Research Summary

Below is a summary of research projects I have worked on over the years (as an undergraduate and graduate researcher). Relevant papers, posters, notes, and slides are provided for each respective project. Please email me with any comments, questions, or corrections.

Spacecraft Attitude Control under Thrust Uncertainty

The motivation for this project arose in experimental studies looking at relationship between thrust magnitude and propagation of uncertainty in the produced moment. Other motivations included the European Space Agency’s LISA Pathfinder mission which required small tolerances on the produced thrust error due to the objective of the mission.
The first phase of the project was focused on deriving the stochastic nonlinear optimal control for the rigid body rotational dynamics. The uncertainty in the rotational equations of motion was modeled through stochastic differential equations in control multiplicative form where the noise was multi-dimensional Brownian motion. To derive the optimal control, approximations of the optimal value function and consequently of the optimal control in the ring of formal power series over ℝ were found. This series-based technique is due to Al’brekht’s work in 1961 where it has also been studied and extended by other mathematicians in the past two decades. Due to the noisy nature of our application, the Hamilton-Jacobi-Bellman’s partial differential equation inherited an extra diffusive term which needed to be accounted for when formulating the Al’brekht’s method. Shown in [1], the rigid body rotational equations of motion were stabilized using the discussed method.
The second phase of the project included the addition of the attitude kinematics which extended the state space to 6 dimensions. The kinematics were parametrized using the parametrization introduced by Tsiotras et al. which allowed the linearization of the dynamics for control derivation. The singularity properties of this parametrization are desirable and are discussed in [2]. In the same work, the existence and uniqueness of the linear control as well as the nonlinear control under certain conditions were proven. The Monte Carlo simulations in [2] showed the desirable control-energy saving properties of the stochastic nonlinear controller. Reference [2] also provides the conditions for which the stochastic nonlinear controller outperforms the linear quadratic regulator. As a future direction of research, I am interested in a similar analysis for underactuated attitude systems, as well as online efficient methods that would extend the local control to a larger domain in the state-space.

References

[1] Golpashin, A. E., Yeong, H. C., Ho, K., & Namachchivaya, N. S. (2018). Stochastic attitude control of spacecraft under thrust uncertainty. In AAS/AIAA Astrodynamics Specialist Conference, 2017 (pp. 205-218). Univelt Inc..
[2] Golpashin, A. E., Yeong, H. C., Ho, K., & Sri Namachchivaya, N. (2020). Spacecraft attitude control: a consideration of thrust uncertainty. Journal of Guidance, Control, and Dynamics, 43(12), 2349-2365.

Related Resources

Practical Methods for Global Solutions of High-dimensional Hamilton-Jacobi-Bellman Equation

This project is currently being investigated and is in its preliminary stages. In this study, the global solutions of the high-dimensional Hamilton-Jacobi-Bellman (HJB) partial differential equation (PDE) associated with stochastic (or deterministic) differential equations are to be studied. Through methods of averaging (homogenization) the HJB equation will be reduced to a lower dimensional PDE. The viscosity (or classical) solution of the HJB is to be solved using an accurate and efficient grid-based method. This study aims to greatly reduce the space complexity of the problem by providing a practical framework for computing the HJB solution for feedback optimal control problems when the state-space is large, and the dynamics may be noisy. Several works have previously aimed to alleviate the curse of dimensionality by carefully choosing the grid topology as well as the interpolation algorithms, i.e. see references [1] and [2]. The practical goal of this study is to make the HJB formulation of high dimensional models such as the ones discussed in [3] feasible.

References

[1] Chilan, C. M., & Conway, B. A. (2020). Optimal nonlinear control using Hamilton–Jacobi–Bellman viscosity solutions on unstructured grids. Journal of Guidance, Control, and Dynamics, 43(1), 30-38.
[2] Kang, W., & Wilcox, L. C. (2017). Mitigating the curse of dimensionality: sparse grid characteristics method for optimal feedback control and HJB equations. Computational Optimization and Applications, 68(2), 289-315.
[3] Gugercin, S., Antoulas, A. C., & Bedrossian, N. (2001, December). Approximation of the International Space Station 1R and 12A models. In Proceedings of the 40th IEEE Conference on Decision and Control (Cat. No. 01CH37228) (Vol. 2, pp. 1515-1516). IEEE.

Non-Gaussian Data Assimilation for Volumetric Network Anomaly Detection

This project is an ongoing effort which I am currently working on. As it has been shown and proven in the literature, network dynamics are best modeled by heavy tailed stochastic processes (as opposed to Gaussian ones [1][2]). The idea behind this project is to explore the methods of non-Gaussian filtering to improve detection in volumetric (i.e. DDoS) cyber anomalies. For simulation purposes, network process is artificially generated using a chaotic map model [3] which exhibits the realistic network properties such as LRD, self-similarity, and impulsiveness. The filter assumes an autoregressive form measuring the dependence of history data points to the current value of the time series. To the best of my knowledge, very few works have been focused on studying methods of impulsive Kalman fileting with non-symmetric (skewed) α-stable error distributions. Moreover, the α parameter has been kept the same in the prior and posterior terms. For a good example, see reference [4].

References

[1] Willinger, W., Paxson, V., & Taqqu, M. S. (1998). Self-similarity and heavy tails: Structural modeling of network Tra c. R. Adler, R. Feldman, and MS Taqqu, editors, A Practical Guide to Heavy Tails: Statistical Techniques for Analyzing Heavy Tailed Distributions, Birkhauser Verlag, Boston.
[2] Willinger, W., Govindan, R., Jamin, S., Paxson, V., & Shenker, S. (2002). Scaling phenomena in the Internet: Critically examining criticality. Proceedings of the National Academy of Sciences, 99(suppl 1), 2573-2580.
[3] Arrowsmith, D. K., Barenco, M., Mondragon, R. J., & Woolf, M. (2004, July). The statistics of intermittency maps and dynamical modeling of networks. In 6th Int. Symp. on Mathematical Theory of Networks and Systems, Leuven, Belgium.
[4] Sornette, D., & Ide, K. (2001). The Kalman Lévy filter. Phys. D, 151(2-4), 142174.

Related Resources

Summer 2021 presentation

Optimized Rendezvous of a Quadrotor

This was an undergraduate robotics project I worked on for about two years. At the time, the lab I was working with was interested in autonomous docking and undocking of quadrotors (UAVs) for applications in construction sites [1]. I and Mihir Patel were tasked with developing a path planning algorithm to minimize the time-to-dock while increasing the probability of docking (the Markov chain figure below). The motivation for the proposed algorithm was drawn from the work of Prof. Bruce Donald in Error Detection & Recovery [2]. My contribution was to develop an algorithm which included two additive quantities as its cost functional: one, minimizing the shortest distance between the docking point, and the other increasing the curvature of the path. The increase in curvature resulted in the desired robustness when disturbances were present. This path planning algorithm was then used along with the PID controller for flight control. Through Monte Carlo simulations and test flights, the robustness of the algorithm was shown. The task of increasing the probability of docking was accomplished as docking was made possible from any initial condition in the vicinity of the docking port.

References

[1] DeGol, J., Hanley, D., Aghasadeghi, N., & Bretl, T. (2015, September). A passive mechanism for relocating payloads with a quadrotor. In 2015 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) (pp. 4337-4342). IEEE.
[2] Donald, B. R. (1987). Error detection and recovery in robotics. Springer-Verlag.