Rensselaer Polytechnic Institute
To: Professor John T Wen.
From: Team 2: Adjima Mareira, Teresa Bernardi, Brian Lewis, Matthew Rosmarin
Subject: Control System Design – Conceptual Design
Date: 1/31/2006

Executive Summary

The goal of the proposed project is to create a tilt-tilt control system that will balance an inverted pendulum in 3D space. The inverted pendulum’sbalancing arm will be attached to another arm by a universal joint. It is this second arm that will be actuated. Each axis of the actuated arm will be driven by a motor. Optical encoders will be mounted to sense the rotation of both bodies.

The system will be capable of swinging the balancing arm up from a resting position. The design will allow for perturbations to the system. If this happens, the inverted pendulum will be able to compensate and rebalance itself. The best available pan-tilt system for implementation with the proposed system is the ball-on-plate system. Of the seven available mechanisms, the ball-on-plate is the only one that is tilt-tilt instead of pan-tilt. Since the system is already in the desired configuration, no major modifications will be necessary.

In order for the system to be developed, additional hardware must be acquired. The pendulum arms need be manufactured, as well as the specialized joints to allow for the desired range of motion. In addition, two additional sensors, preferably encoders, need be obtained to measure the angles of the balancing arm relative to the actuated arm.

The initial step to designing a stable system is to begin with a simplified version of the system; a two dimensional inverted pendulum. Initially the system will function with one axis locked. Once this system has been mastered, the second axis will be unlocked. The ultimate goal is to have the system be able to balance stick-on-stick in 3D.

Specifications:

  • Range of Motion:
  • Both tilt axes will require at least 90 degrees of movement.
  • Speed Requirement:
  • Ultimately, the speed of the system dictates the ability to balance a lighter/shorter object at the end the stick.
  • Accuracy:
  • The accuracy of the system will be paramount because of the limited range of motion. Any significant error will prohibit balancing.
  • Payload:
  • The Payload of the system will be variable because it has significant impact on how easy to control the system is. We will size the payload from the results of our simulations.
  • Noise Tolerance:
  • The noise tolerance will depend on the components used. The use of slip-rings may be a significant source of noise in a fast moving system.
  • Cost:
  • Raw Material for stick and manufacturing of the joints: $30
  • Rotational Fasteners/ bearings $30
  • Encoders (cost relative to accuracy) $20 – $40 each x 2

Design Constraints and past system Selections

To achieve our goal we will need a tilt – tilt system. Ideally, the current Ball on Plate system would be the most optimal system. It is already in the configuration to perform the action we wish to achieve. The only modifications needed would be on the payload area.

Figure 1 Ball on Plate System

Several of the other systems could be used, but they would require various amounts of modification to convert from “pan – tilt” to “tilt – tilt”. Though many are feasible, we do not wish to choose another specific system without first inspecting it first.

Once we have a viable “tilt – tilt” system, we can construct our payload. The specifications (lengths, weight) of the payload will be selected from our simulation results. We will size the payload to insure that it will be controllable with the current hardware.

Design Approach:

Figure 2 Engineering System Investigation

Because of the inherent instability of the system, we will first start with a simpler system. By first locking down one of the axes of rotationof the balancing arm, we can represent the system in 2D.

Figure 3Structure of a Pendubot

Figure 4“Pendubot-Like” System

This system is very similar to a system know as a “Pendubot”. The Pendubot is a very well documented system. There are literally hundreds of papers written up on it implementing many different modeling and control approaches. Below are some examples of Pendubotsright here at RPI.


Figure 5 K. Craig /
Figure 6J. Wen

The main difference between our 2D system and a normal Pendubot is that our system will have one equilibrium point instead of two (due to the restricted movement). By first mastering the Pendubot-like system, we will be in a much better position to work on the full 3D system.

Modeling:

Modeling of the constrainedPendubot-like System will be relatively straight forward. The dynamics are relatively simple and it is already a well documented system. Modeling of the full 3D system will not be so simple. We have not found any similar systems as of yet, and the dynamics will be quite complex. Because it is so complex, we will run simulations in three ways.

  1. We will use the NI Labview Simulation Toolkit to Simulate the Differential Equations that we derived mathematically.
  1. We will use the Simulink SimMechanics Toolkit to Simulate the Mechanical system.
  1. We will use also MSC.visualNastran to Simulate the Mechanical System.

All three of these options will allow us to also simulate our control system in conjunction with the Mechanical system.

Control System:

The Control design for this system will use a linearized version of our model. We will treat the two axes of rotation as decoupled and design each one independently. We will then modify the control system and/or the hardware to reduce the coupling effects of the two axes of the full 3D system.

Control Implementation:

Prototyping of the control system will be done primarily using the National Instruments compactRIO system. Current control of the motors as well as the interface with any sensors will be dictated by the characteristics of the system.Additionally, we hope to create an embedded self-contained version of the control system if time permits.

Challenges:

The main challenge for this project is that all inverted pendulums are inherently unstable in an open loop system. In the design of this system, the objective is not about sending in a step response, and improving the response by designing a better control. The objective is to create a feedback system that will force the system to be stable and robust. This ultimate goal is the challenge.

The initial challenge will be to create a two dimensional system by fixing one of the axis. The final challenge will be the transition to the three dimensional system Creating a non-linear model to characterize the dynamics of the three dimensional inverted pendulum will be one of the greatest challenges in this transition between the two models.

Conclusion:

Taking an unstable system and creating a control system to make it stable is a tremendous undertaking. The challenge of this should prove the end result to be a very rewarding experience. As our final project of our undergraduate education at Rensselaer we are looking forward to this challenge, and plan on gaining significant knowledge in dynamic systems and control design.

References:

Yi J., Zhao D., Swing Up Pendubot with a GA-tuned Bang-bang Controller, Proceedings of the 2003 IEEE, Intemational Conference on Robotics,Jntelligent Systems and Signal Processing, October 2003

Hurst J., Mechatronics System Case Study: Link on Link Pendulum—The Pendubot, September 2003

Team Contributions:

All members contributed to the brainstorming of ideas at two group meetings on January 28th and 30th

The paper was typed up equally by all members. Matt Rosmarin and Brian Lewis worked on layout and proofing.

Adjima Moreira / Teresa Bernardi
Brian Lewis / Matthew Rosmarin