0.1. The initial true anomaly of the leader is 0°. The follower has the initial values \( {\varvec{\uprho}}_{0} = \left[ { - 100,900,150} \right]^{\text{T}} \;{\text{m}} \), and it ought to track the sinusoidal trajectories \( {\varvec{\uprho}}_{d} = \left[ {500\sin \left( {nt} \right),} \right. \)\( \left. {1000\cos \left( {nt} \right),500\sqrt 3 \sin \left( {nt} \right)} \right]^{\text{T}} \;{\text{m}} \), where \( n \) is the mean angular velocity of the leader. The disturbances are assumed to be known, since these do not belong to the main scope of the present work. In this simulation, \( {\mathbf{D}} = \left[ {2\sin \left( {nt} \right),2\cos \left( {nt} \right),2\sin \left( {nt} \right)} \right]^{\text{T}} \times 10^{ - 5} {{\text{m}} \mathord{\left/ {\vphantom {{\text{m}} {{\text{s}}^{2} }}} \right. \kern-0pt} {{\text{s}}^{2} }} \). The simulating time of the controller is set as \( t_{f} = {{2\pi } \mathord{\left/ {\vphantom {{2\pi } n}} \right. \kern-0pt} n} \).
Simulation parameters of M-BBO algorithm
Parameter | Value |
---|---|
Population size N_{P} | 30 |
Maximum generation number N_{G} | 25 |
Maximum immigration rate I | 1 |
Maximum emigration rate E | 1 |
Maximum mutation rate σ_{max} | 0.01 |
Control parameters obtained by M-BBO algorithm
Parameter | K_{1,1} | K_{1,2} | K_{1,3} | K_{2,1} | K_{2,2} | K_{2,3} |
Value | \( 1. 8 4 2\times 10^{{{ - }5}} \) | \( 1. 9 9 5\times 10^{{{ - }5}} \) | \( 1. 6 4 0\times 10^{{{ - }5}} \) | \( 1. 1 1 4\times 10^{ - 2} \) | \( 9. 2 8 2\times 10^{ - 3} \) | \( 6. 0 4 2\times 10^{ - 3} \) |
6 Conclusion
In this paper, the problem of optimizing the control parameters for SFF missions is addressed. A nonlinear Lyapunov-based control law is adopted, and the key of the optimization is tuning the feedback matrix parameters of the controller. A modified version of BBO, i.e., M-BBO, is presented to deal with this parameter optimization problem. By virtue of the new migration operator, which assimilates the exploration of GOA without destroying the exploitation of BBO, M-BBO has an appropriate balance between solution diversification and intensification. With the help of M-BBO, the generated SFF controller can guarantee an excellent performance index that performs an overall evaluation of tracking error and fuel cost. Comparative results suggest that the proposed M-BBO is a feasible and superior method for determining the SFF control parameters.
It has to be pointed out that the limitation of the developed method is that the objective function during the optimization considers a weighted fitness function, that is, the method is only targeted at the single-objective optimization. In the future work, applying multi-objective optimization algorithms to the research field of SFF control is worthy of investigation.
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