Controlling the Movement of a TRR Spatial Chain With Coupled Six-Bar Function Generators for Biomimetic Motion

This paper describes a synthesis technique that constrains a spatial serial chain into a single degree-of-freedom mechanism using planar six-bar function generators. The synthesis process begins by specifying the target motion of a serial chain that is parameterized by time. The goal is to create a mechanism with a constant velocity rotary input that will achieve that motion. To do this we solve the inverse kinematics equations to find functions of each serial joint angle with respect to time. Since a constant velocity input is desired, time is proportional to the angle of the input link, and each serial joint angle can be expressed as functions of the input angle. This poses a separate function generator problem to control each joint of the serial chain. Function generators are linkages that coordinate their input and output angles. Each function is synthesized using a technique that finds 11 position Stephenson II linkages, which are then packaged onto the serial chain. Using pulleys and the scaling capabilities of function generating linkages, the final device can be packaged compactly. We describe this synthesis procedure through the design of a biomimetic device for reproducing a flapping wing motion

Homotopy Directed Optimization to Design a Six-Bar Linkage for a Lower Limb With a Natural Ankle Trajectory

This paper presents a synthesis method for the Stephenson III six-bar linkage that combines the direct solution of the synthesis equations with an optimization strategy to achieve increased performance for path generation. The path synthesis equations for a six-bar linkage can reach as many as 15 points on a curve; however, the degree of the polynomial system is 10 46 . In order to increase the number of accuracy points and decrease the complexity of the synthesis equations, a new formulation is used that combines 11 point synthesis with optimization techniques to obtain a six-bar linkage that minimizes the distance to 60 accuracy points. This homotopy directed optimization technique is demonstrated by obtaining a Stephenson III six-bar linkage that achieves a specified gait trajectory

Re-evaluating Augustinian Fatalism through the Eastern and Western Distinction between God's Essence and Energies

In this dissertation, I will examine the problem of theological fatalism in St. Augustine and, specifically, whether or not Augustine was philosophically justified in his belief that his views on divine grace and human freedom could be harmonized. As is well-known, beginning with his second response To Simplician (ca. 396) and continuing through his works against the semi-Pelagians (ca. 426–429), Augustine espoused the Pauline doctrine of all-inclusive grace: that the fallen will’s ability to accomplish the good is totally a function of God’s elective grace. What, then, does the fallen will do to work out its own salvation? There is the further issue of how to reconcile Augustine’s rather extreme emphasis on grace in his later works with the more balanced picture we receive in his sermones ad populum, written throughout his forty-year preaching career. In many of these sermons, even those written during the Pelagian controversy, Augustine is careful to leave space for both divine and human initiative in the process of our justification within the totus Christus, or ‘whole Christ.’ How we can understand Augustine in his role as doctor gratiae and as preacher of human freedom will be a major inquiry of this dissertation. The most serious obstacle to moving forward on these problems has been and remains the essentialist interpretation of Augustine’s Trinitarian theology by most commentators. On their interpretation, Augustine thought that there were no real distinctions within the Trinity, with each of the three divine persons and their actions sharing in the absolute unity of the divine essence. Holding this interpretation not only does away with the distinctness of each of the persons, but also requires all of God’s different powers and attributes, including willing and foreknowing, to be coalesced into one another without distinction in the divine essence. God’s foreknowledge is thereby identified with God’s will, which necessarily leads to theological fatalism: God would have to will everything that He foreknows, and God would have to foreknow everything that He wills. Since God is omniscient, He wills everything that will happen, including the future willings of the fallen human will. It cannot be denied that there are texts in the Augustinian corpus that seem to point to a reading of the Trinity as absolutely simple. But this study will endeavor to show that there are also other largely overlooked texts in On the Trinity, the Confessions, and his Commentaries on the Literal Interpretation of Genesis (among others) that argue for various distinctions within the Trinity to make sense of the relation between Creator and creature, and the differences between the divine processions of generation/spiration, and the act of creation. These texts will be shown to parallel very closely the position of the Eastern Orthodox Christian tradition, which consistently uses the real distinction between God’s essential being and energetic activities (also known as the essence-energy distinction) to avoid the problem of theological fatalism. This rich theological and philosophical tradition, from the time of the fourth-century Greek Fathers to the Byzantine tradition that followed, differs less with Augustine concerning the essentials of Trinitarian theology and its practical implications for solving the problem of making human freedom and divine grace compatible than has been hitherto thought

Controlling the movement of a trr spatial chain with coupled six-bar function generators for biomimetic motion

This paper describes a synthesis technique that constrains a spatial serial chain into a single degree-of-freedom mechanism using planar six-bar function generators. The synthesis process begins by specifying the target motion of a serial chain that is parameterized by time. The goal is to create a mechanism with a constant velocity rotary input that will achieve that motion. To do this we solve the inverse kinematics equations to find functions of each serial joint angle with respect to time. Since a constant velocity input is desired, time is proportional to the angle of the input link, and each serial joint angle can be expressed as functions of the input angle. This poses a separate function generator problem to control each joint of the serial chain. Function generators are linkages that coordinate their input and output angles. Each function is synthesized using a technique that finds 11 position Stephenson II linkages, which are then packaged onto the serial chain. Using pulleys and the scaling capabilities of function generating linkages, the final device can be packaged compactly. We describe this synthesis procedure through the design of a biomimetic device for reproducing a flapping wing motion

Numerical Synthesis of Six-Bar Linkages for Mechanical Computation

This paper presents a design procedure for six-bar linkages that use eight accuracy points to approximate a specified input-output function. In the kinematic synthesis of linkages, this is known as the synthesis of a function generator to perform mechanical computation. Our formulation uses isotropic coordinates to define the loop equations of the Watt II, Stephenson II, and Stephenson III six-bar linkages. The result is 22 polynomial equations in 22 unknowns that are solved using the polynomial homotopy software BERTINI. The bilinear structure of the system yields a polynomial degree of 705,432. Our first run of BERTINI generated 92,736 nonsingular solutions, which were used as the basis of a parameter homotopy solution. The algorithm was tested on the design of the Watt II logarithmic function generator patented by Svoboda in 1944. Our algorithm yielded his linkage and 64 others in 129 min of parallel computation on a Mac Pro with 12±2.93 GHz processors. Three additional examples are provided as well

Kinematic synthesis of Stephenson III six-bar function generators

This paper presents a direct solution of the kinematic synthesis equations for Stephenson III six-bar function generators to achieve as many as 11 accuracy points. The approach is similar to that used to design Stephenson II function generators, except additional algebraic manipulations reduce the system to a multihomogeneous degree of 55,050,240. A numerically general multihomogeneous homotopy was used to obtain 834,441 nonsingular solutions, which were then used to construct an efficient parameter homotopy for specific tasks consisting of 11 accuracy points. The thousands of linkage solutions found by this parameter homotopy are sorted and analyzed to verify nonbranching movement through the specified task positions. An example is presented of a function generator that creates a specified torque-angle profile for a dynamic wrist splint that cancels the effects of spasticity in the wrists of stroke survivors