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
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