Experimental Investigation of a Passive Compliant Torsional Suspension for Curved-Spoke Wheel Stair Climbing

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

 Comments on “experimental investigation of a passive compliant torsional suspension for curved-spoke wheel stair climbing”

 

 

This paper proposes a passive Compliant Spiral Torsional Suspension, which is attached to the wheel’s drive axis, to make curved-spoke wheels climb stairs with improved stability. The main contribution of the paper is in section 3.2, the details about the design procedure may be illustrated furthermore.

 

The following questions need to be addressed:

 

• Is it possible to prove the results theoretically, that the C-STS can definitely improve dynamic stability for curved-wheel-based robotic systems? Can the C-STS be improved furthermore，Is it optimized?

 

• What is the gap between maximum and minimum velocities during low-,medium-, and high-speed climbing?

 

• “Experimental findings revealed that the proposed suspension system significantly improves dynamic stability”. What about the stability metrics? How the conclusion be drawn without explicit validation in theory or extensive data analysis considering some slipping scenarios? From my humble point of view, more discussions are needed about stability issues.

Comments for author File: Comments.pdf

Author Response

Answer to reviewer 1

Comments 1: This paper proposes a passive Compliant Spiral Torsional Suspension, which is attached to the wheel’s drive axis, to make curved-spoke wheels climb stairs with improved stability. The main contribution of the paper is in section 3.2, the details about the design procedure may be illustrated furthermore.

Response 1:

Thank you for this constructive suggestion. We have expanded the design procedure in section 3.3., specifically discussing how we determined the stiffness values. In the revised manuscript, we further elaborate on the spiral spring’s theoretical stiffness (Equation (6)) and compare theoretical and empirical stiffness measurements (lines 249–274). Table 3 has been added to clarify both theoretical and experimentally obtained stiffness values, offering concrete insights into our design methodology.

 

Comments 2: Is it possible to prove the results theoretically, that the C-STS can definitely improve dynamic stability for curved-wheel-based robotic systems? Can the C-STS be improved furthermore，Is it optimized?

Response 2:

We greatly appreciate this valuable comment. To theoretically substantiate our claim, we have added Section 3.1, explicitly defining dynamic stability through deceleration during the Discontinuous Contact State (DCS) phase. We have further clarified in the Introduction (lines 76–80) and Section 3.1 (lines 142–160) that the C-STS mechanism improves stability by reducing the abrupt deceleration at contact transitions. Additionally, we have revised illustrations and an expanded explanation of the theoretical principle behind how the C-STS enhances stability (Figures 6).

In the discussion (Section 5.1, lines 335–379), we acknowledge that the current design is not fully optimized. We propose that future work could focus on further optimization of the C-STS to achieve improved synchronization of energy release timing with wheel dynamics, thereby maximizing the mechanism’s effectiveness in enhancing dynamic stability.

 

Comments 3: What is the gap between maximum and minimum velocities during low-, medium-, and high-speed climbing?

Response 3:

Following your recommendation, we explicitly defined the dynamic stability metric as deceleration during the DCS phase, clearly stated in Section 3.1 (lines 142–160). Additionally, Tables 6–8 provide detailed measurements of velocity gaps at low, medium, and high speeds, including comparative analyses of maximum and minimum velocities under each stiffness condition, allowing readers to clearly understand velocity fluctuation magnitudes across different scenarios.

 

 

Comments 4: “Experimental findings revealed that the proposed suspension system significantly improves dynamic stability”. What about the stability metrics? How the conclusion be drawn without explicit validation in theory or extensive data analysis considering some slipping scenarios? From my humble point of view, more discussions are needed about stability issues.

Response 4:

We appreciate this critical insight. In our revised manuscript, the stability metrics have been explicitly defined (Section 3.1), and detailed data analysis from experimental validations (Tables 6–8 and Figures 15–17) has been provided to demonstrate the suspension’s effectiveness.

Additionally, we added discussion in Section 5 (Analysis and Discussion, Lines 487-492), addressing slip scenarios, stating that although slip was not observed in the experiments, our theoretical analysis (Section 3.1 and 3.2) explains clearly how the C-STS theoretically mitigates instability and reduces the likelihood of slip by attenuating abrupt deceleration.

Moreover, in response to the reviewer’s request for deeper exploration of dynamic stability and potential slip issues, we substantially revised and expanded the Discussion (Section 5). There, we offer a step-by-step analysis of how abrupt deceleration can lead to slip in curved-spoke wheels, and we present new findings on the suspension’s ability to prevent such instability under various speed and stiffness conditions. Specifically, we detail how lower speeds occasionally trigger premature spring release, whereas higher speeds better synchronize the spring’s energy discharge with the robot’s torque demands—thus attenuating sharp velocity drops and reducing the likelihood of slip. We believe this restructured discussion now provides a clearer, more comprehensive rationale for the C-STS’s effectiveness in enhancing stair-climbing stability.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The authors present an interesting paper on passive compliant torsional suspensions for curved-spoke wheel stair climbing. In general, the manuscript is well structured, progressing logically through modelling, simulation, and experimental validation.

However, while the kinematic model is sound, the treatment of the suspension’s stiffness K remains overly simplified. The authors determine a baseline value k_t empirically, but they do not account for the fact that, in real operation, only the single contacting curved spoke and the spiral spring act as two elastic elements in series. At each step transition, the next spoke may land at radii r>r_min (due to stair height/run tolerances, wheel phasing, spoke bending under load, or dynamic impacts), so the arm itself also contributes compliance. Because only the one contacting arm carries the load, the total torsional stiffness should be modelled as

1/K_total=1/K_arm(r)+1/K_spiral  

where K_arm(r) is the spoke’s torsional stiffness (itself a function of the contact radius r).

I recommend the authors:

1. Characterise K_arm(r) via beam theory, FEA, or simple torque–deflection tests at several radii.

2. Incorporate this series‐spring formulation into their dynamic model.

Doing so will capture the compliance variation when spokes land away from the root, improving prediction of velocity discontinuities, energy storage/release, and overall climbing performance.

Author Response

Answer to reviewer 2

Comment 1: The authors present an interesting paper on passive compliant torsional suspensions for curved-spoke wheel stair climbing. In general, the manuscript is well structured, progressing logically through modelling, simulation, and experimental validation.

Response 1: We sincerely appreciate the reviewer’s positive feedback regarding our manuscript structure and content. We aimed to present the development of our passive compliant torsional suspension in a clear, step-by-step manner so that readers can readily follow the underlying concepts and technical details. We are glad that the reviewer found our approach logical and well organized. If there are any further refinements that could enhance clarity or coherence, we would be pleased to address them.

 

Comment 2: However, while the kinematic model is sound, the treatment of the suspension’s stiffness K remains overly simplified. The authors determine a baseline value kt empirically, but they do not account for the fact that, in real operation, only the single contacting curved spoke and the spiral spring act as two elastic elements in series. At each step transition, the next spoke may land at radii r>rmin (due to stair height/run tolerances, wheel phasing, spoke bending under load, or dynamic impacts), so the arm itself also contributes compliance. Because only the one contacting arm carries the load, the total torsional stiffness should be modelled as

1/K_total=1/K_arm(r)+1/K_spiral  

where K_arm(r) is the spoke’s torsional stiffness (itself a function of the contact radius r).

I recommend the authors:

1. Characterise K_arm(r) via beam theory, FEA, or simple torque–deflection tests at several radii.
2. Incorporate this series‐spring formulation into their dynamic model.

Doing so will capture the compliance variation when spokes land away from the root, improving prediction of velocity discontinuities, energy storage/release, and overall climbing performance.

 

Response 2: We appreciate the reviewer’s insight regarding the potential influence of the spoke’s elasticity. To address this concern, we conducted a SolidWorks simulation to determine the spoke’s torsional stiffness at three representative contact points (P1, P2, and P3). We have added a new table (Table 3), “Spoke Torsional Stiffness Calculation Based on SolidWorks Simulation,” and a new figure (Figure 9), “Stiffness Simulation Setup and Results: (a) P1, (b) P2, (c) P3,” to illustrate the boundary conditions and computed stiffness values. Specifically, we fixed the wheel’s center of rotation (CoR) and applied a 20 N normal load at each point, using PLA material properties for the spoke. By measuring the resulting deformation and calculating the torsional stiffness from torque–angle relationships, we then combined the spoke’s stiffness with that of the spiral spring to determine the total system stiffness.

The results show that, at most, the spoke’s flexibility contributes 1.7% to the overall torsional stiffness. Notably, point P3, located near the contact region at the moment of transition, displayed a slightly higher spoke stiffness, but still remained negligible relative to the spiral spring. Hence, the spoke can be treated as effectively rigid in our model without sacrificing accuracy. This outcome supports our initial assumption that ignoring the spoke’s elasticity in the main analysis is valid. We have summarized these findings in the revised manuscript (Section 3.3, lines 275–287) to clarify why the series-spring effect from the spoke is minimal in our design.

Author Response File: Author Response.pdf

Round 2

Reviewer 2 Report

Comments and Suggestions for Authors

No additional comments, the authors have addressed the comments provided in the previous round of revision adequately.