A Switchable Parallel Elastic Actuator and its Application to Leg Design for Running Robots

Transcription

1 A Switchable Parallel Elastic Actuator and its Application to Leg Design for Running Robots Xin Liu, Anthony Rossi and Ioannis Poulakakis Member, IEEE Abstract Motivated by the role of compliant elements in animal motion, springs are introduced in the driving train of legged robots to improve their locomotion performance. This paper presents the working principle, hardware realization and experimental evaluation of a Switchable Parallel Elastic Actuator (Sw-PEA) design actuating the monopedal robot SPEAR. In the proposed design, a mechanical switch engages the parallel spring only during the stance phase, when it is needed to support body weight and promote energy recovery. During flight, the spring is disengaged to allow for unobstructed joint movement. Furthermore, the proposed design enables online leg stiffness adjustments simply by changing the landing configuration of the knee joint. Experimental results demonstrate the effectiveness of the design in improving energy efficiency without compromising mobility. SPEAR can run with an electrical cost of transport of.86 at.5m/s, as well as reach a toe clearance of more than 45% of its leg length. The overall design is compact and reliable, and can be easily scaled for legged robots with different sizes. Index Terms Switchable parallel elastic actuators, legged robots, running, mobility and efficiency, cost of transport. Fig. 1. The Switchable Parallel Elastic Actuator Robot (SPEAR) employs a Switchable Parallel Elastic Actuator (Sw-PEA) at its knee. I. INTRODUCTION Legged robots have the potential to extend our reach to terrains that challenge the traversal capabilities of traditional wheeled platforms [1]. To realize this potential, diverse legged robot designs have been proposed, and a number of these robots achieved impressive indoor and outdoor terrain mobility. However, combining mobility with energy efficiency is a challenging task due to the inherently dissipative nature of legged locomotion [2]. Furthermore, legged robots typically operate in regimes where the natural dynamics of the mechanical system imposes strict limitations on the capability of the actuators to regulate its motion [3]. To address these challenges, a series of actuator designs has been introduced, the majority of which combine elastic energy storage elements with motors to generate and sustain locomotion. Focusing on robots powered by electrical motors, a widely adopted approach is to introduce compliance into the driving The material in this paper was partially presented at the IEEE/RSJ International Conference on Intelligent Robots and Systems, Hamburg, Germany, Sep. 28 Oct. 2, 215. This work is supported in part by National Science Foundation (NSF) grant CMMI and CAREER award IIS , and by the Army Research Office (ARO) under contract W911NF (Corresponding author: Ioannis Poulakakis). X. Liu is with Verb Surgical Inc., Mountain View, CA 9443, USA. This work was performed while he was with the Department of Mechanical Engineering, University of Delaware, Newark, DE 19716, USA ( xinliu@udel.edu). A. Rossi and I. Poulakakis are with the Department of Mechanical Engineering, University of Delaware, Newark, DE 19716, USA ( avrossi@udel.edu, poulakas@udel.edu). This paper has supplementary downloadable material. train in series with the motor, forming Series Elastic Actuators (SEAs) [4]. A variety of robots successfully employ this actuator architecture, including bipeds [5], [6] and humanoids [7], as well as quadrupeds [8]. A less common actuator configuration is to insert a compliant element in parallel with the motor, resulting in Parallel Elastic Actuators (PEAs) [9]. An attractive feature of this configuration is that the spring provides most of the torque required at the joint to maintain a desired motion, while the motor can modify the torque profile as needed [1]. However, introducing springs in parallel with the actuators may limit joint dexterity, since the spring may interfere with the motor even when it is not needed [11]. This fact possibly explains the scarcity of robots using PEAs. To overcome this drawback, the idea of inserting a switchable spring in parallel with an actuator has been mentioned in [1]. Although a few prototypes that combine parallel elasticity with switching mechanisms have been proposed as standalone actuation units [11] [14], the effectiveness of Switchable Parallel Elastic Actuators (Sw-PEAs) in realizing the advantages of PEAs without compromising control authority has not been assessed experimentally in dynamic legged locomotion. This paper focuses on evaluating Sw-PEAs as a design choice for a leg actuation unit in running robots. To this end, we present the conceptual design, hardware realization and experimental evaluation of the Switchable Parallel Elastic Actuator Robot (SPEAR), a two-dof leg that employs a Sw- PEA at its knee joint and is suitable for dynamic locomotion; see Fig. 1. SPEAR features a passive mechanical switch at its foot to engage an energy-storing (relatively stiff) spring only

2 when the leg is in contact with the ground, thereby recycling part of the energy during leg compression and decompression. With the mechanical switch placed at the foot, the Sw-PEA is automatically synchronized with the hopping motion, without requiring additional energy and sensing. An additional feature of the proposed design is that the stiffness of the virtual leg connecting the foot with the hip joint can be adjusted online in a step-by-step fashion without engaging the physical spring, simply by modifying the targeted landing configuration of the leg during flight. To assess the implications of these properties on energy efficiency, mobility and control design, a series of leaping, and periodic hopping in place and running forward experiments with the SPEAR are analyzed. Preliminary results on SPEAR have appeared in [15], and a simulation analysis of the implications of Sw-PEAs in dynamic locomotion can be found in [16]; see [17] for an overview. II. BACKGROUND The design of energy efficient legged robots without impairing mobility and control authority is an active research area. In the context of electrically actuated systems, one way to achieve energy efficiency is to use regenerative braking to capture some of the energy when the motor is performing negative work [18]. With this approach, the MIT cheetah realized impressive locomotion efficiency using customized high torque density motors [19] and light legs [2]; see [18] for an overview of the design. Note though that electrical energy, including the part recaptured through specially designed motor driving circuitry, needs to be converted back to mechanical energy [12]. Particularly in legged robots, this conversion occurs at low efficiency because of the large torques involved and the ensuing losses due to Joule heating; in [18], a 24% conversion efficiency is reported with optimally designed motors. Inspired by compliant structures such as tendons and muscle fibers in the legs of animals [21], mechanical springs have been incorporated into legged robots to recycle energy [22]. Beyond energy recycling, suitably inserted compliant elements can reduce the peak power and torque of the actuators [23]. This further improves efficiency as the actuators operate in a low torque region where they are more efficient. One way to introduce compliance in a legged robot s structure is the SEA architecture, in which a spring is placed in series with an actuator [4]; see also [22], [24] and references therein. Focusing on legged systems, an early implementation of series compliance can be found in Raibert s robots, in which air springs were placed in series with hydraulic actuators [1]. To realize spring-mass walking [25] and running [26], the bipedal robot MABEL has been designed with large leaf springs connected in series with the actuators [5], [27]. More recently, ATRIAS used a series-elastic parallelogram mechanism to achieve spring-mass 3D hopping and walking motions [6]. Along a different philosophy, the humanoid COMAN [7] is powered by intrinsically compliant knee and ankle joints, while the quadrupedal robot StarlETH comprises 12 SEAs actuating its joints [28]. In these systems, besides passive mechanical energy storage [5], [6], [27] and torque control capabilities [24], [28], SEAs provide protection by filtering out impulsive loads at collisions. On the other hand, the motors in SEAs must be capable of producing forces and torques that are comparable to those developed by the springs [29]. In the context of dynamic legged robots this feature translates to large motors and gear reduction ratios. Furthermore, SEAs typically increase the number of degrees of freedom (DOF) of the system, requiring special design and control considerations [24], [3]. Finally, as was mentioned in [31], SEA designs may limit the range of behaviors that can be realized by the system. Being able to actively modify the stiffness of the passive component as in variable stiffness actuators (VSA), can help mitigate this restriction. However, many current VSA designs tend to increase the complexity of the system [22]. An alternative way to introduce compliance is to insert the spring in parallel with the motor so that the spring and the actuator work in an additive fashion [1]. An example of using PEAs can be found in [29], where springs are used to improve energy efficiency and safety of known maneuvers in passiveassist devices for active joints. In the context of legged robots, the biped ERNIE utilizes springs in parallel with its knee actuators to generate walking motions [9]. PEAs may reduce both power and torque requirements, as suggested in simulations of bipedal [23], [32] and quadrupedal [33] running. In general, however, introducing springs in parallel with the actuators may limit joint dexterity, since the actuator needs to work against the spring [11]. Recently, numerical optimization has been used in [34] to find the optimal actuation configuration among different combinations of SEAs and PEAs actuating a planar hopping model. It was found that the optimal in terms of positive electrical work, a metric of energy efficiency actuation configuration is velocity dependent. Recently, there has been an increasing number of actuator designs that incorporate discrete coupling elements such as clutches or brakes into the SEA paradigm [35]. These coupling elements can increase the output performance of the actuation unit. For example, in [36], a clutchable SEA is developed, where the clutch is in parallel with the motor. When the clutch engages, it connects one end of the spring to a fixture, allowing it to deform. The motor is bypassed and does not need to produce any torque. The design in [36] is used to power an active knee prosthetic device, and the observed energy consumption is an order of magnitude less than previous results. However, the motor is unable to provide additional energy when the clutch is active. A general purpose SEA that employs discrete coupling elements to achieve multimodal operation for versatile applications can be found in [37]. The prototype actuator is.67m long and weights 4.5kg, which makes it suitable for relatively large systems. Discrete coupling elements can also be used to overcome the shortcomings associated with PEAs. Based on analyzing a planar bipedal walking model, [1] advocates the use of a Switchable Parallel Elastic Actuator (Sw-PEA) with positiondependent clutch function so that the spring is engaged when a leg is in stance that is, when it is mostly needed but not during flight. A similar observation is made in [33] in the context of modeling high-speed quadrupedal running. The authors of [33] mention, however, that commercially available clutches are generally slow, and their size and weight make

3 them unsuited for light-weight legs [2]. Beyond simulation studies, only a few hardware prototypes exist that combine PEAs with switches. One such prototype, which uses a clutch to realize the switch is described in [11]. Experiments in which the actuator is constrained to mimic the torque and motion pattern of the knee extensor muscle of a human hopping task suggest that the energy consumption is reduced by 8% and the peak torque requirement decreases by about 66% compared to the case where no springs are used. A different approach is proposed in [13], which develops a prototype of an actuator that uses a single motor to recruit several parallel elastic elements in sequence with mutilated gears. This arrangement increases the maximum output torque of the actuator, and in a more recent version [14], it also allows for stiffness adjustments, although the complexity of the current prototypes is relatively high. Recently, [12] presented a bidirectional Sw-PEA prototype, which uses a differential with two locking mechanisms to load and unload the spring in a controlled manner, achieving reduction in energy consumption. Although the aforementioned prototypes indicate that Sw- PEAs offer potential advantages in terms of energy efficiency and motor torque reduction, their use has not been evaluated in the context of dynamic legged robots, an application that would benefit from such capabilities. Only a few results are available in exoskeleton design, which employ a mechanical clutch [38] or electrostatic forces [39] to engage and disengage the parallel spring. To the best of the authors knowledge, only [4] presented a dynamic bipedal robot design that employs a Sw-PEA at its knee joint. These observations set the stage of this research which aims at evaluating the contribution to energy efficiency and mobility of a Sw-PEA in dynamic running motions. The proposed design of a Sw-PEA takes advantage of the leg s geometry to realize a reliable and compact mechanical switch at the foot that engages the energystoring spring only when it is needed, i.e., during stance. Another example of a mechanical stiffness switch can be found in [41], where through it is used to regulate the ground reaction forces (GRF) of a quadrupedal robot. With the switch passively operated by the GRF, these designs are suitable for light legs while they can be easily modified for larger and heavier robots. III. SW-PEA MECHANICS & INTEGRATION INTO SPEAR This section presents the basic principle underlying the Sw- PEA design, as well as its implementation in SPEAR. A. Working Principle of Sw-PEA Part of the challenge of introducing compliance into running robots is that different objectives must be satisfied by the mechanical system as these robots go through different locomotion phases. To address this challenge, we propose the Sw-PEA, a new actuator design, which incorporates a discrete coupling element a switch to engage compliance in parallel with a motor when elastic energy storage is needed. For illustration, Fig. 2 presents the underlying principle in a prismatic joint setting. In Fig. 2, Link A is assumed to be the base, and is connected to Link B through an actuator and a compliant element inserted in parallel with the actuator. Fig. 2. Link A Key S 2 S 1 actuator chain Link B A schematic of the general idea underlying the design of a Sw-PEA. The compliant element contains two springs S 1 and S 2 with different stiffnesses: the spring S 1 has stiffness K 1, much larger than the stiffness K 2 of S 2 ; i.e., K 1 K 2. The two springs are connected in series through a mechanical switch realized by a key and a chain. The effective stiffness of the combined springs is determined by the status of the key. When the key is inserted in the chain, the left side of the hard spring S 1 is restrained; hence, in this configuration, the mechanical switch engages S 1, leading to a large joint stiffness K 1 that favors energy storage as Link B moves. When the key is not in the chain, the springs S 1 and S 2 are connected in series, and the effective stiffness of the joint is mainly determined by the softer spring S 2. With K 2 negligible, the hard spring S 1 is effectively switched off, favoring precise joint motion control via the actuator as if no spring were present. B. SPEAR: A leg design that uses the Sw-PEA The monopedal robot SPEAR shown in Fig. 1 consists of a torso and two links corresponding to the thigh and the shank of a kneed leg that terminates at a point foot (toe). To reduce the overall weight of the system, the thigh is composed of two lightweight aluminum plates and the shank is formed by a carbon fiber tube. The thigh is connected to the torso via the hip joint with a range of motion [ 85,85 ], while the shank is connected to the thigh by the knee joint with a range of motion[ 1,14 ]. The robot s torso is connected to a boom, as described in Section III-E below; see Fig. 8 for the overall setup. The assembly is actuated by two brushless motors, one for the hip and one for the knee joint. To further reduce the moment of inertia with respect the hip axis, the knee motor is placed in proximity to the hip joint, and a two-way cablepulley system is used to transmit the motor torque to the knee. Table I provides some key parameters of SPEAR. The knee joint of SPEAR is driven by an Sw-PEA. In our implementation of the Sw-PEA concept to SPEAR, Link A and Link B of Fig. 2 correspond to the shank and thigh in Fig. 3, respectively. One end of a hard spring S 1 is rigidly attached to the thigh by a steel cable. The other end of S 1 is attached to another cable, which first wraps around a circular knee spacer that is rigidly mounted on the shank, and then connects to one end of a roller chain that passes through the toe. The other end of the chain is then attached to the shank via a soft return spring S 2. The knee spacer serves to make the prismatic spring S 1 function as a rotational spring acting at the knee joint; see Section III-D for more details. As shown in Fig. 3, a two-way cable-pulley system is used to transfer the torque of the knee motor to the knee joint in a way that allows bi-directional motion so that the leg can be shortened and lengthened in a fully controllable manner.

4 - Fig. 3. (a) The mechanical realization of a Sw-PEA at the knee of SPEAR. The proximal end of the stiff spring S 1 is attached to the thigh by a steel cable; the other end first passes through the knee spacer, then passes through the foot (where it can be locked or unlocked depending on the state of the toe) terminating at the shank via the soft spring S 2. (b) Bi-directional motion transmission from the knee motor to the knee joint: there are two pairs of pulleys on the left and right of the median plane of the leg. At the knee, the two pulleys are on the left and right of the knee spacer so that during flight the knee actuator can shorten and lengthen the leg without interfering with S 1. (c) A schematic of SPEAR; the virtual leg connecting the toe with the hip is also depicted. In more detail, to shorten the leg, the knee motor applies a clockwise torque which engages the pair of pulleys on the right side of the thigh, marked by red in Figs. 3(a) and 3(b); to lengthen the leg, a counterclockwise toque is applied engaging the pair of pulleys on the left, marked by blue in Figs. 3(a) and 3(b). As a result, the knee angle changes in response to the motor, deforming the springs S 1 in stance and S 1 in series with S 2 in flight through the knee spacer by a corresponding amount of rotation; hence, the parallel configuration of the springs with respect to the knee motor. We do not show the two-way pulleys on Fig. 3(c) to avoid clutter; instead τ m,knee is used to represent the effect of the motor. The Key block depicted in the conceptual Fig. 2 is realized at the foot of the robot, as shown in Fig. 4. The chain in Fig. 4 represents one part of a mechanical switch, the other part of which is the foot itself. The foot is designed to have TABLE I SPEAR DESIGN PARAMETERS Parameter (Symbol) Value Units Torso mass (incl. boom & CPU) (M) 4.91 kg Thigh/shank mass (M 1,M 2 ) (2.43,.73) kg Thigh/shank inertia (incl. motor) (J 1,J 2 ) (.6,.2) kgm 2 Thigh/shank length (L 1,L 2 ) (.32,.327) m Stiffness of spring S 1 (K 1 ) 39, N/m Stiffness of spring S 2 (K 2 ) 31 N/m Torque constant (hip & knee motors) (K t).414 Nm/A Coil resistance (hip & knee motors) (R).72 Ω Knee motor gear ratio 25:1 n/a Hip motor gear ratio 6:1 n/a Knee joint max torque (max current) 12.4 Nm Hip joint max torque (max current) 29.8 Nm Knee joint max velocity 46 rad/s Hip joint max velocity 19.2 rad/s a tooth shape at one end, and is connected to the shank via a prismatic joint, which allows the foot to move in the direction of the shank with a maximum displacement of 1cm. During the stance phase, the GRF pushes the tooth of the foot inside the chain, thereby engaging the spring S 1, as shown in Fig. 4(a). During the flight phase, an additional small spring S 3 is used to push the foot outside the chain, thereby disengaging the spring S 1. As a result, this mechanical realization of the switch does not require additional control effort to synchronize the engagement of the energy storing spring S 1 with the state of the leg; a similar mechanical stiffness switch has also been used in [41] for the purpose of regulating the GRF. As a final remark, note that other types of locking devices such brakes or clutches could be used to implement the switch in the Sw-PEA concept of Fig. 2. In the case of running robots, however, due to the large ground reaction Foot(Key) To S 2 S 3 (a) GRF To S 1 S 3 (b) Contact Switch Fig. 4. (a) The section view of the foot which realizes the Key of Fig. 2. The foot is inserted into the chain by the GRF and the spring S 1 is engaged. S 3 is a small spring used to keep the foot unplugged from the chain when there is no force applied as in the flight phase. The contact switch is used to determine if the foot is inserted in the chain. (b) The manufactured foot.

5 forces involved, a relatively high locking torque capability is required. While locking devices capable of reliably holding such torques are commercially available, they tend to be heavy and would have the undesirable effect of increasing leg inertia. Contrary to these solutions, the proposed passive self-locking mechanical switch makes the design compact and reliable. Finally, note that the larger the spring deformation is, the larger the GRF that pushes the key into the chain will be, and thus the proposed design can be easily scaled to larger robots as long as a stronger chain and suitably selected springs are used. C. Energy Flow of Sw-PEA in SPEAR To explain how the Sw-PEA improves the performance of SPEAR in terms of energy consumption, Fig. 5 presents the energy flow diagram of SPEAR s knee joint. Assuming that the energy lost due to resistive (Ohmic) heating in the electrical circuitry and due to switching effects in the motor drivers is small, three principal sources of energy dissipation are identified. These are associated with Joule heating in the electromagnetic motor, friction losses in the mechanical transmission system primarily in the gearbox and interaction losses at the interface between the leg and the ground. As indicated in Fig. 5, the proposed Sw-PEA design improves the energetic performance of SPEAR mainly in two ways. First, the spring stores part of the mechanical energy during leg compression in the stance phase when negative work is performed and recycles this energy to power the cyclic motion during subsequent leg decompression. Note however that, in recycling energy, part of it is lost due to spring damping and part of it flows back through the transmission system to the motor, which is back-driven during stance compression. The latter is eventually lost too through fiction in the gearbox and in the absence of regenerative braking circuitry Joule heating in the actuator. Despite these losses, the energy calculations of Section IV demonstrate that the Reduce Torque Recycle Energy Leg Segments Impact Battery Motor Transmission (1-η 1) Joule Heating (1-η 2) Friction Spring Damping Fig. 5. Energy flow of the Sw-PEA in SPEAR s knee in stance. Red arrows denote energy that is lost or eventually lost; green arrows denote energy input and the bi-directional arrow denotes recycling. Arrow sizes indicate the relative magnitudes of the corresponding energy flow. The red dashed boxes show the two main ways torque reduction and energy recycling by which the Sw-PEA improves locomotion efficiency in SPEAR. activation of the parallel spring in the Sw-PEA during stance results in significant improvements of locomotion efficiency. The second aspect of the Sw-PEA that contributes to reduced energy consumption is associated with the configuration of the spring S 1, which is connected in parallel with the actuator. As a result, the torques developed by the spring and the combined actuator/transmission system are added during stance so that the need for the motor unit to develop large torques is lessened. This is beneficial for two reasons. First, reducing the torque allows the motor to convert electrical energy to mechanical more efficiently. Indeed, neglecting rotor friction, the efficiency η 1 of a DC motor can be approximated by η 1 1 I2 R VI = 1 R K τ tv m, where K t is the torque constant,v is the supply voltage,i is the current in the motor s coils with resistance R, and τ m is the motor torque. Thus, for a given motor and voltage applied, the energy dissipated due to Joule heating decreases with the motor torque. Second, due to the reduced torque requirement, a gearbox with a relatively smaller ratio can be chosen, which further improves the efficiency η 2 of the transmission system [6]. Hence, for a given specification of output torque, different combinations of motors and gearboxes can be selected depending on whether optimizing η 1 or η 2 is the objective of the specific design. It should be emphasized though that choosing a smaller gear ratio is advantageous, for it decreases the reflected moment of inertia and it enhances joint back-drivability, which is particularly important in our implementation of the Sw-PEA concept 1. As shown in Table I, SPEAR uses a 25 : 1 reduction ratio at the knee actuator. D. Stiffness Properties of the Leg 1) Stiffness of the Knee Joint: Due to the geometry of the leg and the knee spacer, the prismatic springs S 1 and S 2 have the equivalent effect of a rotational spring located at the knee joint. Indeed, let K Lin be the combined stiffness of the prismatic (physical) springs; its value depends on the phase of motion and will be discussed below. To relate K Lin with the corresponding stiffness K Rot consider Fig. 6. Keeping the shank fixed and rotating the thigh from D to D causes the knee joint to rotate by an angle θ, thereby deforming the prismatic spring by L = r θ, where r is the radius of the knee spacer. As a result, the stiffness K Rot of the rotational spring at the knee joint can be computed by K Rot = τ spring θ = (K Lin L) r L r = K Lin r 2, (1) where, as was mentioned above, K Lin depends on the phase of the leg. In more detail, suppose that K 1 and K 2 represent the spring constants of S 1 and S 2, respectively; see Table I. Then, when the leg is in stance and the mechanical switch engages S 1, the stiffness is K s Lin = K 1, and (1) results in K s Rot = K 1 r 2. (2) 1 Note that contrary to the SEA architecture in PEAs the combined motor and transmission unit is not totally isolated from the leg segments, and thus it is exposed to impacts. In SPEAR, however, the back-drivability of the knee joint and the cable driven system adopted provide sufficient protection. Additional protection of the Sw-PEA could be achieved by incorporating a torsion bar or a belt transmission system to isolate impact.

6 D θ S 1 D knee spacer S 2 (a) A thigh C C O r B shank D Virtual Leg l K Rot F leg (b) A L 1 L 2 knee θ knee Fig. 6. (a) The relation between the linear spring and the equivalent knee rotational spring. (b) The spring K Rot captures the effect of S 1 or S 2. In our design, K 1 = 39,N/m and r =.38m, resulting in KRot s = 56.67Nm/rad. When, on the other hand, the leg is in flight and S 1 and S 2 are connected in series, the stiffness is KLin f = K 1 K 2 /(K 1 +K 2 ), and (1) results in ( ) KRot f K1 K 2 = r 2 < K 2 r 2. (3) K 1 +K 2 With K 2 = 31N/m and r =.38m, the knee joint during flight has a stiffness KRot f <.3Nm/rad, which is very small and will be neglected. To summarize, due to the switching, the knee joint of SPEAR has stiffness KRot s = 56.67Nm/rad when the leg is on the ground, acting primarily as an elastic energy storage element, developing torque τ spring = K s Rot ( θ knee θ knee ) (4) where θ knee is the touchdown angle of the knee. When the leg is in the air, however, the knee joint stiffness is negligible, so that the motors can efficiently control the leg s motion. 2) Adjusting the Virtual Stiffness of the Leg: The implementation of the Sw-PEA in SPEAR offers a simple way to adjust the virtual stiffness of the leg on line and in an energy efficient manner. To illustrate this capability, we begin by noting that, due to the multiple locking positions available at the chain, the rest angle θ knee in (4) can be adjusted easily as follows. During flight when the springss 1 ands 2 are connected in series and the effective compliance of the joint is negligible the knee actuator can set the knee angle at different desired values in anticipation of landing. Hence, when touchdown occurs and the energy storing spring S 1 engages, the torque developed by the knee spring is given by (4) with θ knee being the angle that has been set cheaply during the preceding flight phase. The ability to change θ knee during flight significantly affects the stiffness properties of the leg during the subsequent stance phase. To illustrate this behavior, suppose that during flight, the control law selects a knee angle θ knee. Then, at touchdown, the length of the line that connects the hip joint with the foot that is, the length of the virtual leg; see Fig. 6(b) is l = L 2 1 +L2 2 +2L 1L 2 cos θ knee, (5) where L 1 and L 2 are the lengths of the thigh and shank, respectively; see Fig. 6(b). During the subsequent stance phase, the change θ knee = θ knee θ knee in the knee angle results in the torque τ spring by (4). As in [42], assuming for simplicity that the inertia forces are small, τ spring results in a force F leg acting along the direction of the virtual leg, and F leg = 1 L 1 L 2 l(θ knee) sinθ knee τ spring, (6) where τ spring is computed by (4) and l(θ knee ) = L 2 1 +L2 2 +2L 1L 2 cosθ knee (7) is the current value of the length of the virtual leg. Hence, the knee spring τ spring ( θ knee ) given by (4) corresponds to a leg force-compression relationship F leg ( l), which represents a nonlinear spring; the stiffness of this nonlinear spring depends on the value θ knee of the knee angle at landing. Figure 7 presents the relationship between the virtual spring force F leg computed by (6) and the corresponding displacement, computed as the difference l = l l between (7) and (5) for different values θ knee of the knee touchdown angle. As can be seen, the force-displacement relationship of the leg is changed, and the more straight the leg is at touchdown, the stiffer the leg spring becomes in the subsequent stance phase. This mechanism of adjusting the effective stiffness of the leg will be explored further in the following sections. Virtual Spring Force(N) Virtual Spring Deformation(m) Fig. 7. Leg stiffness adjustment by modifying the value θ knee of the knee angle at touchdown. The x-axis is the deformation l computed by (7) and (5); the y-axis is the corresponding force F leg by (6). As θ knee varies from 13 to 1, the virtual spring becomes stiffer. E. Test Setup and Electronic System To realize hopping motions in SPEAR, a support system consisting of a vertical and a horizontal boom arranged as shown in Fig. 8 has been constructed. SPEAR s torso is mounted at one end of the two-meter long horizontal boom, which restricts the motion of the hip joint on the surface of a sphere. In what follows, however, due to the relatively long horizontal boom, we will assume that the motion of the system occurs in the sagittal plane. No counter weight is used, and the boom adds to the mass of the torso. A safety cable is used to offer protection, but it does not prevent the leg from falling. To implement the locomotion controllers see Section IV below for details the position and velocity of the leg with

7 leg at a desired configuration in anticipation of touchdown. On the second level, the parameters introduced by the continuoustime controller are updated in an event-based fashion by the discrete-time controller Γ d to regulate task-level objectives, such as hopping height and forward velocity. 1) Continuous-time control: During stance, the continuoustime controller applies desired knee and hip torques as in [45] instead of trajectories. However, compared to [45], the objective here is to inject energy in the system in a way that preserves the natural compliance of the leg due to the action of the springs 1. For simplicity in parameter tuning, the torque applied at the knee during stance is Fig. 8. The testbed used for the experimental evaluation of SPEAR. respect to the world frame are required. To obtain this information, two incremental encoders placed as shown in Fig. 8 are used to measure the rotation of the boom. Note that due to the relatively large length of the horizontal boom, a 3, counts per revolution encoder is used to estimate the horizontal displacement of the robot with accuracy.1mm, which is sufficient for controller implementation. To monitor its motion, SPEAR is equipped with a collection of sensors to measure and estimate the robot s state, and the contact state of the leg. At the joint level, measurements from the encoders are combined with data from joint potentiometers to provide accurate information regarding the joint position and spring deformation. Finally, a snap-acting switch is placed at the foot to detect the state of the leg, stance or flight. IV. EXPERIMENTAL RESULTS To evaluate the performance of the proposed implementation of a Sw-PEA, this section presents experiments with hopping in place and forward running motions of SPEAR. Here, we focus on experimental results; for simulation analysis see [16]. A. Hopping Controller Similar to [43], [44], the controller used in SPEAR is organized on two levels, as shown in Fig. 9. On the first level, a continuous-time feedback control law Γ c is employed. During stance, the purpose of Γ c is to apply desired torques to the hip and knee actuators; during flight, its purpose is to drive the Fig. 9. Γ d Γ c motion and force control discrete update for stability The framework of the controller used for experiments. τ knee (t)= { uknee, if t t TD < T st 2 u knee, if t t TD T st 2 where u knee is kept constant over a step but can vary from one step to the next, as explained below in the event-based controller (1), t TD is the touchdown instant of the current step, and T st is a constant that captures, on average, the stance duration at a given speed. The prescription (8) ensures that the knee actuator performs mainly positive work, injecting energy by further compressing the springs 1 during the first half of the stance phase and assisting with spring recoil during the second half in preparation for takeoff. Finally, throughout stance the hip actuator applies a small constant torque (8) τ Hip (t) = τ Hip. (9) During flight, on the other hand, the objective of the controller is to drive the leg to a desired configuration prior to landing. This is achieved by specifying desired target angles θ knee and θ hip for the knee and hip joints, respectively. Recall that θ knee specifies the rest position of the knee joint compliance (4), and thus the equivalent stiffness of the virtual leg connecting the foot with the hip; see Section III-D. Both angles( θ knee, θ hip ) are needed to specify the touchdown angle γ defined in Fig. 3(c), which will be used in the eventbased controller (11) below to regulate forward velocity. With the ability to switch off the energy-storing spring S 1 during flight, the target joint angles are achieved simply, without interfering with compliance. In SPEAR, a PD controller at the hip and knee actuators is used to impose the desired angles ( θ knee, θ hip ) sufficiently fast, prior to touchdown. 2) Event-based control: The event-based controller updates certain parameters in a step-by-step fashion to stabilize the motion and achieve task-level objectives such as desired hopping height and forward velocity. In more detail, to regulate hopping height, the parameter u knee in (8) is updated at the apex height of the k-th step based on feedback from the current hopping height y[k] as follows u knee [k] = τ knee +c 1 (ȳ y[k]), (1) where τ knee and ȳ are the nominal values of the knee torque and apex height, respectively, and c 1 is a positive gain. To regulate the forward velocity, the touchdown angle γ of the virtual leg connecting the foot and the hip joint see Fig. 3(c) is updated according to a modified version of Raibert s velocity controller [1] as γ[k] = γ c 2 ( x hip ẋ hip [k]), (11)

8 where ẋ hip and x hip are the current and nominal values of the horizontal velocity of the hip joint right after liftoff, γ is the nominal touchdown angle, and c 2 is a positive gain. Note that in experiments, x hip is estimated by the boom encoders see Section III-E for details using the known geometry of the boom and the leg. Furthermore, to realize γ in (11), different combinations of ( θ knee, θ hip ) can be used. As was described in Section III-D, θ knee determines the stiffness of the virtual leg during the subsequent stance phase, and has a great influence both on the stability and energy efficiency of the motion; its effect will be briefly explored in Section IV-C below. Given θ knee and γ the target value of θ hip can be computed easily, and the continuous-time flight controller can be applied to achieve the target touchdown values of the knee and hip angles. 3) Implementation details: To implement the control law, first the flight controller s gains are computed to obtain a sufficiently fast step-response when the leg is not in contact with the ground. Then, the values of the parameters { τ hip, τ knee, γ, θ knee, T st } corresponding to desired nominal velocity x hip and hopping height ȳ are specified through simulations and trial-and-error experiments. Table II provides the values of these parameters for running with an average velocity x hip =.5m/s and hopping height ȳ =.68m. The gains c 1 and c 2 are tuned to achieve stable motion. In implementing (11), the average velocity after liftoff, instead of the instantaneous one, has been used to filter out the noise. Finally, to initiate hopping, the leg first bends its knee with its maximum torque to store energy in the spring, then the torque is inverted to push against the ground. B. Leaping TABLE II CONTROLLER PARAMETERS FOR RUNNING AT.5 m/s τ hip τ knee γ θknee T st 1Nm 5.54Nm -.14rad 1.2rad.36s The mobility of a legged robot is largely related to the height its toe can clear during flight. The proposed Sw-PEA can increase toe clearance in two ways. First, as indicated by the knee controller (8) in stance, the actuator can inject energy into the spring during the first half of the stance phase compressing it further, much like loading a catapult mechanism. With this energy released later during stance, a much larger vertical liftoff velocity can be achieved causing the COM of the system to reach a higher position. Second, the leg is able to flex its knee freely during flight without interfering with the parallel spring, thereby further increasing the maximum toe clearance; see [16, Section VI] for a detailed simulation analysis. To demonstrate SPEAR s vertical jumping capability, two sets of leaping experiments are carried out, one with active knee flexion and one without. In both experiments, an initialization step is used, after which the controller (8) is engaged with τ knee = 12.4Nm, the maximum torque the actuator can provide within the specified current limit (approximately 12A). The results are presented in Fig. 1, in which it is seen that without active knee flexion during flight the robot can jump with a toe clearance of 14cm, corresponding to 25% of the leg s rest length. With active knee flexion, on the other hand, knee angle (rad) Vertical Position (m) knee current (A) stance 1st flight 1st stance 2nd flight 2nd Time (s) hip toe (a) knee flexion Time (s) (b) Time (s) (c) Fig. 1. Leaping experiments with (black continuous) and without (blue dashed) knee flexion. The first step is an initialization step and the gray area represents flight. (a) Knee angle; red line is the touchdown (rest) angle of the knee spring. (b) Toe and COM vertical position. (c) Knee actuator input. the leg is shortened during flight and SPEAR reaches a 25cm toe clearance, corresponding to 45% of the leg s rest length with the vertical height of the hip joint virtually unchanged. In both experiments, the contribution of the parallel spring during stance is significant. For example, in the initialization step, when the knee spring is deformed about.5rad the corresponding spring toque is 27Nm, about 2.3 times larger than the maximum torque 12.4Nm that the knee actuator can deliver. This advantage is typical of PEA designs. However, if the switch is not present, the energy-storing spring is always engaged, causing the actuator to perform unnecessary work during flight to actively bend the knee. By way of contrast, the proposed Sw-PEA combines the advantages of the parallel spring during stance with unobstructed knee joint control during flight to achieve sufficient toe clearance, thereby enhancing mobility without compromising efficiency. C. Periodic Hopping in Place Using the controller described in Section IV-A, periodic hopping in place motions have been realized experimentally. To quantify the spring s contribution, the hopping efficiency η hop can be defined as in [3] by W + motor η hop := 1 W motor +, (12) +W+ spring

9 where W motor + and W+ spring represent the positive work done by the motors and the spring, respectively, computed by W + motor = i {knee,hip} T max{(τ m,i θ m,i ), } dt T W + spring = max{(τ spring θ knee ), } dt, (13) where T is the duration of the stride, τ m,i = K T I i is the torque delivered at the shaft of the i-th motor,i {knee,hip}, when I i is the current flowing in its coil, and θ m,i is the corresponding shaft velocity. In (13), τ spring is computed by (4) and θ knee the knee joint velocity. The efficiency η hop defined by (12) measures how much of the positive mechanical work in one hopping stride is recycled. For the hopping experiments reported in [15],η hop =.642, implying that64% of the positive mechanical energy in one stride is generated by the spring. Furthermore, as was shown experimentally in [15] and in simulations in [16], the parallel spring reduces both the torque and power required by the motor. In these experiments, the peak mechanical power of the knee motor was 8W, much lower than the power delivered by the spring S 1, i.e., 2W in compression and 4W in decompression. Finally, to investigate the effect of varying leg stiffness on energy consumption, three hopping experiments have been conducted, each corresponding to a different target value of the knee angle at touchdown. As was explained in Section III-D2, landing with different knee angles effectively changes the stiffness of the virtual leg connecting the foot with the hip. Figure 11 presents the results corresponding to the values.74rad,1.7rad, and1.4rad of the angle θ knee, while keeping the maximum toe clearance at.8m. It can be seen from Fig. 11(a) that the leg uses less energy during a stride when landing with more straight leg configurations, in which, as Fig. 11(b) shows, the virtual spring between the foot and the hip becomes stiffer. Note that this relationship between the stiffening of the leg and the decrease in energy consumption is due to the specific operating conditions considered here; in general, there is an optimal leg stiffness that results in minimum energy consumption, as discussed in [5]. D. Periodic Running Forward Modifying the value of x hip in (11), running motions with forward velocities ranging from.1m/s to.5m/s have been realized. To characterize the energetic performance of Energy/Stride (J) Knee TD Angle(rad) (a) Virtual Spring Force(N) Virtual Spring Deformation (m) Fig. 11. (a) Electrical energy consumption for hopping-in-place with different touchdown angles. (b) Virtual leg stiffness for the knee touchdown angles in (a). Landing at more straight leg configurations results in a stiffer leg. (b) SPEAR, the cost of transport (COT) is computed based on these experiments. The COT measures the energy required to transport a unit weight over a unit distance [2]; mathematically, COT = W M g x, (14) where M is the total mass of the robot, x is the distance travelled, and W is the corresponding energy consumption. Depending on how W in (14) is computed, different values for the COT are obtained revealing different aspects of how energy is utilized by the system. In the case of the positive mechanical cost of transport COT + me, W in (14) corresponds to the positive mechanical work of the motors, W = W + motor, (15) where W motor + is given by (13). To incorporate both positive and negative mechanical work, COT me uses W in (14) by W = i {knee,hip} T τ m,i θ m,i dt, (16) where T, τ m,i and θ m,i have the meaning discussed below (13). If the difference between (15) and (16) is small, the negative work performed by the motors is small, implying that the controller makes effective use of the mechanical system in sustaining the motion. This is the case for SPEAR, as can be seen from Fig. 12(a), which shows that the mechanical COTs computed by using (15) and (16) in (14) are close. Hence, the stiff springs 1 stores most of the negative work associated with compression in stance, and the motors contribute by providing mostly positive work to regulate the motion. According to Fig. 12(a), the mechanical cost of transportcot + me of SPEAR decreases as the running speed increases, and COT + me =.45 (17) is the lowest value achieved at the highest speed.54m/s. The COT (14) defined on the basis of mechanical energy focuses on the locomotion task from an output perspective, without considering the actuation technology used to input energy in a specific system. As such, the mechanical COT does not capture the total energy required to sustain locomotion. In the context of the electrically actuated SPEAR, the total cost of transport COT ele can be computed with W in (14) given, as in [46], by the equation W ele = = T i {knee,hip} T i {knee,hip} max{(v i I i ), } dt max{(i 2 i R i +τ m,i θ m,i ), } dt, (18) where V i and I i are the supply voltage and current at the terminals of the i-th motor, R i is the motor s terminal resistance, and τ m,i and θ m,i have the meaning described above. The cost (18) captures the energy at the input of the motors 2, as shown 2 Note that in SPEAR the COT ele does not include the energy lost due to the CPU, motor driver logic and wireless router, which are powered separately.

10 in Fig. 5. As can be seen in Fig. 12(a), the performance is best when the robot runs at its highest speed.54m/s, where COT ele =.86. (19) This value is smaller than the COT ele in [8], in which SEAs are employed, but still about 7% larger than the corresponding value for the MIT cheetah that uses customized high torque-density motors and regenerative braking [18]. Figure 12(a) also shows that the ratio COT me+ /COT ele over the range of speeds reported is about 5%, which is a rough indicator of the efficiency by which electrical energy is transformed to mechanical energy in SPEAR. This is evident from Fig. 12(b), which shows that nearly half of the electrical energy input to the motors, as computed by (18), is lost due to Joule heating; the hip motor does not contribute much to the energy lost due to the small torques developed at the hip. Regarding the knee joint, as was mentioned in Section III-C, inserting the spring in parallel with the actuator decreases the torque required by the knee motor, thereby reducing the energy dissipated due to Joule heating at the coils of the knee motor. In the absence of springs, the energy lost due to Joule heating at the knee motor would have been much higher. Finally, Fig. 13 places SPEAR amongst alternative robotic designs in terms of COT ele ; it can also be seen that SPEAR s COT ele is on the line of the metabolic COT of land animals. Cost of Transport COT me COT me+ COT ele Velocity(m/s) (a) Energy Comsumption (%) knee hip mechanical joule heating Velocity(m/s) Fig. 12. (a) SPEAR s COT for speeds ranging from.1m/s to.54m/s. Each point corresponds to a commanded speed x hip, and represents the COT averaged over 1 steady-state strides. (b) Electrical energy consumption (red) of the knee and hip joints showing that the knee consumes most of the energy input; total mechanical energy and Joule heating (black) of both motors. V. CONCLUSION This paper presented the principles of a Switchable Parallel Elastic Actuator (Sw-PEA), as well as SPEAR, a monopedal robot used to evaluate the concept in dynamic legged locomotion. Using a passive mechanical switch at the foot, the spring at the knee joint is engaged at stance to recycle energy and support body weight, and is disengaged at flight to allow precise motion control. The proposed design is compact and reliable, and can be scaled for heavier robots. Experimental results demonstrate that the Sw-PEA improves energy efficiency both by recycling mechanical energy and by reducing the torque requirements. It also affords a convenient way to adjust the equivalent stiffness of the leg. 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Remy, A comparison of series and parallel elasticity in a monoped hopper, in Proc. IEEE Int. Conf. on Robotics and Automation, 215. [47] D. Campbell and M. Buehler, Preliminary bounding experiments in a dynamic hexapod, in Experimental Robotics VIII. New York, NY: Springer, 23, pp [48] I. Poulakakis, J. Smith, and M. Buehler, Modeling and experiments of untethered quadrupedal running with a bounding gait: The Scout II robot, Int. J. Robot. Res., vol. 24, no. 4, pp , 25. [49] J. Reher, E. A. Cousineau, A. Hereid, C. M. Hubicki, and A. D. Ames, Realizing dynamic and efficient bipedal locomotion on the humanoid robot DURUS, in Proc. IEEE Int. Conf. on Robotics and Automation, 216, pp Xin Liu received his Ph.D. in Mechanical Engineering from the University of Delaware, Newark, DE, in 217. He is currently a Research Engineer with Verb Surgical Inc., Mountain View, CA. His research interests include robot design and modeling and control of nonlinear dynamic systems. Recently his work focuses on surgical robotic arms. Anthony Rossi received his Bachelors Degree in Mechanical Engineering from the University of Delaware, Newark, DE, in 214. He is currently a MS student in the Department of Mechanical Engineering at the University of Delaware, Newark, DE. His research interests include mechanism design, modeling and control of nonlinear systems, and experimentation of legged robotics. Ioannis Poulakakis received his Ph.D. in Electrical Engineering (Systems) from the University of Michigan, Ann Arbor, MI, in 29. From 29 to 21 he was a post-doctoral researcher with the Department of Mechanical and Aerospace Engineering at Princeton University, Princeton, NJ. Since September 21 he has been with the Department of Mechanical Engineering at the University of Delaware, Newark, DE, where he is currently an Associate Professor. His research interests lie in the area of dynamics and control with applications to robotic systems, particularly to dynamically dexterous legged robots. Dr. Poulakakis received the National Science Foundation CAREER Award in 214.