CN118036341A - Pose-dependent on-site robot milling mode coupling flutter modeling method - Google Patents
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Abstract
A pose-dependent on-site robot milling modal coupling flutter modeling method comprises the following steps of S10, establishing a robot system dynamics model of modal coupling flutter; s20, establishing a cutting force model of displacement feedback; s30, deriving and utilizing a vibration form to solve and simplify a stability criterion; s40, providing a deformation ratio coefficient depending on the pose. Aiming at the milling processing of the in-situ robot of the overflow surface of the top cover of the water turbine, the model obtained by the modeling method provides a theoretical basis for the optimization of the stability constraint posture of the subsequent robot in the full operation space, so that the robot is guided to change processing paths or processing technological parameters such as axial cutting depth and the like, and the robot is prevented from modal flutter.
Description
Technical Field
The invention relates to the technical field of industrial robot milling, in particular to a pose-dependent on-site robot milling modal coupling flutter modeling method.
Background
The top cover is used as one of the most important overcurrent components of the hydroelectric generating set, is subjected to the cavitation erosion action of water flow for a long time in the running process of the set, and can form a plurality of cavitation pits on the overcurrent surface, so that great potential safety hazards are brought to the safe running of the set. At present, a damaged area of a top cover overflow surface of a water turbine is milled by a robot in a top cover repairing task, but because the robot has poor rigidity, obvious coupling phenomena exist between the degrees of freedom of the robot, namely, the deformation in three directions can be caused by the stress in a single direction, and the phenomenon of displacement feedback can cause instability of a milling system of the top cover overflow surface robot, modal coupling flutter occurs, so that the processing quality is damaged. In addition, the hinge structure of the robot causes pose dependence of dynamic characteristics, and the characteristic further increases the prediction difficulty of robot modal coupling chatter, so that accurate prediction of the pose dependence robot milling modal coupling chatter is a difficulty in the current robot milling stability research.
Disclosure of Invention
The invention aims to solve the technical problems that: the model obtained by the modeling method provides a theoretical basis for optimizing the stability constraint posture of the subsequent robot under the full operation space aiming at the milling processing of the in-situ robot of the overflow surface of the top cover of the water turbine, thereby guiding the robot to change the processing path or change the processing technological parameters such as axial cutting depth and the like, and leading the robot to avoid modal chatter.
In order to solve the technical problems, the invention adopts the following technical scheme: a pose-dependent on-site robot milling mode coupling flutter modeling method, which comprises the following steps,
S10, establishing a robot system dynamics model of modal coupling flutter;
s20, establishing a cutting force model of displacement feedback;
s30, deriving and utilizing a vibration form to solve and simplify a stability criterion;
s40, providing a deformation ratio coefficient depending on the pose.
In S10, in combination with the cross-robot frequency response characteristics, a kinetic equation of the following robot system is established:
；（1）
wherein M, C, K is mass, damping and rigidity coefficient respectively, Representing the cross mass, cross damping and cross stiffness coefficient of the composite material、/>And/>Respectively representing the function of the deformation in X, Y, Z direction and the time t,/>、/>And/>Respectively representing the derivative of deformation in X, Y, Z direction with respect to time t,/>、/>And/>The second derivative of the deformation amount in X, Y, Z direction with respect to time t is shown, fx is shown as the force in the x direction, fy is shown as the force in the Y direction, and Fz is shown as the force in the Z direction.
In S20, in combination with the cutting force model formula, according to the change of the cutting force caused by the displacement feedback, the following expression is obtained:
；（2）
the above method comprises the steps of:
；（3）
the second order small amount is omitted:
；（4）
Wherein:
represents the instantaneous radial contact angle;
、/> And/> The function of the deformation amount in X, Y, Z directions and the time t are respectively expressed;
wherein:
；（5）
；（6）
Wherein:
represents the instantaneous radial contact angle;
Representing a three-way cutting force coefficient;
taking the force balance position as the initial position, and therefore, the cutting force caused by axial cutting depth is not considered, the final cutting force expression is as follows:
；（7）
Wherein:
represents the instantaneous radial contact angle;
Representing a three-way cutting force coefficient;
、/> And/> The function of the deformation amount in X, Y, Z directions and the time t are respectively expressed;
In S30, deriving and using the vibration form solution stability criterion comprises the steps of:
s31: substituting the final cutting force representation form into a dynamic equation of the robot system, and simplifying to obtain:
；（8）
Wherein:
Representing a three-way cutting force coefficient;
、/> And/> The second derivatives of the deformation amounts in X, Y, Z directions with respect to time t are respectively represented;
representing the cross-over quality;
represents axial depth of cut;
S32: is provided with The vibration pattern of (a) is solved as follows:
；（9）
the vibration pattern is solved into the formula in S31, resulting in:
；（10）
Wherein:
Representing a three-way cutting force coefficient;
And (3) finishing to obtain:
；（11）
Wherein:
Representing a three-way cutting force coefficient;
Representing instantaneous radial contact angle
Representing axial depth of cut
Representing cross damping
If in the equationThen/>Formal solution/>The term converges to zero over time, and the system is stable; otherwise, the system diverges with time and is unstable;
s33: the root formula is used for obtaining:
；（12）
Wherein:
Representing a three-way cutting force coefficient;
represents the instantaneous radial contact angle;
represents axial depth of cut;
Wherein, Constant greater than zero, when: /(I)The system is stable, at this time there is/>I.e./>The following stability criteria were obtained:
；（13）
Wherein:
Representing a three-way cutting force coefficient;
represents the instantaneous radial contact angle;
represents axial depth of cut;
And (3) finishing to obtain:
；（14）
Wherein:
representing the feed per tooth;
represents the instantaneous radial contact angle;
Representing a three-way cutting force coefficient;
Similarly, the final three-way stability criterion is:
； （15）
Wherein:
Represents the modal stiffness of the robot;
representing the feed per tooth;
represents the instantaneous radial contact angle;
represents axial depth of cut;
Representing a three-way cutting force coefficient;
representing coefficients related to the pose, instantaneous radial contact angle, feed direction of the robot, hereinafter referred to as deformation ratio coefficients.
In S40, the pose-dependent deformation ratio coefficient is proposed to include the following steps:
s41: the deformation ratio coefficient is expressed by a cutting force and flexibility matrix, and the formula in the step S33 can be known:
；（16）
Wherein:
representing coefficients related to the pose, instantaneous radial contact angle and feed direction of the robot;
Will be Expressed as/>Is a function of:
；（17）
Wherein:
Representing instantaneous radial contact angle
And (3) finishing to obtain:
；（18）
Wherein:
Representing instantaneous radial contact angle
And (3) the same principle:
；（19）
Wherein:
Representing instantaneous radial contact angle
Wherein,The robot is characterized in that the robot is provided with six joint corners, C is a robot flexibility matrix, and the expression form is as follows:
；（20）
Wherein: is a robot stiffness matrix;
the robot stiffness matrix is provided by the robot manufacturer.
The invention has the following beneficial effects:
1. The milling mode coupling flutter model established by the invention has clear process, simple formula calculation and better model accuracy. Aiming at the in-situ robot milling of the overflow surface of the top cover of the water turbine, the modeling method can provide theoretical support for the optimization of the milling gesture of the robot and the planning of a tool path.
2. The model obtained by the modeling method provides a theoretical basis for stability constraint attitude optimization under the follow-up full operation space of the robot, so that the robot is guided to change a processing path or change processing technological parameters such as axial cutting depth, the robot is prevented from modal flutter, and the development of milling of the robot is promoted.
Drawings
The invention is further illustrated by the following examples in conjunction with the accompanying drawings:
FIG. 1 is a flow chart of the method of the present invention.
Detailed Description
Referring to fig. 1, a pose-dependent in-situ robot milling mode coupling chatter modeling method includes the steps of,
S10, firstly, establishing a robot system dynamics model of modal coupling flutter.
The weak rigidity of the robot determines that the robot has obvious cross rigidity and cross damping characteristics, and after the force is applied to the robot in one direction, the robot can deform in three directions.
In S10, in combination with the cross-robot frequency response characteristics, a kinetic equation of the following robot system is established:
；（1）
wherein M, C, K is mass, damping and rigidity coefficient respectively, Representing the cross mass, cross damping and cross stiffness coefficient of the composite material、/>And/>Respectively representing the function of the deformation in X, Y, Z direction and the time t,/>、/>And/>Respectively representing the derivative of deformation in X, Y, Z direction with respect to time t,/>、/>And/>The second derivative of the deformation amount in X, Y, Z direction with respect to time t is shown, fx is shown as the force in the x direction, fy is shown as the force in the Y direction, and Fz is shown as the force in the Z direction.
S20, secondly, establishing a displacement feedback cutting force model.
Due to modal coupling effects, a single direction of force will be affected by multiple directions. And combining a cutting force model formula, and obtaining the following expression according to the change of the cutting force caused by displacement feedback:
；（2）
the above method comprises the steps of:
；（3）
the second order small amount is omitted:
；（4）
Wherein:
represents the instantaneous radial contact angle;
、/> And/> The function of the deformation amount in X, Y, Z directions and the time t are respectively expressed;
wherein:
；（5）
；（6）
Wherein:
represents the instantaneous radial contact angle;
Representing the three-way cutting force coefficient.
Taking the force balance position as the initial position, and therefore, the cutting force caused by axial cutting depth is not considered, the final cutting force expression is as follows:
；（7）
Wherein:
represents the instantaneous radial contact angle;
Representing a three-way cutting force coefficient;
、/> And/> The amount of deformation in the X, Y, Z direction is shown as a function of time t, respectively.
S30, deducing and utilizing vibration modes to solve the stability criterion.
S31: substituting the final cutting force representation form into a dynamic equation of the robot system, and simplifying to obtain:
；（8）
Wherein:
Representing a three-way cutting force coefficient;
、/> And/> The second derivatives of the deformation amounts in X, Y, Z directions with respect to time t are respectively represented;
representing the cross-over quality;
Representing axial depth of cut.
Because of the displacement feedback effect, the rigidity coefficient of each degree of freedom consists of two parts, namely the rigidity coefficient and the displacement feedback coefficient of the robot system, which are equivalent to changing the rigidity value of the robot system, and displacement feedback in different forms causes different types and degrees of change.
S32: is provided withThe vibration pattern of (a) is solved as follows:
；（9）
the vibration pattern is solved into the formula in S31, resulting in:
；（10）
Wherein:
Representing a three-way cutting force coefficient;
And (3) finishing to obtain:
；（11）
Wherein:
Representing a three-way cutting force coefficient;
represents the instantaneous radial contact angle;
represents axial depth of cut;
representing cross damping.
If in the equationThen/>Formal solution/>The term converges to zero over time, and the system is stable; otherwise, the system diverges with time and is unstable;
s33: the root formula is used for obtaining:
；（12）
Wherein:
Representing a three-way cutting force coefficient;
represents the instantaneous radial contact angle;
represents axial depth of cut;
Wherein, Constant greater than zero, when: /(I)The system is stable, at this time there is/>I.e./>The following stability criteria were obtained:
；（13）
Wherein:
Representing a three-way cutting force coefficient;
represents the instantaneous radial contact angle;
Representing axial depth of cut.
And (3) finishing to obtain:
；（14）
Wherein:
representing the feed per tooth;
represents the instantaneous radial contact angle;
Representing the three-way cutting force coefficient.
Similarly, the final three-way stability criterion is:
； （15）
Wherein:
Represents the modal stiffness of the robot;
representing the feed per tooth;
represents the instantaneous radial contact angle;
represents axial depth of cut;
Representing a three-way cutting force coefficient;
representing coefficients related to the pose, instantaneous radial contact angle, feed direction of the robot, hereinafter referred to as deformation ratio coefficients.
And S40, finally, providing a deformation ratio coefficient depending on the pose.
The deformation ratio coefficient refers to the ratio of vibration amplitudes in different directions under the current cutting parameters and is used for quantifying the milling vibration of the robot.
Although a specific value of the current vibration amplitude cannot be obtained in the built model, a certain relation exists between the stress of the tail end and the deformation of the tail end of the robot, and a flexibility matrix of the robot can be introduced to represent a deformation ratio coefficient by using a cutting force and flexibility matrix.
S41: the deformation ratio coefficient is expressed by a cutting force and flexibility matrix, and the formula in the step S33 can be known:
；（16）/>
Wherein:
representing coefficients related to the pose, instantaneous radial contact angle and feed direction of the robot;
Will be Expressed as/>Is a function of:
；（17）
Wherein:
Representing instantaneous radial contact.
And (3) finishing to obtain:
；（18）
Wherein:
Representing the instantaneous radial contact angle.
And (3) the same principle:
；（19）
Wherein: represents the instantaneous radial contact angle;
Wherein, The robot is characterized in that the robot is provided with six joint corners, C is a robot flexibility matrix, and the expression form is as follows:
；（20）
Wherein: Is a robot stiffness matrix.
Wherein the robot stiffness matrix is provided by a robot manufacturer.
The invention has the characteristics of simple formula deduction and calculation, is convenient for the position and posture optimization of the robot or the performance of tool path planning work, and provides theoretical support for the position and the optimization of the milling processing posture of the robot and the tool path planning. The model obtained by the modeling method provides a theoretical basis for stability constraint posture optimization under the follow-up full operation space of the robot, so that the robot is guided to change a processing path or change processing technological parameters such as axial cutting depth, the robot is prevented from moving to a theoretical vibration area of a modal vibration model in the processing process of the robot, the robot is prevented from generating modal vibration, and the development of milling processing of the robot is promoted.
Claims (4)
1. The pose-dependent on-site robot milling mode coupling flutter modeling method is characterized by comprising the following steps of: comprises the steps of,
S10, establishing a robot system dynamics model of modal coupling flutter;
s20, establishing a cutting force model of displacement feedback;
s30, deriving and utilizing a vibration form to solve and simplify a stability criterion;
S40, extracting a deformation ratio coefficient of pose dependence;
In S10, in combination with the cross-robot frequency response characteristics, a kinetic equation of the following robot system is established:
；（1）
wherein M, C, K is mass, damping and rigidity coefficient respectively, Representing the cross mass, cross damping and cross stiffness coefficient of the composite material、/>And/>Respectively representing the function of the deformation in X, Y, Z direction and the time t,/>、/>And/>Respectively representing the derivative of deformation in X, Y, Z direction with respect to time t,/>、/>And/>The second derivative of the deformation amount in X, Y, Z direction with respect to time t is shown, fx is shown as the force in the x direction, fy is shown as the force in the Y direction, and Fz is shown as the force in the Z direction.
2. The pose-dependent on-site robot milling modal coupling chatter modeling method of claim 1, wherein the method comprises the steps of: in S20, in combination with the cutting force model formula, according to the change of the cutting force caused by the displacement feedback, the following expression is obtained:
；（2）
the above method comprises the steps of:
；（3）
the second order small amount is omitted:
；（4）
Wherein:
Represents the instantaneous radial contact angle;
、/> And/> The function of the deformation amount in X, Y, Z directions and the time t are respectively expressed;
Represents axial depth of cut;
g represents a unit step function;
h represents the cutting thickness;
Representing xyz three-way dynamic cutting force;
Representing xyz three-way static cutting force;
wherein:
；（5）
；（6）
Wherein:
represents the instantaneous radial contact angle;
Representing a three-way cutting force coefficient;
Representing xyz three-way static cutting force;
taking the force balance position as the initial position, and therefore, the cutting force caused by axial cutting depth is not considered, the final cutting force expression is as follows:
；（7）
Wherein:
represents the instantaneous radial contact angle;
Representing a three-way cutting force coefficient;
、/> And/> The amount of deformation in the X, Y, Z direction is shown as a function of time t, respectively.
3. The pose-dependent on-site robot milling modal coupling chatter modeling method of claim 1, wherein the method comprises the steps of: in S30, deriving and using the vibration form solution stability criterion comprises the steps of:
s31: substituting the final cutting force representation form into a dynamic equation of the robot system, and simplifying to obtain:
；（8）
Wherein:
Representing a three-way cutting force coefficient;
represents cross damping;
、/> And/> The second derivatives of the deformation amounts in X, Y, Z directions with respect to time t are respectively represented;
representing the cross-over quality;
represents axial depth of cut;
S32: is provided with The vibration pattern of (a) is solved as follows:
；（9）
the vibration pattern is solved into the formula in S31, resulting in:
；（10）
Wherein:
Representing a three-way cutting force coefficient;
And (3) finishing to obtain:
；（11）
Wherein:
Representing a three-way cutting force coefficient;
represents the instantaneous radial contact angle;
represents axial depth of cut;
represents cross damping;
if in the equation Then/>Formal solution/>The term converges to zero over time, and the system is stable; otherwise, the system diverges with time and is unstable;
s33: the root formula is used for obtaining:
；（12）
Wherein:
Representing a three-way cutting force coefficient;
Representing instantaneous radial contact angle
Represents axial depth of cut;
Wherein, Constant greater than zero, when: /(I)The system is stable, at this time there is/>I.e./>The following stability criteria were obtained:
；（13）
Wherein:
Representing a three-way cutting force coefficient;
represents the instantaneous radial contact angle;
represents axial depth of cut;
And (3) finishing to obtain:
；（14）
Wherein:
representing the feed per tooth;
represents the instantaneous radial contact angle;
Representing a three-way cutting force coefficient;
Similarly, the final three-way stability criterion is:
； （15）
Wherein:
Represents the modal stiffness of the robot;
representing the feed per tooth;
represents the instantaneous radial contact angle;
represents axial depth of cut;
Representing a three-way cutting force coefficient;
representing coefficients related to the pose, instantaneous radial contact angle, feed direction of the robot, hereinafter referred to as deformation ratio coefficients.
4. The pose-dependent on-site robot milling modal coupling chatter modeling method of claim 1, wherein the method comprises the steps of: in S40, the pose-dependent deformation ratio coefficient is proposed to include the following steps:
s41: the deformation ratio coefficient is expressed by a cutting force and flexibility matrix, and the formula in the step S33 can be known:
；（16）
Wherein:
representing coefficients related to the pose, instantaneous radial contact angle and feed direction of the robot;
Will be Expressed as/>Is a function of:
；（17）
Wherein: represents the instantaneous radial contact angle;
And (3) finishing to obtain:
；（18）
Wherein: represents the instantaneous radial contact angle;
And (3) the same principle:
；（19）
Wherein: represents the instantaneous radial contact angle;
Wherein, The robot is characterized in that the robot is provided with six joint corners, C is a robot flexibility matrix, and the expression form is as follows:
；（20）
Wherein:
C is a robot compliance matrix;
The robot is a six-joint corner;
is a robot stiffness matrix;
the robot stiffness matrix is provided by the robot manufacturer.
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- 2024-04-11 CN CN202410430887.1A patent/CN118036341A/en active Pending
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| CN104493636A (en) * | 2014-11-12 | 2015-04-08 | 华中科技大学 | Metallic cryogenic tempering method for improving milling stability |
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