Lagrangian Derivation of Variable-Mass Equations of Motion using an Arbitrary Attitude Parameterization

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.


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 an appropriately dimensioned identity matrix 0 an appropriately dimensioned matrix of zeros

Physical Vectors and Reference Frames
A physical vector u − → ∈ P is an element of physical space, where physical space is denoted P. Physical vectors often represent physical quantities, such as position, velocity, and acceleration. Consider the orthornormal, dextral, physical basis vectors a − → 1 , a − → 2 , and a − → 3 that may be used to define a reference frame, denoted F a . A physical vector may be resolved in, for example, F a as [13] where the vectrix F − → a ∈ P 3 and the column matrix u a ∈ R 3 are defined as and u a = [u a1 u a2 u a3 ] T , respectively. A physical vector may be resolved in any frame, , and the relationship between u a and u b is given by [13]. DCMs are orthonormal, meaning that C ba = C T ab and C T ba C ba = 1. The skew-symmetric cross-product matrix is defined for any column matrix u a holds in any reference frame.
The position of point z relative to point w is denoted by r − → zw . The velocity of point z relative to point w with respect to F a is denoted by v − → zw/a = r − → zw ·a . A physical vector's time rate-of-change with respect to an arbitrary frame F a can be related to its time rate-of-change with respect to another arbitrary frame F b using the Kinematic Transport Theorem [13], where ω − → ba is the angular velocity of F b relative to F a .
The attitude of a body can be globally and uniquely described by a DCM, C ba , where F b is a frame fixed to the body and F a is a datum reference frame.
There are many ways to parameterize a DCM, such as quaternions, Euler angles, and Gibbs parameters [13]. An arbitrary attitude parameterization of C ba will be denoted q ba , which can be the quaternion elements, a set of three Euler angles, or simply the columns of the DCM stacked in a 9 × 1 column matrix. The angular velocity differs from the attitude parameterization's time rate-of-changeq ba , yet they can be related by a mapping matrix S ba b (q ba ), where [13] ω ba b = S ba b (q ba )q ba , and the inverse mapping is given bẏ The matrices S ba b and ba b can be shown to be orthogonal complements, that is where ba (q ba ) is a constraint matrix associated with the attitude constraint. The columns of ba b lie in the null-space of ba (q ba ), that is ba (q ba ) ba b = 0. For any vector u − → that is not a function of q ba , the following three identities (and their transposes) [24], are of crucial importance in forthcoming derivations. Note that the argument q ba has been suppressed in Eq. 2, Eq. 3, and Eq. 4, and will continue to be suppressed in forthcoming derivations.

Derivation of the Equations of Motion
Consider a constant-mass system S. Consider also a region with volume V (t) and boundary B(t) defined such that it encloses the mass of S at all times. The volume V (t) and boundary B(t) are therefore time-varying quantities. The system S can be arbitrarily composed of mass that is rigid, and some that is not. Next, consider an arbitrary time-varying mass systemS(t) that has a known, constant volumeV with boundaryB. Note that systemS has time-varying mass but constant volume and boundary, whereas system S has constant mass but time-varying volume and boundary. Let S be defined such that at a specific instantt the system S conincides exactly withS, and consequently so does V (t) =V , B(t) =B. For any other instant,V will generally differ from V , andB will generally differ from B.
Referring to Fig. 1, let w be an unforced particle and F i be an inertial frame [1]. Let z be a reference point fixed to any rigid portion of S, and F b be a frame fixed to the same rigid portion of S, as shown in Fig. 1. In order to properly define z and F b , there must exist some sort of reference rigid-body, and hence the requirement for S to possess at least some portion that is considered rigid. The enabling theorem in this derivation is Reynold's Transport Theorem, which states d dt where f − → is a scalar-, vector-, or tensor-valued property of a system of interest [4,13,15,16,19,21,29,30]. The term v − → dSz/b refers to the velocity of an area element dS, which can be assumed to be equivalent to the velocity of a mass element at the boundary, v − → dmz/b . The notation is an alternate way to write the time-derivative of u − → with respect to F b , in order to clarify the meaning of the time derivative on the left-hand-side of Eq. 5.

Generalized Coordinates
The generalized coordinates are q = r zw T i q bi T T where q bi is an arbitrary attitude parameterization describing the attitude of F b relative to F i . The reader should be careful not to confuse the attitude parameterization q bi with the generalized coordi- , which is related to the generalized coordinates by Notice that the matrices S and are orthogonal complements, that is, S = 1.

Kinetic Energy and the Lagrangian
For simplicity, potential energy sources are not considered, and their effects can be included as external forces. The kinetic energy of the constant-mass system S, relative to w, with respect to F i is where, using the Kinematic Transport Theorem, the velocity of a mass element dm relative to w, with respect to F i can be shown to be

Resolving in F i , the kinetic energy is
where dm are the zeroth, first, and second moments of mass of S about point z. Equation 6 can be written compactly as which is alternatively written using the generalized coordinates, Notice that the kinetic energy expression in Eq. 7 is not strictly quadratic inq, but includes first and zeroth order terms. In particular, the termsq T S T β and T 0 Sw/i stem from the fact that v dmz/b b is non-zero, which is not the case for rigid bodies. Since there is no potential energy considered, the Lagrangian simply reduces to the kinetic energy, L Sw/i = T Sw/i .

Lagrange's Equation
Lagrange's equation is d dt where f is the generalized forces and moments and is a constraint matrix such that q = 0. Currently, the only constraint is the attitude constraint, and as such = 0 bi . Moreover, it is straightforward to confirm that = 0. The terms of Lagrange's equation will now be evaluated one-by-one.
The termsṀ andβ arė Recall that the system S is a constant-mass system, but is not necessarily rigid. As such,ṁ S = 0 whileċ Sz b = 0 andJ Sz b = 0.

The Second Term,
where the second term can be expanded as

Virtual Work and the Generalized Forces
For simplicity, only discrete forces acting at specific points will be considered. Consider a discrete force f − → p acting at point p. The position of p relative to w resolved Since the total virtual work done on S is the sum of virtual work done by each force, the sum of generalized forces and moments would be a summation of terms identical to f p . That is, f = f p 1 +f p 2 +· · ·+f p N , and thus this is the term that is substituted into Lagrange's equation.

Lagrange's Equation
Substituting Eq. 9, Eq. 11, and the generalized forces described by Eq. 12 into Eq. 8 gives (13) where a non has been defined in Eq. 11. Recalling the identities Eq. 2, Eq. 3, Eq. 4, as well as the fact that T S T = 1 and T T = 0, pre-multiplying Eq. 13 by T and simplifying yields The terms T a non and T f are where f S i is the sum of external forces acting on S and m Sz b is the sum of external moments acting on S about point z. The mass matrix M in Eq. 14 is symmetric positive definite, regardless of the attitude parameterization, and is therefore invertible. Equation 14 constitutes the equation of motion of the constant-mass system S. To convert this to variable-mass equations of motion, Eq. 14 will be expanded into its constituent translational and rotational dynamics and Reynold's Transport Theorem will be invoked. This will also make the equivalence to traditional forms clear.

Translational Dynamics
Expanding the first three rows of Eq. 14, whilst staying in matrix form, yields m Sv Recalling that where r Sz b is the instantaneous center of mass of S, Eq. 15 becomes Using Reynold's Transport Theorem given in Eq. 5, it can be shown that where n b is the components of an outwards-pointing normal unit vector to the boundary B, resolved in F b , and ρ is the mass density of the element dm. By defining, the well-known variable-mass translational equations reported in [4-7, 15, 16, 19] emerge, that being Recall that, in general, neither V (t) or B(t) are known since they correspond to the constant-mass system, and the state of the mass that has left the primary volume of interest is unknown. Only at the specific time t =t where the systems S andS coincide exactly, areV andB known, where by design V (t) =V and B(t) =B. Since Eq. 16 has the same form for any general instantt, the bounds of integration may now be replaced withS,V ,B. This is the exact result seen in [4-7, 15, 16, 19], amongst others, when resolved in F i . Although Eq. 16 is written in a form that can be recognized, it is not necessarily convenient to resolve ω − → bi and r − → Sz in F i . Terms in Eq. 16 can be resolved in any frame with an appropriate use of DCMs.

Rotational Dynamics
Expanding the last three rows of Eq. 14, whilst staying in matrix form, yields The terms −(C bi v resulting in multiple terms in Eq. 17 to cancel out. Equation 17 now becomes, It can be shown by use of Reynold's Transport Theorem that and that Therefore, which is again the exact result obtained in [4-7, 15, 16, 19], amongst others, resolved in F b . As before, since the system S coincides exactly withS at the general instant t =t, and since the form of Eq. 18 is identical for allt, the bounds of integration can be changed to the knownV ,B.

Conclusion
This note presents a Lagrangian approach to deriving the translational and rotational dynamics of variable-mass systems. The advantage of the approach presented is that any arbitrary attitude parameterization can be used to derive equations of motion using the standard form of Lagrange's equation. Moreover, the equations of motion are maintained in matrix form, leading to concise representations of the equations of motion.