Monte Carlo Methods and Skewed Kalman Filters for Orbit Estimation Louis Tonc,∗ David Geller,y Geordie Richardsz April 20, 2018 Abstract in state space, far from the Gaussian class, and results in a wide discrepancy between the uncertainty of the filter Under reasonable orbital conditions, the Extended algorithm and the measurement noise. Kalman Filter (EKF) and Unscented Kalman Filter (UKF) both diverge after several updates from optical measure- POSITION (LVLH) 400 ments. We identify the source of divergence by studying Downrange (filter) a simplified model, and correct this problem by imple- 300 Downrange (monte carlo) menting a Monte Carlo based particle filter. However, the 200 computational cost of the particle filter is high, and future 100 work will implement a Skewed Unscented Kalman Filter 0 as a substitute for the particle filter. We present evidence that this skewed unscented Kalman filter will avoid diver- Position (km) -100 gence at a reduced cost compared to the particle filter, and -200 discuss Monte Carlo methods for validating this claim. -300 -400 0 1 2 3 4 5 6 7 8 9 1 Introduction Time (hours) The increasing abundance of orbital debris in Geostation- Figure 1: EKF Divergence in Downrange Position ary Orbit (GEO) represents a significant challenge for [LEO Orbit & Position Measurement at t = 3.75 hr] blue curve = filter variance, red curve = true variance the future of space travel and satellite operation. Avoid- ing collision events with high probability in real time will require orbit estimation algorithms that can utilize In particular, when tracking a debris object in GEO, sparse observations while maintaining computational ex- if optical measurements from a ground or space based pediency [2]. For example, limited resources require observer have long durations between them, standard tracking of objects in GEO with long time increments Kalman filters such as the Extended Kalman Filter (EKF) between observations. This long time increment allows and Unscented Kalman Filter (UKF) may diverge, incor- for nonlinear dynamics to significantly deviate an initially rectly predicting the variance of the underlying distribu- Gaussian distribution, which estimates the debris location tion. This phenomenon can be illustrated (see Figure 1) through a simplified example of an object propagating ∗Ph.D. student, Department of Mechanical and Aerospace Engineer- with small process noise in a Low Earth Orbit (LEO - ing, Utah State University the low altitude amplifies nonlinear effects), with slightly y Associate Professor, Department of Mechanical and Aerospace En- noisy observations of the full position vector taken at a gineering, Utah State University zAssistant Professor, Department of Mechanical and Aerospace En- time of 3.75 hours. Figure 1 illustrates that, after the first gineering, Utah State University observation, the EKF predicts a downrange variance that is too small. Modified Kalman filters, such as the second tween two measurements the nonlinearity of the dynamics order EKF and UKF exhibit the same divergent behavior begins to make more of an impact. The equation for prop- for this simplified LEO model. agation of the covariance matrix fPig used in the EKF, Monte Carlo methods, such as the bootstrap particle called the discrete Ricatti equation, is given by filter, can produce detailed non-Gaussian statistics evolv- P = Φ P ΦT ; (1) ing under Keplerian dynamics and incorporating nonlin- i+1 i i i ear measurements, but these algorithms are more expen- where Φi is the state transition matrix of the dynamical sive than Kalman filter options. Moreover, methods such system. Equation (1) is based on a linear approximation as the EKF are accompanied by a priori estimates of er- for the state dynamics. When a measurement is available ror through linear covariance analysis, which are absent to process, the Kalman gain when using particle filters. T T −1 In an effort to understand the source of the EKF fail- K = PH (HPH + R) (2) ure, this divergence is observed to occur when the dis- is computed, and used to update the filter mean and co- tribution of the underlying truth has non-zero skewness, variance, which cannot be represented by a Gaussian approxima- tion. Moreover, we observe that when the covariance ma- P + = (I − KH)P − (3) trix propagated by a particle filter is used to update the x^+ =x ^− + K(~z − z^); (4) EKF when a measurement is available, the divergence is removed completely. In this way, the divergence appears where H is the measurement geometry matrix, R is the due to skewed non-Gaussian statistics of the truth after a covariance matrix of the measurement noise, and z~ is the long duration of dynamical evolution, not due to the struc- measurement obtained. ture of the Kalman update itself. For short durations between measurements, the linear The skewed Kalman filter [4] is a recursive filter for approximation behind (1) is accurate, and yields a covari- tracking statistics through observation, which replaces ance matrix such that when implemented into calculat- the Gaussian approximation of the Kalman filter with a ing the Kalman Gain matrix, the mean and covariance are Closed Skew Normal (CSN) approximation. The CSN correctly updated after a measurement. However, when distribution generalizes the normal distribution to a larger the duration between measurements is long, equation (1) class which permits the consideration of non-zero skew- loses fidelity, and the propagated covariance matrix is not ness. The skewed Kalman filter can track a distribution sufficiently accurate. When this happens, the updated with non-zero skewness while maintaining the computa- mean value can be moved farther from the truth and the tion expediency of a Kalman filter, and has suitable al- covariance matrix can become overly confident about the ternative formulations for our purposes, such as an un- wrong mean value. As we will see below, this leads to scented version for incorporating nonlinear dynamics [5]. skewed filter statistics after the measurement update, and Implementation of an unscented skewed Kalman filter subsequent filter divergence. for our problem will be the subject of a future paper. Figure 1 shows a filter error plot with a single measure- Here we remark on challenges inherent to validating this ment taken at 3.75 hours, after which the filter diverges. scheme, namely the implementation of Markov Chain The cause of this divergence is demonstrated in Figure 2, Monte Carlo (MCMC) techniques for the purpose of sam- which shows a shift in filter statistics immediately after pling from a closed skew normal distribution. the measurement update such that estimation of the state becomes skewed in the velocity component. Since the measurement model is a full position vector observation, 2 Divergence of the Kalman Filter the statistics for the position vector behave as expected post measurement, however the velocity vector becomes The Extended Kalman Filter has traditionally been the de- skewed. fault filtering algorithm used in tracking and navigation Through careful simulated experiments of the model for orbital objects, however, if enough time is elapsed be- presented in Figure 1, we have observed that divergence 2 of Kalman-type filters within this context, which is caused by skewness of the filter statistics of the velocity post- measurement (as evidenced in Figure 2), is dependent on the time difference between measurements. More pre- cisely, we have observed that this divergence occurs when the measurement is taken at a time when the underlying true distribution, which can be carefully tracked using a large collection of simulated trajectories, exhibits skew- ness itself. Although it is difficult to infer visually, the statistics computed from the scatter plot of simulated tra- jectories at a measurement time that leads to divergence, depicted in Figure 3, are skewed. This is supported by Figure 2: Filter Error Following Measurement Update Time of Measurement = 3.75 hours Figure 4: Skewness of Truth State as Function of Time Figure 4, which plots the skewness of the true distribution as a function of time. Note that, although varying peri- odically, the amplitude of the skewness is increasing over time. Indeed, the EKF effectively approximates the true statistics after measurement update, and avoids diver- gence, if the measurement is taken at a time when the true distribution is nearly Gaussian (with zero skewness). Fig- ure 5 depicts a convergent EKF when the measurement is taken at a shorter time with underlying statistics that are approximately normal. This motivates us to consider modified filtering schemes which can track a distribution Figure 3: Truth Statistics prior to Measurement of non-zero skewness evolving under Keplerian dynam- Update Time of Measurement = 3.75 hours ics. To demonstrate amplification of skewness of filter ve- locity statistics after a measurement from skewed true statistics, Figure 6 depicts the skewness of filter statistics as a function of time, where the first measurement in the 3 model is taken at that particular time. Clearly, skewness 3 Remediation Using a Particle Filter in the filter velocity statistics is exacerbated the longer the duration between measurements. We have observed that divergence is not due to the struc- The commonly used alternative filtering algorithm for ture of the Kalman update methodology, but rather the handling nonlinear dynamics, the UKF, also diverges propagation of the mean and covariance using linear ap- when updated at these measurement times. The problem proximations. Particle filters require generating a large with the UKF is that it reassigns a Gaussian approxima- number of state vectors fxig sampled from initial condi- tion at each time step, and error in the covariance ma- tions, referred to as particles. These particles are propa- trix can still accumulate over long time intervals between gated forward in time and the filter statistics are the sam- measurements. The breakdown of traditional Kalman fil- ple statistics computed from these particles, namely the tering methods is the motivator to obtain some alternative sample mean means to acquire proper statistics.