## 3 Proposed Redundancy Resolution  
This section proposes a new redundancy resolution technique to generate energy-efficient trajectories for the vehicle and the manipulator. It is known that infinite solutions of the UVMS inverse kinematics can be obtained by inverting the mapping (1). The solution using the pseudo inverse of the Jacobian matrix is expressed as [2]  
\[\mathbf{\zeta}=\mathbf{J}^{+}(\mathbf{R}_{B}^{I},\mathbf{q})\hat{\mathbf{x}}_{E} \tag{6}\]  
where \(\hat{\mathbf{x}}_{E}\) is the end-effector velocity vector. \(\mathbf{J}^{+}(\mathbf{R}_{B}^{I},\mathbf{q})\) is the pseudo inverse of the Jacobian matrix and \(\mathbf{J}^{+}(\mathbf{R}_{B}^{I},\mathbf{q})=\mathbf{J}^{T}(\mathbf{R}_{B}^{I},\mathbf{q})(\mathbf{J}( \mathbf{R}_{B}^{I},\mathbf{q})\mathbf{J}^{T}(\mathbf{R}_{B}^{I},\mathbf{q}))^{-1}\). However, this solution does not exploit the redundant DOFs of the system, and it is not suitable from the perspective of energy consumption. Therefore, a new task-priority redundancy resolution technique is proposed in this section. In the proposed technique, the primary task is to map the end-effector variables into the joint-space variables, and two secondary tasks are provided to explore the kinematic redundancy for energy savings, joint limit avoidance and small roll and pitch angles kept for the vehicle, as shown in (7). \[\zeta_{d}=\mathbf{J}_{W}^{+}(\hat{\mathbf{x}}_{Ed}-\mathbf{K}_{f}\epsilon_{E})+(\mathbf{I}- \mathbf{J}_{W}^{+}\mathbf{J}_{W})[\mathbf{J}_{s}^{+}(\mathbf{\eta},\mathbf{q})(\hat{\mathbf{\eta}}_{sd} -\mathbf{K}_{s}\epsilon_{s})-\alpha\mathbf{K}_{f}\mathbf{J}_{\xi}(\eta_{2})\hat{\xi}] \tag{7}\]  
where \(\mathbf{J}_{W}^{+}=\mathbf{W}^{-1}\mathbf{J}^{T}(\mathbf{J}\mathbf{W}^{-1}\mathbf{J}^{T})^{-1}\) is the weighted pseudo-inverse Jacobian matrix. \(\mathbf{J}\) is considered as the primary task Jacobian matrix and \(\mathbf{W}\in\mathbb{R}^{(6+n)\times(6+n)}\) is the motion distribution matrix with elements belonging to \([0,1]\). When the diagonal elements of the former three rows of \(\mathbf{W}\) are close to 1, the diagonal elements of the later \(n\) rows of \(\mathbf{W}\) will be close to 0. This results in greater movement of the vehicle and less movement of the manipulator. Otherwise, when the diagonal elements of the former three rows of \(\mathbf{W}\) are close to 0, the diagonal elements of the later \(n\) rows of \(\mathbf{W}\) will be close to 1. This results in less movement of the vehicle and greater movement of the manipulator. The diagonal elements of the middle three rows of \(\mathbf{W}\) correspond to the movement of the vehicle's attitude. The larger they are, the greater the movement of the vehicle's attitude. The off-diagonal elements of \(\mathbf{W}\) describe the degrees of the coupling effects between the DOFs of the UVMS, which can refer to our previous work [15]. The closer the off-diagonal element of \(\mathbf{W}\) to \(1\), the greater the corresponding coupling motion. \(\mathbf{J}_{s}(\mathbf{\eta},\mathbf{q})\) and \(\mathbf{J}_{\xi}(\mathbf{\eta}_{2})\) are the secondary task Jacobian matrices. It can be recognized that the secondary tasks are fulfilled in the null space, which will not affect the motion of the primary task. Moreover, the two secondary tasks have the same lower priority relative to the primary task. \(\hat{\mathbf{x}}_{Ed}\) is the primary-task vector and \(\mathbf{e}_{E}=\mathbf{x}_{E}-\mathbf{x}_{Ed}\) is the error of the primary task. \(\hat{\mathbf{y}}_{sd}\) is a secondary-task vector to achieve system coordination between its rotational subsystem and translational subsystem, including the vehicle attitudes and joint angles, and \(\mathbf{e}_{s}=\hat{\mathbf{y}}_{s}-\hat{\mathbf{y}}_{sd}\) is its error. \(\mathbf{K}_{f}\) and \(\mathbf{K}_{s}\) are positive definite matrices. The other secondary task vector is the velocity vector of the UVMS \(\hat{\mathbf{\xi}}=[\hat{\mathbf{y}}_{1}^{T},\hat{\mathbf{y}}_{2}^{T},\hat{\mathbf{q}}^{T}]^{T}\), which contributes to the system's self-motion utilized for reducing energy requirements.