LinkageDesigner is a Mathematica application package for virtual prototyping of linkages. It is designed to analyze, synthesize, and simulate linkages with serial chain, tree, and graph structures. Using the symbolic calculation capabilities of Mathematica , it supports fully parametrized linkage definition and analysis. In LinkageDesigner , kinematic structures (such as linkages, mechanisms, etc.) are represented by kinematic graphs of the linkage, where links are vertices and joints are edges.
Key Features
Automatic generation of non-redundant loop-closing equations for single and multiple loops
Detection of non-feasible and lock-up mechanisms while defining the model
Parametrized linkage definitions, letting 2D and 3D mechanisms be treated identically
New flexible and user-extensible LinkageData data type, which stores all information associated with a mechanism
Easy copying of LinkageData objects between mechanisms
LinkageDesigner can calculate the velocity, angular velocity, acceleration, angular acceleration, and higher order derivatives of any links in closed form. The package has extensive support for visualizing and animating linkages, which can be displayed or animated in Mathematica notebooks, or exported to Dynamic Visualizer , LiveGraphics3D , or VRML97.
The package comes with electronic documentation, in both PDF and Mathematica Help Browser formats. LinkageDesigner 1.0 requires Mathematica 5.0 or higher and is available for all Mathematica platforms .
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Unified handling of 2D and 3D linkages with serial chain, tree, and graph structures
Support for parametrized linkage definitions
Handling of open-chain and graph-structured mechanisms
In the first case, the mechanism is modeled with transformation between the consecutive links. In the second case, only the minimum number of constraint equations are generated (the non-redundant equations) that are needed to model the mechanism (equivalent to calculating the position and orientation of every link). Graph based mechanisms are started as open-chain mechanisms, and they become graph based as soon as one loop closing kinematic pair is defined. At that moment constraint equations are generated and stored in the LinkageData object of the mechanisms.
The length of the $DrivingVariables record equals the mobility of the mechanism
During the mechanism definition phase the mobility of the actual mechanism is always correct. For instance, in four-bar mechanisms the mobility increases from 1 to 3 until the first three rotational joints are defined. After the fourth (this is a loop-closing kinematic pair) rotational joint is defined, the mobility of the mechanism is dropped to 1, and two constraint equations are automatically generated.
Inverse kinematic problem formulation
The inverse kinematic problem is formulated as follows: given the desired position and orientation of a tool relative to the reference coordinate frame, how do we compute the set of joint values of the mechanism to position the tool in this posture? The template equation-based solution technique was originated by Pieper and Paul, who found that the solution of the inverse kinematic equation of typical industrial robots leads to the solution of trigonometric polynomials conforming to some simple pattern. The solution of these simple template equations (sometimes called prototype equations) is known; therefore, if one can identify an equation matching the template, only the parameters need to be extracted, and the solution can be generated symbolically. The template equations can be considered as knowledge representation, which speeds up solution of the inverse kinematic problem in case of certain special linkages.
Calculation of translational velocity, angular velocity, and higher-order derivatives of any links in a closed-form linkage
Visualization and animation of linkages in Mathematica notebooks
Export to Dynamic Visualizer , LiveGraphics3D, or VRML97
Examples
Join Linkages
Very often linkages or mechanisms used in practice share the same kinematic principles. Parametrized linkages give finer control over link dimensions. But you might want to reuse simple linkages as building blocks of more-complicated mechanisms. LinkageDesigner supports this kind of "copy and paste" linkage by providing the AttacheLinkage function, which allows you to "glue" two or more linkages together to achieve more-complex mechanisms with minimal effort. AttacheLinkage is especially useful in defining mechanisms with replicate base mechanisms, like an engine. To illustrate this linkage definition technique, we will build a V-engine with four pistons.
Whitworth's Quick Return Mechanism
The Whitworth quick return mechanism converts rotary motion into reciprocating motion, but unlike the crank and slider, the forward reciprocating motion is at a different rate than the backward stroke. At the bottom of the drive arm, the peg only has to move through a few degrees to sweep the arm from left to right, but it takes the remainder of the revolution to bring the arm back. This mechanism is most commonly seen as the drive for a shaping machine.
