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Title:
DEVICE AND METHOD, BASED ON NEURAL NETWORKS, FOR ESTIMATING THE LATERAL SPEED OF VEHICLES
Document Type and Number:
WIPO Patent Application WO/2023/166536
Kind Code:
A1
Abstract:
The present invention refers to a device and a method for estimating the lateral speed (v) of terrestrial vehicles, using new observers based on neural networks, with architectures designed starting from kinematic principles. The device, adapted to reduce the sensitivity to noise on the measures, in particular the longitudinal acceleration (αx) and lateral acceleration (αy) measures, to improve the estimation of the lateral speed, the device not depending on dynamic characteristics of the vehicles, is of the type that provides means for measuring the yaw rate (ω), the longitudinal acceleration (αx), the lateral acceleration (αy), and the longitudinal speed (um) and processing means of these quantities, equipped with a software based on neural networks based on the kinematic laws of relative motions, using the measured quantities for determining an estimation of the lateral speed v̂ and of the longitudinal speed û, expressed in a reference system, mobile and integral with the vehicle, wherein said neural network is based on modules (M).

Inventors:
DA LIO MAURO (IT)
BIRAL FRANCESCO (IT)
PICCININI MATTIA (IT)
Application Number:
PCT/IT2023/050058
Publication Date:
September 07, 2023
Filing Date:
February 22, 2023
Export Citation:
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Assignee:
UNIV DEGLI STUDI DI TRENTO (IT)
International Classes:
B60W40/10
Other References:
KONG DEBAO ET AL: "Vehicle Lateral Velocity Estimation Based on Long Short-Term Memory Network", WORLD ELECTRIC VEHICLE JOURNAL, vol. 13, no. 1, 23 December 2021 (2021-12-23), pages 1, XP055965442, DOI: 10.3390/wevj13010001
GRABER TORBEN ET AL: "A Hybrid Approach to Side-Slip Angle Estimation With Recurrent Neural Networks and Kinematic Vehicle Models", IEEE TRANSACTIONS ON INTELLIGENT VEHICLES, IEEE, vol. 4, no. 1, 1 March 2019 (2019-03-01), pages 39 - 47, XP011711726, ISSN: 2379-8858, [retrieved on 20190222], DOI: 10.1109/TIV.2018.2886687
KONG DEBAO ET AL.: "Vehicle Lateral Velocity Estimation Based on Long Short-Term Memory Network", WORLD ELECTRIC VEHICLE JOURNAL
Attorney, Agent or Firm:
GARAVELLI, Paolo (IT)
Download PDF:
Claims:
CLAIMS 1. Device for the estimation of a lateral speed (^) of terrestrial vehicles, designed to reduce the sensitivity to noise on measures, in particular longitudinal acceleration (ax) and lateral acceleration (ay) measures, to improve the estimation of the lateral speed, said device not depending on dynamic characteristics of said vehicles, of a type that provides means for measuring the following quantities: - yaw rate (ω); - longitudinal acceleration (ax); - lateral acceleration (ay); - longitudinal speed (um); and processing means of said quantities, characterized in that said processing means are equipped with a software based on neural networks built with an internal structure complying with kinematic laws, using said measured quantities, namely the measure of yaw rate (ω), longitudinal acceleration (ax), lateral acceleration (ay) and longitudinal speed (um) for determining an estimation of the lateral speed and of the longitudinal speed ^^ , expressed in a reference system, mobile and integral with the vehicle, wherein said neural network is based on modules (M). 2. Device for the estimation of the lateral speed (^) of terrestrial vehicles according to claim 1, characterized in that said modules (^) have the following recursion scheme: where the parameters are capable of being optimized, while are linear or non-linear functions of the quantities moreover, such functions being capable of containing parameters which can be optimized, the functions being introduced for collecting some terms in (1a,1b), in particular and enabling a representation of the module (^). 3. Device for the estimation of the lateral speed (v) of terrestrial vehicles according to claim 1, characterized in that said neural network is obtained by assembling two or more modules (M) in parallel (local models), each one trained for a particular yaw rate (ω), and by combining the output of the modules (M) as function of (ω). 4. Device for the estimation of the lateral speed (^) of terrestrial vehicles according to any one of claims 1 to 3, characterized in that in equation (1a) the factor is introduced, or any infinitesimal function of in the term to force (1a) to be uncoupled from (1b) during the rectilinear drive. 5. Device for the estimation of the lateral speed (^) of terrestrial vehicles according to any one of claims 1 to 4, characterized in that in equations (1a, 1b) the functions which can be learnt are sub-neural networks, A ∈ {1,2}, the functions being built with an architecture with two branches, wherein: - the first branch computes with the parameters which can be optimized - in the second branch, a linear layer “Lin2i”, with G neurons and an activating function FA, is connected to the layer “Lin3i” (with 1 neuron), to compute the global expression of becoming: 6. Device for the estimation of the lateral speed (^) of terrestrial vehicles according to claim 5, characterized in that said activating function FA is a hyperbolic tangent tanh. 7. Device for the estimation of the lateral speed (^) of terrestrial vehicles according to any one of claims 1 to 6, characterized in that said sub-networks are combined with a functional layer adapted to compute the remaining terms in (1a,1b). 8. Device for the estimation of the lateral speed (^) of terrestrial vehicles according to any one of claims 1 to 7, characterized in that the outputs of the models are weighed by activating functions said activating functions being functions of with local support, in such a way that a model learns the system dynamics in the neighborhood of the centers being defined in such a way as to partition the operating interval of 9. Device for the estimation of the lateral speed of terrestrial vehicles according to claim 8, characterized in that said activating functions are functions of other variables in addition to ω, for example functions also of the speed 10. Device for the estimation of the lateral speed of terrestrial vehicles according to any one of claims 1 to 9, characterized in that the global formulation of the model is written as follows wherein are the estimations computed with the model using (1a,1b), the equation (3) defining the interpolation between the local models the maximum number of parameters which can be trained for the network 11. Device for the estimation of the lateral speed (^) of terrestrial vehicles according to previous claim, realized with interpolating functions according to claim 9. 12. Device for the estimation of the lateral speed (v) of terrestrial vehicles according to claim 9, characterized in that said global formulation of the model is implemented on a network that uses a channel coding technique to create ' parallel models of a type the outputs of the models or weighed by the activating functions {1, … , P} (that depend on |^| following the assumption that the system noise depends on |^|. 13. Device for the estimation of the lateral speed of terrestrial vehicles according to any one of claims 1 to 12, characterized in that the module of the neural network C is adapted to recursively compute starting from an hypothesis of the initial statuses. 14. Device for the estimation of the lateral speed (^) of terrestrial vehicles according to any one of claims 1 to 13, characterized in that the non-linear branch in the functions is de-activated by setting {1,2, … , G}, and/or in that the parameters are both set equal to 1. 15. Method for the estimation of a lateral speed (v) of terrestrial vehicles, designed to reduce the sensitivity to noise on longitudinal acceleration and lateral acceleration measures, of a type that provides for measuring the following quantities: - yaw rate (ω); - longitudinal acceleration (ax); - lateral acceleration (ay); - longitudinal speed (um); characterized in that the method employs a device according to any one of claims 1 to 14.
Description:
DEVICE AND METHOD, BASED ON NEURAL NETWORKS, FOR ESTIMATING THE LATERAL SPEED OF VEHICLES The present invention refers to a device and to a method, for estimating the lateral speed of vehicles, using new observers based on neural networks, with architectures designed starting from kinematic principles. Networks are generalizations of the traditional kinematic equation-based observers They improve the noise rejection capacity and solve the well-known of observability for low yaw rates. Several families of linear and non-linear networks are taken into account: all have a limited number of parameters which can be learnt and a structure which can be physically interpreted. The estimation of the lateral speed plays an essential role in advanced control and stability systems for vehicles. Since the direct measure of the lateral speed is feasible only with costly optical sensors or double-antenna GPS, that are not convenient for the majority of commercial vehicles, many estimators have been developed. Among them, the model-based approaches assume a major role. In literature, model-based estimators fall into two major categories: - estimators based on a model of the vehicle dynamics; - estimators based on universal kinematic principles. Kinematic estimators do not depend on dynamic characteristics, that can change among different vehicles. However, the kinematic estimation is subjected to observability problems at low yaw rates (for example, on straights) and of problems of strong sensitivity to noise. Two scientific documents are known in literature and they deal with the problem, namely: D1 - Kong Debao et al. – “Vehicle Lateral Velocity Estimation Based on Long Short-Term Memory Network” (World Electric Vehicle Journal); D2 - Graber Torben et al. – “A Hybrid Approach to Side-Slip Angle Estimation With Recurrent Neural Networks and Kinematic Vehicle Models” (IEEE Transactions on Intelligent Vehicles) The cited documents D1 and D2 refer to the use of neural networks with generic internal structures, well known in the art, such as the “Long Short-Term Memory Networks” (LSTM) used in D1 and the “Gated Recurrent Units” (GRU) used in D2, which could model relationships not complying with kinematic principles. In particular, the scientific publication D2 uses a kinematic relationship with the sole purpose of obtaining a further quantity, namely the derivative of the side-slip angle, that is then used as input for a generic neural network GRU. The internal structure of such neural network GRU is not built in such a way as to comply with kinematic principles. Should the skilled person use the kinematic relationship of D2 as input to the neural network LSTM proposed in D1, the resulting neural network would still have an internal structure of the LSTM type, and such internal structure is not designed in order to comply with kinematic principles. A further inventive activity would then be required, to combine D1 and D2 and build a new neural network, whose internal structure is constrained to observe the kinematic laws. The present invention consists in a neural network based on a model, namely a neural network having a structure which implicitly complies with kinematic principles. Such physical structure makes the network sturdy and efficient. Prior art The design of estimators of lateral speed is a research argument active since over 30 years. Examining the methods published till 2018, two big categories can be defined: - a first category is based on models of vehicles; - a second category, data-driven, based on neural networks and non- parametrical approaches. The first category can be divided into three sub-groups: - dynamic models of the vehicle; - kinematic models; - combinations of kinematic and dynamic models. The observers that adopt dynamic models need to know many vehicle parameters, including the tire sub-models, making the specific estimators for a given vehicle. For example, in literature a Kalman filter (KF) has been used with a single-track model of the lateral dynamics, and a look-up table for the no-linear Pacejka model of tires. At the same time, a decision tree has been implemented to program the co-variance matrixes of a KF filter (based on a single-track model) according to pre-defined classes of slip conditions. Several works have performed comparisons between linear, non-linear observers and a sliding mode, using dynamic models of vehicles limited to moderate accelerations. For example a sliding mode estimator has been adopted in combination with a KF, to compute the lateral slip angle using a dynamic model of the vehicle. An inconvenience, due to the use of models of the vehicle dynamics, consists in that the observer performances are affected by variations of tire- road friction and by the non-modeled dynamics. The kinematic observers exploit the planar or three-dimensional kinematics to correlate the time evolution of the components of the vehicle speed with measured acceleration and yaw rate. A Luenberger parameter-varying observer has been developed, such that the estimation error converges to zero for a non-null yaw rate. This approach is rather robust to variations of physical parameters of the vehicle, that do not appear in the formulation. Another inconvenience, due to the use of kinematic observers, consists in that the lateral speed becomes not capable of being observed for a low yaw rate. Moreover, the estimation degrades when there is a bias on measured accelerations (for example, due to the gravity acceleration when the chassis is subjected to roll or pitch, and when the banking and the road slope angles are not negligible). Afterwards, the Luenberger estimator has been extended with an adaptive dead-zone in terms of feedback, allowing to select greater gains for compensating the lateral acceleration bias, without increasing the sensitivity to noise of measure of the longitudinal speed. A kinematic observer of disturbances has been presented for the on-line compensation of the offset of a sensor IMU (Inertial Measurement Unit). The estimations of bias have been used to improve the performances of a lateral slip estimator based on a linear single-track model To overcome the limits of purely kinematic estimators, mixed kineto- dynamic method have been introduced. According to these mixed methods, the dynamic models of lateral acceleration improve the observability of the lateral speed for low yaw rates. Another inconvenience, due to the use of dynamic models, consists in that, compared with an improvement of observability of the lateral speed, sensitivity to changes unavoidably increases in the friction conditions between tires and road and in vehicle parameters. Some authors have used the neural networks (NN) in methods of status estimation purely based on data. For example, multi-layer GRU (Gated Recurrent Units) have been employed to estimate the lateral speed of a racing car with an end-to-end approach, using standard sensors. Such method has resulted better than a mixed KF. Other authors have used a kinematic relationship to compute the derivative of the side-slip angle, to be then used as input for a generic neural network GRU. However, the generic neural networks (not constrained by physical considerations) have the same limitations of dynamic models: they tend to learn the dynamics of a specific vehicle and they do not generalize well to others. No work in literature has proposed a neural network with an internal structure complying with kinematic laws. Summary of the invention Object of the invention is solving the above inconveniences through a device and a method for estimating the lateral speed of vehicles, using new observers based on neural networks, with architectures designed starting from kinematic principles. This object is obtained with a device and a method as claimed in the respective independent claims. The device for estimating the lateral speed (ν) of vehicles, adapted to strongly reduce the sensitivity to noise on the measures of longitudinal and latera acceleration, is of the type that provides means for measuring the following quantities: - yaw rate (ω); - longitudinal acceleration (ax); - lateral acceleration (a y ) - longitudinal speed (u m ); and means for processing said quantities, wherein these processing means are equipped with a software based on neural networks based on the kinematic laws of relative motions, using said measured quantities, namely the measure of yaw rate (ω), longitudinal acceleration (ax), lateral acceleration (a y ) and longitudinal speed (u m ) for determining an estimation of the lateral speed and of the longitudinal speed u^, expressed in a reference system, mobile and integral with the vehicle, said neural network being based on modules (M) having the following recursion scheme: where the parameters are those which can be optimized, while are linear or non-linear functions of the quant ities moreover, such functions can contain parameters which can be optimized, the functions being introduced for collecting some terms in (1a,1b), in particular , enabling a representation of the module (M). The method provides for measuring the following quantities: - yaw rate (ω); - longitudinal acceleration (a x ); - lateral acceleration (a y ); - longitudinal speed (u m ); using the above-described device. From performed evaluations, these neural observers have been proven better in terms of accuracy and sturdiness with respect to known solutions. The original contribution is the description and validation of neural models with discrete time. The architecture of such networks is based on the kinematic laws of relative motions, resulting in a structure that can be interpreted with few parameters which can be learnt. The new estimators improve accuracy and sturdiness of the forecasts, decreasing the sensitivity to noise. The neural networks with generic structure (different from the structure of the present invention) are used in literature as surrogates of models of vehicle black-box. They cannot be physically interpreted, usually subjected to overfitting and are data-hungry (they produce accurate estimations only if the training data set contains many examples). Otherwise, the neural architectures of the invention have a structure that can be interpreted, based on the vehicle kinematics. Such structure that can be interpreted reduces the risk of overfitting, requires less data during training and improves the estimation robustness. Preferred embodiments and non-trivial variations of the present invention are the subject matter of the dependent claims. It is intended that all enclosed claims are an integral part of the present description. It will be immediately obvious that numerous variations and modifications (for example related to shape, sizes, arrangements and parts with equivalent functionality) could be made to what is described, without departing from the scope of the invention, as appears from the enclosed claims. The present invention will be better described by some preferred embodiments thereof, provided as a non-limiting example, with reference to the enclosed drawings, in which: - FIG. 1 (a, b, c) show the structure (with blocks and sub-blocks) of a neural observer: (a) combination of ' models having type M, to define the neural module ( (in the example P = 3); (b) internal structure of the neural model M; (c) internal structure of the sub-networks that compose the module M ; - FIG. 2 shows the use of the neural module ( of FIG. 1a as recursive neural network with infinite impulse response (IIR) wherein the operators z -1 represent unitary delays; - FIG. 3 shows the use of the neural module C of FIG. 1a as recursive neural network with finite impulse response (FIR); - FIG. 4 shows examples of activating functions to interpolate the local neural models. The device according to the invention comprises means for measuring the following quantities: - yaw rate (ω); - longitudinal acceleration (a x ); - lateral acceleration (a y ); - longitudinal speed (u m ); and processing means for these quantities. Neural networks with structure inspired to kinematics Measuring principle The direct measure of the lateral speed of a vehicle (^) cannot be accessed easily, but the estimators can provide indirect measures. The kinematic estimators exploit the measure of yaw rate (ω), acceleration (a x ), lateral acceleration (a y ) and longitudinal speed (u m ) with the following principle. Be: the speed vector expressed in a mobile reference system {i, j, k}, where u, v, w are the components of . in such system. The time derivative of (2) is: where the Poisson formulas are used to express the derivatives of {i, j, k} (for example, After having computed the crossed products, the projection of (3) on axes {i, j} is: where are the components of If the three-dimensional movement of the vehicle is approximated in two dimensions where i is the longitudinal axis of the vehicle on a plane ground and j is leftward oriented (k upwards), finally the known planar kinematic model is obtained: When a y , u and ω can be measured by sensors on-board the vehicle, formula (5a) provides means for computing v, and therefore, in principle, an estimation could be obtained through integration. However, in addition to the approximation from (4a) to (5a), a y , u and ω are signals affected by noise, and their integration according to (5a) will diverge following a partially casual evolution. The drift can be attenuated using both equations (5a) and (5b): the speed ^^ integrated by the first one is used in the second one for computing the expected longitudinal speed Since the actual longitudinal speed can be measured (u m ), the quantity becomes an indicator of the accumulated drift, that can be introduced in formulas (5a) and (5b) as follows: where are suitable feedback corrections. The equations (5a) and (5b) are the prototype of kinematic observers. The remaining description deals with neural implementations of such observers. A last remark deals with the fact that formulas (5a) and (5b) are coupled through the yaw rate ω. When a car travels on straight roads, this coupling disappears and the status v cannot be observed any more. In traditional observers of Luenberger, a chance is zeroing when the yaw rate is below a given threshold. In particular, as will be shown in the following sections, the implementations of the neural networks learn an optimum transition between the driving conditions in a road curve and those in a road straight. Neural architecture a. Neural module custom Starting from the kinematic model (6a, 6b), a neural network module M (FIG. 1b) can be defined using the previously-defined recursion scheme in (1a, 1b). In there are the estimations of at the time step with index k. In the described examples, reference is made to 3 modules M, however the number of such modules can be generic. The module M represents a local model (namely valid in the neighborhood of a certain yaw rate) and many local models can be used, depending on needs. The module ^ obtains a one-step forecast, namely . In the more general case, the parameters which can be optimized in and the parameters of the functions However, some variations are also proposed wherein the parameters of and the {^ @ , ^ @ } can be set to 1 or 0. The equations (1a,1b) can be seen as a generalization of the discretization with Euler explicit of (6a, 6b), wherein the integration step is τ. The factor ^ ^ , or any infinitesimal function of in the term is introduced to force the uncoupling of (1a) from (1b), during the rectilinear drive. The functions which can be learnt are neural networks. A preferred embodiment is proposed, subjected to further modifications and variations, in which the functions are neural networks, and in particular are built with an architecture with two branches, as shown in FIG. 1c. The first branch computes 1 with the parameters which can be optimized In the second branch, the linear layer “Lin2 i ”, with G neurons (in the example G = 3) and an activating function F I , is connected to the layer “Lin3 i ” (with 1 neuron), to compute with the parameters which can be optimized The global expression of is: In the preferred embodiment, the function F I is a hyperbolic tangent tanh. Any other activating function constrained between two ends is also possible. The functions in this example have (at a maximum) 3G + 3 parameters which can be learnt (namely . The number G of neurons in the linear layer “Lin2 i ” could be varied to change the complexity of the (a number of neurons G equal to 3 is enough in many practical applications). However, the number G can be changed to increase or decrease the complexity of the . The (maximum) number of parameters which can be learnt in M is U V = 6G + 10. b. Combination of neural models in parallel In ^, the parameters are constant. However, physical considerations suggest that the noise on the dynamics could change depending on the module of the yaw rate and therefore also the optimized parameters could change with A channel coding technique is used to create ' parallel models of the type The outputs of the models ^ @ are weighed by the activating functions (that depend on |^| following the assumption that the system noise depends on However, the functions could depend also on other status variables, for example on the speed in this example are linear functions piecewise of with local support (FIG. 4), namely are different from zero only in a neighborhood of and partition the unit, namely In this way, a model learns the system dynamics in the neighborhood of The centers ω i are defined in such a way that they partition the operating interval of The functions can be predefined, like in this example, or in turn learnt. Alternatively, the interpolating activating functions can be functions of other variables in addition to ω, for example functions also of the speed In this way, the device for the estimation of the lateral speed of terrestrial vehicles is realized with the above interpolating functions. The complete approximation of the local model is written as follows and implemented in the network shown in FIG. 1a: where are the estimations computed with the model M i , using formulas (1a,1b). The maximum number of parameters which can be trained for the network The functions are also called receptive fields, validity functions or membership functions in different contexts. For the above case (P = 3) the network for ( is shown in FIG. 1a. Structured IIR (Infinite Impulse Response) Model The neural network module ( can be used for recursively computing given a choice of initial statuses and measures (FIG. 2). Therefore, un indefinitely long set of measures can be used as input for the network, that operates as a filter with Infinite Impulse Response (IIR). The following variations are studied: 1) Linear feedback models (S-IIR-L): The models called S-IIR-L inherit the structure (S) of (1a, 1b) and the operation as filters with Infinite Impulse Response (IIR). Moreover, they are linear (L) because the non-linear branch in the functions is de-activated by setting Moreover, the parameters β 1 and γ 2 are set to 1. This family of networks has therefore 6 parameters for local model Since P = 3, the total number of parameters is 6' = 18. Inside every local model, the term operates as a leaky integrator, accumulating a first estimation of the lateral speed while it forgets the less recent noises on the input β 2 plays a similar role for noise rejections. The objective of learning is finding optimum parameters for noise rejection and optimum linear feedback When driving on a straight road, the problem of observability of the traditional Luenberger observer is solved, because the leaky integrator in (1a) is uncoupled from(1b), and therefore The noise rejection and feedback parameters can differ among the three local models, in order to better represent the dependency on the yaw rate. 2) Non-linear feedback models (S-IIR-NL). The model called S-IIR-NL allows a non-linear feedback f i allowing the learning of The coefficients are still unitary. Therefore, this family of models tries to find the optimum noise rejection but with a non- linear feedback. 3) Nonlinear feedback with plant dynamics (S-IIR-NLM): The models called S-IIR-NLM, in addition to the non-linear feedback, allow to change in the learning process. In this case, the status assumes a different meaning, because it cannot converge to and is exploited to learn part of the system dynamics, as will be described below. In (1a), the expression can be considered as the contribution of the kinematic model with reduced weight represents the contribution of a model learnt with the new status dynamics Structured FIR (Finite Impulse Response) Model The same networks complying with the structured IIR model can be used as Finite Impulse Response (FIR) filters (namely, as a recursive, memoryless neural network). At every discrete time step k, a window of past measures is used to compute the statuses with (1a,1b), starting from initial standard statuses (FIG. 3). In other words, the same networks are used in a different way, showing the last U observations and requiring the networks to provide the current status. The input in the time instants which precede the index > − U + 1 are neglected (this operation is permissible if the system has a finite memory). Correspondingly, the network parameters can change to produce the best estimation in U finite steps. The main advantage of FIR networks is their intrinsic stability in the BIBO sense. For each one of linear and non-linear observers complying with the structured IIR model, the corresponding FIR versions are created, and are called S-FIR-L, S-FIR-NL, and S-FIR-NLM. In addition to these models, a new variation has been developed, called S-FIR-LM, that restores the S-FIR-NLM but with linear feedback functions The models S-FIR-LM allow optimizing for a total of 8 trainable parameters per local model. With respect to their non- linear counterparts (S-FIR-NLM), the linearity of the models (S-FIR-LM) provides a better interpretability. The above-described variations are examples of application of the general principle. They can result more or less convenient one with respect to the other in different applications. Conclusions The invention refers to new neural network models for the estimation of the lateral speed of a vehicle. The neural observers are built with a connection graph which can be physically interpreted, that incorporates and generalizes the kinematic principles. Such structure inspired to the kinematics makes the networks sturdier, decreasing the risk of learning characteristics depending on the vehicle and on the adherence. The proposed observers require the measures of longitudinal speed, of yaw rate, of longitudinal and lateral accelerations. Such quantities can be obtained with standard odometers, gyroscopes and bi-axial accelerometers. The new neural models improve the estimation accuracy with respect to the Luenberger observer in literature, while they improve accuracy and sturdiness with respect to generic recurrent neural networks. The neural observers solve the non-observability problem of the planar kinematic model for low yaw rates, by means of a forgetting factor that softly zeroes ^^ during the rectilinear drive. It is possible to demonstrate that the non-linear models S-IIR/FIR-NLM and the linear model S-FIR-LM internally join three different estimations of the lateral speed, reducing the sensitivity to noise of acceleration measures with respect to the Luenberger observer.