Penguin Kinematic Analysis

To model the radius of the cam as a function of its angular position, five points on the cam were measured, starting with the bottom of the drop-off and continuing each 90 degrees until reaching the top of the drop-off. Then, cubic regression analysis was performed to obtain a cubic function representing its radius as it changes over that interval. The reference position for the body of the penguin was taken at the minimum radius of the cam.

To calculate velocity and acceleration of the vertical position of the body, the position function was differentiated twice. Angular velocity was estimated at a constant rate of 1 revolution per second counterclockwise (the physical mechanism's cam is flipped horizontally from the image shown above).

Geometrical analysis was used to determine the relationship between the position of the penguin body and the angle of the wings. As the horizontal distance from the base of the wings to the slots stayed constant, the angle could be calculated using trigonometry. From this point, the angular velocity and acceleration of the wings was directly proportional to the linear velocity and acceleration by the inverse of the same horizontal distance.

To find the vertical position of the wingtips, the vertical component of the wings' length was subtracted from the height of the wing bases using the cosine of the wings’ angle. Then, the linear velocity and acceleration of the wingtips were calculated by multiplying the corresponding angular quantities by the constant length of the wings.