This work seeks to advance methods of spacecraft trajectory design to include elements of robust and reliability design. This refers to the incorporation of uncertainty in the trajectory design process as opposed to post-processing to ensure that mission requirements are satisfied. We leverage methods of Optimization Under Uncertainty (OUU) to perform this task. Some of the software developed through this research is used for intra-formation conjunction assessment for NASA’s Magnetospheric Multi-Scale (MMS) mission. Active areas of research include:

- Computationally efficient orbit state and uncertainty propagation
- Quantifying spacecraft risks (e.g., collision probabilities)
- Leveraging OUU in trajectory design
- Application to new and ongoing NASA and DoD missions

*Uncertainty Propagation*

The first step in leveraging OUU for trajectory design requires propagation of uncertainty. We are developing new methods of representing the uncertainty based on stochastic expansions such as polynomial chaos expansions and separated representations.

**Balducci, M.**,**B. A. Jones**, and A. Doostan, “Orbit Uncertainty Propagation and Sensitivity Analysis With Separated Representations”,*Celestial Mechanics and Dynamical Astronomy*, Vol. 129, Number 1-2, pp. 105-136, 2017.**Feldhacker, J. D.**, J. Smith,**Jones, B. A.**, and A. Doostan, “Multi-Element Trajectory Models for Satellite Tour Missions” AIAA/AAS Astrodynamics Specialist Conference, Long Beach, CA, September 13-16, 2016.**Jones, B. A.**, N. Parrish, and A. Doostan, “Postmaneuver Collision Probability Esti- mation Using Sparse Polynomial Chaos Expansions,”*Journal of Guidance, Control, and Dynamics*, Vol. 38, Number 8, pp. 1425-1437, 2015.**Balducci, M.**,**J. D. Feldhacker**, J. Smith,**B. A. Jones**, “Interplanetary Orbit Un- certainty Propagation Using Polynomial Surrogates”, AAS/AIAA Astrodynamics Specialist Conference, Vail, CO, August 9-13, 2015.**Balducci, M.**,**B.A. Jones**, and A. Doostan, “Orbit Uncertainty Propagation with Sep- arated Representations”, AAS/AIAA Astrodynamics Specialist Conference, Hilton Head, SC, August 11 – 15, 2013.- Wawrzyniak, G.G., J.R. Carpenter, D.J. Mattern, T.W. Williams, N.A. Ottenstein, and
**B.A. Jones**, “Conjunction Assessment Concept of Operations for the Magnetospheric Multi-Scale (MMS) Mission”, AAS/AIAA Astrodynamics Specialist Conference, Hilton Head, SC, August 11 – 15, 2013. **Jones, B. A.**and A. Doostan, “Satellite Collision Probability Estimation Using Polynomial Chaos,”*Advances in Space Research*, Vol. 52, Number 11, pp. 1860-1875, 2013.**Jones, B. A.**, A. Doostan, and G. H. Born “Nonlinear Propagation of Orbit Uncertainty Using Non-Intrusive Polynomial Chaos,”*Journal of Guidance, Control, and Dynamics*, Vol. 36, Number 2, pp. 415-425, 2013.

*Robust and Reliability Design*

This work leverages methods of uncertainty quantification for robust and reliability design. Such methods incorporate probabilistic quantities, e.g., variance of the posterior state Probability Density Function (PDF) and/or collision probability, into a given optimization problem.

**Balducci, M.**, and**B. A. Jones**, “Probability of Collision Estimation and Optimization Under Uncertainty Utilizing Separated Representations” AAS/AIAA Astrodynamics Specialist Conference, Stevenson, WA, August 20-24, 2017.**Feldhacker, J.**,**B. A. Jones**, A. Doostan, and J. Hampton, “Reduced Cost Mission Design Using Surrogate Models,”*Advances in Space Research*, Vol. 57, Number 2, pp. 588-603, 2016.**Feldhacker, J. D.**,**B. A. Jones**, and A. Doostan, “Trajectory Optimization Under Uncertainty for Rendezvous in the CRTBP” AAS/AIAA Space Flight Mechanics Meeting, Napa, CA, February 14-18, 2016.

*Multi-Fidelity Orbit State Propagation*

Recent work in orbit state uncertainty quantification focuses on the representation and propagation of the state uncertainty given a dynamics model. High-fidelity models are not require in certain scenarios, particularly in the case of orbital debris given sparse measurements. We are researching methods for autonomous selection of force models to satisfy accuracy requirements.

**Jones, B. A.**and R. Weisman, “Multi-Fidelity Orbit Uncertainty Propagation”*1st IAA Conference on Space Situational Awareness*, Orlando, FL, November 13-15, 2017.

*Asteroid Mitigation*

Collisions between asteroids and the Earth can pose a risk to human life. Asteroid mitigation refers to the prevention of such collisions, with several concepts proposed in the literature. In the case of a kinetic impactor, the resulting change in velocity varies with impact location, surface topography, surface materials, and other factors. This researched developed methods for quantifying the uncertainty resulting delta-V given such input uncertainties.

**Feldhacker, J. D.**, M. B. Syal,**B. A. Jones**, A. Doostan, J. McMahon, and D.J. Scheeres, “Shape Dependence of the Kinetic Deflection of Asteroids,”*Journal of Guidance, Control, and Dynamics*, Vol. 40, Number 10, pp. 2417-2431, 2017.**Feldhacker, J. D.**,**B. A. Jones**, A. Doostan, D. J. Scheeres, and J. W. McMahon, “Shape Dependence of Kinetic Deflection for a Survey of Real Asteroids,” AAS/AIAA Astrodynamics Specialist Conference, Vail, CO, August 9-13, 2015.**Feldhacker, J. D.**,**B. A. Jones**, A. Doostan, D. J. Scheeres, and J. W. McMahon, “Kinetic Deflection Uncertainties for Real Asteroid Shapes”, 4th IAA Planetary Defense Conference – PDC 2015, Frascati, Roma, Italy, April 13-17, 2015.

*Computationally Efficient Orbit Propagation*

While not a current focus of our research, previous work leveraged advancements in scientific computing for faster orbit state propagation. This includes parallelization and using changes in computer architectures (e.g., available memory) to reduce overall runtime for orbit state propagation.

- Bradley, B. K.,
**B. A. Jones**, G. Beylkin, K. Sandberg, and P. Axelrad, “Bandlimited Implicit Runge-Kutta Integration for Astrodynamics,”*Celestial Mechanics and Dynamical Astronomy*, Vol. 119, Number 2, pp. 143-168, 2014. **Jones, B. A.**, “Orbit Propagation Using Gauss-Legendre Collocation” AIAA/AAS As- trodynamics Specialist Conference, Minneapolis, MN, August 13 – 16, 2012.**Jones, B. A.**and R. L. Anderson, “A Survey of Symplectic and Collocation Integration Methods for Orbit Propagation,” 22nd Annual AAS/AIAA Space Flight Mechanics Meeting, Charleston, SC, January 29 – February 2 2012.**Jones, B. A.**, G. H. Born, and G. Beylkin, “Sequential Orbit Determination with the Cubed-Sphere Gravity Model,”*Journal of Spacecraft and Rockets*, Vol. 49, Num- ber 1, pp. 145-156, 2012.**Jones, B. A.**, G. Beylkin, G. H. Born, and R. S. Provence, “A Multiresolution Model for Small-Body Gravity Estimation,”*Celestial Mechanics and Dynamical Astronomy*, Vol. 111, Number 3, pp. 309-335, 2011.**Jones, B. A.**, G. H. Born, and G. Beylkin, “Comparisons of the Cubed-Sphere Gravity Model with the Spherical Harmonics,”*Journal of Guidance, Control, and Dynamics*, Vol. 33, Number 2, pp. 415-425, 2010.