Dynamic Underactuated Flying-Walking (DUCK) Robot

Christopher J. Pratt and Kam K. Leang∗

Abstract— This paper describes the development of a flyingand walking robot, called the dynamic underactuated flying-walking (DUCK) robot. The DUCK robot combines a high-mobility flying platform, such as a quadcopter (quadrotorhelicopter), with passive-dynamic legs to create a versatilesystem that can fly and walk. One of the advantages of passive-dynamic legs for walking is that additional actuators are notneeded for terrestrial locomotion. Herein, a mathematical modelis presented and simulations are used to help design a prototyperobot. Experimental results demonstrate the feasibility of com-bining an aerial platform with passive-dynamic legs to create aneffective flying and walking robot. In particular, two modes ofwalking are demonstrated: (1) passive walking down inclinedsurfaces for low-energy terrestrial locomotion, and (2) active(powered) walking by leveraging the capabilities of the flyingplatform, where thrust from the quadcopter’s rotors enablesthe DUCK robot to take steps and walk on flat surfaces or upinclined surfaces.

I. INTRODUCTION

This paper describes the development of the dynamic un-deractuated flying-walking (DUCK) robot. The robot, whichis created by combining a multi-rotor aerial platform (such asa quad-rotor helicopter or quadcopter) with passive-dynamiclegs, is capable of both aerial and terrestrial locomotion.Recently, some attention has focused on developing aerialrobots with the ability to walk, swim, roll, etc., to enhanceversatility and/or offer energy-efficient modes of travel tosupplement high-energy and high-mobility flight [1]. Suchdesigns have advantages in situations where the robot mayneed to fly to overcome large obstacles, yet have the abilityto slowly traverse terrain and operate over a long period oftime. As depicted in Fig. 1, the proposed DUCK robot hasthree basic modes of operation: (a) flying in situations thatdemand it, (b) low-energy passive walking down inclinedsurfaces (motors turned off), and (c) active (powered) walk-ing where the quadcopter’s rotors provide the needed forcesto enable the robot to take steps and walk on flat, or upinclined, surfaces. The contributions of this work include:(1) mathematical modeling of the DUCK robot; (2) designof the robot through simulation; (3) creation of a prototypeto demonstrate flying and passive/active walking; and (4) anenergy analysis to compare the power consumption of flightto that of active walking.

II. PRIOR WORK

There have been many successful attempts to create robotswith aerial and terrestrial locomotive capabilities. For exam-ple, some robots use legs, such as the Multi-Tentacle Air

Authors are with the DARC (Design, Automation, Robotics and Control)Lab, University of Utah Robotics Center and Department of MechanicalEngineering, University of Utah, Salt Lake City, Utah 84112.

∗Corresponding author (E-mail: kam.k.leang@utah.edu).

Flying modePassive walking modeActive walking mode

(a)(b)(c)

(a)

(a)

(b)

(c)

(c)

Fig. 1. The modes of operation for the proposed passive-dynamic flying-walking DUCK robot, showing (a) flying mode, then transitioning to (b)passive walking mode, followed by flying again, and finally, (c) active(powered) mode where the quadcopter’s rotors provide the needed forceto enable the robot to take steps and walk on flat or up inclined surfaces.

Vehicle [2]. This design has three limbs made of poweredjoints that can walk, perch, and grasp. Other robots usevariations on crawling, such as the MMALV which usespowered wheel-legs [3], or one that converts downward thrustinto forward motion [4]. The DALER crawls by rotatingits wings [5], and the BOLT and the DASH+WINGS usesynchronized legs and flapping wings [6], [7]. Still othersuse motorized wheels [8], or roll on exoskeletons whichleverage the propulsion mechanism of the aerial platform [1],[9], [10]. Although effective, many of these designs re-quire additional actuators for ground locomotion. Addingactuators increases weight and design complexity, whichincreases power consumption, especially during flight. Theaerial-terrestrial robot (ATR) design described in this paperuses passive-dynamic walking to achieve ground locomotion,which requires no additional actuators to function, resultingin a relatively low-power system with added versatility.The combination of passive-dynamic walking with an aerialplatform is a new concept.

Passive-dynamic walkers are mechanical devices consist-ing of rigid links connected by joints, which can walk downa slope without the use of motors or controllers [11]. Passive-dynamic walkers, both as passive and semi-powered walkingapparatuses, have already been explored as a means of purelyterrestrial locomotion [12] – [18]. Herein, a passive walkingmechanism is combined with a quadcopter flying platformto create an ATR. The resulting ATR can both walk usingthe low-power passive-dynamic walking mechanism and fly.Such a design is advantageous in a wide range of temporal-spacial applications, where high mobility is required butconstant slow movement over long periods of time wouldplace huge energy demands on conventional purely-flying

2016 IEEE International Conference on Robotics and Automation (ICRA)Stockholm, Sweden, May 16-21, 2016

978-1-4673-8025-6/16/$31.00 ©2016 IEEE 3267

(a) RobotFlying platform

Feet

Hip joint

(b) Passive walking

φ(c) Active (powered) walking

γ

Passive-dynamicwalker

Fig. 2. Key features of the DUCK robot: (a) key components of the robot,(b) illustration of passive walking without power down a slope of angle φ,and (c) illustration of active (powered) walking on flat ground or up a slopeof angle γ. During active walking rear thrusters provide forward thrust androcking motion to enable the robot to take steps and walk.

platforms. The design that is proposed is novel and uniquebecause in addition to flying, it has the ability for passivewalking, as well as active or powered walking where theflying platform’s rotors can provide small thrust forces toenable the robot to walk on level surfaces as well as upinclined surfaces. This design extends the applicability ofpassive-dynamic walkers from just traversing down slopesto traversing flat surfaces or up inclined slopes.

III. DYNAMICS MODELING

Key features of the DUCK robot are shown in Fig. 2. Therobot consists of a hover-capable flying platform connectedto passive-dynamic walking legs as shown in Fig. 2(a).The robot has two modes of walking: (1) passive walkingdown an inclined slope (thrusters turned off) as illustrated inFig. 2(b) or (2) active (powered) walking where the flyingmechanism provides the needed force to enable walkingalong flat or up inclined surfaces as shown in Fig. 2(c).

The design of the passive-dynamic legs is accomplished bymodeling the dynamics of the walking mechanism coupledwith the flying platform, then using simulation to tune theparameters of the design. The modeling and simulationadapted details presented in [14]–[16], where the 3D walkingmotion is separated into two, 2D models of the lateral (side-to-side) and sagittal (front-to-back) motions as illustrated inFig. 3(a).

To begin, scalars are represented by regular math script(e.g., a or A), vectors are denoted by a harpoon above avariable (e.g., ⇀a or

⇀

A), unit vectors are represented by ahat above a lower case variable (e.g., x̂), matrices are boldface capitol letters (e.g., A or Ω), reference joints/frames

z0W0W

y0W0W

y 2Lz 2L

FF 2LFF 2L

Initialposition

FL2L F

R 2LFR 2L

Lateral plane

cww+

(b)

(a)

z1L

y1Ly1L

x 4S4Sz 4S4S

z3Sz3Sz3Sz3SFB 3S FF 3 3S

z2S

x2S

z0W0W

x0Wx0W0W

Initialposition

z1Sz1Sx1S

(c)

Sagittal plane

FF 2SFF 2S

x3Sx3S

z0W0Wy0W0W

x0W0W

Sagittal planeLateral

plane

φφ

Fig. 3. Depiction of how the 3D walking motion is split into separatelateral and sagittal motions for modeling [14]–[16]. (a) shows the 3D motionprojected onto the 2D lateral and sagittal planes. (b) and (c) are free bodydiagrams for the lateral and sagittal motions, respectively.

are indicated by subscript numbers in brackets (e.g.,⇀

F1{2}denotes vector

⇀

F1 as viewed by joint/frame 2), and cross anddot products are denoted with × and •, respectively.

A. The Equations of Motion

The equations of motion are obtained using the 2DNewton-Euler method, where a dynamic system is mod-eled as connected rotational or linear joints as depicted inFig. 3(b) and (c). A recursive process is applied to thismodel, producing a system of equations that solve for theaccelerations of each joint given a starting state. The first stepin this process is to create equations for the angular velocitiesand accelerations of each joint, constructed for i = 1 . . .N ,where N denotes the total number of joints. Thus,

⇀ωi{i} =Ri−1,i

⇀ωi−1{i−1} + θ̇ir̂i{i}, (1)

⇀̇ωi{i} =Ri−1,i

⇀̇ωi−1{i−1} + θ̈ir̂i{i}, (2)

where ⇀ωi{i} and⇀̇ωi{i} are the angular velocity and

acceleration of joint i relative to the inertial frame,respectively; θ̇i and θ̈i are the angular velocity andacceleration scalars of joint i relative to the previous joint(measured in radians), respectively; Rm,n is the rotationmatrix to change the reference frame of a vector from jointm to joint n; and r̂i is the rotation axis (x̂ for the lateral caseand ŷ for the sagittal case). For the lateral case Ri−1,i =[1, 0, 0, ; 0, cos(θi),− sin(θi); 0, sin(θi), cos(θi)] andRi−1,i = [1, 0, 0; 0, cos(θi),− sin(θi); 0, sin(θi), cos(θi)] forthe sagittal case.

Next, equations for the linear velocities and accelerations

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of each joint are constructed for i = 1 . . .N , hence

⇀̇vi{i} =Ri−1,i

⇀̇vi−1{i−1} +

⇀̈

di{i} + 2⇀ωi{i} ×

⇀̇

di{i}+⇀̇ωi{i} ×

⇀

di{i} +⇀ωi{i} × (⇀ωi{i} ×

⇀

di{i}), (3)⇀̇vcg,i{i} =

⇀̇vi{i} +

⇀̇ωi{i} ×

⇀

dcg,i{i}+⇀ωi{i} × (⇀ωi{i} ×

⇀

dcg,i{i}), (4)

where ⇀̇vi and⇀̇vcg,i are the linear accelerations of the ith

joint and its center of gravity relative to the inertial frame,

respectively;⇀

di,⇀̇

di, and⇀̈

di are the linear distance, velocityand acceleration, respectively, from joint i−1 to joint i; and⇀

dcg,i is the linear distance from joint i to its center of gravity.For joint i = 0 (the inertial reference frame), ⇀ω0,

⇀̇ω0, and

⇀̇v0

are⇀

0 � [0; 0; 0].Next, equations representing the forces and torques on

each joint are constructed for i = N . . . 1, thus⇀

Fi{i} =Ri+1,i⇀

Fi+1{i+1} +mi⇀̇vcg,i{i} −miR0,i⇀g{0}−∑ ⇀

Fapplied,i{i} , (5)

⇀ui{i} =Ri+1,i

⇀ui+1{i+1} −

⇀

Fi{i} × (⇀

di{i} +⇀

dcg,i{i})+

(Ri+1,i⇀

Fi+1{i+1})×⇀

dcg,i{i} − Ii⇀̇ωi{i}+m∑

n=1

⇀

Fapplied,i{i} ×⇀

d⇀Fapplied,i{i}

+⇀uFric,i{i}, (6)

whereRi+1,i =R

Ti,i+1, (7)

R0,i =R0,1R1,2 . . .Ri−1,i, (8)

Ii =

⎡⎣Ixi 0 00 Iyi 0

0 0 Izi

⎤⎦ , (9)

and⇀

Fi and⇀ui are the total forces and torques on joint i

(always⇀

0), respectively; mi is the mass of joint i;⇀g{0} is the

gravity vector ([0,−g, 0, ]); ⇀uFric,i is the torsional friction onjoint i; Ii is the inertial matrix of joint i; and

⇀

Fapplied,i and⇀

d⇀Fapplied,i

are an applied force on joint i and its distance from

the center of gravity of joint i, respectively. Also,⇀

FN+1 and⇀uN+1 are

⇀

0.Next, the torques or forces on each joint is extracted for

i = 1 . . .N , therefore

Ei = l̂i{i} •⇀

Fi{i} for linear joints, (10)= r̂i{i} • ⇀ui{i} for rotational joints, (11)

where Ei is the equation for joint i, and l̂i is the linear joint’saxis (ŷi for the lateral case and x̂i for the sagittal case).

Lastly, to make all angles be measured relative to theirstarting point, and not the previous joint, the followingsymbolic substitution is made into each Ei equation, forj = N . . . 1,

θj = θj −( j−1∑

k=1

θk

), θ̇j = θ̇j −

( j−1∑k=1

θ̇k

),

θ̈j = θ̈j −( j−1∑

k=1

θ̈k

), (12)

where θi, θ̇i, and θ̈i for linear joints are zero. A systemof N equations with N unknowns (the acceleration of eachjoint) is created using Ei = 0 for each joint. This system ofequations describes the instantaneous motions of the robotgiven any starting state.

To create a simulation of the DUCK robot’s walkingbehavior, the equations of motion for both the lateral andsagittal movements are needed. The lateral motion equationis obtained with the Newton-Euler method using two jointsand the sagittal motion equation requires four joints as shownin Fig. 3. The dimensions of the robot are shown in Fig. 4.Information describing the setup for the lateral motion isprovided in Tables I through III, while Tables IV through VIdescribe the sagittal motion. The resulting equation for thelateral motion is given by

θ̈2L =(2rL(Ftot −mtotacgθ̇22L)− 2mtotacgg) sin(θ2L)−wL(FBL + FFL − FBR − FFR), (13)

where FBL, FFL, FBR, FFR, and Ftot are the back left,front left, back right, front right, and total thruster forces,respectively; rL and wL are the lateral rolling radius andwingspan, respectively; mtot is the total mass; and acg is thedistance from the lateral center of curvature to the collectivecenter of gravity. The equation for the sagittal motion isfound using the same process as for the lateral motion andis omitted for brevity.

Equation (13) does not describe the lateral motion whilethe robot is on the inside edge of its feet (|θ2L| < vin).Adapting details presented in [14] and [15], using

∑T = Iθ̈

to sum torques about the foot’s inside edge yields

θ̈2L =

∑T

I, (14)

where∑T =∓mtotg(acg sin(±θ2L) + rL sin(vin ∓ θ2L))+

(wL/2± rL sin(vin))(FBR + FFR)−(wL/2∓ rL sin(vin))(FBL + FFL), (15)

I =ItotL +mtot(a2cg + r

2L − 2rLacg cos(vin)). (16)

In the above equations, the top signs of “∓” and “±” areused when the robot is on its left leg, and the bottom signswhen on its right leg.

Centers of rotation

φφ

wL w

S

acg d

bcgbcg

ccg

rL rS

vinvin

vout v

toe

Lateral plane

(a)

Sagittal plane

(b)

Fig. 4. Outline of dimensions used in the DUCK robot. (a) The lateraldimensions and (b) the sagittal dimensions.

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TABLE I

NEWTON-EULER JOINTS FOR THE LATERAL MOTION

Joint mi Ixi θi⇀

di{i}⇀

dcg,i{i}0 0 0 0

⇀0

⇀0

1 0 0 0 2πrLθ2Lŷ1{1}⇀0

2 mtot ItotL θ2L⇀0 −acg ẑ2{2}

TABLE II

APPLIED FORCES FOR THE LATERAL MOTION

Force Joint Vector⇀

d⇀Fapplied,i{i}

⇀

FF2L 2 FF2LR1,2ŷ1{1} −rLR1,2ŷ1{1}⇀

FL2L 2 (FBL + FFL)ẑ2{2} −0.5wLŷ2{2}⇀

FR2L 2 (FBR + FFR)ẑ2{2} 0.5wLŷ2{2}

TABLE III

TORSIONAL FRICTION FOR THE LATERAL MOTION

Torsion Joint Vector⇀uFric,i{i} 1, 2, 3

⇀0

TABLE IV

NEWTON-EULER JOINTS FOR THE SAGITTAL MOTION

Joint mi Iyi θi di{i}⇀

dcg,i{i}0 0 0 0

⇀0

⇀0

1 0 0 φ 2πrSθ2S x̂1{1}⇀0

2 ml IlS θ2S −bcg ẑ2{2} −bcg ẑ2{2}3 mq IqS θ3S

⇀0 −cẑ3{3}

4 ml IlS θ4S⇀0 −bcg ẑ4{4}

TABLE V

APPLIED FORCES FOR THE SAGITTAL MOTION

Force Joint Vector⇀

d⇀Fapplied,i{i}

⇀

FF2S 2 FS2FR1,2x̂1{1} (rS − d)R1,2 ẑ1{1}−bcg ẑ2{2}

⇀

FB3S 3 (FBL + FBR)x̂3{3} 0.5wS ẑ3{3}⇀

FF3S 3 (FFL + FFR)x̂3{3} −0.5wS ẑ3{3}

TABLE VI

TORSIONAL FRICTION FOR THE SAGITTAL MOTION

Torsion Joint Vector⇀uFric,i{i} 1, 2

⇀0

⇀uFric,3{3} 3 −.5μb(ml +mq)dbsign(θ̇3)ẑ⇀uFric,4{4} 4 −.5μb(ml +mq)dbsign(θ̇4)ẑ

B. Handling Collisions

A collision between the feet and the ground is assumed tooccur when the robot transitions between leaning left or right(the angle θ2L changes sign). Collisions in the lateral planeare handled by adapting details presented in [14] and [15].Here, it is assumed that an inelastic collision occurs betweenthe leg and the ground, thus

θ̇+2L = θ̇−2L cos

[2 tan

(rL sin(vin)

rL cos(vin)− acg

)], (17)

where the superscripts “−” and “+” mean before and afterthe collision, respectively.

In previous works collisions in the sagittal plane weresolved using the conservation of angular momentum. Thisapproach is made difficult in this work by the addition of thequadcopter and this technique ignores energy loss due to feetscuffing. This paper takes a new approach to solving thesecollisions by leveraging the equations of motion presentedabove. When there is a transition between which leg is onthe ground, if the system’s angular velocities are kept thesame there will appear to be a discontinuity in the system’smomentum. This occurs since the leg describing the system’stranslational velocity changes. To correct this it is assumedthat for one simulation step the robot is sliding with a largefrictional force (

⇀

FF2S in Fig. 3). This sliding ends oncethe system’s velocity predicted by Newton’s Second Lawmatches the system’s velocity predicted by the Newton-Eulerequations of motion. The frictional force which causes this issolved for, and then applied for one simulation step. Only thecomponent of velocity parallel to the slope is considered. Todo this first and equation for the center of masses’ velocityafter an applied frictional force is predicted with Newton’sSecond Law using

vfin = vini +Ffmtot

Δt, (18)

where vini and vfin are the components of initial and finalvelocities of the system’s center of mass parallel to theslope, respectively; Ff is the frictional force; and Δt is theduration of one simulation time step. Next, in the Newton-Euler equations which leg’s θ, θ̇, and θ̈ define joints 2 and 4in Fig. 3 is switched, changing which leg is on the ground.Then an equation that finds vfin after a frictional forceFf has been applied to this state is generated. This is setequal to equation 18 to solve for Ff . Returning to the mainsimulation, the leg defining the grounded leg is switched,and the frictional force Ff is applied for one time step. Thesimulation then continues normally.

IV. SIMULATION AND DESIGN

The mechanical design of the Duck robot’s legs wasdone using the solid modeling program Solidworks. The finetuning of the leg’s dimensions and shape were done throughsimulation of the the equations of motion as presented inSec. III. Unlike many of the previous works in passivewalking, the design focused heavily on weight reduction,leaving only a few parameters open to tune the walkingbehavior. However, it was found that the simulated walkingcould be effectively tuned using only rL (lateral rollingradius) and rS (sagittal rolling radius). Tuning was onlyperformed for passive-dynamic walking, as it is harder toachieve than active walking. It was found that rS neededto be minimized, but not so much that the DUCK robotwould fall over while walking (θ2S − φ < vtoe). A largerS increases the step size, causing the robot to lurch forwardduring steps. This induces large variations in the leg’s sagittalfree swinging frequency and makes walking difficult to tune.Next, rL was used to tune the lateral rocking frequency tothe leg’s sagittal swinging frequency. The objective was to

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make the DUCK robot enter a stable limit cycle, such asthe one depicted in Fig. 5, from any position starting fromrest. A smaller rL was found to be preferred as it producesa more consistent lateral rocking frequency over a widerange of starting positions. The gap between the feet v inand the distance between the hip’s center of rotation and thequadcopter’s center of gravity ccg were also both minimized.A small vin minimizes the energy loss as described byEq. (17). A small ccg minimizes the effects the swingingquadcopter has on the walking, since the quadcopter’s largemass can swing at a different frequency than the legs.

It is important to note that in other passive-dynamic walkerdesigns the hip joint is kept at or above the sagittal centerof rotation, preventing the swinging foot from unexpectedlycolliding with the ground [16], [17]. Due to weight andgeometric constraints the DUCK robot does have this inits design. Thus, the simulation’s assumption that transitionsbetween legs only happen when θ2L changes sign is notalways true. However, a comparison between simulated andexperimental data suggests that this was not a major issue.

V. PROTOTYPE MANUFACTURING

A prototype DUCK robot was created based on themodeling and simulation results. The robot’s components,dimensions, and mass properties are shown in Fig. 6 and7. A commercially available Iris (3D Robotics) quadcopter

0 5 10 15 20−150

−100

−50

0

50

100

Left leg angle (deg.)

Left

leg

angu

lar s

peed

(deg

./s.)

Leg on ground

Leg freely swinging

Ground collision

Starting point

Fig. 5. Simulated phase portrait for the left leg, showing a stable limitcycle. The robot was started from rest standing straight up with θ2L (laterallean) = 2.3◦ and φ (slope angle) = -2.3◦. No thruster forces were used.

Fig. 6. Prototype of the DUCK robot. Feet are 3D printed ABS treatedwith an acetone vapor process for smoothness. Hip joints consists of a shaftand ball bearings. The flying platform is a commercially available Iris (3DRobotics) quadcopter.

was used as the flying platform. The passive-dynamic legsconsist of a custom-designed hip joint, lightweight aluminumlegs, and 3D printed ABS plastic feet. Each hip joint ismade from two cartridge bearings fitted inside an aluminumhousing. The dual bearing design increased weight, but wasnecessary for rigidity which prevented energy loss whilewalking. The hip and leg shaft were joined via a tolerancefit and clamp. The angle of the foot about the ẑ axis wasadjusted by rotating the shaft within the feet. This was foundto be important for adjusting against imbalances that wouldcause the DUCK robot to walk sideways, instead of straightdown the slope.

VI. PERFORMANCE EVALUATION

A. Passive-Dynamic Walking

Testing of the DUCK robot and validation of the simu-lation results were performed using the experimental setupshown in Fig. 8. In the experiment the DUCK robot walkeddown a sloped treadmill while having its position and poserecorded by a motion capture system (VICON). The VICONsystem tracks the infrared (IR) markers attached to the robotas shown in Fig. 8. It was found that the DUCK robot wasable to maintain a passive walk on slopes with angle between-0.6◦and -3.1◦. Steeper slopes caused the robot to fall over,and shallower slopes could not sustain a passive-dynamicwalk. The DUCK robot’s passive walking was limited byoscillations in the walking direction to the left and right

d = 4.17 cm.

bcg

= 13.21 cm.

vtoe

= 23 deg.

rS

= 24.6 cm.rS

= 24.6 cm.

rL

= 32.5 cm.rL

= 32.5 cm.vout

= 19 deg.vout

= 19 deg.

acg

= 43.2 cm.acg

= 43.2 cm.

Cartridge bearing

Retaining clip

Quadcopter mount

3D printed ABS feet

Bearing mount

Centers of rotation

Centers of gravityCenters of gravity

Aluminum tube

(a)

(b)

Fig. 7. Mechanical drawings of the passive-dynamic walking legs, where(a) shows the dimensions and (b) shows an exploded view of the mechanism.Parameters not shown in the lateral plane are: angle to inner foot vin(0.23◦), wingspan wL (49.9 cm), total mass mtot (1.89 kg), and totalrotational inertia ItotL (122 kgcm2). Parameters not shown in the sagittalplane are: distance from hip to quadcopter c.g. ccg (5.4 cm), wingspanwS (14.0 cm), mass of leg ml (0.194 kg), rotational inertia of leg IlS(14.0 kgcm2), mass of quadcopter and mount mq (1.48 kg), rotationalinertia of quadcopter and mount IqS (103 kgcm2), bearing coefficient offriction μb (0.0015), and bearing bore diameter db (0.95 cm).

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Fig. 8. Experimental setup for testing passive-dynamic walking. TheDUCK robot’s position is recorded by a Vicon motion capture system whileit walks down the sloped treadmill. Cameras are the Vicon MX F20 andMX T160 series running at 100 Hz, treadmill is a Cadence 70e series. Hereφ (slope angle) = -2.3◦.

which would increase in size until the robot walked off thetreadmill.

The simulation was validated using the motion capturedata from the experiments. A time was selected from theexperimental data when the DUCK robot’s walking hadreached steady state, and then this information was usedas the initial state in the simulation. The experimental andsimulated responses are compared in Fig. 9. The methodproposed for handling sagittal collisions as described inSection III-B appeared to yield good results, suggested bythe good agreement between the experimental and simulatedresponses. Specifically, the sharp changes in the left leg’sangular velocity between transitions measured experimen-tally matched the simulated behavior. The lateral rockingand sagittal swinging frequencies were predicted to within9% of the measured frequency. The final amplitudes ofthe lateral rocking and quadcopter’s sagittal swing werepredicted to within 12%, but the leg’s final sagittal swingingamplitude was predicted to be roughly 3.5 times largerthan the experimental value. Overall, the simulation yieldgood prediction of the response in terms of overall shapeof the angular motion and frequency, but the amplitude ofmotion was not in good agreement. A possible reason for theobserved inconsistencies in the data is that the hip joints hadmuch more friction than was anticipated. Another possiblereason is that the interactions between the lateral and sagittalmotions were not modeled.

B. Active Walking

One of the key contributions of this paper is the demon-stration of active walking. In active walking thrust forcesfrom the quadcopter enable the passive-dynamic legs to walkalong flat surfaces or up inclined surfaces. This mode oflocomotion is desirable as it requires less energy in manysituations. During tests active walking was achieved throughrocking the robot laterally while providing a forward force byrolling and pitching the quadcopter, respectively. As picturedin Fig. 10, under human control the DUCK robot was able

−5

0

5

θ 2L

(deg

.)

0 1 2 3 4 5 60

5

10

15

20

25

Time (s.)

Lef

t leg

ang

le (d

eg.)

0

5

θ 3S

(deg

.) Experimental dataSimulated data

Fig. 9. Comparison of experimental and simulated results of the robotpassively walking. The DUCK robot was allowed to reach a steady walkingstate on a treadmill as in Fig. 8 while having its position recorded by amotion capture system. An starting point was chosen from this data andused to define the initial state in the simulation. Sagittal quadcopter angleis θ3S , and lateral lean angle is θ2L. Here φ (slope angle) is -2.3◦.

to passively walk down a slope, actively walk up a slope,and then fly. Additionally, it was found that during passivewalking the quadcopter’s stabilization could be turned on,stopping the rocking motions and causing the DUCK robot tostop and wait on the slope. The passive-dynamic walk couldbe restarted by rolling the quadcopter, then turning off thethrusters. The DUCK robot could be steered left and rightby yawing the quadcopter. The video where the snapshotswere taken can be found on the DARC Lab website1.

C. Flying

The added weight and dynamics of the legs affected thequadcopter’s ability to fly. Though the quadcopter is still ableto fly, the legs would oscillate during flight, requiring addi-tional attitude control. Landing is also made more difficult asthe legs were less supportive than a fixed landing gear. It isnoted that the internal stabilization of the quadcopter was notaltered to account for the weight and dynamics of the legsfor all experiments. Future work will consider mechanismsand control algorithms to prevent or minimize swinging ofthe legs during take-off, landing, and flight.

VII. ENERGY ANALYSIS

An energy analysis is performed to compare the energyefficiency of flying to that of active walking. The energyefficiency of level flight is determined by finding the thrustforce (and implied power) needed to negate air friction andgravity. The energy efficiency of active walking is determinedby using the walking simulation described in Section III totest the effects of various thruster patterns while on a flatsurface (φ = 0).

To find the energy efficiency of flight the forces shown inFig. 11 need to be determined. The force of drag is estimated

1http://www.kam.k.leang.com/academics/robotics

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Fig. 10. Sequence shots of the DUCK robot flying, walking, and then flying again: (a) flying, (b) active (powered) walking on flat surface, (c) passivewalking with motors turned off down a -3.1◦ slope, (d) active walking up a 2.7◦ slope, and finally (e) take off and flight.

Fweight

Fweight

Fdrag

Fthrust

vflightvflight

µattackµattack

ventryventry

vexitvexit

vpropvprop

Fig. 11. Free body diagram of flying. Depicted are the forces and velocitiesused for calculating the energy efficiency of flying.

by assuming the robot is a 14-cm radius sphere, i.e.,

Fdrag = 0.5cdρπr2v2flight, (19)

Fweight = mg, (20)

where Fdrag is the drag force; cd is the coefficient of dragfor a sphere; r is the radius of the sphere; vflight is the flyingvelocity; Fweight is the force of weight; m is the mass of therobot; and g is the acceleration due to gravity. To maintaina given speed the total thrust of the four propellers needs tocounteract the drag and weight forces through the equation

Ftotal = 4Fthrust =√F 2weight + F

2drag, (21)

where Ftotal is the total thrust and Fthrust is the thrustprovided by one propeller. To estimate the required powerto produce a thrust force during flight, actuator disk theory(Momentum Theory) with ideal flow is used, which provides

10−2 10−1 100 101 102

500

1000

1500

2000

2500

Speed (m/s)

Theo

. max

trip

leng

th w

ith 3

0kJ (

m)

Active walkingFlying without weight of legsFlying with weight of legs

Fig. 12. Simulation result of estimated maximum distance traveled on 30 kJ(energy in a 2.5 Ah, 11.1 volt battery) v.s. speed for flying and walking.The DUCK robot cannot actively walk faster than 0.15 m/s without fallingover.

the equations

Fthrust =π

4D2ρvprop(vexit − ventry), (22)

Preq = Fthrustvprop, (23)

vprop = (ventry + vexit)/2, (24)

where D is the propeller diameter; ρ is the air density;ventry , vprop, and vexit are the velocity of the air entering,at, and exiting the propeller, respectively; and Preq is therequired power. The velocity of the air entering the propelleris assumed to be the component of the flight velocityperpendicular to the propeller, resulting in

ventry = vflight sin(θattack), (25)

θattack = tan−1(Fweight/Fdrag), (26)

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0 90 180 270 3600

0.2

0.4

0.6

0.8

θ2L (lateral lean angle) phase (deg.)

Thru

st fo

rce

(N)

Back left thruster (F

BL)

Back right thruster(FBR

)

Front left & right thruster(FFL

,FFR

)

Fig. 13. Example thruster pattern used for active walking in the simulation.The pattern follows the repetitive motion of θ2L. The rear thrusters are usedto provide forward thrust and alter the walking frequency.

where θattack is the angle of attack.The combination of Eqs. (19) through (26) predicts the

power needed for the DUCK robot to fly at a given velocity.Next, the energy efficiency of active walking is determinedthrough simulation. In the simulation a thruster pattern isapplied, the robot is given 8 seconds to reach steady state,and then its average velocity and power consumption is foundover an additional 12 seconds. Equations (22) through (25)are used to convert the thruster forces to power consumption.

The results from the energy analysis are shown in Fig. 12,which compares the maximum theoretical distance the robotcould travel on a single battery by walking or flying. It showsthat for our design, flying is the most energy efficient, butonly at relatively high speeds. At low speeds, flying’s energyefficiency approaches zero, and active walking becomes moreefficient. At best, flying is between 13-56 percent moreefficient than walking, depending on if the weight of thelegs is added. Note that real-world inefficiencies in activewalking such as uneven ground are not modeled. Future workwill investigate these effects in the modeling process andcompare to experimental results. Finally, the most efficientmethod found for active walking is to elongate and sustainthe lateral rocking motion (θ2L) using only the rear thrusters,pitching the quadcopter forward resulting in forward motion.An example of the thruster pattern used to produce this isshown in Fig. 13. Future work will validate this claim.

VIII. CONCLUSION

This paper described the development of a flying and walk-ing robot, called the dynamic underactuated flying-walking(DUCK) robot. The DUCK robot combines a high-mobilityquadcopter flying platform with passive-dynamic legs. Adetailed mathematical model was presented and the modelwas used to simulate the walking motion and help designthe prototype. Two modes of walking were experimentallydemonstrated: (1) passive walking down inclined surfacesand (2) active (powered) walking where thrust from thequadcopter’s rotors enables the robot to take steps and walkon flat surfaces or up inclined surfaces. An energy analysiswas performed which estimated that on flat ground therobot is more energy efficient at low speeds using activewalking, and more efficient at high speeds using flying.Both simulation and experimental results demonstrated thefeasibility of combining an aerial platform with passive-dynamic legs to create a robot that can fly and walk.

IX. ACKNOWLEDGMENTSThis material is based upon work supported, in part, by

the National Science Foundation, Partnership for InnovationProgram, Grant No. 1430328. Any opinions, findings, andconclusions or recommendations expressed in this materialare those of the author(s) and do not necessarily reflectthe views of the National Science Foundation. Authors alsothank Prof. Sanford Meek for providing the treadmill for theexperiments.

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