Unconstrained Direct Optimization of Spacecraft Trajectories Using Many Embedded Lambert Problems

Abstract

Direct optimization of many-revolution spacecraft trajectories is performed using an unconstrained formulation with many short-arc, embedded Lambert problems. Each Lambert problem shares its terminal positions with neighboring segments to implicitly enforce position continuity. Use of embedded boundary value problems (EBVPs) is not new to spacecraft trajectory optimization, including extensive use in primer vector theory, flyby tour design, and direct impulsive maneuver optimization. Several obstacles have prevented their use on problems with more than a few dozen segments, including computationally expensive solvers, lack of fast and accurate partial derivatives, unguaranteed convergence, and a non-smooth solution space. Here, these problems are overcome through the use of short-arc segments and a recently developed Lambert solver, complete with the necessary fast and accurate partial derivatives. These short arcs guarantee existence and uniqueness for the Lambert solutions when transfer angles are limited to less than a half revolution. Furthermore, the use of many short segments simultaneously approximates low-thrust and eliminates the need to specify impulsive maneuver quantity or location. For preliminary trajectory optimization, the EBVP technique is simple to implement, benefiting from an unconstrained formulation, the well-known Broyden–Fletcher–Goldfarb–Shanno line search direction, and Cartesian coordinates. Moreover, the technique is naturally parallelizable via the independence of each segment’s EBVP. This new, many-rev EBVP technique is scalable and reliable for trajectories with thousands of segments. Several minimum fuel and energy examples are demonstrated, including a problem with 6143 segments for 256 revolutions, found within 5.5 h on a single processor. Smaller problems with only hundreds of segments take minutes.

Appendix

The appendix contains unconstrained optimizer algorithms, a sensitivity discussion, and performance tables.

1.1 Unconstrained Optimizer and the BFGS Search Direction

The unconstrained optimizer $\text {FMINUNC}_{\mathrm{UT}}$ is presented in Algorithm 1 along with the BFGS search direction in Algorithm 2 for a line search. The computation of the search direction leverages the efficient BLAS [57] subroutines. There are (s)ymmetric and (p)acked versions of some subroutines where packed takes approximately half the memory as the symmetric version counterpart, enabling the computational feasibility of larger problems but with no parallelization runtime improvements for an increase in thread count as shown in Sect. 4.2. The BLAS subroutines are: ddot for the dot product of two vectors; dsymv or dspmv for the multiplication of a matrix and vector; dsyr and dsyr2 or dspr and dspr2 for two different forms of an outer-product; and dnrm2 for the 2-norm of a vector. The search direction is always one unit in magnitude to conveniently limit the step size in the line search. The quadratic growth of the computation shown in Sect. 4.1 is from dspmv, dspr, and spr2 that require $2 n^2$, $n^2$, and $2 n^2$ floating-point operations, respectively.

Every line search performs three primary computations. First, the minimum is bracketed but with a bounded max step size. Second, the bracketed space is shrunk via a golden ratio method. Third, the minimization is finalized with quadratic interpolation of the three points closest to the minimum. If the denominator in the formula for the quadratic interpolation equals or is close to zero, then the lowest point of the three is chosen as the actual minimum. The algorithm for the line search is not explicitly included for brevity.

1.2 Sensitivities

The computational sensitivities are essential for a gradient-based, direct optimization method. These partials can be obtained via many methods of varying accuracy, including finite difference methods, complex step methods, automatic differentiation, and more. The variational equations approach in Ref. [71] or Battin’s variational STM implementation [9] on page 467 (implemented in detail in Ref. [4]) is accurate and common in practice. In the current work, the closed-form ivLam for two-body dynamics is faster due to the absence of an integrated solution to an ODE system as shown in examples 1.c and 1.d and visualized in Fig. 4. A variational equation approach to EBVPs is useful to model perturbed two-body dynamics. A faster, but less accurate, alternative is to approximate two-body perturbations as impulsive $\varDelta \mathbf {v}$ maneuvers [43].

Table 3 EBVP partials in terms of the variational STM

Table 4 Performance of examples 1.a, 1.b, 1.c, and 1.d for the search direction and EBVP solver comparisons

Table 5 Performance of examples 1.e and 1.f for the initial guess demonstrations

Table 6 Performance of ex. 2 that demonstrates a continuation method for a large number of nodes

1.2.1 Cost Partials and Differentials

The following derivation of the cost partials is based off differentials $d(\cdot )$ to organize dependencies within the system. As mentioned previously, the cost J in Eq. (1) is a function of the impulsive $\varDelta \mathbf {v}_i$ maneuvers at each node i, so $J = J ( \varDelta v_1 \left( \varDelta \mathbf {v}_1 \right) , \cdots , \varDelta v_n \left( \varDelta \mathbf {v}_n \right) )$. Using the chain-rule, expand J as a cost differential for nodes 1 to n

$$\begin{aligned} d J = \frac{\partial J}{\partial \varDelta v_1} \frac{\partial \varDelta v_1}{\partial \varDelta \mathbf {v}_1} d \varDelta \mathbf {v}_1 + \cdots + \frac{\partial J}{\partial \varDelta v_n} \frac{\partial \varDelta v_1}{\partial \varDelta \mathbf {v}_n} d \varDelta \mathbf {v}_n \end{aligned}$$

(6)

where the cost partial of node i in terms of the impulsive $\varDelta \mathbf {v}_i$ maneuver is $\frac{\partial J}{\partial \varDelta v_i} = 1$ and $\frac{\partial \varDelta v_i}{\partial \varDelta \mathbf {v}_i} = \frac{\varDelta \mathbf {v}_i^T}{\varDelta v_i}$ for minimum fuel [$k=1$ in Eq. (1)] or $\frac{\partial J}{\partial \varDelta v_i} = \varDelta v_i$ and $\frac{\partial \varDelta v_i}{\partial \varDelta \mathbf {v}_i} = \varDelta \mathbf {v}_i^T$ for minimum energy [$k=0$ in Eq. (1)] or a hybrid combination for $0<k<1$ in Eq. (1). The cost differential Eq. (6) currently terminates at the impulsive $\varDelta \mathbf {v}_i$ maneuver differential $d \varDelta \mathbf {v}_i$ for each node i, but as previously mentioned, the impulsive $\varDelta \mathbf {v}_i$ maneuver is a function of the velocity before $\mathbf {v}_{i-}$ and after $\mathbf {v}_{i+}$ node i written as $\varDelta \mathbf {v}_i = \mathbf {v}_{i+} - \mathbf {v}_{i-}$ and these velocities are a function of position and time written as $\mathbf {v}_{i-} = \mathbf {v}_{i-}(\mathbf {r}_{i-1},\mathbf {r}_i,t_{(i-1)+},t_{i-}) \text {and} \mathbf {v}_{i+} = \mathbf {v}_{i+}(\mathbf {r}_i,\mathbf {r}_{i+1},t_{i+},t_{(i+1)-})$. Thus, the differential of the impulsive $\varDelta \mathbf {v}_i$ maneuver is $d \varDelta \mathbf {v}_i = d \mathbf {v}_{i+} - d \mathbf {v}_{i-}$, and the differential of the velocity before $d \mathbf {v}_{i-}$ and after $d \mathbf {v}_{i+}$ node i is

$$\begin{aligned} d \mathbf {v}_{i-} = {}&\left\langle \frac{\partial \mathbf {v}_{i-}}{\partial \mathbf {r}_{i-1 }} \right\rangle d \mathbf {r}_{i-1 } + \left\langle \frac{\partial \mathbf {v}_{i-}}{\partial \mathbf {r}_{i }} \right\rangle d \mathbf {r}_{i } + \left\langle \frac{\partial \mathbf {v}_{i-}}{\partial t_{(i-1)+}} \right\rangle d t_{(i-1)+} + \left\langle \frac{\partial \mathbf {v}_{i-}}{\partial t_{i- }} \right\rangle d t_{i- } \end{aligned}$$

(7)

$$\begin{aligned} d \mathbf {v}_{i+} = {}&\left\langle \frac{\partial \mathbf {v}_{i+}}{\partial \mathbf {r}_{i }} \right\rangle d \mathbf {r}_i + \left\langle \frac{\partial \mathbf {v}_{i+}}{\partial \mathbf {r}_{i+1 }} \right\rangle d \mathbf {r}_{i+1 } + \left\langle \frac{\partial \mathbf {v}_{i+}}{\partial t_{i+ }} \right\rangle d t_{i+ } + \left\langle \frac{\partial \mathbf {v}_{i+}}{\partial t_{(i+1)-}} \right\rangle d t_{(i+1)-} \end{aligned}$$

(8)

where the $\left\langle \cdot \right\rangle$ represents the EBVP partials that assume position continuity is enforced by design by using EBVPs, not the optimizer. Before substitution of Eqs. (7) and (8) in Eq. (6), careful consideration must be made toward the first node 1 and the last node n. The initial and final nodes are constrained by design, not the optimizer, because they are prescribed by a user-defined path, in this current work this path is a two-body orbit. Thus, the differentials for position at the initial and final orbits are $d \mathbf {r}_1 = \mathbf {v}_{1-} d \tilde{t}_{1-}$ and $d \mathbf {r}_n = \mathbf {v}_{n+} d \tilde{t}_{n+}$ and the differentials for velocity are $d \mathbf {v}_{1-} = \mathbf {a}_{1-} d \tilde{t}_{1-}$ and $d \mathbf {v}_{n+} = \mathbf {a}_{n+} d \tilde{t}_{n+}$. The time-like $\tilde{t}$ to parameterize an orbit is not associated with the time of flight of the transfer trajectory. Furthermore, for every segment $(i+1)i$ from node i to $i+1$, the time of flight is defined to be $\varDelta t = t_{(i+1)-} - t_{i+}$ and the time of flight differential is $d \varDelta t = d t_{(i+1)-} - d t_{i+}$. For convenience, the initial time $t_{i+}$ of segment $(i+1)i$ is designed to be zero, so the time of flight $\varDelta t$ is the final time $t_{(i+1)-}$ of segment $(i+1)i$, i.e., $\varDelta t = t_{(i+1)-}$. The initial time $t_{i+}$ can conveniently be zero because time does not explicitly show up in the dynamics. In other words, for time-free unconstrained spacecraft trajectory optimization problems, only the time of flight $\varDelta t$ of each segment $(i+1)i$ is needed, not the initial and final times of each segment. Thus, when computing the velocity differentials at node i shown in Eqs. (7) and (8), the time of flight differential is $d \varDelta t = d t_{i-} = d t_{(i+1)-}$ because $d t_{(i-1)+} = d t_{i+} = 0$.

Now, substitute Eqs. (7) and (8) in Eq. (6), the aforementioned differentials for time and position, and group similar differential terms to get a cost differential of the form

$$\begin{aligned} d J = \frac{\partial J}{\partial \tilde{t}_{1-}} d \tilde{t}_{1-} + \frac{\partial J}{\partial \tilde{t}_{n+}} d \tilde{t}_{n+} + \frac{\partial J}{\partial \varDelta t} d \varDelta t + \frac{\partial J}{\partial \mathbf {r}_2} d \mathbf {r}_2 + \cdots + \frac{\partial J}{\partial \mathbf {r}_{n-1}} d \mathbf {r}_{n-1} \end{aligned}$$

(9)

where the cost partials are the coefficients of each differential of the initial time-like variable on the initial orbit $\tilde{t}_{1-}$, the final time-like variable on the final orbit $\tilde{t}_{n+}$, the time of flight $\varDelta t$ of each segment $(i+1)i$, and all intermediate positions $\mathbf {r}_i$ from node 2 to $n-1$. The cost partials are

$$\begin{aligned} \frac{\partial J}{\partial \tilde{t}_{1-}} = {}&\frac{\partial J}{\partial \varDelta v_1} \frac{\partial \varDelta v_1}{\partial \varDelta \mathbf {v}_1} \left( \left\langle \frac{\partial \mathbf {v}_{1+}}{\partial \mathbf {r}_1} \right\rangle \mathbf {v}_{1-} - \mathbf {a}_{1-} \right) + \frac{\partial J}{\partial \varDelta v_2} \frac{\partial \varDelta v_2}{\partial \varDelta \mathbf {v}_2} \left( -\left\langle \frac{\partial v_{2-}}{\partial r_1} \right\rangle \mathbf {v}_{1-} \right) \\ \frac{\partial J}{\partial \tilde{t}_{n+}} = {}&\frac{\partial J}{\partial \varDelta v_{n-1}} \frac{\partial \varDelta v_{n-1}}{\partial \varDelta \mathbf {v}_{n-1}} \left( \left\langle \frac{\partial \mathbf {v}_{(n-1)+}}{\partial \mathbf {r}_n} \right\rangle \mathbf {v}_{n+} \right) + \frac{\partial J}{\partial \varDelta v_n} \frac{\partial \varDelta v_n}{\partial \varDelta \mathbf {v}_n} \left( \mathbf {a}_{n+} - \left\langle \frac{\partial \mathbf {v}_{n-}}{\partial \mathbf {r}_n} \right\rangle \mathbf {v}_{n+} \right) \\ \frac{\partial J}{\partial \varDelta t} = {}&\frac{\partial J}{\partial \varDelta v_1} \frac{\partial \varDelta v_1}{\partial \varDelta \mathbf {v}_1} \left\langle \frac{\partial \mathbf {v}_{1+}}{\partial \varDelta t} \right\rangle + \sum _{i=2}^{n-1} \frac{\partial J}{\partial \varDelta v_i} \frac{\partial \varDelta v_i}{\partial \varDelta \mathbf {v}_i} \left( - \left\langle \frac{\partial \mathbf {v}_{i-}}{\partial \varDelta t} \right\rangle + \left\langle \frac{\partial \mathbf {v}_{i+}}{\partial \varDelta t} \right\rangle \right) \\ {}&+ \frac{\partial J}{\partial \varDelta v_n} \frac{\partial \varDelta v_n}{\partial \varDelta \mathbf {v}_n} \left( -\left\langle \frac{\partial \mathbf {v}_{n-}}{\partial \varDelta t} \right\rangle \right) \\ \frac{\partial J}{\partial \mathbf {r}_i} = {}&\frac{\partial J}{\partial \varDelta v_{i-1}} \frac{\partial \varDelta v_{i-1}}{\partial \varDelta \mathbf {v}_{i-1}} \left\langle \frac{\partial \mathbf {v}_{(i-1)+}}{\partial \mathbf {r}_i} \right\rangle + \frac{\partial J}{\partial \varDelta v_i} \frac{\partial \varDelta v_i}{\partial \varDelta \mathbf {v}_i} \left( \left\langle \frac{\partial \mathbf {v}_{i+}}{\partial \mathbf {r}_i} \right\rangle - \left\langle \frac{\partial \mathbf {v}_{i-}}{\partial \mathbf {r}_i} \right\rangle \right) \\ {}&+ \frac{\partial J}{\partial \varDelta v_{i+1}} \frac{\partial \varDelta v_{i+1}}{\partial \varDelta \mathbf {v}_{i+1}} \left( - \left\langle \frac{\partial \mathbf {v}_{(i+1)-}}{\partial \mathbf {r}_i} \right\rangle \right) \end{aligned}$$

Note the following useful identities between velocity partials that relates flight time $\varDelta t$ and time t before and after node i: $\left\langle \frac{\partial \mathbf {v}_{i-}}{\partial \varDelta t} \right\rangle = \left\langle \frac{\partial \mathbf {v}_{i-}}{\partial t_{i-}} \right\rangle \text { and } \left\langle \frac{\partial \mathbf {v}_{i+}}{\partial \varDelta t} \right\rangle = \left\langle \frac{\partial \mathbf {v}_{i+}}{\partial t_{(i+1)-}} \right\rangle$. Also note, if the problem is fixed-state to fixed-state, not orbit-to-orbit, then the initial and final time-like variables that parameterize the initial and final orbits do not vary, i.e., $d \tilde{t}_{1-} = 0$ and $d \tilde{t}_{n+} = 0$ and the associated cost partials are zero, i.e., $\frac{\partial J}{\partial \tilde{t}_{1-}} = 0 \text {and} \frac{\partial J}{\partial \tilde{t}_{n+}} = 0$. Likewise, though not explored in this current work, the time of flight cost partial is zero, $\frac{\partial J}{\partial \varDelta t} = 0$, if the total time of flight is fixed.

1.2.2 Variational State Transition Matrix

The EBVP partials, designated by $\left\langle \cdot \right\rangle$, can be derived either directly when using ivLam, or through the variational STM via integration for rk7sh and rk78sh or by forming the Keplerian submatrices, e.g., via Battin’s Kepler variational STM implemented in Arora et al. [4], or other analytic forms of two-body dynamics by Shepperd [86], Goodyear [37], Pitkin [77], Glandorf [34], and more.

The variational STM solves the following variational equations of motion for a segment $\delta \dot{\mathbf {x}} = \mathbf {F} \delta \mathbf {x}$ for dynamics of the form $\dot{\mathbf {x}} = \mathbf {f}(\mathbf {x})$ so $\mathbf {F} = \partial \mathbf {f}/\partial \mathbf {x}$. The variational STM can then be written as:

$$\begin{aligned} \delta \mathbf {x}_{(i+1)-} = \varvec{\varPhi }(t_{(i+1)-},t_{i+}) \delta \mathbf {x}_{i+} \end{aligned}$$

(10)

for segment $(i+1)i$ from $t_{i+}$ to $t_{(i+1)-}$ or node i to $i+1$. The variational STM $\varvec{\varPhi } = \varvec{\varPhi }(t_{(i+1)-},t_{i+})$ describes the effect of the sensitivity of the initial state, time-fixed variation $\delta \mathbf {x}_{i+} = \delta \mathbf {x}(t_{i+})$ on a final state, time-fixed variation $\delta \mathbf {x}_{(i+1)-} = \delta \mathbf {x}(t_{(i+1)-})$.

The variational STM is then either numerically integrated with $\dot{\varvec{\varPhi }} = \mathbf {F} \varvec{\varPhi }$ or analytically known. The integrated solution is decomposed into sub-matrix form as:

$$\begin{aligned} \varvec{\varPhi } = \frac{\partial \mathbf {x}_{(i+1)-}}{\partial \mathbf {x}_{i+}} = \begin{bmatrix} \frac{\partial \mathbf {r}_{(i+1)-}}{\partial \mathbf {r}_{i+}} &{} \frac{\partial \mathbf {r}_{(i+1)-}}{\partial \mathbf {v}_{i+}}\\ \frac{\partial \mathbf {v}_{(i+1)-}}{\partial \mathbf {r}_{i+}} &{} \frac{\partial \mathbf {v}_{(i+1)-}}{\partial \mathbf {v}_{i+}} \end{bmatrix} = \begin{bmatrix} \mathbf {A}_{(i+1)i} &{} \mathbf {B}_{(i+1)i} \\ \mathbf {C}_{(i+1)i} &{} \mathbf {D}_{(i+1)i} \end{bmatrix} \end{aligned}$$

(11)

where the sub-matrices $\mathbf {A}_{(i+1)i}$, $\mathbf {B}_{(i+1)i}$, $\mathbf {C}_{(i+1)i}$, and $\mathbf {D}_{(i+1)i}$ are defined for notational simplicity. A closed-form variational STM for two-body dynamics by Battin is not reproduced here for compactness but can be found in Ref. [4] via Eqs. 38–50. A similar form of the variational STM can be made for segment $i(i-1)$. The EBVP partials in terms of the variational STM sub-matrices are shown in Table 3. The expressions in Table 3 are primarily for perturbed two-body dynamics; however, the submatrices can be found analytically as previously mentioned for the two-body problem to give the partials of the Lambert problem.

1.2.3 Mass-Leak

The partials of Eq. (1) of minimum fuel include a singularity because the partial $\partial \varDelta v_i/\partial \varDelta \mathbf {v}_i = \varDelta \mathbf {v}_i^T/\varDelta v_i$ for node i has a denominator that can go to zero. A “mass-leak" is termed and eliminates the singularity by including a small constant quantity $\epsilon$

$$\begin{aligned} \varDelta v_i = \sqrt{\varDelta v_{x_i}^2 + \varDelta v_{y_i}^2 + \varDelta v_{z_i}^2 + \epsilon ^2} \end{aligned}$$

(12)

where $\varDelta v_{x_i}$, $\varDelta v_{y_i}$, and $\varDelta v_{z_i}$ are the x, y, and z-components of the impulsive $\varDelta \mathbf {v}_i$ maneuver. Practical choices for $\epsilon$ are approximately $\epsilon = 10^{-6}$ to $10^{-4}$ when the magnitude of velocity is near unity. In practice, the optimizer decreases the value of the impulsive $\varDelta \mathbf {v}_i$ maneuver magnitudes, modified according to Eq. (12), to approximately $\epsilon$ if the maneuver is not needed for minimum fuel solutions. Minimum energy solutions do not need the mass-leak.

1.3 Performance Tables for Examples 1 and 2

The performance tables are for examples 1 and 2 (see Fig. 4). Tables 4, 5 and 6 show total runtime per iteration, total runtimes, number of iterations that update the decision state, function calls, cost, and optimality tolerances. Function call counts are for the computation of the cost, the cost plus partials, and the search direction, respectively.