Legged locomotion is attracting the interest of researchers of a broad range of areas. Engineers, biologists and neurologists are all pooling their knowledge for the success of a legged locomotion device [1]. The major impetus for this technology is coming from the government of the United States of America, which has given a boost to the topic through a significant number of programs sponsored by the Defense Advanced Research Projects Agency (DARPA) in the last ten years [2]. Some of these programs started in 2001 and ended recently while others are still open. The more relevant examples are the Learning Locomotion Program [3]; The BigDog Program [4]; The Exoskeletons for Human Performance Augmentation Program [5] and the more recent Legged Squad Support System Program [6]. However, despite the strong impulse in the research and the forecasted demand for these robots [7], very few advances in real applications exist. The challenges of autonomy and the large power-to-weight ratio demanded for the actuators make conventional actuation and control technology, inherited from industrial robotics, inadequate for the most promising field and service applications of legged locomotion which exhibit significant expected impact on the future society [7]. The HADE (Hybrid Actuator Development) project [8] developed at the Centre for Automation and Robotics at the Spanish National Research Council (CSIC) aims primarily at solving this problem by establishing a new line of research focused on specific actuation and control technologies for the new generation of legged robots: Agile-locomotion robots.

Quadruped robots, emulating their biological counterparts, are the best choice for field missions in a natural environment. However, it is well known that current legged-locomotion devices feature high complexity and very low speed particularly if high payloads have to be transported, and are far from reaching the performance of biological quadrupeds in natural environments.

Table 1 lists most relevant quadruped robots developed in the last 12 years, showing their payload-to-weight ratios and maximum dimensionless forward speed, which has been computed from robot dimensions and speed published by the robot’s authors as u ^ = FR, where FR is the Froude number, obtained from [18]: (1) FR = v 2 gLbeing v, the forward speed; g = 9.81 m s⁻²; and L, the characteristic leg length.

A legged vehicle designed to perform in a natural terrain should be provided with optimum performance against mobility, payload, and endurance. Such specifications were imposed by DARPA for an Agile Ground Vehicle in the UGCV program [19]. A similar denomination is here used for a legged ground vehicle, which we call “Agile” if it is able to reach a dimensionless speed of u = 0.54 and features a payload-to-weight ratio larger than 1, to be comparable with biological quadrupeds. Besides, a quadruped robot to perform using high-speed gaits (i.e., trot and gallop) must make a thrust to the ground yielding dynamic impact loads that could exceed three times the static load on the supporting leg [20]. In a trot or in a dynamic walk, where two legs thrust the ground simultaneously, both legs share the body weight and payload, so the static load on each leg is one half the robot’s weight and added payload. Therefore, in a trot gait dynamic loads on each leg can reach 1.5 times the robot’s weight and payload. For a dynamic walk, dynamic impact loads at each leg approximately equal the robot’s weight and payload. Therefore, the structural design of an agile robot leg should make sure a load capacity-to-robot’s weight ratio of 1–1.5 depending on the envisaged gait.

Biological quadrupeds transition from walking to running gaits at a dimensionless speed between 0.54 to 0.7. Concretely, horses transition from walk to trot at û = 0.59 [21]. As exceeding this dimensionless speed requires the quadruped to run (trot or gallop) and some complex terrains could impede the use of those high-speed gaits, we have considered the dimensionless speed of 0.54 as the lower speed limit for a legged robot to be considered agile. By having a look to Table 1 it is noticed that very few quadrupeds achieve agile locomotion performance, because those robots featuring dimensionless speed above 0.54 have almost negligible payload-to-weight ratio. Although some research labs are working in this direction (Stanford University [22], Italian Institute of Technology [23]), the only existing robot reaching those targets is BigDog [4], a quadruped under development at Boston Dynamics (USA) (see Figure 1). The BigDog project is sponsored by DARPA, the US Marine Corps and the US Army. The goal of the BigDog Project is to build a class of agile unmanned ground vehicles with rough-terrain mobility superior to existing wheeled and tracked vehicles. The BigDog robots currently built “have taken the steps toward these goals, though there remains significant work to be done” [4]. Unfortunately, the ins and outs of the technology underlying BigDog are not available to the research community.

Supplying power to a high-speed legged machine using currently available actuation technology is a challenge. This is particularly true if the machine is expected to be energetically autonomous. In dynamic locomotion, the load experienced by each leg is at least three times the static load on that leg, and it may be much more for running gaits. The cost of building a structure and actuation system capable of providing the performance needed for the dynamic locomotion of a mid- to large-sized machine is prohibitive, even without considering payload.

Considering the mammalian muscle as a reference, direct measurements of muscle function have yielded insight into the versatile way muscles operate. It has been discovered that muscles act as motors, brakes, springs, dampers and struts [24]. The multifunctionality of natural muscle distinguishes it from any human-made actuator and it may hold the key to the success of legged locomotion. In many biological tissues it is hard to distinguish between material and structure. The use of viscoelastic materials can give the robot the spring-mass energy-cycling capacities of legged-animal locomotion, which also reduces the computational complexity of the control. Viscoelastic materials greatly simplify the mechanics of the robot, serving simultaneously as shock absorbers, springs and complete joints.

The spring-mass energy-cycling capabilities can play a key role in the dynamic locomotion of a legged vehicle. Kinetic energy can only be put into the system when the foot is on the ground. It is necessary to keep the mechanical energy in the system by using internal energy storage, that is, compliant actuation. Moreover, the leg is a mechanical oscillator, and it is energetically expensive to drive it at a frequency significantly different to its natural frequency [20]. Any means to modify the natural frequency on the leg would help to make it oscillate at different frequencies with optimal energy expenditure. Thus, inherent adaptable compliance is required [25]. Therefore, to efficiently run a dynamic legged vehicle, high power-density high force-density fast actuators with adaptable compliance are required. Added to this, energetic autonomy is expected. It is evident that these requirements are not met together by conventional technology [26].

HADE is a long-term project [8] aimed at designing energy efficient, large power-to-weight ratio actuators and energy-efficient-locomotion control schemes for the new generation of legged robots following natural muscle multifunctionality. This multifunctionality is approached by means of merging different technologies (smart materials and conventional technologies) in order to extract the best properties of each one. Some prototypes have already been tested and characterized [27].

This paper presents the development of a biomimetic model of a leg for agile locomotion of quadruped robots. The key principles underlying the superior capabilities of strength, speed, agility and endurance of cursorial mammals, like horses, are analyzed in Section 2 and transferred to technological instantiations in Section 3, where a model of a biomimetic leg for agile locomotion is presented. The proposed concept has been implemented on a real prototype. Section 4 describes the leg design, actuation system and sensorial system. Section 5 describes how variable compliance is achieved at the joints of the leg. Experimental analysis of the leg performance to achieve agile locomotion is analyzed in Section 6, and finally Section 7 presents a discussion on the technological barriers that have been encountered in the technological instantiation of the biomimetic leg model and concludes with some proposals.

Taking into consideration that building a quadruped with the size of a horse would be difficult to handle in the laboratory, scaling of the leg length making sure that the effective leg length is the 60% of the body size would comply with the specifications. For a trade-off between reproducing horses dimensions and having a reliable prototype, a scaling factor of 65% has been applied to the design, therefore a robot length of 1.2 m was considered, having an effective leg length of 0.8 m.

The complexity of controlling a planar 4-DoF redundant kinematic chain added to the cost of electronics and actuators and the direct consequence of increasing leg mass as the number of degrees of freedom increases make unfeasible the development of an exact horse-like leg. However, the election of redundant kinematics favors reducing joint torques and thus actuator requirements and power consumption. A possible solution is to use passive elastic elements to drive one or more joints, however, the analysis of joint power requirements for a slow trot (see Section 3.4) advises against purely passive actuation. As a trade-off, a planar 3-DoF leg has been outlined composed of three links: thigh, crus and hoof, connected through 1-DoF joints: the hip, knee and fetlock joints. The lengths of thigh, crus and hoof are proportional to real horse leg’s plus a scaling factor to reach the desired effective leg length, taking into consideration that the use of a 3-DoF model of leg shortens the total leg length in a 34% compared to a horse leg. The 35% reduction in effective leg length plus the increase of 34% in limb length results in a final 1% decrease in each leg link length. Table 3 lists the final link lengths.

Figure 3 and Table 4 show Denavit–Hartenberg parameters for the leg kinematics, which correspond to a conventional three-link planar structure. Following this convention, the direct kinematic model provides hoof position and orientation from joint angles as follows: (2) [ x 0 y 0 φ ] = [ a 1 C 1 a 2 C 12 a 3 C 123 a 1 S 1 a 2 S 12 a 3 S 123 θ 1 θ 2 θ 3 ]where (x₀, y₀, ϕ) are hoof x and y position and orientation respectively in the leg’s base reference frame, and θ_i with i = 1..3 are joint angles numbered from hip to fetlock joint. Parameters a_i are the respective link lengths measured as the distance between adjacent joint axes, and correspond to the values listed in Table 3. In Equation (2) C_i and S_i mean cos(θ_i) and sin(θ_i) respectively, while expression C_ijk means cos(θ_i + θ_j + θ_k) and S_ijk means sin(θ_i + θ_j + θ_k).

Published work on the experimental determination of equine limb inertial properties show wide ranges of average values for leg segment masses for different horse breeds. Table 5 summarizes average results of an experimental work performed on six Dutch Warmblood horses [36]. Considering that our leg model accounts with three links, the selection of link masses cannot be directly extracted from the biological inertial data, and therefore it was performed in an iterative optimization approach for a later comparison with the average values listed in Table 5. In the optimization approach, the link masses which minimized mechanical power in a locomotion cycle at an average nondimensional leg speed of 0.54 were searched for. The cost function is given by the sum of the mechanical power at the leg joints, given by the product of joint torque and joint speed as follows: (3) C W = ∑ i = 1 3 ∫ 0 T τ i ( t ) ⋅ θ ˙ i ( t ) d twhere joint torque is a nonlinear function of all limb masses m_i, lengths a_i, inertia moments I_i and joint angles, speeds and accelerations, given by the inverse dynamics model of the leg: (4) τ = D ( m i , a i , I i , θ , θ ˙ , θ ¨ )

Numerically solving the above optimization problem is computationally unavoidable. Therefore, it has been solved by means of an iterative process through dynamic simulation of the leg model using Yobotics! Simulation Construction Set software [37]. This dynamics simulation package was developed at the MIT Leg Laboratory for the analysis of control algorithms in legged locomotion, and it was later commercialized by Yobotics Inc., spin off company from the MIT. The programmed robot simulation provides joint position, speed and torque based on link inertial properties such as mass, center of mass position and inertia tensor by implementing the Featherstone algorithm for deriving the equations of motion. Figure 4 shows results of the iterative process for hip and knee joints, whose variation resulted more significant, and its convergence for the final optimal leg mass distribution.

After iteration, the resulting leg mass distribution was 2.5 kg, 1.9 kg and 0.6 kg for thigh, crus and hoof respectively, which represent a 50%, 38% and 12% of the final leg weight which results 5 kg. From this results and by comparison with biological mass distribution shown in Table 5 it seems that the mass corresponding to the suppressed metatarsus has been distributed between crus and hoof in order to resemble the inertial properties of horse legs. The resulting mass distribution maintains most of the leg mass in the upper leg segment, as in biological horse legs.

The technological counterpart of the mammalian muscle is the joint actuation system. In order to determine actuator requirements for each joint, the leg motion was simulated again using the Yobotics! Simulation Construction Set. In order to reach a nondimensional speed of 0.54 for an effective leg length of 0.8 m, an average leg speed of 1.5 m/s was commanded, following Equation (1), and an additional payload of 12 kg was added over the leg. These 12 kg account for one half of the robot weight, which has been assumed to be around 12 kg, and one half of a 12-kg payload carried by the robot. Therefore, the leg was assumed to move in agile locomotion (nondimensional speed of 0.54, supporting dynamic loads and robot payload-to-weight of 1). The details on the locomotion controller used in the simulation can be found in Reference [38].

Figure 5 shows screen-shots of the simulated locomotion cycle. Joint torque, speed and power requirements obtained from the dynamic simulation are shown in Figure 6.

Figure 6 shows a peak demand on hip joint torque near 150 Nm at the beginning of the stance phase. The torque imposed on the hip is mainly used for supporting the robot weight and payload and to propel the robot forward in order to achieve the forward speed of 1.5 m/s. Therefore, the hip accelerates the leg structure, demanding an instantaneous mechanical power of 300 Watts. The knee also contributes to this propulsion, however in a less demanding manner. During toe off, the knee and fetlock do most of the work lifting the hoof. The figure shows large joint speeds required for such motion, as the fetlock has a power requirement of 220 Watts. The swing phase of the leg seems to be the less power-demanding motion.

In order to analyze the suitability of commercially available actuators for the power requirements found, Figure 7 shows the speed-torque diagram of a brushless DC motor by Moog model Silencer BN23-23ZL-03LH. The diagram also shows, overlapped, the joint requirements obtained from the simulation plotted in the form of speed vs. torque. The diagram shows that the selected motor fits within the specifications, provided that reduction ratios of 1:290, 1:180 and 1:190 are applied for hip, knee and fetlock joints respectively. Although some parts of the joint trajectories fall within the intermittent operation range of the motor, from Figure 6 we can observe that the duration of the large-power spans is almost instantaneous, lasting less than 50 milliseconds, which is supported by the intermittent motor operation. Therefore, the selected motor is suitable for the application in terms of power requirements. Besides, a light weight (0.5 kg) and a compact design provide suitable power-to-weight and power-to-volume ratios.

In biological quadrupeds energy is stored in tendons during some phases of the locomotion cycle and later released to provide kinetic energy to the leg in those phases which demand a larger power consumption. In a horse, this energetic conversion occurs in several parts of its body, however here we will concentrate on the use of elastic recoil in tendons of the hind legs, which is indeed the principal source of energy in horses. The flexing and then straightening of the hind legs is the mechanism that first stores and then releases energy making use of the tendons. In a horse, energy is stored in the musculotendinous system during the first half of the support phase by stretching of the viscoelastic connective tissues. During the second half of support much of the stored energy is recovered as the connective tissues shorten [39].

In order to analyze the energetic advantage of introducing elastic energy recovery in the robot leg, a spring has been attached to the simulated leg from hip to fetlock joint. The leg motion is again commanded at an average leg speed of 1.5 m/s. The spring stiffness is increased in each cycle. Figure 8 shows the effect of increasing spring stiffness on the vertical propulsion force exerted by the actuators on the leg’s base reference frame. Although simulations were performed for spring stiffness varying up to 650 N/m, a linear tendency line has been plotted in order to show the behavior for larger stiffness. As it can be observed, as long as the spring stiffness increases, the vertical force exerted by the spring increases, what results in a lower force demand provided by the actuators. The tendency lines cross at a spring stiffness of 980 N/m and for larger stiffness the propulsion force exerted by the spring exceeds the propulsion force required by the actuators. Figure 9 shows how the peak power requirement at each joint diminishes as the spring stiffness increases. It also shows a larger dependency on spring stiffness at hip and fetlock, while the knee power shows a minor variation. For the maximum simulated spring stiffness of 650 N/m a 31% of power is reduced due to the elastic energy recovery. However, it is worthwhile noting that the spring stiffness should not be increased too much, because a too large stiffness would dominate the leg dynamics, neglecting the action of the actuators thus yielding a pure spring-mass system without possibility of adaptation to the environment. The effect of leg elasticity on leg speed is shown in Figure 10. In this figure it can be observed that, although a leg speed of 1.5 m/s is commanded, the increase in leg stiffness produces a progressive rise in leg speed by reducing the support time.

As larger values of spring stiffness would dominate the leg dynamics interfering in the locomotion controller, for this biomimetic leg model we have considered a spring stiffness of 650 N/m as a trade-off between leg control performance and power efficiency by reducing the power consumption in a 31%.

Following the biomimetic leg concept herein presented, a real leg prototype has been designed, developed and tested. Section 4 presents the design process and final prototype, while the following sections will go in deep into the experimental locomotion. Conclusions on the effectiveness of translating biomimetic features to robot prototypes are finally given.

The design of the first prototype of the HADE leg is shown in Figure 12. It is a planar 3-DoF leg composed of three links: thigh, shank (including metatarsal) and hoof, connected through the hip, knee and fetlock joints. The mechanical structure of each link has been designed in order to achieve large payload-to-weight ratio and provide impact tolerance. The hoof is endowed with a rubber pad which provides shock absorption and damping at ground contact and also increases friction between foot and ground improving horizontal propulsion.

The large payload-to-weight requirements of the structure could be reached by manufacturing the mechanical structure using Alumec 89, a high strength aluminium alloy which undergoes a special cold stretching operation for maximum stress relief. This material is being used in the aerospace industry and it shows the best payload to weight properties. Unfortunately, these special alloys are only at the reach of big companies of the aerospace industry and it was not possible to obtain the Alumec 89 for the manufacturing of a robotic leg prototype. Therefore, for the purpose of testing the performance of the biomimetic leg concept, the leg has been manufactured in Aluminium 7075, obviously keeping in mind the increase in weight that the final prototype will suffer due to the different material mechanical properties. Table 6 compares mechanical properties for Aluminium 7075 and Alumec89.

Table 7 shows link dimensions and masses of the first prototype. The position of center of mass and inertia tensor are referred to the joint reference frame following the Denavit–Hartenberg convention as shown in Figure 3. Joint ranges of motion and leg kinematics have been detailed in the Appendix.

In order to select the actuators for the HADE leg, the observed joint requirements were transferred back to the motors taking the transmission system into account.

The state of the HADE leg system is monitored by means of torque and position measurements at each joint, and the measurement of ground-reaction forces. Joint torques and angular positions are measured by force and position sensors respectively, which are enclosed in each SEA, while the ground-reaction force is acquired by a force sensor placed inside the pad of the hoof. The force provided by the actuators is measured using linear encoders which measure the spring deflection, thus computing the force on the load using Hook’s law. The position sensors at the actuators are also linear encoders that measure the displacement of the shaft. Specifications of these encoders are listed in Table 8.

The force sensor at the pad to measure ground-reaction forces is a Honeywell miniature precision load cell model 34 (P/N AL312CR). This sensor is placed inside the pad (see Figure 16) to measure vertical ground forces ranging from 9.8 N to 2200 N. The sensor signal is acquired by a 16-bit A/D converter which provides a resolution of 2.10⁻¹³ N. High accuracies of 0.15%–0.25% full scale are achieved. Residual effects of off-axis loads are minimized.

Vertical ground reaction forces are used by the locomotion controller to maintain a desired ground contact force during the stance phase. Figure 17 shows vertical ground forces at the hoof during one locomotion cycle for the leg walking at 0.5 m/s and compares them with the simulated forces for the robot walking at 1.5 m/s assuming a lightweight leg design. The figure shows a sharp impact to the ground at the beginning of the cycle coinciding with the beginning of the stance phase. This impact seems to soften at lower speeds, such as 0.5 m/s although larger loads are involved considering the increase in link weight in the current leg prototype.

Series elasticity provides good force control at the joints and adds some shock tolerance to the mechanism. Also, it decouples the motor inertia from the load, so that slight position errors will be absorbed by the elastic element, preventing the control system from reacting to unexpected impacts. A compliance controller programmed in the SEA imitates a desired dynamic behavior inside the actuator response bandwidth. As in this case the controlled actuator bandwidth is 35 Hz, then in the range 0–35 Hz the controller allows the actuator to behave like a spring-damper. The block diagram of the compliance control scheme is shown in Figure 18, where G_z is the model corresponding to a spring-damper system, G_c is the transfer function of the inner loop controller, G_p is the actuator’s transfer function, K_s is the die compression spring constant, f i d, f_i and f i r are the desired force, exerted force, and the force reference respectively. Finally, s i d, s_i and s i a are the desired position, the load position and the actuator position. Choosing G_z = K (1 + bs) and by modifying the desired spring-damper coefficients K and b, the mechanical impedance is modified and it can be adapted to different leg velocities and ground stiffness. Figure 19 shows a Bode diagram of the system behavior (thin line) overlapped with an ideal spring damper (thick line) system, which shows that for frequencies below the controlled actuator bandwidth (35 Hz), the controlled actuator behaves like a spring-damper. These properties largely improve the adaptation to the ground in high-speed locomotion.

The inner force controller in Figure 18 is based on conventional PID control acting directly on the current command of the SEA’s power amplifier. Figure 20 shows a block diagram of the inner force control scheme.

Active damping is specially required at the knee in order to achieve a natural motion [41]. Such active damping could be provided by means of active compliance control of the actuators, however, at a cost of energy consumption considering that the knee is mostly dissipating power during the gait cycle. However, it is strongly required to reduce the energy consumed by the actuators during the damping motion in these applications where energy efficiency is a primary goal. Therefore, the use of a passive damping device of controllable damping coefficient is desirable.

In this project a LORD RD-2087-01 Magneto-Rheological (MR) rotary damper has been placed at the knee to test the efficiency of actively controlling knee damping along the gait cycle (see Figure 21(a)). Magneto-rheological brakes and dampers are resistive actuators based on Magneto-Rheological Fluids, a kind of smart material of the Magnetoactive family [26]. The MR rotary damper is designed as shown in Figure 21(b): An outer housing attached to the thigh contains a rotor joined to the knee axis. The MR fluid fills the space between the rotor and the outer housing. There is a coil attached to the housing, leaving a small space for the MR fluid between the rotor and the coil. As electric current circulates by the coil, a magnetic field is applied to the MR fluid, which yields a significant change in the rheologic characteristics of the MR fluid which is due to the alignment of the ferromagnetic particles of the MR fluid into chains. As a result, an increased shear stress results between rotor and housing, and consequently an increased resistive torque output is achieved in less than 10⁻² s response time. The increase of torque that damps the rotor depends directly on the magnetic field applied and increases linearly with rotor speed, thus providing controllable viscous rotary damping. A detailed background on the principle of operation of Magneto-rheological Fluid (MRF) devices can be found in Reference [42]. Figure 12 shows the integration of the MR rotary damper inside the knee of the HADE leg. The maximum power consumption of the MR rotary damper is 1 A at 12 VDC for a maximum resistive torque of 4 Nm. Figure 22 shows the typical torque-current curve for the MR rotary damper provided by the manufacturer.

Using the MR rotary damper, the SEAs are used for active motion, while passive damping is performed by the MR rotary damper. The MR rotary damper is commanded through a device controller kit provided by the MRF damper manufacturer. This controller device provides closed loop current control to compensate for changing electrical loads up to the limits of the power supply. The controller device is operated as an interface device for computer control of the MRF damper. When a continuous 0–5 VDC signal is sent to the controller device it commands a proportional pulse-width modulated current signal, where 0% duty cycle matches 0 A and 100% duty cycle matches 1A. A finite-state machine switches between stance and swing states and changes the voltage input to the controller device which in turn modifies the knee damping coefficient. The finite state machine switches from stance to swing when the foot/ground reaction force sensed becomes 0 N and returns to stance state when the ground reaction force reaches 10 N. A block diagram of the MRF control scheme is shown in Figure 23.

The controller gains have been tuned manually to achieve a natural-looking motion, which are 1.45 Nm rad⁻¹ s during the support phase and 0.9 Nm rad⁻¹ s during the leg swing phase.