3. Kinematic Analysis (GC)

Figure 8 motion:

Desired motion showcased in 'Linkage' software and checked with MATLAB:

Mobility:

M = 3(4-1) - 2(4) = 1, L = 4; J1 = 4

Position Analysis:

To perform kinematic analysis, the above model was mirrored and simplified to the following:

The unknown angles theta 3 and theta 4 were found by solving the system as a 4-bar linkage mechanism. Once all theta values were found, the position of point P (point p being the position of the cleaning sponge) was determined using the following equations:

After finding the position of point P, a coordinate transformation was applied to the position of P to account for theta 1 being nonzero. These equations were derived from the figure above and shown below:

A MATLAB script (found in appendix) was then written to repeat this process for every input angle (theta 2). The full range of motion of theta 2 is 0 to 360 degrees. The MATLAB script generates the plot below which agrees with the path generated by the ‘Linkage’ software. The path was then scaled to the necessary size (i.e. link lengths were adjusted such that the motion covered the glasses area).

Velocity Analysis:

Vector Loop:

Velocity analysis was performed to determine the velocity at point P. After finding the angular velocities of omega 3 and 4 using the vector loop equations, the velocities in the x and y direction were found by taking the derivative of Rx and Ry and plugging them into the equations below: 

The speed of point P was then plotted against the full range of motion of theta 2 with an angular velocity of 2pi rad/s: