Force-Impedance Control: a new control strategy of
robotic manipulators
1
1
Fernando Almeida , António Lopes , Paulo Abreu
1
1
IDMEC - Pólo FEUP, Faculdade de Engenharia da Universidade do Porto,
Rua dos Bragas, 4099 Porto codex, Portugal
{fga, aml, pabreu} fe.up.pt
Abstract. A novel control strategy of robotic manipulators is presented in this
paper: the force-impedance controller. This controller enables two kinds of
behaviour: force limited impedance control and position limited force control.
The type of behaviour only depends on the chosen manipulator trajectories.
Free space error dynamics and post-contact manipulator dynamics may be
independently chosen if a new impedance control architecture is used.
Simulation results of a force-impedance controlled parallel manipulator,
executing tasks that involve end-effector contact with uncertain environments
of unknown stiffness, are presented.
1. Introduction
Position control strategies have been successfully used on robotic tasks involving a
null or week interaction between the manipulator and its environment. Good
examples are provided by spray painting, welding and palletising tasks [1], [2], [3].
On the other hand, although the set of tasks requiring a strong interaction between
the manipulator and the environment is very large [1], [4], [5], the use of robots on
assembly, polishing, grinding and deburring tasks, as well as on the field of medical
surgery [6], is still low due to control difficulties.
The two main approaches to the control of the interaction of the manipulator and
its environment are [4], [7]:
− Hybrid Force-Position Control [8], [9];
− Impedance Control [1], [4], [10].
Hybrid control is motivated by the task analysis made by Mason [11]: on each
direction of the task space, the environment imposes a force or a position constraint.
These natural constraints are originated by the task geometry. Only the unconstrained
variables may be controlled, their reference values being artificial constraints
imposed by the task execution strategy.
Hybrid control enables the tracking of position and force references, the task space
being decomposed into force and position controlled directions. As the decomposition
is based on an ideal model of the environment, the ever present modelling errors
always lead to unwanted movements along the force controlled directions and
unwanted forces along position controlled directions. This problem is more acute
during the transition between free space and interactive movements, necessitating the
use of some kind of controller switching strategy based on contact force information.
Unfortunately, the switching process may induce an unstable manipulator behaviour
[12], [13], [14], [15].
The impedance control objective is to control neither force nor position but their
dynamic relation, the desired impedance, along each direction on the task space.
Impedance controllers possess some inherent robustness to environment modelling
errors [16], [17], [7]. Nevertheless, as contact forces cannot be directly imposed, they
may grow in an uncontrolled manner due to modelling errors of the environment
impedance.
Controllers combining the hybrid and the impedance control approaches have been
developed, their main objective being a robust behaviour under environment
modelling errors [18].
The Hybrid Impedance Controller proposed by Anderson and Spong [19] uses an
inner control loop for manipulator dynamics linearization and decoupling. Impedance
control substitutes for the position control used on hybrid controllers position
subspace. This way, a desired manipulator dynamic behaviour is imposed.
Chiaverini and Sciavicco [16] combine a PD position control loop in parallel with
a PI force control loop. The controller achieves the typical robustness of an
impedance controller and the ability to follow position and force references of an
hybrid controller. Position and force references must be specified for each task space
direction. Control action is obtained as the sum of the two parallel loops control
actions. Conflicting situations, as an unexpected contact, are naturally solved as force
control always rules over position control, due to the integral control action.
Unfortunately this may lead to an undesirable manipulator drift, along the contact
surface, in the presence of environment modelling errors.
Neither of the above controllers enable the definition, over a force controlled
direction, of a free space trajectory up to the contact surface without the use of some
sort of force measure based switching strategy.
The novel force-impedance controller presented in this paper behaves itself as an
impedance (position) controller up to contact set up. Afterwards, impedance or force
control is achieved. This behavioural change is performed without the use of any
supervising switching strategy.
Section 2 presents a new impedance controller structure that enables a definition of
different free space error dynamics and post-contact impedance. The novel forceimpedance controller is developed in section 3. Simulation results of an experimental
parallel manipulator under force-impedance control are presented in section 4.
Conclusions are drawn in section 5.
2. Impedance Control
The control objective of an impedance controller is to impose, along each direction of
the task space, a desired dynamic relation between the manipulator end-effector
position and the force of interaction with the environment, the desired impedance.
Usually, the desired impedance is chosen linear and of second order, as in a massspring-damper system. Higher order impedances have a less well known behaviour
and require additional state variables.
In order to fulfil the task requirements, the user chooses a desired end-effector
impedance that may be expressed by equation 1,
M d (x − x d ) + B d (x − x d ) + K d ( x − x d ) = − f e
(1)
where Md, Bd and Kd are constant, diagonal and positive definite matrices
representing the desired inertia, damping, and stiffness system matrices. Vectors x
and xd represent the actual and the desired end-effector positions, and fe represents the
generalised force the environment exerts upon the end effector.
If the manipulator is able to follow an acceleration reference given by
[
x r = x d + M −1
d ⋅ − f e + B d (x d − x ) + K d (x d − x )
]
(2)
it will behave as described by equation 1. So, x r is the reference for an inner loop
acceleration tracking controller, that linearizes and decouples the manipulator
nonlinear dynamics.
xd
fe
xd
Controlled System
Bd
+
xd +
−
+
+
Kd
−
+
+
M −1
d
−
xr
G aa (s)
x
1
s
x
1
s
x
Bd
Fig. 1. Block diagram of an impedance controller
Fig. 1 shows a block diagram of an impedance controller. The transfer function
matrix Gaa(s) represents the non-ideal behaviour of the inner loop acceleration
controller, such as finite bandwidth and incomplete decoupling. At low frequency, up
to a maximum meaningful value, the acceleration controller must ensure that Gaa(s)
may be taken as the identity matrix, I. Hence, and without loss of generality, the
force-impedance controller will be presented for a single dof system.
Under these conditions, and taking all values from Fig. 1 as scalar quantities, the
end-effector position on the Laplace domain, X(s), is given by
X (s) = X d (s) − G f (s) ⋅ Fe (s)
(3)
where
G f (s ) =
1
M d s + Bd s + K d
2
(4)
represents the desired manipulator admittance.
The initial error between x and xd decays to zero, in free space, according to a
characteristic equation equal to the desired impedance.
The desired manipulator dynamics, expressed by equation 1, may also be
implemented using a new, and more general, structure presented in Fig. 2. x ∆
represents the position changes, relative to the free space trajectory, due to the contact
force fe.
In this case,
G f (s ) =
1
⋅
M f s2 + B f s + K f
(5)
This new structure enables the independent definition of free space error dynamics,
and post-contact, Gf(s), manipulator dynamics. In free space, the error between x and
xd decays to zero with a dynamic behaviour given by the following characteristic
equation:
M x s 2 + Bx s + K x = 0 ⋅
(6)
This extra degree of freedom may be used to improve free space error dynamics,
without compromising the desired impedance under contact.
Using the conventional structure presented in Fig. 1, a similar behaviour could
only be obtained by switching controller gains (impedance). With the new control
structure, the need for a supervising switching strategy is avoided.
From equations 5 and 6 it is easily concluded that the impedance controller is
stable as long as parameters Mx, Bx, Kx, Mf, Bf and Kf are chosen strictly positive.
However, it should not be forgotten that the inner acceleration loop finite bandwidth
(Gaa(s) ≠ I) may lead to instability if the desired manipulator dynamics are chosen too
fast.
xd
xd
Bx
+
xd
+
+
1
Mx
Kx
−
−
+
+
x
−
∫
x
∫
x
+
Bx
+
+
+
fe
+
1
Mf
−
x∆
∫
x∆
∫
x∆
+
Bf
+
Kf
Fig. 2. New impedance controller structure: free space error dynamics and manipulator
impedance may be independently defined
3. The new Force-Impedance Controller
The force-impedance controller presented on this paper combines the robustness
properties of an impedance controller with the ability to follow position and force
references of an hybrid controller. The proposed controller has two cascaded control
loops (see Fig. 4): an impedance controller, as the one presented on the previous
section, is implemented as a inner loop controller. An integral force controller is used
as an external loop controller.
This force controller acts by modifying the position reference, given by xd, in order
to limit the contact force to a specified maximum value.
For each task space direction the user must specify a, possible time variant, force
reference, in addition to the desired position trajectory, impedance, and free space
dynamics. The force reference has the meaning of a limiting value to the force the
end-effector may apply to the environment.
The static force-displacement relation imposed by the force-impedance controller
is presented in Fig. 3. Positive force values are obtained with a “pushing”
environment and the negative values with a “pulling” one. The contact force
saturation behaviour is obtained by the use of limited integrators on the force control
loop.
While the manipulator is not interacting with the environment the controller
ensures reference position tracking with the specified free space error dynamics. After
contact is set up the controller behaviour may be interpreted in two different ways:
− as a force limited impedance controller;
− as a position limited force controller.
If contact conditions are planed in such a way that the force reference (limit) is not
attained, the manipulator is impedance controlled. If environment modelling errors
result in excessive force, the contact force is limited to the reference value.
If contact conditions are planed in order to ensure that the force reference is
attainable, the manipulator is force controlled. When environment modelling errors
result in excessive displacement, manipulator position is limited to its reference value.
This kind of manipulator behaviour ensures a high degree of robustness to
environment uncertainty.
fe
fd
x∆+
Kd
x∆−
x∆
−α ⋅ f d
Fig. 3. Static force-displacement relation imposed by the force-impedance controller
Fig. 4 shows the force-impedance controller under a contact situation with a purely
elastic environment. Its contact surface is positioned at xe and presents a stiffness Ke.
In order to allow a choice of integrator gain, KF, that is not dependent on the desired
manipulator stiffness, Kf, the force error is divided by it. When the force control loop
is in action the end-effector position, on the Laplace domain, is described by the next
equation,
X (s) = X d (s) + Gx (s)U ( s) − G f (s)Fe (s)
(7)
where
[
]
U (s) = Fd (s) − Fe (s) ⋅
KF 1
⋅
s Kf
(8)
and
Kx
⋅
M x s 2 + Bx s + K x
G x ( s) =
(9)
Impedance Controller
Force Controller
xd
+
fd
1
−α
−
+
+
+
u
+
1
Kf
+
KF
s
−
xe
0
KF
s
x
Gx
−
+
fe
Ke
−
0
Gf
Fig. 4. Block diagram of the novel force-impedance controller interacting with an elastic
environment of stiffness Ke
Substituting equations 5, 8 and 9 in equation 7 the following relation results:
X (s ) =
PX (s)
P ( s)
P ( s)
⋅ X d ( s) + F
⋅ Fd (s) + E
⋅ X e ( s)
Q X (s )
Q X ( s)
Q X ( s)
(10)
where,
)(
(
PX (s) = K f s M x s2 + Bx s + K x M f s 2 + B f s + K f
(
PF (s) = K F K x M f s 2 + B f s + K f
(
)
)
(11)
)
(12)
(
PE (s) = K F K x K e M f s 2 + B f s + K f + sK e K f M x s2 + Bx s + K x
(
)(
)
Q X (s) = K f s M x s 2 + Bx s + K x M f s 2 + B f s + K f +
(
)
(
)
)
K f K e s M x s + Bx s + K x + K x K F K e M f s 2 + B f s + K f ⋅
2
(13)
(14)
If the desired manipulator free space error dynamics is made equal to the desired
impedance, then Mx = Mf = Md, Bx = Bf = Bd, and Kx = Kf = Kd. Under this assumption
the force-impedance controller can be simplified (see Fig. 5) and the manipulator
position may be expressed as
X (s) =
(M s
2
d
(
2
(
2
(
2
)
+ Bd s + K d s
)
s M d s + Bd s + K d + K e s + K F K e
KF
)
s M d s + Bd s + K d + K e s + K F K e
K F K e + sK e
)
s M d s + Bd s + K d + K e s + K F K e
X d ( s) +
Fd (s) +
(15)
X e ( s) ⋅
If the Routh stability criteria is applied to equation 15, a maximum value of the
force control loop integral gain, KF, is obtained as
KF <
Bd ( K d + K e ) Bd
<
⋅
Md Ke
Md
(16)
So, KF may be chosen in a manner that is independent of the environment stiffness Ke,
increasing the controller robustness to environment modelling uncertainty.
A similar stability analysis for the case when different free space error dynamics
and post-contact impedance are desired is still under study.
Impedance Controller
xd
xd
+
fd
1
−α
KF
s
−
+
−
KF
s
Bd
xd
Force Controller
xe
Controlled System
0
+
0
1
Kf
+
Kd
−
−
+
+
+
+
+
+
M d−1
+
xr
Gaa (s)
−
x
x
x
1
s
1
s
+
−
fe
Ke
Bd
Fig. 5. Force-impedance controller: same free space error dynamics and post-contact
impedance
4. Force-Impedance Controller Simulation Results
A simulation of the force-impedance controller, when applied to the control of a 6 dof
parallel manipulator, was performed. An experimental set-up is under construction:
the Robotic Controlled Impedance Device (RCID). This is a fully parallel minimanipulator with a Merlet platform architecture [20], [21]. The RCID is coupled to
an industrial manipulator: the latter one performs the large amplitude movements
while the RCID is only used for the fine and high bandwidth movements needed for
force-impedance control.
The simulation program was developed on ACSL (Advanced Continuous
Simulation Language) [22] and implements a discrete time version of the controller.
Fully dynamic models of the manipulator, actuators, and transducers were included.
The simulation also takes into account position, acceleration, and force measurement
noises, as well as input and output quantization. The inner loop acceleration controller
uses a 1 kHz sampling frequency and achieves a closed loop bandwidth of 400 rad/s.
The force-impedance controller runs with a 500 Hz sampling frequency and includes
a velocity observer having acceleration and position as input variables.
The two simulation cases that are presented next were chosen in order to enable an
easy performance evaluation of the force-impedance controller.
The first case shows the force-impedance controller working as a force limited
impedance controller. The task includes an approach to a contact surface, followed by
a contact phase and, in the end, a pull back movement. During the contact phase, the
end-effector must follow the surface xe, presented in Fig. 6(a), and the contact force
must be limited to fd = 100 N. The desired trajectory xd was planed without taking into
account a 2 mm hump on the contact surface and requires the end-effector to move
1
with a speed of 5 mm s− .
Simulation results, with impedance matrices Md = diag([200 200 200 1 1 1]) (Kg;
4
4
4
1
1
Kg m), Bd = diag([1.2×10 1.2×10 1.2×10 40 40 40]) (N s m− ; N m s rad− ), and
5
5
5
1
1
Kd = diag([1.8×10 1.8×10 1.8×10 400 400 400]) (N m− ; N m rad− ), force
1
controller gain KF = 40 s− on all dof, and force environment stiffness
6
1
Ke = 2×10 N m− , are presented in Fig. 6(b), position tracking error norm, and
Fig. 6(c), contact force.
The manipulator tracks xd up to the contact surface. After impact, impedance is
regulated. When the unexpected hump is attained, the contact force grows up quickly
and is limited to its reference value. After the hump, and to the end of the contact
phase, the controller returns to impedance control. Position tracking is regained when
the end-effector goes back into free space.
In the second case the force-impedance controller is used as a position limited
force controller, tracking a reference force profile. After a free space approach to the
contact surface, the end-effector must track a desired contact force profile and, in the
end, do a pull back movement. The desired trajectory xd (Fig. 6(d)) was planned on
the basis of an uncertain model of the environment, and requires the end-effector to
1
move with speed of 20 mm s− .
Simulation results, using the same set of parameters as on the previous case, are
presented in Fig. 6(e), position tracking error norm, and Fig. 6(f), reference force
profile and actual contact force. During the approach phase, the manipulator follows
the desired position trajectory up to the contact surface. After impact, a short period
of impedance control is observed. When the desired force becomes attainable, force
control starts. The desired force profile is followed up to the beginning of the pull out
movement. During the pull out phase the controller returns to impedance control,
enabling the manipulator to follow the desired free space position trajectory.
12
Environment / Reference Position (z − axis) (mm)
Environment / Reference Position (z − axis) (mm)
12
10
8
6
4
2
0
−2
0
50
100
10
8
6
4
2
0
−2
0
150
50
Position (x − axis) (mm)
2.5
2.5
2
2
1.5
1
0.5
0
0
1
0.5
50
100
0
0
150
50
(b)
100
150
Position (x − axis) (mm)
200
250
200
250
(e)
140
140
120
120
Contact force / Reference force (N)
Contact force / Reference force (N)
250
1.5
Position (x − axis) (mm)
100
80
60
40
20
0
0
200
(d)
Position error (mm)
Position error (mm)
(a)
100
150
Position (x − axis) (mm)
100
80
60
40
20
50
100
Position (x − axis) (mm)
(c)
150
0
0
50
100
150
Position (x − axis) (mm)
(f)
Fig. 6. Simulation results: (a), (b), and (c) - force limited impedance control case;
(d), (e), and (f) - position limited force control case.
(References are represented by dashed lines)
5. Conclusions
The novel force-impedance controller presented on this paper enables the follow up
of position and force trajectories, nevertheless achieving the robustness properties
inherent to impedance controllers. If the new impedance control structure is used,
different free space error dynamics and post-contact dynamics may be used, without
the need of a switching strategy.
Force limited impedance control and position limited force control may be
implemented with the proposed force-impedance controller. The type of behaviour is
only dependent on the chosen trajectories and does not necessitates any other decision
mechanism.
The simulation of a force-impedance controlled parallel mini-manipulator, the
RCID, shows that good position, impedance, and force tracking is obtained
The robust behaviour of the proposed controller is also enhanced by the fact that
its parameters may be fully defined without the knowledge of the environment
stiffness.
This research was supported by Ministério da Ciência e Tecnologia - FCT, under the project nº
PBIC/C/TPR/2552/95.
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