Initial Heading Error Research Articles

Performance analysis of three-dimensional differential geometric guidance law against low-speed maneuvering targets

The performance of the three-dimensional differential geometric guidance law with proportional navigation formation against a target maneuvering arbitrarily with time-varying normal acceleration is thoroughly analyzed using the Lyapunov-like approach. The validation of this guidance law is firstly proved, and then the performance issues such as capturability, heading error control efficiency, line of sight rate convergence, and commanded acceleration requirement are analyzed, under the condition that the missile is initially flying toward the target with a speed advantage. It is proved that an intercept can occur and the line of sight rate and missile commanded acceleration can be limited in certain ranges, if the initial heading error is small and the navigation gain is sufficiently large. The nonlinear relative dynamics between the missile and the target is taken into full account, and the analysis process is simple and intuitive, due to the use of a convenient line of sight rotating coordinate system. Finally, the new theoretical findings are validated by numerical simulations.

Time-varying sliding mode guidance scheme for maneuvering target interception with impact angle constraint

In this paper, a guidance scheme for impact angle control against maneuvering targets with unknown target acceleration is proposed. In this scheme, the unknown target acceleration is estimated via a linear extended state observer; a novel time-varying global slide mode control technique is presented to eliminate the reaching phase and enforce a desired impact angle exactly at the time of interception with finite-time convergence, good robustness, high precision and smooth guidance command. Moreover, feasible guidance logics are developed to achieve all-aspect interception with the tolerance of large initial heading errors. Numerical simulations in various scenarios are performed to verify the performance of the proposed guidance scheme.

Impact angle constrained guidance for all-aspect interception with function-based finite-time sliding mode control

In this paper, the impact angle constrained guidance problem, focusing on the nonlinear engagement dynamics, is considered. In order to fulfill the terminal constraints, a novel function-based finite-time sliding mode control methodology is introduced to design the guidance law. The main feature of this guidance law is that it achieves the desired impact angle exactly at the time of interception. In addition, it is capable of ensuring a continuous and smooth control action. To improve the tolerance of initial heading errors and broaden the application, a feasible guidance logic is also developed. Numerical simulations in various scenarios have shown that the proposed guidance scheme can realize all-aspect interception against stationary, constant-velocity and maneuvering targets. Furthermore, comparison studies with respect to nonsingular terminal sliding mode control-based methods demonstrate the superiority of the proposed method.

Analytical impact time and angle guidance via time-varying sliding mode technique

To concretely provide a feasible solution for homing missiles with the precise impact time and angle, this paper develops a novel guidance law, based on the nonlinear engagement dynamics. The guidance law is firstly designed with the prior assumption of a stationary target, followed by the practical extension to a moving target scenario. The time-varying sliding mode (TVSM) technique is applied to fulfill the terminal constraints, in which a specific TVSM surface is constructed with two unknown coefficients. One is tuned to meet the impact time requirement and the other one is targeted with a global sliding mode, so that the impact angle constraint as well as the zero miss distance can be satisfied. Because the proposed law possesses three guidance gain as design parameters, the intercept trajectory can be shaped according to the operational conditions and missile׳s capability.To improve the tolerance of initial heading errors and broaden the application, a new frame of reference is also introduced. Furthermore, the analytical solutions of the flight trajectory, heading angle and acceleration command can be totally expressed for the prediction and offline parameter selection by solving a first-order linear differential equation. Numerical simulation results for various scenarios validate the effectiveness of the proposed guidance law and demonstrate the accuracy of the analytic solutions.

Long-Term Supplementation with Beta Serum Concentrate (BSC), a Complex of Milk Lipids, during Post-Natal Brain Development Improves Memory in Rats.

We have previously reported that the supplementation of ganglioside-enriched complex-milk-lipids improves cognitive function and that a phospholipid-enriched complex-milk-lipid prevents age-related cognitive decline in rats. This current study evaluated the effects of post-natal supplementation of ganglioside- and phospholipid-enriched complex-milk-lipids beta serum concentrate (BSC) on cognitive function in young rats. The diet of male rats was supplemented with either gels formulated BSC (n = 16) or blank gels (n = 16) from post-natal day 10 to day 70. Memory and anxiety-like behaviors were evaluated using the Morris water maze, dark–light boxes, and elevated plus maze tests. Neuroplasticity and white matter were measured using immunohistochemical staining. The overall performance in seven-day acquisition trials was similar between the groups. Compared with the control group, BSC supplementation reduced the latency to the platform during day one of the acquisition tests. Supplementation improved memory by showing reduced latency and improved path efficiency to the platform quadrant, and smaller initial heading error from the platform zone. Supplemented rats showed an increase in striatal dopamine terminals and hippocampal glutamate receptors. Thus BSC supplementation during post-natal brain development improved learning and memory, independent from anxiety. The moderately enhanced neuroplasticity in dopamine and glutamate may be biological changes underlying the improved cognitive function.

A new homing guidance law to reduce sensitivity on initial heading errors

In this paper, a new guidance law is developed to reduce sensitivity on heading errors at the initial homing phase. The key enabler to propose such a guidance law was to design a new performance index by introducing distribution functions to the input weighting. The distribution functions in the new performance index are expressed in terms of a second-order polynomial of time-to-go and inverse weighted time-to-go. This allows distribution of the relative input weighting over the entire homing phase. Then, a homing guidance law is derived by applying the optimal control theory to the linear quadratic optimal problem with the new performance index. Unlike the conventional proportional navigation (PN) guidance law with a constant navigation gain, the proposed guidance law has a time-varying gain to ease the heading error sensitivity at the initial homing phase. The characteristics of the proposed guidance algorithm are analytically investigated and its performance is demonstrated through numerical simulations.

IM-filter for INS/GPS-integrated navigation system containing low-cost gyros

This paper proposes a new interacting multiple (IM) filter containing simplified unscented Kalman filter (UKF) -based subfilters with different heading initializations for a low-performance inertial sensors-based inertial navigation system (INS)/global positioning system (GPS) -integrated navigation system tolerant toward large initial heading error. Since each individual subfilter of the IM filter is updated adaptively using the combined information of the estimates from the subfilters, it can converge into a true steady state irrespective of the initial heading of a vehicle containing the navigation system. Thereby the IM filter can provide a stable navigation solution. For the subfilters of the IM filter, a simplified UKF is presented. This has a mixed structure of the extended Kalman filter and UKF and has a lighter computational load than the UKF in the multirate INS/GPS integration. Also, simplified UKF-based subfilters for the INS/GPS-integrated navigation system are designed. Monte Carlo simulations are performed to validate the performance of the proposed IM filter, and an experiment is carried out to confirm the simulation results.

In-motion Alignment of a Low-cost GPS/INS under Large Heading Error

This paper presents a new type of Global Positioning System/Inertial Navigation System (GPS/INS) providing higher navigation accuracy under large initial heading error. The mechanization introduced is applicable to low cost GPS/INS systems and enhances the performance when the heading error is large. The proposed approach has the capability to decrease large heading errors very quickly and can start the strapdown navigation computations under poor heading accuracy without any special alignment procedure. Although the design is applicable to land, sea and aerial vehicles, a land vehicle is used for the performance tests. The test is conducted around a closed path and the proposed system is compared to a GPS/INS system based on small attitude error assumption. The performance of both systems is given in this paper.

Sinusoidal function weighted optimal guidance laws

In this paper, new optimal guidance laws for a stationary or a slowly moving target are proposed by solving the optimal control problem with the energy performance index weighted by sinusoidal functions. This weighting function is adopted to shape the specific guidance command: introducing a small acceleration at the initial and final time. This property is desirable for reducing the sensitivity against the initial heading error and improving terminal guidance performance. Numerical simulations are performed to determine the performance of the proposed guidance laws.

Optimal guidance of extended trajectory shaping

To control missile’s miss distance as well as terminal impact angle, by involving the time-to-go-nth power in the cost function, an extended optimal guidance law against a constant maneuvering target or a stationary target is proposed using the linear quadratic optimal control theory. An extended trajectory shaping guidance (ETSG) law is then proposed under the assumption that the missile-target relative velocity is constant and the line of sight angle is small. For a lag-free ETSG system, closed-form solutions for the missile’s acceleration command are derived by the method of Schwartz inequality and linear simulations are performed to verify the closed-form results. Normalized adjoint systems for miss distance and terminal impact angle error are presented independently for stationary targets and constant maneuvering targets, respectively. Detailed discussions about the terminal misses and impact angle errors induced by terminal impact angle constraint, initial heading error, seeker zero position errors and target maneuvering, are performed.

Command-Shaping Guidance Law Based on a Gaussian Weighting Function

In this paper, a new homing guidance law is introduced based on the energy cost weighted by a Gaussian function in order to shape the missile's trajectory and distribute the acceleration command during the engagement. This law can improve guidance performance by alleviating sensitivity with respect to initial heading error and by reducing the possibility of command saturation through distributing acceleration demand properly.

Differential evolution based 3-D guidance law for a realistic interceptor model

This paper presents a novel, soft computing based solution to a complex optimal control or dynamic optimization problem that requires the solution to be available in real-time. The complexities in this problem of optimal guidance of interceptors launched with high initial heading errors include the more involved physics of a three dimensional missile–target engagement, and those posed by the assumption of a realistic dynamic model such as time-varying missile speed, thrust, drag and mass, besides gravity, and upper bound on the lateral acceleration. The classic, pure proportional navigation law is augmented with a polynomial function of the heading error, and the values of the coefficients of the polynomial are determined using differential evolution (DE). The performance of the proposed DE enhanced guidance law is compared against the existing conventional laws in the literature, on the criteria of time and energy optimality, peak lateral acceleration demanded, terminal speed and robustness to unanticipated target maneuvers, to illustrate the superiority of the proposed law.

Supplementation of complex milk lipid concentrate (CMLc) improved the memory of aged rats

ObjectivesThe socio-economic impact from age-related mental decline is escalating. Supplementation of functional foods for sustaining mental health is desirable. We examined the effect of long-term supplementation of complex milk lipid concentrate (CMLc), mixed dairy phospholipids, on memory and associated vascular and neuronal changes in aged rats.MethodsFisher/Norway Brown rats were used. Two groups of aged rats (24 months) were fed with either gelatin-formulated CMLc or blank gelatin as the control, for 4 months. To determine age-related changes, a young group (5 months) was also fed with blank gelatin. Morris water maze tests were carried out after the supplementation and brain tissues were collected for biological analysis.ResultsThe aged control rats learnt to locate the platform slower than the young control rats during acquisition trials (*P < 0.05), and made fewer entries to and more initial heading errors from the platform zone during testing trials (*P < 0.05). The CMLc supplementation improved memory by showing the reduced initial heading errors in a delayed probe trial (#P < 0.05). We also found that the aged rats with CMLc supplementation improved vascular density, dopamine output, and neuroplasticity (#P < 0.05) in the brain regions involved in memory compared with that of the aged control rats.DiscussionThe data suggested that the supplementation of CMLc during the early stage of brain aging may prevent memory decline possibly through improving vascular and neuronal function.

Optimal Guidance-to-Collision Law for an Accelerating Exoatmospheric Interceptor Missile

A guidance-to-collision law is presented. The guidance law is derived, using the optimal control methodology, for an accelerating exoatmospheric interceptor. It is dependent on a unique zero-effort heading angle error and on a variable denoted as the zero-effort angle of attack. The guidance law enables an accelerating interceptor to fly toward the interception point along a straight line, after the initial heading error is nulled. In addition, the guidance-to-collision law enables interception of the target with a terminal body angle. Moreover, the trajectory imposed by this guidance law enables an accelerating interceptor to estimate the target’s acceleration by using bearings-only measurements. This is possible due to the imposed line-of-sight rotation, making the range between the two vehicles observable. Using a nonlinear two-dimensional simulation, the performance of the guidance law is analyzed, showing superior performance compared to that of classical proportional navigation.

A Novel Scheme for DVL-Aided SINS In-Motion Alignment Using UKF Techniques

In-motion alignment of Strapdown Inertial Navigation Systems (SINS) without any geodetic-frame observations is one of the toughest challenges for Autonomous Underwater Vehicles (AUV). This paper presents a novel scheme for Doppler Velocity Log (DVL) aided SINS alignment using Unscented Kalman Filter (UKF) which allows large initial misalignments. With the proposed mechanism, a nonlinear SINS error model is presented and the measurement model is derived under the assumption that large misalignments may exist. Since a priori knowledge of the measurement noise covariance is of great importance to robustness of the UKF, the covariance-matching methods widely used in the Adaptive KF (AKF) are extended for use in Adaptive UKF (AUKF). Experimental results show that the proposed DVL-aided alignment model is effective with any initial heading errors. The performances of the adaptive filtering methods are evaluated with regards to their parameter estimation stability. Furthermore, it is clearly shown that the measurement noise covariance can be estimated reliably by the adaptive UKF methods and hence improve the performance of the alignment.

In-Flight Alignment Algorithm Based on ADD2 for Airborne POS

The Position and Orientation System (POS) is a special Strapdown Inertial Navigation System (SINS)/Global Positioning System (GPS) integrated system, widely employed in airborne remote sensing. In-Flight Alignment (IFA) is an effective way to improve the accuracy and speed of initial alignment for an airborne POS. IFA is normally accomplished with references from the position and velocity of GPS for SINS, so that unstable GPS measurements will result in poor alignment accuracy. To improve alignment accuracy under unstable GPS conditions, an adaptive filtering algorithm of the Second-order Divided Difference filter (DD2) based on adaptive innovation estimation is proposed, which introduces calculated innovation covariance directly into computation of the filter gain matrix. Then, the adaptive DD2 algorithm is used for the IFA of the POS with a large initial heading error. To validate the proposed algorithm, simulations are undertaken, followed by IFA experiments for the prototype of the airborne POS (TX-F30) under a turning manoeuvre in a car-mounted experiment, and under an “8” manoeuvre in-flight. The simulations and experimental results show that the proposed algorithm can reach better alignment accuracy under unknown statistical characteristic of GPS measurement noises.

Exo-atmospheric guidance of an accelerating interceptor missile

An exo-atmospheric interception scenario between an accelerating missile and its target is investigated. It is assumed that the maneuvering acceleration is obtained by instantaneous rotation of the missile's body to the required attitude. Two different guidance laws are derived for such an interceptor using the sliding mode control methodology. The difference is in the definition of the sliding surface enforcing different trajectories for the interceptor. It is shown that if this surface is chosen as the zero-effort-miss of the well-known proportional navigation guidance law, then the missile is commanded to point its acceleration vector along the line-of-sight and consequently fly along a curved trajectory. For the second guidance law, a unique sliding surface is chosen enforcing the missile to fly on a straight line towards collision, after the initial heading error is nulled. The performance of the guidance laws is analyzed and compared using a nonlinear two dimensional simulation. It is shown that on top of enforcing a different flight geometry for the interceptor, the use of the new guidance to collision sliding mode guidance law can enhance the capture zone of the interceptor.

Study on Innovation Adaptive EKF for In-Flight Alignment of Airborne POS

The Position and Orientation measurement System (POS) is a dedicated Strapdown Inertial Navigation System (SINS)/Global Positioning System (GPS) integrated system for airborne remote sensing. In-flight alignment (IFA) is an effective way to improve the accuracy and speed of initial alignment for an airborne POS. During IFA, the GPS provides the position and velocity references for the SINS, so the alignment accuracy will be degraded by unstable GPS measurements. To improve the alignment accuracy under unstable GPS measurement, an adaptive filtering algorithm of an extended Kalman filter (EKF) combined with innovation-based adaptive estimation is proposed, which introduces the calculated innovation covariance into the computation of the filter gain matrix directly. Then, this innovation adaptive EKF algorithm is used for the IFA of the POS with a large initial heading error. Moreover, it is optimized by blocked matrix multiplication to reduce the computational burden and improve the real-time performance. To validate the proposed algorithm, the car-mounted IFA experiment is carried out for the prototype of the airborne POS (TX-D10) under a turning maneuver, taking Applanix's POS/AV510 as a reference and changing the GPS measurement artificially. The experiment results demonstrate that the proposed algorithm can reach a better alignment accuracy than the EKF under unknown GPS measurement noises.

A Novel Initial Alignment Scheme for Low-Cost INS Aided by GPS for Land Vehicle Applications

This paper proposes a novel mechanism for the initial alignment of low-cost INS aided by GPS. For low-cost INS, the initial alignment is still a challenging issue because of the high noises from low-cost inertial sensors. In this paper, a two-stage Kalman Filtering mechanism is proposed for the initial alignment of low-cost INS. The first stage is designed for the coarse alignment. To solve the problems encountered by the general coarse alignment approach, an INS error dynamic accounting for unknown initial heading error is developed, and the corresponding observation equation, taking into account the unknown heading error, is also developed. The second stage is designed for the fine alignment, where the classical INS error dynamics based on small attitude error is used. Experimental results indicate that the proposed alignment approach can complete the initial alignment more quickly and more accurately compared with the conventional approach.

Error Calibration of Magnetometer Using Nonlinear Integrated Filter Model With Inertial Sensors

This paper presents an onboard heading estimation algorithm using IMU and strapdown magnetometer without other external heading references. To calibrate the magnetic deviation, sensor errors caused by hard iron effect and initial heading of strapdown magnetometers are considered. In our approach, sensor output distortion due to the soft iron effect is ignored, which is relatively small. First, for the estimation of heading angle, system and measurement model is presented. Then particle filter and extended Kalman filter is introduced for performance comparison. The proposed algorithm for the integration of IMU and magnetometer is verified via numerical simulation using Matlab. Simulation result demonstrates accurate heading estimation error under 1 degree for both algorithms when there exists a small initial heading error and hard iron effect, yet particle filter provides more robust and accurate result than the extended Kalman filter in case the initial heading error and biases are large.