**6. Inverse problem of photogravitational dynamics**

Considering the inverse problem for finding forces for a known or given motion in the case of two interacting bodies, we find the control accelerations that can realize the trajectory in the central field

$$f\_i = \frac{d^2 \mathbf{x}\_i}{dt^2} + \frac{\mu}{r^3} \mathbf{x}\_i = u\_i(t), \quad i = \mathbf{1,2,3}, \tag{21}$$

when the parametric functions *xi*ð Þ*t* are predefined for *t* ∈ [0,T].

The results of solving the inverse problem of dynamics are presented as examples. Simulation of closed trajectories *<sup>x</sup>* <sup>¼</sup> *<sup>r</sup>*<sup>0</sup> cos <sup>2</sup>*t*, *<sup>y</sup>* <sup>¼</sup> *<sup>r</sup>*<sup>0</sup> sin *<sup>t</sup>* cos*t*, *<sup>z</sup>* <sup>¼</sup> *<sup>h</sup>*<sup>3</sup> sin *<sup>t</sup>* of a spacecraft with a controlled solar sail is presented to reach the heliopolar regions (**Figure 1**, Viviani curve), and fly over the North and South poles of the Sun and return to near-earth orbit [2]. Assuming that the required control accelerations are created by the position of the solar sail [5], we write them in the form

$$\begin{cases} \quad u\_1 = -\frac{q}{r^2} \cos \gamma\_1 \cos \gamma\_2, \\ \quad u\_1 = \frac{q}{r^2} \sin \gamma\_1 \cos \gamma\_2, \\ \quad u\_1 = \frac{q}{r^2} \sin \gamma\_2 \end{cases} \tag{22}$$

where γ<sup>1</sup> and γ<sup>2</sup> are the angles of the three-dimensional orientation of the normal vector to the shadow side of the mirror sail on both sides, and the dimensionless factor *k* depends on the solar radiation pressure on the surface of the spacecraft sail (**Figure 2**). From here, we obtain the formulas for finding the angles and the construction of the control.

**Figure 2.** *Graph of γ<sup>1</sup> and γ<sup>2</sup> for Viviani curve at h3 = 0.3r0,T3 = 1.03 years.*

*Astrodynamics in Photogravitational Field of the Sun: Space Flights with a Solar Sail DOI: http://dx.doi.org/10.5772/intechopen.102005*

**Figure 3.** *The spiral trajectory is defined by expressions in the orbital coordinate system.*

The spiral trajectory (**Figure 3**) of the possible motion of the CASP is determined by the equations

$$\begin{cases} \quad \varkappa = r\_0 \cos \alpha\_0 t, \\ \quad y = r\_0 \sin \alpha\_0 t + r\_1 \cos \alpha\_1 t, \\ \quad z = r\_1 \sin \alpha\_1 t. \end{cases} \tag{23}$$

The helical trajectory can be supplemented by the selected law of control over the position of the sail elements.

The solution of the inverse problem of dynamics when moving along a given trajectory makes it possible to obtain an initial approximation for control and to evaluate the possibility of implementing a system of spacecraft sail elements for the selected model. With this solution, you can also refine the rotation or steering of the entire structure with respect to the center of mass.
