Design of a Kalman filter for rotary shape memory alloy actuators

Transcription

1 INSIUE OF PHYSICS PUBLISHING Smart Mater. Struct. 3 (4) SMAR MAERIALS AND SRUCURES PII: S964-76(4)778-X Design of a Kalman filter for rotary shape memory alloy actuators Mohammad H Elahinia,Mehdi Ahmadian and Hashem Ashrafiuon Mechanical Engineering Department, Virginia ech, Blacksburg, VA 46, USA Mechanical Engineering Department, Villanova University, Villanova, PA 985, USA elahinia@vt.edu Received 4 July 3, in final form 6 February 4 Published 8 May 4 Online at stacks.iop.org/sms/3/69 DOI:.88/964-76/3/4/6 Abstract Measuring the state variables of systems actuated by shape memory alloys (SMAs) is normally a difficult task because of the small diameter of the SMA wires. In such cases, as an alternative, observers are used to estimate the state vector. his paper presents an extended Kalman filter (EKF) for estimation of the state variables of a single-degree-of-freedom rotary manipulator actuated by an SMA wire. his model-based state estimator has been chosen because it works well with noisy measurements and model inaccuracies. he SMA phenomenological models, that are mostly used in engineering applications, have both model and parameter uncertainties; this makes the EKF a natural choice for SMA-actuated systems. A state space model for the SMA manipulator is presented. he model includes nonlinear dynamics of the manipulator, a thermomechanical model of the SMA, and the electrical and heat transfer behavior of the SMA wire. In an experimental set-up the angular position of the arm is the only state variable that is measured besides the voltage applied to the SMA wire. he other statevariables of the system are the arm s angular velocity and the SMA wire s stress and temperature, which are not available experimentally due to difficulty in measuring them. Accurate estimation of the state variables enables design of a control system that provides better system performance. At each time step, the estimator uses the SMA wire s voltage measurement to predict the state vector which is corrected as necessary according to the measured angular position of the arm. he input and output of the model are used for the EKF simulations. he state variables collected through model simulations are also used to evaluate the performance of the EKF. Several EKF simulations presented in this paper show accurate and robust performance of the estimator, for different control inputs.. Introduction SMAs 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. he shape memory effect is hysteretic and occurs due to atemperature- and stress-dependent shift in the materials crystalline structure between two different phases called martensite and austenite, which are the low and high temperature phases, respectively. SMA actuators have several advantages for miniaturization such as excellent power to mass ratio, maintainability, reliability, and clean and silent actuation. he disadvantages are low speed and low energy efficiency due to conversion of heat to mechanical energy and control difficulties due to hysteresis, nonlinearities, parameter uncertainties, and difficulty in measuring the state variables such as the temperature /4/469+7$3. 4 IOP Publishing Ltd Printed in the UK 69

2 MHElahinia et al dθ/dt dε/dt Kinematic em d/dt σ d/dt dε/dt d/dt σ dσ/dt σ dθ/dt θ Output em dσ/dt d/dt Arm Dynamics Constitutive Phase ransformation Input Amplifier I d/dt Heat ransfer Figure. Block diagramof the-dofsma-actuatedarm model. Figure. he -dof SMA arm, actuated by Nii wire and a bias spring. (his figure is in colour only in the electronic version) here are two classes of SMA actuators. he one-way actuators (bias type) arecomposed of an SMA element and a bias spring []. he two-way actuators (differential type) are made of two SMA elements [ 5]. With regards to control of SMA actuators, in addition to linear control methods, several nonlinear control schemes such as fuzzy logic, feedback linearization, and variable structure control have also been applied [6 ]. Control difficulties are one of the main disadvantages of the SMA actuators. his can be attributed to the nonlinear behavior of the material and difficulties in measuring the state variables of the SMAs. Within the significant toolbox of mathematical tools that can be used for stochastic estimation from noisy sensor measurements, one of the most well known and often used tools is what is known as the Kalman filter [3, 4]. he Kalman filter has been extensively used to help control the motion of robotic manipulators [5 9]. he Kalman filter has also been shown to perform well in vibration control using smart materials such as piezoelectrics [, ]. Lively et al [] have investigated filtering techniques to perform dynamic shape estimation of structures and have shown that the Kalman filter is the superior technique. o the best of our knowledge, the Kalman filter has not been used to control the motion resulting from shape memory alloys (SMAs). Knowing the state variables is essential for developing more effective control algorithms. In most cases, however, it is not possible to measure these variables for the SMAactuated systems. In this work, we have developed an extended Kalman filter (EKF) for estimatingthe state variables of a rotary manipulator. he arm is actuated by a biastype actuator constructed with SMA wire, pulleys, and a linear spring which provides the bias torque. A nonlinear model is developed that has three interconnected parts: the arm kinematic/dynamic model, the SMA thermomechanical model, and the heat transfer model. his model has been experimentally verified in our previous work [9]. he EKF utilizes the wire s input voltage and the angular position of the arm to predict the system s state vector. he filter predicts an accurate estimate of the state vector in the presence of both measurement and process noises. Several simulation results are presented to verify the effectiveness and robustness of the filter. Figure represents the block diagram of the system showing the interdependency of its subsystems.. ing the system he one-degree-of-freedom rotary SMA-actuated arm that is used for this study is shown in figure. It uses a 5 µm diameter Ni i SMA wire, to rotate the arm upward when the SMA wire is electrically heated. he bias spring provides therelativelysmall torque that is needed to elongate the SMA wire at lower temperatures and for the downwards rotation to take place. he experimental data for model verification are collected using dspace M.Inaddition to the three major parts of the model that are explained in the following sections, the experimental set-up consists of an amplifier, an encoder, and a signal conditioner that are also modeled. he nonlinear dynamic model of the arm including spring and payload effects is represented by I e θ = τ w (σ ) τ g (θ) τ s (θ) c θ () where τ w,τ g,andτ s are the resulting torques from SMA wire, gravitational loads, and the bias spring, respectively, and σ is the wire stress. I e is the effective mass moment of inertia of the arm, and the payload, and c isthetorsional damping coefficient approximating the net joint friction. he SMA wire strain rate ε and joint angular velocity θ are related kinematically as ε = r p θ () l where r p is the pulley s radius and l is the initial length of SMA wire. 69

3 Design of a Kalman filter for rotary shape memory alloy actuators he wire constitutive model shows the relationship between stress (σ), strain (ε), martensite fraction (), and temperature ( )[3]: σ = D ε + θ Ṫ + (3) where D = DM+DA is the (average) Young modulus, D A is the austenite Young modulus, D M is the martensite Young modulus, θ is the thermal expansion factor, = Dε is the phase transformation contribution factor, and ε is the initial (i.e. maximum) strain [4]. Martensite fraction << indicates the amount of the material that exists in the martensite phase. Due to hysteretic behavior of the SMA wire, the phase transformation equations are different for heating and cooling, as noted by Liang [4]. Heating results in reverse transformation from martensite to austenite: = M {cos[a A( A s ) + b Aσ]+} (4) where isthemartensite fraction coefficient, M is the maximum martensite fraction obtained during cooling, is the SMA wire temperature, A s and A f are austenite phase start and final temperatures, a A =, b A = aa C A are curve π A f A s fitting parameters, and C A shows the effect of stress on the transformation temperatures. Cooling results in the forward transformation equation from austenite to martensite: = A cos[a M ( M f ) + b Mσ]+ + A where A is the minimum martensite fraction obtained during heating, M s and M f are martensite phase start and final temperatures, a M = and b M = am C M are curve π M s M f fitting parameters, and C M shows the effect of stress on the transformation temperatures. he SMA wire heat transfer equation consists of electrical heating and natural convection: mc p d dt (5) = V R ha c( ) (6) where R is the resistanceperunit length, c p is the specific heat, m is the mass per unit length and A c is the circumferential area of the SMA wire. Also, V is the applied voltage, is the ambient temperature, and h is the heat convection coefficient. We have approximated h by a second order polynomial of temperature to improve the heat transfer model, 3. Extended Kalman filter h = h + h. (7) he Kalman filter addresses the general problem of trying to estimate the state x R n of a discrete-time controlled process that is governed by a linear stochastic difference equation. As an extension to the same idea, the extended Kalman filter (EKF) is used if the dynamic of the system and/or the output dynamic is nonlinear. EKF is based on linearization about the current estimation error mean and covariance. Let us assume that the process has a state vector x R n and a control vector u. he system s dynamic is modeled by a nonlinear stochastic difference equation x k = f (x k, u k,w k ) (8) with the (measurable) output z R m that could also be a nonlinear function of the state variables z k = h(x k,v k ) (9) where the random variables w k and v k represent the process and measurement noise, respectively. hey are assumed to be independent (of each other), white, and with normal probability distributions with covariance matrices Q and R, i.e., p(w) N(, Q) () p(v) N(, R). () Defining ˆx k as the a posteriori estimate of the state (from previous time step k) one can approximate the state and measurement vector without the noise effects x k = f ( ˆx k, u k, ) () and z k = f ( x k, ). (3) he linear governing equations can be obtained by linearizing an estimate about equations () and (3) x k x k + A(x k ˆx k ) + W w k (4) z k z k + H(x k ˆx k ) + V v k (5) where x k (to be estimated) and z k (measured by the sensors) are the actual state and measurement vectors and A, W, H, and V are Jacobian matrices defined at each iteration. Let us define the prediction error and the measurement residual ẽ xk x k x k (6) ẽ zk z k z k. (7) Now we can write the governing equations for an error process as ẽ xk A(x k ˆx k ) + ε k (8) ẽ zk Hẽ xk + η k (9) where ε k and η k present new independent random variables p(ε k ) N(, WQ k W ) () p(η k ) N(, VR k V ). () It can be shown that the time update equations of the EKF are ˆx k = f ( ˆx k, u k, ) P k = A k P k A k + W () k Q k Wk where ˆx k is the aprioristate estimate [4]. hese time update equations project the state and covariance estimate (P k ) 693

4 MHElahinia et al mesured angular position 3 input voltage ime Update a priori state estimate covariance estimate Measurement Update updated state estimate updated covariance estimate Figure 3. he schematic diagram of the extended Kalman filter designed to predict the state vector of the SMA-actuated manipulator. θ (degree) Experiment (V = 7. v) Experiment (V = 7. v) Simulation (V = 7. v) Simulation (V = 7. v) from the previous time step k tothecurrent time step k. Moreover, the measurement update equations of the EKF are K k = P k H k (H k P k H k + V k R k V k ) ˆx k =ˆx k + K k(z k h( ˆx k, )) P k = (I K k H k )P k (3) where K is the correction gain vector and ˆx k is the a posteriori estimate. he gain is found in a such a way as to minimize the a posteriori error covariance. hese measurement update equations correct the state and covariance estimate using the current measurement z k.figure 3 shows a schematic diagram of the extended Kalman filter for the SMA actuator; the design process of this filter is explained next. 3.. EKF for SMA actuator he SMA manipulator model state vector is defined as x = [ θ θ σ ]. (4) he martensite fraction is not a state variable since it is a function of the stress and the temperature. he model can be discretized by converting the differential equations presented in section to their corresponding backward difference equations. he angular position of the arm at the current time may be predicted using previous values of the angular position and velocity as ˆx k () =ˆx k () + ˆx k () (5) where is the sampling time. he angular velocity is predicted using the previous two angular position samples ˆx k () = ˆx k () ˆx k (). (6) herefore, the estimated angular velocity is directly related to the angular position, which is the only measured state variable, instead of integrating the angular acceleration in equation (). Furthermore, since the SMA wire s stress is afunction of strain, temperature and phase transformation, using equation (3) would create an algebraic loop in the model. Instead, the stress is predicted using equation () ˆx k (3) = f 3 = [I e θ +τ g ( ˆx k ())+τ s( ˆx k ())+c ˆx k ()]/(r p A w ) (7) where A w is thesma wire s cross-sectional area. he angular acceleration of the arm in the above equation is computed using the previous two values of the angular velocity θ = ˆx k () ˆx k (). (8) ime (sec) Figure 4. Comparing open-loopsimulationand experimental results. Finally, the temperature estimate is simply based on the convection heat transfer equation (6) which depends only on the previous temperature value: ˆx k (4) =ˆx k (4) + V /R ha c ( ˆx k (4) ). (9) mc p he Jacobian matrix at each iteration can be derived by using the state equations (5) (9): A = s (3) A 3 A 3 A 44 where A 3 = f 3 (x k (), x k (), x k (), ) x A 3 = c r p A w A 44 = f 4 x 4 (x k (4), ). (3) he angular position of the arm is measured hence z k = Hx k + v k = [ ]x k + v k. (3) Initially, the SMA wire is prestressed at room temperature and fully martensetic while the arm is at the lower position. herefore, the initial state vector is [ ] π x initial = 4 (rad) (rad s ) 98. (MPa) ( C). We add uncertainty by selecting the initial state error covariance P = I 4 4 and the initial measurement noise covariance Q = e I 4 4. hus we developed all the necessary elements of the EKF for the SMA rotary actuator. In the next section the results of simulating the filter are presented. 4. Results he EKF is derived based on the nonlinear dynamics of the SMA actuator. he nonlinear model has been 694

5 Design of a Kalman filter for rotary shape memory alloy actuators θ (rad) θ (rad/s) 5 EKF 4 EKF θ (rad) θ (rad/s) 4 ( o C) ( o C) Figure 5. Comparing the simulation results of the SMA-actuated arm model and the extended Kalman filter for a switching step input..5 Figure 7. Comparing the simulation results of the SMA-actuated arm model and the extended Kalman filter for a sinusoidal input (ω = 5rads ). U(V) EKF e x. 5 e x. 3 e x3 e x e x5.5 Figure 6. he state vector estimation error of the extended Kalman filter for a switching step input (e x = x x, e x = x x, e x3 = x 3 x 3, e x4 = x 4 x 4, e x5 = x 5 x 5 ). previously verified through several experiments [9]. A sample comparison of experiment and nonlinear simulation is shown in figure 4. he simulation results with the extended Kalman filter are presented here and compared with the model. For each EKF simulation, the input (SMA wire s applied voltage, V )and measured output (angular position of the arm, θ) are initially collected by simulating the nonlinear model. he data are then used for the EKF simulation. he estimation error, at each time step (iteration), is generated by comparing the filter s predicted state vector with the previously collected model s state vector. Figure 5 shows the comparison between the EKF predicted and the nonlinear model state variables for a step input voltage with changing amplitude. Figure 6 presents the prediction errors for the same input for each state variable. When the high voltage is applied, after the wire reaches the austenite start transformation temperature, the wire starts contracting and hence the arm moves upward. he downward motion occurs when the input voltage is cut off and the wire cools down below the martensite start transformation temperature. For this input, it can be seen that all the state variables are estimated ε (m/m) Figure 8. Comparing the simulation results of the SMA-actuated arm model and the extended Kalman filter in predicting the hysteresis effect in the SMA wire for a sinusoidal input (ω = 5rads ). with bounded errors. he angular position s estimation error remains zero for the entire simulation. his is not unexpected since the angular position is the only measured state variable. Estimation errors for the next three state variables are relatively large at four different time instants. hese times are associated withbeginning of the actuator s upward motion, switching the input voltage and hence stopping the upward motion, starting the downward motion, and stopping the downward motion. For all these cases there exists a sudden change in the velocity of the actuator. he large temperature estimates occur when the voltage switches to zero. Figure 7 depicts the estimation performance by the EKF for a sinusoidal input. he sinusoidal voltage amplitude is V. Since the Joule heating occurs regardless of the polarity of the applied voltage to the wire, the actuator moves upward and stops when the voltage crosses the V line. Similar to the previous case, the estimation errors are bounded and the peak estimation errors occur when the actuator stops or starts moving. 695

6 MHElahinia et al e x U(V) =. sec =.5 sec =.sec e x e x e x3 e x Figure 9. he effect of sampling time on the performance of the extended Kalman filter. θ (rad/s) 5 EKF EKF θ (rad) x4 ( o C) Figure. heeffect of the initial state vector (x initial )onthe performance of the extended Kalman filter: EKF and the model have the same initial state vector; EKF has a different initial state vector. An essential phenomenon of SMA wire behavior is hysteresis. his behavior is accurately predicted by the EKF as shown in figure 8. In this simulation the same sinusoidal input voltage is applied to the wire and the actuator repeatedly moves up and stops. Although the wire only undergoes martensite to austenite phase transformation, the stress follows hysteric increasing and decreasing paths. he estimation error is small, again withthe exception of when the arm changes the direction of motion. he ultimate use of the filter would be in control of the arm. It can be clearly seen that the filter demonstrates good performance for both stabilization and tracking tasks. Since the differential equations representing the model have been approximated by difference equations for the filter, the sampling time has an effect on the performance of the filter. Figure 9 shows a simulation in which a constant voltage is applied to the wire. After reaching the austenite transformation temperature the arm moves upward and eventually stops. he larger estimation errors again occur when the arm either starts or stops moving. Increasing the sampling time reduces the computational load but results in larger estimation error. Figure shows that even if the initial state vector used for e x e x e x x 9 σ v = x σ v = x 3 σ v = Figure. he effect of the measurement noise variance σ v on the arm angular position prediction error of the extended Kalman filter, σ w =. U(V) e x e x e x x 5 σ w = x σ w = x σ w = Figure. he effect of the process noise covariance σ w I 4 on the arm angular position prediction error of the extended Kalman filter, σ v =. the EKF does not match the true state vector the EKF still performs well. In this case a constant voltage is applied to the wire and the filter converges quickly (within five time steps, for =. s) to the true states. At each time step, the extended Kalman filter finds the optimal estimate by adding the previously calculated predicted value, based on the dynamics of the system, to a correction term. he correction term consists of an optimal gain multiplied by the difference of the predicted value and the measurement. he gain is calculated based on the measurement and dynamic system noises. If the measurement noise variance is large then the gain is smaller and as a result the filter tends to place smaller confidence in the noisy measurement. Figure presents the effect of the variance of the measurement noise. Clearly, larger measurement noise variance causes the filter to depend more on its predictions, which leads tolarger prediction error. If the dynamic system (process) noise is large, then the filter will put more weight on the measurement. Figure presents the effect of the 696

7 Design of a Kalman filter for rotary shape memory alloy actuators process noise covariance on the EKF prediction error. In this case, larger process noise causes the filter to rely more on the measured data, that leads to smaller errors. 5. Conclusion he Kalman filter, as an optimal recursive filter, incorporates all the available information including dynamics of the system and sensors, the statistical description of the model uncertainty and noises, and initial conditions of the system. It is difficult to measure the state variables of the SMA-actuated manipulators that are needed by the control algorithms. his is mostly because of the small diameter of the SMA wires (5 µm in this case). An extended Kalman filter, the nonlinear version of the Kalman filter, was designed for estimation of the state variables of an SMA-actuated manipulator and has been tested through simulations. he angular position of the arm and the input voltage to the SMA wire were the only measured variables used by thefilter. In designing the filter, the model was discretized in such a way as to maximize the effect of the measured variables. Satisfactory results have been shown in terms of the filter estimating a state vector that closely matches the state vector generated by a nonlinear model of the system. Since the model had been previously verified against experimental data and the filter is designed based on this model, it is expected that the filter can accurately predict the state vector of the SMA-actuated arm. he sensitivity of the filter to the measurement and process noise and sampling time were discussed. It was shown that that regardless of the initial conditions the filter can converge to the true state variables. he next step in this research will be applying the filter in designing a control system. Since the full state vector is accessible through EKF, the control system should be able to perform better compared to other control algorithms that depend only on the measured state variables. References [] Honma D, Miwa Y and Iguchi N 984 Micro robots and micro mechanisms using shape memory alloy he 3rd oyota Conf. on Integrated Micro Motion Systems, Micro-machining, Control and Applications (Nissan, Aichi, Japan, 984) [] Gharaybeh M A and Burdea G C 995 Investigation of a shape memory alloy actuatorforce-feedbackmastersadv. Robot [3] Grant D and Hayward V 995 Design of shape memory alloy actuator with high strain and variable structure control Proc. IEEE Int. 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