A time-optimal velocity planning method with anti-windup linear constraints and applications

Abstract

This invention relates to the field of intelligent equipment technology, and particularly to a time-optimal speed planning method and application with anti-envelope linear constraints. The method involves solving for control point deviations using the anti-envelope interval as a constraint condition, and then superimposing these deviations with the initial control points to obtain combined control points. An analytical model of the control points with linear constraint factors is constructed and converted into linear equations. Iterative solutions are used to obtain the linear constraint factors and corresponding linear control points. These linear control points are compared and adjusted with the initial control points and the control points corresponding to the maximum feed rate to obtain the final control points. Based on these final control points, an optimal feed rate curve satisfying all constraints is generated. By compensating for order reduction errors through anti-envelope calculations, and combining analytical solutions of linear constraint factors with iterative correction of control points, the method fully exploits constraint margins, achieving time-optimal, more uniform speed, and higher processing quality for large curvature sections while satisfying trajectory error and motor dynamic performance constraints.

Description

Technical Field

[0001] This invention relates to the field of intelligent equipment technology, and in particular to a time-optimal speed planning method and its application with anti-envelope linear constraints. Background Technology

[0002] In multi-axis linkage machining systems, speed planning can reduce mechanical inertial shocks and vibrations, and shorten trajectory motion time, thereby improving machining quality and efficiency. For example, CN110948488A discloses a time-optimal adaptive trajectory planning algorithm for robots. Using the maximum speed step of a joint as a benchmark, the algorithm calculates the shortest necessary time required for each joint to complete the pose offset ΔP. It then determines the optimal planning time by iterating through the time adjustment coefficient K within the (1,2) interval, and finally corrects the set speed of joints that do not meet the standard using an adaptive coefficient K2. However, this scheme does not consider the acceleration and jerk of the motion, making it difficult to guarantee the compliance, safety, and positioning accuracy of multi-axis motion. For example, CN120196054A discloses a time-optimal feed rate planning method for multi-axis CNC machine tools. This method uses NURBS step trajectory segmentation and B-spline velocity modeling, employs simulated annealing and pattern search in multiple rounds of alternating optimization, and combines smoothing of connected regions to achieve feed rate planning for multi-axis CNC machine tools while satisfying constraints such as speed, acceleration, jerk, and chord error. However, the speed, acceleration, and jerk are not pushed to the upper limit of the constraints, resulting in incomplete constraint space utilization. To ensure continuity between segments and smooth curves, the algorithm actively reduces speed in connected regions to avoid impact, further reducing the overall average feed rate.

[0003] In view of this, it is necessary to provide a speed planning method that makes the feed rate closer to the theoretical optimum, shortens the processing time, and takes into account mechanical impact. Summary of the Invention

[0004] The purpose of this invention is to propose a time-optimal speed planning method and its application with anti-envelope linear constraints. By compensating for order reduction errors through anti-envelope calculation, and combining analytical solution of linear constraint factors with iterative correction of control points, the method fully explores the constraint margin and achieves time-optimal, more uniform speed, and higher processing quality of feed speed planning in large curvature sections while satisfying trajectory error and motor dynamic performance constraints.

[0005] To achieve this objective, the present invention adopts the following technical solution:

[0006] A time-optimal velocity planning method with anti-envelope linear constraints is used to plan the feed velocity curve of a multi-degree-of-freedom motion control mechanism.

[0007] The method includes the following steps:

[0008] Step 1: Calculate the inverse envelope interval corresponding to the reduced-order error based on the high-order velocity planning model, solve the control point deviation with the inverse envelope interval as the constraint condition, and superimpose the control point deviation with the initial control point to obtain the combined control point;

[0009] Step 2: Construct an analytical model of control points with linear constraint factors. The analytical model of control points transforms matrix equations into linear equations and obtains the linear constraint factors and corresponding linear control points through iterative solution.

[0010] Step 3: Compare and adjust the linear control point with the initial control point and the control point corresponding to the maximum feed rate to obtain the final control point. Generate the optimal feed rate curve that satisfies all constraints based on the final control point.

[0011] Furthermore, step 1 includes:

[0012] Step 11: Establish the pseudo-jerk constraint equations for the high-order speed programming model, as well as the actual constraint equations for the actual motor speed, acceleration, and jerk; calculate the order reduction error between the pseudo-jerk and the actual jerk.

[0013] Step 12: Calculate the difference between the allowable value and the actual value based on the order reduction error to obtain the inverse envelope interval of velocity, acceleration, and jerk. Solve for the control point deviation using the inverse envelope interval as a constraint condition.

[0014] Step 13: Limit the control point deviation to be less than the initial control point, and superimpose the control point deviation and the initial control point according to the same matrix operation rules as the basis function to obtain the combined control point.

[0015] Furthermore, in step 11, the pseudo-acceleration constraint equation is:

[0016] ,

[0017] in, As the initial control point, , , These are the maximum allowable values ​​for the motor's speed, acceleration, and jerk, respectively. It is a reduction factor. These are the B-spline basis functions for velocity, acceleration, and jerk, respectively.

[0018] The actual constraint equations for the motor speed, acceleration, and jerk are:

[0019] ,

[0020] in, , , These are the actual values ​​of the motor's speed, acceleration, and jerk, respectively.

[0021] Order reduction error The formula for calculation is:

[0022] ;

[0023] In step 12, the formula for calculating the anti-envelope interval is:

[0024] ,

[0025] If an inverse envelope interval is formed, then the control point deviation... The formula for calculation is:

[0026] ;

[0027] In step 13, restrictions are imposed. Combined control points The formula for calculation is: .

[0028] Furthermore, in step 1, the relational expression under the combined control points is:

[0029] ;

[0030] When using the same set of control points to represent the actual jerk:

[0031] ,

[0032] Therefore, the relationship between the combined control point and the jerk is:

[0033] .

[0034] Furthermore, step 2 includes:

[0035] Step 21: Introduce the linear control points to be determined and construct an analytical model of the control points with linear constraint factors;

[0036] Step 22: Transform the matrix equation into a linear equation through matrix transformation, introduce linear constraint factors, and obtain the analytical expression for the linear control points;

[0037] Step 23: Solve for the linear constraint factors through iterative calculations to obtain the corresponding linear control points.

[0038] Furthermore, in step 21, the matrix equation is:

[0039] ,

[0040] in, The final control point to be determined. These are linear control points.

[0041] Furthermore, step 22 includes:

[0042] The matrix equation simplifies to:

[0043] ,

[0044] in, Let be the linear control points to be solved. Let i be the assumed matrix with 1 row and 1 column;

[0045] Substitute into matrix The simplified calculation formula is expressed as:

[0046] ,

[0047] in, Let i be a matrix with 1 row and 1 column. For combined control points;

[0048] By the associative law of matrix multiplication:

[0049] ,

[0050] Removing the square root simplifies to:

[0051] ,

[0052] Therefore, we can conclude that:

[0053] ,

[0054] make Therefore, we can conclude that:

[0055] ,

[0056] in, This is the linear constraint factor.

[0057] Furthermore, the adjustment rule for the final control point in step 3 is as follows:

[0058] If the linear control point is smaller than the initial control point, then the final control point at that position is replaced with the initial control point;

[0059] If the linear control point is greater than the control point corresponding to the maximum feed rate, then the final control point at that position will be pulled back to the control point corresponding to the maximum feed rate.

[0060] Furthermore, the formula for calculating the adjustment rule of the final control point is:

[0061] ,

[0062] in, It is the control point corresponding to the maximum feed rate constraint.

[0063] An application of a time-optimal velocity planning method with anti-envelope linear constraints is described above. This time-optimal velocity planning method with anti-envelope linear constraints is applied to a multi-degree-of-freedom motion manipulation mechanism for trajectory planning in multi-axis linkage machining.

[0064] The multi-degree-of-freedom motion manipulation mechanism includes, but is not limited to, industrial robots and multi-axis CNC equipment.

[0065] The technical solution provided by this invention may include the following beneficial effects:

[0066] The method of this invention can be used for multi-degree-of-freedom motion manipulation mechanisms, such as robots, trajectory planning of multi-axis CNC equipment, and multi-axis linkage machining. While satisfying trajectory error constraints and motor dynamic performance constraints, it plans the optimal feed rate curve and the speed, acceleration, and jerk of each motor, reducing mechanical inertial impact and vibration, and shortening trajectory motion time, thereby improving machining quality and efficiency. Attached Figure Description

[0067] Figure 1 This is a flowchart illustrating a time-optimal velocity planning method with anti-envelope linear constraints according to an embodiment of the present invention.

[0068] Figure 2 This is a schematic diagram illustrating the relationship between the combined control point and the jerk in one embodiment of the present invention;

[0069] Figure 3 This is a schematic diagram showing the relationship between the linear control point and the jerk in one embodiment of the present invention;

[0070] Figure 4 This is a schematic diagram showing the relationship between the final control point and the jerk in one embodiment of the present invention. Detailed Implementation

[0071] The present invention will be further illustrated below with reference to specific embodiments. It should be understood that the specific embodiments described herein are for illustrative and explanatory purposes only and are not intended to limit the present invention.

[0072] An embodiment of the present invention provides a time-optimal velocity planning method with anti-envelope linear constraints, used for planning the feed velocity curve of a multi-degree-of-freedom motion manipulation mechanism;

[0073] The method includes the following steps:

[0074] Step 1: Calculate the inverse envelope interval corresponding to the reduced-order error based on the high-order velocity planning model, solve the control point deviation with the inverse envelope interval as the constraint condition, and superimpose the control point deviation with the initial control point to obtain the combined control point;

[0075] Step 2: Construct an analytical model of control points with linear constraint factors. The analytical model of control points transforms matrix equations into linear equations and obtains the linear constraint factors and corresponding linear control points through iterative solution.

[0076] Step 3: Compare and adjust the linear control point with the initial control point and the control point corresponding to the maximum feed rate to obtain the final control point. Generate the optimal feed rate curve that satisfies all constraints based on the final control point.

[0077] This method first precisely quantifies the unused potential for velocity, acceleration, and jerk through anti-envelope calculation, effectively compensating for the waste of constraint margin caused by pseudo-jerk replacement, making velocity planning closer to the theoretical optimal boundary. Then, it transforms complex nonlinear matrix equations into a linearly solvable form, significantly improving solution stability and computational efficiency while ensuring jerk does not exceed constraints, avoiding the black-box nature and local optima problems of heuristic optimization. Finally, it ensures that the final control point simultaneously satisfies trajectory error constraints and motor dynamic performance constraints through optimal correction rules, achieving a balance between maximizing feed rate and curve smoothness. Ultimately, this increases the velocity, acceleration, and jerk of each axis motor, reducing mechanical inertial impact and vibration. Simultaneously, the increased motor speed synchronously increases the feed rate, shortening trajectory motion time and thus improving machining efficiency. The synchronous increase in feed rate in the large curvature range makes the machining speed more uniform, improving machining quality in the large curvature range. It should be noted that the control point in this invention refers to the feed rate control point.

[0078] The method of this invention has outstanding advantages such as more efficient use of constraints, more optimal speed planning, more controllable solution process, and stronger engineering applicability. It is suitable for high-precision and high-efficiency trajectory planning of multi-degree-of-freedom motion mechanisms such as multi-axis CNC machine tools and robots.

[0079] Understandably, the trajectory error constraint involves calculating the maximum allowable feed rate at each point along the path based on the contour error, chord height error, and curvature, and mapping this to the upper limit of the control points. The motor dynamic constraint involves converting the maximum speed, acceleration, and jerk into the upper limit of the control points using a linear relationship based on basis functions. Therefore, the feed rate control points obtained using this solution satisfy both the trajectory error constraint and the motor dynamic constraint.

[0080] Reference Figure 1In the process of solving the final control point (FCP), the range of the initial control point (ICP) is first set in step 1; in step 2, when solving the linear constraint factor iteratively, the linear constraint factor and the corresponding linear control point are output when the linear control point (LCP) does not exceed the driving constraint; in step 3, the LCP is compared with the ICP and the Max-FCC (maximum feed rate constraint control point) to obtain the final control point (FCP).

[0081] Step 1 of this invention first calculates the pseudo-jerk error caused by the order reduction, obtaining the remaining lift space (inverse envelope interval) for velocity, acceleration, and jerk. The control point deviation is then solved and superimposed with the original control points to obtain the combined control points. In one embodiment, step 1 includes:

[0082] Step 11: Establish the pseudo-jerk constraint equations for the high-order speed programming model, as well as the actual constraint equations for the actual motor speed, acceleration, and jerk; calculate the order reduction error between the pseudo-jerk and the actual jerk.

[0083] Step 12: Calculate the difference between the allowable value and the actual value based on the order reduction error to obtain the inverse envelope interval of velocity, acceleration, and jerk. Solve for the control point deviation using the inverse envelope interval as a constraint condition.

[0084] Step 13: Limit the control point deviation to be less than the initial control point, and superimpose the control point deviation and the initial control point according to the same matrix operation rules as the basis function to obtain the combined control point.

[0085] For ease of solution, high-order velocity programming uses pseudo-jerk to reduce the model to a first-order linear model, resulting in actual jerk being less than the allowable jerk. This leads to an unused boost range for velocity and acceleration, known as the inverse envelope error. This proposed solution first calculates this error to obtain the control point deviation, which is then superimposed with the initial control points to form a combined control point. Using the same basis functions as the original planning model ensures consistency in matrix operations, naturally placing the velocity and acceleration corresponding to the combined control point within the constraint range, thus guaranteeing constraint compliance without additional penalty terms.

[0086] Reference Figure 2 As an example, in step 11, the pseudo-jerk constraint equation is:

[0087] ,

[0088] in, As the initial control point, , , These are the maximum allowable values ​​for the motor's speed, acceleration, and jerk, respectively. It is a reduction factor. These are the B-spline basis functions for velocity, acceleration, and jerk, respectively. For control points obtained using only velocity and acceleration, For order reduction factor basis functions;

[0089] The actual constraint equations for the motor speed, acceleration, and jerk are:

[0090] ,

[0091] in, , , These are the actual values ​​of the motor's speed, acceleration, and jerk, respectively.

[0092] Order reduction error The formula for calculation is:

[0093] ;

[0094] Order reduction error This means that the actual jerk did not reach the allowable limit, so there is room for improvement in both speed and acceleration.

[0095] In step 12, the formula for calculating the anti-envelope interval is:

[0096] ,

[0097] Constructing an anti-envelope interval, and using the values ​​of the anti-envelope interval as constraints for the new planning function, then the control point deviation... The formula for calculation is:

[0098] ;

[0099] In step 13, restrictions are imposed. Combined control points The formula for calculation is: .

[0100] In step 1, the relational expression under the combined control points is:

[0101] ;

[0102] Since the same basis functions are used, the velocity and acceleration formed by the combined control points will not exceed the limit, indicating that the solution method of using the inverse envelope to solve the control point deviation and form the combined control points is reasonable.

[0103] When using the same set of control points to represent the actual jerk:

[0104] ,

[0105] If the actual jerk caused by the combination of control points exceeds the allowable limit, then the relationship between the limit combination of control points and the jerk is as follows:

[0106] .

[0107] Figure 2 'a' is a schematic diagram illustrating the principle of combined control point calculation, where the control point deviation is limited to be less than the initial control point. Figure 2 b is the j-curve formed by all control points, i.e., the jerk curve. It can be seen that the combined control points (CCP) still cause the jerk to exceed the limit. This invention proposes an analytical control point model with a linear constraint factor, which transforms the control point matrix equation into a linear equation. The final linear constraint control point can be obtained through iteration, so that the final velocity, acceleration, and jerk all meet the allowable range.

[0108] In one embodiment, step 2 includes:

[0109] Step 21: Introduce the final control points to be determined and construct an analytical model of the control points with linear constraint factors. The analytical model of the control points is a matrix equation.

[0110] Step 22: Transform the matrix equation into a scalar equation through matrix transformation. After simplifying by squaring both sides of the equation, the analytical expression of the linear control point is obtained, which includes the linear constraint factor to be determined.

[0111] Step 23: Solve for the linear constraint factor through iterative calculation to obtain the corresponding linear control points. The iteration ends when the velocity, acceleration, and jerk curves formed by the current linear control points are within the constraint range.

[0112] Step 2 addresses the over-constraint problem of jerk at combined control points. By introducing a linear constraint factor and performing a linearization transformation, the matrix equations, which are difficult to solve analytically, can be transformed into linearly solvable equations. This enables precise correction of jerk constraints, ensuring that the acceleration and jerk remain within acceptable ranges throughout the entire process. Linearizing the nonlinear, highly coupled control point solution problem significantly reduces the solution complexity. The linear constraint factor can be adaptively adjusted based on trajectory characteristics and machine tool dynamics, making it versatile for various multi-degree-of-freedom motion mechanisms and applicable to speed planning scenarios for multi-axis CNC machine tools, robots, and other equipment.

[0113] Reference Figure 3 As an example, in step 21, the matrix equation is:

[0114] ,

[0115] in, Let be the linear control point to be found, and e be the Hadamard product, which represents the element-wise multiplication of matrices.

[0116] As an example, step 22 includes:

[0117] The matrix equation simplifies to:

[0118] ,

[0119] Where f represents the original result of the matrix operation, These are matrices with i rows and 1 column respectively;

[0120] Substitute into matrix The simplified calculation formula is expressed as:

[0121] ,

[0122] in, These are matrices with i rows and 1 column, used to simplify the original matrix equations. For combined control points;

[0123] By the associative law of matrix multiplication:

[0124] ,

[0125] Removing the square root simplifies to:

[0126] ,

[0127] Therefore, we can conclude that:

[0128] ,

[0129] make Therefore, we can conclude that:

[0130] ,

[0131] in, This is the linear constraint factor.

[0132] Figure 3 a is a schematic diagram illustrating the principle of linear control point calculation. Figure 3b is the j-curve formed by the combined control points, linear control points, and the initial control point, i.e., the jerk curve. The darker shaded area represents the under-constrained region of the j-curve caused by the linear control points. During iterative solving of the linear constraint factor, control points in certain regions may exceed the maximum feed rate constraint limit or fall below the initial control point (ICP), thus generating a curve with insufficient feed rate. To obtain the optimal feed rate with sufficient control points, the linear control points are adjusted by replacing the insufficient linear control points (LCP) with the initial control points and by pulling back the excessive linear control points with control points at the maximum feed rate. Therefore, the adjustment rule for the final control point in step 3 is:

[0133] If the linear control point is smaller than the initial control point, then the final control point at that position is replaced with the initial control point;

[0134] If the linear control point is greater than the control point corresponding to the maximum feed rate, then the final control point at that position will be pulled back to the control point corresponding to the maximum feed rate.

[0135] Reference Figure 4 As an example, the formula for calculating the adjustment rule of the linear control point is:

[0136] ,

[0137] in, It is the control point corresponding to the maximum feed rate constraint.

[0138] Figure 4 'a' is a schematic diagram for obtaining the final control point. Figure 3 b is the j-curve formed by each control point. It can be seen that the final determined FCP can ensure that the velocity, acceleration and jerk are all within their constraint limits, while achieving the optimal feed rate.

[0139] Accordingly, the present invention also provides an application of the time-optimal velocity planning method with anti-envelope linear constraints, which is applied to a multi-degree-of-freedom motion manipulation mechanism for trajectory planning of multi-axis linkage machining; the multi-degree-of-freedom motion manipulation mechanism includes, but is not limited to, industrial robots and multi-axis CNC equipment.

[0140] As an example, executable instructions can be deployed to execute on a single computing device, or on multiple computing devices located in one location, or on multiple computing devices distributed across multiple locations and interconnected via a communication network.

[0141] It should be noted that, in this document, the terms "comprising," "including," or any other variations thereof are intended to cover non-exclusive inclusion, such that a process, method, article, or system that comprises a list of elements includes not only those elements but also other elements not expressly listed, or elements inherent to such a process, method, article, or system. Unless otherwise specified, an element defined by the phrase "comprising one..." does not exclude the presence of other identical elements in the process, method, article, or system that includes that element.

[0142] The sequence numbers of the embodiments in this application are for descriptive purposes only and do not represent the superiority or inferiority of the embodiments.

[0143] Through the above description of the embodiments, those skilled in the art can clearly understand that the methods of the above embodiments can be implemented by means of software plus necessary general-purpose hardware platforms. Of course, they can also be implemented by hardware, but in many cases the former is a better implementation method. Based on this understanding, the technical solution of this application, in essence, or the part that contributes to the prior art, can be embodied in the form of a software product. This computer software product is stored in a storage medium (such as read-only memory / random access memory, magnetic disk, optical disk) and includes several instructions to cause a multimedia terminal device (which may be a mobile phone, computer, television receiver, or network device, etc.) to execute the methods described in the various embodiments of this application.

[0144] The above are merely preferred embodiments of this application and do not limit the patent scope of this application. Any equivalent structural or procedural transformations made using the content of this application's specification and drawings, or direct or indirect applications in other related technical fields, are similarly included within the patent protection scope of this application.

Claims

1. A time-optimal velocity planning method with anti-envelope linear constraints, characterized in that, Used for planning the feed rate curve of a multi-degree-of-freedom motion control mechanism; The method includes the following steps: Step 1: Calculate the inverse envelope interval corresponding to the reduced-order error based on the high-order velocity planning model, solve the control point deviation with the inverse envelope interval as the constraint condition, and superimpose the control point deviation with the initial control point to obtain the combined control point; Step 2: Construct an analytical model of control points with linear constraint factors. The analytical model of control points transforms matrix equations into linear equations and obtains the linear constraint factors and corresponding linear control points through iterative solution. Step 3: Compare and adjust the linear control point with the initial control point and the control point corresponding to the maximum feed rate to obtain the final control point. Generate the optimal feed rate curve that satisfies all constraints based on the final control point.

2. The method according to claim 1, characterized in that, Step 1 includes: Step 11: Establish the pseudo-jerk constraint equations for the high-order speed programming model, as well as the actual constraint equations for the actual motor speed, acceleration, and jerk; calculate the order reduction error between the pseudo-jerk and the actual jerk. Step 12: Calculate the difference between the allowable value and the actual value based on the order reduction error to obtain the inverse envelope interval of velocity, acceleration, and jerk. Solve for the control point deviation using the inverse envelope interval as a constraint condition. Step 13: Limit the control point deviation to be less than the initial control point, and superimpose the control point deviation and the initial control point according to the same matrix operation rules as the basis function to obtain the combined control point.

3. The method according to claim 2, characterized in that, In step 11, the pseudo-acceleration constraint equation is: ， in, As the initial control point, , , These are the maximum allowable values ​​for the motor's speed, acceleration, and jerk, respectively. It is a reduction factor. These are the B-spline basis functions for velocity, acceleration, and jerk, respectively. The actual constraint equations for the motor speed, acceleration, and jerk are: ， in, , , These are the actual values ​​of the motor's speed, acceleration, and jerk, respectively. Order reduction error The formula for calculation is: ； In step 12, the formula for calculating the anti-envelope interval is: ， If an inverse envelope interval is formed, then the control point deviation... The formula for calculation is: ； In step 13, restrictions are imposed. Combined control points The formula for calculation is: .

4. The method according to claim 3, characterized in that, In step 1, the relational expression under the combined control points is: ； When using the same set of control points to represent the actual jerk: ， Therefore, the relationship between the combined control point and the jerk is: 。 5. The method according to claim 1, characterized in that, Step 2 includes: Step 21: Introduce the linear control points to be determined and construct an analytical model of the control points with linear constraint factors; Step 22: Transform the matrix equation into a linear equation through matrix transformation, introduce linear constraint factors, and obtain the analytical expression for the linear control points; Step 23: Solve for the linear constraint factors through iterative calculations to obtain the corresponding linear control points.

6. The method according to claim 5, characterized in that, In step 21, the matrix equation is: ， in, The final control point to be determined. These are linear control points.

7. The method according to claim 6, characterized in that, Step 22 includes: The matrix equation simplifies to: ， in, For linear control points, It is a column vector; Substitute into matrix The simplified calculation formula is expressed as: ， in, , For combined control points; By the associative law of matrix multiplication: ， Removing the square root simplifies to: ， Therefore, we can conclude that: ， make Therefore, we can conclude that: ， in, This is the linear constraint factor.

8. The method according to claim 1, characterized in that, The adjustment rule for the final control point in step 3 is as follows: If the linear control point is smaller than the initial control point, then the final control point at that position is replaced with the initial control point; If the linear control point is greater than the control point corresponding to the maximum feed rate, then the final control point at that position will be pulled back to the control point corresponding to the maximum feed rate.

9. The method according to claim 8, characterized in that, The formula for calculating the adjustment rule of the final control point is: ， in, It is the control point corresponding to the maximum feed rate constraint.

10. An application of a time-optimal velocity planning method with anti-envelope linear constraints, characterized in that, The time-optimal velocity planning method with anti-envelope linear constraints as described in any one of claims 1-9 is applied to a multi-degree-of-freedom motion manipulation mechanism for trajectory planning in multi-axis linkage machining. The multi-degree-of-freedom motion manipulation mechanism includes, but is not limited to, industrial robots and multi-axis CNC equipment.

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