In order to facilitate the subsequent analysis, we propose the following lemmas, explanations and hypotheses for the formula derivation of the research.

Remark 1

For clarify, the notation ( ∗ ) ′ = ∂ ( ∗ ) / ∂ t , ( ∗ ) ″ = ∂ ( ∗ ) / ∂ t , ( ∗ ) x = ∂ ( ∗ ) / ∂ x , ( ∗ ) x x = ∂ 2 ( ∗ ) / ∂ x 2 , ( ∗ ) x x x x = ∂ 4 ( ∗ ) / ∂ x 4 , ( ∗ ) x ′ = ∂ 2 ( ∗ ) / ∂ t ∂ x , ( ∗ ) x x ′ = ∂ 3 ( ∗ ) / ∂ t ∂ x 2

Lemma 1

The length of the flexible manipulator is L, let φ 1 ( x , t ) , φ 2 ( x , t ) ∈ R with x ∈ [ 0 , L ] and t ∈ [ 0 , ∞ ] . Then the following inequalities hold as (Rahn, 2002) | φ 1 ( x , t ) φ 2 ( x , t ) | ≤ 1 δ φ 1 2 ( x , t ) + δ φ 2 2 ( x , t ) (1)

Lemma 2

Let ϕ ( x , t ) ∈ R be a a function defined on x ∈ [ 0 , L ] and t ∈ [ 0 , ∞ ] . Then the following inequalities hold as (Rahn, 2002) { ∫ 0 L ϕ 2 ( x , t ) ≤ L 2 ∫ 0 L ϕ x 2 ( x , t ) d x ϕ 2 ( x , t ) ≤ L ∫ 0 L ϕ x 2 ( x , t ) d x (2)

Assumption 1

In this paper, the effect of gravity is ignored in the established physical model. Since both the rotational movement and elastic vibration of the flexible manipulator occurs to the horizontal plane, and the length of the mechanical manipulator is much larger than its cross-sectional width and height, it is assumed to be an Eider-Bernoulli beam.

As shown in Figure 1, X 0 O Y 0 is defined as the inertial coordinate of the system, the coordinate system X 1 O Y 1 is a follow-up coordinate system fixed on the flexible manipulator, and the X ₁ axis is always tangent to the root of the flexible manipulator. The offset of the flexible manipulator during the movement is z ( x , t ) , and the displacement of the flexible manipulator is z ( x , t ) = y ( x , t ) + x ⋅ θ ( t ) , where y(x, t) and θ(t) respectively indicate the elastic deflection of the flexible manipulator and the angular position of the flexible manipulator.

The kinetic energy E k of the manipulator includes the rotational kinetic energy E k 1 of the central rigid body, the kinetic energy E k 2 of the flexible manipulator and the kinetic energy E k 3 of the mass of the flexible manipulator. The relationship between them can be given as E k = E k 1 + E k 2 + E k 3 , E k 1 , E k 2 , E k 3 can be written as { E k 1 ( t ) = 1 2 I h θ ′ ( t ) 2 E k 2 ( t ) = 1 2 ∫ 0 L ρ L A ( x ) z ′ 2 ( x , t ) d x E k 3 ( t ) = 1 2 m z ′ 2 ( L , t ) (3) Where I h represents the moment of inertia of the central rigid body, ρ L is the density of the flexible manipulator, A ( x ) is the cross-sectional area of the flexible manipulator, which also changes with the length x. θ ( t ) represents the actual rotation angle of the flexible manipulator, and z ( x , t ) is the absolute displacement of t the flexible manipulator in the X 0 O Y 0 coordinate system.

The potential energy Ep of the flexible arm system is expressed as follows E p ( t ) = 1 2 ∫ 0 L E L I ( x ) y x x 2 ( x , t ) d x + 1 2 T L ∫ 0 L [ y x ( x , t ) ] 2 d x (4) Where E L is the elastic modulus of the flexible manipulator, T L is tension of the flexible manipulator, and I ( x ) is the moment of inertia that changes with the length x. y ( x , t ) is the elastic vibration displacement of the flexible arm in the X 1 O Y 1 coordinate system.

The control moment u(x,t) is added to drive the flexible manipulator system, F(L,t) is the input torque of the mass of the flexible manipulator, and its non-conservative force work W c ( t ) can be expressed as W c ( t ) = u ( x , t ) ⋅ θ ( t ) + F ( L , t ) z ( L , t ) (5)

The continuous mass distribution and continuous stiffness distribution characteristics of the flexible manipulator are considered. According to Hamilton’s principle Eq. 37, the variational equation of the flexible manipulator is defined as ∫ t 1 t 2 ( δ E k ( t ) − δ E p ( t ) + δ W c ( t ) ) d t = 0 , ∀ ( x , t ) ∈ ( 0 , L ) × [ 0 , t max ] (6)

Combining Eqs 3, 6, the variational formula for the kinetic energy E k ( t ) of the flexible manipulator is simplified based on the rule of integration as ∫ t 1 t 2 δ E k ( t ) d t = − ∫ t 1 t 2 I h θ ′′ ( t ) δ θ ( t ) d t − ∫ t 1 t 2 ∫ 0 L ρ L A ( x ) z ′′ ( x , t ) δ z ( x , t ) d x d t − ∫ t 1 t 2 m z ′′ ( L , t ) δ z ( L , t ) d t (7)

Combining Eqs 4, 6, the variational formula for the potential energy E p ( t ) of the flexible manipulator is simplified based on the rule of integration as ∫ t 1 t 2 δ E p ( t ) d t = ∫ t 1 t 2 [ T L y x ( L , t ) − E L I ( x ) y x x x ( L , t ) ] δ y ( L , t ) d t + ∫ t 1 t 2 [ E L I ( x ) y x x x ( 0 , t ) − T L y x ( 0 , t ) ] δ y ( 0 , t ) d t + E L I ( x ) ∫ t 1 t 2 y x x ( L , t ) δ y x ( L , t ) d t + ∫ t 1 t 2 ∫ 0 L [ E L I ( x ) y x x x x ( x , t ) − T L y x x ( x , t ) ] δ y ( x , t ) d x d t (8)

Combining Eqs 5, 6, the variational formula for the non-conservative force work W c ( t ) of the flexible manipulator is simplified based on the rule of integration as ∫ t 1 t 2 δ W c d t = δ ∫ t 1 t 2 ( u ( x , t ) θ ( t ) d t + F ( x , t ) z ( L , t ) ) d t (9)

Based on Eqs 6, 7, the boundary conditions of the flexible manipulator are processed. The central rigid body is the fixed end, and its elastic displacement and elastic angular displacement are both zero. The end of the flexible manipulator is in a free state, and its bending moment and shear force are both zero. Then the boundary conditions of the flexible manipulator can be obtained as { y ( 0 , t ) = 0 y x ( 0 , t ) = 0 z ′′ x ( 0 , t ) = θ ′′ ( t ) E L I ( L ) y x x ( L , t ) = 0 m z ′′ ( L , t ) + T L y x ( L , t ) − E L I ( L ) y x x x ( L , t ) − F ( x , t ) = 0 (10)

Property 1: If ∀ ( x , t ) ∈ ( 0 , L ) × [ 0 , t max ) , the kinetic energy of the system described by Eq. 3 is bounded, then the state z ′ ( x , t ) , z ′ x ( x , t ) , z ′ x x ( x , t ) , z ′ x x x ( x , t ) related to it is also bounded in the corresponding range.

Property 2: If ∀ ( x , t ) ∈ ( 0 , L ) × [ 0 , t max ) , the kinetic energy of the system described by Eq. 3 is bounded, then the state y x x ( x , t ) , y x x x ( x , t ) , y x x x x ( x , t ) related to it is also bounded in the corresponding range.

According to the analytical model without external disturbance, the following variable-stiffness flexible manipulator system with governing equation as E L I ( x ) ⋅ z x x x x ( x , t ) + ρ L A ( x ) ⋅ y ′′ ( x , t ) = T L ⋅ y x x ( x , t ) (11) Where E L is the elastic modulus of the flexible manipulator, T L is tension of the flexible manipulator, and I ( x ) is the moment of inertia that changes with the length x.

The control objective is to build a distributed control u(t) to ensure that system state y(x, t) can track the variable reference trajectory θ d ( t ) without violation of desired constraint. The force balance relationship of the boundary can be expressed as u ( t ) = I h θ ′′ ( t ) − E L I y x x ( 0 , t ) − T L y ( L , t ) (12)

A nonlinear boundary input is applied to the end of the manipulator, and control is performed at the end of the manipulator to adjust the vibration of the manipulator, so that the system tends to stabilize faster. According to the boundary Eq. 10, the nonlinear boundary input F(x,t) can be obtained as F ( x , t ) = m z ′′ ( L , t ) + T L y x ( L , t ) − E L I ( L ) y x x x ( L , t ) (13)

When the kinetic energy, the potential energy of the flexible manipulator, and the kinetic energy of the mass are the smallest, the elastic deformation y(x, t) of the flexible manipulator is the smallest. Through considering tracking error and tracking error rate of change, the Lyapunov function is constructed as V ( t ) = V 1 ( t ) + V 2 ( t ) + V 3 ( t ) (14) Where V 1 ( t ) is the sum of the kinetic energy and potential energy of the flexible manipulator, and represents an index for restraining the bending deformation and bending change rate of the flexible manipulator. V 2 ( t ) represents the control error index and the kinetic energy of the mass. V 3 ( t ) is the cross auxiliary term. Then V 1 ( t ) , V 2 ( t ) and V 3 ( t ) are defined as { V 1 ( t ) = ( ∫ 0 L ρ L A ( x ) z ' 2 ( x , t ) d x + E L I ( x ) ∫ 0 L y x x 2 ( x , t ) d x + T L ∫ 0 L y x 2 ( x , t ) d x ) / 2 V 2 ( t ) = ( I h e ' ( t ) 2 + k 1 e ( t ) 2 + m z ' 2 ( L , t ) ) / 2 V 3 ( t ) = α A ( x ) ∫ 0 L y ( x , t ) z ' ( x , t ) d x + β A ( x ) ∫ 0 L ( L − x ) y x ( x , t ) z ' ( x , t ) d x (15) Where k ₁ is the gain related to the controller, and k 1 > 0 , α > 0 , β > 0 . Then the vibration suppression control strategy design is shown in Figure 2.

In the absence of external disturbances, the controller u(t) is designed to suppress vibration and track the trajectory θ d ( t ) . Combining the control equations, boundary conditions and candidate Lyapunov function, through calculation and deduction, the controller of the variable-stiffness flexible manipulator system is designed as u ( t ) = − k 1 ( θ ( t ) − θ d ( t ) ) − k 2 ( θ ' ( t ) − θ ' d ( t ) ) − k 3 y ( L , t ) (16) Where k ₂ is the gain related to the controller, and k 2 > 0 , k 3 > 0 .

Lemma 3

The boundedness of the Lyapunov function Eq. 15 is given as 0 ≤ λ 1 ( V 1 ( t ) + V 2 ( t ) ) ≤ V ( t ) ≤ λ 2 ( V 1 ( t ) + V 2 ( t ) ) (17) where λ 1 , λ 2 > 0 .

Proof

For the cross-phase Eq. 15, the following inequality can be obtained as | V 3 ( t ) | ≤ α A ( x ) ( ∫ 0 L y 2 ( x , t ) d x + ∫ 0 L z ' 2 ( x , t ) d x ) + β A ( x ) L ( ∫ 0 L y x 2 ( x , t ) d x + ∫ 0 L z ' 2 ( x , t ) d x )          ≤ ( α A ( x ) L 2 + β A ( x ) L ) ∫ 0 L y x 2 ( x , t ) d x + ( α A ( x ) + β A ( x ) L ) ∫ 0 L z ' 2 ( x , t ) d x          ≤ ξ 1 V 1 ( t ) (18) Where ξ 1 = max { α L 2 + β L T L , α + β L ρ } .

Then V 3 ( t ) can be obtained as 0 ≤ ( 1 − ξ 1 ) V 1 ( t ) + V 2 ( t ) ≤ V ( t ) ≤ ( 1 + ξ 1 ) V 1 ( t ) + V 2 ( t ) (19)

Therefore, V(t) can be obtained as 0 ≤ λ 1 ( V 1 ( t ) + V 2 ( t ) ) ≤ V ( t ) ≤ λ 2 ( V 1 ( t ) + V 2 ( t ) ) (20) Where λ 1 = min ( 1 − ξ 1 , 1 ) and λ 2 = max ( 1 + ξ 2 , 1 ) are two positive constants.

Lemma 4

The time derivative of the Lyapunov function Eq. 14 is proved to be bounded as V ( t ) ≤ − λ V ( t ) (21)

Proof

Differentiating Eq. 14 with respect to time, V ′ ( t ) is obtained as V ′ ( t ) = V ′ 1 ( t ) + V ′ 2 ( t ) + V ′ 3 ( t ) (22)

The error information e ( t ) , e ′ ( t ) , e ″ ( t ) of the angle can be obtained as { e ( t ) = θ ( t ) − θ d ( t ) e ′ ( t ) = θ ′ ( t ) − θ ′ d ( t ) e ″ ( t ) = θ ″ ( t ) − θ ″ d ( t ) (23)

Substituting boundary condition Eqs 10–15 into Eq. 22, V ′ 1 ( t ) can be obtained as V ′ 1 ( t ) = − E L I ′ ( x ) z ′ ( L , t ) y x x x ( L , t ) − E L I ′ ( x ) y x x ( 0 , t ) θ ′ ( t ) + T L ∫ 0 L θ ′ ( t ) x y x x ( x , t ) d x (24)

Then V ′ 2 ( t ) can be obtained as V ′ 2 ( t ) = e ′ ( t ) [ I h e ′′ ( t ) + k 1 e ( t ) ] + z ′ ( L , t ) m z ′′ ( L , t ) (25)

Then V ′ 3 ( t ) can be obtained as V ' 3 ( t ) = α A ′ ( x ) ∫ 0 L [ y ′ ( x , t ) z ′ ( x , t ) + y ( x , t ) z ′′ ( x , t ) ] d x + β A ′ ( x ) ∫ 0 L ( L − x ) [ y ′ x ( x , t ) z ′ ( x , t ) + y x ( x , t ) z ′′ ( x , t ) ] d x (26)

Combining Eqs 12, 16, 22, 24–26, based on Lemma 1 and Lemma 2, V ′ ( t ) can be obtained as V ′ ( t ) ≤ − γ 1 ∫ 0 L z ' 2 ( x , t ) d x − γ 2 ∫ 0 L y x x 2 ( x , t ) d x − k 1 e 2 ( t ) − γ 3 e ' 2 ( t ) − γ 4 y x x 2 ( 0 , t ) − γ 5 ∫ 0 L y x 2 ( x , t ) d x (27) Where { γ 1 = − β / 2 − β L σ 5 − L β σ 6 − α − α L / σ 7 γ 2 = E I ( x ) α / ρ - T L L 3 / σ 1 − ( T − k 3 ) L 3 / σ 4 − 3 β E L I ( x ) / 2 ρ γ 3 = k 2 − T σ 1 − E L I ( x ) / σ 2 − β L / σ 5 − α L 2 σ 7 γ 4 = β E L I ( x ) L / 2 ρ − E L I ( x ) / σ 3 − α E L I ( x ) σ 2 γ 5 = α / ρ − β T / 2 ρ (28)

The parameters are chosen appropriately to make sure that σ n > 0 , n = 1–6.

Then combining Lemma 3 and Eq. 27, V ′ ( t ) can be obtained as V ′ ( t ) ≤ − λ 3 [ V 1 ( t ) + V 2 ( t ) ] ≤ − λ V ( t ) (29) where λ 3 = min { 2 γ 1 ρ , 2 γ 2 E L I ( L ) , 2 k 1 k 1 + k 2 } , and λ = λ 3 / λ 2 > 0 .

With Lyapunov direct method and based on Lemmas 1 and 2, the stability of the system with the proposed control law is analyzed. According to the analysis result, it can be found that the control system is a closed loop system and the system is stable. When the appropriate control gain parameters are selected, the system vibration state and angle tracking error will eventually converge. So as to achieve the purpose of restraining the elastic vibration during the movement towards the system and driving the arm of a predetermined angle.

In order to explore the superiority in the variable-stiffness manipulator, the control effects of two different rigid-flexible coupling manipulator models can be compared. The flexible robotic manipulators connected with a cantilever manner on the central rigid body include a uniform-stiffness robotic manipulator and a variable-stiffness flexible manipulator. The length and quality of the manipulator remains equal. The cross-section height H of the uniform-stiffness flexible manipulator is 2 . 5 × 10 − 3  m and the width B is 5 × 10 − 2  m. The elastic displacement changes of the middle and end of the uniform and variable-stiffness flexible manipulator are shown in Figure 4.

With the same controller and the same material properties of the manipulator, the maximum amplitude of the end of the variable-stiffness manipulator is 0.058 m, while the maximum amplitude of the end of the uniform-stiffness manipulator is 0.065 m. It can be found that the maximum elastic displacement of the variable stiffness manipulator is smaller.

According to the dynamic model of the variable-stiffness flexible manipulator system, it can be seen that the elastic vibration of the flexible manipulator is related to the joint angular displacement, angular velocity and angular acceleration. The trajectory planning of the flexible manipulator is the process of moving from the initial state to the target state within a certain period of time. According to the dynamic equation of the flexible manipulator system, the relationship between elastic vibration and motion trajectory is established. Therefore, the vibration suppression control of the flexible manipulator system can be performed by the method of joint trajectory planning, and the residual vibration of the system can be reduced.

In order to avoid excessive elastic vibration during the movement towards the flexible manipulator system, not only the trajectory of the system is required to be continuous, but also the first and second derivatives of the trajectory function are also continuous. When trajectory planning is carried out, the following three conditions must be met on the premise of meeting the specified time for the starting position on the target position:

1) The trajectory is smooth and continuous and maintains a monotonous increase or decrease.

At present, the common motion trajectory curves that meet the above constraints mainly include polynomial of degree five, cycloid and exponential functions (Biagiotti and Melchiorri, 2009). The polynomial of degree five, cycloid and exponential functions were taken as the motion trajectory, and compare the elastic vibration generated by the flexible manipulator under different motion trajectories. { θ 1 ( t ) = ( θ b − θ 0 ) [ 6 ( t T E ) 5 − 15 ( t T E ) 4 + 10 ( t T E ) 3 ] + θ 0 θ 2 ( t ) = ( θ b − θ 0 ) [ t T E − 1 2 π sin ( 2 π t T E ) ] + θ 0 θ 3 ( t ) = ( θ b − θ 0 ) [ 1 2 + v c ∫ 0 τ e − σ ⋅ sin 2 ⁡ π τ | cos ⁡ π τ | λ d τ ] + θ 0   (31) Where T E is the trajectory movement time, σ and λ are free parameters, and τ = ( t T E − 0.5 ) .

It is assumed that t b = 20 s , T E = 5 s , θ 0 = 0 , and θ b = 0.5 , the joint angular displacement, angular velocity and angular acceleration of the flexible manipulator under different trajectories can be obtained. As shown in Figure 5, the designed trajectory meets the above constraints.

Since there is no design movement trajectory, as shown in Figure 6, it can be found that the variable-stiffness flexible manipulator system obviously has greater vibration, where the deflection of the variable-stiffness flexible manipulator system reaches to 0.058 m. The system reached a steady state after about 10 s.

The central rigid body joint performs trajectory tracking based on the designed controller according to the trajectory Eq. 31, and the corresponding joint angle and acceleration changes can be obtained. As shown in Figure 7, the greater the maximum acceleration of the motion trajectory, the greater the fluctuation of the joint angle. The smaller the acceleration when near the target position, the faster the system will stabilize.

The elastic displacement changes of the middle and end of the variable-stiffness flexible manipulator under the designed trajectory are shown in Figure 8. The flexible manipulator has large elastic vibrations in the process of following the trajectory, and there is still a certain degree of residual vibration after the movement. Under different motion trajectories, the elastic vibration changes of the flexible mechanical manipulator are different. The greater the maximum acceleration of the motion trajectory, the greater the maximum elastic displacement generated. The smaller the acceleration when near the target position, the smaller the residual vibration displacement. When there is no trajectory planning, a large elastic displacement will be produced during the movement, and vibration will be produced during the movement, especially after the movement, the vibration takes a long time to recover to a stable state. This situation not only reduces the stability of the system, but also shortens the service life of the flexible manipulator. According to the comparison results, the maximum elastic displacement of the end under the fifth-order polynomial motion trajectory, cycloid motion trajectory and exponential motion is 0.040, 0.044 and 0.056 m. It can be found that the maximum elastic displacement of the flexible manipulator is the smallest when the movement is planned according to the polynomial of degree five trajectory.

As shown in Figure 9, the distributed elastic deformation and change rate of the manipulator after the polynomial of degree five trajectory movement can be obtained. It can be found that the vibration of the variable-stiffness flexible manipulator system is obviously reduced, where the maximum deflection rate of the variable-stiffness flexible manipulator system reaches to 0.028 m. The system reached a steady state after about 5 s.

In view of the characteristics of the flexible manipulator system, considering the conservation of energy, the non-conservative force is used to express the total energy consumed by the system during the movement. In order to measure the amount of elastic vibration of the flexible manipulator during the movement and after the movement, a suppression indexed including the elastic displacement of the end during the movement and the residual vibration displacement of the end after the movement is proposed. Therefore, the objective function can be obtained as J = r 1 ∫ 0 4 T E | u ( x , t ) θ ′ ( t ) + F ( x , t ) z ′ ( L , t ) | d t + r 2 ∫ 0 4 T E y T ( l L , t ) y ( l L , t ) d t + r 3 ∫ 0 4 T E | d i s ( t ) | d t (32) Where y ( l L , t ) is the elastic displacement of the end of the flexible manipulator, r 1 , r 2 and r 3 are the weighting factors of the three terms in the formula, and r 1 + r 2 + r 3 = 1 . u(x,t) is the control input signal, and dis(t) is the distance between the actual trajectory and the ideal trajectory. When trajectory planning is performed to ensure that y ( l L , t ) is minimum, the purpose of vibration suppression of the flexible manipulator system can be achieved through trajectory planning.

The differential evolution (DE) optimization algorithm is a bionic intelligent algorithm that simulates the biological evolution mechanism of nature. The realization mechanism is to randomly reorganize the temporary individuals generated by the individual differences in the population to complete the population evolution. However, the mutated individuals are selected randomly, which increases the randomness of the algorithm, which leads to randomness in the optimization direction, and reduces the convergence speed. The cloud adaptive differential evolution (CADE) optimization algorithm uses cloud mutation operation, and new individuals are generated near the best individuals produced by the previous generation, which not only improves the convergence speed, but also maintains the characteristics of the best individuals. To ensure the randomness of basic mutation operations and the stable tendency of cloud mutation operations, the two methods are combined to perform mutation operations. The optimization algorithm has stronger robustness and convergence, and has a good effect on solving numerical optimization problems. The CADE algorithm flow chart is shown in Figure 10.

In order to obtain the optimal trajectory, the ideal trajectory (polynomial of degree five) is optimized to minimize the objective function Eq. 32 based on the CADE optimization algorithm. The algorithm introduces a cloud model that can generate cloud droplets with a stable tendency. During the evolution process, it can target the optimal individual, perform adaptive positioning of the global optimal solution, and improve the convergence speed. The mutation operation of the CADE algorithm is completed by the normal cloud generator, and the mutation factor and crossover factor are adaptively adjusted during the evolution process to ensure the diversity of new individuals in the early stage and the convergence in the later stage.

After the ideal trajectory is optimized by the CADE algorithm, a set of best deviations can be obtained, and then the best discrete trajectory is obtained as θ ¯ o p = [ θ ¯ o p , 0 , θ ¯ o p , 0 , ... , θ ¯ o p , 2 n − 1 , θ ¯ o p , 2 n ] (33)

In order to obtain the continuous optimal trajectory, the cubic spline interpolation method is used to interpolate the discrete trajectory. The interpolation condition can be defined as { θ o p ( 0 ) = θ ¯ o p , 0 = θ 0 θ o p ( T E ) = θ ¯ o p , 2 n = θ d θ ′ o p ( 0 ) = θ ′ 0 = 0 θ ′ o p ( T E ) = θ ' d = 0 (34) Where the interpolation node is θ o p ( t j ) = θ ¯ o p , j ,    t j = j 2 n T E ,     j = 1,2 , ... , 2 n − 1 (35)

The continuous function obtained by interpolation is used as the optimal trajectory of the joint. The designed controller Eq. 16 is used to track the optimal trajectory. As shown in Figure 11, the optimal vibration suppression trajectory curve and its speed curve can be obtained through optimization. Through comparing with the ideal trajectory, it can be seen that the optimized trajectory and its speed after optimization meet the boundary constraints, so the optimized trajectory meets the vibration suppression requirements. By comparing with the ideal motion trajectory, it can be seen that the maximum speed of the optimized vibration suppression trajectory are less than that of the ideal trajectory (polynomial of degree five).

Then the effect of trajectory vibration suppression after unverified optimization can be obtained. As shown in Figure 12, it can be seen that the maximum elastic displacement of the end of the flexible manipulator under the optimized vibration suppression trajectory is 0.033 m, the maximum elastic deflection stabilizes after 4 s.

The vibration suppression effect after the optimized trajectory and the ideal trajectory are compared. As shown in Figure 13, it can be seen that he flexible manipulator moves under the optimized trajectory, and the elastic displacement of the end is smaller than the ideal trajectory during the movement, and the maximum elastic displacement ratio between the optimal trajectory and the ideal trajectory is 0.8:1. Compared with the elastic vibration during the movement of the flexible manipulator, the vibration at the end of the flexible manipulator is suppressed to a greater extent of the optimal vibration suppression trajectory. After the movement is completed, the flexible manipulator can quickly return to stable state, which is 1 s faster than the ideal trajectory.