Space collision probability computation based on on-board optical cues

Collision Probability with Gaussian Mixture Orbit Uncertainty

Fast and accurate collision probability computations are essential for protecting space assets. Monte Carlo (MC) simulation is the most accurate but computationally intensive method. A Graphics Processing Unit (GPU) is used to parallelize the computation and reduce the overall runtime. Using MC techniques to compute the collision probability is common in literature as the benchmark. An optimized implementation on the GPU, however, is a challenging problem and is the main focus of the current work. The MC simulation takes samples from the uncertainty distributions of the Resident Space Objects (RSOs) at any time during a time window of interest and outputs the separations at closest approach. Therefore, any uncertainty propagation method may be used and the collision probability is automatically computed as a function of RSO collision radii. Integration using a fixed time step and a quartic interpolation after every Runge Kutta step ensures that no close approaches are missed. Two orders of magnitude speedups over a serial CPU implementation are shown, and speedups improve moderately with higher fidelity dynamics. The tool makes the MC approach tractable on a single workstation, and can be used as a final product, or for verifying surrogate and analytical collision probability methods.

We propose a method for a fast and accurate estimation of the collision probability in a short window following the generation of a debris cloud. A multirevolution Lambert targeting problem is employed to determine all the locations where the collision probability is nonnegligible. Then, the Lambert problem and state transition matrix are used to obtain the image of the target body in the spread velocity space, where the collision probability computation is performed. To perform and control the accuracy of the integration of the collision rate needed to obtain collision probabilities, we apply high-order Taylor expansion and automatic domain splitting techniques. The method is validated using a Monte Carlo simulation and a literature-based test case. Test cases demonstrate the validity of the approach for Keplerian and perturbed dynamics.

Collision probability calculation is critical to space situational awareness or space traffic management. To determine the collision probability, the conjunction time needs to be determined first, but both collision probability and conjunction time usually exist large uncertainties. In this paper, a continuous representation of the polynomial chaos expansion (PCE)-based surrogate model is used to describe the uncertainty of space objects along with time. This PCE model facilitates the propagation of sampling points representing orbital uncertainty at any time. A machine learning-based Bayesian optimization method is employed to determine the conjunction time and then the collision probability. A data-driven Gaussian process regression model is built to approximate the Hausdorff distance, which is a distance metric between two space objects with uncertainty and is used as an objective function for Bayesian optimization. The proposed PCE model and Bayesian optimization do not require the Gaussian assumption and provide a general framework to calculate the conjunction time and collision probability. Two numerical experiments are used to show the effectiveness of the proposed algorithm and its close performance to the traditional Monte Carlo sampling-based method.

A new tool for conjunction analysis of ISRO's operational satellites, Close Approach Prediction Software: CLAPS

This paper presents a probabilistic threat assessment method for reasoning about the safety of robot trajectories in uncertain and dynamic environments. For safety evaluation, the overall collision probability is used to rank candidate trajectories by considering the collision probability of known objects as well as the collision probability beyond the planning horizon. Monte Carlo sampling is used to estimate the collision probabilities. This concept is applied to a navigation framework that generates and selects trajectories in order to reach the goal location while minimizing the collision probability. Simulation scenarios are used to validate the overall crash probability and show its necessity in the proposed navigation approach.

Collision probabilities for plate geometry cells with an approximate treatment of plate edge regions

About Part of Space Experiment in Solving Problem to Protect the Earth against Collision with Asteroid

d = cross-sectional radius of torus dz = subvolume height h = distance above x–y plane of torus P = collision probability PR = collision probability rate R = radius of torus r = polar coordinate in symmetrized encounter plane t = time Vz = in-track velocity in symmetrized encounter frame x, y, z = coordinates used in probability density function = contour integration parameter = three-dimensional probability density function = standard deviation in symmetrized encounter frame x;y;z = standard deviations of relative position uncertainty for each axis in diagonal frame

Airborne self-collisions occur primarily in military aircraft because of external stores and are frequently experienced by personnel operating these aircraft. In most cases, objects causing self-collisions are irregularly shaped and unstable. Consequently, the trajectories of these objects are uncertain. A framework for the probabilistic risk analysis of aircraft self-collisions is proposed in this study. Based on the probabilistic trajectory prediction model, methods for estimating the probability of collision (POC) and the corresponding risks were developed. Subsequently, a self-collision event involving an ejected gun shell was analyzed as a case study. A model considering random shell rotation, which continuously changes the drag characteristics and trajectories, was developed. Other uncertain factors associated with the aircraft and shell cases were considered. The most influential factors were selected based on the sensitivity analysis and were then used to calibrate the likelihood of the event using historical data. A Monte Carlo simulation, in conjunction with the probabilistic ballistic model, was performed to evaluate the POC. The POC was used to reflect the risk of engine failure up to the operational limit. The calculated risk indices were objective functions used for the design or operation optimization. Various risk measures were evaluated to reduce the incidence of failure and extend the aircraft’s flight envelope.

Bayesian model updating provides a rigorous framework to account for uncertainty induced by lack of knowledge about engineering systems in their respective mathematical models through updates of the joint probability density function (PDF), the so-called posterior PDF, of the unknown model parameters. The Markov chain Monte Carlo (MCMC) methods are currently the most popular approaches for generating samples from the posterior PDF. However, these methods often found wanting when sampling from difficult distributions (e.g., high-dimensional PDFs, PDFs with flat manifolds, multimodal PDFs, and very peaked PDFs). This paper introduces a new multi-level sampling approach for Bayesian model updating, called Sequential Gauss-Newton algorithm, which is inspired by the Transitional Markov chain Monte Carlo (TMCMC) algorithm. The Sequential Gauss-Newton algorithm improves two aspects of TMCMC to make an efficient and effective MCMC algorithm for drawing samples from difficult posterior PDFs. First, the statistical efficiency of the algorithm is enhanced by use of the systematic resampling scheme. Second, a new MCMC algorithm, called Gauss-Newton MCMC algorithm, is proposed which is essentially an M-H algorithm with a Gaussian proposal PDF tailored to the posterior PDF using the gradient and Hessian information of the negative log posterior. The effectiveness of the proposed algorithm for solving the Bayesian model updating problem is illustrated using three examples with irregularly shaped posterior PDFs.

The decision for the International Space Station (ISS) to maneuver to avoid a potential collision with another space object will be based on the probability of collision, P C. The calculation of P C requires the covariance of both objects at conjunction. It is well known that the covariance computed by US Space Command is optimistic (too small), especially at altitudes where atmospheric drag is the dominant perturbation, because its computation assumes there are no dynamic model errors. In this paper the effect of errors in the covariance on P C and the sensitivity of P C to the encounter geometry are investigated.

A new method for the approximation of multivariate scalar probability density functions (PDFs) in turbulent reacting flow by means of a joint presumed discrete distribution (jPDD) is presented. The jPDDs can be generated with specified mean values and variances as well as covariances. Correlations between variables – e.g. fluctuating mixture fractions and/or reaction progress – can thereby be taken into account. In this way the new approach overcomes an important limitation of ordinary presumed PDF methods, where statistical independence between the variables is often assumed. Different methods are presented to generate discrete distributions, based either on biased random number generators or on mixing models familiar from PDF transport models. The new approach is extensively validated on a turbulent flow configuration with simultaneous mixing and reaction. Large eddy simulation data as well as results from a transported PDF model are used for the validation of the jPDD approach. The comparison shows that in particular distributions generated with mixing models are able to predict mean reaction rates accurately. For the configuration considered, the neglect of correlations results in significant underestimation of reaction rates. Moreover it is found that higher statistical moments (e.g. the skewness) can influence reaction rates. The consequences for the generation of jPDDs are discussed. In summary, the new jPDD model has the potential to be significantly more accurate than established presumed PDF methods, because correlations between fluctuating variables can be taken into account. At the same time, the new approach is nearly as efficient as standard presumed PDF formulations, since mean rates are computed in a pre-processing step and stored in look-up tables as a function of the first and second moments of the relevant variables.

The explicit expression of the collision probability is deduced under the assumption that the orbit is a circle. The collision probability is expressed as an explicit function of the encounter geometry (crossing altitude difference and time difference of the line of intersection of the two orbital planes, orbital planes included angle, etc) and position error variance in RSW coordinates. With the help of the explicit expression, the influencing factors of the collision probability were analyzed. Effects of factors such as altitude difference, time difference, orbital planes included angle, position predicted error variances and the equivalent radius of two objects were discussed. Some significant conclusions were obtained.

The influence of the concatenation of arbitrary optical multiplexers/demultiplexers (MUX/DEMUXs) on the probability density function (PDF) of the output current of a transparent optical network is assessed. All PDF results obtained analytically are compared with estimates from Monte Carlo simulation and an excellent agreement is achieved. The non-Gaussian behavior of the PDFs, previously reported by other authors for square-law detectors, is significantly enhanced with the number of nodes increase due to the noise accumulation along the cascade of MUX/DEMUXs. The increase of the MUX/DEMUXs bandwidth and detuning also enhances the PDFs non-Gaussian behavior. The PDF shape variation with the detuning depends strongly on the number of nodes. Explanations for the Gaussian approximation (GA) accuracy on the assessment of the performance of a concatenation of optical MUX/DEMUXs are also provided. For infinite extinction ratio and tuned MUX/DEMUXs, the GA error probabilities are, in general, pessimistic, due to the inaccurate estimation of the error probability for both bits. For low extinction ratio, the GA is very accurate due to a balance between the error probabilities estimated for the bits “1” and “0.” With the detuning increase, the GA estimates can become optimistic.