**1. Introduction**

Inertial measurement units provide continuous and accurate estimates of motion states in between sensor measurements. Future unmanned naval vessels depicted in Figure 1a require very accurate motion measurement units including active sensor systems and inertial algorithms when active sensor data is unavailable. State observers are duals of state controllers used for establishing decision criteria to declare accurate positions and rates and several instantiations are studied here when fused with noisy sensors, where theoretical analysis of the variance resulting from noise power is presented and validated in over ten-thousand Monte Carlo simulations.

The combination of physical sensors and computational models to provide additional information about system states, inputs, and/or parameters, is known as virtual sensoring. Virtual sensoring is becoming more and more popular in many sectors, such as the automotive, aeronautics, aerospatial, railway, machinery, robotics, and human biomechanics sectors. Challenges include the selection of the fusion algorithm and its parameters, the coupling or independence between the fusion algorithm and the multibody formulation, magnitudes to be estimated, the stability and accuracy of the adopted solution, optimization of the computational cost, real-time issues, and implementation on embedded hardware [1].

The proposed methods stem from Pontryagin's treatment of Hamiltonian systems, rather than utilization of classical or modern optimal estimation and control concepts applied to future naval vessels as depicted in (Figure 1) [2–4].

**Figure 1.** Representative motion measurement units for future ships depicted in (**a**) with measurement bases depicted in (**b**) are proposed to be augmented by virtual sensoring by minimization of Hamiltonian systems by the principles of Pontryagin depicted in (**c**). Future unmanned U.S. Navy vessels [3] Medium Unmanned Surface Vessel (MUSV) concept renderings in (**a**) from shipbuilder Austal USA. Photo Credit: Austal USA. Boat motion monitoring [4] uses measurement bases depicted in (**b**) whose graphic is from cited reference modified by author. Photo (**c**) of Lev Pontryagin from the archive of the Steklov Mathematical Institute [2] used with permission (30 June 2021).

Typical motion reference units conveniently have accuracies on the order of 0.05 (in meters and degree for translation and rotation, respectively, as depicted in Figure 1b for representative naval vessels as depicted in Figure 1a). These figures of merit are aspirational for the virtual sensor that must provide accurate estimates whether active measurements are available to augment the algorithm. Lacking active measurements, the algorithm is merely an inertial navigation unit, while with active measurements, the algorithm becomes an augmented virtual sensor. This manuscript investigates virtual sensoring by evaluating several options for algorithms, resulting estimated magnitudes, accuracy of each solution, optimization of resulting costs of motion, and sensitivity to variations like noise and parameter uncertainty of the translational and rotational motion models investigated (both simplified and high-fidelity). Algorithms are compared using various decision criteria to compare approaches for consideration of usage as motion reference units potentially aided by global navigation systems.

Noting the small size of motion measurement units, simple algorithms are preferred to minimize computational burdens that can increase unit size. Motion estimation and control algorithms to be augmented by sensor measurements are based on well-known mathematical models of translation and rotation from physics, both presented in equations. In 1834, the Royal Society of London published two celebrated papers by William R. Hamilton on Dynamics in the Philosophical Transactions. Ref. [5] The notions were slowly adopted, and not presented relative to other thoughts of the age for nearly seventy years [6], but quickly afterwards, the now-accepted axioms of translational and rotational motion were self-evidently accepted by the turn of the twentieth century [7–10] as ubiquitous concepts. Half a century later [11,12], standard university textbooks elaborate on the notions to the broad scientific community. Unfortunately, the notions arose in an environment already replete with notions of motion estimation and control based on classical proportional, rate, and integral feedback, so the fuller utilization of the first principals languished until exploitation by Russian mathematician Pontryagin [13]. Pontryagin proposed to utilize the first principles as the basis for treating motion estimation and control as the classical mathematical feedback notions were solidifying in the scientific community. Decades later, the first-principal utilization proposed by Pontryagin are currently rising in prominence as an improvement to classical methods [14]. After establishing performance benchmarks [15] for motion estimation and control of unmanned underwater vehicles, the burgeoning field of deterministic artificial intelligence [16,17] articulates the assertion of the first-principles as "self-awareness statements" with adaption [18,19] or optimal learning [20] used to achieved motion estimation and control commands. The key difference between the usage of first principals presented here follows. Classical methods impose the form of the estimation and control (typically negative feedback with gains) and they have very recently been applied to railway vehicles [21], biomechanical applications [22], and remotely operated undersea vehicles [23], electrical vehicles [24], and even residential heating energy consumption [25]

and multiple access channel usage by wireless sensor networks [26]. Deterministic artificial intelligence uses first principals and optimization for all quantities but asserts a desired trajectory. Meanwhile the proposed methods in this manuscript leave the trajectory "free" and calculate an optimal state and rate trajectory for fusion with sensor data and calculates optimal decision criteria for estimation and controls in the same formulation.

This manuscript seeks to use the same notion, assertion of the first principals (via Pontryagin's formulation of Hamiltonian systems) in the context of inertial motion estimation fused with sensor measurements (that are presumed to be noisy). Noise in sensors is a serious issue elaborated by Oliveiera et al. [27] for background noise of acoustic sensors and by Zhang et al. [28] for accuracy of pulse ranging measurement in underwater multi-path environments. Barker et al. [29] evaluated impacts on doppler radar measurements beneath moving ice. Thomas et al. [30] proposes a unified guidance and control framework for Autonomous Underwater Vehicles (AUVs) based on the task priority control approach, incorporating various behaviors such as path following, terrain following, obstacle avoidance, as well as homing and docking to stationary and moving stations. Zhao et al. [31] very recently pursued the presently ubiquitous pursuit of optimality via stochastic artificial intelligence using particle swarm optimization genetic algorithm, while Anderlini et al. [32] used real-time reinforcement learning. Sensing the ocean environment parallels the current emphasis in motion sensing, e.g., Davidson et al.'s [33] parametric resonance technique for wave sensing and Sirigu et al.'s [34] wave optimization via the stochastic genetic algorithm. Motion control similarly mimics the efforts of motion sensing and ocean environment sensing, e.g., Veremey's [35] marine vessel tracking control, Volkova et al.'s [36] trajectory prediction using neural networks, and the new guidance algorithm for surface ship path following proposed by Zhang et al. [37]. Virtual sensory will be utilized in this manuscript where noisy state and rate sensors are combined to provide smooth, non-noisy, accurate estimates of state, rate, and acceleration, while no acceleration sensors were utilized. A quadratic cost was formulated for acceleration, since accelerations are directly tied to forces and torques and therefore fuels.

*"* ... *condition of the physical world can either be "directly" observed (by a physical sensor) or indirectly derived by fusing data from one or more physical sensors, i.e., applying virtual sensors"*. [38]

Thus, the broad context of the field is deeply immersed in a provenance of classical feedback driving a current emphasis on optimization by stochastic methods. Meanwhile this study will iterate options utilizing analytic optimization including evaluation of the impacts of variations and random noise in establishing the efficacy of each proposed approach. Analytical predictions are made of the impacts of applied noise power, and Monte Carlo analysis agrees with the analytical predictions. Developments presented in this manuscript follow the comparative prescription presented in [39], comparing many (eleven) optional approaches permitting the reader to discern their own preferred approach to fusion of sensor data with inertial motion estimation:


ification and manually tuned proportional plus integral plus derivative feedback topologies, where these classical methods are utilized as benchmarks for performance comparisons when fused with sensor data that is assumed to be noisy.


Appendix A, Table A1 contains a consolidated list of variables and acronyms in the manuscript.

#### **2. Materials and Methods**

Inertial navigation algorithms use physics-based mathematics to make predictions of motion states (position, rate, acceleration, and sometimes jerk). The approach taken here is to utilize the mathematical relationships from physics in a feedforward sense to produce optimal, nonlinear estimates of states that when compared to noisy sensor measurements yield corrected real-time optimal, smooth, and accurate estimates of state, rate, and acceleration. Sensors are modeled as ideal with added Gaussian noise and the smooth estimates will be seen to exhibit none of the noise. The optimization of the estimates will be derived using Pontryagin's optimization.

Motion control algorithms to be augmented by sensor measurements are based on well-known mathematical models of translation and rotation from physics, both presented in Equation (1), where both high-fidelity motion models are often simplified to identical double-integrator models where nonlinear coupling cross-products of motion are simplified, linearized, or omitted by assumption. The topologies are provided in Figure 2. Centrifugal acceleration is represented in Equation (1) by −*mω* × (*ω* × *r* ). Coriolis acceleration is represented in Equation (1) by −2*mω* × *v* . Euler acceleration is represented in Equation (1) by *<sup>m</sup>* . *ω* × *r* . In this section, double-integrator models are optimized by Pontryagin's treatment of Hamiltonian systems, where the complete (not simplified, linearized, or omitted) nonlinear cross-products of motion are accounted for using feedback decoupling. Efficacy of feedback decoupling of the full equations of motion is validated by disengaging this feature in a single simulation run to reveal the deleterious effects of the coupled motion when not counteracted by the decoupling approach.

$$\pi = I\dot{\omega} + \underbrace{\omega \times I\omega}\_{\text{mass with }\omega} \iff F = m\dot{a}' + \underbrace{m\dot{\omega} \times r' - 2m\omega \times \sigma' - m\omega}\_{\text{mass with }\omega} \times \underbrace{(\omega \times r')}\_{\text{mass with }\omega} \tag{1}$$

*rotation due to rotating re f erence*
