A.I.Memo No. 1331
November, 1991
David L. Brock
In 1950 W. Kuhn and A. Katchalsky produced a fiber which contracts in
response to changes in pH (Kuhn 1950). In their demonstration a
single fiber immersed in a fluid repeated raised and lowered a weight
proportional to the hydrogen ion content of the solvent. It was
proposed at the time that such a device could be used as a linear
actuator. One of the major drawbacks however was the exceeding slow
response time, on the order of minutes. Since then a number of
innovations have made this artificial ``muscle'' worthy of further
study. First, fibers have been produced which contract in seconds and
even tenths of seconds (DeRossi 1987; Suzuki 1989). Second, some
fibers support considerable loads, on the order of 100N/ (DeRossi 1988). The
contraction rates and forces have become comparable, and in some cases
even exceeding, that of a human muscle. Third, the physics of fiber
contraction (or gels as they are more generally called) have become
well understood (Tanaka 1973a - 87b). Finally, technological
innovations such as robotics and implantable artificial biological
organs have created a demand for such devices. A number of
contractile gel devices have already been constructed, such as a robot
gripper (Caldwell 1990), a multifingered hand (Toyota 1990), and an
artificial urethral sphincter (Chiarelli 1988; DeRossi 1985; 86).
The design of these devices from a practical engineering perspective, particularly with regard to dynamic modeling and control has only been considered recently (Genuini 1990).
In this paper we will consider the design of a simple linear actuator based on contractile gel fibers as well as the design and simulation of single mechanical linkage controlled by two antagonist muscles. The first section will discuss the design of a simple linear actuator and the second will present a dynamic model along with approximate parameter estimates. The third section will introduce a nonlinear sliding mode controller which achieves desired trajectory tracking for model inputs.
Figure 1-1. Two antagonist artificial muscles control a single link.
Although there are many possible actuator designs, this paper will focus on the direct chemical to mechanical energy conversion through the control of the hydrogen ion content in the muscle fibers. The basic concept is to use high performance miniature valves to the control the inflow of acid or base to modulate pH and control contraction. A schematic diagram of the proposed actuator is shown in figure 2-1. Two input lines containing 0.1M HCL and 0.1M NaOH enter the muscle at the base. A miniature value placed into the base material controls the inflow of fluid. An irrigation system design to facilitate mixing is composed of numerous small tubes which are interspersed among the fibers. Finally, a drainage tube allows the waste fluid (i.e. salt water) to be removed.
The tubes and wetting surfaces of the valves are constructed from teflon which is chemically inert and the fibers are made from Poly-Vinyl Alcohol PVA. Although many different contractile materials exist (Tanaka 1990), PVA has been used successfully by a number of researchers and has a relatively high tensile strength. The tendons are made of Spectra and the tendon to PVA connections are machined Delran. The fibers are affixed at their terminations with epoxy which is also chemically inert.
The dynamic model of the system is composed of three basic parts: the fluid conveyance system, the hydrogel contractile fibers and the mechanical linkage. The following sections will address each of these systems in detail.
A schematic diagram of the fluid system is shown in figure 3-1. It is
assumed the inlet fluid line is under a moderate pressure and a microvalve with controllable
resistance
modulates the
inflow of fluid. The system of irrigation lines is modeled as a fluid
resistance
and an inertance
I. Finally, the compliant sheath into which the fluid flows is
modeled as a capacitance C and the exit line, a resistance
.
Valve
Although there are commercially available metering valves which continuously modulate flow, they are generally too large from this application. Commercial piezo-electric or solenoid fluid values also exist in the appropriate size, although they have only two states, open and closed. It may be possible to use binary state valves in a pulse width modulation scheme to meter fluid, but this may introduce undesirable affects such as water hammer, excessive part wear and slower response. Alternatively there are some experimental metering valves and pumps which are of the correct dimension (in fact some are significantly smaller than this application demands creating the possibility of very small linear actuators). It may also be possible to use molecular valves in the form of biological or artificial membranes whose porosity is controlled by small voltages.
In any case it is assume some mechanical valve of the appropriate dimension can be constructed, either a commercial two-state valve or an experimental multi-state valve. Based on commercial data sheets for fluid resistance we can approximate the resistance of a valve by
where is fluid resistance in N
sm
,
N
sm
,
m/v, and
are constants. For two-state valves we will assume
and for multi-state valves
, where
is some small value and u is the
control signal input, 0 to 12 v. These parameters where roughly based
on values given for the LEE Interface Fluidic 2-way (normally closed)
microvalve. This valve is energized with 12 v at 250mW, switches in
1.5 msec and operates in a 0 to 7 psi pressure range. The relation of
pressure to flow is given by
where is
flow rate in m
/s,
is pressure in N/m
and
is fluid resistance N
sm
.
Irrigation System
Multiple porous tubes attempt to
uniformly enter and mix the fluid to increase the response time of the
actuator. Consider a single tube as shown in figure 3-2. The tube has
an inner radius mm and a
straight length l = 10 cm after a 90
elbow of radius
cm. Small holes of radius
mm are punctured in the teflon tubes. The holes are
small enough so that an initial pressure
is necessary to induce flow. This prevents back flow
into the acid-base lines, but for this initial model we will assume
is zero. The holes are place
at 90
intervals about the
cross section and every 1 mm along the length.
Suppose the tube
as we describe is non-porous and open at the end. If we assume a
fully developed flow under an entrance pressure of 1 kPa (above atmospheric), the fluid flow is
where
kg/m s (for water, 25
C ),
m, and l = 0.1 m.
This yields a flow of
m
/s and an average fluid velocity
of
m/s. The Reynolds number
is Re
, where
,
m and
m
/s, which is below the critical
value Re
; therefore we may
assume the flow is laminar. For higher pressures, equation
1 yields Reynolds numbers in the
turbulent range, but the significant resistance offered by the sealed
porous tube reduce the flow rates and maintains laminar conditions. As
we will later show the maximum flow rate is approximately
m
/s, which yields
m/s and
which is certainly laminar. Finally, the entrance region before the
fully developed flow is approximately
Re
cm;
therefore we can assume the boundary layer is fully develop within the
entire length of the tube.
The wall shear force will result in a head loss and thereby create non-uniformity flows through the small irrigation holes. However, since the flow will decrease along the length due to seepage, the head loss will not be linear as in the case of homogeneous circular tubes. To approximate the head loss we will assume continuous porosity and a uniform fully developed flow at each point x of tube. The wall shear stress is given by
which results in a pressure gradient
If we assume a flow loss due to porosity
we have the resulting differential equation
where . If we solve this differential
equation with boundary conditions
and
, we
have the head loss as a function of length
where
m
. The head loss in the tube
is not significant in this case as shown in figure 3-2.
For the purposes of the dynamic model, it may be more more useful to
represent the entire porous tube system as a fluid resistance . Following a simplified
convention, we will assume the fluid resistance is given by
where R is fluid
resistance is the resistance
due to the entrance,
is the
resistance per unit length and l is the length of the tube.
The entrance resistance is given by
where d is the inner diameter of the tube in meters, and
. For the tubes
cm in diameter
N
sm
. The resistance per unit length
where N
sm
,
, and d is the inner
diameter. For the tube described above, the resistance per unit
length is
N
sm
,
thus the total resistance for the tube is
N
sm
. Using equation 2 to compute the resistance of the
irrigation holes, we have a resistance
N
sm
for holes
mm in diameter. However with 80 holes/cm and
10 cm of tube, the total resistance is
N
sm
.
For the acid and base
systems we will have six irrigation tubes each, resulting in a total
fluid resistance of N
sm
The relation between pressure and flow is again given
by,
Inertance
Although the inertance of the fluid is small, fast switching times may cause its effects to be important. The inertance is given by
where kg/m
is the fluid density,
m is the length of the tubes and
m
is the total cross sectional area. For this system,
the fluid inertance is
kg/m
and
the dynamics are given by
Suppose the fluid accelerates to m
/s in 10 ms under a pressure of 1 kPa. Then the
inertance I would be
kg/m
. This is comparable to
the value given for the model. So for switching times on the order of
1 ms, inertance effects become significant.
Capacitance
In this particular design, the fiber system is covered with a
compliant sheath. Since it is flexible, it can be approximated as a
fluid capacitance; that is, when fluid is forced into the chamber the
sheath will stretch and resist the flow. If we model the sheath as a
thin-walled uniform cylinder with closed ends, the elemental stress
tensor is given by ,
,
and
,
where P is the pressure, t is the wall thickness and
r is the radius. Given the generalized stress-strain relations,
and the strain-displacement equations
we can solve for the change in volume as a function of pressure
where E is the Elastic Modulus and is Poisson's ratio. Therefore the fluid capacitance
is
If r = 1
cm, l = 10 cm,
N/m
(approximate for
rubber),
and
mm,
With a fluid flow of m
/s, this
capacitance will become significant if the pressure changes more than
1 MPa in a 1 ms. Therefore we will ignore the capacitance.
Drain
The drain line consists of a single tube cm and a check valve. Employing
equation 2, this results in a
fluid resistance of
N
sm
, for a drain line 20 cm in length.
Fluid
dynamic equations
The elemental pressures are related by
and the flows by
Substituting the elemental relations, we have
where Q is the flow into the fibers. Note the steady state flow rate for the open valve is
For a fully open value m
/s. Note the total volume of injected fluid is
Mixing and
Diffusion
When the fluid (either acid or base) is
injected into the fiber network there is a certain lag time before the
entire system uniformly equilibrates to the new concentration. We
could model this mixing as the diffusion of HCl or NaOH in water. If
the tubes are uniformly spaced as in figure 3-3, we can draw
concentric circles around each with radius equal to half the distance between the tubes. The
mixing time could be approximated, as the time for the fibers at
radius
to experience a
certain concentration of fluid. As a simplification we can
approximate the diffusion with the one dimension diffusion equations
where J is the diffusion flux per unit area, c is the concentration in moles per volume and D is a constant. The solution is the standard diffusion model
where
The boundary
concentration , of course,
decreases as particles diffuse into the liquid. However, as a first
order approximation to the diffusion time, we assume
is constant. Then the concentration at
is one-half its initial value
when
ms for
m
/s (0.1M HCL at 25
C ) and
m.
The diffusion model, however, is highly inaccurate since the exit velocity from the pores in the teflon tubes is relatively high and together with the motion of the fibers will cause turbulent mixing, not diffusion. Therefore as a simple approximation, we will assume the fluid mixing and diffusion is a first order lag, hence
where c is the concentration of acid or base in the solvent,
is a time constant which we
will take to be
ms (based
roughly on the previous discussion) and
ratio of solute to solvent, where
is the total volume of the actuator,
m
.
pH
The contractile fibers respond roughly linearly to pH, so it is important to calculate the pH in the solution based on the amount of acid or base entered into the chamber. An acid is a chemical species having the tendency to lose a proton and a base is a species tending to accept a proton. The quantitative measure of acid strength is the acid dissociation constant; that is the equilibrium constant for the reaction
Along with this reaction there is always the self dissociation of water
If the original amount of acid A added to the solution is [HA]o, then the material balance relation is given by
Also the solution must be electrically neutral, hence
Solving these equations for the hydrogen ion concentration yields the cubic
However, we may assume that the from water is negligible
compared to that from the acid, hence
. We may
assume that for strong acids there is complete dissociation; that is
, which yields the trivial result
Similarly, for base a B we have equilibrium
mass balance,
and electric neutrality
Now we can assume total dissociation
which is a valid except for very high concentrations. Also we
can assume the contribution of from self-ionization of water is
negligible, hence
Considering the strong acid and base together we can assume
Since pH is , we have
Diffusion
Besides the lag due to mixing among the fibers, there is a lag due to the diffusion within the fibers themselves. This lag is due to bulk macromolecular rearrangement of the polymers in the network and is approximately proportional to the square of the cross sectional diameter. The time constant is
where d is the diameter in meters and s/m
.
Thus for fibers 10
m in diameter contract in approximately 0.1 s.
This can also be approximated to the first order by
where , pH is the pH in the fiber
and pHs is that in the
solvent. Notice this time constant is substantially slower than the one
given in equation 3. Therefore we will ignore the lag due
to mixing and consider only the diffusion within the fiber as the major
delay.
Hydrogel Physics
A gel is a cross-linked network of polymers immersed in a fluid, which can undergo a volume phase transition in response to changes in external conditions such as temperature, solvent, pH, light and electric fields. Three competing forces act on the polymer gel network: rubber elasticity, polymer-polymer affinity and hydrogen ion pressure. Competition between these forces, collectively called the osmotic pressure, determine the equilibrium volume of the network. The phase transition has been analyzed in terms of the mean field theory of swelling equilibrium of gels. The osmotic pressure of a gel is given by Flory's equation,
where N is Avogadro's number, k is the Boltzmann constant,
represents the difference
between the free energies of a polymer segment-segment
and polymer-solvent interaction, and
and
are the enthalpy and entropy respectively.
In equilibrium, the osmotic pressure is zero.
Thus by solving the above equation, we can theoretically
compute the fiber force. However because of its complexity and
because of the availability of empirical data, we can describe the
contraction of the fiber with the simple model shown in figure 3-4.
The intrinsic force per unit area is
is approximately
assuming a total cross sectional area m
,
the intrinsic force is
N(7-pH
).
Also the intrinsic stiffness is
and viscous damping is roughly estimated as
A model of the mechanical system is shown in figure 3-5. Assuming linear components for this part of the model, we have
where
Using the model as outlined above we have the following system equations
However before we design a controller for this system let us consider a number of
simplifications. First, the lag due to the inertance of the fluid is negligible
compared with the dynamics of the rest of the system. This was verified in a full
dynamic simulation of the system, although it is fairly obvious since the time
constant of the fluid flow is on the order of 10 ms while the diffusion lag is
about 100 ms. This assumption allows us to compute the flow Q directly from the
input u. Thus we can reduce the order of the system by two and take the
input to be the flow rate . In order simplify this problem further let us
assume a single-input single-output system, by taking the co-contraction
of the actuators as an offset force and controlling position through feedback
through a single actuator. With these initial assumptions let us write the system
equations in the form
Specifically,
where
and
We wish to control the joint angle using the simplified model
presented above. We would also like the controller to
maintain zero tracking error in the presence of modeling errors. Therefore,
we will design a non-linear sliding mode controller.
However before we attempt a construct a controller
for this fourth order system, let us initial consider one further simplification.
Based on the numerical estimates of the model parameters the damping ratio of
the linear mechanical dynamics is . Therefore let us initially
assume the inertia can be neglected, and produce a controller based on the
reduce third order system,
where
and
Now the controllability canonical form,
where ,
can be derived by differentiating the equation for the joint
angle three times.
Let be the tracking error on
and
define the sliding surface to be
, where
for some positive . Hence
Thus maintaining zero tracking error is equivalent to the first-order stabilization
problem in s. In addition a bound on s implies a bound on the tracking error
vector; that is if , then
. We must now design a controller
such that for all
,
for some positive .
The estimation of f is given by
and the bounds on the error is
Similarly, the term b is estimated by
where
. If we let
, then
and the control law
for s given above,
and
In this case we chose s
to be approximately equal to the natural
frequency of the system while the estimate errors to be approximately 10% the nominal values.
In order to get a sense of the model dynamics a 1Hz sinusoidal input was given to the system. A plot of the pH as a function of time is shown in figure 5-1 and the joint angle in figure 5-2. The sliding mode controller discussed in the previous section was implemented and simulated for a desired joint trajectory
Figure 5-3 shows the pH and figure 5-4 shows the desired and actual joint position. Notice
desired tracking is achieved after approximately 0.5 seconds. Figure 5-5 shows the effect
of neglected link inertia. Note the joint trajectory is offset by approximately 10 Fourth and fifth order sliding mode controllers were attempted to compensate for the link
inertia and fluid inertance, but were found to be numerically unstable in the calculation
of the higher order derivatives. Numerical filters corrected this problem, though the
simulation times became unacceptable long. A more judicious choice of numerical parameters
could possibly correct this problem.
A sliding mode controller produced reasonable tracking performance for input
sine waves on the order of 1-2 Hz, but deviated significantly at higher frequencies.
This seems reasonable based on the relatively long lag times introduced by the
polymer diffusion. It has been possible more recently, to produce even smaller
fibers (about 1 m), while using UV induced transectional cross-linking to
increase strength. These and other innovations should
allow faster response times for practical actuator development and controller
design.