QDD Actuator: Gear Ratio, Inertia, Bandwidth and Heating

The inertia and contact model follows Wensing et al. (2017), §III-A–B, DOI 10.1109/TRO.2016.2640183. The attached manuscript text has reflowed pages; its pp. 4–5 contain the reduced-leg equations. With $I=N^2J_m+J_g$,

This is the directional IMF of the one-dimensional rigid-drive model. The ideal instantaneous elastic-drive value is $m_b/(m_b+m_l)$ because a finite series spring initially decouples the rotor. The rigid value is used for ratio screening; finite-duration impact loads require a contact transient.

For the compliant model, $\alpha=Ir^2m_bm_l$, $\beta=k_s[I(m_b+m_l)+r^2m_bm_l]+Ir^2k_i m_b$, and $\gamma=k_s k_i(m_b r^2+I)$. The two squared modal frequencies solve $\alpha\omega^4-\beta\omega^2+\gamma=0$. A cancellation-resistant expression evaluates the smaller root. The mechanical bandwidth is the first mode. A chosen modal damping ratio is added to each second-order factor, and the normalized torque-to-contact-force path is multiplied by $1/(1+s/(2\pi f_i))$. A frequency scan brackets the first downward half-power crossing before Brent root refinement. This crossing is distinct from the mechanical design ceiling.

Output friction is $T_f=N T_c+N^2b_m|\omega_j|+T_g$. Available torque is $\eta_g N K_t I_q[1-s_{\rm sat}(I_q/I_{\rm pk})^2]-T_f$ below the peak current. Newton iteration inverts this monotone torque law. Above the current limit the requested current is reported using the constant peak torque constant; it is not clipped to make a failed duty appear deliverable. The hot motoring check uses $V_q=R(T_{\rm lim})I_q+(2/3)K_tN\omega_j$, $|V_d|=pN|\omega_j|L_qI_q$ and $\sqrt{V_q^2+V_d^2}\le V_{\rm dc}/\sqrt3$ (SVPWM, no field weakening).

The two-node thermal model is derived from energy balance:

Here $P_h=P_{\rm aux}+|\omega_j|[T_{\rm load}(1/\eta_g-1)+T_f/\eta_g]$. Thus mechanical loss agrees with the torque/power balance. Each constant-duty interval is an affine two-state ODE; an exact matrix-exponential advance resolves every stance, swing, jump and recovery boundary. Analytical stationary points and interval ends give both peak temperatures independently of plot sampling. Mean-duty equilibrium includes copper's positive thermal feedback; a missing equilibrium is shown as a gap. The continuous stall torque includes auxiliary housing heat and hot winding resistance.

At zero speed the same hot dq voltage balance reduces to $V_q=R(T_{\rm lim})I_q$. Peak stall current is limited to $\min(I_{\rm pk},V_{\rm avail}/R_{\rm hot})$; continuous stall current also respects the thermal-current limit. Both reported stall torques subtract breakaway friction and are floored at zero.

Thermal advancement limits every positive modal exponent to 0.5 before evaluating its exponential. Each finite segment is checked for a winding-limit crossing or the 200 °C housing reporting cutoff. Prefix-maximum bisection locates the first event, and the resulting finite history is resampled at 1,601 points. Reported peaks apply only up to this calculated endpoint; the design verdict and history-end time identify early termination.

• A single local contact direction with constant moment arm, linear contact and transmission springs, and the body/leg mass distribution of Wensing's reduced rack-and-pinion model. The normalized mechanical response is a torque-to-contact-force surrogate, not a measured joint torque servo.
• Amplitude-invariant FOC: $I_q$ is peak phase amplitude, $P_{\rm Cu}=3R_{\rm phase}I_q^2/2$, and $K_e=2K_t/3$. All duty torques and speeds are nonnegative motoring magnitudes. Rotor acceleration torque, regeneration and bus absorption are excluded.
• Saturation is an illustrative quadratic interpolation between the supplied low-current torque constant and its peak-current drop, not a digitization of a measured curve. The current loop is an identified first-order small-signal fit; PWM, delay and PI gains are not simulated.
• Katz (2018), pp. 25, 27, 38, 44, 53–54, supports the 6:1 scale, roughly 0.082 N·m/A low-current constant, 12% saturation drop, 40 A peak phase current, 21 pole pairs, about 30 µH inductance, usual 1 kHz current-loop setting, aggregate 1.23 K/W thermal resistance and 32 J/K effective thermal mass. The 0.13 Ω phase resistance is inferred from his rounded 60 W, 15.4 A and 30% resistance-rise discussion.
• The default rotor/gear split (0.000060 and 0.000140 kg·m²) is assigned to match the published 0.0023 kg·m² total at 6:1. The friction, efficiency, stiffnesses, damping, duty, 0.35/0.88 K/W thermal split and 120 J/K housing capacity are editable example assumptions. Katz's measured effective thermal mass is assigned to the winding node for illustration; the separate nodes require identification on real hardware.
• Copper's linear resistance coefficient is 0.00385/K at the 25 °C resistance reference in the example; replace it for a different winding material or resistance fit.

Use the results for motor and ratio screening, then identify the assembled actuator and test the proposed motion. “Screening targets met” means only that the stated model targets pass; it is not a stability, fatigue or impact-force check.

The ratio sweep uses mean-duty thermal equilibrium, not a transient peak at every ratio. Its bounds are the lowest/highest passing samples on a 1…20 grid at 0.125 intervals; inspect the full candidate set for disconnected ranges. Apply the selected-ratio pulse-resolved thermal check before choosing a reducer. Blank equilibrium values denote electrothermal runaway in the averaged constant-current model.

The thermal history uses 1,601 uniformly spaced display times over the calculated horizon, with exact advances at all intervening duty boundaries. Temperature maxima also include analytical within-interval extrema, so they do not depend on the display grid. The play indicator uses shape-preserving cubic interpolation between display times and may smooth a very short pulse; the computed peak does not. Current settling, inductive energy at transitions, spatial hot spots, temperature-dependent magnets, iron losses varying with speed, and cooling changes with airflow are not resolved. Add measured omitted heat through the housing-loss input.

Bandwidth and IMF are configuration dependent. The rigid-drive directional IMF assumes the drive remains inertially coupled during contact; with a finite series spring, the ideal instantaneous impulse sees the elastic-drive IMF instead. Neither is the full three-dimensional determinant IMF of an articulated robot. Modal damping is assumed, and torque response is linearized below current/voltage saturation. A high half-power crossing beyond a resonance does not imply flat or stable torque tracking. Backdrive friction here is an unpowered no-load Coulomb/viscous approximation and does not model wedging or load-dependent reverse efficiency.

Screening always evaluates both running and jumping block definitions; their durations control the transient history. Omit a load from both screening and the transient by setting its torques to zero.

The thermal history stops at the first winding-limit crossing, or if the housing reaches a 200 °C reporting cutoff. The latter is a numerical/model reporting boundary, not a housing rating. The history-end result gives the crossing time; peaks and temperature margin cover only that finite history. An early-stop verdict means the remainder of the requested schedule was not calculated. Playback ends at the same event. The positive-exponent step bound prevents exponential overflow before detecting it; no nonfinite values are passed to the charts or thermal drawing.