Process Study on Removal of Recast Layer and Orifice Repairing Based on Electrochemical Finishing Machining

Revision as of 10:48, 25 August 2025

Abstract

For open quantum systems, a short-time evolution is usually well described by the effective non-Hermitian Hamiltonians, while long-time dynamics requires the Lindblad master equation, in which the Liouvillian superoperators characterize the time evolution. In this paper, we constructed an open system by adding suitable gain and loss operators to the Chern insulator to investigate the time evolution of quantum states at long times by numerical simulations. Finally, we also propose a topolectrical circuits to realize the dissipative system for experimental observation. It is found that the opening and closing of the Liouvillian gap leads to different damping behaviours of the system and that the presence of non-Hermitian skin effects leads to a phenomenon of chiral damping with sharp wavefronts. Our study deepens the understanding of quantum dynamics of dissipative system.

Keywords: Open quantum system, chiral damping, topolectrical circuits

Jilian Zhong Department of Physics, Jiangsu University, Zhenjiang 212013, People’s Republic of China zhongjilian0532@163.com

Xiaoyue Li Department of Physics, Jiangsu University, Zhenjiang 212013, People’s Republic of China

DOI: 10.23967/j.rimni.2024.05.008

1. Introduction

With the laboratory advances in modulating dissipation and quantum coherence,the theory of open and nonequilibrium systems has received renewed attention [1,2]. Non-Hermitian Hamiltonians have been used to describe a large number of non-conservative systems, such as classical waves with gain and loss [3-8], solids with finite quasiparticles lifetimes [9-11], and open quantum systems [12-14]. The unique features of non-Hermitian systems have been recognized in a variety of physical settings, in particular the non-Hermitian skin effect (NHSE) [15,16], where the eigenstates of the system are exponentially localized on the boundary. In recent years, the impact of NHSE has been extensively studied [17-29].

NHSE was also found in open quantum systems [30]. For open quantum systems, the non-Hermitian effective Hamiltonian describes the time evolution of the wavefunction under post-selection conditions, while the time evolution of the density matrix (without post-selection) is driven by the Liouvillian superoperator in the master equation [2,31-33]. It has been found that the Liouvillian superoperator can also exhibit non-Hermitian skin effects and that such effects can significantly affect the dynamical behaviour of the system at long times [30,34-43]. In a large class of open quantum systems, the quantum state in the long time limit converges to the steady state by algebraic damping under periodic boundary conditions and exponential damping under open boundary conditions [30].

In recent years, it has been discovered that topolectrical circuits can be used as platform to simulate the lattice systems, thus enabling the study of topological states in topolectrical circuits and gradually developing the field of topological circuitry [44-46]. Some of the early experiments and theories were extensively studied in Hermitian systems [45,47]. Since the phenomena of non-Hermitian systems are more rich than that of Hermitian systems, increasing attentions are contributed into the non-Hermitian physics, and some interesting phenomena have also been realized by topolectrical circuits [48-52].

Previous studies on open quantum dynamics and topolectrical circuits have mainly focused on one-dimensional non-Hermitian models, and relatively few studies on higher-dimensional non-Hermitian models. In this paper, we consider a two-dimensional open quantum system based on Chern insulators. Following the method developed in Song et al. [30], we study the dynamics of this system in terms of the damping matrix derived from the Liouvillian superoperator, and give a model of topolectrical circuit realization of the damping matrix based on Kirchhoff’s theory. It is found that due to the NHSE of the damping matrix, the long-time dynamics of the system under open boundary conditions is significantly different from that under periodic boundary conditions.

Our paper is organized as follows: in section 2, we briefly review the general framework on how to convert Liouvillian operators with linear jumps to non-Hermitian damping matrix. In sections 3 and 4, we compute and numerically simulate the long-time evolution of the model. In section 5, we give the circuit model of the non-Hermitian damping matrix . Finally, we conclude in section 6.

2. General formalism of damping matrix

An open quantum system undergoing Markovian damping satisfies the Lindblad master equation

(1)

where

is the density matrix of the system,

is the Hamiltonian that represents unitary evolution of the system, and

are Lindblad dissipation operators describing the quantum jumps induced by the coupling to the environment. The above equation can be abbreviated as

, where

is called the Liouvillian superoperator. By regarding the density matrix

as a vector that consists of matrix elements

,

is represented as a matrix whose elements are given by [53]

(2)

These representations enable one to treat the Lindblad equation as a linear equation. In other words, the dynamics of the system can be understood in terms of the eigenvalue problem of the Liouvillian matrix:

The Hamiltonian and dissipators can be expressed in terms of 2n Majorana fermions [54]

(3)

where

are Majorana fermions satisfying

. The matrix

is chosen to be an antisymmetric matrix,

. Defining

,

, we have

. Under the third quantization [54,55], the Liouvillian superoperator is expressed as a quadratic form of the 2n complex fermions (4n Majorana fermions)

(4)

where

, and

are third quantized complex fermions. Through the above expression, we can obtain the Liouvillian eigenspectrum [54,55]

(5)

with

, where

is the eigenspectrum of

. Here

contains valuable information of the full density-matrix dynamics, and it can be easily obtained from the damping matrix

with

[36]. Rewriting

as

, where

are real matrices, we have

.

can be further written as

{\begin{aligned}Z=&{\frac {1}{4}}(h_{r}\otimes {\sigma }_{y}+ih_{i}\otimes 1)+i{\frac {1}{4}}(M_{r}^{T}\otimes 1-iM_{i}^{T}\otimes {\sigma }_{y})\\=&{\frac {1}{4}}(h_{r}+M_{i}^{T})\otimes {\sigma }_{y}+i{\frac {1}{4}}(h_{i}+M_{r}^{T})\otimes 1\end{aligned}}

(6)

Therefore,

{\begin{array}{c}\det(4iZ-\lambda E)=\det \left({\begin{array}{cc}-(h_{i}+M_{r}^{T})-\lambda &h_{r}+M_{i}^{T}\\-(h_{r}+M_{i}^{T})&-(h_{i}+M_{r}^{T})-\lambda \end{array}}\right)\\=\det(X-\lambda E)\det(X^{\ast }-\lambda E)\end{array}}

(7)

The eigenvalue of

are the union of the eigenvalues of

and

, which gives the Liouvillian eigenspectrum.

Then we outline the general form of the Lindblad damping matrix in open quantum systems [30]. We consider tight-binding models whose Hamiltonian can generally be written as $H=\sum {}_{ij}h_{ij}c_{i}^{\dagger }c_{j}$ , where $c_{i}^{\dagger },c_{i}$ are the creation and annihilation operators on lattice site $i$ , and $h_{ij}=h_{ij}^{\ast }$ is the hopping amplitude between the lattice points of the system ( $i\not =j$ ) or onsite potential ( $i=j$ ). It is convenient to define the single-particle correlation function ${\Delta }_{ij}(t)=Tr[c_{i}^{\dagger }c_{j}\rho (t)]$ to observe the time evolution of the density matrix. Each cell is coupled to the environment through the gain jump operator $L_{\mu }^{g}=\sum {}_{i}D_{\mu i}^{g}c_{i}^{\dagger }$ and loss jump operator $L_{\mu }^{l}=\sum {}_{i}D_{\mu j}^{l}c_{i}$ . Substituting the Lindblad quantum master equation into the time evolution of the single-particle correlation function, we can obtained

(8)

where

is the damping matrix with

and

. The steady state correlation

, to which the long-time evolution of any initial state converges, is determined by

or

. Focusing on the deviation towards the steady state

, whose time evolution is

, we can integrate it with Eq. (1) to obtain

(9)

Therefore, the dynamical behaviour of the system can be characterized by the damping matrix.

3. Model

In this paper, we consider the Chern insulator model with the Hamiltonian in momentum space as

(10)

where

. Let each unit cell contain a single loss and gain dissipator,

{\begin{aligned}L_{x}^{l}=&{\frac {\sqrt {2\gamma }}{2}}\left(e^{-\displaystyle {\frac {\pi }{4}}i}c_{xA}+e^{\displaystyle {\frac {\pi }{4}}i}c_{xB}\right)\\L_{x}^{g}=&{\frac {\sqrt {2\gamma }}{2}}\left(e^{\displaystyle {\frac {\pi }{4}}i}c_{xA}^{\dagger }+e^{-\displaystyle {\frac {\pi }{4}}i}c_{xB}\right),\end{aligned}}

(11)

where

denotes the lattice site,

refer to the sublattice. The Fourier transformation of

is

. The gain and loss dissipators are intra-cell, so these

matrices are independent of

,

. Then, the damping matrix in momentum space is

(12)

It can be written in the form of left and right eigenvectors,

(13)

where

. It is worth noting that our

and

satisfy

, guaranteeing that

is a steady state solution, where

,

is the system size, and

are the size in

direction, respectively. We assume that the initial state of the system is the completely filled state, i.e.,

is an identity matrix. Therefore, Eq.(9) can be re-expressed as

{\begin{aligned}{\tilde {\Delta }}(\mathrm {t} )&={\frac {1}{2}}\displaystyle \sum _{n,n^{'},l}\exp[(\lambda _{n}+\lambda _{n^{'}}^{*})t]{\widetilde {u_{R}}}(i,n){\widetilde {u_{L}}}(l,n){\widetilde {u_{L}^{*}}}(l,n^{'}){\widetilde {u_{R}^{*}}}(j,n^{'})\\&=\displaystyle {\frac {1}{2}}\displaystyle \sum _{n,n^{'}}\displaystyle {\frac {\displaystyle \sum _{l}\exp[(\lambda _{n}+\lambda _{n^{'}}^{*})t]u_{R}(i,n)u_{L}(l,n)u_{L}^{*}(l,n^{'})u_{R}^{*}(j,n^{'})}{\displaystyle \sum _{k}u_{R}(k,n)u_{L}(k,n)\displaystyle \sum _{m}u_{L}^{*}(m,n^{'})u_{R}^{*}(m,n^{'})}}\end{aligned}}

(14)

According to the dissipative property,

always holds. The Liouvillian gap

plays a decisive role in long-time dynamics. The opening gap (

) implies an exponential rate of convergence to the steady state, while the closing gap (

) implies algebraic convergence [34].

4. Chiral damping

For simplicity, the parameters of our model are taken as

,

. We first study the dynamical behaviour under the periodic boundary conditions. Diagonalizing

, we obtain the energy spectrum as shown in Figure 1. It is found that the Liouvillian gap vanishes at

, while the gap opens at

. So we expect the damping rate to be algebraic and exponential in each case, respectively.


Figure 1. Eigenvalues of the damping matrix X. Blue: periodic boundary; Red: open boundary . The Liouvillian gap under periodic boundary condition vanishes for (a) and (b), while it is nonzero for (c) and (d). Under open boundary condition, the Liouvillian gap is nonzero in all four cases. This significant difference between open and periodic boundary comes from the NHSE of ${\textstyle X}$ . (a) $\lambda =0.1,m=1.5$ . (b) $\lambda =0.5,m=1.5$ . (c) $\lambda =0.1,m=2.5$ . (d) $\lambda =0.5,$ $m=2.5$

To verify this, we define the site-averaged fermion number deviation from the steady state

R(t)={\sqrt {{\frac {1}{N_{X}N_{Y}}}{\sum _{x}R_{x}(t)}^{2}}}={\sqrt {{\frac {1}{N_{X}N_{Y}}}{\sum _{x}\left({\frac {n_{x}(t)-n_{x}(t-{\delta }_{t})}{{\delta }_{t}}}\right)}^{2}}}

, where

, and

. The numerical results are shown in Figure 2. As anticipated, it is observed that the damping of

is algebraic for cases black and red lines with

, while exponential for blue and green lines with

under the periodic boundary condition.


Figure 2. Damping of site-averaged fermion number towards the steady state under periodic boundary condition with size $L=30\times 30$ . ${\textstyle m=1.5}$ (black and red) exhibits a slow algebraic damping, while ${\textstyle m=2.5}$ (blue and green) is an exponential damping. The initial state is the completely filled state

Next we turn to the open boundary conditions. Since the damping matrix

has NHSE, its energy spectrum is no longer that of the periodic boundary conditions. At this point all the energy spectrums have a non-zero energy gap (red part of Figure 1), therefore, we expect an exponential long-time damping of

. The numerical simulation in Figure 3 confirms this exponential behaviour with

having a period of algebraic damping before entering into the exponential damping. The time of the algebraic damping increases with the size

(Figure 3(a)). To better understand this feature, we plot the damping in several unit cells in the same x dimension (

), as shown in Figure 3(b). It can be seen that the left end () enters the exponential damping immediately, and the other sites enter the exponential damping in turn according to their different distances to the left end.Due to a process of algebraic damping that occurs before entering the exponential stage,there is a "damping wavefront" from left (

) to right (

). This phenomenon is known as "chiral damping".


Figure 3. (a) Site-averaged particle number damping under periodic boundary conditions (solid line) and open boundary conditions (dashed line) for several sizes ${\textstyle L}$ . The long-time damping of ${\textstyle R(t)}$ follows a power law under periodic boundary condition, while the damping follows an exponential law after an initial power law stage under open boundary condition. (b) Particle number damping on several sites. The system size is ${\textstyle 30\times 30}$ , and the left end ( $ix=1,iy=1$ ) enters the exponential phase from the beginning, followed by the other sites in turn. For (a) and (b), the initial state is completely filled state, $m=1.5,\lambda =0.1$

The phenomenon of chiral damping can be observed more intuitively as shown in Fig. 4(a) where the colour shades indicate the value of

. Under the periodic boundary condition, the time evolution follows a slow power law while under the open boundary condition, a wavefront moving to the upper right is observed. This can be intuitively linked to the phenomenon that all eigenstates of are localized in the upper right corner, which arises from the non-Hermitian skin effect of the damping matrix . If the matrix does not have NHSE under the open boundary condition, the fermion number of the system should have a similar behaviour of damping under different boundary conditions. Therefore, the non-Hermitian skin effect plays an important role in open quantum systems and significantly affects the dynamical behaviour of open quantum systems.


Figure 4. Evolution of ${\textstyle R(t)}$ at each lattice site under open boundary conditions (a) and periodic boundary conditions (b)

5. Experiment realized

Next we give the scheme of topolectrical circuits to simulate the damping matrix. Based on the similarity between the Kirchhoff equation and the Schrödinger equation, it is possible to simulate the Hamiltonian of the system using different circuit components, and the different parameters in the Hamiltonian can be adjusted independently by various components. The circuit Laplacian corresponding to the Hamiltonian can be written as

(15)

where and

are diagonal matrices containing the total conductance from each node to the ground and to the rest of the circuit, respectively.

is the adjacency matrix of conductances [44].


Figure 5. Structure of topolectrical circuit to realize damping matrix under periodic boundary conditions. (a) Connection relations between the nodes. The blue solid line box containing two “sublattice” nodes A (red) and B (blue) simulates a unit cell of ${\textstyle X}$ . The black (grey) solid line indicates the coupling between nodes in the ${\textstyle x(y)}$ -direction. (b) Circuit element structure is detailed for the green dashed framed rectangle in (a). (c) Internal circuit diagram of the INIC element, consisting of an operational amplifier and impedances . The impedance $Z$ is the target element, and different conductance in different directions of $V_{i,r}$ can be achieved by connecting the INIC in series. satisfies $Z_{+}=Z_{-}$ . (d) Grounding module of the nodes. The resistances $R_{A,}R_{B}$ and capacitance ${\textstyle C}$ are used to simulate the onsite potential, and inductance L allows the Laplacian eigenvalue spectrum to be shifted uniformly as desired

Considering the periodic boundary conditions first, the topolectrical circuit for realizing the damping matrix X is illustrated in Figure 5. Figure 5 depicts the schematic diagram of the overall circuit structure, which gives the connection relationship between the nodes. Figure 5 shows the detailed circuit component of the unit which is the green dashed box in Figure 5(a). The blue box in Figure 5(a) represents a unit cell in the system, and the two nodes inside it correspond to sublattices A (red) and B (blue). The circuit connections in the x and y directions are distinguished by black and gray. From Figure 5(b) we can obtain the matrices C and D in Eq. (9), so that

{\begin{array}{c}D-C=-i\omega \left({\begin{array}{cc}-i\displaystyle {\frac {2}{\omega R_{2}}}\cos k_{x}-i\displaystyle {\frac {2}{\omega R_{4}}}\cos k_{y}-C_{1}-i\displaystyle {\frac {1}{\omega R_{0}}}&i2C_{2}\sin k_{x}+\displaystyle {\frac {2}{\omega R_{6}}}\sin k_{y}-i\displaystyle {\frac {1}{\omega R_{1}}}+C_{1}\\i2C_{2}\sin k_{x}+\displaystyle {\frac {2}{\omega R_{5}}}\sin k_{y}+i\displaystyle {\frac {1}{\omega R_{1}}}+C_{1}&-i\displaystyle {\frac {2}{\omega R_{3}}}\cos k_{x}-i\displaystyle {\frac {2}{\omega R_{7}}}\cos k_{y}-C_{1}+i{\frac {1}{\omega R_{0}}}\end{array}}\right),\\{\mbox{    }}\end{array}}

(16)

with

,

, ,

,

, ,

Comparing it with the damping matrix, we need to add grounding elements to match the onsite potential. The grounding elements of nodes A and B are shown in Figure 5(d), where the resistors $R_{A,}R_{B}$ and capacitors $C$ simulate the lattice potential, and $R_{A,}R_{B}$ satisfies $R_{A}=-R_{B}$ . So the diagonal matrix ${\textstyle W}$ is

(17)

From Eq. (10) we get the conductance matrix of the circuit of Figure 5(a) at

frequency

{\begin{array}{c}J(\omega )=-i\omega \left({\begin{array}{cc}-i\displaystyle {\frac {2}{\omega R_{2}}}\cos k_{x}-i\displaystyle {\frac {2}{\omega R_{4}}}\cos k_{y}-(C+C_{1})-i\left(\displaystyle {\frac {1}{\omega R_{0}}}+\displaystyle {\frac {1}{\omega R_{A}}}\right)&i2C_{2}\sin k_{x}+\displaystyle {\frac {2}{\omega R_{6}}}\sin k_{y}-i\displaystyle {\frac {1}{\omega R_{1}}}+C_{1}\\i2C_{2}\sin k_{x}+\displaystyle {\frac {2}{\omega R_{5}}}\sin k_{y}+i\displaystyle {\frac {1}{\omega R_{1}}}+C_{1}&-i\displaystyle {\frac {2}{\omega R_{3}}}\cos k_{x}-i\displaystyle {\frac {2}{\omega R_{7}}}\cos k_{y}-(C+C_{1})+i\left(\displaystyle {\frac {1}{\omega R_{0}}}+\displaystyle {\frac {1}{\omega R_{A}}}\right)\end{array}}\right)+\displaystyle {\frac {1}{i\omega L}}\epsilon \\=-i\omega J_{P}+\displaystyle {\frac {1}{i\omega L}}\epsilon ,\end{array}}

(18)

where

,

. Comparing this Laplacian matrix with the damping matrix, the mapping relationship can be established by

.


Figure 6. Negative impedance module [22]. (a) A single-port circuit to ground. The input impedance is $Z_{g}=-Z$ . (b) Free-port circuit. Its input impedance at both ends is $Z_{ij}=Z_{ji}=-Z$ . The markings on the ideal amplifier indicate the output voltage versus the input voltage

Notice that the circuit requires a negative component,which is implemented as shown in Figure 6. Figures 6(a) and (b) show the equivalent negative impedance modules for a single port to ground and a free two-terminal port, respectively. They achieve the equivalent negative impedance through an amplifier. According to Kirchhoff's law, the input impedance of the single-port circuit to ground (Figure 6(a)) can be obtained as

(19)

The input impedance at both ends of the free port circuit (Figure 6(b)) are

{\begin{array}{c}Z_{ij}=\displaystyle {\frac {V_{i}-V_{j}}{I_{i}}}=\displaystyle {\frac {V_{i}-V_{j}}{[V_{i}-2(V_{i}-V_{j})]/Z}}=-Z,\\Z_{ji}=\displaystyle {\frac {V_{j}-V_{i}}{I_{j}}}=\displaystyle {\frac {V_{j}-V_{i}}{[V_{j}-2(V_{j}-V_{i})]/Z}}=-Z.\end{array}}

(20)

That is

.

Under the open boundary condition, the hopping amplitude of the cells located at the boundary weakens, leading to fewer branches connected to the boundary nodes in the circuit model, as shown in Figure 7(a). Figure 7(a) gives the connection relationship between the nodes of the circuit under the open boundary condition, and the circuit nodes can be classified into body nodes (in the black dashed box), edge nodes (in yellow) and corner nodes (in green). Changes in the branch circuit of the nodes at the boundary will cause variations of the matrices ${\textstyle C}$ and ${\textstyle D}$ . The matrix ${\textstyle C}$ corresponds to the hopping amplitude between the lattice points, which is allowed to change. Whereas the change of D is not desired due to the same onsite potential under different boundary condition.

Therefore, we need to design specific grounding elements to eliminate the effects of variations in

. Owing to the asymmetry of the coupling strengths under periodic boundary condition, the types of the edge and corner nodes are different for each of the four orientations, so there are a total of 16 different grounding modules, as shown in Figure 7(b). The additional grounding elements keep the diagonal matrix D+W unchanged, i.e., the onsite potential is unchanged, which achieves the mapping of the circuit Laplacian in Figure 7 to the damping matrix

under the open boundary condition.


Figure 7. Schematic diagram of the circuit of the damping matrix ${\textstyle X}$ under open boundary conditions. (a) Schematic diagram of the connection relations among the nodes. The black, yellow and green dashed boxes correspond to the body, edge and corner nodes, respectively. The circuit connections of the body node are the same as those of the periodic boundary, while the edge and corner nodes require additional grounding elements to regulate the onsite potential. (b) Grounding modules for edge and corner nodes. The grounding elements for the edge and corner nodes are different for each of the four orientations, where the negative impedance elements can be realized by Figure 6(a). Note that in addition to these grounding elements, all nodes need to be connected to the elements in Figure 5(d)

6. Conclusion

In summary, we study the dynamical properties of a two-dimensional open system. The open quantum system is constructed by introducing appropriate gain and loss to the Chern insulator, and then using the damping matrix derived from the Liouvillian superoperator explore its long-time evolution. It is found that the site-averaged fermion number deviation from the steady state under periodic boundary conditions shows a slow algebraic damping when the energy gap closes and an exponential damping when the energy gap opens. Under open boundary conditions, due to the non-Hermitian skin effect of the damping matrix, the system exhibits the phenomenon of chiral damping that the fermion number at each site undergoes a period of algebraic damping before entering an exponential damping, and the transition time that is proportional to the distance from that site to the boundary. Finally, we map the damping matrix in terms of the circuit Laplacian to give a model diagram of the topolectrical circuit implementation of the system.

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Revision as of 10:48, 25 August 2025

Abstract

1. Introduction

2. General formalism of damping matrix

3. Model

4. Chiral damping

5. Experiment realized

6. Conclusion

References

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 <!-- metadata commented in wiki content
-<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-<big>'''Information release right of scientific research institutions in health emergencies: Taking China as the research object'''</big></div>
-<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
-Shuai Li<sup>a</sup>''<sup>*</sup>''</div>
-''<sup>a </sup>Law School, Beijing Foreign Studies University, Beijing, 100089, People’s Republic of China''
+<span id='_GoBack'></span>
-<span id='OLE_LINK28'></span>''<sup>*</sup>Corresponding address: [mailto:Norbertlau2019@163.com Norbertlau2019@163.com]''
--->==Abstract==
-The integration of autonomous robotic systems is pivotal for advancing agricultural mechanization. A critical challenge impeding their widespread adoption is the achievement of reliable, self-sufficient navigation, particularly in environments where conventional positioning systems are compromised. While Global Navigation Satellite Systems (GNSS), often fused with auxiliary sensors, represent a primary solution for outdoor robot guidance, their susceptibility to signal occlusion necessitates alternative, stable methodologies for consistent operation. Addressing this limitation, this paper presents a novel, model-based navigation algorithm engineered specifically for orchard robots operating under prolonged GNSS denial. The core methodology leverages a deterministic kinematic model, deliberately neglecting higher-order dynamic effects justified by the inherently low operational speeds mandated in precision agricultural settings. This model directly processes commanded trajectory coordinates and real-time vehicle state estimates derived solely from incremental wheel encoders and a steering angle sensor. Within the MATLAB/Simulink environment, this transformation is implemented to generate precise longitudinal velocity and angular steering rate commands necessary for path tracking. Empirical validation was conducted through rigorous simulation and field trials employing a non-holonomic mobile robot platform in representative outdoor conditions. Performance evaluation confirmed the robot’s capability for satisfactory autonomous navigation. Quantitatively, the normalized root mean square error (NRMSE) for lateral path deviation during turning maneuvers ranged between 0.2 and 0.4, while straight-line travel exhibited minimal steering offset, typically within ±0.05 degrees. Furthermore, the correlation coefficient between the model’s predicted steering output and the actual commanded input consistently approached 0.99. Collectively, the results demonstrate that the proposed sensor-minimized, model-driven approach provides a viable and realistic foundation for achieving resilient vehicle navigation in structurally complex agricultural domains where GNSS reliability cannot be assured.
-'''Keywords''': Autonomous Navigation; Agricultural Robotics; GNSS-Denied Environments; Kinematic Modeling; Path Tracking Control
+==Title Page==
-==1. Introduction==
+<div class="center" style="width: auto; margin-left: auto; margin-right: auto;">
+<big>'''Fragmentation to the Holistic: A Study of the Inner Logic of Smart Governance of Public Cultural Space'''</big></div>
-The deployment of self-governing robotic platforms within agriculture has accelerated, driven by the need for novel vehicle solutions that elevate farm output, operational efficiency, and the capacity to execute diverse tasks within inherently variable agricultural settings. This trend offers a promising mitigation strategy for the growing scarcity of agricultural labor. Fundamental to the realization of such autonomous mobile systems is the capability for self-directed navigation. While the conceptual foundation for agricultural robotics was established as early as the mid-1980s [1], practical implementation has surged significantly in recent times. This acceleration is largely attributable to concurrent breakthroughs in computational capabilities, sensor technologies [2], and the advent of powerful, affordable processors. Progress in electronics and the commercialization of cost-effective sensors have rendered the development of economically viable robotic platforms for farm environments increasingly feasible. Nevertheless, the aspiration for fully autonomous robotic or vehicular movement remains partially unrealized within contemporary robotics, constrained by persistent technical hurdles. Outdoor navigation for agricultural robots presents particular difficulties, stemming from the challenge of obtaining precise environmental perception amidst fluctuating weather patterns, diverse terrain, and dynamic vegetation. Consequently, developing robust sensing and control strategies capable of accommodating these environmental characteristics is paramount. Supporting this, research by Bac et al. specifically investigated the detrimental impact of environmental variability on the operational efficacy of a harvesting robot [3]. Achieving a precise interpretation of environmental features and deploying resilient technologies for navigation and environmental detection under demanding conditions is crucial for striking an optimal balance between system cost and technological sophistication [4].
-Affordability is a critical consideration in agricultural technology development, given the primary end-users are farmers operating within constrained financial frameworks compared to other vehicle markets. System complexity inherently drives cost escalation, making the specific operational environment a vital factor in design philosophy. Orchards, characterized by their semi-structured spatial organization and defined biological architecture, present a highly conducive environment for the initial large-scale adoption of robotics and automation in farming [5]. This suitability is further amplified by a pressing concern: the diminishing availability of skilled seasonal labor poses a substantial risk to orchard sustainability, thereby establishing orchard automation as a critical priority [5].
+-->==Abstract==
+For open quantum systems, a short-time evolution is usually well described by the effective non-Hermitian Hamiltonians, while long-time dynamics requires the Lindblad master equation, in which the Liouvillian superoperators characterize the time evolution. In this paper, we constructed an open system by adding suitable gain and loss operators to the Chern insulator to investigate the time evolution of quantum states at long times by numerical simulations. Finally, we also propose a topolectrical circuits to realize the dissipative system for experimental observation. It is found that the opening and closing of the Liouvillian gap leads to different damping behaviours of the system and that the presence of non-Hermitian skin effects leads to a phenomenon of chiral damping with sharp wavefronts. Our study deepens the understanding of quantum dynamics of dissipative system.
-Ongoing research and development efforts signal positive growth indicators for the agricultural robotics sector and the vital transition of these technologies from laboratory prototypes to practical field deployment, irrespective of current manufacturing volumes. Indeed, the pace of innovation in autonomous navigation specifically within agricultural robotics parallels trends observed in the broader automotive market, though the absolute scale of development may currently differ.
+'''Keywords''': Open quantum system, chiral damping, topolectrical circuits
-Accurate localization constitutes a fundamental pillar for autonomous navigation in mobile robots, demanding sophisticated sensor integration and algorithmic processing. It remains a highly active area demanding continued advancement. Wang et al. explored the utilization of machine vision for robotic positioning and pathfinding [6]. Blok et al. demonstrated a combined Particle Filter and Kalman Filter approach for probabilistic localization, employing a 2D LIDAR scanner to enable in-row navigation for orchard robots [7]. Their empirical findings indicated superior performance of the Particle Filter model over the Kalman Filter in this specific context. The critical influence of the environment on localization efficacy is well-recognized; Schwarting et al. detailed the principal localization challenges encountered by terrestrial robots operating in diverse settings [8]. They offered a comprehensive analysis of existing methodologies tackling these challenges, their inherent limitations, and prospective future directions. Bechar and Vigneault provided a parallel overview of design, development, and operational considerations for agricultural robots, including their constraints, emphasizing the necessity for sensor fusion to achieve adequate localization and environmental awareness, alongside the development of adaptive path planning and navigation algorithms for variable conditions [9]. Contemporary research frequently focuses on enhancing navigation reliability through the fusion of Global Positioning System (GPS) data with complementary sensor modalities. For instance, Winterhalter et al. proposed an approach combining GNSS-referenced maps with GNSS signals for in-field localization and autonomous traversal along crop rows [10]. Thomasson et al. documented progress in automatic guidance and steering control, highlighting commercially available systems primarily reliant on Global Navigation Satellite Systems (GNSS) [11]. However, these commercial solutions are predominantly tailored for tractors and often necessitate human intervention when encountering unexpected field or navigation anomalies. Alternative configurations include the use of dual RTK-GPS receivers for navigating 4-wheel-steering agricultural rovers, or localization relying on laser scanners combined with wheel and steering encoders, albeit without integrated obstacle avoidance [12,13]. Zaidner and Shapiro presented a model-based state estimation approach for a vineyard robot utilizing sensor fusion. Recognizing the complexity of real-world testing [14], Linz et al. developed a 3D simulation environment using Gazebo for preliminary virtual validation of robotic navigation algorithms employing image-based sensors, laser scanners, and GPS, prior to outdoor deployment [15].
+Jilian Zhong  Department of Physics, Jiangsu University, Zhenjiang 212013, People’s Republic of China zhongjilian0532@163.com
-As evidenced by the literature, agricultural machinery and automated guidance systems extensively leverage GNSS for navigation and positioning. However, a significant limitation of GNSS-centric approaches is their inability to dynamically perceive and respond to the immediate environment [16]. Furthermore, sole reliance on GPS proves unreliable within orchards due to signal degradation and multipath interference caused by dense tree canopies during in-row operation [17]. Standard GPS offers limited positional accuracy (typically 1-2 meters), insufficient for precise autonomous navigation. While centimeter-level accuracy solutions (e.g., RTK-GNSS) exist, their high cost presents a barrier. The economic viability of systems is further compromised when integrating other expensive sensors.
+Xiaoyue Li  Department of Physics, Jiangsu University, Zhenjiang 212013, People’s Republic of China
+DOI: 10.23967/j.rimni.2024.05.008
+==1. Introduction==
+With the laboratory advances in modulating dissipation and quantum coherence,the theory of open and nonequilibrium systems has received renewed attention [1,2]. Non-Hermitian Hamiltonians have been used to describe a large number of non-conservative systems, such as classical waves with gain and loss [3-8], solids with finite quasiparticles lifetimes [9-11], and open quantum systems [12-14]. The unique features of non-Hermitian systems have been recognized in a variety of physical settings, in particular the non-Hermitian skin effect (NHSE) [15,16], where the eigenstates of the system are exponentially localized on the boundary. In recent years, the impact of NHSE has been extensively studied [17-29].
-Beyond localization, achieving full autonomy necessitates integrated environmental mapping and motion planning modules. Understanding the seamless interaction between these sub-systems is essential for the effective verification of overall control software. The inherent complexity of agricultural robotic vehicles, arising from diverse subsystems and demanding operational requirements, suggests that a model-based design (MBD) methodology offers a promising framework for current development efforts. While MBD is well-established and continually evolving within the automotive industry, particularly for embedded systems, its application in agricultural contexts presents distinct modeling and simulation challenges. Therefore, this paper details the development of a navigation control system for an orchard mobile robot utilizing a model-based design paradigm. The core objective is to create a robot capable of autonomous navigation in GNSS-restricted zones, employing minimal sensor interactions, computationally efficient algorithms, and requiring only minor modifications to its base platform configuration. The system leverages a localization strategy predicated on the provision of a pre-existing environmental map.
+NHSE was also found in open quantum systems [30]. For open quantum systems, the non-Hermitian effective Hamiltonian describes the time evolution of the wavefunction under post-selection conditions, while the time evolution of the density matrix (without post-selection) is driven by the Liouvillian superoperator in the master equation [2,31-33]. It has been found that the Liouvillian superoperator can also exhibit non-Hermitian skin effects and that such effects can significantly affect the dynamical behaviour of the system at long times [30,34-43]. In a large class of open quantum systems, the quantum state in the long time limit converges to the steady state by algebraic damping under periodic boundary conditions and exponential damping under open boundary conditions [30].
-==2. System modelling and simulation==
+In recent years, it has been discovered that topolectrical circuits can be used as platform to simulate the lattice systems, thus enabling the study of topological states in topolectrical circuits and gradually developing the field of topological circuitry [44-46]. Some of the early experiments and theories were extensively studied in Hermitian systems [45,47]. Since the phenomena of non-Hermitian systems are more rich than that of Hermitian systems, increasing attentions are contributed into the non-Hermitian physics, and some interesting phenomena have also been realized by topolectrical circuits [48-52].
-The robotic system depicted in '''Figure 1''' is configured as a four-wheel steering (4WS) vehicle, conceptually akin to a conventional automobile. Its operation is constrained to a two-dimensional (2D) planar domain, wherein idealized rolling motion is presumed in the absence of skidding. The robot’s state is formally defined within a configuration space <math>C</math> described by:
+Previous studies on open quantum dynamics and topolectrical circuits have mainly focused on one-dimensional non-Hermitian models, and relatively few studies on higher-dimensional non-Hermitian models. In this paper, we consider a two-dimensional open quantum system based on Chern insulators. Following the method developed in Song et al. [30], we study the dynamics of this system in terms of the damping matrix derived from the Liouvillian superoperator, and give a model of topolectrical circuit realization of the damping matrix based on Kirchhoff’s theory. It is found that due to the NHSE of the damping matrix, the long-time dynamics of the system under open boundary conditions is significantly different from that under periodic boundary conditions.
-{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;"
+Our paper is organized as follows: in section 2, we briefly review the general framework on how to convert Liouvillian operators with linear jumps to non-Hermitian damping matrix. In sections 3 and 4, we compute and numerically simulate the long-time evolution of the model. In section 5, we give the circuit model of the non-Hermitian damping matrix . Finally, we conclude in section 6.
+==2. General formalism of damping matrix==
+An open quantum system undergoing Markovian damping satisfies the Lindblad master equation
+{| class="formulaSCP" style="width: 100%; text-align: center;"
 |-
 |
-{| style="margin:auto;width: 100%;"
+{| style="text-align: center; margin:auto;"
 |-
-| <math>C = (x,y,\psi ,\delta )</math>
+|   <math>\frac{d\rho }{dt}=-i[H,\rho ]+\sum \left(2L_{\mu }\rho L_{\mu }^{\dagger }-\right. </math><math>\left. \lbrace L_{\mu }^{\dagger }L_{\mu },\rho \rbrace \right),</math>
 |}
-|  style="width: 5px;text-align: right;white-space: nowrap;"|(1)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (1)
+|}where  <math>\rho </math> is the density matrix of the system,  <math>H</math> is the Hamiltonian that represents unitary evolution of the system, and  <math>L_{\mu }</math> are Lindblad dissipation operators describing the quantum jumps induced by the coupling to the environment. The above equation can be abbreviated as  <math>\frac{d\rho }{dt}=L\rho </math>, where  <math>L</math> is called the Liouvillian superoperator. By regarding the density matrix  <math>\rho </math> as a vector that consists of matrix elements  <math>{\rho }_{i,j}</math>,  <math>L</math> is represented as a matrix whose elements are given by [53]
+{| class="formulaSCP" style="width: 100%; text-align: center;"
+|-
+|
+{| style="text-align: center; margin:auto;"
+|-
+| <math>L_{ij,kl}=\sum_{\mu }2L_{\mu ;i,k}L_{\mu ;l,j}^{\dagger }-i{\left(H-i\sum_{\mu }L_{\mu }^{\dagger }L_{\mu }\right)}_{i,k}{\delta }_{l,j}+i{\left(H+i\sum_{\mu }L_{\mu }^{\dagger }L_{\mu }\right)}_{l,j}{\delta }_{ik.}</math>
 |}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (2)
-<div id='img-1'></div>
+|}These representations enable one to treat the Lindblad equation as a linear equation. In other words, the dynamics of the system can be understood in terms of the eigenvalue problem of the Liouvillian matrix: <math>L{\rho }^{\left(i\right)}={\lambda }_i{\rho }^{\left(i\right)}.</math> The Hamiltonian and dissipators can be expressed in terms of 2n Majorana fermions [54]
-{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
+{| class="formulaSCP" style="width: 100%; text-align: center;"
-|-style="background:white;"
-|style="text-align: center;padding:10px;"| [[Image:Draft_Wu_280690876-image3.png|330px]]
 |-
-| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 1'''. Schematic representation of the kinematic configuration employed in the four-wheel steering robotic platform
+|
+{| style="text-align: center; margin:auto;"
+|-
+| <math>H=\sum_{i,j=1}^{2n}{\gamma }_iH_{{}_{ij}}^M{\gamma }_j, \quad L_{\mu }=\sum_{i=1}^{2n}l_{{}_{\mu ,i}}^M{\gamma }_i</math>
 |}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (3)
-In this study, a kinematic representation of the 4WS robotic platform has been adopted. Considering the regime of low-velocity traversal, and under the assumption that wheel velocity vectors align with the instantaneous direction of travel while tire forces remain negligible, a non-dynamic kinematic framework has been employed. The resultant formulation is expressed as:
+|}where  <math>{\gamma }_i</math> are Majorana fermions satisfying  <math>\lbrace {\gamma }_i,{\gamma }_j\rbrace =2{\delta }_{ij}</math>. The matrix  <math>H^M</math> is chosen to be an antisymmetric matrix, <math>{\left(H^M\right)}^T=-H^M</math>. Defining  <math>M_{ij}={\sum }_{\mu }l_{\mu ,i}^\ast l_{\mu ,j}^{}</math>,  <math>M_{{}_{ij}}^M={\sum }_{\mu }{\left(l_{\mu ,i}^M\right)}^\ast l_{\mu ,j}^M</math>, we have  <math>M_{{}_{}}^M=\frac{1}{4}M\otimes (1+{\sigma }_y)</math>. Under the third quantization [54,55], the Liouvillian superoperator is expressed as a quadratic form of the 2n complex fermions (4n Majorana fermions)
+{| class="formulaSCP" style="width: 100%; text-align: center;"
-{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;"
 |-
 |
-{| style="margin:auto;width: 100%;"
+{| style="text-align: center; margin:auto;"
 |-
-| <math>\begin{array}{*{20}{c}}
+| <math>L\mbox{=}\frac{2}{i}\left(\begin{array}{cc}
-{}&{\dot x = vcos\left( {\psi  + \beta } \right)}\\
+c^{\dagger } & c
-{}&{\dot y = vsin\left( {\psi  + \beta } \right)}\\
+\end{array}\right)\left(\begin{array}{cc}
-{}&{\dot \psi  = \frac{{vcos\beta }}{{{l_f} + {l_r}}}(tan{\delta _f} - tan{\delta _r})}\\
+-Z^T & Y\\
-{}&{\beta  = arctan\left( {\frac{{{l_f}tan{\delta _r} + {l_r}tan{\delta _f}}}{{{l_f} + {l_r}}}} \right)}\\
+& Z
-{}&{\delta  = arctan\left( {\frac{{\omega ({l_f} + {l_r})}}{v}} \right)}\\
+\end{array}\right)\left(\begin{array}{c}
-{}&{{\delta _f} =  - {\delta _r}}
+c\\
-\end{array}</math>
+c^{\dagger }
+\end{array}\right),</math>
 |}
-|  style="width: 5px;text-align: right;white-space: nowrap;"|(2)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (4)
+|}where <math>Z=H^M+iRe{\left(M^M\right)}^T,Y=2Im{\left(M^M\right)}^T</math>, and  <math>c=(c_1,c_2,...,c_{2n})</math> are third quantized complex fermions. Through the above expression, we can obtain the Liouvillian eigenspectrum [54,55]
+{| class="formulaSCP" style="width: 100%; text-align: center;"
+|-
+|
+{| style="text-align: center; margin:auto;"
+|-
+| <math>\lambda ={\sum }_iE_iv_i</math>
 |}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (5)
-Here, <math>\dot x</math> and <math>\dot y</math> designate the Cartesian coordinates of the geometric center located along the rear axle, while  ''v'' denotes the linear forward speed. The angular orientation of the vehicle is symbolized by  <math>\dot \psi </math>, with  <math>\dot \psi  = \omega </math> corresponding to the yaw rate. The angular displacements of the front and rear wheels are represented by <math>{\delta _f}</math> and  <math>{\delta _r}</math>, respectively, and their combined effect is consolidated in the equivalent steering angle <math>\delta</math>. These steering angles are designed to be equal in magnitude but directed oppositely across the front and rear wheel assemblies. The parameters <math>{l_f}</math> and <math>{l_r}</math> indicate the longitudinal distances from the vehicle’s center of mass to the respective axles. The parameter <math>\beta </math>, denoting the slip angle, is regarded as negligible in the context of low-speed locomotion, with similar assumptions applied to individual wheel side-slip angles.
+|}with  <math>v_i\in \lbrace 0,1\rbrace </math>, where  <math>\left\{E_i\right\}</math> is the eigenspectrum of  <math>4iZ</math>. Here  <math>\lambda </math> contains valuable information of the full density-matrix dynamics, and it can be easily obtained from the damping matrix <math>X</math> with   <math>X_{ij}=ih_{ji}-{\sum }_{\mu }l_{\mu ,j}^\ast l_{\mu ,i}^{}</math> [36]. Rewriting <math>M</math> as  <math>M=M_r+iM_i</math>, where  <math>M_r,M_i</math> are real matrices, we have  <math>M^M=\frac{1}{4}(M_r+iM_i)\otimes (1+{\sigma }_y)</math>.   <math>Z</math> can be further written as
+{| class="formulaSCP" style="width: 100%; text-align: center;"
-The range of admissible values for the steering configuration is constrained within the following interval:  <math>\delta  \in [ - {45^ \circ }, + {45^ \circ }]</math>, which defines the upper and lower bounds of the equivalent steering angle. This angle serves as a surrogate for representing the aggregate steering effect of both front and rear wheels as if a single wheel were positioned at each end. The equivalent angle is formulated as:
-{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;"
 |-
 |
-{| style="margin:auto;width: 100%;"
+{| style="text-align: center; margin:auto;"
 |-
-| <math>{\delta _{(r,f)}} = \frac{{{\delta _{iw}} + {\delta _{ow}}}}{2}</math>
+| <math>\begin{align}
+Z=&\frac{1}{4}(h_r\otimes {\sigma }_y+ih_i\otimes 1)+i\frac{1}{4}(M_r^T\otimes 1-iM_i^T\otimes {\sigma }_y)\\
+=&\frac{1}{4}(h_r+M_i^T)\otimes {\sigma }_y+i\frac{1}{4}(h_i+M_r^T)\otimes 1
+\end{align}</math>
 |}
-|  style="width: 5px;text-align: right;white-space: nowrap;"|(3)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (6)
+|}Therefore,
+{| class="formulaSCP" style="width: 100%; text-align: center;"
+|-
+|
+{| style="text-align: center; margin:auto;"
+|-
+| <math>\begin{array}{c}
+\det(4iZ-\lambda E)=\det\left(\begin{array}{cc}
+-(h_i+M_r^T)-\lambda  & h_r+M_i^T\\
+-(h_r+M_i^T) & -(h_i+M_r^T)-\lambda
+\end{array}\right)\\
+=\det(X-\lambda E)\det(X^\ast -\lambda E)
+\end{array}</math>
 |}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (7)
+|}The eigenvalue of  <math>4iZ</math> are the union of the eigenvalues of  <math>X</math> and  <math>X^\ast </math>, which gives the Liouvillian eigenspectrum.
+Then we outline the general form of the Lindblad damping matrix in open quantum systems [30]. We consider tight-binding models whose Hamiltonian can generally be written as  <math>H=\sum {}_{ij}h_{ij}c_i^{\dagger }c_j</math>, where  <math>c_i^{\dagger },c_i</math> are the creation and annihilation operators on lattice site  <math>i</math>, and  <math>h_{ij}=h_{ij}^\ast </math> is the hopping amplitude between the lattice points of the system (<math>i\not =j</math>) or onsite potential (<math>i=j</math>). It is convenient to define the single-particle correlation function  <math>{\Delta }_{ij}(t)=Tr[c_i^{\dagger }c_j\rho (t)]</math> to observe the time evolution of the density matrix. Each cell is coupled to the environment through the gain jump operator  <math>L_{\mu }^g=\sum {}_iD_{\mu i}^gc_i^{\dagger }</math> and loss jump operator <math>L_{\mu }^l=\sum {}_iD_{\mu j}^lc_i</math>. Substituting the Lindblad quantum master equation into the time evolution of the single-particle correlation function, we can obtained
-where  <math>{\delta _{iw}}</math> and  <math>{\delta _{ow}}</math> denote the steering angles of the robot’s inner and outer wheels, respectively. This formulation remains valid under the assumption of symmetrical steering inputs, and the value of  <math>\delta</math> is constrained to lie within a specified range, expressed as:
+{| class="formulaSCP" style="width: 100%; text-align: center;"
-{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;"
 |-
 |
-{| style="margin:auto;width: 100%;"
+{| style="text-align: center; margin:auto;"
 |-
-| <math>{\delta _{min}} \le \delta  \le {\delta _{max}}</math>
+| <math>\frac{d\Delta (\mbox{t})}{dt}=X\Delta (\mbox{t})+\Delta (\mbox{t})X^{\dagger }+2M_g,</math>
 |}
-|  style="width: 5px;text-align: right;white-space: nowrap;"|(4)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (8)
+|}where  <math>X=ih^T-(M_l^T+M_g)</math> is the damping matrix with  <math>{\left(M_g\right)}_i{}_j=\sum {}_{\mu }D_{\mu i}^{g\ast }D_{\mu j}^g</math> and  <math>{\left(M_l\right)}_i{}_j=\sum {}_{\mu }D_{\mu i}^{l\ast }D_{\mu j}^l</math>. The steady state correlation  <math>{\Delta }_s=\Delta (\infty )</math>, to which the long-time evolution of any initial state converges, is determined by  <math>d{\Delta }_s/dt=0</math> or  <math>X{\Delta }_s+{\Delta }_sX^{\dagger }+2M_g=0</math>.  Focusing on the deviation towards the steady state  <math>\tilde{\Delta }(t)=\Delta (t)-{\Delta }_s</math>, whose time evolution is  <math>d\tilde{\Delta }(t)/dt=X\tilde{\Delta }(t)+\tilde{\Delta }(t)X^{\dagger }</math>, we can integrate it with Eq. (1) to obtain
+{| class="formulaSCP" style="width: 100%; text-align: center;"
+|-
+|
+{| style="text-align: center; margin:auto;"
+|-
+| <math>\tilde{\Delta }(\mbox{t})=e^{Xt}\tilde{\Delta }(\mbox{0})e^{X\dagger t}.</math>
 |}
-==3. Controller design architecture==
+| style="width: 5px;text-align: right;white-space: nowrap;" | (9)
+|}Therefore, the dynamical behaviour of the system can be characterized by the damping matrix.
-The controller software architecture delineates the high-level structure of constituent modules and their inter-module interactions. As illustrated in '''Figure 2''', the robot’s navigational control system is fundamentally structured around three core computational modules: the path planner, the motion planner, and the vehicle controller. This integrated navigation algorithm is implemented exclusively within the MATLAB® and Simulink® simulation environment.
+==3. Model==
+In this paper, we consider the Chern insulator model with the Hamiltonian in momentum space as
-<div id='img-1'></div>
+{| class="formulaSCP" style="width: 100%; text-align: center;"
-{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
+|-
-|-style="background:white;"
+|
-|style="text-align: center;padding:10px;"| [[Image:Draft_Wu_280690876-image40.png|480px]]
+{| style="text-align: center; margin:auto;"
 |-
-| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 2'''. Hierarchical control architecture devised for autonomous navigational decision-making
+| <math>h(k)=l_x\sin k_x{\sigma }_x+l_y\sin k_y{\sigma }_y+{\epsilon }_k{\sigma }_z,</math>
 |}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (10)
-The path planning module processes environmental imagery, translating it into a binary probabilistic map representing navigable space. Specifically, satellite imagery sourced from providers like Google is converted into this occupancy grid format, which subsequently serves as the foundational spatial representation for both robot navigation and self-localization tasks. This generated map explicitly demarcates traversable regions from non-drivable areas (e.g., obstacles, boundaries) within the operational environment. Leveraging this grid map, the module derives an optimal navigable trajectory for the robot. Detailed exposition of the specific path generation algorithms employed is omitted here, as it constitutes a distinct research domain beyond the present article’s scope. Upon successful path computation, this module outputs the requisite sequence of driving coordinates essential for guiding the robot through the designated environment.
+|}where  <math>{\epsilon }_k=m+t_x\cos k_x+t_y\cos k_y</math>. Let each unit cell contain a single loss and gain dissipator,
+{| class="formulaSCP" style="width: 100%; text-align: center;"
-The vehicle controller actuates the commands generated by the motion planner module. It translates the computed linear and angular velocity setpoints into specific translational velocity and steering angle commands appropriate for the target robotic platform. These actionable directives are transmitted to the physical robot via a dedicated data communication interface. Crucially, sensor feedback – including wheel encoder data and steering angle measurements – is relayed back to the path tracking controller, enabling continuous closed-loop correction to maintain the desired trajectory under kinematic constraints.
-===3.1 Lateral control design strategy===
-To ensure strict adherence to the pre-defined navigation path, the implementation of an effective lateral control strategy is indispensable for governing vehicular steering dynamics. In the present study, robotic trajectory tracking has been facilitated through a pure pursuit control scheme, integrated with vehicle odometry data and a two-dimensional reference map. The realization of accurate navigation necessitates the simultaneous orchestration of both lateral and longitudinal control actions. Specifically, the steering control law—mathematically represented in Eq. (5)—is composed of a heading error correction (yaw/orientation compensation) and lateral position deviation suppression relative to the designated path:
-{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;"
 |-
 |
-{| style="margin:auto;width: 100%;"
+{| style="text-align: center; margin:auto;"
 |-
-| <math>{\delta _{str}} = Heading\;{\rm{error}} + Latera{l_\;}_{}{\rm{error}}</math>
+| <math>\begin{align}
+L_x^l= & \frac{\sqrt{2\gamma }}{2}\left(e^{-\displaystyle\frac{\pi }{4}i}c_{xA}+e^{\displaystyle\frac{\pi }{4}i}c_{xB}\right)\\
+L_x^g= & \frac{\sqrt{2\gamma }}{2}\left(e^{\displaystyle\frac{\pi }{4}i}c_{xA}^{\dagger }+e^{-\displaystyle\frac{\pi }{4}i}c_{xB}\right),
+\end{align}</math>
 |}
-|  style="width: 5px;text-align: right;white-space: nowrap;"|(5)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (11)
+|}where  <math>x</math> denotes the lattice site,  <math>A,B</math> refer to the sublattice. The Fourier transformation of  <math>X</math> is  <math>X(k)=ih^T(-k)-M_l^T(-k)-M_g(k)</math>. The gain and loss dissipators are intra-cell, so these  <math>M(k)</math> matrices are independent of  <math>k</math>,  <math>M_l(k)=\frac{\sqrt{2}}{2}\lambda +\frac{1}{2}{\sigma }_x-</math><math>\frac{1}{2}{\sigma }_y,M_g(k)=\frac{\sqrt{2}}{2}\lambda +</math><math>\frac{1}{2}{\sigma }_x+\frac{1}{2}{\sigma }_y</math>. Then, the damping matrix in momentum space is
+{| class="formulaSCP" style="width: 100%; text-align: center;"
+|-
+|
+{| style="text-align: center; margin:auto;"
+|-
+| <math>X(k)=i[l_x\sin k_x{\sigma }_x+l_y\sin k_y{\sigma }_y+{\epsilon }_k{\sigma }_z+</math><math>i[\lambda {\sigma }_x+\lambda {\sigma }_y]-\sqrt{2}\lambda .</math>
 |}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (12)
+|}It can be written in the form of left and right eigenvectors,
-The heading error  <math>{\psi _e}</math> is defined as the angular deviation between the target path orientation  <math>{\psi _{desired}}</math> and the robot’s actual heading  <math>{\psi _{vehicle}}</math>, whereas the lateral error quantifies the perpendicular displacement between the vehicle’s center of gravity and the reference trajectory:
+{| class="formulaSCP" style="width: 100%; text-align: center;"
-{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;"
 |-
 |
-{| style="margin:auto;width: 100%;"
+{| style="text-align: center; margin:auto;"
 |-
-| <math>\left\{ {\begin{array}{*{20}{l}}
+| <math>X=\sum_n{\lambda }_n\vert u_{Rn}\rangle \langle u_{Ln}\vert ,</math>
-{{\psi _e} = {\psi _{desired}} - {\psi _{vehicle}}}\\
-{{e_{ra}} = {V_x}sin\psi  + {V_y}cos\psi }
-\end{array}} \right.</math>
 |}
-|  style="width: 5px;text-align: right;white-space: nowrap;"|(6)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (13)
-|}
+|}where  <math>X^{\dagger }\vert u_{Ln}\rangle ={\lambda }_n^\ast \vert u_{Ln}\rangle ,X\vert u_{Rn}\rangle =</math><math>{\lambda }_n\vert u_{Rn}\rangle</math>. It is worth noting that our  <math>M_l</math> and   <math>M_g</math> satisfy  <math>M_l^T+M_g=2M_g</math>, guaranteeing that  <math>{\Delta }_S=\frac{1}{2}I_{2L\times 2L}</math> is a steady state solution, where  <math>L=N_x\times N_y</math>, <math display="inline"> L </math> is the system size, and  <math>N_x,N_y</math> are the size in <math display="inline"> x,y </math> direction, respectively. We assume that the initial state of the system is the completely filled state, i.e., <math>\Delta (0)</math> is an identity matrix. Therefore, Eq.(9) can be re-expressed as
+{| class="formulaSCP" style="width: 100%; text-align: center;"
-Here,  <math>{V_x}</math> and  <math>{V_y}</math> respectively represent the vehicle’s longitudinal and lateral velocity components. As detailed in Section 2, under the employed four-wheel steering (4WS) configuration—illustrated in '''Figure 3'''—the lateral velocity component is considered negligible at low speeds to facilitate optimal steering precision. Consequently, Eq. (7) may be reexpressed as:
-{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;"
 |-
 |
-{| style="margin:auto;width: 100%;"
+{| style="text-align: center; margin:auto;"
 |-
-| <math>{\delta _{str}} = {V_x}sin\psi  + {\psi _{desired}} - {\psi _{vehicle}}</math>
+| <math>\begin{aligned}
+\tilde{\Delta}(\mathrm{t})& =\frac{1}{2}\displaystyle\sum_{n,n^{'},l}\exp[(\lambda_{n}+\lambda_{n^{'}}^{*})t]\widetilde{u_{R}}(i,n)\widetilde{u_{L}}(l,n)\widetilde{u_{L}^{*}}(l,n^{'})\widetilde{u_{R}^{*}}(j,n^{'})  \\
+&=\displaystyle\frac{1}{2}\displaystyle\sum_{n,n^{'}}\displaystyle\frac{\displaystyle\sum_{l}\exp[(\lambda_{n}+\lambda_{n^{'}}^{*})t]u_{R}(i,n)u_{L}(l,n)u_{L}^{*}(l,n^{'})u_{R}^{*}(j,n^{'})}{\displaystyle\sum_{k}u_{R}(k,n)u_{L}(k,n)\displaystyle\sum_{m}u_{L}^{*}(m,n^{'})u_{R}^{*}(m,n^{'})}
+\end{aligned}</math>
 |}
-|  style="width: 5px;text-align: right;white-space: nowrap;"|(7)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (14)
-|}
+|}According to the dissipative property,  <math>Re\lbrace {\lambda }_n\rbrace \leq 0</math> always holds. The Liouvillian gap  <math>\Lambda =\min[2Re(-{\lambda }_n)]</math> plays a decisive role in long-time dynamics. The opening gap (<math>\Lambda \not =0</math>) implies an exponential rate of convergence to the steady state, while the closing gap (<math>\Lambda =0</math>) implies algebraic convergence [34].
+==4. Chiral damping==
-<div id='img-1'></div>
+For simplicity, the parameters of our model are taken as  <math>l_x=l_y=1</math>, <math>t_x=t_y=-1</math>. We first study the dynamical behaviour under the periodic boundary conditions. Diagonalizing  <math>X(k)</math>, we obtain the energy spectrum as shown in [[Zhong Li 2024a#img-1|Figure 1]]. It is found that the Liouvillian gap vanishes at  <math>m=1.5</math>, while the gap opens at  <math>m=2.5</math>. So we expect the damping rate to be algebraic and exponential in each case, respectively.<div id="img-1"></div>
-{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
+{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:75%;"
-|-style="background:white;"
+|- style="background:white;"
-|style="text-align: center;padding:10px;"| [[Image:Draft_Wu_280690876-image49.png|360px]]
+| style="text-align: center;padding:10px;" | [[Image:Draft_Zhong_847600978-image84.png|600px|link=https://www.scipedia.com/public/File:Draft_Zhong_847600978-image84.png]]
 |-
-| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 3'''. Depiction of the elWobot’s inverse four-wheel steering configuration under zero lateral slip conditions
+| style="background:#efefef;text-align:justify;padding:10px;font-size: 85%;" | '''Figure 1'''. Eigenvalues of the damping matrix X. Blue: periodic boundary; Red: open boundary . The Liouvillian gap under periodic boundary condition vanishes for (a) and (b), while it is nonzero for (c) and (d). Under open boundary condition, the Liouvillian gap is nonzero in all four cases. This significant difference between open and periodic boundary comes from the NHSE of <math display="inline">  X</math>. (a) <math>\lambda =0.1,m=1.5</math>. (b) <math>\lambda =0.5,m=1.5</math>. (c)  <math>\lambda =0.1,m=2.5</math>. (d) <math>\lambda =0.5,</math>  <math>m=2.5</math>
+|}To verify this, we define the site-averaged fermion number deviation from the steady state  <math>R(t)=\sqrt{\frac{1}{N_XN_Y}{\sum_xR_x(t)}^2}=\sqrt{\frac{1}{N_XN_Y}{\sum_x\left(\frac{n_x(t)-n_x(t-{\delta }_t)}{{\delta }_t}\right)}^2}</math>, where  <math>R_x(t)=n_x(t)-n_x(\infty )</math>, and <math>n_x(t)={\Delta }_{xA,xA}(t)+{\Delta }_{xB,xB}(t)</math>. The numerical results are shown in [[Zhong Li 2024a#img-2|Figure 2]]. As anticipated, it is observed that the damping of <math display="inline">  R(t)</math> is algebraic for cases black and red lines with <math display="inline">m=1.5</math>, while exponential for blue and green lines with <math display="inline">m=2.5</math> under the periodic boundary condition.<div id="img-2"></div>
+{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:55%;"
+|- style="background:white;"
+| style="text-align: center;padding:10px;" | [[Image:Draft_Zhong_847600978-image93.png|348px|link=https://www.scipedia.com/public/File:Draft_Zhong_847600978-image93.png]]
+|-
+| style="background:#efefef;text-align:justify;padding:10px;font-size: 85%;" | '''Figure 2'''. Damping of site-averaged fermion number towards the steady state under periodic boundary condition with size  <math>L=30\times 30</math>. <math display="inline">m=1.5</math> (black and red) exhibits a slow algebraic damping, while <math display="inline">m=2.5</math> (blue and green) is an exponential damping. The  initial state is the completely filled state
+|}Next we turn to the open boundary conditions. Since the damping matrix <math display="inline"> X </math> has  NHSE, its energy spectrum is no longer that of the periodic boundary conditions. At this point all the energy spectrums have a non-zero energy gap (red part of [[Zhong Li 2024a#img-1|Figure 1]]), therefore, we expect an exponential long-time damping of  <math>\tilde{\Delta }(t)</math>. The numerical simulation in [[Zhong Li 2024a#img-3|Figure 3]] confirms this exponential behaviour with <math display="inline">  R(t)</math> having a period of algebraic damping before entering into the exponential damping. The time of the algebraic damping increases with the size <math display="inline"> L </math> ([[Zhong Li 2024a#img-3|Figure 3]](a)). To better understand this feature, we plot the damping in several unit cells in the same x dimension (<math display="inline">ix = 1</math>), as shown in [[Zhong Li 2024a#img-3|Figure 3]](b). It can be seen that the left end () enters the exponential damping immediately, and the other sites enter the exponential damping in turn according to their different distances to the left end.Due to a process of algebraic damping that occurs before entering the exponential stage,there is a "damping wavefront" from left (<math>ix=1,iy=1</math>) to right (<math>ix=1,iy=N_y</math>). This phenomenon is known as "chiral damping".<div id="img-3"></div>
+{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:80%;"
+|- style="background:white;"
+| style="text-align: center;padding:10px;" | [[Image:Draft_Zhong_847600978-image98.png|700px|link=https://www.scipedia.com/public/File:Draft_Zhong_847600978-image98.png]]
+|-
+| style="background:#efefef;text-align:justify;padding:10px;font-size: 85%;" | '''Figure 3'''. (a) Site-averaged particle number damping under periodic boundary conditions (solid line) and open boundary conditions (dashed line) for several sizes <math display="inline"> L </math>. The long-time damping of <math display="inline">  R(t)</math> follows a power law under periodic boundary condition, while the damping follows an exponential law after an initial power law stage under open boundary condition. (b) Particle number damping on several sites. The system size is <math display="inline">  30\times 30</math>, and the left end (<math>ix=1,iy=1</math>) enters the exponential phase from the beginning, followed by the other sites in turn. For (a) and (b), the initial state is completely filled state, <math>m=1.5,\lambda =0.1</math>
+|}The phenomenon of chiral damping can be observed more intuitively as shown in Fig. 4(a) where the colour shades indicate the value of <math display="inline">  R(t)</math>. Under the periodic boundary condition, the time evolution follows a slow power law while under the open boundary condition, a wavefront moving to the upper right is observed. This can be intuitively linked to the phenomenon that all eigenstates of  are localized in the upper right corner, which arises from the non-Hermitian skin effect of the damping matrix . If the matrix  does not have NHSE under the open boundary condition, the fermion number of the system should have a similar behaviour of damping under different boundary conditions. Therefore, the non-Hermitian skin effect plays an important role in open quantum systems and significantly affects the dynamical behaviour of open quantum systems.<div id="img-4"></div>
+{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:60%;"
+|- style="background:white;"
+| style="text-align: center;padding:10px;" | [[Image:Draft_Zhong_847600978-image101-c.png|456px|link=https://www.scipedia.com/public/File:Draft_Zhong_847600978-image101-c.png]]
+|-
+| style="background:#efefef;text-align:justify;padding:10px;font-size: 85%;" | '''Figure 4'''. Evolution of <math display="inline">  R(t)</math> at each lattice site under open boundary conditions (a) and periodic boundary conditions (b)
 |}
+==5. Experiment realized==
-This specific control regime, wherein the lateral slip is eliminated, is termed the Zero-side-slip maneuver, a condition characteristically achieved under reduced velocity conditions. Under such circumstances, the heading angle  <math>\gamma </math> is equated with the vehicle’s course direction  <math>\psi</math>, i.e., the actual trajectory orientation:
+Next we give the scheme of topolectrical circuits to simulate the damping matrix. Based on the similarity between the Kirchhoff equation and the Schrödinger equation, it is possible to simulate the Hamiltonian of the system using different circuit components, and the different parameters in the Hamiltonian can be adjusted independently by various components. The circuit Laplacian corresponding to the Hamiltonian can be written as
+{| class="formulaSCP" style="width: 100%; text-align: center;"
-{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;"
 |-
 |
-{| style="vertical-align: top;margin:auto;width: 100%;"
+{| style="text-align: center; margin:auto;"
 |-
-| <math>\gamma  = \psi  + \beta </math>
+| <math>J=D-C+W,</math>
 |}
-|  style="vertical-align: top;width: 5px;text-align: right;white-space: nowrap;"|(8)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (15)
-|}
+|}where  and <math display="inline"> D </math> are diagonal matrices containing the total conductance from each node to the ground and to the rest of the circuit, respectively. <math display="inline"> C </math> is the adjacency matrix of conductances [44].<div id="img-5"></div>
+{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:70%;"
+|- style="background:white;"
-However, due to physical constraints imposed by steering geometry and the attainable side-slip angle  <math>\beta </math> , the permissible steering angle  <math>{\delta _{str}}</math> required for trajectory realization is bounded as follows:
+| style="text-align: center;padding:10px;" | [[Image:Draft_Zhong_847600978-image103.png|600px|link=https://www.scipedia.com/public/File:Draft_Zhong_847600978-image103.png]]
+|-
-{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;"
+| style="background:#efefef;text-align:justify;padding:10px;font-size: 85%;" | '''Figure 5'''. Structure of topolectrical circuit to realize damping matrix under periodic boundary conditions.  (a) Connection relations between the nodes. The blue solid line box containing two “sublattice” nodes A (red) and B (blue) simulates a unit cell of <math display="inline">  X</math>. The black (grey) solid line indicates the coupling between nodes in the <math display="inline"> x(y) </math>-direction. (b) Circuit element structure is detailed for the green dashed framed rectangle in (a).    (c) Internal circuit diagram of the INIC element, consisting of an operational amplifier and impedances  . The impedance  <math>Z</math> is the target element, and different conductance in different directions of  <math>V_{i,r}</math> can be achieved by connecting the INIC in series.   satisfies  <math>Z_+=Z_{-}</math>. (d) Grounding module of the nodes. The resistances <math>R_{A,}R_B</math> and capacitance <math display="inline"> C </math> are used to simulate the onsite potential, and inductance L allows the Laplacian eigenvalue spectrum to be shifted uniformly as desired
+|}Considering the periodic boundary conditions first, the topolectrical circuit for realizing the damping matrix X is illustrated in [[Zhong Li 2024a#img-5|Figure 5]].  [[Zhong Li 2024a#img-5|Figure 5]] depicts the schematic diagram of the overall circuit structure, which gives the connection relationship between the nodes. [[Zhong Li 2024a#img-5|Figure 5]] shows the detailed circuit component of the unit which is the green dashed box in [[Zhong Li 2024a#img-5|Figure 5]](a). The blue box in [[Zhong Li 2024a#img-5|Figure 5]](a) represents a unit cell in the system, and the two nodes inside it correspond to sublattices A (red) and B (blue). The circuit connections in the x and y directions are distinguished by black and gray. From [[Zhong Li 2024a#img-5|Figure 5]](b) we can obtain the matrices C and D in Eq. (9), so that
+{| class="formulaSCP" style="width: 100%; text-align: center;"
 |-
 |
-{| style="vertical-align: top;margin:auto;width: 100%;"
+{| style="text-align: center; margin:auto;"
 |-
-| <math>\psi  - {\beta _{\max }} \le {\delta _{str}} \le \psi  + {\beta _{max}}</math>
+| <math>\begin{array}{c}
+D-C=-i\omega \left(\begin{array}{cc}
+-i\displaystyle\frac{2}{\omega R_2}\cos k_x-i\displaystyle\frac{2}{\omega R_4}\cos k_y-C_1-i\displaystyle\frac{1}{\omega R_0} & i2C_2\sin k_x+\displaystyle\frac{2}{\omega R_6}\sin k_y-i\displaystyle\frac{1}{\omega R_1}+C_1\\
+i2C_2\sin k_x+\displaystyle\frac{2}{\omega R_5}\sin k_y+i\displaystyle\frac{1}{\omega R_1}+C_1 & -i\displaystyle\frac{2}{\omega R_3}\cos k_x-i\displaystyle\frac{2}{\omega R_7}\cos k_y-C_1+i\frac{1}{\omega R_0}
+\end{array}\right),\\
+\mbox{    }
+\end{array}</math>
 |}
-|  style="vertical-align: top;width: 5px;text-align: right;white-space: nowrap;"|(9)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (16)
-|}
+|}with <math>C_1=-\lambda</math>, <math>C_2=l_x/2</math>, , <math>R_1=-1/\left(\omega \lambda \right)</math>, <math>R_2=-R_3=-2/t_x</math>,  ,
-Alternatively, Eq. (10) provides the steering angle condition necessary to achieve the desired orientation alignment for trajectory following.
-{| class="formulaSCP" style="width: 100%;border-collapse: collapse;width: 100%;text-align: center;"
+Comparing it with the damping matrix, we need to add grounding elements to match the onsite potential. The grounding elements of nodes A and B are shown in  [[Zhong Li 2024a#img-5|Figure 5]](d), where the resistors  <math>R_{A,}R_B</math> and capacitors  <math>C</math> simulate the lattice potential, and  <math>R_{A,}R_B</math> satisfies  <math>R_A=-R_B</math>. So the diagonal matrix <math display="inline"> W </math> is
+{| class="formulaSCP" style="width: 100%; text-align: center;"
 |-
 |
-{| style="margin:auto;width: 100%;"
+{| style="text-align: center; margin:auto;"
 |-
-| <math>\psi  - {\beta _{max}} \le {\delta _{str}} \le \psi  + {\beta _{max}}</math>
+| <math>W=\left(\begin{array}{cc}
+i\omega C+\displaystyle\frac{1}{R_A}+\displaystyle\frac{1}{i\omega L} & \\
+  & i\omega C+\displaystyle\frac{1}{R_B}+\displaystyle\frac{1}{i\omega L}
+\end{array}\right).</math>
 |}
-|  style="width: 5px;text-align: right;white-space: nowrap;"|(10)
+| style="width: 5px;text-align: right;white-space: nowrap;" | (17)
-|}
+|}From Eq. (10) we get the conductance matrix of the circuit of [[Zhong Li 2024a#img-5|Figure 5]](a) at  <math>\omega </math> frequency
+{| class="formulaSCP" style="width: 100%; text-align: center;"
-The configuration illustrated in '''Figure 3''' represents the elWobot in a 4WS-negative alignment during the Zero-side-slip maneuver.
+|-
+|
-Due to limitations in sensor-based localization, the robot’s actual heading '''𝜓''' is inferred indirectly via its steering angle and velocity. The heading error is regulated by the PID controller, which is structured as a nonlinear, discrete-time single-input single-output (SISO) system constrained by predefined saturation bounds:
+{| style="text-align: center; margin:auto;"
-===3.2 Model-based PID controller tuning===
-Effective controller implementation necessitates precise calibration of gain parameters to achieve optimal system performance. Post-design PID tuning constitutes a critical phase for determining ideal proportional, integral, and derivative coefficients that govern controller behavior. While conventional manual tuning remains prevalent, this approach demands considerable expertise from control engineers and frequently evolves into a protracted, iterative process. To circumvent these limitations, our research adopts an automated model-based tuning framework leveraging MATLAB/Simulink’s computational ecosystem. Specifically, we employ Simulink Control Design™ alongside the System Identification Toolbox™ to implement this data-driven methodology. The technique requires a linearized plant approximation, presenting implementation challenges for our inherently nonlinear robotic system. To address this, operational open-loop response datasets are acquired under controlled input conditions. These experimental datasets enable localized linearization within defined operational envelopes, permitting the PID tuner to construct region-specific transfer function approximations. Our implementation utilizes reference trajectory tracking with bounded output constraints during optimization, applying this standardized approach to both steering and velocity control subsystems. '''Figure 4''' illustrates this model-based workflow for steering control, where a dedicated plant model is empirically derived from robotic steering response characteristics. Subsequent system identification procedures generate the foundational transfer function model essential for PID coefficient optimization.
-<div id='img-1'></div>
-{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
-|-style="background:white;"
-|style="text-align: center;padding:10px;"| [[Image:Draft_Wu_280690876-image66.png|510px]]
 |-
-| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 4'''. Steering controller calibration via model-informed PID parameter tuning
+| <math>\begin{array}{c}
+J(\omega )=-i\omega \left(\begin{array}{cc}
+-i\displaystyle\frac{2}{\omega R_2}\cos k_x-i\displaystyle\frac{2}{\omega R_4}\cos k_y-(C+C_1)-i\left(\displaystyle\frac{1}{\omega R_0}+\displaystyle\frac{1}{\omega R_A}\right) & i2C_2\sin k_x+\displaystyle\frac{2}{\omega R_6}\sin k_y-i\displaystyle\frac{1}{\omega R_1}+C_1\\
+i2C_2\sin k_x+\displaystyle\frac{2}{\omega R_5}\sin k_y+i\displaystyle\frac{1}{\omega R_1}+C_1 & -i\displaystyle\frac{2}{\omega R_3}\cos k_x-i\displaystyle\frac{2}{\omega R_7}\cos k_y-(C+C_1)+i\left(\displaystyle\frac{1}{\omega R_0}+\displaystyle\frac{1}{\omega R_A}\right)
+\end{array}\right)+\displaystyle\frac{1}{i\omega L}\epsilon \\
+=-i\omega J_P+\displaystyle\frac{1}{i\omega L}\epsilon ,
+\end{array}</math>
 |}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (18)
-The velocity control subsystem undergoes specialized tuning within the navigational velocity spectrum (0.45–2.25 m/s), reflecting operational boundary conditions. Strategic tuning prioritizes lower velocity regimes to prevent destabilization during slow-speed maneuvers while maintaining acceptable higher-speed responsiveness. As documented in '''Table 1''', the finalized gain parameters reflect platform-specific optimization for both control dimensions. Analysis of tuned responses reveals an intentional design compromise: Minor overshoot (observable in steering response plots) is deliberately preserved to enhance transient performance characteristics. Complete overshoot elimination induces excessive system lethargy, extending stabilization periods beyond functional requirements. Consequently, controller calibration embodies a deliberate equilibrium between dynamic responsiveness and stabilization precision—a fundamental tradeoff in transient performance optimization. This balance ensures adequate reference tracking agility without compromising operational stability, particularly crucial during trajectory execution under variable inertial conditions.
+|}where  <math>C=(1-\sqrt{2})\lambda </math>, <math>\frac{1}{R_A}=-\frac{1}{R_B}=-\omega m-\frac{1}{R_0}</math>. Comparing this Laplacian matrix with the damping matrix, the mapping relationship can be established by  <math>J_P\Leftrightarrow X</math>.<div id="img-6"></div>
+{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:70%;"
-'''Table 1.''' Optimized gain parameters for PID-based regulation of velocity and steering dynamics
+|- style="background:white;"
+| style="text-align: center;padding:10px;" | [[Image:Draft_Zhong_847600978-image123-c.png|600px|link=https://www.scipedia.com/public/File:Draft_Zhong_847600978-image123-c.png]]
-{| style="width: 100%;border-collapse: collapse;"
 |-
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;vertical-align: top;"|'''Control'''
+| style="background:#efefef;text-align:justify;padding:10px;font-size: 85%;" | '''Figure 6'''. Negative impedance module [22]. (a) A single-port circuit to ground. The input impedance is  <math>Z_g=-Z</math>. (b) Free-port circuit. Its input impedance at both ends is  <math>Z_{ij}=Z_{ji}=-Z</math>.  The markings on the ideal amplifier indicate the output voltage versus the input voltage
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;vertical-align: top;"|'''P'''
+|}Notice that the circuit requires a negative component,which is implemented as shown in [[Zhong Li 2024a#img-6|Figure 6]]. [[Zhong Li 2024a#img-6|Figures 6]](a) and (b) show the equivalent negative impedance modules for a single port to ground and a free two-terminal port, respectively. They achieve the equivalent negative impedance through an amplifier. According to Kirchhoff's law, the input impedance of the single-port circuit to ground ([[Zhong Li 2024a#img-6|Figure 6]](a)) can be obtained as
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;vertical-align: top;"|'''I'''
+{| class="formulaSCP" style="width: 100%; text-align: center;"
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;vertical-align: top;"|'''D'''
 |-
-|  style="border-top: 1pt solid black;vertical-align: top;"|'''Velocity'''
+|
-|  style="border-top: 1pt solid black;vertical-align: top;"|0.5106
+{| style="text-align: center; margin:auto;"
-|  style="border-top: 1pt solid black;vertical-align: top;"|2.153
-|  style="border-top: 1pt solid black;vertical-align: top;"|-0.005
 |-
-|  style="border-bottom: 1pt solid black;vertical-align: top;"|'''Steering'''
+| <math>Z_g=\frac{V_g}{I_g}=\frac{V_g}{(V_g-2V_g)/Z}=-Z.</math>
-|  style="border-bottom: 1pt solid black;vertical-align: top;"|1.35
-|  style="border-bottom: 1pt solid black;vertical-align: top;"|3.537
-|  style="border-bottom: 1pt solid black;vertical-align: top;"|0.30
 |}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (19)
+|}The input impedance at both ends of the free port circuit ([[Zhong Li 2024a#img-6|Figure 6]](b)) are
+{| class="formulaSCP" style="width: 100%; text-align: center;"
+|-
+|
+{| style="text-align: center; margin:auto;"
+|-
+| <math>\begin{array}{c}
+Z_{ij}=\displaystyle\frac{V_i-V_j}{I_i}=\displaystyle\frac{V_i-V_j}{[V_i-2(V_i-V_j)]/Z}=-Z,\\
+Z_{ji}=\displaystyle\frac{V_j-V_i}{I_j}=\displaystyle\frac{V_j-V_i}{[V_j-2(V_j-V_i)]/Z}=-Z.
+\end{array}</math>
+|}
+| style="width: 5px;text-align: right;white-space: nowrap;" | (20)
+|}That is  <math>Z_{ij}=Z_{ji}=-Z</math>.
+Under the open boundary condition, the hopping amplitude of the cells located at the boundary weakens, leading to fewer branches connected to the boundary nodes in the circuit model, as shown in [[Zhong Li 2024a#img-7|Figure 7]](a).  [[Zhong Li 2024a#img-7|Figure 7]](a) gives the connection relationship between the nodes of the circuit under the open boundary condition, and the circuit nodes can be classified into body nodes (in the black dashed box), edge nodes (in yellow) and corner nodes (in green). Changes in the branch circuit of the nodes at the boundary will cause variations of the matrices <math display="inline"> C </math> and <math display="inline"> D </math>. The matrix <math display="inline"> C </math> corresponds to the hopping amplitude between the lattice points, which is allowed to change. Whereas the change of D is not desired due to the same onsite potential under different boundary condition.
-===3.3 Model integration===
+Therefore, we need to design specific grounding elements to eliminate the effects of variations in <math display="inline"> D </math>. Owing to the asymmetry of the coupling strengths under periodic boundary condition, the types of the edge and corner nodes are different for each of the four orientations, so there are a total of 16 different grounding modules, as shown in [[Zhong Li 2024a#img-7|Figure 7]](b). The additional grounding elements keep the diagonal matrix D+W unchanged, i.e., the onsite potential is unchanged, which achieves the mapping of the circuit Laplacian in [[Zhong Li 2024a#img-7|Figure 7]] to the damping matrix <math display="inline"> X </math> under the open boundary condition.<div id="img-7"></div>
+{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:75%;"
-The comprehensive navigation framework synthesizes unit-level and subsystem components into an integrated operational model. This architectural integration establishes deterministic data exchange protocols across functional segments, enabling holistic system validation. A significant advantage of this modular design lies in its cross-project reusability: Pre-validated components accelerate development cycles for future autonomous platforms. Debugging efficiency is substantially enhanced through fault isolation capabilities, allowing targeted module inspection rather than full-system analysis. As visualized in '''Figure 5''', the implemented robotic navigation system partitions functionality into four principal domains:
+|- style="background:white;"
+| style="text-align: center;padding:10px;" | [[Image:Draft_Zhong_847600978-image129-c.png|750px|link=https://www.scipedia.com/public/File:Draft_Zhong_847600978-image129-c.png]]
-<div id='img-1'></div>
-{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
-|-style="background:white;"
-|style="text-align: center;padding:10px;"| [[Image:Draft_Wu_280690876-image67.png|342px]]
 |-
-| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 5'''. Fully integrated and operational model of the robotic navigation framework
+| style="background:#efefef;text-align:justify;padding:10px;font-size: 85%;" | '''Figure 7'''.  Schematic diagram of the circuit of the damping matrix <math display="inline"> X </math> under open boundary conditions. (a) Schematic diagram of the connection relations among the nodes. The black, yellow and green dashed boxes correspond to the body, edge and corner nodes, respectively. The circuit connections of the body node are the same as those of the periodic boundary, while the edge and corner nodes require additional grounding elements to regulate the onsite potential. (b) Grounding modules for edge and corner nodes. The grounding elements for the edge and corner nodes are different for each of the four orientations, where the negative impedance elements can be realized by [[Zhong Li 2024a#img-6|Figure 6]](a). Note that in addition to these grounding elements, all nodes need to be connected to the elements in [[Zhong Li 2024a#img-5|Figure 5]](d)
 |}
+==6. Conclusion==
+In summary, we study the dynamical properties of a two-dimensional open system. The open quantum system is constructed by introducing appropriate gain and loss to the Chern insulator, and then using the damping matrix derived from the Liouvillian superoperator explore its long-time evolution. It is found that  the site-averaged fermion number deviation from the steady state under periodic boundary conditions shows a slow algebraic damping when the energy gap closes and an exponential damping when the energy gap opens. Under open boundary conditions, due to the non-Hermitian skin effect of the damping matrix, the system exhibits the phenomenon of chiral damping that the fermion number at each site undergoes a period of algebraic damping before entering an exponential damping, and the transition time that is proportional to the distance from that site to the boundary. Finally, we map the damping matrix in terms of the circuit Laplacian to give a model diagram of the topolectrical circuit implementation of the system.
+==References==
+<div class="auto" style="text-align: left;width: auto; margin-left: auto; margin-right: auto;font-size: 85%;">
-(1) Sensor Fusion Interface: Processes both live sensor streams and simulated inputs essential for localization. This includes ingestion of trajectory waypoint sequences generated by path planning algorithms.
+[1]  Bergholtz E.J.,  Budich J.C.,  Kunst F.K. Exceptional topology of non-Hermitian systems. Rev. Mod. Phys., 93, 015005,  2021.
-(2) Motion Control Core: Embeds the Pure Pursuit tracking controller that computes steering directives using egocentric localization estimates and navigational waypoints. Pose data originates from odometric measurements correlated against prior path planning operations on occupancy grid maps. These environmental representations derive from georeferenced 2D raster workspace models.
+[2]  Ashida Y.,  Gong Z.,  Ueda M. Non-Hermitian physics. Adv. Phys., 69(3):249-435, 2020.
-(3) Kinematic Output Hub: Transmits derived velocity setpoints and steering directives to drive systems. Continuously monitors Euclidean distance to target coordinates, issuing null motion commands when the robot enters predefined terminal proximity thresholds.
+[3]  Zhu X.,  Ramezani H.,  Shi C.,  Zhu J.,  Zhang X. PT- symmetric acoustics. Phys. Rev. X, 4(3), 031042, 2014.
-(4) CAN Communication Gateway: Translates kinematic directives into Controller Area Network messages for actuator control. Incorporates diagnostic telemetry channels and failsafe triggers for real-time system health monitoring.
+[4]  Popa B.I.,  Cummer S.A. Non-reciprocal and highly nonlinear active acoustic metamaterials. Nat. Commun., 5, 3398, 2014.
-All computational processes execute within a rigorously enforced 10ms deterministic cycle. Since native Simulink® lacks inherent real-time capability, temporal synchronization is achieved through dedicated Real-Time Sync blocks. These enforce hardware-timed execution intervals that emulate embedded deployment conditions during simulation. The fail-safe subsystem autonomously triggers protective shutdown protocols upon detecting critical anomalies, including actuator faults or deviation beyond safe operating envelopes. Diagnostic messaging provides continuous state observability through standardized SAE J1939 telemetry frames. This temporal formalization ensures controller outputs maintain phase alignment with sensor inputs - a critical requirement for closed-loop stability during high-speed navigation. The synchronized execution environment further enables valid performance benchmarking against real-world timing constraints prior to physical deployment.
+[5] Regensburger A.,  Bersch C.,  Miri M.-A.,  Onishchukov G.,  Christodoulides D.N.,  Peschel U. Parity-time synthetic photonic lattices. Nature (London), 488:167-171, 2012.
-==4. Simulation results and discussion==
+[6]  Feng L., Wong Z.J., Ma R.-M.,  Wang Y., Zhang  X. Single-mode laser by parity-time symmetry breaking. Science, 346:972-975, 2014.
-This segment presents empirical validation results for the elWObot robotic platform’s navigation controller through co-simulation and field trials. The integrated development framework leveraged MATLAB/Simulink for model-based design, incorporating both virtual simulation and hardware-in-the-loop (HIL) verification. Successful algorithm deployment occurred in unstructured outdoor environments, specifically cobblestone pathways and densely vegetated terrain engineered to simulate agricultural field irregularities and off-road conditions. Under these challenging substrates, the system exhibited consistently reliable operational performance and trajectory adherence.
+[7] Zhou  H.,  Peng C.,  Yoon Y.,  Hsu C.W., Nelson K.A.,  Fu L.,  Joannopoulos J.D., Soljacic M.,   Zhen B. Observation of bulk Fermi arc and polarization half charge from paired exceptional points. Science, 359:1009-1012, 2018.
-===4.1 Robot velocity response===
+[8]  Cerjan A.,  Huang S.,  Wang M.,  Chen K.P.,  Chong Y.,   Rechtsman M.C. Experimental realization of a Weyl exceptional ring. Nat. Photonics, 13:623-628, 2019.
-Performance characterization was conducted at a target operational velocity of 1 m/s within the Wenzhou Vocational College of Science and Technology. While orchard validation remains pending platform refinements, the AST terrain provided sufficient stochastic disturbances for preliminary evaluation. '''Figure 6''' demonstrates the optimized velocity tracking behavior, with model execution at 10ms intervals contrasting with 200ms CAN bus update cycles. The PID controller exhibits characteristic transient dynamics: an intentional overshoot to 1.1 m/s occurs within the initial 200ms, subsequently decaying exponentially to converge within ±1% of the setpoint at t=1s. This overshoot magnitude constitutes a deliberate design parameter to overcome stiction-induced torque requirements at the wheel-terrain interface. Actual velocity feedback (measured via wheel encoders) demonstrates second-order following behavior, achieving asymptotic stability at t=1.8s with near-zero steady-state error. Segments A-C confirm consistent tracking fidelity during sustained operation without observable oscillation. Though enhanced disturbance rejection could theoretically be achieved through gain elevation, such tuning induces hypersensitivity to inertial perturbations and reduces stability margins. Consequently, the implemented PID configuration represents an optimal compromise between transient acceleration capability and robust velocity regulation.
+[9]  Papaj M., Isobe H., Fu L. Nodal arc of disordered dirac fermions and non-hermitian band theory. Phys. Rev. B, 99, 201107, 2019.
-<div id='img-1'></div>
+[10]  Shen H.,  Fu L. Quantum oscillation from in-gap states and a non-hermitian landau level problem. Phys. Rev. Lett., 121, 026403, 2018.
-{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
-|-style="background:white;"
-|style="text-align: center;padding:10px;"| [[Image:Draft_Wu_280690876-image68.png|492px]]
-|-
-| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 6'''. Temporal response of the robot’s velocity following PID gain optimization
-|}
-===4.2 Robot steering response===
+[11]  Cao Y.,  Li Y.,  Yang X., Non-hermitian bulkboundary correspondence in a periodically driven system. Physical Review B, 103, 075126, 2021.
-To evaluate directional control fidelity, the elWObot executed pre-mapped trajectories at its optimized operational velocity of 1 m/s (established in Section 4.5 following comparative velocity trials). All turning maneuvers were digitally recorded and synchronously compared against commanded steering inputs and actuator responses. During critical testing sequences observed from the vehicle’s frontal perspective, the robotic system initiated right-turn maneuvers. Leveraging its four-wheel-independent-steering (4WS) architecture, the platform implemented negative-phase coordination where front and rear axles counter-steered to minimize turning radius. As evidenced in '''Figure 7''', the steering controller effectively translated reference inputs (Str_PID_input) into wheel-angle outputs while maintaining Ackermann-compliant kinematics. This necessitates greater deflection of inner wheels relative to outer wheels during curvature negotiation - a behavior confirmed by rear-left wheel (RL_deg) consistently exceeding rear-right wheel (RR_deg) angular displacement.
+[12] Rotter I.  A non-hermitian hamilton operator and the physics of open quantum systems. Journal of Physics A: Mathematical and Theoretical, 42, 153001, 2009.
-Initial turn execution demonstrated exceptional tracking performance across all wheels. Subsequent maneuvers, however, revealed a systematic discrepancy at extreme steering demands: the inner wheel plateaued at 42°±0.5° tolerance versus the 45° command input ('''Figure 7'''). This limitation stems from inherent kinematic boundaries hard-coded during developmental modeling, where ±42° constitutes the mechanical steering stop. Consequently, the control algorithm intentionally clamps inputs at 45° to emulate physical stops with operational margin. Additional signal phase offsets observed between commanded and achieved angles derive from deterministic CAN bus communication latency operating at 200ms cycles. Comprehensive wheel-angle trajectories during transitional states appear in '''Figure 8''', further validating the 42° saturation threshold. Crucially, sideslip angle (β) variations remained below 1.2° throughout testing ('''Figure 9'''), exhibiting negligible influence on vehicle orientation or steering dynamics per Eqs. 2 and 15. This insensitivity confirms the theoretical advantage of negative-phase 4WS over conventional 2WS systems in slip mitigation. Persistent β offsets of 0.3°±0.1° originated from minor wheel-alignment inconsistencies at neutral positions, though these remained within operational tolerances for agricultural contexts.
+[13]  Xiao L.,  Deng T.,  Wang K.,  Zhu G., Wang Z.,  Yi W.,  Xue P. Observation of non-Hermitian bulk-boundary correspondence in quantum dynamics. Nat. Phys., 16:761-768, 2020.
-<div id='img-1'></div>
+[14]  Xiao L.,  Deng T.,  Wang K.,  Wang Z., Yi W., Xue P. Observation of non-Bloch parity-time symmetry and exceptional points. Phys. Rev. Lett., 126, 230402,  2021.
-{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
-|-style="background:white;"
-|style="text-align: center;padding:10px;"| [[Image:Draft_Wu_280690876-image69.png|444px]]
-|-
-| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 7'''. Steering angle dynamics in response to PID-regulated input signals
-|}
-<div id='img-1'></div>
+[15] Yao S.,  Wang Z.  Edge states and topological invariants of non-hermitian systems. Phys. Rev. Lett., 121, 086803,  2018.
-{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
-|-style="background:white;"
-|style="text-align: center;padding:10px;"| [[Image:Draft_Wu_280690876-image70.png|444px]]
-|-
-| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 8'''. Multi-wheel steering angle response elicited by the applied control input
-|}
-<div id='img-1'></div>
+[16] Yao S., Song F.,   Wang Z.  Non-hermitian chern bands. Physical Review Letters, 121, 136802, 2018.
-{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
-|-style="background:white;"
-|style="text-align: center;padding:10px;"| [[Image:Draft_Wu_280690876-image71.png|444px]]
-|-
-| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 9'''. Quantitative evaluation of the robot’s sideslip angle (β) during maneuvering
-|}
-===4.3 Robot yaw response===
+[17] Lee C.H.,  Thomale R.  Anatomy of skin modes and topology in non-hermitian systems. Phys. Rev. B, 99, 201103, 2019.
-'''Figure 10''' demonstrates close correlation between simulated yaw predictions and experimentally measured yaw rates during trajectory execution. Initial conditions established the robot’s heading at ψ₀ = 180° (π radians) relative to magnetic north. Throughout linear navigation from t=0–200s, both datasets maintained directional stability with angular deviations below ±0.5°. A deliberate 10° yaw rate perturbation introduced at t=200s was accurately replicated in the simulation model within 0.3° RMS error. This tracking fidelity persisted during subsequent maneuvers, with synchronized responses observed at t=480s when commanded directional changes occurred. Here, the simulation registered a 90° orientation shift while physical constraints limited the actual yaw rate to 45° – a discrepancy attributable to steering linkage saturation.
+[18] Wang H., Ruan J.,  Zhang H.  Non-hermitian nodal-line semimetals with an anomalous bulk-boundary correspondence. Phys. Rev. B, 99, 075130, 2019.
-<div id='img-1'></div>
+[19]  Lee C.H., Li L.,   Gong J.  Hybrid higher-order skin-topological modes in nonreciprocal systems. Phys. Rev. Lett., 123, 016805, 2019.
-{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
-|-style="background:white;"
-|style="text-align: center;padding:10px;"| [[Image:Draft_Wu_280690876-image72.png|408px]]
-|-
-| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 10'''. Comparative analysis of simulated versus empirical yaw rate responses
-|}
-Fundamental to this analysis is the established sign convention: positive values denote counterclockwise rotation (viewed top-down), while negative values indicate clockwise motion. The simulation’s negative yaw manifestation at t=760s correctly represented the vehicle’s reversed directional state upon path completion. Notably, the platform returned to its initial geospatial coordinates but with inverted orientation (heading = -180°), confirming successful loop closure despite directional reversal. This terminal state aligns with conventional vehicle dynamics frameworks where magnetic north corresponds to 0° yaw. The persistent 180° offset throughout testing originated from intentional initialization parameters rather than measurement drift, as confirmed by post-mission inertial validation. Kinematic discrepancies at extreme steering demands (e.g., t=480s) highlight the model’s capacity to capture saturation effects inherent to physical systems.
+[20]  Kunst F.K.,   Dwivedi V.  Non-hermitian systems and topology: A transfer-matrix perspective. Phys. Rev. B, 99, 245116, 2019.
-===4.4 Robot navigation testing===
+[21] Borgnia D.S., Kruchkov A.J., Slager R.J. Non-Hermitian Boundary Modes. Phys. Rev. Lett., 124, 056802, 2019.
-Conventional navigation strategies for car-like robotic platforms typically operate within reduced-configuration planar spaces defined solely by Cartesian x-y coordinates. '''Figure 11''' demonstrates trajectory execution within such an environment, simulating agricultural row-traversal behaviors including linear inter-row navigation and headland turning maneuvers. Path planning was implemented via Probabilistic Road Map (PRM) methodology on occupancy grid maps, where white regions denote navigable terrain and black zones represent obstructions. Notably, this approach disregards non-holonomic constraints inherent in wheeled platforms. The triangular representation in '''Figure 11''' illustrates the robot’s orientation during traversal at Wenzhou Vocational College of Science and Technology’ test facility. Velocity trials (0.5, 1.0, 1.5, 2.0 m/s) revealed optimal tracking fidelity at 1.0 m/s ('''Figure 11'''f), with comprehensive velocity-dependent analysis detailed in Section 4.5. Trajectory adherence was quantified through superimposed path visualization (blue trajectory), though Turn 4 exhibited corner-cutting behavior—manifesting as increased turning radius relative to waypoint positioning. This phenomenon stems from discretized path planning and non-holonomic limitations preventing instantaneous directional changes, necessitating minimum rotational radii. Such constraints are particularly pronounced in Ackermann-steered platforms (2WS/4WS) compared to differential/skid-steer systems capable of zero-radius turns. Mitigation strategies include implementing continuous-curvature path planners and optimizing pure pursuit controllers through velocity-adaptive lookahead distances.
+[22]  Wang B.X.,   Zhao C.Y.  Topological phonon polaritons in one-dimensional nonhermitian silicon carbide nanoparticle chains. Phys. Rev. B, 98, 165435, 2018.
-<div id='img-1'></div>
+[23]  Fu Y.,  Hu J.,   Wan S. Non-Hermitian second-order skin and topological modes. Physical Review B, 103(4), 045420, 2021.
-{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
-|-style="background:white;"
-|style="text-align: center;padding:10px;"| [[File:20250620 174112.png|408x408px]]
-|-
-| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 11'''. Real-world trajectory tracking of the robot along a predefined path at the AST test facility, Wenzhou Vocational College of Science and Technology
-|}
-Beyond qualitative assessment, rigorous quantitative evaluation examined linear tracking (Start → Turn 1) and turning performance across four maneuvers. '''Figure 12''' documents straight-line traversal errors bounded within ±0.05° tolerance, with transient overshoots attributable to surface-induced disturbances challenging controller robustness. Turning characteristics ('''Figure 11''') were statistically analyzed in '''Table 2''', revealing a maximum instantaneous peak error of 27% at Turn 1. Crucially, this metric represents localized deviation rather than holistic steering fidelity—a consequence of modeling simplifications where steering angle was computed as the absolute average of front-wheel displacements. Physical systems exhibit asymmetric wheel angles due to Ackermann kinematics, constraining achievable yaw rates. This approximation propagates to yaw estimation (Eq. 2), where velocity-dependent transients precede steady-state convergence. True maneuvering precision is better quantified by average angular offsets below 5° across all turns ('''Table 2'''). Negative steering values indicate leftward deflection, with kinematic saturation evidenced by the front-right wheel’s inability to achieve the 42.97° command angle—validating modeled mechanical limits.
+[24]  Yang L., et al. Topological energy braiding of non-Bloch bands. Physical Review B, 10, 195425, 2022.
-<div id='img-1'></div>
+[25] Ezawa M.  Braiding of majorana-like corner states in electric circuits and its non-hermitian generalization. Phys. Rev. B, 100, 045407, 2019.
-{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
-|-style="background:white;"
-|style="text-align: center;padding:10px;"| [[Image:Draft_Wu_280690876-image74.png|450px]]
-|-
-| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 12'''. Navigation performance of the robot during linear path traversal
-|}
-'''Table 2. '''Comparative metrics of steering input angles and corresponding yaw responses across multiple turning scenarios
+[26] Yang X., Cao Y., Zhai Y.  Non-Hermitian Weyl semimetals: Non-Hermitian skin effect and non-Bloch bulk–boundary correspondence.  Chinese Physics B, 31, 010308, 2022.
-{| style="width: 100%;border-collapse: collapse;"
+[27]  Ge Z.-Y.,  Zhang Y.-R., Liu T.,  Li S.-W., Fan H.,   Nori F.  Topological band theory for non-hermitian systems from the dirac equation. Phys. Rev. B, 100, 054105, 2019.
-|-
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;vertical-align: top;"|'''Turn'''
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;vertical-align: top;"|'''Peak Input angle (deg.)'''
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;vertical-align: top;"|'''Peak yaw angle (deg.)'''
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;vertical-align: top;"|'''Peak error (%)'''
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;vertical-align: top;"|'''Avg.offset (deg.)'''
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;vertical-align: top;"|'''FL angle'''
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;vertical-align: top;"|'''FR angle'''
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;vertical-align: top;"|'''RL angle'''
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;vertical-align: top;"|'''RR angle'''
-|  style="border-top: 1pt solid black;border-bottom: 1pt solid black;vertical-align: top;"|'''Equivalent Steering angle'''
-|-
-|  style="border-top: 1pt solid black;vertical-align: top;"|1st
-|  style="border-top: 1pt solid black;vertical-align: top;"|28.28
-|  style="border-top: 1pt solid black;vertical-align: top;"|20.50
-|  style="border-top: 1pt solid black;vertical-align: top;"|27.5
-|  style="border-top: 1pt solid black;vertical-align: top;"|1.19
-|  style="border-top: 1pt solid black;vertical-align: top;"|-24.10
-|  style="border-top: 1pt solid black;vertical-align: top;"|-18.11
-|  style="border-top: 1pt solid black;vertical-align: top;"|24.18
-|  style="border-top: 1pt solid black;vertical-align: top;"|18.15
-|  style="border-top: 1pt solid black;vertical-align: top;"|22.12
-|-
-|  style="vertical-align: top;"|2nd
-|  style="vertical-align: top;"|38.42
-|  style="vertical-align: top;"|33.04
-|  style="vertical-align: top;"|14.0
-|  style="vertical-align: top;"|3.06
-|  style="vertical-align: top;"|-38.27
-|  style="vertical-align: top;"|-25.57
-|  style="vertical-align: top;"|38.45
-|  style="vertical-align: top;"|25.60
-|  style="vertical-align: top;"|32.18
-|-
-|  style="vertical-align: top;"|3rd
-|  style="vertical-align: top;"|42.97
-|  style="vertical-align: top;"|36.44
-|  style="vertical-align: top;"|15.2
-|  style="vertical-align: top;"|4.26
-|  style="vertical-align: top;"|-41.95
-|  style="vertical-align: top;"|-27.83
-|  style="vertical-align: top;"|41.59
-|  style="vertical-align: top;"|27.82
-|  style="vertical-align: top;"|34.70
-|-
-|  style="vertical-align: top;"|4th
-|  style="vertical-align: top;"|42.97
-|  style="vertical-align: top;"|36.44
-|  style="vertical-align: top;"|15.2
-|  style="vertical-align: top;"|5.04
-|  style="vertical-align: top;"|-41.95
-|  style="vertical-align: top;"|-27.83
-|  style="vertical-align: top;"|41.59
-|  style="vertical-align: top;"|27.82
-|  style="vertical-align: top;"|34.71
-|-
-|  style="vertical-align: top;"|5th
-|  style="vertical-align: top;"|15.95
-|  style="vertical-align: top;"|12.69
-|  style="vertical-align: top;"|20.4
-|  style="vertical-align: top;"|1.15
-|  style="vertical-align: top;"|-13.95
-|  style="vertical-align: top;"|-12.26
-|  style="vertical-align: top;"|14.13
-|  style="vertical-align: top;"|12.22
-|  style="vertical-align: top;"|13.63
-|-
-|  style="vertical-align: top;"|6th
-|  style="vertical-align: top;"|42.97
-|  style="vertical-align: top;"|36.44
-|  style="vertical-align: top;"|15.2
-|  style="vertical-align: top;"|5.04
-|  style="vertical-align: top;"|-41.95
-|  style="vertical-align: top;"|-27.83
-|  style="vertical-align: top;"|41.59
-|  style="vertical-align: top;"|27.82
-|  style="vertical-align: top;"|34.47
-|-
-|  style="border-bottom: 1pt solid black;vertical-align: top;"|7th
-|  style="border-bottom: 1pt solid black;vertical-align: top;"|38.23
-|  style="border-bottom: 1pt solid black;vertical-align: top;"|32.28
-|  style="border-bottom: 1pt solid black;vertical-align: top;"|15.5
-|  style="border-bottom: 1pt solid black;vertical-align: top;"|3.43
-|  style="border-bottom: 1pt solid black;vertical-align: top;"|-38.25
-|  style="border-bottom: 1pt solid black;vertical-align: top;"|-25.55
-|  style="border-bottom: 1pt solid black;vertical-align: top;"|38.43
-|  style="border-bottom: 1pt solid black;vertical-align: top;"|25.55
-|  style="border-bottom: 1pt solid black;vertical-align: top;"|32.00
-|}
+[28]  Ji X.,   Yang X.  Generalized bulk-boundary correspondence in periodically driven non-Hermitian systems.  Journal of Physics: Condensed Matter, 36, 243001, 2024.
-===4.5 Controller tracking performance===
+[29] Li Y., et al.  Universal characteristics of one-dimensional non-Hermitian superconductors.  Journal of Physics: Condensed Matter, 35, 055401, 2022.
-The path-tracking efficacy of the pure pursuit controller was rigorously assessed through steering angle discrepancy analysis during turning maneuvers. Systematic testing evaluated velocity profiles (0.5, 1.0, 1.5, 2.0 m/s) coupled with preview distances (0.75, 1.0, 1.2, 1.5, 2.2 m), with the maximum lookahead constrained to the robotic platform’s inter-axle dimension. Optimal operational parameters were determined through minimization of steering offset error – defined as the angular deviation between commanded steering input and realized yaw response. '''Figure 13''' presents quantile distribution plots of angular tracking discrepancies across turning sequences. Minimal median errors emerged at 1.0 m/s velocity with 0.75 m preview distance, attributable to velocity-dependent orientation dynamics: at this specific velocity, kinematic constraints yielded near-ideal alignment between steering commands and platform heading. Suboptimal velocities induced systematic overcompensation (>1.0 m/s) or underresponsiveness (<1.0 m/s), degrading trajectory adherence. Notably, reduced preview distances consistently outperformed longer horizons across velocity regimes.
+[30] Song F., Yao S.,  Wang Z.  Non-Hermitian skin effect and chiral damping in open quantum systems.  Physical review Letters, 123, 170401, 2019.
-<div id='img-1'></div>
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-{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
-|-style="background:white;"
-|style="text-align: center;padding:10px;"| [[Image:Draft_Wu_280690876-image75.png|450px]]
-|-
-| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 13'''. Distributional analysis of steering offset errors across seven distinct turning maneuvers under varying velocity and look-ahead configurations: (a) 0.75 m look-ahead, 0.5 m/s velocity; (b) 1.0 m look-ahead, 0.5 m/s velocity; (c) 1.2 m look-ahead, 0.5 m/s velocity; (d) 1.5 m look-ahead, 0.5 m/s velocity; (e) 2.2 m look-ahead, 0.5 m/s velocity; (f) 0.75 m look-ahead, 1.0 m/s velocity; (g) 0.75 m look-ahead, 1.5 m/s velocity; (h) 0.75 m look-ahead, 2.0 m/s velocity
-|}
-Analysis of the optimal configuration ('''Figure 13'''f) reveals near-zero median errors for most turns, with exceptions at Turns 4 and 6. These anomalies stem from Ackermann steering limitations during acute maneuvers, where physical saturation prevented attainment of required wheel angles. The resultant kinematic discontinuity generated accumulated heading offsets and statistical outliers. Additional error variance in '''Figure 13'''f derives from sensor latency and transient noise artifacts during high-curvature transitions.
+[32]  Carmichael H.J. Quantum trajectory theory for cascaded open systems. Phys. Rev. Lett., 70, 2273, 1993.
-'''Figure 14''' employs Taylor’s statistical framework to quantify controller performance during turning sequences, correlating reference steering inputs with measured yaw responses [18]. Controller efficacy was benchmarked against three criteria: (1) Normalized root-mean-square difference (RMSD) approaching zero, (2) Normalized standard deviation approximating unity, and (3) Correlation coefficients nearing maximum values.
+[33] Weimer H.,  Kshetrimayum A., R. Orús Simulation methods for open quantum many-body systems. Rev. Mod. Phys., 93, 015008, 2021.
-<div id='img-1'></div>
+[34] Cai Z., Barthel T.  Algebraic versus exponential decoherence in dissipative many-particle systems.  Physical review letters, 111, 150403,  2013.
-{| class="wikitable" style="margin: 0em auto 0.1em auto;border-collapse: collapse;width:auto;"
-|-style="background:white;"
-|style="text-align: center;padding:10px;"| [[Image:Draft_Wu_280690876-image76.png|450px]]
-|-
-| style="background:#efefef;text-align:left;padding:10px;font-size: 85%;"| '''Figure 14'''. Normalized Taylor diagram illustrating controller efficacy across diverse turning conditions during autonomous navigation
-|}
-Experimental data demonstrates robust performance: RMSD values cluster within 0.2–0.4 across maneuvers, while standard deviations converge near 0.85. Correlation coefficients exceeding 0.99 confirm exceptional command-response synchronization. This statistical constellation – minimal RMSD, near-unity standard deviation, and maximal correlation – validates the controller’s proficiency in negotiating curvilinear paths despite kinematic limitations and environmental disturbances.
+[35] Zhou Z.,  Yu Z.  Non-Hermitian skin effect in quadratic Lindbladian systems: An adjoint fermion approach.  Physical Review A, 106, 032216, 2022.
-==5. Conclusion==
+[36] Li T.,  Zhang Y.-S., Yi W.  Engineering dissipative quasicrystals.  Physical Review B, 105, 125111, 2022.
-This study addresses the critical challenge of robotic navigation in dense orchard settings where GNSS/GPS signal degradation impedes localization capabilities. We present a model-driven framework for developing navigation architectures tailored to Ackermann-constrained mobile platforms operating under such conditions. The solution employs a modular decomposition strategy, with dedicated subsystems for: (1) Probabilistic motion planning; (2) Nonlinear vehicle control; (3) Sensor-fused localization; (4) Robust data telemetry.
+[37] He P., et al.  Damping transition in an open generalized Aubry-André-Harper model.  Physical Review A, 105, 023311,  2022.
-This compartmentalized design accelerated iterative development and validation cycles. Our integrated navigation model synthesizes trajectory generation outputs with vehicular kinetic constraints to produce executable motion primitives. The core innovation lies in decoupling velocity regulation from heading control - requiring only proprioceptive inputs from wheel encoders and steering resolvers. This sensor-minimal approach significantly reduces exteroceptive sensing dependencies while maintaining navigation integrity in constrained environments. Rigorous verification was conducted through co-simulation in MATLAB/Simulink environments, followed by field trials on a scaled robotic platform. The architecture demonstrated exceptional path-tracking fidelity during both linear transits and curvilinear maneuvers across varied terrain. Systematic velocity sweeps revealed optimal operational performance at 1 m/s, where platform yaw dynamics exhibited near-perfect correspondence with steering geometry kinematics. This velocity-specific synchronization minimized cumulative heading deviations observed at other test speeds (0.5, 1.5, 2.0 m/s).
-Current investigations focus on enhancing environmental adaptability through tight coupling of inertial navigation systems with online path optimization routines. This sensor-fusion strategy aims to boost robustness against orchard-specific disturbances including: canopy occlusion effects, terrain-induced wheel slip, and foliage multipath artifacts. Subsequent research phases will validate reliability metrics under extended operational durations in commercial orchard deployments.
-==Acknowledgement:==
-'''Funding Statement: '''None.
-'''Author Contributions: '''The authors confirm contribution to the paper as follows: Study conception and design: Li Tian, Xingjia Pan; Data collection: Li Tian; Analysis and interpretation of results: Li Tian, Xingjia Pan; Draft manuscript preparation: Li Tian, Xingjia Pan. All authors reviewed the results and approved the final version of the manuscript.
-'''Ethics Approval: '''Not applicable.
-'''Conflicts of Interest: '''The authors declare no conflicts of interest to report regarding the present study.<span id="RefSection"></span>
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+'''<nowiki/>'''