Research on the Methods for Correcting Helicopter Position on Deck Using a Carrier Robot

Abstract

When the landing position of a shipborne helicopter on the deck does not meet the requirements for towing it into the hangar, its position must first be corrected before towing can proceed. This paper studied the methods for using Shipborne Rapid Carrier Robots (SRCRs) to correct helicopter positions on the deck and proposed two correction methods, the stepwise correction method and the continuous correction method, aiming to improve the efficiency of the position adjustment process. Firstly, the actual helicopter landing position deviation was divided into two components—lateral offset and fuselage yaw angle—to quantitatively assess the deviations. Then, a mathematical model of the SRCR traction system was established, and its traction motion characteristics were analyzed. The kinematic characteristics and control processes of the two proposed position correction methods were subsequently studied, revealing the coordinated control relationships between key control elements. Finally, simulations were conducted to validate the feasibility of the proposed correction methods and compare their efficiencies. The results indicated that both the stepwise and continuous correction methods effectively achieved the position correction objectives. The stepwise method was more efficient when the initial yaw angle was small, while the continuous method proved more efficient when the initial yaw angle was large and the lateral offset was minimal. The results of this study may provide a valuable reference for correcting the positions of helicopters on deck.

1. Introduction

A shipborne helicopter needs to carry out on-board recovery operations after completing its mission at sea. This generally includes the processes of landing assistance, securing the helicopter, correcting the helicopter’s deck position, and towing the helicopter to the ship’s hangar. Shipborne special carrier robots are used for securing and transferring helicopters during the recovery process. Compared to the traditional manual methods of mooring and transferring, shipborne special carrier robots greatly improve the safety and efficiency of helicopter landing operations [1,2,3,4,5]. However, the landing position of the helicopter is affected by the motion of the deck [6] and the ship’s airwakes [7]. Helicopter deck position correction is a prerequisite for helicopter warehousing. Correcting the helicopter’s position in advance is necessary. This allows the helicopter traction and warehousing operations to be carried out smoothly and quickly [8,9,10]. Therefore, studying helicopter deck position correction methods is of great significance for ensuring helicopter warehousing and improving the efficiency of shipborne recovery operations.

Various types of shipborne helicopter traversing systems can be broadly classified into two categories; one type is the towing cart, which only possesses a traction function and needs to be used in conjunction with a harpoon grid system or recovery assist, securing, and traversing (RAST) system [11]. The other type integrates helicopter-assisted landing and towing functions, such as the Aircraft Ship Integrated Secure and Traverse System (ASIST) [12] and the Twin Claw Aircraft Ship Integrated Secure and Traverse System (TC-ASIST) [13].

Ground support equipment (GSE) [14,15] is a helicopter towing cart introduced by Curtiss-Wright, used for towing helicopters on land or aircraft carrier decks. Baury Alexandre enhanced the safety of the GSE towing system by adding a guide rail [16] and vacuum discs [17]. For traversing helicopters secured by the RAST system, Baekken Asbjorn et al. [18] designed the RAST helicopter towing system. Pesando Mario [19] designed a towing device for helicopters using both harpoon grid landing systems and RAST landing systems and used the Sea Lynx helicopter and the Sea King helicopter as examples to illustrate the towing process. However, the towing systems for harpoon grids and RAST require crew members to collaborate on the deck during landing aid and towing processes, resulting in high operational difficulty and low efficiency. To achieve automated operations for carrier-based helicopter landing and towing, Pesando Mario et al. [20] designed the ASIST system. This system completes securing and traversing operations by capturing and towing a rod on the helicopter underbelly. Zhang Z et al. [21] proposed the EASIST system with an asynchronous motor, enhancing shipborne helicopter traction efficiency and combat effectiveness. Zhang Z has enhanced ASIST performance across multiple research fields, including dynamic modeling [22], energy consumption characteristics [23], traversing performance [24], and system reliability [25].

Due to the requirement for helicopters to have rods installed on their underbellies for the ASIST system, which limits its compatibility with all helicopter models for landing and towing, Curtiss-Wright introduced the Two Claw Aircraft Ship Integrated Secure and Traverse System (TC-ASIST) [13] after launching the ASIST [12]. This system features two mechanical arms capable of capturing the probes on both sides of the helicopter’s tail wheel, eliminating the need for helicopter structural modification. It is suitable for a wider range of helicopter models and can tow heavier helicopters.

In terms of control strategies for helicopter towing systems, there has been considerable research on towing carts, including towing stability control and path planning methods. In the area of towing stability control, Liu H et al. [26] proposed a method using active four-wheel steering and variable structure control to improve the stability of shipborne aircraft traction. Wang Y X [27] proposed an adaptive backstepping controller to enhance deck tractor–airplane stability, compensating for ship motion and vehicle parameter uncertainties. Neng Jian W et al. [28] proposed using rear steering control with variable structure control for robust tractor–helicopter stability. In the area of path planning methods, Meng X et al. [29] proposed the KS-RRT* algorithm for safe and time-optimal path planning in autonomous aircraft towing on carrier decks. Liu J et al. [30] propose offline trajectory planning and online tracking methods for efficient and safe dispatching of towed aircraft systems.

Despite the extensive research on trajectory planning for towing carts, the complexity of ship motion control [31,32] makes it impossible for carts to be used for helicopter towing operations in high sea conditions. It is necessary to use integrated securing and traversing helicopter systems to complete the towing work. Due to the simplicity of the RAST and ASIST towing structures, research on the towing process focuses on modeling methods [33], helicopter motion characteristics, and towing system energy consumption characteristics [34]. The shipborne special carrier robot studied in this paper, similar to the TC-ASIST, uses two claws for towing helicopters. It is suitable for a wider range of helicopter models and can tow heavier helicopters. However, the unique structural design and traction motion mode limit the flexibility of helicopter traction motion and increase the difficulty of helicopter deck position correction. There is limited research on rapid helicopter deck position correction methods.

This paper aims to add an automatic helicopter deck position correction process into the TC-ASIST system to improve the efficiency and accuracy of helicopter hangar operations. This paper proposes a stepwise correction method and a continuous correction method to correct the landing position during the transferring of helicopters. In Section 2, the composition and calibration principles of the shipborne rapid carrier robots (SRCRs) are introduced, along with an analysis of the traction motion characteristics. Section 3 and Section 4, respectively, propose the stepwise calibration method and the continuous calibration method, while deriving the velocity-coordinated control equations. In Section 5, simulation studies are conducted on the correction motion trajectory and velocity characteristics. Under various traction conditions (different fuselage yaw angle θ₀ and lateral offset d₀), simulation test comparisons of the stepwise and continuous methods for correction are conducted to validate the feasibility of the proposed correction methods and analyze their simulation performance across different scenarios. The conclusion is presented in Section 6. The Nomenclature is shown in

Table 1. Nomenclature.

Symbol	Description
θ	The helicopter fuselage yaw angle
d	The helicopter lateral offset
v1	The traction velocity of the left winch
v2	The traction velocity of the right winch
V1	The traction speed provided by the left winch
V2	The traction speed provided by the right winch
va	The velocity of the left traction point a of the SRCR
vb	The velocity of the right traction point b of the SRCR
ω	The rotational angular velocity of the SRCR
L	The distance between the left and right traction tracks
Δva	The traction velocity difference between the left and right winches
Δva1	The velocity of the left traction point (shaft a) moving within the sliding groove
Δva2	The rotational velocity of the left traction point a on the SRCR when it performs rigid body motion around the right traction point b
vz	The capturing point velocity of the right robotic arm
vs	The lateral correction velocity of the right robotic arm
rz	The distance from the right robotic arm capturing point cR to the right traction point b
γ	The steering angle of the helicopter steerable nose wheel
xs	The sliding displacement of the right robotic arm
L0	The distance from the initial position of the right robotic arm (i.e., axis y1) to the center of the helicopter’s main wheel
L1	The distance between the right traction point of the SRCR (i.e., shaft b) and the initial position of the right robotic arm
L2	The distance between the helicopter main wheel shaft and the line connecting the left and right traction points (i.e., line a–b)
vB	The rotational velocity of the right robotic arm’s capture point around the right traction point of the SRCR (i.e., shaft b)
δ	The central angle of the continuous correction trajectory
r	The arc radius of the continuous correction trajectory
L3	The distance from the center of the helicopter’s main wheel shaft to the center of the helicopter steerable nose wheel
s	The length of the continuous correction trajectory

.

2. Mathematical Model and Kinematic Characteristic Analysis

2.1. The Structure and Correction Principle of Helicopter Deck Traction System

The research object in this paper is the SRCR, focusing on the helicopter deck towing operation. The simplified system principle is shown in Figure 1 and Figure 2. Figure 1 is a schematic of the three-dimensional layout of the traction system. Figure 2 shows the schematic diagram of the towing system principle, and Figure 3 shows the three-dimensional structure of the SRCR.

As shown in Figure 1 and Figure 2, the deck traction system includes left and right traction winches, the winch hydraulic power system, the traction cable pretightening device, the shipborne rapid carrier robot, and the left and right tracks. The winch hydraulic power system controls the hydraulic motor, thereby driving the winch. The track tractor and the SRCR are connected, as shown in Figure 3. In Figure 3, it can be seen that there are two types of motion pairs between the left track tractor and the SRCR; one is a rotational pair around shaft a, and the other is a sliding pair along the groove. There is only one rotational pair around shaft b between the right track tractor and the secure device. The left and right robotic arms slide along lateral slide rails at the front end of SRCR and are controlled by the winch hydraulic power system. The left and right robotic arms capture and hold the helicopter by clamping the securing rods of the helicopter’s main wheels.

The traction principle of the SRCR can be summarized as follows. The left and right traction winches, respectively, tow the SRCR for translational or rotational movement, and then SRCR robotic arms tow the helicopter for translational or rotational movement.

The ideal landing position of the helicopter is characterized by two key factors. The center point of the helicopter’s main axle should be located on the center line of the two traction rails, and the fuselage should be parallel to the traction rails, as shown in Figure 4. However, the actual landing position is often uncertain due to factors such as the unpredictable motion of the ship, airwake disturbances, and the pilot’s control accuracy.

To quantitatively evaluate the deviation between the actual and ideal landing positions, this paper decomposes the offset of the actual landing position relative to the ideal landing position. The offset can be categorized into two types: lateral position offset and fuselage direction angle offset. These are referred to as lateral offset d₀ and fuselage yaw angle θ₀, as shown in Figure 5. Therefore, the objective of the helicopter deck position is to eliminate the lateral offset and the fuselage yaw angle. The SRCR is equipped with angle sensors and displacement sensors, allowing for real-time monitoring of the helicopter’s fuselage yaw angle θ and lateral offset d.

2.2. Analysis of Traction Motion Characteristics of SRCR

Figure 6 is a schematic of the SRCR structure, simplifying the body structure and the complex internal dynamic system for traction motion analysis.

According to the connection relation between SRCR and traction tracks, the SRCR can only produce relative rotation around shaft b. In relation to the connecting shaft a, it can produce not only relative rotation around shaft a but also relative sliding along the groove. Therefore, the traction motion of the SRCR can be simplified to the planar motion of a rigid body. If the SRCR is considered a rigid body, its traction motion can be decomposed into the translational motion along the direction of the right traction track and rotational motion around shaft b.

Through the kinematic analysis of the SRCR, the schematic is shown in Figure 6. The motion trajectory of the SRCR is determined by the traction velocities of the left and right winches, denoted as v₁ and v₂, respectively. The traction speeds are assumed to be positive in the direction shown in Figure 6. The left and right winches transmit power through shafts a and b on the SRCR, with the velocities of a and b denoted as v_a and v_b, respectively. Shaft b is the base point of the SRCR’s planar motion, its velocity v_b is equal to the right traction velocity v₂, and v_a is equal to the left traction velocity v₁. The relationship between the velocities v_a and v_b can be expressed as follows:

$$\overrightarrow{v_{\mathrm{a}}}=\overrightarrow{v_{\mathrm{b}}}+\overrightarrow{\omega \cdot r},$$

(1)

In Formula (1), ω represents the rotational angular velocity of the SRCR around shaft b, and r is the distance between shaft a and shaft b.

As shown in Figure 6, r changes with the rotation angle θ and can be expressed as:

$$r={\displaystyle \frac{\mathrm{L}}{\cos \theta }},$$

(2)

In Formula (2), L is the distance between the left and right traction tracks, and θ is the fuselage yaw angle of the SRCR. Through dynamic analysis of the robot, it is known that the rotational torque of the robot comes from the difference in left and right traction velocities, Δv_a. According to the velocity decomposition theorem, Δv_a can be decomposed into velocity components Δv_a1 and Δv_a2, as shown in Figure 6. Δv_a1 is the relative motion velocity of point a in the groove direction relative to the base point b, and Δv_a2 is the linear velocity of point a rotating around shaft b. The expressions for the traction velocity difference Δv_a and velocity component Δv_a2 are shown in Formulas (3) and (4).

$$\Delta v_{\mathrm{a}}=v_2-v_1,$$

(3)

$$\Delta v_{\mathrm{a}2}=\omega \cdot r,$$

(4)

The relationship between the velocity component Δv_a2 and velocity difference Δv_a can be expressed as:

$$\Delta v_{\mathrm{a}2}=\Delta v_{\mathrm{a}}\cos \theta ,$$

(5)

According to Formulas (2) through (5), the angular velocity of the SRCR, ω, can be expressed as:

$$\omega ={\displaystyle \frac{\left(v_2-v_1\right){\cos }^2\theta }{\mathrm{L}}},$$

(6)

According to Formula (6), the rotation angular velocity ω is related to v₁, v₂ and the yaw angle θ. When the angular velocity ω is positive, it indicates counterclockwise rotation. Conversely, when ω is negative, it indicates clockwise rotation.

After analyzing the motion of the SRCR, it is also necessary to analyze the motion of the left and right robotic arms along the SRCR’s lateral sliding rail. This analysis will determine the motion trajectory of the helicopter as captured by the left and right robotic arms. The left and right robotic arms move synchronously along the lateral sliding rail during the helicopter’s movement. For example, consider the capture point c_R on the right robotic arm, which secures the right main wheel of the helicopter. Assume that the sliding velocity of the robotic arm along the lateral sliding rail is v_s (i.e., the lateral velocity of the right robotic arm). According to the velocity synthesis theorem, the vector expression of the capturing point velocity v_z of the right robotic arm is as follows:

$$\overrightarrow{v_{\mathrm{z}}}=\overrightarrow{v_{\mathrm{b}}}+\overrightarrow{\omega \cdot r_{\mathrm{z}}}+\overrightarrow{v_{\mathrm{s}}},$$

(7)

In Equation (7), r_z is the distance from the right robotic arm capturing point c_R to base point b. Analysis of Equations (1)–(7) reveals that the capturing point velocity, v_z, is related to the left winch traction velocity (v₁), right winch traction velocity (v₂), and the right robotic arm lateral velocity (v_s).

During the helicopter traversing process, the motion of the helicopter on deck completely depends on the traction applied by the robotic arm to the main wheel capturing probe. The fuselage yaw angle (θ) and lateral offset (d) for helicopter deck position correction must be managed through the coordinated velocity control of left and right traction winches. These factors significantly increase the difficulty of the path planning for helicopter deck position correction. Therefore, the path planning of the helicopter correction must fully consider the traction motion characteristics of SRCR.

3. The Stepwise Method of Helicopter Deck Position Correction

Based on the analysis of the SRCR traction motion characteristics, this paper proposes a stepwise correction method for helicopter deck position and investigates the cooperative control relationship between the winch traction speeds (v₁, v₂) and right robotic arm lateral velocity (v_s).

3.1. The Principle of Stepwise Correction Method

The steering angle of the helicopter’s steerable nose wheel, denoted as γ, determines the helicopter’s motion trajectory. There are two special steering angles: 0° and 90°. When the steering angle is 0°, the helicopter will translate along the fuselage direction. When the steering angle is 90°, the helicopter will rotate around the center point of the main wheel shaft. The stepwise correction method proposed in this paper leverages the particularity of the two steering angles, allowing the helicopter to alternate between translation and rotation. This method aims to eliminate the helicopter’s lateral offset and fuselage yaw angle. Due to its step-by-step nature, it is referred to as the stepwise correction method.

Assuming that the helicopter’s lateral offset and fuselage yaw angle at the initial landing position are not zero, the stepwise correction method can be summarized as follows: first, stepwise translation correction, then stepwise rotation correction.

Firstly, the translation motion of the helicopter is shown in Figure 7a. The angle of the steerable nose wheel is adjusted to 0°. Then, the left and right winch traction speeds and the correction speeds of the robotic arm are cooperatively controlled to move the helicopter along the direction of the steerable nose wheel. When the helicopter’s lateral offset d becomes 0, it marks the completion of the stepwise translation correction. This means that the center point O of the helicopter’s main axle has reached the centerline of the left and right traction tracks, indicated as point O′ in Figure 7a.

Next, the rotating motion of the helicopter is shown in Figure 7b. After adjusting the angle of the steerable nose wheel γ to 90°, the left and right winch traction speeds and the correction speed of the robotic arm are cooperatively controlled to rotate the helicopter around the center point O′ of the main wheel shaft. When the direction of the helicopter fuselage is parallel to the tracks’ centerline, the fuselage yaw angle is reduced to zero, indicating the completion of the stepwise rotational correction.

After completing these stepwise correction operations, the helicopter’s steerable nose wheel angle can be set to 0° to tow the helicopter into the hangar. However, the qualitative motion analysis of the stepwise translational and rotational corrections does not reveal the coordinated control relationship between the left and right winch traction speeds and the robotic arm correction speed during the correction process. Section 3.2 will provide a detailed explanation of this aspect.

If the lateral offset of the helicopter is zero but the yaw angle of the fuselage is not, only the stepwise rotational correction is implemented, as shown in Figure 7b.

If the yaw angle of the helicopter’s initial landing position is zero and the lateral offset is not, the fuselage direction will be parallel to the track direction. This paper defines this condition as the helicopter parallel landing position, as shown in Figure 7c. During the stepwise correction for a parallel landing position, the helicopter first undergoes rotational correction by rotating around point O by a certain fuselage yaw angle θ. Next, the SRCR sequentially completes the stepwise translational correction and the stepwise rotational correction, as shown in Figure 7d. Since the correction angles for the two stepwise rotational corrections are equal in magnitude but opposite in direction, the time required for each stepwise rotational correction is the same. Therefore, the stepwise correction for a parallel landing position with a lateral offset of d₀ can be considered as two stepwise corrections for cross landing positions, each with a lateral offset of 0.5d₀ and a fuselage yaw angle of θ. Different from cross landing positions, in the stepwise correction process for parallel landing positions, the fuselage yaw angle θ is set by the crew. Therefore, under different lateral offsets d₀, it is necessary to find an optimal θ to achieve the best correction performance.

3.2. The Relationship of Speed Cooperative Control in the Stepwise Correction Method

Based on the above analysis of the traction motion characteristics, there is a cooperative matching relationship between the left and right traction speeds (v₁, v₂) and the correction speed of the right robotic arm (v_s). Therefore, establishing a cooperative control relationship is a necessary prerequisite for achieving helicopter traction correction.

Based on the traction motion analysis of SRCR (as shown in Figure 7), the system kinematics of the helicopter during translational motion is analyzed, as shown in Figure 8. The reference coordinate system on the deck is assumed to be x₀–y₀, with the transverse axis of the deck as the x₀ axis, the longitudinal axis as the y₀ axis, and the centerline of the left and right traction tracks as the zero position of the x₀ axis. Additionally, the robotic arm’s lateral sliding rail is taken as the x₁ axis, and the initial position of the right robotic arm is taken as the y₁ axis, with the origin located at the rightmost end of the robotic arm’s lateral sliding rail.

The initial positions of the left and right robotic arms are at the left and right ends of the robotic arm’s lateral sliding rail, respectively. The physical meanings of the marked parameters in Figure 8 are as follows. x_s represents the distance between the current position and the initial position of the right robotic arm (the sliding displacement of right robotic arm); L₀ represents the distance from the initial position of the right robotic arm (i.e., axis y₁) to the center point O of the helicopter’s main wheel; L₁ represents the distance between the right traction point of the SRCR (i.e., shaft b′) and the initial position of the right robotic arm; L₂ is the distance between the main wheel shaft and the line connecting the left and right traction points (i.e., a′–b′); points p₀~p₄ are auxiliary points, and the dotted line is the auxiliary line. It is assumed that θ counterclockwise is positive and clockwise is negative.

The length of $\overline{{\mathrm{p}}_1{\mathrm{p}}_4}$ is expressed as:

$$\overline{{\mathrm{p}}_1{\mathrm{p}}_4}={\displaystyle \frac{\mathrm{L}}{2\cos \theta }}+{\mathrm{L}}_1,$$

(8)

The auxiliary line segments defined by points p₁ through p₄ satisfy the following relation:

$$\overline{{\mathrm{p}}_1{\mathrm{p}}_2}=\overline{{\mathrm{p}}_1{\mathrm{p}}_4}-\overline{{\mathrm{p}}_3{\mathrm{p}}_4}-\overline{{\mathrm{p}}_2{\mathrm{p}}_3},$$

(9)

The length expression of the auxiliary line segment $\overline{{\mathrm{p}}_2{\mathrm{p}}_3}$ is:

$$\overline{{\mathrm{p}}_2{\mathrm{p}}_3}={\mathrm{L}}_2\tan \theta ,$$

(10)

$$\overline{{\mathrm{p}}_3{\mathrm{p}}_4}=x_{\mathrm{s}},$$

(11)

After connecting Equations (8), (10) and (11) with Equation (9):

$$\overline{{\mathrm{p}}_1{\mathrm{p}}_2}={\displaystyle \frac{\mathrm{L}}{2\cos \theta }}+{\mathrm{L}}_1-x_{\mathrm{s}}-{\mathrm{L}}_2\tan \theta ,$$

(12)

Th e length of $\overline{{\mathrm{op}}_0}$ is expressed as:

$$\overline{{\mathrm{op}}_0}=\overline{{\mathrm{p}}_1{\mathrm{p}}_2}-{\mathrm{L}}_0,$$

(13)

The formula for calculating the lateral offset d is:

$$d=\overline{{\mathrm{op}}_0}\cos \theta ,$$

(14)

$$d={\displaystyle \frac{\mathrm{L}}{2}}+\left({\mathrm{L}}_1-x_{\mathrm{s}}-{\mathrm{L}}_0\right)\cos \theta -{\mathrm{L}}_2\sin \theta ,$$

(15)

If d is positive, it means that the center point of the main wheel is located on the right side of the central axis of the traction track; if d is negative, it means that the center point is located on the left side.

During the helicopter correction process, the direction of the SRCR right traction velocity v₂ is related to lateral offset d and fuselage yaw angle θ. Define the speed value provided by the right winch system as V₂, then the SRCR right traction velocity v₂ can be expressed as:

$$v_2={\displaystyle \frac{\theta d}{\left|\theta d\right|}}{\mathrm{V}}_2,$$

(16)

Equation (16) indicates that the direction of right traction velocity v₂ is determined by yaw angle θ and lateral offset d. During the traction process, the SRCR and the helicopter share the same rotational angular velocity. Therefore, when the helicopter is in translational motion, the SRCR must also move translationally. Based on the previous analysis of SRCR traction motion characteristics, when the SRCR moves translationally, the difference between the left and right traction speeds (v₁ and v₂) is zero. Consequently, v₁ should equal v₂ during the helicopter’s translational traction.

Since the synthesis of the translational velocity v₂ and the correction velocity v_s of the right robotic arm results in the translational velocity v_z along the fuselage direction, the correction velocity v_s of the right robotic arm can be expressed according to the velocity synthesis theorem as:

$$v_{\mathrm{s}}={\displaystyle \frac{\theta d}{\left|\theta d\right|}}{\mathrm{V}}_2\sin \theta ,$$

(17)

Equation (17) indicates that when the helicopter is in translational motion, the correction velocity v_s is determined by yaw angle θ, lateral offset d, and the absolute value of the translational traction velocity V₂. The correction velocity v_s is directly proportional to the traction translational velocity V₂.

Next, the cooperative speed control relationship during helicopter rotational correction is analyzed. Figure 9 illustrates the system kinematic analysis for the helicopter’s rotating motion. The angle β, marked in Figure 9, represents the angle between the line (connecting the right robotic arm capture point c_R and rotation base point b′) and fuselage direction. According to the previous analysis of SRCR traction motion characteristics, the velocity v_z at the right robotic arm capture point can be decomposed into three velocity components: right winch traction velocity v₂, rotation velocity v_B around base point b’, and correction velocity v_s of the right robotic arm.

The rotation velocity v_B of the right robotic arm’s capturing point c_R around b′ is:

$$v_{\mathrm{B}}={\displaystyle \frac{{\mathrm{L}}_2\omega }{\cos \beta }},$$

(18)

Since the velocity of the capturing point v_z is perpendicular to the direction of the main wheel shaft, the sum of the projections of the three velocity components (v₂, v_B, v_s) in the direction of the main wheel shaft must be zero. Therefore, the corrected velocity v_s of the right robotic arm can be expressed as:

$$v_{\mathrm{s}}=v_{\mathrm{B}}\cos \beta +v_2\sin \theta ,$$

(19)

$$v_{\mathrm{z}}=v_2\cos \theta -v_{\mathrm{B}}\sin \beta ,$$

(20)

The capturing point c_R of the right robotic arm can also be considered as undergoing rotational motion around the midpoint O′ of the helicopter main wheel shaft. Therefore, the capturing point speed v_z can be expressed as:

$$v_{\mathrm{z}}=\omega \cdot {\mathrm{L}}_0,$$

(21)

The relationship between the left and right traction velocities v₁ and v₂ and the relationship between the right robotic arm correction velocity v_s and traction velocity v₂ are shown in Equations (22) and (23), respectively.

$$v_1=\left(1-{\displaystyle \frac{1}{\left(k_1+k_2\tan \beta \right)\cos \theta }}\right)v_2,$$

(22)

$$v_{\mathrm{s}}=\left(\sin \theta -{\displaystyle \frac{k_2\cos \theta }{k_1+k_2\tan \beta }}\right)v_2,$$

(23)

In the formula, $k_1={\displaystyle \frac{{\mathrm{L}}_0}{\mathrm{L}}},\begin{array}{c}\end{array}{k}_2={\displaystyle \frac{{\mathrm{L}}_2}{\mathrm{L}}},\begin{array}{c}\end{array}\tan \beta ={\displaystyle \frac{x_{\mathrm{s}}-{\mathrm{L}}_1}{{\mathrm{L}}_2}}$.

Equations (22) and (23) comprehensively describe the cooperative control relationship between the traction velocities (v₁, v₂) and the correction velocity (v_s) of the right robotic arm when the helicopter rotates around the center point of the main wheel shaft. If the right traction speed v₂ is taken as the reference speed, the ratios of the left traction speed v₁ and the correction speed of the right robotic arm v_s to the reference speed v₂ are nonlinear functions of the sliding displacement of the right robotic arm x_s and the helicopter fuselage yaw angle θ.

In summary, the speed cooperative control relationship for helicopter traction correction has the following characteristics. During translational correction, the left and right traction velocities (v₁, v₂) are equal, and the correction speed (v_s) of the right robotic arm is directly proportional to the right traction velocity (v₂). During rotational correction, the left traction velocity (v₁) and the correction speed (v_s) of the right robotic arm are time-varying nonlinear functions.

4. The Continuous Method of Helicopter Deck Position Correction

Drawing inspiration from the circular arc trajectory method commonly used in auto-parking systems, this paper designs a continuous correction method for SRCR. The trajectory is a single circular arc, allowing the helicopter correction process to proceed without stepwise operations.

4.1. The Principle of Continuous Correction Method

Based on whether fuselage yaw angle is zero, the helicopter landing position can be classified into parallel landing position and cross landing position. For parallel landing positions, where the fuselage direction is parallel to the track direction, two tangential arcs can be used as the correction track. For cross landing positions, where the fuselage direction is at an angle to the track direction, a single arc track can be used for correction. The following will detail the correction principles for both landing positions.

4.1.1. Parallel Landing Position

The correction principle for parallel landing position is illustrated in Figure 10. The green thick solid line curve in the figure represents the correction trajectory, which is composed of two identical and tangent arcs, $\stackrel{⏜}{{ \mathrm{OO}}_1}$ and $\stackrel{⏜}{{ \mathrm{O}}_1{\mathrm{O}}_2}$. R is the center of correction trajectory, δ is the center angle of correction trajectory, and γ is the helicopter steerable nose wheel angle. The specific expression is as follows:

$$\gamma ={\displaystyle \frac{d_0}{\left|d_0\right|}}\left|\gamma \right|,$$

(24)

The operation process of continuous correction method under parallel landing position is as follows. Firstly, based on the transverse offset d₀ and Formula (24), the pilot sets the helicopter’s steerable nose wheel angle γ. Then, the helicopter moves along the first arc $\stackrel{⏜}{{ \mathrm{OO}}_1}$ in Figure 10 under the traction of SRCR. When the current lateral offset d is reduced to half of the initial lateral offset d₀, it indicates that the center point O of the helicopter’s main wheel shaft has reached the symmetry center point O₁ of the two correction trajectories. At this time, the pilot changes the steerable nose wheel angle to the same magnitude in the opposite direction, and the helicopter moves along the second arc $\stackrel{⏜}{{ \mathrm{O}}_1{\mathrm{O}}_2}$. Finally, when current lateral offset d returns to 0, the center point O of the helicopter reaches O₂, completing the correction operation.

Based on the geometric relationship in Figure 10, the correction arc radius r can be deduced as follows:

$$r={\displaystyle \frac{{\mathrm{L}}_3}{\left|\tan \gamma \right|}},$$

(25)

L₃ is the distance from the center point of the main wheel shaft to the center of the helicopter’s steerable nose wheel.

The central angle δ corresponding to the arc of the correction trajectory is:

$$\delta =\arccos {\displaystyle \frac{2{\mathrm{L}}_3-d_0\tan \gamma }{2{\mathrm{L}}_3}},$$

(26)

The correction length s in the continuous correction method is:

$$s=2r\sin \delta ,$$

(27)

By Formulas (25)–(27), it can be determined:

$$s=\sqrt{{\displaystyle \frac{4{\mathrm{L}}_3{d}_0}{\tan \gamma }}-d_0^2},$$

(28)

As can be seen from Formula (28), the correction length s is determined by the helicopter steerable nose wheel angle γ.

The two continuous correction trajectories for a parallel landing position are symmetric about the trajectory’s center point, meaning that the time required for each correction segment is equal. The correction trajectory along arc $\stackrel{⏜}{{ \mathrm{O}}_1{\mathrm{O}}_2}$ is the same as a single continuous correction trajectory for a cross landing position. The following section will provide a detailed explanation of the continuous correction method for the cross landing position.

4.1.2. Cross Landing Position

The continuous correction method for the cross landing position is similar to that for the parallel landing position, but it requires only one arc. The principle diagram is shown in Figure 11. In Figure 11, the green arc represents the motion trajectory of the center point of the main wheel shaft, γ′ is the helicopter steerable nose wheel angle, and r′ is the correction arc radius.

Unlike the parallel landing position, the helicopter’s steerable nose wheel angle for a cross landing position cannot be arbitrarily set; it must be calculated based on the lateral offset and yaw angle of the helicopter’s initial landing position. By analyzing the geometric relationship in Figure 11, it can be seen that the radius of the arc r′, the lateral offset d₀, and the center angle δ′ of the circle satisfy the following relationships:

$$r\prime \cos \delta \prime +d_0=r\prime ,$$

(29)

The radius r′ of the arc trajectory is:

$$r\prime ={\displaystyle \frac{{\mathrm{L}}_3}{\left|\tan \gamma \prime \right|}},$$

(30)

The central angle δ′ of the arc is exactly equal to the initial yaw angle θ₀.

$$\delta \prime ={\theta }_0,$$

(31)

Using Formulas (25)–(27), the angle of the helicopter’s steerable nose wheel γ′ can be determined:

$$\gamma \prime =\arctan {\displaystyle \frac{{\mathrm{L}}_3\left(1-\cos {\theta }_0\right)}{d_0}}\cdot {\displaystyle \frac{-d_0}{\left|d_0\right|}},$$

(32)

According to Equation (32), γ′ is related to θ₀ and d₀.

To summarize, when the continuous correction method is adopted, the helicopter performs only rotating motion. In the parallel landing position, the two correction trajectories are solely related to helicopter steerable nose wheel angle, γ. This angle is set by the pilot, and the correction trajectory will vary accordingly with different angles.

4.2. The Relationship of Speed Cooperative Control in the Continuous Correction Method

When using the continuous correction method, assuming that the tires do not slip sidesways, the helicopter engages only in rotational motion. The instantaneous center of rotation is located at the intersection of the vertical lines in the direction of each wheel’s speed, as shown in point R in Figure 12. The angle of the helicopter’s steerable nose wheel is γ, the angular speed of the helicopter’s rotating motion is ω′, and the fuselage yaw angle is θ.

Similar to the speed analysis of the stepwise correction method, the speed of the robotic arm capturing point v_z′ can be decomposed into three speed components: the traction translational speed v₂, the rotation speed v_B2′ around the base point b, and the robotic arm correction speed v_s2′, as shown in Figure 12. According to the analysis of the traction motion characteristics of SRCR in the above section, the rotation angular velocity of SRCR is ω′.

$$\omega \prime ={\displaystyle \frac{\left(v_2-v_1\right){\cos }^2\theta }{\mathrm{L}}},$$

(33)

$$v{\prime }_{\mathrm{B}2}={\displaystyle \frac{{\mathrm{L}}_2\omega \prime }{\cos \beta }},$$

(34)

The sum of the projections of the three velocity components (v₂, v_B2′, v_s2′) in the vertical direction is zero.

$$v{\prime }_{\mathrm{s}2}=v_2\sin \theta -v{\prime }_{\mathrm{B}2}\cos \beta ,$$

(35)

The helicopter is only undergoing rotational motion, so the speed of robotic arm capturing point v_z′ is expressed as:

$$v{\prime }_z=\left({\mathrm{L}}_3\cot \gamma +{\mathrm{L}}_0\right)\omega \prime ,$$

(36)

According to the velocity synthesis theorem, the right robotic arm capturing point velocity v_z′ can also be expressed as:

$$v{\prime }_z=v_2\cos \theta -v{\prime }_{\mathrm{B}2}\sin \beta ,$$

(37)

According to the above equations, the cooperative control relationship between the two traction speeds v₁ and v₂, as well as the cooperative control relationship between the robotic arm correction speed v_s2′ and the right traction speed v₂, can be determined, as shown in Equations (38) and (39).

$$v_1=\left(1-{\displaystyle \frac{1}{\left(k_1+k_2\tan \beta +k_3\cot \gamma \right)\cos \theta }}\right)v_2,$$

(38)

$$v_{\mathrm{s}2}\prime =\left(\sin \theta -{\displaystyle \frac{k_2\cos \theta }{k_1+k_2\tan \beta +k_3\cot \gamma }}\right)v_2,$$

(39)

In the formula, $k_1={\displaystyle \frac{{\mathrm{L}}_0}{\mathrm{L}}},\begin{array}{c}\end{array}{k}_2={\displaystyle \frac{{\mathrm{L}}_2}{\mathrm{L}}},\begin{array}{c}\end{array}{k}_3={\displaystyle \frac{{\mathrm{L}}_3}{\mathrm{L}}},\begin{array}{c}\end{array}\tan \beta ={\displaystyle \frac{x{\prime }_{\mathrm{s}2}-{\mathrm{L}}_1}{{\mathrm{L}}_2}}$.

Equations (38) and (39) show that if the right traction velocity v₂ is fixed, the left traction velocity v₁ and the robotic arm correction speed v_s2′ are nonlinear functions related to the robotic arm correction displacement x_s2′ and the fuselage yaw angle θ.

5. Simulation Verification of the Correction Method

5.1. Simulation of the Stepwise Correction Method

To verify the feasibility of the stepwise correction method, this paper utilizes Adams 2020 software [35] and MATLAB 2019b [36] to simulate and analyze the corrected motion trajectory of the helicopter. The simplified three-dimensional model of the traction system is shown in Figure 13. This study takes the controllable steering wheel helicopter as an example for simulation research.

Table 2. Parameters of several medium and heavy helicopters.

Helicopter Type	Width	Rear Wheelbase	Weight
CH-53K	5.33 m	3.52 m	25,000 kg
Sea hawk (SH-60B)	2.36 m	2.8 m	10,400 kg
Sea king (SH-3)	4.98 m	4.70 m	9707 kg
MI-171helicopter	2.50 m	4.28 m	11,000 kg
SH-90	3.8 m	2.7 m	10,600 kg

presents the parameters of several medium and heavy helicopters that are widely used and produced in large quantities worldwide. To ensure that the SRCR can tow all types of helicopters listed in Table 2, the relevant parameters of the simulation model are shown in

Table 3. Main simulation parameters of the stepwise correction method.

Parameter	Value
The length of track spacing for left and right traction tracks L	5000 mm
The length of capturing point to fuselage center line L0	2000 mm
The distance between the right traction point of the SRCR and the initial position of the right robotic arm L1	150 mm
The distance between the helicopter main wheel shaft and the line connecting the left and right traction points L2	300 mm
Initial helicopter yaw angle θ0	10°
Initial helicopter lateral offset d0	200 mm
Right traction speed v2	10 mm/s

.

Firstly, the translation stepwise correction is simulated. With the helicopter steerable nose wheel deflection angle set to 0° and the right winch traction speed set to 10 mm/s, the main wheel motion trajectory obtained by simulation during the translational step is shown in Figure 14. This figure displays the motion trajectories of the left and right main wheel and the center point of the main wheel shaft. It can be seen that these trajectories are three parallel straight lines, and after the translational motion, the center point O of the main wheel shaft reaches the center line of the track. This indicates that the helicopter has achieved the correction target of the translation motion step. Figure 15 shows the velocity characteristic curves of the SRCR during the translational motion process. It also presents the mathematical velocity characteristic curves based on the above collaborative velocity governing equations and the simulation velocity characteristic curves based on the Adams dynamics model. By comparison, it can be seen that Figure 15a,b are in good agreement; the left and right traction velocities are equal, and the correction velocity of the right robotic arm is proportional to right traction velocity. This demonstrates mutual verification of the stepwise correction velocity cooperative control theory and the Adams dynamic simulation model at the kinematic level.

Simulation analysis of the rotating motion was conducted. The deflection angle of the helicopter’s steerable nose wheel was set to 90°, and the right traction speed was set to −10 mm/s. The resulting movement trajectory of the main wheel under stepwise rotational motion of the helicopter is shown in Figure 16. The motion trajectories of the left and right wheels form two symmetrical arcs. After rotating motion of 10°, the shaft of the main wheel aligns horizontally, indicating that the helicopter’s fuselage direction is now parallel to the traction track. This confirms that the rotational motion has successfully achieved the correction target. Figure 17 presents the velocity characteristic curves during the rotational motion. Comparison of Figure 17a,b reveals that the mathematical velocity characteristic curve derived from the cooperative velocity governing equation is in good agreement with the curve obtained from the Adams simulation model, thus validating the accuracy of both the simulation model and the mathematical model.

Through the above analysis, the correctness of the simulation model is verified by comparing the simulation results of the velocity characteristic curve with the mathematical solution results. This comparison also illustrates the correctness of the mathematical model of cooperative control relationship Equations (22) and (23). The simulation results of the helicopter trajectory based on the stepwise correction method demonstrate that the proposed method can quickly achieve the correction target, verifying its feasibility.

5.2. Simulation of the Continuous Correction Method

To verify the feasibility of the continuous correction method, Adams software is used in this paper to simulate and analyze the corrected motion trajectory of the helicopter. To ensure that SRCR can tow all types of helicopters listed in Table 2, the relevant parameters of the simulation model are shown in

Table 4. Main simulation parameters of the continuous correction method.

Parameter	Value
The length of track spacing for left and right traction L	5000 mm
The length of grab point to fuselage center line L0	2000 mm
The distance between the right traction point of the SRCR and the initial position of the right robotic arm L1	150 mm
The distance between the helicopter main wheel shaft and the line connecting the left and right traction points L2	300 mm
The distance between helicopter control wheel and the main wheel shaft L3	8000 mm
Initial helicopter yaw angle θ0	−20°
Initial helicopter lateral offset d0	−200 mm
Right traction speed v2	30 mm/s
Helicopter control wheel angle γ	67.48°

.

The helicopter’s arc correction track simulated in Adams is shown in Figure 18. As can be seen, the tracks of the left and right wheels and the main wheel center point are three concentric arcs. After a 20° rotation, the center point O of the main wheel shaft aligns with the centerline of the traction track. Additionally, the main wheel shaft is perpendicular to the centerline of the trajectory, indicating that the helicopter has successfully completed the landing position correction task.

The speed characteristic curves of SRCR are shown in Figure 19. Figure 19a represents the velocity characteristic curves based on the mathematical model, while Figure 19b shows the velocity characteristic curves based on Adams simulation. The SRCR angular velocity and yaw angle curves during helicopter position correction are illustrated in Figure 20. It can be observed that the rotational angular velocity is not constant but follows a curved shape, reflecting the complex speed matching relationship between the left winch traction velocity, right winch traction velocity, and right robotic arm lateral correction velocity (v₁, v₂, v_s) during rotational motion. By comparing Figure 20a,b, it is evident that the mathematical velocity characteristic curve obtained from the collaborative velocity governing equation aligns well with the curve derived from the Adams simulation model, further validating the accuracy of both the simulation model and mathematical model.

5.3. Comparative Analysis of the Efficiency of the Stepwise Correction Method and the Continuous Correction Method

Through the analysis of traction motion characteristics of SRCR in Section 2 and the simulation tests in Section 5.1 and Section 5.2, it is evident that the main factors affecting the traction motion characteristics are (1) lateral offset d₀; (2) initial yaw angle θ₀; (3) right traction speed v₂. To explore the influence of these variables on operational performance, two sets of tests were conducted using both the stepwise and continuous methods.

In the cross landing position, both the fuselage yaw angle θ₀ and the landing offset d₀ of the helicopter can be measured using the angle sensors and displacement sensors on the SRCR. However, in the parallel landing position, while the landing offset d₀ can be measured using the displacement sensors, the fuselage yaw angle θ must be set through the first stepwise rotational correction shown in Figure 7d (i) or the first trajectory of the continuous correction shown in Figure 10 (i). Therefore, it is necessary to find the optimal fuselage yaw angle θ for different lateral offsets d₀ to minimize the correction time.

To analyze the impact of these factors on the correction efficiency of the stepwise and continuous methods, two sets of simulations were performed for each method: one focusing on the impact of initial yaw angle θ₀ and right traction speed v₂, and the other on the impact of initial yaw angle θ₀ and lateral offset d₀. The factors affecting correction performance under different conditions were analyzed, and finally, the performance of the stepwise correction method and the continuous correction method was compared.

5.3.1. Efficiency Analysis of the Stepwise Correction Method on the Impact of Initial Yaw Angle and Traction Speed

The performance comparison test is shown in

Table 5. Analysis results data on the impact of initial yaw angle and traction speed on efficiency of the stepwise correction method.

Serial Number	Initial Fuselage Yaw Angle θ0/deg	Lateral Offset d0/mm	Right Traction Speed v2/mm/s	Translation Time t1/s	Rotation Time t2/s	Total Correction Time t/s
1	−10°	−200	10	108.71	43.80	152.51
2	−10°	−200	20	54.56	21.80	76.36
3	−10°	−200	30	36.42	14.60	51.02
4	−20°	−200	10	78.29	91.00	169.29
5	−20°	−200	20	39.18	45.60	84.78
6	−20°	−200	30	26.16	30.40	56.56
7	−30°	−200	10	95.56	149.20	244.76
8	−30°	−200	20	47.82	74.60	122.42
9	−30°	−200	30	31.90	49.80	81.70

and Figure 21. By analyzing the data of different right traction speeds v₂ at initial fuselage yaw angles of −10°, −20°, and −30°, the following conclusions can be drawn:

(1)

By comparing serial numbers 1-2-3, 4-5-6, 7-8-9, it can be seen that increasing v₂ can effectively improve the efficiency of the correction operation and reduce the total correction time t.

(2)

By comparing serial numbers 1-4-7, 2-5-8, 3-6-9, it can be seen that during the translational operation, a larger initial yaw angle |θ₀| results in a shorter translation time t₁; however, during the rotational operation, a larger initial yaw angle |θ₀| results in a longer rotation time t₂.

5.3.2. Efficiency Analysis of the Stepwise Correction Method on the Impact of Initial Yaw Angle and Lateral Offsets

The performance comparison test is shown in

Table 6. Analysis results data on the impact of initial yaw angle and lateral offsets on efficiency of the continuous correction method.

Serial Number	Initial Fuselage Yaw Angle θ0/deg	Lateral Offset d0/mm	Right Traction Speed v2/mm/s	Translation Time t1/s	Rotation Time t2/s	Total Correction Time t/s
1	−10°	−600	30	109.26	14.60	123.86
2	−10°	−400	30	72.84	14.60	87.44
3	−10°	−200	30	36.42	14.60	51.02
4	−10°	−150	30	27.32	14.60	41.92
5	−10°	−100	30	18.21	14.60	32.81
6	−10°	−50	30	9.11	14.60	23.71
7	−20°	−600	30	78.48	30.40	108.88
8	−20°	−400	30	52.32	30.40	82.72
9	−20°	−200	30	26.16	30.40	56.56
10	−20°	−150	30	19.62	30.40	50.02
11	−20°	−100	30	13.08	30.40	43.48
12	−30°	−600	30	95.70	49.80	145.50
13	−30°	−400	30	63.80	49.80	113.60
14	−30°	−200	30	31.90	49.80	81.70
15	−30°	−150	30	23.88	49.80	73.68
16	−30°	−100	30	15.85	49.80	65.65

and Figure 22. By analyzing the data for different lateral offsets d₀ when the initial yaw angle θ₀ is −10°, −20°, and −30°, respectively, the following conclusions can be drawn:

(1)

By comparing serial numbers 1-2-3-4-5-6, 7-8-9-10-11, 12-13-14-15-16 and referring to Figure 22, it can be seen that when the initial yaw angle |θ₀| is fixed, the smaller the lateral offset |d₀|, the shorter the translation time t₁, with no effect on rotation time t₂.

(2)

According to Figure 22, in the stepwise correction for the cross landing position, when the lateral offset d₀ = −600 mm and −400 mm, the total correction time t is minimized when the initial yaw angle θ₀ is −20°. For lateral offset d₀ = −200 mm, −150 mm, and −100 mm, the total correction time t is minimized when the initial yaw angle θ₀ is −10°.

(3)

According to Figure 22 and Table 6, in the correction process for the parallel landing position, when the lateral offset d₀ is −1200 mm or −800 mm, the total correction time t is minimized when the fuselage yaw angle θ for the first stepwise rotational correction is −20°. For lateral offsets d₀ of −400 mm, −300 mm, and −200 mm, the total correction time t is minimized when the fuselage yaw angle θ for the first stepwise rotational correction is −10°.

The following conclusions can be drawn regarding the stepwise correction method:

(1)

In the correction process for the cross landing position, a larger initial yaw angle θ₀ results in a longer time t₂ required for stepwise rotational correction. However, for the same initial lateral offset d₀, the stepwise translational correction time t₁ will be shorter. Moreover, with the same initial yaw angle θ₀, the variation in stepwise correction time t is solely dependent on the initial lateral offset d₀.

(2)

In the correction process for the parallel landing position, according to the different lateral offset d₀, weighing the translation time t₁ and the rotation time t₂ and choosing the optimal fuselage yaw angle θ for the first stepwise rotational correction is the key to improving the efficiency of the stepwise correction method.

5.3.3. Efficiency Analysis of the Continuous Correction Method on the Impact of Initial Yaw Angle and Traction Speed

The performance comparison test is shown in

Table 7. Analysis results data on the impact of initial yaw angle and traction speed on efficiency.

Serial Number	Initial Fuselage Yaw Angle θ0/deg	Lateral Offset d0/mm	Right Traction Speed v2/mm/s	Helicopter Control Wheel Angle γ/deg	Helicopter Correction Radius r/mm	Total Correction Time t/s
1	−10°	−200	10	31.29	13162.86	271.60
2	−10°	−200	20	31.29	13162.86	135.80
3	−10°	−200	30	31.29	13162.86	90.60
4	−20°	−200	10	67.48	3316.98	200.60
5	−20°	−200	20	67.48	3316.98	100.40
6	−20°	−200	30	67.48	3316.98	67.20
7	−30°	−200	10	79.43	1492.82	207.60
8	−30°	−200	20	79.43	1492.82	103.80
9	−30°	−200	30	79.43	1492.82	69.20

and Figure 23. The following conclusions can be drawn by analyzing the data of different right traction speeds v₂ when the initial yaw angle θ₀ is −10°, −20°, and −30°, respectively:

(1)

By comparing serial numbers 1-2-3, 4-5-6, 7-8-9, it can be seen that an increase in right traction speed v₂ can effectively improve correction efficiency and reduce the total correction time t.

(2)

By comparing serial numbers 1-4-7, 2-5-8, it can be seen that the larger the initial yaw angle |θ₀|, the larger the helicopter control wheel angle |γ|, and the smaller the helicopter correction radius r in the continuous correction method.

5.3.4. Efficiency Analysis of the Continuous Correction Method on the Impact of Initial Yaw Angle and Lateral Offsets

The performance comparison test is shown in

Table 8. Analysis results data on the impact of initial yaw angle and lateral offsets on efficiency.

Serial Number	Initial Fuselage Yaw Angle θ0/deg	Lateral Offset d0/mm	Right Traction Speed v2/mm/s	Helicopter Control Wheel Angle γ/deg	Helicopter Correction Radius r/mm	Total Correction Time t/s
1	−10°	−200	30	31.29	13162.86	90.60
2	−10°	−150	30	39.01	9875.65	71.60
3	−10°	−100	30	50.55	6582.97	52.80
4	−10°	−50	30	67.63	3292.46	33.80
5	−20°	−200	30	67.48	3316.98	67.00
6	−20°	−150	30	72.73	2487.13	57.60
7	−20°	−100	30	78.29	1658.18	48.40
8	−30°	−200	30	79.43	1492.82	69.20
9	−30°	−150	30	82.03	1120.06	63.40
10	−30°	−100	30	84.67	746.36	57.40

and Figure 24. By analyzing the data of different lateral offsets d₀ when the initial yaw angle θ₀ is −10°, −20°, and −30°, the following conclusions can be drawn:

(1)

As the initial yaw angle |θ₀| increases and the lateral offset d₀ decreases, the length of the helicopter correction radius r also decreases, as indicated by the downward trend of the red line in Figure 24.

(2)

Comparing serial numbers 1-2-3-4, 5-6-7, 8-9-10 and Figure 24, it is evident that with a fixed initial yaw angle |θ₀|, a larger helicopter control wheel angle |γ| results in a smaller helicopter correction radius r and a shorter correction time t.

(3)

According to Figure 24, in the continuous correction for the cross landing position, although the helicopter correction radius r varies from large to small under working conditions from 1 to 10, a comparative analysis of serial numbers 1-5-8, 2-6-9, and 3-7-10 shows that with lateral offsets d₀ of −200 mm, −150 mm, and −100 mm, the shortest correction time t occurs with the initial yaw angle θ₀ of −20°.

(4)

According to Figure 24, in the continuous correction for the parallel landing position, when the lateral offset d₀ is −400 mm, −300 mm, and −200 mm, the total correction time t is minimized when the fuselage yaw angle θ for the first trajectory of the continuous correction is −20°.

The following conclusions can be drawn regarding the stepwise correction method:

(1)

In the correction process for the cross landing position, the continuous correction time t is negatively correlated with the initial yaw angle |θ₀| and positively correlated with the initial lateral offset |d₀|.

(2)

In the correction process for the parallel landing position, it is crucial to balance the helicopter correction radius r and choose the optimal helicopter control wheel angle γ according to different lateral offset d₀ to improve the efficiency of the continuous correction method.

5.3.5. Comparison of Performance between Stepwise and Continuous Correction Methods

Figure 25 shows the performance comparison between the stepwise correction method and the continuous correction method under different initial yaw angles θ₀ and lateral offsets d₀.

As can be seen from Figure 25, when the initial yaw angles are θ₀ = −10° and θ₀ = −20°, the total correction time t for the stepwise correction method is less. However, when the initial yaw angle is θ₀ = −30°, the total correction time t for the continuous correction method is less.

According to the conclusions in Section 5.3.2 and Section 5.3.4, when the lateral offset d₀ is constant and the initial yaw angle |θ₀| is small, the rotation time t₂ for the stepwise correction method is shorter. In contrast, the continuous correction method takes more time because the helicopter control wheel angle |γ| is small.

However, when |θ₀| is large, the increase in rotation time for the stepwise correction method is greater than the decrease in translation operation time. In the continuous correction method, a larger helicopter control wheel angle |γ| results in a smaller helicopter correction arc radius r. Thus, the helicopter’s continuous correction path becomes smaller, reducing the time spent.

The performance comparison results indicate that when the initial yaw angle |θ₀| is small, the stepwise correction method is faster. Conversely, when the initial yaw angle |θ₀| is large and the lateral offset |d₀| is small, the continuous correction method is faster.

6. Conclusions

This paper studies the methods for correcting the position of a helicopter on a deck, which is a critical operational process to ensure smooth towing into the hangar. This process is of great importance for improving the efficiency of helicopter towing operations. The paper proposes two methods for correcting a helicopter’s position based on SRCR towing: the stepwise method and the continuous method. The effectiveness and correction efficiency of these methods are studied, leading to the following conclusions.

(1)

To address the helicopter deck position correction of SRCR, the actual landing position deviation is decomposed into two components: lateral offset d₀ and fuselage yaw angle θ₀. This allows for a quantitative evaluation of the landing position deviation.

(2)

To correct the helicopter landing deviation, this paper proposes both the stepwise correction method and the continuous correction method, based on the motion characteristic model of the SRCR. The feasibility of these two correction methods is verified through mathematical analysis and simulation research.

(3)

The three key factors affecting correction efficiency are lateral offset d₀, fuselage yaw angle θ₀, and right traction speed v₂. The stepwise method and continuous method were simulated and analyzed. Results indicate that in the correction process for the parallel landing position, improving the efficiency of the stepwise correction method hinges on selecting the optimal fuselage yaw angle θ for the first stepwise rotational correction based on the different lateral offsets d₀. Similarly, enhancing the efficiency of the continuous correction method requires choosing the optimal helicopter control wheel angle γ according to the varying lateral offsets d₀.

(4)

Comparing the stepwise method with the continuous method, results show that when the initial yaw angle |θ₀| ≤ 20°, the stepwise correction method has a significant advantage. However, when the initial yaw angle |θ₀| > 20° and the lateral offset |d₀| ≤ 200 mm, the efficiency of continuous correction is superior to the stepwise correction method because the helicopter’s correction arc radius r is smaller.

The limitations of this study lie in it not accounting for trajectory deviations caused by environmental disturbances during pilot operations and control system processes. Future research will address these trajectory errors resulting from environmental disturbances, pilot operation accuracy, and towing speed control during real-time correction to design controllers for real-time deviation correction.