2. Problem statement

The problem of attitude reconfiguration of a team of communicating spacecraft with avoidance constraints can be stated as follows. Given a set of communicating spacecraft with initial positions at xi (t0) ∈ R<sup>3</sup> i = 1⋯n, initial attitudes represented by quaternions qi (t0), generate a sequence of consensus trajectories that drive the team to a consensus attitude q(tf) while satisfying exclusion, avoidance and norm constraints.

computes a new set of quaternion vectors that avoid collision and the cycle repeats until

To understand the avoidance aspect, we begin with a simpler illustration of the spacecraft Q-CAC problem with a single spacecraft and a single obstacle (exclusion) vector, as shown in

nion representing the obstacle to be avoided (e.g. the Sun, as shown in Figure 1). It is

always, while maintaining a minimum angular separation of ∅. The requirement can there-

∀t ∈ t0; tf

The constraint is non-convex and quadratic and should be convexified for it to be represented as a LMI. The convexification was provided in [4], using the quaternion attitude constraints formulation developed in [3] for a single-spacecraft single-obstacle scenario. For that solution, vI

makes it incomplete for practical implementation because, in reality the obstacle and space-

In [7–9, 16], we extended the previous avoidance solution to multiple spacecraft. Then we developed a consensus theory of quaternions and appended the new avoidance protocols. We further solved the problem for spacecraft and dynamic obstacles in different coordinate frames to make the solutions more suitable for practical implementation. Next, we present the basic

In this section, we consider the two basic mathematical theories relevant to this chapter.

ble to the problems of singularities inherent in using Euler angles [17].

It is convenient to use unit quaternions to represent the attitude of a rigid body rotating in three-dimensional space (such as spacecraft or satellite) because quaternions are not suscepti-

ð Þ<sup>t</sup> denote the unit camera vector in <sup>F</sup><sup>I</sup>

Multi-Spacecraft Attitude Path Planning Using Consensus with LMI-Based Exclusion Constraints

cami

ð Þt ≤ cos ∅,

ð Þt was evolving, and both vectors were in the same coordinate frame. This

obsi

cami tf

θð Þt ≥ ∅ (1)

(2)

ð Þ <sup>t</sup><sup>0</sup> to <sup>v</sup><sup>I</sup>

SCi corresponding

obsi ð Þt 49

obs

ð Þt be the attitude quater-

should avoid vI

http://dx.doi.org/10.5772/intechopen.71580

cami

vI cami ð Þt T vI obsi

to the SCi's attitude q<sup>i</sup> (see Table 1 for definitions). Also, let vI

desired that the time evolution of camera vector vI

consensus is achieved.

Let SCi denote spacecraft i, and v<sup>I</sup>

Figure 1.

fore be stated as

was static, vI

cami

mathematical preliminaries.

3. Mathematical background

3.1. Quaternion-based rotational dynamics

craft are in different coordinate frames.

or

There are two aspects of the problem stated above: the first is a consensus problem, wherein it is desired to drive the attitudes to a collective consensus attitude, or to various formation attitudes. For bare consensus, the final consensus is that each spacecraft should eventually point to the average of the initial attitudes. However, relative offset quaternions can be applied so the consensus attitude can be a desired formation attitude, e.g. each spacecraft can point at 5 away from each other about the z axis. The second problem is that of avoidance constraints. This is also important for the team, because spacecraft usually have appendages, some have thrusters that emit plumes, and some have instruments that can be damaged by blinding celestial objects or by the appendage or plume of a team member.

However, the ordinary consensus protocol was not developed for quaternion dynamics. It violates the non-linearity of quaternion kinematics and the quaternion norm preserving requirement. Moreover, the ordinary consensus algorithm also does not incorporate collision avoidance in adversarial situations; this is a Q-CAC problem. Thus, in this paper, we present aspects of our previous works [7–9, 16], where we combined consensus theory with constrained optimization to solve the problems stated above. We cast the problems as a semidefinite program (SDP), which is augmented with some convex quadratic constraints written as linear matrix inequalities (LMI).

We present a quaternion consensus protocol that computes a consensus attitude trajectory each time step, and a Q-CAC optimization procedure, which decides whether it is safe to follow the computed attitude trajectory or not. When generated trajectories are unsafe, it

Figure 1. Constrained attitude control problem for a single-spacecraft single-exclusion scenario.

computes a new set of quaternion vectors that avoid collision and the cycle repeats until consensus is achieved.

To understand the avoidance aspect, we begin with a simpler illustration of the spacecraft Q-CAC problem with a single spacecraft and a single obstacle (exclusion) vector, as shown in Figure 1.

Let SCi denote spacecraft i, and v<sup>I</sup> cami ð Þ<sup>t</sup> denote the unit camera vector in <sup>F</sup><sup>I</sup> SCi corresponding to the SCi's attitude q<sup>i</sup> (see Table 1 for definitions). Also, let vI obsi ð Þt be the attitude quaternion representing the obstacle to be avoided (e.g. the Sun, as shown in Figure 1). It is desired that the time evolution of camera vector vI cami ð Þ <sup>t</sup><sup>0</sup> to <sup>v</sup><sup>I</sup> cami tf should avoid vI obsi ð Þt always, while maintaining a minimum angular separation of ∅. The requirement can therefore be stated as

$$\Theta(t) \succeq \bigotimes \tag{1}$$

or

2. Problem statement

48 Advanced Path Planning for Mobile Entities

satisfying exclusion, avoidance and norm constraints.

ten as linear matrix inequalities (LMI).

celestial objects or by the appendage or plume of a team member.

positions at xi

The problem of attitude reconfiguration of a team of communicating spacecraft with avoidance constraints can be stated as follows. Given a set of communicating spacecraft with initial

(t0) ∈ R<sup>3</sup> i = 1⋯n, initial attitudes represented by quaternions qi

sequence of consensus trajectories that drive the team to a consensus attitude q(tf) while

There are two aspects of the problem stated above: the first is a consensus problem, wherein it is desired to drive the attitudes to a collective consensus attitude, or to various formation attitudes. For bare consensus, the final consensus is that each spacecraft should eventually point to the average of the initial attitudes. However, relative offset quaternions can be applied so the consensus attitude can be a desired formation attitude, e.g. each spacecraft can point at 5 away from each other about the z axis. The second problem is that of avoidance constraints. This is also important for the team, because spacecraft usually have appendages, some have thrusters that emit plumes, and some have instruments that can be damaged by blinding

However, the ordinary consensus protocol was not developed for quaternion dynamics. It violates the non-linearity of quaternion kinematics and the quaternion norm preserving requirement. Moreover, the ordinary consensus algorithm also does not incorporate collision avoidance in adversarial situations; this is a Q-CAC problem. Thus, in this paper, we present aspects of our previous works [7–9, 16], where we combined consensus theory with constrained optimization to solve the problems stated above. We cast the problems as a semidefinite program (SDP), which is augmented with some convex quadratic constraints writ-

We present a quaternion consensus protocol that computes a consensus attitude trajectory each time step, and a Q-CAC optimization procedure, which decides whether it is safe to follow the computed attitude trajectory or not. When generated trajectories are unsafe, it

Figure 1. Constrained attitude control problem for a single-spacecraft single-exclusion scenario.

(t0), generate a

$$\begin{aligned} \left[\boldsymbol{v}\_{cam\_i}^{l}(t)\right]^T \boldsymbol{v}\_{obs\_i}^{l}(t) & \leq \cos \mathfrak{Q},\\ \forall t \in \left[t\_0, t\_f\right] \end{aligned} \tag{2}$$

The constraint is non-convex and quadratic and should be convexified for it to be represented as a LMI. The convexification was provided in [4], using the quaternion attitude constraints formulation developed in [3] for a single-spacecraft single-obstacle scenario. For that solution, vI obs was static, vI cami ð Þt was evolving, and both vectors were in the same coordinate frame. This makes it incomplete for practical implementation because, in reality the obstacle and spacecraft are in different coordinate frames.

In [7–9, 16], we extended the previous avoidance solution to multiple spacecraft. Then we developed a consensus theory of quaternions and appended the new avoidance protocols. We further solved the problem for spacecraft and dynamic obstacles in different coordinate frames to make the solutions more suitable for practical implementation. Next, we present the basic mathematical preliminaries.
