## 3 Control Architecture
### Description of Tasks within an Action
Consider a planar parking problem with a goal at the planar absolute coordinates and absolute yaw angle represented by given and/or measured \(x^{*},y^{*},y^{*}\), as depicted in Figure 4. Then, for constructing a planar horizontal velocity field asymptotically converging towards the goal with the required orientation approach, the angular and linear velocities that are required to drive the position error \(e\) (defined in Figure 4) towards zero are of the form  
\[\left\{ \begin{array}{l}\mathbf{\Omega}\doteq\lambda\mathbf{\kappa}=\lambda \alpha\mathbf{k}_{\alpha}\\ \boldsymbol{w}\doteq\gamma\boldsymbol{e}+(\boldsymbol{e}\wedge\mathbf{\Omega} )=\gamma|\boldsymbol{e}|\boldsymbol{i}_{e}+\lambda\boldsymbol{\kappa}| \boldsymbol{e}|\boldsymbol{j}_{e}\\ \doteq\boldsymbol{w}_{e}+\boldsymbol{w}_{\Omega}\end{array} \right., \tag{15}\]  
where \(\alpha\) represents the angle between the position error vector \(\boldsymbol{e}\) and goal orientation vector \(\boldsymbol{i}_{g}\), and \(\boldsymbol{k}_{\alpha}\) represents the unit vector along the axis of angle \(\boldsymbol{\alpha}\). The linear and angular gains denoted by \(\gamma\) and \(\lambda\) can be modulated for consequently modulating the rate of convergence of the field by using a parameter \(\sigma\) as \(\lambda=\sigma\bar{\lambda}\) and \(\gamma=\sigma\bar{\gamma}\). Corresponding to the current position of the vehicle, the associated velocity \(\boldsymbol{w}\) consequently represents the velocity reference to be tracked by the origin of the vehicle frame \(\langle b\rangle\), as well as the heading to be tracked by the same vehicle frame. Thus, the current reference rate \(\boldsymbol{\hat{x}}\) for the origin of the vehicle body frame simply becomes \(\boldsymbol{\hat{x}}=\boldsymbol{w}\). Moreover if we need to definitely converge towards a moving docking station, the reference rate will become  
\[\boldsymbol{\hat{x}}=\boldsymbol{w}+\boldsymbol{w}_{\boldsymbol{t}}, \tag{16}\]  
where \(\boldsymbol{w}_{\boldsymbol{t}}\) is the linear velocity of the docking station. For simplicity, we assume that the heading of the docking station remains constant as it is typically the case. Moreover, as our vehicle is also required to follow this reference velocity by aligning its heading in the velocity direction, let us consider a case when the vehicle is initially misaligned at an angle of \(\beta\) with the reference velocity vector. An appropriate angular velocity \(\boldsymbol{\omega}_{\beta}=\kappa_{\beta}\boldsymbol{\theta}n_{\beta}\) must be applied to reduce the misalignment, where \(\kappa_{\beta}\) is the gain required to align to the reference velocity vector and \(\boldsymbol{n}_{\beta}\) is the unit vector normal to the  
Figure 3: Parking maneuvers from (a) short-range and (b) long-range distances. plane containing \(\mathbf{\omega}_{\beta}\). Recalling the _Velocity Alignment_ Task described earlier, it can also be used to achieve such required alignment. Thus, combining the _Field Velocity_ and _Velocity Alignment_ as current references to be tracked, in case of a reliable tracking of both, will allow the vehicle to successfully carry out homing/parking with minimum energy requirements, as the vehicle will eventually move along its minimal drag direction. For extending the 2D planar parking maneuver to a 3D parking maneuver, the 3D parking problem can be reduced to a 2D planar parking problem where the linear field velocity always lie on the plane formed between the linear error vector \(\mathbf{e}\) and goal orientation vector \(\mathbf{i}_{\text{g}}\) as depicted in Figure 4.  
For the very short-range terminal docking, we can exploit the holonomic behavior of the vehicle using a translation maneuver at the final step. As the translation maneuver is used only if absolutely necessary, this can eliminate a lot of drag and energy consumption. The dock location is assumed to be a point (dock position) in free space with an associated orientation (dock orientation) that the vehicle needs to reach with its heading aligned to the dock orientation.