BALLBOTPROTOTYPE DESIGN REVIEW

Brian Kosoris

Jeroen Waning

Bahati Gitego

Yuriy Psarev

MTRE 4400 10/11/2011

System Overview• Mechanical Structure

• Base• Vertical structure• Landing gear

• Electronics• Sensors• Actuators/motors• MCU• CPU

• Control System• State-space variable model• MatLab/Simulink model• Synthesis of 3D motion

Mechanical Design (CAD)

• Base• Critical mechanism

• Mechanical function impacts success

• Aluminum vs. steel?• Feasibility

• Cost• Workability• Aesthetics

• Strength/rigidity vs. weight• Two perpendicular pairs

of motors (@ 45’s)• Built in damper for

vertical disturbances

Wheel Base• Design Concept

– Four compound wheels perpendicular to each axis

– Each wheel oriented 45 Degrees – Metal chassis– Motors mounted and coupled to the wheels

• Design and Material Requirements– Robust– Load Bearing, material cannot deform under

any circumstance– Fairly practical and inexpensive to fabricate– Effectively transfer load from wheels to

chassis– Minimal vibrations– Lightweight

Materials Specifications• Fasteners

– Aluminum Alloy ¼” D, 3/8” L, Dome head, Mil-spec Rivets

• Structural 6061 T6 Aluminum– Available Locally– Very strong for its weight– Easy to work with and machinable– Available in beams, channels, angles,

square and rectangular beams, pipes, etc

– ultimate tensile strength of at least 42,000 psi (290 MPa)

– yield strength of at least 35,000 psi (241 MPa)

Wheel Base (CAD Model)

Base Assembly (CAD Model)

Bottom View

Top View

Structural Design

• Vertical structure• Simple aluminum frame• Multiple modular-plateau design

• Houses main CPU, IMU board, power supply, etc.• Modular/adjustable for optimization• Facilitates testing phase• Adjustable center of mass

• Serves as a three-dimensional inverted pendulum• Bolt-able design for quick adjustments

Structural Design• Design Concept

– 2ft tall– Structural tier system with 5 levels– Each level will be octagonal shaped sheet metal– Metal chassis

• Design Requirements– Robust– Load Bearing– Fairly Practical and inexpensive to fabricate– Modular and Accessible

• For ease of adding, removing and modifying components

– Very accessible– Lightweight

Structural Design• Materials

• .125” 6061 T-6 Aluminum Sheet metal

• 1.5”x 1” x .125” Aluminum Channel

• 2”x2” Angle Aluminum

• Specifications– 9”x9”x24” tall

• Minimum success criteria• Balance structure on ball• Wheel base goes here

Fabrication Progress

** NOTE: Only a sample of what has been done

Extraneous Concept Design (CAD Model)

• Landing gear• Supplemental ‘fail-safe’

design• Protects investment• Backup if minimum

success criteria is not met• i.e. BallBot topples over

• Simple & effective• Worm-screw actuator

design• Encapsulates ball when

BallBot is balanced

Electrical Components

• New components• Micro ITX gigabyte board

• High-level CPU to run MatLab• Processes integer data from IMU board• Runs control algorithm to digest sensor data• Provides output to motor controllers• 100% onboard control for self-sufficiency

• A321 batteries x 30 for onboard power supply• Provides 12-16.5V (3-5A) to motors• Provides 5V for digital logic (IMU board and CPU)

Micro ITX onboard Computer

• 1.6GHz CPU• 4GB DDR3• Windows 7• MatLab 2010

• Rotational matrix manipulation

• State-space matrix processing

IMU Board

• Arduino ATmega2560• Microcontroller/microprocessor

• ADXL345 Accelerometer• Three-axis acceleration measurement unit

• IDG500 Gyroscope• Two-axis angular velocity measurement unit

• Provides real-time feedback of inertial orientation/rotation in 3D space

IMU Board

Sensor Data Processing

• IMU data will be relayed to onboard computer• MatLab will process complex state-space

equations and rotational matrices• Control system theory is used to model the

system for analysis of stability • Robotics synthesis

• Rotational matrices synthesize the robots orientation and angular velocity

• MatLab will process the matrices to provide feedback to the Arduino which sends signals to the motor controllers

Electronics Overview

Mathematical Modeling

• To model the inverted spherical pendulum we will first have to simplify and linearize the BallBot model

• The angle of the body makes with the vertical axis relatively close to zero.

• With this assumption, the inverted spherical pendulum can be decoupled into two identical inverted pendulum measurable in the XZ and YZ plane

Mathematical Modeling

Mathematical Modeling • A mathematical model of the Spherical Inverted

pendulum will be derived using the Lagrange-Euler method.

• Lagrange-Euler Method relies on the energy property of the system

• The Lagrange-Euler equation is defined as follow: ,

• L is the Lagrange function.• T is the kinetics energy of the mechanical system.• U is the potential energy of the system.

Deriving the Position and Velocity Equation of the Spherical Inverted pendulum

• Equation 1

• Equation 2

• Equation 3

• Equation 4

Kinetic Energy And Potential Energy of the System

• Equation 5• Equation 6• Equation 7

• Equation 8• = mass of the spherical wheel.• = mass of the body• Inertial moment of the spherical wheel• = inertial moment of the body (pitch) with respect to the body center

of mass• = inertial moment of the wheels.

Deriving the force Equation• Taking the derivative of the Euler- Lagrange function express the formula as a

force function• Equation 9• Equation 10

• computing the Lagrange Equation as depicted in equation 9 and equation 10, the following results are obtained:

• Equation 11

• Equation 12

.

Relationship between the Force and DC motor

Considering the dc motor torque and viscous friction, the following force equation can be derived:

Equation 13

Equation 14

= dc motor internal inductance

= dc motor back emf constant

Assuming the motor inductance to be negligible, current could be expressed as follow:

Equation 16

Equation 17

 

Linearization of The Force Equation• As Mentioned early we will first linearize our force Equation before expressing them as State Space

function.• This is done by taking the limit as • Applying the stated method to Equation11 and Equation 12 we get the following:• Equation 20• Equation 21• Equation 20 and Equation 21 can further be expressed in the following notation:• Equation 22

• Next step in the state space

Expression of Mathematical Equation in State Space• Next step in the state space modeling process will be defining X as a state, and U as the input with.

Next we will express X and U as follow:•

• Equation 24

• , Equation 25

Controller Simulation

• State-space modeling• x’ = Ax + Bu; y = Cx + Du

• MatLabA =

0 0 1.0000 0 0 0 0 1.0000 0 -198.9738 -0.0567 0.0567 0 42.8060 0.0092 -0.0092

B =

0 0 0 1

C =

1 0 0 0

D =0

State-space model (cont.)

controllability_matrix =

0 0.61661 -0.040635 19.844 0 -0.099717 0.0065714 -4.2689 0.61661 -0.040635 19.844 -2.6754 -0.099717 0.0065714 -4.2689 0.5025

Controllable_Rank_is =

4

observability_matrix =

1 0 0 0 0 0 1 0 0 -198.97 -0.056727 0.056727 0 13.715 0.0037383 -198.98

State-space model (cont.)Observable_Rank_is =

4

Poles =

0 -6.5686 6.5168 -0.014085

Kd =

-36.621 -1698.7 -40.986 -423.26

pole_placement =

-14 -5 -240 -180

L =

438.93

-4372.5

51264

-26241

K_f =

-0.14086 -886.74 -1.4844 -141.81

K_i =

-0.0071253

State-space model (cont.)

K_LQR =

-0.14086 -886.74 -1.4844 -141.81

-0.0071253

new_A_by_K_gain =

0 0 1 0 0

0 0 0 1 0

0.086857 347.8 0.85855 87.502 0.0043935

-0.014046 -45.618 -0.13884 -14.151 -0.00071051

1 0 0 0 0

Controller Overview

• State-space subsystem block diagram

Controller Simulation

• Subsystem• Block-diagram representation of inside subsystem

Robot Motion Synthesis

• BallBot’s orientation/angular motion can be represented with rotational matrices• Euler angles indicate

roll, pitch, and yaw of the BallBot due to disturbances (gravity, wind, push)

• Simplifies balancing/stability algorithm

Robot Motion Synthesis

• Frame 0 = 00X0Y0Z0 • Frame 1 = 01X1Y1Z1

• Position vector 0 = 3x1 matrix = [0 0 1] T

• Position vector 1 = [

] = 3x1 matrix• The angular velocities ωψ, ωϕ, ωθ

represent the data provided by the IMU board and are integrated to find position

Robot Motion Synthesis

• The rotational matrix

is very complex in terms of possible orientation synthesis• The axes of frame 0 and frame

1 are compared with the dot product of the components of position vector 0 and position vector 1

Robot Motion Synthesis

• All possible orientations:• C1 = Cos(ψ), C2 = Cos(ϕ), C3 = Cos(θ)

• S1 = Sin(ψ), S2 = Sin(ϕ), S3 = Sin(θ)

Design Requirements – Major milestones

• In this phase of the design:• The mechanical structure must be completed by

October 20th, 2011• Electronics can then be integrated into assembly

(October 27th)• Arduino and MatLab communication algorithm

(November 2nd)• Begin preliminary testing (October 27th – November

10th)• Finalize complete algorithm (November 16th)• Optimization, aesthetics, minor revisions (November

27th)

Gantt Chart

Trade Study – IMU Board

Trade Study – Accelerometer Filter

• ADXL345• Capacitor bandwidth filter – band-limiting filter

• Noise reduction – (dispose of anomalous data)• Anti-aliasing – (prevent data loss due to resolution change)

• X & Y max bandwidth – 1650Hz• Z bandwidth – 550Hz• Minimum

capacitance= 0.0047μF

Trade Study – Accelerometer Filter

• Bandwidth filter - capacitor selection• Capacitance decides bandwidth• Bandwidth indicates data resolution

Table 1 – Bandwidth vs. Capacitance

• Cx, Cy, Cz pins on ADXL345• Low-pass filtering• Noise reduction• 3-dB bandwidth equation

• F−3 dB = 1/(2π(32 kΩ) × C(X, Y, Z))

Trade Study – Accelerometer Filter

• F−3 dB = 1/(2π(32 kΩ) × C(X, Y, Z))• Approximates to F–3 dB = 5 μF/C(X, Y, Z)• 1650 Hz = 5 μF/0.00303 μF

• Cx = Cy = 0.00303 μF• 550 Hz = 5 μF/0.0091μF

• Cz = 0.0091 μF• These capacitor values will provide the highest data

resolution for• 1650 readings per second for X and Y acceleration

• Detect smallest possible acceleration in planar motion• 550 readings for Z acceleration

• The Z axis will thus represent the vertical axis of the BallBot from the center of the ball to the top of the BallBot• Then Z-axis data does not require high resolution

Trade Study – Accelerometer Filter

• Rms noise = Noise Density x sqrt(BW)• Noise is thus a factor of bandwidth

Table 2 – Noise Density

Trade Study – AccelerometerOperating Voltage

• The ADXL345 output is ratio-metric• The output sensitivity (or scale factor) varies

proportionally to the supply voltage.• VS = 3.6 V - output sensitivity = 360 mV/g• VS = 2 V - output sensitivity =195 mV/g.

• Arduino’s 3v3 pin supplies 3.3V• Sensitivity thus approximates to 320 mV/g to 340 mV/g (or

330 mV/g average)• Sensitivity estimated to be adequate for BallBot

• Arduino’s built-in serial monitor read consistent data• Only real-time testing will confirm

• ADXL345 can absorb up to 10,000g of force without being damaged – Hopefully not a concern!

Trade Study – AccelerometerOperating Voltage

X-Y-Z sensitivity (voltage/gravity)

Data Sheet

Extraneous Facilities/Resources

• Jeroen’s family workshop• Industrial

bandsaw• Computerized

lathe

• Machine wheel

shafts• Complex• Critical

Extraneous Facilities/Resources

• Jeroen’s family workshop• Drillmills• 20 ton press• Pneumatic

tools• Workbench, vice, small tools,

bits, materials, etc. (not seen)

Expenses – Bill of Materials

References

• Arduino – microcontroller (libraries/tutorials)• www.arduino.cc

• SparkFun – sensors/electronics (datasheets)• www.sparkfun.com

• MatLab resource (control system toolbox, etc.)• http://www.mathworks.com/matlabcentral/fileexchange/

23931

• SolidWorks helpfile• http://help.solidworks.com/2012/English/SolidWorks/sld

works/r_welcome_sw_online_help.htm• Robot Modeling and Control (textbook)• Control Systems Engineering (textbook)

Questions? Comments? Criticism?

• Notes:

Thank you!