Neural-Learning Trajectory Tracking Control of Flexible-Joint Robot Manipulators with Unknown Dynamics

John T. Wen; Shuyang Chen

REVIEW 3 major objections 5 minor 1 cited by

Neural-Learning Trajectory Tracking Control of Flexible-Joint Robot Manipulators with Unknown Dynamics

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Learned neural feedforward cuts flexible-joint robot tracking error by 30–50% without a dynamics model.

desk verdict A useful empirical comparison of RNN-based feedforward schemes on a Baxter, but the main claim that learned dynamics drive the improvement is undercut by an untested preview baseline. read the letter →

arxiv 1908.03269 v1 pith:3TODG5VY submitted 2019-08-08 cs.RO

Shuyang Chen , John T. Wen This is my paper · ORCID

classification cs.RO

keywords trajectorytrackingflexible-jointrobotmanipulatorrecurrentneuralnetworkbidirectionalRNNiterativelearningcontrolfeedforwardcompensationinversedynamicsBaxter

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper claims that neural networks trained only on observed command–response data can compensate for the unknown, coupled, flexible-joint dynamics of a robot arm. It builds two feedforward controllers for a Baxter arm — one offline iterative-learning scheme using a forward-dynamics RNN, and one direct inverse-dynamics filter using a bidirectional RNN — and reports that both cut tracking error by roughly 30–50% relative to a proportional feedback baseline on unseen trajectories. The point is to show that precise trajectory tracking can be recovered without an analytic robot model or system identification, and that the bidirectional inverse method works in real time for teleoperation.

What carries the argument

The carrying object is a pair of recurrent networks sharing one dataset: a 4-layer unidirectional GRU network that maps a 50-step window of commanded joint positions to the joint configuration 50 steps later, and a 2-layer bidirectional GRU network (BRNN) that maps a 50-step window of measured joint positions centered at time $t$ back to the command at time $t$. The BRNN is what makes the inverse-dynamics approach work: because the stable inverse of a strictly proper or non-minimum-phase system is non-causal, each feedforward command must look both backward and forward along the desired trajectory, and the centered window supplies that context. The forward RNN feeds a gradient-based ILC update $u^{k+1} = u^k - \alpha_k G^*(s) e_q$, where $G^*(s)$ is the adjoint of the learned forward model; the BRNN simply filters the desired trajectory to produce the feedforward command. Both outputs are added to a proportional feedback term to form the commanded joint setpoint.

What would settle it

Record a set of held-out command-response pairs, feed each measured response into the BRNN to get a predicted command, replay that command on the robot, and measure the resulting error; if the replayed output deviates as much as or more than the baseline proportional controller, the learned inverse is not doing the work claimed. A lighter check is to run a test trajectory whose manipulability falls outside the light-blue training region in Fig. 4 and see whether the 30–50% improvement disappears.

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Extended reading notes

Core claim

On its own terms, the paper's claim is that the inner-loop command-to-response dynamics of a flexible-joint manipulator can be learned directly from recorded command and response trajectories, and the learned map can be used for feedforward compensation in two ways. First, a unidirectional RNN approximates the forward dynamics and supplies the model for an offline gradient-based iterative learning control that refines the command before execution. Second, a bidirectional RNN trained on the same data with output and command swapped approximates the (generally non-causal) inverse dynamics, so the desired trajectory can be filtered directly to produce feedforward commands in real time. Experiments on the Baxter left arm show average reductions in $\ell^2$ tracking error of more than 50% for sinusoidal and Cartesian trajectories, over 40% for random joint trajectories, and over 30% for teleoperated trajectories, compared with a proportional feedback baseline; the BRNN method is slightly more accurate than RNN-ILC in most of the reported cases.

Load-bearing premise

The whole scheme rests on the assumption that the command-to-response map of the robot is invertible over the tested region and that a 50-step bidirectional window plus the training trajectories capture that inverse well enough; the paper never separately verifies the learned inverse's prediction accuracy.

Editorial extensions

If this is right

A robot's unknown joint-flexibility dynamics can be compensated for from measurement data alone, without torque-level models or system identification.
A single offline-trained BRNN can produce feedforward commands in real time for trajectories that are not known in advance, such as teleoperated motion.
The learned forward RNN makes iterative learning control practical offline, avoiding slow iterations on the physical robot.
The reported reductions in $\ell^2$ tracking error generalize across unseen sinusoidal, random, and Cartesian trajectories, with further improvement available from additional physical ILC iterations.
Between the two approaches, the inverse-dynamics BRNN is preferred for online use since it needs only a forward pass, while RNN-ILC requires the entire trajectory ahead of time.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

A testable extension is to measure the BRNN's inverse-model prediction error on held-out command–response pairs; if the inverse mapping is not accurate, the feedforward can amplify error instead of reducing it, which the paper does not assess.
The same bidirectional-inversion recipe should transfer to other actuation chains with delay or flexibility—hydraulic actuators, cable-driven arms, or soft robots—where the stable inverse is likewise non-causal.
The reported 0.24 s look-ahead delay at 100 Hz suggests a tunable trade-off: a shorter BRNN window would reduce teleoperation latency but may lose the non-causal information the inverse needs.
Combining the learned feedforward with a more sophisticated feedback law (the paper uses pure proportional gain) is the obvious next step; stability margins of the closed loop with the learned inverse are not analyzed here.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

You'll want to know about this paper: it reports a head-to-head comparison of two RNN-based feedforward compensation schemes for a Baxter arm with flexible joints, evaluated on a physical robot. The experiments are real and the reported tracking improvements (30–50% reductions in l2 error vs a proportional-only baseline) are substantial. That is the good news.

The genuinely new bits are the systematic comparison of an offline RNN+ILC scheme with a bidirectional-RNN inverse-dynamics scheme on the same hardware, and the teleoperation demo where the BRNN runs in real time. The authors are also honest about the coupling of the Baxter joints—they train one coupled network across all seven joints rather than per-joint models—and they provide their middleware bridge code. The data-collection effort (500 trajectories of command-response data) is non-trivial.

Now the soft spots. First, the paper reports no repeated trials, no error bars, no statistical significance. For a claimed 30–50% improvement, we need to know how much of that is run-to-run variability. Second, the ILC update in Eq. (4) is written for a linear adjoint G*(s), but the learned forward model is a nonlinear RNN. The authors never say how the adjoint of the RNN is computed or whether the gradient is accurate. That makes the first method under-specified and hard to reproduce. Third, and most important, the BRNN inverse is never validated on its own, and no trivial baseline is tested. Both RNN inputs contain a substantial look-ahead window: a 50-step desired-trajectory window for the forward RNN, and a ±25-step window for the BRNN. The baseline controller only uses qd(t+1). So part of the improvement likely comes from having future reference information, not from learned nonlinear dynamics. In the teleoperation experiment, the authors deliberately insert a 0.24 s delay to gather future inputs, and they do not compare against a simple time-advanced feedforward. That is a real confound. The central claim as written—that neural learning of unknown dynamics drives the improvement—is not fully supported without a preview-control baseline.

This is a well-written empirical study, and I'd send it to review—a serious referee could push for the missing controls and statistics. It would be a solid contribution if the authors add multi-trial data, a lead/lag feedforward baseline, separate validation of the learned inverse, and a clear description of the ILC gradient computation. As it stands, it is valuable mainly to robotics practitioners who want to see what RNN feedforward looks like on real hardware, but it is not yet a convincing proof that the learned dynamics are the cause.

For yourself: worth a quick look for the experimental setup, but I would not cite the main quantitative claim until the preview confound is addressed.

Referee Report

3 major / 5 minor

Summary. This paper addresses trajectory tracking of a flexible-joint Baxter robot whose inner torque servo has unknown dynamics. The authors propose two feedforward compensation schemes: an offline iterative learning control (ILC) that uses a unidirectional RNN trained to predict the forward dynamics, and a direct inversion scheme in which a bidirectional RNN (BRNN) trained on (actual response, commanded setpoint) pairs is used to filter the desired trajectory into a feedforward command. The feedforward is added to a proportional feedback baseline. Experiments on sinusoidal, random joint, Cartesian square, and teleoperated trajectories report a 30-50% reduction in l2 tracking error relative to the baseline. The paper argues that the learned models capture the coupled, flexible-joint dynamics and that the BRNN's bidirectional window captures the noncausality of the inverse system.

Significance. If the improvement is causally attributable to the learned internal dynamics, the paper would provide a practical, data-driven way to improve tracking of collaborative robots that expose only a joint-setpoint interface, without requiring torque-level control or an accurate simulator. The study has genuine strengths: experiments on a real Baxter with series-elastic actuators, joint coupling explicitly demonstrated, a comparison of two learning-based architectures for the same data set, and a large training corpus (500 trajectories, 1.2M samples). It also makes a conceptually interesting use of a BRNN as a noncausal inverse filter. However, the causal attribution is currently under-supported: the BRNN is not validated as an inverse model, the teleoperation comparison does not control for a 0.24 s preview advantage, and the ILC update for the nonlinear RNN is stated only for a linear adjoint. These issues must be resolved before the central claim can be accepted.

major comments (3)

[IV.B.3, Eq. (4)] The ILC update in Eq. (4) is written for the adjoint G*(s) of a linear time-invariant system, but the forward model used to generate the gradient is the trained nonlinear RNN. The paper does not derive the corresponding adjoint for the RNN, nor does it explain how G*(s)eq is computed (e.g., by backpropagation through time or by linearization). Because the first approach's feedforward command is the output of this ILC, the method is under-specified and not reproducible as it stands. Please provide the gradient computation for the RNN or justify that the linear ILC update applies.
[IV.C.1 and V (Tables I-III)] The BRNN is trained on pairs (q, qd) in which the input is the measured response and the output is the commanded setpoint, but at inference it is applied to desired trajectories qd to produce qf. This presumes the command-to-response map is invertible and that the learned network generalizes from response windows to desired windows. No held-out prediction error for the BRNN is reported, and no ablation is given against a simpler noncausal feedforward (e.g., qf(t)=qd(t+delta)). Since the BRNN sees a +-25-step window (Eq. (5)) whereas the baseline controller (Eq. (2)) uses only qd(t+1), the reported 30-50% improvements may be largely due to preview rather than to a learned inverse. Please validate the inverse model and compare with a matched-preview baseline.
[V.D] In the teleoperation experiment, the BRNN requires 24 future user inputs, which the authors obtain by delaying the commanded trajectory by 0.24 s. Table IV compares this delayed, preview-enhanced controller against a baseline feedback controller that does not have access to future setpoints and is not delayed in the same way. The 30% improvement therefore cannot be unambiguously attributed to the learned inverse dynamics. The authors should compare against a baseline that has the same 0.24 s preview (e.g., a time-shifted desired trajectory) or otherwise control for latency.

minor comments (5)

[V.C] The claim that additional ILC iterations on the physical robot further improve performance is not supported by any reported results; please report these experiments or remove the claim.
[IV.C.2] There is a notational slip: the sentence 'for each input qd and output trajectory q' should read 'for each input q and output trajectory qd', since the BRNN maps response to command. Also, the sample counts 1225000 and 1225500 differ from the formula (2500-T or 2501-T times 500); please check the arithmetic.
[IV.B.2 and IV.C.2] The choice T=50 is justified only qualitatively; since T determines the preview length and the network memory, a sensitivity study (or at least a statement of the tradeoff) would help the reader understand the role of this hyperparameter.
[Fig. 4] The four testing trajectories are hard to distinguish from the training-trajectory background; using distinct colors or markers would improve clarity.
[V (Tables I-IV)] All tables report single-run errors without error bars or statistical tests; over repeated trials this would make the improvement claims more robust.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the learned feedforward is evaluated against an external baseline on separate test trajectories.

full rationale

Walked the derivation chain. The forward RNN plus ILC and BRNN inverse-dynamics feedforwards are trained on offline collected commanded/response pairs, while the four test trajectories are explicitly unseen and the manipulability plot (Fig. 4) distinguishes training from testing data. The baseline is an external proportional-feedback controller (Eq. 2), so the improvements in Tables I-IV are empirical and not forced by construction: no fitted parameter is renamed as a prediction, and no test trajectory is used as a training label. The self-citations to [22], [26], and [27] are methodological references for gradient-based ILC and resolved-velocity teleoperation; the ILC update is stated in Eq. (4) rather than being justified solely by citation, and the teleoperation controller is an input-generation tool rather than the source of the tracking-error reduction. The lack of a separate validation of the learned inverse and the absence of a simple lead/preview feedforward baseline are experimental-design or correctness concerns, not circularity. No step in the paper reduces by definition or by fitted input to its own output.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The central claims rest on the assumption that the Baxter's closed-loop command-to-response dynamics are learnable by finite-memory RNNs, that the learned forward model supports gradient-based ILC convergence, that the BRNN inverse is well-posed and generalizes, and on unspecified controller and model hyperparameters. No new physical entities are introduced; all modeling burden is carried by the RNN approximation, the choice of T=50, and the feedback gain k. The paper does not provide formal convergence or stability guarantees for these assumptions.

free parameters (3)

Feedback gain k = not reported
The control laws (1) and (2) use a proportional gain k; its numerical value is never given, and it determines both the baseline and the compensated closed-loop behavior, so the reported improvements are tied to an unreported tuning choice.
RNN memory window T = 50 time steps (0.5 s at 100 Hz)
T=50 is chosen by hand in Section IV.B.2 and IV.C.2 as a trade-off between dynamics information and delay; for teleoperation it directly causes the reported 0.24 s delay because 25 future samples are needed.
Network architecture and training hyperparameters = 4-layer unidirectional GRU RNN; 2-layer bidirectional GRU BRNN; dropout 0.5; Adam lr 1e-3; batch 256; 10000 iterations
These are selected empirically (GRU over LSTM after comparison) rather than derived; hidden-layer sizes and exact training and validation details are missing, which matters because overfitting to the 500 training trajectories could affect generalization.

assumptions (3)

domain assumption The Baxter closed-loop dynamics are deterministic, time-invariant, and representable by a 50-step recurrent model.
Section IV.B.2 and IV.C.2 choose T=50 and train a single RNN/BRNN over all joints, implicitly assuming the dynamics are stationary and have finite memory within this window.
ad hoc to paper The gradient-based ILC update (4), derived for a linear system, converges when applied through the nonlinear trained RNN.
Section IV.B.3 states that the MIMO gradient-based ILC is easily implemented with the trained RNN and cites [22], [26] for the linear adjoint G*(s), but provides no convergence proof or explanation of how the adjoint is obtained for the nonlinear model.
ad hoc to paper The command-response pairs (qd, q) define an invertible map, so the BRNN trained on (q -> qd) can be used as an inverse filter (qd -> qf).
Section IV.C.1 asserts this relation directly; no uniqueness, existence, or stability analysis is given for a flexible-joint system with time delays, and the learned inverse is never validated in isolation.

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Cite this review

Pith. "Pith review of Neural-Learning Trajectory Tracking Control of Flexible-Joint Robot Manipulators with Unknown Dynamics." pith.science (2026). https://pith.science/paper/3TODG5VY

@misc{pith2026190803269,
  author       = {Pith},
  title        = {Pith review of: Neural-Learning Trajectory Tracking Control of Flexible-Joint Robot Manipulators with Unknown Dynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/3TODG5VY}},
  note         = {Machine review of arXiv:1908.03269}
}

read the original abstract

Fast and precise motion control is important for industrial robots in manufacturing applications. However, some collaborative robots sacrifice precision for safety, particular for high motion speed. The performance degradation is caused by the inability of the joint servo controller to address the uncertain nonlinear dynamics of the robot arm, e.g., due to joint flexibility. We consider two approaches to improve the trajectory tracking performance through feedforward compensation. The first approach uses iterative learning control, with the gradient-based iterative update generated from the robot forward dynamics model. The second approach uses dynamic inversion to directly compensate for the robot forward dynamics. If the forward dynamics is strictly proper or is non-minimum-phase (e.g., due to time delays), its stable inverse would be non-causal. Both approaches require robot dynamical models. This paper presents results of using recurrent neural networks (RNNs) to approximate these dynamical models-forward dynamics in the first case, inverse dynamics (possibly non-causal) in the second case. We use the bi-directional RNN to capture the noncausality. The RNNs are trained based on a collection of commanded trajectories and the actual robot responses. We use a Baxter robot to evaluate the two approaches. The Baxter robot exhibits significant joint flexibility due to the series-elastic joint actuators. Both approaches achieve sizable improvement over the uncompensated robot motion, for both random joint trajectories and Cartesian motion. The inverse dynamics method is particularly attractive as it may be used to more accurately track a user input as in teleoperation.

Figures

Figures reproduced from arXiv: 1908.03269 by the authors.

**Figure 3.** Overall control architecture. either by implementing ILC with the trained unidirectional RNN that approximates the manipulator forward dynamics, or directly filtering by the dynamical inversion approximated by the BRNN. qc(t + 1) is the resulting commanded joint input for robot manipulator at time t + 1. On the other hand, the stable baseline feedback controller is: qc(t + 1) = qd(t + 1) − k · (q(t) − qd(t)) (2) B. … view at source ↗

**Figure 4.** Manipulability measure over 500 training joint tra [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗

Figures from the paper (4 more)

**Figure 5.** Figure 5: Unrolled RNNs structures. hi represents the internal states of each time step. We choose T = 50 to guarantee that the input of RNN contains enough dynamics information for the RNN to model. In total we have 1225000 samples, and we use 80 % of the samples for training a…

**Figure 6.** Figure 6: Comparison of tracking performance without and with the RNN/BRNN feedforward controllers of sinusoidal joint [PITH_FULL_IMAGE:figures/full_fig_p006_6.png]

**Figure 7.** Figure 7: Comparison of tracking performance without and [PITH_FULL_IMAGE:figures/full_fig_p007_7.png]

**Figure 8.** Figure 8: Comparison of tracking performance without and with the BRNN feedforward controller of a user teleoperated [PITH_FULL_IMAGE:figures/full_fig_p008_8.png]

Discussion (0). Continue with ORCID to comment.

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Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Qu and D

Z. Qu and D. M. Dawson, Robust tracking control of robot manipu- lators. IEEE press, 1995

work page 1995

[2] [2]

On the adaptive control of robot manip- ulators,

J.-J. E. Slotine and W. Li, “On the adaptive control of robot manip- ulators,” The international journal of robotics research , vol. 6, no. 3, pp. 49–59, 1987

work page 1987

[3] [3]

Learning control theory for robotic motion,

S. Arimoto, “Learning control theory for robotic motion,” Interna- tional Journal of Adaptive Control and Signal Processing , vol. 4, no. 6, pp. 543–564, 1990

work page 1990

[4] [4]

Feedback- error-learning neural network for trajectory control of a robotic ma- nipulator,

H. Miyamoto, M. Kawato, T. Setoyama, and R. Suzuki, “Feedback- error-learning neural network for trajectory control of a robotic ma- nipulator,” Neural Networks , vol. 1, no. 3, pp. 251 – 265, 1988

work page 1988

[5] [5]

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B. Brogliato, R. Ortega, and R. Lozano, “Global tracking controllers for ﬂexible-joint manipulators: a comparative study,” Automatica, vol. 31, no. 7, pp. 941–956, 1995

work page 1995

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D. Wang, “A simple iterative learning controller for manipulators with ﬂexible joints,” Automatica, vol. 31, no. 9, pp. 1341–1344, 1995

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O. Ogunmolu, X. Gu, S. Jiang, and N. Gans, “Nonlinear Systems Identiﬁcation Using Deep Dynamic Neural Networks,” arXiv e-prints, p. arXiv:1610.01439, Oct 2016

arXiv 2016