Impedance Control of a Pneumatic Actuator for Contact Tasks
Proceedings of the 2005 IEEE International Conference on Robotics and Automation Barcelona, Spain, April 2005
Impedance Control of a Pneumatic Actuator for Contact Tasks Yong Zhu and Eric J. Barth Department of Mechanical Engineering Vanderbilt University Nashville, Tennessee, 37235, USA [email protected] reflected inertia, more accurate and stable force control, less damage during inadvertent contact, and the potential for energy storage.” [1]
Abstract – This paper presents a method for the impedance control of a pneumatic linear actuator for tasks involving contact interaction. The method presented takes advantage of the natural compliance of pneumatic actuators such that a load cell, typically used in impedance control, is not required. The central notion of the method is that by departing from a stiff actuation system, low-bandwidth acceleration measurements can be used in lieu of high-bandwidth force measurements. The control methodology presented contains an inner loop to control the pressure on two sides of a pneumatic cylinder, while an outer loop enforces an impedance relationship between external forces and motion and commands desired pressures to the inner loop. The inner loop enforces the natural compliance of the pneumatic actuator by controlling both the sum and difference of the pressures on both sides of the pneumatic actuator. This is accomplished by utilizing two three-way proportional spool valves instead of a four-way valve typically used in fluid power control. Experimental results are shown demonstrating the pressure tracking control of the inner loop. Experimental results are also shown that demonstrate the impedance tracking of the outer loop for free motion and the transition from free motion to contact.
For tasks requiring high accuracy position control, it is not intuitive to turn to a pneumatic actuator. However, for tasks requiring force control, such as interaction, it is logical to look to a method of actuation that is compliant. Tasks that require a high degree of interaction with the environment require the actuator to be an impedance (in the language of Hogan [2]) – that is, the output is more characteristically force than motion. Typical industrial robot actuators (hydraulic and motor/gearhead systems) are fundamentally admittances possessing high stiffness. The stiffness of a hydraulic actuator is apparent. The stiffness of a motor/gearhead combination is due to a large degree of non-backdrivability resulting from high reflected inertia and friction. While stiff actuation is well suited to accurate positioning tasks, it is a mismatch for kinematically constrained interaction tasks with hard surfaces. Active accommodation techniques applied to hydraulic and motor/gearhead systems, such as impedance control, attempt to transform admittances into impedances. Such approaches require the use of highbandwidth force sensors (load cells) and extremely highbandwidth control [3] [4]. Essentially, these approaches transform the admittance into an impedance, but have their tradeoffs. Passive accommodation techniques, such as the use of Remote Center Compliance (RCC) devices, add a physical device that transforms the admittance into an impedance. While the device provides a robust and guaranteed compliance, it does not allow high bandwidth force control and is only well suited to the one specific task for which it was designed.
Index Terms – Impedance control, contact forces, pneumatic actuation, pressure tracking, proportional spool valve.
I. INTRODUCTION Industrial robots are good at tasks like welding or spray painting, which involve precision positioning but little interaction with the environment. People still assemble most of the products on an assembly line because humans can easily feel the force exerted by the environment and adjust the orientation or position of the parts to make them fit into each other. Compliant manipulation, as in assembly tasks, requires the manipulator to have accurate position tracking and soft collision while making contact with an uncertain environment, whether the uncertainty is with regard to position of a constraint surface, or stiffness of the environment.
Pneumatic actuators, by contrast, are natural impedances with true mechanical compliance. Forces are controlled by manipulating the difference of pressures in the two chambers of the actuator, and compliance is provided by the compressibility of air. By providing high-bandwidth control of the pressures in the chambers of the actuator, highbandwidth force control is achieved while preserving true mechanical compliance. Pneumatic actuation therefore presents the potential for advancement in the state of the art regarding force control and controlled interaction tasks [5].
The most apparent property of a pneumatic system is that of compliant actuation. By virtue of the compressibility of air, a pneumatic actuator offers compliant actuation. As properly noted by Pratt et. al. [1] in their work regarding series elastic actuators: “Most robot designers make the mechanical interface between an actuator and its load as stiff as possible. This makes sense in traditional positioncontrolled systems, because high interface stiffness maximizes bandwidth and, for non-collocated control, reduces instability. However, lower interface stiffness has advantages as well, including greater shock tolerance, lower
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In this paper, we present a method for the impedance control of a pneumatic linear actuator for tasks involving
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contact interaction. The method presented takes advantage of the natural compliance of pneumatic actuators such that a load cell, typically used in impedance control [6] [7], is not required. The central notion of the method is that by departing from a stiff actuation system, low-bandwidth acceleration measurements can be used in lieu of highbandwidth force measurements. With regard to impedance control of interaction tasks, the literature contains a number of approaches to a number of actuation systems. Impedance control of hydraulically actuated excavator arms is presented in [8]. Force measurements are used to modify the bucket trajectory for a desirable compliant behavior. The transient forces are lower than those resulting from utilizing a pure position control technique, but the results are based on appropriate selection of impedance parameters, prior identification of environment parameters and careful study of contact stability. An impedance controller for an industrial robot is presented in [9]. Externally measured torque is fed back through an impedance filter to guarantee the desired impedance relationship. In [10], a dynamic relationship between position error and the internal force is proposed for the impedance control of an industrial manipulator. This approach requires the use of a load cell to measure the external interaction forces (and torques). In general the literature contains a gap when it comes to utilizing non-stiff actuation systems for impedance control and also presents a gap regarding avoiding the use of a load cell. This paper presents a method that allows the use of acceleration feedback as opposed to interaction force feedback as applied to an actuation system (pneumatic in this case) that behaves as a natural impedance as opposed to an admittance. Consider the following simple 1 DOF example regarding the impedance control of an actuator. Suppose the end point of the actuator is governed by the following dynamics: M&x& + Bx& = F + Fe (1) where M and B represent the inertia and damping inherent in the actuator, F represents the force generated by the actuator, and Fe represents the force of the environment on the actuator. In a typical implementation of impedance control of this system, the interaction force Fe is measured and F is commanded such that the following dynamic relationship between motion errors and external forces is enforced: m( &x& - &x&d ) + b( x& - x&d ) + k ( x - xd ) = Fe (2) This is accomplished by commanding the actuation force to be the following value, F = - Fe + Bx& + q (3) where: M (Fe - b( x& - x&d ) - k ( x - xd ) + m&x&d ) q= (4) m It should be noted that it is not necessary to measure the acceleration using this method (a true asset given that highbandwidth low-noise acceleration is difficult to obtain using either position or velocity sensors). Also note that this
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method requires a good estimation of the plant parameters in order to replace the dynamics with the desired dynamics. As an alternative to the commonly employed method utilizing force measurements above, consider the following alternative method. In order to obtain the desired dynamic relationship of Equation (2), measure the acceleration and set the actuation force to the following value: F = M&x& + Bx& - m( &x& - &x&d ) - b( x& - x& d ) - k ( x - xd ) (5) As previously noted, it is difficult to obtain high-bandwidth low-noise acceleration without the use of an expensive dedicated sensor (accelerometer). However, if the actuator in use possesses compliance (i.e. is a natural impedance) as opposed to being stiff and fairly non-backdrivable (i.e. a natural admittance), it is not necessary to measure such a high-bandwidth acceleration. Put another way, whatever phase lag is present in the measurement of acceleration, the actuators compliance will make up for by presenting an open-loop compliance while the closed-loop impedance tracking “catches up”. It should be noted, however, that such an approach still requires a good estimation of the plant parameters for cancellation purposes equivalent to that of the typical implementation of impedance control. This paper presents the impedance control of a pneumatic actuation system utilizing the impedance approach described above requiring the low-bandwidth measurement of acceleration. Although a stability proof of this technique for compliant actuation is not presented, an experimental implementation provides a proof-of-existence of the method. Specifically, this paper first presents a pressure tracking controller followed by an outer-loop impedance controller. Experimental results indicate high bandwidth pressure tracking control, good motion tracking in free space, and controllable contact forces when transitioning from non-contact to contact. II. PRESSURE CONTROL The basic control schematic diagram of the pneumatic impedance controller is shown in Fig. 1. xd,xddot Impedance Control
Fd
Desired Pressure
Pad Pbd
Pressure Control
Pa Pb
Pneumatic System
x,xdot
Fig. 1 Diagram of the pneumatic impedance controller.
As a prerequisite to specifying the force necessary to uphold a desired impedance relationship between the interaction force and the motion errors of the actuator, it is necessary to be able to specify the actuation force at a high bandwidth. In a pneumatic system, this amount to being able to control the pressure in each side of the actuator. Figure 2 shows a schematic of the pneumatic actuation system. In order to control both the difference in pressure between the two sides of the actuator (in order to specify the actuation force), and to control the sum of the pressures of the two sides (in order to specify the compliance of the actuator), two three-way proportional spool valves are used to control the pressures in
the two sides of the actuator separately. This section will consider the control of pressure in one side of the actuator using its respective three-way proportional valve. Pneumatic supply
Combining Equations (10) and (11) results in the following typical sliding mode control law: 1 & u= Pd - f (x) - 2le - l2 ò e - K sgn( s ) (12) b( x) Equation (12) indicates the required mass flow rate for robust pressure tracking. To achieve this mass flow rate by controlling the flow orifice area of the valve, the mass flow rate equation relating the upstream and downstream pressures across the valve needs to be utilized. The mass flow rate can be expressed as, m& = AY ( Pu , Pd ) (13) where A is the high-bandwidth controlled orifice area of the valve and Y ( Pu , Pd ) is the area normalized mass flow rate relationship as a function of the pressure upstream and downstream of the valve. By virtue of the physical arrangement of the valve, the driving pressures of Y ( Pu , Pd ) are dependent on the sign of the “area”. A positive area A indicates that the spool of the proportional valve is positioned such that a flow orifice of area A connects the high pressure pneumatic supply to one side of the pneumatic cylinder, and thereby promotes a positive mass flow rate into the cylinder. A negative area A indicates that the spool of the proportional valve is positioned such that an orifice of area A connects one side of the pneumatic cylinder to atmospheric pressure, and thereby promotes a negative mass flow rate (exhaust from the cylinder). Using this convention, the area normalized mass flow rate can be written as: ì Y ( Ps , P) for A ³ 0 (14) Y ( Pu , Pd ) = í îY ( P, Patm ) for A < 0
[
3-way proportional spool valves VC
VC
a
(10) s& = - K sgn( s )b( x) Taking the derivative of Equation (8) and substituting e& = P& - P&d and P& from Equation (6), this gives: s& = b(x)u + f (x) - P&d + 2le + l2 ò e (11)
b
Pneumatic Cylinder Actuator
x
Fig. 2 Schematic of the pneumatic actuation system. The pressure in each side of the actuator is separately controlled with a three-way proportional valve.
A mathematical model of a pneumatic actuator has been well described in [11] and [12]. Assuming that the gas is perfect, the temperature and pressure within the two chambers are homogeneous, and the kinetic and potential energy of fluid are negligible, the rate of change of pressure within each pneumatic chamber can be expressed as, rRT rPV& (6) P& = m& V V where r is the thermal characteristic coefficient, with r = 1 for isothermal case, R is the ideal gas constant, T is the temperature, V is the control volume, and P is the pressure. Given the highly non-linear nature of the pressure dynamics, a model-based nonlinear controller is required to achieve adequate tracking. Treating the mass flow rate m& as the rPV& control variable u, and defining f (x) = , V rRT , and x T = [V V& P] , the pressure dynamics b ( x) = V can be stated more conveniently as: P& = f (x) + b(x)u (7) Using this standard notation, sliding mode control can be utilized to establish pressure tracking control. The sliding surface [13] is selected as: s = e + 2l ò e + l2 òò e (8) where e = P - Pd is the pressure tracking error and Pd is the desired pressure. The forcing term, s, of this desired error dynamic can be driven to zero by defining the standard 1 positive-definite Lyapunov function V = s 2 , and ensuring 2 that the derivative of this function is negative semi-definite. rRT Given that b(x) = > 0 , where the volume of the V chamber V is physically always greater than zero, the derivative of the Lyapunov function is enforced to be the following, & = ss& = - K s b(x) £ 0 V (9) where K > 0 . Solving for s& , this requires:
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]
A common mass flow rate model used for compressible gas flowing through a valve [9] is the following: C1C f Pu ì ï ï T Y ( Pu , Pd ) = í C C P ï 2 f u ( Pd ) (1 / k ) 1 - ( Pd ) ( k -1) / k ï Pu Pu T î
if
Pd £ C r (choked) Pu
(15)
otherwise (unchoked)
where Pu and Pd are the upstream and downstream pressures, C f is the discharge coefficient of the valve, k is the ratio of specific heats, Cr is the pressure ratio that divides the flow regimes into choked and unchoked flow and C1 and C2 are constants defined as:
C1 =
k 2 ( k +1) /( k -1) ( ) and C2 = R k +1
2k R(k - 1)
(16)
Finally, the required valve area is found by the following relationship: ì u / Y ( Ps , P) for u ³ 0 A=í (17) îu / Y ( P, Patm ) for u < 0 The control law specified by Equations (12-17) applied to each valve separately will enable the independent pressure tracking of each side of the pneumatic actuator. It should be noted that Equation (8) is an unconventional choice for the
sliding surface. Given that the control influence appears in the first derivative of pressure as indicated in Equation (7), more conventional choices for the sliding surface are s = e or s = e + l ò e . However, these choices provided inferior
pressure tracking experimentally.
III. IMPEDANCE CONTROL The equation of motion for the pneumatic cylinder can be expressed as: M&x& + Bx& + F f = F + Fe (18) where F = Pa Aa - Pb Ab - Patm Ar is the force provided by the pneumatic actuator with the pressures in each side of the cylinder Pa and Pb acting on their respective areas Aa and Ab along with atmospheric pressure Patm acting on the area of the rod Ar . Other forcing terms include Coulomb friction F f , viscous friction Bx& , and the force of the environment acting on the actuator Fe . As a matter of control philosophy, this control problem is atypical and interesting in that you want the control system to reject all disturbances except that coming from the environment Fe . The desired dynamic impedance behaviour relating the motion to external forces due to contact with the environment (or in free space when Fe = 0 ) can be expressed as m( &x& - &x&d ) + b( x& - x&d ) + k ( x - xd ) = Fe (19) where m, b, k are the target inertia, damping and spring constant, and xd is the desired position. To enforce this impedance behaviour, it is necessary to specify the actuation force such that Equation (18) becomes Equation (19). As discussed, there are two ways of achieving this objective. The typical impedance control approach requires a measurement of the interaction force but avoids requiring a measurement of the acceleration. The approach taken here is to avoid using a load cell in favour of utilizing the acceleration. The desired actuation force is therefore required to be: ~ ~ ~ Fd = ( M - m) &x& + ( B - b) x& - k ( x - xd ) + m&x&d + bx& d + F f (20) ~ ~ ~ where M , B , and F f are the estimated inertia, damping and coulomb friction of the pneumatic actuator. Assuming that the actual force F can be driven to the desired force Fd at a sufficiently high bandwidth through rapid and accurate pressure control of each side of the actuator, the dynamics become: m( &x& - &x&d ) + b( x& - x& d ) + k ( x - xd ) (21) ~ ~ ~ = Fe + ( M - M ) &x& + ( B - B) x& + ( F f - F f ) Equation (21) relates the importance of adequately modelling the system such that the residual modelling error terms can be kept small so that Equation (21) is a close approximation of the desired impedance behaviour given by Equation (19). To achieve the desired force profile specified in Equation (20), and to simultaneously specify the compliance of the actuator, the following constraint relationship can be
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established between the desired pressure in chamber A and B, Pad + Pbd = Psum (22) where Pad and Pbd are the desired pressure in chamber A and B, respectively, Psum is a constant parameter. A linearized analysis reveals that the sum of the pressures specifies the compliance of the actuator. By maintaining a particular open-loop compliance, and specifying an appropriate target damping, low-bandwidth acceleration feedback may be utilized to enforce the desired impedance behaviour while maintaining stability. The actuation force provided by the cylinder can be expressed as: Pad Aa - Pbd Ab - Patm Ar = Fd (23) Combining Eq. (22) and Eq. (23) gives the desired pressure of chamber A and B to be feed to the pressure tracking controller: F + Psum Ab + Patm Ar Pad = d and Pbd = Psum - Pad (24) Aa + Ab Since the dynamics of a pneumatic actuator are largely influenced by Coulomb friction, and since this Coulomb friction needs to be adequately cancelled to enforce the desired impedance behaviour, Coulomb friction is modelled as follows: ~ F f = - Fc sgn( x& d ) (25) Experimental results for the system under consideration showed asymmetric friction when moving in different directions, and therefore Fc is defined asymmetrically as:
ìï Fcpos ( x&d > 0) Fc = í (26) ïî Fcneg ( x&d < 0) For our system, Fcneg turns out to be crucial and must be compensated, but Fcpos and static friction are set to be zero since they had no obvious influence on tracking. IV. EXPERIMENTAL RESULTS Experiments were conducted to show a) the pressure tracking performance, b) the motion control of the actuator in free-space and c) the force when transitioning from noncontact to contact when hitting an unpredicted stiff wall. A photograph of the experimental setup is shown in Fig. 3. The setup is based on a Festo two degree-of-freedom pick and place pneumatic system. For this experiment, only one degree-of-freedom is used, which is a double acting pneumatic cylinder (Festo SLT-20-150-A-CC-B). The cylinder has a stroke length of 150 mm, inner diameter of 20mm and piston rod diameter of 8mm. A linear potentiometer (Midori LP-150F) with 150 mm maximum travel is used to measure the linear position of the cylinder. The velocity was obtained from position by utilizing a differentiating filter with a 20 dB roll-off at 100 Hz. The acceleration signal was obtained from the velocity signal with a differentiating filter with a 20 dB roll-off at 30 Hz. Two four-way proportional valves (Festo MPYE-5-1/8-LF-
010-B) are attached to the chambers, but they function as two three-way valves. Two pressure transducers (Omega PX202-200GV) are attached to each cylinder chamber, respectively. Control is provided by a Pentium 4 computer with an A/D card (National Instruments PCI-6031E), which controls the two proportional valves through two analog output channels. The load cell (Transducer Techniques MLP-500) is mounted at the end of the cylinder to measure the impact force when it hits the wall, but is not used for control.
performance and contact force profiles for each case are shown in Figures 6, 7, and 8. Figure 6 has target inertia, damping and stiffness of m = 0.5 kg, b = 200 N/(m/s), and k = 800 N/m. Figure 7 has target inertia, damping and stiffness of m = 1 kg, b = 400 N/(m/s), and k = 1600 N/m. Figure 8 has target inertia, damping and stiffness of m = 2 kg, b = 800 N/(m/s), and k = 3200 N/m. Position tracking is good for all three cases, while the peak contact force is seen to increase as the target inertia of the system is increased (17 N for m = 0.5 kg, 23 N for m = 1 kg, and 44 N for m = 2 kg). The final steady state contact force is not zero because there is error between the actual wall position and estimated one. 80
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Fig. 3. The experimental setup of the pneumatic actuation servo system.
A. Pressure Tracking Figure 4 shows the closed-loop pressure tracking on one side of the actuator at 10 Hz utilizing the control law specified in section II.
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Fig. 5. Motion tracking at 1.5 Hz under closed-loop impedance control. The desired position is shown in blue and the actual position is shown in green.
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Fig. 4. Experimental pressure tracking at 10 Hz. The desired pressure is shown in blue and the actual pressure is shown in green.
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C. Non-contact to contact transition A ramp with slope 30mm/sec was commanded to run the cylinder toward the wall, which is at about 55.6mm. The estimated wall position is at position 60mm (i.e. the ramp is commanded to stop at 60mm). The results of experiments are reported for three sets of impedance parameter combinations. Each set of parameters has the same damping ratio x = 5 and natural frequency w n = 40 rad / sec . Psum = 400 Kpa for all three cases. The position tracking
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B. Motion Tracking in Free Space Figure 5 shows closed-loop impedance control of the actuation system in free-space under tracking a 1.5 Hz sinusoidal motion of ±50 mm.
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Fig. 6. Non-contact to contact transition showing the commanded motion and tracking and the contact force. Target values: m = 0.5 kg, b = 200 N/(m/s), and k = 800 N/m.
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Fig. 8. Non-contact to contact transition showing the commanded motion and tracking and the contact force. Target values: m = 2 kg, b = 800 N/(m/s), and k = 3200 N/m.
Contact Force 30 25
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REFERENCES
15 10 5 0 -5
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Fig. 7. Non-contact to contact transition showing the commanded motion and tracking and the contact force. Target values: m = 1 kg, b = 400 N/(m/s), and k = 1600 N/m.
V. CONCLUSIONS A method for the impedance control of a pneumatic linear actuator for tasks involving contact interaction was presented. By exploiting the natural compliance properties of a pneumatic actuator, the impedance control method presented does not require the use of a load cell to measure the interaction force, but rather allows the use of a lowbandwidth acceleration feedback signal instead. A controller to achieve desired pressure tracking in each side of the pneumatic cylinder was also presented. Experimental results show good pressure tracking, good motion tracking in freespace, and a predictable trend of lower contact forces for lower target inertias of the system.
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[1] Pratt, G. A., Williamson, M. M., Dillworth, P., Pratt, J., Ulland, K., Wright, A., “Stiffness Isn’t Everything,” 1995, Fourth International Symposium on Experimental Robotics, ISER ’95, Stanford, California, June 30 – July 2. [2] Hogan, N., 1985, “Impedance Control: An Approach to Manipulation: Part I - Theory,” ASME Journal of Dynamic Systems, Measurement, and Control, vol. 107, March, pp. 1-7. [3] S. Jung, T.C.Hsia and R. Bonitz, "Force Tracking Impedance Control of Robot Manipulators Under Unknown Environment," IEEE Trans. Contr. Syst. Technol., Vol. 12, pp. 474-483, May 2004. [4] H. Seraji and R. Colbaugh, "Adaptive Force-based Impedance Control," in Proc. IEEE Int.Conf. Intelligent Robots Syst., 1993, pp1537-1544. [5] K. Al-Dakkan, “Energy Saving Control for Pneumatic Servo Systems,” Ph.D. dissertation, Dept. Mech. Eng., Vanderbilt Univ., Nashville, TN, Aug. 2003. [6] G. Ferretti, G. Magnani and P. Rocco, "Force Oscillations in Contact Motion of Industrial Robots: An Experiment Investigation," IEEE/ASME Trans. Mechatron., Vol. 4, pp.86-91, Mar. 1999. [7] J. Bobrow and B. W. McDonell, "Modeling, Identifying, and Control of a Pneumatically Actuated, Force Controllable Robot," IEEE Trans. Robot. Automat., Vol. 14, pp732-742, Oct. 1998. [8] S. Tafazoli, S. E. Salcudean, K Hashtrudi-zaad and P. D. Lawrence, "Impedance Control of a Teleoperated Excavator," IEEE Trans. Contr. Syst. Technol., Vol. 10, pp. 355-366, May 2002. [9] G. Ferretti, G. Magnani and P. Rocco, "Impedance Control for Elastic Joints Industrial Manipulators," IEEE Trans. Robot. Automat., Vol. 20, pp488-498, June 2004. [10] A. AlYahmadi and T.C. Hsia, “Internal Force-based Impedance Control of Dual-arm Manipulation of Flexible Objects,” Proc. IEEE Int. Conf. Robot. Automat., April 2000. [11] E. Richer and Y. Hurmuzlu, “A High Performance Pneumatic Force Actuator System: Part I-Nonlinear Mathematical Model” ASME Journal of Dynamic Systems, Measurement and Control, vol. 122, no. 3, pp. 416-425, 2000. [12] E. Richer and Y. Hurmuzlu, “A High Performance Pneumatic Force Actuator System: Part II-Nonlinear Control Design” ASME Journal of Dynamic Systems, Measurement and Control, vol. 122, no. 3, pp. 426434, 2000. [13] J. Slotine and W. Li, “Applied Nonlinear Control”, Prentice Hall, 1991
Impedance Control of a Pneumatic Actuator for Contact Tasks Yong Zhu and Eric J. Barth Department of Mechanical Engineering Vanderbilt University Nashville, Tennessee, 37235, USA [email protected] reflected inertia, more accurate and stable force control, less damage during inadvertent contact, and the potential for energy storage.” [1]
Abstract – This paper presents a method for the impedance control of a pneumatic linear actuator for tasks involving contact interaction. The method presented takes advantage of the natural compliance of pneumatic actuators such that a load cell, typically used in impedance control, is not required. The central notion of the method is that by departing from a stiff actuation system, low-bandwidth acceleration measurements can be used in lieu of high-bandwidth force measurements. The control methodology presented contains an inner loop to control the pressure on two sides of a pneumatic cylinder, while an outer loop enforces an impedance relationship between external forces and motion and commands desired pressures to the inner loop. The inner loop enforces the natural compliance of the pneumatic actuator by controlling both the sum and difference of the pressures on both sides of the pneumatic actuator. This is accomplished by utilizing two three-way proportional spool valves instead of a four-way valve typically used in fluid power control. Experimental results are shown demonstrating the pressure tracking control of the inner loop. Experimental results are also shown that demonstrate the impedance tracking of the outer loop for free motion and the transition from free motion to contact.
For tasks requiring high accuracy position control, it is not intuitive to turn to a pneumatic actuator. However, for tasks requiring force control, such as interaction, it is logical to look to a method of actuation that is compliant. Tasks that require a high degree of interaction with the environment require the actuator to be an impedance (in the language of Hogan [2]) – that is, the output is more characteristically force than motion. Typical industrial robot actuators (hydraulic and motor/gearhead systems) are fundamentally admittances possessing high stiffness. The stiffness of a hydraulic actuator is apparent. The stiffness of a motor/gearhead combination is due to a large degree of non-backdrivability resulting from high reflected inertia and friction. While stiff actuation is well suited to accurate positioning tasks, it is a mismatch for kinematically constrained interaction tasks with hard surfaces. Active accommodation techniques applied to hydraulic and motor/gearhead systems, such as impedance control, attempt to transform admittances into impedances. Such approaches require the use of highbandwidth force sensors (load cells) and extremely highbandwidth control [3] [4]. Essentially, these approaches transform the admittance into an impedance, but have their tradeoffs. Passive accommodation techniques, such as the use of Remote Center Compliance (RCC) devices, add a physical device that transforms the admittance into an impedance. While the device provides a robust and guaranteed compliance, it does not allow high bandwidth force control and is only well suited to the one specific task for which it was designed.
Index Terms – Impedance control, contact forces, pneumatic actuation, pressure tracking, proportional spool valve.
I. INTRODUCTION Industrial robots are good at tasks like welding or spray painting, which involve precision positioning but little interaction with the environment. People still assemble most of the products on an assembly line because humans can easily feel the force exerted by the environment and adjust the orientation or position of the parts to make them fit into each other. Compliant manipulation, as in assembly tasks, requires the manipulator to have accurate position tracking and soft collision while making contact with an uncertain environment, whether the uncertainty is with regard to position of a constraint surface, or stiffness of the environment.
Pneumatic actuators, by contrast, are natural impedances with true mechanical compliance. Forces are controlled by manipulating the difference of pressures in the two chambers of the actuator, and compliance is provided by the compressibility of air. By providing high-bandwidth control of the pressures in the chambers of the actuator, highbandwidth force control is achieved while preserving true mechanical compliance. Pneumatic actuation therefore presents the potential for advancement in the state of the art regarding force control and controlled interaction tasks [5].
The most apparent property of a pneumatic system is that of compliant actuation. By virtue of the compressibility of air, a pneumatic actuator offers compliant actuation. As properly noted by Pratt et. al. [1] in their work regarding series elastic actuators: “Most robot designers make the mechanical interface between an actuator and its load as stiff as possible. This makes sense in traditional positioncontrolled systems, because high interface stiffness maximizes bandwidth and, for non-collocated control, reduces instability. However, lower interface stiffness has advantages as well, including greater shock tolerance, lower
0-7803-8914-X/05/$20.00 ©2005 IEEE.
In this paper, we present a method for the impedance control of a pneumatic linear actuator for tasks involving
999
contact interaction. The method presented takes advantage of the natural compliance of pneumatic actuators such that a load cell, typically used in impedance control [6] [7], is not required. The central notion of the method is that by departing from a stiff actuation system, low-bandwidth acceleration measurements can be used in lieu of highbandwidth force measurements. With regard to impedance control of interaction tasks, the literature contains a number of approaches to a number of actuation systems. Impedance control of hydraulically actuated excavator arms is presented in [8]. Force measurements are used to modify the bucket trajectory for a desirable compliant behavior. The transient forces are lower than those resulting from utilizing a pure position control technique, but the results are based on appropriate selection of impedance parameters, prior identification of environment parameters and careful study of contact stability. An impedance controller for an industrial robot is presented in [9]. Externally measured torque is fed back through an impedance filter to guarantee the desired impedance relationship. In [10], a dynamic relationship between position error and the internal force is proposed for the impedance control of an industrial manipulator. This approach requires the use of a load cell to measure the external interaction forces (and torques). In general the literature contains a gap when it comes to utilizing non-stiff actuation systems for impedance control and also presents a gap regarding avoiding the use of a load cell. This paper presents a method that allows the use of acceleration feedback as opposed to interaction force feedback as applied to an actuation system (pneumatic in this case) that behaves as a natural impedance as opposed to an admittance. Consider the following simple 1 DOF example regarding the impedance control of an actuator. Suppose the end point of the actuator is governed by the following dynamics: M&x& + Bx& = F + Fe (1) where M and B represent the inertia and damping inherent in the actuator, F represents the force generated by the actuator, and Fe represents the force of the environment on the actuator. In a typical implementation of impedance control of this system, the interaction force Fe is measured and F is commanded such that the following dynamic relationship between motion errors and external forces is enforced: m( &x& - &x&d ) + b( x& - x&d ) + k ( x - xd ) = Fe (2) This is accomplished by commanding the actuation force to be the following value, F = - Fe + Bx& + q (3) where: M (Fe - b( x& - x&d ) - k ( x - xd ) + m&x&d ) q= (4) m It should be noted that it is not necessary to measure the acceleration using this method (a true asset given that highbandwidth low-noise acceleration is difficult to obtain using either position or velocity sensors). Also note that this
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method requires a good estimation of the plant parameters in order to replace the dynamics with the desired dynamics. As an alternative to the commonly employed method utilizing force measurements above, consider the following alternative method. In order to obtain the desired dynamic relationship of Equation (2), measure the acceleration and set the actuation force to the following value: F = M&x& + Bx& - m( &x& - &x&d ) - b( x& - x& d ) - k ( x - xd ) (5) As previously noted, it is difficult to obtain high-bandwidth low-noise acceleration without the use of an expensive dedicated sensor (accelerometer). However, if the actuator in use possesses compliance (i.e. is a natural impedance) as opposed to being stiff and fairly non-backdrivable (i.e. a natural admittance), it is not necessary to measure such a high-bandwidth acceleration. Put another way, whatever phase lag is present in the measurement of acceleration, the actuators compliance will make up for by presenting an open-loop compliance while the closed-loop impedance tracking “catches up”. It should be noted, however, that such an approach still requires a good estimation of the plant parameters for cancellation purposes equivalent to that of the typical implementation of impedance control. This paper presents the impedance control of a pneumatic actuation system utilizing the impedance approach described above requiring the low-bandwidth measurement of acceleration. Although a stability proof of this technique for compliant actuation is not presented, an experimental implementation provides a proof-of-existence of the method. Specifically, this paper first presents a pressure tracking controller followed by an outer-loop impedance controller. Experimental results indicate high bandwidth pressure tracking control, good motion tracking in free space, and controllable contact forces when transitioning from non-contact to contact. II. PRESSURE CONTROL The basic control schematic diagram of the pneumatic impedance controller is shown in Fig. 1. xd,xddot Impedance Control
Fd
Desired Pressure
Pad Pbd
Pressure Control
Pa Pb
Pneumatic System
x,xdot
Fig. 1 Diagram of the pneumatic impedance controller.
As a prerequisite to specifying the force necessary to uphold a desired impedance relationship between the interaction force and the motion errors of the actuator, it is necessary to be able to specify the actuation force at a high bandwidth. In a pneumatic system, this amount to being able to control the pressure in each side of the actuator. Figure 2 shows a schematic of the pneumatic actuation system. In order to control both the difference in pressure between the two sides of the actuator (in order to specify the actuation force), and to control the sum of the pressures of the two sides (in order to specify the compliance of the actuator), two three-way proportional spool valves are used to control the pressures in
the two sides of the actuator separately. This section will consider the control of pressure in one side of the actuator using its respective three-way proportional valve. Pneumatic supply
Combining Equations (10) and (11) results in the following typical sliding mode control law: 1 & u= Pd - f (x) - 2le - l2 ò e - K sgn( s ) (12) b( x) Equation (12) indicates the required mass flow rate for robust pressure tracking. To achieve this mass flow rate by controlling the flow orifice area of the valve, the mass flow rate equation relating the upstream and downstream pressures across the valve needs to be utilized. The mass flow rate can be expressed as, m& = AY ( Pu , Pd ) (13) where A is the high-bandwidth controlled orifice area of the valve and Y ( Pu , Pd ) is the area normalized mass flow rate relationship as a function of the pressure upstream and downstream of the valve. By virtue of the physical arrangement of the valve, the driving pressures of Y ( Pu , Pd ) are dependent on the sign of the “area”. A positive area A indicates that the spool of the proportional valve is positioned such that a flow orifice of area A connects the high pressure pneumatic supply to one side of the pneumatic cylinder, and thereby promotes a positive mass flow rate into the cylinder. A negative area A indicates that the spool of the proportional valve is positioned such that an orifice of area A connects one side of the pneumatic cylinder to atmospheric pressure, and thereby promotes a negative mass flow rate (exhaust from the cylinder). Using this convention, the area normalized mass flow rate can be written as: ì Y ( Ps , P) for A ³ 0 (14) Y ( Pu , Pd ) = í îY ( P, Patm ) for A < 0
[
3-way proportional spool valves VC
VC
a
(10) s& = - K sgn( s )b( x) Taking the derivative of Equation (8) and substituting e& = P& - P&d and P& from Equation (6), this gives: s& = b(x)u + f (x) - P&d + 2le + l2 ò e (11)
b
Pneumatic Cylinder Actuator
x
Fig. 2 Schematic of the pneumatic actuation system. The pressure in each side of the actuator is separately controlled with a three-way proportional valve.
A mathematical model of a pneumatic actuator has been well described in [11] and [12]. Assuming that the gas is perfect, the temperature and pressure within the two chambers are homogeneous, and the kinetic and potential energy of fluid are negligible, the rate of change of pressure within each pneumatic chamber can be expressed as, rRT rPV& (6) P& = m& V V where r is the thermal characteristic coefficient, with r = 1 for isothermal case, R is the ideal gas constant, T is the temperature, V is the control volume, and P is the pressure. Given the highly non-linear nature of the pressure dynamics, a model-based nonlinear controller is required to achieve adequate tracking. Treating the mass flow rate m& as the rPV& control variable u, and defining f (x) = , V rRT , and x T = [V V& P] , the pressure dynamics b ( x) = V can be stated more conveniently as: P& = f (x) + b(x)u (7) Using this standard notation, sliding mode control can be utilized to establish pressure tracking control. The sliding surface [13] is selected as: s = e + 2l ò e + l2 òò e (8) where e = P - Pd is the pressure tracking error and Pd is the desired pressure. The forcing term, s, of this desired error dynamic can be driven to zero by defining the standard 1 positive-definite Lyapunov function V = s 2 , and ensuring 2 that the derivative of this function is negative semi-definite. rRT Given that b(x) = > 0 , where the volume of the V chamber V is physically always greater than zero, the derivative of the Lyapunov function is enforced to be the following, & = ss& = - K s b(x) £ 0 V (9) where K > 0 . Solving for s& , this requires:
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]
A common mass flow rate model used for compressible gas flowing through a valve [9] is the following: C1C f Pu ì ï ï T Y ( Pu , Pd ) = í C C P ï 2 f u ( Pd ) (1 / k ) 1 - ( Pd ) ( k -1) / k ï Pu Pu T î
if
Pd £ C r (choked) Pu
(15)
otherwise (unchoked)
where Pu and Pd are the upstream and downstream pressures, C f is the discharge coefficient of the valve, k is the ratio of specific heats, Cr is the pressure ratio that divides the flow regimes into choked and unchoked flow and C1 and C2 are constants defined as:
C1 =
k 2 ( k +1) /( k -1) ( ) and C2 = R k +1
2k R(k - 1)
(16)
Finally, the required valve area is found by the following relationship: ì u / Y ( Ps , P) for u ³ 0 A=í (17) îu / Y ( P, Patm ) for u < 0 The control law specified by Equations (12-17) applied to each valve separately will enable the independent pressure tracking of each side of the pneumatic actuator. It should be noted that Equation (8) is an unconventional choice for the
sliding surface. Given that the control influence appears in the first derivative of pressure as indicated in Equation (7), more conventional choices for the sliding surface are s = e or s = e + l ò e . However, these choices provided inferior
pressure tracking experimentally.
III. IMPEDANCE CONTROL The equation of motion for the pneumatic cylinder can be expressed as: M&x& + Bx& + F f = F + Fe (18) where F = Pa Aa - Pb Ab - Patm Ar is the force provided by the pneumatic actuator with the pressures in each side of the cylinder Pa and Pb acting on their respective areas Aa and Ab along with atmospheric pressure Patm acting on the area of the rod Ar . Other forcing terms include Coulomb friction F f , viscous friction Bx& , and the force of the environment acting on the actuator Fe . As a matter of control philosophy, this control problem is atypical and interesting in that you want the control system to reject all disturbances except that coming from the environment Fe . The desired dynamic impedance behaviour relating the motion to external forces due to contact with the environment (or in free space when Fe = 0 ) can be expressed as m( &x& - &x&d ) + b( x& - x&d ) + k ( x - xd ) = Fe (19) where m, b, k are the target inertia, damping and spring constant, and xd is the desired position. To enforce this impedance behaviour, it is necessary to specify the actuation force such that Equation (18) becomes Equation (19). As discussed, there are two ways of achieving this objective. The typical impedance control approach requires a measurement of the interaction force but avoids requiring a measurement of the acceleration. The approach taken here is to avoid using a load cell in favour of utilizing the acceleration. The desired actuation force is therefore required to be: ~ ~ ~ Fd = ( M - m) &x& + ( B - b) x& - k ( x - xd ) + m&x&d + bx& d + F f (20) ~ ~ ~ where M , B , and F f are the estimated inertia, damping and coulomb friction of the pneumatic actuator. Assuming that the actual force F can be driven to the desired force Fd at a sufficiently high bandwidth through rapid and accurate pressure control of each side of the actuator, the dynamics become: m( &x& - &x&d ) + b( x& - x& d ) + k ( x - xd ) (21) ~ ~ ~ = Fe + ( M - M ) &x& + ( B - B) x& + ( F f - F f ) Equation (21) relates the importance of adequately modelling the system such that the residual modelling error terms can be kept small so that Equation (21) is a close approximation of the desired impedance behaviour given by Equation (19). To achieve the desired force profile specified in Equation (20), and to simultaneously specify the compliance of the actuator, the following constraint relationship can be
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established between the desired pressure in chamber A and B, Pad + Pbd = Psum (22) where Pad and Pbd are the desired pressure in chamber A and B, respectively, Psum is a constant parameter. A linearized analysis reveals that the sum of the pressures specifies the compliance of the actuator. By maintaining a particular open-loop compliance, and specifying an appropriate target damping, low-bandwidth acceleration feedback may be utilized to enforce the desired impedance behaviour while maintaining stability. The actuation force provided by the cylinder can be expressed as: Pad Aa - Pbd Ab - Patm Ar = Fd (23) Combining Eq. (22) and Eq. (23) gives the desired pressure of chamber A and B to be feed to the pressure tracking controller: F + Psum Ab + Patm Ar Pad = d and Pbd = Psum - Pad (24) Aa + Ab Since the dynamics of a pneumatic actuator are largely influenced by Coulomb friction, and since this Coulomb friction needs to be adequately cancelled to enforce the desired impedance behaviour, Coulomb friction is modelled as follows: ~ F f = - Fc sgn( x& d ) (25) Experimental results for the system under consideration showed asymmetric friction when moving in different directions, and therefore Fc is defined asymmetrically as:
ìï Fcpos ( x&d > 0) Fc = í (26) ïî Fcneg ( x&d < 0) For our system, Fcneg turns out to be crucial and must be compensated, but Fcpos and static friction are set to be zero since they had no obvious influence on tracking. IV. EXPERIMENTAL RESULTS Experiments were conducted to show a) the pressure tracking performance, b) the motion control of the actuator in free-space and c) the force when transitioning from noncontact to contact when hitting an unpredicted stiff wall. A photograph of the experimental setup is shown in Fig. 3. The setup is based on a Festo two degree-of-freedom pick and place pneumatic system. For this experiment, only one degree-of-freedom is used, which is a double acting pneumatic cylinder (Festo SLT-20-150-A-CC-B). The cylinder has a stroke length of 150 mm, inner diameter of 20mm and piston rod diameter of 8mm. A linear potentiometer (Midori LP-150F) with 150 mm maximum travel is used to measure the linear position of the cylinder. The velocity was obtained from position by utilizing a differentiating filter with a 20 dB roll-off at 100 Hz. The acceleration signal was obtained from the velocity signal with a differentiating filter with a 20 dB roll-off at 30 Hz. Two four-way proportional valves (Festo MPYE-5-1/8-LF-
010-B) are attached to the chambers, but they function as two three-way valves. Two pressure transducers (Omega PX202-200GV) are attached to each cylinder chamber, respectively. Control is provided by a Pentium 4 computer with an A/D card (National Instruments PCI-6031E), which controls the two proportional valves through two analog output channels. The load cell (Transducer Techniques MLP-500) is mounted at the end of the cylinder to measure the impact force when it hits the wall, but is not used for control.
performance and contact force profiles for each case are shown in Figures 6, 7, and 8. Figure 6 has target inertia, damping and stiffness of m = 0.5 kg, b = 200 N/(m/s), and k = 800 N/m. Figure 7 has target inertia, damping and stiffness of m = 1 kg, b = 400 N/(m/s), and k = 1600 N/m. Figure 8 has target inertia, damping and stiffness of m = 2 kg, b = 800 N/(m/s), and k = 3200 N/m. Position tracking is good for all three cases, while the peak contact force is seen to increase as the target inertia of the system is increased (17 N for m = 0.5 kg, 23 N for m = 1 kg, and 44 N for m = 2 kg). The final steady state contact force is not zero because there is error between the actual wall position and estimated one. 80
60
Position(mm)
40
20
0
-20
-40
-60
0
1
2
3
4
Fig. 3. The experimental setup of the pneumatic actuation servo system.
A. Pressure Tracking Figure 4 shows the closed-loop pressure tracking on one side of the actuator at 10 Hz utilizing the control law specified in section II.
5 Time(s)
6
7
10
Position Tracking 60
Pressure Tracking
desired position actual position
40
500
Position(mm)
20
400 Pressure(Kpa)
9
Fig. 5. Motion tracking at 1.5 Hz under closed-loop impedance control. The desired position is shown in blue and the actual position is shown in green.
600
300
0
-20
200
-40
100
0
8
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0
0.2
0.4
0.6
0.8
1 Time(s)
1.2
1.4
1.6
1.8
2
3
4
2
5
6 Time(s)
7
8
9
10
7
8
9
10
Contact Force 20
Fig. 4. Experimental pressure tracking at 10 Hz. The desired pressure is shown in blue and the actual pressure is shown in green.
15
C. Non-contact to contact transition A ramp with slope 30mm/sec was commanded to run the cylinder toward the wall, which is at about 55.6mm. The estimated wall position is at position 60mm (i.e. the ramp is commanded to stop at 60mm). The results of experiments are reported for three sets of impedance parameter combinations. Each set of parameters has the same damping ratio x = 5 and natural frequency w n = 40 rad / sec . Psum = 400 Kpa for all three cases. The position tracking
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10 Force(N)
B. Motion Tracking in Free Space Figure 5 shows closed-loop impedance control of the actuation system in free-space under tracking a 1.5 Hz sinusoidal motion of ±50 mm.
5
0
-5
2
3
4
5
6 Time(s)
Fig. 6. Non-contact to contact transition showing the commanded motion and tracking and the contact force. Target values: m = 0.5 kg, b = 200 N/(m/s), and k = 800 N/m.
Position Tracking
Contact Force
60
45 40
desired position actual position
40
35 30 25 Force(N)
Position(mm)
20
0
20 15
-20
10 5
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3
4
5
6 Time(s)
7
8
9
-5
10
2
3
4
5
6 Time(s)
7
8
9
10
Fig. 8. Non-contact to contact transition showing the commanded motion and tracking and the contact force. Target values: m = 2 kg, b = 800 N/(m/s), and k = 3200 N/m.
Contact Force 30 25
Force(N)
20
REFERENCES
15 10 5 0 -5
2
3
4
5
6 Time(s)
7
8
9
10
Fig. 7. Non-contact to contact transition showing the commanded motion and tracking and the contact force. Target values: m = 1 kg, b = 400 N/(m/s), and k = 1600 N/m.
V. CONCLUSIONS A method for the impedance control of a pneumatic linear actuator for tasks involving contact interaction was presented. By exploiting the natural compliance properties of a pneumatic actuator, the impedance control method presented does not require the use of a load cell to measure the interaction force, but rather allows the use of a lowbandwidth acceleration feedback signal instead. A controller to achieve desired pressure tracking in each side of the pneumatic cylinder was also presented. Experimental results show good pressure tracking, good motion tracking in freespace, and a predictable trend of lower contact forces for lower target inertias of the system.
Position Tracking 60 desired position actual position
40
Position(mm)
20
0
-20
-40
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3
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5
6 Time(s)
7
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10
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[1] Pratt, G. A., Williamson, M. M., Dillworth, P., Pratt, J., Ulland, K., Wright, A., “Stiffness Isn’t Everything,” 1995, Fourth International Symposium on Experimental Robotics, ISER ’95, Stanford, California, June 30 – July 2. [2] Hogan, N., 1985, “Impedance Control: An Approach to Manipulation: Part I - Theory,” ASME Journal of Dynamic Systems, Measurement, and Control, vol. 107, March, pp. 1-7. [3] S. Jung, T.C.Hsia and R. Bonitz, "Force Tracking Impedance Control of Robot Manipulators Under Unknown Environment," IEEE Trans. Contr. Syst. Technol., Vol. 12, pp. 474-483, May 2004. [4] H. Seraji and R. Colbaugh, "Adaptive Force-based Impedance Control," in Proc. IEEE Int.Conf. Intelligent Robots Syst., 1993, pp1537-1544. [5] K. Al-Dakkan, “Energy Saving Control for Pneumatic Servo Systems,” Ph.D. dissertation, Dept. Mech. Eng., Vanderbilt Univ., Nashville, TN, Aug. 2003. [6] G. Ferretti, G. Magnani and P. Rocco, "Force Oscillations in Contact Motion of Industrial Robots: An Experiment Investigation," IEEE/ASME Trans. Mechatron., Vol. 4, pp.86-91, Mar. 1999. [7] J. Bobrow and B. W. McDonell, "Modeling, Identifying, and Control of a Pneumatically Actuated, Force Controllable Robot," IEEE Trans. Robot. Automat., Vol. 14, pp732-742, Oct. 1998. [8] S. Tafazoli, S. E. Salcudean, K Hashtrudi-zaad and P. D. Lawrence, "Impedance Control of a Teleoperated Excavator," IEEE Trans. Contr. Syst. Technol., Vol. 10, pp. 355-366, May 2002. [9] G. Ferretti, G. Magnani and P. Rocco, "Impedance Control for Elastic Joints Industrial Manipulators," IEEE Trans. Robot. Automat., Vol. 20, pp488-498, June 2004. [10] A. AlYahmadi and T.C. Hsia, “Internal Force-based Impedance Control of Dual-arm Manipulation of Flexible Objects,” Proc. IEEE Int. Conf. Robot. Automat., April 2000. [11] E. Richer and Y. Hurmuzlu, “A High Performance Pneumatic Force Actuator System: Part I-Nonlinear Mathematical Model” ASME Journal of Dynamic Systems, Measurement and Control, vol. 122, no. 3, pp. 416-425, 2000. [12] E. Richer and Y. Hurmuzlu, “A High Performance Pneumatic Force Actuator System: Part II-Nonlinear Control Design” ASME Journal of Dynamic Systems, Measurement and Control, vol. 122, no. 3, pp. 426434, 2000. [13] J. Slotine and W. Li, “Applied Nonlinear Control”, Prentice Hall, 1991
