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					DESIGN OF A BIOLOGICALLY-INSPIRED CLIMBING HEXAPOD

          ROBOT FOR COMPLEX MANEUVERS



                              by

                      Eric David Diller




      Submitted in partial fulfillment of the requirements
             For the degree of Master of Science.



              Thesis Adviser: Dr. Roger Quinn



           Department of Mechanical Engineering
        CASE WESTERN RESERVE UNIVERSITY



                        January 2010




                              1
Contents


   List of Tables                                                                         5


   List of Figures                                                                        6


   Acknowledgments                                                                        9


   Abstract                                                                              10


1 Introduction                                                                           11

   1.1   CWRU Climbing Robots . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

   1.2   DIGbot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

   1.3   Document Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12


2 Background                                                                             14

   2.1   Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

         2.1.1   Search and Rescue . . . . . . . . . . . . . . . . . . . . . . . . . . 14

         2.1.2   Industry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

   2.2   Biological Inspiration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

         2.2.1   Walking Gait . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

         2.2.2   Body Joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

         2.2.3   Surface Adhesion . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

   2.3   Distributed Inward Gripping . . . . . . . . . . . . . . . . . . . . . . . . . 20

                                            2
         2.3.1   Climbing Robots using DIG . . . . . . . . . . . . . . . . . . . . . 22


3 Design and Fabrication of DIGbot                                                        27

   3.1   Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

         3.1.1   Leg Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

         3.1.2   Body Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

   3.2   Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

   3.3   Body Joint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

   3.4   Foot Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38


4 Modeling & Simulation                                                                   42

   4.1   Static Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

   4.2   Planar Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

   4.3   Transition Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

         4.3.1   Effect of Body Joint Use. . . . . . . . . . . . . . . . . . . . . . . . 49


5 Control                                                                                 51

   5.1   DIGbot Gait . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

   5.2   Servomotor Feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

         5.2.1   Foot Force from Servo Torque . . . . . . . . . . . . . . . . . . . . 53

         5.2.2   Butterworth Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

   5.3   Kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

         5.3.1   Planar Climbing . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

         5.3.2   Transitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

   5.4   Foot Placement Optimization . . . . . . . . . . . . . . . . . . . . . . . . . 59

   5.5   Transition Trajectory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

   5.6   Foot Attachment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

         5.6.1   Leg control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

         5.6.2   Natural Environments . . . . . . . . . . . . . . . . . . . . . . . . 63

                                             3
6 Results                                                                               65
   6.1   Tarsus Angle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
   6.2   Body Motion During a Step . . . . . . . . . . . . . . . . . . . . . . . . . . 66
   6.3   Minimum DIG Force . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

         6.3.1   DIG During Attachment . . . . . . . . . . . . . . . . . . . . . . . 71
         6.3.2   DIG During Stance . . . . . . . . . . . . . . . . . . . . . . . . . . 74


7 Conclusions and Future Work                                                           76
   7.1   Summary of Contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 76
   7.2   Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77


A Old DIGbot Designs                                                                    79


B DIGbot Drawings                                                                       83


C Inverse Kinematics                                                                    89


D Foot Designs                                                                          91

E Leg Force-Torque Solution                                                             93


Bibliography                                                                            94




                                            4
List of Tables

 2.1   Animal foot forces. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

 3.1   Component masses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31




                                          5
List of Figures

 2.1   Cockroach transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

 2.2   Insect attachment mechanisms . . . . . . . . . . . . . . . . . . . . . . . . 17

 2.3   Cockroach and DIGbot claws. . . . . . . . . . . . . . . . . . . . . . . . . 18

 2.4   Gecko setae and artificial adhesive. . . . . . . . . . . . . . . . . . . . . . . 19

 2.5   Beetle clinging to a glass ceiling. . . . . . . . . . . . . . . . . . . . . . . . 21

 2.6   Cockroach climbing on a screen mesh. . . . . . . . . . . . . . . . . . . . . 22

 2.7   Screenbot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

 2.8   Stickybot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

 2.9   RiSE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

 3.1   DIGbot on fence. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

 3.2   4-DOF Leg geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

 3.3   Mass proportions of DIGbot. . . . . . . . . . . . . . . . . . . . . . . . . . 32

 3.4   TS7260 Single Board Computer . . . . . . . . . . . . . . . . . . . . . . . 33

 3.5   DIGbot control structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

 3.6   Servomotor stiffness characteristics. . . . . . . . . . . . . . . . . . . . . . 35

 3.7   Half-duplex schematic. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

 3.8   DIGbot Transition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

 3.9   Tarsus deflection angles. . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

 3.10 Exploded foot design. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

                                           6
4.1   Static DIGbot normal forces. . . . . . . . . . . . . . . . . . . . . . . . . . 43

4.2   Leg lengths during turning. . . . . . . . . . . . . . . . . . . . . . . . . . . 45

4.3   Body turn achievable from available leg strokes. . . . . . . . . . . . . . . . 46

4.4   Planar forward motion and turn in place. . . . . . . . . . . . . . . . . . . . 47

4.5   Hexapod transition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.6   Effects on hip height during transition. . . . . . . . . . . . . . . . . . . . . 50

5.1   Leg kinematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

5.2   Body geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.3   Transition geometry. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

5.4   Foot axial angle σ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

5.5   Foot placement optimization space. . . . . . . . . . . . . . . . . . . . . . 60

5.6   Leg attachment algorithm. . . . . . . . . . . . . . . . . . . . . . . . . . . 63

6.1   Cockroach tarsus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

6.2   Calculated Tarsus angles during a step. . . . . . . . . . . . . . . . . . . . . 67

6.3   Body motion during forward steps and a stationary turn. . . . . . . . . . . 68

6.4   Body angle (a) and forward progress (b) during climbing. . . . . . . . . . . 69

6.5   Leg angle and length during a forward step. . . . . . . . . . . . . . . . . . 70

6.6   Screen angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72

6.7   Minimum DIG force required for climbing. . . . . . . . . . . . . . . . . . 72

6.8   DIG verification. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

6.9   DIG during stance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

7.1   DIGbot in natural environments. . . . . . . . . . . . . . . . . . . . . . . . 77

A.1 3-DOF Leg geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

A.2 ServoPodTM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

A.3 First DIGbot body design. . . . . . . . . . . . . . . . . . . . . . . . . . . 81

                                         7
A.4 3-DOF Leg geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

B.1 DIGbot body, dimensioned. . . . . . . . . . . . . . . . . . . . . . . . . . . 84
B.2 Frame holder, dimensioned. . . . . . . . . . . . . . . . . . . . . . . . . . . 85
B.3 4-DOF leg, dimensioned. . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

B.4 DIGbot frame, dimensioned. . . . . . . . . . . . . . . . . . . . . . . . . . 87
B.5 Foot, dimensioned. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

C.1 Leg kinematics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

D.1 Old foot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

D.2 Old foot mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
D.3 Foot design with retracting spines. . . . . . . . . . . . . . . . . . . . . . . 92




                                        8
                                Acknowledgments
   Luther Palmer for allowing me so much freedom on this project, and for teaching,
mentoring and guiding me along the way.
   Kati Daltorio for getting me started on climbing robots and for helping me in my own

research.
   Philip Dunker, Richard Bachmann and Alex Hunt for teaching me how to use the won-
derful CNC milling machine in the lab.
   All the members of the Biorobotics Lab for their help, good will and humor during this

project.
   Professor Roger Quinn for helping me find a project which interested me and in advis-
ing me.
   Special thanks to Professors Joseph Mansour and Roy Ritzmann for serving on my

thesis committee.
   My parents for supporting me since I started “engineering” projects in elementary
school.
   This work was funded by the Case Western office for the support of undergraduate re-
search and creative endeavors (SOURCE) and by the United States Intelligence Community

Postdoctoral Fellowship.




                                           9
               Design of a Biologically-Inspired Climbing Hexapod Robot
                                  For Complex Maneuvers




                                          Abstract
                                             by

                                  ERIC DAVID DILLER




   Some insects are able to climb on nearly any surface and in many orientations using
a variety of attachment mechanisms. In addition, these insects are capable of performing
complex maneuvers such as transitioning between orthogonal surfaces, behaviors which are
desirable in a climbing robot. A hexapod robot which climbs using biologically-inspired

strategies was built for this purpose. DIGbot, a 2.8 kg robot, uses distributed inward grip-
ping (DIG) to actively engage its spines with the surface. This enables the robot to climb
vertical and inverted surfaces of high roughness, such as wire mesh, carpet or tree bark.
DIGbot is the first robot able to perform tight turns on vertical as well as inverted surfaces.
This thesis presents the design, construction and operation of DIGbot. The performance of

the hexapod is quantified by analyzing its motion during these maneuvers.




                                             10
Chapter 1

Introduction

Overcoming large obstacles is one of the most difficult tasks for current mobile robots.
Looking to nature, many animals demonstrate tremendous ability to not only go around
obstacles, but also to go over them. The possibilities for robots which can scale obstacles
much larger than themselves are tremendous. While climbing robot technology is still

in its infancy, significant advances have been made which are exciting. One of the first
climbing robots to use adhesives was Mecho-gecko [1] which climbed using adhesive balls
on the end of spoked legs. While its performance was limited, this robot set the stage for
a number of other climbers. More recently, robots such as Spinybot [2] and RiSE [3] have
demonstrated reliable climbing on a variety of surfaces. In particular, challenges such as

climbing on rough surfaces, overcoming obstacles and transitioning over an edge are now
being undertaken, with promising results.



1.1 CWRU Climbing Robots

The first climbing robots in the Biorobotics Lab of Case Western Reserve University are a

continuation of the WhegsTMline of robots, which combine wheels and legs into a spoked
wheel-leg. These versions of WhegsTM, called Climbing Mini-WhegsTM, not only demon-
strate the versatility of the Whegs platform but also were the first robots to climb using

                                            11
biologically-inspired structured adhesives. Other versions of Climbing Mini-WhegsTMwere
built which used spines, VelcroTMand pressure sensitive adhesives(PSA) such as ScotchTM tape.
Different versions of Climbing Mini-WhegsTMare able to climb vertically and inverted on
glass surfaces and also navigate interior or exterior transitions between two orthogonal

surfaces.
   After Climbing Mini-WhegsTMcame Screenbot, which climbs using only spines. Screen-
bot demonstrates the viability of DIG for vertical and inverted climbing, but is limited to the
simplest of motion on these surfaces. A robot was desired which would use the same basic
principles as Screenbot, but accomplish much more. DIGbot was born to further explore

the progresses made on Screenbot [4].



1.2 DIGbot

The primary purpose of this work is to use distributed inward gripping (DIG) in a robot
capable of performing complex manuevers. As this is not an adhesion strategy project, the

adhesion mechanism used was chosen for its relative ease of implementation on a robust
platform. While the foot and spine design was a fair amount of the project, the bulk of the
project was developing the body design and control strategies required to implement DIG.



1.3 Document Overview

This thesis presents the design, construction, testing and results of the DIGbot project.
   Chapter 2 presents a background on climbing insects and climbing robots and sets the
stage for DIGbot. Chapter 3 presents the mechanical design of DIGbot. During its devel-
opment, DIGbot went through several significant design changes, and the most recent is

the primary focus of most of this document. For some of the past versions of DIGbot, see
appendix A. Chapter 4 presents the simulation which was done to determine the proper
design and control of DIGbot. This includes kinematic simulations to determine body joint

                                              12
and hip positions as well as calculating optimized trajectories for body motion. Chapter
5 presents the control strategies used on DIGbot, including the implementation of the tra-
jectories generated by the methods of chapter 4 as well as strategies by which successful
climbing was achieved. Chapter 6 presents the results of DIGbot, including character-

ization of the climbing of DIGbot. In addition, some comparison is made with climbing
insects. Chapter 7 presents a summary of the contributions made by the DIGbot project and
presents work which can still be done on DIGbot. The appendices contain additional infor-
mation concerning the design of DIGbot, control strategy details and detailed drawings of
the body.




                                           13
Chapter 2

Background


2.1 Applications

There are very few natural terrains that cannot be traversed by legged animals. The abil-
ity to use obstacles as stepping stones is a valuable locomotion tool that few wheeled and
tracked vehicles can match. Mobility is multiplied further if the animal can scale or climb

up obstacles too large to step onto. A legged robot that can climb unstructured obstacles
or vertical surfaces, combined with the ability to work in remote and hazardous environ-
ments, would be valuable for planetary exploration, military reconnaissance, and time-
critical search and rescue missions. Robots provide many advantages over a human pres-

ence in many of these situations in terms of safety, cost and at times, effectiveness. Robots
are currently widely used in space and deep sea exploration due to the high cost and danger
associated with human presence. However, it is becoming more of a reality to use robots
in less extreme environments on earth as cost and effectiveness go up.



2.1.1 Search and Rescue

The 2001 World Trade Center disaster provided the first real search and rescue task in
which robots were used. However, the robots sent into the rubble of the towers generally

                                             14
failed to complete their missions due to a general lack of mobility[5]. The primarily treaded
robots got caught in piles of rubble and were unable to proceed. Any advances in robots’
ability to overcome obstacles help them be more effective in future emergency situations.
While robots currently used for search and rescue applications are relegated to the ground,

the ability to climb over obstacles many times larger than the robot itself would classify the
robot as a climbing robot.



2.1.2 Industry

Other tasks for climbing robots include industrial applications in cleaning and inspection
of pipes, building exteriors, storage tanks and other hazardous areas. Some robots have
been built to address these needs directly, with some success in applications such as tank
crack inspection [6], oil tank labelling [7] and building glass cleaning [8].

   Operation inside hazardous nuclear reactor sites is the only current widely used ap-

plication for climbing robots. Currently, climbing robots which use suction are capable
of inspecting reactor pressure vessels and even performing some maintenance tasks [9].
These tethered tele-operated robots are designed to climb on a specific surface and must be
manually placed on the surface, which presents hazards to the robot operators.




2.2 Biological Inspiration

2.2.1 Walking Gait

Cockroaches typically walk in an alternating tripod gait in which the front and rear legs
on one side works with the middle leg on the opposite side to support the body [10]. This

tripod of support alternates with the tripod formed by the remaining three legs. This is
one of their fastest gait, and is commonly used when walking or climbing at high speeds.
However, cockroaches do not always use a tripod gait. When they climb obstacles they

                                             15
alter their gait such that their climbing ability and stability are improved [11]. Because
of the complexity of the movement during maneuvers such as transitions, some legs need
to be moved more frequently than others, which makes an organized gait less efficient.
Therefore, during these maneuvers, the organized gait pattern breaks down and the legs

move when they need to.


2.2.2 Body Joint

In performing maneuvers like a transition between orthogonal surfaces as well as when

climbing over large obstacles, insects such as cockroaches have demonstrated the use of the
body joint [12]. In Fig. 2.1, five picture captures from high speed video show a cockroach
climbing over an external transition. The cockroach uses its claws on the climbing surface,
which is made of porous polystyrene. The cockroach uses the body joint to keep the body

close to the surface and to provide an easy reach for its front legs as they reach across
the transition. In the first three pictures, the cockroach extends its front leg onto the top
surface as far as possible and uses its body joint to get the maximum reach. This behavior
minimizes the number of steps the cockroach must take in performing the transition. This is

helful because it increases the speed of the maneuver and reduces the chances for climbing
failure. In addition, by flexing its body forward during the transition, the center of gravity of
the cockroach is moved forward which reduces the tipping moment through the transition.
This reduces the adhesive normal force required to maintain stability. Cockroaches exhibit
this same behavior to reduce the tipping moment when climbing over obstacles, which

prevents them from falling backwards.


2.2.3 Surface Adhesion

Insects use several different attachment mechanisms when clinging vertically or inverted.
Some of the attachment mechanisms of two insects are shown in Fig. 2.2. This figure
shows a combination of spines, claws and hairy pads which all are designed for different

                                              16
   Figure 2.1: Cockroach performing a transition over a Styrofoam block [13]. As the cock-
   roach appreoaches the transition, it extends its front feet forward and bends its body joint for
   maximum reach. This allows the cockroach to complete the transition in only a few steps.




   Figure 2.2: Insect attachment mechanisms [14] as viewed from above. Notice the combina-
   tion of attachment mechanisms, including claws, spines and hairy pads.


surfaces. The key here is the variety of mechanisms shown on a single foot. This allows
these insects to cling to a wide variety of surfaces, which a single mechanism could not

allow them to do.

   One method of adhesion used by climbing insects is claws or spines. These claws are
typically at the distal end of the legs and are intended for climbing on relatively rough

surfaces. A cockroach claw is shown in Fig. 2.3(a) and the claw of DIGbot in Fig. 2.3(b).
Notice that both have a claw at the very end of the foot and that the claw angle relative to
the foot is similar in each case. The cockroach claw served as inspiration for the DIGbot
claw.

   A handful of animals including geckos, spiders and beetles have attachment mecha-
nisms which can be classified as dry adhesives pads. These pads enable the animals to

                                                 17
   Figure 2.3: Cockroach (Periplaneta) (a) and DIGbot (b) claws. The cockroach claw served
   as inspiration for the DIGbot claw.


attach to surfaces without picking up contaminants like traditional ”wet” adhesives do.
Spiders and beetles [15] use small hairs for this, while geckos have hierarchical pads with
structures at the nanometer scale. Work by Kellar Autumn [16] has shown that the geome-

try of the gecko setae is the key to its amazing adhesive properties, not its chemical makeup,
as was previously suspected. Gecko setae adhere using Van der Walls forces, the weakest
of intermolecular forces. These forces are dominant in gecko setae adhesion because of
the large surface area in contact with the substrate. On the nanometer scale, a gecko setae
appears as a forest of tiny hairs, each which can deform independently such that when the

pad is pressed against a surface, many hairs contact the surface simultaneously. One im-
portant attribute of gecko pads is their directional nature. When attaching to a surface, a
gecko setae only supports normal loads after it has been loaded tangentially [17]. Unlike
many traditional adhesives, very little normal force is required to preload the adhesion.

This is advantageous because any applied normal force tends to push the animal away from
the surface being attached to. In addition, detachment with these directional adhesives is
accomplished with minimal energy expenditure by simply reversing the tangential loading.

   Recent artificial adhesives which mimic the pads of geckos have been used with some
success for climbing robot adhesion. Several groups have developed manufacturing tech-

niques to make these adhesives, which are typically made of a polymer material and are
comprised of an array of small stalks on the nanometer scale [18] [19] [20]. These stalks
individually contact the surface and provide attachment via Van der Walls forces just like

                                             18
                   (a) Gecko setae [21]                 (b) Structured dry adhesive [19]

   Figure 2.4: Gecko setae are composed of a hierarchical structure which allows for the si-
   multaneous contact of many fibrillar structures. Artificial structured dry adhesives attempt to
   mimic the geometry of gecko setae using polymer materials.



their biological equivalent. This type of adhesive, while not as effective as gecko setae,
do provide suffucient adhesion for some uses. One group has also used carbon nanotubes

arranged in a similar manner on a silicone backing [22]. Using this microstructure, shear
forces of up to ten times that supported by gecko setae with the same surface are can be sup-
ported, and normal forces roughly equal to that of a gecko setae can be supported. While
this strategy gains adhesion on smooth surfaces, adhesion to rough surfaces is the largest

difficulty with these materials and is the factor which currently limits the applications of
these materials.

   Several climbing robots have been built which use artificial dry adhesives [13] [23]
[20]. These robots typically only climb on smooth surfaces such as glass. While these
robots perform fairly well on vertical surfaces, the adhesives are generally not yet strong

enough to support inverted climbing. In addition, these robots are not yet as capable of
performing the variety of manuevers shown by robots using other attachment strategies,
such as tight turns and climbing over obstacles. The adhesives still have problems with

                                                19
fouling, although not nearly to the extent of similar designs which use “wet” adhesives.
However, new adhesives and robots are being developed which hope to address all of these
problems and soon, dry adhesives are expected to outperform wet adhesives as a climbing
adhesion mechanism. It is clear that dry adhesives will play a large part in many future

climbing robot designs.




2.3 Distributed Inward Gripping

Magnets [24], tape [25], suction [26] and microfiber pads [23] have been used on robots to

generate the normal adhesive and shear tangential forces necessary to climb. Biologically-
inspired directional attachment strategies have also recently yielded promising results on
a variety of rough surfaces [12] [2] [26]. When hooks or spines are used (cockroach [12]
SpinybotII [2]), an initial tangential force positions the hook into place such that large

normal adhesive forces can be generated. In the case of directional adhesive pads (gecko
[16], Stickybot [27]), the normal adhesive force is a function of the tangential pulling force,
so a significant pulling force must be maintained throughout the stride. In either case, the
nominally small force required to detach is not a function of the magnitude of the adhesion
forces developed during the leg stride. This helps the animal or robot to detach its foot

quickly, and without having to exhaust energy overcoming significant attachment forces.

   Distributed Inward Gripping (DIG) applies the concept of directional attachment by

directing legs on opposite sides of the body to pull tangentially inward toward the body.
Shear contact forces are actively opposed by each other rather than by the pull of gravity,
allowing an animal or robot to climb on surfaces of any orientation with respect to gravity,
including inverted. DIG is best suited to use directional adhesives because their adhesion

can be activated using the DIG force. Directional adhesive mechanisms which can be used
include directional dry adhesives as well as spines and claws. All of these methods of ad-
hesion only activate when pulled tangentially in a certain direction. When this direction is

                                              20
   Figure 2.5: Chrysolina fastuosa beetles [28] clinging to a glass ceiling with six feet (in A and
   B), three feet (in C and D) and 2 feet (in E and F). The blue arrows indicate the orientation of
   the shear force applied at the attached feet.


aligned with the DIG direction, climbing performance can be enhanced. One main advan-
tage of using DIG when climbing is that the adhesion force can be controlled actively by

the animal or robot by adjusting the magnitude of the DIG force.

   Fig. 2.5 shows a beetle clinging statically to an inverted glass surface. This experiment
[28] shows that the beetles generally attempt to distribute their contacting feet around the
center of mass. For example, when all but two adjacent ipsilateral feet are severed, the
beetle quickly falls from the ceiling. However, if the beetle has two intact non-adjacent
feet, it will stretch its legs to place the feet so that they are directed opposite each other

across the center of mass, as in subfigure (E). From these experiments, it appears that the
animal is using the opposing tension of the legs to increase the shear at the foot of the
animal so that larger tensile normal forces can be supported. This is DIG in action.

   Fig. 2.6 of a cockroach climbing on a screen mesh shows the movement of the cock-
roach body and middle leg when climbing. The head area of the cockroach is circled in

white to highlight the motion of the body. The yellow line is the distal leg linkage of a
middle leg. Note that the orientation of this linkage is unchanged throughout the motion of
the body.

   Table 2.1 compares the forces at the front legs for two well-documented vertical-climbing

                                                 21
   Figure 2.6: Cockroach climbing on a screen mesh, showing the movement of the cockroach
   body and middle leg when the claw is engaged. The head area of the cockroach is circled
   to highlight the direction of motion. The other highlighted area is the distal leg linkage of a
   middle leg, whose orientation with respect to the screen remains unchanged throughout the
   motion of the body, suggesting that the direction of force also remains constant.


animals: cockroaches and geckos. Both of these animals push outward with their legs when
walking on the flat ground but switch to pulling inward (DIG) on vertical walls [29][30].
For these animals, as well as for blowflies [28], the normal forces are about one quarter of
the tangential (shear) foot forces during stance. The shear forces have significant compo-

nents in the lateral direction and in the fore-aft direction.



2.3.1 Climbing Robots using DIG

A few robots have been developed previous to DIGbot which utilize DIG in some fashion to
aid in climbing. These robots all use a form of directional adhesion, and include Screenbot,

Stickybot and RiSE.


Screenbot

Screenbot, the CWRU Biorobotics Lab precursor to DIGbot, is a hexapod which climbs
vertically and inverted on screen mesh using spines. Screenbot was designed to show the

feasibility of DIG on a simple platform [4]. Using a simple 1 degree of freedom (DOF)
drive system, Screenbot’s leg linkages move the legs through a predefined gait which causes
each foot to pull inwards once it has made contact with the screen. All six legs are driven

                                                 22
                                 Geckosa          Cockroachesb          Fliesc
                             running vertically running vertically walking inverted
       Mass                         3g                 2g                0.1g
       Length                      5cm                5cm               1.3cm
       Tensile Normald             5mN                7mN              0.3mN
       Fore-afte                  20mN               24mN              1.3mN
       Inward Lateralf           0–13mN            12–30mN          Not measured
       Normal/Shear               22-26%            17–25%          At least 23%
       Lateral/Fore-aft           0–64%             50–120%

   Table 2.1: Climbing animal foot forces, measured during stance.
   a Gecko Hemidactylus garnotii data from [30].
   b Cockroach Blaberus discoidalis data from [29].
   c Blowfly Calliphora vicina data from [28], note that because fly is walking on the ceiling,

   the fore-aft force is not in the direction of the weight but due to opposing distributed inward
   gripping in the fore-aft direction. Lateral forces were not measured in this experiment.
   d Forces normal to the substrate supported by the adhesive attachment mechanism.
   e On a vertically climbing animal, these forces are directed up to keep the animal from slipping

   down the substrate.
   f Forces that pull the animal laterally towards the same side that the foot is on.




by a single motor via chains and sprockets. Possessing only a single DOF, Screenbot is

constrained to move in a straight line, and tuning of the leg linkages is required when
changing climbing surface or orientation. In addition, the lack of on-board processing
limits Screenbot to remote-control operation. Screenbot climbs fairly slowly, at speeds of
.08 body lengths per second.


Stickybot

Stickybot, a hexapod climber which uses directional adhesive pads, is capable of climbing
on smooth surfaces such as glass. Some lateral DIG force is exerted when climbing due to
the splayed arrangement of the toes [20]. In addition, Stickybot exerts an opposing force
in the fore-aft direction when climbing in the slightly inverted orientation. This is because

when Stickybot climbs in this orientation, the rear feet of the robot are reversed, such that
they oppose the front feet. When inverted, and the robot weight becomes in the normal
direction, but Stickybot is still able to create an inward gripping force which activates its

                                                 23
   Figure 2.7: Screenbot, the CWRU Biorobotics Lab precursor to DIGbot, is a hexapod which
   climbs vertically and inverted on screen mesh using spines.


directional adhesion. The focus of the Stickybot project is on creative adhesion peeling

for better performance. Stickybot climbs slowly, at speeds of up to .05 body lengths per
second.



RiSE


The RISE project [3] is a hexapod robotic platform which has been used with a variety
of feet attachments and control strategies, and has served as testbed for novel foot designs
using spines. Depending on the surface, RiSE climbs using either a single spine per foot or

using a compliant array of spines on each foot. RiSE sometimes directs leg forces toward
the body to increase the directional adhesion of its spines, and is able to climb vertically on
trees, brick, and stone surfaces. RiSE uses a simplified actuation scheme, and achieves the
proper climbing motion using only two DOFs per leg. RiSE has demonstrated significant

climbing ability on planar and cylindrical surfaces, and has the ability to make slight turns.
In addition to planar climbing, RiSE has executed an external transition between orthogonal
surfaces. During any climbing, novel control strategies are used to regulate forces on each

                                              24
   Figure 2.8: Stickybot [20] climbs on vertical and slightly inverted glass surfaces using struc-
   tured artificial dry adhesive feet. Its novel toe peeling allows it to attach and detach large pads
   with relatively little strength.


foot in both the vertical and inward directions [3]. In some of these control strategies, the
inward gripping force is purposefully increased to aid in adhesion. However, because RiSE
depends primarily on the weight of the robot to create the tangential force necessary to
engage its spines, it does not have the ability to climb on inverted surfaces. RiSE is the

robot most similar to DIGbot, although its focus is more on adhesion strategies rather than
complex maneuvers. Using separate foot attachments, RiSE can also walk quickly on the
ground, with its feet tucked under the body. RiSE climbs at speeds of up to .1 body lengths
per second but can also run horizontally at .5 body lengths per second.




                                                  25
Figure 2.9: RiSE [3] climbs vertical surfaces using spines. This versatile climbing testbed
has demonstrated its ability to climb on carpet, tree bark, stone among other surfaces.




                                            26
Chapter 3

Design and Fabrication of DIGbot


3.1 Structure

DIGbot accomplishes DIG through the use of pointed spines on each of its feet. This
allows for robust and simple attachment on the screen mesh surface for which the robot was
designed. Spines were chosen because they were determined to be the easiest method of

directional adhesion possible. This allows the project to focus more on complex maneuvers
rather than adhesion mechanics. Despite this choice, foot and spine design was still a
significant challenge.

   DIGbot is a hexapod robot with four active degrees of freedom per leg. Several ver-
sions of DIGbot were built during the course of the project, many of which are presented in

appendix A. The body of DIGbot measures 35 cm between the fore and hind hip locations
and has a width of 6 cm between hips. The leg joints are powered by Dynamixel servo-
motors from Robotis, Inc. Servomotors were chosen over DC motors, pneumatic cylinders
or other options because of their versatility and because they can easily be powered by

batteries for untethered operation. Using servomotors, the DIGbot design could be made
somewhat modular, with servomotors adapting easily to new designs. In addition, servo-
motors allow simple connectivity with a microcontroller and provide adequate torque and

                                            27
   Figure 3.1: DIGbot is shown posed statically on chain-link fence. This 25-DOF hexapod is
   designed to perform complex maneuvers on a vertical surface using the Distributed Inward
   Gripping (DIG) attachment strategy.


speed for the application. The serially controlled servomotors chosen have advanced com-
munication capabilities including load, velocity and position feedback as well as the ability
to enforce a user-set torque limit. The AX-12 servomotors used for adduction/abduction

are capable of delivering 16 kg · cm of torque in a 55 g package and the RX-28 used for
leg swing and body joint actuation are capable of delivering 28 kg · cm of torque in a 72 g
package.

   Custom DIGbot parts are made primarily from 6061 aluminum and Delrin. These ma-
terials were chosen for their light weight and workability. Parts were machined manually or
using a CNC mill. As many parts as possible were purchased off-the-shelf to reduce time

and effort spent on design and construction. The design of DIGbot went through several
stages. Initially, a one-leg model was built to test different servomotors and leg configura-
tions. In addition, servo control and feedback techniques were tested on this simple model.

                                             28
Next, a two-leg model was built to further test leg configurations and control before the full
hexapod was designed. Pictures and descriptions of these previous models can be found in
appendix A. The shape of DIGbot (specifically, the spacing and alignment of hip locations)
was decided by comparing the possible motion achievable in different configurations. In

order to prevent leg collisions, the hips were spaced fairly far apart in the axial direction.
However, by placing the hips as close together as possible, tighter turns can be achieved
without sacrificing forward motions. Therefore, a compromise was made and with the 4-
DOF legs, leg collision is kinematically possible. This situation must be avoided using
software control.



3.1.1 Leg Design

The six legs of DIGbot are all identical, for ease of design, construction, control and main-

tenance. The current design uses four servomotors as shown in Fig. 3.2. The four servo
motors on each leg control fore-aft swing and adduction/abduction. The fore-aft RX-28
swing servos, controlling angle ω0 , have a sweep limit of approximately 130 deg. The
legs can adduct completely underneath the body, which is necessary for complex maneu-

vers. Extensive testing of the AX-12 servos has been conducted and the characteristics of
the motors have been investigated. Specifically, the ability of the servomotors to generate
force feedback data has been verified (see section 5.2.) The brackets used for the legs are
a combination of custom-made aluminum brackets and purchased off-the-shelf injection-
molded plastic brackets from Dynamixel. The leg dimensions shown in Fig. 3.2 were

chosen after several design iterations, and can achieve a large workspace and allow for a
wide range of motions.

   By taking advantage of the internal bearings in the servomotors, the design avoids the

use of custom bearing assemblies outside of the servomotors. This allows for a very simple
leg design which consists only of servomotors and mounting brackets. This highly modu-
lar design can easily be reconfigured for different purposes and makes servicing and part

                                             29
   Figure 3.2: 4-DOF Leg geometry. Three AX-12 servomotors control adduction/abduction
   and foot approach angle (ω1 , ω2 and ω3 ) and one RX-12 servomotor controls fore/aft swing
   (ω0 ).


replacement very easy. The only exception to this use of internal bearings is in the ω0 joint
which, because of the desired geometry, requires the use of one external bearing. However,
the design of this bearing is very simple, consisting only of a steel shoulder bolt which ro-

tates in a recess in the Delrin body frame. This recessed Delrin bearing sleeve is adequate
because of the low angular speed encountered at this joint. After many hours of operation,
this bearing shows only negligible wear.



3.1.2 Body Design

The total mass of DIGbot in its current form is 2.9 kg, including on-board power and
control. A previous, simpler version of DIGbot with only three DOF per leg had a mass of

                                              30
2.1 kg. The three-DOF version is mentioned here because it was used to take some of the
data presented in the results section of this thesis. The masses of the individual components
of DIGbot are shown in Table 3.1. Notice that the mass of actuators comprise over 50% of
the total mass of DIGbot. The mass of DIGbot was kept low by using lightweight materials

such as Delrin and aluminum and by eliminating excess material where possible. However,
the mass of DIGbot is not a significant concern in this project because of the robustness of
spines as an attachment strategy. By increasing the DIG force, the adhesion force can be
increased to whatever level desired, within the torque limits of the actuators. In general, the
mass of DIGbot has not created problems with climbing. In fact, with each design iteration,

DIGbot tended to become more complex and heavier without affecting performance.

                      component                 mass (g) quantity
                      servo battery                 190         1
                      SBC battery                     76        1
                      AX-12 servo                     56      18
                      RX-28 servo                     72        7
                      mounting brackets/leg           67        6
                      frame                         105         1
                      serial cable                     4      25
                      foot                            12        6
                      SBC and extra electronics     217         1
                      other electronics               12        1
                      fasteners                     150         1
                      total                      2875 g

                  Table 3.1: Component masses for four-DOF per leg DIGbot.



   The frame of DIGbot is made of four custom brackets made from 6 mm thick Delrin,
held together by a central block. This arrangement results in a fairly compliant body, which
can flex about any axis. The six RX-28 swing servomotors bolt directly to these frame
pieces and the external joint discussed in section 3.1.1 is cut into the bottom brackets. The

batteries, power switch, indicator light, and SBC mounting plates connect directly to the
top brackets. The SBC mounting plates are an extension of the frame intended to protect
the board from damage due to falling. It consists of a rigid Delrin backing for the SBC and

                                              31
                                       batteries


                                electronics


                                                    servos

                                    body




                          Figure 3.3: Mass proportions of DIGbot.


a flexible connection to the robot frame. This allows the SBC assembly to deflect a large

amount on impact without damaging the board.




3.2 Control

Control of DIGbot is performed on-board using a single-board computer (SBC) from Tech-
nologic Systems, Inc. The SBC used is the TS7260, shown in Fig. 3.4. The TS7260 runs

on a 40MHz EP9302 processor from Cirrus and has a typical power consumption of 2W
[31]. The board features 64Mb of SDRAM, 30 digital input/output pins, 12 analog in-
put/output pins, 3 RS-232 and 2 TTL serial ports, 1 RS-485 port, 2 USB ports with wi-fi
support, and Ethernet capabilities. The SBC runs a version of the Debian Linux operating

system called TS-LinuxTMand can communicate wirelessly with a laptop computer. This
operating system compiles and runs C code, although it is capable of running code written
in other languages as well. The communication setup for DIGbot is illustrated in Fig. 3.5.
Wireless (or wired through RS-232 or Ethernet cable) interface with a laptop computer is

done using the Token2 terminal emulator from Choung Networks. File transfer is done
using the Core FTP LE software [32]. C code is written on the laptop PC in the Microsoft
Visual Studio environment, compiled using the compiling software provided by Techno-

                                              32
          Figure 3.4: TS7260 Single Board Computer from Technologic Systems, Inc.



logic Systems and then transferred to the SBC using FTP. Code is run on the SBC in Token
2. As it is collected, feedback and debugging information is transmitted back to the ter-
minal on the laptop for inspection or analysis. Alternately, large amounts of data gathered
during operation is stored in a text file on the SBC, and is transferred to the PC after the
experiment is run using FTP. Wireless communication is performed by creating an ad-hoc

wireless network, hosted by the SBC. The wi-fi hardware is in the form of a USB dongle,
which consumes about 1W during normal operation.

   DIGbot can also be controlled directly from a PC using Matlab. Serial data from the
USB port is converted to TTL levels using the USB2Dynamixel adapter from Robotis. As

an integrated environment, Matlab is helpful when running experiments on DIGbot which
produce data. However, wireless control is not possible in this configuration. Previously,
DIGbot was controlled using a different microcontroller, the ServoPodTM from NewMicros
Inc. This microcontroller was chosen for its support of a large number of hobby servo

motors. The original DIGbot design used traditional hobby servos, which are controlled
by individual pulse-width-modulation (PWM) signals. The ServoPodTM is unique in that
it has 26 PWM outputs. However, when the DIGbot design was changed to incorporate

                                            33
                            Figure 3.5: DIGbot control structure.


the Dynamixel serially-controlled servos, the PWM capability of the ServoPodTM was no
longer necessary. The computation power and additional capabilities of the SBC from

Technologic Systems made it worth the switch.

   When climbing untethered, two 3-cell, 11.1 V lithium polymer batteries are used to
power the SBC and servomotors. The batteries used are the eXtreme V2 model from Thun-

der Power Inc. Two batteries are used concurrently on isolated circuits in order to isolate
the motor power from the SBC power. This is done to prevent electrical noise from the dc
servomotors from being introduced to the SBC. The servo battery has a capacity of 2600
mAh and a mass of 190 g, while the SBC battery has a capacity of 700 mAh and a mass of
76 g. As seen in Fig. 3.3 the mass of the batteries accounts for approximately 10% of the

total robot mass. The servo battery typically lasts for 60-90 min of continuous climbing
motion, while the SBC battery lasts for 2-3 hours.

   Communication between the SBC and servomotors occurs over a pair of serial lines,

each running at 115200 bits per second (bps) [33]. The AX-12 servomotors use 5 V TTL
communication, which only requires a single data wire. The RX-28 servomotors use 10 V
differential RS-485 levels, which requires two data wires. Both types of communication

                                             34
   Figure 3.6: Servomotor stiffness characteristics. The torsional compliance of the Dynamixel
   actuator is defined by setting the parameters shown.



are half-duplex, meaning that data is sent and received on the same line. This requires that

the transmitting and receiving of data be coordinated such that the data directions do not
overlap. Each Dynamixel servomotor is assigned a unique id number between 1 and 25,
and this is used to tag information packets sent and received. Because all of the servomotors
on a particular line receive all of the data packets sent, they must only interpret the packets

with their unique id attached. Command data packets sent to a Dynamixel take the form
[id, command, value, checksum]. The command is the data register you’d like to write
to, the value is the value of that command (typically with 8-bit precision, on a scale of
0-1023) and the checksum is a mathematical calculation used to verify that the packet was
received properly. Return packets can be requested, and take a similar form. They contain

information requested as well as any errors, such as notice that excessive temperature or
load or an invalid data packet have been encountered.

   Data packets are generally 5-10 bytes long, such that at 115200bps, packets can the-
oretically be sent at 1000 Hz. This provides sufficient bandwidth for control, even when
half of the packets are status packets returning from the servos.

   The Dynamixel servomotors have a built-in parameter allowing for user-set stiffness

characteristics. The stiffness profile, shown in Fig. 3.6, outlines the behavior of the servo-
motor in the region of the commanded (goal) position. The parameters B and C define the
width of the deadband zone, in which no torque is applied. If this region is set too small,

                                               35
   Figure 3.7: Half-duplex external circuit schematic. The DIRECTION PORT, TXD and RXD
   lines each connect to a pin on the SBC, and the DATA, 9.6 V and GND (ground) lines connect
   to the servos.




the servomotors vibrate when unloaded. If the region is set too large, the servo exhibits a
large amount of slop. In the regions A and D, the applied torque increases linearly with the
position error. The height E and the slope of the lines in A and D can also be set by the user.

These parameters A–E are essentially defining the torsional stiffness of the servomotor and
have a large impact on the control and response of the legs.



   Because the Dynamixel servos use half-duplex serial communication, the full-duplex

signal from the SBC, which has separate transmit and receive pins, must be converted to
half-duplex using an external circuit board. The schematic of this board is shown in Fig.
3.7. The DIRECTION PORT, TXD and RXD lines each connect to a pin on the SBC, and
the DATA, 9.6 V and GND (ground) lines connect to the servos. The DIRECTION PORT

line specifies if data is being transmitted or received and is controlled manually in software.
This circuit prevents data from corrupting the improper port on the SBC TXD and RXD
pins. This circuit was built on a breadboard and mounted under the SBC frame on DIGbot.

                                              36
   Figure 3.8: DIGbot maneuvering over an external transition. The body joint is deflected
   which allows all of the hips to remain close to the surface.


3.3 Body Joint

DIGbot is designed to make interior and exterior transitions between orthogonal surfaces.
Taking inspiration from biology, the body joint flexes forward during external transitions
to lower the front legs toward the surface. During interior transitions, the body joint flexes

back to bring the middle legs closer to the surface. Lowering the head also moves much of
the body mass closer to the surface, reducing the tipping moment and therefore the normal
forces needed at the feet. In the pose shown in Figure 3.8, the fore legs are reaching forward
over the transition.

   The placement of the body joint is optimized in section 4.3, which suggests that the
optimal placement is near the center hips. The current DIGbot design incorporates this
idea by placing the body joint just 6 cm from the center hip location. With the body
joint in this location, DIGbot is able to achieve nearly 90 deg of body joint rotation in

each direction. This allows for greater flexibility and optimization during transitions. The
frame of DIGbot was designed in such a way that the front and rear halves are completely
independent and are stiff enough to allow rotation about the body joint. The body joint

                                             37
itself was made to take advantage of the internal bearings of the RX-28 servomotor. This
means that no external bearings were used, and the two body halves are connected through
the servomotor horn.

    In an ideal system, climbing using DIG should not exert any torque on the body joint
because inward gripping forces are in the lateral direction only. Therefore, the body joint

actuator will be bearing only the weight of the robot and imbalanced forces resulting from
a deflected body. The resulting torque is within the range of the RX-28 servomotor. While
the RX-28 body joint motor is not capable of lifting the entire front of the body by itself,
this is never required during actual operation because the stance legs aid in the body joint

deflection. Body joint designs using more than one servomotor or a stronger DC motor
were considered which would enable DIGbot to lift its front section using only the body
joint, but the additional weight and space required were not deemed worth it.

    The only DIGbot maneuver currently utilizing body joint motion is the transition. For
planar climbing in any orientation with respect to gravity, the body joint stays in the straight

position. For this reason, the body joint has a locking feature which disables any motion and
eliminates power consumption from the body joint servomotor, which is otherwise loaded
nearly continuously. The locking mechanism consists of a Delrin strip which is bolted into
place between the fore and mid swing servomotors. This bar prevents body joint rotation

due to its axial stiffness.




3.4 Foot Design

Much of this section was adapted from [34]. Foot design is critical to the climbing per-
formance of DIGbot. Directional attachment stipulates that the foot should only produce

a gripping force when pulled from a single direction, and DIG further stipulates that the
direction of pull be in opposition to contralateral legs. In order to properly maintain the
desired inward gripping direction, the axis of the foot must be angled perpendicular to the

                                              38
sagittal plane of the robot, regardless of the angle of the leg. The leg is pulled inward
during the stance phase, causing the spine to seek the inward wire and develop a gripping
force. As the body moves through a step, as in Fig. 3.9, the angle of the leg changes with
respect to the desired inward force and the spine rotates about its ankle to maintain the

desired orientation. When the spine is removed from the screen at the end of stance, the
spring returns the spine to its original angle. A number of ankle, foot and spine designs
were prototyped to satisfy these restraints. The current design allows for passive multi-axis
ankle rotation. This simple design results in proper attachment and maintenance of proper
inward gripping forces during the entire stance phase. Design and fabrication of the final

foot design was done by an EMAE283 undergraduate group [35].

   The two tarsus degrees of freedom are shown in Fig. 3.9. The arrow in each subfigure
shows the inward gripping direction. Subfigure (a) indicates rotation through the tarsus ver-

tical angle to two positions encountered during walking and turning. subfigure (b) shows
the rotation through the tarsus swing angle in two positions. Notice that in each subfigure,
the foot remains oriented with the inward gripping direction in both of the shown leg ori-
entations. These two angles represent the relative position of the foot and resulting inward
gripping force with respect to the leg.

   The foot design allows for passive spine reorientation using a sprung tarsus joint. The
flexible two-DOF tarsus joint is made of three key segments, shown in Fig. 3.10; a stainless
steel spine which is embedded in an aluminum foot is attached with a sprung joint to the
rest of the leg. A braided steel cable, held with set screws, runs through the middle of the

assembly and bears the axial load on the foot. The aluminum foot geometry is such that
the face of the foot containing the spine can be presented prone to the substrate for any leg
approach angle. This allows the spine length and orientation to be optimized for a single
foot-substrate angle.

   An earlier foot design, shown previously in Fig. 2.3 features a completely stiff design.
This simple design is used when the compliance in the new foot is not needed. For example,

                                             39
   Figure 3.9: As the body moves through a step, the angle of the leg changes with respect to
   the desired inward force, and the spine rotates about its pivot to maintain the desired inward
   gripping direction. The arrow shows the direction of inward gripping in each subfigure.
   Subfigure (a) shows rotation through the tarsus vertical angle to two positions encountered
   during walking and turning. The type of flexibility is similar to what is observed in the
   cockroach (Fig. 6.1). Subfigure (b) shows the rotation through the tarsus swing angle.


when climbing on tree bark, a stiff spine is required to allow the spine to penetrate the bark.

However, the stiff spine fails frequently due to the problems discussed above, and has never
resulted in robust climbing.




                                                40
Figure 3.10: Exploded foot design. The foot design allows for passive spine reorientation
using a sprung tarsus joint. A braided steel cable, held with set screws, runs through the
middle of the assembly and bears the axial load on the foot. The aluminum foot geometry is
such that the face of the foot containing the spine can be presented prone to the substrate for
any leg approach angle.




                                              41
Chapter 4


Modeling & Simulation


4.1 Static Model

A static analysis of DIGbot is performed assuming 3-point contact with the surface. The

main failure mode of a climbing robot is tipping away from the surface during climbing.
This mode is particularly prevalent in inverted climbing, where the normal adhesion forces
must support most of the weight of the robot. A static analysis is justified due to the
relatively slow speed of DIGbot and because the body of DIGbot is still during the most

critical climbing phases, foot attachment and detachment. The other failure mode is in
shear, which is typically not a problem for a robot climbing using spines because the spines,
once inserted, tend to resist tangential motion very well. Therefore, the shear failure mode
is not analyzed here.

   The geometry and free body diagram of DIGbot are shown in Fig. 4.1. The length of
the body is 2l and the weight mg of the robot is at a distance d from the screen surface.

The normal forces FN 1 , FN 2 , and FN 3 represent the normal adhesion forces exerted by
each of the three feet in stance. Shear forces are neglected, as previously stated. The
moments caused by these three forces about the bottom foot contact point are summed for

                                             42
   Figure 4.1: Static DIGbot normal forces during for arbitrary screen angle. This figure shows
   a free body diagram of DIGbot, with forces in red. Surface tangential forces are neglected.


the condition when the normal forces are on the cusp of failure:


                       mg · d cos θ + mg · l sin θ = FN 1 · 2l + FN 2 · l                   (4.1)


and assuming the middle and top normal forces FN 1 and FN 2 to be the same (FN 1 = FN 2 =
FN ),
                                                        3
                           mg · d cos θ + mg · l sin θ = FN 1 · l.                          (4.2)
                                                        2

This relationship, which describes when pitch-back failure will occur, is further explored
in section 6.3.1.

   One possible flaw in this model is that it assumes that the forces on the fore and mid
legs are equal. However, this is justified by noting that the body is fairly flexible, and that

the rear leg is not included in this assumption. In reality, the normal force on the front leg
FN 1 would usually be the largest, due to its larger moment arm. Many climbing robots try
to reduce this normal force on the front leg by using a tail, which helps to distribute the

                                               43
normal forces evenly among all three feet. A tail was not considered with DIGbot because
in general, the required normal force can be generated by increasing the DIG force. In
addition, this model assumes that the feet contact the surface directly below the hips. In
reality, the legs are capable of reaching forward and back from these locations.




4.2 Planar Simulation

Forward walking and turning require multiple trajectories and a comprehensive strategy for
computing these trajectories. Fig. 4.2 shows the computed leg length trajectories of the left

front leg for left and right turns. The top subfigure shows the leg lengths needed to generate
straight forward walking and turns to the left of 5, 10, 15, and 20 deg. A 20 deg turn means
that the body’s orientation turns 20 deg during one step, which is the motion produced by
one tripod support period. The bottom subplot shows the leg lengths needed to generate the

same body turns to the right. A genetic search algorithm generated these trajectories with
the limitation that the leg length has a finite available stroke. Available stoke is determined
by the kinematic limits of the leg servo system, as discussed in section 3.1.1. From Fig.
4.2, the front left leg needs an available stroke of approximately 2.5 cm, measured from the
minimum length in either turn direction to the maximum length in either turn direction, to

execute turns from -20 deg to 20 deg.

   The search for these trajectories is computed off-line and the equations for the trajec-

tories are approximated by a neural networks. For any offline calculations of leg angles, a
large number of data is generated. In order to lessen the storage required to implement this
data on DIGbot, the leg angles are stores in a series of neural networks. Neural networks
are a method of function approximation using linear combinations of a few simple func-

tions into one network. The differentiable functions used here are the constant f = b, linear
function f = kx and the log-sigmoid function f = 2/(1 + e−2x ) − 1. For a single motion
type, such as a left turn with radius 100 cm, the entire set of data for three legs can be saved

                                              44
   Figure 4.2: Leg lengths during turning. A genetic search algorithm computed the front left
   leg lengths off-line that turn the body to a choice of angles during one step. The function for
   the leg lengths is approximated online with a neural net.


simply as 78 constants expressing the relationship between these functions in the network.
The two inputs for each neural net are the desired turn angle per step and the normalized
phase of the support period. The normalized phase is a timer-generated counter from zero

to one during each step. Once the neural net is trained for the 9 turn angles between -20
deg and 20 deg shown in Fig. 4.2, the leg trajectory for any turn angle in that range can be
calculated using the inverse kinematics solution found in appendix C. However, the inverse
kinematics solutions were not used in Fig. 4.2 because only leg length is plotted.

   The optimal initial foot placement positions are found by a brute force method which
tests all of the possible foot positions and determines which generates the furthest body

motion per step. This method is further discussed in section 5.4.

   The search algorithm was expanded to compute the maximum turn angle the body could
achieve for a given available leg stroke. Fig. 4.3 shows this data for strokes between 2.25
cm and 10 cm, fitted with a quadratic trend line. DIGbot can usually achieve at least 5 cm

                                                 45
                                 45

                                 40

                                 35
           turn per step (deg)
                                 30

                                 25

                                 20

                                 15

                                 10
                                   2   4             6           8              10
                                           available stroke (cm)

   Figure 4.3: Body turn achievable from available leg strokes. The available stroke of the leg
   length primarily dictates how much turn the body can undergo in a single step.




of stroke, allowing for up to 25 deg per step, which is comparable to cockroach turns which
were measured to peak at approximately 20 deg per step [36]. In the best cases, the 4-DOF
per leg version of DIGbot can achieve up to 12 cm of stroke, corresponding to a turn angle
greater than 45 deg. A simulated forward step and turn in place are shown in Fig. 4.4. The

initial body position is shown as solid blue and the final body position is dotted blue. In
this figure, the shaded patches represent the planar workspace of each leg. The black dotted
line shows the bound of a 2 cm space reserved for inward gripping space. Therefore, only
the space inside the black dotted lines are valid initial foot placement positions. Subfigure

(a) shows a forward step of 14 cm, or 45% of the body length, while subfigure (b) shows a
turn in place of 35 deg. Although the figure only shows the initial and final body positions,
the search algorithm also verifies the achievability of intermediate positions.

                                                  46
              40                                           40

              30                                           30

              20                                           20

              10                                           10
    x (cm)




                                                 x (cm)
               0                                            0

             −10                                          −10

             −20                                          −20

             −30                                          −30

             −40                                          −40
                   20      0     −20                            20      0     −20
                        y (cm)                                       y (cm)


   Figure 4.4: Planar 14 cm forward motion and 33 deg turn in place. Red dots are foot touch-
   down positions and green dots are leg joint locations. The initial body position is in solid
   blue while the final body position is dotted blue. The shaded patches represent the planar
   workspace of each leg. The dotted line shows the bound of a 2 cm inward gripping space so
   that only the space inside the dotted lines are valid foot placement positions.


4.3 Transition Simulation

The complexity of a climbing transition between orthogonal surfaces warranted a full sim-

ulation of the motion. Most of this work was done by Luther Palmer using Matlab and
Robot Builder [37].

   Taking inspiration from biology, the body joint flexes forward during external transi-
tions to lower the front legs toward the surface. During interior transitions, the body joint

flexes back to bring the middle legs closer to the surface. Lowering the head also moves
much of the body mass closer to the surface, reducing the normal forces needed at the feet.
The strategy to compute the desired body motion for an orthogonal transition attempts to
reduce the leg lengths necessary for the maneuver. As stated previously, available leg stroke

is limited, and keeping the body hip locations close to the surface decreases the leg length
needed to maintain contact with the surface. Figure 4.5 shows the saggital plane view of
the body as it ascends over an exterior transition. Hip locations are shown as squares and

                                               47
              10




               0




             −10



                                     λ = 0.00
             −20
         y




                                     λ = 0.16
                                     λ = 0.33
                                     λ = 0.50
             −30
                                     λ = 0.66
                                     λ = 0.83
                                     λ = 1.00
             −40




             −50
              −50         −40        −30         −20        −10           0          10
                                                   x


   Figure 4.5: Hexapod transition. Seven of the computed body positions between the initial
   vertical position (λ = 0.00) and the final horizontal position (λ = 1.00). Hip locations are
   shown as squares and the body joint is shown as a green circle. The maximum hip-to-wall
   separation, or hip height, for this transition is approximately 8 cm and occurs in position 4.
   Dimensions are in cm.



the body joint is shown as a green circle. The legs are not shown because the goal of this

analysis is only to compute hip locations — leg trajectories and foot positions are gener-
ated after the body motion has been optimized. An iterative brute-force search algorithm
computed the trajectory of Fig. 4.5 given the body length, location of the body joint and
the maximum joint angle. The progress of DIGbot over the transition is parameterized by

λ, which varies from λ = 0.0 at the beginning of the transition to λ = 1.0 at the end of the
transition. The angle of the body joint is 0 deg when the system is flat, and the maximum
joint angle is a design parameter based upon kinematic constraints. In this simulation, the

                                                48
body joint is increased to 70 deg linearly during 0.00 < λ < 0.33, and decreased back to
0 deg during 0.66 < λ < 1.00.. For clarity, only a fraction of the computed positions are
shown. In Fig. 4.5, DIGbots 35 cm length is used, 70 deg is the maximum joint angle,
and the body joint is located at a distance 30% of the way from the head to the rear. This

routine records values of body position and orientation through the entire transition, which
are stored in a file and used as presented in section 5.5.


4.3.1 Effect of Body Joint Use.

The DIGbot design takes inspiration from a cockroach which flexes the body joint forward

during an external transition to lower the front legs toward the body.
   Figure 4.6 shows the maximum hip height during an exterior transition for a range of
maximum body joint angles (15, 30, 45, 60, 75, 90 deg) and body joint locations. The
location of the body joint is moved from the head (no joint) at a = 0 to the body’s fore-aft
bisector line at a = .5, where the middle legs are mounted, and the resulting maximum hip

height is shown as a percentage of the body length. The maximum hip height is calculated
using the methods previously presented. It is clear from this plot that for any maximum
joint angle, placing the body joint toward the center of the body reduces the maximum
hip height during the transition. This matches the results from another wall-climbing ve-

hicle, Climbing Mini-Whegs [12], which was able to maneuver over interior and exterior
transitions with a body joint.




                                             49
                                              0.34
                                                                                                   15 deg
                                              0.32                                                 30 deg
              maximum hip (shoulder) height




                                               0.3                                                 45 deg
                                                                                                   60 deg
                                              0.28                                                 75 deg
                                              0.26                                                 90 deg

                                              0.24

                                              0.22

                                               0.2

                                              0.18

                                              0.16
                                                0.1   0.15   0.2   0.25     0.3     0.35     0.4     0.45   0.5
                                                                   a (body joint position)


   Figure 4.6: Effects of body joint position and maximum angle on hip height during transition.
The location of the body joint is varied from a = .1 near the head to a = .5 at the center of
the body for different maximum body joint angles. In all cases, increasing a decreases the
maximum hip height.




                                                                           50
Chapter 5


Control


5.1 DIGbot Gait


DIGbot is capable of moving in two main gaits, depending on the type of climbing. Like a
cockroach, the primary gait used for planar walking is the tripod gait, which is the fastest of
the gaits used. In this gait, DIGbot always has at least three feet in contact with the surface.

When any of these feet reach the end of its workspace, DIGbot stops moving and the new
tripod is planted. When the new tripod is planted properly, the old tripod disengages and
DIGbot moves through a new step. However, for more complex maneuvers, DIGbot used
a more stable gait which has six legs in stance. This gait does not have an organized

structure or timing, but rather emphasizes stability. In it, DIGbot only moves a foot when
it is required e.g. when it has reached the edge of it’s workspace. During typical walking,
the attachment and detachment phases are the most time consuming, often occupying up
to 90% of the time. By minimizing the number of times feet are reset, DIGbot minimizes

the number of foot resets and therefore increases the speed of walking. As opposed to the
tripod gait, which moves all three feet in a tripod together, and is limited by the foot capable
of the least motion, the more complex gait means that each foot operates independently.

                                              51
5.2 Servomotor Feedback

DIGbot uses feedback from the servomotors in several ways. First, position feedback is
used during the flight phase to determine when and if the legs have reached their com-
manded positions. This is required in order to minimize the time spent waiting for the
legs to move and also to determine when a foot has failed to detach properly from the sub-

strate. Position feedback is read from the Dynamixel servomotors as discussed in section
3.2. The servos determine the position of the joint using a built-in potentiometer mounted
inline with the output shaft of the servo, as is common with servomotors. Position output
data for a moving servo requires filtering before use. Some noise in the data is suspected

to be caused by inaccuracies in the physical potentiometer contacts while other data noise
originates from the motor. The same power is used for the motor as to effect the voltage
drop across the potentiometer, so any noise resulting from brush contact is fed directly into
the potentiometer input.

   Position data is also used to estimate the torque on the servomotor. By comparing the
                                            ¯
commanded position θ to the actual position θ, a rough estimate is obtained. In general,
this method assumes a linear torsional motor stiffness k about the commanded position.
This stiffness can be adjusted in software as in section 3.2. Using this model, the torque on

the motor is
                                                 ¯
                                       τ = k(θ − θ).                                    (5.1)

The accuracy of this model depends largely on the nature of the PID control used by the

servo to regulate current to the motor as well as on other factors like friction. Verification
of this model, done by comparing the torque output, shows that torque data obtained using
this approximation agrees with that obtained using the other method, described next.

   The Dynamixel servomotors also are capable of directly outputting load data. This data
is not true load, and is simply proportional to the current drawn by the motor. Some form of
internal filtering of this data is inferred because the data is only updated at approximately

                                             52
7Hz, while other feedback such as position can be read at much higher speeds, (as fast
as the serial data transmission will allow.) If the load is read faster than 7Hz, the result-
ing output takes the form of a step function, ”idling” between each sample. Despite this
prefiltering, the load data received is not smooth enough for use, and must be further fil-

tered externally. Because the current through a DC motor is inherently discontinuous, it is
understandable that this data requires a large amount of filtering.

    While neither of the methods used for torque measurement result in highly accurate
data, the degree of accuracy required for climbing is low. Load data is typically only used in

a threshold application, where the load on a leg is measured to determine if surface contact
has been made or if the inward gripping force is sufficient. Details on these applications
are presented in section 5.6.




5.2.1 Foot Force from Servo Torque
                                                        T
For a given ground reaction force F = Fx Fy Fz              at a foot, the corresponding torques
on each servo are given by:                   
                                         τ1 
                                            
                                    τ =  τ2  = AF =
                                                                                         (5.2)
                                            
                                          τ3
                                                                                           
               v                            −u                          0             Fx 
                                                                                         

           w cos ω1                     w sin ω1                 r − proximal        F 
                                                                                      y 
                                                                                         
    distal cos(ω2 + ω3 ) cos ω1 distal cos(ω2 + ω3 ) sin ω1    −distal sin(ω2 + ω3 )     Fz

where τ is the torque exerted on each servomotor and F is the equivalent ground reaction
force at the foot. The conversion matrix A is computed by summing moments about the

axis of each servo, where u and v are the x and y coordinates of the foot in the leg plane,
as in Fig. 5.1. Given the servo torques τ , equation 5.2 can be solved for the equivalent foot
reaction forces F (see appendix E for the full solution of F.) The lateral– (y) component

                                              53
of the force (Fy ) is usually only component required, as this is the inward gripping force.

   In the four-DOF version of DIGbot, the fourth joint torque τ3 is not used in these equa-
tions. This is because the tarsus leg segment is always at a small angle relative to the
ground. Only three motor torques are required for the solution of three unknowns F; the

addition of a fourth equation would be algebraically redundant. Because the angle between
the foot and the ground is generally small, the torque on this motor is also small.



5.2.2 Butterworth Filter

Filtering of position and load data is performed using a software-implemented second-order
Butterworth filter. A Butterworth filter is a low-pass filter which is simple to implement in

software[38]. The filter has a transfer function whose magnitude is given by

                                                   1
                                |H(jω)| =                     ,                         (5.3)
                                              1 + (ω/ωc )2n

which implies a cutoff frequency of ωc . Using this filter with a cutoff frequency of one
third the sampling frequency ωc = 1/3 · ωsampling , the filtered data lags the original data
by about 4 data points. The Butterworth filter is used for both on-board live processing

and data post-processing. All of the data presented in this document is filtered as such.
The implementation of the Butterworth filter onboard requires storing only two previous
data values in addition to the current value, which results in a computationally inexpensive
implementation.




5.3 Kinematics

In the following kinematic analysis, rotation matrices are generated from three orthogonal
Euler angle rotations in the following manner. From the three rotations α about the x-
axis, β about the y-axis and γ about the z-axis which define the Euler rotation vector

                                             54
   Figure 5.1: Planar view of leg kinematics. The positions of the servomotors are parameterized
   by ω1 , ω2 , and ω3 (the ω0 swing angle is not shown here).

                   T
e =     α β γ          , the rotation matrix is given by the multiplication



           R = Rz (γ)Ry (β)Rx (α) =
                                                         
                cosγ sinγ 0 cosβ 0 −sinβ  1    0      0 
                                                         
           R = −sinγ cosγ 0  0
                                   1   0  0 cosα sin α
                                                                                           (5.4)
                                                         
                   0      0    1 sinβ 0 cosα   0 − sin α cosα


XYZ Euler angles contain a singularity at β = ±90 deg. However, the pitch angle during
climbing is generally kept less than 45 deg, which avoids the singularity region.



5.3.1 Planar Climbing

For the case of DIGbot climbing on a planar or near planar surface, the geometry is fairly

simple. Because the body is kept parallel to the surface, the only body parameter is the
body height from the surface, h, as shown in Fig. 5.1. From Fig. 5.1 we can obtain
the forward and inverse kinematics of the leg. These relationships are obtained using a

                                                 55
   Figure 5.2: The body of DIGbot, showing the ground and body frames. rb is the position of
   the ground in body coordinates. The green squares show the hip locations of DIGbot.


purely geometric approach, solving for the appropriate variables algebraically. The tarsus
angle relative to the ground is fixed at 30 deg . This allows the foot to maintain the same

attachment angle at all times. The full inverse kinematics solution is given in appendix C.



5.3.2 Transitioning

A representation of DIGbot is shown in Fig. 5.2. rb is the position of the ground in body

coordinates. The green squares show the hip locations of DIGbot.

   In Fig. 5.3, the body and ground from Fig. 5.2 are now overlaid on a transition. Here,

a hind foot is placed on plane 0. For this geometry, the transformation from ground to hip
coordinates is
                               fb = Rbj (Rbg fg + rb ) − phip,b ,                           (5.5)

where fb is the touchdown position in hip coordinates, Rbj is the rotation about the body
                                                             T
joint obtained from the Euler angles, ebj =      0 αbj 0         , Rbg is the rotation of the body
                                                                                             T
relative to the ground, given by the three Euler angles ebg =           roll pitch yaw           , fg

is the touchdown position in ground coordinates, rb is the position of the ground origin in
body coordinates and phip,b is the position of the hip relative to the body origin, in body
coordinates.

                                              56
Figure 5.3: The body and ground from Fig. 5.2 are now overlaid on a transition. Here, a hind
foot is placed on plane 0. Using the transformation of equation 5.5 (and its inverse, which
gives fg as a function of fb ) the foot position can be expressed in either ground coordinates as
fg or hip coordinates as fb .




                                               57
   Using this transformation (and its inverse, which gives fg as a function of fb ) the foot
touchdown position can be expressed in either ground or hip coordinates. The transition
motion by definition requires a pitch angle change of 90 deg as DIGbot moves between
orthogonal surfaces. However, when a foot touchdown position changes from plane 0 to

plane 1, the ground axis reorients such that the ground z-axis is always normal to the plane
being touched. Therefore, the magnitude of β rarely exceeds 45 deg.



Foot Plane Choice

When navigating a transition, DIGbot must choose whether a foot should be placed on
plane 0 or plane 1. This is done simply by calculating the pitch angle relative to the ground
of the hip in question. The hip pitch angle is the angle by which the hip pitches from its

nominal flat value. For planar climbing, the hip pitch angle is always 0 deg. If the hip pitch
angle is less than 45 deg relative to plane 0, the foot is placed on plane 0. If however the
hip pitch angle relative to plane 0 is greater than 45 deg, then that foot is placed on plane
1 (the hip pitch angle relative to plane 1 would then be less than 45 deg.) This ensures that
the foot axial angle σ (as seen in Fig. 5.4) does not become exceedingly large. σ is the roll

angle of an individual foot relative to the ground plane. This angle is important for spine
attachment because it determines the approach angle of the spine. Large σ values mean that
the spine cannot appropriately reach the surface and occur when the body is not parallel to
the ground.




   Figure 5.4: Foot axial angle σ as viewed head-on. σ represents the rotation about the axis
   of the foot, and becomes non-zero when the body is not parallel to the ground, as while
   executing a transition.



                                              58
5.4 Foot Placement Optimization

DIGbot is capable of calculating online the optimal foot placement for arbitrary body ori-
entation and motion. For translation and rotation on a planar surface, the body orientation
and motion can be input in real time by a user (remote control.) This input takes the form of
a translational velocity vector ∆r in body coordinates and a set of three angular velocities

∆e as well as the desired change in body joint orientation ∆ bj. The translational and
                                                                                       T
angular velocities are actually expressed as finite changes Γ = ∆r ∆e ∆ bj                  . If the
user does not utilize the roll, yaw and vertical (α, γ and ∆rz ) degrees of freedom, the body
will stay parallel to the ground at a fixed distance h. In this case, it is impossible for a user

to give an invalid input because within magnitude bounds, DIGbot is capable of perform-
ing any planar motion desired. In other words, if DIGbot maintains its body parallel to the
surface at a fixed distance, it is always capable of calculating inverse kinematics solutions
for all its feet.

    For more complex motion like transitioning, it is difficult for a user to input a trajec-
tory through which DIGbot can calculate valid inverse kinematics solutions. In general,
with the added roll and yaw degrees of freedom, the hips easily become too far from the
surface. Therefore, for complex movements, body trajectories are calculated off-line using

the methods presented in section 4.3.

    Optimal foot placement is found by searching the ground plane for the foot placement
which will allow DIGbot to move the furthest in the direction of motion. In other words,

given a desired body angular and translational velocity Γ, a foot position is found which
yields the most body motion before the leg reaches the end of its workspace. This is
performed without actually moving the body; motion is simulated on the processor. First,
the position of the hip of the leg being optimized is projected onto the ground xy plane.
                                                                      T
This point is found by converting the hip location fb = 0 0 0             to ground coordinates
using equation 5.5. The x and y components of the hip position in ground coordinates are
then used as a starting point p0 for the search algorithm. The search algorithm searches

                                              59
   Figure 5.5: The ground is searched in the red shaded region for the optimal foot positions.
   The point p0 is the position of the hip projected onto the ground plane. The search area is
   divided in increments of 2 cm radially and 5 deg circumferentially. The space searched is
   parameterized as points (r,θ) from p0 , with bounds of r = 15 to 29 cm.




the space on the ground plane for the optimal position using a brute force search. The
space searched is parameterized as points (r,θ) from p0 , with bounds of r = 15 to 29 cm
as shown in Fig. 5.5. The radial bounds represent the closest and furthest from the hip that

the foot can reach, respectively, as described in section 3.1.1. The search point spacing is 2
cm radially and 5 deg circumferentially, such that 504 points are searched for each foot. At
each potential foot placement point searched, a simulation is run to determine how many
                                                T
times the body position η =       r e      bj       can be incremented by the planned motion

before an inverse kinematics solution for the leg angles is not possible. The simulation also
checks that there is adequate space to allow for inward gripping at the beginning and end
of stance. The algorithm cycles through each starting point and determines which yields
the highest number of valid body position increments. Because the search point spacing

is fairly large in some places (searched points are up to 3 cm apart), the area adjacent to
positions which yield a high number of valid body position increments is searched more
finely.

                                                60
5.5 Transition Trajectory


The transition trajectory is given in section 4.3 and is optimized off-line by minimizing the

hip height from the screen during the transition. A resulting file containing body position
η(λ) through the entire transition is read by DIGbot and the planned path is followed. To
follow the trajectory given by simulation, DIGbot increments its position in finite amounts
through the simulated values, λ = 0 − 1. λ is the dimensionless phase of the transition,

such that the beginning of the transition is λ = 0 and the end is λ = 1. For each transient
step ∆λ, new body position changes Γ are calculated iteratively:
                                             
                                      ∆r 
                                          
                                 Γ =  ∆e  = η new − η
                                                                                       (5.6)
                                          
                                      ∆ bj


In general, finite Euler angles cannot be added as vector quantities as is done in equation 5.6

because the order in which the rotations are applied is significant. However, during a simple
transition, the only nonzero Euler angle is the pitch angle (β), so the Euler angles can safely
be added as vectors. In other words, because the Euler angles only represent rotation about
a single axis, there are no complications associated with order of angle application.


   By following the calculated body trajectory using equation 5.6 we ensure that the hip

distance from the surface has been optimized. While this does not ensure that inverse
kinematics solutions are possible for all body positions during the transition, in practice,
limiting the hip height results in a continuously solvable trajectory for all λ. Onboard, Γ
is used in a function which calculates the change in feet position required for the specified

body motion. This function works by simply calculating forward kinematics for all of the
legs, incrementing the body position η by the desired amount Γ and then calculating the
inverse kinematics to find the desired leg angles.

                                              61
5.6 Foot Attachment

Foot attachment mechanisms, while not the focus of the DIGbot project, are something

which must be done well to climb properly. DIGbot was designed to climb primarily on
screen mesh. The primary mesh climbed on has a spacing of 2 cm and is constructed of
zinc plated steel wire. This mesh is laid out as a planar surface on a wooden frame. How-
ever, surfaces such as carpet, tree bark and a telephone pole were also climbed on with

some degree of success, achieving up to five consecutive steps without failure. Leg kine-
matics, surface morphology, foot and ankle design and control strategies all play a large
role in attachment and many different attachment strategies were attempted to achieve ro-
bust climbing. In addition, proper leg detachment was also a significant challenge because

the spine often gets caught in the surface. The successful strategies are presented here.



5.6.1 Leg control

A number of techniques to get proper attachment are used to achieve robust climbing. A
simple algorithm, shown graphically in Fig. 5.6 was developed which attaches the spine

and checks for proper inward gripping force. First the foot is moved to a position above
the desired attachment point on the surface. The foot is then moved down along the axis
of the spine tip to the desired depth. This is path 1 in Fig. 5.6. This motion is conducted
along a sinusoidal, rather than straight path to decrease the chances that the spine tip gets

planted directly on a wire. The amplitude of this sinusoidal motion is 1 cm. Once inserted,
the foot is moved inward through position control by a set inward gripping distance, which
is set to slightly more than the average screen gap size. At this point, motion pauses briefly
as the inward griping force is measured using the methods of section 5.2.1. If this value is

above a threshold gripping value, then the foot is assumed to have attached. The finding of
and testing for this threshold is discussed in section 6.3. However, if the inward gripping
force is not sufficient, as is the case for path 1, it is assumed that the spine either has not

                                             62
   Figure 5.6: Leg attachment algorithm. The foot searches for the screen along the dotted paths
   shown. If screen contact is not verified (as in cases 1 and 2) the foot searches for a new
   location. The inset details the sinusoidal motion of the spine during attachment to prevent the
   spine from becoming stuck on a wire.


contacted the surface or otherwise hasn’t found an adequate foothold location. In this case,
the spine is pulled back out along its original trajectory and a new location is attempted.
Searching takes place in the x, y and z directions (but only the z direction is shown in Fig.
5.6 for clarity.) Path two shows another failed attempt at a deeper z-level. However, in path

three, a proper foothold is found and adequate inward gripping force is attained. At this
point, attachment is considered successful and DIGbot continues with its normal climbing
routine. In practice, the search for a foothold appears as a ”pawing” motion and resembles
that of many animals searching for a foothold. Results for actual foot motion are presented

in section 6.3.1.



5.6.2 Natural Environments

Limited climbing has been performed on tree trunks and wooden telephone poles. Because

these surfaces are cylindrical, the normal planar walking control was altered slightly. In
order for the feet to reach the surface of the cylinder, the legs must adduct further under
the body than usual. This was accomplished by simply changing the default height value

                                                 63
h. By doing this, the body of the robot no longer interferes with the motion. Additionally,
the natural surfaces, being much more irregular than the screen mesh surface, required that
DIGbot fully utilize its feedback. While DIGbot is capable of climbing (albeit poorly) on
the screen mesh without any feedback, on tree bark or a telephone pole, each foot placement

must be validated to achieve proper climbing. In many cases, the foot must search a wide
area for a proper place. While this slows the attachment process, the overall climbing
behavior is no different on these surfaces. However, because the search area is often very
large for climbing in natural environments, the leg workspace which is left for productive
body motion is limited. This reduces the distance travelled or turned during each step.




                                            64
Chapter 6

Results

Experiments were run during a number of climbing tests over the life of DIGbot. All of the

data presented is for a full hexapod version of DIGbot, although the specific arrangement
of DIGbot varies. The primary variant is the leg design, which changed several times since
the full hexapod was built. Unless stated otherwise, all data is taken with DIGbot climbing
in the tripod stance on a flat steel screen mesh substrate. Tests were performed in laboratory

conditions. In many cases, the data shown is only a representative sample of the complete
data set. This is done for clarity. Much of this chapter is adapted from [34].



6.1 Tarsus Angle

The tarsus vertical angle of a cockroach moving through a step is shown in Fig. 6.1 [39]. In

subfigure (a), the tarsus vertical angle is at 145 deg, which changes to 118 deg in subfigure
(b) as the cockroach moves through a step. These subfigures are taken from high speed
video of a cockroach walking on a flat surface. The angle is measured relative to the
ground. Just like DIGbot, the tarsus deflects considerably because the foot remains parallel

to the ground during the step.

   Calculated DIGbot tarsus deflection angles for a tripod set during a turn in place are
shown in Fig. 6.2. These values are found using the inverse kinematics solution in ap-

                                             65
   Figure 6.1: Cockroach tarsus deflecting during a forward step. The cockroach tarsus deflects
   up to 60 deg during normal walking [39].


pendix C. Subplot (a) shows the tarsus vertical angle for all three feet in stance, which is
kinematically measured from the nominal straight position. This angle is indirectly pro-

portional to the radial length of the leg. Longer leg lengths require smaller amounts of
tarsal bending (see Fig. 3.9(a)). For foot positions closer to the hip, the tarsus vertical
angle becomes larger as the tarsus spring bends more sharply (Fig. 3.9(a)). Similarly, Fig.
6.2(b) shows the tarsus swing angle. This angle varies as the leg protracts or retracts while

in contact with the screen (Fig. 3.9(b)). These two angles represent the two orthogonal
components of the angle between the leg and the inward gripping direction. Fig. 6.2 shows
that for planar walking the tarsus angle bends as much as 50 deg during a step. This is
consistent with the corresponding cockroach deflection angles, which vary up to 60 deg
during walking [39]. More complex maneuvers such as transitions over orthogonal sur-

faces requires up to 90 deg of motion range, which highlights the need for the compliant
tarsus.



6.2 Body Motion During a Step

Body motions during two steps and a turn are presented in Fig. 6.3. Forward motion in

two different orientations with respect to gravity is shown in subfigures (a) and (b) with
a stationary turn shown subfigure (c). The data was obtained through video analysis of
DIGbot climbing, recording the position of the body at eight points during the stance phase.

                                              66
                                           20                                                  80
                                                      fore                                                          fore
                                                      mid                                      60                   mid




             tarsus vertical angle (deg)




                                                                   tarsus swing angle (deg)
                                                      hind                                                          hind
                                           15                                                  40

                                                                                               20

                                           10                                                   0

                                                                                              −20

                                           5                                                  −40

                                                                                              −60

                                           0                                                  −80
                                            0        0.5       1                                 0        0.5              1
                                                stance phase                                         stance phase
                                                     (a)                                                  (b)


   Figure 6.2: Calculated tarsus angles measured from the straight orientation for a stationary
   turn. The angles are shown for all three legs of a tripod during a single stance phase. Subfigure
   (a) shows the vertical angle which is related to the radial distance from hip to foot. Subfigure
   (b) shows the tarsus swing angle which varies as the leg protracts and retracts. At its peak
   flexure, the tarsus bends 75 deg to maintain the correct gripping force angle.


Bright white dots were attached to the fore and aft hip locations of DIGbot and these points

were measured manually on video. In subfigure (a), data is presented for eight forward
steps up a vertical screen surface. The data mean and the desired forward motion are
overlaid on the figure. The forward motion does not reach the desired motion for several
primary reasons: 1) the swing motors are not powerful enough to overcome the weight of
the robot, which oppose the desired motion. 2) spines shift out of their initial screen space

to a different space, and the control system does not compensate for this. 3) Body and leg
frame flexibility and the stiffness of the servomotors all lead to a difference between the
desired and actual body motion when the system is loaded. 4) Because the body is kept
close to the screen to reduce the tipping moment, the belly of DIGbot often catches on

surface asperities, causing its motion to be inhibited.

   Errors due to the weight of the robot do not occur during a horizontal step on the vertical

surface, results of which are shown in subfigure 6.3(b). Only three steps are shown because
the relative success is clearly evident. During this motion, the vertical forces used to oppose
gravity and maintain posture are mostly decoupled from the horizontal forces that generate

                                                                   67
                                                                                                                                                    35
                                               desired                                                     desired                                        desired
                                        0.3                                                         0.3
                                               vertical                                                    horizontal                                     actual
                                               vertical mean                                               horizontal mean                          30    mean
                                       0.25                                                        0.25

       forward motion (body lengths)




                                                                   forward motion (body lengths)
                                                                                                                                                    25




                                                                                                                                 turn angle (deg)
                                        0.2                                                         0.2
                                                                                                                                                    20

                                       0.15                                                        0.15
                                                                                                                                                    15

                                        0.1                                                         0.1
                                                                                                                                                    10


                                       0.05                                                        0.05                                             5


                                         0                                                           0                                              0
                                          0        0.5         1                                      0        0.5           1                       0        0.5       1
                                              stance phase                                                stance phase                                   stance phase
                                                   (a)                                                         (b)                                            (c)


   Figure 6.3: Body motion during forward steps and a stationary turn. The desired values
   are kinematically calculated and show, along with the mean values from the trials shown.
   Subfigure (a) shows the motion during a forward step up a vertical surface. In this orientation,
   the weight of the robot works against the motion and causes the actual motion to fall short
   of the desired motion. Subfigure (b) shows the motion during a forward step with the robot
   oriented horizontally on a vertical surface. The weight of the robot does not act against the
   motion and the actual motion approaches the desired motion. Subfigure (c) shows the rotation
   of DIGbot during a turn in place on a vertical surface. Problems result from spine slipping,
   but the legs maintain adhesion with the surface and the body moves through controlled turns.



the desired motion e.g., the weight of DIGbot is not working against its progress. The
motors achieve their desired position and result in more accurate motion.

   In subfigure 6.3(c), data is presented for a 33 deg stationary turn on a vertical screen.
This motion results in a large deviation among the trials, which also doesn’t correlate well
with the desired turn angle. All of the turns fall short of the desired turn. Spine slipping is
more common during turns than straight walking,because the leg angle changes are large

during a step, which results in large tarsus angle deflections. Despite these errors, during
all of the tests shown in Fig. 6.3, the body successfully executes the motion and the feet
remain in contact with the surface.

   In order to further explore where the body position errors arise, the desired position
can be compared with the actual position, obtained using forward kinematics with angles

                                                                                                           68
                             10                                                                         14
                                                         vertical (DIG)
                                                                                                        12
                                                         horizontal
                              0
                                                         desired                                        10


     body angle (degrees)




                                                                                   body position (cm)
                            −10                                                                          8

                                                                                                         6
                            −20                                                                          4

                                                                                                         2                          vertical (DIG)
                            −30
                                                                                                                                    horizontal
                                                                                                         0
                                                                                                                                    desired
                            −40                                                                         −2
                               0   0.2   0.4       0.6       0.8          1                               0   0.2   0.4       0.6       0.8          1
                                         stance phase                                                               stance phase

                                           (a)                                                                        (b)

   Figure 6.4: Body angle and forward progress during climbing. These angles and positions
   are from the servo data. Body position is calculated using forward kinematics. The results
   from vertical and horizontal motion are shown in each subfigure. In horizontal motion, an
   active DIG force is not required for proper motion, so it is not used.



as measured by the servos. This measurement does not take into account errors from body
frame deflection and slipping and only represents the error due to the servos not reaching

their commanded positions. Fig. 6.4 presents such data. In subfigure (a) the body angle
during a stationary turning step is presented, while in subfigure (b) the position of the
body during a forward step is presented. In both subfigures, the body position or angle is
calculated using forward kinematics, given the servo angles. Just as in Fig. 6.4, the actual
motion fails to perfectly match the desired motion. In both (a) and (b) the vertical climbing

motion has larger error due to the coupling of the robot weight with the forward motion.

   Fig. 6.5 uses the same servo data as Fig. 6.4, but presents positions of a single leg

instead of the body position. Subfigure 6.5(a) shows the leg swing angle and subfigure
6.5(b) shows the leg length during a single forward step. The leg angle, shown in subfigure
(a) is simply the angle of the swing motor on each leg (angle ω0 in Fig. 3.2.) The leg length
shown in subfigure 6.5(b) is calculated using the angles of the remaining servomotors.

Notice in Figs. 6.4 and 6.5 that the data for vertical climbing shows much more error from
the desired position than the data for horizontal climbing. This is in agreement with the
results of Fig. 6.3.

                                                                              69
                             140                                                                       12
                                                                                                                                   vertical (DIG)
                                                                                                                                   horizontal
                             120                                                                       11
                                                                                                                                   desired


     leg 1 angle (degrees)




                                                                                   leg 1 length (cm)
                             100                                                                       10


                             80                                                                         9


                                                         vertical (DIG)
                             60                                                                         8
                                                         horizontal
                                                         desired
                             40                                                                         7
                               0   0.2   0.4       0.6       0.8          1                              0   0.2   0.4       0.6       0.8          1
                                         stance phase                                                              stance phase

                                           (a)                                                                       (b)

   Figure 6.5: Leg angle (a) and length (b) during a forward step. These angles and positions
   are calculated from the servo angle data. Results from both vertical and horizontal motion
   are shown.


6.3 Minimum DIG Force

The minimum required DIG force for consistent climbing at different screen angles θ was
obtained by testing DIGbot on an inverted screen mesh surface as shown in Fig. 6.6. It is
advantageous to minimize the DIG force because for the same body motion, a lower DIG
force consumes less energy. In addition, on certain substrates such as carpet, too large a
gripping force can be damaging to the substrate, potentially compromising its potential as

a foot hold. DIGbot was to perform trials of 3 forward steps and 3 turning steps at a given
screen angle with a changing DIG threshold. For each screen angle, the DIG threshold was
decreased after each 6 step test until DIGbot fell from the surface. Results, normalized
by the weight of DIGbot (21N) are shown in Fig. 6.7 and represent the minimum DIG

threshold for a given inverted angle. The overlaid dotted line, FN , is the normal force on
the front foot during climbing as calculated using equation 4.2. The front foot reaction
force was chosen because this foot is loaded the most heavily during vertical climbing due
to the tip-back moment. By multiplying FN by a constant c (here c = 1.45, shown as

the solid line), FN is seen to be linearly related to the DIG force data. This means that
any increase in the required normal force requires a proportional increase in DIG force to
support it. This provides an easy way to predict the required gripping force for an arbitrary

                                                                              70
robot weight.

   For a completely inverted screen angle of θ = 90 deg the minimum DIG force was
found to be half the weight of DIGbot. This is the worst case of screen angles and so, in
general, if robust climbing is required at an arbitrary screen angle θ, this normalized force

of .5 is used as a minimum DIG force. If screen angle values are expected to remain small
during the duration of climbing, the minimum DIG force could be set lower, as per Fig.
6.7. It would also be possible to vary the minimum DIG force as the robot climbs if the
screen angle is known at all times.

   In the regime of low and negative θ values, the minimum DIG force model value loses
validity because the tangential forces begin to dominate the reaction equation, and equation
4.2 ignores these tangential forces. For values of θ < −7deg, the normal reaction forces of
DIGbot start to reverse direction, and the required minimum DIG force goes to zero. This

represents the case where only tangential forces need to be supported. For screen angles
θ > 90deg, DIGbot begins to climb downwards. In this case, the limiting foot contact
becomes the rear foot. The direction of travel doesn’t effect the model, so this regime
overlaps the 0 < θ < 90deg regime and doesn’t need to be considered separately.



6.3.1 DIG During Attachment

In section 6.3 it was established that a minimum DIG force of half the body weight is nec-
essary to maintain proper adhesion during a step using the tripod gait. This DIG force is

simply a threshold value during the initial foot attachment and does not take into account
measurements of the DIG force during the movement portion of stance. The DIG force dur-
ing for all three legs during the attachment phase is presented in Fig. 6.8. The attachment
method presented in section 5.6 is seen here.

   The attachment phase shown is much slower (taking 9 sec total) than a normal attach-
ment so that force data can be gathered with sufficient bandwidth to show the force as a
function of time. In real operation, the inward gripping phase takes less than a tenth of

                                             71
Figure 6.6: Screen angle for testing the minimum DIG force at different climbing angles.
Results are presented in Fig. 6.7.




                             0.5
   force/body weight (N/N)




                             0.4


                             0.3


                             0.2

                                                                 FN
                             0.1                                 c⋅F
                                                                      N

                                                                 minimum DIG
                              0
                               0   15   30           45         60        75        90
                                             screen angle (deg)

Figure 6.7: Minimum DIG force required for climbing. For each screen angle, the DIG
threshold was decreased after each 6 step test until DIGbot fell from the surface. Results are
normalized by the weight of DIGbot (21N). The overlaid dotted line, FN , is the normal force
on the front foot during climbing. By multiplying FN by a constant c (here c = 1.45, shown
as the solid line), FN is seen to be linearly related to the DIG force data.



                                                   72
                          1.2

                           1

       DIG force (N/N)    0.8

                          0.6
                                 DIG threshold

                          0.4

                          0.2                                                    fore
                                                                                 mid
                           0                                                     aft
                                                                                 verify
                         −0.2
                             0                   3                6                       9
                                                     time (sec)

   Figure 6.8: DIG verification. The DIG force is measured during the attachment phase for
   all three legs in a tripod. At the points denoted by the large dots, the DIG force is checked
   against the threshold DIG value. If the force is less than the threshold, that leg is released and
   re-attached.



a second for each grip. The force data here was obtained using the servo torque readings
which are transformed into an inward foot force by the methods of section 5.2. The “verify”

dots in Fig. 6.8 signify the points at which the DIG force is actually measured and com-
pared with the threshold value (.55 in this example.) During actual operation, the gripping
force is only measured at these points (not during the entire process as shown here.) The
figure shows that after the first grip (2.7sec), only the mid leg had surpassed the minimum

DIG force. As a result, the mid leg is kept stationary during the remainder of the gripping
process. The fore and aft feet did not achieve the threshold DIG force after the first grip,
and must reset. The reset process can be seen as the DIG force going to zero at 3.0sec. The
second grip, completed at 5.8sec is successful for the aft leg, but not for the fore leg, which

must reset again. The fore leg only achieves the proper grip on attempt three (at 8.8sec). At
this point, all three legs have achieved satisfactory grip and DIGbot removes the opposing
tripod and moves through stance.

                                                       73
6.3.2 DIG During Stance

During normal operation, the DIG force is not monitored during stance. However, data was
collected for two consecutive vertical forward steps, and is presented in Fig. 6.9. Notice

the similarity in shape between data in the two subplots. The large changes in DIG force on
a particular leg during stance are largely due to the changing leg orientation during stance.
The moment arm which contributes to the inward gripping force for each servo changes
significantly, and in certain orientations the leg is not capable of delivering large inward
gripping forces. Also notice that the DIG force is not always above the the DIG threshold

(here set at .55*body weight.) This does not mean that DIGbot experienced failure at these
locations, because the DIG threshold is only significant during the attachment phase.
   The attachment and detachment phases are included in Fig. 6.9, and represent the first
and final portions of each subplot, where the DIG force increases (or decreases) rapidly.

Unlike in Fig. 6.8, DIGbot achieves a proper grip during attachment on the first try for both
of these steps. This is evidenced by noting that the DIG force does not drop rapidly back
to zero in the first portion of either subplot, as it does in Fig. 6.8.
   Feedback during stance was at one point used to control the DIG force during stance.

The DIG force was kept at a constant level by adjusting the inward gripping force during
stance in a proportional control loop. This method required that simultaneous data from
up to 18 servomotors to be gathered and processed. The gripping force was adjusted by
moving the commanded foot position in or out in the lateral direction. An attempt at this

method was fairly successful, but was ultimately found to be unnessessary. First, the dura-
tion during which DIGbot is in stance (.5sec) means that DIGbot rarely, if ever falls falls
from the screen during stance. Even if a foot starts to lose its grip during stance, the oppos-
ing tripod will attach before DIGbot falls. In addition, the energy savings possible through
DIG reduction in stance are negligible due to the small fraction of climbing time spent in

stance. The majority of climbing time is spent in the attachment and detachment phases.




                                               74
             1.4                                                                   fore
             1.2                                                                   mid
                                                                                   aft
              1
 DIG (N/N)




             0.8

             0.6
                   DIG threshold
             0.4

             0.2

              0
               0    0.2 0.4 0.6 0.8        1          0     0.2 0.4 0.6 0.8               1
                      stance phase                            stance phase
                           (a)                                     (b)

Figure 6.9: DIG during stance for two consecutive vertical forward steps. The gripping force
for all three legs is presented, and the shape of the force curve for each leg can be seen to be
similar between the two steps (a) and (b).




                                               75
Chapter 7

Conclusions and Future Work


7.1 Summary of Contributions

DIGbot has successfully demonstrated the biologically-inspired climbing technique of DIG
for some complex manuevers. This robot is able to climb vertical surfaces and is the first to
perform turns on inverted surfaces. In addition, DIGbot is capable of motion in any direc-

tion, including making turns. It performs these tasks reliably through the use of feedback
forces at each contact point. The motion of the body during climbing was analyzed and
the requirements for robust climbing were identified. A control strategy which ensures that
these requirements are met was designed and implemented, resulting in robust climbing

behavior.

   In a manner similar to climbing insects, DIGbot uses the inward gripping force to in-
crease the adhesion holding it to the surface. It was also demonstrated that for the claws
used, the normal force which can be supported is proportional to the DIG force. In this
way, DIGbot can achieve any required adhesion by simply increasing the DIG force. Also,

by monitoring the DIG force during foot attachment, the robot searches for an adequate
foohold, rejecting inadequate ones. This behavior manifests itself as a “pawing” of the
surface, a behavior similar to that seen in animals.

                                             76
   Figure 7.1: DIGbot climbs in natural environments on a tree trunk and a telephone pole.
   Using stiff spines, the robot climbed five consecutive steps on each surface.


   By optimizing the initial foot placement positions, DIGbot is able to move the greatest
distance possible during each step and therefore minimize the amount of steps which need

to be taken. This optimization was demonstrated for the planar as well as general geometric
configurations.




7.2 Future Work

The main goal yet to be achieved is performing a transition between two orthogonal sur-
faces. While the body motion has been calculated and DIGbot is able to go through the
motions, there are several problems which need to be addressed for a transition to be suc-

cessfull. Because the position r of the body relative to the ground is important during a
transition (unlike in planar climbing) it is imperative that DIGbot either use position feed-
back from its legs which are in contact with the surface or that DIGbot follows its desired
trajectory very closely. Currently, without using body position feedback, DIGbot is unable

to follow the desired trajectory close enough to navigate the transition. In addition, the high
maneuverability of DIGbot could allow for even more difficult tasks to be executed, such
as transitioning over two orthogonal surfaces when the body is not squaredly aligned with

                                              77
the transition edge or moving over obstacles.
   DIGbot can be an ongoing robot platform for testing any type of directional adhesives.
New feet using claws can be put on DIGbot to test their effectiveness. Foot designs utilizing
multiple spines can also being developed to decrease the amount of searching which must

be done to find a proper foot hold or to spread the gripping load over several spines. It is
possible for multiple spines on the same foot to be simultaneously engaged with the screen
because no rotation of the foot with respect to the screen occurs during stance. Eventually,
it is a goal to use DIGbot with biologically-inspired dry directional adhesives, which would
enable climbing on smooth surfaces. Using a combination of claws and these dry adhesives,

DIGbot would be able to climb on a wide variety of surfaces.
   Other improvements can be made to DIGbot to improve general climbing such as more
finely tuned control techniques or implementing new gaits such as the wave gait. In certain
situations it might be advantageous to maintain a constact body velocity while climbing.

Currently, DIGbot pauses during attachment and detachment for maximum stability. How-
ever, it would be possible to merge the attachment and detachment phases into the stance
phase in a gait such as the tripod. Just like insects, the body would continue to move while
new the feet are attaching.

   Overall, the versatality of DIGbot means that many research topics related to climbing
can be explored on this platform.




                                             78
Appendix A

Old DIGbot Designs

DIGbot has gone through several designs during its life. Originally, a one-leg model was
built to test different servomotor arrangements and servomotor types. The first version used
a combination of rotational servomotors and linear servoactuators. The linear actuators,

from Firgelli Inc. [40] were chosen because they fit well with the concept of DIG. The goal
was to develop a simple leg in which the majority of the gripping force would be controlled
by moving the linear servoactuator, and the leg angle would be controlled by rotational
servomotors. The leg length would be primarily composed of the Firgelli actuator itself.

   A two-leg prototype was then built, shown in Fig. A.1. Using this prototype, some ba-
sic inward gripping methods were explored and the leg trajectories from section 4.2 were
tested. Simple walking behavior was possible with the two-leg model, and the planar walk-
ing methods were first verified using this model. As can be seen in Fig. A.1, carbon fiber

was used for several body parts. This was done to minimize weight but was ultimately
abandoned because a Delrin frame allows for more mounting and connection options. Fab-
rication is also easier and faster with Delrin.

   The leg geometry can also be seen in Fig. A.1. Two rotational servomotors controlled
the swing and adduction/abduction of the Firgelli actuator. In this way, the leg length is
almost completely independent of the motion of the rotational servomotors, and mostly

                                              79
  Figure A.1: An early 2-leg prototype of DIGbot which used linear actuators from Firgelli Inc.




  Figure A.2: The ServoPodTM microprocessor from NewMicros, Inc. was originally used to
  control DIGbot. This controller has 26 PWM outputs for controlling a large number of hobby
  servos.


dependant on the Firgelli actuator. The rotational servomotors used at this point were
standard Futaba hobby servos. These were chosen for their simple interface and general
strength and speed. Communication with the microcontroller was performed using a pulse-
width modulation (PWM) signal for each servo. This required that each servo have its own

communication connection to the microcontroller. For this reason, the ServoPodTM from
NewMicros microcontroller was chosen (Fig. A.2). This microcontroller has outputs for
up to 26 PWM signals, and so can control up to 26 hobby servos. The microcontroller is
programmed using the MaxForthTMprogramming language, which is a variant of the Forth
language.

   A full hexapod was designed which used the Firgelli actuators, as in Fig. A.3. This

                                               80
  Figure A.3: The first full hexapod design, which was never built. Notice the use of Firgelli
  actuators for legs.


body, which was never built, is basically three of the sets from Fig. A.1 connected by an
aluminum frame.
   Eventually, the disadvantages of the Firgelli actuators became apparent. Primarily, the

actuators were too slow to be effective. This became a problem when it was realized that
a large leg length change during steps is desired for the best walking behavior. With a
typical unloaded speed of 5 cm/sec, a step could take up to 4 sec, which is unacceptable.
Therefore, DIGbot was redesigned to use all rotary servomotors. The updated leg design,
shown in Fig. A.4 has three actuators for each leg. This was also eventually replaced by a

similar version which has four servomotors per leg.




                                              81
Figure A.4: Leg geometry of the updated leg, using AX-12 servomotors. This 3-DOF leg
was used in successful climbing on a variety of surfaces.




                                        82
Appendix B

DIGbot Drawings

Several dimensioned drawings are presented for some of the custom-made DIGbot parts.
All dimensions shown are in inches. Unless otherwise specified, parts are made from Delrin
and were machined on a Hurco CNC mill.




                                           83
Figure B.1: This figure shows the entire body of DIGbot. The SBC, electronics and batteries
are removed for clarity.




                                           84
Figure B.2: Subfigure (a) shows a Delrin part which holds the two frame pieces together.
Subfigure (b) shows an aluminum mounting bracket which mounts to the bottom of the AX-
12 servomotor controlling the ω1 axis. This bracket rotates about the ω0 axis.




                                          85
Figure B.3: This figure shows the leg dimensions of the four-DOF DIGbot leg.




                                    86
Figure B.4: This figure shows the four pieces of the frame. The top two pieces are the top
halves of the frame and the bottom two are the bottom halves. The swing RX-28 motors
mount in the large square-shaped holes in the top halves, and cause rotation about the large
holes in the bottom halves.




                                            87
Figure B.5: This figure shows the most recent foot design. Subfigure (a) shows the foot itself,
with the spine removed for clarity. Subfigure (b) shows the stalk of the leg, which is attached
to the foot via a spring and braided steel cable inside. Subfigure (c) shows the mounting
bracket to connect the leg stalk to the large plastic braces of the leg. The entire assembly is
shown in subfigure (d).




                                              88
Appendix C

Inverse Kinematics

The inverse kinematics for the geometry shown in Fig. 5.1 is given here. Fig. C.1 is the
same as Fig. 5.1 but with more detail. The tarsus-ground angle φ is held constant at 30
deg. We solve for the three servo angles ω1 , ω2 and ω3 using algebra:


                                       ¯        ˆ
                                       d = lˆ − hk

                                  h = H − tarsus sin ω

                                   l = l − tarsus cos ω

                                 ac2 = upper 2 + lower2

                                                   lower
                                    cab = arctan
                                                   upper



                                                             ˆ
                      c = ac · cos(ω1 + cab)ˆ − sin(ω1 + cab)k
                      ¯                     

                                      ad2 = l2 + h2

                      ad2 = ac2 + distal2 − 2ac · distal · cos acd

                                         ac2 + distal2 − ad2
                             cos acd =
                                              2acdistal
                                            89
        Figure C.1: Leg kinematics.




  ω2 + acd + π − (π/2 − cab) = 2π

       ω2 = 3π/2 − acd − cab



                            h
            dam = arcsin
                            ad

distal2 = ac2 + ad2 − 2 · ac · ad cos cad



                   distal2 − ac2 − ad2
    cad = arccos
                          2acad

          cam = cad + dam

           ω1 = cam − cab

        ω3 = π/2 + ω1 − ω2 + φ




                    90
Appendix D

Foot Designs

Proper foot design is crucial to the performance of any climbing robot. DIGbot’s premier
foot design, shown in Fig. 3.9 combines several key elements, including a pointed spine
and a flexible passive tarsus joint. Before this design we developed, several other designs
were built.
   The main goal of these feet designs is to allow the spine to stay still relative to the

ground while the leg angle changes, as in Fig. D.1 . Original designs accomplished this
through the use of pin joints, as in Fig. D.2. This foot has a spine made of stainless steel
wire which can pivot about a vertical axis. The spine includes a torsion spring to return
the spine to the default position between steps. This design was used with the two-leg

prototype seen in Fig. A.1 and was fairly successful. However, it could not compensate for
leg rotations about the horizontal axis, so a new foot was designed.
   For surfaces with irregular or widely spaced footholds, the single-spine foot often has
a hard time finding a proper grip location in the desired region. A foot with multiple

spines was designed to help with this problem. Seen in Fig. D.3, this design features four
passively-retracting spines which each have the opportunity to engage the substrate. Spines
which are obstructed from entering a gap in the surface simple retract back into the foot.
This design could result in more reliable climbing.



                                            91
Figure D.1: Leg motion through the support period. As the body moves through a step, the
angle of the leg changes with respect to the desired inward force and the spine rotates about
its pivot to maintain the desired orientation.




Figure D.2: Model of the foot mechanism. The leg is pulled left during the stance phase,
causing the spine to seek the inward wire and develop a gripping force. When the spine is
removed from the screen, the torsion spring returns the spine to its original angle.




Figure D.3: Cross section illustration of a multi-spine foot with passive retractable spines.
Due to uneven screen spacing (1), a spine (2) retracts to allow the other spines to engage and
apply the inward gripping force (3).


                                             92
Appendix E

Leg Force-Torque Solution

                                                           T
DIGbot reads torque data τ =             τ1 τ2 τ3              from the servos and converts this to an
equivalent force as in section 5.2.1. The full torque solution, solved from equation 5.2

using Mathematica, is given here.


                                                                                                                      
                    distal sin ω1 [(r−proximal) cos(ω2 +ω3 )−z sin(ω2 +ω3 )]τ1 +xdistal sin(ω2 +ω3 )τ2 +(r−proximal)τ3
     Fx                                distal(proximal−r−z) cos(ω2 +ω3 )[x cos(ω1 )+y sin(ω1 )]                       
                                                                                                                      
F =  Fy  =                                          −distal cos(ω2 +ω3 )τ2 −zτ3                                       
                                     distal(proximal−r−z) cos(ω2 +ω3 )[x cos(ω1 )+y sin(ω1 )]                         
                                                                                                                      
                                                       −distal cos(ω2 +ω3 )τ2 −zτ3
      Fz                                           distal(proximal−r−z) cos(ω2 +ω3 )
                                                                                                               (E.1)




                                                     93
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                                             98

				
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