4 - Kinematic Analysis

As described in Section 2, dipping an Oreo cookie is a difficult position and velocity problem. Accordingly, our primary objective we are mainly interested in the motion of the end effector and joint 6. 

Mobility Calculation

According to Grubler's equation for planar mechanisms, our mechanism has 1 degree of freedom.

Grubler's Equation: M=3(L-1)-2J₁-J₂

_{Figure 1. Mobility calculation with links (blue) and joints (red) labeled}

Kinematic Analysis

For our mechanism, we found it difficult to find a closed-form solution using the vector loop methods introduced in class. For instance, we realized that by chaining known solutions together, we would end up with a 4 bar in series with a 5 bar in series with another 4 bar. Since the five-bar solution is long and complicated, we found it difficult to implement it into code. A vector loop from scratch for the mechanism also proved to be complicated to simplify. Instead, our team used an iterative root-finding method called the Newton-Raphson method. The Newton-Raphson method algebraically solves the solution of nonlinear solutions using its derivatives in a Jacobian matrix. With an initial guess, the algorithm converges towards the solution iteratively and allows positional and velocity analysis. However, in some cases, the method may not yield a convergence, but rather diverge away from a solution. This was not an issue for our mechanism since we started with a clear set of known values of link lengths and angles of each joint through the use of Motiongen.

Positional Analysis:

The initial analysis comes from a motion analysis on Solidworks.

              Figure 2. Position profile of joint 6                           Figure 3. Position profile of end effector  

Velocity Analysis:

Figure 4. Velocity profile of the end effector

Acceleration Analysis:

Figure 5. Acceleration of end effector versus time of one input rotation

Figure 6. Mechanical Advantage of end effector

After verifying the mechanism through simulation programs such as MotionGen and SolidWorks, we performed a Newton-Raphson method analysis on MATLAB for numerical data and predicted the following. The algorithm currently does not fully accommodate whether a loop is closed or open at certain points during the cycle and therefore has an error, resulting in inaccurate analysis. We aim to adopt restrictions in the calculations to resolve this issue and be able to accurately analyze the mechanism and produce a gif of the full mechanism performing a full input cycle in the future. However, even with our current data and our functioning prototype, we can determine that the mechanical advantage is the lowest immediately after the 'dip' phase of the end effector and the final design will therefore have to be fit to perform the given load at the lowest mechanical advantage point.

Figure 8. Mechanical advantage at joint 6, generated on MATLAB

Figure 8. Full Range of Mechanism in operation