Multidisciplinary Senior Design Conference
Kate Gleason College of Engineering
Rochester Institute of Technology
Rochester, New York 14623
Project Number: P13211
REALIZATION OF A WIRED RIMLESS WHEEL
Owen Accas
Department of Mechanical Engineering
Kate Gleason College of Engineering
Rochester Institute of Technology
Rochester, NY 14623
Daniel Crossen
Department of Mechanical Engineering
Kate Gleason College of Engineering
Rochester Institute of Technology
Rochester, NY 14623
Rebecca Irwin
Department of Electrcal Engineering
Kate Gleason College of Engineering
Rochester Institute of Technology
Rochester, NY 14623
Maddeline Liccione
Department of Electrical Engineering
Kate Gleason College of Engineering
Rochester Institute of Technology
Rochester, NY 14623
Hao Shi
School of Physics and Astronomy
Department of Mechanical Engineering
Rochester Institute of Technology
Rochester, NY 14623
ABSTRACT
In this project, we demonstrate the physical realization of a robotic walker. Our robot is consisted of a pentagon
shaped composite frame and a bicycle wheel coupled by two linear springs. Compared to the previous prototype, our
system is more rigid due to the composite nature of the frame. In order to compensate for the energy loss incurred
from collision with the ground and from friction between the mechanical components, we have added an electrical
motor in order to actuate the bicycle wheel periodically. However, due to time constraint, we were not able to test
the system with periodic actuation. Without electrical actuation, our system is capable of walking down a x degree
ramp for y steps.
INTRODUCTION
A key challenge in the development of robotic walker is to reduce the energy consumption without
compromising the requirement for the periodicity of the motion. A previous project in Prof. Gomes’ Energy &
Motion Lab has successfully demonstrated the feasibility of a robotic walker that can walk down a ramp at about 8˚
incline without external energy source other than gravity. Under more realistic conditions, a robotic walker should
ideally be able to walk across level ground. However, without external energy source, the robotic walker would
eventually come to a stop without walking a substantial amount of steps (~20 steps). Our project is aimed to address
this issue by adding electrical components to actuate the system periodically. In order to achieve stable periodic
motion, the distance between each foot of the frame must be as close to exactly the same as possible. However, the
frame of the previous robotic walker is made of wood, which is relatively easy to deform. Therefore, our second
objective is to produce frames that are rigid enough to maintain the distance between each foot under normal
Copyright © 2012 Rochester Institute of Technology
operation. Finally, the system must be as energy efficient as possible, which is measured by the cost of transport.
The cost of transport is defined as
Added Energy
COT =
.
(System Weight)∗(Distance traveled)
NUMERICAL SIMULATION
Equation of Motion
The rimless wheel can be in two state of motion, namely the single stance state and the double stance state. The
equation of motion for these two states can be written down separately. The conventions for the angles are shown in
Fig. 1. In subsequent discussion, we label any variables related to the frame with subscript “1”, and the wheel
subscript “2”. A subscript of the form “/a” means “with respect to a”.
Fig. 1. The convention for the angles. The leg length is
𝑙.
Fig. 2. The free body diagram for the system in single
stance.
We first examine the single stance state. In this situation, it is convenient to first analyze the system as a whole.
As shown in Fig. 2, the system experiences gravity, the normal force, and the friction force. From Newton's second
law, the equation of motion for the system can be written as
∑ 𝐹𝑥 = (𝑚1 + 𝑚2 )𝑎𝑥
∑ 𝐹𝑦 = (𝑚1 + 𝑚2 )𝑎𝑦
̈
1 ⃗⃗⃗⃗̈
2 ⃗⃗⃗⃗̈
∑ 𝜏⃗/𝑎 = 𝐼/𝑎
𝜃1 + 𝐼/𝑎
(𝜃1 + ⃗⃗⃗⃗⃗
𝜃2 ),
where the subscript in
The most direct way to calculate the angular acceleration of the frame is to use Eq. 3. The explicit sum of the
torques is
∑ 𝜏⃗/𝑎 = 𝑟⃗ × [(𝑚1 + 𝑚2 )𝑔⃗],
from which we conclude that
̈
1 ⃗⃗⃗⃗̈
2 ⃗⃗⃗⃗̈
𝐼/𝑎
𝜃1 + 𝐼/𝑎
(𝜃1 + ⃗⃗⃗⃗⃗
𝜃2 ) = 𝑟⃗ × [(𝑚1 + 𝑚2 )𝑔⃗].
For convenience, we rearrange Eq. 5 in to the following scalar form
1
2
2
[(𝐼𝑐𝑚
+ 𝐼𝑐𝑚
+ (𝑚1 + 𝑚2 )𝑙 2 ]𝜃̈1 + 𝐼𝑐𝑚
𝜃̈2 = 𝑟⃗ × [(𝑚1 + 𝑚2 )𝑔⃗]𝑧 .
Now, we examine the equation of motion for the wheel alone with the free body diagram shown in Fig. 3.
Following Newton's second law, we write
∑ 𝐹𝑥 = 𝑚2 𝑎𝑥
∑ 𝐹𝑦 = 𝑚2 𝑎𝑦
2 ̈
∑(𝜏/𝑎 ) = 𝐼/𝑎
𝜃2
𝑧
Proceedings of the Multi-Disciplinary Senior Design Conference
Fig. 3. Free body diagram of the wheel.
Page 3
Fig. 4. Free body diagram of the system in double
stance.
Again, the most useful equation is the torque equation, which can be explicitly written as
2
𝐼/𝑎
(𝜃̈1 + 𝜃̈2 ) = −𝜅𝜃2 ,
where κ is the effective torsional spring constant.
Combining Eq. 9 and Eq. 10, we can write the equation of motion for 𝜃̈1 and 𝜃̈2 in the following form
2
2
[(𝐼1 + 𝐼𝑐𝑚
+ (𝑚1 + 𝑚2 )𝑙 2 𝐼𝑐𝑚
𝜃̈1
𝑟⃗ × [(𝑚1 + 𝑚2 )𝑔⃗]𝑧
( 𝑐𝑚
).
2
2 )( ̈ ) = (
−𝜅𝜃2
𝐼𝑐𝑚
𝐼𝑐𝑚 𝜃2
In a double stance state, we assume there is friction only between the pivot foot and the ground. The free body
diagram is shown in Fig. 4. The system experiences gravity, normal force, and the friction force. For the wheel alone,
the acceleration in the x andy direction is zero, which means that 𝑁𝑥 = 0. In this state, we wish to calculate the
normal force at the non-pivot foot in order to establish whether the system should be in single stance state or double
stance. We now isolate the frame from the wheel. Newton's second law gives
∑ 𝐹𝑥 = 0
∑ 𝐹𝑦 = 0
∑(𝜏/𝑎 ) = 0
𝑧
From Eq. 12 and 13, we conclude that 𝑅𝑥 = 0 and 𝑁𝑦 = 𝑚2 𝑔. Therefore, Eq. 14 reads
𝑧
(𝑚1 + 𝑚2 )𝑔 + 𝜅𝜃2
2
𝑅𝑦 =
.
𝑧
Equation 15 is then used to determine the state of the system.
Actuation Algorithm
It is straight forward to implement our actuation mechanism in the numerical simulation. Our energy addition is
equivalent to applying a torque as the wheel move past its neutral position. Assuming (a) the motor power is
constant and (b) the angular speed of the wheel near neutral position is nearly constant, we can simply add a
constant torque to the equation of motion of the wheel in double stance. The equation of motion of the wheel in
double stance with actuation is therefore
−(𝜅𝜃2 + 𝑇0 )
𝜃̈2 =
.
2
𝐼𝑐𝑚
Simulation Results
We implemented our computational simulation in MATLAB by modifying the simulation code written by Prof.
Mario Gomes. The physical parameter for our system is approximately
𝑁𝑚
1
2
𝑚1 = 1.5 𝑘𝑔, 𝐼𝑐𝑚
= 0.061 𝑘𝑔𝑚2 , 𝑚2 = 1.25 𝑘𝑔, 𝐼𝑐𝑚
= 0.116 𝑘𝑔𝑚2 , 𝜅 = 4
, 𝑙 = 0.43 𝑚, 𝑇0 = 1 𝑁𝑚.
𝑟𝑎𝑑
We started with the following initial condition:
𝜃1 = 126∘ , 𝜃̇1 = 0, 𝜃2 = 180∘ , 𝜃̇2 = 0.
As shown in Fig. 6, our simulation indicates that periodic motion with driving is possible with the physical
parameters of our system. In Fig. 7, we plot various dynamical variables during one step of the periodic motion. We
Copyright © 2008 Rochester Institute of Technology
actuate when the wheel is passing through its neutral position with the system in single stance as shown in Fig. 8.
Energy loss can be seen when the system collides with the ground as shown in Fig. 9. We note that in all the figures,
the dynamical variables have been scaled so that the maximum value is near unity. We have confirmed that the
system can sustain stable periodic motion for more than 25 steps.
Fig. 5. Computational simulation of the dynamics of the
system. Periodic motion is clearly possible with our
system parameters.
Fig. 6. Computational simulation of the dynamics of the
system. Periodic motion is clearly possible with our
system parameters. The total energy has been rescaled to
reflect the percent change from the initial condition.
Fig. 7. The detailed view of the various dynamical
variables during one step of the motion. The
discontinuity in the angular acceleration for the wheel is
due to the actuation. The total energy has been rescaled
to reflect the percent change from the initial condition.
Fig. 8. Energy loss during the collision. The
discontinuity occurs when the frame collides with the
ground. The total energy has been rescaled to reflect the
percent change from the initial condition.
In our simulation, we have completely ignored the issue of friction. According to other individual who has had
experience with this project, we expect friction to be a major loss of energy, in addition to the collisional energy loss.
Therefore, the simulation results have only been used as a guideline to the design our system.
Cost of Transport
Based on our simulation result, we can estimate the cost of transport for our system with driving. From Fig. 8,
the actuation time is about 0.01 seconds. With an angular speed of 𝜔 = 20 𝑟𝑎𝑑/𝑠 (this is the angular speed of the
wheel near its neutral position), and a constant torque of about 1 𝑁𝑚, the power of the motor is simply given by
𝑃 = 𝑇0 𝜔 ≈ 20 𝑊,
which amounts to ∼ 0.2 𝐽 per actuation. Since we are actuating once per step, the cost of transport can be calculated
using
0.2𝐽
𝐶𝑂𝑇 =
= 0.02.
𝑚
(2.75 𝑘𝑔)(9.8 2 )(0.43 𝑚)
𝑠
We note that the energy required for the electronics and the sensors has not been included in the calculation.
Proceedings of the Multi-Disciplinary Senior Design Conference
Page 5
MECHANCIAL DESIGN
Frame Design
In order to ensure the rigidity of the frame, we have designed our plates as two snowflake-shaped composite
plates. The core of the frame is made out of structure foam, which is cut by a Waterjet machine. The core is then
surrounded by carbon fiber and fiber glass via vacuum bagging.
Spring
We have adopted the energy storage mechanism using two linear springs. Our springs were ordered from
McMaster Carr, each with a spring constant of 10 N/m.
Pulley Systems
The pulley systems used to guide the strings attached to the spring, including the pulley housings, pulleys, and
the central shafts, were machined in house using the design from the previous project with slight modifications to
accommodate our use of a thicker string than the one used previously. The high precision bearings, ordered from
McMaster Carr, are press fitted to the central shaft, which is then press fitted to the aluminum pulley housing.
High Friction Feet
In order to avoid slippage during operation, we have added high friction rubber to the feet of the frame. The
length of the feet can be modified to ensure equal length between each point of contact with the ground, which is
essential for the periodic operation of our system to occur.
ELECTRICAL DESIGN
Control System
Actuation will take place when two conditions are met: (a) the frame is in double stance, (b) the relative angle
between the wheel and the frame is small (~0.2 rad). The state of double-stance is determined by a velocity reading
of zero from the gyroscope attached to the frame. The relative angle is determined by the readings from the encoder.
Motor
We have used a 12 volts brushed DC motor purchased from phidgets.com. The motor is capable of generating
0.54 kg-cm of torque at 670 RPM. The rated current for the motor is about 700 mA. We selected this motor based on
the estimated required torque from the numerical simulation and the efficiency and the torque-speed curve of the
motor.
Motor Driver
We drive the motor with a MC33926 Motor Driver Carrier purchased from polulu.com. We selected this model
based on the input voltage, its ability to operate based on a 3.3 V logic signal, and the fact that the maximum current
the device can handle is higher than the motor can ever draw.
Encoder
The encoder is an E4P optical encoder from US Digital. It came with the motor, and met our specifications for
resolution and speed in the collection of data on the relative angle between the frame and inner wheel.
Gyroscope
The gyroscope is a L3GD20 3-Axis Gyro Garrier with Voltage Regulator from Pololu. We chose this model
because it met our requirement for data sampling rate and accuracy. The gyroscope logic output is 3.3V so that it is
compatible with the microcontroller input.
Microcontroller
The TMS320F28027 microcontroller from Texas Instruments was chosen based on its ability to communicate
through SPI, SCI, and output PWM signals as well as perform high-speed calculations necessary for our control
algorithm. The TMS320F28027 is contained within a development board, the LAUNCHXL-F28027 package, which
is also from TI. This allows for emulation without the need for a custom design or expensive external emulation
hardware and software.
Batteries
We chose AA sized NiMH batteries based on the longevity, available amp-hours, and common size, which
means that mounting hardware was cheap and easy to find.
RESULTS AND DISCUSSION
At the time of writing this paper, we have not been able to test the system with electrical actuation. However,
we did successfully have our system walk down a ramp for at most four steps.
While we all enjoyed working on this team, we believe that a different project structure could help future teams
working on this topic. Our suggested project layout involves two MSD teams. The first team involves three MEs and
one EE. This team would be responsible for everything we have been able to accomplish mechanically, which
includes designing the frame, ordering parts, building the frame, and having the system walk down a ramp
Copyright © 2008 Rochester Institute of Technology
successfully and consistently. The one EE on the team would be responsible for communicating electrical needs to
MEs, making sure that the MEs know what needs to go on the frame, and ordering parts for MEs to attach to frame.
This team would perform no programming, and would only ensure that the frame is able to walk without falling
apart, and is able to hold all required electronics.
A second MSD team with 2-3 EEs and 1-2 MEs would then take over the finished mechanical assembly. The
MEs would be responsible for fixing the frame in the case of a failure, adding any more necessary components to the
system, and testing the whole system. The EEs would be responsible for programming the microcontroller,
acquiring all electronics reading and sending data, and programming the motor controller to read sensor data to
actuate at the proper time.
Our suggestion would enable the EEs to have a full 2 quarters to program and get the motor actuating at the
proper times, which would solve many of the problems we have been facing such as the mutual waiting between the
MEs and EEs.
CONCLUSIONS
We have successfully built a pentagon shaped robotic walker that is capable of walking down a 9 degree slope
for at most four steps. Our mechanical system meets the customer’s requirement of rigidity and reliability. We have
integrated the electrical components required to actuate the system and collect dynamical data. However, we have
not been able to test the system behavior under electrical actuation due to time constraint. We have documented the
difficulties we have encountered during the execution of the project and our suggestions for future teams working on
this topic.
RECOMMENDATIONS
1. Order parts strategically to reduce the shipping costs.
2. Learn from breaking things.
3. Slightly overdesign the components, such as the pulleys and the strings, to ensure the rigidity and the
reliability of the system. If possible, outsource the pulley manufacturing.
ACKNOWLEDGMENTS
We would like to thank Prof. Mario Gomes, Prof. Edward Hanzlik for their insightful susggestions and
guidance throughout the project. We would also like to thank John Bonzo for his help with the Waterjet machine,
Prof. Carl Lundgren and Mr. Rob Aldi for their help with the composite plates, and Mr. Jan Maneti for his help with
the machining of the aluminum housing used in the pulley assembly. We would also like to thank Mr. Rob Kraynik
has offered his invaluable machining expertise to us during our machining process.