CDPMs

In the summer of 2017, I submitted my thesis and presented my findings. The work was challenging but very rewarding. My thesis involved reducing unwanted vibration in the motion of Cable Driven Parallel Manipulators (CDPMs) or more commonly known as “Cable Robots”.

⊕A Cable Robot

A parallel robot is a type of mechanism whose base is connected to the end-effector ⊕An end-effector is the device at the end of the robot’s arm through independently controlled serial chains. This is different from a serial robot such as a Fanuc. A serial robot has its’ end-effector connected to the base through links and motorized joints.

⊕Serial Robot from Kitmondo PPM

Cable robots are a special type of parallel manipulator and offer interesting advantages over serial robots. A big advantage is that a cable robot’s workspace is limited only to your cable length and motor power. However, with these large workspaces cable dynamics become very complicated. Additionally, cables cannot push limiting our control of the end-effector. This distinction breaks CDPMs into two camps: fully constrained and under-constrained. A formal definition was created¹ which stipulates for a CDPM to be fully constrained the number of cables has to be more than the degrees of freedom.

⊕Fully-Constrained on left and Under-Constrained on the right

I created two models for the under-constrained robot. The first has two cables connected to a pendulum end-effector at point P.

⊕Model 1, Pendulum

The “big picture” was to move the pendulum across the workspace while limiting the sway of the pendulum. This was accomplished using the command shaping control known as input shaping. If you have a reasonable estimation of the natural frequency you want to suppress, input shaping can reduce the vibrations experienced. The problem with input shaping for CDPMs is the changing natural frequency and damping ratios as the robot moves. For example, below is a heatmap of the low mode frequency of a 3 meter long pendulum for several thousand points across a 20 meter wide workspace. The frequency does not change much due to the dominance of the pendulum motion for this mode. For a 3 meter pendulum the natural frequency in Hertz can be found using the formula \(\omega = 2\pi \sqrt{\frac{gravity}{length}}\) which is 0.29Hz.

⊕Low Mode Frequency of Model 1. This is a front view of the workspace traversed by the Model.

The cables become more horizontal as the robot moves upwards. This allows the pendulum sway to dominate and as shown on the heatmap the values approach 0.29Hz. Moving down, the cables will sway more contributing to the lower frequencies towards the bottom⊕There’s no reason why the heatmap has two colors, that was my bad..

The nature of under-constrained robots present another interesting problem. In order to determine the pose or orientation of the robot we must use geometry and force-equilibrium equations. This was counterintuitive and strange to consider.

My work involved developing input shaping control for a two degree of freedom CDPM. Input shaping is a fascinating command shaping technique which I’ll cover in another article. Basically, input shaping alters the commands given to the robot to reduce vibration without feedback. I started with a simple CDPM where the device

… to be continued