Mark M. Plecnik

Numerical Synthesis of Six-Bar Linkages for Mechanical Computation

© 2014 by ASME. 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.

Five Position Synthesis of a Slider-Crank Function Generator

In this paper, we present a synthesis procedure for the coupler link of a planar slider-crank linkage in order to coordinate input by a linear actuator with the rotation of an output crank. This problem can be formulated in a manner similar to the synthesis of a five position RR coupler link. It is well-known that the resulting equations can produce branching solutions that are not useful. This is addressed by introducing tolerances for the input and output values of the specified task function. The proposed synthesis procedure is then executed on two examples. In the first example, a survey of solutions for tolerance zones of increasing size is conducted. In this example we find that a tolerance zone of 5% of the desired full range results in a number of useful task functions and usable slider-crank function generators. To demonstrate the use of these results, we present an example design for the actuator of the shovel of a front-end loader.Copyright

Computational Design of Stephenson II Six-Bar Function Generators for 11 Accuracy Points

© 2016 by ASME. This paper presents a design methodology for Stephenson II six-bar function generators that coordinate 11 input and output angles. A complex number formulation of the loop equations yields 70 quadratic equations in 70 unknowns, which is reduced to a system of ten eighth degree polynomial equations of total degree 810 = 1:07 × 109. These equations have a monomial structure which yields a multihomogeneous degree of 264,241,152. A sequence of polynomial homotopies was used to solve these equations and obtain 1,521,037 nonsingular solutions. Contained in these solutions are linkage design candidates which are evaluated to identify cognates, and then analyzed to determine their input-output angles in each assembly. The result is a set of feasible linkage designs that reach the required accuracy points in a single assembly. As an example, three Stephenson II function generators are designed, which provide the input-output functions for the hip, knee, and ankle of a humanoid walking gait.

Design of a 5-SS Spatial Steering Linkage

This paper presents the kinematic synthesis of a steering linkage that changes track, wheelbase, camber, and wheel height in a turn, while maintaining Ackermann geometry. Each wheel is controlled by a 5-SS platform linkage, which consists of a moving platform connected by five SS chains to the vehicle chassis. Ackermann steering geometry ensures all four wheels will travel on circular arcs that share the same center point. S denotes a spherical or ball-in-socket joint.The kinematic synthesis problem is formulated using seven spatial task positions. The procedure computes the SS chains that guide the platform through the seven task positions, and examines all combinations of five that form a single degree-of-freedom linkage. A kinematic analysis identifies the performance of each design candidate, and eliminates functional defects.In the design process, the task positions are modified randomly within constraints in order to find a useful mechanism design. Mechanisms are deemed useful if they travel smoothly through all seven task positions. Upon analyzing 1000 sets of task positions, only 10 useful mechanisms were found. A second iteration produced 22 useful mechanisms from 1000 task sets. An example of the design of a steering linkage is presented. A video of this linkage can be seen at http://www.youtube.com/watch?v=hEvbDiyQMiw.Copyright

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.

Kinematic Synthesis of a Watt I Six-Bar Linkage for Body Guidance

This chapter formulates the synthesis equations for a Watt I six-bar linkage that moves through \(N\) specified task positions. For the maximum number of positions, \(N=8\), the resulting polynomial system consists of 28 equations in 28 unknowns, which can be separated into a nine sets of variables yielding a nine-homogeneous Bezout degree of \(3.43\times 10^{10}\). We verify these synthesis equations by finding isolated solutions via Newton’s method, but a complete solution for \(N=8\) seems beyond the capability of current homotopy solvers. We present a complete solution for \(N=6\) positions with both ground pivots specified.

Dimensional synthesis of six-bar linkage as a constrained RPR chain

In this paper, five positions of a planar RPR serial chain are specified and the synthesis equations for two RR constraints are solved to obtain a six-bar linkage. Analysis of the resulting linkage determines if it moves the end-effector smoothly through the five task positions without a branch defect. The design procedure presented randomly selects variations to the positions of the RPR chain in order to obtain new six-bar linkages. This dimensional synthesis algorithm yields a set of six-bar linkages that move the end-effector near the original task positions. This synthesis procedure is applied to the design of a linkage that generates a square pattern. The procedure yielded 122 defect-free linkages for one million iterations.

Vehicle Suspension Design Based on a Six-Bar Linkage

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Synthesis of a Stephenson II Function Generator for Eight Precision Positions

This paper presents a synthesis methodology for a Stephenson II six-bar function generator that provides eight accuracy points, that is it ensures the input and output angles of the linkage match at eight pairs of specified values. A complex number formulation of the two loop equations of the linkage and a normalization condition are used to form the synthesis equations, resulting in 22 equations in 22 unknowns. These equations are solved using the numerical homotopy continuation solver, Bertini. The approach is illustrated with an example. Copyright