Passivity-Based Impact and Force Control of a Pneumatic Actuator

Passivity-Based Impact and Force Control of a Pneumatic Actuator Yong Zhu Eric J. Barth e-mail: [email protected] Department of Mechanical Engineering, Vanderbilt University, Nashville, TN 37235

tion of the impedance control approach is that it requires accurate environment and stiffness information for good force control. An actuator can be stable in free space but can become unstable when it is coupled to a not well-characterized environment. A stable interaction can be achieved either in a passive way, by using a suitable compliant mechanical device, or in an active way, by designing an interaction control strategy. A passive system can interact stably with any strictly passive environment 关8兴. A safe and stable interaction can be achieved if the actuation system is passive with respect to the environment and/or a human operator. A system is said to be passive if the energy absorbed over any period of time is greater than or equal to the increase in the energy stored over the same period 关9兴. Consider a system with a general form 关10兴 of V˙共t兲 = y Tu − g共t兲

To carry out stable and dissipative contact tasks with an arbitrary environment, it is critical for a pneumatic actuator to be passive with respect to a supply rate consisting of the spool valve position input and the actuation force output. A pseudo-bond graph model with the inner product between spool valve position input and actuation force output as a pseudo-supply rate is developed. Using this pseudo-bond graph model, an open-loop pneumatic actuator controlled by a four-way proportional valve can be proven to not be passive with respect to the pseudo-supply rate. Conversely, it can also be proven to be passive with respect to the pseudo-supply rate under a closed-loop feedback control law. The passivity of the closed-loop pneumatic actuator is verified in impact and force control experiments. The experimental results also validate the pseudo-bond graph model. The pseudo-bond graph model is intended for passivity analysis and controller design for pneumatic actuation applications where contact stability (such as robotic assembly) and/or stable interaction with a passive environment (such as human-robot interaction) is required. 关DOI: 10.1115/1.2837430兴

1

Introduction

Control of the interaction force between a robot manipulator and its environment is critical for the successful execution of many industrial tasks such as polishing, assembly, deburring, etc. Additionally, there is a newfound interest—and research is being conducted—regarding the interaction and coexistence of robots and humans not only on the shop floor, but also for applications at home and in the medical industry 共i.e., entertainment, service robotics, and rehabilitation兲. Maintaining a stable and safe interaction force is the key aspect among these applications. Tasks that require a high degree of interaction with the environment require the actuator to be an impedance as opposed to an admittance 关1兴. Many approaches have been taken to have an actuator contact an environment and maintain a certain contact force for electrical systems 关2兴, hydraulic systems 关3,4兴, and pneumatic systems as well 关5兴. Most of these approaches divide the task into three modes: free space mode, constrained mode, and transition mode. Different switching control strategies are used to guarantee stability and minimize bouncing upon contact. Another widely used approach for contact force control is impedance control 关6,7兴. The key point of the impedance control method is that one controller deals with all stages of the contact task. Hogan 关6兴 proposed stable contact tasks using impedance control. Hogan 关7兴 also showed that if an actuation system has the behavior of a simple impedance, then the stability of the system is preserved when it is coupled to a stable environment. The limitaContributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT, AND CONTROL. Manuscript received March 24, 2006; final manuscript received July 19, 2007; published online February 29, 2008. Review conducted by Mark Costello.

共1兲

where V共t兲 and g共t兲 are scalar functions of time. Generally, V共t兲 is called a storage function and g共t兲 is the dissipative term, u is the system input and y is the system output. A supply rate can be represented as the scalar s共u , y兲 = y Tu. The system is termed passive if V is lower bounded and g共t兲艌 0. Furthermore, a passive system is dissipative if

冕

⬁

g共t兲dt ⬎ 0

共2兲

o

Integrating Eq. 共1兲 gives V共t兲 − V共0兲 =

冕

t

s共u,y兲d␶ −

0

冕

t

g共t兲d␶ 艋

0

冕

t

s共u,y兲d␶

共3兲

0

Since V共t兲 is lower bounded and V共0兲 is the lower bounded initial value of the storage function, 兰t0s共u , y兲d␶ is lower bounded. Thus, passivity is generally defined as follows: A system is said to be passive with respect to the supply rate s共u , y兲 if, for a given initial condition, there exists a c共兩c兩 ⬍ ⬁兲 so that for all time t and for all input u,

冕

t

s共u,y兲d␶ 艌 − c2

共4兲

0

This result merely states that the net energy that has flowed into the system is bounded below or, equivalently, that the net work done by the system 共net energy that has flowed out兲 is bounded above for all time. The directional control valve is the only nonpassive device in a hydraulic actuation system. Li 关11兴 proved that if appropriate first order or second order spool dynamics can be implemented, the spool valve can become passive. The same dynamic passive valve method has been used in a bilateral teleoperation of a hydraulic actuator 关12兴. The passivity concept is also used to design a hydraulic backhoe/force feedback joystick system 关13兴. Other than hydraulic systems, some other passive systems, such as cobots 关14兴 and smart exercise machines 关15兴, have also been designed for different human-robot interaction tasks. As will be shown, the fundamental energetic properties of pneumatic actuation can be exploited to provide stable interaction forces with any passive environment. Another convenient property of a pneumatically actuated system that will be exploited is the ability to measure actuation forces using pressure sensor feedback. The passivity properties of a pneumatic system have not previously been investigated explicitly. The objective of this paper is to investigate the passivity properties of a pneumatic actuator subject to closed-loop feedback control and to apply passivity properties to impact and force control. The paper is organized as follows. First, in Sec. 2, a pseudo-bond graph model of a pneumatic actuator is proposed. Then, a closed-loop feedback control system is proven to be passive in Sec. 3. Section 4 shows the experimental

Journal of Dynamic Systems, Measurement, and Control Copyright © 2008 by ASME

MARCH 2008, Vol. 130 / 024501-1

Downloaded 29 Feb 2008 to 129.59.78.59. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms_Use.cfm

1 V = kx2 2

共8兲

Taking the time derivative of Eq. 共8兲, substituting in Eq. 共5兲, and proceeding gives, Fig. 1 Bond graph models for „a… a hydraulic actuator and „b… a pneumatic actuator

冕

results of impact and force control utilizing the methods and tools developed. Finally, Sec. 5 contains concluding remarks.

2 Pseudo-bond Graph Model for a Pneumatic Actuator A bond graph model for hydraulic systems with incompressible fluid 关11兴 is shown in Fig. 1共a兲. The pressure difference 共P兲 and volumetric flow rate 共Q兲 are directly associated with the actuator work output 共Fx˙兲 through a transformer. The inner product between the effort and flow, generally called the supply rate, represents the true hydraulic power input to the hydraulic system. Since the pressure difference is directly associated with the output actuation force and the volumetric flow rate can be directly specified by the spool valve position input, the bond graph representation of a hydraulic system lends good insight into its passivity analysis. In a pneumatic actuator, specific enthalpy 共h兲 is the effort vari˙ 兲 is the flow variable for the bond able and mass flow rate 共m graph, as shown in Fig. 1共b兲. Because of the capacitance associated with the compressibility of a gas, the behavior of a pneumatic actuator is fundamentally different from the behavior of a strictly ˙ 兲 = hm ˙ is incompressible hydraulic system. The supply rate s共h , m no longer directly related to the actuator work output 共Fx˙兲. This makes the passivity analysis of a pneumatic actuator less straightforward than for a hydraulic actuator. To design a passive pneumatic actuation system, it is useful to first establish a pseudo-bond graph model to directly associate the spool valve position input with the actuation force output. A simple passivity analysis of a spring-mass system is first given to help elucidate this idea through analogy. The dynamic equation of the spring-mass system, as shown in Fig. 2, is mx¨ + kx = F

共5兲

The total energy of the system is 1 1 V = mx˙2 + kx2 2 2

共6兲

Taking the time derivative of Eq. 共6兲 and utilizing Eq. 共5兲 yields V˙ = mx˙x¨ + kxx˙ = x˙共mx¨ + kx兲 = Fx˙

共7兲

This spring-mass system is passive with respect to the supply rate s共F , x˙兲 = Fx˙ according to Eq. 共1兲, where g共t兲 = 0. Or, more explic1 itly, to satisfy Eq. 共4兲: 兰t0s共F , x˙兲d␶ = 兰t0共mx˙x¨ + kxx˙兲d␶ = 2 mx˙2共t兲 1 1 1 1 1 − 2 mx˙2共0兲 + 2 kx2共t兲 − 2 kx2共0兲艌 − 2 mx˙2共0兲 − 2 kx2共0兲 = −c2. The same analysis can be done via a different approach by choosing the storage function as

V˙ = kxx˙ = 共F − mx¨兲x˙ = Fx˙ − mx¨x˙

共9兲

kxx˙ + mx¨x˙ = Fx˙

共10兲

t

0

冕

t

0

kxx˙d␶ +

冕

t

0

mx¨x˙d␶ =

冕

t

s共F,x˙兲d␶

共11兲

0

1 1 1 1 s共F,x˙兲d␶ = kx2共t兲 − kx2共0兲 + mx˙2共t兲 − mx˙2共0兲艌 − c2 2 2 2 2 共12兲

where the right hand side of Eq. 共12兲 is bounded below. The spring-mass system is still shown to be passive with respect to the same supply rate s共F , x˙兲 = Fx˙. It will be shown that a pneumatic system can be analyzed in a manner similar to Eqs. 共9兲–共12兲 by using a storage function more similar to Eq. 共8兲 than Eq. 共6兲. A mathematical model of a pneumatic actuator has been well described 关16,17兴. Assuming that the gas is perfect, the temperature and pressure within the two chambers are homogeneous, and the kinetic and potential energies of the fluid are negligible, the rate of change of pressure within each pneumatic chamber a and b of the actuator can be expressed as rRT rP V˙ ˙ a,b − a,b a,b m P˙a,b = Va,b Va,b

共13兲

where r is the thermal characteristic coefficient, with r = 1 for the isothermal case and r = ␥ for the adiabatic case 共␥ is the specific heat ratio兲, R is the ideal gas constant, T is the temperature, Va,b is the volume of chambers a and b, respectively, Pa,b is the pressure ˙ a,b is the mass flow rate in chambers a and b, respectively, and m into chambers a and b, respectively. All analyses carried out in this paper are based on an isothermal assumption 关17兴. Strictly speaking, this assumption introduces a thermal bond to the pneumatic actuation system not shown in Fig. 1共b兲. Since this thermal bond is always dissipative, it can be neglected from the sufficient passivity conditions that will follow. The actuation force of a pneumatic actuator can be expressed as follows using absolute pressures: Fa = PaAa − PbAb − PatmAr

共14兲

where Patm is the atmospheric pressure and Ar = Aa − Ab. Since the bandwidth of the valve is typically much higher than the bandwidth of the closed-loop system, it will be assumed that the spool valve position xv is proportional to the control voltage. Therefore, in the analysis that follows, the spool valve position is considered as the control input, and xv = 0 corresponds to the center spool position and is defined as the zero mass flow rate. To model a pneumatic actuator as a one-port device, the following admissible storage function is chosen: 1 V1 = 共PaAa − PbAb − PatmAr兲2 2

共15兲

Taking the time derivative of Eq. 共15兲 and combining with Eq. 共14兲 yields V˙1 = Fa共P˙aAa − P˙bAb兲 Fig. 2 A spring-mass system

024501-2 / Vol. 130, MARCH 2008

共16兲

Substituting Eq. 共13兲 into Eq. 共16兲 yields Transactions of the ASME


冊冉

冉

RTAb V˙ A V˙ A RTAa ˙a− ˙ b + Fa − a a Pa + b b Pb V˙1 = Fa m m Va Vb Va Vb

冊共17兲

Referring to Liu and Bobrow’s work 关18兴 on the linearized characteristics of a four-way spool valve, if the valve operates within its mechanical operating range and is under choked flow conditions, the mass flow rate can be accurately characterized in a linear form as being proportional to the valve spool position xv, ˙ a = C 1x v m

共18兲

˙ b = − C 1x v m

共19兲

where C1 is a positive constant depending on the valve design. Regarding the modeling of the mass flow as choked flow, the maximum mass flow rate occurs under choked flow conditions, and the unchoked flow rate is bounded by the choked flow rate. Therefore, if passivity can be shown for the choked flow rate condition, it will also hold for unchoked flow. Additionally, from a practical standpoint, most pneumatic actuation systems operate in the choked flow regime most of the time. The rate of change of volume in each chamber of a pneumatic actuator is related to the actuator velocity x˙ by V˙a = Aax˙

共20兲

V˙b = − Abx˙

共21兲

Substituting Eqs. 共18兲–共21兲 into Eq. 共17兲 yields

冉

冊

冉

A2a A2b RTAa RTAb + C1xvFa − Fax˙ Pa + Pb V˙1 = Va Vb Va Vb

冊

A comparison of the spring-mass, pneumatic, and hydraulic systems are summarized in Table 1, showing the analogous role of the pseudo-bond for a pneumatic actuator. This section can be concluded with the following two lemmas: LEMMA 1. A pneumatic actuator controlled by a four-way proportional valve can be considered using a pseudo-bond with a supply rate s共xv , Fa兲 ª −xvFa for passivity analysis. LEMMA 2. A pneumatic actuator controlled by a four-way proportional valve in an open-loop manner is not passive with respect to the supply rate s共xv , Fa兲 ª −xvFa. Lemma 1 is based on the previous explanation in this section. The proof of Lemma 2 is shown here. Proof of Lemma 2. Rewriting Eq. 共23兲,

共22兲

Letting k1 = 共RTAa / Va + RTAb / Vb兲C1 and k2 = A2a Pa / Va + A2b Pb / Vb, where k1 ⬎ 0 and is upper bounded since Va , Vb ⬎ 0, given that either volume in the actuator is physically limited to some minimum, and k2 ⬎ 0, Eq. 共22兲 can be rewritten as V˙1 = k1xvFa − k2Fax˙

Fig. 3 Passive valve-actuation schematic. The system produces power when the product of the actuation force Fa and spool position xv is positive „as shown…. Feedback is indicated as a virtual link between the actuator force and the valve spool position.

V˙1 k2 = xvFa − Fax˙ k1 k1

and utilizing the supply rate definition s共xv , Fa兲 ª −xvFa, the fact that in freespace Fa = Mx¨, where M is the mass of the moving mass, and integrating yields

冕

共23兲

Comparing Eq. 共23兲 with Eq. 共9兲, we can see that although a pneumatic system has very nonlinear dynamics, it can be closely modeled in a similar structure as a spring-mass linear system using the new storage function 共pseudo-energy兲. The pseudo-bond graph model based on Eq. 共15兲 can help us associate the spool valve position input and the actuation force output for passivity analysis. Furthermore, in the pseudo-bond graph model, the supply rate is the inner product of the input and output of the pneumatic actuator, and the active term xv only appears in the supply rate. The term −k2Fax˙ will be shown to be a passive term associated with the actuator’s kinetic energy. Its role is similar to −mx¨x˙ in Eq. 共9兲 for the spring-mass system, which is also a passive term. This behavior and the springlike role of the storage function of Eq. 共15兲 is intuitive, given the compressibility of the working fluid in pneumatic systems. The power form of the storage function shown in Eq. 共23兲 is 1 central to the idea of this paper. Since V1 = 2 共PaAa − PbAb 2 − PatmAr兲艌 0 and is therefore lower bounded, we can consider V1 as the pseudo-energy of a pneumatic actuator. Defining Fa and xv in the same sense 共see Fig. 3兲, positive power is produced by the system when Faxv ⬎ 0. Given the convention to define the supply rate as positive when the system absorbs power, the supply rate is therefore defined as s共xv , Fa兲 ª −xvFa. This is similar to the approach taken in Ref. 关10兴 in defining the supply rate as s共PL , Q兲 ª −PLQ for a hydraulic actuation system. The perspective taken will be to find a control law that passifies this system. The passified actuation system will then be capable of interacting with any passive environment in a stable manner. Journal of Dynamic Systems, Measurement, and Control

共24兲

t

0

V˙1 d␶ + k1

冕

t

0

k2 Mx¨x˙d␶ = − s共xv,Fa兲 k1

共25兲

For k1 ⬎ 0 and k2 ⬎ 0, the left-hand side of Eq. 共25兲 is bounded below, yielding − c2 艋 − s共xv,Fa兲

共26兲

s共xv,Fa兲艋 c2

共27兲

or

Therefore, the open-loop pneumatic actuator influenced in an open-loop manner by a proportional valve is not passive with 䊐 respect to the supply rate s共xv , Fa兲 ª −xvFa.

Table 1 Comparison of spring-mass, pneumatic, and hydraulic systems

Effort Flow Supply rate Admissible Storage function

Spring mass 共true bond兲

Pneumatic actuator 共pseudo-bond兲

Hydraulic actuator 共true bond兲

F x˙ Fx˙

x␯ Fa −x␯Fa

PL Q −PLQ

1 V = kx2 2

1 V = 共PaAa − PbAb − PatmAr兲2 2

1 V = x2␯ 2

MARCH 2008, Vol. 130 / 024501-3


Fig. 4 Closed-loop feedback control structure Fig. 5 Illustration of passive force control

3

Passive Pneumatic Actuator

To passify the pneumatic actuator in free space, the actuation force is fed back using pressure sensor measurements. If a closedloop pneumatic system can be proven to be passive in free space, it will be able to interact in a stable manner with any passive environment. This will enable stable and well-behaved impact and force control. Therefore, we can assume Fa = Mx¨ 共free space case兲 in the passivity analysis, where Fa is the actuation force, M is the actuator moving mass, and x is the actuator position. This is summarized by the following lemma. LEMMA 3. A pneumatic actuator controlled by a four-way proportional valve is passive with respect to the supply rate s共xv , Fa兲 ª −xvFa under the closed-loop feedback control law: x˙v = −k0Fa, where k0 is a positive feedback gain. Proof of Lemma 3. The same storage function is chosen as Eq. 共15兲 for the pneumatic actuator, with the associated power form given by Eq. 共23兲. The feedback system is constructed, as shown in Fig. 4. The feedback control law is 共28兲

x˙v = − k0Fa

A storage function is chosen for the feedback control law dynamics and is given by 1 V2 = xv2 2

共29兲

Taking the time derivative results in V˙2 = xvx˙v = − k0Faxv

共30兲

Including the storage function associated with the actuator from Eq. 共15兲, the total storage function for the closed-loop system is then given as 共31兲

V = V1 + V2 Taking the time derivative of Eq. 共31兲 gives

V共t兲 − V共0兲 = ␦

冕

t

s共xv,Fa兲d␶ −

0

Rewriting this as 1

␦

冋

V共t兲 − V共0兲 +

冕

t

0

冕

t

k2Mx¨x˙d␶

共37兲

s共xv,Fa兲d␶

共38兲

0

册

k2Mx¨x˙d␶ =

冕

t

0

and considering the fact that k2 ⬎ 0 and ␦ ⬎ 0, the left-hand side of Eq. 共38兲 is bounded below. Therefore,

冕

t

s共xv,Fa兲d␶ 艌 − c2

共39兲

0

proves that the closed-loop pneumatic actuation system controlled by a four way proportional valve is passive with respect to the supply rate given by s共xv , Fa兲 = −xvFa under the feedback control law x˙v = −k0Fa. Furthermore, the original term k2Fax˙ in Eq. 共35兲, and its free-space equivalent k2Mx¨x˙ in Eq. 共37兲, can be seen to be a passive term associated with the kinetic energy of the actuator. Obvious similarities exist between the closed-loop pneumatic system and the mass-spring archetype by comparing Eq. 共38兲 with Eq. 共11兲. 䊐

4

Passive Impact and Force Control

The pneumatic actuator and valve closed-loop system has been shown to be passive using pressure feedback to obtain the actuation force and using the control law of Eq. 共28兲. This is not particularly useful if the valve can only passively dissipate to zero mass flow rate. Another input and output port should be added to the passive control structure so that the actuator can output a desired nonzero actuation force. The control law is modified as follows: x˙v = − k0共Fa − Fd兲

共40兲

Since k1 is upper bounded, there always exists a positive feedback gain k0 ⬎ k1 to make

where Fd is the desired actuation force and Fa is the actuation force, as determined by pressure feedback measurements and Eq. 共14兲. The new control law is illustrated in Fig. 5 and will be proven to be passive. It can be seen from Fig. 5 that the input does not change the passive structure of the closed-loop system, given that Fd is an exogenous input. Proof. The same storage function is chosen as Eq. 共15兲. Substituting Eq. 共40兲 into Eq. 共30兲 gives

V˙ = − ␦Faxv − k2Fax˙

V˙2 = xvx˙v = − k0Faxv + k0Fdxv

V˙ = V˙1 + V˙2

共32兲

Substituting Eqs. 共23兲 and 共30兲 into Eq. 共32兲 gives V˙ = 共k1 − k0兲Faxv − k2Fax˙

共33兲

共34兲

where ␦ is a positive constant. The supply rate to the pneumatic system is s共xv , Fa兲 ª −xvFa, resulting in V˙ = ␦s共xv,Fa兲 − k2Fax˙

共41兲

Substituting Eqs. 共41兲 and 共23兲 into Eq. 共32兲 gives V˙ = − ␦Faxv + k0Fdxv − k2Fax˙ = s„共xv,xv兲,共Fa,Fd兲… − k2Fax˙

共35兲

共42兲

To analyze the passivity of the closed-loop system, the scaled actuator work term k2Fax˙ must be considered in more detail. Integrating Eq. 共35兲 yields

Now, the pneumatic actuator closed-loop control system can be seen as a two-port system with supply rate s(共xv , xv兲 , 共Fa , Fd兲) = −␦Faxv + k0Fdxv. The closed-loop system is still passive based on the previous proof of Lemma 3. 䊐 Experiments were carried out using the experimental setup shown in Fig. 6. The experimental results will verify the passivity of the closed-loop system and show the application of the passivity properties to impact and force control of a pneumatic actuator.

V共t兲 − V共0兲 = ␦

冕

t

s共xv,Fa兲d␶ −

0

冕

t

k2Fax˙d␶

0

Assuming free-space dynamics Fa = Mx¨, this becomes 024501-4 / Vol. 130, MARCH 2008

共36兲

Transactions of the ASME


Fig. 6 Experimental setup for impact and force control

The setup consists of the horizontal actuator of a Festo two degree-of-freedom pick and place pneumatic system 共the vertical degree of freedom has been removed兲. The double acting pneumatic actuator 共Festo SLT-20-150-A-CC-B兲 has a stroke length of 150 mm, an inner diameter of 20 mm, and a piston rod diameter of 8 mm. A linear potentiometer 共Midori LP-150F兲 with 150 mm

maximum travel is used to measure the linear position of the actuator. One four-way proportional valve 共Festo MPYE-5-M5010-B兲 is attached to the chambers of the actuator. Two pressure transducers 共Festo SDE-16– 10 V / 20 mA兲 are attached to each actuator chamber, respectively. Control is provided by a Pentium 4 computer with an analog/digital 共A/D兲 card 共National Instruments PCI-6031E兲, which controls the proportional valve through an analog output channel. A load cell 共Transducer Techniques MLP-25兲 is mounted at the end of the actuator to measure the impact and contact force when it hits a rigid environment 共also shown in Fig. 6兲, but is not used for control purposes. For a desired interaction force Fd = 40 N, the passified actuation system is driven by an input of Fd plus a constant Coulomb friction compensation 共assumed to be a constant兲 toward an unknown environment, which is a stiff wall. The control gain k0 was simply tuned while satisfying the condition k0 ⬎ k1 set out by the proof of Lemma 3. Experimental results are shown below in Fig. 7. Although the velocity in free space is not explicitly controlled, it is typical of impact experiments, such as those of Ref. 关19兴, in terms of approach velocity and the magnitude of the desired contact force. As can be seen in Fig. 7, the unexpected impact is stable and dissipative. The velocity of the actuator suddenly drops to zero

Fig. 7 Impact and force control without the dissipative term. „a… Sensor force, „b… velocity, „c… pressure, and „d… control voltage.

Journal of Dynamic Systems, Measurement, and Control

MARCH 2008, Vol. 130 / 024501-5


Fig. 8 Impact and force control with the dissipative term. „a… Sensor force, „b… velocity, „c… pressure, and „d… control voltage.

right after the impact because of the passivity properties of the closed-loop system. The problem with the experimental results shown in Fig. 7 is that the valve control voltage 共proportional to the valve spool position兲 oscillates for quite a long period of time after the impact because there is no dissipative term in the control law. The valve response finally damps out because of the natural damping of the valve, which is not enough to absorb the energy quickly after the impact occurs. A dissipative term is therefore added to the valve dynamics to cure this problem, and the desired valve dynamics 共control law兲 can be modified as x˙v = − k0共Fa − Fd兲 − kdxv

共43兲

where kd is a positive constant dissipative term. The control gains can therefore be freely tuned subject to the mild conditions of k0 ⬎ k1 and kd 艌 0 with ensured closed-loop system passivity. Since a purely dissipative term is added, it is like adding a damper to the analogous spring-mass system of Sec. 2, and the closed-loop pneumatic actuator can be easily proven to be still passive. It is shown in Fig. 8 that the dissipative term can be increased to almost eliminate the valve oscillations after contact. Although the approach velocity is appreciable at 0.15 m / s and the desired contact force is 40 N, the passive system can suppress the impact force and dissipate the kinetic energy effectively. For all the experimental results, note that a 5 V input to the valve corresponds to zero mass flow rate. 024501-6 / Vol. 130, MARCH 2008

5

Conclusions

Passivity analysis and control design of a pneumatic actuator controlled by a four-way proportional valve is presented in this paper. A pseudo-bond graph model is presented and used to prove the closed-loop passivity of a pneumatic actuator. The resulting passive closed-loop pneumatic system is able to interact with any passive environment in a safe and stable manner. Additionally, the formulation is able to produce a desired interaction force by using pressure sensors in the actuator chambers instead of external force sensing 共load cell兲. The contribution of the pseudo-supply rate introduced in this paper is to allow for extremely simple control laws with guaranteed theoretical assurances for the difficult problem of interacting with constrained 共or, more generally, passive兲 environments. It should be emphasized that control approaches for pneumatic systems are typically nonlinear. Although generally performing well with well modeled loads, nonlinear pneumatic control methods have not supplied stability guaranties for interacting with not well-characterized environments. This paper presents an analysis and synthesis technique capable of providing such guaranties for interaction with environments requiring no model as long as they are shown to be passive environments. With respect to application, the passivity properties of the closed-loop pneumatic feedback control system are advantageous for impact and force control by providing nonoscillatory contact forces with Transactions of the ASME


little overshoot. This passivity methodology is appropriate for application to many human-robot interaction tasks given that humans are typically modeled as passive.

References 关1兴 Hogan, N., 1985, “Impedance Control: An Approach to Manipulation: Part I. Theory,” ASME J. Dyn. Syst., Meas., Control, 107, pp. 1–7. 关2兴 Wu, Y., Tarn, T., Xi, N., and Isidori, N., 1996, “On Robust Impact Control Via Positive Acceleration Feedback for Robot Manipulators,” Proceedings of IEEE International Conference on Robotics and Automation. 关3兴 Niksefat, N., Wu, C. Q., and Sepehri, N., 2001, “Design of a Lyapunov Controller for an Electro-Hydraulic Actuator During Contact Tasks,” ASME J. Dyn. Syst., Meas., Control, 123, pp. 299–307. 关4兴 Sekhavat, P., Wu, Q., and Sepehri, N., 2005, “Impact Control in Hydraulic Actuators,” ASME J. Dyn. Syst., Meas., Control, 127, pp. 197–205. 关5兴 Bobrow, J. E., and McDonell, B. W., 1998, “Modeling, Identification, and Control of a Pneumatically Actuated, Force Controllable Robot,” IEEE Trans. Rob. Autom., 14, pp. 732–742. 关6兴 Hogan, N., 1987, “Stable Execution of Contact Tasks Using Impedance Control,” Proceedings of IEEE International Conference on Robotics and Automation, pp. 1047–1054. 关7兴 Hogan, N., 1988, “On the Stability of Manipulators Performing Contact Tasks,” IEEE J. Rob. Autom., 4, pp. 677–686. 关8兴 Vidyasagar, M., 1993, Analysis of Nonlinear Dynamic Systems, 2nd ed., Prentice-Hall, Englewood Cliffs, NJ. 关9兴 Khalil, H. K., 2001, Nonlinear Systems, 3rd ed., Prentice-Hall, Englewood Cliffs, NJ.

Journal of Dynamic Systems, Measurement, and Control

关10兴 Slotine, J., and Li, W., 1991, Applied Nonlinear Control, Prentice-Hall, Englewood Cliffs, NJ. 关11兴 Li, P. Y., 2000, “Toward Safe and Human Friendly Hydraulics: The Passive Valve,” ASME J. Dyn. Syst., Meas., Control, 122, pp. 402–409. 关12兴 Krishnaswamy, K., and Li, P., 2003, “Passive Teleoperation of a Multi Degree of Freedom Hydraulic Backhoe Using Dynamic Passive Valve,” Proceedings of IMECE’03, Washington, D.C., 15–21. 关13兴 Krishnaswamy, K., and Li, P., 2006, “Bond Graph Based Approach to Passive Teleoperation of a Hydraulic Backhoe,” ASME J. Dyn. Syst., Meas., Control, 128, pp. 176–185. 关14兴 Colgate, J. E., Wannasuphoprasit, W., and Peshkin, M. A., 1996, “Cobots: Robots for Collaboration With Human Operators,” Proceedings of ASME Dynamic Systems and Control Division, DSC-Vol. 58, ASME, New York, pp. 433–439. 关15兴 Li, P. Y., and Horowitz, R., 1997, “Control of Smart Exercise Machines: Part I. Problem Formulation and Non-Adaptive Control,” IEEE/ASME Trans. Mechatron., 2共4兲, pp. 237–247. 关16兴 McCloy, D., and Martin, H. R., 1980, Control of Fluid Power: Analysis and Design, 2nd ed., Wiley, New York. 关17兴 Richer, E., and Hurmuzlu, Y., 2000, “A High Performance Pneumatic Force Actuator System: Part I—Nonlinear Mathematical Model,” ASME J. Dyn. Syst., Meas., Control, 122共3兲, pp. 416–425. 关18兴 Liu, S., and Bobrow, J. E., 1988, “An Analysis of a Pneumatic Servo System and Its Application to a Computer-Controlled Robot,” ASME J. Dyn. Syst., Meas., Control, 110, pp. 228–235. 关19兴 Youcef-Toumi, K., and Gutz, D. A., 1994, “Impact and Force Control: Modeling and Experiments,” ASME J. Dyn. Syst., Meas., Control, 116, pp. 89–98.

MARCH 2008, Vol. 130 / 024501-7
