Series Elastic Actuator: Spring Stiffness, Impact and Bandwidth

SEA mechanics. In a common linear output coordinate, $F_s=k_s(x_m-x_l)$. The unpowered impact equations are $m_m\ddot x_m=-b_m\dot x_m-F_s$ and $m_l\ddot x_l=F_s-F_c$, with downward displacement positive. Both masses start at speed $v_0$, with zero spring strain and zero contact penetration. The rigid comparison uses the combined mass; its transmission force follows the motor's inertia balance.

Nonlinear contact. For penetration $\delta>0$, $F_c=K_c\delta^{3/2}\max(0,1+\alpha\dot\delta)$; otherwise $F_c=0$. The nonzero root of $\ln[(1+\beta)/(1-e\beta)]=(1+e)\beta$, where $\beta=\alpha v_0$, calibrates isolated-contact restitution. This relation is derived by integrating $v/(1+\alpha v)$ over compression and restitution; it avoids a small-loss approximation. At $e=1$, $\alpha=0$. Nonnegative clipping prevents adhesive contact during coupled motion.

Force control. The unity-feedforward PD structure follows Robinson et al. (1999), equations 5–6. With $a=1+K_p$, $T(s)=k_s(a+K_ds)/[m_ms^2+(b_m+k_sK_d)s+k_sa]$. The exact positive −3 dB crossing gives the small-signal bandwidth. Dynamic stiffness is $D(s)=k_s(m_ms^2+b_ms)/[m_ms^2+(b_m+k_sK_d)s+k_sa]$; mechanical impedance is $D(s)/s$.

Motor limit and spring selection. The amplitude input $A$ is the achieved sinusoidal output force, held constant as frequency changes by shaping the command. It is not the force-command amplitude. At the target frequency, the displayed required command amplitude is $A/|T(j\omega)|$. For achieved output amplitude $A$, fixed-output motor velocity is $A\omega/k_s$. Maximizing the symmetric force–speed constraint over phase gives $U=(A/F_{stall})\sqrt{(1-m_m\omega^2/k_s)^2+[(b_m+F_{stall}/v_{free})\omega/k_s]^2}$. The first crossing $U=1$ defines amplitude-limited bandwidth. The conservative range combines both bandwidth lower bounds, achieved-output travel $A/k_s$, and passive energy bounds $k_s\ge2E_0/d_{allow}^2$, $k_s\le F_{allow}^2/(2E_0)$, with $E_0=(m_m+m_l)v_0^2/2$.

Numerical roots and resonances. Quadratic inversions use the rationalized positive root when the linear coefficient is positive. Frequency charts combine a logarithmic background with points spaced at one twentieth of each narrow resonance's half-width, geometric tails, and the analytically determined force and dynamic-stiffness peak frequencies.

Constant reflected masses and transmission ratio; linear bidirectional series spring; zero initial preload; frictionless normal ground contact with exponent 3/2. Motor and foot initially move together. Impact has no motor command and no gravity; viscous mechanical damping remains. The separate force-control calculation has a clamped output, ideal force sensing, an instantaneous current loop, and PD differentiation of force error. The force-amplitude input specifies achieved output maintained by sinusoidal command-amplitude shaping. Stall force and no-load speed define a symmetric triangular force–speed envelope. Contact stiffness and restitution defaults are illustrative design inputs, not measured material properties.

Results apply to the entered effective coordinate, impact speed and calibrated contact pair. Restitution is calibrated for an isolated unforced contact; coupled-leg rebound generally differs. The spring stores energy and returns it; only contact hysteresis and motor damping dissipate energy. Contact-force peaks and transmission-force peaks are different quantities.

Adaptive Dormand–Prince integration uses the engine's 10⁻⁶ relative tolerance. Peaks are sampled at 1,201 times over an automatically scaled window, and 61 stiffnesses from one tenth to ten times the selected value are solved together. Very brief pulses can be under-resolved; the input condition identifies when the output-mass contact scale has fewer than 12 sample intervals. Small peaks between samples and later impacts outside the observation window can be missed. Playback uses PCHIP interpolation of the solved trajectory and exaggerated displacement in the schematic.

The conservative capacity window combines spring-force and spring-travel energy bounds with the two clamped-output bandwidth limits for the specified achieved output amplitude. It does not impose a contact-force bound; check the nonlinear contact result separately. A selected spring outside that conservative window may still meet the simulated limits because energy is dissipated. The verdict checks achieved-output travel and force as well as passive impact limits. “Capacity limits met” is a sizing assessment for achieved output, not approval of an unshaped force command.

No hard-stop contact, spring stress, fatigue, backlash, Coulomb friction, sensor noise, electrical delay, active impact control, preload, gravitational work, or changing robot-leg geometry is included. High ideal PD gains can predict bandwidth that hardware cannot deliver. The amplitude bandwidth is an achieved-output capacity envelope, not a nonlinear closed-loop tracking simulation. An unshaped command of amplitude A produces A|T(jω)|; resonant amplification can exceed the specified output amplitude and invalidate force, travel and motor limits. For example, a 740 N command at 8 Hz with otherwise default parameters predicts about 815 N output. Use the displayed required command amplitude when targeting an achieved output; amplitude shaping does not remove phase lag, transient overshoot or model error.

Robinson's Table I lists 7.9 Hz, Ω = 4 and 35 Hz; the first two imply about 31.6 Hz, so the check uses the consistent 315 kN/m, 128 kg and 7.9 Hz entries. Williamson's 48.0 rad/s theoretical example excludes gearbox inertia; the measured value is lower.

Frequency-response peaks are evaluated from stationary points of the exact rational transfer functions and inserted directly into the plotted grid. A damping-scaled grid resolves the narrow neighborhood; no interpolation is used to invent an unresolved resonance. Ideal linear resonance gains may exceed motor capacity, so inspect them together with the separate capacity limit.