Constrained RPR Chains

Six-bar mechanisms that include a prismatic joint can be synthesized as constrained RPR chains. These single degree of freedom constrained RPR chains are capable of reaching five task positions. The synthesis process begins by first specifying a serial, open, 3 DOF chain. The inverse kinematics are solved to find five positions of the RPR that correspond to five specified task positions.

Then two RR constraints are solved for as shown in the figures. The solution to the second RR constraint depends on the solution to the first. Various combinations of RR constraints provide for new design candidates to be analyzed.

Below is a bolt insertion linkage that was designed using this method.

A portion of this transplanting linkage was also designed using this method.

M. Plecnik and J. M. McCarthy, 2013. “Dimensional synthesis of six-bar linkage as a constrained RPR chain,” New Trends in Mechanism and Machine Science, Springer Netherlands, 273-280.

Spatial 5-SS Linkages

An algorithm has been created for the design of 5-SS linkages that reach 7 spatial task positions. 5-SS refers to the 5 spherical to spherical binary links that connect a moving platform to the fixed platform of these single degree of freedom mechanisms. Attached to that moving platform is an end effector frame. The goal of these motion generators is to move that end effector frame in a particular manner.

The algorithm consists of five main steps:

  1. Specification of task positions
  2. Generating new sets of task positions
  3. Synthesis of 5-SS mechanisms
  4. Analysis of mechanisms
  5. Evaluation of mechanism configurations

At the heart of the synthesis routine is the algebraic solution to the SS dyad equations. This system of 6 bilinear equations in 6 unknowns is solved with a generalized eigenvalue method. Formulation of these equations was introduced by Chen and Roth and an algebraic solution was given by Innocenti.

As an example design, the algorithm has been applied to the design of a steering linkage. The motion objective is to change the track, wheelbase, camber, and wheel height in a turn, while maintaining Ackermann geometry.

P. Chen, and B. Roth, 1969. “Design equations for the finitely and infinitesimally separated position synthesis of binary links and combined link chains,” Journal of Manufacturing Science and Engineering, 91(1): 209-219.

C. Innocenti, 1995. “Polynomial solution of the spatial Burmester problem,” Journal of Mechanical Design 117(1): 64-68.

M. Plecnik and J. M. McCarthy, 2012. “Design of a 5-SS spatial steering linkage,” Proceedings of the ASME 2013 IDETC/CIE Conference, Paper No. DETC2012-71405, August 12-15, 2012, Chicago, Illinois, USA.

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.

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.

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.