...
| Latex formatting |
|---|
After testing by opening bags by hand, we determined that the force exerted on the bag by the $L_3$ linkage needed to be twice of that required to pull the bag under normal conditions due to the clamping force required to prevent slip. In order to help increase the friction of the $L_3$ linkage (and thereby reduce the amount of clamping force required), a rubber strip was applied to the inside of the wooden box’s hook mechanism. Additionally, gearing for the input motor with a 1:7 ratio was added to ensure the motor would be able to apply enough torque for the given additional friction. After these informal tests, we shifted our views to a virtual MATLAB and SolidWorks simulation, where we performed the motion analysis for several significant parameters in the mechanism, mostly in regards to $\theta_2$, the angle of the crank gear. |
A complete document containing all of the MATLAB code and the SolidWorks assembly can be found within the Appendix section of this project page.
Figure #3.2: Diagrams showcasing a counter-clockwise rotation of the machine, demonstrating how the mechanism pulls the slider bar backwards, in addition to defining many of the significant parameters used during the MATLAB analysis
...
Figure #3.3: Complete compilation of all MATLAB motion analysis graphs, featuring θ3, θp, d, b, ω3, ddot, and bdot as a function of θ2, in addition to the position of point P (the end of the claw) in relation to O2
Due to the nature of our machine having two degrees of freedom rather than simply just having one, the mechanism's full range linkage operation could not be rendered in a tool such as Planar Mechanism Kinematic Simulator. Instead, the linkage operation was animated within SolidWorks using the Motion Analysis add-on, as seen in the annotated video below:
...
As seen in the video provided in Figure #3.4, the mechanism travels between four different parts over the course of one single cycle. Part A being the beginning of the counter-clockwise rotation with the motor pin starting at a 0° angle until the slider above the motor gear reaches its leftmost position in order to have the "claw" part of the arm clamp down on the bag. From here, Part B begins, consisting of the slider arm pulling backwards until the motor pin reaches an 180° angle. Part C and Part D are close to being inverses of Part A and Part B respectively, with Part C involving the motor gear slider moving back to its rightmost position in order to release the "claw" part of the arm from its clamping position and Part D moving the slider arm back forward. After Part D, Part A begins once again, repeating the entire process. These four parts are annotated within all eight graphs in Figure #3.3. More in depth motion analysis calculations can be seen below.
Analysis - In Detail
The following section details our motion analysis of the system. The dimensions used in this analysis is shown below in Figure #3.5
Figure #3.5: Diagram showing the basic dimensions of the mechanism
The system's motion is divided up into 4 parts.
Latex formatting Part A: The internal slider is the only linearly moving component and effectively lengthens the connecting rod. The slider is stationary. This opens the gripper until $b=s+l_{min}$ .Latex formatting Part B: The internal slider bottoms out at its maximum, and the connecting rod’s length becomes constant at $b=s+l_{min}$. The mechanism acts as a slider crank. This pulls back the gripper until $d=s+l_{min}-a$ when the crank is at $180\degree$.
Latex formatting Part C: The internal slider is the only linearly moving component and effectively shortens the connecting rod. The slider is stationary. This closes the gripper until $b=l_{min}$ .Latex formatting Part D: The internal slider bottoms out at its minimum, and the connecting rod’s length becomes constant at $b=l_{min}$. The mechanism acts as a slider crank. This pushes the gripper forward until $b$d=a+l_{min}$ when the crank is at $0\degree$.
Thus, for Parts A and C, length b changes, and length d is constant. For Parts B and D, the mechanism is a simple slider crank with length b constant and length d changing. Thus, Parts B and D can be analyzed using the slider crank motion equations while Parts A and C can be solved using vector loop analysis.
Boundary Conditions
The boundary conditions governing the transitions between Parts A and B and between Parts C and D are based on the geometry of the mechanism when the internal slider bottoms out. For this, the Law of Cosines can be used.
For the boundary between Parts A and B:
Figure #3.6: Diagram showing the basic dimensions of the mechanism at the boundary between Parts A and B
| Latex formatting |
|---|
$$\theta_2=\arccos(\frac{a^2+(a+l_{min})^2-(s+l_{min})^2}{2a(a+l_{min})})$$ |
For the boundary between Parts C and D:
Figure #3.7: Diagram showing the basic dimensions of the mechanism at the boundary between Parts C and D
| Latex formatting |
|---|
$$\theta_2=360\degree-\arccos(\frac{a^2+(s+l_{min}-a)^2-(l_{min})^2}{2a(s+l_{min}-a)})$$
Note that the boundaries between Parts D and A and between Parts B and C are 0 degrees and 180 degrees respectively since the joints $O_2$ and $O_4$ are collinear with the global X axis. |
Vector Loop Analysis
Parts A and
...
C
For Parts A and C, the vector loop analyses are the same, but the lengths are different. So, to more easily solve them, a generic system can be solved once and the dimensions for Parts A and C applied to it.
Figure #3.8: Diagram showing the vector loop used for analyzing Parts A and C
Position Analysis
| Latex formatting |
|---|
$$\vec{R_2}-\vec{R_3}-\vec{R_1}=0$$
$$ae^{j\theta_2}-be^{j\theta_3}-de^{j\theta_1}=0$$
$$a(\cos{\theta_2}-j\sin{\theta_2})-b(\cos{\theta_3}-j\sin{\theta_3})-d=0$$
$$\begin{bmatrix}
b\cos{\theta_3}=a\cos{\theta_2}-d \\
b\sin{\theta_3}=a\sin{\theta_2}
\end{bmatrix}$$
$$\begin{bmatrix}
\theta_3=\arctan(\frac{a\sin{\theta_2}}{a\cos{\theta_2}-d})\\
b=\sqrt{(a\cos{\theta_2}-d)^2+(a\sin{\theta_2})^2}
\end{bmatrix}$$ |
...