Spacecraft formation flying in the port-Hamiltonian framework

Abstract

The problem of controlling the relative position and velocity in multi-spacecraft formation flying in the planetary orbits is an enabling technology for current and future research. This paper proposes a family of tracking controllers for different dynamics of Spacecraft Formation Flying (SFF) in the framework of port-Hamiltonian (pH) systems through application of timed Interconnection and Damping Assignment Passivity-Based Control (IDA-PBC). The leader–multi-follower architecture is used to address this problem. In this regard, first we model the spacecraft motion in the pH framework in the Earth Centered Inertial frame and then transform it to the Hill frame which is a special local coordinate system. By this technique, we may present a unified structure which encompasses linear/nonlinear dynamics, with/without perturbation. Then, using the timed IDA-PBC method and the contraction analysis, a new method for controlling a family of SFF dynamics is developed. The numerical simulations show the efficiency of the approach in two different cases of missions.

Appendices

Appendix A

Consider that the position vector of the point mass—follower spacecraft $s_j$—is described with the vector $Q_j\in {\mathbb {R}}^3$ in the ECI frame. The kinetic energy of the point mass unit is given by

$$\begin{aligned} {\mathcal {T}}_j\left( {Q_j,\dot{Q}_j} \right) = \frac{1}{2}{\left\| {\dot{Q}_j} \right\| ^2}. \end{aligned}$$

(50)

The potential energy⁵ of this particle which is including the effect of $J_2$ as presented below [1]:

$$\begin{aligned} {\mathcal {U}}_j(Q_j)=- \frac{\mu }{R_j}\left[ {1 - \frac{{{J_2}}}{2}\frac{{R_e^2}}{{{R_j^2}}}\left( {3\frac{{{Z_j^2}}}{{{R_j^2}}} - 1} \right) } \right] . \end{aligned}$$

(51)

So, the Lagrangian function is

$$\begin{aligned} {{{{\mathcal {L}}}}_j(Q_j,\dot{Q}_j)} = {\mathcal {T}}_j - {\mathcal {U}}_j = \frac{1}{2}{\left\| {\dot{Q}_j} \right\| ^2} - {\mathcal {U}}_j\left( Q_j \right) .\nonumber \\ \end{aligned}$$

(52)

and the forced Euler-Lagrange equation for the point mass dynamics is given by

$$\begin{aligned}&\frac{{\mathrm{d}}}{{{\mathrm{d}}t}}\left( {\frac{{\partial {{{{\mathcal {L}}}}_j}}}{{\partial \dot{Q}_j}}} \right) - \frac{{\partial {{{{\mathcal {L}}}}_j}}}{{\partial Q_j}} \nonumber \\&\quad = {u_j} \Rightarrow {\ddot{Q}}_j = -{\nabla _{Q_j}}{\mathcal {U}}_j\left( Q_j \right) + {u_j}, \end{aligned}$$

(53)

where $u_j$ is the control signal for the follower spacecraft $s_j$. These dynamics can be expressed in the Hamiltonian framework. To this aim, we define $P_j\triangleq \dot{Q_j}$. The Hamiltonian function—the sum of the kinetic and potential energy—is defined

$$\begin{aligned} {\mathcal {H}}_j(Q_j,\dot{Q}_j) = {\mathcal {T}}_j + {\mathcal {U}}_j = \frac{1}{{2}}{P_j^T}P_j + {\mathcal {U}}_j\left( Q_j \right) . \end{aligned}$$

Accordingly, the Hamiltonian structure of this system in the ECI frame is as follows:

$$\begin{aligned} \begin{aligned} \left[ { \begin{array}{*{20}{c}} {\dot{Q}_j}\\ {\dot{P}_j} \end{array}} \right]&= \left[ { \begin{array}{*{20}{c}} {P_j}\\ { - {\nabla _{Q_j}}{\mathcal {U}}_j\left( Q_j \right) } \end{array}} \right] + gu_j \\&= \left[ { \begin{array}{*{20}{c}} 0&{}\quad {{I_3}}\\ { - {I_3}}&{}\quad 0 \end{array}} \right] \left[ { \begin{array}{*{20}{c}} {{\nabla _{Q_j}}{{ {\mathcal {H}}}_j} }\\ {{\nabla _{P_j}}{{{\mathcal {H}}}_j} } \end{array}} \right] + gu_j, \end{aligned} \end{aligned}$$

where $\small g=\left[ { \begin{array}{*{20}{c}}0\\ {{I_3}} \end{array}} \right]$. For more information refer to [14, Chapter 3].

Appendix B

The position vector of the point mass in the ECI frame is related to the position vector of it in the LVLH frame by

$$\begin{aligned} Q_j=T{\bar{q}}_j. \end{aligned}$$

(54)

According to the (14), the velocity vector of the point mass in the ECI frame is defined as below:

$$\begin{aligned} \dot{Q}_j = T\,\dot{{\bar{q}}}_j + \dot{T}\,{\bar{q}}_j = T(\dot{{\bar{q}}}_j + S(\omega ){\bar{q}}_j). \end{aligned}$$

(55)

So, the kinetic energy (50) can be defined in the LVLH frame by

$$\begin{aligned} {\mathcal {T}}_j({\bar{q}}_j,\dot{{\bar{q}}}_j) = \frac{1}{2}{\left\| {\dot{Q}_j} \right\| ^2} = \frac{1}{2}{\left\| {\dot{{\bar{q}}}_j + S(\omega ){\bar{q}}_j} \right\| ^2}. \end{aligned}$$

(56)

Also, the potential energy of the spacecraft without $J_2$ perturbation is obtained from (51) by considering $J_2=0$ and is defined in the LVLH frame as follows:

$$\begin{aligned} {{{\mathcal {E}}}_j}(T{\bar{q}}_j)=- \frac{\mu }{r_j}, \end{aligned}$$

(57)

where $r_j = {\left\| {{{\bar{q}}_j}} \right\| _2}$. Thus, the Lagrangian function is (52) redefined as

$$\begin{aligned} {{{{\mathcal {L}}}}_j({\bar{q}}_j,\dot{{\bar{q}}}_j)}= & {} {\mathcal {T}}_j -{\mathcal {E}}_j \nonumber \\= & {} \frac{1}{2}{\left\| {\dot{{\bar{q}}}_j + S(\omega ){\bar{q}}_j} \right\| ^2} - {\mathcal {E}}_j\left( {T{\bar{q}}_j} \right) . \end{aligned}$$

(58)

Dynamics equations of the spacecraft motion in the LVLH frame can be expressed by forced Euler-Lagrange equation in the form of below:

$$\begin{aligned} \begin{aligned}&\frac{{\mathrm{d}}}{{{\mathrm{d}}t}}\left( {\frac{{\partial {{{{\mathcal {L}}}}_j} }}{{\partial \dot{ {\bar{q}}}_j }}} \right) - \frac{{\partial {{{{\mathcal {L}}}}_j} }}{{\partial {\bar{q}}_j}} = {u_j}\\&\quad \Rightarrow \ddot{ {\bar{q}}}_j + 2S(\omega )\dot{ {\bar{q}}}_j + S(\omega )S(\omega ){\bar{q}}_j\\&\quad + S({{\dot{\omega }}} ){\bar{q}}_j = -{\nabla _{{\bar{q}}_j}}{\mathcal {E}}_j\left( {T{\bar{q}}_j} \right) {+u_j}.\ \end{aligned} \end{aligned}$$

(59)

By defining ${\bar{p}}_j\triangleq \dot{{\bar{q}}}_j$ and applying (59), the pH structure can be obtained as follows:

$$\begin{aligned} \begin{aligned} \left[ { \begin{array}{*{10}{c}} {\dot{{\bar{q}}}_j}\\ {\dot{{\bar{p}}}_j} \end{array}} \right]&=\left[ { \begin{array}{*{10}{c}} {{{{{\bar{p}}}}_j}}\\ { - 2S(\omega ){{\dot{{{\bar{q}}}}}}_j - S(\omega )S(\omega ){{{{\bar{q}}}}_j} - S({{\dot{\omega }}} ){{{{\bar{q}}}}}_j - {\nabla _{{{{{\bar{q}}}}_j}}}{\mathcal {E}}_j\left( {T\,{{{{\bar{q}}}}_j}} \right) } \end{array}} \right] \\&\quad +{gu_j}= \left[ { \begin{array}{*{10}{c}} 0&{}{{I_3}}\\ { - {I_3}}&{}{-2S(\omega )-{\bar{R}}_j} \end{array}} \right] \left[ { \begin{array}{*{10}{c}} {{\nabla _{{{{{\bar{q}}}}_j}}}{{{H}_j} }}\\ {{\nabla _{{{{{\bar{p}}}}_j}}}{{H}_j} } \end{array}} \right] + {gu_j}. \end{aligned} \end{aligned}$$

where $\small g=\left[ { \begin{array}{*{20}{c}}0\\ {{I_3}} \end{array}} \right]$. The Hamiltonian function and ${\bar{R}}_j$ are defined in (18), (19), receptively. For more information refer to [14, Chapter 3].

Appendix C

Similar to“Appendix B”, we can choose ${\bar{p}}_j\triangleq \dot{ {{\bar{q}}}}_j+S(\omega ){\bar{q}}_j$ and by considering Eq. (59) for the potential energy ${\mathcal {U}}_j(T{\bar{q}}_j)$, the Hamiltonian form of these dynamics is as follows:

$$\begin{aligned} \begin{aligned} \left[ { \begin{array}{*{20}{c}} {\dot{{\bar{q}}}_j}\\ {\dot{{\bar{p}}}_j} \end{array}} \right]&= \left[ { \begin{array}{*{20}{c}} {{\bar{p}}_j - S(\omega ){\bar{q}}_j}\\ {{S^T}(\omega ){\bar{p}}_j - {\nabla _{{\bar{q}}_j}}{\mathcal {U}}_j\left( {T\,{\bar{q}}_j} \right) } \end{array}} \right] +gu_j \\&= \left[ { \begin{array}{*{20}{c}} 0&{}\quad {{I_3}}\\ { - {I_3}}&{}\quad 0 \end{array}} \right] \left[ { \begin{array}{*{20}{c}} {{\nabla _{{\bar{q}}_j}}{{H}_j} }\\ {{\nabla _{{\bar{p}}_j}}{{H}_j} } \end{array}} \right] +gu_j, \end{aligned} \end{aligned}$$

and the Hamiltonian function is

$$\begin{aligned} {{H}_j} = \frac{1}{2}{{\bar{p}}_j^\mathrm{T}}{\bar{p}}_j - {{\bar{p}}_j^\mathrm{T}}S(\omega ){\bar{q}}_j + {\mathcal {U}}_j(T\,{\bar{q}}_j) \end{aligned}$$

where the potential energy with $J_2$ effects in the LVLH frame can be obtained from (51) as below:

$$\begin{aligned} {\mathcal {U}}_j(T{\bar{q}}_j)=- \frac{\mu }{r_j}\left[ {1 - \frac{{{J_2}}}{2}\frac{{R_e^2}}{{{r_j^2}}}\left( {3\frac{{\varphi _z{_j}^2}}{{{r_j^2}}} - 1} \right) } \right] .\ \end{aligned}$$