# Kinematic Analysis

Using forward and inverse kinematics (as demonstrated in our Design Review II Report), it is possible to determine the resolution and workspace of a delta robot. These parameters depend entirely on four variables: upper arm length, lower arm length, end effector radius, and motor base radius. By varying each of these parameters while holding the others constant, and observing the effects on the resolution and workspace, it is possible to determine an optimum delta robot geometry. The tables shown below demonstrate this procedure.        For our purposes, we valued workspace radius over resolution, as even the smallest calculated resolutions would be sufficient for our applications. Hence, we aimed to minimize the size of the motor radius, and optimize the end effector radius based on weight. However, the results for the arm lengths are polytonic, suggesting an optimum value at some intermediate combination of arm lengths. In order to find this point, we plotted the variation of each arm length simueltaneously as a surface. # Dynamic Analysis

A dynamic analysis was performed on our robot using the optimized geometry determined from kinematics as described above in order to determine the required torques for a particular end effector acceleration rate. This analysis was performed manually, using the principle of virtual work and the Jacobian matrix, as well as through a simulation of our CAD. The results from each method agreed very well with one another, and thus we felt confident selecting motors based on our results. A plot of the torque requirements on all three motors at an acceleration of 10 Gs is shown below. Using the same dynamic CREO Model, we also measured the joint stresses at the elbows for the same acceleration to ensure that our components would have sufficient strength and durability, remaining rigid during operation. Based on the same joint stresses above, we performed a simple calculation to ensure our carbon fiber tubes would have sufficient strength, and found that they would in fact have a factor of safety approaching 50. Finally, we performed a simple force balance calculation to ensure that our PIAB vacuum pump would be able to provide sufficient suction force to hold a ball at the accelerations and velocity we were expecting to reach. We considered the inertial force, frictional force, drag force, vacuum force, and the force of gravity in this analysis, and found that the PIAB pump would perform admirably.  # PID Tuning

Our delta robot is driven by a Delta Tau Geo Brick Drive controller that using control structure shown in the control diagram below. Note that at this point, we are only using PID control, but we aim to integrate velocity feed forward, acceleration feed forward, and friction feed forward gains. We are also considering the use of a notch filter and a deadband zone to improve stability, and reduce high frequency oscillations. Through careful tuning of the PID loop, we were able to gradually improve the response of our robot, reducing the settling time, decreasing the oscillations, and working towards a fast, stable control law. To do so, we used the Ziegler-Nicols method, first adding proportional gain to increase the speed of the response, then adding integral gain to eliminate steady state error, and finally adding derivative gain to reduce oscillations. The images below demonstrate our progress.      ## Delta Force

Columbia University - Fu Foundation School of Engineering and Applied Science - Department of Mechanical Engineering - Senior Design 2015