Variable Impedance Force Tracking with a Collaborative Joint Based on Second-Order Momentum Observer Under Uncertain Environment
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Variable Impedance Force Tracking
with a Collaborative Joint Based
on Second-Order Momentum Observer
Under Uncertain Environment
Jie Wang1 , Yisheng Guan1(B) , Haifei Zhu1 , and Ning Xi2
1
Biomimetic and Intelligent Robotics Lab (BIRL),
Guangdong University of Technology, Guangzhou 510006, China
ysguan@gdut.edu.cn
2
Department of Industrial and Manufacturing Systems Engineering,
The University of Hong Kong, Hong Kong SAR, China
Abstract. Collaborative robots are the focus of the development of
next-generation industrial robot. In order to improve the flexibility and
force control performance of collaborative robot joint, this paper pro-
poses a variable impedance force tracking algorithm based on a second-
order momentum observer for the collaborative joints developed in our
laboratory. The algorithm first designs a second-order feedforward gen-
eralized momentum external force observation method for force percep-
tion, which is used as the torque outer loop of the force tracking algo-
rithm. Secondly, a simplified robot-environment variable impedance sys-
tem is constructed through the dynamic transfer function, which is used
as a position inner loop to optimize the gain adjustment process and
dynamic response speed by establish a differential error model. The sim-
ulation shows the robustness of the scheme under unknown environment
stiffness and moving environment. The experimental results support the
claim that this method can track the desired force in real time under
uncertain environment.
Keywords: Collaborative joint · External torque observer · Variable
impedance control · Force tracking
1 Introduction
In recent years, collaborative robots have begun to enter people’s field of vision,
and are mainly used in various human-robot interaction scenarios [1]. collabora-
tive robots usually come into force contact with unknown environment (human
The work in this paper is in part supported by the Frontier and Key Technology Inno-
vation Special Funds of Guangdong (Grant No. 2017B050506008, 2017B090910008),
and the Key Research and Development Program of Guangdong Province (Grant No.
2019B090915001).
c Springer Nature Switzerland AG 2021
X.-J. Liu et al. (Eds.): ICIRA 2021, LNAI 13014, pp. 563–574, 2021.
https://doi.org/10.1007/978-3-030-89098-8_53
564 J. Wang et al.
body), such as polishing, precision assembly and other tasks. The error caused
by pure position control will cause excessive contact force between the contact
surfaces [2]. Therefore, the sensitivity of the robot/compliance control problem
is very important to improve the performance of the robot.
Force perception is the prerequisite for achieving compliance control of col-
laborative robots. Olsson proposed a terminal load force sensing method applied
to industrial robots, which realized force sensing by solving the parameters of
the zero point and installation angle of the six-dimensional torque sensor [3].
Park proposed a sensor zero compensation method, which realizes external force
observation by adjusting the robot’s posture [4]. Kebria first applied the joint
sensor to the UR robot, and adopted a double-encoding structure, with high
force sensing accuracy [5]. In addition, there are traditional external force sens-
ing methods based on electric current, piezoelectric effect, and inverse dynamics.
Impedance control method is widely used at home and abroad, which Often
used as a strategy for force tracking. Impedance control was proposed by
Hogan [6]. Jeon built a position control inner loop based on the traditional
impedance idea, and adjusted the trajectory of the robot’s end position by using
the acquired torque information [7]. Reference [8] designed an impedance con-
troller, which not only considers the impedance relationship between the end
effector and the object, but also considers the impedance relationship between
the end and the environment. Reference [9] proposed an adaptive master/slave
control strategy for hybrid force/position tracking to compensate for trajectory
tracking errors. Erhart proposed an impedance-based multi-robot collaborative
dual-arm mobile operation control architecture [10]. Another effective method
of collaborative joint impedance control is to use a cascade structure of internal
torque control loop and external impedance control loop [11,12].
The rest of this paper is organized as follows. In Sect. 2, the dynamic model
is given and proposes an external force detection algorithm based on the second-
order momentum observer. In Sect. 3, a variable impedance force tracking algo-
rithm based on momentum observation is proposed, which can realize variable
impedance control according to the observation threshold and impedance gain
under uncertain environments. Simulation experiments are presented in Sect. 4 to
verify the robustness of the algorithm. In Sect. 5, we conduct verification exper-
iments on the control algorithm proposed in this paper based on self-developed
collaborative joint system. Finally, we give a conclusion in Sect. 6.
2 External Force Detection Based on Second-Order
Momentum Observer
2.1 Integrated Collaborative Joint Dynamic Model
The physical and transmission structure of the collaborative joint developed in
this research is shown in Fig. 1. According to the flexible joint dual-mass spring
system proposed by Spong [13], The dynamic model designed in this paper is as
follows:
Variable Impedance Force Tracking Algorithm 565
Fig. 1. Collaborative joint and transfer model
M (q)q̈ + C(q, q̇)q̇ + G(q) = J −1 τd − τext − τf (1)
where q, q̇, q̈ ∈ Rn×1 are the joint angle, joint angular velocity and joint
angular acceleration of the manipulator respectively, M (q) ∈ Rn×n is the sym-
metrical inertia matrix of the manipulator, C(q, q̇) ∈ Rn×n is the centrifugal
inertia matrix, which including coriolis force and centrifugal force, G(q) ∈ Rn×1
is the gravity term, τd is the joint output torque, J −1 is the inverse matrix of
the joint, and τext is the external disturbance torque.
2.2 Second-Order Momentum External Force Observer
In order to sensitively and accurately determine the direction and magnitude of
the external force at the joint level. this section proposes a second-order momen-
tum observer’s external force detection algorithm. Consider the concentrated
observation of disturbance moments, observation torque can be expressed as
τc = τext − τf , Eq. 2 can be further transformed into the external force distur-
bance equation as follows:
τc = J −1 τd − M (q)q̈ + C(q, q̇)q̇ + G(q) (2)
The generalized momentum equation that defines the collaborative joint is as
follows:
p = M (q)q̇ (3)
where M (q̇) is a positive definite symmetric matrix, and M (q̇) − 2C(q, q̇) is an
antisymmetric matrix [14], and the derivative can be obtained:
Ṁ (q) = C(q, q̇) + C T (q, q̇) (4)
According to the momentum decoupling characteristics of flexible collaborative
joints, the disturbance vector is defined as r = τc , and the disturbance external
torque observation algorithm is designed as follows:
ṙ(t) = Ka [−Kb r(t) + [p(t) − p̂(t)]] (5)
566 J. Wang et al.
where Ka , Kb = dia (ka,b,i ) > 0 is a constant that depends on the system, p̂(t)
is the real-time estimated value of the generalized momentum. Based on Eq. 6,
construct the dynamic feedforward optimization factor Of as follows:
t
Of = p̂(t) − (ṗ(t) − r̂(t))dt (6)
0
Combining Eq. 7, the Eq. 6 after introducing gain constant optimization is as
follows:
t
ṙ(t) = −Ka Kb r̂ + Ka p(t) − Ka Kc τc − Ka τd + Ĉ T (q, q̇)q̇ − G(q) + r̂ dt (7)
0
The block diagram of the external force observation algorithm is shown in Fig. 2,
compared with the classic first-order momentum observer r̂˙ = K (τc − r), the
feedforward optimization factor Of in the algorithm proposed in this paper can
eliminate the suppression system overshoot and high-frequency noise.
Fig. 2. Framework of external force observation algorithm
3 Variable Impedance Force Tracking Algorithm Based
on External Force Observer
3.1 Model of Variable Impedance Force/position Tracking
Impedance control does not directly control position or force, but adjusts the
relationship between force and position. The classic second-order mass-damping-
stiffness constant impedance model is:
Me ë + Be ė + Ke e = fe (8)
where fe is the force error, e is the position error, Me , Ke , Be are the environ-
mental quality, environmental stiffness and damping respectively. This study
takes the two-axis variable impedance model as the research object, as shown in
Variable Impedance Force Tracking Algorithm 567
Fig. 3. Robot-environment impedance contact model
Fig. 3, suppose the variable stiffness of the environment is Kv , the expected
contact force between the end of the manipulator and the environment is
Fd = Kv (qe − qc ), M, K, B are the variable inertia matrix, the variable stiff-
ness matrix and the variable damping matrix of the end of the manipulator.
According to [15], it is impossible to directly observe and compensate the
force tracking error in the interactive environment in real time. Therefore, it is
necessary to construct the variable impedance function by constructing the force
error model based on Kv , and then integrating it into the impedance model. Sup-
pose qe , qd , qa , qc respectively as the environment position, the reference position
of the end of the robot arm, the command (corrected) position and the actual
position, Fc is the real-time external disturbance force. When the end of the
robotic arm is in contact with the environment, a force error model and a posi-
tion error model are generated as follows:
ef = Fd − Fc = Kv (qe − qd − Kef ) − Fc (9)
eq = Kef (10)
Considering that it is difficult to directly compensate the position error in prac-
tice, so choose to indirectly compensate the torque observation error to construct
the target variable impedance model:
M (q̈a − q̈d ) + B (q̇a − q̇d ) + Kv (qa − qd ) = μėf + ηef (11)
where q̈a is the actual end acceleration value, q̇a is the end actual speed value,
q̈d is the end reference acceleration, q̇d is the end reference speed, and μ, η are
system constant.
3.2 Force Tracking Control Method
When the robot moves freely in space, the expected contact force is Fd = 0,
the actual external disturbance force is Fc = 0, and the force error model is
ef = Fd − Fc = 0, Eq. 13 can be expressed as follows:
M (q̈a − q̈d ) + B (q̇a − q̇d ) + Kv (qa − qd ) = 0 (12)
When the robot is in contact with the environment, it is expected that FC ⇒ Fd .
We assume K = 0, the amount of impedance change depends on Kv . Similar
568 J. Wang et al.
methods are mentioned in [15], the conclusion is that the steady state contact
force is constant. And the force error model and the steady-state position are
ef ⇒ 0. Substituting the corrected force model into Eq. 13, and solving the
corrected trajectory as follows:
q̈c = q̈d + [ef − B (qa − qd )] M −1 (13)
Considering the real-time adjustment of the communication cycle T of the con-
troller, the corrected trajectory is substituted into Eq. 11, the real-time tracking
torque after position compensation is inversely solved:
Fc = Kv (qe − qd − Kef ) − ef (14)
The above-mentioned position loop derivation process is used as the inner loop
of variable impedance control to modify the trajectory, and the external force
detection algorithm based on the second-order momentum observer designed
in the previous section is used as the outer moment of the moment loop. The
implementation process of the variable impedance force tracking algorithm based
on external force detection designed in this paper is shown in Fig. 4.
Fig. 4. Framework of variable impedance force tracking algorithm
4 Simulation Verification
In this section, the proposed control algorithm is tested by simulating the track-
ing performance under different environment conditions in MATLAB, the sam-
pling period of the controller is selected as T = 2 ms.
4.1 External Force Observation with Variable Force Environment
The observation result of the external torque r will directly affect the accuracy
of force tracking. In this section, we apply a constant force and a time-varying
force to simulate the actual system being disturbed by constant force and sudden
collision, see Fig. 5. External disturbance force is defined as
3 t ∈ [1, 2]
F = (15)
5t − 15 t ∈ [3, 4]
Variable Impedance Force Tracking Algorithm 569
4.2 Force Tracking with Time Varying Desired Force
the decoupled external torque r is the input of the variable impedance inner loop,
that is Fc = r. Kv is set as a constant in this section, we apply a sinusoidal curve
to simulate a time-varying desired force, see Fig. 6. Simultaneously compared
with the force tracking algorithm [16].
4.3 Force Tracking with Time Varying Impedence Environment
In order to verify the performance of the algorithm in a variable impedance
environment, we set different impedance coefficients Kv at different periods of
time, see Fig. 7. Fc is set as a constant in this section.
⎧
⎨ 10000 t ∈ [1, 2]
Kv = 8000 t ∈ [2, 3] (16)
⎩
2000 t ∈ [3, 4]
Figure 5 shows that the external force observation algorithm can effectively
track the external disturbance force, and there will be no overshoot and oscil-
lation under the time-varying force. The external disturbance observation can
be completed by setting a smaller threshold Ft = 0.3 Nm. The algorithm can
realize the collision stop function at the same time. Figure 6 shows that under
time varying desired force, the initial overshoot of the algorithm proposed in
this paper is obviously smaller than that [16], and the force tracking error is also
smaller. It can be seen that the error [16] reached Fe = −2 Nm. Figure 7 shows
that under the time-varying impedance environment, compared with [16], the
tracking performance of this algorithm at the time of impedance change is more
stable, and the overshoot [16] is reached Fe = −0.7 Nm.
Fig. 5. External force observation with variable force environment
570 J. Wang et al.
Fig. 6. Force tracking with time vary- Fig. 7. Force Tracking with time vary-
ing desired force ing impedence environment
5 Experiments with a Collaborative Joint
In order to verify the effectiveness of the variable impedance force tracking con-
troller proposed in this paper, a collaborative joint development system is built
based on the integrated joint developed by our laboratory, as shown in Fig. 8.
The control system uses Ethercat bus to communicate with the host computer,
the communication period is 2 ms. This experiment is divided into three parts.
The first is external force disturbance experiment under different environments
(human, soft sponge, balloon), the second is force tracking experiment in fixed
unknown environment (foam, cartons, metal part), and the last is force tracking
experiment in moving environment (human).
5.1 Joint External Force Observation (sensitive Collision Detection)
Experiment
As show in Fig. 9, the robot starts to move freely from the initial point q0 = 150◦ ,
and the external force observation gain factor is set Ka = 2000 (the same as in
the simulation). The human hand appears on different trajectories of the robotic
arm multiple times at random. Whenever the robotic arm touches the palm, it
can be stopped suddenly with only a small contact force. When the palm leaves
the end, the robotic arm immediately resumes free movement.
Fig. 8. Experiment platform
Variable Impedance Force Tracking Algorithm 571
Fig. 9. Robot-human variable force observation
Figure 10 shows the low-density sponge and balloon collision experiment,
represents objects of different stiffness and unknown environment respectively.
As shown in Fig. 12, when contact occurs, the observer responds immedi-
ately. The threshold of the human hand collision experiment is 0.2 Nm, and the
threshold of the variable stiffness environment is 0.3 Nm. Verified the sensitivity
and accuracy of the external force observation algorithm.
Fig. 10. Robot-variable stiffness environment force observation
5.2 Joint Force Tracking with Stationary Environment
Figure 11 is a tracking comparison with the desired force under objects with dif-
ferent stiffness (foam, cartons, metal part) and different desired force. Figure 13
shows that the Collaborative joint has a shorter adaptation time when it comes
into contact with objects with greater stiffness. It is obviously that force track-
ing can be achieved under different expected torques. When Fd = 0.2 Nm, the
error is the smallest, but the adaptive period is longer, when Fd = 0.8 Nm, the
adaptive period is the shortest, which is 2.3 s.
Figure 14 shows the force tracking error of three different stiffness objects in
the same expected force tracking experiment. It can be seen that the force errors
ef < 0.04 Nm.
Fig. 11. Robot-variable stiffness objects force tracking
572 J. Wang et al.
Fig. 12. Performance analysis of external force observation
5.3 Joint Force Tracking with Moving Environment
Figure 15 shows the Variable impedance force tracking in a moving environment.
Human hand represent the moving environment. The robot starts to move from
the initial position, and the human hand randomly appear in the trajectory and
interact with the robot. We repeat the experiment several times and take the
average value for calculation. The robot responds immediately when a human
hand is observed, it reaches a steady state (Fd = 2.9 Nm) through adaptive
force tracking in a small time (t < 0.5s), and ef < 0.15 Nm, see Fig. 16. Table 1
shows the parameter settings in this experiment and Table 2 gives the values of
performance index.
Fig. 13. Force tracking performance analysis
Fig. 14. Force tracking error analysisVariable Impedance Force Tracking Algorithm 573
Fig. 15. Robot-palm variable impedance force tracking
Fig. 16. performance analysis
Table 1. Parameters setting.
Parameters Values
Load m 1.5 kg
Link length L 250 mm
Communication cycle T 2 ms
Variable impedance coefficient M 1.2
Variable impedance coefficient B 45
Table 2. Experimental results.
Performance index Values
Desired force Fd 2.9 Nm
Average error ef 0.15 Nm
Average response time t 0.5 s
6 Conclusion
In this paper, a collaborative joint variable impedance force tracking algorithm
based on the second-order momentum observer is presented. The stability and
robustness of the algorithm are verified through analysis and simulation, The
experimental results support the claim that this method can track the desired
force in real time under variable impedance and unknown environment. The
future, we will implement the algorithm proposed in this paper on a 7-DOF
collaborative robot, and improve the performance of the algorithm.574 J. Wang et al.
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