Methods for inertial parameters identification of manipulator based on multibody dynamics and neural networks

Abstract

In this paper, two methods for directly identifying inertial parameters of manipulator are proposed, which solves the problem that inertial parameters are difficult to identify due to coupling. First, a Model Identification Method (MIM) is proposed based on multibody dynamics. MIM realizes the decoupling of inertial parameters by the mathematical description of graph theory, and has extremely high identification accuracy. Since complex modeling and calculation will affect the real-time performance of the algorithm, MIM is suitable for offline identification. Second, a numerical identification method (NIM) is proposed based on neural networks. NIM avoids complex mechanical modeling process and obtains the approximate value of inertial parameters by numerical fitting method. Compared with MIM, the identification results lose some accuracy. However, NIM has excellent real-time performance and is suitable for online identification. Finally, the effectiveness of the method is verified by simulation experiments. MIM and NIM have complementary characteristics and different application scenarios, which can meet the vast majority of identification needs. The obtained results shed light on achieving precise control of manipulators.

Keywords

Multibody dynamics parameters identification neural networks inertia parameters graph theory

Introduction

Identifying the inertia parameters (mass and principal moment of inertias of each rigid body) of robots is of great significance for their control,¹ the accuracy of the inertia parameters largely determines the control accuracy, especially for control based on dynamic models.² However, current research is difficult to directly identify inertial parameters and can only identify their coupling terms, especially for multi degree-of-freedom (DOF) robots. Dynamics studies the relationship between pose (position and posture), velocity (linear and angular velocity), acceleration (linear and angular acceleration), and force (force and moment), while force and kinematic parameters (i.e. the first three) are mathematically related by inertial parameters. Therefore, whether controlling kinematic parameters by force or controlling forces by kinematic parameters, the accuracy of control algorithms largely depends on the accuracy of inertial parameters.^3–5

The dynamic model of a robot includes inertia parameters, while the kinematic model does not. Therefore, to identify inertia parameters, it is necessary to establish a dynamic model of the robot. It should be noted that there is a difference between inertia parameters and dynamic parameters. Inertia parameters refer to the mass and principal moment of inertia, while dynamic parameters refer to the position of centroid, inertia tensor, and mass. Therefore, inertia parameters can be considered as a subset of dynamic parameters. Most existing papers on dynamic parameters identification does not identify strictly defined dynamic parameters, but rather identifies coupling terms of dynamic parameters, such as the product of mass and center of mass,⁶ the mixed terms of inertia tensor, mass, and centroid position,⁷ etc., and in literature,⁸ the coupling term also includes friction parameters, and some papers also refer to the above mixed terms as inertial parameters.⁹ The identification of inertial parameters in a strict sense is still very difficult at present. Due to the parameters coupling is difficult to eliminate in multi-DOF mechanism, the research of inertial parameters identification is mostly limited to the mechanism with few DOFs. For example, the research object of Sharma et al.¹⁰ is a single rigid body with one rotational DOF, and a method to identify the dynamic parameters of 2-DOF rigid body robots is proposed in Gautier et al.¹¹ In addition, some research results on parameters identification of parallel mechanisms have also been obtained. An algorithm to identify the dynamic parameters of parallel robots was proposed in Díaz-Rodríguez et al.,⁶ and the weighted least squares method was adopted to identify the dynamic model established by the Gibbs-Appell equation. An algorithm based on inverse dynamic equation to identify the dynamic parameters of parallel robots was proposed in Briot and Gautier.¹² IDIM-TLS technology was used to enable the robot to track some loaded and unloaded trajectories, in this process, data of motor current and joint position were sampled to calculate the dynamic parameters.

These studies are very meaningful, but usually only customized parameters identification is realized for robots with specific structures or models. The identification method lacks universality and portability, and usually cannot be directly applied to other robots. The reason for this problem is that the coupling term of the dynamic parameters cannot reflect the most essential inertia properties of the robot, and directly identifying the inertia parameters can solve this difficulty. The challenge in identifying inertial parameters in dynamic models lies in inertial parameters decoupling, which is laborious to achieve in traditional Newton Euler and Lagrange methods. Newton Euler’s method^13–16 is a recursive method in which the recursive relationship results in inertial parameters coupling. In the Lagrange method,^17–19 the process of taking partial derivatives of Lagrangian can also lead to inertial parameters coupling. Even if traditional dynamic methods are used with great effort to establish a dynamic model, its lengthy expressions and numerous parameters will bring great difficulties to the calculation, which makes it impossible to guarantee the real-time performance of the identification algorithm.

In view of the shortcomings of traditional dynamics, it is necessary to explore the dynamic modeling method that can realize inertial parameters decoupling, and adopt the method such as least squares^20,21 for identification. This method is the Model Identification Method (MIM). The advantage of MIM is that the identified results are accurate and stable. The disadvantage is that as the DOF increase, dynamic modeling becomes more complex and difficult, and computation becomes more time-consuming, resulting in a decrease in real-time performance.

For multi-DOF mechanisms, considering the aforementioned drawbacks of MIM, it is necessary to propose an algorithm that can improve real-time performance and reduce solution difficulty, at the cost of reducing identification accuracy to a certain extent. The feasible direction of this algorithm is to use numerical analysis methods, which are used to process complex data and obtain approximate solutions. Although some accuracy is lost, it can ensure stable real-time performance. This method is called Numerical Identification Method (NIM).

In this paper, the identification method of inertia parameters of 6-DOF manipulator, which is the most versatile industrial robot, is studied in order to ensure the method as practical as possible. In this paper, MIM and NIM are proposed respectively. For MIM, an identification method based on multibody dynamics is proposed, which can realize the decoupling of inertial parameters of each rigid body, so as to achieve the purpose of direct identification of inertial parameters. For NIM, an identification model based on neural network is proposed. Neural network is a numerical method of fitting, which inputs a certain number of parameters for fitting training to output results approximating the true value. The manipulator has many DOFs, so the modeling and solving of the MIM in this paper is complicated, which affects the real-time performance of the algorithm, but the advantage is that the inertial parameters can be accurately identified. NIM, on the contrary, has the advantage of good real-time performance, but has the disadvantage of precision loss. In the simplest terms, MIM is suitable for offline identification and NIM is suitable for online identification.

The innovative methods are proposed to identify the inertial parameters of the manipulator in this paper, which can directly identify the mass and the principal moment of inertias. The methods have universality and portability. MIM has extremely high identification accuracy and NIM has excellent real-time performance, which can meet the needs of different application scenarios of the manipulator and provide data support for precise control.

MIM based on multibody dynamics

In this paper, an inertial parameters identification method based on multibody dynamics, namely MIM, is proposed. An accurate identification model is established in this method, and the identification results have high precision. However, the modeling and calculation process are complicated, which will affect the real-time performance. Firstly, the decoupling of inertial parameters of each rigid body is realized by the mathematical description of graph theory, which lays a mathematical foundation for the separation and identification of inertial parameters. Secondly, the multibody dynamics model of the manipulator is established, and there is no coupling relationship between the inertial parameters in the model. Finally, the dynamic model is written in least squares form, and the inertial parameters are separated for identification.

Graph theory description

From the perspective of graph theory, the manipulator is a single chain tree system composed of connected rigid bodies. The tree system has a relationship of separation and combination with the single rigid body, and the dynamic model of the manipulator and the dynamic equation of the single rigid body established on this mathematical basis also have a relationship of separation and combination. In geometric description, individual rigid bodies are independent of each other, correspondingly, in algebraic description, the inertial parameters between rigid bodies are decoupled from each other.

It should be noted that both traditional and multibody dynamics methods need to make assumptions about the irregular rigid body, simplify the irregular rigid body into a regular shape, and determine the position of the centroid on this basis, which is the premise of parameter identification. In this paper, the rigid body is assumed to be a homogeneous cylinder, and the center of mass of the rigid body is assumed to be located at the centroid of the cylinder, that is, the midpoint of the axis of the cylinder. In fact, the centroid can be set to be located at any point on the axis, not necessarily at the midpoint of the axis. As long as the centroid is on the axis, we can reasonably establish a coordinate system in the centroid, so that each axis of the coordinate system is the principal axis of inertia, in this case there is only the principal moment of inertias, and the product of inertia is eliminated. There is a one to one correspondence between the position of centroid and its corresponding principal moment of inertias. As long as the dynamical model, parameters identification model and control model all adopt the same centroid position and principal moment of inertias, precise control can be achieved without caring about the precise position of the centroid. Because the real centroid and the principal moment of inertias cannot be measured by disassembly (the robot arm is a complex precision instrument, which is usually not allowed to disassemble), the position of the centroid can only be assumed, and assuming the centroid in the center of the cylinder will simplify the derivation and calculation.

The 6-DOF manipulator is composed of six rigid bodies $B_{i} (i = 1, 2, \dots, 6)$ of length $l_{i} (i = 1, 2, \dots, 6)$ and radius $R_{i} (i = 1, 2, \dots, 6)$ , with six rotational DOFs. The base of the manipulator is B₀. The manipulator is shown in Figure 1. The coordinate base is established at the centroid of each rigid body, and the coordinate base is firmly connected with the rigid body, denoted as $(O_{i}, {\underline{e}}^{(i)}) (i = 1, 2, \dots, 6)$ (bold represents the vector, underline _ represents the matrix form), and the three components $e_{1}^{(i)}, e_{2}^{(i)}, e_{3}^{(i)}$ are the base vectors along the three coordinate axes, pointing respectively to the direction of the three central principal inertia axes, and the world coordinate base $(O_{0}, {\underline{e}}^{(0)})$ is established on the $B_{0}$ . The mass of each rigid body is m_i, and the central inertia tensor is ${\underline{J}}_{i}^{(i)} = diag (J_{i 1}^{(i)} J_{i 2}^{(i)} J_{i 3}^{(i)})$ (with superscript means expressed in $(O_{i}, {\underline{e}}^{(i)})$ , without superscript means expressed in $(O_{0}, {\underline{e}}^{(0)})$ ).

Figure 1.

6-DOF manipulator and its coordinate base diagram: (a) 6-DOF manipulator and (b) coordinate base diagram.

The pose of B_i is recorded as:

{\underline{r}}_{i} = (x_{i} y_{i} z_{i})^{T}, {\underline{θ}}_{i} = (α_{i} β_{i} γ_{i})^{T}

(1a)

{\underline{q}}_{i} = ({\underline{r}}_{i}^{T} {\underline{θ}}_{i}^{T})^{T} = (x_{i} y_{i} z_{i} α_{i} β_{i} γ_{i})^{T}

(1b)

where, ${\underline{r}}_{i}$ is the centroid position coordinate of $B_{i}$ , ${\underline{θ}}_{i}$ is the generalized angle coordinate of $B_{i}$ , that is, the Euler Angle of x-y-z in turn, and the Euler Angle is used as the generalized angle coordinate in this paper, ${\underline{q}}_{i} = ({\underline{r}}_{i}^{T} {\underline{θ}}_{i}^{T})^{T}$ is the generalized coordinate of $B_{i}$ . The generalized velocity ${\underline{\overset{\cdot}{q}}}_{i}$ and the generalized acceleration ${\underline{\overset{\cdot\cdot}{q}}}_{i}$ of $B_{i}$ can be obtained by taking the derivative of ${\underline{q}}_{i}$ .

The incidence matrix $\underline{S}$ and the body-joint vector matrix $\underline{C}$ of the manipulator can be defined by graph theory, which are the key matrices for deriving the multibody dynamics model.

The structural diagram of the tree system of the manipulator is shown in Figure 2(a). The incidence matrix is defined as

\underline{S} = [\begin{matrix} S_{11} & \dots & S_{1 n} \\ ⋮ & ⋱ & ⋮ \\ S_{n 1} & \dots & S_{nn} \end{matrix}]

(2)

Figure 2.

Structure diagram and body-joint vector diagram of the tree system of 6-DOF manipulator: (a) structure diagram and (b) body-joint vector diagram.

Its element $S_{ij}$ is defined as

S_{ij} = {\begin{matrix} 1 & O_{j} has incidence with B_{i} and starts with B_{i} \\ - 1 & O_{j} has incidence with B_{i} and ends with B_{i} \\ 0 & O_{j} has no incidence with B_{i} \end{matrix}

(3)

Therefore, the incidence matrix of the 6-DOF manipulator is

\underline{S} = {[\begin{matrix} - 1 & 1 & 0 & 0 \\ 0 & ⋮ & ⋮ & 0 \\ 0 & 0 & - 1 & 1 \\ 0 & 0 & 0 & - 1 \end{matrix}]}_{6 \times 6}

(4)

The weighted body-joint vector $C_{ij}$ of the manipulator is the vector connecting joint $O_{j}$ and the centroid $O_{ci}$ of the adjacent rigid bodies, and the directions are all away from $B_{0}$ and pointing outwards, as shown in Figure 2(b). The $n \times n$ square matrix composed of $C_{ij}$ is called the body-joint vector matrix of the system and is denoted as $\underline{C}$ . $\underline{C}$ reflects the distribution of joints associated with each rigid body.

The body-joint vector matrix of the 6-DOF manipulator is

\begin{matrix} \underline{C} = [\begin{matrix} \frac{l_{1}}{2} e_{1}^{(1)} & \frac{l_{1}}{2} e_{1}^{(1)} & 0 & 0 & 0 & 0 \\ 0 & \frac{l_{2}}{2} e_{1}^{(2)} & \frac{l_{2}}{2} e_{1}^{(2)} & 0 & 0 & 0 \\ 0 & 0 & \frac{l_{3}}{2} e_{1}^{(3)} & \frac{l_{3}}{2} e_{1}^{(3)} & 0 & 0 \\ 0 & 0 & 0 & \frac{l_{4}}{2} e_{1}^{(4)} & \frac{l_{4}}{2} e_{1}^{(4)} & 0 \\ 0 & 0 & 0 & 0 & \frac{l_{5}}{2} e_{1}^{(5)} & \frac{l_{5}}{2} e_{1}^{(5)} \\ 0 & 0 & 0 & 0 & 0 & \frac{l_{6}}{2} e_{1}^{(6)} \end{matrix}] \end{matrix}

(5)

In graph theory, six independent rigid bodies are combined and described as a manipulator tree system, which can be used to decouple inertial parameters and is the key to dynamic modeling and parameters identification.

Multibody dynamics modeling

The multibody dynamics model of the manipulator is established on the basis of the graph theory description. There is no coupling relationship between the inertial parameters in the model, so the model can be rewritten to the least squares form for parameters identification.

The Newton Euler equation of B_i is

m_{i} {\overset{\cdot\cdot}{r}}_{i} = F_{i} (i = 1, 2, \dots, 6)

(6a)

J_{i} \cdot {\overset{\cdot}{ω}}_{i} + ω_{i} \times (J_{i} \cdot ω_{i}) = M_{i} (i = 1, 2, \dots, 6)

(6b)

where the sum of the external forces $F_{i}$ and the sum of the external moments $M_{i}$ on the rigid body $B_{i}$ are

F_{i} = F_{i}^{g} - \sum_{k = i}^{i + 1} S_{ik} F_{k}^{e}

(7a)

M_{i} = - \sum_{k = i}^{i + 1} (C_{ik} \times F_{k}^{e} + S_{ik} \times M_{k}^{a})

(7b)

$F_{i}^{g}$ is the gravity of B_i, $F_{k}^{e}$ is the active force of O_i on $B_{i}$ , and $M_{i}^{a}$ is the active moment of O_i on $B_{i}$ .

For ease of calculation, Newton Euler’s equation equation (6) in vector form is written in matrix form.

m_{i} {\underline{\overset{\cdot\cdot}{r}}}_{i} = {\underline{F}}_{i} (i = 1, 2, \dots, 6)

(8a)

{\underline{J}}_{i}^{(i)} {\underline{\overset{\cdot}{ω}}}_{i}^{(i)} + {\underline{\tilde{ω}}}_{i}^{(i)} {\underline{J}}_{i}^{(i)} {\underline{ω}}_{i}^{(i)} = {\underline{M}}_{i}^{(i)} (i = 1, 2, \dots, 6) .

(8b)

where the matrix symbol $\underline{\tilde{a}}$ with tilde is represented as a third-order antisymmetric matrix.

The rotation matrix ${\underline{A}}^{(0 i)}$ of $(O_{i}, {\underline{e}}^{(i)})$ relative to $(O_{0}, {\underline{e}}^{(0)})$ is

{\underline{A}}^{(0 i)} = [\begin{matrix} c β_{i} c γ_{i} & - c β_{i} s γ_{i} & s β_{i} \\ c α_{i} s γ_{i} + s α_{i} s β_{i} c γ_{i} & c α_{i} c γ_{i} - s α_{i} s β_{i} s γ_{i} & - s α_{i} c β_{i} \\ s α_{i} s γ_{i} - c α_{i} s β_{i} c γ_{i} & s α_{i} c γ_{i} + c α_{i} s β_{i} s γ_{i} & c α_{i} c β_{i} \end{matrix}]

(9)

where s represents the sine function $\sin$ and c represents the cosine function $\cos$ .

The inverse matrix of ${\underline{A}}^{(0 i)}$ is denoted as ${\underline{A}}^{(i 0)}$ . In general, there is no calculus relation between angle and angular velocity, that is, angular coordinates cannot be obtained by integrating angular velocity. Therefore, the Euler Angle is chosen as the generalized angular coordinate in this paper. The equation (8b) is written in generalized angular coordinate form.

{\underline{J}}_{i}^{(i)} {\underline{D}}_{i} {\underline{\overset{\cdot\cdot}{θ}}}_{i} + ({\underline{J}}_{i}^{(i)} {\underline{\overset{\cdot}{D}}}_{i} + \tilde{{\underline{D}}_{i} {\underline{\overset{\cdot}{θ}}}_{i}} {\underline{J}}_{i}^{(i)} {\underline{D}}_{i}) {\underline{\overset{\cdot}{θ}}}_{i} = {\underline{M}}_{i}^{(i)}

(10)

where

{\underline{D}}_{i} = [\begin{matrix} c β_{i} c γ_{i} & s γ_{i} & 0 \\ - c β_{i} s γ_{i} & c γ_{i} & 0 \\ s β_{i} & 0 & 1 \end{matrix}]

(11a)

{\underline{\overset{\cdot}{D}}}_{i} = [\begin{matrix} - {\overset{\cdot}{β}}_{i} s β_{i} c γ_{i} - {\overset{\cdot}{γ}}_{i} c β_{i} s γ_{i} & {\overset{\cdot}{γ}}_{i} c γ_{i} & 0 \\ {\overset{\cdot}{β}}_{i} s β_{i} s γ_{i} - {\overset{\cdot}{γ}}_{i} c β_{i} c γ_{i} & - {\overset{\cdot}{γ}}_{i} s γ_{i} & 0 \\ {\overset{\cdot}{β}}_{i} c β_{i} & 0 & 1 \end{matrix}]

(11b)

The dynamic equation of the rigid body expressed in generalized coordinates is obtained by the combination of equations (8a) and (10).

\begin{matrix} [\begin{matrix} m_{i} \underline{E} & \underline{0} \\ \underline{0} & {\underline{J}}_{i}^{(i)} {\underline{D}}_{i} \end{matrix}] [\begin{matrix} {\underline{\overset{\cdot\cdot}{r}}}_{i} \\ {\underline{\overset{\cdot\cdot}{θ}}}_{i} \end{matrix}] = [\begin{matrix} \underline{0} \\ - ({\underline{J}}_{i}^{(i)} {\underline{\overset{\cdot}{D}}}_{i} + \tilde{{\underline{D}}_{i} {\underline{\overset{\cdot}{θ}}}_{i}} {\underline{J}}_{i}^{(i)} {\underline{D}}_{i}) {\underline{\overset{\cdot}{θ}}}_{i} \end{matrix}] \\ + [\begin{matrix} {\underline{A}}^{(0 i)} {\underline{F}}_{i}^{(i)} \\ {\underline{M}}_{i}^{(i)} \end{matrix}] (i = 1, 2, \dots, 6) \end{matrix}

(12a)

$\underline{0}$ is a matrix with all elements of 0, and for the convenience of expression, ${\underline{0}}_{3 \times 1}$ and ${\underline{0}}_{3 \times 3}$ are abbreviated as $\underline{0}$ in this paper. Equation (12a) is abbreviated as

{\underline{A}}_{i} {\underline{\overset{\cdot\cdot}{q}}}_{i} = {\underline{B}}_{i} (i = 1, 2, \dots, 6)

(12b)

The relationship between angular velocity ${\underline{ω}}_{i}^{(i)}$ and angular acceleration ${\underline{\overset{\cdot}{ω}}}_{i}^{(i)}$ in Cartesian coordinates and angular velocity ${\underline{\overset{\cdot}{θ}}}_{i}$ and angular acceleration ${\underline{\overset{\cdot\cdot}{θ}}}_{i}$ in Euler angles is as follows.

{\underline{ω}}_{i}^{(i)} = {\underline{D}}_{i} {\underline{\overset{\cdot}{θ}}}_{i} (i = 1, 2, \dots, 6)

(12c)

{\underline{\overset{\cdot}{ω}}}_{i}^{(i)} = {\underline{D}}_{i} {\underline{\overset{\cdot\cdot}{θ}}}_{i} + {\underline{\overset{\cdot}{D}}}_{i} {\underline{\overset{\cdot}{θ}}}_{i} (i = 1, 2, \dots, 6)

(12d)

Equations (12c) and (12d) can be expanded as

ω_{i 1}^{(i)} = \overset{\cdot}{α} c β_{i} c γ_{i} + \overset{\cdot}{β} s γ_{i}

ω_{i 2}^{(i)} = - \overset{\cdot}{α} c β_{i} s γ_{i} + \overset{\cdot}{β} c γ_{i}

ω_{i 3}^{(i)} = \overset{\cdot}{α} s β_{i} + \overset{\cdot}{γ}

{\overset{\cdot}{ω}}_{i 1}^{(i)} = - \overset{\cdot}{α} ({\overset{\cdot}{β}}_{i} s β_{i} c γ_{i} + {\overset{\cdot}{γ}}_{i} c β_{i} s γ_{i}) + \overset{\cdot}{β} {\overset{\cdot}{γ}}_{i} c γ_{i} + \overset{\cdot\cdot}{α} c β_{i} c γ_{i} + \overset{\cdot\cdot}{β} s γ_{i}

{\overset{\cdot}{ω}}_{i 2}^{(i)} = \overset{\cdot}{α} ({\overset{\cdot}{β}}_{i} s β_{i} s γ_{i} - {\overset{\cdot}{γ}}_{i} c β_{i} c γ_{i}) - \overset{\cdot}{β} {\overset{\cdot}{γ}}_{i} s γ_{i} - \overset{\cdot\cdot}{α} c β_{i} s γ_{i} + \overset{\cdot\cdot}{β} c γ_{i}

{\overset{\cdot}{ω}}_{i 3}^{(i)} = \overset{\cdot}{α} {\overset{\cdot}{β}}_{i} c β_{i} + \overset{\cdot}{γ} + \overset{\cdot\cdot}{α} s β_{i} + \overset{\cdot\cdot}{γ}

(12e)

The conversion between ${\underline{ω}}_{i}^{(i)}, {\underline{\overset{\cdot}{ω}}}_{i}^{(i)}$ and ${\underline{\overset{\cdot}{θ}}}_{i}, {\underline{\overset{\cdot\cdot}{θ}}}_{i}$ can be realized by equation (12e).

The dynamic model of 6-DOF manipulator is derived by combining the dynamic equation equation (12b) of the single rigid body.

\underline{A} \underline{\overset{\cdot\cdot}{q}} = \underline{B}

(13)

where

\underline{A} = diag ({\underline{A}}_{1} {\underline{A}}_{2} \dots {\underline{A}}_{6})_{36 \times 36}

(14a)

\underline{B} = {({\underline{B}}_{1}^{T} {\underline{B}}_{2}^{T} \dots {\underline{B}}_{6}^{T})^{T}}_{36 \times 1}

(14b)

Since the dynamic model equation (13) of the manipulator is obtained by combining the dynamic equation equation (12b) of the rigid body, the matrix ${\underline{A}}_{i} (i = 1, 2, \dots, 6)$ reflecting the inertial parameters of each rigid body exists independently in the diagonal position of $\underline{A}$ , so there is no coupling of the inertial parameters of each rigid body, which provides conditions for parameters identification.

Parameters identification modeling

The multibody dynamics model is rewritten to the least squares form, and the inertial parameters in the model are separated as parameters to be identified, which can be identified by the least squares method. This is the MIM of this paper.

Since the inertial parameters are calculated by precise mathematical model, the MIM has extremely high accuracy.

First, equation (8a) is rewritten as

\begin{matrix} [\begin{matrix} {\overset{\cdot\cdot}{x}}_{i} \\ {\overset{\cdot\cdot}{y}}_{i} \\ {\overset{\cdot\cdot}{z}}_{i} + g \end{matrix}] [\begin{matrix} m_{i} \\ m_{i} \\ m_{i} \end{matrix}] = [\begin{matrix} - \sum_{k = i}^{i + 1} S_{ik}^{e} F_{k 1}^{e} \\ - \sum_{k = i}^{i + 1} S_{ik}^{e} F_{k 2}^{e} \\ - \sum_{k = i}^{i + 1} S_{ik}^{e} F_{k 3}^{e} \end{matrix}] \end{matrix}

(15a)

where $F_{k 1}^{e}$ , $F_{k 2}^{e}$ , and $F_{k 3}^{e}$ are the scalar components of $F_{k}^{e}$ on $e_{1}^{(0)}$ , $e_{2}^{(0)}$ , and $e_{3}^{(0)}$ , respectively. Equation (15a) is abbreviated as

{}^{id}{\underline{R}}_{i} {}^{id}{\underline{m}}_{i} = {\underline{F}}_{i}^{R} .

(15b)

Second, equation (10) is rewritten as

[\begin{matrix} \begin{matrix} - {\overset{\cdot}{α}}_{i} ({\overset{\cdot}{β}}_{i} s β_{i} c γ_{i} + {\overset{\cdot}{γ}}_{i} c β_{i} s γ_{i}) + \\ {\overset{\cdot}{β}}_{i} {\overset{\cdot}{γ}}_{i} c γ_{i} + {\overset{\cdot\cdot}{α}}_{i} c β_{i} c γ_{i} + {\overset{\cdot\cdot}{β}}_{i} s γ_{i} \end{matrix} & \begin{matrix} - (- {\overset{\cdot}{α}}_{i} c β_{i} s γ_{i} + {\overset{\cdot}{β}}_{i} c γ_{i}) \cdot \\ ({\overset{\cdot}{α}}_{i} s β_{i} + {\overset{\cdot}{γ}}_{i}) \end{matrix} & \begin{matrix} (- {\overset{\cdot}{α}}_{i} c β_{i} s γ_{i} + {\overset{\cdot}{β}}_{i} c γ_{i}) \cdot \\ ({\overset{\cdot}{α}}_{i} s β_{i} + {\overset{\cdot}{γ}}_{i}) \end{matrix} \\ \begin{matrix} ({\overset{\cdot}{α}}_{i} c β_{i} c γ_{i} + {\overset{\cdot}{β}}_{i} s γ_{i}) \cdot \\ ({\overset{\cdot}{α}}_{i} s β_{i} + {\overset{\cdot}{γ}}_{i}) \end{matrix} & \begin{matrix} {\overset{\cdot}{α}}_{i} ({\overset{\cdot}{β}}_{i} s β_{i} s γ_{i} - {\overset{\cdot}{γ}}_{i} c β_{i} c γ_{i}) - \\ {\overset{\cdot}{β}}_{i} {\overset{\cdot}{γ}}_{i} s γ_{i} - {\overset{\cdot\cdot}{α}}_{i} c β_{i} s γ_{i} + {\overset{\cdot\cdot}{β}}_{i} c γ_{i} \end{matrix} & \begin{matrix} - ({\overset{\cdot}{α}}_{i} c β_{i} c γ_{i} + {\overset{\cdot}{β}}_{i} s γ_{i}) \cdot \\ ({\overset{\cdot}{α}}_{i} s β_{i} + {\overset{\cdot}{γ}}_{i}) \end{matrix} \\ \begin{matrix} - ({\overset{\cdot}{α}}_{i} c β_{i} c γ_{i} + {\overset{\cdot}{β}}_{i} s γ_{i}) \cdot \\ (- {\overset{\cdot}{α}}_{i} c β_{i} s γ_{i} + {\overset{\cdot}{β}}_{i} c γ_{i}) \end{matrix} & \begin{matrix} ({\overset{\cdot}{α}}_{i} c β_{i} c γ_{i} + {\overset{\cdot}{β}}_{i} s γ_{i}) \cdot \\ (- {\overset{\cdot}{α}}_{i} c β_{i} s γ_{i} + {\overset{\cdot}{β}}_{i} c γ_{i}) \end{matrix} & {\overset{\cdot}{α}}_{i} {\overset{\cdot}{β}}_{i} c β_{i} + {\overset{\cdot}{γ}}_{i} + {\overset{\cdot\cdot}{α}}_{i} s β_{i} + {\overset{\cdot\cdot}{γ}}_{i} \end{matrix}] [\begin{matrix} J_{i_{11}}^{(i)} \\ J_{i_{22}}^{(i)} \\ J_{i_{33}}^{(i)} \end{matrix}] = {\underline{M}}_{i}^{(i)}

(16a)

Equation (16a) is abbreviated as

{}^{id}{\underline{S}}_{i}^{(i)} {}^{id}{\underline{J}}_{i}^{(i)} = {\underline{M}}_{i}^{(i)}

(16b)

Finally, the combination of equations (15b) and (16b) is expressed in the following form, that is, the parameters identification equation of $B_{i}$ .

[\begin{matrix} {}^{id}{\underline{R}}_{i} \\ {}^{id}{\underline{S}}_{i}^{(i)} \end{matrix}] [\begin{matrix} {}^{id}{\underline{m}}_{i} \\ {}^{id}{\underline{J}}_{i}^{(i)} \end{matrix}] = [\begin{matrix} {\underline{A}}^{(0 i)} {\underline{F}}_{i}^{R (i)} \\ {\underline{M}}_{i}^{(i)} \end{matrix}]

(17a)

Equation (17a) is abbreviated as

{}^{id}{\underline{Y}}_{i} {}^{id}{\underline{X}}_{i} = {}^{id}{\underline{Z}}_{i}

(17b)

Equation (17b) is in the least squares form, and its least squares solution is

{}^{id}{\underline{X}}_{i} = {({}^{id}{\underline{Y}}_{i}^{T} {}^{id}{\underline{Y}}_{i})}^{- 1} {}^{id}{\underline{Y}}_{i}^{T} {}^{id}{\underline{Z}}_{i} (i = 1, 2, \dots, 6)

(18)

The inertia parameters ${}^{id}{\underline{X}}_{i}$ of $B_{i}$ can be identified by equation (18).

In order to improve the readability and contribution of this paper, and to facilitate the application and implementation of the method in this paper, the specific expressions of the multibody dynamics equations and identification equations of each rigid body $B_{i} (i = 1, 2, \dots, 6)$ are described in detail, and the specific model of the 2-DOF manipulator is provided, as shown in Appendix A.

The 6-DOF manipulator is the combination of each rigid body, and its identification model can be written as

{[\begin{matrix} {}^{id}{\underline{Y}}_{1} \\ ⋱ \\ {}^{id}{\underline{Y}}_{6} \end{matrix}]}_{36 \times 36} {[\begin{matrix} {}^{id}{\underline{X}}_{1} \\ ⋮ \\ {}^{id}{\underline{X}}_{6} \end{matrix}]}_{36 \times 1} = {[\begin{matrix} {}^{id}{\underline{Z}}_{1} \\ ⋮ \\ {}^{id}{\underline{Z}}_{6} \end{matrix}]}_{36 \times 1}

(19a)

Equation (19a) is abbreviated as

{}^{id}{\underline{Y}} {}^{id}{\underline{X}} = {}^{id}{\underline{Z}}

(19b)

Equation (19b) is in the least squares form, and its least squares solution is

{}^{id}{\underline{X}} = {({}^{id}{\underline{Y}}^{T} {}^{id}{\underline{Y}})}^{- 1} {}^{id}{\underline{Y}}^{T} {}^{id}{\underline{Z}}

(20)

The inertia parameters ${}^{id}{\underline{X}}$ of the manipulator can be identified by equation (20).

The MIM in this paper can identify the inertial parameters of each rigid body separately, and can also identify the inertial parameters of the manipulator as a whole. The parameters to be identified have clear mathematical analytic expression, which ensures the high accuracy of the identification results.

NIM based on neural networks

An inertial parameters identification method based on neural networks is proposed in this paper, namely NIM. This method avoids the tedious modeling and calculation process of MIM, and has good real-time performance. However, as a numerical fitting method, it inevitably loses some accuracy. When the number of input/output data parameters is large, the data fitting problem presents a strong nonlinear relationship, and the neural networks is a powerful tool to deal with this problem.

The NIM in this paper is a data-driven identification algorithm, which uses the collected experimental data to train a neural network, and can quickly output the results of inertial parameters after training. NIM avoids complex dynamic modeling and computational processes, and has the advantage of real-time performance compared to MIM. The design of NIM in this paper is based on the following design ideas: a basic fact is that the number of collected data, the number of hidden layers, and the number of neurons in the hidden layer all affect the accuracy of the results and the real-time performance of the algorithm. Considering the time consumption of data collection in practical applications, the number of data should be as small as possible. However, a small number of data may lead to overfitting, so it is necessary to set a small number of hidden layers and neurons to eliminate the effect of overfitting, ensure the generalization of data, and further improve the real-time performance. The NIM algorithm provides a design idea for neural networks. The number of hidden layers and the number of data can be adjusted according to the structure of different robots, with the aim of achieving the desired result. Different requirements for results (e.g. accuracy, real-time, etc.) lead to different choices of strategies (number of hidden layers, number of neurons, and number of data).

Specifically, the neural networks consists of one input layer, five hidden layers, and one output layer. There are six data items in the input layer, which are Cardano Angle $(α_{i} β_{i} γ_{i})^{T}$ , force vector $F_{i}^{e}$ , moment vector $M_{i}^{a}$ , linear acceleration vector $({\overset{\cdot\cdot}{x}}_{i} {\overset{\cdot\cdot}{y}}_{i} {\overset{\cdot\cdot}{z}}_{i})^{T}$ , angular velocity vector $(ω_{i 1} ω_{i 2} ω_{i 3})^{T}$ , and angular acceleration vector $({\overset{\cdot}{ω}}_{i 1} {\overset{\cdot}{ω}}_{i 2} {\overset{\cdot}{ω}}_{i 3})^{T}$ . There are one data item in the output layer, which is the mass and the principal moment of inertia $(m_{i} J_{i 1}^{(i)} J_{i 2}^{(i)} J_{i 3}^{(i)})^{T}$ . There are 18 parameters in the input layer and 4 parameters in the output layer, which means that finding the fitting curve between the input layer and the output layer is a strong nonlinear problem. The five hidden layers between the input layer and the output layer are designed to fit this nonlinear relationship. Each hidden layer contains 128 neurons to construct a nonlinear function with good fitting effect and good real-time computation performance. The neural networks structure is shown in Figure 3.

Figure 3.

The neural networks structure.

The activation function of each layer is ReLu function.

Relu (z) = f (z) = \max (0, z)

(21)

Relu function is a nonlinear function, and the nonlinearity of neural networks comes from the nonlinearity of Relu function. The expression for layer L_i is

h_{i} = Relu ({\underline{w}}_{i} h_{i - 1} + b_{i}) = f ({\underline{w}}_{i} h_{i - 1} + b_{i})

(22)

where ${\underline{w}}_{i}$ and $b_{i}$ are weight matrix and bias vector.

The data of the input layer is

h_{0} = x = (x_{1} x_{2} x_{3} x_{4} x_{5} x_{6})^{T}

(23a)

where

x_{1} = (α_{i} β_{i} γ_{i})^{T}

(23b)

x_{2} = F_{i}^{e} = (F_{ix}^{e} F_{iy}^{e} F_{iz}^{e})^{T}

(23c)

x_{3} = M_{i}^{a} = (M_{i 1}^{a} M_{i 2}^{a} M_{i 3}^{a})^{T}

(23d)

x_{4} = ({\overset{\cdot\cdot}{x}}_{i} {\overset{\cdot\cdot}{y}}_{i} {\overset{\cdot\cdot}{z}}_{i})^{T}

(23e)

x_{5} = (ω_{i 1} ω_{i 2} ω_{i 3})^{T}

(23f)

x_{6} = ({\overset{\cdot}{ω}}_{i 1} {\overset{\cdot}{ω}}_{i 2} {\overset{\cdot}{ω}}_{i 3})^{T}

(23g)

The data of the output layer is

h_{6} = \hat{y} = (m_{i} J_{i 1}^{(i)} J_{i 2}^{(i)} J_{i 3}^{(i)})^{T}

(24)

The purpose of training neural networks is to make the calculated output $\hat{y}$ as close as possible to its true value y .

For this purpose, the cost function of the error is set, namely, the Mean Squared Error (MSE) of the output $\hat{y}$ .

MSE = \frac{1}{N} \sum_{j = 1}^{4} {({\overset{\land}{y}}_{j} - y_{j})}^{2}

(25)

N is the number of samples.

The algorithm is as follows.

Algorithm 1. NIM based on neural networks.
Input: Training Set $x_{training}$ , Label y , Test Data $x_{test}$ ,
Output: Test Label $y_{test}$ ; 1: $lr = 2.5 \times 10^{- 4}$ , $ε = 10^{- 5}$ , Random initialization: wights $w_{i} (i = 1, 2, \dots, m)$ and bias $b_{i} (i = 1, 2, \dots, m)$ in function equation (25). 2: while not converge do 3: Computing $\hat{y}$ using equation (22). 4: Computing Loss function using equation (25). 5: Check convergence condition: $MSE < ε$ . 6: Update wights $w_{i}$ and bias $b_{i}$ in function equation (22). 7: end while
8: Computing $y_{test}$ with $x_{test}$ using equation (22).

The algorithm sets the learning rate (step size) to $lr = 2.5 \times 10^{- 4}$ , and stops the calculation when the cost function MSE is less than the threshold $ε = 10^{- 5}$ . The algorithm divides the data into two parts, one is the training data $x_{training}$ and y , which is used to train the weights and biases of the neural networks. The other part is the test data $x_{test}$ , which is input to the trained neural networks to output the prediction data $y_{test}$ . In this neural networks, $m = 6$ .

Compared with MIM, the design of NIM based on neural network in this paper is uncomplicated, which transforms the complex mechanical modeling and mathematical calculation process into a data fitting problem that can be solved by computer.

Numerical simulation

In order to verify the correctness of MIM and NIM in this paper, the simulation model of 6-DOF manipulator was established, the motion trajectory was designed, and the data during the motion was collected, which was used as the basis for calculation/fitting of the identification algorithm.

ADAMS software is used to establish the simulation model of the manipulator. The physical parameters of each rigid body are set as shown in Table 1.

Table 1.

The physical parameters of each rigid body.

Rigid body	$B_{1}$	$B_{2}$	$B_{3}$	$B_{4}$	$B_{5}$	$B_{6}$
Length ( $m$ )	1.045	1.245	1.21	0.315	0.29	0.3
Radius ( $m$ )	0.5	0.3	0.2	0.2	0.2	0.3
$m_{i}$ ( $kg$ )	1429.5806	613.1464	264.8486	68.9482	63.4761	100.0000
$J_{i_{11}}^{(i)}$ ( $kg \cdot m^{2}$ )	178.6976	27.5916	5.2970	1.3790	1.2695	4.5000
$J_{i_{22}}^{(i)}$ ( $kg \cdot m^{2}$ )	219.4436	92.9952	34.9622	1.2596	1.0796	3.0000
$J_{i_{33}}^{(i)}$ ( $kg \cdot m^{2}$ )	219.4436	92.9952	34.9622	1.2596	1.0796	3.0000

In the simulation, the control of the manipulator is realized by setting the drive in Adams software. The drive is set at the joint position, which can make the joint rotate according to a certain angular velocity or angular acceleration. This is consistent with the fact that the manipulator is driven by the joint motors in the actual scenario. The monitoring of the manipulator is realized by measuring the parameters in Adams software. The parameters to be measured include force and moment, Euler Angle, angular velocity, acceleration, and angular acceleration.

The initial state is static, and the initial posture is shown in Figure 1. The end time of the simulation is 2 s and the number of steps is 100. The simulation process is shown in Figure 4. ADAMS software is used to measure various parameters during the movement, and the corresponding data of the parameters can be obtained. In order to compare the identification results of MIM and NIM, the experiment and data are used in both the two identification algorithms.

Figure 4.

Posture at each moment in ADAMS simulation of 6-DOF manipulator: (a) 0 s, (b) 0.3 s, (c) 0.6 s, (d) 0.9 s, (e) 1.2 s, (f) 1.5 s, (g) 1.8 s, and (h) 2 s.

$B_{1}$ is set to move along $e_{1}^{(0)}$ direction (i.e. the direction of guide rail) with the linear acceleration of $2 m / s^{2}$ in the simulation, while $O_{1}$ and $O_{2}$ rotate with the angular acceleration of $30^{°} / s^{2}$ , and other joints do not rotate.

Numerical simulation of MIM

The data of the simulation process is input into the identification model equations (18) or (20) to calculate the accurate inertia parameters, and the MIM is verified. According to equations (18) or (20), the parameters to be identified are inertial parameters, and there are six known parameters obtained by measurement, which are ${\underline{θ}}_{i}$ , $F_{i}^{e}$ , $M_{i}^{a}$ , ${\underline{\overset{\cdot\cdot}{r}}}_{i}$ , ${\underline{\overset{\cdot}{θ}}}_{i}$ and ${\underline{\overset{\cdot\cdot}{θ}}}_{i}$ .

The known parameters of 1.5 s (or any other time at random) are extracted, and the inertial parameters are identified by equation (18). The identification results are shown in Table 2, and the absolute errors between the identification results and the true values are shown in Table 3.

Table 2.

The identified inertia parameters of MIM.

Rigid body	$B_{1}$	$B_{2}$	$B_{3}$	$B_{4}$	$B_{5}$	$B_{6}$
$m_{i}$ ( $e_{1}^{(0)}$ )( $kg$ )	$1429.5805$	$613.1462$	$264.8488$	$68.9482$	$63.4761$	$100.0000$
$m_{i}$ ( $e_{2}^{(0)}$ )( $kg$ )	–	$613.1469$	$264.8485$	$68.9482$	$63.4761$	$100.0000$
$m_{i}$ ( $e_{3}^{(0)}$ )( $kg$ )	$1429.5809$	$613.1463$	$264.8486$	$68.9482$	$63.4761$	$100.0000$
$J_{i_{11}}^{(i)}$ ( $kg \cdot m^{2}$ )	$178.6975$	$27.5966$	$5.2968$	$1.3789$	$1.2695$	$4.5000$
$J_{i_{22}}^{(i)}$ ( $kg \cdot m^{2}$ )	$219.4436$	$92.9892$	$34.9628$	$1.2595$	$1.0796$	$3.0000$
$J_{i_{33}}^{(i)}$ ( $kg \cdot m^{2}$ )	$219.4436$	$92.9981$	$34.9626$	$1.2595$	$1.0796$	$3.0000$

Table 3.

The absolute errors of MIM.

Rigid body	Absolute error
Rigid body	$m_{i}$ ( $e_{1}^{(0)}$ )( $kg$ )	$m_{i}$ ( $e_{2}^{(0)}$ )( $kg$ )	$m_{i}$ ( $e_{3}^{(0)}$ )( $kg$ )	$J_{i_{11}}^{(i)}$ ( $kg \cdot m^{2}$ )	$J_{i_{22}}^{(i)}$ ( $kg \cdot m^{2}$ )	$J_{i_{33}}^{(i)}$ ( $kg \cdot m^{2}$ )
$B_{6}$	0	0	0	0	0	0
$B_{5}$	0	0	0	0	0	0
$B_{4}$	0	0	0	−0.0001	−0.0001	−0.0001
$B_{3}$	0.0002	−0.0001	0	−0.0002	0.0006	0.0004
$B_{2}$	−0.0002	0.0005	−0.0001	0.0050	−0.0060	0.0029
$B_{1}$	−0.0001	–	0.0003	−0.0001	0	0

where, $m_{i}$ ( $e_{1}^{(0)}$ ), $m_{i}$ ( $e_{2}^{(0)}$ ), and $m_{i}$ ( $e_{3}^{(0)}$ ) are the mass parameters identified in the $e_{1}^{(0)}$ , $e_{2}^{(0)}$ , and $e_{3}^{(0)}$ directions respectively. Since $B_{1}$ has no acceleration in the $e_{2}^{(0)}$ direction, the mass cannot be identified in this direction.

According to Table 3, the absolute errors of the identified inertial parameters are extremely small, the maximum absolute error of the mass parameter is 0.0005 kg and the maximum absolute error of the principal moment of inertia is −0.0060 kg m², which verifies the accuracy of the MIM proposed in this paper.

Numerical simulation of NIM

The NIM based on neural networks can achieve the effect of data fitting by training weights and bias, and the NIM after training can be used to predict the parameters to be identified. In order to compare with the MIM identification results, the NIM experiment is exactly the same as that of MIM, with 100 simulation steps and a total of 100 sets of parameters. It should be noted that we can collect more than 100 sets of data, and by changing the time and step size in the Adams simulation, any amount of data can be obtained. The first 80 sets of data are used to train the neural networks model, and the last 20 sets of data are used to validate the model. The training process: Input layer and output layer parameters of the first 80 sets of data are input to the neural networks model for training each weight and bias. The verification process: The parameters of the input layer of the last 20 sets of data are input into the neural networks model to obtain the predicted value of the output layer. There are 20 predicted values of the output layer, and the algebraic average is taken as the identification parameters. The identified parameters of each rigid body are shown in Table 4, and the absolute errors with the true values are shown in Table 5.

Table 4.

The identified inertia parameters of NIM.

Rigid body	$B_{1}$	$B_{2}$	$B_{3}$	$B_{4}$	$B_{5}$	$B_{6}$
$m_{i}$ ( $kg$ )	$1441.8040$	$615.8533$	$262.2815$	$68.9220$	$63.6023$	$100.3631$
$J_{i_{11}}^{(i)}$ ( $kg \cdot m^{2}$ )	$180.4628$	$27.1413$	$5.1695$	$1.3671$	$1.2634$	$4.4764$
$J_{i_{22}}^{(i)}$ ( $kg \cdot m^{2}$ )	$223.1567$	$93.4262$	$34.5954$	$1.2087$	$1.0801$	$3.0770$
$J_{i_{33}}^{(i)}$ ( $kg \cdot m^{2}$ )	$221.7422$	$93.7425$	$34.3432$	$1.3774$	$1.0737$	$3.0805$

Table 5.

The absolute errors of NIM.

Rigid body	Absolute error
	$m_{i}$ ( $kg$ )	$J_{i_{11}}^{(i)}$ ( $kg \cdot m^{2}$ )	$J_{i_{22}}^{(i)}$ ( $kg \cdot m^{2}$ )	$J_{i_{33}}^{(i)}$ ( $kg \cdot m^{2}$ )
$B_{6}$	0.3631	−0.0236	0.0770	0.0805
$B_{5}$	0.1262	−0.0061	0.0005	−0.0059
$B_{4}$	−0.0262	−0.0119	−0.0509	0.1178
$B_{3}$	−2.5671	−0.1275	−0.3668	−0.6190
$B_{2}$	2.7069	−0.4503	0.4310	0.7473
$B_{1}$	11.4998	1.7652	3.7131	2.2962

As can be seen from Table 5, the maximum absolute error of the identified mass parameter is 11.4998 kg, and the maximum absolute error of the principal moment of inertia is 3.7131 kg m². Compared with Tables 5 and 3, there is a gap between the absolute error of NIM and MIM, which reflects the difference of accuracy of the two. However, if the absolute error in Table 5 is converted into a percentage form, NIM still maintains a relatively high accuracy. NIM also has the advantage of very fast training and calculation speed, so it must have targeted application value.

The relationship between MSE and the number of training steps of each rigid body is shown in Figure 5. When the number of training steps reaches 400, MSE reaches below the threshold, the neural networks is trained and the algorithm converges, which can be used for the identification of inertial parameters.

Figure 5.

The relationship between MSE and training times: (a) $B_{1}$ , (b) $B_{2}$ , (c) $B_{3}$ , (d) $B_{4}$ , (e) $B_{5}$ , and (f) $B_{6}$ .

Discussion

Accurate identification of inertial parameters is only one of the ways to improve the control accuracy of the manipulator. As a problem involving many factors that need to be comprehensively considered, there are other ways to improve the control accuracy. This paper only puts forward an innovative solution from the aspect of accurate identification of inertial parameters.

There are five main factors affecting the control accuracy, which are the real-time performance of the control algorithm, the accuracy of the control algorithm, the accuracy of the inertial parameters, the accuracy of the trajectory planning algorithm, and the accuracy of the hardware sensor. (1) The real-time performance of the control algorithm depends on the computation cost. The lower the computation cost, the better the real-time performance and the higher the control accuracy.²² (2) The more accurate the control model is, and the more variable factors are considered, including external forces and disturbance factors, the higher the accuracy of the control algorithm.^23,24 (3) The accuracy of inertial parameters depends on the parameters identification algorithm. The identification algorithm can directly identify inertial parameters, and the more accurate the identification algorithm is, the higher the accuracy of the identified inertial parameters. (4) The velocity and acceleration parameters of trajectory planning based on kinematics should be consistent with their expected parameters as far as possible. The more accurate the trajectory planning algorithm is, the higher the position accuracy of the controlled end effector is.^25,26 (5) The more types of sensors (six-dimensional force sensors, pose sensors, speed and acceleration sensors) and the higher the accuracy, the faster the response speed, and the higher the control accuracy.^23,27

In this paper, only the influence of the accuracy of inertial parameters on the control accuracy is considered, and the study of the control accuracy from the other four aspects is the future research direction of this paper.

Conclusion

In this paper, MIM and NIM are proposed, which can realize the direct identification of the inertia parameters of the manipulator and are of great importance for precise control.

MIM is derived based on multibody dynamics, and graph theory is used to decompose the robotic arm into individual rigid bodies as the analysis object, and the decoupling of inertial parameters between each rigid body is realized. MIM has clear mathematical analytical expressions and extremely high identification accuracy, the maximum absolute error of the identified mass parameter is 0.0005 kg, and the maximum absolute error of the principal moment of inertia is −0.0060 kg m². However, the complex modeling and computation process has an impact on real-time performance. NIM is trained by neural networks and has excellent real-time performance. NIM avoids tedious mechanical modeling and calculations, the maximum absolute error of the identified mass parameter is 11.4998 kg, and the maximum absolute error of the principal moment of inertia is 3.7131 kg m². Although the fitting results inevitably lose some accuracy compared to MIM, the accuracy still has a relatively high level. The MIM and NIM inertial parameters identification methods proposed in this paper have complementary advantages, which can meet the vast majority of the identification requirements and provide an important basis for the accurate control of the manipulator.

In this paper, the inertial parameters identification of the manipulator has been realized. In the future work, on the one hand, the application range of this method will be extended. The main application directions are parallel robots with complex structure and multi-DOF and variable structure robots with real-time changed inertia parameters. The identification difficulty of the two is greater than that of serial robots, and the implementation of inertia parameters identification is very beneficial for their precise control and extension of application range. On the other hand, the improvement of control accuracy will be studied from four aspects: real-time control algorithm, control algorithm modeling, trajectory planning algorithm, and sensor development and layout.

Footnotes

Appendix A

In order to improve the readability of this paper, the concrete expressions of the multibody dynamic equation and identification equation of rigid body $B_{i} (i = 1, 2, \dots, 6)$ are derived.

Equation (7) deduces the sum of the external forces $F_{i}$ and the sum of the external torque $M_{i}$ of the rigid body $B_{i}$ . For the 6-DOF manipulator, its specific expression is

(A1)

\begin{matrix} \underline{F} = [\begin{matrix} F_{1} \\ F_{2} \\ F_{3} \\ F_{4} \\ F_{5} \\ F_{6} \end{matrix}] = [\begin{matrix} F_{1}^{g} - F_{1}^{e} + F_{2}^{e} \\ F_{2}^{g} - F_{2}^{e} + F_{3}^{e} \\ F_{3}^{g} - F_{3}^{e} + F_{4}^{e} \\ F_{4}^{g} - F_{4}^{e} + F_{5}^{e} \\ F_{5}^{g} - F_{5}^{e} + F_{6}^{e} \\ F_{6}^{g} - F_{6}^{e} \end{matrix}] \end{matrix}

$\underline{F} = {[\begin{matrix} F_{1}^{T} & F_{2}^{T} & F_{3}^{T} & F_{4}^{T} & F_{5}^{T} & F_{6}^{T} \end{matrix}]}^{T}$ is a matrix of $F_{i} (i = 1, 2, \dots, 6)$ .

(A2)

\begin{matrix} \underline{M} = [\begin{matrix} M_{1} \\ M_{2} \\ M_{3} \\ M_{4} \\ M_{5} \\ M_{6} \end{matrix}] = [\begin{matrix} (F_{1}^{e} + F_{2}^{e}) \times \frac{l_{1}}{2} e_{1}^{(1)} \\ (F_{2}^{e} + F_{3}^{e}) \times \frac{l_{2}}{2} e_{1}^{(2)} \\ (F_{3}^{e} + F_{4}^{e}) \times \frac{l_{3}}{2} e_{1}^{(3)} \\ (F_{4}^{e} + F_{5}^{e}) \times \frac{l_{4}}{2} e_{1}^{(4)} \\ (F_{5}^{e} + F_{6}^{e}) \times \frac{l_{5}}{2} e_{1}^{(5)} \\ F_{6}^{e} \times \frac{l_{6}}{2} e_{1}^{(6)} \end{matrix}] + [\begin{matrix} M_{1}^{a} - M_{2}^{a} \\ M_{2}^{a} - M_{3}^{a} \\ M_{3}^{a} - M_{4}^{a} \\ M_{4}^{a} - M_{5}^{a} \\ M_{5}^{a} - M_{6}^{a} \\ M_{6}^{a} \end{matrix}] \end{matrix}

$\underline{M} = {[\begin{matrix} M_{1}^{T} & M_{2}^{T} & M_{3}^{T} & M_{4}^{T} & M_{5}^{T} & M_{6}^{T} \end{matrix}]}^{T}$ is a matrix of $M_{i} (i = 1, 2, \dots, 6)$ .

For rigid body $B_{6}$ (end effector), only the inner side is forced, and the outer side is not forced. Bringing the parameters of $B_{6}$ into equations (8a) and (11), the dynamic equation for $B_{6}$ is

(A3)

m_{6} {\underline{\overset{\cdot\cdot}{r}}}_{6} = {\underline{F}}_{6}^{g} - {\underline{F}}_{6}^{e}

(A4)

{\underline{J}}_{6}^{(6)} {\underline{D}}_{6} {\underline{\overset{\cdot\cdot}{θ}}}_{6} + ({\underline{J}}_{6}^{(6)} {\underline{\overset{\cdot}{D}}}_{6} + \tilde{{\underline{D}}_{6} {\underline{\overset{\cdot}{θ}}}_{6}} {\underline{J}}_{6}^{(6)} {\underline{D}}_{6}) {\underline{\overset{\cdot}{θ}}}_{6} = {\underline{F}}_{6}^{e} \times \frac{l_{6}}{2} {\underline{e}}_{1}^{(6)} + {\underline{M}}_{6}^{a}

${\underline{e}}_{1}^{(i)} = [\begin{matrix} e_{1}^{(i)} & 0 & 0 \end{matrix}] (i = 1, 2, \dots, 6)$ is the scalar matrix of ${\underline{e}}_{1}^{(i)} = [\begin{matrix} e_{1}^{(i)} & 0 & 0 \end{matrix}] (i = 1, 2, \dots, 6)$ . By bringing the parameters of $B_{6}$ into equation (18a), the inertial parameter identification model of $B_{6}$ is obtained.

(A5)

\begin{matrix} [\begin{matrix} {}^{id}{\underline{R}}_{6} \\ {}^{id}{\underline{S}}_{6}^{(6)} \end{matrix}] [\begin{matrix} {}^{id}{\underline{m}}_{6} \\ {}^{id}{\underline{J}}_{6}^{(6)} \end{matrix}] = [\begin{matrix} - {\underline{F}}_{6}^{e} \\ {\underline{F}}_{6}^{e} \times \frac{l_{6}}{2} {\underline{e}}_{1}^{(6)} + {\underline{M}}_{6}^{a} \end{matrix}] \end{matrix}

Equation (28a) is abbreviated as

(A6)

{}^{id}{\underline{Y}}_{6} {}^{id}{\underline{X}}_{6} = {}^{id}{\underline{Z}}_{6}

The least squares solution is

(A7)

{}^{id}{\underline{X}}_{6} = ({}^{id}{\underline{Y}}_{6} {}^{T id}{\underline{Y}}_{6})^{- 1 id} {\underline{Y}}_{6}^{T id} {\underline{Z}}_{6}

From this, the inertia parameter ${}^{id}{X_{6}}$ can be identified.

For rigid body $B_{5}$ , the force is applied on both the inner and outer sides. Bringing the parameters of $B_{5}$ into equations (8a) and (11), the dynamic equation for $B_{5}$ is

(A8)

m_{5} {\underline{\overset{\cdot\cdot}{r}}}_{5} = {\underline{F}}_{5}^{g} - {\underline{F}}_{5}^{e} + {\underline{F}}_{6}^{e}

(A9)

\begin{matrix} {\underline{J}}_{5}^{(5)} {\underline{D}}_{5} {\underline{\overset{\cdot\cdot}{θ}}}_{5} + ({\underline{J}}_{5}^{(5)} {\underline{\overset{\cdot}{D}}}_{5} + \tilde{{\underline{D}}_{5} {\underline{\overset{\cdot}{θ}}}_{5}} {\underline{J}}_{5}^{(5)} {\underline{D}}_{5}) {\underline{\overset{\cdot}{θ}}}_{5} = {\underline{F}}_{5}^{e} \times \frac{l_{5}}{2} {\underline{e}}_{1}^{(5)} \\ - {\underline{F}}_{6}^{e} \times \frac{l_{5}}{2} {\underline{e}}_{1}^{(5)} + {\underline{M}}_{5}^{a} - {\underline{M}}_{6}^{a} \end{matrix}

By bringing the parameters of $B_{5}$ into equation (18a), the inertial parameter identification model of $B_{5}$ is obtained.

(A10)

\begin{matrix} [\begin{matrix} {}^{id}{\underline{R}}_{5} \\ {}^{id}{\underline{S}}_{5}^{(5)} \end{matrix}] [\begin{matrix} {}^{id}{\underline{m}}_{5} \\ {}^{id}{\underline{J}}_{5}^{(5)} \end{matrix}] \\ = [\begin{matrix} - {\underline{F}}_{5}^{e} + {\underline{F}}_{6}^{e} \\ {\underline{F}}_{5}^{e} \times \frac{l_{5}}{2} {\underline{e}}_{1}^{(5)} - {\underline{F}}_{6}^{e} \times \frac{l_{5}}{2} {\underline{e}}_{1}^{(5)} + {\underline{M}}_{5}^{a} - {\underline{M}}_{6}^{a} \end{matrix}] \end{matrix}

Equation (31a) is abbreviated as

(A11)

{}^{id}{\underline{Y}}_{5} {}^{id}{\underline{X}}_{5} = {}^{id}{\underline{Z}}_{5}

The least squares solution is

(A12)

{}^{id}{\underline{X}}_{5} = {({}^{id}{\underline{Y}}_{5}^{T} {}^{id}{\underline{Y}}_{5})}^{- 1} {}^{id}{\underline{Y}}_{5}^{T} {}^{id}{\underline{Z}}_{5}

From this, the inertia parameter ${}^{id}{X_{5}}$ can be identified.

Rigid body $B_{1}, B_{1}, \dots, B_{4}$ are both inner and outer side forced cases, the inertia parameter identification method is the same as $B_{5}$ .

The multibody dynamic modeling and inertial parameters identification methods in this paper are extensible and can be applied to both 2-DOF manipulators and multi-DOF manipulators. In order to facilitate the implementation and generalization of the method in this paper, a specific mathematical model of the 2-DOF manipulator is provided. The rigid bodies of the 2-DOF manipulator are homogeneous cylinders, and the lengths of $B_{1}, B_{2}$ are $l_{1}, l_{2}$ , respectively, as shown in Figure A1.

From the dynamic equation equation (13a) of A single rigid body, the dynamic equation of $B_{1}$ can be obtained as

(A13)

[\begin{matrix} m_{1} \underline{E} & \underline{0} \\ \underline{0} & {\underline{J}}_{1}^{(1)} {\underline{D}}_{1} \end{matrix}] [\begin{matrix} {\underline{\overset{\cdot\cdot}{r}}}_{1} \\ {\underline{\overset{\cdot\cdot}{θ}}}_{1} \end{matrix}] = [\begin{matrix} \underline{0} \\ - ({\underline{J}}_{1}^{(1)} {\underline{\overset{\cdot}{D}}}_{1} + \tilde{{\underline{D}}_{1} {\underline{\overset{\cdot}{θ}}}_{1}} {\underline{J}}_{1}^{(1)} {\underline{D}}_{1}) {\underline{\overset{\cdot}{θ}}}_{1} \end{matrix}] + [\begin{matrix} {\underline{F}}_{1}^{g} - {\underline{F}}_{1}^{e} + {\underline{F}}_{2}^{e} \\ A^{(10)} (({\underline{M}}_{1}^{(a)} - {\underline{M}}_{2}^{(a)}) + ({\underline{F}}_{1}^{e} + {\underline{F}}_{2}^{e}) \times \frac{l_{1}}{2} {\underline{e}}_{1}^{(1)}) \end{matrix}]

The dynamic equation of $B_{2}$ is

(A14)

[\begin{matrix} m_{2} \underline{E} & \underline{0} \\ \underline{0} & {\underline{J}}_{2}^{(2)} {\underline{D}}_{2} \end{matrix}] [\begin{matrix} {\underline{\overset{\cdot\cdot}{r}}}_{2} \\ {\underline{\overset{\cdot\cdot}{θ}}}_{2} \end{matrix}] = [\begin{matrix} \underline{0} \\ - ({\underline{J}}_{2}^{(2)} {\underline{\overset{\cdot}{D}}}_{2} + \tilde{{\underline{D}}_{2} {\underline{\overset{\cdot}{θ}}}_{2}} {\underline{J}}_{2}^{(2)} {\underline{D}}_{2}) {\underline{\overset{\cdot}{θ}}}_{2} \end{matrix}] + [\begin{matrix} {\underline{F}}_{2}^{g} - {\underline{F}}_{2}^{e} \\ A^{(20)} ({\underline{M}}_{2}^{(a)} + {\underline{F}}_{2}^{e} \times \frac{l_{2}}{2} {\underline{e}}_{1}^{(2)}) \end{matrix}]

By combining equations (A13) and (A14), the dynamic model of 2-DOF manipulator is obtained.

(A15)

{\underline{A}}_{2 - DOF} {\underline{\overset{\cdot\cdot}{q}}}_{2 - DOF} = {\underline{B}}_{2 - DOF}

where,

(A16)

{\underline{A}}_{2 - DOF} = [\begin{matrix} m_{1} \underline{E} & \underline{0} & \underline{0} & \underline{0} \\ \underline{0} & {\underline{J}}_{1}^{(1)} {\underline{D}}_{1} & \underline{0} & \underline{0} \\ \underline{0} & \underline{0} & m_{2} \underline{E} & \underline{0} \\ \underline{0} & \underline{0} & \underline{0} & {\underline{J}}_{2}^{(2)} {\underline{D}}_{2} \end{matrix}]

(A17)

{\underline{\overset{\cdot\cdot}{q}}}_{2 - DOF} = {[\begin{matrix} {\underline{\overset{\cdot\cdot}{r}}}_{1} & {\underline{\overset{\cdot\cdot}{θ}}}_{1} & {\underline{\overset{\cdot\cdot}{r}}}_{2} & {\underline{\overset{\cdot\cdot}{θ}}}_{2} \end{matrix}]}^{T}

(A18)

{\underline{B}}_{2 - DOF} = [\begin{matrix} \underline{0} \\ - ({\underline{J}}_{1}^{(1)} {\underline{\overset{\cdot}{D}}}_{1} + \tilde{{\underline{D}}_{1} {\underline{\overset{\cdot}{θ}}}_{1}} {\underline{J}}_{1}^{(1)} {\underline{D}}_{1}) {\underline{\overset{\cdot}{θ}}}_{1} \\ \underline{0} \\ - ({\underline{J}}_{2}^{(2)} {\underline{\overset{\cdot}{D}}}_{2} + \tilde{{\underline{D}}_{2} {\underline{\overset{\cdot}{θ}}}_{2}} {\underline{J}}_{2}^{(2)} {\underline{D}}_{2}) {\underline{\overset{\cdot}{θ}}}_{2} \end{matrix}] + [\begin{matrix} {\underline{F}}_{1}^{g} - {\underline{F}}_{1}^{e} + {\underline{F}}_{2}^{e} \\ A^{(10)} (({\underline{M}}_{1}^{(a)} - {\underline{M}}_{2}^{(a)}) + ({\underline{F}}_{1}^{e} + {\underline{F}}_{2}^{e}) \times \frac{l_{1}}{2} {\underline{e}}_{1}^{(1)}) \\ {\underline{F}}_{2}^{g} - {\underline{F}}_{2}^{e} \\ A^{(20)} ({\underline{M}}_{2}^{(a)} + {\underline{F}}_{2}^{e} \times \frac{l_{2}}{2} {\underline{e}}_{1}^{(2)}) \end{matrix}]

By rewriting the dynamic equations equations (A13) and (A14) into the parameters identification form, the inertial parameters are separated as the parameters to be identified, and the identification can be realized by least square method.

According to the inertia parameters identification equation equations (16a) and (17a) of A single rigid body, the mass parameters identification equation of $B_{1}$ is

(A19)

[\begin{matrix} {\overset{\cdot\cdot}{x}}_{1} \\ {\overset{\cdot\cdot}{y}}_{1} \\ {\overset{\cdot\cdot}{z}}_{1} + g \end{matrix}] [\begin{matrix} m_{1} \\ m_{1} \\ m_{1} \end{matrix}] = [\begin{matrix} F_{21}^{e} - F_{11}^{e} \\ F_{22}^{e} - F_{12}^{e} \\ F_{23}^{e} - F_{13}^{e} \end{matrix}]

Abbreviated as

(A20)

{}^{id}{\underline{R}}_{1} {}^{id}{\underline{m}}_{1} = {\underline{F}}_{1}^{R}

The equation for identifying the principal moment of inertia parameters of rigid body $B_{1}$ is

(A21)

{}^{id}{\underline{S}}_{1}^{id} {\underline{J}}_{1}^{(1)} = {\underline{M}}_{1}^{(1)}

where,

(A22)

{}^{id}{\underline{S}}_{1}^{(1)} = [\begin{matrix} \begin{matrix} - {\overset{\cdot}{α}}_{1} ({\overset{\cdot}{β}}_{1} s β_{1} c γ_{1} + {\overset{\cdot}{γ}}_{1} c β_{1} s γ_{1}) + \\ {\overset{\cdot}{β}}_{1} {\overset{\cdot}{γ}}_{1} c γ_{1} + {\overset{\cdot\cdot}{α}}_{1} c β_{1} c γ_{1} + {\overset{\cdot\cdot}{β}}_{1} s γ_{1} \end{matrix} & \begin{matrix} - (- {\overset{\cdot}{α}}_{1} c β_{1} s γ_{1} + {\overset{\cdot}{β}}_{1} c γ_{1}) \cdot \\ ({\overset{\cdot}{α}}_{1} s β_{1} + {\overset{\cdot}{γ}}_{1}) \end{matrix} & \begin{matrix} (- {\overset{\cdot}{α}}_{1} c β_{1} s γ_{1} + {\overset{\cdot}{β}}_{1} c γ_{1}) \cdot \\ ({\overset{\cdot}{α}}_{1} s β_{1} + {\overset{\cdot}{γ}}_{1}) \end{matrix} \\ \begin{matrix} ({\overset{\cdot}{α}}_{1} c β_{1} s γ_{1} + {\overset{\cdot}{β}}_{1} c γ_{1}) \cdot \\ ({\overset{\cdot}{α}}_{1} s β_{1} + {\overset{\cdot}{γ}}_{1}) \end{matrix} & \begin{matrix} {\overset{\cdot}{α}}_{1} ({\overset{\cdot}{β}}_{1} s β_{1} s γ_{1} - {\overset{\cdot}{γ}}_{1} c β_{1} c γ_{1}) - \\ {\overset{\cdot}{β}}_{1} {\overset{\cdot}{γ}}_{1} s γ_{1} - {\overset{\cdot\cdot}{α}}_{1} c β_{1} s γ_{1} + {\overset{\cdot\cdot}{β}}_{1} c γ_{1} \end{matrix} & \begin{matrix} - ({\overset{\cdot}{α}}_{1} c β_{1} c γ_{1} + {\overset{\cdot}{β}}_{1} s γ_{1}) \cdot \\ ({\overset{\cdot}{α}}_{1} s β_{1} + {\overset{\cdot}{γ}}_{1}) \end{matrix} \\ \begin{matrix} - ({\overset{\cdot}{α}}_{1} c β_{1} c γ_{1} + {\overset{\cdot}{β}}_{1} s γ_{1}) \cdot \\ (- {\overset{\cdot}{α}}_{1} c β_{1} s γ_{1} + {\overset{\cdot}{β}}_{1} c γ_{1}) \end{matrix} & \begin{matrix} ({\overset{\cdot}{α}}_{1} c β_{1} c γ_{1} + {\overset{\cdot}{β}}_{1} s γ_{1}) \cdot \\ (- {\overset{\cdot}{α}}_{1} c β_{1} s γ_{1} + {\overset{\cdot}{β}}_{1} c γ_{1}) \end{matrix} & {\overset{\cdot}{α}}_{1} {\overset{\cdot}{β}}_{1} c β_{1} + {\overset{\cdot}{γ}}_{1} + {\overset{\cdot\cdot}{α}}_{1} s β_{1} + {\overset{\cdot\cdot}{γ}}_{1} \end{matrix}]

(A23)

{}^{id}{\underline{J}}_{1}^{(1)} = {[\begin{matrix} J_{1_{11}}^{(1)} & J_{1_{22}}^{(1)} & J_{1_{33}}^{(1)} \end{matrix}]}^{T}

(A24)

{\underline{M}}_{1}^{(1)} = A^{(10)} (({\underline{M}}_{1}^{(a)} - {\underline{M}}_{2}^{(a)}) + ({\underline{F}}_{1}^{e} + {\underline{F}}_{2}^{e}) \times \frac{l_{1}}{2} {\underline{e}}_{1}^{(1)})

The mass parameters identification equation of $B_{2}$ is

(A25)

[\begin{matrix} {\overset{\cdot\cdot}{x}}_{2} \\ {\overset{\cdot\cdot}{y}}_{2} \\ {\overset{\cdot\cdot}{z}}_{2} + g \end{matrix}] [\begin{matrix} m_{2} \\ m_{2} \\ m_{2} \end{matrix}] = [\begin{matrix} - F_{21}^{e} \\ - F_{22}^{e} \\ - F_{23}^{e} \end{matrix}]

Abbreviated as

(A26)

{}^{id}{\underline{R}}_{2} {}^{id}{\underline{m}}_{2} = {\underline{F}}_{2}^{R}

The equation for identifying the principal moment of inertia parameters of rigid body $B_{2}$ is

(A27)

{}^{id}{\underline{S}}_{2}^{id} {\underline{J}}_{2}^{(2)} = {\underline{M}}_{2}^{(2)}

where,

(A28)

{}^{id}{\underline{S}}_{2}^{(2)} = [\begin{matrix} \begin{matrix} - {\overset{\cdot}{α}}_{2} ({\overset{\cdot}{β}}_{2} s β_{2} c γ_{2} + {\overset{\cdot}{γ}}_{2} c β_{2} s γ_{2}) + \\ {\overset{\cdot}{β}}_{2} {\overset{\cdot}{γ}}_{2} c γ_{2} + {\overset{\cdot\cdot}{α}}_{2} c β_{2} c γ_{2} + {\overset{\cdot\cdot}{β}}_{2} s γ_{2} \end{matrix} & \begin{matrix} - (- {\overset{\cdot}{α}}_{2} c β_{2} s γ_{2} + {\overset{\cdot}{β}}_{2} c γ_{2}) \cdot \\ ({\overset{\cdot}{α}}_{2} s β_{2} + {\overset{\cdot}{γ}}_{2}) \end{matrix} & \begin{matrix} (- {\overset{\cdot}{α}}_{2} c β_{2} s γ_{2} + {\overset{\cdot}{β}}_{2} c γ_{2}) \cdot \\ ({\overset{\cdot}{α}}_{2} s β_{2} + {\overset{\cdot}{γ}}_{2}) \end{matrix} \\ \begin{matrix} ({\overset{\cdot}{α}}_{2} c β_{2} c γ_{2} + {\overset{\cdot}{β}}_{2} s γ_{2}) \cdot \\ ({\overset{\cdot}{α}}_{2} s β_{2} + {\overset{\cdot}{γ}}_{2}) \end{matrix} & \begin{matrix} {\overset{\cdot}{α}}_{2} ({\overset{\cdot}{β}}_{2} s β_{2} s γ_{2} - {\overset{\cdot}{γ}}_{2} c β_{2} c γ_{2}) - \\ {\overset{\cdot}{β}}_{2} {\overset{\cdot}{γ}}_{2} s γ_{2} - {\overset{\cdot\cdot}{α}}_{2} c β_{2} s γ_{2} + {\overset{\cdot\cdot}{β}}_{2} c γ_{2} \end{matrix} & \begin{matrix} - ({\overset{\cdot}{α}}_{2} c {\overset{\cdot}{β}}_{2} c γ_{2} s + β_{2} s γ_{2}) . \\ ({\overset{\cdot}{α}}_{2} s β_{2} + {\overset{\cdot}{γ}}_{2}) \end{matrix} \\ \begin{matrix} - ({\overset{\cdot}{α}}_{2} c β_{2} c γ_{2} + {\overset{\cdot}{β}}_{2} s γ_{2}) \cdot \\ (- {\overset{\cdot}{α}}_{2} c β_{2} s γ_{2} + {\overset{\cdot}{β}}_{2} c γ_{2}) \end{matrix} & \begin{matrix} ({\overset{\cdot}{α}}_{2} c β_{2} c γ_{2} + {\overset{\cdot}{β}}_{2} s γ_{2}) \cdot \\ (- {\overset{\cdot}{α}}_{2} c β_{2} s γ_{2} + {\overset{\cdot}{β}}_{2} c γ_{2}) \end{matrix} & {\overset{\cdot}{α}}_{2} {\overset{\cdot}{β}}_{2} c β_{2} + {\overset{\cdot}{γ}}_{2} + {\overset{\cdot\cdot}{α}}_{2} s β_{2} + {\overset{\cdot\cdot}{γ}}_{2} \end{matrix}]

(A29)

{}^{id}{\underline{J}}_{2}^{(2)} = {[\begin{matrix} J_{2_{11}}^{(2)} & J_{2_{22}}^{(2)} & J_{2_{33}}^{(2)} \end{matrix}]}^{T}

(A30)

{\underline{M}}_{2}^{(2)} = A^{(20)} ({\underline{M}}_{2}^{(a)} + {\underline{F}}_{2}^{e} \times \frac{l_{2}}{2} {\underline{e}}_{1}^{(2)})

The inertia parameters identification model of 2-DOF manipulator is obtained by combining the inertia parameters identification equations equations (A20), (A21), (A26) and (A27) of $B_{1}, B_{2}$ .

(A31)

[\begin{matrix} {}^{id}{\underline{R}}_{1} & \underline{0} & \underline{0} & \underline{0} \\ \underline{0} & {}^{id}{\underline{S}}_{1}^{(1)} & \underline{0} & \underline{0} \\ \underline{0} & \underline{0} & {}^{id}{\underline{R}}_{2} & \underline{0} \\ \underline{0} & \underline{0} & \underline{0} & {}^{id}{\underline{S}}_{2}^{(2)} \end{matrix}] [\begin{matrix} {}^{id}{\underline{m}}_{1} \\ {}^{id}{\underline{J}}_{1}^{(1)} \\ {}^{id}{\underline{m}}_{2} \\ {}^{id}{\underline{J}}_{2}^{(2)} \end{matrix}] = [\begin{matrix} {\underline{F}}_{1}^{R} \\ {\underline{M}}_{1}^{(1)} \\ {\underline{F}}_{2}^{R} \\ {\underline{M}}_{2}^{(2)} \end{matrix}]

Abbreviated as

(A32)

{}^{id}{\underline{Y}}_{2 - DOF} {}^{id}{\underline{X}}_{2 - DOF} = {}^{id}{\underline{Z}}_{2 - DOF}

Equation (A32) is in the least squares form, and its least squares solution is

(A33)

{}^{id}{\underline{X}}_{2 - DOF} = {({}^{id}{\underline{Y}}_{2 - DOF}^{T} {}^{id}{\underline{Y}}_{2 - DOF})}^{- 1} {}^{id}{\underline{Y}}_{2 - DOF}^{T} {}^{id}{\underline{Z}}_{2 - DOF}

The inertia parameters ${}^{id}{\underline{X}}_{2 - DOF}$ of the 2-DOF manipulator can be identified by equation (A33).

Handling Editor: Sharmili Pandian

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: This work was supported by “the National Key R&D Program of China (No. 2022YFC3800405).”

ORCID iD

Jun Cheng

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