Design and performance analysis of position-based impedance control for an electrohydrostatic actuation system

Yongling FU,Xu HAN,Nrimn SEPEHRI,Guozhe ZHOU,Jin FU,*,Liming YU,Rongrong YANG

aSchool of Mechanical Engineering and Automation,Beihang University,Beijing 100083,China

bDepartment of Mechanical Engineering,University of Manitoba,Manitoba R3T 2N2,Canada

cChina Academy of Launch Vehicle Technology,Beijing 100076,China

dFlying College,Beihang University,Beijing 100083,China

1.Introduction

The aerospace industry is in search of innovative technologies to realize greener,safer,and cheaper commercial air transport with environmental,competitive,and economic benefits.1Many research activitieshave been conducted in the development of power-by-wire(PbW)actuators,2,3including electrohydrostatic actuators(EHAs)and electromechanical actuators,to replace conventional hydraulic ones in various applications of electrical actuation systems,such as flight control,landing gears,thrust reversers,thrust vector control,and space robots.4,5EHAs are considered the most promising PbW actuators in the medium term for all hydraulic and all electric evaluations.Thus,the use of EHAs is deemed attractive.For EHAs applied in aviation,continuous effort is still being made to improve the performance of a hydraulic system.Nonlinearities6,7and parameter uncertainties8should be considered during controller design.In addition,the parasitic(mechanical)stiffness should also be considered,which originates from the actuator itself and the environment that affects the airframe;the driven load(non-infinite anchorage and attachment stiffness)should be mitigated.The effect of stiffness-induced resonance on the driven load is serious(i.e.,chattering of flight control surface or vibration of landing gear).Several studies have been conducted to realize the force control9–12or vibration control13,14of an actuator,and effective reactions to external force disturbances have been reported.EHAs can also be used in space flight applications.With the continuous development of human space activities,including the establishment of spacecraft and space stations,in-space assembly,space experiments,and space repair work,relying solely on astronauts is inadequate.Consequently,space robotic manipulators have attracted increasing attention,and EHAs may be proven useful in such applications.However,in consideration of environment stiffness uncertainties,relative motion should be considered for precise position control.Thus,force control or feedback should be introduced for these specific applications.

Impedance control allows manipulators to interact with environments in a controlled manner.It is a unified method that allows manipulators to work in both constraint and unconstraint environments.Impedance control has been extensively applied to robotic manipulators15–17and hydraulic actuation systems.18–20Unlike position control or force control,impedance control can adjust the apparent dynamics of an actuated system.The merits of impedance control become evident in situations wherein an’actuated system is required to interact with the environment and expected to be compliant to avoid damage due to undesirable collision.Impedance control achieves desirable dynamic interaction responses by modifying the relationship between the end-effector motion and the external force.Conventional methods adopt two control algorithms to handle unconstrained and constrained motion states.As a result,algorithms have to detect any occurrence of contact and switch schemes accordingly.Impedance control can be used in the two motion states without incorporating a switching algorithm.

The two types of impedance control are torque-based impedance control(TBIC)and position-based impedance control(PBIC).In TBIC,a torque/force controller is required in the inner loop.Based on the impedance model,the actual position signal adjusts the required torque/force to achieve the desired impedance.By contrast,the inner loop in PBIC is a position loop,and the required position is modified based on the measured interaction force.In robotic applications,TBIC demands an accurate dynamics model of the system,including friction and nonlinearity,and it is sensitive to uncertainties and time-variant parameters.21PBIC,however,only requires robotic inverse kinematics,which can be easily computed.Furthermore,PBIC is ideal for hydraulic systems due to the difficulty of force control.22

In previous studies,the concept of impedance control was implemented in the contact task control of a manipulator23and various robots in manufacturing,24rehabilitation,25–27cooperation,28and service tasks.29,30In these applications,all the systems were completely driven by electric motors.However,several hydraulically actuated systems in the industry can benefit from the application of impedance control.Hydraulic systems with high power/weight ratios are appropriate in situations requiring a high torque/force.31Heinrichs et al.32developed an impedance controller for an industrial hydraulic manipulator with valve-controlled actuators.Impedance control was also implemented in a hydraulic hexapod robot to render the robot adaptable to uneven and soft terrain.33As a machine for digging soil or lifting heavy objects,excavators frequently interact with unstructured environments.The implementation of impedance control in autonomous hydraulic excavators has also been reported in the literature.34,35For improved vehicle handling and passenger comfort,active suspension systems are used to accommodate the dynamics under different road surface conditions.Robust and adaptive impedance controllers have been introduced to isolate vibration by regulating the interaction between the hydraulic suspension system and the road surface.36–38

However,much of previous research on hydraulic impedance control,including the aforementioned studies,focused on valve-controlled actuators/hydraulic servo actuators(HSAs),which suffer from high throttling losses in servovalves.In particular,the application of position-controlled impedance to EHAs is limited and needs immediate attention because most hydraulically actuated machines are moving from valve-controlled actuations to pump-controlled ones.Any development in that direction can increase energy efficiency and benefit the industry.Consequently,studying impedance control applied to EHAs is important due to many applications,in which a system interacts with an unstructured environment and a controlled motion is needed.Kaminaga et al.39–41designed backdrivable EHAs and applied them to a humanoid robot,robot hand,and knee power assist device with impedance control.However,these inherently flexible EHAs are passive compliant actuators.Their incapability to achieve high positioning accuracy renders them inappropriate in applications that require precise impedance relationships between the position and the force.To date,impedance control of a general EHA without flexibility has not yet been developed.Further research on the application of PBIC to general EHAs is clearly needed.

In this study,we implement the concept of PBIC in a general EHA.An EHA model is firstly constructed.Knowledge of the model is essential for examining position controller stability and the implementing stable and compliant impedance control.A particle swarm optimization(PSO)algorithm42is employed because of the difficulty of identifying the four coefficients in the model.In this algorithm,massive particles are generated to search for the optimal result via iterative calculation.The foundation of PBIC is an accurate and reliable position controller.Friction,inertia,and inevitable leakage are detrimental to an accurate implementation of the position control of the EHA.The nonlinear proportional-integral(PI)position controller introduced by Sepehri et al.43for a valvecontrolled hydraulic actuator is adopted to achieve high performance.The position accuracy and robustness of the position controller for various external loads are investigated experimentally.In impedance control,simultaneous adjustments of three independent target impedance parameters to achieve stability and compliance may be challenging.Thus,the root-locus technique is used for parameter selection;this technique can present the variation tendencies of the dynamics in a diagram form.Experiments are conducted to validate the benefits of applying PBIC to the EHA.PBIC is firstly examined in free space.The behavior of the actuator shows that the apparent impedance matches the target impedance well.The compliance of the actuator is exhibited in an impact test,and itsmotion isstable during interaction with the environment.

The remainder of this article is organized as follows.Section 2 presents the mathematical schematics of the EHA and the PSO algorithm.Section 3 introduces the test rig and identifies coefficients.Section 4 demonstrates the stepwise development of the position controller,from a linear PI model to a nonlinear one,whose stability is analyzed via a simulation using the identified model.Section 5 discusses the selection of target impedance parameters and the experimental results of thePBIC system.Conclusionsarefinally presented in Section 6.

2.EHA system description

A fixed-displacement pump and variable-speed motor(FPVM)-type EHA is used in this study.A schematic of the EHA system is depicted in Fig.1.A controller is constructed(i.e.,integrated motor drive electronics)for the electric motor with position and force control loops.The motor is coupled to a hydraulic pump,which induces oil flow and pressure changes in the hydraulic cylinder.The pressure difference in the chambers results in rod extension that drives the external load,which is considered the effect of a stiffness-dominant environment.A pneumatic source supplies the oil tank to charge the low-pressure side(through the check valve)of the loop to avoid cavitation.In some cases,two relief valves are used to limit the maximum pressure of the hydraulic system for safety.

2.1.Mathematical model

A mathematical model of the EHA is derived for the pre-implementation analysis of the selection of controller parameters.Electric power from the power source is modulated by the EHA controller.In order to track the desired position xd(m),the motor speed ωm(rad/s)is regulated by the control signal uc(V),and it is immune to a variable torque load because of the high-performance motor controller.The current loop in the motor is introduced by considering the winding and inductance effects,which can be described by a first-order transfer function.Therefore,the relationship between the motor speed and control signal can be simplified as an inertial element:

where Kmis the rotary speed gain(rad/(s·V-1)),τmis the motor time constant(s),ucis the control signal(V),and s is the Laplace variable.

On the assumption of an inner rigid in the assembly of the EHA,the pump flow considering internal leakage can be described as follows:

where QLis the load flow(m3/s),Dpis the pump displacement(m3/rad),Kplis the pump leakage coefficient(m3·s-1·Pa-1),and PLis the load pressure(Pa).

In accordance with Newton’s second law,EHA friction is linearized as a viscous section,and load dynamics can be expressed by the following equation:

where A is the piston area(m2);MLis the equivalent total mass of the piston,rod,and load(kg);Kfvis the viscous friction coefficient(N/(m·s-1));FLis the load force(N);and xpis the piston displacement(m).

The equation for the load flow QLis derived by considering hydraulic compression,leakage,and piston motion as follows44:

where Vtis the total actuator volume(m3),βeis the effective bulk modulus(N/m2),Kclis the cylinder leakage coefficient(m3/(s·Pa-1)),˙PLis the rate of load pressure(Pa/s),and˙xpis the velocity of the piston(m/s).

For simplicity,both pump and cylinder leakages are neglected in subsequent analysis.The piston displacement of the EHA is obtained by combining Eqs.(1)–(4)and taking the Laplace transform as follows:

Fig.1 Schematic of a typical FPVM-EHA system.

where Kin,Kdp,and Khsare the model parameters and given by

If the actuator interacts with a stiffness-dominant environment,the relationship between the piston position and the load force is

where keis the environment stiffness(N/m).

Substituting Eq.(7)into Eq.(5)yields the following transfer function of the EHA:

where the identified coefficients A1to A4are as follows:

The transfer function in Eq.(8)shows that the EHA is a fourth-order system.This function is for an ideal system,and non-idealities such as friction are ignored.

2.2.PSO algorithm

According to the obtained mathematical model of the EHA,the values of A1to A4in Eqs.(8)and(9)should be determined firstly.A PSO algorithm is adopted to identify these coefficients.PSO is based on the behavior of social systems,such as bird flocking.45A number of particles are positioned in the solution space of a problem,and each particle’s position represents a solution.These particles then move in the space to search for the optimal solution.Each particle determines its next movement according to the best position found by the particle itself,as well as the best position found by the entire swarm.The velocity of each particle denotes the direction and step of the next movement.This method has been successfully employed in some parameter optimization problems,such as modeling of fluid dampers46and electric power systems.47

Mathematically,each particle is characterized by threedimensional vectors as follows:

where i is the index of the i-th particle,D is the dimension of the solution space,xiis the vector describing the i-th particle’s current position(a solution),viis the velocity vector of the i-th particle,and piis the best position vector found by the i-th particle.

Another vector for the entire swarm is

where pgis the best position vector found by the entire swarm.

In each iteration,the quality of the current position,xi,is assessed by a fitness function.If this position is better than the previous best position,pi,the best position is updated using the current position.When all vectors,pi,are obtained,the best one is compared to the global best position,pg.If the best piis better,then the vector pgis updated.Subsequently,the velocity,vi,to move xi,the i-th particle,is determined,and a new position for the next iteration is obtained.After all the iterations,the final global best position,pg,is the best solution found by the algorithm.

On each dimension j,the iterative equations for the i-th particle are

where k is the iteration number,j is the dimension,c1and c2are the acceleration constants,and r1jand r2jare the random coefficients.

The acceleration constants,c1and c2,which determine the convergence speed of the algorithm,are both set to 0.1.The random coefficients,r1jand r2j,which are set from 0 to 1,introduce some disturbances to prevent the particle from falling into the local optimal area.The specific process of the algorithm is illustrated in Fig.2.

3.Test rig configuration and coefficient identification

Fig.2 PSO flow chart.

Fig.3 Electrohydrostatic actuation system test rig.

The developed approach is implemented on an experimental platform shown in Fig.3.The test rig consists of four parts:an oil source system,a mechanical bench,an EHA system,and a loading system.Extra loads and resistive cylinders for additional inertia and damping can be optionally installed on the main actuator to examine the robustness of the controller.Various springs are used to emulate stiffness-dominant environments.

The position signal is acquired by a digital encoder sensor,which has a resolution of 0.031 mm.The velocity signal is calculated using the 20-point linear regression method and position data.24The analog signal from a force sensor is used in the impedance control loop.A servo drive(Teknic:SSt-6000-R)is employed to control the permanent magnet synchronous motor’s(Teknic:M-4650)rotary speed.The motor position,speed,torque loops,and compensation are integrated in the servo drive.For the hydraulic pump,a fixed-displacement piston pump is used(Parker:F11-005HUCV).The controller is implemented using a personal computer with Simulink and Quanser’s hardware-in-the-loop control board.The control cycle is 1 ms,and the range of the control signal transmitted to the motor servo drive is±10 V.

The number of particles(i),dimension of the solution space(j),and maximum iteration times(k)are initially generated(Table 1)to obtain the values of the four coefficients of the EHA system test rig.

Thus,the four coordinates of each particle’s position,xi,represent the values of the four coefficients A1to A4.The EHA model obtained using each particle,the transfer function Gi EHA,is

A fitness function is constructed to evaluate the quality of each particle.The actual frequency response of the EHAsystem is obtained firstly via experiments.The stiffnessdominant environment is emulated using ke.A linear PI controller with a proportional gain KPand an integral gain KIis employed to constitute a position loop with the EHA.These gains are determined by trial and error,and their values are shown in Table 1.

Table 1 EHA controller and load parameters.

A total of 24 groups of different frequencies and amplitudes are applied to a 13-mm sinusoidal wave for the input signals.The frequency response data of the EHA are obtained by measuring the displacement of the hydraulic cylinder piston rod.The transfer function in Eq.(14)is used to simulate the frequency response of the identified system with the same environment stiffness and controller in MATLAB/Simulink.The errors between the simulated and experimental results are used to assess the fitness of the particles.The fitness function f(λ)is given by

where ampaand phsaare the actual amplitude(dB)and phase(°),respectively;ampsand phssare the simulated amplitude(dB)and phase(°),respectively; λ is the weight coefficient,which increases the weight of phase accuracy in the fitness function.

To observe the curves of various identification results and experimental data points,the weight coefficient,λ,is set to 10.During the identification,the fitness value of the global best position does not change after the maximum iteration number of 100.Therefore,the termination criterion is the maximum iteration number of 100.The convergence of the fitness value of the global best position is shown in Fig.4.The fitness value eventually converges to 0.702.The identified values of the coefficients are listed in Table 2.

Fig.4 Fitness value of global best position.

Table 2 Identified values of four coefficients.

Fig.5 Frequency response of identified system.

Finally,the simulated frequency response and experimental data points are acquired using the identified coefficients of A1to A4for the EHA model.The data points are plotted in Fig.5,in which the solid line represents the response of the simulated model and the circles indicate the experimental data.The analytical simulation results show a good agreement with the corresponding experimental data.This actuation system model is used in the subsequent investigation of position controller stability and the selection of target impedance parameters.

4.Position controller design

4.1.Linear PI controller

After the EHA system is studied and the fourth-order model is identified,position control is subsequently investigated.The conventional linear PI controller is expressed as

where e(t)is the position error with time varying(m),and I(t)is the error integral with time varying(m·s).xcis the command position(m).uc,xc,and xpare time-varying,which can be noted as uc(t),xc(t),and xp(t),respectively.t is the time moment(s).Δt is the sampling time interval(s).

The controller gains are shown in Table 1.Both the transient response and steady-state accuracy are acceptable.Tests on the multiple-step(i.e.,200,100,50,25,and 10 mm)input response and decreasing sine wave(0.8 Hz)response are then conducted to examine the performance of the linear PI controller in free motion.The results are shown in Fig.6.However,as shown in Fig.6(a)and (c),although the performance of the controller is good in the small-step response,integral saturation in the integral signal causes an overshoot and prolonged settling time in the large-step response.In Fig.6(b)and(d),the decreasing sine wave frequency is 0.8 Hz,and the maximum amplitude is less than 0.75 mm.In the initial phase,the controller can follow the command signal.However,with a decrease in the signal amplitude due to the dead zone and static friction effects in the EHA,the delay of the actual displacement increases,and then the controller fails to track the trajectory.

Furthermore,the Nyquist plot of the position loop is obtained using the identified model of the EHA to assess the stability margin.The plot is shown in Fig.7,in which the stability margin is sufficient.The gain margin is 14.9 dB,and the phase margin is 84.6°.

4.2.Nonlinear PI controller

Morse’s method48is developed to solve the integral saturation effect when controlled by a variable-step signal,and this method is improved by Sun et al.13with the following formula:

where α is the weight coefficient and˙e is the velocity error(m/s).

The nonlinear adjustment factor can regulate the intensity of the integral based on the velocity error.For instance,in the transient process of a step response,the adjustment factor approaches zero due to a high velocity error.The integral term dramatically weakens to avoid integral saturation and overshoot.In the steady state,the adjustment factor is unity as the velocity error equals zero.The full-power integral term can remove the steady-state error effectively.The weight coefficient determines the sensitivity of the integral term to the velocity error.The weight coefficient is linked to the command velocity to maintain a consistent adjustment effect at different velocities:

where αstaticis the static weight(m2/s2)and μ is the command velocity(m/s).Decreasing αstaticcan enhance the effect of the adjustment factor.The relationship between the weight coefficient and the command velocity is determined by μ.

In this study,αstaticand μ are determined by trial and error(Table 3).The test results of the modified Morse integrator controller in free motion are demonstrated in Fig.8.The tests on the multi-step(i.e.,200,100,50,25,and 10 mm)input response and decreasing sine wave(0.8 Hz)response are conducted following the same procedure used for the linear PI controller.

Integral saturation is effectively addressed,as shown in Fig.8(a).Even in the large-step response,no overshoot occurs during the transient process.Fig.8(c)exhibits less integral effort because the nonlinear control terms added to the PI controller only allow the integral to work.The system presents a better performance with less effort compared to the results in Fig.6(a)and(c).Despite the improvement in the step response,the modified controller still fails to track smallamplitude command trajectories,as shown in Fig.8(b)and(d).

Fig.6 Responses of linear PI controller in free motion.

Fig.7 Nyquist diagram for stability analysis.

Table 3 Parameters for nonlinear controller design.

Real-time detection of whether relative motion is halted by stiction is critical to compensate for the effect of static friction.Thus,an indicator that significantly changes from a normal state to a halted state should be established.The method of Heinrichs13compares the position error,its derivative,and its integral during a halted state.Velocity error,which is the derivative of the position error,supplies a strong signal.Therefore,the velocity error is utilized to construct the indicator(vst),which is established by multiplying the velocity error by a nonlinear factor as follows:

where vstis the velocity indicator(m/s).β is the parameter increasing selectivity.is the command velocity(m/s).is the piston velocity(m/s).When relative motion is halted by stiction(=0),the nonlinear factor equals unity,and the indicator increases withUntil the actuator starts to move,the indicator is zero due to the increasing

With this indicator,a threshold of motion state vth(m/s),is introduced to assess the motion state.Once the indicator exceeds the threshold,the actuator is assumed to be halted by stiction,and the integral part of the control signal is augmented instantly.Shortening the time in a halted state without waiting for the accumulation of error is beneficial.Consequently,the modified integral term is

where Ucomis the compensation voltage(V).Through this modification,the control signal will be directly increased to the compensation voltage when a halted state is detected.

In Eq.(19),β determines the sensitivity of the indicator to the piston velocity,and it should be high to amplify the distinction between normal and halted states.The threshold can be reasonably set by observing the value of the indicator to ensure a correct judgment of the motion state.The compensation voltage should be adequate to activate the actuator halted by stiction;excessive compensation can result in an overshoot.

The values of β,vth,and Ucomare also determined by trial and error(Table 3).The effect of stiction compensation is verified by a decreasing sine wave response test,the results of which are shown in Fig.9.The effect of stiction is alleviated,and the function of integral augmentation is evident in the plot of the control signal.The command position inputs of variable-frequency decreasing sine waves,the step,and the ramp are used in the tests to examine the performance of the final controller in free motion with various situations.The results are shown in Fig.10.

Fig.8 Responses of modified Morse integrator controller in free motion.

Fig.9 Responses for modified Morse integrator controller-introduced velocity error-triggered integral augmentation in free motion(0.8 Hz decreasing sine input).

In the tests,the step and ramp responses in the transient process and steady state are satisfactory.During the tracking of decreasing sine waves,the response delay increases with increasing frequency,but the stiction of the actuator is well compensated for the entire time.

Further tests are conducted to assess the robustness of the controller.Various combinations of load,damping,and environmental stiffness are applied.In subsequent tests,an inertia load with an equivalent translational mass of 10 kg is carried by the spring to present the environmental stiffness,ke.The actuator is also connected to resistive cylinders,which increase damping.Over 20 experiments are performed using the final controller.For brevity,only typical results are presented in Fig.11.

The maximum position state error is about 0.08 mm in thefirst phase of the step command of Fig.11(a).The load situation is shown through plots of the load pressure.The performances of the nonlinear controller in constrained motion are almost consistent with those in free motion.The robustness of the nonlinear position controller in PBIC should be proven using various types of load.

5.Experimental evaluation of PBIC

The experiments on the designed position controller for the EHA system show that various environmental conditions,that is,varied stiffness of the load,result in a deviation of the real control accuracy from the desired one.Therefore,adjustable mass,damping,and stiffness of the target impedance block are introduced to develop a PBIC architecture.The block diagram of a typical PBIC scheme is depicted in Fig.12.The inner loop of PBIC is a general position control loop.The position controller generates a control signal,uc,to regulate the piston position,xp,of the EHA,which interacts with the environment.Depending on the target impedance,the command position,xc,is produced by the desired position,xd(m),desired force,Fd(N),and load force,FL(N),which originate from the environment.To realize expected dynamics,the target impedance transfer function(GTI)is

where GTI(s)is the target impedance transfer function.Mt,Bt,and ktrepresent the target mass(kg),damping(N·m-1·s-1),and stiffness(N/m),respectively.Based on Eq.(21),the target impedance relationship is

Fig.10 Responses offinal controller in free motion.

Fig.11 Responses of final controller in constrained motion.

Fig.12 Block diagram of developed PBIC scheme.

The load force signal is introduced as the force feedback for PBIC,as shown in Fig.12.However,in the test rig,the technical imperfection of the force sensor with white noise detrimentally affects the signal quality of the load force and significantly deteriorates control.Thus,a second-order Butter worth filter49is introduced in the feedback loop.The Bode diagram in Fig.5 shows that the cut-off frequency of the EHA position control loop is 2.5 Hz.The cut-off frequency of the filter is set to fbf=3 Hz(ωbf=2πfbf),and the filter damping ratio is set to ξbf=0.7 to reduce the influence on the system dynamic performance.The transfer function of the Butter worth filter,GF,is

where GF(s)is the Butter worth filter transfer function,fbfis the Butter worth filter cut-off frequency(Hz),ωbfis the Butter worth filter cut-off angular frequency(rad/s),and ξbfis the Butter worth filter damping ratio.

The measured and filtered noise signals are shown in Fig.13.A comparison of the spectra shows that the noise,especially at high frequencies,is remarkably restrained by the filter.The smooth force signal is used in impedance control.

5.1.PBIC in free space

A free-space response test is conducted to verify the validity of PBIC,as reflected by matching the apparent actuator impedance with the target impedance.In this experiment,the actuator operates in free space without any obstacle.The desired position(xd)is set to zero,and the desired force(Fd)is a step signal of 200 N at 2 s.The behavior of the actuator should be equivalent to suddenly release a spring pressed with a force of 200 N,and the dynamics after that moment is determined by the target impedance.The target mass,Mt,is 200 kg,and the stiffness,kt,is 103N/m.

For comparison,three values of target damping(Bt=270,890,and 2680 N·s/m)are adopted to produce the damping ratios(ξ)0.3,1,and 3.The responses are plotted in Fig.14.The position results demonstrate the step-response differences between under damping,critical damping,and over damping systems.The 200 mm displacement is correct for the 200 N force step with a target stiffness of 1000 N/m.The oscillation period of the under damping system matches the desired value of 3 s.Various target impedances are achieved in the tests.

Fig.13 Comparison between measured and filtered noises.

Fig.14 Step responses of PBIC of actuator in free space.

5.2.Selection of target impedance parameters

In applications of PBIC,the operating modes of the EHA are divided into free motion and constrained motion.In free motion,the load force is zero,and the dynamics and stability of the system depend on the position controller.Target impedance parameters influence constrained motion only.

In this section,the mechanism by which target impedance affects the dynamics of the system is shown firstly.The parameters of the linear PI position controller(KPand KI)are utilized,and a stiffness-dominant environment is considered.Moreover,the EHA is in contact with the spring stiffness(ke).The desired position(xd)is set to zero,and the desired force(Fd)is set as a step of 2000 N.The target mass and stiffness are fixed(i.e.,Mt=150 kg and kt=120 kN/m),and various target damping values(Bt)are used.The step responses are shown in Fig.15.

As shown in Fig.15(a),motion overshoot occurs when the target damping Btis 21.4 kNs/m.When the target damping Btis greater than 97.3 kNs/m,motion is overdamped,and the response speed declines.As shown in Fig.15(b),when the target damping Btis 2 kN·s/m,motion experiences sustained oscillation.For a target damping Btbetween 2 and 97.3 kN·s/m,motion is under damped.The dominant poles in the right-half plane indicate that the system is unstable.In this example,a target damping Btof 97.3 kN·s/m is adopted because of the compromise between speed and oscillatory motion.

Fig.15 Step responses of PBIC of actuator with selected target impedance parameters.

Fig.16 Block diagram of transfer function of PBIC system.

In this analysis,the linear PI position controller is employed.The nonlinear controller obtained in Section 4.2 is established on the basis of this linear controller.Thus,these results are used in subsequent PBIC experiments,which adopt the nonlinear controller.

5.3.Impact test

Fig.17 Root locus of controlled actuator with various target damping Bt.

These effects can be illustrated via the root-locus technique.Based on Fig.12,the block diagram of the transfer functions of the PBIC system is depicted in Fig.16,in which GTI(s),GPC(s),GEHA(s),GEI(s),and GF(s)are the transfer functions of the target impedance,position controller,EHA system,environment impedance,and filter,respectively.

The open-loop transfer function of the system is obtained using the identified model of the EHA.Nine closed-loop poles exist in the system.Root-locus analysis demonstrates how these poles change with varying target damping(Bt).Only the root locus of the two dominant poles is shown in Fig.17.

As shown in Fig.17,as the target damping Btapproaches 2 kN·s/m,the dominant poles move closer to the imaginary axis;as a result,motion does not converge.When the target damping Btis higher than 97.3 kN·s/m,the dominant poles are both located on the real axis,and motion is overdamped.

The position-tracking capability in free motion and the compliance in constrained motion of the PBIC system can be examined with an impact test.In the impact test,the desired position,xd,is set as a sine wave so that the actuator is commanded to track the sine wave trajectory.However,the spring stiffness(ke)as an obstacle is placed at zero position,cutting the sine trajectory in half.The desired force(Fd)is set to zero.The target impedance setting of damping is Bt=97.3 kN·s/m,and previous values of Mtand Ktare adopted.The test results are shown in Fig.18(a)and(c).

The EHA tracks the position trajectory well in free motion,and compliance is achieved in constrained motion.For comparison,the target damping Btis modified to 2 kN·s/m,which is used in Fig.15(b).The same test is conducted without any otherchanges,and theresultsareshown in Fig.18(b)and(d).During this test,position trajectory tracking is acceptable in free motion,thereby showing the applicability of PBIC to the EHA system.

6.Conclusions

(1)PBIC was achieved for an industrial electrohydrostatic

Fig.18 Impact test of target damping Bt.

actuation system in this paper.An EHA mathematical model was established and used in the stability investigation of the control system with different position controller gains and target impedance parameters.The coefficients of the model were identified using the PSO algorithm and the experimental frequency response of the EHA.Optimization was highly effective in finding an optimal solution to fit all the experimental data.A nonlinear PI position controller was developed stepwise to meet the positioning requirement of PBIC.The final controller effectively addressed the problem of integral saturation and compensated for the adverse effect of stiction.A position accuracy of 0.08 mm was attained,and robustness to various external loads was experimentally examined.

(2)The validity of PBIC was initially tested in free space using the developed position controller.The apparent impedance of the actuator matched the target impedance well in experiments.In an impact test,the target impedance not only determined the compliance of the actuator but also in fluenced the stability of the control system.Subsequently,the selection of target impedance parameters was investigated using the identified model of the EHA.During the impact test,the actuator interacting with the environment was compliant,and motion was stable during the entire time.Therefore,the effectiveness of EHA position control using PBIC to remove the tracking error even when the environment stiffness varied was verified.

(3)The equivalency between PBIC and explicit force control can be further studied for the EHA system.It can show how force control will be achieved using the proposed PBIC method to remove the tracking error.To make the EHA system adaptive to varying environmentalcharacteristics, the PBIC parameters can beidentified via root-locus analysis and used for onlinereal-time adjustment of target impedance through further studies.

This work was completed in the Fluid Power and Tele-Robotics Research Laboratory at the University of Manitoba.The authors would like to acknowledge the supports of the Natural Sciences and Engineering Research Council(NSERC)of Canada,China Scholarship Council(CSC),and the NationalNaturalScienceFoundation ofChina (Nos.51275021 and 61327807).

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