Dynamic wind-up locomotion enabled by embodied intelligence

Abstract

Repetitive subtasks of locomotion are offloaded from a conventional computer-actuator-sensor set-up to automatic mechanical processes. The subtasks considered are: (1) when out-of-contact with the environment, move a leg to a ready position in preparation for step contact, and (2) when contact is detected, push off the ground. Using conventional closed-loop control, subtask (1) would be accomplished by programming logic and a feedback loop onto a computer-motor-encoder system, and subtask (2) would be accomplished by sensing contact, then commanding the leg motor to push-off via programmed computer logic. We demonstrate how to transition this programmed logic from a computer processor to a mechanical processor. The mechanical processor performs preprogrammed actions based on combinations of states of components, some of which are internal and some that interact with the environment. Because signals are not digital, but rather mechanical quantities of energy, position, and force; transitioning to a mechanical processor enables a third subtask not possible by the computer alone: that is, (3) the accumulation of elastic energy while out-of-contact with the environment, and its automatic release upon contact for a more powerful push-off motion. Migrating processing out of the computer reduces the number of transduction steps, allows for faster responses to dynamic events, and instantiates a high-powered reflex triggered by ground contact. To illustrate these benefits, a robot with built-in onboard mechanical processing is compared to a conventional robot with logic executed by an offboard computer.

Keywords

actuation and joint mechanisms hardware-software integration in robotics natural machine motion computer architecture for robotics and automation

Introduction

The foot of a robot makes and breaks contact with the ground often. When not in contact, it should prepare for the next contact, positioning itself appropriately. When the next contact occurs, it should push-off the ground to locomote the robot. Without this control, the foot would make contact with the ground in an unpredictable configuration. This control is conventionally accomplished by a sophisticated arrangement of actuators, sensors, and computers. For example, the position of the leg is controlled by motors commanded according to a feedback control law programmed onto a computer taking signals from encoders. Contact may be detected by reading spikes from encoders installed directly on the leg, or by estimating the ground force from motor current sensors, or by installing a discrete contact sensor at the foot. Once the computer is informed that stance phase has begun, it sends signals to the leg motors so that the foot pushes off the ground under force or position control enabled by current or encoder readings. This paradigm, although conventional, requires high-speed communication between motors, sensors, and a computer in order to react to fast dynamic events, particularly contact detection. It would be beneficial if these low-level subtasks were performed automatically by the leg itself without the need of high bandwidth computer control.

The goal of this work is to migrate these low-level subtasks from a computer processor to a mechanical processor. A mechanical processor is a mechanism that responds to a set of component states by performing a set of component actions. For example, when the ground force acting on the foot is null, and the leg is not yet positioned for oncoming stance, the leg should move to the ready position, then stay until the onset of stance phase. A mechanical processor would perform these subtasks automatically without the need of motor control by reacting to various component states. When the foot is in a no contact state, and the state of the leg is not at the ready position, then the action of the motor is to advance. Once the state of the leg is at the ready position, the action of the motor is to stay. Then once the foot switches to a contact state, the action of the motor is to push-off.

Mechanically this is accomplished by designing the leg mechanism to have a kinematic singularity (more specifically, an input singularity, see Bohigas et al. (2017)) at its ready position. At the singularity, the torque of the motor (no matter how large) cannot advance the mechanism, so the motor and leg stays at the ready position. However, by the designed properties of this singularity, it only takes a small contact force to advance the mechanism past the singularity, enabling the motor torque to transform into a foot force at the onset of stance, pushing off the ground. These actions can all proceed as powered by a motor receiving a constant signal, for example, a brushed motor connected to a battery with no other electronics.

A key difference between a computer processor and a mechanical processor is that the signals of the former are digital while the signals of the latter are mechanical quantities of force/torque, position, and energy. Unlike digital signals, these mechanical signals are not only used for communication, but are consequential to environmental interaction. Large magnitude signals can be exploited to enhance task goals and add functionality. In this work, we exploit the large magnitude torque that builds up at the singularity present at the leg’s ready position. By placing a torsion spring in series after the motor, this large torque deflects the spring, resulting in substantial elastic energy storage while the foot is out-of-contact with the environment. The event of the foot contacting the ground moves the mechanism out from the kinematic singularity, and the elastic energy is released into a push-off motion with greater power than if the motor were to act alone. This results in a high-powered push-off reflex triggered by contact with the environment. The key principle is that the motor is able to add energy to the system both when the foot is not in contact (by doing work on the spring), and when the foot is in contact (by doing work on the robot directly). Giving the motor extra time to power the system is central to enhancing its energetic output during a step. This extra functionality is not possible with a conventional computer processor unless an additional actuator or clutch is installed and wired to the computer. Commercial-off-the-shelf (COT) clutches tend to be heavy. For example, the AMC electromagnetic clutch that could hold 2–4 Nm weighs 400–700 g. Enabling out-of-contact spring charging within the weight limits and without the addition of a clutch may have advantages.

Mechanical processor (this paper)

The mechanical processor developed in this paper is shown in Figure 1(top). It is powered by a brushless motor receiving a constant signal. For electronic commutation, the brushless motor requires a driver board which we categorize as a lower level computer. Hardware specifications are given in Construction of prototypes. The brushless motor powers the mechanical processor which performs all control logic for the three low-level subtasks mechanically programmed in this paper:

(1) While out-of-contact with the environment, move the leg to the ready position.

(Displayed as “Position ctrl loop” in Figure 1(a))

(2) Detect contact and react by pushing off the ground.

(Displayed as “Contact detection/reaction ctrl loop” in Figure 1(a))

(3) While out-of-contact, charge a spring for use during the next push-off.

(Displayed as “Spring charging ctrl loop” in Figure 1(a))

The control logic of the mechanical processor is shown in Figure 1(b), which we refer to as a truth table. The mechanical processor responds to the state of three different components:

• Spring – (1) not fully energized, (2) fully energized

• Leg linkage – (1) not at the ready position, (2) at the ready position

• Foot – (1) no contact, (2) contact

For each of the eight state combinations (as shown in Figure 1(b)), the following two components take various actions:

• Motor – (1) stay, (2) advance, (3) charge spring

• Leg linkage – (1) stay, (2) advance, (3) push-off

The control logic of the first two columns of the truth table in Figure 1(b) can be interpreted as:

(1) If the spring is not fully energized, the leg is not at the ready position, and the foot is not in contact, then the motor and leg should advance.

(2) If the spring is not fully energized, the leg is at the ready position, and the foot is not in contact, then the motor should charge the spring and the leg should stay put.

…and so forth

A kinematic diagram corresponding to Figure 1(d) is displayed in Figure 2(a).

Computer controlled mechanism (for comparison)

For comparison, a second computer controlled robot was built, shown in Figure 1(bottom). Its leg mechanism possesses no input singularity, making it incapable of inherently performing the three subtasks described above. The computer takes on this role, and the mechanism performs no mechanical processing. Control loops are formed between a high-level computer and sensors installed on the robot’s body. The control logic of the computer processor is shown in the truth table of Figure 1(g). Compared to the mechanical processor, the logic is simpler because the spring charging control loop is not present, leaving the motor and leg to always act in unison (due to negligible spring deflection). To recover this functionality in a computer controlled robot, a clutch could be installed that fixes the leg when not in contact with ground to allow spring charging, then releases the leg and elastic energy after contact is detected. This strategy is illustrated in Figure 3(a), and would require high-speed communication.

As an additional experimental control, the computer controlled mechanism was implemented with no programmed logic at all, simply sending the motor a constant signal such that it rotated nominally at 40 rad/s (384 r/min) under no load. We refer to this as the no processor case.

In this paper, we compare the use of mechanical processor to a computer processor as well as using no processor at all. Experimental and simulation studies were conducted with various gravitational constants to assess the energetic output and consistency of locomotion for each case. To conduct these studies, small robots were built that hop in the x-y plane (using a planarizing boom) with their body rotation axis fixed, see Figure 4(top).

The novelty of this work is the design of the mechanical processor (which uses high-speed mechanical logic to invoke a high-powered mechanical reflex capable of energizing with an undersized motor while out-of-contact with the environment). The contribution is the theoretical and experimental characterization of the mechanical processor as an example of embodied intelligence. The generalizability is the principle of dynamic wind-up locomotion, which we believe has applications to other types of legged robots or arms.

A Literature review is provided in the upcoming section. In Methods, design principles are detailed in the section Programming a mechanical processor, details on how the control is accomplished for each hopper is in the section Hopping c ontrol . Furthermore, details are expanded upon in regards to Construction of p rototypes shown in Figure 1, set-up of the Reflex energy experiment, the Gait consistency experiment, the Hop h eight m odulation e xperiment , and formulation of the Dynamic s imulation studies performed in this work. In the Results, outcomes of the experiments and simulation studies are conducted to assess Reflex e nergy , Gait c onsistency , and Hop h eight m odulation. Hardware demonstrations of robot locomotion on complex terrain Demonstrations on complex terrain are presented.

Literature review

Mechanical computation has been discussed extensively in modern works. A perspective paper by Yasuda et al. (2021) provides the example of a soft finger conforming around an object as a form of mechanical computation. In a more recent perspective paper by He et al. (2024), forms of mechanical computing are framed (noncomprehensively) as combinations of stimuli-responsive materials with metamaterial mechanisms to achieve autonomous responses such as locomotion, manipulation, morphing, and sensing. Examples of using stimuli-responsive materials for locomotion, albeit without the use of metamaterials, include Yan et al. (2023) and Meng et al. (2020). Several works frame the goal of mechanical computing as performing general computing tasks, see Yasuda et al. (2021), Mei et al. (2021), El Helou et al. (2022), Lee et al. (2022), and Mei and Chen (2023). The last four of these built and tested mechanical computers, demonstrating the ability to produce logic gates, add numbers, and emulate neural networks. This work differs in that we do not aim to replace electronic computers with mechanical components. A major contrast is our emphasis on mechanical energy enhancement, a priority which is omitted from past perspective papers. Our work is better framed by the perspective paper of Sitti (2021), who describes embodied intelligence as a coupling of physical intelligence and computational intelligence. This understanding originates from Pfeifer et al. (2007). Mengaldo et al. (2022) focuses less on coupling to the brain in their interpretation, perhaps leaning more toward what Sitti calls “physical intelligence.” Nonetheless, these works stress the importance of the body’s roles in information processing.

The field of prosthetic design has shown interest in mechanically automating energy storage subtasks during locomotion. Tran et al. (2022) reduced the weight of a knee and ankle prosthesis by incorporating key functionalities into mechanical elements rather than increasing the number or size of actuators. They built a “torque-sensitive” knee, capable of adjusting its mechanical advantage based on loading conditions. Mazzarini et al. (2023) designed a module that retrofits onto an off-the-shelf passive ankle-foot prosthesis to control the storage and release of energy in its leaf spring structure in order to enhance push-off. The robotic ankle incorporated a contact sensor in the foot to detect gait events. Cherelle et al. (2017) accomplished a similar energetic enhancement via a locking mechanism that enabled energy storage during early stance.

Past works have shown hopping/skipping gaits to be an efficient choice for low gravity environments, see Ackermann and van Den Bogert (2012); Pavei et al. (2015). This has inspired the current work to investigate performance while varying the gravitational constant. Arm et al. (2019) built SpaceBok, capable of exploring extraterrestrial environments using a pronking gait. Montminy et al. (2008) built a hopping robot for exploring Mars that uses SMA (shape memory alloy) actuators powered by diurnal temperature variations.

While many past works have incorporated springs to increase power output, few have mechanically automated contact detection/reaction or out-of-contact position control. Brown and Zeglin (1998); Zeglin (1999) achieved a thrust reflex by flexing a bow leg using a revolving pulley that pushed on a string, then subsequently slipped off the pulley through impact with the ground, releasing the bow energy. They referred to their design as a “programmable mechanism.” Similarly, Nichol et al. (2004) achieved automatic energy release by designing a capstan with a special release bevel cut into it such that its winding cable slipped off upon impact. Inspired by spinal reflex loops in animals, Masuda et al. (2020) developed an artificial receptor-muscle-neuron system from pneumatic components capable of an autonomous running gait in the absence of electronics. A review by Hawkes and Cutkosky (2018) cites further examples where responsive mechanisms appear in nature and engineering for the purposes of energy storage/release, decreased response times to sudden events, and passive open-loop behaviors, among others.

Various past works have employed springs to move beyond actuator power limits. The Salto robot (Haldane et al. (2016); Plecnik et al. (2017)) actuated a variable mechanical advantage leg linkage with a series-elastic actuator. A low mechanical advantage at the beginning of stroke allowed for a transient accumulation of spring energy that amplified the robot’s jump height. Hawkes et al. (2022) built a 22.5 g jumper capable of accumulating 24.2 J of strain energy over 32 s, producing a single jump of 30 m upon release. Inspired by the extreme accelerations of the mantis shrimp’s striking appendage, Steinhardt et al. (2021) and Wells et al. (2022) designed a micro-robot that uses the overcenter position of a four-bar to mediate the storage and release of elastic energy. Similarly, in this work we employ input singularities to enable the storage of elastic energy. But this work differs from past robots that employ a slow wind for a powerful burst in that our robot spends no time dwelling on the ground while winding. Shin et al. (2022) used a cam to store and release energy in a repeated hopping cycle. De and Koditschek (2015) used an actuated tail to power hops of a robot with springy legs that were unactuated in the radial thrust direction. Tsuda et al. (2012) used the phenomena of snap-through buckling of closed elastica to create a hopping robot.

Fewer works have exploited the time periods out-of-contact with the environment to energize a system. The hopper of Brown and Zeglin (1998); Zeglin (1999) was able to flex its bow leg in the air by pushing on its connecting string with a revolving pulley. Similarly, Nichol et al. (2004) used a capstan to flex a spring-loaded leg. This design was later modified by Estremera and Waldron (2008) to incorporate pneumatics for additional control of energy storage and release. Scarfogliero et al. (2006) packaged this functionality into a 10 g robot.

Other researchers have focused on increasing force/torque output, rather than energy. Inspired by running emus and ostriches, Badri-Spröwitz et al. (2022) arranged springs and tendons to functionally form an automatic clutching/unclutching system. Clutching engaged during stance to perform weight compensation, and disengaged during swing to ease leg recycling. Using electrohydraulic artificial muscles as actuators, Buchner et al. (2024) demonstrated a conventional 2-link robot leg hopping on a variety of complex terrains. In this paper, we demonstrate how to make gains by embedding intelligence into the body rather than seeking out stronger actuators.

It is advantageous for a system, such as that studied in this paper, to exhibit a stable gait given an open-loop signal. Cham and Cutkosky (2007) analyzed a pneumatically powered vertical hopper’s motions through open-loop control without sensory feedback, concluding that outside of a certain bound of parameters, unpredictable and non-periodic motions emerge. Hexapedal legged robots known to produce periodic motions with open-loop control include iSprawl (Kim et al. (2006)), RHex (Saranli et al. (2001)), and VelociRoACH (Haldane et al. (2013)). These robots were inspired by cockroach locomotion which uses mechanical feedback without altering neural signals to rapidly traverse complex terrain, see Sponberg and Full (2008).

Although not a focus of this paper, several past works have focused on mechanically automating balance control, or in the same vein, relieving the control effort to perform balance control. Fastrunner (Cotton et al. (2012)) and the KAIST Raptor (Park et al. (2014)) achieved stability in the sagittal place during high-speed running. Blickhan et al. (2007) constructed hoppers that self-stabilize under oscillatory fore-aft movements at the hip using the effects of nonlinear mechanics to increase robustness without the need to sense disturbances. Passive-dynamic walkers, see Collins et al. (2005), that walk down a slope without control and actuation also exhibit self-stabilization. The Steinkamp hopper (Stiesberg et al. (2017)) demonstrated the ability to hop down slopes under gravity power, exploiting the passive dynamics of its spring leg. Compared to passive-dynamic hoppers and walkers (McGeer (1990)), the current work consigns embodied intelligence into both passive and active aspects of the robot. As opposed to passive-dynamic walkers, here we exploit an actuation system. The mechanical processor takes advantage of the series-elastic actuator to increase the robot’s capacity to perform positive work tasks such as stair climbing, which is in contrast to gravity-powered robots.

We term the emergent behavior of our new device as dynamic wind-up locomotion. Although this article fixates on hopping, the concept of dynamic wind-up locomotion serves as a transferable principle with applications to powered prosthetics, hexapedal robots, or in the knee, ankle, or foot joints of a humanoid—all of which can benefit from greater periodic bursts without upsizing the motor. Current interest on the mantis shrimp’s striking appendage motivates employing such a strategy on other types of limbs as well. Tangentially, the mechanical processor shares commonalities with a pneumatic system, in that a “pump” motor constantly adds mechanical energy while valves (or environmental stimuli) meter it out. Implementing this functionality without an air tank or pneumatic peripheries might have advantages.

Methods

Programming a mechanical processor

The logic to perform the three subtasks listed in the Introduction was programmed into a two degree-of-freedom (DOF) series-elastic driven linkage-based mechanical processor. The two DOF are the angle of the motor and the configuration of the leg, which correspond to the output actions performed by the mechanical processor in the truth table of Figure 1(b). The actions to be performed by each DOF are determined by energy, position, and loading states, which correspond to the input states of the truth table in Figure 1(b). Namely, the actions of the motor and the leg (which are coupled by a spring but are nonetheless independent DOF) are defined by the current energy state of the spring, the current pose of the leg, and the current force enacted by the environment on the foot. The system’s dynamics are continuous, but since the linkage possesses a noninstantaneous kinematic singularity,¹ the mechanism changes its DOF during its motion and discrete behaviors emerge.

The spring, leg, and foot are classified to have two states each, leading to 2³ = 8 combinations of input states (see Figure 1(b)). The motor and the leg are classified into three actions each, leading to 3² = 9 combinations of output actions. The truth table of Figure 1(b) maps all combinations of input states to resulting actions to be performed by the leg and the motor. For a constant signal sent to the motor, the kinematic singularity mediates motion between the motor and leg. Based on the system states, it dictates which DOF should move and which should stay. At the singularity (which corresponds to the leg’s upstroke), series-elastic torque has no authority over the mechanism’s motion, and the foot contact force has full authority.

When the robot does not contact the environment, only series-elastic torque acts on the mechanism, advancing the leg toward its ready position (Figure 4(a)). Once it reaches this position, the singularity stops the leg. The interplay between the motor, leg, and singularity forms a stable position control loop with the ready position hardcoded as the setpoint. At the singularity the motor continues to turn, but since the leg cannot advance, instead the spring begins to deflect (Figure 4(b)). Deflection continues until spring torque matches the torque setpoint sent to the motor driver, forming a stable spring charging control loop with an adjustable setpoint for elastic energy storage. Once both the position and spring charging control loops have achieved their respective setpoints, the singularity dictates that both motor and leg remain stationary until contact occurs. At this state combination, the processor prohibits motion due to series-elastic torques, but is sensitive to vertical contact forces at the foot. Once the contact force appears, it advances the leg beyond its prohibitory singular configuration (Figure 4(c)), opening the way for the series-elastic energy to rapidly unload into a downward foot push-off motion (Figure 4(d)). This forms the stable contact detection/reaction control loop. The mechanical processor is set-up for periodic motions. At the conclusion of push-off, it automatically and immediately enters its mechanical position control loop (Figure 4(a)), with no need to reverse the motor. The description above details nominal cyclic operation and corresponds to the first six columns of the truth table shown in Figure 1(b). High-speed footage of the various phases of the locomotion cycle can be found in Supplemental Video 1 and Supplemental Video 2.

Outside of nominal operation, the mechanical processor reacts to other situations as well. In case the foot were to make contact before spring charging is complete, the processor dictates that the leg should push-off with whatever elastic energy has been accumulated thus far. In case contact is made while the spring is completely depleted, then the leg should push-off as directly driven by the gear motor. Both correspond to column 7 of Figure 1(b). In case contact is made and the leg is not yet at the ready position, then similarly the resulting push-off motion will be directly driven by the gear motor, corresponding to column 6 of Figure 1(b). Column 8 practically does not occur.

The kinematic singularity plays a major role in the mapping between states and actions performed by the mechanical processor. Singularities can be observed in the linkage geometry exactly at points when the three binary links share a common intersection (that is links AB, CD, and GH shown in Figure 2(a)). Intuitively, with the input link fixed (link ACF of Figure 2(a)), the linkage would still have mobility about this common intersection point. Linkages practically exhibit singular behavior not just at singularities but in neighborhoods around singularities. This linkage was designed by Liu and Plecnik (2021), Liu and Plecnik (2022) to have a large nearly singular neighborhood that spans the upstroke of an approximate straight line foot point, and regular (nonsingular) configurations that correspond to its downstroke. This can be seen in the configuration curve plotted in Figure 2(b) that shows the series-elastic rotational input versus the foot’s vertical coordinate. Portions of this curve with (nearly) vertical tangents indicate (nearly) singular configurations. During stance phase the foot cycles up in a singular fashion (dominated by the contact force) then pushes back down in regular configurations (dominated by the series-elastic torque). The rotational axis of the configuration plot of Figure 4 exhibits 2π periodicity, such that the motion logic repeats without need for the motor to reverse and reset. Demonstrations of conditions for kinematic singularity can be found in Supplemental Video 1.

The design of embodied intelligence was performed using the procedures of Plecnik and Fearing (2020) and Liu and Plecnik (2020). The processes begin by conducting a search for kinematic attributes which are dynamically useful. Utility is assessed by performing dynamic simulations on the kinematic attributes in the absence of a physical mechanism that instantiates those kinematics. These are a special sort of custom simulation where the number of links, their connectivity, and their inertias are all left unspecified. Instead what is known is an actuator input-output function and surrogate inertias. After advantageous kinematic attributes are discovered, there is no guarantee such attributes are feasible with a real mechanism. A best approximation is found through kinematic synthesis using the tools of numerical homotopy continuation such as the Finite Root Generation technique (Plecnik and Fearing (2017)) or the Random Monodromy Loops technique (Baskar and Plecnik (2020)). Numerical homotopy continuation enables a more complete design space exploration by finding all roots of the polynomial systems that comprise synthesis equations. This is in contrast to local optimizers which require a starting guess and return a single design candidate. By these tools and techniques, the process of programming a mechanical processor moves beyond tuning a few design parameters.

The linkage used for the computer processor and no processor cases possesses no input singularities. In the absence of mechanical processing, no meaningful deflection occurs in the series-elastic actuator. Therefore, the motor and leg actions are always the same. For this reason, the control block diagram of Figure 1(e) possesses no spring charging loop and the number of output actions reduces to one.

Construction of prototypes

The robotic device of Figure 1(c), incorporating a mechanical processor, was fabricated with a combination of manufacturing processes and materials. The links of the mechanical processor were machined from steel or aluminum, depending on strength requirements determined by joint forces. Links ACF, AB, GH, BDGP were machined from 4140 alloy steel. Links CD was machined from 6061 aluminum. Leg links KL and LP were 3D printed with PLA (polylactic acid). Link LP also included a 50 mm radius curved foot comprised of 6.21 mm thick polycarbonate to add a bit of compliance to foot-ground interaction. All links were attached to a body (link FH), which was 3D printed from PLA as two halves that bolted together with a 4140 alloy steel reinforced section at its bottom. Depending on space constraints, a variety of bushings and bearings were used. Joint F used ball bearings of inner ø6.35 mm (R188-2Z). Joint K used ball bearings of inner ø3.18 mm (R144-2Z). Joints A , B , and C used bronze bushings of inner ø6.375 mm (SDP/SI A 7B 4-F106). Joint D used bronze bushings of inner ø3.213 mm (SDP/SI A 7B 4-P092). Joints G and H used bronze bushings of inner ø2.385 mm (SDP/SI S99BP4-FB061204). Joints L and P used plastic sleeve bushings of inner ø3.175 mm (Igus JFI-0204-06). A small linear spring (Lee Spring LEM090DB01M) was installed between links CD and FH to sharpen the set point of the mechanical processor’s position control loop.

The scale of the designed mechanism was chosen to accommodate off-the-shelf components, but biased towards making the size smaller. The series-elastic actuator comprised of a brushless motor (Maxon ECXSP13L 18V) with a 243:1 gearbox (GPX16HP) coupled to a torsion spring (Lee Spring LTR095G30 M). 3D printed parts interfaced the motor to the spring, and the spring to link ACF. The motor was electronically commutated using a Maxon EPOS4 Compact 50/15 performing Field-Oriented Control (FOC). The motor encoder was a Maxon ENX13 EASY INT. A second encoder, the leg encoder, that reads out rotation of link ACF was a CUI AMT102-V. The same series-elastic actuator and leg encoder were used for both robotic devices produced for this work. The mass of the device shown in Figure 1(c), including the boom’s contribution, was 575 g. Mass was measured by placing a scale under the robot while connected to the boom set-up.

The second robotic device (Figure 1(g)) was used to test the computer processor and no processor cases. It was fabricated mostly from 3D printed parts as its joints endured smaller forces. Leg links K′O, K′L′, and the main body were 3D printed with PLA. Leg link L′P′ comprised of a plastic clevis, a ø6.35 aluminum rod, with an attached 50 mm radius C foot comprised of 6.21 mm thick polycarbonate. Joints O used ball bearings of inner ø6.35 mm (R168-2Z). Joints K′ and L′ used plastic sleeve bushings of inner ø3.175 mm (Igus JFI-0204-06) The sliding joint used recirculating ball bearings (McMaster-Carr 60595K11). The same series-elastic actuator was used, and a AMT102-V leg encoder read out the rotation of leg link K′O. The mass of the device shown in Figure 1(g), including the boom’s contribution, was 550 g.

Hopping control

For the hopping task investigated in this paper, the mechanical processor executes a spring charging control loop, a position control loop, and a contact detection control loop, by simply sending a constant torque setpoint to the motor. The torque setpoint was adjusted at slow speeds in between hops to modulate hop height. Torque control was implemented by Maxon’s EPOS4 software and hardware. Voltage could have been used instead of torque as the control signal, but it was more straightforward to interface with Maxon’s current control interface rather than the brushless motor’s three windings. The EPOS4 implements field-oriented control. Since the EPOS4 measures current, not torque, a calibration was conducted to correlate the two.

The computer processor implemented the state machine shown in Figure 1(f). A motor encoder was mounted on the motor before the gearbox and spring. A leg encoder was mounted on the input link of the leg, after the gearbox and spring (see Figure 1(e)). When the foot is not in contact, the computer communicates to the motor driver to set the leg in a predetermined position, called the ready position, most ideal for contact detection and push-off. Contact is detected once the computer senses a spike in the leg encoder’s readings. The computer then immediately sends a signal to the motor driver to push-off the ground and subsequently return the leg to the ready position for the next cycle. The commanded speed during push-off was used to regulate the height of the robot. Motor position control was implemented by Maxon’s EPOS4 software and hardware.

The computer processor’s performance was mostly influenced by the selection of the ready position. The best ready position was determined empirically, and could be identified uniquely by the angle of the input link. When the input link points down, the leg is maximally elongated. We called this 0°. All possible angles were tested in 10° increments by measuring reflex energy (see Reflex Energy Experiment) and gait consistency for 10 s (see Gait Consistency Experiment). The best ready position was at 40° clockwise from the maximally elongated configuration at 0°, as it was empirically the most repeatable, and exhibited the most reflex energy.

The no processor case sent a constant torque signal to the motor and implemented no closed-loop control.

Reflex energy experiment

As an initial experiment, the reflex of each robotic leg was tested. Experiments were conducted for each of the three cases described in the Introduction: reflex via a mechanical processor, reflex via a computer processor, and with no processor (no reflex). The purpose of this test was to measure the power of each reflex.

Reflex energy is a measure of the kinetic energy added by the actuator during contact detection/reaction. It is measured from the apex of center of mass (CoM) rebound over two drop tests: one powered and one with the motor shaft fixed, performed over multiple trials. The latter is negated from the former to remove the effect of passive rebound from the energy calculation:

Reflex energy = m g (h_{powered} - h_{fixed}) .

(1)

All drops used a planarizing boom, and were initiated at a height so the CoM travels 32 cm vertically before reaching its lowest point. The planarizing boom was a ø20 mm carbon fiber tube of length 1.613 m that attached to the lab bench via a universal joint. The mass of the boom by itself was 224 g. Experiments were performed for different gravity constants, which was adjusted by connecting a long bungee at a measured moment arm from the boom to the ceiling to provide weight compensation. The experimental apparatus yielded approximations of Earth, Mars, and Moon gravitational constants that varied with height due to the use of a bungee as an approximate constant-force spring and the radius of the boom tracing an arc. Figure 5 shows each gravity constant’s change with height. The shaded region along each line shows the hop height range reached during dynamic hopping cycles for all cases. Four trials were performed for each device configuration for each gravity constant.

Drops were initiated by releasing robots from a mounted clamp released by a drawn cable. The initial configuration of the mechanical processor was set at the singularity, with the motor commanded to produce 2.3 Nm (75% of the motor nominal torque T_nom). The initial configuration of the other device was set with the input link at 40° after the bottom of stroke (0° marks the bottom of stroke). Intuitively, just like the mechanical processor, the conventional singularity-free leg mechanism has some stroke to absorb impact, moving the foot upwards, then a long downwards push-off distance. The no processor case was additionally tested without a spring to confirm that it did not impose any detrimental effects. For measuring the motion of the CoM, it was estimated to be a fixed point on the body. This fixed point on the body was found by hanging the robot from a single string against a flat vertical surface, letting it settle, then drawing an extension of the string line onto the body (like a plumb bob). This was repeated a second time with a different single string mounting point. The intersection of the two resulting lines located the planar CoM.

Each experiment was performed with the robot set in a nominal configuration. The CoM was marked, and its time trajectory was recorded using video footage at 1000 frames per second (Chronos 2.1-HD, Kron Technologies) for 1 second, post-processed using the software Tracker, see Brown et al. (2023). This footage was used to measure the time evolution of gravitational potential and vertical kinetic energy (Figure 9(b)). The reference for the former was taken as the lowest point of the CoM.

Spring potential was estimated by measuring spring deflection using both the motor and leg encoders, installed on either side of the series-elastic element. Spring potential was calculated by $1 / 2 k Δ θ_{spr}^{2}$ , where torsional stiffness k was measured using a universal testing machine (Admet eXpert 7603). Electrical energy was measured through the time integration of P_elec = VI sampled at 1000 Hz. Voltage and current values were read from the q-axis of FOC commutation. Voltage was calculated from PWM duty values recorded onboard the EPOS4, then multiplied by the supply voltage. Current was provided by the Maxon EPOS4 as an integer. A calibration experiment was performed to correlate these values to torque: an arm was attached to the motor shaft and set to push on a load cell at various torques. Current readouts were correlated to torque measurements with a linear fit. The mechanical energy added to the system was measured by the time integration of $P_{mech} = T_{mot} {\dot{θ}}_{mot}$ .

Finite differences were used to obtain all time derivatives. Apex and ground contact events were used to synchronize data from different sources. Demonstrations of the reflex energy experiments can be found in Supplemental Video 3.

Gait consistency experiment

Gait consistency was evaluated for each of the three cases. The motivation is that, like a hub motor per se, if a constant signal is sent to the motor, the leg will automatically locomote in a predictable way. The leg should exhibit a consistent gait without the need for high bandwidth motor coordination. For this experiment, each robot was filmed hopping vertically in place.

The gait consistency experiments involved measuring the trajectory of each robot as it performed repeated vertical hops. For these experiments, the planarizing boom was constrained to only rotate vertically. The trajectory of the CoM of each robot was recorded at 1000 frames per second (Chronos 2.1-HD, Kron Technologies) for 5 seconds. 3 trials were performed for each device for each gravity constant, with gravity being adjusted as described above. Demonstrations of the gait consistency experiments can be found in Supplemental Video 3.

Hop height modulation experiment

The motion of the mechanical processor unfolds by sending a constant torque setpoint to the motor. That setpoint can be adjusted at slow speeds in between hops to modulate hop height. An experiment was set up to correlate the torque setpoint to hop height.

The hop height modulation experiments were conducted by measuring the steady state height of the mechanical processor while it is hopping repeatedly under 3 different torque setpoints: {0.6, 1.2, 1.8} Nm. The no processor robot received the same torque setpoint signals. The computer processor was implemented through position control of the motor using EPOS4. The robot’s take-off power was throttled by setting the push-off speed of the motor after contact detection. Three different speeds were experimented: {7.6, 15.1, 22.7} rad/s. The three torque setpoints and speed setpoints were designated as low, medium, and high, with low and high indicating the lowest and highest hops the robot could perform. Each robot hopped at each setting for 3 s in one single go, updating its hop height in real time. For these experiments, the robots were constrained to only hop up and down. The footage of each robot hopping was recorded at 1000 frames per second (Chronos 2.1-HD, Kron Technologies) for 10 s. Three trials were performed for each robot for each gravity constant, with gravity being adjusted as described above. The average steady state hop height under each setpoint is obtained by taking the average of the steady state hop heights after three trials. Demonstrations of the hop height modulation experiments can be found in Supplemental Video 3.

Dynamic simulation

All experiments were reproduced using planar rigid body dynamics simulation. Simulation data appears alongside experimental data in Figures 9 and 10. A purpose built simulation environment was written for this work. The simulator begins by specifying a number of planar rigid bodies, set in a reference configuration at t = 0. Rigid bodies are connected to each other through either revolute joints or prismatic joints (Figure 6), specified in a reference configuration.

The coordinates of a revolute joint connecting bodies i and j in the reference configuration are given by $A_{i j}^{0}$ . After t = 0, the joint moves to a displaced configuration A_ij. The coordinates of a prismatic joint connecting bodies i and j in the reference configuration are given by $S_{i j}^{0}$ and $s_{i j}^{0}$ , where $S_{i j}^{0}$ is some point on the joint axis and $s_{i j}^{0}$ is its direction. After t = 0, the joint moves to a displaced configuration, S_ij and s_ij. The slide of a prismatic joint is measured by scalar s_ij. The CoM of each body in the reference configuration is given by $X_{i}^{0}$ , which is displaced to X_i after t = 0. The rotation of each body from its reference configuration is measured by θ_i.

For every body introduced, a generalized coordinate vector is populated,

q = {θ_{1}, X_{1}, \dots, θ_{n}, X_{n}},

(2)

where

q \in R^{3 n}

for n bodies. For every joint introduced a constraint vector f(q) = 0 is populated. A revolute joint between bodies i and j adds the following two components to f (two vector components),

X_{j} + [R (θ_{j})] (A_{i j}^{0} - X_{j}^{0}) - X_{i} - [R (θ_{i})] (A_{i j}^{0} - X_{i}^{0}) .

(3)

A prismatic joint between bodies i and j adds the following two components to f,

\begin{align} (1 - η) & [i] [R (θ_{j})] s_{i j}^{0} \cdot (X_{j} - X_{i}) \\ + η & [i] [R (θ_{i})] s_{i j}^{0} \cdot (X_{j} - X_{i}) - [i] s_{i j}^{0} \cdot (X_{j}^{0} - X_{i}^{0}), \\ θ_{j} - θ_{i}, \end{align}

(4)

where

[R (θ)]

is a planar rotation matrix by θ, and

[i] = [R (π / 2)]

. Parameter η adjusts the location (and thus the magnitude) of the prismatic joint’s normal force. These constraints are set-up so that in the expression [∂f/∂q]^T λ , where λ is a vector of Lagrange multipliers, the two Lagrange multipliers associated with the two constraints (equation (3)) of a revolute joint are exactly the x and y components of the reaction force F_ij of body i acting on body j. The two Lagrange multipliers associated with the two constraints (equation (4)) of a prismatic joint are exactly the reaction force F_ij of body i acting on j normal to the prismatic joint, and the reaction moment M_ij of body i acting on body j.

The dynamics equations are simply:

[M] \ddot{q} = {[\frac{\partial f}{\partial q}]}^{T} λ + Q + Q_{fric} (λ),

(5)

where the mass matrix takes a diagonal form populated by rotational & linear inertias [M] = diag (I₁, m₁, m₁, …, I_n, m_n, m_n), Q is a generalized force vector, and Q_fric( λ ) is a generalized force vector composed of joint friction terms. Note the inclusion of Q_fric( λ ) makes the equations of motion nonlinear in λ , which make these equations computationally more difficult than the frictionless case. The friction torque in the revolute joint between bodies i and j is modeled as,

τ_{i j} = - μ_{i j} r_{i j} \sqrt{λ_{k}^{2} + λ_{k + 1}^{2}} sigm ({\dot{θ}}_{j} - {\dot{θ}}_{i}),

(6)

where μ_ij is the coefficient of friction, r_ij is the joint radius, (λ_k, λ_k+1) are the reaction force components, and sigm(v) is the sigmoid function defined by sigm(v) = 2/(1 + e^−χv) − 1. The sigmoid function is implemented to remove the discontinuity of the Coulomb friction model. The friction force in the prismatic joint between bodies i and j is similarly modeled as

f_{i j} = - μ_{i j} | λ_{k} | sigm ({\dot{s}}_{i j}) .

(7)

Forces and torques are cast into generalized forces in the usual way. Gravity acts on all bodies. The contact force of the ground acting on the foot point P = {P_//, P_⊥} was modeled as a nonlinear spring $F_{⊥} = k_{⊥} Δ P_{⊥}^{2} - c_{⊥} {\dot{P}}_{⊥}$ in the normal direction with ΔP_⊥ as penetration, and with a friction term F_// = −μ_grF_⊥sigm (P_//) in the tangential direction.

The motor torque model used was

T_{mot} = - (\frac{T_{s}}{ω_{f}} + c_{mot}) {\dot{θ}}_{mot} + T_{s} \frac{V}{V_{nom}},

(8)

where T_s is theoretical stall torque after the gearbox, ω_f is free-running speed after the gearbox, c_mot accounts for bearing & gearbox losses, V_nom is nominal voltage, and V is the control input signal. Additionally, a torque (current) cut-off was implemented at a value T_cut. Depending on the case simulated, V was defined according to a position or torque control law.

Combining equation (5) with the second derivative of the constraints,

[\frac{\partial f}{\partial q}] \ddot{q} + \frac{d}{d t} ([\frac{\partial f}{\partial q}]) \dot{q} = 0,

(9)

yields a square system that can be solved in terms of unknown accelerations

\ddot{q}

and reaction forces λ . Accelerations

\ddot{q}

appear linearly and can easily be eliminated, yielding a nonlinear system in λ which is solved by Newton’s method during each integrator step. Subsequently,

\ddot{q}

is easily back-solved at each integrator step. The dynamics described above were packaged into the form of an initial value problem

\ddot{q} = h (q, \dot{q})

and time integrated using the submethod “StiffnessSwitching” of Wolfram Mathematica’s NDSolve. State positions and velocities were corrected to exactly adhere to constraints at the beginning of every time step.

The model of the robot equipped with a mechanical processor is shown in Figure 7, and the other comparative robot is shown in Figure 8. For all cases, the gravity constant was not varied, but instead the weight compensation bungee was modeled as a vertical spring acting through X₁. The simulations account for energy loss through joint friction, ground friction, spring damping, motor damping, motor Joule heating, and negative work performed by the motor.

Inertial parameters were determined from CAD (computer aided design) models. However, the rotational inertia of the first links, i = 1, was set to infinity so θ₁ = 0 for all time. Motor parameters were calculated from manufacturer data sheets, or tuned by the authors during commissioning of the motors. Spring stiffness and damping coefficients were measured with a universal testing machine (Admet eXpert 7603). Bearing friction coefficients were estimated from manufacturer data. The sigmoid sharpness parameter χ, ground stiffness & damping were selected as the largest powers of 10 for which the numerical integrator did not fail (due to the system being too stiff). Values of physical parameters are shown in Table 1.

Table 1.

Parameters used to conduct dynamic simulations. The notation matches Figures 6 –8.

Robot with mechanical processor (Figure 7)
Joints	Reference coordinates		Diameter (mm)	Coefficient of friction
Joints	x (mm)	y (mm)	Diameter (mm)	Coefficient of friction
A₁₂ =	−52.7	−16.2	3.18	0.10
A₁₃ =	−75.6	51.1	3.18	0.20
A₁₄ =	−65.5	−7.29	6.35	0.20
A₂₇ =	−47.8	−10.2	3.18	0.20
A₄₅ =	−62.2	−3.55	6.35	0.20
A₄₆ =	−55.7	−5.46	6.35	0.20
A₅₇ =	−62.5	−20.4	6.35	0.20
A₆₇ =	−51.1	−7.98	6.35	0.40
A₇₈ =	−19.9	−32.4	3.18	0.20
A₃₈ =	−75.6	51.1	3.18	0.20
B₁ =	−105.7	10.6	N/A	N/A
B₅ =	−62.5	−20.4	N/A	N/A

Bodies	Center of mass			Moment of inertia (kg⋅m²)
	Reference coordinates		Mass
	x (mm)	y (mm)	(g)
X₁ =	−50.0	30.0	489	∞
X₂ =	−50.2	−13.1	2.5	0.06 × 10⁻⁶
X₃ =	−44.7	42.0	5	3.71 × 10⁻⁶
X₄ =	−62.3	8.43	20	6.95 × 10⁻⁶
X₅ =	−53.4	1.26	13.5	2.56 × 10⁻⁶
X₆ =	−41.4	53.7	1	0.08 × 10⁻⁶
X₇ =	−45.3	−3.53	19	6.34 × 10⁻⁶
X₈ =	−16.6	−31.6	25	15.3 × 10⁻⁶
Total			575

Robot for comparison (Figure 8)
Joints	Reference coordinates		Diameter	Coefficient of friction
Joints	x (mm)	y (mm)	(mm)	Coefficient of friction
A₁₂ =	0	0	6.35	0.10
A₂₄ =	−19.9	−12.0	3.18	0.20
A₄₃ =	0	−55.51	6.35	0.20
S₁₃ =	0	−70.6	N/A	N/A
s₁₃ =	0	−1.00	N/A	0.10

Bodies	Center of mass		Mass (g)	Moment of inertia (kg⋅m²)
	Reference coordinates
	x (mm)	y (mm)
X₁ =	0	0	480	∞
X₂ =	11.6	0	5	0.79 × 10⁻⁶
X₃ =	0	−70.6	59	164 × 10⁻⁶
X₄ =	11.6	−22.6	6	2.71 × 10⁻⁶
Total			550

Parameters
Description	Symbol	Value
Motor
Gear ratio	G	243:1
Stall torque	T _s	49.8852 Nm
Nominal torque (cut-off)	T _nom	3.0645 Nm
Free-running speed	ω _f	40.3154 rad/s
Nominal speed	ω _nom	37.8459 rad/s
Damping coefficient	c _mot	8.7415 × 10⁻¹ Nm/(rad/s)
Torsion spring
Stiffness coefficient	k _spr	0.3579 Nm/rad
Damping coefficient	c _spr	8.1363 × 10⁻³ Nm/(rad/s)
Ground
Stiffness coefficient	k _⊥	10⁶ N/m
Damping coefficient	c _⊥	10³ N/(m/s)
Friction coefficient	μ _{/ /}	0.8
Sigmoid sharpness	χ	10²
Stroke length	l _stroke	0.04648 m

Results

Reflex energy

To measure the power of each reflex, each robot was initiated out-of-contact with the ground and at a height so the CoM travels 32 cm vertically before reaching its lowest point. Robots were powered on, then released from height. Upon contact with the ground, each robot’s control logic triggered (either mechanically, electronically, or not at all), and the robot pushed-off the ground. The apex of its ensuing potential energy was measured. To negate the effect of the elastic foot and torsion spring, these experiments were repeated, but with the motor shaft fixed. Potential energy measured from the fixed tests was negated from the powered tests to form what we term as reflex energy, see Figure 9(a). Reflex energy was measured for multiple gravitational constants. More details on the measurement procedure are given in Methods (Reflex Energy Experiment). Reflex energy for all cases measured are presented in Figure 9(c).

The energy of the reflex enacted by the mechanical processor was greater across the board, due to its low latency and by enacting the spring charging loop (Figure 1(a), Introduction), leading to greater series-elastic power generation. The reflex enacted by the computer processor exhibited negative reflex energy, meaning the powered push-off is inferior to the passive rebound. This is because of the electronic and mechanical latency associated with its contact detection/reaction control loop. From high-speed video, the leg begins to push-off 125 ms after the impact event between the foot and ground. At this point, the passive recoil of the foot has nearly completed and contact is almost broken. Some of leg stroke is spent reestablishing contact, although this phenomenon is observed for all cases.

Mechanical processors are better suited for handling high-speed mechanical events. To understand the events that occur during reflex tests, we plot the time evolution of energy in both experiment and simulation for all cases. Experimental measurement techniques are described in Reflex Energy Experiment, and the simulation formulation is described in Dynamic Simulation of the Methods section. Figure 9(b) shows that the spring charging loop of the mechanical processor enables 8 J storage in the series-elastic element while out-of-contact with the ground. This comes at the cost of a high motor-spring holding torque as the mechanical processor waits for contact detection, leading to large electrical power consumption. This situation can be fixed by the inclusion of an additional singularity (an output singularity, see Bohigas et al. (2017)) to the mechanical processor to reduce holding torque to zero when the spring is fully charged. This feature was not implemented in this work. Upon impact, the force signal detected at the foot sends through the kinematic singularity, triggering the release of elastic potential and a downward push-off motion of the foot, leading to the superior reflex energy exhibited in Figure 9.

As shown in Figure 9(c), in experiments, the mechanical processor enabled around 1.08 J (105%) more reflex energy on average as compared to the computer processor in experiment under Earth gravity. For Mars and Moon gravity, the energetic output was 91% and 97% greater, respectively. All experiments used the same series-elastic actuator, described in Construction of prototypes.

Gait consistency

For each of the three cases, the time evolution of hop heights for each case across various gravitational constants is shown in Figure 10. The no processor case exhibited an inconsistent trajectory, as evidenced by a large variability in hop heights. Without position control or contact detection, the leg contacts the ground in some unplanned configuration with every step, since the motor only received a constant signal. Tuning the constant signal or devising a more complicated open-loop signal would most likely improve gait consistency, but the robustness of this consistency across speeds and heights would warrant investigation in itself.

Equipped with position control and contact detection, the computer processor yielded a consistent gait, but no improvement to hop height. However, this approach concedes the motivation to locomote with a constant signal. This goal is accomplished by the mechanical processor, which yielded the most consistent gait and highest energy push-off.

As shown in Figure 10, in experiments, hop heights driven by the mechanical processor varied with standard deviations less than 5% of their average. Under Earth gravity, the mechanical processor exhibited an average hop height of 0.140 m and standard deviation of 0.007 m (5% of average), notated as 0.140 m ± 5%. In comparison, the computer processor produced hops of 0.0537 m ± 0.5%, and the no processor case produced 0.060 m ± 5%. With the gravitational constant decreased to resemble Mars, the mechanical processor, computer processor, and no processor cases produced hops of 0.277 m ± 3.6%, 0.0673 m ± 1.1%, and 0.072 m ± 36%, respectively. For moon gravity, hops of 0.356 m ± 2.2%, 0.108 m ± 7.2%, and 0.114 m ± 38.6% were produced, respectively.

Hop height modulation

The mechanical processor is able to regulate its hop height by adjusting the torque setpoint of its motor. The robot was tested with three torque settings: low (0.6 Nm), medium (1.2 Nm), and high (1.8 Nm). The average steady state hop height at each torque setpoint across various gravitational constants are shown in Figure 11. Under each gravitational constant, average steady state hop height increases as the torque setpoint increases.

The mechanical processor at medium and high settings exhibited consistent hops. The high end was limited by the structural integrity of the robot. The low end began to exhibit inconsistencies. Either the ground impact force was not always enough to trigger the mechanical reflex, or the energy storage setpoint while airborne was not achieved. The former issue is caused by joint friction. The latter issue might have been alleviated by adjusting the motor controller gains to reduce settling time. Motor gains were chosen by Maxon EPOS4’s automatic regulation tuning procedure, which can be argued is ill-prepared to handle a mechanism with a variable transmission ratio.

Videos of the experiments are found in Supplemental Video 3 . Equipped with position control and contact detection, the computer processor case was able to regulate its hop height at low, medium, and high settings, albeit at hop heights considerably lower than the mechanical processor. The no processor case was not able to regulate its hop height, as it lacks closed-loop control, either electronically or mechanically.

As shown in Figure 11, in experiments, hop heights driven by the mechanical processor varied with torque. At the high setting, hops at approximate Earth, Mars, and Moon gravities were 0.27 m, 0.37 m, and 0.44 m, respectively.

Demonstrations on complex terrain

Hopping was demonstrated for all three cases (mechanical processor, computer processor, no processor) at all three gravity levels (Earth, Mars, Moon) for five different terrains (Level, Rough, Stairs Tall, Stairs Short, Compliant). The results can be found in Supplemental Video 3 and a summary is shown in Figure 12. In all videos, the hoppers move from right to left. Level flat terrain served as a baseline. A “Rough Terrain” added some complexity, including a granular media rock pit and a 23° downsloping ramp. Two sets of stairs were tested, one with 40 mm steps designated as “Tall” and with 20 mm steps designated as “Short.” Finally, a “Compliant Terrain” was constructed from upside down opened soft cover books. As can be seen in the videos, these books have a lot of give and move when hopped upon.

The takeaway from these demonstrations is qualitative: that is, legs with greater reflex energy (see Figure 9) had greater success when traversing complex terrains. From the perspective of high-level control, all robots were run open-loop, blind to the terrains, and did not make any decisions. From the perspective of low-level control, the mechanical processor ran fast mechanical closed-loop control, taking nearly instant actions based on environmental stimuli and its internal state (see Figure 1(b)). Surely more terrains can be concocted, perhaps with overhangs or pedestals requiring precise foot placement, in which the mechanical processor would fail. Fast, energetic, mechanical, low-level control is an enabler to be combined with vision-based, terrain-aware, higher-level loops, and not an end-all in itself. But on the converse, if a step is too high for the robot, this problem is not solved by greater terrain-awareness. The brain enables the body, and the body enables the brain.

Discussion

The study shows the mechanical processor is capable of exhibiting higher powered motions (see Reflex energy ), greater consistency in gait (see Gait consistency ), and is able to modulate its hop height by adjusting a low bandwidth control input (see modulation). Additionally, the mechanical processor was able to traverse rough & compliant terrains, and climb stairs (see Demonstrations on complex terrain).

Offloading electronic control tasks to morphology, while impressive, does not necessarily offer new capabilities. However, the inclusion of a spring in the mechanical processor enables high discharge rates of mechanical energy invoked by high-speed mechanical logic that are not possible with electronic components within weight limits. While batteries and capacitors do have high discharge rates, transducing this energy from electrical to mechanical must go through a pinchpoint—the motor. Widening this pinchpoint means using a heavier motor, which has adverse consequences. By raising the robot’s energetic profile without upsizing the motor, this work shows how mechanical processing offers new capabilities beyond alleviating control tasks. Furthermore, by avoiding extra sensors, other overheads are avoided, such wiring, mounting, and housing. As well, mounting a sensor at a distal, high impact location can be problematic.

For a wheel to locomote, its driving motor simply needs to receive a constant signal. No mechanical processing is needed. In the simplest case, the driving motor can be brushed, and the constant signal can be the voltage delivered from a directly connected battery. A leg mechanism, such as the linkage of Jansen (2017) (among many others), can be driven in the same way. Both a wheel and the Jansen linkage interact with the environment through contact, but neither machine intelligently alters its behavior based on this interaction. For example, if a motor-driven wheel were lifted from the ground, it will continue to turn, as it does not know whether it is in contact or not. The same goes for the Jansen linkage.

A more intelligent wheel (or Jansen linkage) would incorporate a force sensor to determine contact, then react accordingly. This would necessitate a closed loop in the robot’s control block diagram (like those shown in Figures 1 and 3). A computer processor could execute the control logic: if contact is sensed, move the wheel, or else, stop the wheel. Instead, the same logic could be programmed into a mechanical processor, allowing the leg to react to its environment while receiving a constant signal. The problem with the mechanical approach is that design methods are bespoke, see Hawkes and Cutkosky (2018), and each component introduced raises concerns of weight, volume, friction, and mechanical latency.

This mechanical nature is both a curse and a blessing. One advantage is that mechanical signals (force, position, elastic energy) require less transduction to perform mechanical tasks. For example, consider the low-level task of contact detection. For a computer processor to detect contact and push-off, the contact force is transduced to electronic pulses, interpreted and processed onboard a computer, then electrical signals sent to the motor are transduced to torque, and finally motor torque is transduced to foot force by the mechanism. In contrast, a mechanical processor involves a single transduction step, simultaneously transducing contact force and series-elastic torque into foot force, avoiding electronic latency which could be a bottleneck in cost-effective systems.

A second advantage of functioning with purely mechanical signals is that large magnitude signals can be exploited to unlock new functionalities, exemplified by the spring charging control loop presented in Figure 1. In particular, the large magnitude torque build-up within the series-elastic actuator when configured at the singularity enables elastic energy storage when out-of-contact that boosts power output during the push-off reflex. This extra functionality enabled by the mechanical processor approximately doubled the energetic output for the tested cases.

Mechanical processors excel at tasks which are mechanical, fast, low-level, repetitive, and specialized. Each characteristic is expanded below. A mechanical processor excels at mechanical tasks by limiting the number of transduction steps, as described above. An example of non-mechanical tasks are floating point operations. Although beautiful, see Rojas-Sola et al. (2021), there is little justification for mechanical calculators in modern robots. Fewer transduction steps reduce latency, which makes mechanical processing amenable to fast tasks such as reactions. But if a slow reaction is sufficient, then computer control is usually easier to implement. Low-level tasks are basic primitives from which higher-level tasks are built. In the context of this paper, taking a step is low-level, while producing a gait, steering, or balance are higher-level tasks that involve taking multiple steps. Fast mechanical processing relieves the computer from low-level minutiae, which is often repetitive. The number of components in a mechanical processor should be limited due their weight and volume versus electronic components. Therefore, the number of subtasks that can be handled is generally limited, lending mechanical processors to perform specialized subtasks at a lower loop of the overall control architecture. An effective mechanical processor will perform many subtasks with few components.

Conclusion

In the end, this paper does not make the argument that mechanical processing is better than computer processing or vice versa. They are different tools to be considered not separately but blended into the control architecture of a robot. Mechanical processors excel at controlling low-level, dynamic subtasks that avoid extra transduction steps. Computer processors excel at higher-level planning, and stand to benefit from reduced communication requirements by outsourcing control of the fastest mechanical events to a mechanical processor. Additionally, mechanical processors may unlock new functionalities (such as the spring charging loop in this paper) to enhance robot performance, but ultimately these functionalities should be directed by a high-level computer.

Supplemental Material

Supplemental Material - Dynamic wind-up locomotion enabled by embodied intelligence

Supplemental Material for Dynamic wind-up locomotion enabled by embodied intelligence by Chang Liu and Mark Plecnik in The International Journal of Robotics Research.

Supplemental Material

Footnotes

Acknowledgments

The authors thank Boardwalk Robotics Inc. for advice on this work.

ORCID iDs

Chang Liu

Mark Plecnik

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This material is based upon work supported by the National Science Foundation under Grant No.’s CMMI-2041789 and CMMI-2144732.

Declaration of conflicting interests

The authors declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Supplemental Material

Supplemental material for this article is available online.

Note

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