IEEE/ASME Transactions on Mechatronics, No. xx, Vol. xx
1
Stable Walking Pattern for a SMA Actuated Biped
E. Tarkesh Esfahani, Mohammad H. Elahinia, Member, IEEE
I.
Abstract—In this paper, a walking pattern filter for Shape
Memory Alloy (SMA) actuated biped robots is presented. SMAs
are known for their high power to mass ratio as well as slow
response. When used as actuators, SMA speed limitation can
potentially lead to stability problems for biped robots. The
presented filter adapts the human motion such that a SMA biped
robot maintains a stable walking pattern. The Zero Moment
Point (ZMP) is used as the main criterion of the filter to
guarantee the stability of the motion. The SMA actuators are
designed based on the dynamics and kinematics of the motion.
The response time of each SMA actuator is modeled in order to
estimate the behavior of the actuator in realizing the given
trajectory. After applying the delay times to the motion, the new
trajectories are generated and evaluated by the filter for the
ZMP criterion. Using simulations it is shown that the filter can
generate smooth trajectories for the SMA-actuated biped robots.
The filter furthermore guarantees the stability of a robot
mimicking human walking motion.
Index Terms—Biped Robot, SMA, Walking Pattern, ZMP
xa , ya , xh , yh Ankle and hip joint coordinates in xy plane
θ a ,θ h
Ankle and hip relative sagital rotational angles
Physical Parameters:
Lth , Lsh , L f
Length of thigh, shank and foot segments
Lab
Laf
Distances from back of the foot to the ankle (x-axis)
Distances from front of the foot to the ankle (x-axis)
Lan
Distances along the y-axis from foot to the ankle
Gait Parameters:
Tc
The Instant in which one walking cycle is completed
Td
Tm
The instant in which double support phase is finished
The instant in which x a and y a reach Lam and H am
Ds
Length of each step
H am
Lam
Maximum height of the ankle during its motion
Corresponding x position for H am
hmin , hmax
Min and Max height of the hip during its motion
qb , q f
Foot departing and foot landing angle
Control Parameters:
xed , x sd Distances along the x-axis from the hip to the ankle of
the support foot
E. Tarkesh Esfahani is a graduate student at the Mechanical, Industrial and
Manufacturing Engineering Department, in the University of Toledo, Toledo,
OH 43606 USA (e-mail: [email protected]).
M. Elahinia is an Assistant Professor at the Mechanical, Industrial and
Manufacturing Engineering Department, in the University of Toledo, Toledo,
OH 43606 USA (e-mail: [email protected]).
INTRODUCTION
A. Shape Memory Alloy Actuators
S
hape Memory Alloy (SMA) composites are a class of
smart materials that exhibit extremely large recoverable
strains. The shape memory effect occurs due to a temperature
and stress dependent shift in the materials’ crystalline
structure between two different phases called martensite and
austenite. The use of SMAs in applications involving
actuation has several advantages such as large deformation,
excellent power to mass ratio, maintainability, reliability, and
clean and silent actuation. The disadvantages are slow
response, low energy efficiency due to conversion of heat to
mechanical energy and also motion control difficulties due to
hysteresis, nonlinearities, parameter uncertainties, and
unmodeled dynamics [1].
When a SMA wire is at a high temperature without stress, it
will be in the austenitic phase. As the temperature of the wire
decreases, the phase will become martensitic. In shape
memory effect, a specimen exhibits a large residual strain after
loading and unloading. This strain can be fully recovered
upon heating the material. In pseudoelastic effect, the SMA
material provides a large strain upon loading that is fully
recovered in a hysteresis loop upon unloading [2, 3].
Large deformation and excellent power to mass ratio of
SMA wires have encouraged researchers to develop different
types of SMA actuator for robotic applications [4-9]. Honma
et al. proposed that the small size and the large displacement
of SMA actuators were ideal for developing small-size and
micro robots [10]. SMAs have been used in many different
robotic systems as actuators [11-13]. Tu et al. used SMA
wires in a small biped robot [14]. They designed a fuzzy
walking pattern for the biped to achieve the stable motion. It
was shown that one of the main challenges of the SMA
actuated biped was stabilizing the motion.
B. Biped Trajectory Generation
Planning walking patterns is one of the most important
parts in stabilizing the motion of biped robots. Well-designed
walking patterns can guarantee the stability of the motion. As
a way to collect walking pattern, researchers have been
placing markers on human joints to recording human walking
patterns [15]. Seyfarth et al. proposed a trajectory generation
method based on series of sinusoidal functions [16]. Through
sampling of some key points of human walking patterns and
developing new joint trajectories on those points, Hung et al.
generated all the possible trajectories for a biped robot in a
IEEE/ASME Transactions on Mechatronics, No. xx, Vol. xx
sagital plane for various length steps and velocities [17]. The
same method was used for lateral plane by Azimi et al. [18].
Depending on the analysis method, two different
approaches are used for stability verification of the walking
robots. In the first approach the, Center of Gravity (COG) is
used as a stability criterion for static walkers with low speed.
Such a system can be modeled as a static system at each
instant. The similar concept for dynamic analysis is the Zero
Moment Point (ZMP), which was introduced by Vukobratovic
and Borovac [23]. In some studies, a desired ZMP is first
generated and then a trajectory is found to maintain this
pattern [19-21]. Others find the desired ZMP among several
smooth trajectories. The second method has been shown to be
more applicable because it results in smoother trajectories
[17].
Although stability is the critical criterion in walking pattern
generation, in few researches the effect of actuator on the
stability has been investigated. Using an actuator with
complex nonlinear behaviors and speed limitation, such as
Shape Memory Alloy (SMA), will affect the stability of the
motion of a biped robot. This effect can be studied and
predicted by including the actuator model in walking pattern
generation. This prediction can further be used as a filter for
modifying the walking pattern. Such a filter is a computational
engine that takes a motion sequence as its input, and outputs
another sequence, which preserves the characteristics of the
original motion while satisfying some physical criteria such as
stability. The input motion to the filter can be from the motion
captured data or hand-drawn. While this input motion may
satisfy a physical criterion, when applied to a biped robot, it
may not satisfy a different criterion, such as the stability.
Yamani and Nakamora proposed the concept of a dynamic
filter. A dynamic filter transforms a physically inconsistent
motion into a consistent motion. The authors provided an
example of its implementation using feedback control and
local optimization [22].
In this paper, a walking pattern filter for Shape Memory
Alloy (SMA) actuated biped robots is presented. The recorded
data of a human walking is used as a source input. The filter
adapts the source input in such a way that the SMA biped can
follow a stable trajectory. This way the output of the filter will
be a smooth trajectory that is dynamically stable on a SMAactuated biped. The filtered motion is also similar to the initial
recorded human motion.
II.
TRAJECTORY GENERATION
A biped with two legs and a trunk is considered here as
shown in Figure 1. The robot has 6 degrees of freedom
(DOF). Each leg consists of 3 DOF with one degree at each
joint: the ankle, the knee and the hip. Most of the parameters
used for formulating the motion of the robot are shown in
Figure 1.
The general motion of a biped consists of several sequential
steps each composed of two phases: the single support phase
and the double support phase. During the single support
phase, one leg is on the ground and the other is in the
swinging motion. The double support phase starts as soon as
2
the swinging leg meets the ground and ends when the support
leg leaves the ground. To be similar to human walking cycle,
the period of double support phase for the biped is considered
as 20% of the whole cycle [15]. It can be shown that having
ankle and hip trajectories is sufficient for uniquely
formulating the trajectories of all leg joints [17].
Xed
Xsd
Lan
Laf
Lab
qb
Hao
Lao
qf
Ds
Ds
Figure. 1. The gait and geometrical parameters of the biped robot
In this paper, all calculations are carried out for one leg but
they can be repeated for the other leg. Using experimental
data, captured from real human motion, it is possible to
consider each joint's location at specific points. Interpolating
these points by a particular function that both matches human
motion and is compatible with geometrical and boundary
conditions, it is possible to plan trajectories for the ankle and
hip. These functions must be smooth, which means, the
functions should be differentiable and their second derivative
be continuous. For this purpose, polynomial spline and
sinusoidal functions are used here that satisfy these
requirements. In the following sections the trajectory for the
ankle and the hip joints are explained and compared to human
captured data.
A. Ankle Trajectory
A fourth degree polynomial is used for the ankle trajectory
when the foot is in the swinging motion, while a sinusoidal
function is used for the interval of foot-ground contact as
shown in Equation 1. The use of the latter function makes the
foot trajectory compatible with geometrical constraints.
Applying transitional conditions (1), these two curves are
joined smoothly as shown in Figure 2. In these trajectories, the
foot angle is designed to vary to better match the human
motion. If the foot is assumed to be always level with the
ground, (like in most previous works), the biped’s speed will
be reduced due to shorter steps. In this equation, F (t ) and
G (t ) are 4th and 5th order polynomial functions, respectively,
and p is used for adjusting swing acceleration. Function
Fk (t ) and Gk (t ) in the Equation 2 are calculated using the
constraints given by Equation 1.
IEEE/ASME Transactions on Mechatronics, No. xx, Vol. xx
⎧⎪ xa = Laf (1 − cos qb ) + Lan sin qb
t = Td : ⎨
⎪⎩ y a = Laf sin qb + Lan cos qb , θ a = − qb
t = Tm : {xa = Lam , y a = ham , θ a = qm
(
)
⎧ xa = 2 Ds − Lab 1 − cos q f − Lan sin q f
⎪⎪
t = Tc : ⎨ y a = Lab sin q f + Lan cos q f
⎪
⎪⎩θ a = q f
(1)
3
Fx (0) = xed
Fx′ (0) = G x′ (Tc )
G x (Td ) = Ds − x sd
G x (Tc ) = Ds + xed
Fx′′(Td ) = G x′′ (Td )
Fx′′(0) = G x′′ (Tc )
C&D
Fy (T f ) = hmax
Fy′ (Ts ) = G ′y (To )
Fy (Ts ) = hmin
G y (T f ) = hmax
Fy′′(Ts ) = G ′y′ (To )
G y (To ) = hmin
Fy′′(T f ) = G ′y′ (T f )
Fy′ (T f ) = G ′y (T f )
(5)
Ts = 0.5Td , T f = 0.5(Tc − Td ), To = Tc + 0.5Td
⎧ xa = Laf (1 − cosθ a ) − Lan sin θ a
⎪
0 < t < Td : ⎨
t p
⎪ y a = − Laf sin θ a + Lan cosθ a , θ a = − qb ( T )
d
⎩
Td < t < Tm : {xa = F1 (t ), y a = F2 (t ), θ a = G1 (t )
Y-hip
(mm)
X-hip
(mm)
Tm < t < Tc : {xa = F3 (t ), y a = F4 (t ), θ a = G2 (t )
⎧
⎪ x = 2 D − L (1 − cosθ ) − L sin θ
s
ab
a
an
a
⎪⎪ a
Tc < t < Tc + Td : ⎨ y a = Lab sin θ a + Lan cosθ a
⎪
T + Td − t p
⎪θ a = q f ( c
)
Td
⎩⎪
(2)
(II)
Time (s)
(I)
Time (s)
Figure 3. (I)-Hip Trajectory in X-Direction, (II)-Hip Trajectory in YDirection (dashed line is the human recorded data and the solid line is the
generated trajectory)
Tc + Td < t < 2Tc : {xa = 2 Ds , y a = Lan , θ a = 0
Y-Ankle
(mm)
X-Ankle
(mm)
(II)
Time (s)
(I)
Time (s)
Figure 2. (I)-Ankle Trajectory in X-Direction, (II)-Ankle Trajectory in YDirection (dashed line is the human recorded data and the solid line is the
generated trajectory)
B. Hip Trajectory
The hip trajectory is interpolated with third–order splines
assuming to have its highest position midway in the single
support phase and its lowest midway in the double support
phase. This choice of trajectory minimizes the energy
consumption of the body. This is because of the fact that with
this trajectory the body has the minimum height while it has
maximum velocity and vice versa. This trajectory is
represented in Equation 3 and is shown in the corresponding
curves in Figure 3. Transitional conditions for hip equations
are listed in Equations 4-5. The equations for the hip motion
in sagital plane in x and y positions are given as:
⎧⎪ x h = X [1,1]t 3 + X [ 2,1]t 2 + X [3,1]t + X [ 4,1] = Fx
0 < t < Td : ⎨
3
2
⎪⎩ y h = Y[1,1]t + Y[ 2,1]t + Y[3,1]t + Y[ 4,1] = Fy
⎧⎪ x h = X [5,1]t 3 + X [ 6,1]t 2 + X [ 7,1]t + X [8,1] = G x
Td < t < Tc : ⎨
3
2
⎪⎩ y h = Y[5,1]t + Y[ 6,1]t + Y[ 7,1]t + Y[8,1] = G y
(3
)
−1
X 8 x1 = A8 X 8 × B8 x1 ,
−1
Y8 x1 = C8 X 8 × D8 x1
A, B, C, D matrixes are formed according to the
constrained equations, which are given as:
Fx (Td ) = Ds − x sd
Fx′ (Td ) = G x′ (Td )
(4)
A&B
C. Experimental Method
Based on the proposed method for trajectory generation, the
joint trajectories are dependent on three sets of parameters.
The first set is the geometrical parameters, such as the length
of each segment, that can be calculated based on the captured
human motion. In Table 1, all the physical properties of the
segments are given with respect to the mass and height of the
robot. M and H are the mass and height of the robot,
respectively, while L denotes length of the segment. The
second set of parameters is a set of the gait parameters, which
are Tc , Tm , qb , q f , Lao , H ao , hmin , hmax . The last group of
parameters is a set of control parameters, which are xed
and x sd .
TABLE I
PHYSICAL PROPERTIES OF THE SEGMENTS OF THE BIPED ROBOT
Mass
Radius of
Gyration
Length
Foot’s
Length
Trunk
0.678 M
Thigh
0.1M
Shank
0.0465M
Foot
0.0145M
1.142Ltr
0.323Lth
0.302Lsh
0.475Lf
Ltr =0.52H
Lth = 0.245H
Lsh = 0.246H
Lf = Lab +Laf
Lf = 0.152H
Lab = .33 Lf
Laf = .67 Lf
Lan = .39H
It is worth noting that the segment properties are
proportional to the total height and total mass of the body and
can be calculated easily. The gait parameters are calculated by
capturing the human motion and then normalizing the
parameters with respect to the total height and the time cycle.
To capture the human motion, a subject who had reflective
markers on the lateral second metatarsal, the lateral malleolus,
the heel, the center of rotation of the knee and the greater
trochanter was used. Eight infrared cameras captured the
motion at 120 Hz. These data were passed through a fourthorder Butterworth filter. Using the filtered data, each joint
angle and the orientation of the segments were calculated at
each time step. The gait parameters are estimated and are used
IEEE/ASME Transactions on Mechatronics, No. xx, Vol. xx
in the filter.
The filter that is the focus of this paper adjusts the control
parameters (the third group of parameters) and highlights the
time cycle limit (speed limit).
III.
There are several methods to ensure the stability of a
particular walking trajectory. As described above, we utilized
the Zero Moment Point (ZMP) criterion. ZMP is a wellknown concept for the synthesis of a walking pattern. For the
biped robot, the ZMP is the point where the resultant of the all
reaction forces is applied. In other words, it is a point about
which the total moment of all the external forces zero. The
ZPM criterion states that as long as the zero moment point is
within the convex hull of all contact points, the stability of the
robot is guaranteed and the robot can walk [23]. In the biped,
the stable region is the surface of the support foot (feet). The
location of the ZMP is given by Equation 6.
x zmp =
n
n
i =1
i =1
∑ mi ( &y&i + g ) xi − ∑ mi &x&i yi − ∑ I iθ&&i
i =1
n
∑ mi ( &y&i + g )
σ 0 and σ 2 are Coulomb and viscous friction parameters and
σ 1 provides damping for the tangential compliance, also
another function ‘ α ( x, z ) ’ is used to achieved stiction. z& and
α ( x, z& ) are governed by Equations 8 and 10 respectively.
i
STABILITY OF BIPED
n
4
(6)
i =1
The distance between the boundaries of the stable region
and the ZMP is called the stability margin. The stability is
directly affected by this distance. To find the safest trajectory,
it is necessary to evaluate all possible trajectories planned by
the method described above to satisfy this stability criterion.
The pattern in which the largest stability margin is gained
should be selected as the optimum walking pattern.
IV. DYNAMIC SIMULATION
The foot-ground reaction forces (contact forces) are
important items in the modeling process of walking robots.
Using the equations that best describe the physics of contact
and reduce the frequency of switching between equations is
essential in avoiding high frequency switching between
differential equations, which are usually hard to solve
numerically. It is well known that classical friction models,
such as Coulomb and Karnopp in which the relation between
friction forces and the relative velocity between contact
surfaces are discontinuous, generate discontinuity in the biped
robot model. This could result in difficulty in integrating
equations of motion into one equation. In this work after
considering different types of friction laws, the Elasto-Plastic
law which renders both pre-sliding and stiction is selected
[25]. In order to remove the discontinuity of the contact
equation in Elasto-Plastic Model, the parameter ‘z’ is used to
present the state of strain in frictional contact. The difference
of this model with the LuGre friction model, is an extra
parameter, which provides stiction and excludes pre-sliding.
This way the Elasto-Plastic model is more comprehensive.
Using this model, the equation of friction may be written as:
f f = σ 0 z + σ 1 z& + σ 2ν
σi > 0
(7)
⎛
⎞
σ0
σ0
z& = x& ⎜⎜1 − α ( x, z& )
sgn( x& ) z ⎟⎟ i ∈ Ζ,
>0
&
&)
f
x
f
(
)
ss
ss ( x
⎝
⎠
f ss (x& ) is expressed by:
(8)
f ss ( x& ) = f c + ( f ba − f c ) e H , H = − x& / vs
(9)
‘ f c ' is obtained by Coulomb law. ‘’ is the relative velocity
between the two contact surfaces and ‘ fba ’ is the breakaway
force. Finally, ‘ vs ’ is the characteristic velocity.
⎧0
, z < zba
⎪
z
(
0
.
5
(
z
z
))
−
+
⎪
max
ba
α ( z, x& ) = ⎨0.5 sin(π
) , zba < z < z max
z max − zba
⎪
⎪1
, z > z max
⎩
(10)
‘ zba ’ is a breakaway displacement and is defined such that
the models behave elastically for z < zba .
For the normal reaction of the contact surface, the Young's
modulus was used to account for the elasticity of foot and
ground material and is given by Equation 11.
(11)
N = Kδ + cδ&
‘ K ’ is the modulus of elasticity; ‘ c ’ is the structural
damping coefficient of contact material; and ‘ δ ’ is the
penetration depth of the contact foot into the ground.
In order to apply the above-mentioned model to the foot
ground interaction, the foot is divided into segments. The
described model is employed for each segment by using
Equations 7-11. Finally, summing the effects of all the
segments, total surface reaction on the foot is obtained.
Equation 7 changes the traditional two-state friction force to a
single state function, which avoids the switching problems in
the model.
V. SMA MODEL
SMA actuators provide an interesting alternative to
conventional actuation methods. Their advantages include
drastically reduced size, weight, and mechanical complexity.
SMAs also have disadvantages, which must be thoroughly
considered prior to application. Compared to more
conventional actuators, they operate with a low efficiency, at
low bandwidths, and with small displacements. To overcome
the small displacement of SMA actuators, the mechanism used
by Elahinia and Ashrafiuon, as shown in Figure 4 was used at
each joint [1].
IEEE/ASME Transactions on Mechatronics, No. xx, Vol. xx
Pulley
SMA Wire
Bias Spring
Body rotates
as the stress of
SMA wires
change
Body
Pulley
The dynamic model of the arm including the spring and the
load of the segments is represented by:
(12)
‘ τ w ’, ‘ τ g ’ and ‘ τ s ’ are the resulting torques from the SMA
wire, gravitational loads, and the bias spring, respectively, and
‘ σ ’ is the wire stress. ‘ I e ’ is the effective mass moment of
inertia of the segment, and ‘ cr ’ is the torsional damping
coefficient approximating the joint friction. The details on the
assumption and constants, used in Equation 12 can be found
in [1]. For simplicity, the above dynamic equation can be
expressed as:
I eθ&& + h(θ ,θ&) = 0.25π (n d 2σ rp )
(13)
Function ‘h’ includes viscous damping ,the spring and
gravitational terms; ‘n’ is the number of SMA wires; ‘d’ is the
radius of the SMA wire; and ‘ rp ’ is the radius of the pulley.
The strain of the wire causes the rotation of the segment and
the two pulleys attached to it. Therefore the SMA wire strain
‘ ε ’ and joint angle ‘ θ ’ are kinematically related as shown in
Equation 14.
ε=
−2r p (θ − θ 0 )
l0
+ εl
(14)
‘ l 0 ’ is the initial length of SMA wire, ‘ θ 0 ’ is the initial
position of the manipulator, and ‘ ε l ’ is the maximum strain of
the SMA wire.
The thermo-mechanical behavior of SMAs can be described
in terms of strain ‘ ε ’, martensite fraction ‘ ξ ’, and
temperature ‘ T ’. In the most general form, the thermomechanical constitutive equation is [2, 26]:
dσ = D(ε , ξ , T )dε + Ω(ε , ξ , T )dξ + Θ(ε , ξ , T )dT
desired temperature. Therefore the heat transfer model
represents the dynamics of the actuator as a first order system
and can give the response time of the SMA actuator.
The heat transfer equation consists of electrical heating and
natural convection as shown in Equation 16. The response
time of the SMA actuator therefore is shown in Equation 17.
mc p
dT V 2
=
− hc Ac (T − T∞ )
dt
R
ρ cp
τ=
d
4hc
(16)
(17)
‘R’ is resistance, ‘ c p ’ is the specific heat, ‘m’ is mass, ‘ ρ ’
Figure 4. A simple model of SMA actuator used at each joint
I eθ&& + c rθ& + [τ g (θ ) + τ s (θ )] = τ w (σ )
5
(15)
D (ε , ξ , T ) represents the modulus of the SMA material,
Ω(ε , ξ , T ) is the transformational tensor, and Θ(ε , ξ , T ) is
related to the thermal coefficient of expansion.
The shape memory effect as the actuation mechanism of
the SMAs is caused by the phase transformation of the
molecular structure between martensite and austenite, and can
be defined by two models: one for martensite to austenite
transformation and the other for austenite to martensite
transformation [2, 3, 27]. Equations 12-15 show that in order
to reach a desired position, the SMA wire should reach the
is the density, and ‘ Ac ’ is circumferential area of the SMA
wire. ‘ V ’ is the applied voltage, ‘ T∞ ’ is the ambient
temperature, and ‘ hc ’ is heat convection coefficient.
To adjust the actuator speed, the actuator at each joint will
be selected by specifying the geometry and number of
required SMA wires. This way, using Equation 10, we can
find the maximum required stress based on the maximum
required torque. By increasing the diameter of the SMA wire
or the number of wires it is possible to produce more torque. It
should however be noted that the maximum stress of the wire
is limited and by increasing the number of wires the required
actuation voltage will increase.
Introducing ‘ M max ’ and ‘ σ max ’ as the maximum required
torque and the maximum allowed stress, we can rewrite
Equation 13 as Equation 18.
M max = 0.25π (n d 2σ rp ) ⇒
nd 2 =
4 M max
πσ max rp
(18)
The length of the wire can be calculated based on the range
of the joint angles and the maximum stress of the SMA wires.
The last two calculations can be done by comparing the
reference trajectory and Equation 11, where the maximum
strain is 8%. It is worth noting that the length of wires has no
effect on the response time.
VI.
FILTER ALGORITHM
As the flowchart of the filter shows in Figure 5, the only
input of the filter is the total height and weight of the robot.
By using the proportional data provided in Table 1, all the
physical properties of the segments are calculated. The gait
properties are calculated and fixed from the captured data of
the human body. Next, the control parameters are set to zero
and the whole trajectory is calculated. This trajectory is
checked by the ZMP criterion. If the trajectory is inside the
stable region, the filter will go to the next step; otherwise, the
control parameters will be increased.
The required torque is estimated in the dynamic simulation.
This parameter and the maximum rotational angle are used to
specify the geometry of the actuator at each joint. In the next
step based on the selected actuator the delay time of each
SMA actuator is calculated and the real trajectory of each joint
is determined. Again, the new trajectory is checked by the
ZMP criterion and if the trajectory is stable, the controlled
parameters, as well as the minimum distance of ZMP to the
IEEE/ASME Transactions on Mechatronics, No. xx, Vol. xx
6
stable margins, will be recorded. After this step, the loop is
continued by increasing the control parameters. This
procedure continues until the control parameters overflow half
of the step length. At the end, using a search algorithm, the
most stable trajectory is selected among the recorded control
parameters. If there is no stable trajectory, the time cycle will
be highlighted which means that the SMA biped cannot have a
stable walking at the specified speed. To solve this problem
the time cycle should be increased.
Robot’s height
and mass
Segment
Properties
(I)
Captured data of
Human Motion
(II)
Time (s)
(I) Torque (Nm) at ankle
joint vs Time
Gait parameter
Trajectory
Generation
ZMP
Criterion
satisfied?
Control Parameters
(xed,xsd)
(III)
N
Increase (xed,xsd)
Dynamic
Simulation
Actuator
Selection
Calculating the
delay time
Store
xed and xsd
N
N
xed > Ds/2
xsd >Ds/2
Y
Search
Algorithm
Figure 5. Flowchart of the Filter for finding stable walking trajectories for a
SMA actuated biped robot
Display xed, xsd
VII.
at knee
(III) Torque (Nm)
joint vs Time
at hip
Time (s)
Figure 6. Required torque in 2.5 cycles for joints of the biped right leg
M max
Real Trajectory
ZMP
Criterion
satisfied?
Y
(II) Torque (Nm)
joint vs Time
It was shown in equation 14 that the actuator response time
is linearly proportional to the diameter of the SMA wires. In
order to reduce the delay, SMA wires with smaller diameter
are preferred. To maintain the same level of torque at the
joints, the number of wires should increase according to
Equation 18. It can be shown that the applied voltage to the
SMA actuator has the following relationship with the number
of wires:
πσ max rp 14 54
V = l0 2 ρhc (T − T∞ ) (
) n
(19)
Y
SMA Actuator
Time (s)
RESULTS
The filter is simulated for a SMA actuated biped with a
height of 40 cm and weight of 1kg. As shown in Figure 5, the
generated stable trajectory goes through the dynamic
simulation, which calculates the maximum torque required at
each joint. Figure 6 illustrates the required torque for ankle,
knee, and hip joints for 2.5 walking cycles. The sudden jumps
in this figure are because of the inaccuracy in the contact force
model and the numerical method used for solving the dynamic
equations. Neglecting these sudden jumps the maximum
required torque at the ankle, knee and hip are 0.5, 0.8 and 0.2
Nm, respectively.
Based on Equation 16, the required number of wires is
depended on the maximum applied voltage, which is assumed
to be 18 volts. Therefore, based on Equations 17-19, the
response time of the actuators will be 0.9, 1.14, and 0.57
seconds for ankle, knee and hip joints, respectively. The
output of the filter, as shown in Figure 7, is a surface that
shows the minimum distance of the ZMP trajectory to the
stability margin. The figure shows this quantity as a function
of the control parameters. Negative distance in this figure
indicates instability of the trajectory. Maximum stability is
obtained with xed of 2.5 cm and x sd of 1.5 cm.
Minimum distance to the
stability margin (mm)
xed (cm)
xsd (cm)
Figure 7. Minimum stability with respect to control parameters
Simulation results are shown in Figures 8-12. In these
IEEE/ASME Transactions on Mechatronics, No. xx, Vol. xx
figures, the solid line represents the reference trajectory while
the dashed line indicates the real trajectory obtained by SMA
actuation. In the simulations, the cycle time and step length
are 3 seconds and 12 centimeters, respectively. The effect of
the SMA actuator delay for the joints and the segments is
shown in these figures. For better comparison of the reference
and actual segment orientation, Figures 11-12 are plotted with
respect to the time cycle. It is worth noting that the thigh
angle, which is not shown, is exactly the same as the hip
angle, but with a shorter delay with respect to the reference
angle.
Using these segment angles, the real Cartesian joint
coordinates, resulted from the SMA actuation, are calculated.
Figure 13 illustrates the most stable ZMP, which is calculated
based on the real joint coordinates.
__ Reference angle
--- Real angle
(Time cycle is 3 s)
Ankle angle
(rad)
Time (s)
Figure 8. Reference ankle angle compared to a SMA actuation angle
__ Reference angle
--- Real angle
(Time cycle is 3 s)
Knee angle
(rad)
7
__ Reference angle
--- Real angle
(Time cycle is 3 s)
Foot angle
(rad)
Time (s)
Figure 11. Reference foot angle compared to its real angle after SMA
actuation
__ Reference angle
--- Real angle
(Time cycle is 3 s)
Shank angle
(rad)
Time
Figure 12. Reference shank angle compared to its real angle after SMA
actuation
X-ZMP
(cm)
Stable Margin
Stable Margin
Time (s)
Figure 9. Reference knee angle compared to a SMA actuation angle
__ Reference angle
--- Real angle
(Time cycle is 3 s)
Hip angle
(rad)
Time
Figure13. The ZMP trajectory in the X direction (time cycle is 3 s)
VIII. CONCLUSION
Time (s)
Figure 10. Reference knee angle compared to a SMA actuation angle
In this paper, we described a filter to adapt the walking
pattern of SMA actuated biped robots. The filter combines the
SMA actuator dynamics with ZMP stability criterion to
generate a stable walking pattern. All the physical parameters
of the robot as well as the actuation response limitation of the
SMA materials are included in the calculation. SMA actuators
have an excellent power to mass ratio. As a result, the ratio of
the physical parameters of the SMA-actuated biped robot is
closer to the ratio of parameters of a human body than those of
similar motor-actuated biped robots. This way, a SMA-
IEEE/ASME Transactions on Mechatronics, No. xx, Vol. xx
actuated biped robot can potentially be used to simulate the
human walking behavior more accurately. In converting the
human trajectory to the desired trajectory for the simulation, a
hip motion is formulated based on two control parameters.
This makes it possible to derive a highly stable, smooth hip
motion without first designing the desired ZMP trajectory.
In this work, the control of the SMA actuator is not
considered and the dynamics of the SMA actuator is modeled
with a delay time. In the following work, the control problem
of the actuator will be investigated.
APPENDIX
Physical Properties of SMA wires used in this study
Parameter
Description
Unit
Value
hc
Heat convection coefficient
J / m .°C Sec
150
cp
Specific Heat of Wire
Kcal / Kg .°C
0.2
2
ρ
Density
gr / cm
T∞
Ambient Temperature
°C
3
6.5
20
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Ehsan Tarkesh Esfahani (M’06) is a Graduate
student in department of Mechanical, Industrial
and Manufacturing Engineering at the University
of Toledo (OH). His main research interest is
humanoid robots and application of smart
materials in active orthosis/prosthesis. He received
his B.Sc. in Mechanical Engineering from Isfahan
University of Technology (04).
Mohammad H. Elahinia (M’01) is an Assistant
Professor in the Mechanical, Industrial, and
Manufacturing Engineering at The University of
Toledo. His main research interest is in modeling
and control of smart material systems. He received
his M.S. from Villanova University (01) and his
Ph.D. from Virginia Polytechnic Institute and
State University (04).