CALIFORNIA STATE UNIVERSITY, NORTHRIDGE
CLIMBING WHEEL CHAIR, DESIGN AND CONTROL
A graduate project submitted in partial fulfillment of the requirements
For the degree of Masters of Science
In Mechanical Engineering
By
Harout Markarian
December 2014
This graduate project of Harout Markarian is approved:
--------------------------------------------------------------Dr. Michael Kabo
------------------------Date
--------------------------------------------------------------Prof. Aram Khatchatourian
------------------------Date
--------------------------------------------------------------Dr. C.T. Lin, Chair
------------------------Date
California State University, Northridge
ii
ACKNOWLEDGMENT
I would like to thank Dr. C.T. Lin, my graduate project advisor. Dr. Lin helped me
understand the real challenges revolving around this particular project. Not only did he
provide different approaches, but he has been a source of guidance throughout the
difficulties that I encountered during this period. I would also like to thank Dr. Kabo and
Prof. Khachatourian as committee members for this project who supported and guided
me throughout my academic career at this university.
I want to thank Brian Straight for contributing to the paper and testing the control system,
which helped me complete this project successfully.
Additionally, I want to thank my parents for their unconditional support throughout my
life and throughout my academic years. I also want to thank my sisters for putting up
with me during my tough times.
iii
TABLE OF CONTENTS
Signature Page.................................................................................................................... ii
Acknowledgements .......................................................................................................... iii
List of Figures ....................................................................................................................vi
List of Tables......................................................................................................................ix
Abstract............................................................................................................................... x
1.0 Introduction
a. Anatomy of stairs………………………………………………………….1
b. Stairs Safety……………………………………………………………….2
c. Wheels and Stairs………………………………………………………….3
d. Common stair climbing assistive devices…………………………………3
e. Modern stair climbing wheelchair designs...…………………...…………4
f. Thesis Outline……………………………………………………………..9
2.0 Proposed Design
2.1 Force estimation Method
3.0 Dynamic Model and Control Equations
a. Jacobians: velocities and static forces……………………………………13
b. Dynamic Equations of the design………………………………………..19
c. Outward iteration to compute velocities and acceleration……………….23
d. Inward Iteration…………………………………………………………..27
4.0 Dynamics of the system
5.0 Design Study
a. Adaptive methods………………………………………………………..35
iv
b. Finite Element analysis of the Link 1.…………………………………...36
c. Loads and Fixtures……………………………………………………….37
d. Mesh Information………………………………………………………...39
e. Study Results…………………....……………………………....……….41
f. Factor of Safety calculation.......................................................................48
g. Design Study Conclusion.....…...………………………………….……..49
6.0 Programming and Control
a. Microcontroller..........................................................................................50
b. LabVIEW program and servo wiring diagram..........................................50
7.0 Prototype and test results
8.0 Conclusion
9.0 Future Work
References
Appendix A – Prototype and test pictures
Appendix B – Relevant Data
v
LIST OF FIGURES
Figure 1.1 – A slope suitable for a manual wheelchair…………………………………...1
Figure 1.2 – Representation of a typical stair...............................................................…...2
Figure 1.3 – stair descent………………………………...………………………………..3
Figure 1.4 – Stair lifts……………………………………………………………………..4
Figure 1.5 – Curb assistive mechanism…………………………………………………...5
Figure 1.6 – Curb wheelchair & curb 4WD Scooter..…………………………………….5
Figure 1.7 – TAQT design………………………………………………………………...6
Figure 1.8 – Stair climbing experiment…………………………………………………...6
Figure 1.9 – XEVIUS deformable tracks………………………………………………….7
Figure 1.10 – Dual cluster – front articulated stair climber, “Freedom”………………….7
Figure 1.11 – iBOT ……………………………………………………………………….8
Figure 2.1 – Climbing wheelchair proposed design……………………………………..10
Figure 2.2 – Force estimation method…………………………………………………...11
Figure 3.1a – Frame assignments for the 2-link manipulator..…………………………..13
Figure 3.1b – Static equilibrium link 2 and 3 (  A )………………………………………13
Figure 3.1c –FBD of proposed design link2 and link3 …….……………………………13
vi
Figure 3.2a – Static equilibrium link 2 and 3 ( B )………………………………………14
Figure 3.2b – Link 2 and Link 3 with applied force at its tip……………………………19
Figure 3.3 – Free body diagram of dynamic manipulator……………………………….20
Figure 4.1 – Static equilibrium of wheelchair……….……………………..……………30
Figure 4.2 – System Acceleration...................……………...……………………………32
Figure 4.3 – Forces during initial climb............................................................................33
Figure 5.1 – SolidWorks Assembly of proposed design…………………………………34
Figure 5.2 –Representation of loads and fixtures..………………………………………36
Figure 5.3 – Fixed Geometry restraints….………………………………………………37
Figure 5.4 – forces applied on the Link 1 .………...…………………………………….38
Figure 5.5 – Parabolic solid elements………….………………………………………...39
Figure 5.6 – Meshed Link 1……………………………………………………………...40
Figure 5.7 – Von Mises Stress Results (Link 1)…………………..…….……………….41
Figure 5.8 – Displacements Results (Link 1)..…...…………...………………………….42
Figure 5.9 – Strain Results (Link 1)………….................………………………………..42
Figure 5.10 – Link 2 with loads and fixtures.....................................................................43
Figure 5.11 – Fixed Geometry restraints (Link 2)……………………………………….44
vii
Figure 5.12 – Forces applied on Link 2 .....................................................................…...44
Figure 5.13 – Meshed Link 2 …...….....….......………………………………………….45
Figure 5.14 – Von Mises Stress Results (Link 2)………….…………………………….46
Figure 5.15 – Displacements Results (Link 2)...………......……………………………..47
Figure 5.16 – Strain Results (Link 2).................…..……………………………………..47
Figure 6.1 – Arduino Mega 2560.......................................................................................50
Figure 6.2 – Code and wiring diagram..............................................................................50
Figure 6.3 – LabVIEW code..............................................................................................51
Figure 6.4 – Complete Program, Front Panel....................................................................51
Figure 6.5 – Complete Program, Block Diagram..............................................................52
Figure 6.6 – PWM pins in block diagram ……………………………………………….52
Figure 6.7 – DC gear motor controller...............................................................................53
viii
List of Tables
Table 4.1 – Average value of tire friction coefficient .......................................................31
Table 5.1 – Link 1 Specification .......................................................................................36
Table 5.2 – P-adaptive Options .........................................................................................37
Table 5.3 – Material Properties – Link 1..……………………………………………….37
Table 5.4 – Mesh Information – Link 1..………………………………………..……….40
Table 5.5 – Link 2 Specifications……………………..…………………………………43
Table 5.6 – Material Properties – Link 2………………………………………………...44
Table 5.7 – Mesh Information – Link 2……………………………………………….…45
Table 5.8 – FOS results......................................................................................................48
Table 5.9 – Test results......................................................................................................54
ix
ABSTRACT
Climbing Wheel Chair, Design and Control
By
Harout Markarian
Masters of Science in Mechanical Engineering
Wheelchair users face difficulties while crossing rugged terrains and sidewalks, as
well as climbing up and down stairs. With the safety of the user as a main concern, the
following concept will allow the occupant of the wheelchair to ascend and descend stairs
while remaining safely on the seat. This conceptual design consists of two differential
drive wheels and a two legged mechanism, in addition, it uses light weight material.
Using Solidworks simulations, the dynamic analysis of the wheelchair is presented while
it climbs the stairs. Finite element analysis is also performed on the wheelchair’s frame,
along with DC motor power analysis and control system design. This design uses a
closed loop control system. This wheelchair will be entirely controlled by the user. A
prototype was built to demonstrate the concept discussed in this thesis.
x
1.0 Introduction
a. Anatomy of Stairs
The main focus of this paper revolves around the design and controls of a stair
climbing wheelchair. This conceptual design negotiates stairs that allow ascent and
descent. It is important to note the reason behind the preference of using stairs. Ramps
could be used as alternatives for stairs, and there are of course various options when
powered assertive mechanisms are available. However, even though a slope is beneficial
since it does not create a major obstacle for wheeled vehicles, it does have some
disadvantages, two of which are “the space used compared to a set of stairs and the
presence of sufficient traction” (Lawn, 2002).
It is not very feasible to add ramps to most architecture that is already present since it
affects the architecture’s functionality and can be quite expensive. In addition, ramps are
usually avoided in multi-level buildings and hence are rarely seen [5]. A 4.8° ramp is the
maximum angle for negotiation by the average user of manual wheelchair. In the case of
powered wheelchair the recommended maximum angle is 7.1° [10]. Some tests that were
carried out in ideal conditions demonstrated successful climb and descent rates of up to
20°. A comparison between ramps and stairs is illustrated in Figure 1 and Figure 2.
Figure 1.1 - A slope suitable for a manual wheelchair
1
For the same rise of 24 inches (2 feet) the difference of the length between the
ramp and the stairs is huge. For a 2 feet rise at 4.8° climb the length of the ramp is almost
23.8 feet. As for a set of stairs with 2 feet rise (24 inches) and a standard step size, the
length of the stairway is only 27 inches, a little over 2 feet as illustrated in the figure
below:
Figure 1.2 - representation of typical stair (step height- riser 8 in, step depth- tread 9 in)
b. Stairs safety
Compared to slopes, stairs represent the least amount of risk in regards to
slipping. At the same time stairs can be seen as obstacles for many wheelchair
applications (Lawn, 2002). Any movement can be seen as risk. And any movement
around obstacles represents a higher level of risk. There are a lot of considerations that a
normal human being takes for granted while wandering about stairs. For instance, the
location of the stair edge must be recognized and the height of the step must be estimated
to plant the feet accordingly. There’s also the shift in Center of Gravity that becomes
complex compared to walking on a flat surface. It is common knowledge that the task of
climbing takes more energy than descent; however the control in stair descent becomes
more difficult. The stairs are easier to locate when they have a positive slope (upwards),
2
therefore easier to locate steps. Descending stairs represents effort in regard to control
(Lawn, 2002). Here, the visual distance is compromised and negotiating becomes more
difficult and more dangerous in case of a fall (Lawn, 2002) [5].
c. Wheels and stairs
It is often believed that stairs create environments not suited for vehicles. A
couple of ways to negotiate stairs is to increase the wheel diameter or use a tracked
operation. The advantage of these two approaches is that the stepping places weight on
the stair’s treads, which is where it should be to avoid any risk of slip. But some
disadvantages involve locating the stair’s edge and the vehicle weight, which is applied
on the edge and requires the use of robust stair edges (Lawn, 2002) [5].
d. Common stair climbing assistive devices
In the early stages of wheelchairs the most common assistance was people themselves.
Figure 1.3 – Stair descent [6]
3
However, carrying a person in a wheelchair can be hectic and may require more than two
people to be able to move the wheelchair up and down stairs and over curbs. Lifts are
also one way of enabling a wheelchair user to climb stairs. Lifts are typically very
expensive and consume significant amount of space.
Figure 1.4 – Stair lifts [1]
Independence is regarded highly in today’s society but remains largely unrealized for
disabled people.
e. Modern stair climbing wheelchair designs
Many designs of stair climbing wheelchair already exist. Below are some designs along
with their advantages and disadvantages.
4
 Curb assistive mechanism for wheelchairs
Figure 1.5 - Curb assistive mechanism [2]
The advantage of this type of assistive device is the ability to raise the curb negotiating
ability. It’s a low cost and light weight addition to manually propelled or powered
wheelchairs [2]. However, this increases the area required for turning, cannot operate
backwards and is not available for all designs.
 Curb capable powered wheelchair
Figure 1.6 - curb wheel chair and curb 4WD scooter [3]
This design provides high curb negotiating ability, a good level of mobility in most
environments, and a high level of stability. But they have a large turning radius and are
very heavy.
5
 Train adaptive Quadru-Track based wheelchair
Figure 1.7 - TAQT design [5]
The 4WD mechanism of this wheelchair provides improved curb negotiation compared to
2WD. But it lacks the necessary traction, and the change of vehicle angle during the stair
climb reduces the vehicles stability [5].
 Track based stair climbing wheelchairs
Figure 1.8 – Stair climbing experiment [4]
These devices have stair climbing ability, are stable due to the large contact area with the
stairs, and the variable geometry single track mechanism (VGSTM) can actively control
the robot shape and the track tension to adapt to obstacles (stairs). However, they must
climb stairs backwards, are unsuitable for most indoor stairs, and the track tension
optimization is absolutely necessary to avoid slip or excessive load on the track and the
6
mechanism [4]. The most fundamental track based problem is the high pressure exerted
on the stair edges; therefore a deformable track has been proposed and modeled [4].
 20 Xero-Viscous Upstair Service
Figure 1.9 - XEVIUS deformable tracks [5]
The XEVIUS is a proposed design to solve the high stress pressure problem on stair
edges [5].
 Dual wheel cluster stair-climber
Figure 1.10 - Dual cluster – front articulated stair climber, “Freedom” [5]
This design has a stair climbing ability suitable for most stairs. It can operate
autonomously, and as a general purpose powered wheelchair. But like the track based
model, it must climb stairs backwards. It is also very large and very heavy [5].
7
 iBot Mobility System
Figure 1.11 – iBOT [9]
The iBOT Mobility System was the first marketable wheelchair produced. Freedom for
people with disabilities is what was in mind when the independence iBOT Mobility System was
developed in 1995. The iBOT was not the first wheelchair that can elevate the user to a higher
position, and it isn’t the first to use four-wheel drive to tackle uneven surfaces and go over curbs.
It was however the first on the market that can climb stairs, and the first to combine all three
functions. The iBOT is able to make the chair’s front wheel rotate on top of its back wheels. The
chair is then able to balance itself on two wheels and put the user at eye level while still seated.
This “standing” feature is also able to enable a user to reach high shelves, cupboards and other
spaces beyond the grasp of traditional wheelchair users. The iBOT is able climb up and down
stairs while the occupant’s seat remains level. It uses a complex system of sensors, gyroscopes
and electronics to simulate human balance [9]. The iBOT was tested with many different types of
users, including people with neuromuscular diseases. The iBOT is recommended for users based
on their functional ability to operate the device rather than on specific disabilities. It’s better
suited to those who have reasonable hand control, which is needed to operate the device. This
8
device was discontinued in 2009 due to the fact that its demand was not proven sufficient to
create a sustainable market [8].
f. Thesis outline
This thesis focuses on the design and controls of a stair negotiating wheelchair.
There are two issues when negotiating stairs, the stability of the mechanism, and the
actual climb or descent. In order to provide an assistive mobility device suitable for stair
negotiation, a mechanism capable of negotiating stairs must be provided. The conceptual
design presented here will make this possible. Stability is provided by four points of
contact when the wheelchair is on flat ground, and a minimum three points of contact
while negotiating stairs. Even though this design can negotiate both high and low steps,
the prototype was only tested on regular size steps.
9
2.0 Proposed design
The proposed design focuses on the challenges of climb and descent. The mechanism
uses 4 wheels when wandering on flat ground the same as any wheelchair. The design
entails differential drive wheels to allow a zero radius turn and two caster wheels.
Steering is achieved by independently controlling the rear wheels. The drive wheels are
mounted to two independent columns (link 1) that hold the seat. The seat is positioned at
a convenient height and bridges the two columns (link 1) for more rigidity. At the top of
each column two servo motors are attached holding two 16 inch arms (link 2). At the
other end of these arms two smaller servo motors are attached that in turn hold link 3 that
has the caster wheels at its ends.
Figure 2.1 - Climbing wheelchair proposed design
The modeling process consisted of two major parts, the numerical modeling to
confirm design viability and the building of a prototype model to confirm dimensional
practicality and understand the controllability.
10
2.1 Force estimation method
In order to successfully estimate the force seen by the caster during the climb, a
scale was put on the surface of the platform designed for this project. The caster of
the robot was then fixed on the scale and the robot was manually pushed up the stairs
and the value given by the scale during this process was considered to be the highest
force seen by the caster while climbing. The weight of the user is incorporated in this
design. The figure below shows this process.
Figure 2.2: Force approximation method
The force acting on the caster wheel during the climb turned out to be F=3 lbs.
11
Five measurements were taken prior to determining the average force acting on the caster
wheel during the climb. This was the value of the force used to calculate the component
forces on link 4 shown in the FBD (figure 3.3).
3.0 Dynamic Model and Control Equations
In this scenario links 2 and 3 are treated as a two-link manipulator with rotational
joints. Each link is considered as a rigid body with linear and angular velocity vectors
describing its motion. These velocities will be expressed with respect to the link frame
itself rather than with respect to the base coordinate system. It is worth mentioning here
that linear velocity is associated with a point and angular velocity with a body. The term
“velocity of a link” here means the linear velocity of the origin of the link frame and the
rotational velocity of the link. First, we start by determining the velocity at the origin of
each frame, starting from frame {0}, which has zero velocity. Rotational velocities can be
added when both ω vectors are written with respect to the same frame. Therefore, the
angular velocity of link i+1 is the same as that of link i plus a new component caused by
rotational velocity at joint i+1. These calculations make use of the link transformations
shown below. In this case link 1 is not considered. (Refer to figure 2.1 for link notations).
12
a. Jacobians: Velocities
and static forces
Figure 3.1a – frame assignments for the
Figure 3.1b – static equilibrium links 2
2-link manipulator (Intro. to robotics,
and 3 (
Mechanics and Control, 2005)
13
Figure 3.1c – FBD of proposed design link2 and link3
In order to compute the velocity of the origin of each frame starting from frame {0},
which has zero velocity, we first need to calculate the link transformations.
Eq. 1
Where
represents the measured angle between joint A and link 2 at static equilibrium
of the wheelchair. This was modeled and measured in SolidWorks.
B
Figure 3.2a - Static equilibrium links 2 and 3 ( B )
We know the position of the end effector (caster wheel) relative to the second joint of the
robot arm, this is determined by the measured angle
.
is the angle created between
link 2 and link 3 when the robot is in static equilibrium. In Figure 3.2
14
. We’re
interested in where it is relative to reference frame {A}, which is Link 1’s reference
frame in this case. This means that we need a transformation matrix, from the reference
frame {B} (link 3) back to reference frame {A} (link 2).
Eq. 2
We know the position of the end effector relative to the caster wheel of the robot arm.
We’re interested in where it is relative to reference frame B, which is Link 3’s reference
frame in this case. This means that we need a transformation matrix, from the reference
frame B (link 3) back to reference frame C (the caster wheel).
Eq. 3
The angular velocity of each link is then calculated along with their linear
velocities. Angular velocities can be added when both ω vectors are written with respect
to the same frame. Therefore, the angular velocity of link “i+1” is the same as that of link
15
“i” plus a new component caused by rotational velocity at joint “i+1”. (Refer to figure
3.1a for link notations). (Craig J. J. introduction to robotics Mechanics and Control,
2005)
Eq. 4
In the above equation we have made use of the rotation matrix relating frame {A} and
{B} (refer to figure 3.2 for frame notation) in order to represent the added rotational
component due to motion at the joint frame {A}. Here, the rotation matrix rotates the axis
of rotation of joint B into its description in frame {A}, so that the two components of
angular velocity can be added. Equation 4 is the description of the angular velocity of
link B with respect to frame {B}, where
is the rotation of B with respect to A. and
is the transpose of the rotation of A with respect to B.
The linear velocity of the origin of frame {B} is the same as that of the origin of
frame {A} plus a new component caused by rotational velocity of link A. Therefore, we
have
Eq. 5
16
is the position of B with respect to A.
The cross product from the equation above is shown below
Eq. 6
So far joint torques are found that will exactly balance forces at the caster in the static
situation. When forces act on a mechanism, work is done if the mechanism moves
through displacement. Work is defined as a force acting through a distance and is a scalar
with units of energy. We can equate the work done in Cartesian terms with work done in
joint-space terms.
In the field of robotics, Jacobians are generally used because it relates joint velocities to
Cartesian velocities of the tip of the arm. The Jacobian is a multidimensional form of the
derivative. The number of rows equals the number of degrees of freedom in the Cartesian
space being considered. The number of columns in a Jacobian is equal to the number of
joints of the manipulator. In this case (two-link arm) a 2 x 2 Jacobian can be written that
relates joint rates to end effector velocity. So the Jacobian becomes as follows:
Eq. 7
Where
17
After calculating the Jacobian for the static case, the required torque can be determined:
WL 2  1.083lbf
WL 3  0.538lbf
Wwheel  0.45lbf
Wtotal  mL1  mL 2  mwheel  2.071lbf  Ftotal  2.071lbf
Eq. 8
Eq. 9
and
are the torques required at joints A and B when the wheelchair is in static
equilibrium.
Where
is the weight in lbf of link 3 plus weight of the caster wheel.
An ideal factor of safety for our model is 1.3 [4]. Torque calculations were increased by
another 10% for extra safety. Final torque values were increased by 40%, which gave us
a minimum torque requirements of
18
Figure 3.2b – Link 2 and link 3 with applied force at its tip
b. Dynamic Equations of the design
These calculations are based on the iterative Newton-Euler dynamic formulation. The
problem here is to compute the torques required of the servo motors corresponding to a
given manipulator trajectory. In this case the position is assumed, the velocity is taken
from the servo specifications (see Appendix), and the acceleration is calculated. The joint
torques can be calculated with the above knowledge along with knowledge of the
kinematics and the mass-distribution of the robot. Below is a free diagram of the system.
19
Figure 3.3 - Free body diagram of dynamic manipulator
The local frames follow the Denavit-Hartenberg approximation for robotics application
(Refer to figure 3.3)
The component forces acting on the end effector (caster wheel) are calculated first
Eq. 10
Eq.11
M1 , M2, and M3 are the corresponding link mass
20
The vectors that locate the center of mass for each link are:
l  5 


1
1
Pc1  l1 ' X 1   0   0
 0  0
2
3
l2  8

Pc 2  l2 ' X 2   0   0
 0  0
 l3  3
Pc3  l 3' X 3   0   0
 0  0
{Ci}: Imaginary frame attached to each link with its origin being at the center of mass of
the link. Same orientation as the link frame {i}.
{li}: distance from origin of a link to its center of mass
The forces at the end-effector (caster wheel) are as follows:
 f x4 
 f 4   f y 4 
 0 
4
There are no torques at joint 4  4 n4  0
First, the angular velocity and acceleration of both servos need to be calculated.
21
The angular velocity was calculated using the purchased servo motor specifications.
Appendix B contains all parameters of the servo motors. The base of the robot is rotating
about z,
2
 1.05rad / s
3


 1.05  0
 1  1 

 0.525rad / s 2
t
2
 1  10rpm  10 
The effect of gravity loading on the links can be included simply by setting
,
where G has the magnitude of gravity vector but points in the opposite direction. This is
equivalent to saying that the base of the robot is accelerating upward with 1 g
acceleration. This causes exactly the same effect on the links as gravity would.
To include the gravity forces the following was used:
0
0 

0 




2
 0  g Y 0   0   g   386.4in / s 
 0  

0
0 
Eq. 12
The rotation between successive link frames is given by:
ci 1

i
i 1 R   s i 1
 0
 si 1
ci 1
0
0
c1
0 21R   s1
 0
1
 s1
c1
0
0 0.866  0.5 0
0   0.5 0.866 0
1  0
0
1
Here, c1 is cosine of ϑ1 and s1 is sine of ϑ1
22
Eq. 13
c. Outward iterations to compute velocities and accelerations.
The complete algorithm for computing joint torques from the motion of the joints is
composed of two parts. First, link velocities and accelerations are iteratively computed
from link 1 out to link n (outward iteration) and Newton Euler equations are applied to
each link. Second, forces and torques of interaction and joint actuator torques are
computed recursively form link n back to link 1 (inward iteration).
The propagation of rotational velocity from link to link is given by

i 1 

0 
 1 
0

i1 ii1R ii   i1 Z i1 11 10R 00   1 Z 1   Z   0    0  Eq. 14
1 1
  1.05
i 1
 1

Where 0  0 and  0  0

Angular acceleration of link 1:
i 1 

i 
i 1
i 1 i
 i 1  i R  i  i R i   i 1
i 1 

Z i 1  i 1
i 1

Z i 1 
0  0 
  

 1  R  0  R  0   1 Z 1  1 Z1  1 Z1   0    0 

  0.525
1 

1
0
0 
1
0
0
 1 




1
1
Eq. 15
Linear acceleration of link frame:
 gs1  386.4

  

 i 1  R(  i  Pi 1  i ( i  Pi 1 )  i ) R  0   gc1    0   1 1
 0   0 
i 1 
i 1
i
i 
i
i
i
i 
i
23
1
0
0 

i 1 
i 1 
i 1
i 1
i 1
i 1
i 1
 C  i 1  PC  i 1 ( i 1  PC )  i 1 
i 1
1
1 
i 1
i 1
1

1 

1 

1 
0 
1
1
 C  1  PC  1 ( 1  PC )  1   1 Z1  PC   1 Z1  ( 1 Z1  PC ) 0 R  0
218.6
1 
  C   294.3
 0 
1
1
1
1
1
1
1
1
1
1
1

i 1
i 1 
1
Fi 1  mi 1  Ci 1  F1  m1  C1
1
 1.524 
0 


1
 0.0105, N1  0
 0 
0
Eq. 16
: denotes angular acceleration of joint 1 same as
: denotes angular velocity of joint 1 same as
: denotes linear acceleration of joint 0
denotes angular velocity of link 1 with respect to frame 1
denotes angular velocity of joint 1 along Z1 axis of rotation
Because of the point mass assumption, the inertia tensor written at the center of mass for
each link is the zero matrix:
The outward iteration for link 2:

Angular velocity and acceleration:
2

2 
2 12R 11   2 Z 2
24
2
0 0   0 
 2   0    0    0 


1  2  2.31

Where 2  2  1.26rad / s (from motor specification)

And 2 
2 
1.26  0
 0.63rad / s 2
2
1 

2 

 2  R 1  R 1   2 Z 2   2
2
1
2
1
1
 0   0 

 

Z 2   2    0     0 
1  2  1.155
2 
2 
 Linear acceleration:

 

s



l
l
1 1 2
1 1 c 2  gs12 
 194.75 

2 
1 
1




2
1
1
1
1


 2  1 R(  1  P2  1 ( 1  P2 )  1 )  l11 s 2  l11 s 2  gc12   0354.01
0

  0 








2

l
(

)

l
s

l
c2  gs12 




2
1
2
1
1
2
1
1


2 
2 
2 





  C2   2 2 PC2  22 ( 22 2 PC2 )  2   l2 (1  2 )  l1 1 c2  l1 12 s2  gc12 


0




2 
  C2

2
139.25 
 363.25
 0 
2 
F2  m2  C 2



 2


 m2 l2 (1  2 )  m2 l1 1 s2  m2 l1 1 c2  m2 gs12 



 2


  m2 l2 (1  2 )  m2 l1 1 c2  m2 l1 1 s2  m2 gc12 


0




25
0.557
 F2  1.453 
 0 
2

2
0
N 2  I 2 2  2  I 2 2  0
 
0
C2
2
2 
2
C2
N represents the torque about the center of mass.
The outward iteration for link 3:

Angular velocity and acceleration
Using the same approach the results of link 3 parameters are as follows:
 0 
3   0 
3.46
3
 0 
3   0 
1.73
3 

Where 3  3  1.15rad / s (from motor specification)

And 3 

1.15  0
 0.575rad / s 2
2
Linear acceleration
 231.21
  3   321.4 
 0 
3
3
C
3
 266.41
  326.6 
 0 
26
 0.56
 F3   0.69 
 0 
3
3
0 
N 3  0
0
d. Inward Iteration:
Having computed the forces and torques acting on each link, the joint torques must
now be calculated. The joint torques will result in these net forces and torques being
applied to each link.
By summing the forces acting on link 3, the force-balance relationship can be
written as follows:

fi i 1iRi 1fi 1 iFi
i
Inward iteration for link 3
1 0 0 2.6  0.56 2.04
f 3  R f 4  F3  f 4  F3  0 1 0  1.5    0.69   2.19
0 0 1  0   0   0 
3
3
4
4
3
4
3
Eq. 17
Results from equations 10 & 11 were applied above.
The torque acting on link 3 is:

i
ni i N i  i 1iR i 1ni 1  iPCi i Fi  iPi 1 i 1i R i 1f i 1
is the torque at the joint.
3
n3 3N3  43R 4n4 3PC3 3 F3 3P4 43 R 4f 4
3
N 3 is equal to zero because
3
4
R 4n4 is equal to zero because there are no torques acting at joint 4,
27
Eq. 18
3
3  0.56 6 2.6  0 
n3  0   0.63   0  1.5    0 
0  0  0  0  11.07

T
T3 3n3 3Z 3
Eq.19
f i  Force exerted on link i by link i-1
ni  Torque exerted on link i by link i-1
3
 0 
n3   0 
11.07
Inward iteration for link 2

2

2
 0.5  .0866 0 2.04 0.557  0.32
f 2  R f 3  F2  0.866
0.5
0  2.19  1.453    4.313 
 0
0
1  0   0   0 
2
3
3
2
n2  2N 2  32R 3n3  2PC 2 2 F2  2P3 32 R3f 3
 0  0.557 8 6  0.5  0.866 0 2.04  0 
 0   1.453   0  0  0.866
0.5
0 2.19   0 

 
     
11.07  0  0 0  0
0
1  0  39.9
 0 
 n2   0 
39.9
2
And similarly the inward iteration of link 1
1
0.866 0  0.32 1.524  5.124
 0.5

f1  R f 2  F1   0.866 0.5 0  4.313   4.313   2.45 
 0
0
1  0   0   0 
1
2
2
1
28
1
n1 1N1  21R 2n2 1PC11F1 1P2 21R 2f 2
0.866 0  0.32  0 
 0  5  1.524  10  0.5
  0   0  0.0105   0    0.866 0.5 0  4.313    0 
39.9 0  0   0   0
0
1  0  64.3
 0 
 n1   0 
64.3
1
These equations are evaluated link by link, starting from link 3 and working inward
toward the base of the robot.

The required joint torques are found by taking the Z component of the torque applied by
one link to the other:


 i  i ni T i Z i
  1 1n1
T 1
Eq. 20

Z1  64.3in.lb


 2  2 n2 T 2 Z 2  39.9in.lb

 3  3 n3T 3 Z 3  11.07in.lb

 1 ,  2 and  3 are the torques required at the actuators as a function of joint position,
velocity, and acceleration. The calculations are performed for one side of the system
29
4.0 Dynamics of the system
Two gear motors were chosen to be able to drive the wheelchair and overcome the
minimum torque requirements (64.3 in.lb) while negotiating stairs. The design is
symmetrical about the mid-plane; therefore the analysis can be performed on one side
and assumed to be true for the other one.
The specification of the gear motor chosen is listed in the appendix.
Below is the free body diagram of the system in static equilibrium.
O
Figure 4.1 – static equilibrium of the wheelchair
In the figure above:
M’ is the Mass of the system (wheelchair)
A is the acceleration of the system
N1, N2 are the reaction forces
30
µ is the coefficient of static friction
Summation of the forces along the y – direction results in the following:
F  0 W  N  N
 M  W (2.89)  N (18)
  M  0.052(2.89)  N
y
1
O
Eq. 21
2
2
O
2
(18)  0
 N 2  0.0084lb f
 N1  19.9916lb f
  M O  302.05in.lb
Here the friction force needs to be considered. The table below displays the average value
of tire friction coefficients on different terrains.
Road Surface
Peak Value
Sliding Value
Asphalt and concrete (dry)
.080-0.90
0.75
Asphalt (wet)
0.50-0.70
0.45-0.60
Concrete (wet)
0.80
0.70
Gravel
0.60
0.55
Earth road (dry)
0.68
0.65
Earth road (wet)
0.55
0.40-0.50
Table 4.1 – Average value of tire friction coefficient
The Engineering toolbox, 11/02/2014, http://www.engineeringtoolbox.com/frictioncoefficients-d_778.html
31
Acceleration of the wheelchair on flat ground about its center of mass:
Figure 4.2 – System acceleration
is the angular velocity of each link. The subscript “i" represents the link number.
Is the angular velocity of the system
The angular acceleration for each link is calculated and the sum of the links is equated to
the angular acceleration of the system about its center of mass. It is argued here that the
sum of angular acceleration of the links is equal to the angular acceleration of the system
at its center of mass.


  I i  i  I C .G .  C . G .

Eq. 22



( I1  m1r1 ) 1  ( I 2  m2r2 )  2  ( I 3  m3r3 )  3  I  CG
2
2
2
Where r1, r2, r3 are the distances from CG of link to CG of system
m1, m2, and m3 are the corresponding link mass
I1, I2, I3 are the moment of inertia of each link calculated from SolidWorks
32
The parallel axes theorem was used for each link.
The drive motor velocity is 10rpm = 1.05rad/s


251.5  970.28 CG   CG  0.259rad / s 2
Newton’s second law for rotation:

   I CGCG Where  C.G.  C.G.  0.259rad / s 2
Eq. 23
   570.28  0.259  147.7in.lb
ICG of the system was given by SolidWorks
In order for the system to climb up the stairs it requires 147.7 in.lb of torque at its center
of mass. If we multiply τ1 from eq. 20 by 2 (64.3 x 2 = 128.6 in.lb) to encompass the
entire system we notice that both results are fairly close to each other.
The figure below shows the most required torque during initial ascent
Figure 4.3 – Forces during initial climb
33
5.0 Design Study
This conceptual design was modeled using SolidWorks. The final assembly of the design
is illustrated in the figure below:
Figure 5.1 - SolidWorks Assembly of proposed design
After finalizing the assembly, SolidWorks Simulation was used to conduct finite element
analysis and ensure quality, performance, and safety. SolidWorks Simulation uses the
displacement formulation of the finite element method to calculate component
displacements, strains, and stresses under internal and external loads. The geometry under
analysis is discretized using tetrahedral (3D), triangular (2D), and beam elements, and
solved by either a direct sparse or iterative solver.
34
a) Adaptive methods
The H-method
The concept of the h-method is to use smaller elements in regions with high errors. After
running the study and estimating errors, the software automatically refines the mesh
where needed to improve the results.
The P-method
The p-method uses more efficient elements in regions with high errors. The program
increases the order of elements in regions with errors higher than a user-specified level
and reruns the study. This method does not change the mesh. It changes the order of the
polynomials used to approximate the displacement field. The use of unified polynomial
order throughout the elements is not efficient. The software increases the order of the
polynomial only where it is needed. This approach is called the selective adaptive pmethod. The program runs as many times as necessary. After each loop, it assesses the
global and local errors and decides whether to make another run. The loop is stopped
when one of the following conditions is met:

The global criterion converges

All local errors converge

The maximum number of loops is reached
The convergence check is based on total strain energy, root mean square (RMS) of Von
Mises stresses, or RMS of resultant displacements. After running the static analysis using
the p- adaptive method convergence plots can be generated.
35
b) Finite Element Analysis of the link 1 - using P-adaptive method.
Figure 5.2 - Representation of loads and fixtures
Description
Treated as
Volumetric properties
Link 1
Solid Body
Table 5.1 – Link 1 specification
36
Mass:1.03045 lb
Volume:10.564 in^3
Density:0.0975437 lb/in^3
Weight:1.02975 lbf
1 % or less
Stop when RMS von Mises Stress change is
2 % or more
Update elements with relative Strain Energy
error of
2
Starting p-order
5
Maximum p-order
4
Maximum no. of loops
Table 5.2 – P-adaptive options
Name
Model type
Yield strength
Tensile Strength
Elastic modulus
Poisson’s ratio
Mass density
Shear modulus
6061-T6 (SS)
Linear Elastic Isotropic
39885.4 psi
44961.7 psi
1.00076e+007 psi
0.33
0.0975437 lb/in^3
3.77098e+006 psi
Table 5.3 – Material Properties – Link 1
c) Loads and Fixtures
Link 1 was constrained through the middle slot and the lower hole as shown in the
pictures below:
Figure 5.3 – Fixed geometry restraints
37
To simulate the forces acting on the column (link 1) a normal force was applied through
the plate which had a greater impact than the actual load seen by link 1 through the servo
motor’s weight. The figure shows the forces the way they were applied. Link 1
experiences a 0.9 lbf force applied as shown simulating the motor’s weight applied on the
column (link 1).
High stress at
these locations
Figure 5.4 - forces applied on the Link 1
38
d) Mesh Information:
To mesh a part with solid elements the software generates either a draf quality mesh or
high quality mesh. Linear (or lower order) tetrahedral solid elements is generated by the
mesher to achieve the draft quality mesh, and a parabolic (oe higher order) tetrahedral
solid elements is generated to achieve a high quality mesh. Having no complex shaped
parts in this design, it was possible to perform a draft quality mesh but to make this study
more reliable a high quality mesh was used. The parabolic tetrahedral elements is defined
by four corner nodes, six mid-side nodes, and six edges. The following shows a
schematic of parabolic tetrahedral solid elements:
Figure 5.5 - Parabolic solid elements
For the same mesh density the parabolic elements yield better results because they
represent curved boundaries more accurately, and they produce better mathematical
approximations. However, they require a greater computational resources.
Below are the mesh information used for the purposes of this design.
39
Mesh type
Solid mesh
Mesher used
Standard mesh
Jacobian points
4 Points
Element size
0.219483 in
Tolerance
0.0109742 in
Mesh Quality
High
Total Nodes
15813
Total Elements
8774
Maximum Aspect Ratio
4.3486
% of elements with Aspect Ratio < 3
97.6
Table 5.4 - Mesh information – Link 1
Figure 5.6: Meshed Link 1
40
e) Study Results:
A Von Mises Stress analysis was performed and the results are shown in the figure
below. The highest stress was seen at the two corners of the U-shaped plate. The reason
of these accumulated stresses is due to the sharp corners. This could’ve been avoided,
but since this is just a prototype and the load is not very high we can manage to have
sharp corners at these critical locations. The highest experienced load was registered at
214.6 psi (910 nodes) and the yield strength is 39,800 psi.
Figure 5.7 - Von Mises Stress Result (Link 1)
41
As for the displacement, it is obvious that with the applied loads and fixtures the
displacement is not significant and the column (link 1) won’t experience any permanenet
or undesirable deformation. The maximum displacement was at the top of link 1 with
0.00113041 inches (749 nodes).
Figure 5.8 - displacement Result (Link 1)
Figure 5.9 - Strain results (Link 1)
42
Moreover, the strain analysis shows that the highest amount of strain was experienced
around the sharp corners with a negligible maximum value of
. Based on
these readings it was concluded that link 1 won’t fail at any point during the operation of
the robot.
Application of loads and fixtures (Link 2)
Figure 5.10 - Link 2 with loads and fixtures
Description
Link 2
Treated As
Solid Body
Volumetric Properties
Mass:1.10291 lb
Volume:10.8643 in^3
Density:0.101518 lb/in^3
Weight:1.10217 lbf
Table 5.5 - Link 2 specifications
43
Name
Model type
Yield strength
Tensile Strength
Elastic modulus
Poisson’s ratio
Mass density
Shear modulus
AL 7075-T6
Linear Elastic Isotropic
73244.1 psi
82671.5 psi
1.04427e+007 psi
0.33
0.101518 lb/in^3
3.90152e+006 psi
Table 5.6 - Material Properties – Link 2
Loads and fixtures:
In order to simulate the loads and fixtures of this link the following approach was taken;
Link 2 was restrained as fixed geometry through the half inch hole. The reason fixed
geometry was used is to verify that when the link is not rotating about the hole axis it is
going to withstand the load on its opposite end due to the servo motor and link 3.
Figure 5.11 - fixed geometry restraints (Link 2)
Loads were applied at the extended side of the bar where the motor is attached along with
link 3. The figure below shows how the load was applied:
Figure 5.12 – Forces applied on Link 2
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Mesh information
A high quality solid mesh was used to perform this analysis.
Mesh type
Solid mesh
Mesher used
Standard mesh
Jacobian points
4 Points
Element size
0.221543 in
Tolerance
0.0110772 in
Mesh Quality
High
Total Nodes
16626
Total Elements
9598
Maximum Aspect Ratio
9.3049
% of elements with Aspect Ratio < 3
99.4
Table 5.7 – Mesh information - Link 2
Figure 5.13 - Meshed Link 2
45
Study Results
The stress analysis shows that the area with the highest amount of accumulated stress is
experienced at the edge of the 1 x 1 extrusion where the attached plate starts to bend. It
looks like a cantilever beam. The yield strength is 73,244.1 psi and the highest recorded
stress is 747.5 psi (15414 nodes)
Figure 5.14 - Von Mises stress results (Link 2)
In addition, the design analysis shows that the maximum amount of displacement occurs
at the edge of the attached plate at 0.00736465 in (1693 nodes). The over hanged plate is
going to see a very little deflection.
46
Figure 5.15 - displacement results (Link 2)
Finally the strain analysis shows that the bent area is where maximum strain occurs at
2.99798e-005 (7552 nodes), which is very small and will not affect the design.
Figure 5.16 - Strain results (Link 2)
47
f. Factor of Safety
The factor of safety is evaluated at each node of the model based on a failure criterion.

A factor of safety less than 1.0 at a location indicates that the material at that
location has failed.

A factor of safety of 1.0 at a location indicates that the material at that location
has just started to fail

A factor of safety larger than 1.0 at a location indicates that the material at that
location is safe.
Factor of safety was calculated using SolidWorks simulations data. For the two parts of
this design that FEA was performed on, the factor of safety is listed in the table below:
Description
FOS
Link 1
185.6
Link 2
97.9
Table 5.8 – FOS results
The material used for this project was 6061 and 7075 Aluminum alloy. Since there are no
significant stresses on these parts we have a very high safety factor. The ratio of the yield
strength to the highest von misses stress was taken in order to calculate the factors of
safety of these parts.
FOS
yieldstren gth 39885

 185.6 >>1
max stress
215
FOS
yieldstren gth 73244

 97.9 >>1
max stress
747.5
48
g. Design Study Conclusion
In conclusion, the design analysis shows that the critical components of the robot will
withstand the loads that they are exposed to. There are no significant concerns about any
of the locations that were tested in the design analysis. The design can now be considered
ready for manufacturing and assembly.
h. Materials and Methods
6061 and 7075 Aluminum alloys was used to build this prototype. The material was
acquired from and machined in the CSUN machine shop. Prior to machining the
prototype was modeled in SolidWorks. After finalizing the design, 2D drawings of
different components were prepared to assist in manufacturing. The manufacturing
process took a little over two weeks. All components were weighed using a weight scale
from the robotics lab. After all components were manufactured and electrical components
were purchased, the robot was assembled and tested. In addition a scaled stairs was built
in order to perform the test on. The stairs was made from wood and composed of two
steps.
49
6.0 Programming and control
a. Microcontroller
In order to control the different links of the wheelchair a microcontroller was required
with PWM (pulse width modulation) pins. For the purposes of this project the Arduino
Mega 2560 microcontroller was used. It has 15 PWM pins of which 4 of them were used
to control the different servo motors at the joints.
Figure 6.1 – Arduino Mega 2560
The Arduino was powered by the laptop and each servo had its own 6V battery pack. The
reason for the separate battery packs for the servos was to maximize the torque output of
each servo motor.
b. LabView Program and servo wiring diagram
Servos
Arduino
Figure 6.2 – Wiring diagram
50
Figure 6.3 – LabVIEW Code
Figure 6.3 is the LabVIEW code used to test two servo motors. In order to test the servos
we wired each servo separately and try to operate it with the written LabView code. After
ensuring the proper behavior of each servo separately, we wired all four servos and tested
the program. The test was successful and Links 2 and 3 were controlled through the code
below.
The code used to control all the servos is illustrated in the figures below:
Indicates which key is
being pressed on the
keyboard
Neutral Position
Figure 6.4 – Complete program Front panel
Each servo had a slider bar in the program as displayed in the figure above. The right side
had two servos hence the two slider bars below “RIGHT SIDE” and the left side had two
51
servos hence the two slider bars below “LEFT SIDE”. Each slider bar represents one
servo connected to a specific PWM output on the Arduino. We decided to name each
slider with the appropriate PWM number to the left of the figure below. The cursor is
initially set to its neutral position at 1500. Above 1500 the motors were rotating
counterclockwise and below 1500 the motors were rotating clockwise, so we named each
side of the cursor as “up” and “down” accordingly.
Figure 6.5 – Complete program Block diagram
Figure 6.6 – PWM Pins in block diagram
Each PWM in figure 6.6 relates to a single slide bar in figure 6.4 that controls a single
servo motor. This program allows the user to utilize the keyboard to move the sliders left
and right in order to facilitate the control of links 2 and 3 on either side.
52
Notice the program does not include any DC gear motor control because they were
controlled separately. Figure 6.7 shows the controller used for the two DC gear motors.
The gear motors were used to accelerate and decelerate the robotic wheelchair while
maneuvering on flat ground or while climbing and descending stairs. This controller
provides proportional forward and reverse control and it entails an H-bridge, capacitors,
resistors, and potentiometer.
Figure 6.7 – DC gear motor controller
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7.0 Prototype Test Results
As mentioned previously the prototype success was measured by whether or not the robot
was able to make it up and down the stairs. The results from each trial are listed the table
below:
Trial
Climb
success
1
2
3
4
5
6
7
8
9
10
descent
fail
x
x
x
x
success
fail
x
x
x
x
x
x
x
x
x
x
x
x
x
x
time of climb
time of descent
min:sec
N/A
12:50
12:45
13:20
N/A
10:32
9:26
9:18
10:40
10:02
min:sec
N/A
N/A
8:01
7:05
N/A
5:50
6:00
5:42
5:50
5:36
total
run
min:sec
N/A
N/A
20:46
20:25
N/A
16:22
15:26
15:00
16:30
15:38
Table 5.9 – Test Results
The test was timed and the best total time achieved was 15 minutes. The most difficult
part was the ascent where the robot was taking a lot of time to climb. This is due to the
slow RPM drive motors and also the difficulty to control while climbing. The robot never
tipped over it was very stable at ascent and descent. The stability was maintained by
having three points of contact at all times determined by the 2 drive wheels and one
caster wheel. The center of gravity was maintained within the triangle defined by the
three points of contact.
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8.0 Conclusion
The conceptual design discussed in this thesis has the ability to climb and descend stairs
successfully. With its given design it is believed it has the ability to negotiate all types of
stairs. However, it was discovered that during its descent it has the tendency to move
forward when the drive wheel is trying to come down a step. This is due to having caster
wheels in the front with no braking system.
The proposed design’s length to width ratio is .669, which is comparable to a standard
wheelchair length to width ratio of .625.
Since the thesis focused mostly on negotiating stairs, the seat of the wheelchair is not
ideal for this design.
As for the control part of this thesis, the manual controller in figure 6.7 was able to
control the drive wheels very well with no failure. The equations that were used to
calculate torque for different motors were incorporated in the LabView program and used
to manipulate the links of the robot.
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9.0 Future Work
The recommended future work for this project would be a new and improved seat to
keep the user safe at all times. The current seat is not ideal; the new seat design
should balance itself while the wheelchair climb and descend stairs. This prototype
can also be improved by adding a braking system to the casters to stabilize the
descent.
56
References
[1] ACRON Stairlifts [Online image]. 2008. Retrieved April 14, 2014 from
http://www.drfix-it.net/acorn-stairlifts.html
[2] Curb Skipper [Online image]. 1999. Retrieved March 16, 2014 from
http://www.arabmedicare.com/curbskipper.htm
[3] Jazz 1120 Electric wheelchair [Online image]. 2014. Retrieved March 20, 2014 from
http://wheelchairchic.com/the-versatile-capability-of-the-jazzy-1120-electricwheelchair/
[4] Kumar, V., Rahman, T., and Krovi, V. (1997). Assistive Devices For People With
Motor Disabilities. Assessment of international research and development in
robotics. Retrieved from http://www.wtec.org/robotics/us_workshop/June22/
[5] Lawn, M. (2002). Study of stair-climbing assistive mechanisms for the disabled.
Retrieved from
http://ieeexplore.ieee.org/xpl/login.jsp?tp=&arnumber=1231243&url=http%3A%
2F%2Fieeexplore.ieee.org%2Fiel5%2F7333%2F27583%2F01231243.pdf%3Farn
umber%3D1231243
[6] Showing the correct way to assist the fire evacuation of a wheelchair user in an
evacuation staircase [Online image]. 2002. Retrieved April 14, 2014 from
http://www.cjwalsh.ie/tag/irish-building-regulations
[7] Suyan, Y., Wang, T., Wang, Z., Wang, Y. and Li, X. Track Tension Optimization for
Stair-Climbing of a wheelchair Robot with Variable Geometry Single Tracked
Mechanism. IEEE Xplore digital library. Retrieved from
http://ieeexplore.ieee.org/xpl/
[8] Watanabe, L. (2003). Independence technology discontinues the iBOT. Mobility
Management. Retrieved from
http://www.mobilitymgmt.com/Articles/2009/02/01/Independence-TechnologyDiscontinues-the-iBOT.aspx
[9] Wechsler, K. (2003). Reaching new heights with the iBOT. Quest. Muscular
Dystrophy Association. Retrieved from http://quest.mda.org/article/reaching-newheights-ibot
57
[10]
Wheelchair ramp information. Retrieved March 4, 2014 from
http://www.brainline.org/content/2008/07/wheelchair-rampinformation.html
[11]
Robert L. Norton. “Design of Machinery”, Fourth Addition. Pearson Education,
Inc.: New Jersey, 2011
58
Appendix A – Prototype and test pictures
18”
12”
Figure a – Prototype size
Figure b – standard wheelchair size
Wheelchair measurement, 11/17/2013,
http://suntran.com/pop_access_measurements.htm
59
Servos link 2
Breadboard
Arduino
Drive motors
Servos link 3
Figure c – Completely assembled prototype
60
Scaled stairs
Figure d – prototype while testing (ascent)
61
Appendix B – Relevant Data
Terms
1
Description
PC1
Position of frame C1 with respect to link 1
C1
frame attached to link 1 having its origin located at the center of
mass of the link and having the same orientation as the link frame
2
PC 2
Position of frame C2 with respect to link 2
C2
frame attached to link 2 having its origin located at the center of
mass of the link and having the same orientation as the link frame
3
PC 3
Position of frame C3 with respect to link 3
C3
frame attached to link 3 having its origin located at the center of
mass of the link and having the same orientation as the link frame
M1
Mass of link 1 located at the C.G. of link 1
M2
Mass of link 2 located at the C.G. of link 2
M3
Mass of link 3 located at the C.G. of link 3
f x4
Component force at the end effector (caster wheel)
f y4
Component force at the end effector (caster wheel)
Table 1 – Symbols definition
62
Servo Motor
Description
Weight
4.8V Speed
(sec/60°)
6.0V Speed
(sec/60°)
13.6 oz.
.95 sec/60°
.72 sec/60°
4.75 oz.
1 sec/60°
.85 sec/60°
HS-805BB
5:1 ratio
Unlimited rotation
HS-5485HB
5:1 ratio
Unlimited rotation
Table 2 - Servo motor specifications
Trial
1
2
3
4
5
Reading
2.98
2.99
2.98
2.97
2.96
(lbf)
Table 3 - Force estimation trials
Operating range
No Load speed
Weight
6-12 VDC
10 RPM
0.504 lbs
Table 4 - Gear Motor Specifications
63