An Adaptive Line-of-Sight (ALOS) Guidance Law for Path Following of Aircraft and Marine Craft

This brief presents a novel nonlinear adaptive line-of-sight (ALOS) guidance law for path following, compensating for drift forces due to wind, waves, and ocean currents. The ALOS guidance law is proven to have uniform semiglobal exponential stability (USGES) properties during straight-line path following at constant speed. The ALOS guidance law performs similar to the classical integral line-of-sight (ILOS) and adaptive ILOS guidance laws when the sideslip angle is nearly constant. The ALOS guidance law, however, has better tracking capabilities when compensating for rapidly varying sideslip caused by a time-varying disturbance. This is because the integral state of the ALOS guidance law is additive to the unknown sideslip angle (disturbance matching). In contrast, the ILOS guidance laws must compensate sideslip through a saturating arctangent function. The study also includes an input-to-state stable (ISS) reduced-order extended state observer for estimation of the line-of-sight (LOS) crab angle, known as the ELOS guidance law. The performance of the ALOS, ILOS, and ELOS guidance laws is compared by simulating rapid changes in the sideslip angle to stress the critical assumptions of the algorithms. Finally, a case study of the Remus 100 autonomous underwater vehicle (AUV) exposed to stochastic ocean currents is used to compare the performance of the ILOS, ALOS, and ELOS algorithms during normal operation.

Vehicle path-following control systems can be implemented using both heading and course autopilot systems in cascade with a guidance law.This article studies LOS guidance laws for path following using a heading autopilot command ψ d .The system under consideration is where y p e is the cross-track error expressed in a path-tangential reference frame {p} rotated an azimuth angle π h with respect to the North-East reference frame {n}, ψ is the yaw (heading) angle, U = (u 2 + v 2 ) 1/2 is the speed where (u, v) denotes the surge and sway velocities, and β c = tan −1 (v/u) is the T. I. Fossen is with the Department of Engineering Cybernetics, Norwegian University of Science, Trondheim, Norway (e-mail: thor.fossen@ntnu.no).
vehicle's crab angle; see Section II for details.It is assumed that the heading autopilot achieves ψ = ψ d where ψ d is the LOS yaw angle command (2), which can be tuned by the proportional gain K p > 0. The crab angle β c is assumed to be unknown, and the control objective is to cancel β c by using the control signals θ 1 or θ 2 to drive the cross-track error y p e to zero.Previous studies assumes that θ 1 = θ 2 = 0 or that θ 1 can be designed to cancel β c .The main contribution of this article is an adaptive law for θ 2 , which achieves disturbance matching.The different design techniques and their root in the literature are presented below.
A. Line-of-sight (LOS) guidance (θ 1 = θ 2 = 0) The study in this article builds on the concept of path following using proportional guidance where K p = 1/∆, and ∆ is the look-ahead distance.The proportional guidance law, ψ d = π h − tan −1 (y p e /∆), mimics the heading angle command of an experienced sailor (Healey and Lienhard [14]).Applications to marine craft are discussed by Pettersen and Lefeber [26], and Fossen et al. [7].A similar approach has been applied to small unmanned aerial vehicles (UAVs) by Nelson et al. [23].This work uses a vector field surrounding the path generating course commands to guide the UAV toward the desired path.A comparative study of the LOS and vectorfield guidance laws with application to AUVs are found in Charaija et al. [5].Proportional guidance techniques can also be used to guide missiles, see Siouris [29] and Yanushevsky [32].Model-based predictive control (MPC) has been applied successfully to LOS guidance path following by numerous authors; see Liu et al. [19], Oh and Sun [24], Pavlov et al. [25], and Rout and Subudhi [28].
Uniform global asymptotic stability (UGAS) and uniform local exponential stability (ULES) of the proportional LOS guidance law were first proven by Pettersen and Lefeber [26].This is also referred to as global κ-exponential stability as defined by Sørdalen and Egeland [30].More recently, Fossen and Pettersen [11] have shown that the proportional guidance law, when applied to course and heading control, is uniformly semiglobally exponentially stable (USGES).This guarantees strong convergence and robustness properties to perturbations; see Pettersen [27].An immediate consequence of Fossen and Pettersen [11] is that global exponential stability (GES) of the proportional LOS guidance law cannot be achieved due to a saturating sinusoidal function in the cross-track error dynamics.

B. Integral line-of-sight (ILOS) guidance
Despite the effectiveness and popularity of proportional LOS guidance laws, they have limitations when the vehicle is exposed to drift forces caused by wind, waves, and ocean currents.In addition, underactuated vehicles cannot produce a sway force.Hence, convergence to the desired path under the influence of drift is a non-trivial task.The consequence can be large cross-track errors during path following.The standard solution to this problem has been to add an integral state θ 1 to the guidance law in an ad-hoc manner and rely on linear superposition.This is referred to as proportionalintegral (PI) guidance.Unfortunately, the PI guidance laws can be challenging to tune, and no global stability results exist.
In 2008, a nonlinear ILOS guidance law was proposed by Børhaug et al. [2].In this work, global κ-exponential stable for straight-line path following at constant speed was proven using Lyapunov stability analysis.Extensions to path following for curved paths using monotone cubic Hermite splines were made by Lekkas and Fossen [17].The ILOS guidance law has been successfully implemented in many applications, and excellent performance has been demonstrated by Charija [4] and Caharija et al. [6].
An adaptive ILOS guidance law replacing the integral state with a parameter for sideslip compensation has been proposed by Fossen et al. [10], while Fossen and Lekkas [9] present indirect and direct adaptive control methods for LOS path following.Alternative methods using observer theory have been proposed by Liu et al. [20] and [21].
The article's main contribution is a novel adaptive LOS guidance law with USGES stability properties.The ALOS guidance law is proven to have USGES properties during straight-line path following at constant speed.Unknown disturbances due to wind, waves, and ocean currents are modeled as drift, and the ALOS guidance law compensates for this by parameter adaptation.The main difference between the classical and adaptive ILOS guidance laws to the ALOS guidance law is that the integral state θ 1 is replaced by θ 2 , which is additive to the unknown crab angle.In contrast, the ILOS guidance law must compensate the crab angle through a saturating arctangent function using the integral state θ 1 .The guidance laws have similar tracking performance during normal operations.However, the ALOS guidance law has better tracking capability for rapidly varying crab angles caused by environmental disturbances.

D. Organization of the article
Section II contains the kinematic preliminaries, including the cross-track error dynamics expressed in a path-tangential reference frame.Sections III and IV present the classical and adaptive ILOS guidance laws represented by the control signal θ 1 .Section V discusses the new ALOS guidance law defined by the control signal θ 2 , while the extended state observer for LOS path following is presented in Section VI.A comparative study of the guidance laws is presented in Section VII, and Section VIII is a case study based on a high-fidelity model of the Remus 100 AUV.Finally, Section IX concludes the results.II.KINEMATIC PRELIMINARIES This section presents the coordinate frames, tracking error dynamics, and kinematic differential equations.

A. Coordinate systems
For marine craft and aircraft, the six different motion components in the body frame {b} are defined as surge, sway, heave, roll, pitch and yaw.The North-East coordinate system is denoted by {n}.Consider a straight-line segment given by two waypoints (x n i , y n i ) and (x n i+1 , y n i+1 ) expressed in {n} where i = 1, 2, . . ., N .Assume that the path-tangential coordinate system {p} has its origin located at (x n i , y n i ) and the x p -axis is pointing towards the next waypoint (x n i+1 , y n i+1 ).Hence, the path-tangential coordinate system can be obtained by rotating the North-East coordinate system {n} an angle π h about the downwards z n axis as shown in Fig. 1.

B. Tracking errors
The alongand cross-track errors (x p e , y p e ) expressed in {p} are given by where and This can be extended to path following for curved paths using the approach of Fossen and Pettersen [11].

C. Kinematic differential equations
Let p n = [x n , y n ] be the position vector expressed in {n} and v b = [u, v] be the velocity vector expressed in {b}.Then it follows that (Fossen [12]) where ψ is the yaw angle.

D. Amplitude-phase representation
It is advantageous to express the kinematic differential equations ( 5)-( 6) in amplitude-phase form where the amplitude U is the speed over ground (SOG) and β c is the crab angle given by The course over ground (COG) is recognized as, χ := ψ + β c .

E. Influence of wind, waves and ocean currents
Aircraft operate in the wind, while marine craft can be exposed to ocean currents, waves, and wind.The aerodynamic and hydrodynamic forces are functions of the relative velocity vector is the flow velocity vector due to wind, waves, and currents expressed in {b}.This implies that lift will be perpendicular and drag will be parallel to the relative flow.The linear relative velocities can be expressed by (Fossen [12]) where U r = (u 2 r + v 2 r )1/2 is the relative speed and is the sideslip angle.Note that the crab angle 1 defined by ( 10) is equal to the sideslip angle when u f = v f = 0. We can express the crab angle as a function of the flow velocities by This formula is used in Section VI when simulating vehicles exposed to environmental disturbances.

F. Tracking-error differential equations
The tracking-error dynamics expressed in {p} is found by time differentiation of (3) and substitution of ( 7)- (8).This gives the formulas

III. THE CLASSICAL ILOS GUIDANCE LAW
The control objective is to choose the yaw angle ψ in ( 16) such that the cross-track error y p e → 0. The LOS algorithms for path following are usually employed at a kinematic level under the assumption that the heading autopilot guarantees that ψ = ψ d .The classical ILOS guidance law (Børhaug and Pettersen [2]) assumes that β c is unknown for a given azimuth angle π h .Hence, where y p int is the integral state used to compensate β c as defined by ( 14), ∆ > 0 is the user specified look-ahead distance, and K py = 1/∆.The integral gain κ > 0 is a tunable parameter.Substitution of ( 17) into (16) gives Hence, β c must be compensated for by the integral state y p int to satisfy the control objective.In the stability analysis, the following assumptions will be made: Assumption 2: The crab angle β c is constant during path following such that βc = 0.
Assumption 3: The crab angle estimation error βc = β c − βc where βc is the parameter estimate, is small during path following.When applying integral control instead of parameter adaption, βc ≡ 0 and thus βc = β c is small.Remark 1: Vehicle control systems are designed to keep the relative speed U r constant and the sideslip angle β small.Because of ( 14), the crab angle β c will be nearly constant for a vehicle traversing straight lines and circular paths if the flow velocity vector is nearly constant.Also note the sway velocity v (and thus the yaw rate r) will be small during straight-line path following.Straight lines and circles are the main segments used to construct Dubins paths [13].The switching between the segments will appear as steps in the integral state.For vehicles traversing a non-circular feasible path, i.e., a small curvature path, β c will vary slowly.However, the dynamics of β c will be much slower than the control bandwidth; thus, integral control will track the changes.Also, note that although the crab angle is relatively small, it primarily affects the path-following properties of the vehicle.Not adequately compensated, this results in significant deviations from the desired path.
For the ILOS guidance law ( 17)-( 18), βc = β c is small by Assumption 3. Hence, sin(β c ) ≈ β c and cos(β c ) ≈ 1.Furthermore, application of the trigonometry identity, sin(a − b) = sin(a) cos(b) − cos(a) sin(b), to (19) gives where d is an additive disturbance defined by Assumption 4: The additive disturbance d is nearly constant and upper bounded, so integral action can be applied.
Proof: The origin (y p e , y p int ) = (0, ȳp int ) is UGAS/ULES as shown by Børhaug and Pettersen [2].This can be extended to USGES following the approach of Fossen and Pettersen [11].

IV. THE ADAPTIVE ILOS GUIDANCE LAW
Assumption 4 stating that d given by ( 22) must be constant can be removed.The adaptive ILOS guidance law of Fossen et al. [10] replaces the integral state (18) with a parameter estimate βc according to Theorem 2: The adaptive ILOS guidance law ( 23)-( 24) applied to the system (16) renders the origin (y p e , βc ) = (0, 0) USGES under Assumptions 1-3 if γ > 0 and the heading autopilot guarantees that ψ = ψ d .

V. ADAPTIVE LOS (ALOS)
The main result of the paper is the ALOS guidance law where γ is the adaptation gain and βc is the parameter estimate.Note that βc in ( 25) is additive to the unknown crab angle in (16).This is referred to as disturbance matching.In contrast, the classical and adaptive ILOS guidance laws must compensate for the crab angle through a saturating arctangent function.Also note that the parameter update law (26) does not include the additional term ∆ βc of the adaptive ILOS parameter update law (24) nor the speed U .
Proof: See Appendix A.

VI. EXTENDED STATE OBSERVER FOR LINE-OF-SIGHT (ELOS) PATH FOLLOWING
Liu et al. [21] has derived a reduced-order extended state observer (ESO) for estimation of the crab angle, given by with ) ĝ = p + ky p e (32) where k > 0 and p(0) = −ky p e (0).The estimation error g = g − ĝ satisfies the differential equation ġ = −kg − ġ.Liu et al. [21] has derived conditions for the state g to be inputto-stable (ISS) with the input being ġ.The ESO-based LOS (ELOS) guidance law is chosen as (23).The observer (30)-( 32) is capable of estimating a time-varying crab angle β c .

VII. COMPARATIVE STUDY OF THE GUIDANCE LAWS
The goal of the case study is to stress test the critical assumptions of the guidance laws by computer simulations.The following algorithms are tested: ELOS (Liu, Wang and Peng [21])  Classical ILOS (Børhaug and Pettersen [2]) Adaptive ILOS (Fossen, Pettersen and Galeazzi [10]) ALOS (main result of Section V) The properties and assumptions of the different LOS algorithms are summarized in Table I.The guidance laws were implemented using ∆ = 20.0 m.The ILOS integrator gain was tuned to obtain maximum performance, and the chosen value was κ = 3.0.A similar approach was used to find the adaptive ILOS and ALOS adaptation gain γ = 0.2, while the ELOS observer used k = 0.5.The initial states were chosen as y p e (0) = 0 m, y p int (0) = 0, βc (0) = 0, and p(0) = 0.The sampling time was selected as 20 Hz.It is assumed that the vehicle control system keeps the speed at U = 2.0 m/s during path following.Hence, the environmental disturbances can be simulated by perturbing β c according to (14).Three phases are considered:   [20], the adaptive guidance laws may suffer from peaking and oscillation behaviors during the transient state since a large initial tracking error may deteriorate the learning process.The largest oscillations are observed for the adaptive ILOS algorithm, but there are also some oscillations in the ALOS algorithm, particularly when | βc | is large.The overall conclusion is that the ALOS and ELOS algorithm outperform the other two when | βc | is large.The ELOS algorithm removes the oscillations to the price of degrading the USGES stability property to ISS.However, during normal operation (Phase 1), the performance of the four algorithms is quite similar.The case study also confirms that the ALOS assumption that β c is constant can be relaxed since | βc | ≤ 2.0 deg/s gives accurate regulation of the cross-track error to zero.

VIII. CASE STUDY WITH THE REMUS 100 AUV
Since the ALOS guidance law outperforms the adaptive ILOS guidance law, it was decided to compare the ALOS guidance law with the classical ILOS and ELOS guidance laws using a high-fidelity model of the Remus 100 AUV; see Fig. 3.The goal is to evaluate the efficiency and robustness of the AUV during normal operation when the AUV is exposed to a stochastic ocean current.
The three guidance laws were implemented with ∆ = 10 m.The ILOS integrator gain was chosen as κ = 0.1.The ALOS The mathematical model of the AUV is available in the MATLAB MSS toolbox; see Fossen and Perez [8].The script remus100.mdescribes an AUV of length 1.6 m, cylinder diameter of 19.0 cm, mass 31.9 kg, and trim weight of 1.0 kg.The vehicle's maximum speed is 2.5 m/s, obtained by running the propeller at 1525 rpm in zero currents.The endurance is 22 hours at the optimum speed of 1.5 m/s.Depth is controlled using the stern plane δ s , and the heading angle is controlled using a tail rudder δ r .In the case study, the following control systems have been implemented (see Fig. 5): Propeller revolution: The propeller revolution is increased linearly from 1000 to 1300 rpm (see Fig. 5) until the AUV reaches its cruise speed.Depth: the depth is changed from 0 to 20 m as shown in Fig. 6 using successive-loop closure where θ d is the desired pitch angle, z n is the heave position, θ is the pitch angle and q is the pitch rate.The controller gains and time constant can be computed using pole placement (Beard and McLain [1, Section 6.4]).However, trial and failure gave excellent performance for K pz = 0.1, T z = 100.0s, K p θ = 2.0, K d θ = 3.0 and K i θ = 0.1.The maximum allowed stern-plane deflection is chosen as ±30 deg. Heading: The ILOS guidance law ( 17)-( 18) and ALOS guidance law ( 25)-( 26) are used to compute the desired yaw angle A switching mechanism for selecting the next waypoint is used when moving along the piecewise linear path.Waypoint (x n i+1 , y n i+1 ) is selected based on whether or not the vehicle lies within a circle of acceptance with radius R = 10 m around (x n i+1 , y n i+1 ).In other words, if the vehicle's position (x n , y n ) at time t satisfy the next waypoint (x n i+1 , y n i+1 ) is selected.The case study is: Phase 1: The Remus 100 AUV starts at an position (x n (0), y n (0)) = (0 m, 0 m) and heading ψ(0) = 0 deg at time t = 0.During the first phase (0-900 seconds), the vehicle accelerates up to its cruise speed while exposed to a  Phase 2: After 900 seconds an extreme stochastic current is simulated.The current's direction β Vc is changed from 180.0 deg to −180.0 deg in one minute.At the same time, the magnitude V c of the current is increased from 1.0 m/s to 2.0 m/s.The ILOS and ALOS algorithms handle the timevarying current quite well, even though both assume that β c and U are constant during path following.As expected, the performance of the ELOS algorithm is best in extreme situations with a rapidly varying β c .It is interesting to notice that the ALOS algorithm has a slightly better tracking performance than the ILOS algorithm when studying the cross-track error p e during rapidly changing current speed and direction (time 900-1400 seconds).Also note that the ELOS algorithm has larger overshoots than the other algorithms even though the estimate of β c is smoother.
IX. CONCLUSIONS A novel nonlinear adaptive line-of-sight (ALOS) guidance law for vehicle path following, which compensates for drift forces due to wind, waves, and ocean currents, has been presented.The equilibrium points of the cross-track and parameters estimation errors are shown to be uniformly globally exponentially stable (USGES) during straight-line path following at constant speed.This guarantees strong convergence and robustness properties to perturbations.Further, it was demonstrated that the ALOS guidance law performs similarly to the classical integral line-of-sight (ILOS) guidance law by Børhaug et al. [2] when the sideslip angle is nearly constant.The ALOS guidance law, however, has better tracking capabilities when compensating for rapidly-varying sideslip caused by a time-varying disturbance.This is because the integral state of the ALOS guidance law is additive to the unknown sideslip angle (disturbance matching).The case study used a high-fidelity model of the Remus 100 autonomous underwater vehicle (AUV) exposed to a severe stochastic ocean current.It was concluded that both the ALOS and the classical ILOS guidance laws give excellent performance and robustness for normal AUV operations.In contrast, the ILOS guidance law must compensate sideslip through a saturating arctangent function.The case study also compared the ILOS and ALOS guidance laws with a disturbance observer for β c , which can handle rapidly-varying sideslip.As expected, the observerbased guidance law (ELOS) gave smoother estimates of β c and better accuracy for large values of the crab angle rate βc .However, the ILOS and ALOS guidance law had less overshoot than the ELOS algorithm when considering the cross-track error y p e .Future studies should analyze the stability of the feedback interconnection of the ALOS guidance law and the heading autopilot for varying plant parameters and environmental conditions.The assumption that the crab angle is small should be relaxed by extending the stability proofs to arbitrary crab angles.

Fig. 1 .
Fig. 1.North-East and path-tangential coordinate systems {n} and {p}, respectively.The along-and cross-track errors are denoted by (x p e , y p e ).
This article has been accepted for publication in IEEE Transactions on Control Systems Technology.This is the author's version which has not been fully edited and content may change prior to final publication.Citation information: DOI 10.1109/TCST.2023.3259819This work is licensed under a Creative Commons Attribution 4.0 License.For more information, see https://creativecommons.org/licenses/by/4.0/

Fig. 2 .
Fig. 2. Performance of the classical ILOS, adaptive ILOS, ALOS and ELOS algorithms when βc (deg) is allowed to vary as a function of time.The green vertical lines separate Phases 1-3.

Fig. 2
Fig. 2 shows the exponential convergence of y p e to zero during Phases 1-3 for the four guidance laws when the vehicle is exposed to stochastic disturbances.The classical ILOS algorithm shows a performance reduction compared to the other guidance laws during Phases 2 and 3 when the crab angle is allowed to vary rapidly.During these phases | βc | is 1.0 to 2.0 deg/s.The adaptive ILOS and ALOS algorithms, and the ELOS algorithm keep the cross-track error y p e close to zero even when β c is time-varying.As shown by Liu et al. [20], the adaptive guidance laws may suffer from peaking and oscillation behaviors during the transient state since a large initial tracking error may deteriorate the learning process.The largest oscillations are observed for the adaptive ILOS algorithm, but there are also some oscillations in the ALOS algorithm, particularly when | βc | is large.The overall conclusion is that the ALOS and ELOS algorithm outperform the other two when | βc | is large.The ELOS algorithm removes the oscillations to the price of degrading the USGES stability property to ISS.However, during normal operation (Phase 1), the performance of the four algorithms is quite similar.The case study also confirms that the ALOS assumption that β c is constant can be relaxed since | βc | ≤ 2.0 deg/s gives accurate regulation of the cross-track error to zero.
This article has been accepted for publication in IEEE Transactions on Control Systems Technology.This is the author's version which has not been fully edited and content may change prior to final publication.Citation information: DOI 10.1109/TCST.2023.3259819This work is licensed under a Creative Commons Attribution 4.0 License.For more information, see https://creativecommons.org/licenses/by/4.0/
can be expressed as ẋ = Ω(x)Ax (46) This article has been accepted for publication in IEEE Transactions on Control Systems Technology.This is the author's version which has not been fully edited and content may change prior to final publication.Citation information: DOI 10.1109/TCST.2023.3259819This work is licensed under a Creative Commons Attribution 4.0 License.For more information, see https://creativecommons.org/licenses/by/4.0/where x = [x 1 , x 2 ] := [y p e , βc ] and , det(A − λI 2 ) = 0, becomes λ 2 + (U/∆)λ + γU = 0. Since U > 0 and ∆ > 0, the matrix A is Hurwitz if the adaptive gain γ > 0. Consider the nominal system ẋ = Ω(x)Ax and Lyapunov function W (x) = (1/2)x P x with P = P > 0. Consequently, Ẇ (x) = Ω(x)x (P A + A P )x (48) Let Q = diag{q 1 , q 2 } > 0. Then P and Q satisfy the Lyapunov equation P A + A P = −Q and Ẇ (x) = −Ω(x)x Qx < 0, ∀x = 0 (49) This article has been accepted for publication in IEEE Transactions on Control Systems Technology.This is the author's version which has not been fully edited and content may change prior to final publication.Citation information: DOI 10.1109/TCST.2023.3259819Using, cos(tan −1 ((x + a)/d)) = d/ d 2 + (x + a) 2 and sin(tan −1 ((x + a)/d)) = (x + a)/ d 2 + (x + a) 2 , the expression for (20) is further simplified such that This work is licensed under a Creative Commons Attribution 4.0 License.For more information, see https://creativecommons.org/licenses/by/4.0/ This article has been accepted for publication in IEEE Transactions on Control Systems Technology.This is the author's version which has not been fully edited and content may change prior to final publication.Citation information: DOI 10.1109/TCST.2023.3259819 This work is licensed under a Creative Commons Attribution 4.0 License.For more information, see https://creativecommons.org/licenses/by/4.0/