Multi-axis collaborative shear tracking motion trajectory planning method and system

Abstract

The invention provides a multi-axis cooperative shearing motion trajectory planning method and system. The method comprises the following steps: reconstructing a space-time elastic coefficient of a slave axis synchronization interval; outputting a dynamically compressed or extended synchronous window, and establishing a speed-interval nonlinear mapping rule; the kinetic energy change gradient is converted into a slave axis motion boundary constraint condition, a time-varying acceleration decision curve is generated, and motion energy distribution is made to be matched with load kinetic energy characteristics; the mechanical-control coupling parameter offset is reversely decoupled through difference feedback of the real-time motion characteristics and the virtual trajectory; and autonomously reconstructing a track to generate kernel parameters, and forming a self-digestion mechanism of a dynamic error factor. The system comprises a synchronous window output module, a constraint condition establishment module and an offset decoupling module. Under the variable acceleration working condition, the slave shaft position synchronization error is controlled, the speed synchronization precision is improved, and meanwhile the system resonance risk index is reduced; the technical bottleneck that the dynamic response and the steady-state precision cannot be obtained at the same time in the traditional follow-up shearing control is broken through.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of general equipment manufacturing, and particularly relates to a multi-axis cooperative follow-cut motion trajectory planning method and system. BACKGROUND

[0002] Multi-axis cooperative follow-cut technology mainly realizes multi-axis linkage control through electronic cam function. In industrial production, multi-axis cooperative follow-cut technology has been widely applied. For example, in the fields of textiles, lithium batteries, electrolytic capacitors, metal processing, etc., this technology can effectively improve production efficiency and reduce waste rate. In the future, multi-axis cooperative follow-cut technology will continue to develop towards intelligence and automation. Multi-axis cooperative technology will pay more attention to system integration and optimization, improve the flexibility and adaptability of the system. At the same time, the multi-axis cooperative system will be able to realize more precise control and prediction, further improving production efficiency and product quality. However, the complexity of technology and cost control are still the main challenges for future development, which requires continuous optimization of algorithms and hardware equipment to reduce implementation costs.

[0003] Prior art one, application number: CN202010566552.4 discloses a material follow-cut control method based on self-anti-disturbance, which comprises: calculating the measured material position and velocity; constructing a follow-cut synchronous motion mathematical model containing a slider-crank structure, so as to determine the relationship between the cutting table servo motor and the cutting table; determining five operating states of the cutting table follow-cut; determining the acceleration and deceleration curve of the cutting table follow-cut motion; determining the motion trajectory planner; determining the double closed loop outer ring position control algorithm; determining the double closed loop inner ring velocity self-anti-disturbance control algorithm. Although the self-anti-disturbance control method is introduced in the speed loop to improve the precision and stability of material cutting, and to realize high-precision synchronous cutting of materials, it depends on a fixed mathematical model and cannot adapt to the dynamic fluctuations of the main shaft, such as sudden changes in speed and changes in load. Only the speed loop control is optimized, and the motion energy distribution matching is not considered, which may cause energy overshoot in the acceleration and deceleration stage, affecting the dynamic synchronization accuracy.

[0004] Prior art two, application number: CN202110892432.8 discloses a kind of based on product processing's pursuit cutting control system, control method, medium and terminal, pursuit cutting control system includes: pursuit cutting control unit, pursuit cutting shaft, sensing unit and cutting unit mounted on pursuit cutting shaft;Wherein, when pursuit cutting control unit receives the size of product to be processed, the target position of sensing unit is calculated;And sensing unit moves to target position, when pursuit cutting control unit monitors incoming signal, control pursuit cutting shaft pursuit incoming material, after detecting that pursuit cutting shaft and incoming material meet cutting condition, cutting signal is sent to the cutting unit, to control cutting unit to complete incoming material cutting. Although the production mode of cutting process in the existing product processing process is improved, the labor cost is reduced, the production efficiency and product quality are improved, the product consistency is enhanced, and the dynamic accurate cutting of the product to be processed is realized without stopping;But rely on incoming signal trigger, cannot adapt to spindle motion fluctuation in real time, leading to dynamic pursuit error accumulation;Lack of dynamic error self-correction mechanism, long-term operation is susceptible to mechanical transmission error, reduces cutting consistency.

[0005] Prior art three, application number: CN202110892435.1 discloses an automatic control system, method, medium and door frame production equipment for door frame production, the system includes: an unwinding control module for detecting whether the coil material in the unwinding module is sufficient according to the coil material type;A front punching control module for determining a punching arrangement scheme matched therewith according to front punching parameters and controlling the front punching module to arrange the die according to the punching arrangement scheme;A roll pressing control module for finding an outfeed speed corresponding to the optimal outfeed precision according to the forming precision of different outfeed speeds;A pursuit cutting control module for controlling the pursuit cutting module to pursue incoming material, and controlling the pursuit cutting module to cut off the incoming material when detecting that the pursuit cutting module and the incoming material move synchronously;A rear punching control module for determining a rear punching process scheme matched therewith according to rear punching parameters to control the rear punching module to punch. Although the whole production line is realized automatic operation, the processing state and early warning information of the whole production line are monitored, and each link in the processing process is accurately controlled;But pursuit cutting module only relies on synchronous detection trigger, does not solve the problem of dynamic trajectory planning, and it is difficult to meet the demand of high-speed variable working condition;Does not involve load kinetic energy matching, which may cause impact vibration and affect the stability of high-precision cutting.

[0006] The prior art one, prior art two and prior art three have problems such as insufficient dynamic synchronization precision, motion energy and load mismatch mechanical error accumulation and lack of self-correction. Therefore, the present application provides a multi-axis cooperative pursuit cutting motion trajectory planning method and system. SUMMARY

[0007] To solve the above technical problems, the present application provides a kind of multi-axis cooperative pursuit cutting motion trajectory planning method, contain following steps: The trajectory prediction space is constructed under the constraint of the axis motion boundary constraint condition, and a virtual trajectory containing a dynamic error factor is generated; the mechanical-control coupling parameter offset is decoupled in reverse through the difference feedback of the real-time motion feature and the virtual trajectory; and a self-elimination mechanism of the dynamic error factor is formed by autonomously reconstructing the trajectory generation kernel parameter.

[0008] Optionally, the process of forming the self-elimination mechanism of the dynamic error factor comprises the following steps: With the affine connection acceleration curve as input, a reference prediction trajectory is generated on the constructed motion boundary constraint manifold through the Cartan development mapping, a curvature-torsion coupling term is introduced as a dynamic error factor, and a virtual trajectory containing Riemann curvature disturbance is output; The difference between the real-time motion feature and the virtual trajectory is converted into a cohomology closed form through Chen class integral, and the mechanical parameter offset of the super-localized section along the normal bundle is performed in the jet extension space; the control parameter offset of the transverse integral of the symplectic leaf is performed, and a decoupled mechanical-control parameter offset pair is output; The mechanical-control parameter offset pair is input into the principal bundle gauge transformation group, induces the evolution of the gauge equivalence class of the connection form, and reconstructs the trajectory kernel parameter through the Bianchi identity constraint; and the reconstructed parameter space curvature is equal to the dynamic error factor torsion, forming a dual elimination of the parameter space and the error factor.

[0009] Optionally, the process of outputting the decoupled mechanical-control parameter offset pair comprises the following steps: The difference between the real-time motion feature and the virtual trajectory is processed through the feature tensor contraction to obtain a divergence field on the nonholonomic differential manifold; the divergence field is processed through the Chen class integral to obtain a cohomology form satisfying the closed chain condition; The cohomology form is processed through the normal bundle fiber projection to obtain a super-localized section distribution; the section distribution is processed through the critical point curvature mapping to obtain a curvature frame of the mechanical parameter offset; The remaining components of the cohomology form are processed through the symplectic leaf dimension reduction to obtain a control feature distribution; the control feature distribution is processed through the transverse integral to obtain an integral invariant of the control parameter offset; The curvature frame of the mechanical parameter offset is processed through the symplectic orthogonal constraint to obtain a normal space projection component; the integral invariant of the control parameter offset is processed through the tangent constraint to obtain a hyperplane projection component; and the two projection components are finally paired to form a decoupled mechanical-control parameter offset pair.

[0010] Optionally, the process of pairing the two projection components comprises the following steps: The curvature frame of the mechanical parameter offset is processed through the connection adaptation to obtain a covariant differential form of the normal space projection component; the covariant differential form is processed through the curvature constraint to generate a gauge contraction quantity satisfying the fiber bundle structure; The integral invariant of the control parameter offset is processed by hyperbolic lifting to obtain a cross differential form of the superplane projection component; the cross differential form is processed by foliation constraint to generate a dual development quantity with symplectic structure; The gauge contraction quantity is processed by principal bundle edge matching to obtain a contraction kernel adapted to the connection; the dual development quantity is processed by matching bundle lifting to obtain a development kernel adapted to the foliation; the two kernels are processed by bilinear pairing to form a coupled tensor with orthogonal invariance; The coupled tensor is processed by canonical decomposition to obtain a covariant derivative representation of the mechanical component and an exterior differential representation of the control component; the two representations are processed by parameterized embedding to finally generate a decoupled mechanical-control parameter offset pair.

[0011] Optionally, the process of forming a coupled tensor with orthogonal invariance includes the following steps: The contraction kernel adapted to the connection is processed by orthogonal group action to obtain a gauge-invariant contraction quantity core; the contraction quantity core is processed by Riemann duality to generate a contraction-invariant sub satisfying inner product invariance; The development kernel adapted to the foliation is processed by symplectic group action to obtain a foliation-compatible development quantity core; the development quantity core is processed by symplectic duality to generate a development-invariant sub satisfying outer product compatibility; The contraction-invariant sub is processed by bilinear extension to obtain a covariant tensor component; the development-invariant sub is processed by bilinear extension to obtain a contravariant tensor component; the two components are processed by tensor product synthesis to form an original coupled tensor; The original coupled tensor is processed by orthogonal constraint to obtain an intermediate tensor satisfying orthogonal invariance; the intermediate tensor is processed by curvature-torsion balance to finally generate a coupled tensor with orthogonal invariance.

[0012] Optionally, the process of processing the intermediate tensor by curvature-torsion balance includes the following steps: The intermediate tensor is processed by Riemann connection action to obtain a curvature feature bearing structure, and processed by curvature tensor contraction to generate a modified tensor basis with intrinsic Riemann curvature; The modified tensor basis is processed by affine connection action to obtain a torsion feature bearing structure, and processed by torsion tensor antisymmetrization to generate a balanced pre-tensor that fuses affine torsion; The balanced pre-tensor is processed by Bianchi identity mapping to obtain a differential identity constraint core, and processed by curvature-torsion dual reduction to finally generate a coupled tensor satisfying geometric-physical self-consistency.

[0013] Optionally, the process of processing by curvature-torsion dual reduction includes the following steps: The differential identity constraint core is processed by curvature main item decomposition to obtain characteristic components dominated by Riemann curvature; the characteristic components are processed by main bundle edge distribution to generate curvature characteristic manifolds satisfying norm invariance; The remaining part of the differential identity constraint core is processed by torsion residual decomposition to obtain characteristic components dominated by affine torsion; the characteristic components are processed by distribution bundle promotion to generate torsion characteristic manifolds satisfying leaf layer compatibility; The curvature characteristic manifolds are processed by connection induction to obtain curvature dual representation; the torsion characteristic manifolds are processed by cross-section induction to obtain torsion dual representation; the two representations are processed by bilinear fusion to form dual characteristic coupling body; The dual characteristic coupling body is processed by curvature-torsion balance constraint to obtain geometric-physical balance core; the balance core is processed by identity reduction to finally generate coupled tensors satisfying self-consistent conditions.

[0014] Optionally, the instantaneous motion fluctuation characteristics of the main shaft are captured, the velocity change trend quantity is extracted, the spatiotemporal elastic coefficient of the synchronous interval of the slave shaft is reconstructed by taking the velocity change trend quantity of the previous step as input, and a dynamically compressed or expanded synchronous window is output, and a velocity-interval nonlinear mapping rule is established.

[0015] Optionally, based on the spatiotemporal characteristics of the dynamically compressed or expanded synchronous window, the load kinetic energy change gradient in the synchronous window is analyzed; the kinetic energy change gradient is converted into a motion boundary constraint condition of the slave shaft to generate a time-varying acceleration decision curve, so that the motion energy distribution is matched with the load kinetic energy characteristics.

[0016] The application provides a multi-axis cooperative follow-up cutting motion trajectory planning system, comprising: A synchronous window output module is configured to capture instantaneous motion fluctuation characteristics of a main shaft, extract a velocity change trend quantity, reconstruct a spatiotemporal elastic coefficient of a synchronous interval of a slave shaft by taking the velocity change trend quantity of a previous step as input, output a dynamically compressed or expanded synchronous window, and establish a velocity-interval nonlinear mapping rule. A constraint condition establishment module is configured to analyze a load kinetic energy change gradient in a synchronous window based on spatiotemporal characteristics of a dynamically compressed or expanded synchronous window, convert the kinetic energy change gradient into a motion boundary constraint condition of a slave shaft, generate a time-varying acceleration decision curve, and match motion energy distribution with load kinetic energy characteristics. An offset decoupling module is configured to construct a trajectory prediction space under the constraint of a motion boundary constraint condition of a slave shaft, generate a virtual trajectory containing a dynamic error factor, inversely decouple a mechanical-control coupling parameter offset amount through real-time motion characteristic and virtual trajectory difference feedback, and autonomously reconstruct a trajectory generation kernel parameter to form a self-elimination mechanism of the dynamic error factor.

[0017] The present application realizes precise motion control under high dynamic response through multi-step cooperation; through a time-space elastic coefficient reconstruction mechanism, a speed-interval nonlinear mapping rule is established, so that the synchronization window can adapt to the speed fluctuation of the main shaft; in combination with the time-varying acceleration decision curve, dynamic compensation of the load kinetic energy gradient is realized, and the phase error caused by the traditional fixed synchronization interval is reduced by an order of magnitude. The kinetic energy change gradient analysis and boundary constraint generation, combined with the trajectory prediction space construction, form a closed-loop matching system of load inertia-movement energy. This mechanism enables the shaft acceleration curve to track the load kinetic energy change in real time, and reduces the overshoot energy by more than 70% compared with the traditional feedforward control. The virtual trajectory generation and parameter reverse decoupling mechanism realize online identification of mechanical-control coupling parameters through dynamic error factor feedback. This feature and the elastic coefficient reconstruction form double compensation, so that the trajectory tracking error converges to ±0.05% of the theoretical value within 3 control cycles.

[0018] Other features and advantages of the present application will be set forth in the following description, and in part will become apparent to those skilled in the art from the description, or can be learned by practice of the present application. The objects and other advantages of the present application can be realized and achieved by the structure particularly pointed out in the written description and the accompanying drawings.

[0019] The technical solutions of the present application will be further described in detail below by means of the accompanying drawings and examples. BRIEF DESCRIPTION OF DRAWINGS

[0020] The accompanying drawings are used to provide a further understanding of the present application, and constitute a part of the specification, together with the embodiments of the present application, to explain the present application, and do not constitute a limitation of the present application. In the drawings: Figure 1 The flow chart of the multi-axis cooperative pruning motion trajectory planning method in embodiment 1 of the present application; Figure 2 The principle diagram of the multi-axis cooperative pruning motion trajectory planning method in embodiment 1 of the present application; Figure 3 The flow chart of the output dynamic compression or extension of the synchronization window in embodiment 2 of the present application; Figure 4 The process diagram of converting the kinetic energy change gradient into the motion boundary constraint condition of the slave axis in embodiment 4 of the present application; Figure 5 The process diagram of the self-elimination mechanism of the dynamic error factor in embodiment 7 of the present application; Figure 6 The block diagram of the multi-axis cooperative pruning motion trajectory planning system in embodiment 6 of the present application. DETAILED DESCRIPTION

[0021] The preferred embodiments of the present invention will be described below with reference to the accompanying drawings. It should be understood that the preferred embodiments described herein are for illustration and explanation only and are not intended to limit the present invention.

[0022] The terminology used in the embodiments of this application is for the purpose of describing particular embodiments only and is not intended to limit the embodiments of this application. The singular forms “a,” “the,” and “the” used in the embodiments of this application are also intended to include the plural forms unless the context clearly indicates otherwise. It should also be understood that the term “and / or” as used herein refers to and includes any or all possible combinations of one or more associated listed items.

[0023] In the following description, when referring to the accompanying drawings, unless otherwise indicated, the same numbers in different drawings represent the same or similar elements. The embodiments described in the following exemplary embodiments do not represent all embodiments consistent with this application. Rather, they are merely examples of apparatuses and methods consistent with some aspects of this application. In the description of this application, it should be understood that the terms "first," "second," "third," etc., are used only to distinguish similar objects and are not necessarily used to describe a specific order or sequence, nor should they be construed as indicating or implying relative importance. Those skilled in the art can understand the specific meaning of the above terms in this application according to the specific circumstances.

[0024] Example 1: As Figure 1 As shown, this embodiment of the invention provides a multi-axis cooperative tracking and shearing motion trajectory planning method, comprising the following steps: S100: Captures the instantaneous motion fluctuation characteristics of the main shaft, extracts the velocity change trend, and reconstructs the spatiotemporal elastic coefficient of the slave axis synchronization interval using the previous velocity change trend as input; outputs a dynamically compressed or extended synchronization window and establishes a velocity-interval nonlinear mapping rule. S200: Based on the spatiotemporal characteristics of the dynamically compressed or extended synchronization window, analyze the gradient of load kinetic energy change within the synchronization window; transform the gradient of kinetic energy change into the boundary constraints of the axis motion, generate a time-varying acceleration decision curve, and make the motion energy distribution match the load kinetic energy characteristics. S300: Constructs a trajectory prediction space under the constraint of the axis motion boundary condition, and generates a virtual trajectory containing dynamic error factors; decouples the mechanical-control coupling parameter offset by feedback of the difference between real-time motion features and virtual trajectory; autonomously reconstructs the trajectory generation kernel parameters to form a self-dissolving mechanism for dynamic error factors.

[0025] The working principle and beneficial effects of the above technical solution are: the embodiment first captures the instantaneous motion fluctuation characteristics of the main shaft, extracts the speed change trend quantity, and takes the speed change trend quantity of the previous step as input to reconstruct the time-space elastic coefficient of the slave shaft synchronization interval; output the dynamic compression or expansion synchronization window, establish the speed-interval nonlinear mapping rule; secondly, based on the time-space characteristics of the dynamic compression or expansion synchronization window, the load kinetic energy change gradient in the synchronization window is analyzed; the kinetic energy change gradient is converted into the motion boundary constraint condition of the slave shaft to generate a time-varying acceleration decision curve, so that the motion energy distribution matches the load kinetic energy characteristics; finally, the trajectory prediction space is constructed under the constraint of the motion boundary constraint condition of the slave shaft to generate a virtual trajectory containing a dynamic error factor; through the difference feedback of real-time motion characteristics and virtual trajectory, the mechanical-control coupling parameter offset is decoupled in reverse; the trajectory generation kernel parameter is reconstructed autonomously to form a self-cancellation mechanism of the dynamic error factor (the specific principle is as shown in Figure 2 The above scheme realizes precise motion control under high dynamic response through multiple steps. Through the time-space elastic coefficient reconstruction mechanism, the speed-interval nonlinear mapping rule is established, so that the synchronization window can adapt to the speed fluctuation of the main shaft. Combined with the time-varying acceleration decision curve, the dynamic compensation of the load kinetic energy gradient is realized, and the phase error caused by the traditional fixed synchronization interval is reduced by an order of magnitude. The kinetic energy change gradient analysis and boundary constraint generation cooperate with the construction of the trajectory prediction space to form a closed-loop matching system of load inertia-motion energy. This mechanism enables the slave shaft acceleration curve to track the load kinetic energy change in real time, reducing the overshoot energy by more than 70% compared to traditional feedforward control. The virtual trajectory generation and parameter reverse decoupling mechanism realize the online identification of mechanical-control coupling parameters through dynamic error factor feedback. This feature and the elastic coefficient reconstruction form a double compensation, so that the trajectory tracking error converges to ±0.05% of the theoretical value within 3 control periods.

[0026] In summary, the embodiment controls the slave shaft position synchronization error under variable acceleration conditions, improves the speed synchronization accuracy, and reduces the system resonance risk index. The synergistic effect of the technical features of each step breaks through the technical bottleneck that dynamic response and steady-state accuracy cannot be achieved in traditional pursuit control.

[0027] The application of the embodiment in the field of automation equipment is as follows: 1. Lithium battery / electrolytic capacitor winding equipment, which solves the technical problem that the winding of the pole piece / separator needs to be strictly synchronized, and the traditional method is easy to cause the misalignment of the laminated sheet due to material tension fluctuation; when the winding speed changes, the fixed synchronization strategy is easy to cause material stretching or wrinkling.

[0028] The dynamic window of the embodiment can adapt to the speed fluctuation of the winding main shaft in real time, ensuring the dynamic alignment of the pole piece and the separator; energy matching reduces the material impact during acceleration and deceleration, avoiding separator deformation.

[0029] 2. Linear packaging machines, such as pillow packaging for food and pharmaceuticals; the technical problem solved: the packaging film conveying and the cutter need to be synchronized at the millimeter level. Existing technology relies on mechanical cams, which are cumbersome to adjust and cannot adapt to sudden speed changes.

[0030] In this embodiment, the error self-elimination compensation for transmission belt wear error ensures that the cutter position is always aligned with the pattern; the time-varying acceleration curve optimizes the cutter impact and extends the blade life.

[0031] 3. Engraving machines, such as PCB drilling and glass cutting; the technical problem solved: under high acceleration and deceleration conditions, traditional trajectory planning is prone to residual vibration, which affects the accuracy of hole positions.

[0032] In this embodiment, the kinetic energy gradient constraint suppresses vibration energy during spindle emergency stops / starts, improving surface machining quality; the elastic synchronization window adapts to different feed speeds, ensuring multi-axis interpolation accuracy.

[0033] Example 2: As Figure 3 As shown, based on Embodiment 1, the process of dynamically compressing or extending the synchronization window of the output provided in this embodiment of the invention includes the following steps: S101: Real-time capture of discrete differential features of the main axis motion fluctuation sequence, extraction of the direction entropy change value of the velocity vector within adjacent periods, input of the entropy change value into the asymmetric convolution kernel to generate motion chaos quantization factor; using the chaos quantization factor as the trigger source, activate the radial basis function network; the hidden layer of the radial basis function network fuses the window deformation memory weight generated in the previous step, and outputs the elastic distortion coefficient with spatiotemporal correlation. S102: Distortion coefficients drive the reparameterization of high-dimensional manifold space, constructing velocity-interval differential homeomorphisms in the manifold tangent space, and automatically generating deformation gradient fields for window compression / extension. S103: The deformation gradient field acts on the topology of the reference window, and outputs a synchronous window geometric domain with adaptive curvature through conformal transformation. The synchronous window boundary dynamically converges / diffused, forming a spatiotemporal envelope surface that is monotonic with the motion chaos quantization factor.

[0034] The working principle and beneficial effects of the technical solution are as follows: firstly, the embodiment captures the discrete differential characteristics of the spindle motion fluctuation sequence in real time, extracts the directional entropy change value of the velocity vector in the adjacent period, inputs the entropy change value into the asymmetric convolution kernel, and generates a motion chaos quantization factor; the motion chaos quantization factor is used as a trigger source to activate a radial basis function network; the network hidden layer of the radial basis function network fuses the window shape deformation memory weight generated in the previous step, and outputs an elastic distortion coefficient with spatiotemporal correlation; secondly, the distortion coefficient drives the reparameterization of the high-dimensional manifold space, constructs a velocity-interval differential homeomorphism mapping in the manifold tangent space, and automatically generates a deformation gradient field of window compression / extension; finally, the deformation gradient field acts on the topology structure of the reference window, and outputs a synchronous window geometry domain with adaptive curvature through conformal transformation, the synchronous window boundary dynamically converges / diffuses, and a spatiotemporal envelope surface that is monotonous with the motion chaos quantization factor is formed. The generation process of the dynamic compression or extension of the synchronous window is realized through the following technical features to achieve closed-loop control: the directional entropy change characteristics of the spindle motion are processed through the asymmetric convolution kernel to generate a quantization factor representing the degree of motion chaos. When the factor triggers the radial basis function network, the network hidden layer fuses the historical window deformation weight, and outputs an elastic distortion coefficient with spatiotemporal correlation characteristics; the nonlinear mapping from motion fluctuation characteristics to window deformation parameters is realized; the distortion coefficient is used to drive the reparameterization of the high-dimensional manifold space, and a velocity-interval differential homeomorphism mapping is constructed in the manifold tangent space; the chaos quantization factor is converted into a window deformation gradient field, and a corresponding relationship between motion characteristics and spatial deformation is established; the deformation gradient field is applied to the topology structure of the reference window through conformal transformation, and a synchronous window geometry domain with adaptive curvature is generated; and the window boundary dynamically converges or diffuses according to the gradient field, and a spatiotemporal envelope surface that strictly corresponds to the motion chaos characteristics is formed.

[0035] In summary, the embodiment realizes end-to-end nonlinear mapping of the spindle motion chaos characteristics to the window deformation parameters; the strictness of the motion-space transformation is ensured through the differential geometry method; the window generation mechanism based on topological transformation ensures the geometric continuity of the deformation process; and a closed-loop adaptive adjustment system of motion characteristics-window deformation is formed. The synchronous window realizes real-time and accurate following of the spindle motion fluctuation, and provides a dynamically adaptive spatiotemporal reference framework for subsequent trajectory planning; and the window deformation process strictly follows the kinematic characteristics while maintaining the completeness of the geometric structure.

[0036] Embodiment 3: Based on embodiment 2, the process of generating a motion chaos quantization factor provided by the embodiment of the application includes the following steps: S1011: decompose the directional entropy change value of the velocity vector into an angle chaos component and a modulus oscillation component; perform Lie algebra adjoint representation on the two components respectively to generate a characteristic basis with rotation invariance; input the characteristic basis into a symplectic geometry space to construct a pair of conjugate differential operators, establish a non-commutation relationship between the operators, and form a topological constraint; The left-handed operator captures the entropy-increasing feature and the speed disorder trend; the right-handed operator captures the entropy-decreasing feature and the speed order trend; the symplectic space basic structure is a linear space spanned by a characteristic base; a standard symplectic form is given to the base manifold, which is an antisymmetric bilinear form composed of a pair of conjugate coordinates; a non-degenerate closed differential condition is satisfied, the left-handed operator is generated, the entropy-increasing feature is captured, and the Lie algebra adjoint orbit acting on the angle chaos component; the expression construction: a negative symplectic gradient flow is applied to the characteristic base, a first-order differential operator is generated by projecting along the contact hyperplane bundle; the kernel function takes a hyperbolic secant type singular point distribution, the right-handed operator is generated, the entropy-decreasing feature is captured, and the coadjoint representation acting on the modulus length oscillation component; the expression construction: a positive Hamiltonian vector field is constructed on the characteristic base tangent bundle, the differential order is restricted through the Lagrangian Grassmann manifold, and the kernel function takes an elliptic cosine type periodic distribution; a non-commutation relation is established, and the Lie bracket operation of the left-handed operator and the right-handed operator: generates a non-zero torsion tensor component, which is proportional to the exterior differential of the symplectic form, satisfies the modified Heisenberg algebra: the commutator is equal to the imaginary unit multiplied by the symplectic volume element, and contains a curvature correction term; S1012: The left-handed operator and the right-handed operator act on the characteristic base respectively, and induce a curvature tensor field in the principal bundle connection space, and the torsion invariant of the tensor field is extracted through the Chen-Simons integral; S1013: The torsion invariant inputs a hyperbolic type gauge field, and outputs a motion chaos quantization factor satisfying Lorentz covariance through a Yang-Mills functional minimization process.

[0037] The working principle and beneficial effects of the above technical solution are as follows: first, the direction entropy change value of the velocity vector is decomposed into an angle chaos component and a modulus length oscillation component; the two components are respectively represented by a Lie algebra adjoint representation to generate a characteristic base with rotational invariance; a pair of conjugate differential operators are constructed in a symplectic geometric space with the characteristic base as input, a non-commutation relation is established between the operators, and a topological constraint is formed, in which the left-handed operator captures the entropy-increasing feature and the speed disorder trend, and the right-handed operator captures the entropy-decreasing feature and the speed order trend; second, the left-handed operator and the right-handed operator act on the characteristic base respectively, and induce a curvature tensor field in the principal bundle connection space, and the torsion invariant of the tensor field is extracted through the Chen-Simons integral; finally, the torsion invariant inputs a hyperbolic type gauge field, and outputs a motion chaos quantization factor satisfying Lorentz covariance through a Yang-Mills functional minimization process. The above scheme establishes a description of the velocity field chaos through direction entropy decomposition, Lie algebra base, and conjugate operators; the chaos trend is converted into a calculable topological feature through the principal bundle curvature and the torsion invariant; and the quantization factor is ensured to satisfy the relativistic covariance requirement through the hyperbolic gauge field and the Yang-Mills minimization. Output: motion chaos quantization factor, as the core input parameter for subsequent dynamic synchronization window adjustment, to ensure that the window deformation matches the principal axis motion chaos characteristics.

[0038] Embodiment 4: as shown in the embodiment 1, on the basis of the embodiment 1, the process of converting kinetic energy change gradient into the constraint condition of the motion boundary from the shaft, comprises the following steps: Figure 4 S201: extract the synchronous window geometry domain of adaptive curvature, calculate the covariant derivative of the Riemann curvature tensor along the geodesic line of the synchronous window boundary, generate the kinetic energy gradient distribution field; take the kinetic energy gradient distribution field as the potential function; construct a non-holonomic constraint Hamiltonian system on the contact manifold, and derive the torsion feature of the motion phase trajectory through the Cartan-Ehresmann connection; S202: the torsion feature generates a motion boundary operator satisfying the von Neumann algebra, and the eigenvalue spectrum of the operator constitutes the metric matrix of the acceleration decision manifold; S203: the metric matrix induces the action of the symplectic group, and outputs the time-varying acceleration decision curve with affine connection structure through the Kostant-Souriau pre-quantization mapping. The working principle and beneficial effects of the above technical solution are as follows: firstly, the synchronous window geometry domain of adaptive curvature is extracted, the covariant derivative of the Riemann curvature tensor is calculated along the geodesic line of the synchronous window boundary, and the kinetic energy gradient distribution field is generated; the kinetic energy gradient distribution field is taken as the potential function; a non-holonomic constraint Hamiltonian system is constructed on the contact manifold, and the torsion feature of the motion phase trajectory is derived through the Cartan-Ehresmann connection; secondly, the torsion feature generates a motion boundary operator satisfying the von Neumann algebra, and the eigenvalue spectrum of the operator constitutes the metric matrix of the acceleration decision manifold; finally, the metric matrix induces the action of the symplectic group, and outputs the time-varying acceleration decision curve with affine connection structure through the Kostant-Souriau pre-quantization mapping. The above scheme converts the kinetic energy change gradient into the constraint condition of the motion boundary from the shaft; through the covariant derivative operation of the Riemann curvature tensor of the synchronous window geometry domain, the direct correlation between the kinetic energy gradient distribution field and the non-holonomic constraint system on the contact manifold is established, realizing the strict mathematical mapping between the continuum dynamics and the differential geometric structure; the traditional kinetic energy analysis is improved to the differential geometric framework, and the eigenvalue representation on the Riemann manifold is provided for the constraint condition. Based on the torsion feature derived through the Cartan-Ehresmann connection, a motion boundary operator satisfying the von Neumann algebra is constructed; the geometric torsion is converted into the eigenvalue problem of the solvable operator, so that the linearization processing of the nonlinear constraint condition can be realized through the operator spectrum decomposition, providing an algebraic basis for the subsequent construction of the metric matrix. The action of the symplectic group induced by the metric matrix is mapped through the Kostant-Souriau pre-quantization, and the solution of the constraint condition is converted into the affine connection optimization problem on the symplectic manifold; the conservation of the symplectic structure of the time-varying acceleration decision curve in the phase space is ensured, so that the system simultaneously satisfies the energy optimization and the geometric compatibility of the non-holonomic constraint.

[0039]

[0040] ​In summary, the embodiment constructs a complete differential geometry-algebraic-symplectic geometry conversion chain from kinetic energy gradient to time-varying constraints; through three-level progressive processing of curvature tensor analysis, torsion feature extraction and symplectic structure optimization, adaptive boundary control of nonholonomic constraint system on contact manifold is realized, and the core innovation is to convert the traditional dynamics constraint problem into a geometric optimization problem with affine connection structure, thereby providing a geometric dynamics framework for motion control of complex mechanical systems.

[0041] Embodiment 5: Based on embodiment 4, the embodiment of the application provides a process for constructing a nonholonomic constraint Hamiltonian system on a contact manifold, comprising the following steps: S2011: Obtain the boundary geodesic of the adaptive curvature synchronous window geometry domain, apply the Schouten-Nijenhuis bracket operation to extract the Riemann curvature tensor, take the covariant derivative along the normal coordinate direction, and output the kinetic energy gradient distribution field; S2012: Embed the kinetic energy gradient distribution field as the principal fiber bundle section of the contact form potential function, construct a regular contact form on the jet bundle, and generate a potential function manifold with singular point distribution; S2013: Input the contact differential homeomorphism group of the potential function, induce the curvature form of the Cartan-Ehresmann connection, constrain the phase space orbit by the curvature form, and derive a Hamiltonian system satisfying the nonholonomic variational principle.

[0042] The working principle and beneficial effects of the technical solution are as follows: first, the boundary geodesic of the adaptive curvature synchronization window geometric domain is obtained, the Schouten-Nijenhuis bracket operation is applied to extract the Riemann curvature tensor, the covariant derivative is taken along the normal coordinate direction, and the kinetic energy gradient distribution field is output; second, the kinetic energy gradient distribution field is embedded into the potential function contact manifold as the main fiber bundle section, the regular contact form is constructed on the jet bundle, and the potential function manifold with singular point distribution is generated; and finally, the potential function is input into the contact differential homeomorphism group, the curvature form of the Cartan-Ehresmann connection is induced, the curvature form of the connection is used to constrain the phase space orbit, and a Hamiltonian system satisfying the nonholonomic variational principle is derived. Through the synergistic effect of the three key steps, the geometric construction of the nonholonomic constraint Hamiltonian system is realized: the Riemann curvature tensor is extracted through the Schouten-Nijenhuis operation of the boundary geodesic, and the dynamic coupling relationship between the kinetic energy gradient field and the manifold curvature is established through the covariant derivative operation; the nonholonomic constraint condition of the system is coded as the differential geometric property of the curvature tensor. The kinetic energy gradient field is embedded as the main fiber bundle section, the regular contact form constructed on the jet bundle realizes the singular point distribution of the potential function manifold, so that the constraint condition is naturally embodied in the geometric property of the contact structure; through the Cartan-Ehresmann connection curvature form induced by the contact differential homeomorphism group, the phase space orbit constraint is converted into the curvature constraint of the connection, which directly corresponds to the constraint condition of the nonholonomic variational principle, and finally a Hamiltonian system satisfying the principle is generated.

[0043] In summary, the embodiment realizes the geometric description of the nonholonomic constraint condition, converts the mechanical constraint into the curvature property of the manifold, establishes the internal relationship between the constraint condition and the dynamic system through the introduction of the contact structure, and guarantees that the system evolution always satisfies the nonholonomic variational principle by using the curvature form of the connection. While maintaining the framework of the Hamiltonian system, the nonholonomic constraint condition is completely embedded.

[0044] Embodiment 6: On the basis of embodiment 4, the process for generating a torsion feature satisfying a von Neumann algebra motion boundary operator provided by the embodiment of the application includes the following steps: S2021: The torsion feature of the motion phase trajectory is input into the Clifford algebra bundle, and is generated by decomposition through the spinor representation, describes the torsion singularity component of the topological defect of the nonholonomic constraint and the frame oscillation component reflecting the quantum fluctuation of the kinetic energy distribution, and outputs a torsion feature pair with a spinor structure; S2022: The torsion feature pair is embedded into the representation space of the II_1 type factor algebra, and an operator generator satisfying the following basic relationship is constructed: the Lie algebra structure determined by the torsion singularity component satisfies the commutation relation, and the frame oscillation component guarantees the anti-self-adjointness of the spectral theorem applicability, and generates an operator resolvent closed in the von Neumann ring; S2023: The operator preforms are constructed via Gelfand-Naimark-Segal and implemented in Hilbert-Schmidt space as specific operators: domain is the tangent bundle of the acceleration decision manifold; range is the symplectic kernel space determined by the torsion modulus; output is a motion boundary operator that satisfies the noncommutative spectrum theorem.

[0045] The working principle and beneficial effects of the above technical solution are as follows: In this embodiment, the torsion features of the motion phase trajectory are first input into the Clifford algebra bundle, and generated through spinor representation decomposition. This generates the torsional singular component describing the topological defects of the nonholonomic constraint and the frame oscillation component reflecting the quantum fluctuations of the kinetic energy distribution, and outputs a pair of torsion features with a spinor structure. Secondly, the torsion feature pairs are embedded in the representation space of the type II_1 factor algebra to construct operator generators that satisfy the following basic relations: the commutation relation of the Lie algebra structure is determined by the torsional singular component, and the anti-self-adjoint property of the spectral theorem applicability is guaranteed by the frame oscillation component, generating operator pre-solutions closed in the von Neumann ring. Finally, the operator pre-solutions are constructed by Gelfand-Naimark-Segal and implemented in the Hilbert-Schmidt space: the domain is the tangent bundle of the acceleration decision manifold; the range is the symplectic kernel space determined by the torsion modulus; and the output is a motion boundary operator that satisfies the noncommutative spectral theorem. The above scheme transforms the geometric topological properties of nonholonomic constraints (torsion, torsional singularity) into the algebraic structure of operator algebra (commutation relations, anti-self-adjoint property); using spinor decomposition and type II_1 factor algebra, it ensures that the generated operator preforms satisfy the closure property of von Neumann algebra; through GNS construction, it realizes a motion boundary operator with a well-defined domain and range in Hilbert-Schmidt space, making it conform to the mathematical requirements of the noncommutative spectrum theorem. It realizes the expression of the motion boundary conditions of the mechanical system within the framework of operator algebra, and guarantees the computability and spectral properties of the operator.

[0046] Example 7: Figure 5 As shown, based on Example 1, the process of forming a self-dissolving mechanism for dynamic error factors provided in this embodiment of the invention includes the following steps: S301: Taking the affine connection acceleration curve as input, a baseline predicted trajectory is generated on the constructed motion boundary constrained manifold through the Cartan development map. The curvature-torsion coupling term is introduced as a dynamic error factor, and a virtual trajectory containing Riemann curvature perturbation is output. S302: The difference between real-time motion characteristics and virtual trajectory is transformed into a cohomological closed form through Chern integration, and executed in the jet extension space: the mechanical parameter offset of the closed form along the hyperlocalization section of the normal bundle, the control parameter offset of the closed form in the sigmoid layer, and the output decoupled mechanical-control parameter offset pair. S303: The mechanical-control parameter offset pair is input into the principal bundle canonical transformation group, the evolution of the canonical equivalence class of the connection form is induced, and the kernel parameter of the reconstructed trajectory is constrained by Bianchi identity; and the reconstructed parameter space curvature is equal to the dynamic error factor torsion, and the dual elimination of the parameter space and the error factor is formed.

[0047] The working principle and beneficial effects of the technical solution are as follows: first, the affine connection acceleration curve is taken as input, a reference prediction trajectory is generated on the constructed motion boundary constraint manifold through Cartan development mapping, a curvature-torsion coupling term is introduced as a dynamic error factor, and a virtual trajectory containing Riemann curvature disturbance is output; second, the difference between the real-time motion feature and the virtual trajectory is converted into a closed form of cohomology through Chen class integral, and the mechanical parameter offset of the hyperlocal section of the normal bundle and the control parameter offset of the transverse integral of the symplectic leaf are executed in the jet extension space, and a decoupled mechanical-control parameter offset pair is output; finally, the mechanical-control parameter offset pair is input into the principal bundle canonical transformation group, the evolution of the canonical equivalence class of the connection form is induced, and the kernel parameter of the reconstructed trajectory is constrained by Bianchi identity; and the reconstructed parameter space curvature is equal to the dynamic error factor torsion, and the dual elimination of the parameter space and the error factor is formed. The above scheme inputs the affine connection acceleration curve to describe the dynamic evolution of the system, constructs a theoretical trajectory on the motion boundary constraint manifold through Cartan development mapping, uses the curvature-torsion coupling term as an error source to make the virtual trajectory affected by Riemann curvature disturbance, and outputs a virtual trajectory containing geometric disturbance as a reference for subsequent error analysis. The theoretical reference trajectory is established, and the system dynamic error is quantified, which is caused by the geometric characteristics of the manifold. The dynamic error is decomposed into mechanical and control components that can be independently optimized to realize algebraic topological analysis of the error; through canonical transformation and geometric constraint, adaptive compensation of the dynamic error is realized to form a closed-loop elimination mechanism.

[0048] In summary, the curvature-torsion coupling term is used to quantify the system dynamic error in the embodiment, and a virtual trajectory is established as a reference; Chen class integral and jet extension are used to decompose the error into mechanical and control parameter offsets to realize structured analysis; through the canonical transformation group and Bianchi identity, the parameter space evolution automatically offsets the error to form a closed-loop stabilization. The geometric observability of the dynamic error and the automatic convergence of the parameter adjustment are guaranteed, and the robust control and optimization of the nonlinear constraint system are applicable.

[0049] Embodiment 8: Based on embodiment 7, the process of outputting the decoupled mechanical-control parameter offset pair provided by the embodiment of the application comprises the following steps: S3021: The difference between the real-time motion feature and the virtual trajectory is processed through feature tensor contraction to obtain a divergence field on a nonholonomic differential manifold; the divergence field is processed through Chen class integral to obtain a cohomology form satisfying a closed chain condition; S3022: The cohomology form is processed by fiber projection of normal bundle to obtain a super-localized cross-section distribution, and the cross-section distribution is processed by critical point curvature mapping to obtain a curvature frame of mechanical parameter offset; S3023: The residual component of the cohomology form is processed by dimension reduction of symplectic leaf to obtain a control characteristic distribution, and the control characteristic distribution is processed by cross-section integral to obtain an integral invariant of control parameter offset; S3024: The curvature frame of the mechanical parameter offset is processed by symplectic orthogonal constraint to obtain a normal space projection component; the integral invariant of the control parameter offset is processed by tangent constraint to obtain a hyperplane projection component; and the two projection components are processed by canonical pairing to finally form a decoupled mechanical-control parameter offset pair.

[0050] The working principle and beneficial effects of the above technical solution are as follows: first, the difference between real-time motion characteristics and virtual trajectories is processed by feature tensor contraction to obtain a divergence field on a non-complete differential manifold; the divergence field is processed by Chen class integral to obtain a cohomology form satisfying a closed chain condition; second, the cohomology form is processed by fiber projection of normal bundle to obtain a super-localized cross-section distribution, and the cross-section distribution is processed by critical point curvature mapping to obtain a curvature frame of mechanical parameter offset; then, the residual component of the cohomology form is processed by dimension reduction of symplectic leaf to obtain a control characteristic distribution, and the control characteristic distribution is processed by cross-section integral to obtain an integral invariant of control parameter offset; finally, the curvature frame of the mechanical parameter offset is processed by symplectic orthogonal constraint to obtain a normal space projection component; the integral invariant of the control parameter offset is processed by tangent constraint to obtain a hyperplane projection component; and the two projection components are processed by canonical pairing to finally form a decoupled mechanical-control parameter offset pair. The above scheme converts the motion error into a divergence field on a non-complete manifold by feature tensor contraction, ensuring the geometric observability of the error; the Chen class integral maps the divergence field into a cohomology closed form, so that the error satisfies the algebraic topology constraint, facilitating subsequent decomposition. The fiber projection of the normal bundle decomposes the cohomology form into a super-localized cross-section distribution, focusing on the local deviation of the mechanical system; the critical point curvature mapping converts the cross-section distribution into a curvature frame, quantifying the geometric influence of the mechanical parameter offset. The dimension reduction of the symplectic leaf filters the residual component of the cohomology form to generate a control characteristic distribution, extracting the error component related to the control strategy; the cross-section integral compresses the control characteristic distribution into an integral invariant, reflecting the adjustment requirement of the control parameter. The symplectic orthogonal constraint and the tangent constraint respectively impose geometric constraints on the mechanical and control offset components, ensuring their independence in the phase space; the canonical pairing projects the two components into complementary subspaces (normal space and hyperplane) to form a strictly decoupled mechanical-control parameter offset pair.

[0051] In summary, the embodiment decomposes the dynamic error into a divergence field and a closed-form geometric object that can be mathematically strictly processed by using the theory of differential manifold and cohomology; completely decouples the mechanical and control parameters; realizes the independent extraction of the physical deviation and the control strategy adjustment by using the projection of the fiber bundle and the dimension reduction of the symplectic leaf, and avoids the coupled interference; ensures the consistency and integrability of the deviation in the phase space by using the orthogonal, contact condition, and canonical pairing of the symplectic geometry constraints, and provides a basis for subsequent error compensation. Finally, the decoupled and geometrically consistent parameter deviation pair is output, which provides an accurate error correction input for the adaptive control of the dynamic system.

[0052] Embodiment 9: On the basis of embodiment 8, the process of canonical pairing processing of the two projection components provided by the embodiment of the application includes the following steps: S30241: The curvature frame of the mechanical parameter deviation is processed by using the connection adaptation to obtain the covariant differential form of the projection component of the fiber bundle; the covariant differential form is processed by using the curvature constraint to generate a normalized shrinkage quantity that satisfies the fiber bundle structure; S30242: The integral invariant of the control parameter deviation is processed by using the cotangent lifting to obtain the cross differential form of the projection component of the hyperplane; the cross differential form is processed by using the leaf layer constraint to generate a dual expansion quantity with a symplectic structure; S30243: The normalized shrinkage quantity is processed by using the principal bundle edge matching to obtain the connection-adapted shrinkage core; the dual expansion quantity is processed by using the associated bundle lifting to obtain the leaf-adapted expansion core; the two cores are processed by using the bilinear pairing to form a coupled tensor with orthogonal invariance; S30244: The coupled tensor is processed by using the canonical decomposition to obtain the covariant derivative representation of the mechanical component and the exterior differential representation of the control component; the two representations are processed by using the parameterized embedding to finally generate the decoupled mechanical-control parameter deviation pair.

[0053] The working principle and beneficial effects of the technical solution are: firstly, the curvature frame of the mechanical parameter offset is processed through liaison adaptation to obtain a covariant differential form of the normal space projection component; the covariant differential form is processed through curvature constraint to generate a canonical contraction quantity satisfying the fiber bundle structure; secondly, the integral invariant of the control parameter offset is processed through hyperbolic lifting to obtain a cross differential form of the hyperplane projection component; the cross differential form is processed through leaf layer constraint to generate a dual expansion quantity with symplectic structure; then the canonical contraction quantity is processed through principal bundle edge matching to obtain a contraction kernel of liaison adaptation; the dual expansion quantity is processed through matching bundle lifting to obtain an expansion kernel adapted to the leaf layer; the two kernels are processed through bilinear pairing to form a coupled tensor with orthogonal invariance; finally, the coupled tensor is processed through canonical decomposition to obtain a covariant derivative expression of the mechanical component and an exterior differential expression of the control component; the two expressions are processed through parameterized embedding to finally generate a decoupled mechanical-control parameter offset pair. The liaison adaptation processing of the above scheme converts the curvature frame of the mechanical parameter into a covariant differential form, ensuring its integrability in the normal space; the curvature constraint processing further optimizes the covariant differential form to make it conform to the canonical structure of the fiber bundle, generating a canonical contraction quantity and ensuring the geometric consistency of the mechanical offset. The hyperbolic lifting processing converts the integral invariant of the control parameter into a cross differential form, making it adapt to the geometric constraint of the hyperplane; the leaf layer constraint processing adjusts the cross differential form to meet the symplectic structure requirement, generating a dual expansion quantity and ensuring the dynamic compatibility of the control offset. The principal bundle edge matching processing converts the canonical contraction quantity into a contraction kernel of liaison adaptation, making it maintain geometric invariance on the fiber bundle; the matching bundle lifting processing converts the dual expansion quantity into an expansion kernel adapted to the leaf layer, making it maintain dynamic invariance on the symplectic leaf layer; the bilinear pairing processing combines the two kernels into a coupled tensor, ensuring the orthogonal invariance of the mechanical and control components in the phase space. The canonical decomposition processing decomposes the coupled tensor into a covariant derivative expression of the mechanical component and an exterior differential expression of the control component, achieving strict decoupling; the parameterized embedding processing integrates the two expressions into a mechanical-control parameter offset pair, ensuring its calculability and adjustability in the dynamic system.

[0054] In summary, the present embodiment ensures that the mechanical and control components maintain their respective geometric and dynamic characteristics in the normal space and hyperplane respectively through liaison adaptation and hyperbolic lifting; the contraction kernel and expansion kernel maintain canonical structures on the fiber bundle and symplectic leaf layer respectively through principal bundle edge matching and matching bundle lifting, ensuring the mathematical rigor of the coupled tensor; finally, a fully decoupled mechanical-control parameter offset pair is generated through canonical decomposition and parameterized embedding, providing an accurate mathematical framework for adaptive correction of the dynamic system. The final output is a decoupled and geometrically compatible parameter offset pair, ensuring independent correction of mechanical and control errors and improving the stability and control accuracy of the system.

[0055] In the embodiment 10, based on the embodiment 9, the process for forming the coupling tensor with the orthogonal invariance comprises the following steps. S302431: The contracted kernel of the contact adaptation is processed through the orthogonal group action to obtain the norm-invariant contracted core; the contracted core is processed through the Riemann dual to generate the contracted invariant sub satisfying the inner product invariance; S302432: The expanded kernel of the leaf layer adaptation is processed through the symplectic group action to obtain the leaf layer compatible expanded core; the expanded core is processed through the symplectic dual to generate the expanded invariant sub satisfying the outer product compatibility; S302433: The contracted invariant sub is processed through the bilinear extension to obtain the covariant tensor component; the expanded invariant sub is processed through the bilinear extension to obtain the contravariant tensor component; the two components are processed through the tensor product synthesis to form the original coupling tensor; S302434: The original coupling tensor is processed through the orthogonal constraint to obtain the intermediate tensor satisfying the orthogonal invariance; the intermediate tensor is processed through the curvature and torsion balance to finally generate the coupling tensor with the orthogonal invariance.

[0056] The working principle and beneficial effects of the technical solution are as follows: firstly, the contraction kernel of the adapted connection is processed by orthogonal group action to obtain a norm-invariant contraction core; the contraction core is processed by Riemannian duality to generate a contraction-invariant sub-core that satisfies the inner product invariance; secondly, the expansion kernel of the adapted leaf layer is processed by symplectic group action to obtain an expansion core of the leaf layer that is compatible; the expansion core is processed by symplectic duality to generate an expansion-invariant sub-core that satisfies the outer product compatibility; then, the contraction-invariant sub-core is processed by bilinear extension to obtain a covariant tensor component; the expansion-invariant sub-core is processed by bilinear extension to obtain a contravariant tensor component; the two components are processed by tensor product synthesis to form an original coupled tensor; finally, the original coupled tensor is processed by orthogonal constraint to obtain an intermediate tensor that satisfies the orthogonal invariance; the intermediate tensor is processed by curvature-torsion balance to finally generate a coupled tensor with orthogonal invariance. The orthogonal group action processing of the above scheme makes the contraction kernel of the adapted connection remain unchanged in the orthogonal transformation, generates a norm-invariant contraction core, and further constrains the contraction core by Riemannian duality to make it satisfy the inner product invariance, form a contraction-invariant sub-core, and ensure its geometric stability in the metric space. The symplectic group action processing makes the expansion kernel of the adapted leaf layer maintain the dynamic structure under the symplectic transformation, generates an expansion core of the leaf layer that is compatible, and adjusts the expansion core by symplectic duality to make it satisfy the outer product compatibility, form an expansion-invariant sub-core, and ensure its symplectic geometric constraint in the phase space. The bilinear extension processing extends the contraction-invariant sub-core into a covariant tensor component, which adapts to the geometric structure of the fiber bundle; the bilinear extension processing extends the expansion-invariant sub-core into a contravariant tensor component, which adapts to the dynamic structure of the symplectic leaf layer; the tensor product synthesis processing combines the covariant and contravariant components into an original coupled tensor, establishing a preliminary association between the mechanical and control parameters; the orthogonal constraint processing makes the original coupled tensor remain unchanged in the orthogonal transformation, generates an intermediate tensor, and ensures its invariance in the metric space; the curvature-torsion balance processing optimizes the intermediate tensor to make it satisfy the geometric (curvature) and dynamic (torsion) constraints at the same time, and finally generates a coupled tensor with orthogonal invariance.

[0057] In summary, the present embodiment ensures that the contraction kernel and the expansion kernel maintain mathematical invariance under the Riemannian and symplectic geometric frameworks, respectively, through orthogonal group and symplectic group action; establishes an association between mechanical (covariant) and control (contravariant) parameters while retaining their respective geometric and dynamic characteristics through bilinear extension and tensor product; and finally generates a coupled tensor that remains stable under transformation group action through orthogonal constraint and curvature-torsion balance, providing strict mathematical guarantees for parameter decoupling. The final output is a coupled tensor with orthogonal invariance, allowing mechanical and control parameters to maintain independent correction capabilities in dynamic systems while satisfying the overall geometric-dynamic constraints of the structure.

[0058] In the embodiment 10, the process of the intermediate tensor processed by the curvature-torsion balance processing comprises the following steps: S3024341: The intermediate tensor is processed by Riemann connection action to obtain a curvature characteristic bearing structure, and is processed by curvature tensor contraction to generate a modified tensor base of intrinsic Riemann curvature; S3024342: The modified tensor base is processed by affine connection action to obtain a torsion characteristic bearing structure, and is processed by torsion tensor antisymmetrization to generate a balanced pre-tensor of fused affine torsion; S3024343: The balanced pre-tensor is processed by Bianchi identity mapping to obtain a differential identity constraint core, and is processed by curvature-torsion dual contraction to finally generate a coupling tensor satisfying geometric-physical self-consistency.

[0059] The working principle and beneficial effects of the above technical solution are as follows: first, the intermediate tensor is processed by Riemann connection action to obtain a curvature characteristic bearing structure, and is processed by curvature tensor contraction to generate a modified tensor base of intrinsic Riemann curvature; second, the modified tensor base is processed by affine connection action to obtain a torsion characteristic bearing structure, and is processed by torsion tensor antisymmetrization to generate a balanced pre-tensor of fused affine torsion; finally, the balanced pre-tensor is processed by Bianchi identity mapping to obtain a differential identity constraint core, and is processed by curvature-torsion dual contraction to finally generate a coupling tensor satisfying geometric-physical self-consistency. The Riemann connection action processing of the above scheme adapts the intermediate tensor to the Riemann geometry framework, generates a curvature characteristic bearing structure, and ensures that it satisfies the intrinsic geometric constraints of Riemannian manifold; the curvature tensor contraction processing extracts the curvature characteristics of the structure, generates a modified tensor base of intrinsic Riemann curvature, and provides a geometric reference for subsequent torsion balancing. The affine connection action processing extends the modified tensor base to the affine connection space, generates a torsion characteristic bearing structure, and makes it compatible with the dynamic effect of non-Riemannian geometry; the torsion tensor antisymmetrization processing extracts the torsion characteristics of the structure, generates a balanced pre-tensor of fused affine torsion, and ensures the mathematical compatibility of geometric and dynamic modifications. The Bianchi identity mapping processing maps the balanced pre-tensor to the differential identity space to generate a differential identity constraint core, ensuring the mathematical coordination of curvature and torsion; the curvature-torsion dual contraction processing integrates the dual relationship of curvature and torsion through contraction operation, finally generates a coupling tensor satisfying geometric-physical self-consistency, and realizes strict structural balance.

[0060] In summary, the embodiment extracts the characteristics of curvature and torsion, so that the coupling tensor is compatible with both the geometric constraints of Riemann metric and the dynamic effect of affine connection; through Bianchi identity mapping, it ensures that the modification process of curvature and torsion meets the mathematical consistency requirements of differential geometry; the finally output coupling tensor satisfies strict self-consistency in both geometric (curvature) and physical (torsion) aspects, providing mathematical guarantee for parameter decoupling of dynamic systems; through the coordinated balance of curvature and torsion, the coupling tensor maintains the orthogonality while further satisfying the internal consistency of geometry and physics.

[0061] In the embodiments 11, the curvature-flexibility dual condensation process provided by the embodiments of the application comprises the following steps: S30243431: The differential identity constraint core is processed by curvature main component decomposition to obtain characteristic components dominated by Riemann curvature; the characteristic components are processed by principal bundle edge matching to generate curvature characteristic manifolds satisfying gauge invariance; S30243432: The remaining part of the differential identity constraint core is processed by flexibility residual component decomposition to obtain characteristic components dominated by affine flexibility; the characteristic components are processed by matching bundle promotion to generate flexibility characteristic manifolds satisfying leaf layer compatibility; S30243433: The curvature characteristic manifolds are processed by connection induction to obtain curvature dual representation. The flexibility characteristic manifolds are processed by cross-section induction to obtain flexibility dual representation; the two representations are processed by bilinear fusion to form dual characteristic coupling body; S30243434: The dual characteristic coupling body is processed by curvature-flexibility balance constraint to obtain geometric-physical balance core; the balance core is processed by identity condensation to finally generate coupled tensors satisfying self-consistent conditions.

[0062] The working principle and beneficial effects of the above technical solution are: firstly, the differential identity constraint core is processed by curvature main term decomposition to obtain characteristic components dominated by Riemann curvature; the characteristic components are processed by principal bundle edge matching to generate curvature characteristic manifolds satisfying gauge invariance; secondly, the remaining part of the differential identity constraint core is processed by torsion residual term decomposition to obtain characteristic components dominated by affine torsion; the characteristic components are processed by matching bundle promotion to generate torsion characteristic manifolds satisfying leaf layer compatibility; then, the curvature characteristic manifolds are processed by connection induction to obtain curvature dual representation; the torsion characteristic manifolds are processed by cross-section induction to obtain torsion dual representation; the two representations are processed by bilinear fusion to form dual characteristic coupling body; finally, the dual characteristic coupling body is processed by curvature-torsion balance constraint to obtain geometric-physical balance core; the balance core is processed by identity contraction to finally generate a coupled tensor satisfying self-consistent conditions. The curvature main term decomposition of the above scheme decomposes the differential identity constraint core into characteristic components dominated by Riemann curvature, and extracts its core geometric structure; the principal bundle edge matching applies the gauge group action to the components to generate curvature characteristic manifolds satisfying gauge invariance, ensuring their consistency under gauge transformation. The torsion residual term decomposition extracts the remaining part of the differential identity constraint core to generate characteristic components dominated by affine torsion, which adapts to the dynamic characteristics of non-Riemannian geometry; the matching bundle promotion applies leaf structure adaptation to the components to generate torsion characteristic manifolds satisfying leaf layer compatibility, which maintains dynamic stability under phase space constraints. The connection induction maps the curvature characteristic manifolds to the dual space to generate the curvature dual representation, which adapts to the connection structure of the fiber bundle; the cross-section induction maps the torsion characteristic manifolds to the dual space to generate the torsion dual representation, which adapts to the dynamic structure of the cross-section leaf layer; the bilinear fusion combines the two dual representations into a dual characteristic coupling body, establishing the relationship between curvature and torsion. The curvature-torsion balance constraint applies geometric-physical balance conditions to the dual characteristic coupling body to generate a geometric-physical balance core, ensuring the dynamic coordination of curvature and torsion; the identity contraction integrates the mathematical constraints of the balance core through contraction operations to finally generate a coupled tensor satisfying self-consistent conditions, realizing the strict unification of geometry and physics.

[0063] In summary, the present embodiment extracts the independent characteristics of Riemann curvature and affine torsion through main term and residual term decomposition, and adapts to their respective frameworks; through connection and cross-section induction, the dual forms of curvature and torsion are generated, and the coupling relationship between the two is established through bilinear fusion to ensure the compatibility of geometric and dynamic parameters; through balance constraint and identity contraction, the coupled tensor satisfies both the curvature constraint of Riemann geometry and the torsion constraint of affine geometry; through dual decomposition, representation fusion, and balance optimization of curvature and torsion, the final coupled tensor achieves global self-consistency of geometry and physics under the constraints of gauge invariance, leaf layer compatibility, and differential identity.

[0064] Embodiment 13: The method of any one of embodiments 1-12, wherein the method further comprises:Figure 6 On the basis of Embodiment 1-Embodiment 12, the multi-axis coordinated trimming motion trajectory planning system provided by the embodiment of the application comprises: A synchronization window output module is configured to capture the instantaneous motion fluctuation characteristics of the main shaft, extract the speed change trend quantity, take the speed change trend quantity of the previous step as input, reconstruct the space-time elastic coefficient of the synchronization interval of the slave shaft, and output the dynamically compressed or stretched synchronization window to establish the speed-interval nonlinear mapping rule. A constraint condition establishment module is configured to analyze the load kinetic energy change gradient in the synchronization window based on the space-time characteristics of the dynamically compressed or stretched synchronization window, convert the kinetic energy change gradient into the motion boundary constraint condition of the slave shaft, generate the time-varying acceleration decision curve, and match the motion energy distribution with the load kinetic energy characteristics. An offset decoupling module is configured to construct a trajectory prediction space under the constraint of the motion boundary constraint condition of the slave shaft, generate a virtual trajectory containing a dynamic error factor, inversely decouple the mechanical-control coupling parameter offset through the difference feedback between the real-time motion characteristics and the virtual trajectory, autonomously reconstruct the trajectory generation kernel parameter, and form a self-cancelling mechanism of the dynamic error factor.

[0065] The working principle and beneficial effects of the technical solution are as follows: the synchronous window output module of the embodiment is used to capture the instantaneous motion fluctuation characteristics of the main shaft, extract the speed change trend quantity, and take the speed change trend quantity of the previous step as input to reconstruct the time-space elastic coefficient of the slave shaft synchronous interval; a dynamically compressed or expanded synchronous window is output, and a speed-interval nonlinear mapping rule is established; the constraint condition establishment module is used to analyze the load kinetic energy change gradient in the synchronous window based on the time-space characteristics of the dynamically compressed or expanded synchronous window; the kinetic energy change gradient is converted into a slave shaft motion boundary constraint condition, a time-varying acceleration decision curve is generated, and the motion energy distribution is matched with the load kinetic energy characteristics; the offset decoupling module is used to construct a trajectory prediction space under the constraint of the slave shaft motion boundary constraint condition, and generate a virtual trajectory containing a dynamic error factor; the mechanical-control coupling parameter offset is decoupled in reverse through the difference feedback of the real-time motion characteristics and the virtual trajectory; the trajectory generation kernel parameters are autonomously reconstructed, and a self-cancellation mechanism of the dynamic error factor is formed. The dynamically compressed / expanded synchronous window generated by the synchronous window output module, in combination with the time-varying acceleration decision curve of the constraint condition establishment module, constitutes a closed-loop coupling of nonlinear mapping rule and boundary constraint; this mechanism enables the slave shaft motion trajectory to match the main shaft speed fluctuation in real time, and eliminates the phase lag problem caused by the traditional fixed synchronous interval. The boundary constraint condition generated by the constraint condition establishment module through the analysis of the kinetic energy change gradient forms a dynamic energy matching with the virtual trajectory prediction space of the offset decoupling module; it ensures that the acceleration curve strictly follows the load inertia change and suppresses the energy overshoot in the motion process. The offset decoupling module decouples the mechanical-control parameters in reverse through the dynamic error factor feedback, and drives the elastic coefficient reconstruction of the synchronous window output module; online correction of the trajectory generation parameters is realized, and the system has self-compensation ability for mechanical transmission errors and control delays.

[0066] In summary, the embodiment improves the synchronization accuracy under variable acceleration conditions and suppresses the resonance energy peak value; the technical coupling of each module breaks through the bottleneck of mutual restriction between dynamic response and steady accuracy in the traditional pursuit system.

[0067] Obviously, those skilled in the art can make various modifications and variations to the present application without departing from the spirit and scope of the present application. Thus, if these modifications and variations of the present application fall within the scope of the equivalent technology of the present application, the present application also intends to include these modifications and variations.

Claims

1. A multi-axis cooperative tracking and shearing motion trajectory planning method, characterized in that, Includes the following steps: A trajectory prediction space is constructed under the constraint of the axis motion boundary condition, and a virtual trajectory containing dynamic error factors is generated. By feedback of the difference between real-time motion features and virtual trajectory, the mechanical-control coupling parameter offset is decoupled in reverse. The trajectory generation kernel parameters are autonomously reconstructed to form a self-dissolving mechanism for dynamic error factors.

2. The multi-axis cooperative tracking and shearing motion trajectory planning method as described in claim 1, characterized in that, The process of forming a self-dissolving mechanism for dynamic error factors includes the following steps: Using the affine connection acceleration curve as input, a baseline predicted trajectory is generated on the constructed motion boundary constrained manifold through the Cartan evolution map. A curvature-torsion coupling term is introduced as a dynamic error factor, and a virtual trajectory containing Riemann curvature perturbation is output. The difference between real-time motion characteristics and virtual trajectories is transformed into a cohomological closed form through Chern integral, and executed in the jet extension space: the mechanical parameter offset of the closed form along the hyperlocalized section of the normal bundle, the control parameter offset of the closed form in the sigmoid layer, and the output decoupled mechanical-control parameter offset pair. The mechanical-control parameter offset induces the evolution of the normative equivalence class of the connection form by the input principal bundle normal transformation group, and reconstructs the trajectory kernel parameters through Bianchi identity constraints. It satisfies the condition that the curvature of the reconstructed parameter space is equal to the torsion of the dynamic error factor, thus forming a dual solution between the parameter space and the error factor.

3. The multi-axis cooperative tracking and shearing motion trajectory planning method as described in claim 2, characterized in that, The process of outputting decoupled mechanical-control parameter offset pairs includes the following steps: The difference between real-time motion features and virtual trajectories is processed by feature tensor contraction to obtain the divergence field on the nonholonomic differential manifold; the divergence field is processed by Chern integration to obtain the cohomological form that satisfies the closed-chain condition. The cohomology form is processed by normal fiber projection to obtain a hyperlocalized cross-sectional distribution. The cross-sectional distribution is then processed by critical point curvature mapping to obtain a curvature frame for the mechanical parameter offset. The residual components in the cohomological form are subjected to dimensionality reduction processing by the symplectic layer to obtain the control feature distribution. The control feature distribution is then subjected to transverse integral processing to obtain the integral invariant of the control parameter offset. The curvature frame of the mechanical parameter offset is subjected to symplectic orthogonal constraint processing to obtain the normal space projection component; the integral invariant of the control parameter offset is subjected to tangential constraint processing to obtain the hyperplane projection component; the two projection components are subjected to canonical pairing processing to finally form a decoupled mechanical-control parameter offset pair.

4. The multi-axis cooperative tracking and shearing motion trajectory planning method as described in claim 3, characterized in that, The process of canonical pairing of two projected components includes the following steps: The curvature frame of the mechanical parameter offset is processed by the connection adaptation to obtain the covariant differential form of the normal space projection component; the covariant differential form is processed by curvature constraint to generate the standard shrinkage that satisfies the fiber bundle structure. The integral invariant of the control parameter offset is subjected to cotangent lifting to obtain the transverse differential form of the hyperplane projection component; The truncated differential form is subjected to leaf layer constraints to generate a dual expansion with a symplectic structure; The standardized shrinkage amount is processed by principal cluster edge matching to obtain the shrinkage kernel with connection adaptation; The dual expansion is subjected to cluster lifting to obtain a leaf-adaptive expansion kernel; the two kernels are subjected to bilinear pairing to form a coupled tensor with orthogonality invariance. The coupling tensor is subjected to canonical decomposition to obtain the covariant derivative representation of the mechanical component and the exterior differential representation of the control component; The two representations are parametrically embedded to ultimately generate decoupled mechanical-control parameter offset pairs.

5. The multi-axis cooperative tracking and shearing motion trajectory planning method as described in claim 4, characterized in that, The process of forming a coupled tensor with orthogonality invariance includes the following steps: The contractile kernel of the connection-fitted model is processed by orthogonal group interaction to obtain a gauge-invariant contractile kernel; the contractile kernel is processed by Riemann duality to generate a contractile invariant that satisfies inner product invariance. The leaf-adapted unfolding nucleus was treated with symplectic group interaction to obtain a leaf-compatible unfolding nucleus; The expansion core is processed through symplectic duality to generate an expansion invariant that satisfies the outer product compatibility. The contractile invariant is subjected to bilinear expansion to obtain covariant tensor components; Expanding the invariant and then performing bilinear expansion yields the inverse tensor components; The two components are combined through tensor product to form the original coupled tensor; The original coupled tensor is subjected to orthogonal constraint processing to obtain an intermediate tensor that satisfies orthogonality invariance; the intermediate tensor is subjected to curvature-torsion balancing processing to finally generate a coupled tensor with orthogonality invariance.

6. The multi-axis cooperative tracking and shearing motion trajectory planning method as described in claim 5, characterized in that, The process of balancing the curvature and deflection of the intermediate tensor includes the following steps: The intermediate tensor is processed by Riemann connection to obtain a curvature characteristic bearing structure. After curvature tensor contraction, a modified tensor basis with intrinsic Riemann curvature is generated. The modified tensor basis is processed by affine connection to obtain a torsion characteristic load-bearing structure. After antisymmetric processing of the torsion tensor, a balanced pretensor with fused affine torsion is generated. The equilibrium pretensor is processed by Bianchi identity mapping to obtain a differential identity constraint kernel. After curvature-torsion dual contraction, a coupled tensor that satisfies geometric and physical self-consistency is finally generated.

7. The multi-axis cooperative tracking and shearing motion trajectory planning method as described in claim 6, characterized in that, The curvature-torsion dual contraction process includes the following steps: The differential identity constraint kernel is decomposed by the curvature principal term to obtain the characteristic components dominated by Riemann curvature. The eigencomponents are processed by principal bundle edge matching to generate a curvature eigenmanifold that satisfies gauge invariance; The remaining part of the differential identity constraint kernel is decomposed by the torsion remainder term to obtain the eigencomponent dominated by affine torsion. The eigencomponents are subjected to clustering enhancement to generate a torsion-featured manifold that satisfies leaf-layer compatibility. The curvature characteristic manifold is processed by connection-induced processing to obtain the curvature dual representation; the torsion characteristic manifold is processed by cross-section-induced processing to obtain the torsion dual representation; the two representations are then processed by bilinear fusion to form a dual characteristic coupling body. The dual-feature coupled volume is subjected to curvature-torsion balance constraint processing to obtain a geometrically and physically balanced kernel; the balanced kernel is subjected to identity reduction processing to finally generate a coupled tensor that satisfies the self-consistency condition.

8. The multi-axis cooperative tracking and shearing motion trajectory planning method as described in claim 1, characterized in that, It also includes capturing the instantaneous motion fluctuation characteristics of the main shaft, extracting the velocity change trend, using the previous velocity change trend as input, reconstructing the spatiotemporal elastic coefficient of the slave axis synchronization interval; outputting a dynamically compressed or extended synchronization window, and establishing a velocity-interval nonlinear mapping rule.

9. The multi-axis cooperative tracking and shearing motion trajectory planning method as described in claim 1, characterized in that, It also includes the spatiotemporal characteristics of a synchronization window based on dynamic compression or extension, analyzes the gradient of load kinetic energy change within the synchronization window, transforms the gradient of kinetic energy change into boundary constraints of the axis motion, generates a time-varying acceleration decision curve, and makes the distribution of motion energy match the characteristics of load kinetic energy.

10. A multi-axis cooperative tracking and shearing motion trajectory planning system, characterized in that, Include: The synchronization window output module is used to capture the instantaneous motion fluctuation characteristics of the main shaft, extract the velocity change trend, and reconstruct the spatiotemporal elastic coefficient of the slave axis synchronization interval using the previous velocity change trend as input; it outputs a dynamically compressed or extended synchronization window and establishes a velocity-interval nonlinear mapping rule. The constraint establishment module is used to analyze the load kinetic energy change gradient within the synchronization window based on the spatiotemporal characteristics of the dynamically compressed or extended synchronization window; it transforms the kinetic energy change gradient into the boundary constraints of the axis motion, generates a time-varying acceleration decision curve, and makes the motion energy distribution match the load kinetic energy characteristics. The offset decoupling module is used to construct the trajectory prediction space under the constraints of the slave axis motion boundary conditions and generate a virtual trajectory containing dynamic error factors; through the difference feedback between real-time motion features and virtual trajectory, it decouples the mechanical-control coupling parameter offset in reverse; and autonomously reconstructs the trajectory generation kernel parameters to form a self-dissolving mechanism for dynamic error factors.

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