Time-dependent autonomous task planning of agile imaging satellites

Abstract

This study was intended to investigate the time-dependent autonomous task planning of agile imaging satellites. Based on the analysis of the major constraints of the problem, a model was established by integrating the satellite task planning with time-dependent scheduling. According to the dynamic observation requirements, the mode and framework of on-board autonomous task planning were designed by combining planning, decision-making, execution and feedback. A strategy of simultaneous planning and execution and a heuristic algorithm of rolling planning were proposed in order to replace the one-time global planning with successive local planning. Results demonstrated that the proposed method can not only respond quickly to the dynamic observation requirements, but also allow more tasks to be imaged around the optimal time point.

Keywords

Time-dependent agile imaging satellites autonomous task planning heuristic

1 Introduction

As mankind has been increasingly dependent on imaging satellites, user requirements for them are becoming increasingly demanding, for instance, requirements for quick responses in dynamic observation and high quality of satellite images. However, in the traditional management and control model, the execution plans of satellite task are given by the ground measurement and control system. Limited by factors such as time window, even the agile imaging satellites [1 –4] are unable to quickly respond to the dynamic observation requirements. In addition, the quality of images taken by the agile imaging satellites dynamically changes over the imaging time. If the time-dependent relationship between image quality and imaging time is neglected, image quality can be hardly guaranteed. The most effective way to improve the response speed and image quality of the agile imaging satellite is to make the satellite more intelligent, enabling it to carry out autonomous task planning on board, according to the dynamic requirements, the satellite status, and the time-dependent features [5, 6].

Previous studies [7, 8] on the investigation of the feasibility of on-board autonomous task planning proposed a reaction-deliberation structure of autonomous task planning and designed an iterative, random greedy algorithm, but the impacts of execution on planning were not taken into account. Other studies [9 –13] deeply explored the autonomous task planning of EO-1, but the emergency response of the planning method was undesirable as it was time-consuming. Chien et al. [14] employed iterative repair to conduct the autonomous task planning, but the timing of repair was difficult to grasp, and the duration of each planning could not be determined. Verfaillie and Bornschlegl [15] proposed a method of reverse dynamic planning, but its model was too simple. Moreover, none of these methods took into account the time-dependent relationship between the image quality and imaging time. The time-dependent scheduling was first put forward in solving the problem of job shop scheduling [16 –20]. In the stand-alone environment, Voutsinas and Pappis [21] investigated the fact that the workpiece value decreased exponentially over time and proposed the optimal algorithm for particular cases. On this basis, Janiak et al. [22] further verified that the problem had the NP-hard property and proposed the heuristic algorithm. Studies [23, 24] investigated the fact that the workpiece value decreased over time, as described by a piecewise function, and proposed three types of heuristic algorithm.

This paper will describe and analyze the difficulties in the time-dependent autonomous task planning of agile imaging satellite. On the basis of the above literature, we will establish a model by integrating the satellite task planning with time-dependent scheduling and design the mode and framework of on-board autonomous task planning. Finally, we will put forward a strategy for simultaneous planning and execution and propose a heuristic algorithm for rolling planning.

2 Problem description and modeling

Dynamic observation requirement means that users may propose a new task requirement or cancel an observation requirement at any time. The satellite image quality is measured by the ground sampling distance (GSD); the smaller the value, the higher the quality. Let the focal length of image camera be f, the line width be DP, the distance between satellite and target be r, and the angle between the line connecting satellite and target and the horizontal surface of ground be elev, then GSD is calculated as follows [25]: $GSD = \frac{DP \cdot r}{f \cdot \sqrt{sin (ele v)}}$ (1)

The agile imaging satellite has the mobility of three-axis stabilization: pitch, yaw, and roll. Therefore, for the same target, the observation can be started at any time point within a relatively long time window, as shown in Fig. 1. Since the imaging time point determines elev, the closer to the central point of the time window an imaging time point is, the better the image quality, and vice versa.

The problem can be described as follows: according to the dynamic changes of the image quality over the start time point, the satellite assigns observation period and data feedback period for the task in an autonomous, fast, reasonable manner, without or with few interventions by the ground crew. By doing this, it can not only provide maximal observation benefits with minimal resource consumptions, but also allow more tasks to be imaged around the optimal time point.

The difficulties are as follows:

Considered as a complex combinatorial optimization problem, the task planning of the non-agile imaging satellite [26 –29] is NP-hard [30]. The problem we are facing is not merely dealing with the agile imaging satellites that have more complicated constraints, but also taking the time-dependency into consideration, hence, making the modeling even moredifficult.

The mode and framework of autonomous task planning are the basis for on-board autonomous task planning. However, it is difficult to properly design the mode and framework so that real-time response and optimization of task planning can be integrated on board.

Since the computing resources are very limited on board, the algorithm must adapt to the mode and framework of autonomous task planning while its solving speed directly influences the satellite’s quickness in response to environmental changes. For this reason, it is difficult to design a logical and efficient algorithm.

In this paper, we extend the results of a previous study [28, 31] and establish a model with the following assumptions:

Considering only a single satellite with one remote sensor;

Excluding the maneuvering imaging of satellite;

Assuming the on-board memory is capable of randomly accessing data;

No distinction between point target and regional target;

Considering only fixed ground station;

Disregarding (or ignoring) the impacts of meteorological conditions on the observation task. Excluding the three-dimensional imaging tasks;

Excluding or ignoring the energy constraints of satellite.

Model Input: J denotes the set of dynamic observation tasks. Each task j has its corresponding priority w_j, i.e., the maximum profit of the task; c_j denotes the size of memory; d_j denotes the observation and return duration; M_j denotes the set of laps in which the satellite can observe task j; and for each lap k, s_j,k and e_j,k respectively denote the start and end time points of its corresponding observation time window. I denotes the set of time window of data return. b_i and r_i respectively denote the start and end time points each return time window i. C denotes the maximum memory of satellite.

Model Output: x_j,k and y_j,i denotes the decision-making variables of the model, j ∈ J, k ∈ M_j, i ∈ I. If the satellite observes task j in lap k, then x_j,k = 1, or x_j,k = 0. If the observed data of task j is returned during the return time window i, then y_j,i = 1, or y_j,i = 0. $s_{j, k}^{'}$ and $e_{j, k}^{'}$ respectively denote the start and end time points of each observation task, and b_j,i and r_j,i respectively denote the start and end time points of each return task.

The objective of optimization is to achieve the maximum accumulated profit of the observed and returned tasks. Function f_j (t) is used to describe the profit value of task j imaged at t time point, and the objective function is:

$max \sum_{j \in J} \sum_{k \in M_{j}} \sum_{i \in I} f_{j} (t) x_{j, k} y_{j, i}$ (2)

Constraints:

1) Each target can only be observed once: $\sum_{k \in M_{j}} x_{j, k} \leq 1$ (3)

2) Data of each observation task can only be returned once: $\sum_{i \in I} y_{j, i} \leq 1$ (4)

3) Only the data of the finished observation task can be returned: $x_{j, k} \geq y_{j, i}$ (5)

4) Each observation task must be observed within its observation time window: $s_{j, k} \leq s_{j, k}^{'} < e_{j, k}^{'} \leq e_{j, k}$ (6)

5) Each observation task must be returned within the return time window: $b_{i} \leq b_{j, i} < r_{j, i} \leq r_{i}$ (7)

6) The start time of returning the observed data cannot be earlier than the end time of observing the target: $e_{j, k}^{'} \leq b_{j, i}$ (8)

7) The satellite memory cannot exceed the maximum memory limit at any time, and C_t denotes the size of satellite memory at any time: $0 \leq C_{t} \leq C$ (9) 8) S_j and E_j respectively denote the start and end time points of task j. When tasks j and j′ are successively executed in lap k, there must be enough time interval between the two tasks to ensure that the satellite remote sensor has sufficient time to transform its attitude, TS_j,j′ denotes the attitude transformation time between tasks j and j′: $E_{j} + {TS}_{j, j^{'}} \leq S_{j^{'}}$ (10)

(9) Any two tasks cannot be executed at the same time: $[S_{j}, E_{j}] \cap [S_{j^{'}}, E_{j^{'}}] = φ$ (11)

Since the center of time window is the optimal imaging time, the profit function f_j (t) is designed in the piecewise form, according to references [23, 24]. The domain of f_j (t) is [t₀, t₆], t₀ = s_j,k, t₆ = e_j,k, and each observation time window is evenly divided into 5 sections, as shown in Fig. 2. $f_{j} (t) = {\begin{matrix} 0.6 \cdot w_{j} & t_{0} \leq t < t_{1} \\ 0.8 \cdot w_{j} & t_{1} \leq t < t_{2} \\ w_{j} & t_{2} \leq t < t_{3} \\ 0.8 \cdot w_{j} & t_{3} \leq t < t_{4} \\ 0.6 \cdot w_{j} & t_{4} \leq t < t_{5} \end{matrix}$ (12)

3 Mode and framework of satellite autonomous task planning

In the conventional planning mode [32 –37], the satellite has to keep implementing the planning program with no interruption. For emergency tasks, He et al. [28] proposed a planning method based on rolling horizon optimization (RHO) strategy, which was characterized by alternative planning and execution. In this paper, we propose the designed mode of on-board autonomous task planning to adopt a strategy of simultaneous planning and execution, referred to as SPE strategy for short. This approach can not only respond quickly to environmental changes through rolling planning, but also ensure the consistency of task execution, as shown in Fig. 3. The autonomous task planning mode includes four stages: rolling, planning, decision-making, and execution. In the rolling stage, the dynamic task information is determined by establishing the rolling window, which refers to the set of all the unexecuted tasks except for the decision-making results. In the planning stage, the tasks in the rolling window are planned. In the decision-making stage, decisions are made about the tasks to be executed by the satellite. The decision-making results refer to the top tasks among the planning results during the execution period. In the execution stage, the satellite executes the decision-making results, and the decision-making stage begins once the execution stage ends. The rolling, planning and decision-making will be carried out orderly, while the planning and execution are conducted simultaneously. The process of autonomous task planning is shown in Fig. 4.

In this paper, decision-making, planning, execution, and data feedback are integrated to put forward the framework of on-board autonomous task planning, composed by 4 functional modules, as shown in Fig. 5.

Decision-making module: Decision-making starts when planning and execution end simultaneously. Exact trigger on timing can be divided into normal decision-making and emergent decision-making. The normal decision-making refers to the decision-making triggered after each planning stage. The emergent decision-making refers to the decision-making triggered when the decision-making results cannot be executed. After the decision-making stage, the planning and execution module is triggered.

Planning module: According to user’s requirements, satellite status and various constraints, the tasks in the rolling window are planned. Before the planning stage ends, the iterative solution is performed. After the planning stage, the planning program with the highest profit will be selected as the final result.

Status detection module: The current satellite status is detected and provided real-time. If the decision-making result does not meet the current on-board constraints, the emergent decision-making will be triggered.

Execution module: The hardware execution is controlled according to the action instructions, and the execution results and parameter information are timely fed back.

4 Algorithm design

Since the on-board computing resources are limited, the algorithm must feature small size and simple program. In this paper, we propose a heuristic algorithm LSS combining the random mechanisms and the idea of roulette based on references [8 , 38]. Searching is an iterative process consisting of two phases: a construction phase and a local search phase. The construction phase randomly selects one rule out of various rules and sorts the tasks with the idea of roulette to generate a task queue. By following the various constraints the local search phase assigns the period for the task according to the sequence of the queue and generates a feasible solution. Each iteration is accompanied by these two phases. Before the start of iteration, the solution set is empty. For each upcoming iteration, the current feasible solution will be compared with the one in the solution set by the objective function, and the solutions in the set will be updated to the best value of the two. After each iteration, the feasible solution in the set is the output of the search.

Algorithm LSS consists of the following 5 steps:

Step 1. Task sorting. There are 7 types of sorting rules of LSS: sorting by the priority, PF; sorting by giving priority to task of short observation time, ST; sorting by giving priority to task of long observation time task, LT; sorting by the sequence of time window, OT; sorting by the ratio of priority and total duration (the total duration of all available observation time window), which can be further divided into smaller ratio priority, SP, and larger ratio priority, LP, and random sorting RS. The sorting strategies are shown in Table 1. First, a sorting rule was randomly selected, and the task’s sorting index f_j and selection probability P_j were calculated. Then, a random number r between 0 and 1 was generated based on the idea of roulette. If P₁ + P₂ + ·· · + P_j-1 < r < P₁ + P₂ + ·· · + P_j, then task j would be selected. The sorting was completed by constantly selecting tasks.

Step 2. Time window clipping. The occupied periods are clipped off from the available time window, to avoid overlapping of different tasks. Suppose the occupied period is [T_s, T_e], the time window to be clipped is [W_s, W_e], then the clipping methods are shown in Fig. 6.

Step 3. Task period determination. After time window clipping, the available time window of each task can be clipped into multiple small time windows. First, the earliest available time window is selected. Then, the start time point of this time window is set as the start of the task. At last, the end of task is determined by the task duration.

Step 4. Constraint checking. To determine whether the memory constraint and attitude transformation constraint are violated.

Step 5. Profit calculation. First, the profit of each observation task is calculated by the time-dependent function. Then, the total profit of the planning program is calculated. The algorithm flow is shown in Fig. 7.

5 Experimental analysis

Referring to the orbit and attitude of IKONOS satellite, 4 groups of task data with different sizes were obtained by Satellite Tool Kit. In this study, the autonomous task planning of the satellite within 24 hours and the dynamic changes of task and satellite status were simulated, with a task profit range of 1–8. The experimental computer was configured with Intel(R) Core i7-3517U, 2800 MHz, 4 G of memory, an operating system of Windows 7, a programming environment of Microsoft Visual Studio 2010, and programming language of C.

In this study, the step-size was defined as the number of tasks contained in each decision-making result. As shown in Fig. 8, when the step-size was gradually extended, the task completion rate (the percentage of finished data return tasks in all tasks) dropped after a rise. When the step-size was particularly small, the time for execution and planning was short, resulting in fewer numbers of iteration. Therefore, both the planning results and decision-making results were undesirable, ultimately reducing the task completion rate. With the increase in step-size, the time for execution and planning was gradually extended, resulting in increased number of iteration. As a result, both the planning results and decision-making results were improved, thereby promoting the task completion rate. However, if the step-size was too large, the time for continuous execution would be too long. Hence, when entering the next planning stage, many tasks could not be included in the planning program because the available time window had been missed or the resources had been occupied. For this reason, the task completion rate was lowered.

In Fig. 9, the task completion rate refers to the ratio of the tasks that completed data return among each priority task. As shown in Fig. 9, the advantage of a large step-size is the capability to finish more tasks with higher priority. Since the task with higher priority will increase the total profit of the plan, the planning program generated in each planning stage must contain many tasks with higher priority. However, due to the limits of time window, the execution time of tasks with higher priority tend to be the middle or bottom ones, instead of the top one. So when the step-size is small, there might be no task with higher priority in the decision-making results while the large step-size will increase the probability of tasks with higher priority showing in the decision-making results.

To improve the satellite response speed, the small step-size should be selected because the intervals between decision-making are short and the satellite can make plan in accordance with the latest information of task and satellite status. However, in order to obtain high task completion rate, the medium step-size should be selected. To improve the completion rate of task with higher priority the large step-size should be selected. In engineering practice, not only the satellite’s response speed should be taken into account, but also the satellite’s observation efficiency should be fully used, so as to complete as many observation tasks as possible. For this reason, the medium step-size would be most appropriate.

The algorithm LSS was modified so that the planning stage would not be triggered until the execution stage ended, thereby achieving the RHO strategy. The maximum number of iterations for each planning solution in RHO strategy was set to 100. The step-sizes of the two strategies were set to 5, a medium size. As shown in Table 2, SPE strategy was significantly better than RHO strategy. This is because RHO strategy requires alternative planning and execution, inevitably leading to overlapping of the time occupied by the execution stage and some available time windows. Therefore, after the planning stage, some tasks might lose their available time window. When the period occupied by the planning stage overlapped with the return time window, this period certainly could not return task data, thereby reducing the use rate of return time window (the percentage of the period used for data return in the total duration of all return time window). Since SPE strategy allows simultaneous planning and execution with no contradiction in time, its outcome is better.

The optimized target of algorithm LSS was modified into max ∑_j∈J∑_{k∈M_j}∑_i∈Iw_jx_j,ky_j,i. The modified algorithm did not consider the time-dependency. As shown in Fig. 10, if the time-dependency was not taken into consideration, the imaged tasks in period 3 accounted for around 30%, while the imaged tasks in periods 1 and 5 accounted for nearly 19%. After considering time-dependency, the tasks imaged in period 3 were drastically increased, while those in periods 1 and 5 were drastically decreased. This is because if time-dependency was considered, then lower task profit could be obtained when it was imaged in period 1 or 5, while higher task profit could be obtained when it was imaged in period 3. Hence, the algorithm ensured more tasks to be imaged in period 3, while selecting the planning program with higher profit.

The solving approach of algorithm LSS is to replace the one-time global planning with successive local planning. Each local planning is conducted for one task in the rolling window. As a static planning problem, it can be solved by using the IBM ILOG CP Optimizer. In this study, we respectively used LSS and CP Optimizer to make plan for the tasks in the same rolling window. The solving time of CP Optimizer was set to 600 seconds, and the maximum number of iterations of LSS was set to 100. As shown in Fig. 11, CP Optimizer did not allow more tasks to be imaged around the optimal imaging time point. The imaging time of observation task mainly focused on period 1. This is because CP Optimizer adopts a generic algorithm that lacks specificity. Due to the extraordinary amount of area to search in space, it was difficult to obtain a satisfactory solution in a limited period of time. Based on the characteristics of the problem, LSS introduced the idea of roulette and the clipping of time window to factor in the impacts of sorting rules on the task sequence while guaranting the diversity of task sequences at the same time. By clipping the time window, the search space was narrowed and the solving outcome was improved.

6 Conclusions

This study was intended to investigate the time-dependent autonomous task planning of agile imaging satellite by establishing the model of the problem. We put forward the mode and framework of on-board autonomous task planning and designed the heuristic algorithm LSS to achieve the following results:

By combining planning, decision-making, execution and data feedback, we proposed a strategy of simultaneous planning and execution and designed the mode and framework of on-board autonomous task planning.

On the basis of meeting the time window constraint, storage constraint and attitude transformation constraint, the model was established by integrating the satellite task planning with time-dependent scheduling. The idea of roulette and the clipping of time window were introduced into the solving approach, and a heuristic algorithm of rolling planning was proposed.

Through our experiments, we verified the effectiveness of the mode, framework and algorithm of the on-board autonomous planning and compared with the RHO strategy and CP Optimizer. The results demonstrated that the proposed method can not only respond quickly to dynamic observation requirements, but also allow more tasks to be imaged around the optimal imaging time point.

Footnotes

Acknowledgments

The authors express their heartfelt gratitude to the anonymous reviewer for such excellent comments and suggestions which have enormously enhanced the quality and presentation of this paper.

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