cobot Cobot 2752-5813 F1000 Research Limited London, UK 10.12688/cobot.17444.1 Review Articles A review of dynamic parameters identification for manipulator control

[version 1; peer review: 1 approved, 1 approved with reservations]

 Huang Wenhui Investigation Resources Validation Writing – Original Draft Preparation https://orcid.org/0000-0001-7305-0816 1 Min Huasong Funding Acquisition Project Administration Supervision Writing – Review & Editing https://orcid.org/0000-0003-4845-0097 a 1 Guo Yixuan Writing – Review & Editing 1 Liu Mingxin Writing – Review & Editing 1 Institute of Robotics and Intelligent Systems, Institute of Robotics and Intelligent Systems, College of Information Science and Engineering, Wuhan University of Science and Technology, Wuhan, 430000, China mhuasong@wust.edu.cn

Competing interests: The article processing charge for this article was funded by AUBO.

 21 12022 2022 1 5 5 12022 Copyright: © 2022 Huang W et al. 2022 This is an open access article distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Due to the important role of the manipulator dynamic model in manipulation control, the identification of the dynamic parameters of manipulators has become a research hotspot once again. In this paper, we present an overview of the modeling of manipulator dynamics, the optimization methods of excitation trajectory, the identification methods for dynamic parameters, and the identification of friction model parameters. First, the process and basic methods of identification of manipulation dynamic parameters are summarized, and the optimization methods for excitation trajectory are analyzed in detail. Further, friction model parameter identification and the physical feasibility of dynamic parameters are discussed. These are research hotspots associated with the identification of dynamic parameters of manipulators. The backgrounds and solutions of the problems of physical feasibility and identification of friction parameters are reviewed in this paper. Finally, neural networks and deep learning methods are discussed. The neural networks and deep learning methods have been used to improve the accuracy of identification. However, deep learning methods and neural networks need more in-depth analysis and experiments. At present, the instrumental variable method with complete physical feasibility constraints is an optimal choice for dynamic parameter identification. Moreover, this review aims to present the important theoretical foundations and research hotspots for the identification of manipulation dynamic parameters and help researchers determine future research areas.

 Manipulator dynamic dynamic parameters identification friction model physical feasibility deep learning National Natural Science Foundation of China No.62073249 Hubei Province Technology Innovation No.2019AAA071 This work was partly supported by National Natural Science Foundation of China (Grant No. 62073249), and is a major project of Hubei Province Technology Innovation (Grant No. 2019AAA071). The funders had no role in study design, data collection and analysis, decision to publish, or preparation of the manuscript. 1 Introduction

Force perception and force control have been the key factors for the development of human-robot collaboration. In practical terms, the mobile manipulators are the next step after the collaborative manipulator, and the base of the manipulator is flexible, which varies the contact force between the manipulator and the environment 1 . Thus, a high precision dynamic model is necessary for dynamic control of manipulator dynamics, and it is fundamental for many force perception and force control methods, such as sensorless collisions detection and impedance control 2, 3 . The precision of the dynamic parameters, which describe the dynamic model, are the key factor influencing the accuracy of the dynamic model. However, dynamic parameters are not always provided by the robot manufacturers, or else those parameters provided by the manufacturers might not be precise enough. The manipulator operators must therefore master the methods of obtaining the dynamic parameters, which is the key skill for operators to develop a high-performance manipulator dynamic control system.

The methods of obtaining the dynamic parameters can be divided into three categories 4 : CAD (Computer-Aided Design) technology, physical experiments, and identification methods that estimate dynamic parameters with special experiments. Due to the manufacturing error, the accuracy of the estimated parameter of CAD technology is low 4 . The physical experiments detect the dynamic parameters of links separately, which ignores the assembly errors and joint characteristics. The identification methods can estimate the dynamic parameters of manipulators simply and accurately, and identification methods have been used and researched extensively 2– 4 .

Usually, the process of dynamic parameter identification can be divided into five steps 4 : I model formulation, II excitation trajectory design, III signal processing, IV parameter identification, V verification. The dynamic model equation is converted to an identifiable form in the process of formulation, and the regressor matrix in the identified form is a linear matrix 5, 6 . The purpose of excitation trajectory design is to excite dynamic parameters and reduce the negative impact of noise 7, 8 . In signal processing, filtering is a process that removes noise in the sampled encoder or current sensor data. Moreover, some singular values in the sampled data set must be filtered to avoid identification errors. Then, numerical methods have been used to estimate dynamic parameter values during parameters identification 6, 9 . Finally, the accuracy of identified dynamic parameters is verified with test trajectory.

In dynamic parameters identification, there are another two research areas, namely friction parameter identification and physical feasibility of dynamic parameters. Friction is an additional force in the dynamic model, which can affect the accuracy of the dynamic model. However, the friction force is nonlinear, and the friction force cannot be modeled directly by the analytical method. Moreover, indirect identification methods or specific identification experiments must be used to identify the friction parameters. For the physical feasibility problem, additional constraints are used to ensure the physical feasibility of dynamic parameters, to improve the reliability of the identified dynamic model.

This paper reviews the works on dynamic parameter identification of serial manipulation. The paper is organized as follows: after a summary of the dynamic identification process ( Section 2), the physical feasibility problems of dynamic parameters are discussed in Section 3. The parameter identification methods of the friction model are surveyed in Section 4. In Section 5, deep learning and neural network methods are discussed. Finally, a summary is presented.

 2 Model and identification algorithm 2.1 Dynamic model and dynamic parameters

An identifiable mathematical formulation of the dynamic model of manipulators is necessary for dealing with problems of identification of dynamic parameters. First, an essential dynamic model must be calculated. The dynamic model can be calculated by several methods such as the Newton-Euler method, Lagrangian method, or energy model 2 . The Newton-Euler method and Lagrangian method are commonly used. No matter which of these methods is used, a standard form of manipulator dynamic in joint space could be calculated, as shown in Equation (1) 2 (the friction force is considered in Section 4).

τ = M ( q ) q • • + V ( q , q • ) + G ( q ) ( 1 )

Where, M( q) is the inertial matrix, V( q, q ˙ ) is the Coriolis and centripetal matrix, G( q) is the gravity force, q is the generalized joint position, and τ is the generalized joint force. Equation (1) cannot be directly used to identify the dynamic parameters, and the dynamic parameters must be extracted separately as vectors. Thus, Atkeson 9 proposed a standard identified expression of the dynamic model, as Equation (2).

τ = Y ( q , q • , q • • ) X ( 2 )

Where X is the manipulator dynamic parameters vector, n is the number of links, Y is the regressor matrix calculated from the Equation (1).

The dynamic parameters of link i contains: the inertial matrix, as shown in Equation (3) 2 , m i is the weight of link i, ( mx i , my i , mz i ) is the first moment of link i, and ( x i , y i , z i ) is the centroid of link i.

r i = [ r i , x r i , y r i , z ]  

The regressor matrix is a linear matrix, which makes the dynamic parameters redundant. Thus, dynamic parameters have been divided into two categories: identified parameter and unidentified parameter 5, 6 . Unidentified dynamic parameters do not affect the torque calculation of dynamic model of manipulators. Conversely, when the identifiable parameters change, the calculation results of the dynamic model will change with the values of identifiable dynamic parameters. Different identifiable dynamic parameters determine different dynamic models. Therefore, a dynamic model can be uniquely determined by a set of identifiable dynamic parameters, which are called base parameters 10 . The set that contains all dynamic parameters are called standard dynamic parameters, expressed as δ. There are some columns of regressor matrix that have all zero elements or can be expressed linearly by other columns, thus, Equation (2) can be rewritten as Equation (3) 11 , where δ b T is the set of base parameters, δ d T is the set of unidentified dynamic parameters. Y b has n b linearly independent columns, and Y d has n d remaining null and dependent columns. n = n b + n d is the total number of columns of the regression matrix Y, which is also the total number of dynamic parameters.

τ = [ Y b , Y d ] [ δ b T , δ d T ] T , δ = [ δ b , δ d ] ( 3 )

The Y d can be written as a linear combination of Y b , as shown in Equation (4), K d is a constant matrix. Thus, the base parameters can be calculated as Equation (5), where β is the set of base parameters.

Y d = K d Y b ( 4 ) β = δ b + K d δ d ( 5 )

Moreover, the base parameters are the linear combination of the standard dynamic parameters, as Equation (6), where, P b and P d are the permutation matrix of Y b and Y d , Y b = YP b , Y d = YP d .

β = K δ , K = P b T + K d P d T ( 6 )

Numerical analysis methods are used to extract the base parameters from standard dynamic parameters in research, such as QR decomposition method or Singular Value Decomposition(SVD) method 6 . We respectively describe the process of QR decomposition and SVD method, and briefly describe the application of numerical methods in the least-squares problem, to provide a reference for research and engineering of dynamic parameters identification.

Least square problem 9 :

‖ ε ‖ 2 = ‖ τ − Y X ‖ 2 ( 7 )

Where the ε is the error between sampled torque and simulated torque.

QR decomposition:

The basic representation of QR decomposition is shown in Equation (8) 6, 9 , Q is an orthogonal matrix. Therefore, the least squares problem can be rewritten as Equation (10) according to the Equation (9), where Q 2 T τ is constant. The Equation (11) is the simplified form of Equation (10).

Y = Q R = [ Q 1 Q 2 ] [ R 1 0 ] ( 8 ) ‖ Q T ε ‖ 2 = ( Q T ε ) T ( Q T ε ) = ε T Q Q T ε = ε T ε = ‖ ε ‖ 2 ( 9 ) ‖ ε ‖ 2 = ‖ Q T τ − Q T Y β ‖ 2 = ‖ [ Q 1 T Q 2 T ] τ − [ R 1 0 ] β ‖ 2 ( 10 ) ‖ ε ‖ 2 − ‖ Q 2 T τ ‖ 2 = ‖ Q 1 T τ − R 1 β ‖ 2 ( 11 )

Where the Q 1 and Q 2 is the sub-matrix of Q, R 1 is the singular value of regression matrix.

SVD methods:

Equation (12) is a basic expression of SVD method 6 , and the ∑ 2 is zero when the measured data is noise-free. Thus, the least square problem can be rewritten as Equation (13), the U and V are orthogonal matrix. Finally, a decomposition expression can be calculated, as Equation (14)

Y V = U ∑ , ∑ = [ ∑ 1 0 n − n b , n b 0 n b , n − n b ∑ 2 ] ( 13 ) ‖ ε ‖ 2 = ‖ τ − Y X ‖ 2 = ‖ τ − U [ ∑ 1 0 0 0 ] V T [ β α ] ‖ 2 = ‖ U τ − [ ∑ 1 0 0 0 ] [ V 1 V 2 ] T [ β α ] ‖ 2 ( 14 ) ‖ τ − Y β ‖ 2 = ‖ U τ − ∑ V 1 β ‖ 2 ( 15 )

The classification of dynamic parameters is the prerequisite for the identification of dynamic parameters. It is only possible to identify the base parameters since the regressor matrix is linear. Thus, the research on the identification of dynamic parameters of the manipulators revolves around the base parameters, since the accuracy of the manipulator dynamic model is determined by the base parameters. The numerical decomposition methods are mostly used for calculating the base parameters. Numerical decomposition methods can calculate base parameters directly, and many auxiliary tools can realize numerical decomposition methods easily, such as the ‘svd’ command in MATLAB. In addition, the symbolic calculation method can also be used to calculate the base parameters by merging the linear correlation columns of the regression matrix. However, the symbolic calculation method of calculation is complicated when the number of degrees of the manipulator is more than three.

 2.2 Excitation trajectory design

There is lots of noise in sampled position data, speed data, and torque data. This noise will reduce the precision of the identified dynamic parameters and introduce the over-determined problem 12 . When the amount of sampled data is insufficient, the dynamic parameters cannot be fully identified. When the amount of sampled data is too large, the identified result may be overfitting. To ensure the accuracy of the identification result, it is necessary to ensure the quality of sampled data, which is related to the convergence rate and the noise immunity of the estimation of dynamic parameters. The objective of excitation trajectory design is to provide a high-quality sampled data set for identification in the presence of measurement noise and various interferences 12 . The excitation trajectory algorithm is divided into two parts: excitation trajectory optimization and excitation trajectory parameterization.

The excitation trajectory optimization aims to obtain a trajectory that can fully excite identified dynamic parameters. The statistical process of the manipulator dynamic is not stable due to the nonlinear relationship between the input and output of the manipulator system. Thus, the linear system trajectory optimization method may not be implemented directly, which optimizes the excitation trajectory by minimizing the condition number of the regressor matrix (observation matrix). An improved criterion must be presented. Some criteria have been used: (I) The sum of the condition number of the observation matrix and the equilibrium value of the observation matrix 8 , as shown in Table 1 No.1; (II) The condition number of the observation matrix and the smallest singular value derivative of the observation matrix, as shown in Table 1 No.2; (III) The condition number of the weighted observation matrix, as shown in Table 1 No. 3, where the Z is the weighted matrix.

Criterion for optimization <sup> <xref ref-type="bibr" rid="ref-12">12</xref>, <xref ref-type="bibr" rid="ref-15">14</xref>, <xref ref-type="bibr" rid="ref-15">15</xref> </sup>.
No. Criterion for optimization Framework
1 k 1 Cond( Y) + k 2 Y max Y min deterministic
2 Cond( Y) + 1 λ min ⁡ ( Y T Y ) deterministic
3 Cond( Ydiag( Z)) deterministic
4 –log(det( Y T ∑ –1 Y)) Statistical(D-optimality) 14
5 det( Y TY) ≤ Π g = 1 N b Y g s statistical

Notation: k 1 and k 2 are constant,

Y max is the max element of regressor matrix

Y min is the minimum element of the regressor matrix

λ min ( Y TY) is the minimum singular value

Y g s is the summation of elements of gth columns of the regressor matrix.

There are other methods based on the statistical framework in addition to the above methods based on the deterministic framework 13, 14 . The method based on the statistical framework can only be carried out when having information on the statistical characteristics of measurement noise. For example, reference 12 proposed a method to optimize the excitation trajectory and identification results simultaneously. The bound of covariance and the Fisher information matrix is used to design the criterion. Reference 14 proposed a hybrid criteria method so that the excitation trajectory satisfies the criteria of the deterministic framework and criteria of the statistical framework at the same time. Reference 15 used the Hadamard inequality based on the statistical framework as the criterion to optimize the excitation trajectory. The equation of Hadamard inequality is shown in Table 1 No. 5, where the Y g s is the summation of elements of gth columns of regressor matrix. Table 1 shows the different criteria for optimization. k 1 and k 2 are constant, Y max is the max element of regressor matrix, Y min is the minimum element of regressor matrix, λ min ( Y TY) is the minimum singular value.

min ( F ( Y ) ) s t . q I i ≤ q i ≤ q F i | q • i | ≤ q • M i | q i • • | ≤ q Mi • • ( 15 )

Optimization methods should be given after presenting the criteria. The excitation trajectory optimization problem involves a nonlinear optimization problem with motion constraints 12 . The basic criterion form is given by Equation (15), where q Ii is the lower bound of the position trajectory, q Fi is the upper bound of the position trajectory, and q Fi is the velocity, q ˙ Mi is the limit of the acceleration. Sometimes, additional constraints should be given depending on the difference in the parameterization method of the excitation trajectory. Many scholars use intelligent optimization algorithms to solve this nonlinear optimization problem. Reference 16 uses sequential quadratic programming algorithm, reference 17 uses a genetic algorithm (GA) with binary coding to solve this nonlinear optimization problem, reference 18 uses a memetic algorithm (memetic algorithm, a hybrid evolutionary algorithm), and reference 19 uses the artificial bee colony algorithm (ABC).

The excitation trajectory parameterization is to interpolate the position, velocity, and acceleration curve between two points of optimized trajectory to ensure continuity of joints of manipulators. The most used trajectory interpolation methods are the fifth polynomial method and finite Fourier series method. Reference 8 uses the fifth polynomial to interpolate the position, velocity, and acceleration trajectory in joint space, as in Equation (16). Reference 20 uses Fourier series to generate periodic trajectories to further suppress measurement noise, as shown in Equation (17), where ω f is the fundamental frequency of the Fourier series, and q io is the offset value of the position trajectory. The periodic trajectory can average the measured noise, so the Fourier series method has a better noise suppression effect. Some scholars have given an improved finite Fourier series method 21 , which optimizes the problem of the discontinuity of the velocity of the finite Fourier series at the starting point of the trajectory.

Fifth polynomial method：

q i ( t ) = a 0 + a 1 t + a 2 t 2 + a 3 t 3 + a 4 t 4 + a 5 t 5 q • i ( t ) = a 1 + 2 a 2 t + 3 a 3 t 2 + 4 a 4 t 3 + 5 a 5 t 4 q i • • ( t ) = 2 a 2 + 6 a 3 t + 12 a 4 t 2 + 20 a 5 t 3 ( 16 )

Finite Fourier series method：

q i ( t ) = ∑ l = 1 N i a l i ω f l sin ⁡ ( ω f l t ) − b l i ω f l cos ⁡ ( ω f l t ) + q i O q • i ( t ) = ∑ l = 1 N i a l i cos ( ω f l t ) + b l i sin ( ω f l t ) q i • • ( t ) = ∑ l = 1 N i − a l i ω f l cos ( ω f l t ) + b l i ω f l sin ( ω f l t ) ( 17 )

The signal processing method is another part of the identification of dynamic parameters which ensure the quality of the sampled data. The value of speed and acceleration needs to be calculated by differential equations, since the manipulator may only contain encoders in the joints. The joint toque is also calculated from the motor current or a six-dimensional force/torque sensor. Some manipulators have torque sensors in joints, such as Franka Emika Panda Manipulator 3 . The joint positions obtained from the encoders are discrete. Thus, it is impossible to filter the position signal by analog filtering. Generally, a Butterworth low-pass filter is used to filter the noise in the position signal 19 . To avoid phase shifts in the differential calculation, it is necessary to use the central differential method to calculate the value of speed and acceleration from encoder data 22, 23 . The use of motor current to calculate the torque also requires filtering. Reference 15 proposed a regression method-Robust LOcal polynomial regression(RLOESS), which uses a moving average filter and residual analysis to remove the singular value in motor current.

Based on the analysis of the above references, we have summarized some conclusions. The optimization of excitation trajectory is the prerequisite to ensuring the accuracy of the identification result. The noise in sampled data reduces the accuracy of the result of the dynamic parameters identification, which is consistent with the previous analysis. The noise in sampled data is the main source of errors in parameters identification. The objective of optimization of excitation trajectory is to better excite dynamic parameters and suppress data noise. The main method of optimization is to minimize the condition number of the regression matrix with constraints. The constraints of optimization of excitation trajectory can be divided into two categories: deterministic framework and statistical framework, and the problem of the optimization is nonlinear. Many scholars solve this nonlinear problem by using intelligent algorithms. However, trajectory optimization cannot completely avoid the effect of noise, and additional filtering methods are needed to further process sampled data.

 2.3 Parameter identification method and verification

The problem of identification of dynamic parameters of the manipulators can be described as how to calculate the base parameters with position, speed, acceleration, and torque of joints. The commonly used identification methods are the least-squares method 24, 25 , the maximum likelihood estimation 12, 26 , and instrumental variable method 27 . There are other methods, such as the weighted least squares method 28 , extended Kalman filtering 29 , nonlinear least-squares methods 30 , particle swarm optimization (PSO) 31 , etc., as shown in Figure 1.

The identification methods of dynamic parameters <sup> <xref ref-type="bibr" rid="ref-12">12</xref>, <xref ref-type="bibr" rid="ref-24">24</xref>, <xref ref-type="bibr" rid="ref-26">26</xref>, <xref ref-type="bibr" rid="ref-27">27</xref>, <xref ref-type="bibr" rid="ref-29">29</xref>, <xref ref-type="bibr" rid="ref-31">31</xref> </sup>.

Orange block diagrams represent the methods mostly discussed in research papers.

Nowadays, the least square method is the most used method to identify the dynamic parameters, because the least square method can directly minimize the root mean square error and the implementation of the least square method is simple. This error refers to the difference between the measured joint torque and the predicted joint torque. However, the least square method has two problems 26 : a. The value of parameters identified by the least square method is not the optimal local minimum, b. The least-squares method is sensitive to measurement noise. Maximum likelihood estimation is used to reduce the estimation error caused by measurement noise since this method estimates parameters with the known statistical characteristics of measurement noise of joint torque and joint position. However, the rate of convergence of the maximum likelihood method is slow. The instrumental variable method is to use the instrumental variables to reconstruct the instrumental matrix, and then use the least square method to estimate the dynamic parameters. The instrumental variable method reduces the identification error of measurement noise and accelerates the rate of convergence.

We present the linear least square method to illustrate the basic process of parameter identification. The least-squares problem is shown in Equation (18) 9 , which is to find a set of dynamic parameters to minimize the residual error. τ i * is the estimated torque of the joint i, and τ i is the measured torque of the joint i. Then, the cost function ( 19) is given by solving Equation (18) with a matrix 24, 25 . According to the principle of the least square method, the value of the derivative of the cost function should be zero. Thus, a general form of the least-squares method based on matrix form can be presented 24, 25 , as Equation (20). The block diagram of the linear least square method is shown in Figure 2.

min ⁡ ∑ i = 1 n ( τ i * − τ i ) 2 ( 18 ) J ( θ ) = 1 2 ( Y X − τ ) T ( Y X − τ ) ( 19 ) X = ( Y T Y ) − 1 Y T τ ( 20 ) The block diagram of the linear least-squares method.

q r , q • r , q • • r is the reference position, speed and acceleration; q s , q • s , q • • s , τ s is the sample position, speed, acceleration, and torque; q s F , q s F • , q s F • • , τ s F is the filtered sample position, speed and acceleration; β is the identified dynamic parameters.

2.3.1 Maximum likelihood estimation. Maximum likelihood estimation is the statistically correct estimate 32 . The reference 32 first used maximum likelihood estimation method to identify dynamic parameters of the manipulator. The maximum likelihood estimation method is used to reduce the negative effect of noise in sampled data set. The detailed process of maximum likelihood estimation has been described in 32: Using n measured points x i ∈ R p i = 1,..., n to estimate parameters β = ( β 1,..., β r ) T related to a given model G k ( β, x) = 0, k = 1,..., m. To be more precise, G k ( β, x) ≈ 0, G k ( β, x) – ε( β, x) = 0, where ε( β, x) represents the integration of model error and measurement error. However, the characteristics of ε( β, x) is not clear. For the convenience of calculation, suppose the mean value of ε( β, x) is 0, and the covariance is δ ii . For dynamic parameter identification of manipulators, G k ( β, x i ) can be represented as Equation (21) 33 . The maximum likelihood estimation can be expressed as the Equation (22). Then, an objective function is calculated by using the Lagrange multiplier, as Equation (23). Finally, this objective function needs to be minimized to find the optimal identification result. However, the calculation of solving the minimization of Equation (24) is difficult, especially to solve the second derivative and the third derivative of the objective function (23). Thus, a linear approximation of G k ( β, x i ) must be given, as Equation (24) 32 .

G k ( β , x i ) = Y ( q , q • , q • • ) β − τ ( 21 ) { min 1 2 ∑ i , j = 1 n ( x i − ψ i ) T [ δ i j 2 ] − 1 ( x i − ψ i ) G k ( β , ψ i ) = 0 ∀ i , k ( 22 ) 1 2 ∑ i , j = 1 n ( x i − ψ i ) T [ δ i j 2 ] ( x i − ψ i ) + ∑ i , k = 1 n , m λ i , k G k ( β , ψ i ) ( 23 ) { min 1 2 ∑ i , j = 1 n ( x i − ψ i ) T [ δ i j 2 ] ( x i − ψ i ) [ B ( i ) ( β ) ] ( x i − ψ i ) = b ( i ) ( β ) ( 24 )

Where, B ( i)( β) is a m× p matrix, the element of kth row of is B ( i) ( β) is [∇ x G k ( β, x i )] T , b ( i) is a vector, which elements are – G k ( β, x i ). The analytical solution of Equation (24) can be represented as Equation (25), and the optimization problem can be rewritten as Equation (26) 32 .

( x i − ψ i ) = δ i j 2 [ B ( i ) ( β ) ] T { [ B ( i ) ( β ) ] δ i j 2 [ B ( i ) ( β ) ] T } T b ( i ) ( β ) ( 25 ) min 1 2 ∑ i , j = 1 n [ b ( i ) ( β ) ] T { [ B ( i ) ( β ) ] δ i j 2 [ B ( i ) ( β ) ] T } T b ( i ) ( β ) ( 26 )

The maximum likelihood estimation method avoids part of errors of the parameters identification result caused by measurement noise, but the statistical characteristics of the noise must be determined. Compared with the least square method, the negative effects of noise are suppressed. However, the calculation amount of the maximum likelihood estimation method is larger than the least square method, and the increase of the amount of parameter identification does not greatly improve the accuracy of the parameter identification results.

2.3.2 Instrumental variable method. The instrumental variable method 22, 27 is iterative. The instrumental variable method reconstructs an instrumental matrix Z to estimate dynamic parameters. When the instrument matrix fulfills the following conditions, a. Z T X is a full column rank, b. the instrument matrix must be uncorrelated with the error, the dynamic parameters can be calculated as Equation (29) 22 . Thus, how to calculate the instrument matrix is important. The direct dynamic model is the most auxiliary model to calculate the instrument matrix., The direct dynamic model can be calculated as Equation (27) with the previous moment estimated instrumental variable β t -1, and the simulated data ( q, q ˙ , q ¨ ) can be calculated from the direct dynamic model 27 . Then, a reconstructed instrumental matrix can be calculated as Equation (28). Finally, the β t can be calculated, as shown in Equation (29)

M ( q s , β t − 1 ) q s • • = τ s − N ( q s , q s • , β t − 1 ) ( 27 ) Z = X ( q s , q s • , q s • • , β t − 1 ) ( 28 ) β t = ( Z T X ) − 1 Z T y ( 29 )

The instrumental variable method is a closed-loop method, which is more stable than the least square method. Compared with the maximum likelihood estimation method, it can better guarantee the convergence of the data. The instrumental variable method and the least square method are also the most widely used and discussed methods for the identification of manipulator dynamic parameters.

2.3.3 Closed-loop output error method. The closed-loop output error method uses the torque error between actual torque and the simulated torque to estimate the dynamic parameters, the structure of this method is shown in Figure 3. First, the direct dynamic model, such as Equation (30) 34 , and the estimated value of the measured torque and dynamic parameters are used to calculate the position, velocity, and acceleration. Then, the values of position, velocity, and acceleration obtained by simulation are compared with the actual measured values of position, velocity, and acceleration. Generally, the identification just uses the position error 34 . For convenience, the positive dynamic model is transformed to a state-space model, as Equation (31) 34 .

The block diagram of the closed-loop output error method.

q r , q • r , q • • r is the reference position, speed and acceleration; q s , q • s , q • • s , τ s is the sample position, speed, acceleration and torque; q s F , q s F • , q s F • • is the filtered sample position, speed and acceleration; q d d m , q • d d m , q • • d d m is the simulated position, speed, and acceleration; β is the identified dynamic parameters.

 M ( q d d m , β ) q • • d d m = τ d d m − N ( q d d m , q • d d m , β ) ( 30 ) G ( x s ) x • s = f ( x s , u s ) y s = C s x s + D s u s ( 31 ) G ( x s ) = [ I n 0 n × n 0 n × n M ( q d d m , β ) ] , f ( x s , u s ) = [ q • d d m τ d d m − N ( q d d m , q • d d m , β ) ] C s = [ I n × n 0 n × 2 n ] ,   D = 0 n × n ( 32 )

Where, G( x s ), f( x s,u s ), C s , D s are shown in Equation (32), I n is the identity matrix, x s = [ q ddm , q ˙ ddm ], u s = τ ddm and y s = q ddm . Finally, the estimated values of dynamic parameters are corrected according to the position error. The closed-loop output error method is also an iterative process, and to ensure the optimal identification result, a quadratic standard such as Equation (33) is given, which is a nonlinear least-squares problem. Y s and Y are obtained by filtering the vectors of samples Y fm and Y sfm , where Y f m = [ Y f m 1 , … , Y f m n ] T , Y f m j = [ q f m 1 , … , q f m n ] T and Y s f m = [ Y s f m 1 , … , Y s f m n ] T , Y s f m j = [ q s f m 1 , … , q s f m n ] T .

J ( x ) = ‖ Y s − Y ‖ 2 ( 33 )

In general, the Newton method, the Lewinberg Marquardt method, etc. are used to calculate the minimized result of the Equation (33) 34 . The closed-loop output error method also has a deformation method, the closed-loop input error method, which turns the position error into a torque error, that is, the error between the simulated model torque and the actual measured torque.

2.3.4 Particle swarm optimization algorithm. The particle swarm optimization method is an intelligent evolution algorithm. The particle swarm optimization method is used to solve the negative effect of measurement noise and model error on the identification of dynamic parameters. The structure of the particle swarm optimization identification method is similar to the closed-loop input error method. The error of the measured torque and the simulated torque is used to identify the dynamic parameters, as shown in Figure 4. Similarly, the particle swarm algorithm also needs a cost function or constraint 31 . The cost function given in 31 is as Equation (34).

The block diagram of particle swarm optimization algorithm.

q r , q • r , q • • r is the reference position, speed and acceleration; τ s is the sample torque; β PSO is the estimated parameters.

 E ( k ) = 1 M ∑ i = 1 M e 1 2 ( i ) + ... + e j 2 ( i ) + ... + e n 2 ( i ) ( 34 )

Where M is the number of sampling points, n is the number of joints, and e j ( i) is the error between the actual measured torque and the simulated torque of the j joint. Then, many particles perform the swarm movement to find the optimal solution. That is, change the velocity or acceleration of each particle according to the local optimal solution and the global optimal solution, as in formula (35).

V i ( t ) = λ ( V i ( t − 1 ) + a 1 r 1 ( p b e s t i − P i ( t − 1 ) ) ) + a 2 r 2 ( g b e s t i − P i ( t − 1 ) ) ) ( 35 )

Where pbest i is the local optimal solution of particle i, gbest i is the global optimal solution of particle i, and V i ( t – 1) is the speed of the particle i at moment t-1, V i ( t) is the speed of the particle i at moment t, a 1 and a 2 are constants bigger than 0, r 1 and r 2 are random numbers between 0 and 1. λ is the construction factor, as Equation (36). Thus, the position function of particle i as Equation (37). Finally, take the cost function ( 34) as the position of the particles, and then, given the global optimal solution and the local optimal solution of the cost function, the iterative process is used to estimate the optimal solution of the dynamic parameters.

λ = 2 | 2 − b − b 2 − 4 b | , b > 4 ( 36 ) P i ( t ) = P i ( t − 1 ) + V i ( t ) ( 37 )

The intelligent optimization algorithm can obtain a better identification result under the premise of sufficient data and can ensure the global optimum while ensuring the local optimum. However, there is also a problem that the data cannot be converged, which also causes the number of iterations of the particle swarm algorithm to be uncertain.

The confidence of the dynamic model with identified dynamic parameters should be verified 4 . The specific trajectory is used to implement verification experiments. Then, the difference between the curve of actual joint torque and the curve of estimated joint torque from the dynamic model will be intuitively presented, such as reference 35 . Numerically, there are three methods to illustrate the difference between the actual torque and estimated torque: Root means square error (RMS), residual standard deviation (RSD), and protocol index (AI) 36 . The expressions of these methods are shown in Equation (38).

R M S = ∑ i = 1 N ( o i − p i ) 2 N R S D = ∑ i = 1 N ( o i − p i ) 2 ∑ i = 1 N o i 2 A I = 1 − ∑ i = 1 N ( o i − p i ) 2 ∑ i = 1 N ( | o i ' | − | p i ' | ) 2 ( 38 )

Different identification methods will generate different results. Thus, how to choose an identification method is a basic issue. The Reference 37 presented many comparative experiments and discussions on several identification methods, such as least squares, maximum likelihood estimation, extended Kalman filtering, instrumental variable method, and recurrent neural networks. These experiments were implemented on TX40 and RV2SQ manipulators. Three evaluation indicators are given: noise immunity, estimation accuracy, and convergence speed. A summary from these three aspects 37 : a. all methods are very sensitive to the noise of the joint position signal, on the contrary, the joint torque measurement noise has a relatively small impact on the estimated statistical characteristics of these methods; b. Under the same noise environment, the instrumental variable method has higher identification accuracy; c instrumental variable method also has the fastest convergence. A comparison of typical identification methods is shown in Table 2.

Identification methods comparison <sup> <xref ref-type="bibr" rid="ref-37">37</xref> </sup>.
Methods Theory Advantage Disadvantage
Linear least square method. Least square easy to implement; Minimize the root mean square error The identification result is not accurate enough
Closed-loop output error method. Non-linear least square the identification accuracy is high; The experiment is complicated
Instrumental variable method. Least square The identification accuracy is high; the rate of convergence is fast. A high detailed direct dynamic simulation model is necessary
Partial swarm optimization Evolutionary algorithm The identification accuracy is high. The rate of convergence is uncertain.
Maximum. Likelihood estimation. Statistical Noise immunity is good. The rate of convergence is slow.
3 The physical feasibility and consistency of dynamic parameters

Reference 38 first mentioned that the physical feasibility of the dynamic parameters would affect the accuracy of the dynamic model in simulation in 2000. However, scholars did not pay more attention to this problem until reference 39 . The physical feasibility became a research hotspot in the dynamic parameters identification with the development of human-robot interaction. When dynamic parameters are estimated by the numerical method, the value of dynamic parameters may be less than 0, so the dynamic parameters are physically infeasible. This problem reduces the accuracy of the dynamic model when we use the Newton-Euler recursion method in simulation.

The physical feasibility of dynamic parameters is that the values of mass and inertia matrix of links must be greater than or equal to 0. Thus, the criteria of physical feasibility must be discussed first. Reference 39 optimized the identification of dynamic parameters with constraints in 2013 IEEE/RSJ International Conference on Intelligent Robots and Systems(IROS). Reference 39 uses CAD parameters as prior values to optimize the identified dynamic parameters so that the physical feasibility of dynamic parameters can be ensured. Reference 40 also uses CAD dynamic parameters as prior values to optimize the identification parameters and uses the simulated annealing algorithm (SA) to achieve an optimal identification of the standard dynamic parameters. However, the accuracy of CAD parameters is limited by the CAD software and the mechanical accuracy of the manipulators. Therefore, reference 11 proposed a method based on linear matrix inequality, the inequality constraints as Equation (39). Moreover, the semi-definite programming method is used to estimate optimal dynamic parameters, which are physical feasibility. The dynamic parameter identification method can only identify base dynamic parameters. Thus, the optimization problem can be described as how to map from the physical feasibility set of standard dynamic parameters to the physical feasibility set of base parameters. Figure 5 shows the physical feasibility sets in standard dynamic parameter set and base dynamic parameter set, and an intermediate virtual space D β is used to translate the space D β to space D βB , which is illustrated in 11.

Physical feasibility set in standard dynamic parameter set and physical feasibility set in base dynamic parameter.

δ is the standard dynamic parameter set; D is the physical feasibility set in space δ; β is the base dynamic parameter base; D βB is the physical feasibility set in space β. D β is the intermediate virtual physical feasibility space described in reference 11 .

 { m i > 0 L i − 1 m i S ( l i ) T S ( l i ) > 0 ( 39 )

Where l i is first moment of link i, l i = [ l i,x l i,y l i,z ] T – m k r k – [ m i r i,x m i r i,y m i r i,z ] T , r i is the centroid of link, r i = [ r i,x r i,y r i,z ]. S( x): Skew symmetric matrix operator，as Equation (40).

S ( x ) = [ 0 x 3 − x 2 − x 3 0 x 1 x 2 − x 1 0 ] , x = [ x 1 x 2 x 3 ] ( 40 )

To further optimize the accuracy of identification results, reference 41 proposed another physical consistency constraint, a triangular inequality of the inertial matrix. Such as Equation (41). I i,x , I i,y , I i,z are the characteristic roots of the inertial matrix, which guarantees the physical consistency of identified dynamic parameters. The problem of dynamic identification is described as a manifold optimization problem. Moreover, the constraint proposed in 41 is a relatively complete physical meaning constraint. A novel optimization problem is introduced by these constraints.

{ I − i , x + I − i , y > I − i , z I − i , y + I − i , z > I − i , x I − i , z + I − i , x > I − i , y ( 41 )

This optimization problem of physical feasibility and physical consistency is to find a local minimum value. Reference 42 combines the inertial matrix triangle inequality to the LMI-SDP (linear matrix inequality-semi-definite programming) framework, and gives the specific implementation process. In addition, under the framework of linear matrix inequalities and inertial triangular matrix inequalities, reference 43 gives an optimization method based on the penalty mechanism, and reference 44 uses a global optimization method.

The physical feasibility and consistency of dynamic parameters is a hotspot of current dynamic parameter identification research. By ensuring the physical feasibility and consistency of the dynamic parameters, the dynamic parameters can be identified more accurately. At present, the most discussed constraint is linear matrix inequality and the inertial triangular matrix inequality. However, the physical feasibility and consistency problems have multiple minima value in a local area, and it is necessary to choose a suitable optimization method to calculate the optimal value.

 4 Friction force model and identification method

Equation (1) is a basic dynamic mathematical model in joint space without considering friction, but a complete dynamic model must include friction. The friction force is a key factor that affects the accuracy of the dynamic model of the manipulator, especially in the process of joint commutation. However, the friction force is a typical nonlinear force, and there are greater difficulties in modeling friction, including the research on plastic deformation, fluid mechanics, wave transmission, and material science 45 . The methods of friction model generally use empirical modeling methods. The friction modeling method can be divided into two categories: static friction model and dynamic friction model.

The structure of the static friction model is simple and intuitive, and the static friction model is a static function of the relative speed of the contact surfaces, so it is mostly used in engineering applications. In robotics, the most familiar static friction model is the Coulomb friction model combined with the viscous friction model, as Equation (42) 45 , f c is the Coulomb friction and f v is the viscous friction. The relationship between the friction force and friction model parameter is linear. Thus, linear least squares can also be used for identifying friction model parameters, and the set of dynamic parameters can be expanded, as Equation (43) 45 . The static friction model has been used in several papers, such as reference 46, 47 .

τ f = f c s i g n ( q • ) + f v q • ( 42 ) X i = [ X X i , X Y i , X Z i , Y Y i , Y Z i , Z Z i , m i , m i x i , m i y i , m i z i , f c , f v ] ( 43 )

However, the static friction model cannot fully illustrate the dynamic characteristics of the friction, especially the transition from static process to sliding friction, that is, the pre-sliding process. Therefore, many scholars have studied the dynamic friction models 45 . The dynamic friction models are described by a set of state variables that not only considers the current state of the system, but also consider the past state of the system. The methods of dynamic friction models include the Dahl model 48 , Bristle model 49, 50 , Bliman-Sorine model 47 , LuGre model 48 , and maxwell sliding model 51 . The LuGre model is an improvement of the Dahl model, combined with other dynamic characteristics of friction, such as the Strebeck effect. Adding additional dynamic characteristics can also form an extension of the LuGre model, such as elasto-plastic model 52 . These models have been used to compensate for the joint friction force errors, such as reference 53 and reference 54 . However, neither the Dahl model nor the LuGre model specifies the characteristics of friction. The reference 55 proposed a generalized maxwell sliding friction model. Moreover, the reference 55 compares several different dynamic friction models and describes some specific characteristics of friction force. These characteristics are also the evaluation index for describing the quality of a friction model. Sliding process, Stribeck characteristics, friction hysteresis, separation force, and drift characteristics. References 56, 57 use the generalized Maxwell slip model (GMS) to design an adaptive friction compensation control to improve the accuracy of friction force compensation.

Dahl model:

The Dahl model 48, 58 is as in Equation (44), where x is the relative sliding position of the two metal surfaces, F is the friction force, F c is the Coulomb friction, σ is the stiffness coefficient, and α is the coefficient that determines the stress-strain curve.

d F d x = σ ( 1 − F F c sgn ⁡ v ) α ( 44 )

The Dahl model describes the change in friction between two metal surfaces. Compared with the static friction model, it additionally considers the effect of strain on friction. However, the Dahl model neither considers the Stribeck effect nor the sticking problem.

LuGre model:

The LuGre model 48 is shown in Equation (45), where z represents the deformation of the Bristles, σ 0 is the stiffness of the brush, σ 1( v) is the damping of the Bristles, v is the relative velocity of the contact surface, and f( v) is the viscous friction. The LuGre model regards the contact points of two contact surfaces as the bristles.

{ d z d t = v − σ 0 | v | g ( v ) z F = σ 0 z + σ 1 ( v ) d z d t + f ( v ) ( 45 )

Generalized Maxwell slip model (GMS):

The generalized Maxwell slip model 51 is composed of multiple essential friction models connected in parallel, as shown in Figure 6, and the LuGre model is used to give the expression of essential friction, as in Equation (46), where z is the deformation of the Bristles, k i is the stiffness, W i is slip force limit. Comparing Equation (46) and Equation (45), the middle term and viscous friction of Equation (45) are omitted. First, the middle term of Equation (45) has no corresponding physical explanation. Second, viscous friction is additional superimposed friction and is not a component of the essential friction models. The general form of the first derivative of z is shown in formula (47), where g*( t) and h*( t) need to be defined. Under the premise of steady state, dz / dt = 0, g*( t) = z ss can be calculated, z ss is the stiffness of the Bristles in the steady-state, therefore, Equation (47) can be rewritten as Equation (48), v is the relative velocity of the contact surface, and s( v) is the Stribeck curve. h( t) can be approximated as C/ | v|, C is a constant.

The model of Generalized Maxwell slip friction model.

The z is the deformation of the Bristles; k i is the stiffness; W i is the slip force.

 F = σ 0 z ( 46 ) d z d t = v ( 1 − z g * ( t ) ) h * ( t ) ( 47 ) d F d t = v ( 1 − sgn ⁡ ( v ) F f s ( v ) ) h ( t ) ( 48 )

Then, it is necessary to merge the pre-sliding model and the sliding model to give the expression of the essential friction model, such as Equation (49). The expression of the complete generalized Maxwell sliding friction force model can be expressed as formula (50), which σ 2 is the viscous friction coefficient.

d F i d t = sgn ⁡ ( v ) C ( α i − F i s ( v ) ) ( 49 ) F f ( t ) = ∑ i = 1 N F i ( t ) + σ 2 f ( v ) ( 50 )

The dynamic parameters identification process of friction can be divided into two types: 1. Identify the friction model parameters individually; 2. The friction parameters are identified as part of dynamic parameters. For much research, the simplest static friction model, Coulomb friction + viscous friction, is often used, and the friction model parameters are identified as part of dynamic parameters, such as reference 36, 38 . However, when accurate force detection and force control are required, this indirect identification strategy will degree the accuracy of the dynamic model. Identifying the parameters of the friction model separately is a way to use a specific trajectory for friction, which can avoid the interference caused by other dynamic parameters.

The friction force is also influenced by some parameters which are speed weakness dependent, such as load, temperature 59 . The reference 60– 62 has discussed the effect of temperature parameters on the friction force with many experiments. The results of experiments show that the temperature has a large impact on the accuracy of the friction model. The modeling method of temperature parameters of friction force usually adopts the four-parameters method and six-parameters method 61 . Reference 61 also compared these two modeling methods experimentally. Under sufficient experimental data, the fitting effect of the six-parameter method is significantly better than that of the four-parameters method. However, this will cause obvious overfitting in the actual robot arm parameter identification, which may degree the accuracy of the friction model. In addition, the four-parameters method and six-parameters method are linear modeling methods. Thus, there is a linear relationship between temperature parameters and friction force. The linear least square method can be used to identify temperature parameters of the friction model directly. The detail of the design of the four-parameters method can be found in reference 62 .

In robotics, the complex dynamic friction models are not used very commonly, due to the typical nonlinear characteristics of some friction force models which make identification and control more difficult. Reference 62 compared the identification results of nonlinear regression methods, linear regression methods, and polynomial linear fitting methods by using GMS models. The experimental results clearly illustrate the better accuracy of predicted torque of dynamic models by using nonlinear regression methods and nonlinear dynamic models. Thus, it is very necessary to model the friction force model with nonlinear characteristics completely. At present, the influence of the weakness speed-dependent factors on the friction force and identification methods are also the main research topics 62 . Improving the accuracy of friction force model will significantly improve the accuracy of predicted torque.

 5 Problems and future works

Based on the above review of the identification of dynamic parameters of manipulators, the identification errors are mainly caused by the noise in sampled data. Moreover, the non-linear characteristics of friction force also complicate parameters identification, since it is important and complicated to choose a proper modeling method for friction force in manipulators. Besides, analytical modeling methods inevitably result in identification errors, and it is difficult to reduce the identification error by adding details to the dynamic model. Moreover, a more complicated dynamic model does not fully guarantee that the identification accuracy can be improved, and it greatly complicates the identification.

For structure errors, some scholars reduced the structure errors without using deep learning and neural networks. This type of method is also called the non-parametric regression method, which is used to learn the inverse dynamic 63 . Reference 64 proposed a polynomial approximation method by using Jacobian polynomials (hypergeometric polynomials). This method used the iterative methods to make the Jacobian matrix approximate the actual regression matrix Y, and then the identification methods were used to identify dynamic parameters with the estimated regression matrix. For robot dynamic model learning and parameter identification optimization, many scholars are trying to use deep learning and neural networks for research 37 . In research 37 , the application of basic neural networks in dynamic parameters identification is introduced, involving Adaline neural networks and recurrent neural networks.

ADALINE neural network:

Adaline neural network is an adaptive linear neural network based on the least square method, and the calculation model of the Adaline neural network is shown in Figure 7. For the manipulator, the cost function used by the Adaline model can be the square of the torque error between the actual torque and the simulated torque, as shown in Equation (51). The network weighted value changes according to the partial derivative of the cost function, as shown in Equation (52), where η is the learning rate. Thus, the gradient descent method can be used to find the weight which makes the torque error smallest.

Calculation model of Adaline neural network.

X i represents the state, W i represents the weights.

 J = e T e ( 51 ) w i = w i − η Δ J ( 52 )

Recurrent neural network:

The recurrent neural network is very effective for data with sequential characteristics, and it can mine the timing information and semantic information in the data. The reference 65 first uses recurrent neural network for parameter identification of system, the calculation model of the recurrent neural network as shown in Figure 8, and the mathematical model is as Equation (53).

Calculation model of Recurrent neural network.

X i is intermediate state; u is the input; w ij is the weight between i neuron and j neuron.

 X • i = − X j + ∑ j = 1 n w i j δ ( x j ) + γ i u ( 53 )

Where, X i is the intermediate state, u is the input, w ij is the weight between neuron i and neuron j. Reference 37 mentioned a method that use hyperbolic tangent function as the activation function, as Equation (54).

δ ( x i ) = α i tanh ⁡ ( x i ϑ i ) ( 54 )

Where α i and ϑ i are constants bigger than 0. Moreover, the state of the recurrent neural network will naturally converge to the minimum value of the energy function by using Equation (55) as an active function. The energy function is shown as Equation (53).

E = − 1 2 ∑ i = 1 N ∑ j = 1 N w i j δ ( x i ) δ ( x j ) + ∑ i = 1 N γ i u δ ( x j ) ( 55 )

Recurrent neural networks have the problem of difficulty in training. Therefore, we can consider using an improved type of recurrent neural network, such as long-short-term memory network. The long-short-term memory network adds a new state to the original recurrent neural network, which is used to save the state for a long time and solve the problem of gradient disappearance.

The application of these neural networks is also to optimize the results of the dynamic parameters identification of the manipulator. By converting and transferring the output error to the weight of the neural network, the optimal identification of the dynamic parameters is realized. However, there is no specific change in the structure of the identification algorithm, only the traditional optimization algorithm is replaced by the neural network methods, and the problem of structural errors of parameter identification has not been solved. Therefore, many scholars have used the data-driven characteristics of neural networks to propose methods for system identification and dynamic model learning, e.g. reference 66, 67 . They are no longer limited to optimizing the parameter identification results, but also consider the problems of the model itself.

Reference 66 uses a recurrent neural network, using a data-driven approach to improve the dynamic model's ability to respond to environmental changes in real-time. This method dynamically adapts to the input and output characteristics of the dynamic model and avoids the limitations of theoretical modeling. Reference 67 proposed a neural network-based uncertainty compensation model to compensate for the errors caused by uncertain factors in the process of dynamic parameter identification and to solve the problems caused by structural errors. However, there is no standard data set to train the model. The actual collected data often contains a lot of noise. Thus, reference 68 proposes a "Gen2Real" method to transfer the pre-training model to the actual robot, which can more accurately realize the learning of the dynamic model. The pre-training model is trained in the simulation environment. These methods all require a lot of time to collect and process data. Thus, reference 69 uses the Generative Adversarial Networks (GAN) to approximate the dynamic characteristics as much as possible under a limited data set. The adversarial neural network can generate additional data based on the existing data set. Reference 70 also uses the characteristics of GAN to generate data to solve the problem of traditional excitation trajectory design over-reliance on evolution and gradient-based optimization methods. These optimization methods generate a single long trajectory which reduces identification efficiency. In addition, a GAN-based method for simultaneously generating multiple trajectories is proposed, which improves the efficiency of trajectory optimization.

The above methods are iterative regression methods based on a large amount of data to obtain dynamic models and optimize the result of dynamic parameter identification. However, the dynamic model and parameters of the manipulators are not always constant, and the value of dynamic parameters may change with the execution of the task 71 . For traditional parameters identification methods or model learning methods, there is not enough time and data to obtain an accurate dynamic model or dynamic parameters in real-time. Many scholars have proposed solutions to this problem, which can be divided into two categories: 1. Incremental learning methods 72, 73 , 2. Error model learning methods 74, 75 . For dynamic model and dynamic parameter identification, the changing characteristics of the dynamic model should be studied further. A more flexible dynamic model can make the manipulator better interact with the environment.

The nature of neural networks is data-driven, which allows researchers to identify the characteristics of dynamic model of the manipulator directly. This makes it possible to obtain a higher precision dynamic model than analytical modeling methods. Thus, many scholars have proposed many methods on the research of system identification and dynamic model learning, but, these methods lack systematical theoretical analysis, especially for the problem of structural errors. How to use neural networks to reduce the structural errors of dynamic models has become a research hotspot. Moreover, calibrating the dynamic model and dynamic parameters of the manipulator through real-time identification methods is also an interesting solution for structural errors.

 6 Conclusions

This paper presents an overview of excitation trajectory optimization, identification methods, and hotspots in dynamic parameters identification. The main problem is estimating the dynamic parameters from the measured data optimally. First, the design of the excitation trajectory is an important part of suppressing the measurement noise. This paper divides excitation trajectory design into two parts, excitation trajectory optimization, and excitation trajectory parameterization. The use of finite Fourier series and intelligent optimization algorithms can reduce the noise level in sampled data set and accelerate the convergence of the dynamic parameter identification. Moreover, the identification of dynamic parameters must filter the noise and singular values in the sampled data set to ensure the accuracy of the identification result. However, the estimated dynamic parameters may be also physically unfeasible by using the numerical identification methods. Thus, many scholars have proposed constraint-based optimization methods for optimizing the estimated dynamic parameters. Linear matrix inequalities and triangular inequalities are commonly used to ensure the physical feasibility and consistency of dynamic parameters. In addition, the identification of the parameters of the friction model also has an impact on the accuracy of the dynamic model of manipulators, and many scholars have conducted in-depth research on the model of friction force. Among them, the static friction models have been widely used. However, according to many surveys by scholars, the accuracy of the static friction model is not enough when manipulators interact with humans or the environment. Now, a more precise dynamic friction model should be studied, and some nonlinear factors should be considered, such as temperature.

The traditional identification methods of dynamic parameters based on analytical models require special trajectories, long identification times, and complex optimization methods. The most important problem is that there are unavoidable structural errors in analytical modeling methods, which limit the accuracy of the identification methods that can be achieved. Therefore, some scholars try to use deep learning and neural network methods to reduce structural errors. At the same time, some real-time identification methods have been proposed to calibrate dynamic parameters online.

Identification of dynamic parameters of manipulators is very important for manipulator dynamic control, since an accurate dynamic model is the fundamental part of a collaborative manipulators system.

Nowadays, the manipulators are mobile manipulators, which are combined with a mobile trolley or a lifting platform. This hybrid system is a floating base system. Thus, the manipulators will control the contact force during the movement. First, there is a dynamic modeling problem in joint space and interactive space. Secondly, just identifying the base parameters may not satisfy the demand of the dynamic control for mobile manipulators, since the base of the manipulator is mobile.

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Competing interests: No competing interests were disclosed.

 27 102022 Copyright: © 2022 Roveda L 2022 This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. recommendation approve-with-reservations

The paper reviews the state of the art related to dynamics parameters identification for robot control purposes.

Comments:

Matrices and vectors should be bold;

The paper should consider also some other approaches based on ML, such as these papers 1 - 3 . This is important to provide more insights to the last  generation data-driven methods;

Some other papers such as Beschi et al., (2015) 4  and Pedrocchi et al., (2014) 5 should be discussed. The three papers I mentioned above are related to the topic covered by the proposed paper and their discussion and, together with the other two mentioned papers, would improve the state of the art analysis that is performed by the authors;

In general, it seems that the review is not taking into consideration all the approaches in the state of the art (see comment 1). A more clear description on the selection procedure for the paper included in the review has to be provided;

Check the English for typos.

Is the review written in accessible language?

Yes

Are all factual statements correct and adequately supported by citations?

Yes

Are the conclusions drawn appropriate in the context of the current research literature?

Yes

Is the topic of the review discussed comprehensively in the context of the current literature?

Yes

Reviewer Expertise:

Control; modelling; robotics; machine learning

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard, however I have significant reservations, as outlined above.

References : Robot joint friction compensation learning enhanced by 6D virtual sensor. International Journal of Robust and Nonlinear Control .2022;32(9) : 10.1002/rnc.61085741-5763 10.1002/rnc.6108 : Iterative Learning Procedure With Reinforcement for High-Accuracy Force Tracking in Robotized Tasks. IEEE Transactions on Industrial Informatics .2018;14(4) : 10.1109/TII.2017.27482361753-1763 10.1109/TII.2017.2748236 : Robot control parameters auto-tuning in trajectory tracking applications. Control Engineering Practice .2020;101: 10.1016/j.conengprac.2020.104488 10.1016/j.conengprac.2020.104488 : A general analytical procedure for robot dynamic model reduction. 2015 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS) .2015; 10.1109/IROS.2015.73539604127-4132 10.1109/IROS.2015.7353960 : Robot-dynamic calibration improvement by local identification. 2014 IEEE International Conference on Robotics and Automation (ICRA) .2014; 10.1109/ICRA.2014.69077425990-5997 10.1109/ICRA.2014.6907742 10.21956/cobot.18719.r26831 Reviewer response for version 1 Jiang Quansheng 1 Referee https://orcid.org/0000-0002-4148-3438 School of Mechanical & Electronical Engineering, Suzhou University of Science and Technology, Suzhou, China

Competing interests: No competing interests were disclosed.

 8 32022 Copyright: © 2022 Jiang Q 2022 This is an open access peer review report distributed under the terms of the Creative Commons Attribution Licence, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. recommendation approve

The identification of the dynamic parameters of manipulators in robot field is a research hotspot. This paper give an overview of the modeling of manipulator dynamics, the optimization methods of excitation trajectory, the identification methods for dynamic parameters, and the identification of friction model parameters. The work is interesting. The framework of the manuscript is reasonable and the overall quality is good. Specific recommendations are as follows:

In the abstract, the authors point out that the instrumental variable method with complete physical feasibility constraints is an optimal choice for dynamic parameter identification. Please give specific instructions in the content.

In the introduction, please add the reviews about deep learning methods based parameters identification for manipulator control.

The future works of parameters identification should be further analysis.

To improve the readability of the submitted manuscript, the editing of typos and grammar expression should be checked.

Is the review written in accessible language?

Partly

Are all factual statements correct and adequately supported by citations?

Yes

Are the conclusions drawn appropriate in the context of the current research literature?

Yes

Is the topic of the review discussed comprehensively in the context of the current literature?

Yes

Reviewer Expertise:

Robot and automation; Mechanical signal processing.

I confirm that I have read this submission and believe that I have an appropriate level of expertise to confirm that it is of an acceptable scientific standard.