Configuration Space and Linkage Defects

The design of linkages presents many challenges.  Most notably the nonlinearity of the design space. However, another challenge is created by the presence of defects in linkage designs. The regularly studied linkage defects are named order, branch, and circuit. See Balli and Chand. An effective method to eliminate linkage designs that possess branch and circuit defects is to construct the complete configuration space of candidate mechanism designs. Constructing this space is a two part process:

  1. solve the forward kinematics equations for a set of input joint parameters to obtain all configurations at every input and
  2. sort all configurations into their proper solution branches and circuits.

Part (1) requires a solution method that finds all solutions to the forward kinematics equations. This can be accomplished by resultant elimination methods and continuation. Generally, a linkage’s forward kinematics equations are much simpler to solve than its synthesis equations. Part (2) involves sorting configurations (solutions to the fk equations) into branches and circuits. This is accomplished by tracing the branch curves with a Newton-Raphson based method. Particularly challenging is dealing with the singularities present in the configuration space, those are the points where solution branches would intersect.  A strategy for dealing with singular points is established in Plecnik and McCarthy.

Once the configuration space of a mechanism is constructed, one now simply checks whether the entirety of a motion requirement is contained on a single branch or circuit, depending on the application’s requirement. Here is an example:

The forward kinematics equations of a Watt II function generator are solved and solutions are sorted. The configuration space is constructed. Singularities are marked with purple points.

The motion requirement is a set of nine input-output points (in blue below). Taking a slice of the configuration space, we construct the mechanism’s input-output function.

We conclude that the analyzed mechanism is defective. Zooming in on the problematic area of the configuration space, we see two singular points and two mechanism circuits.

S. S. Balli and S. Chand, 2002. “Defects in link mechanisms and solution rectification,” Mechanism and Machine Theory, 37(9):851-876.

M. Plecnik and J. M. McCarthy, 2014. “Numerical synthesis of six-bar linkages for mechanical computation,” Journal of Mechanisms and Robotics.

Synthesis of Six-bars

The six-bar linkage is the next simplest 1-DOF planar mechanism after the four-bar. However, designing these mechanisms presents a challenging set of synthesis equations. For example, a four-bar motion generator is determined by a polynomial system that has 4 isolated roots (of which one is the origin), where each root represents a potential design. On the other hand, a six-bar function generator is determined by a polynomial system that could have as many as 4.13×108 roots. Chances are the actual number of roots of this system is much smaller (by 2 or 3 magnitudes), but that number is unknown. It is much harder to find complete solution sets for general six-bar design problems than it is for four-bars.

From a practical point of view, six-bars have 2 more moving parts than four-bars. With the increased complexity comes advantages:

1. More design options. As mentioned above, each root above represents a design candidate, and surely it is better to have thousands of design candidates than 3. Especially because most of these design candidates will suffer from linkage defects rendering them useless.

2. Six-bars do more complex stuff. Four-bars can move through 5, 5, and 9 positions for function, motion, and path generation, respectively. On the other hand, six-bars can move through 11, 8, and 15 positions. The maximum degree of the curve traced by a point on the coupler link of a four-bar versus a six-bar is 6 to 18. Six-bars have even been designed that can perform 2:1 gear ratios.

3. You can put things where they need to be. That is there is much more design freedom shaping links and choosing the locations of pivots. This is useful if your linkage needs to mount to specific hard points or fit within a certain envelope. Four and six-bar motion generators provide a good example. There will be a maximum of 6 four-bars that move through 5 task positions with no pivots specified. Meanwhile, there will be a maximum of 5,735 six-bars that move through 6 task positions with both ground pivots specified!

Here is a bit on my research with six-bars. Six-bars are divided into 5 kinematic inversions. For simplicity, we’ll just call these types. There are three common motion requirement tasks. The table below provides a breakdown of the size of the synthesis problem for each linkage type over the three motion tasks.

This table shows six-bar info

Function Motion Path
Watt I Max positions: 5
Degree: 4
Max positions: 8
Degree: 3.43×1010
Max positions: 15
Degree: 2.28×1046
Watt II Max positions: 11
Degree: 5.50×107
Max positions: 5
Degree: 4
Max positions: 9
Degree: 268,720
Stephenson I Max positions: 5
Degree: 4
Max positions: 5
Degree: 4
Max positions: 15
Degree: ?
Stephenson II Max positions: 11
Degree: 4.13×108
Max positions: 5
Degree: 4
Max positions: 15
Degree: ?
Stephenson III Max positions: 11
Degree: 4.13×108
Max positions: 5
Degree: 4
Max positions: 15
Degree: ?

 

Path Generators

Designing a path generating linkage means designing a linkage to guide a point path in some particular manner.  We specifically have worked on designing six-bars.  Much work has been completed on four-bar path generators, but it wasn’t until 1992 that Wampler et al. [1] found a complete solution set of 4,326 to the general 9-point problem. We generalized their work to six-bars which can achieve as many as 15 points, however, the number of solutions to the design equations greatly increased. The smallest degree computed thus far is 2.28×1046.

This work diverges from solving the most general case and instead focuses on inverting six-bar function generators into six-bar path generators. How this works is by reformulating the problem as the control of a 2R chain. A 2R chain can be specified and its end effector can be moved to any point within its workspace where its joint angles can be calculated. Moving the the end effector through 11 points provides 11 joint angle pairs which can be coordinated by six-bar function generators. Attaching these six-bar function generators onto the 2R chains is equivalent to kinematic inversion. Specifically, we focus on Stephenson linkages due to their ability to create 11 point function generators. There are four ways to invert Stephenson function generators into Stephenson path generators which appears in the figures below.

Stephenson III function generator inverted to a Stephenson I path generator

Stephenson III function generator inverted to a Stephenson II path generator

Stephenson II function generator inverted to a Stephenson II path generator

Stephenson II function generator inverted to a Stephenson III path generator

This work makes use of the large parameter homotopy solution sets that we calculated during research on six-bar function generators. The kinematic requirements for path generation are formulated as a function generation problem and then solved as such. Solutions to function generator synthesis equations are then inverted back to path generation. This synthesis technique was used for the design of six-bar walking linkages.

[1] C. W. Wampler, A. J. Sommese, and A. P. Morgan, 1992. “Complete solution of the nine-point path synthesis problem for four-bar linkages,” Journal of Mechanical Design 114(1):153-159.

Creative Design

A massive piece of kinetic artwork was designed to be installed on wall. A simulation of this artwork appears below.

 

For creative design inquiries, fill the contact form below:

Shape Changing Parabolic Reflector

Here is a bit from my undergraduate honors thesis work at the University of Dayton. The objective was to design a mechanically adaptable light reflector that moves five panels through five parabolic design profiles. The video below also shows designs for a shape-changing seat, cam, and face.

M. Plecnik, Design of a Shape-Changing Rigid-Body Parabolic Light Reflector, Honors Thesis, University of Dayton, 2010. (pdf)

Rotary Weaver

A hydraulic hose manufacturer in the US approached UC Irvine in order to design a linkage for their rotary weaver equipment. They were trying to equip their machines so that they could change out the current 2 over 2 braid pattern into a 3 over 3 braid pattern. They wanted to avoid the use of cams but could not find a linkage that created a dwell motion from a constant input that wouldn’t result in the end effector hitting the spools.

I designed them the following linkage that solved their problem.

Rice Transplanter

A novel six-bar linkage was designed to move transplant rice seedlings. The idea is to create a manual device that eliminates very labor intensive hand transplantation.  For more information on this subject click here.

The next phases of this project are to design and test end effector geometry and feed indexing. This work is being completed by a team of undergraduate mechanical engineers that I advise. Progress on our end effector designs appears below.

Wing Mechanism

A wing mechanism was designed to reproduce a complex accelerative flapping gait from a single constant RPM motor. The flapping gate was deciphered from data obtained by Tobalske and Dial, 1996, of black-billed magpies. The synthesis process begins by specifying a 4R spatial serial chain that resembles a magpie’s anatomy. Moving the spatial chain through the desired flapping gait defines a function of joint angle over time at each of the four joints. A six-bar Stephenson II function generator was designed for each joint angle function and the whole system was coupled together such that it can be run by a motor spinning at constant RPM.

Finally, compliant joints were added between the wrist and wing tip to mimic this portion of a bird’s anatomy. This joints utilize hard stops in order to limit their compliance to one direction creating aerodynamic check values such that control surfaces remain rigid during downstroke and deflect during backstroke.

The resulting motion has a long, stretched out downstroke followed by a quick, compressed backstroke.

B. W. Tobalske and K. P. Dial, “Flight Kinematics of Black-billed Magpies and Pigeons Over A Wide Range of Speeds,” The Journal of Experimental Biology, 199(2): 263-280, 1996.

Six-bar Suspension

For Long Travel

A design algorithm was created that synthesizes suspension linkages that feature the Watt I six-bar mechanism. Watt I mechanisms offer motion capabilities beyond four-bar double wishbone designs, however their design in not intuitive so we depend on the mathematics to find linkage designs for us. The algorithm was applied to the design of a long travel suspension for use on an SAE Baja vehicle. The resulting linkage has the following specifications:

  • 16 inches of travel
  • Less than 1 inch of track change
  • 62 inch track
  • 12 inch ride height
  • Roll center height about 1.68 inches below ground
  • 1.13 deg/in camber gain for upper 8 inches of motion
  • Minimal camber gain for bottom 8 inches of motion
  • Neutral camber when the wheels are fully drooped

The roll characteristics of the vehicles have been designed such that the outside tire is perpendicular to the ground during cornering loads.  Shocks can be mounted to the small inboard links eliminating the need for additional pushrod and rocker links to move the shocks inboard and manipulate mechanical advantage for whatever spring size and travel. With these links eliminated, the suspension showcased here has the same number of links as a traditional double wishbone and rocker setup.

A fifth-scale prototype of this suspension was built.  It’s geometry is exhibited in the video below.

At the heart of the design algorithm is a synthesis method for four position motion generation of Watt I linkages. This is a simplification of general motion synthesis for Watt I’s that allows eight positions. The simplification comes in the form of prescribing the positions of the ground pivots and end effector pivots. This has the particular advantage in suspension linkage design of allowing us to locate the chassis mounting points and upright mounting points. Since many of the algorithm’s results suffer from packaging problems or linkage defects, the algorithm includes the ability to search for design candidates within zones of chassis pivots and upright pivots.

A common solution for achieving a long travel suspension is to design a long parallelogram four-bar linkage. However, these linkage suffer from track change and packaging issues.

For Racing

The same synthesis algorithm was used to design a novel racing suspension. The goal is to maintain near vertical alignment of the wheels to the road during cornering. The complete suspension is analyzed as a symmetric planar 12-bar linkage with ground pivots located at the contact patches. The design procedure specifies the vehicle chassis orientation and the tire camber angles of the vehicle when cornering.  As well, two task positions of the wheels with respect to the chassis are specified for suspension movement in straightaways.  The result is 18 design equations with 18 unknowns that have a total degree of 2,097,152, though only 336 roots.

Fifth-scale prototype of racing suspension

M. Plecnik and J. M. McCarthy, 2014. “Vehicle suspension design based on a six-bar linkage,” Proceedings of the ASME 2014 IDETC/CIE Conference, Paper No. DETC2014-35374, August 17-20, 2014, Buffalo, New York, USA.