Mechanical arm imitation learning method and device based on DMP

Abstract

The invention provides a mechanical arm imitation learning method and device based on DMP, and belongs to the technical field of robots. The method comprises the steps that a trajectory is fitted through a motion primitive method; according to the teaching or simulation track, the weight of the motion primitive is obtained through a least square method; a new trajectory regeneration method is provided, and new trajectories with the same geometric features are generated by setting a new starting point, a new ending point or a new scaling coefficient; a new track passing point strategy is provided, so that the generated new track can pass through a specific key point; applying a Gaussian process to generate a high-order smooth trajectory passing through a specific point; generating a high-order smooth curve meeting the geometric characteristics by using a weighted average method; substituting the generated new trajectory into the new DMP model, and establishing a dynamic system model; and according to the dynamic system model, the mechanical arm joints are driven through the joint speed track, and speed control is achieved. According to the method, a large amount of similar industrial programming work can be reduced, and the flexibility and efficiency of trajectory planning and control are improved.

Description

Technical Field

[0001] The present invention relates to the technical field of robotics, in particular to a trajectory planning and control method for a robotic arm, and more particularly to a method and device for robotic arm imitation learning based on DMP (Dynamic Movement Primitives). Background Art

[0002] With the rapid development of industrial automation, industrial robotic arms, as an efficient and flexible execution device, have been widely used in multiple fields such as manufacturing, rescue, and equipment maintenance. For the maintenance task of the first wall of a fusion reactor device, the specific tasks performed by the robotic arm mainly include detection, disassembly and assembly, welding, etc. For the above tasks, the teaching method is often used in industry for trajectory planning, enabling the robotic arm to learn a specified trajectory and controlling the robotic arm to move along the trajectory through a controller. Compared with industrial robots participating in assembly line operations, the robotic arm used for fusion device maintenance is transported into the interior of the fusion device by a heavy-duty arm, its base is not fixed, and the maintenance target is not fixed relative to the base coordinates of the robotic arm. If the trajectory planning method of industrial robots is applied, a large amount of industrial programming work will be faced. Moreover, there are a large number of tasks that are similar but not exactly the same in the maintenance tasks, or special tasks passing through specific points. The traditional teaching method does not have flexible generalization ability and cannot plan trajectories that meet new geometric conditions or special conditions; although the existing DMP (Dynamic Movement Primitives) imitation learning algorithm has generalization ability, it will fail when facing teaching trajectories with the same or similar start and end positions and cannot effectively handle all planning situations.

[0003] Considering the above problems, it is necessary to propose an imitation learning algorithm with strong generalization ability that can handle special trajectory situations such as special passing points and close start and end points, which can efficiently plan trajectories that meet engineering requirements and retain the geometric features of the teaching trajectory, greatly reducing the workload of industrial programming while completing the maintenance task and significantly improving production efficiency. Summary of the Invention

[0004] To solve the above-mentioned technical problems, the present invention proposes a DMP-based robotic arm imitation learning method and device, specifically a robotic arm trajectory planning method based on the DMP model that learns and imitates the geometric features of the teaching trajectory and has flexible generalization ability according to engineering requirements. In the present invention, a multi-level planning strategy is adopted, and a new trajectory regeneration algorithm for different engineering requirements is proposed. When there is no specific passing point requirement, the direct regeneration algorithm is used, which can effectively handle the special case where the starting and ending positions of the teaching trajectory are close while imitating the geometric features of the teaching trajectory, and generate a new trajectory with both geometric features and close starting and ending points; when there is a specific passing point requirement in the middle of the target trajectory, the segmented planning strategy is adopted, the target trajectory is divided into several trajectory segments at the passing point, and the segments are generated and spliced into a complete trajectory one by one, and the splicing points of the trajectory are softened to achieve high-order continuity of the trajectory; the planned trajectory is used as a reference trajectory and brought into the DMP dynamic equation, and the ordinary differential equation is solved and integrated over time to obtain the control quantity; finally, the obtained control quantity is input into the robotic arm controller to drive the robotic arm joints, and ultimately achieve more flexible and efficient trajectory planning and smooth and stable control of the robotic arm.

[0005] To achieve the above object, the technical solution adopted by the present invention is as follows:

[0006] A DMP-based robotic arm imitation learning method, comprising the following steps:

[0007] Step 1, fitting the teaching trajectory by using the method of motion primitives;

[0008] Step 2, obtaining the weights of the motion primitives by the least squares method according to the teaching trajectory;

[0009] Step 3, setting a new starting point, ending point and scaling factor according to the new trajectory imitation method, and generating a new trajectory with the same geometric features as the original teaching trajectory;

[0010] Step 4, making the generated new trajectory pass through specific passing points through the new trajectory passing point strategy;

[0011] Step 5, applying the Gaussian process to generate a high-order smooth trajectory passing through specific passing points;

[0012] Step 6, generating a high-order smooth curve that satisfies the geometric features and passes through specific passing points by using a weighted method based on the curves in Steps 4 and 5;

[0013] Step 7, substituting the generated new trajectory into the new DMP model to establish a dynamic system model;

[0014] Step 8, generating a control quantity according to the dynamic system model in Step 7 to drive the robotic arm joints.

[0015] A DMP-based robotic arm imitation learning device, comprising:

[0016] A fitting module for fitting a teaching trajectory using the method of motion primitives;

[0017] A weight calculation module for obtaining the weights of motion primitives by the least squares method according to the teaching trajectory;

[0018] A first generation module for setting a new starting point, ending point or scaling factor according to a new trajectory imitation method and generating a new trajectory with the same geometric features as the original teaching trajectory;

[0019] A strategy module for enabling the generated new trajectory to pass through specific passing points through the new trajectory passing point strategy;

[0020] A second generation module for applying a Gaussian process to generate a high-order smooth trajectory passing through specific passing points;

[0021] A third generation module for generating a high-order smooth curve that satisfies geometric features and can pass through specific passing points by a weighting method based on the curves in steps four and five;

[0022] An establishment module for substituting the generated new trajectory into a new DMP model to establish a dynamic system model;

[0023] A driving module for generating a control quantity according to the dynamic system model in step seven to drive the robotic arm joints.

[0024] The innovation points of the present invention are:

[0025] An improved trajectory imitation learning algorithm is proposed, which includes using a direct regeneration algorithm when there is no requirement for specific passing points, which can effectively process trajectories with similar starting and ending positions while retaining the geometric features of the teaching trajectory; for target trajectories with specific passing point requirements in the middle section, a segmented planning strategy is introduced to implement the generation and splicing of trajectory segments one by one, and the trajectory splicing is softened to ensure the smoothness of the generated complete trajectory in high-order continuity and meet the engineering requirements of robotic arm movement; the planned reference trajectory is brought into the DMP dynamics equation, and the control quantity is generated by solving the ordinary differential equation and integrating with respect to time, realizing trajectory tracking control at the speed level, and improving the control accuracy and stability of the robotic arm in combination with the ROS framework.

[0026] The beneficial effects of the present invention are:

[0027] The present invention proposes a new trajectory imitation learning algorithm, which realizes the learning of trajectories with similar starting and ending points, enhancing the flexibility and generalization ability of this type of imitation learning algorithm. At the same time, a segmented planning and trajectory softening method based on Gaussian processes is proposed, which realizes the planning of special trajectories that need to pass through specific points and ensures the smoothness of the trajectories. Finally, the planned reference trajectory is brought into the DMP dynamics equation, and through the trajectory tracking of the manipulator speed level, more stable control is achieved. The present invention proposes a more flexible trajectory imitation learning algorithm that can cover most engineering requirements, realizes efficient trajectory planning and real-time control, greatly reduces the complexity of industrial programming, reduces manual intervention and industrial programming workload, and at the same time ensures the accuracy of trajectory planning, providing technical support for flexible and efficient manipulator operation.

[0028] The present invention can imitate the position geometry and speed trajectory of the taught trajectory, and generate new, smooth, high-order continuous trajectories passing through specific passing points according to the requirements of new work tasks, with excellent generalization ability. The trajectories generated based on the DMP model not only have stable convergence, but also can reduce a large amount of similar industrial programming work, improving the flexibility and efficiency of trajectory planning and control. Brief Description of the Drawings

[0029] Figure 1 is the flow chart of the manipulator imitation learning method based on DMP of the present invention;

[0030] Figure 2 is the shape of the taught trajectory;

[0031] Figure 3 is the of the trajectory curve of the component changing with the phase variable s;

[0032] Figure 4 is the of the trajectory curve of the component changing with the phase variable s;

[0033] Figure 5 is the comparison between the generated new trajectory and the original taught trajectory;

[0034] Figure 6 is the finally generated trajectory that satisfies the geometric characteristics of the imitation taught trajectory and passes through specific passing points, and its comparison with the original taught trajectory;

[0035] Figure 7 is the of the final trajectory and the position corresponding to the specific passing point on the component;

[0036] Figure 8 is the of the final trajectory component and the specific passing point passed through on the The corresponding position on the component. DETAILED DESCRIPTION

[0037] Exemplary embodiments of the present disclosure will be described in more detail below with reference to the accompanying drawings. Although exemplary embodiments of the present disclosure are shown in the accompanying drawings, it should be understood that the present disclosure can be implemented in various forms and should not be limited by the embodiments described herein. Rather, these embodiments are provided to enable a more thorough understanding of the present disclosure and to fully convey the scope of the present disclosure to those skilled in the art.

[0038] In this application, the end of the robot arm draws a two-dimensional graphic as an example, that is, the end of the robot arm The axis coordinates remain unchanged, only considering , Coordinates; use the "∞" shaped trajectory as the teaching trajectory for imitation learning to complete the planning task. The shape of the trajectory is as follows Figure 2 As shown; Figure 3 and Figure 4 Respectively represent the trajectory , The curve of the component changing with the phase variable s can be seen from the figure. , The starting point and end point of the taught trajectory of the component are the same. For the above example, the traditional DMP method cannot directly imitate and generate a new trajectory.

[0039] The technical problem to be solved by the present invention is: the planned new trajectory can be scaled based on the shape of the teaching trajectory "∞", and the horizontal coordinate The component is magnified to twice the original trajectory, the vertical coordinate The component is reduced to the original trajectory At the same time, the planned new trajectory is required to pass through the specified point. The coordinates of the specified point and the corresponding phase variables are shown in Table 1.

[0040] Table 1 Coordinates of passing points

[0041] Figure 1 Flowchart of the DMP-based robot arm imitation learning method of the present invention. Figure 1 As shown, the method includes the following steps:

[0042] Step 1: Fit the teaching trajectory using the motion primitive method. Step 1 may include:

[0043] Step S1-1: fitting the teaching trajectory using the motion primitive method;

[0044] Using the motion primitive method, the trajectory is represented as a combination of a set of weighted basis functions:

[0045] (1)

[0046] where is a set of basis functions, is the weight vector of the basis functions.

[0047] Step S1-2: Introduce a phase variable;

[0048] To make the system not explicitly dependent on time, introduce a phase variable to replace the time evolution. The phase variable is derived from a canonical system, and its dynamics are:

[0049] (2)

[0050] where is the total time of motion duration, is the maximum value of the phase variable, which is 1; is the time scaling factor used to control the phase evolution speed. In the specific implementation, can be set, (3).

[0051] Step S1-3: Construct a basis function matrix;

[0052] Select a Gaussian function as the basis function, and the basis function matrix is specifically:

[0053] (4)

[0054] where: (5), is the number of basis functions for fitting; is the center of the -th Gaussian basis function, is the width of the corresponding Gaussian basis function, which is calculated by the following formula:

[0055] (6)

[0056] where is the scaling factor for controlling the overlap degree between basis functions, which is set to 1 in this example.

[0057] Step Two: According to the taught trajectory, obtain the weights of the motion primitives by the least squares method;

[0058] According to the trajectory fitting method described in Step One, each basis function has a weight to achieve the fitting of the geometric features of the trajectory. It is necessary to obtain from the taught trajectory Collect data points and construct a motion primitive fitting model Using the least squares method, obtain the weight vector value. Specifically:

[0059] S2-1, Collect data points from the teaching trajectory and construct a motion primitive fitting model where, represents the overall teaching trajectory, represents the th sampling point on the teaching trajectory, is the corresponding phase variable, is and is the weight vector to be solved.

[0060] S2-2, Solve the optimization problem by minimizing the following loss function:

[0061] (7)

[0062] where, is the number of sampling points;

[0063] Solve to obtain the weight vector:

[0064] (8)

[0065] where: (9).

[0066] Step 3: According to the new trajectory regeneration method, generate a new trajectory with the same geometric characteristics as the original teaching trajectory by setting a new starting point, ending point, or scaling factor. This process may include:

[0067] Step S3-1: According to actual requirements, determine the starting point of the new trajectory to be obtained , ending point and static scaling factor ;

[0068] Step S3-2: Set the dynamic scaling factor , that is, the dynamic scaling factor is a proportional function of the phase variable, which can ensure , ;

[0069] Step S3-3: According to the new trajectory generation formula, generate a new trajectory that meets the conditions:

[0070] (10)

[0071] The comparison between the generated new trajectory and the original taught trajectory is as Figure 5 shown

[0072] Step 4: Through the new trajectory passing point strategy, enable the generated new trajectory to pass through specific passing points; the relative position between the specific passing point position and the trajectory is as Figure 5 shown, and this step may include:

[0073] Step S4-1: Decompose the coordinates of the specific passing point required for the trajectory to each degree-of-freedom axis, and correspond the specific passing point to the phase variable;

[0074] Decompose the coordinates of the specific passing point required for the trajectory to each degree-of-freedom axis. For the position of the robotic arm, decompose the specific passing point position into coordinates on three axes. In this example the axis coordinate remains unchanged, and only decompose the specific passing point position to axis and axis, and correspond it to the phase variable, as , where is the th specific passing point, represents the coordinate of this specific passing point on axis, represents the phase variable corresponding to this specific passing point, represents that at the moment corresponding to the phase variable , the component of the required trajectory on axis passes through the specific position .

[0075] Step S4-2: Generate segmented trajectories;

[0076] For the trajectory that needs to pass through specific passing points, the starting point of the trajectory, the specific passing points, and the ending point of the trajectory together points divide the whole trajectory into segmented trajectories. According to the trajectory generation formula in Step 3, the generation method of the trajectory segment between the th specific passing point and the th specific passing point is:

[0077] (11)

[0078] Among them, is the basis function matrix corresponding to the trajectory segment, which is a sub-matrix of the original basis function matrix , is the th coordinate value of the specific passing point, is the scaling ratio of each segmented curve, ; is the phase variable corresponding to the th specific passing point;

[0079] Based on formula (11), the generation methods of the first and last section trajectories are respectively:

[0080] (12)

[0081] (13)

[0082] Step S4-3: Generation of the complete passing point trajectory;

[0083] Connect the segmented trajectories generated in step S4-2 end to end to form a continuous complete trajectory ;

[0084] Step Five: Apply the Gaussian process to generate a high-order smooth trajectory passing through specific passing points;

[0085] Step S5-1: Set the specific passing points and the starting and ending points of the trajectory as training points;

[0086] Step S5-2: Select the kernel function of the Gaussian process;

[0087] Select the Gaussian kernel function as the kernel function of the Gaussian process to ensure the smoothness of the trajectory. For two input vectors and , the expression of the Gaussian kernel function is:

[0088] (14)

[0089] where is the bandwidth parameter, which determines the smoothness of the generated trajectory and is set to 1.2; is the amplitude parameter and is set to 1.

[0090] Step S5-3: Generate the covariance matrix:

[0091] Sample N points on the trajectory generated in step four as prediction points. The N phase variables corresponding to these N prediction points form the prediction point input vector , corresponding to the prediction point output vector . According to step S5-2, calculate the covariance matrix , , , ; where is the training point input vector composed of the phase variables corresponding to the training points determined in step S5-1.

[0092] Step S5-4: setting the prior means of the training data and the prediction points;

[0093] A set of training points determined by step S5-1, the values of which constitute a vector ; The trajectory generated in step 4 The value corresponding to the same phase as the specific passing point is used as the prior mean of the training point data ; The predicted point input vector determined by step S5-3 , the trajectory generated in step 3 The corresponding phase value in is used as the prior mean of the prediction point .

[0094] Step S5-5: Calculate the posterior distribution and predict the value of the new data point;

[0095] The training point data and the prediction point data obey the joint Gaussian distribution. According to the conditional Gaussian distribution, the posterior distribution is obtained as follows:

[0096] (15)

[0097] in, is the vector of values corresponding to the predicted points:

[0098] (16)

[0099] Input vector from prediction point and the formula ( ) calculated Construct a high-order smooth trajectory passing through a specific passing point ( ), recorded as .

[0100] Step 6: Based on the curves in steps 4 and 5, a weighted method is used to generate a high-order smooth curve that meets the geometric characteristics;

[0101] Step S6-1: Calculate weight;

[0102] From step 4, a segmented trajectory is generated that imitates the geometric features of the taught trajectory and passes through specific passing points. , a set of uniform sampling with a large number of points is performed on the trajectory, and each sampling point obtained corresponds to a weight, which is calculated as follows:

[0103] (17)

[0104] Normalized weights:

[0105] (18)

[0106] in, is the curve smoothness parameter, is the phase variable corresponding to the th sampling point, is the phase variable corresponding to the th specific passing point, and there are

[0107] According to Step 5, a high-order smooth trajectory passing through the specific passing points is generated , and a relatively large number of uniform samplings are performed on this trajectory with the same number of points as the previous process. The weight corresponding to each sampling point obtained is calculated as:

[0108] (19)

[0109] Perform weighted fusion on and these two curves:

[0110] (20)

[0111] A final high-order smooth trajectory that meets the geometric characteristics of the imitation teaching trajectory and passes through the specific passing points is obtained , as shown in Figure 6 ; the components of the trajectory on the axis and the axis and the positions of the specific passing points passed through are on the axis and the axis as shown in Figure 7 , Figure 8 . It can be seen from the figure that in the special case where the starting and ending points of the teaching trajectory are the same or similar, this method can well imitate the teaching trajectory and generate a new trajectory that meets the requirements and has the same geometric characteristics. At the same time, this method can also make the trajectory pass through the specific passing points and ensure that the trajectory has the same geometric characteristics and high-order smoothness.

[0112] Step 7: Substitute the generated new trajectory into a new DMP model to establish a dynamic system model;

[0113] The dynamic expression of the DMP system is:

[0114] (21)

[0115] Among them, is a positive scalar, is the trajectory calculated in the previous step, and each discrete trajectory point in it, , where it is the corresponding velocity and acceleration, , , They are the current position, speed, and acceleration values of the real robotic arm system respectively.

[0116] Step Eight: Generate a control quantity according to the dynamic system model in Step Seven to drive the robotic arm joints;

[0117] Solve the dynamic equation of the DMP system in Step Seven to obtain the acceleration trajectory, and perform time integration on it to obtain the speed trajectory, which is sent to the robotic arm speed controller as a control quantity, and realize speed-level trajectory tracking control based on the ROS framework.

[0118] The present invention also provides a robotic arm imitation learning device based on DMP, including the following modules:

[0119] The fitting module is used to fit the teaching trajectory by using the method of motion primitives;

[0120] The weight calculation module is used to obtain the weights of the motion primitives by the least squares method according to the teaching trajectory;

[0121] The first generation module is used to set a new starting point, ending point, or scaling factor according to the new trajectory imitation method, and generate a new trajectory with the same geometric characteristics as the original teaching trajectory;

[0122] The strategy module is used to make the generated new trajectory pass through specific passing points through the new trajectory passing point strategy;

[0123] The second generation module is used to apply the Gaussian process to generate a high-order smooth trajectory passing through specific passing points;

[0124] The third generation module is used to generate a high-order smooth curve that satisfies the geometric characteristics and can pass through specific passing points by using a weighted method based on the curves in Steps Four and Five;

[0125] The establishment module is used to substitute the generated new trajectory into the new DMP model to establish a dynamic system model;

[0126] The driving module is used to generate a control quantity according to the dynamic system model in Step Seven to drive the robotic arm joints.

[0127] For the parts not detailed in the above device, please refer to the description of the above method embodiment.

[0128] The above embodiments only represent the implementation manners of the present invention, but should not be construed as limiting the scope of the present invention. It should be noted that for those skilled in the art, without departing from the concept of the present invention, several deformations and improvements can still be made, and these all belong to the protection scope of the present invention.

Claims

1. A robotic arm imitation learning method based on DMP, characterized in that, It includes the following steps: Step 1: Fit the teaching trajectory using the method of motion primitives; Step 2: Obtain the weights of the motion primitives by the least squares method according to the teaching trajectory; Step 3: Set a new starting point, ending point, and scaling factor according to the new trajectory imitation method, and generate a new trajectory with the same geometric characteristics as the original teaching trajectory; Step 4: Make the generated new trajectory pass through specific passing points through the new trajectory passing point strategy; Step 5: Apply the Gaussian process to generate a high-order smooth trajectory passing through specific passing points; Step 6: Generate a high-order smooth curve that satisfies the geometric characteristics and passes through specific passing points by the weighted method based on the curves in Steps 4 and 5; Step 7: Substitute the generated new trajectory into the new DMP model to establish a dynamic system model; Step 8: Generate a control quantity according to the dynamic system model in Step 7 to drive the robotic arm joints.

2. The method for robotic arm imitation learning based on DMP according to claim 1, wherein, Step 1 includes: Step S1-1: Fit the teaching trajectory using the method of motion primitives; Using the method of motion primitives, the taught trajectory is represented as a combination of a set of weighted basis functions: （1） Among them, is a set of basis functions, is the weight vector of the basis functions; Step S1-2: Introduce a phase variable ; To make the system not explicitly dependent on time, a phase variable is introduced to replace time evolution. The phase variable is derived from a canonical system, and its dynamics are as follows: （2） Among them, is the total duration of the movement, is the maximum value of the phase variable, which is 1; is the time scaling factor used to control the phase evolution speed. In specific implementation, it can be set to , (3) Step S1-3: Construct the basis function matrix; Select the Gaussian function as the basis function, and the basis function matrix is as follows: （4） in: (5), is the number of basis functions used for fitting, It is The center of the Gaussian basis function, is the width of the corresponding Gaussian basis function, calculated by the following formula: （6） Among them, is a scaling factor for controlling the overlap degree between basis functions.

3. The robotic arm imitation learning method based on DMP according to claim 2, wherein, Step 2 includes: Step S2-1: Collect data points from the teaching trajectory and construct a motion primitive fitting model , where represents the entire teaching trajectory, represents the th sampling point on the teaching trajectory, is the phase variable corresponding to the sampling point , and is the weight vector to be solved; Step S2-2: Solve the optimization problem to minimize the following loss function: （7） Among them, is the number of sampling points; Obtain the weight vector by solving; （8） Wherein: (9).

4. A DMP-based robotic arm imitation learning method according to claim 3, characterized in that Step 3 includes: Step S3-1: Determine the starting point, ending point, and static scaling factor of the new trajectory to be obtained according to actual requirements. of the starting point , the ending point and the static scaling factor ; Step S3-2: Set the dynamic scaling factor , that is, the dynamic scaling factor is a direct proportional function of the phase variable, which can ensure , ; Step S3-3: Generate a new trajectory that meets the conditions according to the new trajectory generation formula: （10）。 5. A method for robotic arm imitation learning based on DMP according to claim 4, characterized in that, Step 4 includes: Step S4-1: Decompose the coordinates of the specific passing points that the required trajectory passes through to each degree-of-freedom coordinate axis, and map the specific passing points to phase variables; Step S4-2: Generate segmented trajectories: For a trajectory that needs to pass through specific passing points in total, the trajectory starting point, specific passing points, and trajectory ending point in total points divide the entire trajectory into segmented trajectories. According to the trajectory generation formula in step three, the generation method of the trajectory segment between the th specific passing point and the th specific passing point is as follows: （11） Among them, is the basis function matrix corresponding to the trajectory segment, and is the submatrix of the original basis function matrix . is the coordinate value of the th specific passing point, is the scaling ratio of each piecewise curve, ; is the phase variable corresponding to the th specific passing point; Based on formula (11), the generation methods of the first and last segments of the trajectory are respectively: （12） （13） Step S4-3: Generate the complete passing point trajectory; Connect the segmented trajectories generated in step S4-2 end to end to form a continuous and complete trajectory .

6. A method for robotic arm imitation learning based on DMP according to claim 5, characterized in that Step 5 includes: Step S5-1: Set specific passing points and trajectories The starting and ending points are training points; Step S5-2: Select the kernel function of the Gaussian process; Select the Gaussian kernel function as the kernel function of the Gaussian process to ensure the smoothness of the trajectory. For two input vectors and , the expression of the Gaussian kernel function is: （14） Among them, is the bandwidth parameter, which determines the smoothness of the generated trajectory and is set to 1.2; is the amplitude parameter and is set to 1; S5-3: Generate the covariance matrix; Sample N points on the trajectory generated in Step 4 as prediction points, and the N phase variables corresponding to these N prediction points form the prediction point input vector , corresponding to the prediction point output vector . According to Step S5-2, calculate the covariance matrix by the Gaussian kernel function , , , ; where is the training point input vector composed of the phase variables corresponding to the training points determined in Step S5-1 Step S5-4: Set the prior means of the training point and prediction point data; A set of training points determined by step S5-1, the values of the training points form a vector ; Use the values corresponding to the same phase as the specific passing points in the trajectory generated in step four as the prior mean of the training point data ; The prediction point input vector determined by step S5-3, use the values corresponding to the phase in the trajectory generated in step three as the prior mean of the prediction point ; Step S5-5: Calculate the posterior distribution and predict the values of new data points; The training point data and prediction point data follow a joint Gaussian distribution. According to the conditional Gaussian distribution, the posterior distribution obtained is: （15） Among them, is a vector composed of the values of the corresponding prediction points: （16） Input vector from the prediction point and the one calculated by formula (16) constitute a high-order smooth trajectory passing through a specific passing point ( ), denoted as .

7. A method for robotic arm imitation learning based on DMP according to claim 6, characterized in that Step 6 includes: Step S6-1: Calculate the weights; In step four, a segmented trajectory that mimics the geometric features of the teaching trajectory and passes through specific passing points is generated. , a set of relatively large numbers of uniform samplings are performed on this trajectory, and each obtained sampling point corresponds to a weight, and the calculation method is as follows: （17） Normalize the weights: （18） Among them, is the curve smoothness parameter, is the phase variable corresponding to the th sampling point, is the phase variable corresponding to the th specific passing point, and there are specific passing points in total; In step five, a high-order smooth trajectory passing through specific passing points is generated , and uniform sampling with a relatively large number of points the same as the previous process is performed on this trajectory. The weight corresponding to each sampling point obtained is calculated as follows: （19） Pair and These two curves are weighted and fused as follows: （20） Obtain a high-order smooth trajectory that finally satisfies the geometric characteristics of the imitation teaching trajectory and passes through specific passing points .

8. A DMP-based robotic arm imitation learning method according to claim 7, characterized in that Step 7 includes: The dynamic expression of the DMP system is: （21） wherein, is a positive scalar, is the trajectory calculated in the previous step at each discrete trajectory point, , which are the corresponding speed and acceleration, , , which are the current position, speed, and acceleration values of the real robotic arm system, respectively.

9. The method for robotic arm imitation learning based on DMP according to claim 8, wherein, Step 8 includes: Solve the dynamic expression of the DMP system in Step 7 to obtain the acceleration trajectory, and perform time integration on it to obtain the velocity trajectory, which is sent to the robotic arm speed controller as a control quantity to achieve velocity-level trajectory tracking control based on the ROS framework.

10. A robotic arm imitation learning algorithm based on DMP, characterized in that, It includes the following steps: A fitting module for fitting the teaching trajectory using the method of motion primitives; A weight calculation module for obtaining the weights of the motion primitives by the least squares method according to the teaching trajectory; A first generation module for setting a new starting point, ending point, or scaling factor according to the new trajectory imitation method and generating a new trajectory with the same geometric characteristics as the original teaching trajectory; A strategy module for making the generated new trajectory pass through specific passing points through the new trajectory passing point strategy; A second generation module for applying the Gaussian process to generate a high-order smooth trajectory passing through specific passing points; A third generation module for generating a high-order smooth curve that satisfies the geometric characteristics and can pass through specific passing points by the weighted method based on the curves in Steps 4 and 5; A building module, which is used to substitute the generated new trajectory into a new DMP model to build a dynamic system model; A driving module, which is used to generate a control quantity according to the dynamic system model in Step Seven to drive the joints of the robotic arm.

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