Design Space and Motion Development for a Pole Climbing

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i


Design Space and Motion Development for a Pole Climbing
Serpentine Robot Featuring Actuated Universal Joints


Gabriel Jacob Goldman


Thesis submitted to the faculty of the Virginia
Polytechnic Institute and State University in partial
fulfillment of the requirements for the degree of


Master of Science

In

Mechanical Engineering


Dr. Dennis Hong, Chair

Dr. Mary Kasarda

Dr. Robert Sturges





February 24, 2009

Blacksburg, Virginia


Keywords:
Robot, Serpentine, Climbing, Kinematics, Universal Joints


Copyright

2009, Gabriel Goldman

ii


Design Space and Motion Development for a Pole Climbing Serpentine Robot
Featuring Actuated Universal Joints

Gabriel Jacob Goldman

(
A
BSTRACT)


Each year,
falls

from elevated structures, like scaffolding, kill or

seriously injure over a thousand
construction workers. To prevent such falls, the development of a robotic system is proposed that can
climb and navigate on the complex structures, performing hazard
ous inspection and maintenance
in place
of humans. In this paper, a

serpentine robotic system is developed

that will be able to climb pole
-
like
structures, such as scaffolding and trusses, commonly found on worksites. Serpentine robots have been
proven to

be effective at traversing unstructured terrains and manipulating complex objects. The work
presented in this paper adds a new method of mobility for serpentine robots, specifically those with
actuated universal joint structures. Movement is produced by i
nducing a wobbling motion between
adjacent modules through oscillatory motions in the actuated axis of the universal joint. Through the
frictional interactions between the modules of the serpentine and the surface of the pole, the wobbling
motion lets the
serpentine effectively roll up the pole’s surface.

This paper investigates how to make a serpentine robot climb a pole structure. It discusses the structure
and design parameters of the robot and develops relationships between them. These geometric and
per
formance
-
based relationships are then used to create a design space that provides a guide for choosing
combinations of module dimensions for a desired set of performance parameters. From this, cases are
shown which give examples of how the design space can

be used for several different applications.

Based on the design space procedure, a serpentine robot, HyDRAS (Hyper
-
Dynamic Discrete Robotic
Articulated Serpentine) was designed and built. The robot was used to prove the validity of the design
space proced
ure and to validate the climbing motion algorithms. Several tests were performed with
HyDRAS that showed the practicality of the helical rolling motion, as well as the feasibility of serpentine
pole climbing. Observations and discussion based on the experi
ments are given, along with the plans for
future work involving pole
-
climbing serpentine robots.


iii


Acknowledgments

I want to thank my family
.
They have shared every high and low point I have had. Anytime when I was
knocked down, they gave me the motivation
to get back up. They have always given me love and
support in endeavors I have pursued. Dad
, your knowledge is
sagely and your advice is spot on. Mom,
you will always be there to help me stay centered, and provide
me with the motherly advice that I need
to

get through any situation. Eliza, I can always depend on you to take time to talk to me about what is
on my mind.

I also want to thank

my
advisor who has guided me through the graduate research process. It has been a
great experience being part of the
Wor
ld Famous

Robotics and Mechanisms Lab.
Dr. Hong, thank you for
providing me with plenty of advice, expertise, mentorship, and free food during the last two years. You
will always be the only person who can make me believe that a robot

is
sexy
.

Another than
k you goes out to all of m
y friends who have stuck around even after declining to go to
many social gatherings due to part machining, programming, robot testing, or thesis writing. I owe you
all a drink of your choice (assuming that you bring me a copy of
this thesis as proof that you read it).

One person who I would not be able to do all this work without is Dr. Brand. Thank you for your support
throughout my studies so far and giving me the opportunity to continue my involvement with FIRST.

Finally, I
wanted to thank my other committee members, Dr. Kasarda and Dr. Sturges. Dr. Kasarda, you
have been an amazing influence to work with and you provide a much needed change of perspective.
Dr. Sturges, please always have Mozart or Bach playing in the backgro
und when you talk to students. It
makes the words you say float on air with such artistry they will never leave my mind.






iv


Contents

1

Introduction

................................
................................
................................
..

1

2

Literature Review
................................
................................
..........................

4

2.1

Biolog
ical Serpentine Motion

................................
................................
................................
.......

4

2.2

Biologically Mimicking Serpentine Robots

................................
................................
....................

6

3

Geometric Relationships

................................
................................
.............

12

3.1

Nomenclature

................................
................................
................................
.............................

12

3.2

Assumpti
ons

................................
................................
................................
................................

13

3.3

Module Geometry

................................
................................
................................
.......................

14

3.3.1

Angle Between Modules

................................
................................
................................
.....

15

3.3.2

Module Contact Region

................................
................................
................................
......

15

3.3.3

Determination of Contact and Endpoint Locations

................................
............................

18

3.4

Design Space Analysis

................................
................................
................................
.................

21

3.4.1

General Bounding Conditions

................................
................................
.............................

21

3.4.2

Maximum Range of Motion

................................
................................
................................

25

3.4.3

Maximum Number of Modules Per Wrap

................................
................................
..........

26

3.4.4

Climbing Distance per Roll

................................
................................
................................
..

27

3.4.5

Minimum Helical Pitch

................................
................................
................................
........

29

3.4.6

Maximum Motor Torque

................................
................................
................................
....

30

3.4.7

General Bounding Limits

................................
................................
................................
.....

36

3.4.8

Summary of bounding conditions

................................
................................
.......................

36

3.5

Case Study

................................
................................
................................
................................
...

37

3.5.1

Case 1:
Maximum Volume Design Space

................................
................................
............

37

3.5.2

Case2: Low
-
Gravity/Space Operation

................................
................................
.................

39

3.5.3

Case 3: Fast Climb Rate

................................
................................
................................
.......

41

4

Development of the Climbing Motion

................................
.........................

46

4.1

Universal Joint Definitions

................................
................................
................................
..........

46

4.2

Kinematic Structure

................................
................................
................................
....................

49

4.3

Forward Kinematics

................................
................................
................................
....................

50

4.4

In
verse Kinematics

................................
................................
................................
......................

56

4.5

Simulation

................................
................................
................................
................................
...

70

v


4.5.1

Overview

................................
................................
................................
.............................

70

4.5.2

“Donut” Wrap Cases

................................
................................
................................
...........

70

4.5.3

Helical Wrap Cases

................................
................................
................................
..............

72

5

Mechanical Design

................................
................................
......................

78

5.1

Physical Limitations

................................
................................
................................
.....................

78

5.2

Concepts
................................
................................
................................
................................
......

78

5.3

Prototype Design

................................
................................
................................
........................

80

6

Experiments, Results and Discussion

................................
...........................

88

6.1

Helical Configurations

................................
................................
................................
.................

88

6.
2

Donut Climb

................................
................................
................................
................................

94

7

Conclusions and Summary

................................
................................
..........

97

8

Appendix A: Images of Constructed Prototype

................................
............

98

9

Appendix B: Simulation Code

................................
................................
....

100

10

Works Cited

................................
................................
...........................

107



vi


Table of Figures

Figure 1.1: Artist rendering of how a serpentine robot could be used in hazardous situations inspecting
scaffolding
structures (a) and bridge piers (b)

2

Figure 2.1: Diagrammatic reconstruction of trunk vertebra of scolecophidian.
Top left
, posterior view;
top right

dorsal view;
middle left
, anterior view;
mid
dle right
, lateral view;
bottom,
ventral view.

4

Figure 2.2: Concertina locomotion is usually used in confined areas (Jayne, 1988
).

5

Figure 2.3: Shan's Mechanical Snake, MS
-
1

8

Figure 2.4: Choset's USAR ETR performing an inspection exercise in rubble.

10

Figure 2.5: Choset's series of Snake
-
Bots

11

Figure 3.1: Geometry of a single modul
e shown with




14

Figure 3.2: General contact case, shown with zero helical pitch.

16

Figure 3.3: Detail of two adjacent links in the midpoint contact case

17

Figure 3.4: Module geometry projected onto the x
-
y plane

19

Figure 3.5: x
-
y projection o
f module geometry including Φ angle

20

Figure 3.6: Cartesian (Hooke’s) Universal Joint Model

21

Figure 3.7: Plot of mechanical advantage required to actuate the rolling motion for a given β angle

22

Figure 3.8: Increasing helical pitch for fixed module radius, pole radius, and module length.

23

Figure 3.9: Varying helical pitc
h for a serpentine with
N
=12 and fixed module geometry

23

Figure 3.10: Height between module




and






24

Figure 3.11: Increasing module length for fixed module radius, pole radius, and helical pitch.

25

Figure 3.12: Design space for limited range of motion,




26

Figure 3.13: Design space bounded by the maximum number of module per wrap limit,






is
shown.

27

Figure 3.14: Example difference in the climbing rate for two configurations (zero pitch case)

28

Figure 3.15: Design space for bounded climbing rate

29

Figure 3.16: Design
space that is bounded by the minimum helical pitch condition.

29

Figure 3.17: Free body diagram of an end module

30

Figure 3.18: Mechanical Advantage for





31

Figure 3.19: Mechanical Advantage for





32

Figure 3.20: Design space that shows valid combina
tions of parameters to keep the assumptions in the toque
analysis to produce real results

34

3.21: Relationship between the
module frame and the frame of the motors in their critical

35

Figure 3.22: Design space that is bounded by a maximum motor torque value,



.

36

Figure 3.23: Maximum volume design spa
ce for the values shown in table 4.1

38

Figure 3.24: Example case which fits within the design space for


















39

Figure 3.25: Design Space for the low gravity or space operation

40

Figure 3.26: Example design case for the space/low gravity operating case

41

Figure 3.27: Design space for configurations that satisfy a climb rate


.

42

Figure

3.28: Example configuration for the design space for a minimum climbing rate with

43

Figure 3.29: Design case for the parameters
shown in table 3.4.

44

Figure 3.30: Example configuration from within the combined condition design space shown in 3.29

45

4.1: Rotation error between an input and output rotation for 10 axles connected by universal joints with the
arrows pointing in the dire
ction of axles closer to the output axle (





.

47

Figure 4.2: Comparison between single and double Cardian universal joints.

48

Figure 4.3: Thompson Coupling universal joint

48

vii


Figure 4.4: Rotation of the interm
ediate and output axles of a double Cardian universal joint for a constant rotation
about the input axle.

49

Figure 4.5:
Structure of a single module of the serpentine robot.

50

Figure 4.6: Two modules joined together to form a universal joint.

50

Figure 4.7: Example of a homogeneous transformation between two sucessive frames.

52

Figure 4.8: Geometric structure of the first module in contact with the pole structure.

53

Figure 4.9: Kinematic structure of two modules of serpentine robot

54

Figure 4.10: Relationship between




and

for varying helical pitches,


58

Figure 4
.11: Relationship between




and

for varying helical pitches,


58

Figure 4.12: Relationship between the maximum amplitude for the motor rotations and the helical pitch for
constant module and pole geometry.

59

Figure 4.13: Relationship between




and

for varying module length,
L

60

Figure 4.14: Relationship between




and

for varying module length,
L

60

Figure 4.15: Comparison between the maximum amplitude of motor rotation for varying module lengths

61

Figure 4.16: Relationship between




and

for varying module radius,
r
m

62

Figure 4.17: Relationship between




and

for varying module radius,
r
m

62

Figure 4.18:
Comparison between the maximum amplitude of motor rotation for varying module radius

63

Figure 4.19: Relationship between




and

for varying pole radius,
r
p

64

Figure 4.20: Relationship between




and

for varying pole radius,
r
p

64

Figure 4.21:
Comparison between the maximum amplitude of motor rotation for varying module radius

65

Figure 4.22: Motor rotation angles for the modules 1
-
3 (a) and 4
-
6(b) for the geometry given in table 4.5

66

Figure 4.23: Motor rotation angles for the modules 7
-
9 (a) and 10
-
12(b) for the geometry given in table 4.5

67

Figure 4.4.24: Resulting motor rotation angles for






for varying pitch angles from




68

Figure 4.25: Resulting motor rotation angles for






for varying pitch angles from




69

4.26: Maximum number of modules for a donut wrap for a set of module dimensions

70

4.27: Simulation of donut roll for


















for roll angles from






71

4.28: Simulation of donut roll for


















for roll angles from







72

4.29: Minimum
helical pitch required for varying module geometry with the pole radius held at unity

73

4.30: Simulation results for the minimum

helical pitch case for module and pole geometry
















for roll angles of






74

4.31: Simulation resul
ts for the minimum helical pitch case for module and pole geometry
















for roll angles of







75

4.32: Simulation results for helical pitch




case for module and pole geometry
















for roll angles of






76

4.33: Simulation results for helical pitch




case for module and pole geometry
















for roll angles of







77

Figure 5.1: Prototype of an actuated universal joint module featuring a flexible chain drive train

78

Figure 5.2: Prototype of an actuated universal joint module featuring miter gear drive train

79

5.3: Prototype featuring a kin
ematic inversion of the actuated universal joint

80

Figure 5.4: Design space used in the mechanical design for helical pitch great
er than zero

81

Figure 5.5: Design space for the mechanical design without the minimum helical bounding condition

82

5.6: Placement and orientation of motors for actuated universal joint

83

5.7: Detail of the motor assembly in line with the z
-
axis (a), x
-
axis (b), and y
-
axis (c)

83

5.8: Assembly of the b
earing block

84

5.9: Full assembly of a single motor module

84

5.10: Module link shown drawn with






85

viii


5.11: Assembly of the module body with body ring and skin cut a
way for clarity with module links in configuration
to the z
-
axis (a) and the y
-
axis (b) of the motor module

86

5.12: Two body
modules connected to each output axis of a motor assembly (shown with the skin removed from
the front body module)

86

5.13: Range
of motion of two modules connected to a single motor module

87

5.14: Constructed version of robot in a donut configuration

87

Figure 6.1: Experimental set
-
up for testing the helical configuration of the robot

88

Figure 6.2: Testing configuration of the robot shown in a holding rig with eight module links and seven motor
assemblies.

89

Figure 6.3: Simulated configuration of robot wrapped in the minimum helical pitch configuration

90

Figure 6.4: Robot configured in a 7.5
° helical pitch

91

Figure 6.5: Simulated configuration for a helical wrap of


91

Figure 6.6: Robot configured in a 15
° helical pitch

92

Figure 6.7: Simulated wrap of robot using a helical pitch of


93

Figure 6.8: Robot configured in a 30
° helical pitch.

94

Figure 6.9: Experimental setup of donut climbing case with a safety plate lowered beneath the robot.

95

Figure 6.10: Donut roll test using the parameters shown in table 6.5.

96

Figure 8.1: HyDRAS concet 2 with miter gears wrapped around a 4 inch pole

98

Figure 8.2: Structure
of HyDRAS version 2, with the acrylic body rings removed

99

Figure 8.3: HyDRAS version 2 shown in a more common serpentine configu
ration

99




ix



Table of Tables


Table 3.1: Maximum Volume Design Space

38

Table 3.2: Case 2, Minimum Overall Size/ Weight

40

Table 3.3: Case 3 for Specified Climbing Rate

42

Table 3.4: Case for combined conditions

44

Table 4.1: Relationship between helical pitch and maximum amplitude
of motor rotations

59

Table 4.2: Relationship between changing module length and maximum amplitude of motor rotations.

61

Table 4.3: Relationship between changing module radius and maximum amplitude of motor rotations.

63

Table 4.4: Relationship between changing pole radius and maximum amplitude of motor rotations.

65

Table 5.1: Design Space for Mechanical Design

80

Table 6.1: Robot configuration for helical tests

89

Table 6.2: Motor actuation angles for minimal pitch configuration (shown in initial position without roll)

90

Table 6.3: Motor actuation angles for minimal pitch configuration (shown in initial position without roll)

92

Table 6.4: Motor actuation angles for minimal pitch configuration (shown in initial position without roll)

93

Table 6.5: Configuration of robot for donut climb tests

94

Table 6.6: Holding torque for a donut wrap

95


1


1

Introduction

According to the most recent National Census of Fatal Occupational Injuries, 835 fatal falls were
reported, representing the series all time high since 1992.
F
alls from ladders, roofs, or scaffolding

accounted for

40% of occupational fatalitie
s by construction trade workers, who had an overall fatality
rate of 10.3 per 1
00,000 in 2007 alone. O
ther workplace hazards,
such as

falling and flying objects or
equipment and exposure to harmful environments
,

also increase the risk of fatal injury. Of t
he 933
workers killed by contact with objects or equipment, 386 were struck by either falling or flying objects or
equipment (representing 41% of the deaths in that category). Additionally, of the 488 reported deaths
due to exposure to harmful environments
, 367 (or 75% of the deaths in that category) can be attributed
to contact with overhead power lines, contact with temperature extremes, exposure to caustic, noxious
or allergenic substances, or oxygen deficiency. All together, 1
,
588 deaths
resulted

in 200
7

alone from
these workplace hazards
(Bureau of Labor Statistics, 2007)
.

Obviously, a system that could determine in advance whether given tasks are safe enough to be
performed by humans would reduce exposure to these dangers.

One
such system might be a network of

sensors, cameras, thermistors, or chemical detectors
installed
throughout the worksite.
Such
measures
are
usually
either too costly or unavailable for construction site managers to implement.
Another

solution
might b
e

to introduce an autonomous
, robotic

vehicle that
could

be kept at a worksite
,

on call
to

inspect any
potentially
hazardous situation before a human worker is put at risk. This
vehicle

would
need to be able to traverse the full worksite, including all of the s
caffolding and other s
tructures in place
to allow human workers acce
ss to the interior and exterior.
In addition,
it

would need to transfer
between those structures into
areas

that

could have a variety of unknown environments, surfaces,
obstacles, or hazards. Because of this, many conventional wheeled robots would be ineffective at
searching the worksite without the aid of elevators or lifts operated by human workers. An autonomous
air vehicle could be used
,

in theory
,

to fly around the extremities of the structure, but would have
difficulty entering
and navigating
the inside
,

where a hazard could be. Instead, a robotic vehicle is
needed with mobility in a variety of modes
to
propel
it
self
through unstructured
interior
environments
while still maintaining an ability
to
climb common workplace structures, like scaffolding.

W
orkers who help to perform search and rescue tasks on structures that have catastrophically failed, or
were subje
ct to malicious attacks
, are exposed to other hazards
. In the case of a building collapse, it is
not always possible to reach some levels of the building wi
thout the aid of a bucket truck

or other lifting
2


equipment

(National Inst
itute for Occupational Health, 2001)
. A problem arises
,

though
,

when
rubble
caused by the failure prevents the equipment from getting

close enough to provide an access point to
upper levels.

T
he same mobile robot that can scale common worksite structures
, like scaffolding,
could
possibly

scale pole
-
like structures
such as

piping, support beams, or fallen structural pieces to reach
inaccessible areas too hazardous for humans.

The proposed solution in both cases is to implement a serpentine, or snake
-
like,
robot to inspect those
hazardous environments
, as shown in Figure 1.1
. Serpentine robots have the ability to propel
themselves through many different unstructured environments
,

either by using snake
-
like motions, or
with

more novel
,

non
-
snake like
,

gaits.
One special property of serpentine robots is the
ir

ability
to use a
novel, whole
-

body rolling gai
t to wrap around and subsequently climb
pole
-
like structure
s
, like
scaffolding.



(a)







(b)




Figure
1
.
1
: Artist rendering of how a serpentine robot could be used in hazardous situations inspecting
scaffolding structures (a) and bridge piers (b)


All though this gait has in
itially been studied by Choset and Dowling,
there is no current process to
methodolo
gically design a snake robot
to

effectively climb a po
le structure. Moreover, most
research in
serpentine robot field uses fairly arbitrary measures to develop the geometric dimensions used for their
designs.
These dimensional restrictions limit a serpenti
ne robot's mechanical characteristics and,
3


consequently, the range of its climb
ing motions for inspecting hazardous environments
. This paper will
answer the question of how to effectively design a serpentine robot specifically for the task of climbing a
po
le
-
like structure, like scaffolding, so that serpentine robots can eventually be used to reduce the fatal
risks
to

construction trade workers.

This paper will first discuss the history of serpentine robots.
Discussion

will include previous milestones
in se
rpenoid research
,

as well as the current collection of gaits that they can perform. Next,
it will
present
an overview of the proposed solution, followed by the geometric and kinematic analyses that
allow for the novel serpentine climbing gait to be develop
ed.
It will then give a

method for

determining
the bounded design space

of a climbing serpentine robot and
a set of cases which prove its use
. Finally,
simulation and experimentation results are presented that validate the method used for both operating
a

serpentine as well as
choosing an appropriate design within its design space.




4


2

Literature Review

2.1

Biological Serpentine Motion

Biological snakes are unique
among
other land animals. Unlike legged animals that can walk or crawl
through an environment,
snakes use strictly their body motions to react against the ground to

propel
themselves
.
Their

unique skeletal structure
makes this possible
.
Comprising

of only three types of bones
-

a skull, vertebra, and ribs
-

the snake utilizes a naturally simple repe
ating structure to form a long
backbone that can be made up of 100 to 400 verte
brae. Even though each vertebr

alone can allow only
10 to 20 degrees of lateral and 2 to 3 degrees of ventral motion, the combination of multiple vertebra
e

produce
s

very large
overall angles. To achieve this motion, the bone structure of the vertebrae
Figure
2.1
form
s ball
-
and
-
socket joints with added projections that prevent most torsional motion
(Gray &
Lissmann, 1949)

while also protecting the spi
nal cord. This achieves the same structure as a mechanical
universal joint
,

since the torsional degree of freedom about the centerline of the snake can be
ne
glected. Additionally, the ball
-
and
-
socket joints in the vertebra
e

keep the backbone from stretchi
ng,
maintaining a constant body length.



Figure
2
.
1
:
Diagrammatic reconstruction of trunk vertebra of scolecophidian.
Top left
, posterior view;

top
right

dorsal view;
middle left
, anterior view;
middle
right
, lateral view;
bottom,
ventral view.

5





In nature, snakes perform a limited number of gaits to traverse their environment. The most
frequent of these gaits is
lateral undulation
. When performing this gait, a snake moves all of its
vertebrae simultane
ously while continuously sliding along the ground. All parts of the snake move at the
same speed through propagations of waves from front to rear. According to
(Walton, Jayne, & Bennett,
1990)
, the energy that is consumed from
this motion is comparable to the gaits of legged animals.
Walton concluded that this efficiency is high due to the snake’s ability to push

itself
off objects in its
environment, like rocks or tree trunks, to gain extra forward momentum. Additionally, Walto
n
concluded that the lateral undulation motion requires a minimum of
three

contact points for continued
forward progress. Two of the contact points actually generate the force,
while

the third balances the
forces.


Other common gaits that are used by snake
s in nature are the
concertina
,
sidewinding
, and
rectilinear

motions. The concertina gait, as shown in

Figure 2.2
, is a repeatable sequence of motions
where part of the snake base at a certain point while other parts move forward. This gait is normally
see
n in snakes that are in confined spaces, like tunnels. What makes this motion achievable in snakes is
the difference between the high forces and static friction experienced by the parts of the snake that are
based and the low forces and dynamic friction ex
perienced by the forward moving parts of the snake.
Since this gait changes momentum frequently, highly utilizes static friction forces, and travels at lower
speeds, it has been found to be a
n

inefficient mode of locomotion

(Walt
on, Jayne, & Bennett, 1990)
.




Figure
2
.
2
: Concertina locomotion is usually used in confined areas
(Jayne, 1988)
.



The rectilinear gait is performed when the movement of the
snake’s skin with respect to the
skeleton to
propel

the body forward along the ground. Unlike other gaits, several portions of the snake’s
6


body are in contact with the ground at any given time. The body moves forward due to a series of
symmetrical waves th
at progress through the body. The detailed movement of the rectilinear gait
requires that sections of the skin of the belly be drawn forward such that the scales on the skin are
bunched. The bunched group of scales
is

then pressed down, allowing the ventra
l edges to engage the
surface. Next, the body slides forward within the skin until it regains its original alignment.


Sidewinding motion, much like the undulating gait, utilizes a series of continuous and alternating
waves of lateral bending. These waves
react against the surface to propel forward by taking advantage
of rolling static contacts on surfaces with low shear. This gait minimizes slippage and is more effic
ient
than the undulating motion, and

is most commonly used on ground surfaces that have loo
se soils or
sand.


O
ther known
snake gaits
are not very common. The
slide pushing
gait is performed when the
snake propagates waves more quickly backwards than the snake moves forward.
Saltation

is an
interesting mode of locomotion where the snake performs a jumping or leaping motion through the
storage and release of energy in
its

body. This also inclu
des a free
-
fall stage where full control of the
snake is difficult. Some Asian tree snakes are
able to actually glide through t
he air by positioning their
bodies

in specific ways, with some even being able to expand their rib cage
s to form
gliding surface
s

(Socha & LaBarbera, 2005)
.


2.2


Biologically Mimicking Serpentine
Robots

The earliest conception of an automaton that could perform a snake
-
like movement was from Russian
artist Petr Miturich. His series of designs for mechanical undulators, termed
volnoviki
, moved primarily
through wiggling motions
(Lodder, 1983)
. Al
though he tried to patent his ideas for the
volnoviki
, no
patents
were granted, and his designs lacked any actuators or control.


Hirose and Umetami were the first to move beyond pure mimicry and derived the expressions for force
and power for a snake as a function of distance and torque along the curve described by the snake. Their
work was the first to truly develop limbless mobile

robots. Hirose continued their work and built several
serpentine robots which he called Active Cord
Mechanisms (ACM). H
e
also coined the term
serpenoid
curve
to describe the shape of a snake with curvatures that vary sinusoidally along its body axis. He f
irst
7


derived the expressions of force and power as a function of distance and torque along the serpentine’s
body length. These equations, termed the
serpenoid

equations, are shown in e
quation
s

2.1

and
2.2
.



(






(








(




(



)







(2.1)






(






(










(







(







)







(2.2)


These derivations were then compared to emperical results from natural snake locomotion.
Additionally, he developed models for
normal, tanegential, and power

distribu
tion of the serpentine’s
muscular forces along the body. He found that snakes can quickly adapt locally to changes in the terrain
they travel

on. One

important finding

from his studies showed that snake locomotion is not limited to
purely two dimensions. He found that during high speed movement, snakes utilize ventral motions
which allow better weight distribution without disrupting the forward movement.


After calcula
ting the required torques, power, and velocities from these equations, Hirose gave a set of
design guideli
nes for the actuators and drive
trains for serpentine robots performing undulating motion.
Hirose was the first to look at the control form for a serpe
ntine robot using angle commands at each
joint. The variables for control were closely related to the amplitude, wavelength, and velocity of the
body axis. Hirose was able to steer by biasing the control patterns to adjust the amount of curvature in
the bo
dy.


The first ACM that Hirose developed had 20 links, weighed 28 kg (later reduced to 13 kg with different
actuators), and featured DC motors and potentiometers for feedback. Small contact switches along the
sides of the ACM provided tactile feedback abou
t its environment. Using these sensors, the ACM was
able to navigate and propel itself through a series of winding tracks. Initially, Hirose’s work with the
ACM was limited to 2D motion, and his mechanisms were limited to only lateral undulation.


Shan dev
eloped a mechanism that utilized a form of concertina motion to bolster his work in obstacle
accommodation. His device used a series of single degree of freedom links and solenoids to drive
vertical pins into the surface it was travelling on to establish f
ixed contact points, shown in
Error!
eference source not found.
2.3
. From those contact points, the rest of the machine moves towards its
8


target. This

motion had a greater efficiency than true concertina motions in biological snakes, since most
of the friction that is experienced on the underside of snakes during the motion is provided by the linear
solenoid’s contact points
(
Shan & Koren, 1993)
.



Figure
2
.
3
:
Shan's Mechanical Snake, MS
-
1


Burdick and Chirikjian were able to take the field of serpentine robotics in the third dimension.
Chirikjian’s thesis work gave a framewo
rk for the kinematics and motion planning of a serpentine
mechanism. He described curve shapes in R
3

and allowed for roll distribution, extensions, and
contractions along curve segments to specify the serpentine shape. To fit the rigid devices to the curve

in R
3
, a model approach was used that resolved excess. The approach was able to characterize the
required shapes and paths without having to
resort

to full inverse kinematics. From this, some optimal
techniques for minimizing the measures of bending, exte
nsion,
etc.

were developed by using the
calculus of variation. Using this method, Burdick developed some novel patterns of geometry: the
travelling wave and the stationary wave. The travelling wave is similar to a caterpillar’s motion, or
rectilinear gait,

whereas the stationary wave is reminiscent of an inchworming motion.


Charikijan also introduced the idea that snake robots could be used to provide grasping and
manipulation possibilities while, at the same time,
maintaining

its mobility. In his work, h
e posed that a
9


snake could wrap around an object and use wave propagation to simultaneously grasp, move and
manipulate the object.


A variable geometry mechanism was built which was
composed

of linear actuators. The initial
implementation of a robot, Snakey, linked several of these mechanisms together. Even though the
mechanism was able to investigate some unique gaits, it was unable to do much more than prove only a
limited number of them
,

s
ince it was primarily a fixed based machine. Burdick was able to achieve
sidewinding motions by combining piecewise continuous curves to the mechanisms. The motion that
Snakey achieved was not
completely
snake
-
like, but the overall form of the motion was
indeed identical
to that of natural snakes.


Dowling’
s work also mentions some three
-
dimensional gaits. His work focused on finding optimal gaits
for serpentine robots based on
specific resistance
, which is the measure of the energetic cost of
locomotion a
s described in Equation
2.3
. Dowling states that it is misleading to compare specific
resistances across a variety of different systems, but for a single vehicle, it is a useful measure of how
well the vehicle is performing. In general, systems with a spec
ific resistance of
zero

are achieving pure
horizontal motion, where systems with a specific resistance of 1 are achieving pure vertical motion.











(
2
.3)









Using both genetic algorithms and probabilistic
-
based incremental learning (PBIL), Dowling’s
optimization resulting in the production of a variety of snake
-
like and non
-
snake like gaits for a 20 DOF
serpentine. The goal of his optimization was to not only
focus on locomotion, but also
to
pay special
attention to the robot dimensions


in the cases of pass
ing through a bent tube passage
way. Some of
the re
sults produced some novel three
-
dimensional gaits that include
d lateral rolling, travel
ing wave
rotor, wh
eel, flapping, and rolling collar gaits


which will be further discussed in the following section.
Most of these results were reproduced
only
in simulations
,

though, and not fully conceived with his
serpentine robot
(Dowling, 19
98)
.


Howie Choset’s research roots are with Burdick in sensor based motion planning for mobile robots. His
first known work was on the development of the hierarchical generalized voronoi graph
,

which is a
roadmap that can serve as a basis for sensor ba
sed robot motion planning
(Choset & Burdick, 1996)
. His
10


work in path planning continued on multiple robotic platfor
ms including the AERcam (a free
-
flying
inspection robot in space)
(Choset, et al., Ma
y 1999)
. More recently, Choset’s work branched into the
realm of serpentine robotics, first with the development of some novel 2 and 3 degree of freedom joint
design, shown in ,then his Urban Search and Rescue Elephant Trunk
-
like Robot (USAR ETR) shown i
n
Error!
Reference source not found.
, and most recently his series of serpentine robots, shown in
Error!
ference source not found.
.


Figure
2
.
4
:
Choset's USAR ETR performing an inspection
exercise

in rubble.


11



Figure
2
.
5
:
Choset's series of Snake
-
Bots


Dowling outlines some fundamental advantageous and disadvantages of snakes compared to other
animals. Unlike animals that walk or crawl in their environment, snakes are inherently stable creatur
es
due to their low state of potential energy. This can also translate into higher survivability in the case of
free falls from elevated surface
s
. Since the center of gravity and potential energy of the snake are low,
there is less chance
of

critically dam
age to a single critical connection point, like a leg joint in a walking
animal.



12


3

Geometric Relationships


The geometric analysis of the relationship between the physical parameters presented gives a method
for determining the spatial locations for all m
odule endpoints and contact points on the pole. These
locations are based on a number of variables, including those involving the pole geometry, module
geometry, and configuration of the serpentine. The relationships between those variables are presented
s
o that they can

be used in the creation of a design space for module parameters
, and the kinematic
analysis
for the climbing motion
. Additionally, the method for determining the angle between modules,
which will be proven to relate to the required range of

motion, is presented.

The nomenclature for the analysis is presented first and will be used throughout the paper. The
assumptions used for the analysis are presented next. Then, the definition of the module geometry and
the derivation of the full serpent
ine helical backbone curve will be given. Finally, the method for
determining the spatial location of module endpoints and contact points is given.

3.1


Nomenclature


Module


A single unit of the serpentine robot that is connected by actuated universal joints





Pole radius





Module radius




Module length, distance between the two axes of rotation of a single module

Linearized Helical Backbone Curve





The set of line segments in



that represent the centerlines of all





modules when helically wra
pped around a pole

α


Helical pitch of the linearized helical backbone curve




Length of the region on a module which can contact the pole surface




Number of modules in the serpentine robot

13





Module number,
(














Roll angle about the helical backbone curve for module

.








Roll angle about actuated axis
m

of the universal joint,
(











Angle between the centerlines of modules


and











Spatial point and its corresponding position ve
ctor in



that represent the location



where module


connects with module


+1






Orthogonal projection of



onto


{x,y}









Spatial point and its corresponding position vector in



that represent the location



where module


con
tacts the pole’s surface.






Orthogonal projection of



onto


{x,y}








Spatial point and its corresponding position vector in


that represent



projected to



the centerline of the module.






Orthogonal projection of



onto


{x
,y}





Interior angle between




and







3.2


Assumptions




The serpentine is designed to climb a pole with a circular cross
-
section and a constant
radius of





The pole is positioned such that at its base, its centerline is at the global origin,










The z
-
axis is parallel to the vertical direction of the pole



The x
-
axis is perpendicular to the centerline of module 1



Each module has the same geometric parameters



Each module has equivalent helical pitch

14




Modules have circular cross sections at t
he points where it can come in contact with the
pole to allow for them to roll up its surface



Each module contacts the pole at a single point



Each module contacts the pole at its midpoint since this is the ideal case where the
weight loads are evenly distr
ibuted

3.3

Module Geometry

Due to the unique way in which the serpentine robot will climb the pole structure, each module is
required to have a certain overall cylindrical shape. Each module contains two actuators, each of which
act about a single axis of two
separate universal joints. One axis of two separate universal joints is at
opposite ends of the module. The module’s shape is therefore defined by a module radius,


, and a
module length,

. The module length describes the distance between the two axes
of rotation for the
two actuators at opposite ends of the module. To prevent interference between adjacent links, the
cylindrical part of the module is limited to a defined area called the module contact region, as shown in
Figure
3
.
1
. Contact with the pole is limited to this region of the robot. Determination of the size of the
module contact region is discussed later in Section 3.6.


Figure
3
.
1
:
Geometry of a single module shown with





15



The endpoints of the module,


, and module contact point with the pole,



, are defined for
each module. Module endpoints





and



describe the endpoin
ts for module

. The location of the
contact point of the module that is projected onto the centerline of the module’s axis is defined as


.


3.3.1

Angle Between Modules


A parameter that is needed to effectively design and operate a pole climbing serpentine

robot is the
angle in between adjacent modules. This angle can be synonymous with the range of motion for each
joint. This parameter directly affects the size of the module contact region that can be achieved. The
angle between two adjacent modules, defin
ed as


, is measured from the neutral position where the
modules would otherwise be in a straight line. The angle can be determined using the law of cosines,
using the vectors from endpoint to endpoint of module

,





















, and the vector from endpo
int to
endpoint of module



,





















. The calculation of



is shown in equation 3.1.











































































































(3.1)

For the more specific case where every contact point is at the midpoint of the module a
geometric
approach provides a simpler derivation for β, as shown in equation 3.2.








(





(
(




)


(





)

)
)






(3.2)

3.3.2

Module Contact Region


As discussed in the assumptions, each module has a cylindrical shape to allow for it to roll up a pole
surf
ace during climbing. Limits to the dimensions of each module are based on the module’s length and
radius, the required range of motion, and the area of contact with the pole. Even though each module
makes only a single point contact with the pole, that con
tact point location can vary during operation,
which in turn can change the resulting


angle significantly. The ideal case would be when the projected
16


contact point location on the centerline is at the midpoint of the module. This case, as demonstrated
in
figure 3.2, results in a single β angle that is consistent among all of the modules of the serpentine when
wrapped around a circular structure.


Figure
3
.
2
:
General contact case, shown with zero helical
pitch.



When



is not at the midpoint location, two separate β angles are produced, as shown in figure 3.3.
These angles are repeated in an alternating pattern throughout the modules of the serpentine. The
pattern is determined from the alternating location of


, ca
using every other


angle to be equivalent,
as shown in equation 3.2.













(3.2)

17


All though having a middle contact point case is ideal, in real use the robot will not always make contact
at that location. Because of this, a module contact regi
on length,

, needs to be accounted for. This
length, in theory, has no lower limit beyond that a point contact will be difficult to achieve with standard
building methods. The upper limit, however, is bound by the desired range of motion for each module.

Selecting the correct module contact region length can avoid physical interference during operation.




For the non center contact case, one of the two β angles will have a larger magnitude. In the case shown
in figure 3.2, it is apparent that






is
the larger than



. This larger value will be used as the critical β
value in determining the maximum range of motion allowed for the wrap around the pole.


Figure
3
.
3
:
Detail of two adjacent links in the

midpoint contact case


The module contact region should be chosen such that the required range of motion for each module
can be achieved, as determined by the critical β angle. This relationship is shown in figure 3.3. The
length module contact region ,

, must follow the limit where



. As



, the maximum



.
Similarly, as




the maximum allowable



. The relationship between module length, module
18


radius, range of motion, and maximum contact region length is described in equation 3.4. This
assumes
that at the maximum β angle the modules make physica
l contact as shown in figure 3.3
.









(




)





(3.4)

After the module contact area is accounted for in the module design, it is assumed that all modules
contact the pole at the midpoin
t of the pole, constraining






















for all
n
.

3.3.3

Determination of Contact and Endpoint Locations


A geometric method is used to determine the spatial contact and endpoint location for the modules of
the serpentine. The initial contact point location can

be arbitrarily chosen on the pole’s surface, but for
the purpose of this research, the location is assumed to occur at Q
1
, spatially equivalent to {r
p
,0,0}, as
shown in figure 3.4

. By projecting the geometry of the modules onto the x
-
y plane, as shown i
n figure
3.4
, a simple trigonometric approach can be used to determine any contact point on the serpentine. The
goal of this approach is to determine the required maximum β for a given helical configuration. The first
step in determining the angle is to p
roject the module geometry onto the x
-
y plane. Each endpoint ,


,
contact point,


, and contact point projected onto the module centerline,


, is projected onto the x
-
y
plane, with the projected point being denoted as



,



, and



, respe
ctively, as shown in figure 3.5.


1
9



Figure
3
.
4
:
Module geometry projected onto the x
-
y plane


For the initial pole contact at


, the spatial location is
{









}
. Since both


and



are on the x
-
y
p
lane, both







and






. The following module endpoint,


, is defined geometrically by the
overall helical pitch, α, and the module length,

, as described in equation 3.5.







{











}




(3.5)





is determined by projecting its spatial location onto the x
-
y plane as shown in equation 3.6.























(3.6)


A geometric relationship is used to determine the locations of the following module’s contact and
endpoints projected onto t
he x
-
y plane
, as shown in figure 3.5
. This relationship includes
Ф
as









with










as defined in equation 3.7.

20








(




(






)





(3.7)



Figure
3
.
5
:
x
-
y projection of module
geometry including Φ angle




A general equation is defined for module endpoints,


, contact points,



, and contact points
projected onto the module centerline,


, as shown in equations 3.8 through 3.10 for modules with



. The vectors are show
n in cylindrical coordinates for clarity.




{




(





(







}




(3.8)






















(3.9)

21









{





(





)



(










}



(3.10)


3.4

Design Space Analysis


A procedure is developed in this research that gives a design space
that can be used to choose
mod
ule geometry that satisfies
a desired set of performance criteria. From the
previous
section, relationships between the module geometry and the pole geometry were developed.
Additional performance relationships are presented in this section, which in turn will lead to
developing the design space. The relationships provide a series

of inequalities that lead to a
constrained design space.
For all of the relationships, the value of the pole radius,


, is held to
a constant value of 1. Therefore all of the remaining values used in the design space analysis,








, are cons
idered a ratio of the pole radius.

3.4.1

General Bounding Conditions


When utilizing actuated unive
rsal joints, shown in figure 3.6
, to produce a rotational motion, there is a
limit to the maximum angle between the input and output shafts.


Figure
3
.
6
:
Cartesian (Hooke’s) Universal Joint Model


22


This limit for both axes of the universa
l joint demonstrated in figure 3.7

is




. At that point, the
universal joint is physically unable to turn about the input axes and produce a roll about the output axis.
An infinite mechanical advantage is needed to achieve a roll with





.

(the derivation of the
mechanical advantage is highli
ghted later in the chapter).

Additionally, there is no mechanical
advantage when



. Therefore, an actuated universal joint can only create the rolling motion when β
is with
in the range shown in equation 3
.1
1
.


Figure
3
.
7
:
Plot of mechanical advantage required to actuate the rolling motion for a given β angle




|


|










(3
.1
1
)

There is also a physical limit to the helical pitch, α. As the angle approaches





the serpentine’s
modules are arranged
in a near vertical configuration. At that point, it would take an infinite amount of
modules to make a single wrap around the pole, making the design unfeasible. The physical limit
is
therefore given in equation 3
.
12 and described in figure 3.8
.



|

|











(3.12
)

23



Figure
3
.
8
:
Increasing helical pitch for fixed module radius, pole radius, and module length.



In the case where there are enough modules to make at least one full wrap around the pole, there
is a
lower limit helical pitch constraint that will keep the modules from physically i
nterfering, as shown in
figure 3.9
. In the case of the configurations shown in figure
3.9
a
,
3.9
b
, and
3.9
c
, the modules physically
intersect with one another. In figure
3.9
d

the helical pitch is large enough to allow clearance. The
minimum helical pitch is determined for the case where there are more modules than needed for a
single
wrap (as defined in equation 3.13
).














(
3
.13
)

(a)




(b)




(c)





(d)

Figure
3
.
9
:
Varying helical pitch for a serpentine with
N
=12 and fixed module geometry


24


As shown in figure 3.10
, there is a minimum



, as defined in equation 3.14
, that keeps from physical
co
ntact from occurring.















(3.14
)


Figure
3
.
10
:
Height between module




and








The vertical distance between the two modules,


, can also be defined using the z
-
component
from
the derivation of a module’s contact point location (equation 3.10). Using the substitution for the z
-
component for module





’s contact point for


, a resulting limiting relationship between the
helical pitch, α, link length,

, module ra
dius,


, and angle between contact points, φ is de
veloped as
shown in equation 3.15
.

(



)














(3.15
)

Another constraint used limits the length of the modules when compared to the module radius, pole
diameter, and helical pitch. As the

module length increases while the module radius, pole radius, and
helical pitch are held constant, there is a critical value where the β angle between modules approach



, limiting the ability for the universal joints to rota
te as demonstrated in figur
e 3.11
.

25



Figure
3
.
11
:
Increasing module length for fixed module radius, pole radius, and helical pitch.


The constraining value of the modul
e length is given in equation
3
.16
. Using the center contact case,
a
ratio of the module length to the total of the pole and module radius greater than or equal to 2 causes
this condition to be violated. Increasing the helical pitch satisfies this condition can be satisfied for ratios
larger than 2.




(





)







(
3
.16
)

Each module must be able to physically perform the rolling motion without interference between
modules. As
mentioned earlier in section

3, a designer will be able to choose the length of the module
contact area for a given set of module dimensions.

Those relationships described in equation 3.4 are
also considered as a constraint in the
development of the design space
.


3.4.2

Maximum Range of Motion


The first relationship limits the design space bounds the range of motion of each axis of the
actuated univ
ersal joint to a maximum value,


. The relationship uses the derivation for



shown
earlier in

section

3, and is shown in equation 3.17
.












(3.17)

This relationship bounds the design space to the values shown in the volume
in figu
re 3.12.


26



Figure
3
.
12
: Design space for limited range of motion,







At very small values of


, the module length is limited to values that maintain the range of motion
constraint. As the module radius is increased, it effectively decreases the angle between adjacent
modules. This would have the same effect as increasing the pole radius. Similarly,

increasing the helical
pitch reduces the range of motion. The distance between adjacent projected to the x
-
y plane gets closer


3.4.3

Maximum Number of Modules Per Wrap


The
next

relationship that is used gives a limit to the module geometry based on a desired

maximum number of modules per wrap

around the pole structure. If this case is left
unconstrained, the design space will include geometries that would cause for physical
configurations that
could become impractical due to the very small module size compare
d to
the pole radius.

The number of modules per wrap is calculated by using the derivation for


derived in
a

previous section, as shown in equation
3.7.


The

bounding case
shown in equations
3.18 and 3.19

limits the design space to values that have numbers of modules per wrap less
than a maximum number of modules per wrap,




.

27















(3.18)














(3.19)

The design space for module geometries with th
e maximum module per wrap bounding case of





is shown in figure 3.13
.


Figure
3
.
13
: Design space bounded by the maximum number of module per wrap limit,







is
shown.

3.4.4

Climbing
Distance per Roll


A designer can also find the design space that will give configurations that can achieve a desired
height climbed per roll
, a
s shown in figure 3.14
. This value could be equated to the climbing
rate of the serpentine, but only in the case

where the torque required to roll up the pole is not
considered and it is purely a geometric relationship. T

28



Figure
3
.
14
: Example difference in the climbing rate for two configurations (zero pitch case)


T
he climbing speed is related to only the vertical component of the robot’s velocity. The horizontal (or
tangential) component results in an overall twisting motion about the center of the pole. For a single
rotation, the module will travel a distance of




. The vertical and tangential components of velocity,


and


, respectfully, are adjusted by the

helical pitch, as equations 3.20 and 3.21

show.















(3.20
)















(3.21
)

To estimate the climbing rate, only the vertical component of the motion will be considered,
setting





.
By limiting the amount climbed per roll to be







, a design
space can be created which gives acceptable module
config
urations, as shown in figure 3.15
.

29



Figure
3
.
15
: Design space for bounded climbing rate


3.4.5

Minimum Helical Pitch

As demonstrated in the previous section, there is a minimum pitch value that will keep the
modules from physically interfering with each other for multiple wraps. This condition, derived
in equation
3.15

is used as a bounding case for the design space, for

the specific cases where





and



. The resulting design space for this
condition is shown in figure 3.16
.


Figure
3
.
16
: Design space that is bounded by the minimum helical pitch condition.

30


Th
e resulting design space shows that the minimum
helical pitch condition is effected mostly by the
module radius and helical pitch, as shown in figure 3.16.

3.4.6

Maximum Motor Torque

The final bounding conditions on the design space result from limits on the act
uation torque. In
order to fully determine the exact motor torque required during operation, a full grasp analysis
would be needed. This would include developing a contact model to characterize the soft
contact between the robot’s surface and the pole’s su
rface. Since developing a new contact
model is outside the scope of this phase of the research, a simpler torq
ue model, as shown in
figure 3.17
, is used to determine an approximate torque required for the motors to hold a
single end module in the helical c
onfiguration. For this analysis, it is assumed that the universal
joint is in a configuration that results in maximum loading on the motors.


Figure
3
.
17
:
Free body diagram of an end module


The worst case scenario for the motors in which they experience the most loading can be
estimated by determining the total mechanical advantage for the universal joint. The
mechanical advantage of each motor can be estimated by the ratio of the actuation t
orque of
the motor compared to the desired rolling torque, as shown in
equation 3.22
.

31














̇




̇













(3.22)

This ratio can also be equivalent to the ratio of the actuator velocity compared to the desired
rolling veloc
ity of the module. For the climbing motion, a constant climbing velocity is desired,
therefore, the mechanical advantage of each actuator can be equated to the magnitude of its
velocity only. The derivation of the velocity of the actuated axes of the unive
rsal

joint is shown
in equations 3.23 Through 3.24
.







(





(

(





)



(






(

(





)

(3.23)


̇




(



(








)
(






(




)




(3.24)


̇







(

(







(









(3.25)

Figures 3.18 And 3.19

show the velocity of each motor for a desired roll angle for







.
The results show that the minimum mechanical advantage for each motor occurs at















.


Figure
3
.
18
: Mechanical Adv
antage for





32



Figure
3
.
19
: Mechanical Advantage for







To determine the motor torques,







and


, required to hold the end module in the helical
shape, an estimation of the module’s weight is used, as shown in equation 4.9, which calculates
its volume (modeled as a cylinder and two cones) and determines weight through an
approximation of the density,


, of the module. The weight is then split into two components,
that act in the x and z direction of the module,







, respectively
, as described in
equations 3.26 and 3.27
.



̃

(





)















(3.26
)














(3.27
)













(3.28
)

Coulomb’s friction law is used to form a relationship between the required gripping force,


,
and the vertical and tangential friction forces,



and



respect
ively, as shown in equation
3.29
.





















(3.29
)

33


Due to the complexities of determining the exact gripping force required from contact models,
the force is estimated by a designer and is considered an input to the system. For the purposes
of
the development of the design space
, the maximizin
g case shown

in equation 3.30

is used.





















(3.30
)

The reaction from the connected modules is assumed to be in the direction of the next module
in the chain. As previously mentioned in the geometric analysis section, each module is
connected at
an angle equal to β. Since the loads that act on the structure of the module are
assumed to be much smaller in magnitude than those acting on the motor, the loads that are
absorbed by the structure are assumed to be negligible. Therefore, the forces that a
ct in the z
-
axis of the module are considered negligible as well, and the resulting forces acting on the
universal joint are assumed to be a two
-
force member acting in the direction of the next
module with an angle of β and a magnitude of


.

The follow
ing system of equations
, shown in equations 3.31 through 3.36

is used to determine
each individual force and moment acting on the module, as described in the

free body diagram
in figure 3.17
.
























(3.31
)





















(3.32
)




















(3.33
)





















(3.34
)

























(3.35
)

























(3.36
)

34


With the set of assumptions that were made to allow for a solvable system
,

there is a critical

relationship that must be held true, as shown in equation
3.37
. When the condition is not met,
resulting values for



is complex and has non
-
real components.




































(3.37
)

The effect that this
bounding condition has on the de
sign space is shown in figure 3.20
.



Figure
3
.
20
: Design space that shows valid combinations of parameters to keep the assumptions in the toque
analysis to produce real resul
ts


The torques acting on the x,y, and z axes o
f the module shown in figure 3.17

are not the same
as the directions of the actuated axes of the universal joint. Therefore, the torques are
translated to the axes of the universal joint for when the mechanica
l advantage of the universal
joint is maximum,






. Figure 3.20

shows the motor frames and the module frames in the
configuration where







and is used to determine the relationships shown in equations
4.19 and 4.20.















(3.38
)

35












(










(3.39
)



3
.
21
: Relationship between the module frame and the frame of the motors in their critical

configuration when






.


The final bounding condition limits the design space based on the magnitude of the torque
acting on




. The resulting torque,






is bounded to the case where it is less than the
maximum torque that the motor can produce,




The design space that give the allowable
configurations for this condition is shown in figure
3.22
.

36



Figure
3
.
22
: Design space that is bounded by a maximum motor torque value,




.


3.4.7

General Bound
ing Limits

In addition to these constraints, there are physical limits to the module length and radius. Both
the module length and radius must be large enough to accommodate the size of the actuators.
Therefore, the bounding conditions












. An example of this is shown in the
mechanical design section, since the physical orientation of the motors limits the minimum
module radius.

3.4.8

Summary of bounding conditions


The following list summarizes all of the bounding conditions that are us