Event-Triggered Optimal Attitude Consensus
of Multiple Rigid Body Networks with
Unknown Dynamics

Throughout this paper, $\mathbb{R}$ represents the set of all real numbers, $\mathbb{R}_{>0}$ represents the set of all positive real numbers, $\mathbb{N}$ represents the set of all non-negative integers, and $\mathbb{N}_{>0}$ is the set of all positive integers, i.e., $\mathbb{R}=(-\infty,+\infty)$, $\mathbb{R}_{>0}=(0,+\infty)$, $\mathbb{N}=\{0,1,2,...\}$ and $\mathbb{N}_{>0}=\{1,2,...\}$. $x\in\mathbb{R}^{n}$ indicates an $n$-dimensional vector, $I_{n}$ indicates an $n$-dimensional identity matrix, $A\in\mathbb{R}^{n\times m}$ indicates an $n\times m$ dimensional matrix. For a vector $x$, its Euclidean norm is defined as $\lVert x\rVert=\sqrt{x^{T}x}$. For a square matrix $B=[b_{ij}]\in\mathbb{R}^{n\times n}$, its trace is defined as $\textrm{tr}(B)=\sum_{i=1}^{n}b_{ii}$, its Frobenius norm is defined as $\lVert B\rVert=\sqrt{\sum_{i=1}^{n}\sum_{j=1}^{n}|b_{ij}|^{2}}$. Define ${\lambda}_{\textrm{min}}(B)$ and ${\lambda}_{\textrm{max}}(B)$ as the minimum eigenvalue and maximum eigenvalue, respectively. $B>0$ $(B\geq 0)$ indicates that $B$ is positive (semi-positive) definite.

For any two vectors $\xi=[x_{1},y_{1},z_{1}]^{\top}\in\mathbb{R}^{3}$ and $\zeta=[x_{2},y_{2},z_{2}]^{\top}\in\mathbb{R}^{3}$, their cross product is expressed as follows:

Let $\mathcal{G}=(\mathcal{V},\mathcal{E})$ represent the directed communication graph among $N\in\mathbb{N}_{>0}$ rigid bodies, where $\mathcal{V}=\{1,2,...,N\}$ indicates the set of all rigid bodies and $\mathcal{E}\subseteq\mathcal{V}\times\mathcal{V}$ indicates the communication relationship between any two rigid bodies. For the rigid body $i$, we use $\mathcal{N}_{i}=\{j\in\mathcal{V}|(j,i)\in\mathcal{E}\}$ to represent the set of its neighbors. For any two rigid bodies, if there is always a direct path between them, we call this communication graph strongly connected. In this paper, we suppose that all directed communication graphs are strongly connected.

In order to express the communication relationship between all rigid bodies more clearly, the weighted adjacency matrix $A=[a_{ij}]\in\mathbb{R}^{N\times N}$ is introduced. If the rigid body $i$ can receive the data transmitted by the rigid body $j$, $a_{ij}>0$ and $a_{ij}=0$ otherwise. The in-degree matrix of the directed communication graph can be expressed as $D=\text{diag}\{d_{1},d_{2},...,d_{N}\}\in\mathbb{R}^{N\times N}$, where $d_{i}=\sum_{j\in\mathcal{N}_{i}}a_{ij}$. Let $\mathcal{L}=D-A=[l_{ij}]$ represent the Laplacian matrix, where $l_{ii}=d_{i}$, and $l_{ij}=-a_{ij}$ when $i\neq j$.

We consider a multiple rigid body network with $N$ nodes, where the attitude of each node can be expressed by Modified Rodriguez Parameters (MRPs) [35]. For the rigid body $i$, the attitude is represent by $\sigma_{i}=[\sigma_{i}^{1},\sigma_{i}^{2},\sigma_{i}^{3}]^{\top}=\Psi_{i}\textrm{tan}\frac{\Phi_{i}}{4}\in\mathbb{R}^{3}$, where $\Psi_{i}\in\mathbb{R}^{3}$ indicates the Euler axis, and $\Phi_{i}\in[0,\pi)$ denotes the angle respective to the Euler axis.

Then, the attitude dynamics of each rigid body is given in the following form:

where $\omega_{i}\in\mathbb{R}^{3}$, $J_{i}\in\mathbb{R}^{3\times 3}$ and $\tau_{i}\in\mathbb{R}^{3}$ indicate the angular velocity vector, the inertia matrix and the control input torque, respectively. The matrix $G(\sigma_{i})=\frac{1}{2}(\sigma_{i}^{\times}+\sigma_{i}\sigma_{i}^{\top}+\frac{1-\sigma_{i}^{\top}\sigma_{i}}{2}I_{3})\in\mathbb{R}^{3\times 3}$, where

Definition 1: Given that the communication topology of a multiple rigid body network (1) is strongly connected, the attitude consensus is said to be achieved when the following conditions hold:

Considering the communication topology among these $N$ rigid bodies, we can define the following form of consensus error for the rigid body $i$:

where $\alpha_{i}\in\mathbb{R}_{>0}$. When $\lim_{t\to\infty}\delta_{i}=0$, $i=1,2,...,N$, we can easily obtain that $\sigma_{1}=\sigma_{2}=...=\sigma_{N}$ and $\omega_{1}=\omega_{2}=...=\omega_{N}$ with $t\to\infty$. That is to say, the attitude consensus is achieved.

The dynamics of $\delta_{i}$ can be obtained by taking the derivative of Eq. (3), which is described as follows:

where $\varGamma_{i}(\delta_{i})=\alpha_{i}\sum_{j\in\mathcal{N}_{i}}a_{ij}(\dot{\sigma}_{i}-\dot{\sigma}_{j})+\sum_{j\in\mathcal{N}_{i}}a_{ij}\Big{[}-J_{i}^{-1}\Big{(}\big{(}G^{-1}(\sigma_{i})\dot{\sigma_{i}}\big{)}\times\big{(}J_{i}G^{-1}(\sigma_{i})\dot{\sigma_{i}}\big{)}\Big{)}+J_{j}^{-1}\Big{(}\big{(}G^{-1}(\sigma_{j})\dot{\sigma_{j}}\big{)}\times\big{(}J_{j}G^{-1}(\sigma_{j})\dot{\sigma_{j}}\big{)}\Big{)}\Big{]}$.

In order to overcome the dependence on model information, a compensator is introduced, which can be expressed by the following affine differential equation:

where $f(\cdot):\mathbb{R}^{3}\to\mathbb{R}^{3}$, $g(\cdot):\mathbb{R}^{3}\to\mathbb{R}^{3\times 3}$ are two functions to be designed later, and $u_{i}\in\mathbb{R}^{3}$ is the control input of the compensator. We need to choose appropriate functions $f(\cdot)$ and $g(\cdot)$ to ensure that the compensator is controllable. In this paper, a feasible pair of $f(\cdot)$ and $g(\cdot)$ is given as follows:

By combining the consensus error $\delta_{i}$ and the control input torque $\tau_{i}$, we define an augmented consensus error, which is expressed as $e_{i}=[\delta_{i}^{\top},\tau_{i}^{\top}]^{\top}\in\mathbb{R}^{6}$. According to Eq. (III-A) and Eq. (5), we can use the following augmented system to describe the dynamics of the augmented consensus error:

where $X_{i}(e_{i})$ and $Y_{i}(e_{i})$ are represented as follows:

Assumption 1: The matrix $X_{i}(e_{i})$ is bounded, i.e., $\forall i\in\mathcal{V}$, $\lVert X_{i}(e_{i})\rVert\leq X_{M}\lVert e_{i}\rVert$ is satisfied, where $X_{M}\in\mathbb{R}_{>0}$.

In order to measure the performance cost of implementing the attitude consensus, a performance function is defined in the following form:

where $u_{-i}=\{u_{j}|j\in\mathcal{N}_{i}\}$ indicates the set of control inputs for the neighbors of the rigid body $i$, $Q_{i}\in\mathbb{R}^{6\times 6}$, $Q_{i}\geq 0$, $R_{i}\in\mathbb{R}^{3\times 3}$ and $R_{i}>0$. According to Eq. (7), we can conclude that $e_{i}$ is driven by $u_{i}$ and $u_{-i}$. Therefore, the left side of Eq. (8) also contains $u_{-i}$.

According to (8), the value function can be defined as

By taking the derivative of Eq. (9), we can obtain the Hamiltonian function in the following form:

where $\nabla V_{i}=\frac{\partial V_{i}}{\partial e_{i}}.$

The implicit solution of the model-free optimal controller $u_{i}^{*}$ can be derived from $\frac{\partial H_{i}}{\partial u_{i}}=0$, which is represented as

where $V_{i}^{*}$ indicates the optimal value function and $\nabla V_{i}^{*}=\frac{\partial V_{i}^{*}}{\partial e_{i}}$.

Combining (III-A) and (11), we can derive the HJB equation for the rigid body $i$ as follows:

From Eq. (11), we can observe that the optimal controller contains $Y_{i}$ and $V_{i}^{*}$. According to the definition of the augmented system (7), we can obtain $Y_{i}$ which overcomes the dependence on system dynamics. Therefore, we only need to obtain the optimal value function $V_{i}^{*}$ from Eq. (III-A) to achieve the optimal controller.

For the purpose of reducing the computational burden when updating the controller, the event-triggered mechanism is introduced. Under the event-triggered mechanism, we only update the controller at a series of discrete instants $\{t_{i}^{h}\}_{h\in\mathbb{N}}$, where $t_{i}^{h}<t_{i}^{h+1}$ holds with $\forall h\in\mathbb{N}$ and the initial instant is set as $t_{i}^{0}=0$.

To define the difference between the augmented consensus error at the last event-triggered instant and in real-time as the measurement error $E_{i}(t)\in\mathbb{R}^{6}$, which is expressed as

For the rigid body $i$, the controller $u_{i}$ is only updated at the event-triggered instants $t_{i}^{h}$, and remains unchanged until a new event is triggered. During the event-triggered intervals $[t_{i}^{h},t_{i}^{h+1})$, the neighbor $j$ of the rigid body $i$ might update its controller at some instants $t_{j}^{h^{\prime}}$, which performs as a piecewise constant. Letting $\mu_{j}$ indicate the event-triggered times of the neighbor $j$, we have $t_{j}^{h^{\prime}}\in\{t_{j}^{0},t_{j}^{1},...,t_{j}^{\mu_{j}}\}$ and $t_{j}^{0}=t_{i}^{h}$.

Therefore, the $e_{i}$-dynamics (III-A) during $[t_{i}^{h},t_{i}^{h+1})$ is represented as follows:

where $\hat{u}_{i}=u_{i}(t_{i}^{h})$ and $\hat{u}_{j}=u_{j}(t_{j}^{h^{\prime}})$ are the control input vectors at the event-triggered instants.

Definition 2 (Event-Triggered Admissible Control): If the following conditions are met: $u_{i}$ is piecewise continuous, $u_{i}(0)=0$, $e_{i}$-dynamics (14) is stable, and the performance function (8) is finite, then $u_{i}$ is called event-triggered admissible control.

Depending on the form of optimal controller (11), we can obtain the event-triggered optimal controller as follows:

where $\nabla\hat{V}_{i}^{*}=\frac{\partial V_{i}^{*}}{\partial e_{i}}(t_{i}^{h})$.

By combining (III-A) and (III-B), the event-triggered HJB equation for the rigid body $i$ is given as follows:

The following assumption is proposed for proving the stability of system (14).

Assumption 2 [36]: The controller $u_{i}$ is Lipschitz continuous during the time interval $[t_{i}^{h},t_{i}^{h+1})$, and there exists a constant $P$ that satisfies the following inequality:

where $P$ indicates the Lipschitz constant. In the actual engineering applications, the selection of $P$ should not be smaller than the maximum value of $\lVert\partial u_{i}/\partial e_{i}^{\top}\rVert$.

For the event-triggered control, it is necessary to ensure that Zeno-behavior does not occur in the proposed event-triggered mechanism. Therefore, we introduce a dynamic event-triggered mechanism inspired by [31] to exclude the Zeno-behavior implicitly. Firstly, a dynamic variable $y_{i}(t)$ is defined as follows:

where $y_{i}(0)\in\mathbb{R}_{\geq 0}$, $\gamma_{i}\in\mathbb{R}_{>0}$, $\kappa_{i}\in[0,\frac{1}{2}]$ and $\varpi_{i}\in[0,1]$. Then, we can obtain the event-triggered instants through the following dynamic event-triggered condition:

where $\theta_{i}\in\mathbb{R}_{>0}$ will be determined later in the proof analysis.

Lemma 1: Assuming that the event-triggered instants $t_{i}^{h}$ are determined by (III-B), $y_{i}(t)\geq 0$ always holds for $\forall t\in[0,+\infty)$ if the given initial value $y_{i}(0)\geq 0$.

Proof: For $\forall t\in[0,+\infty)$, the event-triggered condition (III-B) guarantees the following inequality:

Since the selection of $\theta_{i}$ must satisfy $\theta_{i}>0$, the inequality (20) becomes

By combining (III-B) and (21), we can easily obtain that for $\forall t\in[0,+\infty)$,

According to the comparison lemma in [37], we can deduce that

Therefore, $y_{i}(t)\geq 0$ always holds for $\forall t\in[0,+\infty)$. The complete proof is given. $\hfill\blacksquare$

Theorem 2: Consider a multiple rigid body network with $N$ nodes under a strongly connected communication topology. Supposed that Assumption 2 holds, the performance function and the event-triggered optimal controller are given by (8) and (III-B), respectively. If the event-triggered instants are determined by the dynamic event-triggered condition (III-B), then the following two conclusions can be obtained:

1) The $e_{i}$-dynamics (14) is asymptotically stable, i.e., the optimal attitude consensus is achieved.

2) The Zeno behavior is excluded, i.e., the interval between $t_{i}^{h+1}$ and $t_{i}^{h}$, $\forall i\in\mathcal{V}$ has a positive lower bound.

Proof: 1) Firstly, we prove that the $e_{i}$-dynamics (14) is asymptotically stable. Choosing $\Pi_{i}(t)=V_{i}^{*}\big{(}e_{i}(t)\big{)}+y_{i}(t)$ as the Lyapunov function, which contains the optimal value function $V_{i}^{*}\big{(}e_{i}(t)\big{)}$ in (9) and the dynamic variable $y_{i}(t)$ is governed by (III-B).

By taking the first-order derivative of $V_{i}^{*}(e_{i})$ with respect to $t$ along with the consensus error $e_{i}$, we derive

During the event-triggered intervals $[t_{i}^{h},t_{i}^{h+1})$, assuming that the neighbors of the rigid body $i$ execute $\hat{u}_{j}^{*}=u_{j}^{*}(t_{j}^{h^{\prime}})$. According to (11) and (III-A), it can be easily obtained that

and

Thus, Eq. (III-B) can be redescribed as

According to (III-B) and (III-B), we can obtain the first-order derivative of $\Pi_{i}(t)$ as follows:

Substituting the dynamic event-triggered condition (III-B) into (III-B), $\dot{\Pi}_{i}(t)$ becomes

Since $\varpi_{i}\in[0,1]$, we have $\dot{\Pi}_{i}(t)\leq 0$ if $\theta_{i}\in[\frac{1-\kappa_{i}}{\gamma_{i}},+\infty)$. Therefore, we can select appropriate $\gamma_{i}\in\mathbb{R}_{>0}$, $\kappa_{i}\in[0,\frac{1}{2}]$, $\varpi_{i}\in[0,1]$ and $\theta_{i}\in[\frac{1-\kappa_{i}}{\gamma_{i}},+\infty)$ to ensure that the $e_{i}$-dynamics (14) is asymptotically stable under the dynamic event-triggered condition (III-B).

2) Then, we prove that the Zeno behavior is excluded.

According to (23), we can deduce a sufficient condition of the dynamic event-trigger condition (III-B) when $\varpi$ is selected as zero, which is expressed as follows:

According to the definition of $Y_{i}$, we can conclude that $Y_{i}$ is bounded. That is to say, $Y_{i}\leq Y_{M}$ is satisfied, where $Y_{M}\in\mathbb{R}_{>0}$. With Assumption 1 and the definition of measurement error $E_{i}(t)$, we can obtain that for $\forall t\in[t_{i}^{h},t_{i}^{h+1})$,

Since $E_{i}(t_{i}^{h})=0$ is satisfied at the event-triggered instants, we can derive the following inequality by using the comparison lemma [37]:

Let $\tilde{t}_{i}^{h+1}$ indicate the next event-triggered instant determined by the sufficient condition (III-B). According to (III-B), the sufficient condition (III-B) can be divided into two situations during the time interval $[t_{i}^{h},\tilde{t}_{i}^{h+1})$, which contains: 1) there is no events occuring for all rigid bodies in $\mathcal{N}_{i}$, and 2) there exists at least one event for the rigid body $j\in\mathcal{N}_{i}$.

Situation 1: For all rigid bodies in $\mathcal{N}_{i}$, there is no event-triggered instants during $[t_{i}^{h},\tilde{t}_{i}^{h+1})$. Therefore, we can deduce the following inequality: