**Appendix A. Influence of the Forward Differentiation Scheme to the Linearization of the EOMs**

In Section 3, the single step backward differentiation formula (BDF-1) was introduced to derive the fully explicit discrete-time linearized form of the EOMs (Equation (19)). In this appendix, the influence of another differentiation scheme, namely the single step forward differentiation scheme, is reported, demonstrating the limitations introduced by such differentiation formula to the proposed linearization approach.

The single step forward differentiation scheme can be written as

$$\begin{cases} \dot{v}\_k = \frac{1}{\hbar} (v\_{k+1} - v\_k) \\ v\_k = \frac{1}{\hbar} (q\_{k+1} - q\_k) \end{cases} \tag{A1}$$

with *h* representing the constant time step size as shown in Figure A1.

**Figure A1.** Graphical interpretation of the forward and backward linearization for a generic state-time evolution *x*.

By substituting Equation (A1) into Equation (8) the discrete-time EOMs *g <sup>d</sup>* are obtained:

$$\begin{cases} \mathbf{g}'\_{d1} = \mathbf{v}\_k - \frac{1}{\hbar} (q\_{k+1} - q\_k) = 0\_v\\ \mathbf{g}'\_{d2} = \mathcal{M}(q\_k) \frac{\mathbf{v}\_{k+1} - \mathbf{v}\_k}{\hbar} + f\_{nl}(q\_k) + f^T(q\_k) \lambda\_k = 0\_\eta\\ \mathbf{g}'\_{d3} = \phi(q\_k) = 0\_\lambda \end{cases} \tag{A2}$$

Similarly to Equation (21) the Jacobian for the forward Euler time-integrator can be computed as:

$$\mathbf{G}'\_{x\_{k+1}} = \frac{\partial \mathbf{g}'\_{d}}{\partial x\_{k+1}} = \begin{bmatrix} \frac{\partial \mathbf{g}'\_{d1}}{\partial v\_{k+1}} & \frac{\partial \mathbf{g}'\_{d1}}{\partial q\_{k+1}} & \frac{\partial \mathbf{g}'\_{d1}}{\partial \lambda\_{k+1}} \\ \frac{\partial \mathbf{g}'\_{d1}}{\partial v\_{k+1}} & \frac{\partial \mathbf{g}'\_{d2}}{\partial q\_{k+1}} & \frac{\partial \mathbf{g}'\_{d2}}{\partial \lambda\_{k+1}} \\ \frac{\partial \mathbf{g}'\_{d1}}{\partial v\_{k+1}} & \frac{\partial \mathbf{g}'\_{d1}}{\partial q\_{k+1}} & \frac{\partial \mathbf{g}'\_{d1}}{\partial \lambda\_{k+1}} \end{bmatrix} = \begin{bmatrix} 0\_{v,v} & -\beta' & 0\_{v,\lambda} \\ \beta' M'\_t & -\gamma' M'\_t + \beta' \mathbf{C}'\_t & 0\_{q,\lambda} \\ 0\_{\lambda,v} & 0\_{\lambda,q} & 0\_{\lambda,\lambda} \end{bmatrix};\tag{A3}$$

Here, *C <sup>t</sup>* and *M <sup>t</sup>* are the tangent damping and mass matrices obtained from the partial derivatives of the continuous *g*<sup>2</sup> equations in Equation (8) evaluated at time step *k*:

$$\mathbf{C}'\_{t} = \left. \frac{\partial \mathcal{g}\_2}{\partial v} \right|\_{k}; \qquad \qquad \qquad \qquad \qquad M'\_{t} = \left. \frac{\partial \mathcal{g}\_2}{\partial \dot{v}} \right|\_{k} \tag{A4}$$

and *β* and *γ* are matrix coefficients function of the defined integration rule, which are given for the forward Euler scheme by:

$$\beta' = \frac{\partial v\_k}{\partial q\_{k+1}} = \frac{\partial \dot{v}\_k}{\partial v\_{k+1}} = \frac{1}{\hbar} I\_q; \qquad \qquad \qquad \gamma' = \frac{\partial \dot{v}\_k}{\partial q\_{k+1}} = \frac{1}{\hbar^2} I\_{q'} \tag{A5}$$

$$\frac{\partial \upsilon\_k}{\partial q\_k} = \frac{\partial \psi\_k}{\partial v\_k} = -\theta'; \tag{A6}$$

As it can be seen from Equation (A3), the matrix *G xk*+<sup>1</sup> is singular and therefore not invertible to compute the linearized system matrix *A* of Equation (24) required for the KF-based estimation framework. Based on the above demonstration, the forward Euler differentiation scheme is not applicable to the proposed linearization approach.

This can be better understood looking at the fundamental assumptions behind the choice of the differentiation scheme. If a forward differentiation scheme is chosen, it practically means that Equation (19) will be derived starting from *xk* looking forward in time to *xk*<sup>+</sup><sup>1</sup> as graphically depicted in Figure A1, but due to the implicit nature of the problem there are not enough information in *xk* to invert the problem with respect to *xk*+1.

#### **References**

