The theory of asynchronous relative motion II: universal and regular solutions

Abstract

Two fully regular and universal solutions to the problem of spacecraft relative motion are derived from the Sperling–Burdet (SB) and the Kustaanheimo–Stiefel (KS) regularizations. There are no singularities in the resulting solutions, and their form is not affected by the type of reference orbit (circular, elliptic, parabolic, or hyperbolic). In addition, the solutions to the problem are given in compact tensorial expressions and directly referred to the initial state vector of the leader spacecraft. The SB and KS formulations introduce a fictitious time by means of the Sundman transformation. Because of using an alternative independent variable, the solutions are built based on the theory of asynchronous relative motion. This technique simplifies the required derivations. Closed-form expressions of the partial derivatives of orbital motion with respect to the initial state are provided explicitly. Numerical experiments show that the performance of a given representation of the dynamics depends strongly on the time transformation, whereas it is virtually independent from the choice of variables to parameterize orbital motion. In the circular and elliptic cases, the linear solutions coincide exactly with the results obtained with the Clohessy–Wiltshire and Yamanaka–Ankersen state-transition matrices. Examples of relative orbits about parabolic and hyperbolic reference orbits are also presented. Finally, the theory of asynchronous relative motion provides a simple mechanism to introduce nonlinearities in the solution, improving its accuracy.

Appendices

Appendix 1: The Stumpff functions

Stumpff (1947) introduced a family of functions defined in terms of the convergent series:

$$\begin{aligned} \mathscr {C}_k(z) = \sum _{i=0}^\infty \frac{(-{z})^i}{(2i+k)!}. \end{aligned}$$

The Stumpff functions are intimately related to the universal variables (Everhart and Pitkin 1983; Battin 1999, Chap. 4). They allow the solution to Keplerian motion to be generalized, so the formulation is unique no matter the eccentricity of the orbit. In the present work the argument of the Stumpff functions is $z=\omega ^2s^2$, with $\omega ^2=-2{\mathcal {E}}$.

When the orbital energy vanishes (in the parabolic case) it follows $z=0$ and the Stumpff functions reduce to

$$\begin{aligned} \mathscr {C}_k(0) = \frac{1}{k!}. \end{aligned}$$

The first Stumpff functions admit simple closed-form expressions shown in Table 4.

Table 4 Explicit expressions for the first Stumpff functions $\mathscr {C}_k(z)$, with $z=\omega ^2s^2$

Increasing the degree k of the Stumpff functions yields:

$$\begin{aligned} \mathscr {C}_{k+1}(z) = \sum _{i=0}^\infty \frac{(-{z})^i}{(2i+k+1)!},\quad \mathscr {C}_{k+2}(z) = \sum _{i=0}^\infty \frac{(-{z})^i}{(2i+k+2)!}. \end{aligned}$$

From the later it follows that

$$\begin{aligned} \mathscr {C}_{k+2}(z) = \sum _{i=0}^\infty \frac{(-{z})^i}{\big (2(i+1)+k\big )!} = \frac{1}{z}\sum _{i=0}^\infty \frac{(-1)^i{z}^{i+1}}{\big (2(i+1)+k\big )!} = \frac{1}{z}\left[ \frac{1}{k!} - \mathscr {C}_k(z)\right] , \end{aligned}$$

which provides the recurrence formula

$$\begin{aligned} \mathscr {C}_k(z) + z\,\mathscr {C}_{k+2}(z) = \frac{1}{k!}. \end{aligned}$$

(54)

Fig. 5

Analysis of the growth-rate of the terms $S_k^i(z)$

Techniques for computing the derivatives of the Stumpff functions can be found, for instance, in the book by Bond and Allman (1996, Appx. E). Some useful relations are:

$$\begin{aligned} s\, \dfrac{\partial {\mathscr {C}_k(z)}}{\partial {s}}= & {} \mathscr {C}_{k-1}(z)-k\mathscr {C}_k(z), \qquad \dfrac{\partial {\mathscr {C}_k(z)}}{\partial {s}} = - \omega ^2 s \mathscr {C}_{k+2}^*(z) \end{aligned}$$

(55)

$$\begin{aligned} \omega \,\dfrac{\partial {\mathscr {C}_k(z)}}{\partial {\omega }}= & {} \mathscr {C}_{k-1}(z)-k\mathscr {C}_k(z), \qquad \dfrac{\partial {\mathscr {C}_k(z)}}{\partial {\omega }} = - s^2 \omega \mathscr {C}_{k+2}^*(z) \end{aligned}$$

(56)

Note that Eqs. (55) and (56) are only valid for $k>0$. The auxiliary term $\mathscr {C}_{k+2}^*(z)$ corresponds to:

$$\begin{aligned} \mathscr {C}_{k+2}^*(z) = \mathscr {C}_{k+1}(z) - k\mathscr {C}_{k+2}(z). \end{aligned}$$

Danby (1992, p. 171) discussed in detail the computational aspects of handling the Stumpff functions. It is more convenient to compute the highest-degree functions via the series, and then to apply the recurrence formula—Eq. (54)—to obtain the remaining functions. If the highest-degree functions were to be computed from the lower-degree ones, the recurrence relation may become singular for $z=0$. Usual applications of the Stumpff functions are restricted to $k\le 3$, and do not consider higher-degree functions (Danby 1987; Sharaf and Sharaf 1997). In this work, Stumpff functions up to $k=5$ appear, and the required formulas are given explicitly in the following lines.

The numerical stability of the convergent series deserves a dedicated analysis. Let $S_k^i\in {\mathbb {R}}$ denote the i-th term of the series defining $\mathscr {C}_k(z)$:

$$\begin{aligned} S_k^i(z) = \frac{(-z)^i}{(2i+k)!}. \end{aligned}$$

Since the Stumpff functions converge absolutely the factorial term compensates the power for i sufficiently large (Spivak 1994, p. 308) although $S_k^i$ may suffer strong changes. Figure 5 shows the evolution of $S_4^i(z)$ and $S_5^i(z)$ for different values of the argument z, and for increasing i. The terms $S_k^i$ experience changes of several orders of magnitude. As the argument grows, the amplitude of the this variation increases. This phenomenon may lead to important losses of accuracy provided that the least significant digits of the series are lost when added or subtracted from large quantities. Performing the computations in quadruple precision floating-point arithmetic delays the appearance of these problems since truncation errors are reduced. However, as the computation advances the loss of accuracy will eventually appear. To fully overcome this issue the argument of the Stumpff functions is reduced making use of the so called half-angle relations:

$$\begin{aligned} \begin{array}{llll} \displaystyle \mathscr {C}_0(4z) &{}= \big [\mathscr {C}_0(z)\big ]^2 -1, &{}\qquad \displaystyle \mathscr {C}_1(4z) = \mathscr {C}_0(z)\mathscr {C}_1(z) \\ \displaystyle \mathscr {C}_2(4z) &{}= \dfrac{1}{2}\big [\mathscr {C}_1(z)\big ]^2, &{}\displaystyle \qquad \mathscr {C}_3(4z) = \dfrac{1}{4}\big [\mathscr {C}_2(z) +\mathscr {C}_0(z)\mathscr {C}_3(z)\big ] \\ \displaystyle \mathscr {C}_4(4z) &{}= \dfrac{1}{8}\big \{ \big [\mathscr {C}_2(z)\big ]^2+2\mathscr {C}_4(z) \big \}, &{}\displaystyle \qquad \mathscr {C}_5(4z) = \dfrac{1}{32}\big \{ \mathscr {C}_3(z) + 2\mathscr {C}_5(z) + 2\mathscr {C}_1(z)\mathscr {C}_4(z)\big \} \end{array} \end{aligned}$$

Note that these expressions do not require the computation of higher-order terms. They can be applied repeatedly to reduce the value of z below a certain threshold $z_{\mathrm {crit}}$. Danby (1992, p. 173) summarized the algorithm even though he only provided half-angle formulas  up to $k=3$.

There are two ways of computing the Stumpff functions by series. First, the series can be truncated when a certain accuracy has been reached. In this case each term is computed sequentially. A possible way to compute each term in the series is:

$$\begin{aligned} S_k^{i} = -\frac{z}{(2i+k)(2i+k-1)}S_k^{i-1}, \qquad \text {with} \,\,S_k^0 = \frac{1}{k!}. \end{aligned}$$

Convergence will have been reached when $|S_k^{i}|<\varepsilon _{\mathrm {tol}}$, where $\varepsilon _{\mathrm {tol}}$ denotes the tolerance. Second, the series can be truncated a priori and a nested expression for the Stumpff function can be constructed. Different forms of nesting the terms in the Stumpff functions can be found in the cited works.

Appendix 2: The inverse KS map

For the sake of brevity, we omit the properties of the inverse KS transformation and simply present the expressions that transform the initial conditions ${\mathbf {x}}_0^\top =[{\mathbf {r}}_0^\top ,{\mathbf {v}}_0^\top ]$ into $\mathbf {y}_0^\top =[\mathbf {u}_0^\top ,{\mathbf {u}^\prime _0}^\top ]$. Depending on the sign of $x_0$ one should use:

$$\begin{aligned} x_{0}\ge 0:\qquad u_{10}= & {} 0, \, u_{20} = - \frac{z_{0}}{ p},\,\,\, u_{30} = \frac{y_{0}}{ p}, \,\,\, u_{40} = -\frac{ p}{2}\\ x_{0}<0:\qquad u_{10}= & {} - \frac{z_{0}}{ p}, \,\,\, u_{20} =0,\,\,\, u_{30} = - \frac{ p}{2}, \,\,\, u_{40} = \frac{y_{0}}{ p} \end{aligned}$$

where $p=\sqrt{2 q}$ and $q=r_0+|x_0|$. The initial value of $\mathbf {u}^\prime$ is obtained from:

$$\begin{aligned} \mathbf {u}^\prime _0 = \frac{1}{2}{{\mathbf {\mathsf{{L}}}}}^{\top }(\mathbf {u}_0)\, {\mathbf {v}}_0. \end{aligned}$$

Here ${\mathbf {v}}_0=[\dot{x}_0,\dot{y}_0,\dot{z}_0,0]^\top$ is the extension of the velocity vector to ${\mathbb {R}}^4$, defined by its components in the inertial frame. It follows

$$\begin{aligned} x_{0}\ge & {} 0:\quad u^\prime _{10} = \frac{y_0\dot{z}_0-z_0\dot{y}_0}{2 p}, \;\; u_{20}^\prime = \frac{z_0\dot{x}_0-\dot{z}_0 q}{2 p},\\&\quad u_{30}^\prime = \frac{\dot{y}_0 q-y_0\dot{x}_0}{2 p}, \;\; u_{40}^\prime = -\frac{\dot{x}_0 q +y_0\dot{y}_0 + z_0\dot{z}_0}{2 p} \\ x_{0}< & {} 0:\quad u_{10}^\prime = -\frac{\dot{z}_0 q+z_0\dot{x}_0}{2 p}, \;\; u_{20}^\prime =\frac{y_0\dot{z}_0-z_0\dot{y}_0}{2 p},\\&\quad u_{30}^\prime = \frac{\dot{x}_0 q - y_0\dot{y}_0 - z_0\dot{z}_0}{2 p}, \;\; u_{40}^\prime =\frac{y_0\dot{x}_0+\dot{y}_0 q}{2 p} \end{aligned}$$

The different definitions for the initial conditions in the KS space differ by the selection of the reference frame.

Appendix 3: The transformation ${{\mathbf {\mathsf{{T}}}}}(\mathbf {X}_0)$

The transformation $\mathbf {y}_0\mapsto {\mathbf {x}}_0$ (Appendix 2) has been defined in two different ways depending on the sign of the component $x_0$. This gives rise to two alternative definitions of the matrix ${{\mathbf {\mathsf{{T}}}}}(s)$

$$\begin{aligned} \delta \mathbf {y}_0 = {\left\{ \begin{array}{ll} \; {{\mathbf {\mathsf{{T}}}}}^{+}({\mathbf {x}}_0)\,\delta {\mathbf {x}}_0,\qquad &{} \text {if}\, x_0\ge 0 \\ \; {{\mathbf {\mathsf{{T}}}}}^{-}({\mathbf {x}}_0)\,\delta {\mathbf {x}}_0,\qquad &{} \text {if}\, x_0<0 \end{array}\right. } \end{aligned}$$

In the first case it is

$$\begin{aligned} {{\mathbf {\mathsf{{T}}}}}^{+}({\mathbf {x}}_0) = k\left[ \begin{array}{llllll} 0 ,&{} 0 ,&{} 0 ,&{} 0 ,&{} 0 ,&{} 0 \\ +2z_0 ,&{} +2\dfrac{y_0z_0}{ q} ,&{} 2\dfrac{z_0^2}{ q}-4r_0 ,&{} 0 ,&{} 0 ,&{} 0 \\ -2y_0 ,&{} 4r_0-2\dfrac{y_0^2}{ q} ,&{} -2\dfrac{y_0z_0}{ q} ,&{} 0 ,&{} 0 ,&{} 0 \\ -2 q ,&{} -2y_0\, ,&{} -2z_0,&{} 0 ,&{} 0 ,&{} 0 \\ - f\, ,&{} 2r_0\dot{z}_0 - y_0\dfrac{ f}{ q} ,&{} - 2r_0\dot{y}_0 - z_0\dfrac{ f}{ q},&{} 0 ,&{} -2r_0z_0 ,&{} 2r_0y_0\\ - q\dot{z}_0 - z_0\dot{x}_0 ,\,&{} -y_0\Big ( \dot{z}_0 + \dfrac{z_0\dot{x}_0}{ q} \Big ) ,&{} 2\dot{x}_0r_0-z_0\dot{z}_0 -\dfrac{ z_0^2\dot{x}_0 }{ q},&{} 2r_0z_0 ,&{} 0 ,&{} -2r_0 q\\ q\dot{y}_0+y_0\dot{x}_0 ,&{} y_0\dot{y}_0 - 2\dot{x}_0r_0+ \dfrac{y_0^2\dot{x}_0}{ q} ,\,&{} z_0\Big ( \dot{y}_0 + \dfrac{y_0\dot{x}_0}{ q} \Big ),&{} -2r_0y_0 ,&{} 2r_0 q ,&{} 0\\ h - \dot{x}_0 q ,&{} y_0\dfrac{ h}{ q} - y_0\dot{x}_0 - 2\dot{y}_0r_0 ,&{} z_0\dfrac{ h}{ q} - z_0\dot{x}_0 - 2\dot{z}_0r_0,&{} -2r_0 q ,&{} -2r_0y_0 ,&{} -2r_0z_0 \end{array}\right] \end{aligned}$$

having introduced the auxiliary variables:

$$\begin{aligned} f=y_0\dot{z}_0-z_0\dot{y}_0, \quad h=y_0\dot{y}_0+z_0\dot{z}_0, \qquad q=r_0+|x_0|,\qquad k = \dfrac{\sqrt{2}}{8}\dfrac{1}{r_0\sqrt{ q}}. \end{aligned}$$

For $x_0<0$ it is more convenient to use

$$\begin{aligned} {{\mathbf {\mathsf{{T}}}}}^{-}({\mathbf {x}}_0) = k\left[ \begin{array}{llllll} -2z_0 ,&{} 2\dfrac{y_0z_0}{ q} ,&{} -4r_0 +2\dfrac{z_0^2}{ q} ,&{} 0 ,&{} 0 ,&{} 0 \\ {0} ,&{} {0} ,&{} {0},&{} 0 ,&{} 0 ,&{} 0 \\ 2 q ,&{} -2y_0 ,&{} -2z_0,&{} 0 ,&{} 0 ,&{} 0\\ 2y_0 ,&{} 4r_0 - 2\dfrac{y_0^2}{ q} ,&{} -2\dfrac{y_0z_0}{ q},&{} 0 ,&{} 0 ,&{} 0\\ q\dot{z}_0-z_0\dot{x}_0 ,&{} -y_0\Big (\dot{z}_0-\dfrac{z_0\dot{x}_0}{ q} \Big ) ,&{} \dfrac{z_0^2\dot{x}_0}{ q}- 2r_0\dot{x}_0 - z_0\dot{z}_0 ,\,\,&{} -2r_0z_0 ,&{} 0 ,&{} -2r_0 q \\ f\, ,&{} 2\dot{z}_0r_0 - y_0\dfrac{ f}{ q} ,&{} - 2\dot{y}_0r_0 - z_0\dfrac{ f}{ q} ,&{} 0 ,&{} -2r_0z_0 ,&{} +2r_0y_0 \\ - q\dot{x}_0 - h ,&{} y_0\dot{x}_0 -2\dot{y}_0r_0 +y_0\dfrac{ h}{ q} ,&{} z_0\dot{x}_0 -2\dot{z}_0r_0 +z_0\dfrac{ h}{ q} ,&{} 2r_0 q ,&{} -2r_0y_0 ,&{} -2r_0z_0 \\ y_0\dot{x}_0- q\dot{y}_0 ,\,&{} y_0\dot{y}_0 +2\dot{x}_0r_0 -\dfrac{y_0^2\dot{x}_0}{ q} ,\,\,&{} z_0\Big ( \dot{y}_0-\dfrac{y_0\dot{x}_0}{ q} \Big ) ,&{} 2r_0y_0 ,&{} 2r_0 q ,&{} 0 \end{array}\right] \end{aligned}$$