13.3 - Kinematic Analysis

Proposed Path:

Click the figure below to see animation.

Mobility:

M = 3*(L-1) - 2*J1 -J2 where L = 5, J1 = 5, J2 = 0

M = 3*(5-1) - 2*5 - 0 = 2 

Therefore we have 2 degrees of freedom

Position Analysis:

Below are the equations that lead to finding the unknown thetas; numbers 3 and 4. After all the theta values were found, we need to find the position of point B (also seen as P above). 

Since we knew the desired path, we knew what the position plot would look like for our data. 

Velocity Analysis:

Let's start off with our Vector Loop:

We performed a velocity analysis to determine what the velocity at point B is. From the vector loop above we solved for omega 3 and omega 4 values. From the newly calculated angular velocities around each point we then solved for the velocities of point A, B relative to A, and C. Using our relative velocity formula we then calculated for the velocity at point B. With our velocity of point B in hand we could now plot it versus our theta 2. Below are the equations that were solved and used to create the plot. 

The speed at point B was plotted against the range of theta 2 that is expected to occur within our mechanism when it is in action at an angular velocity of 10 degrees/sec.

Swept Area Analysis:

In order to ensure efficiency with our robot we wanted to determine the total area that would be cleaned by the moving squeegee arm. Below is a drawing of the expected area that is to be cleaned.

To calculate this we have to determine the dimensions of the ellipse that is formed around our figure 8 path. To do this we used measuring tape to determine the major and minor axis lengths with our final prototype. We moved our gears to simulate the path and made markings at the edges of the ellipse. From here we were able to easily take our ellipse dimensions. 

Ellipse Dimensions:

Now we have to consider the extra area that is cleaned by our extended squeegee arm. For this we can just add the squeegee arm length to the major and minor axis lengths. With a squeegee arm length of 4 inches we can add that to our the axes lengths.

Ellipse Dimensions with Squeegee arm

Now we just have to use the area of an ellipse formula to solve for our area. Here a = major axis and b = minor axis. 

Solving for the area, our robot should theoretically clean 345.17 in² of a given surface.