The kinematic challenge of this project was deriving a set of 3D vector loop equations to describe the motion of the slider along the helical guide. As shown in the diagram below, the motion of the slider is defined by two counteracting movements: the clockwise procession of the slider along the helix and the counterclockwise rotation of the helix. The counterclockwise rotation of the helix is because the slider is trying to leave the y-z plane as it processes along the helix; however, it is kept in plane by link R4. This means that for the slider to move along the helix, the helix itself has to rotate.
...
...
Figure 1. Diagram of Mechanism Figure 2. MATLAB Animation of Mechanism
Below is a brief summary of the derivation of the vector loop equations for our mechanism. Note that c0 is the distance form the elbow to the back end of the helix and t is the angular displacement of the slider along the helix.
After setting up our vector loop, we had a three dimensional vector loop equation. We used MATLAB’s symbolic toolbox to solve for the th1 and c variables in terms of th2. The kinematics analysis was very valuable here, because we were able to explore the feasible range of dimensions for this mechanism. These results were also useful for analyzing the position, velocity, and acceleration profiles of our mechanism.
In order to perform a reasonable motion study of our mechanism, we needed data from a normal human elbow bend. We searched around the literature and found a study on human hand motion [1]. The study had recorded the position, velocity, and acceleration data for a normal human hand raise over a 0.6 second period. Based on the length of the arm in our model, we converted this data into angular positions, velocities, and accelerations of an elbow joint, theta 2. These values were then fed into our earlier derived expressions to determine the motion profiles of link R4's rotation (theta 1) and the slider's sliding (c).
Figure 3. Joint Motion Profiles Through a Normal, Human, Elbow-Bend
...
In addition to deriving the vector loop equations for our slider, we also derived the expression for the Coriolis acceleration of the slider.
This was used to plot the X,Y, and Z projections of the Coriolis acceleration for this range of motion. Note that the X component of this acceleration is 0.
Figure 4. Coriolis Acceleration Profiles Through a Normal, Human, Elbow-Bend
...
Figure 1. Mechanism Diagram
...