Lagrangian Derivation of Variable-Mass Equations of Motion using an Arbitrary Attitude Parameterization
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Lagrangian Derivation of Variable-Mass Equations of Motion using an Arbitrary Attitude Parameterization Charles Champagne Cossette1
· James Richard Forbes1
· David Saussie´ 2
© American Astronautical Society 2020
Abstract Lagrange’s equation is a popular method of deriving equations of motion due to the ability to choose a variety of generalized coordinates and implement constraints. When using a Lagrangian formulation, part of the generalized coordinates may describe the attitude. This paper presents a means of deriving the dynamics of variable-mass systems using Lagrange’s equation while using an arbitrary constrained attitude parameterization. The equivalence to well-known forms of the equations of motion is shown. Keywords Lagrange’s equation · variable-mass dynamics · constrained attitude parameterizations
Introduction Deriving the equations of motion of a dynamic system that expels mass is a complex problem with historical roots in rocketry [5]. Numerous technical reports and papers present different means to arrive at the now familiar equations of motion of a variablemass system. A Newton-Euler approach is presented in [10, 15, 16, 22, 25], while Charles Champagne Cossette
[email protected] James Richard Forbes [email protected] David Saussi´e [email protected] 1
Department of Mechanical Engineering, McGill University, 817 Sherbrooke St. W., Montr´eal, QC, H3A 0C3, Canada
2
Department of Electrical Engineering, Polytechnique Montr´eal, 2500 Chemin de Polytechnique, Montr´eal, QC, H3T 1J4, Canada
The Journal of the Astronautical Sciences
Kane’s Equations are used in [4–6]. Hamilton’s Principle is adapted to the variablemass dynamics problem in [11, 14], and Lagrange’s equation is commonly seen with an appended term to account for mass-variablility, as discussed in [9, 14, 18]. Lagrange’s equation is a popular method of deriving equations of motion due to it’s ability to accommodate different generalized coordinates, as well as its ease of handling constraints. Lagrange’s equation can be applied to systems where a subset of the chosen generalized coordinates is an attitude parameterization. Attitude parameterizations will often have constraints, such as the quaternion possessing a unit-norm constraint. In [24], identities are derived for general attitude parameterizations that enable a streamlined application of Lagrange’s equation, without having to resort to the use of the Boltzmann-Hamel equations [24]. These identities greatly enhance the tractability of the equations derived by maintaining the equations in matrix form. Similar identities appear in [3, 8, 17, 20, 23], when attitude is parameterized using Euler angles, a quaternion, or axis/angle parameters, whereas the identities in [24] collectively consider any attitude parameterization. The identities allow access to the analysis of new and more sophisticated systems using a Lagrangian framework, such as those seen in [2, 12, 26–28]. However, these identities have only been used to model constant-mass systems.
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