# LAMDA Controller Applied to the Trajectory Tracking of an Aerial Manipulator
## 2 Aerial Manipulator Robot Model
### Kinematic Model  
The aerial manipulator robot consists of a robotic arm located on hexacopter, as shown in Figure 1. In the configuration of the aerial manipulator robot the position and orientation of all its points with respect to an inertial reference system \(\mathcal{R}(\mathcal{X},\mathcal{Y},\mathcal{Z})\) is known. The configuration of the aerial manipulator robot is defined by a vector \(q(t)\) of \(n\) independent coordinates, which are called the generalized coordinates of the aerial manipulator robot, defined as \(q(t)=\left[q_{a}\;q_{Q}\right]\) where \(q_{a}\) are the velocities of hexacopter, and \(q_{Q}\) are velocities of the robotic arm. The configuration of \(q(t)\) is an element of the configuration space of the aerial manipulator robot represented by \(\mathcal{N}\). The position of the aerial manipulator robot end-effector \(\xi(t)=\left[\xi_{1}\;\xi_{2}\dots\;\xi_{m}\right]\) is defined by an \(m\)-dimensional vector representing the orientation and position of the robot's point of interest with respect to \(\{\mathcal{R}\}\). The position of the end-effector with respect to \(\{\mathcal{R}\}\) defines the operating space of the aerial manipulator robot, denoted by \(\mathcal{M}\).  
The direct kinematics of the aerial manipulator robot defines the position and orientation of the end-effector \(\xi(t)\) as a function of the configuration of the robotic arm and the  
Figure 1: Aerial manipulator robot schematic.  
hexacopter (or its operational coordinates as functions of the generalized coordinates of the robotic arm and the operational coordinates of the hexacopter).  
\[f:\ \mathcal{N}_{a}x\mathcal{N}_{Q}\rightarrow\mathcal{M} \tag{1}\]  
\[\big{(}q_{a}(t),\ q_{Q}(t)\big{)}\rightarrow\check{\xi}(t)=f\big{(}q_{a},\ q_{Q} \big{)} \tag{2}\]  
The kinematic model of the aerial manipulator robot sets the derivative of the end-effector position as a function of the derivative of the configuration of the robotic arm and the hexacopter (or their operational velocities as functions of their generalized velocities).  
\[\dot{\check{\xi}}(t)=\frac{\partial\big{(}q_{a},\ q_{Q}\big{)}}{\partial q} \mu(t) \tag{3}\]  
where \(\dot{\check{\xi}}(t)=[\dot{\xi}_{1}\ \dot{\dot{\xi}}_{2}\dots\ \dot{\dot{\xi}}_{m}]\in R^{m}\) represents the velocity vector of the end-effector with respect to the inertial reference system \(\{\mathcal{R}\}\). Let \(\mu(t)=\big{[}\dot{q}_{a}\ \dot{u}\big{]}\) be the maneuverability vector of the aerial manipulator robot. Considering \(J(q)=\frac{\partial\check{J}}{\partial q}T(q)\) in the previous equation, one has:  
\[\dot{\check{\xi}}(t)=J(q)\mu(t) \tag{4}\]  
where the Jacobian matrix \(J\big{(}q_{q},\ q_{Q}\big{)}\) defines the linear mapping between the aerial manipulator robot maneuverability vector \(\mu(t)\) and the end-effector velocity vector \(\dot{\check{\xi}}(t)\). \(T(q)\) is the transformation matrix relating the velocities of \(\dot{q}(t)\) and the velocities of the aerial manipulator robot \(\mu(t)\), in such a way that: \(\dot{q}(t)=T(q)\mu(t)\).  
It is essential to emphasize that the dimension of the operational space \(m\) is smaller than the aerial manipulator robot degree of freedom \(n\ (m<n.\ )\), so the aerial manipulator robot is called a redundant system.