3) Kinematic Analysis and Synthesis

Kinematic Analysis:

        The figure below shows our drawing used for the kinematic analysis. The figure contains the variables we used along with their definitions. Θ_2, Θ₃, Θ₄, c, r, and d all change with respect to time. The mechanism’s degrees of freedom was calculated with the equation:

M = 3*L - 3 - 2*J_1 - J_2 (1)

where L is the number of linkages, J_1 is the number of full joints, and J_2 is the number of half joints. In this mechanism we have four linkages, three full joints (O2, A, and O3), and two half joints (O1, and D). Using equation one and the values given we have a degree of freedom of one. 

Figure 3.1. Overall Drawings for the Kinematic Analysis

        To perform the analysis we used two separate vector loops to solve for all of the unknown variables. Loop one contains the points O2, A, B, and O1. Vector loop two contains the points C, D, E, O3, and O1. Using these two vector loops, shown in figures 3.2 and 3.3, we were able to solve for all the unknowns for position, velocity, and acceleration. We performed hand calculations to set up our two equations and two unknowns in each case. From there we turned to MATLAB to symbolically solve those equations for our desired unknowns. Then we turned those symbolic equations into MATLAB functions we could call in our simulation. After that we created our point position vectors for A, B, C, D, and E with O2 being at the origin. By taking the time derivatives of the position vectors we also calculated the magnitude of velocity and acceleration over time for each point. Then to end our hand calculations we grouped together the terms for the present Coriolis acceleration.

Figure 3.2. Vector Loop 1

Figure 3.3. Vector Loop 2

        For our simulation we used a for loop to go through all of the possible values of Θ₂, from 0 to two pi, and calculating the resulting Θ₃, c, ω₃, c_dot, α₃, c_Ddot, Θ₄, r, d, ω₄, d_dot, α₄, and d_Ddot. With those resulting values we could then calculate the position, magnitude of velocity, and magnitude of acceleration for each point of interest. All of the MATLAB functions, MATLAB code, and the resulting figures are located in the appendix. Figure 3.4 is the hand calculations we performed in the kinematic analysis. The figures past 3.4 were created in MATLAB and are some of the important figures showing we met our goals. The single PDF of the hand calculations is also in the appendix. 

Figure 3.4. Kinematic Analysis

Figure 3.5 Path of Each Point of Interest

Figure 3.6. GIF of Simulation

Figure 3.7 Magnitude of Velocity for Each Point of Interest

Figure 3.8 Magnitude of Acceleration for Each Point of Interest

Figure 3.9 Magnitude of Coriolis Acceleration for Each Rotating Slot

        From our plots and simulation we can see that we achieved some desired paths that we can use to collect and pour water. We also have a non-constant velocity profile on link three which is used in the collecting and tipping process. When the water would be collected and poured it is a smooth and constant velocity. After the pour we have a rapid acceleration and the item quickly returns back to collect the water. 

        One challenging part of the analysis was the variable r(t). One reason was because of its dependence on c(t) and the length from point B to O1. Another reason was because depending on theta2, c(t) and r(t) could overlap each other. While coding the analysis in MATLAB we needed to make sure r(t) was correctly handled because without it our mechanism would not simulate properly. A second challenge we encountered was when solving for Θ₃, c(t), Θ₄, and d(t) since we had two possible solutions to choose from. This meant we had to decide which one was the correct solution when simulating our mechanism. We used trial and error to determine which solutions were the correct ones.