—We present electric energetic consequences for me- chanical design trade-offs in lower-extremity powered prostheses. There are four main hardware components commonly imple- mented in these devices that can be tuned to achieve desired performance: motor, reduction ratio N, series spring stiffness K s, and parallel spring stiffness K p. The allowed joint range of mo- tion is a ﬁfth parameter that can also drastically change energy consumption. We apply a kinematically clamped analysis to the system equations to map the electric cost of transport (COT) for knee and ankle level-ground walking, in addition to ankle stair ascent and descent. We also utilize an optimization procedure to identify minimum energy hardware conﬁgurations. The energy map provides insight into consequences of variance from optimal parameters. Our results support the contribution of the series elastic element for improved power output. Parallel stiffness can provide 8% improvements in walking with minimal negative effect with varied terrain, and a varying ankle transmission ratio can similarly improve COT by 8% from level-ground to stair ascent. Limited dorsiﬂexion can further improve COT by 30%. These observations can provide the designer clarity to how design decisions modulate hardware performance. Index Terms —robotics, prostheses, actuator, ankle, knee. I. I NTRODUCTION THE motivation to build powered ankle prostheses is to generate the roughly 0.2 J/kg net positive work done by the ankle during walking that cannot be provided by a traditional ankle-foot prosthesis [1], [2]. Providing this energy can reduce the energetic cost of walking [3]–[5], improve qualitative feel and ground control, reducing falls, and nor- malizing gait to reduce social stigma of pathological gaits as well as early onset of co-morbidities [6]. Though the knee operates primarily as an energy absorber during walking, co- morbidities and pathologies, similar to the ankle, can poten- tially be avoided with the dynamic control of powered devices throughout swing and stance phases. Mimicking nature with synthetic hardware is a capability that our ﬁeld is closely approaching. The mechanical de- sign trade-offs pose challenges that differ between academic and commercial applications. The commercial-off-the-shelf powered ankle and knee systems are not able to provide biologically accurate kinetics with kinematics, nor do they enable access to the underlying control systems [7], [8]. To Manuscript received Month, Year. Financial support: this work was supported in part by the MIT Media Lab Consortia, the U.S. Army Medical Research and Materiel Command (W81XWH-14-C-0111) and the Defense Advanced Research Projects Agency (W911NF-17-2-0043). 1MIT Center for Extreme Bionics and MIT Media Lab - Biomechatronics Group, Massachusetts Institute of Technology, Cambridge, MA 02139 USA. e-mail: mcarney@mit.edu.improve upon the commercial options, a growing handful of academic research groups around the world have been building individualized platforms for study, all with their own trade- offs. The ﬁrst powered ankle, published by Au, Weber, Herr, used a series elastic actuator (SEA) [9] conﬁgured with a parallel spring to improve torque bandwidth in the transition from controlled dorsiﬂexion to powered-push-off. Au et al. showed that their actuator’s ability to contribute energy during powered push-off improved the metabolic cost of walking [10]. Since then numerous actuators have been designed following a similar actuator topology [11]–[15]. Many of these designs have focused around ﬂat, level-ground walking, where primary kinetic behavior occurs as a large power maneuver during powered plantar ﬂexion. The series elastic actuator (SEA) [9] has been the core actuator technology for the design of powered prostheses due to its energetically favorable features of a contractile element in series with an elastic element. This conﬁguration of spring is a biomimetic representation of the biological muscle and tendon unit [16]. The SEA has a number of beneﬁts that include: reduced impact loading on drivetrain, force control from position measurement and control, decoupling of driv- etrain impedance from output impedance, and the ability to deliver more power than a motor alone when operated as slingshot [17]. The parallel element has been used to improve control bandwidth and energetics in level-ground walking [3], [12], [18]. However, the parallel element also increases system complexity and mass. To determine the efﬁcacy of the parallel spring for our application we analyzed its energetic contributions during walking and extended these effects to uneven terrain. Reductions in metabolic Cost of Transport (COT) for peo- ple with lower-extremity limb differences wearing powered prostheses compared to those with conventional passive de- vices has been accomplished [3]. However, differing results of metabolic energy beneﬁts from such devices have also been reported [5], [19]–[21]. Even in results that have shown metabolic improvements, the energy cost of walking still remains greater than that of un-affected persons, particularly at common self-selected walking speeds and faster [4], [5], [22]; the ﬁeld has struggled to reduce the metabolic cost of walking with powered prostheses, tethered or not. These limitations in clinical studies may be due to differ- ences between machine and biological kinetic and kinematic capabilities, along with the distal mass of such devices. Many clinical energetic studies have used the commercial product BiOM from BionX which has limited range-of-motion (ROM) 2 Fig. 1. The TF8 Actuator is designed to operate as either a knee or ankle actuator – shown here conﬁgured as an ankle prosthesis. Borrowed with permission from [24]. from zero dorsiﬂexion to twenty-ﬁve degrees plantar ﬂexion and a mass of 2.3kg [7]. Though well designed for energy efﬁcient level-ground walking, these systems quickly hit kine- matic limits at higher walking velocities and varied terrain, where joint angle extension increase [23]. In this study we aim to elucidate how the common trade- offs in the mechanical design of lower-extremity powered prostheses affect overall system performance. We provide our process for evaluating the design space and searching for minimum energy combinations of motors, reduction-ratios, series and parallel stiffness, and joint range of motion limits. We then apply these tools to evaluate ankle prostheses walking on stairs in addition to level-ground, as well as knee actuators during level-ground walking. Application of these results in built hardware, Fig. 1 are presented in an accompanying mechatronic design paper [24], separated for clarity. II. M ETHODS Understanding the energetic consequences of design vari- ables leads to higher performance hardware and improved de- sign for application speciﬁcity. For this application we initiated the design assuming series elasticity would be included as a mechanical energy storage mechanism, as has been done by many researchers of ambulating robotics. Following a process similar to [25]–[27] we evaluated the performance of a SEA kinematically clamped to mean biological gait data while searching for drivetrain component speciﬁcations for a series spring, motor, reduction ratio and in some cases unidirectional parallel springs that satisfy the search objective and system constraints. We also performed wider design space evaluation of the energetic consequences each of these hardware compo- nents to help understand the consequences of design decisions on stairs in addition to level-ground walking. Fig. 2. Power spectral analysis of knee torque contributions up to 70 % of the joint power during a gait cycle of a 90 kg person walking at 2 m/s. Knee torque frequency contributions amount to 73 Nm at 6 Hz with a maximum torque of 118 Nm. A. Design Speciﬁcation To determine the design speciﬁcation for range of motion, torque, power, and system bandwidth, we normalized and scaled by body mass mean gait data from a total of one thousand unique gait cycles of walking data from nine able- bodied subjects collated from [28]. In addition to walking, mean stair ascent and descent ankle trajectories from [29] are used to evaluate energetics of stairs. We set a performance target and evaluated component contributions based on a 90 kg user walking at a near jogging pace of 2.0 m/s. The widely accepted measure of an actuator’s ability to achieve a desired performance is the torque bandwidth: a measure of the ability of an actuator to assert a speciﬁed torque at a speciﬁed velocity. Much of the literature states the ankle bandwidth design requirement is 2-4 Hz at 50-140 Nm torque for human level-ground walking. This speciﬁcation originates with [10], where they deﬁned bandwidth as a frequency range within which 70 % (3 dB range) of the system energy is contained when evaluated with Parseval’s Theorem. To evaluate torque bandwidth for more complex trajectories other than the ankle such as the knee, or other arbitrary joint trajectories we used the [10] method and expanded it to determine a generalized bandwidth magnitude requirement. The sum of peaks over 2.5 % of the maximum absolute torque spectral density were then summed and the maximum frequency of these torque contributions were taken as the frequency bandwidth. This method was veriﬁed against the [10] method applied to a 75 kg 1.25 m/s walking able-bodied subject. Shown in Fig. 2is collected from the [28] knee torque trajectories scaled to a 90 kg user walking at 2 m/s. The result is a desired frequency bandwidth of 73 Nm at 6 Hz. The speciﬁcity of this method is somewhat misleading due to its frequency domain approach to analysis of an entire gait cycle, rather than true time-domain features. Nonetheless, this method provides an effective estimation point. 3 FsprFs τm Jlmm,Jm br θm xmθlτl rs Δspring Kpθp0Ks Fig. 3. Schematic of the series elastic actuator and parallel elastic element used for the design process (rack-pinion model adapted from [31]). Our coordinate frame assumes ankle angle and torque do positive work on the environment – plantarﬂexion is positive; for knee ﬂexion angle and torque is also positive. B. Modeling and Optimization To model the mechanical and electrical systems dynamics we used a kinematically clamped analysis method where the actuator output explicitly tracks the prescribed joint load torque, angle and velocity trajectories [25], [27], [30]. Lin- earized kinematic and electric dynamic models clamped to these vectors output: motor, power electronics and spring behavior. These models do not include non-linear viscous friction terms but the search method does include saturation constraints on motor torque, velocity, current, and voltage. The result is a model that generally trends correctly but is operating in an open-loop dynamic condition with minimal damping, thereby potentially exhibiting large velocity response at impact conditions e.g. parallel spring engagement or ground contact. Mean torque, velocity, and angle trajectories aggregated from [28] were mass normalized, scaled, and fed into the dynamic equations for a series elastic actuator. Kinematic analysis from free body and mass acceleration diagrams of the joint components give the following equations that reference Fig. 3: l= s+ pJll (1) where, s=rsFs=rsKs(m Nrsl) (2) p=( Kp(lp0); l<p0 0;  l>=p0(3) and where, joint load is l, the unidirectional parallel spring contribution is p, the parallel spring rate is Kp, the parallel spring engagement point is p0, the series elastic actuator torque contribution is s, the load inertia is Jland load acceleration is l. In the case of the knee Jlassumes a point mass of an equivalent leg segment [32] located one third of a leg distance away from the knee joint. Positive joint angle and torque is in the plantarﬂexion direction, based on a reference that positive work is done to the environment. The load inertia is not necessarily known as it changes through stance and swing phases; for this analysis inertia was assumed operatingin free-space and used equivalent leg segment mass as deﬁned in [24] for the ankle or knee analyses depending on the mass of the next distal link. Here ris the moment arm acted on by the linear actuator force Fsat the joint axis. This moment armrswas initially held constant, but later as the actuator topology evolved to include a linkage geometry, the moment arm became a function rs(l). The motor position is m, and the drivetrain reduction ratio is N=Nsrs, where for the case of the ballscrew Ns=2 lead. Plugging sand pinto equation (1), and rearranging, the force in the series spring actuator is Fs: Fs=1 rs[ lKp(lp0) +Jll] (4) Passing this force through the transmission results in an effective torque that the motor sees: m= (Jm+Jtr)m+bm_m+Fs Ns(5) bm=KtInl !nl+br (6) where,Jmis motor rotor inertia and Jtris transmission inertia. Along with the motor and transmission inertias, the damping termbmgiven by [33] represents viscous friction in the motor, whereKtis motor constant, Inlis no-load current, !nlis no- load speed, and bris viscous damping in the drivetrain and rolling elements estimated based on [3]. Rearranging (2) to solve for motor angle mgives: m=N(Fs Ks+rsl) (7) In accounting for drivetrain inefﬁciencies motor torque is adjusted by an estimated transmission efﬁciency that biases motor torque by the bidirectional power ﬂow [33]. Four quadrant motor control is assumed and the power electronic efﬁciency is included in the piecewise continuous power efﬁciency term, : Pmech = m_m (8) m=( m; Pmech<0 m1 ; Pmech>0(9) Torque and angle at the motor are then fed into the linearized electromagnetic model of the motor to determine required current and voltage in the motor windings, shown below. The electrical power ﬂow through the motor is integrated over a gait cycle to determine the electric energy consumed by the motor to produce the speciﬁed torque and velocity trajectory: im= m Kt(10) vm=imRm+Kt_m+di dtL (11) Pm=imvm (12) where,im,vm,Pmare motor current, voltage, and power respectively. Rmis motor winding resistance, and Lis winding inductance. To identify the necessary operating parameters for powered prosthesis joint actuators, we used an optimization process 4 similar to [25], [26]. The search objective was to minimize the electrical COT per gait cycle, subject to physical constraints such as motor and power electronics capabilities. To identify ideal component parameters we applied a gradient descent optimization procedure with non-linear constraints imposed by the motor and power electronics: COT =R Pm(t)dt 1 2mgx(13) min COT (14) s.t.j mj allow0 (15) j!nj!ndes0 (16) where,!nis lumped-mass natural frequency of the motor and drivetrain [9], [31]. For motors with limited manufacturer data availability the following relations are used to estimate motor capabilities: allow = o !oj_mj+ o (17) where, oand!oare stall torque and no-load speed, or the maximum values given by the manufacturer. Finally, we aggregated the results and sorted them by minimum electric energy consumption across all the design parameters. C. Static Evaluation An initial guess at potential motor candidates and parallel spring stiffness can be estimated by inspection of static pa- rameters. A ﬁrst estimate of a parallel spring stiffness comes from a static inspection of 1 and 3. In a static condition we can neglect the inertial contribution. With the goal of reducing load on the motor, the torque contribution sis negated, and we set pto equal l, such that: Kp=min( l) min(lpo)(18) . To compare motor capabilities we need to evaluate the motor torque generating efﬁciency in relation to the effect of rotor inertia and the expected reduction ratio. The motor constant : Km=Ktp R(19) with units [Nmp W], can be used to compare motor torque gener- ation efﬁciency, where, Ris motor winding resistance, and Kt is torque constant with units [Nm=A ]. Though, because this measure misses the effects of rotor inertia, Sensinger deﬁned theSpeed Rate as a benchmark to use in evaluating motor dynamic performance by normalizing the motor constant with rotor inertia [34]. However, this term does not consider the effect of drivetrain on reﬂected inertia, making it difﬁcult to compare motor/drive-train architectures for a given applica- tion. To improve comparison across motors we normalize speed rate by the application-speciﬁc reﬂected inertia, expanding thespeed rate into a relation we deﬁne as the Reﬂected Speed Rate (RSR): RSR =Km JmN2max(20) where, units are [Nmp WKgm2],Nmax =max (j lj) ois the maximum gear reduction required to achieve the maximum target output torque l, and ois still motor stall torque. D. Energetic Studies The series spring in the SEA improves the power and energetic response of the motor [9], [35]. To understand how the series spring affects the actuator response, we took the nominal spring stiffness output from the optimization and compared it against a spring 50% stiffer and an effectively rigid spring with three orders of magnitude greater stiffness. Sweeping across transmission ratio N, series spring stiffness Ks, parallel spring stiffness Kp, and parallel spring engage- ment point Xp0and evaluating motor voltage and current response we generated COT surface and contour maps for both knee and ankle target trajectories, similar to the method in [3]. The graphical mapping of parameters builds understanding of the relationships between actuator components. We utilized the RSR (20) to identify candidate motors for these studies, ultimately settling on the U10 Plus Kv100 from T-motor [36] for our analyses. We built a powered ankle prosthesis, described in [24] as TF8, and tested it with a human subject to validate our simulation models. Human subject testing followed a protocol approved by an independent review board, the MIT Committee On Use of Human Experimental Subjects. A male, 75 kg, human subject was asked to walk on an instrumented treadmill at a set speed of 1.5 m/s wearing our TF8 ankle actuator. We measured joint angle, joint torque, motor voltage and motor current. We calculated the cumulative power consumption across a series of gait cycles to evaluate system energetics. A total of 28 gait strides were collected, their mean and standard deviation were then compared to our simulation data. The conﬁguration tested included a 378 kN/m series spring, and no parallel spring. III. R ESULTS A. Series Stiffness Series elasticity can improve power delivery at the output by storing and releasing energy in cyclical maneuvers. In Fig. 4 (a) we plot the motor torque/speed (top) and power (bottom) constraints. The solid red line is the physical limitation of the motor, below this line is technically achievable though not including thermal behavior considerations. The dotted lines of increasing density represent motor trajectories when three springs of increasing stiffness are each coupled in series with the actuator: Ks= 271kNm m;344kNm m, and 271MNm m. In Fig. 4(b) joint kinetics are shown in solid red, and again increasing stiffness series springs are represented as the increasing dot density plots. The right axis and red lines are the mechanical kinetics of the joint. The top ﬁgure showing travel the screw, followed below by the mechanical output 5 (a) (b) Fig. 4. Effects of series spring stiffness on motor performance and ankle gait energetics for 90kg user walking at 2.0 m/s. (a) The linearized model of motor capacity is shown against the performance of the actuator with varying series elasticity. Increasing stiffness begins to saturate motor capabilities. (b) Spring deﬂection affect on both electrical and mechanical energetics. power of the joint, and the cumulative mechanical energy of the joint and series springs. The left blue axes and associated dotted lines are from top to bottom: the spring displacement of the actuator, electric power at the motor, and cumulative electric energy at the motor (including dissipative losses). Three nominal spring stiffnesses are shown, a light, heavy and effectively rigid structure. The optimal spring according to Fig. 4(a) reduces motor power and distance traveled re- quirements by 50% and overall energy consumption by about 30% compared to an actuator without series stiffness. Preliminary data from level-ground walking experiments with the TF8 actuator show alignment with estimates. Fig. 5 is cumulative electric energy measured at the motor leads compared to simulated data for an equivalent mass subject. Simulated data expected 35 J per gait cycle, while measured mean data resulted in 28 J with a standard deviation of 4.9 J. We expect this variation between experiment and simulation is due to the difference in utilizing a ﬁnite-state machine controller, rather than true biological waveform. Additionally, this being preliminary data, we expect further tuning of the low-level control algorithms to improve torque tracking performance. B. Parallel Stiffness Contour and surface maps of reduction ratio and series stiffness affects on system energetics for stair descent, level- ground and stair ascent of a biological ankle joint are shown in Figs. 6. Sub-ﬁgures (a-c) do not include a parallel spring, while (d-f) include a nominal parallel spring of KP= 3:59Nm radkg. This parallel spring value came out of a parameter search and was found to be more energetically favorable than larger value from (18). The larger Kpvalues are found to be effective at level-ground walking, but counter-productive when the motor % Gait CycleFig. 5. Preliminary validation of the model with our ankle actuator described in [24] with a human test subject. User mass is 75 kg, walking 1.5 m/s, andKs= 378kNm m. Solid line is mean, with purple shading showing one standard deviation from 28 strides. Estimated data is shown as dotted line. must overcome the force of the spring for changing stride lengths and terrains. The parallel spring does provide up to 8% improvement in energy expenditure. Optimal transmission ratio is about 60:1 but shifts towards higher ratios for stair ascent. C. Knee Level-ground walking of the knee joint in Fig. 7(a) shows a near zero electric energy cost and optimal reduction ratio of 35:1. One might expect greater energy regeneration in the knee with (b) showing negative mechanical energy at the joint being as much as -0.15J kg. The kinematically clamped analysis forces the knee to fully accelerate and decelerate itself, the 6 00.10.2 00.10.2 (a) 00.10.2 00.10.2 (b) 00.10.2 00.10.2 (c) 00.10.2 00.10.2 (d) 00.10.2 00.10.2 (e) 00.10.2 00.10.2 (f) Fig. 6. Contour plots of the COT for one gait cycle are affected by series spring and parallel spring stiffness and overall transmission ratio. These analyses are based on the ﬁnal choice of motor: TMotor U10 kV100 Plus. (a-c) show stair descent, level-ground walking, and stair ascent, respectively with no parallel spring. (d-f) show stair ascent, level-ground, and stair ascent, respectively with a parallel spring Kp= 3:59Nm rad kg, where we normalize by subject mass. large spikes in the power curve in (b), however, in reality this motion is coupled with hip swing in addition to the knee. D. Limited Range of Motion The single largest improvement in actuator energy consump- tion is the case when dorsiﬂexion is restricted, as is the case with the BiOM. A 35% energy reduction is realizable when dorsiﬂexion is limited to 1in energy conﬁguration map Fig. 8. IV. D ISCUSSION The largest affects on system energetics are reducing range of motion, on-the-ﬂy adjustment of reduction ratio for varied tasks and the inclusion of a moderately stiff parallel spring. Spring stiffness changes are also beneﬁcial but are not as task dependent as we initially expected. Adjusting series spring stiffness between larger and smaller mass walking subjects can produce an improved energetic outcome, but across moderate changes in mass the energetic improvement is not substantial: e.g. a 90kg person walking on a spring speciﬁed for a 75kguser results in a 7 % increase in COT. Limiting range of motion has the single largest reduction in energy cost, but also affects the natural kinematics of a user. The trade-off in exchanging walking time for user feel is a consideration that should be considered based on desired user outcomes. A varying reduction ratio could beneﬁt task changes from level-ground walking to stair ascent. In level-ground walking an optimal reduction is about 60:1. In stair ascent the optimal reduction shifts to about 85:1 for the given motor. This shift is even more pronounced when a parallel spring is also included where the shift is from 45:1 to about 80:1 where there could be about 10% energy improvement between level walking and stair ascent. Screw driven actuators naturally vary their reduction ratio as the moment arm passes along the swept arc of the rotary output joint. Designs should be careful to recognize that decreasing screw ratios quickly ramp into steep energetic consequences. This should be considered in designs, or the linkage geometry itself should also be optimized according to gait trajectories. The unidirectional parallel spring at the ankle can provide about 8% electric COT improvement across level-ground and 7 00.10.2 00.10.2 (a) (b) Fig. 7. (a) Energetic map for a knee actuator, and (b) the cumalative energy along a gait cycle for level-ground walking. stair ambulation. Selection of the parallel spring stiffness is crucial. Too stiff of a spring is actually more energetically costly when the motor must overcome the parallel spring as in large range of motion maneuvers or slower walking speeds. Too soft of a spring does not cost energy, but also does not provide substantial energetic improvement considering the increased complexity and mass. The knee does not beneﬁt from a parallel spring. Fig. 7(b) helps show that the ﬁrst order approximation of the knee is a damper: net energy is negative for each gait cy- cle. In mechatronics there is a chance this energy can be redistributed through the power bus, however, due to power transfer inefﬁciencies and the energy required to accelerate and decelerate the knee shank this behavior is not observed in 00.10.2 00.10.2Fig. 8. When an actuator is limited to full plantarﬂexion but only one degree of dorsiﬂexion as is the BiOM, there is 30% energy savings for level-ground walking compared to biological range of motion. this analysis. The series spring contributes to ﬂattening of the motor energy for the ﬁrst half of the gait cycle, but electric energy is consumed during the acceleration and deceleration of the knee in swing. The parallel spring contributes a load reduction during a negative energy portion of a gait cycle, quickly followed by a large positive energy gait phase, such as in the ankle. The knee, though, does not express this type of behavior during level-ground walking leaving the parallel spring a hindrance to motion. Potentially a clutched parallel spring could be utilized in stair ascent or descent, this could be evaluated in future analyses. A set of maps that was not included in this paper are the infeasible operating zones for these energetic maps. Infea- sibility is deﬁned by the constraints deﬁned in (15,16) and Fig. 4 and can be used to disallow conﬁgurations that can not be achieved. These results severely limit the operating regimes and clearly deﬁne operating speciﬁcations, but they limit clarity in visualizing the energetic maps, so they are left for the reader to apply their own design process. V. C ONCLUSIONS These simulations help to understand the design space and to specify hardware conﬁgurations that can achieve speciﬁc task targets. The kinematic clamped analysis is a good starting point however we would recommend future design attempts to utilize a more sophisticated actuator model that includes controller effort. The process of disqualifying designs based on behavior that fails search constraints may be limiting when controller effort could allow generally better agreed behavior across the wider trajectory with torque, velocity or motor current and voltage saturation at only a few data points. Further, inclusion of a dynamic system model with 8 control effort could potentially give better understanding of ﬁnal system response. Realization of hardware can force variance from target com- ponent speciﬁcations. Limitations in commercially available components such as ideal gear ratios and springs can force a design away from strictly optimal conﬁgurations. In this analysis we provide a means to understand the design trade- offs in addition to a means to search for ideal components. Strict adherence to optimization results can leave one blind to feasible or infeasible design regimes. These two tools together, the energetic map and optimization procedures can provide insight into design trade-offs for future actuator architectures. Our own application of these methods are demonstrated in hardware in an accompanying paper [24]. ACKNOWLEDGMENT The authors would also like to thank our funding sources for enabling this research, and our colleagues in the ﬁeld contributing to the greater knowledge.