Team 26 - Position and Velocity Kinematic Analysis

The mechanism analysis was split into two parts - the analysis of the bottom linkage; driven by the sliding joint, and the analysis of the top linkage, which includes the gripper. 

The bottom portion of the mechanism can be modeled as a slider-crank which receives its power from the linear motion of the slider rather than the rotary motion of the crank. The angles and angular velocities of the joint can be derived as functions of slider position d and its derivative, the sliding velocity. Both of these are directly proportional to the motor output and the pitch of the screw. 

The position analysis solves for the angles of the proximal finger (a, shown in cyan in the CAD) and the secondary slider input arm (b, shown in yellow in the CAD).

The velocity analysis solves for the angular velocities of the same two links as above.

The top portion of the mechanism can be modeled as a closed loop composed of one ground link (green), a grounded sliding joint, and two links. The angles and angular velocities of the joints can be derived as a function of the input angle, from the cyan link, which is solved for in the analysis of the bottom linkage. 

The position analysis solves for the linear position of the slider on a (red in the CAD) and the angular position of the distal finger (a, shown as red in the CAD). The input link here is c, which forms a triangular link with the output link a of the previous sub-mechanism. Using the known angular difference between these two effective links on the same body, we can provide the correct inputs to these equations.

Because of their complexity, these equations were solved using a symbolic solver to yield the following solutions:

The velocity analysis was then performed by differentiating the same vector loop, this time solving for the slider linear velocity and distal finger angular velocity.

Shown below are plots of the angular positions and velocities of the two gripper finger links (color-coded to match the CAD figures) for the full range of input slider displacements where 0mm of input displacement corresponds to the gripper in its maximally closed grasp state and roughly 80mm of slider displacement corresponds to the gripper in its most open, widest stance landing-gear state. 

A video showing the mechanism moving through its full range of motion is shown below. A qualitative comparison between the above plots and the mechanism's motion in the video confirms the accuracy of the derived equations.