Richard Linares

University of Minnesota

205B Akerman Hall

110 Union St. SE

Minneapolis, MN 55455-0153

Email:rlinares(at)umn.edu

Phone: 612-625-2540

Fax: 612-626-1558

Mail: 117C Akerman Hall

Although the general public may not realize it, modern life is deeply reliant on satellite technology. For example, one such satellite technology is the Global Positioning System (GPS) used for navigation (land, water, and air), for synchronize banking transactions worldwide, our smart phones, and the internet as well as the power grid. Interruption to the GPS’s accessibility could cause catastrophic effects on energy distribution networks and financial markets. Satellites are indispensable for modern life and this need motivates the development of methods for protecting them from threats that would upset their proper functioning.

One threat that is becoming a concern is space debris. Presently, there are more than half a million debris objects orbiting the Earth. Even the smallest debris objects such as paint flakes, traveling at seven times the speed of a bullet, can collide with active satellites and cause severe damage or completely destroy them. As more debris is created from collisions, the density of debris in orbit could reach and pass a threshold, where random collisions between debris objects could cause a cascade, otherwise known as the Kessler syndrome. The Kessler syndrome could leave much of space inaccessible for many generations due to the orbital debris hazard.

In order to avoid collisions, it is imperative for us to know to the best of our abilities where debris objects presently are, and where they will be in the future. Predicting collisions requires knowledge of multiple physical parameters and models that affect the path of these objects. In general, these parameters and models are impossible to know precisely. Therefore, we must characterize the limits of our knowledge (i.e. the uncertainties) mathematically, and understand the impact of these limits on making predictions of the possible or likely locations of debris objects. The ability to understand uncertainties allows us to calculate the likelihood or the probability of a collision between objects in space.

The objectives of our work are the development of uncertainty quantification methods, high fidelity debris models, and the most detailed orbital debris catalog to date. Out work focuses on developing new methods for understanding and forecasting uncertainty applied to the orbital debris problem. The approach we use is based on Polynomial Chaos, a mathematical method for handling nonlinear problems in a very computationally efficient way. Our work focuses on developing and applying this method to understanding positional and environmental uncertainties for the orbital debris population.

Building the most detailed catalog requires observations. Therefore, we are also building an optical telescope network with sites at Australia’s Royal Melbourne Institute of Technology and University of Arizona. With these two telescopes, the network will have large global coverage for telescope-based orbital debris tracking. Figure to the left shows one of the first debris objects observed with our telescope at Arizona.

- Kessler, D. J., et al. “Collision frequency of artificial satellites.” J. Geo. Research (1978).

Operations of unmanned aerial vehicles (UAV) in a dense urban environment has always been a challenging issue due to the presence of multiple moving and stationary objects (obstacles). The UAV can pose a safety hazard to itself as well as to its environment. For the UAV to be able to safely navigate itself in a dense and dynamic environment, it needs to be able to “see” and understand what is happening around it. In other words, the UAV needs to be able to detect and track these obstacles that might pose a collision hazard and reroute its trajectory as it sees fit.

Traditionally, detecting and tracking of targets are done using visual cues, either static images taken at specific interval or videos. Targets are isolated at each time step using detection algorithm and then tracked across multiple time step to generate its probable trajectory in the near instance. However, getting the range and dimensions properties of these targets in not straight forward and can be computationally heavy.

With the recent popularity and advancement in the Light Detection and Ranging (LiDAR) scanner, it is now possible to equip a puck size LiDAR scanner aboard an UAV and obtain direct range measurement of objects in the surroundings. LiDAR scanner emits a laser pulse to the environment and measure the time of flight for the laser to return back to the sensor after being reflected by the surrounding objects. Working with LiDAR scanner poses a different set of issues, such as each target can produce multiple returns (in the range of 100s depending on the LiDAR sensor resolution and proximity to the scanner) and extended object tracking (the detected shape of the same object changes depending on the viewing angle from the scanner).

Our research aims to allow the UAV to detect and track both moving and stationary targets in the environment using LiDAR data fused with traditional visual cues, and achieve fully autonomous flight with collision avoidance in the near future. Attached below are some preliminary results that we have obtained from using probability hypothesis density (PHD) filter on LiDAR data.

Precise modelling of uncertainties is essential to obtaining an accurate estimate of a spacecraft’s future state by propagating its known initial states. These error estimates of the future state are vital for advancements in the field of space situational awareness, as they enable us to make informed decisions on tracking, updating and avoiding collision risks. In our work, we use Gaussian Mixture Models (GMMs) and Polynomial Chaos Expansion (PCE) in a novel and hybrid manner to efficiently capture complete shape of the true non-Gaussian distribution using low-order polynomials to achieve a desired accuracy.

The Gaussian Mixture Model-Polynomial Chaos (GMM-PC), first converts the initial distribution into a GMM with means of the weighted Gaussian distribution along most influential directions. Splitting the initial distribution into a GMM reduces the size of the covariance associated with each new element. Then, the PCE is used to propagate the state uncertainty represented by each of the elements through the nonlinear dynamics.

The initial GMM split helps reduce the domain of approximation thereby allowing for lower-order polynomials to be used. This method has shown to be effective in cases when the PCE method doesn’t converge. The GMM-PC is an alternate form of the multi-element polynomial chaos expansion. For extending a PCE implementation to the GMM-PC, only the weights, means, and standard deviations of a univariate splitting library are required, which are previously archived and stored in a tabular manner.

- Vittaldev, V., Russel, R., and Linares, R. Spacecraft Uncertainty Propagation using Gaussian Mixture Models and Polynomial Chaos Expansions, AIAA Journal of Guidance, Control, and Dynamics, 2016.
- Jones, B. A., Doostan, A., and Born, G. H., “Nonlinear Propagation of Orbit Uncertainty Using Non-Intrusive Polynomial Chaos,” Journal of Guidance, Control, and Dynamics, Vol. 36, No. 2, 2013, pp. 430–444. doi:10.2514/1.57599.
- Vittaldev, V., and Russell, R. P., “Uncertainty Propagation Using Gaussian Mixture Models,” Proceedings of the SIAM Conf. on Uncertainty Quantification, Savannah, GA, March–April 2014.
- Wan, X., and Karniadakis, G. E., “An Adaptive Multi-Element Generalized Polynomial Chaos Method for Stochastic Differential Equations,” Journal of Computational Physics, Vol. 209, No. 2, 2005, pp. 617–642.

We are developing a new methodology based on proper orthrogonal decomposition (POD) with a vision of a quasi-physical, predictive, reduced order model that combines the speed of empirical and the predictive/forecasting capabilities of physics-based models. The methodology is developed to reduce the high-dimensionality of physics-based models while maintaining its capabilities. The approach uses data-driven modal analysis to extract the most energetic modes of variation for neutral thermospheric density using proper orthogonal decomposition, where the time-independent modes or basis represent the dynamics and the time-depedent coefficients or amplitudes represent the model parameters. We are working towards incorporation of a dynamic systems approach for modeling of the model coefficients.

The video above shows an intuitive simulation of the POD modes combining to reproduce the true density distribution. For more information, please read the article below.

- Mehta, P.M., and Linares, R. A methodology for reduced order modeling and calibration of the upper atmosphere, AGU space weather, 2017.

Professor Linares, in collaboration with Professor Seiler and ASTER Labs, Inc are developing innovative technology for Satellite Swarm Localization And Control Via Random Finite Set Statistics. It demonstrates a new approach to perform real-time relative vehicle localization within a swarm formation with application to communication-less coordination. Each vehicle estimates its location in the formation by tracking the vehicles around it. The algorithms make it possible to localize indistinguishable vehicles within a formation, without communication, using a probabilistic modeling of the swarm to account for low measurement accuracy, and uses probabilistic models for control. All of this is achieved by using Random Finite Sets statistics theory to solve the problem of multiple object tracking.

The ability to control the movement and track the location of a group of objects provides a significant advantage. Whether it be used to track and control a group of aircraft drones as they survey a crop, to alleviate the infamous congestion of rush hour by coordinating movement of cars and trucks, or to accurately direct nanobots as they work together during an otherwise complicated surgery, this technology offers a new approach to challenges across many diverse fields. In the field of aerospace especially, this innovation allows for better mission precision and reduced risk. Safety is increased for the same price of better mission results in both Earth environments and space.

NASA applications consist of enabling autonomous precision swarm coordination for satellites traveling in Earth orbit or eventually into deep space, including greater precision for vehicle control. The swarm formation coordination and control algorithms and software will provide expanded mission planning and analysis capabilities, reduction of communication requirements, and reduction of mission risk. Non-NASA applications for this technology include increased coordination and control for units of multiple unmanned aerial systems performing search and rescue operations, for the Department of Homeland Security and other government agencies or local municipalities. Robotic or autonomous land, sea, and air vehicle coordination for the Department of Defense, and reduction of communication and relay requirements is an added application. Commercial telecommunication satellite providers that desire to transmit large data rate information between multiple vehicles, such as imaging or internet-like inter-satellite networks, could realize the formation control benefits through this enabling technology.

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