\$% & *translation due to rotating re f erence*

where

*F*, *τ* external force and torque, respectively *m*, *I* mass and mass moment of inertia, respectively *<sup>ω</sup>*, . *ω* angular velocity and acceleration, respectively *r* , *v* , a position and velocity, and acceleration relative to rotating reference *<sup>τ</sup>* <sup>=</sup> *<sup>I</sup>* . *ω* and *F* = *m* a are double integrator plants *ω* × *Iω* cross-product rotational motion due to rotating reference frame *m* . *ω* × *r* cross-product translation motion due to rotating reference frame −2*mω* × *v* cross-product translation motion due to rotating reference frame

−*mω* × (*ω* × *r* ) cross-product translation motion due to rotating reference frame.

**Figure 2.** SIMULINK simulation program topologies used to generate the results in Section 3: (**a**) Overall system topology used to simultaneously produce state and rate estimates integrated with noisy sensors and additionally optimal control calculations; (**b**) Euler's moment from Equation (1) elaborated in [5–12] describing rotational motion (notice the nonlinear coupled motion).

#### *2.1. Problem Scaling and Balancing*

Consider problems whose solution must simultaneously perform mathematical operations on very large numbers and very small numbers. Such problems are referred to as poorly conditioned. Scaling and balancing problems are one potential mitigation where equations may be transformed to operate with similarly ordered numbers by scaling the variables to nominally reside between zero and unity. Scaling problems by common, well-known values permits single developments to be broadly applied to a wide range of state spaces not initially intended. Consider problems simultaneously involving very large and very small values of time (*t*), mass (*m*)/mass moments of inertia (*I*), and/or length (*r*). Normalizing by a known value permit variable transformation such that newly defined variables are of similar order, e.g., *<sup>t</sup>* <sup>≡</sup> *<sup>t</sup> tf* , *<sup>I</sup>* <sup>≡</sup> *<sup>I</sup> Isystem* <sup>=</sup> *<sup>J</sup>* <sup>≡</sup> *<sup>J</sup> Jsystem* , *<sup>m</sup>* <sup>≡</sup> *<sup>m</sup> msystem* ,*<sup>r</sup>* <sup>≡</sup> *<sup>r</sup> <sup>r</sup>* where *r* indicates generic displacement units like *x*, *y*, *or angle*. Such scaling permits problem solution with a transformed variable mass and inertia of unity value, initial time of zero and final time of unity, and state and rate variables that range from zero to unity making the developments here broadly applicable to any system of particular parameterization.
