Sensors for Evaluation of Thermodynamical Model of pMA

Lukas Kopecny Central European Institute of Technology

Brno University of Technology Brno, Czech Republic

lukas.kopecny@ceitec.vutbr.cz

Ludek Zalud Department of Control and Instrumentation

Faculty of Electrical Engineering and Communication, BUT Brno, Czech Republic zalud@feec.vutbr.cz

Abstract— In this paper a mathematical model of Pneumatic Muscle Actuator (pMA) is developed and verified. The focus is put on thermodynamical behavior of pMA during charging and discharging process. Problems with selection of temperature sensor are discussed and evaluation of a mathematical model is done.

Keywords-pMA, temperature sensor, thermodynamic

I. INTRODUCTION Modern haptic interfaces and other applications in robotics

require high performance force actuators, with high force output per unit weight. Traditional geared electrical motors cannot provide these characteristics [1].

Many concepts of “uncommon” actuators are employed in latest robotics. Undoubtedly one of the most promising actuator is pMA – Pneumatic Muscle Actuator [2]. It has a number of exceptional properties:

Actuators have exceptionally high power and force to weight volume ratios.

The pMA can be made in any diameter and length.

The structure of pMA makes it comparable in shape, properties and performances to human muscles which makes it easy to implement a human/computer interface.

The actual achievable displacement (contraction) is typically 30% of the dilated length.

The muscles are highly flexible, soft in contact and have excellent safety potential (by limited contraction).

Controllers developed for the muscle systems have shown them to be controllable to an accuracy of 1% of displacement. Bandwidth of muscles of up to 5 Hz can be achieved.

The contractile force of actuator can be over 300 N/cm2 for pMA (compared to 20-40 N/cm2 for natural muscles).

Accurate smooth motion from start to stop.

Low cost, powerful actuation, lightweight compact device.

High safety – it works in wet or explosive environment.

Although this actuator was invented in the fifties of last century, it is not up-to-date commonly used. The difficulty of control mainly due to the high nonlinear structure still keeps the pMA at the edge of interest of industrial applications. However there are a lot of phenomenons influencing a tricky behavior of pMA, one of the most important features is natural damping of pMA caused by very complex inner friction [3]. This friction is highly temperature dependant. That is the main reason why thermodynamical model was developed.

II. THERMODYNAMICAL MODEL OF PMA In this section is developed a differential equation that links

the muscle pressure to the mass flow through the system, muscle contraction and contraction speed.

In similar works describing mathematical model of pneumatic cylinder the authors derived this equations using assumption that both, the charging and the discharging processes, were adiabatic. In [4] authors found experimentally, that the temperature inside the cylinder lies between the theoretical adiabatic and isothermal curves. The temperature was close to the adiabatic curve only for the charging process.

There is an assumption, that similar principle should be valid also for pneumatic Muscle Actuator.

The most general model for volume of gas consists of three equations:

1. equation of state (ideal gas law)

2. conservation of mass (continuity) equation

3. energy equation.

Let us assuming that:

the gas is perfect

pressure and temperature inside the muscle is homogenous

kinetic and potential energy are negligible (pMA is a pneustatic device).

This work was supported by project CEITEC (CZ.1.05/1.1.00/02.0068) from European Regional Development Fund.

2013 Seventh International Conference on Sensing Technology

978-1-4673-5221-5/13/$31.00 ©2013 IEEE 440

RTP (1)

VVVdtdmmm outin (2)

UWTmTmkCqq outininvoutin (3)

Where: P is pressure, V volume with density ρ, R ideal gas constant, T thermodynamic temperature, m mass, outinm , mass flows entering resp. leaving the muscle, outinq , are heat transfer terms (heat flows), k specific heat ratio, vC specific heat at

constant volume, inT temperature of incoming gas flow, W the rate of change in the work, U the change of internal energy.

The total change in internal energy is

VPPVk

PVdtd

kmTC

dtd

U v 11

11

(4)

Substituting VPW and Eq. (4) into Eq. (3) and asuming, that the incoming flow is already at the temperature of the gas in the muscle, the energy equation become

PkPVVmmqq

kPk

outinoutin11 (5)

If the process is considered to be adiabatic ( 0outin qq ), the time derivative of the muscle pressure is

VVPkmm

VRTkV

VPkmm

VPkP outinoutin

(6)

If the process is considered to be isothermal, then Eq. 5 can be writen as

outinoutin mmPVPqq (7)

and the rate of change in pressure is

VVPmm

VRTP outin (8)

The only difference between Eq. 6 and Eq. 8 is in the specific heat ratio term k. Thus both equation can be rewritten as

VVPmm

VRTP outoutinin (9)

with outin ,, taking values between 1 and k. The volume V of a pMA can be expressed [5] as

2

22

4 nLbLV (10)

and its time derivative V as

2

22

43nLbLV (11)

where L is current muscle length, b muscle thread length and n number of thread turns.

Substituing Eq. (10) and (11) into (9), the time derivative for the muscle pressure becomes

LLbLLbPmm

LbLRTnP outoutinin )(

3422

22

22

2

(12)

Figure 1. Determining of coefficients n and b

In this form the pressure equation accounts different heat transfer characteristics of the charging and discharging processes and air compression and expansion due to muscle contraction resp. extension.

III. VERIFICATION OF THE MATHEMATICAL MODEL The mathematical model of the system based on the

equations from the previous chapter was created for the numerical simulation in Matlab Simulink. The model is verified first with simulated working cycle and then compared with the waveforms of the real system.

Isotonic working cycle is simulated. Pneumatic muscle is inflated during half period by constant mass flow of gas and than other half of the period again deflated with the same mass flow.

The progress of the temperature measured in the actual muscle and the temperature derived from the model is shown in the graph on Fig. 5.

IV. TEMPERATURE SENSORS In chapter II we assumed, that the temperature inside the

muscle is homogenous. As can be seen on the thermo image of pneumatic muscle actuator on picture 2, the temperature is far away from homogenous distribution.

2013 Seventh International Conference on Sensing Technology

441

Figure 2. Thermo image of pMA after 20 working strokes.

Scale 20 to 40 °C.

It is caused by better gas exchange on the muscle air port side.

Figure 3. Temperature sensor integrated into muscle wall

This looks like a serious problem for temperature measuring of filling gas. But on the other side, we are interesting mostly in changes in temperature, not so in its absolute value.

For appropriate measurement of muscle internal temperature we look for a temperature sensor with:

1. fast time response

2. ability of spatial temperature averaging to suppress the temperature non-homogenity

Figure 4. Temperature sensor placed inside the pneumatic

muscle

We try to solve the temperature measuring problem in two ways:

1. We tried to integrate the temperature sensor into braided sleeving of the muscle wall (Fig. 3). This placement proved a bad solution. The sensor gives data more about a rubber muscle wall temperature (heated by material friction) instead of inside gas temperature changes.

2013 Seventh International Conference on Sensing Technology

442

2. We had to place the temperature sensor inside the pneumatic muscle. We used fast response low mass foil Pt sensor with light carbon housing against damage (see figure 4) and we placed it in the centre inside of pneumatic muscle.

With this arrangement we obtain result, which can be seen on Figure 5.

Figure 5. Instantaneous temperature in pneumatic muscle -

real system (blue), model (red)

Let us take into account the inertia of the temperature sensor and the time shift caused by rapid filling and venting of the muscle in the real system against the system model, we can see a rough shape match between signal obtained from a mathematical simulation model and measured on the real system. The reason for the slower growth of real muscle temperature is cooling by ambient air flowing under the thermal insulation when muscle is moving.

V. CONCLUSION The main objective of this work is to extend the

mathematical model of pneumatic muscle especially in the field of thermodynamics. The paper applies the method originally designed for pneumatic linear drives [4] on pneumatic muscles, creating a new thermodynamic model and the newly proposed mathematical model includes the heat generated by friction, which has significant influence on the behaviour of the actuator.

If the task is a description of the physical (especially thermomechanical) phenomena taking place during the working cycle of the muscle, not the precise identification of a specific muscle, we obtained a good degree of consistency in the behaviour of the model and the real system. The aim of verification of the model is rather qualitative comparison with the actual behaviour of the real system, rather than a quantitative assessment of its accuracy.

The paper is focused also on temperature sensor selection and especially on its placement. Measurements are compared with simulations and obtained results discussed.

REFERENCES [1] Richer, E., and Hurmuzlu, Y.: "A High Performance Pneumatic Force

Actuator System: Part I-Nonlinear Mathematical Model," ASME J. Dyn. Syst., Meas. Control, 122, pp. 416-425, 1999.

[2] Tondu, B.,Lopez, P.: Modeling and Control of McKibben Artificial Muscle Robot Actuators, IEEE Control Systems Magazine, pages 15 – 38, April 2000.

[3] Doumit, M., Fahim, A., Munro, M.: Analytical Modeling and Experimental Validation of the Braided Pneumatic Muscle, IEEE Transactions on Robotics, vol. 25, no. 6, pages 1282 - 1291, December 2009.

[4] AI-Ibrahim, A. M., and Otis, D. R.: "Transient Air Temperature and Pressure Measurements During the Charging and Discharging Processes of an Actuating Pneumatic Cylinder," Proceedings of the 45th National Conference on Fluid Powe., 1992.

[5] Chou, C.P., Hannaford, B.: Static and Dynamic Characteristic of McKibben Pneumatic Artificial Muscles, In: IEEE International Conference on Robotics and Automation, pages 281-286 vol. 1, 1994.

0 10 20 30 40 50 60 70 80 90 100 298

298.2

298.4

298.6

298.8

299

299.2

299.4

299.6

299.8

300 Muscle temperature (real/model)

time [s]

temperature [K]

2013 Seventh International Conference on Sensing Technology

443