KR20170008486A - Parameter identification for robots with a fast and robust trajectory design approach - Google Patents

Parameter identification for robots with a fast and robust trajectory design approach Download PDF

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KR20170008486A
KR20170008486A KR1020150099727A KR20150099727A KR20170008486A KR 20170008486 A KR20170008486 A KR 20170008486A KR 1020150099727 A KR1020150099727 A KR 1020150099727A KR 20150099727 A KR20150099727 A KR 20150099727A KR 20170008486 A KR20170008486 A KR 20170008486A
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robot
parameter
estimation
trajectory
dynamic
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김윤구
이동하
김경복
갠스 니콜라스
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재단법인대구경북과학기술원
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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
    • B25J9/16—Programme controls
    • B25J9/1628—Programme controls characterised by the control loop
    • B25J9/1653—Programme controls characterised by the control loop parameters identification, estimation, stiffness, accuracy, error analysis
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1602—Programme controls characterised by the control system, structure, architecture
    • B25J9/1605—Simulation of manipulator lay-out, design, modelling of manipulator
    • B—PERFORMING OPERATIONS; TRANSPORTING
    • B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
    • B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
    • B25J9/00—Programme-controlled manipulators
    • B25J9/16—Programme controls
    • B25J9/1679—Programme controls characterised by the tasks executed
    • B25J9/1692—Calibration of manipulator

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  • Robotics (AREA)
  • Mechanical Engineering (AREA)
  • Automation & Control Theory (AREA)
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Abstract

Disclosed is a robots parameter estimation method through trajectory design, wherein the method comprises: (S110) a step of collecting position data or torque data of the robot; (S120) a signal processing step including reducing noise of the collected data to enhance accuracy of the collected data; (S130) a step of performing kinetic estimation modeling of the robot; and (S140) a step of optimizing a trajectory used for kinetic parameter estimation of the robot using a minimum square parameter estimator to the result acquired from the dynamics estimation modelling. Accordingly, the present invention provides a new optimal standard which is effective and intuitive in calculation to design an excitation trajectory a robot follows.

Description

[0001] PARAMETER IDENTIFICATION FOR ROBOTS WITH A FAST AND ROBUST TRAJECTORY DESIGN APPROACH [0002]

Model-based torque-level robot control can guarantee the advantages of higher accuracy and speed than speed-level or position-level robot control, but the kinetic parameters of the robot must be accurately estimated. The kinematic parameter estimation includes steps of kinetic modeling of the system, robot joint / torque data acquisition and filtering, experimental design, estimation and evaluation of kinetic parameters.

Robotic arm applications such as modern advanced manufacturing and multi-robot system control require high precision and speed. These applications typically require a model-based control algorithm or a torque-input-based control algorithm. Such a control system scheme requires accurate information of the kinetic parameters of the robot arm. Experimental estimation or calibration is therefore a reliable approach to obtaining such information.

Many models of robot dynamics have been proposed in the context of dynamics parameter estimation. Gautier proposed an energy estimation model and a power model. There have also been studies using an inverse kinematic model of a robotic arm to estimate kinetic parameters. An inverse dynamics model provides more information than an energy or power model. This additional information can produce well-conditioned and over-determined regression matrices.

Other methods of estimating the kinetic parameters include the Least Squares Estimation Method (LSE) and the Maximum Likelihood Estimation Method (MLE). Other approaches include an extended Kalman filter and an approach that utilizes a total least squares method, an online recursive total least squares method, a weighted least squares method, nonlinear least squares optimization, and tool parameters. In general, robot joint angle and torque / current data can be measured directly, but the speed and acceleration of the robot joint must be estimated. Observer / Estimators, zero-phase bandpass filters, low-pass filters, Kalman filters, and so on can be used to estimate velocity and acceleration. Filter (KF)).

The design of the trajectory is an essential and important part of improving the estimation accuracy. The trajectory of the 5th order polynomial in the joint space is proposed. To enable iterative estimation experiments and to improve the signal-to-noise ratio (SNR), periodic excitation trajectories based on finite sum of Fourier series, modified Fourier series and harmonic sine functions are proposed, respectively. Two optimal conditions for finding optimal periodic trajectories are widely used. One is to minimize the number of conditions in the regression matrix, and the other is to minimize the log {det (o)} of the Fisher information matrix. Since each Fourier series contains 2 * N i +2 parameters, solving the optimization problem can be difficult. In addition, each Fourier series must satisfy the constraints of the trajectory, such as the initial and final conditions and range of position, velocity, and acceleration. Model verification is another important procedure for confirming parameter estimation results.

Korean Patent Publication No. 10-2010-0105143, filed on September 29, 2010, discloses a method and system for estimating a robot kinematic parameter using a Kalman filter.

The present invention is intended to provide a computationally efficient and intuitive new optimum criterion for designing the excitation trajectory followed by the robot.

A method for estimating a parameter of a robot through a trajectory design, comprising: acquiring (S110) position data or torque data of a robot according to an embodiment; reducing noise of the collected data to improve accuracy of the collected data; (S130) of performing a kinetic estimation modeling of the robot (S130), optimizing a locus used for estimating a kinetic parameter of the robot using a least squares parameter estimator for the result obtained from the kinetic estimation modeling S140).

In the signal processing step (S120), the positions are calculated by a zero-phase low-pass filter, the velocity is calculated by a center difference method, the acceleration is calculated by a center difference method and is performed by a robust LOcal polynomial regression (RLOESS) smoother And the torque can be removed by a smoothing process performed by the Robust LOcal polynomial regression (RLOESS) smoother.

In a step (S140) of optimizing a locus used for estimating a dynamic parameter of the robot using a least squares parameter estimator for the result obtained from the dynamic estimation modeling, reducing the number of conditions defining the locus, By applying an inequality, we can make the determinant of the guard arc matrix equal to or smaller than the product of its diagonal elements.

In the step (S140) of optimizing the trajectory used for estimating the dynamic parameter of the robot using the least squares parameter estimator for the result obtained from the dynamic estimation modeling, the excitation trajectory q * (t) same.

Figure pat00001

Here, the conditional expression is as follows.

Figure pat00002

remind

Figure pat00003
Is an objective function sequentially determined, and the conditional expression
Figure pat00004
Wow
Figure pat00005
Can represent the position, velocity, and acceleration limits of the joint, respectively.

The objective function

Figure pat00006
Is as follows.

Figure pat00007

 The observation matrix W composed of the position of the joint, the estimated velocity and the estimated acceleration samples measured by the Hadamard inequality and measured by the parameter estimation method is as follows.

Figure pat00008

From the observation matrix W, the following equation can be obtained.

Figure pat00009
Figure pat00010

Wherein W kg is the g th column, k th element of the regression matrix W, and the sum of W 2 kg is defined as W s g ,

Figure pat00011
From
Figure pat00012
To maximize
Figure pat00013
Can be maximized.

In a step (S140) of optimizing a locus used for estimating a dynamic parameter of the robot using a least squares parameter estimator for the result obtained from the dynamic estimation modeling, the least squares parameter estimator performs a kinetic estimation modeling (S130) To solve the obtained over-determined metrics, the matrix notation is as follows.

Figure pat00014

 Here, the estimated basic parameters are as follows,

Figure pat00015
Wow
Figure pat00016

Figure pat00017
The estimated error covariance matrix < RTI ID = 0.0 >
Figure pat00018
Lt;
Figure pat00019
Is an error injection,
Figure pat00020
Is the following equation.

Figure pat00021

 The covariance matrix of the estimation error is given by the following equation,

Figure pat00022

Figure pat00023
Of the j < th >
Figure pat00024
The relative standard deviation (RSD)

Figure pat00025

The present invention provides a computationally efficient and intuitive new optimal criterion for designing the excitation trajectory the robot follows.

1 is a flowchart of a method of estimating parameters of an industrial robot using fast and robust locus design.
2 is a path of a typical joint location of a robot estimated according to one embodiment.
3 is a path of a typical joint velocity of a robot estimated according to one embodiment.
4 is a path of a typical joint acceleration of a robot estimated according to one embodiment.

Hereinafter, embodiments will be described in detail with reference to the accompanying drawings. The following description is one of many aspects of the embodiments and the following description forms part of a detailed description of the embodiments.

In the following description, well-known functions or constructions are not described in detail to avoid unnecessarily obscuring the subject matter of the present invention.

1 is a flowchart of a method of estimating parameters of an industrial robot using fast and robust locus design.

The present invention proposes a new, compact and intuitive optimal criterion for designing the trajectory of the excitation. The proposed approach reduces the number of conditions that define the trajectory and simplifies the optimization problem by applying Hadamard inequality. Single

Figure pat00026
For the square matrix W, the complexity of the upper bound, which computes the determinant using Hadamard inequality,
Figure pat00027
to be. But
Figure pat00028
The complexity of calculating the determinant of
Figure pat00029
, And the complexity of computing the condition number of W is
Figure pat00030
to be. The use of Hadamard inequality greatly reduces complexity and computation time for finding optimal parameters.

The present invention compares the results with two well-known optimization functions in terms of computational complexity. The trajectory proposed in the present invention not only performs well as the trajectory found from the existing optimization function on the basis of the RMSE (root mean square error), but also can generate the trajectory with a calculation time which is 10 times smaller than the proposed trajectory.

In the present invention, an inverse dynamic model and a least squares (LS) estimation method are applied to estimate inertial parameters of a robot arm. We also use a zero-phase low-pass filter to process the position data and calculate the velocity with a central difference algorithm. The acceleration is calculated by the center difference method and smoothed by the Robust LOcal polynomial regression (RLOESS) smoother. RLOESS has gained wide acceptance in statistics as a very good solution for fitting noise data to smooth curves.

Referring to FIG. 1, in step S110 of collecting position data or torque data of a robot, data of a position and a torque of the robot are collected. The data collected in the above process may have some influence of errors or noise.

In order to improve the accuracy of the collected data, a signal processing step (S120) of reducing noise of the collected data includes:

Figure pat00031
Wow
Figure pat00032
Which is an essential step to improve the accuracy of the parameter estimation result. Positions are calculated by forward and reverse IIR Butterworth filters. The velocity is calculated by the center difference method. The acceleration is calculated by a central difference method and smoothed by RLOESS implemented using the MATLAB smooth function. RLOESS is a regression method that uses a moving average filter and performs residual analysis to remove outliers before smoothing. It also uses RLOESS to eliminate noise or torque ripple in the collected torque data. In order to remove samples without information, downsampling using a decimate filter
Figure pat00033
Wow
Figure pat00034
.

In step S130, the dynamic model of the rigid robot of the n-link is calculated by Euler-Lagrange or Newton-Euler in the step of performing the dynamics estimation modeling of the robot based on the collected data. Can be derived using the formula. The mathematical model in the robot joint space is as follows.

Figure pat00035

here,

Figure pat00036
Is a joint position vector,
Figure pat00037
and
Figure pat00038
Are the joint velocity vector and the acceleration vector, respectively.
Figure pat00039
Is the mass or inertia matrix of the robot.
Figure pat00040
Include Coriolis, centrifugal force and gravity conditions.
Figure pat00041
Is a frictional force,
Figure pat00042
Represents the joint torque vector which is the input of the system.

The frictional force is modeled as follows.

Figure pat00043

here,

Figure pat00044
and
Figure pat00045
Is a constant representing the viscosity and coulomb friction parameters, respectively
Figure pat00046
Diagonal matrix.

The modified DH model (Denavit-Hartenberg model (MDH)) rules

Figure pat00047
In a linear parameterization format with standard parameters, a mathematical model
Figure pat00048
If you write again, it is as follows.

Figure pat00049

here,

Figure pat00050
Is a regressor matrix,
Figure pat00051
Is a standard parameter vector. Rigid robots have 13 standard parameters for each link and joint. This means that at the origin of frame j the six inertia matrix elements of link j
Figure pat00052
, The first moment of link j
Figure pat00053
, Mass of link j
Figure pat00054
, The total inertial moment for the rotor and gear of the actuator j
Figure pat00055
, And viscosity and coulomb friction coefficient
Figure pat00056
to be.

The basic parameters are the minimum set of identifiable parameters for parameterizing the kinematic equations. The dynamic equation with Nb identifiable basic parameters can be expressed as:

Figure pat00057

here,

Figure pat00058
Is a base parameter,
Figure pat00059
The
Figure pat00060
Is a subset of the independent columns of.

Trajectories with a single reference must be used to operate the given system continuously. The present invention uses a periodic trajectory. Joint position and motor torque

Figure pat00061
Is assumed to be measured at a sampling frequency of < RTI ID = 0.0 >
Figure pat00062
. If the fundamental frequency of the trajectory is
Figure pat00063
, We have one cycle
Figure pat00064
During
Figure pat00065
Can be collected. These measurements were performed using the equation
Figure pat00066
Can be used to obtain an over-determined metric of. Here, the observation matrix is as follows.

Figure pat00067

here,

Figure pat00068
and
Figure pat00069
Is a vector of errors due to complex friction, modeling errors, measurement noise, and the like. Therefore, the measurement of torque / force will show a difference from the actual motor torque. The dimension of the observation matrix W depends on the number of samples collected,
Figure pat00070
to be.

In the step (S140) of optimizing the locus used for the dynamic parameter estimation of the robot using the least squares parameter estimator to the result obtained from the dynamic estimation modeling,

Figure pat00071
We use the LS predictor to solve the decision matrix of. The matrix notation is as follows.

Figure pat00072

here,

Figure pat00073
Wow
Figure pat00074
Are the estimated basic parameters.
Figure pat00075
The estimated error covariance matrix < RTI ID = 0.0 >
Figure pat00076
, And
Figure pat00077
Is the variance of the error.
Figure pat00078
Is a generally unknown value,
Figure pat00079
Is estimated by the following equation.

Figure pat00080

The covariance matrix of the estimation error is given by the following equation.

Figure pat00081

Figure pat00082
The
Figure pat00083
Of the jth element,
Figure pat00084
The relative standard deviation (RSD)

Figure pat00085

In addition to LS, dynamic parameters were calculated using the Weighted Least Squares (WLS) and the Least Squares (TLS). In this case, however, the identification results were not significantly improved over LS. As a result, LS is adopted because LS is more compact.

The present invention proposes a new and modified Fourier series that can generate a persistent excitation trajectory that greatly reduces complexity and computation time required for trajectory parameter optimization.

With respect to trajectory parameterization, the trajectory for each joint is the finite sum of the N harmonic sine and cosine functions. joint position of the n-link robot with respect to the i-th joint

Figure pat00086
, speed
Figure pat00087
, And acceleration
Figure pat00088
The trajectories are as follows.

Figure pat00089

Figure pat00090

Figure pat00091

here,

Figure pat00092
Is the fundamental frequency,
Figure pat00093
Is the joint position offset of the reference trajectories. All joints share the same fundamental frequency to ensure the periodicity of the trajectory. On the other hand, each trajectory has only one reference trajectory
Figure pat00094
≪ / RTI > parameter
Figure pat00095
Wow
Figure pat00096
Can determine the amplitude of the cosine and sine functions, and can be determined through optimization and trial and error. The trade-off for determining the fundamental frequency was discussed in Swevers' study.

With respect to trajectory optimization,

Figure pat00097
Can be expressed as the following equation.

Figure pat00098

The conditional equation is as follows.

Figure pat00099

here,

Figure pat00100
Is an objective function that is determined sequentially
Figure pat00101
Wow
Figure pat00102
Indicates the limit of joint position, velocity, and acceleration, respectively. if
Figure pat00103
,
Figure pat00104
, It will cause unexpected behavior at the start and end points. That is,
Figure pat00105
And equation
Figure pat00106
Are added to solve this drawback.

Figure pat00107
(One)

Figure pat00108
(2)

Substituting equation (1) into equation (2)

Figure pat00109
Figure pat00110
(3)

(3). Likewise,

Figure pat00111
Figure pat00112
(4)

Figure pat00113
Figure pat00114
(5)

(4) and (5), and furthermore,

Figure pat00115
(6)

Figure pat00116
(7)

Figure pat00117
(8)

The constraints of Eqs. (6), (7), and (8)

Figure pat00118
(9)

Figure pat00119
(10)

Figure pat00120
(11)

Can be rewritten as Eqs. (9), (10), (11). In particular, the locus of excursion is the observation matrix W or

Figure pat00121
Which is optimized by minimizing the number of conditions.

With respect to the proposed objective function using Hadamard's inequality, according to Hadamard inequality, the determinant of a constant-order matrix is equal to or smaller than the product of its diagonal elements.

Figure pat00122
The complexity of computing the upper bound of the determinant using the Hadamard inequality for the square matrix W
Figure pat00123
as,
Figure pat00124
The complexity of calculating the determinant of
Figure pat00125
And the complexity of computing the condition number of W is
Figure pat00126
to be. More than one thousand samples are collected to calculate kinetic parameters. In this case, the size of the observation matrix W is 11250 x 52. Obviously, Hadamard inequality reduces complexity and computation time in finding optimal parameters. Hadamard inequality can be applied to obtain the following equation.

Figure pat00127
Figure pat00128

here,

Figure pat00129
Is the gth column, kth element of the regression matrix W.
Figure pat00130
Sum
Figure pat00131
And the above equation is rearranged as shown in the following equation.

Figure pat00132

therefore,

Figure pat00133
To maximize
Figure pat00134
Will ideally maximize the upper bound and will make the determinant larger. Experimental results show that the proposed method works as well as other methods in terms of the root-mean-square (RMS) error of torque prediction and greatly reduces the computation time and complexity required to find the optimal parameters.

 Therefore, the objective function is selected as follows.

Figure pat00135

At this time, the condition equation is as follows.

Figure pat00136

In the experiment, the persistent excitation trajectory

Figure pat00137
and
Figure pat00138
Condition. The optimization problem is a predetermined offset
Figure pat00139
And can be resolved in any suitable way, such as the family fmincon of the MATLAB optimization toolbox. The physical limits of the joint position, velocity, and acceleration of the Staubli TX-90 robot are shown in Table 1.

Figure pat00140

The initial conditions used for optimization are randomly generated on each link. Then, a referenceexcitation trajectory with optimal parameters can be generated. Examples of typical joint positions, velocities, and acceleration paths are shown in Figs. 2-4. In Figures 3 and 4, the start and end points of the reference speed and acceleration are close to zero or nearly zero. This result satisfies the following constraint.

Figure pat00141

Figure pat00142

 Although the present invention has been described in connection with what is presently considered to be the most practical and preferred embodiment, it is to be understood that the invention is not limited to the disclosed embodiments. The present invention is not limited to the above-described embodiments, and various modifications and changes may be made thereto by those skilled in the art to which the present invention belongs. Therefore, the spirit of the present invention should not be construed as being limited to the above-described embodiments, and all of the equivalents or equivalents of the claims, as well as the following claims, are included in the scope of the present invention.

Claims (6)

A method for estimating parameters of a robot through trajectory design,
Collecting position data or torque data of the robot (S110);
A signal processing step (S120) of reducing noise of the collected data to improve the accuracy of the collected data;
Performing dynamic modeling of the robot (S130);
(S140) optimizing a trajectory used for estimating a dynamic parameter of the robot using a least squares parameter estimator for the result obtained from the dynamic estimation modeling;
/ RTI > The method according to claim 1,
In the signal processing step (S120), the positions are calculated by a zero-phase low-pass filter, the velocity is calculated by a center difference method, the acceleration is calculated by a center difference method and is performed by a robust LOcal polynomial regression (RLOESS) smoother Wherein the noise is removed by a smoothing process performed by a Robust LOcal polynomial regression (RLOESS) smoother. The method according to claim 1,
In a step (S140) of optimizing a locus used for estimating a dynamic parameter of the robot using a least squares parameter estimator for the result obtained from the dynamic estimation modeling, reducing the number of conditions defining the locus, And applying an inequality to make the determinant of the guard arc matrix equal to or less than the product of its diagonal elements. The method according to claim 1,
In the step (S140) of optimizing the trajectory used for estimating the dynamic parameter of the robot using the least squares parameter estimator for the result obtained from the dynamic estimation modeling, the excitation trajectory q * (t) Can be expressed as,
Figure pat00143

The conditional equations are as follows,
Figure pat00144

remind
Figure pat00145
Is an objective function determined sequentially,
The conditional expression
Figure pat00146
Wow
Figure pat00147
Respectively represent the limitations of the position, velocity and acceleration of the joints. 5. The method of claim 4,
The objective function
Figure pat00148
The
Figure pat00149
ego,
The observation matrix W composed of the position of the joint, the estimated velocity, and the estimated acceleration samples measured by the Hadamard inequality and measured by the parameter estimation method is as follows,
Figure pat00150

From the observation matrix W,
Figure pat00151
Figure pat00152
Lt; / RTI >
The W kg is the g th column, k th element of the regression matrix W,
The sum of W 2 kg is defined as W s g ,
Figure pat00153
From
Figure pat00154
To maximize
Figure pat00155
Wherein the upper limit of the parameter estimation is maximized. The method according to claim 1,
In the step (S140) of optimizing the trajectory used for estimating the dynamic parameter of the robot using the least squares parameter estimator for the result obtained from the dynamic estimation modeling, the least squares parameter estimator performs the dynamic estimation modeling To overcome the over-determined metrics obtained in the above-
Matrix notation ego,
here,
Figure pat00157
Wow
Figure pat00158
Are the estimated basic parameters,
Figure pat00159
The estimated error covariance matrix < RTI ID = 0.0 >
Figure pat00160
Lt;
remind
Figure pat00161
Is the variance of the error,
remind
Figure pat00162
The estimate of
Figure pat00163
Lt;
The covariance matrix of the estimation error is
Figure pat00164
Lt;
Figure pat00165
Of the j < th >
Figure pat00166
The relative standard deviation (RSD)
Figure pat00167
/ RTI >
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