Frontiers in Computing and Intelligent Systems 
ISSN: 2832-6024 | Vol. 12, No. 1, 2025 

 

144 

Identification	of	Robot	Dynamic	Parameters	Based	on	
BP	Neural	Network	
Yingjie Zhong 

School of Mechanical Engineering, Taiyuan University of Science and Technology, Taiyuan, Shanxi, China 

 

Abstract: This paper proposes a dynamic parameter identification method based on the BP neural network to improve the 
accuracy of dynamic modeling and control performance of industrial robots. The method employs Fourier polynomial trajectory 
planning and constructs a BP neural network model for parameter identification. A six-degree-of-freedom (DOF) robotic arm is 
simulated using the Matlab Robotics Toolbox for validation. The results demonstrate that the proposed method can achieve 
accurate dynamic parameter identification under complex conditions, exhibiting robustness against environmental noise and 
friction disturbances. Furthermore, it enhances the efficiency and accuracy of dynamic parameter identification. 

Keywords: BP Neural Network; Robot Dynamics; Parameter Identification Introduction. 
 

1. Introduction 
New-quality productive forces have emerged in practice, 

demonstrating strong driving and supporting capabilities for 
high-quality development. Industrial robots, known for their 
high precision, flexibility, and adaptability, have been widely 
applied in various fields, particularly in interactive 
application scenarios involving rigid contact between the 
robot and the environment. Such scenarios include processing 
robots engaged in cutting, grinding, milling, and polishing 
tasks [1]. Model-based torque control enables real-time 
adjustments to the robot's motion, allowing it to respond 
promptly to rigid collisions and meet production requirements, 
thereby necessitating an accurate dynamic model. Neural 
networks, with their ability to learn from large volumes of 
input-output data, offer superior identification and 
generalization under complex conditions. This paper aims to 
investigate the problem of dynamic parameter identification 
using neural networks. 

Researchers worldwide have conducted extensive studies 
on dynamic parameter identification. Early approaches relied 
on generating dynamic parameters directly from CAD models, 
but these methods were significantly affected by external 
disturbances, making it difficult to obtain accurate dynamic 
models. Gautier et al. [2] improved parameter identification 
accuracy by optimizing excitation trajectories to obtain 
smooth and continuous motion paths. However, measurement 
randomness often led to local optimal solutions. Among 
existing studies, offline identification methods are considered 
more reliable. The general process of parameter identification 
includes establishing a dynamic model, obtaining a minimal 
dynamic parameter set and linearizing the system, 
determining and optimizing excitation trajectories, collecting 
motion and torque data, performing data preprocessing, and 
solving the dynamic equations to compute torques. The 
collected data is then fed into the dynamic parameter 
identification equations, followed by model validation. 

Traditional identification methods, such as the least squares 
method, are commonly used. For example, Zhou Jun et al. [3] 
applied linear least squares for dynamic parameter 
identification of flexible joint modular robots. However, the 
identified parameters were not sufficiently accurate and were 
highly sensitive to noise. Feng Limin et al. [4] adopted a 

weighted least squares approach to mitigate the effects of 
noise interference to some extent. Zhu Zhenbiao et al. [5] 
generated excitation trajectories using a combination of 
Fourier series and fifth-order polynomials, achieving precise 
torque tracking based on joint angle and current 
measurements, though issues such as trajectory 
discontinuities and accuracy remained. Zhang Shiyuan et al. 
[6] optimized excitation trajectory signals using genetic 
algorithms, effectively reducing errors. Chen Zhihuan et al. 
[7] proposed an improved version of the whale optimization 
algorithm. Liu Yu et al. [8] enhanced the genetic algorithm to 
mitigate premature convergence and local optima, 
significantly improving parameter identification accuracy. 
Zhang Luming et al. [9] utilized a neural network-based 
torque compensation approach for dynamic parameter 
identification, reducing the impact of nonlinear factors. 
Zhang Di et al. [10] established a neural network-based 
trajectory tracking model for robotic arms, enabling real-time 
trajectory adjustments, though their study did not delve 
deeply into dynamic parameter identification. 

Among these methods, identification approaches derived 
from inverse dynamic models, particularly the traditional 
least squares regression-based parameter identification, are 
highly susceptible to environmental noise and friction 
disturbances, leading to reduced identification efficiency. To 
improve parameter identification accuracy, enhance 
efficiency, and mitigate the issue of local optimal solutions, 
this paper proposes a dynamic parameter identification 
method based on the BP neural network. The proposed 
method utilizes Fourier polynomial trajectory planning and 
constructs a neural network model. Finally, a six-degree-of-
freedom robot is simulated using the Matlab Robotics 
Toolbox for parameter validation, confirming the accuracy of 
the identified model. 

2. Robotic Arm Identification Model 

2.1. Robot Dynamics Model 
The Newton-Euler iterative method is used to construct the 

dynamic equations, with outward iteration for velocity and 
acceleration, followed by inward iteration for force and 
torque. 



 

145 

 
Fig 1. Dynamic parameters of the connecting rod 

 
The outward iteration is used to calculate the angular 

velocity, linear acceleration, and angular acceleration of each 
link. The calculation of angular velocity can start from link 1 
and proceed to link n. Given that the angular velocity of link 
i+1i+1i+1 in the iii-coordinate system is: 

 1
1 1 1 1

i i i i
i i i i iR Z   
       

Where: 1i 
  is the angular velocity of joint i+1, and 1

i
i R  

is the rotation matrix from coordinate system {i+1} to {i}. 
By left-multiplying Equation (1) by the rotation matrix 

1i
i R
 , the angular velocity of link i+1 in the {i+1} coordinate 

system is given by: 

 1 1 1
1 1 1

ˆi i i i
i i i i iR Z    
      

Where: i
i   is the angular velocity of link I in the {i} 

coordinate system, and 1i
i R
  is the rotation matrix from the 

{i} to the {i+1} coordinate system. 
The angular acceleration transformation equation between 

the links can be obtained, which gives the angular 
acceleration of link i to i+1 as: 

 1 1 1 1 1
1 1 1 1 1

ˆ ˆi i i i i i i
i i i i i i i i iR R Z Z        
           

Where 1i 
  is the angular velocity of joint i+1, and 1i 

  is 

the angular acceleration of joint i+1. 
The formula for calculating the linear acceleration of the 

link in a rotational joint manipulator is given by:v 

  A B A A A B A A B
Q BORG B B B B BV V R Q R Q            

The linear acceleration at the origin of each link's 
coordinate system is given by: 

   1 1
1 1 1

i i i i i i i i
i i i i i i iR P P     
          

Similarly, the linear acceleration of the link's center of mass 
is given by: 

  
i i i

i i i i i i i
c i c i i c iP P             

Where 
i

i
cP  is the vector of the manipulator. 

The inertial force and torque at the center of mass of each 

link can be determined using the Newton-Euler equations: 


ii cF m   

 i ic c
i i i iN I I      

After determining the forces and torques on each link, the 
torques at the joints need to be calculated, which correspond 
to the forces and torques applied to the links by the joint 
motors. 

By iterating inward from link nnn to the base coordinate 
system, the inertial forces and torques acting on each link can 
be obtained: 

 1
1 1

i i i i
i i i if R f F

    

 1 1
1 1 1 1 1i

i i i i i i i i i
i i i i c i i i in N R n P F P R f 

          

The joint torque can be obtained by calculating the 
component of the torque applied by one link to the adjacent 
link along the Z-axis: 

 ˆi T i
i i in Z   

 
Fig 2. The link model of a robot with nnn degrees of freedom 

 
As shown in the figure, the robot joints are typically 

connected in a series configuration. Where joint i (i≠0) is the 
i-th revolute joint, and link i is the i-th link.Units 

In the absence of external forces, the torques acting on the 
center of mass of the link are summed, and their sum is set to 
zero, thus obtaining the torque balance equation. The dynamic 
equation is generally expressed as [11]: 

      ,M V G          

Here,  1 2 3, , , ,
T

n       represents the joint torque, 

 M  	∈ n nR   is the inertia matrix, , ,      denote the joint 

position, angular velocity, and angular acceleration, 
respectively.  V  ∈ n nR   represents the Coriolis and 

centrifugal force matrix, and  G  ∈ n nR    represents the 

gravitational torque. 



 

146 

2.2. Linearization of the Dynamic Model and 
Solution of The Minimal Set of Dynamic 
Parameters 

In the above dynamic model, the dynamic parameters are 
in a nonlinear form, which makes parameter identification 
difficult. Therefore, the dynamic model can be represented as 
follows: 

  , , fullY X       

Where:  , ,Y     ∈ n nR    is the function matrix of the 

joint position, angular velocity, and angular acceleration of the 

robotic arm, and fullX
 represents all dynamic parameters. 

For a robotic arm with n degrees of freedom, the dynamic 
inertia parameters of the i-th link of the joint can be expressed 
as follows: 

 , , , , , , , , ,fulli xx yy zz xy xz yzX m mx my mz I I I I I I     

Where: fulliX  represents the dynamic parameter of the i-

th joint, , ,xx yy zzI I I  represents the moment of inertia of the 

joint, and x, y, z represent the position of the joint's center of 
mass, , ,xy xz yzI I I  represents the inertia product of the joint. 

The dynamic parameters can be expressed as: 
 1 2 3, , , , nX X X X X   

The dynamical model can be expressed as: 



11 12 1 1
1

21 22 2 2

1 2

n

n

n
n n nn n

Y Y Y X

Y Y Y X
YX

Y Y Y X






   
     
            
      

   





    



 

However, the observation matrix of the robotic arm Y 

 , ,Y       is rank-deficient, with some rows and columns 

being constantly zero, while others are linear combinations of 
certain other rows and columns. It is necessary to reorganize 
and identify the set of dynamic parameters to obtain the 
minimal dynamic parameter set, which is required for the 
identification of dynamic parameters. In this way, the 
relationship between the joint torque and the minimal 
dynamic parameter set is: 

 min( , , )Y X       

By linearizing the dynamical model, the identification of 
nonlinear dynamic parameters is transformed into the 
identification of linear dynamic parameters. 

2.3. Excitation Trajectory Design and Data 
Collection 

To activate all parameters in the robotic dynamics model 
and ensure that all parameters are sufficiently "activated" 
during the identification process, thereby allowing for precise 
collection of motion, torque, and other related datasets, it is 
necessary to design appropriate excitation trajectories for the 
robot. Fourier series-based trajectories [12] can ensure 
periodic input, with data collected from different cycles being 
consistent. This makes data collection convenient and highly 
feasible. Therefore, Fourier series can be used to construct 
excitation trajectories for collecting joint positions, angular 
velocities, and angular accelerations of the robotic arm. In the 
Fourier series, the angular displacement, angular velocity, and 
angular acceleration of the i-th rotational joint of the robotic 
arm are given by: 



     

     

     

0
1

1

1

sin cos

cos sin

sin cos

N
il il

i f f i
l f f

N

i il f il f
l

N

i f il f il f
l

a b
q t w lt w lt q

w l w l

q t a w lt b w lt

q t w a l w lt b l w lt







 
   

  

   

    











 

Where: ila , ilb  represents the amplitude of the cosine and 

sine components for the I -th joint at different frequencies, 

0iq  denotes the positional deviation of the i -th joint, N is the 

number of harmonic terms contained in the Fourier series, and 

fw   is the fundamental angular frequency of the excitation 

frequency. 
For the UR5 robotic arm studied in this paper, a 5th-order 

Fourier series can be employed, specifically N = 5, where ila ,

ilb  , 0iq   are the parameters to be optimized. Each joint has 

2 1N    parameters that require optimization for the 
excitation trajectory. The 5th-order Fourier excitation 
trajectory parameters were obtained using the Fmincon 
optimization toolbox in MATLAB. 

 
Fig 3. Fourier series excitation trajectories 



 

147 

 
During the real-time operation of the robotic arm along the 

excitation trajectory, the joint angles and joint angular 
velocities can be collected. By differentiating the Fourier 
series representation, the joint angular acceleration can be 

obtained. Additionally, the motor current can be used to infer 
the motor output torque, leading to the calculation of the joint 
torque. 

 
Fig 4. The Dynamic parameter identification process 

 

3. Neural Network Modeling 

3.1. BP Neural Network Model 
A Back-Propagation Neural Network (BPNN), or back-

propagation neural network, constitutes a multi-layer feed-
forward neural network with the capacity to accommodate 
intricate non-linear correlations through the implementation 
of a back-propagation algorithm for the purposes of network 
training. A BP neural network typically comprises three layers: 
an input layer, a hidden layer and an output layer. Each layer 
is comprised of multiple neurons, which are connected to one 
another by means of weights. Each neuron performs a 
weighted sum of the received signals and then performs a 
non-linear transformation through an activation function to 
produce an output. The input signals of a BP neural network 
originate in the input layer and are conveyed to the output 
layer in a successive manner. Subsequently, the output of the 
network is compared with the true value, and the loss function 
is calculated. The weights of the network are then adjusted 
according to the loss function with the objective of reducing 
the error and updating the weights. Finally, the iterations are 
repeated in order to achieve the predetermined accuracy. 

From the relationship in Equation (16) between the joint 
torque and the minimal set of dynamic parameters, it can be 
inferred that by assigning physical meaning to each weight, 
the trained weights correspond to the dynamic parameters of 
the robotic arm. 

In BP neural networks, a multitude of neurons are 
interconnected in this manner to form a sophisticated network 
structure, enabling the network to discern intricate patterns 
and characteristics within the data. 

Neural networks construct more complex models by 
introducing hidden layers, which have their own internal 
weights and biases. These hidden layers enable the network 
to learn nonlinear relationships and deeper features within the 
data. The hidden layers are one or more intermediate layers 
that are not directly connected to the input or output. Neurons 
in the hidden layers receive signals from the input layer, and 
after processing with weights and biases, the activation 
function transforms these signals into new representations. 

The inclusion of hidden layers increases the network's 
parameters, weights, and biases, enhancing its parameter 

capacity and allowing it to fit more complex functions. Each 
layer further abstracts and generalizes from the previous layer, 
which helps the neural network better understand and process 
the data. 

3.2. Neural Network Construction 

 
Fig 5. BP Neural network construction 

3.3. The Selection of the Activation Function 
In neural network models, the choice of activation function 

has a crucial impact on the model's learning performance and 
nonlinear fitting ability. The Sigmoid function is the most 
traditional and commonly used nonlinear activation function, 
and its mathematical expression is: 

   1

1 x
x

e
 


 

The Sigmoid function works well for logistic regression 
and binary classification problems, but it suffers from the 
vanishing gradient problem when the input values approach 
infinity or negative infinity. Therefore, this paper uses the 
ReLU function, which is commonly used in modern deep 
learning: 

   max(0, )x x   

The ReLU function has a simple expression, involving only 
a basic threshold operation. It requires less computation 



 

148 

during optimization, is more efficient, and offers strong 
flexibility and generalizability. 

3.4. The Selection of Training and 
Optimization Methods for Neural Network 
Models 

To measure the difference between the predicted values and 
the actual values of the neural network model, and to 
accurately predict or classify new data, the loss function J is 
defined using the Root Mean Square Error RMSE of the joint 
torque [13]: 

  2

1

1
ˆ

N

i i
i

RMSE
N

 


   

where N is the total number of sample data, i  is the actual 

value of data point i, and î  is the predicted value of data 

point i. 
Typically, gradient descent is chosen as the optimization 

algorithm to minimize the loss function. By randomly 
initializing the model's weights and biases, the derivative of 
the loss function with respect to each parameter is computed 
to obtain the gradient of the loss function. The parameters are 
then updated based on the gradient and a predefined step size:  

  J       

where θ represents the parameters, ∝ is the learning rate, 
and  J    is the gradient of the loss function J with 

respect to the parameter θ 
By continuously calculating and updating the gradient until 

the minimum value of the loss function is found, and the 
termination criteria are met, the training of the BP neural 
network model is completed. 

4. Validation of the BP Neural 
Network-Based Parameter 
Identification Model 

4.1. Experimental Design 
This study validates the accuracy of the identification 

model through simulation. A six-degree-of-freedom serial 
robotic arm is constructed using the Matlab R2024a Robotics 
Toolbox as the parameter identification object, with 
theoretical dynamic parameters defined. Input data for the 
identification model is generated through data construction. 
The process is as follows: 

(1) Define the excitation trajectory and obtain the 
corresponding joint sequence for executing the trajectory 
through inverse kinematics of the robotic arm, then save the 
data. 

(2) Input the dynamic parameters and joint sequence into 
the dynamic model to compute the joint output torques, and 
save the results. 

(3) Feed the joint angle and torque data into the proposed 
identification network. Use the trained neural network to 
predict the identification results, compare them with the 
ground truth, and output the identification error. 

(4) Provide an arbitrary validation motion trajectory and 
compare the actual joint output torque with the theoretical 
torque based on the identification results. 

In the neural network model identification experiment, the 
more trajectory data available, the more accurate the 
identification model becomes. Multiple sets of trajectory data 
were collected for the experiment, with each set containing a 
large number of joint data points. These data sets were divided 
into a training set (80%) and a validation set (20%), where the 
training set is used for training the neural network, and the 
validation set is used to assess the model's prediction 
performance. During the training process, the size of the loss 
function and training accuracy are monitored in real time. As 
the model iterates, its accuracy continuously improves, 
reaching a maximum value, while the loss function steadily 
decreases and approaches zero. 

4.2. Identification Results 
 

 

Fig 6. Predicted moments for each joint at the validation trajectory 



 

149 

 
The model output torques and desired output torques are 

visualized as shown in the figure. The results show that, on 
the validation trajectory, the torques predicted by the BP 
neural network are highly consistent with the theoretical 
values. The prediction performance on complex motion 
trajectories is excellent. This can be attributed to the fact that 
the neural network does not require explicit modeling of 
nonlinear factors and is capable of handling nonlinear 
relationships. Additionally, the network demonstrates 
stronger robustness to noise, leading to better control 
performance and higher accuracy.   

In addition, the torque curve obtained from the model is 
smooth, with small torque errors and peak errors. Compared 
to traditional methods, the BP neural network-based dynamic 
parameter identification method demonstrates stronger 
robustness and generalization capabilities, making it more 
suitable for high-precision control scenarios in industrial 
robots. Future work could focus on further optimizing the 
network structure and training algorithms to enhance 
efficiency and adapt to more complex task requirements. 

5. Conclusion 
This paper proposes a BP neural network-based dynamic 

parameter identification method aimed at improving the 
accuracy of robot dynamic parameter identification, thereby 
achieving high-precision torque control. By utilizing Fourier 
polynomial trajectory planning and constructing the BP 
neural network, the identification accuracy is effectively 
enhanced. Finally, the dynamic parameter identification 
simulation is conducted using a six-degree-of-freedom 
robotic arm model constructed in the Matlab Robotics 
Toolbox, validating the accuracy and effectiveness of the 
model. 

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