Symposia organized for the 2016 Mechanisms and Robotics Conference

The 2016 Mechanisms and Robotics conference is part of International Design Engineering Technical Conferences organized by ASME International in Charlotte, North Caroline, August 22-24.

MotionGen is a planar four-bar linkage simulation and synthesis app that helps users synthesize planar four-bar linkages by assembling two of the planar RR-, RP- and PR-dyad types, (R refers to a revolute or hinged joint and P refers to a prismatic or sliding joint).

The input task is a planar motion given as a set of discrete positions and orientations and the app computes type and dimensions of synthesized planar four-bar linkages, where their coupler interpolates through the given poses either exactly or approximately while minimizing an algebraic fitting error. The algorithm implemented in the app extracts the geometric constraints (circular, fixed-line or line-tangent-to-a-circle) implicit in a given motion and matches them with corresponding mechanical dyad types enumerated earlier. In the process, the dimensions of the dyads are also computed. By picking two dyads at a time, a planar four-bar linkage is formed. Due to the degree of polynomial system created in the solution, up to a total of six four-bar linkages can be computed for a given motion.

MotionGen display

MotionGen also lets users simulate planar four-bar linkages by assembling the constraints of planar dyads on a blank- or image-overlaid screen. This constraint-based simulation approach mirrors the synthesis approach and allows users to input simple geometric features (circles and lines) for assembly and animation. As an example of the Simulation capabilities of the app, Figure shows a walking robot driven by two sets of planar four-bar linkages where the foot approximately traces a trajectory of walking motion. The users can input two dyads on top of an imported image of a robot or machine to verify the motion and make interactive changes to the trajectories.

Anurag Purwar describes MotionGen and its applications in this video:

Algebraic curve. An algebraic curve is the set of points P=(x, y) with coordinates that satisfy an algebraic equation,

Algebraic curve

where u_{i} and v_{i} are integer exponents, and c_{i} are real coefficients.

Forward kinematics. In order to draw this curve, we use an RR serial chain shown in Figure 1. All that is needed is to coordinate the angles of this chain so its end-point P traces the curve.

Figure 1. The joints of an RR serial chain are coordinated to guide the point P along a specified curve.

The relationship between the joint angles and the coordinates P=(x, y) is given by the forward kinematics equations of this RR chain,

RR forward kinematics

Kempe’s formula. In his 1876 paper, “On a general method of describing plane curvesof nth degree by linkwork,” A. B. Kempe substituted this into the equation of the curve and simplified the result to obtain,

Kempe formula

Kempe interpreted this formula to be the x-coordinate of an N link serial chain, where each link has the length A_{j} and is positioned at the angle,

Kempe link angles

The end-point of Kempe’s serial chain maintains the value x=C, which means it only moves along the y-direction.

Mechanical computation. Now connect cable drives to the two joints of the RR chain so they can be used as inputs to N mechanical computers that calculate the Psi_{j} angular values. The output of the mechanical computers are connected by cable drives to the joints of Kempe’s serial chain.

The operations needed in the mechanical computers are simply multiplication and addition. Multiplication of an angle by a given factor is achieved by using a cable drive with the ratio of the pulley sizes set to the specified factor. The sum of two angular values is achieved using a bevel gear differential.

The result is a mechanical system that constrains the angles of an RR chain so it draws the specified algebraic curve within the workspace of RR serial chain.

Elliptic cubic curve. As an example, consider the elliptic cubic curve,

Elliptic cubic curve

Let the RR chain have link lengths L_{1}=L_{2}=1, and obtain Kempe’s formula for this curve,

Elliptic cubic formula

This equation has 12 terms, so Kempe’s serial chain has 12 links. The 12 mechanical computers connecting the angles of Kempes serial chain to the driving joints for the RR chain, include six additions and eight multiplications, Figure 2.

Figure 2. The Kempe’s serial chain and the mechanical computations that constrain the RR serial chain to draw an elliptic cubic curve.

Rather than use differentials and cable drives, Kempe used linkages to perform the mechanical computations and couple the results. The outcome is a lot of links as can be seen in this animation provided by Alex Kobel.

http://mechanicaldesign101.com/wp-content/uploads/2016/08/Elliptic-M5.jpg14082780Prof. McCarthyhttp://mechanicaldesign101.com/wp-content/uploads/2016/07/mechanical-design-101LOGOf.pngProf. McCarthy2016-08-03 16:34:322016-08-14 12:29:10Mechanical computation and algebraic curves

This is the latest from Yang Liu. It is a mechanical synthesis of the Fourier expansion of the batman logo found on Wolfram.com. The video below shows the operation of this harmonic synthesis.

This article by Jon Hauenstein with me for SIAM News (Society for Industrial and Applied Mathematics) describes research by Mark Plecnik in the computer-aided design of linkages to provide mechanical movement of a bird’s wing. Here is Mark’s video of the his wing flapping mechanism.

This mechanical system was designed by Yang Liu to trace the shape of a heart. The work is inspired by the mechanical 32-element harmonic synthesizer described by Dayton Miller in the 1916 article in the Journal of the Franklin Institute.

Michael Sutherland and his team at Zennen Engineering designed this full-size folding bicycle that has a dramatically different folding action from current designs.

Zennen Engineering has a new concept that they are kind enough to say was inspired by our UCI Folding Structure. This new design rotates the rear wheel support around the bottom bracket, and folds the front forks against the down tube and seat tube to form a compact package. It is a unique movement.

Jon Stokes, in our Robotics and Automation Lab, helped by adding the four-bar function generator to combine the two folding actions. It is a great concept, and it will be interesting to see if it achieves commercial success.

Montague Bikes provides a popular line folding full sized bicycles, which fold sideways around the seat tube.

Lawrence Funke and Prof. James Schmiedeler of the University of Notre Dame Locomotion and Biomechanics Lab show that the movement of a morphing linkage through its target profiles can be improved by coordinating actuation of the sub-chains. This was presented at the Mechanisms and Robotics Conference which was part of the 2015 ASME Design Engineering Technical Conferences, August 2-5, in Boston, MA. The video below shows the improvement obtained by moving from 1 to 3 coordinated actuators.

Mechanical engineering students in Prof. Larry Howell’s Compliant Mechanisms Research Group designed and constructed this kinetic structure for the BYU Museum of Art. It illustrates paper folding known as origami.

http://mechanicaldesign101.com/wp-content/uploads/2015/08/Origami-Art-BYU.png698862Prof. McCarthyhttp://mechanicaldesign101.com/wp-content/uploads/2016/07/mechanical-design-101LOGOf.pngProf. McCarthy2015-08-09 09:17:252016-07-31 09:35:16Origami Art at BYU