A Second Torque Port for Series Elastic Actuators: Parallel-Integrated Design and Time-Scale Torque Allocation

Donghao Jia; Xiubo Xia; Junhang Liu; Zeyu Zhu; Xiaoyu Geng; Jian Sun

cs.ROSep 25, 2026

A Second Torque Port for Series Elastic Actuators: Parallel-Integrated Design and Time-Scale Torque Allocation

Authors: Donghao Jia, Xiubo Xia, Junhang Liu, Zeyu Zhu, Xiaoyu Geng, Jian Sun

Organizations: School of Mechanical Engineering and Automation, Beihang University, Beijing 100191, China

Abstract

A series elastic actuator has a single torque port and pays for it twice: the geared motor must swing its own reflected inertia through the spring, so the amplitude it delivers collapses as $ω^{-2}$ in the command frequency $ω$ once it saturates, while commands below the transmission's breakaway friction never arrive at all. This letter opens a second torque port on the load side, placing a frameless direct-drive micro motor in parallel with a fixed-stiffness spring -- a parallel-integrated SEA, or Pi-SEA, whose delivered torque is read from spring deflection and micro current without a sensor -- and dividing the commanded torque between the two channels by time scale rather than by filter design. The micro torque loop is the fast subsystem, which makes the closed loop singularly perturbed and turns the separation the channels need into a bound to check rather than a crossover to tune; a leaky mid-ranging integrator returns the steady load to the spring; and the amplitude ceiling, read backwards, becomes a closed-form sizing rule that matches spring, geared motor and micro motor to the amplitudes and frequencies an application asks for. Against SEAs, the Pi-SEA widens the tracked band at small amplitudes and lowers the residual the joint imposes on its environment, each by an order of magnitude.

Figures & tables

Fig. 1: Single-joint model of the parallel-integrated SEA. The slow subsystem is the compliant SEA–load interaction, the fast subsystem the torque loop of the load-side micro motor, whose torque reaches the load without passing through the spring or the reduction.

Fig. 2: Dual-time-scale allocation: the macro channel is force-controlled on the spring deflection with a leaky mid-ranging integral, the micro channel renders the residual $\tau_{d}-F_{s}$ , and both act on the common load.

Fig. 3: The sizing rule ( 16 ) as a design surface for the 10 N m at 20 Hz specification. Height and colour are the micro torque $T_{\mu}$ the rule requires at each $(K_{s},\tau_{m}^{\max}/J_{m})$ , falling to zero where the macro channel already covers the specification alone; a motor serves everything below its own magenta iso-line. The orange curves section the surface at the as-built joint; the shortfall they show is a prediction at the joint’s peak torque, quantified in the text.

Panel of Figure 4 — Fig. 4: The prototype as built. (a) Cross-section: the torsion spring ( $K_{s}$ ) and the frameless micro motor ( $\tau_{\mu}$ ) act in parallel on the load, and an absolute encoder across the spring measures $\theta_{s}$ . (b) The assembled joint and its bench. One EtherCAT master closes both loops at 1000 Hz over three slaves; the spring doubles as the torque sensor, so no external transducer is in the loop.

Symbol	Value	Symbol	Value
$J_{m}$ [kg m 2 ]	1.56	$K_{ps}$ [N m/rad]	8000
$B_{m}$ [N m s/rad]	12.2	$K_{ds}$ [N m s/rad]	160
$\tau_{b}$ [N m]	9.0	$K_{I}$ , $\lambda$ [1/s]	10, 0.1
$K_{s}$ [N m/rad]	780	$N_{d}$ , $\omega_{Q}$ [rad/s]	300, 50
$k_{t}$ (meas.) [N m/A]	0.105	$J_{l}$ bar / link [kg m 2 ]	0.071 / 0.30
$\omega_{\mu}/2\pi$ [Hz]	745	$a_{\mu}$ [rad/s]	112

TABLE I: Parameters identified on the assembled prototype and the gains and load conditions used.

Fig. 5: Closed-loop

T_{\rm id}/\tau_{d}

, load clamped, identical gains. (a) 0.5 N m and (b) 2 N m chirps, 0.3–20 Hz, drawn where the coherence exceeds 0.5; (a) adds the fixed command split at 2 and 18.1 Hz, (b) the Pi-SEA split into

F_{s}

and

\tau_{\mu}

, with

\tau_{\mu}

under the frictionless model dashed and both

-3

dB points dotted—the gap is the allocation responding to delivered torque, which no fixed filter of

\tau_{d}

does. (c) 5 N m, with the splits at 10 Hz, and (d) 15 N m stepped sines; hollow markers were cut short by the spring-energy guard.

Fig. 6: Where each controller tracks the command (

|1-T_{\rm id}/\tau_{d}|\leq 0.5

) with the load clamped, under the

\tau_{m}^{\max}=100

N m software clamp of this campaign, not the joint’s 170 N m peak. Orange, the region the macro-only loop already tracks; blue, the band the micro motor adds; white, neither. Markers are the four measured amplitudes, the 0.5 N m Pi-SEA edge from stepped sines past the chirp; the boundaries between them are interpolated. Dotted, the macro amplitude ceiling

\tau_{m}^{\max}K_{s}/(J_{m}\omega^{2})

of ( 16 ); dashed, that ceiling plus

T_{\mu}

; dash–dot, the spring-energy guard of the test rig, which is what stops the 15 N m row at 7 Hz.

Fig. 7: Leaky mid-ranging, load clamped. (a) 1 N m step with

K_{I}=10

: the micro motor carries the edge and the spring channel takes the load over. (b) Micro torque after the same step with the DOB off for

\lambda=0.1,0.2,0.5

s -1 (purple) and with the DOB on for

\lambda=0.1

(grey).

Fig. 8: Free symmetric bar ( $J_{l}=0.071$ kg m 2 ). (a) Closed-loop gain $|T_{\rm id}/\tau_{d}|$ of the 2 N m chirp with ( 5 ) (dark) and with $\tau_{\rm ff}=\tau_{d}$ (light); dotted, the low-frequency share of the latter. (b), (c) Residual output torque under $\tau_{d}=0$ while the bar is pushed by hand, slow and fast, against the bar rate: raw samples faint, fitted loops dark (median torque in equal-count rate bins on each branch of the stroke).

Fig. 9: Torque tracking on the gravity link (bar with 2.5 kg at 0.3 m, hanging start): (a) sine 1 N m at 0.3 Hz, (b) triangle $\pm 5$ N m at 0.25 Hz, (c) sine 12 N m at 4 Hz with ( 5 ). Rows: command and delivered torque; error $\tau_{d}-T_{\rm id}$ (raw faint, filtered dark); Pi-SEA split into $F_{s}$ and $\tau_{\mu}$ . One run of each controller is shown.

Robotics

Robotics studies the design and control of machines that perceive, move, and act in physical environments.

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Robotics Haptic Feedback

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Designing robots for high-torque, high-fidelity haptic interaction is challenging. Parallel Elastic Actuators (PEAs) use elastic elements in parallel to smaller motors to complement torques, and Series Elastic Actuators (SEAs) use elastic elements in series to decouple motor impedance and improve force control. Recent work combines SEAs and PEAs to obtain both benefits but requires separate elastic elements or clutching. This paper presents the Series Parallel Integrated Nonlinear Elastic Actuator (SPINEA), which merges SEA and PEA such that a single elastic element takes on dual roles simultaneously, parallel and series. This is achieved by a nonlinear transmission in which the motor and load have misaligned rotation axes and are elastically connected. This geometry enables both high peak torque and precise torque tracking. We apply SPINEA to actuate lean of a haptic bicycle simulator, which requires high moments and precise rendering for safe and realistic rider interactions. We realized a prototype and performed experiments, both with an external excitation setup and with riders cycling. Our results confirm SPINEA's low impedance and precise torque tracking, up to 4.25 Hz with the bicycle frame fixed and up to 4 Hz with riders. The benefits may transfer to other applications requiring compact, high-performance actuation.

Abstract

Robotics

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