Humanoid Robot Teleoperation for Nonprehensile Transportation: A Multiple-Constraint Safety-Critical Control Framework

Round 1

Reviewer 1 Report

Comments and Suggestions for Authors

The authors propose a 3-layer architecture CBF+MPC+QP IK for humanoid robot teleoperation. The document is well structured and results are clearly described. However, I have some concerns:

Mayor concerns:

1. The contribution appears to be mainly an integration of existing methods rather than a fundamentally novel proposal. Authors are encouraged to clarify the contribution against existing frameworks.
2. The authors propose an CBF with acceleration restriction and a piece-wise function, but there is no formal stability proof, and there is no comparison against standard CBF.
3. Related to the discrete map and FCNN, what is the physical meaning of the 0.15 error? and 97% accuracy? Also there is no comparison with existing methods.
4. Although experimental results are very valuable, there is no comparison with a baseline and standard metrics like tracking error or constraint violation are not reported. 
5. The proposal could be well improved if computational load, scalability and noise robustness are considered or at least commented.

Minor concerns:

6. Please revise consistency of variables and subindexes along the document.

7. Some parameters are mentioned but not justified, a brief explanation would help the clarification.

8. Details of the FCNN like activation functions, training dataset set size and training procedure are limited.

9. A brief paragraph discussing limitations would be very valuable to improve the paper.

Author Response

Author's Reply to the Review Report (Reviewer 1)

Comments No. 1:

The contribution appears to be mainly an integration of existing methods rather than a fundamentally novel proposal. Authors are encouraged to clarify the contribution against existing frameworks.

Response:

Thank you very much for this valuable comment. We have carefully considered this concern and realized that the distinction between the proposed contribution and existing studies was not sufficiently clear in the original manuscript. This may have led to the impression that the paper mainly integrates existing methods.

To address this issue, we have substantially rewritten and expanded the Introduction section. In particular, we clarified the conflict relationships among multiple constraints in humanoid robot teleoperation for nonprehensile transportation, distinguished the proposed hierarchical decoupling framework from conventional priority-based constraint-handling methods, and reorganized the contribution statements to better highlight the motivation and novelty of this work.

In the revised Introduction, we have explicitly clarified that this work is not a simple integration of existing techniques such as CBF, MPC, and QP. Instead, the proposed framework is designed around the conflict relationships among workspace boundary constraints, object dynamic constraints, and intrinsic kinematic constraints of the robot. The corresponding revised text is provided below:

Although teleoperation enables robots to exploit human cognitive capabilities in complex environments, transferring nonprehensile transportation tasks to humanoid robot platforms still poses major challenges to control system design. On the one hand, nonprehensile transportation relies on friction, gravity, and inertial forces to maintain object stability. The object must be prevented from slipping or toppling during motion, which imposes strict coupling among the position, orientation, velocity, and acceleration of the end-effector trajectory. In particular, during highly dynamic teleoperation, the object orientation is highly sensitive to variations in linear and angular accelerations. Any abrupt change or relaxation of constraints may directly lead to task failure. On the other hand, the upper body of a humanoid robot has a highly redundant dual-arm structure. Intrinsic constraints, including dual-arm coordination, joint limits, singularity avoidance, self-collision avoidance, and workspace boundaries, coexist and become dynamically coupled as the robot configuration changes.

Traditional robot safety control methods usually assign high priority to constraints related to the robot’s intrinsic safety, while treating task constraints as low-priority or soft constraints[2-5]. This design can prioritize the prevention of dangerous robot motions in general manipulation tasks. For nonprehensile transportation, however, object non-slipping, non-toppling, trajectory smoothness, and orientation stability are not secondary objectives that can be arbitrarily relaxed. Instead, they are hard conditions that determine whether the task can be completed. Thus, if the conventional priority allocation is directly adopted, and constraints such as workspace boundaries, joint limits, and singularity avoidance are combined with object dynamic constraints in a single optimization problem, or if their conflicts are handled only by weight tuning, two problems often arise. First, the abrupt activation of robot intrinsic constraints may induce acceleration discontinuities at the end-effectors, thereby disrupting object balance. Second, excessively conservative restrictions on robot motion may be imposed to maintain object stability, which greatly compresses the effective operational workspace.

When a humanoid robot executes the task, workspace boundary constraints are particularly prone to conflicts with task dynamic constraints. Workspace boundary evaluation for dual-arm coordinated tasks is fundamentally different from that in single-arm scenarios. Because the dual end-effectors must maintain a relative pose relationship and the two kinematic chains share a movable waist joint, even if a target pose lies within the single-arm reachable regions of the left and right arms separately, there is no guarantee that a common waist configuration exists such that both arms can reach their target poses simultaneously. Therefore, the workspace boundary in nonprehensile transportation is not merely a geometric envelope approached by a single end-effector. Instead, it results from the combined effects of dual-arm cooperative reachability, waist configuration, object orientation, and task constraints.

To address these issues, the core idea of this paper is to hierarchically decouple multiple constraints according to their physical roles and conflict relationships in nonprehensile transportation. Specifically, the constraints are divided into three levels. The top layer handles the cooperative workspace boundary constraints of the robot. The middle layer addresses task dynamic constraints, such as object non-slipping and non-toppling. The bottom layer processes intrinsic kinematic constraints of the robot, including dual-arm coordination, trajectory tracking, singularity avoidance, self-collision avoidance, and joint limits. This hierarchy is not a simple priority setting that places robot safety above task constraints. Instead, it reorganizes the order in which constraints take effect for nonprehensile transportation. The top layer first constrains user inputs through a safe velocity filter, allowing the system to decelerate smoothly with controlled acceleration before approaching the cooperative workspace boundary. This prevents direct conflicts between workspace boundary constraints and the object dynamic constraints handled in the middle layer. Based on the safe velocity output from the top layer, the middle layer generates pose-coupled trajectories that satisfy multi-object non-slipping and non-toppling constraints. The bottom layer then handles the remaining kinematic constraints, including dual-arm coordinated motion, self-collision avoidance, singularity avoidance, and configuration optimization within the null space of the task trajectory.

In this framework, the top-level safe velocity filter plays a key role in decoupling constraints. Unlike conventional hard boundary truncation methods, the proposed safe velocity filter based on an improved CBF explicitly introduces a maximum acceleration constraint during deceleration. This allows the robot to achieve smooth and controlled deceleration when approaching workspace boundaries, thereby reducing the risk of abrupt orientation changes and object toppling. Compared with conventional linear CBFs, this method avoids the tradeoff between overly conservative deceleration and excessive acceleration peaks at the activation instant. It maintains a larger effective operational workspace while keeping the object forces smooth during nonprehensile transportation. In addition, to address the difficulty of analytically describing the cooperative workspace boundaries of the robot, this paper designs a reachability determination method based on a differentiable reachability surrogate model. This method estimates the cooperative workspace boundary distance along the current motion direction in real time, providing boundary information for the top-level velocity filter.

Based on the above hierarchical decoupling mechanism, the bottom-level controller no longer needs to simultaneously handle the direct conflict between workspace boundary constraints and object dynamic constraints. Instead, it can track the object trajectory and achieve dual-arm coordinated motion through quadratic programming while satisfying the task trajectory generated by the middle layer. Self-collision avoidance, singularity avoidance, joint limits, and configuration optimization are then realized in the task null space. Accordingly, this paper constructs a Multiple-Constraint Safety-Critical Control Framework (MC-SCCF) for humanoid robot teleoperation in nonprehensile transportation, as shown in Figure 1. The proposed framework enables coordinated processing of workspace boundary constraints, task dynamic constraints, and intrinsic kinematic constraints of the robot.

Figure 1. Block diagram of the proposed MC-SCCF.

The primary contributions of this paper are summarized as follows:

1. A MC-SCCF is proposed for humanoid robot nonprehensile transportation. Different from conventional priority-based designs that simply place robot intrinsic safety constraints above task constraints, this paper decouples multiple constraints into three hierarchical levels according to the conflict relationships among workspace boundary constraints, object dynamic constraints, and intrinsic kinematic constraints of the robot. The top layer is responsible for the intrinsic safety of the robot under workspace constraints. It handles cooperative workspace boundaries through a safe velocity filter and isolates the direct impact of boundary constraints on task dynamic constraints. The middle layer generates pose-coupled reference trajectories that satisfy non-slipping and non-toppling requirements, thereby enforcing object dynamic constraints and ensuring task safety. The bottom layer handles the remaining kinematic constraints and task execution. While satisfying joint and manipulability constraints, it achieves self-collision avoidance, coordinated motion, and configuration optimization. This framework provides a systematic method for coordinating multiple hard constraints in humanoid robot nonprehensile transportation.
2. A top-level safe velocity modulation strategy is proposed to handle the workspace boundaries of humanoid robots. To address the acceleration discontinuities caused by conventional hard boundary truncation and the tradeoff between conservative deceleration and orientation jitter in conventional linear CBFs, this paper designs an improved CBF based on acceleration constraints. This method enables the robot to decelerate smoothly with controlled acceleration when approaching workspace boundaries, thereby keeping the object forces and orientation variations more stable. In addition, to address the difficulty of analytically describing the cooperative workspace boundaries of humanoid dual arms, this paper proposes a boundary determination method based on a differentiable reachability surrogate model, which enables real-time estimation of the cooperative workspace boundary distance.

Comments No. 2:

The authors propose an CBF with acceleration restriction and a piece-wise function, but there is no formal stability proof, and there is no comparison against standard CBF.

Response:

Thank you very much for this valuable comment. We have carefully considered this issue and revised the manuscript from two aspects: the formal proof of the proposed CBF formulation and the clarification of the baseline comparisons.

First, regarding the theoretical analysis of the improved CBF, we have added a formal proof in Section 3.2 of the revised manuscript. In this proof, we show that the proposed piecewise class function is continuously differentiable and that the safe set remains forward invariant when the improved CBF constraint is satisfied. We also further derive that, during the deceleration process near the workspace boundary, the acceleration is always bounded by the prescribed maximum acceleration. This demonstrates that the proposed method can ensure safety while avoiding excessive acceleration peaks. The corresponding revision has been added in Section 3.2 of the revised manuscript.

Second, regarding the comparison with standard CBF methods, we agree that the baseline settings were not clearly presented in the original manuscript. Although baseline comparisons were included, the previous naming and explanation may have made them difficult to identify. Therefore, we have revised the beginning of Sections 6.1 and 6.3 to make the baseline methods clearer. In the simulation study, CBF1 and CBF2 are both standard CBF-based methods and are used as baselines for comparison with the proposed improved CBF. In addition, in the hardware experiments in Section 7, we have also revised the naming and description of the control modes. Specifically, the CBF mode denotes the baseline method using a conventional CBF, the No-SVF mode denotes the method without the safe velocity filter, and the MC-SCCF mode denotes the proposed method.

With these revisions, the comparison between MC-SCCF and CBF is used to demonstrate the advantages of the improved CBF over the standard CBF in reducing abrupt acceleration changes, suppressing attitude disturbance, and maintaining a larger effective operational workspace. The comparison between MC-SCCF and No-SVF is used to demonstrate the importance of the safe velocity filter in handling workspace boundary constraints. We have revised the relevant descriptions in the simulation and experimental sections to make the baseline comparisons and their purposes more explicit.

3.2. Improved CBF Based on Acceleration Constraints

Proposition: Consider the system state in the safe set . If the control input always satisfies the improved control barrier function constraint , where

and , then the safe set  is forward invariant.

Proof: When ​, we have . Since , this segment is strictly increasing and satisfies .

When ​, we have . Since , its derivative is . Thus, this segment is also strictly increasing.

At the switching point ​, the left and right limits of the two segments and their derivatives satisfy

Therefore,  is continuously differentiable for . It also satisfies , and  holds for any .

Consider the comparison system:

Since  and  for any ,  cannot cross from the nonnegative half-axis to the negative half-axis. According to the comparison lemma, we have

Thus, as long as the initial state lies in the safe set, the system trajectory remains in the safe set, and the workspace boundary is not violated. This completes the proof.

Furthermore, when the object approaches the workspace boundary and the barrier constraint is active, we have

Therefore, the acceleration magnitude during deceleration can be written as

When ​, we obtain .

When ​, we obtain .

Thus, throughout the boundary-approaching process, the acceleration during deceleration always satisfies .

The improved class  function partitions the boundary approach process into a constant deceleration phase and an exponential convergence phase:

Constant Deceleration Phase (): When , the system undergoes constant deceleration subject to the constraint of .

Exponential Convergence Phase (): When , the constraint switches to the linear CBF form, and the system velocity decays exponentially.

Assuming a constant velocity direction for the object, when , the object undergoes two stages of deceleration, and the total deceleration distance is:

.                              

When , the deceleration process remains in the exponential convergence phase, and the deceleration distance is:

.                                                   

6.1. Verification of the Reachability Determination Method

To evaluate the reachability determination algorithm based on the differentiable surrogate model, we conduct comparative tests using both the benchmark method and the proposed method.

Benchmark method: For a given dual-arm target pose, after confirming that no interference occurs with the mobile base, all possible configurations within the waist joint limits are traversed at a resolution of 0.5°. For each waist configuration, the analytical inverse kinematic solutions of both arms are solved. Since the employed manipulators have an SRS configuration, analytical inverse kinematic solutions are available. If analytical solutions exist for both arms under any waist configuration, the target pose is determined to be truly reachable. If no solution exists after all waist configurations are traversed, the target pose is determined to be truly unreachable. The results obtained by the benchmark method are used as the ground truth for the reachability of dual-arm target poses. The selection of bounding volumes is shown in Figure 5.

6.3. Evaluation of the Safe Velocity Filter

To evaluate the advantages of the proposed acceleration-constrained piecewise CBF, we compare it with two standard linear CBF baseline methods. Both baselines adopt the conventional class  function , but with different gains. All other components of the control framework, including the middle-layer trajectory smoothing module and the low-level inverse kinematics controller, remain unchanged. Therefore, this comparison allows the effect of replacing the standard CBF with the proposed improved CBF to be evaluated independently. The parameter settings of the control schemes are listed in Table 4.

7.2. Experimental Conditions and Evaluation Metrics

This section defines the following three control modes:

• No-SVF: No Safe Velocity Filter Mode (Hard Boundary Mode). The middle-layer MPC (ensuring multi-object dynamic constraints) and low-level kinematic hard limits are enabled, whereas the top-layer safe velocity filter is deactivated.
• CBF: Linear CBF Mode (Standard CBF Baseline Mode). The middle-layer MPC is enabled, and a conventional linear CBF () is employed at the top layer as the safe velocity filter.
• MC-SCCF: Proposed Multiple-Constraint Safety-Critical Control Framework. The middle-layer MPC is enabled, and the improved CBF proposed in this study is utilized at the top layer (with algorithm parameters identical to those in the Section 6 simulations).

Comments No. 3:

Related to the discrete map and FCNN, what is the physical meaning of the 0.15 error? and 97% accuracy? Also there is no comparison with existing methods.

Response:

Thank you very much for this valuable comment. We have carefully considered this issue and revised the manuscript to further clarify the meaning of the surrogate-model fitting error, the  metric, and the comparison method.

First, regarding the fitting error of the FCNN surrogate model shown in Figure 3, its definition is:

where  denotes the true reachability index at position  in the discrete reachability map, and  denotes the predicted reachability value given by the FCNN surrogate model at the same position. Therefore, the statement that the error is less than 0.15 means that, in the cross-sections shown in Figure 3, for a given spatial position, the absolute difference between the reachability index predicted by the surrogate model and the corresponding true reachability index in the discrete reachability map is less than 0.15. In other words, this error reflects the numerical approximation accuracy of the surrogate model for the discrete single-arm reachability index. A smaller error indicates that the predicted reachability value at the corresponding spatial position is closer to the true value in the discrete reachability map.

Second, the meaning of  on the test set indicates that the FCNN surrogate model can explain 97.1% of the variance of the reachability index in the discrete reachability map. This means that the model can well preserve the overall trend and boundary variation of the original discrete reachability distribution. It should also be noted that, in the online computation, the surrogate model is mainly used to provide a continuous and differentiable reachability gradient to guide waist-configuration optimization. The final reachability decision of the target pose is still obtained through nearest-neighbor query in the true discrete reachability map. Therefore, the role of the surrogate model is to improve the efficiency of waist-configuration search and boundary approximation, rather than directly replacing the final decision process based on the true discrete reachability map.

Finally, regarding the comparison with existing methods, we have rewritten Section 6.1 according to the reviewer’s suggestion. In the revised manuscript, we introduce a benchmark method for comparison. This benchmark method traverses the waist-joint configurations and solves the inverse kinematics of both arms under each waist configuration to determine whether a target pose is reachable. Since this method exhaustively searches the waist configurations and directly uses inverse-kinematics results for reachability determination, it provides a reliable reference for evaluation. We have also added a comparison of computational time between the benchmark method and the proposed method. The results show that the proposed method maintains high reachability-determination accuracy while significantly reducing the online query time, making it more suitable for real-time teleoperation control.

6.1. Verification of the Reachability Determination Method

To evaluate the reachability determination algorithm based on the differentiable surrogate model, we conduct comparative tests using both the benchmark method and the proposed method.

Benchmark method: For a given dual-arm target pose, after confirming that no interference occurs with the mobile base, all possible configurations within the waist joint limits are traversed at a resolution of 0.5°. For each waist configuration, the analytical inverse kinematic solutions of both arms are solved. Since the employed manipulators have an SRS configuration, analytical inverse kinematic solutions are available. If analytical solutions exist for both arms under any waist configuration, the target pose is determined to be truly reachable. If no solution exists after all waist configurations are traversed, the target pose is determined to be truly unreachable. The results obtained by the benchmark method are used as the ground truth for the reachability of dual-arm target poses. The selection of bounding volumes is shown in Figure 5.

Table 3. Computational Efficiency Results for the Two Methods

The computational efficiency results for the two methods in determining target pose reachability are listed in Table 3. The benchmark method requires traversing the waist configurations and solving the inverse kinematics of both manipulators under each waist configuration. Moreover, its determination depends on the sampled waist configurations. For unreachable targets and some poses close to joint limits, the benchmark method must traverse all waist configurations before producing a result. Therefore, its query time is much higher than that of the proposed method.

Comments No. 4:

Although experimental results are very valuable, there is no comparison with a baseline and standard metrics like tracking error or constraint violation are not reported.

Response:

Thank you very much for this valuable comment. We have carefully considered this issue. We realized that, although baseline methods and control-performance metrics were included in the original manuscript, their naming, organization, and explanations were not sufficiently clear. As a result, the comparison between the proposed method and the baseline methods, as well as the relationship between the reported metrics and tracking/constraint performance, may not have been immediately clear to readers. Therefore, we have reorganized and revised the simulation and experimental sections in the revised manuscript.

First, regarding the baseline comparison, we have rewritten Section 6 to make the comparison settings more explicit. In the simulation study, the proposed improved CBF method is compared with two standard CBF-based baseline methods, denoted as CBF1 and CBF2, to demonstrate the advantages of the improved CBF during the boundary-approaching process. In addition, for the reachability-determination problem, we introduce a benchmark method that traverses the waist configurations and solves the inverse kinematics of both arms. We have added comparisons between this benchmark method and the proposed method in terms of reachability-determination results and computational time.

Second, to make the comparison in the hardware experiments clearer, we have renamed and clarified the control modes in Sections 7.2 and 7.3. Specifically, Mode CBF denotes the baseline method using a conventional CBF, Mode No-SVF denotes the method without the safe velocity filter, and Mode MC-SCCF denotes the proposed method. The comparison between Mode MC-SCCF and Mode CBF is used to demonstrate the advantage of the improved CBF over the conventional CBF. The comparison between Mode MC-SCCF and Mode No-SVF is used to demonstrate the importance of the safe velocity filter in handling workspace boundary constraints.

Finally, regarding standard metrics, we have provided a concentrated presentation of the control-performance indicators in Section 7.3.3. These metrics cover task-execution performance, trajectory-tracking accuracy, and constraint satisfaction. Specifically, we report the object position, object velocity magnitude, object tilt angle, object position tracking error, relative position error between the two end-effectors, relative attitude error between the two end-effectors, robot manipulability, minimum distance between bounding volumes, waist-joint angle, and inverse-kinematics computation time. These indicators allow the tracking performance, object stability, dual-arm coordination, self-collision avoidance, singularity avoidance, and real-time performance of the proposed method to be evaluated more clearly.

Following the reviewer’s suggestion, we have revised the manuscript to make the baseline settings, comparison purposes, and control-performance metrics more explicit. We sincerely appreciate this comment, which helped us improve the readability and persuasiveness of the experimental section.

6.1. Verification of the Reachability Determination Method

To evaluate the reachability determination algorithm based on the differentiable surrogate model, we conduct comparative tests using both the benchmark method and the proposed method.

Benchmark method: For a given dual-arm target pose, after confirming that no interference occurs with the mobile base, all possible configurations within the waist joint limits are traversed at a resolution of 0.5°. For each waist configuration, the analytical inverse kinematic solutions of both arms are solved. Since the employed manipulators have an SRS configuration, analytical inverse kinematic solutions are available. If analytical solutions exist for both arms under any waist configuration, the target pose is determined to be truly reachable. If no solution exists after all waist configurations are traversed, the target pose is determined to be truly unreachable. The results obtained by the benchmark method are used as the ground truth for the reachability of dual-arm target poses. The selection of bounding volumes is shown in Figure 5.

6.3. Evaluation of the Safe Velocity Filter

To evaluate the advantages of the proposed acceleration-constrained piecewise CBF, we compare it with two standard linear CBF baseline methods. Both baselines adopt the conventional class  function , but with different gains. All other components of the control framework, including the middle-layer trajectory smoothing module and the low-level inverse kinematics controller, remain unchanged. Therefore, this comparison allows the effect of replacing the standard CBF with the proposed improved CBF to be evaluated independently. The parameter settings of the control schemes are listed in Table 4.

7.2. Experimental Conditions and Evaluation Metrics

This section defines the following three control modes:

• No-SVF: No Safe Velocity Filter Mode (Hard Boundary Mode). The middle-layer MPC (ensuring multi-object dynamic constraints) and low-level kinematic hard limits are enabled, whereas the top-layer safe velocity filter is deactivated.
• CBF: Linear CBF Mode (Standard CBF Baseline Mode). The middle-layer MPC is enabled, and a conventional linear CBF () is employed at the top layer as the safe velocity filter.
• MC-SCCF: Proposed Multiple-Constraint Safety-Critical Control Framework. The middle-layer MPC is enabled, and the improved CBF proposed in this study is utilized at the top layer (with algorithm parameters identical to those in the Section 6 simulations).

7.3.3 Control Performance Metrics

To verify the effectiveness of the low-level controller and more intuitively demonstrate the experimental process, a set of trials using Mode MS-SCCF was selected. Figure 12 illustrates the variations of relevant variables during the experiment, where the red regions indicate periods of operational pauses.

During the experiment, the maximum motion velocity of the object exceeded 0.25 m/s, and its maximum tilt angle exceeded 6°. Throughout the entire motion, the trajectory position error remained below  m, the relative position error between the dual-arm end-effectors was below  m, and the relative orientation error was below  rad. This achieved the objectives of trajectory tracking and maintaining a constant relative pose between the dual-arm end-effectors.

Although the robot’s configuration underwent considerable changes during transportation, safety was consistently guaranteed. The robot’s manipulability remained above 0.04, and the minimum distance between its bounding volumes was greater than 0.29 m. The goals of evading singular configurations and avoiding self-collisions were realized. During the lifting of the tray and obstacle crossing, driven by the waist regulation objective, the waist pitch axis automatically returned to its upright state after providing the necessary forward-reaching compensation. This not only prevented the waist joint from remaining in a high-load configuration for extended periods but also rendered the robot's overall operational posture more natural and anthropomorphic.

Within each control period during the task, the inverse kinematics solving time was approximately 3 ms, with a peak below 6 ms. This computational overhead is lower than the control period of the teleoperation system, meets the real-time requirements of teleoperation control.

The data indicate that the low-level controller can effectively achieve the design objectives of multiple-constraint coordinated processing and configuration optimization while fulfilling real-time performance requirements.

Figure 12. Variation curves of relevant variables during the experiment. (a) Object position. (b) Object velocity. (c) Object tilt angle. (d) Object position error. (e) Relative position error between dual-arm end-effectors. (f) Relative orientation error between dual-arm end-effectors. (g) Robot manipulability. (h) Minimum distance between bounding volumes. (i) Robot waist joint angles. (j) Inverse kinematics solving time.

Comments No. 5:

The proposal could be well improved if computational load, scalability and noise robustness are considered or at least commented.

Response:

Thank you very much for this valuable suggestion. We have carefully considered this comment and added a new Section 8, Discussion, in the revised manuscript. In this section, we specifically discuss the computational load, scalability, and noise robustness of the proposed framework. With these additions, we hope to provide a clearer discussion of the computational requirements, scalability, and robustness of the proposed framework in practical applications. We sincerely appreciate the reviewer’s suggestion, which helped us improve the completeness of the discussion on the engineering applicability and limitations of the proposed method.

8. Discussions

The proposed Multiple-Constraint Safety-Critical Control Framework decouples offline computation from online control in its design, thereby reducing the computational burden during online control. For workspace boundary determination, both the discrete reachability map and the differentiable reachability surrogate model are constructed offline. During online operation, only the surrogate model is used for gradient-guided waist configuration search, followed by nearest-neighbor queries in the true discrete reachability map. Therefore, the computational overhead of online reachability determination is low. The simulation results show that the average computation time for a single target pose reachability query is approximately 60μs, while the average computation time for workspace boundary detection is approximately 1.18ms, with the maximum detection time remaining below 2.5ms. In the hardware experiments, the teleoperation controller runs at 20 Hz, the master device data acquisition frequency is 40 Hz, the robot joint state feedback frequency is 100 Hz, and the video feedback frequency is 24 Hz. These results indicate that the proposed method satisfies the real-time control requirements of the current experimental platform.

From the perspective of scalability, the proposed framework adopts a modular structure consisting of top-level safe velocity filtering, middle-layer task constraint processing, and bottom-level inverse kinematics optimization. Information is trans-mitted among different layers through desired velocities, reference trajectories, and task-space constraints. Thus, each module can be replaced or extended according to the specific robot platform and task requirements. For example, when the robot structure changes, the corresponding reachability map and reachability surrogate model can be reconstructed. When the task objects or contact conditions change, the middle-layer virtual object model and dynamic constraints can be adjusted. When the ro-bot degrees of freedom or the number of collision objects increases, the corresponding kinematic constraints or optimization objectives can be added to the bottom-level QP. It should be noted that the scalability of the proposed method is still affected by the robot degrees of freedom, the number of collision bounding volumes, the number of constraints, and the scale of the QP problem. For whole-body humanoid robots with higher degrees of freedom, hierarchical solving or parallel implementation strategies require further investigation.

From the perspective of noise robustness, the proposed framework provides a buffering capability against a certain level of input noise. First, the middle-layer MPC smooths the operator input and generates high-order continuous reference trajectories, thereby suppressing high-frequency jitter in the master input. Second, the top-level safe velocity filter processes user commands before they enter the task trajectory generation module. This prevents noisy velocity commands from causing sudden boundary violations or emergency stops when the robot approaches the workspace boundary.

Although the proposed MC-SCCF has been validated through simulations and hardware experiments, several limitations remain and require further study. First, the current experiments mainly focus on dual-arm upper-body manipulation of a human-oid robot with a fixed base. The framework has not yet been jointly validated with mo-bile base motion or whole-body motion control. Future work will extend this frame-work to mobile manipulation scenarios while considering base motion, whole-body balance maintenance, and dual-arm coordinated control. Second, the accuracy of the reachability surrogate model depends on the resolution and coverage of the offline reachability map. When the robot platform or kinematic configuration changes, the reachability map must be regenerated and the surrogate model must be retrained. Future work will explore reachability representations with better generalization capability, as well as efficient model transfer and adaptation strategies for different humanoid robot platforms. Third, the current task modeling does not explicitly consider communication or control delays during teleoperation. Since delays may affect operator intent mapping, safe velocity filtering, and task stability, future work will introduce delay-aware control strategies and further evaluate the robustness of the framework under different delay conditions.

Comments No. 6:

Please revise consistency of variables and subindexes along the document.

Response:

Thank you very much for this careful comment. We agree that the notation of some variables and subscripts in the original manuscript was not sufficiently consistent, especially in the connection between the safe velocity filter in Section 3 and the dynamic constraint processing method in Section 4. This may have caused ambiguity for readers. Following your suggestion, we have carefully checked and revised the relevant variables and subscripts throughout the manuscript to improve the consistency and readability of the notation.

Specifically, in Section 3,  denotes the actual position of the transported object in the task space,  denotes the desired velocity input by the operator, and  denotes the safe velocity output by the top-level safe velocity filter. Section 4 focuses on the processing of task-level dynamic constraints. To avoid confusion between the virtual reference object and the virtual offset object, we have removed the potentially ambiguous single-letter subscripts o and a in the revised manuscript. We also avoid using complicated subscript expressions related to the virtual offset object in Section 4. After the revision, Eq. (3) mainly represents the dynamic constraint that should be satisfied by the virtual reference object.

In addition, in Section 4.3, we replaced the previously ambiguous single-letter subscripts with clearer three-letter subscripts. For example, ​ is used to denote the desired position of the virtual reference object. Consistent with the notation in Section 3,  denotes the actual position of the virtual reference object,  denotes the operator-input velocity of the reference object, and  denotes the safe velocity of the reference object generated by the top-level safe velocity filter. The trajectory smoother in Section 4 is then used to achieve smooth tracking of ​ by ​.

We believe that these revisions make the meanings of the variables in different control layers clearer and improve the consistency of variables and subscripts throughout the manuscript.

Comments No. 7:

Some parameters are mentioned but not justified, a brief explanation would help the clarification.

Response:

Thank you very much for this valuable comment. We agree that the justification of some parameters in the original manuscript was not sufficiently clear, which may have affected the reader’s understanding of the algorithm implementation and experimental settings. Following your suggestion, we have added further explanations for the relevant parameters in the revised manuscript.

Specifically, in Section 3.3.1, we have added explanations for the parameters used in constructing the discrete reachability map, including the spatial grid resolution , the number of sampled rotation axes , and the number of sampled rotation angles . These parameters were selected to balance reachability determination accuracy, storage cost, and computational efficiency.

In addition, we have added explanations after Table 1 in Section 6.1 to clarify the meanings and selection considerations of the remaining algorithm parameters. These explanations further describe the roles of the parameters in safe velocity filtering, trajectory smoothing, and task constraint processing. For the hardware experiments, we have also added Table 5 in Section 7.3 to list and explain the main parameters used in the middle-layer controller in Section 7.

We believe that these additions make the parameter settings clearer and improve the completeness and reproducibility of the experimental description.

3.3.1. Reachability Determination

The discrete reachability map divides the space into three-dimensional voxels with an edge length of ​, and the centroid of each voxel represents its position. The selection of the grid resolution  requires a tradeoff between storage cost and determination accuracy. An excessively large  introduces notable discretization errors, leading to inaccurate boundary determination. In contrast, an excessively small  causes the number of voxels to increase cubically, which increases both the storage burden and the size of the training dataset. Considering the robot scale, task accuracy requirements, and storage cost,  is set to 0.05m.

The axis-angle representation is used to specify orientations. Specifically,  unit rotational axes uniformly distributed in space are defined, and  rotational angles uniformly distributed within  are specified. These axes and angles jointly constitute  orientations. At the position of each voxel, the inverse kinematics for all orientations are calculated sequentially. All poses are encoded, and the existence of their inverse kinematic solutions is recorded. Compared with uniform longitude-latitude sampling, spherical sampling ensures an approximately uniform distribution of directions on the sphere and avoids statistical bias caused by excessive sampling density near the polar regions. In this section,  and  are used, meaning that each voxel position covers 720 discrete orientations.

6. Simulations

Table 1. Algorithm Parameters.

The parameters  and  in the table are used in the middle-layer task constraint processing module. Here,  and  denote the base edge length and height of the virtual reference object and the virtual offset object, respectively, and  denotes the maximum radius of the tray. In this section, the virtual object is set as a rectangular prism with a base edge length of 5 cm and a height of 25 cm, corresponding to an aspect ratio of 5:1. This setting results in a high center of mass and a small support area, making the object prone to toppling. The friction coefficient  is set to 0.15 to simulate low-friction contact conditions. These settings impose stricter task dynamic constraints and are more likely to induce conflicts between the dynamic constraints and other constraints.  and  are the state error weight and input weight in the trajectory smoother, respectively, which balance trajectory tracking accuracy and trajectory smoothness. The parameters  and  are used in the top-level safe velocity filter, affecting the convergence rate of the boundary constraint, numerical tolerance, and upper limit on online search iterations, respectively. The relevant parameters are selected through preliminary simulation tests and are validated in hardware experiments, showing that they meet the combined requirements of real-time performance, constraint safety, and smooth velocity modulation.

7.1. Experimental Platform and Task Setup

The dimensions of the tray are 560 mm375 mm80 mm. 14 objects to be transported are randomly positioned on the tray. These objects comprise common items, including vases, water bottles, wooden blocks, and glassware, with heights ranging from 100 mm to 235 mm, as exhibited in Figure 8. According to the dimensions of the tray and the objects placed on it, the relevant parameters of the middle-layer controller are listed in Table 5. Because objects inevitably slide during motion, fiducial markers are attached to their tops, and a camera is employed to detect their sliding distances. The camera is fixed above the robot platform and is exclusively utilized to detect sliding distances for result analysis; the positions of the objects are practically under open-loop control.

Table 5. Parameters of the Middle-Layer Controller.

Comments No. 8:

Details of the FCNN like activation functions, training dataset set size and training procedure are limited.

Response:

Thank you very much for this valuable comment. We agree that the implementation details of the FCNN were not sufficiently described in the original manuscript, especially regarding the network architecture, activation functions, training dataset size, and training procedure. Following your suggestion, we have added the relevant descriptions in Section 3.3.1 of the revised manuscript.

Specifically, we have added details on the FCNN architecture, including the number of hidden layers, the number of neurons in each layer, the activation function, and the use of batch normalization. We have also supplemented the description of the training dataset construction, the number of valid samples, the training/validation split, and the training procedure, including the optimizer, learning rate setting, mini-batch size, maximum number of epochs, learning-rate decay strategy, regularization, gradient clipping, and early stopping.

We believe that these additions provide a clearer explanation of the training process of the reachability surrogate model and improve the understandability and reproducibility of the proposed method.

3.3.1. Reachability Determination

We employ a fully connected neural network (FCNN) to train the reachability surrogate model, which learns the nonlinear mapping  from the end-effector position  to the reachability index , where . The input is the three-dimensional end-effector position expressed in the corresponding shoulder base frame, and the output is the continuous reachability prediction at that position. In each shoulder coordinate frame, the original sampling space is discretized into  three-dimensional grid points, where , given that the manipulator length is 1m. Therefore, each single-arm reachability dataset contains 68921 sampled points.

Before training, the robot joints are set to the zero configuration. The end-effector positions in the left- and right-arm reachability maps are first transformed into their corresponding shoulder base frames. Invalid samples with a reachability index of zero are then removed, and only spatial points with valid inverse kinematic solutions are retained for training. Each sample consists of a three-dimensional end-effector position and the corresponding reachability index. The valid datasets contain 30757 samples for the left arm and 30753 samples for the right arm, which are randomly divided into training and validation sets at a ratio of 70% and 30%, respectively.

The FCNN contains three hidden fully connected layers with 256, 128, and 64 neurons, respectively. The Swish activation function is used in all hidden layers, and batch normalization layers are added after the first two hidden layers to improve training stability. The network is trained using the Adam optimizer with a maximum of 150 epochs, a mini-batch size of 512, and an initial learning rate of 0.001. A piecewise learning-rate decay strategy is adopted, in which the learning rate is reduced by a factor of 0.5 every 30 epochs. During training,  L2 regularization, gradient clipping, and validation-based early stopping are applied to reduce the risk of overfitting and improve the generalization capability of the model. After training, the surrogate model provides a continuous fit to the discrete reachability data, with  on the test set. This continuously differentiable model supports the subsequent reachability gradient calculation and waist configuration optimization. The fitting results are shown in Figure 3. In the illustrated cross-sections, the fitting error is less than 0.15.

Comments No. 9:

A brief paragraph discussing limitations would be very valuable to improve the paper.

Response:

Thank you very much for this valuable suggestion. We agree that adding a discussion of the limitations would help readers better understand the scope of applicability of the proposed framework and would further improve the completeness of the paper.

Following your suggestion, we have added a new “Discussion” section as Section 8 in the revised manuscript, where the limitations of the proposed method are discussed in more detail. Specifically, we have supplemented the discussion regarding the remaining limitations of the current method in terms of the accuracy of the reachability surrogate model, parameter dependence, adaptability to complex dynamic environments, and the scope of experimental validation. We also discuss the potential influence of these limitations and outline possible directions for future improvement.

We believe that the newly added discussion section makes the manuscript more objective and complete, and helps readers more clearly understand both the advantages and limitations of the proposed method in practical applications.

8. Discussions

The proposed Multiple-Constraint Safety-Critical Control Framework decouples offline computation from online control in its design, thereby reducing the computational burden during online control. For workspace boundary determination, both the discrete reachability map and the differentiable reachability surrogate model are constructed offline. During online operation, only the surrogate model is used for gradient-guided waist configuration search, followed by nearest-neighbor queries in the true discrete reachability map. Therefore, the computational overhead of online reachability determination is low. The simulation results show that the average computation time for a single target pose reachability query is approximately 60μs, while the average computation time for workspace boundary detection is approximately 1.18ms, with the maximum detection time remaining below 2.5ms. In the hardware experiments, the teleoperation controller runs at 20 Hz, the master device data acquisition frequency is 40 Hz, the robot joint state feedback frequency is 100 Hz, and the video feedback frequency is 24 Hz. These results indicate that the proposed method satisfies the real-time control requirements of the current experimental platform.

From the perspective of scalability, the proposed framework adopts a modular structure consisting of top-level safe velocity filtering, middle-layer task constraint processing, and bottom-level inverse kinematics optimization. Information is trans-mitted among different layers through desired velocities, reference trajectories, and task-space constraints. Thus, each module can be replaced or extended according to the specific robot platform and task requirements. For example, when the robot structure changes, the corresponding reachability map and reachability surrogate model can be reconstructed. When the task objects or contact conditions change, the middle-layer virtual object model and dynamic constraints can be adjusted. When the ro-bot degrees of freedom or the number of collision objects increases, the corresponding kinematic constraints or optimization objectives can be added to the bottom-level QP. It should be noted that the scalability of the proposed method is still affected by the robot degrees of freedom, the number of collision bounding volumes, the number of constraints, and the scale of the QP problem. For whole-body humanoid robots with higher degrees of freedom, hierarchical solving or parallel implementation strategies require further investigation.

From the perspective of noise robustness, the proposed framework provides a buffering capability against a certain level of input noise. First, the middle-layer MPC smooths the operator input and generates high-order continuous reference trajectories, thereby suppressing high-frequency jitter in the master input. Second, the top-level safe velocity filter processes user commands before they enter the task trajectory generation module. This prevents noisy velocity commands from causing sudden boundary violations or emergency stops when the robot approaches the workspace boundary.

Although the proposed MC-SCCF has been validated through simulations and hardware experiments, several limitations remain and require further study. First, the current experiments mainly focus on dual-arm upper-body manipulation of a human-oid robot with a fixed base. The framework has not yet been jointly validated with mo-bile base motion or whole-body motion control. Future work will extend this frame-work to mobile manipulation scenarios while considering base motion, whole-body balance maintenance, and dual-arm coordinated control. Second, the accuracy of the reachability surrogate model depends on the resolution and coverage of the offline reachability map. When the robot platform or kinematic configuration changes, the reachability map must be regenerated and the surrogate model must be retrained. Future work will explore reachability representations with better generalization capability, as well as efficient model transfer and adaptation strategies for different humanoid robot platforms. Third, the current task modeling does not explicitly consider communication or control delays during teleoperation. Since delays may affect operator intent mapping, safe velocity filtering, and task stability, future work will introduce delay-aware control strategies and further evaluate the robustness of the framework under different delay conditions.

Author Response File: Author Response.pdf

Reviewer 2 Report

Comments and Suggestions for Authors

The authors present a structured and effective framework for addressing safety‑critical multi‑constraint challenges in humanoid robot teleoperation for nonprehensile transportation tasks. The proposed hierarchical MC‑SCCF architecture is well motivated, technically sound, and validated through both simulations and hardware experiments.

In Table 1, the typographical error concerning the epsilon (ε) value should be corrected.
In Figure 10 and Figure 11, only the bar graph values are presented; however, the numerical values corresponding to the error bars are not specified and should be included for clarity.
Additionally, Section heading 7.3.3 should begin on a new page.

Author Response

Author's Reply to the Review Report (Reviewer 2)

Comments No. 1:

In Table 1, the typographical error concerning the epsilon (ε) value should be corrected.

Response:

Thank you very much for carefully pointing out the typographical error concerning the epsilon (ε) value in Table 1. We have rechecked the corresponding parameter value in Table 1 and corrected this error in the revised manuscript. In addition, to avoid similar issues, we further reviewed the parameter symbols, mathematical expressions, figure and table captions, table entries, and related descriptions throughout the manuscript to ensure consistency between the text and the reported values. The corresponding correction has been made in Table 1 of the revised manuscript.

Table 1. Algorithm Parameters.

Comments No. 2:

In Figure 10 and Figure 11, only the bar graph values are presented; however, the numerical values corresponding to the error bars are not specified and should be included for clarity.

Response:

Thank you very much for this valuable comment. We fully agree that presenting only the bar charts and error bars, without explicitly providing the corresponding numerical values, may reduce the clarity and completeness of the results.

Following your suggestion, we further revised the presentation of Figure 10 and Figure 11. Specifically, we redrew both figures and added the numerical information corresponding to the error bars in Figure 10, so that readers can more clearly understand the variability of the results. In addition, considering that directly placing too many numerical values in Figure 11 might affect the visual clarity of the figure, we added (or supplemented) a corresponding table in the revised manuscript to explicitly report the mean values and standard deviations of the subjective evaluation metrics shown in Figure 11. These revisions make the presentation of the results more complete, clearer, and easier for readers to interpret and compare.

We sincerely thank the reviewer for this careful suggestion. Your comment helped us further improve the rigor and readability of the manuscript in terms of result presentation. The corresponding revisions have been incorporated into Figure 10, Figure 11, and the related table in the revised manuscript. In the current revised version, Figure 10 presents the experimental results under the three control modes, Figure 11 presents the subjective feedback results, and Table 7 reports the means and standard deviations of the subjective evaluation metrics.

Figure 10. Experimental results under the three control modes.

Figure 11. Subjective feedback results of the subjects after using the three control modes. Results were compiled using a 7-point Likert scale, where higher scores represent better subjective experiences.

Table 7. Means and Standard Deviations of Subjective Evaluation Metrics under the Three Control Modes.

Comments No. 3:

Additionally, Section heading 7.3.3 should begin on a new page.

Response:

Thank you very much for carefully checking the formatting details of the manuscript and for pointing out this issue. Since the manuscript has been supplemented and revised during the revision process, the pagination has changed accordingly compared with the previous version. In response to your comment regarding the heading of Section 7.3.3, we carefully rechecked the page layout of this section in the revised manuscript and adjusted the relevant page break and heading position. This ensures that the subsection heading is properly arranged together with the subsequent text in the updated layout, and avoids situations where the heading is separated from its content, placed at the bottom of a page, or presented in an inconsistent format.

In addition, we conducted a systematic formatting check throughout the manuscript, including the placement of section headings, font style and size, line spacing, spacing before and after paragraphs, figure and table placement, table formatting, and figure/table captions. We have made the necessary corrections to ensure that the overall formatting of the revised manuscript is consistent with the author template provided by the journal. The corresponding revision has been reflected in Section 7.3.3 Control Performance Metrics and in the formatting adjustments throughout the manuscript.

Author Response File: Author Response.pdf

Reviewer 3 Report

Comments and Suggestions for Authors

Please refer to the uploaded word document.

Comments for author File: Comments.pdf

Author Response

Author's Reply to the Review Report (Reviewer 2)

Comments No. 1:

In Table 1, the typographical error concerning the epsilon (ε) value should be corrected.

Response:

Thank you very much for carefully pointing out the typographical error concerning the epsilon (ε) value in Table 1. We have rechecked the corresponding parameter value in Table 1 and corrected this error in the revised manuscript. In addition, to avoid similar issues, we further reviewed the parameter symbols, mathematical expressions, figure and table captions, table entries, and related descriptions throughout the manuscript to ensure consistency between the text and the reported values. The corresponding correction has been made in Table 1 of the revised manuscript.

Table 1. Algorithm Parameters.

Comments No. 2:

In Figure 10 and Figure 11, only the bar graph values are presented; however, the numerical values corresponding to the error bars are not specified and should be included for clarity.

Response:

Thank you very much for this valuable comment. We fully agree that presenting only the bar charts and error bars, without explicitly providing the corresponding numerical values, may reduce the clarity and completeness of the results.

Following your suggestion, we further revised the presentation of Figure 10 and Figure 11. Specifically, we redrew both figures and added the numerical information corresponding to the error bars in Figure 10, so that readers can more clearly understand the variability of the results. In addition, considering that directly placing too many numerical values in Figure 11 might affect the visual clarity of the figure, we added (or supplemented) a corresponding table in the revised manuscript to explicitly report the mean values and standard deviations of the subjective evaluation metrics shown in Figure 11. These revisions make the presentation of the results more complete, clearer, and easier for readers to interpret and compare.

We sincerely thank the reviewer for this careful suggestion. Your comment helped us further improve the rigor and readability of the manuscript in terms of result presentation. The corresponding revisions have been incorporated into Figure 10, Figure 11, and the related table in the revised manuscript. In the current revised version, Figure 10 presents the experimental results under the three control modes, Figure 11 presents the subjective feedback results, and Table 7 reports the means and standard deviations of the subjective evaluation metrics.

Figure 10. Experimental results under the three control modes.

Figure 11. Subjective feedback results of the subjects after using the three control modes. Results were compiled using a 7-point Likert scale, where higher scores represent better subjective experiences.

Table 7. Means and Standard Deviations of Subjective Evaluation Metrics under the Three Control Modes.

Comments No. 3:

Additionally, Section heading 7.3.3 should begin on a new page.

Response:

Thank you very much for carefully checking the formatting details of the manuscript and for pointing out this issue. Since the manuscript has been supplemented and revised during the revision process, the pagination has changed accordingly compared with the previous version. In response to your comment regarding the heading of Section 7.3.3, we carefully rechecked the page layout of this section in the revised manuscript and adjusted the relevant page break and heading position. This ensures that the subsection heading is properly arranged together with the subsequent text in the updated layout, and avoids situations where the heading is separated from its content, placed at the bottom of a page, or presented in an inconsistent format.

In addition, we conducted a systematic formatting check throughout the manuscript, including the placement of section headings, font style and size, line spacing, spacing before and after paragraphs, figure and table placement, table formatting, and figure/table captions. We have made the necessary corrections to ensure that the overall formatting of the revised manuscript is consistent with the author template provided by the journal. The corresponding revision has been reflected in Section 7.3.3 Control Performance Metrics and in the formatting adjustments throughout the manuscript.

Author Response File: Author Response.pdf