A Flexible Manipulator Trajectory Tracking Control Method for a Predetermined Time

Abstract

A flexible robotic arm trajectory tracking control method for a predetermined time, which is used to control the trajectory tracking control and vibration problems of the flexible robotic arm; First, a Fourier trajectory planning function is designed for the flexible robotic arm joint system model, which belongs to an open-loop control and can eliminate the error of the residual vibration of the robotic arm by adjusting the coefficient parameters; On this basis, considering the inner dynamic loop of the motor, a nonsingular predetermined-time sliding surface is proposed. Finally, based on the proposed sliding surface, a predetermined-time sliding mode controller is designed to make the motor trajectory track the motor trajectory planned by Fourier within a predetermined time; This control method is not affected by the initial values of the system and does not require trial-and-error parameter tuning. The control design is simple and the performance is more stable and reliable.

Description

Technical Field

[0001] The present invention relates to the field of control technologies, and in particular to a trajectory tracking control method for a flexible robotic arm at a predetermined time. Background Art

[0002] Due to characteristics such as flexibility, rapidity, and accuracy, flexible joint robotic arms are widely used in industries such as medical treatment, aerospace, and machinery, as well as in social practices. However, they can generate difficult-to-control vibration problems, thereby reducing the operating accuracy and production efficiency of the robotic arms. Through research, it has been found that trajectory planning is one of the core technologies for controlling vibrations of robotic arms. Trajectory planning is an effective open-loop control method for suppressing vibrations. Common trajectory planning functions include cubic interpolation and quintic interpolation polynomials, and there are also some composite functions and other functions. Trajectory planning functions need to satisfy certain initial conditions. Cubic polynomials can ensure the smooth continuity of joint angles and angular velocities, but cannot satisfy the boundary conditions of accelerations; quintic polynomials can satisfy the boundary conditions of angles, angular velocities, and angular accelerations, but contain high-order functions and are prone to causing vibrations.

[0003] Finite-time sliding mode controllers are closely related to the initial conditions of the system and cannot control systems with unknown initial conditions. To overcome the difficulties faced by finite time, a fixed-time stability method has been proposed, which can be controlled to a stable state without being affected by the initial values of the system. However, it still requires multiple parameter adjustments to achieve the best control effect.

[0004] Patent application CN108789418A discloses a control method for a flexible robotic arm, which combines a trajectory tracking controller and a vibration suppression controller, and can achieve trajectory tracking and vibration suppression of the flexible robotic arm. First, a sliding mode variable structure controller is designed to track the robotic arm trajectory; then, relevant algorithms are used for parameter optimization to suppress vibrations. However, the algorithm implementation has many and complex parameters. Summary of the Invention

[0005] In order to solve the defects existing in the above-mentioned prior art, the purpose of the present invention is to provide a trajectory tracking control method for a flexible robotic arm at a predetermined time, and propose Fourier trajectory planning, which belongs to open-loop control. Compared with closed-loop control, the control design is simple, the performance is more stable and reliable, and it can enable the flexible robotic arm to quickly track the planned trajectory during movement and will not generate chattering phenomena after movement.

[0006] In order to achieve the above purpose, the technical solution adopted by the present invention is as follows:

[0007] A trajectory tracking control method for a flexible robotic arm at a predetermined time, the specific steps are as follows:

[0008] Step 1: Based on the complex dynamic behavior of the flexible manipulator, assuming that the flexible joint is a linear spring, establish the dynamic model of the motor rotor, the dynamic model of the joint end, and the error tracking dynamic model;

[0009] Step 2: Without considering the inner dynamics of the motor, design a Fourier trajectory planning for the manipulator and use an open-loop control method to eliminate chattering;

[0010] Step 3: On the basis of Steps 1 and 2, considering the inner dynamics of the motor, in order to make the motor trajectory track the planned Fourier trajectory, design a nonsingular prescribed-time sliding mode control, and finally make the position tracking error of the manipulator converge to zero within the prescribed time.

[0011] The specific method of Step 1 is as follows:

[0012] Based on the complex dynamic behavior of the flexible manipulator, assuming that the flexible joint is a linear spring, the dynamic model of the motor rotor and the dynamic model of the joint end are established as follows

[0013]

[0014] where, θ represents the rotation angle of the joint, q represents the motor angle, τ m represents the driving torque input by the motor, K represents the torsional stiffness coefficient, J l represents the total inertia at the joint end, τ f represents the viscous friction torque, J m represents the inertia of the motor rotor;

[0015] Let x1 = θ, x3 = q, Express the differential equation form of Equation (1) as the state equation form

[0016]

[0017] Assume that the desired trajectory of the manipulator motor is The error between the motor trajectory and the desired trajectory is expressed as

[0018]

[0019] Then, define The error dynamic equation of the flexible manipulator is defined as

[0020]

[0021] The specific method of Step 2 is as follows:

[0022] Without considering the internal dynamics of the motor and only considering the impact of trajectory planning on the flexible joint, the manipulator system model for trajectory planning is

[0023]

[0024] Let Equation (5) can be expressed as:

[0025]

[0026] The trajectory planning of the manipulator satisfies the following boundary conditions:

[0027]

[0028] Under the influence of trajectory planning theory and initial values, the trajectory planning function based on Fourier series is designed as follows:

[0029]

[0030] Wherein, represents the expected trajectory of the joint, ω n = nπ / t f , n = 1, 2, 3, q f , q0 is the initial position and the target position of the joint, t f is the time of trajectory planning.

[0031] The specific method of step 3 is as follows:

[0032] Considering the internal dynamics of the motor, in order to enable the motor to track the planned Fourier trajectory within a predetermined time, the predetermined-time sliding surface of Equation (2) is designed as follows:

[0033]

[0034] Where T s is the predetermined-time parameter, 0 < α < 1, and the non-singular function is defined as

[0035]

[0036] The designed non-singular function is differentiable. Therefore, The derivative of is expressed as:

[0037]

[0038] Where And δ = α,

[0039] According to the sliding surface, i.e., Equation (10), the controller is designed as:

[0040]

[0041] where 0 < α s < 1.

[0042] Compared with the prior art, the present invention has the following advantages:

[0043] (1) Compared with the fifth-order interpolation function trajectory planning, the proposed Fourier can eliminate the residual vibration error of the flexible manipulator.

[0044] (2) The Fourier trajectory planning belongs to open-loop control. Compared with closed-loop control, the control design is simple and the performance is more stable and reliable.

[0045] (3) On this basis, considering the inner dynamic loop of the motor, a new non-singular prescribed-time sliding surface is proposed, which can make the motor trajectory track the Fourier trajectory planning within a prescribed time. BRIEF DESCRIPTION OF THE DRAWINGS

[0046] Figure 1 It is a system model diagram of the flexible joint manipulator of the present invention.

[0047] Figure 2 It is a schematic diagram of the trajectory planned by the motor of the present invention.

[0048] Figure 3 It is a schematic diagram of the joint and motor trajectory errors of the present invention.

[0049] Figure 4 It is a simulation diagram of the sliding surface s of the present invention.

[0050] Figure 5 It is a simulation diagram of the controller u of the present invention.

[0051] Figure 6 It is a simulation diagram of the error e1 of the present invention.

[0052] Figure 7 It is a simulation diagram of the error e2 of the present invention.

[0053] Figure 8 It is a simulation diagram of the motor trajectory q tracking the desired trajectory of the present invention.

[0054] Figure 9 It is a flow chart of the implementation steps of the present invention. DETAILED DESCRIPTION OF THE INVENTION

[0055] The following further describes the present invention in detail with reference to the drawings.

[0056] A method for tracking the trajectory of a flexible manipulator with a prescribed time specifically includes the following steps:

[0057] Step 1. Due to the complex dynamic behavior of the flexible manipulator, assuming that the flexible joint is a linear spring, establish the dynamic model of the motor rotor, the dynamic model of the joint end, and the error tracking dynamic model:

[0058] Figure 1 The system model of the flexible joint manipulator is shown. Due to the complex dynamic behavior of the flexible manipulator, assuming that the flexible joint is a linear spring, the dynamic model of the motor rotor and the dynamic model of the joint end are established as follows:

[0059]

[0060] where, θ represents the rotation angle of the joint, q represents the motor angle, τ m represents the driving torque input by the motor, K represents the torsional stiffness coefficient, which can usually be obtained through experimental methods, J l represents the total inertia at the joint end, τ f represents the viscous friction torque, J m represents the inertia of the motor rotor;

[0061] Let x1 = θ, x3 = q, Express the differential equation form of Equation (1) as the state equation form

[0062]

[0063] Assume that the desired trajectory of the manipulator motor is The error between the motor trajectory and the desired trajectory can be expressed as:

[0064]

[0065] Then, define The error dynamic equation of the flexible manipulator can be defined as:

[0066]

[0067] Step 2: Without considering the internal dynamics of the motor, design a Fourier trajectory planning for the manipulator and use an open-loop control method to eliminate chattering:

[0068] Without considering the internal dynamics of the motor, only consider the influence of the trajectory planning on the flexible joint. The manipulator system model of the trajectory planning is:

[0069]

[0070] Let Equation (13) can be expressed as:

[0071]

[0072] The trajectory planning of the robotic arm satisfies the following boundary conditions:

[0073]

[0074] Under the influence of the trajectory planning theory and the initial value, the trajectory planning function based on the Fourier series is designed as follows:

[0075]

[0076] where represents the expected trajectory of the joint, ω n = nπ / t f , n = 1, 2, 3, q f , q0 is the initial position and the target position of the joint, t f is the time of trajectory planning.

[0077] Taking the first derivative and the second derivative of formula (8) can obtain the angular velocity and angular acceleration of the motor trajectory

[0078]

[0079]

[0080] Combining formulas (7), (8), (8-1) and (8-2) can obtain:

[0081]

[0082] The parameters α0, α1, α3, α5 can be designed as:

[0083]

[0084] Step 3: On the basis of Step 1 and Step 2, considering the internal dynamics of the motor, in order to make the motor trajectory track the planned Fourier trajectory, a non-singular prescribed-time sliding mode control is designed, and finally the position tracking error of the robotic arm converges to zero within the prescribed time:

[0085] Considering the internal dynamics of the motor, in order to make the motor track the planned Fourier trajectory within the prescribed time, the prescribed-time sliding mode surface of formula (2) can be designed as follows:

[0086]

[0087] where T s is the prescribed-time parameter, 0 < α < 1. The non-singular function is defined as:

[0088]

[0089] The designed non-singular function is differentiable. Therefore, The derivative of

[0090]

[0091] can be expressed as: and δ = α.

[0092] Taking the derivative of Equation (9) gives

[0093]

[0094] It can be seen from the error equations (4) and (9) that:

[0095]

[0096] According to the sliding mode surface, i.e., Equation (9), the predefined-time controller can be designed as:

[0097]

[0098] Proof of predefined-time stability:

[0099] Step 1: Proof of sliding mode surface stability:

[0100] Select a Lyapunov function as follows:

[0101]

[0102] Taking the derivative of Equation (5) gives:

[0103]

[0104] Therefore, according to the predefined-time stability theorem, it can be known that:

[0105]

[0106] v = s = 0, t ≤ T s (16)

[0107] Step 2: Proof of motor trajectory tracking stability:

[0108] Combining Equations (9) and (16) gives

[0109]

[0110] When |e1| ≥ δ

[0111]

[0112] where

[0113] When |e1| < δ, then there is

[0114] |e1| ≤ δ, if t ≥ εT s +γ(T e -T s ) (19)

[0115] where

[0116] Let ε < γ, scale the time

[0117] |e1| ≤ δ, if t ≥ γT s (20)

[0118] Combined with (20), it can be known that

[0119] s = 0, t ≥ γT s (21)

[0120] Then, combined with formulas (9) and (10), it can be known that:

[0121]

[0122] The Lyapunov function is as follows:

[0123]

[0124] Taking the derivative of formula (23) gives:

[0125]

[0126] When δ ∈ (0, 1 / e), α ∈ (0, 1), then there is:

[0127]

[0128]

[0129] Considering formula (20) and it can be obtained that:

[0130]

[0131] Then, combined with formulas (23), (24), (25), (26) and (27), it can be obtained that

[0132]

[0133] where g e1 =(δ -2α / 2(1 - α - δlnδ))δ > 0

[0134] From formula (28), it can be concluded that H is less than zero

[0135]

[0136] Therefore, under the condition of |e1| < δ, when t ≥ γT s H(∞) = 0.

[0137] Proof completed

[0138] The simulation results are as follows Figure 2-8 shown Figure 2 For the designed Fourier motor planning Figure 3 For the residual error situation at the joint end, compared with the quintic interpolation trajectory planning respectively, it can be seen that the designed planning can eliminate the residual vibration error; Figure 4 Represents the designed non - singular prescribed - time sliding surface, and it can be seen that the sliding surface reaches zero within the prescribed time t = 1s; Figure 5 Represents the designed controller; Figure 6 and Figure 7 Represents the error situation of the motor trajectory tracking the Fourier planning. It can be seen that the error reaches zero at the prescribed time t = 1s; Figure 8 Is the state diagram of the motor trajectory tracking. It can be clearly seen that the motor trajectory can track the Fourier trajectory planning at t = 1s.

[0139] In summary, the design of the present invention aims at the trajectory tracking control and vibration problems of flexible manipulators. First, a Fourier trajectory planning function is designed for the joint system model, and the error of the manipulator residual vibration can be eliminated by adjusting parameters. Then, considering the inner - dynamic loop of the motor, a non - singular prescribed - time sliding surface is proposed. Finally, based on the proposed sliding surface, a prescribed - time sliding - mode controller is designed. Like the existing fixed - time sliding - mode controller, the proposed prescribed - time sliding - mode controller has a fast convergence speed independent of the system initial conditions. Different from the fixed - time sliding - mode controller, the convergence time of the proposed method is predefined, which means that the proposed method can directly reach the expected convergence time without using the trial - and - error method to select control parameters. It has a wide range of applications and can be applied in fields such as medical treatment, aerospace, and industrial production.

Claims

1. A trajectory tracking control method for a flexible robotic arm at a predetermined time, characterized in that: Specifically, it includes the following steps: Step 1: Based on the complex dynamic behavior of the flexible robotic arm, assuming that the flexible joint is a linear spring, establish the dynamic model of the motor rotor, the dynamic model of the joint end, and the error tracking dynamic model; the dynamic models of the motor rotor and the joint end are established as follows: where, θ represents the rotation angle of the joint, q represents the motor angle, τ m represents the driving torque input to the motor, K represents the torsional stiffness coefficient, J l represents the total inertia at the joint end, τ f represents the viscous friction torque, J m represents the inertia of the motor rotor; Let \(x_1 = \theta\), \(x_3 = q\), Express the differential equation form of Equation (1) as the form of the state equation Suppose the desired trajectory of the robotic arm motor is The error between the motor's trajectory and the desired trajectory is expressed as: Then, define The error dynamic equation of the flexible manipulator is defined as: Step 2: Without considering the internal dynamics of the motor and only considering the influence of trajectory planning on the flexible joint, design a Fourier trajectory planning for the robotic arm and use an open-loop control method to eliminate chattering; The robotic arm system model for trajectory planning is Let Equation (5) can be expressed as The trajectory planning of the robotic arm satisfies the following boundary conditions: Based on the trajectory planning theory and the influence of the initial value, the trajectory planning function based on Fourier series is designed as follows: Among them, represents the expected trajectory of the joint, ω n = nπ / t f , n = 1, 2, 3, q f , q0 is the initial position and the target position of the joint, t f is the time of trajectory planning; Step 3: On the basis of Steps 1 and 2, considering the internal dynamics of the motor, in order to make the motor trajectory track the planned Fourier trajectory within a predetermined time, design a non-singular prescribed-time sliding mode control, and finally make the position tracking error of the robotic arm converge to zero within a predetermined time; the prescribed-time sliding surface of Equation (2) is designed as follows: where T s is a predetermined time parameter, 0 < α < 1, and the non-singular function is defined as: The designed non-singular function is differentiable. Therefore, the derivative of f e1 is expressed as: Among them and δ = α, According to the sliding surface, i.e., Equation (10), the controller is designed as: where 0 < α s < 1.

Images