Ankle Torque During Mid-Stance Does Not Lower Energy Requirements of Steady Gaits Mike Hector, Kevin Green, Burak Sencer, Jonathan Hurst AbstractIn this paper, we investigate whether applying ankle torques during mid-stance can be a more effective way to reduce energetic cost of locomotion than actuating leg length alone. Ankles are useful in human gaits for many reasons including static balancing. In this work, we specifi cally avoid the heel-strike and toe-off benefi ts to investigate whether the progression of the center of pressure from heel-to-toe during mid-stance, or some other approach, is benefi cial in and of itself. We use an Ankle Actuated Spring Loaded Inverted Pendulum model to simulate the shifting center of pressure dynamics, and trajectory optimization is applied to fi nd limit cycles that minimize cost of transport. The results show that, for the vast majority of gaits, ankle torques do not affect cost of transport. Ankles reduce the cost of transport during a narrow band of gaits at the transition from grounded running to aerial running. This suggests that applying ankle torque during mid-stance of a steady gait is not a directly benefi cial strategy, but is most likely a path between benefi cial heel-strikes and toe-offs. I. INTRODUCTION While ankles are not strictly necessary for bipedal lo- comotion, they are ubiquitous in both bipedal animals and robots 1. Ankles enable a low effort method of stabilized standing by controlling the position of the center of pressure. Roboticists designing bipedal robots take advantage of this to allow their robots to stand still; however, it is unclear how ankles should be used during locomotion 2. Work in biomechanics has shown that ankles have an important role in locomotion including impacting the ground with the heel to soften ground impacts and pushing off at the toe to accelerate the leg into swing. This is seen in humans along with a transition of the center of pressure from heel to toe during mid-stance. In addition to being a path between benefi cial heel-strikes and toe-offs, the progression of the center of pressure also results in the application of mid-stance ankle torque, which affects the dynamics of walking and running 3. However, it is unknown whether this progression is benefi cial to the gait itself or if it is an emergent feature of heel-strike and toe-off. This paper presents a fi rst principles approach to under- standing mid-stance ankle function. Mid-stance ankle actu- ation, not toe-off or heel-strike, is examined to understand how the forces of the leg and ankle can work together to reduce the cost of locomotion. The leg model is as simple as possible, while retaining the hybrid dynamic and nonlinear This work was supported by DARPA contract W911NF-16-1-0002 and NSF Grant No. 1314109-DGE. AllauthorsarewiththeSchoolofMechanical,Industrial, but in general ankles are responsible for applying forces on the body during stance. We defi ne the ankle as the most distal active joint which enables the center of pressure to be shifted within the support polygon under the foot. Two common anatomical classifi cations of leg morphology, plantigrades and digitigrades, have this defi nition of ankles. Plantigrades, such as humans, walk with their feet fl at on the ground through most of stance and shift their COP within the foot. In contrast, digitigrades, such as ostriches, walk on their toes, or digits, with raised feet throughout most of stance and shift their COP using their toe 2. While these ankles seem very different, they can both shift the center of pressure to provide a force on the body during stance 3. Ankles in humans and animals are understood to make walking and running more effi cient primarily by softening 2019 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) Macau, China, November 4-8, 2019 978-1-7281-4003-2/19/$31.00 2019 IEEE5491 ground impacts, propelling the leg forward at lift-off, extend- ing the stride length, and maintaining balance while standing 2. Heel-strikes in humans reduce ground impact forces by using soft feet while tendons store energy. As the foot comes in contact with the ground, an impact occurs. About 20% of the collision energy is dissipated through the soft footpad in the human heel 4. Additionally, elastic energy is stored in the Achilles tendon 5. The elastic energy is released during toe-off, which helps accelerate the leg into swing. The push-off of the foot corresponds to the largest acti- vation of the ankle during stance, as energy is released in a catapult that launches the foot 6. That energy is then transferred up the leg, through the knee and hip, and to the trunk, resulting in forward motion of the body 7. Both heel- strike and toe-off are benefi cial to walking gaits, but to do both in a single stride, the center of pressure must be shifted during mid-stance. Shifting the center of pressure (COP) is benefi cial in humans because it extends the length of a stride which lowers the cost of transport. For humans, shifting the COP can be thought of as continuously changing leg placement. Landing at the heel and leaving at the toe effectively lengthens the stride 8. To this advantage, the roll-over model was developed, which imagines an arc of a wheel at the bottom of the leg 9. This rolling progression of the COP from heel to toe shows a decrease in metabolic cost of locomotion in humans 10. While this can explain the benefi ts of shifting the COP, it does not give good intuition for the effect of the ankle actuation on the body or the interaction between leg and ankle dynamics. The fi rst principles approach to modeling legged loco- motion commonly uses either the Inverted Pendulum (IP) or Spring Loaded Inverted Pendulum (SLIP) reduced order model to implement stance leg dynamics and to study energy cycles 11. The IP and SLIP models encode nonlinear hybrid passive dynamics with few parameters and simple leg morphologies. SLIP models can demonstrate steady state lo- comotion through equilibrium gaits, which can be produced by the correct choice of touchdown angle 12. In legged locomotion, when a gait has the same height and velocity at the start and end of a cycle (Fig. 1), a limit cycle called an equilibrium gait has been achieved 13. To apply these models to legged robots, damping and actuation can be added so that leg actuation strategies can be investigated. Many variants of these models have been created to understand the dynamics of locomotion, some of which also include models of feet and ankles 14. Some leg models have been extended to include ankles by modeling them as rolling wheels and passive torsion springs 10, 15. However, there has been no previous work examining how an actuated ankle contributes to energetically effi cient locomotion in the context of one of these reduced order models. This paper investigates how the ankle torque and resulting force on the body affects the overall leg and ankle trajectories to improve the energy effi ciency of locomotion. We seek to understand if the resulting force during stance contributes to energy effi ciency or if heel to toe COP progression is simply a path between benefi cial heel strikes and toe offs with net stride length benefi ts. III. MODELINGMID-STANCEDYNAMICS OFANKLES We aim to understand the mid-stance dynamics of legs and ankles by combining the actuated spring loaded inverted pendulum leg model with an ankle actuator. This model, shown in Fig. 2, combines features of actuated SLIP models with a simple implementation of mid-stance ankle actuation to gain insight into how shifting COP can improve energy effi ciency during stance. A. SLIP and Actuated SLIP models To understand the trade-offs between leg and ankle ac- tuation, we use a leg model that includes an actuator in series with a damped leg spring. The actuated SLIP (ASLIP) dissipates energy through damped spring compression and actuates leg extension through the spring set point. To reduce parameters and complexity of the model, we model the leg actuator by bounding the acceleration of leg extension 12. These limits create a sense of actuator dynamics without modeling an explicit motor inertia or gear ratio. This pro- duces leg dynamics of the form Fleg= k(r0r)+c( r0 r)(1) where k, c, r, r0are the stiffness, damping, length of the leg, and leg spring set point, respectively. B. Actuated SLIP with Ankle An ankle is added so that torque at the joint will change the COP and cause a net force on the body (Fig. 2). This combined model is called the Ankle Actuated SLIP or AASLIP. The ankle torque, ankle, shifts the COP between the heel and toe, but it is strictly limited from applying more torque. This keeps the foot fl at on the ground and models the force produced by a shifting COP. The ankle torque is limited by the COP, which is dependent on leg force, Fleg, length of the foot, lf, and the leg angle, , by xCOP= Ankle FLegsin (2) with limits lf 2 xCOP lf 2 .(3) No ankle torque results in the COP remaining at the ankle axis; thus, the AASLIP model collapses to the ASLIP model when Ankle= 0. This constrained torque at the ankle can be translated to an ASLIP pinned to the ground with the ability to apply a constrained force on COM perpendicular to leg length direction with magnitude FAnkle=Ankle/r as shown in Fig 2. This produces the full dynamics equations, x = xFLeg mr yFAnkle mr (4) y = yFLeg mr + xFAnkle mr g(5) 5492 Fig. 2.Combined dynamics of the leg and ankle are implemented as a spring mass model with an actuated ankle. The torque at the ankle creates a constrained force on the center of mass that can help propel the model forward through stance. where m is the body mass and g is the gravitational accel- eration. Because this model is pinned to the ground, friction cone constraints are not included. Generally, trajectories generated in this study do not have shallow leg angles that would cause feet to slip during stance. C. Nominal Model Parameters Model parameters are chosen to refl ect human charac- teristics in a non-dimensionalized parameter space. The parameters (m = 1, g = 1, k = 20, c = 0.4, l0= 1, lf= 0.15) are non-dimensionalized by the Froude number to make them dimensionless and applicable to many walking systems 12. Stiffness, k, and damping, c are chosen to refl ect values found in human locomotion studies, and l0is the nominal length of the leg 6. D. Model Strengths and Limitations The AASLIP model captures some hypothesized features of leg-ankle mid-stance dynamics while ignoring heel-strike, toe-off, and energy storage. For instance, the ankle can apply torque to extend the distance traveled during stance. By using the ankle to push the body through stance, a larger leg angle, , at touchdown can be chosen resulting in farther distance travelled during stance. However, this is different from the stride length increase in humans, which is due to changing contact from heel to toe. In the AASLIP, the stride length change is caused by the force on the body produced by the ankle. The AASLIP is a representative model of midstance leg- ankle actuation in both plantitgrades and digitigrades. Since it is focused on the net force from the ankle on the body with the foot fi rmly on the ground, rather than modeling the extension of the ankle, it can be used to analyze the mid- stance dynamics of any leg-ankle system in stance. Using the AASLIP model allows us to analyze the in- terplay of leg and ankle forces on the body. Additionally, because it is an actuated locomotion model, it can be used to analyze the energetics of steady gaits to give insights into how legs and ankles work together to make locomotion more effi cient. IV. CALCULATINGMID-STANCEANKLEUTILITY USING OPTIMIZATION We calculate the energetic benefi t of using ankles during mid-stance by using trajectory optimization to fi nd and compare the optimal equilibrium gaits of both the model with the ankle (AASLIP) and the model without an ankle (ASLIP). Energy effi ciency is used because there is evidence that animals minimize power in steady gaits to increase energy economy over long distances 16. The energy re- quired to complete an equilibrium gait is commonly non- dimensionalized to a quantity which is applicable to all kinds of locomotion called the Cost of Transport (CoT), which is defi ned as CoT = Ereq mgd (6) where Ereqis the energy input required for the gait, m is the body mass, g is the gravitational acceleration, and d is the distance traveled in one cycle 5. Ankle benefi t is deter- mined by comparing equilibrium gait CoTs of the AASLIP and ASLIP with matching model and gait parameters. This new quantity called ankle utility, U, is defi ned as the percent decrease in CoT to complete the cycle when the ankle is present or U = 100 CoTNo AnkleCoTAnkle CoTNo Ankle .(7) An ankle utility, U, greater than 0 means that the ankle is energetically benefi cial. Since no ankle actuation collapses the AASLIP model to the ASLIP model, the ankle utility must always be greater than 0. To calculate ankle utility, a trajectory optimization is posed to determine minimal CoT trajectories which produce equilibrium gaits. A. Objective Function The optimization objective is to minimize CoT using a cost function which describes both positive actuator work and thermal losses. This is implemented as Ereq= Z T 0 max(0, (1)PLeg+RLegF2 Leg) dt + Z T 0 max(0, (1)PAnkle+RAnkle2 Ankle) dt (8) where PLeg= FLeg r0(9) PAnkle= Ankle( xyx y) r2 .(10) Positive mechanical actuator work is modeled because nega- tive work is diffi cult to recover 17. Integrating the positive instantaneous mechanical power of the actuator, Pi, over time gives the positive work done by the actuator. A smoothing function for max(0, x) of the form max(0, x) = x+ x2 +2 2 (11) where = 106is used to eliminate negative contributions of mechanical power and maintain a differentiable objective function. 5493 Thermal losses or electrical losses are used to penalize high force in the actuator by using a model based on DC motors. In a DC motor thermal loss is the power that is dissipated in the resistance of the motor windings. Since resistive power losses are proportional to the square of current and current is proportional to the actuator force or torque output, the thermal losses are proportional to the square of actuator effort. We use RLegand RAnkleto represent the thermal loss coeffi cient in the leg and ankle, respectively. When representing a system with both leg and ankle motors there exists a balance between each actuators elec- trical losses which is infl uenced by motor selection and gear ratio 18. The selection of actuator and transmission is infl uenced by many factors beyond the thermal losses in the system, such as peak torque, refl ected inertia, maximum velocity, and maximum acceleration. We choose this balance to match the bipedal robot Cassie, designed by the Oregon State University Dynamic Robotics Laboratory. This is a conservative selection because the ankle motor has a much higher gear ratio compared to the leg extension motor, which reduces the thermal losses of the ankle. The thermal loss coeffi cients are determined by non-dimensionalizing a function of Cassies actuator forces and power losses into the reduced order model space. This is done for both the leg and ankle motors to calculate RLeg=0.028 m1l3/2 0 g3/2 and RAnkle= 0.50 m1l1/2 0 g1/2. Because RLegtranslates force to power loss, and RAnkletranslates torque to power loss, they cannot be directly compared. In addition to setting the magnitude of the actuator elec- trical losses relative to each other, the balance of total mechanical losses to total electrical losses is also important because it describes many confi gurations of actuators and gearboxes at different motor scales. To explore this balance, we introduce a convex combination parameter, , with the property that varying from 0 to 1 represents the cost function changing from only mechanical power to only thermal losses. Only the one parameter is needed because the total magnitude of the objective function has no effect on the optimal solution. Additionally, when =0.5 the mechanical power and thermal loss terms are balanced to match the nominal case with parameters that match Cassie. B. Constraints Constraints are imposed on this problem to ensure feasible dynamics, ankle torques, leg accelerations, and equilibrium gaits. Dynamics are implemented as constraints through trapezoidal direct collocation with 30 knot points 19. To avoid modeling a hybrid system, the fl ight phases are formed as constraints on the initial and fi nal conditions of stance. These constraints guarantee ballistic trajectories into and out of stance such that apex conditions are reached. Periodic equilibrium gait constraints are created by match- ing the states at the initial and fi nal apexes of the trajectories where there is no vertical velocity, thus creating a single stride that could be repeated to create a stable gait. During the trajectory, ankle torques are constrained by Eq. 2 and Eq. 3 so that the COP stays between the heel and toe. Similarly, the acceleration of the leg motor, length of the leg, and force in the leg are bounded between 1 r0 1, .5 r 1, and 0 FLeg 5, which respects values found in other locomotion studies 12. C. Implementation The nonlinear optimization problem posed can be solved effi ciently using a standard nonlinear program solver. Matlab Symbolic Math Toolbox was used to generate analytical gradients of the objective and constraints. These are sent to fmincon, a nonlinear constrained optimization program, and solved using a sequential quadratic programming algorithm. Resulting trajectories are consistently found in less than 10 seconds on a desktop computer and are within a constraint tolerance of 109. With this fr