analytical dynamic model and experimental...

1 Copyright © 2002 by ASME

ANALYTICAL DYNAMIC MODEL AND EXPERIMENTAL ROBUST AND OPTIMAL CONTROL OF SHAPE-MEMORY-ALLOY BUNDLE ACTUATORS

Chunhao Joseph Lee1

R & D and Planning, General Motors M/C: 480-106-354

30500 Mound Road, Box 9055 Warren, MI 48090

Constantinos Mavroidis2 Robotics and Mechatronics Laboratory

Department of Mechanical and Aerospace Engineering Rutgers University, The State University of New Jersey

98 Brett Road, Piscataway, NJ 08854-8058

1 Senior Research Engineer, ASME Member 2 Associate Professor, ASME Member, Corresponding Author

ABSTRACT In this paper, the analytical dynamic model derivation and the robust and optimal position control of Shape Memory Alloy (SMA) bundle actuators using the LQR and H2 techniques are presented. SMA bundle actuators, composed of multiple SMA wires placed in parallel, have been recently proposed as a means to considerably increase the lifting capabilities of SMA actuators. Robust and optimal linear controllers could provide the desired robustness in the performance of these non-linear and highly sensitive actuators combined with the simplicity of these control schemes. The novel contributions of the present research are: a) the derivation of a generic, linearized, time-invariant analytical system model for SMA Bundle actuators that is used in the design of the LQR and H2 based controllers; b) the development of a new improved estimator in discrete-time H2 optimal control design based on the Kalman Filter predictor form for use in the control of SMA bundle actuators; c) the experimental study of two control design methods using state-space models, LQR and H2 Optimal Design, in discrete-time domain, using an experimental SMA bundle actuator consisting of 48 Flexinol SMA wires and able to apply up to 100 lbs. (445 N). As demonstrated in the experiments, the designed controllers provide satisfactory results in accuracy, stability and speed.

1. Introduction In many applications of robotic and mechanical systems, such as space exploration, medical operations, entertainment industry and military tasks, there is an increasing need for developing small size and lightweight devices that will be able to apply large forces, develop high speeds, achieve large displacements and be highly energy efficient. Advancing such robotic systems, in part, requires using new actuators, since classical forms of actuators, such as DC motors, hydraulics, or pneumatics, are heavy and cumbersome. Utilizing advanced actuators based on smart materials can make possible the development of innovative robotic systems that would satisfy many of the requirements stated above. In this research, the key methodology in drastically reducing the weight and size of robotic systems is the use of Shape Memory Alloy (SMA) wires as actuators of the robot joints. SMA wires, such as

Nickel-Titanium (Ni-Ti) wires, have the property of shortening when heated and thus are able to apply forces. This phenomenon, called the Shape Memory Effect (SME), occurs when the material is heated above a certain transition temperature changing its crystalline phase from martensite to austenite. Heating, and thus actuation, of an SMA wire is easily accomplished by applying a voltage drop across the wire causing current to flow through the material, resulting in joule heating. Ease of actuation is not the only advantage of SMA actuators. Other advantages are their incredibly small size and weight, their high force to weight ratio, their low cost and their noiseless operation. Their limitations include a relatively small bandwidth and low energy efficiency. Despite of these limitations, SMAs have one of the highest payload to weight ratios among "smart material" based actuators. Therefore, SMAs are one of the few "smart materials" that, at the present time, can be used in applications that require small size and large forces from the actuators. While design, modeling and dynamics of SMA actuators have been studied extensively, very little work has been done in the area of control. This is a very difficult problem to solve for three main reasons: a) SMA actuators present complex thermal-electrical-mechanical dynamics that are difficult to model; b) due to their temperature dependency, SMA actuators are very sensitive in temperature changes; c) due to the flexible characteristics of SMA actuators, substantial vibrations can be excited when these actuators are used to power the joints of robotic systems. Controllers for SMA actuators need to be robust in system and environmental changes and modeling errors. They also need to have vibration suppression characteristics. In addition the controllers need to be able to handle both position and force control tasks and be simple in implementation. Achieving accurate and robust performance of SMA actuators, is very important since it will allow their use in many important applications. Hashimoto, Takeda, Sagawa, Chiba and Sat [1] applied a PD control scheme to the SMA wires used as actuators of a biped walking robot. Ikuta, Tsukamoto and Hirose [2] used active PID control on a segmented active endoscope made with SMA springs. Troisfontaine, Bidaud and Dario [3] applied PI control on SMA actuators with an additional thermal sensor. Madill and Wang [4] used a very simple

2 Copyright © 2002by ASME

Proportional control to verify the SMA system model they adapted and discuss the system stability. The control gains are tuned either on-line or through simulations with trial and error method. The drawback of linear P, PI or PID control is that the controller may perform well in the range where the control gains are tuned, but deteriorates dramatically once outside the range. Various adaptive control algorithms have been proposed for use in SMA actuators. Dickinson used adaptive control to compensate directly the hysteresis of SMA when these actuators are used in vibration suppression applications [5], [6]. Webb, Wilson, Lagoudas and Rediniotis [7] took into account in their adaptive control algorithm the estimation of thermal changes between the SMA actuators and the environment and calculated the compensating input using an established SMA model. The drawback of this method is that the model and the calculations are very complicated, thus increasing the burden for on-line computation in experimental implementations. Another adaptive/nonlinear control of SMA is Pulse Width Modulation (PWM) [1], [8], [9], [10]. Grant and Hayward ([11], [12]) applied variable structure/sliding mode control methods under PWM to perform force control of a SMA actuator made for a robotic eye. The control gains were varied on-line. Tebbe, Schroeder and Butler [13] studied state-space multivariable control in large flexible smart structures actuated by SMA wires. The controllers were designed using either Eigenvalue pole placement or LQR methods. The model of the flexible structure with force inputs and flexible displacement was studied and applied in the controller design, but the model that describes the force/current relation between the SMA wires and the structure was established experimentally by an open-loop input/output diagram and was not further studied. In this paper, the analytical dynamic model derivation and the robust and optimal position control of SMA bundle actuators using the LQR and H2 techniques are studied. SMA bundle actuators, composed of multiple SMA wires placed in parallel, have been recently proposed as a means to considerably increase the lifting capabilities of SMA actuators [14-16]. The design of SMA bundle actuators has been studied in [14] and [15] and the non-linear, open loop dynamics have been studied in [16]. Initial control studies using PID controllers enhanced with an exponential input preconditioning function have been performed in [14]. Due to the inherent non-linear behavior of SMA actuators and their high sensitivity to changes in environmental and operating conditions, the PID based controllers could not perform well when different or varying position inputs are used. Robust and optimal linear controllers could provide the desired robustness combined with the simplicity of these control schemes. The novel contributions of the present research are: a) the derivation of a generic, linearized, time-invariant analytical system model for SMA Bundle actuators that is used in the design of the LQR and H2 based controllers; b) the development of a new improved estimator in discrete-time H2 optimal control design based on the Kalman Filter predictor form for use in the control of SMA bundle actuators; c) the experimental study of two control design methods using state-space models, LQR and H2 Optimal Design, in discrete-time domain, using an experimental SMA bundle actuator consisting of 48 Flexinol SMA wires and able to apply up to 100 lbs. (445 N). As demonstrated in the experiments, the designed controllers provide satisfactory results in accuracy, stability and speed.

2. System Model of Shape Memory Alloy Wires Shape memory alloys consist of a group of metallic materials that demonstrate the ability to return to some previously defined shape or size when subjected to the appropriate thermal procedure. These phenomena are defined as the Shape Memory Effects (SME), which occur due to a temperature and stress dependent shift in the material’s crystalline structure between two different phases called Martensite

and Austenite. Martensite, the low temperature phase, is relatively soft whereas Austenite, the high temperature phase, is relatively hard. The change that occurs within a SMA’s crystalline structure during the SME is not a thermodynamically reversible process. As a result, a temperature hysteresis occurs. This temperature hysteresis between Martensite ratio ξm (or Austenite ratio ξa) and temperatures AS, AF, MS, and MF translates directly into hysteresis in the strain/temperature relationship. The hysteresis behavior makes it challenging to develop modeling and control schemes for SMA actuators. For a given SMA, the hysteresis is dependent on the composition of the alloy and the manufacturing processes. Most Shape Memory Alloys (such as nickel-titanium (Ni-Ti)) have a hysteresis loop width of 27.8 to 67.8°C (50 to 122 ºF), with the exception of some wide hysteresis alloys used for joining applications such as couplings. Additional information on the principle of operation of shape memory alloys can be found in [17-20]. Developing a mathematical model that captures the behaviors of a Shape Memory Alloy as it undergoes temperature, stress, and phase changes is a complicated and challenging problem. Researchers continue to study what are the best ways to model and control actuators that use this unique family of materials. As discussed earlier, it is the significant hysteresis loop that causes the problems. Therefore, this section focuses on developing a new model, which will be used to design feedback controllers. This model assembles different features of several existing models for SMA actuators. The methods to establish the mathematical models for SMA actuators can be separated into two realms: a) Using experimental data to find the relations between forces, deformation and inputs (voltage or temperature changes), such as [8] and [21]; and b) Introducing a variable ξ, the Martensite (or Austenite) ratio which is shown in the relation to temperature change in the hysteresis effect. The model of Shu, Lagoudas, Hughes and Wen [22], later adapted in Dickinson [5] and [6], and the model of Ikuta, Tsukamoto and Hirose [23] are in this category. The new linearized model developed in this paper, which fits in the second category above, includes three relations: a) the temperature change inside the SMA due to the electric power input and the heat convection with the ambient environment; b) the relation between the temperature of the SMA and the Austenite ratio (ξa); c) the relation between the Martensite ratio and the mechanical properties which is mainly based on the sub-layer model by Ikuta et al. [23] with the addition of important features from the deformation model by Shu et al. [22].

HEAT CONDUCTION The temperature of the SMA wire is considered to be independent of strain and Martensite/Austenite fraction, and to be only a function of the heat transfer. The general solid heat conduction equation is derived by applying the conservation of mass, the first law of thermodynamics and Fourier’s law [24]. Using the electrical charge as the input power, the equation is written as:

( ) ( )dTc k T h T T Pe

dtρ ∞= ∇ ∇ − − +i

(1) where: ρ is the material density; c is the specific heat coefficient; T(t,x,y,z) is the temperature as a function of time (t) and of geometry

(x,y,z); ∇ is the gradient vector ˆˆ ˆi j k∂ ∂ ∂+ +

∂ ∂ ∂; h represents the heat

transfer coefficient which depends on the surrounding fluids/gas; T∞ is the ambient temperature; and Pe is the electrical power where

22VPe i R

R= =

. V represents voltage; i electric current; and R electric resistance.


If the direction x is set to be along a thin SMA wire, and the directions y and z are on the cross section (assuming a circular one), the

variations, the partial derivatives y∂∂ and z

∂∂ in both y and z

directions, are negligible because the cross section is very small

compared with the variation x∂∂ in x direction. In addition, the

transient temperature response of the wire heat conduction in the direction x can be considered to be much faster than the heat convection. Therefore, Equation (1) can be simplified into the following form [4]: ( )v a

dTcV hA T T Pedt

ρ ∞= − − + (2)

where Vv and Aa are the total volume and surface area of the SMA wire.

MARTENSITE/AUSTENITE RATIO Ikuta et al. [23] established the variable sub-layer model. It can be assumed that there are two main phases (Martensite and Austenite) existing in the SMA. An extra phase (Rhonbohedral phase, or R-phase) exists also in between, but most of the time it can be neglected [4]. This is an assumption that is adopted in this paper. The two main phases can be considered as being two parallel sub-layers connected, even though the phases are distributed randomly in the bulk metal. The macroscopic view of the two layers acting as a composite material with two different properties is shown in Figure 1. When the Martensite/Austenite ratio changes, the cross sectional areas and the volumes of the two phases change accordingly.

FIGURE 1: Variable Sub-layer Model of SMA.

In the figure above, ξm, ξa represent Martensite and Austenite ratios σm, σa and σ denote stresses of Martensite and Austenite and total stress respectively. The relation of the two ratios are simply: 1m aξ ξ+ = (3)

The Martensite ratio ξm decreases from one to zero when the temperature rises from the m-phase start temperature Ms to the finish temperature Mf. On the other hand, the Austenite ratio ξa increases from 0 to 1 when the temperature rises. All the relations can be written either in terms of ξm or ξa and the Austenite ratio is used in this work. In Ikuta et al. [23], the Austenite ratio ξa (or Martensite ratio ξm) is an exponential function of temperature T, but this ratio can be simplified further. Using the hysteresis loop the ratio ξa, as a function of temperature T, can be approximated in two linear relations: in heating process 0,

,

1,

s

sa s f

f s

s

T AT A A T AA A

T A

ξ

≤

−= ≤ ≤ − ≥

(4)

in cooling process 1 ,

,

0

s

fa s f

s f

f

T MT M

M T MM M

T M

ξ

≥

−= ≥ ≥ − ≤

(5)

STRESS/STRAIN RELATION In one-dimensional deformation, Shu et al. [22] applied the generalized Hooke’s law which has the form:

0[ ( )]e t

E E T T (6)

where: σ, εe, ε and εt are the uniaxial stress, elastic strain, total strain and transformation strain respectively; T is the current temperature and T0 is the ambient temperature (or the temperature without any

heating/cooling process); E is the Young’s Modulus; and α is the coefficient of thermal expansion. Both E and α are assumed to be functions of ξ during the transformation process and have the relation: ( )A M AE E E Eξ= + − ; ( )A M Aα α ξ α α= + − (7) where EA, αA and EM, αM are the material properties at pure Austenite and Martensite states, respectively. The rate of change of the transformation strain is assumed to be proportional to the rate of change of the Martensite fraction: tε ξ= Λ (8) where Λ in one dimension is defined to be:

( ) 0 ( ) 0

( ) <0 ( ) <0t t

t t

H H

H H

σ σξ ξσ σ

ε εξ ξε ε

> >Λ =

(9)

Equation (9) provides the directions in which the transformation strains develop. Parameter H is the maximum axial transformation strain (εt). From this equation, it can be shown that if the SMA is

under tension only, Λ is a positive constant equal to H for both 0ξ >

and 0ξ < . The total strain ε is defined as:

0

0SMA SMA

SMA

L L HL

ε−

= + (10)

in which SMAL is the current length of the SMA wire and 0SMA

L is the

original length of the wire. According to the one-dimensional deformation model in Shu et al. [22], the total strain ε of the deformation contains the elastic strain εe, the transformation strain εt and the thermal expansion. The thermal expansion can be neglected because it is 10-3 order smaller than εt. Therefore, the total strain can be written as: t eε ε ε= + (11) The transformation strain is a function of the Martensite ratio as it is shown in Equation (8). Without losing generality, the rate-change relation (8) can be rewritten as: (1 )t

m aε ξ ξ= Λ = Λ − (12)

If the process is only under tension, Λ is equal to H where H is the maximum axial transformation strain (prestrain) and it is equal to approximately 5%. The elastic strain can be related to stress using Equation (6). It can be represented in the Austenite ratio as: ( )

[ ( ) ]

e M A em a

M A M ea

E E E

E E E

σ ε ξ ξ ε

ξ ε

= = +

= + − (13)

3. Discrete-Time Robust and Optimal Control Design In this paper, the discrete time Linear Quadratic Regulator (LQR) [25] and the H2 robust optimal control [26] design techniques will be used for the control of SMA actuators. In this section, both methods are presented briefly. A novel feature, presented here, is the development of an improved estimator in discrete-time H2 optimal control design based on the Kalman Filter predictor form. Both LQR and H2 methods are using the separation theory to determine the state feedback control gain matrix K and estimator gain matrix L independently. The discrete-time feedback control loop is described using the discretized linear dynamical system in Equation (14)

1n d n d nx A x B u+ = + , n d n d ny C x D u= + (14)

where in general x is the (p×1) state vector; u is the (r×1) input vector; y is (q×1) output vector; Ad is a (p×p) matrix; Bd is a (p×r) matrix; Cd is a (q×p) matrix; and Dd is a (q×r) matrix. The subscript n represents each time step.


At each time step n, the control input u is calculated based on the full state vector xn and the control matrix K (r×p):

n nu Kx= − (15) Because the full state space vector xn cannot be measured directly from the sensors, instead at each time step the estimated state space vector ˆnx is calculated using the measured data y and the estimator

matrix L (p×q):

1 1ˆ ˆ ˆ( ( ( ) ))n d n d n n d d n d n d nx A x B u L y C A x B u D u+ += + + − + + (16) If a dynamic system is required to follow a reference or path yd, the controller design becomes a servo problem. At steady state ( t →∞ or n →∞ ), the system output y is equal to the reference yd. Given the system equations in discrete-time domain with reference yd, the system Equation (14) at steady state can be rewritten as:

ss d ss d ssx A x B u= + , d d ss d ssy C x D u= + (17)

where the constants xss and uss are the state vector and input vector at steady state respectively. It is desired to find xss and uss with yd given from Equation (17):

1 0ss d d

ss d d d

x A I Bu C D y

−− =

(18)

where I is the identity matrix. In order to make the steady-state Equation (17) and the original system Equation (14) to one regulator system Equation, a new set of system equations at each iteration n is established by subtracting the two equations (14) and (17). With the introduction of a new state vector nx′ , new input vector nu′ , and new output vector ny′ , the system of equations becomes:

1n d n d nx A x B u+′ ′ ′= + , n d n d ny C x D u′ ′ ′= + (19)

where

n n ssx x x′ = − , n n ssu u u′ = − ,

n n dy y y′ = −

LINEAR QUADRATIC REGULATOR AND KALMAN ESTIMATOR METHOD

Both LQR and Kalman estimator require optimization for which the Discrete-time Algebraic Ricatti equations are composed [25]. If the system in Equation (14) (Ad,Bd) is controllable, the design criterion for the best control gain K is to minimize errors from outputs (or state variables) and energy consumption of inputs. The control gain K is derived as: ( )( ) ( )1T T T T

d K d d d d K d d dK B X B D D R B X A D QC−

= + + + (20)

where both Q and R are weighting matrices. The associated Discrete Algebraic Riccati Enquation (DARE) is:

( ) ( ) ( )10T T T T T

K K K K K K K K K K K K K K K K KA X A X A X B S B X B R B X A S Q−

− − + + + + =

K dA A= ,

K dB B= , TK d dS C QD= , T

K d dR D D R= + , TK d dQ C QC= (21)

If the system (Ad,Cd) is observable, the Kalman estimator gain L can be calculated as [25] 1( )T T T

L d d L d w wm wL X C C X C D R D −= + (22) Where Rwp and Rwm are the covariances of process and measurement noises accordingly. With the DARE: ( ) 1

0T T T T T Td L d L d L d d L d w wm w d L d w wp wA X A X A X C C X C D R D C X A B R B

−− − + + = (23)

H 2 OPTIMAL ROBUST CONTROL DESIGN METHOD While H2 optimal robust control design in continuous time system has been used frequently, its implementation in discrete time systems has been given less attention. The implementation of H2 control of discrete time system is presented in this subsection and the full derivation can be found in Lee [27]. The robustness of the controller is determined by taking into account the disturbances of the system. The disturbances of inputs wd and/or

outputs v are weighted and integrated into the system of equations continuous-time domain. The system is discretized and is written as:

1 1 2n d n d n d nx A x B w B u+ = + + , 1 11 12n d n d n d nz C x D w D u= + + ,

2 21 22n d n d n d ny C x D w D u= + + (24) T

dw w v=

where z is the output criterion defined by users and vector w represents all the weighted disturbances. For H2 optimal robust control method, the H2 norm (norm in Hardy space) is used to form the cost function for optimization. The H2 norm of Linear Time Invariant systems in discrete time domain is defined as in [26]:

2 * * * *2

( ) o cG z Trace D D B L B Trace DD CLC= + = + , 2( )

A BG z RH

C D

= ∈

(25)

where Lo and Lc are the controllability and observability Gramians respectively. Similarly to the LQR/Kalman Estimator method, the separation theory is applied so that optimal gains based on full-state feedback and optimal gains based on Kalman state estimation are calculated separately and then combined together without losing stability. The objective functions for both gain matrices K and L are formulated using the H2 norm from disturbance w to state vector output z. If the system matrices (Ad,Bd2) are controllable, the control gain matrix K of full-state feedback has the form [27]: ( ) ( )1

2 2 12 12 2 12 1T T T Td K d d d d K d d dK B X B D D B X A D C

−= + + (26)

The corresponding Discrete-time Algebraic Ricatti Equation is ( )( ) ( )1

0T T T T TK K K K K K K K K K K K K K K K KA X A X A X B S B X B R B X A S Q

−− − + + + + =

K dA A= ,

2K dB B= , 1 12

TK d dS C D= ,

12 12T

K d dR D D= , 1 1

TK d dQ C C= (27)

The gain matrix L is calculated using the method of Kalman Filtering instead of the regular format described in [26]. If the system (Ad,Cd2) is observable, the system (Ad,Cd2Ad) is also observable thus the solution of the estimator is guaranteed [25]. The gain matrix L is calculated as in [27]:

( ) ( )( ) ( ) ( ) ( )( )( ) 1

2 1 2 1 21 2 2 2 1 21 2 1 21T T T T

d L d d d d d d d d L d d d d d d d dL AX AC B C B D C A X C A C B D C B D−

= + + + + + (28)

The associated Algebraic Ricatti Equation is: ( ) ( ) ( )1

0T T T T TL L L L L L L L L L L L L L L L LA X A X A X B S B X B R B X A S Q

−− − + + + + =

L dA A= , ( )2T

L d dB C A= , ( )1 2 1 21T

L d d d dS B C B D= + ,

( )( )2 1 21 2 1 21T

L d d d d d dR C B D C B D= + + , 1 1

TL d dQ B B= (29)

4. SMA Bundle Actuator Experimental Set-Up The SMA Wire Bundle actuator experimental setup, shown in Figure 2, is used in this paper to apply the robust and optimal controllers. It consists of 48, 30.5cm (12in.) long, 150µm (0.006in.) diameter wires. The wires making up the bundle are fabricated from a Nickel-Titanium (Ni-Ti) alloy made by Dynalloy, Inc. and are manufactured such that they undergo a maximum length contraction of 8% and can apply a considerable amount of force compared to their weight [28]. The bundle was designed to apply a maximum of 445N (100lbs.) over a distance of approximately 1.27cm (0.5in.). The 48 wires were connected mechanically in parallel between two 0.635cm (0.25in.) thick, 3.81cm (1.5in.) diameter virgin Teflon end plates. WinReC, a software developed in our laboratory, provides deterministic fast timers, 200 Hz in this experiment, on Windows NT platforms [29]. IIR (Infinite Impulse Response) Butterworth digital filters are used in WinReC to filter the sensor signals. The cutoff frequency of each filter is chosen equal to 25Hz because the bandwidth of the SMA wire is lower than that. The Power Supply for bundle actuation was a Tellabs® 48volt (nominal), 10 amp DC power source. The 480Watt supply was controlled using a custom designed 10x operational amplifier circuit, which used a Burr Brown OPA 512 Power Operational Amplifier. The


reaction force between the load and bundle wires was sensed using a Transducer Techniques® MLP-100 Mini Low Profile Load Cell which has a capacity of 445N (100lbs.) and an accuracy of ±0.445N (±0.1lbs.). Linear displacement of the payload was sensed by a Space Age Control® Series 150 Analog-Output Ultra-Small Subminiature Position Transducer, which was fastened to an adjustable bracket on the base of the test rig and the extension cable was attached to the bottom of the load. Its range of measurement is 0 to 3.81cm (0 to 1.5in.) with a maximum linearity error of ±0.361% of full range. The total current passing through the bundle set was sensed using a Hewlett Packard® HP34330A 30Amp Current Shunt, which contained a precision 0.001Ω resistor. This precision resistor provided an output signal of 1mV per Amp. The recommended maximum electric current passing through the SMA wire is 400mA. The 48-wire bundle usage was restricted carefully to the total power consumption when it was constructed. Measured experimentally, the resistance of the whole bundle varies in the range from 10 to 13Ω and the maximum current with 8 sets of wires connected electrically in parallel is 3.2A. Thus the corresponding maximum voltage between the bundled wires should not exceed 32V. Therefore the maximum control voltage sent by the computer to the bundle is limited to 30V.

FIGURE 2: SMA Bundle Experimental Setup.

In addition to the linear load configuration shown in Figure 2, we have also developed a modified version of the setup that is able to power a revolute joint. Due to space limitation, the description and experimental results obtained with the revolute load configuration have not been included in this paper and can be found in [27].

5. Dynamics and Control of the SMA Bundle Actuator Set-Up

DYNAMICS OF THE SMA BUNDLE ACTUATOR In this sub-section we present the motion dynamics of the SMA Bundle actuator described in Section 4. While only a simplified version of the derived model will be used in the controller design, we present it here in its full form in case that will be needed in future works. It is assumed that the temperatures of all 48 bundled SMA wires change homogeneously under the same conditions. Equations (2), (4), and (5), which describe the heat transfer effects and the relation of temperature and Austenite ratio, are used directly in this bundle-wire dynamics. The relation between the inertia and load forces of the bundle wires and the stresses in the wires needs to be established and applied to Equation (13). In addition, the relation of the length change

of the bundle SMA wires and the strain in the wires is also needed for Equation (13). This dynamic model of the SMA bundle actuator is further linearized in order to be applied in the controller design. The system dynamics of the SMA bundle actuator in the linear load configuration can be derived from the stress/strain relation of Equation (13). The stress (σ) applied to the bundle wires is approximated as: F

Aσ ≈ ,

48tFF = (30)

where: A is the cross section area of a single SMA wire; Ft is the total reaction force; and F is the reaction force on each wire, the average of Ft out of 48 wires, assuming that all the forces are evenly distributed to each wire. The Newton’s equation of the total reaction force Ft is written as:

t w w fF m g m x F= + + (31)

in which mw is the mass of the load that the actuator lifts; g is the gravitational constant; x is the acceleration of the weight displacement; and Ff are the disturbances such as friction. It should be noted that the mass of the SMA wires is neglected because it is small compared to the load. The linear displacement is the length change from the original to the final position that corresponds to the temperature change from room temperature to the final temperature after the electrical charge is applied. L0 is set to represent the original length of the bundle SMA wires under no external loads; ε0, ε0

e, and ε0t are the total strain, the

elastic strain, and the transformation strain at room temperature. Thus the initial elongation of the wire after the load is applied and the final elongation of the wire with loading along with the temperature changes, are presented accordingly:

0 0 0 0 0( )e tL Lε ε ε= + , 0 0 ( )e tL Lε ε ε= + (32)

The total displacement ∆x as moving upward (wires contract) in the positive direction can be calculated using Equations (11) and (12):

0 0

0 0 0

0 0 0

( )( ) ( )

( )

e e t

e ea

x LL L H

L L H

ε ε

ε ε ε

ε ε ξ

∆ = −

= − + −

= − +

(33)

Substituting Equations (30), (31) and (33) into equation (13), the system dynamic model can be written as: 0( )

[ ( ) ]48

ew w a f M A M e

a

m g m L H FE E E

Aξ ε

ξ ε+ − +

= + − (34)

If a new variable:

0e e eε ε ε′ = −

is introduced, its initial condition when the wires are at room temperature is zero. As the system is in static mode at room temperature, the gravity can be presented as

0( )M ewm g A E ε′= (35)

where A′ is equal to 48A. The system dynamics (34) can be

rewritten using the new strain variable eε ′ :

00

[ ( )( ) ]fe M e A M e ea a

w

FA E E E Hm L A

ε ε ε ε ξ ξ′′ ′ ′= − − − + + +

′ (36)

with ε0e a constant.

CONTROL OF THE SMA BUNDLE ACTUATOR In this section, the general guidelines are set in order to design the feedback controller using a state-space variable representation of the system equation. Through these guidelines, the linearized system of equations of the SMA bundle actuator can be established and used in the controller design by the LQR or H2 methods. Due to the complexity of the system dynamics shown in the previous section, the controller for this one-way actuation device will be designed by neglecting the dynamics of the linear load configuration. This approximation is acceptable because the double time derivative


of the elastic strain eε ′ in Equation (36) is assumed relatively small compared with other significant terms from the stresses. Therefore this equation is reduced into linear static relations and will not be included in the model for controller design. The linear system of state equations is constructed using only the heat conduction Equation (2). This first order system has a (1×1) state vector x and input u as follows: [ ]( )x T T∞= − , 2u V= where V is the voltage applied to the SMA wires. This continuous-time system of equations is: x Ax Bu= + ; a

v

hAAcVρ

= −

, 2a

v

KBcV Rρ

=

(37)

in which Ka is the amplifier gain of the input voltage. It should be noted that the initial condition of the above equation at room temperature is zero. The output measures the displacement of the payload/weight using the potentiometer. Because the bundle actuator is a one-way actuation system, only the heating process of Equation (4) is taken into account. From Equation (33), it can be found through linear approximation:

0

0 ( ),

a

s s ff s

x L HL H T A A T A

A A

ξ∆ ≈

= − ≤ ≤−

(38)

The output equation can be written as:

oy Cx CT ′= − , sT A T∞′ = −

where: 0

f s

L HCA A

=−

(39)

in which yo is the output measurement for the linear displacement. Equation (39) is written into the form: oy y CT

Cx′= +

= (40)

where T’ is assumed to be a constant; and y is the output vector in the linear system of equations. Because the bundle actuator is required to follow reference inputs to move the loads, the position control of the system is a servo problem as described in Section 3. The reference yd that is required in Equation (18) and corresponds to Equation (40) is written as:

d ody y CT ′= + (41) where yod is the desired position of the linear displacement. It should be noted that variable T ′ is eliminated when a different variable y′ is used in Equation (19), where:

dy y y′ = − The parameters of Ni-Ti SMA wires that are needed to determine the coefficients of the system equations are listed in Table 1 [28], [12].

TABLE 1: Parameters of 150µm Ni-Ti SMA Wire from Flexinol®

Parameter (unit) Symbol Value Wire Diameter (µm) r 150 Linear Electrical Resistance (10-8 Ω/m) R 80 - 89 Density (Kg/m3) ρ 6450 Heat Capacity (Joul/Kg-°C) c 235 (single phase)

2695 (during phase change) Heat Transfer Coefficient (Joul/sec.-°C) h 80 – 110 Austenite Start Temperature (°C) As 68 Austenite Finish Temperature (°C) Af 78 Martesite Start Temperature (°C) Ms 52 Martesite Finish Temperature (°C) Mf 42 Prestrain (%) H 5 Original SMA wire Length (m) L0 0.305

It should be noted that the ambient (room) temperature T∞ around the SMA-wire bundle ranges roughly from 20 to 30°C. The constant T ′ in Equation (41) can be calculated from the As; therefore T ′ is found to be equal to 40°C. However, when the initial experiment was conducted with the desired position chosen as zero, the bundle moved too much at this value of T ′ . For this reason, Equation (41) is modified to:

fT M T∞′ = − (42)

and the new constant T ′ is chosen experimentally to be equal to 25°C to confine the bundle movement within a tolerable range.

6. Experimental Results

LQR/KALMAN-ESTIMATOR CONTROL A weight of 4.99Kg (11lb) is applied to the bundle actuator as shown in Figure 2. The linear model of the system in Equations (37) and (40) is used in the controller design. Following the standard procedure described in Section 3, the LQR/Kalman-Estimator controller is designed using the weighting matrices as: 50Q = , 0.1R = , 0.01T

w wp wB R B = , 50,000vR =

Due to fact that the precision of the control performance is more important than the input saturation, the values of the penalty weighting matrices Q and Rwm, which are associated with precision errors, are chosen much larger than the values of R and T

w wp wB R B , which are

associated with input limits. The calculated gain matrices from Equations (20) and (22) are: 18.01356K = , 1.110429L = The results of the LQR controller for a unit step input (1mm) are shown in Figure 3-(1). This unit step test has been used to fine-tune the control gains. The rising time from 0.1 to 0.9mm is 0.53sec and the steady-state error is within 10% of the reference input. Figure 4-(1) shows the corresponding control voltage to the actuator, the interaction force between the weight and the actuator, and the error difference between the estimated output from the linear model and the real measurement from the potentiometer. The input voltage reaches the saturation limit 30V when the motion begins because the controller tries to drive the weight to the desired position as fast as possible. At the steady state, the voltage is maintained at around 15V due to the heat convection because the temperature of the SMA wires should be a constant to hold the bundled wire position steady. The interaction force between the bundle and weight is quite steady between 5 to 5.4Kg throughout the rectilinear motion. The error between the estimated output, due to the Kalman estimator, and the real measurement is very small with a standard deviation of 2.9×10-5mm.

0 2 4 6 8 10 0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

time ( sec.)

posit

ion

( mm

.)

measure reference

0 2 4 6 8 10 0

0.2

0.4

0.6

0.8

1

1.2

1.4

time ( sec. )

posit

ion (

mm. )

measure reference

FIGURE 3: Unit Step Response of (1) LQR, (2) H2 Controlled SMA

Bundle Actuator

The controller is tested using a series of step inputs: from 0 to 2, 4, 2 and back to 0mm. Figure 5-(1) shows the displacement of the linear bundle actuator under the LQR control, and the error between the guidance step inputs and the measured position of the weight. Even


though the SMA bundle is only a one-directional (upward) actuator, the controller still works very well as the reference inputs change from a higher position to a lower one. The control voltage is reduced (exactly to zero volts) to allow the wires to cool down so that the weight stretches the SMA bundle downward until it reaches the desired position.

0 2 4 6 8 10 0

10

20

30

time ( sec.)

input

vol

tage

( volt.)

0 2 4 6 8 10 4.8

5

5.2

5.4

5.6

time ( sec.)

reac

ting

forc

e (

Kg.)

0 2 4 6 8 10 -5

0

5

10 x 10 -4

time ( sec.)

estim

atio

n er

ror(

mm.)

0 2 4 6 8 10 0

10

20

30

time ( sec. ) in

put v

olta

ge (

volt.

)

0 2 4 6 8 10 4.5

5

5.5

6

time ( sec. )

reac

ting

forc

e (

Kg.

)

0 2 4 6 8 10 -5

0

5

10

15 x 10 -4

time ( sec. )

estim

atio

n er

ror(

mm. )

FIGURE 4: Corresponding Voltage, Force, and Error of Position

Estimation of (1) LQR, (2) H2 Control for Unit-Step Linear Actuation.

The repetitiveness of the controlled bundle is tested using a squared wave function of 0.08Hz between 1 and 3mm. The time span of relaxation is 1.5 times larger than that of contraction to guarantee that the weight reaches the lower position before the next cycle starts. The second timer of the square wave trajectory in WinReC is set to be 20Hz as it is described in [29]. Figure 6-(1) shows the results for the squared wave function.

0 10 20 30 40 50 60 70 80 -1

0

1

2

3

4

5

time ( sec.)

posit

ion

( mm

.)

measure reference

0 10 20 30 40 50 60 70 80 -3

-2

-1

0

1

2

3

time ( sec.)

posit

ion

erro

r( mm

.)

0 10 20 30 40 50 60 70 80 -1

0

1

2

3

4

5

time ( sec. )

posi

tion

( mm. )

0 10 20 30 40 50 60 70 80 -3

-2

-1

0

1

2

3

time ( sec. )

posi

tion

erro

r( mm. )

measure reference

FIGURE 5: Step Responses of LQR Controlled Linear Bundle

Actuator.

H 2 CONTROL In this section, an H2 controller is designed and applied to the SMA bundle. Because the velocity (acceleration) of the weight should be limited, it is considered in the optimization objective vector z. Therefore the new customized objective function following the description in Section 3 becomes: Tz y x u=

The values for the weightings are chosen so that the error of the displacement is penalized prior to the input saturation. The control gains K and L are: 18.65029K = , 1.109502L = Because the model of the SMA wire bundle is a first order plant, the H2 control gains can be tuned to have the same order as the ones in the LQR control.

0 10 20 30 40 50 60 70 0 0.5

1 1.5

2 2.5

3 3.5

time ( sec.)

posit

ion ( mm

.)

0 10 20 30 40 50 60 70 -3 -2 -1 0 1 2 3

time ( sec.)

posit

ion e

rror(

mm.)

measure reference

0 10 20 30 40 50 60 70 0 0.5

1 1.5

2 2.5

3 3.5

time ( sec. )

posit

ion (

mm. )

0 10 20 30 40 50 60 70 -3 -2 -1 0 1 2 3

time ( sec. )

posit

ion er

ror(

mm. )

measure reference

FIGURE 6: Squared Wave Guidance of (1) LQR, (2) H2 Linear

Bundle Control. The experiments follow the same procedure as in the LQR/Kalman-Estimator control. Figure 3-(2) and Figure 4-(2) show the unit step test of the H2 controller. The rising time of the unit step response from 0.1 to 0.9mm is 0.53sec, which is the same as the response of the LQR controller. The steady-state error however is better, within 4% of the input reference. Also the interaction force measurement between the SMA wire bundle and the weight, and the input voltage of the H2 controller are very similar to the ones of the LQR control. The error of estimated and measured outputs is very small, again with a standard deviation of 4.8×10-5mm. The responses of the bundle for a series of step inputs and for a square wave function are shown in Figure 5-(2) and Figure 6-(2), in which the robustness of the controller is demonstrated. There are some differences between the PID controller and LQR or H2 controllers, which make the latter ones superior. In PID control, gains of the controllers are tuned only based on experimental performance. If the gains get too high, the closed-loop system tends to be fragile. However, in LQR/H2 control, the gains are tuned within the Lyapunov stability criteria. Even though the feedback gain matrix K can be increased as the weighting of output precision becomes higher, the observer gain matrix L has a limit and remains at the limit no matter how large the weighting increases. This guarantees the stability of the system. The other advantage of LQR/H2 control is its ability to calculate the steady-state inputs, which act as feedforward control. PID control without feedforward cannot get the steady-state inputs directly; therefore a higher integral gain Ki is needed. But the higher the integral gain is, the more sensitive the closed-loop system is.

7. Conclusions

In this paper, we studied the analytical dynamic model derivation and the robust and optimal position control of SMA bundle actuators using the LQR and H2 techniques. We developed a generic, linearized, time-invariant system model for SMA Bundle actuators that can be used in the design of the LQR and H2 based controllers. We also proposed a new improved estimator in discrete-time H2 optimal control design based on the Kalman Filter predictor form for use in the control of SMA bundle actuators. We performed detailed experiments of the two control design methods using state-space models, LQR and H2 Optimal Design, in discrete-time domain, using an experimental SMA bundle actuator consisting of 48 Flexinol SMA wires and able to apply up to 100 lbs. (445 N). As demonstrated in the experiments, the designed controllers provide satisfactory results in accuracy, stability and speed.


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