Robot Leg Design: A Constructive Framework

Erik Jonsson School of Engineering and Computer Science

Robot Leg Design: A Constructive Framework

UT Dallas Author(s):

Siavash Rezazadeh

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Open Access Policy: Personal use is permitted, but republication/redistribution requires IEEE permission.©2018 IEEE

Citation:

Rezazadeh, S., A. Abate, R. L. Hatton, and J. W. Hurst. 2018. "Robot LegDesign: A Constructive Framework." IEEE Access 6: 54369-54387, doi:10.1109/ACCESS.2018.2870291

This document is being made freely available by the Eugene McDermott Libraryof the University of Texas at Dallas with permission of the copyright owner. Allrights are reserved under United States copyright law unless specified otherwise.

Received July 9, 2018, accepted August 28, 2018, date of publication September 17, 2018, date of current version October 17, 2018.

Digital Object Identifier 10.1109/ACCESS.2018.2870291

Robot Leg Design: A Constructive FrameworkSIAVASH REZAZADEH 1, (Member, IEEE), ANDY ABATE2, ROSS L. HATTON 3, (Member, IEEE),AND JONATHAN W. HURST4, (Member, IEEE)1Department of Bioengineering, The University of Texas at Dallas, Richardson, TX 75080, USA2Agility Robotics, Albany, OR 97321, USA3Collaborative Robotics and Intelligent Systems Center, Oregon State University, Corvallis, OR 97331, USA4School of Mechanical, Industrial, and Manufacturing Engineering, Oregon State University, Corvallis, OR 97331, USA

Corresponding author: Siavash Rezazadeh ([email protected])

This work was supported in part by DARPA under Grant W91CRB-11-1-000221 and in part by NSF under Grant 1462555.

ABSTRACT Most robot leg designs are either based on biomimetics of humans’ or animals’ leg morpholo-gies or on being mechanically optimized for specific task(s). In the first approach, differences betweenthe actuation of the mechanical leg and the original biological structure usually result in inefficiencies andcontrol malfunction, and legs in the second group are often limited to good performance for a single taskand would fail when used for others. In this paper, we present a constructive framework for robot leg design,which tries to take advantage of the positive factors of both aforementioned approaches. For this purpose,we first, through selection of a template whose biological relevance for a wide range of tasks has beenproven, establish a foundation on which mechanical design can be built. Then, we present a general theoremfor designing a mechanism based on a template in order to maximize efficiency. In the final step and once themechanism is designed, we address the problem of selecting the actuators and formulate it as a constrainedoptimization problem. In a case study with experimental walking data, we show how the mechanism designtheorem and the formulated optimization problem can be used together to improve the walking energyefficiency by more than 50%. The proposed three-step approach is not limited to any template and shouldprovide amore structured procedure for leg design, result in optimal energy economy, andmaintain importantbioinspired factors vital for control and versatility of legged robots.

INDEX TERMS Legged locomotion, legged robots, mechanism design, design optimization, reduced ordersystems.

I. INTRODUCTIONLeg design is critical to the success of walking and run-ning robots. A well-designed leg will enable the systemto locomote with little energy dissipation and to passivelyaccommodate terrain irregularities with minimal requirementfrom a controller to sense them and devise a response. Con-versely, even the best software control cannot extract efficientperformance from a leg whose actuators are in oppositionto each other. Similarly, control design cannot be of muchhelp when the dynamics of the leg require unreasonablyhigh-bandwidth controllers to stabilize against impacts andother perturbations and uncertainties. In other words, due tohigh complexity of the problem and limitations of the actu-ator technology, design of a legged robot should be seen asan interconnected problem between mechanism and controldesign. Since it is neither logical nor practical to considerall possible paradigms for control of a legged robot a priori,we rely on reduced order models that capture the uniqueand dominating characteristics of these systems. In addition

to this, reduced order models (or templates as introducedin [1]) are capable of simulating and explaining complex andhyper-redundant phenomena such as biological locomotion ina wide range of tasks. This can potentially lead to leg designswhich can accomplish not only routine tasks such as walkingand running on flat ground, but also more complicated oper-ations, including walking on soft or uneven ground or moredemanding actions, such as jumping.

The reduced order models are typically energy-conservative or have minimal energy dissipation (usuallyimpact losses). As a result, they are not suitable for detailedenergy optimization studies of robots, where, for example,the actuators contribute to a significant part of the losses.For such analyses, a more complicated model, with moredegrees of freedom and characteristics sufficiently close tothose of the final leg design is required. But here, the trade-off between versatility and efficiency arises. A historicalexample is the so-called straight-line mechanisms whichbecame popular in the early days of legged robots for the

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S. Rezazadeh et al.: Robot Leg Design: A Constructive Framework

FIGURE 1. The proposed three-stage design procedure.

purpose of complete decoupling of gravity from the directionof motion [2], [3]. The small workspace and limited abilitiesof such mechanisms led to considering articulated mecha-nisms as a next-step development. However, with the addeddegrees of freedom at the knee, the problem of actuatorsworking against each other arose, which in turn inspiredthe design of different decoupling leg mechanisms [4], [5]including pantographs [6], [7]. In a way, these early worksfor designing an efficient leg mechanism through restrictingthe motion to the horizontal direction, can be viewed as firstexamples of intuitive template-based leg design, relying on atemplate which was theorized years later for control purposein the form of Linear Inverted Pendulum Model (LIPM) [8].

Works such as [9] investigate the problem of finding engi-neering strategies for considering factors known to affectpower efficiency in the leg design process. Such factorsinclude low-inertia legs (to minimize impact energy loss),energy reharvesting ability, and high torque-density motors(i.e., minimizing actuator weight). Although consideringthese factors certainly improves the efficiency of leggedrobots, still several fundamental questions about leg designremain unanswered. What behavior is the leg to be designedfor? What is the optimal mechanism for this behavior? Howare the actuators to be selected for the optimal performanceof this mechanism?

In this paper, we aim at laying out a philosophy for design-ing robotic legs, through maximizing efficiency and at thesame time maintaining the ability of the leg for stable, robust,

and dynamic locomotion. For this goal and based on the abovequestions, we propose to divide the leg design problem intothree stages, namely template selection, mechanism synthe-sis, and actuator design. As can be seen in Fig. 1, each stagebuilds upon the previous stage with more details and withdigging into lower-level design factors. This also providesa nice scheme in terms of energy usage in the standardterminology of locomotion; from a conservative or nearlyconservative level (template) to considering Mechanical Costof Transport (MCOT) in the mechanism level, and ending atthe Total Cost of Transport (TCOT).

Breaking the design problem in multiple stages has alsothe advantage of avoiding leg designs specifically optimizedfor a single task or objective, which are prone to fail in theface of deviations from the nominal conditions. For example,leg mechanisms such as Cornell Ranger [10], or Salto [11]have been highly successful in terms of the objectives theyhave been designed for (cost of transport and hopping height,respectively), but they are essentially unable to conduct anytask or maneuver other than that. We aim to avoid this issuethrough establishing design on a template that can explainvarious modes of locomotion in the biological systems, andthen design the optimized leg mechanism based on this tem-plate. On the other hand and in addition to these advantages,such designs can be utilized for better understanding of bio-logical locomotion, as suggested for instance in [12] or [13].

In addition to the design framework, another contributionof this paper is providing a mathematical tool for design of

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an energy-optimal mechanism based on a template (step 2).This tool serves as a link between template selection andactuator optimization. We develop step 2 and mathematicallyprove it independently from step 1 and step 3. Thereby, it canbe readily used with other templates and different actuatorselection approaches.

The sections of the paper are organized according to thethree-stage paradigm proposed above. In §II, we talk aboutreduced-order models, the model we use for walking, and itsbiological and mechanical implications. Next, in §III, basedon the selected template we mathematically show how theefficiency and performance can be optimized by appropriatemechanism design. In the last step and once the leg mech-anism is designed, in §IV, we define a strategy for optimalselection of actuators for best efficiency and performance.As a case study, in the next section, we useATRIAS’s walkingdata in order to examine the proposedmethods in the potentialimprovement of a real robotic leg through corrections in itsmechanism design and actuator selection. This completesour proposed framework, which will be summarized anddiscussed in the Conclusion section.

II. TEMPLATE-BASED LEG DESIGNAs the first step towards the optimal leg design, in thissection, we discuss the basics of template-based design andthe template upon which we build our proposed mechanism.In this way, we start by taking a look at the concept ofnatural/passive dynamics and its use in legged robots. Basedon this, we continue by showing how the natural dynamics ofa spring-mass system can be immensely beneficial for leggedlocomotion. We conclude the section by providing guidelinesabout mechanical design of a leg to be closely characterizedby the proposed template.

A. ENGINEERING PASSIVE-DYNAMICSPassive dynamics design means creating a mechanism withunactuated motions and behaviors (physics) that support andenable the desired operational motions of the robot. By usingwell-placed passive elements and connecting bodies, desiredbehaviors can occur naturally in the hardware. The sys-tem’s response in the absence of feedback will resemblethe response that would otherwise have been enforced bycontrol, reducing the necessary participation of active con-trol to high-level regulations. In a way, this can be thoughtof as encoding efficient and fast feedback control into thehardware [1]. Robot designs which rely solely on full actu-ation and rigid joints may result in passive dynamics whichconflict with the desired dynamics. In such systems, discrep-ancies need to be corrected by the control system, and inmany cases the two sets of dynamics cannot be reconcileddue to bandwidth, torque, and other limitations. Enforcingdynamics with actuation currently comes at high energeticcost when compared to passive elements, and for thesereasons we believe that the future of legged locomotionlies in the use of passive dynamics integrated with activecontrol.

As a leading example from the work that coined ‘‘passivedynamics’’, McGeer’s passive machines [14], [15] were ableto naturally excite walking patterns without any actuationor active control. Further evolution of this concept led tohigh-efficiency actuated robots such as Cornell Ranger [10].As a result of the fact that the main motion of the robotwas dictated by the passive dynamics, Ranger was able towalk nonstop for 40.5 miles over flat ground, with a TCOTof 0.28 [10], which is comparable to that of humans. Theparticular sets of dynamics based on which these mechanismswere designed led to highly efficient machines, but on thedown side, they lacked robustness and offered only marginalstability.

In a contrasting approach for designing legged machines,the robots are essentially fully-actuated mechanical systems.By virtue of this, robots such as Honda’s ASIMO [16],HRP [17], andHUBO [18], have been capable of demonstrat-ing impressive locomotion in well-controlled environments.Walking control in these robots is normally done throughregulation of the Zero Moment Point (ZMP) [19] of an LIPMat each instant. This leads to walking with flat feet andbent knees, and hence a ‘‘robotic-looking’’ gait. Naturally,canceling the dynamical effects for the purpose of achievingLIPM stability results in low energy economy which makesthe untethered application of these robots problematic. Forinstance, TCOT of ASIMO is estimated to be 3.5 which isby comparison 17.5 times more than that of humans [10].Hydraulically actuated robots have even higher TCOTs thanASIMO, with Boston Dynamics’ BigDog at an estimatedTCOT of 15 [9].

Passive dynamics design becomes increasingly impor-tant in unstructured environments, where the robot hasan incomplete or inaccurate map, making motion plan-ning and stabilization a highly challenging task. Unexpectedchanges require instantaneous response from the actuators,which is generally not possible due to rigid gearing andactuator inertia. Robots such as HRP-4C [20] and BostonDynamics machines [21] have tried to push the limits ofresponse and have indeed exhibited successful demonstra-tions. However, these achievements are essentially results ofnew high-bandwidth actuation systems [22] and novel fastonline optimization and control algorithms [23] that allowbetter enforcement of dynamics. As mentioned above, thesemethods are energetically costly, and only asymptoticallyapproach the stabilization capabilities that humans and ani-mals exhibit in dynamic environments despite their latenciesand much slower feedback rates [24].

In an effort to find dynamical behaviors which lead toefficient and robust locomotion, we can look to animals asthe current best example. There is certainly a common patternamong legged creatures, with features including articulatedlimbs, elastic tissues in series with actuators, and dynamicgaits. It is impossible to say whether these commonalities aresimply artifacts of an evolutionary process or if there is someunderlying reason for specific features. Given the impressiveagility of animals in unstructured and changing environments,

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however, there must be a tie between animal morphologyand a set of ‘‘rules’’ which ‘‘solve’’ the problem of dynamicbalance. Behaviors in this set are not obvious, especiallyconsidering themassive complexity of the biological tie to theset, but we can find clues by applying engineering reasoningand analysis. This is the reason that we use the concept of‘‘templates’’ [1] to trim away extraneous features and distillthe key behavior of the full animal in the absence of anymorphology or physiology. The goal is to find features whichbenefit efficiency and/or stability, and can be implementedwith passive elements.

Based on the above discussion, we establish our designon using passive dynamics of a template. The template thatwe believe is the most prominent for legged locomotion andarguably the future of this field is the spring-mass system,as it will be discussed in the next subsection.

B. SPRING-MASS AS A TEMPLATE FOR LOCOMOTIONDespite the immense diversity in the leg morphologies andthe motion of the animals, it has been long known that theSpring-Loaded Inverted Pendulum (SLIP) is a good templatefor capturing the essential characteristics of running [25].More recently, Geyer showed that walking can also bedescribed using the bipedal version of this template [26].Now, the Spring-Mass Model is considered as an essentialtemplate describing all modes of locomotion (running, walk-ing, grounded running, and jumping) [27].

Spring-mass template consists of a point mass, repre-senting the center of mass of the animal or robot, and anumber of massless springs extending toward the groundto represent compliant leg(s) (Fig. 2). By encapsulating acomplex biological system into a simple model, spring-masssystems explain several fundamental phenomena in a wayreplicable for legged robots. This template and its varia-tions such as spring-mass with torso (Asymmetric SLIP -ASLIP) [28], spring-mass with actuation (Energy-StabilizedSLIP - ES-SLIP) [29], and spring-mass with leg mass [30]provide several benefits for efficient, stable, and robust loco-motion of the robots. Some of these benefits include:

1) ENERGYRECYCLING: First and foremost, the highefficiency of mechanical springs is a great advantagefor the legged robots. Essentially, in all locomotion

FIGURE 2. The spring-mass model. ϕ and l are called leg angle and leglength, respectively.

modes, energy has to be cycled between kinetic andpotential forms. Therefore, it is reasonable to keep thecircular path (Fig. 3) free of inefficiencies as much aspossible. Ideally, and since the actuators are typicallyinefficient in the regeneration of power,1 it is desir-able that all mechanical energy be kept in mechanicalform once it is injected into the system. Using highlyefficient mechanical springs, the springy legs of thespring-mass template can be realized in robots to enablecycling of energy in all different modes of locomotion.Note that, again, the regeneration efficiency (or lackthereof) is one of the reasons against using a softwarespring (i.e., mimicking a spring through actuators) andmethods such as stiffness control [31] and impedancecontrol [32].

FIGURE 3. The continuous conversion of kinetic and potential energiesduring cyclic locomotion for a spring-mass system. The spring addsanother level of energy conversion between elastic and gravitationalpotential energies.

2) POWER AMPLIFICATION: The passive seriesspring can act as a power amplifier, allowing low-poweractuators to increase the energy stored in the springsover time and release it in a very short interval througha high-power burst. Conversely, high-power impulsesapplied to the leg (impacts, etc.) can be attenuatedand absorbed gradually, which can prevent damageto actuators and other components. Again, this ideais biologically-inspired, as through use of elastic ele-ments (tendons, etc.), the peak powers produced invarious activities of animals are higher than the abilityof their pertaining muscles [33]. In robotic locomotionthis idea becomes important in order for allowing the

1Despite recent efforts for improving the regeneration efficiency of elec-tric motors (see [9], for example), it is still considerably lower than mechan-ical springs.

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use of smaller, cheaper actuators, which is beneficial inreducing the cost and the weight (which in turn affectsthe performance) of robots [34], [35].

3) IMPULSE ATTENUATION: Walking and runningare, by their nature, hybrid dynamical phenomena;i.e., each of them has two separate operation phaseswith different sets of dynamics. The impact at legtouchdown and the corresponding jump in the veloc-ities make the hybrid system a ‘‘system with impulseeffect’’, which is typically problematic to control [36].Hybrid Zero Dynamics (HZD) introduced by Grizzleet al. [37] is an endeavor to solve the problem of statejumps through matching the zero dynamics manifoldsof the two phases. However, with springy legs and smallunsprung mass, the velocity jumps can be essentiallyeliminated, allowing smooth switching between thetwo phases and thus making the system simpler tocontrol.In addition to this, it is well-known that reducinginelastic impacts of large unsprung inertias reduces lossof (kinetic) energy at each touchdown. In the templatelevel, this property can be observed in the compari-son between McGeer’s passive walker, which needs adownward slope to compensate the energy loss of theimpact, versus the spring-mass system, which is com-pletely energy-conservative. Energetically, the exis-tence of the spring becomes evenmore prominent in thefull-order robot (compared to the template), whereinthe large reflected inertias of the actuators add evenmore to the kinetic energy lost at each impact andcause large force spikes that can damage the sensitivecomponents of the system.

4) INHERENT STABILITY: An interesting character-istic of the spring-mass template is its stability indifferent modes of operation. Seyfarth et al. [38] andGeyer et al. [39] investigated and showed the stabil-ity of spring-mass running by choosing an appropri-ate leg angle at touchdown. Later, they found thata simple swing leg retraction helps the spring-masssystem overcome larger disturbances [40].2 Likewise,Ernst et al. [42] suggested a feedforward control ontouchdown angle scheme to maintain the running speedon uneven ground. From a different point of view,Schmitt [43], based on [44] obtained a simple feedbacklaw for stabilization via leg angle correction. Later,Schmitt and Clark [45] proposed a robust clock-basedleg length feedforward control scheme for stability onuneven ground. More recently Vejdani et al. [46] foundasymptotically stable control schemes for stable walk-ing and for moving between different fixed points/gaitsin the state space.Qualitatively, series compliance allows changes in theexpected terrain to be handled by spreading the distur-

2As an example of implementation of ideas from template level to therobots, this idea helped better control of ATRIAS, as reported in [41].

bance out across a number of steps and giving the con-troller time to adjust, as it is observed in animals [47].The existence of these controllers and stability schemesin the template level can be remarkably helpful inthe control design for the robot. Again, this approach,as mentioned before, is in total agreement with thebiological observations and studies on locomotion.

5) GROUND IMPEDANCE CHANGE: Spring-in-series actuation allows force control [34], which witha very simple mechanism can yield the same resultsas variable compliance systems [48]. This allows thespring-mass robots to be able to walk and run in gravel,mud, and other variations in ground composition usingthe same control scheme, simply by regulating theGround Reaction Force (GRF) profile. The force con-trol scheme can also justify the biological observationsindicating the change in the leg stiffness in orderto maintain the total leg+surface stiffness constant[49]–[51].Based on the benefits discussed in the above, we estab-lish the spring-mass template as the foundation of ourdesign. In the next subsection, the essential consider-ations for robot design in order to preserve the maincharacteristics of spring-mass template in the real sys-tem will be discussed.

C. DESIGNING ROBOT LEGS BASED ONSPRING-MASS PHYSICSThe interesting traits of the spring-massmodel (as listed in theprevious subsection) motivated the design of several robotswith various levels of complexity over the years. In severalmonopods with literal translations of the spring-mass model’s‘‘pogo-stick’’ morphology [52]–[57], the prismatic actuatorsset the neutral length for the axial leg spring, and a rotaryhip actuator sets the leg angle. The original Raibert monopodhopper [58] was able to demonstrate spectacular dynamicalmaneuvers using a compressed-air spring and fixed-impulseinput combined with simple leg-angle control. The abilityfor this robot and the related Raibert active-balance robotsis impressive on its own, but the simplicity of the controlscheme makes them a standing example of elegant designeven today.

A number of articulated leg designs have also imple-mented spring-mass physics with varying degrees of fidelity.Papantoniou attempted to remedy the lack of efficiencyin the air-spring actuators of Raibert [59]. He argued thatpneumatics and hydraulics are excellent prototyping tools,but have much reduced efficiencies compared to electricmotors. To this end, they created an articulated leg designwhich approximated a straight line linkage, with mechan-ical springs in series with an electric motor. By includinga self-locking transmission, the leg spring would performnearly all of the necessary gait energy cycling. Other exam-ples of articulated spring-mass-based leg design can be foundin [60]–[68].

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In a similar attempt as [59], the compliant bipedMABEL [69], [70] decoupled leg-length and leg-angle actu-ation through cable differential systems. The essential flawsof this robot were its large unsprung mass in the leg whichresulted in large impacts (hence the name Thumper forits monopod version) and the substantial complexity of itsdifferential system. These lessons applied to the design ofATRIAS [71], wherein the leg mechanism was simplified tovery light carbon-fiber four-bar linkages. However, as it willbe discussed in the next section, this simplification was notwithout associated costs.

With the experience taken from the previous endeav-ors, we define four main features for design based on thespring-mass template:

1) ARTICULATED LEG: Although telescoping legsseem to be the most immediate descendants of thespring-mass template, several advantages motivatechoosing an articulated leg. Linear guides robustagainst impacts are typically heavy and inefficient.Moreover, having a knee helps the leg be suited forother purposes beside walking and running, for exam-ple sitting on a chair or clearing obstacles during walk-ing.

2) SPRING IN LEG LENGTH DIRECTION: Tomatch the dynamics of a spring-mass system, a springbetween the toe and the hip is necessary. The spring canbe a virtual spring, i.e., a spring in another place whichhas the same dynamical effect, or combination of twoor more springs (such as in ATRIAS [71]).

3) MINIMIZED UNSPRUNG MASS: As discussedbefore, a major part of the spring-mass characteristicscomes from the fact that the legs in this template aremassless. Therefore, in order to be able to take advan-tage of these characteristics in the full-order robot, it isdesirable to reduce the unsprung mass of the robots’legs as much as possible. For this purpose, it is usefulto place the actuators in the proximity of the hip andabove the springs to minimize the effect of their largemass and reflected inertia on the dynamics of the sys-tem. This has several advantages including providingthe ability to reach high accelerations in swing phase,as well as having less effect on the dynamics of thetorso, which facilitates template-based control design.

4) ELECTRIC ACTUATORS: Although the swiftresponse of hydraulic actuators is highly advantageousfor control purposes (part of success of Boston Dynam-ics’ robots definitely owes to this), their very lowefficiency makes their use for untethered walking lessappealing. Therefore and in order to compensate fortheir lower bandwidth, we suggest the use of hightorque-density motors (as in [9]) together with carefulselection of the motor-transmission characteristics forthe specific application (as it will be discussed in §IV).Having established these strategies, in the next sec-tions, we build a mathematical framework for optimaldesign of robot legs based on the spring-mass template.

III. MECHANISM DESIGN AND ACTUATOR PLACEMENTBASED ON A TEMPLATEOnce the template is selected, the next step is designinga mechanism that embodies the essence of the template’sdynamics and adds DoF and actuators to it, as required.Naturally, for each template there exist an infinite numberof mechanisms with the same number of added DoF andactuators. Dynamically, all tasks can be mapped to the actu-ators using the mechanism’s Jacobian matrix, regardless ofits form (so long as it is not singular). However, as we showin this section, the Jacobian is important for other objectives,and primarily efficiency. In what follows, through derivingconditions on the Jacobian matrix, we introduce some criteriato limit our choices of mechanisms embodying the template.As the first criterion, we obtain conditions for the placementof the actuators in order to maximize efficiency. We do thisthrough elimination of actuators working against each other,or formally, minimization of ‘‘geometric work’’, which as wewill show, is related to minimizing MCOT.

Once the geometric work condition is satisfied, anothercondition is required to fully define the final mechanism.We show how maximizing manipulability can serve as sucha tool and what the associated benefits are.

A. GEOMETRIC WORKOne can define geometric work as the power lost due to aninternal power loop among two or more actuators. In otherwords, for the existence of geometric work we must have:1) more than one actuator; and 2) at least one actuator doingpositive work and at least one doing negative work. Thisphenomena is schematically illustrated in Fig. 4.

The unnecessary cost of negative powers has been knownalmost since the advent of legged robots and leg design-ers have tried to eliminate or minimize some variationof it. One way was restricting the actuation to horizon-tal direction, as Waldron and Kinzel proposed [72] . Usingthe same concept, Hirose [73] developed his well-known‘‘gravitationally decoupled actuator’’ (GDA) based on aCartesian-coordinated pantograph. Designs such as ‘‘Adap-tive Suspension Vehicle’’ presented by Pugh et al. [6] andODEX by Russel [74] are other examples of leg mechanismsstemming from the same concept.

This idea was later extended to the concept of geometricwork. Grieco et al. [75] used a pantograph leg mechanismto avoid geometric work, while Kar et al. [76] and Chouand Song [77] tried to minimize this parameter by finding anoptimum force trajectory and path planning. Alexander [78]describes how geometric work affects linear-leg models andshows that it can be created or avoided with different combi-nations of leg designs and walking schemes.

The original definition of geometric work considers allnegative works as loss. But upon further investigation, thisproves to be an incomplete definition. This is especiallyapparent when being back-driven is part of the characteristicsof the task; i.e., when the total output power itself is negative,

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FIGURE 4. Schematic of geometric work when one motor is doing positive work (driving)and the other is doing negative work (braking). Some of the braking power is regeneratedand goes back to the driving motor, but most of it is lost.

for instance in the case of walking downhill. To solve thisproblem, Song and Lee [79] proposed a new definition forgeometric work and defined it as the sum of absolute valuesof the input energy minus absolute value of the output energy:

Wg = (p∑i=1

|Ei|)− |Em|

= (p∑i=1

|Ei|)− |p∑i=1

Ei| (1)

where Ei is the mechanical work of the ith actuator, and Emis the total output mechanical energy. With this definition,one can check that Wg ≥ 0; i.e., the minimum value of thegeometric work is zero. In a similar definition, in [80] the sumof the squares of the joint powers were used as a measure ofgeometric work.

The common basis for all these definitions is penalizing thenegative Eis, pertaining to the actuators doing negative work.In the case of legged robots, this concept is closely related toMCOT, for computation of which negative powers are set tozero [10]. Motivated by this, we redefine the geometric workas the sum of all positive works minus the output work:3

Wg = (p∑i=1

Ei)− |Em| (2)

3Note that although TCOT is a more popular measure for the efficiencyof locomotion, it depends on parameters such as gear ratio and motorcharacteristics (i.e. parameters pertaining to things other than the mechanismmorphology). Therefore, for the study and design of the mechanism -whichis the subject of the present section-, MCOT is preferred and thus the abovedefinition for geometric work is adopted as a useful tool and a starting pointfor this analysis. In the next section, the formulation will be extended toconsider TCOT.

where

Ei =

{Ei Ei ≥ 00 Ei < 0

(3)

Using this definition, and having (see [10]):

MCOT =

p∑i=1

Ei

mgx(4)

where m, g, and x are the robot mass, gravitational acceler-ation, and the forward distance traveled, respectively, thenfor a given task (i.e., given Em), minimization of geometricwork becomes mathematically equivalent to minimization ofMCOT. Moreover, this definition imposes a more realisticpenalty compared to the original Song and Lee definitionin (1), which essentially assumes a negative regenerationefficiency, and as a result, leads to an unnecessarily strongerpenalty on negative powers.

In the general case and for arbitrary force and velocity pro-files in the task space, it is not possible to design a mechanismthat always avoids geometric work. However, note that wehave founded our design on a template-based approach. Thismeans that despite the leg being capable of more generalmaneuvers, most of the time it is following the dynamics ofthe reduced-order model. Therefore, we limit our approach tomaximizing efficiency when the leg is following this mode ofbehavior. In what follows, we present and prove a methodol-ogy for the general case of template-based mechanism designand then show its implication for our specific design problem(i.e., the spring-mass template).

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B. TEMPLATE-BASED DESIGN APPROACH FORELIMINATION OF GEOMETRIC WORK: GENERAL CASEConsider the spring-mass template as shown in Fig. 2. Thegoal is to start from this template and obtain a mechanismwith more capabilities (actuators and/or more DoF), whichin ideal conditions can embody the same dynamics as thoseof the template. ATRIAS’s leg mechanism (Fig. 5) is a goodexample of this process, in which two actuators and oneDoF (in leg length direction) are added to the system. In thissection, we present a methodology for the placement of theactuators in order to avoid geometric work and thereby max-imize the efficiency.

FIGURE 5. Design of ATRIAS’s mechanism based on a spring-masstemplate.

Consider a general energy-conservative template withnt degrees of freedom. We assume that the template canhave energy-storing elements (i.e., springs), but no actua-tors. The template-based mechanism design problem consistsof adding additional degrees of freedom to the template,as well as adding actuators and finding their best placement.We define four steps to start from a template and design amechanism based on that:

1) Decide how many degrees of freedom are to beadded (m), and the kinematic constraints to be removedin order to add those degrees of freedom.

2) Decide how many actuators are to be added (p).Naturally, the number of the actuators should be at leastequal to the number of the released constraints in theprevious step (p ≥ m); otherwise, the extra degreesof freedom of the mechanism cannot be controlled tomatch the template’s dynamics.

3) Choose the manifold that the actuators span (hereafteractuated space/manifold). Analogous with the previousstep, the span of the released constraints should be asubmanifold of the actuated manifold.

4) Choose the placement of the actuators.Out of these steps, steps 1 to 3 are decisions made consider-

ing higher level aspects such as the nature of the mechanism,the tasks it is designed for, and the disturbances and theuncertainties that deviate the mechanism’s performance fromthe ideal behavior.4 Step 4 is the problem which is addressedin this subsection.

4We briefly address these steps in the next subsection, in the case ofmechanism design based on the spring-mass template.

As stated before, the goals we consider for actuator place-ment are:

1) Provide the ability of matching to the template’sdynamics when the actuators work ideally as the per-taining (released) constraints or degrees of freedomof the template. Naturally, in addition to matchingthe template, the actuators add other capacities to thesystem as well.

2) Avoid geometric work among the actuators.The following theorem, which is the main result of this

section, provides the conditions for these objectives.Theorem 1: Assume a mechanism has been designed

based on an energy-conservative template by releasing mconstraints and adding p ≥ mactuators, and it is to match thegeneralized forces and velocities of that template. Also, if ntrepresents the DoF of the template, assume that p ≤ nt + m,i.e., the mechanism is not overactuated. Then the geometricwork in the mechanism vanishes if and only if there are mactuators whose displacements do not have any projection onthe span of the (original) degrees of freedom of the template.

Proof: Let n = p − m. It is assumed that them-dimensional manifold of the released constraints is a sub-manifold of the actuated manifold (otherwise the mechanismcannot match the dynamics of the template). As a result,n (with 0 ≤ n ≤ nt ) can be considered as the dimensionof a vector space spanned by a set of vectors in the spaceof degrees of freedom of the template, whose union with thespan of the released degrees of freedom forms the actuatedspace. Based on this, we can write:

θ = Aq (5)

where θ ∈ Rp and q ∈ Rp are the coordinates of joint spaceand actuated space, respectively, and A is the correspondingJacobian. Note that actuated space and joint space have thesame dimension and thus A is a square matrix. Now, let uspartition q as:

q =[qrqb

](6)

where qr ∈ Rm and qb ∈ Rn respectively refer to thereleased constraints and degrees of freedom coming from thetemplate (to form the actuated space). Then, the Jacobian canbe correspondingly decomposed as:[

θ1θ2

]=

[A1 A2A3 A4

] [qrqb

](7)

with θ1 ∈ Rm, θ2 ∈ Rn, and Ais appropriately sized. SinceAp×p is to be designed to be full-rank, there will be a full-rankn× n submatrix in it. Without loss of generality, we partitionthe actuator displacements such that A4 is this submatrix.

Similarly, for mapping the generalized forces Q to theactuator torques/forces (τ ), one can write:[

τ1τ2

]=

[B1 B2B3 B4

] [QrQb

](8)

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where τ1 ∈ Rm and τ2 ∈ Rn. Note that B = A−T , orBTA = Ip, where I is the identity matrix. This is equivalentto a condition on A and B that5[

BT1 A1 + BT3 A3 BT1 A2 + B

T3 A4

BT2 A1 + BT4 A3 BT2 A2 + B

T4 A4

]= Ip (9)

To show the sufficiency of the theorem, we need to showthat if A2 = 0 (i.e., when the displacement ofm actuators rep-resented by θ1 depends only on qb, degrees of freedom of thetemplate), then there will be no geometric work in the system.Note that since the system is energy conservative, the sumof the actuator powers is zero, for which the elimination ofgeometric work is equivalent to having all actuator powers tobe zero. Now, if A2 = 0, the actuator velocities in the firstgroup depend only on the released velocities:

θ1 = A1qr (10)

In order to match the dynamics of the template, equivalentvelocities should be matched. Therefore, the displacementof the released constraints (which are fixed in the originaltemplate) should be set to zero, and thus their velocities willalso be zero, qr = 0. This condition results in θ1 = 0, whichin turn means that powers of all the actuators in the first groupare zero.

For the actuators in the second group we check the con-dition of matching the mechanism’s generalized forces tothose of the template. Since the template does not have anyactuator, there is no force applied to its degrees of freedom,i.e., Qb = 0, and thus:

τ2 = B3Qr (11)

Now, from the identity structure in (9) and the condition thatA2 = 0, the second term in the upper-right element of thematrix in (9) must also be zero:

BT3 A4 = 0m×n (12)

Because we assumed A4 is full-rank, this condition requiresthat B3 ≡ 0. Inserting this result into (11) then resultsin τ2 = 0. This shows that the powers of the second setof actuators are also zero, which completes the sufficiencyproof.

To prove the necessity of keeping m actuators out of thespan of original degrees of freedom, we start by multiplyingthe transpose of (11) into the lower half of (7) and incorpo-rating the condition that qr = 0, which gives an expressionfor power in the template degrees of freedom:

τT2 θ2 = QTr BT3 A4qb (13)

Because the total power is zero (the template is conservative),according to (2), in order to eliminate the geometric work, allpowers have to be zero. Therefore, τT2 θ2 = 0. Now, notingthat there is no restriction on Qr and qb, (13) requires that

BT3 A4 = 0m×n (14)

5The same result can be obtained using Banachiewicz’s inversion formula,but what we present here is a more concise path to the result.

which, again, results in B3 = 0. Substituting this result into(9) we get:

(BT1 )m×m(A1)m×m = Im (15)

which, because rank of the right hand side is m, implies thatrank(A1) = rank(B1) = m. Again from (9):

BT1 A2 = 0m×n (16)

which, because B1 has been shown to be full-rank, requiresthat A2 ≡ 0. This concludes the proof. �Remark: Theorem 1 considers under- and fully-actuated

cases. Note that the elimination of geometric work becomessimpler with an overactuated mechanism, as the redundancycan be leveraged to obtain more than a unique solution.For example, extra actuators can be simply set to have zerovelocity (in case of serial redundancy) or zero torque (incase of parallel redundancy) when matching the template,which leads to vanishing their powers. In [81], overactuationwas used in an optimization framework in order to minimizethe geometric work. Nevertheless, overactuation is usuallyaccompanied with increase in weight, cost, and often TCOT(because of added losses) and normally is not recommendedfor legged robots.

Theorem 1 essentially divides the actuators into two sets;one set applying force with no displacement when matchingthe template, and the other set with displacement but no force.In addition to avoiding geometric work, this method has theadvantage of decoupling the tasks and providing the option ofusing different types of actuators for different tasks. This canimprove both the TCOT and the control performance throughmore specific selection of the actuators.

C. MECHANISM DESIGN BASED ON THESPRING-MASS TEMPLATEHaving the results of Theorem 1 for the general case, in thissection we address the design of a mechanism based on thechosen template for leg design, i.e., the spring-mass system.

1) ELIMINATION OF GEOMETRIC WORK IN THE CASE OFTHE SPRING-MASS TEMPLATEBased on the methodology proposed in the previous subsec-tion, the first three steps for design of a mechanism based onthe spring-mass template are as follows:

1) Other than the spring, the template has one degree offreedom (leg angle, ϕ).6 Another degree of freedom isto be added in leg length direction, which will con-trol the neutral length of the spring in stance phase,as well as extending/flexing the leg during swing phase.Fig. 6(a) depicts this added degree of freedom. There-fore, the number of released constraints, m, is 1.

2) Two actuators are to be added (p = 2) in order for theleg to be multi-purpose and to be able to sweep the

6Note that we are referring to the degrees of freedom of the leg, not thewhole system.

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FIGURE 6. (a) Spring-mass model with an additional DoF (releasedconstraint) in leg length direction. (b) Spring-mass model with an addedactuator in leg length direction, which controls the rest length of thespring, lm.

whole plane through swinging and extending/flexing.Thus n = p− m = 1.

3) The actuated space is two-dimensional: the span of leglength and leg angle degrees of freedom.

For the fourth step, the results of Theorem 1 states that inorder to eliminate geometric work, we need θ1 = A1qr . In thiscase, since the only released constraint, qr , is in leg lengthdirection, one of the actuators is required to work strictly inthis direction. There is no condition on the other actuatorother than satisfying non-singularity of the Jacobian. Thusfor deciding on the placement of the second actuator we needanother metric.Note: In this particular case (spring-mass template),

we have: [θ1θ2

]=

[a11 a12a21 a22

] [lmϕ

](17)

where ϕ and lm are as in Fig. 6(b). Then if Ql and Qθrespectively represent the generalized forces in leg length andleg angle directions:[

QlQθ

]=

[a11 a21a12 a22

] [τ1τ2

](18)

Or:[τ1τ2

]=

1a11a22 − a12a21

[a22 −a21−a12 a11

] [QlQθ

](19)

From (17) and (19) one can compute the actuator powers:p1 =

a11 lm + a12ϕa11a22 − a12a21

(a22Ql − a21Qθ )

p2 =a21 lm + a22ϕa11a22 − a12a21

(−a12Ql + a11Qθ )

(20)

Noting that the ideal operation of the mechanism is to matchthe characteristics of spring-mass model, we have: lm = 0and Qθ = 0. Then the powers, p1 and p2 will be:

p1 =a12a22

a11a22 − a12a21Ql ϕ

p2 =−a12a22

a11a22 − a12a21Ql ϕ

(21)

Therefore, p1 = −p2, which is expected as the system isconservative. In order to avoid geometric work, we have tohave p1 = p2 = 0. This can be satisfied by a12 = 0 ora22 = 0. Either of these choices leads to one actuator workingstrictly in leg length direction, which is in agreement withTheorem 1.

2) EXAMPLESBased on the result of Theorem 1, let us investigate thegeometric work associated with some of the mechanismsdesigned based on spring-mass template. The first example isa pogo stick (Fig. 7(a), which is the most intuitive realizationof the spring-mass template. In this mechanism, a linearactuator works in leg length direction and a rotary actuatorin leg angle direction. This means that in the format of (7)and (17), the Jacobian matrix has the form:

A =[1 00 1

](22)

which satisfies the condition of Theorem 1 (A2 = 0). There-fore, there is no geometric work for this mechanism.

FIGURE 7. (a) A pogo stick mechanism with one linear actuator and onerotary actuator at the hip. (b) A conventional leg mechanism with anactuator at each joint.

Another interesting example is the popular mechanismof an articulated leg with two actuators, one at each joint(Fig. 7(b)). In this mechanism, leg length can be related tothe knee angle, ql , using:

l = 2ls sin(ql2) (23)

where ls is the length of thigh and shank, which, for the sakeof simplicity, are assumed to be equal here. The mapping of

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velocities for this mechanism can be obtained as:

[θ1θ2

]=

1

ls cos(ql2)

0

−1

2ls cos(ql2

)1

︸︷︷︸

A

[lmϕ

](24)

the form of which again indicates that the geometric work isavoided, as the first actuator (knee actuator) strictly works inthe leg length direction.

The ATRIAS leg mechanism (Fig. 5) is an interesting caseof a mechanism with nonzero geometric work. Intuitively,one can observe that during stance phase, one actuator appliespositiveworkwhile the other actuator performs negativework(Fig. 8). Analytically, with ls and ql as before, the Jacobianmatrix for this case can be obtained as:

A =

−

1

2ls cos(ql2

)1

12ls cos(

ql2 )

1

(25)

in which, both actuators work in combinations of leg lengthand leg angle directions. As a result, the geometric work isnonzero.

FIGURE 8. Geometric work in ATRIAS. Depending on the direction ofmotion, one of the motors (here: motor 2) has to apply negative power,i.e., torque and speed in opposite directions.

3) PLACEMENT OF THE SECOND ACTUATORAs stated before, Theorem 1 does not provide a condition forthe placement of the second set of the actuators (the ones notstrictly working in the manifold of the released constraints).An interesting example is the articulated leg with an actuatorat each joint (Fig. 7(b)), as investigated previously. As canbe seen from (24), in this mechanism, any placement forthe second actuator (corresponding to the second row of theJacobian) results in elimination of geometric work.

In order to use this redundancy as an extra design parameterto improve a different objective, we turn our attention to theconcept of manipulability. We use manipulability as a toolfor ensuring the maximization of the actuators’ capability tocomplement the natural dynamics of the template in unex-pected maneuvers. Yoshikawa [82] suggested the product ofthe singular values of the Jacobian as a measure for manipu-lability. This definition, although useful, is an absolute quan-tity and does not consider scaling factors among differentmechanisms and also among different degrees of freedom ofa specific mechanism. To solve this problem, Togai [83] pro-posed to use the condition number (maximum singular valuedivided by minimum singular value) of the Jacobian matrixas another measure of distance from singularity (degeneracyof the Jacobian). Note that condition number is a relativequantity and it is not affected by scalings. Furthermore, Togaishowed the interesting equivalence of condition number ofthe Jacobian to sensitivity, and as a result, error propagationfrom joint space to task space [83]. Inspired by these reasons,minimization of condition number for ql and ϕ is chosenas an additional factor for fully defining the placement ofthe second actuator.

For a 2 × 2 matrix, A, the condition number γ can becalculated as:

γ =

√√√√√√δ1 +

√δ21 − 4δ2

δ1 −

√δ21 − 4δ2

(26)

where δ1 = tr(ATA) and δ2 = |ATA|. Note that minimumcondition number in (26) is γ = 1, which takes placewhen δ21 = 4δ2. Now, assuming a12 = 0 (according to thegeometric work analysis), and substituting in (26), we obtain:

a211 + a221 + a

222 = ±2a11a22 (27)

whose sole solution is a21 = 0 and a11 = ±a22. In otherwords, a diagonal Jacobian (decoupled actuation; one actu-ator working in leg length direction and the second one inleg angle direction) results in elimination of geometric workfor the spring-mass template, as well as minimization ofthe condition number. From (22), the pogo stick is such amechanism. However, as mentioned in §II-C, it is not the bestpractical choice.

A possible articulated realization of this Jacobian form isdepicted in Fig. 9. The actuators drive the pulleys, with thelarge pulley moving in leg angle direction, and the small oneat the knee (fixed to the shank) moving in the leg lengthdirection. The pulley ratio is 2:1 and the lengths of thigh andshank are equal. For this mechanism:[

θ1θ2

]=

[1 00 1

] [qlϕ

]

=

1

ls cos(ql2)

0

0 1

[lmϕ

](28)

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FIGURE 9. An articulated leg mechanism decoupling leg length and legangle directions. For the sake of clarity, the actuators have not beenshown. The leg angle actuator works between mass (torso) and the largerpulley at the hip. The leg length actuator works between the thigh andthe smaller pulley (fixed to the shank). The pulley ratio is 2:1.

Note that actuator 1 of this mechanism is the same as in themechanism of Fig. 7(b), but here, actuator 2 drives the largerpulley instead of the thigh.

IV. ACTUATOR SELECTIONThe method proposed in §III essentially relies on avoidinggeometric work in the actuators for producing the requiredground reaction forces. Note that when the leg is not in con-tact with the ground, the ground reaction force is zero. As aresult, the swing phase does not affect the presented analysis.However, it is well-known that once inertias are considered,the energy losses due to accelerations and decelerations inthe swing phase are not negligible. In early leg design workssuch as [84], it was tried to avoid these losses using mech-anisms that keep the actuator speeds constant for a givenswing profile. This method, although potentially effective indecreasing the losses, because of limiting the system to followa specific trajectory is not suitable for a multi-purpose leggedrobot. As a result, these losses are usually minimized throughappropriate selection of the actuators and transmissions. Thisis in complete accordance with the framework we proposedfor leg design procedure (Fig. 1). Our analysis in the previoussection provided a basis for design of morphology of theleg mechanism and actuator placement. The final step inleg design is the selection of the motor and the associatedtransmission.

The classical approach for motor and transmission selec-tion has been picking the smallest motor that can providethe necessary joint power (dictated by the expected load)and choosing the transmission ratio based on matching thereflected inertia of the motor to that of the load as in [85].It can be shown that this transmission ratio maximizes theenergy efficiency under certain assumptions [86]. Works byChen and Tsai [87], [88] introduce another factor in actuatorselection, which is specifically important for robotic sys-tems, namely agility (‘‘acceleration capacity’’ in their terms).

Although several developments of these pioneering workshad been proposed [89]–[92], specific methods for leggedrobot applications started to appear in the literature onlyrecently [86], [93]. In what follows, we briefly discuss howthe approach proposed in [86] can fit within our philosophyof leg design and development. The interested reader can findfurther details in [86].

As discussed in §I, while MCOT is a measure of themechanical energy used in locomotion and is attributed tothe mechanism design (a la §III), TCOT is a measure of thetotal power drawn from the energy source including windinglosses of the electric motors. As such, an optimized selectionof the actuators can substantially decrease the energy drawnfrom the power source and thereby increase the untetheredwalking time. The importance of total power for actuatorselection is in accordance with the majority of research workson the topic of actuator selection, which have consideredpower minimization as their sole objective. However, follow-ing Chen and Tsai [87], [88], the agility factor can be just ascrucial for a highly dynamic system such as a legged robot.Inspired by this, we propose to formulate the problem as amulti-objective optimization scheme.

Unlike total power, which has a fairly clear definition,it is not straightforward to mathematically define agility.The agility measure of Chen and Tsai [87], [88], the accel-eration ability, considers merely actuator torque and inertiaas the determining factors. As proposed in [86], bandwidth,as a parameter which encapsulates more attributes of thedynamics of the system, can be a better metric for agility.Also, note that the rise time of mechanical systems (as anessential measure of the response speed) typically has a directrelationship with the inverse of bandwidth frequency.

With these premises, the multi-objective optimizationscheme can be formulated as:[

n∗

j∗

]= argmin

n,j

{[E−ωb

]}(29)

where E is the total consumed energy from the power source(containing all electrical and mechanical losses), ωb is thebandwidth, n is the transmission ratio, and j is the index ina given set of motors. Similar to (4), E is related to TCOTwith TCOT = E/(mgx).For a DC motor, the primary constraint associated with the

optimization problem of equation (29) is the linear charac-teristic of this type of electric motors (Fig. 10). This limitis usually neglected and instead is replaced by maximumpower ability of the motor [86]. However, for a low-voltagepower source (as it is inevitable for untethered legged robots),this constraint is dominant. Therefore, we define the firstconstraint as follows:

C1: For a given task, themotor operation point shouldnot exceed the linear limit of its torque-speeddiagram.

This constraint arises from the following set of inequalities:

|τ | ≤ −k2m|θm| +k2mktV (30)

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FIGURE 10. Linear characteristic of a DC motor (Allied MotionMF0150010, ATRIAS’s sagittal plane motors) with default winding. Theregion inside the diamond is the feasible region for the motor based onits voltage (48 V) and torque constant. Note that this does not includeheating, maximum torque, and maximum current limits.

where τ is motor torque, θm is motor speed, km is motorconstant, kt is torque constant, and V is the input voltage.Note that for a given motor, km is constant and thus from (30),maximum torque (at zero speed) andmaximum speed (at zerotorque) drop with decrease of voltage and/or increase oftorque constant. Also, note that satisfying the maximumpower condition, which is usually considered as the primarycondition for motor selection, is a special case (a single pointin the set) of C1.The second condition that needs to be satisfied is the max-

imum torque condition (regardless of speed). The maximumtorque should never reach the demagnetization torque, τdemag,which damages the motor. Also, if there is a restriction on theelectrical current supplied by the motor driver, imax , it shouldbe considered as well. Hence:

C2: τ < min {kt imax , τdemag}The next constraint is the continuous operation of the

motor, which is constrained by its thermal characteristics.Because the dissipated heat in the motor windings is pro-portionally related to the integral of τ 2 [86], the RMS(Root Mean Square) torque (or current) becomes the deter-mining factor here. That is:

C3: τrms ≤ τcwhere τc is the maximum continuous torque of the motor.

An additional option which has recently become availableby the motor manufacturing companies is customized wind-ings. The change in the motor winding does not affect motorconstant km and since the winding loss Ew is obtained from:

Ew =1k2m

∫τ 2dt, (31)

the thermal dissipation characteristics do not change. How-ever, using a different winding can be leveraged to modifytorque constant kt , and as a result, the range of operation

of the motor (quantified in constraints C1-C3). Consideringthis, the optimization problem of (29) can be reformulated inthe more general form of:n∗k∗t

j∗

= argminn,j

{[E−ωb

]}, subject to: C1-C3 (32)

which summarizes our approach to actuator selection.

V. CASE STUDYIn this section, we present a quantitative investigation of theproposed approach for leg mechanism design and actuatorselection. For this purpose, we use experimental data from awalking test of ATRIAS and show how MCOT and TCOTcan be reduced through appropriate changes in the mech-anism and the actuators. As presented in [94], a template-based controller has been designed for walking and runningof ATRIAS, which provides the ability of tracking any com-manded speed with good precision. Using the joint trajectorydata from an experiment (instead of a gait obtained from asimulation) has the advantage of realistic consideration of allfactors and not relying on ideal-case gaits that can be inher-ently unstable and/or infeasible.We pick motors with weightsapproximately equal to the original ATRIAS’s sagittal planemotors, in order not to change the inertial properties of themechanism. For the present case study, we have picked thejoint velocity and torque data from an interval of 30 secondsof an experiment for the steady state walking of ATRIASwiththe commanded velocity of 1.35 m/s. The torques and speedsof the motors were calculated using these data and parameterssuch as gear ratio, motor inertia, and efficiencies.7 The testset-up and the velocity tracking performance are depictedin Fig. 11, and the joint torques and velocities for one of thestrides are shown in Fig. 12.

A. OPTIMIZATION OF THE LEG MECHANISMUsing the formulation presented in §III, in this subsectionwe show that by preserving all the parameters and merelychanging the actuator placement (i.e., the Jacobian of themechanism), stance MCOT8 can be significantly reduced.We compute the geometric work for three different mecha-nisms:

1) ATRIAS mechanism;2) Mechanism 1: actuators at knee and leg angle direction,

as in Fig. 9;3) Mechanism 2: the articulated mechanism of actuators

at knee and hip joints, as in Fig. 7(b).Note that according to our analysis in §III, both second caseand third case eliminate geometric work in an ideal system

7Since the motors are of the same size and the changes in gear ratios arelimited, in the present study we assume that the efficiencies remain constantfor all cases. In a more accurate approach, one can, for example, penalizemechanical efficiency for the higher gear ratios.

8Since geometric work is only considered for the stance phase (externalforces are zero during swing phase), only the stance phase part of the gait isused for this analysis. Swing phase will be included in the next subsectionfor computation of the TCOT.

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FIGURE 11. (a) ATRIAS test set-up on a treadmill. The gantry is merely for safety reasons and does not interfere in the normal operation ofthe robot (as can be seen, the safety cable is slack). The speed of the robot is controlled remotely such that the robot maintains anapproximately fixed position on the treadmill. (b) Experimental data for ATRIAS’s velocity tracking test. The controller is able to maintainthe steady-state for a long duration, enough for cost of transport studies. The measured velocity is computed from joint velocities of thestance leg and kinematics of the system. The small discrepancy between the treadmill and the average measured speeds can be attributedto errors such as slipping of the stance foot.

(ideal actuators, no leg mass, etc.), but an analysis withtorques obtained from a set of experimental data can pointto the differences of these mechanisms in real conditions.

Computation of geometric work is performed as pro-posed in (2). With the position and torque data from the

aforementioned experiment, the stance MCOT for ATRIASis obtained as 0.553, while for Mechanism 1 and Mecha-nism 2 the stance MCOT is computed as 0.303 and 0.327,respectively. It is interesting to note the relatively significantimprovement of stance MCOT (45% and 41%, respectively)

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FIGURE 12. Joint torques, velocities, and powers of one leg during arepresentative stride in the experiment. The torques were obtained usingthe precise measurement of the springs’ deflections. Due to low mass ofthe leg compared to the motors’ reflected inertias, the joint torques (andnot the motor torques) are small during swing phase and thus wereneglected. Note the negative power of the second joint (p2 is negative)due to geometric work.

in Mechanism 1 and Mechanism 2 compared to the originalATRIAS mechanism. On the other hand, while these twoimproved mechanisms exhibit relatively close performances(which is expected by our geometric work analysis, as men-tioned before), still Mechanism 1 shows a slightly betterresult. In other words, although this mechanismwas preferredand proposed on the ground of better manipulability, it seemsthat the decoupled nature of the mechanism helps reducethe geometric work in non-ideal cases as well. A subset ofthe authors observed similar mechanism-based differences ingeometric work for non-ideal gaits in [80].

B. OPTIMIZATION OF THE ACTUATORS FORMINIMUM TCOTIn the next analysis, we adopt the optimization problemformulated in §IV to optimize the TCOT for the threeaforementioned mechanisms with three different motors.The first motor is ATRIAS’s base motor, Allied MotionMF0150010, and the two other motors are RoboDrive ILM115 × 25 and T-Motor U13, respectively. As mentionedbefore, these motors were chosen because their weights wereapproximately equal to ATRIAS’s base motor, and as a result,the dynamics of the robot would not be affected. See Table 1for a list of the parameters of these motors used for thecomputation of TCOT and its optimization.

The optimization problem was formulated similar to (32)and solved using MATLAB’s fmincon function. Note thatas the purpose of this case study is maximizing the efficiency

FIGURE 13. Optimization results for minimizing TCOT with three differentmotor and three different mechanisms. Mechanism 1 refers to Fig. 9 andMechanism 2 to Fig. 7(b). Base system refers to ATRIAS without anychange in its motor or transmission.

of the robot, energy (or equivalently, TCOT) is considered asthe sole objective. See [86] for an example of simultaneousoptimization of bandwidth and energy.

Fig. 13 depicts the optimized TCOTs for the nineconsidered cases (three motors and three mechanisms). Thecorresponding optimal values of gear ratios and torque con-stants for each permutation of mechanism-motor are reportedin Table 2. The TCOT of the base system (ATRIAS withits current mechanism, gear ratio, and motor winding) wascomputed as 1.24. It is slightly less than the measured valueof 1.35 (obtained from measuring voltage and current of thebattery set during the walking experiment), but is still a closeapproximation and indicates the acceptable accuracy of themodel. Some observations from the optimization results:

1) ILM 115x25 with Mechanism 1 represents the mostefficient case with TCOT = 0.61. This is equivalentto more than 50% improvement compared to the basecase.

2) The difference between TCOTs of Mechanism 1 andMechanism 2 for all three motors (16.3%, 14.8%,and 14.5%, respectively) is higher than the differencebetween MCOTs of the same mechanism obtained inthe previous subsection (7.9%). The role of the decou-pled nature of Mechanism 1 in reducing MCOT wasmentioned in the previous subsection. It seems thischaracteristic is even more effective in reducing TCOT.

3) The optimized MF0150010 for the ATRIAS mech-anism exhibits a very close TCOT compared to thebase (unoptimized) system (1.23 vs. 1.24). The opti-mized parameters of the two actuators of the leg for thiscase are {n∗, k∗t } = {57.3, 0.084} and {54.0, 0.078}.Indeed, these values are close to the base values of{50, 0.121}. In other words, although motor windingand transmission ratio are well-chosen for ATRIAS,

VOLUME 6, 2018 54383


TABLE 1. Parameters of the three motors used for optimization.

TABLE 2. Optimized gear ratios and torque constant ({n∗, k∗t }) for permutations of mechanisms and motors. Each cell corresponds to the two legactuators for one permutation. The order of the actuators in each cell is the same as in (25), (28), and (24).

still the geometric work caused by the mechanismdesign leads to a significant increase in TCOT. Forcomparison, Mechanism 1 with the same motors canimprove the TCOT by 26%.

4) The optimized torque constants are fairly low. This isbecause of the fact that for a constant voltage, decreas-ing kt increases the range of both speed and torque(constraintC1, (30)). Therefore, the optimization tendsto reduce this parameter, so long as constraint C2 isnot violated. The maximum current imax for ATRIAS’smotor drivers (Elmo Gold Drum) is 200 A.

VI. CONCLUSIONIn this paper, we addressed the problem of building up aframework for design of robotic legs and provided a pro-cedure for this purpose based on the proposed framework.The idea serving as the starting point for this investigationwas to look at legged locomotion as a phenomenon thatcan be described by a reduced-order model. Based on thisperspective, we divided the design problem into three levels,each level adding more complexity and more details to the legdesign. Among other advantages, this approach enabled us toutilize and incorporate bioinspired concepts into the designproblem. Specifically, in the first/highest level, we discussedhow using compliant elements is essential for achieving theextreme maneuvers that animals demonstrate as well as theremarkable efficiency that they exhibit. Based on this andother interesting traits of the spring-mass model, we used thistemplate as a basis for leg design.

An important contribution of this work was propos-ing a method for the intermediate step between templateselection and actuator design. For this purpose, we proveda general theorem for designing mechanisms embodyingan energy-conservative template without causing geometric

work. The generality of the theorem makes it independentfrom the first and third steps and suitable for using withdifferent templates and actuator selection methods. Further-more, the theorem is not limited to legged robots and canbe used for design of any mechanism stemming from anenergy-conservative template with the purpose of havingzero geometric work. We used this theorem together withmaximum manipulability condition to determine the Jaco-bian for the optimal leg mechanism designed based on thespring-mass model.

At the lowest level of our leg design framework,we addressed the problem of actuator selection. We for-mulated the actuator selection as an optimization problemwith all of the constraints relevant to the present applica-tion included. For instance, we considered and formulatedthe limited voltage supply (inherent to untethered leggedrobots), which was almost invariably neglected in the actuatorselection literature due to the assumption of availability of ahigh-voltage source for a general electromechanical system.

As a case study and using a set of experimental data fromATRIAS, we performed an analysis to investigate how theproposed methods would improve the walking efficiency ofthe robot. First, we showed that designing a leg mechanismbased on Theorem 1 would lead to a 45% decrease in MCOTcompared to the base system. Next, we compared permuta-tions of three leg mechanisms with three sets of optimizedactuators in terms of TCOT. Again, the proposed mechanism(Mechanism 1) had the best TCOT. The combination of thismechanism with the optimized actuator (transmission andwinding) resulted in a 50% improvement in TCOT comparedto the base ATRIAS system. This is equivalent to a twicelonger walking time with the same battery set.

Although we primarily investigated the design of a bipedallegged robot (using ATRIAS as a basis), the spring-mass

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template can be used for robots with different number oflegs, including monopods, quadrupeds, and hexapods. Fur-thermore, bothmechanism design and actuator selection wereformulated as general problems and not specifically for thespring-mass template. This approach facilitates leg designbased on other and more complex reduced-order models.For example, as suggested in [95], a rotational spring athip can potentially benefit efficiency and control. An inter-esting extension of the present work can be studying thisreduced-order model (as well as others) and the optimal legdesigns based on that template.

Note that engineering design of the leg is outside the scopeof this paper. Realization of the guidelines, paradigms, andmechanisms discussed and developed in this paper as a finalengineering product is by itself an interesting area with itsown challenges. Such problems include material selection,packaging of the actuators, reduction of the legmass, optimiz-ing the electronic systems, etc. The interested reader can referto works such as [96] and [97] for instance, as good examplesof the state of the art in this field.

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SIAVASH REZAZADEH received the B.Sc. andM.Sc. degrees from the Sharif University of Tech-nology and the Ph.D. degree from the Universityof Alberta. He was with the Dynamic RoboticsLaboratory, Oregon State University, where hewas involved in control of ATRIAS, a bipedalrobot, for DARPA Robotics Challenge. He is cur-rently a Research Scientist with the LocomotorControl Systems Laboratory, The University ofTexas at Dallas, Richardson, TX, USA, where he

is involved in design and control of prosthetic legs. His research interestsinclude using fundamental concepts of mechanics for design and control ofnovel robots and mechanisms.

ANDY ABATE received the B.Sc. and Ph.D.degrees from Oregon State University. He wasa member of the Dynamic Robotics Laboratory,Oregon State University, where he was involved inthe ATRIAS and Cassie biped projects.

ROSS L. HATTON received the M.S. and Ph.D.degrees in mechanical engineering from CarnegieMellon University and the S.B. in mechanicalengineering from the Massachusetts Institute ofTechnology. He is currently an Associate Profes-sor of robotics and mechanical engineering withOregon State University, where he directs the Lab-oratory for Robotics and Applied Mechanics. Hisgroup also works with local industry to transfermodern developments in robotics from the lab to

the factory. His research focuses on understanding the fundamental mechan-ics of locomotion and sensory perception, making advances in mathematicaltheory accessible to an engineering audience, and finding abstractions thatfacilitate human control of unconventional locomotors. He was a recipient ofthe NSF CAREER Award.

JONATHAN W. HURST received the B.S. degreein mechanical engineering and the M.S. andPh.D. degrees in robotics from Carnegie MellonUniversity. He is currently an Associate Profes-sor of robotics and the College of EngineeringDean’s Professor with the School of Mechani-cal, Industrial, and Manufacturing Engineering,Oregon State University, where he is also theChief Technology Officer and the Co-Founder ofAgility Robotics. His university research focuses

on understanding the fundamental science and engineering best practicesfor legged locomotion. Investigations range from numerical studies andanalysis of animal data to simulation studies of theoretical models, designing,constructing, and experimenting with legged robots for walking and running.Agility Robotics is taking this research to toward commercial uses for roboticlegged mobility, working toward a day when robots can go where people go,generate greater productivity across the economy, and improve quality of lifefor all.

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Robot Leg Design: A Constructive Framework