Table of Contents:
Introduction: ........................................................................................................................ 2
Objectives: .......................................................................................................................... 4
Plan of Action and Verification .......................................................................................... 5
Plan and Testing Procedures ........................................................................................... 5
Tolerance Analysis.......................................................................................................... 7
Design Strategy ................................................................................................................... 8
Modeling ......................................................................................................................... 9
Parameter Identification ................................................................................................ 10
Control Design .............................................................................................................. 11
Implementation and Integration .................................................................................... 12
Cost: .................................................................................................................................. 13
Schedule: ........................................................................................................................... 14
Bibliography: .................................................................................................................... 14
Appendix A: Terms........................................................................................................... 15
Appendix B: SimMechanics Modeling ............................................................................. 16
Appendix C: Schedule ...................................................................................................... 19
Table of Figures:
Figure 1: Front and Side views of a 3D Arm Driven Inverted Pendulum .......................... 2
Figure 2: Pan and Tilt System ............................................................................................. 3
Figure 3: National Instruments cRIO .................................................................................. 3
Figure 4: Plan of Action ...................................................................................................... 5
Figure 5: Design Strategy ................................................................................................... 8
Figure 6: Sim Mechanics Simulink Diagram..................................................................... 9
Figure 7: SimMechanics 3D Model ................................................................................. 10
Figure 8: SimMechanics 2D System................................................................................. 16
Figure 9: Simulink 2D System .......................................................................................... 16
Figure 10: SimMechanics 3D System............................................................................... 17
Figure 11: Simulink 3D System ........................................................................................ 18
1
Introduction:
The inverted pendulum is a popular control problem which is frequently used and studied
throughout the scientific community. The significance behind the inverted pendulum is
that it is inherently unstable. The ability to stabilize an unstable system has many useful
and interesting applications. Cutting edge military fighters and spaceships harness the
ability to stabilize unstable systems to achieve tremendous performance gains. Many
robotic applications such as biped walking robots also use similar aspects in their
operation [4].
There are many different forms of inverted pendulums. Linear, Rotary, and Spherical are
some of the more traditional implementations. A Two Dimensional Arm-Driven Inverted
Pendulum, also known as a Pendubot, is the most similar design to the proposed project
[2] [3].
The goal of this project is to design and stabilize a Three Dimensional Arm-Driven
Inverted Pendulum. The simplest way to visualize this system is to imagine a stick
balancing another stick. The lower stick, called the Actuated Arm, is driven by a motor.
The upper stick, called the Balanced Arm, is not actuated. The two arm ends are
connected with a universal joint. The picture below illustrates this system.
Figure 1: Front and Side views of a 3D Arm Driven Inverted Pendulum
2
Initially, the three dimensional pendulum will be modeled in Simulink using
SimMechanics. Then, using the control design features of Matlab, a controller to stabilize
the system can be generated. For the physical system design, the arm driven inverted
pendulum apparatus will have to be built on top of a modified pan and tilt system that
was supplied.
Figure 2: Pan and Tilt System
A National Instruments CompactRIO (Compact Reconfigurable I/O, cRIO) will be used
to realize and implement a stabilizing control system.
Figure 3: National Instruments cRIO
3
Objectives:
The primary objective of the project is to design and balance a Three Dimensional ArmDriven Inverted Pendulum. Since the size and dimensions of this system greatly influence
how easy or difficult it will be to control, four different mechanical configurations will be
created. These configurations will range from being very easy to control to being very
difficult to control. Part of the project will be to identify what mechanical characteristics
cause the system to be easy or difficult to control. Once the configurations are decided
upon, then they will have to be implemented in hardware.
The overall success of the project will be defined by how well a system configuration can
be balanced. The two areas of analyzing system performance will be first in simulation
and then in final implementation. In simulation, there will be a vast amount of flexibility
in setting up initial conditions and tests. Unfortunately, evaluations of the physical
implementation must be both realizable and repeatable. The current plan is to use torque
disturbance as a way if monitoring the performance of the physical system.
The project has two secondary objectives. The first is to implement the entire controller
on the FPGA section of the cRIO. This will require special attention to numerical issues
and filtering. The second is to add a human interface that will allow the user to act as the
control system. This will add a human vs. computer competition aspect to the project.
The scope of this project has changed somewhat from its inception. The focus is now
primarily on balancing rather than both balancing and swing up. This change was
primarily based on the availability issues associated with acquiring a slip ring.
4
Plan of Action and Verification
The figure below is an overview of the plan of action for designing, building and
analyzing the proposed Three Dimensional Arm-Driven Inverted Pendulum balancing
system. There are a total of ten steps that will be taken in order to accomplish a controlled
three dimensional arm-driven inverted pendulum. While the steps are listed in a
chronological order, there may be some overlap between stages, as well as complete
changes in order, depending on the presence of circumstances that are out of the realm of
human control. A general synopsis of each block and potential verification procedures
may be found in the subsequent section.
Plan and Testing Procedures
A model of the intended system will be created and analyzed for
feasibility before the system will be implemented. The Simulink
SimMechanics toolkit has been used to do the initial model. First a two
dimensional model was formed, and then the three dimensional system
was modeled. As a check of the accuracy of the SimMechanics
equations, modeling will also be done for the two dimensional system
by hand. If the two methods of modeling agree for the two dimensional
system, then it will be assumed that the three dimensional model
created in SimMechanics is accurate as well. The system model will
then be linearized and a state space controller will be designed. The
SimMechanics modeling has been done by Matt Rosmarin and Teresa
Bernardi. The hand calculations will be done by Brian Lewis.
The parameters of the system base will be identified before the control
design is completed. The properties of the motors and corresponding
assembly will affect the system’s response and capabilities. Specific
parameters that need to be identified are the friction, inertia, and
backlash. A Labview interface has been created in order to directly
measure the friction of the motors by Matt Rosmarin and Brian Lewis.
The inertia of the motors and assembly will be found using SolidWorks
and verified experimentally. The most likely experimental verification
of the assembly inertias is a series of pendulum tests in which one axis
of rotation is locked and a pendulum is allowed to swing freely about
the other axis. From the frequency of the oscillations, the inertia about
that axis of rotation may be determined. An analysis of the amount of
backlash also needs to be accomplished. Additionally, the physical
assembly of the base has some compliance. This compliance can be
measured by applying a torque to each of the motors separately, and
determining the amount and direction of any deflections in the
assembly. Should the compliance be established as excessive for the
system application, the assembly will be refashioned out of a more
rigid material.
Figure 4: Plan of Action
5
Each component of the balancing subsystem needs to be designed. The main components
of the subsystem are the actuated arm, balanced arm, universal joint, and end mass. The
design should take into account the center of mass of each arm and the speed necessary to
respond to perturbations in the system balance. The control design will then be adapted to
the system as each of the balancing subsystem components are sized.
The sensors and motors will also be selected and/or designed to work well with the
designed system. For example, the proposed encoders need to be of high enough
resolution to detect an offset angle that is an order of magnitude less than the specified
maximum allowable oscillation. The gearing of the motors may also have to be
redesigned to attain the required torque and speed to accurately control the system.
The construction and fabrication of the physical system will be set into motion once the
sizing of the balancing subsystem is complete. The machining will be accomplished in
one of the machine shops on-campus by several of the team members. Some of the
components will most likely be commercially available. The construction of the physical
system will be a team effort; however Brian Lewis will be the lead.
While the system is being constructed the various aspects of the interaction between the
mechanical and control systems will be tested and any problems identified. The
integration of the systems will be monitored by all team members. An additional
controller may also be added at this stage to allow for human control of the balancing
system, so that the control algorithm response may be compared to human reflexes.
Once the integration of the physical and control systems is complete, the control
algorithms may be tuned. Since the purpose of our system is to balance, the optimization
of the system will be focused on this goal. Tuning will also focus on the speed and
robustness of the overall system.
The overall performance of the system will be analyzed as to its ability to acquire and
maintain a balancing state, and compensate for slight disturbances in both the two and
three dimensional configurations. The analysis will most likely be done in tandem with
the tuning of the controls, as a bad performance review indicates the need for further
tuning. While the speed of system response will be analyzed in the system performance,
the target speed will vary between the different size systems.
If there is time, existing swing up techniques will be applied to the system so that human
initialization is not necessary. The controller design for the swing up of pendulums
already exists for a two dimensional system. For this reason, the swing up may be
restricted to two dimensions.
6
Tolerance Analysis
The components that most affect the performance of the system are the sensors. Should
the sensors that detect either the motor rotation or the rotation of the balanced arm be
inadequate, the system will continuously go unbalanced. The reaction of the system to
disturbances will also be affected in that the system will be very likely to either under- or
over-compensate due to erroneous sensor readings.
The balancing subsystem parameters will also affect the system performance. While the
controller should be designed to accommodate for small changes in the system, any large
changes will cause difficulties in controlling the balanced arm. For example, adding an
additional small mass to the tip of the balancing arm should not affect the system
performance; while adding an object of significant mass, with respect to the original
system, will cause the system to be unable to adequately compensate.
The base parameters will also have an affect on the performance of the system. The
specific parameters that will impact the performance are the friction of the motors,
backlash in the gear train, and the compliance in the assembly. Since the controller will
be linearized, these aspects of the system will not necessarily be taken into account. If
there is a significant amount of friction and backlash, then the system’s ability to balance
will be compromised.
7
Design Strategy
Physical Model
Modeling
SimMechanics
Hand Derived
Parameter Identification
Friction
Backlash
/
Compliance
Inertia
Control
Parameter Selection
Arm Lengths
Arm Weights
Linearization
Simulink
Linearization
Hand
Linearization
Control Design
Root
Locus
Design
LQR
Generation
Non Linear Simulation / Evaluation
Simulink / Labview
Implementation / Integration
Mechanical Fabrication
Arms
Universal Joint
Fixtures
Electrical
Sensors
Actuators
Human Interface
Controller
cRIO
(Onboard Controller / FPGA)
Figure 5: Design Strategy
8
Modeling
The primary method of modeling the system will be utilizing SimMechanics.
SimMechanics is an add-on for Matlab Simulink. SimMechanics allowed the team to
very quickly create a full non-linear model of a mechanical system. Below is the
Simulink model diagram the full 3D system.
Figure 6: Sim Mechanics Simulink Diagram
In addition, SimMechanics also automatically allows for 3D Visualization of a created
model.
9
Figure 7: SimMechanics 3D Model
The key advantage of using SimMechanics is that model can be quickly created, and later
the model can be edited. Large scale changes which would normally require total
reworking of a hand derived model can be implemented with a few keystrokes in
SimMechanics. To verify that SimMechanics is correct, a hand derived model of the two
dimensional system will be compare against a two dimensional version of the
SimMechanics model. Finally, because it is already part of Simulink, it will be very easy
to include the control system loop, and nonlinearities such as saturation, quantization,
fixed point calculations, loop rates, and time delays in the simulation.
Parameter Identification
The next major step in the project will be to identify all the parameters of the system
base. The main parameters of interest are the Friction Parameters, the Inertia Parameters,
and the Backlash Parameters.



Friction:
To identify the coulomb static friction, the coulomb dynamic friction, and the
viscous friction, the team has created Labview VI that automates this
parameter identification. Specifically, the test plots the applied torque vs.
steady-state velocity through a predefined profile.
Inertia:
Finding the inertia of the base will be a slightly more complicated. To find it
experimentally, a pendulum will be rigidly attached to the axis of rotation. By
finding the frequency of oscillation and measuring the weights and sizes of
several other parts, the inertia of the system can be determined with respect to
the axis of rotation. Solidworks will be used to verify our experimental results.
Backlash:
Due to the significant amount of backlash in the system, the team is concerned
that it will severely impact the balancing of the system. To better identify its
effects, the backlash needs to be clearly characterized. It will be measured
10
through the use of two high resolution optical encoders. Both of these
encoders are already mounted. One is mounted directly to the axis of rotation
and the other is mounted to the back of the motor. From several experiments
there will be a clearer picture of how much backlash there is and what needs
to be done to reduce it.
Control Design
Since of there is a large amount of flexibility in the selection of the system parameters
above the base (arm length and weight), there will be four different configurations. These
mechanical configurations will range from easy to control to difficult to control. The
control design process will require several iterations of the following steps for all four
configurations.
1. Parameter Sizing:
The length and weight parameters will be chosen for the actuated arm and the
balanced arm. These sizes will be based on past iterations and experiences.
Certain combinations should be easy to control while others may be
impossible or very difficult.
2. Linearization:
Linearization of the SimMechanics model will be done by using the
linearization feature of Simulink. Once an operation point is selected, then
linearized equations of motion will automatically be generated. Linearization
of the hand derived model will be done by hand, but will only be done a
limited number of times because its purpose is only to verify the
SimMechanics model.
3. State Space Control Design:
Once the state matrices are known, then Matlab’s LQR (linear-quadratic statefeedback regulator) command will be used to generate a controller.
4. Evaluation:
The entire system will be simulated with all nonlinearities to evaluate the
systems performance. Evaluation will based on:
 How well it recovers from a non-zero starting angle.
 How well it recovers from a non-zero starting velocity
 How well it recovers when arm parameters (length, weight) are varied by
± 10% to get an idea of how robust the system will be.
 How well it recovers from a torque disturbance.
By setting up the following steps in an automated batch file, the team will be able to view
and compare the performance of hundreds of different system configurations and
controllers.
11
Implementation and Integration
System Implementation and Integration is the process of bringing the Mechanical,
Electrical, and Control portions together.



Mechanical Fabrication:
The primary mechanical part of the project will be the design and fabrication
of the arms, the universal joint, and any mounting fixtures. These features will
be designed in Solidworks and manufactured using the fabrication facilities
accessible to the team. The key mechanical issues will be to keep the universal
joint friction very low and to also allow for the easy swapping of components.
Electrical:
The electrical portion will consist of sensors, motors, amplifiers and
interconnection. The electrical portion will also need to allow for a user
interface such as a joystick. All in all, the electrical section of this project
should be quite straightforward.
Control Issues:
The encoders that will be used have very high resolutions, but since they are
incremental, it is necessary to set their zero position manually. To solve this
issue, the team will to use inexpensive accelerometers to accurately calibrate
the incremental encoders.
12
Cost:
Parts:
Part Number:
S2-2048-B
GM8724S017
MSC: 74924887
MMA7260Q
Raw Materials:
Part Number:
Description
US Digital Encoder
Pittman Motor
Flange Bearings
Pan gear A
Pan gear B
Tilt gear A
Tile gear B
Pan belt
Tilt belt
Triple Axis Accelerometer with Breakout
Description
MSC: 02629335
Alloy 3003
Aluminum Material:
Aluminum - Alloy 3003
Outside Diameter: 1/2
Wall Thickness: 0.035 In.
Alloy 6061
Aluminum Material:
Aluminum - Alloy 6061
Diameter: 1 Length: 12
MSC: 32012254
Alloy 6061
Aluminum Material:
Aluminum - Alloy 6061
Length: 72 Length: 6
Thickness: 1/4 Width: 1
MSC: 32000820
Labor Costs
Name:
Brian Lewis
Adjima Moreira
Teresa Bernardi
Matthew Rosmarin
Quantity
4
2
4
1
1
1
1
1
1
2
Quantity
Title
Engineer
Engineer
Engineer
Engineer
Price
$ 85.55
$ 210.00
$ 7.44
$ 9.97
$ 22.02
$ 7.95
$ 22.02
$ 3.92
$ 4.00
$ 34.95
Price
Total Cost
Our Cost
$
$
$
$
$
$
$
$
$
$
$
$
342.20
420.00
29.76
9.97
22.02
7.95
22.02
3.92
4.00
69.90
Parts Total
$931.74
Total Cost
$
$
$
$
$
$
$
$
29.76
69.90
$
99.66
Our Cost
2
$9.66
$19.32
$19.32
1
$7.56
$7.56
$7.56
1
$12.38
Hours
135
135
135
135
Cost
$40
$40
$40
$40
$12.38
Raw Materials Total
$39.26
$12.38
$39.26
Total Cost
Our Cost
$5,400 $
$5,400 $
$5,400 $
$5,400 $
Labor Costs Total
$21,600 $
-
Final Cost
$22,418.84
13
-
Our Final Cost
$ 138.92
Schedule:
See Appendix C
Bibliography:
[1] CJC, CTM Example: Inverted Pendulum Modeling
http://www.engin.umich.edu/group/ctm/examples/pend/invpen.html (c) 1997
[2] Craig, Kevin Inverted Pendulum Systems: Rotary and Arm-Driven, A Mechatronic
System Design Case Study
[3] Kajiwara, Hiroyuki, Wide Range Stabilization of an Arm-Driven Inverted Pendulum
Using Linear Parameter-Varying Techniques February 1998
[4] Kajita, Shuuji A Realtime Pattern Generator for Biped Walking IEEE Robotics &
Automations Conference May 2002
14
Appendix A: Terms
Abbreviations:
2D = two dimensional
3D = three dimensional
cRIO = Compact Remote Input Output or Compact RIO
FPGA = Field Programmable Gate Array
Device Parts:
15
Appendix B: SimMechanics Modeling
Figure 8: SimMechanics 2D System
Figure 9: Simulink 2D System
16
Figure 10: SimMechanics 3D System
17
Figure 11: Simulink 3D System
18
Appendix C: Schedule
See next page.
19
20
Team Contributions:
Work on this paper was shared equally between, Teresa Bernardi, Brian Lewis, and
Matthew Rosmarin.
Teresa Bernardi
Brian Lewis
Matthew Rosmarin
21