## 6 Extended Generalized Jacobian for a 3-DOF Redundant Manipulator with Full Base Movement  
### Derivation of Extended Jacobian  
For a redundant manipulator (\(n_{e}>2\)), i.e., with more than 2 joints actuated in a 2D plane and a 2D velocity tracking task \(v_{e}\in\mathbb{R}^{2}\), we have infinite solutions and formulate the inverse kinematics problem as a minimization of joint velocities, as illustrated in Section 2.2. We can also leverage the redundancy of the manipulator to add additional tasks/constraints, for example, to minimize the torque disturbance from the manipulator to the base. The general form of the additional constraint for a 3-DOF manipulator is \(f_{1}q_{1}+f_{2}\dot{q}_{2}+f_{3}\dot{q}_{3}=b\). Adding this constraint to the generalized Jacobian for a 3-DOF serial link manipulator allows us to find a unique solution:  
\[ \begin{bmatrix}v_{x}\\ v_{z}\\ b\end{bmatrix} = \begin{bmatrix}&J_{G}\\ f_{1}&f_{2}&f_{3}\end{bmatrix}\begin{bmatrix}\dot{q}_{1}\\ \dot{q}_{2}\\ \dot{q}_{3}\end{bmatrix}  \tag{29}\]  
As outlined earlier in (20) and (21), the change in angular momentum is caused by an external torque. The forces/torques acting on the manipulator are shown in Figure 16. The equation of the derivative of the angular momentum of the manipulator about the base CoG is:  
\[\frac{dK_{man}}{dt}=-v_{B}\times L_{man}+\tau_{B}+\left(\tau_{m}+r_{m}\times \begin{bmatrix}R_{x}\\ R_{y}\\ R_{z}\end{bmatrix} \right) \tag{30}\]  
where \(v_{B}\) is the velocity of the base CoG (the pole of the angular momentum), \(L_{man}\) is the linear momentum of the manipulator, \(\tau_{B}\) is the torque about the base CoG due to the gravity force (\(m_{m}g\), where \(m_{m}\) is the manipulator mass) and due to the external force on the end-effector (\(F_{e}\)). The term in the brackets is the change in angular momentum of the manipulator caused by the reaction torques exerted by the base on the manipulator and, as this term is equal and opposite to the torque disturbance transmitted from the manipulator to the UAV, we impose that it is null. So, we obtain:  
\[\frac{dK_{man}}{dt}=-v_{B}\times L_{man}+\tau_{B} \tag{31}\]  
Figure 16: Forces and torques on the manipulator for the extended Jacobian formulation. Centers of gravity are the circles with crosses (black for the UAV, green for the manipulator), external forces are in red, reaction forces/torques in orange. Similarly to the approach used in (23) and (24), we can express the manipulator angular momentum in terms of the generalized coordinates of the base \(\dot{q}_{b}\) and manipulator \(\dot{q}_{m}\):  
\[K_{man}=H_{man} \begin{bmatrix}\dot{q}_{b}\\ \dot{q}_{m}\end{bmatrix} = \begin{bmatrix}H_{b,man}&H_{m,man}\end{bmatrix}\begin{bmatrix} \dot{q}_{b}\\ \dot{q}_{m}\end{bmatrix}  \tag{32}\]  
where \(H_{man}\) is partitioned into \(H_{b,man}\in\mathbb{R}^{3\times 3}\) and \(H_{m,man}\in\mathbb{R}^{3\times n_{e}}\). We discretize Equation (31): we write the derivative of the angular momentum as \(\left(K_{man,k}-K_{man,k-1}\right)/\Delta t\) and we isolate \(K_{man,k}\) which can be written as a function of the unknown joint velocities \(\dot{q}_{m,k}\):  
\[K_{man,k}= \begin{bmatrix}H_{b,man}(q_{k-1})&H_{m,man}(q_{k-1})\end{bmatrix} \begin{bmatrix}\dot{q}_{b,k}\\ \dot{q}_{m,k}\end{bmatrix} =K_{man,\;k-1}+\left(\tau_{B,k-1}-v_{B,k-1}\times L_ {man,k-1}\right)\Delta t \tag{33}\]  
In (33), we take all joint angles and manipulator momenta for timestep \(k-1\) with the assumption that these quantities do not change significantly over one timestep. As in the derivation of the generalized Jacobian with external forces, we express the generalized velocities of the UAV \(\dot{q}_{b,\;k}\) in terms of \(\dot{q}_{m,\;k}\) and of the angular momenta of the UAV to obtain the inverse kinematics equation for the extended Jacobian, dropping the discretized notation for joint angles \((q_{k-1})\) for brevity:  
\[ \begin{bmatrix}H_{b,man}\;H_{b}^{-1}\;H_{m}+H_{m,man}\end{bmatrix} \dot{q}_{m,k} =