Active tracking coverage method based on channel knowledge map in mobile antenna system

Abstract

The invention discloses an active tracking and covering method based on a channel knowledge map in a mobile antenna system, which aims to maximize the total spectrum efficiency in a service period by expressing an active tracking and covering task of mobile user equipment as a long-term trajectory optimization problem of a mobile antenna. Accurately modeling a dependency relationship of a channel on each surface freedom degree of the antenna by using a control parameter obtained based on a Rodrigues rotation matrix; key physical feasibility constraint conditions are systematically integrated, so that the feasibility of an optimization result in engineering is ensured; a channel knowledge map is introduced to realize proactive planning, and traditional reactive control is converted into active planning by utilizing prediction information, so that performance loss caused by channel information lag is reduced; on the basis of a manifold optimization framework, rotation parameters of the movable antenna system are mapped to a special orthogonal group manifold for optimization, rotation constraints are processed through Lie group operation, and singularity and calculation redundancy caused by traditional parameterization are avoided.

Description

Technical Field

[0001] This invention belongs to the field of communication technology, specifically relating to an active tracking and coverage method based on a channel knowledge map in a movable antenna system. Background Technology

[0002] To address the ever-increasing demand for communication capacity in future wireless networks, MIMO technology has evolved from its initial form to massive MIMO and even ultra-massive MIMO. The industry has achieved significant success in improving spectral efficiency and communication reliability by continuously increasing the number of antenna elements. However, as systems migrate to higher frequency bands and antenna sizes continue to expand, this traditional paradigm, which relies solely on stacking antenna hardware, is facing severe bottlenecks: high hardware costs, enormous energy consumption, and diminishing marginal performance gains are becoming increasingly prominent. Against this backdrop, movable antenna (MA) technology emerged and has rapidly evolved into its more advanced form—the six-degree-of-freedom movable antenna (6D Movable Antenna, 6DMA). 6DMA technology endows base stations with an unprecedented capability: not only can the position of the antenna array be translated in three-dimensional space, but its three-dimensional rotational attitude can also be adjusted in real time. This mechanical geometrical reconfiguration capability allows the system to actively adapt to dynamically changing channel environments, significantly enhancing beamforming gain and spatial multiplexing capabilities without increasing the number of active antennas by seeking physically superior channel conditions.

[0003] Existing technical solutions propose a snapshot-based optimization method, which suffers from kinematic feasibility issues. This method decomposes the dynamic tracking task into a series of independent static optimizations, completely ignoring the physical motion constraints of the antenna. This short-sighted optimization approach may generate a series of theoretically optimal but physically impossible-to-achieve consecutive attitude jumps, leading to uneven or even infeasible trajectories. Secondly, there are issues with channel information acquisition and operating system latency. The traditional passive control loop of "measurement-optimization-execution" inherently has delays. By the time the system completes calculations based on the current Channel State Information (CSI) and drives the antenna to the target attitude, the user has already moved, and the actual channel has changed. This latency results in a continuous mismatch between the antenna attitude and the actual channel, severely weakening the quality and stability of the communication link. Summary of the Invention

[0004] To address the aforementioned problems in the prior art, this invention provides an active tracking and coverage method based on a channel knowledge map in a mobile antenna system. The technical problem to be solved by this invention is achieved through the following technical solution: This invention provides an active tracking and coverage method based on a channel knowledge map in a mobile antenna system, the method comprising: A position control model is constructed based on control parameters; a hyperplane is constructed based on the cumulative rotation matrix, and the constraints of the support rod and antenna surface in the movable antenna system are confirmed based on the hyperplane and the position control model; based on the field response channel model and motion relationship, and according to the position control model, the channel vectors corresponding to the user equipment and the base station in each time slot are constructed; using the channel knowledge map, the sum and capacity of the movable antenna system are obtained based on the channel vectors corresponding to the user equipment and the base station in each time slot; and based on the constraints and the sum and capacity of the movable antenna system, the sum and capacity optimization problem model is determined. The control parameters are represented as an unconstrained product manifold that preserves smoothness and inherits Riemannian metrics; the exact penalty method with a hybrid strategy is used to reconstruct the constraints, resulting in a smooth unconstrained problem on the manifold; based on the smooth unconstrained problem, the sum capacity optimization problem model is reconstructed into a smooth unconstrained optimization problem model on the unconstrained product manifold. A two-layer loop structure is used to optimize the smooth unconstrained optimization problem model on the unconstrained product manifold. The outer loop iteratively updates the solution, penalty weight, and smoothing factor in the smooth unconstrained optimization problem model until the preset outer loop stopping condition or the number of outer loop iterations is met, and outputs the optimized control parameters. In each outer loop, the inner loop solves the subproblem by configuring the parameters in the current outer loop until the preset inner loop stopping condition or the number of inner loop iterations is met, and outputs the solution of the subproblem as the solution of the smooth unconstrained optimization problem model in the next outer loop. Active tracking and coverage are completed based on the optimized control parameters.

[0005] The beneficial effects of this invention are: The solution provided by this invention describes the active tracking and coverage task of mobile user equipment as a long-term trajectory optimization problem for a mobile antenna, aiming to maximize the total spectral efficiency during the service period. It utilizes control parameters obtained based on the Rodrigues rotation matrix to accurately model the channel's dependence on the degrees of freedom of each antenna surface. A series of key physical feasibility constraints are systematically integrated to ensure the engineering feasibility of the optimization results. A channel knowledge map is introduced to achieve proactive planning, using predictive information to transform traditional "reactive" control into "active" planning, eliminating performance loss caused by channel information lag. Based on a manifold-based optimization framework, the rotation parameters of the mobile antenna system are mapped to a special orthogonal group manifold for optimization. Lie group operations are used to handle rotation constraints, avoiding the singularity and computational redundancy introduced by traditional parameterization. Attached Figure Description

[0006] Figure 1 This is a schematic diagram illustrating the steps of an active tracking and coverage method based on a channel knowledge map in a mobile antenna system provided by an embodiment of the present invention. Figure 2 This is a schematic diagram of a mobile antenna system in an active tracking and coverage method based on a channel knowledge map provided in an embodiment of the present invention. Figure 3 This is a schematic diagram of the position control model of the mobile antenna system in an active tracking and coverage method based on a channel knowledge map provided in an embodiment of the present invention. Figure 4 This is a schematic diagram of electromagnetic characteristic constraints in an active tracking and coverage method based on a channel knowledge map in a mobile antenna system provided by an embodiment of the present invention. Figure 5 This is a schematic diagram of the mechanical structure constraints in an active tracking and coverage method based on a channel knowledge map in a mobile antenna system provided in an embodiment of the present invention. Figure 6 A schematic diagram of Lie group embedding calculation in an active tracking and coverage method based on a channel knowledge map in a mobile antenna system provided in an embodiment of the present invention; Figure 7 This is a schematic diagram comparing the cumulative achievable spectral efficiency and time slot of different antenna orientation strategies in an active tracking coverage method based on a channel knowledge map in a mobile antenna system provided by an embodiment of the present invention. Figure 8 This is a schematic diagram comparing the peak instantaneous angular velocity and time slot of different antenna orientation strategies in an active tracking and coverage method based on a channel knowledge map in a mobile antenna system provided in an embodiment of the present invention. Figure 9 This is a schematic diagram illustrating the relationship between the sum of spectral efficiencies of different strategies and the number of user equipment in an active tracking and coverage method based on a channel knowledge map in a mobile antenna system provided by an embodiment of the present invention. Figure 10 This is a schematic diagram showing the relationship between the sum of spectral efficiencies of different strategies and the number of antenna surfaces in an active tracking and coverage method based on a channel knowledge map in a mobile antenna system provided by an embodiment of the present invention. Figure 11 This diagram illustrates the relationship between the sum of spectral efficiencies of different strategies and the maximum angular velocity limit in an active tracking and coverage method based on a channel knowledge map in a mobile antenna system provided by an embodiment of the present invention. Detailed Implementation

[0007] The present invention will be further described in detail below with reference to specific embodiments, but the implementation of the present invention is not limited thereto.

[0008] This invention provides an active tracking and coverage method based on a channel knowledge map in a mobile antenna system, such as... Figure 1 As shown, it may include: S1. Construct a position control model based on control parameters; construct a hyperplane based on the cumulative rotation matrix; determine the constraints of the support rod and antenna surface in the movable antenna system based on the hyperplane and the position control model; construct the channel vectors corresponding to the user equipment and base station in each time slot based on the field response channel model and motion relationship, according to the position control model; use the channel knowledge map to obtain the sum and capacity of the movable antenna system based on the channel vectors corresponding to the user equipment and base station in each time slot; determine the sum and capacity optimization problem model based on the constraints and the sum and capacity of the movable antenna system. S2 represents the control parameters as an unconstrained product manifold that preserves smoothness and inherits the Riemannian metric; the exact penalty method with a hybrid strategy is used to reconstruct the constraints, resulting in a smooth unconstrained problem on the manifold; based on the smooth unconstrained problem, the sum capacity optimization problem model is reconstructed into a smooth unconstrained optimization problem model on the unconstrained product manifold. S3 employs a two-layer loop structure to optimize the smooth unconstrained optimization problem model on the unconstrained product manifold. The outer loop iteratively updates the solution, penalty weights, and smoothing factors in the smooth unconstrained optimization problem model until the preset outer loop stopping condition or the number of outer loop iterations is met, and outputs the optimized control parameters. In each outer loop, the inner loop solves the subproblem by configuring the parameters in the current outer loop until the preset inner loop stopping condition or the number of inner loop iterations is met, and outputs the solution of the subproblem as the solution of the smooth unconstrained optimization problem model for the next outer loop. S4 completes active tracking coverage based on the optimized control parameters.

[0009] This invention aims to address the bottlenecks faced by existing 6DMA technology in continuously tracking and serving mobile users. Overcoming the contradiction between the physical infeasibility and suboptimal performance of existing technologies, it designs a 6DMA trajectory that can be both proactively planned and physically continuously realized. This allows for the full release of the performance potential of 6DMA technology while providing stable, high-quality service to mobile users. The core idea is to fundamentally change the modeling and solution paradigm of the problem. This invention abandons the traditional approach of discretizing dynamic tasks into independent static problems, constructing the user tracking task as a single, complete long-period trajectory optimization problem. The goal of this problem is to maximize the overall spectral efficiency of the system throughout the entire service cycle, and the kinematic and physical constraints of the antenna (such as maximum speed, collision avoidance, etc.) are directly incorporated into the model as endogenous variables and constraints to ensure the physical realizability of the final solution. Secondly, to overcome system latency, this invention introduces a Channel Knowledge Map (CKM). By utilizing the predicted information of the user's future location and corresponding channel provided by the CKM, the system can shift from passive "reactive" control to proactive "proactive" planning, thereby eliminating the performance loss caused by channel information lag. Finally, addressing the inherent high dimensionality, non-convexity, and complex geometric constraints of the constructed problem, this embodiment of the invention designs an innovative manifold optimization algorithm. This algorithm maps the antenna's rotation state onto the SO(3) Lie group manifold for optimization, fundamentally avoiding the singularity and complexity brought about by the traditional Euler parameterization method. Within the manifold optimization framework, an adaptive penalty method is integrated, and gradients are efficiently calculated using backpropagation in the tangent space, thereby stably and efficiently handling various complex physical constraints and solving for high-quality continuous motion trajectories.

[0010] A schematic diagram of a movable antenna system, such as Figure 2 As shown, this embodiment of the invention considers a multi-user uplink communication system, in which one base station serves a service period. Inside A mobile user device provides services. The service period is discretized into... Each time slot, composed of a set Index, the length of each time slot is The base station is equipped with A 6DMA surface, composed of Index. Each surface constitutes a uniform planar array containing Each antenna array element is composed of a set of Index. Each user device is indexed. The index is equipped with a single antenna in a fixed position. A key assumption in the model proposed in this embodiment of the invention is that the base station can perfectly know the spatiotemporal trajectories of all users a priori.

[0011] The spatial configuration of each 6DMA surface, including its three-dimensional position and orientation, can be dynamically reconfigured across different time slots to provide enhanced channel conditions for trajectory-aware services. Specifically, each 6DMA surface installed at the base station... A retractable, rotatable support rod with an embedded flexible cable connects to the central processing unit located in the base station. The assembly of the support rods is as follows: .

[0012] Each support rod supports a 6DMA surface, with its end connected to the surface via a gimbal, and the indices are one-to-one. The central processing unit controls the extension, rotation, and three-dimensional movement of these support rods, allowing for flexible adjustment of the position coordinates of each 6DMA surface in three-dimensional space and its deflection angle relative to the plane normal. This system supports precise position adjustment and attitude control of the 6DMA surfaces in omnidirectional space, providing a hardware foundation for dynamic optimization of wireless transmission links.

[0013] For ease of understanding, the following describes each step of the active tracking and coverage method based on a channel knowledge map in a mobile antenna system proposed in the embodiments of the present invention.

[0014] Regarding step S1, in order to describe the effect of the antenna support rod rotation on the antenna position, this embodiment of the invention uses the connection point between the base station and the antenna support rod as the origin. In this embodiment of the invention, a global Cartesian coordinate system is constructed. In the time slice... , No. The center position of the antenna surface of the 6DMA, i.e. the first The end of the support rod is denoted as ,in , and These represent the antenna surface of the 6DMA along the global Cartesian coordinate system. The coordinates of the axis. The axis of rotation of the root support rod is determined by a unit vector. Definition. In this global coordinate system, the angular velocity of rotation about this axis is denoted as . The corresponding rotation angle From the relation Decision. Referring to the rotation method based on the Rodrigues rotation matrix, for the vector to be rotated... Rotation operations can be represented in matrix-vector form: .

[0015] Similarly, a local Cartesian coordinate system for the antenna is established with the connection point between the antenna support rod and the surface of the 6DMA antenna as the origin, denoted as... In this coordinate system, the first... The first 6DMA surface The coordinates of each antenna are represented as follows: The rotation of the 6DMA antenna surface is determined by a unit vector. and angular velocity To characterize. In time slots During this period, it corresponds to the first The rotation transformation of the 6DMA antenna surface is determined by the rotation matrix. describe.

[0016] Control parameters may include: rotation control parameters for the support rod, rotation control parameters for the antenna surface, and scaling factor for the support rod. These parameters take into account the rotation of the antenna support rod, the rotation of the 6DMA antenna surface, and the scaling factor. The extension and retraction of the antenna support rod is characterized by the following: The time slot in the first The first 6DMA antenna surface The global position of an antenna can be represented by an cumulative control model. The effects of global and local scale control on the global position can be defined as follows: , , .

[0017] Subsequently, for The expression for the position control model can be derived as follows: ; in, Indicates the first The time slot in the first The expansion factor of each support rod Indicates the first The time slot in the first The rotation control parameters corresponding to each support rod Indicates the first The initial position of each support rod Indicates the first The time slot in the first Rotation control parameters corresponding to each antenna surface Indicates the first On the surface of the first antenna... The initial position of each antenna. A schematic diagram of the position control model is shown below. Figure 3 As shown, , Together with the initial rotation parameters, the first... The initial state of the surface of a 6DMA antenna.

[0018] For S1, a hyperplane is constructed based on the cumulative rotation matrix. The constraints on the support rod and antenna surface in the movable antenna system are determined based on the hyperplane and the position control model. These constraints may include: The cumulative rotation matrices corresponding to the support rod and the antenna surface are determined based on the rotation control parameters of the support rod and the antenna surface in the control parameters. The normal vectors of each antenna surface in the global coordinate system after rotation are determined based on the cumulative rotation matrix. A hyperplane is constructed using the center of each antenna surface and its corresponding rotated normal vector; The constraints on the support rod and antenna surface in the movable antenna system were confirmed based on the hyperplane and position control model.

[0019] Specifically, the complex multi-degree-of-freedom motion of a 6DMA system requires its antenna trajectory to adhere to strict physical constraints. Therefore, embodiments of this invention construct geometric constraints to ensure the safe and efficient operation of the system. To prevent signal reflection between any two 6DMA surfaces, a schematic diagram of electromagnetic characteristic constraints is provided, as shown below. Figure 4 As shown, a rotational direction constraint must be applied to the surface of each pair of 6DMA antennas.

[0020] In the Assuming the local and global coordinate systems of the 6DMA surface are consistent in orientation and scale, the local rotation matrix has an equivalent effect on the global and local normal vectors of the antenna surface. For the sequential rotation process of the antenna support rod and the antenna surface, the... The normal vectors of a 6DMA surface after transformation in the global coordinate system can be characterized by the cumulative product of rotation matrices across time slots. Specifically, the expression for the normal vectors after sequential rotation is: ; The expression for the cumulative rotation matrix is ​​as follows: ; The cumulative rotation matrix encapsulates the cumulative effect of the support rod rotation and surface rotation on the final orientation of the 6DMA antenna surface in each time slot, while This represents the initial outward unit normal vector at the global reference point. Then, the first... The centers of the 6DMA surfaces and their outward normal vectors in the global coordinate system construct a hyperplane, which can be represented as: ; For any point in space The hyperplane is composed of all vectors that satisfy the condition. With normal vector orthogonal points Composition. This hyperplane divides three-dimensional space into two half-spaces, one of which is a closed half-space, defined as: ; The closed half-space contains the point And any point within it With normal vector The vector formed An obtuse or right angle is formed between them. When When set to the center position of any different 6DMA antenna surface, for all and The constraints can be expressed as: ; This constraint ensures that the center of the other 6DMA antenna surfaces is not located along the first... Within an open half-space extending in the direction of the surface normal vector of a 6DMA antenna.

[0021] In addition, a schematic diagram of mechanical structure constraints, such as... Figure 5 As shown, robust constraints are necessary in a 6DMA system to prevent... Figure 5 The mechanical collisions and disturbances depicted in the text. Embodiments of the present invention employ a more efficient strategy based on geometric angle constraints. These angle constraints can serve as a feasible alternative to physical clearances and inherently follow the kinematic limits of the system. This approach ensures that the optimized trajectory is both physically achievable and operationally safe. As... Figure 5 As shown, to avoid collisions between the support rod on the 6DMA antenna surface and the 6DMA antenna surface itself, and to reduce the mutual coupling between different 6DMA surfaces, this embodiment of the invention imposes the following constraints on each pair of 6DMA antenna surfaces and support rods: For any ,in The surface constraints of paired 6DMA antennas can be expressed as: ; in, This indicates the angle threshold that ensures no collisions or coupling between 6DMA surfaces.

[0022] To prevent collisions between the 6DMA antenna surface and the base station (BS) structure, the following constraints must be met: ; in, This represents the minimum angle required to ensure that the 6DMA antenna surface does not collide with the base station (BS) structure. The outward unit normal vector represents the base station (BS) structure.

[0023] For any The constraint between the 6DMA antenna surface and its support rod can be expressed as: ; in, This indicates the angle at which the 6DMA antenna surface and its support rod may collide.

[0024] The constraints defined above are based on the geometric relationships between the centers of the 6DMA antenna surfaces. Due to the minimum angular constraint between the antenna centers, combined with the defined closed half-space constraint, severe signal reflections between the 6DMA antenna surfaces are effectively mitigated. Simultaneously, the defined constraints can ensure operational safety through sufficient safety margins. It is worth noting that compared to modeling all antenna element-level interactions, considering only the relationships between the centers of the 6DMA surfaces significantly reduces the number of constraints, thereby significantly reducing computational complexity and improving practical feasibility. This advantage is particularly pronounced in scenarios with large-scale antenna arrays.

[0025] Understandably, the above constraints are intended to maintain spatial separation, control relative orientation, and preserve functional integrity during movement.

[0026] For S1, based on the field response channel model and motion relationship, and according to the position control model, the channel vectors corresponding to the user equipment and the base station in each time slot are constructed, which may include: Based on the elevation and azimuth angles of the arrival angles of each channel path corresponding to each user equipment, determine the global pointing vectors corresponding to each channel path from the user equipment to the base station reference position in the global coordinate system. The guiding vectors corresponding to each antenna surface are obtained based on the global pointing vector and the position control model; Projecting the global pointing vector onto the local coordinate system yields the local pointing vector; Based on the local pointing vector, the local pitch angle and local azimuth angle are obtained; Based on the local elevation angle and local azimuth angle, a linear scale of the effective antenna gain of each antenna surface is obtained; The channel vectors corresponding to the user equipment and the base station in each time slot are determined based on the steering vector and linear scale.

[0027] Specifically, throughout the service period, the channel response between each 6DMA antenna surface and any user equipment (UE) within the network is determined by the spatial coordinates of the UE and the three-dimensional positioning and orientation parameters of the 6DMA antenna surface. This embodiment of the invention focuses on the uplink transmission phase and describes a channel model that controls signal propagation from each UE to all antenna elements distributed across multiple 6DMA antenna surfaces.

[0028] For a given time slot ,make and They represent from the first The first user equipment (UE) to the 6DMA base station (BS) reference location The elevation and azimuth (AOA) angles of arrival for each channel path, where , Indicates the number of receive channel paths. The direction vectors corresponding to each channel path in the global coordinate system are: ; No. The steering vector of each 6DMA antenna surface is determined by the phase difference between its antenna element and the base station (BS) reference position. Given... The positional relationships are defined through scaling and rotation. The phase difference between each antenna element and the reference position of the base station (BS) is: ,in This represents the carrier wavelength. Therefore, the steering vector is expressed as a function of rotation and scaling components: ; Effective antenna gain depends on the radiation pattern and the orientation of the 6DMA surface. For the first... Each surface, a global pointing vector Projecting its rotation matrix onto its local coordinate system, it can be represented as: ; because and Both are orthogonal matrices, therefore, using To simplify the calculation, the local pitch angle and azimuth angle (AOD) can be obtained as follows: ; in, It is a symbolic function that depends on the vector The sign of the component is adjusted by... The angle calculated by the function is defined as: ; Each antenna element has a half-power beamwidth in both the vertical and horizontal dimensions. and ,in, Therefore, the gain of the three-dimensional antenna element can be determined. It can be represented as: ; in, It includes vertical and horizontal radiation patterns, while This represents the maximum directivity gain of each antenna element in the main lobe direction. The vertical radiation pattern is defined as: ; in, This indicates sidelobe level limitation. The horizontal radiation pattern is given by the following formula: ; in, Indicates a comparison before and after, therefore, for the th The linear scale of the effective antenna gain of a 6DMA surface can be expressed as: .

[0029] Under the considered far-field channel model, the distance difference between the signal propagation path and each 6DMA antenna surface is negligible compared to the total signal transmission distance. Therefore, assuming that the path response coefficient for the same signal propagation path is the same across all antenna elements, the user equipment (UE)... With base station BS in the The channel vector between time slots can be represented as: ; in, , and They represent the first time. Control information for all antenna support rods and the surface of the 6DMA antenna in each time slot. From User Equipment (UE) The reference point within the coverage area of ​​the base station (BS) The path gain coefficient of each path.

[0030] The channel knowledge map (CKM) is essentially a location-space-rooted database that associates UE locations with channel characteristics to form a location-channel information mapping system. As large-scale propagation parameters determined by environmental geometry and UE locations, path loss, angle of attack (AOA), and angle of deviation (AOD) exhibit sparsity and slow temporal variation, making it feasible to pre-store a discretized angle set in the CKM. Assuming the existence of a perfect CKM containing the path loss and AOA information for each UE, the channel information constructed above, given all antenna control information, can be converted into a UE-specific channel mapping, expressed as: ; in, This represents the location-channel knowledge mapping function. and Representing time slots Environmental information and user equipment (UE) Location information. By utilizing a pre-built CKM, the base station (BS) can obtain channel information for all potential paths in advance, transforming traditional blind optimization that relies on real-time feedback into environment-aware communication planning based on prior knowledge.

[0031] Consider from Uplink transmission from a mobile user equipment (UE) to a base station (BS), the base station (BS) in the first... The signal received in each time slot can be represented as: ; in, Indicates from all Multiple access channels from each user equipment (UE) to all 6DMA antenna surfaces of the base station (BS). Denotes the transmitted signal vector, where each Indicates User Equipment (UE) The transmitted signal has an average power normalized to 1. This indicates that all user equipment (UE) has the same transmit power. The covariance matrix is Zero-mean additive white Gaussian noise. From the defined control model, it can be observed that the time slot... The effective antenna gain and steering vector are fundamentally determined by the cutoff time slot. The sum and capacity of the mobile antenna system are determined by all rotational components. Therefore, under the conditions of perfect CSI, optimal Gaussian signaling, and multi-user joint decoding at the base station (BS), the sum and capacity can be expressed as: ; It is worth noting that, unlike traditional fixed phased array systems, the achievable rate of a 6DMA system is determined by the effective channel variation introduced by the rotation of the 6DMA antenna surface and the rotation and extension / retraction motion of its support rod. To maximize the service performance of the 6DMA system throughout the entire operating cycle while ensuring stable system operation, this embodiment of the invention jointly optimizes the extension / retraction component of the antenna support rod, the rotation control parameters of the antenna support rod, and the rotation control parameters of the antenna plane to maximize the spectral efficiency of the 6DMA wireless system. Furthermore, physical motion constraints are incorporated to ensure engineering feasibility. The system's sum and capacity optimization problem model proposed in this embodiment of the invention is as follows: ; in, and These represent the maximum angular velocity of the antenna support rod and the single-slot angular velocity of the 6DMA antenna surface, respectively. Constraints P1(1)-(2) ensure that the rotational angular velocity of each support rod and the 6DMA antenna surface remains within mechanical limits. Constraint (3) restricts the spatial domain of the center of the 6DMA antenna surface to the 3D operating space predefined by the base station (BS). By limiting the incremental displacement of the 6DMA antenna surface within each time slot, constraints (4)-(5) ensure smooth, controllable motion and avoid unrealistic positional fluctuations. Constraint (6) utilizes the half-space projection of the antenna surface normal to mitigate inter-surface signal reflections caused by unfavorable directions. Geometric constraints (7)-(9) enforce minimum angular separation between antenna components to prevent mechanical collisions and reduce coupling.

[0032] The core of problem P1 lies in maximizing the total system capacity over the entire cycle, which is represented as a matrix logarithmic determinant function involving multidimensional rotation parameter programming. Although the logarithmic determinant function is convex under fixed linear channel parameters, the calculation of the steering vector and antenna gain depends on the trigonometric functions of the coupling angle variables and unit vectors, which excludes simple convex optimization methods. Furthermore, the dependence of the relative position and normal vector relationship on the cumulative rotation of the support rod and the cumulative rotation of the surface makes the objective function nonconvex in the joint space of position and rotation parameters, and the complex interaction of variables exacerbates the nonlinearity. Moreover, the geometric constraints (7)–(9) involve nonlinear boundaries formed by the combination of dot products, norms, and inverse cosine functions, which cannot be represented by linear or convex quadratic constraints, thus introducing an additional layer of nonconvexity.

[0033] To address the challenges of high-dimensional optimization space, non-convex variables, and complex geometric constraints in the constructed 6DMA optimization problem, this invention proposes a manifold-based optimization framework. This invention utilizes Lie group theory to reconstruct the problem onto a product manifold composed of a special orthogonal group and a set of positive real numbers. This manifold naturally contains the rotational and translational degrees of freedom of the mechanical structure. This method conforms to the Riemannian optimization principle, where rotations reside on the SO(3) manifold, while translations are embedded in the Euclidean subspace. By utilizing the inherent geometric properties of the manifold, problems such as singularities, computational redundancy, and geometrical distortions encountered in traditional parameterization methods like Euler angles can be avoided. Furthermore, by reconstructing the problem onto the manifold, the complex geometric constraints defining the rotation group can be intrinsically satisfied by the manifold structure itself. This method transforms the original constrained problem in Euclidean space into an unconstrained problem on the manifold, where physical constraints such as angular velocity limits and collision avoidance can be treated as smooth inequality constraints, thus avoiding the complexity and potential instability of traditional constrained optimization methods. Based on the above factors, this paper maps the problem to the manifold space for optimization. The following section will elaborate on how to represent and solve this problem in a manifold space.

[0034] Unprocessed optimization control parameters are addressed in this embodiment by mapping them to a manifold to reconstruct the optimization problem. For a special orthogonal group SO(3), its corresponding Lie algebra... The geometric features can be described as group identity elements. The tangent space at any point. For any and The polar rotation parameter set can be expressed as ,in, and Through antisymmetric isomorphism operators This can be mapped to Lie algebra form: ; in, Let Lie algebra be the sum of all third-order antisymmetric matrices. Expressing opposition to isomorphism operators, ( ) represents the rotation vector The Each component. Through exponential mapping. It can convert any element in a Lie algebra elements in its Lie group The exponential mapping is defined as: ; in, express The norm of the exponential map. As the inverse operation of the exponential map, the logarithmic map maps elements on a special orthogonal group manifold to its Lie algebra. Defined as: ; in, ; Similarly, surface rotation parameters Elements can be represented as Lie group elements in the local coordinate system using the same mechanism. and its corresponding Lie algebra elements For the other parameter , This can be viewed as a one-dimensional positive real number manifold. In this case, the total parameter space constitutes an unconstrained product manifold, defined as: .

[0035] The product manifold preserves smoothness and inherits Riemannian metrics. Thereafter, a notational approach can be used. Let represent any set of feasible solutions that conform to the manifold structure. The advantage of this reparameterization method is that the orthogonality constraints defining the effective rotation matrices can be intrinsically satisfied by the geometric properties of the SO(3) space. Therefore, the original problem involving complex matrix constraints in Euclidean space is transformed into an optimization problem on the manifold, where these rotation constraints are implicitly handled by the manifold structure, significantly simplifying the problem.

[0036] For S2, the exact penalty method using a hybrid strategy is used to reconstruct the constraints, resulting in a smooth, unconstrained problem on the manifold, which can include: For constraints that define simple boundaries in the constraint conditions, a reparameterization method is used to reconstruct them; The geometric and kinematic constraints in the constraints are reconstructed using algebraic transformation methods; For the state-dependent set constraints in the constraints, a logarithmic summation exponential function is used to smooth the penalty term of the state-dependent set constraints, so as to reconstruct the constraints and obtain a smooth unconstrained problem on the manifold.

[0037] Specifically, although the manifold optimization framework can elegantly handle the inherent geometric constraints of rotation, the constructed optimization problem is still limited by a series of physical and kinematic inequalities. These constraints are inherently nonsmooth due to their involvement in norms and conditional logic, and this nondifferentiability makes them incompatible with the gradient-based Riemannian optimization method employed in this embodiment of the invention. To address this issue, this embodiment of the invention employs a smoothing exact penalty method to reconstruct the constrained problem into a smooth, unconstrained problem on the manifold. A hybrid strategy is used to efficiently handle different types of constraints.

[0038] First, for constraints with simple boundaries, a reparameterization method is employed. This method improves numerical stability by constructing constraints that are enforced. Specifically, scaling factor constraints are processed through the following transformation: the unconstrained vector... Mapping to effective scaling factor : ; Secondly, for more complex geometric and kinematic constraints (from the index set) (represented), it is first transformed into its standard form through simple algebraic transformations. In this form, Indicates the first One constraint was violated. For example, the maximum angular velocity constraint. It can be refactored as: ; However, precise penalty items This approach is not suitable for standard Riemann gradient descent because the max operator is not differentiable. Therefore, this embodiment of the invention employs its continuously differentiable approximation—a logarithmic summation exponential function. For more complex state-dependent geometric constraints, a smoothing technique is directly applied to the penalty term. The constraint is reconstructed as the following penalty function: ; in, For global smoothing parameters, This is the penalty weight vector. The choice of these parameters involves a key trade-off: when When the summation is logarithmic, the exponential function approximates the true penalty more accurately, but the gradient exhibits numerical instability; conversely, when the penalty weights are... At this point, the solution is driven to strictly satisfy the corresponding constraints, but often at the cost of slow convergence or oscillation, because the optimization process prioritizes feasibility over improvement of the objective function. Furthermore, the physical constraints considered in the embodiments of this invention exhibit significant heterogeneity in dimension and scale, making the use of a single uniform penalty weight inefficient. To address this multi-scale challenge, the proposed method employs a vector-based approach... Each penalty weight A strategy of independent updates is adopted to achieve a more targeted and efficient convergence process.

[0039] By integrating the smoothing penalty term into the original objective function, the constrained optimization problem can be reconstructed into a product manifold. The expression for the smooth unconstrained optimization problem on the unconstrained product manifold is as follows: ; in, Describe the penalty objective function. This represents the original capacity optimization problem without considering constraints. Represents an index set. Indicates the first Each penalty weight, Represents the smoothing factor. Indicates the first Whether the constraint has been violated.

[0040] in Let represent the penalty objective function. The logarithmic summation exponential approximation ensures the differentiability of the penalty term, thus enabling the application of Riemann gradient-based optimization algorithms. In the unsmooth case, similar to the case in Euclidean space, Riemannian manifolds possess similar properties: finite penalty weights. It is sufficient to precisely satisfy the constraints, and when the penalty parameter is large enough, the solution to the penalized problem is strictly consistent with the solution to the original problem. This form transforms the constrained optimization problem on the manifold into an unconstrained problem by embedding equality and inequality constraints into the penalty term of the objective function, thus enabling the application of unconstrained optimization algorithms on Riemannian manifolds for solution.

[0041] A schematic diagram of Lie group embedding computation, such as Figure 6 As shown, Figure 6 A computation graph update mechanism with Lie group embeddings is demonstrated. This is achieved using the rotation matrix during forward propagation. For example, its gradient is not calculated in traditional Euclidean space, but in its corresponding tangent space. Backpropagation and gradient calculation are performed. The update vector obtained from the optimizer... Located in this tangent space, and then through Map back to SO(3) manifold for update .

[0042] For step S3, before executing the first outer loop in the double-loop structure, it is necessary to initialize the solution, penalty parameters, and smoothing factor in the smooth unconstrained optimization problem model to obtain the initial solution, initial penalty weight, initial smoothing factor, and initial gradient tolerance.

[0043] In a nested loop structure, the process of one inner loop can include: S321, based on the current solution of the current second outer loop. Set the hot start point for subproblems ; S322, using a logarithmic mapping to transform the points on the manifold Map to point The corresponding tangent space; S323, Calculating the penalty objective function based on tangent space The Riemann gradient; S324, by updating the Riemann gradient of the tangent space to the manifold through the exponential mapping, a new solution is obtained. ; S325, with a new solution replace Repeat steps S322-S325 until the gradient norm satisfies the current gradient tolerance or the number of inner loop iterations reaches the number of inner loop iterations. Output the solution of the subproblem as the solution of the smooth unconstrained optimization problem model for the next outer loop iteration.

[0044] Specifically, the long-term optimization problem proposed in this embodiment of the invention involves the computation of a three-dimensional transformation group. By constructing the capacity optimization problem of a 6DMA system within a Lie group manifold framework, gradient calculation and parameter updates are performed using the differential geometry properties of Lie groups. When the computation graph involves three-dimensional transformations, the common practice in traditional solution frameworks of embedding the manifold into Euclidean space can lead to critical problems: backpropagation may encounter singularities, resulting in gradient explosion; and complex group computation expressions can cause computation graph bloat. To solve this problem, this embodiment of the invention employs a tangent space-based backpropagation method. This method preserves the group structure of the three-dimensional transformation and directly propagates gradients in the tangent space of the manifold, avoiding the differential problems related to singularities in the embedding strategy and ensuring numerical stability. However, the interactions between Lie group elements cannot be solved using conventional addition and subtraction operations in Euclidean space. Therefore, based on logarithmic and exponential mappings, for any two elements... and vectors representing the elements of Lie algebras The addition and subtraction operations are redefined as follows: ; in, Operators utilize vectors in Lie algebras Lie group elements To cause a disturbance. Map the differences on the Lie group back to the tangent space. It is a binary operator, indicating that two group elements can be combined to form a third element. For functions on a Lie group In this embodiment of the invention, the differential is generalized to a perturbation in the tangent space, and its gradient is expressed as: ; in, It is a point Vectors in the tangent space. This formula establishes the vectors from... The disturbance in The mapping of perturbations in the equation. Substituting the orthonormal basis vectors into the above equation yields the Jacobian matrix. : ; in, Indicates that it is defined in The inner product over tangent space. It's important to note that in Euclidean space, exponential and logarithmic mappings are identity mappings, and group multiplication is used. It degenerates into vector addition. When both the input and output of the function are variables in Euclidean space, the above equation is consistent with the definition of the standard directional derivative. Therefore, this framework can handle both scaling and rotation components simultaneously.

[0045] To optimize the capacity of a 6DMA system, the entire calculation process, from initial control parameters to the final position and attitude of antenna elements and the channel matrix, involves... From construction to system capacity The computation—conceptualized as a computation graph, and Figure 6 The descriptions are consistent; some nodes in the graph represent elements on a Lie group manifold, and edges represent operations on those elements. Each node in the computational graph represents a Lie group. elements on Each edge represents a mapping function between Lie groups. To efficiently handle computational graphs involving manifold variables in practice, this embodiment of the invention utilizes the LieTorch extension library to perform backpropagation operations in the tangent space of each group element. During forward propagation, each function is evaluated in topological order, and the output is stored for backpropagation. Backpropagation propagates gradients in the reverse topological order. The computation method for functions not involving Lie group manifold elements is consistent with the traditional method of obtaining Euclidean gradients by calculating partial derivatives. Therefore, this embodiment of the invention focuses on how to obtain the gradients of functions involving Lie group manifold elements. It is assumed that there exist arbitrary functions containing manifold elements. Gradient backpropagation follows the chain rule: ; in, Is with Row vectors with the same dimension in the tangent space It is the Jacobian matrix. The above formula is used to calculate the Jacobian-vector product.

[0046] The algorithm proposed in this embodiment first initializes key parameters, which may include initial solutions on the manifold. Initial penalty weights Smoothing factor Gradient tolerance And iteration counters, etc., lay the foundation for subsequent optimizations. The optimization algorithm provided in step S3 adopts a two-layer loop structure: the outer loop (with...) The counter is responsible for dynamically adjusting the penalty parameter and smoothing factor, and the inner loop (with) (If it is a counter), then the subproblem is solved based on the current parameter configuration.

[0047] In the inner loop, the solution is first determined based on the current solution from the outer loop. Set the hot start point for subproblems To accelerate convergence, a logarithmic mapping is then used to map the points on the manifold. Mapping to its tangent space, we get ; The penalty objective function is calculated based on the tangent space. The Riemann gradient; Then, by using an exponential mapping, the gradient update of the tangent space is mapped back to the manifold to obtain a new solution. The inner loop iterates continuously until the gradient norm satisfies the current gradient tolerance. Or reach the maximum number of iterations This completes the solution of the subproblems.

[0048] After the inner loop finishes, the outer loop updates the iteration point. Solution to the subproblem And adjust key parameters according to the strategy: smoothing factor According to attenuation factor Reduce (not lower than the minimum value) (in order to gradually approximate the original constraints); Penalty weight By dynamically updating the two-stage strategy, a balance is achieved between goal optimization and constraint satisfaction; Gradient tolerance According to attenuation factor Reduce (not lower than the minimum value) This is done to gradually improve the accuracy of the solution.

[0049] The outer loop termination condition is: the manifold distance between two consecutive iterations is less than a threshold. Gradient tolerance and smoothing factor both reach their minimum values, or the number of iterations exceeds [a certain threshold]. The final output is the optimized 6DMA control parameters. .

[0050] This algorithm addresses geometric constraints such as rotations through manifold optimization, and balances optimization performance and physical feasibility by combining dynamic penalty and smoothing mechanisms, achieving efficient capacity optimization of a 6DMA system with complex constraints. The core challenge of backpropagation in Lie groups stems from the noncommutativity of group multiplication, which requires the use of adjoint operators to transform tangent space vectors between different reference frames. For Lie group elements... Accompanying operator It is a linear mapping used for parameterizing the right and left actions of the associated tangent space, satisfying: ; in, yes The tangent space vector in the local coordinate system. Specifically, for The adjoint operator is the rotation matrix itself, i.e. For group multiplication (in ),gradient Backpropagation to and .about The differential is: ; about The differential involves the adjoint operator to handle noncommutativity: ; Therefore, the gradient of backpropagation is: .

[0051] In Euclidean space, the gradient is directly represented as coordinate increments. However, in tangent space, the gradient must be represented as a local perturbation on the manifold. Increments in tangent space are mapped to a Lie group via an exponential map to achieve parameter updates. This is achieved using a defined exponential map and a defined differential function. This process can be represented as: ; Based on the Beck-Campbell-Hausdorff formula, we can derive: ; Specifically, for , Having a closed expression: .

[0052] When dealing with constrained optimization problems on Riemannian manifolds, the exact penalty method transforms the constrained problem into an unconstrained problem by introducing a penalty term. However, the penalty parameter... The choice is crucial. When When a certain threshold is exceeded, the local minimum solution of the penalty function problem coincides with the local minimum solution of the primal problem. Since this threshold is usually unknown, setting an excessively large initial value is problematic. This may worsen the condition number and slow down the convergence speed. Therefore, in the specific execution of S3, this embodiment of the invention proposes a two-stage adaptive penalty weight update strategy to dynamically balance the value function and constraint satisfaction. In the warm-up phase, initial penalty weights are allocated based on the proportion of constraint violations to ensure that the constraint terms and the objective function have comparable magnitudes: ; in, Indicates the first A measure of the smoothness of a constraint violation. This represents the current objective function value. In the formal phase, it is based on historical weights. The weights are adaptively adjusted based on the severity of the current constraint. ; in, It is the update step size and smoothing factor. The update method is as follows: ; This represents the minimum allowable value for the smoothing parameter. This represents the smoothing decay factor. Because... and Different values ​​of correspond to different solutions, in updating and Subsequently, in this embodiment of the invention, the newly formed problem is considered a sub-problem. The gradient tolerance determines the accuracy threshold for the convergence of the optimization algorithm, and its changes directly affect the iteration termination condition and the accuracy of the sub-problem solution. Reducing the gradient tolerance can encourage the algorithm to approach a more rigorous stationary point, improving the accuracy of the solution, but it increases the number of iterations and computation time. In the initial iteration phase, the algorithm tends to explore feasible solutions without requiring extremely high solution accuracy; therefore, this embodiment of the invention adopts a similar dynamic adjustment strategy for the gradient tolerance, and the update process can be expressed as: ; in, This is the gradient tolerance decay factor. Using exponential mapping, logarithmic mapping, and differential functions, the increment in the tangent space can be obtained, and then mapped onto the Lie group via exponential mapping to achieve parameter updates. The solution to the subproblem... It can be obtained through backpropagation in the tangent space. The update process can be represented as: ; in, express logarithmic mapping, The penalty objective function is expressed as follows: The Riemann gradient, It's the learning rate. It's the rate at which the maximum number of exploration steps is reached or a certain condition is met. When the iteration of the subproblem terminates, the iteration of the subproblem ends. In each update... , and Then, a warm start method was used to continue optimization. When the gradient norm satisfies: The problem is considered to have been approximately solved, and the iteration point is obtained. When the first The algorithm terminates when the following condition is met in the next iteration: ; in, express and The Riemann distance between them.

[0053] Representing the tangent space The Riemann norm on [the given value]. This indicates that the problem is approaching a local minimum, and the approximation is sufficiently close to the original problem. Theoretically, under exact computation, if the minimum step size [is such that]... Minimum smoothing parameter and minimum gradient tolerance If all values ​​are set to 0, and there exists a feasible limit point that satisfies the linear independent constraint specification, then this point satisfies the KKT conditions of the original problem. The optimization algorithm proposed in this embodiment integrates manifold optimization and an adaptive penalty mechanism to handle complex kinematic constraints, forming the core of the solution. For ease of subsequent performance evaluation, this embodiment refers to this overall method as kinematically-constrained manifold optimization (KC-MO) supported by CKM.

[0054] To evaluate the performance of the 6DMA system, this embodiment of the invention simulates detailed parameter configurations and scenario structures. At the parameter level, system and communication parameters include a 0.5-second time slot length, 200mW transmit power per user equipment (UE), a carrier wavelength of 0.125m, an average noise power of -50dBm, an initial distance range of 20-200m between the UE and the base station, and a moving speed of 1-6m / s. The 6DMA mechanical constraints include a 2-path receive channel, 4 antennas per surface, a scaling factor range of [0.95, 1.05], a support rod length range of [0.8, 1.5]m, a maximum displacement of 0.3m per time slot surface, and geometric constraints such as a minimum angle of π / 15 rad between any two surfaces, a minimum angle of π / 8 rad between a surface and the base station structure, and a minimum angle of π / 12 rad between a surface and the support rod. Based on this, a channel knowledge map (CKM) was constructed. Its structure example includes key information such as user equipment (UE) location, angle of arrival (AOA), and path loss mapping, providing realistic scenario support for the performance simulation of 6DMA systems.

[0055] The following embodiments of the invention evaluate the performance of the proposed 6DMA-BS design and algorithm in maximizing the total spectral efficiency through numerical results. The CKM used was generated using the intelligent ray tracing method. City maps, including those of Ankara and Berlin, were obtained from OpenStreetMap.

[0056] To simulate real-world operation, each 6DMA surface is initialized with a random position and orientation within a predefined operating space; for clarity, it is assumed that... To verify the effectiveness of the proposed solution, several benchmark solutions are introduced for performance comparison: FPA: The antenna is deployed in a fixed three-sector configuration with a standard downtilt angle of 15 degrees.

[0057] Ideal Adaptive Optimization (I-AO): In each communication time slot, the optimal position and orientation of all 6DMA surfaces are jointly determined using an adaptive optimization (AO) algorithm.

[0058] Kinematically Constrained Adaptive Optimization (KC-AO): The optimal pose is calculated for each time slot using the AO algorithm, and then the required motion is linearly scaled to meet the maximum angular velocity constraint.

[0059] Ideal line-of-sight tracking (I-LT): Each time slot uses the k-means algorithm to cluster the user equipment (UE) so that the normal vector of each antenna is aligned with the assigned cluster center.

[0060] Kinematic constraints without CKM manifold optimization (KC-WCKMMO): A manifold optimization framework is used to determine the antenna pose of the next time slot using the channel state information of the current time slot.

[0061] Ideal manifold optimization (I-MO): Applying a manifold optimization framework, there are no kinematic constraints on antenna motion.

[0062] A diagram comparing the cumulative achievable spectral efficiency with time slots for different antenna directional strategies, as shown below. Figure 7 As shown in the diagram, the peak instantaneous angular velocity versus time slot is compared for different antenna orientation strategies. Figure 8 As shown, the system is configured as fixed. User Equipment (UE) and Each 6DMA surface has a maximum angular velocity constraint of 0.1 rad / slot. Figure 7 and Figure 8 The performance comparison of various schemes is presented. The static FPA and I-LT methods constitute the baseline performance boundary; however, they cannot utilize available spatial degrees of freedom due to their inability to adapt to real multipath channel structures. The reactive KC-WCKMMO scheme initially performs well, but its performance subsequently declines sharply, a consequence of its short-sighted optimization strategy—this method leads to the antenna pose falling into a kinematically unrecoverable suboptimal state. In contrast, while the unconstrained I-AO has excellent theoretical performance, its required angular velocity is physically difficult to achieve. The KC-AO scheme enforces constraints through trajectory truncation and linear scaling, and its performance is even lower than the static baseline. The proposed KC-MO framework, while strictly satisfying all engineering constraints, successfully achieves near-optimal results comparable to the I-AO scheme. This is due to its end-to-end manifold optimization, which can solve the trajectory problem globally, avoiding the performance loss caused by iterative approximations or simple constraint processing.

[0063] A diagram illustrating the relationship between the sum of spectral efficiencies for different strategies and the number of user equipment, as shown below. Figure 9 As shown, the system is configured as fixed. Each time slot and For each surface, the maximum angular velocity is constrained to 0.25 rad / time slot. For example... Figure 9As shown, the sum of the spectral efficiency of all schemes improves with the increase of the number of user equipment (UE) K, thanks to multi-user diversity. Crucially, the performance gap between the proposed active KC-MO and reactive KC-WCKMMO widens with increasing K. Increased channel complexity amplifies the short-sighted decision-making drawbacks of KC-WCKMMO, as kinematic constraints make it increasingly difficult to correct past suboptimal poses. In contrast, FPA, I-LT, and KC-AO exhibit irregular performance fluctuations, lacking the adaptability to continuously utilize spatial degrees of freedom in dense user scenarios. This result highlights the superior robustness and scalability of the KC-MO algorithm proposed in this embodiment.

[0064] A schematic diagram illustrating the relationship between the sum of spectral efficiencies for different strategies and the number of antenna surfaces, as shown below. Figure 10 As shown, the system is configured as fixed. Each time slot and For each user equipment (UE), the maximum angular velocity constraint is 0.25 rad / timeslot. Figure 10 As can be seen, although increasing the number of 6DMA surfaces improves the performance of all schemes, the degree of improvement varies significantly. Benchmark methods such as I-LT and KC-AO only show small or irregular gains and cannot effectively utilize the additional degrees of freedom. Notably, the performance gap between KC-MO and KC-WCKMMO exhibits a non-monotonic trend: when A is small, KC-MO's long-term planning has a significant advantage in resource-constrained environments; when A is large, the rich degrees of freedom allow even the short-sighted KC-WCKMMO to find an effective configuration with less kinematic cost, thus narrowing the gap. This dynamic highlights the higher spatial resource efficiency of the proposed framework in complex kinematically constrained scenarios.

[0065] A schematic diagram illustrating the relationship between the sum of spectral efficiencies for different strategies and the maximum angular velocity limit, as shown below. Figure 11 As shown, the system is configured as fixed. Each surface Each time slot and One UE. Figure 11 The impact of the maximum angular velocity limit on system performance is evident. The performance of the KC-MO scheme proposed in this embodiment improves with increasing speed, but the gain gradually decreases at higher speeds. This indicates that once the system has sufficient flexibility to track optimal channel changes, the marginal benefit of further increasing the speed diminishes. In stark contrast, the short-sighted KC-WCKMMO scheme exhibits an almost linear performance improvement—this reactive strategy continuously utilizes greater maneuverability to compensate for the short-sighted decisions of previous time slots. Meanwhile, the KC-AO scheme consistently performs poorly and is unstable because its simple trajectory truncation method struggles to generate coherent and effective paths regardless of available speeds.

[0066] Therefore, the simulation results above demonstrate that the KC-MO framework proposed in this embodiment of the invention can achieve near-optimal performance even under moderate, physically achievable speed constraints, significantly outperforming the benchmark scheme.

[0067] Understandably, this invention describes the active tracking and coverage task of mobile user equipment (UE) as a long-term 6DMA trajectory optimization problem, aiming to maximize the total spectral efficiency during the service period. A novel long-period kinematic architecture is proposed, utilizing the Rodrigues rotation matrix to accurately model the channel's dependence on the six degrees of freedom of the antenna. A series of key physical feasibility constraints are systematically integrated, including maximum angular velocity, displacement limits, minimum safe distance between antennas, and avoidance of signal self-reflection, thereby ensuring the engineering feasibility of the optimization results. Proactive planning is achieved by introducing a channel knowledge map, using pre-stored UE location-channel characteristic mapping and predictive information to transform traditional "reactive" control into "active" planning, eliminating performance loss caused by channel information lag. Based on a manifold-based optimization framework, the rotation parameters of the 6DMA system are mapped to a special orthogonal group manifold for optimization. Lie group operations such as exponential and logarithmic mappings are used to handle rotation constraints, avoiding the singularity and computational redundancy introduced by traditional parameterization.

[0068] The above description is merely a preferred embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention are included within the scope of protection of the present invention.

Claims

1. A method for active tracking coverage based on channel knowledge map in a mobile antenna system, characterized in that, The application relates to a method for optimizing control parameters of a movable antenna system. The method comprises the following steps: constructing a position control model according to the control parameters; constructing a hyperplane based on a cumulative rotation matrix, and confirming constraint conditions of support rods and antenna surfaces in the movable antenna system according to the hyperplane and the position control model; constructing channel vectors of user equipment and a base station corresponding to each time slot according to the position control model based on a field response channel model and a motion relationship; obtaining sum capacity of the movable antenna system based on the channel vectors of the user equipment and the base station corresponding to each time slot by using a channel knowledge map; and determining a sum capacity optimization problem model according to the constraint conditions and the sum capacity of the movable antenna system; expressing the control parameters as an unconstrained product manifold which maintains smoothness and inherits a Riemann metric; reconstructing the constraint conditions by using a mixed strategy precise penalty method to obtain a smooth unconstrained problem on the manifold; and reconstructing the sum capacity optimization problem model into a smooth unconstrained optimization problem model on the unconstrained product manifold based on the smooth unconstrained problem; optimizing the smooth unconstrained optimization problem model on the unconstrained product manifold by using a double-layer circulation structure, iteratively updating a solution, a penalty weight and a smooth factor in the smooth unconstrained optimization problem model through an outer loop until a preset outer loop stop condition or an outer loop iteration number is met, and outputting the optimized control parameters; and in each outer loop, solving a parameter configuration sub-problem in the current outer loop through an inner loop until a preset inner loop stop condition or an inner loop iteration number is met, and outputting a solution of the sub-problem as a solution of the smooth unconstrained optimization problem model in the next outer loop; 2. The method of claim 1, wherein, completing active tracking coverage according to the optimized control parameters. The control parameters comprise:

3. The method of claim 1, wherein, a rotation control parameter of the support rod, a rotation control parameter of the antenna surface and an extension factor of the support rod. ； in, Indicates the first The time slot in the first The expansion factor of each support rod Indicates the first The time slot in the first The rotation control parameters corresponding to each support rod Indicates the first The initial position of each support rod Indicates the first The time slot in the first Rotation control parameters corresponding to each antenna surface Indicates the first On the surface of the first antenna... The initial position of each antenna.

4. The method of claim 2, wherein, An expression of the position control model is as follows: The method comprises the following steps: determining a cumulative rotation matrix corresponding to the support rod and the antenna surface according to the rotation control parameter of the support rod and the rotation control parameter of the antenna surface in the control parameters; confirming a rotated normal vector of each antenna surface in a global coordinate system according to the cumulative rotation matrix; constructing a hyperplane by using a center of each antenna surface and the corresponding rotated normal vector; 5. The method of claim 1, wherein, confirming the constraint conditions of the support rod and the antenna surface in the movable antenna system according to the hyperplane and the position control model. The method comprises the following steps: confirming global pointing vectors of each channel path of each user equipment to a reference position of the base station in the global coordinate system according to a pitch angle and an azimuth angle of an angle of arrival of each channel path corresponding to each user equipment; obtaining a steering vector corresponding to each antenna surface according to the global pointing vectors and the position control model; projecting the global pointing vectors into a local coordinate system to obtain local pointing vectors; obtaining a local pitch angle and a local azimuth angle according to the local pointing vectors; Based on the local elevation angle and the local azimuth angle, a linear scale of the effective antenna gain of each antenna surface is obtained; According to the steering vector and the linear scale, a channel vector corresponding to each time slot is confirmed between the user equipment and the base station.

6. The method of claim 1, wherein, The expression of the sum capacity of the movable antenna system is as follows: ； wherein denotes a control parameter , and corresponding and capacities, denotes a covariance matrix of an additive white Gaussian noise with zero mean, denotes a transmit power of a user equipment, denotes an average noise power, denotes a multi-address access channel of a user equipment to all antenna surfaces of a base station, denotes a Hermitian matrix.

7. The method of claim 1, wherein, The exact penalty method with the hybrid strategy is used to reconstruct the constraint condition, and a smooth unconstrained problem on the manifold is obtained, which includes: For the constraint condition defining a simple boundary, a reparameterization method is used to reconstruct the constraint condition; For the geometric and kinematic constraints in the constraint condition, an algebraic change method is used to reconstruct the constraint condition; For the state-dependent set constraint in the constraint condition, a logarithmic sum exponential function is used to smooth the penalty term of the state-dependent set constraint, so as to reconstruct the constraint condition and obtain a smooth unconstrained problem on the manifold.

8. The method of claim 1, wherein, The expression of the smooth unconstrained optimization problem model on the unconstrained product manifold is as follows: ； wherein, represents a penalized objective function, represents an original capacity optimization problem without considering constraints, represents a set of indices, represents the th penalty weight, represents a smoothing factor, represents whether the th constraint is violated.

9. The method of claim 1, wherein, Before performing the first outer loop in the double-loop structure, the solution, the penalty parameter and the smoothing factor in the smooth unconstrained optimization problem model need to be initialized to obtain an initial solution, an initial penalty weight, an initial smoothing factor and an initial gradient tolerance.

10. The method of claim 1, wherein, In the process of one inner loop in the double-loop structure, it includes: S321, the current solution based on the current outer loop iteration Setting a warm start point for a subproblem ; S322, mapping points on the manifold to points by a log map corresponding tangent spaces; S323, compute a penalized objective function based on tangent space Riemannian gradient; S324, mapping the Riemannian gradient update of the tangent space to the manifold by exponential mapping to get a new solution ; S325, replace the new solution S325, replace the new solution The steps S322-S325 are executed in a loop until the gradient norm meets the current gradient tolerance or the inner loop iteration number reaches the inner loop iteration number, and the solution of the sub-problem is output as the solution of the smoothing unconstrained optimization problem model in the next outer loop.

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