A backup system of a satellite orientation based on radiative inverse problems approach

Highlights

• An approach to solve Inverse Heat Transfer Problem for distributed and lumped parameter systems is suggested.

• A new technology to estimate the orientation of spacecraft is proposed.

• Experimental approving of suggested approach was executed.

Abstract

A reliable control of current orientation of a spacecraft is a very important task in aerospace engineering. This makes important to elaborate a backup or replacement system which can be used for the verification or correction of the orientation. It seems natural to use various conditions of radiative heating of the design elements of different orientations with respect of the Sun and strongly radiating or reflecting planets of the Solar system. In the case of an ordinary slow variation of the vehicle orientation in space, the temperature measurements can be used to distinguish different integral (over the spectrum) radiative flux to various surfaces of a specially designed thermal sensor with several facets. An analysis of a possibility of such an engineering solution is a subject of the present study. The mathematical problem to be solved is one of the so-called inverse heat transfer problems, and its solution is not a simple task. Therefore, the main efforts of the authors are focused on solution of this ill-posed problem. The known methods of the inverse problem regularization are modified to take into account special features of the heat transfer problem under consideration. The resulting algorithm is verified using the typical case problems. It was shown that one can obtain sufficiently accurate results on the bases of a limiting set of relatively simple temperature measurements. The latter enables us to consider the method suggested as a promising way to elaborate a series simple backup/replacement system of an approximate retrieval of a spacecraft current orientation. This statement is confirmed by calculation for one of the typical trajectory of a vehicle in the Solar system.

Introduction

One of the main problems of spacecraft design is the development of control systems and particularly orientation system, which provides angular directions to the space objects (Sun, planets, stars). A promising way to develop such systems is based on measuring the radiative flux from the environment. The latter approach has been suggested for the first time in paper [1]. Unfortunately, in majority of practical situations the direct measurements of heat flux are also problematic. These difficulties can be overcome with the use of some indirect thermal measurements combined with an inverse problem technique. The problem of retrieval of the angular orientation of a spacecraft demands to solve two inverse problems sequentially. The first one is the estimating of heat fluxes absorbed by spacecraft surface. The second one is the determining of angles of orientation based on the estimated values of radiative heat fluxes.

In the general case the orientation of an arbitrary surface element of a spacecraft can be determined by the following nine angles:

• Three angles determine the relative position of the equatorial XYZ and orbital coordinate systems: Ω is the longitude of ascending node, i is the inclination of orbit, u is the argument of latitude (Fig. 1a). The planetocentric equatorial coordinate system can be considered as an inertial coordinate system for the most engineering problems. We assume that the corresponding angles are always known from the predetermined space flight control program.

• Angles α_N, β_N , γ_N determine the direction of the unit vector $\overline{N}$, which is a normal to an arbitrary surface element of a spacecraft. Thus, these angles directly define the relative position of the arbitrary surface element in the coordinate system X_CY_CZ_C associated with the spacecraft (Fig. 1b). These angles are known from the spacecraft design.

• The relative position between orbital coordinate system and associated with the spacecraft coordinate system can be easily determined by using next three angles: ϑ is the pitch angle, ψ is a yaw angle, γ is a roll angle. These angles provide the current orientation of the spacecraft during a space flight. We can define this triple (ϑ,ψ,γ) by using its typical definition from [2] ϑ is the angle between the longitudinal axis of the spacecraft and the plane of the local horizon; ψ – the angle between the transversal direction O_Cn of the orbital coordinate system (nrb) and the projection of the longitudinal axis of the spacecraft to the plane of the local horizon; γ – the angle between the normal axes O_CY_Cof the coordinate system X_CY_CZ_C and the plane O_CrX_C [Fig. 2].

As it is known, if we want to represent the same vector in the different coordinate systems we should define the corresponding transition matrices. So to determine coordinates of the normal vector $\overline{N}$ in the orbital coordinate system (nrb) we can use the transition matrix A [3]:

$$\begin{array}{c}\hfill A=\left|\begin{array}{ccc}\cos \vartheta \cos \psi & \sin \vartheta & -\cos \vartheta \sin \psi \\ \sin \psi \sin \gamma -\sin \vartheta \cos \psi \cos \gamma & \cos \vartheta \cos \gamma & \cos \psi \sin \gamma +\sin \vartheta \sin \psi \cos \gamma \\ \sin \psi \cos \gamma +\sin \vartheta \cos \psi \sin \gamma & -\cos \vartheta \sin \gamma & \cos \psi \cos \gamma -\sin \vartheta \sin \psi \sin \gamma \end{array}\right|\end{array}$$

This matrix is orthogonal and it defines the linear link between the corresponding coordinates of the arbitrary vector, written in the coordinate systemX_CY_CZ_C, associated with the spacecraft, and the orbital one. Due to the orthogonality A^-1=A^T the inverse matrix is equal to the transpose one. Considering this we can now write the components of the vector $\overline{N}$ in the orbital coordinate system (nrb):

$$\begin{array}{c}\hfill \left(\begin{array}{c}{N}_n\\ N_r\\ N_b\end{array}\right)=A^{-1}\left(\begin{array}{c}{N}_{X_C}\\ N_{Y_C}\\ N_{Z_C}\end{array}\right)\end{array}$$

or in the explicit form:

$$\begin{array}{cc}\hfill N_X& =N_n(-\sin u\cos \Omega -\cos u\sin \Omega \cos i)\hfill \\ & +N_T(\cos u\cos \Omega -\sin u\sin \Omega \cos i)-N_b\sin \Omega \sin i\hfill \\ \hfill N_Y& =N_n(-\sin u\sin \Omega +\cos u\cos \Omega \cos i)\hfill \\ & +N_T(\cos u\sin \Omega +\sin u\cos \Omega \cos i)+N_b\cos \Omega \sin i\hfill \\ \hfill N_Z& =N_n\cos u\sin i+N_T\sin u\sin i-N_b\cos i\hfill \end{array}$$

Thereby one can define the orientation of the analysed surface element in space by using nine specified angles.

We as well need to know next parameters of the spacecraft orbit to determine its current centre of the mass position: the apocenter altitude – Hα, the pericenter altitude – Hπ, and the argument of pericenter ω. Using them we can easily define the shape of the instant spacecraft orbit, determine the current angular position of the spacecraft and link it with the time variable. So to define the shape of the instant spacecraft orbit, one can use next simple geometrical relations:

$$\begin{array}{c}\hfill \begin{array}{c}a=\frac{2R_{pl}+H_{\pi }+H_{\alpha }}{2},e=\frac{H_{\alpha }-H_{\pi }}{H_{\alpha }+H_{\pi }},\hfill \\ p=a\left(1-e^2\right)=\frac{2R_{pl}+H_{\pi }+H_{\alpha }}{2}\left(1-{\left(\frac{H_{\alpha }-H_{\pi }}{H_{\alpha }+H_{\pi }}\right)}^2\right)\hfill \end{array}\}\end{array}$$

where R_pl – the radius of a planet (gravity attraction centre), a – semi-major axes of the spacecraft orbit, p – semi latus rectum, e – eccentricity.

In general case the total radiative heat flux absorbed by an arbitrary surface element of the spacecraft consists of three absorbed heat fluxes and one irradiated:

$$q_{abs}=A_s\left(q_s+q_R\right)+\epsilon q_e-\epsilon \sigma T^4(0,\tau )$$

where q_s(Ω,i, u, α_N,β_N,γ_N,ψ, ϑ, γ) is an integral (over the spectrum) solar radiative flux, q_R(Ω,i, u, α_N,β_N,γ_N,ψ, ϑ, γ) is an integral solar radiative flux reflected by a planet, q_e(Ω,i, u, α_N,β_N,γ_N,ψ, ϑ, γ) is an integral radiative flux emitted by a planet, A_s, ε are absorptivity and emissivity of an optically grey surface.

Therefore if we can correctly estimate total absorbed heat flux (q_abs) and temperature of surface (T) in an experiment. Then it can be used to determine searched angles ϑ, ψ,γ using the actual estimations of A_s, ε [6] and adequate mathematical models for calculations of q_s, q_R, q_e.

As has been mentioned above, the direct measurements of absorbed heat flux from the Sun and a planet are impossible. Therefore, we suggested to use the inverse problems technique [5] based on temperature measurements. Following this approach, the problem of determination of the current orientation of the spacecraft demands to solve two inverse problems sequentially. Firstly, we need to estimate heat fluxes absorbed by the spacecraft surface. Secondly, we should determine the angles of orientation based on the estimated values of radiative heat fluxes. The first one can be solved using radiative heat flux sensors (RHFS), which can be installed on the non-self-irradiated spacecraft surface Fig. 1b). Theoretically the minimal number of such sensors is three. But if two of them are in the shadow of a planet, the third sensor will not provide sufficient uniqueness of the second inverse problem solution, because it provides estimation of the total integral absorbed heat flux ((3). Thus, for practical applications six sensors installed on the orthogonal surfaces (Fig. 1b) will be sufficient. It should be mentioned here, that in this paper just convex shape of the satellite (like in Fig. 1b) is considered. If the satellite will have a complex shape, the sensors should be installed at the external elements (antennas, etc.). Two main approaches to develop such sensors have been analyzed.

• The first one is a thin metal slab which is thermally insulated from the spacecraft structure. In this case it can be considered as an isothermal element or as the lumped parameters system. The corresponding mathematical model is:

  $$\begin{array}{cc}& d_m{\rho }_m{c}_{m}d{T}_m/d\tau ={A_s}_m\left(q_{sm}\left(\tau \right)+q_{Rm}\left(\tau \right)\right)\hfill \\ & +{\epsilon }_m{q}_{em}\left(\tau \right)-\epsilon {}_m\sigma T_m^4,m=1,2,...M\hfill \end{array}$$

  $$T_{ml}\left({\tau }_{\min }\right)=T_{l0},m=1,2,...M$$

  where T is temperature, T₀ is initial temperature, τis time, m is a number of sensor, M is total number of sensors, d_m, ρ_m, c_m, A_sm, ε_m are the thickness, density, heat capacity, the absorptivity and emissivity of an optically grey m-th sensor, respectively, q_sm(Ω,i, u, α_Nm,β_Nm,γ_Nm,ψ, ϑ, γ) is an integral (over the spectrum) solar radiative flux, q_Rm(Ω,i, u, α_Nm,β_Nm,γ_Nm,ψ, ϑ, γ) is an integral solar radiative flux reflected by a planet, q_em(Ω,i, u, α_Nm,β_Nm,γ_Nm,ψ, ϑ, γ) is an integral radiative flux emitted by a planet.

• The distributed parameters system:

  $$\begin{array}{cc}& {\rho }_m{c}_m\left(T_m\left(\tau ,x\right)\right)\frac{\partial T_m\left(\tau ,x\right)}{\partial \tau }=\frac{\partial }{\partial x}\left({\lambda }_m\left(T_m\left(\tau ,x\right)\right)\frac{\partial T_m\left(\tau ,x\right)}{\partial x}\right),\hfill \\ & 0<x<d_m,m,{\tau }_{\min }\le \tau \le {\tau }_{\max },m=1,2,...,M\hfill \end{array}$$

  $$T_m\left({\tau }_{\min },x\right)=T_{l0}\left(x\right),0\le x\le d_m,m=1,2,...,M$$

  $$\begin{array}{c}\hfill -{\lambda }_m\left(T\left(0,\tau \right)\right)\frac{\partial T_m\left(0,\tau \right)}{\partial x}=A_{Sm}\left(q_{sm}+q_{Rm}\right)\\ \hfill +{\epsilon }_m{q}_{em}-{\epsilon }_m\sigma T_m^4(0,\tau )\end{array}$$

  $$-{\lambda }_m\left(T_m\left(d_m,\tau \right)\right)\frac{\partial T_m\left(X_m,\tau \right)}{\partial x}=q_{2m}\left(\tau \right)$$

  where x is spatial coordinate, λ_m is the thermal conductivity of m-th sensors.

In the first case one should solve an ill-posed problem [4] of differentiation of experimental function Т to estimate heat fluxes absorbed by heat flux sensors:

$$q_{\mathrm{m}}^{\exp }=d_m{\rho }_m{c}_{m}d{T}_m/d\tau +\epsilon {}_m\sigma T_m^4,m=1,2,...M$$

In the second case, the boundary inverse heat transfer problem should be solved [5]:

$$q_{\mathrm{m}}^{\exp }=-{\lambda }_m\left(T\left(0,\tau \right)\right)\frac{\partial T_m\left(0,\tau \right)}{\partial x}+{\epsilon }_m\sigma T_m^4(0,\tau )$$

Therefore, we can find some estimates for the heat flux absorbed by heat flux sensor, which can be used then to estimate angles ψ, ϑ,γ using an adequate calculated models for q_sm(Ω,i, u, α_Nm,β_Nm,γ_Nm,ψ, ϑ, γ), q_Rm(Ω,i, u, α_Nm,β_Nm,γ_Nm,ψ, ϑ, γ), and q_em(Ω,i, u, α_Nm,β_Nm,γ_Nm,ψ, ϑ, γ) for  m = 1, 2....,M.