A Three-Stage Offline SDRE-Based Control Framework for Human Motion Reproduction on a Suspended Bipedal Robot

Chien-Wu Lan; Ching-Kai Lin; Chin-Tien Wu; Ping-Kong Huang

arxiv: 2506.04680 · v2 · submitted 2025-06-05 · 💻 cs.RO · math.OC

A Three-Stage Offline SDRE-Based Control Framework for Human Motion Reproduction on a Suspended Bipedal Robot

Ping-Kong Huang , Chien-Wu Lan , Chin-Tien Wu , Ching-Kai Lin This is my paper

Pith reviewed 2026-05-19 11:21 UTC · model grok-4.3

classification 💻 cs.RO math.OC

keywords bipedal robothuman motion reproductionSDREexoskeleton evaluationoffline controlmotion capture dataPID-LQR controllertracking error minimization

0 comments

The pith

A three-stage offline control framework enables accurate human motion reproduction on a suspended bipedal robot using SDRE, optimization, and PID-LQR.

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

The paper develops a method to let robots safely test exoskeletons by copying human movements instead of using people. It proposes a three-stage process that starts with calculating optimal torques from the robot's model, then creates command sequences that respect actuator limits, and finally adjusts those commands to match observed motion closely. This is tested on squatting and walking data from motion capture. The approach matters because it reduces safety risks in exoskeleton development by providing repeatable robot-based evaluations. If successful, it shows that offline command generation can bridge the gap between human data and robot hardware.

Core claim

The paper claims that combining an SDRE controller for torque trajectories, parameterized optimization for velocity and acceleration commands under constraints, and a data-driven PID-LQR controller for error minimization allows the suspended bipedal robot to reproduce human motions with average RMSE below 3 degrees.

What carries the argument

The three-stage offline SDRE-based control strategy that uses motion-capture data and robot dynamics to synthesize and refine control commands.

If this is right

The method supports systematic testing of gravity-counteracting exoskeletons.
High tracking fidelity is achieved despite structural and actuator differences.
Offline command synthesis can overcome mismatches with human motion-capture data.
Experimental validation on squatting and walking tasks confirms effectiveness.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

This framework could be adapted for other types of robots used in human motion emulation.
Future work might explore online versions of the control strategy for dynamic adjustments.
Such testing platforms may accelerate the safe development of wearable devices by allowing more extensive trials.

Load-bearing premise

The robot's dynamic model and actuator constraints are representative enough that offline commands can compensate for any differences from human motion data.

What would settle it

Running the three-stage controller on the robot with new human motion data and measuring if the average RMSE stays below 3 degrees or rises substantially would test the claim.

Figures

Figures reproduced from arXiv: 2506.04680 by Chien-Wu Lan, Ching-Kai Lin, Chin-Tien Wu, Ping-Kong Huang.

**Figure 1.** Figure 1: Dynamic Model of the Lower Limb TABLE I: DESCRIPTION OF THE PARAMETERS IN THE DYNAMIC MODEL OF THE LOWER LIMB Symbol Description P1 Hip joint P2 Knee joint Pc1 Center of mass of thigh Pc2 Center of mass of lower leg l1 Thigh length l2 Lower leg length m1 Mass of hip servo motor m2 Mass of knee servo motor mc1 Mass of thigh mc2 Mass of lower leg θ1 Hip joint angle θ2 Knee joint angle In the model, we define… view at source ↗

**Figure 2.** Figure 2: Piecewise linear representation is applied to simplify [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 4.** Figure 4: Torque of the hip/knee joint with SDRE method [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

**Figure 5.** Figure 5: Torque error of the right hip/knee joint with proposed [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

**Figure 3.** Figure 3: Torque of the right hip/knee joint with proposed [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗

read the original abstract

During the development of wearable exoskeletons, evaluations involving human subjects pose inherent safety risks. Therefore, systematic testing is often conducted using robots that emulate human motion. However, reproducing human movements is challenging due to differences in robot structure and actuator characteristics. This study proposes a three-stage offline control strategy that uses motion-capture data and robot-specific properties to generate control commands for accurate motion replication. First, an optimal torque trajectory is generated via a State-Dependent Riccati Equation (SDRE) controller based on the dynamic model of the bipedal system. Second, joint velocity and acceleration command sequences are synthesized through parameterized optimization under actuator constraints. Finally, a data-driven PID-LQR offline controller refines these commands by minimizing the tracking error between the desired and executed motions. Experimental validation is performed on a suspended bipedal robot platform designed for the evaluation of gravity-counteracting exoskeletons. Motion-capture data collected from squatting and walking tasks are used for system assessment. The experimental results demonstrate high tracking fidelity, with an average root mean square error (RMSE) below 3 degrees. These results verify the effectiveness of the proposed three-stage control strategy for robot-based systematic testing of exoskeletons.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper delivers a practical three-stage offline pipeline for motion reproduction on this suspended bipedal platform, but the experimental support for the central RMSE claim is thinner than it needs to be.

read the letter

The main thing to know is that the authors put together an offline three-stage controller for a suspended bipedal robot meant to stand in for human subjects during exoskeleton tests. Stage one uses SDRE on the robot's dynamic model to get torque trajectories, stage two runs constrained optimization to produce feasible velocity and acceleration commands, and stage three tunes a PID-LQR loop on actual tracking error to clean up the result. They report average RMSE below 3 degrees on squatting and walking motions captured from humans. That combination is new for this specific platform and task set, and it directly targets the safety problem of testing exoskeletons without putting people at risk right away. The approach is straightforward to implement once the model and constraints are in place, and the offline nature makes it repeatable for systematic evaluation work. The soft spot is the experimental section. The abstract states the RMSE number but gives no error bars, no comparison against simpler baselines like pure inverse dynamics or basic PID, and no separate validation of how well the SDRE model captures the suspension cables, variable tension, or pendulum effects. Without those checks or an ablation that removes the first stage, it is hard to tell whether the low error comes from the full pipeline or mostly from the final data-driven correction. The actuator constraints in stage two are mentioned but not quantified in detail. This is a solid application paper for people working on rehabilitation robotics and human-robot interaction testbeds. Readers who need a concrete method for safe, repeatable motion replication on similar suspended platforms will find it useful. It is not reshaping control theory, but the implementation is honest and the platform is real. I would send it to peer review with a request for more model-validation residuals and baseline comparisons; the core idea is worth referee time.

Referee Report

3 major / 2 minor

Summary. The paper presents a three-stage offline control framework for reproducing human motion-capture trajectories on a suspended bipedal robot intended for exoskeleton testing. Stage 1 generates optimal torque trajectories via a State-Dependent Riccati Equation (SDRE) controller derived from the robot's dynamic model. Stage 2 synthesizes joint velocity and acceleration commands through constrained optimization respecting actuator limits. Stage 3 applies a data-driven PID-LQR controller tuned on experimental tracking error to refine the commands. Validation on squatting and walking tasks reports average RMSE below 3 degrees, concluding that the framework enables accurate, safe robot-based evaluation of gravity-counteracting exoskeletons.

Significance. If the central experimental claim holds under fuller validation, the work offers a practical offline pipeline that combines model-based optimal control with data-driven refinement to bridge kinematic and dynamic mismatches between humans and robots. This could reduce reliance on human-subject trials during early exoskeleton development and provide a reproducible testbed for systematic performance assessment.

major comments (3)

[Abstract / Experimental validation] Abstract and experimental results: The headline claim of average RMSE below 3 degrees is stated without error bars, number of trials, standard deviations, or explicit data-exclusion criteria. This information is required to evaluate whether the reported fidelity is statistically reliable or sensitive to outlier runs.
[SDRE torque generation / Dynamic model] Dynamic model and SDRE stage: No model-validation residuals, torque-prediction errors, or comparison of simulated versus measured suspension dynamics (cable compliance, tension variation, pendulum modes) are reported. Because the first-stage Riccati solution and second-stage optimization rely on this model to produce commands that the third stage can correct, the absence of such diagnostics leaves the weakest assumption untested.
[Experimental results] Ablation and baseline comparisons: The manuscript provides no ablation that removes the SDRE stage or direct comparison against simpler baselines (e.g., pure optimization or PID without LQR). Without these, the incremental benefit of the three-stage architecture over existing offline command-generation methods cannot be quantified.

minor comments (2)

[Third-stage controller] The description of the PID-LQR gains as 'data-driven' should clarify whether the tuning data are disjoint from the final evaluation runs or drawn from the same experimental sessions.
[Figures] Figure captions and axis labels for tracking-error plots should explicitly state the number of overlaid trials and whether shaded regions represent standard deviation or min/max envelopes.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We appreciate the referee's thorough review and valuable suggestions for improving our manuscript on the three-stage offline SDRE-based control framework. We address each major comment in detail below, indicating the revisions we plan to make.

read point-by-point responses

Referee: [Abstract / Experimental validation] Abstract and experimental results: The headline claim of average RMSE below 3 degrees is stated without error bars, number of trials, standard deviations, or explicit data-exclusion criteria. This information is required to evaluate whether the reported fidelity is statistically reliable or sensitive to outlier runs.

Authors: We agree that additional statistical details are needed to substantiate the experimental claims. In the revised manuscript, we will report the number of trials for each task (squatting and walking), include standard deviations with the average RMSE, add error bars to the relevant plots, and explicitly state the data collection and exclusion criteria used. revision: yes
Referee: [SDRE torque generation / Dynamic model] Dynamic model and SDRE stage: No model-validation residuals, torque-prediction errors, or comparison of simulated versus measured suspension dynamics (cable compliance, tension variation, pendulum modes) are reported. Because the first-stage Riccati solution and second-stage optimization rely on this model to produce commands that the third stage can correct, the absence of such diagnostics leaves the weakest assumption untested.

Authors: The referee correctly notes the lack of explicit model validation in the original submission. Although the dynamic model follows standard Lagrangian mechanics for the suspended biped, we will add a dedicated subsection presenting model-validation residuals, torque-prediction errors, and comparisons of simulated versus measured suspension dynamics (including cable compliance and pendulum effects) to strengthen the foundation of the SDRE stage. revision: yes
Referee: [Experimental results] Ablation and baseline comparisons: The manuscript provides no ablation that removes the SDRE stage or direct comparison against simpler baselines (e.g., pure optimization or PID without LQR). Without these, the incremental benefit of the three-stage architecture over existing offline command-generation methods cannot be quantified.

Authors: We recognize the benefit of quantitative comparisons. A complete ablation removing the SDRE stage would require a fundamentally different pipeline, which is outside the scope of the current integrated framework. However, we will add comparisons against a baseline that uses only the constrained optimization stage and against a standard PID controller (without the LQR refinement) to better illustrate the incremental value of the full three-stage approach. revision: partial

Circularity Check

0 steps flagged

No significant circularity in three-stage offline control derivation

full rationale

The paper presents a three-stage framework where the first stage computes optimal torques from an external bipedal dynamic model via SDRE, the second synthesizes velocity/acceleration commands via constrained optimization, and the third applies a data-driven PID-LQR refinement explicitly minimizing measured tracking error against motion-capture data. The headline RMSE result is an experimental measurement on the physical suspended robot, not a quantity that reduces by construction to fitted parameters or prior outputs. No self-definitional loop, fitted-input-as-prediction, or load-bearing self-citation chain appears in the described chain; the dynamic model and actuator constraints remain independent inputs whose fidelity is tested rather than assumed into the final metric.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The framework rests on a standard dynamic model assumption for the bipedal system and on tunable gains inside the optimization and PID-LQR stages; no new physical entities are postulated.

free parameters (2)

PID-LQR controller gains
Gains are tuned via data-driven refinement to minimize tracking error on the experimental platform.
optimization constraint bounds
Actuator velocity and acceleration limits are chosen to match robot hardware and directly affect the synthesized command sequences.

axioms (1)

domain assumption The dynamic model of the bipedal system is sufficiently accurate to support SDRE-based optimal torque generation.
Invoked explicitly in the first stage to produce the initial torque trajectory from motion-capture data.

pith-pipeline@v0.9.0 · 5765 in / 1432 out tokens · 68471 ms · 2026-05-19T11:21:02.882723+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel echoes

?

echoes
ECHOES: this paper passage has the same mathematical shape or conceptual pattern as the Recognition theorem, but is not a direct formal dependency.

SDRE cost function J = ∫ (x^T Q x + u^T R u) dt with state-dependent A(x), B(x) from double-pendulum model (Eqs. 18-20)
IndisputableMonolith/Foundation/BranchSelection.lean branch_selection unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

Piecewise-linear velocity profile and fmincon optimization under motor speed/acceleration bounds (Eqs. 28-33)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

20 extracted references · 20 canonical work pages

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[1] [1]

Lee, H.-D

B.-K. Lee, H.-D. Lee, J.-Y . Lee, K. Shin, J.-S. Han, and C.- S. Han, ”Development of dynamic model-based controller for up- per limb exoskeleton robot,” in *2012 IEEE International Confer- ence on Robotics and Automation*, 2012, pp. 3173–3178. doi: 10.1109/ICRA.2012.6224675

work page doi:10.1109/icra.2012.6224675 2012

[2] [2]

Sanz-Merodio, M

D. Sanz-Merodio, M. Cestari, J.C. Arevalo, and E. Garcia, ”Control Motion Approach of a Lower Limb Orthosis to Reduce Energy Con- sumption,” in *International Journal of Advanced Robotic Systems*,

work page

[3] [3]

F. Chen, Y . Yu, Y . Ge, and Y . Fang, ”WPAL for human power assist during walking using dynamic equation,” in *2009 International Conference on Mechatronics and Automation*, 2009, pp. 1039–1043. doi: 10.1109/ICMA.2009.5246270

work page doi:10.1109/icma.2009.5246270 2009

[4] [4]

N. L. Tagliamonte, F. Sergi, G. Carpino, D. Accoto, and E. Guglielmelli, ”Human-robot interaction tests on a novel robot for gait assistance,” in *2013 IEEE 13th International Conference on Rehabilitation Robotics (ICORR)*, 2013, pp. 1–6. doi: 10.1109/ICORR.2013.6650387

work page doi:10.1109/icorr.2013.6650387 2013

[5] [5]

Sun, J.-W

W. Sun, J.-W. Lin, S.-F. Su, N. Wang, and M. J. Er, ”Reduced Adaptive Fuzzy Decoupling Control for Lower Limb Exoskeleton,” *IEEE Transactions on Cybernetics*, vol. 51, no. 3, pp. 1099–1109,

work page

[6] [6]

doi: 10.1109/TCYB.2020.2972582

work page doi:10.1109/tcyb.2020.2972582 2020

[7] [7]

Srivastava, P.-C

S. Srivastava, P.-C. Kao, S. H. Kim, P. Stegall, D. Zanotto, J. S. Higginson, S. K. Agrawal, and J. P. Scholz, ”Assist-as-Needed Robot-Aided Gait Training Improves Walking Function in Individ- uals Following Stroke,” *IEEE Transactions on Neural Systems and Rehabilitation Engineering*, vol. 23, no. 6, pp. 956–963, 2015. doi: 10.1109/TNSRE.2014.2360822

work page doi:10.1109/tnsre.2014.2360822 2015

[8] [8]

Zhou and J

K. Zhou and J. C. Doyle, *Essentials of Robust Control*, vol. 104. Upper Saddle River, NJ: Prentice Hall, 1998, pp. 233–244

work page 1998

[9] [9]

P. J. Antsaklis and A. N. Michel, *Linear Systems*, vol. 8. New York: McGraw-Hill, 1997, pp. 272–274

work page 1997

[10] [10]

J. R. Cloutier and J. C. Cockburn, ”The state-dependent nonlinear regulator with state constraints,” in *Proceedings of the 2001 American Control Conference (Cat. No. 01CH37148)*, 2001

work page 2001

[11] [11]

B. D. O. Anderson and J. B. Moore, *Optimal Control: Linear Quadratic Methods*. Upper Saddle River, NJ: Prentice Hall, 1990

work page 1990

[12] [12]

Available: https://www.parker.com/parkerimages/emn/ServoFundamentals.pdf

Parker Hannifin Corporation, ”Servo Fundamentals,” Parker Hannifin Corporation, 2010.[Online]. Available: https://www.parker.com/parkerimages/emn/ServoFundamentals.pdf

work page 2010

[13] [13]

H. Shao, D. Liao, and J. W. F. Cheung, ”System Identification and Profile Planning for High Accuracy Servo System,” in *Intelligent Robotics and Applications (ICIRA 2008)*, C. Xiong et al., Eds., Lecture Notes in Artificial Intelligence, vol. 5314. Springer-Verlag Berlin Heidelberg, 2008, pp. 410–419

work page 2008

[14] [14]

Z. Yu, C. Han, and H. Mu, ”A novel approach of tuning trapezoidal velocity profile for energy saving in servomotor systems,” in *Proceed- ings of the 2015 34th Chinese Control Conference (CCC)*, 2015, pp. 7028–7033. doi: 10.1109/ChiCC.2015.7260775

work page doi:10.1109/chicc.2015.7260775 2015

[15] [15]

Lan, S.-S

C.-W. Lan, S.-S. Lin, B.-S. Chen, M.-F. Lo, C.-T. Ku, and M.-C. Chien, ”Design and Development of a Biped Robot for a Knee Ex- oskeleton Analysis,” in *2021 International Symposium on Intelligent Signal Processing and Communication Systems (ISPACS)*, IEEE, Taoyuan, Taiwan, 2021

work page 2021

[16] [16]

C. -W. Lan, S. -S. Lin, C. -T. Ku, B. -S. Chen, M. -F. Lo and M. -C. Chien, ”Using Biped Robot on Knee Exoskeleton for Stair Climbing Assistance Analysis,” 2021 International Automatic Control Conference (CACS) , Chiayi, Taiwan, 2021, pp. 1-6, doi: 10.1109/CACS52606.2021.9639061

work page doi:10.1109/cacs52606.2021.9639061 2021

[17] [17]

Fazel, A

R. Fazel, A. M. Shafei, and S. R. Nekoo, ”Dynamic modeling and closed-loop control design for humanoid robotic systems: Gibbs–Appell formulation and SDRE approach,” Multibody System Dynamics, 2024, pp. 1–30

work page 2024

[18] [18]

S. O. Schrade et al. , ”Knee Compliance Reduces Peak Swing Phase Collision Forces in a Lower-Limb Exoskeleton Leg: A Test Bench Evaluation,” IEEE Transactions on Biomedical Engineering , vol. 68, no. 2, pp. 535-544, Feb. 2021, doi: 10.1109/TBME.2020.3006787

work page doi:10.1109/tbme.2020.3006787 2021

[19] [19]

C. A. Laubscher and J. T. Sawicki, ”Gait Guidance Control for Damping of Unnatural Motion in a Powered Pediatric Lower-Limb Orthosis,” 2019 IEEE 16th International Conference on Rehabilitation Robotics (ICORR) , Toronto, ON, Canada, 2019, pp. 676-681, doi: 10.1109/ICORR.2019.8779437

work page doi:10.1109/icorr.2019.8779437 2019

[20] [20]

Y . He, G. Zhou, H. Jiang, and J. Zhang, ”Risk management and reg- ulations for lower limb medical exoskeletons: a review,” in *Medical Devices: Evidence and Research*, vol. 10, 2017, pp. 89–107

work page 2017