## 4 Thick Ellipsoid State Estimation Algorithm
### Thick Ellipsoid Prediction Step  
For state estimation purposes, typically thick ellipsoids have to be considered in the prediction step according to Section 3.1 which have a non-zero midpoint. Their propagation then consists of separating the state equation as illustrated in Figure 7 into a part that depends on an ellipsoid centered at the origin and an offset term that accounts for the non-zero center according to  
\[\mathbf{x}_{k+1}=\mathbf{A}(\mathbf{x}_{k},\mathbf{p}_{k})\cdot\tilde{\mathbf{ x}}_{k}+\tilde{\mathbf{A}}_{k}\cdot\boldsymbol{\mu}_{k}+\left(\mathbf{A}( \mathbf{x}_{k},\mathbf{p}_{k})-\tilde{\mathbf{A}}_{k}\right)\cdot\boldsymbol{ \mu}_{k}\enspace, \tag{66}\]  
Figure 5: Union of ellipsoids \(\mathcal{E}_{1,k}\) and \(\mathcal{E}_{2,k}\) with identical and different midpoints. Inner bound of the resulting thick ellipsoid: \(\mathcal{E}_{UL,k}^{\mathrm{U}}\); outer bound: \(\mathcal{E}_{UL,k}^{\mathrm{O}}\) (* symbols denote the solution enclosure according to Equation (65)). (**a**) Union of ellipsoids with identical midpoints, cf. (55). (**b**) Union of ellipsoids with the initial ellipsoid parameterization (55), (56), enclosure definition (65). (**c**) Union of ellipsoids with the initial ellipsoid parameterization (56), (57), enclosure definition (60). (**d**) Union of ellipsoids with the initial ellipsoid parameterization (56), (57), enclosure definition (65). Figure 6: Thick ellipsoid state estimation algorithm consisting of prediction and correction stages. where  
\[\mathbf{x}_{k}\in\big{(}\mathcal{E}\big{)}_{\!k}\!\left(\mu_{k}, \Gamma_{k},\left[\underline{\rho}_{k}\;;\,\overline{\rho}_{k}\right]\right)\, \tag{67}\] \[\tilde{\mathbf{x}}_{k}\in\big{(}\tilde{\mathcal{E}}\big{)}_{\!k} \!\left(\mathbf{0},\mathbf{\Gamma}_{k}\left[\underline{\rho}_{k}\;;\, \overline{\rho}_{k}\right]\right)\,\] (68) \[\tilde{\mathbf{A}}_{k}=\mathbf{A}(\mu_{k},\mathrm{mid}([\mathbf{p }_{k}]))\,\ \ \ \mathrm{and}\] (69) \[\mathbf{p}_{k}\in[\mathbf{p}_{k}]=\left[\underline{\rho}_{k}\;;\, \overline{\rho}_{k}\right]\ \ \ \mathrm{with}\ \ \ \mathrm{mid}([\mathbf{p}_{k}])=\frac{1}{2}\cdot\left(\underline{\rho}_{k}+ \overline{\rho}_{k}\right). \tag{70}\]  
Then, the following steps are executed during the state prediction:  
1. Propagate the inner bound of the thick ellipsoid (68) and extract the inner hull of the image set that is obtained by applying the mapping \[\tilde{\mathbf{x}}_{k+1}=\mathbf{A}(\mathbf{x}_{k},\mathbf{p}_{k})\cdot\tilde {\mathbf{x}}_{k}\.\] (71) Denote the corresponding shape matrix of the inner ellipsoid by \(\tilde{\mathbf{Q}}_{k+1}^{\mathrm{I}}\). This matrix is related to the shape matrix \(\alpha_{\mathrm{I},k+1}^{2}\cdot\mathbf{Q}_{k+1}\) in Equation (25) by \[\tilde{\mathbf{Q}}_{k+1}^{\mathrm{I}}=\alpha_{\mathrm{I},k+1}^{2}\cdot\underline {\rho}_{k}^{2}\cdot\mathbf{\Gamma}_{k}\cdot\mathbf{\Gamma}_{k}^{T}\ \.\] (72)
2. Propagate the outer bound of the thick ellipsoid (68) and extract the outer hull of the image set that is obtained by applying the mapping (71). Denote the corresponding shape matrix of the outer ellipsoid by \(\tilde{\mathbf{Q}}_{k+1}^{\mathrm{O}}\). This matrix is related to the shape matrix \(\alpha_{\mathrm{O},k+1}^{2}\cdot\mathbf{Q}_{k+1}\) in Equation (9) by \[\tilde{\mathbf{Q}}_{k+1}^{\mathrm{O}}=\alpha_{\mathrm{O},k+1}^{2}\cdot\overline {\rho}_{k}^{2}\cdot\mathbf{\Gamma}_{k}\cdot\mathbf{\Gamma}_{k}^{T}\ \.\] (73)
3. Compute interval bounds for the term Figure 5:0 (74) with \(\mathbf{x}_{k}\), \(\tilde{\mathbf{A}}_{k}\), and \(\mathbf{p}_{k}\) defined according to (67), (69), and (70) and deflate the inner ellipsoid bound as well as inflate the outer bound according to the procedure described in [20; 24] with Figure 5:1 (75) and Figure 5:2 (76) Note, for \(\rho_{1}\geq 1\), the inner bound becomes the empty set. Figure 5:3 Separation of the state equations according to (66)–(70) into the mapping of an origin-centered ellipsoid and the verified treatment of non-zero offset terms. 4. Compute the updated ellipsoid midpoint as Figure 5:4 (77)
5.