However, the conflicting force in the rigid closed chain is stored in the virtual spring of the impedance controller especially in microgravity, where even small disturbances cause a sig
Trang 1Ifnternational Journal of Advanced Robotic Systems
Adaptive Impedance Controller
for a Robot Astronaut to
Climb Stably in a Space Station
Regular Paper
Bo Wei1,2,3, Zhihong Jiang1,2,3*, Hui Li1,2,3, Que Dong1,2,3, Wencheng Ni1,2,3 and
Qiang Huang1,2,3
1 IRI, School of Mechatronic Engineering, Beijing Institute of Technology, Beijing, China
2 Key Laboratory of Biomimetic Robots and Systems, Ministry of Education, China
3 Key Laboratory of Intelligent Control and Decision of Complex System, China
*Corresponding author(s) E-mail: jiangzhihong@bit.edu.cn
Received 18 December 2015; Accepted 09 April 2016
DOI: 10.5772/63544
© 2016 Author(s) Licensee InTech This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the
original work is properly cited
Abstract
Maintaining stability is a significant challenge during the
control of a robot astronaut while climbing with
human-like dual-arm action in a space station This challenge is
caused by conflicting force generated by dynamic internal
forces in the closed chain during dual-arm climbing In
general, an impedance controller is suitable for solving this
problem However, the conflicting force in the rigid closed
chain is stored in the virtual spring of the impedance
controller (especially in microgravity), where even small
disturbances cause a significant change in robot astronaut
movements As such, it is difficult to select suitable control
parameters for the stable climbing of a robot astronaut This
paper proposes an adaptive algorithm to optimize the
impedance controller parameters This eliminates conflict‐
ing force disturbances, with one arm in compliance with
the motion of the other It provides scope for achieving
stable motion without the need for precise control param‐
eters Finally, the stability of the proposed algorithm is
demonstrated by Lyapunov theory using a robot called
ASTROBOT The experimental results show the validity of
the proposed algorithm
Keywords Adaptive Impedance Controller, Dual-arm Coordination, Multipoint Impact Dynamics, Robot Astro‐ naut
1 Introduction
Space robots were developed to assist or replace astronauts for tasks in outer space Space robots alleviate human astronaut workloads and can also reduce work-based risks
in the international space station (ISS); furthermore, they are time-saving and economical [1,2] However, existing robots, such as CanadaⅡ and SPDM [3,4], cannot cover all extravehicular areas, especially at the base of the ISS, where the earth observation equipment is located Therefore, significant research and development have gone into producing human-like robot astronauts, such as Robo‐ naut2 [5] and DLR Justin [6]
The dynamics and control methods of a robot astronaut are significantly different from an earth-based robot [7] Also, due to microgravity and narrow room in the space station, uncertain dynamic interactions and collision forces are serious threats to control stability, where even small disturbances may lead to significant movement changes
Trang 2This makes robot astronaut dynamics very complex, with
stable movement difficult to maintain Furthermore, the
moving modes of a robot astronaut are no longer solely
based on biped dynamics; they now involve dual arms to
climb or even glide (including rolling and reversing) by
using ergonomic handrails in the space station (Fig 1)
Therefore, control methods for earth-based robots cannot
be directly used in the control of space-based robots
Figure 1 Motion model of climbing astronaut
Dual-arm coordinated locomotion control approaches can
be classified as hybrid position/force control and impe‐
dance control Raibert and Craig (1980) [8] formulated the
hybrid position/force control method for interactive tasks
between a mechanical arm and the environment; this
included such tasks as peg-in-hole assembly and wiping
glass Uchiyama et al (1988) [9] extended hybrid position/
force control to the coordination of two robots, which
performed well on the coordinated control scheme
However, this approach required a real-time switch control
mode in accordance with the selection matrix, which may
cause disturbances in the robot astronaut system Further‐
more, it reduced the robustness of the system under
microgravity conditions
In order to maintain stability against external environment
disturbances and overcome force/position hybrid control
method defects, Hogan (1985) [10] applied impedance
control to the robot control, which established an expected
dynamic constraint relationship between the manipulator
and the environment Schneider and Cannon (1992) [11]
proposed a dual-arm cooperative theory based on object
impedance control, where the desired impedance was
aimed at the object itself, rather than on the end effector
Platt (2011) [12] employed an extension of multi-priority
impedance control for a controller, where the first-priority
impedance was defined at the end effector in the Cartesian
space and the second-priority impedance was defined in
the joint space [13] NASA applied this method to their
Robonaut2 simulation Typically, the impedance control‐
ler, which is mentioned above, is used for docking with peg
and hole [14], assembly [15] and moving objects [16]
Besides, a variety of control techniques has been developed
for flexible manipulators [24], linear motor with
high-frequency dynamics [25] and space robots for capturing a
non-cooperative target [26-28]
In addition to the impedance controller tasks mentioned above, conflicting forces exist in the rigid closed chain of the robot astronaut, where its arms are required to correct deviations while climbing As a result, the deformation of the virtual spring is significant (approximately 30 mm), which means that the spring stiffness has to be small in order to limit the conflicting force in the closed chain However, small spring stiffness is vulnerable to instability Also, the dynamics of a robot astronaut are very different and more complex compared to an earth-based robot This paper proposes an adaptive impedance controller to adaptively adjust the impedance parameters based on a forgetting factor function This strategy includes a differ‐ ent control method for each arm In order to ensure stability
of the robot astronaut, position control was adopted in the master arm and the proposed adaptive impedance control‐ ler was adopted in the control of the slave arm This reduced (and even eliminated) the conflicting force, thereby correcting the moving deviations by adaptively changing the static equilibrium position of the virtual spring Section 2 presents the development of a multipoint impact dynamics model and also derives force optimization for the robot astronaut Section 3 presents an analysis of the conflicting force, as well as introduces the master-slave arm selection method Section 4 proposes a novel, improved impendence control method based on a forgetting factor function, along with demonstrating the stability of the system under the Lyapunov stability theorem The last section presents the experimental results, which validate the proposed approach, with the final concluding remarks presented prudently
2 Multipoint Impact Dynamics and Force Optimization
in the Robot
When climbing in narrow or cluttered spaces (common in
a space station), impact regularly occurs when the robot astronaut makes contact (including multipoint impact) with its environment Due to microgravity, the forces generated upon impact greatly influence the robot; there‐ fore, a multipoint impact dynamics model was established
as the basis for the stable control of a robot astronaut in a space station Furthermore, where two or more contacts occur when climbing, the optimization provided a solution for obtaining the force distribution during impact
Figure 2 Multipoint impact model for robot astronaut
Trang 3The Newton-Euler method established the multipoint
impact dynamics model for the robot astronaut through
kinematical constraint (Fig 2) Assuming that the robot
astronaut has n contact points with the station, the kine‐
matic constraints are imposed as:
The dynamical equation for the robot astronaut in micro‐
gravity is:
1
0
i i i
M q q C q q J f
é ù + =ê ú+
ë û å
where q is the joint angle, M (q) is the inertial matrix in the
joint space, C(q,q˙) is the resultant force vector of the Coriolis
and centrifugal forces, τ is the joint torque f i is the impact
force of the ith contact point, and J i is the Jacobian matrix
of the ith contact point.
f zext is defined as the desired resultant external force; it is
the vector sum of all the contact forces
1
n zext i i
=
Force optimization is used to simplify the influence of
contact impact According to the dynamic model, the
resultant force on the z-axis direction is:
1
2
ˆ
z
r zext n
zn
f
f
f
é ù
(4)
set
n
(5)
So, the optimization formula can be written as:
1
# 2
1
z
r z
z r zext zn
f
x f
D y f f
é ù
é ù
ê ú
ê ú
ê ú = ê ú
ê×××ú
ê ú
ê ú
ë û
(6)
where D z = D z T (D z D z T)−1 The optimization formula mini‐
mizes the sum of squared component forces on the z-axis
direction Similarly, contact forces on the x-direction and
y-direction can also be obtained by the same method Multipoint impact and force optimization are the basis of the conflicting force analysis The next section presents the use of conflicting force analysis to determine the master and slave arms
3 Conflicting Force Analysis
During the climbing process, the robot arms form a rigid closed chain and affect each other through the body Here,
a small error may lead to a significant waggle or even a tilt (Fig 3)
Figure 3 During the climbing process, the robot arms form a rigid closed
chain and affect each other through the body Here, a small error may lead
to a significant waggle or even a tilt.
The control of a robot climbing in a space station is com‐ plicated due to the holding errors and closed chain system There are various kinds of holding error that can occur because of the unexpected multipoint impacts, mechanism deformation and vision errors (Fig 4) In addition, the closed chain formed by the two arms should be treated as
a combination of subsystems given that errors in the closed chain cause an unexpected internal force and intensify instability In this section, on the basis of the conflicting force analysis, the two arms are defined as the master and slave arms
Figure 4 Grasping state
The force analysis was developed on forces between the arms and handrails because there were only two six-axis force/torque sensors installed at the end of the arms The forces acting on the shoulders were decomposed into two
Trang 4parts: parallel to climbing direction and perpendicular to
climbing direction (Fig 5)
The robot astronaut was designed to move in the XOY
plane; as such, forces were projected onto the XOY plane
The force analysis is shown below:
Figure 5 Robot statics analysis
Forces acting on the shoulders can be derived from the
contact force and torques using a statics analysis [23]
Forces and torques passing from the i+1th joint to the ith
joint are:
1
i i i
i i i
f R f+
i i i i i
i i i i i i
n = + R n+ + + P+ ´ f+ (8)
where f i is the force linkage i+1 acting on linkage I, while
n i is the torque linkage i+1 acting on linkage i.
The applied force of shoulder A ( F A) is derived and
projected onto the XOY plane; it is then decomposed into
perpendicular and parallel forces
A Ax A Ay A
Ax A Ay A A
Similarly, the applied force of shoulder B (F B) is decom‐
posed as follows:
B Bx B By B
Bx B By B B
The arm with the larger internal force needs to be adjusted because it causes a conflicting force in the closed chain As
a result, the arm with the larger perpendicular force is defined as the slave arm, while the other is defined as the master arm Comparing F Α// with F Β// : if F A// < F Β//, then arm
A is the master and arm B is the slave; if F Α// > F Β//, then arm
B is the master and arm A is the slave Different control strategies were used for arms A and B (see Section 4)
4 Control of Robot Astronaut Climbing with One Arm in Compliance with the Other
In Section 2, the expected forces and torques exerted on the robot body were calculated using force optimization In Section 3, the master and slave arms were defined with conflicting force analysis In this section, an adaptive algorithm is proposed to optimize the parameters of the impedance controller, which can reduce (and even elimi‐ nate) conflicting force disturbances in the slave arm
4.1 Master-slave adaptive impedance controller
The slave arm changed the equilibrium position of the virtual spring (Fig 6) corresponding to the main arm motion in order to prevent the robot astronaut from shaking It correctly adjusted the robot position and orientation, then released the conflict force
Figure 6 The controller policy for master slave coordination
When the robot holds the handrails, the desired pose of the robot involves two arms that are symmetrical to each other Eventually, arms in such a motion posture will move stably
As is shown in Fig 6, the green line is the desired symmetric line Due to the errors, such as unexpected multipoint impacts, mechanism deformation and vision errors, however, the robot is often asymmetric and the centre line
of the robot is deviated (red line in Fig 6) As a result, there will be an internal force in the virtual spring of the impe‐ dance controller, which is generated from the distance
Trang 5between the red line and the green line The internal force
is the source of conflicting force in the closed chain To solve
this problem, the master arm’s path-planning strategy is to
control the robot towards the expected symmetrical pose
Meanwhile, the slave arm gradually adjusts the equilibri‐
um position of the virtual spring in the impedance control
by the forgetting factor function in order to eliminate
conflicting force under the precondition of the stably
moving robot
Figure 7 Adaptive impedance control law
The configuration of both arms was symmetrical, while
their rigid damping coefficients were the same; hence, the
impedance equation of both arms can be written as:
s des- + s des- + s des- = des- ext
M c&& &&c B c& c& K c c Q Q (13)
where χdes describes the expected trajectory of the slave
arm, χ describes its actual trajectory, Ms is the inertia
matrix, B s is the damping matrix, Ks is the rigidity matrix,
Qdes represents the expectation forces and torques of the
slave arm shoulder, and Qext represents the detecting forces
and torques by the six-dimensional (6D) force/torque
sensor at the end effector Qext is written as:
[ T, T T]
ext= f n
where 7f
7 and 7n7 are derived from Eqs (7) and (8),
respectively
For impedance control, Eq (13) was derived in six direc‐
tions: three position directions and three angle directions
In order to simplify the calculation, the following analysis
addresses only one direction (as the other five directions
are identical to it)
Set e as the error between the equilibrium position, χdes, and
the actual position, χ, of the virtual spring:
(k) (k) (k)
s des
Set ΔF as the difference between the expected contact
force., F des, and the actual contact force, F ext :
F( ) = F ( ) - F ( )k des k ext k
Now, the impedance equation can be changed to:
M ( )e ( ) + B ( )e ( ) + K ( )e ( ) = F( )k && k k & k k k D k (17) The spring damping system should operate in the critical damping state:
B ( ) = 2 K ( )M (k)k k (18) Bring Eq (18) into Eq (17),
e ( ) + 2 e ( ) + e ( ) =
D
In order to simplify the equation, it may be defined as:
s s
( ) ( )
M ( )
K k
H k
k
Now Eq (19) can be written as:
2
s
F( )
e ( ) e ( ) ( ) 2e ( ) ( )
M ( )
k
k
D
The classical impedance control method for robot climbing (as described above) has a large conflicting force stored in the closed chain Hence, the classical impedance controller cannot eliminate the inherent internal force Inspired by the astronaut climbing process, the following subsection proposes an improved method based of impedance control
4.2 Forgetting factor function for adaptive impedance controller
This paper introduces a forgetting factor function [17-19] into the impedance controller; see Eq (22) When the robot astronaut is climbing, the virtual spring position can be adjusted to the expected motion state in the impedance controller using the provided forgetting factor function This can eliminate the conflicting force in the rigid closed chain when the slave arm is controlled using the proposed impedance controller in order to follow the motion of the master arm
( ) ( ) (1 ( )) ( ) ( ) ( )
( )
n
n
e t
u t r n u t r n u t
e t
+
&
(22)
where r(n)∈ 0, 1) describes the forgetting factor function,
n is the amount of iterations and e n (t) is the deviation of the
controlled variable
Based on the forgetting factor function, Eq (13), which describes the slave arm impedance control, can be written as:
Trang 6des ext
M e + B e + K e = Q - Q&&% &% % (23)
As per the previous section, the following analysis only
addresses one direction Here, e˜ s (n) describes the difference
between the desired posture and the actual posture in Eq
(15), which can be written as:
p
s des
e n = n - n
where χp(n) describes the actual posture of the slave arm,
while χ˜desp (n) defines the expected posture using the
forgetting factor function:
( 1) (1 ( )) ( ) ( ) ( )
des n+ = -r n a n r n+ des n
where χ#
des (k) describes the expected posture of the slave
arm shoulder, while χ#
a (k) describes the posture of the slave
arm shoulder calculated from the master arm Here, χ#
a (k)
can be calculated as:
#a( )n T n= #a( )× a( )n
where χa (n) is the actual posture of the master arm.
Defining the coordinate transformation from the master
arm to the slave arm shoulder as T#
a (k), the following
equation is obtained:
p
( ) (1 ( )) ( ) ( ) ( ) ( ) ( )
s a a des
e n% = -r n T n ×c n r n+ c n -c n (27)
The forgetting factor function, r(n)∈ 0, 1 is a monotone
decreasing function in the climbing period At first, r(n) is
closed to 1, so the equilibrium position of the virtual spring
is nearer to the expected position of the slave Then, r(n)
decreasingly approaches 0 and the equilibrium position
approaches the position corresponding to the master arm
When the climbing ceases, the master arm determines the
equilibrium position
The forgetting factor function is written as:
1 ( )
ln( 2.9) 10
r n = n
As a result, the acceleration change of the slave arm e¨ s is
obtained from the adaptive impedance controller
4.3 Stability analysis
Lyapunov's direct method is a reliable and robust tool for
system stability analysis [20] From an energy perspective,
it is used for determining the stability of a system’s
equilibrium state by the Lyapunov function, V(x) The
stability criterion shows that the key of Lyapunov's direct
method is finding the function V(x).
Due to special compliance control for robotic dual-arm climbing, the Lyapunov function is selected in the error space:
( ), T T
V e e = e M e + e K e
& & % & %
where the matrices M ˜ des and K˜ s are both positive definite
matrices that satisfy Silvester conditions Hence, V is a
positive definite matrix that provides:
( ),
( s s)
The time derivative of the Lyapunov function is:
V e ,e = e M e + e K e& % %& &% % && &% % % % (32)
It can be converted to:
V e ,e = e M e + K e& % %& &% % &&% % % (33) Next, based on Eq (23), the following equation is obtained:
6×1 6×1
0 0
=
M e K e% &&% % % % &% % % (34)
By substituting Eq (34) with Eq (33), the following equation is obtained:
( ) Téë- D - ùû
V e ,e = e& % %& &% B e + Q% &% % Q )% (35)
As Q ˜ des and Q ˜ ext only account for a small proportion in practical applications, Eq (35) can be simplified as:
( ) - T
V e ,e = e B e& % %& &% % &% (36)
where V˙ is the negative definite from Eq (36); it indicates that the proposed impedance controller algorithm, based
on the forgetting factor function, ought to enable the stable control of a robot astronaut when climbing in a space station
5 Experimental Results and Analysis
5.1 System description of a robot astronaut
In order to simulate an astronaut climbing in a space station, a humanoid space robot system named “ASTRO‐ BOT” was developed The robot astronaut consisted of a
Trang 7head, two arms and a body (Fig 8) A binocular stereo
vision system was built into the head in order to obtain
position information There were two end effectors at the
end of each arm
Figure 8 ASTROBOT system
Gravity compensation equipment is necessary for motion
research involving space robots Various schemes have
been employed to compensate for gravity on earth,
including air flotation, hanging wire and buoyancy
methods [21] However, these methods are expensive and
complex The air flotation method can only accommodate
testing in two dimensions The hanging wire method is
complex and has difficulty achieving real-time control The
buoyancy method needs to seal the robot, while there is also
a factor in water tension during testing Therefore, this
paper proposes a passive mechanism to simulate a micro‐
gravity environment for the ASTROBOT research [22] The
mechanism comprises three main parts: a horizontal tuning
mechanism, a motion mechanism and a gravity compen‐
sation mechanism (Fig 9)
Figure 9 Microgravity simulation system: ① to ⑥ horizontal tuning
mechanism; ⑦ to ⑧ motion mechanism; ⑨ handrail; and ⑩
gravity-compensation mechanism.
5.2 Robot astronaut parameters
At the climb commencement, the actual speed of a robot is
V A, the detecting speed is V C and the speed error is V B ; hence, the relationship between them is V→A =V→B + V→C (Fig
10) For planning, the robot climbing distance is L and the
acceleration is a(t)=v C2/2L
Figure 10 Initial motion status of ASTROBOT
Table 1 lists the basic information of each joint
Joint initial
angle (rad) -0.51 0 -0.60 -1.57 1.57 -0.47 -0.47 The robot mass was m b =255.8kg
Table 1 Joint base parameters
Figure 11 shows that, where the robot was only controlled
in position, some of the joint driven torques were too large for the motors Therefore, the position controller could not
be used to complete climbing
Figure 11 Joint torque in position control
Based on practical situations and experience, the total control time was set to 5 s and the control cycle was set to 0.001 s Table 2 lists the other control parameters
The damping factor was B s =2 K s m because the impe‐ dance control system was set to work in critical damping conditions
Trang 85.3 Experiment results and analysis
After experimental analysis, arm A was selected as the
main arm and arm B as the slave arm The experimental
results and analysis are provided below
A Climbing position status
In order to demonstrate the effectiveness of the adaptive
impedance controller for slave arm compliance with master
arm motion, some of the experimental tracking data were
plotted, showing the trajectory error of the x, y coordinates
(Δx, Δy) (Figs 12 and 13) Moving errors in Fig 12 are
caused by the robot using classical impedance control
When the robot’s two end effectors hold handrails, they
form a closed chain, such that even small distractions
(unexpected multipoint impacts, mechanism deformation
and vision errors) can cause a big moving error In the last
stage of climbing, the robot body kept shaking, with the
shaking curve showing divergence trends This means that
it was difficult to keep stable In Fig 13, the moving error
curves become flat after the climbing finished because the
robot is controlled by the adaptive impedance control
method By this method, the conflicting force is eliminated
and the robot body becomes stable
Figure 12 Moving error of robot body using classical impedance control
Figure 13 Moving error of robot body using adaptive impedance control
Expected position
Expected initial body
Virtual spring stiffness
Table 2 Decoupling control parameters
B Torques of joints
Figs 14 and 15 show the joint output torques of robots’ arm using classical impedance control Figs 16 and 17 show the torques of arms using adaptive impedance control At the beginning, the torques in Figs 14 and 15 are similar to torques in Figs 16 and 17 In the later stage, the fluctuation
in the classical method’s diagrams becomes bigger and bigger in Figs 14 and 15 However, the joint torques in Figs
16 and 17 respond to a reasonable extent and finally fall within a normal range using the adaptive controller
In this experiment, the joints of 2, 4 and 7 have maximum output torques Table 3 lists their maximum values (excluding torques in the initial period) From the table, we can see that the joint output force torques’ maximum values
in relation to the adaptive method were much smaller than the maximum values for the classical method This shows that the adaptive impedance control method is able to prevent joint damage and robot shaking when climbing
Classical impedance control A 36.57 18.43 31.35
Adaptive impedance control
Table 3 Maximum output torques of joints of 2, 4 and 7 after 2.2 s
Figure 14 Joint output torques of arm A using classical impedance control
Figure 15 Joint output torques of arm B using classical impedance control
Trang 9Figure 16 Joint output torques of master arm A using adaptive impedance
control
Figure 17 Joint output torques of slave arm B using adaptive impedance
control
C Contact forces between end effector and handrails
Due to errors, such as unexpected multipoint impacts, joint
positioning errors and vision inaccuracy, the robot deviates
in terms of both pose and position from the desired ones
These deviations cause a conflicting force in the rigid closed
chain formed by two arms, which may lead to big force/
torque between the end effector and the handrail, as well
as damage the joints As is shown in Figs 18 to 21, the forces
and torques between two arms and handrails controlled by
the classical impedance control method started fluctuating
in the final stage and showed a trend of amplification in the
end To overcome this problem, an adaptive algorithm to
eliminate the conflicting force during the motion of one
arm, in compliance with the other one, is proposed It is
based on the impedance control and forgetting factor function, which are applied in order to adjust the virtual spring As is shown in Figs 22 to 25, the contact forces and torques of two arms, controlled using the adaptive impe‐ dance controller, started fluctuating and were alleviated in the end Table 4 lists the maximum contact forces under the control of each of the two different methods Obviously, the contact forces and torques of the arms controlled using the classical method are much bigger than their counterparts when using the adaptive method These show that the adaptive impedance controller that we proposed can reduce the connecting force/torque between the end effector and the handrail, as well as eliminate the conflict‐ ing force in a rigid closed chain
Figure 18 Contact force of arm A using classical control
Figure 19 Contact torque of arm A using classical control
Classical
impedance
control
Adaptive
impedance
control
Table 4 Contact force/torque
Trang 10Figure 20 Contact force of arm B using classical control
Figure 21 Contact torque of arm B using classical control
Figure 22 Contact force of master arm A using adaptive impedance control
Figure 23 Contact torque of master arm A using adaptive impedance control
Figure 24 Contact force of slave arm B using adaptive impedance control
Figure 25 Contact torque of slave arm B using adaptive impedance control
6 Conclusion
This paper proposed a feasible impedance controller with
a forgetting factor function for a robot astronaut to climb stably in a space station It also defined the master and slave arms of a dual-arm robot based on the kinematics and multipoint dynamics of a robot astronaut The master arm moved with a path-planning algorithm based on a position controller, while the slave arm was compliant with the motion of the master arm when using the proposed impedance controller This eliminated the conflicting force disturbance in the closed chain during the dual-arm robot astronaut’s climbing tasks, ensuring good stability Com‐ parison experiments were conducted using ASTROBOT; the following conclusions were reached:
a During ASTROBOT climbing, it was difficult for the
robot astronaut to stably move with just the position controller in both arms Here, undesired trajectory movement occurred (and even caused damage to the joints) When the classical impedance controller was used, there were large conflict forces between the end effecters and the handrails The maximum joint output torques of the two arms were 36.57 Nm and 14.91 Nm; the maximum contact force/torque was 145.22 N/49.47 Nm
b The proposed control strategy with the impedance
controller, based on the forgetting factor function, achieved better results The slave arm was sufficiently