A.I.Memo No. 1331
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 .
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 Nsm, Nsm, 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 Nsm.
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, 25C ), 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 Nsm. The resistance per unit length
where Nsm, , and d is the inner diameter. For the tube described above, the resistance per unit length is Nsm, thus the total resistance for the tube is Nsm. Using equation 2 to compute the resistance of the irrigation holes, we have a resistance Nsm for holes mm in diameter. However with 80 holes/cm and 10 cm of tube, the total resistance is Nsm.
For the acid and base systems we will have six irrigation tubes each, resulting in a total fluid resistance of Nsm The relation between pressure and flow is again given by,
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.
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.
The drain line consists of a single tube cm and a check valve. Employing equation 2, this results in a fluid resistance of Nsm, for a drain line 20 cm in length.
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
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
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 25C ) 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.
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
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
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.
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
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
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,
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,
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.