**5. Control and stabilization capabilities**

The ability to control the orientation of the CASP and individual elements of the sails system has a special character to form the best control of the main thrust vector and the moment of forces for turning the CASP relative to the center of mass or maintaining the desired position.

Heliocentric motion of a spacecraft with a solar sail during flights to the Sun, planets, asteroids, or comets; and also to create special orbits of motion in the vicinity of the Sun, taking into account the force of light pressure, leads to the equations of motion in the form of a system

$$\frac{d^2\mathbf{x}\_i}{dt^2} = -\frac{\mu}{r^3}\mathbf{x}\_i + \frac{\partial U}{\partial \mathbf{x}\_i} + f\_i(\mathbf{x}) + u\_i(t, \mathbf{x}), \quad i = 1, 2, 3,\tag{16}$$

where the notation is used: *xi* are the Cartesian coordinates of the spacecraft, *r* is the modulus of the radius vector, *μ* is the gravitational parameter of the central body, the function *U* is determined by the influence of disturbances from potential forces, and the functions *fi* ð Þ *x* , *ui*ð Þ *t*, *x* are the components of the acceleration vectors non-potential forces and the vector control contribution *u t*ð Þ , *x* , including the action of light pressure forces or jet engines on active sections of motion, when expanded on the axes of the orbital coordinate system [20]. In this case, the forces of light pressure on the elements of the sail system and the moments relative to the center of mass of the system determine the vector quantities:

$$\mathbf{F} = \sum \mathbf{F}\_i = \sum k\_i \mathbf{S}\_i \frac{b\_i}{r^2} \mathbf{n}\_i \ (\theta\_i), \ \mathbf{M} = \sum\_i \mathbf{p}\_i \times \mathbf{F}\_i(\theta\_i). \tag{17}$$

The contribution of the light pressure is determined by the angle of deviation of the normal vector *n* from the direction of the flux of the sun's rays *e*. If the flat mirror sail is angled *θ<sup>i</sup>* to the beams, then the transmitted pulse will be directed almost perpendicular to the reflective surface. The photons will retain a part of the impulse directed parallel to the sail, so that the sail will get less than with full opening to the rays. The magnitude of the light pressure decreases, and the direction will almost coincide with the normal to the sail, laid down from its shadow side. By turning the sail, we get the opportunity to change the direction of the thrust vector and control the spacecraft. However, this changes the value. If the normal of a flat sail is perpendicular to the stream of rays, then the sail exerts no thrust at all. In the general case, the vector projections onto the radial and transverse directions will affect the change in the parameters of the motion orbit. Projection to the normal to the orbital plane will allow you to change its inclination relative to its original position. Acceleration also depends on the ratio of the sail area *S* to the mass of the entire structure and on the surface properties or surface reflection coefficient. Here, the designations are additionally used when summing over all structural elements.

The design features of the spacecraft in the problems under consideration make it possible to use a solar sail to control motion. By turning the sail, taking into account the pressure of sunlight, it is possible to control the orbital dynamics and the relative position of the spacecraft in space, as well as perform stabilization.

For rotational motion relative to the center of mass, the kinetic energy of the body is determined by the moments of inertia and angular velocity

$$T = \frac{1}{2} \left( \dot{I}\_x a\_x^2 + I\_y a\_y^2 + I\_z a\_x^2 \right).$$

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

Differential equations can be written under the action of the moment of forces in the form

$$\begin{cases} I\_x \dot{o}\_{\mathbf{x}} = \left(I\_{\mathbf{y}} - I\_x\right) o\_{\mathbf{y}} \alpha\_{\mathbf{z}} + M\_{\mathbf{x}}, \\\ I\_{\mathbf{y}} \dot{o}\_{\mathbf{y}} = \left(I\_{\mathbf{z}} - I\_{\mathbf{x}}\right) o\_{\mathbf{x}} \alpha\_{\mathbf{z}} + M\_{\mathbf{y}}, \\\ I\_{\mathbf{z}} \dot{o}\_{\mathbf{z}} = \left(I\_{\mathbf{x}} - I\_{\mathbf{y}}\right) o\_{\mathbf{y}} \alpha\_{\mathbf{x}} + M\_{\mathbf{z}} \end{cases} \tag{18}$$

Taking into account the effect of light pressure on the spacecraft sail leads to the appearance of special stability conditions that can be used to control the movement. Correctly chosen shape of the sail allows you to keep the spacecraft in the desired position. The effect of disturbances can be compensated for by changing the size or reflective properties of the elements of the sail of the spacecraft, as well as their relative position. This creates the additional moments of force that can be used as controls.

If we introduce into consideration the control moments *u<sup>i</sup>* (*i* = 1,2,3), relative to the main axes of inertia, and use the projections of the angular momentum as unknowns *x*<sup>7</sup> ¼ *Ixωx*, *x*<sup>8</sup> ¼ *Iyωy*, *x*<sup>9</sup> ¼ *Izωz*, then the system of equations of motion (1) and (4) for the considered set of generalized coordinates can be represented in the normal form:

$$\begin{cases} \dot{\boldsymbol{x}}\_{i} = \boldsymbol{x}\_{i+3}, \quad \quad i = 1, 2, 3, \\ \dot{\boldsymbol{x}}\_{i+3} = -\mu \ \boldsymbol{r}^{-3} \ \boldsymbol{x}\_{i} + \boldsymbol{f}\_{i}(t, \ \boldsymbol{x}\_{i}, \ \boldsymbol{u}\_{i}), \\ \dot{\boldsymbol{x}}\_{7} = \beta\_{1} \ \boldsymbol{x}\_{8} \boldsymbol{x}\_{9} + \boldsymbol{u}\_{4}, \\ \dot{\boldsymbol{x}}\_{8} = \beta\_{2} \ \boldsymbol{x}\_{9} \boldsymbol{x}\_{7} + \boldsymbol{u}\_{5}, \\ \dot{\boldsymbol{x}}\_{9} = \beta\_{3} \ \boldsymbol{x}\_{8} \boldsymbol{x}\_{7} + \boldsymbol{u}\_{6}, \\ \beta\_{1} + \beta\_{2} \ \boldsymbol{+} \beta\_{3} = \mathbf{0}. \end{cases} \tag{19}$$

In the case of possible oscillations [13], while maintaining the orientation of one of the main axes orthogonal to the plane of motion with angular orbital velocity, the change in the angular momentum taking into account the action of the geopotential can be investigated using the equation

$$I\_x \dot{o}\_x = -ko\_0^{-2} \left(I\_x - I\_y\right) \sin \phi = \mathcal{M}\_x. \tag{20}$$

Taking into account the additional force leads to new, different from the classical, formulations of optimal control problems, and to new mathematical models. In these models, additional equations appear and other control functions are considered. To solve such problems, the methods of Pontryagin or the Bellman equation are used. There are analytical and numerical methods of research and analysis of the main properties of new equations, which allow obtaining exact or approximate solutions that deliver an extremum to the quality criterion and satisfy the necessary conditions [8, 21–23].
