Per Hägg, FOI Swedish Defence Research Agency
Keywords: Sensor Planning, Sensor Tasking, Space Situational Awareness, Convex Optimization, System Identification
Abstract:
The number of space objects in Earth’s orbit is growing rapidly. This growth drives a need for improved space situation awareness, both in terms of increased sensor capacity and a more efficient use of sensor resources. The orbits of space objects are predominantly determined using data from networks of radar and optical sensors, but other techniques such as passive radio and laser sensors are used as well. Intergovernmental organizations, governments and commercial companies continuously expand their sensors networks to cope with the rapid growth. This provides a necessary increase in sensor capacity, but it also add further complexity to prioritization and optimization of the sensors network for efficient use.
In this paper, we study the problem of optimal sensor planning, with the objective of finding efficient methods for the scheduling of space object measurements using a set of different sensors in a sensor network, in order to retain sufficient accuracy in the orbital parameters of the object. The network could consist of multiple sensors of different types, different operational modes, and with different accuracy and performance.
The main contribution of this paper is to apply the idea of application-oriented input design from the field of system identification to the sensor-planning problem. The field of system identification handles the problem of building a model of a dynamical system based on experimental data.
We assume that an initial estimate of the state of each object is known; the method does not cover the discovery of new objects or the reacquisitions of objects for which custody has been lost. Without any knowledge of the state, no planning is possible. The initial states are used to calculate when each object is in range of a sensor and can be observed. Each observation by a sensor is associated with a cost. The cost could be financial but any other measures of cost can be used . The sensor-planning problem can be stated as follows: Find the combination of possible observations of a group of objects such that the total observation cost is minimized while the orbital parameters, estimated with the observed data, are accurate enough for their intended use.
In application-oriented input design the objective is to find the cheapest possible identification experiment such that the identified model satisfies the performance specification given by the intended use of the model. The optimization is performed over the properties of the input signal, used to excite the system during the experiment, such as signal spectrum, power or length. In the sensor-planning problem, instead of designing the input signal, we select at which time a certain sensor should be used to make an observation. Transferred to the sensor-planning task this can be seen as selecting the least costly combination of possible observations such that the estimated orbital parameters are accurate enough for their intended application.
To quantify if the orbital parameters, estimated using sensor data, are sufficiently accurate, we define an application set. The application set consists of all parameters that, when used in the intended application, satisfy the performance criterion. It is possible to define unique application sets for each object to reflect different requirements or importance of different objects. The application set could for example be a confidence ellipsoid of the estimated parameters or described by a more general function of the orbital parameters. The sensor-planning problem can now be solved, simply put, by calculating the expected covariance of the estimated orbital parameters for each combination of possible observations, and verify whether they lie within the application set or not. The optimal solution is considered as the combination of observations that satisfies the application constraints with the lowest observation cost. However, the number of combinations to evaluate grows exponentially and this approach can only be used on small problems with a few objects and possible observations. Instead, we show how to approximate this combinatorial problem with a convex optimization problem.
To this end, we first show that, under certain conditions, the estimated orbital parameters lie within a confidence ellipsoid that we denote as the estimation ellipsoid. By approximating the application set by an ellipsoid, the problem of estimating the orbital parameters to be within the application set can hence be posed as the problem of fitting the estimation ellipsoid within the application ellipsoid. We then show that this constraint can be formulated as a linear matrix inequality, convex in the observation selection parameters. The cost function is the weighted sum of the observation selection parameters, using the defined observation cost. A nice property of the weighted sum, or weighted 1-norm, is that it often produces sparse solutions, that is, only a few of the possible observations will be selected. The complete optimization problem is a convex semidefinite program and can be solved efficiently, even for large problems.
The proposed method is demonstrated in a simulation example. The example illustrates how the method is used and aims to gain insight into the inner workings of the method.
As this paper presents early results of a novel way of formulating the sensor-planning problem, the last part of the paper focuses on discussions on the strengths and weaknesses of the method, and outlines directions for further research.
Date of Conference: September 27-20, 2022
Track: Optical Systems & Instrumentation