A robot end vibration suppression method based on acceleration optimization

Abstract

The application belongs to the field of robot motion trajectory planning, and discloses a robot end vibration suppression method based on acceleration optimization. The method introduces an adjustable jerk trajectory, divides the constructed model trajectory into more fine segments, and can significantly improve the flexible adaptability under various working conditions. The jerk trajectory is designed as a smooth parameter-adjustable function, which can be adjusted according to experience or a multi-objective optimization algorithm, aiming to solve the problem of fixed trajectory curve parameters, so as to ensure the vibration suppression of the mechanical arm under various complex working conditions, and as far as possible to play the performance of the driver to improve the working efficiency of the mechanical arm.

Description

Technical Field

[0001] This invention belongs to the field of robot motion trajectory planning, specifically relating to a method for suppressing robot end effector vibration based on acceleration optimization. Background Technology

[0002] Currently, robotic arms are used in various scenarios, with industrial robotic arms and collaborative robotic arms being widely used, primarily for processes such as handling, sorting, and welding. In addition, the rapid development of humanoid robots in recent years has placed higher demands on the trajectory planning of dual-arm collaborative robots and the improvement of vibration suppression and efficiency when grasping various workpieces. Under increasingly complex task requirements, these robotic arms are no longer satisfied with simply grasping and placing a single load. The items that robotic arms need to grasp vary in size and weight, and the changing load at the end effector causes severe vibration during operation. This necessitates that robotic arms flexibly switch planning parameters when grasping various loads to efficiently and safely complete the specified tasks, and trajectory planning technology is one of the key technologies to ensure the safe and efficient operation of robotic arms.

[0003] Traditional robotic arm trajectory planning methods use polynomial interpolation and spline interpolation, which can typically achieve smooth vibration suppression of position, velocity, and acceleration. However, these methods cannot strictly guarantee velocity, acceleration, and jerk constraints, and struggle to ensure optimal trajectory timing, making them unsuitable for scenarios with high efficiency requirements. In contrast, another type of trajectory planning method often employs S-curve velocity curves or trigonometric function velocity planning algorithms. These methods can ensure compliance with the kinematic and dynamic constraints of the robotic arm, and offer simple models and strong real-time performance. However, traditional S-curve velocity curve algorithms have limited trajectory smoothness, and their parameters are usually fixed, making it impossible to adjust the smoothness. Therefore, there is an urgent need to propose a velocity planning curve that balances efficiency and vibration suppression, allowing for real-time parameter adjustment, to achieve a balance between robotic arm efficiency and vibration suppression performance under varying load scenarios while satisfying various motion constraints. Summary of the Invention

[0004] The purpose of this invention is to provide a robot end effector vibration suppression method based on acceleration optimization, which is a robotic arm trajectory planning method. This method introduces an adjustable jerk trajectory, dividing the constructed model trajectory into more refined segments, which can significantly improve the flexibility and adaptability under various working conditions. The jerk trajectory is designed as a smooth parameter adjustable function, which can be adjusted based on experience or multi-objective optimization algorithms. This aims to solve the problem of fixed trajectory curve parameters, so as to ensure the vibration suppression of the robotic arm under various complex working conditions, while maximizing the performance of the actuator to improve the working efficiency of the robotic arm.

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

[0006] A method for suppressing robot end-effector vibration based on acceleration optimization includes the following steps:

[0007] Based on the homogeneous transformation matrix between the joints of the robotic arm and the forward kinematic equations of the end effector relative to the base coordinate system, a kinematic model of the robotic arm is established, and the hard constraints of the robotic arm are determined.

[0008] Construct a unit combination cosine function, the curve shape of which is determined by a shape adjustment factor. Determine whether there is a flat-top segment with constant acceleration by judging the peak value of the unit combination cosine function. Obtain the critical shape adjustment factor when a flat-top segment exists. There is a mapping relationship between the shape adjustment factor and the normalized shape factor.

[0009] Based on the rigid constraints of the robotic arm, the acceleration curve is constructed using a unit combination cosine function. The acceleration curve is divided according to the flat-top segment to obtain a continuous function of the acceleration curve. The continuous function is then applied to the acceleration segment, deceleration segment, acceleration-deceleration segment, and deceleration-deceleration segment to obtain seven complete acceleration curve segments.

[0010] Based on the seven complete acceleration curves obtained, the acceleration time, uniform acceleration time, and uniform velocity time are solved to determine the final motion type of the acceleration curve.

[0011] The final trajectory curve is obtained based on the set normalized shape factor; or the normalized shape factor and the dynamic constraints of each joint of the robotic arm are used as variables to be optimized, and the optimal normalized shape factor and the dynamic constraints of each joint of the robotic arm are obtained by using the objective optimization algorithm, thereby obtaining the final trajectory curve of each joint of the robotic arm.

[0012] Several alternative methods are provided below, but they are not intended as additional limitations on the overall solution above. They are merely further additions or optimizations. Provided there are no technical or logical contradictions, each alternative method can be combined individually with respect to the overall solution above, or multiple alternative methods can be combined with each other.

[0013] Preferably, the unit combination cosine function is expressed as follows:

[0014]

[0015] in, For unit combination cosine functions, Represented as a dimensionless time parameter, and Corresponding to angular frequencies and The cosine component, It is a shape adjustment factor, and .

[0016] Preferably, the determination of the peak value of the unit combination cosine function determines whether a flat-top segment with constant acceleration exists, and the resulting critical shape adjustment factor when a flat-top segment exists includes:

[0017] Unit combination cosine function Transform into a quadratic function and in the interval Extreme value analysis was performed to obtain the critical shape adjustment factor when a flat-top segment exists.

[0018] When the shape adjustment factor exceeds the critical shape adjustment factor, the unit combination cosine function The value exceeds the upper bound of the standardization target; at this point, the unit combination cosine function is used. The value is restricted to the upper bound of the standardized target, thus obtaining a flat-top segment whose value remains constant over a continuous period of time.

[0019] Preferably, there is a mapping relationship between the shape adjustment factor and the normalized shape factor, and the mapping relationship is constructed as follows:

[0020] Using the unit combination cosine function with a flat top segment as the execution unit limiting function, and integrating the execution unit limiting function, we obtain the area enclosed by the single-segment accelerometer curve under the current shape adjustment factor.

[0021] The area enclosed by the single-segment acceleration curves under the current shape adjustment factor is normalized by using the area enclosed by the single-segment acceleration curves under the maximum and minimum shape adjustment factors, thus obtaining the normalized shape factor.

[0022] The shape adjustment factor is sampled uniformly within the maximum and minimum values ​​of the shape adjustment factor.

[0023] For each sampling point, calculate the area enclosed by the single-segment accelerometer curve under the current shape adjustment factor, and obtain the normalized shape factor. Thus, the mapping relationship between each sampling point and the corresponding normalized shape factor is obtained.

[0024] Preferably, the hard constraints based on the robotic arm, using a unit combination of cosine functions to construct the jerk curve, include:

[0025] Take the maximum jerk from the rigid constraints of the robotic arm;

[0026] Use the unit combination cosine function with a flat top segment as the execution unit limiting function;

[0027] The jerk curve is taken as the product of the maximum jerk and the execution unit constraint function.

[0028] Preferably, the step of dividing the accelerometer curve according to the flat-top segment to obtain a continuous function of the accelerometer curve includes:

[0029] When a flat-top segment exists, the acceleration curves are as follows: between the start time of the acceleration segment and the start time of the flat-top segment, and between the end time of the flat-top segment and the end time of the acceleration segment. Between the start and end times of the flat-top segment, the acceleration curve remains constant. ,in, The maximum jerk within the rigid constraints of the robotic arm. Angular frequency, For the current time, Shape adjustment factor;

[0030] When there is no flat-top segment, the acceleration curve between the start and end times of the acceleration segment is as follows: .

[0031] Preferably, applying the continuous function to the acceleration segment, deceleration segment, acceleration-deceleration segment, and deceleration-deceleration segment yields seven complete acceleration curves, including:

[0032] In the acceleration segment, the continuous function is substituted into the current time offset from the start time of the acceleration segment to obtain the acceleration segment curve;

[0033] In the deceleration phase, substitute the current time offset from the start time of the deceleration phase into the continuous function and take the negative value to obtain the deceleration phase curve;

[0034] During the acceleration / deceleration phase, substitute the current time offset from the start time of the acceleration / deceleration phase into the continuous function and take the negative value to obtain the acceleration / deceleration phase curve;

[0035] During the deceleration phase, substitute the current time offset from the start time of the deceleration phase into the continuous function to obtain the deceleration phase curve;

[0036] Set the acceleration value to 0 for the uniform acceleration segment, uniform speed segment, and uniform deceleration segment.

[0037] Preferably, the process of solving for acceleration time, uniform acceleration time, and uniform velocity time includes:

[0038] When the acceleration time for solving is Based on the displacement constraints, the maximum velocity constraint among the rigid constraints of the robotic arm, and the maximum acceleration constraint among the rigid constraints of the robotic arm, the uniform acceleration time is obtained. and constant speed time ;

[0039] When the acceleration time for solving is At that time, set the uniform acceleration time. Based on the displacement constraints and the maximum velocity constraint in the rigid constraints of the robotic arm, the uniform velocity time is obtained. ;

[0040] When the acceleration time for solving is At that time, set the uniform acceleration time. At the same time, uniform time ,in The maximum acceleration constraint is one of the hard constraints for the robotic arm. The maximum jerk constraint is one of the hard constraints of the robotic arm. The parameter is adjusted for the shape adjustment factor. The maximum speed constraint is one of the hard constraints on the robotic arm. Let be the magnitude of the displacement vector.

[0041] Preferably, the shape adjustment factor adjustment parameter is calculated as follows:

[0042] If the shape adjustment factor is less than or equal to the critical shape adjustment factor, then the shape adjustment factor adjustment parameter... It is the sum of the shape adjustment factor and the preset growth gradient;

[0043] If the shape adjustment factor is greater than the critical shape adjustment factor, then the shape adjustment factor adjustment parameter... for ,in For shape adjustment factor, This is the normalized value of the start time of the first flat-top segment in the seven complete acceleration curves.

[0044] Alternatively, the normalized shape factor and the dynamic constraints of each joint of the robotic arm can be used as variables to be optimized, and an objective optimization algorithm can be used to solve for the optimal normalized shape factor and the optimal dynamic constraints of each joint of the robotic arm, including:

[0045] The dynamic constraints of each joint of the robotic arm are taken as the maximum speed, maximum acceleration and maximum jerk of each joint, which together with the normalized shape factor form the variables to be optimized;

[0046] The multi-objective gray wolf optimization algorithm is used for optimization to obtain the elite file collected at the end of the iteration, and the elite file contains several candidate solutions;

[0047] Calculate the trajectory execution time evaluation value and trajectory impact evaluation value for each candidate solution. After normalizing the trajectory execution time evaluation value and trajectory impact evaluation value, fuse them to obtain the weighted comprehensive evaluation value for each candidate solution.

[0048] The candidate solution with the smallest weighted comprehensive evaluation value in the elite files is selected as the final solution. The final solution is then substituted into the seven complete acceleration curves to obtain the final trajectory.

[0049] Compared with existing technologies, this invention proposes a robot end effector vibration suppression method based on acceleration optimization. This method solves the problems of discontinuity and fixed, unadjustable parameters in ordinary S-shaped jerk (acceleration) curves. It enables the trajectory to have both continuous and doable characteristics and low vibration energy input. Compared with methods that only improve the smoothness of jerk, this method can also adjust the peak impact and time performance as needed. Thus, it can fully utilize the performance of the actuator while ensuring vibration suppression and smoothness, achieving both smoothness and engineering controllability. Moreover, the flexible selection of parameters allows for a trade-off between efficiency and vibration suppression. The added adjustable shape degree of freedom can be adapted to multi-objective optimization algorithms, making it better able to cope with complex task scenarios. Attached Figure Description

[0050] Figure 1 This is a flowchart of a robot end-effector vibration suppression method based on acceleration optimization according to the present invention;

[0051] Figure 2 This is a schematic diagram of the acceleration segment of the S-curve under different curve smoothing parameters in an embodiment of the present invention;

[0052] Figure 3 This is a comparison diagram of the combined cosine acceleration curve and the sine acceleration curve in an embodiment of the present invention;

[0053] Figure 4 This is the acceleration curve obtained by multi-objective optimization of each joint in the joint space of a four-joint palletizing robot in this embodiment of the invention;

[0054] Figure 5 The acceleration curves are obtained by multi-objective optimization of each joint in the joint space of a four-joint palletizing robot arm in this embodiment of the invention. Detailed Implementation

[0055] The technical solutions of the embodiments of the present invention will be clearly and completely described below with reference to the accompanying drawings. Obviously, the described embodiments are only some embodiments of the present invention, and not all embodiments. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0056] Unless otherwise defined, all technical and scientific terms used herein have the same meaning as commonly understood by one of ordinary skill in the art to which this invention pertains. The terminology used herein in the description of the invention is for the purpose of describing particular embodiments only and is not intended to limit the invention.

[0057] Example 1:

[0058] To overcome the shortcomings of existing technologies, this embodiment provides a method for suppressing robot end effector vibration based on acceleration optimization, such as... Figure 1 As shown, the specific steps include the following:

[0059] Step 1: Set the corresponding Cartesian space starting position and target position according to the planning method of the robotic arm, and determine the conversion relationship between the robotic arm's Cartesian space and joint space.

[0060] This embodiment uses a four-joint palletizing and handling robot as an example. The joint axes include the waist joint, upper arm joint, forearm joint, and wrist joint. The linkages in the robotic arm kinematic model are... Coordinate system relative to the link The homogeneous transformation matrix of the coordinate system is:

[0061]

[0062] in, Link Coordinate system relative to the link The homogeneous transformation matrix of the coordinate system, the first three times three of which is the coordinate system... Relative to coordinate system The rotation matrix, with the first three rows of the fourth column representing the coordinate system. Origin in coordinate system Position vector at the origin; Indicates the first A joint around The rotation angle of the shaft, Representing the coordinate system Origin along The distance from the axis to the common normal. Indicates circling Shaft from The axis rotates to The included angle of the axis.

[0063] Let the starting and target positions of the Cartesian space be respectively... and The inverse kinematics of the starting and target positions are solved to obtain the joint angles of each point.

[0064] Based on the kinematic model of the robotic arm, the boundary parameters of the curvilinear kinematics are set, and the maximum speed is set respectively. For the curve velocity boundary and maximum acceleration For the curve acceleration boundary, maximum acceleration Add jerk boundary to the curve and set target displacement based on start and stop positions. The specific settings for curvilinear motion constraints are as follows:

[0065]

[0066] in, For the velocity curve during the motion, For acceleration curves, The acceleration curve is shown; the corresponding velocity constraints, acceleration constraints, and jerk constraints are as follows: , , ; This represents the time variable during the trajectory motion process.

[0067] Based on the type of motion trajectory planned by the curve and the starting position With the target location Determine the target displacement of the curve .

[0068] Step 2: Construct a unit combinatorial cosine function and set the shape adjustment factor of the function within it. The effect of this parameter on the function is determined, and the specific characteristics of the function are obtained. That is, the magnitude of the peak value of the function determines whether there is a flat-top segment with constant acceleration, and the critical shape adjustment factor when a flat-top segment exists is obtained.

[0069] Based on the characteristics of the S-shaped velocity curve trajectory planning of the robotic arm in joint space, this embodiment constructs the jerk or equivalent shape control term as a class of combined cosine functions, according to... Figure 2 The combined cosine function curve shown is used to construct an acceleration segment curve, and a unit combined cosine function is constructed, whose normalized expression is... for:

[0070]

[0071] in, For unit combination cosine functions, It is represented as a dimensionless time parameter, used to describe the change of the combined cosine function within a standardized time interval; and Cosine components corresponding to different angular frequencies are used to construct periodic components with different deformation rates, thereby achieving controllable changes in jerk over time; This is the shape adjustment factor for the function, used to adjust the overall shape characteristics of the combined cosine function and determine whether the function has a flat-top segment. (Parameter) The value satisfies ,when Peak values ​​can be smoothed or broadened when... This can enhance the concentration of the curve, thereby enabling compliance or time compression adjustments based on the dynamic characteristics of the mechanical structure.

[0072] This embodiment utilizes a shape adjustment factor. To determine if a function has a flat-top segment, the specific method is as follows: parameters When the value exceeds a preset threshold range, the combined term causes the function to reach a value exceeding the upper bound of the standardization target of 1 in a local region. At this point, the output value is restricted to 1, resulting in a flat-top segment where the value remains constant for a continuous period. The specific execution unit limits the function. for:

[0073]

[0074] By introducing a limiting operator (which is set to 1 in this embodiment), the output of the modified function is made possible. When the upper bound of the target is exceeded, the value is fixed at 1 according to saturation logic; at the same time, when... When the value is less than -1, the function value will be lower than 0, which is meaningless in the description of acceleration in the velocity curve. Therefore, the shape adjustment parameter needs to be limited. .

[0075] The maximum value of the unit combination cosine function was calculated. ,right Perform analytical analysis to obtain Shape adjustment factor when it is 1 According to specific Solve, which is to Transform it into a quadratic function, and perform extremum analysis on its interval to obtain... To satisfy the critical value when the amplitude does not overshoot by 1.

[0076] In summary, this embodiment incorporates a unit combination cosine function with a shape adjustment factor. When the shape adjustment factor satisfies Sometimes, The acceleration curve never exceeds the normalization upper limit of 1, and no saturation flat-top segment is produced; when Sometimes, If the combined cosine acceleration curve exceeds the upper limit within a certain time interval, the interval needs to be saturated and pruned to form a flat-top segment.

[0077] Regarding the shape adjustment factor in this embodiment In order to quickly determine the shape adjustment factor currently in actual use In this embodiment, a lookup table is pre-constructed (containing a mapping relationship between shape adjustment factors and normalized shape factors), and the shape adjustment factors are... Normalization is performed based on the acceleration area to obtain a practically adjustable normalized shape factor. The specific process is as follows:

[0078] To simplify the final optimization variables, the shape adjustment factor is... Normalization is required to obtain the normalized shape factor that actually needs to be optimized. ; Execute the unit-limiting function according to step 2 The area enclosed by a single acceleration segment is calculated as follows:

[0079]

[0080] This area, on a real time scale, is Based on this, a normalized shape factor is defined. for:

[0081]

[0082] in, The area enclosed by the single acceleration segment. The lower limit of the shape adjustment factor is -1, when When the value is less than -1, the function value may be lower than 0, which is meaningless in the description of acceleration in a velocity curve. ; To set the upper limit of the shape adjustment factor, and to ensure the effectiveness of the algorithm while avoiding wasting the adjustment range in the critical region, the shape adjustment factor is defined here as 95% of the square wave signal area. The upper limit, calculated here under the unit function. Therefore, take .

[0083] To achieve normalized shape factor With shape adjustment factor A fast mapping between them, with normalized accelerometer area pre-built in the offline phase and The corresponding table; shape adjustment factor Samples are uniformly sampled within a preset value range to obtain a sample sequence. For each sampling point The normalized area coefficient of the combined cosine acceleration segment is obtained by numerical integration. Based on the formula for calculating the normalized shape factor, the area interval is mapped to... dimensionless normalized shape factor within ;

[0084] This results in a set of binary correspondence data:

[0085]

[0086] in, To maximize the number of sampling points, during online execution, the system normalizes the shape factor based on the input target. The corresponding value is retrieved from the lookup table using a linear interpolation method. Value. Implement from The interval is quickly mapped to the actual control parameter interval, thereby simplifying the optimization variables and avoiding real-time integral or root calculation.

[0087] Since the calculated area is a unit area, the lookup table can be universal without changing the definition of the acceleration curve and the reference value for area normalization, so only one discrete sampling is required.

[0088] Step 3: Based on the given motion parameters, construct the acceleration segment curve using the unit combination cosine function, subdivide the curve into three finer segments, and generalize to the entire motion curve to obtain the final seven-segment jerk curve.

[0089] Step 3.1: The value of the unit combination cosine function is amplified to the target peak value. This then creates a smooth accelerometer input segment, and the unit combination cosine function output reaches its peak value. At this time, the output of this segment is maintained at a constant value, forming a limited maximum plateau, maintaining saturated jerk for a certain duration, specifically resulting in an acceleration segment. for:

[0090]

[0091] in, For maximum jerk limit, To accelerate the start time of the acceleration phase, The acceleration segment ends at time; when the combined cosine function... When a flat-top segment exists, the entire acceleration segment is divided into three continuous stages: the first segment is a gradual rise segment generated by a combination of cosine functions; the middle segment remains constant after reaching the maximum allowable acceleration; and the last segment is a symmetrical gradual fall segment generated by a combination of cosine functions.

[0092] Step 3.2: Calculate the start and end times of the flat-top segment based on whether a flat-top segment exists. , When the shape adjustment factor At this time, there is no flat-top section. and Recorded as 0; when the shape adjustment factor At that time, there was a flat-topped section. The duration of the acceleration phase and the shape adjustment factor The specific value is obtained, and then obtained through the symmetry of the combined cosines. Specifically:

[0093]

[0094]

[0095] Step 3.3: Establish a mathematical model of the acceleration curve. In a palletizing robot arm, the curve can be applied to the Cartesian trajectory planning of the end effector, or to the joint trajectory planning of the four joints; a combined cosine acceleration segment curve is established based on the acceleration curve. It can be defined as a piecewise continuous function.

[0096] When the flat-top segment exists, that is hour, Defined as a piecewise continuous function:

[0097]

[0098] in, To limit the maximum acceleration of the curve, the first and third segments correspond to the mapping of the front and back parts of the normalized interval, respectively, thus constructing a symmetrical acceleration gradually changing segment; This indicates the time of the acceleration segment of the curve, corresponding to... Representing different time periods; parameters and Used to define the start and end positions of the maximum value holding interval, the position of which is determined by the jerk saturation determination, and used to control the duration range of the saturation flat-top segment; To control the angular frequency parameter of the period of cosine variation, Indicates the curve's first The duration of the segment Its function is to limit the rate of change of the combined cosine function within a unit time interval.

[0099] When the flat-top segment does not exist, that is At this point, the unit combination cosine function is a complete curve segment, and the acceleration segment curve is... Defined as:

[0100]

[0101] Step 3.4: After constructing the combined cosine jerk for a single acceleration segment, extend the segmented structure to a complete acceleration, constant velocity, and deceleration process, forming a jerk time history spliced ​​from multiple continuous stages, and construct a seven-segment trajectory planning model. for:

[0102]

[0103] in, For the acceleration phase, It is a uniform acceleration segment. For the deceleration / acceleration phase, For the uniform speed segment, For acceleration and deceleration, It is a uniform deceleration segment. This is the deceleration phase. For each joint, the acceleration during the acceleration phase of the curve adopts a gradual ascent and hold phase constructed using a unit combination cosine function; upon entering the uniform velocity phase, the acceleration can be maintained at 0; upon reaching the deceleration phase, a gradual descent phase with the same structure as described above is used to achieve a continuous decrease in acceleration, thus continuously splicing the entire acceleration curve on the time axis to obtain a continuous acceleration curve. Integrating this curve yields the corresponding acceleration curve. Velocity curve Displacement curves .

[0104] Step 4: Based on the kinematic constraints and the seven Jerk curves obtained in Step 3, solve for the acceleration time, uniform acceleration time, and uniform velocity time of the curves to determine the type of curvilinear motion.

[0105] The seven-segment trajectory planning model with acceleration was completed. Following the construction, this embodiment further solves for the duration of the seven stages to satisfy the requirements of maximum velocity constraints, maximum acceleration constraints, and target displacement during joint movement. Seven time segments Instead of being predetermined, the trajectory is adaptively determined based on the trajectory planning objectives and constraints to ensure that the final generated velocity and displacement curves meet the industrial robot drive limitations throughout the entire process.

[0106] Step 4.1: Integrate the continuous acceleration curve obtained above to obtain the corresponding acceleration curve. Velocity curve and displacement curve .

[0107] Step 4.2: Based on the obtained acceleration curve, assuming the initial acceleration is zero, calculate the maximum acceleration after the entire acceleration phase. The final acceleration to maximum is determined by the shape adjustment factor. There are two scenarios; when there is no flat-top segment, that is... At that time, for any point on the motion curve All meet The maximum acceleration it can achieve for:

[0108]

[0109] When shape adjustment factor At that time, there exists a first time. Make The maximum acceleration it can achieve for:

[0110]

[0111] In both of the above cases, the final maximum acceleration can be expressed as the maximum jerk and... And a shape adjustment factor The product of the expressions, which Represented as:

[0112]

[0113] Among them, parameters This represents the normalized time of the first arrival at the summit. The overall maximum acceleration can be expressed as:

[0114]

[0115] The jerk function is time-symmetric about the midpoint of the interval during the acceleration phase. The deceleration and acceleration segments are opposite in time, therefore the acceleration increases from 0 to... The process of falling back to 0 forms a pair of symmetrical slopes with equal areas, and their combined area is always equal to... .

[0116] Further consider the duration of the uniform acceleration phase The internal acceleration remains constant. Furthermore, based on the time-reverse symmetry of acceleration, we can conclude that the areas of the slopes on both sides are equal, and the combined area of ​​the accelerations corresponding to the ascending and descending segments is always equal. Therefore:

[0117]

[0118] The specific waveform changes of accelerometer are only through The value of can be represented by the values ​​obtained above. To complete the maximum speed calculation, the obtained maximum speed for:

[0119]

[0120] Because the acceleration and deceleration phases are mirror images of each other in time, and the speed monotonically increases from 0 to [a certain value] during the acceleration phase... During the deceleration phase, The velocity decreases monotonically to 0; the velocity trajectory is symmetrical about the midpoint of time, and the average velocity during the acceleration phase remains constant. The deceleration phase is symmetrical to the acceleration phase, and the constant velocity phase is in time. Internal speed The total displacement during the entire process is:

[0121]

[0122] in, For the total time of the acceleration phase, .

[0123] The deceleration phase is symmetrical to the acceleration phase, and its displacement is... Similarly, in the uniform velocity phase, time... Internal speed If the motion continues, then the total displacement during the entire motion process is:

[0124]

[0125] in, This is the displacement during the deceleration phase, the same as during the acceleration phase; The displacement during the uniform motion phase is . ;

[0126] Furthermore, based on the relationship between maximum acceleration and maximum velocity, the final displacement can be derived from the maximum acceleration as follows:

[0127]

[0128] This invention, at a given maximum acceleration Time allocation , , In this case, the maximum speed and the displacement at the end of the motion can be directly obtained through the formula; at the same time, the maximum acceleration, maximum speed, and final displacement can also be used as constraints to obtain the minimum time required for each segment of the current curve.

[0129] Step 4.3: Further, based on the obtained kinematic constraints, solve for the duration of each segment without violating any constraints. Using the above formulas for maximum acceleration, velocity, and final displacement, obtain the duration without violating any constraints. The optimal value is:

[0130]

[0131] in, To limit maximum acceleration, To limit the maximum speed; through the above constraint superposition method, the determined... Under no circumstances will it cause acceleration overload, speed exceeding limits, or terminal displacement distortion, ensuring that the trajectory operates within the hardware's tolerance range.

[0132] Step 4.4: Based on the obtained acceleration period time To solve for the remaining time of uniform acceleration and the time of uniform velocity, in order to ensure that the required duration is meaningful, when in the above formula... When the acceleration limit first reaches its saturation value, determine whether the uniform acceleration time is first constrained by the displacement. If so, there is no uniform acceleration segment and:

[0133]

[0134] If not, then there exists a uniform velocity segment and:

[0135]

[0136] Step 4.5, regarding the time of the uniform velocity phase When the uniform acceleration time in the above formula is... At that time, because the maximum permissible speed had already been reached during the acceleration phase. The existence of the uniform velocity phase is only used to compensate for the remaining displacement, therefore The determination of is only affected by displacement constraints, and can be given as:

[0137]

[0138] When solving for a uniformly accelerated time interval When this condition is met, it indicates that the target displacement has been fully achieved under the current parameters, and there is no need to configure a constant velocity segment. Therefore, the setting is... The maximum achievable speed is recalculated to be 0. .

[0139] When solving for the acceleration period At this point, the maximum permissible speed has been reached at the end of the acceleration phase. Any positive acceleration at this point will only lead to exceeding the speed limit; therefore, there is no effective uniform acceleration phase. And the maximum acceleration is recalculated as Further calculations The method is the same as described above.

[0140] When solving the equation for acceleration time At this point, the acceleration phase parameters have reached the limit of the allowable travel. Continuing to maintain the peak acceleration will lead to the accumulation of terminal errors. Therefore, the uniform acceleration and constant velocity phases should be canceled, and the parameters should be set... , Recalculate the maximum acceleration and maximum velocity as follows: , .

[0141] Step 5: Set the weights of efficiency and vibration suppression during motion, and adjust the normalized shape factor of the curve empirically according to actual needs. At the same time, set the total time and the curve impact equation as objective functions, and set the normalized shape factor and the boundary of the curve motion parameters as optimization variables to adapt to multi-objective optimization adjustment.

[0142] Step 5.1: In this embodiment, the weighting coefficients for time and vibration suppression are set to adjust the curve shape factor. Based on the required vibration suppression smoothness or time efficiency, [the following steps are taken]. Adjustment between The value is used to achieve flexible adaptation; such as Figure 2 The image shown is after adjustment. The corresponding jerk curve was then obtained.

[0143] Simultaneously, based on the characteristics of the unit combination cosine function motion curve, an optimization objective function can be established with the goal of minimizing the curve's time and impact; the total time spent by each curve can be established separately. The average acceleration value of the entire curve :

[0144]

[0145]

[0146] in, This represents the time for each segment of the curve. Indicates time period exist The magnitude of the jerk at any given moment.

[0147] Based on the characteristics of the S-curve and the above calculation of time segments according to kinematic constraints, the variable to be optimized for a segment of the curve can be set as... Simultaneously, to increase the optimization of impact by one degree of freedom, the normalized shape factor is... Also included in the optimization variables, at this point the optimization variables for a curve are: In step 1 , , The maximum speed is a hard constraint on the robotic arm, and it is one of the variables to be optimized. Maximum acceleration and maximum jerk For dynamic constraints, which are the optimization results, the value is less than that of hard constraints.

[0148] Step 5.2: Optimize the time and impact of the curve using a multi-objective optimization algorithm to obtain the non-dominated Pareto front. Then, select the optimal solution on the front surface based on time and damping weight.

[0149] This embodiment employs the multi-objective gray wolf optimization algorithm to optimize the curve parameters. The gray wolf optimization algorithm parameters are initialized, and candidate individuals are generated within a preset population size. Each candidate is randomly initialized using a uniform distribution within the decision space. Based on the curve cost calculated at each position, non-dominated solutions are selected to form an elite solution set. Specifically, the selection method is as follows: for any member within the set… If for any other solution Both can be satisfied:

[0150]

[0151] in, Indicates the first The objective function value corresponding to each optimization objective (i.e., the subsequent...) );symbol Indicates for any; symbol This indicates that at least one member exists; if true, then the member is considered to exist. For a non-dominated solution in this population, store it in the elite archive to construct a set of non-dominated solutions.

[0152] Subsequently, the three leaders of the gray wolves were selected from the constructed multi-objective spatial grid. , , After the triple-pull update, the position of the candidate solution at the next iteration time. It is obtained by averaging the three candidate positions, specifically:

[0153]

[0154] in, , , The leader is given three candidate updated positions.

[0155] After obtaining a new generation of population through the above process, the iteration count is incremented, the elite archive is updated again, and the above steps are iterated repeatedly until the maximum number of iterations is reached. After the termination condition is met, the current Pareto front solution set is output as the Pareto optimal solution set.

[0156] Step 5.3: Based on the final solution of the curve obtained by adjustment or multi-objective optimization, substitute it into the curve parameters, solve for the running time of each segment, and obtain the final running trajectory.

[0157] After obtaining the final non-dominated Pareto solution file, in order to use it in the trajectory planning execution stage, it is necessary to determine the required vibration suppression level based on whether the palletizing robot arm is holding goods and the weight of the goods being held, and further determine a set of single decision vectors in the multi-objective boundary. To this end, this implementation constructs a final selection mechanism based on objective normalization and weighted decision rules.

[0158] The Elite Archive contains The candidate solution, the th The multi-objective cost vector corresponding to each candidate solution Represented as:

[0159]

[0160] in This represents the trajectory execution time evaluation value. This represents the trajectory impact evaluation value.

[0161] To eliminate the difference in dimensions, the two objectives are normalized to intervals. Dimensionless indices within:

[0162]

[0163]

[0164] Based on the normalized objective, the weighted comprehensive evaluation function is constructed as follows:

[0165]

[0166] in, This is the normalized trajectory execution time evaluation value. This is the normalized trajectory impact evaluation value. The minimum value for trajectory execution time evaluation. The maximum value is the trajectory execution time evaluation. This is the minimum value for trajectory impact evaluation. This represents the maximum value of the trajectory impact evaluation. This is a weighting factor used to adjust time sensitivity and shock sensitivity. When When the value is large, the system tends to select the first... The candidate solutions correspond to the solutions with shorter execution times; when When the value is small, the system tends to choose the solution with lower motion impact. To obtain the weighted comprehensive evaluation value, the candidate solution with the smallest weighted comprehensive evaluation value in the Elite Archive is selected as the final solution. This solution is then substituted into the curve parameters to calculate the running time of each segment, thus obtaining the final running trajectory.

[0167] In the experiment, the normalized shape factor was modified manually. To verify its effect on time and vibration damping balance; here is set , , , The palletizing robot arm is designed to travel along a Cartesian straight line at its end. Experiments are conducted using acceleration curve planning improved by a combination of cosine functions and sine functions, respectively. The results are as follows: Figure 3 The acceleration curve shown here requires setting the normalized shape factor of the combined cosine function. When the time intervals of the two curves are approximately equal, The curves show that, under the same running time and kinematic constraints, the combined cosine function exhibits significantly gentler acceleration during the start-stop phase, verifying its suitability for suppressing end-effector vibrations.

[0168] Based on the structure of the four-joint palletizing robot arm, given the starting and ending points of the robot arm in the Cartesian coordinate system, the starting and ending points are converted into the corresponding angles in the joint space by solving the inverse kinematic equations of the four-joint palletizing robot arm. Since this invention optimizes curves, it can be used for both Cartesian space planning and joint space planning of the robot arm. Here, the planning is performed in the joint space.

[0169] Here, the limitations for each joint are set to be... , , The angles traversed from joint 1 to joint 4 are 60°, 36°, 25°, and 40°, respectively. Motion curves are constructed for each of the four joints, and their respective kinematic limits are constrained. The aforementioned multi-objective gray wolf optimization algorithm is then used for optimization to obtain the Pareto solution set, and the weight values ​​are... Set to 0.8; based on the above settings, the optimal solutions for the four joint optimization variables are shown in Table 1:

[0170] Table 1: Variables after optimization of the four joints

[0171]

[0172] Substitute the optimized kinematic constraint parameters and the normalized shape adjustment factor into the first three time segments of the solution curve. , , The time of the joint with the longest running time was used to standardize the time of the remaining joints, resulting in seven complete curves. The acceleration and jerk curves of each joint after standardization are shown below. Figure 4 and Figure 5 As shown.

[0173] Substituting the obtained time into the joint trajectory, the optimized combined cosine accelerator resulted in the following joint performance metrics: Joint 1 time increased by 47.5%, impact decreased by 64%; Joint 2 time increased by 42%, impact decreased by 54.7%; Joint 3 time increased by 47.3%, impact decreased by 56%; Joint 4 time increased by 39%, impact decreased by 53%. The results demonstrate that using a multi-objective optimization algorithm to optimize the curve effectively suppresses its vibration.

[0174] Example 2:

[0175] This embodiment provides a computer-readable storage medium storing a computer program thereon, which, when executed by a processor, implements the steps of the acceleration-optimized robot end-effector vibration suppression method described in Embodiment 1.

[0176] Those skilled in the art will understand that all or part of the processes in the methods of the above embodiments can be implemented by a computer program instructing related hardware. The computer program can be stored in a non-volatile computer-readable storage medium, and when executed, it can include the processes of the embodiments of the above methods. Any references to memory, storage, databases, or other media used in the embodiments provided in this application can include non-volatile and / or volatile memory. Non-volatile memory can include read-only memory (ROM), programmable ROM (PROM), electrically programmable ROM (EPROM), electrically erasable programmable ROM (EEPROM), or flash memory. Volatile memory can include random access memory (RAM) or external cache memory. By way of illustration and not limitation, RAM is available in various forms, such as static RAM (SRAM), dynamic RAM (DRAM), synchronous DRAM (SDRAM), dual data rate SDRAM (DDRSDRAM), enhanced SDRAM (ESDRAM), synchronous link DRAM (SLDRAM), Rambus direct RAM (RDRAM), direct memory bus dynamic RAM (DRDRAM), and memory bus dynamic RAM (RDRAM), etc.

[0177] The technical features of the above embodiments can be combined in any way. For the sake of brevity, not all possible combinations of the technical features in the above embodiments are described. However, as long as there is no contradiction in the combination of these technical features, they should be considered to be within the scope of this specification.

[0178] The embodiments described above are merely illustrative of several implementations of the present invention, and while the descriptions are specific and detailed, they should not be construed as limiting the scope of the invention. It should be noted that those skilled in the art can make various modifications and improvements without departing from the concept of the present invention, and these modifications and improvements all fall within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the appended claims.

Claims

1. A method for suppressing robot end-effector vibration based on acceleration optimization, characterized in that, Includes the following steps: Based on the homogeneous transformation matrix between the joints of the robotic arm and the forward kinematic equations of the end effector relative to the base coordinate system, a kinematic model of the robotic arm is established, and the hard constraints of the robotic arm are determined. Construct a unit combination cosine function, the curve shape of which is determined by a shape adjustment factor. Determine whether there is a flat-top segment with constant acceleration by judging the peak value of the unit combination cosine function. Obtain the critical shape adjustment factor when a flat-top segment exists. There is a mapping relationship between the shape adjustment factor and the normalized shape factor. Based on the rigid constraints of the robotic arm, the acceleration curve is constructed using a unit combination cosine function. The acceleration curve is divided according to the flat-top segment to obtain a continuous function of the acceleration curve. The continuous function is then applied to the acceleration segment, deceleration segment, acceleration-deceleration segment, and deceleration-deceleration segment to obtain seven complete acceleration curve segments. Based on the seven complete acceleration curves obtained, the acceleration time, uniform acceleration time, and uniform velocity time are solved to determine the final motion type of the acceleration curve. The final trajectory curve is obtained based on the set normalized shape factor; or the normalized shape factor and the dynamic constraints of each joint of the robotic arm are used as variables to be optimized, and the optimal normalized shape factor and the dynamic constraints of each joint of the robotic arm are obtained by using the objective optimization algorithm, thereby obtaining the final trajectory curve of each joint of the robotic arm.

2. The method for suppressing robot end-effector vibration based on acceleration optimization according to claim 1, characterized in that, The unit combination cosine function is expressed as follows: in, For unit combination cosine functions, Represented as a dimensionless time parameter, and Corresponding to angular frequencies and The cosine component, It is a shape adjustment factor, and .

3. The method for suppressing robot end-effector vibration based on acceleration optimization according to claim 2, characterized in that, The determination of the peak value of the unit combination cosine function determines whether a flat-top segment with constant acceleration exists, and the resulting critical shape adjustment factor when a flat-top segment exists includes: Unit combination cosine function Transform into a quadratic function and in the interval Extreme value analysis was performed to obtain the critical shape adjustment factor when a flat-top segment exists. When the shape adjustment factor exceeds the critical shape adjustment factor, the unit combination cosine function The value exceeds the upper bound of the standardization target; at this point, the unit combination cosine function is used. The value is restricted to the upper bound of the standardized target, thus obtaining a flat-top segment whose value remains constant over a continuous period of time.

4. The method for suppressing robot end-effector vibration based on acceleration optimization according to claim 1, characterized in that, There is a mapping relationship between the shape adjustment factor and the normalized shape factor. The mapping relationship is constructed as follows: Using the unit combination cosine function with a flat top segment as the execution unit limiting function, and integrating the execution unit limiting function, we obtain the area enclosed by the single-segment accelerometer curve under the current shape adjustment factor. The area enclosed by the single-segment acceleration curves under the current shape adjustment factor is normalized by using the area enclosed by the single-segment acceleration curves under the maximum and minimum shape adjustment factors, thus obtaining the normalized shape factor. The shape adjustment factor is sampled uniformly within the maximum and minimum values ​​of the shape adjustment factor. For each sampling point, calculate the area enclosed by the single-segment accelerometer curve under the current shape adjustment factor, and obtain the normalized shape factor. Thus, the mapping relationship between each sampling point and the corresponding normalized shape factor is obtained.

5. The method for suppressing robot end-effector vibration based on acceleration optimization according to claim 1, characterized in that, The rigid constraints based on the robotic arm, using a unit combination of cosine functions to construct the jerk curve, include: Take the maximum jerk from the rigid constraints of the robotic arm; Use the unit combination cosine function with a flat top segment as the execution unit limiting function; The jerk curve is taken as the product of the maximum jerk and the execution unit constraint function.

6. The method for suppressing robot end-effector vibration based on acceleration optimization according to claim 1, characterized in that, The process of dividing the accelerometer curve based on the flat-top segment to obtain a continuous function of the accelerometer curve includes: When a flat-top segment exists, the acceleration curves are as follows: between the start time of the acceleration segment and the start time of the flat-top segment, and between the end time of the flat-top segment and the end time of the acceleration segment. Between the start and end times of the flat-top segment, the acceleration curve remains constant. ,in, The maximum jerk within the rigid constraints of the robotic arm. Angular frequency, For the current time, Shape adjustment factor; When there is no flat-top segment, the acceleration curve between the start and end times of the acceleration segment is as follows: .

7. The method for suppressing robot end-effector vibration based on acceleration optimization according to claim 1, characterized in that, The continuous function is applied to the acceleration, deceleration, acceleration-deceleration, and deceleration-deceleration segments to obtain seven complete acceleration curves, including: In the acceleration segment, the continuous function is substituted into the current time offset from the start time of the acceleration segment to obtain the acceleration segment curve; In the deceleration phase, substitute the current time offset from the start time of the deceleration phase into the continuous function and take the negative value to obtain the deceleration phase curve; During the acceleration / deceleration phase, substitute the current time offset from the start time of the acceleration / deceleration phase into the continuous function and take the negative value to obtain the acceleration / deceleration phase curve; During the deceleration phase, substitute the current time offset from the start time of the deceleration phase into the continuous function to obtain the deceleration phase curve; Set the acceleration value to 0 for the uniform acceleration segment, uniform speed segment, and uniform deceleration segment.

8. The method for suppressing robot end-effector vibration based on acceleration optimization according to claim 1, characterized in that, The process of solving for acceleration time, uniform acceleration time, and uniform velocity time includes: When the acceleration time for solving is Based on the displacement constraints, the maximum velocity constraint among the rigid constraints of the robotic arm, and the maximum acceleration constraint among the rigid constraints of the robotic arm, the uniform acceleration time is obtained. and constant speed time ; When the acceleration time for solving is At that time, set the uniform acceleration time. Based on the displacement constraints and the maximum velocity constraint in the rigid constraints of the robotic arm, the uniform velocity time is obtained. ; When the acceleration time for solving is At that time, set the uniform acceleration time. At the same time, uniform time ,in The maximum acceleration constraint is one of the hard constraints for the robotic arm. The maximum jerk constraint is one of the hard constraints of the robotic arm. The parameter is adjusted for the shape adjustment factor. The maximum speed constraint is one of the hard constraints on the robotic arm. Let be the magnitude of the displacement vector.

9. The method for suppressing robot end-effector vibration based on acceleration optimization according to claim 8, characterized in that, The shape adjustment factor adjustment parameter is calculated as follows: If the shape adjustment factor is less than or equal to the critical shape adjustment factor, then the shape adjustment factor adjustment parameter... It is the sum of the shape adjustment factor and the preset growth gradient; If the shape adjustment factor is greater than the critical shape adjustment factor, then the shape adjustment factor adjustment parameter... for ,in For shape adjustment factor, This is the normalized value of the start time of the first flat-top segment in the seven complete acceleration curves.

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