A dynamic parameter identification method of industrial robots considering joint elasticity

Abstract

Considering the joint elasticity, a novel dynamic parameter identification method is proposed for general industrial robots only with motor encoders. Firstly, the unknown parameters of the elastic joint dynamic model are analyzed and divided into two types. The first type is the motion-independent parameter only including the joint stiffness, which can be identified by the static force/torque-deformation experiments without the dynamic model. The second type is the motion-dependent parameter composed of the rest of the parameters, which needs the dynamic excitation experiments. Therefore, these two types of parameters can be identified separately. Meanwhile, it is found that the rotor inertia parameters can be obtained from the manufacturer, which reduces the identification difficulty of other parameters. After obtaining the rotor inertia and joint stiffness, an approximate processing algorithm is proposed considering the motor friction to establish the linear identification model of other parameters. Hence, the least squares can be employed to identify the parameters, and the independence of the inertia and joint viscous friction parameters are not affected. Meanwhile, the exciting trajectories can be optimized throughout the robot workspace, which reduces the effect of measurement noise on identification accuracy. With the proposed separated identification strategy and approximate processing algorithm, the dynamic parameters can be obtained precisely without double encoders on each joint. Finally, a series of simulations are conducted to evaluate the good performance of the proposed method.

Keywords

Dynamic parameter identification elastic joint dynamic model separated identification strategy approximate processing algorithm least squares

Introduction

With the increasing applications of industrial robots in modern manufacturing field, a lot of advanced motion control methods based on the torque input have been proposed and applied to satisfy the demand for high speed and high accuracy, which requires the complete and accurate dynamic model of robot.^1

–4 Meanwhile, the dynamic model is the basis for the characteristic analysis of robot.⁵ However, the parameters of dynamic model are usually not provided from the robot manufacturer. In addition, the parameters obtained from the computer-aided design (CAD) model are not accurate due to the manufacturing and assembly error. Therefore, it is critical to develop a rational and practical parameter identification method for industrial robots.

The rigid body dynamic model (RBDM) is wildly used to describe the robot. And a lot of parameter identification methods for RBDM have been proposed. One of the methods most used is the Inverse Dynamic Identification Model + Least Squares (LS) (IDIM + LS) scheme.^6

–11 An inverse dynamic model that is linear with respect to the dynamic parameters is established. Then, the excitation trajectory is designed with various methods.^6,9 The robot input and output signals are sampled during the dynamic excitation experiment. Finally, the parameters are estimated by the LS. Based on this idea, the Direct Inverse Dynamic Identification Model (DIDIM) method^12,13 is developed without calculating the velocity and acceleration through the band pass filter of the position. In addition, the extended Kalman filter and weighted LS^14,15 are employed to the parameter identification. However, the extended Kalman filter is very sensitive to initial conditions, and its convergence velocity is slower. Another alternative identification method is the maximum-likelihood estimation.¹⁶ Nevertheless, it should only be considered in cases where both measurements, position and torque, are noisy.¹⁷ In summary, the IDIM + LS method is still a good choice to identify the parameters of RBDM.¹¹

For the parameter identification of RBDM by these methods, the motor encoders were used to sample the angle as the joint positions. However, the robot is an elastomer, and the elasticity is mainly concentrated on the joints due to reducers and other transmission components. Hence, the neglect of joint elasticity can cause large errors to identification. To cope with this problem, the elastic joint dynamic models (EJDM) of robots are established, and some identification methods have been proposed. Zollo et al.¹⁸ established the dynamic model based on the Lagrangian formulation where the torque vector is expressed as the product of a regressor matrix and defined by the vector of dynamic parameters. The LS method is also employed. Miranda-Colorado¹⁹ built the filtered model and used the LS with forgetting factor (LSFF) to identify the parameters. Gautier et al.²⁰ extended the DIDIM method to the identification of flexible system. However, these methods are only suitable for the specially designed robots with double encoders on each joint, where both the motor and joint positions can be measured directly. For general industrial robots, only the motor encoders are designed and mounted. Hence, these methods have low practicality for industrial application.

To realize the parameter identification of the general robots only with motor encoders and elastic joints, Gautier et al.²¹ proposed a joint stiffness identification method. Because of the unknown rotor inertia, joint stiffness, and motor friction parameters, the linear IDIM is difficult to establish. Hence, the experiments were conducted in a small range to reducing the influence of nonlinearity on linear regression identification. However, the identification accuracy is sensitive to measurement noise because the exciting trajectories cannot be optimized. Meanwhile, the base parameter set was reconstructed where the joint stiffness and motion friction parameters were coupled with the inertia and joint friction parameters.

To cope with these problems, a novel dynamic parameter identification method considering the joint elasticity is proposed for general industrial robots. Firstly, the characteristics of the unknown parameters are analyzed and divided into two types called motion-dependent and motion-independent parameters. For the motion-independent parameters, which only includes the joint stiffness, the static force/torque-deformation experiments are sufficient to identify them without the dynamic model. For the motion-dependent parameters composed of the rest of the parameters, the dynamic excitation experiments are needed. Therefore, these two types of parameters can be identified separately. Meanwhile, it is found that the rotor inertia parameters can be obtained from the manufacturer, which reduces the identification difficulty of other parameters. After obtaining the rotor inertia and joint stiffness, an approximate processing algorithm is proposed considering the effect of motor friction parameters to establish the linear minimal identification model. Hence, the LS still can be employed and the independence of inertia and joint viscous friction parameters are not affected. Meanwhile, similar to the rigid body situation, the exciting trajectories can be optimized throughout the robot workspace, which can reduce the influence of measurement noise. With the proposed separated identification strategy and approximate processing algorithm, the dynamic parameters can be obtained precisely without double encoders on each joint. Finally, a series of simulations are conducted to evaluate good performance in accuracy of the proposed method.

The remainder of this article is organized as follows. In the second section, the EJDM of robot is established. Third section describes the proposed parameter identification method. The identification results are illustrated and compared with previous works in fourth section. And, the conclusions are given in fifth section.

Modeling

Dynamic model of robot with elastic joint

As shown in Figure 1, the elastic joint is always modeled as a linear torsional spring with stiffness K and reduction gear ratio R. Meanwhile, since the stiffness of link is much larger than joint,²² the link is assumed as rigid body. Therefore, the EJDM of an n-degree-of-freedom (n-DOF) robot can be established as follows based on Newton–Euler or Lagrange formulations

M (q_{e}) {\ddot{q}}_{e} + C (q_{e}, {\dot{q}}_{e}) + f_{e} ({\dot{q}}_{e}) + G (q_{e}) = τ_{e}

τ_{e} = K (R^{- 1} q_{m} - q_{e})

J_{m} {\ddot{q}}_{m} + f_{m} ({\dot{q}}_{m}) + R^{- 1} τ_{e} = τ_{m}

with

q_{e} = {[q_{e 1} ... q_{e n}]}^{T} \in ℝ^{n},

τ_{e} = {[τ_{e 1} ... τ_{e n}]}^{T} \in ℝ^{n},

q_{m} = {[q_{m 1} ... q_{m n}]}^{T} \in ℝ^{n},

τ_{m} = {[τ_{m 1} ... τ_{m n}]}^{T} \in ℝ^{n},

J_{m} = diag {J_{m 1} ... J_{m n}} \in ℝ^{n \times n,}

K = diag {K_{1} ... K_{n}} \in ℝ^{n \times n,}

R = diag {R_{1} ... R_{n}} \in ℝ^{n \times n,}

where $q_{e}$ , ${\dot{q}}_{e}$ , ${\ddot{q}}_{e}$ are the vectors of joint position, velocity, and acceleration, respectively, $τ_{e}$ is the vector of joint torque, $M (q_{e}) \in ℝ^{n \times n}$ is the inertia matrix, $C (q_{e}, {\dot{q}}_{e}) \in ℝ^{n}$ is the vector of centrifugal and Coriolis forces, $f_{e} ({\dot{q}}_{e}) \in ℝ^{n}$ is the vector of joint friction torque, $G (q_{e}) \in ℝ^{n}$ is the vector of gravitational torque or force, $q_{m}$ , ${\dot{q}}_{m}$ , ${\ddot{q}}_{m}$ are the vectors of motor position, velocity, and acceleration, respectively, $J_{m}$ is the matrix of rotor inertia, $τ_{m}$ is the vector of motor torque, $K$ is matrix of joint stiffness, $R$ is the matrix of reduction gear ratio and $f_{m} ({\dot{q}}_{m}) \in ℝ^{n}$ is the vector of motor friction torque.

Figure 1.

Elastic joint model.

There are several typical models to describe the friction characteristics of joint and motor including the Coulomb, Coulomb + viscous and Stribeck models.^23,24 For the joint friction, Coulomb + viscous model is widely used both in rigid body and in flexible dynamic models. Therefore, the Coulomb + viscous model is also employed in this article as follows

f_{e} ({\dot{q}}_{e}) = F_{v e} {\dot{q}}_{e} + F_{s e} sign ({\dot{q}}_{e})

F_{v e} = diag {F_{v e 1} ... F_{v e n}} \in ℝ^{n \times n}

F_{s e} = diag {F_{s e 1} ... F_{s e n}} \in ℝ^{n \times n}

where $F_{v e}$ and $F_{s e}$ are the matrix of joint viscous and Coulomb friction coefficients.

For the motor friction, the viscous friction torque is small and the Coulomb friction always plays a major role.²⁵ Meanwhile, to simplify the parameter identification process of EJDM, the Coulomb model is used in this article as follows

f_{m} ({\dot{q}}_{m}) = F_{s m} sign ({\dot{q}}_{m})

F_{s m} = diag {F_{s m 1} ... F_{s m n}} \in ℝ^{n \times n}

where $F_{s m}$ is the matrix of motor Coulomb friction coefficients.

For ith link, there are 10 unknown inertia parameters which need to be identified as follows

{}^{i}P_{iner} = {[I_{i x x}^{​} I_{i y y}^{​} I_{i z z}^{​} I_{i x y}^{​} I_{i y z}^{​} I_{i x z}^{​} H_{i x}^{​} H_{i y}^{​} H_{i z}^{​} m_{i} ​^{​}]}^{T}

where $I_{i x x}^{}$ – $I_{i x z}^{}$ are the elements of inertia tensor matrix, $H_{i} = [H_{i x}^{} H_{i y}^{} H_{i z}^{}] = m_{i} \times r_{i c}^{} = m_{i} [r_{i c x}^{} r_{i c y}^{} r_{i c z}^{}]$ , $r_{i c}^{}$ is the vector of centroid position and $m_{i}$ is the mass. Meanwhile, there are two unknown friction parameters in ith joint corresponding ith link

{}^{i}P_{fric} = {[F_{v e i}^{​} F_{s e i}^{​}]}^{T}

Considering the joint elasticity, another three parameters including the joint stiffness $K_{i}$ , rotor inertia $J_{m i}$ and friction coefficient $F_{s m i}$ are unknown. Therefore, there are 15 unknown parameters in each DOF as follows

{}^{i}P ​_{dyna}^{​} = (\begin{matrix} {}^{i}P_{iner} \\ \begin{array}{l} {}^{i}P_{E} \\ K_{i} \\ J_{m i} \\ F_{s m i} \end{array} \end{matrix})

Problem of the minimal identification model for EJDM

Owing to the linear relation between the parameter ${}^{i}P _{iner}^{}$ , ${}^{i}P _{fric}^{}$ , and joint torque $τ_{e}$ , equation (1) can be rewritten as

τ_{e} = Φ_{r i g i d} (q_{e}, {\dot{q}}_{e}, {\ddot{q}}_{e}) X_{r i g i d}

where $Φ_{rigid} (q_{e}, {\dot{q}}_{e}, {\ddot{q}}_{e})$ is the observation matrix that is nonlinear with $q_{e}$ , ${\dot{q}}_{e}$ , and ${\ddot{q}}_{e}$ , $X_{r i g i d}$ is the base parameter set composed by ${}^{i}P_{fric}$ and the linear combination of the elements in ${}^{i}P_{iner}$ . Equation (12) is also called the minimal identification model of RBDM.²¹ For the parameter identification of RBDM with the IDIM + LS method, the joint elasticity is neglected and the joint motion parameters $q_{e}$ , ${\dot{q}}_{e}$ , ${\ddot{q}}_{e}$ can be obtained by the encoders. Therefore, $Φ_{r i g i d} (q_{e}, {\dot{q}}_{e}, {\ddot{q}}_{e})$ is a known matrix and the LS estimation method can be employed to identify $X_{r i g i d}$ .

However, it is difficult to establish the linear minimal identification model for EJDM. For general industrial robots, each joint has only one encoder which is usually mounted on the motor. Hence, only the motor motion parameters $q_{m}$ , ${\dot{q}}_{m}$ , ${\ddot{q}}_{m}$ can be sampled and calculated. Correspondingly, the motor torque $τ_{m}$ can be obtained without the joint torque $τ_{e}$ . Based on equations (2) and (3), $τ_{e}$ and $q_{e}$ can be expressed as follows

τ_{e} = R (τ_{m} - J_{m} {\ddot{q}}_{m} - F_{s m} s i g n ({\dot{q}}_{m}))

q_{e} = R^{- 1} q_{m} - K^{- 1} R (τ_{m} - J_{m} {\ddot{q}}_{m} - F_{s m} s i g n ({\dot{q}}_{m}))

${\dot{q}}_{e}$ , ${\ddot{q}}_{e}$ can be obtained by the first and second derivatives of equation (14), where $s i g n ({\dot{q}}_{m})$ is removed because its derivative is not defined

{\dot{q}}_{e} = R^{- 1} {\dot{q}}_{m} - K^{- 1} R ({\dot{τ}}_{m} - J_{m} {\overset{⃛}{q}}_{m})

{\ddot{q}}_{e} = R^{- 1} {\ddot{q}}_{m} - K^{- 1} R ({\ddot{τ}}_{m} - J_{m} {\overset{⃜}{q}}_{m})

By substituting equations (14) to (16) into equation (1), the linear minimal identification model cannot be obtained due to the nonlinear relation between $K$ , $J_{m}$ , $F_{s m}$ , and $Φ_{r i g i d} (q_{e}, {\dot{q}}_{e}, {\ddot{q}}_{e})$ . Hence, the parameters cannot be identified by the typical IDIM + LS method and the new identification scheme needs to be developed to cope with this problem, which will be given in “Parameter identification method for EJDM” section.

Parameter identification method for EJDM

In this section, the proposed parameter identification method is described in detail. The characteristics of the unknown parameters in ${}^{i}P_{dyna}$ are analyzed firstly in subsection “Characteristic analysis of the unknown parameters”. Then, subsections “Identification of the motion-independent parameters” and “Identification of the motion-dependent parameters” illustrate the identification procedures step by step.

Characteristic analysis of the unknown parameters

As described in “Problem of the minimal identification model for EJDM” subsection, it is difficult to identify the unknown parameters through IDIM + LS method although the data sampled during the motion process contain all the information of these parameters. Actually, the motion and sampling process are not necessarily essential for the identification of each parameter. As shown in equation (11), the 15 unknown parameters of ith DOF can be divided into two types as follows:

(1) Motion-dependent parameters

This type of parameters consists of ${}^{i}P_{iner}$ , ${}^{i}P_{fric}$ , $J_{m i}$ , and $F_{s m i}$ , which can only be identified based on the sampled data and the dynamic model. Thus, the dynamic excitation experiment is necessary.

(2) Motion-independent parameters

This type of parameters only includes $K_{i}$ .The static force/torque-deformation experiment is sufficient to identify $K_{i}$ without the dynamic model. Hence, the dynamic excitation and sampling process are not needed.

It should be noted that $J_{m i}$ is a special parameter. Actually, $J_{m i}$ does not need to be identified. For most industrial robots, servo motors are used as the drivers and their rotor inertias are provided by the manufacturer. Servo motor is a kind of relatively precise driver with higher manufacturing accuracy. Meanwhile, the wear of rotor is small during the use of servo motor. Therefore, the value of rotor inertia provided by the manufacturer can be used directly without identification. However, the corresponding friction parameter $F_{s m i}$ is not provided because it is closely related to the working conditions. Hence, $F_{s m i}$ needs to be identified.

Based on the analysis results, these two types of dynamic parameters can be identified separately, which is given in the following subsections.

Identification of the motion-independent parameters

The static force/torque-deformation experiments and identification are wildly used to obtain the joint stiffness of industrial robots,^26
–28 which can be described briefly as follows. During the static experiments, the determined force/torque is applied to the robot links or end effector. Meanwhile, the displacement sensors can measure the steady state deformation of the end effector. Hence, the Cartesian stiffness of robot can be calculated. Then, the analytical relation between the joint and Cartesian stiffness can be established based on the kinematic model of robot. Finally, many kinds of identification methods such as the LS and genetic algorithms can be employed to obtain the joint stiffness. As can be seen, the robot dynamic model is not used, which also indicates that there is no need to identify the joint stiffness through dynamic excitation experiments with other parameters.

Using these static force/torque-deformation experiments and identification methods, various external displacement sensors such as the machine vision, grating, and laser sensors can be employed to measure the deformation of the end effector. Therefore, the double encoders of each joint, which is not designed and mounted on general industrial robots, are not required. In addition, the identified joint stiffness has higher accuracy owing to the static experiments and the measured data in steady state.

Identification of the motion-dependent parameters

Based on subsections “Characteristic analysis of the unknown parameters” and “Identification of the motion-independent parameters”, $J_{m}$ and $K$ are obtained and can be used to identify the motion-dependent parameters. To guarantee the identification accuracy under the various measurement noise, the idea of typical IDIM + LS method is also employed in this article. However, as can be seen from equations (13) and (14), the joint position $q_{e}$ and torque $τ_{e}$ still relate to $F_{s m}$ and the linear identification model cannot be established. To cope with this problem, an approximate processing algorithm is proposed to reconstruct the linear minimal identification model.

Equation (14) can be rewritten as follows

q_{e} = R^{- 1} q_{m} - K^{- 1} R (τ_{m} - J_{m} {\ddot{q}}_{m}) + K^{- 1} R F_{s m} sign ({\dot{q}}_{m})

For general industrial robots, the order of magnitude of reduction gear ratio R_i is 10², which is much smaller than the joint stiffness.^26,27,29 Meanwhile, the Coulomb coefficient $F_{s m i}$ of motor is small. Therefore, $q_{e}$ can be approximated as follows by neglecting the friction part

q_{e} \approx R^{- 1} q_{m} - K^{- 1} R (τ_{m} - J_{m} {\ddot{q}}_{m}) = q_{e}^{*}

As can be seen from equations (14) and (15), ${\dot{q}}_{e}$ and ${\ddot{q}}_{e}$ are not affected by $F_{s m}$ . Correspondingly, equation (13) can be rewritten as follows

τ_{e} = R (τ_{m} - J_{m} {\ddot{q}}_{m}) - R F_{s m} sign ({\dot{q}}_{m})

Different from $q_{e}$ , the motor friction is magnified due to the reduction gear ratio $R$ and has a certain effect on $τ_{e}$ , which cannot be ignored. But it can be found that $τ_{e}$ is linear with $F_{s m}$ . Thus, by substituting equations (18) and (19) into equation (1), it can be obtained as follows

Φ_{r i g i d} (q_{e}^{*}, {\dot{q}}_{e}, {\ddot{q}}_{e}) X_{r i g i d} = R (τ_{m} - J_{m} {\ddot{q}}_{m}) - R F_{s m} sign ({\dot{q}}_{m})

Through shifting the motor friction part, equation (20) can be rewritten as follows

\underset{left}{\underset{︸}{Φ_{r i g i d} (q_{e}^{*}, {\dot{q}}_{e}, {\ddot{q}}_{e}) X_{r i g i d} + R F_{s m} sign ({\dot{q}}_{m})}} = \underset{right}{\underset{︸}{R (τ_{m} - J_{m} {\ddot{q}}_{m})}}

As shown in equation (21), the left part is the linear expression of the unknown parameters $X_{r i g i d}$ and $F_{s m}$ . Meanwhile, the right part is the torque which can be calculated by the sampled data and $J_{m}$ .Therefore, equation (21) can be used as the new identification model. Correspondingly, the new base parameter set $X_{e l a s}$ should be reconstructed. It should be noted that ${\dot{q}}_{e}$ always has same sign with ${\dot{q}}_{m}$ in the sampled data because of the large joint stiffness. Hence, the coefficients of $F_{s e}$ and $R F_{s m}$ are the same. To ensure that the observation matrix is full column rank, $X_{e l a s}$ should be equal to $X_{r i g i d}$ , but the $F_{s e}$ should be updated by $F_{s e} + R F_{s m}$ . The observation matrix is still $Φ_{r i g i d} (q_{e}^{*}, {\dot{q}}_{e}, {\ddot{q}}_{e})$ and does not need to be reconstructed. Therefore, the approximate minimal identification model can be established as follows and the LS can be used to identify $X_{e l a s}$

Φ_{r i g i d} (q_{e}^{*}, {\dot{q}}_{e}, {\ddot{q}}_{e}) X_{e l a s} = R (τ_{m} - J_{m} {\ddot{q}}_{m})

In summary, the procedures of the proposed parameter identification method can be illustrated in Figure 2. As can be seen, the parameters of joint stiffness and rotor inertia can be obtained independently without double encoders on each joint, which is essential for the general industrial robots. Meanwhile, the proposed approximate identification model does not affect the independence of inertia and joint viscous parameters in $X_{r i g i d}$ . In addition, similar to the rigid body situation, the exciting trajectories can be optimized throughout the workspace of robot to reduce the effects of measurement noise and guarantee the identification accuracy.

Figure 2.

The procedures of the proposed parameter identification method.

Results and validation

In this section, simulation tests are conducted to evaluate the good performance of the proposed dynamic parameter identification method for the EJDM. Analysis and comparisons are also performed with the traditional IDIM + LS method without considering joint elasticity.

Test case

Figure 3 shows a typical 6-DOF industrial robot, and its modified Denavit–Hartenberg (D-H) parameters are illustrated in Table 1. Since the inertia and friction are mainly concentrated on the first three links and joints, the first three DOFs are selected as the case studies. By locking the last three joints under the gesture shown in Figure 3, they become part of the third link. Hence, the base parameter set $X_{e l a s}$ can be obtained with the algorithm given by Gautier³⁰ and equation (22), which is illustrated in Table 2 with the real value. In addition, the real values of joint stiffness and rotor inertia are given in Table 3, where the setting of rotor inertia value is referenced to Panasonic MSME series motors with approximately 2–4 kW power.

Figure 3.

Model of typical 6-DOF industrial robot. DOF: degree of freedom.

Table 1.

The modified D-H parameters of 6-DOF industrial robot.

Joint i	$α_{i - 1}$	$a_{i - 1}$	$d_{i}$	$θ_{i}$
1	0	0	0	$θ_{1}$
2	90°	0.18 m	0	$θ_{2}$
3	0	0.59 m	0	$θ_{3}$
4	90°	0.115 m	0.625 m	$θ_{4}$
5	−90°	0	0	$θ_{5}$
6	90°	0	0	$θ_{6}$

D-H: Denavit–Hartenberg; DOF: degree of freedom.

Table 2.

Definition of the base parameter set.

Parameters	Definition	Value	Parameters	Definition	Value
x₁ (kgċm²)	$(m_{2} + m_{3}) a_{1}^{2} + m_{3} a_{2}^{2} + I_{2 y y}^{} + I_{3 y y}^{} + I_{1 z z}^{}$	18.0574	x₁₂ (kgċm²)	$I_{3 y z}^{}$	−1.0624
x₂ (kgċm²)	$- m_{3} a_{2}^{2} + I_{2 x x}^{} - I_{2 y y}^{}$	−14.6053	x₁₃ (kgċm²)	$I_{3 z z}^{}$	3.5057
x₃ (kgċm²)	$I_{2 x y}^{}$	0.4436	x₁₄ (kgċm)	$m_{3} r_{3 c x}^{}$	0.8681
x₄ (kgċm²)	$I_{2 x z}^{} - a_{2} m_{3} r_{3 c z}^{}$	−2.4431	x₁₅ (kgċm)	$m_{3} r_{3 c y}^{}$	−8.4047
x₅ (kgċm²)	$I_{2 y z}^{}$	−0.07682	x ₁₆	$F_{s e 1}^{} + R_{1} F_{s m 1}^{}$	40.0000
x₆ (kgċm²)	$m_{3} a_{2}^{2} + I_{2 z z}^{}$	14.8343	x ₁₇	$F_{s e 2}^{} + R_{2} F_{s m 2}^{}$	38.0000
x₇ (kgċm)	$m_{2} r_{2 c x}^{} + a_{2} m_{3}$	27.3428	x ₁₈	$F_{s e 3}^{} + R_{3} F_{s m 3}^{}$	38.0000
x₈ (kgċm)	$m_{2} r_{2 c y}^{}$	1.6203	x ₁₉	$F_{v e 1}^{}$	20.0000
x₉ (kgċm²)	$I_{3 x x}^{} - I_{3 y y}^{}$	3.3578	x ₂₀	$F_{v e 2}^{}$	17.0000
x₁₀ (kgċm²)	$I_{3 x y}^{}$	−0.2641	x ₂₁	$F_{v e 3}^{}$	17.0000
x₁₁ (kgċm²)	$I_{3 x z}$	0.09713

Table 3.

Real values of the joint stiffness and rotor inertia.

Joint i	$K_{i}$ (Nċm/rad)	$J_{m i}$ (kgċm²)
1	12,000.00	0.001
2	9000.00	0.001
3	9000.00	0.0005

The Fourier serious equation³¹ is utilized to generate the exciting trajectories in joint space. The minimum condition number principle³² is employed to optimize the coefficients of the finite sum of sine and cosine functions, which can obtain the trajectories throughout the workspace and reduce the effect of measurement noise. The fmincon function in Matlab [version 2018b] is used to conduct the optimization. Therefore, the fifth-order Fourier series equation, which is shown in Appendix 1, is employed to describe each joint position. Correspondingly, the trajectory of each motor can be obtained according to equations (1) and (2) and is shown in Figure 4. During the simulations, the motor angular positions are sampled at 2 kHz with high-resolution encoders (8,388,608 counts/round). The sampled data are filtered through a low-pass Butterworth filter. To consider the effect of measurement noise on identification accuracy, the Gaussian noise with zero mean value is added to the motor motion parameters and torque with different levels of signal-to-noise ratio.

Figure 4.

Exciting trajectory of each motor.

Identification results and comparisons

By neglecting the measurement noise, the torque data of each motor are sampled and shown in Figure 5. For the joint stiffness and rotor inertia, they can be obtained by static experiments and motor manufacturer precisely. Therefore, we assume that they have 1% and 0.5% error, respectively. The identification results of $X_{e l a s}$ in the absence of measurement noise by the proposed identification method and the traditional IDIM + LS method without considering the joint elasticity are shown in Table 4. As can be seen, the estimate values obtained by the proposed identification method are very close to the real values. The error comes from the inaccuracy of joint stiffness and rotor inertia and the approximate processing for joint position. But the small error indicates the effectiveness of the proposed approximate processing algorithm and the whole identification method. However, due to the neglect of joint elasticity, the joint motion parameters in traditional IDIM + LS method are obtained directly by the sampled motor parameters and the reduction gear ratios, which has large error. Thus, the estimate values are far from the real values.

Figure 5.

Sampled torque of each motor.

Table 4.

Identification results by the two methods without measurement noise.

Parameters	Real value	The proposed method		Traditional IDIM + LS method
Parameters	Real value	Estimated value	Error (%)	Estimated value	Error (%)
x₁ (kgċm²)	18.0574	18.0527	0.02627	30.8784	71.0013
x₂ (kgċm²)	−14.6053	−14.6033	0.01361	−17.2989	18.4426
x₃ (kgċm²)	0.4436	0.4438	0.05695	0.7557	70.3561
x₄ (kgċm²)	−2.4431	−2.4424	0.02900	−3.4473	41.1035
x₅ (kgċm²)	−0.07682	−0.07678	0.02934	−0.9654	1157.0312
x₆ (kgċm²)	14.8343	14.8361	0.01237	23.8328	60.6601
x₇ (kgċm)	27.3428	27.3431	0.00094	27.7302	1.4168
x₈ (kgċm)	1.6203	1.6195	0.04974	1.5545	4.0610
x₉ (kgċm²)	3.3578	3.3623	0.1337	1.8413	45.1635
x₁₀ (kgċm²)	−0.2641	−0.2645	0.1346	−0.2932	11.0186
x₁₁ (kgċm²)	0.09713	0.09653	0.5894	0.6161	534.5005
x₁₂ (kgċm²)	−1.0624	−1.0612	0.1100	−2.0062	88.8366
x₁₃ (kgċm²)	3.5057	3.5036	0.06042	7.1834	104.9063
x₁₄ (kgċm)	0.8681	0.8682	0.01291	0.7024	19.0876
x₁₅ (kgċm)	−8.4047	−8.4042	0.006394	−8.9190	6.1192
x ₁₆	40.0000	40.0017	0.004140	39.8430	0.3925
x ₁₇	38.0000	38.0022	0.005798	42.4298	11.6573
x ₁₈	38.0000	38.0082	0.02168	26.6045	29.9882
x ₁₉	20.0000	19.9989	0.005282	17.8494	10.7530
x ₂₀	17.0000	17.0052	0.03100	19.1784	12.8141
x ₂₁	17.0000	16.9994	0.003711	20.2812	19.3012

IDIM: inverse dynamic identification model; LS: least squares.

The direct validation is also conducted where the torque of each motor is estimated based on the identified parameters given in Table 4. As shown in Figure 6, the estimated torque by the proposed method is closer to the measured one, which indicates its higher accuracy.

Figure 6.

Measured and estimated torque of each motor by the proposed method and the traditional IDIM + LS method. IDIM: inverse dynamic identification model; LS: least squares.

The identification results by the two methods with different levels of signal-noise-ratio (i.e. SN = 30, SN = 50, SN = 80) are shown in Tables 5 and 6, respectively. As can be seen, the measurement noise has a certain influence on the accuracy of the proposed method. But with the increase of signal-noise-ratio, the identification accuracy is improved obviously. The error with SN = 50 is small, and the accuracy with SN = 80 is close to the absence of measurement noise. However, there is no obvious change in accuracy of the traditional IDIM + LS method with different levels of signal-noise-ratio because the neglect of joint elasticity is the key factor to cause error.

Table 5.

Identification results by the proposed method with measurement noise.

Parameters	SN = 30		SN = 50		SN = 80
Parameters	Estimate value	Error (%)	Estimate value	Error (%)	Estimate value	Error (%)
x₁ (kgċm²)	17.4714	3.2452	18.0890	0.1749	18.0553	0.0116
x₂ (kgċm²)	−14.8231	1.4912	−14.7256	0.8236	−14.6043	0.0068
x₃ (kgċm²)	−0.3258	173.4445	0.4077	8.0928	0.4436	0
x₄ (kgċm²)	−2.1022	13.9535	−2.4089	1.3998	−2.4436	0.0205
x₅ (kgċm²)	−0.2477	222.5260	−0.0664	13.5416	−0.0781	1.6927
x₆ (kgċm²)	15.2144	2.5623	14.7979	0.2453	14.8348	0.0033
x₇ (kgċm)	27.1702	0.6312	27.3301	0.0464	27.3425	0.0011
x₈ (kgċm)	1.6357	0.9504	1.6509	1.8885	1.6201	0.0123
x₉ (kgċm²)	4.4737	33.2331	3.3933	1.0572	3.3579	0.0030
x₁₀ (kgċm²)	−0.2886	9.2767	−0.2785	5.4524	−0.2645	0.1515
x₁₁ (kgċm²)	0.2007	106.6941	0.0987	1.6477	0.0975	0.4119
x₁₂ (kgċm²)	−1.0684	0.5647	−1.0691	0.6306	−1.0621	0.0282
x₁₃ (kgċm²)	3.6025	2.7612	3.5055	0.0057	3.5049	0.0228
x₁₄ (kgċm)	1.0015	15.3668	0.8837	1.7970	0.8690	0.1037
x₁₅ (kgċm)	−8.5025	1.1636	−8.4107	0.0713	−8.4044	0.0036
x ₁₆	38.6999	3.2502	39.8837	0.2907	39.9977	0.0058
x ₁₇	36.1085	4.9776	38.113	0.2973	38.0043	0.0113
x ₁₈	37.9454	0.1436	37.9902	0.0257	38.0118	0.0311
x ₁₉	21.966	9.8312	20.2728	1.3640	19.9980	0.0100
x ₂₀	18.3387	7.8747	16.9343	0.3864	17.0091	0.0535
x ₂₁	16.5823	2.4571	17.0801	0.4711	16.9947	0.0312

SN: signal-to-noise ratio.

Table 6.

Identification results by the traditional IDIM + LS method with measurement noise.

Parameters	SN = 30		SN = 50		SN = 80
Parameters	Estimate value	Error	Estimate value	Error	Estimate value	Error
x₁ (kgċm²)	30.0852	66.6087	30.8046	70.5926	30.8806	71.0135
x₂ (kgċm²)	−17.9636	22.9937	−17.2810	18.3200	−17.297	18.4296
x₃ (kgċm²)	0.5396	21.6411	0.7243	63.2777	0.7543	70.0405
x₄ (kgċm²)	−3.5907	46.9731	−3.4584	41.5578	−3.4470	41.0912
x₅ (kgċm²)	−0.8515	1008.7239	−0.9532	1141.1458	−0.9645	1155.8593
x₆ (kgċm²)	23.4144	57.8396	23.8166	60.5508	23.8335	60.6648
x₇ (kgċm)	27.6379	1.0792	27.7283	1.4098	27.7301	1.4164
x₈ (kgċm)	1.7888	10.3993	1.5367	5.1595	1.5542	4.0794
x₉ (kgċm²)	3.0400	9.4645	1.8741	44.1866	1.8365	45.3064
x₁₀ (kgċm²)	−0.5115	93.6766	−0.2987	13.1010	−0.2918	10.4884
x₁₁ (kgċm²)	0.4163	328.7332	0.6315	550.3604	0.6165	534.9124
x₁₂ (kgċm²)	−1.7144	61.3704	−2.0182	89.9661	−2.0063	88.8460
x₁₃ (kgċm²)	7.4672	113.0016	7.1845	104.9376	7.1831	104.8977
x₁₄ (kgċm)	0.6749	22.2555	0.7023	19.0991	0.7023	19.0991
x₁₅ (kgċm)	−9.0469	7.6409	−8.9149	6.0704	−8.9188	6.1168
x ₁₆	38.3204	4.199	39.8297	0.4257	39.8420	0.3950
x ₁₇	44.2557	16.4623	42.479	11.7868	42.4242	11.6426
x ₁₈	27.8241	26.7786	26.6167	29.9560	26.6034	29.9910
x ₁₉	20.1809	0.9045	17.8555	10.7225	17.8477	10.7615
x ₂₀	18.3658	8.03411	19.1588	12.6988	19.1797	12.8217
x ₂₁	18.8857	11.0923	20.3295	19.5852	20.2779	19.2817

IDIM: inverse dynamic identification model; LS: least squares; SN: signal-to-noise ratio.

Conclusion and future works

In this article, a novel dynamic parameter identification method is proposed for general industrial robots only with motor encoders and elastic joints. The main conclusions of the study are as follows:

The unknown parameters of the EJDM are analyzed and divided into two types and named motion-dependent and motion-independent parameters that can be identified separately.

The joint stiffness, which belongs to the motion-independent parameter, can be identified precisely based on the static force/torque-deformation experiments and kinematic model of robot. In addition, the rotor inertia can be obtained from the motor manufacturer without identification.

To identify the motion-dependent parameters, an approximate processing algorithm is proposed to establish the linear identification model, which can employ the LS. Hence, the independence of the inertia and joint viscous friction parameters are not affected and the exciting trajectories can be optimized to reduce the effect of measurement noise.

With the proposed separated identification strategy and approximate processing algorithm, the parameters of general industrial robots can be identified without double encoders on each joint.

A series of simulations are performed with previous works to validate the good performance in accuracy of the proposed method.

Future works will take the motor viscous friction parameters into consideration. Other methods will be developed and used to establish the linear minimal identification model with higher accuracy.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The work is supported by Special Foundation for National Integrated Standardization and New Model of Intelligent Manufacturing, China (grant no. Z135060009002-132) and National Natural Science Foundation of China (grant no. 51875323).

ORCID iD

Chengrui Zhang

Chao Chen

Appendix 1

After the optimization based on the minimum condition number principle and Fmincon function in Matlab, the formulation of each joint position is shown in equation (A1) and the corresponding coefficients are illustrated in Table A1.

where $i = 1, 2, 3$ indicates the joint number and $f = 0.1 H z$ . $q_{e i 0}$ is the initial position and $q_{e 10} = 0.03549 rad$ , $q_{e 20} = - 0.3500 rad$ , $q_{e 30} = - 0.4240 rad$ .

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