# Kinematic Singularity and Bifurcation Analysis of Sidestay Landing Gear Locking Mechanisms

Authors: Kui Xu; Yin Yin; Yixin Yang; Hong Nie; Xiaohui Wei
Journal: International Journal of Aerospace Engineering
 (2021)

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## Abstract

A dual-sidestay landing gear is prone to locking failure in the deployed state due to the restriction of movement between two sidestays. However, the principle of its locking movement still remains unclear. The present study is aimed at investigating the synchronous locking performance of the dual-sidestay landing gear based on the singularity and bifurcation theory. From the perspective of the kinematic mechanism, the reason for high sensitivity to structural dimensions in the locking process is explained, and the locked position is investigated by employing the numerical continuation method in the case of a single-sidestay landing gear. Afterwards, the reason for the locking failure of the dual-sidestay landing gear is analyzed, and a kinematic optimization method for the synchronous locking is proposed. The results reveal that the lock links must reach the lower overcenter singular point to fully lock the landing gear, and the singular point is sensitively affected by structural parameters. Owing to the different positions of singular points, the movements of fore and aft sidestays seriously restrict each other, causing locking failure of the dual-sidestay landing gear. The singular points of two sidestays can be optimized to be approximately identical, making their movements more coordinated.


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## Body

## 1. Introduction

A supercritical wing is extensively used in large aircraft for attaining better aerodynamic performance [1, 2]. For the supercritical wing, however, the stowage for the landing gear is considerably smaller than that for the conventional wing. Narrow space signifies that a conventional planar (2D) landing gear can hardly comply with the design requirements, since the sidestay links are located above the main strut after folding, which occupy a large vertical space, as shown in Figure 1(a). In contrast, for a three-dimensional (3D) landing gear, the sidestay plane can be rotated about a specific axis, so that it is overall folded over the side of the main strut. As a result, the 3D landing gear excellently reduces the size of the vertical space, as shown in Figure 1(b).Figure 1
Vertical storage space of 2D versus 3D landing gear.The extensive use of composite material has contributed hugely to the reduction in aircraft weight. A composite wing, however, does not carry as much ground load as a metallic one. From the load transfer perspective, the ground impact load should not be borne entirely by the wing. Thus, a dual-sidestay (DS) landing gear is used to transfer ground load to the wing and fuselage [3, 4]. Compared to a single-sidestay (SS) landing gear, the DS landing gear has an additional sidestay, as shown in Figure 2, and the motion coordination between fore and aft sidestays appears particularly important. In the case of an unreasonable design of structural parameters, the two sidestays can hinder and constrain each other, disallowing its complete locking. Hence, it is imperative to investigate the motion coordination between two sidestays.Figure 2
Changes in the DS versus SS landing gears.The multidisciplinary time-domain cosimulation technology [5–7] has often been adopted to research landing gear mechanisms. However, this approach requires substantial iterations in handling highly nonlinear problems, which is time- and labour-consuming. Numerical continuation [8, 9] has been applied to aircraft design for several years [10], which can enable fast and efficient exploration of highly nonlinear problems of the landing gear. Following the initial proposal of this method for analyzing landing gear mechanisms by Knowles et al. [11], the research approach integrating numerical continuation and bifurcation analysis has become increasingly mature [12]. Exploiting this method, Knowles et al. [13–16] and Yin et al. [17] analyzed the bifurcation characteristics of different landing gear mechanisms sequentially and concluded that the landing gear would inevitably undergo a jump phenomenon at bifurcation points in the locking process. However, when solving the dynamic equations of the DS landing gear using the numerical continuation method, it has been found that the appearance of bifurcation points does not necessarily imply that two sidestays can be locked synchronously, which indicates the limitation of using the dynamic equations in solving DS mechanisms with extremely high motion sensitivity.The DS landing gear has several singular points in its retraction cycle, which cause extremely high sensitivity of mechanism motion. Hence, the DS mechanism is analyzed from a kinematic singularity perspective in this paper. Existing methods concerning singularity include the Jacobian matrix method [18], the Grassmann line geometry [19], and the differential geometric theory [20], while attempting to avoid the singular position in the mechanism as far as possible [21, 22]. For instance, Xie et al. [23] and Li and Herve [24] used the above methods to explore the singularity characteristics of different mechanisms. Nevertheless, little literature focuses on the singularity of landing gear mechanisms, particularly the complex spatial DS mechanisms. Therefore, continuation analysis on the kinematics equations of the DS landing gear is performed, and the locked position as well as the underlying jump phenomenon of the mechanism is analyzed from the kinematic singularity and bifurcation perspective in this paper. Furthermore, the fundamental causes of mutual hindrance and restriction between fore and aft sidestays are explained, which can provide crucial guidance for the design of DS landing gear mechanisms.Initially, the locking performance of a 3D SS mechanism is analyzed. On this basis, the reasons for incomplete locking caused by mutual restriction between two sidestays are investigated. Finally, a novel method for analyzing the synchronous locking of the DS landing gear is developed, which is based on the singularity and bifurcation theory. The structure of this paper is organized as follows: In Section 2, the kinematic model of a 3D SS landing gear mechanism is built, after analyzing its bifurcation and singularity characteristics, and the jump phenomenon and the locked position are analyzed in Section 3. Meanwhile, the effects of structural parameters on the locked position are explored. In Section 4, the principle of mutual hindrance between fore and aft sidestays for the 3D DS landing gear mechanism is investigated based on the foregoing analysis, and the fundamental causes of asynchronous locking are identified. Furthermore, a set of optimal solutions satisfying constraints are given with the aid of multiobjective particle swarm optimization (PSO). The application of the proposed method can effectively achieve the synchronous locking at the initial design phase of DS landing gear mechanisms.


## 2. Kinematic Model of the Landing Gear Mechanism

In Figure3, a certain 3D landing gear mechanism is illustrated, which consists of five links (two sidestay links, two lock links, and a main strut), as well as three nodes (two sidestay nodes and a lock node). These various components are connected via planar revolute joints. To achieve the 3D retraction function, the axes of sidestay nodes (axes 2 and 4 in Figure 3), the axis of the lock node (axis 1), and the rotation axis of the main strut (axis 3) must intersect at one point (point o) [25]. In this way, the motion of sidestay can be always within the same plane. During the retraction cycle of the landing gear, the main strut rotates along the axis 3, while the sidestay not only performs the folding motion but also rotates about the axis 2, so that the rotation plane of the main strut is not in the same plane as the sidestay. The entire retraction and extension processes can be summarized as follows: When retracting the gear, the unlock actuator is needed to unlock the lock links. Then, the retraction actuator can be engaged to allow the main strut to rotate about axis 3, and the sidestay moves accordingly. Finally, the landing gear reaches the deployed position and gets locked. When extending the gear, after upper unlocking, the retraction actuator supplies a pushing force to enable slow extension of the main strut around axis 3. When approaching the lower locked position, the lock links jump from the upper to the lower overcenter state by the action of locking springs, thereby reaching a downlock state. The lock links are equipped with a stop block to prevent excessive downward folding. Eventually, the landing gear stops at the deployed position to complete the locking procedure.Figure 3
A certain 3D landing gear mechanism.


### 2.1. Coordinate Transformation

Since the rotation plane of the main strut is not in the same plane as the sidestay, it can hardly analyze the mechanism in the same coordinate system. Owing to the fact that the motion of the sidestay is always within the same plane [6, 26], the mechanism model can be divided into different coordinate systems to describe the motion of the main strut and sidestay. The transformation matrix is employed to accomplish the data exchange between them.Figure3 presents the establishment of the coordinate systems. In the global coordinate system O−XYZ, the origin O is the intersection between the main strut and the axis 3; the X-axis is parallel to axis 3 (positive forward); the Z-axis is aligned with the gravity vector (positive down); and the direction of the Y-axis is determined by the right-hand rule. In the local coordinate system o−xyz, the origin o is the intersection point of four axes (axes 1-4); the y-axis is the direction of axis 2; the z-axis is defined to be perpendicular to the y-axis within the sidestay plane; and the direction of the x-axis is determined by the right-hand rule.With the operation of the landing gear, the main strut rotates about theX-axis, and the sidestay plane rotates about the y-axis. Accordingly, the total transformation matrix T for the global-to-local coordinate system is
(1)T=T0⋅T1,where T0 is the transformation matrix of the global to the local coordinate system in the initial state (extended, locked position) and T1 is the transformation matrix of the local coordinate system rotating about the y-axis.In the initial state, given the presence of both translational and rotational transformations during the global-to-local system conversion, the homogeneous coordinate transformation is adopted herein. Lete1=e11e12e13T and e2=e21e22e23T be the base vectors in the global and local systems, respectively, and oO be the vector of origin O in the local system; then,
(2)T0=e1⋅e2ToOT01.Assumingα is the rotation angle of local system about the y-axis, then
(3)T1=cosα0sinα00100−sinα0cosα00001.


### 2.2. Motion Constraint Equations

In Figure4, the schematic of the landing gear mechanism is presented. The constraint relations between various links can be derived from eight variables Xi,Yi,Zi,Θi,xi,yi,zi,andθi, where Xi,Yi,Zi and xi,yi,ziare the global and local coordinates of the gravity center of the i-th link, respectively; Θi denotes the angle between the i-th link and the Z-axis (specifically, Θ1 is the retraction angle of the main strut); and θi is the angle between the i-th link and the y-axis.Figure 4
Schematic diagram of the 3D landing gear mechanism.Since the global and local coordinates can be interconverted through the coordinate transformation, the position of various links can be identified independently withXi,Yi,Zi,Θi or xi,yi,zi,θi. There are twenty independent geometric variables for the SS mechanism. As the analysis reveals, the mechanism has nineteen independent constraint equations (see Equation (4) and Table 3 in the Appendix), so the degree of freedom (DOF) is one.
(4)X1Y1‐0.5l1⋅cosΘ1Z1‐0.5l1⋅sinΘ1x2y2−0.5l2⋅cosθ2−yAz2−0.5l2⋅sinθ2−zAx3y3−0.5l3⋅cosθ3‐y2−0.5l2⋅cosθ2z3−0.5l3⋅sinθ3‐z2−0.5l2⋅sinθ2x4y4+0.5l4⋅cosθ4‐y2−0.5l2⋅cosθ2z4−0.5l4⋅sinθ4‐z2−0.5l2⋅sinθ2x5y5+0.5l5⋅cosθ5‐y4+0.5l4⋅cosθ4z5−0.5l5⋅sinθ5‐z4+0.5l4⋅sinθ4y5‐0.5l5⋅cosθ5‐yEz5−0.5l5⋅sinθ5‐zEy3+0.5l3⋅cosθ3‐yCz3+0.5l3⋅sinθ3‐zC=0,where li denotes the length of the i-th link; y∗,z∗ denote the local coordinates of the points ∗ (points A/C/E in Figure 4), which can be derived from the node parameters as follows:
(5)yAzAyCzCyEzE=yF+lAF⋅cosφAFlAF⋅sinφAFyG‐lCG⋅cosφCGzG‐lCG⋅sinφCGyH‐lEH⋅cosφEHzH‐lEH⋅sinφEH,where lAF,lCG,lEH denote the node lengths and φAF,φCG,φEH denote the angles between the node axes and the y-axis. Meanwhile, the local coordinates of points G and H are obtained through coordinate transformation as follows:
(6)xGyGzG1=T⋅XGYGZG1,(7)xHyHzH1=T⋅XHYHZH1.


### 2.3. Bifurcation Equation of Mechanism Motion

Given the one DOF of the 3D mechanism,Θ1 is selected as the independent control parameter, while the other variables can be regarded as the state variables. Accordingly, Equation (4) can be rewritten as
(8)Fx,λ=0,where x=X1,Y1,Z1,x2,y2,z2,θ2,⋯,x5,y5,z5,θ5 and λ=Θ1.Equation (8) expresses the variation of state parameter x with the bifurcation control parameter λ, which is called the bifurcation equation of the 3D mechanism. Utilizing this equation, the bifurcation characteristics of the mechanism can be analyzed.


## 2.1. Coordinate Transformation

Since the rotation plane of the main strut is not in the same plane as the sidestay, it can hardly analyze the mechanism in the same coordinate system. Owing to the fact that the motion of the sidestay is always within the same plane [6, 26], the mechanism model can be divided into different coordinate systems to describe the motion of the main strut and sidestay. The transformation matrix is employed to accomplish the data exchange between them.Figure3 presents the establishment of the coordinate systems. In the global coordinate system O−XYZ, the origin O is the intersection between the main strut and the axis 3; the X-axis is parallel to axis 3 (positive forward); the Z-axis is aligned with the gravity vector (positive down); and the direction of the Y-axis is determined by the right-hand rule. In the local coordinate system o−xyz, the origin o is the intersection point of four axes (axes 1-4); the y-axis is the direction of axis 2; the z-axis is defined to be perpendicular to the y-axis within the sidestay plane; and the direction of the x-axis is determined by the right-hand rule.With the operation of the landing gear, the main strut rotates about theX-axis, and the sidestay plane rotates about the y-axis. Accordingly, the total transformation matrix T for the global-to-local coordinate system is
(1)T=T0⋅T1,where T0 is the transformation matrix of the global to the local coordinate system in the initial state (extended, locked position) and T1 is the transformation matrix of the local coordinate system rotating about the y-axis.In the initial state, given the presence of both translational and rotational transformations during the global-to-local system conversion, the homogeneous coordinate transformation is adopted herein. Lete1=e11e12e13T and e2=e21e22e23T be the base vectors in the global and local systems, respectively, and oO be the vector of origin O in the local system; then,
(2)T0=e1⋅e2ToOT01.Assumingα is the rotation angle of local system about the y-axis, then
(3)T1=cosα0sinα00100−sinα0cosα00001.


## 2.2. Motion Constraint Equations

In Figure4, the schematic of the landing gear mechanism is presented. The constraint relations between various links can be derived from eight variables Xi,Yi,Zi,Θi,xi,yi,zi,andθi, where Xi,Yi,Zi and xi,yi,ziare the global and local coordinates of the gravity center of the i-th link, respectively; Θi denotes the angle between the i-th link and the Z-axis (specifically, Θ1 is the retraction angle of the main strut); and θi is the angle between the i-th link and the y-axis.Figure 4
Schematic diagram of the 3D landing gear mechanism.Since the global and local coordinates can be interconverted through the coordinate transformation, the position of various links can be identified independently withXi,Yi,Zi,Θi or xi,yi,zi,θi. There are twenty independent geometric variables for the SS mechanism. As the analysis reveals, the mechanism has nineteen independent constraint equations (see Equation (4) and Table 3 in the Appendix), so the degree of freedom (DOF) is one.
(4)X1Y1‐0.5l1⋅cosΘ1Z1‐0.5l1⋅sinΘ1x2y2−0.5l2⋅cosθ2−yAz2−0.5l2⋅sinθ2−zAx3y3−0.5l3⋅cosθ3‐y2−0.5l2⋅cosθ2z3−0.5l3⋅sinθ3‐z2−0.5l2⋅sinθ2x4y4+0.5l4⋅cosθ4‐y2−0.5l2⋅cosθ2z4−0.5l4⋅sinθ4‐z2−0.5l2⋅sinθ2x5y5+0.5l5⋅cosθ5‐y4+0.5l4⋅cosθ4z5−0.5l5⋅sinθ5‐z4+0.5l4⋅sinθ4y5‐0.5l5⋅cosθ5‐yEz5−0.5l5⋅sinθ5‐zEy3+0.5l3⋅cosθ3‐yCz3+0.5l3⋅sinθ3‐zC=0,where li denotes the length of the i-th link; y∗,z∗ denote the local coordinates of the points ∗ (points A/C/E in Figure 4), which can be derived from the node parameters as follows:
(5)yAzAyCzCyEzE=yF+lAF⋅cosφAFlAF⋅sinφAFyG‐lCG⋅cosφCGzG‐lCG⋅sinφCGyH‐lEH⋅cosφEHzH‐lEH⋅sinφEH,where lAF,lCG,lEH denote the node lengths and φAF,φCG,φEH denote the angles between the node axes and the y-axis. Meanwhile, the local coordinates of points G and H are obtained through coordinate transformation as follows:
(6)xGyGzG1=T⋅XGYGZG1,(7)xHyHzH1=T⋅XHYHZH1.


## 2.3. Bifurcation Equation of Mechanism Motion

Given the one DOF of the 3D mechanism,Θ1 is selected as the independent control parameter, while the other variables can be regarded as the state variables. Accordingly, Equation (4) can be rewritten as
(8)Fx,λ=0,where x=X1,Y1,Z1,x2,y2,z2,θ2,⋯,x5,y5,z5,θ5 and λ=Θ1.Equation (8) expresses the variation of state parameter x with the bifurcation control parameter λ, which is called the bifurcation equation of the 3D mechanism. Utilizing this equation, the bifurcation characteristics of the mechanism can be analyzed.


## 3. Analysis of the SS Landing Gear Mechanism

### 3.1. Bifurcation Analysis

The retraction and extension processes of the landing gear can be simulated through numerically continuing Equation (8), and the variation of state parameters (e.g., the overcenter angle θoc=θ4‐θ5 of lock links) with the control parameter Θ1 can be derived, as shown in Figure 5. For convenience of observation, the kinematic sketches of the mechanism are given in Figure 6, which correspond to various points in Figure 5.Figure 5
Bifurcation curve of the 3D SS landing gear ((b) presents an enlarged view of the rectangular frame in (a)).
(a)(b)Figure 6
Sketches of the retraction and extension motion of a landing gear: (a) U1; (b) U2; (c) S1; (d) S2; (e) D1; (f) D2; (g) S3.
(a)(b)(c)(d)(e)(f)(g)In Figure5, the motion bifurcation curve is symmetrical about θoc=0. The upper half of the curve represents the upward folding of lock links (U1, U2, and S1), while the lower half represents their downward folding (D1, D2, and S3). Three bifurcation points (S1, S2, and S3) appear in the figure, at which two types of motion trends of links are present. As the landing gear extends, it moves along the curve from U1 to U2. At the beginning, Θ1 changes drastically, while θoc changes a little. On the contrary, when the landing gear gets close to the deployed state, Θ1 is almost unchanged, while θoc changes drastically. Bifurcation occurs when the landing gear moves to the point S1, and two motion branches appear, which correspond to the two motion trends of sidestay links (the dashed lines in Figure 6(c)). The point S1 corresponds to the collinear position of upper and lower sidestay links. At this time, Θ1 is the smallest, implying that the main strut is extended to the farthest position. By the favorable action of locking springs, the landing gear will move to point S2, at which the upper and lower lock links are collinear. Similarly, two motion branches appear when the mechanism moves to S2, which correspond to the two motion trends of lock links (the dashed lines in Figure 6(d)). When moving to S3 by the action of springs, the upper and lower sidestay links are collinear again, and the lock links are folded down. The appearance of two motion branches of sidestay links is noted (the dashed lines in Figure 6(g)). Generally, the landing gear is provided with a stop block here to prevent further downward motion of lock links, so it is impossible to reach the states D1 and D2 in reality.


### 3.2. Singularity Analysis

Derivation of Equation (8) with respect to time yields is as follows:
(9)Ax.+Bλ.=0,where A=∂F/∂x and B=∂F/∂λ. Let J=‐A‐1⋅B; then,
(10)x.=J⋅λ..When the mechanism is in a singular position,detA=0 [27], i.e., detJ⟶∞. Since the bifurcation points of a mechanism must be in a singular position [28], the points S1, S2, and S3 are the singular positions of the landing gear mechanism. When considering the locking status of lock links, Equation (10) can be rewritten as
(11)θ.oc=J⋅Θ.1.It can be seen from Equation (11) that in the vicinity of singular positions, due to J⟶∞, the lock link speed θ.oc is considerably high even if Θ.1 is very low. Accordingly, the lock links undergo a quick jump when the landing gear is close to locking. The slope of the curve in Figure 5 can be derived as
(12)k=∂θoc∂Θ1=∂θoc/∂t∂Θ1/∂t=θ.ocΘ.1.The equation ofk=J can be derived from Equations (11) and (12). That is, the value of J corresponds to the slope of motion trajectory in Figure 5. Meanwhile, using the principle of virtual work and combining with Equation (11), Equation (13) can be obtained.
(13)M1=J⋅Moc,where M1 denotes the external moment of the main strut and Moc denotes the internal moment of lock links (centered on the intersection of the lock links).In the unlocking process of the landing gear, the sidestay links and lock links must be folded. Since the stop block restricts the downward folding of lock links, the lock links can only be folded upwards. Unlocking is impossible to achieve without unlocking the actuator force at the singular points (Figure7). If the unlocking force is not provided, overcoming the drag load from the upward folding of lock links will be required. This drag load is the equivalent moment Moc of locking springs. From Equation (13), it is clear that J⋅Moc⟶∞ when J⟶∞. In theory, the landing gear can be unlocked only when the external moment M1 is infinite. In other word, the landing gear will never be unlocked at the singular point. Therefore, the singular feature can be exploited to lock the landing gear near the singular point.Figure 7
Load analysis at the singular position of landing gear.


### 3.3. Analysis of Locked Position

As is clear from the above analysis, the landing gear should be locked near a singular position. A total of three singular positions are present. This section analyzes at which singular position the landing gear should be locked.The overcenter angle of sidestay links is defined asθd=θ2‐θ3. Figure 8 shows the curves obtained by numerically continuing Equation (8), where θd is used as the state parameter and Θ1 is used as the control variable. The black line represents the upward folding of lock links (U1, U2, and S1), whereas the grey line represents the downward folding of lock links (D1, D2, and S3). The black and grey lines overlap each other. Corresponding to Figures 5 and 6, there are three singular points on the motion trajectories. The point S2 is at the intersection of black and grey lines.Figure 8
Bifurcation curves for variation ofθd with Θ1.
(a)(b)The landing gear is highly likely to vibrate during the aircraft take-off, landing, and taxiing [29, 30], so the downlock mechanism must possess a certain function of perturbation resistance. The stop block, as the important component of the downlock mechanism, is mainly used to prevent the excessive motion of lock links. Eventually, the landing gear is locked by the stop block. The locked position can be controlled by setting up the stop block, which is generally arranged at the lock links. To achieve complete locking, the motion of lock links must be restricted in both upward and downward directions. The stop block restricts the downward motion of lock links, while the upward motion is restricted by the structural design and locking springs. In this process, the unlocking of the downlock mechanism can be avoided.The locked position of landing gear can be analyzed based on Figure8 combined with Figure 6.
(1)
In case the landing gear is locked at the point S1, the stop block will restrict the further downward folding of the lock links and sidestay links. At this time, the upper and lower sidestay links are collinear, while the lock links are in the upper overcenter state. When the sidestay links are subjected to a perturbation to leave the collinear state, they will drive the lock links to fold upward and get unlocked. As is clear from Figure8, when the sidestay links are subjected to a perturbation of Δθd>0, the landing gear will leave the singular point S1, thereby resulting in unlocking(2)
In case the landing gear is locked at the point S2, the stop block will restrict the downward folding of the lock links. At this time, the upper and lower lock links are collinear. It is not difficult to find that when the lock links are subjected to a perturbation to leave the collinear state, they will drive the sidestay links to makeΔθd>0. Thus, the landing gear will leave the singular point S2, resulting in unlocking. Furthermore, according to the force transmission relationship, the lock links need to bear ground load, which will compromise the reliability of the downlock mechanism [31](3)
In case the landing gear is locked at the point S3, the stop block will restrict the downward folding of lock links and the upward folding of sidestay links. At this time, the upper and lower sidestay links are collinear again, while the lock links are in the lower overcenter state. When the sidestay links are subjected to a downward perturbation ofΔθd<0, the lock links will leave point S3 and move to point S2. Although the landing gear will not be unlocked since the lock links will not go beyond point S2, the main strut may rotate due to the perturbation. To diminish the rotation angle, the value of θocS3 should not be excessively large in the deployed state. According to Figure 5, the landing gear is designed to have θocS3=2°.To sum up, the locked position of the landing gear should be at the point S3. As described in the foregoing analysis, when the landing gear is at the singular position, the main strut is unable to move regardless of how large the load of the retraction actuator is. If the landing gear is locked at the point S3, it needs to pass the singular points S1 and S2. At this time, the action of locking springs assists in passing the singular points, so that the landing gear does not return along the original trajectory.


### 3.4. Effects of Structural Parameters on the Locked Position

Once the installation location of the landing gear is decided, the link length parameters will directly affect its locked position (point S3). With changes in the link lengths, the number of singular points also changes accordingly. Thus, selecting appropriate link lengths to ensure the smooth and complete locking of the landing gear is necessary. When considering the effects of link lengths on the point S3,Θ1 can be treated as a fixed parameter (Θ1=0), where θoc is used as the state parameter, and the link lengths l2 and l4 are used as the control parameters. Figure 9 presents the variation curves of the singular point S3 through numerically continuing Equation (8).Figure 9
Variations of points S1 and S3 with the link length parameters.
(a)(b)The singular points exhibit a consistent variation trend with the link lengths in the two subfigures. Asl2/l4 decreases, the points S1 and S3 come together and disappear at the point SN. A new bifurcation point SN appears. As described in the previous section, the landing gear should be locked at the point S3. If the link lengths are decreased past the point SN, the point S3 will disappear, which means failure of locking. Therefore, the feasible range of l2>549.9mm or l4>249.9mm should be selected to accomplish the locking of the landing gear. In addition, it is clear that l4 is more sensitive to the influence of θoc.In actuality, the point SN corresponds to the state ofθoc=θd=0, as shown in Figure 10(b). The state wherein link lengths are not in the feasible range is shown in Figure 10(a). In this scenario, the landing gear cannot be locked in the deployed position. The point SN is determined jointly by the link lengths. When one of the parameters is changed, the remaining several parameters may also undergo corresponding changes. Their mutual coupling together influences the landing gear locked position. For a more detailed analysis of the effects of link lengths, it can be computed directly by using two-parameter continuation, thereby obtaining a two-parameter continuation of the point SN locus, as shown by the black line in Figure 11. Figure 11(a) is a 3D view, and Figure 11(b) is the projections of the black curve in the l2‐l4 plane. With changes in the parameters, the bifurcation point SN changes accordingly, which in turn affects the point S3 as well as the locked position.Figure 10
Landing gear mechanism corresponding to the vicinity of point SN.
(a)(b)Figure 11
Bifurcation point SN variation under two-parameter continuation.
(a)(b)From Figure11(b), it is clear that when the values of l2 and l4 locate at the left side of the black curve, points S1 and S3 are absent, implying that the landing gear cannot be locked at the S3 position. In contrast, when the values of l2 and l4 locate at the right side of black curve, points S1 and S3 are present, and the landing gear can be fully locked. Consequently, the right side of the curve is the feasible region of link lengths.The research on the SS landing gear suggests that the lock links must be folded down, in order to achieve full locking. For the SS landing gear, the downward folding motion of lock links will not be hindered due to the promotion by locking springs. In the case of the DS landing gear, however, the sidestay links on both sides may affect each other. Even with the beneficial effect of locking springs, it is highly likely that lock links on the one side undergo locking failure. Hence, the next section carries out research focusing on the DS landing gear.


## 3.1. Bifurcation Analysis

The retraction and extension processes of the landing gear can be simulated through numerically continuing Equation (8), and the variation of state parameters (e.g., the overcenter angle θoc=θ4‐θ5 of lock links) with the control parameter Θ1 can be derived, as shown in Figure 5. For convenience of observation, the kinematic sketches of the mechanism are given in Figure 6, which correspond to various points in Figure 5.Figure 5
Bifurcation curve of the 3D SS landing gear ((b) presents an enlarged view of the rectangular frame in (a)).
(a)(b)Figure 6
Sketches of the retraction and extension motion of a landing gear: (a) U1; (b) U2; (c) S1; (d) S2; (e) D1; (f) D2; (g) S3.
(a)(b)(c)(d)(e)(f)(g)In Figure5, the motion bifurcation curve is symmetrical about θoc=0. The upper half of the curve represents the upward folding of lock links (U1, U2, and S1), while the lower half represents their downward folding (D1, D2, and S3). Three bifurcation points (S1, S2, and S3) appear in the figure, at which two types of motion trends of links are present. As the landing gear extends, it moves along the curve from U1 to U2. At the beginning, Θ1 changes drastically, while θoc changes a little. On the contrary, when the landing gear gets close to the deployed state, Θ1 is almost unchanged, while θoc changes drastically. Bifurcation occurs when the landing gear moves to the point S1, and two motion branches appear, which correspond to the two motion trends of sidestay links (the dashed lines in Figure 6(c)). The point S1 corresponds to the collinear position of upper and lower sidestay links. At this time, Θ1 is the smallest, implying that the main strut is extended to the farthest position. By the favorable action of locking springs, the landing gear will move to point S2, at which the upper and lower lock links are collinear. Similarly, two motion branches appear when the mechanism moves to S2, which correspond to the two motion trends of lock links (the dashed lines in Figure 6(d)). When moving to S3 by the action of springs, the upper and lower sidestay links are collinear again, and the lock links are folded down. The appearance of two motion branches of sidestay links is noted (the dashed lines in Figure 6(g)). Generally, the landing gear is provided with a stop block here to prevent further downward motion of lock links, so it is impossible to reach the states D1 and D2 in reality.


## 3.2. Singularity Analysis

Derivation of Equation (8) with respect to time yields is as follows:
(9)Ax.+Bλ.=0,where A=∂F/∂x and B=∂F/∂λ. Let J=‐A‐1⋅B; then,
(10)x.=J⋅λ..When the mechanism is in a singular position,detA=0 [27], i.e., detJ⟶∞. Since the bifurcation points of a mechanism must be in a singular position [28], the points S1, S2, and S3 are the singular positions of the landing gear mechanism. When considering the locking status of lock links, Equation (10) can be rewritten as
(11)θ.oc=J⋅Θ.1.It can be seen from Equation (11) that in the vicinity of singular positions, due to J⟶∞, the lock link speed θ.oc is considerably high even if Θ.1 is very low. Accordingly, the lock links undergo a quick jump when the landing gear is close to locking. The slope of the curve in Figure 5 can be derived as
(12)k=∂θoc∂Θ1=∂θoc/∂t∂Θ1/∂t=θ.ocΘ.1.The equation ofk=J can be derived from Equations (11) and (12). That is, the value of J corresponds to the slope of motion trajectory in Figure 5. Meanwhile, using the principle of virtual work and combining with Equation (11), Equation (13) can be obtained.
(13)M1=J⋅Moc,where M1 denotes the external moment of the main strut and Moc denotes the internal moment of lock links (centered on the intersection of the lock links).In the unlocking process of the landing gear, the sidestay links and lock links must be folded. Since the stop block restricts the downward folding of lock links, the lock links can only be folded upwards. Unlocking is impossible to achieve without unlocking the actuator force at the singular points (Figure7). If the unlocking force is not provided, overcoming the drag load from the upward folding of lock links will be required. This drag load is the equivalent moment Moc of locking springs. From Equation (13), it is clear that J⋅Moc⟶∞ when J⟶∞. In theory, the landing gear can be unlocked only when the external moment M1 is infinite. In other word, the landing gear will never be unlocked at the singular point. Therefore, the singular feature can be exploited to lock the landing gear near the singular point.Figure 7
Load analysis at the singular position of landing gear.


## 3.3. Analysis of Locked Position

As is clear from the above analysis, the landing gear should be locked near a singular position. A total of three singular positions are present. This section analyzes at which singular position the landing gear should be locked.The overcenter angle of sidestay links is defined asθd=θ2‐θ3. Figure 8 shows the curves obtained by numerically continuing Equation (8), where θd is used as the state parameter and Θ1 is used as the control variable. The black line represents the upward folding of lock links (U1, U2, and S1), whereas the grey line represents the downward folding of lock links (D1, D2, and S3). The black and grey lines overlap each other. Corresponding to Figures 5 and 6, there are three singular points on the motion trajectories. The point S2 is at the intersection of black and grey lines.Figure 8
Bifurcation curves for variation ofθd with Θ1.
(a)(b)The landing gear is highly likely to vibrate during the aircraft take-off, landing, and taxiing [29, 30], so the downlock mechanism must possess a certain function of perturbation resistance. The stop block, as the important component of the downlock mechanism, is mainly used to prevent the excessive motion of lock links. Eventually, the landing gear is locked by the stop block. The locked position can be controlled by setting up the stop block, which is generally arranged at the lock links. To achieve complete locking, the motion of lock links must be restricted in both upward and downward directions. The stop block restricts the downward motion of lock links, while the upward motion is restricted by the structural design and locking springs. In this process, the unlocking of the downlock mechanism can be avoided.The locked position of landing gear can be analyzed based on Figure8 combined with Figure 6.
(1)
In case the landing gear is locked at the point S1, the stop block will restrict the further downward folding of the lock links and sidestay links. At this time, the upper and lower sidestay links are collinear, while the lock links are in the upper overcenter state. When the sidestay links are subjected to a perturbation to leave the collinear state, they will drive the lock links to fold upward and get unlocked. As is clear from Figure8, when the sidestay links are subjected to a perturbation of Δθd>0, the landing gear will leave the singular point S1, thereby resulting in unlocking(2)
In case the landing gear is locked at the point S2, the stop block will restrict the downward folding of the lock links. At this time, the upper and lower lock links are collinear. It is not difficult to find that when the lock links are subjected to a perturbation to leave the collinear state, they will drive the sidestay links to makeΔθd>0. Thus, the landing gear will leave the singular point S2, resulting in unlocking. Furthermore, according to the force transmission relationship, the lock links need to bear ground load, which will compromise the reliability of the downlock mechanism [31](3)
In case the landing gear is locked at the point S3, the stop block will restrict the downward folding of lock links and the upward folding of sidestay links. At this time, the upper and lower sidestay links are collinear again, while the lock links are in the lower overcenter state. When the sidestay links are subjected to a downward perturbation ofΔθd<0, the lock links will leave point S3 and move to point S2. Although the landing gear will not be unlocked since the lock links will not go beyond point S2, the main strut may rotate due to the perturbation. To diminish the rotation angle, the value of θocS3 should not be excessively large in the deployed state. According to Figure 5, the landing gear is designed to have θocS3=2°.To sum up, the locked position of the landing gear should be at the point S3. As described in the foregoing analysis, when the landing gear is at the singular position, the main strut is unable to move regardless of how large the load of the retraction actuator is. If the landing gear is locked at the point S3, it needs to pass the singular points S1 and S2. At this time, the action of locking springs assists in passing the singular points, so that the landing gear does not return along the original trajectory.


## 3.4. Effects of Structural Parameters on the Locked Position

Once the installation location of the landing gear is decided, the link length parameters will directly affect its locked position (point S3). With changes in the link lengths, the number of singular points also changes accordingly. Thus, selecting appropriate link lengths to ensure the smooth and complete locking of the landing gear is necessary. When considering the effects of link lengths on the point S3,Θ1 can be treated as a fixed parameter (Θ1=0), where θoc is used as the state parameter, and the link lengths l2 and l4 are used as the control parameters. Figure 9 presents the variation curves of the singular point S3 through numerically continuing Equation (8).Figure 9
Variations of points S1 and S3 with the link length parameters.
(a)(b)The singular points exhibit a consistent variation trend with the link lengths in the two subfigures. Asl2/l4 decreases, the points S1 and S3 come together and disappear at the point SN. A new bifurcation point SN appears. As described in the previous section, the landing gear should be locked at the point S3. If the link lengths are decreased past the point SN, the point S3 will disappear, which means failure of locking. Therefore, the feasible range of l2>549.9mm or l4>249.9mm should be selected to accomplish the locking of the landing gear. In addition, it is clear that l4 is more sensitive to the influence of θoc.In actuality, the point SN corresponds to the state ofθoc=θd=0, as shown in Figure 10(b). The state wherein link lengths are not in the feasible range is shown in Figure 10(a). In this scenario, the landing gear cannot be locked in the deployed position. The point SN is determined jointly by the link lengths. When one of the parameters is changed, the remaining several parameters may also undergo corresponding changes. Their mutual coupling together influences the landing gear locked position. For a more detailed analysis of the effects of link lengths, it can be computed directly by using two-parameter continuation, thereby obtaining a two-parameter continuation of the point SN locus, as shown by the black line in Figure 11. Figure 11(a) is a 3D view, and Figure 11(b) is the projections of the black curve in the l2‐l4 plane. With changes in the parameters, the bifurcation point SN changes accordingly, which in turn affects the point S3 as well as the locked position.Figure 10
Landing gear mechanism corresponding to the vicinity of point SN.
(a)(b)Figure 11
Bifurcation point SN variation under two-parameter continuation.
(a)(b)From Figure11(b), it is clear that when the values of l2 and l4 locate at the left side of the black curve, points S1 and S3 are absent, implying that the landing gear cannot be locked at the S3 position. In contrast, when the values of l2 and l4 locate at the right side of black curve, points S1 and S3 are present, and the landing gear can be fully locked. Consequently, the right side of the curve is the feasible region of link lengths.The research on the SS landing gear suggests that the lock links must be folded down, in order to achieve full locking. For the SS landing gear, the downward folding motion of lock links will not be hindered due to the promotion by locking springs. In the case of the DS landing gear, however, the sidestay links on both sides may affect each other. Even with the beneficial effect of locking springs, it is highly likely that lock links on the one side undergo locking failure. Hence, the next section carries out research focusing on the DS landing gear.


## 4. Analysis of DS Landing Gear Mechanism

Compared to the SS landing gear, the DS landing gear will encounter difficulty in synchronous locking of fore and aft sidestays if the additional sidestay is unreasonably designed. An obvious conclusion is that if the fore and aft sidestays are fully symmetrical, their motion trajectories will completely overlap. However, since the ground load needs to be transferred separately to the fuselage and wing via the two sidestays, full symmetry of sidestays can hardly be achieved due to the difference in attachment points. In case the unreasonable design of structural parameters causes mutual restriction between the fore and aft sidestays, incomplete locking will be unavoidable. Thus, the motion coordination between the two sidestays appears particularly important.


### 4.1. Constraint Equations of DS Landing Gear

Figure12 plots the schematic diagrams of a DS landing gear mechanism. Clearly, the mechanism comprises nine links, which has thirty-six independent geometric variables, and thirty-five independent constraint equations, as shown in the appendix. Compared to the SS mechanism, the number of constraint equations increases by sixteen. The superscripts f and a represent the relevant parameters of fore and aft sidestays, respectively, and the definitions of other parameters are consistent with those of the SS landing gear.Figure 12
Schematic diagrams of the DS mechanism.


### 4.2. Singularity Analysis of DS Landing Gear

The retraction and extension processes of the DS landing gear can be simulated through numerically continuing the constraint equations, and the singular points and bifurcation curves of two sidestays can be derived, as shown in Figure13. The ordinate θoc denotes the difference between angles of lock links, which is θoc=θ4−θ5 for the fore lock links and θoc=θ8−θ9 for the aft lock links.Figure 13
Bifurcation curves of DS landing gear.Figure13 shows the relationship between θoc and the retraction angle, where the grey and black lines represent the motion trajectories of the fore and aft lock links, respectively. The fore sidestay has three singular points (S1f, S2f, and S3f). In contrast, the aft sidestay has only one singular point (S1a). It can be seen intuitively that the fore lock links can move to the area of θoc<0, while the aft lock links only move within the area of θoc>0. This indicates that the fore lock links can be locked, while the aft lock links cannot be locked. To better explain the motion of the studied landing gear, Figure 14 provides the kinematic sketches of mechanism that correspond to various points in Figure 13.Figure 14
Sketches of DS mechanism motions: (a) U1f(U1a); (b) U2f(U2a); (c) S1f(S1a); (d) S1f(A2a); (e) D1f(U1a); (f) D2f(U2a); (g) S3f(S1a).
(a)(b)(c)(d)(e)(f)(g)In Figure14, the grey lines represent the fore sidestay, whereas the black lines represent the aft sidestay. It is not difficult to find that the sketches of fore and aft sidestays are similar in Figures 14(a)–14(c). When the fore sidestay reaches the singular point S2f, the aft sidestay moves to point A2a, as shown in Figure 14(d). At this time, the fore lock links are collinear, while the aft lock links are still in the upper overcenter state. By the beneficial action of locking springs, the fore sidestay reaches the singular point S3f, while the aft sidestay can only return to the singular point S1a, as shown in Figure 14(g). As a result, only the fore sidestay is locked. Due to stop block, the landing gear does not move to the states shown in Figures 14(e) and 14(f) in reality.To explain the reason that the aft sidestay cannot be fully locked, numerical continuation independently on the fore and aft sidestays is performed, as shown in Figure15.Figure 15
Independent computational results of the fore and aft sidestays.According to Figure15, the motion trajectory of the aft sidestay also has three singular points when the two sidestays are computed separately, with S2a appearing later than S2f. When the landing gear reaches the point S1f(S2a), the main strut rotates to the farthest end, then begins to move backwards. At the time the fore sidestay reaches the point S2f, the main strut has moved backwards to the farthest position (Figure 14(d)). To allow the aft sidestay to reach the S2a point, the main strut needs to move continuously backwards to a position farther than S2f. However, such motion is restricted by the fore sidestay, thereby creating a contradiction in motion coordination. Precisely, this contradiction of mutual restriction between the fore and aft sidestays is the fundamental reason why the DS landing gear is unable to be fully locked. To achieve complete locking, the retraction angle corresponding to the singular points S2f and S2a must be identical (i.e., Θ1S2f=Θ1S2a).


### 4.3. Effects of Link Length Parameters on the Synchronous Locking

In this section, the effects of link lengths on the singular points S2f and S2a are analyzed. It can be seen from Figure 15 that θocf and θoca are equal to zero at the points S2f and S2a. Hence, when considering the effects of points S2f and S2a, the parameters θocf and θoca can be used as the fixed parameters (θocf=θoca=0), where Θ1 is used as the state parameter, and the link lengths are used as the control parameters. Figure 16 plots the analytical results.Figure 16
Curves describing the effects of link lengths on the second singular point.
(a)(b)In Figure16, the grey solid lines represent the influence of l2 and l4 on the singular point S2f, while the black dashed lines represent the influence of l6 and l8 on the singular point S2a. It is clear that the singular points exhibit generally consistent trends of variation with the link lengths. As described in the previous section, the synchronous locking can only be achieved when Θ1S2f=Θ1S2a. So once the value of Θ1 is determined (e.g., Θ1=0.1°), the link lengths can be obtained intuitively, as shown by the dotted lines in Figure 16. The effects of variations in other link lengths can be analyzed in a similar way, which are thus not detailed herein.Although changing the link lengths can yieldΘ1S2f=Θ1S2a, it also changes the θocf and θoca in a deployed state. Hence, a comprehensive consideration of the link lengths is needed based on multiple parameters.


### 4.4. Optimization of Synchronous Locking

In this section, an optimization is carried out to accomplish fully locking of the DS landing gear via the PSO, which is achieved by integrating ISIGHT with MATLAB for cosimulation analysis. Through optimization of link lengths, the fore and aft sidestays can be locked synchronously. PSO is a population-based optimization algorithm proposed by Kennedy and Eberhart [32] in 1995, who got inspiration from birds hunting for food. It initializes a system with a substantial number of random solutions and performs search for the optimal solution by updating generations. The PSO algorithm features very high probability of convergence to a global optimal solution, very fast computation, and strong global search ability. Existing research and applications indicate that PSO is a promising and effective optimization method [33–35]. Hence, the PSO algorithm is used for the optimization purpose in this paper.As suggested in the foregoing analysis,θocf and θoca should not be excessively large when the landing gear is locked. Thus, the invariant overcenter angles of the lock links is used as one of the constraints, i.e., θocfS3f=θocaS3f=2°. Let ΔΘ=Θ1S2f‐Θ1S2a; then, the synchronous locking of fore and aft sidestays necessitates ΔΘ=0. The smaller the value of ΔΘ, the lesser the mutual restraint effect between the fore and aft sidestays. Accordingly, the optimization objective is set as min ΔΘ. The optimization variables are the lengths of various links. Due to the constraints, there are only four independent optimization variables, i.e., lii=2,4,6,8.In Figure17, the cosimulation-based optimization process is displayed, where the operators include the ISIGHT algorithm module and the MATLAB numerical continuation module. As a first step, the ISIGHT generates a set of initial parameters li based on the PSO algorithm and then passes the parameters to the MATLAB module. Afterwards, the MATLAB calculates the bifurcation points of the fore and aft sidestays based on the numerical continuation, as well as ΔΘ, θocfS3f, and θocaS3f, which are then transferred back to the ISIGHT. Finally, the ISIGHT identifies the optimal results that conform to the constraints.Figure 17
Computational flow of cosimulation-based optimization.Since the change in link lengths leads to changed solutions of constraint equations, the starting point that satisfies the constraint equations is solved initially based on the PSO algorithm. To reduce the computational burden, the solution atΘ1=5° is chosen as the starting point. Afterwards, the singular points S2f and S2a are obtained through the parameter continuation analysis for reducing Θ1, and the corresponding Θ1 is identified. Finally, ΔΘ is solved.Based on the consideration of the mechanism motion interference, the optimization variables are set tol2∈450mm,650mm and l6∈450mm,650mm, and the lock link lengths are set to l4∈200mm,300mm and l8∈200mm,300mm. The optimization results are shown in Tables 1 and 2 and Figure 18.Table 1
Synchronous locking optimization results.
l2l4l6l8ΔΘBefore550 mm250 mm550 mm250 mm7.72e−08°Optimization532.4655 mm258.8501 mm538.6727 mm288.9462 mm2.24e−10°Table 2
Synchronous locking optimization results (other parameters).
l3l5l7l9Before550 mm260 mm550 mm260 mmOptimization567.5345 mm261.2052 mm561.3273 mm227.5027 mmFigure 18
Singular points of dual-sidestay links after optimization.According to the optimization results in Figure18, the points S2f and S2a are almost completely coincident, which allow simultaneous locking of the fore and aft sidestays, respectively, at S2f and S2a. When the two sidestays are in a locked state, the angle between lock links remains constant at 2°.


## 4.1. Constraint Equations of DS Landing Gear

Figure12 plots the schematic diagrams of a DS landing gear mechanism. Clearly, the mechanism comprises nine links, which has thirty-six independent geometric variables, and thirty-five independent constraint equations, as shown in the appendix. Compared to the SS mechanism, the number of constraint equations increases by sixteen. The superscripts f and a represent the relevant parameters of fore and aft sidestays, respectively, and the definitions of other parameters are consistent with those of the SS landing gear.Figure 12
Schematic diagrams of the DS mechanism.


## 4.2. Singularity Analysis of DS Landing Gear

The retraction and extension processes of the DS landing gear can be simulated through numerically continuing the constraint equations, and the singular points and bifurcation curves of two sidestays can be derived, as shown in Figure13. The ordinate θoc denotes the difference between angles of lock links, which is θoc=θ4−θ5 for the fore lock links and θoc=θ8−θ9 for the aft lock links.Figure 13
Bifurcation curves of DS landing gear.Figure13 shows the relationship between θoc and the retraction angle, where the grey and black lines represent the motion trajectories of the fore and aft lock links, respectively. The fore sidestay has three singular points (S1f, S2f, and S3f). In contrast, the aft sidestay has only one singular point (S1a). It can be seen intuitively that the fore lock links can move to the area of θoc<0, while the aft lock links only move within the area of θoc>0. This indicates that the fore lock links can be locked, while the aft lock links cannot be locked. To better explain the motion of the studied landing gear, Figure 14 provides the kinematic sketches of mechanism that correspond to various points in Figure 13.Figure 14
Sketches of DS mechanism motions: (a) U1f(U1a); (b) U2f(U2a); (c) S1f(S1a); (d) S1f(A2a); (e) D1f(U1a); (f) D2f(U2a); (g) S3f(S1a).
(a)(b)(c)(d)(e)(f)(g)In Figure14, the grey lines represent the fore sidestay, whereas the black lines represent the aft sidestay. It is not difficult to find that the sketches of fore and aft sidestays are similar in Figures 14(a)–14(c). When the fore sidestay reaches the singular point S2f, the aft sidestay moves to point A2a, as shown in Figure 14(d). At this time, the fore lock links are collinear, while the aft lock links are still in the upper overcenter state. By the beneficial action of locking springs, the fore sidestay reaches the singular point S3f, while the aft sidestay can only return to the singular point S1a, as shown in Figure 14(g). As a result, only the fore sidestay is locked. Due to stop block, the landing gear does not move to the states shown in Figures 14(e) and 14(f) in reality.To explain the reason that the aft sidestay cannot be fully locked, numerical continuation independently on the fore and aft sidestays is performed, as shown in Figure15.Figure 15
Independent computational results of the fore and aft sidestays.According to Figure15, the motion trajectory of the aft sidestay also has three singular points when the two sidestays are computed separately, with S2a appearing later than S2f. When the landing gear reaches the point S1f(S2a), the main strut rotates to the farthest end, then begins to move backwards. At the time the fore sidestay reaches the point S2f, the main strut has moved backwards to the farthest position (Figure 14(d)). To allow the aft sidestay to reach the S2a point, the main strut needs to move continuously backwards to a position farther than S2f. However, such motion is restricted by the fore sidestay, thereby creating a contradiction in motion coordination. Precisely, this contradiction of mutual restriction between the fore and aft sidestays is the fundamental reason why the DS landing gear is unable to be fully locked. To achieve complete locking, the retraction angle corresponding to the singular points S2f and S2a must be identical (i.e., Θ1S2f=Θ1S2a).


## 4.3. Effects of Link Length Parameters on the Synchronous Locking

In this section, the effects of link lengths on the singular points S2f and S2a are analyzed. It can be seen from Figure 15 that θocf and θoca are equal to zero at the points S2f and S2a. Hence, when considering the effects of points S2f and S2a, the parameters θocf and θoca can be used as the fixed parameters (θocf=θoca=0), where Θ1 is used as the state parameter, and the link lengths are used as the control parameters. Figure 16 plots the analytical results.Figure 16
Curves describing the effects of link lengths on the second singular point.
(a)(b)In Figure16, the grey solid lines represent the influence of l2 and l4 on the singular point S2f, while the black dashed lines represent the influence of l6 and l8 on the singular point S2a. It is clear that the singular points exhibit generally consistent trends of variation with the link lengths. As described in the previous section, the synchronous locking can only be achieved when Θ1S2f=Θ1S2a. So once the value of Θ1 is determined (e.g., Θ1=0.1°), the link lengths can be obtained intuitively, as shown by the dotted lines in Figure 16. The effects of variations in other link lengths can be analyzed in a similar way, which are thus not detailed herein.Although changing the link lengths can yieldΘ1S2f=Θ1S2a, it also changes the θocf and θoca in a deployed state. Hence, a comprehensive consideration of the link lengths is needed based on multiple parameters.


## 4.4. Optimization of Synchronous Locking

In this section, an optimization is carried out to accomplish fully locking of the DS landing gear via the PSO, which is achieved by integrating ISIGHT with MATLAB for cosimulation analysis. Through optimization of link lengths, the fore and aft sidestays can be locked synchronously. PSO is a population-based optimization algorithm proposed by Kennedy and Eberhart [32] in 1995, who got inspiration from birds hunting for food. It initializes a system with a substantial number of random solutions and performs search for the optimal solution by updating generations. The PSO algorithm features very high probability of convergence to a global optimal solution, very fast computation, and strong global search ability. Existing research and applications indicate that PSO is a promising and effective optimization method [33–35]. Hence, the PSO algorithm is used for the optimization purpose in this paper.As suggested in the foregoing analysis,θocf and θoca should not be excessively large when the landing gear is locked. Thus, the invariant overcenter angles of the lock links is used as one of the constraints, i.e., θocfS3f=θocaS3f=2°. Let ΔΘ=Θ1S2f‐Θ1S2a; then, the synchronous locking of fore and aft sidestays necessitates ΔΘ=0. The smaller the value of ΔΘ, the lesser the mutual restraint effect between the fore and aft sidestays. Accordingly, the optimization objective is set as min ΔΘ. The optimization variables are the lengths of various links. Due to the constraints, there are only four independent optimization variables, i.e., lii=2,4,6,8.In Figure17, the cosimulation-based optimization process is displayed, where the operators include the ISIGHT algorithm module and the MATLAB numerical continuation module. As a first step, the ISIGHT generates a set of initial parameters li based on the PSO algorithm and then passes the parameters to the MATLAB module. Afterwards, the MATLAB calculates the bifurcation points of the fore and aft sidestays based on the numerical continuation, as well as ΔΘ, θocfS3f, and θocaS3f, which are then transferred back to the ISIGHT. Finally, the ISIGHT identifies the optimal results that conform to the constraints.Figure 17
Computational flow of cosimulation-based optimization.Since the change in link lengths leads to changed solutions of constraint equations, the starting point that satisfies the constraint equations is solved initially based on the PSO algorithm. To reduce the computational burden, the solution atΘ1=5° is chosen as the starting point. Afterwards, the singular points S2f and S2a are obtained through the parameter continuation analysis for reducing Θ1, and the corresponding Θ1 is identified. Finally, ΔΘ is solved.Based on the consideration of the mechanism motion interference, the optimization variables are set tol2∈450mm,650mm and l6∈450mm,650mm, and the lock link lengths are set to l4∈200mm,300mm and l8∈200mm,300mm. The optimization results are shown in Tables 1 and 2 and Figure 18.Table 1
Synchronous locking optimization results.
l2l4l6l8ΔΘBefore550 mm250 mm550 mm250 mm7.72e−08°Optimization532.4655 mm258.8501 mm538.6727 mm288.9462 mm2.24e−10°Table 2
Synchronous locking optimization results (other parameters).
l3l5l7l9Before550 mm260 mm550 mm260 mmOptimization567.5345 mm261.2052 mm561.3273 mm227.5027 mmFigure 18
Singular points of dual-sidestay links after optimization.According to the optimization results in Figure18, the points S2f and S2a are almost completely coincident, which allow simultaneous locking of the fore and aft sidestays, respectively, at S2f and S2a. When the two sidestays are in a locked state, the angle between lock links remains constant at 2°.


## 5. Conclusion

A bifurcation theory-based method for analyzing the kinematics of sidestay landing gear locking mechanisms is proposed in this paper. Initially, the simple single-sidestay (SS) mechanism is analyzed, and then, the complex dual-sidestay (DS) mechanism is investigated based on SS. The main conclusions are as follows:(1)
Based on the bifurcation theory and singularity, the locked position of the SS landing gear is analyzed, clarifying that the lock links must be at the singular point of a lower overcenter angle, in order to achieve fully locking. Besides, the effect of link lengths on the locked position is analyzed, concluding that the link lengths should be within a feasible range(2)
For DS, the presence of an additional sidestay poses an adverse impact on the downward folding motion of lock links. The kinematic cause underlying the adverse impact is analyzed. Owing to the different positions of singular points, the movements of fore and aft sidestays seriously restrict each other, causing locking failure of the DS landing gear(3)
A kinematic optimization method for the synchronous locking of the DS landing gear is proposed. This study indicates that the full locking of the DS gear is achievable on condition that the overcenter angle of lock links is invariant.



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