CN113276118B - Robot low-speed motion nonlinear dynamics characteristic analysis and modeling method - Google Patents
Robot low-speed motion nonlinear dynamics characteristic analysis and modeling method Download PDFInfo
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- CN113276118B CN113276118B CN202110567965.9A CN202110567965A CN113276118B CN 113276118 B CN113276118 B CN 113276118B CN 202110567965 A CN202110567965 A CN 202110567965A CN 113276118 B CN113276118 B CN 113276118B
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- B—PERFORMING OPERATIONS; TRANSPORTING
- B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
- B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
- B25J9/00—Programme-controlled manipulators
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Abstract
The invention provides a robot low-speed motion nonlinear dynamics characteristic analysis and modeling method, which comprises the following steps: designing a robot single-joint rotation experiment, and acquiring joint motor torque, joint angle and joint speed data; performing cubic interpolation value resampling after low-pass filtering the torque data, calculating motor fluctuation deterministic torque and joint friction torque according to the resampled torque data, and obtaining the motor fluctuation deterministic torque and the joint friction deterministic torque by adopting a fitting method; analyzing nonlinear torque data, analyzing the difference value of the motor fluctuation moment and the motor fluctuation certainty moment, and the difference value of the joint friction moment and the joint friction certainty moment, researching the nonlinear fluctuation characteristics of the two groups of moment difference values, and defining the nonlinear fluctuation characteristics moment and the joint friction chaos moment as the motor fluctuation chaos characteristic moment and the joint friction chaos moment; and modeling the low-speed motion nonlinear dynamics characteristics of the robot based on a phase space reconstruction theory and a neural network.
Description
Technical Field
The invention belongs to the field of robot dynamics characteristic research, and particularly relates to a robot low-speed movement nonlinear dynamics characteristic analysis and modeling method.
Background
Robot dynamics models are used for robot motion control algorithm design and contact force control algorithm design. The more accurate the dynamic model is, the more beneficial the realization of the robot control algorithm is. The Newton-Euler method can only model rigid body dynamics of the robot generally and does not contain complex nonlinear dynamics of low-speed motion of the robot. For an assembled robot, it is difficult to model each of the factors that contribute to the dynamic non-linearity independently. In low-speed application scenes such as robot shaft hole assembly and the like, the joint movement speed is generally not more than 10% of full speed, and acceleration and deceleration changes are smooth in the operation process. The non-linear dynamics of a low-speed moving robot are due to several aspects. This is because the robot is an organic whole composed of a motor, a reducer and other transmission mechanisms, and the nonlinearity of all parts is coupled together and finally appears as motor signal fluctuation.
In practice robots have complex non-linear dynamic models. The nonlinear dynamics of the robot low-speed motion is generally considered to be caused by low-speed nonlinear friction. In order to improve the control performance of the robot, the Xiao et al compensates the nonlinear friction torque of the robot by using a second-order Fourier series model based on an angle and a cubic polynomial model based on an angular velocity. The effect of joint loading on friction torque was studied by Hamon et al, bittencourt et al (Bittencourt AC, gunnarsson S.Static vibration in a robot joint-modeling and identification of load and temperature effects [ J ]. Journal of Dynamic Systems, measurement, and Control,2012,134 (5): 051013.) and Gao et al, which also discuss the effect of temperature on friction torque. They expressed some parameters in the friction model as dependent variables of temperature or load and concluded that the effect of temperature and speed on viscous friction is non-linear and that the effect of load on coulomb friction is linear and causes a slight Stribeck effect. Meanwhile, complex nonlinear factors also exist in a gear and bearing transmission system in the robot rotating mechanism. Al-Shyyab and Kahraman adopt a multi-scale harmonic balancing method and a numerical method to analyze the nonlinear vibration response and amplitude jumping characteristic of the multi-meshing gear train. Guilbault et al studied the time-varying mesh damping between gears, performed a dynamic modeling of cylindrical gears, and analyzed the jump characteristics of gears. Li and Kahraman consider the lubrication action between the gear teeth, and analyze the nonlinear vibration characteristics of the gear including the tooth disengaging phenomenon and the amplitude jump by adopting a numerical method. Lu et al analyzed the complex motion state of gears under different parameters using a global bifurcation map and a maximum Lapunov index (Lu J W, chen H, zeng F L, et al. Infiluence of system parameters on dynamic bearing of gear pair with a stored kinetic backlash [ J ]. Meccanica,2014,49 (2): 429-440). In addition, the low-speed motion characteristics of the robot joint motor, such as low-speed crawling, ripple torque phenomenon and the like, also cause the nonlinear fluctuation of the output torque of the robot joint. For compensating the ripple torque of the motor, petrovic et al propose a torque ripple suppression method of a permanent magnet synchronous motor with an adaptive feedback structure. Chen et al analyzed motor torque ripple caused by controller causes such as calculation error, current sensor error, PWM modulation frequency error, etc., and proposed and verified a servo controller parasitic torque compensation mathematical model (Chen S, namulri C, mir S.controller-induced parasitic torque compensations in a PM synchronous motors [ J ]. IEEE transactions on industrial applications,2002,38 (5): 1273-1281.). Ji et al artificially overcome the influence of speed measurement delay on system stability, and perform adaptive compensation on the controller gain according to the damping index rule, thereby improving the stability and tracking accuracy of the permanent magnet synchronous motor at low speed. Bin et al quickly compensate for torque ripple through motion controller PMAC based on steady state error analysis using spectral analysis and least squares.
At present, researches on nonlinear dynamics of robots generally carry out independent modeling analysis on various influencing factors, but the robots are organic whole bodies formed by various components, and all the nonlinear characteristics are necessarily coupled together and finally show irregular fluctuation of motor output torque signals. For the assembled robot, it is difficult to model each factor causing the dynamic non-linear characteristic independently. If the robot is disassembled and the dynamics characteristic analysis is performed on each part, the possible nonlinear coupling characteristics between the parts cannot be studied. In addition, in the process of assembly and disassembly, the characteristics of the whole robot joint are difficult to maintain unchanged. For example, when the robot is reassembled, the relative positions of some parts are inevitably deviated within the tolerance range, and the dynamic characteristics of the robot can be changed. The chaos theory is a method for researching a nonlinear power system, and is used for researching the nonlinear dynamics characteristics of a robot. The bifurcation characteristic and the chaos characteristic of the 2-degree-of-freedom nonlinear gear transmission system are analyzed by Farshidianfar and Saghafi through a Melnikov analytical method, and the accuracy of the analytical method is verified by combining a numerical method with a Poinc-line phase plane graph, a time domain frequency domain curve and the like. Wang and Wu propose an analysis method to study the chaotic vibration in the space robot, and establish a single-step prediction model of the chaotic time sequence and the chaotic vibration by using an SVR prediction model with an RBF kernel function. Chen et al performed chaotic characterization of a spatial 4-UPS-RPU parallel mechanism with spherical gaps and indicated that as the play value increased, the stability of the mechanism decreased and chaotic motion occurred on the moving platform of the mechanism as the gap value increased to 2.1mm (Chen X, gao W, deng Y, et al. El Arem explores the dynamic response of the cross-sectional fracture rotation axis, and reveals the oscillation characteristics of the system using poincare and bifurcation diagrams. The chaos theory provides a method for analyzing the nonlinear characteristics of a dynamic system from experimental data. For the nonlinear dynamics characteristics of the robot caused by multi-factor coupling, it is quite difficult to construct a complete analytical solution. By means of the chaos theory, the evolution rule of the nonlinear dynamical system of the robot can be restored from the experimental data.
Robots have complex nonlinear dynamics. The nonlinear fluctuation of the joint moment is particularly obvious in low-speed movement. Based on the Newton Euler method, only the rigid body dynamic equation of the robot can be deduced, and the complex nonlinear moment at low speed cannot be described. Therefore, when the robot moves at a low speed, the difference between the calculation result of the obtained dynamic model and the actual joint moment is large. The invention aims to analyze and model-compensate the nonlinear dynamics characteristics of the low-speed motion of the robot by combining the chaos theory, curve fitting and a neural network method.
Disclosure of Invention
The invention provides a robot low-speed motion nonlinear dynamics characteristic analysis and modeling method, which aims to describe the nonlinear fluctuation characteristic of joint torque when a robot moves at low speed by a method of combining curve fitting and a neural network by means of the joint position, the joint speed and some calculation derivative quantities of the joint position and the joint speed. The invention analyzes the nonlinear fluctuation of the joint torque by introducing a chaos theory, determines the characteristic quantity related to the change rule of the joint torque, and restores the evolution rule of the nonlinear dynamic characteristic of the robot from experimental data by means of a phase space reconstruction theory. The method does not carry out independent modeling research on various factors causing the nonlinear fluctuation of the moment, but synthesizes curve fitting, a phase space reconstruction theory and a neural network method to carry out modeling on the nonlinear dynamics of the robot during low-speed motion.
The invention is realized by at least one of the following technical schemes.
A robot low-speed movement nonlinear dynamics characteristic analysis and modeling method mainly comprises a robot single joint rotation experiment, original data processing, nonlinear moment data analysis and robot low-speed movement nonlinear dynamics characteristic modeling, and respectively corresponds to the following steps:
step 3, nonlinear torque data analysis is carried out, the difference value of the motor wave dynamic torque and the motor wave deterministic torque and the difference value of the joint friction torque and the joint friction deterministic torque are analyzed based on a chaos theory, the nonlinear fluctuation characteristics of the two groups of torque difference values are obtained, and the two groups of torque difference values with the nonlinear fluctuation characteristics are defined as the motor wave chaotic torque and the joint friction chaotic torque;
and 4, analyzing characteristic quantities related to respective nonlinear fluctuation characteristics of the motor fluctuation chaotic moment and the joint friction chaotic moment, and modeling the nonlinear dynamic characteristics of the low-speed motion of the robot based on a phase space reconstruction theory and a neural network.
Preferably, the robot dynamics equation is:
in the formula: q, q,
Respectively representing the joint angle, the angular velocity and the angular acceleration of the robot; m (q) represents an inertia matrix;
representing a matrix of coriolis forces and centrifugal forces; g (q) represents a gravitational moment term;
a term representing joint friction torque; τ represents joint drive torque.
Preferably, the calculation formula of the motor fluctuation torque and the joint friction torque is as follows:
in the formula: tau. + Representing the positive rotation moment after resampling; tau is - Representing the negative rotation moment after resampling; g 0 Representing theoretical joint moment calculated by the gravity moment term expression; t is f0 And T r0 The joint friction torque and the motor fluctuation torque calculated from the forward and reverse rotation torque are indicated.
Preferably, the step 2 adopts a curve fitting method for fitting, specifically, a cubic polynomial is selected as a fitting line type, the joint position is taken as an independent variable, and the joint friction torque and the motor fluctuation torque at each expected speed are respectively fitted; obtaining a set of polynomials describing the variation trend of the joint friction torque, which are called joint friction deterministic torque; and simultaneously obtaining a group of polynomials for describing the variation trend of the motor wave dynamic torque, which are called motor wave deterministic torque.
Preferably, a Stribeck-viscous model is adopted to fit the constant term so as to represent the Stribeck characteristic of the joint friction torque:
in the formula:
constant terms corresponding to joint friction torque polynomials; f c Coulomb friction torque; f s Maximum static friction moment;
stribeck speed; alpha is Sribeck nonlinear index; f. of v Is the viscous friction coefficient.
Preferably, step 3 is based on a phase space reconstruction method in the chaos theory, the change rule of the nonlinear difference moment is recovered from the time series, for the series { x (k), k =1,2, \8230;, N }, x (k) represents a time series value at the time of k, N is the number of elements in the time series, and the reconstructed phase space is:
Y(t i )＝[x(t i ),x(t i -t d ),x(t i -2t d ),…,x(t i -(m-1)t d )]
in the formula, t i I =1,2, \ 8230for the time corresponding to the phase point of the reconstruction phase space, wherein M and M are the number of the phase points of the reconstruction phase space; y (t) i ) Phase points of the reconstructed phase space; t is t d Is time delay; m is an embedding dimension; x (t) i ) Is t i The original time series value of the time of day.
Preferably, the time delay t d According to the autocorrelation solution, the autocorrelation calculates a sequence { x (k), k =1,2, \8230;, N } with respect to the time span jt d The autocorrelation function of (a):
fixing the parameter j, selecting R xx Time t corresponding to the time when the initial value is reduced to (1-1/e) times d Time delay t as reconstruction phase space d (ii) a x (t) is an original time series value at the time t; x (t + jt) d ) Is t + jt d An original time series value of a time;
the embedding dimension m is solved according to a pseudo-neighbor method, which assumes each phase point Y (t) in an m-dimensional space i ) And nearest neighbor point Y NN (t i ) A distance of R m (t i ) The calculation method comprises the following steps:
R m (t i )＝||Y(t i )-Y NN (t i )||
when it becomes m +1 dimensional space, if R m+1 Far greater than R m If so, the two points are considered as the false nearest neighbors; specifically, Y is considered to be Y when the following inequality is satisfied NN (t i ) Is Y (t) i ) False neighbors of (2):
when the ratio of the embedding dimension m to the false neighbor points is increased to be less than 5% or the false neighbor points are not reduced along with the increase of m, the m at the moment is the embedding dimension of the reconstruction phase space.
Preferably, whether the time sequence is chaotic or not is judged according to a maximum Lyapunov exponential method, and each point Y (t) is calculated firstly i ) Nearest neighbor of (2)
And the distance is calculated as:
in the formula:
representing the reconstructed phase space time corresponding to the nearest neighbor point; p is the average period of the time sequence and is obtained by fast Fourier transform;
and then the distance of the two phase points after j discrete time steps is obtained:
in the formula:
for each discrete time step j, x (j) is solved according to the following formula:
in the formula: q is non-zero
The number of (2); Δ t is the sample period; x (j) is an inter-phase distance index corresponding to the discrete time step j;
obtaining a regression line of x (j) relative to j by a least square method, wherein the slope of the line is the maximum Lyapunov index, and when the maximum Lyapunov index is greater than 0, judging that chaos exists;
if the chaotic characteristics of the joint friction difference value moment and the motor fluctuation difference value moment at the low speed are judged to exist, the joint friction chaotic moment and the motor fluctuation chaotic moment are renamed, the change trend of the motor fluctuation chaotic moment is approximately in the form of a sum of 3 sine functions with the joint position as an independent variable and is recorded as T t (ii) a The coefficients of the sinusoidal function are then a function of joint velocity:
in the formula:
coefficients of a general formula of a sine function.
Preferably, a phase space reconstruction neural network is obtained by combining a phase control reconstruction theory and a neural network, joint friction chaotic torque characteristic quantity and motor fluctuation chaotic torque characteristic quantity are respectively used as input, joint friction chaotic torque and motor fluctuation chaotic torque are used as output, a random gradient descent method and a back propagation algorithm are adopted to carry out neural network training, and an approximate model of the joint friction chaotic torque and the motor fluctuation chaotic torque is obtained; and obtaining an approximate model of the low-speed motion nonlinear dynamics characteristic of the robot by combining the approximate expression of the joint friction deterministic torque and the motor fluctuation deterministic torque and the approximate models of the joint friction chaotic torque and the motor fluctuation chaotic torque.
Preferably, the characteristic quantities comprise shutdown friction chaotic torque and motor fluctuation chaotic torque characteristic quantities, and the shutdown friction chaotic torque characteristic quantities comprise joint angles q and joint side angle characteristic coefficients q f Characteristic coefficient of joint velocity
Characteristic of fluctuations for characterizing joint velocity, q f And
the calculation formula of (c) is:
in the formula: p is f The method is selected according to the generalized period characteristic expressed in shutdown friction chaotic torque data; r g Is a joint reduction ratio;
representing the average value of the joint speed corresponding to the group of experimental data;
the motor fluctuation chaotic moment characteristic quantity comprises a joint angle q and a joint side angle characteristic coefficient q r1 Characteristic coefficient q of motor side angle r2 And the variation trend value T of the motor fluctuation chaotic moment t ，q r1 And q is r2 The calculation formula of (c) is:
in the formula: p is r The parameters are selected according to the generalized period characteristics expressed in the motor fluctuation chaotic torque data; pi is the circumference ratio; mod is the remainder symbol.
Compared with the prior art, the invention has the beneficial effects that:
according to the robot low-speed movement nonlinear dynamics characteristic analysis and modeling method provided by the invention, the robot low-speed movement nonlinear dynamics characteristic is analyzed by means of a chaos theory based on joint torque data of a single joint rotation experiment, joint angle derivative quantity related to joint torque fluctuation characteristic is determined, and the overall evolution rule of the nonlinear torque is described through curve fitting, a phase space reconstruction method and a neural network.
The method has the advantages of small root mean square error when the nonlinear moment of the joint is estimated, good overall estimation performance, capability of better describing the nonlinear dynamics characteristics of the robot in low-speed motion and improvement of the precision of the robot dynamics model in low-speed motion occasions.
Drawings
FIG. 1 is a flow chart of a robot low-speed motion nonlinear dynamics characteristic analysis and modeling method according to an embodiment of the invention;
FIG. 2 illustrates a phase space reconstruction method according to an embodiment of the present invention;
FIG. 3 is a phase space reconstruction neural network for calculating joint friction chaotic moment according to an embodiment of the present invention;
fig. 4 is a phase space reconstruction neural network for calculating a motor fluctuation chaotic moment according to an embodiment of the present invention.
Detailed Description
For a better understanding of the present invention, reference is made to the following further description taken in conjunction with the accompanying drawings.
Example 1
The embodiment is a method for analyzing and modeling the low-speed motion nonlinear dynamics characteristics of a robot, as shown in fig. 1, and comprises the following steps:
1) Low-speed and uniform-speed rotation experiment of single joint of robot
And (3) performing dynamic modeling on the robot according to the Newton Euler method, completing dynamic parameter identification, and arranging the dynamic parameter identification into a general form shown in the formula (1).
In the formula: q, q,
Respectively representing the joint angle, the angular velocity and the angular acceleration of the robot; m (q) represents a 6 × 6 inertia matrix;
representing a 6 x 6 coriolis force and centrifugal force matrix; g (q) represents a 6 × 1 gravitational moment term;
represents a joint 6 x 1 friction torque term; τ represents joint drive torque.
When a certain joint rotates independently, a group of fixed position values of other static joints can be analyzed according to the symbolic expression of the gravity moment item, so that the influence of the gravity moment on the rotating joint is minimum; when part of the joints rotate independently, the gravity moment item is changed into 0, and the calculation expression of the gravity moment item of the rest joints is simplified.
And setting the motion range of the single-joint rotation experiment for each robot joint under the condition of being far away from the limit position and not interfering with the outside. The maximum rotation speed which can be reached by the single joint is measured, and the low-speed movement is defined as the maximum rotation speed which is less than 10 percent. The desired speed is set at 0.001rad/s between 0 and 0.010 rad/s. The desired speed is set to 0.002rad/s between 0.010 and 0.020 rad/s. The desired speed is set to 0.004rad/s between 0.020 and 0.060 rad/s. The desired speed is set at 0.008rad/s between 0.060 and 0.100 rad/s. The desired speed is set at 0.016rad/s between 0.100 and 0.180 rad/s. Thereafter the desired speed is set at a common multiple of 0.250rad/s up to the maximum rotational speed.
And (3) carrying out 3 back-and-forth rotation experiments on each joint in a set rotation range after one forward uniform rotation and one reverse uniform rotation according to each set expected speed. And (4) carrying out conversion processing and low-pass filtering on the experimental data of the three times of forward rotation, and then taking an average value to obtain the joint torque, the joint position (angle) and the joint speed of the forward motion at the speed. The joint torque is converted by motor current, the shutdown position is converted by a motor encoder value, and the joint speed is converted by a motor speed signal. Similarly, the three times of experiment data of negative rotation are converted, low-pass filtered and then averaged to obtain the joint torque, the joint position and the joint speed of negative movement at the speed.
2) Raw data processing
And performing low-pass filtering processing on the positive rotating torque and the negative rotating torque. The number of nodes of the joint position is redistributed in the motion range of each joint by taking 0.005 degrees as an angle interval, and the nodes are integrated into a set as Q r,i And i represents a joint number. And resampling the data such as the positive rotation moment, the negative rotation moment, the joint speed and the like by using a cubic interpolation resampling method, so that the lengths of the experimental data at different expected speeds are consistent.
Through qualitative analysis of experimental data, obvious motor fluctuation moment can be found in joint moment, so that formula (1) is modified,
in the formula:
representing the motor ripple torque, abbreviated as T r ；
Representing the friction torque of the joint, abbreviated as T f (ii) a The change rule of the joint is related to joint position and joint speed.
When the single joint rotates at low speed and even speed, the assumption is that
And
approximately 0, it can be considered that in the formula (2)
And
also approximately 0. The gravity moment item and the motor fluctuation moment item are only related to the joint position and are unrelated to the rotation direction; when the motor rotates in the forward direction and the reverse direction, the joint friction torque and the motor fluctuation torque which are obtained by calculating the experimental data when the motor rotates at a certain expected speed can be obtained.
In the formula: tau. + Representing the positive rotation moment after resampling; tau is - Representing the negative rotation moment after resampling; g 0 Expressing theoretical joint moment calculated by the gravity moment term expression in the formula (2); t is f0 And T r0 And the joint friction torque and the motor fluctuation torque calculated by the forward and reverse rotation torque are shown.
A set of experimental data calculated joint friction torque and motor ripple torque can be obtained corresponding to a set of set desired speeds. Selecting a cubic polynomial as a fitting linear type, and fitting joint friction torque and motor fluctuation torque at each expected speed by taking a joint position as an independent variable; obtaining a set of polynomials describing the variation trend of the joint friction torque, which are called joint friction deterministic torque; and simultaneously obtaining a group of polynomials for describing the variation trend of the motor wave dynamic torque, which are called motor wave deterministic torque.
From a set of joint friction deterministic moment polynomials, four corresponding sets of polynomial coefficients can be obtained. And taking the expected speed as an independent variable, selecting a proper line type to fit each group of polynomial coefficients, and obtaining an approximate expression of each coefficient relative to the speed.
Particularly, the constant term is fitted by the following Stribeck-viscous model to represent the Stribeck characteristic of the joint friction torque.
In the formula:
constant terms corresponding to joint friction torque polynomials; f c Coulomb friction torque; f s Maximum static friction moment;
stribeck speed; alpha is Sribeck nonlinear index and is set to be 2; f. of v Is the viscous friction coefficient; the coefficients other than α are obtained by a fitting method.
Similarly, with the fitting method, approximate expressions of the coefficients of each polynomial in the motor fluctuation deterministic moment polynomial with respect to the speed can be obtained.
The fitting function with the angle as the independent variable and the fitting function with the speed as the independent variable are combined, and finally an approximate expression of the joint friction deterministic torque and an approximate expression of the motor fluctuation deterministic torque can be obtained. Both approximation expressions are cubic polynomials with respect to joint position, while the polynomial coefficients are functions with respect to joint velocity.
3) Nonlinear moment data analysis
And subtracting the joint friction deterministic torque from the joint friction torque to obtain a group of nonlinear joint friction difference value torques. And subtracting the motor fluctuation deterministic torque from the motor fluctuation torque to obtain a group of nonlinear motor fluctuation differential value torques.
By means of a phase space reconstruction method in the chaos theory, the change rule of the nonlinear difference value moment can be recovered from the time sequence. For the sequence { x (k), k =1,2, \ 8230;, N }, x (k) represents the time-series value at time k, N is the number of elements in the time series, and the reconstructed phase space is:
Y(t i )＝[x(t i ),x(t i -t d ),x(t i -2t d ),…,x(t i -(m-1)t d )] (5)
in the formula, t i I =1,2, \ 8230for the time corresponding to the phase point of the reconstruction phase space, wherein M and M are the number of the phase points of the reconstruction phase space; y (t) i ) Phase points of the reconstructed phase space; t is t d Is time delay; m is an embedding dimension; x (t) i ) Is t i The original time series values of the time of day.
Fig. 2 shows a diagram of the original time series versus the reconstructed phase space.
Time delay t d Solving according to an autocorrelation method, and solving the embedding dimension m according to a false neighbor method. The autocorrelation calculation sequence { x (k), k =1,2, \8230;, N } is related to the time span jt d The autocorrelation function of (a):
fixing the parameter j, selecting R xx Time t corresponding to the time when the initial value is reduced to (1-1/e) times d Time delay t as reconstruction phase space d 。
The pseudo-neighbor method assumes each phase point Y (t) in the m-dimensional space i ) And nearest neighbor point Y NN (t i ) A distance of R m (t i ) The calculation method comprises the following steps:
R m (t i )＝||Y(t i )-Y NN (t i )|| (7)
when it becomes m +1 dimensional space, if R m+1 Far greater than R m Then the two points are considered to be false nearest neighbors.
Specifically, Y can be considered to be when the following inequality is satisfied NN (t i ) Is Y (t) i ) False neighbors of (2).
When the ratio of the embedding dimension m to the false neighbor points is increased to be less than 5% or the false neighbor points are not reduced along with the increase of m, the m at the moment is the embedding dimension of the reconstruction phase space.
Further, according to the mostThe large Lyapunov exponent method can judge whether the time sequence is chaotic or not. First, each point Y (t) is calculated i ) Nearest neighbor of (2)
And the distance is calculated as:
in the formula:
representing the reconstructed phase space time corresponding to the nearest neighbor point; p is the average period of the time series and can be obtained by fast Fourier transform.
And then the distance of the two phase points after j discrete time steps is obtained:
in the formula:
for each discrete time step j, x (j) is solved according to the following formula:
in the formula: q is nonzero
The number of (2); Δ t is the sample period; and x (j) is an inter-phase distance index corresponding to the discrete time step j.
And obtaining a regression line of x (j) relative to j by a least square method, wherein the slope of the line is the maximum Lyapunov exponent. When the maximum Lyapunov exponent is greater than 0, the existence of chaos can be judged.
By this method, it is possible to determine the time of low speedChaotic characteristics exist in the joint friction difference value moment and the motor fluctuation difference value moment. Therefore, they are renamed as joint friction chaotic torque and motor fluctuation chaotic torque. Particularly, the variation trend of the motor fluctuation chaotic moment can be approximated to a form of a sum of 3 sine functions taking the joint position as an independent variable, and is marked as T t . The coefficients of the sinusoidal function are then a function of joint velocity:
in the formula:
coefficients of a general formula of a sine function.
4) Robot low-speed motion nonlinear dynamics characteristic modeling
Through further analysis of the shutdown friction chaotic torque, 3 characteristic quantities related to the fluctuation characteristic of the shutdown friction chaotic torque are determined, and are respectively as follows: joint angle q, joint side angle characteristic coefficient q f Characteristic coefficient of joint velocity
Characteristic of fluctuations for characterizing joint velocity, q f And
calculated from the following formula,
in the formula: p f The method is selected according to the generalized period characteristic expressed in shutdown friction chaotic torque data; r g Is a joint reduction ratio;
the mean joint velocity values for this set of experimental data are shown.
By fluctuating chaotic force to the motorFurther analysis of the moments determines 4 characteristic quantities related to their fluctuation characteristics, respectively: joint angle q, joint side angle characteristic coefficient q r1 Motor side angle characteristic coefficient q r2 And the variation trend value T of the motor fluctuation chaotic moment t 。q r1 And q is r2 Calculated from the following formula,
in the formula: r is g Is a joint reduction ratio; p r The method comprises the following steps of selecting parameters according to the generalized period characteristics expressed in motor fluctuation chaotic torque data; pi is the circumference ratio; mod is a remainder symbol, meaning the value of the left variable divided by the fractional part of the right variable.
Setting the hidden layer node as 10, and performing phase space reconstruction on each characteristic quantity to be used as an input quantity to obtain a corresponding chaotic torque T for calculating joint friction fc Phase space reconstruction neural network model and method for calculating motor fluctuation chaotic moment T rc The neural network model is reconstructed from the phase space as shown in fig. 3 and 4. The inputs of the neural network in FIG. 3 are feature quantities q, q of the shutdown frictional chaotic torque respectively calculated according to equation (5) f And
the reconstructed phase space value is output as joint friction chaotic torque T fc . The input of the neural network in FIG. 4 is the characteristic quantities q, q of the motor fluctuation chaotic moment respectively calculated according to the formula (5) r1 、q r2 And T t Reconstructing a phase space value and outputting the reconstructed phase space value as a motor fluctuation chaotic torque T rc . The neural network training adopts a random gradient descent method and a back propagation algorithm.
And obtaining an approximate model of the nonlinear dynamic characteristic of the robot low-speed motion by combining the approximate expressions of the joint friction deterministic torque and the motor fluctuation deterministic torque and the approximate models of the joint friction chaotic torque and the motor fluctuation chaotic torque.
Example 2
The embodiment is a robot low-speed motion nonlinear dynamics characteristic analysis and modeling method, and compared with embodiment 1, the difference is that a LuGre model is adopted to fit a constant term of a joint friction deterministic moment polynomial.
Example 3
The embodiment is a robot low-speed motion nonlinear dynamics characteristic analysis and modeling method, and compared with embodiment 1, the difference is that the change trend value T of motor fluctuation chaotic torque is used t Approximately in the form of a fourier series.
Example 4
Compared with the embodiment 1, the method is different in that the RBF radial basis function neural network is used for modeling the joint friction chaotic torque and the motor fluctuation chaotic torque.
Example 5
The embodiment is a robot low-speed motion nonlinear dynamics characteristic analysis and modeling method, and compared with the embodiment 1, the difference is that the time delay t is d And solving according to a mutual information method, solving the embedded dimension m according to a Cao method, and calculating the maximum Lyapunov index through a Wolf method.
The preferred embodiments of the invention disclosed above are intended to be illustrative only. The preferred embodiments are not intended to be exhaustive or to limit the invention to the precise embodiments disclosed. Obviously, many modifications and variations are possible in light of the above teaching. The embodiments were chosen and described in order to best explain the principles of the invention and the practical application, to thereby enable others skilled in the art to best utilize the invention. The invention is limited only by the claims and their full scope and equivalents.
Claims (9)
1. A robot low-speed movement nonlinear dynamics characteristic analysis and modeling method is characterized by mainly comprising a robot single joint rotation experiment, original data processing, nonlinear moment data analysis and robot low-speed movement nonlinear dynamics characteristic modeling, which respectively correspond to the following steps:
step 1, setting an expected speed, setting the position of a static joint according to a robot kinetic equation, performing a robot single-joint low-speed and uniform-speed rotation experiment, and acquiring joint motor torque, joint angle and joint speed data;
step 2, processing original data, performing cubic interpolation value resampling after low-pass filtering of the torque data, calculating motor fluctuation moment and joint friction moment according to the resampled torque data, and obtaining motor fluctuation certainty moment and joint friction certainty moment through fitting; the calculation formula of the motor fluctuation torque and the joint friction torque is as follows:
in the formula: tau. + Representing the positive rotation moment after resampling; tau. - Representing the negative rotation moment after resampling; g 0 Representing theoretical joint moment calculated by the gravity moment term expression; t is f0 And T r0 Representing the joint friction torque and the motor fluctuation torque calculated by the forward and reverse rotation torque;
step 3, nonlinear torque data analysis is carried out, the difference value of the motor wave moment and the motor wave deterministic moment and the difference value of the joint friction moment and the joint friction deterministic moment are analyzed based on a chaos theory, the nonlinear fluctuation characteristics of the two groups of torque difference values are obtained, and the two groups of torque difference values with the nonlinear fluctuation characteristics are defined as the motor wave chaotic moment and the joint friction chaotic moment;
and 4, analyzing characteristic quantities related to respective nonlinear fluctuation characteristics of the motor fluctuation chaotic moment and the joint friction chaotic moment, and modeling the nonlinear dynamic characteristics of the low-speed motion of the robot based on a phase space reconstruction theory and a neural network.
2. The method for analyzing and modeling the low-speed motion nonlinear dynamics of the robot according to claim 1, wherein the robot dynamics equation is as follows:
in the formula:
respectively representing the joint angle, the angular velocity and the angular acceleration of the robot; m (q) represents an inertia matrix;
representing a matrix of coriolis forces and centrifugal forces; g (q) represents a gravitational moment term;
a term representing joint friction torque; τ represents joint drive torque.
3. The analysis and modeling method for the low-speed motion nonlinear dynamics characteristics of the robot as claimed in claim 2, characterized in that step 2 adopts a curve fitting method for fitting, specifically, a cubic polynomial is selected as a fitting line type, and joint positions are used as independent variables to respectively fit joint friction torque and motor ripple torque at each expected speed; obtaining a set of polynomials describing the variation trend of the joint friction torque, which are called joint friction deterministic torque; and simultaneously obtaining a group of polynomials describing the variation trend of the motor wave moment, namely the motor wave deterministic moment.
4. The method for analyzing and modeling the nonlinear dynamics of low-speed motion of the robot as claimed in claim 3, wherein a Stribeck-viscous model is adopted to fit a constant term to characterize the Stribeck characteristic of the joint friction torque:
in the formula:
constant terms corresponding to joint friction torque polynomials; f c Coulomb friction torque; f s Maximum static friction moment;
stribeck speed; alpha is Sribeck nonlinear index; f. of v Is the viscous friction coefficient.
5. The method for analyzing and modeling the nonlinear dynamics characteristics of the low-speed motion of the robot according to claim 4, wherein the step 3 is based on a phase space reconstruction method in the chaos theory, the change rule of the nonlinear difference moment is recovered from the time series, for the sequence { x (k), k =1,2, \8230;, N }, x (k) represents the time series value at the moment k, N is the number of elements in the time series, and the reconstructed phase space is:
Y(t i )＝[x(t i ),x(t i -t d ),x(t i -2t d ),...,x(t i -(m-1)t d )]
in the formula, t i I =1,2, \ 8230for the time corresponding to the phase point of the reconstruction phase space, wherein M and M are the number of the phase points of the reconstruction phase space; y (t) i ) Phase points of the reconstructed phase space; t is t d Is time delay; m is an embedding dimension; x (t) i ) Is t i The original time series value of the time of day.
6. The method as claimed in claim 5, wherein the time delay t is a time delay t d According to the autocorrelation solution, the autocorrelation calculates a sequence { x (k), k =1,2, \8230;, N } with respect to the time span jt d The autocorrelation function of (a):
fixing the parameter j, selecting R xx Time t corresponding to the time when the initial value is reduced to (1-1/e) times d Time delay t as reconstruction phase space d (ii) a x (t) is an original time sequence value at the time t; x (t + jt) d ) Is t + jt d An original time series value of a time;
the embedding dimension m is solved according to a pseudo-neighbor method, which assumes each phase point Y (t) in an m-dimensional space i ) And nearest neighbor point Y NN (t i ) A distance of R m (t i ) The calculation method comprises the following steps:
R m (t i )＝||Y(t i )-Y NN (t i )||
when it becomes m +1 dimensional space, if R m+1 Far greater than R m If the two points are false nearest neighbors, the two points are considered as false nearest neighbors; specifically, Y is considered to be Y when the following inequality is satisfied NN (t i ) Is Y (t) i ) False neighbors of (2):
when the ratio of the embedding dimension m to the false neighbor points is increased to be less than 5% or the false neighbor points are not reduced along with the increase of m, the m at the moment is the embedding dimension of the reconstruction phase space.
7. The robot low-speed motion nonlinear dynamics characteristic analysis and modeling method according to claim 6, characterized in that whether the time sequence is chaotic is judged according to the maximum Lyapunov exponent method, and each point Y (t) is calculated first i ) Nearest neighbors of
And the distance is calculated as:
in the formula:
representing the reconstructed phase space time corresponding to the nearest neighbor point; p is the average period of the time sequence and is obtained by fast Fourier transform;
and then the distance of the two phase points after j discrete time steps is obtained:
in the formula:
for each discrete time step j, x (j) is solved according to the following formula:
in the formula: q is nonzero
The number of (2); Δ t is the sample period; x (j) is an inter-phase point distance index corresponding to the discrete time step j;
obtaining a regression line of x (j) relative to j by a least square method, wherein the slope of the line is the maximum Lyapunov index, and when the maximum Lyapunov index is greater than 0, judging that chaos exists;
if the joint friction difference value moment and the motor fluctuation difference value moment at low speed are judged to have chaotic characteristics, the joint friction difference value moment and the motor fluctuation difference value moment are renamed to be joint friction chaotic moment and motor fluctuation chaotic moment, the change trend of the motor fluctuation chaotic moment is approximate to the form of the sum of 3 sine functions taking the joint position as an independent variable and is recorded as T t (ii) a The coefficients of the sinusoidal function are then a function of joint velocity:
in the formula:
coefficients of a general formula of a sine function.
8. The method for analyzing and modeling the low-speed movement nonlinear dynamics characteristics of the robot according to claim 7, characterized in that a phase space reconstruction neural network is obtained by combining a phase space reconstruction theory and a neural network, joint friction chaotic torque characteristic quantity and motor fluctuation chaotic torque characteristic quantity are respectively used as input, joint friction chaotic torque and motor fluctuation chaotic torque are used as output, a random gradient descent method and a back propagation algorithm are adopted for neural network training, and an approximate model of the joint friction chaotic torque and the motor fluctuation chaotic torque is obtained; and obtaining an approximate model of the low-speed motion nonlinear dynamics characteristic of the robot by combining the approximate expression of the joint friction deterministic torque and the motor fluctuation deterministic torque and the approximate models of the joint friction chaotic torque and the motor fluctuation chaotic torque.
9. The robot low-speed motion nonlinear dynamics characteristic analysis and modeling method according to claim 8, wherein the characteristic quantities comprise shutdown friction chaotic torque and motor fluctuation chaotic torque characteristic quantities, and the shutdown friction chaotic torque characteristic quantities comprise joint angles q and joint side angle characteristic coefficients q f Characteristic coefficient of joint velocity
Characteristic of fluctuations for characterizing joint velocity, q f And
the calculation formula of (2) is as follows:
in the formula: p f The method is selected according to the generalized period characteristics expressed in shutdown friction chaotic torque data; r g Is a joint reduction ratio;
representing the average value of the joint speed corresponding to the group of experimental data;
the motor fluctuation chaotic moment characteristic quantity comprises a joint angle q and a joint side angle characteristic coefficient q r1 Characteristic coefficient q of motor side angle r2 And the variation trend value T of the motor fluctuation chaotic moment t ，q r1 And q is r2 The calculation formula of (2) is as follows:
in the formula: p r The parameters are selected according to the generalized period characteristics expressed in the motor fluctuation chaotic torque data; pi is the circumference ratio; mod is the remainder symbol.
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