Modeling Load-, Velocity-, and Temperature-Dependent Transmission Errors of Cycloidal Drives for Industrial Robots Using Fourier Series

I. INTRODUCTION

• Industrial-robot path accuracy is 100 to 1000 times lower than that of standard machine tools, limiting machining applications across high- and low-force processes.Source: §I · p. 1
• Drive trains are the main contributor to path errors; gearbox compliance and transmission errors, plus motor torque ripple, affect angular position error and accumulate along serial kinematics.The errors are amplified as axis length increases.Source: §I · p. 1
• Load-side encoders increase cost and complexity; heavy-payload industrial robots commonly use cycloidal drives, where prior compliance compensation reduced tracking error by 81% but did not model transmission errors.The paper focuses on state-dependent cycloidal-drive transmission errors; closed-loop evaluation and combining compensation strategies are planned for future work.Source: §I · p. 1

A. State of the Art

• Transmission errors are mainly periodic, angle-dependent deviations between motor- and load-side angles caused by manufacturing and assembly tolerances.Prior work also reports load-torque and velocity dependence in strain wave gears and lubricant-temperature dependence of gear errors due to viscous friction.Source: A. State of the Art · pp. 1–2
• Transmission-error compensation has used model-based methods –, acceleration-sensor peak filters, adaptive notch filters, and load-side sensors –, mostly for strain wave gears –.Source: A. State of the Art · p. 2
• For cycloidal drives, prior work models angle-dependent transmission errors with Fourier series, disturbance observers –, analytical geometry models,, and a combined transmission-error and lost-motion model.Source: A. State of the Art · p. 2
• Prior state-dependent approaches include rack-and-pinion load and velocity effects, regression-tree compensation, neural-network and regression-tree combinations, and cycloidal-drive Fourier amplitude and phase models for velocity, temperature, and load torque.These compensation approaches use load-side angle as an input, which cannot be measured in a standard robot.Source: A. State of the Art · p. 2

• The proposed cycloidal-drive transmission-error model uses a Fourier series with amplitude and phase polynomially dependent on load torque, velocity, and temperature.Source: B. Contribution · p. 2
• The approach uses motor-encoder input-side angle and velocity but currently still relies on sensors for load-side torque and lubricant temperatures.Source: B. Contribution · p. 2
• The work analyzes the relationship between cycloidal-drive mechanical properties and transmission-error harmonic frequencies.Source: B. Contribution · p. 2
• In the planned concept, compensation parameters would be identified end-of-line using load-side encoder, torque, and lubricant-temperature sensors, while operation would require only the robot’s input-side encoders.An observer-based implementation is planned.Source: B. Contribution · p. 2

II. EXPERIMENTAL SETUP AND METHODS

• Experiments used a setup for analyzing transmission-error effects on an isolated robot joint, followed by development and parameter determination of a polynomial Fourier-series formulation.The setup is shown in Fig. 2.Source: §II · p. 2

A. Isolated Robot Joint for Experiments

• The Nabtesco RH380-N drive has a planetary stage with a 24-tooth input gear driving three 96-tooth gears, followed by a cycloidal stage with 45 disc lobes and 46 pins.The drive's rated output torque is 3724 N m.Source: A. Isolated Robot Joint for Experiments · p. 2
• Input position was measured by a 13-bit encoder, while a PT100 sensor measured lubricant temperature and a 16-bit-digitized torque sensor measured gearbox output torque.The torque sensor has a 10 N m rated torque.Source: A. Isolated Robot Joint for Experiments · p. 2
• A load-side motor connected through a 15 kN m/rad coupling provided loading, and its 25-bit encoder measured output position; control ran on Speedgoat over EtherCAT at 2 kHz.The load-side motor is rated at 1650 N m and can exert up to 3400 N m.Source: A. Isolated Robot Joint for Experiments · p. 2, Figure 2 · p. 2

B. Modeling of the Transmission Errors

• Transmission errors are assumed periodic and modeled by an amplitude–phase Fourier series with harmonic amplitudes, phases, and a constant offset.The series uses spatial frequencies defined per load-side revolution.Source: B. Modeling of the Transmission Errors · p. 3
• Harmonic frequencies fk are dimensionless fractions of a full load-side revolution, specifying how often each transmission-error component occurs per revolution.The load-side period is θperiod = 2π.Source: B. Modeling of the Transmission Errors · p. 3
• Amplitude, phase, and offset depend on load-side torque, input-side velocity, and lubricant temperature through equivalent second-order polynomials.The amplitude polynomial has 10 parameters, includes two-factor interactions, and is chosen to balance nonlinear coverage against parameter count.Source: B. Modeling of the Transmission Errors · p. 3

C. Parameter Identification

• Measurements used constant low input velocity and applied load, with lubricant temperature raised between runs by high-speed gearbox heating to mimic internal heating during operation.Separate polynomial parameter sets were estimated for forward and backward motion because transmission errors differed by direction.Source: C. Parameter Identification · p. 3
• Harmonic amplitudes came from spatial-frequency spectra after interpolation to equidistant θ samples, detrending, Hamming windowing, and zero-padding to improve FFT frequency resolution.The FFT spectrum was window-normalized and converted to a single-sided amplitude spectrum.Source: C. Parameter Identification · p. 3
• For each measurement, the phase was derived from the Fourier series’ equivalent sine-cosine coefficients using a two-argument arctangent, while the constant offset was the mean transmission error.The equivalent form uses ak · cos (fk · θ) + bk · sin (fk · θ).Source: C. Parameter Identification · p. 3
• Standardized load torque, velocity, and temperature were used to fit each harmonic’s polynomial parameters by linear least squares, then coefficients were rescaled to raw inputs.Only harmonics with high amplitudes linked to drive mechanical properties were retained, avoiding unreliable components that could cause overcompensation or phase error.Source: C. Parameter Identification · p. 3

• The section examines measured transmission errors in relation to cycloidal-drive mechanical properties and compares model output with independent validation data.Source: §III · p. 4

A. Transmission Errors

• Load strongly shifted the constant offset through gear compliance; velocity also changed amplitudes, phases, and offset less distinctly, while temperature mainly affected offset and phase rather than amplitude.The temperature-related amplitude changes were described as minor, whereas changes in constant offset and phase were more relevant.Source: A. Transmission Errors · p. 4
• Selected harmonics implicated output-shaft periodic/angular errors (f = i for i ∈ {1, 3, 6, 9, 13, 14, 18, 19, 27}), two planetary-stage tolerances, six ambiguous cycloidal-stage tolerances, and one input-gear tolerance.Cycloidal-stage frequencies may also relate to deformations and tolerances in the cycloidal disc.Source: A. Transmission Errors · p. 4

B. Model Validation

• Nineteen validation measurements sampled load torque, velocity, and temperature with Sobol’s sequence within identification bounds, ensuring independence; extrapolation suitability was not evaluated.Source: B. Model Validation · p. 4
• In the example validation, the reference Fourier series captured general error behavior but omitted some peaks; its RMSE was 1.83 × 10−5 rad and MAE was 1.49 × 10−5 rad.Source: B. Model Validation · p. 5, Figure 6 · p. 6
• Positive-direction mean RMSE was 2.40×10−5 rad and mean MAE was 1.92 × 10−5 rad; negative-direction mean RMSE was 2.82 × 10−5 rad and mean MAE was 2.26 × 10−5 rad.Source: B. Model Validation · p. 5
• A quadratic dependence of harmonic amplitudes and phases prevents the polynomial model from perfectly matching values derived from a single measurement.Source: B. Model Validation · p. 5

IV. DISCUSSION

• The mechanically relevant harmonic frequencies of cycloidal drives can inform a Fourier-series model that accounts for load torque, velocity, and temperature.The discussion describes Fourier series as valid for mainly periodic transmission errors and notes that velocity effects may reflect changes in friction torque, making them a form of load-dependent effect.Source: §IV · p. 6
• Non-periodic errors, such as those caused by damaged pins or teeth, prevent even a Fourier series with all frequencies from achieving a perfect fit.Adding higher harmonics could improve fit, but the discussion says they contribute little to the overall result.Source: §IV · p. 6
• Future work could test higher-order polynomial coupling through sensitivity analysis or explore machine learning to capture non-periodic effects, while limiting complexity to avoid overfitting and support real-time computation.The discussion cites a machine-learning approach for rack-and-pinion drives as an alternative that could provide a better transmission-error fit.Source: §IV · p. 6
• After identification, the model requires only the motor encoder’s input-side angle, but its operation still relies on load-side torque and lubricant-temperature sensors.It does not depend on the load-side angle after identification.Source: §IV · p. 6

V. OUTLOOK

• The model output is added to the desired load-side angle, and the correction propagates through the cascaded controller to produce a corrected output angle.The scheme derives velocity from the input-side motor angle and measures load torque and temperature directly.Source: §V · p. 6, Figure 7 · p. 6
• In quasi-static operation, motor torque inferred from current can substitute for measured load torque because τload ≈ τm.The authors also propose replacing load-side torque and lubricant-temperature measurements with observer structures; a load-side torque observer was presented in.Source: §V · p. 6
• The authors plan closed-loop evaluation and intend to combine the transmission-error model with compliance compensation, whose influence is larger.They characterize the Fourier-series model as a partial solution to improving industrial-robot path accuracy.Source: §V · p. 6