## 3 CLIK Algorithms for UAM Control
### Acceleration-Level CIK Algorithm for UAMs  
In this subsection, we present a novel and more sophisticated approach based on the work done for space manipulators kinematic control [22], which allows us to solve flying-base robotic systems inverse kinematics at the acceleration level. We assume that the trajectory of the end-effector is given through its acceleration profile, \(\ddot{\mathbf{x}}_{e,des}\), and that the generalized external forces, \(\mathbf{F}_{ext}\), acting on the system are known. The momentum in Equation (7) can be rearranged to obtain the following acceleration constraint:  
\[[\mathbf{H}_{b}]\ddot{\mathbf{x}}_{b}+[\mathbf{H}_{bm}]\ddot{\mathbf{q}}=\dot{\mathbf{h}}-\left( \left[\dot{\mathbf{H}}_{b}\right]\dot{\mathbf{x}}_{b}+\left[\dot{\mathbf{H}}_{bm}\right] \dot{\mathbf{q}}\right)\doteq\mathbf{z}_{1} \tag{21}\]  
where \(\dot{\mathbf{h}}=\{\dot{\mathbf{p}},\dot{\mathbf{K}}\}^{T}\) is the vector containing the derivatives of the linear and angular momenta. Since we consider the angular momentum about the CoG of the base, which is a moving point, its derivative is \(\dot{\mathbf{K}}=-\mathbf{v}_{B}\wedge m_{UAM}\mathbf{v}_{G_{UAM}}+\mathbf{m}_{ext}\) (where \(\mathbf{v}_{B}\) and \(\mathbf{v}_{G_{UAM}}\) represent the linear velocity of the base CoG and that of the entire system CoG, respectively). Along with (21), we consider the kinematic Equation (4), which can be rearranged as follows:  
\[[\mathbf{J}_{b}]\ddot{\mathbf{x}}_{b}+[\mathbf{J}]\ddot{\mathbf{q}}=\ddot{\mathbf{x}}_{e,des}- \left(\left[\dot{\mathbf{J}}_{b}\right]\dot{\mathbf{x}}_{b}+\left[\dot{\mathbf{J}}\right] \dot{\mathbf{q}}\right)\doteq\mathbf{z}_{2} \tag{22}\]  
Equations (21) and (22) can be put together to form the following linear system:  
\[ \begin{bmatrix}[\mathbf{H}_{b}]&[\mathbf{H}_{bm}]\\ \dot{[}\mathbf{J}_{b}]&[\mathbf{J}]\end{bmatrix}\begin{cases}\ddot{\mathbf{x}}_{b}\\ \dot{\mathbf{q}}\end{cases} = \begin{cases}\mathbf{z}_{1}\\ \mathbf{z}_{2}\end{cases}  \tag{23}\]  
For a UAM with an \(n\)-DOF manipulator, the matrix in the above equation is of dimensions \(12\times(n+6)\). If all the external forces and torques acting on the system can be estimated accurately so that the term \(\mathbf{F}_{ext}\) is sufficiently reliable, the inversion of Equation (23) leads to open-loop inverse kinematic control at the acceleration level:  
\[ \begin{cases}\ddot{\mathbf{x}}_{b}\\ \dot{\mathbf{q}}\end{cases} = \begin{bmatrix}[\mathbf{H}_{b}]&[\mathbf{H}_{bm}]\\ \dot{[}\mathbf{J}_{b}]&[\mathbf{J}]\end{bmatrix} ^{-1} \begin{cases}\mathbf{z}_{1}\\ \mathbf{z}_{2}\end{cases}  \tag{24}\]  
If the manipulator is redundant, the solution can be obtained by using the pseudoinverse instead of the inverse of the matrix in (24). The accurate prediction of the external forces on the system is difficult. Therefore, a more reliable control scheme can be obtained by adding a feedback term into the task-space error in \(\mathbf{z}_{2}\):  
\[\mathbf{z}_{2}\doteq\ddot{\mathbf{x}}_{e,des}-\left(\left[\dot{\mathbf{J}}_{b}\right] \ddot{\mathbf{x}}_{b}+\left[\dot{\mathbf{J}}\right]\dot{\mathbf{q}}\right)+\left([\mathbf{K} _{P}]\mathbf{e}+[\mathbf{K}_{D}]\dot{\mathbf{e}}\right) \tag{25}\]  
As we are operating at the acceleration level, the feedback term can include a derivative action based on the end-effector velocity error, \(\dot{\mathbf{e}}=\dot{\mathbf{x}}_{e,des}-\dot{\mathbf{x}}_{e}\). The gain matrices \([\mathbf{K}_{P}]=k_{P}[\mathbb{I}_{n}]\) and \([\mathbf{K}_{D}]=k_{D}[\mathbb{I}_{n}]\) are diagonal and positive definite. The joint velocities, \(\dot{\mathbf{q}}\)inside \(z_{2}\) can be measured through sensors.