# A Learnable Viewpoint Evolution Method for Accurate Pose Estimation of Complex Assembled Product
## 2 Proposed Method
### Feasible Region Construction and Update
#### 2.2.3 Boundary Projection  
For each \(\mathbf{T}\) of \(vp,vp\in\mathcal{V}\), the projection to the boundary of \(\mathcal{F}_{i}\) can be planned as  
\[argmin_{\mathbf{T}^{\perp}}\Big{\{}f_{d}(\mathbf{R}^{\perp},\mathbf{R},\mathbf{t} ^{\perp},\mathbf{t})\Big{\}},\ s.t.F_{0}^{\delta_{0},W}\Big{(}\mathbf{T}^{\perp} \Big{)}=0, \tag{6}\]  
where \(\left\{\mathbf{R}_{o}^{\perp},\mathbf{t}_{v}^{\perp}\right\}\in\mathbf{T}^{\perp}\) is the projection result. The function \(f_{d}(\cdot)=\left\|\mathbf{q}\otimes\mathbf{q}_{v}{}^{-1}-[\mathbf{0}_{3\times 1},1]^{ \mathrm{T}}\right\|\) is designed to measure the distance between poses, where \(\mathbf{q}\) is the quaternion representation of \(\mathbf{R}\) and \(\otimes\) defines the multiplication of quaternions. The objective function (6) is a small-scale with equality-constrained optimization problem, and the solution of each iteration can be approximately equivalent to  
\[minf_{1}(\mathbf{T})=\frac{1}{2}\Delta\mathbf{T}^{\mathrm{T}}\
abla^{2}f\Big{(} \mathbf{T}^{k}\Big{)}\Delta\mathbf{T}+\
abla f\Big{(}\mathbf{T}^{k}\Big{)} \Delta\mathbf{T},\ s.t.\
abla F_{0}^{\delta_{0},W}\Big{(}\mathbf{T}^{k}\Big{)} \Delta\mathbf{T}+F_{0}^{\delta_{0},W}\Big{(}\mathbf{T}^{k}\Big{)}=0, \tag{7}\]  
according to SQP [43], where \(f_{d}\Big{(}\mathbf{T}^{k}\Big{)}\) is the objective function of (8) and \({}^{\star\mathrm{T}}\) is the transpose operation. Then, (7) can be rewritten as  
\[minL(\Delta\mathbf{T},\mathbf{\lambda})=\frac{1}{2}\Delta\mathbf{T}^{\mathrm{T}} \mathbf{H}_{f}\Delta\mathbf{T}+\mathbf{J}_{f}\Delta\mathbf{T}+\mathbf{\lambda}^ {\mathrm{T}}\Big{(}\mathbf{J}_{F}\Delta\mathbf{T}+F_{0}^{\delta_{0},W}\Big{(} \mathbf{T}^{k}\Big{)}\Big{)}, \tag{8}\]  
at \(\mathbf{T}^{k}\), \(\Delta\mathbf{T}=\mathbf{T}^{k+1}\odot\mathbf{T}^{k}\) is the variable to be solved, and \(\mathbf{\lambda}\) is the Lagrange multiplier. It should be noted that the above calculation involves the derivation of \(\mathbf{T}\) and the update of \(\Delta\mathbf{T}\). However, the \(\mathbf{R}\) of \(\mathbf{T}\) is only closed for multiplication, not for addition and subtraction. Therefore, a logarithmic mapping needs to be performed on \(\mathbf{T}\) to obtain \(\mathfrak{z}=\left[\phi,\mathbf{\rho}\right]\in\mathbb{R}^{\delta},\mathbf{\rho}\in \mathfrak{so}(3)\) based on the representation of a Lie algebra. \(\odot\) is the subtraction defined on \(\mathfrak{z}\). Let \(f(\cdot)=\left\|\mathbf{T}_{q}\right\|+\left\|\mathbf{T}_{t}\right\|\), where \(\hat{\mathbf{T}}_{q},\hat{\mathbf{T}}_{t}\) are their respective unit vectors. Then, the \(\mathbf{J}_{f}\) of (8) can be expanded as  
\[\mathbf{J}_{f}=\left[\frac{\partial f_{1}}{\partial\delta\mathbf{q}},\frac{ \partial f_{1}}{\partial\delta\mathbf{t}}\right]=\left[\bar{\mathbf{T}}_{q}^{ \dagger}\frac{\partial\mathbf{T}_{q}}{\partial\delta\mathbf{q}}+\hat{\mathbf{T}}_{t }^{\intercal}\frac{\partial\mathbf{T}_{t}}{\partial\delta\mathbf{q}},\hat{ \mathbf{T}}_{t}^{\intercal}\frac{\partial\mathbf{T}_{t}}{\partial\delta\mathbf{ t}}\right], \tag{9}\]  
The related research is mature, and hence, the calculation process of \(\mathbf{\rho}\), \(\phi\), and (9) is omitted. After completing the optimization, the results can be recovered from \(\mathfrak{so}(3)\) to \(\mathrm{SE}(3)\) by exponential mapping. The rSQP [44] is recommended if the scale of \(\mathbf{x}^{w\prime}\) and \(\mathbf{x}\) in other studies is still large.  
Figure 4: Pose feasible region construction.