III. Kinematic Analysis and Synthesis - CZ
Mechanism Mobility
Position, Velocity and Acceleration Analysis
The Klann linkage that I decided to use in this robot project consists of 6 links and is a variation on the Stephenson’s 6-bar linkage. In order to solve for the position, velocity, and acceleration of the 6-bar linkage, I split the linkage into a 4-bar and a 5-bar analysis. The 4-bar consisted of the input crank (R2), arm (R3), and the lower rocker (R4). This 4-bar system has two unknowns, theta3 and theta4, for any given input angle. The 5-bar consists of the input crank (R2), arm (R3), leg (R5), and upper rocker (R6). While the 5-bar has three unknowns, theta3, theta5 and theta6, I can use the theta3 calculated in the 4-bar in the 5-bar analysis since it is a shared linkage. With theta3 known, the 5-bar has only two unknowns and can be solved.
Figure 7 - Vector Loop Equations for Linkage
I performed position, velocity, and acceleration analysis on the 4-bar as shown in the calculations below. From that analysis, I found theta3, theta4, omega3, omega4, alpha3, and alpha4 in terms of the system constants.
I copied the equations into Matlab and plotted the arm angle, velocity, and acceleration versus the crank angle.
Figure 8 - Linkage 3 Angle, Velocity, and Acceleration vs Crank Angle
I approached the 5-bar analysis in the same way as the 4-bar but ran into some significant issues calculating the position of the 5-bar analysis. Because of the additional bar, the same trigonometric method used to solve for the angles in the 4-bar cannot be used. I attempted to remove the theta6 term but still ended up with theta6 in the equation for theta5. I attempted to plug the system of equations into Wolfram Alpha but even that service was unable to simplify or solve for the system of equations. After hours on non-productivity, I decided to move on and finish the calculations for the velocity and acceleration of the 5-bar, which I had no issues with.
Without working position analysis for theta5 and theta6, I was unable to solve for the velocity and acceleration of the robot leg. Therefore, to fulfill the graphing and animation requirements for the project, I recreated the mechanism in the PMKS software. The PMKS model shows that the leg walks exactly as intended.
I exported the data from the PMKS software and plotted the leg position, velocity, and acceleration on the x and y axis for a full rotation of the input cranks. As you can see, the velocity and acceleration graphs are very complicated due to the 6-bar mechanism.
Figure 9 - Leg Mechanism Modeled in PMKS
Figure 10 - Plots of Leg Position, Velocity, and Acceleration from PMKS
Animation of linkage operation from PMKS
Green Line = Position of leg
Red Line = Velocity of leg
Yellow Line = Acceleration of leg
Animation of linkage operation from Solidworks
Next → IV. Manufacturing and Assembly