In response to challenges faced in engineering education, this paper presents the Moving Frame Method (MFM) as an alternative to the existing pedagogies. This paper, and a companion paper, present the MFM for the analysis of rigid single-bodies, in the context of an international collaborative senior design project between two engineering schools: The Cooper Union in New York, NY and The Western Norway University of Applied Sciences in Bergen, Norway. Students at Cooper Union analyzed the smart vehicle as a single-body problem; a second team in Norway used the same method to model the problem as a multi-body. Both teams modeled the gyroscopic lift effect using the foundational theory of the MFM. This first paper presents the underlying theory of how the MFM uses the Special Orthogonal Group, SO(3), to model the problem of gyroscopic lift of a smart vehicle. It presents the underlying theory: motion of a reference frame in terms of the reference frame; and the various properties of SO(3) and its associated algebra as applied to a new, undergraduate approach to dynamics. The paper presents two design alternatives to lift a vehicle, the first one uses two gyroscopes and the second a single reaction wheel. The modeling and analysis process led to the design that employs a single reaction wheel. Next, it examines the model-generated estimates for torque generated and subsequent motion of the car. After analysis using the initial model, the Cooper Union Team pivoted a design approach before entering the rapid prototyping phase of the project. Then, it discusses the manufacturing process for the final version of the car and the construction of a test enclosure for ensuring safety by containing the heavy reaction wheel spinning at high speeds. Finally, it describes the testing of the car, which involves measurement of angular impulse due to a braking event. The results from initial experiments on the prototype validate the model and confirm that the car can lift itself using the selected method.

Novel biomarkers of upper airway biomechanics may improve diagnosis of obstructive sleep apnea syndrome (OSAS). Upper airway effective compliance (EC), the slope of cross-sectional area versus pressure estimated using computational fluid dynamics (CFD), correlates with apnea-hypopnea index (AHI) and critical closing pressure (P-crit).The study objectives are to develop a fast, simplified method for estimating EC using dynamic MRI and physiological measurements and to explore the hypothesis that OSAS severity correlates with mechanical compliance during wakefulness and sleep. Five obese children with OSAS and five control subjects with obesity aged 12-17 yr underwent anterior rhinomanometry, polysomnography, and dynamic MRI with synchronized airflow measurement during wakefulness and sleep. Airway cross section in retropalatal and retroglossal section images was segmented using a novel semiautomated method that uses optimized singular value decomposition (SVD) image filtering and k-means clustering combined with morphological operations. Pressure was estimated using rhinomanometry Rohrer's coefficients and flow rate, and EC was calculated from the area-pressure slope during five normal breaths. Correlations between apnea-hypopnea index (AHI), EC, and cross-sectional area (CSA) change were calculated using Spearman's rank correlation. The semiautomated method efficiently segmented the airway with average Dice Coefficient above 89% compared with expert manual segmentation. AHI correlated positively with EC at the retroglossal site during sleep (r(s) = 0.74, P = 0.014) and with change of EC from wake to sleep at the retroglossal site (r(s) = 0.77, P = 0.01). CSA change alone did not correlate significantly with AHI. EC, a mechanical biomarker which includes both CSA change and pressure variation, is a potential diagnostic biomarker for studying and managing OSAS. NEW & NOTEWORTHY This study investigated the dynamics of the upper airway at retropalatal and retroglossal sites during wakefulness and sleep by evaluating the effective compliance (EC) of each site and its correlation with apnea-hypopnea index (AHI) using novel semiautomated image processing. AHI correlated significantly with retroglossal EC during sleep and change of retroglossal EC from wake to sleep. The results suggest EC as a promising noninvasive diagnostic marker for estimating the mechanical properties of various upper airway regions in patients with OSAS.

The moving frame method for multi-body dynamics, established by Murakami in [10] and [11], embodies a consistent notation and mathematical framework that simplifies the derivation of equations of motion of complex systems. The derivation of the equations of motion follows Hamilton’s principle and requires the calculation of virtual angular velocities and the corresponding virtual rotational displacements. The goal of this paper is to present a projection-based approach, which only requires knowledge of Euler’s first and second law, that results in the same equation of motion. The constraints need not satisfy d’Alembert’s principle and the projection is based on a generalization of Gauss’ principle of least constraint [14]. One advantage of the proposed method is that it avoids variational principles and therefore is more accessible to undergraduate students. In addition, the final form of the equation of motion is more easily understood. We motivate our approach using the example of the simple pendulum, derive the main result, and apply the methodology for derivation of the equations of motion for a modified Chaplygin sleigh and a rotary pendulum.

This book is an accessible and comprehensive introduction to the field of data-driven science and engineering. It is unique in the sense that it brings together interdisciplinary concepts from machine learning, dynamical systems, and feedback control and applies them to physical systems arising in science and engineering. The book provides a broad overview of these concepts and develops tools for data-driven modeling, prediction, and control. Overall, it provides a perfect starting point for an aspiring graduate student or researcher in this field. It can also be used as a text for an advanced undergraduate or graduate course on data-driven model reduction and control. A wealth of accompanying online material (such as Matlab/Python code and a variety of YouTube video lectures) makes the book very suitable for self-study

Active noise control (ANC) systems detect incoming sound and generate an anti-noise signal to attenuate it through destructive interference. ANC systems are generally limited to low-frequency sounds because the sensors and actuators are close together, leaving insufficient time to react to high-frequency or impulsive sounds. To compensate, common applications, such as noise-cancelling headphones, rely on complete ear coverings to block out higher-frequency sound indiscriminately, potentially hindering communication and situational awareness. The present work proposes to use an external array of microphones surrounding the user to increase the distance between the sensors and the user to afford more time to generate anti-noise signals and provide better path estimation. Using time delay of arrival (TDOA), the array tracks the locations of the user and the noise sources in real-time. The ANC system uses a feed-forward Filtered-x Least Mean Squared (FxLMS) algorithm that adjusts the weights of the controller based on the TDOA path estimation instead of adapting the filters with an error microphone and feedback loop. Simulations of the proposed system and feedback FxLMS path estimation were conducted in MATLAB. Compared to the feedback FxLMS algorithm, the TDOA system yielded 11 ± 1 dB less root-mean-square error in the generated anti-noise signals.

As demonstrated by the 2014 MV Sewol incident, the prevention of top heavy ship capsize is necessary to protect life and property aboard a ship. The goal of this paper is to prevent the capsize of ships, which lack a restoring torque about the roll axis, by using a feedback-controlled pendulum actuator. A seven degrees-of-freedom (7DOF) model is developed for a ship equipped with a pendulum actuator. The model is used to conduct parameter analyses on the pendulum length, pendulum mast height, pendulum mass, ship center of mass (COM) height, and the pendulum controller's proportional feedback gain. The results of these analyses are depicted via time responses and phase plots. Key points for designing a pendulum actuator summarize simulation results, stating that the pendulum mass should be 3–7% of the total ship mass, and the pendulum moment of inertia should be 0.5–1.0 times the roll moment of inertia of the ship.

Norway conducts operations on a variety of structures in the North Sea; e. g. oilrigs, monopole windmills, subsea trees. These structures often require subsea installation, observation, and maintenance. A remotely operated vehicle (ROV) can assist in these operations. Automation of intended motion is the desired goal. This paper researches the motion of an ROV induced by the motion of the robotic manipulators, motor torques, and added mass of fluid. This project builds upon a previous project that had one robotic arm; this time, there are two, but the method is unchanged. Furthermore, this work explores both the patterns in addressing such challenges, and an improved integration scheme. This research uses the Moving Frame Method (MFM) to carry out this project. In fact, this paper demonstrates the ease with which the MFM is extensible. Notable is that this work represents an international collaboration between an engineering school in Norway and one in the US. This work invites further research into improved numerical methods, solid/fluid interaction and the design of Autonomous Underwater Vehicles (AUV). AUVs beckon an era of Artificial Intelligence when machines think, communicate and learn. Rapidly deployable software implementations will be essential to this task.

Simultaneous localization and mapping (SLAM) has been an emerging research topic in the fields of robotics, autonomous driving, and unmanned aerial vehicles over the past thirty years. State of the art SLAM research is often inaccessible for undergraduate student researchers due to expensive hardware and difficult software setup. We present a cost-friendly vehicle research platform and a robust implementation of SLAM. Our SLAM algorithm fuses visual stereo image and 2D light detection and ranging (Lidar) data and uses loop closure for accurate odometry estimation. Our algorithm is benchmarked against other popular SLAM algorithms using the publicly available KITTI dataset and shown to be very accurate. For educational purposes, we publicly share the models and code presented in this work*.

Most airdropped cargo use a combination of one or more parachutes and an impact attenuation system to land safely. The latter adds cost, weight and complexity. However, by using their legs for impact attenuation, airdropped quadruped robots may avoid the need for such a system. In this paper, various leg configurations for attenuating impact of airborne landings were studied and tested. Using simple lumped element models for simulation and analysis, a quadruped robot with a three-segment leg was designed and built. This model was validated with experiments with a small scale 20 cm-tall test robot. During the experiments, the test robot experienced 7.7 × 10 m/s 2 or 7.9 g-acceleration when dropped from height of 37.85 cm. This result is much better than the result of 1.4 × 10 2 m/s 2 or 14.7g-acceleration when dropped at 10% of the original height with the same robot equipped with rigid legs. Such compliant leg design could be potentially used for impact attenuation of airdrop landings of robots five times larger.

As a precursor to capsize, marginal stability, resulting from incorrect loading conditions and crew negligence, poses a serious danger to ships. Therefore, as a benchmark problem for preventing capsize, the use of an actively controlled pendulum for the stabilization of a marginally stable ship was analyzed. Lyapunov stability criteria and closed loop eigenvalues were used to evaluate the extent to which a proposed pendulum controller could cope with different ship stability conditions. Equations of motion were solved to observe the controller’s performance under different damping conditions. The behavior of the controller yielded the following results: a marginally stable ship can be stabilized, as long as there is no right hand plane zero; energy dissipation is key to the stabilization of a marginally stable ship; the controller must have knowledge of the ship’s stability to prevent controller-induced excitation; and a stabilized tilted ship is more robust to external disturbances than a stabilized upright ship. Nomenclature m Ship Mass including Pendulum Mass Ixx Ship Rotational Inertia mcw Pendulum mass g Gravitational Acceleration φ Roll Angle φ̇ Roll Anglular Velocity of Ship θcw Pendulum Angle θ̇cw Pendulum Angular Velocity θcw,re f Reference Pendulum Angle broll Ship Roll Damping bpend Pendulum Damping Tpend Torque on the Pendulum Tpend,eq Pendulum Torque that Satisfies Equilibirum zg z-coordinate of the Center of Mass (COM) ρw Density of Displaced Fluid A Ship Displaced Volume BoG Distance btw. Centers of Mass and Buoyancy at φ = 0◦ BoMx Distance btw. Buoyancy Center and Metacenter GMx Metacentric Height GZ Righting Arm Lcw Length of Pendulum Lo f f Pendulum Offset k1 Nonlinear Feedforward Reference Gain k2 Linear Feedback Gain kp Proportional Feedback Gain zship Potential Right Half Plane Zero of Ship pship Right Half Plane Pole of Ship Emech Total Mechanical Energy KE Total Kinetic Energy PE Total Potential Energy

As a precursor to capsize, marginal stability, resulting from incorrect loading conditions and crew negligence, poses a serious danger to ships. Therefore, as a benchmark problem for preventing capsize, the use of an actively controlled pendulum for the stabilization of a marginally stable ship was analyzed. Lyapunov stability criteria and closed loop eigenvalues were used to evaluate the extent to which a proposed pendulum controller could cope with different ship stability conditions. Equations of motion were solved to observe the controller’s performance under different damping conditions. The behavior of the controller yielded the following results: a marginally stable ship can be stabilized, as long as there is no right hand plane zero; energy dissipation is key to the stabilization of a marginally stable ship; the controller must have knowledge of the ship’s stability to prevent controller-induced excitation; and a stabilized tilted ship is more robust to external disturbances than a stabilized upright ship.

Accurate simulation of flight beyond normal conditions requires models of the aircraft aerodynamics at high angles of attack, for which the flow over the wing and control surfaces may be rapidly changing and massively separated. This study focuses on large-amplitude forced oscillations of a benchmark commercial aircraft configuration. Existing models are modified to describe the unsteady aerodynamic forces on the aircraft up to stall range. Steady-state and periodically forced unsteady Reynolds-averaged Navier–Stokes simulation data are employed to calibrate the model parameters. The results of the reduced-order models are in good overall agreement with those observed in the unsteady Reynolds-averaged Navier–Stokes simulations.

Accurate simulation of flight beyond normal conditions requires models of the aircraft aerodynamics at high angles of attack, for which the flow over the wing and control surfaces may be rapidly changing and massively separated. This study focuses on large-amplitude forced oscillations of a benchmark commercial aircraft configuration. Existing models are modified to describe the unsteady aerodynamic forces on the aircraft up to stall range. Steady-state and periodically forced unsteady Reynolds-averaged Navier-Stokes simulation data are employed to calibrate the model parameters. The results of the reduced-order models are in good overall agreement with those observed in the unsteady Reynolds-averaged Navier-Stokes simulations.

Originally introduced in the fluid mechanics community, dynamic mode decomposition (DMD) has emerged as a powerful tool for analyzing the dynamics of nonlinear systems. However, existing DMD theory deals primarily with sequential time series for which the measurement dimension is much larger than the number of measurements taken. We present a theoretical framework in which we define DMD as the eigendecomposition of an approximating linear operator. This generalizes DMD to a larger class of datasets, including nonsequential time series. We demonstrate the utility of this approach by presenting novel sampling strategies that increase computational efficiency and mitigate the effects of noise, respectively. We also introduce the concept of linear consistency, which helps explain the potential pitfalls of applying DMD to rank-deficient datasets, illustrating with examples. Such computations are not considered in the existing literature but can be understood using our more general framework. In addition, we show that our theory strengthens the connections between DMD and Koopman operator theory. It also establishes connections between DMD and other techniques, including the eigensystem realization algorithm (ERA), a system identification method, and linear inverse modeling (LIM), a method from climate science. We show that under certain conditions, DMD is equivalent to LIM.

We present an efficient and accurate method for long-time uncertainty propagation in dynamical systems. Uncertain initial conditions and parameters are both addressed. The method approximates the intermediate short-time flow maps by spectral polynomial bases, as in the generalized polynomial chaos (gPC) method, and uses flow map composition to construct the long-time flow map. In contrast to the gPC method, this approach has spectral error convergence for both short and long integration times. The short-time flow map is characterized by small stretching and folding of the associated trajectories and hence can be well represented by a relatively low-degree basis. The composition of these low-degree polynomial bases then accurately describes the uncertainty behavior for long integration times. The key to the method is that the degree of the resulting polynomial approximation increases exponentially in the number of time intervals, while the number of polynomial coefficients either remains constant (for an autonomous system) or increases linearly in the number of time intervals (for a non-autonomous system). The findings are illustrated on several numerical examples including a nonlinear ordinary differential equation (ODE) with an uncertain initial condition, a linear ODE with an uncertain model parameter, and a two-dimensional, non-autonomous double gyre flow.

To date, physically meaningful representations of the nonstationarity in complex 3D flows with converged turbulent statistics are scarce and shed little light on the nonlinear processes in turbulent motion. This study attempts to address part of this deficit by concentrating on the kinematics of larger scales of motion. Two methods are utilized to describe the kinematics of large-scale unsteady motion in the flow around a wall-mounted finite circular cylinder at Reynolds number ReD = 200 000. The first, Proper Orthogonal Decomposition (POD), is a global method resulting in spatial modes defined over the whole domain and their corresponding temporal coefficients. The second, Coherent Structure Tracking (CST), belongs to a class of local methods that extracts connected domains in the flow data. Modes specific for distinct harmonics are extracted by temporal harmonic filtering. Based on time coefficients of the dominant mode pairs provided by POD or harmonic filtering, phase-averaging has been performed. A scalar-field version of CST is proposed, yielding an intuitively more accessible description of the flow. The extent to which POD and CST are complementary is discussed, as well as the extent to which they partially overlap. The combination of POD, filtering, phase-averaging and CST allowed for identification and quantification of important flow patterns in a complex turbulent flow field.