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Actuator design using biomimicry methods and a pneumatic muscle system
ARTICLE IN PRESS
Control Engineering Practice 14 (2006) 999–1009 www.elsevier.com/locate/conengprac
Actuator design using biomimicry methods and a pneumatic muscle system D.W. Reppergera,, C.A. Phillipsb, A. Neidhard-Dollb, D.B. Reynoldsb, J. Berlinc a
b
Air Force Research Laboratory, WPAFB, OH, USA School of Biomedical, Human Factors and Industrial Engineering, Wright State University, Dayton, OH, USA c General Dynamics, Dayton, OH, USA Received 27 May 2003; accepted 11 June 2005 Available online 19 August 2005
Abstract An empirical and theoretical study is conducted on a special actuator termed ‘‘pneumatic muscle’’ (PM) being used in a force control system framework. Such an actuator has similarities to biological systems and has many advantages (extremely high power/ weight, power/volume and power/energy ratios). However, due to its inherent nonlinearities, this actuator suffers from poor position and force control. The study described here accomplishes three main goals. (1) A force control system is developed within an open and closed loop framework to emulate how biological systems work in an agonist–antagonist framework. (2) The PM used in the study has such strength that it excites the frame dynamics. This undesired dynamic response is then effectively cancelled using an impedance model control scheme. (3) The PM is demonstrated to both change length yet still produce force in a controlled manner. r 2005 Elsevier Ltd. All rights reserved. Keywords: Force control; Biomimicry design; Pneumatic muscle actuator
1. Introduction Biomimicry (bio-inspired or emulating biological systems in nature) provides a powerful framework from which to design and analyze systems (Benyus, 1997). It is difficult to argue that 3.8 billion years of evolution have not produced a host of successful designs in nature. By studying and emulating these successful products of the natural world, engineers can develop and improve upon present systems to generate both reliable and robust applications to work in difficult environments. For actuation control, many biological systems (e.g. the eye muscles or arm muscles—bicepts and tricepts) work on a principle of agonist–antagonist control. To describe agonist–antagonist control, in Fig. 1a, linear motion or force control can be achieved. Muscles only generate a force via contraction, i.e. a muscle can Corresponding author. Tel.: +1 937 2558765; fax: +1 937 2558752.
E-mail address: [email protected] (D.W. Repperger). 0967-0661/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2005.06.009
only ‘‘pull’’ and does not ‘‘push.’’ The right muscle (agonist) in Fig. 1a contracts and simultaneously the left muscles relaxes (antagonist, which increases in length), thus producing a force and motion on the mass to the right. In Fig. 1b, the same effect can be realized in a rotational sense by generating a clockwise angular position rotation or torque on the robotic joint through the contraction of the agonist and relaxation of the antagonist muscle. The key to the successful use of agonist–antagonist control lies in the properties of the actuator that can successfully contract and expand (relax). Control systems can be used in numerous ways to effect agonist–antagonist control. These applications are widespread in robotics (Hajian, Sanchez, & Howe, 1997) and bioengineering (Peterson & Chizeck, 1987; Zhou et al., 1996) and have been applied in the nuclear power industry (Caldwell, Medrano-Cerda, & Goodwin, 1995) using PM technology. An important actuator that can be used for agonist– antagonist control is a pneumatic muscle (PM) (Fig. 2)
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which has many similarities to biological systems. Such a device is analogous to animal skeletal muscle for the following reasons: (1) forces are only generated via contraction and (2) in a force or position control mode, such an actuator is highly nonlinear. There are many industrial uses of the PM system thus described. As discussed, applications appear widely in robotics (Caldwell, 1993; Caldwell et al., 1995; Noritsugu & Tanaka, 1997; Sira-Ramirez, Lopez, & Tondu, 1996), in nuclear power plants for position control of uranium rods (Caldwell & Favede, 1998), and for rehabilitation (Inoue, 1988; Schulte, 1961). Other advantages of this form of actuation include the fact that it fails gracefully (termed ‘‘soft actuator’’) and is considered to be safer than electric or hydraulic counterparts generating the same force level. In addition, such devices can be cheaply built and provide little contamination to the environment in which they are designed to work. Finally, in comparison to a hydraulic or electric actuator, the PM has the highest ratios of power/weight (1 wt/gm), power/volume (1 wt/cc), as well as power/ energy. To explain the operation of such a device, Fig. 2 illustrates that as the air pressure rises inside the bladder, enclosed within the outer sheath, the muscle contracts in the longitudinal direction yet expands in the radial direction. This produces a force externally on the outside environment which can perform work. A typical PM system may weight a few ounces and can lift 200–300 pounds which has significant advantages over traditional hydraulic or electrical actuators. One can
easily see how nonlinearities arise in the use of the PM system. Even if the pressure input is a linear time function, the volume resulting from this inflow would vary in a nonlinear manner with respect to the pressure input. Also, from the physics of this process, the contractile force generated on the external environment is proportional to the net change of the cross-section surface area affected via the inflation as follows: (1)
DForce ¼ ðPressureÞ DArea;
where Pressure refers to gauge pressure (air pressure inside the bladder above the atmosphere or external environment) and DArea refers to the change in the cross section area of the muscle as a consequence of the inflation. Of course, the area change in Eq. (1) is nonlinear with pressure input, since the system in Fig. 2 changes shape as it contracts. It is well documented in the literature that the PM has nonlinear models of position change with pressure (Chou & Hannaford, 1996;Repperger, Johnson, & Phillips, 1998). This is biologically similar or ‘‘mimics’’ skeletal muscle (Hill, 1938; Phillips & Petrofsky, 1983; Rome, 1997; Stein & Gordon, 1986; Winters, 1990). Little work, however, is reported in the literature using PM systems in a force control sense. Fig. 3 illustrates the force control regulation problem considered here, i.e. the framework in which it is desired herein to investigate the PM system. Force control methods are of considerable interest in a number of industrial applications including cutting processes (Carrillo & Rotella, 1997), Magnetic levitation systems (Yi et al., 1996), in milling processes (Charbon-
Linear Antagonist muscle B = Relaxation B
Motion x 0 Mass
Net Force on Environment
A Air Pressure Inlet Internal Air Bladder
Agonist muscle A = Contraction
Mass
(a)
Robot Link i-1
Antagonist muscle B = Relaxation B
θ = Rotational Displacement
Protective Outer Sheath Material Fig. 2. Operation of a pneumatic muscle system.
A Agonist muscle A = Contraction (b)
Force Input Command
Force Error Controller
To Robot Link i
Fig. 1. Agonist–antagonist control: (a) linear motion, (b) rotational motion.
-
Pneumatic Muscle System
Force Output
Fig. 3. Force control system for a pneumatic muscle.
ARTICLE IN PRESS D.W. Repperger et al. / Control Engineering Practice 14 (2006) 999–1009
naud, Carrillo, & Ladeveze, 2001), robotics (Siciliano & Villani, 2001), hydraulic actuators (Niksefat & Sepehri, 2000), in deburring applications (Liu, 1995), strip castings (Bernhard, Enning, & Rake, 1994), and pressure control (Alleyne & Liu, 2000). Traditional controllers, as well as intelligent based systems, have been of interest for these force control methods (Mattiazzo, Mauro, Raparelli & Velardocchia, 1995; Naghdy & Nguyen, 1998; Baptista, Sousa, & Sa’ da Costa, 2001). It should be noted that the goal here is to synthesize a force generator, which displaces, yet emulates skeletal muscle. The overall system (muscle and supporting enclosure) must be considered within this analysis which has to be stable and show adequate performance. Next, to demonstrate the relationship of the PM actuator to that of a biological system, a model is examined within the framework of skeletal muscle (Phillips, Repperger, Neidhard-Doll, & Reynolds, 2004).
2. A biomimicry framework to emulate skeletal muscle The operation of animal skeletal muscle is very similar to how the PM system performs. Forces are only generated via contraction, but are a combination of the action of additional factors. First, to discuss biological muscle, four main variables play a prominent role in the generation of a muscle force (Phillips, 2000). The primary variables of interest are first defined. Then, a state variable formulation will convert this problem into a framework amenable to control theory analysis. The latter model will then implement this theory into a PM scenario to synthesize a force generator. 2.1. A biological analogy to build an isometric model Initially, an isometric model will first be constructed. The term ‘‘isometric’’ implies that the muscle generates force (which may vary) with zero position movement. For all the biological muscle analyses, the main variables are: (1) V(t) ¼ Neural action potential, (2) P(t) ¼ SR (sarcoplasmic reticulum) permeability to calcium ions, (3) C(t) ¼ Free sarcoplasmic calcium ion concentration, and (4) F(t) ¼ Isometric force twitch. These variables are defined more precisely by Neidhard–Doll, Phillips, Repperger, and Reynolds, (2004), and Fig. 4 displays these four variables, which are described in more detail below. To review the basics of the biological system in Fig. 4, note: (1) V(t) ¼ Neural action potential is an electrical signal that activates the motor unit. Due to its short duration (4 ms) when compared to the other physiological events in the model, the peak action potential voltage (V0) is simply modeled as an impulse variable (at
V(t)
V(t)
α
P(t)
P(t)
k *1
1001
C(t)
k1
C(t)
k *2
k2
F(t) k3
V(t) - The stimulating neural action potential (impulse function) – units [mvolt] µmeter ] msec C(t) - The free sarcoplasmic calcium ion concentration [Ca2+] – units [mole]
P(t) - The SR permeability – units [
F(t)- The isometric force twitch – units [mg]
Fig. 4. Open four-compartment system.
time t0) that results in the peak SR permeability (Ppeak) Ppeak ¼ a V 0
ðunits of mm=msÞ;
(2)
where a is a proportionality constant E0.000327 (mm/ (mV ms)). The value of a is determined from the ratio of Ppeak to V0 (Phillips, 2000). (2) P(t) ¼ The instantaneous SR permeability to the calcium ions which can be approximated by the following differential equation: d=dt PðtÞ ¼ k1 PðtÞ;
Ppeak ¼ an V 0 ,
(3)
where the exponential decay rate constant k1 is found by fitting SR permeability data to a single exponential function based upon ek1 t approaching zero after P(t) has reached its peak. (3) C(t) ¼ Free sarcoplasmic calcium ion concentration [Ca2+] (units of mmol) which is related to the SR permeability equation (3) via d=dt CðtÞ ¼ k2 CðtÞ þ kn1 PðtÞ;
Cð0Þ 0,
(4) 2+
where d/dt C(t) is the time rate of change of [Ca ] across a unit volume of the SR membrane with units (mmol/ms), kn1 is a phenomenological coefficient that couples SR permeability to [Ca2+] with units (mmol/ mm), and P is defined in Eq. (3) with units (mm/ms). k2 is the exponential decay rate constant (reciprocal time constant associated with the re-sequestering of calcium by the calcium ATP pump), found by fitting free myoplasmic calcium data to a single exponential function based upon ek2 t approaching zero after C(t) has reached its peak, and C(t) is the instantaneous concentration of free sarcoplismic calcium [Ca2+] (mmol). (4) F(t) ¼ The force twitch signal which results as a consequence of the calcium interaction and can be shown to be coupled to Eq. (4) via the following relationship: d=dtF ðtÞ ¼ k3 F ðtÞ þ k2 CðtÞ;
F ð0Þ 0;
(5)
where dF/dt is the time rate of change of force with units (mg/ms), k2 is a phenomenological coefficient that couples [Ca2+] to force with units mg=mmol=ms and k3 is the exponential decay rate constant (reciprocal
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stress–relaxation, i.e. viscoelastic, time constant) found by fitting force data to a single exponential function based upon ek3 t approaching zero after F(t) has reached its peak. The units of k3 are 1/ms. F denotes the instantaneous force with units mg. To summarize the discussion so far, Table 1 displays the various ki variables with their appropriate SI (system international) units and values. Table 2 then summarizes the relevant differential equations discussed. It is noted in Fig. 4 that these first-order diffusiontype processes assumed for the formation of force, viewing it within the context of a calcium pump, are not unlike similar studies in other industrial applications, e.g. involving drug models (Mahfouf, Linkens, & Xue, 2002). Modeling work accomplished using the PM actuator by Reynolds, Repperger, Phillips, and Bandry (2003) also discusses some of these relevant issues.
x2 ¼ CðtÞ;
x2 ð0Þ ¼ 0 ,
(7b)
x3 ¼ F ðtÞ;
x3 ð0Þ ¼ 0.
(7c)
Simplifications can now be made from Fig. 4 and Eqs. (6a)–(6c) by selecting new state variables which are defined as z1 ¼ C(t), and z2 ¼ F(t). Eqs. (6a)–(6c) now simplify to the following state variable representation: " # " #" # " # 0 k2 z_1 z1 kn1 PðtÞ, (8) ¼ þ n k2 k3 z2 z_2 0 where P(t) is now considered as the input variable time function ðPðtÞ ¼ a V 0 ek1 t Þ. The output equation (y(t) ¼ F(t), the force response generated) would be of the form " # z1 yðtÞ ¼ ½0; 1 , (9) z2
2.2. A state variable formulation of the isometric system
d=dt x1 ðtÞ ¼ k1 x1 ðtÞ,
(6a)
d=dt x2 ðtÞ ¼ kn1 x1 ðtÞ k2 x2 ,
(6b)
d=dt x3 ðtÞ ¼ kn2 x2 ðtÞ k3 x3 ,
(6c)
where the presumption is made that the measured output variable y(t) would be the isometric force twitch. Such a formulation (8–9) represents a completely controllable, observable, and stable system. Its limitation, however, is that it embodies an isometric model of muscle dynamics, i.e. the position change must be zero. It is obvious that PM systems and other force generators have only practical applicability if they move and produce some external work on the environment. The isometric muscle system, however, is extremely well studied and selected to be emulated in this investigation. However, in this paper, the PM system will be allowed to move (violate the implicit hypothesis of being isometric, yet behave similarly to Eqs. (8) and (9)).
(7a)
3. Some additional challenges of using the PM system
The bio-inspired system previously described will now be converted to a state variable description which can then be generalized to the PM system. Setting x1 (the SR permeability ¼ P(t)) as the input to the system that we can control through modulation of the a term in Eq. (2), and x3 (the developed isometric force twitch) as the output, the preliminary state equations for the system become
where, from normalization, x1 ¼ PðtÞ;
x1 ð0Þ ¼ 1 ,
Table 1 Decay rate constants and phenomenological coefficients ki variable Type of coefficient
Units
Value of coefficient
k1 k2 k3 k*1 k*2
1/ms 1/ms 1/ms mmol/mm mg/(mmol/ms)
0.1578 0.0628 0.0172 0.29 0.0572
Decay rate constant Decay rate constant Decay rate constant Phenomenological Phenomenological
There are special challenges involving the combined PM system when utilized within an industrial framework. First, as previously mentioned, the PM, itself, is inherently nonlinear. Secondly, such a device must be attached to an object to perform useful work but must also be connected to the stable environment (cf. Figs. 1a,b and 2). The frame and connectors may introduce problems due to resonance in their structure when interacting with the extremely powerful PM. Thus, the
Table 2 Summary of the key differential equations and couplings Physical process
Variable
Variable name
Differential equation relationship
Biophysics Biochemistry Biochemistry Biomechanics
V(t) P(t) C(t) F(t)
Action potential Permeability Free calcium Force twitch
p(0) ¼ (0.000327) V0 ¼ Ppeak d/dt P(t) ¼ (0.1578) P(t) d/dt C(t) ¼ (0.0628) C+(0.29) P d/dt F(t) ¼ (0.0172) F+(0.0572)C
ARTICLE IN PRESS D.W. Repperger et al. / Control Engineering Practice 14 (2006) 999–1009
yt) = F(t)= Force Output of PM System versus time
534 Force output in Newtons
1003
445 356 267 178 Series1
10
89
volts
10 volt step input 0
Fig. 5. Overall structure for the PM system.
force application desired, the frame, and the actuator may all work in a manner to compromise the overall performance when the muscle moves dynamically. To show the extent of these problems, a series of tests were conducted on the system displayed in Fig. 5. The bladder was constructed from common rubber bicycle tire tube and the outer sheath material was ordinary coaxial (polyester) insulation, which makes the cost quite minimal.
0
1
2 4 3 Time in seconds
5
0 6
Fig. 6. Open loop response of the PM system (with frame) to step input y(t) ¼ F(t) ¼ Force output of the PM system versus time.
jω Axis
0
3.1. An open loop characterization of the device and frame (system concept) A step function of pressure was applied to the system (with a load of about 392 N) in Fig. 5. Fig. 6 shows the resulting force profile as a function of time. It is observed that structure resonance occurred. The metal frame was strengthened several times but responses similar to Fig. 6 were seen, quite typically. The sampling rate was at 2048 Hz and the natural frequency of the frame/connectors was determined to be about 5 Hz. Also, since the filling time constant of a PM is about 1 s, the only way resonances of the form of Fig. 6 could occur must be from the frame and the connectors. The time responses in Fig. 6 cannot be avoided due to the extremely high strength and power produced by the PM system. Typically with structures, active control is possible if actuators could be placed appropriately within the frame (Goodall & Austin, 1994), but vibration reduction always presents a challenging and important problem with all control systems (Yu, Chai, & Yuan, 1996). Also, since the target goal of this effort is biomimicry, an examination of Eq. (8) shows that for the skeletal muscle case, the dominant eigenvalue occurs at 15.92 Hz ¼ 1/0.0628 (cf. Fig. 7). Thus, the untoward response depicted in Fig. 6 (5 Hz) is much lower in frequency than even the desired target response (15.92 Hz), producing additional problems. Fig. 6 was a consequence of a step response input. Certainly, this allowed the frame/
Real Axis
1/(.0172) =
1/(.0628) =
58.14 Hz =
15.92 Hz =
365.11 rad/sec.
100.01 rad/sec.
Fig. 7. Pole-zole diagram of skeletal muscle dynamics in Eqs. (8) and (9).
connector dynamics to be excited because of the wide spectral nature of the step input. In an attempt to minimize the spectral content of the input signal sufficiently to produce responses of use to generate a force twitch, an input signal of the lowest possible frequency content is suggested. The next section describes the derivation of a desired force pulse to be generated from the PM system. 3.2. Derivation of a force twitch signal of a low frequency nature The goal in this section is to synthesize an input signal into an open loop PM system and frame which will emulate the output signal F(t) from Eqs. (8) and (9) as it occurs in biological muscle. From prior studies in muscle dynamics, an approximation to a force twitch is displayed in Fig. 8. This is a time domain representation of force (slow twitch; Rome, 1997). Appendix A describes how the force unit variable y(t) ¼ F(t), the
ARTICLE IN PRESS D.W. Repperger et al. / Control Engineering Practice 14 (2006) 999–1009
1004 y(t) = F(t)
R difference of exponentials (10,4)
Zsource
45 R 40
Vinitial
35
Source Vinput
PSI Desired
30 25 20
Zinput impedance looking Inward.
Zinput
Vreflected
15
2 = 10, 1 = 4.0
10 5
Fig. 9. Matching the source impedance to the input impedance.
0 -5
0
0.2
0.4
0.6
t = time axis in seconds
0.8
1
1.2
1.4
1.6
1.8
2
time
Fig. 8. Simulation of force twitch.
solution of (9), can be characterized by the difference of two exponentials, i.e., yðtÞ ¼ Aðeg1 t eg2 t Þ,
(10)
where g2 4g1 40 and Eq. (10) satisfies the boundary conditions y(0) ¼ 0 and y(N) ¼ 0. Eq. (10) is actually the solution of the bioinspired model of Eqs. (8) and (9) and becomes the basis for the force emulation of the PM system, as demonstrated in this paper. It is important to note that a time scale change must be instituted with the PM. Thus, the biomimicry developed will produce a shape similar to Fig. 8 in a temporal sense, but on an expanded time scale. This is a consequence of the fact that the inflation time constant for the PM is 1 s, which is much too slow to exactly replicate the time scale of skeletal muscle. Filling the dead volume of the PM is always a limiting factor in its use. Thus, the goal is to replicate a force pulse that emulates Fig. 8, but on a slower time scale. The next section describes the means of reducing the frame excitation as described in Fig. 6. 3.3. Modifying the input to minimize wave reflections To reduce the undesired response from Fig. 6, a technique from the circuit theory, as illustrated in Fig. 9, uses the concept of wave reflections. This approach is employed commonly in the analysis of transmission lines or other types of distributed systems. The input wave Vinitial indicated in blue and moves to the right. The reflected wave, Vreflected, is indicated in red and moves to the left. The scattering parameter S is the ratio of the magnitude of the reflected wave to the magnitude of the input wave. Thus S¼
jV reflected j ; where 0pSp1 jV initial j
(11)
if the system is stable, which is desired. It can also be shown (Appendix B) that S ¼ 0 and the reflection wave
is zero if the source impedance Zsource in Fig. 9 is matched precisely to the characteristic impedance Zinput looking into the circuit or system (Zsource ¼ Zinput). To implement the concepts in Fig. 9, Zinput is assumed to be first order and characterized by a transfer function (in steady state): Z input ¼
x1 , 1 þ s=x2
(12)
where s is the Laplace transform variable, x1 is a forward gain and x2 represents the bandwidth term. By varying the source impedance Zsource (via a digital controller on the pressure) until it impedance matches Zinput, the source impedance can be tuned to the input impedance of the network under study. Hence, the goal is to make Zsource ¼ Zinput ¼ the characteristic impedance. Thus, for the proper choices of x1 and x2 , the reflected wave can be made zero, S ¼ 0, and the undesired dynamics induced via wave reflections would be minimized. Fig. 10 illustrates this effect when applying the method in Fig. (9) (x2 ¼ 0:8) to the system in Fig. 5, using, in addition, a small amount of rate and position feedback to reduce some of the very high frequency response observed in the force response output. This was accomplished by making Zsource equivalent to Zinput in Eq. (12), but with a PD loop around Zsource to keep its transfer function close to Zinput in a frequency domain sense but still maintaining the quality of the output force signal via a digital controller. Thus, the system now appears reasonably stabilized. The force output in Fig. 10 now sufficiently emulates a force pulse of the type in Fig. 8 above a base line value (approximately 330 N). This is similar to problems that occur in vibration isolation studies. To see the actual force twitch signal (generated above the load value), Fig. 11 illustrates this as a comparison to Fig. 8 above the baseline in Fig. 10. An overall strategy for control of such a PM system will now be synthesized employing knowledge from how biological systems recruit forces using summations of both spatial motor units as well as through temporal summation of these motor units.
ARTICLE IN PRESS D.W. Repperger et al. / Control Engineering Practice 14 (2006) 999–1009 LC1 LC1
Force Output in Newtons
445 356 267
Step Voltage Input
178
10
89
volts
0 0
1.0
2.0 3.0 4.0 TIME IN SECONDS
5.0
0
Fig. 10. Force pulse matching input impedance using wave analysis.
Fig. 11. Force twitch signal extracted from Fig. 10.
4. A strategy for force recruitment derived from biological systems In order to use PM systems for industrial applications in either series or parallel, a review will be made on how such events occur in nature. In biological systems, the control strategy employed to attain a target skeletal muscle force is based upon the superimposition of the mechanical effects of individually stimulated skeletal muscle fibers. This peripheral nervous system control strategy for increasing muscle force amplitude until target acquisition is a function of both the number of new motor units recruited spatially, and the speed or frequency at which these recruited motor units are stimulated. A motor unit is the smallest functional component of the neuromuscular system, and is defined as an alpha motoneuron and the skeletal muscle fibers it innervates. The number of muscle fibers innervated varies, from less than a dozen for fine motor control (e.g. extraocular muscles), to several hundred for gross motor control and muscles responsible for posture. A stimulation parameter $P can be defined to represent the quantity of exciting electrical charge
1005
(delivered in current pulses) to a motor unit by the peripheral motor nerve (Phillips, 2000). An increase in the amplitude or width of the stimulation parameter (a given current pulse) results in the recruitment of additional neighboring motor units, or spatial summation. An increase in the frequency of current pulses (in a current pulse train) results in enhanced force output by the existing recruited motor units, or temporal summation. To model a population of motor units, a set of physiological-based rules (or strategy) must be employed to define recruitment during skeletal muscle contraction. Strength–recruitment curves based upon empirical data for human skeletal muscle have historically indicated that with respect to acquiring target force development, the first 50% of motor units are recruited spatially (Phillips, 2000). Spatial recruitment results in a successive increase in the peak amplitude of the generated muscle twitch, due to the combined effect of the individual force twitches produced by multiple (neighboring) motor units. The remaining increase in force required to capture the desired target is obtained by increasing the stimulation frequency of the already (spatially) recruited motor units. This temporal recruitment results in superimposition (or summation) of the mechanical effects, and is characterized by an increase in the peak amplitude of the generated muscle force (Rack & Westbury, 1969). If the stimulation frequency is increased to approximately 60 Hz, the superimposed mechanical effects fuse completely, resulting in maximum force amplitude (in which tension maximally plateaus) termed tetanus (Fung, 1993), as illustrated in Fig. 12. During tetanus, the stimulation frequency is so rapid that the SR does not have time to reclaim calcium ions from the myoplasm. Consequently, the skeletal muscle endures a sustained (and smooth) contraction, which is useful for performing work on the environment. The goal here is to have the PM system perform a similar type of force generation. To achieve this strategy with the PM system, each force twitch, yi, will be considered an individual element which will be recruited either by temporal summation or by spatial recruitment. The proper use of these control units to synthesize a desired force response is not unlike strategies developed in other industrial applications, e.g. the fiber-yarn production (Sette, Boullart, & Van Langenhove, 1998) where setpoint values and raw materials are designed for optimal quality, for strategies involving the rehabilitation of power plants based on cost and performance issues (Soares, Goncalves, Silva, & Lemos, 1997), and for strategies involving the control of servo-pneumatic actuators (cylinders) with acceleration feedback to improve stability and to mitigate the effects of time delay in Wang, Pu, and Moore (1999).
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1006
LC1 Tetanus 40psi, interval .5 140 LC1
120 100 80 Voltage step inputs
10
60 40
volts
20 0
0 0
2000
4000
6000
8000
10000
12000
Fig. 14. Tetanus emulated with the PM system.
Fig. 12. Tetanus generation via temporal summation (Fung, 1993).
Designed Source Impedance Zsource = yi = A(e−1t −e−2t)
ξ1
1+
y¯ ðtÞ ¼
s ξ1
y(t) in
To Load
Load
Fig. 13. Strategy for temporal and spatial summation.
The tetanus generation can be emulated for the PM system. To describe this action, Fig. 13 describes a summary of what has been determined up to this point. 4.1. Summary of results up to this point A force pulse yi(t) can be generated similar to Eqs. (8) and (9). This signal is then transmitted through an impedance Zsource (via a digital controller) in Fig. 13 to reduce any wave reflections. This new waveform received at the load then produces a force twitch presented in Fig. 11. To generate tetanus using the PM system, it can be realized in two ways. First, at a fixed location, in a time sense (temporal summation) via the summation of yi signals over a fixed time period (n per unit time) but delayed in time Temporal Summation to Tetanus: n X i¼1
yi ðt ti Þ,
m X
yi ðtÞ ¼ m yi ðtÞ.
(14)
i¼1
equation (10)
y¯ i ðtÞ ¼
where ti ¼ 0, Dt, 2Dt, y (n1)Dt and there may be up to n, yi signals in a desired time period. This is a time domain summation. An alternative manner to achieve tetanus would be via a spatial summation (at a specific time point) and in this case more motor units would simultaneously fire. Assuming m motor units would fire, this would be equivalent to: Spatial Summation to Tetanus:
(13)
Thus, by simply multiplying the yi signal by m, a constant, would be equivalent to spatially summing up to m motor units. Fig. 14, which was demonstrated on the PM system, can be generalized by a temporal summation (more frequently adding a force twitch) or by increasing the amplitude of an individual force twitch (recruiting additional motor units). Fig. 14 can be compared to Fig. 12 for similarity. The goal of Fig. 14 is to emulate the small oscillations shown in Fig. 12, where force twitches are synthesized by step voltage inputs into the PM system and the oscillations in both figures correspond to force twitch responses that are desired.
5. Discussion In the human skeletal system, P(t) occurs over less than 10 m/s; C(t) occurs over approximately 30 m/s; and F(t) occurs over 100–300 m/s, depending on fiber type (fast or slow-twitch). Therefore, P(t) has the fastest dynamics, whereas F(t) has the slowest, due to electrochemical and chemomechanical time delays (i.e. kn1 and kn2 , respectively). Fig. 15 portrays the three critical coefficients found from this procedure using the homogenous (free response) terms of Eqs. (3)–(5), i.e., _ ¼ k1 PðtÞ, PðtÞ
(15)
ARTICLE IN PRESS D.W. Repperger et al. / Control Engineering Practice 14 (2006) 999–1009
F (t)
Force Twitch = Slowest time constant
C (t)
P(t) = SR Permeability Fastest Time Constant
Ca 2+ = Intermediate time constant
Time Axis
Fig. 15. Homogenous solutions to Eqs. (15)–(17).
_¯ ¼ k CðtÞ, ¯ CðtÞ 2
(16)
F_¯ ðtÞ ¼ k3 F¯ ðtÞ,
(17)
where the barred quantities refer to the free response terms of Eqs. (3)–(5). The interesting point is in the rank order of the coefficients (time constants) of the above three variables, where k1 ¼ 0.1578, k2 ¼ 0.0628, and k3 ¼ 0.0172 (all units 1/ms). It is no accident that the rank order of these coefficients follows similar to the compartmental model in Fig. 4. What this means, in a physical sense, is that the time history of P(t) (tp ¼ 1/ (0.1578) in ms) is much faster (shorter time constant) ¯ than the time history of CðtÞ (tc ¼ 1/(0.0628)), which, in turn, is much faster than the time history of F¯ ðtÞ (tF ¼ 1/(0.0172)). This makes intuitive sense that the dynamics of the permeability P(t) must be sufficiently faster than the dynamics of the calcium ions to build up. Also, the calcium ion build up is required to be much slower than the permeability dynamics (to have some useful effect) but it is required to be much faster than the force twitch, in order for the force twitch to perform useful work. Recall that the force twitch must include both temporal and spatial summations, so these dynamics must be sufficiently slower than the calcium dynamics yet much, much slower than the permeability dynamics. Thus, the design by nature has carefully constructed the relative time constants of these key variables to enable a force twitch to be created and manipulated for the performance of useful work accomplished on the external environment.
Reynolds, and Berlin (2005) and we present a task showing more advanced results. The goal is to perform force tracking, with a position constraint covering the material presented previously. In Fig. 16 shows the classical work loop diagram used when biological muscle completes a task (full cycle of an agonist–antagonist task). The vertical axis is the force generated by the actuator and the horizontal axis is the length of the muscle. The area A1 in the enclosed curve is the net work delivered by the actuator upon the environment. This has similarities to a Carnot cycle in thermodynamics but here the area is productive in producing work on the environment which is a key output variable. This is a force tracking task because force is generated and the length of the muscle varies during the completion of the mission over one full cycle. Fig. 17 shows empirical data of the actual PM performing this force tracking task. The performance of the system in Fig. (17) can be considered in several ways (Repperger et al., 2005). One key output variable is area A1 in Fig. 17, which shows the net work delivered to the environment from this system. At variance to the method in a Carnot cycle from thermodynamics, here the goal is to maximize area A1, showing the ability of the actuator to perform work on the external environment. Table 3 shows this efficacy for a variety of controllers demonstrating that the area A1 is a key metric for evaluation of this type of controller. Controller 1 is the temporal controller (summation of force pulses in a time sense), controller 2 is the spatial controller (summation of force pulses from diverse locations), and controller 3 is the biologically based as described in Eq. (9).
7. Summary and conclusions Emulating biological signals involving animal skeletal muscle, a pneumatic muscle actuator has been designed Force B
Direction of
6. Implementation of the PM in a closed loop force tracking task To this point, the techniques described herein have been based mainly on an open loop analysis of this actuator and its analogy to biological (skeletal muscle). More recently, implementation results have been demonstrated in Repperger, Phillips, Neidhard-Doll,
1007
Direction of contraction
Relaxation
Area A1 A L1
Muscle Length (normalized as a percent change) L2
Fig. 16. Work loop analysis of biological muscle.
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1008
"
2-Biomimic - 40 PSI B Plot 45 Fmax 40
z2
#
" ;
¯ ¼ A
k2
0
kn2
k3
"
# ;
B¼
kn1 0
# ;
¯ ¼ ½0; 1: C
2-Biomimic - 40 PSI - B Plot
35
(A.3)
Work Loop
30 Psi
x¯ ¼
psi
z1
Since C(0) ¼ F(0) ¼ 0 in Eqs. (7b) and (7c) and from Laplace transforming (A.1) and (A.2) yields
Contraction
25 Area A1
20
¯ ¯ 1 BPðsÞ ¯ ¼ HðsÞPðsÞ, ¯ F¯ ðsÞ ¼ Y ðsÞ ¼ CðsI AÞ
Relaxation
15 10 Fmin 5 0 0
2
1
3 Length
5
4
6
(A.4)
where the bars indicate that Laplace transform quantities are being utilized. The transfer function matrix H(s) is scalar and can be represented via HðsÞ ¼
Fig. 17. PM in a work loop force tracking task.
F¯ ðsÞ kn1 kn2 . ¼ ¯ ðs þ k2 Þðs þ k3 Þ PðsÞ
(A.5)
From Eq. (10), it is desired to reduce (A.5) to the time domain representation Table 3 Performance of the PM system in a full cycle of a work loop task Controller employed Controller Controller Controller Controller Controller Controller
1 1 2 2 3 3
PSI for 10 v signal input
A1 in joules as work produced externally
40 70 40 70 40 70
0.303 0.589 0.319 0.655 0.303 0.543
yðtÞ ¼ F ðtÞ ¼ A½eg1 t eg2 t .
(A.6)
To achieve this goal, H(s) is decomposed via a partial fraction expansion: HðsÞ ¼
D1 D1 . s þ k2 s þ k3
(A.7)
This will occur if D1 k3 D1 k2 ¼ kn1 kn2 or choose the following relationships to realize the form of (A.6): A ¼ D1 ¼
kn1 kn2 1 1 ; with g1 ¼ and g2 ¼ . k2 k3 k3 k 2 (A.8)
Note, k24k3 and A40, resulting in g24g1. to work in a force control sense. The overall system had to be stabilized, including the frame and connector dynamics. Once a force twitch that showed a reasonable semblance to a typical slow-twitch animal muscle response could be produced, the tetanus recruitment paradigms could also be replicated. This provides a method for the pneumatic muscle system to be used in a force control systems mode both in its generated force output and in synthesizing its strategy to accumulate force, much like is accomplished in animal skeletal muscle.
Appendix A It is desired to show the relationship between the state variable representation in Eqs. (8)–(10), which represents a time domain representation of the force twitch signal. Rewriting Eqs. (8) and (9) as ¯ x¯ þ BPðtÞ, x_¯ ¼ A
(A.1)
y ¼ C¯ x, ¯
(A.2)
Appendix B To show the scattering parameter S of Eq. (11) would be zero if Zsource ¼ Zinput, the ratio S ¼ Vr/ Vi ¼ (ZinputZsource)/(Zinput+Zsource) now follows from definitions commonly occurring in circuit analysis via transmission lines. The total voltage across the load is V ¼ Vi+Vr, The total current in the series circuit is given by IT ¼ Ii+Ir. This gives rise to the ratios: Vi/ Ii ¼ Vr/Ir ¼ Zsource. Hence, V i ¼ ð12ÞðV þ Z source I T Þ and V r ¼ ð12ÞðV Z source I T Þ. This results in S ¼ Vr/ Vi ¼ (ZinputZsource)/(Zinput+Zsource) and when Z input ¼ Zsource , then S ¼ 0. This means there is no reflected wave. References Alleyne, A., & Liu, R. (2000). A simplified approach to force control for electro-hydraulic systems. Control Engineering Practice, 8, 1347–1356. Baptista, L. F., Sousa, J. M., & Sa’ da Costa, J. M. G. (2001). Fuzzy predictive algorithms applied to real-time force control. Control Engineering Practice, 9, 411–423.
ARTICLE IN PRESS D.W. Repperger et al. / Control Engineering Practice 14 (2006) 999–1009 Benyus, J. M. (1997). Biomimicry: Innovation inspired by nature. New York: William Morrow and Company. Bernhard, S., Enning, M., & Rake, H. (1994). Automation of a laboratory plant for direct casting of thin steel strips. Control Engineering Practice, 2(6), 961–967. Caldwell, D. G. (1993). Natural and artificial muscle elements as robot actuators. Mechatronics, 3(3), 269–283. Caldwell, D. G., & Favede, C. (1998). Human sensory feedback program at Wright– Patterson AFB. Window on Science Program Report 9th August–21 August, 1998. Caldwell, D. G., Medrano-Cerda, G. A., & Goodwin, M. (1995). Control of pneumatic muscle actuators. IEEE Control Systems, February, 40–48. Carrillo, F. J., & Rotella, F. (1997). A robust adaptive controller for constant turning force regulation. Control Engineering Practice, 5(12), 1673–1682. Charbonnaud, P., Carrillo, F. J., & Ladeveze, D. (2001). Monitored robust force control of a milling process. Control Engineering Practice, 9, 1047–1061. Chou, C.-P., & Hannaford, B. (1996). Measurement and modeling of McKibben pneumatic artificial muscles. IEEE Transactions on Robotics and Automation, 12(1), 90–102. Fung, Y. C. (1993). Biomechanics: Mechanical properties of living tissues (2nd ed). New York: Springer. Goodall, R. M., & Austin, S. A. (1994). Modeling, placing and controlling active elements in structural engineering design. Control Engineering Practice, 2(5), 743–753. Hajian, A. Z., Sanchez, D. S., & Howe, R. D. (1997). Drum roll: Increasing bandwidth through passive impedance modulation. Proceedings of the 1997 IEEE international conference on robotics and automation, ICRA (vol. 3, pp. 2294–2299). Hill, A. V. (1938). The heat of shortening and the dynamic constants of muscle. Proceedings of the Royal Society of Biological Science, 126, 136–195. Inoue, K. (1988). Rubber actuators and applications for robotics. In R. Bolles, & B. Roth (Eds.), Robotics research: The 4th international symposium. Cambridge, MA: MIT Press. Liu, M.-H. (1995). Force-controlled fuzzy-logic-based robotic deburring. Control Engineering Practice, 3(2), 189–201. Mahfouf, M., Linkens, D. A., & Xue, D. (2002). Control Engineering Practice, 10, 67–81. Mattiazzo, G., Mauro, S., Raparelli, T., & Velardocchia, M. (1995). A fuzzy controlled pneumatic gripper for asparagus harvesting. Control Engineering Practice, 3(11), 1563–1570. Naghdy, F., & Nguyen, N. P. (1998). Fuzzy logic compliance of the peg in hole insertion. Control Engineering Practice, 6(1998), 1459–1474. Niedhard-Doll, A. T., Phillips, C. A., Repperger, D. W., & Reynolds, D. B. (2004). Biomimetic model of skeletal muscle isometric contraction: II—A phenomenological model of the skeletal muscle excitation—contraction coupling process. Computers in Biology and Medicine, 34(4), 323–344. Niksefat, N., & Sepehri, N. (2000). Design and experimental evaluation of a robust force controller for an electro-hydraulic actuator via quantitative feedback theory. Control Engineering Practice, 8, 1335–1345. Noritsugu, T., & Tanaka, T. (1997). Application of rubber artificial muscle manipulator as a rehabilitation robot. IEEE/ASME Transactions on Mechatronics, 2(4), 259–267. Peterson, D. K., & Chizeck, H. J. (1987). Linear quadratic control of a loaded agonist–antagonist muscle pair. IEEE Transactions on Biomedical Engineering, BME-34(10), 790–796. Phillips, C. A. (2000). Human factors engineering. New York: Wiley.
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Phillips, C. A., & Petrofsky, J. S. (1983). Mechanics of skeletal and cardiac muscle. Springfield: Charles C. Thomas. Phillips, C. A., Repperger, D. W., Neidhard-Doll, A. T., & Reynolds, D. B. (2004). Biomimetic model of skeletal muscle isometric contraction: I. An energetic viscoelastic model for the skeletal muscle isometric force twitch. Computers in Biology and Medicine, 34(4), 307–322. Rack, & Westbury, (1969). The effects of length and stimulus rate on tension in the isometric cat soleus muscle. Journal of Physiology, 204, 443–460. Repperger, D. W., Johnson, K. R., & Phillips, C. A. (1998). A VSC position tracking system involving a large scale pneumatic muscle actuator. Proceedings of the 37th IEEE conference on decision and control, December (pp. 4302–4307). Repperger, D. W., Phillips, C. A., Neidhard-Doll, A., Reynolds, D. B., & Berlin, J. (2005). Power/energy metrics for controller evaluation of actuators similar to biological systems. Mechatronics, 15, 459–469. Reynolds, D. B., Repperger, D. W., Phillips, C. A., & Bandry, G. (2003). Modeling the dynamics characteristics of pneumatic muscle. Annals of Biomedical Engineering, 31, 310–317. Rome, L. C. (1997). Testing a muscle’s design. American Scientist, 85, 356–363. Schulte, H.F., Jr. (1961). The characteristics of the McKibben artificial muscle. In: The application of external power in prosthetics and orthotics (Publication 874) (Appendix H, pp. 94–115). Lake Arrowhead, Washington, D.C: National Academy of Sciences— National Research Council. Sette, S., Boullart, L., & Van Langenhove, L. (1998). Using genetic algorithms to design a control strategy of an industrial process. Control Engineering Practice, 6, 523–527. Siciliano, B., & Villani, L. (2001). An inverse kinematics algorithm for interaction control of a flexible arm with a compliant surface. Control Engineering Practice, 9, 191–198. Sira-Ramirez, H., Lopez, P., & Tondu, B. (1996). Robust stabilization and tracking for robotic manipulators with artificial muscles. International Journal of Systems Science, 27(11), 1067–1075. Soares, A. O., Goncalves, A., Silva, R. N., & Lemos, J. M. (1997). A methodology for impact evaluation of alternative control strategies in a large-scale power plant. Control Engineering Practice, 5(3), 325–335. Stein, R. B., & Gordon, T. (1986). Nonlinear stiffness-force relationships in whole mammalian skeletal muscles. Canadian Journal of Physiology and Pharmacology, 64, 1236–1244. Wang, J., Pu, J., & Moore, P. (1999). A practical control strategy for servo-pneumatic actuator systems. Control Engineering Practice, 7, 1483–1488. Winters, J. M. (1990). Hill based muscle models: A systems engineering perspective. In J. M. Winters, & S. Woo (Eds.), Multiple muscle systems: Biomechanics and movement organization. New York: Springer. Yi, J. H., Park, K. H., Kim, S. H., Kwak, Y. K., Abdelfatah, M., & Busch-Vishniac, I. (1996). Robust force control for a magnetically levitated manipulator using flux density measurement. Control Engineering Practice, 4(7), 957–965. Yu, W., Chai, T., & Yuan, Y. (1996). Decreased vibration control for centrifuges: A new adaptive hybrid control technique. Control Engineering Practice, 4(12), 1693–1700. Zhou, B. H., Baratta, R. V., Solomonow, M., Ikuvuerm, L. J., Bgytenm, G. T., & D’Ambrosia, R. D. (1996). Evaluation of isometric antagonist coactivation strategies of electrically stimulated muscles. IEEE Transactions on Biomedical Engineering, 43(2), 150–160.
Control Engineering Practice 14 (2006) 999–1009 www.elsevier.com/locate/conengprac
Actuator design using biomimicry methods and a pneumatic muscle system D.W. Reppergera,, C.A. Phillipsb, A. Neidhard-Dollb, D.B. Reynoldsb, J. Berlinc a
b
Air Force Research Laboratory, WPAFB, OH, USA School of Biomedical, Human Factors and Industrial Engineering, Wright State University, Dayton, OH, USA c General Dynamics, Dayton, OH, USA Received 27 May 2003; accepted 11 June 2005 Available online 19 August 2005
Abstract An empirical and theoretical study is conducted on a special actuator termed ‘‘pneumatic muscle’’ (PM) being used in a force control system framework. Such an actuator has similarities to biological systems and has many advantages (extremely high power/ weight, power/volume and power/energy ratios). However, due to its inherent nonlinearities, this actuator suffers from poor position and force control. The study described here accomplishes three main goals. (1) A force control system is developed within an open and closed loop framework to emulate how biological systems work in an agonist–antagonist framework. (2) The PM used in the study has such strength that it excites the frame dynamics. This undesired dynamic response is then effectively cancelled using an impedance model control scheme. (3) The PM is demonstrated to both change length yet still produce force in a controlled manner. r 2005 Elsevier Ltd. All rights reserved. Keywords: Force control; Biomimicry design; Pneumatic muscle actuator
1. Introduction Biomimicry (bio-inspired or emulating biological systems in nature) provides a powerful framework from which to design and analyze systems (Benyus, 1997). It is difficult to argue that 3.8 billion years of evolution have not produced a host of successful designs in nature. By studying and emulating these successful products of the natural world, engineers can develop and improve upon present systems to generate both reliable and robust applications to work in difficult environments. For actuation control, many biological systems (e.g. the eye muscles or arm muscles—bicepts and tricepts) work on a principle of agonist–antagonist control. To describe agonist–antagonist control, in Fig. 1a, linear motion or force control can be achieved. Muscles only generate a force via contraction, i.e. a muscle can Corresponding author. Tel.: +1 937 2558765; fax: +1 937 2558752.
E-mail address: [email protected] (D.W. Repperger). 0967-0661/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.conengprac.2005.06.009
only ‘‘pull’’ and does not ‘‘push.’’ The right muscle (agonist) in Fig. 1a contracts and simultaneously the left muscles relaxes (antagonist, which increases in length), thus producing a force and motion on the mass to the right. In Fig. 1b, the same effect can be realized in a rotational sense by generating a clockwise angular position rotation or torque on the robotic joint through the contraction of the agonist and relaxation of the antagonist muscle. The key to the successful use of agonist–antagonist control lies in the properties of the actuator that can successfully contract and expand (relax). Control systems can be used in numerous ways to effect agonist–antagonist control. These applications are widespread in robotics (Hajian, Sanchez, & Howe, 1997) and bioengineering (Peterson & Chizeck, 1987; Zhou et al., 1996) and have been applied in the nuclear power industry (Caldwell, Medrano-Cerda, & Goodwin, 1995) using PM technology. An important actuator that can be used for agonist– antagonist control is a pneumatic muscle (PM) (Fig. 2)
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which has many similarities to biological systems. Such a device is analogous to animal skeletal muscle for the following reasons: (1) forces are only generated via contraction and (2) in a force or position control mode, such an actuator is highly nonlinear. There are many industrial uses of the PM system thus described. As discussed, applications appear widely in robotics (Caldwell, 1993; Caldwell et al., 1995; Noritsugu & Tanaka, 1997; Sira-Ramirez, Lopez, & Tondu, 1996), in nuclear power plants for position control of uranium rods (Caldwell & Favede, 1998), and for rehabilitation (Inoue, 1988; Schulte, 1961). Other advantages of this form of actuation include the fact that it fails gracefully (termed ‘‘soft actuator’’) and is considered to be safer than electric or hydraulic counterparts generating the same force level. In addition, such devices can be cheaply built and provide little contamination to the environment in which they are designed to work. Finally, in comparison to a hydraulic or electric actuator, the PM has the highest ratios of power/weight (1 wt/gm), power/volume (1 wt/cc), as well as power/ energy. To explain the operation of such a device, Fig. 2 illustrates that as the air pressure rises inside the bladder, enclosed within the outer sheath, the muscle contracts in the longitudinal direction yet expands in the radial direction. This produces a force externally on the outside environment which can perform work. A typical PM system may weight a few ounces and can lift 200–300 pounds which has significant advantages over traditional hydraulic or electrical actuators. One can
easily see how nonlinearities arise in the use of the PM system. Even if the pressure input is a linear time function, the volume resulting from this inflow would vary in a nonlinear manner with respect to the pressure input. Also, from the physics of this process, the contractile force generated on the external environment is proportional to the net change of the cross-section surface area affected via the inflation as follows: (1)
DForce ¼ ðPressureÞ DArea;
where Pressure refers to gauge pressure (air pressure inside the bladder above the atmosphere or external environment) and DArea refers to the change in the cross section area of the muscle as a consequence of the inflation. Of course, the area change in Eq. (1) is nonlinear with pressure input, since the system in Fig. 2 changes shape as it contracts. It is well documented in the literature that the PM has nonlinear models of position change with pressure (Chou & Hannaford, 1996;Repperger, Johnson, & Phillips, 1998). This is biologically similar or ‘‘mimics’’ skeletal muscle (Hill, 1938; Phillips & Petrofsky, 1983; Rome, 1997; Stein & Gordon, 1986; Winters, 1990). Little work, however, is reported in the literature using PM systems in a force control sense. Fig. 3 illustrates the force control regulation problem considered here, i.e. the framework in which it is desired herein to investigate the PM system. Force control methods are of considerable interest in a number of industrial applications including cutting processes (Carrillo & Rotella, 1997), Magnetic levitation systems (Yi et al., 1996), in milling processes (Charbon-
Linear Antagonist muscle B = Relaxation B
Motion x 0 Mass
Net Force on Environment
A Air Pressure Inlet Internal Air Bladder
Agonist muscle A = Contraction
Mass
(a)
Robot Link i-1
Antagonist muscle B = Relaxation B
θ = Rotational Displacement
Protective Outer Sheath Material Fig. 2. Operation of a pneumatic muscle system.
A Agonist muscle A = Contraction (b)
Force Input Command
Force Error Controller
To Robot Link i
Fig. 1. Agonist–antagonist control: (a) linear motion, (b) rotational motion.
-
Pneumatic Muscle System
Force Output
Fig. 3. Force control system for a pneumatic muscle.
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naud, Carrillo, & Ladeveze, 2001), robotics (Siciliano & Villani, 2001), hydraulic actuators (Niksefat & Sepehri, 2000), in deburring applications (Liu, 1995), strip castings (Bernhard, Enning, & Rake, 1994), and pressure control (Alleyne & Liu, 2000). Traditional controllers, as well as intelligent based systems, have been of interest for these force control methods (Mattiazzo, Mauro, Raparelli & Velardocchia, 1995; Naghdy & Nguyen, 1998; Baptista, Sousa, & Sa’ da Costa, 2001). It should be noted that the goal here is to synthesize a force generator, which displaces, yet emulates skeletal muscle. The overall system (muscle and supporting enclosure) must be considered within this analysis which has to be stable and show adequate performance. Next, to demonstrate the relationship of the PM actuator to that of a biological system, a model is examined within the framework of skeletal muscle (Phillips, Repperger, Neidhard-Doll, & Reynolds, 2004).
2. A biomimicry framework to emulate skeletal muscle The operation of animal skeletal muscle is very similar to how the PM system performs. Forces are only generated via contraction, but are a combination of the action of additional factors. First, to discuss biological muscle, four main variables play a prominent role in the generation of a muscle force (Phillips, 2000). The primary variables of interest are first defined. Then, a state variable formulation will convert this problem into a framework amenable to control theory analysis. The latter model will then implement this theory into a PM scenario to synthesize a force generator. 2.1. A biological analogy to build an isometric model Initially, an isometric model will first be constructed. The term ‘‘isometric’’ implies that the muscle generates force (which may vary) with zero position movement. For all the biological muscle analyses, the main variables are: (1) V(t) ¼ Neural action potential, (2) P(t) ¼ SR (sarcoplasmic reticulum) permeability to calcium ions, (3) C(t) ¼ Free sarcoplasmic calcium ion concentration, and (4) F(t) ¼ Isometric force twitch. These variables are defined more precisely by Neidhard–Doll, Phillips, Repperger, and Reynolds, (2004), and Fig. 4 displays these four variables, which are described in more detail below. To review the basics of the biological system in Fig. 4, note: (1) V(t) ¼ Neural action potential is an electrical signal that activates the motor unit. Due to its short duration (4 ms) when compared to the other physiological events in the model, the peak action potential voltage (V0) is simply modeled as an impulse variable (at
V(t)
V(t)
α
P(t)
P(t)
k *1
1001
C(t)
k1
C(t)
k *2
k2
F(t) k3
V(t) - The stimulating neural action potential (impulse function) – units [mvolt] µmeter ] msec C(t) - The free sarcoplasmic calcium ion concentration [Ca2+] – units [mole]
P(t) - The SR permeability – units [
F(t)- The isometric force twitch – units [mg]
Fig. 4. Open four-compartment system.
time t0) that results in the peak SR permeability (Ppeak) Ppeak ¼ a V 0
ðunits of mm=msÞ;
(2)
where a is a proportionality constant E0.000327 (mm/ (mV ms)). The value of a is determined from the ratio of Ppeak to V0 (Phillips, 2000). (2) P(t) ¼ The instantaneous SR permeability to the calcium ions which can be approximated by the following differential equation: d=dt PðtÞ ¼ k1 PðtÞ;
Ppeak ¼ an V 0 ,
(3)
where the exponential decay rate constant k1 is found by fitting SR permeability data to a single exponential function based upon ek1 t approaching zero after P(t) has reached its peak. (3) C(t) ¼ Free sarcoplasmic calcium ion concentration [Ca2+] (units of mmol) which is related to the SR permeability equation (3) via d=dt CðtÞ ¼ k2 CðtÞ þ kn1 PðtÞ;
Cð0Þ 0,
(4) 2+
where d/dt C(t) is the time rate of change of [Ca ] across a unit volume of the SR membrane with units (mmol/ms), kn1 is a phenomenological coefficient that couples SR permeability to [Ca2+] with units (mmol/ mm), and P is defined in Eq. (3) with units (mm/ms). k2 is the exponential decay rate constant (reciprocal time constant associated with the re-sequestering of calcium by the calcium ATP pump), found by fitting free myoplasmic calcium data to a single exponential function based upon ek2 t approaching zero after C(t) has reached its peak, and C(t) is the instantaneous concentration of free sarcoplismic calcium [Ca2+] (mmol). (4) F(t) ¼ The force twitch signal which results as a consequence of the calcium interaction and can be shown to be coupled to Eq. (4) via the following relationship: d=dtF ðtÞ ¼ k3 F ðtÞ þ k2 CðtÞ;
F ð0Þ 0;
(5)
where dF/dt is the time rate of change of force with units (mg/ms), k2 is a phenomenological coefficient that couples [Ca2+] to force with units mg=mmol=ms and k3 is the exponential decay rate constant (reciprocal
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stress–relaxation, i.e. viscoelastic, time constant) found by fitting force data to a single exponential function based upon ek3 t approaching zero after F(t) has reached its peak. The units of k3 are 1/ms. F denotes the instantaneous force with units mg. To summarize the discussion so far, Table 1 displays the various ki variables with their appropriate SI (system international) units and values. Table 2 then summarizes the relevant differential equations discussed. It is noted in Fig. 4 that these first-order diffusiontype processes assumed for the formation of force, viewing it within the context of a calcium pump, are not unlike similar studies in other industrial applications, e.g. involving drug models (Mahfouf, Linkens, & Xue, 2002). Modeling work accomplished using the PM actuator by Reynolds, Repperger, Phillips, and Bandry (2003) also discusses some of these relevant issues.
x2 ¼ CðtÞ;
x2 ð0Þ ¼ 0 ,
(7b)
x3 ¼ F ðtÞ;
x3 ð0Þ ¼ 0.
(7c)
Simplifications can now be made from Fig. 4 and Eqs. (6a)–(6c) by selecting new state variables which are defined as z1 ¼ C(t), and z2 ¼ F(t). Eqs. (6a)–(6c) now simplify to the following state variable representation: " # " #" # " # 0 k2 z_1 z1 kn1 PðtÞ, (8) ¼ þ n k2 k3 z2 z_2 0 where P(t) is now considered as the input variable time function ðPðtÞ ¼ a V 0 ek1 t Þ. The output equation (y(t) ¼ F(t), the force response generated) would be of the form " # z1 yðtÞ ¼ ½0; 1 , (9) z2
2.2. A state variable formulation of the isometric system
d=dt x1 ðtÞ ¼ k1 x1 ðtÞ,
(6a)
d=dt x2 ðtÞ ¼ kn1 x1 ðtÞ k2 x2 ,
(6b)
d=dt x3 ðtÞ ¼ kn2 x2 ðtÞ k3 x3 ,
(6c)
where the presumption is made that the measured output variable y(t) would be the isometric force twitch. Such a formulation (8–9) represents a completely controllable, observable, and stable system. Its limitation, however, is that it embodies an isometric model of muscle dynamics, i.e. the position change must be zero. It is obvious that PM systems and other force generators have only practical applicability if they move and produce some external work on the environment. The isometric muscle system, however, is extremely well studied and selected to be emulated in this investigation. However, in this paper, the PM system will be allowed to move (violate the implicit hypothesis of being isometric, yet behave similarly to Eqs. (8) and (9)).
(7a)
3. Some additional challenges of using the PM system
The bio-inspired system previously described will now be converted to a state variable description which can then be generalized to the PM system. Setting x1 (the SR permeability ¼ P(t)) as the input to the system that we can control through modulation of the a term in Eq. (2), and x3 (the developed isometric force twitch) as the output, the preliminary state equations for the system become
where, from normalization, x1 ¼ PðtÞ;
x1 ð0Þ ¼ 1 ,
Table 1 Decay rate constants and phenomenological coefficients ki variable Type of coefficient
Units
Value of coefficient
k1 k2 k3 k*1 k*2
1/ms 1/ms 1/ms mmol/mm mg/(mmol/ms)
0.1578 0.0628 0.0172 0.29 0.0572
Decay rate constant Decay rate constant Decay rate constant Phenomenological Phenomenological
There are special challenges involving the combined PM system when utilized within an industrial framework. First, as previously mentioned, the PM, itself, is inherently nonlinear. Secondly, such a device must be attached to an object to perform useful work but must also be connected to the stable environment (cf. Figs. 1a,b and 2). The frame and connectors may introduce problems due to resonance in their structure when interacting with the extremely powerful PM. Thus, the
Table 2 Summary of the key differential equations and couplings Physical process
Variable
Variable name
Differential equation relationship
Biophysics Biochemistry Biochemistry Biomechanics
V(t) P(t) C(t) F(t)
Action potential Permeability Free calcium Force twitch
p(0) ¼ (0.000327) V0 ¼ Ppeak d/dt P(t) ¼ (0.1578) P(t) d/dt C(t) ¼ (0.0628) C+(0.29) P d/dt F(t) ¼ (0.0172) F+(0.0572)C
ARTICLE IN PRESS D.W. Repperger et al. / Control Engineering Practice 14 (2006) 999–1009
yt) = F(t)= Force Output of PM System versus time
534 Force output in Newtons
1003
445 356 267 178 Series1
10
89
volts
10 volt step input 0
Fig. 5. Overall structure for the PM system.
force application desired, the frame, and the actuator may all work in a manner to compromise the overall performance when the muscle moves dynamically. To show the extent of these problems, a series of tests were conducted on the system displayed in Fig. 5. The bladder was constructed from common rubber bicycle tire tube and the outer sheath material was ordinary coaxial (polyester) insulation, which makes the cost quite minimal.
0
1
2 4 3 Time in seconds
5
0 6
Fig. 6. Open loop response of the PM system (with frame) to step input y(t) ¼ F(t) ¼ Force output of the PM system versus time.
jω Axis
0
3.1. An open loop characterization of the device and frame (system concept) A step function of pressure was applied to the system (with a load of about 392 N) in Fig. 5. Fig. 6 shows the resulting force profile as a function of time. It is observed that structure resonance occurred. The metal frame was strengthened several times but responses similar to Fig. 6 were seen, quite typically. The sampling rate was at 2048 Hz and the natural frequency of the frame/connectors was determined to be about 5 Hz. Also, since the filling time constant of a PM is about 1 s, the only way resonances of the form of Fig. 6 could occur must be from the frame and the connectors. The time responses in Fig. 6 cannot be avoided due to the extremely high strength and power produced by the PM system. Typically with structures, active control is possible if actuators could be placed appropriately within the frame (Goodall & Austin, 1994), but vibration reduction always presents a challenging and important problem with all control systems (Yu, Chai, & Yuan, 1996). Also, since the target goal of this effort is biomimicry, an examination of Eq. (8) shows that for the skeletal muscle case, the dominant eigenvalue occurs at 15.92 Hz ¼ 1/0.0628 (cf. Fig. 7). Thus, the untoward response depicted in Fig. 6 (5 Hz) is much lower in frequency than even the desired target response (15.92 Hz), producing additional problems. Fig. 6 was a consequence of a step response input. Certainly, this allowed the frame/
Real Axis
1/(.0172) =
1/(.0628) =
58.14 Hz =
15.92 Hz =
365.11 rad/sec.
100.01 rad/sec.
Fig. 7. Pole-zole diagram of skeletal muscle dynamics in Eqs. (8) and (9).
connector dynamics to be excited because of the wide spectral nature of the step input. In an attempt to minimize the spectral content of the input signal sufficiently to produce responses of use to generate a force twitch, an input signal of the lowest possible frequency content is suggested. The next section describes the derivation of a desired force pulse to be generated from the PM system. 3.2. Derivation of a force twitch signal of a low frequency nature The goal in this section is to synthesize an input signal into an open loop PM system and frame which will emulate the output signal F(t) from Eqs. (8) and (9) as it occurs in biological muscle. From prior studies in muscle dynamics, an approximation to a force twitch is displayed in Fig. 8. This is a time domain representation of force (slow twitch; Rome, 1997). Appendix A describes how the force unit variable y(t) ¼ F(t), the
ARTICLE IN PRESS D.W. Repperger et al. / Control Engineering Practice 14 (2006) 999–1009
1004 y(t) = F(t)
R difference of exponentials (10,4)
Zsource
45 R 40
Vinitial
35
Source Vinput
PSI Desired
30 25 20
Zinput impedance looking Inward.
Zinput
Vreflected
15
2 = 10, 1 = 4.0
10 5
Fig. 9. Matching the source impedance to the input impedance.
0 -5
0
0.2
0.4
0.6
t = time axis in seconds
0.8
1
1.2
1.4
1.6
1.8
2
time
Fig. 8. Simulation of force twitch.
solution of (9), can be characterized by the difference of two exponentials, i.e., yðtÞ ¼ Aðeg1 t eg2 t Þ,
(10)
where g2 4g1 40 and Eq. (10) satisfies the boundary conditions y(0) ¼ 0 and y(N) ¼ 0. Eq. (10) is actually the solution of the bioinspired model of Eqs. (8) and (9) and becomes the basis for the force emulation of the PM system, as demonstrated in this paper. It is important to note that a time scale change must be instituted with the PM. Thus, the biomimicry developed will produce a shape similar to Fig. 8 in a temporal sense, but on an expanded time scale. This is a consequence of the fact that the inflation time constant for the PM is 1 s, which is much too slow to exactly replicate the time scale of skeletal muscle. Filling the dead volume of the PM is always a limiting factor in its use. Thus, the goal is to replicate a force pulse that emulates Fig. 8, but on a slower time scale. The next section describes the means of reducing the frame excitation as described in Fig. 6. 3.3. Modifying the input to minimize wave reflections To reduce the undesired response from Fig. 6, a technique from the circuit theory, as illustrated in Fig. 9, uses the concept of wave reflections. This approach is employed commonly in the analysis of transmission lines or other types of distributed systems. The input wave Vinitial indicated in blue and moves to the right. The reflected wave, Vreflected, is indicated in red and moves to the left. The scattering parameter S is the ratio of the magnitude of the reflected wave to the magnitude of the input wave. Thus S¼
jV reflected j ; where 0pSp1 jV initial j
(11)
if the system is stable, which is desired. It can also be shown (Appendix B) that S ¼ 0 and the reflection wave
is zero if the source impedance Zsource in Fig. 9 is matched precisely to the characteristic impedance Zinput looking into the circuit or system (Zsource ¼ Zinput). To implement the concepts in Fig. 9, Zinput is assumed to be first order and characterized by a transfer function (in steady state): Z input ¼
x1 , 1 þ s=x2
(12)
where s is the Laplace transform variable, x1 is a forward gain and x2 represents the bandwidth term. By varying the source impedance Zsource (via a digital controller on the pressure) until it impedance matches Zinput, the source impedance can be tuned to the input impedance of the network under study. Hence, the goal is to make Zsource ¼ Zinput ¼ the characteristic impedance. Thus, for the proper choices of x1 and x2 , the reflected wave can be made zero, S ¼ 0, and the undesired dynamics induced via wave reflections would be minimized. Fig. 10 illustrates this effect when applying the method in Fig. (9) (x2 ¼ 0:8) to the system in Fig. 5, using, in addition, a small amount of rate and position feedback to reduce some of the very high frequency response observed in the force response output. This was accomplished by making Zsource equivalent to Zinput in Eq. (12), but with a PD loop around Zsource to keep its transfer function close to Zinput in a frequency domain sense but still maintaining the quality of the output force signal via a digital controller. Thus, the system now appears reasonably stabilized. The force output in Fig. 10 now sufficiently emulates a force pulse of the type in Fig. 8 above a base line value (approximately 330 N). This is similar to problems that occur in vibration isolation studies. To see the actual force twitch signal (generated above the load value), Fig. 11 illustrates this as a comparison to Fig. 8 above the baseline in Fig. 10. An overall strategy for control of such a PM system will now be synthesized employing knowledge from how biological systems recruit forces using summations of both spatial motor units as well as through temporal summation of these motor units.
ARTICLE IN PRESS D.W. Repperger et al. / Control Engineering Practice 14 (2006) 999–1009 LC1 LC1
Force Output in Newtons
445 356 267
Step Voltage Input
178
10
89
volts
0 0
1.0
2.0 3.0 4.0 TIME IN SECONDS
5.0
0
Fig. 10. Force pulse matching input impedance using wave analysis.
Fig. 11. Force twitch signal extracted from Fig. 10.
4. A strategy for force recruitment derived from biological systems In order to use PM systems for industrial applications in either series or parallel, a review will be made on how such events occur in nature. In biological systems, the control strategy employed to attain a target skeletal muscle force is based upon the superimposition of the mechanical effects of individually stimulated skeletal muscle fibers. This peripheral nervous system control strategy for increasing muscle force amplitude until target acquisition is a function of both the number of new motor units recruited spatially, and the speed or frequency at which these recruited motor units are stimulated. A motor unit is the smallest functional component of the neuromuscular system, and is defined as an alpha motoneuron and the skeletal muscle fibers it innervates. The number of muscle fibers innervated varies, from less than a dozen for fine motor control (e.g. extraocular muscles), to several hundred for gross motor control and muscles responsible for posture. A stimulation parameter $P can be defined to represent the quantity of exciting electrical charge
1005
(delivered in current pulses) to a motor unit by the peripheral motor nerve (Phillips, 2000). An increase in the amplitude or width of the stimulation parameter (a given current pulse) results in the recruitment of additional neighboring motor units, or spatial summation. An increase in the frequency of current pulses (in a current pulse train) results in enhanced force output by the existing recruited motor units, or temporal summation. To model a population of motor units, a set of physiological-based rules (or strategy) must be employed to define recruitment during skeletal muscle contraction. Strength–recruitment curves based upon empirical data for human skeletal muscle have historically indicated that with respect to acquiring target force development, the first 50% of motor units are recruited spatially (Phillips, 2000). Spatial recruitment results in a successive increase in the peak amplitude of the generated muscle twitch, due to the combined effect of the individual force twitches produced by multiple (neighboring) motor units. The remaining increase in force required to capture the desired target is obtained by increasing the stimulation frequency of the already (spatially) recruited motor units. This temporal recruitment results in superimposition (or summation) of the mechanical effects, and is characterized by an increase in the peak amplitude of the generated muscle force (Rack & Westbury, 1969). If the stimulation frequency is increased to approximately 60 Hz, the superimposed mechanical effects fuse completely, resulting in maximum force amplitude (in which tension maximally plateaus) termed tetanus (Fung, 1993), as illustrated in Fig. 12. During tetanus, the stimulation frequency is so rapid that the SR does not have time to reclaim calcium ions from the myoplasm. Consequently, the skeletal muscle endures a sustained (and smooth) contraction, which is useful for performing work on the environment. The goal here is to have the PM system perform a similar type of force generation. To achieve this strategy with the PM system, each force twitch, yi, will be considered an individual element which will be recruited either by temporal summation or by spatial recruitment. The proper use of these control units to synthesize a desired force response is not unlike strategies developed in other industrial applications, e.g. the fiber-yarn production (Sette, Boullart, & Van Langenhove, 1998) where setpoint values and raw materials are designed for optimal quality, for strategies involving the rehabilitation of power plants based on cost and performance issues (Soares, Goncalves, Silva, & Lemos, 1997), and for strategies involving the control of servo-pneumatic actuators (cylinders) with acceleration feedback to improve stability and to mitigate the effects of time delay in Wang, Pu, and Moore (1999).
ARTICLE IN PRESS D.W. Repperger et al. / Control Engineering Practice 14 (2006) 999–1009
1006
LC1 Tetanus 40psi, interval .5 140 LC1
120 100 80 Voltage step inputs
10
60 40
volts
20 0
0 0
2000
4000
6000
8000
10000
12000
Fig. 14. Tetanus emulated with the PM system.
Fig. 12. Tetanus generation via temporal summation (Fung, 1993).
Designed Source Impedance Zsource = yi = A(e−1t −e−2t)
ξ1
1+
y¯ ðtÞ ¼
s ξ1
y(t) in
To Load
Load
Fig. 13. Strategy for temporal and spatial summation.
The tetanus generation can be emulated for the PM system. To describe this action, Fig. 13 describes a summary of what has been determined up to this point. 4.1. Summary of results up to this point A force pulse yi(t) can be generated similar to Eqs. (8) and (9). This signal is then transmitted through an impedance Zsource (via a digital controller) in Fig. 13 to reduce any wave reflections. This new waveform received at the load then produces a force twitch presented in Fig. 11. To generate tetanus using the PM system, it can be realized in two ways. First, at a fixed location, in a time sense (temporal summation) via the summation of yi signals over a fixed time period (n per unit time) but delayed in time Temporal Summation to Tetanus: n X i¼1
yi ðt ti Þ,
m X
yi ðtÞ ¼ m yi ðtÞ.
(14)
i¼1
equation (10)
y¯ i ðtÞ ¼
where ti ¼ 0, Dt, 2Dt, y (n1)Dt and there may be up to n, yi signals in a desired time period. This is a time domain summation. An alternative manner to achieve tetanus would be via a spatial summation (at a specific time point) and in this case more motor units would simultaneously fire. Assuming m motor units would fire, this would be equivalent to: Spatial Summation to Tetanus:
(13)
Thus, by simply multiplying the yi signal by m, a constant, would be equivalent to spatially summing up to m motor units. Fig. 14, which was demonstrated on the PM system, can be generalized by a temporal summation (more frequently adding a force twitch) or by increasing the amplitude of an individual force twitch (recruiting additional motor units). Fig. 14 can be compared to Fig. 12 for similarity. The goal of Fig. 14 is to emulate the small oscillations shown in Fig. 12, where force twitches are synthesized by step voltage inputs into the PM system and the oscillations in both figures correspond to force twitch responses that are desired.
5. Discussion In the human skeletal system, P(t) occurs over less than 10 m/s; C(t) occurs over approximately 30 m/s; and F(t) occurs over 100–300 m/s, depending on fiber type (fast or slow-twitch). Therefore, P(t) has the fastest dynamics, whereas F(t) has the slowest, due to electrochemical and chemomechanical time delays (i.e. kn1 and kn2 , respectively). Fig. 15 portrays the three critical coefficients found from this procedure using the homogenous (free response) terms of Eqs. (3)–(5), i.e., _ ¼ k1 PðtÞ, PðtÞ
(15)
ARTICLE IN PRESS D.W. Repperger et al. / Control Engineering Practice 14 (2006) 999–1009
F (t)
Force Twitch = Slowest time constant
C (t)
P(t) = SR Permeability Fastest Time Constant
Ca 2+ = Intermediate time constant
Time Axis
Fig. 15. Homogenous solutions to Eqs. (15)–(17).
_¯ ¼ k CðtÞ, ¯ CðtÞ 2
(16)
F_¯ ðtÞ ¼ k3 F¯ ðtÞ,
(17)
where the barred quantities refer to the free response terms of Eqs. (3)–(5). The interesting point is in the rank order of the coefficients (time constants) of the above three variables, where k1 ¼ 0.1578, k2 ¼ 0.0628, and k3 ¼ 0.0172 (all units 1/ms). It is no accident that the rank order of these coefficients follows similar to the compartmental model in Fig. 4. What this means, in a physical sense, is that the time history of P(t) (tp ¼ 1/ (0.1578) in ms) is much faster (shorter time constant) ¯ than the time history of CðtÞ (tc ¼ 1/(0.0628)), which, in turn, is much faster than the time history of F¯ ðtÞ (tF ¼ 1/(0.0172)). This makes intuitive sense that the dynamics of the permeability P(t) must be sufficiently faster than the dynamics of the calcium ions to build up. Also, the calcium ion build up is required to be much slower than the permeability dynamics (to have some useful effect) but it is required to be much faster than the force twitch, in order for the force twitch to perform useful work. Recall that the force twitch must include both temporal and spatial summations, so these dynamics must be sufficiently slower than the calcium dynamics yet much, much slower than the permeability dynamics. Thus, the design by nature has carefully constructed the relative time constants of these key variables to enable a force twitch to be created and manipulated for the performance of useful work accomplished on the external environment.
Reynolds, and Berlin (2005) and we present a task showing more advanced results. The goal is to perform force tracking, with a position constraint covering the material presented previously. In Fig. 16 shows the classical work loop diagram used when biological muscle completes a task (full cycle of an agonist–antagonist task). The vertical axis is the force generated by the actuator and the horizontal axis is the length of the muscle. The area A1 in the enclosed curve is the net work delivered by the actuator upon the environment. This has similarities to a Carnot cycle in thermodynamics but here the area is productive in producing work on the environment which is a key output variable. This is a force tracking task because force is generated and the length of the muscle varies during the completion of the mission over one full cycle. Fig. 17 shows empirical data of the actual PM performing this force tracking task. The performance of the system in Fig. (17) can be considered in several ways (Repperger et al., 2005). One key output variable is area A1 in Fig. 17, which shows the net work delivered to the environment from this system. At variance to the method in a Carnot cycle from thermodynamics, here the goal is to maximize area A1, showing the ability of the actuator to perform work on the external environment. Table 3 shows this efficacy for a variety of controllers demonstrating that the area A1 is a key metric for evaluation of this type of controller. Controller 1 is the temporal controller (summation of force pulses in a time sense), controller 2 is the spatial controller (summation of force pulses from diverse locations), and controller 3 is the biologically based as described in Eq. (9).
7. Summary and conclusions Emulating biological signals involving animal skeletal muscle, a pneumatic muscle actuator has been designed Force B
Direction of
6. Implementation of the PM in a closed loop force tracking task To this point, the techniques described herein have been based mainly on an open loop analysis of this actuator and its analogy to biological (skeletal muscle). More recently, implementation results have been demonstrated in Repperger, Phillips, Neidhard-Doll,
1007
Direction of contraction
Relaxation
Area A1 A L1
Muscle Length (normalized as a percent change) L2
Fig. 16. Work loop analysis of biological muscle.
ARTICLE IN PRESS D.W. Repperger et al. / Control Engineering Practice 14 (2006) 999–1009
1008
"
2-Biomimic - 40 PSI B Plot 45 Fmax 40
z2
#
" ;
¯ ¼ A
k2
0
kn2
k3
"
# ;
B¼
kn1 0
# ;
¯ ¼ ½0; 1: C
2-Biomimic - 40 PSI - B Plot
35
(A.3)
Work Loop
30 Psi
x¯ ¼
psi
z1
Since C(0) ¼ F(0) ¼ 0 in Eqs. (7b) and (7c) and from Laplace transforming (A.1) and (A.2) yields
Contraction
25 Area A1
20
¯ ¯ 1 BPðsÞ ¯ ¼ HðsÞPðsÞ, ¯ F¯ ðsÞ ¼ Y ðsÞ ¼ CðsI AÞ
Relaxation
15 10 Fmin 5 0 0
2
1
3 Length
5
4
6
(A.4)
where the bars indicate that Laplace transform quantities are being utilized. The transfer function matrix H(s) is scalar and can be represented via HðsÞ ¼
Fig. 17. PM in a work loop force tracking task.
F¯ ðsÞ kn1 kn2 . ¼ ¯ ðs þ k2 Þðs þ k3 Þ PðsÞ
(A.5)
From Eq. (10), it is desired to reduce (A.5) to the time domain representation Table 3 Performance of the PM system in a full cycle of a work loop task Controller employed Controller Controller Controller Controller Controller Controller
1 1 2 2 3 3
PSI for 10 v signal input
A1 in joules as work produced externally
40 70 40 70 40 70
0.303 0.589 0.319 0.655 0.303 0.543
yðtÞ ¼ F ðtÞ ¼ A½eg1 t eg2 t .
(A.6)
To achieve this goal, H(s) is decomposed via a partial fraction expansion: HðsÞ ¼
D1 D1 . s þ k2 s þ k3
(A.7)
This will occur if D1 k3 D1 k2 ¼ kn1 kn2 or choose the following relationships to realize the form of (A.6): A ¼ D1 ¼
kn1 kn2 1 1 ; with g1 ¼ and g2 ¼ . k2 k3 k3 k 2 (A.8)
Note, k24k3 and A40, resulting in g24g1. to work in a force control sense. The overall system had to be stabilized, including the frame and connector dynamics. Once a force twitch that showed a reasonable semblance to a typical slow-twitch animal muscle response could be produced, the tetanus recruitment paradigms could also be replicated. This provides a method for the pneumatic muscle system to be used in a force control systems mode both in its generated force output and in synthesizing its strategy to accumulate force, much like is accomplished in animal skeletal muscle.
Appendix A It is desired to show the relationship between the state variable representation in Eqs. (8)–(10), which represents a time domain representation of the force twitch signal. Rewriting Eqs. (8) and (9) as ¯ x¯ þ BPðtÞ, x_¯ ¼ A
(A.1)
y ¼ C¯ x, ¯
(A.2)
Appendix B To show the scattering parameter S of Eq. (11) would be zero if Zsource ¼ Zinput, the ratio S ¼ Vr/ Vi ¼ (ZinputZsource)/(Zinput+Zsource) now follows from definitions commonly occurring in circuit analysis via transmission lines. The total voltage across the load is V ¼ Vi+Vr, The total current in the series circuit is given by IT ¼ Ii+Ir. This gives rise to the ratios: Vi/ Ii ¼ Vr/Ir ¼ Zsource. Hence, V i ¼ ð12ÞðV þ Z source I T Þ and V r ¼ ð12ÞðV Z source I T Þ. This results in S ¼ Vr/ Vi ¼ (ZinputZsource)/(Zinput+Zsource) and when Z input ¼ Zsource , then S ¼ 0. This means there is no reflected wave. References Alleyne, A., & Liu, R. (2000). A simplified approach to force control for electro-hydraulic systems. Control Engineering Practice, 8, 1347–1356. Baptista, L. F., Sousa, J. M., & Sa’ da Costa, J. M. G. (2001). Fuzzy predictive algorithms applied to real-time force control. Control Engineering Practice, 9, 411–423.
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