![Muscle Force Control of a Kinematically Redundant
Bionic Arm with Real-Time Parameter Update
Haiwei Dong, Nadia Figueroa
New York University AD
Abu Dhabi, UAE
{haiwei.dong, nadia.figueroa}@nyu.edu
Abdulmotaleb El Saddik
New York University AD and University of Ottawa
Abu Dhabi, UAE and Ottawa, Canada
elsaddik@uottawa.ca
Abstract—Redundant muscle-driven arms have numerous ad-
vantages, such as increased robustness, ability for load distribu-
tion, impedance change etc. However, controlling such a muscle-
driven arm is a difficult task. This is mainly due to its redundancy,
specially when the muscle force is required to follow certain
output constraints and fulfill optimization objectives. In this
paper, a new method for controlling such muscle-like systems
is proposed. By considering both joint and muscle acceleration
contributions, a set of linear equations was constructed. Driving
muscle activation is thus framed as the only unknown vector.
To solve this linear equation set, a pseudo-inverse solution was
used. The null space within this solution represents the internal
force, which was used to evenly distribute the muscle forces,
which is considered as “anti-fatigue” way. Moreover, to make
the proposed method adaptive to modeling errors, an estimated
system model was added to represent the real model. By updating
the parameters of the estimated model based on prediction error,
the estimated model approaches the real model gradually in real
time. The overall method was tested for the case of a bending-
stretching movement. The presented results verify the validity of
the method, and illustrate its useful features and advantages.
I. INTRODUCTION
In nature, animals and human are able to move mainly
through their musculoskeletal systems. The bones constitute
a basic support structure. By linking the bones with different
kinds of joints, we effectively have a number of degrees of
freedom (DOF) that arise. Each muscle has its attachment
points on the bones and can only output the “pull” force
to drive the joint. In comparison with classical motor-driven
systems, there are several advantages for muscle-like systems.
First, as multiple muscles are tied to each joint, the total load
can be distributed, resulting in small loads for each muscle [1].
Thus, each muscle only has a small load. Second, if one muscle
is broken, the other muscles can still drive the joint to rotate
[2]. Third, bi-articular muscles can not only output joint torque
for movement, but also can change the impedance character
of the body [3]. Due to the advantages of a muscle-like
driven system compared with a classical motor-driven system,
researchers recently have been focusing on the biologically-
inspired mechanical system with many successful applications
in robotics. Some examples include: ECCEROBOT-2 Robot
[4], Biorob Arm [3], [5], musculo-skeletal arm [6], etc. Such
robots are often termed “bionic robot”. In this paper, we focus
on controlling a muscle-like bionic robot with bi-articular and
mono-articular muscles.
If we try to approach bionic robot control in the classical
way, the solution would be to initially compute the joint
torque, and then distribute the joint load to the muscles.
The advantage of such an approach is that all the existing
control methods can be used directly, such as sliding control,
impedance control, robust control, etc. Thus, what remains is
to design a proper muscle distribution principle. Dong et al.
designed a composite control where sliding control is used to
achieve robust joint torque. This torque was then distributed
to muscles by virtual work between muscles and joints [7].
The disadvantage of this method is that it artificially divided
a problem into two separate processes. Furthermore, because
the joint torque control methods are designed specifically for
classical motor-driven systems, the computed joint torque may
lead the muscle force change not smooth. Thus, it might be
worthy to follow a novel avenue: model the bionic robot arm
as a muscle-driven system considering its complex dynamics
and propose a novel method for controlling it. In this paper,
we choose to follow this avenue.
Specifically, controlling a system driven by bi-articular
and mono-articular muscles involves many issues. Four im-
portant problems are listed below. The first problem is muscle
coordination. Muscle configurations vary, therefore a muscle
may correspond to one or two joints. Furthermore, for one
specific joint, there are usually several muscles corresponding
to its movement. Coordination of these muscles can be seen
as a general control problem for over-actuated systems. Until
now, many approaches have been proposed. For example,
Yokishawa proposed a method to avoid singularities in the arm
by minimizing a manipulability measure [8]. Maciejewski et
al. used the null-space vector to aid obstacle avoidance [9].
Klug et al. developed a 3 DOF bionic robot arm which is
controlled by a PD controller with feed-forward compensation
[5]. The trajectory of the arm is optimized and adjusted for
a time and energy-optimal motion [10]. Potkonjak et al. built
a humanoid robot with antagonistic drives whose controller is
designed by loop-shaping [4].
A second important problem is enforcing muscle force
constraints. Each muscle can only provide a “pull” force.
Meaning that, the muscle force is nonnegative. In addition,
each muscle has its limit in force output. Thus, the output
force for each muscle is within an interval from 0 to maximum
output force. For many years, researchers have focused on
solving such control problems with input constraints. Tahara
proposed a simple sensor-motor control scheme for internal
force and simulated the overall stability for a cable-driven
2013 IEEE International Conference on Systems, Man, and Cybernetics
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![system [6]. Tsang applied Generalized Predictive Control
(GPC) in DC motor drive while guaranteeing the amplitude
limits of the input currency [11]. Fujimoto et al. proposed an
optimal control method bearing input constraint by interactive
learning [12]. Yamaguchi used gradient iteration to modify the
muscle force until it satisfies the constraint [13].
Yet a third important problem, is designing the system so
that it attempts to decrease muscular fatigue. This is a higher
requirement than the muscle force constraint. The challenge is
that we need to find an optimization solution that separates all
the muscle forces evenly, within the feasible solution space of
muscle forces. For this distribution method, optimization is a
common method. Anderson et al. used dynamic optimization
of minimum metabolic energy expenditure to solve the motion
control of walking [14]. Manal et al. designed a nonlinear
optimal controller to develop a real-time EMG-driven virtual
arm [15]. Neptune evaluated different multivariate optimization
methods in pedaling [16].
Finally, the fourth important problem is adaptivity. For a
real-world control plant, absolute models are not achievable
mainly due to multiple many uncertainties, such as sen-
sor uncertainty, actuator uncertainty, environment uncertainty,
modeling uncertainty, etc. Thus, the control method must have
the ability to tolerate these uncertainties or to update the
parameters of the modeled plant. This issue of adaptivity has
been considered for years. In robotics, the research on adapta-
tion comes from online system identification. The objective is
to estimate the system structure and parameters in real-time,
while the system is in operation. Here, the system is considered
as a black box. By stimulating the system and creating relations
between the inputs and outputs, the parameters of the system
can be adjusted online [17]. Based on this system identification
idea, many control methods with adaptive parameter update
have come out, such as robust control [18], adaptive feedback
control [19], neurofuzzy adaptive control [20], etc. However,
notice that the adaptation ability problem has not been consid-
ered in arm control driven by over-actuated muscles.
In this paper, we propose a new method for controlling a
muscle-driven system, that addresses all of the four important
problems previously explained. As a preparatory step for our
method, the desired joint trajectory points was connected by
a continuous derivative polynomial function for each moment;
resulting with a Jacobian matrix from muscle space to joint
space. For the muscle activation computation, by considering
the acceleration in the joint space and muscle space respec-
tively, we obtained a set of linear equations. The pseudo-
inverse solution to this linear equation set gave us the initial
result for the muscle forces. To make the muscles forces
satisfy the output force constraint and furthermore work in
the feasible output force range, we gave a gradient direction
for the muscle forces to satisfy the mentioned constraint in
the null space of the previous pseudo-inverse solution. As the
body movement load distributes evenly in all the muscles, the
proposed algorithm has a character of “anti-fatigue”. To make
the proposed method adaptive for modeling error, we added an
estimated model to represent the real model. By updating the
parameters of the estimated model based on prediction error,
the estimated model approaches the real model gradually in
real time.
The overall proposed method is tested for a bending-
Fig. 1: Bionic robot arm model.
stretching movement. The results verify the validity of the
method, as well as dealing with the four previously mentioned
problems effectively. We will start by presenting our bionic
arm mathematical model, continue by deriving the steps of
our muscle activation calculation method, and then discuss our
simulation and animation results, before providing a concise
summary and conclusion.
II. BIONIC ROBOT ARM MODEL
A. Dynamic Modeling
We built a 2-dimensional model of an arm in the horizontal
plane (no gravity) based on the upper limb structure of a digital
human. The model includes six muscles (shown as 1 to 6) and
two degrees of freedom (shoulder flexion-extension and elbow
flexion-extension). The range of the shoulder angle is from -20
to 100 degree, and the range of the elbow is from 0 to 170
degree. Four of the muscles are mono-articular, and two are
bi-articular where 1 and 2 cross the shoulder joint; 3 and 4
cross the elbow joint; 5 and 6 cross both joints (Fig.1).
Considering the arm (including upper arm and lower arm)
as a planar, two-link, articulated rigid object, the position of the
hand can be derived by a 2-vector q of two angles. The input
is a 6-vector Fm of muscle forces. The dynamics of the rigid
object are strongly nonlinear. Using the Lagrangian equations
from classical dynamics, we get the dynamic equations of the
ideal upper limb model
H11 (t) H12 (t)
H21 (t) H22 (t)
¨q1
¨q2
(1)
+
C11 (t) C12 (t)
C21 (t) C22 (t)
˙q1
˙q2
=
τ1(t)
τ2(t)
or abbreviated as
H (t) ¨q + C (t) ˙q = τ (2)
with q = [ q1 q2 ]
T
= [ θ1 θ2 ]
T
being the two joint
angles. τ = [ τ1 τ2 ] = f (Fm) is the joint torque which
is considered as a function of muscle force Fm
16411641](https://image.slidesharecdn.com/2013smcmuscleforcecontrol-140523201253-phpapp01/85/Muscle-Force-Control-of-a-Kinematically-Redundant-Bionic-Arm-with-Real-Time-Parameter-Update-2-320.jpg)
![Fm = [ Fm,1 Fm,2 Fm,3 Fm,4 Fm,5 Fm,6 ]
T
(3)
H (q, t) is the inertia matrix which contains information with
regards to the instantaneous mass distribution. C (q, ˙q, t) is
centripetal and coriolis torques representing the moments of
centrifugal forces.
H11 = J1 + J2 + m2d2
1 + 2m2d1c2 cos (q2)
H12 = H21 = J2 + m2d1c2 cos (q2)
H22 = J2
C11 = −2m2d1c2 sin (q2) ˙q2 (4)
C12 = −m2d1c2 sin (q2) ˙q2
C21 = m2d1c2 sin (q2) ˙q1
C22 = 0
where ci is the distance from the center of a joint i to the
center of gravity point of link i. di is the length of link i.
Ji = mid2
i + Ii where Ii is the moment of inertia about the
axis through the center of the mass of link i.
B. Estimated Model
In our research, we consider that the arm model in Eq.1
is influenced by modeling errors and disturbances from the
environment. Hence, we use an estimated arm model to
represent the real one, which is written as
ˆH11 (t) ˆH12 (t)
ˆH21 (t) ˆH22 (t)
¨q1
¨q2
(5)
+
ˆC11 (t) ˆC12 (t)
ˆC21 (t) ˆC22 (t)
˙q1
˙q2
=
ˆτ1(t)
ˆτ2(t)
or abbreviated as
ˆH (t) ¨q + ˆC (t) ˙q = ˆτ (6)
where ˆ· means estimated value of (·). The relation between
the ideal model (Eq.1) and the estimated model (Eq.5) is that
we choose τ = ˆτ. In this paper, we use the estimated model
to generate torque for controlling a real system. In addition,
the parameter adaptation updates the estimated parameters ˆH,
ˆCand ˆG in real-time.
For the convenience of the following derivation, we define
the actual and estimated arm parameter vector
P = PT
H PT
C
T
, ˆP = ˆPT
H
ˆPT
C
T
(7)
where
PH =
⎡
⎢
⎣
H11
H12
H21
H22
⎤
⎥
⎦ , PC =
⎡
⎢
⎣
C11
C12
C21
C22
⎤
⎥
⎦
ˆPH =
⎡
⎢
⎢
⎣
ˆH11
ˆH12
ˆH21
ˆH22
⎤
⎥
⎥
⎦ , ˆPC =
⎡
⎢
⎢
⎣
ˆC11
ˆC12
ˆC21
ˆC22
⎤
⎥
⎥
⎦
then the estimation error vector can be defined as
˜P = ˆP − P = ˜PT
H
˜PT
C
T
(8)
III. MUSCLE FORCE CALCULATION
A. Muscle Force Computation (Step 1): Distributing Muscle
Forces
The general dynamic equation with 6 muscles and 2
degrees of freedom (Eq.1) can be simply written as
H (q, t) ¨q + C (q, t) ˙q = τ (9)
Here, we transform it into the following form
H (q, t) ¨q = Γ + Λ (10)
where Γ = τ, Λ = −C (q, t) ˙q. From this viewpoint, we can
separate the total acceleration contribution by two parts as ¨qΓ
and ¨qΛ, i.e.
¨q = ¨qΓ + ¨qΛ
¨qΓ = H (q, t)
−1
Γ (11)
¨qΛ = H (q, t)
−1
Λ
We define Jm as the Jacobian matrix from joint space
to muscle space, based on which, the joint torque can be
calculated as [7]
τ = JT
mFm (12)
In order to find the maximum acceleration contribution of each
muscle, we define ¨qm,j,max (j = 1, 2, · · · , 6) as the maximum
acceleration contribution of j-th muscle. According to Eq.12,
we have
¨qm,1,max = H−1
JT
m [ Fm,1,max 0 0 0 0 0 ]
T
· · · (13)
¨qm,6,max = H−1
JT
m [ 0 0 0 0 0 Fm,6,max ]
T
As each muscle can be considered to have its own contri-
bution to the acceleration, we can get a set of linear equations
AX = B (14)
where
A = [ ¨qm,1,max ¨qm,2,max · · · ¨qm,6,max ]
X = [ σ1 σ2 · · · σ6 ]
T
B = ¨qΓ
and vector X gives us the muscle activation level. We can use
pseudo-inverse to compute muscle activation level as
X = A+
B (15)
16421642](https://image.slidesharecdn.com/2013smcmuscleforcecontrol-140523201253-phpapp01/85/Muscle-Force-Control-of-a-Kinematically-Redundant-Bionic-Arm-with-Real-Time-Parameter-Update-3-320.jpg)
![where A+
= AT
AAT −1
. Therefore, the muscle force can
be calculated as the multiplication of muscle activation level
with maximum muscle force. By defining a vector
Fm,max = [ Fm,1,max Fm,2,max · · · Fm,6,max ]
T
(16)
as the maximum muscle force, we can give the initial muscle
force based on pseudo-inverse solution as
Fm,ini = Fm,max ⊗ X (17)
where operator ⊗ calculates the dot product of two vectors.
According to the properties of the pseudo-inverse, the initial
muscle force satisfies
min Fm,ini 2 =
⎛
⎝
6
j=1
F2
m,j
⎞
⎠ (18)
i.e., Fm,ini is the minimum muscle force under the sense of
least-squares.
B. Muscle Force Computation (Step 2): Satisfying Constraints
However, the above solution Fm,ini does not consider
physical constraints, for example that the maximum output
force of a muscle is limited, or that muscles can only contract,
etc. To satisfy these constraints, we define Fin as a vector with
the same dimension as Fm which expresses the internal forces
generated by the redundant muscles. We can define the internal
force in Fm space as
g (Fin) = I − JT
m
+
JT
m Fm (19)
where I is an identity matrix having the same dimension with
muscle space. According to Moore-Penrose pseudo-inverse
(Eq.15), the vector g (Fin) is orthogonal with Fm,ini. Thus,
we can choose any vector as Fin. Below, we give a gradient
direction to Fin to make Fm satisfy the boundary constraints.
Here, we assume each muscle force is limited in the interval
from Fm,j,min to Fm,j,max for 1 ≤ j ≤ 6. Our objective is
to choose a gradient direction to make each element of Fm,j
(1 ≤ j ≤ 6) equal or greater than Fm,j,min, and equal or less
than Fm,j,max. Considering the issue of “muscle fatigue”, one
reasonable way is to make the output force of the muscles to
remain in the middle of the range Fm,j,min and Fm,j,max. The
physical meaning of this idea is to distribute load to all muscles
into their proper load interval. Based on these considerations,
we choose a function h as
h (Fm) =
6
j=1
Fm,j − Fm,j,mid
Fm,j,mid − Fm,j,max
2
(20)
where
0 ≤ Fm,j,min ≤ Fm,j ≤ Fm,j,max
Fm,j,mid = (Fm,j,min + Fm,j,max) /2
j = 1, 2, · · · , 6
then we chose Fin as the gradient of the function h, i.e.
Fin = Kin
∂h (Fm)
∂Fm Fm,ini
= Kin ∇h|Fm,ini
(21)
=
⎡
⎢
⎢
⎢
⎢
⎣
2 ·
Fm,ini,1−Fm,1,mid
Fm,1,mid−Fm,1,max
2 ·
Fm,ini,2−Fm,2,mid
Fm,2,mid−Fm,2,max
...
2 ·
Fm,ini,6−Fm,6,mid
Fm,6,mid−Fm,6,max
⎤
⎥
⎥
⎥
⎥
⎦
where Kin is a scalar matrix. It is simple to prove that
the direction of Fin points to Fm,i,mid. According to the
computation in Eq.15 and Eq.21, the final muscle force is
calculated as
Fm = Fm,ini + g (Fin) (22)
C. Dynamic Parameter Update
To make the estimated robot arm model approach the real
model, we use a parameter adapter to adjust the dynamic
parameters in the estimated model. The parameter update is
based on the prediction error between the real and estimated
robot model. From the viewpoint of information flow, the
dynamic equation (Eq.1) can be written in the form
τ (t) = S (t, q, ˙q, ¨q) P (23)
where
S =
¨q1 ¨q2 0 0 ˙q1 ˙q2 0 0
0 0 ¨q1 ¨q2 0 0 ˙q1 ˙q2
, P =
PH
PC
From this equation, τ is the “output” of the system. S is
a signal matrix. P is a vector of real parameters. To avoid
the acceleration terms in S, we can use a filtering technique.
Specifically, by multiplying both sides of S with e−λ(t−r)
and
integrating it, we can get
ˆ t
0
e−λ(t−r)
τ (r) dr (24)
=
ˆ t
0
e−λ(t−r) ¨q1 ¨q2 0 0 ˙q1 ˙q2 0 0
0 0 ¨q1 ¨q2 0 0 ˙q1 ˙q2
·dr
PH
PC
where λ and r are positive numbers. By using partial integra-
tion, the acceleration terms on the right side can be written
as
ˆ t
0
e−λ(t−r) ¨q1 0
0 ¨q2
dr (25)
= e−λ(t−r) ¨q1 0
0 ¨q2
t
0
−
ˆ t
0
d
dr
e−λ(t−r) ˙q1 0
0 ˙q2
dr
16431643](https://image.slidesharecdn.com/2013smcmuscleforcecontrol-140523201253-phpapp01/85/Muscle-Force-Control-of-a-Kinematically-Redundant-Bionic-Arm-with-Real-Time-Parameter-Update-4-320.jpg)
![Therefore, Eq.23 can be written in the form
τ (t) = S (t, q, ˙q) P (26)
We can predict the value of the output τ based on the parameter
estimation, i.e.
ˆτ = S ˆP (27)
where ˆP is the estimation of P. ˆτ is the prediction of τ. Thus,
the prediction error e can be defined as
e = ˆτ − τ = S ˆP − SP = S ˜P (28)
According to this, we formulate the parameter adaptation
method based on the prediction error e as follows
˙ˆP = −Ξ
∂ eT
e
∂ ˆP
(29)
= −Ξ
∂ S ˆP − SP
T
S ˆP − SP
∂ ˆP
= −2ΞST
S ˆP − SP = −2ΞST
e = −2ΞST
(ˆτ − τ)
where Ξ is a diagonal coefficient matrix.
Proof: Parameter Update Convergence
Here we choose a Lyapunov function candidate
V (t) =
1
4
˜PT ˜P (30)
which describes the energy of parameter change. If we consider
that the parameters change slower with respect to the parameter
identification, from Eq.8, we can get
˙˜P =
˙ˆP − ˙P = −2ΞST
S ˜P (31)
then the time-derivative of V (t) is
˙V (t) =
1
2
˜PT ˙˜P =
1
2
˜PT
−2ΞST
S ˜P (32)
= −Ξ S ˜P
T
S ˜P < 0 (33)
which means that the parameter estimation converges to real
values.
Therefore, according to Eq.29, the parameter adaptation
can be obtained by
˙ˆP = −2ΞST
(ˆτ − τ) (34)
Segment Upper arm Lower arm
Length (m) 0.282 0.269
Mass (kg) 1.980 1.180
MCS Pos (m) 0.163 0.123
I11 (kg·m2
) 0.013 0.007
I22 (kg·m2
) 0.004 0.001
I33 (kg·m2
) 0.011 0.006
TABLE I: Anthropological parameter value (MCS Pos is the
position of the mass center).
D. Procedures
The desired trajectory points are connected by a continuous
derivative function for each moment. Based on it, we compute
the muscle forces. These force signals are used in the real
arm model and estimated arm model as inputs. The output
error of the two models, i.e., prediction error, is used to
change the dynamic parameters of the estimated arm model for
approaching the real one. The performance of the algorithm is
shown by a 3D virtual bionic arm (Fig.2).
IV. SIMULATION AND ANIMATION
The performance of the proposed muscle force computation
method was tested through a bending-stretching simulation.
The desired movement is bending the upper arm and lower arm
from 0 rad to π/2 rad and then stretching them back to 0 rad.
The frequency of the movement is 1Hz and the total simulation
time is 10s. To test the performance of the parameter update
method, we set all the estimated dynamic parameters to be zero
(i.e., ˆPH, ˆPC are zero vectors) at the beginning. The estimated
parameters are updated to approach the real parameters with
time afterwards.
A. Parameter Setting
The parameters of the arm model are based on the real
data of a human upper limb. The setting of length, mass, mass
center position and inertia coefficients are shown in Table
1. This anthropological data comes from [21]. Without loss
of generality, the muscle configuration coefficients are set as
aij = 0.1m (1 ≤ i ≤ 6, 1 ≤ j ≤ 2).
In this algorithm, there are very few parameters that need
to be set. These include the parameters relating with estimated
model update (Eq.34), i.e.
2Ξ = 0.001 · Diag ([ 1 1 1 1 1 1 1 1 1 1 ])
(35)
and the parameters relating with muscle force computation
(Eq.16 and Eq.21), i.e.,
Kin = 100 · Diag ([ 1 3 1 1 1 2 ]) (36)
Fm,i,min = 0, Fm,i,max = 500 (1 ≤ i ≤ 6) (37)
where Diag (·) denotes a diagonal matrix with diagonal ele-
ments being as (·).
16441644](https://image.slidesharecdn.com/2013smcmuscleforcecontrol-140523201253-phpapp01/85/Muscle-Force-Control-of-a-Kinematically-Redundant-Bionic-Arm-with-Real-Time-Parameter-Update-5-320.jpg)
![Fig. 2: Block diagram of the proposed method.
B. Muscle Force
According to the bending-stretching movement, the desired
trajectory points of the shoulder joint and elbow joint were
generated [7]. These trajectory points were used to create a
trajectory function for each joint. The muscle force was com-
puted by tracking the trajectory function. By applying these
muscle forces, we controlled the arm to move. The computed
muscle force is shown in Fig.3 where (a) gives the muscle
force curve (from Fm,1 to Fm,6) with respect to time. We
can find that, all the muscle forces satisfy the force constraint,
i.e., they are within the force range of [Fm,i,min, Fm,i,max] for
(1 ≤ i ≤ 6). While (b) concludes the statistical results where
x axis is the muscle index (from m1 to m6) and y axis is the
average force in the simulation time. As we set the maximum
muscle force to 500N, the force in the middle of the force
range is 250N. σi (1 ≤ i ≤ 6) which is above each histogram
is the derivation of the i-th muscle force. From (b), we can
see that the muscles approximately work around the middle of
their force ranges.
C. Parameter Updating Performance
Fig.4 shows the dynamic parameter change in the simula-
tion where (a) indicates the estimated parameter ˆPH, ˆPC and
(b) indicates the true parameter PH, PC. For both of the sub-
plots, the x axis is time index corresponding with simulation
time. It is clear that, the estimated dynamic parameters are not
the same as the real dynamic parameters. The reason for this
is that the desired tracking trajectory only stimulates part of
the whole dynamic features of the system. However, based on
this partly-stimulated parameters, the tracking task can still
be fulfilled (shown in the next Subsection). The estimated
dynamic parameters can be closer to the real ones when the
dynamics are further stimulated by the desired trajectory.
0 2 4 6 8 10
200
300
400
Fm1
(N)
0 2 4 6 8 10
−500
0
500
F
m2
(N)
0 2 4 6 8 10
100
200
300
Fm3
(N)
0 2 4 6 8 10
−500
0
500
F
m4
(N)
0 2 4 6 8 10
200
400
600
Fm5
(N)
0 2 4 6 8 10
0
200
400
Fm6
(N)
time (s)
(a) Muscle force curve.
m1 m2 m3 m4 m5 m6
0
50
100
150
200
250
300
350
400
muscle index
averageforce(N)
σ
1
=73.55N
σ2
=152.85N
σ
3
=35.39N σ
4
=113.36N
σ5
=72.27N
σ
6
=69.43N
F
mid
=250N
(b) Muscle force distribution.
Fig. 3: Computed muscle force.
16451645](https://image.slidesharecdn.com/2013smcmuscleforcecontrol-140523201253-phpapp01/85/Muscle-Force-Control-of-a-Kinematically-Redundant-Bionic-Arm-with-Real-Time-Parameter-Update-6-320.jpg)
![Fig. 6: Arm movement animation in isometric view.
main computation happens through the pseudo-inverse, the
computational scale of the proposed method is mainly related
to the number of the joints, not to the number of muscles. This
means that for the bionic arm, even if we add more muscles,
the computational time would not increase significantly. Fur-
thermore, the proposed muscle activation computation method
can be used as a solution for a general redundancy problem. We
believe that this is an important contribution not only towards
modeling such systems, but most importantly towards creating
the robots of the future, which will exhibit the numerous
advantages of muscle-driven systems. Our future work would
be extending the bionic arm model to 3D and extending the ex-
periments to real human arm movements using measurements
from Kinect Joint Tracking/Motion Capture System.
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16471647](https://image.slidesharecdn.com/2013smcmuscleforcecontrol-140523201253-phpapp01/85/Muscle-Force-Control-of-a-Kinematically-Redundant-Bionic-Arm-with-Real-Time-Parameter-Update-8-320.jpg)
Muscle Force Control of a Kinematically Redundant Bionic Arm with Real-Time Parameter Update
- 1. Muscle Force Control of a Kinematically Redundant Bionic Arm with Real-Time Parameter Update Haiwei Dong, Nadia Figueroa New York University AD Abu Dhabi, UAE {haiwei.dong, nadia.figueroa}@nyu.edu Abdulmotaleb El Saddik New York University AD and University of Ottawa Abu Dhabi, UAE and Ottawa, Canada elsaddik@uottawa.ca Abstract—Redundant muscle-driven arms have numerous ad- vantages, such as increased robustness, ability for load distribu- tion, impedance change etc. However, controlling such a muscle- driven arm is a difficult task. This is mainly due to its redundancy, specially when the muscle force is required to follow certain output constraints and fulfill optimization objectives. In this paper, a new method for controlling such muscle-like systems is proposed. By considering both joint and muscle acceleration contributions, a set of linear equations was constructed. Driving muscle activation is thus framed as the only unknown vector. To solve this linear equation set, a pseudo-inverse solution was used. The null space within this solution represents the internal force, which was used to evenly distribute the muscle forces, which is considered as “anti-fatigue” way. Moreover, to make the proposed method adaptive to modeling errors, an estimated system model was added to represent the real model. By updating the parameters of the estimated model based on prediction error, the estimated model approaches the real model gradually in real time. The overall method was tested for the case of a bending- stretching movement. The presented results verify the validity of the method, and illustrate its useful features and advantages. I. INTRODUCTION In nature, animals and human are able to move mainly through their musculoskeletal systems. The bones constitute a basic support structure. By linking the bones with different kinds of joints, we effectively have a number of degrees of freedom (DOF) that arise. Each muscle has its attachment points on the bones and can only output the “pull” force to drive the joint. In comparison with classical motor-driven systems, there are several advantages for muscle-like systems. First, as multiple muscles are tied to each joint, the total load can be distributed, resulting in small loads for each muscle [1]. Thus, each muscle only has a small load. Second, if one muscle is broken, the other muscles can still drive the joint to rotate [2]. Third, bi-articular muscles can not only output joint torque for movement, but also can change the impedance character of the body [3]. Due to the advantages of a muscle-like driven system compared with a classical motor-driven system, researchers recently have been focusing on the biologically- inspired mechanical system with many successful applications in robotics. Some examples include: ECCEROBOT-2 Robot [4], Biorob Arm [3], [5], musculo-skeletal arm [6], etc. Such robots are often termed “bionic robot”. In this paper, we focus on controlling a muscle-like bionic robot with bi-articular and mono-articular muscles. If we try to approach bionic robot control in the classical way, the solution would be to initially compute the joint torque, and then distribute the joint load to the muscles. The advantage of such an approach is that all the existing control methods can be used directly, such as sliding control, impedance control, robust control, etc. Thus, what remains is to design a proper muscle distribution principle. Dong et al. designed a composite control where sliding control is used to achieve robust joint torque. This torque was then distributed to muscles by virtual work between muscles and joints [7]. The disadvantage of this method is that it artificially divided a problem into two separate processes. Furthermore, because the joint torque control methods are designed specifically for classical motor-driven systems, the computed joint torque may lead the muscle force change not smooth. Thus, it might be worthy to follow a novel avenue: model the bionic robot arm as a muscle-driven system considering its complex dynamics and propose a novel method for controlling it. In this paper, we choose to follow this avenue. Specifically, controlling a system driven by bi-articular and mono-articular muscles involves many issues. Four im- portant problems are listed below. The first problem is muscle coordination. Muscle configurations vary, therefore a muscle may correspond to one or two joints. Furthermore, for one specific joint, there are usually several muscles corresponding to its movement. Coordination of these muscles can be seen as a general control problem for over-actuated systems. Until now, many approaches have been proposed. For example, Yokishawa proposed a method to avoid singularities in the arm by minimizing a manipulability measure [8]. Maciejewski et al. used the null-space vector to aid obstacle avoidance [9]. Klug et al. developed a 3 DOF bionic robot arm which is controlled by a PD controller with feed-forward compensation [5]. The trajectory of the arm is optimized and adjusted for a time and energy-optimal motion [10]. Potkonjak et al. built a humanoid robot with antagonistic drives whose controller is designed by loop-shaping [4]. A second important problem is enforcing muscle force constraints. Each muscle can only provide a “pull” force. Meaning that, the muscle force is nonnegative. In addition, each muscle has its limit in force output. Thus, the output force for each muscle is within an interval from 0 to maximum output force. For many years, researchers have focused on solving such control problems with input constraints. Tahara proposed a simple sensor-motor control scheme for internal force and simulated the overall stability for a cable-driven 2013 IEEE International Conference on Systems, Man, and Cybernetics 978-1-4799-0652-9/13 $31.00 © 2013 IEEE DOI 1640 2013 IEEE International Conference on Systems, Man, and Cybernetics 978-1-4799-0652-9/13 $31.00 © 2013 IEEE DOI 10.1109/SMC.2013.283 1640
- 2. system [6]. Tsang applied Generalized Predictive Control (GPC) in DC motor drive while guaranteeing the amplitude limits of the input currency [11]. Fujimoto et al. proposed an optimal control method bearing input constraint by interactive learning [12]. Yamaguchi used gradient iteration to modify the muscle force until it satisfies the constraint [13]. Yet a third important problem, is designing the system so that it attempts to decrease muscular fatigue. This is a higher requirement than the muscle force constraint. The challenge is that we need to find an optimization solution that separates all the muscle forces evenly, within the feasible solution space of muscle forces. For this distribution method, optimization is a common method. Anderson et al. used dynamic optimization of minimum metabolic energy expenditure to solve the motion control of walking [14]. Manal et al. designed a nonlinear optimal controller to develop a real-time EMG-driven virtual arm [15]. Neptune evaluated different multivariate optimization methods in pedaling [16]. Finally, the fourth important problem is adaptivity. For a real-world control plant, absolute models are not achievable mainly due to multiple many uncertainties, such as sen- sor uncertainty, actuator uncertainty, environment uncertainty, modeling uncertainty, etc. Thus, the control method must have the ability to tolerate these uncertainties or to update the parameters of the modeled plant. This issue of adaptivity has been considered for years. In robotics, the research on adapta- tion comes from online system identification. The objective is to estimate the system structure and parameters in real-time, while the system is in operation. Here, the system is considered as a black box. By stimulating the system and creating relations between the inputs and outputs, the parameters of the system can be adjusted online [17]. Based on this system identification idea, many control methods with adaptive parameter update have come out, such as robust control [18], adaptive feedback control [19], neurofuzzy adaptive control [20], etc. However, notice that the adaptation ability problem has not been consid- ered in arm control driven by over-actuated muscles. In this paper, we propose a new method for controlling a muscle-driven system, that addresses all of the four important problems previously explained. As a preparatory step for our method, the desired joint trajectory points was connected by a continuous derivative polynomial function for each moment; resulting with a Jacobian matrix from muscle space to joint space. For the muscle activation computation, by considering the acceleration in the joint space and muscle space respec- tively, we obtained a set of linear equations. The pseudo- inverse solution to this linear equation set gave us the initial result for the muscle forces. To make the muscles forces satisfy the output force constraint and furthermore work in the feasible output force range, we gave a gradient direction for the muscle forces to satisfy the mentioned constraint in the null space of the previous pseudo-inverse solution. As the body movement load distributes evenly in all the muscles, the proposed algorithm has a character of “anti-fatigue”. To make the proposed method adaptive for modeling error, we added an estimated model to represent the real model. By updating the parameters of the estimated model based on prediction error, the estimated model approaches the real model gradually in real time. The overall proposed method is tested for a bending- Fig. 1: Bionic robot arm model. stretching movement. The results verify the validity of the method, as well as dealing with the four previously mentioned problems effectively. We will start by presenting our bionic arm mathematical model, continue by deriving the steps of our muscle activation calculation method, and then discuss our simulation and animation results, before providing a concise summary and conclusion. II. BIONIC ROBOT ARM MODEL A. Dynamic Modeling We built a 2-dimensional model of an arm in the horizontal plane (no gravity) based on the upper limb structure of a digital human. The model includes six muscles (shown as 1 to 6) and two degrees of freedom (shoulder flexion-extension and elbow flexion-extension). The range of the shoulder angle is from -20 to 100 degree, and the range of the elbow is from 0 to 170 degree. Four of the muscles are mono-articular, and two are bi-articular where 1 and 2 cross the shoulder joint; 3 and 4 cross the elbow joint; 5 and 6 cross both joints (Fig.1). Considering the arm (including upper arm and lower arm) as a planar, two-link, articulated rigid object, the position of the hand can be derived by a 2-vector q of two angles. The input is a 6-vector Fm of muscle forces. The dynamics of the rigid object are strongly nonlinear. Using the Lagrangian equations from classical dynamics, we get the dynamic equations of the ideal upper limb model H11 (t) H12 (t) H21 (t) H22 (t) ¨q1 ¨q2 (1) + C11 (t) C12 (t) C21 (t) C22 (t) ˙q1 ˙q2 = τ1(t) τ2(t) or abbreviated as H (t) ¨q + C (t) ˙q = τ (2) with q = [ q1 q2 ] T = [ θ1 θ2 ] T being the two joint angles. τ = [ τ1 τ2 ] = f (Fm) is the joint torque which is considered as a function of muscle force Fm 16411641
- 3. Fm = [ Fm,1 Fm,2 Fm,3 Fm,4 Fm,5 Fm,6 ] T (3) H (q, t) is the inertia matrix which contains information with regards to the instantaneous mass distribution. C (q, ˙q, t) is centripetal and coriolis torques representing the moments of centrifugal forces. H11 = J1 + J2 + m2d2 1 + 2m2d1c2 cos (q2) H12 = H21 = J2 + m2d1c2 cos (q2) H22 = J2 C11 = −2m2d1c2 sin (q2) ˙q2 (4) C12 = −m2d1c2 sin (q2) ˙q2 C21 = m2d1c2 sin (q2) ˙q1 C22 = 0 where ci is the distance from the center of a joint i to the center of gravity point of link i. di is the length of link i. Ji = mid2 i + Ii where Ii is the moment of inertia about the axis through the center of the mass of link i. B. Estimated Model In our research, we consider that the arm model in Eq.1 is influenced by modeling errors and disturbances from the environment. Hence, we use an estimated arm model to represent the real one, which is written as ˆH11 (t) ˆH12 (t) ˆH21 (t) ˆH22 (t) ¨q1 ¨q2 (5) + ˆC11 (t) ˆC12 (t) ˆC21 (t) ˆC22 (t) ˙q1 ˙q2 = ˆτ1(t) ˆτ2(t) or abbreviated as ˆH (t) ¨q + ˆC (t) ˙q = ˆτ (6) where ˆ· means estimated value of (·). The relation between the ideal model (Eq.1) and the estimated model (Eq.5) is that we choose τ = ˆτ. In this paper, we use the estimated model to generate torque for controlling a real system. In addition, the parameter adaptation updates the estimated parameters ˆH, ˆCand ˆG in real-time. For the convenience of the following derivation, we define the actual and estimated arm parameter vector P = PT H PT C T , ˆP = ˆPT H ˆPT C T (7) where PH = ⎡ ⎢ ⎣ H11 H12 H21 H22 ⎤ ⎥ ⎦ , PC = ⎡ ⎢ ⎣ C11 C12 C21 C22 ⎤ ⎥ ⎦ ˆPH = ⎡ ⎢ ⎢ ⎣ ˆH11 ˆH12 ˆH21 ˆH22 ⎤ ⎥ ⎥ ⎦ , ˆPC = ⎡ ⎢ ⎢ ⎣ ˆC11 ˆC12 ˆC21 ˆC22 ⎤ ⎥ ⎥ ⎦ then the estimation error vector can be defined as ˜P = ˆP − P = ˜PT H ˜PT C T (8) III. MUSCLE FORCE CALCULATION A. Muscle Force Computation (Step 1): Distributing Muscle Forces The general dynamic equation with 6 muscles and 2 degrees of freedom (Eq.1) can be simply written as H (q, t) ¨q + C (q, t) ˙q = τ (9) Here, we transform it into the following form H (q, t) ¨q = Γ + Λ (10) where Γ = τ, Λ = −C (q, t) ˙q. From this viewpoint, we can separate the total acceleration contribution by two parts as ¨qΓ and ¨qΛ, i.e. ¨q = ¨qΓ + ¨qΛ ¨qΓ = H (q, t) −1 Γ (11) ¨qΛ = H (q, t) −1 Λ We define Jm as the Jacobian matrix from joint space to muscle space, based on which, the joint torque can be calculated as [7] τ = JT mFm (12) In order to find the maximum acceleration contribution of each muscle, we define ¨qm,j,max (j = 1, 2, · · · , 6) as the maximum acceleration contribution of j-th muscle. According to Eq.12, we have ¨qm,1,max = H−1 JT m [ Fm,1,max 0 0 0 0 0 ] T · · · (13) ¨qm,6,max = H−1 JT m [ 0 0 0 0 0 Fm,6,max ] T As each muscle can be considered to have its own contri- bution to the acceleration, we can get a set of linear equations AX = B (14) where A = [ ¨qm,1,max ¨qm,2,max · · · ¨qm,6,max ] X = [ σ1 σ2 · · · σ6 ] T B = ¨qΓ and vector X gives us the muscle activation level. We can use pseudo-inverse to compute muscle activation level as X = A+ B (15) 16421642
- 4. where A+ = AT AAT −1 . Therefore, the muscle force can be calculated as the multiplication of muscle activation level with maximum muscle force. By defining a vector Fm,max = [ Fm,1,max Fm,2,max · · · Fm,6,max ] T (16) as the maximum muscle force, we can give the initial muscle force based on pseudo-inverse solution as Fm,ini = Fm,max ⊗ X (17) where operator ⊗ calculates the dot product of two vectors. According to the properties of the pseudo-inverse, the initial muscle force satisfies min Fm,ini 2 = ⎛ ⎝ 6 j=1 F2 m,j ⎞ ⎠ (18) i.e., Fm,ini is the minimum muscle force under the sense of least-squares. B. Muscle Force Computation (Step 2): Satisfying Constraints However, the above solution Fm,ini does not consider physical constraints, for example that the maximum output force of a muscle is limited, or that muscles can only contract, etc. To satisfy these constraints, we define Fin as a vector with the same dimension as Fm which expresses the internal forces generated by the redundant muscles. We can define the internal force in Fm space as g (Fin) = I − JT m + JT m Fm (19) where I is an identity matrix having the same dimension with muscle space. According to Moore-Penrose pseudo-inverse (Eq.15), the vector g (Fin) is orthogonal with Fm,ini. Thus, we can choose any vector as Fin. Below, we give a gradient direction to Fin to make Fm satisfy the boundary constraints. Here, we assume each muscle force is limited in the interval from Fm,j,min to Fm,j,max for 1 ≤ j ≤ 6. Our objective is to choose a gradient direction to make each element of Fm,j (1 ≤ j ≤ 6) equal or greater than Fm,j,min, and equal or less than Fm,j,max. Considering the issue of “muscle fatigue”, one reasonable way is to make the output force of the muscles to remain in the middle of the range Fm,j,min and Fm,j,max. The physical meaning of this idea is to distribute load to all muscles into their proper load interval. Based on these considerations, we choose a function h as h (Fm) = 6 j=1 Fm,j − Fm,j,mid Fm,j,mid − Fm,j,max 2 (20) where 0 ≤ Fm,j,min ≤ Fm,j ≤ Fm,j,max Fm,j,mid = (Fm,j,min + Fm,j,max) /2 j = 1, 2, · · · , 6 then we chose Fin as the gradient of the function h, i.e. Fin = Kin ∂h (Fm) ∂Fm Fm,ini = Kin ∇h|Fm,ini (21) = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ 2 · Fm,ini,1−Fm,1,mid Fm,1,mid−Fm,1,max 2 · Fm,ini,2−Fm,2,mid Fm,2,mid−Fm,2,max ... 2 · Fm,ini,6−Fm,6,mid Fm,6,mid−Fm,6,max ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ where Kin is a scalar matrix. It is simple to prove that the direction of Fin points to Fm,i,mid. According to the computation in Eq.15 and Eq.21, the final muscle force is calculated as Fm = Fm,ini + g (Fin) (22) C. Dynamic Parameter Update To make the estimated robot arm model approach the real model, we use a parameter adapter to adjust the dynamic parameters in the estimated model. The parameter update is based on the prediction error between the real and estimated robot model. From the viewpoint of information flow, the dynamic equation (Eq.1) can be written in the form τ (t) = S (t, q, ˙q, ¨q) P (23) where S = ¨q1 ¨q2 0 0 ˙q1 ˙q2 0 0 0 0 ¨q1 ¨q2 0 0 ˙q1 ˙q2 , P = PH PC From this equation, τ is the “output” of the system. S is a signal matrix. P is a vector of real parameters. To avoid the acceleration terms in S, we can use a filtering technique. Specifically, by multiplying both sides of S with e−λ(t−r) and integrating it, we can get ˆ t 0 e−λ(t−r) τ (r) dr (24) = ˆ t 0 e−λ(t−r) ¨q1 ¨q2 0 0 ˙q1 ˙q2 0 0 0 0 ¨q1 ¨q2 0 0 ˙q1 ˙q2 ·dr PH PC where λ and r are positive numbers. By using partial integra- tion, the acceleration terms on the right side can be written as ˆ t 0 e−λ(t−r) ¨q1 0 0 ¨q2 dr (25) = e−λ(t−r) ¨q1 0 0 ¨q2 t 0 − ˆ t 0 d dr e−λ(t−r) ˙q1 0 0 ˙q2 dr 16431643
- 5. Therefore, Eq.23 can be written in the form τ (t) = S (t, q, ˙q) P (26) We can predict the value of the output τ based on the parameter estimation, i.e. ˆτ = S ˆP (27) where ˆP is the estimation of P. ˆτ is the prediction of τ. Thus, the prediction error e can be defined as e = ˆτ − τ = S ˆP − SP = S ˜P (28) According to this, we formulate the parameter adaptation method based on the prediction error e as follows ˙ˆP = −Ξ ∂ eT e ∂ ˆP (29) = −Ξ ∂ S ˆP − SP T S ˆP − SP ∂ ˆP = −2ΞST S ˆP − SP = −2ΞST e = −2ΞST (ˆτ − τ) where Ξ is a diagonal coefficient matrix. Proof: Parameter Update Convergence Here we choose a Lyapunov function candidate V (t) = 1 4 ˜PT ˜P (30) which describes the energy of parameter change. If we consider that the parameters change slower with respect to the parameter identification, from Eq.8, we can get ˙˜P = ˙ˆP − ˙P = −2ΞST S ˜P (31) then the time-derivative of V (t) is ˙V (t) = 1 2 ˜PT ˙˜P = 1 2 ˜PT −2ΞST S ˜P (32) = −Ξ S ˜P T S ˜P < 0 (33) which means that the parameter estimation converges to real values. Therefore, according to Eq.29, the parameter adaptation can be obtained by ˙ˆP = −2ΞST (ˆτ − τ) (34) Segment Upper arm Lower arm Length (m) 0.282 0.269 Mass (kg) 1.980 1.180 MCS Pos (m) 0.163 0.123 I11 (kg·m2 ) 0.013 0.007 I22 (kg·m2 ) 0.004 0.001 I33 (kg·m2 ) 0.011 0.006 TABLE I: Anthropological parameter value (MCS Pos is the position of the mass center). D. Procedures The desired trajectory points are connected by a continuous derivative function for each moment. Based on it, we compute the muscle forces. These force signals are used in the real arm model and estimated arm model as inputs. The output error of the two models, i.e., prediction error, is used to change the dynamic parameters of the estimated arm model for approaching the real one. The performance of the algorithm is shown by a 3D virtual bionic arm (Fig.2). IV. SIMULATION AND ANIMATION The performance of the proposed muscle force computation method was tested through a bending-stretching simulation. The desired movement is bending the upper arm and lower arm from 0 rad to π/2 rad and then stretching them back to 0 rad. The frequency of the movement is 1Hz and the total simulation time is 10s. To test the performance of the parameter update method, we set all the estimated dynamic parameters to be zero (i.e., ˆPH, ˆPC are zero vectors) at the beginning. The estimated parameters are updated to approach the real parameters with time afterwards. A. Parameter Setting The parameters of the arm model are based on the real data of a human upper limb. The setting of length, mass, mass center position and inertia coefficients are shown in Table 1. This anthropological data comes from [21]. Without loss of generality, the muscle configuration coefficients are set as aij = 0.1m (1 ≤ i ≤ 6, 1 ≤ j ≤ 2). In this algorithm, there are very few parameters that need to be set. These include the parameters relating with estimated model update (Eq.34), i.e. 2Ξ = 0.001 · Diag ([ 1 1 1 1 1 1 1 1 1 1 ]) (35) and the parameters relating with muscle force computation (Eq.16 and Eq.21), i.e., Kin = 100 · Diag ([ 1 3 1 1 1 2 ]) (36) Fm,i,min = 0, Fm,i,max = 500 (1 ≤ i ≤ 6) (37) where Diag (·) denotes a diagonal matrix with diagonal ele- ments being as (·). 16441644
- 6. Fig. 2: Block diagram of the proposed method. B. Muscle Force According to the bending-stretching movement, the desired trajectory points of the shoulder joint and elbow joint were generated [7]. These trajectory points were used to create a trajectory function for each joint. The muscle force was com- puted by tracking the trajectory function. By applying these muscle forces, we controlled the arm to move. The computed muscle force is shown in Fig.3 where (a) gives the muscle force curve (from Fm,1 to Fm,6) with respect to time. We can find that, all the muscle forces satisfy the force constraint, i.e., they are within the force range of [Fm,i,min, Fm,i,max] for (1 ≤ i ≤ 6). While (b) concludes the statistical results where x axis is the muscle index (from m1 to m6) and y axis is the average force in the simulation time. As we set the maximum muscle force to 500N, the force in the middle of the force range is 250N. σi (1 ≤ i ≤ 6) which is above each histogram is the derivation of the i-th muscle force. From (b), we can see that the muscles approximately work around the middle of their force ranges. C. Parameter Updating Performance Fig.4 shows the dynamic parameter change in the simula- tion where (a) indicates the estimated parameter ˆPH, ˆPC and (b) indicates the true parameter PH, PC. For both of the sub- plots, the x axis is time index corresponding with simulation time. It is clear that, the estimated dynamic parameters are not the same as the real dynamic parameters. The reason for this is that the desired tracking trajectory only stimulates part of the whole dynamic features of the system. However, based on this partly-stimulated parameters, the tracking task can still be fulfilled (shown in the next Subsection). The estimated dynamic parameters can be closer to the real ones when the dynamics are further stimulated by the desired trajectory. 0 2 4 6 8 10 200 300 400 Fm1 (N) 0 2 4 6 8 10 −500 0 500 F m2 (N) 0 2 4 6 8 10 100 200 300 Fm3 (N) 0 2 4 6 8 10 −500 0 500 F m4 (N) 0 2 4 6 8 10 200 400 600 Fm5 (N) 0 2 4 6 8 10 0 200 400 Fm6 (N) time (s) (a) Muscle force curve. m1 m2 m3 m4 m5 m6 0 50 100 150 200 250 300 350 400 muscle index averageforce(N) σ 1 =73.55N σ2 =152.85N σ 3 =35.39N σ 4 =113.36N σ5 =72.27N σ 6 =69.43N F mid =250N (b) Muscle force distribution. Fig. 3: Computed muscle force. 16451645
- 7. P_hat_H 0.5 1 1.5 2 2.5 3 x 10 4 H_hat_22 H_hat_21 H_hat_12 H_hat_11 P_hat_C time index 0.5 1 1.5 2 2.5 3 x 10 4 C_hat_22 C_hat_21 C_hat_12 C_hat_11 0 1 2 3 x 10 −3 (a) Estimated Parameter. P_H 0.5 1 1.5 2 2.5 3 x 10 4 H_22 H_21 H_12 H_11 P_C time index 0.5 1 1.5 2 2.5 3 x 10 4 C_22 C_21 C_12 C_11 −0.2 0 0.2 (b) Real parameter. Fig. 4: Parameter update. D. Joint Tracking Performance The computed muscle forces were used to control the arm model. Compared with the desired trajectory, the tracking errors of shoulder joint and elbow joint gradually converge (Fig.5). The big tracking error at the beginning is due to the uncertainty of the initial dynamic parameters. By updating the parameters in real-time, the tracking error decreases with time. The static tracking error of both shoulder joint and elbow joint is smaller than 0.01 rad and thus, the tracking performance is considered acceptable. E. Animation A 3D virtual robot arm was created by Simulink (SimMe- chanics Toolbox). The arm model consists of three parts: right humerus (1 bones), torso (6 bones) and right ulna radius hand (29 bones). The detailed information of the bones is listed in Table II. The polygon files (see Table II) are from OpenSim. The original format of these files are .vtp. To aid SimMechanics on importing these polygons, we changed the format of polygon files from .vtp file to .stl file. We use this 3D virtual arm to visualize the arm movement. Fig.6 shows the arm movement in the bending-stretching movement cycle. V. CONCLUSION This paper presented a novel method for controlling a muscle-driven system. Compared with previous methods, there are numerous advantages. First, there are very few parameters that need to be set – partially, because the pseudo-inverse can be seen as an “optimization” method. Second, as the 0 1 2 3 4 5 6 7 8 9 10 −0.07 −0.06 −0.05 −0.04 −0.03 −0.02 −0.01 0 0.01 0.02 0.03 time (s) trackingerrorofshoulderjoint(rad) (a) Shoulder joint. 0 1 2 3 4 5 6 7 8 9 10 −0.015 −0.01 −0.005 0 0.005 0.01 time (s) trackingerrorofelbowjoint(rad) (b) Elbow joint. Fig. 5: Tracking error. Part name Bone name arm_r_humerus arm_r_humerus torso ground_jaw ground_r_scapula ground_skull ground_r_clavicle ground_ribs ground_spine r_ulna_radius_hand arm_r_1mc arm_r_4mc arm_r_pisiform arm_r_2distph arm_r_4midph arm_r_radius arm_r_2mc arm_r_4proxph arm_r_scaphoid arm_r_2midph arm_r_5distph arm_r_thumbdist arm_r_2proxph arm_r_5mc arm_r_thumbprox arm_r_3distph arm_r_5midph arm_r_trapezium arm_r_3mc arm_r_5proxph arm_r_trapezoid arm_r_3midph arm_r_capitate arm_r_triquetrum arm_r_3proxph arm_r_hamate arm_r_ulna arm_r_4distph arm_r_lunate TABLE II: Bones setup in visualization. 16461646
- 8. Fig. 6: Arm movement animation in isometric view. main computation happens through the pseudo-inverse, the computational scale of the proposed method is mainly related to the number of the joints, not to the number of muscles. This means that for the bionic arm, even if we add more muscles, the computational time would not increase significantly. Fur- thermore, the proposed muscle activation computation method can be used as a solution for a general redundancy problem. We believe that this is an important contribution not only towards modeling such systems, but most importantly towards creating the robots of the future, which will exhibit the numerous advantages of muscle-driven systems. Our future work would be extending the bionic arm model to 3D and extending the ex- periments to real human arm movements using measurements from Kinect Joint Tracking/Motion Capture System. REFERENCES [1] H. Dong, Z. Luo, and A. Nagano, “Distributing joint torques to muscle forces for robot manipulator control,” in Proceeding of 29th Annual Conference of the Robotics Society of Japan, pp. 1Q3–3, 2011. [2] S. Oh and Y. Hori, “Development of two-degree-of-freedom control for robot manipulator with biarticular muscle torques,” in Proceeding of American Control Conference, pp. 325–330, 2009. [3] S. Klug, O. Stryk, and B. Mohl, “Design and control mechanisms for a 3dof bionic manipulator,” in Proceeding of IEEE/RAS-EMBS International Conference on Biomedical Robotics and Biomechatronics, pp. 450–454, 2006. [4] V. Potkonjak, M. Jovanovic, P. Milosavljevic, N. Bascarevic, and O. Holland, “The puller-follower control concept in the multi-jointed robot body with antagonistically coupled compliant drives,” in Proceed- ing of IASTED International Conference on Robotics, pp. 375–381, 2011. [5] S. Klug, B. Mohl, V. Ttryk, and O. Barth, “Design and application of a 3 dof bionic robot arm,” in Proceeding of 3rd International Symposium on Adaptive Motion in Animals and Machines, pp. 1–6, 2005. [6] K. Tahara, Z. Luo, S. Arimoto, and H. Kino, “Sensory-motor control mechanism for reaching movements of a redundant musculo-skeletal arm,” Journal of Robotic Systems, vol. 22, pp. 639–651, 2005. [7] H. Dong and N. Mavridis, “Adaptive biarticular muscle force control for humanoid robot arms,” in Proceeding of 12th IEEE-RAS International Conference on Humanoid Robots, pp. 284–290, 2012. [8] “Robotics research: The first international symposium,” in Analysis and control of robot manipulators with redundancy, pp. 735–748, MIT Press, 1984. [9] A. Maciejewski and C. Klein, “Obstacle avoidance for kinematically redundant manipulators in dynamically varying environment,” Interna- tional Journal of Robotics Research, vol. 4, pp. 109–117, 1985. [10] A. Heim and O. Stryk, “Trajectory optimization of industrial robots with application to computer aided robotics and robot controllers,” Optimization, vol. 47, pp. 407–420, 2000. [11] T. Tsang, “Generalised predictive control with input constraints,” IEE Proceedings on Control Theory and Applications, vol. 135, pp. 451– 460, 1988. [12] K. Fujimoto, T. Horiuchi, and T. Sugei, “Optimal control of hamiltonian systems with input constraints via iterative learning,” in Proceeding of IEEE Conference on Decision and Control, pp. 4387–4392, 2003. [13] G. Yamaguchi, Dynamic Modeling of Musculoskeletal Motion: A Vec- torized Appoach for Biomechanical Analysis in Three Dimensions. Springer, 2005. [14] F. Anderson and M. Pandy, “Dynamic optimization of human walking,” Journal of Biomechanical Engineering, vol. 125, pp. 381–390, 2001. [15] K. Manal, R. Gonzalez, D. Lloyd, and T. Buchanan, “A real-time emg-driven virtual arm,” Computers in Biology and Medicine, vol. 32, pp. 25–36, 2002. [16] R. Neptune, “Optimization algorithm performance in determining opti- mal controls in human movement analysis,” Journal of Biomechanical Engineering, vol. 121, pp. 249–252, 1999. [17] D. Graupe, Identification of Systems. Litton Educational Publishing, 1972. [18] K. Zhou and J. Doyle, Essentials of Robust Control. Prentice Hall, 1997. [19] J. Slotine and W. Li, Applied Nonlinear Control. Prentice Hall, 1991. [20] M. Brown and C. Harris, Neurofuzzy Adaptive Modelling and Control. Prentice Hall, 1994. [21] A. Nagano, S. Yoshioka, T. Komura, R. Himeno, and S. Fukashiro, “A three-dimensional linked segment model of the whole human body,” International Journal of Sport and Health Science, vol. 3, pp. 311– 325, 2005. 16471647