Rutgers, The State University of New Jersey tugraz Halo Orbit Stationkeeping using Nonlinear Outline Introduction MPC and Polynomial Optimization Circular Restricted Three Body Problem Gaurav Misra, Hao Peng, and Xiaoli Bai Global Polynomial Optimization Rutgers, The State University of New Jersey Polynomial MPC Numerical Results Conclusions 28th AAS/AIAA Spaceflight Mechanics Meeting AIAA Scitech 2018, Kissimmee, Florida 10/01/2018 Gaurav Misra, Hao Peng, and Xiaoli Bai Halo Orbit Stationkeeping using Nonlinear MPC and Polynomial Optimization 1 / 31
Rutgers, The State University of New Jersey tugraz Outline Outline Introduction Circular Restricted Three Body Introduction and motivation Problem Problem Formulation Global Polynomial Motion in restricted three body problem Optimization Global polynomial optimization and MPC Polynomial MPC Numerical results Numerical Results Conclusion and future work Conclusions Gaurav Misra, Hao Peng, and Xiaoli Bai Halo Orbit Stationkeeping using Nonlinear MPC and Polynomial Optimization 2 / 31
Rutgers, The State University of New Jersey tugraz Outline Outline Introduction Circular Restricted Three Body Introduction and motivation Problem Problem Formulation Global Polynomial Motion in restricted three body problem Optimization Global polynomial optimization and MPC Polynomial MPC Numerical results Numerical Results Conclusion and future work Conclusions Gaurav Misra, Hao Peng, and Xiaoli Bai Halo Orbit Stationkeeping using Nonlinear MPC and Polynomial Optimization 2 / 31
Rutgers, The State University of New Jersey tugraz Outline Outline Introduction Circular Restricted Three Body Introduction and motivation Problem Problem Formulation Global Polynomial Motion in restricted three body problem Optimization Global polynomial optimization and MPC Polynomial MPC Numerical results Numerical Results Conclusion and future work Conclusions Gaurav Misra, Hao Peng, and Xiaoli Bai Halo Orbit Stationkeeping using Nonlinear MPC and Polynomial Optimization 2 / 31
Rutgers, The State University of New Jersey tugraz Outline Outline Introduction Circular Restricted Three Body Introduction and motivation Problem Problem Formulation Global Polynomial Motion in restricted three body problem Optimization Global polynomial optimization and MPC Polynomial MPC Numerical results Numerical Results Conclusion and future work Conclusions Gaurav Misra, Hao Peng, and Xiaoli Bai Halo Orbit Stationkeeping using Nonlinear MPC and Polynomial Optimization 2 / 31
Rutgers, The State University of New Jersey tugraz Outline Outline Introduction Circular Restricted Three Body Introduction and motivation Problem Problem Formulation Global Polynomial Motion in restricted three body problem Optimization Global polynomial optimization and MPC Polynomial MPC Numerical results Numerical Results Conclusion and future work Conclusions Gaurav Misra, Hao Peng, and Xiaoli Bai Halo Orbit Stationkeeping using Nonlinear MPC and Polynomial Optimization 2 / 31
Rutgers, The State University of New Jersey tugraz Background Outline m 1 = 1.00 m 2 = 0.10 (non-dimensional units) Introduction Circular Restricted 1 L4 Three Body Problem 0 L3 L1 L2 y Global Polynomial L5 -1 Optimization Polynomial -1.5 -1 -0.5 0 0.5 1 1.5 MPC x Numerical Results Conclusions Libration points: Ideal locations for human/robotic space exploration. Several successful past missions: ISEE-3, SOHO. Active station-keeping required. Gaurav Misra, Hao Peng, and Xiaoli Bai Halo Orbit Stationkeeping using Nonlinear MPC and Polynomial Optimization 3 / 31
Rutgers, The State University of New Jersey tugraz Background Outline m 1 = 1.00 m 2 = 0.10 (non-dimensional units) Introduction Circular Restricted 1 L4 Three Body Problem 0 L3 L1 L2 y Global Polynomial L5 -1 Optimization Polynomial -1.5 -1 -0.5 0 0.5 1 1.5 MPC x Numerical Results Conclusions Libration points: Ideal locations for human/robotic space exploration. Several successful past missions: ISEE-3, SOHO. Active station-keeping required. Gaurav Misra, Hao Peng, and Xiaoli Bai Halo Orbit Stationkeeping using Nonlinear MPC and Polynomial Optimization 3 / 31
Rutgers, The State University of New Jersey tugraz Common station-keeping approaches: Outline Discrete Introduction Discrete LQR [Folta and Vaughn, 2004] Circular Restricted Chebyshev-Picard iterations [Bai and Junkins, Three Body 2012] Problem Continuous Global Polynomial Continuous LQR [Nazari et al., 2014] Optimization Nonlinear optimization [Ulybyshev, 2015] Polynomial MPC Linear MPC [ Peng et al., 2017, Kalabi´ c et al., 2015] Numerical Results Nonlinear MPC [Li et al., 2015] Conclusions Goal for this work: Globally optimal constrained receding horizon solution. Gaurav Misra, Hao Peng, and Xiaoli Bai Halo Orbit Stationkeeping using Nonlinear MPC and Polynomial Optimization 4 / 31
Rutgers, The State University of New Jersey tugraz Common station-keeping approaches: Outline Discrete Introduction Discrete LQR [Folta and Vaughn, 2004] Circular Restricted Chebyshev-Picard iterations [Bai and Junkins, Three Body 2012] Problem Continuous Global Polynomial Continuous LQR [Nazari et al., 2014] Optimization Nonlinear optimization [Ulybyshev, 2015] Polynomial MPC Linear MPC [ Peng et al., 2017, Kalabi´ c et al., 2015] Numerical Results Nonlinear MPC [Li et al., 2015] Conclusions Goal for this work: Globally optimal constrained receding horizon solution. Gaurav Misra, Hao Peng, and Xiaoli Bai Halo Orbit Stationkeeping using Nonlinear MPC and Polynomial Optimization 4 / 31
Rutgers, The State University of New Jersey tugraz Motion near libration points Outline Equations of motion with origin at L1 libration point Introduction Circular Restricted y + (1 − γ L ) + x − (1 − µ ) Three Body x = 2 ˙ ¨ (1 − γ L + x ) r 3 Problem 1 Global + µ Polynomial ( γ L − x ) − µ r 3 Optimization 2 Polynomial x + y − y (1 − µ ) − y µ MPC y = − 2 ˙ ¨ r 3 r 3 Numerical 1 2 Results z = − (1 − µ ) z − µ z Conclusions ¨ r 3 r 3 1 2 γ L : Distance between L 1 and primary Gaurav Misra, Hao Peng, and Xiaoli Bai Halo Orbit Stationkeeping using Nonlinear MPC and Polynomial Optimization 5 / 31
Rutgers, The State University of New Jersey tugraz Legendre Polynomial Approximation Outline Equations of motion in terms of legendre Introduction polynomials Circular Restricted Three Body y − (1 + 2 c 2 ) x = ∂ c n ρ n P n ( x Problem � x − 2 ˙ ¨ ρ ) (1) ∂ x Global Polynomial n ≥ 3 Optimization c n ρ n P n ( x x + ( c 2 − 1) y = ∂ � Polynomial y + 2 ˙ ¨ ρ ) (2) MPC ∂ y n ≥ 3 Numerical Results z + c 2 z = ∂ c n ρ n P n ( x � ¨ ρ ) (3) Conclusions ∂ z n ≥ 3 c n = γ − 3 L ( µ + ( − 1) n (1 − µ )( 1 − γ L ) n +1 ) γ L Gaurav Misra, Hao Peng, and Xiaoli Bai Halo Orbit Stationkeeping using Nonlinear MPC and Polynomial Optimization 6 / 31
Rutgers, The State University of New Jersey tugraz Polynomial Approximations Outline Richardson’s third order approximation Introduction y − (1 + 2 c 2 ) x = 3 2 c 3 (2 x 2 − y 2 − z 2 ) Circular x − 2 ˙ ¨ Restricted Three Body + 2 c 4 x (2 x 2 − 3 y 2 − 3 z 2 ) + O (4) Problem Global x + ( c 2 − 1) y = − 3 c 3 xy − 3 Polynomial 2 c 4 (4 x 2 − y 2 − z 2 ) y y + 2 ˙ ¨ Optimization Polynomial MPC + O (4) Numerical z + c 2 z = − 3 c 3 xz − 3 2 c 4 z (4 x 2 − y 2 − z 2 ) Results ¨ Conclusions + O (4) Analytic periodic solution based on Lindstedt-Poincar` e perturbation method Gaurav Misra, Hao Peng, and Xiaoli Bai Halo Orbit Stationkeeping using Nonlinear MPC and Polynomial Optimization 7 / 31
Rutgers, The State University of New Jersey tugraz Polynomial Approximations Analytical methods: qualitatively insightful, but Outline insufficient for dynamical analysis. Introduction Combined with differential correction for trajectory Circular Restricted refinement. Three Body Problem Non-dimensional units 10 -4 Global L 1 Polynomial 10 Analytical solution Optimization 5 Single shooting 0 z Polynomial -5 MPC Numerical 4 Results 2 Conclusions 10 -3 0 -2 y 0.991 -4 0.99 0.989 x Gaurav Misra, Hao Peng, and Xiaoli Bai Halo Orbit Stationkeeping using Nonlinear MPC and Polynomial Optimization 8 / 31
Rutgers, The State University of New Jersey tugraz Polynomial Approximations Analytical methods: qualitatively insightful, but Outline insufficient for dynamical analysis. Introduction Combined with differential correction for trajectory Circular Restricted refinement. Three Body Problem Non-dimensional units 10 -4 Global L 1 Polynomial 10 Analytical solution Optimization 5 Single shooting 0 z Polynomial -5 MPC Numerical 4 Results 2 Conclusions 10 -3 0 -2 y 0.991 -4 0.99 0.989 x Gaurav Misra, Hao Peng, and Xiaoli Bai Halo Orbit Stationkeeping using Nonlinear MPC and Polynomial Optimization 8 / 31
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