Proceedings of the 2006 IEEE International Conference on Robotics and Automation
Orlando, Florida - May 2006
Power Assist System for Sinusoidal Motion
by Passive Element and Impedance Control
Mitsunori Uemura, Katsuya Kanaoka and Sadao Kawamura
Department of Robotics, Ritsumeikan University
Shiga 525-8577, Japan
Email: rr002993@se.ritsumei.ac.jp, kanaoka@se.ritsumei.ac.jp, kawamura@se.ritsumei.ac.jp
Abstract— In this paper, we propose a power assist system
that amplifies sinusoidal human’s torque and attains minimization
of control input requirement using an impedance control and
resonance. This impedance control is designed to realize the
following features: 1) Sinusoidal torque amplification. 2) Minimization of control input requirement by adjusting stiffness. 3) No
requirement for the knowledge of amplitude, frequency and phase
of human’s torque. 4) No requirement for myoelectric signals. 5)
Satisfaction of causality. Convergence of the proposed controller
is proven theoretically. Simulation results verify the validity of
the control scheme including some conditions not discussed in
the theory. Experimental results show the validity of the control
system including real human’s complex dynamics and feedback
loops.
I. I NTRODUCTION
Recently, a lot of human’s force/torque amplification systems
are proposed [1-6] in order to support the elderly, the physically
challenged and people carrying heavy loads.
A. Existing Studies
Kawamoto et al. proposed ”HAL” [1] which enables human
to walk with less effort than usual. This system uses myoelectric
sensors like Fig.1 to estimate human’s motion intention and to
construct a control input. Electric motors with high reduction
gears and high capacity batteries are used to generate enough
torque required for walking assist.
Chu et al. proposed ”BLEEX” [2] which supports people
carrying heavy loads. This system intends to reduce interaction
force by amplifying the interaction force like Fig.2. This type
of systems has to sense the interaction force or equivalent ones
for their feedback controllers. BLEEX uses filtered acceleration
signal to estimate it. Hydraulic actuators and fuel-based engine
are used to amplify the force.
B. Problems in Walking Power Assist Systems
For further usability and practicality, there are some problems
that must be solved for walking power assist systems.
Electric motors with high reduction gears have relatively
large mass and friction [3]. Presenting low impedance in
order not to impede human’s motion has been a challenge
in most power assist systems [4]. High impedance of actuator
systems possibly worsens energy efficiency due to large control
input requirement. Myoelectric controllers are suffered from
its drawbacks [5] such as significant noise [7]. Force/torque
tracking control is rarely discussed in power assist studies.
C. Prostheses
On the other hand, lower limb prostheses are widely used in
practical level [8].
These systems mainly use not active actuators but passive
elements. Thus they can remove excessive dependence on energy sources. Passive elements cannot generate arbitrary force
and they are usually not controlled precisely, but necessary
functions for walking can be realized by well construction of
them. Therefore their users can walk naturally by making full
use of them.
D. Studies based on Passive Walking
A framework based on ”passive walk”, which recently comes
under the spot light, has similar concepts [9].
They utilize characteristics of dynamics so as to realize a
natural and energy efficient walking. They achieved about 10
times energy efficiency compared to humanoids, which uses
electric motors with high reduction gears [9].
Load
Human
Load
Myoelectric
Sensor
Amplifed Force
Human
Interaction
Force
Assist Force
Robot
Fig. 1.
Robot
Power Assist System using Myoelectric Signals
0-7803-9505-0/06/$20.00 ©2006 IEEE
Fig. 2.
3935
Power Assist System Amplifying Interaction Force
E. Support Systems for Hip Joint
Because walking motion requires energy supply, constructing
only passive elements for whole leg support attaining a natural
walking seems to be difficult. In fact, active above-knee prostheses realize a more natural walking than passive ones [10].
Energy analysis [2] using Clinical Gait Analysis (CGA) data
[11] shows that human generates energy on hip joint during
walking. Hence active support systems can be more effective
than passive ones for hip joint support.
On the other hand, dynamic characteristics of hip joint can be
pointed out. Torque and angular motion of hip joint are nearly
sinusoidal. This characteristic is easy to deal with for feedback
controllers. The characteristic is observed in CGA data [11].
generated by the human and τr is actuator’s torque [Nm]. τh is
resultant torque composed of muscle activations, viscoelasticity
of muscles and so on.
B. Assumptions
For constructing a controller, dynamics of hip joint is simplified by setting following assumptions.
One is to assume τn includes elastic torque arose from
gravity and so on. The other is to assume τn includes viscous
torque. These characteristics can be observed in energy analysis
of hip joint in real human walking [2]. Therefore we assume
τn to be composed of viscoelastic torque.
τn = −kh q − dq̇
(2)
where kh is stiffness [Nm/rad] and d is viscosity [Nms/rad].
Thus overall dynamics is given like this.
F. This Paper
In this study, we adopt similar concepts to prostheses and
passive walking based systems to solve some problems of
power assist systems in a certain condition. Utilizing resonance
enables minimization of control input requirement for energy
efficiency. Then, we can select relatively small actuators and
actuator system can have relatively small impedance. Restricting control purpose to sinusoidal torque amplification makes it
easy to achieve torque tracking control without use of noisy
signals. This restriction disables arbitrary torque amplification,
but necessary function for walking assist on hip joint can be
achieved because hip joint mostly requires sinusoidal torque for
walking. Therefore it can be expected that its users can walk
naturally by making full use of it.
Some assumptions are introduced to construct a controller
and we design an impedance control so that torque amplification and stiffness adjustment to minimize control input are
simultaneously realized. Convergence of the impedance control
is proven theoretically.
We conduct numerical simulations to verify the validity of
the proposed scheme and to confirm behaviors in some extra
conditions not discussed in the theory.
Experiments are also conducted to verify the validity of the
control system including real human complexities.
II. P ROBLEM F ORMULATION
In this section, problems of torque amplification on hip joint
are simplified and formulated.
I q̈
= −dq̇ − kh q − kq + τh + τr
It is assumed I, d, kh to be known and q, q̇ to be measured with
adequate accuracy.
The last assumption is human’s torque τh is sinusoidal.
τh
= a sin(ωt + φ)
(4)
where a is amplitude [Nm], ω is angular frequency [rad/s],
t is time [s] and φ is phase [rad]. This characteristic can be
observed in CGA data [11].
τh can be calculated from (3) as follows, but its coefficients
a, ω, φ are assumed to be unknown.
τh = I q̈ + dq̇ + kh q + kq − τr
(5)
This calculated τh can’t be used as control input, because (5)
includes the acceleration signal.
C. Objectives
Then, our objectives are defined by two formulations. One is
to amplify the human torque τh by designing the control input
τr . The other is to minimize the control input by adjusting the
stiffness k.
III. I MPEDANCE C ONTROL
This section shows details of the proposed control scheme.
According to the previous section, we construct a controller
to a system having the dynamics (3) and the human’s torque is
assumed to be sinusoidal (4).
A. Dynamics
Structure of a hip joint power assist system is shown in
Fig.3. The spring is introduced for utilizing resonance and its
stiffness is assumed to be adjustable. Some adjustment methods
of stiffness have been proposed [12].
Dynamics of the hip joint can be described as follows.
I q̈ = τn − kq + τh + τr
(3)
(1)
where I is inertia moment [kgm2 ] composed of the human
and the system, q is joint angle [rad] of the hip joint, τn is
torque [Nm] generated by non-linear effects including gravity,
coriolis and torque from other joints, k is programmable
stiffness [Nm/rad] exerted by the spring, τh is joint torque [Nm]
3936
Human
Link 1 of
Power Assist
System
Actuator
Hip Joint
Spring
Link 2 of
Power Assist
System
Fig. 3.
Structure of Hip Joint Power Assist System Worn by Human
A. Controller Design
Firstly, control input τr is designed to amplify the human
torque τh .
τr = kpa τ̂h
(6)
where kpa is an amplification gain [-] and τ̂h is the estimated
value of τh [Nm].
In consideration of causality, τr is constructed using the
estimated value of τh . This estimation is done using a technique
of adaptive observer [13].
kpa
1
τ̂˙h = −
(q̇ − α) + η̂ + λτh + (k + kh )ξ
γr
I
+β1 (τh − τ̂h ) + β2 (τhi − τ̂hi )
(7)
(8)
η̂˙ = −λη̂ − λ2 τh
(9)
ξ˙ = −λξ − τh
(1 + kpa )
τh
(10)
d
where β1 , β2 , λ, γr are
t gain [-], τh in
t(7)(8)(9)(10) is calculated
from (5) and τhi = 0 τh dt, τ̂hi = 0 τ̂h dt.
τ̂˙h (7) includes acceleration signal in the terms of τh , but τ̂h
does not depend on acceleration signal owing to integral effect.
Hence the control input (6) satisfies causality.
α =
B. Adjustment of Stiffness
Next, adjustment law of the stiffness is designed using
techniques of adaptive observer [13] and impedance control
[14].
1
k̇ = γk γr (q̇ − α)q + ξ(τh − τ̂h )
(11)
I
where γk is adaptation gain [-].
k also does not depend on acceleration signal.
D. Optimality of Resonant Condition
To calculate control input τr at the steady state, τh = a sin ωt
1+k
and q̇ = d pa a sin ωt are substituted into (3).
Iω
∴ τr
η̇
I
Iω 2 2
1
Δq̇ 2 +
Δq +
Δk 2
2
2
2γk
γr
γr β2
1
2
Δτhi
+ Δτh2 +
+ Δη 2
2
2
2
−λη − λ2 τh
=
=
kpa a sin ωt
k + kh 1 + kpa
a cos ωt
+ Iω −
ω
d
(16)
IV. S IMULATION
We conducted numerical simulations in order to verify the
validity of the proposed control scheme.
Since assumptions of human’s torque τh in the proposed
controller are too restricted, we conducted simulations with
following extra conditions, which aren’t discussed in the theory.
•
•
•
•
Amplitude, frequency and phase of τh are changed
halfway.
Wave shape of τh is not sinusoidal.
τh includes bias term.
τh includes set-point feedback control.
A. Method
(12)
(13)
2
where, Δk = Iω − kh − k, Δq̇ = q̇ − α, Δτh = τh − τ̂h ,
Δη = η − η̂ and Δτhi = τhi − τ̂hi .
Then time derivative of V is given like this.
V̇ = −dΔq̇ 2 + γr ΔηΔτh − β1 γr Δτh2 − λΔη 2
=
1 + kpa
a sin ωt
d
(k + kh )(1 + kpa )
+
a cos ωt
ω
+a sin ωt + τr
(15)
−d
Obviously amplitude of τr is minimized when k = Iω 2 − kh
(resonant condition).
Then, τr and q̇ will have the same phase. This means work
done by the actuator τr q̇ will be entirely positive. Hence
unnecessary negative work is removed in resonant condition.
This technique of removing negative work is utilized for energy
efficiency in studies based on passive walking [9]. Therefore
we can expect high energy efficiency in resonant conditions.
C. Proof of Convergence
The following Lyapnov function candidate V can be defined
to prove convergence of the proposed controller.
V
1 + kpa
a cos ωt =
d
(14)
V̇ can be negative semi definite by adequate choice of
β1 , λ, γr . Therefore, q̇ → α, τ̂h → τh , η̂ → η are proven when
t → ∞. This means τr → kpa τh and torque amplification will
be attained.
Next, to consider a maximum invariant set, q̇ = α, τ̂h = τh ,
η̂ = η are substituted into (3). Then, steady state is attained
only if k = Iω 2 − kh . Therefore, k → Iω 2 − kh is proven
when t → ∞ and automatic adjustment of stiffness will be
attained.
Overall dynamics is assumed to be (3) with following
physical parameters: I = 1.0[kgm2], d = 2.5[Nms/rad] and
kh = 50[Nm/rad].
(6) to (11) are adopted as control scheme with following
parameter setting: kpa = 4.0[-], β1 = 50.0[-], β2 = 50.0[-],
λ = 150.0[-], γr = 1.0[-] and γk = 150.0[-].
B. Conditions
Four types of simulations summarized in Table I are conducted with each setting of τh .
In the case 1, τh is set to be 3.0 sin 2.0πt in the first 10.0[s].
Frequency of τh is changed at 10.0[s]. Amplitude and phase
are changed at 20.0[s]. In the case 2, τh is set to be rectangular
wave with amplitude 3.0 and frequency 2.0[Hz]. In the case 3,
τh is set to include bias term like 3.0 sin 2.0πt + 1.0. In the
case 4, τh is changed into PD control at 6.0[s].
3937
τr
kpa τh
20
τr [Nm]
10
0
-10
-20
0
5
10
15
(a) τr in the case 1
20
25
t[s]
20
25
t[s]
k[Nm/rad]
300
k
Iω 2 − kh
200
100
5
10
τr
kpa τh
20
τr [Nm]
0
-10
10
10
0
t[s]
(c) τr in the case 2
0
k
Iω 2 − kh
(f) τr in the case 3
200
100
0
200
5
(d) k in the case 2
t[s]
q[rad]
-0.5
(g) k in the case 3
5
(e) q in the case 2
t[s]
0
q
0.0
t[s]
5
(j) k in the case 4
q
0.5
-0.5
0
200
t[s]
5
0.5
0.0
k
Iω 2 − kh
100
0
q
0.5
t[s]
5
(i) τr in the case 4
300
100
0
0.0
-0.5
0
5
(h) q in the case 3
Fig. 4.
In the first 10[s] in the case 1, τr converged to kpa τh within
4[s] as shown in Fig.4(a). k converged to Iω 2 − kh within 5[s]
as shown in Fig.4(b). In the next 20[s], τr and k also converged
to desired ones within 6[s] after a, ω, φ of τh were changed.
In the case 2, τh did not converge completely to kpa τh , but a
certain degree of convergence to kpa τh was obtained as shown
in Fig.4(c). k converged to vicinity of Iω 2 − kh with small
oscillation as shown in Fig.4(d). Fig.4(e) shows q converged
TABLE I
S IMULATION C ONDITIONS
Overview
Amplitude, frequency and phase of τh were changed halfway
τh was rectangular wave
τh included bias term
τh was changed into PD control halfway
t[s]
0
5
(k) q in the case 4
t[s]
Simulation results
C. Results
case
1
2
3
4
-20
k
Iω 2 − kh
300
k[Nm/rad]
300
0
t[s]
5
k[Nm/rad]
5
τr
kpa τh
-10
τr
kpa τh
-20
0
k[Nm/rad]
20
-10
-20
q[rad]
20
q[rad]
τr [Nm]
10
15
(b) k in the case 1
τr [Nm]
0
to a motion like a sinusoid. In the case 3, τr almost converged
to a biased sinusoid kpa τh as shown in Fig.4(f). k converged
to vicinity of Iω 2 − kh like the case 2 as shown in Fig.4(g).
Fig.4(h) shows q converged to a slightly biased sinusoid by
effects of biased τh and τr . While τh was PD control in the
case 4, τr did not track kpa τh but did not become completely
different from τh as shown in Fig.4(i). k was slightly changed
after the change of τr as shown in Fig.4(j). Fig.4(k) shows the
convergence of q to origin within 2[s] after τr was changed
into PD control.
D. Discussion
Torque amplification and adjustment of stiffness were verified when τh was composed of only sinusoid. Even when τh
included non-sinusoidal term, we can expect enough amplification effect and optimality of resonance, because τr and k
3938
converged to vicinity of desired ones. Instability did not occur
even when τh included feedback control.
Hence the validity of the proposed control scheme was verified and effectiveness was confirmed in some extra conditions
not discussed in the theory.
V. E XPERIMENT
This section shows experimental verifications of the proposed
scheme using 1-dof human arm power assist system as shown
in Fig.5 and Fig.6.
Because real human includes complex dynamics and feedback loops, assumptions regarding human’s torque τh in the
theory may be unrealistic. Thus behaviors of the proposed
controller are experimentally verified. An experiment is also
conducted to verify whether the user can stop the motion while
the control scheme is applied.
The reason why we adopted a simple power assist system as
a experimental system is to confirm behaviors and safety of the
proposed controller. We are planning to apply the method to
human walking assist systems after the adequate effectiveness
and safety will be confirmed.
A. Method
(6) with gravity compensation of the robot and the human
arm was adopted as a control input of the joint actuator. (7) to
(11) were adopted as a torque estimation and an adjustment law
of the stiffness. q and q̇ were measured by a optical encoder
installed in the joint actuator. −kh q and −kq in (3) were
emulated by the control input with parameter setting: kh = 1.0.
Parameters were set to be kpa = 2.0[-], β1 = 12.0[-], β2 =
36.0[-], λ = 3.0[-], γr = 100.0[-] and γk = 0.02[-]. The reason
why kpa was small was the system went unstable when kpa was
more large. All physical parameters were roughly calibrated
beforehand as I = 0.05[kgm2] and d = 0.4[Nms/rad].
B. Conditions
We conducted 2 experiments with following conditions summarized in Table II.
In the case 1, the subject was instructed to make vertical
sinusoidal motion. The instructed motion was 1[Hz] in first
10[s], 1.5[Hz] in next 10[s] and wider range motion of 1.5[Hz]
in last 10[s]. In the case 2, the subject was instructed to make
1.5[Hz] sinusoidal vertical motion in first 6[s] and to stop the
motion in next 4[s].
C. Results
In the case 1, fr converged to vicinity of kpa fh within
2[s] after changes of motion pattern as shown in Fig.7(a).
k converged to vicinity of Iω 2 − kh as shown in Fig.7(b).
Instructed motion was almost achieved as shown in Fig.7(c).
In the case 2, the subject could stop the motion as shown in
Fig.7(f). Divergent behavior did not occurred while stopping
the motion as shown in Fig.7(d) and Fig.7(e)
D. Discussion
Torque amplification was adequately achieved despite the
control system included complexities of real human. Therefore
the effectiveness of our approach to sinusoidal torque amplification was verified. The convergence of k to Iω 2 −kh can achieve
high energy efficiency if restoring force −kq is generated by
real elastic elements.
However, the system went unstable unless gain settings were
carefully chosen. the reason of the unstability seems to be
arose from a backlash of the reduction gears. because the
experimental system is not constructed proficiently, However,
some other causes are conceivable. Therefore we are planning
to improve the experimental system, the controller design, the
gains tuning and so on.
VI. C ONCLUSION
In this paper, we proposed a power assist system that
amplifies sinusoidal human’s torque and attains minimization
of control input utilizing resonance.
The concepts of the proposed method for hip joint support
were discussed in detail.
An impedance control was designed to amplify sinusoidal
human’s torque and to adjust a stiffness to minimize control
input. Convergence of the controller and optimality of resonant
condition were proven theoretically.
Simulation results verified the validity of proposed controller.
Some extra conditions that aren’t discussed in the theory were
also simulated and effectiveness of the proposed controller was
confirmed.
Attachment
Human Arm
q
Actuator
Robot Link
Rotation Center
Fig. 5.
Experimental System
Fig. 6.
3939
Schematic of Experimental System
τr
kpa τh
τr [Nm]
2
1
0
-1
-2
k[Nm/rad]
0
5
10
15
(a) τr in the case 1
4
3
2
20
25
t[s]
20
25
t[s]
20
25
t[s]
k
Iω 2 − kh
1
0
-1
0
5
10
15
(b) k in the case 1
q[rad]
0.5
q
0.0
-0.5
5
τr
kpa τh
k[Nm/rad]
2
τr [Nm]
10
1
0
-1
-2
0
5
(d) τr in the case 2
t[s]
15
(c) q in the case 1
k
Iω 2 − kh
2
1
0
-1
0.0
-0.5
0
5
(e) k in the case 2
Fig. 7.
t[s]
0
5
(f) q in the case 2
t[s]
Experimental results
Experimental results showed the validity of the proposed
approach even when the system includes real human’s complex
dynamics and feedback loops.
In the future, we will apply the proposed method to hip joint
assist system.
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TABLE II
E XPERIMENTAL C ONDITIONS
case
1
2
q
0.5
4
3
q[rad]
0
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Overview
Amplitude and frequency of instructed motion were changed halfway
Stopped motion halfway
3940