arXiv:1605.07834v1  [cs.RO]  25 May 2016
Dynamic analysis of simultaneous adaptation of force, impedance and
trajectory
Y. Li and E. Burdet
When carrying out tasks in contact with the environment,
humans are found to concurrently adapt force, impedance
and trajectory. Here we develop a robotic model of this
mechanism in humans and analyse the underlying dynamics.
We derive a general adaptive controller for the interaction
of a robot with an environment solely characterised by its
stiffness and damping, using Lyapunov theory.
I. SYSTEM DYNAMICS
The dynamics of a n-degree-of-freedom (n-DOF) robot in
the operational space are given by
M(q) ¨x + C(q, ˙q) ˙x + G(q) = u + f
(1)
where x is the position of the robot and q the vector of
joints angle. M(q) denotes the inertia matrix, C(q, ˙q) ˙x the
Coriolis and centrifugal forces, and G(q) the gravitational
force, which can be identified using e.g. nonlinear adaptive
control [1]. u is the control input and f the interaction force.
In [2], we have described the control input u in two parts:
u = v + w ,
(2)
with v to track the reference trajectory xr by compensating
for the robot’s dynamics, i.e.
v = M(q) ¨xe + C(q, ˙q) ˙xe + G(q) −Γε
(3)
where
˙xe = ˙xr −αe ,
e ≡x −xr ,
α > 0 ,
(4)
Γ a symmetric positive-definite matrix with minimal eigen-
value λmin(Γ) ⩾λΓ > 0 and
ε ≡˙e + α e
(5)
the tracking error. w is to adapt impedance and force in
order to compensate for the unknown interaction dynamics.
II. FORCE AND IMPEDANCE ADAPTATION
Suppose that the interaction force can be expanded as
f = F ∗
0 + K∗
S(x −x∗
0) + K∗
D ˙x ,
(6)
where the force F ∗
0 (t), stiffness K∗
S(t) and damping K∗
D(t)
are feedforward components of the interaction force, x∗
0(t)
is the rest position of the environment visco-elasticity and
all of these functions are unknown but periodic with T :
F ∗
0 (t + T )
≡
F ∗
0 (t) ,
K∗
S(t + T ) ≡K∗
S(t) ,
(7)
K∗
D(t + T )
=
K∗
D(t) ,
x∗
0(t + T ) = x∗
0(t) .
(8)
The authors are with the Department of Bioengineering, Imperial College
of Science, Technology and Medicine, London SW72AZ, UK.
To simplify the analysis, we rewrite the interaction force as
f ≡F ∗+ K∗
S x + K∗
D ˙x
(9)
where F ∗≡F ∗
0 −K∗
Sx∗
0 is also periodic with T . w in Eq.(2)
is then defined as
w = −F −KSx −KD ˙x
(10)
where KS and KD are stiffness and damping matrices,
respectively, and F is the feedforward force.
By substituting the control input u into Eq.(1), the closed-
loop system dynamics are described by
M(q) ˙ε + C(q, ˙q) ε + Γε = eF + eKS x + eKD ˙x ,
(11)
eF ≡F ∗−F ,
eKS ≡K∗
S −KS ,
eKD ≡K∗
D −KD .
In this equation, we see that the feedforward force F,
stiffness KS and damping KD ensure contact stability by
compensating for the interaction dynamics. Therefore, the
objective of force and impedance adaptation is to minimise
these residual errors which can be carried out through
minimising the cost function
Jc(t)
≡
1
2
Z t
t−T
eF T Q−1
F eF + vecT ( eKS)Q−1
S vec( e
KS)
+vecT ( eKD)Q−1
D vec( eKD) dτ ,
(12)
where QF, QS and QD are symmetric positive-definite
matrices, and vec(·) stands for the column vectorization
operation. This objective is achieved through the following
update laws:
δF(t)
≡
F(t) −F(t −T ) ≡QF [ε(t) −β(t)F(t)]
(13)
δKS(t)
≡
KS(t) −KS(t −T ) = QS[ε(t)x(t)T −β(t)KS(t)]
δKD(t)
≡
KD(t) −KD(t −T ) = QD[ε ˙x(t)T −β(t)KD(t)]
where F, KS and KD are initialised as zero matrices/vectors
with proper dimensions for t ∈[0, T ).
Now that we have dealt with the interaction dynamics,
stable trajectory control can be obtained by minimising the
cost function
Je(t) ≡1
2ε(t)T M(q) ε(t) .
(14)
Consequently, we use a combined cost function Jce ≡Jc +
Je that yields concurrent minimisation of tracking error and
control effort.
III. TRAJECTORY ADAPTATION
In a typical interaction task, the contact between the
robot and the environment is maintained through a desired
interaction force Fd. Assuming that there exists a desired
trajectory xd yielding Fd, i.e. from Eq.(6)
Fd
=
F ∗
0 + K∗
S(xd −x∗
0) + K∗
D ˙xd
(15)
=
F ∗+ K∗
S xd + K∗
D ˙xd ,
F ∗= F ∗
0 −K∗
S x∗
0 ,
we propose to adapt the reference xr in order to track xd.
However, xd is unknown as the parameters F ∗, K∗
S and K∗
D
in the interaction force are unknown. Nevertheless, we know
that xd is periodic with T as F ∗, K∗
S and K∗
D are periodic
with T and we also set Fd to be periodic with T .
In the following, we develop an update law to learn the
desired trajectory xd. First, we define
ξd ≡K∗
S xd + K∗
D ˙xd ,
ξr ≡KS xr + KD ˙xr .
(16)
Then, we develop the following update law
δξr(t) ≡ξr −ξr(t −T ) ≡L−T Qr(Fd(t) −F(t) −ξr(t))
(17)
where Qr and L are positive-definite constant gain matrices.
This update law minimises the error between ξd and ξr,
which is described by the following cost function
Jr ≡1
2
Z t
t−T
(ξr −ξd)T QT
r (ξr −ξd) dτ .
(18)
Because of the coupling of adaptation of force and
impedance and trajectory adaptation, we modify the adap-
tation of feedforward force Eq.(13) to
δF(t) ≡QF[ε(t) −β(t)F(t) + QT
r δξr(t)] .
(19)
As a result, update laws Eqs.(17) and (19) minimise the
overall cost J = Jc + Je + Jr as shown in Appendix A.
Then, we obtain the update law for trajectory adaptation
δxr ≡xr(t) −xr(t −T )
(20)
by solving
δξr = KS δxr + KD δ ˙xr = KS δxr + KD
d
dt(δxr)
(21)
using δξr(t) from Eq.(17). According to the convergence of
δξr, KS and KD as shown in Appendix A, xr will converge,
as
δξr −ξd = KSδxr + KDδ ˙xr ,
(22)
Upon convergence, the desired interaction force Fd is main-
tained between the robot and the environment according to
Eq.(17). At the same time, the properties with adaptation of
force and impedance are preserved which include trajectory
tracking and control effort minimisation. However, from the
analysis in Appendix A, we cannot draw the conclusion
that F, KS, KD and xr converge to F ∗, K∗
S, K∗
D and xd,
respectively, which will require the condition of persistent
excitation (PE), similar to classical adaptive control theory
[3].
IV. DISCUSSION
A. No contact
In a special case when there is no force applied by the
environment and Fd is also zero, the controller component
w will converge to zero. According to the update law Eq.(17),
the reference trajectory will not adapt, as expected.
B. No damping
If we neglect the damping component in the interaction
force f of Eq.(9), the trajectory adaptation described by
Eqs.(17) and (21) can be simplified to
δxr = L−T Qr(Fd −F −KS xr)
(23)
Correspondingly, the update laws for force and impedance
Eq.(13) needs to be modified as
δF
≡
QF (ε −βF + QT
r δxr) ,
(24)
δKS
≡
QS(ε xT −βKS + xT
r QT
r δxr) .
The stability analysis is similar to the case with damping and
is briefly explained in Appendix B.
C. Force sensing
As in [2], force sensing is not required in the proposed
framework, in contrast to traditional methods for surface
following where the force feedback is used to regulate the
interaction force [4].
In particular, in a first phase force and impedance adap-
tation is used to compensate for the interaction force from
the environment. During this process, the unknown actual
interaction force is estimated when the tracking error ε goes
to zero as can be seen from Eq.(11): when ε = 0, we have
w = −f.
(25)
Using this estimated interaction force, then a desired force
in Eq.(15) can be rendered by adaptation of the reference
trajectory xr.
In this sense, it is important to note that trajectory adap-
tation should be conducted only when force and impedance
adaptation takes effect, which guarantees compensation of
the interaction force and tracking of the current reference
trajectory. Nevertheless, as shown in above stability analysis,
adaptation of force, impedance and trajectory can be realised
simultaneously.
This also suggests that a force sensor should be used if
available, as force and impedance adaptation could then be
replaced by force feedback. In this way, trajectory adaptation
would not depend on the force estimation process and can in
principle happen faster than force and impedance adaptation
is needed. However, the potential advantages of a force
sensor depends on the quality of the signal it could provide,
its cost and the complexity of its installation and use.
V. APPENDIX
A. Proof for minimisation of overall cost J
Considering the definition of Jr in Eq. (18), we have
δJr(t) ≡Jr(t) −Jr(t −T )
=
1
2
Z t
t−T
[ξr(τ) −ξd(τ)]T QT
r [ξr(τ) −ξd(τ)] dτ
−1
2
Z t
t−T
[ξr(τ) −ξd(τ)]T QT
r [ξr(τ −T ) −ξd(τ −T )] dτ
+1
2
Z t
t−T
[ξr(τ) −ξd(τ)]T QT
r [ξr(τ −T ) −ξd(τ −T )] dτ
−1
2
Z t
t−T
[ξr(τ −T ) −ξd(τ −T )]T QT
r ×
[ξr(τ −T ) −ξd(τ −T )] dτ
=
1
2
Z t
t−T
[ξr(τ) −ξd(τ)]T QT
r δξr(τ) dτ
+1
2
Z t
t−T
[ξr(τ −T ) −ξd(τ −T )]T QT
r δξr(τ) dτ
=
Z t
t−T
[ξr −ξd −1
2δξr]T QT
r δξr dτ
(as ξd(t) = ξd(t −T ))
⩽
Z t
t−T
[Qr(ξr(τ) −ξd(τ))]T δξr(τ) dτ .
(26)
According to Eqs.(15) to (17), we rewrite this inequality as
δJr
⩽
Z t
t−T
[Qr(ξr −Fd + F + eF)]T δξr dτ
=
Z t
t−T
(−LT δξr + Qr eF)T δξr dτ.
(27)
Consider the difference between Jc of two consecutive
periods
δJc ≡Jc −Jc(t −T )
(28)
=
1
2
Z t
t−T
[( eF T Q−1
F eF −eF T (τ −T )Q−1
F eF(τ −T ))
+tr( eKT
S Q−1
S
eKS −eKT
S (τ −T )Q−1
S
eKS(τ −T )
+( eKT
DQ−1
D eKD −eKT
D(τ −T )Q−1
D eKD(τ −T ))] dτ
where tr(·) stands for the trace of a matrix. We consider that
eF T (τ)Q−1
F eF(τ) −eF T (τ −T )Q−1
F eF(τ −T )
=
[ eF T (τ)Q−1
F eF(τ) −eF T (τ)Q−1
F eF(τ −T )]
+[ eF T(τ)Q−1
F eF(τ −T ) −eF T (τ −T )Q−1
F eF(τ −T )]
=
−eF T(τ)Q−1
F δF(τ) −eF T (τ −T )Q−1
F δF(τ)
=
−(2 eF T(τ) + δF(τ))Q−1
F δF(τ)
⩽
−2 eF T(τ)Q−1
F δF(τ)
=
−2 eF T(τ)[ε(τ) −β(τ)F(τ) + QT
r δξr(τ)]
(29)
Then, similarly, we have
tr[ eKT
S (τ)Q−1
S
eKS(τ) −eKT
S (τ)(τ −T )Q−1
S
eKS(τ −T )]
⩽−2tr{ eKT
S (τ)[ε(τ)xT (τ) −β(τ)KS(τ)]}
tr[ eKT
D(τ)Q−1
d
eKD(τ) −eKT
D(τ −T )Q−1
D eKD(τ −T )]
⩽−2tr[ eKT
D(τ)(ε(τ) ˙xT (τ) −β(τ)KD(τ))]
(30)
Substituting Ineqs. (29) and (30) into Eq.(28) and considering
Ineq. (27) yields
δJr + δJc ⩽
Z t
t−T
−δξT
r Lδξr −eF T (ε −βF)
(31)
−tr[ eKT
S (εxT −βKS)] −tr[ eKT
D(ε ˙xT −βKD)] dτ .
The rest is to deal with the residual in the above inequality,
which is similar to that in [2]. For completeness, we show
the outline in the following. In particular, we consider the
time derivative of Je
˙Je
=
εT M(q, ˙q) ˙ε + 1
2εT ˙M(q, ˙q)ε
=
εT M(q, ˙q) ˙ε + 1
2εT C(q)ε
(32)
as [5]
zT ˙Mz ≡zT Cz
∀z .
(33)
Considering the closed-loop dynamics Eq.(11), above equa-
tion can be written as
˙Je(t) ≡εT ( eF T + eKT
S x + eKT
D ˙x −Γε) .
(34)
Integrating ˙Je from t −T to t and considering Ineq. (31),
we obtain
δJ
=
δJc + δJr + δJe
⩽
Z t
t−T
−εT Γε −δξT
r Lδξr
+β[ eF T F + tr( eKT
S KS + eKT
DKD)] dτ
=
Z t
t−T
−εT Γε −δξT
r Lδξr
−β[ eF T eF + tr( eKT
S eKS + eKT
D eKD)]
+β[ eF T F ∗+ tr( eKT
S K∗
S + eKT
DK∗
D)] dτ . (35)
A sufficient condition for δJ ⩽0 is
λΓ∥ε∥2 + λL∥δξr∥2 + β(∥eF∥2 + ∥eKS∥2 + ∥eKD∥2)
−β(∥eF∥∥F ∗∥+ ∥eKS∥∥K∗
S∥+ ∥eKD∥∥K∗
D∥) ≥0 .(36)
where λΓ and λL are the minimal eigenvalues of Γ and L,
respectively. Therefore, ∥ε∥, ∥δξr∥, ∥eF∥, ∥eKS∥and ∥eKD∥
are bounded. In particular, they satisfy
λΓ∥ε∥2 + λL∥δξr∥2 + β
2 (∥eF∥2 + ∥eKS∥2 + ∥eKD∥2)
≤β
2 (∥F ∗∥2 + ∥K∗
S∥2 + ∥K∗
D∥2) .
(37)
By choosing large λΓ and λL, ∥ε∥and ∥δξr∥can be made
small.
B. Proof for minimisation of overall cost when neglecting
damping
Consider the cost function
J′
r ≡1
2
Z t
t−T
(xr −xd)T K∗T
S QT
r (xr −xd) dτ .
(38)
Following similar procedures to Ineqs. (26), (27), we obtain
δJ′
r ⩽
Z t
t−T
[−LT δxr + Qr( eF + eKSxr)]T δxr dτ
(39)
Considering further the cost function
J′
c ≡1
2
Z t
t−T
eF T Q−1
F eF + vecT ( e
KS)Q−1
S vec( eKS) dτ . (40)
and following similar procedures from Ineqs.(28) to (31), we
obtain
δJ′
r + δJ′
c
(41)
⩽
Z t
t−T
−δxT
r Lδxr −eF T (ε −βF) −tr[ e
KT
S (ε xT −βKS)] dτ .
The rest is similar to the case with damping and thus
omitted.
REFERENCES
[1] E Burdet, A Codourey and L Rey (1998), Experimental evaluation of
nonlinear adaptive controllers. IEEE Control Systems Magazine 18(2):
39-47.
[2] C Yang, G Ganesh, S Haddadin, S Parusel, A Albu-Schaeffer and
E Burdet (2011), Human-like adaptation of force and impedance in
stable and unstable interactions. IEEE Transaction on Robotics 27(5):
918-30.
[3] KJ Astrom and B Wittenmark (1995), Adaptive control. Reading, MA:
Addison-Wesley.
[4] S Jung, TC Hsia and RG Bonitz (2001), Force tracking impedance
control for robot manipulators with an unknown environment: theory,
simulation, and experiment. The International Journal of Robotics
Research 20(9) 765-74.
[5] C de Wit, B Siciliano and G Bastin (1996), Theory of robot control.
Springer.
arXiv:1605.07834v1  [cs.RO]  25 May 2016
Dynamic analysis of simultaneous adaptation of force, impedance and
trajectory
Y. Li and E. Burdet
When carrying out tasks in contact with the environment,
humans are found to concurrently adapt force, impedance
and trajectory. Here we develop a robotic model of this
mechanism in humans and analyse the underlying dynamics.
We derive a general adaptive controller for the interaction
of a robot with an environment solely characterised by its
stiffness and damping, using Lyapunov theory.
I. SYSTEM DYNAMICS
The dynamics of a n-degree-of-freedom (n-DOF) robot in
the operational space are given by
M(q) ¨x + C(q, ˙q) ˙x + G(q) = u + f
(1)
where x is the position of the robot and q the vector of
joints angle. M(q) denotes the inertia matrix, C(q, ˙q) ˙x the
Coriolis and centrifugal forces, and G(q) the gravitational
force, which can be identified using e.g. nonlinear adaptive
control [1]. u is the control input and f the interaction force.
In [2], we have described the control input u in two parts:
u = v + w ,
(2)
with v to track the reference trajectory xr by compensating
for the robot’s dynamics, i.e.
v = M(q) ¨xe + C(q, ˙q) ˙xe + G(q) −Γε
(3)
where
˙xe = ˙xr −αe ,
e ≡x −xr ,
α > 0 ,
(4)
Γ a symmetric positive-definite matrix with minimal eigen-
value λmin(Γ) ⩾λΓ > 0 and
ε ≡˙e + α e
(5)
the tracking error. w is to adapt impedance and force in
order to compensate for the unknown interaction dynamics.
II. FORCE AND IMPEDANCE ADAPTATION
Suppose that the interaction force can be expanded as
f = F ∗
0 + K∗
S(x −x∗
0) + K∗
D ˙x ,
(6)
where the force F ∗
0 (t), stiffness K∗
S(t) and damping K∗
D(t)
are feedforward components of the interaction force, x∗
0(t)
is the rest position of the environment visco-elasticity and
all of these functions are unknown but periodic with T :
F ∗
0 (t + T )
≡
F ∗
0 (t) ,
K∗
S(t + T ) ≡K∗
S(t) ,
(7)
K∗
D(t + T )
=
K∗
D(t) ,
x∗
0(t + T ) = x∗
0(t) .
(8)
The authors are with the Department of Bioengineering, Imperial College
of Science, Technology and Medicine, London SW72AZ, UK.
To simplify the analysis, we rewrite the interaction force as
f ≡F ∗+ K∗
S x + K∗
D ˙x
(9)
where F ∗≡F ∗
0 −K∗
Sx∗
0 is also periodic with T . w in Eq.(2)
is then defined as
w = −F −KSx −KD ˙x
(10)
where KS and KD are stiffness and damping matrices,
respectively, and F is the feedforward force.
By substituting the control input u into Eq.(1), the closed-
loop system dynamics are described by
M(q) ˙ε + C(q, ˙q) ε + Γε = eF + eKS x + eKD ˙x ,
(11)
eF ≡F ∗−F ,
eKS ≡K∗
S −KS ,
eKD ≡K∗
D −KD .
In this equation, we see that the feedforward force F,
stiffness KS and damping KD ensure contact stability by
compensating for the interaction dynamics. Therefore, the
objective of force and impedance adaptation is to minimise
these residual errors which can be carried out through
minimising the cost function
Jc(t)
≡
1
2
Z t
t−T
eF T Q−1
F eF + vecT ( eKS)Q−1
S vec( e
KS)
+vecT ( eKD)Q−1
D vec( eKD) dτ ,
(12)
where QF, QS and QD are symmetric positive-definite
matrices, and vec(·) stands for the column vectorization
operation. This objective is achieved through the following
update laws:
δF(t)
≡
F(t) −F(t −T ) ≡QF [ε(t) −β(t)F(t)]
(13)
δKS(t)
≡
KS(t) −KS(t −T ) = QS[ε(t)x(t)T −β(t)KS(t)]
δKD(t)
≡
KD(t) −KD(t −T ) = QD[ε ˙x(t)T −β(t)KD(t)]
where F, KS and KD are initialised as zero matrices/vectors
with proper dimensions for t ∈[0, T ).
Now that we have dealt with the interaction dynamics,
stable trajectory control can be obtained by minimising the
cost function
Je(t) ≡1
2ε(t)T M(q) ε(t) .
(14)
Consequently, we use a combined cost function Jce ≡Jc +
Je that yields concurrent minimisation of tracking error and
control effort.
III. TRAJECTORY ADAPTATION
In a typical interaction task, the contact between the
robot and the environment is maintained through a desired
interaction force Fd. Assuming that there exists a desired
trajectory xd yielding Fd, i.e. from Eq.(6)
Fd
=
F ∗
0 + K∗
S(xd −x∗
0) + K∗
D ˙xd
(15)
=
F ∗+ K∗
S xd + K∗
D ˙xd ,
F ∗= F ∗
0 −K∗
S x∗
0 ,
we propose to adapt the reference xr in order to track xd.
However, xd is unknown as the parameters F ∗, K∗
S and K∗
D
in the interaction force are unknown. Nevertheless, we know
that xd is periodic with T as F ∗, K∗
S and K∗
D are periodic
with T and we also set Fd to be periodic with T .
In the following, we develop an update law to learn the
desired trajectory xd. First, we define
ξd ≡K∗
S xd + K∗
D ˙xd ,
ξr ≡KS xr + KD ˙xr .
(16)
Then, we develop the following update law
δξr(t) ≡ξr −ξr(t −T ) ≡L−T Qr(Fd(t) −F(t) −ξr(t))
(17)
where Qr and L are positive-definite constant gain matrices.
This update law minimises the error between ξd and ξr,
which is described by the following cost function
Jr ≡1
2
Z t
t−T
(ξr −ξd)T QT
r (ξr −ξd) dτ .
(18)
Because of the coupling of adaptation of force and
impedance and trajectory adaptation, we modify the adap-
tation of feedforward force Eq.(13) to
δF(t) ≡QF[ε(t) −β(t)F(t) + QT
r δξr(t)] .
(19)
As a result, update laws Eqs.(17) and (19) minimise the
overall cost J = Jc + Je + Jr as shown in Appendix A.
Then, we obtain the update law for trajectory adaptation
δxr ≡xr(t) −xr(t −T )
(20)
by solving
δξr = KS δxr + KD δ ˙xr = KS δxr + KD
d
dt(δxr)
(21)
using δξr(t) from Eq.(17). According to the convergence of
δξr, KS and KD as shown in Appendix A, xr will converge,
as
δξr −ξd = KSδxr + KDδ ˙xr ,
(22)
Upon convergence, the desired interaction force Fd is main-
tained between the robot and the environment according to
Eq.(17). At the same time, the properties with adaptation of
force and impedance are preserved which include trajectory
tracking and control effort minimisation. However, from the
analysis in Appendix A, we cannot draw the conclusion
that F, KS, KD and xr converge to F ∗, K∗
S, K∗
D and xd,
respectively, which will require the condition of persistent
excitation (PE), similar to classical adaptive control theory
[3].
IV. DISCUSSION
A. No contact
In a special case when there is no force applied by the
environment and Fd is also zero, the controller component
w will converge to zero. According to the update law Eq.(17),
the reference trajectory will not adapt, as expected.
B. No damping
If we neglect the damping component in the interaction
force f of Eq.(9), the trajectory adaptation described by
Eqs.(17) and (21) can be simplified to
δxr = L−T Qr(Fd −F −KS xr)
(23)
Correspondingly, the update laws for force and impedance
Eq.(13) needs to be modified as
δF
≡
QF (ε −βF + QT
r δxr) ,
(24)
δKS
≡
QS(ε xT −βKS + xT
r QT
r δxr) .
The stability analysis is similar to the case with damping and
is briefly explained in Appendix B.
C. Force sensing
As in [2], force sensing is not required in the proposed
framework, in contrast to traditional methods for surface
following where the force feedback is used to regulate the
interaction force [4].
In particular, in a first phase force and impedance adap-
tation is used to compensate for the interaction force from
the environment. During this process, the unknown actual
interaction force is estimated when the tracking error ε goes
to zero as can be seen from Eq.(11): when ε = 0, we have
w = −f.
(25)
Using this estimated interaction force, then a desired force
in Eq.(15) can be rendered by adaptation of the reference
trajectory xr.
In this sense, it is important to note that trajectory adap-
tation should be conducted only when force and impedance
adaptation takes effect, which guarantees compensation of
the interaction force and tracking of the current reference
trajectory. Nevertheless, as shown in above stability analysis,
adaptation of force, impedance and trajectory can be realised
simultaneously.
This also suggests that a force sensor should be used if
available, as force and impedance adaptation could then be
replaced by force feedback. In this way, trajectory adaptation
would not depend on the force estimation process and can in
principle happen faster than force and impedance adaptation
is needed. However, the potential advantages of a force
sensor depends on the quality of the signal it could provide,
its cost and the complexity of its installation and use.
V. APPENDIX
A. Proof for minimisation of overall cost J
Considering the definition of Jr in Eq. (18), we have
δJr(t) ≡Jr(t) −Jr(t −T )
=
1
2
Z t
t−T
[ξr(τ) −ξd(τ)]T QT
r [ξr(τ) −ξd(τ)] dτ
−1
2
Z t
t−T
[ξr(τ) −ξd(τ)]T QT
r [ξr(τ −T ) −ξd(τ −T )] dτ
+1
2
Z t
t−T
[ξr(τ) −ξd(τ)]T QT
r [ξr(τ −T ) −ξd(τ −T )] dτ
−1
2
Z t
t−T
[ξr(τ −T ) −ξd(τ −T )]T QT
r ×
[ξr(τ −T ) −ξd(τ −T )] dτ
=
1
2
Z t
t−T
[ξr(τ) −ξd(τ)]T QT
r δξr(τ) dτ
+1
2
Z t
t−T
[ξr(τ −T ) −ξd(τ −T )]T QT
r δξr(τ) dτ
=
Z t
t−T
[ξr −ξd −1
2δξr]T QT
r δξr dτ
(as ξd(t) = ξd(t −T ))
⩽
Z t
t−T
[Qr(ξr(τ) −ξd(τ))]T δξr(τ) dτ .
(26)
According to Eqs.(15) to (17), we rewrite this inequality as
δJr
⩽
Z t
t−T
[Qr(ξr −Fd + F + eF)]T δξr dτ
=
Z t
t−T
(−LT δξr + Qr eF)T δξr dτ.
(27)
Consider the difference between Jc of two consecutive
periods
δJc ≡Jc −Jc(t −T )
(28)
=
1
2
Z t
t−T
[( eF T Q−1
F eF −eF T (τ −T )Q−1
F eF(τ −T ))
+tr( eKT
S Q−1
S
eKS −eKT
S (τ −T )Q−1
S
eKS(τ −T )
+( eKT
DQ−1
D eKD −eKT
D(τ −T )Q−1
D eKD(τ −T ))] dτ
where tr(·) stands for the trace of a matrix. We consider that
eF T (τ)Q−1
F eF(τ) −eF T (τ −T )Q−1
F eF(τ −T )
=
[ eF T (τ)Q−1
F eF(τ) −eF T (τ)Q−1
F eF(τ −T )]
+[ eF T(τ)Q−1
F eF(τ −T ) −eF T (τ −T )Q−1
F eF(τ −T )]
=
−eF T(τ)Q−1
F δF(τ) −eF T (τ −T )Q−1
F δF(τ)
=
−(2 eF T(τ) + δF(τ))Q−1
F δF(τ)
⩽
−2 eF T(τ)Q−1
F δF(τ)
=
−2 eF T(τ)[ε(τ) −β(τ)F(τ) + QT
r δξr(τ)]
(29)
Then, similarly, we have
tr[ eKT
S (τ)Q−1
S
eKS(τ) −eKT
S (τ)(τ −T )Q−1
S
eKS(τ −T )]
⩽−2tr{ eKT
S (τ)[ε(τ)xT (τ) −β(τ)KS(τ)]}
tr[ eKT
D(τ)Q−1
d
eKD(τ) −eKT
D(τ −T )Q−1
D eKD(τ −T )]
⩽−2tr[ eKT
D(τ)(ε(τ) ˙xT (τ) −β(τ)KD(τ))]
(30)
Substituting Ineqs. (29) and (30) into Eq.(28) and considering
Ineq. (27) yields
δJr + δJc ⩽
Z t
t−T
−δξT
r Lδξr −eF T (ε −βF)
(31)
−tr[ eKT
S (εxT −βKS)] −tr[ eKT
D(ε ˙xT −βKD)] dτ .
The rest is to deal with the residual in the above inequality,
which is similar to that in [2]. For completeness, we show
the outline in the following. In particular, we consider the
time derivative of Je
˙Je
=
εT M(q, ˙q) ˙ε + 1
2εT ˙M(q, ˙q)ε
=
εT M(q, ˙q) ˙ε + 1
2εT C(q)ε
(32)
as [5]
zT ˙Mz ≡zT Cz
∀z .
(33)
Considering the closed-loop dynamics Eq.(11), above equa-
tion can be written as
˙Je(t) ≡εT ( eF T + eKT
S x + eKT
D ˙x −Γε) .
(34)
Integrating ˙Je from t −T to t and considering Ineq. (31),
we obtain
δJ
=
δJc + δJr + δJe
⩽
Z t
t−T
−εT Γε −δξT
r Lδξr
+β[ eF T F + tr( eKT
S KS + eKT
DKD)] dτ
=
Z t
t−T
−εT Γε −δξT
r Lδξr
−β[ eF T eF + tr( eKT
S eKS + eKT
D eKD)]
+β[ eF T F ∗+ tr( eKT
S K∗
S + eKT
DK∗
D)] dτ . (35)
A sufficient condition for δJ ⩽0 is
λΓ∥ε∥2 + λL∥δξr∥2 + β(∥eF∥2 + ∥eKS∥2 + ∥eKD∥2)
−β(∥eF∥∥F ∗∥+ ∥eKS∥∥K∗
S∥+ ∥eKD∥∥K∗
D∥) ≥0 .(36)
where λΓ and λL are the minimal eigenvalues of Γ and L,
respectively. Therefore, ∥ε∥, ∥δξr∥, ∥eF∥, ∥eKS∥and ∥eKD∥
are bounded. In particular, they satisfy
λΓ∥ε∥2 + λL∥δξr∥2 + β
2 (∥eF∥2 + ∥eKS∥2 + ∥eKD∥2)
≤β
2 (∥F ∗∥2 + ∥K∗
S∥2 + ∥K∗
D∥2) .
(37)
By choosing large λΓ and λL, ∥ε∥and ∥δξr∥can be made
small.
B. Proof for minimisation of overall cost when neglecting
damping
Consider the cost function
J′
r ≡1
2
Z t
t−T
(xr −xd)T K∗T
S QT
r (xr −xd) dτ .
(38)
Following similar procedures to Ineqs. (26), (27), we obtain
δJ′
r ⩽
Z t
t−T
[−LT δxr + Qr( eF + eKSxr)]T δxr dτ
(39)
Considering further the cost function
J′
c ≡1
2
Z t
t−T
eF T Q−1
F eF + vecT ( e
KS)Q−1
S vec( eKS) dτ . (40)
and following similar procedures from Ineqs.(28) to (31), we
obtain
δJ′
r + δJ′
c
(41)
⩽
Z t
t−T
−δxT
r Lδxr −eF T (ε −βF) −tr[ e
KT
S (ε xT −βKS)] dτ .
The rest is similar to the case with damping and thus
omitted.
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