Robot Joint RMS Torque & S1 Motor Thermal Check

For a rigid rotary link, the parallel-axis theorem gives $J=J_c+m r_c^2+m_p L^2+J_{extra}$. With angle measured counterclockwise from horizontal, inverse dynamics is $T_j=J\ddot q+g(mr_c+m_pL)\cos q+T_C\operatorname{sign}(\dot q)+b\dot q+T_{ext}$. Gravity can be disabled for a horizontal-plane joint.

The smooth out-and-return motion uses the quintic $s(u)=10u^3-15u^4+6u^5$ with zero endpoint velocity and acceleration. The optional triangular-speed move has constant acceleration in each half-move, with integration breaks at the velocity corners. Table mode solves the cyclic cubic-spline system on the actual, potentially unequal, time intervals; the closing position is the starting position. Velocity and acceleration are analytic derivatives of the selected interpolation.

Transmission torque is $T_j/(N\eta_f)$ when output power is nonnegative, and $\eta_bT_j/N$ when the load backdrives. The shaft-equivalent demand adds $J_{drive}N\ddot q$ to cover rotor and input-side acceleration, and motor speed is $N\dot q$. Every dwell retains gravity and applied-load demand. Midpoint quadrature gives $T_{RMS}=\sqrt{\sum T_m^2\Delta t/T}$. Kollmorgen's RMS method motivates the time weighting.

Continuous and peak torque curves are linearly interpolated between the three entered speeds. Peak utilization is $\max |T_m|/T_{pk}(n)$; continuous utilization is $\sqrt{\langle[T_m/T_{S1}(n)]^2\rangle}$. Both this speed-weighted check and RMS torque against the standstill rating must pass, together with the speed and temperature limits.

The entered motor curves are shaft ratings and are compared with shaft-equivalent demand $T_m$. Internal rotational losses are $P_{rot}=P_s(\omega_m/\omega_c)^2$, with signed drag torque $T_{drag}=P_s\omega_m/\omega_c^2$ and zero drag at standstill. Electromagnetic torque is $T_{em}=T_m+T_{drag}$, and signed winding current is $I=T_{em}/K_t$, following maxon's distinction between shaft torque and current including no-load drag (November 2014, printed pp. 43 and 49). Thus rotational heat is counted once, and its associated current also produces copper heat. The one-node RC energy balance is $C_{th}\dot{\theta}=I^2R_{25}[1+\alpha_{Cu}(T_a+\theta-25\,^{\circ}C)]+P_0+P_s(n/n_c)^2-\theta/R_{th}$. Exact exponential propagation with each time cell's frozen midpoint losses builds a cycle map $\theta_{end}=A\theta_0+B$. The indefinite-repetition initial rise is $B/(1-A)$, requiring $A<1$. The warm-up envelope uses the same cycle map: for cycle $k$, $\theta_k(t)=\theta_{per}(t)-\theta_{per}(0)\exp[k\ln A+a(t)]$. It takes the maximum and minimum over every solved phase edge in each displayed cycle, then plots those cycle extrema at the completed-cycle time. RC balance and spline equations are derived here; the simple FIRGELLI steady example independently checks the thermal limit.

• One rigid link and a point payload, with a fixed joint axis. Other joints are stationary; extra output inertia and a constant signed external demand may be included.
• A positive gear ratio uses the same joint and motor sign convention. Forward and backdriving efficiencies are constant. At zero velocity the forward factor is used conservatively, and Coulomb friction supplies no holding assistance.
• Ideal torque control and constant torque constant. Use a matched RMS-current convention: $P_{Cu}=I^2R_{eq}$; a balanced star winding with sinusoidal line RMS current has $R_{eq}=3R_{phase}=1.5R_{line-line}$.
• The entered shaft-rated curves apply to both rotation directions and both motoring and braking, at the actual DC-bus voltage, ambient temperature and mounting condition. They must already include manufacturer derating.
• Cooling is constant and represented by an effective winding-to-ambient RC fit. Copper resistance is referenced to 25 °C; 0.0039 K⁻¹ is maxon's published coefficient. The quadratic rotational-loss model supplies both internal drag and motor heat. Constant motor heat is an independent thermal source, excluding copper and rotational losses. All of these losses enter this effective thermal node.
• Defaults describe an illustrative small geared joint, not a named commercial motor. Replace the motor and thermal inputs with a coherent datasheet or measured set.

This checks indefinite repetition against an entered S1 continuous rating. In IEC 60034-1 terminology S1 means a constant load maintained to thermal equilibrium; a varying robot cycle is not automatically classified S1. S1 terminology is described in ABB's Motor Guide (July 2019). IEC 60034-1:2026 (15th edition, March 2026) supersedes the 2022 edition; check against the standard in force. This calculator does not implement a standard rating test.

Table entry supports 3–8 intervals, with the final time–angle point added automatically. A periodic cubic spline can overshoot sparse or noisy positions and can move through nominally equal-angle rows; inspect the trajectory and derivatives. Use a generated profile for exact dwells. Triangular speed has acceleration steps and infinite ideal jerk; use the smooth quintic for a finite-jerk command. A VLA policy's sampled positions do not uniquely determine accelerations: validate the reconstructed motion against the controller's actual interpolation and limits. Unwrap angles before entry.

The default solves 320 midpoint cells per interval (up to 2560 cells in table mode), with analytic kinematics and exponential thermal propagation. Increase the resolution to test narrow peaks and efficiency reversals; midpoint sampling can slightly underestimate peaks. The triangular profile uses six intervals before zero-duration dwells are removed. Warm-up displays cycle peak/minimum envelopes for up to 121 representative cycles, including the first and final displayed cycles. Every solved phase edge is scanned for each retained cycle. The frozen-coefficient temperature is monotone within each thermal cell, so this preserves its local extrema. From ambient, the temperature at every repeated phase increases with cycle count in this stable positive model; skipped cycles are bounded by the retained envelopes. Each envelope value is placed at that cycle's end time. Animation interpolates cached results and does not rerun the thermal calculation.

A speed-weighted continuous utilization is a sizing screen, not a substitute for the manufacturer's loss map or drive overload timer. A peak curve alone does not establish how long an overload is permitted. The one-node thermal fit cannot resolve separate winding/frame time constants, local hot spots, saturation, magnet weakening or speed-dependent cooling; fit it over the relevant pulse durations and verify marginal designs with measured temperatures or a validated multi-node model. No finite equilibrium is reported when the model's mean resistance feedback is unstable. Temperatures above 300 °C are outside its range and withhold the result cards. Internal thermal propagation stops once a rise exceeds 10⁶ K, before numerical overflow; this cannot affect a reported result within the validity range.

Multi-axis Coriolis and coupling torques, contacts and impacts, backlash, compliance, gearbox temperature and strength, bearing loads, brake release, current-loop voltage limits, drive I²t and regenerative bus absorption are not solved. Check those separately before choosing hardware.