Series Elastic Actuator

A series elastic actuator (SEA) is an actuator with an intentionally compliant element in the mechanical force path between its power source and its load. In a common electric SEA, a motor drives a reduction mechanism, the reduction drives a spring, and the spring drives the robot joint or linear output. The deflection of the spring provides an estimate of transmitted force or torque. A local controller uses that estimate to make the actuator behave as a controlled force source.

Series elasticity differs from incidental drivetrain flexibility. The spring, flexure, or other compliant element is selected, measured, and included in the control model. Its compliance can isolate the transmission from short impact loads, lower the controlled output impedance, and store mechanical energy. These properties have made SEAs useful in robotics systems that make repeated contact with the environment or with people.^[1][3][4]

Compliance also changes the actuator dynamics. A softer spring permits a larger, easier-to-measure deflection for a given force, but it requires more travel and generally narrows the range over which force and position can be controlled without substantial phase lag. An SEA can resonate, saturate its elastic travel, or store enough energy to create a hazard. It is therefore not inherently safe or efficient under every controller and task.^[3][7][8]

Definition and operating principle

The defining series connection means that, neglecting parallel load paths, the same transmitted force passes through the elastic element and the load. A basic rotary SEA contains a motor, speed reducer, elastic element, spring-deflection sensor, output-position sensor, and control electronics. A prismatic design substitutes a linear transmission such as a ball screw and measures spring compression or extension.^[1][4]

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For a linear spring operating within a calibrated linear range, the estimated transmitted force is

F_s = k_s delta_x

where k_s is linear spring stiffness and delta_x is the measured spring deflection. For a torsional spring,

tau_s = k_theta delta_theta

where k_theta is torsional stiffness and delta_theta is angular deflection. The elastic energy stored in the corresponding ideal springs is

E_s = 1/2 k_s delta_x^2

and

E_s = 1/2 k_theta delta_theta^2.

These equations are constitutive models, not guarantees of output accuracy. A real estimate can also contain errors from spring hysteresis, temperature, mounting compliance, encoder quantization, backlash, unmeasured parallel load paths, and calibration drift. Nonlinear elastic elements require a calibrated force-deflection curve, such as F_s = f(delta_x), rather than one constant stiffness.^[4][5]

The spring converts force measurement into displacement measurement. If a linear displacement sensor has a resolution contribution Delta(delta_x), the associated first-order force increment is approximately Delta F = k_s Delta(delta_x). A lower stiffness therefore improves this part of force resolution, provided that the increased deflection stays within the available travel and the spring remains in its calibrated range.^[1][3]

Origins and development

Matthew M. Williamson submitted the MIT S.M. thesis Series Elastic Actuators in January 1995 under Gill A. Pratt. The thesis described the design, control, and experimental evaluation of a force-controlled actuator with a spring between its gearbox and load. It framed the spring as a deliberate trade: less achievable force-control bandwidth in exchange for stable, low-noise force control and protection from shock loads.^[1]

Pratt and Williamson presented the architecture at the IEEE/RSJ International Conference on Intelligent Robots and Systems in Pittsburgh in August 1995. Their paper placed the term series elastic actuator into the robotics literature and described how spring deflection could turn a difficult geared-motor force-control problem into a position-measurement problem.^[2] The work did not introduce mechanical compliance in general. Flexible joints, tendon drives, remote-center compliance, and force sensors already existed. Its contribution was the purposeful integration of a compliant load sensor in series with a high-force drivetrain and a local force controller.

David W. Robinson's 2000 MIT doctoral thesis extended the analysis beyond the early prototype. It treated series elasticity as a general closed-loop force-control architecture and examined bandwidth, output impedance, inertial loading, and implementations in electromagnetic and hydraulic power domains.^[3] Later work expanded the design space to rotary joints, compact prismatic units, cable transmissions, reaction-force-sensing arrangements, nonlinear springs, variable stiffness, and integrated robot joints.^[4][5][17]

Mechanical arrangements

The spring can occupy several locations relative to the motor, transmission, ground, and load. Its location determines what its deflection measures and which dynamics lie between that measurement and the output.

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Paine, Oh, and Sentis used the FSEA and RFSEA distinction when comparing two common architectures. Hyunsoo Kim and Youngjin Choi later generalized spring placement into FSEA, RFSEA, and TFSEA categories.^[4][5] These names describe sensing topology, not the shape of the output. Any of them may be rotary or prismatic.

In a conventional output-side FSEA, impact energy from the load reaches the spring before it reaches the reducer. The spring can therefore act as a mechanical low-pass path for a short collision. In an RFSEA, the reaction spring may remain fixed to the chassis and avoid adding length to the moving output. That packaging advantage comes with a less direct measurement: acceleration of the motor housing or other supported components can contribute to spring force. A controller may need an identified dynamic model and filtered velocity or acceleration estimates to infer output force.^[4]

Rotary SEAs commonly use torsion bars, spiral or planar springs, flexures, or compression springs coupled through a lever. Prismatic units often place coil or die springs around a ball screw or pushrod. Cable and tendon designs can move the motor away from the joint, reducing distal mass, while leaving an elastic element in the force path. Remote actuation adds cable stretch, friction, routing-dependent losses, and hysteresis to the model.

Spring placement before or after a reduction also changes the required stiffness and sensor range. A motor-side spring sees smaller torque but larger angular motion after the quantities are reflected through the gear ratio. An output-side spring sees joint torque directly. High reduction ratios can give compact motors high output torque, but they also increase reflected motor inertia and often add friction, backlash, and torque ripple. Series compliance can filter some of these effects at the output; it does not remove their energy loss or eliminate them from the internal dynamics.^[1][4]

Force estimation and calibration

An SEA usually estimates force from relative displacement across the compliant element. A rotary joint can subtract a load-side encoder reading from a gear-side encoder reading after converting both to the same coordinate system. A dedicated differential encoder or strain-sensitive flexure can make the measurement more direct. The controller then applies the spring's calibrated torque-deflection relation.

Calibration has to cover the installed assembly rather than an isolated spring alone. Bearing preload, fastener compliance, joint housing deformation, temperature, and mechanical stops can change the measured relation. Bidirectional calibration can expose hysteresis and different tension and compression behavior. A nonlinear spring needs enough calibration points to represent its full operating range, and the controller must avoid extrapolating beyond validated deflection.

Force estimation also depends on topology. In an output-side FSEA, the spring force closely approximates output force when its own moving mass and parallel load paths are negligible. In an RFSEA, spring force is a reaction measurement that can contain housing inertia and damping. Paine and colleagues showed that accurate RFSEA output-force estimation requires more model information than the corresponding FSEA estimate.^[4]

A force estimate from spring displacement has several useful properties. It can be more sensitive than estimating output torque from motor current through a high-ratio gearbox, and it measures load after much of the gearbox friction and ripple. It can also remain noisy if a stiff spring deflects by only a few encoder counts. Filtering reduces noise but adds delay, which must be included in the stability and bandwidth analysis.

Force, position, and interaction control

The simplest SEA force controller compares desired force with the force inferred from spring deflection, then commands motor current, torque, velocity, or position to reduce the error. Many implementations use nested loops:

1. An inner current or motor-velocity loop regulates the electrical drive.
2. A spring-force or joint-torque loop tracks the requested load.
3. An outer loop sets position, impedance, admittance, or whole-body behavior.

This separation allows a higher-level controller to treat the joint approximately as a torque source within the validated bandwidth of the inner loops. The approximation fails near saturation, unmodeled resonance, communication delay, or frequencies where the torque loop has substantial phase lag.

Feedforward control can command the motor motion needed to create a desired spring deflection before feedback error develops. Model-based inverse dynamics can compensate for known motor inertia, linkage geometry, and gravity. Disturbance observers estimate the combined effect of friction, uncertain loads, and model mismatch. Kong, Bae, and Tomizuka used a disturbance observer in a rotary SEA intended for physical human-robot interaction, with the aim of reducing apparent motor friction and inertia when commanded torque was near zero.^[6] Paine and colleagues combined PID control, identified models, inverse dynamics, and disturbance observation in a high-power prismatic SEA.^[4]

Position control of an SEA needs both sides of the elastic transmission. If the controller holds only the motor position, load position changes with spring deflection under force. A common approach closes an outer load-position loop around an inner torque loop. This can produce accurate motion while retaining force control, but the compliant mode remains part of the closed-loop plant.

Impedance control commands a relation between output motion and interaction force, such as a virtual mass, damper, and spring. Admittance control measures interaction force and generates a motion command. Both are used in contact-rich and human-interactive systems. The physical spring limits which virtual impedances can be rendered accurately and passively, especially at high frequency.^[7][8]

Passivity analysis asks whether the controlled actuator can deliver net energy through its interaction port. A passive interaction is useful because it can remain stable when coupled to a broad class of passive environments. The metal spring alone is passive, but the controlled SEA may not be. Sampling, filtering, integral action, time delay, virtual stiffness, motor damping, and controller gains all affect the result. Formal studies have derived passivity conditions for common velocity-sourced cascades and have shown that apparently helpful physical damping can change those conditions when integral control is present.^[7][8]

Design parameters and tradeoffs

SEA design is a coupled mechanical, electrical, sensing, and control problem. Choosing a spring from peak torque alone is insufficient because stiffness also sets deflection, resonance, resolution, and stored energy.

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For the simplest clamped-load model in Williamson's thesis, a motor inertia J_m coupled to spring stiffness k_s has the natural frequency

omega_n = sqrt(k_s / J_m).

This expression is useful for intuition, but a robot joint usually has motor and load inertia, reflected gearbox inertia, damping, nonlinear friction, structural modes, and a digital controller. Its relevant resonances should be found from the full model and measured frequency response.^[1][3]

Spring stress and fatigue can set the true torque limit before motor current does. A design must check maximum deflection, yield margin, buckling for compression springs, cycle life, creep for polymers, and the consequences of a failed spring or sensor. Mechanical stops can prevent overtravel, but an abrupt stop temporarily turns the compliant drive into a much stiffer impact path.

Motor and transmission selection remain important. The motor must supply the velocity needed both to move the load and to wind or unwind the spring. A spring can release energy faster than the motor alone, increasing peak output power for a suitable task. Paluska and Herr's model showed that a properly selected series spring can increase actuator work and peak power for some inertial motions.^[9] The result depends on spring constant and motion timing. Off-resonance operation, damping, or a mismatched spring can increase motor motion and energy use instead.

Benefits and limitations

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The central benefit is controllable force, not softness by itself. A passive spring without a deflection sensor and force loop can absorb impacts, but it does not provide the same torque-source behavior. Conversely, a stiff actuator with a load cell can perform force control, but it lacks the SEA's intentional compliant buffer and may be more sensitive to contact delay and drivetrain disturbances.^[1][3]

Safety claims require care. Series compliance can lower collision stiffness and give a controller time to respond, which is useful around people. The complete risk also depends on robot mass, speed, geometry, maximum spring energy, mechanical stops, software limits, and failure behavior. An SEA should therefore be described as a safety-enabling component, not a complete safety system.

Variants and related actuator concepts

A rotary series elastic actuator (RSEA) transmits torque through a torsional or mechanically converted spring. A linear or prismatic SEA transmits force along a line, often through a ball screw or cable. These are geometric variants of the same series-force-path principle.^[4][6]

Fixed-compliance SEAs use one mechanical stiffness during operation. Variable-stiffness series elastic actuators add a mechanism that changes spring preload, lever geometry, active spring length, or another property of the elastic transmission. This allows a controller to trade force sensitivity and impact compliance against deflection and bandwidth as the task changes. The extra mechanism adds mass, control variables, failure modes, and energy cost. Variable-stiffness actuators belong to the broader class of variable-impedance actuators, which also includes mechanisms that regulate damping.^[17]

Nonlinear SEAs use a force-deflection curve that changes with displacement. Progressive stiffness can provide high sensitivity near zero torque and greater load capacity near the ends of travel. The controller must use the nonlinear calibration, and the changing tangent stiffness moves the actuator resonance as load changes.

A series-damped elastic actuator (SDEA) adds a physical damping element in parallel with the series spring. Kenanoglu and Patoglu showed that, under specified linear control conditions, this added damping can relax a passivity bound that prevents a conventional SEA from passively rendering virtual spring stiffness above its physical spring stiffness.^[19] The damper also dissipates energy, so it can reduce efficiency in cyclic tasks.

Clutched and lockable variants can bypass or lock the spring for selected operating modes. They may provide rigid positioning or protect the elastic element when compliance is not wanted, but switching changes the plant dynamics and can create impact if it occurs with nonzero relative motion.

Parallel elastic actuators are different. Their spring shares load with the motor rather than sitting as the measured element in the sole transmitted-force path. A parallel spring can reduce motor torque for a known posture or periodic trajectory, but it does not automatically provide SEA force sensing. Some robots combine series and parallel springs. Quasi-direct-drive actuators take another approach, using a high-torque motor and low reduction to obtain low reflected inertia without a deliberately soft output spring.

Applications

SEAs are most useful when accurate interaction force, repeated impacts, or cyclic energy exchange matter more than perfectly rigid position transmission. Representative systems show different reasons for adopting the architecture.

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In legged robots, a spring can absorb landing energy, reduce torque ripple at ground contact, and temporarily store energy between phases of a gait. Spring Flamingo was an early bipedal example, while ATRIAS designed the leg mechanism around spring-mass dynamics.^[12][13] The controller still has to manage spring state. Unreleased energy, deflection limits, or an unmodeled compliant mode can destabilize foot placement and body control.

Several humanoid robots have used SEA joints for torque-controlled whole-body motion. COMAN used distributed passive compliance, and Valkyrie used local series-elastic torque controllers beneath higher-level robot control.^[10][11] Their implementations also illustrate why an article on SEA should not be a catalog of robots: the same actuator concept supports different mechanics and control hierarchies.

Rehabilitation robots and wearable exoskeletons need controlled physical contact over varying human limb dynamics. LOPES used impedance-controlled joints for patient-in-charge and robot-in-charge gait modes. Vallery and colleagues found that series compliance aided interaction but also bounded the stiffness that the system could passively render, a direct example of the performance and stability tradeoff.^[14][15]

Upper-limb systems use the same principle at smaller scale. Chen, Casas, and Lum reported a two-degree-of-freedom elbow and forearm exoskeleton using two SEAs and a cable differential. Bench tests reported torque-control bandwidth of 3.7 Hz and root-mean-square torque error below 0.19 N m for that device.^[16] These figures describe one mechanism and controller; they are not general SEA performance limits.

Commercial modular hardware also uses the architecture. HEBI Robotics identifies its current T-Series and R-Series modules as smart series-elastic actuators that integrate a brushless motor, gear train, spring, encoders, and control electronics. The same documentation lists the older X-Series as discontinued.^[18]

Selection and evaluation

An SEA specification should begin with the load trajectory and interaction requirement. Peak and continuous force, output speed, required torque bandwidth, acceptable position deflection, impact spectrum, duty cycle, and environment determine whether series compliance is useful. The spring and motor should then be evaluated together rather than selected independently.

Bench characterization commonly includes static force-deflection calibration, hysteresis, force resolution, torque-tracking error, frequency response at several loads, output impedance, backdriving force, efficiency, thermal limits, overtravel behavior, and impact tests. Measurements should distinguish spring force from actual load force when the topology or moving inertia makes them different. Controllers intended for human contact also need coupled-stability testing across plausible user impedances, sampling rates, delays, and failure states.^[4][7][8]

No single stiffness maximizes every property. Soft springs favor measurable deflection and low passive impedance. Stiff springs favor compact travel and position transmission. Mechanical arrangement, sensor quality, and control design decide how much of either advantage survives in the finished robot.

References

1. ^Williamson, Matthew M. *Series Elastic Actuators*. S.M. thesis, Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 1995. groups.csail.mit.edu/...mattw_ms_thesis.pdf
2. ^Pratt, Gill A., and Matthew M. Williamson. *Series Elastic Actuators*. Proceedings of the 1995 IEEE/RSJ International Conference on Intelligent Robots and Systems, vol. 1, 1995, pp. 399-406. doi.org/...IROS.1995.525827
3. ^Robinson, David W. *Design and Analysis of Series Elasticity in Closed-loop Actuator Force Control*. Ph.D. thesis, Massachusetts Institute of Technology, 2000. hdl.handle.net/...54838
4. ^Paine, Nicholas, Sehoon Oh, and Luis Sentis. *Design and Control Considerations for High-Performance Series Elastic Actuators*. IEEE/ASME Transactions on Mechatronics, vol. 19, no. 3, 2014, pp. 1080-1091. doi.org/...TMECH.2013.2270435
5. ^Kim, Hyunsoo, and Youngjin Choi. *Generalization of Series Elastic Actuator Configurations and Dynamic Behavior Comparison*. Actuators, vol. 6, no. 3, 2017, article 26. doi.org/...act6030026
6. ^Kong, Kyoungchul, Joonbum Bae, and Masayoshi Tomizuka. *Control of Rotary Series Elastic Actuator for Ideal Force-Mode Actuation in Human-Robot Interaction Applications*. IEEE/ASME Transactions on Mechatronics, vol. 14, no. 1, 2009, pp. 105-118. doi.org/...TMECH.2008.2004561
7. ^Calanca, Andrea, Riccardo Muradore, and Paolo Fiorini. *Impedance Control of Series Elastic Actuators: Passivity and Acceleration-Based Control*. Mechatronics, vol. 47, 2017, pp. 37-48. doi.org/...j.mechatronics.2017.08.010
8. ^Tosun, Fatih Emre, and Volkan Patoglu. *Necessary and Sufficient Conditions for the Passivity of Impedance Rendering With Velocity-Sourced Series Elastic Actuation*. IEEE Transactions on Robotics, vol. 36, no. 3, 2020, pp. 757-772. doi.org/...TRO.2019.2962332
9. ^Paluska, Daniel, and Hugh Herr. *The Effect of Series Elasticity on Actuator Power and Work Output: Implications for Robotic and Prosthetic Joint Design*. Robotics and Autonomous Systems, vol. 54, no. 8, 2006, pp. 667-673. doi.org/...j.robot.2006.02.013
10. ^Paine, Nicholas, Joshua S. Mehling, James Holley, Nicolaus A. Radford, Gwendolyn Johnson, Chien-Liang Fok, and Luis Sentis. *Actuator Control for the NASA-JSC Valkyrie Humanoid Robot: A Decoupled Dynamics Approach for Torque Control of Series Elastic Robots*. Journal of Field Robotics, vol. 32, no. 3, 2015, pp. 378-396. doi.org/...rob.21556
11. ^Tsagarakis, Nikos G., Stephen Morfey, and Gustavo A. Medrano-Cerda. *Compliant Humanoid COMAN: Optimal Joint Stiffness Tuning for Modal Frequency Control*. Proceedings of the 2013 IEEE International Conference on Robotics and Automation, 2013, pp. 673-678. doi.org/...ICRA.2013.6630645
12. ^Hubicki, Christian, Jesse Grimes, Mikhail Jones, Daniel Renjewski, Alexander Spröwitz, Andy Abate, and Jonathan Hurst. *ATRIAS: Design and Validation of a Tether-Free 3D-Capable Spring-Mass Bipedal Robot*. International Journal of Robotics Research, vol. 35, no. 12, 2016, pp. 1497-1521. doi.org/...0278364916648388
13. ^Pratt, Jerry, Chee-Meng Chew, Ann Torres, Peter Dilworth, and Gill Pratt. *Virtual Model Control: An Intuitive Approach for Bipedal Locomotion*. International Journal of Robotics Research, vol. 20, no. 2, 2001, pp. 129-143. doi.org/...02783640122067309
14. ^Veneman, Jan F., Rik Kruidhof, Edsko E. G. Hekman, Ralf Ekkelenkamp, Edwin H. F. van Asseldonk, and Herman van der Kooij. *Design and Evaluation of the LOPES Exoskeleton Robot for Interactive Gait Rehabilitation*. IEEE Transactions on Neural Systems and Rehabilitation Engineering, vol. 15, no. 3, 2007, pp. 379-386. doi.org/...TNSRE.2007.903919
15. ^Vallery, Heike, Jan F. Veneman, Edwin H. F. van Asseldonk, Ralf Ekkelenkamp, Martin Buss, and Herman van der Kooij. *Compliant Actuation of Rehabilitation Robots: Benefits and Limitations of Series Elastic Actuators*. IEEE Robotics & Automation Magazine, vol. 15, no. 3, 2008, pp. 60-69. doi.org/...MRA.2008.927689
16. ^Chen, Tianyao, Rafael Casas, and Peter S. Lum. *An Elbow Exoskeleton for Upper Limb Rehabilitation with Series Elastic Actuator and Cable-driven Differential*. IEEE Transactions on Robotics, vol. 35, no. 6, 2019, pp. 1464-1474. doi.org/...TRO.2019.2930915
17. ^Vanderborght, Bram, et al. *Variable Impedance Actuators: A Review*. Robotics and Autonomous Systems, vol. 61, no. 12, 2013, pp. 1601-1614. doi.org/...j.robot.2013.06.009
18. ^HEBI Robotics. *Hardware: T-Series and R-Series Actuators*. Official documentation, accessed July 24, 2026. docs.hebi.us/hardware
19. ^Kenanoglu, Celal Umut, and Volkan Patoglu. *A Fundamental Limitation of Passive Spring Rendering With Series Elastic Actuation*. IEEE Transactions on Haptics, vol. 16, no. 4, 2023, pp. 456-462. doi.org/...TOH.2023.3260063