Tendon-driven mechanical arm trajectory tracking control method capable of adjusting preset time

Abstract

The invention provides a tendon-driven mechanical arm trajectory tracking control method capable of adjusting preset time. The tendon-driven mechanical arm trajectory tracking control method comprises the steps that a second-order full-driving mechanical model of a tendon-driven mechanical arm is designed by considering external bounded disturbance and system parameter uncertainty; designing a finite time extended state observer based on the extended error vector so as to perform real-time estimation on uncertain parameters and external disturbance of the system; and based on an estimation result of the limited time extended state observer, introducing an adjustable gain parameter to design an adjustable preset time controller. According to the method, real-time accurate estimation of the composite uncertainty of the system is realized through the construction of the second-order full-driving mechanical model and the innovative design of the finite time expansion state observer; the all-drive preset time control method provided by the invention is combined with an adjustable gain parameter technology, and has global preset time stability, good transient response speed and steady-state precision, and strong robustness for unknown external disturbance and inertial parameters.

Description

Technical Field

[0001] This invention relates to the field of robotic arm control technology, and in particular to a method for trajectory tracking control of a tendon-driven robotic arm with adjustable predetermined time. Background Technology

[0002] As a core execution component for spacecraft on-orbit servicing, robotic arms need to achieve high-precision trajectory tracking control in complex environments. However, robotic arms often face problems such as external bounded disturbances (e.g., microgravity disturbances, changes in operating loads) and unknown inertial parameters (e.g., parameter drift caused by component wear and load changes), which seriously affect trajectory tracking accuracy.

[0003] In existing trajectory tracking control methods, preset time control strategies can ensure system convergence within a pre-defined time, but their convergence time adjustability is poor, making it difficult to adapt to the response speed requirements of different tasks. Traditional extended state observers, while able to estimate system uncertainties, are insufficient in terms of finite-time convergence accuracy and dynamic response. Furthermore, tendon-driven robotic arms, due to their unique transmission structure, exhibit nonlinear characteristics such as rope deformation and tension fluctuations, further increasing the control difficulty. Therefore, it is necessary to develop a trajectory tracking control method that combines strong robustness, high precision, and adjustable preset time convergence characteristics. Summary of the Invention

[0004] To address the shortcomings of existing technologies, this invention provides an adjustable predetermined time trajectory tracking control method for tendon-driven robotic arms. This method aims to solve the problems in existing technologies, such as the inability to adjust the convergence time of trajectory tracking control, low accuracy, and insufficient robustness, when there are external bounded disturbances and unknown inertial parameters. This invention can achieve high-precision trajectory tracking of robotic arms within a preset time.

[0005] The technical solution of the present invention is: a method for trajectory tracking control of a tendon-driven robotic arm with adjustable predetermined time, comprising the following steps: S1) Construct the equations for the tendon-driven robotic arm system; S2) By considering external bounded disturbances and system parameter uncertainties, the system equations of the robotic arm are transformed into a second-order fully driven mechanical model of the tendon-driven robotic arm. S3) Design the extended error vector for the trajectory tracking of the robotic arm, and design a finite-time extended state observer based on the extended error vector to estimate the uncertain parameters of the system and external disturbances in real time. S4) Based on the estimation results of the time-limited extended state observer, an adjustable preset time controller is designed by introducing an adjustable gain parameter.

[0006] Preferably, in step S2), the robotic arm system equations are transformed into a second-order fully driven mechanical model of the robotic arm by introducing external disturbance terms and system parameter uncertainties, i.e.: ; (2-7) in, This is an uncertain term; Motor output control torque; External disturbance term; , , Representing the inertia matrix respectively Coriolis force and centrifugal force matrix Torque loss due to rope deformation Known nominal terms; , and They represent , , The uncertain part; For joint angle; For joint velocity, For joint acceleration, The motor current outputs torque; The Jacobian matrix representing current-torque.

[0007] Preferably, in step S3), the time-limited extended state observer is: (3-10) In the formula, , These represent the first derivatives of the observer's state variables; , These represent the observer state variables respectively; Indicates parameters of the all-wheel drive system; Indicates control input; , For observer gain, ; , The parameter is an exponential parameter, and and .

[0008] Preferably, in step S4), the expression for the adjustable preset time controller is: ; (4-1) In the formula, Indicates control input; Preset time parameters; This is an adjustable gain parameter; Given positive constants; This is the extended error vector for joint angle trajectory tracking; This is the observer's estimated value.

[0009] The beneficial effects of this invention are as follows: 1. This invention achieves real-time and accurate estimation of the complex uncertainties of a system through the construction of a second-order fully driven mechanical model and the innovative design of a finite-time extended state observer; 2. The proposed all-drive preset time control method, which combines adjustable gain parameter technology, has global preset time stability, good transient response speed and steady-state accuracy, and strong robustness to external disturbances and unknown inertial parameters. Attached Figure Description

[0010] Figure 1 This is a schematic diagram of the framework of the method of the present invention; Figure 2 This is a graph showing the position error curves of each joint trajectory tracking at different preset time values ​​in an embodiment of the present invention. Figure 3 This is a graph showing the motor current curves of each joint control at different preset time values ​​in an embodiment of the present invention. Figure 4 This is a graph showing the position error curves of each joint trajectory tracking with different adjustable parameters in an embodiment of the present invention; Figure 5 This is a graph showing the motor current output of each joint control with different adjustable parameters in an embodiment of the present invention. Figure 6 This is a graph comparing the observer's estimated value with the actual value in an embodiment of the present invention. Figure 7 This is a graph showing the position error curves of each joint trajectory tracking with and without ESO in an embodiment of the present invention; Figure 8 This is a graph showing the trajectory tracking position curves of joint 1 with and without ESO in an embodiment of the present invention; Figure 9 This is a graph showing the trajectory tracking position curves of joint 2 with and without ESO in an embodiment of the present invention; Figure 10 This is a graph showing the end effector trajectory tracking curves of the robotic arm with and without ESO in an embodiment of the present invention; Figure 11 The graph shows the estimation error curves of each joint observer of different controllers in the embodiments of the present invention. Figure 12 This is a graph showing the position error curves of the trajectory tracking of each joint of different controllers in an embodiment of the present invention; Figure 13 This is a graph showing the trajectory tracking position curves of joint 1 for different controllers in this embodiment of the invention; Figure 14 This is a graph showing the trajectory tracking position curves of joint 2 for different controllers in this embodiment of the invention; Figure 15 The output motor current curves of joint 1 control for different controllers in this embodiment of the invention are shown. Figure 16 The output motor current curves for joint 2 control of different controllers in this embodiment of the invention are shown. Figure 17 The above are graphs showing the tracking trajectory of the robotic arm end effector for different controllers in this embodiment of the invention. Figure 18 This is a curve diagram of joint trajectory tracking position in an embodiment of the present invention; Figure 19 This is a graph showing the joint trajectory tracking position error in an embodiment of the present invention. Figure 20 This is a cable tension curve diagram in an embodiment of the present invention; Figure 21 This is a diagram showing the control input motor current curve in an embodiment of the present invention. Detailed Implementation

[0011] The specific embodiments of the present invention will be further described below with reference to the accompanying drawings: Example 1 like Figure 1 As shown, this embodiment provides a method for trajectory tracking control of a tendon-driven robotic arm with adjustable predetermined time, including the following steps: S1) Construct the equations for the tendon-driven robotic arm system, the expression of which is: ; (1-1) In the formula, The inertia matrix; Joint angle; Joint velocity; Joint acceleration; The matrix represents the Coriolis force and the centrifugal force. The torque lost due to rope deformation; The motor current outputs torque; The Jacobian matrix representing current-torque.

[0012] S2) By considering external bounded disturbances and system parameter uncertainties, a second-order fully driven mechanical model of the tendon-driven robotic arm is established, specifically including the following steps: S21) Considering external bounded disturbances The equations of the robotic arm system can be rewritten as follows: ; (2-1) In the formula, The externally bounded perturbation term satisfies , For known positive constants; S22), considering the uncertainty of system parameters, namely: ; (2-2) ; (2-3) ; (2-4) in, , , They represent , , Known nominal terms; , and They represent , , The uncertain part; in: ; (2-5) In the formula, Given positive constants; Based on the uncertainties in the above parameters, the equations for the tendon-driven robotic arm system can be described as follows: ; (2-6) S23) Introducing the lumped uncertainty term of the system Thus, a second-order fully driven mechanical model of the robotic arm with external disturbances and uncertainties in system parameters is obtained, namely: ; (2-7) in, This is an uncertain term; Motor output control torque.

[0013] S3) Design the extended error vector for the trajectory tracking of the robotic arm, and design a finite-time extended state observer based on the extended error vector to estimate the uncertain parameters of the system and external disturbances in real time; specifically including the following steps: S31) For a given continuous desired joint angle trajectory and desired joint velocity trajectory; calculate the extended error vector of the robot arm's trajectory tracking. , ;Right now: ; (3-1) ; (3-2) In the formula, Indicates the expanded error coefficient; The desired joint acceleration; Desired joint velocity, , , These are joint angle, joint velocity, and joint acceleration, respectively. S32), Substituting equations (3-1) and (3-2) into the second-order fully driven mechanical model (2-7) of the robotic arm, we get: ; (3-3) make: ,get: ; (3-4)

[0014] S33) The overall control law is designed as follows: ; (3-5) In the formula, Motor output control torque; Indicates parameters of the all-wheel drive system; Indicates control input; Substituting (3-5) into (3-4), we get: ; (3-6) In the formula, This represents the extended error vector for joint angle trajectory tracking; This represents the extended error vector for joint velocity trajectory tracking. This represents the system's uncertainty and external disturbances; S34), let , Substituting these values ​​into equation (3-6) yields the trajectory tracking error model: ; (3-7) In the formula, Represents the extended error vector The first derivative; for The first derivative; Representing the lumped uncertainty of the system The first derivative, ; Indicates the system output quantity; in, It is bounded and satisfies the following inequalities: ; (3-8) In the formula, Represents the ensemble uncertainty term for each joint; Represent an unknown constant; S35) Define the estimation error of the extended state observer, i.e.: ; (3-9) In the formula, , These represent the observer estimation errors, respectively. , These represent the observer state variables respectively; Finally, the time-limited extended state observer is obtained as follows: (3-10) In the formula, , These represent the first derivatives of the observer's state variables; Indicates parameters of the all-wheel drive system; Indicates control input; , For observer gain, ; , The parameter is an exponential parameter, and and .

[0015] S4) Based on the estimation results of the time-limited extended state observer, an adjustable preset time controller is designed by introducing an adjustable gain parameter; that is: ; (4-1) In the formula, Indicates control input; Preset time parameters; This is an adjustable gain parameter; Given positive constants; This is the extended error vector for joint angle trajectory tracking; For the observer's state variables.

[0016] Example 2 This embodiment verifies the designed preset time controller under different preset time parameters. and The convergence characteristics under different preset time parameters were first verified. The convergence characteristics of the system are analyzed. The sampling period for the numerical simulation is set to 1ms, and the simulation time is 10s. The initial states and desired trajectories of each joint of the robotic arm are as follows: ; ; ; The robotic arm dynamics model established in Example 1 was used for simulation verification. The physical parameters are shown in Table 1, and the control parameters are set in Table 2.

[0017] Table 1 Physical parameters of the dual-joint four-cable driven robotic arm Table 2 Control Parameters Numerical simulation results are as follows Figure 2-3 As shown, Figure 2 It describes different preset time values ​​under the all-wheel drive preset time control scheme. Dual-joint trajectory tracking position error The convergence curve, Figure 3 The output current curves of each joint control are described; it can be seen that the tracking error system is stable at a preset time, when When the number increases, the convergence time increases; when When the value is reduced, the convergence time decreases, and the trajectory tracking position error decreases. able to The neighborhood converges to zero. The simulation results above show that, within a preset time... It is possible to determine the upper bound of the system's steady-state time, and a small... This will make the convergence speed very fast. It's worth noting that a faster convergence rate often means a larger control output is needed, which can be seen from... Figure 3 The output current curves of each joint control show that, with the preset time parameter... As the value decreases, the control output gradually increases.

[0018] Then, this embodiment verifies different adjustable parameters. The convergence characteristics of the system were verified by simulation using a dynamic model of a dual-joint, four-cable driven robotic arm. The sampling period was set to 1ms, and the simulation time was 10s. The initial states and desired trajectories of each joint of the robotic arm were the same as described above, and the control parameters are shown in Table 3. Table 3 Control Parameters Numerical simulation results are as follows Figure 4 , 5 As shown, Figure 4 Describes different adjustable parameters under all-wheel drive preset time control. Dual-joint trajectory tracking position error Convergence curve Figure 5 The output current curves for each joint control are described. It can be seen that when... When the value increases, the convergence time decreases; when As the value decreases, the convergence time increases. It's worth noting that a faster convergence rate often means a larger control output is needed, which can be seen from... Figure 5 The output current curves of each joint control show that, with the adjustment parameters... As the value increases, the control output gradually increases.

[0019] Based on the simulation results above, for the adjustable preset time controller (4-1) designed in Example 1, the preset time parameter... Used to predefine the upper bound of the system's convergence time, adjustable parameter Used to adjust the actual convergence time of the system. This is achieved through adjustable parameters. and This not only enables the system to converge within a preset time, but also allows for further adjustment of the system's convergence time, thereby achieving a high-precision trajectory tracking control objective with greater controllability of the robotic arm system's convergence time.

[0020] Example 3 This embodiment compares the adjustable preset time controller FASPTC with and without a finite-time extended state observer (ESO). A dynamic model of a dual-joint, four-cable driven robotic arm is used for simulation verification. The physical parameters are shown in Table 1, the sampling period is set to 0.01s, and the simulation time is 50s. The initial states and desired trajectories of each joint of the robotic arm are consistent with the previous section. External disturbances acting on each joint are... The system parameters are uncertain as follows: The initial state and parameters of the Extended State Observer (ESO) are shown in Table 4, and the control parameters of the Adjustable Preset Time Controller (FASPTC) are shown in Table 5. Table 4 Initial State and Parameters of the Extended State Observer Table 5 Control parameters of the adjustable preset time controller

[0021] Simulation results are as follows Figure 6-10 As shown, Figure 6 The tracking curves between the observer's estimated values ​​and the actual values ​​are described, showing that the observer can achieve fast and accurate estimation of time-varying disturbances and uncertainties. Figure 7 The position error curves for trajectory tracking of each joint are described. Figure 8 , 9 The trajectory tracking position curves of each joint are described. Figure 10The tracking trajectory of the robotic arm's end effector is presented. It can be seen that the adjustable preset time controller FASPTC in Example 1 can achieve error convergence within a preset time. Compared to the adjustable preset time controller FASPTC without an extended state observer (ESO), joints 1 and 2 under the action of the adjustable preset time controller FASPTC with an ESO can achieve transient convergence of system errors faster, while also exhibiting better steady-state accuracy and tracking performance. The adjustable preset time controller FASPTC without an ESO is difficult to achieve accurate and rapid tracking of the desired trajectory due to time-varying disturbances and uncertainties. However, it is not difficult to observe that even under conditions of large disturbances, the adjustable preset time controller FASPTC still possesses a certain degree of robustness against disturbances and uncertainties.

[0022] Secondly, from Figure 11 It can be seen that the transient convergence rate of the observation error of joints 1 and 2 under the action of the adjustable preset time controller FASPTC in Example 1 is significantly faster than that of PTC. The observation error under the action of the adjustable preset time controller can converge to zero more quickly with a smaller overshoot, showing better steady-state accuracy.

[0023] Figure 12 The position error curves for trajectory tracking of each joint of the robotic arm are given; Figure 13 , 14 The position curves for tracking the trajectory of each joint are given. Figure 17 The tracking trajectory of the robotic arm's end effector is presented. It can be seen that, for uncertain robotic arm systems with unknown model parameters and external disturbances, under conditions of large initial errors, the proposed adjustable preset time controller FASPTC exhibits a faster response speed in tracking position errors of each joint trajectory, quickly converging to near zero and demonstrating higher steady-state accuracy. Simulation results also show that the average trajectory tracking errors under PTC are 1.72° and 3.37°, while the average trajectory tracking errors under the adjustable preset time controller FASPTC are 1.43° and 2.57°. The adjustable preset time controller FASPTC method in Example 1 demonstrates higher control accuracy.

[0024] Figure 15 , 16 The control output current comparison curves for each joint are given. It can be seen that in the early stage of control (0-3s), compared with PTC, the proposed FASPTC requires a larger control torque to achieve faster convergence speed and realize accurate trajectory tracking control. However, in the middle and late stages of control (3-50s), the control output required for trajectory tracking control is not much different between the two.

[0025] Example 4 This embodiment demonstrates the effectiveness of the method in Example 1 in practical engineering applications by constructing an experimental platform with physical parameters shown in Table 1; the sampling period is set to 0.1s, and the time is 160s. Initial state of the robotic arm joint position. Expected trajectory The safety protection current is -1.5A to 0.15A. The FASPTC control parameters are shown in Table 6.

[0026] Table 6 FASPTC Control Parameters The tendon-driven robotic arm is composed of multiple cable-driven joint modules. Each joint module employs a novel 180-degree rotating hinge, with a lifting structure designed at the shaft joint. Adjacent links are interconnected via the lifting structure. Cables are positioned on both sides of the link, one end connected to a winch and the other to a tension sensor. Cable drive and control elements are located at the ends of the links, controlling the movement of the links by rotating a motor to contract and release the cables, enabling the joints to achieve a large angular range of ±180 degrees.

[0027] Figure 18 The joint trajectory tracking position curve is given. Figure 19 The joint trajectory tracking position error curve is presented. It can be seen that, with an initial position error of 10°, the joint trajectory tracking position under the proposed all-drive preset time control (FASPTC) can quickly track the desired trajectory, and the system error can converge to near zero within the preset time (convergence is achieved in about 3 seconds), demonstrating high steady-state accuracy. Due to limitations of the experimental equipment, some degree of chattering occurs between 50-70 seconds due to significant frictional resistance during wheel reversal; however, the system can still achieve fast and accurate tracking of the desired trajectory, exhibiting good robustness.

[0028] Figure 20 The cable tension curve of the robotic arm is given. Figure 21 The control input motor current curve is presented. It can be seen that with an initial error of 10°, a larger control torque is required to achieve rapid tracking within the preset time. Therefore, the cable tension is relatively high from 0-10s, and even greater control torque is needed during wheel commutation to overcome frictional resistance and achieve the desired trajectory tracking. Figure 21 The current curves show that, from 0 to 40 seconds, cables 1 and 3 provide the main torque, while cables 2 and 4 provide a smaller preload to maintain cable tension. From 40 to 120 seconds, cables 2 and 4 provide the main torque, while cables 1 and 3 provide a smaller preload to maintain cable tension, which is consistent with the theoretical results.

[0029] The embodiments and descriptions above are merely illustrative of the principles and preferred embodiments of the present invention. Various changes and modifications may be made to the present invention without departing from its spirit and scope, and all such changes and modifications fall within the scope of the present invention as claimed.

Claims

1. A method for trajectory tracking control of a tendon-driven robotic arm with adjustable predetermined time, characterized in that, Includes the following steps: S1) Construct the equations for the tendon-driven robotic arm system; S2) By considering external bounded disturbances and system parameter uncertainties, the system equations of the robotic arm are transformed into a second-order fully driven mechanical model of the tendon-driven robotic arm. S3) Design the extended error vector for the trajectory tracking of the robotic arm, and design a finite-time extended state observer based on the extended error vector to estimate the uncertain parameters of the system and external disturbances in real time. S4) Based on the estimation results of the finite-time extended state observer, an adjustable preset time controller is designed by introducing an adjustable gain parameter.

2. The method for trajectory tracking control of a tendon-driven robotic arm with adjustable predetermined time according to claim 1, characterized in that: In step S1), the equations for the tendon-driven robotic arm system are constructed, and their expression is: In the formula, M(θ) is the inertia matrix; θ is the joint angle; Joint velocity; Joint acceleration; The matrix represents the Coriolis force and the centrifugal force; τ s The torque lost due to rope deformation; i m J represents the motor current and output torque. m The Jacobian matrix representing current-torque.

3. The method for trajectory tracking control of a tendon-driven robotic arm with adjustable predetermined time according to claim 2, characterized in that: In step S2), the second-order fully driven mechanical model of the tendon-driven robotic arm, established by considering external bounded disturbances and system parameter uncertainties, is expressed as follows: in, For uncertainties; I m =J m i m Motor output control torque; d(t) external bounded disturbance term; ΔM(θ), and Δτ s M(θ), respectively τ s The uncertain part.

4. The method for trajectory tracking control of a tendon-driven robotic arm with adjustable predetermined time according to claim 3, characterized in that: In step S2), the second-order fully driven mechanical model of the tendon-driven robotic arm, established by considering external bounded disturbances and system parameter uncertainties, specifically includes the following steps: S21) Considering the external bounded disturbance d(t), the equations of the robotic arm system are rewritten as follows: In the formula, d(t) is an external bounded disturbance term that satisfies ||d(t)|| ≤ d M d M For known positive constants; S22), considering the uncertainty of system parameters, namely: M(θ)=M0(θ)+ΔM(θ); (2-2) t s =t s0 +Δt s (2-4) Where M0(θ), τ s0 M(θ), respectively τ s The known nominal terms; ΔM(θ), and Δτ s M(θ), respectively τ s The uncertain part; in: In the formula, M m C m T m Given positive constants; Based on the uncertainties in the above parameters, the equations for the tendon-driven robotic arm system can be described as follows: S23) Introducing the lumped uncertainty term Ξ of the system, we obtain a second-order fully driven mechanical model of the robotic arm with external disturbances and system parameter uncertainties, namely: in, For uncertainties; I m =J m i m Motor output control torque.

5. The method for trajectory tracking control of a tendon-driven robotic arm with adjustable predetermined time according to claim 4, characterized in that: In step S3), the extended error vector for the robotic arm's trajectory tracking is: In the formula, S r This represents the extended error vector for joint angle trajectory tracking; This represents the expanded error vector for joint velocity trajectory tracking; α represents the expanded error coefficient. The desired joint acceleration; Desired joint velocity, θ These are joint angle, joint velocity, and joint acceleration, respectively.

6. The method for trajectory tracking control of a tendon-driven robotic arm with adjustable predetermined time according to claim 5, characterized in that: In step S3), based on the extended error vector S r Design the following overall control law: In the formula, I m =J m i m Motor output control torque; A0 represents all-drive system parameters; v represents control input; 7. The method for trajectory tracking control of a tendon-driven robotic arm with adjustable predetermined time according to claim 6, characterized in that: In step S3), let: x1=S r The trajectory tracking error model is obtained as follows: In the formula, This represents the first derivative of the extended error vector x1; Π represents the first derivative of x²; Π represents the first derivative of the system's lumped uncertainty Ξ. y represents the system output.

8. The method for trajectory tracking control of a tendon-driven robotic arm with adjustable predetermined time according to claim 5, characterized in that: In step S3), the finite-time extended state observer is: In the formula, Let z1 and z2 represent the derivatives of the observer state variables, respectively; z1 and z2 represent the observer state variables, respectively; A0 represents the parameters of the all-drive system; v represents the control input; c1 and c2 are the observer gains, where c1, c2 > 0; r1 and r2 are power parameters, and And r2 = 2r1 - 1; ∈1 represents the estimation error.

9. The method for trajectory tracking control of a tendon-driven robotic arm with adjustable predetermined time according to claim 8, characterized in that: In step S4), the adjustable preset time controller; that is: In the formula, v represents the control input; T f >0 represents the preset time parameter; 0<ρ<1, λ>0 represent adjustable gain parameters; γ is a known positive constant; S r z1 is the extended error vector for joint angle trajectory tracking; z2 is the state variable of the observer.

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