Kinematic synthesis, at its heart, involves finding the zero-dimensional solution set of a

system of polynomials. The degree of these polynomials increases rapidly as more com

plex designs are considered. The computation time required to find the solution set has

traditionally been the bounding factor for what can been achieved in kinematic synthesis.

Homotopy continuation is typically used to find solutions to these polynomials. Homotopy

continuation is itself an inherently parallelizable method. Graphics processing units (GPUs)

were developed were developed with a structure that makes them optimal to solve problems

in parallel. This dissertation explores the use of homotopy continuation running a GPU in

order to decrease the computation time required for kinematic synthesis.

First, we discuss the development of an algorithm for homotopy continuation that is ideal

to run on a GPU. The traditional path tracking algorithm is analyzed and then modified to

better perform on a GPU. Additionally, the endgame methods are analyzed and the more

ideal method is identified and implemented in CUDA. We outline the drawbacks of such

modifications and discuss why they are admissible in the context of kinematic synthesis.

The implementation of the new homotopy continuation algorithm on a GPU is demon

strated by solving the four-bar linkage synthesis problem. This is the first development of

a GPU-accelerated four-bar linkage design system. Novel (non-Burmester) loop equations

are derived such that the entire mechanism is solved in one computation. These equations

are then reduced and implemented into CUDA. The entire program is outlined and then

demonstrated on a sample design problem. The results are compared with a similar CPU

implemented system and a GPU speedup of around 120 times was observed.

The results of the four-bar linkage design system were then extended to the problem of six

bar linkage synthesis. A system was developed utilizing the same GPU-based path tracker

and endgame. This has resulted in the first known GPU-accelerated six-bar linkage design

system that presents new opportunities for multi-GPU systems capable of designing even

more complicated mechanisms.

This dissertation presents and demonstrates a new design procedure for customized wearable devices that provide joint guidance, support, and increased mobility for individuals with hand or lower-leg impairments. This new procedure uses motion capture data to guide kinematic synthesis of closed-loop linkages that match the specific movement of an individual and is demonstrated in the design and evaluation of a mechanical thumb, a hands-free crutch and a knee-ankle-foot orthosis.

The mechanical thumb was designed using motion capture data of a thumb curling task. A geometric model of the resulting four-bar linkage moved smoothly between the task positions and matched the desired thumb tip physiological movement. Then, a hands-free crutch was designed with a four-bar linkage synthesized to match motion capture data defining the movement of an individual's knee joint during walking. Tests of a prototype demonstrated reduced hip-hiking and increased stride length compared to a fixed-knee hands-free crutch.

A six-bar knee-ankle-foot orthosis was designed to match both the angular movements at the knee and ankle joints and the metatarsal trajectory obtained from motion capture data of a natural walking gait in the sagittal plane. Tests of a prototype showed normal knee

function and stride lengths during walking. Optimization was combined with the kinematic synthesis procedure to match the metatarsal trajectory.

The result is a new design procedure to obtain simple and robust wearable devices for application as hand, knee, and knee-ankle-foot orthoses that match natural human movement kinematics.

This thesis presents an exploration and analysis of the feasibility of two tensegrity systems through a series of optimization studies. First, double-helix tensegrity systems are defined for their application in a package delivery processes as payload carriers that provide impact attenuation for their payload. The architecture of the established double-helix tensegrity payload carrier is defined and modified to enhance flexibility in optimization studies. A dynamic model to simulate ground impact for the payload carriers is derived such that the physical and material characteristics of the carriers are simulated. The dynamic simulation calculates stress and strain in all members of the payload carriers and outputs all member failures. The simulation also performs checks on the carrier to ensure that it has not deformed to flip inwards on itself. Finally, the simulation ensures that the payload within the carrier does not experience forces greater than 10gEto ensure that no damage occurs to the payload from carrier impact. The dynamic simulation is then used to perform a full optimization analysis to optimize the mass and impact attenuation of the payload carrier by varying topological parameters and applying penalties to configurations that experience physical or material failures during the simulation. The MATLAB genetic algorithm is used to conduct the optimization using a fitness score to penalize any configurations that contain material or physical failures as well as any configurations that have a maximum payload deceleration greater than 10gE. An optimization study is conducted using three shapes, cylinder, ellipsoid, and parabola, each carrying a 5 kg payload dropped at a height of 2 m, to find the optimal geometry and topology for a payload delivery application. The parabolic double-helix tensegrity payload carrier obtained the lowest mass of the three carrier types with a structure mass of 2.33 kg while its payload experienced a maximum deceleration during impact of 8.99gE. The ellipsoidal carrier provided the second lightest structure mass at 3.88 kg with a maximum payload deceleration of 6.15gE. Finally, the cylindrical carrier converged to a mass of 5.44 kg with a maximum payload deceleration of 8.71gE. A similar optimization technique is applied to a tensegrity morphing wing that actuates using a twisting tensegrity column design. The tensegrity morphing wing has a greater lift-to-drag ratio than a conventional wing that uses flaps to actuate due to the reduction of induced drag from the elimination of control surfaces. The tensegrity morphing wing replaces control flaps with a tensegrity torsional mechanism to twist the entire wing and create a change in angle of attack. A finite element model of the wing is developed in ABAQUS to simulate a Carl Goldberg Falcon 56 Mk II R/C unmanned aerial vehicle with a wingspan of 142.24 cm and a chord length of 25.31 cm. The finite element model evaluates the maximum twist achievable at an air speed of 20 m/s given that no components of the wing fail from material or structural failure, as determined by ABAQUS. A multi-objective optimization study is conducted to maximize twist angle and minimize wing mass. The multi-objective optimization provided a range of solutions in the form of a Pareto front. Three designs are chosen from the Pareto front to determine the effectiveness of the optimization. Compared to previous design of experiment studies, the optimization was able to provide wing configurations with higher twist angles and lower masses. The highest twist wing design obtained a maximum wing twist of 18.68◦while obtaining a low mass of 1.88 kg.

This dissertation presents the kinematic design of six-bar linkages for function, motion, and path generation by means of polynomial homotopy continuation algorithms. When no link dimensions are specified beforehand, the synthesis formulations for each design objective yield polynomial systems of degrees in the millions and billions, suggesting a large number of solutions. Complete solution sets to these systems have not yet been obtained and is the topic of this dissertation. Function generation for eleven positions is explored in most detail, in particular the Stephenson II and III function generators, for which we calculate multihomogeneous degrees of 264,241,152 and 55,050,240. A numerical reduction using homotopy estimates these systems to have 1,521,037 and 834,441 roots, respectively. For motion generation, the Watt I linkage can be specified for eight positions, producing a system of a multihomogeneous degree over 19 billion. However, for this work we focus on the smaller case of six positions, numerically reducing this system to an estimated 5,735 roots. For path generation we take a different approach. The design of path generators is formulated as RR chains constrained to have a single degree-of-freedom by attaching six-bar function generators to them. This enables us to use our results obtained on Stephenson II and III function generators to create four types of eleven position path generators: the Stephenson I linkage, two types of Stephenson II linkages, and the Stephenson III linkage.

This dissertation describes research into an automated methodology for designing eight-bar linkage systems for specialized tasks. The user specifies the desired motion as five discrete task positions, and a backbone chain robot that can achieve those task positions. The design algorithm then uses the graph theory approach to find all the different ways of constraining the robot using mechanical constraints, which are rigid binary links (RR constraints), to generate single degree of freedom linkages.

Linkage solutions synthesized using this method are only guaranteed to reach the five positions. They can exhibit branch defects which results in discontinuities during their motion in between the task positions. An analysis algorithm is used to discard the defective solutions and ensure that the filtered solutions will move the end-effector smoothly through all the five desired task positions. Once the original task positions are explored for finding linkage solutions, the design system runs the algorithm iteratively over randomized task positions, within acceptable variations provided by the user. This results in the generation of large number of defect-free linkage solutions. These solutions are further checked for a minimum and maximum allowed link length criteria to further filter out linkages. The result is a list of successful linkage solutions that are presented to the user.

The RR constraining strategy for the user specified robot has provided some useful insights in to linkage design. It has been observed that as we increase the complexity of the starting serial or closed chain robot, the number of linkage solutions that could be synthesized increases rapidly. This increases the probability of finding useful solutions and hence more design options can be presented to the designer for selection. The design system has been implemented in a computer aided linkage software system called MechGen for four-bar and six-bar linkages, and has demonstrated its effectiveness to design better devices for several applications.

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