Robotics and Autonomous Systems 149 (2022) 103971
Contents lists available at ScienceDirect
Robotics and Autonomous Systems
journal homepage: www.elsevier.com/locate/robot
A fast robotic arm gravity compensation updating approach for
industrial application using sparse selection and reconstruction
∗
Chenglong Yu , Zhiqi Li, Dapeng Yang, Hong Liu
State Key Laboratory of Robotics and System, Harbin Institute of Technology, Harbin 150001, China
article
info
Article history:
Received 30 June 2021
Received in revised form 8 November 2021
Accepted 7 December 2021
Available online 16 December 2021
Keywords:
Industrial robot arm
Gravity compensation
Feature selection
Model reconstruction
ADMM
a b s t r a c t
With the increasing complexity of robot tasks in the industrial field, sometimes the manipulator needs
to change the payload or replace end-effectors frequently in a period of time, which will cause the
change of gravity model. This paper presents a fast gravity compensation updating method based on
sparse selection and reconstruction. Instead of using the classic model, feature sets in the formulation
of dynamic motion are exploited and the gravity model learning is transformed into a sparse problem.
Then, the Alternating Direction Multiplier Method (ADMM) is used to accelerate the process of solving
the sparse optimization problem and reconstructing the effective features of the gravity model from
the original signal. The merits of our method are that it does not depend on any kinematic and
dynamic parameters, and there is no need to redesign the specific excitation trajectory in the gravity
model updating. Thus, the updating process avoids heavy work of calibration and simplifies the labor
complexity considerably from the conventional analytical methods. The results of various payload
experiments on a real 7-DOF manipulator show that the proposed method can update the gravity
compensation model efficiently and accurately.
© 2021 Elsevier B.V. All rights reserved.
1. Introduction
Many applications in robotics involve operations at slow speed
to ensure different tasks [1]. For example, the robot in the area of
surgery is required to play motions at very low speeds for safety
and repeatability [2,3]. By having robots working in low-speed
processes to ensure high position accuracy, reduce production
cost and improve collision avoidance performance, as arc welding, etc [4,5]. In these cases, the gravity torque generated by
the mass of the connecting rod usually plays a dominant role
in driving torque and is much greater than dynamic torques.
Therefore, gravity compensation is beneficial and necessary for
the design of robot control algorithms. Moreover, as the types of
tasks performed by robots become more and more complex in
many industrial applications, sometimes it is necessary to change
the payload or replace the end-effectors frequently. The change of
load will affect the gravity model of the robot. In this case, gravity
compensation is required to update fast and accurately.
Previous approaches in literature for gravity compensation
can be categorized as passive mechanical compensation, PD-type
feedback and Gravity model estimation. In the passive compensation method, gravity compensation can be performed mechanically. The robot is required to be isolated from each link, and
∗ Corresponding author.
E-mail addresses: yuchenglong@hit.edu.cn (C. Yu), lzq@hit.edu.cn (Z. Li),
yangdapeng@hit.edu.cn (D. Yang), hong.liu@hit.edu.cn (H. Liu).
https://doi.org/10.1016/j.robot.2021.103971
0921-8890/© 2021 Elsevier B.V. All rights reserved.
use counterweight, spring or specific measuring device to compensate gravity, which minimizes the torques required from the
motors to sustain the weight of the robot. For example, the mass
of the connecting rod can be directly weighed, the position vector
of the center of mass can be estimated by determining the equilibrium point of the connecting rod, and the middle equilibrium
point can be obtained by pendulum motion [6,7]. These kinds
of methods are complex and tedious, and the characteristics of
joints are ignored. The compensation accuracy depends on the accuracy of the mechanical device. Therefore, gravity compensation
is considered to be performed at the software level to improve the
accuracy. This does not help to reduce the burden of the motors,
but can greatly improve the control performance.
The PD-type feedback compensation method has proved to
be a practical algorithm for robot gravity compensation due to
its simplicity and clear physical meaning. They are generally
designed including terms that partially (or totally) compensate
for the system dynamics [8]. In these approaches, the saturation
is applied by introducing nonlinear functions in the control laws
and gravity compensation is realized with bounded inputs [9,10]
For better closed-loop performance, full system information is
required in these control laws. However, most conventional PID
control methods have no learning capability. With the change
of gravity terms and operating conditions, the parameters of the
controller are difficult to adapt to. Moreover, the design of these
controllers is a challenge for a multi-input/multi-output (MIMO)
C. Yu, Z. Li, D. Yang et al.
Robotics and Autonomous Systems 149 (2022) 103971
system [11–13]. To achieve an optimal or near-optimal control
signal is still a difficult problem to address.
In the gravity model estimation method, a model can be obtained by minimizing the errors between a function of the robot
variables and an estimation model. The gravity parameters are
regressed based on the physical manifestation restrictions and
the input/output behaviors of the robot on some planned motion
[14–16]. This method has been extensively used since it has the
best performance in the simplicity of the experiment and the
model accuracy [17]. It relates the condition number of regressor
to the reliability of the data and designs the exciting trajectory
so as to minimize the condition number [18,19]. In particular, the
method requires the proper design of exciting trajectories to provide accurate and fast parameter estimation, even in the presence
of measurement noise and external disturbances. However, to
design specific trajectories for a complex robotic system is a challenging task with the growing number of variables. In addition,
the exciting trajectory has to be redesigned when the payload
of the manipulator changes. This increases the complexity of the
work and sometimes leads to unreliable results.
Inspired by the problem of robot gravity compensation, this
paper attempts to explore a novel gravity model updating approach with lower complexity and higher efficiency, which can be
applied in industrial practice. The core principle of this method is
to select the feature sets from the dynamic gravity motion model
and then transform the model learning into a sparse problem.
This method is data-driven, which only needs the measurements
of joint angle and driving torque of the robot in a simple motion.
It does not need any prior knowledge of the physical parameters
of the manipulator, thus simplify the process consistently from
the conventional analytical method. In the process of solving the
gravity model, we use ADMM to solve the sparse optimization
problem and reconstruct the effective features of the gravity
model from the original signal. It can accelerate the process of the
L1-norm regularization optimization while ensuring accuracy, so
as to improve the updating speed of the gravity model. Furthermore, the specific forms of the features are viewed as restrictions
of learning. These restrictions can be viewed as a strong priority
of selection and reconstruction, which makes good performance
in data efficiency and generalization capability.
gravity is about tens to hundreds of times larger than that caused
by even the maximum angular acceleration [23], which indicates
that the inertia tensor has little effect on the driving torque. The
main idea of gravity compensation in our approach is to model
the gravitational term in dynamics which is a linear function of
the joint angle q. Friction at the joints is considered as unmodeled
dynamic errors vulnerable to sensor noise and is not included.
According to the above theoretical basis, the dynamic equation of
a multi-DOF manipulator at low-speed motion without external
force can be simplified. The relationship between the joint driving
torque and the gravity term can be expressed as:
g(q) =
)
mi g
(2)
i
where, g(q) is the joint driving torque generated by the effect of
gravity. mi is the mass of link i. g is the gravitational acceleration
in the base frame. 0 J vi is a submatrix of the Jacobian, which maps
joint velocities to translational velocities of link i. To find the
consistency characteristics of dynamic equation and facilitate the
gravity term modeling, the following property is considered (see
Appendix A for proof).
Property 1. The Jacobian matrix of a kinematic chain with N-DOF
has the feature sets with the following form
F =
{N }
∏
en
(3)
n=1
where,
en ∈ {1, sin(qn ), cos(qn )}
(4)
qn is the angle of joint n. It can be verified that for each link n ∈
[1, N].
• The Jacobian matrix w.r.t the base frame denoted by 0 Jn can be
expressed as a linear combination of features in F .
• The argument also holds for
( the
) Jacobian matrix written in the
end-effector frame, i.e.,
N
0
Jn .
Note that the weighted coefficients of the features in F are
functions of the kinematic parameters in Jacobian matrix such as
link lengths, twist angles, link mass and etc. For a specific serial
manipulator with revolute joints, these parameters are fixed. The
length of the links may be any suitable value and the mass of the
links are constants. Therefore, the following expression could be
obtained based on Property 1
2. Gravity model observation
Considering a serial manipulator composed of rigid links connected through rotational joints, the equations of motion of the
system can be obtained via the recursive Newton–Euler or Lagrange formulation. The compact dynamic model of the robot can
be written in the following form as
τ = M(q)q̈ + C (q, q̇)q̇ + g(q)
0 v T
Ji
∑(
Remark 1. There exists a general expression for the joint torques
caused by gravity:
gn (q) = (g ⊗ f (q))T β
(5)
where, gn (q) is the torque of joint n caused by gravity in Eq. (1).
β is the weighted parameter vector. ⊗ is the Kronecker product.
The features are indexed to obtain the vector function f (q) for
compactness. In Eq. (5), the variables except β can be measured
without difficulties, namely gn (q) and q can be obtained using
joint torque sensors and joint position sensors. Many industrial
robots are equipped with torque sensors, and can solve part or
all of the dynamic models by learning or regression approach
according to the mapping between the inputs (joint positions)
and outputs (joint torques) of the sampled data [24,25]. These
methods are undoubtedly more accurate and convenient [26].
Considering that some robots do not include joint torque sensors,
a motor torque measurement scheme based on DSP is proposed
in [27]. According to the measured motor end waveform and
the dynamic model, the torque can be calculated by a digital
signal processor without any mechanical sensors. We refer to
the interested readers to [27] for a detailed discussion of the
no-mechanical-sensor torque measurement method.
(1)
where, q, q̇, q̈ are the arm joint angle, angular velocity and angular
acceleration of the manipulator, respectively. τ is the joint driving
torques. M(q) is the inertia matrix of the manipulator which
is multiplied to the second derivative of q to obtain the joint
torques. C (q, q̇) donates the centrifugal and Coriolis term, where
the quadratic terms in the first derivatives of q are included and
the coefficients could depend on q. g(q) is the gravity term with
q involved and no derivatives. We use the tensor notation [20]
in the following paper, for instance, k (j Ji ) represents the Jacobian
matrix of frame i respect to (w.r.t) frame j expressed in frame k.
When two superscripts are the same, the outer one is omitted,
i.e. j (j Ji ) = j Ji .
With the manipulator in low-speed motion, the relative and
absolute values of Coriolis moment and centripetal moment are
extremely small and can be ignored along a trajectory in practical
industrial applications [21,22]. In addition, the torque caused by
2
C. Yu, Z. Li, D. Yang et al.
Robotics and Autonomous Systems 149 (2022) 103971
The weighted parameter vector β includes constant coefficient
related to the physical structure of the mechanical compositions
of the arm, such as link lengths and twist angle. These mechanical
compositions are a few and limited and the vector β should only
contain a small portion of the features. For robotic manipulators
with a fixed base, the constant gravity acceleration g can be
absorbed by model parameters β . By now, the expression of
the feature-based gravity model can be updated to the following
form:
τg = F βg
(6)
where, τg is the joint torque measurements of joint n. βg is the
gravity model parameters. And
F = f (1)
f (2)
[
· · · f (M )
( (j) )
]T
(7)
where, f (j) = f q
for j = 1, 2 · · · M and M is the number of
data measurements.
According to Property 1, for an N-DOF manipulator, the total
number of possible feature sets in one measurement is 3N . Since
the coefficient related to the physical structure of the mechanical
compositions are a few and limited, the size of the feature function is much more than the small portion of the effective features.
It indicates that the feature-based gravity model in Eq. (6) is
sparse. In [28,29], an effective method for model simplification
and characterization using sparsity is proposed, which is called
the Lasso. This method has twofold advantages of ridge regression
and subset selection. It can shrink the coefficients continuously in
the process, and give an easily interpretable model with a stable
state. Model complexity can be reduced by forcing less influential
variables to have zero influence. In the process, the same sparse
model is found (with many zero regression coefficients) as a
well-predicting model. Based on the Lasso regression, the gravity
model parameter vector βg could be estimated in our approach
and is expressed as:
2
minimizeπ F βg − τ 2 + λ βg 1


 
Fig. 1. ADMM-based algorithm for solving gravity modeling procedure.
Then, the augmented Lagrangian function in Eq. (10) is defined
as:
1
∥Ax − b∥22 + λ ∥z ∥1 + yT (x − z )
(11)
2
where, y is the Lagrangian multiplier vector. The benchmark
algorithm ADMM splits the subproblem in iteration to avoid the
problem of unknown vectors overlapped. The steps over variables
x and z are read as the following form:
L (x, z, y) =
(8)
x∗ = AT A + ρ E
(
3. Sparse feature recovery
z ∗ = Sλ/ρ x∗ + y
(
The feasible way to find the solution is to apply convex optimization solvers [30,31]. However, the solvers need the secondorder information of the problem. When the size of the matrix is
large, it will increase the computational burden and need more
storage memory and time. As a method that only needs firstorder information, ADMM is considered to be an ideal method to
deal with large-scale L1-regularized least square problems. It can
coordinate solutions of small local subproblems to find solutions
to large global problems. Therefore, ADMM can be used to capture
the complexity of data and deal with high-dimensional data
problems in machine learning. Since this approach accelerates the
processing of convex problems greatly, it is well suited for solving
and updating gravity model while training time is limited.
For the following structured convex optimization problem in
gravity model updating
minimizex
1
2
∥Ax − b∥22 + λ ∥x∥1
subject to
1
2
AT b + ρ (z − y)
)
)
(12)
(13)
where, ρ is the penalty parameter which can improve the convergence properties. The soft thresholding operator S is defined
as [32]:
Sρ (α ) = (α − ρ)+ − (α + ρ)+
(14)
The ADMM algorithm can solve the optimization problem
via splitting it into small subproblems and implement it in a
distributed manner. The subproblems can be easily solved with
global convergence to the optimal solution for any ρ > 0.
Moreover, when the other two sets of variables are held fixed,
it can be verified that there are closed-form solutions for x and z.
Computing them directly without iterations will reduce the time
complexity greatly and is beneficial for storage requirements.
Through the ADMM-based algorithm, the L1-regularized least
squares problem in Eq. (8) is solved and a set of non-zero weighted
coefficients are obtained. Notice that the feature sets in the
gravity model expression only exist in three forms, namely 1,
sin (qn ) and cos (qn ), the position of the trigonometric function
expression of the joint angle can be extracted by the conversion
of decimal numbers to ternary ones. With the positions of the
non-zero weighted coefficients, the expressions for all effective
features in the gravity model are available. The procedure of
the ADMM-based fast gravity modeling method (FGM-ADMM) is
summarized in Fig. 1.
(9)
where, A and b are the model parameters. x is the weighted solution vector. λ is the scalar regularization parameter. With ADMM
algorithm, the problem can be converted in to the following form:
minimizex,z
)−1 (
∥Ax − b∥22 + λ ∥z ∥1
(10)
x=z
3
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Robotics and Autonomous Systems 149 (2022) 103971
Table 1
The physical parameters of the iiwa arm including the mass, the position of the center of mass and the inertia of the links.
Link
mi
mxi
myi
mzi
Ii xx
Ii xy
Ii xz
Ii yy
Ii yz
Ii zz
i
i
i
i
i
i
i
3.456
3.482
4.056
3.482
2.163
2.347
3.129
0
0.0003
0
0
0.0001
0
0
−0.03
0.12
0.042
0.13
0.076
0.076
0.0004
0.02
0.02183
0.02076
0.03204
0.02178
0.01287
0.00651
0.01464
0
0
0
0
0
0
0.00059
0
0.00770
0.02179
0.00972
0.02075
0.00571
0.00626
0.01465
−0.00389
0.02083
0.00779
0.03042
0.00778
0.01112
0.00453
0.00287
=1
=2
=3
=4
=5
=6
=7
0.059
0.03
0.021
0.021
0.0006
0
−0.00363
0
0
0
0
0
0
0.00623
−0.00362
−0.00395
0.00032
0
The units of link mass m are kg. The units of the center of mass mx,my and mz are kg m. The unit of inertia tensor Ixx , Ixy , Ixz , Iyy , Ixz and Izz are kg m2 .
Fig. 3. Periodic sinusoidal trajectories for generating test data sets.
version is implemented based on Numpy 1.0.25 version and
cvxpy 1.16.5. Based on the above information, a total of 1000
sets random joint configurations and calculated joint torque were
obtained as training data. Taking joints 2, 3 and 4 with small
signal-to-noise ratio as examples, the effective features and the
gravity model of the joints can be obtained by using the proposed
method. Because the learning algorithm is implemented based
on the ability of data processing of the computer hardware, it is
believed that the boost should be more significant on more recent
GPU hardware. The following experiments were carried out based
on the GPU and CPU versions.
Fig. 2. Visual structure of the KUKA LBR iiwa 7 R800 robot.
4. Experimental validation
In the following section, experiments were performed to validate the proposed gravity model learning approach. The universality of the method was illustrated with different manipulator
models, including the LBR iiwa 7 R800 manipulator simulation
model and the realistic 7-DOF light-weight robotic arm. A comparative experiment was carried out with the existing methods
to show the advances of the proposed method.
4.1.2. Results and validation
After the gravity model of each joint was obtained by the proposed method, the test data and verification data were generated
to evaluate the effect of the model. Sinusoidal trajectories were
designed for the LBR iiwa R800 robot to generate position data of
joint motion and shown in Fig. 3. The complete trajectory of one
period is given by
4.1. Simulation experiment
4.1.1. Robot description and data acquisition
To validate the efficiency of the proposed approach, simulation
experiments were carried out in the simulation environment of
the KUKA LBR iiwa 7 R800 manipulator. The LBR iiwa 7 R800 is a
7-DOF lightweight redundant robot with a serial structure and
anthropomorphic characteristics. In the simulation experiment,
all data samples were generated based on the Kinematics and
Dynamics Library (KDL, http://www.orocos.org/kdl). The Unified
Robot Description Format (URDF) of the LBR iiwa 7 R800 arm is
available from the ROS iiwa stack (https://github.com/IFL-CAMP/
iiwa_stack). The required KDL data structure was obtained by
processing the URDF description using the python parser (http:
//wiki.ros.org/urdfdom_py).
The visual simulation model of the LBR iiwa 7 R800 robot is
shown in Fig. 2. The physical parameters used to generate simulation data specify the mass, the position of the center of mass
and the inertia for each link are shown in Table 1. In addition,
the joint limits of the manipulator were taken into account and
the relative Gaussian noise n = N (0, 0.01τ ) was added to the
computed torque.
The hardware configuration is Intel Core i7-6850K CPU @
3.60 GHz and NVIDIA GeForce GTX 1080Ti GPU board. The CPU
q1 (t ) = 1.4 sin (0.25π t )
q2 (t ) = 1.4 sin (0.2π t )
q3 (t ) = 1.8 sin (0.167π t )
q4 (t ) = 1.3 sin (0.25π t − 0.167π)
(15)
q5 (t ) = 3 sin (0.33π t )
q6 (t ) = 2 sin (0.167π t )
q7 (t ) = 3 sin (0.2π t )
The joint positions of the sine wave were collected with a
frequency of 10 Hz in 20 s. 200 sets of joint angle–torque data
were obtained from the trajectory as simulation test data. The
corresponding explicit expression of gravity term is derived by
Denavit–Hartenberg (D–H) conventions using the Lagrangian formulation of dynamics (EFD) as the verification model. The relevant parameters for iiwa robot based on the D–H method were
shown in Table 2. The corresponding 200 sets of verification
4
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Robotics and Autonomous Systems 149 (2022) 103971
Fig. 4. The comparison curve of the KDL simulated torque, the predicted torque by EFD and the predicted torque by the proposed method.
Table 2
D–H Parameter of the iiwa robot in [35].
Table 3
Experiment results including feature size, nRMSE and selecting time.
Link
ai−1 (mm)
αi−1 (rad)
di (mm)
θi (rad)
Joint
Feature size
nRMSE
Selecting time (s)
i
i
i
i
i
i
i
0
0
0
0
0
0
0
0
0.5π
0.5π
−0.5π
−0.5π
0.5π
0
340
0
400
0
400
0
0
θ1
θi
θi
θi
θi
θi
θi
i=2
i=3
i=4
246
76
171
0.43%
0.48%
0.57%
0.28
0.44
0.37
=1
=2
=3
=4
=5
=6
=7
4.2. Realistic arm experiment
4.2.1. Robot system setup
In the current section, a light-weight industrial robotic arm
system is established to collect data samples and verify the proposed gravity modeling method. An industrial robot is defined as
a programmable multifunction operator for handling mechanical
parts or workpieces, or a special mechanical device that can
perform various tasks by changing programs [36]. It is typically
some form of jointed structure with various configurations. The
structures are achieved by the linking of a number of rotary
and/or linear motions or joints. Each of the joints provides a
motion that collectively positions the robot in a specific position.
To provide the ability to position the end-effector at the desired
working place at any angle, multiple degrees of freedom are
required (commonly greater than or equal to six) [37].
The serial robotic arm used in the experiment has seven rotational degrees of freedom and is composed of an S-R-S structure.
The payload capacity of the arm is 7 kg. The simple structure of
the system has a large load/self-weight ratio, which enhances its
reliability and safety. For different tasks, different tools can be
mounted on the end of the robot through a flange. The multidegree of freedom design ensures the end-effector is able to reach
the required position and posture. According to the definition and
its characteristics, this robotic arm is considered to be representative of a typical industrial robot. The 7-DOF manipulator system
used in this experiment is shown in Fig. 5.
The joint modular structure includes the motor, harmonic
reducer, magnetic encoders, torque sensor, brake, motor driver,
power supply, and controller board, which is shown in Fig. 6. One
of the encoders is used to measure the position of the motor, and
another is used to measure the position of the joint. The modular
torque data are obtained through the joint position. Finally, 200
sets of predicted torque data were generated from the gravity
model obtained by the proposed ADMM-based learning method.
The normalized root mean squared errors (nRMSE) was introduced as the evaluation standard which can be defined as:
√∑
T
nRMSE =
τ̂ − τi )2 /T
i=1 ( i
(τmax − τmin )
(16)
where, τ̂i is the predicted torque value by the learned model, τi
is the test sample torque data, τmax and τmin are the maximum
and minimum torque values respectively. T is the test data size.
The results including the feature size, the resulting nRMSE value
and selecting time are reported in Table 3. Fig. 4 compares the
predicted torque generated by the learned gravity model based
on the proposed method with the predicted torque based on the
EFD and KDL simulated torque. The experimental results show
that the gravity model learned according to the proposed method
can accurately predict the joint torque for the joints of iiwa robot.
Moreover, the model prediction results are close to the experimental results of the explicit expression of the dynamic equations
derived from Lagrangian formulation and D–H parameters, which
proves the effectiveness of the method. It should be noted that
the predicted torque accuracy of the explicit expression obtained
by the D–H parameter method is relatively lower. The structural
parameters cannot be perfectly matched and the internal kinematics description of the robot is not completely accurate [33,34].
The proposed sparse learning and reconstruction method based
on data solves this problem better.
5
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Robotics and Autonomous Systems 149 (2022) 103971
Fig. 7. Periodic sinusoidal trajectory for data acquisition.
movement of the manipulator, the sine wave data were collected
in 100 s at 5 Hz. The data acquisition cycle was 0.2 s. A total of
500 joint-angle and driving-torque data sets were collected. To
prevent collisions, the rotation of joint 2 and joint 4 was limited.
The complete trajectory of one period is given by
Fig. 5. The 7-DOF industrial robot manipulator.
q1 (t ) = 0.65 sin (0.1t + 1.5)
q1 ∈ (−0.65, 0.65)
q2 (t ) = 0.25 (cos (0.2t + 0.2) − cos(0.2)) + 0.5
q3 (t ) = 0.5 (sin (0.2t + 0.3) − sin(0.3))
q3 ∈ (−0.7, 0.7)
q4 (t ) = 0.9 (sin (0.125t + 0.2) − sin(0.2)) + 0.25
q5 (t ) = 0.25 (sin (0.2t ))
q2 ∈ (−0.03, 0.55)
q4 ∈ (−0.2, 1)
q5 ∈ (−0.3, 0.3)
q6 (t ) = 0.4 (cos (0.25t + 0.4) − cos (0.4)) + 0.25
q7 (t ) = 0.25 (sin (0.2t + 0.4) − sin (0.4))
q6 ∈ (−0.55, 0.55)
q7 ∈ (−0.5, 0.5)
(17)
The periodic sampling trajectory in the experiment is shown
in Fig. 7. The maximum velocity of each joint in motion did not
exceed 0.1 rad/s. The speed of joint motion was very small to
ensure that the Coriolis moment and centripetal moment were
negligible. The joint-angle and driving-torque samples were collected by the trajectory as training data. The gravity equation
was built with data acquisition generated by the sensors and the
effective features could be selected from the training data.
In many industrial applications, friction is being assumed as
dynamic friction in the dynamic model. The friction model including viscous and Coulomb friction can be presented as a function of
q̇ [24,39]. The friction part related to the velocity term has little
influence on the data of the actual robot in low-speed motion.
Therefore, it is reasonable to regard friction as unmodeled dynamic errors. However, in the real condition, friction may have
some effects, especially when the joint motion is reversed. Before
the formal experiment, the joint torque between the low-speed
motion state and the static state was compared. In the process,
100 sets of joint position and joint driving torque measurements
by the trajectory in Eq. (17). Then run the trajectory in reverse
and measure 100 sets of joint torques at the same positions.
Finally, 100 sets of static driving torque data were collected at the
same joint positions. The residual standard error (RSE) between
the low-speed motion state and the static state is defined as
Fig. 6. The mechatronics design of the robotic arm joint.
design is able to minimize the difference in the structure of the
joints. With the embedded design, six wires inside the robot are
used to connect the joints with the outer controller PC and the
power supply. The torque sensor is assembled between the flex
spline of the harmonic drive gear and the link of the joint. With
the sensors, joint angles and driving torques can be collected
easily as training and test data during the working process of the
manipulator.
For the control framework, the joints are serially connected
with the master computer by the EtherCAT bus [38]. It takes full
advantage of Ethernet full-duplex and has a transmission rate
of 100 Mbit/s. The EtherCAT servo drive supports the CANopen
servo protocol. The service data object (SDO) is used to transmit
the machine sensor and configuration information of the robot
system, such as the torque values and joint position values. These
data will be of fundamental importance for the reconstruction of
the gravity model of the robot. The computing hardware settings
are the same as those in the above simulation experiments.
4.2.2. Data acquisition
In order to select the effective feature sets and reconstruct
the gravity model, it is necessary to collect the joint position and
driving torque through sensors during the manipulator motion.
In the experiment, a simple trajectory was designed to make the
manipulator move in the range of the joints. The frequency of
the signal sent from the arm joint sensors was 1000 Hz. In the
√
RSE =
)2 ∑n (
)2
∑n (
+ i=1 τREVi − τSi
i=1 τFW Di − τSi
2n
(18)
where, n the size of sampling data sets. τFWD is the torque data
collected with the joints moving forward. τREV is the torque
collected in reverse motion. τS is the torque data collected with
6
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Robotics and Autonomous Systems 149 (2022) 103971
Fig. 8. Selected features and weight distribution of the gravity model.
Table 4
Experiment results including feature size, RMSE, nRMSE and selecting time.
the robotic arm static. The RSE and nRMSE standardized values
were 0.307 and 0.512%, which were small enough. It is reasonable
that when the joints of the manipulator are in a low-speed state
without external force, the joints are considered only affected by
gravity.
4.2.3. Gravity model reconstruction
Through the analysis of ADMM-based optimization and sparse
reconstruction, the main challenges of the framework are capable
of obtaining a high reconstruction precision of gravity model and
ensuring reasonable processing time. In the experiment, we used
500 sets of joint-position and driving-torque data measurements
in low-speed motion as training data. The sampling data were
used to establish equations and represent sparse signals. The
reconstruction of the gravity model was realized by the proposed
approach. Since there were no payloads at the end of the manipulator, the signal-to-noise ratio of the following joints was low.
Experiments were carried out on the first four joints to verify
the proposed method. According to the positions and weights of
the effective features, the gravity model of the 7-DOF industrial
manipulator of joints 1–4 is given (see Appendix B, Table B.1).
The results of effective features from the original signals are
first reported. Fig. 8 shows the feature weight distribution of the
ADMM-based gravity modeling approach. It can be seen that the
proposed method screened out the effective features of the gravity model of the sampling data, especially for the leading items.
In addition, the solver also produced a large number of regression
coefficients which were strictly equal to zero, thus realized the
sparse signal representation and gravity model reconstruction.
To further validate the effectiveness of the proposed method,
the reconstructed gravity model was used to predict the driving
torque of the joint and compared with the measured torque.
Joint
Feature size
RMSE (N m)
nRMSE
Selecting time (s)
i
i
i
i
55
39
45
57
0.90
1.05
0.47
0.61
0.87%
1.75%
1.99%
1.35%
0.22
0.26
0.38
0.40
=1
=2
=3
=4
In the experiment, 200 sets of joint-position and driving-torque
data generated by the trajectory different from the reference one
were collected as the test data, which was about 40% of the size
of the training data. The results including the feature size, the
resulting RMSE and nRMSE value, and selecting time are reported
in Table 4. The comparison curve of the predicted torque and the
measured torque is given in Fig. 9.
It is shown that the prediction of the joint torque matches
well with the measurement data. The entire gravity model reconstruction process is completed efficiently in less than half a
second. It is noted that the original data are the unfiltered joint
sensor measurement data, which contain noise. The experimental
results show that the proposed method has good adaptability
to data. Since the restriction of the features with specific forms
is viewed as a strong priority of reconstruction, the obtained
gravity model can compensate the noise to a certain extent and
accurately predict the changing trend of the joint torque.
4.2.4. Gravity compensation updating
After preliminary verification of the effectiveness of the proposed method on the realistic robot arm, the experiments to
update the gravity model of the industrial manipulator with
different payloads were carried out. In the experiments, payloads
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Robotics and Autonomous Systems 149 (2022) 103971
Fig. 9. The comparison curve of the predicted torque and the measured torque.
Fig. 10. Different types of payloads attached to the end of the manipulator.
with different structures and weights were attached to the end
of the 7-DOF manipulator. The type of payload included counterweights and a polishing tool, as shown in Fig. 10. Their weight
was 1.8 kilograms, 2 kilograms (for one piece) and 4 kilograms
(for two pieces), respectively.
Since the continuous geometry structure transmits the increasing forces back through the mechanism, the tool or payload
mounted at the end will cause the change of gravity model of the
arm with a series kinematic chain. In this instance, the gravity
compensation of the robot needs to be updated. In the proposed
method, when the gravity model changes due to the change of
end-effector, the form of the basis function dictionary remains
the same. However, the effective features will change as a result
of the force on the end of the arm. Therefore, it is necessary to
relearn the gravity model through updated sampling data.
The simple trajectory in Eq. (17) was still used to collect
training data, and 500 sets of joint-position and driving-torque
sensor measurements were collected. The gravity model with
different payloads was updated by using the ADMM-based selecting method. The feature and weight distribution of the updating
gravity models of joint 1 are shown in Fig. 11. It can be seen
that the gravity model of the manipulator changed due to the
change of the payloads. The algorithm screened out the features
and weights according to the new data. With different payloads,
the gravity reconstruction method produced completely different
features, thus realizing updating of the gravity model.
In the experimental verification, 200 sets of test data independent of the training data were generated, and the driving
torque prediction was obtained based on the updating models.
The results including the feature size, the resulting RMSE value
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Robotics and Autonomous Systems 149 (2022) 103971
Fig. 11. The feature and weight distribution of the updating gravity models of joint 1.
Table 5
Various payloads results including feature size, RMSE and average selecting time.
Payload
type
Polishing
tool
2 kg
counterweight
4 kg
counterweights
Feature size
RMSE (N m)
Average
selecting time
(s)
J1
J2
J3
J4
J5
J6
J7
J1
J2
J3
J4
J5
J6
J7
39
50
46
71
56
58
38
1.08
1.16
0.64
0.71
0.26
0.41
0.22
0.37
45
49
53
64
40
47
32
1.23
1.14
0.72
0.73
0.32
0.53
0.22
0.29
50
52
52
88
65
58
45
1.08
1.32
0.50
0.84
0.37
0.41
0.32
0.41
and average selecting time of joints 1–7 are reported in Table 5.
The comparison curve of predicted torque and measured torque
is given in Fig. 12. According to the positions and weights of
the effective features, the updating gravity model with different
payloads of the 7-DOF industrial manipulator of joints 1 is given
(see Appendix, Table B.2).
The verification results show that the updated model can
predict the trend of torque change with good accuracy. The proposed method provides a fast framework for updating gravity
compensation and it takes no more than 0.45 s to finish the
execution process. It can be conveniently applied to the industrial
tasks of updating the gravity model of the manipulator, especially
updating time is limited. More importantly, although the payloads change, the gravity model can still be updated accurately
using the same simple trajectory sampling data. It indicates that
the proposed method does not need to redesign the excitation
trajectory or reformulate the mechanics, which simplifies the
labor complexity.
method for estimating the gravity parameters of the manipulator
based on kernel technique is proposed. Using the regularized least
square regression technique, the gravity parameters are regressed
based on the configuration parameters of the manipulator, such
as D–H parameters. The kernel ridge regression (KRR) algorithm
is introduced to improve the performance of the algorithm. This
method estimates the relationship between the center of gravity
and the mass of the link with the sensor measurements, so as to
obtain the gravity model. The to-be-determined gravity vector γi
is defined as
γi = [mi , mxi , myi , mzi ]T
(19)
where, mi is the mass of the joint link i. [xi , yi , zi ]T is the coordinate of gravity center relevant to the coordinate system of the
link i.
A comparative experiment was carried out to evaluate the
proposed method and the conventional LR method. In data acquisition, the simple periodic joint trajectory in Eq. (15) was
still used to collect training data for both methods. The signal
frequency and data acquisition cycle were the same as the ones
in Section 4.2. The sampling data by the trajectory independent of
training data was used as test data to verify the results. The D–H
parameters of the 7-DOF manipulator used in the experiment are
shown in Table 6. The parameters of the gravity model of joint
1–4 solved by the method in [42] are presented in Table 7.
The results of the comparative experiment including resulting
the nRMSE value and algorithm execution time are reported in
Table 8. According to the obtained gravity models, the comparison curve of predicted torque is shown in Fig. 13. It is shown from
the results that the gravity models obtained by the methods can
predict the change trend of joint torque. Nevertheless, the gravity
model obtained from the proposed method is better than the LR
method of each joint. Especially in joint three, due to the lack
of constraints on joint characteristics and corresponding training
4.2.5. Comparison with estimation method
The basis of most conventional gravity model estimation methods is that the mapping between gravity parameters and joint
driving torque is linear, which can be estimated based on linear
regression. It has a solid theoretical foundation and various mature implementations [40,41]. The mapping of the analytical form
of the dynamic equation is extracted and a set of recognizable
basic parameters are obtained. Then, the generalized torque and
joint motion data of the manipulator are collected. Linear regression (LR) techniques use these measurements to estimate gravity
parameters. In the implementation process of the methods, the
design of the excitation trajectory is sometimes important.
Unlike the parameter estimation method, the proposed one
uses the feature-based selection framework. The gravity modeling
problem is transformed into a sparse optimization problem, and
the model is reconstructed by ADMM-based algorithm. In [42], a
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Robotics and Autonomous Systems 149 (2022) 103971
Fig. 12. Model updating validation of various payloads based on comparison between predicted and measured torque. From top to bottom: joints 1 to 7.
Table 6
Denavit–Hartenberg parameters of 7-DOF manipulator.
Table 7
Gravity model parameters of the regression method in [30].
Link
ai−1 (mm)
αi−1 (rad)
di (mm)
θi (rad)
Link
mi (kg)
mxi (kg
i
i
i
i
i
i
0
0
275
0
0
0
0
0
0
0
250
0
0
0
i
i
i
i
0
0
0
0
0.037
0.146
0.020
=1
=2
=3
=4
=5
=6
−0.5π
0
0.5π
−0.5π
−0.5π
−0.5π
0.5π
0
−0.5π
0
=1
=2
=3
=4
−0.106
m)
myi (kg
0.170
0.170
0.176
0.176
m)
mzi (kg m)
−0.152
−2.603
−2.603
−1.371
based on sparsity and features has better adaptability to data and
generalization ability. With the constraints of the specific feature
forms, the precise gravity model can be reconstructed by using
data, the regression result by the method in [42] is highly deviant, which is not satisfactory. The gravity model solving method
10
C. Yu, Z. Li, D. Yang et al.
Robotics and Autonomous Systems 149 (2022) 103971
Fig. 13. Experiment results on torque prediction effect by different methods. From top to bottom: joints 1 to 4.
Table 8
The nRMSE of predicted torque from the different methods.
data measurements from a simple trajectory. Consequently, the
proposed method can solve the gravity model efficiently. It does
not need prior kinematics knowledge of geometric parameters,
thus avoiding heavy work of physical parameter calibration and
simplifying manual operation.
5. Conclusion
Joint
nRMSE of the
proposed method
nRMSE of the
method in [42]
i
i
i
i
0.85%
1.57%
1.58%
1.38%
2.39%
1.73%
6.30%
1.49%
0.29
4.09
=1
=2
=3
=4
Average
processing time (s)
In this paper, a novel framework for fast gravity compensation
updating of the industrial robot arm is developed. The core idea
of this paper is to derive the components from the observation
of the dynamic model, and then select and reconstruct a finite
number of effective features according to the sparsity of the
feature sets of the gravity model. The ADMM algorithm is used
to accelerate the process of solving the optimization problem of
gravity model updating while ensuring accuracy. The proposed
data-based method has the advantage that it does not depend
on kinematics or dynamics parameters. Only the joint angle and
driving torque data measurements by a simple trajectory are
needed, thus avoiding the heavy work of calibration and assembly
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Robotics and Autonomous Systems 149 (2022) 103971
Table B.1
Gravity model of 7-DOF industrial robotic arm of joints 1–4.
Joint 1
Joint 2
Joint 3
Joint 4
pg
Features
pg
Features
pg
Features
pg
Features
−20.81
−4.22
−4.17
s1c2
c2s3
c3
c1s3s4
c2c3c4
s1c2c5
c1s3s4c5
c1s3s6
c3s4s6
c3s5s6
c1c3s5s6
c1s3c5s6
s1c2s4c5s6
c3s4c5s6
c2c4c5s6
s1s4c6
c1s3s4c6
s1c3s4c6
c5c6
c2c5c6
s1s4c5c6
c1c3s7
c1c3s4s7
c1c3s4c5s7
s4s6s7
c3s4s6s7
c1c3s4s6s7
c3s4c5s6s7
c1c3s4c5s6s7
s1c2c7
c2s3c7
c3c7
c2c3c7
s4c7
c1s3s4c7
c2c3c4c7
s1c2c5c7
c2s3c5c7
s4c5c7
c1s3s4c5c7
c1s3s6c7
c3s4s6c7
c1c3s5s6c7
c1s3c5s6c7
s1c2s4c5s6c7
c3s4c5s6c7
s1s4c6c7
s3s4c6c7
c1s3s4c6c7
s1c3s4c6c7
s1s4c5c6c7
s3s4c5c6c7
c1s3s4c5c6c7
s1c3s4c5c6c7
−12.81
−11.84
−1.73
−0.61
−9.05
−1.38
−6.25
−2.02
−3.48
s2
c1s2
s2c3
c1s2c3
c3s4
c1c3s4
c2c3s4
s2c4
c1s2c4
c2c4
s2c5
c1s2c5
c3s4c5
c2c3s4c5
c2s6
c4s6
c2c4s6
c4c5s6
c2c4c5s6
s2c6
c1s2c6
c3s4c6
c2c3s4c6
s2c7
c1s2c7
s2c4c7
c1s2c4c7
c2c4c7
s5c7
s2c5c7
c1s2c5c7
c2s6c7
c2s3c4s6c7
c2s3c4c5s6c7
s2c6c7
c2s2c6c7
c2s3c6c7
s1c2c5c6c7
c2s3c5c6c7
−1.59
−125
−2.37
s2
c1c2c3
s2s4
s1c3s4
c1s5
c1c3s5
s1s4s5
s1c2s4s5
s2s4c5
s1c3s4c5
s1c2s6
c2s3s6
s5s6
c2s5s6
c2s3c5s6
s1c3s4c6
c1s5c6
c1c3s5c6
s1c3s4c5c6
s1s7
s2s7
s1c2s7
s2c3s7
s1c5s7
s2c5s7
s1c2c5s7
s2c3c5s7
s1c6s7
s1c2c6s7
s1c5c6s7
s1c2c5c6s7
s2s4c7
s1c3s4c7
s1s5c7
s1c2s5c7
s1s4s5c7
s1c3s4c5c7
c2s3s6c7
s5s6c7
c2s5s6c7
c2s3c5s6c7
c1s5c6c7
c1c3s5c6c7
c5c6c7
c4c5c6c7
−3.51
−0.83
−3.84
−1.03
−4.82
−1.76
−6.43
−3.99
−1.88
−2.74
−1.37
s2
c1s2
s2c3
c1s2c3
s4
c1s4
c2s4
c1c2s4
s1c4
s2c4
c1s2c4
c2c4
s2c3c4
c1s2c3c4
s2c5
s2c3c5
c2s4c5
s1c4c5
s2c4c5
c2c4c5
s2c3c4c5
c1s2c3c4c5
s6
c1s6
c4s6
c2c4s6
c5s6
c1c5s6
c4c5s6
s2c6
c1s2c6
s2c3c6
c1s2c3c6
c2s4c6
s2c4c6
s2c3c4c6
c1s2c3c4c6
s2c5c6
s2c3c5c6
s2c3c4c5c6
s2c7
s2c3c7
s4c7
c2s4c7
c1c2s4c7
s1c4c7
s2c4c7
s2c3c4c7
c1s2c3c4c7
s1c4c5c7
c2c4c5c7
s2c3c4c5c7
s6c7
c5s6c7
s2c6c7
s2c3c6c7
s2c3c4c6c7
3.11
−2.12
−12.38
0.88
−0.27
0.61
−2.38
−3.49
−0.01
0.99
0.69
−2.44
3.55
1.02
1.97
1.89
3.13
1.24
0.67
0.98
2.02
−0.14
−2.42
−0.26
−2.26
−0.04
−22.69
−7.43
−5.10
−2.14
−0.29
6.03
−0.50
−15.29
−3.98
−0.96
3.67
−0.72
0.20
−0.82
−0.35
0.82
0.19
5.01
2.11
3.99
2.81
2.57
0.20
2.29
0.53
1.2
−4.90
−3.88
−2.68
−0.60
−0.16
−1.58
−2.06
−0.32
−0.51
−6.54
−5.72
−5.10
−1.07
−9.61
−8.11
−0.55
−1.48
0.20
2.05
−1.72
−0.20
−0.03
0.69
0.45
−3.70
−2.42
−0.14
−0.41
−2.11
8.84
0.30
0.54
−1.30
−0.51
−0.76
8.69
0.23
1.60
−3.35
−2.29
0.35
6.14
0.66
0.77
5.69
−0.50
0.15
−0.39
0.15
−0.63
0.02
−0.51
0.018
−0.31
−0.26
−0.41
−0.36
−1.26
2.07
−0.80
−0.77
−0.42
2.05
1.53
−1.14
−0.10
0.27
0.13
0.20
0.77
1.13
error. Moreover, it does not need to redesign the excitation trajectory or reformulate the mechanics in model updating, which
simplifies the labor complexity. The effectiveness and universality
of the proposed method are verified by the validation of the
iiwa robot simulation and the experiment of a realistic 7-DOF
industrial robot arm. The superiority of the proposed method
is demonstrated with the comparison to the existing estimation
method. Experiment results show that the gravity model can
be updated with good accuracy through the data of a simple
trajectory in reasonable processing time. The proposed method
can be conveniently applied to the gravity model updating of the
manipulator when the payload and the end-effector are changed
2.08
−4.07
−2.49
−1.03
−1.37
−0.94
−1.94
−0.93
4.39
−2.23
−0.72
−1.03
−0.23
−0.99
−0.58
−1.04
−0.13
−0.29
−2.91
−0.33
−2.66
−0.01
−0.71
−1.41
−2.06
−0.47
−0.58
−0.37
−0.32
−1.05
−1.33
−2.53
−4.44
−1.87
−0.08
−0.89
−2.13
−0.56
−0.19
1.77
−0.35
−0.19
−0.12
−0.65
−0.36
−0.25
frequently according to different tasks in the industrial scene, especially when the updating time is limited. In the future, we plan
to apply the fast gravity compensation updating method to the
advanced controller and continue to focus on the effective feature
and reconstruction approach to further explore the solution of
robot model active learning according to data.
Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared
to influence the work reported in this paper.
12
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Robotics and Autonomous Systems 149 (2022) 103971
Table B.2
Updating Gravity model of various payload of joint 1.
Polishing tool
A counterweight
Two counterweights
pg
Features
pg
Features
pg
Features
−0.88
−23.49
−9.14
−3.63
−4.69
−12.69
c2
s1c2
c2c3
s4
s1c2c4
s1c2c5
c3s4s6
s5s6
c2s5s6
s4s5s6
s1s4c6
s1c2s4c6
s3s4c6
s1c3s4c6
s1c2c3s4c6
c1c3c4c6
s1s4c5c6
s1c2s4c5c6
s1c3s4c5c6
s1c2c3s4c5c6
c1c3c4c5c6
s7
s4s6s7
s4c5s6s7
s1c7
s1c2c7
s4c7
s1c2c4c7
s1c2c5c7
c2s3s4s6c7
s5s6c7
s4s5s6c7
c2s3s4c5s6c7
c2c4c5s6c7
s1s4c6c7
s1c3s4c6c7
c1c3c4c6c7
s1c3s4c5c6c7
c1c3c4c5c6c7
−0.49
−2.37
−0.14
−6.73
−3.58
−3.59
−4.27
−0.23
−0.31
−1.74
s1c2
s1s5
s1c2s5
c4c5
c3c4c5
s2s6
s2c3s6
c1s4s6
s2s4s6
c1c3s4s6
c3s5s6
c3c4s5s6
c1c3c4s5s6
c2c3c4s5s6
c1c2c6
c2c3c6
c1c2c3c6
s3s4c6
c1s3s4c6
c1c2s3s4c6
c1c3s5c6
c1s3s4c5c6
c1c3c6s7
s1c7
s1c2c7
s3c7
c2s3c7
s1c4c7
s1c2c4c7
s1s5c7
s1c2s5c7
s1c5c7
s1c2c5c7
c2s3c5c7
c2s4c5c7
c4c5c7
s1c2c4c5c7
c3c4c5c7
s3s6c7
c3c4s5s6c7
s1c5s6c7
c2c5s6c7
s3c5s6c7
c1s3s4c6c7
c1s3s4c5c6c7
−0.26
−28.16
s1
s1c2
s1s4
s1c3s4
c2c4
s1c2c4
c2c3c4
s1c2c5
s1s4c5
s1c3s4c5
c1c3s4c5
s1c2c4s6
s5s6
c3s5s6
s4s5s6
c2c4c5s6
s1c2c6
s1s4c6
s3s4c6
s1c2c5c6
s1s4c5c6
s3s4c5c6
c1s3s4c5c6
c5s7
c3c5s7
s1c2c7
c2s3c7
s1s4c7
s1c3s4c7
c1c3s4c7
s1c2c4c7
s1c2c5c7
c2s3c5c7
s1s4c5c7
c1s4c5c7
s1c3s4c5c7
c1c3s4c5c7
c2s4s6c7
c2s3s4s6c7
c3s5s6c7
s4s5s6c7
c2s3s4c5s6c7
c2c4c5s6c7
s1c2c6c7
s1s4c6c7
s3s4c6c7
s1c2c5c6c7
s1s4c5c6c7
s3s4c5c6c7
c1s3s4c5c6c7
1.73
−6.74
−2.60
−2.75
11.56
2.58
1.24
9.81
0.65
0.33
10.69
1.65
9.44
0.24
1.86
1.70
−4.17
−4.47
−5.90
−33.00
−2.53
−6.74
−23.66
5.28
−1.37
−2.32
7.55
−6.44
0.97
0.94
1.23
0.33
2.65
1.17
7.99
2.45
1.31
3.43
0.97
4.41
7.50
18.10
6.59
−1.40
10.76
−0.03
−13.25
−23.63
−0.63
−10.83
−3.91
−10.89
−4.58
−2.12
−13.90
−24.52
−8.59
−0.46
−2.42
−6.82
−4.12
2.51
1.39
4.64
1.25
6.61
10.86
3.62
Acknowledgements
3.67
3.56
−4.87
−7.14
−0.43
−19.80
5.76
5.43
−1.30
1.92
−4.10
−5.08
−3.15
−0.59
−8.15
3.50
5.46
−1.26
5.74
6.79
0.35
1.92
2.03
−27.33
−4.78
1.01
2.14
−1.09
−3.28
−20.71
−4.20
2.62
−0.25
3.56
−7.42
−0.02
4.38
−0.44
−3.82
4.39
−4.61
−7.45
3.40
4.90
−2.15
5.14
5.72
0.08
the rotation matrix about x-axis. Note that for revolute joints,
only Rz (qi ) depends on qi in the right-hand side of Eq. (A.1), i.e.
⎤
⎡
cos (qi ) − sin (qi ) 0 0
This research was supported by the National Natural Science
Foundation of China (No. 61803124), the National Basic Research
Program of China (No. 973-2013CB733124), and the Foundation
for Innovative Research Groups of the National Natural Science
Foundation of China (Grant No. 51521003).
Rz (qi ) = ⎣ sin (qi )
cos (qi ) 0 0⎦
⎢
⎥
0
0
1 0
(A.2)
= C1 cos (qi ) + C2 sin (qi ) + C3
Appendix A
where,
i−1
Ti = Rz (qi ) Tz (di ) Tx (ai ) Rx (αi )
1
0
0
0
C1 = ⎣0
1
0
0⎦
0
0
0
0
0
−1
0
0
C2 = ⎣1
0
0
0⎦
0
0
0
0
⎡
Proof of Property 1. Assume the Denavit–Hartenberg convention
is used. The pose matrix of joint i in the frame i - 1 is expressed
as:
⎢
(A.1)
⎡
where, qi the angle of joint i, di , ai , and αi are link parameters.
Rz is the rotation matrix about z-axis. Tz is the translation matrix
about z-axis. Tx is the translation matrix about x-axis. And Rx is
⎢
13
⎤
(A.3)
⎥
⎤
⎥
(A.4)
C. Yu, Z. Li, D. Yang et al.
Robotics and Autonomous Systems 149 (2022) 103971
0
0
0
0
C3 = ⎣0
0
0
0⎦
0
0
1
0
⎡
⎢
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⎤
(A.5)
⎥
The other three namely Tz (di ), Tx (ai ), Rx (αi ) are fixed for a
specific robot. Substitute Eq. (A.2) into Eq. (A.1), it can be verified
that the components of i−1 Ti could be expressed as:
i−1
Ti = C ′ 1 cos (qi ) + C ′ 2 sin (qi ) + C ′ 3
′
where, C 1 , C
0
0
1
′
2
and C
Tn = T1 T2 · · ·
n−1
′
3
(A.6)
are constant matrices. Then from
Tn
1≤n≤N
(A.7)
It can be seen that 0 Tn are linear combinations of features in
F in Eq. (3). In addition, since F in Eq. (3) is closed under partial
derivative w.r.t joint angles q
( j (j)= 1, . . . , N ), therefore, for any
∂
0T
n
1 ≤ n ≤ N and 1 ≤ m ≤ N, ∂ q
is also in the span of F .
m
Considering the first argument in Property 1 based on [43], the
Jacobian 0 Jn (1 ≤ n ≤ N ) could be written as:
∂ tn
⎢ ∂q
0
1
Jn = ⎢
⎣
⎡
z1
∂ tn
∂ q2
···
z2
···
⎤
∂ tn
∂ qn ⎥
⎥
⎦
(A.8)
zn
where, 0 tn is the translation vector of frame i. 0 zi is the direction
vector of the z-axis of frame i. Both are submatrices of 0 Tn .
Therefore, it could be concluded that 0 Jn is a linear combination
of features shown in Eq. (3).
To verify the second argument, we consider the kinematic
chain, reversely. From the first argument, 0 Jn (is in) the span of F
for any 1 ≤ n ≤ N. By noting that n J0 = −n 0 Jn always holds,
the proof is completed.
Appendix B
In the following expressions, the trigonometric functions sin(θ2 )
and cos(θ4 ) are abbreviated as s2 and c4, respectively. The gravity
model of the 7-DOF industrial robotic arm for joints 1–4 including
parameters pg and effective features are given in Table B.1.
The updating gravity model with different payloads for joint
1 including parameters pg and effective features are given in
Table B.2.
The updating gravity model of joints 2–4 obtained in the
experiment are available from the corresponding author upon
reasonable request.
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14
C. Yu, Z. Li, D. Yang et al.
Robotics and Autonomous Systems 149 (2022) 103971
Chenglong Yu received the B.S. degree in automation from Harbin Engineering University (HEU), Harbin,
China, in 2015, and M.S. degree in mechanical engineering from Harbin Institute of Technology (HIT),
Harbin, China, in 2018, where he is currently pursuing
the Ph.D. degree with the State Key Laboratory of
Robotics and System, Harbin Institute of Technology
(HIT). His research interests include industrial manipulator control, robot dynamic modeling and machine
learning techniques.
Dapeng Yang received the B.S. degree in mechanic
engineering, and the M.S. and Ph.D. degrees in mechatronic engineering from the Harbin Institute of Technology (HIT), in 2004, 2006, and 2011, respectively.
Since 2015, he has been an Associate Professor with
the State Key Laboratory of Robotics and System
(SME), HIT. His research interests include dexterous
robotic/prosthetic hand systems, biomedical signal processing and control, human–machine interaction, and
machine learning techniques in robotics.
Zhiqi Li received the B.S. and M.S. degrees in Control Science and Technology, and the Ph.D. degree
in mechatronics from Harbin Institute of Technology
(HIT) in 2006, 2008 and 2016 respectively, where he
is currently a research assistant with the School of
Mechatronics Engineering. His research interests include robot modeling and control, machine vision and
machine intelligence
Hong Liu received the Ph.D. degree from the Harbin
Institute of Technology (HIT), China, in 1993. During
1991–1993, he was a Joint Ph.D. Candidate with the
Institute of Robotics and System Dynamics, German
Aerospace Research Establishment (DLR), Germany,
where he has been a Research Fellow since 1993. He
became one of the First batches of Changjiang Scholars
in 1998. He is currently a Professor with HIT. His
research projects include the development of dexterous
robot hand and space manipulator.
15