51st IEEE Conference on Decision and Control December 10-13, 2012. Maui, Hawaii, USA
Design Of Robust Nonlinear Force And Stiffness Controller For Pneumatic Actuators Behzad Taheri, David Case, and Edmond Richer Abstract— This paper introduces a new backstepping-sliding mode controller designed specifically for pneumatic actuators. Based on a detailed mathematical model of the pneumatic system that included the dynamics of the valves, the algorithm was proven able to track the desired force and stiffness independently without chattering. The global Lyapunov asymptotic stability of the pressure tracking for each chamber was analyzed. Numerical simulations and validating experiments using a real-time platform were performed for a pneumatic actuator suitable for wearable robotics applications.
I. INTRODUCTION Applications in medical robotics, prosthetics and orthotics, haptic interfaces, virtual simulators, and demanding industrial applications require high performance actuators, with rapid dynamics, high force output per unit weight and volume, as well as low mechanical impedance, while minimizing cost. Electrical motors, while easy to model and control, have low power density in applications that require direct drive and low speed high force output. In such applications they require torque amplifying devices, that increase impedance and reduce their back-drivability [1]. While hydraulic actuators have very good power/volume ratio, their high stiffness can be detrimental in some applications, and they have a larger contamination risk. In contrast, pneumatic actuators have large power output, are lightweight and clean, and have a relatively low cost [2], [3]. Unfortunately, due to compressibility of air and highly nonlinear flow through pneumatic system components, high bandwidth position and force control requires intricate, and relatively difficult to implement, nonlinear algorithms [4]–[6]. In addition to force control applications in prosthetics, orthotics, rehabilitation, and locomotion require the ability to control the impedance or stiffness of the actuator [7]– [10]. Pneumatic actuators are inherently suitable for such applications since their stiffness can be adjusted by simply modifying the internal pressure in the antagonist chambers. Shen and Goldfarb showed a simultaneous force and stiffness control of a pneumatic actuator that uses two proportional valves and a sliding mode control algorithm [9]. However, they did not include the dynamics of the valves into their model and controller design. Even though the sliding mode controller exhibit inherent robust properties and is capable of compensating for uncertainties in the mathematical model [11], it is particulary prone to chattering. This behavior can be exacerbated by the delay introduced by the valve dynamics and the hardware limited available sampling rate. B. Taheri, D. Case, and E. Richer are with the Department of Mechanical Engineering, SMU, Dallas, TX, 75205 USA e-mail:
[email protected].
978-1-4673-2066-5/12/$31.00 978-1-4673-2064-1/12/$31.00 ©2012 ©2012 IEEE IEEE
In this paper we present a new backstepping-sliding mode algorithm based on a detailed model of the pneumatic system that incorporates the dynamics of the valves, capable to track the desired force and stiffness independently without chattering. The global Lyapunov asymptotic stability of the pressure tracking for each chamber is analyzed. Numerical simulations were performed and the algorithm was implemented in a real-time controller. Validating experiments were performed for a pneumatic actuator selected for application in a upper limb wearable orthosis designed for essential tremor patients. II. DYNAMIC MODEL OF THE ACTUATOR AND PROPORTIONAL VALVE A. Dynamic Model Of The Cylinder Chambers In order to control the force and stiffness of the pneumatic actuator the pressure of the cylinder chambers should be controlled such that the difference between them determines the desired force and the summation of them governs the desired stiffness of the actuator. This section describes the relationship between the pressure in each chamber and the air mass flow rate and the piston position. As shown in Fig. 1 the pneumatic system consists of a double acting pneumatic cylinder actuated by two proportional 3-way valves. Richer and Hurmuzlu modeled each chamber as a control volume with the following assumptions: (i) the air is an ideal gas, (ii) uniform distribution of pressure and temperature within each chamber, and (iii) the kinetic and potential energy of the air is negligible [3]. The charging process of the chamber is better modeled by an adiabatic process whereas the discharging process of each chamber is closer to an isothermal process. Considering these assumptions and the different thermodynamics process associated with chamber charging and discharging, and applying the conservation of mass, ideal gas model, and the conservation of energy to each chamber we get, P RT αm ˙ − α V˙ (1) P˙ = V V where R is the ideal gas constant, P is the pressure, T is the absolute temperature, V is the volume, and m ˙ is the mass flow rate in or out of the cylinder chamber. Parameter α = γ for the adiabatic process (γ = Cp /Cv is the specific heat ratio) and α = 1 for the isothermal process. In this work, the value of α = 1 was assumed and the controller was designed robust against this parametric uncertainty. The volume of each chamber depends on piston position, 1 V = V0 + A L ± xp (2) 2
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Comparing (4) and (8) the partial derivative of pressure with respect to position is, ∂Pa,b −Pa,b =± ∂xp La,b ± xp
(9)
from (7) and (9) the actuator stiffness will be, K=
A a Pa A b Pb + La + xp Lb − xp
(10)
B. Air Flow Through Proportional Valves The valves used in this study are 3-way proportional spool valves, in which the spool position is controlled by a voice coil motor. As shown in Fig. 2 the spool dynamics can be modeled with a single degree of freedom mass-springdamper system. The equation of motion for the valve spool can be written as, Ms x ¨s + cs x˙ s + 2ks xs = Ki u Fig. 1.
Schematic representation of the pneumatic system
where V0 is the inactive volume at the end of the stroke and includes the volume of the tube connecting the chamber to the pressure sensor, L is the piston stroke and xp is the piston position. The normalized inactive length associated to the inactive volume in each chamber can be defined as: Loa = Voa /Aa , and Lob = Vob /Ab . Therefore the volume of each chamber as a function of the piston position will be, 1 Va,b = Aa,b (La,b ± xp ), La,b = Loa,ob + L (3) 2 Combining (1)and (3) and assuming the isothermal process for charging and discharging of the chambers the pressure equations of the chambers become, RT −Pa,b P˙a,b = m ˙ a,b ± x˙p Va,b La,b ± xp
(4)
In (3) and (4) the plus sign is for chamber a and the minus sign is applied to chamber b. The force produced by the pneumatic the actuator is, F = Pa Aa − Pb Ab − Patm Ar
(5)
The stiffness of the actuator is the rate of change of the actuator force with respect to the piston position, K=−
∂F ∂xp
(6)
Using force from (5) into (6), K = −Aa
∂Pa ∂Pb + Ab ∂xp ∂xp
(7)
Assuming constant temperature, the pressure in each chamber will be a function of the mass of the air inside the chamber (m) and the position of the piston (xp ). The time derivative of pressure becomes, dP ∂P ∂P = m ˙ + x˙p dt ∂m ∂xp
(8)
(11)
where xs is the spool position, Ms is the spool and coil assembly mass, cs is the viscous friction coefficient, ks is the spool springs constant, u is the coil current, and Ki is the force coefficient of the coil. The Coulomb friction force is neglected in this model and will be considered as an uncertainty in the design of the controller. The inertial force Ms x ¨s in (11) is much smaller than the viscous and spring forces. Thus, the response of the valve can be approximated by a first order system with a sufficient precision. The dimensionless position of the spool is defined as z = xs /max(xs ). Thus the model can be represented as, z˙ +
1 1 z= u τv τv
(12)
which is a first order model for spool dynamics with the time constant τv = 0.0015. On the other hand the relationship between the spool position and the area of the valve is an unknown nonlinear function which needs to be identified. Specifically, the area of the valve, and consequently the air flow, has a dead zone when the spool is at the middle position that should be considered in the design of the control system. The flow through the valve can be modeled as compressible flow through an orifices with variable area [3]. Assuming the ideal gas model for the air the mass flow rate of the valve is, m ˙ v (Pu , Pd , z) = Av (z)µ(Pu , Pd ) (13) where Av (z) is the aperture area of the valve (area of the orifice) as a function of the spool position, and µ(Pu , Pd ) can be found by assuming an isentropic process through the valve (orifice), Pu Pd ≤ Pcr C1 Cf √T P u s 1/γ (γ−1)/γ µ= Pd Pd Pd P C2 Cf √u 1− > Pcr Pu Pu T Pu (14) in this equation Pu is the upstream pressure of the valve, Pd is the downstream pressure, Cf is the discharge coefficient of the valve, and C1 and C2 are dependent on the ideal gas
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Fig. 2. valve
Single DOF mass-spring-damper model of the proportional spool
Fig. 3. Experimental values for the valve area as a function of z (dimensionless spool position) and the fitted valve area function
constant and specific heat ratio of the air. Pcr is the critical pressure ratio that differentiate subsonic and sonic flows in the valve (Pcr = 0.528) [3]. A set of experiments were run to identify the Cf and Av (z) based on (13, 14). One pressure sensor was connected to the input port and another one to the output port of the valve and the air flow was measured for the different upstream pressures and different currents applied to the valve’s coil. The identified value of the valve discharge coefficient from the experimental results was Cf = 0.5393. Assuming the symmetry of the spool geometry the following function was used to establish the relationship between the spool position and the aperture area of the valve, Av (z) = sgn(z) a + b|z| + c|z|2 + d|z|3 + f |z|4 (15) 1 − tanh(g|z| + h) The unknown coefficients of this function were found using the least-squares method. The fitted function for the valve aperture area as a function of dimensionless spool position, as well as the values obtained experimentally, are shown in Fig. 3.
in (5,10) are defined as functions of chamber pressures, and piston position. For the system presented in Fig. 1 that contains two pressure sensors and one position sensor, the tracking control of force and stiffness of the actuator is equivalent to the tracking control of chamber pressures Pa and Pb independently. For precise control of chamber pressure especially in higher frequencies the dynamics of the valves should be considered. On the other hand, considering the dynamics of the valves along with the mathematical model of the pressure in each chamber results in a nonlinear system with unmatched uncertainty. Backstepping method relaxes the matching condition and is one of the strong nonlinear control design tools for systems with unmatched uncertainty. The sliding mode control is rather simple to design and robust against the uncertainties that satisfy the matching condition [12]. Accordingly, following the backstepping method in each step a new sliding surface is defined and a robust sliding mode controller is designed for the virtual input of each subsystem [13]. First the desired pressure of each chamber is found based on the desired force-stiffness functions, and two independent pressure tracking controllers are designed using the backstepping-sliding mode algorithm. In the first step the pressure tracking error is defined and the required area of the valve is found as a virtual control, and in the second step the displacement of the valve is controlled to force the area of the valve to track the required area obtained in the first step. Based on (5) and (10) the desired values of the force and stiffness are, Fd (t) = Pad (t)Aa − Pbd (t)Ab − Patm Ar A P (t) Ab Pbd (t) Kd (t) = a ad + La + xp Lb − xp
Solving these equations the desired pressure of each chamber will be described as a function of the Fd (t), Kd (t), and piston position, (L + x ) A P + (L − x ) K (t) + F (t) a p r atm b p d d Pad (t) = Aa (La + Lb ) (Lb − xp ) (La + xp ) Kd (t) − Ar Patm − Fd (t) Pbd (t) = Ab (La + Lb ) (17) The methodology for designing the controller for pressure tracking is presented for chamber A of the pneumatic actuator, and the pressure tracking for chamber B is derived similarly. The dynamic equation of the pressure in chamber A along with the dynamics of valve A can be written as, RT Pa x˙ p P˙a = µa (Pa , za )Ava (za ) − Va La + xp z˙a + 1 za = 1 ua τv τv
III. DESIGN OF FORCE-STIFFNESS CONTROLLER FOR PNEUMATIC CYLINDERS In this section a robust nonlinear controller based on a backstepping-sliding mode technique was developed for the pneumatic system to control the force and stiffness of the actuator simultaneously. Force and stiffness of the actuator
(16)
(18)
where the input of the system is the valve current ua , the output is the pressure measured by the pressure sensor Pa ,
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and the flow function µa is defined as, ( µ(Ps , Pa ) za ≥ 0 µa (Pa , za ) = µ(Pa , Patm ) za < 0
In the next step the error between the actual area of the valve and its desired value is defined as, (19)
with function µ(Pu , Pd ) defined in (14). The uncertainty sources for the pressure equation are: neglecting the dynamics of the air through the tube connecting the valve to the chamber, using the pressure of the chamber for estimating the pressure of output of the valve, assuming the uniform pressure and temperature inside the chamber, assuming the isothermal process for charging and discharging of the chamber, estimating the inactive volume of the chamber (V0a ), and the error in estimating the valve area (Ava (za )). Since there is no feddback sensor for the spool position the state variable za (t) has to be estimated using a numerical observer. Consequently there will be uncertainty in the spool position dynamic model. The aforementioned uncertainties can be included in the dynamic model as δPa (Pa , za ) and δza (Pa , za ). The perturbation functions δPa and δza are assumed to be bounded functions: δPa < b1 |Pa |, and δza < b2 |Pa | + b3 |za |. In the first step of the backstepping-sliding mode design the dynamics of the valve (za ) is neglected. Assume the system with a virtual input β, Pa x˙ p RT µa (Pa , za )β(Pa , t)− +δPa (Pa , za ) (20) P˙a = Va La + xp The tracking error and the first sliding surface are, e1a = Pa (t) − Pad (t) Z s1a = e1a (t) + λa e1a (t)dt
(21)
V˙ 1a = s1a
Va
µa (Pa , za )β −
Pa x˙ p + δPa La + xp − P˙ad + λa e1a (22)
Designing the virtual input β as, βa (Pa , t) =
Va Pa x˙ p + P˙ad − λa e1a RT µa La + xp − η1a Sgn(s1a )
(23)
in which βa (Pa , t) is the desired value for the area of the valve Ava (za ) to ensure the asymptotic pressure tracking. The time derivative of the Lyapunov function V1a becomes, V˙ 1a = s1a − η1a sgn(s1a ) + δPa < −η1a |s1a | + |δPa ||s1a | = |s1a |(|δPa (Pa , za )| − η1a (24)
(25)
The second Lyapunov function is defined as, V2a =
1 1 2 s1a + e22a 2 2
(26)
and, V˙ 2a = s1a s˙ 1a + e2a e˙ 2a = s1a − η1a sgn(s1a ) + δPa RT 1 ∂Ava 1 ∂Ava + e2a s1a µa − za + u Va τv ∂za τv ∂za ∂Ava (27) − β˙ + δza ∂za The control law for the current of the valve A (ua ) will be designed as, RT µa τv + β˙ a (Pa , t) − η2a Sgn(e2a ) ua = za + ∂Ava − s1a Va ∂za (28) Therefore the derivative of the Lyapunov function will be, ∂Ava V˙ 2a = V˙ 1a + e2a − η2a sgn(e2a ) + δza (29) ∂za We can assume an upper bound for the function |
where the desired value for the pressure is defined in (17). For the system with virtual input β we define the Lyapunov 1 function: V1a = s21a . The time derivative of this function 2 is, RT
e2a = Ava (za ) − βa (Pa , t)
∂Ava as, ∂za
∂Ava ∂Ava | < b4 |za | ⇒ | ||δza | < b5 |Pa ||za | + b6 za2 (30) ∂za ∂za
where b5 = b2 b4 , and b6 = b3 b4 . Now we can choose the appropriate value for the switching law gain to ensure the robustness of the controller, ηza − (b5 |Pa ||za | + b6 za2 ) = ρ2a > 0 ⇒ V˙ 2a = V˙ 1a − ρ2a |e2a | ≤ 0
(31)
Therefore the tracking controller is globally asymptotically ˙ a , t) is the time derivative of the desired stable. In (28) β(P area of the valve and can be expressed as, x˙ 2p Va ¨ x˙ p Pad + P˙a − Pa RT µa La + xp (La + xp )2 x ¨p d + Pa − λa e˙ 1a − η1a sgn(s1a ) − La + xp dt Va ∂µa ˙ Pa x˙ p ˙ad −λa e1a −η1a sgn(s1a ) P + P (32) a RT µ2a ∂Pa La+xp β˙ a (Pa , t) =
The time derivative of Pa can be found using numerical methods or can be estimated using (18). The switching parts of the controller, sgn(.), can be replaced with the saturation function within a boundary layer, φ1a and φ2a respectively for s1a and s2a , to avoid chattering. y sgn( ) |y| > |φ| y φ sat( ) = (33) y φ |y| ≤ |φ| φ
If η1a − b1 |Pa | = ρ1a > 0, then V˙ 1a = −ρ1a |s1a | ≤ 0. 1195
(a)
(b)
(c)
(d)
Fig. 4. Numerical simulation results for sinusoidal force and stiffness with different frequencies: (a) and (b) control inputs and valves spool displacements, (c) desired and simulated pressures in the actuator chambers, (d) desired and simulated force and stiffness
IV. NUMERICAL AND EXPERIMENTAL RESULTS A. Numerical Simulation
better reflect the mixed nature of the analog physical system coupled with the discrete computer implemented controller.
The intended application of this pneumatic actuator was the development of a active tremor suppression wearable orthosis for the human arm. Based on the clinical presentation of essential tremor patients that exhibit tremor at 3-11 Hz and the designed parameters of the orthosis, the required force from the actuator was deemed to have an amplitude of up to 20 N and frequency up to 12 Hz. The required stiffness amplitude can vary between 0.7–1.7 N/mm at same frequencies. The continuous time mathematical model was implemented in symbolic format, and numerical simulations were run using Mathematica 8.0 (Wolfram Research Inc., Champaign, IL). In order to mimic the performance of an actual digital control system, a zero-order holder with the sampling rate set to 2 kHz (as available in the experimental setup) was implemented for the control inputs in the numerical model. Thus, the results of the numerical simulation will
As shown in Fig. 4d the desired force was selected as a sinusoidal function with frequency of 5 Hz and amplitude of 8 N. The desired stiffness was chosen as a periodic function with frequency of 1 Hz, amplitude of 0.15 N/mm, and oscillating around the average value of 1.10 N/mm. The desired signals were set to their respective average values up to time t = 0. Figures 4a and 4b show the computed control inputs (valve coil currents) and valves spool displacements. The simulated pressure trajectories, shown in Fig. 4c, closely follow the desired values computed by the controller both in amplitude and phase. It is noticeable that the dynamics of the the pressure of each chamber contains the two frequencies associated with those of the desired force and stiffness. As explained in the previous section the available feedback signals from the physical system are the pressures of the chambers, and in fact we control the chambers’ pressures and estimate the resulting stiffness and force. As a result
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of the excellent chamber pressure control, the force and stiffness estimated from the actual chamber pressures show excellent amplitude and phase tracking of the desired values, as seen in Fig. 4d. Simulations were performed for several frequencies and amplitudes for the desired force and stiffness with excellent results observed up to the maximum selected frequency of 20 Hz. B. Experimental Validation For the purpose of validating the results obtained by the numerical simulation of the force-stiffness pneumatic controller, an experimental setup was designed and constructed as shown in Fig. 6. It included an Airpel E9 low stiction dual action pneumatic cylinder (Airpot Corp., Norwalk, CT), two LS-V05s proportional 3 ways pneumatic control valves, actuated by LS-C21 analog drivers (Enfield Technologies, Trumbull, CT). Two ASCX100AN absolute pressure sensors (Honeywell, Freeport, IL) were employed to measure the cylinder chamber pressures, and a linear position sensor MLT-38000104 (Honeywell, Morristown, NJ) provided the piston position. The load consisted of a metallic disk actuated through a belt mechanism by a brushless DC motor (EDC, Cambridge, MA). An optical encoder (ACCU-Coder 260 NT-02-S-1000, Encoder Products Co., Sandpoint, ID) and a force transducer (MLP-50, Transducer Techniques, Temecula, CA) allowed measurements of the angular position of the disk and the force produced at the cylinder’s rod. A cRIO-9024 modular CompactRIO real-time controller, was employed for controller algorithm implementation using the Real-Time and FPGA modules of LabView 2011 (National Instruments, Austin, TX). Data acquisition was performed using a NI 9205 cRIO module, while a cRIO NI 9263 digital to analog module was used to produce the control command applied to the valve drivers. The velocity and acceleration of the piston were estimated in real time using a customdeveloped algorithm. The desired force and stiffness were selected as sinusoid as described in the numerical simulation results subsection. Figures 5a and 5b show the applied control inputs (valve coil currents) and the estimated valves spool displacements. Since no spool position feedback was available, a first order numerical observer that approximates the dynamics of the spool was used. While the amplitudes, frequencies, and phases of the actual control inputs are very similar to the ones obtained in the numerical simulations, there are differences in the actual signal profiles due to the inherent uncertainties in the mathematical model, as well as unmodeled factors (friction, air tube flow dynamics, air leakage through seals, etc.). The measured cylinder chamber pressures, shown in Fig. 5c, closely mimic the desired values computed by the controller. As a result, the force and stiffness estimated from the actual chamber pressures show excellent amplitude and phase tracking of the desired values, as seen in Fig. 5d.
Fig. 6. Experimental setup for force-stiffness pneumatic controller performance validation.
incorporates the dynamics of the valves was developed. The goal of the controller was to track both the desired force and stiffness independently without chattering. The global Lyapunov asymptotic stability of the pressure tracking for each chamber was proven. Numerical simulations were performed for a class of desired functions that were considered relevant for the intended application, an upper limb wearable orthosis designed for essential tremor patients. The algorithm was implemented in a National Instruments cRIO real-time controller and validating experiments were performed for a compact low stiction pneumatic actuator deemed appropriate for powering the orthosis. The numerical simulation and experimental results showed excellent agreement, validating the suitability of the mathematical model and the performance of the control algorithm. The desired force and stiffness were accurately tracked, with very low error in amplitude and minimum phase shift. This work will be incorporated in the upper limb wearable tremor suppression orthosis developed in the Biomedical Instrumentation and Robotics Laboratory at SMU.
V. CONCLUSIONS A new backstepping-sliding mode algorithm based on a detailed mathematical model of the pneumatic system that 1197
(a)
(b)
(c)
(d)
Fig. 5. Experimental results for sinusoidal force and stiffness with different frequencies: (a) and (b) control inputs and valves spool displacements, (c) desired and measured pressures in the actuator chambers, (d) desired and measured force and stiffness
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