2012 American Control Conference Fairmont Queen Elizabeth, Montréal, Canada June 27-June 29, 2012
A Nonlinear Spring Model of Hydraulic Actuator for Passive Controller Design in Bilateral Tele-operation Venkat Durbha and Perry Li
Abstract— In this paper, the expressions that define compressibility of hydraulic medium are used to develop a nonlinear spring-like model for a hydraulic actuator. Using this model, an energy based controller for passive bilateral tele-operation of hydraulic actuator is designed. The control problem is posed to achieve power amplification from the master to the slave. The unknown external forces on the slave system are assumed to be constant and are estimated using techniques from direct adaptive control. Efficacy of the controller is evaluated by implementing it on a single degree-of-freedom setup. Experimental results demonstrate good co-ordination characteristics both in free space and when encountering a hard surface.
I. INTRODUCTION Bilateral tele-operation is a convenient way to extend human presence to remote and inaccessible locations. Haptic feedback is preferrable in these applications to provide the human operator a ’feel’ for the remote environment. Passive controllers [14] are commonly used to achieve tele-operation as they guarantee safe and stable interaction with unknown environments. As shown in [11], hydraulic systems are not inherently passive. Passification of hydraulic systems can be achieved through appropriate design or by implementing suitable feedback controllers [11]. A few different methods for active passification of hydraulic systems are available in literature. In [5] and [13] casimir functions are used to obtain passive controllers for stabilization of the hydraulic system. In [9] a pseudo energy function is defined to obtain passive controllers for position regulation. In [8], an energetically passive scheme for bilateral tele-operation of hydraulic actuators is investigated. In that work, all the components in the hydraulic system, including the valve, are modeled within passivity framework. Suitable controllers are defined to achieve passivity with respect to a desired supply rate. A bond graph based controller for bilateral tele-operation is presented in [4]. In both [4] and [8], the actuator energy is defined based on linear models and the compressibility of actuator is not modeled. In [7], an energy function based on the nonlinear dynamics of the actuator is presented. A different energy based formulation is presented in the current paper for energetically passive bilateral tele-operation. In [6], a novel passivity based controller was proposed for a hydraulic human power amplifier. The actuator is modeled as a combination of an ideal velocity source and a spring for V. Durbha is a graduate student at University of Minnesota
[email protected] P. Li is with the Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455, USA
[email protected]
978-1-4577-1094-0/12/$26.00 ©2012 AACC
modeling the compressibility affects of the fluid medium. A schematic of this model is shown in Fig. (1). The flow rate to/from the actuator is related to the ideal velocity as, Q1 Q2 x˙ I = =− (1) A1 A2
Fig. 1. Schematic of Hydraulic Actuator illustrating the modeling paradigm adopted in this paper
where x˙ I is the ideal velocity, and Qi and Ai correspond to the flow rate and piston area in chamber i ∈ (1, 2) respectively. In the above equation it is assumed that the flow going to the actuator is positive and flow from the actuator is negative. Passivity framework is achieved by enforcing a passive element, such as a virtual inertia, to be the source of this ideal velocity. This modeling paradigm of the actuator is as shown in Fig. (2).
Fig. 2. Model of hydraulic actuator with the ideal velocity commands provided by a virtual inertia. In a human power amplifier Fd = ρFh
While the actuator modeling paradigm in this paper is similar to [6], a key difference is the formulation of energy of the hydraulic actuator. In [6], a linear spring is used to capture the compressibility affects of the hydraulic medium, and a corresponding energy function is defined. In the current paper it is shown that the dynamics of a hydraulic actuator is more aptly captured by a nonlinear spring. An advantage of the framework for actuator model shown in Fig. (1) is that it can be conveniently extended to achieve bilateral tele-operation. In this paper, the master is simulated to be an electro-mechanically actuated system. In such a case, the virtual mass in Fig. (1) is replaced by the master system. Co-ordination between master and slave is achieved by implementing suitable feedback controllers. In most applications it is desirable that the power input by the
3471
human operator at the master is amplified at the slave while performing the required tasks. In the current work, this is achieved by designing a co-ordination controller with amplified master dynamics. This framework is as illustrated in Fig. (3). If the master system is also actuated by fluid power (hydraulic or pneumatic), the framework described in [2] can be adopted. It should be mentioned that communication delay is not pertinent to our experimental setup and is therefore not addressed in this paper.
Fig. 3. Schematic of Tele-operation for power amplification. The master dynamics are scaled by a factor ρ
In the following section, dynamics of the systems involved is presented. The spring-like model for the hydraulic actuator is presented in section III. Expressions for the actuator energy and the power flow from the actuator, required for controller design, are also presented in section III. In section IV, the control problem for bilateral tele-operation is formulated. The controller design and passivity analysis of the controller are presented in section V. Experimental results are presented in section VI, followed by concluding remarks in section VII. II. SYSTEM MODEL In most current applications of tele-operation the master is usually a servo-motor actuated joystick. For the sake of simplicity, it is therefore assumed that the dynamics of the master can be captured by a mass-spring-damper system. The slave system, as shown in Fig. (4), is hydraulic actuated. The force from the hydraulic actuator is measured using a load cell. There are sensors on board to measure the position of the slave system. The dynamics of master and slave inertias is given by,
where the subscripts m, s represents master and slave entities respectively, M corresponds to mass, x corresponds to position, Fe represent environmental forces and ρ is the scaling factor from master to slave. The actuator force is given by Fa , and Fh is the force exerted by the human operator. The environmental forces on master include the spring force and the damping force. The force feedback from slave to master is the scaled actuator force. The actuator force is given by, F a = P1 A 1 − P2 A 2
(4)
where Pi is the pressure of ith ∈ (1, 2) chamber. The pressure dynamics in the individual actuator chambers are given by [10], β (Q1 − A1 x˙ s ) P˙1 = V1 (xs ) β P˙2 = (Q2 + A2 x˙ s ) (5) V2 (xs ) where β is the bulk modulus of the hydraulic fluid, Q1 and Q2 are the flow rates to chamber 1 and 2 respectively, and the actuator chamber volumes V1 (xs ) and V2 (xs ) are defined as, V1 (xs ) = A1 (L1o + xs ) V2 (xs ) = A2 (Lo − (L1o + xs )) where L1o is the length corresponding to dead volume in chamber 1 and Lo is the sum of actuator stroke length and dead volume corresponding to both chambers of the actuator. In the subsequent sections we will use L2o to represent Lo − L1o . From Eq (4) and Eq (5), the force dynamics are obtained as, F˙a = P˙1 A1 − P˙2 A2 A1 Q1 A2 Q2 A21 A22 =β − + −β x˙ s V1 (xs ) V2 (xs ) V1 (xs ) V2 (xs ) (6) The flowrate to individual actuator chambers is related to ideal velocity x˙ I as given by Eq (1). The ideal actuator speed is related to the valve command u as [12], x˙ I = Kq (Fa , sgn(u))u (7) 1 p 2 r3 w Ps − Fa /A1 if u ≥ 0 A1 r 3 +1 Kq (Fa , sgn(u)) = 2 12 p r w Ps + rFa /A1 if u < 0 A1 r 3 +1 where w is the area gradient, Ps is the supply pressure, and r = A1 /A2 is the area ratio. If pressure sensors are available, the more conventional orifice equation can be used to obtain the valve model without affecting the controller design. In the following section, an expression for the energy in the actuator is proposed to aid in the design of passive controllers.
Fig. 4. The hydraulic system used as the slave system used in this study. The system is actuated to move in the vertical plane.
ρMm x ¨m Ms x ¨s
= ρ(Fh + Fem ) − Fa
(2)
= Fa + Fes
(3)
III. ACTUATOR E NERGY Consider the following relationship between pressure and volume for a given mass of fluid, dVi dPi = −β (8) Vi
3472
where i ∈ (1, 2) On integrating the above equation we get, Vi (xs ) Pi = −β log (9) Ci o
where Ci = Vio ePi /β is the integration constant, and Pio , Vio correspond to the initial pressure and volume. Substituting the above equation in Eq (4), the force dynamics as a function of position are obtained as, C1 C2 Fa (xs ) = β A1 log − A2 log V1 (xs ) V2 (xs ) = −βh(xs ) (10) where h(xs ) is appropriately defined. Note that the above equation is only valid for fixed mass of fluid in each chamber of the actuator. At the equilibrium position x ¯s the actuator force is zero. We therefore have, C2 C1 − A2 log =0 β A1 log V1 (¯ xs ) V2 (¯ xs )
(11)
Intuitively, the equilibrium position should depend on the mass of fluid in each chamber. As the mass changes, the pressure changes and this consequently affects the equilibrium position. The exact expression for equilibrium position is however not required for controller design. From Eq (10) and Eq (11), the actuator force can then be expressed as, V1 (¯ xs ) V2 (¯ xs ) Fa (xs , x ¯s ) = β A1 log − A2 log V1 (xs ) V2 (xs ) = β(h(¯ xs (m)) − h(xs )) (12) where m represents the mass of fluid in the actuator. Above equation shows the actuator force is a function of the actuator position and the mass of fluid in the actuator chamber. Note that actuator force in Eq (12) resembles that of a nonlinear spring element. Expanding h(xs ) about the equilibrium position x ¯s , the actuator force can be approximated as, Fa (xs , x ¯s ) = K(xs )(¯ xs − xs )
(13)
where the nonlinear spring stiffness K(xs ) is given by, A1 A2 K(xs ) = β + (14) L1o + xs L2o − xs On differentiating h(x) we get, A22 A21 ˙h(xs ) = + x˙ s V1 (xs ) V2 (xs )
(15)
The force dynamics in Eq (6) can be expressed as, A1 Q1 A2 Q2 ˙ s) F˙a (xs ) = β − − β h(x (16) V1 (xs ) V2 (xs ) In the above equation x ¯s is dropped for convenience. Energy in the spring for a given mass of fluid in each chamber is given by, Z x¯s Wact = Fa (z) dz (17) xs
where x ¯s and Fa correspond to the equilibrium position of the actuator and the actuator force at different positions z of the actuator respectively, for a given mass of fluid. The
power flow from the actuator is obtained by differentiating Eq (17) and is given by, Z x¯s ˙ xs ) dz ˙ h(¯ (18) Wact = −Fa (xs )x˙ s + β xs
˙ xs ) can be expressed as, From Eq (12) and Eq (15), h(¯ F˙a (z) ˙ xs ) = h(z) ˙ h(¯ + (19) β A1 Q1 A2 Q2 = − (20) V (z) V2 (z) 1 A1 A2 = + x˙ I (21) (L1o + z) (L2o − z) Using the above equation and Eq (7), power flow in the actuator is obtained as, Z x¯s A1 A2 ˙ Wact = −Fa (xs )x˙ s + β + x˙ I dz (L1o + z) (L2o − z) xs = −Fa x˙ s + Fa Kq u (22) The above equation shows that the actuator is a two port system, where mechanical power is extracted from one port and the other port interacts with the fluid input port. The control problem is formally presented in the following section. IV. P ROBLEM F ORMULATION Tele-operation is achieved through position and velocity co-ordination between the master and the slave systems. Passive operation is achieved if the external supply rate satisfies the following condition ∀ t, Z t s(Fh , Fem,s , x˙ m,s ) dτ ≥ −d2o (23) 0
where s(Fh , Fem,s , x˙ m,s ) is the desired external supply rate. The above condition states that the amount of energy that can be extracted from the tele-operator is bounded by d2o . In this work, the desired energetic supply rate is defined as, . s(Fh , Fem,s , x˙ m,s ) = (ρ(Fh + Fem )x˙ m + Fes x˙ s (24) The control problem is therefore to achieve position and velocity co-ordination given by Eq (25) while satisfying the passivity condition in Eq (23). . qE = (xs − xm ) → 0 . VE = (x˙ s − x˙ m ) → 0 (25) As we are interested in co-ordination, it is desirable to study the problem in relative co-ordinates. The system dynamics are therefore transformed using the following transformation [1], VL x˙ s =S (26) VE x˙ m where VL corresponds to the velocity of locked system and VE is the velocity of shape system. The transformation matrix is given by, 1−φ φ S= (27) 1 −1
3473
m where φ = ρM ML , and ML = (ρMm + Ms ) is the inertia corresponding to the locked system. The locked system represents the dynamics of the combined master and slave systems, whereas the shape system represents the relative dynamics and they are given by,
ML V˙ L ME V˙ E
=
ρ(Fh + Fem ) + Fes
(28)
=
Fa − Fe1 + Fe2
(29)
where Fe1 is the transformed external forces acting on master, Fe2 is the transformed external force on slave and they are given by, ((1 − φ)(ρ(Fh + Fem ))
Fe 1
=
Fe 2
= φFes
Therefore to achieve the control objective in Eq (25) the shape system dynamics have to be regulated. From the locked system dynamics it can be inferred that once co-ordination is achieved, the combined system moves under the influence of amplified human force and other environmental forces only. Thus, the proposed formulation of bilateral tele-operation provides power amplification while achieving co-ordination between master and slave systems. V. C ONTROLLER D ESIGN The passive controller to achieve bilateral tele-operation is presented in the following theorem. Theorem 1: For the shape system dynamics given by Eq (29) and actuator dynamics given by Eq (16), passive bilateral co-ordination is achieved asymptotically for the following valve command input, x˙ I (30) u= Kq (sgn(u), Fa ) where the ideal velocity command x˙ I is given by, F˙ad x˙ I = x˙ m + − qE − Kf F˜a − Kf i Zf K(xs )
where xds is the position of the actuator providing the desired force, and F˜ex is the error in estimating the unknown external force. Differentiating the above Lyapunov function and using Eq(31), Eq(33) and Eq(32) we get, ˙ = −(λ−)ME V 2 −Kp q 2 +λME VE qE −Kf F˜ 2 (36) W E E a In the above equation, by selecting the constants λ and ˙ can be made negative semi-definite. appropriately, W Differentiating the above equation we get, ¨ = −(λ − )(Fa − Fe + Fe )VE − Kp qE VE − W 1 2 A Q A21 A22 A Q 2 2 1 1 − − + x˙ s Kf F˜a β V1 (xs ) V2 (xs ) V1 (xs ) V2 (xs ) + λ ME VE2 + (Fa − Fe1 + Fe2 )qE + Kf F˜a F˙ad (37) Assuming that all the external signals and their deriva¨ is bounded. tives are bounded, it can be shown that W Hence, using Barbalat’s lemma [3], it can be shown that the (qE , VE , F˜a ) converges to 0 asymptotically. The control command as given by the Eq (31) results in port interaction as shown in Fig. (5). In the figure, ue represents components of the input other than x˙ m . The figure shows that the control command results in a naturally scaled power flow from master to the slave system. The additional input ue is required to achieve co-ordination. As the feedback term is an active input, it can lead to non-passive operation. By coupling the energy demands of the feedback term with that of a flywheel, passivity is enforced. This is explanined in the following section.
(31)
where Fad is the desired actuator force and is given by, Fda = Fe1 − Fˆe2 − Kp qE − λME VE (32) F˜a = Fa − Fad is the error in force tracking, the estimate for the unknown external force Fˆe2 is obtained from, ˙ Fˆe2 = λ1 (VE + qE ) (33)
Fig. 5. Schematic illustrating port interaction between the master and the slave system
A. Passivity analysis The dynamics of the flywheel are given by,
Zf is the integral of the force tracking error and is given by, Z t Zf = F˜a dt (34)
Mf x ¨f = Tf
Now consider the following storage function, Z x¯s 1 ρ 1 W = Ms x˙ 2s + Mm x˙ 2m + Fa (z) dz + Mf x˙ 2f (39) 2 2 2 xs
0
and Kp , λ, Kf , Kf i , and λ1 are positive constants, x˙ m is the velocity of master, K(xs ) is the stiffness of the nonlinear spring. Proof: Consider the following Lyapunov function, W =
1 1 2 ME VE2 + Kp qE + ME VE qE + 2 2 Z xds + (Fa (z) − Fad ) dz + xs
1 ˜2 F 2λ1 e2 1 Kf i Zf2 2
(38)
Differentiating the above expression and plugging in the system dynamics from Eq (2), Eq (3), Eq (38) and using Eq (22) we get, ˙ = Fe x˙ s + (ρ(Fh + Fe ) + Fa )x˙ m + Fa x˙ I + Tf x˙ f (40) W s m
(35)
Using the expression for control input x˙ I as defined in Eq (31) we get,
3474
˙d ˙ = Fe x˙ s + (ρ(Fh + Fe ))x˙ m + Fa ue + Fa Fa + Tf x˙ f W s m K(xs ) (41) The feedback input and the flywheel torque are coupled such that any energy input into the system is extracted from the flywheel. Therefore two different modes of operation depending on the kinetic energy of the flywheel are defined. The torque input to the flywheel is given by, ˙ Fˆad 1 Tf = − (Fa u ˆe + p(t)Fa ) (42) g(x˙ f ) K(xs ) where (
Fad 0
if x˙ f ≥ fo x˙ f < fo
( 1 p(t) = 0
if x˙ f ≥ fo x˙ f < fo
Fˆad =
( x˙ f g(x˙ f ) = fo
( u ˆf b =
uf b −Fa
if x˙ f ≥ fo x˙ f < fo
if x˙ f ≥ fo x˙ f < fo
at the master. The slave system was expected to follow the master trajectory. In this experiment an obstacle was placed in the path of the slave to investigate stability of interaction with a hard contact. The position and velocity tracking results are presented in Fig. (6) and Fig. (7). In free space the results show good co-ordination characteristics between the master and slave. At the hard contact the slave immediately comes to a stop and the master quickly follows. The actual and desired actuator forces are shown in Fig. (8). From Eq (32), note that the once co-ordination is achieved, the desired actuator force is a scaling of forces acting on the master system. As seen from Fig. (8) and Fig. (6), even as the external forces on the master increase, both the master and the slave are stationary at the hard contact without any interaction instabilities. This demonstrates reflection of the interaction force from the slave to the master. The commanded input during this operation is as shown in Fig. (9).
0.12
where fo is a predefined threshold velocity. When the flywheel velocity is greater than the threshold velocity, appropriate feedback for co-ordination is input to the teleoperator. During this mode of operation Eq (41) changes to,
0.1 0.09 Position (m)
˙ = Fe x˙ s + (ρ(Fh + Fe ))x˙ m W s m
Master Slave
0.11
(43)
0.08 0.07 0.06
On integrating the above equation we achieve the desired passivity condition, Z t Fes x˙ s + (ρ(Fh + Fem ))x˙ m dτ ≥ −W (0) (44)
0.05 0.04 Contact with hard surface
0.03
0
50
55
60
65 70 Time (s)
75
80
85
Fig. 6. Position co-ordination while moving in free space and when encountering hard contact
Master Slave
0.06 0.04 0.02 Velocity (m/s)
Comparing the above equation with Eq (23), one can notice that the bound on the energy that can be extracted from the tele-operator is given by the initial energy in the system. When the flywheel velocity falls below the threshold value, the controller is switched to guarantee stability at the expense of performance. In this mode Eq (41) changes to, ˙ = Fe x˙ s + (ρ(Fh + Fe ))x˙ m − (1 − x˙ f )Fa2 W s m fo < Fes x˙ s + (ρ(Fh + Fem ))x˙ m On integrating the above equation the passivity condition in Eq (44) is again satisfied. VI. I MPLEMENTATION R ESULTS
0 −0.02 −0.04 −0.06
Due to lack of access to a viable haptic interface, the master is simulated as a mass-spring-damper system. To make the experiments more realistic however, the human force to the master system was applied through an external load-cell. It is assumed that the position and velocity of the master are available. It is also assumed that all the external forces acting on the master are known. The slave system is a hydraulically actuated inertia moving in the vertical plane. Position information of the slave is available through a sensor. As mentioned in section V, environmental forces acting on the slave are unknown and are estimated. To evaluate the controller, two sets of experiments were performed. In the first set, the input commands were given
−0.08 −0.1 50
55
60
65 70 Time (s)
75
80
85
Fig. 7. Velocity co-ordination while moving in free space and when encountering hard contact
To investigate bilateral nature of the controller, input commands by the human were applied at the slave instead of the master. The position tracking in this experiment is as shown in Fig. (10) and the velocity tracking is shown in Fig. (11). The plots clearly indicate that the master and slave
3475
300
0.04 0.03
200 Actuator Force (N)
0.02 0.01 Velocity (m/s)
100 0 −100
0 −0.01 −0.02 −0.03
−200
−0.04 Master Slave
−0.05 −300
Actual Desired 50
55
Fig. 8.
60
65
70 Time (s)
75
80
−0.06
85
15
Comparison of actuator force and the desired force
Fig. 11.
0.04 Control Input (m/s)
30
35
40
Velocity tracking with human input on the slave
an energetic supply rate. Experimental results demonstrate bilateral operation and also efficacy of the controller in both free space and when encountering a hard contact. The presented work assumes that the system parameters such as bulk modulus, dead volume are well known. In future work, adaptive control strategies will be employed to estimate these parameters.
0.06
0.02 0 −0.02 −0.04 −0.06 −0.08
R EFERENCES
−0.1 −0.12 50
55
60
Fig. 9.
65 70 Time (s)
75
80
85
Control input to the valve
move together. Bilateral nature of operation can therefore be inferred from the co-ordination results provided in Fig. (6) and Fig. (10).
Master Slave
0.12 0.11 0.1 Position (m)
25 Time (s)
0.08
0.09 0.08 0.07 0.06 0.05 0.04 15
Fig. 10.
20
20
25
30 Time (s)
35
40
[1] D.Lee and P.Li, Passive bilateral control of nonlinear mechanical teleoperator, IEEE Trans. on Robotics 21 (2005), no. 5, 936–951. [2] V. Durbha and P.Y. Li, Passive bilateral tele-operation and human power amplification with pneumatic actuators, ASME Proc. Dynamic Systems and Control (2009). [3] H.Khalil, Nonlinear systems, Prentice Hall, 1995. [4] K. Krishnaswamy and P.Y. Li, Bond graph based approach to passive teleoperation of a hydraulic backhoe, Journal of dynamic systems, measurement, and control 128 (2006), 176. [5] A. Kugi, Non-linear control based on physical models, Springer, 2001. [6] P. Li, A new passive controller for a hydraulic human power amplifier, IMECE (Chicago), ASME, November 2006, pp. 1–11. [7] P. Li and M. Wang, Passivity based nonlinear control of hydraulic actuators based on euler-lagrange formulation, ASME Proc. Dynamic Systems and Control Conference (2011). [8] Perry Li and K. Krishnaswamy, Passive bilateral teleoperation of a hydraulic actuator using an electrohydraulic passive valve, Int. Journ. Fluid Power 5 (2004), no. 2, 43–56. [9] F. Mazenc and E. Richard, Stabilization of hydraulic systems using a passivity property, Systems & Control Letters 44 (2001), no. 2, 111– 117. [10] H.E. Merritt, Hydraulic control systems, John Wiley & Sons Inc, 1967. [11] P.Y.Li, Towards safe and human friendly hydraulics: the passive valve, ASME Journal Dyn. Sys. Meas. and Control 122 (2000), 402–409. [12] P.Y.Li and V.Durbha, Passive control of fluid powered human power amplifiers, Japan Fluid Power Symp. (Toyama, Japan), Int. Fluid Power Symp., September 2008, pp. –. [13] S. Sakai and S. Stramigioli, Passivity based control of hydraulic robot arms using natural casimir functions: Theory and experiments, Intelligent Robots and Systems, 2008. IROS 2008. IEEE/RSJ International Conference on, IEEE, 2008, pp. 538–544. [14] J.C. Willems, Dissipative dynamical systems, part 1: General theory, Arch. Rational Mech. Anal. 45 (1972), no. 22, 321–351.
Position tracking with human input on the slave system
VII. C ONCLUSIONS In this paper a passive scheme for bilateral tele-operation of hydraulic actuators has been presented. Power flow analysis affirms that the controller is passive with respect to 3476
