lambert universal variable algorithm - Arabian Journal for Science ...

M.A. Sharaf, A.N. Saad, and M.I. Nouh
1. INTRODUCTION
LAMBERT UNIVERSAL VARIABLE ALGORITHM
Lambert problem of space researches is concerned with the determination of an orbit from two position vectors and the
time of flight [1]. It has very important applications in the areas of rendezvous, targeting, guidance [2], and
interplanetary mission [3].
Solutions to Lambert’s problem abound in the literature, as they did even in Lambert’s time shortly after his original
formulation in 1716. Examples are Lambert’s original geometric formulation, which provides equations to determine the
minimum-energy orbit, and the original Gaussian formulation, which gives geometrical insight into the problem.
Up to the year 1965, a fairly comprehensive list of references on Lambert’s problem are given in references [4–6].
In 1969, Lancaster and Blanchard [7] (also Mansfield [8]) established unified forms of Lambert’s problem; in 1990
Gooding [9] developed a procedure for the solution; and in 1995, Thorne and Bain [10] developed a direct solution using
the series inversion technique.
Each of the above methods is characterized primarily by: (1) a particular form of the time of flight equation; and
(2) a particular independent variable to be used in an iteration algorithm to determine the orbital elements.
One of the most compact and computationally efficient forms of Lambert’s problem is that of Battin (cited in
reference [11]). In this form, the time of flight equation is universal (i.e., includes elliptic, parabolic, and hyperbolic
orbits) as a well-behaved function of a single, physically significant, independent variable.
In the present paper, we aim at building an algorithm for universal Lambert’s problem on the basic approach of Battin,
extending it to include the following important points: (1) An iterative scheme which converges for all orbit types; and
(2) an efficient procedure to evaluate the Stumpff’s functions. Applications of the algorithm are also given.
2. BASIC FORMULATIONS
2.1. Two-Body Formulations
• The equation describing the relative motion of the two bodies of masses m1 and m2 in rectangular coordinates is:
d µ
v= r =− r 3 , (2.1)
dt r
where µ is the gravitational parameter (universal gravitational constant times the sum of the two masses), r and v
are the position and velocity vectors given in components as:
r = x i + y i + z i , (2.2)
x y z
v= x i + y i + z i , (2.3)
x y z
ix, iy, and iz are the unit vectors along the coordinate axes x, y, and z respectively, and
2 2 2 1 2
r = ( x + y + z ) . (2.4)
Equation (2.1) is unchanged if we replace r with –r. Thus Equation (2.1) gives the motion of the body of mass m2
relative to the body of the mass m1, or the mass m1 relative to m2. Also, if we replace t with –t, Equation (2.1)
remains unchanged.
88 The Arabian Journal for Science and Engineering, Volume 28, Number 1A. January 2003