## 2 Improving the Accuracy of Determination of Work Object's Shape and Location
 \tag{3}\]  
where (\(\cdot\)) is the scalar product of vectors; (\(\times\)) is the vector product of vectors. The trajectory projections consist only of the coordinates of the points \(C_{i}\) and \(C_{i}^{\prime}\) and, therefore, they can be transmitted to the operator's control console even via a hydroacoustic communication channel with a low bandwidth. The operator can visually evaluate the locations of these trajectories relative to each other, and then either confirm execution of the manipulative operation or give a command to re-scan and re-align the clouds. For a UUV having no communication channel with its operator, criteria should be set out to evaluate adequacy of the previously performed alignment. Standard deviation of the coordinates of the projected trajectories' points can be used as such a criterion: \(Q=\sqrt{\sum_{i=1}^{n}\|\overset{\rightarrow}{E_{i}}\|^{2}},\overset{ \rightarrow}{E_{i}}=C_{i}-C_{i}^{\prime}\) (Figure 5). Figure 4: General view of the trajectory, projection vectors and triangulation surfaces. The threshold value of the criterion \(Q\) depends on the dimensions of the object and the resolving power of MVS. Studies have shown that the threshold value for the simulated MVS and object is \(Q_{max}\); if it is exceeded, the alignment can be considered erroneous. In this example of successful alignment of point clouds, the value amounts to \(Q=0.000346\). When the performed test shows that the object model's point cloud is precisely aligned with the object's cloud, the trajectory of the MM working tool built on the original object model can be transferred to the actual object. This trajectory can be initially defined both in an analytical form and as a sequence of points \(X_{j}^{\prime}\in R^{3}\) (\(j=\overline{1,s}\), where \(s\) is the number of points) with the respective single vectors \(\vec{a^{\prime}}_{j}\in R^{3}\) and \(\vec{b^{\prime}}_{j}\in R^{3}\) defining the desired orientation of the working tool at the \(j\)th point. With Equation (2) taken into account, the trajectory transferred to an actual object can be expressed as follows:  
\[X_{j}=H_{D}+M_{r2}M_{r1}\left(X_{j}^{\prime}-H_{K}\right) \tag{4}\]  
\[\vec{a}_{j}=M_{r2}M_{r1}\vec{a}_{j}^{\prime},\] \[\vec{b}_{j}=M_{r2}M_{r1}\vec{b}_{j},\]  
where \(X_{j}\) are the points with the respective orientation vectors \(\vec{a}_{j}\) and \(\vec{b}_{j}\) that define the desired positions of the MM working tool on the object surface, and \(H_{K}=\frac{1}{m}\sum_{q=1}^{m}K_{q}\) is the center of mass of the object model's cloud of initial points \(K_{q}\). In cases where the exact shape of an object cannot be known in advance, or there is a high probability of its deformation, fouling or silting, the desired trajectory of the MM working tool (4) can additionally be projected onto the triangulation surface of this object constructed using MVS. To do this, similarly to Equation (3), points \(X_{j}\) are projected in the positive and negative directions of the vector \(\vec{o}\), because these points can be located both outside and inside the triangulation surface of the object. As a result, a sequence of \(T_{j}\in R^{3}\) points is formed to be located on the surface of the actual object:  
\[T_{j}=X_{j}\pm\frac{\left(\left(X_{j}-V_{10}\right)\times\left(V_{h1}-V_{h0} \right)\right)\cdot\left(V_{h2}-V_{h0}\right)\rightarrow}{\left(\pm\vec{o} \times\left(V_{h2}-V_{h0}\right)\right)\cdot\left(V_{h1}-V_{h0}\right)}o.\]  
To provide smooth movements of the MM working tool along the trajectory, the sequences of points \(X_{j}\) or \(T_{j}\) can be smoothed by parametric 3 rd-order B-splines using the algorithm [32].