Scalable Parametric-Identification Procedure for Kinematics of Automated N-Trailer Vehicles

Kinematic description of a nonholonomic articulated N-Trailer vehicle includes two kinds of parameters: trailer lengths and hitching offsets. In the case of picking up of various trailers by an automated tractor in logistic hubs or transshipment terminals, the kinematic parameters of trailers can be uncertain or even unknown to a control system of the automated tractor. Since an accurate kinematic model is usually required to keep effective functionality of automated vehicles, it seems justified to provide a model-learning capability to the automated or intelligent tractors of N-Trailer vehicles. In this paper, we propose a scalable (with respect to any finite number of trailers) parametric identification procedure applicable to any type of N-Trailer kinematics with non-steerable trailers' wheels. The key idea results from a reformulation of joint-angles kinematics in an iterative form of linear regression models with only two parameters. The proposed estimation algorithm assumes availability of measurements of articulation angles and characteristic velocities of the tractor. Numerical results obtained for the 5-Trailer nonholonomic kinematics and for a high-fidelity TruckSim vehicle model equipped with three trailers illustrate effectiveness of the proposed data-based modeling approach and large-sample statistical properties of the applied estimation procedure.


I. INTRODUCTION
R APID development of flexible high-capacity ground trans- portation allows expecting an increasing number of applications of intelligent articulated N-Trailer vehicles in various domains of freight and public transport services [1], [2].In order to diminish a human-driver's burden caused by execution of maneuvers with articulated vehicles, one can predict increased practical usage of autonomous or semi-automated N-Trailers, at least in isolated workspaces [3], [4], [5] (examples of automated tractors capable of maneuvering with semitrailers are presented in Fig. 1).Maneuvering with N-Trailers during the pick-up-anddelivery tasks in the distribution centres, logistic hubs, or transshipment terminals requires numerous changes of various types (and numbers) of trailers temporarily attached to an automated tractor [5].In this kind of tasks, the N-Trailers permanently The author is with the Institute of Automatic Control and Robotics (IAR), Poznan University of Technology, 60-965 Poznań, Poland (e-mail: maciej.michalek@put.poznan.pl).
In contrast to a kinetic (dynamic) model of the N-Trailer, which for a large number of trailers is very complex and structurally uncertain even for a fixed vehicle chain [13], [14], [15], a simpler kinematic model of the N-Trailer can be effective enough for the motion algorithmization purposes in the low-speed motion conditions [7], [10], [16], [17], [18].Kinematic models of N-Trailers have been largely investigated in the literature, especially in its modular form [19], [20], [21], [22], [23], assuming however that all the kinematic parameters are perfectly known.Although numerous works have been devoted also to the problem of parametric and state estimation of tractor-trailer vehicles, the available results concern mainly N-Trailers with N ≤ 2 (see [24], [25], and references cited therein) where a scalability of the methods with respect to a number of trailers attached to a tractor was not addressed.A main objective of this paper is to fill this gap by proposing a new systematic and well scalable approach to parametric identification of N-Trailers for an arbitrary finite number N .
Kinematics of N-Trailers is characterized by two kinds of geometric parameters -the trailer lengths and the hitching offsets [20], [23].These parameters can be known with a sufficient accuracy for the fixed-structure vehicles.However, in the pickup-and-delivery tasks performed by automated N-Trailers using various unknown trailers in logistic hubs or transshipment ports, there arises a need of intermittent model updating by parametric identification of the N-Trailer's kinematics upon measurement data available on-board.Hence, an attempt to solve the parametric identification problem for the N-Trailer kinematics in a scalable (compact) form seems practically justified.
In the proposed paper, a scalable parametric identification procedure for the N-Trailer kinematic model is introduced, which requires only proprioceptive measurements of articulation angles and tractor's velocities.The data-based modeling procedure proposed in the sequel enables one to sequentially estimate the trailer lengths and hitching offsets of all the vehicle's segments.The main contribution of this work comes from the following properties of the proposed data-based modeling approach: r the identification procedure is generic, being applicable to N-Trailer kinematics comprising any finite number N of trailers with non-steerable wheels, interconnected by any type of hitches, i.e., on-axle or off-axle ones, r thanks to a decomposition of the original N-Trailer kine- matics containing 2N parameters into the iteratively formulated subsystems including only two parameters, a large 2N -dimensional estimation problem (required for the original kinematics) is reduced here to the number of N much simpler 2-dimensional estimation problems (a high scalability of the method with respect to N is obtained in this way), r thanks to the iterative decomposition applied for the iden- tification purposes, satisfaction of a sufficient excitation conditions for the 2-parameter subsystems is relatively simple in practical conditions, r the proposed procedure applies the continuous-time iden- tification from sampled data scheme [26], which preserves physical interpretation of the estimated parameters.This article in an extended version of the conference paper [27].To the authors' best knowledge, any scalable parametric identification procedure for truly N-Trailer kinematics has not been proposed in the literature before.
Notation:Throughout the paper the following notation will be used.Symbol [z(t)] denotes a Laplace transform of z(t), while L −1 {•} denotes an inverse Laplace transform; {z(nT p )} M −1 n=0 denotes a sequence of M samples of a signal z(t), where T p > 0 is a constant sampling interval; I i×i is an identity matrix of dimension i; sα ≡ sin α, cα ≡ cos α; z (j) is a jth time-derivative of z, whereas z E (t) denotes a continuous-time signal resulting from an extrapolation of the discrete-time sequence {z(nT p )} M −1 n=0 ; R, R + , and N denote the real, positive-real, and natural numbers, respectively.

II. KINEMATIC MODEL AND ASSUMPTIONS
Let us consider the nonholonomic N-Trailer kinematics with fixed trailers' wheels presented in Fig. 2. The N-Trailer comprises a tractor -the only one active segment, which can be either a unicycle-like of a car-like unit -and an arbitrary number N of single-axle trailers interconnected with passive rotary joints.A shape-configuration of the vehicle is described by the joint angles where θ i denotes an orientation angle of the ith vehicle's segment, expressed relative to x G axis of a global frame (cf.Fig. 2).
If the tractor is a unicycle-like cart, a kinematic control input to the N-Trailer is a vector u 0 = [ω 0 v 0 ] , which includes an angular speed ω 0 of the tractor body and a longitudinal speed v 0 of a mid-point of the tractor's axle.In a case of the car-like tractor, the kinematic control input ω 0 can be reconstructed, e.g., upon a measurement of a steering angle β 0 and using a tractor's wheels base L 0 (see assumption A3 below).Description of N-Trailer kinematics requires two types of kinematic parameters denoted in Fig. 2: the trailer lengths L i > 0, i = 1, . . ., N, and the hitching offsets L hi ∈ R. If L hi = 0, then a hitching type in the ith joint is called on-axle; whereas if L hi > 0 (i.e., when the joint is placed behind a wheels' axle of a preceding unit) or L hi < 0 (i.e., when the joint is placed in front of a wheels' axle of a preceding unit), then a hitching type in the ith joint is called off-axle.Let us introduce the following parameter vectors (i = 1, . . ., N): According to works [22], [23], one can write kinematics of the joint angles for a nonholonomic N-Trailer vehicle with nonsteerable trailers' wheels in the following form: . . . where Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
for all i = 1, . . ., N, see [22].Since the joint-angle kinematics depends on all, possibly uncertain or unknown, kinematic parameters of the N-Trailer collected in the vector η, the model (3) will be used for the identification purposes.
For the identification purposes the following assumptions are formulated: A1: If the tractor is a car-like unit, the parameter L 0 > 0 is perfectly known (a priori knowledge).A2: All the joint angles β [β 1 . . .β N ] and the steering angle β 0 (for a car-like tractor) are measurable.A3: An instantaneous longitudinal speed v 0 is available (e.g., it is measured), whereas an instantaneous angular speed ω 0 is either available or can be reconstructed for a car-like tractor, e.g., by taking ω 0 = 1 L 0 tan β 0 v 0 upon assumptions A1 and A2.A4: A finite set Z M {u 0 (nT p ), β 0 (nT p ), β(nT p )} M −1 n=0 of data, containing samples of particular measurements recorded with a constant sampling interval T p > 0, is available for computational purposes.Assumption A1 is weak, because geometry of a tractor is constant and well known in practical applications.Assumption A2 seems to be a minimal measurement requirement; it could be replaced, e.g., by an assumption of measuring positions and orientations of all the N-Trailer segments, but it seems to be more difficult (and expensive) in practical applications.A3 can be satisfied in most practical cases, because the longitudinal speed v 0 can usually be measured, e.g., by using encoders mounted in the tractor's wheels.Assumption A4 is essential for the identification problem addressed in this paper, where we admit that all the measurements can be corrupted by measurement noises.

A. Identification Problem Statement
The problem addressed in the paper can be formulated as follows.
Problem 1: Upon assumptions A1-A4, find a scalable (with respect to a number N of trailers) identification procedure for the generic kinematics (3) of N-Trailers, equipped with arbitrary types of hitching, which provides estimates of kinematic parameters η i , for i = 1, . . ., N, using the available finite set Z M of sampled data.
The key property expected by a solution to Problem 1 is its scalability, which means here a polynomial grow of a computational cost of the identification procedure with respect to the number N of trailers attached to the tractor.One can observe that complexity of kinematics (3) substantially grows with a number of trailers (see Appendix A).Hence, to ensure scalability of the identification procedure the model (3) has to be reformulated in a form which enables, to some extent, treating the particular vehicle segments independently from each other -see Section III-B.
It is worth stressing that since the kinematics ( 3) is a differential equation, we will consider here the problem of continuoustime identification from sampled data, which proved to be beneficial in numerous practical applications -see [26], [28] for a detailed discussion.

B. Joint-Angle Kinematics as Iterative Linear Regressions
In order to formulate a parametric identification procedure in a scalable manner, we propose to rewrite kinematics (3) in an iterative way as follows: and further where ω i−1 and v i−1 are the angular and longitudinal speeds related to the (i − 1)st vehicle unit.By introducing an auxiliary output one can rewrite (5) in a linear regression form with the regression variables and a new ith vector of parameters Note that (9) determines a relationship between the original parameter vector η i from (2) and the new vector p i (the inverse relation exists if only p bi = 0).
Observe that for i = 1 the regression variables (8) include the tractor's speeds ω 0 and v 0 which are available upon assumption A3.For i ≥ 2, however, the regression variables (8) include the unavailable speeds ω i−1 and v i−1 .Therefore, one proposes to replace them by the corresponding predicted speeds, ωi−1 and vi−1 , which can be computed by applying the transformation (4) in the following way: (11) where corresponds to the transformation matrix introduced in ( 4), but used here with the estimated parameters Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
An iterative nature of ( 10)- (11) guarantees that a computation of predictions ωi−1 and vi−1 is feasible upon the Assumptions A2 to A4, and by using the estimates (13) obtained on the preceding estimation stages for j = 1 up to j = i − 1.As a consequence, the model ( 5) can be sequentially used to estimate all the parameter vectors p i for i = 1, . . ., N.
When the measurements included in the set Z M (cf.A4) are corrupted by noises, and since we use the predicted speeds ( 10)-( 11) instead of true (unknown) ones, the auxiliary output ( 6) and the regression variables in (8) will be affected by these stochastic noises.Therefore, from the identification point of view, one shall rewrite the model (7) in a data-explanatory form with a resultant (stochastic) equation error ζ i , that is: with a regression vector where is a joint-angle measurement comprising a true (unknown) value β i and a stochastic noise term ξ i , whereas and (for i ≥ 2) ωi−1 vi−1 are the velocity components comprising the true (unknown) values ω j , v j , and the corresponding (stochastic) unknown perturbations ρ j , ν j for j = 0, . . ., N.
Note that an iterative application of the model (20), for i = 1, . . ., N, enables one to reduce an identification problem from the original 2N -dimensional parametric space of kinematics (3) to the number of N simpler identification problems determined in 2-dimensional parametric spaces of models (20).As a consequence, the proposed solution is now well scalable with respect to the number N of trailers, and enables one to solve the Problem 1.
Remark 1: Application of the SVF filtration preserves a continuous-time structure of the joint-angle kinematics under consideration (no model discretization is applied).As a consequence, the model parameters retain their physical interpretation, and the estimation procedure is prevented from a potential numerical ill-conditioning when a sampling interval T p is (very) short, see [26], [28].

C. Scalable Procedure of Parametric Identification
A scalable parametric identification procedure which solves Problem 1 for N-Trailer kinematics (3) can be summarized in a form of six-step algorithm.Before formulating the computational algorithm, two preliminary steps have to be taken: r Select a sufficiently small sampling interval T p > 0 and a time constant T F = γT p , γ ∈ N, for the State Variable Filters (26).
r Design the open-loop control input u 0 (nT p ), excite the N-Trailer kinematics with u 0 (nT p ) for n ∈ [0; M − 1] and a sufficiently large M ; collect and store all the data in the set Z M required by assumption A4.Scalable procedure of parametric identification: n=0 by using (15) and taking for i ≥ 2 the predicted speeds ( 18)- (19).S3: Utilizing the sequences from step S2 and data from the set Z M , compute sequences of samples , and { φbiF (nT p )} M −1 n=0 , according to ( 21)- (23), by applying State Variable Filters (26) with a prescribed time constant T F .S4: By referring to the linear regression (20), apply a selected estimation method to compute pi = [p ai pbi ] , e.g., by using the well-known batch-type LS estimator or its recursive version [31], [32] (for alternative applicable estimation methods see, e.g., [26], [32]), where ȳiF If needed, the estimates ηi can be reconstructed upon the relationship (13).S5: IF(i == N ) THEN STOP.S6: Assign i := i + 1. Compute predictions ωi−1 and vi−1 using ( 18)-( 19) and GOTO S2.It is worth noting that the above proposed identification procedure has a cascade form, that is, the estimates pi are computed with utilization of the estimates pj achieved in the previous iterations for j = 1 up to j = i − 1.As a consequence, the estimation error from the (i − 1)st iteration propagates (through the predictions ( 18)-( 19)) to the regression variables (15) in the ith iteration.
The stopping condition used in S5 assumes that a number of trailers present in a vehicle chain is known a priori.If this assumption cannot be met, the procedure stops if the estimate Li+1 of a trailer's length is close or equal to zero.

Remark 2:
In practical applications, one shall expect the the resultant error component ζ iF in (20) is correlated with regression variables.Hence, an application of the basic LS method will lead in this case to a biased estimation result [26], [28].An additional source of a bias can be caused by an application of the approximated SVF filtration to sampled data (see Appendix B) -this issue has been recently investigated in [33] for the LS identification of continuous-time linear systems.However, it will be shown in the next section that for a sufficiently short sampling interval T p and sufficiently large number M of data collected in the set Z M , the resultant effect of all bias sources can be substantially limited, making the estimation accuracy acceptable in practical applications.
Remark 3: It is well known that in the identification task an excitation level of a plant must be sufficient (or persistent) to enable an acceptable quality of parametric estimation results.In the case of N-Trailer kinematics, the author's experience indicates that to achieve an appropriate excitation of jointangle kinematics the control inputs (i.e., tractor velocity u 0 ) should be designed in a manner which leads to a sufficient amplitude range of joint-angles' evolution during a vehicle's motion (it also ensures a larger signal-to-noise ratio in the case of noisy measurements).Due to a nonlinearity of the joint-angle kinematics (3), even elementary profiles of tractor speeds transform in a nonlinear manner to speeds of subsequent trailers and, as a consequence, allow obtaining a sufficient excitation for particular 2-parameter subsystems (20).Obviously, the sufficient excitation of the ith subsystem (20) must be reflected by the informative data included within a sufficiently long series of M collected measurement samples (the requirement of a large number of samples is also motivated by statistical properties of the estimation method commented in Remark 2 and studied in [33]).Therefore, since the number of samples M = T h /T p + 1 depends on the applied fixed sampling frequency f p = 1/T p and the time-horizon T h of a data-collecting process, it is clear that the time horizon T h should be selected long enough (depending on a tractor's velocity u 0 ) in order to cover an expected amplitude-range of joint-angles' evolution by a set of samples measured during the data acquisition process (see additional comments on this point included in Section IV).

IV. NUMERICAL RESULTS FOR 5-TRAILER KINEMATICS
The proposed parametric identification procedure has been applied to the 5-Trailer kinematics comprising of a unicyclelike tractor and five trailers interconnected by various types of hitching (cf.Fig. 3) -the values of true kinematic parameters prescribed for 5-Trailer kinematics are presented in Table I.The open-loop control (excitation) signals have been selected as follows: the angular speed ω 0 as a trapezoidal wave of a profile explained by the plot shown in Fig. 3, the longitudinal speed v 0 (t) = L −1 {G v (s)0.05/s} as a low-pass filtered steplike signal of amplitude 0.05 m/s, and G v (s) = 0.36/(s 2 + 1.2s + 0.36) (see Fig. 3).The data used for the estimation purposes have been collected with a sampling interval T p = 0.005 s; the autocorrelated (coloured) stochastic perturbations included in the data (see ( 16)-( 17)) have been generated as lowpass filtered white noises, by taking: , where H(s) = (1 + 0.08 s) −1 and e i ∼ N (0, 0.001), i = 1, . . ., 5, ρ ∼ N (0, 10 −5 ), and ν ∼ N (0, 10 −5 ).
Exemplary estimation results obtained for 100 realizations (trials), using a new set of M = 20001 data samples in each trial, are illustrated in Fig. 4. Averaged values of the obtained estimated parameters, and the corresponding standard deviations, are collected in Table II.
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.I.
All 100 estimation trials have been next repeated using new datasets corrupted this time by white (uncorrelated) noises of comparative variances to the coloured noises generated in the previous datasets.The estimation results are presented in Fig. 5.The results (when confronted with those from Fig. 4) reveal how the lack of autocorrelation of noises affect resultant empirical variances of the estimators.Fig. 6 presents the exemplary (but representative) evolutions of estimates pi = [p ai pbi ] for i = 1, . . ., 5, computed sequentially with the recursive LS method.By confronting the time plots in Fig. 3 with the evolution of the estimates one can observe that the estimates pi with larger indexes i (especially the blue ones) converge toward the true values later (i.e., after using a larger number M of data samples used for computations) relative to the estimates with smaller indexes.This trend is well correlated with the sequence of joint angles evolutions, where the angles βi with smaller indexes i reach relatively large amplitudes earlier (that is, for a smaller number of data samples already collected) when compared to the angles with larger indexes (in other words: substantial excitation level for particular trailers is spread in the data record with various lag intervals).Large-sample properties of the LS estimator applied in the proposed identification procedure can be assessed upon the results collected in Figs.7 and 8.
The plots present a change of average (ensemble means) absolute biases and the empirical standard deviations computed upon 100 identification trials for various numbers M of data samples used in the proposed estimation procedure (the noise characteristics were the same as for the results collected in Table II).One observes that an application of the LS estimator leads, in general, to the biased estimation results.However, the bias can be made acceptably small if a sufficiently large number M of data is used in the estimation procedure (all the absolute biases in Fig. 7 are evidently less than 1 mm for M ≥ 20000).On the other hand, if an increased number M of data samples is used for the estimation procedure, all empirical standard deviations almost gradually decrease, reaching in this case a level of about 1 mm just for M ≥ 40000.One may also observe that for M 60000 the obtained deviations are larger for the estimators of parameters with higher indexes (corresponding to the parameters of trailers located farther from the tractor unit along a kinematic chain).Although the estimation accuracy (corresponding to absolute biases) does not improve much for M ≥ M * , where M * > 0 is some minimal data-sample number, using a larger set of data samples (i.e., M M * ) definitely improves estimation precision (repeatability), which is especially important in practical applications.In real vehicles, the jointangle sensors can be imprecisely calibrated, leading to constant but unknown offsets β off,i included in the measurements.In order to check a sensitivity of the proposed identification procedure to the presence of the offsets independent simulations have been  I).I).Fig. 6.Exemplary evolutions of estimates pi = [p ai pbi ] for i = 1, . . ., 5, illustrating the results of computations with the recursive LS method using a single selected data set Z M (the horizontal axis is expressed in MT p seconds in order to confront the above results with the time plots presented in Fig. 3); the black dashed lines denote true values of parameters p i (units here are inherited from Table I).

Fig. 5. Estimation results (denoted by grey markers) of parameters p
Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.conducted for the 5-Trailer, assuming in this case the lack of any stochastic noises in measurements of joint angles and in the control signals applied to the vehicle (we try to check solely an influence of the offsets on the estimation quality).All the other simulation conditions have been set the same as previously.In order to avoid any specific selection of the offsets β off,i , their values have been randomly selected for any of 100 independent trials as realizations of five independent random variables -every one having a uniform distribution within a narrow range of [−1.0; 1.0] deg.Estimation results are presented in Fig. 9.It is clear that the presence of unknown parasitic offsets β off,i = 0 leads to additional biases of estimators; sizes of the biases depend on particular sets of offsets.Because the obtained spreads of estimates are in this case larger than the deviations of estimators obtained for the offset-free but noisy measurements (cf.Fig. 4), the calibration accuracy of joint-angle sensors turns out to be an important issue from the viewpoint of estimation accuracy, especially for the parameters of those trailers which are attached farther from the tractor.
In order to show a tendency how a location of a parasitic offset affects the estimates of parameters particular trailers, an additional set of results has been obtained for three special sets of offsets selected as follows: where the only non-zero offset (taken here from a boundary of the range [−1.0; 1.0] deg) is located, respectively, in the first (A), third (B), and fifth (C) joint-angle sensor.The results illustrated in Fig. 10 indicate that the presence of the non-zero offset in a sensor of the ith joint affects the estimates of parameters only for trailers of indexes from i to N .

V. APPLICATION TO A TRUCKSIM VEHICLE
The results of numerical simulations presented in Section IV assumed perfect knowledge of a model structure, that is, the nonholonomic kinematics (3) was used when generating the dataset Z M .In the current section, we are going to check if the proposed parametric identification approach can be effective in more practical conditions when the dataset Z M is generated by a real (or close to real) articulated vehicle, for which the nonholonomic constraints (build into the model ( 3)) can be violated in general motion conditions (i.e., a structure of the model (3) can be only approximately valid).To this aim, the proposed estimation procedure has been applied to the dataset generated by a high-fidelity TruckSim dynamic (kinetic) model of a truck-trailers vehicle shown in Fig. 11.The selected vehicle comprises a car-like tractor and a chain of semitrailer-dolly-semitrailer units attached to the tractor (from the kinematic point of view, the vehicle has got the Generalized 3-Trailer structure, [22], where the dolly is treated as a second trailer).Since a high-fidelity model of the TruckSim vehicle takes into account, among others, the wheels-ground interactions and can get into the wheels' skid-slip motion regime under high-speed / high-acceleration motion conditions, the proposed  kinematics-based identification approach will be tested only in the low-speed, but still realistic, motion scenarios.
For the excitation purposes, the selected 3-Trailer articulated TruckSim vehicle was forced to follow a roundabout maneuver illustrated in Fig. 12, independently with three different (relatively low, in order to avoid an extensive skid-slip effects) constant longitudinal speeds v 0 ∈ {5, 10, 15} km/h.Constant speeds were selected intentionally to keep a simplicity of excitation signals, however more complex (non-constant) profiles of v 0 (t) are also acceptable.After every maneuver, the set Z M of M = 140001 data samples, collected with a sampling interval T p = 0.001 s, has been used for the parametric estimation purposes (velocity u 0 taken for computations has been reconstructed upon speeds of the right and left wheels of a rear tractor's axle).Analogously to Section IV, the scalable identification algorithm has been applied independently to three collected datasets (every one corresponding to a different speed v 0 ), using the same State Variable Filters (with T F = 100T p = 0.1 s) and the same recursive LS estimator initialized in the same manner as in Section IV.The estimation results obtained for the TruckSim vehicle traveling with three different longitudinal tractor's speeds v 0 are presented in Table III, where the relative parametric errors δ i and Δ i are defined as follows (i = 1, 2, 3): The identified resultant kinematic model (3) for the Truck-Sim vehicle has been next validated using the newly collected validation dataset Z M v including M = 90000 samples of appropriate signals (cf. assumption A4).The validation dataset was obtained during a new low-speed maneuver corresponding to the tractor's velocity u 0 (t) illustrated on the upper plot in Fig. 13.The same dataset Z M v has next been used to validate all three kinematic models previously identified for three speeds computed for every i ∈ {1, 2, 3}, where β i [β i (0) . . .β i ((M − 1)T p )] is a vector of validation samples of the ith joint angle, β im [β im (0) . . .β im ((M − 1)T p )] is a vector of model-response samples (i.e., a response of the ith row of (3) to the validation input from Fig. 13), m[β i ] is a sample-mean computed upon the elements of β i , while The last column of Table III includes values of the measures obtained for the particular identified joint-angle kinematics.
Upon the results presented in Fig. 13 and in Table III one may expect that the proposed parametric identification procedure can be an effective data-based modeling approach for kinematics of real articulated vehicles, at least when using the datasets collected in sufficiently low-speed motion scenarios.Although even for the lowest assumed speed v 0 = 5 km/h one observes the non-zero estimation errors in Table III (all the non-singular relative errors are < 2%), they seem non-critical to a sufficiently accurate predictive capability of the identified kinematic model, which can be assessed upon the model-output errors shown in Fig. 13 (all the absolute errors |β i − β im | for i = 1, 2, 3 do not exceed a level of 10 −3 rad).
In the conditions of higher tractor's speeds v 0 , the presence of skid-slip effects of the vehicle's wheels lead to the deteriorated estimation accuracy, however the resultant predictive capability of the identified models can still be useful in various applications (it is reflected by the values of indexes J i FIT , which are higher than 97% in all the cases).One can conjecture that the acceptable predictive capability of the kinematic model obtained despite the less accurate estimates of parameters is a consequence of a cascade character of the proposed estimation procedure, where the estimators on the jth stage try to compensate, in some extent, the estimation errors from the stage j − 1.More accurate identification results could be probably achieved by modifying a structure of kinematics (3) and directly taking into account the skid-slip velocity components of the wheels.However, the identification computations in this case would require measuring the skid-slip velocity components, and it seems either non-realistic or too expensive in most practical applications.

VI. CONCLUSION
The scalable parametric identification procedure proposed in the paper can provide an effective estimation capability for kinematics of (semi-)automated tractors picking up various chains of unknown trailers, interconnected with any kind of hitching.Thanks to a reformulation of the N-Trailer kinematics in an iterative (cascaded) form, the original 2N -dimensional parametric estimation problem was reduced to the number of N simpler 2-D estimation problems, making the identification procedure scalable with respect to a number of trailers attached in a vehicle chain.
Although an application of the simplest LS estimation method in the stated identification problem leads generally to biased results [26], [28], the accuracy and precision of the estimators can be improved by using a sufficiently large set of low-pass filtered data samples, collected with a sufficiently short sampling interval (contemporary measurement capabilities seem to make this requirement realistic in the case of automated vehicles).The identification results obtained for a high-fidelity Truck-Sim model of a vehicle allow one to expect that the proposed kinematics-based identification procedure can be effective in kinematic modeling of real articulated vehicles, at least for lowspeed motion scenarios where the nonholonomic constraints are satisfied with a good approximation.It is worth emphasizing that acceptable estimation results have been obtained even for straightforward profiles of tractor speeds v 0 (t) and ω 0 (t).It indicates a relative simplicity in ensuring sufficient excitation of vehicle kinematics after its iterative decomposition into the N simpler 2-parameter subsystems.However, conducting a formal analysis for a persistent excitation condition guaranteeing unique parametric identification of the generic N-Trailer kinematics remains an open issue.
An interesting research topic, worth investigating in the future, is to extend the proposed estimation procedure to the case of N-Trailers towing multi-axle trailers of non-steerable wheels, where the reduced parameters of N-Trailer kinematics can be time-varying in the general cornering motion conditions [34].

APPENDIX A
Complexity grow of the joint-angles kinematics (3) with a number N of trailers can be assessed upon explicit forms of the first three rows of the kinematic matrix S β (β 1 , . . ., β N , η) which can be derived by hand, namely: (omitting the arguments and using a shortened notation s i ≡ sin β i , c i ≡ cos β i ): where (omitting the arguments) Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.
Fig. 14.Block scheme explaining the approximate filtration of an original analogue signal χ(t) with a digitally emulated State Variable Filter (SVF); T p > 0 is a sampling interval (computations included in the grey area on the scheme can be performed using the lsim function of Matlab).

APPENDIX B
Identification of continuous-time models from sampled data usually requires application of an approximate filtration with State-Variable Filters (SVF) defined as follows: where F 0 (s) ≡ F (s) = (1 + sT F ) −k , k determines a filter order, and T F > 0 is a prescribed time constant.Approximate filtration of an analogue signal χ(t) is performed just after sampling the signal, and requires extrapolating a series of samples {χ(nT p )} M −1 n=0 in a digital computer by using either the ZOH (Zero Order Hold) or FOH (First Order Hold) extrapolation method, see [28], [33].After extrapolation, the obtained continuous-time signal χ E (t) for t ∈ [0; (M − 1)T p ] is filtered by a numerically emulated filter (26).A result of filtration is again sampled with the sampling interval T p , leading to a resultant sampled sequence (for t ∈ [0; (M − 1)T p ]) {χ (j) which approximates a sampled signal resulting from an analogue filtration of the original signal χ(t) by the filter (26), i.e.: {χ (j) The approximated SVF filtration concept is illustrated by a block scheme in Fig. 14.The approximated SVF filtration can be performed, e.g., in the Matlab environment by utilizing the lsim function.

Manuscript received 9
June 2023; revised 28 November 2023; accepted 21 January 2024.Date of publication 24 January 2024; date of current version 20 June 2024.This work was supported by the research subvention Number 0211/SBAD/0123.The review of this article was coordinated by Dr. Ricardo Pinto de Castro.

Fig. 1 .
Fig. 1.Examples of automated tractors capable of picking up and autonomously maneuvering with semitrailers (left: the Volvo Autonomous Truck Concept, right: the AutoTUG terminal tractor of TERBERG company); source of original photos: images.volvotrucks.com,clarkforklifts.com.au.

Fig. 2 .
Fig.2.Kinematics of the N-Trailer with fixed trailers' wheels; a tractor is either a unicycle-like or a car-like cart (all the hitching offsets L hi are denoted as positive only for a presentation simplicity); θ i for i = 0, . . ., N denotes an orientation angle of the ith vehicle's segment.

Fig. 3 .
Fig. 3. Illustration of the open-loop control signals (corrupted by stochastic noises) applied to the 5-Trailer kinematics (upper plot), the resultant motion of the 5-Trailer in a workspace during the open-loop excitation stage (two middle plots; the vehicle's joints are denoted by circular markers), and the evolution of noisy articulation angles observed during the excitation stage (bottom plot); dimensions of the 5-Trailer result from the parameters included in TableI.

Fig. 4 .
Fig. 4. Estimation results (denoted by grey markers) of parameters p i = [L hi /L i 1/L i ] = [p ai p bi ] , i = 1, . . ., 5, (first row), and the corresponding estimates of parameters η i = [L hi L i ] (second row; the values reconstructed upon the estimates of p i ), obtained from 100 realizations (trials) -each one computed with M = 20001 new data samples corrupted by coloured (autocorrelated) stochastic noises (units inherited from TableI).
Fig. 5. Estimation results (denoted by grey markers) of parameters p i = [L hi /L i 1/L i ] = [p ai p bi ] , i = 1, . . ., 5, (first row), and the corresponding estimates of parameters η i = [L hi L i ] (second row; the values reconstructed upon the estimates of p i ), obtained from 100 realizations (trials) -each one computed with M = 20001 new data samples corrupted by stochastic white (uncorrelated) noises (units inherited from TableI).

Fig. 7 .
Fig. 7. Ensemble means of absolute biases b hi L hi − Lhi and b i L i − Li , i = 1, . . ., 5, for hitching offsets (top) and trailer lengths (bottom), respectively, as a function of a number M of data samples used in the estimation procedure; each mean value has been computed upon the results of 100 estimation procedures.

Fig. 8 .
Fig.8.Empirical standard deviations σ hi and σ i , i = 1, . . ., 5, obtained upon estimates of hitching offsets L hi (top) and trailer lengths L i (bottom), respectively, as a function of a number M of data samples used in the estimation procedure; each empirical deviation has been computed upon the results of 100 estimation procedures.

Fig. 9 .
Fig. 9. Estimation results (grey markers, values in [m]) of parameters η i = [L hi L i ] , i = 1, . . ., 5, obtained for 100 realizations of stochastically selected constant offsets β off,i ∈ [−1; 1] deg added to noise-free measurements of joint angles β i (every estimate has been computed upon M = 20001 data samples not corrupted by any stochastic noise); for comparison purposes the ranges of all the axes in the above plots have been set analogous as in Figs.4-5.

Fig. 11 .
Fig.11.Articulated vehicle selected in the TruckSim environment for the identification experiment and its G3T (Generalized 3-Trailer,[22]) scheme describing the nonholonomic vehicle's kinematics (note: from the kinematic viewpoint a dolly is treated as a trailer).

Fig. 12 .Fig. 13 .
Fig. 12. Roundabout maneuver performed with the TruckSim vehicle for the estimation-data acquisition purposes (a sequence of three selected vehicle views -A, B, and C -are presented).

v 0 ∈
{5, 10, 15} km/h.For validation purposes, open-loop responses of all three identified kinematic models (3) have been computed by applying the validation velocity u 0 (t) from Fig.13to (3), and using the estimated parameters ηi = [ Lhi Li ] from TableIII.Fig.13illustrates the resultant time plots obtained for the best estimated model (cf.

TABLE I PRESCRIBED
(7)UES OF PARAMETERS η i FOR THE 5-TRAILER KINEMATICS (3) AND THE CORRESPONDING TRUE VALUES OF PARAMETERS (9) USED IN THE LINEAR REGRESSION MODELS(7)

TABLE II EMPIRICAL
MEANS AND STANDARD DEVIATIONS OF ESTIMATED PARAMETERS p i AND RECONSTRUCTED ESTIMATES OF PARAMETERS η i , i = 1, . .., 5, OBTAINED UPON 100 REALIZATIONS OF M = 20001 DATA SAMPLES CORRUPTED BY AUTOCORRELATED NOISES (UNITS AS INTABLE I) Table III), obtained upon the dataset Z M corresponding to speed v 0 = 5 km/h.The plots compare the model responses β The model-output errors |β i − β im |, for i = 1, 2, 3, are illustrated in a logarithmic scale on the bottom plot in Fig. 13.Quantitative validation of the ith row of the identified model (3) can be assessed upon the values of the fitness measures im (t) with the TruckSim vehicle's joint angles β i (t) for i = 1, 2, 3 taken from the validation dataset Z M v .