A joint vibration active control method and device for low-frequency chatter suppression of a milling robot

Abstract

The present application belongs to the technical field of robot milling vibration suppression, and discloses a joint vibration active control method and equipment for low-frequency chatter suppression of a milling robot, comprising the following steps: S1, the air gap diameter and axial length of an electromagnetic torsional actuator are calculated preliminarily, and then the structure design of the electromagnetic torsional actuator is completed; S2, the relationship between the input current of the electromagnetic torsional actuator and the output electromagnetic torque thereof is solved by using the AC winding theory and the virtual work principle; S3, the electromagnetic torsional actuator is installed at the joint of the robot, so that the robot joint and the electromagnetic torsional actuator can rotate coaxially; S4, the force received by the end of the robot is detected online, and the required output torque of the electromagnetic torsional actuator is calculated based on the force; and the electromagnetic torsional actuator is controlled to output the output torque based on the relationship to act on the robot joint, so as to suppress the joint vibration. The present application expands the range of low-frequency vibration suppression.

Description

Technical Field

[0001] This invention belongs to the technical field of vibration suppression in robotic milling, and more specifically, relates to an active joint vibration control method and device for suppressing low-frequency chatter in milling robots. Background Technology

[0002] Robotic milling offers advantages such as a large workspace and high flexibility, making it a promising technology for machining large and complex structural parts. However, the multi-joint, serial structure of robots results in poor end-effector stiffness and significant variations in dynamic characteristics throughout the workspace. If large cutting parameters are used during machining, the robot experiences low-frequency chatter, primarily driven by its own structure. This chatter not only damages the surface quality of the machined part but can also lead to damage to the tool and robot structure. Existing milling stability prediction methods struggle to quickly and accurately predict the stability boundaries across the entire workspace. In practical engineering applications, to ensure part quality, only conservative cutting parameters can be selected, significantly limiting milling efficiency. Therefore, suppressing low-frequency chatter during robotic milling is crucial for improving its efficiency.

[0003] Existing research on low-frequency chatter suppression in robotic milling mainly focuses on three aspects: passive damping, semi-active damping, and active damping. Regarding passive damping: In 2016, Kaldestad et al. proposed a method using a passive damping absorber, which reduces chatter generated during milling of hard materials by installing sandbags on the top of the spindle. In 2018, Chen designed a novel eddy current damper (ECD) mounted on the robot spindle to suppress tool tip vibration during milling. In 2024, Xin et al. designed a joint damper based on magnetorheological effects, installing the damper at the robot joints to increase passive damping against joint vibration. Regarding semi-active damping: In 2019, Yuan et al. designed a stiffness-adjustable magnetorheological (MRE) damper, mounted on the spindle to absorb low-frequency vibrations of the robot structure in the 7Hz–20Hz frequency range. In 2023, Wu et al. designed a tuned mass damper (TMD) that can specifically suppress low-frequency chatter in milling robots under different postures by adjusting the damper's installation direction. Regarding active vibration suppression: In 2022, Guo et al. designed a novel cutting tool that generates a contact force during milling due to the tool's deformation while clamping the workpiece. The magnitude of this contact force is controlled to suppress low-frequency chatter in the milling robot. Also in 2022, Zhang et al. mounted an inertial actuator on the spindle head, using the inertial force generated by the actuator to control low-frequency chatter during robot milling.

[0004] Existing methods for suppressing low-frequency chatter in milling robots mainly focus on vibration suppression at the end effector of the milling system, and some research progress has been made. However, the dynamic characteristics of the robot's end effector vary greatly within the workspace, making passive and semi-active vibration suppression methods difficult to adapt to a wide range of machining conditions. Furthermore, installing heavy active vibration suppression devices at the robot's end effector not only introduces additional end-effector loads but also affects its flexibility in confined spaces. For milling robots with a series structure, end-effector vibration during machining is primarily caused by joint flexibility. A few studies have focused on suppressing joint vibration to achieve low-frequency chatter suppression, such as directly controlling the robot's joint motors for active joint vibration control. However, due to the delay between motor execution and control commands, this method can only produce a certain degree of suppression for lower frequency vibrations. Moreover, for safety reasons, most commercial robots do not grant users access to motor control, making this method difficult to apply. Summary of the Invention

[0005] In view of the above-mentioned defects or improvement needs of the existing technology, the present invention provides an active joint vibration control method and device for low-frequency chatter suppression of milling robots, which aims to solve the problem that existing vibration suppression methods are difficult to apply to large-scale robot milling processes.

[0006] To achieve the above objectives, according to one aspect of the present invention, an active joint vibration control method for suppressing low-frequency chatter in a milling robot is provided, the method comprising the following steps:

[0007] S1. Based on the magnitude of the disturbance torque on the robot joint and the actuator shape requirements, the air gap diameter and axial length of the electromagnetic torsion actuator are initially calculated, and then the structural design of the electromagnetic torsion actuator is completed.

[0008] S2. Based on the designed electromagnetic torsion actuator, the relationship between the input current and the output electromagnetic torque of the electromagnetic torsion actuator is obtained by using AC winding theory and the principle of virtual work.

[0009] S3, Install the electromagnetic torsion actuator at the robot joint so that the robot joint and the electromagnetic torsion actuator can rotate coaxially;

[0010] S4. Detect the force on the robot end effector online, calculate the required output torque of the electromagnetic torsion actuator based on the force, and control the electromagnetic torsion actuator to output the output torque to the robot joint based on the relationship, so as to suppress joint vibration.

[0011] Furthermore, assuming the required maximum torque is T, r is the air gap radius, and D... r Where σ is the air gap diameter, l is the axial length, and σ is the axial length. t S is the tangential stress in the air gap. rV is the surface area of ​​the air gap cylinder. r Let the volume of the cylinder enclosed by the air gap be:

[0012]

[0013] Determine V r Then, based on the actual shape requirements, select the required length-to-diameter ratio. Therefore, we can conclude that:

[0014]

[0015] The tangential stress in the air gap is taken as σ. t =17000~59500Pa, estimate the air gap diameter D based on the torque magnitude and length-to-diameter ratio. r Based on the axial length l, the structural design of the electromagnetic torsion actuator is carried out.

[0016] Furthermore, the air gap magnetomotive force generated by a single stator gear winding is:

[0017]

[0018] In the formula, N is the number of turns in a single toothed coil, and i is the magnitude of the current flowing through the coil. Z is the tooth groove angle, Z is the total number of stator teeth, and θ is a variable representing the circumferential angle of the air gap;

[0019] Air gap magnetic flux density B generated by a single gear winding coil (θ) is equal to the product of the air gap magnetomotive force and the magnetic permeability it generates, and thus:

[0020]

[0021] In the formula, δ e =δ air +h m δ e δ is the equivalent air gap length. air h is the actual air gap length. m Let μ be the thickness of the permanent magnet, μ0 be the permeability of free space, and Λ be the value of the magnet. air It is the air gap permeability.

[0022] Furthermore, B coil (θ) Expanded using Fourier series:

[0023]

[0024] In the formula, n = 1, 3, 5...∞ represents the harmonic order. According to AC winding theory, the nth harmonic pitch coefficient of the gear ring winding is:

[0025] The air gap magnetic flux density generated by the Z / 3 toothed windings of the same phase is superimposed to obtain the air gap magnetic flux density B of the single-phase winding. sp (θ):

[0026]

[0027] Let the total number of series turns of a single-phase winding be but:

[0028]

[0029] For the air gap magnetic flux density generated by the stator gear winding, the winding distribution coefficient of its nth harmonic is:

[0030]

[0031] Furthermore, the three-phase currents must satisfy the following relationship:

[0032]

[0033] Among them, i a i b i c θ represents the magnitude of the current in the A, B, and C phase windings, respectively, where I represents the amplitude of the sinusoidal three-phase current, and θ represents the amplitude of the current in the three phase windings. e Let θ represent the electrical angle of the nth harmonic of the stator magnetic field. e =nθ s θ s Given the spatial angle of the stator magnetic field, the air gap magnetic flux density generated by the A, B, and C phase windings are respectively:

[0034]

[0035] The sum of the air gap magnetic flux density generated by the three-phase windings is the combined air gap magnetic flux density of the stator three-phase windings:

[0036] B stator (θ,θ s ) = B sa (θ,θ s )+B sb (θ,θ s )+B sv (θ,θ s (11)

[0037] After simplification, we get:

[0038]

[0039] In the formula, the winding coefficient k wn =k qn k yn ,

[0040] When n≠3k+3 and n≠2k+2, the nth harmonic component of the combined magnetic flux density of the air gap in the three-phase stator windings is expressed as: B n =A n cos[n(θ±θ s )).

[0041] Furthermore, the magnetic flux density waveform generated by the rotor permanent magnet poles in the air gap is a rectangular wave, which, when expanded using Fourier series, yields:

[0042]

[0043] In the formula, p represents the number of pole pairs of the rotor permanent magnet, α i The polar arc coefficient, B, is related to the actual shape of the permanent magnet and the magnetization method. r This represents the residual magnetic flux density on the surface of a permanent magnet;

[0044] Let the rotor harmonic order m = vp = p, 3p, 5p..., then B rotor (θ) is represented in the following form:

[0045]

[0046] in, θ r For a rotor rotating at an angle, the m-th harmonic component in the air gap magnetic flux density generated by the rotor's permanent magnet is represented as: B m =A m cos[m(θ-θ r )).

[0047] Furthermore, the nth harmonic of the combined magnetic flux density in the air gap of the three-phase stator windings and the mth harmonic of the magnetic flux density in the air gap generated by the rotor permanent magnet can be expressed in the following form:

[0048]

[0049] The combined air gap magnetic flux density B is the sum of the magnetic flux density generated by the stator windings and the magnetic flux density generated by the rotor permanent magnets:

[0050] B = B stator (θ,θ s )+B rotor (θ,θ r (17)

[0051] The total energy storage capacity of the air gap is:

[0052]

[0053] W(θ r ,θ s ) for θ rTaking the partial derivative, we obtain the electromagnetic torque generated by the interaction between the nth harmonic of the stator and the mth harmonic of the rotor:

[0054]

[0055] The electromagnetic torque generated by the interaction of the p-th harmonic of the stator and the p-th harmonic of the rotor is the same as the electromagnetic torque generated by the interaction of the stator and rotor magnetic fields, i.e., m = n = p, thus:

[0056]

[0057] Furthermore, by using a six-dimensional force sensor to measure the disturbance force experienced by the end effector during robot milling online, and taking the disturbance force and robot joint angles as inputs, the required output torque of the electromagnetic torsional actuator is calculated. The corresponding formula is:

[0058] M 235 (t)=-J + Q235 (q)F m (t) (27)

[0059] Among them, M 235 (t)=[M 2a (t),M 3a (t),M 5a (t)] T This indicates the output torque, J, of the electromagnetic torsional actuators at joints 2, 3, and 5. + Q235 (q) is J Q235 The left pseudo-inverse of (q), J Q235 The elements of matrix (q) are as follows:

[0060] J Qij (i,j=1~6) represents matrix J Q The element in row i and column j of (q), J Q (q)=J F -1 (q), J F (q) is the Jacobian matrix of the robot, F m (t)=[F xm (t),F ym (t),F zm (t),M xm (t),M ym (t),M zm (t)] T Let q represent the perturbation force at the robot's end effector as measured by a six-dimensional force sensor, q = [q1(t), q2(t), q3(t), q4(t), q5(t), q6(t)]. TThis indicates the angles of each joint of the robot.

[0061] The present invention also provides an active joint vibration control system for suppressing low-frequency chatter in milling robots. The system includes a memory and a processor. The memory stores a computer program. When the processor executes the computer program, it performs the active joint vibration control method for suppressing low-frequency chatter in milling robots as described above.

[0062] The present invention also provides a computer-readable storage medium storing machine-executable instructions, which, when invoked and executed by a processor, cause the processor to implement the active joint vibration control method for low-frequency chatter suppression in a milling robot as described above.

[0063] In summary, compared with the prior art, the active joint vibration control method and equipment for low-frequency chatter suppression in milling robots provided by the present invention have the following beneficial effects:

[0064] 1. This invention designs the size and structure of an electromagnetic torsion actuator, and uses AC winding theory and the principle of virtual work to solve for the relationship between the input current and the output electromagnetic torque of the electromagnetic torsion actuator. It then conducts active joint vibration control to suppress low-frequency chatter during large-scale robot milling and improve the efficiency of robot milling.

[0065] 2. The relationship between the input current and the output electromagnetic torque of the electromagnetic torsion actuator was obtained. The magnitude of the electromagnetic torque can be changed by changing the magnitude of the current flowing through the stator winding.

[0066] 3. The calculation formula for the electromagnetic torque generated by the interaction of the stator and rotor magnetic fields was obtained through analysis. It can be seen that the torque T depends on the variable θ. r θ s Size, θ r By installing an angle measuring device in the electromagnetic torsion actuator, by changing θ s Size (θ) s The size of the current determines the size of the output torque T.

[0067] 4. A joint vibration active control method based on disturbance torque compensation is proposed. By measuring the disturbance force on the robot end effector online, the output torque of the electromagnetic torsion actuator is calculated. The output torque of the electromagnetic torsion actuator is then applied to the robot joint to suppress joint vibration, thereby achieving active joint vibration control. Attached Figure Description

[0068] Figure 1This is a flowchart of an active joint vibration control method for suppressing low-frequency chatter in milling robots, provided by the present invention.

[0069] Figure 2 (a) and (b) in the figure are schematic diagrams of the electromagnetic torsion actuator mentioned in this invention at different angles;

[0070] Figure 3 yes Figure 2 A schematic diagram of the internal structure of the electromagnetic torsion actuator in the image;

[0071] Figure 4 yes Figure 2 A schematic diagram defining the air gap calculation dimension parameters of the electromagnetic torsion actuator in the diagram;

[0072] Figure 5 yes Figure 2 A schematic diagram of a single gear ring winding in an electromagnetic torsion actuator;

[0073] Figure 6 yes Figure 2 A schematic diagram of the single-phase winding of the electromagnetic torsion actuator in the diagram;

[0074] Figure 7 yes Figure 2 A schematic diagram of the stator three-phase windings of the electromagnetic torsion actuator in the diagram;

[0075] Figure 8 (a) and (b) in the diagram are schematic diagrams showing the position and dimensions of the rotor permanent magnet, respectively.

[0076] Figure 9 (a) and (b) in the figure are schematic diagrams of the magnetic flux density waveform and harmonics of the air gap of the permanent magnet, respectively;

[0077] Figure 10 (a) and (b) in the diagram are installation diagrams of the encoder at different angles;

[0078] Figure 11 This is an assembly diagram of an electromagnetic torsion actuator;

[0079] Figure 12 This is a schematic diagram showing the installation positions of the electromagnetic torsion actuator and the six-dimensional force sensor;

[0080] Figure 13 This is a flowchart of an active joint vibration control method based on disturbance torque compensation;

[0081] Figure 14 (a), (b), and (c) are schematic diagrams comparing the vibration displacement of the robot's end effector in the X, Y, and Z directions, respectively.

[0082] In all the accompanying drawings, the same reference numerals are used to denote the same elements or structures, wherein: 1-stator left housing, 2-coil, 3-left tapered roller bearing, 4-left sleeve, 5-stator core, 6-stator right housing, 7-right sleeve, 8-right tapered roller bearing, 9-rotor flange, 10-rotor yoke, 11-tile-type neodymium iron boron permanent magnet, 12-internal hex bolt. Detailed Implementation

[0083] To make the objectives, technical solutions, and advantages of this invention clearer, the invention will be further described in detail below with reference to the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are merely illustrative and not intended to limit the invention. Furthermore, the technical features involved in the various embodiments of this invention described below can be combined with each other as long as they do not conflict with each other.

[0084] This invention provides an active joint vibration control method for suppressing low-frequency chatter in milling robots. The control method implements active joint vibration control to suppress low-frequency chatter during a wide range of robot milling processes, thereby improving the efficiency of robot milling.

[0085] Please see Figure 1 The control method mainly includes the following steps:

[0086] S1. Based on the magnitude of the disturbance torque on the robot joint and the shape requirements, the air gap diameter and axial length of the electromagnetic torsion actuator are initially calculated, and then the structural design of the electromagnetic torsion actuator is completed.

[0087] Please see Figure 2 and Figure 3 The electromagnetic torsion actuator includes a left stator housing 1, a coil 2, a left tapered roller bearing 3, a left sleeve 4, a stator core 5, a right stator housing 6, a right sleeve 7, a right tapered roller bearing 8, a rotor flange 9, a rotor yoke 10, tile-shaped neodymium iron boron permanent magnets 11, and hexagon socket head cap screws 12. To reduce the number of connecting structural components and lower the overall structural weight, most structural components are connected by adhesive, using epoxy resin AB glue. The tile-shaped neodymium iron boron permanent magnets 11 are bonded to the outer ring of the rotor yoke 10. Ten of these magnets are evenly arranged circumferentially on the rotor yoke 10, with adjacent tile-shaped neodymium iron boron permanent magnets 11 having opposite polarities. The rotor yoke 10 is a hollow cylinder, with its opposite ends bonded to the left sleeve 4 and the right sleeve 7, respectively. The left sleeve 4 and the right sleeve 7 are mounted on the rotor flange 9 shaft and are bonded together. The two opposite ends of the rotor flange 9 are respectively supported on the inner rings of the left tapered roller bearing 3 and the right tapered roller bearing 8 to form a rotor.

[0088] The stator core 5 has 12 pairs of gear rings, with a coil 2 wound on each tooth to form a set of gear ring windings. Four gear ring windings are connected in series to form a 1-phase stator winding. The 12 coils 2 form a total of 3-phase windings. The left stator housing 1 and the right stator housing 6 serve as the positioning and support structures for the stator core 5. The contact surfaces are bonded to transmit torque. The outer rings of the left tapered roller bearing 3 and the right tapered roller bearing 8 respectively mate with the bearing seats in the left stator housing 1 and the right stator housing 6. The left stator housing 1 and the right stator housing 6 are connected by six hexagon socket head cap screws 12. There are 18 hexagon socket head cap screws 12 evenly distributed around the circumference.

[0089] In the electromagnetic torsion actuator, there is a small air gap between the stator and the rotor. The tile-shaped neodymium iron boron permanent magnet 11 generates a permanent magnetic field in the air gap. By energizing the three-phase stator windings, an electromagnetic magnetic field is generated in the air gap. The interaction between the electromagnetic magnetic field and the permanent magnetic field forms an electromagnetic torque. The magnitude and direction of the electromagnetic torque can be adjusted by regulating the current flowing through the coil. In use, the axis of the electromagnetic torsion actuator is coaxially mounted with the axis of the robot joint. The stator of the electromagnetic torsion actuator is fixed, and the rotor rotates with the joint. The output torque of the electromagnetic torsion actuator is obtained through online measurement and calculation. The output torque acts on the connecting rod and is then transmitted to the robot joint through the connecting rod, suppressing joint vibration and thus suppressing low-frequency chatter in the milling robot.

[0090] The magnitude of the disturbance torque experienced by different robot joints varies. Based on the magnitude of the disturbance torque and the shape requirements, the key dimensions of the electromagnetic torsional actuator can be preliminarily calculated: air gap diameter D. r And axial length l, the parameters are defined as follows Figure 4 As shown, the specific calculation process is as follows:

[0091] Assuming the required maximum torque is T, r is the air gap radius, and D... r Where σ is the air gap diameter, l is the axial length, and σ is the axial length. t S represents the tangential stress value in the air gap. r V is the surface area of ​​the air gap cylinder. r Let the volume of the cylinder enclosed by the air gap be:

[0092]

[0093] Determine V r Then, based on the actual shape requirements, select the required length-to-diameter ratio. Therefore, we can conclude that:

[0094]

[0095] The tangential stress value in the air gap is usually taken as σ. t=17000~59500Pa, the air gap diameter D can be estimated based on the torque magnitude and length-to-diameter ratio. r The axial length l and other structural dimensions are designed based on these two dimensions.

[0096] S2. Based on the designed electromagnetic torsion actuator, the relationship between the input current and the output electromagnetic torque of the electromagnetic torsion actuator is obtained by using AC winding theory and the principle of virtual work.

[0097] Electromagnetic torque originates from the interaction between the electromagnetic magnetic field generated by the stator windings and the permanent magnetic field generated by the rotor permanent magnets. Changing the magnitude of the current flowing through the stator windings changes the magnitude of the electromagnetic torque. To precisely control the magnitude of the electromagnetic torque, it is necessary to solve the functional relationship between the input current and the electromagnetic torque based on AC winding theory and the principle of virtual work.

[0098] Ignoring nonlinear factors such as magnetic saturation and leakage flux, the magnetic field satisfies the superposition principle, allowing the determination of the air gap magnetic flux density, and subsequently, the aforementioned relationship. Specifically, the process includes the following steps:

[0099] 2.1 Solve for the air gap magnetic flux density of a single gear ring winding.

[0100] A single gear winding, such as Figure 5 As shown, since the air gap reluctance is much greater than the stator core reluctance, the core magnetic voltage drop can be ignored. Therefore, the air gap magnetomotive force generated by a single gear winding is:

[0101]

[0102] In the formula, N is the number of turns of the gear winding, and i is the magnitude of the current flowing through the coil. Z is the tooth groove angle, Z is the total number of stator teeth, and θ is a variable representing the circumferential angle of the air gap;

[0103] Air gap magnetic flux density B generated by a single gear winding coil (θ) is equal to the product of the air gap magnetomotive force it generates and the air gap permeability, and we have:

[0104]

[0105] In the formula, δ e =δ air +h m δ e δ is the equivalent air gap length. air h is the actual air gap length. m Let μ be the thickness of the permanent magnet, μ0 be the permeability of free space, and Λ be the value of the magnet. air It is the air gap permeability.

[0106] B coil (θ) Expanded using Fourier series:

[0107]

[0108] In the formula, n = 1, 3, 5...∞ represents the harmonic order. According to AC winding theory, the pitch coefficient of the nth harmonic of the gear ring winding is...

[0109] 2.2 Solve for the air gap magnetic flux density of a single-phase winding (consisting of four toothed ring windings).

[0110] Four toothed ring windings with specific relative positions constitute a single-phase winding, such as... Figure 6 As shown; the superposition of the air gap magnetic flux density generated by the four toothed windings is the air gap magnetic flux density B of a single-phase winding. sp (θ):

[0111]

[0112] Let the total number of series turns of a single-phase winding be but:

[0113]

[0114] For the air gap magnetic flux density generated by the stator gear winding, the winding distribution coefficient of its nth harmonic is:

[0115]

[0116] 2.3 Solve for the air gap magnetic flux density of the three-phase stator windings.

[0117] Please see Figure 7 The stator three-phase winding consists of three sets of single-phase windings spaced 120° apart in the circumferential direction. The four toothed ring windings A1, X1, A2, and X2 are the A-phase windings; the four toothed ring windings B1, Y1, B2, and Y2 are the B-phase windings; and the four toothed ring windings C1, Z1, C2, and Z2 are the C-phase windings. The coils of the same-phase toothed ring windings are connected in series. The three-phase windings carry sinusoidal alternating currents of equal frequency and phases spaced 120° apart, thereby generating a rotating magnetic field whose direction and magnitude can be adjusted. The three-phase currents must satisfy the following relationship:

[0118]

[0119] Among them, i a i b i c θ represents the magnitude of the current in the A, B, and C phase windings, respectively, where I represents the amplitude of the sinusoidal three-phase current, and θ represents the amplitude of the current in the three phase windings. e Let θ represent the electrical angle of the nth harmonic of the stator magnetic field. e =nθ s θ s Given the spatial angle of the stator magnetic field, the air gap magnetic flux density generated by the A, B, and C phase windings are respectively:

[0120]

[0121] The sum of the air gap magnetic flux density generated by the three-phase windings is the combined air gap magnetic flux density of the stator three-phase windings:

[0122] B stator (θ,θ s ) = B sa (θ,θ s )+B sb (θ,θ s )+B sc (θ,θ s (11)

[0123] After simplification, we get:

[0124]

[0125] In the formula, the winding coefficient k wn =k qn k yn ,

[0126] When n≠3k+3 and n≠2k+2, the nth harmonic component of the combined magnetic flux density of the air gap in the three-phase stator windings is expressed as: B n =A n cos[n(θ±θ s )).

[0127] 2.4 Solving for the air gap magnetic flux density of tile-type neodymium iron boron permanent magnets.

[0128] Ignoring the effects of saturation, leakage flux, and cogging effects, the relative permeability of neodymium iron boron permanent magnets is 1.02–1.05, which can be considered the same as that of air. Concentric radial tile-shaped permanent magnet poles are used and mounted on the rotor yoke surface. The length, width, and thickness of the permanent magnets are l, b, and b, respectively. m h m ,like Figure 8 As shown, the air gap magnetic flux density waveform generated by the permanent magnet is as follows: Figure 9 As shown.

[0129] The rectangular wave of the permanent magnet air gap magnetic flux density, expanded by Fourier series, yields:

[0130]

[0131] in: p represents the number of pole pairs of the rotor permanent magnet, α i The polar arc coefficient, B, is related to the actual shape of the permanent magnet and the magnetization method. r This represents the residual magnetic flux density on the surface of a permanent magnet. Figure 9The figure shows the waveforms of the 1st, 3rd, and 5th harmonics of the air gap magnetic flux density of the permanent magnet.

[0132] Let m = vp = p, 3p, 5p..., then B rotor (θ) is represented in the following form:

[0133]

[0134] in, θ r The rotor rotation angle can be obtained by measuring the rotor rotation angle using an encoder. From formula (15), it can be seen that the air gap magnetic flux density of the permanent magnet is the sum of all harmonic magnetic fields, where the m-th harmonic component is expressed as: B m =A m cos[m(θ-θ r )).

[0135] 2.4 Calculation of electromagnetic torque.

[0136] Both the stator and rotor generate multiple harmonics in the air gap magnetic flux density. To simplify the calculation process, based on the principle of magnetic field superposition, we can first consider the interaction between the stator's nth harmonic and the rotor's mth harmonic separately. The nth harmonic of the combined magnetic flux density of the stator's three-phase winding air gap and the mth harmonic of the rotor's permanent magnet air gap magnetic flux density are expressed in the following form:

[0137]

[0138] The air gap composite magnetic field B is the sum of the stator winding magnetic field and the rotor permanent magnet magnetic field:

[0139] B = B stator (θ,θ s )+B rotor (θ,θ r (17)

[0140] The total energy storage capacity of the air gap is:

[0141]

[0142] W(θ r ,θ s ) for θ r Taking the partial derivative, we obtain the electromagnetic torque generated by the interaction between the nth harmonic of the stator and the mth harmonic of the rotor:

[0143]

[0144] Electromagnetic torque is mainly generated by the interaction of the p-th harmonics in the rotor and stator magnetic fields, i.e., m = n = p. Other higher harmonic components can be further weakened by optimizing the shape of the permanent magnet poles and changing the magnetization method of the permanent magnet, thus achieving smaller torque fluctuations. Therefore, other components can be ignored, and the electromagnetic torque generated by the interaction of the stator and rotor magnetic fields can be obtained. Taking p = 3k + 1 as an example, at this time:

[0145]

[0146] In the formula, θ s θ is the spatial angle of the stator magnetic field, which determines the magnitude of the current; r The rotor rotation angle; from formula (20), it can be seen that the torque T depends on the variable θ. r θ s Size, θ r Measurements need to be taken using an angle measuring device, by changing θ. s The magnitude of θ controls the magnitude of the output torque T. To measure θ online... r A rotary encoder needs to be installed between the stator and rotor, such as Figure 10 As shown, the encoder stator is connected to the encoder connecting plate via a spring clamp. The encoder connecting plate is fixed to the stator housing using screws. The encoder stator is fixedly connected to the stator of the electromagnetic torsion actuator. The encoder rotor is connected to the conical shaft via a thread. The left end face of the conical shaft is bonded to the right end face of the electromagnetic torsion actuator rotor flange using structural adhesive. The encoder rotor rotates together with the electromagnetic torsion actuator rotor, thereby realizing the measurement of the rotation angle of the electromagnetic torsion actuator rotor.

[0147] S3, Install the electromagnetic torsion actuator at the robot joint so that the robot joint and the electromagnetic torsion actuator can rotate coaxially.

[0148] In order to achieve active joint vibration control, the actuator assembly structure requires that the output torque of the electromagnetic torsional actuator be applied to the robot joint from outside the robot. A design is as follows: Figure 11The assembly scheme shown is as follows: The two connecting rods of the joint are connecting rod 1 and connecting rod 2. Connecting rod 1 is fixed, and connecting rod 2 rotates with the joint. The stator of the electromagnetic torsion actuator is fixed to connecting rod 1 through the stator connecting plate, and the rotor of the electromagnetic torsion actuator is fixed to connecting rod 2 through the rotor connecting flange. Positionally, the rotation center of the joint is kept coaxial with the rotation center of the electromagnetic torsion actuator rotor. Therefore, the electromagnetic torsion actuator rotor and connecting rod 2 can rotate coaxially together. While not hindering the rotation of the robot joint, the torque output by the electromagnetic torsion actuator acts on the connecting rod and is transmitted to the robot joint. The stator connecting plate uses a nut-fixed hinge pin with a hole to transmit torque at the bend. The provided sliding through-hole allows for fine adjustment of the connecting plate position during assembly, avoiding assembly interference.

[0149] S4. Detect the force on the robot end effector online, calculate the required output torque of the electromagnetic torsion actuator based on the force, and control the electromagnetic torsion actuator to output the output torque to the robot joint based on the relationship, so as to suppress joint vibration.

[0150] The system detects the force acting on the robot's end effector online and calculates the corresponding disturbance torque based on the disturbance force to obtain the required output torque for the electromagnetic torsion actuator. Then, it calculates the required input current according to the formula and controls the input current of the electromagnetic torsion actuator coil to be the required input current to suppress the robot's joint vibration.

[0151] In one implementation, during robotic milling, the cutting tool tip is subjected to dynamically changing milling forces of varying magnitude and direction, generating disturbance torques on the robot's joints and causing joint vibration. Therefore, electromagnetic torsion actuators are needed to apply compensating torques to the robot joints to suppress joint vibration. A six-dimensional force sensor is installed at the robot's end flange. During milling, the milling force on the cutting tool tip is transmitted to the center of the flange. Based on the force measured at the robot's end flange at each sampling moment by the six-dimensional force sensor, the required output torque for each electromagnetic torsion actuator is calculated in real time. The electromagnetic torsion actuators then execute the required output torque to act on the robot joints to suppress joint vibration, ultimately achieving low-frequency chatter suppression in the milling robot. This is illustrated using the ESTUN ER70 robot as an example. Figure 12 The document describes the installation locations of the electromagnetic torsion actuator and the six-dimensional force sensor in the milling robot.

[0152] Specifically, based on the six-dimensional force at the robot's end flange measured in real time by the six-dimensional force sensor, the output torque of the electromagnetic torsional actuator needs to be calculated using an effective vibration suppression algorithm. The following disturbance torque compensation algorithm is proposed:

[0153] The six-dimensional force acting on the robot's end flange is mapped to the joint torque in joint space through the force Jacobian matrix:

[0154] M(t) = J F (q)F(t) (21)

[0155] In the formula, F(t)=[F x (t),F y (t),F z (t),M x (t),M y (t),M z (t)] T Let M(t) represent the six-dimensional forces acting on the robot's end effector, where M(t) = [M1(t), M2(t), M3(t), M4(t), M5(t), M6(t)]. T This represents the torque experienced by each joint of the robot under the action of a six-dimensional force F(t) at the end effector. q = [q1(t), q2(t), q3(t), q4(t), q5(t), q6(t)] T J represents the angles of each joint of the robot. Q The elements of matrix (q) are as follows:

[0156]

[0157] Under non-singular robot postures, the Leyakubi matrix is ​​invertible, and J is defined as follows: Q (q)=J F -1 (q), then the equivalent six-dimensional force output at the end effector after applying active torque to each joint of the robot is:

[0158] F a (t)=J Q (q)M a (t) (22)

[0159] In the formula, M a (t)=[M 1a (t),M 2a (t),M 3a (t),M 4a (t),M 5a (t),M 6a (t)] T F represents the active torque applied to each joint of the robot. a (t)=[F xa (t),F ya (t),F za (t),M xa (t),M ya (t),M za [t] represents the applied active torque M a (t) is the equivalent six-dimensional force output by the robot's end effector.

[0160] Considering only the active torque applied to joints 2, 3, and 5, i.e., M 1a (t)=M 4a (t)=M 6a (t) = 0; therefore:

[0161]

[0162] After simplification, we get:

[0163] F a (t)=J Q235 (q)M 235 (t) (24)

[0164] In the formula, Among them, J Qij (i,j=1,2...6) is matrix J Q (q) The element in the i-th row and j-th column, M 235 (t)=[M 2a (t),M 3a (t),M 5a (t)] T .

[0165] Let F m (t)=[F xm (t),F ym (t),F zm (t),M xm (t),M ym (t),M zm (t)] T The disturbance force on the robot's end effector, measured by a six-dimensional force sensor, is used to counteract the disturbance torque at each joint caused by the end effector's disturbance force. The electromagnetic torsional actuator outputs torque, and the equivalent six-dimensional force generated at the end effector should satisfy the following condition with respect to the disturbance force on the robot's end effector:

[0166] F m (t)+F a (t)=0 (25)

[0167] Substituting into equation (23), we get:

[0168] J Q235 (q)M 235 (t)+F m (t)=0 (26)

[0169] The linear equation system in equation (26) is an overdetermined system of equations, and the least squares solution of the system can be obtained:

[0170] M 235 (t)=-J +Q235 (q)F m (t) (27)

[0171] In the formula, J + Q235 (q) is J Q235 The left pseudo-inverse of (q).

[0172] Equation (27) yields the required output torques M2, M3, and M5 for the electromagnetic torsion actuators at joints 2, 3, and 5. This process needs to be solved in real time in the controller. Therefore, the controller needs to read the joint angle values ​​of each joint under the robot's processing posture online to calculate the force Jacobian matrix, as well as the end disturbance force measurement value of the six-dimensional force sensor and the rotor rotation angle θ of the electromagnetic torsion actuator measured by the encoder. r The specific implementation process of the active joint vibration control method based on disturbance torque compensation is as follows: Figure 13 As shown.

[0173] Combining equations (23) and (26), we can obtain the equivalent resultant force F on the robot's end effector under the combined action of the end effector disturbance and the output torque of the electromagnetic torsion actuator. c (t) satisfies:

[0174]

[0175] In the formula, F c (t)=[F xc (t),F yc (t),F zc (t),M xc (t),M yc (t),M zc (t)] T

[0176] The simulation model applies the equivalent resultant force as described above to the end effector of the robot milling machine. Through simulation, the milling process is compared between the two scenarios: with and without active joint vibration control. Figure 14 As shown, the tool vibration displacement diverges without active joint vibration control, i.e., low-frequency chatter occurs. However, after active joint vibration control is applied, the vibration displacement tends to stabilize. The surface active joint vibration control method effectively suppresses the low-frequency chatter phenomenon in the robot milling process, verifying the effectiveness of active joint vibration control based on disturbance torque compensation in suppressing low-frequency chatter in robot milling.

[0177] The present invention also provides an active joint vibration control system for suppressing low-frequency chatter in milling robots. The system includes a memory and a processor. The memory stores a computer program. When the processor executes the computer program, it performs the active joint vibration control method for suppressing low-frequency chatter in milling robots as described above.

[0178] The present invention also provides a computer-readable storage medium storing machine-executable instructions, which, when invoked and executed by a processor, cause the processor to implement the active joint vibration control method for low-frequency chatter suppression in a milling robot as described above.

[0179] Those skilled in the art will readily understand that the above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention. Any modifications, equivalent substitutions, and improvements made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A method for active joint vibration control to suppress low-frequency chatter in milling robots, characterized in that, This method Includes the following steps: S1. Based on the magnitude of the disturbance torque on the robot joint and the shape requirements, the air gap diameter and axial length of the electromagnetic torsion actuator are initially calculated, and then the structural design of the electromagnetic torsion actuator is completed. S2. Based on the designed electromagnetic torsion actuator, the relationship between the input current and the output electromagnetic torque of the electromagnetic torsion actuator is obtained by using AC winding theory and the principle of virtual work. S3, Install the electromagnetic torsion actuator at the robot joint so that the robot joint and the electromagnetic torsion actuator can rotate coaxially; S4. Detect the force on the robot end effector online, calculate the required output torque of the electromagnetic torsion actuator based on the force, and control the electromagnetic torsion actuator to output the output torque to the robot joint based on the relationship to suppress joint vibration. Assuming the maximum required torque is , Where is the air gap radius, The air gap diameter is... axial length This represents the tangential stress value in the air gap. Let be the surface area of ​​the air gap cylindrical surface. Let the volume of the cylinder enclosed by the air gap be: （1） Sure Then, based on the actual shape requirements, select the required length-to-diameter ratio. Therefore, we can conclude that: （2） The tangential stress value in the air gap is taken as Based on the torque magnitude and length-to-diameter ratio, the air gap diameter is estimated. and axial length ; The combined magnetic flux density of the air gap in the three-phase stator windings n Subharmonics and the air gap magnetic flux density generated by the rotor permanent magnet m The subharmonic is represented in the following form: （16） Air gap combined magnetic flux density It is the sum of the magnetic flux density generated by the stator windings and the magnetic flux density generated by the rotor permanent magnets: （17） The total energy storage capacity of the air gap is: （18） right Taking the partial derivative yields the stator. n Subharmonics and rotor m Electromagnetic torque generated by the interaction between subharmonics: （19） stator p Subharmonics and rotor p The torque generated by the interaction of subharmonics is the electromagnetic torque generated by the interaction of the stator and rotor magnetic fields, that is... ,get: （20）。 2. The active joint vibration control method for low-frequency chatter suppression in milling robots as described in claim 1, characterized in that: The air gap magnetomotive force generated by a single stator gear winding is: （3） In the formula, The number of coil turns in a single toothed winding. The magnitude of the current flowing through the coil. For the tooth groove angle, Z This represents the total number of stator teeth. The variable representing the circumferential angle of the air gap; Air gap magnetic flux density generated by a single gear winding It is equal to the product of the air gap magnetomotive force and the air gap permeability it generates, then: （4） In the formula, , This is the equivalent air gap length. This is the actual air gap length. The thickness of the permanent magnet; Permeability of free space; It is the air gap permeability.

3. The active joint vibration control method for low-frequency chatter suppression in milling robots as described in claim 2, characterized in that: Will Expand using Fourier series: （5） In the formula, For harmonic orders, according to AC winding theory, the gear ring winding... Subharmonic pitch coefficient ; Same phase Z The air gap magnetic flux density generated by the three gear windings is superimposed to obtain the air gap magnetic flux density of a single-phase winding. : （6） Let the total number of series turns of a single-phase winding be ,but: （7） For the air gap magnetic flux density generated by the stator gear winding, The winding distribution factor for the subharmonic is: （8）。 4. The active joint vibration control method for low-frequency chatter suppression in milling robots as described in claim 3, characterized in that: The three-phase currents must satisfy the following relationship: （9） in, These represent the current magnitudes of the input windings A, B, and C, respectively. This represents the amplitude of a sinusoidal three-phase current. Represents the stator magnetic field n The electrical angle of the subharmonic makes , Given the spatial angle of the stator magnetic field, the air gap magnetic flux density generated by the A, B, and C phase windings are respectively: （10） The sum of the air gap magnetic flux density generated by the three-phase windings is the combined air gap magnetic flux density of the stator three-phase windings: （11） After simplification, we get: （12） In the formula, the winding coefficient , ; when ,and At that time, the combined magnetic flux density of the air gap of the three-phase stator windings is n The subharmonic components are represented as follows: .

5. The active joint vibration control method for low-frequency chatter suppression in milling robots as described in claim 1, characterized in that: The magnetic flux density waveform generated by the rotor permanent magnet poles in the air gap is a rectangular wave, which can be expanded using Fourier series: （14） in, , , Indicates the number of pole pairs of the rotor permanent magnet. This is the polar arc coefficient, which is related to the actual shape of the permanent magnet and the magnetization method. This represents the residual magnetic flux density on the surface of a permanent magnet; Let the rotor harmonic order ,but Write it in the following form: （15） in, , The rotor rotation angle is the air gap magnetic flux density generated by the rotor permanent magnet. m The subharmonic components are represented as follows: .

6. The active joint vibration control method for low-frequency chatter suppression in milling robots as described in claim 1, characterized in that: The disturbance force experienced by the end effector of the robot during milling is measured online using a six-dimensional force sensor. The required output torque of the electromagnetic torsional actuator is calculated by using the disturbance force and the robot joint angle as inputs. The corresponding formula is as follows: （27） In the formula, This indicates the output torque of the electromagnetic torsional actuators at joints 2, 3, and 5. for The left pseudo-reversal, The elements of the matrix are as follows: , Representative matrix The Middle i Line 1 j Column elements, , , For the robot's Jacobian matrix, This represents the disturbance force experienced by the robot's end effector as measured by a six-dimensional force sensor. This indicates the angles of each joint of the robot.

7. An active joint vibration control system for suppressing low-frequency chatter in milling robots, characterized in that: The system includes a memory and a processor. The memory stores a computer program, and when the processor executes the computer program, it performs the active joint vibration control method for low-frequency chatter suppression of milling robots as described in any one of claims 1-6.

8. A computer-readable storage medium, characterized in that: The computer-readable storage medium stores machine-executable instructions, which, when invoked and executed by a processor, cause the processor to implement the active joint vibration control method for low-frequency chatter suppression in a milling robot as described in any one of claims 1-6.

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