FPMC PASSIVITY BASED BACKSTEPPING CONTROL FOR TRAJECTORY TRACKING USING A HYDRAULIC TRANSFORMER

Transcription

1 Proceedings of the ASME/BATH 25 Symposium on Fluid Power & Motion Control FPMC25 October 2-4, 25, Chicago, Illinois, United States FPMC PASSIVITY BASED BACKSTEPPING CONTROL FOR TRAJECTORY TRACKING 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 Throttling loss is a major contributor to the low system efficiency in hydraulic systems. Hydraulic transformers can potentially be an energy efficient, throttle-less control approach for multi-actuators systems powered by a common pressure rail (CPR). The transformer transforms the input CPR pressure to the desired pressure of the actuator instead of throttling it. Regenerative energy can also be captured. For transformers to be useful, they must also have good control performance. This paper presents a a passivity based trajectory tracking controller for a hydraulic actuator driven by a transformer consisting of two mechanically coupled variable displacement pump/motors. In addition to controlling the motion of the actuator, the transformer speed can also be regulated at the most efficient operating speed. The nonlinear controller is designed using a Lyapunov function that is based upon a recently discovered natural energy storage function for hydraulic actuators. Experimental results validate the efficacy of this controller. INTRODUCTION Hydraulic power transmission offers multiple benefits over competing technologies including an order of magnitude higher power density than electric systems, relatively low cost, fast response, and flexible packaging. Thus, hydraulic actuators are often used in applications that demand power, precision and compactness. However, typical hydraulic systems suffer from low system efficiency with the use of throttling valves for control. Various methods to improve the efficiency of hydraulic systems have been researched in recent years. These include load sensing (LS) control and displacement control. Yet another approach is to use a common pressure rail (CPR) and a hydraulic transformer for each individual actuator to transform the CPR pressure to the required pressure of the actuator. Since hydraulic transformers do not rely on throttling for control, it can improve over LS systems especially when the various actuators have very different pressure requirements. A hydraulic transformer consists of a hydraulic pump and a hydraulic motor connected mechanically. By varying the relative displacements of the pump and of the motor, hydraulic power at one pressure/flow at the input port is converted to another pressure/flow at the output port and vice versa. A hydraulic transformer used in place of a servo valve can eliminate throttling loss and can allow for energy regeneration through four-quadrant operation, increasing the overall efficiency of the system. Although a hydraulic transformer can be configured by connecting two pump/motors mechanically, there has been a focused research effort, in the past decade, to develop a different configuration, known as INNAS hydraulic transformer (IHT) (see e.g. 4). By using a rotatable 3-ported port plate, IHT combines the pumping and motoring pistons into a single rotating group and the transformation ratio is determined by the rotation of the port plate. For a transformer to be useful, it must also have good control performance in addition to efficiency. However, most previous works are concerned with the design of hydraulic transformers. Only a few papers discuss the control aspect. For Copyright c 25 by ASME

2 PA FL FL PA D D2 QA P2, A2 x P2, A2 x PT QA PB QB P, A PT D PB D2 QB P, A FIGURE. Schematic of hydraulic transformer: PM- FIGURE 3. Schematic of hydraulic transformer: PM-3 PT PA QA D D2 PB QB FL P2, A2 x P, A figured transformers that differ by port connections. Compared with our previous work 9, this paper uses the natural energy storage function in instead of a quadratic function in the definition of the Lyapunov function to achieve better robustness. The controller performance is also validated experimentally in this paper. In section 2, the dynamics of the transformer controlled hydraulic actuator system is presented. In section 3, the controller design is presented. Experimental results are presented in Section 4. Section 5 contains concluding remarks and future works. FIGURE 2. Schematic of hydraulic transformer: PM-2 example, Werndin and Palmberg 5,6 presented design concepts necessary to control the IHT at low speed. They used a model based estimator and a feed-forward control in parallel with a PI controller to simulate an IHT driving a hydraulic cylinder. Vael et al 7 qualitatively laid out various possible hydraulic systems to be used in their experiment on an excavator. Ahn and Ho 8 presented a robust controller based on disturbance observer for regulating the position of a hydraulic cylinder driven by a traditionally configured transformer where two pump/motors are coupled together. In this paper, we investigate the control performance of a hydraulic actuator controlled using a traditionally configured hydraulic transformer (two pump/motors connected mechanically) in which both displacements can be used as control input. While a transformer can be constructed with only one pump/motor displacement being variable, with both displacements being adjustable, both the transformation ratio and the transformer speed can be controlled simultaneously. We propose a passivity based back-stepping control strategy to enable a hydraulic actuator to precisely track a desired trajectory, and to control the rotational speed of the transformer. This extra degree of freedom can be used to optimize efficiency and to avoid transformer stalling. The proposed controller can be adapted to all three traditionally con- 2 System Description In a typical hydraulic system, a directional servo valve is placed between the pressure source (often constant) and the service to throttle down any excess pressure. This is a major source of hydraulic system inefficiency. Such losses can be eliminated by using a hydraulic transformer. In the circuit shown in Fig., a hydraulic transformer with two pump/motor units (PM transformer) is used in place of a servo valve to control the flow rate into the cylinder chamber carrying a vertical mass load. By controlling the displacement ratio of right hand side pump/motor in the PM transformer, velocity and the position of the cylinder can be tracked. To recover the gravitational potential to the common pressure rail (CPR) while the load is being lowered, the pump/motor can go over-center or run in opposite direction. By controlling the displacement ratio of the left hand side pump/motor, the shaft speed of the transformer can be controlled by varying the power/torque balance within the transformer. If more power is injected to the transformer than what is required by the load trajectory, the transformer speed will increase. When more power is present in the transformer than what is needed, the extra potential can be recovered back to the CPR. In Fig., pump/motor unit serves as a motor and unit 2 serves as a pump when the energy is delivered into the hydraulic cylinder. Their roles reverse when recovering regenerative loads. The two pump/motors need to be controlled simultaneously in or- 2 Copyright c 25 by ASME

3 der to achieve trajectory tracking performance while re-capturing the energy from the gravitational potential to attain high efficiency. Figs. 2 and 3 illustrate two other PM transformers that differ in the ways that the ports are connected. The PM-2 configuration in Fig. 2 is more adept at pressure bucking whereas the PM-3 configuration in Fig. 3 is more adept at pressure boosting. In all three configurations, by adjusting the displacements of the 2 pump/motors, the flow requirements and the energy/torque balance of the transformer can also be simultaneously satisfied. This feature could be used to prevent a transformer from stalling in low speed region and to operate transformer at the most optimal speed. PM- (Fig. ): J ω = (P A P T ) D 2π u (P B P T ) D 2 2π B t ω T loss Q A = ω D 2π u + Q leak Q B = ω D2 2π Q leak λ(u, ) = D u D 2 PM-2 (Fig. 2): ( : P B P T P A P T ) (3) 2. System Dynamics The inertia dynamics of the hydraulic cylinder in Figs. -3 are: mẍ = bẋ + P (t)a P 2 A 2 + F L () where m is the mass of the cylinder rod and load, x is the vertical position of the cylinder load mass, A and A 2 are respectively the cap side and rod side areas of the hydraulic actuator, b is the viscous friction coefficient, and F L (t) is a load force that encapsulates any external load including gravity, environment forces and un-modeled dynamics. The dynamics of the cap-side pressure P are given by the compressibility of the fluid in the cylinder and hose: Ṗ = β(p ) V + A x (Q B A ẋ) (2) where Q B is the flow rate into the cap side chamber to be provided by the transformer, V is the volume in the capside chamber and hose when the actuator is at the position x =, and β(p ) is the pressure dependent bulk modulus. The rod side is connected to the lower common pressure rail so, P 2 = P T, which is assumed to be constant. The capside flow is supplied (or absorbed) by the hydraulic transformer which consists of a pair of variable displacement hydraulic pump/motors. The pump/motors are connected mechanically, and two of the ports, one from each pump/motor, are connected together. The transformer dynamics are governed by the common rotational inertia J and the torque applied by the pump/motors. The input, output and tank ports are labeled as A, B and T. By permuting the port connections, the three configurations in Figs. -3 can be obtained. Each configuration will have different flow capability and efficiency characteristics. The transformer rotational speed (ω) dynamics, input flow (Q A ), output flow (Q B ), and (ideal) pressure transformation ratio (λ) for the three configurations are given by: J ω = (P A P B ) D 2π u (P B P T ) D 2 2π B t ω T loss Q A = ω D 2π u + Q leak ( Q B = ω D 2π u + D ) 2 2π Q leak D u λ(u, ) = D u D 2 PM-3 (Fig. 3): J ω = (P A P T ) D 2π u + (P A P B ) D 2 2π B t ω T loss ( Q A = ω D 2π u + D ) 2 2π + Q leak Q B = ω D2 2π Q leak λ(u, ) = D u + D 2 D 2 where D and D 2 are the maximum volumetric displacements of the pump/motor units in m 3 /rev, u and, are control inputs which are the normalized displacements, B t is the damping coefficient, Q leak, Q leak and T loss are the lumped volumetric loss at the A and B ports and the mechanical loss inside the transformer due to friction. These losses are generally configuration, pressure and speed dependent. λ is the input to output flow transformation ratio, which, in steady state, is also the output to input pressure transformation ratio when losses are absent. 3 Control Strategy The control objective is for the actuator position x(t) to track a reference trajectory x d (t) subjected a load F L, while regulating the hydraulic transformer speed at ω d (t). In all 3 configurations in Figs. -3, P A and P T are the high and low pressures of the common pressure rails which are assumed constant. Overrunning load and cavitation are assumed (4) (5) 3 Copyright c 25 by ASME

4 not to occur as would be the case when the gravity load is sufficiently large and speed is sufficiently slow. Otherwise, a directional control valve can be added between the transformer and the actuator. In the proposed approach, the desired velocity, force, pressure of the actuator, and finally the required flow to the actuator are successively defined and controlled via passivity backstepping. Unlike feedback linearization or backstepping that uses a generically defined quadratic lyapunov function 9 where active cancellation of specific terms are needed, passivitybased approach uses a natural energy inspired Lyapunov function such that the cancellation is done automatically due to the structural property of the system. This results in improved performance, robustness against model uncertainties, and fewer gains to tune. In addition to specifying the required flow (Q B ) to control the cylinder motion, the net torque (U total ) on the transformer will also be specified. The required flow to the cylinder and the net torque are then simultaneously satisfied by decomposing these requirements into appropriate settings for the two displacements of the hydraulic transformer. 3. Cylinder Flow Requirement - Q B In this subsection, we design required Q B in Eq. (2) such that x(t) x d (t), where x d, ẋ d, ẍ d, and... x d are assumed to be smooth and available. The passivity approach in, summarized below, is taken for this purpose. The readers are referred to for details. Let e := x x d be the tracking error and define the reference velocity, and the reference velocity error as r := ẋ d λ p e (6) e v := ẋ r = ė + λ p e (7) where λ p >. Then, by designing the desired pressure to be: := A (mṙ + br F L + A 2 P 2 K p e K v e v ) (8) where K p >, and K v >, the reference velocity error dynamics become: mė v = K p e K v e v + A P (9) where P := P and K v = K v + b. With the Lyapunov (or storage) function W mech := 2 me2 v + 2 K pe 2 Ẇ mech = K v e 2 v λ p K p e 2 + PA e v () the mechanical system can be seen to be passive with respect to the supply rate PA e v. Next, pressure dynamics is taken into account by augmenting the Lypaunov (or storage) function W mech with the pressure error energy: V (x)w V ( P, ) where V (x) := (V + A x), Pd + P W V ( P, ) := e g(+ P,P ) dp () is the volumetric pressure error energy density associated with compressing the fluid from pressure to + P with Pd + P g( + P,P d ) := β(p ) (2) and β(p ) is the bulk modulus at pressure P (see for details). Hence, the augmented Lyapunov function is: W total = 2 me2 v + 2 K pe 2 +V (x)w V ( P, ) (3) Using the property : d (V (x)w V ( P, ) dt = P +W V ( P, ) Q B PA ẋ V (x) and writing Q B = Q d + Q, we have: Ẇ total = K v e 2 v λ p K p e 2 + PAe v + PQ B PAẋ e g(p, ) Ṗd +W V ( P, )Q B V (x)e g(p, ) Ṗ d = K v e 2 v λ p e 2 + P Q d A r V (x) B(P, )Ṗd +W V ( P, )Q d + P + W V ( P, ) Q P }{{} > where B(P, ) is defined from e g(p, ) = By successively applying Eq. (6) and Eq. (8), (4) B(P, ) P. (5) Ṗ d = A m(... x d λ p ë) Ḟ L K p ė K v ė v (6) =... m x d Ḟ L + f (e,e v, P) (7) A } {{} Ṗ d where f (e,e v, P) = α e e + α ev e v + α P P for some α e, α ev, α P. 4 Copyright c 25 by ASME

5 Now compensating only for the terms related to the trajectory, Q d is designed to be: Q d = A r + V (x) β( )Ṗd (8) where r(t) is the reference velocity defined in Eq. (6) and Ṗ d is given in Eq. (7). Using this term: Ẇ total K v e 2 v λ p e 2 + P + W V ( P, ) Q P P V (x) β( ) f (e,e v, P) + (µ(p, )V (x)ṗ d + ε(p, ) Q d ) }{{} κ where µ(p, ) >, ε(p, ) > satisfy: µ(p, ) P /B(P, ) /β( ) ε(p, ) W V ( P, )/ P 2 P 2 (9) Note that the term PA e v from the mechanical system Eq. (4) has been canceled out automatically by the term from the pressure error dynamics. Finally, since it can be shown that +W V ( P, )/ P >, we design Q = λ 3 P such that the overall control law for desired flow into the piston chamber is: Q B = A r + V (x) β( )Ṗd λ 3 P (2) Using the notation V β to denote V (x) 2β( ) gives rise to where Ẇ total ( e e v P ) M pass e e ṽ (2) P λ p K p α e V β M pass := K v α ev V β α e V β α ev V β λ3 κ + 2α P V β with λ 3 = λ 3 ( + W V ( P, ) ). Thus, for λ P 3 > sufficiently large, M pass is positive definite and (e,e v, P) converge to (,,) exponentially. This implies that the bounded un-modeled disturbance would only cause bounded effects. Despite the analysis being a little involved, the control law in (2), (6), (7), (6) and (7) is quite straightforward. Moreover, inaccuracies or ignorance in the estimation of Ṗ d and β( ) in Eq. (2), could be treated as disturbances with negligible effects after proper controller tuning. 3.2 Transformer Speed Control Since the displacements of both pump/motors in the hydraulic transformer can be manipulated, an additional control objective other than controlling the cylinder motion can be specified. Here, we impose that the transformer speed should track an arbitrary profile ω d (t). ω d (t) can be designed to prevent stalling or to optimize the operating efficiency of the transformer. From (3)-(5), the speed dynamics of the three transformer configurations can be written as: J ω = U total B t ω T loss (22) where (P A P T ) D 2π u (P B P T ) D 2 2π U total = (P A P B ) D 2π u (P B P T ) D 2 2π (P A P T ) D 2π u + (P A P B ) D 2 2π PM- PM-2 PM-3 (23) is the total torque acting on the transformer by the pump/motor units. Given the reference shaft speed for transformer ω d (t), an appropriate U total is needed to drive the transformer speed ω to the desired speed. Here we use a simple PI control with feedforward: ω I = ω := ω ω d U total = J ω d K p ω K I ω I (24) Additional terms can also be added to compensate for damping and mechanical loss. 3.3 Displacement inputs Here, we determine u and to work simultaneously to provide the desired torque in Eq. (24) and the desired flow Q B in Eq. (2). For each transformer configuration, u and could be solved simultaneously using the flow equations in Eqs. (3)-(5) and Eq. (23) as follow: PM-: u ω D2 = 2π QB (P A P T ) D 2π (P B P T ) D (25) 2 U 2π total PM-2: u PM-3: u ω D = 2π ω D2 2π QB (P A P B ) D 2π (P B P T ) D 2 2π U total ω D2 = 2π QB (P A P T ) D 2π (P A P B ) D 2 2π U total (26) (27) Notice that mechanical and volumetric losses were ignored in these expressions. Additional terms to compensate for these could also be added to improve performance. 5 Copyright c 25 by ASME

6 FIGURE 5. Prototype transformer based upon two 3.5 cc micropiston pump/motors FIGURE 4. Transformer based control is tested on the pitch axis of this experimental setup. TABLE. Experimentation parameters Parameter Notation Value Cylinder mass m kg Load F L -98 N Viscous damping b 5 N/m s Piston Cap Area A.87 cm 2 Piston Rod Area A 2 5. cm 2 Supply (gauge) pressure P A 5.5 MPa Return (gauge) pressure P T MPa 4 Experimental Results The controller in Section 3 has been experimentally implemented on the pitch axis of the robotic device shown in Fig. 4. The prototype hydraulic transformer used was constructed by mechanically connecting two 3.5cc micro-piston pump/motors (Fig. 5). Lead screws, stepper motors and encoders are added to actuate the swashplates and to adjust the displacements of the pump/motors. The transformer in all three configurations in Fig. -3 have been tested. Various system parameters are summarized in Table. Although the orientation of the hydraulic cylinder in Fig. 4 does vary, the variation is small. This allows us to estimate the effective mass and damping coefficient (referred to the actuator) as constants. 4. Cylinder trajectory Tracking First, the controller is tested with two different sinusoidal trajectories on the PM- setup (Fig. ). One trajectory has a higher amplitude (.4m) and lower frequency (.4π rad/s), and the other has a lower amplitude (.5 m) and higher frequency (.7π rad/s). The transformer speed is to be regulated at 96 TABLE 2. Fast and slow sinusoidal trajectory tracking with fixed transformer speed on PM-. RMS errors in position, pressure and transformer speed. e mm P MPa ω rad/s Slow Fast rad/s and 67 rad/s. Results for these two cases are shown in Fig. 6 and Fig. 7 respectively and RMS errors in position, pressure and transformer speed are shown in Tab. 2. The performance with both trajectories are similar. RMS motion errors of less than mm and transformer speed errors of less than.3% are achieved for both trajectories. 4.2 Transformer Speed Tracking Next, the controller is tested with a smoothed trapezoidal motion trajectory on the PM-, 2, 3 setups while the desired transformer operating speed is varied arbitrarily. Tracking results are plotted on Figs 8-. RMS errors are summarized in table 3. The RMS motion errors are within mm for all three configurations whereas the transformer speed errors are slightly larger than the case when the desired speed is a constant. While the performances of all 3 configurations are very similar, PM- and PM-3 have slightly better motion control performance than PM-2, whereas PM- and PM-2 have slightly better transformer speed control performance than PM-3. These experiments demonstrate that the proposed control algorithm can be used to simultaneously control the trajectory and the transformer operating speed. The latter could be used in the future to optimize the transformer operation. 4.3 Effect of Parameter Uncertainty Effects of uncertainty in the effective mass and damping coefficient are tested. Instead of using the best known parameters in Tab., other parameters in Tab. 4 are used in the controller instead. Control gains are, however, kept the same. Results with 6 Copyright c 25 by ASME

7 . Actuator.6.4 Actuator m.5 m Pressure Pressure MPa.8.6 P b MPa.8.6 P b rad/sec FIGURE 6. PM Transformer Speed Control inputs Low frequency large amplitude trajectory tracking for u rad/sec Transformer Speed Control inputs FIGURE 7. PM- High frequency, small amplitude trajectory tracking for u the PM- configuration are summarized in Tab. 4 and select cases are plotted in Fig.. As expected, uncertainties in mass and damping values result in larger error and the position error tends to be an bias. However, in all cases but one, the RMS position error does not increase by more than.7mm, and in all cases, the RMS transformer speed error does not increase by more than.8 rad/s. To improve on these errors, an adaptive control scheme that estimates the mass and the damping coefficient (or similarly, by adding an integral action) can be a fruitful avenue for further investigation. 5 Conclusion This paper presents a controller for a hydraulic cylinder driven by a pump/motor transformer such that the cylinder tracks a predefined trajectory and the transformer speed is regulated at its desired valve. The controller, which was designed based upon the passivity property of the hydraulic actuator and a recently discovered natural energy storage functions, can be applied to all three configurations of the transformer in Figs. -3. Experimental results show satisfactory cylinder trajectory and transformer speed regulation performance. All three transformer configurations have similar control performance and achieve RMS position errors of less than mm. While transformer controlled system is expected to have better energy efficiency than a servo-valve controlled system, a concern has been whether an adequate control performance can be obtained. Experimental results in this paper shows that indeed good trajectory tracking performance can be achieved. The desired transformer speed has been arbitrarily defined in this paper, the reference speed can be determined to improve efficiency and to avoid stalling. In the next step, the proposed control 7 Copyright c 25 by ASME

8 Actuator Actuator m..5 m..5.4 Pressure.4 Pressure MPa.2 P b MPa.2 P b Transformer Speed 2 Transformer Speed rad/sec Control inputs Time s FIGURE 8. Trapezoidal trajectory tracking with variable desired transformer speed for PM- u rad/sec Control inputs Time s FIGURE 9. Trapezoidal trajectory tracking with variable desired transformer speed for PM-2 u algorithm will be expanded to enable the transformer to operate at its most efficient region. This will involve transformer speed optimization, consideration of energy regeneration capability of a hydraulic transformer, and active switching of the transformer between the three configurations. 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. REFERENCES 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. 2 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 Achten, P., Van den Brink, T., Paardenkooper, T., Platzer, T., Potma, H., Schellekens, M., and Vael, G., 23. Design and testing of an axial piston pump based on the floating cup principle. In The Eighth Scandinavian International Conference on Fluid Power SICFP 3, pp Ouyang, X., 25. Hydraulic Transformer Research. 8 Copyright c 25 by ASME

9 m..5.4 Actuator Pressure TABLE 3. Trapezoidal trajectory tracking with varying desired transformer speed on PM-, PM-2, PM-3. RMS errors in position, pressure and transformer speed. e mm P MPa ω rad/s PM PM PM MPa rad/sec Transformer Speed Control inputs Time s FIGURE. Trapezoidal trajectory tracking with variable desired transformer speed for PM-3 PhD Thesis, Zhejiang University, Hangzhou, China. 5 Werndin, R., and Palmberg, J.-O., 2. Controller design for a hydarulic transformer. In The Fifth International Conference on Fluid Power Transmission and Control ICFP, Vol. 5, pp Werndin, R., and Palmberg, J.-O., 22. Hydraulic transformer in low-speed operation - a study of control strategies. In The 5th International Symposium of Fluid Power, JFPS 2, Nara, Japan. 7 Vael, G., Achten, P., and Potma, J., 23. Cylinder control with the floating cup hydraulic transformer. In The Eighth Scandinavian International Conference on Fluid Power SICFP 3 Tampere, Finland, pp Ho, T. H., and Ahn, K. K., 28. A study on the position control of hydraulic cylinder driven by hydraulic trans- P b u TABLE 4. Tracking results for various assumed effective mass and damping coefficient: RMS erros in position e, pressure P and transformer speed ω. Position m m kg b N/m s e mm P MPa ω rad/s FIGURE. Effect of Parameter Error m:5 b: m:8 b:5 m: b:5 m:2 b:5 Reference Time s Trajectory tracking for various parameter values former using disturbance observer. In International Conference on Control, Automation and Systems 28, IEEE, pp Lee, S., and Li, P. Y., 24. Trajectory tracking control using a hydraulic transformer. 24 International Sympo- 9 Copyright c 25 by ASME

10 sium on Flexible Automation, Awaji Island, Japan. 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 Cho, B.-H., Lee, H.-W., and Oh, J.-S., 22. Estimation technique of air content in automatic transmission fluid by measuring effective bulk modulus. International journal of automotive technology, 3(2), pp Copyright c 25 by ASME