- Email: [email protected]
Energy saving control of a hydraulic manipulator using five cartridge valves and one accumulator*
6th IFAC Symposium on Mechatronic Systems The International Federation of Automatic Control April 10-12, 2013. Hangzhou, China
Energy saving control of a hydraulic manipulator using five cartridge valves and one accumulator ⋆ Lu Lu ∗ Bin Yao ∗∗ Zhibin Liu ∗∗∗ ∗
School of Mechanical Engineering at Purdue University, 585 Purdue Mall, West Lafayette, 47907, IN USA (e-mail: [email protected]). ∗∗ School of Mechanical Engineering at Purdue University, 585 Purdue Mall, West Lafayette, 47907, IN USA (e-mail: [email protected]). ∗∗∗ The State Key Lab of Fluid Power Transmission and Control at Zhejiang University, Yuquan Campus, Hangzhou, 310027, China, (e-mail: [email protected]). 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 control algorithm consisting of three levels is proposed. In level I, an adaptive robust controller is synthesized to generate the desired flow rates for the two chambers of the cylinders Q1m and Q2m so that the joint angle of the manipulator tracks the desired trajectory as accurately as possible, and the offside pressure of the cylinder also follows the desired pressure profile to be generated in level III accurately. In level II, an energy-optimum flow distribution law is designed and implemented to generate the desired flow rates passing through the five programmable valves. In level III, the desired offside pressure of the cylinder is generated so that the total energy consumption during the whole movement cycle is minimized. Experimental study validates that the proposed strategy can indeed achieve both accurate motion tracking and minimum energy consumption simultaneously. Compared to the previous 4-valve scheme and 5-valve flow regeneration scheme without the use of accumulator in Liu and Yao (2008), the proposed strategy has much less total energy consumption and equally good tracking performance. Keywords: Adaptive robust control, Motion control, Electro-hydraulic systems, Energy saving, Optimal control. 1. INTRODUCTION
the system and the control law design need to be properly done.
The hydraulic manipulator has been widely used in heavy machinery such as excavators, cranes and wheel loaders. The significant practical meaningfulness of the control of hydraulic manipulator has drawn the attention of many control engineers to apply the state-of-art control techniques. The control of electro-hydraulic systems is very challenging not only because of the high-order and highly nonlinear nature of the plant, but the hydraulic energy consumption is also a big issue that needs to be taken into account for the controller design. Typically, for the tracking control of a hydraulic manipulator, it is required that the desired joint angle of the manipulator is tracked as accurately as possible while the total energy assumption is as small as possible. To meet these two targets simultaneously, both the hardware configuration used to control
Traditionally, a 4-way directional valve or servo valve is often used to control the position of the hydraulic cylinder of manipulator. 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 can not 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 sides of the cylinder and metering the flows passing through each of the valves. A detailed review of such type of independently valve metering configuration can be found in Eriksson and Palmberg (2011). Typically, most of these current schemes like Hu
⋆ The work is supported in part by the US National Science Foundation (Grant No. CMMI-1052872) and in part by the Ministry of Education of China through a Chang Jiang Chair Professorship.
978-3-902823-31-1/13/$20.00 © 2013 IFAC
84
10.3182/20130410-3-CN-2034.00101
IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China
throttle losses of the valves that connect the chambers to the pump. To reduce this amount of energy waste, loadsensing (LS) pump may be used to reduce the supply pressure ps (t). To provide flow rate into the system to push the cylinder, the supply pressure needs to be higher than the pressures of both sides of the cylinder. 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.
and Zhang (2002, 2003); Book and Goering (1999); Eriksso (2007); Liu and Yao (2004, 2008) 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. Specifically, in our previous work Liu and Yao (2008), 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 headend chamber to the rod-end chamber during the downward movement of the cylinder. An adaptive robust control 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 valves and one accumulator is proposed. By storing the flow into the accumulator when the cylinder moves downward and using such amount of flow to pump the cylinder up, the so-called ’energy recovery’ can be achieved. It is demonstrated that such a hardware configuration is capable of achieving much smaller overall energy consumption than the one in Liu and Yao (2008). 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 Q1m and Q2m so that the angular position of the joint tracks the desired trajectory as accurately as 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 energy-optimum 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 profile of the cylinder is generated so that the total energy consumption is minimized. Experiments are conducted to compare the performances of the proposed 5-valve-withaccumulator approach with the 5-valve flow regeneration approach in Liu and Yao (2008) and the traditional 4valve approach. The results demonstrate that the proposed approach can indeed achieve accurate motion tracking as well as much smaller energy consumption than the other two approaches.
Since the LS pump has complicated structure and high cost, constant pressure pump is sometimes preferred to be used 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. For a single-rod cylinder, since the head-end piston area of the cylinder is much smaller than the rodend piston area, the pressure of the head-end chamber is significantly higher than the pressure of the rod-end chamber for most of the time during the extension of the rod if the acceleration of the cylinder is not very large, which allows flow regeneration from the head-end chamber to the rod-end chamber if the two chambers are directly connected with a cartridge valve between them. This idea has been implemented in Liu and Yao (2008). It is clear that if the desired cylinder position is back-and-forth, the total energy consumption is approximately proportional to the ’head-end liquid volume’, which equals the head-end area multiplied by the maximum difference of the cylinder position for the desired back-and-forth movement. In this paper, a new hardware configuration is proposed 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 headend ram area is much larger than the rod-end piston area, which means that the head-end chamber definitely needs much more flow supply than the rod-end chamber. With the proposed scheme, when the cylinder is moving downward, the pressure of the rod-end chamber is set to be high so that accumulator can be fully charged. When the cylinder is moving upward, the stored flow can be used to push the cylinder if the desired rod-end pressure is set to be low. It is seen that the total energy consumption of the proposed scheme for a back-and-forth movement of the cylinder is roughly proportional to the ’rod-end liquid volume’, which is much lower than the energy consumption in Liu and Yao (2008).
2. MOTIVATION AND HARDWARE CONFIGURATION The energy consumption of the overall hydraulic system during any time period [t0 ; t1 ] is calculated by ∫ t1 E= ps (t)Qs (t)dt; (1)
After proposing the hardware configuration, the control problem will be formulated in the following 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 motion as small as possible.
t0
where ps (t) is the supply pressure of the pump and Qs (t) is the flow rate into the system. From the above 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.
3. CONTROL PROBLEM FORMULATION
When the supply pressure ps (t) is high but the chamber pressures are low, a lot of energy will be wasted on the
The overall dynamics of the boom motion of the cylinder (including both the mechanical and hydraulic dynamics)
85
IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China
described by: Qvi = fxi (∆Pvi ; uvi ); i = 1; 2; · · · ; 5; (4) where ∆Pvi is the pressure drop across the i-th valve, i.e., ∆Pv1 = ps p1 , ∆Pv2 = p1 pt , ∆Pv3 = p1 pac , ∆Pv4 = ps p2 , ∆Pv5 = p2 pt . In the above, ps is the constant pressure of the supply pump, pt is the pressure of the tank, 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 as detailed in Liu and Yao (2006). However, there are always discrepancies between the ideal flow mapping functions and the identified ones. Denote fxim as the identified flow mapping function for the i-the valve, Qvim = fxim (∆Pvi ; uvi ) as the ’desired’ flow rates ˜ vi = obtained from the identified function fxim and Q Qvim Qvi as the discrepancies between Qvim and Qvi . Qvi in (5) can be expressed as as ˜ vi ; i = 1; 2; · · · ; 5; Qvi = Qvim Q (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 2nd and 3rd equations of (2)) can be rewritten as V1 (xL ) ∂xL ˜1; p˙1 = A1 q˙ + Q1m Q βe ∂q (6) V2 (xL ) ∂xL ˜2; p˙2 = A2 q˙ Q2m + Q βe ∂q
Fig. 1. The proposed configuration with five valves and an accumulator. can be described by Liu and Yao (2008): ∂xL (Jc + mL le2 )¨ q= (p1 A1 p2 A2 ) Gc (q) ∂q Df q˙ + T (t; q; q); ˙ V1 (xL ) ∂xL p˙1 = A1 q˙ + Q1 ; βe ∂q V2 (xL ) ∂xL p˙2 = A2 q˙ Q2 ; βe ∂q
mL glg (q)
The dynamic equation of the accumulator is given by V˙ f = Qv3 ; (7) where Vf is the volume of the liquid within the accumulator. Since the gas inside the accumulator undergoes adiabatic process when expanding or being compressed, the relationship between pac and Vf is given by 0; if Vf ≤ 0 ppr pac = ; (8) 1 ( Vf )k ; else Vtot
(2) where q is the joint angle of the boom cylinder, xL is the displacement of the boom cylinder. Jc is the moment of inertia of the boom without payload, mL is the mass of the unknown payload, 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-end 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 hoses connected. 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 lg (q) can be found in Bu and Yao (2000). T (t; q; q) ˙ represents the lumped disturbances and nonlinearities of the system.
in which Vtot is total volume of the accumulator, ppr is the ’preset pressure’ of the gas inside the accumulator when Vf = 0. The desired trajectory to be tracked by the cylinder is denoted as qd (t), t ∈ [0; tf ]. For the purpose of perfect tracking, qd (t) is assumed to be continuously differentiable up to 3rd 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.
When the five programmable valves and one accumulator in Fig. 1 are used to control the boom cylinder, Q1 and Q2 are given by Q1 = Qv1 Qv2 Qv3 ; (3) Q2 = Qv4 + Qv5 ; where Qv1 , Qv2 , Qv3 , Qv4 , Qv5 are the flow rate 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 ∆Pvi and the control voltage uvi are
4. CONTROLLER DESIGN 4.1 Overall Control Structure To achieve simultaneously accurate tracking of the joint angle q with respect to qd (t) and the minimization of the overall energy consumption., the following triple level control structure shown in Fig. 2 is proposed. In level I, an adaptive robust control is designed to generate Q1m and
86
IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China
in which Q1m and Q2m are generated by the level I controller. 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) and Q t ∈ [0; tf ], the values of p1 (t), p2 (t) and Q1m (t), Q2m (t) for all t ∈ [0; tf ] are uniquely determined. In this case, if ∫t ∆ we approximate E in ... by E ≈ Em = ps 0 f (Qv1m (t) + ˜ v1 and Q ˜ v4 are ignored, Qv4m (t))dt, in which the terms Q the energy consumption index Em is also uniquely determined and only depends on how the flow rates are distributed. Treating Qvim , i = 1; 2; 3; 4; 5 as optimization variables, a constrained optimal control problem can be formulated to minimize the energy consumption during the time period [0; tf ]:
Q2m so that: i) q tracks qd (t) as accurately as possible, ii) p2 tracks the desired offside pressure profile p2d (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 through each of the 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 shown 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 p2d (t) is planned so that the total energy consumption is minimized with the given level I controller and level II optimal flow distribution law.
∫ min
Q′vim (t), i=1,2,3,4,5, ∀t∈[0,tf ]
ps (
0
tf
(Q′v1m (t) + Q′v4m (t))dt)
subject to Vf′ (0) = Vf 0 ; ppr V˙ f′ (t) = Q′v3m (t); p′ac = ; V ′ (t) 1 ( Vftot )k Q1m (t) = Q′v1m (t) Q′v2m (t) Q′v3m (t); Q2m (t) = Q′v4m (t) + Q′v5m (t); ′ Qv1m (t); Q′v2m (t); Q′v4m (t); Q′v5m (t) ≥ 0; 0 ≤ Q′v3m (t) ≤ fxim (p1 (t) p′ac (t); uvimax ); if p1 (t) p′ac (t) ≥ 0; ′ fxim (pac (t) p1 (t); uvimax ) ≤ Q′v3m (t) ≤ 0; if p1 (t) p′ac (t) < 0; Vf′ ≥ 0; ∀t ∈ [0; tf ];
Fig. 2. The proposed control structure. 4.2 Level I-desired motion and pressure tracking controller In this level, a discontinuous-projection-based adaptive robust control law is designed and implemented to generate the required desired flow rates Q1m and Q2m such that the joint angle q tracks the desired trajectory qd (t) as accurately as possible and the offside pressure p1 or p2 tracks the desired offside pressure p1d (t) or p2d (t) as accurately as possible. The details of the controller design has been given in ?? and will just be outlined in the following. First, a discontinuous-projection-based adaptation law is constructed to estimate the unknown parameters such as βe and mL to be used in the synthesis of Q1m and Q2m . Next, an offside pressure regulator is designed to generate Q2m (t) (suppose p2 is the offside pressure to be regulated) such that p2 tracks the desired pressure profile p2d (t). Finally, using the values of Q2m (t) already obtained, the desired flow rate of the other chamber Q1m (t) is synthesized so that q(t) tracks the desired angle q2d (t) accurately. The final forms of Q1m and Q2m depend on the measurements of q, q, ˙ p1 , p2 , the desired angle trajectory q2d (t) and the desired offside pressure p1d (t) or p2d (t).
(10)
In the above, Q′vim (t); i = 1; 2; 3; 4; 5; are the optimization variables to be solved for. All the other notations with prime signs are changing variables depending upon the values of Q′vim (t); i = 1; 2; 3; 4; 5;. Since p1 (t), p2 (t) and Q1m (t), Q2m (t) are already fixed, the dynamic constraints for the above optimal control problem only come from the accumulator dynamics in which Vf′ and p′ac change in response to the variations of Q′vim (t); i = 1; 2; 3; 4; 5;. Vf 0 is the initial liquid volume in the accumulator. fxim (p1 (t) p′ac (t); uvimax ) and fxim (p′ac (t) p1 (t); uvimax ) represent the upper and lower limit of the flow rate that can pass through the valve #3 when the pressure drop across the orifice is p1 (t) p′ac (t) (when p1 (t) ≥ p′ac (t)) or p′ac (t) p1 (t) (when p1 (t) < p′ac (t)), in which uvimax is the maximum input voltage of the valve. Theorem 1. If the fxim (•; uvimax ) is a non-decreasing function of •, then the optimal solution variables to the problem (10) at any time t, denoted as Q∗vim (t); i = 1; 2; 3; 4; 5, can be expressed as follows:
4.3 Level II-flow distribution law In this level, a minimum-energy flow distribution law will be synthesized to generate the desired flow rates Qvim , i = 1; 2; 3; 4; 5 for the five cartridges valves satisfying Q1m = Qv1m Qv2m Qv3m ; (9) Q2m = Qv4m + Qv5m ;
87
IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China
If Q1m ≥ 0: { max{ fxim (p∗ac (t) p1 (t); uvimax ); ∗ Q1m (t)}; if p∗ac (t) ≥ p1 (t); Qv3m (t) = 0; else ∗ ∗ Qv1m (t) = Qv3m (t) + Q1m (t); Q∗v2m (t) = 0; { Q2m (t) if Q2m (t) ≤ 0; ∗ Qv4m (t) = 0; { else Q2m (t) if Q2m (t) ≥ 0; ∗ Qv5m (t) = 0; else If Q (t) < 0: 1m { min{fxim (p1 (t) p∗ac (t); uvimax ); ∗ Q1m (t)}; if p∗ac (t) ≤ p1 (t); Q (t) = v3m 0; else ∗ Qv1m (t) = 0; Q∗v2m (t) = {Q∗v3m (t) Q1m (t); Q2m (t) if Q2m (t) ≤ 0; ∗ Qv4m (t) = 0; else { Q2m (t) if Q2m (t) ≥ 0; ∗ Qv5m (t) = 0; else (11)
To avoid the discontinuity problem of the desired pressures, let us fix p2 as the offside pressure to be regulated and generate a continuous p2d to minimize the overall energy consumption. The reason that p2 is fixed to be the offside pressure rather than p1 is that the rod side of the cylinder is not directly connected to the accumulator and thus not directly affected by the uncertain and varying dynamics of the accumulator. 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 rod-end pressure should be low enough so that the resulting head-end pressure is also low to allow the flow to go from the accumulator to the head-end chamber no matter what value the accumulator pressure is. In order to fully charge the accumulator when the arm is moving down (q˙d < 0), the desired rod-end pressure should be high enough so that the resulting head-end pressure is also high to allow the flow to go from the headend chamber to the accumulator no matter what value the accumulator pressure is. On the other hand, when setting the desired pressure p2d ∈ [pt ; ps ], the resulting p1 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 p1 in terms of the desired offside pressure p2d and the desired load force pLda and set up an inequality constraint on p2d .
The above optimal flow distribution law 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 is possible given the pressure drop across the valve #3. The rest of the flow comes from the source through valve #1. If Q1m (t) < 0, then the accumulator is charged either with the total flow rate Q1m or as much as is possible given the pressure drop across the 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 the flow goes to the tank through valve #5. If Q2m (t) < 0, then all the flow is supplied from source through valve #4.
Based on the above analysis, the following simple algorithm which can be implemented online is proposed to generate the desired offside pressure profile p2d (t): ′ p2d (t); pt1 A1 pLda (t) ps1 A1 pLda (t) ; ]; if p′2d (t) ∈ [ A2 A2 pt1 A1 pLda (t) ; p2d (t) = A2 pt1 A1 pLda (t) ; if p′2d (t) < A2 ps1 A1 pLda (t) ; else, A2 ′ where p2d (t) is defined as |q˙d (t)| pt1 + (ps1 pt1 ); if q˙d (t) ≤ 0 p′2d (t) = (12) 0; else,q˙dmax
One good property about the above energy-optimum flow distribution law 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.
4.4 Level III-planning of the desired offside pressure profile After designing the desired flow rates Q1m , 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 the desired offside pressure profile. This extra freedom should be used to minimize the total energy consumption of the closed-loop system. In Liu et al. (2008), 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.
in which q˙dmax =
sup q˙d (t), and pt1 and ps1 are two
t∈[0,tf ]
pressure setpoints satisfying pt < pt1 < ps1 < ps . pt1 is often chosen to be a little higher than pt and ps1 is often chosen to be a little lower than ps . 5. EXPERIMENTAL RESULTS In this section, the proposed control strategy will be applied to our experimental testbed set up at Herrick Lab, Purdue university. This is a 3-DOF robot manipulator system that uses a constant pressure pump as the fluid supply source with the supply pressure about 6:9M P a and the tank pressure is about 0P a. The geometric parameters of the arms and cylinders and the inertias of the arms are given in Bu and Yao (2000). The boom motion of the
88
IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China
robot manipulator can be either controlled by the fivevalve-and-accumulator scheme as proposed in this paper, or by five-valve flow regeneration scheme as done in Liu and Yao (2008). In the following, three algorithms will be compared:
700
Power Usage (W)
500
C1). Adaptive robust control with four-valve flow distribution scheme (only the valves #1, #2 and #4, #5 are used); C2). Energy saving adaptive robust control with five-valve flow regeneration scheme in Liu and Yao (2008); C3). The proposed three-level energy saving adaptive robust control algorithm with five valves and one accumulator;
0 0
tracking error (rad)
0 −1 −2 −3 8
8
10
Book, R. and Goering, C. (1999). Programmable electrohydraulic valve. In SAE Technical Paper. Bu, F. and Yao, B. (2000). nonlinear adaptive robust control of hydraulic actuators regulated by proportional directional control valves with deadband and nonlinear flow gain coefficients. In Proceedings of American Control Conference, 4129–4133. Eriksso, B. (2007). Control strategy for energy efficient fluid power actuators: Utilizing individual metering. Ph.D. thesis, Institutionen Fr Ekonomisk Och Industriell Utveckling, Linkping. Eriksson, B. and Palmberg, J.O. (2011). Individual metering fluid power systems: challenges and opportunities. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, 225(2), 196–211. Hu, H. and Zhang, Q. (2002). Realization of programmable control using a set of individually controlled electrohydraulic valves. International Journal of Fluid Power, 3(2), 29–34. Hu, H. and Zhang, Q. (2003). Multi-function realization using an integrated programmable e/h control valve. Applied Engineering in Agriculture, 19(3), 283–290. Liu, S. and Yao, B. (2004). Adaptive robust control of programmable valves with manufacturer supplied flow mapping only. In the 43rd IEEE Conference on Decision and Control, 1117–1122. Liu, S. and Yao, B. (2006). Automated modeling of cartridge valve flow mapping. IEEE/ASME Transactions
1
6
6
REFERENCES
2
time (s)
time (s)
In this paper, an energy saving control strategy is developed to control the joint angle of the boom motion of the hydraulic manipulator. A novel configuration with 5 programmable valves and one accumulator is first proposed to make the energy recovery possible. Then, a triplelevel control algorithm is 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 is demonstrated to be able to maintain excellent tracking performance while achieving much smaller energy consumption compared to the previous 4valve control scheme and 5-valve flow regeneration control scheme.
3
4
4
6. CONCLUSION
C1 C2 C3
2
2
Fig. 4. Energy usage.
−3
−4 0
300
100
x 10
4
400
200
The desired trajectory qd (t) used is a point-to-point S curve from 0rad to 0:5rad. Since the three algorithms have the same offside pressure regulator and motion tracking controller (the same ’level I’ controller design), the corresponding controller parameters are selected to be the same. After the experimental results are obtained, the tracking errors and the power consumptions are plotted in Fig. 3 and 4, respectively. From the plot of tracking errors, it can be seen that three controllers achieve almost the same level of tracking accuracy, showing the good tracking capabilities of the level-I ARC design. However, the plot of power usages shows that the three algorithms give totally different power consumption profiles. For C1, the energy is consumed for both the upward movement and downward movement of the cylinder. For the flow regeneration scheme C2, the energy is consumed only during the upward movement of the cylinder. For C3, with the proposed system configuration and the corresponding control structure, the energy is only consumed during the downward movement of the cylinder. Since the ram area of the rod end of the cylinder is much smaller than that of the head end, the energy usage of C3 is much less than C2. The energy usages of the three algorithms are calculated as EC1 = 1:9KW , EC2 = 1:2KW and EC3 = 0:78KW . It can be seen that, the energy consumption of C3 is much less than that of C1 and C2, showing the ability of energy saving of the proposed hardware configuration and control solution.
5
C1 C2 C3
600
10
Fig. 3. Tracking errors of C1-C3.
89
IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China
on Mechatronics, 12(4), 381–388. (Part of the paper appeared in the IEEE/ASME Conference on Advanced Intelligent Mechatronics (AIM’05), pp.789-794, 2005, and received the Best Student Paper Competition Award). Liu, S. and Yao, B. (2008). Coordinate control of energy saving programmable valves. IEEE Transactions on Control Systems Technology, 16(1), 34–45. Liu, X., Su, H., Yao, B., and Chu, J. (2008). Adaptive robust control of nonlinear systems with dynamic uncertainties. International Journal of Adaptive Control and Signal Processing, 22.
90
Energy saving control of a hydraulic manipulator using five cartridge valves and one accumulator ⋆ Lu Lu ∗ Bin Yao ∗∗ Zhibin Liu ∗∗∗ ∗
School of Mechanical Engineering at Purdue University, 585 Purdue Mall, West Lafayette, 47907, IN USA (e-mail: [email protected]). ∗∗ School of Mechanical Engineering at Purdue University, 585 Purdue Mall, West Lafayette, 47907, IN USA (e-mail: [email protected]). ∗∗∗ The State Key Lab of Fluid Power Transmission and Control at Zhejiang University, Yuquan Campus, Hangzhou, 310027, China, (e-mail: [email protected]). 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 control algorithm consisting of three levels is proposed. In level I, an adaptive robust controller is synthesized to generate the desired flow rates for the two chambers of the cylinders Q1m and Q2m so that the joint angle of the manipulator tracks the desired trajectory as accurately as possible, and the offside pressure of the cylinder also follows the desired pressure profile to be generated in level III accurately. In level II, an energy-optimum flow distribution law is designed and implemented to generate the desired flow rates passing through the five programmable valves. In level III, the desired offside pressure of the cylinder is generated so that the total energy consumption during the whole movement cycle is minimized. Experimental study validates that the proposed strategy can indeed achieve both accurate motion tracking and minimum energy consumption simultaneously. Compared to the previous 4-valve scheme and 5-valve flow regeneration scheme without the use of accumulator in Liu and Yao (2008), the proposed strategy has much less total energy consumption and equally good tracking performance. Keywords: Adaptive robust control, Motion control, Electro-hydraulic systems, Energy saving, Optimal control. 1. INTRODUCTION
the system and the control law design need to be properly done.
The hydraulic manipulator has been widely used in heavy machinery such as excavators, cranes and wheel loaders. The significant practical meaningfulness of the control of hydraulic manipulator has drawn the attention of many control engineers to apply the state-of-art control techniques. The control of electro-hydraulic systems is very challenging not only because of the high-order and highly nonlinear nature of the plant, but the hydraulic energy consumption is also a big issue that needs to be taken into account for the controller design. Typically, for the tracking control of a hydraulic manipulator, it is required that the desired joint angle of the manipulator is tracked as accurately as possible while the total energy assumption is as small as possible. To meet these two targets simultaneously, both the hardware configuration used to control
Traditionally, a 4-way directional valve or servo valve is often used to control the position of the hydraulic cylinder of manipulator. 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 can not 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 sides of the cylinder and metering the flows passing through each of the valves. A detailed review of such type of independently valve metering configuration can be found in Eriksson and Palmberg (2011). Typically, most of these current schemes like Hu
⋆ The work is supported in part by the US National Science Foundation (Grant No. CMMI-1052872) and in part by the Ministry of Education of China through a Chang Jiang Chair Professorship.
978-3-902823-31-1/13/$20.00 © 2013 IFAC
84
10.3182/20130410-3-CN-2034.00101
IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China
throttle losses of the valves that connect the chambers to the pump. To reduce this amount of energy waste, loadsensing (LS) pump may be used to reduce the supply pressure ps (t). To provide flow rate into the system to push the cylinder, the supply pressure needs to be higher than the pressures of both sides of the cylinder. 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.
and Zhang (2002, 2003); Book and Goering (1999); Eriksso (2007); Liu and Yao (2004, 2008) 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. Specifically, in our previous work Liu and Yao (2008), 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 headend chamber to the rod-end chamber during the downward movement of the cylinder. An adaptive robust control 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 valves and one accumulator is proposed. By storing the flow into the accumulator when the cylinder moves downward and using such amount of flow to pump the cylinder up, the so-called ’energy recovery’ can be achieved. It is demonstrated that such a hardware configuration is capable of achieving much smaller overall energy consumption than the one in Liu and Yao (2008). 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 Q1m and Q2m so that the angular position of the joint tracks the desired trajectory as accurately as 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 energy-optimum 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 profile of the cylinder is generated so that the total energy consumption is minimized. Experiments are conducted to compare the performances of the proposed 5-valve-withaccumulator approach with the 5-valve flow regeneration approach in Liu and Yao (2008) and the traditional 4valve approach. The results demonstrate that the proposed approach can indeed achieve accurate motion tracking as well as much smaller energy consumption than the other two approaches.
Since the LS pump has complicated structure and high cost, constant pressure pump is sometimes preferred to be used 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. For a single-rod cylinder, since the head-end piston area of the cylinder is much smaller than the rodend piston area, the pressure of the head-end chamber is significantly higher than the pressure of the rod-end chamber for most of the time during the extension of the rod if the acceleration of the cylinder is not very large, which allows flow regeneration from the head-end chamber to the rod-end chamber if the two chambers are directly connected with a cartridge valve between them. This idea has been implemented in Liu and Yao (2008). It is clear that if the desired cylinder position is back-and-forth, the total energy consumption is approximately proportional to the ’head-end liquid volume’, which equals the head-end area multiplied by the maximum difference of the cylinder position for the desired back-and-forth movement. In this paper, a new hardware configuration is proposed 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 headend ram area is much larger than the rod-end piston area, which means that the head-end chamber definitely needs much more flow supply than the rod-end chamber. With the proposed scheme, when the cylinder is moving downward, the pressure of the rod-end chamber is set to be high so that accumulator can be fully charged. When the cylinder is moving upward, the stored flow can be used to push the cylinder if the desired rod-end pressure is set to be low. It is seen that the total energy consumption of the proposed scheme for a back-and-forth movement of the cylinder is roughly proportional to the ’rod-end liquid volume’, which is much lower than the energy consumption in Liu and Yao (2008).
2. MOTIVATION AND HARDWARE CONFIGURATION The energy consumption of the overall hydraulic system during any time period [t0 ; t1 ] is calculated by ∫ t1 E= ps (t)Qs (t)dt; (1)
After proposing the hardware configuration, the control problem will be formulated in the following 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 motion as small as possible.
t0
where ps (t) is the supply pressure of the pump and Qs (t) is the flow rate into the system. From the above 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.
3. CONTROL PROBLEM FORMULATION
When the supply pressure ps (t) is high but the chamber pressures are low, a lot of energy will be wasted on the
The overall dynamics of the boom motion of the cylinder (including both the mechanical and hydraulic dynamics)
85
IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China
described by: Qvi = fxi (∆Pvi ; uvi ); i = 1; 2; · · · ; 5; (4) where ∆Pvi is the pressure drop across the i-th valve, i.e., ∆Pv1 = ps p1 , ∆Pv2 = p1 pt , ∆Pv3 = p1 pac , ∆Pv4 = ps p2 , ∆Pv5 = p2 pt . In the above, ps is the constant pressure of the supply pump, pt is the pressure of the tank, 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 as detailed in Liu and Yao (2006). However, there are always discrepancies between the ideal flow mapping functions and the identified ones. Denote fxim as the identified flow mapping function for the i-the valve, Qvim = fxim (∆Pvi ; uvi ) as the ’desired’ flow rates ˜ vi = obtained from the identified function fxim and Q Qvim Qvi as the discrepancies between Qvim and Qvi . Qvi in (5) can be expressed as as ˜ vi ; i = 1; 2; · · · ; 5; Qvi = Qvim Q (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 2nd and 3rd equations of (2)) can be rewritten as V1 (xL ) ∂xL ˜1; p˙1 = A1 q˙ + Q1m Q βe ∂q (6) V2 (xL ) ∂xL ˜2; p˙2 = A2 q˙ Q2m + Q βe ∂q
Fig. 1. The proposed configuration with five valves and an accumulator. can be described by Liu and Yao (2008): ∂xL (Jc + mL le2 )¨ q= (p1 A1 p2 A2 ) Gc (q) ∂q Df q˙ + T (t; q; q); ˙ V1 (xL ) ∂xL p˙1 = A1 q˙ + Q1 ; βe ∂q V2 (xL ) ∂xL p˙2 = A2 q˙ Q2 ; βe ∂q
mL glg (q)
The dynamic equation of the accumulator is given by V˙ f = Qv3 ; (7) where Vf is the volume of the liquid within the accumulator. Since the gas inside the accumulator undergoes adiabatic process when expanding or being compressed, the relationship between pac and Vf is given by 0; if Vf ≤ 0 ppr pac = ; (8) 1 ( Vf )k ; else Vtot
(2) where q is the joint angle of the boom cylinder, xL is the displacement of the boom cylinder. Jc is the moment of inertia of the boom without payload, mL is the mass of the unknown payload, 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-end 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 hoses connected. 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 lg (q) can be found in Bu and Yao (2000). T (t; q; q) ˙ represents the lumped disturbances and nonlinearities of the system.
in which Vtot is total volume of the accumulator, ppr is the ’preset pressure’ of the gas inside the accumulator when Vf = 0. The desired trajectory to be tracked by the cylinder is denoted as qd (t), t ∈ [0; tf ]. For the purpose of perfect tracking, qd (t) is assumed to be continuously differentiable up to 3rd 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.
When the five programmable valves and one accumulator in Fig. 1 are used to control the boom cylinder, Q1 and Q2 are given by Q1 = Qv1 Qv2 Qv3 ; (3) Q2 = Qv4 + Qv5 ; where Qv1 , Qv2 , Qv3 , Qv4 , Qv5 are the flow rate 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 ∆Pvi and the control voltage uvi are
4. CONTROLLER DESIGN 4.1 Overall Control Structure To achieve simultaneously accurate tracking of the joint angle q with respect to qd (t) and the minimization of the overall energy consumption., the following triple level control structure shown in Fig. 2 is proposed. In level I, an adaptive robust control is designed to generate Q1m and
86
IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China
in which Q1m and Q2m are generated by the level I controller. 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) and Q t ∈ [0; tf ], the values of p1 (t), p2 (t) and Q1m (t), Q2m (t) for all t ∈ [0; tf ] are uniquely determined. In this case, if ∫t ∆ we approximate E in ... by E ≈ Em = ps 0 f (Qv1m (t) + ˜ v1 and Q ˜ v4 are ignored, Qv4m (t))dt, in which the terms Q the energy consumption index Em is also uniquely determined and only depends on how the flow rates are distributed. Treating Qvim , i = 1; 2; 3; 4; 5 as optimization variables, a constrained optimal control problem can be formulated to minimize the energy consumption during the time period [0; tf ]:
Q2m so that: i) q tracks qd (t) as accurately as possible, ii) p2 tracks the desired offside pressure profile p2d (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 through each of the 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 shown 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 p2d (t) is planned so that the total energy consumption is minimized with the given level I controller and level II optimal flow distribution law.
∫ min
Q′vim (t), i=1,2,3,4,5, ∀t∈[0,tf ]
ps (
0
tf
(Q′v1m (t) + Q′v4m (t))dt)
subject to Vf′ (0) = Vf 0 ; ppr V˙ f′ (t) = Q′v3m (t); p′ac = ; V ′ (t) 1 ( Vftot )k Q1m (t) = Q′v1m (t) Q′v2m (t) Q′v3m (t); Q2m (t) = Q′v4m (t) + Q′v5m (t); ′ Qv1m (t); Q′v2m (t); Q′v4m (t); Q′v5m (t) ≥ 0; 0 ≤ Q′v3m (t) ≤ fxim (p1 (t) p′ac (t); uvimax ); if p1 (t) p′ac (t) ≥ 0; ′ fxim (pac (t) p1 (t); uvimax ) ≤ Q′v3m (t) ≤ 0; if p1 (t) p′ac (t) < 0; Vf′ ≥ 0; ∀t ∈ [0; tf ];
Fig. 2. The proposed control structure. 4.2 Level I-desired motion and pressure tracking controller In this level, a discontinuous-projection-based adaptive robust control law is designed and implemented to generate the required desired flow rates Q1m and Q2m such that the joint angle q tracks the desired trajectory qd (t) as accurately as possible and the offside pressure p1 or p2 tracks the desired offside pressure p1d (t) or p2d (t) as accurately as possible. The details of the controller design has been given in ?? and will just be outlined in the following. First, a discontinuous-projection-based adaptation law is constructed to estimate the unknown parameters such as βe and mL to be used in the synthesis of Q1m and Q2m . Next, an offside pressure regulator is designed to generate Q2m (t) (suppose p2 is the offside pressure to be regulated) such that p2 tracks the desired pressure profile p2d (t). Finally, using the values of Q2m (t) already obtained, the desired flow rate of the other chamber Q1m (t) is synthesized so that q(t) tracks the desired angle q2d (t) accurately. The final forms of Q1m and Q2m depend on the measurements of q, q, ˙ p1 , p2 , the desired angle trajectory q2d (t) and the desired offside pressure p1d (t) or p2d (t).
(10)
In the above, Q′vim (t); i = 1; 2; 3; 4; 5; are the optimization variables to be solved for. All the other notations with prime signs are changing variables depending upon the values of Q′vim (t); i = 1; 2; 3; 4; 5;. Since p1 (t), p2 (t) and Q1m (t), Q2m (t) are already fixed, the dynamic constraints for the above optimal control problem only come from the accumulator dynamics in which Vf′ and p′ac change in response to the variations of Q′vim (t); i = 1; 2; 3; 4; 5;. Vf 0 is the initial liquid volume in the accumulator. fxim (p1 (t) p′ac (t); uvimax ) and fxim (p′ac (t) p1 (t); uvimax ) represent the upper and lower limit of the flow rate that can pass through the valve #3 when the pressure drop across the orifice is p1 (t) p′ac (t) (when p1 (t) ≥ p′ac (t)) or p′ac (t) p1 (t) (when p1 (t) < p′ac (t)), in which uvimax is the maximum input voltage of the valve. Theorem 1. If the fxim (•; uvimax ) is a non-decreasing function of •, then the optimal solution variables to the problem (10) at any time t, denoted as Q∗vim (t); i = 1; 2; 3; 4; 5, can be expressed as follows:
4.3 Level II-flow distribution law In this level, a minimum-energy flow distribution law will be synthesized to generate the desired flow rates Qvim , i = 1; 2; 3; 4; 5 for the five cartridges valves satisfying Q1m = Qv1m Qv2m Qv3m ; (9) Q2m = Qv4m + Qv5m ;
87
IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China
If Q1m ≥ 0: { max{ fxim (p∗ac (t) p1 (t); uvimax ); ∗ Q1m (t)}; if p∗ac (t) ≥ p1 (t); Qv3m (t) = 0; else ∗ ∗ Qv1m (t) = Qv3m (t) + Q1m (t); Q∗v2m (t) = 0; { Q2m (t) if Q2m (t) ≤ 0; ∗ Qv4m (t) = 0; { else Q2m (t) if Q2m (t) ≥ 0; ∗ Qv5m (t) = 0; else If Q (t) < 0: 1m { min{fxim (p1 (t) p∗ac (t); uvimax ); ∗ Q1m (t)}; if p∗ac (t) ≤ p1 (t); Q (t) = v3m 0; else ∗ Qv1m (t) = 0; Q∗v2m (t) = {Q∗v3m (t) Q1m (t); Q2m (t) if Q2m (t) ≤ 0; ∗ Qv4m (t) = 0; else { Q2m (t) if Q2m (t) ≥ 0; ∗ Qv5m (t) = 0; else (11)
To avoid the discontinuity problem of the desired pressures, let us fix p2 as the offside pressure to be regulated and generate a continuous p2d to minimize the overall energy consumption. The reason that p2 is fixed to be the offside pressure rather than p1 is that the rod side of the cylinder is not directly connected to the accumulator and thus not directly affected by the uncertain and varying dynamics of the accumulator. 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 rod-end pressure should be low enough so that the resulting head-end pressure is also low to allow the flow to go from the accumulator to the head-end chamber no matter what value the accumulator pressure is. In order to fully charge the accumulator when the arm is moving down (q˙d < 0), the desired rod-end pressure should be high enough so that the resulting head-end pressure is also high to allow the flow to go from the headend chamber to the accumulator no matter what value the accumulator pressure is. On the other hand, when setting the desired pressure p2d ∈ [pt ; ps ], the resulting p1 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 p1 in terms of the desired offside pressure p2d and the desired load force pLda and set up an inequality constraint on p2d .
The above optimal flow distribution law 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 is possible given the pressure drop across the valve #3. The rest of the flow comes from the source through valve #1. If Q1m (t) < 0, then the accumulator is charged either with the total flow rate Q1m or as much as is possible given the pressure drop across the 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 the flow goes to the tank through valve #5. If Q2m (t) < 0, then all the flow is supplied from source through valve #4.
Based on the above analysis, the following simple algorithm which can be implemented online is proposed to generate the desired offside pressure profile p2d (t): ′ p2d (t); pt1 A1 pLda (t) ps1 A1 pLda (t) ; ]; if p′2d (t) ∈ [ A2 A2 pt1 A1 pLda (t) ; p2d (t) = A2 pt1 A1 pLda (t) ; if p′2d (t) < A2 ps1 A1 pLda (t) ; else, A2 ′ where p2d (t) is defined as |q˙d (t)| pt1 + (ps1 pt1 ); if q˙d (t) ≤ 0 p′2d (t) = (12) 0; else,q˙dmax
One good property about the above energy-optimum flow distribution law 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.
4.4 Level III-planning of the desired offside pressure profile After designing the desired flow rates Q1m , 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 the desired offside pressure profile. This extra freedom should be used to minimize the total energy consumption of the closed-loop system. In Liu et al. (2008), 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.
in which q˙dmax =
sup q˙d (t), and pt1 and ps1 are two
t∈[0,tf ]
pressure setpoints satisfying pt < pt1 < ps1 < ps . pt1 is often chosen to be a little higher than pt and ps1 is often chosen to be a little lower than ps . 5. EXPERIMENTAL RESULTS In this section, the proposed control strategy will be applied to our experimental testbed set up at Herrick Lab, Purdue university. This is a 3-DOF robot manipulator system that uses a constant pressure pump as the fluid supply source with the supply pressure about 6:9M P a and the tank pressure is about 0P a. The geometric parameters of the arms and cylinders and the inertias of the arms are given in Bu and Yao (2000). The boom motion of the
88
IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China
robot manipulator can be either controlled by the fivevalve-and-accumulator scheme as proposed in this paper, or by five-valve flow regeneration scheme as done in Liu and Yao (2008). In the following, three algorithms will be compared:
700
Power Usage (W)
500
C1). Adaptive robust control with four-valve flow distribution scheme (only the valves #1, #2 and #4, #5 are used); C2). Energy saving adaptive robust control with five-valve flow regeneration scheme in Liu and Yao (2008); C3). The proposed three-level energy saving adaptive robust control algorithm with five valves and one accumulator;
0 0
tracking error (rad)
0 −1 −2 −3 8
8
10
Book, R. and Goering, C. (1999). Programmable electrohydraulic valve. In SAE Technical Paper. Bu, F. and Yao, B. (2000). nonlinear adaptive robust control of hydraulic actuators regulated by proportional directional control valves with deadband and nonlinear flow gain coefficients. In Proceedings of American Control Conference, 4129–4133. Eriksso, B. (2007). Control strategy for energy efficient fluid power actuators: Utilizing individual metering. Ph.D. thesis, Institutionen Fr Ekonomisk Och Industriell Utveckling, Linkping. Eriksson, B. and Palmberg, J.O. (2011). Individual metering fluid power systems: challenges and opportunities. Proceedings of the Institution of Mechanical Engineers, Part I: Journal of Systems and Control Engineering, 225(2), 196–211. Hu, H. and Zhang, Q. (2002). Realization of programmable control using a set of individually controlled electrohydraulic valves. International Journal of Fluid Power, 3(2), 29–34. Hu, H. and Zhang, Q. (2003). Multi-function realization using an integrated programmable e/h control valve. Applied Engineering in Agriculture, 19(3), 283–290. Liu, S. and Yao, B. (2004). Adaptive robust control of programmable valves with manufacturer supplied flow mapping only. In the 43rd IEEE Conference on Decision and Control, 1117–1122. Liu, S. and Yao, B. (2006). Automated modeling of cartridge valve flow mapping. IEEE/ASME Transactions
1
6
6
REFERENCES
2
time (s)
time (s)
In this paper, an energy saving control strategy is developed to control the joint angle of the boom motion of the hydraulic manipulator. A novel configuration with 5 programmable valves and one accumulator is first proposed to make the energy recovery possible. Then, a triplelevel control algorithm is 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 is demonstrated to be able to maintain excellent tracking performance while achieving much smaller energy consumption compared to the previous 4valve control scheme and 5-valve flow regeneration control scheme.
3
4
4
6. CONCLUSION
C1 C2 C3
2
2
Fig. 4. Energy usage.
−3
−4 0
300
100
x 10
4
400
200
The desired trajectory qd (t) used is a point-to-point S curve from 0rad to 0:5rad. Since the three algorithms have the same offside pressure regulator and motion tracking controller (the same ’level I’ controller design), the corresponding controller parameters are selected to be the same. After the experimental results are obtained, the tracking errors and the power consumptions are plotted in Fig. 3 and 4, respectively. From the plot of tracking errors, it can be seen that three controllers achieve almost the same level of tracking accuracy, showing the good tracking capabilities of the level-I ARC design. However, the plot of power usages shows that the three algorithms give totally different power consumption profiles. For C1, the energy is consumed for both the upward movement and downward movement of the cylinder. For the flow regeneration scheme C2, the energy is consumed only during the upward movement of the cylinder. For C3, with the proposed system configuration and the corresponding control structure, the energy is only consumed during the downward movement of the cylinder. Since the ram area of the rod end of the cylinder is much smaller than that of the head end, the energy usage of C3 is much less than C2. The energy usages of the three algorithms are calculated as EC1 = 1:9KW , EC2 = 1:2KW and EC3 = 0:78KW . It can be seen that, the energy consumption of C3 is much less than that of C1 and C2, showing the ability of energy saving of the proposed hardware configuration and control solution.
5
C1 C2 C3
600
10
Fig. 3. Tracking errors of C1-C3.
89
IFAC Mechatronics '13 April 10-12, 2013. Hangzhou, China
on Mechatronics, 12(4), 381–388. (Part of the paper appeared in the IEEE/ASME Conference on Advanced Intelligent Mechatronics (AIM’05), pp.789-794, 2005, and received the Best Student Paper Competition Award). Liu, S. and Yao, B. (2008). Coordinate control of energy saving programmable valves. IEEE Transactions on Control Systems Technology, 16(1), 34–45. Liu, X., Su, H., Yao, B., and Chu, J. (2008). Adaptive robust control of nonlinear systems with dynamic uncertainties. International Journal of Adaptive Control and Signal Processing, 22.
90