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Previous work on time-optimal satellite slewing maneuvers, with one satellite axis (sensor axis) required to obey multiple path constraints (exclusion from keep-out cones centered on high-intensity astronomical sources) reveals complex motions with no part of the trajectory touching the constraint boundaries (boundary points) or lying along a finite arc of the constraint boundary (boundary arcs). This paper examines four cases in which the sensor axis is either forced to follow a boundary arc, or has initial and final directions that lie on the constraint boundary. Numerical solutions, generated via a Legendre pseudospectral method, show that the forced boundary arcs are suboptimal. Precession created by the control torques, moving the sensor axis away from the constraint boundary, results in faster slewing maneuvers. A two-stage process is proposed for generating optimal solutions in less time, an important consideration for eventual onboard implementation.

The problem of reorienting a spacecraft in minimum time, often through large angles (so-called

Before addressing the time-optimal, constrained problem, it is useful to review what is known about the unconstrained problem. In a seminal paper, Bilimoria and Wie [

Several subsequent papers have revisited the unconstrained problem, including such modifications as axisymmetric mass distribution and only two-axis control [

Hablani [

Trajectory of sensor axis between Sun (yellow) and Moon (gray) cones.

This paper presents a preliminary study of boundary arcs and boundary points in this same problem. A full analytical solution is not possible; however, some insight to the problem can be gained by examining instances where the sensor axis is constrained to follow the constraint boundary, and those where the initial and final sensor axis directions lie exactly on the boundary. The paper also addresses the practical challenge of implementing onboard optimal control for this type of maneuver and proposes a means for reducing computation time for the reference trajectory.

The problem is formulated as a Mayer optimal control problem, with performance index

The appendix discusses the relationship between the Euler parameters,

For the

Analytically, the path constraint given by (

For some missions, the reorientation strategy may be altered if the final orientation is not completely specified. An example would be the need to reorient the sensor axis to a desired target direction in minimum time, with no constraints on the orientation of the other body-fixed axes at the final time. In practice, some subsequent rotation about the sensor axis might be required to optimize some other parameter (e.g., maximizing illumination of solar panels, or minimizing solar heating of sensitive components), but the principal reorientation maneuver could be achieved faster. The corresponding optimal control problem is the same as before, but with the final conditions on the Euler parameter values given in (

Consider now the suboptimal approach of a simple

Constrained rotation about the keep-out cone axis.

Four cases are considered: two of these (cases BA_{1} and BA_{2}) constrain the sensor axis to move along the constraint boundary; the initial and final sensor axis directions also lie on the boundary. These cases serve as proxies for boundary arcs, at least in that they provide some indication of the slewing time and other qualitative properties of the motion. The other spacecraft axes are unconstrained at the final time. The other two cases (BP_{1} and BP_{2}) have the sensor axis beginning and terminating on the constraint boundary (and of course, prohibited from entering the constraint cone); the other spacecraft axes are unconstrained at the final time.

The numerical results are generated using a Legendre pseudospectral method, implemented in the software package DIDO [^{−16} (corresponding units) at each node. These cases all require significant computation time (as much as 72 hours on a computer with an Intel Core 2 2.0 GHz processor, with the number of pseudospectral nodes in the range of 100–250.

Note that the Hamiltonian and costate values are not evaluated directly during the pseudospectral solution process, but rather are reconstructed via the covector mapping principle [

A system of nondimensional units is employed, partly to provide somewhat more general results, but chiefly because this system of units provides the kind of scaling needed for the pseudospectral method to perform well. In physical units, the angular velocities, moments of inertia, and control torques about the spacecraft’s principal axes are denoted as

This is a forced boundary arc trajectory of the sensor axis

A notable feature of the motion (Figure

Dynamic response and controls for the case BA_{1}.

Dynamic response and controls [

Figure

Hamiltonian and costates for the case BA_{1}.

For this forced boundary arc, the constraint cone has half-angle

For this case, the solution uses 100 nodes and has a final time of _{1}. It should be noted that in both BA_{1} and BA_{2}, the path of the sensor axis is verified to lie within 10^{−16} radians of the constraint boundary.

Dynamic response and controls for the case BA_{2}.

Hamiltonian and costates for the case BA_{2}.

This case is identical to case BA_{1}, except that only the initial and final directions of the sensor axis are constrained to lie on the constraint boundary, corresponding to two forced boundary points. Two of the control torques (

Dynamic response and controls for the case BP_{1}.

Hamiltonian and costates for the case BP_{1}.

Distance from the sensor axis to the constraint boundary (case BP_{1}).

Nevertheless, the trend is clear: precession created by the control torques, works to reduce the final time to 1.9258, approximately 1% faster than the solution in BA_{1}.

This case is identical to case BA_{2}, except that only the initial and final directions of the sensor axis are constrained to lie on the constraint boundary. Unlike case BP_{1}, the control torques display more intermediate behavior (Figure _{1}, the only points where the sensor axis contacts the constraint boundary (see Figure

Dynamic response and controls for the case BP_{2}.

Hamiltonian and costates for the case BP_{2}.

Distance from the sensor axis to the constraint boundary (case BP_{2}).

Regardless of whether boundary points or boundary arcs exist as part of an optimal trajectory, the numerical determination of the solution via a pseudospectral method frequently requires considerable computation time; indeed, the actual slewing maneuver of the spacecraft can be accomplished in seconds or minutes (depending upon the control authority) while the pseudospectral solver requires from 20 minutes to 72 hours to compute the solution. Experience shows that providing even a rudimentary estimate of the states and controls as an initial guess for the pseudospectral solver can reduce the computation time significantly. A two-stage process is proposed, wherein a random-process algorithm, such as a particle swarm optimizer (PSO) which can rapidly explore the solution space, provides the initial guess to the pseudospectral algorithm. The literature abounds with hybrid methods used in related control problems. For example, Ahmed et al. [

Initial efforts that employ a two-stage process for optimal slewing maneuvers show promising results. A PSO produces a low-quality approximation of the states, controls, and node times for the solution, followed by the pseudospectral solver, which takes the approximate solution as its initial starting point. An efficient method for generating the first-stage solution is to represent each control torque component

As an example, the control torque for a one-dimensional slewing maneuver (with no keep-out cones) is shown in Figure

(a) Approximate solution for 1D slewing maneuver, generated by 50 iterations of a particle swarm optimizer. (b) Optimal control solution for the same maneuver, generated by DIDO (running in accurate mode). Total required cpu time for solution: 148 sec using only DIDO, with no initial guess; 76 sec for the combined PSO-DIDO solution.

This preliminary study indicates that, although boundary arcs and boundary points may exist in time-optimal spacecraft slewing maneuvers with path constraints, they are at best part of a suboptimal solution. The numerical calculations (completed via a Legendre pseudospectral method) show that even if the initial and final states are boundary points, the solution moves away from the constraint boundary, resulting in a lower final time than if the motion is forced to move along the boundary (a forced boundary arc). The necessary conditions lead to an unwieldy set of relations, making it impossible to determine analytically if boundary points or boundary arcs are excluded. Further examination of the problem using a Bellman chain to improve the numerical accuracy may provide additional insight. A two-stage method for generating optimal solutions in less time than that required by the pseudospectral method alone shows some promise, but further work is needed to determine its utility for the three-dimensional constrained problem.

Consider a dextral orthonormal basis set

If one uses a direction cosine matrix

Every direction cosine matrix has one unity-valued eigenvalue with corresponding eigenvector (in matrix form)

By taking dot products of both sides of (

Constraint function

Hamiltonian without the path constraint

Hamiltonian with the path constraint

Principal moment of inertia (nondimensional)

Principal moment of inertia (in physical units)

Performance index

Control torque (nondimensional)

Control torque (in physical units)

Final time

Half-angle of the keep-out cone for object

Euler parameter

Rotation axis

Costate corresponding to state

Lagrange multiplier associated with constraint function

Rotation angle about the

Time unit

Direction of the sensor axis

Direction of the central axis of the keep-out cone for object

Angular velocity component (nondimensional)

Angular velocity component (in physical units).

This paper has been presented at the 6th International Workshop and Advanced School “Spaceflight Dynamics and Control,” Covilha, Portugal, March 28–30, 2011,