# LAMDA Controller Applied to the Trajectory Tracking of an Aerial Manipulator
## 3 Proposed Controllers
### Kinematic Controller  
The kinematic controller receives the desired position and velocity of the aerial manipulator robot end-effector \(\tilde{\xi}_{d}(t)\in R^{m}\) and \(\tilde{\xi}_{d}(t)\in R^{m}\), respectively, and generates the maneuverability velocities of the aerial manipulator \(\mu_{ref}(t)\in R^{n}\). In other words, the desired operational motion of the robot is an application of \(\left(\tilde{\xi}_{d}(t)\middle|t\in\left[t_{0},t_{f}\right]\right)\). Therefore, the motion control problem is to determine the maneuverability vector \(\left(\mu_{ref}(t)\middle|t\in\left[t_{0},t_{f}\right]\right)\) to achieve the desired operational motion of the robot. The corresponding evolution of the entire aerial manipulator robot is given by the generalized coordinate vector of the aerial manipulator robot \(\left(q(t)\middle|t\in\left[t_{0},t_{f}\right]\right)\).  
The proposed controller is based on a minimum norm solution, which means that, at any given time, the aerial manipulator robot will achieve its navigation and manipulation goals with as few movements as possible. In addition, the redundancy of the aerial  
Figure 2: Proposed controller scheme.  
manipulator robot is used to achieve secondary control objectives, i.e., the control of the internal configuration of the robotic arm. Therefore, the following control law is proposed,  
\[\mu_{c}(t)=J(q(t))^{\#}\Big{(}\xi_{d}^{-1}(t)+Wtanh\Big{(}\widetilde{\xi}(t) \Big{)}\Big{)}+\Big{(}I_{nxm}-J(q(t))^{\#}J(q(t))\Big{)}\mu_{0}(t) \tag{11}\]  
where \(J(q(t))^{\#}=J(q(t))^{\mathsf{T}}\Big{(}J(q(t))J(q(t))^{T}\Big{)}^{-1}\in R^{ nxm}\) represents the pseudoinverse matrix; \(W\in R^{nxm}\) is a positive diagonal matrix weighing the control errors \(\widetilde{\xi}(t)\). In the proposed control law (11), the first term on the right describes the main task of the end-effector, which minimizes the term \(\frac{1}{2}\|\mu(t)\|_{2}^{2}\). The second term defines the motion of the aerial manipulator robot configuration in which the matrix \(\big{(}I_{nxm}-J(q(t))^{\#}J(q(t))\big{)}\) projects an arbitrary vector \(\mu_{0}(t)\) onto the null space of the Jacobian matrix \(J(q(t))\) such that the secondary control objectives do not affect the main task of the end-effector. Therefore, any value given to \(\mu_{0}(t)\) will affect only the internal structure of the aerial manipulator robot, but not the final control of the end-effector. The proposed control law (12) considers an analytical saturation of the maneuverability velocities of the aerial manipulator robot. Therefore, the function \(\tanh\) (hyperbolic tangent) bounds the control errors between \([-1,\ 1]\). The redundancy of the aerial manipulator robot can be effectively used for the achievement of additional performances such as avoiding obstacles in the workspace and singular configurations, or to optimize various motion criteria. In this work, it is considered to control the internal configuration of the robotic arm so that the center of mass of the aerial manipulator robot does not affect the motion of the robotic system. Therefore, \(\mu_{0}(t)\) is defined as,  
\[\mu_{0}(t)=Ktanh(q_{d}(t)-q(t)) \tag{12}\]  
where \(q_{d}\) is the desired joint positions of the robotic arm, considering that the center of gravity is located at the center of the aerial manipulator robot, and \(K\in R^{nxm}\) is a diagonal matrix weighing the errors of the joint positions of the robotic arm, defined as \(\widetilde{q}(t)=q_{d}(t)-q(t)\).