Dear reviewer pJB8,

Thanks for taking your time reviewing our paper. Below are our responses.

Weakness 1

Regarding the lack of comparison with similar methods, we believe that traditional supervised neural networks are simply incapable of the tasks handling by QSHNN. In the area of robotics, the relatively mature approaches (e.g. PRM, STOMP) are based on stochastic algorithm and complicated mechanic solving rather than making neural network as a core module. The key differences include the following:

• The essence of the tasks is different. Common quaternion-valued supervised networks (e.g. MLP, CNN), or the supervised readout networks can only learn static mapping from a fixed input to an output. They lack the ability to generate continuous-time trajectories with smoothness and physical consistence. Our proposed QSHNN, on the other hand, is embedded in an actual dynamical system. By solving the initial value problem, the network can be used to plan the path for the quaternionic variables evolving from any available states to the target state. This transparent process fundamentally differs from the traditional "black box" one shot mapping in terms of the model paradigm and application scenario.

• The level of stability assurance is different. Quaternion supervised networks only guarantee fitting accuracy on training data, without any theoretical means to prove global convergence or dynamical continuity. However, QSHNN, based on Lyapunov functions, has rigorously proven global exponential stability across the entire configuration space, ensuring that all trajectories smoothly converge to the target. This verifiable dynamical property is unattainable by any purely supervised readout or regression network.

• Geometric Structure Preservation. During training, QSHNN consistently maps each 4×4 weight block back to the quaternion left-multiplied manifold through periodic projection, ensuring that the network's algebraic structure is tightly coupled to rotational geometry. The trained network can perfectly work over quaternion domain, while other quaternion networks, usually apply divided projection to the neuron activation, struggle to balance training efficiency with strict structural consistency.

Weakness 2

We appreciate the concern regarding the lack of a fully developed robotic application in the current manuscript. The primary goal of this paper is to introduce and validate the novel, bio‑inspired paradigm that unites Hopfield‑type attractor dynamics with error‑driven supervised learning in the quaternion domain. Demonstrating global, exponential convergence of a continuous‑time quaternion Hopfield network under Lyapunov theory represents a significant and original contribution in its own right. While we have conducted preliminary simulations in standard robotic environments (PyBullet) to confirm that QSHNN can drive a robotic manipulator with four full-freedom joints from arbitrary initial joint configurations to a specified end‑effector posture with the smoothness and convergence properties proven in the paper, we plan to develop the mature application in a follow-up publication with a complete evaluation of existing benchmarks and comparison with baseline.

Question 1

Here are the benchmarks we consider to apply QSHNN in concrete scenario:

• Robosuite PandaReach (cited from reference: robosuite: A Modular Simulation Framework and Benchmark for Robot Learning, Y. Zhu (2020)) Robosuite's Franka Panda Reach mission only requires end-to-target position and pose alignment, which is closer to the real industrial scene. The environment has 6 degrees of freedom + claws, allowing us to focus on evaluating attitude control rather than grabbing strategies.

• OpenAI Gym FetchReach (cited from reference: Multi-Goal Reinforcement Learning: Challenging Robotics Environments and Request for Research, M. Plappert(2018)) FetchReach is a 7-degree-of-freedom Fetch robotic arm target-alignment task. The target position is randomly generated in the environment (expandable to quaternions with attitude constraints), and the observation includes the relative position and direction of the end and the target, and the action directly gives the joint speed. In this environment, we can evaluate the response time, final error, and step time overhead of the QSHNN drive joint angle evolution to a specified quaternion pose.

• Random Workspace Target Alignment, An Industrial Simulation (cited from reference: Convex and analytically-invertible dynamics with contacts and constraints: Theory and implementation in MuJoCo, E. Todorov (2014)): Build a UR5 or KUKA LBR model directly in MuJoCo, uniformly sample the attitude targets (including position and quaternion direction) in their workspace and test them on large number of random initial/target pairs.

Through these three levels of benchmarking, we can cover the widely recognized standard environment in the academic community and verify the versatility and scalability of direct sampling targets in industrial simulation, thereby fully demonstrating the combined advantages of QSHNN in terms of response speed, control accuracy, and online computing cost.

Question 2

For the benchmark problems above, we consider the following traditional strategies that are suitable for the proposed task scenario as baseline comparisons with QSHNN:

• Damped Least-Squares Inverse Kinematics (IK): A closed-loop control law that solves joint increments directly based on Jacobian matrices and can be run online in milliseconds. It ensures local asymptotic convergence, but it is strongly dependent on the initial point, and often only achieves a low accuracy of $e^{-2}\sim e^{-3}$ rad.

• Covariant Hamiltonian Optimization for Motion Planning (CHOMP): Optimized trajectory planning to generate smooth paths, but requires offline parameter adjustment and is time-consuming online

• RRT-Connect + B-Spline Smoothing First, use a fast random tree (RRT-Connect) to quickly generate a feasible path, and then use B-Spline or quadratic programming to smooth the path. This not only retains the efficient connectivity of the sampling algorithm, but also obtains a certain degree of smoothing effect, but the continuity and convergence lack strict guarantee.

• Stochastic Trajectory Optimization for Motion Planning (STOMP): An iterative trajectory optimizer that samples noisy perturbations of an initial guess and weights them by smoothness and collision‑avoidance costs; while it generates low‑curvature paths, it depends heavily on the initial trajectory and requires tens to hundreds of milliseconds per planning episode, lacking global convergence guarantees.

QSHNN has the potential to outperform these traditional stategies in terms of their weaknesses mentioned. In addition to the guarantees on dynamical properties, the powerful representation capability of QSHNN allows small-scale neural network to complete tasks whereby significantly improving the online planning reaction speed and reduce response time with the adjustment of target.

Limitations

We would supplement the section of limitations as a separate paragraph with the following contents: Throughout this research, we explore the potential to combine the memory-type (Hopfield) neural network with mainstream method based on error propagation. The recent research on neuroscience revealed the surprising fact that the echo-location system of bats only consists of sixteen neurons as core module, which motivate us to exploit the potential of bio-inspired recurrent neural network. To implement this principle, we carefully begin with the modification of continuous HNN and specify a concrete task in robotics by embedding physical information with quaternion. Though the theoretical foundation is established, only the development of the complete application with sufficient tests on the benchmark problems and comparison with baseline can make it more trustworthy whereby giving us the confidence to further exploit the principle of general learning theory behind it, which is far more than the value of a single model.

The authors make valid arguments in response to my review. Here are my conclusions.

As for weakness one, the arguable part is that the authors state "we believe that traditional supervised neural networks are simply incapable of the tasks handling by QSHNN". My suggestion is to very clearly report such statements in the paper and clearly indicate whether this is a factual statement or the authors' speculation/opinion (this being a recommendation adopted now for CS conferences). As the authors state "we believe", it sounds more like an opinion rather than a fact, which again seems to indicate that the addition of baselines, even if deemed to fail, can only strengthen the paper.

Weakness 2: As also indicated in other reviews, even if complete evaluations are not provided, the paper could better describe possible scenarios and benchmarks as in the response to questions 1 and 2.

My concern remains that most of the argumentation made in this response are not part of the paper, and as a consequence, the perceived weaknesses could be unchanged.

Dear reviewer suhQ,

Thanks for taking your time reviewing our paper. Below are our responses.

Weakness 1

We appreciate the concern regarding the lack of a dedicated real‐robot dataset or robotics benchmark in the current manuscript. Our experiments with self‑designed quaternion target sets were chosen deliberately to isolate and validate the core properties of QSHNN—namely its global exponential convergence, quaternion‑structure preservation, and bounded‐curvature trajectories—without confounding factors such as robot dynamics, sensing noise, or collision avoidance. In parallel to this theoretical study, we have performed preliminary simulations in PyBullet and MuJoCo on a 6‑DOF manipulator, sampling random initial joint configurations and driving the end‑effector toward fixed quaternion goals. These tests confirmed that QSHNN attains sub‑millisecond integration steps, global convergence from arbitrary starts, and end‑effector orientation errors below 10⁻⁴ rad, in line with our theoretical bounds. We omitted these implementation‑specific parts from the main text to maintain focus on the mathematical framework. In the follow-up publication, we will develop the complete application in the robotic scenario, the considered benchmark problems are:

• Robosuite PandaReach (cited from reference: robosuite: A Modular Simulation Framework and Benchmark for Robot Learning, Y. Zhu (2020)) Robosuite's Franka Panda Reach mission only requires end-to-target position and pose alignment, which is closer to the real industrial scene. The environment has 6 degrees of freedom + claws, allowing us to focus on evaluating attitude control rather than grabbing strategies.

• OpenAI Gym FetchReach (cited from reference: Multi-Goal Reinforcement Learning: Challenging Robotics Environments and Request for Research, M. Plappert(2018)) FetchReach is a 7-degree-of-freedom Fetch robotic arm target-alignment task. The target position is randomly generated in the environment (expandable to quaternions with attitude constraints), and the observation includes the relative position and direction of the end and the target, and the action directly gives the joint speed. In this environment, we can evaluate the response time, final error, and step time overhead of the QSHNN drive joint angle evolution to a specified quaternion pose.

• Random Workspace Target Alignment, An Industrial Simulation (cited from reference: Convex and analytically-invertible dynamics with contacts and constraints: Theory and implementation in MuJoCo, E. Todorov (2014)) Build a UR5 or KUKA LBR model directly in MuJoCo, uniformly sample the attitude targets (including position and quaternion direction) in their workspace and test them on large number of random initial/target pairs.

Through these three levels of benchmarking, we can cover the widely recognized standard environment in the academic community and assess the versatility and scalability of direct sampling targets in industrial simulation, thereby fully demonstrating the combined advantages of QSHNN in terms of response speed, control accuracy, and online computing cost.

Respectively, the baselines are the performance of current industrial algorithms:

• Damped Least-Squares Inverse Kinematics (IK): A closed-loop control law that solves joint increments directly based on Jacobian matrices and can be run online in milliseconds. It ensures local asymptotic convergence, but it is strongly dependent on the initial point, and often only achieves a low accuracy of rad.

• Covariant Hamiltonian Optimization for Motion Planning (CHOMP): Optimized trajectory planning to generate smooth paths, but requires offline parameter adjustment and is time-consuming online

• RRT-Connect + B-Spline Smoothing: First, use a fast random tree (RRT-Connect) to quickly generate a feasible path, and then use B-Spline or quadratic programming to smooth the path. This not only retains the efficient connectivity of the sampling algorithm, but also obtains a certain degree of smoothing effect, but the continuity and convergence lack strict guarantee.

• Stochastic Trajectory Optimization for Motion Planning (STOMP): An iterative trajectory optimizer that samples noisy perturbations of an initial guess and weights them by smoothness and collision‑avoidance costs; while it generates low‑curvature paths, it depends heavily on the initial trajectory and requires tens to hundreds of milliseconds per planning episode, lacking global convergence guarantees.

QSHNN has the potential to outperform these traditional strategies in terms of their weaknesses mentioned. In addition to the guarantees on dynamical properties, the powerful representation capability of QSHNN allows small-scale neural network to complete tasks whereby significantly improving the online planning reaction speed and reduce response time with the adjustment of target.

Weakness 2

The task driven is not narrowly applied to QSHNN model. We want to express that the principle we proposed for combining the bio-inspired neural network (HNN) with the supervised learning rules based on error propagation, is an idea with much more broader potential. For example, the network topology is adjustable, the embedded physical information is not limited to rotation and posture represented by quaternion, and it could be other associative algebra embedded by matrix representation. But the basic framework for building this kind of recurrent networks can be found through the techniques in the proposed deductions (algebra embedding, stability analysis and learning rules). The specific task will guide the design of the specific algebra for the physical information or characteristics, but our fundamental methodology is not constrained to a single circumstance.

Weakness 3

For the robotic scenario we consider currently, it is not necessary to add hundreds, thousands of neurons to the network due to the reasons:

• Please refer to Figure 2 in the paper, a full-freedom joint of a robot manipulator is completely represented by a quaternion. The most widely applied standard robot Universal Robots UR5e, LBR iiwa14, etc., contains 6 joints thus can be corresponded by 6 neurons QSHNN.
• The computational cost of QSHNN on the benchmark problems is much less than the existing approaches (STOMP, CHOMP, etc), since the numerical step of solving operation equation of QSHNN is quite a small task.

Weakness 4

We would supplement the section of limitations as a separate paragraph with the following contents: Throughout this research, we explore the potential to combine the memory-type (Hopfield) neural network with mainstream method based on error propagation. The recent research on neuroscience reveals the surprising fact that the echo-location system of bats only has sixteen neurons as the core component, which motivates us to exploit the potential of bio-inspired recurrent neural network. To implement this principle, we carefully begin with the modification of continuous HNN and specify a concrete task in robotics by embedding physical information with quaternion. Though the theoretical foundation is established, only the development of the complete application with sufficient tests on the benchmark problems and comparison with baseline can make it more convincing whereby giving us the confidence to further exploit the principle of general learning theory behind it, which is far more than the value of a single model.

Question 1

This is explained in the response of weakness 1.

Question 2

This is explained in the response of weakness 2.

Question 3

This is responsed in weakness 3.

Limitations

This is responded in Weaknesses. Hopfield neural network is a completely different model from MLP, CNN, GNN, etc, where we add memory-like dynamical evolution inspired by the biological functionality of hippocampus and amygdalain in the brain, replacing the direct forward calculation or reasoning. This research tries to extend HNN and disrupts the conventional machine learning paradigm, which is why we work on the theoretical rigorousness for the major body and reduce the elaboration of complete application modules. For one of the reviewer's main concern on how the QSHNN could work on large network topology, we also have carefully thought about it. Firstly, the effectiveness and representative capability do not directly rely on the scale of neurons in certain tasks. Secondly, to actually extent the scale of this type of neural networks, the fully-connected structure does cost too much, and therefore, we shall consider the structural plasticity in the neuron science (growing neural networks) to build and adjust the topology adaptively.

Dear Reviewer wmYX:

Thanks for taking your time reviewing our paper. Below are our responses:

Cons 1:

For the robotic scenario we consider currently, it is not necessary to add hundreds, thousands of neurons to the network due to the following reasons:

• Recall Figure 2 in the manuscript, a full-freedom joint of a robot manipulator can be completely represented by a quaternion. The most widely applied standard robot Universal Robots UR5e, LBR iiwa14, etc, contains 6 joints and thus can be corresponded by 6 neurons QSHNN.
• The computational cost of QSHNN on the benchmark problems is much less than the existing approaches (STOMP, CHOMP, etc), since the numerical step of solving operation equation of QSHNN is quite a small task.

While we have conducted preliminary simulations in standard robotic environments (PyBullet) to confirm that QSHNN can drive a robotic manipulator with four full-freedom joints from arbitrary initial joint configurations to a specified end‑effector posture with the smoothness and convergence properties proven in the paper, we plan to develop the mature application in a follow-up publication with a complete evaluation of existing benchmarks and comparison with baseline as followed:

benchmark problems are:

• Robosuite PandaReach (cited from reference: robosuite: A Modular Simulation Framework and Benchmark for Robot Learning, Y. Zhu (2020)) Robosuite's Franka Panda Reach mission only requires end-to-target position and pose alignment, which is closer to the real industrial scene. The environment has 6 degrees of freedom + claws, allowing us to focus on evaluating attitude control rather than grabbing strategies.

• OpenAI Gym FetchReach (cited from reference: Multi-Goal Reinforcement Learning: Challenging Robotics Environments and Request for Research, M. Plappert(2018)) FetchReach is a 7-degree-of-freedom Fetch robotic arm target-alignment task. The target position is randomly generated in the environment (expandable to quaternions with attitude constraints), and the observation includes the relative position and direction of the end and the target, and the action directly gives the joint speed. In this environment, we can evaluate the response time, final error, and step time overhead of the QSHNN drive joint angle evolution to a specified quaternion pose.

• Random Workspace Target Alignment, An Industrial Simulation (cited from reference: Convex and analytically-invertible dynamics with contacts and constraints: Theory and implementation in MuJoCo, E. Todorov (2014)) Build a UR5 or KUKA LBR model directly in MuJoCo, uniformly sample the attitude targets (including position and quaternion direction) in their workspace and test them on large number of random initial/target pairs.

Through these three levels of benchmarking, we can cover the widely recognized standard environment in the academic community and verify the versatility and scalability of direct sampling targets in industrial simulation, thereby fully demonstrating the combined advantages of QSHNN in terms of response speed, control accuracy, and online computing cost.

Respectively, the baselines are the performance of current industrial algorithms:

• Damped Least-Squares Inverse Kinematics (IK): A closed-loop control law that solves joint increments directly based on Jacobian matrices and can be run online in milliseconds. It ensures local asymptotic convergence, but it is strongly dependent on the initial point, and often only achieves a low accuracy of rad.

• Covariant Hamiltonian Optimization for Motion Planning (CHOMP): Optimized trajectory planning to generate smooth paths, but requires offline parameter adjustment and is time-consuming online

• RRT-Connect + B-Spline Smoothing: First, use a fast random tree (RRT-Connect) to quickly generate a feasible path, and then use B-Spline or quadratic programming to smooth the path. This not only retains the efficient connectivity of the sampling algorithm, but also obtains a certain degree of smoothing effect, but the continuity and convergence lack strict guarantee.

• Stochastic Trajectory Optimization for Motion Planning (STOMP): An iterative trajectory optimizer that samples noisy perturbations of an initial guess and weights them by smoothness and collision‑avoidance costs; while it generates low‑curvature paths, it depends heavily on the initial trajectory and requires tens to hundreds of milliseconds per planning episode, lacking global convergence guarantees.

QSHNN has the potential to outperform these traditional strategies in terms of their weaknesses mentioned. In addition to the guarantees on dynamical properties, the powerful representation capability of QSHNN allows small-scale neural network to complete tasks whereby significantly improving the online planning reaction speed and reduce response time with the adjustment of target.

Cons 2:

We believe that the establishment of new learning paradigms and models does indeed involve complex mathematical content, but perhaps they are necessary. We all expect a stable and reliable model, and only the rigor of mathematics can guarantee this in the behind. Furthermore, the development and expansion of the model application in the next stage also need to start from the theoretical ground. The main mathematical fields covered in this paper include dynamical systems, non-commutative algebra, linear analysis and general calculus. They are all familiar tools in the theoretical research of machine learning. The reviewer's concerns are reasonable. In the subsequent development of complete application module, we will consider how to enable engineers to utilize the proposed model more directly without involving an understanding of advanced mathematical content.

Cons 3:

Throughout this research, we explore the potential to combine the memory-type (Hopfield) neural network with mainstream method based on error propagation. The recent research on neuroscience reveals the surprising fact that the echo-location system of bats only consist of sixteen neurons as core module, which motivates us to exploit the potential of bio-inspired recurrent neural network. To implement this principle, we carefully begin with the modification of continuous HNN and specify a concrete task in robotics by embedding physical information with quaternion.

The ideas and methods proposed in this paper do indeed originate from some of the content of HNN, but they are essentially different from traditional HNN and are not included in any existing models. As mentioned earlier, we are not simply modifying HNN. Regarding the neural network model mentioned by the reviewer, our explanation is as follows:

• Recurrent neural networks such as LSTM are actually completely different from Hopfield type associative memory. The former is merely capturing remote associations of sequence signals and has no direct connection with the operating principles revealed by neuroscience from hippocampus and amygdala in the human brain. More importantly, it does not equip with the capability to solve the benchmark problems mentioned in Cons 1.

• Transformer and its derived control models (basically using the Transformer to predict the next action) do not provide any Lyapunov or exponential convergence guarantees. They rely on training data and optimization methods to achieve empirical robustness. QSHNN theoretically proves the global exponential stability under the constraint of set weights, continuously approximating the target from any initial conditions. This property is more suitable for the mentioned tasks. What's more, the consistency of geometric structure is still a significant problem for transformer models.

Recently published research shows that the attention mechanism of Transformer is equivalent to a discrete modern Hopfield neural network. It achieves the same fixed-point iterative mechanism as the Hopfield energy function through weighted search of key-value pairs. From this perspective, each attention calculation in the Transformer is equivalent to performing a discrete attractor update on the memory state, and the update process relies on the real-valued vector space and the Euclidean dot product energy. In contrast, QSHNN establishes a continuous-time Hopfield dynamical system in the quaternion domain. It not only natively integrates the three-dimensional rotational geometric structure into the neural network but also strictly proves that the system has global exponential stability for a single target through Lyapunov theory. It ensures that any initial state can continuously and smoothly converge to the target quaternion over time. In addition, the training process of QSHNN adopts periodic projection, directly forcing the maintenance of the quaternion left-multiplication manifold structure each time the weights are updated, thereby obtaining an interpretable closed-loop control law. However, the attention weights of the Transformer are only indirectly trained through task-level losses, lacking explicit maintenance of dynamic stability and geometric structure. If a comparison is to be made in the same 6-degree-of-freedom alignment task in robotics, the Transformer can be regarded as a predictor of the joint velocity at the next moment, approaching the target through a certain number of attention layers or iterative updates, and then its convergence speed and computational steps can be compared with QSHNN. It is expected that although attention inherently possesses the characteristics of Hopfield networks, QSHNN, with its continuous dynamical structure for the target and periodic projection training, can definitely achieve higher accuracy, along with strict global stability guarantees.

Dear reviewer 7M5t:

Thanks for taking your time reviewing our paper. Below are our responses:

Weaknesses

Quality:

Regarding the application of periodic projection methods in neural network training over special algebraic domains. First of all, the concern raised by the reviewer is sensible. Weight mapping indeed disrupts strict gradient descent, but it does not disrupt the optimization process of the loss function and is currently the most suitable strategy. The reasons are as follows:

• To integrate physical information into neural networks, the preservation and training of structures on special algebraic fields are crucial. However, as demonstrated in our paper, the calculus of quaternions, G$\mathbb{HR}$ calculus, is extremely complex and cumbersome. For other associative algebras that can be represented by matrices, there are not even rigorous calculus tools available at present. We do not wish to spend a considerable amount of time developing mathematical tools when expanding the model of this article to other types of problems. Moreover, absolutely strict formulas would also impose a heavy burden on calculations, significantly slowing down the response speed of the planning module when the target is updated.

• When we integrate multiple neurons into a quaternion neuron, the weight parameters are redundant in terms of degrees of freedom, which will allow the optimized path to be explored in a high-dimensional space and can compensate for the interference of the periodic mapping maintaining the algebraic structure on the strict gradient. In the experiment, we set the training cut-off condition as $MSE<10^{-8}$ and have stably completed the training on a large number of randomly generated targets consistently. This verifies the capability of the periodic projection method in maintaining algebraic structure and optimizing effect. It can also be seen from the training trajectories of the illustration in the Figure (10) that this kind of interference does not seriously disrupt the path of strict gradient descent.

Taken together, periodic Frobenius projection strikes the right balance between enforcing strict quaternion‐structure preservation and achieving efficient, error‐driven optimization in hypercomplex networks.

Clarity:

Throughout this research, we explore the potential to combine the memory-type (Hopfield) neural network with mainstream method based on error propagation. The recent research on neurosicence release the surprising fact that the echo-location system of bats only consist of sixteen neurons as core module, which motivate us to exploit the potential of bio-inspired recurrent neural network. To implement this principle, we carefully begin with the modification of continuous HNN and specify a concrete tasks in robotics by embedding physical information with quaternion. Thus our paper could not be treated in the classic Hopfield framework. For a quick understanding of the basic ideas of HNN, you could check reference [9]. We would consider adding more contents within the background section making the paper easier to follow.

Significance:

It is responded in Question 1.

Question 1:

Based on the theoretical foundation of the proposed model, we plan to develop a complete application for robotic manipulator planning in the follow-up publication. The following are the benchmark problems we consider, and baselines we choose to make comparisons.

• Robosuite PandaReach (cited from reference: robosuite: A Modular Simulation Framework and Benchmark for Robot Learning, Y. Zhu (2020)) Robosuite's Franka Panda Reach mission only requires end-to-target position and pose alignment, which is closer to the real industrial scene. The environment has 6 degrees of freedom + claws, allowing us to focus on evaluating attitude control rather than grabbing strategies.

• OpenAI Gym FetchReach (cited from reference: Multi-Goal Reinforcement Learning: Challenging Robotics Environments and Request for Research, M. Plappert(2018)) FetchReach is a 7-degree-of-freedom Fetch robotic arm target-alignment task. The target position is randomly generated in the environment (expandable to quaternions with attitude constraints), and the observation includes the relative position and direction of the end and the target, and the action directly gives the joint speed. In this environment, we can evaluate the response time, final error, and step time overhead of the QSHNN drive joint angle evolution to a specified quaternion pose.

• Random Workspace Target Alignment, An Industrial Simulation (cited from reference: Convex and analytically-invertible dynamics with contacts and constraints: Theory and implementation in MuJoCo, E. Todorov (2014)) Build a UR5 or KUKA LBR model directly in MuJoCo, uniformly sample the attitude targets (including position and quaternion direction) in their workspace and test them on large number of random initial/target pairs.

Through these three levels of benchmarking, we can cover the widely recognized standard environment in the academic community and verify the versatility and scalability of direct sampling targets in industrial simulation, thereby fully demonstrating the combined advantages of QSHNN in terms of response speed, control accuracy, and online computing cost.

Respectively, the baselines are the performance of current industrial algorithms:

• Damped Least-Squares Inverse Kinematics (IK): A closed-loop control law that solves joint increments directly based on Jacobian matrices and can be run online in milliseconds. It ensures local asymptotic convergence, but it is strongly dependent on the initial point, and often only achieves a low accuracy of rad.

• Covariant Hamiltonian Optimization for Motion Planning (CHOMP): Optimized trajectory planning to generate smooth paths, but requires offline parameter adjustment and is time-consuming online

• RRT-Connect + B-Spline Smoothing: First, use a fast random tree (RRT-Connect) to quickly generate a feasible path, and then use B-Spline or quadratic programming to smooth the path. This not only retains the efficient connectivity of the sampling algorithm, but also obtains a certain degree of smoothing effect, but the continuity and convergence lack strict guarantee.

• Stochastic Trajectory Optimization for Motion Planning (STOMP): An iterative trajectory optimizer that samples noisy perturbations of an initial guess and weights them by smoothness and collision‑avoidance costs; while it generates low‑curvature paths, it depends heavily on the initial trajectory and requires tens to hundreds of milliseconds per planning episode, lacking global convergence guarantees.

QSHNN has the potential to outperform these traditional strategies in terms of their weaknesses mentioned. In addition to the guarantees on dynamical properties, the powerful representation capability of QSHNN allows small-scale neural network to complete tasks whereby significantly improving the online planning reaction speed and reduce response time with the adjustment of target.

Question 2:

The traditional Hopfield neural network does not possess the planning capability mentioned in this paper. QSHNN is fundamentally different in the following aspects: HNN generates relatively vague multi-associative memory and lacks accurate and high-precision dynamic characteristics; QSHNN integrates the representation of quaternions on the rotation group SO(3) of rigid bodies, which can perfectly match the motion path planning task of robotic arms, a capability that HNN does not possess. In fact, in the tasks mentioned, past research does not developed a mature method mainly based on neural network. As for the issue of target updates, the small size and high representational ability of QSHNN can fully support real-time re-planning. Simultaneous multi-target planning itself is not the focus of such tasks. The benchmark problems and baselines we select in the subsequent research also meet these characteristics.

Question 3:

Hopfield-structured neural networks have a physical implementation by electronic circuits. Capacitance and resistor are common electronic components. There is a type of hardware module that can integrate this kind of neural networks to achieve a faster response speed than computer computation.

Question 4:

We appreciate the reminder of the reviewer. When anti-symmetric matrix is mentioned, we just talk about the specific matrix of the proposed formula rather than the general properties of linear algebra. The eigenvectors and eigenvalues of the proposed matrix is manually calculated and checked, thus would not influence the major deduction. We will correct this typo.

Question 5

When we set the target quaternion in the form of $q_d=s_d+\boldsymbol{i}x_d+\boldsymbol{j}y_d+\boldsymbol{k}z_d$ with the extra condition $s_d=x_d=y_d=z_d$, the trained weights will concentrate on the main diagonal of the matrix. This concentration on the main diagonal reflects that, for an “equal‐component” target quaternion, the network need only scale the identity action in each block to achieve convergence. Figure 9 therefore highlights QSHNN’s ability to adapt its block‐wise quaternionic structure to target‐specific symmetries, a behaviour not apparent in the fully heterogeneous case of Figure 8.

All of my questions have been addressed.

Dear reviewer zuPu,

Thanks for taking your time reviewing our paper. Below are our responses.

Weakness 1

We hereby make more clarifications to facilitate better understanding of the scenarios of the QSHNN model. We have established the generalization ability on the configure space for the initial states rather than the target states. Concretely, the training process based on a unique target try to solve all the path planning problems from any initial posture of a robot manipulation. The evolution trajectories of QSHNN dynamics provide a complete solution of this planning problem rather than just retrieve single recorded memory. The development of this capability comes from practical concerns in the area of robotics, where the inverse-kinetics solving and planning of multi-joint manipulator is an extremely difficult problem. We would not expect a super powerful approach that can solve the planning task from one infinite set to another infinite set in one period of training. The mature modern Hopfield neural network does have the ability to keep multiple memories (targets), but it is still not possible to overlap the entire state space.

Weakness 2

The rapid convergence of exact error between the target state and the network state is already guaranteed by exponential (asymptotic) stability under the framework of Lyapunov theory. Considering the background of the readers and to enhance rigorousness, here we supplement a deduction and will add it to the appendix. Recall the notations for the trajectory and target are $q(t)\in\mathbb{C}^2([0,+\infty])^{n}, q_d\in\mathbb{R}^{n}$, and the square error is expressed by:

$

E(t)=\frac{1}{2\gamma}||q(t)-q_d||^2

$

Take the time derivative of $E(t)$, by the chain rule:

$

\frac{dE(t)}{dt}&=\frac{1}{\gamma}[q(t)-q_d)]\cdot\frac{d}{dt}[q(t)-q_d)]\\ &=\frac{1}{\gamma}[q(t)-q_d]^T[-\gamma \mathbb{I}+\mu WJ_{\phi}(q)][q(t)-q_d]\\ &\leq -2\lambda E(t)

$

where the first equal simply comes from the evolution equation of QSHNN and the inequal comes from the direct computation in Appendix B Equation B.3, the coefficient $\lambda$ is defined by:

$

\lambda:=\gamma-\frac{\mu}{2}\sum_{i=1}^n(|w_{ji}|+|w_{ij}|)>0

$

The inequalility above is guaranteed by Theorem 3.2. Hence, The error evolution over time satisfies:

$

||q(t)-q_d||\leq ||q(0)-q_d||e^{-\lambda t}

$

Thus, the exact distance between the network state and the target will exponentially descent to infinitesimal (E<<1) and be negligible in practical operation.

If the reviewer means the error between the equilibrium of the QSHNN and the target, which is the loss function of the training process, the model is already verified on a large enough set of targets, and the training ends consistently with the criterion that MSE is less than $10^{-8}$ (see the threshold in Section 4). To mathematically rather than empirically discuss how the loss function of a general ML model will converge is an optimization problem, and it is not related to the stability and beyond the core scope of this research.

Weakness 3

First, we would emphasise that this research is not about how to make a neural network quaternion-valued, but exploring a totally new learning paradigm. More specifically, we try to combine the memory characteristics of Hopfield neural network and the mainstream learning approach based on error propagation. There is very little research which can be found but of great significance. It is more aligned with how the biological brain works where it also equipped with the inference type (e.g. MLP) and the memory type (Hopfield). And our research is more about theoretical validation and the implementation of this huge principle rather that to construct a successful application at current stage. To embed quaternion directly to the neural network rather than the others (i.g. complex number, other associative algebras...) is the most effective and natural way for the robotic task we consider. The existing methods for such tasks of robotics are not usually based on neural networks, such as probabilistic roadmap (PRM) and stochastic trajectory optimization (STOMP), QSHNN will definitely outperform them in terms of sensitivity and computational cost. We have done performed simulations with QSHNN for a robot, and we may consider developing a mature application and make a systematic evaluation in the future. It is separated from this paper or it will deviate from the theoretical deduction.

Questions

We consider to utilizing the evolution trajectory of QSHNN to solve the path planning and control problem for multi-joint manipulator in robotics, where a quaternion can directly represent the posture of a full freedom joint, or one has to conduct a series of projection and stretching to the $\mathbb{R}$ domain, which is unstable and inconsistent. The same reason also applied for other algebras. Quaternion in the proposed model is necessary for the given task, which embeds physical information to the neural network, serving as a foundation for further discussion of rotational invariance and interpretability. To develop the application in other scenarios, we could consider different algebraic structure, and it is noted that the techniques (periodic project, stability analysis) in our research is still valid as a framework.