Attitude Control of Spacecraft Formations subject to Distributed Communication Delays

The multiple spacecraft attitude consensus problem is considered with the states of each spacecraft being the Modified Rodrigues Parameters (MRP) ${\bm{\sigma}}$ and angular velocity ${\bm{\omega}}$ in the body frame. The attitude kinematics and the dynamics is assumed to be identical for all spacecraft in formation and is governed by the following nonlinear ordinary differential equations:

	$\displaystyle\dot{\bm{\sigma}}(t)$	$\displaystyle=$	$\displaystyle{\bm{P}}({\bm{\sigma}}){\bm{\omega}}(t)$		(1)
	$\displaystyle\dot{\bm{\omega}}(t)$	$\displaystyle=$	$\displaystyle{\bm{J}}^{-1}\left(-[\tilde{\bm{\omega}}(t)]{\bm{J}}{\bm{\omega}}(t)+{\bm{\tau}}\right)$		(2)

where ${\bm{P}}({\bm{\sigma}})=\dfrac{1}{4}\left(2[\tilde{\bm{\sigma}}(t)]+2{\bm{\sigma}}(t){\bm{\sigma}}^{T}(t)+(1-{\bm{\sigma}}^{T}(t){\bm{\sigma}}(t))\mathbb{I}_{3\times 3}\right)$, $[\tilde{\bm{a}}]=\begin{bmatrix}0&-a_{3}&a_{2}\\ a_{3}&0&-a_{1}\\ -a_{2}&a_{1}&0\end{bmatrix}$,

${\bm{J}}$ is the symmetric moment of Inertia matrix and ${\bm{\tau}}$ is the control torque.

The MRP is a vector defined as

$${\bm{\sigma}}=\hat{\bm{n}}\tan\left(\frac{\Phi}{4}\right)$$

where $\hat{\bm{n}}$ is the principal axis and $\Phi$ is the principal angle as given by Euler’s rotation theorem. The MRPs yield a unique representation for the attitude of the spacecraft for all principal rotations that lie in the interval $[0,~2\pi)$. At $\Phi=2\pi$, a singularity is encountered that is typically addressed using the shadow MRP set. In this paper, it is assumed that the rotations are limited to principal rotation angles less than $2\pi$.

The system represented by Eqs. (1) and (2) is rewritten into a compact second order nonlinear ode:

$${\bm{M}}({\bm{\sigma}})\ddot{\bm{\sigma}}+{\bm{C}}({\bm{\sigma}},\dot{\bm{\sigma}})\dot{\bm{\sigma}}={\bm{u}}$$

(3)

Let ${\bm{\sigma}}_{1}={\bm{\sigma}}$ and ${\bm{\sigma}}_{2}=\dot{\bm{\sigma}}$. Performing the transformations ${\bm{\omega}}={\bm{P}}({\bm{\sigma}})^{-1}\dot{\bm{\sigma}}$ and ${\bm{P}}({\bm{\sigma}})^{-1}{\bm{\tau}}={\bm{u}}$ the dynamics can be expressed as

	$\displaystyle\dot{\bm{\sigma}}_{1}$	$\displaystyle={\bm{\sigma}}_{2}$
	$\displaystyle\dot{\bm{\sigma}}_{2}$	$\displaystyle={\bm{M}}({\bm{\sigma}}_{1})^{-1}\left(-{\bm{C}}({\bm{\sigma}}_{1},{\bm{\sigma}}_{2}){\bm{\sigma}}_{2}+{\bm{u}}\right)$		(4)

where ${\bm{M}}({\bm{\sigma}}_{1})={\bm{P}}({\bm{\sigma}}_{1})^{-1}{\bm{J}}{\bm{P}}({\bm{\sigma}}_{1})$ and ${\bm{C}}({\bm{\sigma}}_{1},{\bm{\sigma}}_{2})={\bm{P}}({\bm{\sigma}}_{1})^{-1}\left({\bm{J}}\dot{\bm{P}}({\bm{\sigma}}_{1})+\widetilde{\left[{\bm{P}}({\bm{\sigma}}_{1}){\bm{\sigma}}_{2}\right]}{\bm{J}}{\bm{P}}({\bm{\sigma}}_{1})\right)$.

Using full state feedback and a straightforward feedback linearization based control law, ${\bm{u}}={\bm{C}}({\bm{\sigma}}_{1},{\bm{\sigma}}_{2}){\bm{\sigma}}_{2}+{\bm{M}}({\bm{\sigma}}_{1}){\bm{v}}$, the above dynamics are further reduced to that of a double integrator as follows.

	$\displaystyle\dot{\bm{\sigma}}_{1}$	$\displaystyle={\bm{\sigma}}_{2}$
	$\displaystyle\dot{\bm{\sigma}}_{2}$	$\displaystyle={\bm{v}}$			(5)

Now, consider a system of $N$ spacecraft with communication pathways among them forming a strongly connected graph, each of whom is being controlled using the control law ${\bm{u}}(t)$ described earlier. Thus, the attitude consensus problem for $N$ nonlinear spacecraft can be posed using the simpler structure derived earlier in Eq. (5).

Let $\mathbf{x}=[\mathbf{x_{1}}^{T}\ \mathbf{x_{2}}^{T}]^{T}$ be a vector such $\mathbf{x}_{1}$ consists of the attitudes of the $N$ spacecraft while $\mathbf{x_{2}}$ consists of their derivatives i.e. $\mathbf{x_{1}}=\left[{}^{1}{\bm{\sigma}}_{1}^{T}\cdots{}^{N}{\bm{\sigma}}_{1}^{T}\right]^{T}$ and $\mathbf{x_{2}}=\left[{}^{1}{\bm{\sigma}}_{2}^{T}\cdots{}^{N}{\bm{\sigma}}_{2}^{T}\right]^{T}$. This vector defines the state of the $N$ spacecraft system and the combined dynamics can be written as:

	$\displaystyle\mathbf{\dot{x}_{1}}(t)$	$\displaystyle=\mathbf{x_{2}}(t)$
	$\displaystyle\mathbf{\dot{x}_{2}}(t)$	$\displaystyle={\mathbf{u}}(t)$		(6)

where the vector ${\mathbf{u}}=\left[{}^{1}{\bm{v}}^{T}\cdots{}^{N}{\bm{v}}^{T}\right]^{T}$ is the control input obtained by cascading the control inputs of all the spacecraft.

For the system of $N$ spacecraft described above in Eq. (2.3), attitude consensus is said to be achieved when $||{}^{i}{\bm{\sigma}}_{1}-{}^{j}{\bm{\sigma}}_{1}||\rightarrow 0$ and $||{}^{i}{\bm{\sigma}}_{2}-{}^{j}{\bm{\sigma}}_{2}||\rightarrow 0$ as $t\rightarrow\infty$ $\forall\ i,~j,~i\neq j$.

In the case where the spacecraft exchange state information to their connected neighbors with no delay, the following control law drives the system to consensus [13]. For all the analysis from here on, we will use the compact set of equations in Eq. (2.3) for the $N$ connected spacecraft.

$$\mathbf{u}(t)=-\mathbf{L}\mathbf{x_{1}}(t)-\gamma\mathbf{L}\mathbf{x_{2}}(t)$$

where $\mathbf{L}=[l_{ij}]$ is the Laplacian matrix for a given communication topology [5, 13] and $\gamma>0$ is a damping gain.

In this paper, the consensus problem is analyzed when the communication is subject to time delays i.e. the state information of the $i^{\mathrm{th}}$ spacecraft is relayed to the $j^{\mathrm{th}}$ spacecraft after a delay $\tau_{ij}$ that satisfies

$\displaystyle 0~\leq~\tau_{ij}~\leq~h_{ij}\quad\&\quad|\dot{\tau}_{ij}|~\leq~d_{ij}$

(7)

Clearly the time delays are different along different communication paths (spacecraft pairs) and are time-varying.

To achieve consensus, the following control law is investigated.

$\displaystyle\mathbf{u}(t)=-\mathbf{x_{1}}(t)-\gamma\mathbf{x_{2}}(t)-\mathop{\sum_{i=1}^{N}\sum_{j=1}^{N}}_{i\neq j}{}^{ij}\mathbf{K}\mathbf{x_{1}}(t-\tau_{ij})-\mathop{\sum_{i=1}^{N}\sum_{j=1}^{N}}_{i\neq j}\gamma{}^{ij}\mathbf{K}\mathbf{x_{2}}(t-\tau_{ij})$

(8)

For $N$ spacecraft defining the communication topology, then for every pair $(i,~j)$ of spacecraft we have $l_{ij}\leq 0$ (from the Laplacian matrix). ${}^{ij}\mathbf{K}\in\mathbb{R}^{N\times N}$ and ${}^{ij}\mathbf{K}\otimes\mathbb{I}_{3\times 3}$ is the coefficient of the delayed state, where $\tau_{ij}$ is the time delay in sending information from the $i^{\mathrm{th}}$ spacecraft to the $j^{\mathrm{th}}$ spacecraft. The $ji^{th}$ element of ${}^{ij}\mathbf{K}$ is equal to $l_{ij}$. All other elements are zero.
This is illustrated via an example involving three spacecraft with the communication topology specified by figure  1.

The Laplacian matrix specifying the communication topology between the three spacecraft be given by

$$\begin{bmatrix}1&0&-1\\ -\frac{1}{2}&1&-\frac{1}{2}\\ 0&0&0\end{bmatrix}$$

Let the delays in the communication channels be $\tau_{12},\ \tau_{31}$ and $\tau_{32}$. Then, the matrices ${}^{ij}K$ are given by

$${}^{12}K=\begin{bmatrix}0&0&0\\ -\frac{1}{2}&0&0\\ 0&0&0\end{bmatrix}\otimes\mathbb{I}_{3\times 3}\ ^{31}K=\begin{bmatrix}0&0&-1\\ 0&0&0\\ 0&0&0\end{bmatrix}\otimes\mathbb{I}_{3\times 3}\ ^{32}K=\begin{bmatrix}0&0&0\\ 0&0&-\frac{1}{2}\\ 0&0&0\end{bmatrix}\otimes\mathbb{I}_{3\times 3}\$$

Substituting Eq. (8) in Eq. (2.3), the state dynamics are expressed as a linear differential equation with multiple time delays.

$\displaystyle\mathbf{\dot{x}}(t)=\mathbf{A}_{0}\mathbf{x}(t)+\mathop{\sum_{i=1}^{N}\sum_{j=1}^{N}}_{i\neq j}{}^{ij}\mathbf{A}\mathbf{x}(t-\tau_{ij})$

(9)

where
$\mathbf{A}_{0}=\begin{bmatrix}\mathbf{0}_{N\times N}&\mathbb{I}_{N\times N}\\ -\mathbb{I}_{N\times N}&-\gamma\mathbb{I}_{N\times N}\end{bmatrix}\otimes\mathbb{I}_{3\times 3}$  and  ${}^{ij}\mathbf{A}=\begin{bmatrix}\mathbf{0}_{N\times N}&\mathbf{0}_{N\times N}\\ -{}^{ij}\mathbf{K}&-\gamma{}^{ij}\mathbf{K}\end{bmatrix}\otimes\mathbb{I}_{3\times 3}$.

Note that $\mathop{\sum_{i=1}^{N}\sum_{j=1}^{N}}_{i\neq j}{}^{ij}\mathbf{A}=\begin{bmatrix}\mathbf{0}_{N\times N}&\mathbf{0}_{N\times N}\\ \mathcal{A}&\gamma\mathcal{A}\end{bmatrix}\otimes\mathbb{I}_{3\times 3}=\mathbf{A}_{\gamma}$ where $\mathcal{A}$ is the adjacency matrix of the connection topology.

The lemmas stated above are key to obtaining an LMI condition for ensuring stability of the desired consensus.

Proof
Consider the following Lyapunov-Krasovskii candidate funtional with $\mathbf{y}(t)=E\mathbf{x}(t)\ (i.e.,\mathbf{y}_{1}=\left[\mathbf{1}\ -\mathbb{I}_{n-1}\right]\otimes\mathbb{I}_{3\times 3}\ \mathbf{x}_{1},\ \mathbf{y}_{2}=\left[\mathbf{1}\ -\mathbb{I}_{n-1}\right]\otimes\mathbb{I}_{3\times 3}\ \mathbf{x}_{2}$)

	$\displaystyle V=$	$\displaystyle\underbrace{\mathbf{y}(t)^{T}\mathbf{y}(t)}_{V_{1}}+\underbrace{\mathop{\sum_{i=1}^{n}\sum_{j=1}^{n}}_{i\neq j}\int_{t-\tau_{ij}}^{t}\mathbf{y}(s)^{T}Q_{ij}\mathbf{y}(s)ds}_{V_{2}}$
		$\displaystyle+\underbrace{\mathop{\sum_{i=1}^{n}\sum_{j=1}^{n}}_{i\neq j}\int_{t-\tau_{ij}}^{t}\int_{\eta}^{t}\mathbf{\dot{y}}(s)^{T}S_{ij}\mathbf{\dot{y}}(s)dsd\eta}_{V_{3}}$

where $Q_{ij},S_{ij}$ are appropriately dimensioned symmetric positive definite matrices.
Defining $\mathbf{X}=[\mathbf{x}(t)^{T}\ \mathbf{x}^{T}(t-\tau_{12})\ \mathbf{x}^{T}(t-\tau_{13})...\mathbf{x}^{T}(t-\tau_{nn-1})]^{T}$, the derivatives of each individual term of the Lyapunov-Krasovskii functional are obtained as

$$\dot{V}_{1}=\mathbf{X}^{T}\Psi_{1}\mathbf{X}$$

$$\dot{V}_{2}=\mathbf{X}^{T}\Psi_{2}\mathbf{X}$$

	$\displaystyle\dot{V}_{3}$	$\displaystyle\leq\mathop{\sum_{i=1}^{n}\sum_{j=1}^{n}}_{i\neq j}h_{ij}\mathbf{\dot{x}^{T}}(t)E^{T}(S_{ij})L\mathbf{\dot{x}}(t)$
		$\displaystyle-\mathop{\sum_{i=1}^{n}\sum_{j=1}^{n}}_{i\neq j}(1-d_{ij})\int_{t-\tau_{ij}}^{t}\mathbf{\dot{x}^{T}}(s)E^{T}(S_{ij})L\mathbf{\dot{x}}(s)ds$

Using Jensen’s inequality (Lemma 2), the above inequality can be expressed as

$\displaystyle\dot{V}_{3}$

$\displaystyle\leq\mathbf{X}^{T}\Omega\mathbf{X}$

where

$$\Omega=\begin{bmatrix}\Omega_{11}&\Omega_{12}&\ldots&\Omega_{1n}\\ \Omega_{12}^{T}&\ddots&\Omega_{23}\ldots&\vdots\\ \vdots&\vdots&\ddots&\vdots\\ \Omega_{1n}^{T}&\ldots&\ldots&\Omega_{nn-1}\end{bmatrix}$$

and

	$\displaystyle\Omega_{11}$	$\displaystyle=\mathop{\sum_{i=1}^{n}\sum_{j=1}^{n}}_{i\neq j}(h_{ij}A^{T}_{0}E^{T}(S_{ij})EA_{0}-\frac{1-d_{ij}}{h_{ij}}E^{T}(S_{ij})E)$
	$\displaystyle\Omega_{12}$	$\displaystyle=\mathop{\sum_{i=1}^{n}\sum_{j=1}^{n}}_{i\neq j}h_{ij}A^{T}_{0}E^{T}(S_{ij})EA_{12}+\frac{1-d_{12}}{h_{12}}E^{T}(S_{12})E$
	$\displaystyle\Omega_{22}$	$\displaystyle=\mathop{\sum_{i=1}^{n}\sum_{j=1}^{n}}_{i\neq j}h_{ij}A^{T}_{12}E^{T}(S_{ij})EA_{12}-\frac{1-d_{12}}{h_{12}}E^{T}(S_{12})E$
	$\displaystyle\Omega_{23}$	$\displaystyle=\mathop{\sum_{i=1}^{n}\sum_{j=1}^{n}}_{i\neq j}h_{ij}A^{T}_{12}E^{T}(S_{ij})EA_{13}$
	$\displaystyle..\textrm{and so on.}$

To make the elements of the above matrix affine in the unknown variables, Schur’s complement (Lemma 1) is applied using the fact that $\mathop{\sum_{i=1}^{n}\sum_{j=1}^{n}}_{i\neq j}h_{ij}S_{ij}$ is positive definite and invertible. The resulting LMI is $\Psi_{3}=<\mathbf{0}$

The vector $\mathbf{y}(t)$ converges to $\mathbf{0}$ if there exist symmetric positive definite matrices $P,\ Q_{ij},\ S_{ij}$ such that the following LMIs are feasible

	$\displaystyle\Psi_{1}$	$\displaystyle<\mathbf{0}$
	$\displaystyle\Psi_{2}$	$\displaystyle<\mathbf{0}$
	$\displaystyle\Psi_{3}$	$\displaystyle<\mathbf{0}$

Observe that $\mathbf{y_{1i}}(t)=\mathbf{x_{11}}(t)-\mathbf{x_{1\ i+1}}(t)\ \forall\ i>1$ . Thus, if $\mathbf{y}(t)\rightarrow\mathbf{0}$ as $t\rightarrow\infty$, we have $\mathbf{x_{11}}(t)=\mathbf{x_{1i}}(t)$ as $t\rightarrow\infty\ \forall\ i>1$. Hence, consensus is achieved.$\blacksquare$
Note that finding feasible solutions for the above LMIs is a tedious task considering that the LK functional is required to be positive definite on the augmented state space $\mathbf{y}(t)$ while its derivative is required to be negative definite on the actual state space $\mathbf{x}(t)$ which can’t be obtained from $\mathbf{y}(t)$ via an invertible transformation due to the nature of how the time delays appear in the closed loop dynamics. To guarantee feasible solutions, a leader-follower strategy or a constant set point must be specified to obtain an invertible transformation such that the derivatives of $\mathbf{y}(t)$ appear as linear functions of $\mathbf{y}(t)$ itself, i.e, $\dot{\mathbf{Y}}(t)=\bar{\mathbf{A}}_{0}\mathbf{Y}(t)+\mathop{\sum_{i=1}^{N}\sum_{j=1}^{N}}_{i\neq j}{}^{ij}\bar{\mathbf{A}}\mathbf{Y}(t-\tau_{ij})$.
To this end, we also investigate the consensus problem using a frequency domain approach to obtain delay dependent stability criteria.