## 3 Design of the Luenberger Observer  
In this section, a Luenberger observer is designed based on SDRE. The key functions of the observer designed in this section are to estimate the passive side-slipping occurring and to tune the observer gains to adapt to it. First, the observable states are discussed quantitatively. Second, the Luenberger observer is designed, based on SDRE. Finally, an algorithm for state estimation for adaptive side-slipping determination is constructed, and the tuning algorithm of observer gain, corresponding to the result of the state estimation algorithm, is constructed. First, the observable states are discussed. As installation of a rotary encoder in each joint and each elastic part was realized in our previous study [47], the related angles of each joint \(\phi_{j}\ (j=2,\ \cdots,\ 5)\) and twist angles of the springs \(s_{j}(j=2,\ \cdots,\ 5)\) are observable. Additionally, as the installation of a camera on the head link has also been realized in related studies [50; 51], the absolute angle \(\theta_{1}\) and position \((x_{1},y_{1})\) of the robot's head are observable. On the other hand, velocity sensors are not installed,  
By defining the output error \(\mathbf{e}\) and the state of the Luenberger observer \(\mathbf{\hat{x}}\) as follows,  
\[\mathbf{e} =\mathbf{y}-\mathbf{\hat{y}},\] \[\mathbf{\hat{x}} = \begin{bmatrix}y_{\mathbf{\hat{y}}}^{T}&\mathbf{s}^{T}&\mathbf{\hat{q}}^{T} \end{bmatrix} ^{T},\]  
the Luenberger observer is obtained as  
\[\mathbf{\hat{x}} =\mathbf{A}_{o}(\mathbf{\hat{x}})\mathbf{\hat{x}}+\mathbf{B}_{o}(\mathbf{\hat{x}}) \mathbf{u}+\mathbf{L}(\mathbf{\hat{x}})^{T}\mathbf{e},\] \[\mathbf{\hat{y}} =\mathbf{C}_{o}\mathbf{\hat{x}},\] \[\mathbf{A}_{o}(\mathbf{\hat{x}}) = \begin{bmatrix}\mathbf{Z}^{\gamma\times 7}&\mathbf{Z}^{\gamma\times 4}& \mathbf{H}_{y}\\ \mathbf{Z}^{4\times 7}&\mathbf{Z}^{4\times 4}&\mathbf{H}_{s}\\ \mathbf{Z}^{11\times 7}&\mathbf{\hat{K}}&\mathbf{\hat{M}}\end{bmatrix} ,\] \[\mathbf{B}_{o}(\mathbf{\hat{x}}) = \begin{bmatrix}\mathbf{Z}^{\gamma\times 4}\\ \mathbf{Z}^{4\times 4}\\ \mathbf{\hat{N}}\end{bmatrix} ,\] \[\mathbf{C}_{o} = \begin{bmatrix}\mathbf{I}^{11\times 11}&\mathbf{Z}^{11\times 11} \end{bmatrix} ,\]  
where \(\mathbf{Z}^{m\times n}\) represents the zero matrix with \(m\times n\); in addition,  
\[\mathbf{\hat{M}} =(\mathbf{D}^{T}\mathbf{M})^{-1}\mathbf{D}^{T}(F\mathbf{H}-\mathbf{M}\mathbf{D}+\mathbf{\alpha }),\] \[\mathbf{\hat{N}} =(\mathbf{D}\mathbf{M}\mathbf{D})^{-1}\mathbf{D}^{T}(\mathbf{E}+\mathbf{\beta}),\] \[\mathbf{\hat{K}} =(\mathbf{D}^{T}\mathbf{M}\mathbf{D})^{-1}\mathbf{D}^{T}(\mathbf{G}+\mathbf{\gamma}).\]  
Note that an accent mark \(\wedge\) represents the estimated state of the observer. Moreover, \(\mathbf{L}(\mathbf{\hat{x}})\) denotes the observer gain and  
\[\mathbf{L}(\mathbf{\hat{x}})=-\mathbf{R}_{o}(\mathbf{\hat{x}})\mathbf{C}_{o}\mathbf{P}_{o}(\mathbf{\hat{x}}),\]  
where \(\mathbf{P}_{o}(\mathbf{\hat{x}})\) is a symmetric positive definite matrix of following state-dependent Riccati equation,  
\[\mathbf{A}_{o}(\mathbf{\hat{x}})\mathbf{P}(\mathbf{\hat{x}})+\mathbf{P}(\mathbf{\hat{x}})\mathbf{A}_{o}( \mathbf{\hat{x}})^{T}+\mathbf{Q}_{o}(\mathbf{\hat{x}})-\mathbf{P}(\mathbf{\hat{x}})\mathbf{C}_{o}^{T} \mathbf{R}_{o}(\mathbf{\hat{x}})^{-1}\mathbf{C}_{o}\mathbf{P}(\mathbf{\hat{x}})=0,\]  
where \(\mathbf{Q}_{o}(\mathbf{\hat{x}})\) and \(\mathbf{R}_{o}(\mathbf{\hat{x}})\) represent the weighting matrices for the observer, and are chosen as positive semidefinite and positive definite, respectively. Finally, the algorithm for adaptive state estimation of side-slipping determination is constructed. The estimation precision of the designed observer depends greatly on the accuracy of the dynamic model; that is, to appropriately extend the dynamic model according to passive side-slipping is a key issue. To extend the model appropriately, slipping links are discriminated by utilizing the estimated state of the observer. Corresponding to the discriminant result, the weighting matrices of the observer are tuned.