Trajectory-based actuator identification via differentiable simulation

Vyacheslav Kovalev; Ekaterina Chaikovskaia; Egor Davydenko; Roman Gorbachev

cs.ROApr 11, 2026

Trajectory-based actuator identification via differentiable simulation

Authors: Vyacheslav Kovalev, Ekaterina Chaikovskaia, Egor Davydenko, Roman Gorbachev

Organizations: Moscow Institute of Physics and Technology (MIPT), Dolgoprudny, Moscow Region, Russia

Abstract

Accurate actuation models are critical for bridging the gap between simulation and real robot behavior, yet obtaining high-fidelity actuator dynamics typically requires dedicated test stands and torque sensing. We present a trajectory-based actuator identification method that uses differentiable simulation to fit system-level actuator models from encoder motion alone. Identification is posed as a trajectory-matching problem: given commanded joint positions and measured joint angles and velocities, we optimize actuator and simulator parameters by backpropagating through the simulator, without torque sensors, current/voltage measurements, or access to embedded motor-control internals. The framework supports multiple model classes, ranging from compact structured parameterizations to neural actuator mappings, within a unified optimization pipeline. On held-out real-robot trajectories for a high-gear-ratio actuator with an embedded PD controller, the proposed torque-sensor-free identification achieves much tighter trajectory alignment than a supervised stand-trained baseline dominated by steady-state data, reducing mean absolute position error from 14.20 mrad to as low as 7.54 mrad (1.88 times). Finally, we demonstrate downstream impact for the same actuator class in a real-robot locomotion study: training policies with the refined actuator model increases travel distance by 46% and reduces rotational deviation by 75% relative to the baseline.

Figures & tables

Approach	Instrumentation	Data Regime	Optimization	Interpretability	Model complexity
Ideal PD/torque abstraction	None	Simulation assumption	—	High	Low
Domain randomization	None	Sim randomized	—	—	—
Test-stand characterization	Torque sensor + test stand	Test stand (primarily quasi-static)	Supervised	Low-High	Low-High
Offline traj-ID	None	Real rollouts	Self-supervised; gradient-free	High	Low
Learned actuator modules / residual action	None	Real rollouts	Self-supervised; gradient-based	Low	High
Differentiable traj-ID (ours)	None	Real rollouts	Self-supervised; gradient-based	Low–High	Low–High

TABLE I: Comparison of actuator modeling and compensation approaches. Model complexity is a qualitative proxy for the number of trainable parameters and runtime evaluation cost.

Fig. 1: Illustration of the trajectory-matching optimization: The real system follows a trajectory

\{s_{i}\}

under control inputs

\{a_{i}\}

. Simulated states

\{s^{\prime}_{i}\}

are generated using a parameterized simulator. At each timestep, the state prediction error

e_{i}:=s^{\prime}_{i}-s_{i}

is computed, and the total trajectory error is minimized during optimization.

Stand-based supervised baseline
Model	Supervision / data	Model type
Bench-Sup	Supervised.	MLP
Trajectory-based gradient-free baseline
Param-ES	Self-supervised; gradient-free.	PD gains, armature (evolution strategy)
Trajectory-based residual-action (ASAP) baseline
Residual-RL	Self-supervised.	MLP

TABLE II: Comparison of actuator models and training protocol.

Fig. 2: Measured vs simulated motor response under identical control inputs. Motor velocity measured on hardware (black dashed) versus simulated trajectories for each model (colored lines).

Fig. 3: Position MAE comparison across actuator models. Box-and-whisker plot of the mean absolute position error (Eq. 3 ) on the held-out trajectory. Boxes indicate the interquartile range; the central line indicates the median; whiskers show the spread of the observed values.

Fig. 4: Parameter, loss, and gradient norm convergence of TrajID-Param over 25 independent runs. Each run lasts 3000 epochs. Shaded regions indicate mean $\pm$ std.

Fig. 5: Validation MAE versus objective weight $\alpha$ ( $w_{q}=\alpha$ , $w_{\dot{q}}=1-\alpha$ ) for TrajID-Param , TrajID-NN , and Torque-Oracle .

Fig. 6: Validation MAE versus prediction horizon for TrajID-Param , TrajID-NN , and Torque-Oracle .

Fig. 7: Robot Performance Comparison: Refined vs. Baseline RL Policies. The traveled distance is denoted as $d$ , and the rotation deviation from the original direction is represented as $\phi$ . The top image shows robot performance using the baseline model, exhibiting excessive rotation and reduced travel distance. The bottom image shows the robot’s performance using the refined model, demonstrating improvements in both travel distance and rotational alignment.

RL Policy Comparison
Metric	Default Model	TrajID Model
	(mean $\pm$ std)	(mean $\pm$ std)
Rotation deviation (deg) ↓	$109\pm 14.3$	$27\pm 6.4$
Travel distance (m) ↑	$1.12\pm 0.13$	$1.64\pm 0.08$

TABLE III: Comparison of our method vs. baseline

Appendix figures & tables4 assets

Supplementary material from the paper’s appendix.

Appendix

Fig. 8: Motor test-stand setup: Experimental apparatus used to characterize the actuator under controlled loading. The rig couples a test motor and a load (brake) motor through an inline torque transducer, enabling synchronized torque/velocity measurements.

Fig. 9: Measured vs. model-predicted torque. Steady-state motor torque $\tau$ (N m) versus motor velocity (rad/s) for two PWM duty-cycle commands ( $u=0.2$ and $u=0.8$ ). Gray markers show downsampled test-stand measurements; the orange curve shows the learned ( Bench-Sup ) model evaluated on a dense velocity grid.

Fig. 10: Table-top setup with actuator and rigid rod load used to collect identification trajectories.

Term	Formula	Scale
Tracking linear velocity	$\exp\!\big(-\tfrac{(v-v^{\mathrm{cmd}})^{2}}{\sigma^{2}}\big)$ for $v_{x},v_{y}$	1.5
Tracking angular velocity (yaw)	$\exp\!\big(-4\,(\omega_{z}-\omega^{\mathrm{cmd}}_{z})^{2}\big)$	1.5
Base upright (roll/pitch)	$\exp\!\big(-30\,\lVert[\mathrm{roll},\mathrm{pitch}]\rVert_{2}\big)$	0.9
Base height	$\exp\!\big(-500\,(z/z_{\mathrm{target}}-1)^{2}\big)$	0.8
Base acceleration penalty	$\exp\!\big(-\sum_{j}\|a^{\mathrm{root}}_{j}\|\big)$	0.1
Action smoothness	$\lVert a_{t}-a_{t-1}\rVert^{2}+\lVert a_{t}+a_{t-2}-2a_{t-1}\rVert^{2}$	-0.02

Appendix

TABLE IV: Reward terms used in the RL experiments.

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Open paper AskPDF arXiv

Differentiable simulators have advanced policy learning and model-based control, yet actuator dynamics remain an important source of sim-to-real error. This is particularly acute on low-cost platforms, where the linear current-to-torque relation

τ= K_tI

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Abstract

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