Sensorless Torque Estimation using Adaptive Kalman Filter and ...

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I motor K T D input 1 J B J D output 1 1 s I , D s law is applied: u = K P e + K I ∫ edτ + K D de dt (16) The PID control law(15) together with ė 0 = e 1 and error equations (14), error dynamics described as follows: D disturbance. K n Fig. 3. PID 1 J n u D u ˆ total B J n n 1 1 s I s 1 J n B J n n Extraction 1 1 s * s I I , D Estimation * D Disturbance Estimator Block diagram of motor with disturbance 2) Estimation part: The estimation part calculates the final estimation result by using a control loop. The reference signal of the control loop is output of the extraction part(θ D ). Except the torque coefficient(K n ) of the extraction part and a PID controller of the estimation part, the open-loop transfer functions are exactly same. If the estimation part is stable, the feedback signal(θ ∗ D ) follows the reference(θ D), and the output of the PID controller approaches the total disturbance. Consequently, designing the disturbance estimator is concluded by making the estimation part asymptotically stable. In order to obtain a stable region of the PID gains, state space equations is used. Except the PID controller, the open-loop dynamics of the estimation part is described as follows: ẋ 1 = x 2 ẋ 2 = − B n x 2 + 1 u (14) J n J n y = x 1 where x = [ θ ∗ D ω∗ ] T . In order to determine error equations, let e = θ D − θ ∗ D = θ D − x 1 = e 1 . Then error equations obtained as follows: ė 1 = −ẋ 1 = e 2 ė 2 = B n x 2 − 1 u (15) J n J n To determine the control signal(u) in (14), the PID control ė 0 = e 1 ė 1 = e 2 (17) ( ) ( ) ( ) KI KP Bn ė 2 = − e 0 − e 1 − + K D e 2 J n J n J n If the characteristic matrix of the error dynamics is Hurwitz, error approaches zero asymptotically. It indicates θD ∗ and u also asymptotically approach θ D and the total disturbance D total , respectively. In order to calculate stable region of the PID gains, Routh−Hurwitz theorem is used. Calculated region of the PID gains are written as follows : K I J n > 0 , K P J n > 0 , B n J n + K D > 0 (18) As we mentioned above, the estimated total disturbance contains not only exogenous disturbances(D input and D output ) but also all the discordance between the real system model and the model of the extraction part. In addition, simply removing all the viscous friction loops in Fig.3, the viscous friction also estimated as the disturbance. It shows the flexibility of the disturbance estimator. V. EXPERIMENTAL RESULTS To validate proposed two estimation methods, experimental tests were conducted using a master-slave system. The overall setup is shown in Fig.4, and the specifications of motor drivers and motors are shown in Tables I and II. For the tests, the slave manipulator was fixed, and the operator manipulated the master to give an order to the slave device. A. AKF based disturbance observer The experimental test of the AKF based estimation method is conducted by two ways. At the first, we confirmed the improvement of the AKF based method in comparison with the DKF. In order to obtain noisy measurement, same zero mean gaussian noise having 0.5 of the standard deviation is added to the measurements of both the AFK and DKF. In addition, wrong measurement covariance (R k = 0.2 2 ) is given to the DKF. Due to the manipulator is fixed, the mean of the measurement is ideally zero. Fig.5 shows the measurements, i.e., control signal(I) of the slave motor, and the position discordance between the master and slave manipulators. The AKF based observer also tested under the same condition. We set the wrong measurement covariance as (0.2 2 ), and we set the accumulation number(N) in (12) as 5. The estimation results of the DKF and AKF based observer are shown in Fig.6. Due to the DKF dose not have the measurement covariance adaptation algorithm, the
Fig. 5. Measured armature current and Position difference(DKF) Fig. 4. Experimental setup : master-slave manipulator TABLE I SPECIFICATION OF MOTOR DRIVERS Output voltage Output current Maxon - ADS 50/5 VCC min. 12 VDC; max. 50VDC Depending on load, continuous 5A Fig. 6. Disturbance estimation. (a) DKF, (b) AKF (N=5). TABLE II SPECIFICATION OF MOTORS Maxon - RE30 Assigned power rating(W) 60 Max. continuous current(A) 4 Torque constant(mNm / A) 25.9 Max. continuous torque(mNm) 86.2 estimation is based on the given wrong measurement covariance matrix. As the Fig.6(a) demonstrates, the estimation result of the DKF containing a lot of noise. If it used for the haptic feedback and the position control, it will make huge vibration on the manipulator. On the other hand, even the given initial measurement covariance matrix of the AKF was mismatched to the actual noise, the AKF calculated relatively clear estimation result(see Fig.6(b)) based on the covariance uncertainties online estimator(12). Such result demonstrates the AKF based disturbance observer corrects the measurement covariance matrix at each processing time. Moreover, such result indicates possibility of the reliable estimation under the noisy circumstance. As the second test, we applied huge number(R k = 1 2 ) of the standard deviation to the measurement noise. Fig.7(a) shows the estimation result when the number of accumulation(N) is 5. On the other hand, when we used 30 number of measurement data, as shown in Fig.7(b) significantly improved estimation result is obtained. As depicted above, the disturbance estimation using noisy measurement is achieved. B. Disturbance Estimator This subsection validate the disturbance estimator. For the test, same bilateral teleoperation system is used. Two Fig. 7. Disturbance estimation using different number of measurement accumulation (AKF), standard deviation = 1 measurements of the disturbance estimator, the control signal and the angular position of the slave manipulator are depicted in Fig.8. From the two measurements, extraction part calculates the position(θ D ) by using (13), and then estimation part calculates the total disturbance. In the experimental test, PID gains K P = 0.135, K I = 0.01, K D = 0.008 are used to make estimation part asymptotically stable. We can find that a set of PID gains satisfies condition (17) in the continuous time domain. Of course PID gains can be selected by trial and error also. In Fig.9(a), the output of the extraction part(θ D : reference of the estimation part) is denoted as reference, and the feedback signal of the estimation part(θD ∗ ) is represented as tracking. Tracking performance can be used as an estimation performance index in real time. The total estimation result is shown in Fig.9(b). In the test, the angular position is used for the measurement of the disturbance estimator, however, even the designer uses the angular velocity as the measurement but not position, disturbance estimator will precisely estimate the total disturbance just by eliminating the last one integrator at each extraction and estimation part. It shows the flexibility
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