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Hindawi Publishing Corporation Journal of Robotics Volume 2011, Article ID 845685, 8 pages doi:10.1155/2011/845685 Research Article ControlLoopSensorCalibration UsingNeuralNetworksfor Robotic Control 1 2 Kathleen A. Kramer and Stephen C. Stubberud Department of Engineering, University of San Diego, 5998 Alcala ´ Park, San Diego, CA 92110, USA Advanced Programs, Oakridge Technology, Del Mar, CA 92014, USA Correspondence should be addressed to Kathleen A. Kramer, email@example.com Received 15 July 2011; Accepted 8 November 2011 Academic Editor: Ivo Bukovsky Copyright © 2011 K. A. Kramer and S. C. Stubberud. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Whether sensor model’s inaccuracies are a result of poor initial modeling or from sensor damage or drift, the eﬀects can be just as detrimental. Sensor modeling errors result in poor state estimation. This, in turn, can cause a control system relying upon the sensor’s measurements to become unstable, such as in robotics where the control system is applied to allow autonomous navigation. A technique referred to as a neural extended Kalman ﬁlter (NEKF) is developed to provide both state estimation in a control loop and to learn the diﬀerence between the true sensor dynamics and the sensor model. The technique requires multiple sensors on the control system so that the properly operating and modeled sensors can be used as truth. The NEKF trains a neural network on-line using the same residuals as the state estimation. The resulting sensor model can then be reincorporated fully into the system to provide the added estimation capability and redundancy. 1. Introduction with navigation systems where multiple inertial navigation systems (INSs) are employed or for safety considerations to In the target tracking applications, sensor systems are placed provide back-up in case of failure . Sometimes multiple in various locations about the region of interest and provide sensors are employed because more accurate sensors cannot measurements that are combined over time to provide an provide measurements at the necessary update rates. In such improved picture of the region. One of the signiﬁcant cases, less accurate sensors might be used to provide reports problems with target tracking systems is that the sensors at the sampling time between the measurement reports might not be properly calibrated, aligned, or modeled. This of the more accurate system. Finally, additional sensors can wreak havoc with the performance of the involved are added in that they provide measurements that contain tracking systems. When calibration and alignment are the directly observable state information. In mobile robotic main factors, the problem is referred to as sensor registration systems, for example, sensors include position sensors to [1, 2]. Since the real-time activities of a tracking problem determine absolute position, such as a global position system cannot be postponed, the correction of the registration or (GPS), or relative position such as radars, sonars, and the mismodeling must be performed online. This problem imagery equipment, as well as speedometers and possible can extend to other applications, including that of feedback- accelerometers that provide the velocity states more directly control systems. than the indirectly observed values from position. While some control applications use a single sensor to While additional sensors are often used to provide provide measurements that are used in the feedback loop, improved overall accuracy, errors can arise in the accuracy of many control systems use several sensors to provide the nec- the measurements provided to the control law by individual essary measurements. Multiple sensors are used for a variety sensors. This occurs with tracking sensors and is well known of reasons. Often times the sensors add redundancy, such as with gyroscopes in an INS. While a sensor that drifts from its 2 Journal of Robotics calibration point in a nonrandom and identiﬁable manner in Section 4. These examples show the beneﬁts of the NEKF can be recalibrated, there is no remedy other than redun- calibration. dancy or replacing a sensor that fails or breaks completely 2. Recalibration Approach since it is unable to provide a meaningful measurement. For a sensor that can be recalibrated, online calibration is In this eﬀort, a recalibration technique that can be used desired because taking the inaccurate sensor oﬄine may not online is developed. The approach relies upon other sensors be possible in the short term or could have a deleterious eﬀect in the multisensor system operating properly which are used on the performance of the controller. When a closed-loop to provide a level of truth from which to calibrate. The sensor control system relies upon a sensor with an error, the result to be recalibrated may have some inaccuracy but cannot be is that the control signal varies from the value needed to irreparably damaged. achieve the desired response. These eﬀects are evident, even The dynamics of a nonlinear system can be modeled as a in a linear system . Sensor calibration is thus an important set of recursive diﬀerence equations: issue for the control design . For nonlinear systems, these x = f(x , r ) + ν , issues become even more pronounced and can lead to a more k+1 k k k (1) signiﬁcant eﬀect on stability, as the feedback may not fall ( ) z = h x , u + η , k k k within the Lyapunov stability criteria . where x is the state vector representing the system dynamics, In this work, an online calibration technique is proposed f(·) is state-coupling model, r is the reference input vector, for the case where the control law utilizes a state estimator z is measurement vector, also considered the report from the . The approach uses a technique referred to as a neural sensor system, u is an external input to the sensor system extended Kalman ﬁlter (NEKF) [8–10]. The NEKF has the often simply the reference signal, and h(·) is the output- beneﬁt of being able to train a function approximation online coupling function . The vectors ν and η represent noise and provide the state estimation simultaneously. The NEKF on the system states and the measurement states, respectively. provides a uniﬁed approach using the same residual for If the reference input r is assumed to be both the refer- both components. Techniques such as in , in contrast, ence signal to the system and the external input to the sensor, can provide the desired sensor correction for sensors in a then r is considered to be the same as u. Then, the state control loop but are not integrated to the state estimator and estimation model used in the control law would be rewritten corrections are performed outside of the control loop. The as technique applied to this sensor correction problem is based upon an approach that was developed for a multisensor x = f(x , u ), k+1 k k (2) tracking system where the sensor model was in error or a z = h(x , u ). k k k sensor registration was detected . In that development, one sensor was considered to be local while the other was The accuracy of the models in the estimator determines the considered to be oﬀ-board. All corrections were made to the accuracies of state estimates. For this eﬀort, it is assumed that oﬀ-board reports relative to the local reference frame. Unlike the state-coupling function is very accurate while at least one the tracking problem where the estimator is open loop, of the sensor models has some inaccuracy. The error in this control applications require the consideration of closed-loop modelisdeﬁnedas issues including stability. The NEKF is used to recalibrate ε = h (x, u) − h (x, u). (3) true model the sensor model based on online operational measurements while in the control loop. It also provides the state estimation To reduce the measurement error, the function ε needs to for the control law. Thus, the proposed technique estimates be identiﬁed. A neural network that satisﬁes the function the states and provides the training paradigm in a single approximation requirements of the Stone-Weierstrauss The- algorithmic implementation [13, 14], unlike other neural orem  is proposed to approximate the error function ε. network sensor modeling techniques [15–17]. The software Such aneuralnetwork canbedeﬁnedas correction to the actual sensor measurements can be applied NN(x, w) = NN x, ω, β = β g x, ω ,(4) j j j to the case where the sensor is still operational but poorly calibrated. The NEKF algorithm requires a truth measurement from where w are the weights of the neural network, which are which to calibrate the sensor. Using additional sensors on the decomposed into the set of input weights ω and the set of dynamic system, a poorly calibrated sensor can be modeled output weights β. The hidden function g(·) is a sigmoid such that its reporting errors are removed prior to their function: incorporation to the feedback loop. −ω x − ω x k jk k 1 − e 1 − e g(x, ω) = = . (5) In Section 2, the NEKF algorithm and its implementation T −ω x − ω x k jk k 1+ e 1+ e in the control loop is developed. Section 3 provides the This creates a new measurement and more accurate measure- ﬁrst example system and the performance results of the ment model: online calibration using the NEKF for a two-sensor moving platform that has a range-bearing sensor and a miscalibrated ( ) ( ) ( ) e = h x, u − h x, u − NN x, u, w,(6) true model velocity sensor. A navigation problem where an intermittent use of a position sensor such as GPS is available is presented where eε. Journal of Robotics 3 The extended Kalman ﬁlter (EKF) is a standard for The necessary coupling between the weights and the dynamic nonlinear estimation in a control loop [4, 19]. An approach states occurs as a result of the output-coupling Jacobian. This developed in  referred to as a neural extended Kalman coupling permits the weights to be completely observable to ﬁlter provides both the state estimation of the system and the the measurements and to train oﬀ of the same residual as training of the weights  of a neural network using the the system states. This also implies that the NEKF trains the same measurement estimates. neural network online without the need for additional oﬀ- The NEKF is a coupled state-estimation and neural- line training sets. network training algorithm and has similarities to parameter The NEKF deﬁned in (8)–(12)requiresatruthmea- estimation approaches . In such implementations, the surement for the neural network to learn. For a target state vector of the EKF contains both the system states and tracking application where a surveyed target can be used the parameters of the model being estimated. Parameter to provide ground truth, this implementation is appropriate estimation, though, is based on a known model structure. . Control applications are diﬀerent in that such “truth” is The neural network of (4), in contrast, is a general function unavailable. Instead, the state of the system dynamics must with its weights as the parameters. Therefore, the augmented be generated using sensors in the control system that are state of this NEKF is the state estimate and the input and more accurately modeled. This requires a modiﬁcation of the output weights of the neural network: NEKF design to utilize the existing state estimates and the ⎡ ⎤ measurements from the other sensors to correct the output- k coupling function of the mismodeled sensor using the neural ⎢ ⎥ x = = ⎣ ⎦ . (7) w network. For this implementation, the detection of a sensor failure Incorporating this new state and the neural network is assumed to have occurred. Such detection techniques abound in tracking theory and similar techniques exist aﬀects all of the Kalman ﬁlter equations. In the Kalman gain equation and the state error covariance update equation, (8) in dynamic system implementation . The detection of and (10), the Jacobian of the output-coupling function is the faulty sensor initiates two changes in the NEKF. First, the process noise matrix is modiﬁed. The ratio of the properly augmented. In (9), the neural network is incor- porated into the state update. Finally, the prediction equa- process noise Q to the measurement noise R is reduced tions, (11)and (12), must be properly augmented for the to signiﬁcantly favor the dynamic state estimate over the residual. This reduces the eﬀect of the measurements from augmented state. Thus, the NEKF equations for the sensor modeling become the poor sensor on the system estimate. The process noise of the weights is increased to favor the residual over the weight −1 T T K = P H HPH + R,(8) k k|k−1 estimates so that the weights will train to learn the error function. A similar approach of reducing the ratio of Q to ⎡ ⎤ k|k R could be used for the reduction of state-coupling error ⎣ ⎦ x = = x k|k k|k−1 [7, 8]. The second modiﬁcation is that, for the ﬁrst n steps k|k of the correction of the NEKF, the dynamic state estimates are decoupled from the control loop. The resulting modiﬁed + K z − h x , u + NN x , u , w , k k k|k−1 k k|k−1 k k|k−1 NEKF equations are given as (9) −1 T T K = P H HPH + R , (15) P = I − HK P , (10) k k|k−1 k|k k k|k−1 k|k f x k|k x = = x k|k k|k−1 x = f x = , (11) k+1|k k|k k|k k|k + K z − NN x , u , w + h x , u , k k k|k−1 k k|k−1 k|k−1 k ∂f(x) ∂f(x) (16) P = P + Q k+1|k k|k k ∂x ∂x ⎡ ⎤ ⎡ ⎤ (12) F0 F 0 P = I − HK P , (17) k|k k k|k−1 ⎣ ⎦ ⎣ ⎦ = P + Q . k|k k 0I 0I acc k+1|k x = f x = , (18) k+1|k k|k The augmented Jacobian of the sensor function model, k|k ⎡ ⎤ h(x, u, w) = h(x, u) + NN(x, u, w) (13) Block1,2 acc P P k+1|k k|k ⎣ ⎦ P = + Q , (19) is therefore deﬁned as k+1|k k Block2,1 Block2,2 P P k|k k|k ∂h(x) + NN(x, u, w) ∂NN(x, u, w) H = ∂x ∂w x=x ,w where the superscript acc indicates the accurate covariance k|k−1 k|k−1 from the estimator using the other sensors. If more than ∂NN(x, u, w) ∂NN(x, u, w) one poor measurement is produced between accurate mea- = H + . ∂x ∂w x=x ,w k|k−1 k|k−1 surements, only the last measurement was used for the state (14) estimator . 4 Journal of Robotics Two approaches for decoupling have been considered. In Local magnetic pole the ﬁrst, the measurement noise for the faulty sensor is kept artiﬁcially high throughout the experiment. This reduces the eﬀect of the poor sensor even as the neural network trains. In the second approach, as the weights of the neural network Desired vehicle location settle, the measurement noise is reduced from its artiﬁcially high values to values closer to the calibrated sensor noise covariance. This allows the improved model of the sensor to have greater eﬀect on the control loop. 10 m 3. Control Example I Vehicle Initial vehicle location A simulated control example is used to demonstrate the capability of the NEKF technique to model a sensor change while the system is in operation. A small motorized vehicle is operating in a two dimensional space, as seen in Figure 1. Surveyed location The goal is for the vehicle to move from one point in the operational grid to a second point in the operational 10 m grid. The vehicle maintains its position via a sensor system thatprovidesarange-bearing measurementtoasurveyed Figure 1: Example scenario of an autonomous vehicle using two sensors to achieve a speciﬁc end location. location in the operational grid. The platform also has a speedometer and compass system that is used to provide the vehicle’s heading relative to a local magnetic pole. This is similar to INS systems that are GPS denied. The state-space model of the vehicle-dynamics is deﬁned accuracy of the position sensor is given as 1.0 degrees for as bearing and 0.005 m for range: ⎡ ⎤ ⎡ ⎤ 0100 x ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ 2 2 x˙ + y˙ 0000 x˙ ⎢ ⎥ ⎢ ⎥ k k k ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ vel x˙ = f(x, u) = + g(u), (20) ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ z = = , (23) k ⎣ y˙ ⎦ ⎢ 0001⎥ ⎢ y⎥ arctan ⎣ ⎦ ⎣ ⎦ x˙ 0000 y˙ ⎡ ⎤ 2 2 ⎡ ⎤ T x − x + y − y ⎢ k survey k survey ⎥ g(u) = 0 a 0 a pos ⎢ ⎥ x y ⎣ ⎦ z = = ⎢ ⎥ . (24) y − y ⎣ k survey ⎦ (21) arctan x − x k survey = 0 a · cos θ + ψ 0 a · sin θ + ψ , For this eﬀort, the control input has an input rate of where a indicates acceleration, θ denotes the heading angle, 0.1 seconds to synchronize with the sensor reports. The and ψ denotes the change in heading angle. For a sample rate initial location of the vehicle is deﬁned at (−5m, −6m) of dt, the discretized representation becomes with a heading of 0 degrees, while the desired ﬁnal point ⎡ ⎤ ⎡ ⎤ is (0 m, 0 m) with a heading of 45.0 degrees. The range- 1 dt 00 x bearing beacon is placed at (+3 m, −8 m), while the local ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ 0100 x ⎢ ⎥ ⎢ k+1⎥ north pole is deﬁned at (−2 m, +3 m). The position sensor ⎢ ⎥ ⎢ ⎥ x = f(x , u ) = + g (u ). (22) k+1 k k k k ⎢ ⎥ ⎢ ⎥ measurement uncertainty matrix had standard deviations of ⎢ 001 dt⎥ ⎢ y ⎥ ⎣ ⎦ ⎣ ⎦ 0.005 m for the range and 1.0 radian for the bearing. For 0001 y k+1 the velocity sensor, the measurement error covariance had standard deviations of 0.001 m/s for the speed and 2 radians The speed sensor and heading sensor have combined into for the heading. The broken velocity sensor reported speeds a single sensor package modeled in (23). The package increased by a factor of 1.9 and added a 0.5-radian bias to the has an update rate of 0.2 seconds. The accuracy of the heading, in addition to the additive Gaussian noise. compass heading is 2.0 degrees while speedometer accuracy For the NEKF, the process noise for the weights of the is 0.001 m/s. The position sensor, providing a range (r), and neural networks, Q , and its initial error covariance, P , w wts bearing (α) relative to the surveyed location, is modeled were increased to allow for changes based on the residuals in (24), also with an update rate of 0.2 seconds. The using the broken sensor. The process noise for the input position sensor reports are interleaved with the velocity weights was set to 1.0 and for the output weights, it was set to measurements, as shown in the time line of Figure 2.The 2.0. The initial state error covariance, P , was set to 100.0. states Journal of Robotics 5 Velocity Velocity Velocity Position Position Position t t + 0.1 s t + 0.2 s t + 0.3 s t + 0.4 s t + 0.5 s t + 0.6 s o o o o o o o Figure 2: Sensor reports from diﬀerent sensors are interleaved to provide uniform reporting. The process noise, Q, was set to the integrated white noise 4. Control Example II model : The small motorized vehicle operating in a two-dimensional ⎡ ⎤ 3 2 space shown in Figure 1 is also the basis for the second dt dt ⎢ ⎥ control example. In this case, the vehicle maintains its 3 2 ⎢ ⎥ ⎢ ⎥ position via a two-sensor system. One sensor provides a ⎢ ⎥ dt ⎢ ⎥ dt 00 range-bearing measurement to a surveyed location in the ⎢ ⎥ 2 2 ⎢ ⎥ Q = q , (25) operational grid and another, slower reporting, position ⎢ 3 2⎥ dt dt ⎢ ⎥ ⎢ 00 ⎥ sensor provides a linear position report. This would be ⎢ ⎥ 3 2 ⎢ ⎥ similar to a GPS system updating an INS. ⎣ 2 ⎦ dt The state-space model of the vehicle-dynamics is deﬁned 00 dt as in (20)–(22). The range-bearing error is described as in (23). The position sensor has an update rate of 0.6 seconds with a factor q of 0.0017. For the NEKF, the initial value and provides a linear report as seen in (26). The accuracy for P was set to 1000. For comparison purposes, an EKF wts of the position sensor is assumed to be ±0.01 m in both using the same values for the dynamic state components was directions. The other position sensor, providing a range (r), generated as well. and bearing (α) relative to the surveyed location, is modeled Four separate cases, two with the EKF and two with the in (24) and has an update rate of 0.2 seconds. The accuracy of NEKF were implemented. The four cases are (1) an EKF the faster, range-bearing position sensor is given as 1.0 degree with an inﬂation factor of 100 throughout the scenario, (2) of bearing accuracy and 0.005 m of range accuracy: an EKF with an inﬂation factor of 10,000,000 throughout ⎡ ⎤ the scenario, (3) an NEKF with an inﬂation factor of 100 pos throughout the scenario, and (4) an NEKF with an initial ⎣ ⎦ z = . (26) inﬂation factor of 1000 held for 80 iterations and then y decayed by 15% for later iterations down to a minimum value of 1. The sensor reports of positions are interleaved as shown in Inﬂating the uncertainty matrix, R, of the velocity sensor the time line of Figure 4. For this eﬀort, the control input has by a given inﬂation factor allows the state update component an input rate of 0.2 seconds to synchronize with the range- of EKF and of the NEKF to be less aﬀected by these poor bearing sensor report. measurements. For the NEKF, a 4-node hidden layer was The initial location of the vehicle is deﬁned at (−5m, used. The results are shown in Figures 3(a)–3(d). −6 m) with a heading of 0 degrees, while the desired ﬁnal Figure 3(a) indicates that the EKF is unable to remain point is (0 m, 0 m) with a heading of 45.0 degrees. The range- stable with the poor measurements having the reduced, but bearing beacon is placed at (+3 m, −8 m). The position not completely insigniﬁcant, eﬀect on the state estimate. sensor providing linear reports provides measurements with By overcompensating with a signiﬁcant increase in the abiasof(0.5m, −0.25 m) added to the coordinates and is considered to be miscalibrated. measurement covariance, the EKF basically eliminates all of the poor sensor reports. This is seen in Figure 3(b). For the NEKF, the process noise for the weights of the However, the vehicle never settles in at the location. The neural networks, Q , and its initial error covariance, P , w wts NEKF results are shown in Figures 3(c) and 3(d) where it were increased to allow for changes based on the residuals learns sensor errors while on line to provide clearly improved using the broken sensor. The process noise for the input control performance. Even with the same measurement weights was set to 1.0 and for the output weights, it was set to covariance as the EKF in ﬁrst case, as in Figure 3(c), the 2.0. The initial state error covariance, P , was set to 10.0. states NEKF implementation remains stable. In both Figures 3(c) The process noise, Q, was set to the integrated white noise and 3(d), the results are quite similar with more changes after model of (25)withafactor q of 0.0017. For the NEKF, the measurement noise decays. the initial value for P was set to 0.1. The inﬂation factor for wts With multiple sensor systems, the NEKF is able to the broken sensor in both the NEKF and the EKF tests was set provide a fault correcting mechanism for sensors that are to 100. For comparison purposes, the EKF that was generated still providing information, although that information needs used the same values for the dynamic state components. correction. The research of this eﬀort has also shown that the In this example case, when the system had two fully slower the decay rate is on the inﬂation factor, the lower the functional sensors, the vehicle took 3.6 seconds to reach its initial inﬂation factor can be. desired endpoint with a 0.001 m total distance error. 6 Journal of Robotics ×10 1 3 0 2 −1 −2 0 −3 −1 −4 −2 −5 −3 −6 −4 −7 −5 −8 −6 0 100 200 300 400 500 600 700 800 900 0 100 200 300 400 500 600 700 800 Iterations Iterations (a) (b) 3 3 2 2 1 1 0 0 −1 −1 −2 −2 −3 −3 −4 −4 −5 −5 −6 −6 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 Iterations Iterations x-coordinate x-coordinate y-coordinate y-coordinate (c) (d) Figure 3: (a) Position coordinates using the EKF and a low inﬂation rate. (b) Position coordinates using the EKF and a high inﬂation rate. (c) Position coordinates using the NEKF and a low inﬂation rate. (d) Position coordinates using the NEKF and a higher inﬂation rate that is allowed to decay after 80 iterations. Position Position xy xy Position Position Position Position Position Position rα rα rα rα rα rα t t + 0.2 s t + 0.4 s t + 0.6 s t + 0.8 s t + 1 s t + 1.2 s o o o o o o o Figure 4: Sensor reports from diﬀerent sensors are interleaved to provide uniform reporting. A comparison between the EKF with the bad sensor and aﬀected by these poor measurements. For the NEKF, a 4- the NEKF with the bad sensor was generated. Two diﬀerent node hidden layer was used. In this case, both the NEKF and comparisons were run. In the ﬁrst comparison, an inﬂation the EKF went unstable. factor of 10 for measurement error covariance matrix of the In the second comparison, an inﬂation factor of 1000 broken sensor was used. Inﬂating the uncertainty matrix, R, was used. The results of the four states, x-position, x- of the velocity sensor by a given inﬂation factor allows the velocity, y-position, and y-velocity are shown for the EKF in state update component of EKF and of the NEKF to be less Figure 5. While the individual elements are hard to discern, Position Position Position Position Journal of Robotics 7 300 100 −20 −40 −60 −100 −80 −100 0 5 10 15 20 25 30 35 40 45 −200 Iterations Residual −300 0 200 400 600 800 1000 1200 3 times error bound Iterations Figure 7: Comparison of EKF residual and the error covariance weighting. x position y position x velocity y velocity Figure 5: EKF state responses. 3 500 −500 −1000 −1 −1500 −2 0 5 10 15 20 25 30 35 40 45 −3 Iterations −4 Residual 3 times error bound −5 Figure 8: Comparison of NEKF residual and the error covariance −6 0 102030405060708090 weighting. Iterations x position y position x velocity y velocity One of the interesting comparisons is that of the residual Figure 6: NEKF state responses. errors for heading in this example compared to a 3-sigma error based upon the system and sensor covariances. Figure 7 shows the results for the EKF, and Figure 8 shows the results for the NEKF. In both cases, the residuals are within the the important fact is that none of the values converge. This bounds, but the NEKF has a more accurate bound in the causes the vehicle to leave the region of interest. Figure 6 steady-state portion allowing for the Kalman gain to permit depicts the NEKF results for the four states. As is clearly seen, greater residual eﬀect on the state update for the corrected both techniques achieve a result, but the NEKF trajectory is sensor. Thus, without the NEKF, the system would go much smoother and the states are less erratic and require unstable with deleterious eﬀects almost immediately. With signiﬁcantly fewer iterations to converge to the desired result the NEKF, the vehicle remains in the operational area and (84 iterations versus 498 iterations). slowly converges to the desired point. If the position sensor was miscalibrated and not cor- rected but only had a reduced eﬀect on the navigation estimator, as would be the case when using the EKF only 5. Conclusion and assumed it were inaccurate, the vehicle position becomes unstable causing it to leave the grid area within 10 seconds of A new neural extended Kalman ﬁlter algorithm has been beginning the test. When the NEKF was used, the vehicle was developed that can improve sensor modeling for a poorly able to reach its desired endpoint in 38.4 seconds with 0.45 m modeled sensor in the control loop. The technique expanded total distance error and 0.3 m/s velocity error. on an approach originally developed for target tracking Residual Residual 8 Journal of Robotics that relied upon that availability of truth. In the algorithm,  S. C. Stubberud, K. Kramer, and A. R. 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