DSCC PASSIVE CONTROL OF A HYDRAULIC HUMAN POWER AMPLIFIER USING A HYDRAULIC TRANSFORMER

Transcription

1 Proceedings of the ASME 25 Dynamic Systems and Control Conference DSCC25 October 28-3, 25, Columbus, Ohio, USA DSCC PASSIVE CONTROL OF A HYDRAULIC HUMAN POWER AMPLIFIER USING A HYDRAULIC TRANSFORMER Sangyoon Lee ERC for Compact and Efficient Fluid Power Department of Mechanical Engineering University of Minnesota Minneapolis, Minnesota leex626@umn.edu Perry Y. Li ERC for Compact and Efficient Fluid Power Department of Mechanical Engineering University of Minnesota Minneapolis, Minnesota lixxx99@umn.edu ABSTRACT The hydraulic human power amplifier is a tool that uses hydraulic actuation to amplify the force that the human exerts on it. Our control objective and framework are to make the system behave like a passive mechanical tool when interacting with the human and with the work environment with a specified power scaling factor. A virtual velocity coordination control approach casts the human power amplifier problem into one of velocity coordination by generating a fictitious reference mechanical system. Force amplification becomes a natural consequence of velocity coordination. This control has been previously demonstrated using servo valves which is a major contributor to energy loss in hydraulic system. In this paper, a hydraulic transformer, which does not rely on throttling to accomplish its control function is used instead of a servo valve to achieve human power amplification. In addition, a passivity based control approach that makes use of the natural energy storage of the hydraulic actuator is used to define the flow requirement. This approach fully accounts for the non-linearity due to the pressure dynamics. The controller was experimentally validated with good force amplification and velocity coordination performance on a single degree of freedom hydraulic human power amplifier. Introduction Human amplifiers or extenders are tools that humans manipulate directly but with the ability to attenuate or amplify the apparent power applied by a human. The machine operation is more intuitive for the human operator as he/she is physically connected to the task. The control objective is to amplify the power/force that the human exerts on the machine. Since a human power amplifier interacts directly with humans, it is required that the interaction is in an energetically passive for safety. A control structure motivated from connection of passive physical systems was proposed in 2 and 3 for hydraulic human power amplifier. This controller modeled the actuator as a combination of an ideal velocity source and a nonlinear spring for modeling the compressibility effects of the fluid medium. Instead of controlling the actuator to track the desired force directly, the controller coordinates the velocities of the system and of a fictitious virtual mass whose dynamics are influenced by the hydraulic actuator and the human force. Coordination is achieved by first applying a passive decomposition 4 6 to the system into a shape and a locked system. A variety of control laws can then be designed to stabilize the shape system and to coordinate the velocities of the actual system and the virtual mass. This control is more robust as passivity property is enforced by the control structure itself 2,3. While in 2, a frequency domain approach is used to design the shape system control, in this paper, a passivity based control is applied where the recently discovered natural energy storage for the hydraulic actuator is used to define a physically motivated Lyapunov function. In previous works, the flow requirement was achieved using a servo valve which has fast response. Although servo valve control is widely used in hydraulic control due to its simplicity and Copyright c 25 by ASME

2 PA Fhuman+Fenv+Fa P2, A2 Mp xi xp 2 System Description and Control Objective We consider a DOF human power amplifier shown in Fig. with an effective inertia M p that is actuated by the hydraulic actuator force F a, direct human input F human and external environment force F env that includes gravity load, friction and other disturbances: D D2 M p ẍ p = F human + F env + F a () QA PB QB PT P, A FIGURE. Schematic of a hydraulically actuated human power amplifier with a hydraulic transformer in PM- configuration. The position x p is measured with a potentiometer, the hydraulic actuator force F a is measured by a force sensor mounted in series with it, the human interaction force F humaan is measured by a force sensor mounted on a handle. The environment force F env, however, is not measured. The flow to the hydraulic actuator Q B, is supplied/absorbed by a hydraulic transformer connected to a constant pressure supply. Hence, the actuator force and pressure dynamics are given by: performance, servo-valves are major contributors to the low system efficiency in hydraulic systems as they throttle away any excess pressure coming from the main pressure supply. This is especially significant when there are multiple actuators attached to the common pressure rail (CPR). Instead of a servo-valve, a hydraulic transformer could be used to transform hydraulic power in a conservative manner to a desired level. In addition to increasing the system efficiency, energy from braking load can also be recuperated 7. A hydraulic transformer consists of a hydraulic pump and a hydraulic motor connected mechanically as in seen Fig.. By varying the displacements of the pump/motors, hydraulic power at one pressure/flow at the input port is converted conservatively to another pressure/flow at the output port and vice versa. A hydraulic transformer used in place of servo valve can eliminate throttling loss and can allow for energy regeneration through four-quadrant operation, potentially increasing the overall efficiency of the system significantly. While many work has been performed analyzing design aspect of the transformer 7, 8, only few work is found on controlling a system driven by a transformer 9. In this paper, we show that passive human power amplification control could be achieved by a system using hydraulic transformer such that it could be applied to a system like a patient transfer device introduced in. The rest of the paper is organized as follows. System dynamics and control objective are stated in section 2. In section 3, the reformulation of the problem as a velocity coordination is first reviewed, followed by the presentation for the proposed flow requirement and its analysis. The control of the transformer to satisfy the flow command and to regulate its speed is presented in section 4. Experimental results and concluding remarks are given in sections 5-6. F a = P A P 2 A 2 (2) where A and A 2 are the cap side and piston side area, and where Ṗ = β(p ) V (x p ) (Q B A ẋ p ) (3) V (x p ) := V + A x p (4) is the fluid volume in the cap-side chamber and the hose, β(p ) is the pressure dependent fluid bulk modulus. In Fig., the nonlinear spring is a conceptual representation of the fluid compressibility responsible for creating the actuator force. The rod side is connected to the lower common pressure rail so that, P 2 = P T is assumed to be constant. It is assumed that the gravity load (in F env ) is sufficiently large such that overrruning load and cavitation do not occur. Otherwise, a directional control valve can be added between the transformer and the actuator. The transformer consists of two variable displacement pump/motors whose shafts are connected mechanically. Two of the ports, one from each pump/motor, are connected together. In the transformer configuration (which we referred to as PM-) in Fig., the two remaining ports and the common port are connected to supply (A), load (B), and tank (T) respectively. By permuting the port connections, two other configurations, PM-2 and PM-3, can be obtained as seen in Fig. 2. Each configuration has different flow capability and efficiency characteristics. In particular, PM-2 is adept at pressure bucking and PM-3 is adept at pressure boosting. The inertia dynamics and output flow of the three transformer configurations are: 2 Copyright c 25 by ASME

3 PB XI Mp Mp Xv Fhuman+Fenv PA PA Fd=ρFhuman MV MP Fa QA D D2 QA PT PB QB P2, A2 x P, A PT D D2 QB P2, A2 x P, A FIGURE 3. Model of hydraulic actuator with the ideal velocity command provided by the velocity of a virtual inertia, and the virtual inertia affected by the actuator force. In a human power amplifier F d = ρf h FIGURE 2. Left: Transformer in PM-2 configuration; Right: Transformer in PM-3 configuration PM- (Fig. ): J ω = (P A P T ) D u (P B P T ) D 2 B t ω T loss (5) Q B = ω D2 Q leak PM-2 (Fig. 2 Left): J ω = (P A P B ) D u (P B P T ) D 2 B t ω T loss ( Q B = ω D u + D ) (6) 2 Q leak PM-3 (Fig. 2 Right): J ω = (P A P T ) D u + (P A P B ) D 2 B t ω T loss (7) Q B = ω D2 Q leak where J is the inertia is of the pumpu/motors and shaft, D and D 2 are the maximum volumetric displacements of the pump/motors, u and, are the control inputs which are the normalized displacements, B t is the damping coefficient, Q leak and T loss are the volumetric loss at the output port and the mechanical loss inside the transformer due to friction. Note that both Q leak and T loss are configuration, pressure and speed dependent. The control objective is to control the normalized displacements u and (for one of the three transformer configurations in Figs. and 2) in order that the applied human force is amplified by a factor of (ρ + ), resulting in the dynamics of a target passive mechanical tool: M L ẍ p = (ρ + )F human + F env (8) where M L is the apparent inertia of the tool to be designed. The human would feel that he/she is interacting with an inertia and an environment force that are attenuated as M p/(ρ + ) and F env /(ρ + ) respectively. For M p = M p, this can be achieved if the actuator force satisfies: F a ρf human 3 Virtual Coordination Control Approach to Force Tracking 3. General concept As shown in 2, a direct approach to controlling the actuator force F a to track ρf human leads to a positive velocity feedback of ẋ p which is not robust especially in free motion. Instead, the virtual coordination control approach first proposed in 2 is used instead. This approach has a physical interpretation of interconnection of passive components so that it is more robust and safe to operate. Consider conceptually that the hydraulic actuator consists of an ideal kinematic actuator in series with a nonlinear spring representing the compressibility. One end of the spring interacts with the inertia M p and the velocity of the other end is velocity of the ideal actuator given by ẋ I = Q B /A. If the dead volume V in (4) is large compared to the variation in A x p such that V (x p ) = V can be considered a constant, this representation is exact as from (2)-(3), we have the actuator force dynamics: A 2 Ḟ a = β(f a /A ) (ẋ I ẋ p ) (9) V }{{} spring constant K(F a ) The controller is then designed to mimic the mass spring system in Fig. 3 where M p and the spring represent the physical system of the plant inertia and the hydraulic actuator. M v is a virtual mass whose dynamics (implemented as part of the controller) are acted on by the desired actuator force F d = ρf human and the actuator force F a : and let the ideal actuator velocity be: M v ẍ v = F d F a () ẋ I = Q B A = ẋ v + u () 3 Copyright c 25 by ASME

4 where u is some additional control (not represented in Fig. 3). If u, which affects F a, is controlled such that the velocities of the virtual mass and of the plant mass are rigidly coordinated i.e. ẋ v (t) ẋ p (t), then the resulting dynamics becomes: (M v + M p )ẍ p = (ρ + )F human + F env which is the desired target dynamics in (8) with the apparent inertia being M p = M v + M p. The control structure has the additional property that after coordination with ẋ v (t) ẋ p (t), the closed loop system is energetically passive with respect to the supply rate of the scaled human and environment power, (ρ + )F human ẋ p + F env ẋ p, such that there exists c 2 > so that for all F human ( ) and F env ( ), t (ρ + )F human ẋ p + F env ẋ p dτ c 2 (2) The passivity property is useful because it ensures that the physical coupling with any passive system at the environment port and human port (most objects are passive and human can be considered passive) will remain stable. Note that coupling between two stable systems does not guarantee stability (since a feedback loop is formed). In the following, we will derive this virtual coordination controller. Although we have used, as in 2, the approximate representation of the hydraulic actuator as a nonlinear spring to motivate the controller, the derivation below uses a recently discovered natural energy storage for the hydraulic actuator, which does not require this approximation. 3.2 Transformation into locked and shape systems As we are interested in coordination between ẋ p and ẋ v, i.e. V E := ẋ p ẋ v it is desirable to study the problem in relative coordinates. However, we also do not wish to disturb the energetics of the desired target dynamics (8). Thus, we apply the passive decomposition 4 6 to transform the velocities into locked and shape coordinates: ( ) ( ) VL φ φ )(ẋp := (3) V E where φ = M v /M L and M L = M v + M p is the inertia corresponding to the locked system. V L (referred to as the locked system velocity) is the velocity of the center of mass of the combined virtual and actual system, whereas V E (referred to as the shape system velocity) is the coordination error. In these coordinates, the dynamics () and () become: M L V L = F human + F env + F d (4) M E V E = F a + φ(f human + F env ) ( φ)f d (5) }{{} F E ẋ v An interesting property of this decomposition is that it preserves kinetic energy in that: 2 M pẋ 2 p + 2 M vẋ 2 v = 2 M pv 2 L + 2 M EV 2 E so, (4)-(5) can be considered inertial dynamical systems. Notice that we have recovered the target dynamics (8) as the locked system dynamics when F d = ρf human. 3.3 Shape System Control The virtual coordination problem is to regulate the shape system dynamics Eqs. (5), (2), (3): M E V E (t) = F a (t) + F E (t) F a (t) = P (t)a P T A 2 V (x p ) β(p ) Ṗ(t) = Q B (t) A ẋ p (t) (6) such that V E, where F E (t) = φ(f human +F env ) ( φ)f d. A classical linear control design approach was taken in. Here, a passivity based approach using the natural storage energy function in 2 is used to account for nonlinearity due to the hydraulic actuator. It does not require the approximation that the actuator chamber volume is constant. We propose the following control law: Q d B = A ẋ v + V (x) β(p d )Ṗd λ p P (7) where P d is the desired actuator pressure given by P d = A PT A 2 ˆF E (t) λv E (8) and ˆF E (t) is an estimate of F E, P = P P d is the error in pressure dynamics. The estimate for the external force ˆF E is obtained from the adaptation algorithm, ˆF E = σv E + Ḟ E (9) λ, σ and λ 3 are positive constants with λ p sufficiently large. 3.4 Derivation of the control law To derive this control law, notice that the desired pressure in (8) would generate the shape velocity dynamics: Consider a Lyapunov function, M E V E (t) = λv E F E (t) + A P W = 2 M EV 2 E + σ F 2 E (2) 4 Copyright c 25 by ASME

5 Ẇ = λv 2 E V E F E (t) + F E σ ( ˆF E Ḟ E ) +V E A P Thus, the adaptation algorithm (9) gives rise to: Ẇ = λv 2 E V E F E (t) +V E A P Ḟ If Ḟ E is not available, an extra term V E F Eσ E will be present. However, if F E (t) is slowly varying such that Ḟ E /σ is small, this term can be ignored. To account for the pressure error term V E A P, we shall augment the Lyapunov function (2) as: W total = W +V (x p )W V ( P,P d ) (2) where W V ( P,P d ) is the volumetric pressure error energy density associated with compressing the fluid from pressure P d to P d + P defined in 2: Pd + P W V ( P,P d ) := e g(p d+ P,P ) dp (22) P d Pd + P dp g(p d + P,P d ) := P d β(p (23) ) and β(p ) is the bulk modulus at pressure P. Lemma (See 2 for details and proof) The pressure error energy function V (x p )W V ( P,P d ) is positive definite in P and has the property that: d (V (x p )W V ( P,P d ) dt = P +W V ( P,P d ) Q B PA ẋ p }{{} Ψ( P,P d ) where B(P d + P,P d ) = V (x p ) B(P d + P,P d )Ṗd (24) e g(p d+ P,P d ). The hydraulic port effort Ψ( P,P d ) is monotone increasing with P since ( + W V ( P,P d )/ P) >. (24) shows that the hydraulic actuator is a passive 3-port system (hydraulic, mechanical and Ṗ d ) with respect to the supply rate given by the right hand side of (24) and V (x p )W V ( P,P d ) being the storage function. Using Lemma, W V as a shorthand for W V ( P,P d ), and V E = ẋ p ẋ v, we have: Ẇ total = λv 2 E + (ẋ p ẋ v ) PA + ( P + W V )Q B PA x p V (x) B(P d + P,P d )Ṗd = λv 2 E ẋ v PA + ( P + W V )Q B V (x) B(P d + P,P d )Ṗd If Q B = Q d B, the commanded flow, and writing Qd B = Q + Q, Ẇ total = λve 2 + P Q A ẋ v + P + W Ṽ Q P V (x p ) B(P d + P,P d )Ṗd + W V Q Using β(p d ) to approximate B(P d + P,P d ), and the fact that ( + W V / P) >, we define Q and Q as Q = A ẋ v + V (x p ) β(p d ) Ṗd Q = λ p P This recovers the proposed control law in (7). Thus, Ẇ total = λve 2 λ p + W Ṽ P 2 + P B(P d + P,P d ) β(p d ) V (x p )Ṗ d + W V Q The last two terms can be bounded by κ( P,P d ) P 2 where κ = µ( P,P d )V (x)ṗ d + ε( P,P d ) Q µ( P,P d ) P B(P d + P,P d ) β(p d ) Thus, with λ p = λ p ( + W V / P), ε( P,P d ) P 2 W V ( P,P d ) Ẇ total λv 2 E ( λ p κ) P 2 Therefore, when λ p is sufficiently large such that λ p > κ, this analysis shows that V E and P. 4 Transformer Control 4. Transformer speed control Flow control requirement in Eq. (7) is provided by a transformer dynamics described in Eq. (5) where the two normalized displacements u and are the control inputs. In addition to meeting the flow requirements, the transformer can also be controlled to track an arbitrary speed r T (t). The speed dynamics of the three transformer configurations can be written as: J ω = U total B t ω T loss (25) where (P A P T ) D u (P B P T ) D 2 U total = (P A P B ) D u (P B P T ) D 2 (P A P T ) D u + (P A P B ) D 2 PM- PM-2 PM-3 (26) 5 Copyright c 25 by ASME

6 Given the reference shaft speed for transformer r T (t), U total needs to provide appropriate torque dynamics to drive the transformer speed ω to desired speed. This can be provided by a PI controller: ω I = ω := r T (t) ω Utotal = Jṙ (27) T + K p ω + K I ω I 4.2 Displacement inputs Relating back to the transformer dynamics equation, u and must work simultaneously to provide the desired torque in Eq. (27) for the reference speed tracking while providing the desired flow Q d B in Eq. (7) to the attached cylinder. For each transformer configuration, u and could be solved simultaneously using the flow equations in Eqs. (5)-(7) and Eq. (26) as follows: PM-: PM-2: u u = = ω D2 (P A P T ) D (P T P B ) D 2 ω D ω D2 (P A P B ) D (P B P T ) D 2 Q d B Utotal Q d B U total (28) (29) Force N Velocity m/s Transformer Speed rad/sec Control inputs Virtual Time s u PM-3: u = ω D2 (P A P T ) D (P A P B ) D 2 Q d B U total (3) FIGURE 4. Expiermental results of controller implementation using a transformer in PM- configuration. The mechanical and volumetric losses are ignored in these expressions. Additional terms to compensate for these could also be added to improve performance. 5 Results Experimental demonstration of the human power amplifier control is performed with a hydraulic transformer prototype developed in our lab. The transformer consists of two micro-piston pump/motors connected mechanically whose displacements can be varied by stepping motors. The proposed control law has been implemented with all three transformer configurations. The desired transformer speed is 2 rad/s. Figs. 4-6 show the results. Notice that velocities of the virtual mass and the actual system are coordinated, and the actuator force follows closely the amplified human input force. The transformer speed was also regulated. 6 Conclusions In this paper, a hydraulic human power amplifier is controlled using a hydraulic transformer instead of a servo valve. The control law converts a force control objective into a coordination control problem with an interpretation of a (nonlinear) mass-spring system. The natural energy of the hydraulic actuator has been used to derive the passivity based control law. Experimental results show that this control approach could be delivered using hydraulic transformer. ACKNOWLEDGMENT This work is performed within the Center for Compact and Efficient Fluid Power (CCEFP) supported by the National Science Foundation under grant EEC Donation of components from Takako Industries is gratefully acknowledged. 6 Copyright c 25 by ASME

7 Force N Velocity m/s Transformer Speed rad/sec Virtual Force N Velocity m/s Transformer Speed rad/sec Virtual Control inputs Time s u Control inputs Time s u FIGURE 5. Expiermental results of controller implementation using a transformer in PM-2 configuration. FIGURE 6. Expiermental results of controller implementation using a transformer in PM-3 configuration. REFERENCES Li, P. Y., 24. Design and control of a hydraulic human power amplifier. In ASME 24 International Mechanical Engineering Congress and Exposition, American Society of Mechanical Engineers, pp Li, P. Y., 26. A new passive controller for a hydraulic human power amplifier. In ASME 26 International Mechanical Engineering Congress and Exposition, American Society of Mechanical Engineers, pp Li, P. Y., and Durbha, V., 28. Passive control of fluid powered human power amplifiers. In Proceedings of the JFPS International Symposium on Fluid Power, Vol. 28,, pp Lee, D., and Li, P. Y., 25. Passive bilateral control and tool dynamics rendering for nonlinear mechanical teleoperators. Robotics, IEEE Transactions on, 2(5), pp Lee, D. J., and Li, P. Y. Passive decomposition ap- proach to formation and maneuver control of multiple rigidbodies. ASME Journal of Dynamic Systems, Measurement and Control, 29, p Lee, D. J., and Li, P. Y. Passive decomposition of multiple mechanical systems under coordination requirements. IEEE Transactions on Automatic Control, 58, p Achten, P., Fu, Z., and Vael, G., 997. Transforming future hydraulics: a new design of a hydraulic transformer. In The Fifth Scandinavian International Conference on Fluid Power SICFP 97, p. 287ev. 8 Achten, P. A., van den Brink, T., van den Oever, J., Potma, J., Schellekens, M., Vael, G., van Walwijk, M., and Innas, B., 22. Dedicated design of the hydraulic transformer. Vol. 3, pp Lee, S., and Li, P. Y., 24. Trajectory tracking control using a hydraulic transformer. 24 International Symposium on Flexible Automation, Awaji Island, Japan. Humphreys, H. C., Book, W. J., and Huggins, J. D., Copyright c 25 by ASME

8 Hydraulically actuated patient transfer device with passivity based control. In ASME/BATH 23 Symposium on Fluid Power and Motion Control, American Society of Mechanical Engineers, pp. VTA5 VTA5. Li, P. Y., 26. A new passive controller for a hydraulic human power amplifier. In ASME 26 International Mechanical Engineering Congress and Exposition, American Society of Mechanical Engineers, pp Li, P. Y., and Wang, M., 24. Natural storage function for passivity-based trajectory control of hydraulic actuators. IEEE/ASME Transactions on Mechatronics, 9(3), July, pp Copyright c 25 by ASME