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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

arXiv:2609.30873v1cs.ROcs.SYeess.SY

TL;DR

A conventional SEA’s single torque path limits high-frequency output and fails to deliver commands below transmission breakaway friction. The Pi-SEA adds a direct-drive micro motor in parallel with the spring and allocates torque by time scale; single-joint experiments report restored phase, a tracked band below the breakaway window, and reduced zero-torque interaction residual.

  • Problem

    A conventional SEA’s single elastic torque path produces an ω^-2 amplitude ceiling after motor saturation and does not deliver commands below transmission breakaway friction.

    Source: §I · p. 1

  • Method

    The Pi-SEA adds a load-side direct-drive micro motor to a fixed-stiffness spring and allocates torque by time scale, with a leaky mid-ranging law and a closed-form micro-motor sizing rule.

    Source: §I · pp. 1–2, A. Design Rule: Spring, Macro and… · p. 3

  • Results

    Single-joint experiments against an identical-gain macro-only baseline report restored phase, a tracked band below the macro channel’s breakaway window, and reduced zero-torque interaction residual.

    Source: A. Setup, Baseline and Metrics · p. 4, §V · p. 6

  • Takeaways & Limitations

    The load-side torque port supports tracking in the macro channel’s breakaway window while the allocation returns steady load to the spring.

    Source: §I · p. 2, §V · p. 6

    The study covers one joint; its regulation-scope bound depends on state scaling, and it claims no bandwidth advantage when the residual exceeds the micro motor’s short-term peak.

    Source: §I · p. 2

Abstract

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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 $\omega^{-2}$ in the command frequency $\omega$ 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.

I. INTRODUCTION

A conventional SEA’s single elastic torque path limits high-frequency output and blocks commands below breakaway friction; existing alternatives still leave the load with one elastic port. The Pi-SEA adds a direct-drive load-side micro motor, making its torque loop the fast subsystem and the spring–load interaction the slow one.

I. INTRODUCTION

  • The geared motor’s reflected inertia drives an ω^-2 amplitude ceiling after saturation, while commands below 9.03 N m breakaway friction are not delivered.The passage says controller gain cannot lift the ceiling; both limits grow with the reduction ratio.Source: §I · p. 1
  • Existing variable-stiffness, adjustable-equilibrium, parallel, damping, and control approaches retain one elastic load-torque port or do not let both motors contribute simultaneously.The cited design costs include added mass and volume and stiffness rate limits; damping suppresses resonance but cannot supply denied torque.Source: §I · p. 1
  • The Pi-SEA keeps a fixed-stiffness spring and adds a coaxial frameless direct-drive micro motor on the load side.The motor’s stator is on the housing and its rotor on the link; the micro torque is read from current and spring torque from deflection, without a transducer.Source: §I · pp. 1–2

A. Dynamic Model and Time-Scale Separation

  • The single-joint model separates spring-mediated macro torque from the micro motor’s direct load torque, which bypasses both the spring and reduction.The slow subsystem is the compliant SEA–load interaction; the fast subsystem is the load-side micro motor torque loop.Source: Figure 1 · p. 2, A. Dynamic Model and Time-Scale… · p. 2
  • The micro motor’s torque loop is the fast subsystem, while the compliant SEA–load interaction remains slow because the micro motor has neither a reduction nor a spring.The channel bandwidth ratio supplies the small parameter for the singular-perturbation analysis.Source: A. Dynamic Model and Time-Scale… · p. 2

B. Force Controller

The controller force-controls the macro channel using spring deflection and commands the micro channel to render the residual load torque. A leaky mid-ranging integral shifts steady micro torque back to the macro channel while bounding its state.

B. Force Controller

  • The macro channel is force-controlled on spring deflection, with a motor-side disturbance observer and feedforward scaled to the macro torque share reaching the load.For a held load the feedforward share is unity; for a free inertial load below resonance it is Jl/(Jm + Jl), and the gravity correction applies above the pendulum frequency.Source: B. Force Controller · p. 2
  • The mid-ranging state is a leaky integral of micro torque, transferring low-frequency micro torque to the macro channel through KIη.Anti-windup saturation is used, and the leak bounds the state whether or not the loop reaches constant equilibrium.Source: B. Force Controller · pp. 2–3
  • The micro channel renders the residual between commanded load torque and spring torque, so output torque error equals fast-channel tracking error.The macro loop and mid-ranging integral determine how delivered torque is split, rather than whether it is delivered.Source: B. Force Controller · p. 3

C. Singular-Perturbation Stability

The stability analysis concerns the nominal closed loop under a constant commanded torque and zero residual motor and load disturbances; bounded residual disturbances and slowly varying commands enter as inputs giving local practical stability.

C. Singular-Perturbation Stability

  • The nominal stability analysis assumes constant commanded torque and zero residual motor and load disturbances.Bounded residual disturbances and slowly varying commands enter as inputs and give local practical stability.Source: C. Singular-Perturbation Stability · p. 3
  • The stated stability result is scoped to the nominal closed loop with zero residual disturbances, not arbitrary disturbance magnitudes or command variation.The passage specifies bounded residual disturbances and slowly varying commands as inputs for local practical stability.Source: C. Singular-Perturbation Stability · p. 3
  • Bounded residual disturbances and slowly varying commands are treated as inputs that give local practical stability.The nominal closed loop instead takes the command as constant and both residual disturbances as zero.Source: C. Singular-Perturbation Stability · p. 3

1) Reduced and boundary-layer subsystems:

The reduced system describes the SEA–load loop with an ideal micro torque source, while the boundary-layer system captures fast convergence of micro torque to its command. Composite local exponential stability then follows only when the reduced loop is stable and the time-scale parameter meets an explicit bound.

1) Reduced and boundary-layer subsystems:

2) Composite stability:

  • Local exponential stability of the composite equilibrium requires reduced-loop stability and holds for 0 < ϵ < ϵ⋆ under Assumption 1.For the linearized macro loop, the stated Routh–Hurwitz condition also upper-bounds KI.Source: 2) Composite stability · p. 3

3) Steady-state allocation:

The leaky mid-ranging allocation returns steady load to the spring channel, leaving micro torque with a floor proportional to the leak. In the frequency-domain reading, the micro channel complements spring torque not delivered, with its crossover set by the spring channel’s delivery rather than a designed command filter.

3) Steady-state allocation:

  • At steady state, the micro motor renders transient torque while the spring channel takes the steady load, leaving a micro-torque floor proportional to the leak.Whether that floor can be driven to zero as λ → 0 is unresolved.Source: 3) Steady-state allocation · p. 3

D. Frequency-Domain Reading of the Allocation

  • Linearized about the stable equilibrium, the micro channel is the low-passed complement of spring torque not delivered, with DC content O(λ).Unlike a command-splitting filter, its crossover migrates to where the spring channel stops delivering.Source: D. Frequency-Domain Reading of the… · p. 3

III. DESIGN RULE AND PROTOTYPE

The design rule combines the macro channel’s frequency-dependent amplitude ceiling with the micro motor’s direct torque to size hardware against a requested amplitude and frequency. Read in reverse, the same rule gives the reachable bandwidth for a given hardware set.

III. DESIGN RULE AND PROTOTYPE

  • The macro channel’s maximum load torque falls with frequency as τ max_m Ks/(Jmω^2), while the micro channel contributes flat torque Tµ.The macro ceiling follows from the reflected-inertia torque required to swing the motor through spring deflection; its ω^2 term is retained as the high-frequency asymptote above resonance.Source: A. Design Rule: Spring, Macro and… · pp. 3–4
  • For a specified amplitude and frequency, rule (16) sizes Tµ to cover the shortfall when the macro channel alone cannot meet the specification.Figure 3 maps required micro torque over spring stiffness and macro acceleration, with zero required where the macro channel covers the specification.Source: A. Design Rule: Spring, Macro and… · pp. 3–4, Figure 3 · p. 4
  • Read backward, rule (16) gives the force bandwidth of a hardware set: for A > Tµ the macro channel sets reach, while for A ≤ Tµ the micro-loop bandwidth does.The regimes meet at ωcross.Source: A. Design Rule: Spring, Macro and… · p. 4

B. Specification, Selection and Prototype

The prototype pairs a harmonic-drive macro joint and spring with a frameless direct-drive micro motor mounted at the load. The design rule’s 20 Hz specification informed motor selection, while the assembled joint’s usable micro torque and crossover constrain the demonstrated operating range.

B. Specification, Selection and Prototype

  • For the 10 N m at 20 Hz target, the rule requires Tµ ≥4.51 N m, and the selected motor’s 6.30 N m peak rating provides a 1.40× margin.The joint uses a 100:1 harmonic drive, 170 N m peak macro torque, and a spring designed for 750 N m/rad.Source: B. Specification, Selection and… · p. 4
  • The prototype mounts a frameless motor directly between the housing and link, injecting micro torque without a transmission alongside the flange-to-link torsion spring.Joint and micro-rotor encoders provide θm and q.Source: B. Specification, Selection and… · p. 4
  • With usable micro torque limited to 3 N m peak, the as-built joint misses the sizing-rule prediction by a factor of 1.30; it reaches 10 N m up to 17.5 Hz.Its 20.5 Hz crossover lies beyond the 15–16 Hz macro-loop phase limit, so the micro motor operates in the small-amplitude regime.Source: B. Specification, Selection and… · p. 4

A. Setup, Baseline and Metrics

With the load clamped, the second torque port improves tracking most below the macro loop’s breakaway window and improves phase at larger amplitudes. The experiments also check the time-scale separation bound and compare delivered-torque allocation with fixed command splits.

A. Setup, Baseline and Metrics

  • Experiments use an unchanged-gain macro-only baseline and define delivered torque as Tid = Ksθs + τµ, using spring deflection and measured micro current without an external torque sensor.Tracking is defined by |1−Tid/τd| ≤0.5; the logged torque readback trails physical torque by about 2 ms.Source: A. Setup, Baseline and Metrics · p. 4, Figure 4 · p. 5

B. Time-Scale Separation: ϵ against ϵ⋆

  • The measured separation is ϵ = 0.024, below the clamped-load bound ϵ⋆ = 0.093, placing the prototype inside Theorem 1 in the stated coordinates.The clamped loop’s dominant poles give a smaller ϵ = 0.016, making the 0.024 choice conservative.Source: B. Time-Scale Separation: ϵ against ϵ⋆ · pp. 4–5

C. Force Bandwidth and Allocation with the Load Clamped

D. Mid-Ranging

Leaky mid-ranging transfers steady load from the micro motor to the spring, while reducing hand-push residuals and improving torque tracking on the free bar and gravity link. The experiments also identify the tested scope and remaining system-level limits.

D. Mid-Ranging

  • Mid-ranging shortens the 1 N m step handover from the micro motor to the spring from 99.9 to 50.9 ms and leaves a steady load of 0.00017 N m.Handover time is measured to 10% of peak.Source: D. Mid-Ranging · p. 5, Figure 7 · p. 6
  • With the disturbance observer off, residual micro torque is 0.021, 0.063, and 0.129 N m for λ = 0.1, 0.2, and 0.5 s−1, respectively.These values are close to the predicted λη⋆ values of 0.03, 0.066, and 0.130 N m.Source: D. Mid-Ranging · p. 6, Figure 7 · p. 6

E. Free Bar: Authority on a Light Load

  • On a free bar, Pi-SEA delivers 0.943 of a 2 N m command over 3–10 Hz, versus 0.261 for the macro-only loop over 2–10 Hz.The feed-forward recovers most of the macro-only deficit, and the micro motor closes the rest.Source: E. Free Bar: Authority on a Light Load · p. 6

F. Zero-Torque Command and Torque Tracking on a Gravity Link

  • On slow hand pushes, the second port reduces zero-command residual torque 10.4-fold, from 0.456–0.617 to 0.046–0.059 N m rms.Remaining torque is concentrated at stroke reversals, where the macro channel turns its inertia against friction.Source: F. Zero-Torque Command and Torque… · p. 6
  • On the gravity link, rms torque error falls 7.92-fold for a 1 N m sine at 0.3 Hz, 12.0-fold for a ±5 N m triangle at 0.25 Hz, and 3.18-fold for a 12 N m sine at 4 Hz.The 4 Hz result is with feed-forward active and the macro channel already carrying its share.Source: F. Zero-Torque Command and Torque… · p. 6

V. CONCLUSION

  • The evaluation covers one joint; coupled multi-DoF behavior, stability in environmental contact, and a thermal model replacing the fixed clamp remain future work.The regulation-scope bound also depends on state scaling.Source: §V · p. 6

APPENDIX

The appendix gives a sufficient, coordinate-dependent stability bound for the fast micro torque loop and practical tuning conditions for the leaky mid-ranging controller. It also describes torque-tracking and residual-torque figures for free-bar and gravity-link tests.

APPENDIX

  • Fig. 8 plots closed-loop gain and residual torque during hand pushes on a free bar, while Fig. 9 shows tracking and torque allocation on a gravity link.Fig. 9 includes 1 N m at 0.3 Hz, ±5 N m at 0.25 Hz, and 12 N m at 4 Hz; captions do not state comparative outcomes.Source: Figure 8 · p. 7, Figure 9 · p. 7
  • The leaky mid-ranging integral relaxes the practical tuning bound and keeps η bounded, but a free load must be damped or gravity-restored for the assumption to hold.For a free load, the micro motor supplies τd − Fs exactly, so the macro loop no longer damps q.Source: APPENDIX · p. 7
  • The positive-definiteness condition sets the stability bound; the best weight is d = 0.09 here, rather than 1/2 unless L1 = L2.The analysis uses L1 = 10.8 and L2 = 112 in torque-equivalent coordinates.Source: APPENDIX · p. 8
  • The composite-Lyapunov result guarantees local exponential stability only when ϵ is below a sufficient ϵ⋆ bound, whose value depends on state scaling.The torque-equivalent coordinates put slow states in the fast-error unit, N m; the bound is not a plant invariant.Source: APPENDIX · p. 8
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