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: ?


Function Generators

For 11 positions

Function generation refers to the use of a linkage to coordinate angles. This has many useful applications and function generators are often installed as elements of other linkages, such as path generators. Designing function generators is a particular challenge because of the large degree of their design equations.

We have solved the design equations for Stephenson II and Stephenson III six-bar linkages. The total degree for the design equations for each case is 1.07×109. However, we show a more in depth solution count limits the number of roots to 264,241,152 and 55,050,240. Furthermore, a numerical reduction by homotopy results in approx. 1,500,000 and 800,000 solutions for each case. The solution sets we approximated during these numerical reductions can be used as starting points with the homotopy solver Bertini to solve these large systems for specific cases in a couple of hours.

Numerical reductions for the Stephenson II case were completed over 311 hours on 256×2.2GHz cores of the UCI HPC. Following solves take 2 hours on a single 64 core node. Numerical reductions for the Stephenson III case were completed over 40 hours on 512×2.6GHz cores of the San Diego Supercomputing Center. Following solves take 1 hour on a single 64 core node.

For 8 positions

Function generation is a practical and commonly studied motion requirement. Function generation refers to the coordination of input and output joint parameters of a mechanism. For example, in the figure of the four-bar below, function generation involves the coordination of the angles φ and ψ.

The addition of two more bars provides for much more exotic and accurate functions. The six-bars that are most useful for function generation are Watt II, Stephenson II, and Stephenson III.

Six-bars are capable of producing 11 precision positions. The numerical complexity of these systems requires more study. A simpler approach is design for 8 precision positions. The design equations for these systems are highly nonlinear but tractable, which allows for the generation of many candidate designs. For example, when synthesizing for the parabola and ψ=φ2/90°, the results yielded 86, 19, and 73 defect-free linkages for the Watt II, Stephenson II, and Stephenson III designs.

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

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)

5-SS Steering Mechanism

A scaled prototype was built in order to demonstrate the kinematics of a novel steering linkage design. The design adapts the track, wheelbase, camber angles, and wheel height in order to improve the turning radius, body roll, and straightline stability of the vehicle without incurring any trade-offs.

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.