Robot safety cutting trajectory generation method based on neurodynamics optimization

Abstract

The application discloses a kind of robot safety cutting trajectory generation methods based on neural dynamics optimization, belong to robot motion planning and safety control technical field.The method obtains nonlinear forced term weight by training one-dimensional dynamic motion primitive model through offline teaching data;Desired cutting path is reduced to one-dimensional scalar;When generating trajectory on line, use control barrier function to carry out safety constraint to velocity and acceleration, and construct neural dynamics equation by combining Karush-Kuhn-Tucker condition and nonlinear complementary function, solve safety nonlinear forced term;Establish the mapping of one-dimensional scalar to end task space and construct natural orthogonal complementary matrix, obtain the end and joint speed instruction without side slip;Finally, through four-order Runge-Kutta integration, real-time update system state, generate safety cutting trajectory.The application reduces tool damage risk, improves the smoothness and stability of trajectory solving under complex safety constraint.

Description

Technical Field

[0001] This invention relates to the field of robot cutting trajectory generation and safety control technology, specifically to a method for generating safe robot cutting trajectories based on neurodynamics optimization. Background Technology

[0002] In modern industrial manufacturing, the use of multi-axis robotic arms equipped with cutting saws and other tools for contact processing has become commonplace. Unlike non-contact tasks such as spraying and welding, contact cutting has extremely strong dynamic physical interaction characteristics. In actual cutting processes, the internal material of the workpiece is often uneven and has high hardness, which causes the robotic arm's end effector to encounter strong reaction forces during cutting. If the robot's trajectory slips during the cutting process, or if it ignores the underlying physical constraints and maintains an unreasonable propulsion speed or acceleration, it will lead to physical overload of the actuator, resulting in tool damage or mechanical structural damage.

[0003] To achieve active tool breakage prevention, existing technologies typically introduce Dynamic Movement Primitives (DMPs) to generate compliant trajectories and attempt to combine Control Barrier Functions (CBFs) to impose physical limits on the robot's velocity and acceleration. However, existing trajectory generation methods based on DMPs and CBFs face the following technical bottlenecks in practical industrial deployments:

[0004] On the one hand, when traditional quadratic programming solvers incorporate hard constraints of CBF, mathematical conflicts easily occur between DMP and CBF safety boundaries under extreme conditions, leading to matrix rank reduction and singular collapse of the solution. On the other hand, independent trajectory generalization based on task space is difficult to effectively constrain redundant degrees of freedom of the robotic arm, resulting in displacement deviation and attitude jitter in the cutting lateral direction, increasing the risk of tool damage.

[0005] Therefore, there is an urgent need for a robot safe cutting control method that can eliminate sideslip deviation from the kinematic level and still ensure the convergence stability of the solver and the smooth output of commands when facing physical boundary constraint conflicts. Summary of the Invention

[0006] To address the technical problems in the existing technologies, such as the lack of geometric zero sideslip constraints and the tendency of quadratic programming solvers to crash, the present invention aims to provide a robot safe cutting trajectory generation method based on neurodynamic optimization. This method aims to eliminate sideslip degrees of freedom from the kinematic level and ensure stable convergence of the mathematical solver and output of smooth instructions when facing extreme physical boundary constraint conflicts, thereby achieving active anti-cutting protection for the robotic arm.

[0007] To achieve the above objectives, the present invention adopts the following technical solution:

[0008] A method for generating safe cutting trajectories for robots based on neurodynamics optimization includes the following steps:

[0009] Step 1: Train a one-dimensional dynamic motion primitive model using the teaching data obtained offline, and obtain the nonlinear forced term weight matrix of the one-dimensional dynamic motion primitive model.

[0010] Step 2: Determine the starting and ending points of the cutting in the 3D point cloud of the object to be cut, obtain the desired spatial cutting path, and reduce the spatial cutting path to a one-dimensional scalar along the cutting direction;

[0011] Step 3: Transform the second-order ordinary differential equation of the one-dimensional dynamic motion primitive model into an affine control form, set velocity and acceleration safety thresholds, construct a control obstacle function, and use the control obstacle function to apply safety constraints to the nonlinear forcing term weight matrix of the one-dimensional dynamic motion primitive model when generating the trajectory of the one-dimensional scalar online.

[0012] Step 4: Based on the set target optimization function, by establishing the Lagrange function and combining it with the Carlow-Kun-Tucker condition, the inequality constraints are transformed into equality constraints using the nonlinear complementarity problem. In this way, the neurodynamic equation is constructed to solve the safety nonlinear forcing term that satisfies the safety constraints and obtain the safety acceleration.

[0013] Step 5: Establish a mapping from a one-dimensional scalar to the task space of the robotic arm end effector, and introduce the saw blade rotation angular velocity as an additional term to construct a natural orthogonal complement matrix, converting the end effector velocity into a speed command without sideslip and a joint speed command.

[0014] Step 6: Using the safety acceleration, recalculate the safety nonlinear forcing term, safety acceleration, and natural orthogonal complement matrix in each time substep of the fourth-order Runge-Kutta integral, and update the displacement, velocity, joint state, and neurodynamic state vector accordingly to complete the safety cutting trajectory generation task.

[0015] Furthermore, step 1 includes the following sub-steps:

[0016] Step 11: Obtain the offline acquired one-dimensional teaching sequence, including the displacement sequence. velocity sequence With acceleration sequence Where t is time, and the total duration of the teaching trajectory is extracted. Total length of the teaching trajectory ;

[0017] Step 12, define the exponentially decaying phase variable of the control system. , Given the phase decay constant, It is a natural constant;

[0018] Step 13: Based on the second-order ordinary differential system equations of the one-dimensional dynamic motion primitive model, substitute them into the teaching sequence to obtain the ideal target sequence for Gaussian radial basis function fitting. ,That Target value at time for:

[0019]

[0020] In the formula, Let be the system stiffness constant. Let be the system damping constant. The total duration of the teaching trajectory. The total length of the teaching trajectory. This represents the phase variable at the current moment;

[0021] Step 14, Define the normalized nonlinear forcing term The training objective of the one-dimensional dynamic motion primitive model is to make the weighted sum of the Gaussian function at each time step such that... All of them can approximate the target value at that moment. To learn the shape of the control trajectory; when generating the trajectory online, the nonlinear forcing term actually acting on the one-dimensional dynamic motion elementary equations is... ;

[0022] and The expressions are as follows:

[0023]

[0024]

[0025] In the formula, The first one to be solved Weights of nonlinear forcing terms, Let be the index of the Gaussian function, and its range is . , For the first Gaussian radial basis functions The number of Gaussian functions, This represents the length of the cutting path;

[0026] To achieve a uniform distribution of the Gausky function along the time axis, in Generate within the interval A time series is composed of uniformly distributed discrete points. The center point of each Gaussian function in the phase space is obtained by exponential mapping. :

[0027]

[0028] In the formula, For the first The center of Gaussian function;

[0029] Calculate each discrete time point Activation value of each Gaussian function :

[0030]

[0031] In the formula, For the first The width of a Gaussian kernel, This represents the current time step at the discrete moment.

[0032] Obtain normalized activation weights :

[0033]

[0034] In the formula, Let be the index of the Gaussian function, and its range is . , Indicates the first A Gaussian function in The activation value at a given time. This represents the sum of all Gaussian function activation values, used for normalization of the molecule;

[0035] The feature activation matrix is ​​constructed by combining the normalized activation weights at all discrete time points in chronological order. ;

[0036] Using the damped least squares ridge regression algorithm, the nonlinear forcing term weight matrix of the one-dimensional dynamic motion primitive model is obtained. :

[0037]

[0038] In the formula, superscript Indicates matrix transpose. The regularization coefficient is . It is the identity matrix. The ideal objective value at each discrete time point The vector formed by these vectors.

[0039] Furthermore, step 2 includes the following sub-steps:

[0040] Step 21: Calculate the center of the point cloud of the workpiece to be cut, and use the random sampling consensus algorithm to fit the surface of the workpiece to extract its normal vector as the Z-axis of the local directed bounding box; perform boundary estimation on the workpiece point cloud, and perform straight line fitting on the extracted boundary point cloud. The direction of the longest straight line is used as the X-axis of the local directed bounding box. The two are completed by positive cross product relationship to complete the Y-axis. Finally, construct the transformation matrix from the camera coordinate system to the local coordinate system, and transform the workpiece point cloud to the local coordinate system to establish the directed bounding box of the workpiece.

[0041] Step 22: Generate candidate points at preset intervals on the four boundaries of the directed bounding box and calculate their angles relative to the horizontal plane of the camera coordinate system. For each angle, calculate the entry and exit parameters of the ray and the bounding box boundary plane in the local coordinate system using the 3D parallel plate algorithm, and extract the corresponding intersection points. Project the intersection points to the nearest points in the original boundary point cloud through nearest neighbor search, and select the point closest to the robotic arm position as the cutting starting point. The corresponding cutting endpoint is ;

[0042] Step 23, calculate the cutting start point and the finish line The distance between them is defined as the cutting path length. Extract the unit vector from the starting point to the ending point. As the cutting direction vector, the cutting path in space is represented as:

[0043]

[0044] In the formula, To cut points on the path, Let be the displacement scalar along the cutting direction. .

[0045] Furthermore, step 3 includes the following sub-steps:

[0046] Step 31, define the second-order ordinary differential equation for the online generalization stage of the one-dimensional dynamic motion primitive model as:

[0047]

[0048] In the formula, Let be the displacement scalar along the cutting direction. Let be the velocity scalar along the cutting direction. Let be the acceleration scalar along the cutting direction. This refers to the execution time of the online cutting phase. Let be the system stiffness constant. Let be the system damping constant. The length of the cutting path. The phase variable at the current moment. It is a nonlinear forcing term;

[0049] Step 32: Divide both sides of the above second-order ordinary differential equation by the time parameter. The equation is transformed into an affine control form:

[0050]

[0051] In the formula, the nonlinear forcing term Replace with the safety constraints to be solved Among them, state items that depend only on the current system state Defined as:

[0052]

[0053] Control gain term Defined as:

[0054]

[0055] Step 33, set the velocity scalar The safety limit is acceleration scalar The security limit size is For scalar speed Construct the control barrier function:

[0056]

[0057] Set control constraints for scalar acceleration:

[0058]

[0059] Based on the first-order evolution condition of the control barrier function Taking the first-order time derivative of the control barrier function yields the extremum inequality for scalar acceleration:

[0060]

[0061] In the formula, Gain parameters for adjusting the sensitivity to velocity boundary approaches;

[0062] Affine control form Substituting these values ​​into the above extreme value inequalities, we obtain the safety coercion term. The three inequalities:

[0063]

[0064]

[0065]

[0066] Convert to only restrict control quantity The constraint matrix form:

[0067]

[0068] constraint matrix for:

[0069]

[0070] constraint quantity for:

[0071]

[0072] Step 34: Construct a quadratic programming model to solve for the safety coercion terms that satisfy the constraint matrix. :

[0073]

[0074] In the formula, The original expected forcing term is generated for the weight matrix of the nonlinear forcing term obtained during training.

[0075] Furthermore, step 4 includes the following sub-steps:

[0076] Step 41, for the objective function In addition to three inequality constraints, three Lagrange multipliers are introduced. And thus construct the complete Lagrange function. :

[0077]

[0078] In the formula, and The constraint matrix and constraint quantities corresponding to velocity, and The constraint matrix and constraint quantities corresponding to the upper limit of acceleration, and The constraint matrix and constraint quantities corresponding to the lower limit of acceleration;

[0079] Step 42, according to the Carlow-Kun-Tucker conditions, the optimal solution must satisfy the condition that the Lagrangian function is perpendicular to the given condition. The partial derivative of is zero, that is:

[0080]

[0081] Since the control barrier function constraint is an inequality constraint, it cannot be solved directly using partial derivatives. The optimal conditions are found, but the Caro-Kun-Tucker conditions have the following characteristics:

[0082]

[0083]

[0084] Multiplying the two together gives:

[0085]

[0086] Transform the inequality constraints into equality equations and define slack variables. For the first Safety quantity of each constraint:

[0087]

[0088] From the nonlinear complementary function, we know that if and only if hour, If true, the inequalities in the Caro-Kun-Tucker conditions can be equivalently replaced by the following equation:

[0089]

[0090] In the formula, To guarantee continuously differentiable minimum numbers, For the constructed nonlinear complementary function;

[0091] Step 43, define the vector to be determined for neurodynamics. Based on the Lagrange derivative formula and the equation for nonlinear complementary function substitution, an error function is constructed. :

[0092]

[0093] In the formula, The coefficient matrix, For nonlinear vectors, they are defined as follows:

[0094]

[0095]

[0096] In the formula, For time-dependent constraint matrices, for Transpose of;

[0097] For error function Find the time derivative:

[0098]

[0099] Define the error integral term:

[0100]

[0101] Combining neurodynamic formulas ,have to:

[0102]

[0103] In the formula, Positive parameters that control the convergence of the error function;

[0104] Step 44: Multiply both sides of the equation by the matrix. The reverse ,have to The expression is:

[0105]

[0106] Simultaneously update the integral error term:

[0107]

[0108] Iterate until the error function is calculated. Approximating to zero yields the desired safety coercion. And then according to the formula Determine the magnitude of the terminal safety acceleration.

[0109] Furthermore, step 5 includes the following sub-steps:

[0110] Step 51, define the five-dimensional joint variables of the robotic arm as follows: The end effector space of the robotic arm is , The coordinates of the end position, The pitch angle of the cutting tool. This is the tool's rotation angle;

[0111] Step 52, establish a one-dimensional scalar Mapping function to the task space coordinate vector :

[0112]

[0113] In the formula, , , These represent the three-dimensional position coordinates of the cutting point at the end effector of the robotic arm in the task space, and these coordinates vary with... change; This indicates the pitch angle of the cutting saw relative to the workpiece surface; This represents the rotation angle of the cutting saw blade about its own axis, which varies with time. change;

[0114] Find the mapping function with respect to Using the partial derivatives, construct the natural orthogonal complement matrix. :

[0115]

[0116] In the formula, and It is a constant when cutting straight lines. and This is the partial derivative of the plane equation determined by the workpiece surface normal vector along the corresponding coordinate axis; since the tool pitch angle does not change with displacement in plane cutting, its derivative term is 0; the saw blade rotation angle is not determined by a scalar... The decision, regarding The derivative term is 0;

[0117] Step 53, define the saw blade's rotational angular velocity as:

[0118]

[0119] In the formula, This refers to the spindle speed of the saw blade. The angular velocity of the saw blade's rotation;

[0120] The additional speed parameter for constructing the task space is:

[0121]

[0122] The task space velocity is then expressed as:

[0123]

[0124] The velocity direction is along the cutting trajectory and satisfies the nonholonomic constraint that the tool's lateral velocity is zero. This represents the additional velocity term in the task space, which is unaffected by displacement along the cutting direction. The impact is still considered as a change in mission space velocity;

[0125] Step 54: Calculate the Jacobian matrix of the robotic arm and extract the corresponding first three-dimensional translational linear velocity and fourth-dimensional pitch angular velocity to construct the dimensionality-reduced model. Task Jacobian Matrix Simultaneously extract the naturally orthogonal complement matrix of the task space. The first four dimensions construct the vector Establish a method to calculate joint angular velocity from the end-effector velocity. The system of equations:

[0126]

[0127] The damped least squares method is used to solve the pseudo-inverse of the equation system, and the naturally orthogonal complement vectors in the joint space are defined. for:

[0128]

[0129] In the formula, The damping coefficient is... It is the identity matrix;

[0130] Step 55, the final joint speed command sent to the underlying motor is equivalent to:

[0131]

[0132] The saw blade rotation speed command is:

[0133] .

[0134] Furthermore, step 6 includes the following sub-steps:

[0135] Step 61: Set the discrete integral step size of the underlying control system. The cutting displacement, velocity, robotic arm joint angle, neurodynamic state vector, and error integral term are combined to form the state. :

[0136]

[0137] In the formula, Let be the displacement scalar along the cutting direction. Let be the velocity scalar along the cutting direction. For the five-dimensional joint variables of the robotic arm, For the neurodynamic state vector, This is the integral term for the error;

[0138] Step 62: Using the fourth-order Runge-Kutta integral algorithm, the differential slopes of the four stages are calculated sequentially within each control step. ;

[0139] When calculating the slope at each stage, based on the predicted state parameters derived from the current integral microstep, the phase variable, unconstrained forcing term, neurodynamic state vector derivative, safety forcing term, safety acceleration, and natural orthogonal complement matrix mapping are recalculated to obtain the state respectively. Parameters in;

[0140] Step 63: Based on the differential slopes of the four stages, calculate the system state at the next time step using the weighted average formula. :

[0141]

[0142] In the formula, Let be the column vector of the state space at the current moment. This represents the discrete integration step size;

[0143] Step 64: The saw blade rotation angle is not included in the Runge-Kutta integral; perform a first-order integral on it:

[0144]

[0145] In the formula, This indicates the saw blade rotation angle at the current moment. This indicates that after one integration step... Then, the saw blade's rotation angle at the next moment;

[0146] Step 65: Repeat the above integral update process until the running time reaches the generalized total time, and generate the final robot safe cutting execution trajectory.

[0147] The present invention also provides a system for implementing the aforementioned method for generating robot safe cutting trajectories based on neurodynamics optimization, comprising:

[0148] A one-dimensional dynamic motion primitive training module is used to train a one-dimensional dynamic motion primitive model using teaching data acquired offline, and to obtain the nonlinear forced term weight matrix of the one-dimensional dynamic motion primitive model.

[0149] The path extraction and dimensionality reduction module is used to determine the cutting start point and end point in the 3D point cloud of the object to be cut, obtain the desired spatial cutting path, and reduce the spatial cutting path to a one-dimensional scalar along the cutting direction.

[0150] The safety constraint module is used to transform the second-order ordinary differential equation of the one-dimensional dynamic motion primitive model into an affine control form, set safety thresholds for velocity and acceleration, construct a control obstacle function, and apply safety constraints to the nonlinear forcing term weight matrix of the one-dimensional dynamic motion primitive model when generating the trajectory of the one-dimensional scalar online using the control obstacle function.

[0151] The neurodynamics solution module is used to construct neurodynamic equations based on the set target optimization function. By establishing a Lagrangian function and combining it with the Carlow-Kun-Tucker condition, the inequality constraints are transformed into equality constraints using a nonlinear complementary function. The solution is then used to obtain the safety nonlinear forcing term that satisfies the safety constraints and to obtain the safety acceleration.

[0152] The natural orthogonal complement mapping module is used to establish the mapping from the one-dimensional scalar to the task space of the robotic arm end effector, and introduces the saw blade rotation angular velocity as an additional term to construct a natural orthogonal complement matrix, which converts the end effector velocity into a speed command without sideslip and a joint speed command.

[0153] The integral update module is used to recalculate the safe nonlinear forcing term, the safe acceleration, and the natural orthogonal complement matrix in each time substep of the fourth-order Runge-Kutta integral using the safe acceleration, and update the displacement, velocity, joint state, and neurodynamic state vector accordingly to complete the safe cutting trajectory generation task.

[0154] Beneficial effects: By introducing neurodynamics to solve the problem, this invention effectively overcomes the matrix rank reduction and singular collapse issues that are easily caused by traditional quadratic programming (QP) solvers when robotic arms encounter sudden changes in working conditions, ensuring the smooth output of safety control commands. At the same time, by reducing the multidimensional cutting trajectory to one dimension and combining it with the natural orthogonal complement (NOC) matrix, the lateral velocity of the cutting process is limited, realizing nonholonomic geometric constraints. In addition, this invention combines the velocity and acceleration physical limit constraints of the control obstacle function (CBF) with the DMP, reducing the risk of tool breakage in contact cutting processes and improving the safety and operational robustness of robotic arm cutting tasks. Attached Figure Description

[0155] Figure 1 This is the overall flowchart of the robot safe cutting trajectory generation method and system based on neurodynamics optimization of the present invention;

[0156] Figure 2 This is a schematic diagram of a robotic arm in a simulation environment according to an embodiment of the present invention;

[0157] Figure 3 This is a schematic diagram of the desired spatial cutting path in an embodiment of the present invention;

[0158] Figure 4 This is a schematic diagram of generating a cutting trajectory for a floor slab in an embodiment of the present invention;

[0159] Figure 5 This is a schematic diagram showing the lateral position and velocity offset of the cutting saw during the movement of the robotic arm in an embodiment of the present invention;

[0160] Figure 6 This is a schematic diagram showing the displacement, velocity, and acceleration of the end effector of the robotic arm during movement in an embodiment of the present invention. Detailed Implementation

[0161] The invention will now be further explained with reference to the accompanying drawings.

[0162] like Figure 1As shown, the present invention provides a method for generating a robot's safe cutting trajectory based on neurodynamics optimization, comprising the following steps:

[0163] Step 1: Train a one-dimensional dynamic motion primitive (DMP) model using the teaching data obtained offline, and obtain the nonlinear forcing term weight matrix of the one-dimensional dynamic motion primitive (DMP) model, as follows:

[0164] Step 11: Obtain the offline acquired one-dimensional teaching sequence, including the displacement sequence. velocity sequence With acceleration sequence Where t is time, and the total duration of the teaching trajectory is extracted. Total length of the teaching trajectory .

[0165] In this embodiment, the total duration of the teaching trajectory is... The total length of the teaching trajectory is 0.01s. .

[0166] Step 12, define the exponentially decaying phase variable of the control system. , Given the phase decay constant, It is a natural constant.

[0167] In this embodiment, the phase attenuation constant .

[0168] Step 13: Based on the second-order ordinary differential system equations of the one-dimensional dynamic motion primitive (DMP) model, substitute them into the teaching sequence to obtain the ideal target sequence for Gaussian radial basis function fitting. ,That Target value at time for:

[0169]

[0170] In the formula, Let be the system stiffness constant. Let be the system damping constant. The total duration of the teaching trajectory. The total length of the teaching trajectory. This represents the phase variable at the current moment.

[0171] In this embodiment, the system stiffness constant System damping constant .

[0172] Step 14, Define the normalized nonlinear forcing term The training objective of the one-dimensional dynamic motion primitive (DMP) model is to make the weighted sum of the Gaussian function at each time step such that... All of them can approximate the target value at that moment. To learn the shape of the control trajectory; when generating the trajectory online, the nonlinear forcing term actually acting on the one-dimensional dynamic motion elementary equations is... ;

[0173] and The expressions are as follows:

[0174]

[0175]

[0176] In the formula, The first one to be solved Weights of nonlinear forcing terms, Let be the index of the Gaussian function, and its range is . , For the first Gaussian radial basis functions The number of Gaussian functions, This represents the length of the cutting path;

[0177] To achieve a uniform distribution of the Gausky function along the time axis, in Generate within the interval A time series is composed of uniformly distributed discrete points. The center point of each Gaussian function in the phase space is obtained by exponential mapping. :

[0178]

[0179] In the formula, For the first The center of Gaussian function;

[0180] Calculate each discrete time point Activation value of each Gaussian function :

[0181]

[0182] In the formula, For the first The width of a Gaussian kernel, This represents the current time step at the discrete moment.

[0183] Obtain normalized activation weights :

[0184]

[0185] In the formula, Let be the index of the Gaussian function, and its range is . , Indicates the first A Gaussian function in The activation value at a given time. This represents the sum of all Gaussian function activation values, used for normalization of the molecule;

[0186] The feature activation matrix is ​​constructed by combining the normalized activation weights at all discrete time points in chronological order. ;

[0187] Using the damped least squares ridge regression algorithm, the nonlinear forcing term weight matrix of the one-dimensional dynamic motion primitive model is obtained. :

[0188]

[0189] In the formula, superscript Indicates matrix transpose. The regularization coefficient is . It is the identity matrix. The ideal objective value at each discrete time point The vector formed by these vectors.

[0190] In this embodiment, the number of Gaussian functions Gaussian kernel width Regularization coefficient .

[0191] Step 2: Determine the starting and ending points of the cut in the 3D point cloud of the object to be cut, obtain the desired spatial cutting path, and reduce the spatial cutting path to a one-dimensional scalar along the cutting direction, as follows:

[0192] Step 21: Calculate the center of the point cloud of the workpiece to be cut, and use the Random Sample Consensus (RANSAC) algorithm to fit the workpiece surface to extract its normal vector as the Z-axis of the local directed bounding box (OBB); perform boundary estimation on the workpiece point cloud, and perform straight line fitting on the extracted boundary point cloud. The direction of the longest straight line is used as the X-axis of the local directed bounding box (OBB). The two are completed by positive cross product relationship to complete the Y-axis. Finally, construct the transformation matrix from the camera coordinate system to the local coordinate system, and transform the workpiece point cloud to the local coordinate system to establish the directed bounding box of the workpiece.

[0193] In this embodiment, after obtaining the point cloud of the workpiece to be cut, the center of the point cloud is calculated, and normal estimation and boundary estimation are performed on the point cloud. The nearest neighbor count for normal estimation is 20, the nearest neighbor count for boundary estimation is 30, and the boundary angle threshold is set to... The RANSAC algorithm was used to fit the workpiece surface, with a fitting distance threshold of 0.03, and the surface normal vector was extracted as the Z-axis of the local directed bounding box.

[0194] Linear fitting is performed on the boundary point cloud, with a RANSAC distance threshold of 0.02 for the boundary lines. The main direction of the fitted boundary is used as the X-axis of the local coordinate system, and the Y-axis is determined by the positive cross product of the Z-axis and the X-axis. Thus, the transformation matrix from the camera coordinate system to the workpiece local coordinate system is obtained.

[0195] Within the bounding box of the workpiece point cloud, candidate cutting azimuth angles are generated at preset intervals on the four boundaries of the OBB. In this embodiment, the target spacing between adjacent candidate cutting lines is 1.0m: when the boundary segment length is greater than this spacing, candidate lines are arranged at 1.0m intervals; when the remaining length is less than 2.0m, candidate lines are arranged at the midpoint. Then, candidate lines are deleted one by one. If the spacing between adjacent cutting points on the four boundaries still meets the requirements after deletion, the deletion result is retained.

[0196] Step 22: Generate candidate points at preset intervals on the four boundaries of the directed bounding box and calculate their angles relative to the horizontal plane of the camera coordinate system. For each angle, use the 3D Slab algorithm in the local coordinate system to calculate the entry and exit parameters of the ray and the bounding box boundary plane, and extract the corresponding intersection points. Project the intersection points to the nearest point in the original boundary point cloud through nearest neighbor search, and select the point closest to the robotic arm position as the cutting starting point. The corresponding cutting endpoint is .

[0197] Step 23, calculate the cutting start point and the finish line The distance between them is defined as the cutting path length. Extract the unit vector from the starting point to the ending point. As the cutting direction vector, the cutting path in space is represented as:

[0198]

[0199] In the formula, To cut points on the path, Let be the displacement scalar along the cutting direction. .

[0200] Step 3: Transform the second-order ordinary differential equations of the one-dimensional dynamic motion primitive (DMP) model into an affine control form, set velocity and acceleration safety thresholds, construct a control barrier function, and when generating the trajectory of the one-dimensional scalar online, use the control barrier function (CBF) to apply safety constraints to the nonlinear forcing term weight matrix of the one-dimensional dynamic motion primitive model, as follows:

[0201] Step 31, define the second-order ordinary differential equation for the online generalization stage of the one-dimensional dynamic motion primitive (DMP) model as follows:

[0202]

[0203] In the formula, Let be the displacement scalar along the cutting direction. Let be the velocity scalar along the cutting direction. Let be the acceleration scalar along the cutting direction. This refers to the execution time of the online cutting phase. Let be the system stiffness constant. Let be the system damping constant. The length of the cutting path. The phase variable at the current moment. This is a nonlinear forcing term.

[0204] Step 32: Divide both sides of the above second-order ordinary differential equation by the time parameter. The equation is transformed into an affine control form:

[0205]

[0206] In the formula, the nonlinear forcing term Replace with the safety constraints to be solved Among them, state items that depend only on the current system state Defined as:

[0207]

[0208] Control gain term Defined as:

[0209]

[0210] Step 33, set the velocity scalar The safety limit is acceleration scalar The security limit size is For scalar speed Constructing the Control Barrier Function (CBF):

[0211]

[0212] Set control constraints for scalar acceleration:

[0213]

[0214] Based on the first-order evolution condition of the control barrier function Taking the first-order time derivative of the control barrier function yields the extremum inequality for scalar acceleration:

[0215]

[0216] In the formula, Gain parameters for adjusting the sensitivity to velocity boundary approaches;

[0217] Affine control form Substituting these values ​​into the above extreme value inequalities, we obtain the safety coercion term. The three inequalities:

[0218]

[0219]

[0220]

[0221] Convert to only restrict control quantity The constraint matrix form:

[0222]

[0223] constraint matrix for:

[0224]

[0225] constraint quantity for:

[0226]

[0227] Step 34: Construct a quadratic programming model to solve for the safety coercion terms that satisfy the constraint matrix. :

[0228]

[0229] In the formula, The original expected forcing term is generated for the weight matrix of the nonlinear forcing term obtained during training.

[0230] Step 4: Based on the set objective optimization function, by establishing a Lagrangian function and combining it with the Carlow-Kuhn-Tucker conditions, the inequality constraints are transformed into equality constraints using a nonlinear complementarity problem. This allows the construction of a neurodynamic equation to solve for the safety nonlinear forcing term that satisfies the safety constraints, thereby obtaining the safety acceleration. Specifically, as follows:

[0231] Step 41, for the objective function In addition to three inequality constraints, three Lagrange multipliers are introduced. And thus construct the complete Lagrange function. :

[0232]

[0233] In the formula, and The constraint matrix and constraint quantities corresponding to velocity, and The constraint matrix and constraint quantities corresponding to the upper limit of acceleration, and The constraint matrix and constraint quantities corresponding to the lower limit of acceleration;

[0234] Step 42, according to the Caro-Kuhn-Tucker (KKT) conditions, the optimal solution must satisfy the condition that the Lagrangian function is perpendicular to the given condition. The partial derivative of is zero, that is:

[0235]

[0236] Since the control barrier function (CBF) constraint is an inequality constraint, it cannot be solved directly using partial derivatives. The optimal conditions are found, but its Caro-Kun-Tucker (KKT) conditions have the following characteristics:

[0237]

[0238]

[0239] Multiplying the two together gives:

[0240]

[0241] Transform the inequality constraints into equality equations and define slack variables. For the first Safety quantity of each constraint:

[0242]

[0243] From the nonlinear complementary (NCP) function, it can be seen that if and only if hour, If true, the inequalities in the Caro-Kun-Tucker conditions can be equivalently replaced by the following equation:

[0244]

[0245] In the formula, To guarantee continuously differentiable minimum numbers, For the constructed nonlinear complementary function.

[0246] In this embodiment, take .

[0247] Step 43, define the vector to be determined for neurodynamics. Based on the Lagrange derivative formula and the equation for nonlinear complementary function substitution, an error function is constructed. :

[0248]

[0249] In the formula, The coefficient matrix, For nonlinear vectors, they are defined as follows:

[0250]

[0251]

[0252] In the formula, For time-dependent constraint matrices, for Transpose of;

[0253] For error function Find the time derivative:

[0254]

[0255] Define the error integral term:

[0256]

[0257] Combining neurodynamic formulas ,have to:

[0258]

[0259] In the formula, A positive parameter is used to control the convergence of the error function. In this embodiment, , .

[0260] Step 44: Multiply both sides of the equation by the matrix. The reverse ,have to The expression is:

[0261]

[0262] Simultaneously update the integral error term:

[0263]

[0264] Iterate until the error function is calculated. Approximating to zero yields the desired safety coercion. And then according to the formula Determine the magnitude of the terminal safety acceleration.

[0265] Step 5: Establish a mapping from a one-dimensional scalar to the task space of the robotic arm's end effector, and introduce the saw blade's rotational angular velocity as an additional term to construct a Natural Orthogonal Complement (NOC) matrix. Convert the end effector velocity into speed commands and joint speed commands without sideslip, as follows:

[0266] Step 51, define the five-dimensional joint variables of the robotic arm as follows: The end effector space of the robotic arm is , The coordinates of the end position, The pitch angle of the cutting tool. This is the tool's rotation angle;

[0267] Step 52, establish a one-dimensional scalar Mapping function to the task space coordinate vector :

[0268]

[0269] In the formula, , , These represent the three-dimensional position coordinates of the cutting point at the end effector of the robotic arm in the task space, and these coordinates vary with... change; This indicates the pitch angle of the cutting saw relative to the workpiece surface; This represents the rotation angle of the cutting saw blade about its own axis, which varies with time. change;

[0270] Find the mapping function with respect to Using the partial derivatives, construct the natural orthogonal complement matrix. :

[0271]

[0272] In the formula, and It is a constant when cutting straight lines. and This is the partial derivative of the plane equation determined by the workpiece surface normal vector along the corresponding coordinate axis; since the tool pitch angle does not change with displacement in plane cutting, its derivative term is 0; the saw blade rotation angle is not determined by a scalar... The decision, regarding The derivative term is 0;

[0273] Step 53, define the saw blade's rotational angular velocity as:

[0274]

[0275] In the formula, This refers to the spindle speed of the saw blade. ω is the angular velocity of the saw blade's rotation.

[0276] In this embodiment, the spindle speed of the saw blade is set to 800 rpm.

[0277] The additional speed parameter for constructing the task space is:

[0278]

[0279] The task space velocity is then expressed as:

[0280]

[0281] The velocity direction is along the cutting trajectory and satisfies the nonholonomic constraint that the tool's lateral velocity is zero. This represents the additional velocity term in the task space, which is unaffected by displacement along the cutting direction. The impact is still considered as a change in mission space velocity;

[0282] Step 54: Calculate the Jacobian matrix of the robotic arm and extract the corresponding first three-dimensional translational linear velocity and fourth-dimensional pitch angular velocity to construct the dimensionality-reduced model. Task Jacobian Matrix Simultaneously extract the Natural Orthogonal Complement (NOC) matrix of the task space. The first four dimensions construct the vector Establish a method to calculate joint angular velocity from the end-effector velocity. The system of equations:

[0283]

[0284] The damped least squares method is used to solve the pseudo-inverse of the equation system, and the naturally orthogonal complement (NOC) vector in the joint space is defined. for:

[0285]

[0286] In the formula, The damping coefficient is... It is an identity matrix.

[0287] In this embodiment, .

[0288] Step 55, the final joint speed command sent to the underlying motor is equivalent to:

[0289]

[0290] The saw blade rotation speed command is:

[0291] .

[0292] Step 6: Using the aforementioned safety acceleration, recalculate the safety nonlinear forcing term, safety acceleration, and natural orthogonal complement (NOC) matrix within each time substep of the fourth-order Runge-Kutta integral. Based on this, update the displacement, velocity, joint state, and neurodynamic state vectors to complete the safety cutting trajectory generation task, as detailed below:

[0293] Step 61: Set the discrete integral step size of the underlying control system. The cutting displacement, velocity, robotic arm joint angle, neurodynamic state vector, and error integral term are combined to form the state. :

[0294]

[0295] In the formula, Let be the displacement scalar along the cutting direction. Let be the velocity scalar along the cutting direction. For the five-dimensional joint variables of the robotic arm, For the neurodynamic state vector, This is the error integral term.

[0296] In this embodiment, .

[0297] Step 62: Using the fourth-order Runge-Kutta integral algorithm, the differential slopes of the four stages are calculated sequentially within each control step. ;

[0298] When calculating the slope at each stage, based on the predicted state parameters derived from the current integral microstep, the phase variable, unconstrained forcing term, neurodynamic state vector derivative, safety forcing term, safety acceleration, and natural orthogonal complement matrix mapping are recalculated to obtain the state respectively. Parameters in;

[0299] Step 63: Based on the differential slopes of the four stages, calculate the system state at the next time step using the weighted average formula. :

[0300]

[0301] In the formula, Let be the column vector of the state space at the current moment. This represents the discrete integration step size;

[0302] Step 64: The saw blade rotation angle is not included in the Runge-Kutta integral; perform a first-order integral on it:

[0303]

[0304] In the formula, This indicates the saw blade rotation angle at the current moment. This indicates that after one integration step... Then, the saw blade's rotation angle at the next moment;

[0305] Step 65: Repeat the above integral update process until the running time reaches the generalized total time, and generate the final robot safe cutting execution trajectory.

[0306] like Figure 2 As shown in the figure, in this embodiment, the robotic arm used in the simulation environment has a cutting saw installed at its end position.

[0307] like Figure 3 As shown, this embodiment performs candidate cutting lines for the floor slab point cloud. First, a local bounding box is established based on the surface normal vector and the principal direction of the boundary of the point cloud. Then, the starting and ending points of the cutting are determined by dividing the candidate lines and finding the intersection of Slab. The local oriented bounding box is represented by a yellow box, and the desired cutting line is represented by an orange-yellow box, where the green point is the starting point and the red point is the ending point. The green numbers around the bounding box represent the remaining length of the floor slab's length and width after cutting along the desired cutting line, with a maximum requirement of 1 ± 0.1 m. The light blue part below the bounding box is the reference horizontal plane, used to visualize the tilt height of the floor slab.

[0308] like Figure 4 As shown, this embodiment uses DMP combined with CBF constraints, and a neurodynamic solver and NOC mapping to generate the end trajectory. The trajectory in the figure is located above the target floor slab and moves along the determined cutting direction. The blue box is a visualization of the cutting saw, the orange line is the generated cutting trajectory, the green dots are the cutting start points, the red dots are the cutting end points, the light blue point cloud is the target floor slab to be cut, and the gray point cloud is the environmental point cloud during scanning.

[0309] like Figure 5 As shown in the figure, this embodiment illustrates the changes in the lateral position offset and lateral velocity offset of the cutting saw. Since the NOC matrix is ​​recalculated based on the current joint state in each integral substep and participates in the calculation of the joint velocity command, the lateral velocity remains zero during the cutting process, and the lateral position offset also remains zero. The results demonstrate that the NOC mapping can effectively suppress cutting saw slippage, meeting the slippage-free execution requirements in contact cutting.

[0310] like Figure 6As shown in the figure, this embodiment illustrates the changes in displacement, velocity, and acceleration along the cutting direction during the robotic arm's movement. The displacement curve changes continuously until the total generalization time is reached. Since the trajectory length generated for the new floor slab point cloud is greater than the trajectory length during training, the total generalization time exceeds 3 seconds. The velocity curve remains continuous during operation and maintains the constraint at the maximum velocity of the CBF (Constant Velocity Flow). The acceleration curve also remains within the limited range of the maximum acceleration magnitude. These results demonstrate that the DMP trajectory, after being constrained by CBF, can be correctly solved through neurodynamics and generate a safe trajectory.

[0311] The above description is only a preferred embodiment of the present invention. It should be noted that for those skilled in the art, several improvements and modifications can be made without departing from the principle of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention.

Claims

1. A method for generating a safe cutting trajectory for a robot based on neurodynamics optimization, characterized in that, Includes the following steps: Step 1: Train a one-dimensional dynamic motion primitive model using the teaching data obtained offline, and obtain the nonlinear forced term weight matrix of the one-dimensional dynamic motion primitive model. Step 1 includes the following sub-steps: Step 11: Obtain the offline acquired one-dimensional teaching sequence, including the displacement sequence. velocity sequence With acceleration sequence Where t is time, and the total duration of the teaching trajectory is extracted. Total length of the teaching trajectory ; Step 12, define the exponentially decaying phase variable of the control system. , Given the phase decay constant, It is a natural constant; Step 13: Based on the second-order ordinary differential system equations of the one-dimensional dynamic motion primitive model, substitute them into the teaching sequence to obtain the ideal target sequence for Gaussian radial basis function fitting. ,That Target value at time for: In the formula, Let be the system stiffness constant. Let be the system damping constant. The total duration of the teaching trajectory. The total length of the teaching trajectory. This represents the phase variable at the current moment; Step 14, Define the normalized nonlinear forcing term The training objective of the one-dimensional dynamic motion primitive model is to make the weighted sum of the Gaussian function at each time step such that... All of them can approximate the target value at that moment. To learn the shape of the control trajectory; when generating the trajectory online, the nonlinear forcing term actually acting on the one-dimensional dynamic motion elementary equations is... ; and The expressions are as follows: In the formula, The first one to be solved Weights of nonlinear forcing terms, Let be the index of the Gaussian function, and its range is . , For the first Gaussian radial basis functions The number of Gaussian functions, This represents the length of the cutting path. To achieve a uniform distribution of the Gausky function along the time axis, in Generate within the interval A time series is composed of uniformly distributed discrete points. The center point of each Gaussian function in the phase space is obtained by exponential mapping. : In the formula, For the first The center of Gaussian function; Calculate each discrete time point Activation value of each Gaussian function : In the formula, For the first The width of a Gaussian kernel, This represents the current time step at discrete moments; Obtain normalized activation weights : In the formula, Let be the index of the Gaussian function, and its range is . , Indicates the first A Gaussian function in The activation value at a given time. This represents the sum of all Gaussian function activation values, used for normalization of the molecule; The feature activation matrix is ​​constructed by combining the normalized activation weights at all discrete time points in chronological order. ; Using the damped least squares ridge regression algorithm, the nonlinear forcing term weight matrix of the one-dimensional dynamic motion primitive model is obtained. : In the formula, superscript Indicates matrix transpose. The regularization coefficient is . It is the identity matrix. The ideal objective value at each discrete time point The vector formed; Step 2: Determine the starting and ending points of the cutting in the 3D point cloud of the object to be cut, obtain the desired spatial cutting path, and reduce the spatial cutting path to a one-dimensional scalar along the cutting direction; Step 3: Transform the second-order ordinary differential equation of the one-dimensional dynamic motion primitive model into an affine control form, set velocity and acceleration safety thresholds, construct a control obstacle function, and use the control obstacle function to apply safety constraints to the nonlinear forcing term weight matrix of the one-dimensional dynamic motion primitive model when generating the trajectory of the one-dimensional scalar online. Step 4: Based on the set target optimization function, by establishing the Lagrange function and combining it with the Carlow-Kun-Tucker condition, the inequality constraints are transformed into equality constraints using the nonlinear complementarity problem. In this way, the neurodynamic equation is constructed to solve the safety nonlinear forcing term that satisfies the safety constraints and obtain the safety acceleration. Step 5: Establish a mapping from a one-dimensional scalar to the task space of the robotic arm end effector, and introduce the saw blade rotation angular velocity as an additional term to construct a natural orthogonal complement matrix, converting the end effector velocity into a speed command without sideslip and a joint speed command. Step 6: Using the safety acceleration, recalculate the safety nonlinear forcing term, safety acceleration, and natural orthogonal complement matrix in each time substep of the fourth-order Runge-Kutta integral, and update the displacement, velocity, joint state, and neurodynamic state vector accordingly to complete the safety cutting trajectory generation task.

2. The method according to claim 1, characterized in that, Step 2 includes the following sub-steps: Step 21: Calculate the center of the point cloud of the workpiece to be cut, and use the random sampling consensus algorithm to fit the surface of the workpiece to extract its normal vector as the Z-axis of the local directed bounding box; perform boundary estimation on the workpiece point cloud, and perform straight line fitting on the extracted boundary point cloud. The direction of the longest straight line is used as the X-axis of the local directed bounding box. The two are completed by positive cross product relationship to complete the Y-axis. Finally, construct the transformation matrix from the camera coordinate system to the local coordinate system, and transform the workpiece point cloud to the local coordinate system to establish the directed bounding box of the workpiece. Step 22: Generate candidate points at preset intervals on the four boundaries of the directed bounding box and calculate their angles relative to the horizontal plane of the camera coordinate system. For each angle, calculate the entry and exit parameters of the ray and the bounding box boundary plane in the local coordinate system using the 3D parallel plate algorithm, and extract the corresponding intersection points. Project the intersection points to the nearest points in the original boundary point cloud through nearest neighbor search, and select the point closest to the robotic arm position as the cutting starting point. The corresponding cutting endpoint is ; Step 23, calculate the cutting start point and the finish line The distance between them is defined as the cutting path length. Extract the unit vector from the starting point to the ending point. As the cutting direction vector, the cutting path in space is represented as: In the formula, To cut points on the path, Let be the displacement scalar along the cutting direction. .

3. The method according to claim 2, characterized in that, Step 3 includes the following sub-steps: Step 31, define the second-order ordinary differential equation for the online generalization stage of the one-dimensional dynamic motion primitive model as: In the formula, Let be the displacement scalar along the cutting direction. Let be the velocity scalar along the cutting direction. Let be the acceleration scalar along the cutting direction. This refers to the execution time of the online cutting phase. Let be the system stiffness constant. Let be the system damping constant. The length of the cutting path. The phase variable at the current moment. It is a nonlinear forcing term; Step 32: Divide both sides of the above second-order ordinary differential equation by the time parameter. The equation is transformed into an affine control form: In the formula, the nonlinear forcing term Replace with the safety constraints to be solved Among them, state items that depend only on the current system state Defined as: Control gain term Defined as: Step 33, set the velocity scalar The safety limit is acceleration scalar The security limit size is For scalar speed Construct the control barrier function: Set control constraints for scalar acceleration: Based on the first-order evolution condition of the control barrier function Taking the first-order time derivative of the control barrier function yields the extremum inequality of scalar acceleration: In the formula, Gain parameters for adjusting the sensitivity to velocity boundary approaches; Affine control form Substituting these values ​​into the above extreme value inequalities, we obtain the safety coercion term. The three inequalities: Convert to only restrict control quantity The constraint matrix form: constraint matrix for: constraint quantity for: Step 34: Construct a quadratic programming model to solve for the safety coercion terms that satisfy the constraint matrix. : In the formula, The original expected forcing term is generated for the weight matrix of the nonlinear forcing term obtained during training.

4. The method according to claim 3, characterized in that, Step 4 includes the following sub-steps: Step 41, for the objective function In addition to three inequality constraints, three Lagrange multipliers are introduced. And thus construct the complete Lagrange function. : In the formula, and The constraint matrix and constraint quantities corresponding to velocity, and The constraint matrix and constraint quantities corresponding to the upper limit of acceleration, and The constraint matrix and constraint quantities corresponding to the lower limit of acceleration; Step 42, according to the Carlow-Kun-Tucker conditions, the optimal solution must satisfy the condition that the Lagrangian function is perpendicular to the given condition. The partial derivative of is zero, that is: Since the control barrier function constraint is an inequality constraint, it cannot be solved directly using partial derivatives. The optimal conditions are found, but its Caro-Kun-Tucker conditions have the following characteristics: Multiplying the two together gives: Transform the inequality constraints into equality equations and define slack variables. For the first Safety quantity of each constraint: From the nonlinear complementary function, we know that if and only if hour, If true, the inequalities in the Caro-Kun-Tucker conditions can be equivalently replaced by the following equation: In the formula, To guarantee continuously differentiable minimum numbers, For the constructed nonlinear complementary function; Step 43, define the vector to be determined for neurodynamics. Based on the Lagrange derivative formula and the equation for nonlinear complementary function substitution, an error function is constructed. : In the formula, The coefficient matrix, For nonlinear vectors, they are defined as follows: In the formula, For time-dependent constraint matrices, for Transpose of; For error function Find the time derivative: Define the error integral term: Combining neurodynamic formulas ,have to: In the formula, Positive parameters that control the convergence of the error function; Step 44: Multiply both sides of the equation by the matrix. The reverse ,have to The expression is: Simultaneously update the integral error term: Iterate until the error function is calculated. Approximating to zero yields the desired safety coercion. And then according to the formula Determine the magnitude of the terminal safety acceleration.

5. The method according to claim 4, characterized in that, Step 5 includes the following sub-steps: Step 51, define the five-dimensional joint variables of the robotic arm as follows: The end effector space of the robotic arm is , The coordinates of the end position, The pitch angle of the cutting tool. This is the tool's rotation angle; Step 52, establish a one-dimensional scalar Mapping function to the task space coordinate vector : In the formula, , , These represent the three-dimensional position coordinates of the cutting point at the end effector of the robotic arm in the task space, and these coordinates vary with... change; This indicates the pitch angle of the cutting saw relative to the workpiece surface; This represents the rotation angle of the cutting saw blade about its own axis, which varies with time. change; Find the mapping function with respect to Using the partial derivatives, construct the natural orthogonal complement matrix. : In the formula, and It is a constant when cutting straight lines. and The partial derivatives of the plane equation determined by the workpiece surface normal vector along the corresponding coordinate axes are given; since the tool pitch angle does not change with displacement in plane cutting, its derivative term is 0; the saw blade rotation angle is not determined by a scalar. The decision, regarding The derivative term is 0; Step 53, define the saw blade's rotational angular velocity as: In the formula, This refers to the spindle speed of the saw blade. The angular velocity of the saw blade's rotation; The additional speed parameter for constructing the task space is: The task space velocity is then expressed as: The velocity direction is along the cutting trajectory and satisfies the nonholonomic constraint that the tool's lateral velocity is zero. This represents the additional velocity term in the task space, which is unaffected by displacement along the cutting direction. The impact is still considered as a change in mission space velocity; Step 54: Calculate the Jacobian matrix of the robotic arm and extract the corresponding first three-dimensional translational linear velocity and fourth-dimensional pitch angular velocity to construct the dimensionality-reduced [model / structure]. Task Jacobian Matrix Simultaneously extract the naturally orthogonal complement matrix of the task space. The first four dimensions construct the vector Establish a method to calculate joint angular velocity from the end-effector velocity. The system of equations: The damped least squares method is used to solve the pseudo-inverse of the equation system, and the naturally orthogonal complement vectors in the joint space are defined. for: In the formula, The damping coefficient is... It is the identity matrix; Step 55, the final joint speed command sent to the underlying motor is equivalent to: The saw blade rotation speed command is: 。 6. The method according to claim 5, characterized in that, Step 6 includes the following sub-steps: Step 61: Set the discrete integral step size of the underlying control system. The cutting displacement, velocity, robotic arm joint angle, neurodynamic state vector, and error integral term are combined to form the state. : In the formula, Let be the displacement scalar along the cutting direction. Let be the velocity scalar along the cutting direction. For the five-dimensional joint variables of the robotic arm, For the neurodynamic state vector, This is the integral term for the error; Step 62: Using the fourth-order Runge-Kutta integral algorithm, the differential slopes of the four stages are calculated sequentially within each control step. ; When calculating the slope at each stage, based on the predicted state parameters derived from the current integral microstep, the phase variable, unconstrained forcing term, neurodynamic state vector derivative, safety forcing term, safety acceleration, and natural orthogonal complement matrix mapping are recalculated to obtain the state respectively. Parameters in; Step 63: Based on the differential slopes of the four stages, calculate the system state at the next time step using the weighted average formula. : In the formula, Let be the column vector of the state space at the current moment. This is the discrete integration step size; Step 64: The saw blade rotation angle is not included in the Runge-Kutta integral; perform a first-order integral on it: In the formula, This indicates the saw blade rotation angle at the current moment. This indicates that after one integration step... Then, the saw blade's rotation angle at the next moment; Step 65: Repeat the above integral update process until the running time reaches the generalized total time, and generate the final robot safe cutting execution trajectory.

7. A system for implementing the neurodynamics-optimized robot safe cutting trajectory generation method according to any one of claims 1 to 6, characterized in that, include: A one-dimensional dynamic motion primitive training module is used to train a one-dimensional dynamic motion primitive model using teaching data acquired offline, and to obtain the nonlinear forced term weight matrix of the one-dimensional dynamic motion primitive model. The path extraction and dimensionality reduction module is used to determine the cutting start point and end point in the 3D point cloud of the object to be cut, obtain the desired spatial cutting path, and reduce the spatial cutting path to a one-dimensional scalar along the cutting direction. The safety constraint module is used to transform the second-order ordinary differential equation of the one-dimensional dynamic motion primitive model into an affine control form, set safety thresholds for velocity and acceleration, construct a control obstacle function, and apply safety constraints to the nonlinear forcing term weight matrix of the one-dimensional dynamic motion primitive model when generating the trajectory of the one-dimensional scalar online using the control obstacle function. The neurodynamics solution module is used to construct neurodynamic equations based on the set target optimization function. By establishing a Lagrangian function and combining it with the Carlow-Kun-Tucker condition, the inequality constraints are transformed into equality constraints using a nonlinear complementary function. The solution is then used to obtain the safety nonlinear forcing term that satisfies the safety constraints and to obtain the safety acceleration. The natural orthogonal complement mapping module is used to establish the mapping from the one-dimensional scalar to the task space of the robotic arm end effector, and introduces the saw blade rotation angular velocity as an additional term to construct a natural orthogonal complement matrix, which converts the end effector velocity into a speed command without sideslip and a joint speed command. The integral update module is used to recalculate the safe nonlinear forcing term, the safe acceleration, and the natural orthogonal complement matrix in each time substep of the fourth-order Runge-Kutta integral using the safe acceleration, and update the displacement, velocity, joint state, and neurodynamic state vector accordingly to complete the safe cutting trajectory generation task.

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