DESIGN AND CONTROL OF A HYDRAULIC HUMAN POWER AMPLIFIER

Transcription

1 Proceedings of IMECE4 24 ASME International Mechanical Engineering Congress and Exposition November 13-2, 24, Anaheim, California USA IMECE DESIGN AND CONTROL OF 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 This paper describes the design of and some preliminary control results for a hydraulically actuated human power amplifier. The system is in the form of an oar, with its reach and pitch degrees of freedom being hydraulically assisted. A robust PI force controller is proposed so that the hydraulic actuator force tracks a scaled copy of the force exerted by the human. Nonlinearities and uncertainties in the compression spring, as well as parametric uncertainties are taken into account. The passivity property of the closed loop system is also analyzed. The controller has been tested in simulations and experimentally. It is shown to be effective when pushing against an object, and in assisting in bearing static loads. 1 Introduction Human power amplifiers or extenders [1 4] are tools that humans operate directly and which have the ability to amplify the mechanical power or force exerted by the human operators. These tools enable humans to be physically connected to the task being performed and to take advantage of the additional mechanical power provided by the machine. In this way, the operator controls the machine and is informed by it via physical quantities like power, forces and displacements, just as in the use of common mechanical tools like hammer, scissors etc. Potentially, human power amplifiers are more natural and intuitive to use than more autonomous machines for which humans play only supervision or task planning roles. Exoskeleton is a special form of human power amplifiers with anatomically compatible degrees of freedom that humans wear [1]. In the effort to involve undergraduate students in the design and control of fluid power systems, a team of undergraduate mechanical engineering students at the University of Minnesota recently undertook to design and construct a hydraulically powered human amplifier. The University of Minnesota human power amplifier takes the form of a hydraulically assisted oar (phase 1, mounted on a mobile platform (phase 2 - to begin in Fall 24. 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 if anything were to go wrong. This paper describes the design of the hydraulically assisted oar, as well as some preliminary results for the control design. The control objective is to amplify the human exerted force on the oar. A robust force control algorithm, its nonlinear analysis as well as some experimental results will be presented. The nonlinear analysis takes into account possibly nonlinear compressibility, uncertain parameters and measurements, and valve nonlinearities. The proposed control law is effective in tracking the desired force profile during constrained motion, and in sharing in static loads bearing during unconstrained motion. The rest of this paper is organized as follows: Section 2 briefly presents the UMn Power Oar design. Section 3 presents the system models and the control objectives. The control design and analysis are presented in section 4. Section 5 presents the passivity property of the overall control system. Simulation and experimental results are given in sections 6-7. Section 8 contains some concluding remarks. 1 Copyright c 24 by ASME

2 2 The UMn Power Oar The University of Minnesota hydraulic assisted power oar (Fig. 1 pitches in the vertical plane, reaches in and out linearly, as well as yaws in the horizontal plane. It has also a gripper which is actuated by squeezing a trigger via a linkage mechanism. The oar interacts with the human 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 motion in the reach degree of freedom 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 it in the passive mode, as well as with hydraulic assistance. Currently, only the pitch and the reach motions are assisted. The pitch motion is assisted by a clevis mounted single ended (3:2 area ratio 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 76 series servo valves (rated at 2.5gpm. The linear forces generated by the hydraulic actuator, and by the hydraulic motor via the rack are measured by a pair of 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 relief pressure of 1Psi. The modest flow ratings and pressure setting for the valves and pump for this first prototype are chosen to ensure that the system is relatively safe during design and testing. As a consequence the system saturates easily. For example, the maximum pitch velocity is between.2-.3 rad/sec (depending on the direction. This limits controller performance especially during unconstrained motion. 3 System Model and Control Objectives The dynamics of the oar are given by: M(q q+c(q, q q = F f riction + F load + F act + F }{{ human (1 } Goal: (ρ+1f human where F act is the applied force by the hydraulic actuator, F human is the applied force by the human, F f riction is the friction force, and F load is the external load (including gravity acting on the machine. Our objective is to control the hydraulic actuators such that F act ρf human, where ρ >. If this is satisfied, then the human would feel that he/she is (ρ+1 times more powerful, or equivalently, the machine and the external load are ρ+1 times lighter and smaller. 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. To be concrete, let q 1 [m] correspond to the reach displacement in the fore-aft direction, and q 2 [rad] correspond to the pitch angle. The hydraulic actuation forces F act = (F 1,α(q 2 F 2 where F 1 [N] is generated by the hydraulic motor via a rack-and-pinion with a ratio of r m/rad, and the F 2 is force generated by the hydraulic actuator, and α(q 2 is the corresponding pitch moment arm. The control objective is to control the hydraulic motor and actuator so that ( F1 F 2 ( Fd,1 F d,2 ( Fhuman,1 := ρ. F human,2 /α(q 2 To account for the fluid compressibility, we model the actuators as consisting of an ideal kinematic actuator interacting with the system inertia via an equivalent spring (Figure 2. The spring encompasses the compressibility of the fluid in the actu- 2 Copyright c 24 by ASME

3 u Figure 2. Ideal Kinematic actuator Acap Q P s Aannulus T x I Servo valve x p Aannulus A cap Equivalent fluid spring Q Inertia Each hydraulic actuator is modeled as an ideal kinematic actuator and an equivalent fluid spring. ator and in the fluid line, as well as other mechanical compressibility. Thus, the rack-and-pinion actuator is modeled as: ẋ I,1 = r D Q 1 (2 F 1 = F s,1 (x I1 x p,1 (3 where D[m 3 ] is the hydraulic motor displacement, Q 1 is the flow into the hydraulic motor, x p,1 = q 1 is the linear displacement of the oar, and F s,1 ( is the (nonlinear spring force for a compression of. The pitch actuator is modeled as, ẋ I,2 = 1 A(ẋ 2 Q 2; (4 { A cap if ẋ I,2 > A(ẋ I,2 = (5 A annulus if ẋ I,2 F 2 = F s,2 (x I,2 x p,2 (6 where A cap and A annulus are the capside area and the annulus area of the hydraulic actuator, Q 2 is the flow to the pitch actuator, F s,2 ( is the force in the equivalent hydraulic spring when it has a compression of, and x p,2 = l(q 2 is the length of the hydraulic actuator for the pitch angle q 2. We assume that both F s,1 (ε and F s,2 (ε are differentiable, and monotone increasing. The control algorithm to be presented deals with spring functions F s,1, F s,2 that are nonlinear, and possibly unknown. Nevertheless, linear approximations of these functions: F s,i = K s,i i where K s,i [N/m] is the spring constant are useful for gain tuning. For a single ended cylinder (pitch actuator, this is obtained by considering the pressure variation in the actuator chambers as the actuator is displaced and with the valve closed: K s,2 = β( A 2 cap V cap + A2 annulus V piston where V cap and V annulus are the volumes of the capside and annulus side chambers including hose volumes, and β [N/m 2 ] is the bulk modulus of the fluid. The hydraulic motor connected to a rack-and-pinion is equivalent to a double ended actuator with A cap = A annulus = D/(2πr, where D [m 3 ] is the motor displacement, and r [m/rad] is the rack-and-pinion velocity ratio. Therefore, the approximate spring constant for the reach DOF would be: [ K s,1 = βd2 1 4π 2 r ] V m,1 V m,2 where V m,1 and V m,2 are the fluid volumes on either side of the hydraulic motor. For each degree of freedom, a symmetric, matched 4 way directional control two-stage servo valve is used (Moog 76 series, 2.5gpm, 8Hz bandwidth. We assume that the command input u i, i = 1,2 are proportional to the valve openings. Consider first the valve controlling pitch single ended actuator. Using the matched, and symmetric conditions, as well as the relationship between the outlet and return flows: A annulus Q cap = A cap Q annulus, Q 2 = Q cap = { C ( A cap P s F 2 u2 u 2 ; C ( A annulus P s + F 2 u2 u 2 < ; where C is the valve coefficient, F 2 is the force exerted by the hydraulic actuator, and P s is the pump supply pressure. For the valve controlling the hydraulic motor for the reach degree of freedom, using the equivalence between a hydraulic motor and a double ended actuator, we have: (7 ( Q 1 = C D/(2πrPs sgn(uf 1 u 1 (8 where F 1 is the force exerted by the hydraulic motor via the rackand-pinion, and P s is the pump supply pressure. For both cases, we can write, Q i = K q,i (sgn(u i,f i u i In our setup, the displacements of the human amplifier q are measured via potentiometers; and both the applied actuator force F act and the applied human force F human are measured by force sensors. From these, we can compute x p,1 = q 1, x p,2 = l(q 2, and F d,1 = ρf human,1, F d,2 = ρf human,2 /α(q 2. 3 Copyright c 24 by ASME

4 F d 1 d F d K dt s + K p s + K I + s + Controller A K q K q A Actuator d x dt p Rigid body dynamics K s S Spring F Define := d (13 F = F s ( F s ( d (14 (15 Figure 3. setting. Block diagram of basic control structure illustrated in the linear 4 Actuator force control 4.1 Basic control approach Both the linear and pitch actuator systems are of the form: By the mean value theorem, for any, d, there exists K s (, d = df s /d (x for some x [ d, ] such that K s (, d = F = F( F( d. To avoid clutter, we shall use K s (x to denote K s (, d. Consider the P-I control law, F = F s ( F d = x I x p A(ẋ I ẋ I = Q = K q (sgn(u,f s u where F s ( is potentially nonlinear. Our control approach is to produce a robust actuation system such that F s. We assume that both F d (t and F s ( are available, as are the velocities ẋ p. To gain insight, we first develop the control concept using a linear model (i.e. F s ( = K s, A(ẋ I = A, and K q (sgn(u,f = K q. The block diagram of the linear system is shown in Fig. 3 in which the controller consists of a Proportional-Integral force feedback controller, and the feedforward cancellation of the measured ẋ p and Ḟ d. This results in the perfect transfer function: F(s = K sk p s+k s K I s 2 F d (s + K s K p s+k s K I s + s 2 Ḟ d (s = F d (s (9 + K s K p s+k s K I where Ḟ d (s = sf d (s denotes the Laplace transform of Ḟ d. The linear analysis, however, does not apply to the nonlinear system in which the spring compressibility is nonlinear, and the system parameters are uncertain. Thus, in the next section, we shall develop a similar P-I controller in a completely nonlinear setting. 4.2 Nonlinear formulation and analysis Since the equivalent spring is strictly monotone increasing, there exists a unique compression d (t so that F d (t = F( d. Each actuator has dynamics of the form: F = F( F( d (1 = x I x p (11 A(ẋ I ẋ I = Q = K q (sgn(u,fu. (12 ė I = F (16 v = K I e I K p F + Ḟd K s (x (17 1 u = K q (sng(u,f A(ẋ I(v + ẋ p (18 where K s (x = F and v represents the desired spring compression rate. Notice that K q (sgn(u,f > and thus Eq.(18 can be solved by first determining sgn(u. Theorem 1 Consider the actuator system (1-(12 together with the PI controller in Eqs.(16-(18. Let the spring force F s ( be a monotone strictly increasing function of the compression, and the spring rate K s ( := df d be bounded, strictly non-zero and globally Lipschitz. Hence, maxk s K s ( mink s >, and K s ( 1 K s ( 2 < L 1 2. If the controller gains satisfy K p >, and K I >, and Ḟ d is bounded, then, the equilibrium point ( F,e I = (, is globally exponentially stable. Proof: Consider the Lyapunov function given by: V = (F(x F( d dx+ 1 d 2 K Ie 2 I + εe I (19 where ε > is to be determined. Notice that 1 2 mink s 2 (F(x F( d dx 1 d 2 maxk s 2 so that V is radially unbounded and positive definite in ( F,e I for 4 Copyright c 24 by ASME

5 ε sufficiently small. Consider first the case when Ḟ d =, V = F +K I e I ė I + ε(ė I +e I (2 = F( K I e I K p F+K I e I F + ε F εk I e 2 I εk p e I F (21 = K p F 2 εk I e 2 I + ε F (22 = ( e I F ( ( εk I εk p /2 eĩ (23 εk p /2 K p ε/k s F By choosing ε > so that both V and the matrix above are positive definite, and using the upper bound for V, then we have V λv for some λ >. This shows that ( F,e I = (, is exponentially stable. In the case when Ḟ d, the extra terms that show up are: V = ( e I F ( εk I εk p /2 εk p /2 K p ε/k s +( F/K s (xḟ d Ḟ d + ε because F = K s (x, Now V = ( e I F ( εk I εk p /2 εk p /2 K p ε/k s ( 1 + ε K s (x 1 e I Ḟ d. K s ( d Therefore, ( eĩ F ( 1 K s (x 1 K s ( d ( eĩ 1 K s (x 1 K s ( d = K s ( d K s (x K s (xk s ( d L d x K s (xk s ( d L K s (xk s ( d L K s (x 2 K s ( d F L minks 3 F V = ( e I F ( εk I εm/2 εm/2 K p ε/k s F ( eĩ F e I Ḟ d.. (24 where m = K p + L Ḟ minks 3 d. Hence there exists a ε >, such that both V and V are positive definite. This shows that ( F,e I = (, is globally exponentially stable. Remark 1 The analysis in section 4.2 tends to be conservative in the estimation of the convergence rate (dependent on the choice of ε. Linear analysis can provide a better way of tuning K p and K I. To wit, suppose that the spring is linear with F = K s, where K s is a constant. The characteristic equation is s 2 + K p K s s+k I K s =. For a desired natural frequency ω n and damping ratio ζ, K I = ω 2 n/ K s ; K p = 2ω n / K s. 4.3 Robust modification Let K B = k q (sgn(u,f/a(ẋ I, then control proposed above is of the form, u nom = 1ˆK B ( K I e I K p F + ẋ p + 1 ˆK s ( d ˆK B ˆḞ d (25 where ˆ denotes the estimates of the uncertain parameters, K B, K s (x, and Ḟ d. u nom can be written as: where be: u nom = u ideal + δ KB (K I e I + ẋ p +δḟd u ideal = K p F 1 1 (K p F + ẋ p + Ḟ d K B K B K s ( d K B δ KB (t = 1ˆK B 1 K B, (26 δḟd = ˆḞ d Ḟd. (27 ˆK B ˆK s K B K s To combat the effect of uncertainties, u can be augmented to u = u nom sgn( F + εe I [ δkb (K I e I + ẋ p + δḟd] (28 }{{} u rob where ε > is given by the proof of the Theorem 1, and δ KB > δ KB (t, δḟd > δḟd. Theorem 2 In the presence of uncertain parameters K B, K s ( d and Ḟ d, if the control law (16-(18 is modified according to (28 and ˆK B >, then ( F,e I = (, is globally exponentially stable. Proof: Following the Proof of Theorem 1, define the Lyapunov function: V = (F(x F( d dx+ 1 d 2 K Ie 2 I + εe I (29 5 Copyright c 24 by ASME

6 where ε > is to be determined. Let K p = K B / ˆK B K p. Then, V = ( e I F ( ( εk I εm/2 eĩ εm/2 ( K p ε/k s F ] +( F + εe I [u rob + δ KB (K I e I + ẋ p +δḟd L where m = K p + Ḟ minks 3 d. The definitions for δḟd and δ KB ensure that the second term is negative. Since there exists ε > such that V and V are positive definite in (e I, F, the exponential convergence property is established. Remark 2 1. The robust modification term which is proportional to e I may not be needed if K p is sufficiently large. This is because the extra term that this term generates is Fe I (K B ˆK B / ˆK B, which can be compensated for if K p is large enough. 2. By making use of the robust modification, and with K p sufficiently large, the implementation of the force control law can be simplified by using K B as a constant, ˆḞ d =, and ε arbitrarily small. 3. Uncertainty in the estimation of ẋ p can be similarly handled. 4.4 Anti-windup As mentioned earlier, the system has a small flow capability, and hence the system is prone to enter into saturation. To ameliorate the adverse effect of saturation on the PI controller, an anti-windup scheme is proposed. Suppose that the desired control is u but the actual control is sat(u where sat( is the saturation function. Let u = u sat(u. Notice from the original Lyapunov analysis (2 for the PI controller that a key identity is that if ė I = F, then F( K I e I +e I K I ė I = where the first term on the LHS is due to I-action of the control. This identity signifies the exchange of Lyapunov energies V between the storage in the integral state 1 2 K Ie 2 I and in the spring in (19. In the case of saturation, V is modified by V = K B u( F + εe I. When saturations are not too significant, we assume that the extra term is reflected in the modification of the I action. For the purpose of developing anti-windup, we can take ε = for convenience (since the exponential proof is valid for arbitrarily small ε. Thus, if V, no modification is needed. When V > (taking ε =, we choose ė I = α(t F for some α(t > so that F( K I e I K B u+e I K I ė I = 5 Passivity Properties In this section, we analyze the passivity property of the system. Passivity is an important property for machines that interact physically with humans and other environment [5] as it provides a certain level of safety and robust coupling stability. By applying the control approach in 4 to the actuators for the reach and pitch degrees of freedom, we produce an actuator force (in the same coordinates as in (1 F act = ρf human + F where F = ( F 1,α(q 2 F 2 T where F 2 denotes the force error in the pitch error coordinates, and F. Thus, the interaction of the environment and the human operator with the human power amplifier is: M(q q+c(q, q q = (F f riction F+(ρ+1F human + F load (3 Consider a positive storage function W = 1 2 qm(q q, then, using the fact that Ṁ(q 2C(q, q is skew symmetric, Ẇ = q T (ρ+1f human + q T F load q T [ F f riction F ] (ρ+1 q T F human + q T F load =: s ρ (F human,f load, q, if the power dissipated by friction is less than that generated by the force error (e.g. when F is less than Coulomb friction, then the human-amplifier will be passive with respect to the scaled supply rate, s ρ (F human,f load, q := (ρ + 1q T F human + q T F load : there exists c such that for all F human ( and F load (, and for all t, t s ρ (F human,f load, qdτ c 2. In contrast to intrinsically passive controllers such as [6, 7] for teleoperators that structurally enforces passivity, the control proposed in this paper is not intrinsically passive. Consequently, in the presence of uncertainties, if the bounds of the uncertainties are not properly estimated, then F cannot be guaranteed. In this case, the passivity property can be destroyed. 6 Simulation In the simulation, we used a constant ˆK B which is 2% different from the K B (F =. The estimated spring constant K s is 1 times smaller than the actual case. The human is modeled as a poorly tuned PD controller that moves the oar up and down according to the desired trajectory q d (t =.1sin(4πt. The oar is subject to a periodic load of F L (t = 1sin(.667πt+1. 6 Copyright c 24 by ASME

7 8 6 Desired and actual force Desired force Actual Force 25 Desired and actual forces during pressing task Force N Unconstrained Pitch Displacement Force N Desired force Actual force Pitch angle rad Figure 4. Simulation results - see text for details 5 Control starts here Pitch displacement during pressing task Figure 4 shows that the controller is capable of achieving very good force tracking. Also shown is the displacement trajectory of the oar. 7 Experimental Results The experiments are performed for the pitch motion only. The reach displacement is fixed using a proportional controller. In addition, the velocity feedforward term (ẋ p in (18 significantly reduced in experiments to maintain stability. This is an issue of the discrete time control. Simulations shows that the system would be unstable unless the sampling time has to be at least 1 times smaller than the applied sampling time (.8s. The force scaling of the exerted human force and the desired actuator force is ρ = 1 (since they act on different sides of the pivot. Two types of experiments were performed. The first is the constrained task in which the operator presses the oar against a hard constraint (a chair. Figure 5 shows that the controller is able to track the desired force well. Second, the oar is unconstrained and moves in the free space. The human executes a series of point to point motions. Fig. 6 shows the case when the oar is not loaded. Notice that when the oar is stationary, the force tracking is accurate. The slight increase in both the scaled human force (i.e. the desired force and the actuator force is due to the change in the pose and the effective moment arms. When the oar is moving, the transient desired force however is not tracked well. This issue is likely related to limitation in sampling time and the inability of the hardware to generate large flow and flow acceleration. Fig. 7 shows the free motion results when the oar is loaded Displacement rad Figure 5. Control starts here Experimental results - constrained pressing task. Top: Desired and actual force; bottom: Pitch motion by a 2lb weight at the end. Notice that both the human force and the actuator force increase from around 15N to 33N. Again the static force is tracked well. Similar to the unloaded case, the static force decreases as the pitch angle increases. Again, the force during the motion transient is also not tracked well. 8 Conclusions A hydraulically powered human amplifier in the form of an oar is presented. A robust PI force controller is shown to be effective in the constrained situation. In the case of the oar being unconstrained, the force controller stabilizes the force at the desired level (as determined by the static load and the force scaling 7 Copyright c 24 by ASME

8 Unloaded free motion desired and actual forces Desired and actual force in a loaded free motion task Desired force Actual force 45 Desired force Actual force Force N 1 Force N Pitch displacement of a unloaded free motion task.35 Pitch displacement of a loaded free motion task.45.4 Control started here Pitch displacement rad Pitch displacement rad Figure 6. Experimental results - Unloaded unconstrained free motion: Top: Desired and actual force; bottom: Pitch motion Figure 7. Experimental results - unconstrained free motion with a 2lb load. Top: Desired and actual force; bottom: Pitch motion ρ when the oar is stationary. However, when the oar is in motion, the dynamic desired force cannot be tracked. Although, the compensation for static load force still gives the user the sense that the machine is lighter, the system does not generate the desired acceleration. Passivity analysis shows that the control system is passive when the model is accurate. However, the controller itself is not intrinsically passive. From a safety point of view, a controller that structurally enforces passivity would be more advantageous. Future work will investigate the force control issue in free space as well as controller structures that enforce passivity robustly. Acknowledgements The team of University of Minnesota undergraduate students who designed and built the machine consists of: Husaini Hashim, Aaron Hicks, Zaki Hussein, Toni LaPlante and Russell Ryan. Dan Hardy and Josh Friell were responsible for the electronics, computer interface, and the final stage of the hydraulics interface. This project is partially supported by the National Fluid Power Association (NFPA through its Small Grant program. In kind donations by Eaton Corporation are also gratefully acknowledged. 8 Copyright c 24 by ASME

9 REFERENCES [1] H. Kazerooni, Human power amplifier technology at the university of california, berkeley, Robotics and Autonomous Systems, vol. 19, no. 2, pp , [2], 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 [3] H. Kazerooni and J. Guo, Human extenders, ASME Journal of Dynamic Systems, Measurement and Control, vol. 115, no. 2B, pp , [4] K. Kosuge, Y. Fujisawa, and T. Fukuda, Control of a man-machine system interacting with the environment, Advanced Robotics, vol. 8, no. 4, pp , [5] P. Y. Li and R. Horowitz, Control of Smart Exercise Machines: Part 1. Problem Formulation and Non-Adaptive Control, IEEE/ASME Transactions on Mechatronics, vol. 2, no. 4, pp , December [6] D. Lee and P. Y. Li, Towards robust passivity: a passive control implementation structure for mechanical teleoperators, in Proceedings of the 11th Symposium on Haptic Interfaces for Virtual Environments and Teleoperators, Los Angeles, CA, March, 23. [7] P. Y. Li and K. Krishnaswamy, Passive bilateral control of a hydraulic actuator using an electrohydrauliuc passive valve, in Proceedings of 21 American Control Conference, June. Arlington VA., Copyright c 24 by ASME