Modelling mechanical systems using symbolic equations can provide many advantages over the more widely-used numerical methods of modelling these systems. The use of symbolic equations produces more efficient models, which can be used for many purposes such as real-time simulation and control. However, the number, complexity, and computational efficiency of these equations is highly dependent on which coordinate set was used to model the system. One method of modelling a mechanism's topology and formulating its symbolic equations is to model the system using a graph-theoretical approach. This approach models mechanisms using a linear graph, from which spanning trees can be used to define a mechanism's coordinate set. This report develops two tree selection algorithms capable of estimating the tree set, and hence coordinate set, that produces models having the fastest forward dynamic simulation times. The first tree selection algorithm is a heuristic-based algorithm that tries to find the coordinate set containing the minimal possible number of modelling variables. Most of this algorithm's heuristics are based on tree selection criteria found in the literature and on observations of a series of benchmark problems. It uses the topology information provided by a system's graph to find the coordinates set for the given system that produce very low simulation times of the system. The second tree selection algorithm developed in this report also uses graph theory. It bases most of its heuristics on observations of one of the methods developed to obtain a mechanical system's symbolic equations using graph theory. This second algorithm also makes use of, and improves upon, a few of the heuristics developed in the first tree selection algorithm. A series of examples for both algorithms will demonstrate the computational efficiency obtained by using the modelling variables found by the automated tree selection algorithms that are proposed in this report.

Modelling mechanical systems using symbolic equations can provide many advantages over the more widely-used numerical methods of modelling these systems. The use of symbolic equations produces more efficient models, which can be used for many purposes such as real-time simulation and control. However, the number, complexity, and computational efficiency of these equations is highly dependent on which coordinate set was used to model the system. One method of modelling a mechanism's topology and formulating its symbolic equations is to model the system using a graph-theoretical approach. This approach models mechanisms using a linear graph, from which spanning trees can be used to define a mechanism's coordinate set. This report develops two tree selection algorithms capable of estimating the tree set, and hence coordinate set, that produces models having the fastest forward dynamic simulation times. The first tree selection algorithm is a heuristic-based algorithm that tries to find the coordinate set containing the minimal possible number of modelling variables. Most of this algorithm's heuristics are based on tree selection criteria found in the literature and on observations of a series of benchmark problems. It uses the topology information provided by a system's graph to find the coordinates set for the given system that produce very low simulation times of the system. The second tree selection algorithm developed in this report also uses graph theory. It bases most of its heuristics on observations of one of the methods developed to obtain a mechanical system's symbolic equations using graph theory. This second algorithm also makes use of, and improves upon, a few of the heuristics developed in the first tree selection algorithm. A series of examples for both algorithms will demonstrate the computational efficiency obtained by using the modelling variables found by the automated tree selection algorithms that are proposed in this report.

<p>Study of physical phenomena by means of mathematical models is common in various branches of engineering and science. In biomechanics, modelling often involves studying human motion by treating the body as a mechanical system made of interconnected rigid links. Robotics deals with similar cases as robots are often designed to imitate human behavior. Modelling human movements is a complicated task and, therefore, requires several simplifications and assumptions. Available computational resources often dictate the nature and the complexity of the models. In spite of all these factors, several meaningful results are still obtained from the simulations.</p><p>One common problem form encountered in real life is the movement between known initial and final states in a pre-specified time. This presents a problem of dynamic redundancy as several different trajectories are possible to achieve the target state. Movements are mathematically described by differential equations. So modelling a movement involves solving these differential equations, along with optimization to find a cost effective trajectory and forces or moments required for this purpose.</p><p>In this study, an algorithm developed in Matlab is used to study dynamics of several common human movements. The main underlying idea is based upon temporal finite element discretization, together with optimization. The algorithm can deal with mechanical formulations of varying degrees of complexity and allows precise definitions of initial and target states and constraints. Optimization is carried out using different cost functions related to both kinematic and kinetic variables.</p><p>Simulations show that generally different optimization criteria give different results. To arrive on a definite conclusion on which criterion is superior over others it is necessary to include more detailed features in the models and incorporate more advanced anatomical and physiological knowledge. Nevertheless, the algorithm and the simplified models present a platform that can be built upon to study more complex and reliable models.</p>

The high demands of manual wheelchair propulsion put users at risk of additional pain and injury that can lead to further reductions in independence and quality of life. Seat position is an adjustable parameter that has been shown to influence propulsion biomechanics. As a result, a number of studies have attempted to optimize this position. However, due to complexities in quantifying upper extremity demand, seat position guidelines are often based on studies aimed at reducing indirect quantities (e.g., cadence, handrim forces, joint ranges of motion and muscle excitation levels) rather than more direct measures of demand (e.g., muscle stress and metabolic cost). Forward dynamics simulations provide an alternative approach to systematically investigate the influence of seat position on more direct measures of upper extremity demand. The objective of this study was to generate and analyze a set of forward dynamics simulations of wheelchair propulsion across the range of attainable seat positions to identify the optimal seat position that minimizes upper extremity demand (i.e., muscle stress, metabolic cost and muscle antagonism). The optimization results showed both metabolic cost and muscle stresses were near minimal values at superior/inferior positions corresponding to top dead center elbow angles between 110 and 120 degrees while at an anterior/posterior position with a hub-shoulder angle between 10 and 2.5 degrees. These minimal values coincided with a reduction in the level of antagonistic muscle activity, primarily at the glenohumeral joint. Seat positions that deviated from these minimal values increased the level of co-contraction required to maintain a stable, smooth propulsive stroke, and consequentially increased upper extremity demand. These results can provide guidelines for positioning the seat to help reduce upper extremity overuse injuries and pain, and thus improve the overall quality of life for wheelchair users. / text

Study of physical phenomena by means of mathematical models is common in various branches of engineering and science. In biomechanics, modelling often involves studying human motion by treating the body as a mechanical system made of interconnected rigid links. Robotics deals with similar cases as robots are often designed to imitate human behavior. Modelling human movements is a complicated task and, therefore, requires several simplifications and assumptions. Available computational resources often dictate the nature and the complexity of the models. In spite of all these factors, several meaningful results are still obtained from the simulations. One common problem form encountered in real life is the movement between known initial and final states in a pre-specified time. This presents a problem of dynamic redundancy as several different trajectories are possible to achieve the target state. Movements are mathematically described by differential equations. So modelling a movement involves solving these differential equations, along with optimization to find a cost effective trajectory and forces or moments required for this purpose. In this study, an algorithm developed in Matlab is used to study dynamics of several common human movements. The main underlying idea is based upon temporal finite element discretization, together with optimization. The algorithm can deal with mechanical formulations of varying degrees of complexity and allows precise definitions of initial and target states and constraints. Optimization is carried out using different cost functions related to both kinematic and kinetic variables. Simulations show that generally different optimization criteria give different results. To arrive on a definite conclusion on which criterion is superior over others it is necessary to include more detailed features in the models and incorporate more advanced anatomical and physiological knowledge. Nevertheless, the algorithm and the simplified models present a platform that can be built upon to study more complex and reliable models. / QC 20101110

The purpose of this thesis was to determine how shaft stiffness affects clubhead speed and how it alters clubhead orientation at impact. For the first time, a 3D, six-segment forward dynamics model of a golfer and club was developed and optimized to answer these questions. A range of shaft stiffness levels from flexible to stiff were evaluated at three levels of swing speed (38, 45 and 53 m/s). At any level of swing speed, the difference in clubhead speed did not exceed 0.1 m/s across levels of shaft stiffness. Therefore, it was concluded that customizing the stiffness of a golf club shaft to perfectly suit a particular swing will not increase clubhead speed sufficiently to have any meaningful effect on performance. The magnitude of lead deflection at impact increased as shaft stiffness decreased. The magnitude of lead deflection at impact also increased as swing speed increased. For an optimized swing that generated a clubhead speed of 45 m/s, with a shaft of regular stiffness, lead deflection of the shaft at impact was 6.25 cm. The same simulation resulted in a toe-down shaft deflection of 2.27 cm at impact. Using the model, it was estimated that for each centimeter of lead deflection of the shaft, dynamic loft increased by approximately 0.8 degrees. Toe-down shaft deflection had relatively no influence on dynamic loft. For every centimeter increase in lead deflection of the shaft, dynamic closing of the clubface increased by approximately 0.7 degrees. For every centimeter increase in toe-down shaft deflection, dynamic closing of the clubface decreased by approximately 0.5 degrees. The results from this thesis indicate that improvements in driving distance brought about by altering shaft stiffness are the result of altered clubhead orientation at impact and not increased clubhead speed.

The purpose of this thesis was to determine how shaft stiffness affects clubhead speed and how it alters clubhead orientation at impact. For the first time, a 3D, six-segment forward dynamics model of a golfer and club was developed and optimized to answer these questions. A range of shaft stiffness levels from flexible to stiff were evaluated at three levels of swing speed (38, 45 and 53 m/s). At any level of swing speed, the difference in clubhead speed did not exceed 0.1 m/s across levels of shaft stiffness. Therefore, it was concluded that customizing the stiffness of a golf club shaft to perfectly suit a particular swing will not increase clubhead speed sufficiently to have any meaningful effect on performance. The magnitude of lead deflection at impact increased as shaft stiffness decreased. The magnitude of lead deflection at impact also increased as swing speed increased. For an optimized swing that generated a clubhead speed of 45 m/s, with a shaft of regular stiffness, lead deflection of the shaft at impact was 6.25 cm. The same simulation resulted in a toe-down shaft deflection of 2.27 cm at impact. Using the model, it was estimated that for each centimeter of lead deflection of the shaft, dynamic loft increased by approximately 0.8 degrees. Toe-down shaft deflection had relatively no influence on dynamic loft. For every centimeter increase in lead deflection of the shaft, dynamic closing of the clubface increased by approximately 0.7 degrees. For every centimeter increase in toe-down shaft deflection, dynamic closing of the clubface decreased by approximately 0.5 degrees. The results from this thesis indicate that improvements in driving distance brought about by altering shaft stiffness are the result of altered clubhead orientation at impact and not increased clubhead speed.

Understanding the underlying coordination mechanisms that lead to a patient’s poor walking performance is critical in developing effective rehabilitation interventions. However, most common measures of rehabilitation effectiveness do not provide information regarding underlying coordination mechanisms. The overall goal of this research was to analyze the relationship between two potential measures of walking performance (step length asymmetry and module composition) and underlying walking mechanics. Experimental analyses were used to analyze the walking mechanics of hemiparetic subjects grouped by step length asymmetry. All groups had impaired plantarflexor function and the direction of asymmetry provided information regarding the compensatory mechanism used to overcome this plantarflexor impairment. Those subjects who walked with longer paretic than nonparetic steps compensated using increased output from the nonparetic leg, while those with symmetric steps compensated using a bilateral hip strategy. These results suggest that step length asymmetry may provide information regarding underlying coordination mechanisms that can be used to guide rehabilitation efforts. Another way to assess walking performance is to directly analyze deficits in muscle coordination. Recent studies have suggested that complex muscle activity during walking may be generated using a reduced neural control strategy organized around the co-excitation of multiple muscles, or modules, which may provide a useful framework for characterizing coordination deficits. Simulation analyses using modular control were performed to understand how modules contribute to important biomechanical functions of non-impaired walking and how the generation of these functions is altered in groups of post-stroke hemiparetic subjects who commonly merged different sets of non-impaired modules. The non-impaired simulation found that six modules are needed to generate the three-dimensional tasks of walking (support, forward propulsion, mediolateral balance control and leg swing control). When the plantarflexor module was merged with the module controlling the knee extensors and hip abductors, forward propulsion and ipsilateral leg swing were impaired. When the module controlling the hamstrings was merged with the module controlling the knee extensors and hip abductors, forward propulsion, body support and mediolateral balance control were impaired. These results suggest that module analysis may provide useful information regarding the source of walking deficits and can be used to guide rehabilitation efforts. / text

Advances in graphics and robotics have increased the importance of tools for synthesizing humanoid movements to control animated characters and physical robots. There is also an increasing need for analyzing human movements for clinical diagnosis and rehabilitation. Existing tools can be expensive, inefficient, or difficult to use. Using simulated physics and motion capture to develop an interactive virtual reality environment, we capture natural human movements in response to controlled stimuli. This research then applies insights into the mathematics underlying physics simulation to adapt the physics solver to support many important tasks involved in analyzing and synthesizing humanoid movement. These tasks include fitting an articulated physical model to motion capture data, modifying the model pose to achieve a desired configuration (inverse kinematics), inferring internal torques consistent with changing pose data (inverse dynamics), and transferring a movement from one model to another model (retargeting). The result is a powerful and intuitive process for analyzing and synthesizing movement in a single unified framework. / text

Unilateral, below-knee amputees have altered gait mechanics, which can significantly affect mobility. For example, amputees often have asymmetric leg loading as well as higher metabolic cost and an increased risk of falling compared to non-amputees. Below-knee amputees lose the functional use of the ankle muscles, which are critical in non-amputee walking for providing body support, forward propulsion and leg-swing initiation. The ankle muscles also regulate angular momentum in non-amputees, which is important for providing body stability and preventing falls. Thus, compensatory mechanisms in amputee walking are developed to accomplish the functional tasks normally provided by the ankle muscles. In Chapters 2 and 3, three-dimensional forward dynamics simulations of amputee and non-amputee walking were generated to identify compensatory mechanisms and their effects on joint loading and metabolic cost. Results showed that the prosthesis provided body support, but did not provide sufficient body propulsion or leg-swing initiation. As a result, compensations by the residual leg gluteus maximus, gluteus medius, and hamstrings were needed. The simulations also showed the intact leg tibio-femoral joint contact impulse was greater than the residual leg and that the vasti and hamstrings were the primary contributors to the joint impulse on both the intact and residual legs. The amputee simulation had higher metabolic cost than the non-amputee simulation, which was primarily due to prolonged muscle activity from the residual leg gluteus maximus, gluteus medius, hamstrings, vasti and intact leg vasti and ankle muscles. In Chapter 4, whole-body angular momentum in amputees and non-amputees was analyzed. Reduced residual leg propulsion resulted in a smaller range of sagittal plane angular momentum in the second half of the gait cycle. Thus, to conserve angular momentum, reduced braking was needed in the first half of the gait cycle. Decreased residual leg braking appears to be an important mechanism to regulate sagittal plane angular momentum in amputee walking, but was also associated with a greater range of angular momentum that may contribute to reduced stability in amputees. These studies have provided important insight into compensatory mechanisms in below-knee amputee walking and have the potential to guide rehabilitation methods to improve amputee mobility. / text