A NEW PASSIVE CONTROLLER FOR A HYDRAULIC HUMAN POWER AMPLIFIER

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1 Proceedings of IMECE26 26 ASME International Mechanical Engineering Congress and Exposition November 5-1, 26, Chicago, Illinois, USA IMECE A NEW PASSIVE CONTROLLER FOR A HYDRAULIC HUMAN POWER AMPLIFIER Perry Y. Li Department of Mechanical Engineering University of Minnesota 111 Church St. SE Minneapolis, MN pli@me.umn.edu ABSTRACT A new, intrinsically passive controller for hydraulic human power amplifier is presented. The hydraulic human power amplifier is a tool that amplifies (or attenuates) the force that the human exerts on it. The control objective is to cause the system to behave like a passive mechanical tool when interacting with the human and with the work environment with a specified power scaling factor. Although a previous Proportional-integral with velocity feedforward force controller [1] works well in the constrained space, it lacks robustness or performance during freemotion because of the sensitivity to the implementation of velocity feedforward term. The difficulty in implementing the velocity feedforward term also prevents the controller from being intrinsically passive. The new controller recasts the human power amplifier problem into one of velocity coordination by generating a fictitious reference mechanical system. Force amplification become a natural consequence of velocity coordination. This enables the controller to be intrinsically passive and to achieve good performance both in free motion and constrained motion. These properties have been experimentally validated. 1 Introduction Human amplifiers or extenders [2 5] are tools that humans manipulate directly, but have the ability to attenuate or amplify the apparent power that the human exerts. Since the human amplifiers are kinematically connected to the humans, power amplification or attenuation is equivalent to force/torque amplification/attenuation. Figure 1. Handle with force sensor Hydraulic motor Rack and pinion q 2 Force sensor Force sensor Hydraulic actuator Turntable base q 1 Powered Oar Picture and schematic of the University of Minnesota Power Oar, emphasizing the two hydraulic assisted degrees of freedom (fore-aft linear and pitch). These tools enable the humans to be physically connected to the task being performed. Therefore, the operator controls and is informed by the machine via physical quantities like power, forces and displacements, as in the use of common mechanical tools such as hammer, scissors etc. Potentially, human amplifiers are more natural and intuitive to use. Exoskeletons are a special class of human amplifiers with anatomically compatible degrees of freedom so that humans can wear them [2]. They exactly duplicate the anatomical motions of humans. The human power amplifier at the University of Minnesota takes the form of a hydraulic assisted oar. The main advantage of the oar concept over an exoskeleton concept is that of safety. It would be easier for the human operator to let go of the system 1 Copyright c 26 by ASME

2 x I x p contains some concluding remarks. u Figure 2. Ideal Kinematic actuator Acap Q P s Aannulus T Aannulus A cap Servo valve Equivalent fluid spring Q Inertia Each hydraulic actuator is modeled as an ideal kinematic actuator and an equivalent fluid spring. if anything were to go wrong. Hydraulics are used instead of electromechanical actuators because of the intrinsic large force and power densities. We must however consider the fluid compressibility and the different perspective that a hydraulic actuator is mainly a velocity source whereas an electromechanical actuator is typically treated as a torque/force source. The control objective is to amplify the power that the human exerts on the machine. In addition, for safety the control system will be designed such that the human power amplifier interacts passively with the human operator at the handle, and with its physical environment. In a previous paper [1], we presented a robust PI with velocity feedforward control law for a similar system with the objective of controlling the actuator force to be a scaled copy of the human force. Although good results are obtained during constrained tasks, performance or robustness can be improved during free-motion. This is because, in practice, the velocity feedforward term tends to have a destabilizing effect in the presence of uncertainty, slow sampling or time delay. On the other hand, partial implementation of this term (i.e. by a less than unity scaling) sacrifices force tracking performance during unconstrained free motion. Moreover, the passivity property is satisfied only after convergence. The present paper proposes an alternate controller which is intrinsically passive. Instead of considering a force control problem, it converts the control problem into one of coordinating the velocity of the system with that of a fictitious mechanical system with passive dynamics. In essence, the velocity feedforward term is now generated by the velocity of the fictitious mechanical system. The rest of this paper is organized as follows: Section 2 presents the Power Oar design and its model. The coordination control approach is described in section 3. Section 4 presents the passivity analysis of the overall control system. Simulation and experimental results are presented in sections 6. Section 7 2 Modeling The University of Minnesota hydraulic assisted power oar (Fig. 1) can pitch in the vertical plane, slide in and out linearly, as well as yaw in the horizontal plane. The human interacts with the oar through a handle instrumented with two force sensors for sensing the forces parallel and normal (in the plane of the oar) to the oar. Pitching occurs at a horizontal pivot pin. Sliding is achieved by mounting the oar to the vertical post via a rack and pinion arrangement. Yaw motion occurs at the base where the vertical post is mounted on a rotary turntable. The system allows the human to manipulate in the passive mode, as well as with hydraulic assistance. Currently, only the pitch and the sliding motion are assisted. The pitch motion is assisted by a clevis mounted single ended hydraulic actuator between the oar and the vertical post. The sliding motion is assisted by a hydraulic motor mounted at the pinion. Both the hydraulic actuator and the hydraulic motor are controlled by Moog series 7 servo valves (rated at 2.5gpm). The linear force generated by the hydraulic actuator, and by the hydraulic motor via the rack are measured by force sensors. The pitch angle and as well as the pinion angle are measured via rotary potentiometers. The sliding displacement of the rack is then computed using the rack and pinion ratio. Hydraulic power is provided by a 4.1gpm hydraulic power supply with a maximum pressure setting of 1psi. For simplicity, we consider only one degree of freedom motion, i.e. the other DOFs are locked. Extension to fully coupled dynamics can be done but will detract from the key concept. For each degree of freedom: the generalized inertia M p > is acted on by the generalized human the environment force inputs F human and F env, as well as the hydraulic actuator generalized force F s ( ). In the case of the pitch motion, the actuator is a single rod cylinder; in the case of the reach motion, the actuator is a hydraulic motor connecting to a rack and pinion. The model below is appropriate for both situations. To account for the fluid compressibility, the hydraulic actuator consists of an ideal kinematic actuator (with displacement x I ) interacting with the system inertia (with displacement x p ) via an equivalent spring with a compression = x I x p (Figure 2). The possibly nonlinear spring F s ( ) encompasses the compressibility of the fluid in the actuator and the fluid line, as well as other mechanical compressibility. We assume that F s ( ) is differentiable and monotone increasing, and that F s () =. Inclusion of the compressibility effect is essential for modeling force exerted by the hydraulic actuator. In our setup, the displacements of the human amplifier x p is measured via potentiometers; and both the applied actuator force F s and the applied human force F human are measured by 2 Copyright c 26 by ASME

3 F d + Kps s + K I 1 K s Controller d F dt d + + d x dt p A K q K q A Actuator d x dt p Rigid body dynamics K s S Spring F Figure 3. Block diagram of the P-I with velocity feedforward force control structure in [1] illustrated in the linear setting. force sensors. The environment force F env and the ideal actuator velocity ẋ I are not measured. The ideal actuator is controlled by a matched, symmetric and critically lapped four-way valve. We assume that the servo-valve bandwidth is sufficiently high so that the valve command corresponds statically to spool displacement. Under this assumption, the relationship between the ideal actuator speed ẋ I and the valve command u is given as follows (see Appendix for derivation): ẋ I = K q (sgn(u ))u S q (sgn(u ),F L ) u (1) }{{}}{{} u u loss The dynamics of the human power amplifier with compressibility is modeled by: M p ẍ p = F human + F env + F s ( ) (2) = ẋ p + u u loss (3) where u = K q (sgn(u ))u is the unloaded velocity, u loss = S q (sgn(u ),F))u is the shunt velocity which is the decrease in velocity from the unloaded situation to the actual loaded condition. Because u and u are uniquely related, we shall henceforth consider u as the control input instead. It can be shown [6] that because of the monotonic relationship between load pressure and valve flow, the shunt velocity u loss is dissipative in the sense that F u loss u, F. Using the non-negative storage function W = and differentiating it, we can show that F s (δ) dδ+ 1 2 M pẋ 2 p, (4) Ẇ = F s ( )u+(f human + F env )ẋ p F s ( )u loss (5) Ẇ F s ( )u+(f human + F env )ẋ p (6) Integrating w.r.t. time gives W () F s ( ) u+f human ẋ p }{{}}{{} command human + F env ẋ p }{{} environment dτ (7) Thus, the open loop system is passive w.r.t. the supply rate consisting of the command power, the human power and the environment power. In the previous paper [1], the control objective is to control F act (t) F d (t) := ρf human (t). This was achieved using a controller as shown in Fig. 3 in which the controller consists of a Proportional-Integral force feedback controller, and the feedforward cancellation of the measured velocity ẋ p and of Ḟ d (t). In implementation, the feedforward term of Ḟ d (t) is not critical and can be neglected. When the velocity feedforward term is neglected or only partially implemented (by a less than unity scaling), this controller has excellent force tracking performance during constrained pushing tasks or load carrying when the system is not moving very fast (Fig. 4 a) and b)). However, the performance becomes degraded during unconstrained free motion as the velocity feedforward term is critical in this case. Unfortunately, full implementation of the velocity feedforward term has a destabilizing run away effect and thus can only be partially implemented (although this difficulty is somewhat alleviated with a faster sampling time). 3 Coordination control formulation Instead of considering a force control problem of requiring F( (t)) ρf human (t), we formulate the hydraulic human power amplifier as one of coordinating the velocities of two mechanical systems with coupling. Let the dynamics of a virtual mechanical system and its coupling to the hydraulic human power amplifier be given by: M v ẍ v = F d F( )+w u = x v + u 1 (8) 3 Copyright c 26 by ASME

4 Force N ρ = 1 Desired force Actual force where F d (t) is the desired actuator force typically given by F d (t) := ρf human (t), and w is the control input. Notice that the coupling between the virtual mechanical system and the human power amplifier is similar to an integral controller (when w =, u 1 = ). The coupled system is given by: M p ẍ p = F human + F env + F( ) M v ẍ v = F d F( )+w = ẋ v ẋ p u loss + u 1 (9) Force N Time s Desired force Actual force ρ = 5. Varying load Load removed Load removed Load added Load added Time s ρ = 5. No load which has a bond graph representation shown in Fig. 5. The bond graph representation in turn suggests for the coupled system Eq.(9) the storage function, W c = F s (s) ds+ 1 2 M pẋ 2 p+ 1 2 M vẋv 2 (1) }{{} W where W is the storage function for the original system (4). From this, it can be shown that the coupled system (9) satisfies the passivity property: [F d ẋ v +(F human + F env )ẋ p ]dτ+ [F( )u 1 + wẋ v ]dτ c 2. (11) for some c and for all t. Thus, if the control terms w and u 1 are designed such that for some c 1, and for all t, [F( )u 1 + wẋ v F d (t)v E ]dτ c 2 1 (12) Force N Desired force Actual force Time s Figure 4. Results with the P-I force controller in [1]. a) Constrained pressing task; b) Unconstrained task during which a 2lb load is consecutively added and removed. c) Unloaded free-motion. Please note that the force values and the power amplification factor are not physical units as the sensors were not calibrated, also moment arm effects have not been corrected. where v E = ẋ v ẋ p, then, for F d (t) = ρf human (t), the human power amplifier under control will satisfy a passivity property: [(ρ+1)f human + F env ]ẋ p dτ c 2 = (c 2 + c 2 1) (13) for all t. This is the desired passivity property for a rigid passive mechanical tool that interacts with the ρ+1 times amplified human force and the work environment. Furthermore, if it can be guaranteed that v E L 1 (e.g. v E exponentially) then Eq.(12) and Eq.(13) will also be satisfied as long as, c 2 s.t. for all t, [F( )u 1 + wẋ v ]dτ c 2 2. (14) 4 Copyright c 26 by ASME

5 3.1 Coordinate transformation into locked and shape systems To see the effect of the control (u 1,w) on the dynamically coupled system and the coordination error, consider the coordinate transformation: ( vl v E ) = ( Mp /M L M v /M L 1 1 ) )(ẋp ẋ v (15) M L := M v + M p (16) ( 1 M E := + 1 ) 1 = M pm v M p M v M p + M v (17) (v L,v E ) are referred to as the locked and shape system velocities, since v E measures the difference between the speeds of the virtual and actuator manipulator, and v L = ẋ v = ẋ p when v E =. Hence, v L is the speed of the manipulator and the virtual manipulator when they are locked in place. M L and M E are respectively the locked system inertia, and the shape system inertia. Eqs. (15)-(17) is an instance of decomposing the velocity space of the coupled system (i.e. R 2 = {(ẋ p,ẋ v )}) into a component given by the coordination error v E = ẋ p ẋ v and its orthogonal complement v L with respect to diag(m p,m v ) being the metric. This decomposition approach can be generalized to nonlinear multi-dof mechanical systems as well [7, 8]. The dynamics in the transformed coordinates are given by: M L v L = F human + F env + F d + w (18) M E v E = M v (F human + F env ) M p F d (t) +F s ( ) M p w (19) M L M }{{ L M } L F E = v E u loss + u 2 (2) We can associate storage functions for the locked and shape systems respectively as, W L = 1 2 M Lv 2 L; W S = 1 2 M Ev 2 E + F s (δ)dδ (21) Notice that the coordinate transformation Eqs.(15)-(17) preserves kinetic energies and storage functions in that the sum of the kinetic energies and storage functions, respectively, of the locked and shape systems are exactly those of the coupled system [9, 1] κ := 1 2 M pẋ 2 p M vẋ 2 v = 1 2 M Lv 2 L M Ev 2 E (22) W c := W L +W S (23) This means that by ensuring that the shape and locked systems are passive, the coupled system in Eq.(9) will also be passive. Notice that the locked and shape systems are coupled only when w. 3.2 Shape system control The shape system control objective is to make v E in: M E v E = F E (t)+f s ( ) M p w M L (24) = v E u loss + u 1 (25) While a variety of control algorithms can be applied, here we choose to use a linear control design approach. In the design process, we consider w, u loss and Ḟ E (t) as disturbances (i.e. F E (t) is assumed to be constant) and the spring is linear F s ( ) = K s, and let u 1 be the primary control actuator. Since F( ) is measured and v E is available for feedback, we can increase the natural frequency and add damping to the undamped system by: u 1 = γ 1 v E γ 2 F( )+(1+γ 1 )u 2 (26) This increases the natural frequency by a factor of (1+γ 1 ) and K introduces a damping ratio of ζ = γ s 2 ω n where ω n is the natural frequency of the undamped system. w can also be used to introduce further damping if necessary (e.g. by setting w = κ w v E if desired). The transfer function from u 1 to v E becomes: G(s) = ω 2 n s 2 + 2ζω ns+ω 2 n which has a steady state gain of 1. We use the approximation G(s) 1 as the nominal plant and design an integral plus leadlag control, u 2 (s) = C(s)v E (s) = p2 s(s+2p) v E(s) (27) where p >. This places the characteristic equation of the nominal closed loop system to be ψ(s) = (s+ p) 2. Since the oscillatory mode at ω n is ignored in the design, p << ω n to maintain stability. To increase the bandwidth of the system, we improve the complementary sensitivity of the nominal closed loop system via innovation feedback [11] u 2 (s)= C(s)v E (s) Q(s)(v E (s) u 2 (s))= C(s)+Q(s) 1 Q(s) v E(s) 5 Copyright c 26 by ASME

6 I : M v C : F s ( ) I : M p ẋ v F s ( ) ẋ p S e : F d 1 1 S e : F human u loss S e : w S f : u1 R : S q S e : F env Figure 5. Bond graph of hydraulic human power amplifier with a virtual mechanical system since v E (s) = u 2 (s) for the nominal system G(s) 1, the stability of the nominal system will not be affected as long as Q(s) is stable. However, Q(s) affects the robustness property of the system. In fact, Q(s) modifies the complementary sensitivity modified affinely: T o (s) = C(s)+Q(s) 1+C(s) We choose Q(s) to maintain the T o (s) 1 for low frequency but also to introduce a notch filter at s = ± jω n. One choice is: which gives Q(s) = p 2 (r/ω 2 n)s 1 (s+2p)(s+r) T o (s) = rp2 ω 2 n s 2 + ω 2 n (s+ p) 2 (s+r) as desired. Experimentally, ω n is found to be approximately 88rad/s. This limits p < 25 to remain stable without the Q(s) compensation. Using the Q(s) compensation with r = p, the system remains stable for p < Locked system control If F d := ρf human, the locked system dynamics becomes M L v L = (ρ+1)f human + F env }{{} +w (28) F total which is the desired dynamics of a common passive mechanical tool without using the additional control w. In future work, we can explore using w to implement guidance and obstacle avoidance similar to that presented in teleoperation [12]. In this paper, we only utilize w for introducing additional damping for the shape system. We can thus assume that w(v E ). When v E, we also have the RHS in Eq.(19) being zero on average: = F E + F s ( ) M p M L w = M v M L F total + F s ( ) F d (t) M p M L w Since F total is the total forcing term for the locked system (28) scaled by M v /M L (which is small if M v is small), thus unless the locked system undergoes large acceleration, F total is typically small. Therefore, we have the original force control objective being approximately satisfied: F s ( ) F d (t) This is despite the specified objective (because of the dynamics of the added M v inertia) being a coordination control problem. 4 Passivity Analysis As mentioned earlier, if the passivity property Eq.(12) is satisfied by the control system, then the human power amplifier would satisfy the desired passivity property (13). In the case when w =, Eq.(12) reduces to: for some c 1, [F( )u 1 + wẋ v F d (t)v E ]dτ c 2 1 (29) where the designed control for u 1 and w are: u o 1 = γ 1 v E γ 2 F( )+(1+γ 1 )u 2 (3) w o = (31) 6 Copyright c 26 by ASME

7 Here, we actively enforce this condition using a fictitious flywheel energy storage element to keep track of the net energy flow, M f v f = [F( )u 1 + wẋ v + F d (t)v E ]/v f (32) where the flywheel energy is initialized by 1 2 M f v 2 f () = W f(). At any point the integral in (12) equals W f () 1 2 M f v 2 f (t). By ensuring that v f (t) is greater than some threshold f stop >, the integral in (12) will be less than W f (). The integration of v f (t) can be saturated to ensure that v f (t) f max (t) which in turn limits future energy flow. The use of fictitious flywheel to enforce passivity was first proposed in [13] and is similar to passivity observer proposed by Hannaford and co-workers. Based on v f (t) and its history, several operating modes can be defined: Normal mode: This occurs when v f (t) f cap > f stop, we set u 1 = u o 1, w =. Shut-down mode: This occurs when v f (t) falls below f stop, u 1 = γf( ) w = λ(t)ẋ v (33) F d will also be set to be F d =. Notice that the modified control extracts energy from the system: d 1 dt 2 M f v 2 f γf 2 ( )+λ(t)ẋv 2 In this case, the control does not create any energy at all, so that the system satisfies the passivity property with power scaling (ρ = ): [[F human + F env ]ẋ p ]dτ c 2 (34) The system does not emerge from the shut-down mode until v f (t) f cap. Energy capture mode: This occurs when v f (t) falls below f cap. u 1 = u o 1 w = λ(t)ẋ v (35) The different operating modes are designed to robustly maintain the passivity property of the closed loop system in unforseen situation. In normal operating condition, the control system should be designed such that the system can indefinitely operate in the normal mode. For this to be the case, the energy balance must be such that the energy drain by the control is bounded. Assuming that v E, then F s ( (t)) F E (t). Let E (t) be such that F( E (t)) = F E (t). Then the required control input would be u(t) u loss + E The energy associated with this input is: F s ( E )u loss dτ+ E (t) F s (δ)dδ. F E (t) can be assumed to be bounded so that the second term is bounded. The first term is positive and thus represents constant energy drain. To compensate for this we can augment the normal mode by w = εẋ v where ε is a small number. In practice, this is not necessary because of the small amount of energy associated with this loss. 5 Experimental results We set ρ = 3 for these experiments. Figures 6, 7 and 8 show the fictitious velocity and actual velocity, as well as the force and desired force, respectively, for the constrained moving tasks, for the task when a 2 lb was loaded and unloaded, and for the unconstrained free motion case. Good results are obtained. Notice that the force tracking results are better than the results using the PI+feedforward controller, especially in the unconstrained free motion case. 6 Conclusions In this paper, a new control law for hydraulic human power amplifier is proposed. The control law converts a force control objective into a coordination control paradigm. In particular, the difficulty of introducing a velocity feedforward signal is alleviating by using a virtual mechanical system. The control law can be intrinsically passive as well. This will enhance safety of operation. F d (t) = ρf human is also not modified. During this mode, energy is being deposited into the controller using the w term. However, since control is still active it cannot be guaranteed that the flywheel energy must increase, although it is very likely to be so, except for faulty or un-modeled situations. 7 Conclusions REFERENCES [1] P. Y. Li, Design and control of a hydraulic human power amplifier, in Proceedings of the 24 ASME IMECE. Paper #IMECE FPST Divisiion, Copyright c 26 by ASME

8 Actual and virtual velocities 1.8 Actual and Virtual Velocities when Loaded rho = rad/s Pushing task Time (sec) Loaded rho = 3 rad/s.1.2 Torque (Nm) Time s Actuator torque and desired torque Time (sec) Figure 7. Loading and unloading: Top: Virtual mechanical system velocity and actual velocity; Bottom: Actual force and desired force (3 times human force) 2 1 Nm Figure 6. Pushing task Time s Constrained pushing task: Top: Virtual mechanical system velocity and actual velocity; Bottom: Actual force and desired force (3 times human force) [2] H. Kazerooni, Human power amplifier technology at the university of california, berkeley, Robotics and Autonomous Systems, vol. 19, no. 2, pp , [3], Human power extender: an example of humanmachine interaction via the transfer of power and information signals, in Proceedings of the 1998 International Workshop on Advanced Motion Control, AMC, 1998, pp [4] H. Kazerooni and J. Guo, Human extenders, ASME Journal of Dynamic Systems, Measurement and Control, vol. 115, no. 2B, pp , [5] K. Kosuge, Y. Fujisawa, and T. Fukuda, Control of a manmachine system interacting with the environment, Advanced Robotics, vol. 8, no. 4, pp , [6] P. Y. Li, Towards safe and human friendly hydraulics: the passive valve, ASME Journal of Dynamic systems, Measurements and Control, vol. 122, no. Sept, pp , [7] D. Lee and P. Y. Li, Passive decomposition of multiple mechanical systems under coordination requirement, in Proceedings of the IEEE CDC 24, vol. 2, 24, pp [8] D. J. Lee, Passive decomposition and control of interactive mechanical systems under motion coordination requirements, Ph.D. dissertation, Department of Mechanical Engineering, University of Minnesota, May 24. [9] P. Y. Li, Passive control of bilateral teleoperated manipulators, in Proceedings of the 1998 American Control Conference, 1998, pp [1] D. J. Lee and P. Y. Li, Passive feedforward approach for linear dynamically similar bilateral teleoperated manipulators, IEEE Transactions on Robotics and Automation, vol. 19, no. 13, pp , June 23. [11] G. Goodwin, S. F. Graebe, and M. E. Salgado, Control System Design. Prentice Hall, 22. [12] D. Lee and P. Y. Li, Passive bilateral control and tool dynamics rendering for nonlinear mechanical teleoperators, IEEE Transactions on Robotics, vol. 21, no. 5, pp , October 25. [13] P. Y. Li and R. Horowitz, Control of smart exercise machine: Part 1. problem formulation and non-adaptive control, IEEE/ASME Transactions on Mechatronics, vol. 2, no. 4, pp , Copyright c 26 by ASME

9 .3.2 Free motion Actual and virtual velocities w is the valve coefficient which can be determined from the rated flow equation: w = Q rated P rated : typically 5psi (38) Prated rad/s Pushing If the actuator is under a load F L, then P 1 A c P 2 A p = F L (39) where A c and A p are the cap side and piston side areas. Let A c : A p = r : 1, then, Time s Actuator torque and desired torque P s = P 1 + r 2 P 2 (4) Denote P L := F L /A c, and substituting (4) into (39), Free motion Free motion P 2 = r r (P s P L ) (41) P 1 = 1 r (P s r 3 P L ) (42) Nm Figure 8. Pushing Time s Unconstrained free motion task: Top: Virtual mechanical system velocity and actual velocity; Bottom: Actual force and desired force (3 times human force) Appendix: Modeling of the ideal actuator Consider an ideal actuator without compressibility connected to a four way spool valve. Let u be the normalized valve spool displacement so that u = ±1 correspond to the valve being fully open, u > corresponds to extension and u < corresponds to retraction. ( r Q 2 = u w r ) 1 2 Ps P L ( ) 1 r Q 1 = u 3 2 w Ps r 3 P L + 1 The actuator extension speed for x > is then given by: ẋ I = u w ( ) 1 r 3 2 Ps A c r 3 P L. + 1 For u <, we conduct a similar analysis. Let the inlet (piston side) and outlet (cap side) pressures be P 1 and P 2, respectively. Then, the piston side and capside flows are Q 1 = x w P s P 1 = A c r ẋp (43) Q 2 = x w P 2 = A c ẋ p (44) Moreover, the force balance gives P 2 P 1 r = P L so that Q 1 = u w P s P 1 (36) Q 2 = u w P 2 (37) P 1 = r3 P s rp L 1+r 3 P 2 = r2 1+r 3 (P s + rp L ) 9 Copyright c 26 by ASME

10 ẋ I = u w ( ) 1 r 2 2 Ps A c r 3 + rp L + 1 In summary, we have: ( ) 1 u wa r 3 2 ẋ I = c r 3 Ps P +1 L, u ( ) 1 u wa r 2 2 Ps c + rp L, u < r 3 +1 ẋ I = K q (sgn(u ),F L )u ( ) 1 w r 3 2 K q (sgn(u A ),F L ) = c r 3 Ps F +1 L /A c, u ) 1 w 2 Ps + rf L /A c, u < ( r 2 A c r 3 +1 (45) where K q (sgn(u ),F L ) is the loaded velocity gain. Eq. (45) can be further written as a no-load velocity reduced by an amount due to load: ẋ I = K q (sgn(u ))u S q (sgn(u ),F L ) u (1) where the no-load velocity gain is: K q (sgn(u )) = w r 3 A c w r 2 A c r 3 +1 P s u r 3 +1 P s u < (46) and the shunt velocity gain is: S q (sgn(u ),F L ) = w r 3 A c w r 2 A c r 3 +1 ( P s P s F L /A c ) u r 3 +1 ( P s P s rf L /A c ) u (47) This shows that the no-load extension speed is higher than the no-load retraction speed. This is slightly counter intuitive. For a linear actuation setup consisting of a hydraulic motor with displacement D [m 3 ] and a rack-and-pinion with a pitch of r p [m/rad], [ ] 2πrp ẋ I = Q 1 = D [ 2πrp D ] Q 2 It is equivalent to a double ended linear actuator with r = 1, and A c = D 2πr p. 1 Copyright c 26 by ASME