A mathematical model of wheelchair racing

Wheelchair racing strokes are very complicated movements, which involve a coupling between the athlete and his or her racing chair. Each body segment, as well as the wheel, follows a distinct trajectory as the motion is performed. Understanding the kinematics and kinetics of various stroke techniques would provide the athletes and their coaches with information, which could help guide the racers toward improved performances. In this thesis, a mathematical model is developed, which is capable of providing such valuable kinematic and kinetic information. This two-dimensional model represents the body segments as a coupled pendulum system of point masses and the wheel as a distributed-mass disk. Furthermore, the model incorporates several fundamental assumptions, including that the stroke cycle can be divided into an arbitrary number of consecutive ballistic phases such that segment positions are continuous at phase boundaries. Each phase is mathematically a second-order, nonlinear ordinary boundary value problem (BVP). Numerical methods are used to solve the BVPs independently, resulting in velocity discontinuities at the phase boundaries. These instantaneous velocity increases or decreases must be caused by impulsive forces. In turn, these impulsive forces are interpreted as muscular input and/or physical impacts. In this research, the model is used to produce numerous stroke techniques, which are consistent with a given racer\u27s structural parameters and prescribed stroke characteristics (racing speed, cycle time, recovery cycle time, the athlete\u27s orientation in the racing chair, and wheel contact and release angles). The kinematics of these different techniques are contrasted. In addition, the muscular mechanical energy costs of these strokes are determined and an interpretation as to the mechanical energy efficiency of each technique is given. The model is used to provide insight into the intricacies of an actual wheelchair racing stroke. In this thesis, the kinematics and energetics of model-produced techniques guide the analysis of these characteristics of an empirical stroke. One conclusion of this analysis is that this model may be able to provide more mechanically efficient alternative strokes from which the athlete can choose. Finally, suggestions are offered toward improving the model

Methods of applied dynamics

The monograph was prepared to give the practicing engineer a clear understanding of dynamics with special consideration given to the dynamic analysis of aerospace systems. It is conceived to be both a desk-top reference and a refresher for aerospace engineers in government and industry. It could also be used as a supplement to standard texts for in-house training courses on the subject. Beginning with the basic concepts of kinematics and dynamics, the discussion proceeds to treat the dynamics of a system of particles. Both classical and modern formulations of the Lagrange equations, including constraints, are discussed and applied to the dynamic modeling of aerospace structures using the modal synthesis technique

A Rule Based Biped Dynamic Walking

Dynamic walking approach has got its significance because of its energy efficiency in walking. Walking models are made using this approach which would consume energy as low as the energy required for human being walking. The basis of this dynamic walking is purely passive walking which takes no energy for walking. For a simple compass model passive walking can be achieved only for particular initial conditions (angular positions and velocities) which are found by trial and error or from previous experience.Various ways are derived to make the model walk on a level ground by supplying external energy through some means i.e torques at hip joint and ankle joints which is called active walking. Two approaches are available for active walking, one is creating virtual slope and then by applying equivalent torques at ankle and hip as the functions of virtual slope;other approach is using torsional springs and dampers at hip as well as ankles such that the torques are given in terms of springs' stiffness coefficient and damping coefficient. The stability is analyzed based on ZMP position. When ZMP of the system falls within the foot support area then system is said to be stable

Dynamic Balance and Gait Metrics for Robotic Bipeds

For legged robots to be useful in the real world, they must be able to balance and walk reliably. Both of these abilities improve when a system is more effective at moving itself around relative to its contacts (i.e., its feet). Achieving this type of movement depends both on the controller used to perform the motion and the physical properties of the system. Although much work has been done on the development of dynamic controllers for balance and gait, only limited research exists on how to quantify a system’s physical balance capabilities or how to modify the system to improve those capabilities. From the control perspective, there are three strategies for maintaining balance in bipeds: flexing, leaning, and stepping. Both stepping and leaning strategies typically depend on balance points (critical points used for maintaining or regaining balance) to determine whether or not a step is needed, and if so, where to step. Although several balance point estimators exist, the majority of these methods make undesirable assumptions (e.g., ignoring the impact dynamics, assuming massless legs, planar motion, etc.). From the physical design perspective, one promising approach for analyzing system performance is a set of dynamic ratios called velocity and momentum gains, which are dependent only on the (scale-invariant) dynamic parameters and instantaneous configuration of a system, enabling entire classes of mechanisms to be analyzed at the same time. This thesis makes four key contributions towards improving biped balancing capabilities. First, a dynamic bipedal controller is proposed which uses a 3D balance point estimator both to respond to disturbances and produce reliable stepping. Second, a novel balance point estimator is proposed that facilitates stepping while combining and expanding the features of existing 2D and 3D estimators to produce a generalized 3D formulation. Third, the momentum gain formulation is extended to general 2D and 3D systems, then both gains are compared to centroidal momentum via a spatial formulation and incorporated into a generalized gain definition. Finally, the gains are used as a metric in an optimization framework to design parameterized balancing mechanisms within a given configuration space. Effectively, this enables an optimization of how well a system could balance without the need to pre-specify or co-generate controllers and/or trajectories. To validate the control contributions, simulated bipeds are subjected to external disturbances while standing still and walking. For the gain contributions, the framework is used to compare gain-optimized mechanisms to those based on the cost of transport metric. Through the combination of gain-based physical design optimization and the use of predictive, real-time balance point estimators within dynamic controllers, bipeds and other legged systems will soon be able to achieve reliable balance and gait in the real world

Multiple Objective Function Optimization and Trade Space Analysis

Optimization can assist in obtaining the best possible solution to a design problem by varying related variables under given constraints. It can be applied in many practical applications, including engineering, during the design process. The design time can be further reduced by the application of automated optimization methods. Since the required resource and desired benefit can be translated to a function of variables, optimization can be viewed as the process of finding the variable values to reach the function maxima or minima. A Multiple Objective Optimization (MOO) problem is when there is more than one desired function that needs to be minimized concurrently. In MOO, Pareto Solutions are defined as the set of solutions that are not worse than any single solution of all objective functions simultaneously. In other words, MOO is a process of applying algorithms to find Pareto solutions to a certain problem. Using Tradespace analysis, we can further identify the optimal Pareto Solution with the highest utility at a fixed cost. The combination of MOO and tradespace analysis can evaluate hundreds of designs simultaneously to select the optimal one. Mechanical system design is the process of devising a procedure to accomplish the given task, for which a design engineer\u27s role is to optimize resource consumption. With recent advancements in multi-functional systems, the complexity of machines has been increasing. This presents a great challenge for design engineers, who must contend with optimizing systems with several functions in tandem. It is thus essential to develop methods that can simplify the design process. Tower clocks, as a classical type of machine, were extensively used for public time display during the period when watches and home clocks iii were rare. These mechanical movements once played essential roles in society and industry. They could be found at churches, courthouses, and universities/schools to visually and auditorily record the passage of time for residents and students. They were also used to regulate railroad schedules and workforce hours for the emerging industrial sector. Although mainly used for decorative purposes today, the components of such movement, including the assembled gears, escapement, and pendulum with weight drive, provide insight into optimization and tradespace analysis problems. In this research, computational methods plus experimental observations were used to investigate the optimal designs of the E. Howard Clock Model 00 - a movement was manufactured by E. Howard Clock Company. First, A computer-aided-design (CAD) model of this movement was created using the SolidWorks® software package to illustrate the working principle of the pendulum clock and facilitate engineering optimization studies. Next, the mathematical model of this clock was developed and simulated to explore the operation behaviors and conversion of potential-to-kinetic energy. The experimental process to validate this model was also described in detail. After that, A Single Objective Optimization (SOO) algorithm (i.e., simulated annealing) was applied to the model to optimize the pendulum subsystem for accuracy, quality factor, and mass. Numerical results show the desired quality factor can be achieved by varying the pendulum length and bob radius/thickness. Compared to the original, the optimized design added 15% to the mass of the pendulum while maintaining the clock\u27s accuracy. Tradeoffs between quality factor, pendulum properties, and period were investigated and discussed with representative experimental and computational results. Lastly, two Multiple Objective Optimization iv (MOO) approaches (i.e., Multi-objective Genetic Algorithm (MOGA-II) and Multiobjective Simulated Annealing (MOSA) were applied to the developed mathematic model. The optimal movement designs in terms of pendulum mass and time accuracy were further explored for a range of clock periods. Numerical results demonstrated a 0.7% increase in the quality factor and a 0.56% reduction in the mass while maintaining the designed period by modifying the above-mentioned pendulum\u27s parameters. More importantly, these changes can provide material cost savings in a mass production scenario. Overall, this study highlighted the optimization design engineers have considered for decades which can now be visualized using computer tools for greater insight. This methodology has the potential to be applied in the designing of other complex systems as well

A DYNAMICS-BASED FIDELITY ASSESSMENT OF PARTIAL GRAVITY GAIT SIMULATION USING UNDERWATER BODY SEGMENT BALLASTING

In-water testing is frequently used to simulate reduced gravity for quasi-static tasks. For dynamic motions, however, the assumption has been that drag effects invalidate any data, and in-water testing has been dismissed in favor of complex and restrictive techniques such as counterweight suspension and parabolic flight. In this study, motion-capture was used to estimate treadmill gait metrics for three environments: underwater and ballasted to 1 g and to 1/6th g, and on dry land at 1 g. Ballast was distributed anthropometrically. Motion-capture results were compared with those for a simulated dynamic walker/runner, and used to assess the effect of the in-water environment on simulation fidelity. For each test case, the model was tuned to the subject's anthropometry, and stride length, pendulum frequency, and hip displacement were computed. In-water environmental effects were found to be sufficiently quantifiable to justify using in-water testing, under certain conditions, to study partial-gravity gait dynamics

Control Systems Approach to Balance Stabilization during Human Standing and Walking.

Humans rely on cooperation from multiple sensorimotor processes to navigate a complex world. Poor function of one or more components can lead to reduced mobility or increased risk of falls, particularly with age. At present, quantification and characterization of poor postural control typically focus on single sensors rather than the ensemble and lack methods to consider the overall function of sensors, body dynamics, and actuators. To address this gap, I propose a controls framework based on simple mechanistic models to characterize and understand normative postural behavior. The models employ a minimal set of components that typify human behavior and make quantitative predictions to be tested against human data. This framework is applied to four topics relevant to daily living: sensory integration for standing balance, limb coordination for one-legged balance, momentum usage in sit-to-stand maneuvers, and the energetic trade-offs of foot-to-ground clearance while walking. First, I demonstrate that integration of information from multiple physiological sensors can be modeled by an optimal state estimator. I show how such a model can predict human responses to conflict between visual, vestibular, and other sensors and use visual perturbation experiments to test this model. Second, I demonstrate that feedback control can model multi-limb coordination strategies during one-legged balance. I empirically identify a control law from human subjects and investigate how reducing stance ankle function necessitates greater gains from other limbs. Third, I show the advantages of momentum usage in sit-to-stand maneuvers. Counter to many human movements, this strategy is not performed with energetic economy, requiring excess mechanical work. However, with optimization models, I demonstrate that momentum serves to balance effort between knee and hip. Fourth, I propose a cost model for preferred ground clearance during swing phase of walking. Walking with greater foot lift is costly, but inadvertent ground contact is also costly. Therefore the tradeoff between these costly measures, modulated by movement variability, can explain expected cost of ground clearance. These controls-based models demonstrate the mechanisms behind normative behavior and enables predictions under novel situations. Thus these models may serve as diagnostic tools to identify poor postural control or aid design of intervention procedures.PhDMechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/116654/1/amyrwu_1.pd

Kinematic and dynamic analysis for biped robots design

En esta tesis un nuevo método para encontrar sistemas dinámicamente equivalentes es propuesto. El objetivo es el de crear una herramienta para el análisis de robots bípedos. La herramienta consiste en modelos simplificados obtenidos del principio de equivalencia dinámica, que dice que si dos sistemas poseen la misma masa, el mismo centro de masa y el mismo momento de inercia, entonces son dinámicamente equivalentes. Este concepto no es nuevo y es comúnmente utilizado en el diseño de máquinas alternativas, o para encontrar el sweet spot de objetos esbeltos tales como bates o espadas. Con la aplicación del principio de equivalencia dinámica se encuentra el centro de percusión. La aportación en esta tesis es la aplicación de este concepto al análisis de robots bípedos, y la extensión del centro de percusión a cadenas cinemáticas. La herramienta fundamental para la obtención de resultados de investigación en esta tesis hace uso del lenguaje de simulación Modelica®. Las simulaciones son altamente detalladas gracias a la librería estándar Multibody incluida en las especificaciones del mismo. Como consecuencia de los trabajos desarrollados se crearon nuevas clases para extender la capacidad de la librería y aplicarla a m´aquinas caminantes. El desarrollo de esta tesis está centrado en el desarrollo de dos modelos. El primero es un péndulo invertido equivalente, con la característica que posee las mismas propiedades dinámicas del robot que modela. Dichas propiedades son la masas total, el centro de masa y el momento de inercia. Este modelo es luego utilizado para generar el caminar de un bípedo simple. El bípedo es simulado con un volante de inercia como cuerpo, y pies de contacto puntual. Posee rodillas y está totalmente actuado. Los eslabones del robot poseen propiedades de sólido rígido y ninguna simplificación ha sido considerada. El segundo modelo tiene el objetivo de imitar la topología del bípedo que representa, por lo tanto tiene un grado mayor de complejidad que el anterior. Este modelo es construido al dividir al robot en tres grupos: Las dos piernas, y otro grupo compuesto por la cabeza, los brazos y el torso (Denominado HAT por sus siglas en inglés). Este modelo es denominado modelo de cuatro masas puntuales. Este modelo es posteriormente validado utilizándolo para desacoplar la dinámica del sistema, la única información utilizada para llevar a cabo esta tarea es proporcionada por dicho modelo. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------In this thesis a method to find dynamically equivalent systems is proposed. The objective is to provide a tool to analyze biped robots by simplifying their dynamics to simpler models. The equivalent models are obtained with the concept of dynamic equivalence that states that if two systems share the same total mass, the same center of mass, and the same moment of inertia then they are considered to be dynamically equivalent. This concept is not new and it is used in the design of alternative machines, or to find the sweet spot of long object like swords or bats. The result of the application of the dynamic equivalence principle is the point known as the center of percussion. The novelty in this thesis is to apply this concept to the analysis of biped robots, and the extension of the center of percussion to kinematic chains. The work in this thesis developed with the help of the simulation language Modelica®. The simulations are very detailed by implementing elaborated rigid body dynamics provided by the multibody standard library included in the language specifications. New classes were created in order to be able to simulate walking machines. Those classes introduce contact objects at ground foot interactions and mechanical stops for knee joints. The development of this thesis is centered around the proposal of two models. The first model is an equivalent inverted pendulum with the characteristic that it has the same dynamic properties, i.e., total mass, center of mass and moment of inertia, of the biped that models. This model is later used to synthesize gait in a simple, but realistic biped. The biped is simulated with a flywheel body, and point feet. It has knees and it is fully actuated. Also all the links have complete rigid body properties and no simplifications were done. The second model has the objective to resemble the topology of the biped it represents, therefore it is slightly more complex than the equivalent inverted pendulum. This model is constructed by grouping the components of the robot in three groups: Two legs and the HAT group (HAT stands for head, arms and trunk). This model is denominated four point masses model. The model is later validated by decoupling the dynamics of the system only with the information provided by the four point masses model