# Analytical Attitude Guidance Planner for Multiple Ground Targets Acquisitions
## 2 Problem Description and Proposed Solution
### Physical Constraints
#### 2.3.2 Mapping the Control Constraint to a Kinematic Constraint  
Considering the limited capabilities of control actuators and the presence of perturbations, the restriction on control torque is expressed as an inequality constraint for each component in \(BF\)  
\[|u_{c_{i}}(t)+u_{p_{i}}(t)|\leq u_{max},\quad i=x,y,z, \tag{5}\]where \(u_{max}\) is a parameter designed to assure that the control torque \(u_{c_{i}}(t)\) is consistent with the planned satellite kinematic also considering the perturbations. Since the goal of this work is to find an analytical solution without using attitude dynamics, it is useful to translate the control constraint in Equation (5) in a limitation of the angular acceleration. The \(u_{c_{i}}(t)\) components, with \(i=x,y,z\), are written as a function of angular velocity and angular acceleration by using Equation (2):  
\[\begin{split}|I_{x}\dot{\omega}_{x}+(I_{z}-I_{y})\omega_{y} \omega_{z}|&\leq u_{max}\\ |I_{y}\dot{\omega}_{y}+(I_{x}-I_{z})\omega_{x}\omega_{z}|& \leq u_{max}\\ |I_{z}\dot{\omega}_{z}+(I_{y}-I_{x})\omega_{y}\omega_{x}|& \leq u_{max}\end{split} \tag{6}\]  
By applying the triangle inequality and substituting \(\omega_{y}=\omega_{z}=\omega_{max}\), Equation (6) can be rewritten as follows  
\[\begin{split}|I_{x}\dot{\omega}_{x}+(I_{z}-I_{y})\omega_{y} \omega_{z}|&\leq|I_{x}\dot{\omega}_{x}|+|(I_{z}-I_{y})\omega_{max }^{2}|&\leq u_{max}\\ |I_{y}\dot{\omega}_{y}+(I_{x}-I_{z})\omega_{x}\omega_{z}|& \leq|I_{y}\dot{\omega}_{y}|+|(I_{x}-I_{z})\omega_{max}^{2}|& \leq u_{max}\\ |I_{z}\dot{\omega}_{z}+(I_{y}-I_{x})\omega_{y}\omega_{x}|& \leq|I_{z}\dot{\omega}_{z}|+|(I_{y}-I_{x})\omega_{max}^{2}|& \leq u_{max}\end{split} \tag{7}\]  
Imposing the equality condition in the last component of Equation (7), the following equations are obtained  
\[\begin{split}\dot{\omega}_{max_{x}}&=\frac{u_{max}-|( I_{z}-I_{y})\omega_{max}^{2}|}{I_{x}}\\ \dot{\omega}_{max_{y}}&=\frac{u_{max}-|(I_{x}-I_{z}) \omega_{max}^{2}|}{I_{y}}\\ \dot{\omega}_{max_{z}}&=\frac{u_{max}-|(I_{y}-I_{x}) \omega_{max}^{2}|}{I_{z}},\end{split} \tag{8}\]  
where \(\dot{\omega}_{max_{x}}\triangleq|\dot{\omega}_{i}|\) (\(i=x,y,z\)) by definition is the value of \(\dot{\omega}_{i}\) such that, for any combination of achievable satellite angular velocities, the control constraint in Equation (5) is satisfied. Hence, given \(u_{max}\), we choose \(\dot{\omega}_{max_{i}}\) as the maximum allowed angular acceleration  
\[|\dot{\omega}_{i}(t)|\leq\dot{\omega}_{max_{i}},\quad i=x,y,z. \tag{9}\]  
For simplicity, a unique value is chosen by taking \(\dot{\omega}_{max}=\min(\dot{\omega}_{max_{i}})\), i.e.,  
\[|\dot{\omega}_{i}(t)|\leq\dot{\omega}_{max},\quad i=x,y,z. \tag{10}\]  
Equations (4) and (10) are kinematic constraints and will be used in the \(APPG\) phase to generate a feasible path by means of an analytical approach. A polynomial approximation is exploited for each component of the angular velocity vector. With this technique, both the boundary conditions and the dynamical constraints are directly satisfied and the guidance is obtained in a closed-form analytical solution. It is worth pointing out that the attitude dynamics will no longer be taken into consideration, in fact, the manuscript operates with kinematics only to design a reference guidance policy in terms of quaternion, angular velocities, and angular accelerations.