# Four-Bar Linkage Optimization

This example shows a four-bar linkage modeled in Simscape Multibody that is optimized using MATLAB.

Mechanical designers often wish to design a four-bar linkage that will enable an end effector to follow a certain path. The lengths of the links and the position of the end effector influence the trajectory of the end effector in a complex kinematic relationship. Optimization algorithms can be used to tune those lengths to achieve the desired motion.

In this example, a parameter sweep is performed to see which trajectories are possible when varying a subset of the lengths. Then those lengths are tuned using MATLAB optimization algorithms until the resulting trajectory is within tolerances of the desired trajectory.

## Contents

## Model

## Simulation Results from Simscape Logging

The plot below shows the path of a pointer on the end of a four-bar linkage. Varying the lengths of the bars will change the trajectory of this point.

## Results from Parameter Sweep

Four-bar linkages can be grouped into various cases based on the lengths of their links. See http://en.wikipedia.org/wiki/File:Linkage_four_bar_fixed.svg The resulting trajectories vary quite widely. We will limit the scope of our problem to a crank-rocker mechanism. This means:

- Link a (driven link) is the shortest link (a <= min(b,c,d))
- Link b (connecting link) is the longest link (b >= max(a,c,d))
- Link a must be able to rotate 180 degrees (a+b <= c+d)

## Results from Optimization

Adhering to the same conditions as in the parameter sweep, optimization algorithms are used to find the lengths of Bar A and Bar B that permit the point on the four-bar linkage to follow the desired trajectory. Note that the trajectories are translated so that the minimum x and y values of the trajectories are 0. This makes visual inspection of the curves slightly easier.

Iter Func-count f(x) MeshSize Method 0 1 0.383738 0.215 1 1 0.383738 0.05375 Refine Mesh 2 2 0.00661575 0.1075 Successful Poll 3 2 0.00661575 0.02687 Refine Mesh 4 5 0.00272637 0.05375 Successful Poll 5 9 0.000112725 0.1075 Successful Poll 6 11 0.000112725 0.02687 Refine Mesh 7 15 2.05831e-05 0.05375 Successful Poll 8 19 2.05831e-05 0.01344 Refine Mesh 9 23 1.67366e-06 0.02687 Successful Poll 10 27 1.67366e-06 0.006719 Refine Mesh 11 31 1.47636e-07 0.01344 Successful Poll 12 35 1.47636e-07 0.003359 Refine Mesh 13 39 1.47636e-07 0.0008398 Refine Mesh Optimization terminated: mesh size less than options.MeshTolerance. Elapsed Time = 50.5413 Link Initial Final ____ ________ ________ 'a' '0.1800' '0.1195' 'b' '0.2500' '0.2651'