*2.8. Feedback Decoupling of Nonlinear, Coupled Motion Due to Cross Products*

The real-time feedback update of boundary value problem optimum solutions is often used in the field of real-time optimal control, but a key unaddressed complication remains the nonlinear, coupling cross-products of motion due to rotating reference frames. Here, a feedback decoupling scheme is introduced, allowing the full nonlinear problem to be addressed by the identical scaled problem solution presented, and such is done without simplification, linearization, or reduction by assumption. In proposition 2, feedback decoupling is proposed to augment the optimal solution already derived. The resulting modified decision criteria in Equation (42) is utilized in simulations presented in Section 3

of this manuscript, but a single case omitting Proposition 2 is presented to highlight the efficacy of the approach.

**Proposition 2.** *The real-time optimal guidance estimation and/or control solution may be extended from the double-integrator to the nonlinear, coupled kinetics by feedback decoupling as implemented in Equation (41).*

$$
\pi = \operatorname{l\dot{\omega}} + \underbrace{\omega \times \operatorname{I\omega}}\_{\begin{subarray}{c} \text{rotation due} \\ \text{to rotating} \\ \text{reference} \end{subarray}} \tag{41}
$$

**Proof of Proposition 2.** For nonlinear dynamics of translation or rotation as defined in Equation (1), where the double-integrator is augmented by cross-coupled motion due to rotating reference frames, the same augmentation may be added to the decision criteria in Equation (40) using feedback of the current motion states in accordance with Equation (42). The claim is numerically validated with simulations of "cross-product decoupling" that are nearly indistinguishable from open loop optimal solution, and a single case "without cross-product decoupling" is provided for comparison.

$$
\mu^\* \equiv \mathfrak{a}t + \mathfrak{a} + \omega \times I\omega \tag{42}
$$

### *2.9. Analytical Prediction of Impacts of Variations*

Assuming Euler discretization (used in the validating simulations) for output *y*, index *i* and integration solver timestep *h* Equation (43) would seem to indicate a linear noise output relationship. Equation (44) indicates the relationship for quiescent initial conditions indicating the results of a style draw. In a Monte Carlo sense (to be simulated) of a very large number *n*, Equation (45) indicates expectations from theory Equation (46) in simulation for scaled noise entry to the simulation to correctly reflect the noise power of the noisy sensors in the discretized computer simulation. Equation (46) was used to properly enter the sensor noise in the simulation (Figures 2a and 3a).

$$\dot{y}(t) = \frac{y\_{i+1} - y\_i}{h} = n\_i \to y\_{i+1} = y\_i + h n\_i \tag{43}$$

$$y\_1 = \underbrace{y\_0}\_{0} + h n\_0 \tag{44}$$

$$\frac{1}{N}\sum\_{i=1}^{N}y\_{1i}^{2} = \sigma\_{y}^{2} \to \frac{1}{N}\sum\_{i=1}^{N}(hn\_{o,i})^{2} = h^{2}\frac{1}{N}\sum\_{i=1}^{N}n\_{o,i}^{2} \to \sigma\_{y}^{2} = h^{2}\sigma\_{n}^{2} \tag{45}$$

$$\det \sigma\_{sim}^2 = \frac{\sigma\_n^2}{h} \to \sigma\_y^2 = h^2 \sigma\_{sim}^2 = h^2 \frac{\sigma\_n^2}{h} = h \sigma\_n^2 \to \sigma\_{sim}^2 = \frac{\sigma\_n^2}{h} \tag{46}$$

Assuming this implementation of noise power for a given Euler (ode1) discretization in SIMULINK, 1 − *σ* error ellipse may be calculated as Equation (47) for the system in canonical form in accordance with [40] and was implemented in Figure 3a and depicted on "scatter plots" in Section 3's presentation of results of over ten-thousand Monte Carlo simulations.

$$
\sigma\_{\eta\_{\text{M\_{\text{star}}}}} = \sqrt{\frac{\omega\_n^2 + 4\zeta^2}{4\zeta\omega\_n}} \sigma\_{\eta\_{\text{star}}} = \sqrt{\frac{\omega\_n^3 + 4\zeta^2\omega\_n}{4\zeta}} \tag{47}
$$
