Analysis of the trigger mechanism began by construct an equation that represented the system at each possible instant during its motion. Vector representation was the best method for this because the system was 3-dimensional. For the vector loop equation below to accurately describe the system, a consistent definition of geometry must be set by carefully chosen axes.
Figure 2. Trombone F-Attachment
To simplify the system, the axes were defined in a way that made the input link easy to describe. The x-axis is defined as the axis about which the input link rotates, the x-z plane being the plane that contains both the x-axis and the joint between the input link (Link A) and the middle link (Link B). The y-axis was then perpendicular to the x-z plane. By setting this relationship, the vector to describe Link A was:
where 2.3 and 4.9 are measurements in centimeters. Within this set of axes, the "ground" link (Link G, the straight line from the origin of Link A to the end of link C) was measured to be:
The next challenge in describing the system was characterizing the plane in which Link C traveled. Link C traveled in a plane 18º about the y-axis offset from the x-y plane, rotating about an axis which was the same amount offset from the z-axis. Defining this relationship between the planes of motion was critical because it provided the final relationship needed to solve the following system of equations (generated from the vector loop equation). From these equations, the 'solve' function in MATLAB was used to determine the vectors for all links at each possible position as a function of the input angle, Θi, across its measured range [0º, 14.32º].
