Closed-Form Solutions for Single-Axis Slew Maneuvers Under a Fixed Final Time Constraint

Abstract

This paper addresses a single-axis rotation problem in diverse space applications. In the problem, spin-to-spin boundary conditions, a finite jerk constraint, and a fixed final time constraint are considered. The objective is to find a path which maximizes the stabilization period under the given state boundary conditions and path inequality constraints. The optimal control problem is converted into a switching time design problem, and the unknown parameters are calculated analytically by solving the quartic, the cubic, or the quadratic equation. The proposed computational procedures are explained through numerical examples and the closed-form solutions are verified.

Appendix

1.1 Formula of Quartic Equations

In this Appendix, a formula for the below quartic equation is summarized.

$$ ax^{4} + bx^{3} + cx^{2} + dx + e = 0, $$

(A.1)

where $a = 1$. First, a cubic equation is formulated as follows:

$$ y^{3} + \frac{f}{2}y^{2} + \left( {\frac{{f^{2} - 4h}}{16}} \right)y - \frac{{g^{2} }}{64} = 0, $$

(A.2)

where f, g, and h are as follows:

$$ f = c - \frac{{3b^{2} }}{8}, $$

(A.3)

$$ g = d + \frac{{b^{3} }}{8} - \frac{bc}{2}, $$

(A.4)

$$ h = e - \frac{{3b^{4} }}{256} + \frac{{b^{2} c}}{16} - \frac{bd}{4}. $$

(A.5)

Let $y_{1}$, $y_{2}$, and $y_{3}$ be the roots of the cubic equation in Eq. (A.2), and $y_{2}$ and $y_{3}$ be the non-zero roots. Using $y_{1}$, $y_{2}$, and $y_{3}$, the following p, q, r, and s are calculated.

$$ p = \sqrt {y_{2} } , $$

(A.6)

$$ q = \sqrt {y_{3} } , $$

(A.7)

$$ r = - \frac{g}{8pq}, $$

(A.8)

$$ s = \frac{b}{4a}. $$

(A.9)

Four roots denoted by $x_{1}$, $x_{2}$, $x_{3}$, and $x_{4}$ for the quartic equation in Eq. (A.1) are derived as follows:

$$ x_{1} = p + q + r - s, $$

(A.10)

$$ x_{2} = p - q - r - s, $$

(A.11)

$$ x_{3} = - p + q - r - s, $$

(A.12)

$$ x_{4} = - p - q + r - s. $$

(A.13)