ATRIAS: Design and Validation of a Tether-free

3D-capable Spring-Mass Bipedal Robot

Christian Hubicki∗, Jesse Grimes∗, Mikhail Jones∗, Daniel Renjewski†,Alexander Sprowitz‡, Andy Abate∗ and Jonathan Hurst∗§

Abstract

ATRIAS is a human-scale 3D-capable bipedal robotdesigned to mechanically embody the spring-massmodel for dynamic walking and running. To helpbring the extensive work on this theoretical modelfurther into practice, we present the design and val-idation of a spring-mass robot which can operate inreal-world settings (i.e., off-tether and without pla-narizing restraints). We outline the mechanisms anddesign choices necessary to meet these specifications,particularly ATRIAS’ four-bar series-elastic leg de-sign. We experimentally demonstrate the followingrobot capabilities, which are characteristics of thetarget model. 1) We present the robot’s physical ca-pability for both grounded and aerial gaits, includingplanar walking and sustained hopping, while beingmore efficient than similarly gait-versatile bipeds. 2)The robot can be controlled by enforcing quantitiesderived from the simpler spring-mass model, suchas leg angles and leg forces. 3) ATRIAS replicatesthe center-of-mass dynamics of human hopping and(novelly) walking, a key spring-mass model feature.Lastly, we present dynamically stable stepping in 3Dwithout external support, demonstrating that thistheoretical model has practical potential for real-world locomotion.

1 Introduction

ATRIAS (Figure 1) is a human-scale 3D-capablebipedal robot designed to mechanically embody a

∗Dynamic Robotics Laboratory, School of Mechanical, In-dustrial and Manufacturing Engineering, Oregon State Uni-versity, USA.†Robotics and Embedded Systems Group, Technische Uni-

versitat Munchen, Germany‡Physical Intelligence Department, Max Planck Institute

for Intelligent Systems, Stuttgart, Germany§Corresponding author. Jonathan Hurst can be contacted

at jonathan.hurst@oregonstate.edu.

Figure 1: ATRIAS (Assume The Robot Is ASphere) is a highly dynamic bipedal robot whosepassive dynamics are designed to approximate anagile and efficient reduced-order math model: thespring-mass model. Standing 171 cm (5’7”) talland 62 kg (137 lb) with batteries, the human-scalebiped is designed to both walk and run, and has theactuation, on-board power, real-time control, andwireless communication necessary to trek untetheredthrough real environments. At present, there arethree ATRIAS bipeds in circulation functioning ascommon research platforms.

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far simpler set of dynamics, the spring-mass model.The bipedal spring-mass model is a unifying mathmodel that reproduces dynamics for walking andrunning as seen in animal gaits. We aim to phys-ically reproduce the dynamics of bipedal spring-mass locomotion, specifically characterized by 1)running/hopping with “single-humped” ground re-action forces, and novelly 2) walking with “double-humped” ground-reaction forces (Geyer et al. 2006)(Figure 2). Our approach is to design the passivedynamics of ATRIAS (Assume The Robot Is ASphere) such that the machine’s dynamics naturallyrespond as if it were an ideal spring mass model,which we abbreviate a “spring-mass robot”. Bytargeting spring-mass dynamics by design, ATRIASdemonstrates that this locomotion theory can berendered on a practical machine which is 3D-capable(i.e. it requires no power tether or planarizing re-straint). We present the design of ATRIAS and ex-perimentally demonstrate the advantages of this de-sign approach, which we categorize: 1) engineering,2) control synthesis, 3) and biomechanical relevance.

This spring-mass approach has the potential toameliorate some of the long-standing engineeringproblems in bipedal robotics: particularly, thatrobots are either too energy consuming or overlyspecialized to a particular gait. We posit thatrobots are often held back by inherent dynamicallimits which are imposed by their mechanical de-sign. While humanoid robots have long been amongthe most motion-versatile bipeds, with robots likeASIMO able to walk, run, and hop (Hirose andOgawa 2007), their design approach does have draw-backs. Many humanoid robots have stiff connec-tions at every joint between their motor rotors, geartrains, and the ground. This means that all posi-tive and negative work in each gait cycle must flowthrough actuators and transmissions, creating ineffi-ciencies. In contrast, passive dynamic walkers withvery limited actuation are paragons of energy econ-omy (Collins et al. 2005), but are too restricted toparticular walking speeds, gaits, and motions to beversatile in application. No controller, no matterhow clever, can achieve ambitious agility and econ-omy goals if the hardware is too encumbering. Assuch, ATRIAS aims to find an effective blend of pas-sive dynamics and controlled actuation to achievelocomotion that is both efficient and versatile.

Bipedal robots are also challenging from a con-trol synthesis standpoint, in part due to their in-

Figure 2: Spring-mass walking and running, as de-scribed in (Geyer et al. 2006), which unifies the dy-namics of animal walking and running with a singlemath model. These are the characteristic dynamicsof spring-mass gaits, as defined by ground reactionforces: a “double-humped” profile for walking and a“single-humped” profile for running and hopping.

herent nonlinearities, hybrid dynamics, numerousdegrees of freedom, and occasional underactuation.This approach of building ATRIAS to embody anideal model provides a reduced-order structure forexploring the control of the complex full-order ma-chine. Further, the spring-mass model has beenwidely studied in the dynamics and control litera-ture. The results include self-stabilizing techniques(Seyfarth and Geyer 2002), step-planning methods(Piovan and Byl 2013), and dynamic gait speedchanges (Ernst et al. 2012). As such, ATRIAS hasthe potential to be controlled by implementing thesame policies, or structurally similar policies (Reza-zadeh et al. 2015; Martin et al. 2015), as those de-veloped in the literature.

This design approach also offers an opportunityfor achieving locomotion with greater biomechani-cal relevance. The spring-mass model was first in-troduced in the context of legged locomotion as abiomechanical descriptor, particularly of the center-of-mass running and hopping dynamics in humans(Blickhan 1989). In the decades since, this spring-legconcept has become cemented in the biomechanicscommunity to the point that it is common practiceto report “leg stiffness” as a standard quantity inanimal locomotion dynamics (Lee et al. 2014). No-tably, the spring-mass model has been suggested asa means of unifying running dynamics with walking

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(Geyer et al. 2006). We present the ATRIAS robotas the first machine to demonstrate and report thesebiological walking dynamics.

In overview, we present this case in the followingstages. For background, Section 2 catalogs the suc-cesses and limitations of bipedal robots and spring-legged robot designs to date, and describes thespring-mass model (ATRIAS’ target math model).Section 3 outlines the design features which werekey to approximating the spring-mass model, theirdesign impact, and our solutions to constructing a3D-capable spring-mass biped. Section 4 reports anumber of hardware experiments which validate thatATRIAS functions as a spring-mass robot. Thesepassive-dynamic experiments also demonstrate thatdropping, pushing, or throwing ATRIAS yielded 3-5passive hops or 18 passive walking steps before halt-ing1.

We further demonstrate the capacity of ATRIAS(Figure 3e), and its monopod predecessor (Figure3d), to execute a variety of dynamic maneuverswhen actuation is unleashed: hopping on one leg(a precursor to running), and reflexively recoveringfrom a hop into an unexpected 16.5-cm-deep gravelpit. Particularly, we report walking with human-likecenter-of-mass dynamics (as measured by ground-reaction forces), which we believe is a novel observa-tion of a walking machine. We measure the economyof our tested walking gait, reporting an electricalcost of transport of 1.13, less than a third the en-ergy cost estimated for some humanoid robots (e.g.ASIMO) (Collins et al. 2005). We also demonstrateits mechanical capability of practical 3D locomotionby showing ATRIAS march in place without pla-nar restraints. Section 5 surveys future work withATRIAS, particularly in terms of design, control,and biological investigations. In summary, Section6, we take inventory of ATRIAS’ capabilities andassess spring-mass robot design as an approach toenabling agile legged robots that are ready for thereal world.

2 Background

The concept of legged locomotion is inspired bynature, but engineered systems, especially bipedalrobots, often do not go beyond the stage of mor-phological biomimetics. Bipedal robots, for exam-

1Tested by either tightly constraining or entirely eliminat-ing positive actuator work.

ple, are frequently built with joints, actuation, andlink lengths approximating human morphology, butinfrequently approximate biological dynamics. Thechallenges of keeping balance, managing the dy-namic interaction with the environment and adapt-ing trajectories according to terrain changes are ad-dressed in a number of ways.

In varying combinations, existing bipedal robotshave exhibited versatile behaviors, been off-tethercapable, tackled the challenge of human-scale im-plementation, and even approximated dynamicalmodel-based behavior like spring-mass running.With ATRIAS, we draw from this array of ideas toachieve spring-mass walking and hopping on a 3D-capable machine that can operate off tether. We out-line some key approaches to designing bipedal ma-chines, the robots that emerged as a consequence,and argue why spring-mass model dynamics are aworthwhile design target for achieving a variety ofefficient bipedal gaits.

2.1 Fully-Actuated Humanoids

Fully articulated humanoid robots have been themost practical and publicly visible representativesof bipedal locomotion. Notable examples such asHonda’s ASIMO (Hirose and Ogawa 2007), AIST’sHRP series (Kaneko et al. 2011), KAIST’s HUBO(Park et al. 2007) are electromechanically driven,fully actuated machines capable of versatile, au-tonomous motion carrying their energy source.These high-DOF robots address the challenge ofbipedal balance by careful regulation of their zero-moment point (ZMP) (Vukobratovic and Borovac2004). The ZMP framework has been a hallmarkof humanoid control for decades thanks to its math-ematical tractability.

However, one consequence of ensuring controlla-bility of the ZMP is that it calls for actuators andstiff mechanical connections at every joint. In suchgear-driven robot designs without mechanical com-pliance (as is typically the case with humanoids),there are significant rotor inertias that inherentlylimit the joint accelerations. We argue this preventsthese humanoids from exhibiting the bouncy, highlydynamic locomotion mastered by animals. Bipedswith high leg stiffness also cannot absorb a signifi-cant amount of the energy from impacts and mustmitigate damage by slowing down the impact ve-locity (limiting potential dynamic gaits). With fullactuation, every joint motion must be accompanied

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(a) (b) (c) (d) (e)

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Figure 3: The design evolution of ATRIAS from simple model to single-legged prototype to full biped.(a) spring-mass model for legged locomotion (b) ATRIAS model with naming convention of the four-barlinkage members (c) first-generation prototype monopod (Grimes et al. 2014) (d) final prototype monopod(used for experiments in sections 4.2.4 and 4.3.3) (e) ATRIAS, a bipedal instantiation of the spring-massmodel, complete with lateral actuation and inertial measurement for fully 3D locomotion.

by the controlled acceleration of its associated mo-tor, which includes the significant inertia of its rotorand transmission. These factors, among others, arepotential contributors to the large power draws inmany humanoids. So while quadruped robots haverecently made significant economy gains (Seok et al.2013; Sprowitz et al. 2013), versatile bipeds, such ashumanoids, still have energy transport costs an orderof magnitude greater than their human counterparts(Collins et al. 2005).

2.2 Passive-Dynamic Robots

Another class of bipedal robots locomotes with onlylittle or no actuation utilizing the passive dynamicsof the mechanical system. While exhibiting very effi-cient locomotion, their action is limited to few gaitsand very specific environmental conditions. Thisclass comprises the passive dynamic walkers (Collinset al. 2005) and their motorized offspring, the de-sign of which was driven by the inverted pendulummodel for walking (McGeer 1990). Delft’s robotsFlame and TUlip are the largest scale implemen-tations of this approach, standing 1.2 m tall andweighing 15 kg, both were able to walk at 0.45 m/s(Hobbelen et al. 2008). The most energy economi-cal example is the Cornell Ranger, which bears thepseudo-biped inverted-pendulum configuration com-mon among passive dynamic walkers, but is suffi-ciently actuated and economical that it has walkedover 64 km (40 miles) on a single battery charge

(Bhounsule et al. 2012).

2.3 Compliant Actuated Robots

Few robots incorporated compliance to enhance dy-namic capabilities alongside energy efficiency. Thesebipeds show more or less versatile behavior and alarge range of gaits2. Among them are the Raiberthoppers (Raibert 1986), the ARL monopod (Ahmadiand Buehler 1999), the CMU bowleg (Zeglin 1999),the Spring Turkey and Spring Flamingo (Pratt etal. 2001), the tendon-like hopping mechanism ofKenKen (Hyon and Mita 2002), and the Jena Fox(Renjewski 2012). Other monopedal testbed hop-pers have been constructed for the purpose of test-ing model-based control techniques (Andrews et al.2011; Byl et al. 2012). Planar bipeds have been builtwith distal prismatic springs for the purpose of im-pact reduction (Hyon and Emura 2005). Further,MABEL (Grizzle et al. 2009), a robotic precursor ofATRIAS, has been able to both walk and run whileattached to a boom (Sreenath et al. 2013).

ATRIAS differs from these previous compliantmachines in two key ways: 1) ATRIAS’ design hasadditional features that particularly suit it for repli-cating the dynamics of the spring-mass model 2) andis fully capable of 3D locomotion (enables practi-cal locomotion). 1) Most of these prior examples

2A notable multi-bipedal example is the hexapod, RHex(Altendorfer et al. 2001), which crawls about on six springylegs.

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have mechanical compliance in one degree of free-dom per leg (the ARL monopod is a notable excep-tion). This lack of compliance hinders spring-massmodel behaviors, which assume no rigid degrees offreedom (leg rotation is zero-torque and leg exten-sion is compliant). This rigid degree of freedomalso impedes the bandwidth necessary for creatingthese forces virtually via stiffness or impedance con-trol (Kemper et al. 2010). ATRIAS is built withseries compliance in both planar actuators for eachleg, which allows for more effective force control torender the spring-mass model’s zero-torque hip joint(i.e. a pin joint). We demonstrate, with data andmodel comparisons, that ATRIAS’ design, with sim-ple control, replicates spring-mass dynamics in aerialgaits, and more notably from a scientific locomotionstandpoint, grounded gaits. 2) Each of these com-pliant prototypes have showcased their agility whileleashed via power tethers, motivating us to empowerATRIAS with the ability to cut the cord.

2.4 Spring-Mass Model: Dynamics andControl

The spring-mass model was originally proposed as asimplified model for running and hopping (Blickhan1989). It has been used to model and explain animallocomotion by reducing the system complexity whilepreserving the general dynamics (Farley et al. 1993;Blickhan and Full 1993; Dalleau et al. 1998). Theuse of highly reduced-order models to describe dy-namical systems for legged locomotion on land andto investigate control strategies was formalized by(Full and Koditschek 1999), termed “templates andanchors.”

One of the advantageous features of the spring-mass-models is its passive-dynamic stability (Sey-farth and Geyer 2002). Stability analysis has beencombined with spring-mass model dynamics to testhypotheses of neuromechanical functions (Full et al.2002). The use of stability as an objective functionfor control gave rise to a number of control strategiesto respond to various perturbations during locomo-tion (Seyfarth et al. 2003; Geyer et al. 2005; Seipeland Holmes 2005; Ghigliazza et al. 2003; Daley etal. 2006; Blum et al. 2010; Peuker et al. 2012). Thesame model has been able to produce bipedal walk-ing gaits (Geyer et al. 2006), reproducing a numberof features of animal walking (Lipfert et al. 2012),and could be extended toward control strategies fortrunk stabilization (Maus et al. 2010).

Spring-mass locomotion dynamics bear other at-tractive features, appealing to varied scientific audi-ences. To roboticists, spring-mass locomotion has atheoretical energy cost of zero, teasing a solution tothe cost-of-transport problem in legged robotics. Itis also a low-order dynamical system, typically re-quiring only four state variables, rendering locomo-tion planning a more computationally-feasible prob-lem. Such advantages have spurred controls re-searchers to develop an abundance of specializedspring-mass control techniques which could be plau-sibly employed on a spring-mass robot (Ernst et al.2012; Andrada et al. 2012; Piovan and Byl 2013).For biologists, spring-mass dynamics serve as dy-namical predictors of animal locomotion, model-ing organisms from cockroaches (Blickhan and Full1993), to lizards (Full and Koditschek 1999), to hu-mans (Blickhan 1989), to quail (Ferris et al. 1998).This general scientific interest in spring-mass loco-motion suggests that a spring-mass robot could be ablank canvas for implementing long-studied controlsconcepts and testing biological hypotheses alike.

3 System Design

Rendering ATRIAS into both a dynamically spring-mass and practical robot requires meeting two setsof specifications: 1) approximating key model fea-tures and 2) reconciling practical robot design reali-ties. Approximating ideal model features facilitatesthe dynamic behavior we require, while “design real-ities” ensure that the robot is sufficiently actuated,states are measured, components are protected andhoused accessibly, and the robot can maneuver out-side the sagittal plane (the typical domain of thespring-mass model). We then discuss mechanicalphenomena present in the design that notably im-pact model adherence or overall robot control: geo-metric power, drive-train efficiency, reflected inertia,and point feet.

3.1 Model-Feature Implementation

Equipping ATRIAS with the dynamical advantagesof the spring-mass model entails approximating fourof its key mechanical features: 1) a massless leg,2) compliance between the ground contact and hipjoint, 3) restricting leg forces to the virtual leg axis(i.e. zero hip torque), and 4) positioning the robotmass center near the hip joint. These model features

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Lateral actuator motors

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Figure 4: Range of motion and “keep-out” volume of the leg in the (a) sagittal plane and (b) frontal plane.(c) The range of motion for leg length and (d) the maximum kinematic spring deflection until the hardstops are reached.

are primarily achieved as a consequence of ATRIAS’leg mechanism, a two-degree-of-freedom compliantleg (Figure 3b).

3.1.1 Massless Leg

To approximate the massless leg of the spring-massmodel, we designed ATRIAS’ leg mechanism to be alightweight four-bar linkage (schematic illustrated inFigure 3b). The linkage has two degrees of freedom,allowing for a large range of motion for the toe in thesagittal plane, as drawn in Figure 4a. The four-barlinkage is constructed of lightweight and stiff carbonfiber with aluminum joints, totaling less than 4% ofthe robot mass that inelastically impacts the groundwith every step3.

3.1.2 Compliance

Compliance on ATRIAS is achieved mechanically byinstalling fiberglass plate springs (often called leafsprings), akin to those used in archery bows. By em-ploying mechanical compliance, ATRIAS enjoys thehigh restitution rates of mechanical springs. In con-trast to using virtual compliance approaches (Hogan1985; Boaventura et al. 2012), mechanical springs al-leviate concerns of actuator bandwidth by creating

3While a full impact analysis would be intricately state-and posture-dependent, the mass of the components distal tothe series plate springs offers a simple estimation, which weigh2.4 kg, 3.9% of the 62 kg robot mass. This assumes that formost gaits, only one leg impacts per step.

passive dynamics similar to our target dynamics, andspare the inefficiencies of electrical regeneration thatwould accompany each cycle.

ATRIAS’ plate springs bridge the connection be-tween the four-bar leg and the rest of the robot,as visualized in Figures 3b, and 5b. These springswere constructed from fiberglass bar stock becauseof its high energy density, high coefficient of resti-tution and stable spring stiffness with temperature,aging, and humidity changes. The springs them-selves exhibit an approximately linear spring behav-ior4. When loaded via the four-bar leg, the effectiveleg spring exhibits a “softening” behavior with re-spect to the virtual leg length (the mechanical ad-vantage of the linkage wanes as the leg crouches, asillustrated in Figure 4d and later measured in section4.2.1 and plotted in Figure 9a). This effective non-linearity in compliance is not intentional, but alsono obstruction to ATRIAS’ dynamical goals. Thesprings are mounted such that they can be quicklyreplaced as needed, or swapped for springs with dif-ferent stiffness profiles.

3.1.3 Axial Leg Forces

All leg forces in the spring-mass model act exclu-sively in the axial direction. This feature is a con-tributing factor for the model’s zero-cost gaits and

4This roughly linear force-length relationship was mea-sured experimentally using a custom-built load-deflectionmeasuring device.

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its characteristic underactuated agility. ATRIASmust be able to swing its legs and thus occasion-ally exert non-axial forces on the leg, but while onthe ground (or “during stance” as it is sometimesreferred), it must also have the capability to restrictits leg forces to the leg axis. The legs’ series-elasticparallelogram mechanism configuration provides ameans for ensuring only axial forces that is ratherintuitive. If the net torques on each of the prox-imal leg links are equivalent, there is no tangen-tial force on the leg. So, via a simple controllerwhich regulates the two spring torques into equiv-alence during stance, ATRIAS’ forces are renderedeffectively axial. The two degrees of elastic actu-ation make the linkage better suited for minimizetangential forces via control, because small errors inmotor position result in lower forces errors when thesystem is compliant (Kemper et al. 2010). This man-ner of restricting forces is implemented in two latter-described hardware tests: passive walking (Section4.2.3) and the passive throw (Section 4.2.4)

3.1.4 Hip-Centered Body Mass

In the spring-mass model, all mass is lumped into ahip-centered body. As the center-of-mass drifts awayfrom the position of the hip joint, axial leg forces willinduce increasing net moments on the torso, causingthe body to pitch. For ATRIAS, we required thecenter of mass to be near the hip point to mitigatetorso pitching.

Two design choices most affect this mass location:the proximal location of the leg drive assembly andthe arrangement of torso components. First, the legdrive assemblies for each leg weigh 14 kg (28 kg intotal), significantly more than weight of the entiretorso (22 kg), and the drives’ mass centers are just2.9 cm above the hip point. This distribution makesthe mass-center difficult to offset significantly. Sec-ond, as shown in Figure 6b, the torso is shaped suchthat several components can be mounted at or belowthe hip point, partly balancing the necessarily highplacement of the heavy lateral actuators. Conse-quently, the center of mass of ATRIAS is only 17 cmabove the hip even with the torso included, whichhas proven to be a manageable location in otherwalking experiments with ATRIAS (Ramezani et al.2013).

3.2 Robot Realities

While essential math model properties are approxi-mated with the above features, there is a necessaryset of requirements for the robot to function, unteth-ered and unsupported, in the real world. These in-clude sensing and measuring forces, the software andelectrical system, actuation, and component housingand protection. Further, we describe compromiseswith the spring-mass model dynamics that emergefrom practical robot design, particularly the pres-ence of large rotor inertias.

3.2.1 Actuation

ATRIAS is driven by six actuators in total, i.e., threeactuators dedicated to each of its legs. Within theseactuator trios, two “leg motors” drive the sagittalmotion of the leg through an elastic connection tothe leg linkages, while the third “hip motor” directlycontrols the leg’s abduction and adduction (Figure5a).

The paired leg motors each drive one of the an-terior and posterior thigh members through an in-series plate spring. Movement can be coordinatedto achieve changes in leg length or leg angle, Figure4 (a) and (c). This configuration combines torquessynergistically between the motors when extend-ing/retracting the leg during high-axial-force maneu-vers, such as jumping.

The leg motor pairs are rendered series-elastic byplate-spring connections to their respective linkages(Figure 5b). ATRIAS’ series-elastic actuator designdeviates from preceding examples (Robinson et al.1999) in that ATRIAS’ springs are designed to storea significant amount of energy relative to the totalmechanical energy in the gait. For perspective, dur-ing a single-legged hopping test (Section 4.3.1), theplate springs have stored as much as 90 J in one leg,the equivalent mechanical energy of the robot hop-ping with 15 cm of total vertical travel.

The leg motors were selected specifically for theirhigh torque density (Emoteq Corp.). High motortorque was important in order to limit the size of thenecessary gear reduction. The higher the gear reduc-tion, the higher the effective rotor inertia (Section3.3.3) as well as the potential footprint of the trans-mission. The leg motors were paired with 50:1 com-pact harmonic drive transmissions (Harmonic Drive,LLC), further amplifying torque (assembly depictedin Figure 5c). This transmission has minimal back-

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Wave Generator

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Figure 5: (a) The assembled ATRIAS leg mecha-nism, actuated in-plane by a two-motor stack assem-bly and rotated in the frontal plane through a pul-ley connection to a third motor. (b) The explodedleg assembly, diagramming the kinematic connec-tions from the central motor stack, through the se-ries springs, to the four-bar leg mechanism. (c) Across-sectional view of one of the paired leg motors,including the transmission via harmonic drive.

lash and also makes the actuator assembly very com-pact, allowing more room around the hip joint tomount leg assemblies to the torso.

For lateral motion, the “hip motors” extend theplanar leg into a 3-dimensional workspace, as illus-trated in Figure 4b. Each hip motor is placed highup in the torso and drives a pulley segment on a largelever arm, creating a 56:1 gear ratio. A timing beltis used between the pulleys for the benefits of highforce capacity, ease of assembly, and zero backlash.

3.2.2 State Measurement

Control of ATRIAS is most feasible when all statesare measured, both actuated and unactuated. Inaddition to the standard practice of mounting en-coders to each motor rotor, ATRIAS employs highresolution sensors to measure the spring deflections

and the state of the robot in the world frame, all de-grees of freedom which on ATRIAS are underactu-ated (Spong 1998). For a comprehensive accountingof ATRIAS’ sensing components, see Table A.1.

Due to ATRIAS’ series elasticity, the force gener-ated by spring deflections is paramount to the gaitdynamics, effectively dictating the robot’s trajec-tory. This makes spring deflections high-value stateinformation to capture. Deflections of each fiber-glass plate spring are measured by a set of two high-resolution absolute encoders (32-bit), which conse-quently enables accurate force measurements.

Position and velocity in the world frame are domi-nant in defining gaits, but for ATRIAS, these degreesof freedom are non-trivial to measure. Many hu-manoid robots move using flatly planted feet, whichgives them an absolute point of reference to measuretheir velocities in world coordinates. In contrast,ATRIAS has point feet which can be at a range of an-gles during stance, and thus foot orientation cannotbe used to easily infer the robot’s state in the worldframe (with only proprioceptive measurements). Weinclude a high-precision inertial measurement unit toestimate the robot pitch, angular velocities, and in-tegrate the cumulative Cartesian motion during un-restrained locomotion.

3.2.3 Software and Electronics Specifica-tions

The design of the integrated electronics and soft-ware subsystem was driven by a design decision touse available commercial hardware and open-sourcesoftware. Sensor processing and motor control isfacilitated by a number of microcontroller-enabled(ATmega128, Atmel, San Jose, CA, USA) electronicstacks that are connected via Ethercat-bus to acommercial computer (Mini ITX, i1000A, OEM Pro-duction, San Francisco, CA, USA). The control sys-tem runs at 1000 Hz on a real-time linux kernel5

and was developed using Robot Operating System(ROS) and the Orocos framework. A detailed de-scription of the electronics/software system design isavailable in (Peekema et al. 2013). Each motor andthe associated sensors, among them 32-bit linear en-coders (RL32BAT, Renishaw, Wotton-under-Edge,UK), limit switches and thermal sensors (See TableA.1), is controlled by one electronic stack, placed in

599.9% of control cycles are clocked within 41 µs of themean 1 ms sampling period, demonstrating low jitter.

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the body (see Sec. 3.2.4).

3.2.4 Component Housing and Protecting

In the case of a potentially damaging fall, internalcomponents are protected by a shell-like body struc-ture. This body is constructed using composites andcomposed of two halves. To bear the impact, a stiffand strong structure is made from carbon fiber withbalsa core and fiberglass inserts for mounting points,Figure 6a. The other half of the body shell is a non-structural and lightweight cover made from carbonfiber with no core. The resulting shell is designed tobe strong enough to shield internal components fromincidental damage and prevent wires from snaggingwhen disturbed by kicks, pushes, etc.

The components are bolted to the structural halfof the body and are accessible when the cover half isremoved. Stiffness of this body structure was impor-tant to ensure any body-mounted inertial measure-ment unit would suffer minimal kinematic flexibility,which would add noise to sensitive measurements.The top of the body features a lifting eye for usewith the robot support boom (described in Section4.1) or an overhead gantry system for catching therobot in the case of falls during troubleshooting andexperimentation.

In the event of large lateral forces on the leg be-yond the design limit, the knee allows for a con-trolled break of the leg by way of a mechanical fuse,as shown in Figure 7. The four-bar leg is inher-ently strong in plane and more flexible when loadedout-of-plane. Use of nylon pins and screws allowthe lower limbs (the anterior and posterior shins) tobreak away cleanly to the left or right. These pinsand screws are easily and quickly replaced. Straingauges are installed at the knee which measure theselateral forces on the leg. By measuring this lateralforce, the lateral actuator could be used to activelyreduce this force, keeping the toe forces in plane withthe leg where it is strong.

3.3 Design Discussion

3.3.1 Geometric Power

A fundamental consequence of ATRIAS’ leg config-uration is a wasteful expense of internal power, so-called geometric power (Waldron and Kinzel 1981).With two motors driving the four-bar linkage leg,swinging the leg while loaded requires each motor

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Shear pins intact

Nylon bolt

(break-away)

Strain gauge Str

ain

gau

ge

(a)

Lateral force

applied at toe

Figure 7: When a lateral force, F, is applied at thetoe that exceeds the designed failure point a set ofplastic shear pins break allowing the leg fall awayand prevent higher loads from damaging other, moreexpensive components. Strain gauges are included tomeasure and control the magnitude of these lateralforces.

to perform nontrivial positive and negative work re-spectively, even if no net work is performed on therobot’s mass center. In essence, geometric poweris purely overhead, i.e. does not contribute to themechanical energy of the gait. This phenomenon,plotted in Figure 8, results from the leg-load sharingof the two motors and is inherent to the ATRIASconfiguration. Electromechanical regeneration miti-gates this overhead to a degree, as ATRIAS’ regen-eration efficiency is calculated to be between 30 and40%. For measurement methods and more in-depthpower analysis, see Appendix A.

3.3.2 Drive Train Efficiency

The choice of a harmonic drive transmission alsocomes with a mechanical efficiency cost. A prioriteration of the ATRIAS leg included an epicycliccable drive (Grimes et al. 2014), but was removeddue to its large keep-out volume (total volume sweptby the mechanism, corresponding to all points inthe end-effector workspace) which overly limited thespace available for torso volume. Torso volume wasparticularly important for housing all the compo-nents necessary to make ATRIAS 3D-capable. Whileharmonic drives are sufficiently compact, they alsohave lower mechanical efficiencies, typically around50-80% (compared to the cable drives of predeces-

9

0.5 in (12.7 mm)

balsa wood core

3 ply plain weave

carbon fiber

inside and outside

Lifting eye

Fiberglass insert

connection to boom

4x Fiberglass insert

electronics mounts

4x fiberglass insert

lateral actuator motor

mounts

Aluminum insert

hip joint connection

Flange for connection

to modular tether-free

fall protection system

(all around)

Wire conduit

(to each leg)

Lateral actuator

motor (2x)

Inertial

measurement

unit (IMU)

Nettop computer

3x liPo

batteriesPower amplifier

+ distributed

microcontroller

stack (6x)

Lifting eye

Off-board power

access (from Boom)

Hip joint axis

Cooling fans (4x)

17 cm

Hip axis

Robot mass center

(a) (b)

Figure 6: (a) Construction details of the body to make it a strong protective shell to house the variouscomponents of the robot (cover half not shown for clarity). (b) Layout of components secured to thestructure half of the protective body shell.

sors MABEL and ATRIAS 1.0 of 94%). Theselosses likely exacerbate ATRIAS’ mechanical trans-port costs.

3.3.3 Reflected Inertia & Torso Eccentricity

When regulating its angular orientation, ATRIASmust contend with both the lever arm of its torsomass and the inertia of its spinning rotors. Ac-celeration of the sagittal motors exerts significantreaction torques on the torso as a consequence ofso-called reflected inertia. While the rotor inertiafor each of the sagittal rotor assemblies is rathersmall, 0.0019 kgm2, when transmitted through a50:1 harmonic drive, the reflected inertia scales withthe square of the gear ratio, i.e. a 2,500-fold in-crease. Consequently, when the two leg motors accel-erate in concert (such as in swinging motions), theircombined reflected rotor inertia sums to 9.50 kgm2.This effective inertia is massive, totaling twice thehip-centered inertia of all non-rotor components(4.65 kgm2). Such inertial effects are a considerabledrawback of large gear reductions in general, andharmonic drives is particular (which are commonlymanufactured with gear ratios of 150:1 or higher).

However, effects of rotational inertias on torsopitching are mitigated by leg recirculation. It is gen-erally uncommon for a gait to require both legs toswing in the same direction, thereby canceling at

least a portion of the inertial effects. Further, theresulting reaction torques have not prevented suc-cessful walking in practice, as multiple controllershave regulated the torso position of ATRIAS whilewalking (e.g., both with torso angle controlled steady(Ramezani et al. 2013) and with the torso pitchingsignificantly (Hereid et al. 2014)).

Reflected inertia also brings deleterious effects onactuator performance for certain tasks, such as forcecontrol. There are occasions in which it is likely use-ful for a robot to regulate its leg forces (as demon-strated by “hopping in gravel” in Section 4.3.3),which high rotor inertia is known to impede. Specif-ically, this inertia enforces an effective motor accel-eration limit, which curtails the actuator bandwidthfor accurately rendering desired forces (Kemper etal. 2010). While the ATRIAS leg is still capable ofsome demonstrative and useful feats of force control(Figure 16), such large inertia is likely the main lim-iting factor of ATRIAS’ actuation scheme.

3.3.4 Feet, in absentia

Unlike many of its robotic counterparts, ATRIAShas no actuated feet, nor feet of any meaningful sort.Its legs terminate with simple rubber nubs, approx-imating point feet. This, of course, is by design asthe spring-mass model is point-footed. However, thismakes standing still non-trivial as point feet have no

10

-0.5

0.0

0.5

-1.0

Pow

er (

kW

)

Time (% of gait cycle)50 60 70 80 90 10040

Sum

DifferenceGeometric Power

Pow

er (

kW

)

-0.5

0.0

0.5

-1.0

50 60 70 80 90 10040

Motor A

Motor B

(a)

(b)

Figure 8: (a) Instantaneous power generated by mo-tors A and B during stance phase of a walking gait,as described in Section 4.3.2. Motor B is produc-ing negative power for the duration. (b) The sum(|A| + |B|) and difference (|A| − |B|) of the abso-lute value of mechanical power produced by motorA and B. The shaded region represents geometricpower, a byproduct of the four-bar leg and motorconfiguration. When swinging the leg while axiallyloaded (i.e., during stance phase), geometric poweracts as power overhead, exacerbating energy costs.These power estimates were calculated by measur-ing spring deflections and motor velocities, and aredescribed in further detail in Appendix A.

effective polygon of support. This means ATRIASmust adopt more dynamic approaches to stability,such as taking a stepping to recover from falls (Prattet al. 2012; Stephens and Atkeson 2010).

For traversing a fully three-dimensional environ-ment, some form of contact pad can be helpfulfor resisting yaw accelerations. This would pre-vent ATRIAS from spinning like a top and has beenshown to be usefully stabilizing in 3D simulations(Hamed and Grizzle 2014). As such, we prototypedlightweight passively-pivoting contact pads for usein our 3D-stepping experiment (Section 4.3.4). Thepad is 10-cm-long, 2.5-cm-wide fiberglass sheet withrubber nubs attached under each end. The pad

freely pivots in the sagittal plane to conform to theorientation of the ground upon touch down, and elas-tic bands act as a return spring to reset the pad angleupon lift-off. Each pad is sufficiently lightweight andloosely sprung that it can easily flicked with a fin-ger. As such, it has no apparent effect on sagittalplane spring-mass dynamics, but effectively curtailsyaw accelerations. In short, this addition in 3D ex-perimens (Section 4.3.4) does not compromise ourspring-mass design intent.

4 Experiments

Assessing the design of ATRIAS requires validationon two fronts: 1) the degree to which its passive dy-namics match the spring-mass model, and 2) its dy-namical capabilities, or agility, when actuated. Af-ter explaining our experimental setup in Section 4.1,Section 4.2 validates ATRIAS’ spring-mass behaviorwith a number of passive tests. In these passive tests,actuation is either thoughtfully limited or eliminatedentirely in order to isolate ATRIAS’ passive dynam-ics, and compare them to the spring-mass model.Section 4.3 demonstrates the capacity of ATRIASto perform a variety of sustained dynamic maneuverswhile retaining key features of its passive motions.For additional details about the controllers for theseexperiments, see Appendix C.

4.1 Experimental Setup

In pointedly testing ATRIAS’ dynamical features,experiments were conducted while constraining itsmotion to a roughly planar workspace. Such restric-tion facilitates a more controlled test of ATRIAS’ ac-tuation capacity and spring-mass properties (whichprimarily manifest in the sagittal locomotion plane).This approximately sagittal constraint was achievedby connecting the torso to a pivoting boom.

The boom offers a 2.0-meter turning radius andthree degrees of freedom: boom azimuth, boomaltitude, and robot pitch (each can be individu-ally locked or freed). This spherically constrainedmotion approximates sagittal-plane locomotion forlarge radii (Colett and Hurst 2011). Further, thesethree rotational axes are measured with encoders, al-lowing measurement of the world-frame for both ex-perimental analysis and feedback for real-time con-trol. All state information from the boom is collectedusing the same software/electronics infrastructure as

11

Figure 9: (a) ATRIAS’ force-length curve, measured via a system identification test, where the robot isbounced on one leg and forces are measured by a ground-mounted force plate. The experiment shows aclose match between the loading edge of axial leg force measurements (thin, solid line) and the theoreticalforce-length curve derived from leg kinematics (thick dashed line). The rest length of the leg is 0.9mduring the test. The figure also includes a labeled photograph of the experimental setup. (b) The decayingoscillation of the dropped robot and its fitted curve (computed using a second-order oscillation model)which show close agreement. Figures 9a and 9b present data from the same drop test.

the robot, recorded at the same 1 kHz control rate.

To minimize unintended interference in experi-ments, the boom is designed with minimal rota-tional inertia and rotational friction. The boom’stwo protruding rods, for constraint and catching re-spectively, are constructed with carbon fiber tubes,helping limit the boom’s rotational inertia to just4.3% of the boom-robot system. To limit friction,the rotational joint is a four-point contact ball bear-ing, with any routed cables connected through a low-friction slip ring6. The resulting drag force againstthe robot is estimated to be less than 1 N, only 0.2%of ATRIAS’ body weight.

Further, the boom is designed with other prac-tical functions for robot experiments. To raise therobot for drop tests, the boom provides a motor-driven hoisting line, allowing for both easy liftingand more precise test replication. To provide safeand clean drops, a quick-release mechanism connectsto the robot and is disconnected via a swiftly pulledpin. The boom also serves as a safety mechanism,

6While the presented experiments used an off-board powertether, the robot can function fully untethered, using on-boardlithium-polymer batteries and computing, with wireless com-munication to a user-interface computer.

catching the robot on an independent slack line ifthe torso dips too low.

When possible, ground reaction forces are mea-sured with a force plate that is mounted flush withthe laboratory floor. Typically used in biomechanicslabs for gait analysis, this force plate (OR6-7-4000,AMTI, Watertown, MA, USA) measures forces (x,y and z) and moments (Mx, My and Mz) at a sam-pling rate of 1 kHz. Force plate measurements aresynchronized to the robot’s data collection systemby means of an electronic triggering signal. Mea-suring spring deflections provides another mode forforce reporting, which is particularly apt for loggingmulti-step maneuvers for which the single force platespans an insufficient surface area. Each experimentindicates which manner of force measurement wasemployed.

Some experiments and demonstrations were per-formed using the monopod prototype of ATRIASshown in Figure 3d. The monopod leg design isnearly identical to the ATRIAS biped, differing pri-marily in torso configuration (and of course, includesonly one leg). Most relevantly, the monopod weighsonly 30 kg, just under half the biped’s 62 kg weight,better suiting throwing experiments (Section 4.2.4)

12

and testing the most dynamic maneuvers (Section4.3.3).

4.2 Model-matching Validation

4.2.1 Identifying Spring-Mass Behavior

One straightforward way to assess ATRIAS’ spring-mass dynamics is by identifying a force-length rela-tion and a decayed oscillation model for the robot.Using mechanical stops, we locked the ATRIAS’ mo-tors in place and disabled control, permitting a fullypassive robot test. We configured the support boomto restrict torso rotation and horizontal motion, al-lowing only for the vertical oscillation of a spring-mass system. Using the boom’s winching cable, weraised the robot slightly off the ground and droppedit.

In Figure 9a, we see the resulting force-lengthcurve, resembling a nonlinear spring with a softeningbehavior. This softening curve is an expected fea-ture of ATRIAS’ leg configuration, where the appar-ent falling-rate leg length stiffness results from thenonlinear geometric relations inherent to the fourbarlinkage. We can see the previously derived theoreti-cal force-length relation (Grimes et al. 2014) matchesvery closely to the loading side of the experimentalcurve. On the unloading side, we see hysteresis as-sociated with dissipative losses that must be offsetthrough actuation.

We can determine the magnitude of this dissipa-tion by examining the decay of the spring-mass oscil-lations. We fitted the experimental data to a lineardamped spring-mass model with one degree of free-dom. Figure 9b illustrates a comparison to the fittedmodel parameters7, m = 59.9 kg, k = 6.543 kN/m,c = 38.0 Ns/m, and ζ = 0.061. This damping ra-tio is relatively small, suggesting ATRIAS’ springsare reasonably efficient energy storage mechanisms.However, this analysis does not include the lossesfrom an impact.

4.2.2 Passive Drop

While the previous analysis estimates the dissipationin ATRIAS’ springs, a number of other phenomenain the passive system can contribute to energy lossesin legged robots (e.g. impacts, friction, backlash). In

7Reported values for robot mass that vary slightly acrossexperiments are due to the inclusion/exclusion of the onboardbattery pack.

Time (s)

Hip

Hei

ght

(m)

0.600.0 1.0 2.00.5 1.5 2.5

0.90

0.85

0.80

0.75

0.70

0.65

Robot Trajectory

Stance Phase

Touch-down Height0.95

1.00

Hip

Height

Passive Drop

Figure 10: A passive drop experiment, designed toelicit the robot’s passive spring-mass dynamics as ex-pected in hopping gaits. ATRIAS is dropped onto itsunpowered leg with its motors mechanically locked,isolating the passive plate springs as the single ofspring-mass dynamics. The center of mass position(solid line) exhibits a “bouncing” trajectory charac-teristic of a spring-mass system. Periods of groundcontact, i.e. the stance phase, (shaded region), andthe threshold height where ATRIAS makes groundcontact (dashed line) are also visualized. An illus-tration depicts the passive drop experimental setup,the support boom constrains the body rotation andall non-vertical motion, and ground-reaction forcesare measured by a force plate. Video shown in Ex-tension 1a.

this test scenario, we evaluate the magnitude of thesecombined losses from the passive system (includingspring dissipation) through passive bouncing. Weconstrained movement to the vertical direction anddropped ATRIAS from a fixed height (5 cm) with amechanically fixed leg length (84 cm), allowing thesprings to absorb the impact.

After the drop, ATRIAS bounced into an airbornestate three times before settling into a groundedstate, as shown in Figure 10. When dropped from5 cm, the robot returns to a height of 2.4 cm on thenext apex. By examining the potential energy lostbetween these subsequent apexes, we determine a

13

coefficient of restitution of 0.69. This suggests thatATRIAS’ leg mechanism has the capability to returnas much as 48% of the energy to a dynamic gait (suchas hopping) that would be otherwise lost in a plasticcollision.

4.2.3 Passive Walking

FZ

FX

Left Leg Right LegFZ

FX

PushRelease "Passive" Walk

17 steps (6.3 meters) in 8.1 secondsHalt

1.0 m/s

Right Leg FootholdLeft Leg Foothold

0.0

0.5

1.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Forc

e (b

ody w

eights

) Experiment

0.0

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1.0

0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4

Forc

e (b

ody w

eights

)

Time (seconds)

Simulation

Spring-MassModel

x

z

ATRIAS

x

z

Figure 11: A “passive” walking experiment, whereATRIAS is pushed, released, and walks until it stopsfrom energy dissipation (cartooned at top with mea-sured foothold locations). Ground-reaction forces forATRIAS (middle) are compared to a simulation ofthe spring-mass model (bottom), showing a numberof qualitative similarities, including the characteris-tic double-humped vertical force. Using a controllerdesigned to reciprocate its legs while avoiding per-forming work on the robot CoM, ATRIAS walked anotable distance of 6.3 meters over 17 steps beforestopping. Video shown in Extension 1b.

This experiment evaluates whether ATRIAS’ pas-sive dynamics exhibit spring-mass walking dynam-ics in low-energy gaits. Particularly, we are look-ing to see if ATRIAS will exhibit the characteris-

tic double-humped ground reaction force associatedwith spring-mass walking, and by extension, humanwalking (Geyer et al. 2006).

Unlike the passive drop test, in order to walk,ATRIAS’ motors must be unlocked and able toswing. However, in order to approximate a passiverobot test, we employed a controller that recipro-cates the legs while avoiding performing any instan-taneous net work on the robot. During a leg’s stancephase, the phase with the most potential for signif-icant energy exchange, the two leg motors regulatetwo key outputs to facilitate passive spring-mass dy-namics: the net leg torque (which is driven to zero)and the set point of the virtual spring (which main-tains a nominal position). The net leg torque is ze-roed by controlling the spring deflections to be equiv-alent, as suggested in Section 3.1.3. A second con-trollers simultaneously controls the position of thevirtual spring set point. This was accomplished inthe prior experiments (Sections 4.2.1 and 4.2.2) bymechanically locking the motors at a specified sep-aration distance, which we now also achieve usingposition control on the motor angles (See AppendixC for mathematical formulation of the controller).

During swing phase, the controller positionsthe leg to a touchdown angle which would yieldlimit-cycle walking for the spring-mass model.Such touchdown-angle-driven control has been usedto control spring-mass models (Ghigliazza et al.2003), incidentally demonstrating ATRIAS’ poten-tial amenability to spring-mass control techniques.At the beginning of leg swing, a position controllerdrives the swing-leg’s toe position along a spline,tracing a path between the current position and thedesired leg angle. Again, for this test, the robot wasmounted to the boom with the torso rotation lockedas this is a test of spring-mass dynamics, not torsoregulation or frontal stabilization.

After a human operator pushed ATRIAS to a1 m/s walking speed, ATRIAS was released and con-tinued to walk for 6.3 m over 17 steps before slowingto a stop as illustrated in Figure 11. Propulsion wasalso added by tracking errors in our zero-leg-torquecontrol (averaging approximately 31 N (7 lbf) of nethorizontal force at the toe).

The passive walk test produced the typical double-humped ground reaction forces of the spring-massmodel (Figure 11). More notably, the experimen-tal data fits well with a simulation of the spring-mass model, simulated using ATRIAS’ mass, spring-

14

61

Time (s)

En

erg

y (J

)

61.560.5600

10

20

30

40

50

Spring Energy Storage

Kinetic Energy

Gravitational Potential

Total

Figure 12: A plot of the energy being cycled throughthe gait during the pushed-release-stop walking test,showing that significant energy is exchanged betweenspring storage, gravitational potential, and its mo-tion. Total energy decreases as the gait slows toa halt. Non-conservative effects result from dissi-pative losses (likely primarily molecular damping inthe spring, ground impacts and friction) and the er-ror dynamics of the actuators from attempting totrack zero stance torque.

function, and initial conditions. This demonstratesthat spring-mass walking dynamics can be the dom-inant dynamics of ATRIAS. Further, the correspon-dence between the robot and model by enforcingleg angles used in the simple model suggests thatATRIAS is a well-suited platform for spring-masscontrol policies.

This walking test also shows that the springs playa significant role in the energetics of the walkinggait. In Figure 12, we plot the kinetic, gravitational,and stored spring energy of the robot when walking.Within a step, ATRIAS’ kinetic energy fluctuateswith an amplitude of 14.6±2.44J during the spring-mass gait cycle, and gravitational potential swings11.7 ± 1.96J with each rise and fall of the robot’smass. The springs store and release 10.4± 1.81J ineach step, a very similar magnitude to both gravita-tional and kinetic energy. This demonstrates thatthe springs are a significant energetic participantin the cycling between kinetic and potential energyduring locomotion, evidencing ATRIAS’ embrace ofspring-mass locomotion dynamics.

Spring-Mass Model

FZ

FX

x

z Simulation

ATRIAS Monopod (thrown)

FZ

FX

Experimentx

z

ThrownRobot

0.20.0

2

4

Time (s)

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e (B

W)

2nd

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2

41st

Time (seconds)

Forc

e (b

ody

wei

gh

ts)

Time (s)

Forc

e (B

W)

0.20.0

2

43rd

Figure 13: The ATRIAS monopod prototype be-ing tossed to observe its passive dynamics whilehopping or running. The monopod completed fivehops before falling. For the first three hops, theground reaction forces of the robot (solid lines)closely matched a simple spring-mass model (dashedlines) with the same parameters and initial condi-tions. Video shown in Extension 1c.

4.2.4 Passive Throw (Monopod)

To elicit ATRIAS’ passive dynamics in aerial gaitssuch as hopping and running, we connected theATRIAS monopod to the boom and tossed it, ob-serving its subsequent bounding gait. Like the pas-sive walking test (Section 4.2.3), the leg was activelyreciprocated during swing phases, in this case re-setting to a constant attack angle. During stance,again, the plate springs were regulated to enforcezero net hip torque while the effective rest lengthof the virtual spring was held constant, therebyattempting to impart zero net work on the robot(See Appendix C for mathematical implementation).Due to its lighter weight, this test was better suitedfor the monopod as it is far easier for a human ex-perimentalist to throw. The torso rotation was me-chanically locked to the boom.

Upon throwing, the monopod completed five hopsbefore dissipative effects led to a fall. Such successivepassive hops provides evidence of ATRIAS’ mechan-ical efficiency even with higher energy gaits, whereimpacts losses are typically exacerbated. Like in thepassive walking experiment, the thrown monopodshows marked similarity to the spring-mass model,as plotted in Figure 13. The ground-reaction forces

15

display the characteristic single-humped force profileof spring-mass running (as well as animal running).When directly compared to a simulated spring-massmodel with the same mass, spring function, and ini-tial throwing conditions, the ground-reaction forcesmatch very closely to the monopod over the firstthree hops. This degree of model matching indicatesthat ATRIAS’ engineering was significantly success-ful in rendering spring-mass passive dynamics forboth walking and hopping gaits.

4.3 Performance Demonstration

4.3.1 One-Legged Hopping

Hopping represents a high-power mode of locomo-tion, requiring fast push-off while legs are heavilyloaded. For ATRIAS, hopping also serves as a me-chanical precursor for the strenuous dynamical de-mands of running. To validate ATRIAS can handlethe rigorous power requirements of hopping and run-ning, we controlled ATRIAS to hop on one leg.

In this test, ATRIAS starts from standing posi-tion and begins an uninterrupted sequence of hopswith one leg for support. For hopping control, wecommanded the springs to generate ground-reactionforce trajectories (see Appendix C) associated witha spring-mass hopping model. The boom was con-figured to lock the torso angle to facilitate this test.Video shown in Extension 1d.

Upon startup, ATRIAS demonstrated notablyhigh hopping heights, which we claim are on-par-with or greater than human capabilities. The bipedreached a height of over 9cm within the first hop,and continued to hop on the same leg for dozensmore cycles with an average toe clearance of 9.6 cm(±0.9 cm, N = 55), as shown in Figure 14. How-ever, a more apt measurement of hopping prowess isthe center-of-mass height, which cannot be cheatedby rapidly retracting legs during flight. ATRIAS’center of mass, as approximated by the hip point,also consistently clears 9cm over its takeoff height.This is higher than humans hop (less than 4.4 cm)when asked to do so continuously on one foot (Changet al. 2008) (See Appendix B for detailed analy-sis). The rate of hopping was also similar as therobot hopping frequency was 2 Hz, very close to thepreferred human hopping frequency of 2.2 Hz. Ar-guably, this hopping frequency was “comfortable”for ATRIAS as well, since the hopping frequency wasthat of ATRIAS’ passive spring-mass model, thereby

Hip Height Toe Height Stance PhaseTD Height

0.0 0.5 1.0 1.5 2.0 2.5

Time (seconds)

Hei

ght

(met

ers)

0.0

0.2

0.4

0.6

0.8

1.0

9.6 cm

Apex Touch down Nadir Take off Apex

Figure 14: ATRIAS hopping on one leg over morethan 50 hops. The toe (dash-dotted line) consis-tently clears a height of 9.6 cm (±0.9 cm) and thehopping robot achieved a sustained body trajectory(solid line). Biomechanics measurements suggestthat this hopping height is higher than humans typ-ically exhibit on one leg when hopping continuously.The stance-phase (shaded region) duty factor for thehopping cycle is 50% (±2%), demonstrating non-trivial flight times. Further, five points of interestin the hopping cycle (apex, touch down, nadir, takeoff, apex) are labeled and shown as performed byATRIAS. Video shown in Extension 1d.

approximating maximum efficiency. The robot per-formed this test multiple times, stopping only oncommand, showing that the hardware is sufficientlyhigh-powered for continuous hopping on just oneleg. We claim that this sustained human-height-matching hopping capability is novel for a human-scale robot that can function untethered.

4.3.2 Sustained Walking

While Section 4.2.3 demonstrated spring-mass walk-ing in the short term with a nearly passive machine,we seek to demonstrate that such walking can besustained on ATRIAS. To do so, we implemented acontroller based on a simple state machine.

The walking controller alternates between stanceand swing phases, with each leg occupying the op-posite phase of its counterpart. The phase switchis triggered by the leg currently in stance reach-

16

Sustained

Walking

0 1 2 3 4

0

200

400

600

Time (seconds)

Ax

ial

Leg

Fo

rce

(N)

Left Leg

Right Leg

5 meters

Figure 15: A stroboscopic image of a sustained ATRIAS walking gait and its associated axial leg forces,demonstrating that sustained walking is viable for ATRIAS while sporting the characteristic “double-humped” force profile of both spring-mass walking and human walking. Video shown in Extension 1e.

ing a predefined extreme angle. During stance, theleg has to support and propel the body. The mo-tors generate holding torques allowing the springs tobe loaded and to redirect the trunk’s vertical mo-tion, thus supporting the robot’s weight during thestride. A hip torque is generated by distributing theholding torque unevenly between the motors rotat-ing the leg, propelling the robot forward. During theswing phase, the leg is driven forward, first shorten-ing to ensure sufficient ground clearance and extend-ing again towards the end of swing, culminating inleg retraction when nearing touchdown. Timing ofmotion and phasing is based on the stance leg mo-tion, introducing a virtual constraint (see Appendixfor mathematical control description).

The robot exhibited compliant walking, generat-ing ground reaction force profiles similar to thoseobserved in human walking, as plotted in Figure15. To our knowledge, this characteristic double-humped force pattern has not yet been reported insustained robotic walking.

4.3.3 Hops in a Box of Rocks (Monopod)

Nonrigid surfaces and uncharacterized terrain areinevitable in natural environments. As a test ofATRIAS’ reflexive capability to handle extreme andunmodeled terrain changes, we challenged the hop-ping ATRIAS monopod with a pit of pea gravel, i.e.a box of rocks. We commanded the monopod tohop vertically on the laboratory floor, and while air-borne, pushed it over the pit. This “rocks test” as-sesses how closely ATRIAS can maintain its nomi-nal mass trajectory despite an extreme disturbance,testing whether ATRIAS’ mechanical system is ca-pable of responding to surprises in the real world.

Admittedly, a box of gravel is an unusual robottest, but is an appropriate reflexive challenge inboth magnitude of disturbance and unpredictabil-ity of terrain dynamics. Extending 16.5 cm beneaththe floor, this box of rocks is hazardously deep for ahopping robot, a full 17% of ATRIAS’ nominal leglength. Further, while an open-air fall is simple tomodel and easy to plan for, granular media is noto-riously not (Zhang et al. 2013), making a gravel trapan apt analogue for the unknowns of real-world ter-

17

rain, less befitting of intricate model-informed con-trol design. Further, this gravel is somewhat looselypacked, meaning that ATRIAS will likely sink someportion of the entire 16.5-cm depth of the trap, re-quiring significant energy injection from the actua-tors to maintain its hopping height. For this hard-ware test, the boom was locked to constrain therobot to only vertical motion.

The test controller was designed to regulate theground-reaction forces, as measured by the springs,tracking the force trajectory associated with its nom-inal hopping gait. By controlling these forces, ineffect, the robot mass trajectory is regulated irre-spective of the terrain beneath it. In essence, thecontroller does not “switch” when on the rocks asopposed to the floor, but merely regulates force asdescribed in Appendix C. Properties of this “forcecontrol” technique are elaborated upon other stud-ies (Koepl and Hurst 2013), including it being work-optimal disturbance rejection for dissipative surfaces(Hubicki and Hurst 2012). The rocks were not mod-eled and neither the controller nor its gains weretuned for these terrain dynamics, so as not to meddlewith the reflexive nature of this rocks test.

After over a dozen cycles of steady-state hopping,the ATRIAS monopod was nudged over the rocks.The robot toe plunged into the gravel pit, as shownin the film strip in Figure 16a. ATRIAS regulatedits forces by rapidly extending its leg, thereby in-jecting lost energy. After eight complete hops in thebox, ATRIAS jumped out of the pit and resumed itsprior steady hopping on the laboratory floor. Fig-ure 16b shows the hopping apex height was largelymaintained despite the sizable terrain change. Thisdisturbance is dynamically extreme for legged loco-motion, but was handled by the ATRIAS monopodwithout planning or switching controllers.

4.3.4 Dynamic Balancing in 3D

In this balancing experiment, we demonstrate thatATRIAS is mechanically capable of stable 3D loco-motion. In particular, this means the robot mustbe able to move and balance without external sup-port, external power, or communication cables. Four22V lithium polymer batteries (two in series, twoin parallel) were mounted inside the torso, and awireless bridge provided communication to a user-interface computer. The only connection to any sup-port structure was a slack safety cable, which onlybecomes taut when if the robot falls due to malfunc-

1 2 3 4 5 6 7 8

Time (seconds)

0.0

0.2

0.4

0.6

0.8

1.0

Hip Height

Stance Phase

Hei

gh

t (m

eter

s)

(a)

(b)

Figure 16: As a test of reflexive actuation in a sur-prise terrain change, the ATRIAS monopod hoppedinto, and out of, a 16.5-cm-deep gravel pit. The con-troller was not switched depending on terrain andthe robot was unaware of the impending trap. (a)Images of the monopod plunging into the rock trap.(b) The hip height of the monopod before, during,and after the terrain change, showing it can largelymaintain its apex hopping height despite the 16.5 cmdepth of the trap. Video shown in Extension 1f.

tion.

Using a specially-developed 3D locomotion con-troller (described in (Rezazadeh et al. 2015)),ATRIAS was commanded to step in place and main-tain balance by stepping in place from right footto left foot repeatedly (Figure 17). During the ex-periment, ATRIAS was able to balance continuouslywithout signs of instability. This simple experimentdemonstrates that ATRIAS’ spring-mass design ismechanically capable of practical locomotion, whereexperimental restraints, tethered power, and wiredcommunication is not possible. Further, the groundreaction forces also manifest in a “double-humped”profile as per spring-mass walking, all while balanc-ing in three dimensions.

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Figure 17: A sequence of tiled images showingATRIAS marching in place in 3D. Using a controllerdescribed in (Rezazadeh et al. 2015), the robot con-tinually steps in place to hold position. Batteriesand computers are all mounted onboard, with nocommunication cables or connection to the experi-ment boom. A slack safety cable is attached to thetop of the torso which would catch the robot in caseof a fall. Plotting the vertical leg forces also shows“double-humped” ground reaction forces of spring-mass walking. Video shown in Extension 1g.

5 Future Work

5.1 Leg Configurations and GeometricPower

ATRIAS’ four-bar leg mechanism has certain advan-tages such as being very lightweight, but also cleardisadvantages like geometric power. However, thereis no inherent reason why other lightweight leg de-signs must bear these inherent losses. By explor-ing the space of possible leg configurations, one canidentify a kinematic layout that consumes no geo-metric power for the desired motions that can alsoaccommodate low-weight and compliance (Abate etal. 2015). Given such a leg mechanism, the totalmechanical power consumed by the actuators maybegin to approach that of the mechanical energy lostduring the gait cycle.

5.2 Control

In a broad view, various control frameworks are po-tentially suited to take advantage of ATRIAS’ under-actuated dynamics. Hybrid-zero dynamics (HZD)

(Westervelt et al. 2003; Ames 2014), receding-horizon differential dynamic programming (Erez etal. 2013), and sum-of-squares verified trajectory li-braries (Manchester et al. 2011) are a few examplesof control frameworks that can support ATRIAS’high degree of underactuation. More specifically, ap-proaches have also been developed to tackle morebiped-specific issues such as torso and frontal-planestabilization.

While the presented hardware performance testswere performed with locked torso rotation, many so-lutions exist for handling torso dynamics (some ofwhich have already achieved walking with ATRIAS).Within the framework of spring-mass techniques, thetorso angle can be regulated by constraining the di-rection of resultant leg forces through a predeter-mined intersection point, i.e., virtual pivot point(Maus et al. 2010). ATRIAS has this capabilityto arbitrarily constrain leg forces, as demonstratedin Sections 4.2.3 and 4.2.4, and thus is amenableto virtual pivot techniques. Also, hybrid zero dy-namics (HZD) has long offered various solutions totorso management which have been implemented onATRIAS in-plane (Ramezani et al. 2013; Hereid etal. 2014).

Further, HZD formulations have been positedto formally collapse robot dynamics into a sim-pler spring-mass model using virtual constraints(Poulakakis and Grizzle 2009). With ATRIAS, suchdynamics-matching techniques may be most appli-cable for two reasons. First, the passive dynamicsalready approximate a spring-mass model, likely re-quiring much less control effort to formally match thetarget model. Second, its impact-minimizing springconfiguration is beneficial in satisfying hybrid invari-ance, a key control constraint for HZD methods. Ineffect, the discontinuous jump in state space is verysmall for the robot mass velocity and rotor veloc-ity, which are normally quite sensitive variables forgenerating an invariant manifold.

While dynamic locomotion outside the sagittalplane remains an open controls challenge, sev-eral strategies already exist which can feasiblybe implemented on ATRIAS. Despite traditionalzero-moment point strategies being rendered mootby ATRIAS’ point feet, strategically analogouspush-recovery methods can be adapted to stabilizeATRIAS in three dimensions (Stephens and Atke-son 2010; Pratt et al. 2012). Hip abduction an-gles have been used to heuristically stabilize the

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Raibert hoppers (Raibert 1986), and more formalabduction-control strategies have also been devel-oped for the spring-mass model (Seipel and Holmes2005; Peuker et al. 2012; Maus and Seyfarth 2014;Sullivan and Seipel 2014) and more have been de-veloped in the context of HZD (Sinnet and Ames2009; Hamed and Grizzle 2014; Buss et al. 2016). Astrategy that works within the framework of spring-mass locomotion may help maximally leverage themachine’s passive dynamics to achieve speed andefficiency. In fact, policies adapted from spring-mass studies, in combination with other components,are now exhibiting successful 3D locomotion withATRIAS in simulation (Martin et al. 2015) and ex-periment (Rezazadeh et al. 2015).

5.3 Biological Investigation

Due the general similarities between spring-massmodel and biological locomotion (Section 2), we cantune ATRIAS’ parameters to more quantitativelymatch animal dynamics. Humans, birds, and otherbipeds have long been characterized by an equivalentleg-spring stiffness, which ATRIAS can approximateby installing plate springs with stiffness to match.

In addition to effective stiffness, recent systemidentification work on ground-running birds from os-trich (120 kg) to quail (0.2 kg) has also estimatedtheir effective leg dissipation (Birn-Jeffery et al.2014). Reduced-order modeling of these species’ run-ning gaits suggested that the shape of their ground-reaction forces is consistent with work-optimal run-ning control with actuated-spring-mass-damper legs,a math model highly similar to ATRIAS’ legs.Specifically, the degree of dissipation in the reduced-order model was modulated to tightly fit to bird dy-namical measurements, yielding an effective leg dis-sipation for each species’ running gait. Interestingly,running ostriches (the species closest to ATRIAS inmass) exhibited a damping ratio of 0.058, which isnotably similar to ATRIAS’ damping ratio of 0.061.This suggests that ATRIAS can embody both theenergy-cycling and energy-dissipating dynamics ofanimal gaits. Put more directly, ATRIAS may bevery dynamically suited to run with ostrich-like gaitdynamics.

This quantitative similarity to measured animaldynamics can be used to investigate animal runningin a particularly comparable way. Specifically, if theunderlying dynamics of ATRIAS are similar to ani-mals, we hypothesize that energy-optimal control of

ATRIAS should yield similar gaits to animals, whichalso likely minimize energy costs (Srinivasan andRuina 2006). Tight correlations between optimalATRIAS running and animal running would providestrong evidence for a more predictive reduced-orderbiological model. Such a model could be used toquantitatively predict animal maneuvers; a poten-tially important biological finding.

6 Conclusion

We present ATRIAS as an example of apply-ing a spring-mass design approach to a versatile,human-scale bipedal robot. By leveraging a low-mass fourbar leg mechanism and series-elastic platesprings, ATRIAS exhibits spring-mass locomotionwhen dropped, pushed, or thrown. Thoughtfullyactuating ATRIAS allows for both sustained walk-ing and hopping on one leg (higher than humans),demonstrating a capability for executing a variety ofgaits, and demonstrated preliminary walking with aspecific cost of transport of 1.13 (approximately athird of energy cost estimates for ASIMO (Collinset al. 2005)). This versatility included hoppinginto, and out of, an unseen 16.5-cm-deep gravel pit,demonstrating notable control authority. Further,ATRIAS was able to step in place in 3D, untetheredand unsupported, showing that ATRIAS is mechan-ically capable of locomotion in practical settings.

To our knowledge, ATRIAS is the first docu-mented bipedal machine to replicate the human-like ground reaction forces of spring-mass walking.This characteristic “double-humped” force profilewas measured both during a 17-step minimally con-trolled “passive” walk and a sustained steady walk-ing (using a simple proof-of-concept walking con-troller). Further, the results also show hopping withthe single-humped ground-reaction forces similar toanimal running.

These biology-matching results advance a broaderpoint regarding bioinspired robot design. Robotmechanisms need not be morphologically biomimetic(e.g. ATRIAS’ spring-loaded fourbar leg mecha-nism) in order to produce biologically relevant dy-namics (human ground-reaction forces). Further,these human-like dynamics emerge from efficientcontrol of appropriately designed passive dynamics,and do not necessarily need to be the explicit targetof a feedback control loop.

ATRIAS owes its dynamical capabilities to its

20

spring-mass design approach. The extent to whichit performs varied motions are predicted by the sim-ple model and the springs play a clear energetic rolein the exhibited gaits. As such, these results em-body an extension of theoretical compliant gait dy-namics to more practical machines. Ultimately, thepresented design and experiments demonstrate thatinsights from theoretical models are not limited tolaboratory platforms, and in fact, can be a means forachieving the efficiency, versatility, and dynamismdemanded of bipeds that the world will want.

Acknowledgments

This work was supported by the Defense AdvancedResearch Projects Agency [grant number W91CRB-11-1-0002]; the Human Frontier Science Program[grant number RGY0062/2010]; and the NationalScience Foundation [grant number CMMI-1100232].

The authors also thank their DARPA project col-laborators: Dr. Jessy Grizzle and the Control Sys-tems Laboratory at the University of Michigan, andDr. Hartmut Geyer’s research group at the RoboticsInstitute at Carnegie Mellon University. The au-thors also thank their HFSP project collaborator,Dr. Monica Daley of the Structure and Motion Lab-oratory at the Royal Veterinary College, London.

The authors further thank Kit Morton, Ryan VanWhy, Soo-Hyun Yoo, Ethan Shepard, Michael An-derson, Ross McCullough, and Ryan Skeele for theirtechnical contribution to building ATRIAS, its elec-tronics and software systems, its supporting equip-ment, and their assistance in performing experi-ments.

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[77] Eric R. Westervelt, Jessy W. Grizzle, andDaniel E. Koditschek. “Hybrid Zero Dynamicsof Planar Biped Walkers”. In: IEEE Transac-tions on Automatic Control 48.1 (Jan. 2003),pp. 42–56.

[78] Garth Zeglin. “The Bow Leg Hopping Robot”.PhD thesis. Pittsburgh, PA: Robotics Insti-tute, Carnegie Mellon University, Oct. 1999.

[79] Tingnan Zhang, Feifei Qian, Chen Li,Pierangelo Masarati, Aaron M. Hoover, PaulBirkmeyer, Andrew Pullin, Ronald S. Fear-ing, and Daniel I. Goldman. “Ground fluidiza-tion promotes rapid running of a lightweightrobot”. In: The International Journal ofRobotics Research 32.7 (July 2013), pp. 859–869.

A Power Analysis

Using a preliminary example gait controller (as perSection 4.3.2), we were able to identify ATRIAS’ typ-ical, main actuator power profile. ATRIAS’ load sce-nario is given through its four main actuator veloci-ties, their in-series spring/actuator torques, and themeasured power. This was particularly importantfor assessing the empirical effect of the leg mech-anism’s geometric power on the electrical economyon a walking gait (See Section 3.3.1 and its asso-ciated power data, Figure 8). Observing a walk-ing gait with a speed of 0.85 m/s, the mean dataof 12 consecutive steps were recorded and analyzed.

25

A motor model was implemented to fill gaps fromnon-measurable, otherwise missing dynamical mo-tor characteristics (See Table A.1 for a list of avail-able sensors). Power “components,” such as ratio ofregeneration, electrical power losses, and individualmotor powers were non-measurable, mostly becausethey were not accessible in the ATRIAS setup. Onlythe instantaneous summed power of all four actua-tors could be measured by a power clamp. Conse-quently, we present an ATRIAS motor model thatprovides otherwise unaccessible system parameterssuch as a) instantaneous power applied to acceler-ate actuators, b) electrical power losses, c) actuatorefficiencies, d) and the instantaneous regeneration ofactuators and amplifiers. The model further allowsfor precise estimation of the swing and stance phasepower characteristics of the ATRIAS system giventhe example gait controller.

A.1 Experimental Setup

Full dynamics in a legged robotic system can be de-rived if both kinematic and dynamic data are avail-able. ATRIAS’ sensor infrastructure provides suf-ficient information to calculate the instantaneousload on each actuator continuously during the swingand stance phases. Axial leg forces and tangen-tial leg torques were measured indirectly throughthe robot’s deflecting leg springs, one for each ofATRIAS leg motors (Section 3.2.2). Leg forces andtorques are derived through the robot’s kinematicsand the measured actuator forces and torques. In asystem without internal actuator torque estimation,swing phase dynamics cannot be estimated. Fewrobots are equipped with a full set of joint torquesensors, and only sparse data of robot swing phasedynamics is available from literature. ATRIAS’s in-ternal leg dynamic measurements directly enable usto measure those values and apply them in the mo-tor model to gain insights into details of the system’smotor power.

All sensor data was recorded at a sampling fre-quency of 1000 Hz. The data of the walking ATRIASrobot was recorded over 12 full locomotion cycles. Afull cycle was arbitrarily defined between the begin-ning of the left leg swing phase, and the end of theleft leg stance phase. The average gait cycle timewas 1.1sec, at a gait cycle duty factor of 0.6 (Fig-ure 18, stance duration per gait cycle). The motor-torque crossing from loaded to unloaded, of the leftleg motor-B, was used to define the onset of swing

phase, and the end of stance phase again. All cy-cles were cut manually, and trajectories were inter-polated with 1000 samples.

A.2 Motor Model

We implemented a variant of the motor modelby (Roos et al. 2006), for each of ATRIAS’ four mainactuators. Only the sum of the applied current forall four motors (ABAB) could be measured in theATRIAS setup, with the help of a single currentclamp (Pclamp). The load-motor model allowed usto approximate the individual, instantaneous currentof each motor. Further, we extended Roos’ motormodel with the capability of identifying actuators ingenerator mode; in phases of negative power, thoseactuators would push electrical power back into therobot’s electrical grid. We found that this regener-ated power was almost always instantly re-used byone of the other motors, and did not charge the bat-teries. This a consequence of the geometric powerinherent to the mechanism as described in Section3.3.1.

Motor power Pm is described as the sum of elec-trical losses Pelec and mechanical losses Pmech (Rooset al. 2006, equation 10-13) for a system withoutpower regeneration of mechanical energy. Mechani-cal losses depend on the instantaneous power appliedthrough the load (Tl, θl) and through acceleratingthe motor and gear components (Jm motor inertia,Jg gearbox inertia).

Pm = Pelec + Pmech (1)

Pmech = (Jm + Jg)θlθln2 +

Tlµgθl (2)

Pelec = RmI2 = Rm

T 2m

k2T(3)

θl indicates the position of the load, here the actu-ator position in [rad], θl its velocity and θl its ac-celeration. Tl in [Nm] is the load torque measuredthrough spring deflection, µg (50%) is the approxima-tion of the harmonic drive gearbox efficiency. Elec-trical losses are calculated through the motor wind-ing resistance (Rm in [Ω]), and the motor coefficientkT. n is the gear ratio of the harmonic drive. We ex-ternally measured ATRIAS’ electrical power Pclamp

and compared it to the model-predicted motor powerPm,robot, as sum of its four main leg actuators [motorID’s: (L) left and (R) right side, A and B motor]:

Pm,robot = Pm,LA + Pm,LB + Pm,RA + Pm,RB (4)

26

Measured State Type of Sensor Source Model Number

Motor rotor (transmissiveencoder)

3,500 Lines/Rev. IncrementalQuadrature

US Digi-tal

3500-EM1-0-360-I

Motor rotor (hall effect) 3 sensors (1 per phase) EmoteqMotor winding thermistor temperature DigikeySpring + Limb Angles 1nm Absolute Optical w/ RTLA-S

martensitic steel scale (294µRad/bit)Renishaw RL32BAT001B30F

Leg Abduction 13-bit Absolute Magnetic Encoder Renishaw RMB30SI13BC1Ground Contact Custom Full-Bridge Load Cell Omega 120 Ohm Linear Foil

GaugeKnee Sideload Custom Quarter-Bridge Load Cell ’ ’ ’ ’R.O.M. Limits SPDT NC Omron D2F-01FL-DRobot pitch (via Boom) 17-bit absolute (geared to

12.6µRad/bit)Hengstler AC36/0017AR.41SBB

Robot horizontal (viaBoom)

17-bit absolute (geared to107.3µm/bit)

’ ’ ’ ’

Robot vertical (via Boom) 17-bit absolute (geared to 26.8µm/bit) ’ ’ ’ ’

Table 1: Detailed list of sensors to measure state information on ATRIAS.

The above motor model, however, does not accountfor the imperfect generator mode of the actuatorsand power amplifier in the case of negatively ap-plied power. Only externally applied loads, or loadthrough acceleration can produce negative power(equation 2). We include a generator efficiency termµregen into the equation for the motor power Pm:

Pmech,regen,motorID = −µregenPmech,motorID (5)

Pm = Pelec + Pmech + Pmech,regen (6)

and we identified the efficiency of regeneration be-tween µregen = 30 % and 40 %.

A.3 Power Analysis Results

The average, minimal and maximal leg forces and legtorques per full gait cycle of this example gait aregiven in Figure 18 (top and bottom, respectively).The leg force profile shows the walking-gaits-specificdouble hump profile, with a maximum leg force of650 N. Leg torques reached from −130 Nm up to100 Nm, and showed much higher variations, com-pared to the robot’s leg forces (Figure 18).

Quantitative results from the ATRIAS motormodel are presented in Table A.3, separated bystance phase values (40 % of cycle time), swing phasevalues, and full cycle values (1.1 sec in average). Themotor model is based on individual sensor data ofthe left and right ATRIAS leg, and indicates that

both legs are not working in complete symmetry.Because the robot is walking in a circle, forces andtorques applied are different between its inner (left)and outer (right) leg. Here shown are only motormodel power values for the left robot leg. Powervalues are provided, hence the mean of stance andswing, weighted by the duty factor, gives the full cy-cle power values. The externally applied load (Pl,A

and Pl,B) during stance phase is caused by impactand weight of the robot, and during swing phase bythe mass and inertia of the legs (Pl,AB,swing ≈ 15 W).Swing power values are very low, due to the low-weight leg design of ATRIAS. During stance phasemotor-A shows a large positive power consumptionPl,A,stance = 273 W, as it is supporting the weightof the robot. Motor-B is showing an almost equallylarge, but negative power, because the direction oftorque is opposite to that of motor A (four-bar con-struction), while the direction of movement is iden-tical (backwards). Parts of this negative power arebeing pushed back through the amplifiers, into other,power-draining motors (Pregen,B,stance = −57 W). Asa consequence of a large stance phase torque at lowermotor speed, the electrical power losses of motor-Aare significant: Pelec,A,stance = 99 W, or almost 1/3of the load power.

Mechanical swing phase power (instantaneous sumof load and acceleration) is roughly equal betweenmotor-A and motor-B actuators. Though leg masses

27

Type Motor Side Swing Stance Full

Pm A L 135 363 265Pm B L 119 -57 19Pm ABAB LR 584 517 546

Pmech A L 151 265 216Pmech B L 157 0 67Pmech ABAB L 606 510 551

Pl A L 14 273 162Pl B L 16 -268 -146Pl ABAB LR 19 22 21

Pacc A L 12 -9 0Pacc B L -31 25 1Pacc ABAB LR -24 20 1

Pelec A L 20 99 65Pelec B L 13 0 6Pelec ABAB LR 137 128 132

Pregen A L -36 0 -15Pregen B L -51 -57 -55Pregen ABAB LR -159 -121 -137

Pclamp ABAB LR 559 573 567

Table 2: Pm indicates the cummulative, modeledmotor power (Equation 1), Pmech the modeled, in-stantaneous sum of the external load power Pl andthe internal load power Pacc (both with positive andnegative values). Latter is caused by the accelera-tion of motor and gearbox components. Swing timeof this gait covers approximatively 40 %. The exter-nally measured power over all four motors (ABAB),is shown as Pclamp(ABAB). All values are reported in[W].

are low, both actuators require larger accelerationsduring swing phase, which is 20 % shorter than therobot’s stance phase. Further, leg length is short-ened in addition to the swing-forward movement,which adds an additional acceleration component tothe actuators. For the full cycle and both legs, themotor model predicts an electrical power consump-tion of Pm = 546 W, versus a measured power con-sumption of Pclamp = 567 W. For the ATRIAS sys-tem, this corresponds to an electrical cost of trans-port of 1.13 for this preliminary example gait.

B Comparative Hopping Analysis

While one-legged hopping is a common diagnos-tic test for various human medical conditions, thecenter-of-mass hopping height is rarely measured

-600

-400

-200

0

Fo

rce

(N)

20 40 60 80 1000

-100

-50

Cycle Duration (%)

0

50

100

Mo

men

t (N

m)

Figure 18: ATRIAS’ leg forces (top) and leg torques(bottom), x-values are normalized by a full gait cycleof a walking gait with an average forward speed of0.85 m/s. Values of the left leg only are shown. Theplot starts with the swing phase, stance phase be-gins at 40 % of the gait cycle. Continues, bold linesshow average force and torque values over 12 strides,grey shaded areas indicate the maximum and mini-mum band. Maximum leg forces during the walkingexperiment were around 650 N. Stance phase of theleg torques indicates an initial braking phase (av-erage −25 Nm), followed by a longer pushing phase(average 55 Nm), and a final, short braking torqueburst (−30 Nm). Strong oscillations of the leg inswing phase are visible both as leg forces, and as legtorque during the first 40 % of the gait cycle.

and reported in human studies. We estimate one-legged hopping height in humans using more com-monly reported measurements, which we analyzeusing work-energy methods. Specifically, we infercenter-of-mass hopping height using subject mass(m), peak force measurements (Fpk) and center-of-mass compression while on the ground (hc). We usedmeasurements of these quantities from (Chang et al.2008), which included an experimental control groupwhich hopped on one foot at a “comfortable” 2.2 Hzhuman hopping tempo.

We begin by defining work-energy equations at thehighest position of the center of mass (apex), and thelowest point while the ground (zenith).

Eapex = Ezenith +Wnonconserved (7)

28

At both apex and zenith, the center-of-mass ve-locity is zero, reducing the energy terms to the dif-ference in gravitational potential. Further, since theforce-length relation of human hopping was highlylinear (r2 = 0.99 and visible in (Chang et al. 2008,Figure 3a)), we can accurately approximate the legwork as the area under the force-length curve, whichis a triangle:

mg (hhop + hc) =1

2Fpkhc (8)

where hhop is the desired quantity, the vertical dis-tance the center of mass travels after take off. Givenquantities are: m, the subject mass, g, gravitationalacceleration, Fpk, the peak leg force, and hc, the dis-tance the leg compresses during stance. This equa-tion simplifies to:

hhop =

(Fpk

2mg− 1

)hc (9)

where (Chang et al. 2008, Table 1)) givesm=59.1±9.2 kg (very close to ATRIAS’ 62 kgwith batteries), g=9.81 m/s2, Fpk=1570±207 N, andhc=8.18±0.36 cm.

Further, we employ an error propagation analysison peak force and leg compression (the parametersthat change from hop to hop) given their reportedstandard deviations. We derive a standard deviationfor hhop, σhhop

as:

σhhop=

√√√√(δhhopδFpk

)2

σ2Fpk+

(δhhopδhc

)2

σ2hc(10)

σhhop=

√(hc

2mg

)2

σ2Fpk+

(Fpk

2mg− 1

)2

σ2hc(11)

which evaluates to hhop=2.9±1.5 cm, significantlylower than ATRIAS’ hopping height of 9.6±0.9 cm.In the main text (Section 4.3.1), we report the highend of this computed range, which is 4.4 cm. Fur-ther, we can employ a maximally generous statis-tical interpretation and assume the best-case val-ues within the human statistical range (high legcompression, 8.54 cm, high peak force, 1777 kg, andlow mass, 49.9 kg), and compute a human hoppingheight of 7.0 cm, which is also eclipsed by ATRIAS.

We believe that our presented center-of-mass mea-surements demonstrate ATRIAS’ capability for sus-

tainable one-legged hopping on par with human ca-pability. We believe this is the first documenta-tion of sustainable human-level hopping performancewith a human-scale biped that can operate off-tether. While other tether-free-capable humanoidrobots can hop (Cho et al. 2010), and ASIMO canhop on one leg, it is difficult to estimate exactly howfar the center of mass is vaulted from presented data.Further, the high-powered HRP3L-JSK (Urata etal. 2010) has been shown to jump a very-significant44 cm off the ground. This HRP3L-JSK test at-tempts a single hop using both legs, does not at-tempt a safe landing, and rapidly retracts its legsduring flight, thereby reporting a hopping heightthat differs from the height of the center of mass.

C Control for Experiments

The ATRIAS validation controllers are designed toregulate quantities in the axial and rotational di-rections in the plane. The purpose of these proof-of-concept controllers is to embody the dynamicalproperties of the spring-mass model. For instance,the passive spring mass model has a fixed rest lengthof the spring and no net torques about the center ofmass. On ATRIAS, these properties can be imple-mented as targets of control.

First, we define our coordinate system as per Fig-ure 19. The planar ATRIAS has four motor angles(θm = [θ1sm, θ2sm, θ1nsm, θ2nsm]T ) as depicted in Fig-ure. The robot also has access to the angles of theleg linkages, which are on the distal side of the spring(θl = [θ1sl, θ2sl, θ1nsl, θ2nsl]

T ). Using these angles, wecan construct functions for all of our key quantitieswe wish to regulate, including leg angles, leg force,and hip torque.

For our general controller, we define a simple PDcontrol law that regulates the position of each motorangle

τ = KP

(θdesm − θm

)−KDθm (12)

where τ = [τ1s, τ2s, τ1ns, τ2ns] are commanded mo-tor torques to their matching motor angles (θm) andKP and KD are proportional and derivative gains

respectively. θdesm =[θdes1sm, θ

des2sm, θ

des1nsm, θ

des2nsm

]Tare

the desired position of the motor angles and are de-termined differently for each experiment. In generalform, θdesm is the solution to the system of equations:

gsaxial(θdesm , θl) = ysaxial

29

x

y

Motor-side

coordinates

Leg-side

coordinates

Figure 19: Coordinate system for the ATRIAS bipedal robot in the sagittal plane. The middle drawingpoints to angles of the motors and the rightmost illustration points to angles of the deflected springs.

gsrot(θdesm , θl) = ysrot

gnsaxial(θdesm , θl) = ynsaxial

gnsrot(θdesm , θl) = ynsrot

at any given instant, where g(θdesm , θl) =[gsaxial, g

srot, g

nsaxial, g

nsrot] are functions defining our key

regulated quantities and y = [ysaxial, ysrot, y

nsaxial, y

nsrot]

defines the target value of the quantity. The sub-scripts axial and rot indicate axial and rotationalquantities and the superscripts s and ns denotethe stance and nonstance legs respectively. Thesefunctions are defined below and apportioned to eachexperiment in Table 3.

We now define our key quantities to be used in theexperiments. The rest length of the virtual spring issolely a function of motor positions, θm. The deflec-tions of each rotational spring can be calculated fromthe difference between motor and leg angles (θm−θl).We use this information to compute the zero-forcelength of the leg mechanism measured from the hipposition (px, pz) via:

ls0 (θm) = Lsh cos (θ1sm) + Lth cos (θ2sm) (13)

lns0 (θm) = Lsh cos (θ1nsm) + Lth cos (θ2nsm) (14)

where ls0 and lns0 are the zero force leg lengths ofthe stance and non-stance legs respectively, and the

lengths of the thigh and shin (Lth and Lsh) are both0.5m. We further require the deflected length of thevirtual spring (lsk) to compute axial forces:

lsk (θm) = Lsh cos (θ1sl) + Lth cos (θ2sl) (15)

F s = ks

√1− lsk

2 − (arccos lsk − arccos ls0) lk

2LshLth3

√1− lsk

2(lsk − ls0)(16)

where F s (a function of lsk and ls0) is the force inthe axial direction of the leg when given a rotationalspring stiffness of the plate springs (ks). Rotationalzero-force leg angles and deflected angles reduce tomere averages of the mechanism’s angles:

θs0 (θm) =1

2(θ1sm + θ2sm) (17)

θns0 (θm) =1

2(θ1nsm + θ2nsm) (18)

θsk (θl) =1

2(θ1sl + θ2sl) (19)

θnsk (θl) =1

2(θ1nsl + θ2nsl) (20)

where θs0 and θns0 are the zero-force leg angles andθsk and θnsk are the deflected leg angles for the stanceand non-stance legs respectively. This allows us to

30

Experiment Phase Axial (gaxial → yaxial) Rotational (grot → yrot)

#1 Passive Drop N/A N/A N/A

#2 Passive Walking Stance Leg Constant Rest Length (ls0 → 0.9m ) Zero-torque (δsk → 0)Non-stance Rest-length trajectory (lns0 → l (α)) Angle (θns0 → θ (α))

#3 Passive Throw Stance Constant Rest Length (ls0 → 0.9m) Zero-torque (δsk → 0)Flight Nominal rest length (ls0 → 0.9m) Leg angle (θs0 → θTD)

#4 Hopping Stance Regulate force profile (F s → f (t)) Hold vertical (θs0 → π)Flight Nominal rest length (ls0 → 0.9m) Hold vertical (θs0 → π)

#5 Sustained Walking Stance Leg Constant Rest Length (ls0 → 0.9m) Small torque (δsk → ε)Non-stance Rest length trajectory (lns0 → l (α)) Angle traj. (θns0 → θ (α))

#6 Hops in Rocks Stance Regulate force profile (F s → f (t)) Hold vertical (θs0 → π)Flight Nominal rest length (ls0 → 0.9m) Hold vertical (θs0 → π)

#7 Balancing in 3D (Rezazadeh et al. 2015)

Table 3: A comparative summary of the controllers used for the ATRIAS validation experiments. Eachleg has two motors, and thus has two target quantities for the controller to regulate at all times (one inthe axial direction and one in the rotational direction). These quantities are driven via PD controllersto the value indicate by the → symbol. ε is a small torque offset, α is the angle of the stance leg withrespect to vertical, θTD is an angle determined by simulations with the spring-mass model, and f (t) is atime-dependent force trajectory derived from numerical simulations of the spring-mass model.

compute the deflection of the mechanisms’ virtualrotational spring

δsk (θm, θl) = θs0 − θsk (21)

δnsk (θm, θl) = θns0 − θnsk (22)

where δsk and δnsk are the rotational spring deflectionsfor the stance and non-stance legs respectively.

Example. Let’s take the example of the passivewalking test (Section 4.2.3 and Experiment #2 inTable 3). For the stance leg, the axial target is aconstant rest length (ls0) of 0.9m and the rotationaltarget is a leg torque (δsk) of zero. For the non-stanceleg, the axial target is a nonstance leg length (lns0 )that is a function of the stance leg angle, α, which isdesigned to lift and lower the leg through swing. Thenonstance rotational target is nonstance leg angle,which that too varies with α, which in this case isdesigned to swing the leg forward as α increases. Wethen solve for the set of four desired motor anglesθdesm which satisfies all four of these conditions:

ls0(θdesm , θl) = Lsh cos

(θdes1sm

)+ Lth cos

(θdes2sm

)= 0.9m

δsk(θdesm , θl) = θs0 − θsk

Extension Type Description

1 Video a) Figure 10 test (0:06)b) Figure 11 test (0:17)c) Figure 13 test (0:43)d) Figure 14 test (0:52)e) Figure 15 test (1:01)f) Figure 16 test (1:19)g) Figure 17 test (2:06)

Table 4: List of multimedia extensions.

=1

2

(θdes1sm + θdes2sm

)− 1

2(θ1sl + θ2sl)

= 0

ls0(θdesm , θl) = Lsh cos

(θdes1nsm

)+ Lth cos

(θdes2nsm

)= l(α)

θns0 (θdesm , θl) =1

2

(θdes1nsm + θdes2nsm

)= θ(α).

These conditions fully describe, for that instant, allfour target motor angles, θdes1sm, θdes2sm, θdes1nsm, andθdes2nsm. These θdesm are then fed to the controller (12)to compute commanded motor torques, τ .

31