The Design and Performance of DASH

Paul Birkmeyer

Electrical Engineering and Computer SciencesUniversity of California at Berkeley

Technical Report No. UCB/EECS-2010-75

http://www.eecs.berkeley.edu/Pubs/TechRpts/2010/EECS-2010-75.html

May 14, 2010

Copyright © 2010, by the author(s).All rights reserved.

Permission to make digital or hard copies of all or part of this work forpersonal or classroom use is granted without fee provided that copies arenot made or distributed for profit or commercial advantage and that copiesbear this notice and the full citation on the first page. To copy otherwise, torepublish, to post on servers or to redistribute to lists, requires prior specificpermission.

Acknowledgement

The author would like to thank Ron Fearing for his brainstorming, supportand guidance, and Alexis Birkmeyer for her mental support and willingnessto share her time and Illustrator skills. Thanks to Bob Full for his advice andsupport. Thanks also to Aaron Hoover and the rest of the BiomimeticMillisystems Lab for their support, Kevin Peterson for his help with theIROS paper, the Center for Integrative Biomechanics in Education andResearch for their advice and the use of their force platform, and the CRABLab for their collaboration. This work is supported by the NSF Center ofIntegrated Nanomechanical Systems.

Abstract

DASH, the Dynamic Autonomous Sprawled Hexapod, is a small, lightweight, power autonomous

robot capable of running at speeds up to 15 body lengths per second. Drawing inspiration from

biomechanics, DASH has a sprawled posture and uses an alternating tripod gait to achieve dy-

namic open-loop horizontal locomotion. The kinematic design which uses only a single drive

motor and allows for a high power density is presented. The design is implemented using a scaled

Smart Composite Manufacturing (SCM) process. Two different means of turning are presented,

one of which is actuated. Actuated turning results from altering the kinematics via body defor-

mation, a method supported by a simple 2-D dynamic model. Evidence is given that DASH runs

with a gait that can be characterized using the spring-loaded inverted pendulum (SLIP) model.

In-situ power measurements are performed to give cost of transport values both on hard ground

and granular media. In addition to having high maximum forward velocities, DASH is also well

suited to surviving falls from large heights due to the uniquely compliant nature of its structure.

Contents

1 Introduction 1

2 Mechanical Design 4

2.1 Design Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2.2 Hip Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.3 Differential Drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 Leg Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.5 Body Design and Steering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Hexapod Simulation 17

3.1 Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4 Results 27

4.1 Running Performance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

4.2 Turning Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

4.3 Step Climbing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

4.4 Surviving Falls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

5 Discussion of Results and Conclusions 39

i

List of Figures

1.1 Photo of DASH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 Simulation of kinematics in MATLAB model . . . . . . . . . . . . . . . . . . . . 6

2.2 Kinematics illustrations of the hip mechanism . . . . . . . . . . . . . . . . . . . . 8

2.3 SolidWorks model of SCM hip joint . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.4 SolidWorks model of SCM differential drive mechanism . . . . . . . . . . . . . . 10

2.5 Complete SolidWorks model of DASH . . . . . . . . . . . . . . . . . . . . . . . . 11

2.6 Horizontal and compliant leg designs . . . . . . . . . . . . . . . . . . . . . . . . 12

2.7 Stiff leg rotation versus four-bar linkage leg rotation . . . . . . . . . . . . . . . . . 13

2.8 Two designs using a four-bar linkage leg . . . . . . . . . . . . . . . . . . . . . . . 14

2.9 Kinematic illustration showing effect of body deformation to produce turns . . . . 15

2.10 Four-bar ankle to provide normal compliance and turning behavior . . . . . . . . . 16

3.1 2-D hexapod model used in dynamic simulations . . . . . . . . . . . . . . . . . . 18

3.2 Simulation results of straight running showing center of mass motion and instan-taneous velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.3 Simulation results of straight running showing relative heading, orientation, andangular velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.4 Simulation results showing center of mass motion during S-turn maneuver . . . . . 23

3.5 Simulation results showing forward velocity and angular velocity during left turn . 25

3.6 Simulation results showing relative heading and orientation before and after right-turning input is applied . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.1 Images of dynamic running in DASH . . . . . . . . . . . . . . . . . . . . . . . . 28

ii

4.2 Forward velocity vs motor duty cycle when using parallel leg design . . . . . . . . 29

4.3 Tracked center of mass during a horizontal run . . . . . . . . . . . . . . . . . . . 30

4.4 SLIP-like behavior of DASH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

4.5 Cost of transport on hard ground and granular media . . . . . . . . . . . . . . . . 33

4.6 Experimental results of S-turn maneuver in DASH . . . . . . . . . . . . . . . . . 35

4.7 Turning rate vs duty cycle using toe-extension turning method . . . . . . . . . . . 36

4.8 Step-climbing behavior in DASH . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

4.9 Images of high-velocity impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

iii

List of Tables

3.1 Parameters used in simulations of 2-D rigid-body hexapod . . . . . . . . . . . . . 19

5.1 Select properties of comparable robots . . . . . . . . . . . . . . . . . . . . . . . . 40

iv

Chapter 1

Introduction

Highly mobile, small robotic platforms offer several advantages over larger mobile robots. Their

smaller size allows them to navigate into more confined environments that larger robots would be

unable to enter or traverse, such as caves or debris. Small robots are easily transported by either

vehicles or humans to be deployed to remote locations as needed. One example includes field

workers who need access to otherwise dangerous, inaccessible areas such as collapsed buildings

or ones damaged in earthquakes. Small, inexpensive robots are also a key component for rapid

installation of ad-hoc networks.

As robot size decreases, however, maintaining mobility can become a challenge if climbing

means are not available. As objects are larger relative to body dimensions, legs gain advantages

over traditional wheels or treads by being able to overcome obstacles greater than hip height.

Reduced size also introduces challenges involving controls and power; reduced volume leaves less

space for the multiple actuators per leg typically seen on larger robots.

Biology offers a wealth of examples of small creatures with remarkable locomotion abilities,

such as the cockroach. Studying these animals has revealed lessons that guide the design of legged

robots. One such example is ”preflexes” wherein passive mechanical elements, such as tendons,

1

contribute stabilizing forces faster than the neurological system can respond [1, 2]. Another passive

design element that imparts stability is the sprawled alternating tripod gait. In addition to static

stability, this limb arrangement has been shown to lend robustness to both fore-aft and lateral

perturbations while running [3], suggesting application in legged robots.

The gaits of running animals suggest an additional template useful for stable running. The

spring-loaded inverted pendulum (SLIP) model can explain the ground reaction forces during run-

ning seen in almost all legged systems with the relative stiffness being nearly uniform across all

species [4, 5]. The SLIP model has also been shown to impart dynamic stability to horizontal

running. Possessing SLIP-like behavior may benefit the mobility of any legged robotic system.

Much work has been done in highly mobile legged robots, some of which incorporate these

lessons derived from biology. Examples include Sprawlita [6], iSprawl [7], and RHex [8]. These

robots incorporate passive compliance to achieve preflex-like behavior and SLIP-like motion. They

also all use some form of feed-forward controls, mimicking the central-pattern generator feature

from biology [2, 5].

10 cm

Figure 1.1: DASH: a power autonomous hexapedal robot.

2

The Dynamic Autonomous Sprawled Hexapod, or DASH, uses design principles from biology

and employs a differential drive using only a single motor. It is capable of power autonomous, ro-

bust dynamic locomotion at speeds of approximately 15 body-lengths per second (1.5 m/s). DASH

measures 10 cm in length, weighs 16.2 grams, and uses wireless communication for feed-forward

commands. It is constructed using a scaled Smart Composite Manufacturing (SCM) process that

enables fast build times, scalable designs, and lightweight systems [9]. The SCM process also en-

ables falling survivability through an energy absorbing structure and high surface-to-weight ratio.

3

Chapter 2

Mechanical Design

The primary challenge in the development of DASH was to create a robot that is capable of

efficiently and effectively coupling the output of a single actuator to the legs to create a dynamic,

feed-forward locomotion. Though long-term goals include developing the ability to run vertically,

initial efforts were focused on developing a robot that could first run effectively on a horizontal

surface. Once the drive mechanism and legs are refined from their original designs, they can then

be adapted to more complicated terrain as well as climbing.

The first step in development is determining the manufacturing process. Previous work has

developed the Smart Composite Microstructures (SCM) manufacturing process [10], made by laser

micro-machining carbon fiber laminates to form rigid linkages with flexible, polymer joints. This

process was later adapted to a prototyping process which uses cardboard rather than carbon fiber

laminates as the material for the rigid linkages [9]. This process easily produces lightweight robots

with lengths on the order of centimeters and allows for a manufacturing time several orders of

magnitude shorter than with the original SCM process.

To create a dynamic, high-power density robot, there should be a minimal number of power-

4

dense actuators that serve as the main drive actuators. For example, if there is one actuator per

tripod, the tripod in swing contributes no work to the system. For the design of DASH, a single

DC motor was chosen as the main drive actuator for all six legs. DC motors, compared to other

actuators such as piezoelectric actuators and shape memory alloy (SMA) wires, provide longer

stroke lengths, higher efficiencies, greater power densities, easier electrical interfacing, and op-

erate in a region of frequencies comparable to those observed in insect legged locomotion when

appropriately geared [11].

A sprawled alternating tripod gait was chosen for DASH. This gait is often seen in insects,

and it functions well when locomoting on horizontal, inclined, and vertical surfaces [4, 5, 12, 13].

By keeping the center of mass above the support polygon, an alternating tripod is conveniently

statically stable. A sprawled stance with an alternating tripod gait has also been shown to impart

stability in the presence of fore-aft and lateral perturbations [3]. Alternating tripod gaits can be

seen in prior legged robots, including iSprawl [7], RHex [8], and Mini-Whegs [14], and several

also exhibit sprawled postures [6, 15].

5

2.1 Design Overview

!!"

"

!"

!!"!#!

"#!

!"

!$"

!#"

"

#"

E1 (mm)E2 (mm)

E3

(mm

)

Figure 2.1: Simulated foot trajectory for DASH. The circles are traces of the feet as they movethrough one stride. The two different colors represent the two different tripods which are 180degrees out of phase.

The coupling of the DC motor output to the legs functions much like the oar of a rowboat: the cir-

cular input from the motor drives the end of the oar-like leg in an amplified circular motion through

a mechanical coupling. By orienting the output of the motor in the sagittal plane of DASH and

coupling it to the oar mechanism, the resulting circular foot trajectories are aligned roughly with

the sagittal plane. This creates leg trajectories which have both fore-aft and vertical displacements.

To ensure that the hips never contact the ground, the circular orbit is biased slightly downwards.

Figure 2.1 shows a MATLAB model of the desired kinematics. The model offers a quick method

of varying design parameters and seeing the resultant changes in kinematics in 3-D space. The

circular foot trajectories traced by each tripod are marked with different colors.

6

2.2 Hip Design

In the drive mechanism of DASH, the hip linkages provide both the fore-aft and vertical motion of

the legs. These two degrees of freedom are largely decoupled and can be understood separately.

Figure 2.2(a) shows a kinematic drawing of the vertical degree of freedom and Figure 2.2(b) shows

the fore-aft degree of freedom in the hip. In both figures, the grounded linkages are those mounted

to the motor casing, and the linkages adjacent to the input arrows are those mounted to the output

of the motor gearbox. In Figure 2.2(a), looking at the front of DASH, the vertical motion of the top

beam relative to the grounded beam causes the hips to rotate in opposite directions. This rotation

results in vertical motion of the feet, and the anti-phase motion of the opposing hips is required

for alternating tripod gaits. Looking from the top of DASH, Figure 2.2(b) shows how fore-aft

motion of the middle horizontal beam moves the two opposing four-bar linkages, swinging one leg

forward and the other leg backward. The hip mechanism appropriately couples the input beams of

Figure 2.2 to the vertical and fore-aft displacement of the final stage of the motor gearbox to create

the desired leg trajectory.

7

Top

Bottom

FRONTVIEW

(a)

TOPVIEW

Front

Back(b)

Figure 2.2: Visualization of the degrees of freedom that enable (a) vertical motion and (b) lateralmotion of the legs. In both images, the beams to which the hips are attached are grounded.

The realization of the kinematic drawings of Figure 2.2 in a SCM manufacturing process is

shown in Figure 2.3. Viewed horizontally from the side of DASH, one can see how a circular input

applied to the hip linkages create the circular motion of the feet. The grounded linkages and input

linkages from Figure 2.2 are the transparent beam and black beams in Figure 2.3, respectively.

The opaque beige linkages are the hip four-bar mechanisms and their attached legs. The output

of the motor gearbox is applied through mechanical constraints discussed in Figure 2.4, and the

resulting motion of the black input linkages is shown with a green dashed line. Motion of the end

of the black linkages along this circular dashed line result in the motion the legs as shown. Two

adjacent hips are shown to illustrate that they do, in fact, move 180 degrees out of phase with each

other. When these two hip designs are propagated appropriately to the remaining hips, it creates

8

the desired alternating gait.

(a) (b)

(c) (d)

Figure 2.3: Four different positions of two adjacent hip joints from DASH presented from a lateralview. The black beams are attached to a circular input. The beams move in the dotted paths,causing the motion of the legs. The axes of rotation for the hips lie on opposite sides relative totheir respective inputs, creating an anti-phase motion. The image in the top left shows when theinput is in the top left position of its circular trajectory and the image in the top right shows whenthe input is in the top right position the circular trajectory, and likewise for the (c) and (d). Theinput motion prescribed here would move the robot to the right.

2.3 Differential Drive

The hip design of Figure 2.3 requires the motion of the black input beams relative to the motion of

the transparent beam. Moreover, this relative motion is designed to be the circular motion of the

motor output. We created the differential drive mechanism of Figure 2.4 to ensure that the single

output of the drive motor drives each of the six hips with a similar circular input. Zoomed out and

9

(a) (b)

(c) (d)

Figure 2.4: Looking from the side of DASH, linkages connect the top beams with the bottombeams and force them to remain parallel. The linkages form two parallel four-bar mechanismsthat share the horizontal beam that runs between the bottom and top beams. The round motor ismounted rigidly to the bottom beams, which are transparent to reveal the motor. The motor outputis rigidly connected to all of the dark beams. The parallel constraint means that every point on thetop beams move in the same circular path as the motor output, keeping the motion of the legs ineach tripod identical. Note that the positions correspond with the position of the input in Figure2.3.

omitting the hip mechanisms and input beams from Figure 2.3, one can see how this mechanism

functions. The transparent beam on the bottom is the same as in Figure 2.3, and the vertical black

beams of Figure 2.3 are attached to the horizontal black beams of Figure 2.4. The final stage of

the drive motor gearbox is shown as the grey circle, and the motor itself is rigidly mounted on

the transparent bottom beam. The dark diagonal beams are a mechanical linkage that connects the

motor output to the black input beams.

The five thin linkages between the top and bottom horizontal linkages in Figure 2.4 enforce

the constraint that the top and bottom linkages remain parallel. Original designs did not have the

thin horizontal linkage, and while the remaining four thin beams did prevent the top structure from

rolling relative to the bottom structure, the constraint did not enforce the desired parallel constraint.

The thin horizontal linkage in the middle of Figure 2.4 enforces the parallel constraint by forming

10

coupled four-bar linkages. Figure 2.5 shows how all of these linkages combine together to form

DASH. Beyond verifying how the SCM design assembles together, this SolidWorks model can be

used as a design tool. The effect of adding and removing constraints can be observed in this model,

as well as the effects of varying link lengths on the kinematics. The model also allows one to see

how new components will integrate into the existing structure.

This differential drive mechanism has a floating ground where only the relative motion of the

top and bottom structures of the robot create the desired output of the legs. The motor could be

mounted to either structure and a similar circular leg motion would be achieved. The motor was

mounted to the bottom structure to keep the center of mass (COM) close to the ground to reduce

pitching moments during climbing. This differential drive differs from other robots such as iSprawl

and RHex wherein the entire drive train is grounded to the same structure to which the hips are

attached [7, 8].

Figure 2.5: Fully-constrained, functional SolidWorks model of DASH used to verify how SCMlinkages assemble together and create the resulting kinematics. The color of beams correspondwith colors seen in Figures 2.3 and 2.4.

11

2.4 Leg Design

Several different leg designs have been used for horizontal locomotion in DASH. The most simple

leg design is a rigid, nearly straight leg design shown in Figure 2.6(a). It angles down slightly and

then has a rigidly constrained flexure which creates a flat underside. A second leg is a compliant

leg design seen in Figure 2.6(b). This leg has one flexure joint that is free to rotate when pressed

into the ground. The flexure joint serves as a torsional spring as it is deformed from the presence

of normal forces, and the equivalent spring constant can be controlled by adjusting the flexure

material. To achieve larger forward accelerations and more repeatable behavior during locomotion,

polydimethylsiloxane (PDMS) boots were molded to fit over the ends of these leg designs. These

press-fit rubber boots increased friction generated with the surface and reduced slipping.

(a) (b)

Figure 2.6: Two different leg designs used by DASH. (a) is the stiff, horizontal design and (b) isthe compliant design.

Both of the aforementioned legs are rigidly mounted to the hip, and this causes the legs and

feet to rotate through the same angular deflection as that of the hip, as seen in Figure 2.7(a). While

these leg designs seem to function well on horizontal ground where ground engagement is not

difficult, foot rotation may cause loss of adhesion during climbing due to twisting the foot contact

12

relative to the ground. Thus, the leg design of Figure 2.7(b) was created to eliminate the rotation

of the feet during stance phase. A foot is mounted to the end of the four-bar leg, and during the

stance the foot still moves through an arc but its orientation relative to the robot does not change.

The SCM implementation of this design can be seen in Figure 2.8.

Back Foot

TOPVIEW

Foot

Front

(a)

TOPVIEW

Back

Front

Foot

Foot

(b)

Figure 2.7: The foot orientation of the original rigid-legs rotates with the fore-aft rotation of theleg, twisting the foot contact along the ground, as seen in (a). To prevent the rotation of the foot,a parallel four-bar leg linkage was created. This linkage maintains the orientation of the foot withrespect to the body during stance, as seen in (b). The vertical arrows indicate direction of actuation.

The press-fit PDMS boots created for DASH were designed to increase lateral and fore-aft

forces and were not concerned with generating adhesive normal forces. Subsequent foot designs

include various efforts to include controlled compliances and claw structures, all of which can

be attached to the four-bar leg design. One such design uses a polymer-based spring element.

This design, shown in Figure 2.8(a), uses a loop formed from the polymer flexure layer extending

from the bottom of the leg. This polymer loop deforms and elastically stores energy during a foot

touchdown, adding compliance while adding minimal mass and inertia to the leg. The polymer

13

is cut to create small claw structures to better engaged surfaces such as carpet. During construc-

tion, the polymer is plastically deformed to have a flatter bottom to promote more surface contact.

A different design replaces the polymer loop spring with a four-bar linkage that serves a similar

purpose. This design has the advantage of being more repeatable across build iterations and more

easily adapted to new ground engagement mechanisms due to having rigid mounting structures.

The flexures in the four-bar mechanism act as torsional springs when deflected through some an-

gular displacement (see Figure 2.10), creating an effective stiffness when the four-bar linkage is

deflected. These four-bar linkages could also be actuated for turning, of which a proof-of-concept

can be seen later in Section 4.2. Figure 2.8(b) shows an example where simple claw structures are

added.

(a) (b)

Figure 2.8: Two different leg-foot combinations using the parallel four-bar leg design. (a) has afoot made of a polyethylene terephthalate (PET) loop etched with claws to create a compliant footmechanism. (b) replaces the PET loop with a mechanical four-bar ankle that is compliant in thenormal direction and is attached with a rigid claw structure.

14

2.5 Body Design and Steering

The cardboard beams used to construct DASH are rigid when forces are directed along the face

of the beam; however, because the beams are only approximately 900 microns thick, they lack

rigidity when subjected to forces directed into the face of the beam. All the beams in DASH are

oriented to present the most stiffness in both the fore-aft and vertical axes, since these are the two

directions which experience the greatest forces during locomotion. Stiffness in these directions

is required for good power transmission to the legs. There are instances, however, when loading

occurs in the weak axes of the beams. The lack of off-axis rigidity introduces compliance in the

structure as the beams deform under loads. This is in addition to the small amounts of compliance

already present due to polymer hinges not perfectly emulating pin joints.

Figure 2.9: A model of how the kinematics change when the body is contorted. The midpoint ofthe fore-aft strokes are biased forward on one side of the body and backwards on the opposite side.This induces a moment on the body and results in turning.

During straight-ahead running, the arc swept out by the legs’ fore-aft motion is centered about

a midpoint perpendicular to the body, as in Figure 2.2(b). These leg motions are balanced on both

sides of the body and propel the robot straight. To steer, skewing the frame of DASH biases the

midpoint of the arcs swept by the legs as shown in Figure 2.9. This shifts the location and direction

15

of the ground reaction forces, imparting a moment on the body. A one gram shape-memory alloy

(SMA) SmartServo RC-1 from TOKI Corporation, mounted to the rear of DASH, pulls on the front

corners of DASH’s frame, resulting in a skewed frame and an induced turn. The direction of the

turn is dependent upon which corner of DASH is pulled toward the SmartServo. No control input

to the SmartServo results in straight-ahead motion. This behavior is explored both in simulation in

Chapter 3 and experimentally in Section 4.2.

Alternative methods to the body distortion method of turning mentioned above are also being

developed. One such method leverages the four-bar ankles on the end of the parallel leg design

shown in 2.8(b). By extending either the front-left or front-right foot and applying a force to keep

it extended, the robot can be made to turn either right or left, respectively. The extended leg not

only touches the ground earlier than the other feet, on average, but also has a higher stiffness than

the other feet and creates larger impulses. These effects combine to induce turning behavior in

DASH. When no foot is extended, the feet should be balanced and the robot will run straight.

Ankle

Leg

Leg

Ankle

Foot

Foot

(a)

Ankle

Leg

Leg

Ankle

Foot

Foot

(b)

Figure 2.10: The four-bar ankle linkage is normally horizontal with no loading (a). During loco-motion, the four-bar ankle acts as a spring when normal loads deflect the four-bar linkage from itsneutral position. To induce turning, the four-bar linkage of either the front-right or front-left leg islocked into the position shown in (b). This induces a left or right turn, respectively. This extensionof the four-bar ankle could be done with SMA actuation in the future.

16

Chapter 3

Hexapod Simulation

A model was created that is capable of emulating the dynamic running of a hexapedal robot in the

horizontal plane. The model can simulate straight running as well as the turning method described

in Figure 2.9. Though the model is simple, it proves to be fairly effective at predicting the behavior

of DASH. It also serves to verify that the modification of the leg trajectories shown in Figure 2.9

actually induces moments on the body and results in turning, even under the assumptions outlined

below.

3.1 Construction

A reflection-symmetric rigid body of mass m and moment of inertia I was created. The model has

massless legs and uses an alternating tripod gait with a 50% duty cycle in which one tripod enters

stance as soon as the other tripod exits. The model and design parameters are shown in Figure

3.1. Touchdown angle β, angular sweep during stance φ, stride frequency f , turning input angle,

and individual foot friction values can be varied between simulations. All legs are assumed to be

17

rigid and coupled so that defining the position of one leg defines the position of all of the legs and

feet. For later reference, the left tripod is defined as the tripod using legs 1, 3, and 5, and the right

tripod is defined as the tripod using legs 2, 4, and 6. The parameters listed in Table 3.1 are used

by the simulations shown in this chapter to emulate DASH as configured for turning experiments

in Section 4.2.

d

rcom

1

2

3

4

5

6

Figure 3.1: The 2-D model used in the turning simulation showing various parameters used todefine the motion of the rigid body through the plane. Though all six legs are shown, only a singletripod is engaging the surface at any time.

Note that the turning input does not effect the total angular deflection of the legs during stance,

but does change the touchdown angle β. When turning, all of the legs on one side will have their

touchdown position moved either forward or backward, and the legs on the opposite side will have

their touchdown position moved backward or forward, respectively, as shown in Figure 2.9. Legs

4, 5, and 6 have their touchdown position moved forward if β is increased while legs 1, 2, and

3 have their touchdown position moved backward. β, as defined in Figure 3.1, is reduced by the

18

Mass m 16gMoment of inertia I 0.0001 kg m2

Leg length 0.035 m

µd of legs 2 and 5 0.3µd of legs 1, 3, 4, and 6 0.1

Touchdown angle β3.3 π12

Sweep during stance φ5.4 π12

Center of mass offset d 0.015 m

Turning input angular offset π10

Table 3.1: Parameters used in simulations of 2-D rigid-body hexapod.

turning input angular offset to turn right, and increased by the turning offset to turn left.

Ground reaction forces are determined using a Coulomb friction model. Due to the planar

assumptions, the lack of roll and pitch of the rigid body dictates that the normal forces generated

by each foot are equivalent. When also considering that the opposing legs move in opposite lateral

directions, it emerges that the system is dominated by dynamic friction with coefficient of friction

µd. A time-varying normal force, similar to those seen in experimental trials and modeled by SLIP

(see Figure 4.4), was dictated at each foot during every stance. Friction forces are generated at

the end of each leg and are directed along the relative motion between the foot and ground. The

friction reaction forces from the engaged tripod sum to impart accelerations and moments to the

body. The center of mass position and velocity, the orientation and heading, as well as the stride

phase of the system are tracked.

Model simulations are done using the Runge-Kutta integrator ode45 in MATLAB. Two similar

sets of equations of motion are created, one describing the motion during each of the two tripod

stances. Integration is performed through the duration of a single stance, and the event functionality

of ode45 signals the end of stance and the transition between tripods. Between integrations of the

sequential tripods, final state vectors become the initial state vectors of the subsequent integration.

19

3.2 Simulation Results

Initial trials were conducted without an applied turning input to serve as a control. The system is

operated at a stride frequency of 17 Hz, which is near the upper limit of operating frequencies of

DASH. Results can be seen in Figures 3.2 and 3.3. The system accelerates quickly and levels off at

a top-speed of approximately 1.6 m/s. During running, the system decelerates at the beginning of

stance, accelerates through the middle of stance, and decelerates again near the end of the stroke.

These fore-aft accelerations have a frequency of half of the stride frequency. The orientation of

the body oscillates at the stride frequency, as does the relative heading of the system. The abrupt

acceleration in heading and angular velocity seen in Figure 3.2 occurs at the middle of the stance

phase. More precisely, it occurs when the legs are positioned perpendicular to the body. It is at

this point when the horizontal component of the net reaction forces change direction. Prior to this

point, the net reaction forces push the body away from the side with two feet in stance (pushes left

when the right tripod is in stance and right when the left tripod is in stance); after this time, the net

reaction forces pull the body towards the side with two feet in stance.

The behavior seen during straight-ahead running matches expectations and results from more

complicated hexapedal models. The deceleration and accelerations match the behavior seen in

many legged animals, the SLIP model, and also more complicated hexapedal models [5, 16].

Likewise, the orientation oscillations are similar to those exhibited in legged animals, the lat-

eral leg spring (LLS) model, as well as previous models [5, 16, 17]. The angular acceleration Θ̇

also closely resembles the results from [16] with the decelerations near the change of tripods. The

abrupt jerk of the orientation at mid-stance in Figure 3.3(c) is likely due to the rigidity of the legs;

the results in [16] have smaller jerk at mid-stance but benefit from having compliance modeled in

the legs.

One may note that the orientation of the body is not directed vertically in Figure 3.2(a). This is

20

20 15 10 5 0 5 100

5

10

15

20

Tracked Center of Mass

Distance (m)

Dis

tanc

e (m

)

(a)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.70

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Velocity vs Time

Velo

city

(met

ers

per s

econ

d)

Time (s)

(b)

1.14 1.16 1.18 1.2 1.22 1.24 1.26 1.281.61

1.62

1.63

1.64

1.65

1.66

1.67

1.68

1.69

Velocity vs Time

Velo

city

(met

ers

per s

econ

d)

Time (s)

(c)

Figure 3.2: Results of the simulated system running straight without turning inputs operating atthe fastest stride frequencies seen in DASH. (a) shows the trajectory of the system across 2-DEuclidean space. Though initially the system was oriented straight-ahead, the system had sometransients at the beginning that turned the system slightly. The arrow indicates the motion of thesystem as time progresses. (b) shows the acceleration from a stop to a steady-state velocity. Once ata steady-state velocity, the forward velocity oscillates slightly (c). The dashed line in (c) indicatesthe steady-state mean forward velocity. Red circles indicate the state when a transition betweentripods occurs.

21

because there are some transients that occur during startup. The robot has no initial velocity, and

during the acceleration to steady-state velocity, the body experiences unbalanced moments created

from the different tripods at different points in the acceleration.

5.4 5.5 5.6 5.70.05

0

0.05Relative Heading vs Time

(rad

ians

per

sec

ond)

Time (s)

(a)

5.35 5.4 5.45 5.5 5.55 5.6 5.65 5.7

0.385

0.38

0.375

0.37

0.365

0.36Orientation vs Time

Orie

ntat

ion

(radi

ans)

Time (s)

(b)

5.4 5.5 5.6 5.7

0.4

0.2

0

0.2

0.4

Angular Velocity vs Time

Angu

lar V

eloc

ity (r

adia

ns p

er s

econ

d)

Time (s)

(c)

Figure 3.3: The rigid body exhibits slight changes in orientation and heading during straight run-ning. (a) shows how the heading changes during running, with the greatest heading accelerationoccurring when the legs reach the middle of stance. The orientation oscillates with the same fre-quency as the stride frequency (b). The angular velocity is seen in (c). Dashed lines indicate themean values at steady-state. The average relative heading is 0 radians, and the average angularvelocity is 0 radians per second. Red circles indicate the state of the system when a transitionbetween tripods occurs.

Results of a simulation of a single run with multiple turning inputs applied can be seen in

Figures 3.4, 3.5, and 3.6. The simulation consists of a period of running straight, followed by a

22

period of applying a left turning input, then another period of running straight, and then a period of

applying a right turning input. This creates a S-turn maneuver. The system was operated at 10Hz,

near the experimentally-determined best stride frequency for turning in DASH of 11.5 Hz. Figure

3.4 tracks the center of mass as the simulation evolves over time, and the regions where the turning

inputs are applied is apparent.

6 5 4 3 2 1 00

1

2

3

4

5

Tracked Center of Mass

Distance (m)

Dis

tanc

e (m

)

Figure 3.4: Simulation results of running at 10Hz stride frequency with turning input of π/10radians and higher friction mid-legs. The simulation has four distinct regions of operation: the firstwith no turning input; the second with a left turning input; the third with no turning input again;and the final with a right turning input. The COM is plotted over 2-D Euclidean space. Arrowsindicate the direction of motion and are adjacent to the sections during which no turning input isapplied.

When a turning input is applied, the instantaneous forward velocity of the system changes from

a smooth, SLIP-like motion with equivalent behaviors for both tripod stances to a behavior with

decidedly different behaviors between the two tripod stances. During a left turn in Figure 3.5(a),

the body has a larger acceleration during the left tripod stance phase and a smaller acceleration

during the right tripod stance phase. During a right turn, the larger acceleration occurs during the

right tripod stance phase and the smaller acceleration occurs during the left tripod stance phase.

The angular velocity exhibits similar behavioral differences between the two tripod stances, as

seen in Figure 3.5(b). In this figure, the hexapod is turning left and has an average negative angular

23

velocity of approximately -0.28 radians per second, or 16 degrees per second for a stride frequency

of 10Hz. When the right tripod is in stance during a left turn, the body actually begins to turn right

with an angular velocity as high as 0.4 radians per second. It is during the left stance that the body

has a negative angular velocity for nearly the entire duration of stance. The pattern is reversed

during right turns, when the right tripod induces the largest right angular velocity.

Figure 3.6 shows how the heading and orientation change when a right-turn turning input is

applied. Prior to t = 7s, the system is running straight with no turning input. At t = 7s, the right-

turn input is applied, and the relative heading δ of the robot quickly changes. After several strides,

it settles to a steady-state oscillation centered around 0.33 radians, implying that the orientation is

always lagging behind the instantaneous velocity. The quick change in heading explains why the

center of mass appears to abruptly change directions at the transition between turning and straight

running in Figure 3.4. The orientation, shown in Figure 3.6(b), continues to oscillate after the

turning input is applied, but now begins to steadily increase as the body turns. The cockroach

Blaberus discoidalis does have these oscillations and changes in heading and orientation, but the

magnitudes are significantly larger and less uniform between strides [18].

24

5.4 5.45 5.5 5.55 5.6 5.65 5.7 5.750.94

0.95

0.96

0.97

0.98

0.99

1Velocity vs Time

Velo

city

(met

ers

per s

econ

d)

Time (s)

(a)

2.4 2.5 2.6 2.7 2.8

1

0.5

0

0.5

1

Angular Velocity vs Time

Angu

lar V

eloc

ity (r

adia

ns p

er s

econ

d)

Time (s)

(b)

Figure 3.5: Simulation results of running at 10Hz stride frequency with turning input of π/10radians and higher friction mid-legs. These results track the velocity (a) and angular velocity (b) ofthe orientation while a left turning input is applied. The average velocities are marked with dashedlines. Red circles indicate the state when a transition between tripods occurs.

6.8 7 7.2 7.40.05

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Relative Heading vs Time

(rad

ians

)

Time (s)

(a)

6.8 7 7.2 7.41.3

1.28

1.26

1.24

1.22

1.2

Orientation vs Time

Orie

ntat

ion

(radi

ans)

Time (s)

(b)

Figure 3.6: Simulation results of running at 10Hz stride frequency with turning input of π/10radians and higher friction mid-legs. These figures track the changes to the heading (a) and theorientation (b) before and while a turning input is applied. Prior to t = 7s, there is no turninginput; After t = 7s, there is a right-turning input applied. Red circles indicate the state when atransition between tripods occurs.

25

It should be noted that turning could be induced in the model using means other than those

used to generate the results above. For example, increasing the friction on only one foot creates

an imbalance and turns the robot away from the side with the foot with increased friction, as one

would expect. Moving the high friction feet from feet 2 and 4 to 1 and 6 still produces turns in the

same direction when the turning input is applied; the high friction on the mid-legs was used only

to compare directly to experimental results from Section 4.2. Reducing the frequency below 10Hz

increases the turning rate but also results in more hard accelerations and jerky motions.

26

Chapter 4

Results

DASH is capable of running at 15 body-lengths per second (1.5 meters per second). This speed was

observed on a DASH with no steering actuator mounted, but the addition of this actuator appears to

have little effect on the top speed. Turning is achieved using a single actuator to deform the body,

though an alternative turning method has been demonstrated as a proof of concept. In addition to

running and turning on smooth terrain, DASH has been shown to be able to climb over relatively

large obstacles and survive falls from large heights [19].

The entire system weighs only 16.2 grams with battery and electronics. Its body is 10 cm long

and 5 cm wide. Including legs, it measures approximately 10 cm wide and 5 cm tall. The electron-

ics package includes a microcontroller, motor driver, and Bluetooth communication module for

wireless operation [20]. Running at full power with no steering control, its 3.7V 50mAh lithium

polymer battery provides approximately 40 minutes of battery life. Larger batteries, such as a 2.6

gram 90mAH lithium polymer battery, can easily replace the 1.8 gram 50 mAh battery to augment

the battery life with minimal effect on the performance metrics described below.

27

4.1 Running Performance

Figure 4.1 shows DASH running at 11 body-lengths per second. From the sequence of images, the

alternating tripod gait produced by the structure can be seen. In general, there are aerial phases

between the touchdown of the alternating tripods with aerial phases taking up to 30 percent of a

full cycle [19].

(a) t = 0 ms (b) t = 16.7 ms (c) t = 33.3 ms

(d) t = 50 ms (e) t = 66.7 ms (f) t = 83.3 ms

Figure 4.1: DASH exhibits dynamic horizontal locomotion. (a) depicts the beginning of one tripodstance and (b) shows the end of the tripod stance. (c) captures the middle of the second tripodstance. (e) shows the middle of the first tripod stance. (d) and (f) both show aerial phases. Theends of the legs are painted white for better visibility.

Performance is highly dependent upon on stride frequency, leg geometry, and foot design. The

rigid legs of Figure 2.6(a) are most effective for running on smooth terrain, and with these legs

DASH was able to reach speeds of 1.5 m/s [19]. There is an observed, roughly-linear relationship

between stride frequency and forward velocity [21]. The leg design of Figure 2.8(b) was able to

reach speeds near 1.5 m/s, and the relationship between the forward velocity and duty cycle applied

to the drive motor is shown in Figure 4.2. Note that this plot shows the relationship between

velocity and duty cycle, not the relationship between velocity and stride frequency. The duty cycle

28

applied to the drive motor does not have a linear relationship to stride frequency due to mechanical

stiffness in the differential drive mechanism. The compliant leg in Figure 2.6(b) was not able to

achieve as high velocities as these other legs.

The simulation results of Figure 3.2 seem to suggest that the maximum forward velocity is just

above 1.6 m/s, which is only slightly higher than DASH manages to achieve at the same stride

frequency. Indeed, foot designs haven’t been shown to increase the maximum forward velocity

above 1.6 m/s, though they can significantly impede performance if not designed well. Foot design

does have a dramatic impact on the acceleration of the system, however. Figure 3.2 shows the

simulated system accelerates to full speed in roughly 11 complete strides. However, using a similar

stride frequency and configuration of friction on the feet of DASH, it takes nearly 30 strides to reach

1.5 m/s. Using either the flexure-loop foot with claws or the cardboard claws on the four-bar ankle

on a rough surface such as carpet or rigid foam, DASH was able to accelerate to 1.5 m/s within four

strides. The simulation also shows a transient effect during startup that results in a small deflection

in orientation during acceleration, which is seen in DASH during acceleration on smooth surfaces

with very high probability.

20 40 60 80 100

0.8

1

1.2

1.4

1.6

Duty Cycle (%)

Velo

city

(met

ers

per s

econ

d)

Forward Velocity vs Time

Figure 4.2: Forward velocity of DASH with the parallel leg design and four-bar ankle on tile floor.There is a monotonic relationship of velocity with applied motor duty cycle.

29

Figure 4.3: Center of mass tracked over nearly one half second. The center of mass follows anapproximate sinusoidal trajectory. The grid pattern marks 5 cm increments. The dashed line showsthe resting position of the center of mass.

When DASH runs on hard ground, the center of mass follows a roughly sinusoidal trajectory,

which is indicative of the SLIP model. Fig. 4.3 shows the trajectory during another example

of horizontal running. The center of mass is tracked throughout the run, and it follows a stable

sinusoidal motion. This trajectory of the center of mass is similar to those seen in running animals

[4, 5].

The SLIP model of ground reaction forces is shown in Fig. 4.4(a). In the model, when a system

makes initial contact with the ground, it decelerates as it stores energy in the system. As the normal

force peaks, however, the fore-aft force becomes positive as the energy is returned to the system

and it accelerates forward. Analysis of the SLIP model shows that these ground reaction forces lend

dynamic stability when running on a level surface [5]. The ground reaction forces of a cockroach

are shown in Figure 4.4(b) [22]. The cockroach data show a cockroach running in an alternating

tripod gait where the animal never enters an aerial phase, causing the normal ground reaction force

to never reach zero. DASH was tested on a force platform to see if it exhibits similar ground

reaction forces to those of natural systems, including the cockroach. A representative sample of

filtered data showing DASH running on a horizontal force platform is in Fig. 4.4(c). Each large

increase in normal force corresponds with a single tripod making contact with the ground. Just

30

as in the SLIP model and with the cockroach, there is a deceleration upon initial contact followed

by an acceleration. The phase relationship between the fore-aft and normal forces are the same as

both the model and what is observed in the cockroaches. The shapes of the force curves are also

very similar, with some slight differences between the measured forces and the sinusoidal forces

of the SLIP model. The fore-aft accelerations were also predicted in the simulation (Figure 3.2),

though the simulation had steeper accelerations than decelerations while the experimental results

were generally more balanced.

(a) (b)

(c)

Figure 4.4: Dynamic locomotion in the horizontal plane has been shown to be similar across manylegged systems as modeled by SLIP. In the figures, red solid lines indicate fore-aft forces and bluedotted lines indicate normal forces into the ground. Negative fore-aft forces represent deceleratingforces and positive fore-aft forces represent accelerating forces. Positive normal forces indicateforces pushing down on the surface. The SLIP model is shown in (a), the ground reaction forcesof a cockroach are shown in (b), and the ground reaction forces of DASH are shown in (c). Resultsin (b) are borrowed from [22] with permission.

31

DASH has also been shown to be able to run on loose granular media [21]. In early experiments

done jointly with Li and Hoover, DASH has shown only partial performance loss when running

through loosely-packed media (velocity decreased to approximately 5 body-lengths per second).

The performance loss is much less than was been observed in earlier trials with Sandbot (velocity

went down to 0.1 body-lengths per second [23]). The sprawled posture and low mass of DASH

keep the body above the surface of the granular media, and the robot appears to operate in a

type of swimming mode. Figure 4.5(a) shows the velocity of DASH running on a hard surface

(drywall), and on a granular media (loosely-packed poppy seeds) [21]. One may note that the

hard surface performance wasn’t as fast as in other hard ground trials. This was due to using

motor-driver electronics that enabled real-time power measurements but were unable to provide

the instantaneous current levels required for the fastest forward velocities.

The real-time power measurements taken from the trials in Figure 4.5 enabled accurate cost

of transport calculations for DASH. The mechanical power is determined from Pmechanical = τω,

where τ = KtI is the motor torque, and ω = 2πVEMF /Ke is the motor angular velocity. The motor

torque constant was previously computed empirically to be Kt = 0.00683 N/A. The motor back

EMF constant was also found to be Ke = 0.15 V/Hz. The current to the motor can be computed

by I = (Vref − VEMF )/R, where Vref is the battery voltage and R is the motor resistance. The

metabolic power is simply the electrically power consumed by the motor, or Pmetabolic = I2R. The

battery voltage was measured before and after each run to aid in the calculation of the current.

The cost of transport is then given by COT = P/mv, where m is the mass of DASH, and v

is the forward velocity, and P is either Pmechanical or Pmetabolic. The forward velocity and stride

frequency is obtained from slow-motion video and the back EMF VEMF is measured from the

onboard electronics.

32

0 5 10 150

20

40

60

80

100

f (Hz)

v (c

m/s

)

Hard groundLoosely packed

(a)

0 5 10 150

2

4

6

8

10

12

14

f (Hz)

CO

T mec

hani

cal (J

/kg*

m)

Hard groundLoosely packed

(b)

0 5 10 150

5

10

15

20

25

30

f (Hz)

CO

T met

abol

ic (J

/kg*

m)

Hard groundLoosely packed

(c)

Figure 4.5: Performance metrics of DASH on hard ground and loosely-packed granular media.Hard ground results are indicated by red squares and granular media results are indicated by bluecircles. (a) shows the velocity of DASH as a function of stride frequency in both media, and theresults show a roughly linear relationship between stride frequency and velocity. (b) shows themechanical cost of transport as a function of stride frequency and (c) shows the metabolic cost oftransport.

On both media, the mechanical cost of transport tends to increase with the stride frequency,

though the highest velocity data points on hard ground and granular media show a dip in the

cost of transport. The hard ground metabolic cost of transport of DASH also seems to increase

with stride frequency but shows a decrease at the highest stride frequency tested. The decrease

33

in the hard ground metabolic cost of transport above a stride frequency of 15Hz is large and may

represent the system operating at a resonant frequency or other preferred mechanical operating

frequency. The metabolic cost of transport values computed on hard ground have a trend similar to

the mechanical cost of transport values, but the metabolic cost of transport values on granular media

vary dramatically. Because the metabolic cost of transport depends on the square of the battery

voltage and the battery voltage cannot be measured instantaneously during a run but instead before

and after a run, there is some uncertainty about the accuracy of these values. Future electronics

boards may allow for direct measurement of battery voltage during a run, which may observe

decreases in voltage during loaded conditions and thus lend more credibility to the metabolic cost

of transport values. The metabolic cost of transport of DASH on hard ground determined in [21]

is close to the earlier values estimated in [19], which used the total electrical power of the system

including not only the motor electrical power but also wireless communications power.

4.2 Turning Results

DASH demonstrates that turning can be achieved with relatively simple modifications of kinemat-

ics. The top speed of DASH is negligibly affected by the turning actuator, but turning is most

successful when operating at a 20 percent duty cycle applied to the main DC motor. Figure 4.6

shows a turning maneuver of DASH, first turning left and then immediately turning right. DASH

turns approximately 50 degrees per second to left and 55 degrees per second to the right when

applying a maximum turning input and a 20 percent duty cycle to the main drive, which in this

case corresponds with stride frequency of 11.5Hz.

34

Figure 4.6: Controlled S-turn maneuver using a single turning actuator.

The simulation from Figure 3.4 underestimated the ability of the turning input to induce turns

in DASH. The simulation, which was run at a 10Hz stride frequency as opposed to the 11.5Hz

stride frequency from the experiment in Figure 4.6, suggested a turning rate of approximately 16

degrees. Increasing the friction in the simulation did increase the turning rate, but the friction

values used were fairly close to the conditions on the floor used in Figure 4.6. This is turning is

significantly slower than in Blaberus discoidalis, which can have average turning rates of a couple

hundred degrees per second [18].

The extended-toe turning method described in Section 2.5 and Figure 2.10 also resulted in

turning behavior. In fact, it was able to produce turning rates higher than using the method used

in Figure 4.6. This method was only a proof-of-concept and did not rely on actuators, instead

mechanically locking a single toe downward. If the toe extension were actuated rather than locked

mechanically, it would likely have less downward displacement and reduced stiffness and thus have

a lower turning rate. The turning rate in Figure 4.7 is taken from right-turn trials. When no toe

was locked downward, the robot ran straight with the forward velocities of Figure 4.2. At lower

duty cycles without a turning input, there was slight turning, but this was likely due to construction

error. There was no discernible turning above a 40% duty cycle when there was no applied turning

35

input. When applying the turning input, DASH was exhibiting a sort of skipping motion at the

lowest duty cycles. Though these achieved the highest turning rates, the skipping behavior was

unlike the smooth dynamics seen when turning at higher duty cycles or when performing high-

speed straight running. Increasing the duty cycle with a turning input applied resulted in decreased

turning rates, eventually getting to nearly straight running despite having a toe rigidly extended.

All measurements were determined from video captured by overhead slow-motion video cameras.

20 40 60 80 10020

0

20

40

60

80

100

120

Duty Cycle (%)

(deg

ree/

s)

Angular Velocity vs Duty Cycle

Figure 4.7: Turning performance of DASH with the parallel leg design and four-bar ankle on tilefloor. Turning is achieved by extending a single front toe and the turning rate is dependent uponduty cycle applied to the drive motor. These data are from observed right turns only.

4.3 Step Climbing

DASH is able to mount obstacles greater than its own body height using only feed-forward controls

[19]. The compliance throughout the structure can compensate for the differences in foot position

when running on uneven terrain, and power is still successfully delivered to the feet. Though

lacking control of individual foot placement, DASH is eventually able clear the 5.5 cm obstacle in

Figure 4.8. This is near the upper limit of step heights DASH can surmount.

36

Figure 4.8: Sequential images of DASH climbing over a stack of acrylic. The obstacle is approxi-mately 5.5 cm tall.

4.4 Surviving Falls

Several tests were carried out to show the capability of DASH to survive falls from large heights.

After multiple drops of 7.5 meters, 12 meters, and 28 meters (75, 120 and 280 body lengths) on

to concrete, the robot remained operational, with no damage [19]. From 28 meters, the velocity

at impact is approximately 10.3 m/s. This has been verified to be the terminal velocity using

wind tunnel experiments, indicating it can survive falls from any height. Despite this high impact

velocity, the impact energy is only 0.795 J due to the lightweight construction. As can be seen

from Fig. 4.9, the body contorts dramatically upon impact, in the process absorbing a significant

amount of the impact energy. In these images, the body was falling at approximately 6.5 meters

per second just prior to impact.

37

(a) t = 0 ms (b) t = 10 ms

(c) t = 16.7 ms (d) t = 23.3 ms

Figure 4.9: Sequential images of DASH impacting the ground at a velocity of approximately 6.5m/s. (b) and (c) show the great contortion that the body undergoes during a high velocity impact.By (d), the body has almost fully recovered its original shape. The compliance of the structureenables it to withstand high impact velocities and survive without damage.

38

Chapter 5

Discussion of Results and Conclusions

With a top speed of approximately 15 body-lengths a second, DASH meets the goal of achieving

efficient high-speed running using a single drive actuator. It does so using a novel differential drive

and the circular output of a DC motor to create two anti-phase circular leg motions which can

later be adapted to climbing. The resulting kinematics create an alternating tripod gait that exhibits

dynamic behavior and matches the ground reaction forces seen in natural systems.

These kinematics were modeled using a 2-D dynamic model constructed to emulate DASH.

The model predicted a maximum forward velocity very near the current maximum forward veloc-

ity of DASH given the maximum stride frequency of DASH. The model also demonstrated fore-aft

accelerations similar to those exhibited by the SLIP model and lateral accelerations similar to those

exhibited by the LLS model [5, 17]. The body dynamics from the straight-running simulation re-

sults also matched the straight-running results of a more complicated model previously published

[16]. The 2-D model also predicted the emergence of turning when the leg trajectories are modified

as shown in Figure 2.9, though it underestimated the real-world performance. This suggests the

existence of a more complicated dynamic or mechanical behavior augmenting the turning perfor-

39

Table 5.1: Select properties of comparable robots1

DASH iSprawl [7] Mini-Whegs [14] RHex [8, 24]Size (cm) 10 x 5 x 10 15.5 x 11.6 x 7 9 x 6.8 x 7.2 53 x 20 x 15Mass (g) 16.2 300 146 7500

Speed (body-lengths/second) 15 15 10 5Leg Frequency (Hz) 17 14 22 5

Density (g/cc) 0.03 0.24 0.33 0.48Battery life (min) 40 5 N/A 30

Max Power Density2 (W/kg) 20 8.6 8.2 8Electrical Cost of Transport (J/kg/m) 8 17.4 8.9 20

1Several values in this table were not explicitly published in the literature. They were calculated usingthe available published data.2The power density requires knowing the force delivered to the ground during locomotion. An upperbound for this value is calcutated for iSprawl, Mini-Whegs, and Rhex using the rated output power forthe motors. The value for DASH was calculated using the experimental output power of the motor.

mance.

Two methods of turning were successfully demonstrated. The first method uses an SMA servo-

motor to deform the body of DASH an alter the touchdown angle of the feet. The second method

demonstrated the ability to turn by simply modifying a single foot, but it was never actuated.

When running, the presence of serial compliance mechanically between the foot contact and

the drive motor positively contribute to horizontal locomotion. Without this compliance, the gait is

much more unforgiving and less smooth and results in poor ground contact. The compliance likely

creates a more SLIP-like gait, improving the robot’s overall locomotion.

DASH compares favorably to other small legged robots in terms of running speed in body

lengths per second (Table 5.1). At 15 body lengths per second, DASH is faster than both Mini-

Whegs and RHex, and is comparable to iSprawl. Its dynamic locomotion exhibits SLIP behavior

similar to that of natural systems. In addition to being fast, DASH also offers several unique

characteristics. While DASH is not significantly different in volume from Mini-Whegs, it is ap-

40

proximately 10 times lighter. This is a consequence of the SCM process which replaces traditional

solid structural elements with hollow, folded structures. The light weight enables a battery life of

40 minutes at top speed with only a 1.8 gram 50mAh battery. Upgrading to a larger battery, such as

a 2.6 gram 90mAh battery, could dramatically increase battery life while having a minimal impact

on locomotion speed.

The low mass and high-power density enabled by the scaled SCM process combine with the

specific design of DASH to create a platform that is capable of surviving falls from large heights in

addition to fast horizontal locomotion. Previous studies have shown that velocity at impact, impact

surface compliance, orientation at impact, the number and compliance of limbs, and the impact

load per unit area all contribute to the ability of a system to survive a fall [25]. Many of these

properties are present in DASH. The low mass and density of DASH result in reduced impact ve-

locities and low impact energies. The design of DASH has high stiffness in the directions required

for power transmission from the drive motor, but it retains compliance in off-axis directions. The

design of DASH also enables the electronics to be housed in the interior of the robot, reducing

the chance that they will suffer direct impacts with the ground. These properties combine to allow

DASH to survive falls from large heights onto concrete. MiniWhegs has survived falls of 0.9 me-

ters [14] without damage as did RHex from a 6 meter height [26]. In general, however, there are

very few claims of robustness to falling in legged robots. Since DASH survives high falls without

damage and is low cost, it can take risks with falls which another robot might not survive.

Future work will progress from studying dynamics in the horizontal plane to instead focus on

locomotion on inclined surfaces as well as vertical surfaces. The author will study foot designs

and how to precisely manage ground interactions to generate the ground reaction forces needed for

climbing. The dynamics of running on inclined and vertical surfaces will be explored and lever-

aged in future designs of DASH. The differences between horizontal and vertical locomotion will

need to be managed to maintain a high level of performance on horizontal surfaces while adding

41

the ability to scale walls. Future designs will strive for passive stability in both the horizontal and

vertical planes. The transition between horizontal and vertical locomotion will also be explored.

Work towards these goals will be aided by interactions from colleagues in the Poly-PEDAL Labo-

ratory as well as the Center Interdisciplinary Bio-Inspiration in Education and Research who share

interests in system dynamics, engagement, and performance on surfaces with various angles of

inclination.

Acknowledgment

The author would like to thank Ron Fearing for his brainstorming, support and guidance, and

Alexis Birkmeyer for her mental support and willingness to share her Illustrator skills. Thanks to

Bob Full for his advice and support. Thanks also to Aaron Hoover and the rest of the Biomimetic

Millisystems Lab for their support, Kevin Peterson for his help with the IROS paper, the Center

for Integrative Biomechanics in Education and Research for their advice and the use of their force

platform, and the CRAB Lab for their collaboration. This work is supported by the NSF Center of

Integrated Nanomechanical Systems.

42

Bibliography

[1] I. E. Brown and G. E. Loeb, “A reductionist approach to creating and using neuromuscu-

loskeletal models,” in Biomechanics and Neural Control of Posture and Movement, J. M.

Winters and P. E. Crago, Eds. New York: Springer, 2000.

[2] S. Sponberg and R. J. Full, “Neuromechanical response of musculo-skeletal structures in

cockroaches during rapid running on rough terrain.” The Journal of Experimental Biology,

vol. 211, no. 3, pp. 433–446, Feb 2008.

[3] T. M. Kubow and R. J. Full, “The role of the mechanical system in control: a hypothesis of

self-stabilization in hexapedal runners,” Phil. Trans. R. Soc., vol. 354, pp. 849–851, 1999.

[4] R. Blickhan and R. J. Full, “Similarity in multilegged locomotion: Bouncing like a

monopode,” Journal of Comparative Physiology A: Neuroethology, Sensory, Neural, and

Behavioral Physiology, vol. 173, no. 5, pp. 509–517, 11 1993.

[5] P. Holmes, R. J. Full, D. Koditschek, and J. Guckenheimer, “The dynamics of legged lo-

comotion: Models, analyses, and challenges,” SIAM Review, vol. 48, no. 2, pp. 207–304,

2006.

[6] J. G. Cham, S. A. Bailey, J. E. Clark, R. J. Full, and M. R. Cutkosky, “Fast and robust:

Hexapedal robots via shape deposition manufacturing,” International Journal of Robotics

Research, vol. 21, no. 10, 2002.

43

[7] S. Kim, J. E. Clark, and M. R. Cutkosky, “iSprawl: Design and tuning for high-speed au-

tonomous open-loop running,” International Journal of Robotics Research, vol. 25, no. 9, pp.

903–912, 2006.

[8] U. Saranli, M. Buehler, and D. E. Koditschek, “RHex: A simple and highly mobile hexapod

robot,” International Journal of Robotics Research, vol. 20, no. 7, pp. 616 – 631, July 2001.

[9] A. Hoover and R. Fearing, “Fast scale prototyping for folded millirobots,” IEEE International

Conference on Robotics and Automation, pp. 886–892, May 2008.

[10] R. S. F. Aaron M. Hoover, Erik Steltz, “RoACH: An autonomous 2.4g crawling hexapod

robot,” in International Conference International Conference on Intelligent RObots and Sys-

tems, 2008.

[11] E. E. Steltz, “Redesign of the micromechanical flying insect in a power density context,”

Ph.D. dissertation, EECS Department, University of California, Berkeley, May 2008.

[12] D. I. Goldman, T. S. Chen, D. M. Dudek, and R. J. Full, “Dynamics of rapid vertical climbing

in cockroaches reveals a template,” The Journal of Experimental Biology, vol. 209, pp. 2990–

3000, May 2006.

[13] J. Schmitt and S. Bonnono, “Dynamics and stability of lateral plane locomotion on

inclines,” Journal of Theoretical Biology, vol. 261, no. 4, pp. 598 – 609, 2009.

[Online]. Available: http://www.sciencedirect.com/science/article/B6WMD-4X2DD79-4/2/

fa73720938da5af21f577b31681d1f95

[14] J. M. Morrey, B. Lambrecht, A. D. Horchler, R. E. Ritzmann, and R. D. Quinn, “Highly mo-

bile and robust small quadruped robot,” in Proceedings of the 2003 IEEE/RSJ International

Conference on Intelligent Robots and Systems, Las Vegas, NV, 2003, pp. 82–87.

[15] H. Komsuoglu, K. Sohn, R. J. Full, and D. E. Koditschek, “A physical model for dynamical

arthropod running on level ground,” ScholarlyCommonsPenn, Tech. Rep., 2008.

44

[16] J. E. Seipel, P. J. Holmes, and R. J. Full, “Dynamics and stability of insect locomotion: a

hexapedal model for horizontal plane motions,” Biol. Cybern., vol. 91, no. 2, pp. 76–90,

2004.

[17] J. Schmitt and P. J. Holmes, “Mechanical models for insect locomotion: Dynamics and sta-

bility in the horizontal plane II. Application,” Biol. Cybern., vol. 83, no. 6, pp. 517–527,

2000.

[18] D. L. Jindrich and R. J. Full, “Many-legged maneuverability: Dynamics of turning in

hexapods,” The Journal of Experimental Biology, no. 202, pp. 1603–1623, 1999.

[19] P. Birkmeyer, K. Peterson, and R. Fearing, “DASH: A dynamic 16g hexapedal robot,” in

International Conference on Intelligent Robots and Systems (IROS), 2009.

[20] F. G. Bermudez and R. Fearing, “Optical flow on a flapping wing robot,” in IROS, 2009.

[21] C. Li, A. M. Hoover, P. Birkmeyer, P. B. Umbanhowar, R. S. Fearing, and D. I. Goldman,

“Systematic study of the performance of small robots on controlled laboratory substrates.”

SPIE DSS, 2010.

[22] R. Full and M. Tu, “Mechanics of six-legged runners,” J Exp Biol, vol. 148, no. 1, pp. 129–

146, 1990.

[23] C. Li, P. B. Umbanhowar, H. Komsuoglu, D. E. Koditschek, and D. I. Goldman, “Sensitive

dependence of the motion of a legged robot on granular media,” PNAS, vol. 106, no. 9, pp.

3029–3034, March 2009.

[24] D. Campbell, “Bounding and stair descent in the hexapod rhex,” Master’s thesis, McGill

University, February 2004.

[25] J. M. Cameron and R. C. Arkin, “Survival of falling robots,” W. J. Wolfe and W. H. Chun,

Eds., vol. 1613, no. 1. SPIE, 1992, pp. 91–102.

45

[26] E. Z. Moore, “Leg design and stair climbing control for the RHex robotic hexapod,” Master’s

thesis, McGill University, January 2002.

46