Aerospace Mission Design on Quotient Manifolds

Michael J Sparapany, Purdue University

Abstract

Conceptual aerospace mission design has typically been performed in a computationally intensive and iterative manner. The introduction of modern computing has resulted in the widespread adoption of various numerical methods. As a result, useful information associated with the optimal solution is largely ignored. Optimization through indirect methods, while still computationally intense, leverages this information and also reveals a much deeper mathematical structure. This mathematical structure provides the gateway to reformulating the problem definition to one with certain desirable properties. In the presence of symmetries and constants-of-motion, the dynamical systems of indirect methods live in a reduced dimensional quotient manifold. Studies leveraging this reduced dimensional quotient manifold may benefit in performance by using fewer operations per iteration.Many limitations prevent the use of these quotient manifolds in practical aerospace mission design. The five main issues include (1) rephrasing indirect methods in terms of differential geometry in an efficient manner, (2) Pontryagin’s minimum principle generating a large number of valid dynamical systems, (3) implementing reduction in a global manner for highly non-linear systems, (4) numerical boundary-value problem solvers not supporting missions on quotient manifolds, and (5) scalability of the methods to real aerospace missions. This work addresses all five issues.In previous studies, computer algebra systems have been proven to be an effective tool for automating complex indirect methods. However, when posed in the language of differential geometry, the majority of support from digital software is lost. A version of indirect methods is recasted using differential geometry that effectively retains all information of so-called traditional methods. Exploitation of the anti-symmetric differential structure enables large-scale problems to be studied.Root-solving the stationary Hamiltonian condition may generate several, potentially valid dynamical systems. Each system must be evaluated at every point along the trajectory using Pontryagin’s minimum principle. This process prohibits later analytical derivations on the dynamical system. In the Integrated Control Regularization Method, the control law is posed as a state of the dynamical system with an equation-of-motion, thereby moving complicated root-solving to the boundary where it is solved once. Introduction of the control law as a new state is done using geometric adjoining methods where the original mathematical structure of the problem is preserved.Reduction is traditionally studied in a topological context where there is a wealth of information. In terms of aerospace missions, there are very few applications in existence. These traditional studies rely on the quotient of entire global spaces. This is impossible to apply on non-integrable, non-linear dynamical systems. To get around this, a compact procedure is developed where the Lie algebra identifies dynamical sub-systems that may be effectively eliminated. This removes the reliance of integrability on the symmetry space.Conceptual aerospace mission design has typically been performed in a computationally intensive and iterative manner. The introduction of modern computing has resulted in the widespread adoption of various numerical methods. As a result, useful information associated with the optimal solution is largely ignored. Optimization through indirect methods, while still computationally intense, leverages this information and also reveals a much deeper mathematical structure. This mathematical structure provides the gateway to reformulating the problem definition to one with certain desirable properties. In the presence of symmetries and constants-of-motion, the dynamical systems of indirect methods live in a reduced dimensional quotient manifold. Studies leveraging this reduced dimensional quotient manifold may benefit in performance by using fewer operations per iteration.Many limitations prevent the use of these quotient manifolds in practical aerospace mission design. The five main issues include (1) rephrasing indirect methods in terms of differential geometry in an efficient manner, (2) Pontryagin’s minimum principle generating a large number of valid dynamical systems, (3) implementing reduction in a global manner for highly non-linear systems, (4) numerical boundary-value problem solvers not supporting missions on quotient manifolds, and (5) scalability of the methods to real aerospace missions. This work addresses all five issues.In previous studies, computer algebra systems have been proven to be an effective tool for automating complex indirect methods. However, when posed in the language of differential geometry, the majority of support from digital software is lost. A version of indirect methods is recasted using differential geometry that effectively retains all information of so-called traditional methods. Exploitation of the anti-symmetric differential structure enables large-scale problems to be studied.Root-solving the stationary Hamiltonian condition may generate several, potentially valid dynamical systems. Each system must be evaluated at every point along the trajectory using Pontryagin’s minimum principle. This process prohibits later analytical derivations on the dynamical system. In the Integrated Control Regularization Method, the control law is posed as a state of the dynamical system with an equation-of-motion, thereby moving complicated root-solving to the boundary where it is solved once. Introduction of the control law as a new state is done using geometric adjoining methods where the original mathematical structure of the problem is preserved.Reduction is traditionally studied in a topological context where there is a wealth of information. In terms of aerospace missions, there are very few applications in existence. These traditional studies rely on the quotient of entire global spaces. This is impossible to apply on non-integrable, non-linear dynamical systems. To get around this, a compact procedure is developed where the Lie algebra identifies dynamical sub-systems that may be effectively eliminated. This removes the reliance of integrability on the symmetry space.In reduction, various dimensions desirable to a designer may be eliminated from the system defined on a reduced dimensional quotient manifold. Crucial to satisfying mission requirements, present day numerical solvers do not have the capability to perform the necessary reconstruction. A modified collocation and shooting algorithm with this functionality is given and a numerical example of a problem on a reduced dimensional quotient manifold is explored.

Degree

Ph.D.

Advisors

Grant, Purdue University.

Subject Area

Design|Computer science|Mathematics|Systems science|Transportation

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