7046
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 12, DECEMBER 2014
Energy-Saving Adaptive Robust Control of a Hydraulic Manipulator Using Five Cartridge Valves With an Accumulator Lu Lu, Member, IEEE, and Bin Yao, Senior Member, IEEE
Abstract—In this paper, a novel energy-saving control strategy is proposed for the accurate motion tracking of a hydraulic manipulator. To achieve independent pressure regulation for each chamber of the cylinder, as well as energy recovery during the back-and-forth movement of the cylinder, a hardware configuration with five low-cost programmable cartridge valves and an accumulator is developed to control the motion of the cylinder. Based on the hardware configuration, a novel multilevel control algorithm is proposed, which consists of the desired motion and pressure tracking control level (level I), the flow distribution level (level II), and the offside pressure profile planning level (level III). The proposed approach is theoretically shown to be able to minimize the energy consumption of the system while maintaining good motion tracking performance. Experimental studies confirm that the proposed strategy can indeed achieve both accurate motion tracking and minimum energy consumption simultaneously. Compared with the previous four-valve scheme and five-valve flow regeneration scheme without the use of accumulator by Liu and Yao, the proposed strategy has much less total energy consumption and equally good tracking performance. Index Terms—Adaptive robust control (ARC), electrohydraulic systems, energy saving, motion control, optimal control.
I. I NTRODUCTION
H
YDRAULIC actuators have been widely used in heavy machinery such as excavators, cranes, and wheel loaders. The widespread applications of hydraulic actuators have drawn the attention of the engineers to apply the state-of-art system design, control, and fault detection techniques [2]–[10]. The control of electrohydraulic actuator systems is very challenging not only because of the high-order and highly nonlinear nature of the plant but also because the hydraulic energy consumption is a big issue that needs to be taken into account for the controller design. Typically, for the tracking control of a manipulator driven by hydraulic actuators, it is required that the desired joint angle of the manipulator is tracked as accurately Manuscript received May 15, 2013; revised October 21, 2013; accepted February 13, 2014. Date of publication March 27, 2014; date of current version September 12, 2014. This work was supported in part by the National Basic Research and Development Program of China under 973 Program Grant 2013CB035400 and in part by the Science Fund for Creative Research Groups of the National Natural Science Foundation of China under Grant 51221004. The authors are with the State Key Laboratory of Fluid Power Transmission and Control, Zhejiang University, Hangzhou 310027, China, and also with the School of Mechanical Engineering, Purdue University, West Lafayette, IN 47907 USA (e-mail:
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TIE.2014.2314054
as possible while the total energy consumption is as small as possible. To meet these two targets simultaneously, both the hardware configuration for the control of the system and the control law design need to be properly done. Traditionally, for the valve-controlled hydraulic systems, a four-way directional valve or servo valve is often used to control the position of the hydraulic cylinder [11], [12]. By regulating the spool position of the valve, accurate tracking of the desired cylinder motion may be achieved. However, such a scheme does not allow independent regulation of the pressures of both chambers of the cylinder. Once the desired cylinder motion is fixed, the required flow rates to the two chambers are then uniquely determined. Thus, the total energy consumption is also fixed and cannot be reduced through the control law design. In order to save energy as well as accurately tracking the desired motion trajectory, independent pressure regulation of both chambers of the cylinder has to be made. On the hardware side, this can be achieved by using separate valves connecting to both chambers of the cylinder and metering the flows passing through each of them. A detailed review of such types of independent valve metering configuration can be found in [13]. Typically, most of these current schemes achieve energy saving by recycling the flow from one chamber of the cylinder to another when the cylinder is moving, which is referred to as “flow regeneration” technique [14]–[18]. Specifically, in our previous work [1], a five-valve configuration is used to control the boom motion of the hydraulic manipulator in which the fifth valve connects the two chambers of the cylinder and permits the flow regeneration from the head-end chamber to the rodend chamber when the piston is moving downward. An adaptive robust control (ARC) law with a novel flow distribution strategy is then designed to achieve accurate motion tracking of the manipulator and minimum hydraulic energy consumption simultaneously. In this paper, a more advanced hardware configuration which consists of five cartridge valves and one diaphragm-type accumulator is studied. By storing the flow into the diaphragm-type accumulator when the piston of the cylinder moves downward and using such amount of flow to pump the piston up, the socalled “energy recovery” can be achieved. Based on this hardware configuration, a three-level control strategy is designed. In the desired motion and pressure tracking level (level I), an adaptive robust controller is synthesized to generate the desired flow rates for the two chambers of the cylinder so that the position of the cylinder tracks the desired trajectory as accurately as
0278-0046 © 2014 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.
LU AND YAO: ARC OF HYDRAULIC MANIPULATOR USING FIVE CARTRIDGE VALVES WITH AN ACCUMULATOR
possible, and the offside pressure of the cylinder also follows the desired pressure profile to be generated in level III accurately. In the flow distribution level (level II), an energyoptimum flow distribution algorithm is designed to generate the desired flow rates passing through each of the five programmable valves knowing Q1m and Q2m and the pressures of the two chambers and the accumulator. In the offside pressure profile planning level (level III), the desired offside pressure of the cylinder is generated so that the total energy consumption is minimized. Various experiments are conducted to compare performance of the proposed five-valve-with-accumulator approach with those of the five-valve flow regeneration approach in [1] and the traditional four-valve approach. The results demonstrate that the proposed approach can indeed achieve accurate motion tracking of the manipulator while the total energy consumption is much smaller than the other two approaches. Compared to the approach in [1], the energy saving of the proposed approach is 40% when a constant-pressure pump is used and 30% when a load-sensing (LS) pump is used. II. M OTIVATION AND H ARDWARE C ONFIGURATION D ESIGN The energy consumption of the hydraulic system during any time period [t0 , t1 ] is calculated by t1 (1) E = ps (t)Qs (t)dt t0
where ps (t) is the supply pressure of the pump and Qs (t) is the flow rate into the system. From the aforementioned formula, it is clear that, in order to save energy, one can either reduce the supply pressure of the pump or minimize the flow rate into the system. When the supply pressure ps (t) is high but the chamber pressures are low, a lot of energy will be wasted on the throttle losses of the valves that connect the chambers to the pump. To reduce this amount of wasted energy, an LS pump may be used to reduce the supply pressure ps (t) [19]. To provide flow rate into the system to push the cylinder, the supply pressure needs to be higher than the pressure of the chamber into which the flow is pumped. The LS pump works by setting the supply pressure of the pump to be only a little bit higher than the maximum chamber pressure through the use of certain pressure feedback mechanisms. If the desired pressure of one chamber of the cylinder is set as low as possible, the maximum chamber pressure is also reduced. Then, the LS pump only needs to use a very low pressure to supply the whole system. The throttle losses on the valves are thus minimized, and the total energy consumption is very low. Since the LS pump has complicated structure and high cost, a constant-pressure pump is sometimes preferred as supply source to the hydraulic systems. In this case, ps (t) is a constant value. The only way to save energy is to reduce the total flow rate supplied to the system. In [1], a cartridge valve is placed on the flow path directly connecting the two chambers of the cylinder, allowing precision motion control with flow regeneration during the extension of the rod. In particular,
7047
Fig. 1. Hardware configuration with five valves and one accumulator. A1 is the ram area of the head-end chamber, and A2 is the ram area of the rod-end chamber.
when the piston of the cylinder is moving downward (the part of the rod inside the cylinder is extending) with constant velocity, the desired rod-end chamber pressure is set to zero. The headend chamber pressure will be controlled to a positive value to balance the gravity of the load. In this situation, the head-end chamber pressure is greater than the rod-end chamber pressure, allowing flow regeneration from the head-end chamber to the rod-end chamber, which saves energy since the pump does not need to supply flow into both chambers. Experimental results in [1] show that, if the desired piston movement is back and forth, the total energy consumption is approximately proportional to the “head-end liquid volume,” which equals the head-end piston area multiplied by the maximum difference of the piston position for the desired back-and-forth movement. In this paper, a new hardware configuration is studied, which uses five valves with an accumulator to control the cylinder, as shown in Fig. 1. The accumulator is directly connected to the head-end chamber because the head-end piston area is normally much larger than the rod-end piston area for a singlerod cylinder, which means that the head-end chamber definitely needs much more flow supply than the rod-end chamber. With this set of hardware, when the piston of the cylinder is moving downward, the pressure of the head-end chamber is set to be higher than the accumulator pressure so that the accumulator can be fully charged. When the piston is moving upward, the flow stored in the accumulator can be used to push the piston if the desired head-end pressure is set to be lower than the accumulator pressure. It is seen that the total energy consumption of the proposed scheme for a back-and-forth movement is roughly proportional to the “rod-end liquid volume,” which would be much lower than the energy consumption in [1]. After presenting the hardware configuration, the control problem will be formulated in the next section, and a novel control law will be designed to minimize the energy consumption of the system while keeping the tracking error of the manipulator joint angle as small as possible.
7048
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 12, DECEMBER 2014
III. C ONTROL P ROBLEM F ORMULATION The overall dynamics of the boom motion of the cylinder (including both mechanical and hydraulic dynamics) can be described by [1]
∂xL (p1 A1 − p2 A2 ) − Gc (q) − mL glg (q) Jc + mL le2 q¨ = ∂q − Df q˙ + T (t, q, q) ˙
˜ vi , Qvi = Qvim − Q
i = 1, 2, . . . , 5.
(5)
Defining Q1m = Qv1m − Qv2m − Qv3m , Q2m = −Qv4m + ˜1 = Q ˜ v1 − Q ˜ v2 − Q ˜ v3 , and Q ˜ 2 = −Q ˜ v4 + Q ˜ v5 , the Qv5m , Q flow-pressure dynamic equations of the two chambers [the second and third equations of (2)] can be rewritten as ∂xL V1 (xL ) ˜1 q˙ + Q1m − Q p˙ 1 = −A1 βe ∂q
∂xL V1 (xL ) q˙ + Q1 p˙ 1 = −A1 βe ∂q V2 (xL ) ∂xL q˙ − Q2 p˙ 2 = A2 βe ∂q
(2)
where q is the joint angle of the boom cylinder and xL is the displacement of piston of the boom cylinder. Jc is the moment of inertia of the boom without payload, mL is the mass of the unknown payload, and p1 and p2 are the pressures of the head end and rod end of the cylinder, respectively. Q1 and Q2 are the supply and return flows for the two chambers. A1 and A2 are the head- and rod-end ram areas of the cylinder. V1 (xL ) = Vh1 + A1 xL and V2 (xL ) = Vh2 − A2 xL are the total volumes of the two chambers, including the volume of the connecting hoses. Df is the equivalent viscous friction coefficient of the boom motion. Gc (q) is the gravitational load of the boom without payload, and mL glg (q) and mL le2 are terms associated with the unknown payload mL . The expressions of Gc (q) and ˙ represents the lumped lg (q) can be found in [11]. T (t, q, q) disturbances and unmodeled nonlinearities of the system such as the frictions. With the flow circuit shown in Fig. 1 to control the boom cylinder, Q1 and Q2 are given by Q1 = Qv1 − Qv2 − Qv3 Q2 = − Qv4 + Qv5
(3)
where Qv1 , Qv2 , Qv3 , Qv4 , and Qv5 are the flow rates through the five valves with the directions shown in Fig. 1. Ignoring the valve dynamics, for a particular valve, the relationship between the flow rate Qvi , the pressure drop across the orifice of the valve ΔPvi , and the control voltage uvi is described by Qvi = fxi (ΔPvi , uvi ),
Qvi as the discrepancy between Qvim and Qvi . Qvi can be expressed as
i = 1, 2, . . . , 5
(4)
where ΔPvi is the pressure drop across the ith valve, i.e., ΔPv1 = ps − p1 , ΔPv2 = p1 − pt , ΔPv3 = p1 − pac , ΔPv4 = ps − p2 , and ΔPv5 = p2 − pt . In the above, ps is the pressure of the supply pump, pt is the pressure of the tank, and pac is the pressure of the accumulator. To derive the control command uvi from the flow rate Qvi , the flow mapping function fxi needs to be obtained in advance [18]. However, there are always discrepancies between the actual flow mapping function and the identified one. Denote fxim as the identified flow mapping function for the ith valve, Qvim = fxim (ΔPvi , uvi ) as the “desired” flow rate ˜ vi = Qvim − obtained from the identified function fxim , and Q
V2 (xL ) ∂xL ˜2. q˙ − Q2m + Q p˙ 2 = A2 βe ∂q
(6)
The accumulator used in our system is a diaphragm-type accumulator. The gas inside the accumulator can store and release energy and is separated from the liquid by an elastic diaphragm. When the liquid flows into the accumulator, the gas is compressed and stores energy. When the liquid flows out of the accumulator, the gas expands and releases energy. The dynamic equation of the accumulator is given by V˙ f = Qv3
(7)
where Vf ≥ 0 is the volume of the liquid inside the accumulator. Since the gas within the elastic diaphragm undergoes adiabatic process when expanding or being compressed, the relationship between pac and Vf is given by pac =
ppr 1−
Vf Vtot
k ,
for Vf ≥ 0
(8)
in which Vtot is the total volume of the accumulator and ppr is the “preset pressure” of the gas inside the accumulator when Vf = 0. The desired trajectory to be tracked by the joint angle of the manipulator is denoted as qd (t), t ∈ [0, tf ]. For the purpose of perfect tracking, qd (t) is assumed to be continuously differentiable up to the third order. The control objective is to generate the valve control commands uvi for i = 1, 2, 3, 4, 5 such that the joint angle q follows the desired trajectory qd (t) as accurately t as possible, while the energy usage E = ps 0 f Qs (t)dt = t ps 0 f (Qv1 (t) + Qv4 (t))dt is as small as possible. IV. C ONTROLLER D ESIGN A. Overall Control Structure Defining the load force as pL = p1 A1 − p2 A2 , it is seen from (2) and (6) that, to achieve accurate motion tracking, the desired value of the load force (denoted as pLd ) should be chosen at first such that q accurately tracks qd (t). Then, the desired flow rates into the two chambers Q1m and Q2m are chosen such that the actual load force tracks the desired one accurately. This scheme is known as the backstepping control and has been widely applied to the control of hydraulic systems. For the system proposed in this paper and some of the previous
LU AND YAO: ARC OF HYDRAULIC MANIPULATOR USING FIVE CARTRIDGE VALVES WITH AN ACCUMULATOR
7049
1) Offside Pressure Controller Design: The accumulator is directly connected to the head-end chamber. To charge (or discharge) the accumulator, the head-end pressure must be higher (or lower) than the accumulator pressure, which means that the head-end pressure needs to be controlled during the cycles of movement. Thus, p1 will be the offside pressure to be actively controlled. Define ep1 = p1 − p1d (t), where p1d (t) is the desired offside pressure for p1 . From (6), the dynamics of ep1 can be written as e˙ p1 =
Fig. 2.
Proposed control structure.
publications, Q1m and Q2m can be independently specified. So, there is only one variable pL to be regulated but two inputs Q1m and Q2m . To make full use of this extra degree of freedom, it is often required that the pressure of one chamber tracks a prespecified offside pressure profile which is designed such that certain optimal performance (e.g., minimum energy consumption) is achieved by the closed-loop system. After this step, Q1m and Q2m can be uniquely determined. To obtain the control commands for the five valves, the desired flow rates through each of the valves, i.e., Qvim , i = 1, 2, 3, 4, 5, need to be determined based on Q1m and Q2m , which again provides us an extra design freedom. Under the constraints Q1m = Qv1m − Qv2m − Qv3m and Q2m = −Qv4m + Qv5m , a suitable flow distribution scheme should be chosen to minimize the total energy consumption of the system. Based on the aforementioned analysis, the following triplelevel control structure as shown in Fig. 2 is proposed. In level I, an adaptive robust controller is designed to generate Q1m and Q2m so that: 1) q tracks qd (t) as accurately as possible; and 2) p1 tracks the desired offside pressure profile p1d (t) as accurately as possible. In level II, based on Q1m , Q2m and p1 , p2 , a flow distribution scheme is developed to generate the desired flow rates to all five valves Qvim , i = 1, 2, 3, 4, 5, and, subsequently, the control commands uvi , i = 1, 2, 3, 4, 5, using the inverse flow mapping functions. The proposed flow distribution scheme is proved to be able to minimize the total energy consumption with the given profiles of Q1m , Q2m , p1 , and p2 . In level III, the desired offside pressure profile p1d (t) is planned so that the total energy consumption is minimized with the given level I controller and level II optimal flow distribution law. B. Level I—Desired Motion and Pressure Tracking Controller The ARC [20], [21] has been shown to be an effective control strategy in dealing with the tracking control problems of physical systems with both parametric uncertainties and uncertain nonlinearities through various applications [22]–[26]. As such, it will be used in level I to generate the required desired flow rates Q1m and Q2m such that both joint angle q and offside pressure p1 or p2 track their desired trajectory accurately as follows.
θQ βe A1 βe ∂xL q˙ − Q1m − − p˙ 1d (t) − ΔQ1 V1 V1 ∂q V1
(9)
˜ 1n and Q ˜ 1n is the nominal value of Q ˜1. where θQ = βe Q ˜ ˜ ΔQ1 = βe (Q1 − Q1n )/V1 ). Define θ p = [βe , θQ ]T as the vector of uncertain parameters for the ep1 dynamics. θ p is estimated online using the following discontinuous projection law [21]: ˆ˙ p = Proj[−Γp ϕp ep1 ] θ
(10)
where ϕp = [(Q1ma /V1 ) − (A1 /V1 )(∂xL /∂q)q, ˙ −(1/V1 )]T in which Q1ma = A1 (∂xL /∂q)q˙ + (θˆQ /βˆe ) − (p1d (t)V1 /βˆe ). Γp > 0 is a positive definite gain matrix. The discontinuous projection mapping Proj is defined in element as ⎧ ⎨ 0, if θˆpi = θp max i and •i > 0 Proji (•i ) = 0, (11) if θˆpi = θp min i and •i < 0 ⎩ otherwise •i , where θp max i and θp min i denote the upper and lower bounds for the ith vector of θ p . The desired control input Q1m is chosen to be Q1m = Q1ma + Q1ms Q1ms = −
kp1 V1 ep1 + Q1msn θp min 1
(12)
where Q1ma is the model compensation term defined previously and Q1ms is the robust feedback term. kp1 > 0 is the linear feedback gain, and Q1msn is a nonlinear feedback term satisfying the following conditions: θβ ˜ p − ΔQ1 ≤ εp Q1msn − ϕT θ (i) ep1 p V1 (ii) Q1msn ≤ 0
(13)
ˆ p − θ p is the parameter estimation error. ˜p = θ where θ 2) Motion Tracking Controller Design: After using Q1m to actively control the pressure p1 , the next step is to synthesize a control law for Q2m so that the joint angle q tracks the desired trajectory qd (t) accurately. The ARC backstepping design is used to achieve this goal. First, define the vector of unknown parameters as θ q = [θq1 , . . . , θq6 ]T , where θq1 = 1/(1 + (mL le2 /Jc )), θq2 = Df /(Jc + mL le2 ), θq3 = Tn /(Jc + mL le2 ), θq4 = βe , θq5 = ˜ 1n , and θq6 = βe Q ˜ 2n , where Tn , Q ˜ 1n , and Q ˜ 2n are the βe Q
7050
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 12, DECEMBER 2014
˜ 1 , and Q ˜ 2 , respectively. The dynamics nominal values of T , Q (2) can be rewritten as
θq1 ∂xL glg θq1 (p1 A1 − p2 A2 ) − Gc + 2 − qθ ˙ q2 + θq3 q¨ = Jc ∂q le glg − 2 +Δ le A1 ∂xL θq4 θq5 qθ ˙ q4 + Q1m − − ΔQ1 p˙ 1 = − V1 ∂q V1 V1 A2 ∂xL θq4 θq6 qθ ˙ q4 − p˙ 2 = Q2m + + ΔQ2 (14) V2 ∂q V2 V2 ˜1 −Q ˜ 1n )/V1 , where Δ = 1/(Jc +mL le2 )(T −Tn ), ΔQ1 = βe (Q ˜ ˜ and ΔQ2 = βe (Q2 − Q2n )/V2 are the time varying portions of the lumped uncertainties. Define z1 = q − qd (t) as the tracking error. Also, define a switching-function-like quantity z2 = z˙1 + k1 z1 = q˙ − q˙r , where q˙r = q˙d − k1 z1 . The dynamics of z2 can be calculated as
1 ∂xL glg glg pL −Gc + 2 − 2 − q¨r − θq2 q+θ ˙ q3 +Δ z˙2 = θq1 Jc ∂q le le (15) where pL = p1 A1 − p2 A2 is the load force. Treating pL as the control input to the aforementioned dynamics of z2 , we can design a virtual control law pLd to minimize z2 . Specifically, let pLd = pLda + pLds ∂q Jc gl θˆq1 glg ˆ g pLda = ˙ θˆq3 + 2 + q¨r Gc + − 2 + θq2 q− ∂xL le le θˆq1 Jc ∂q k2 z2 + pLdsn (16) pLds = − θ1 min ∂xL ˆ q = [θˆq1 , . . . , θˆq6 ]T is the vector of the estimated pawhere θ rameters, which will be synthesized later. pLda is the model compensation term, and pLds is the robust feedback term. k2 > 0 is the linear feedback gain, and pLdsn is a nonlinear feedback term satisfying the following conditions: θq1 ∂xL T˜ pLdsn − ϕq θ q + Δ ≤ ε2 (i) z2 Jc ∂q (ii) z2
∂xL PLdsn ≤ 0 ∂q
(17)
˜ = [θ˜q1 , . . . , θ˜q6 ]T , where ε2 > 0 is a design parameter. θ and θ˜qi = θˆqi − θqi is the parameter estimation error. ϕq = [ϕq1 , . . . , ϕq6 ]T is the vector of regressors with ϕq1 = ˙ ϕq3 = 1, (((∂xL /∂q)PLda − Gc )/Jc ) + (glg /le2 ), ϕq2 = −q, and ϕq4 = ϕq5 = ϕq6 = 0. Let z3 = pL − pLd be the discrepancy between the actual pL and the desired one pLd . The derivative of z3 is calculated as z˙3 = p˙ L − p˙ Ld 2 A1 A22 ∂xL A2 θq4 Q2m A1 θq5 A2 θq6 qθ ˙ q4 + =− + − − V1 V2 ∂q V2 V1 V2 − p˙ Ldc +
A1 θq4 Q1m −A1 ΔQ1 −A2 ΔQ2 − p˙ Ldu V1
(18)
where p˙ Ldc and p˙ Ldu represent the calculable and incalculable parts of the time derivative of pLd [1]. Treating Q2m as the control input to the z3 dynamics, the following control law is designed to minimize z3 : Q2m = Q2ma + Q2ms V2 θˆ ∂xL z2 Q2ma = − Jc ∂q A2 θˆq4 2 A1 A1 A2 ∂xL q˙ − + θˆq4 + 2 Q1m V1 V2 ∂q V1 A1 ˆ A2 + θˆq5 + θq6 + p˙ Ldc V1 V2 V 2 k 3 z3 + Q2msn (19) Q2ms = − A2 θq4min where Q2m is the model compensation term and Q2ms is the robust feedback term. k3 > 0 is the linear feedback gain, and Q2msn is a nonlinear feedback term satisfying the following conditions: A2 ∂pLd T ˜ Δ−A1 ΔQ1 −A2 ΔQ2 z3 θq4 Q2msn − θ q φq − V2 ∂ q˙ ≤ ε3 z3 Q2msn ≤ 0 (20) is a design constant. φq = where ε3 > 0 T is the vector of regressors with [φq1 , . . . , φq6 ] ˙ φq1 = (z2 /Jc )(∂xL /∂q)−(∂pLd /∂ q)[(∂x L /∂q)(pL /Jc ) − ˙ q, ˙ φq3 = −(∂pLd /∂ q), ˙ (Gc /Jc ) + (glg /le2 )], φq2 = (∂pLd /∂ q) φq4 = −((A21 /V1 ) + (A22 /V2 ))(∂xL /∂q)q˙ + (A2 /V2 )Q1m + (A1 /V1 )Q2ma , φq5 = −(A1 /V1 ), and φq6 = −(A2 /V2 ). ˆ p , the vector of parameter estimates θ ˆ q is updated Like θ online by the following discontinuous projection law: ˆ˙ q = Proj −Γq (ϕq z2 + φq z3 ) (21) θ where Γq > 0 is a positive definite gain matrix. The discontinuous projection mapping Proj is defined in the same way as in (11). 3) Theoretical Results: The same as in [1], the following theoretical results can be obtained. Theorem 1: With the control law (12), the tracking error of the offside pressure is bounded by εp (1 − e−2kp1 t ). (22) e2p1 (t) ≤ e2p1 (0)e−2kp1 t + kp1 Furthermore, if, after a finite time, ΔQ1 = 0, asymptotic pressure tracking is achieved, i.e., ep1 → 0 as t → ∞. Theorem 2: With the control law (19), the closed-loop system is stable, and the transient performance and final tracking accuracy is quantified by εq (1 − e−2λt ) (23) V3 (t) ≤ V3 (0)e−2λt + 2λ where V3 (t) = (1/2)(k2 z22 + k3 z32 ), λ = min{k2 , k3 }, and εq = ε2 + ε3 . Furthermore, if, after a finite time, Δ = 0, ΔQ1 = 0, and ΔQ2 = 0, asymptotic motion tracking is achieved, i.e., z1 → 0 as t → ∞.
LU AND YAO: ARC OF HYDRAULIC MANIPULATOR USING FIVE CARTRIDGE VALVES WITH AN ACCUMULATOR
C. Level II—Flow Distribution Law In this level, a minimum-energy-consumption flow distribution law will be synthesized to generate the desired flow rates Qvim , i = 1, 2, 3, 4, 5, for the five programmable cartridge valves satisfying Q1m = Qv1m − Qv2m − Qv3m Q2m = −Qv4m + Qv5m
(24)
in which Q1m and Q2m are given in the level I controller designs. It is seen that, for a particular set of initial states of the system and particular values of the uncertainty terms ˜ vi (t), i = 1, 2, 3, 4, 5, during the time period t ∈ T (t) and Q [0, tf ], the values of p1 (t), p2 (t) and Q1m (t), Q2m (t) for all t ∈ [0, tf ] are uniquely determined. Thus, noting E ≈ t Δ Em = ps 0 f (Qv1m (t) + Qv4m (t))dt when neglecting valve ˜ v4 , minimizing the ˜ v1 and Q flow mapping modeling errors Q total energy consumption E during the time period [0, tf ] is the same as the optimization problem of minimizing Em with Qvim , i = 1, 2, 3, 4, 5, as the optimization variables subject to the constraints (24). This leads to the following constrained optimal control problem: ⎛ min
Qvim (t), i=1,2,3,4,5, t∈[0,tf ]
ps⎝
tf
⎞ (Qv1m (t) + Qv4m (t)) dt⎠
0
subject to Q1m (t) = Qv1m (t) − Qv2m (t) − Qv3m (t) Q2m (t) = − Qv4m (t) + Qv5m (t) Qv1m (t), Qv2m (t), Qv4m (t), Qv5m (t) ≥ 0 0 ≤ Qv3m (t) ≤ fx3m (p1 (t) − pac (t), uv3 max ) , if p1 (t) − pac (t) ≥ 0 − fx3m (pac (t) − p1 (t), uv3 max ) ≤ Qv3m (t) ≤ 0, if p1 (t) − pac (t) < 0 ppr pac = V (t) k 1 − Vftot V˙ f (t) = Qv3m (t), with Vf (0) = Vf 0 Vf (t) ≥ 0 ∀t ∈ [0, tf ].
(25)
In (25), Qvim (t), i = 1, 2, 3, 4, 5, t ∈ [0, tf ], denotes the optimization variables to be solved for. All other notations with prime signs are intermediate variables depending upon the values of Qvim (t). Since p1 (t), p2 (t), and Q1m (t), Q2m (t) are known functions from the level I design, the dynamic constraints for the aforementioned optimal control problem come only from the accumulator dynamics in which Vf and pac change in response to the variations of Qvim (t), i = 1, 2, 3, 4, 5, with Vf 0 being the initial liquid volume in the accumulator. fx3m (p1 (t) − pac (t), uv3 max s) and fx3m (pac (t) − p1 (t), uv3 max ) are the maximal flow rates that can pass through valve #3 when the pressure drop across the orifice is |p1 (t) − pac (t)|, in which uv3 max is the maximum input voltage to the valve. Theorem 3: If the fx3m (•, uv3 max ) is a nondecreasing function of •, then the optimal solution variables to the problem
7051
(25) at any time t, denoted as Q∗vim (t), i = 1, 2, 3, 4, 5, can be expressed as follows: If Q1m (t) ≥ 0 ⎧ ⎧ if p∗ (t) ≤ p1 (t); otherwise ⎨ 0, ⎪ ⎪ ⎪ Q∗ (t) = − min {f ac (p∗ (t) − p (t), u ⎪ ⎪ x3m 1 v3 max ) ac v3m ⎪ ⎩ ⎪ ⎪ Q (t)} ⎪ 1m ⎪ ⎪ ⎪ ⎨ Q∗v1m (t) = Q∗v3m (t) + Q1m (t) 0 Q∗v2m (t) = ⎪ ⎪ −Q2m (t), if Q2m (t) ≤ 0 ⎪ ∗ ⎪ Qv4m (t) = ⎪ ⎪ 0, else ⎪ ⎪ ⎪ ⎪ ⎪ Q (t), if Q2m (t) ≥ 0 2m ⎪ ⎩ Q∗v5m (t) = 0, else. If Q1m (t) < 0 : ⎧ ⎧ if p∗ac (t) ≥ p1 (t); otherwise ⎨ 0, ⎪ ⎪ ⎪ Q∗ (t) = min {f ∗ ⎪ ⎪ x3m (p1 (t) − pac (t), uv3 max ) ⎪ ⎩ ⎪ v3m ⎪ −Q1m (t)} ⎪ ⎪ ⎪ ∗ ⎪ ⎨ Qv1m (t) = 0 ∗ Q∗v2m (t) = −Q v3m (t) − Q1m (t) ⎪ ⎪ −Q2m (t), if Q2m (t) ≤ 0 ⎪ ⎪ Q∗v4m (t) = ⎪ ⎪ 0, else ⎪ ⎪ ⎪ ⎪ ⎪ Q (t), if Q2m (t) ≥ 0 2m ⎪ ⎩ Q∗v5m (t) = 0, else (26) where p∗ac (t) is the pressure of the accumulator at time t after applying the optimal solution Q∗vim (t ), i = 1, 2, 3, 4, 5, to the system for t ∈ [0, tf ]. The aforementioned optimal flow distribution can be explained as follows: For the head-end flow, if, at any time t, Q1m (t) ≥ 0, then the accumulator supplies either the total flow rate of Q1m or as much as possible given the pressure drop across valve #3. The rest of the flow comes from the pump through valve #1. If Q1m (t) < 0, then the accumulator is charged either with the total flow rate Q1m or as much as possible given the pressure drop across valve #3. The rest of the flow goes to the tank through valve #2. For the rod-end flow, if Q2m (t) ≥ 0, then all flow goes to the tank through valve #5. If Q2m (t) < 0, then all flow is supplied from the source through valve #4. One good property about the aforementioned energyoptimum flow distribution strategy is that the flow distribution at certain time instance depends neither on the future values of p1 , p2 , Q1m , and Q2m nor on the initial pressure (or liquid volume) of the accumulator, which makes it very convenient and straightforward to be implemented online. The proof of Theorem 3 is omitted due to page limit. However, it can be easily worked out by using the Hamilton–Jacobi–Bellman equation for optimality. D. Level III—Planning of the Desired Offside Pressure Profile After designing the desired flow rates Q1m and Q2m in level I and scheduling an energy efficient flow distribution law for them in level II, the only freedom left for the overall controller deign is to generate the desired offside pressure profile p1d (t).
7052
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 12, DECEMBER 2014
This extra freedom should be used to minimize the total energy consumption of the closed-loop system. In [1], a mode switching technique was developed to determine which side should be the offside and what value the desired pressure should be based on the calculated load force and the desired velocity of the cylinder. However, this mode switching technique has some drawbacks. 1) The overall energy consumption is not minimized. 2) Jumping from one mode to another results in discontinuity of the desired pressures for both sides of the cylinder, which could cause problems such as control input chattering or oscillation of the tracking error if the bandwidths of the cartridge valves are not high enough. To avoid the discontinuity problem of the desired pressures, in this paper, p1 will be fixed as the offside pressure to be actively controlled, and a continuous p1d will be generated to minimize the overall energy consumption. Intuitively, in order to make full use of the flow volume stored in the accumulator when the arm is moving up (q˙d > 0), the desired head-end pressure should be low enough so that the flow can go from the accumulator to the head-end chamber. In order to fully charge the accumulator when the arm is moving down (q˙d < 0), the desired head-end pressure should be high enough so that the flow can go from the head-end chamber to the accumulator. Furthermore, when setting the desired pressure p1d ∈ [pt , ps ], the resulting p2 should also be within the range [pt , ps ]. Since the desired load force pLda is calculable and only depends on the desired motion of the cylinder, assuming sufficiently small tracking error, one can use the approximate relationship p1 A1 − p2 A2 ≈ pLda to express p2 in terms of the desired offside pressure p1d and the desired load force pLda and set up an inequality constraint on p1d . Based on the aforementioned analysis, the following simple algorithm, which can be implemented online, is proposed to generate the desired offside pressure profile p1d (t): ⎧ pt A2 +pLda (t) ps A2 +pLda (t) ⎪ p (t), if p (t) ∈ , ⎪ 1d A1 A1 ⎨ 1d p1d (t) = pt A2 +pLda (t) , if p1d (t) < pt A2 +pLda (t) A1 A1 ⎪ ⎪ ⎩ ps A2 +pLda (t) , else A1 (27) where p1d (t) is defined as p1d (t) = pac (t) −
q˙d (t) pdiff q˙d max
(28)
in which pdiff > 0 is the maximum pressure difference between the desired offside pressure and the accumulator pressure and q˙d max is the maximum desired speed of the joint angle during the whole cycle of movement. V. E XPERIMENTAL R ESULTS In this section, the proposed control strategy will be applied to our experimental test-bed setup, in which the arm is the same as described in [1]. This 3-DOF robot manipulator system uses a constant-pressure pump as the flow supply source, with the supply pressure being 6.9 MPa. The tank pressure is about 0 Pa. The geometric parameters of the arms and cylinders and the inertias of the arms are given in [1]. The boom motion of the
Fig. 3.
Desired angular position, velocity, and acceleration for the joint angle.
Fig. 4.
Tracking errors for C1–C3.
robot manipulator can be controlled either by the five-valvewith-accumulator scheme as proposed in this paper or by fivevalve flow regeneration scheme as done in [1]. In the following, three algorithms will be compared: 1) C1: ARC with four-valve flow distribution scheme (only valves #1, #2 and #4, #5 are used); 2) C2: energy-saving ARC with five-valve flow regeneration scheme in [1]; 3) C3: the proposed three-level energy-saving ARC algorithm with five valves and one accumulator. The desired trajectory qd (t) used is a point-to-point S curve [27] from 0 to 0.4 rad, with a maximum velocity of 0.16 rad/s, a maximum acceleration of 0.4 rad/s2 , and a maximum jerk of 1 rad/s3 , as shown in Fig. 3. The corresponding controller gains for the offside pressure regulators and motion tracking controllers are selected to be the same for the three algorithms. The accumulator preset pressure ppr is set to be 1.8 MPa for C3. After the experimental results are obtained, the tracking errors, pressures of the two chambers, the power usages, and the accumulator pressure for C3 are plotted and shown in Figs. 4–7. From Fig. 4, it is seen that the tracking error of C3 when the motion starts and stops is a bit more undershoot. This is because
LU AND YAO: ARC OF HYDRAULIC MANIPULATOR USING FIVE CARTRIDGE VALVES WITH AN ACCUMULATOR
Fig. 5.
Cylinder pressures for C1–C3.
Fig. 6.
Power usages for C1–C3.
valve #3 in our system (the one connecting the accumulator to the head-end chamber) has much lower bandwidth than the other four valves and the flow regeneration valve of C2. However, in general, the tracking errors of the three controllers are at the same level. However, Fig. 6 shows that the power consumption profiles of C1–C3 during each cycle of movement are totally different. For C1, since neither the flow regeneration nor the accumulator is employed, the power is consumed for both upward and downward movements of the manipulator. For C2, due to the use of flow regeneration, the power is only consumed during the upward movement of the manipulator. For C3, the accumulator is connected to the head-end chamber and is charged and discharged alternately for the downward and upward movements of the manipulator, which can be seen from the pressure change of the accumulator in Fig. 5. As a result, the power is only consumed during the downward movement of the manipulator. The total energy consumptions for C1–C3 are calculated as 1901.8, 1242.3, and 758.76 J, respectively. Due to the big area difference between the headend piston area and rod-end piston area, the total energy consumption of C3 during one cycle of movement is about 40% less than C2 and 60% less than C1.
7053
Fig. 7. Accumulator pressure of C3.
Fig. 8. Power usages of C1–C3 with LS pump.
In some hydraulic applications, an LS pump may be used instead of the constant-pressure pump to achieve further energy saving. Although our experimental setup does not have an LS pump, a simulation study is performed to test the energy consumption characteristics of C1–C3. To mimic the LS effect of the pump, the supply pressure is modeled as p˙ s + ωLS ps = ωLS (max(p1 , p2 ) + ΔpLS ) − ωLS ps
(29)
where ΔpLS is the difference between the maximum chamber pressure and the pump pressure. ΔpLS is set to be 1 MPa to allow sufficient flow to follow the trajectory. ωLS represents how fast the pump pressure changes in response to the change in maximum chamber pressure and is set as 100 rad/s. The power usages for C1–C3 are plotted in Fig. 8. It can be seen that, during the downward movement, the power consumption of C3 is bigger than that of C1. This is due to the fact that, in order to fully charge the accumulator, the maximum chamber pressure of C3 is greater than that of C1, resulting in a little bit higher ps and, thus, higher power consumption. The total energy consumptions for C1–C3 are calculated as 509.47, 355.12, and 249.05 J, respectively. It is seen that the energy consumption of
7054
IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 12, DECEMBER 2014
C3 is still much less than the energy consumptions of C1 and C2 (30% less than C2 and 50% less than C1). VI. C ONCLUSION In this paper, an energy-saving control strategy has been developed to control the joint angle of the boom motion of the hydraulic manipulator. A novel configuration with five programmable valves and one accumulator has first been proposed to make the energy recovery possible. Then, a triplelevel control algorithm has been developed, which consists of the motion tracking level (level I), the flow distribution level (level II), and the offside pressure tracking level (level III). The overall scheme has been demonstrated to be able to maintain excellent tracking performance while achieving much less energy consumption compared to the previous four-valve control scheme and the five-valve flow regeneration control scheme. R EFERENCES [1] S. Liu and B. Yao, “Coordinate control of energy saving programmable valves,” IEEE Trans. Control Syst. Technol., vol. 16, no. 1, pp. 34–45, Jan. 2008. [2] T. A. Minav, J. J. Pyrhonen, and L. I. E. Laurila, “Permanent magnet synchronous machine sizing: Effect on the energy efficiency of an electrohydraulic forklift,” IEEE Trans. Ind. Electron., vol. 59, no. 6, pp. 2466– 2474, Jun. 2012. [3] A. Mohanty and B. Yao, “Indirect adaptive robust control of hydraulic manipulators with accurate parameter estimates,” IEEE Trans. Control Syst. Technol., vol. 19, no. 3, pp. 567–575, May 2011. [4] A. Mohanty and B. Yao, “Integrated direct/indirect adaptive robust control of hydraulic manipulators with valve deadband,” IEEE/ASME Trans. Mechatronics, vol. 16, no. 4, pp. 707–715, Aug. 2011. [5] H.-H. Liao, M. J. Roelle, J.-S. Chen, S. Park, and J. C. Gerdes, “Implementation and analysis of a repetitive controller for an electro-hydraulic engine valve system,” IEEE Trans. Control Syst. Technol., vol. 19, no. 5, pp. 1102–1113, Sep. 2011. [6] X. Lu and M. Huang, “System-decomposition-based multilevel control for hydraulic press machine,” IEEE Trans. Ind. Electron., vol. 59, no. 4, pp. 1980–1987, Apr. 2012. [7] A. Mohanty, S. Gayaka, and B. Yao, “An adaptive robust observer for velocity estimation in an electro-hydraulic system,” Int. J. Adaptive Control Signal Process., vol. 26, no. 12, pp. 1076–1089, Dec. 2012. [8] A. Y. Goharrizi and N. Sepehri, “A wavelet-based approach for external leakage detection and isolation from internal leakage in valve-controlled hydraulic actuators,” IEEE Trans. Ind. Electron., vol. 58, no. 9, pp. 4374– 4384, Sep. 2011. [9] J. Yao, Z. Jiao, and D. Ma, “Extended-state-observer-based output feedback nonlinear robust control of hydraulic systems with backstepping,” IEEE Trans. Ind. Electron., vol. 61, no. 11, pp. 6285–6293, Nov. 2014. [10] W. Sun, H. Gao, and B. Yao, “Adaptive robust vibration control of full-car active suspensions with electrohydraulic actuators,” IEEE Trans. Control Syst. Technol., vol. 21, no. 6, pp. 2417–2422, Nov. 2013. [11] B. Yao, F. Bu, J. Reedy, and G. Chiu, “Adaptive robust control of singlerod hydraulic actuators: Theory and experiments,” IEEE/ASME Trans. Mechatronics, vol. 5, no. 1, pp. 79–91, Mar. 2000. [12] B. Yao, F. Bu, and G.T.C. Chiu, “Nonlinear adaptive robust control of electro-hydraulic systems driven by double-rod actuators,” Int. J. Control, vol. 74, no. 8, pp. 761–775, 2001. [13] B. Eriksson and J.-O. Palmberg, “Individual metering fluid power systems: Challenges and opportunities,” Proc. Inst. Mech. Eng., Part I: J. Syst. Control Eng., vol. 225, no. 2, pp. 196–211, 2011. [14] H. Hu and Q. Zhang, “Realization of programmable control using a set of individually controlled electrohydraulic valves,” Int. J. Fluid Power, vol. 3, no. 2, pp. 29–34, 2002. [15] H. Hu and Q. Zhang, “Multi-function realization using an integrated programmable e/h control valve,” Appl. Eng. Agric., vol. 19, no. 3, pp. 283– 290, 2003. [16] R. Book and C. Goering, “Programmable electrohydraulic valve,” presented at the SAE Technical Paper, Warrandale, PA, USA, 1999.
[17] B. Eriksso, “Control strategy for energy efficient fluid power actuators: Utilizing individual metering,” M.S. thesis, Institutionen of Ekonomisk Och Industriell Utveckling, Linkoping, Sweden, 2007. [18] S. Liu and B. Yao, “Automated modeling of cartridge valve flow mapping,” IEEE/ASME Trans. Mechatronics, vol. 11, no. 4, pp. 381–388, Aug. 2006. [19] J. A. Aardema and D. W. Koehler, “System and method for controlling an independent metering valve,” U.S. Patent 5 947 140, Sep. 7, 1999. [20] B. Yao and M. Tomizuka, “Adaptive robust control of SISO nonlinear systems in a semi-strict feedback form,” Automatica, vol. 33, no. 5, pp. 893– 900, 1997. [21] B. Yao, “Desired compensation adaptive robust control,” ASME J. Dyn. Syst., Meas., Control, vol. 131, no. 6, pp. 1–7, 2009. [22] Z. Chen, B. Yao, and Q. Wang, “Adaptive robust precision motion control of linear motors with integrated compensation of nonlinearities and bearing flexible modes,” IEEE Trans. Ind. Informat., vol. 9, no. 2, pp. 965– 973, May 2013. [23] Z. Chen, B. Yao, and Q. Wang, “Accurate motion control of linear motors with adaptive robust compensation of nonlinear electromagnetic field effect,” IEEE/ASME Trans. Mechatronics, vol. 18, no. 3, pp. 1122–1129, Jun. 2013. [24] L. Lu, Z. Chen, B. Yao, and Q. Wang, “A two-loop performance oriented tip tracking control of a linear motor driven flexible beam system with experiments,” IEEE Trans. Ind. Electron., vol. 60, no. 3, pp. 1011–1022, Mar. 2013. [25] C. Hu, B. Yao, and Q. Wang, “Performance oriented adaptive robust control of a class of nonlinear systems preceded by unknown dead-zone with comparative experimental results,” IEEE/ASME Trans. Mechatronics, vol. 18, no. 1, pp. 178–189, 2013. [26] B. Yao, C. Hu, and Q. Wang, “An orthogonal global task coordinate frame for contouring control of biaxial systems,” IEEE/ASME Trans. Mechatronics, vol. 17, no. 4, pp. 622–634, Aug. 2012. [27] K.-H. Rew and K.-S. Kim, “A closed-form solution to asymmetric motion profile allowing acceleration manipulation,” IEEE Trans. Ind. Electron., vol. 57, pp. 2499–2506, Jul. 2010.
Lu Lu (S’10–M’14) received the B.Eng. degree in mechatronic engineering from Zhejiang University, Hangzhou, China, in 2008, and the M.S. and Ph.D. degrees in mechanical engineering from Purdue University, West Lafayette, IN, USA, in 2010 and 2013, respectively. Since September 2013, he has been a Postdoctoral Research Associate with the Center for Automation Technologies and Systems, Rensselaer Polytechnic Institute, Troy, NY, USA. His research interests include control theory, advanced motion control, manufacturing process control, sensor fusion, robotics, and human–machine interaction.
Bin Yao (S’92–M’96–SM’03) received the B.Eng. degree in applied mechanics from Beijing University of Aeronautics and Astronautics, Beijing, China, in 1987; the M.Eng. degree in electrical engineering from Nanyang Technological University, Singapore, in 1992; and the Ph.D. degree in mechanical engineering from the University of California, Berkeley, CA, USA, in 1996. Since 1996, he has been with the School of Mechanical Engineering, Purdue University, West Lafayette, IN, USA, where he has been a Professor since 2007 (currently on leave). He is currently a Chang Jiang Chair Professor with Zhejiang University, Hangzhou, China. He was honored as a Kuang-Piu Professor in 2005 and a Changjiang Chair Professor at Zhejiang University by the Ministry of Education of China in 2010. Dr. Yao was the recipient of the O. Hugo Schuck Best Paper (Theory) Award from the American Automatic Control Council in 2004, the Outstanding Young Investigator Award of the American Society of Mechanical Engineers (ASME) Dynamic Systems and Control Division (DSCD) in 2007, and the Best Conference Paper Awards on Mechatronics of ASME DSCD in 2012. He is a fellow of ASME.
