Negative Collision Measurements in Position/Velocity State Estimation

A VARIETY of aerospace applications such as orbital rendezvous, powered descent to the surface of a celestial object, aircraft landing, and missile intercept require some form of a desired collision. Other aerospace applications such as orbital satellite inspection, formation flying, and air traffic control require collision avoidance. A key element of each of all these applications is position/velocity state estimation. Because a collision can often times be detected with a simple inexpensive accelerometer, this Note considers the addition of an accelerometer-based collision measurement to the state estimation process and provides evidence that the information associated with a negative collision measurement (i.e., a collision has not occurred) can provide useful state information and reduce state uncertainties. A positive collision measurement (e.g., an abrupt change in an accelerometer output) will clearly provide some useful state information. During aircraft landing or powered descent, a positive collision measurement provides information about the altitude of the vehicle. During orbital rendezvous, a positive collision measurement provides information about the relative position of two spacecraft and indicates, at aminimum, that the two vehicles are in contact with each other. Although a positive collision measurement clearly has value, this is not the topic of this Note. The absence of an abrupt change in acceleration, a negative collision measurement, is the topic of this Note. Because a negative collision is the absence of a measurement, one may initially believe that it cannot provide any useful information. However, consider the position covariance of an aircraft during landing. As the aircraft approaches the surface, the equiprobability ellipsoid begins to extend into the ground. A negative collision measurement, if processed properly, should eliminate the portion of the ellipsoid below the surface from consideration, move the mean or expected value of the altitude up away from the surface, and produce a smaller altitude error covariance. Consider also the relative position error covariance ellipsoid associated with orbital rendezvous. In this case, a negative collision measurement should eliminate the portion of the error ellipsoid associated with the position and size of the spacecraft from consideration, move themean position away from the spacecraft, and again produce a smaller position error covariance. If subsequent negative collision measurements are processed properly, the effect of coupled dynamics and state correlations may reduce the uncertainly further. Because a negative collision measurement is the absence of a measurement, it can be considered negative information. Applications of negative information have been successfully incorporated and used in the estimation process. Koch [1] and Blanding et al. [2] investigated several radar tracking problems in which information associated with range and bearing measurements, aswell as the absence of expected range and bearingmeasurements in jammed environments or low sensor resolution environments, is used to maintain better track of targets. Tishler and Vogt [3] also successfully used negative information in a cooperative multivehicle environment. They used the absence of expected radarmeasurements to aid in the collaborative perception of the environment. In both of these applications, negative information was derived from the absence of an expected traditional range and bearing measurement. The negative collision measurements considered in this Note may also be considered negative information, but here themeasurement is a nontraditional binary measurement (positive when a collision occurs and negative if a collision has not occurred), as opposed to the traditional range and bearing measurements considered in [1,3]. Although positive measurements will obviously provide very useful information, they will not be considered in this Note. The objective of this Note is to conduct a preliminary assessment of the usefulness of negative collision measurements and determine whether or not negative collision measurements can provide real and useful information for improved position/velocity state estimation. In the future, a more detailed analysis will be required to fully assess the viability of negative collision measurements in real-time aerospace applications. Because a collision measurement is a binary (positive or negative) random variable, a nonlinear sequential Monte Carlo “bootstrap filter” will be employed. A review of the bootstrap filter is provided next in Sec. II. In Sec. III, the bootstrap filter is applied to a constrained pendulum problem where collisions with side walls are possible. More complex satellite inspection and orbital rendezvous problems will be considered in Secs. IVand V. Final conclusions and remarks are presented in Sec. VI.