A Global Tracking Control Method for Flexible Joint Robots Based on Obstacle Functions

Abstract

This invention belongs to the field of robot control technology, specifically relating to a global tracking control method for flexible joint robots based on obstacle functions. The method includes: S1: constructing and reconstructing the dynamic model of the flexible joint robot to adapt it to a reverse-engineering control framework; S2: designing a nonlinear filter that integrates obstacle functions, and using the obstacle functions to limit the filtering error within a preset range; S3: based on the nonlinear filter, recursively designing a controller using a reverse-engineering method, constructing an energy function, and performing stability analysis to ensure that all signals in the closed-loop system are globally consistent and bounded. This invention enables the control method to significantly improve tracking accuracy and system robustness while maintaining low computational complexity. It is suitable for robot systems with compliant actuation characteristics and has good engineering application value.

Description

Technical Field

[0001] This invention belongs to the field of robot control technology, specifically relating to a global tracking control method for a flexible joint robot based on an obstacle function. Background Technology

[0002] With the accelerating aging of the global population, the demand for service, assistive and rehabilitation robots is constantly growing. In application scenarios involving direct human-robot interaction, robot safety has become a key factor in the design process. Although traditional rigid joint robots can achieve high-speed and accurate trajectory tracking, they may pose safety hazards when human-robot contact occurs. To improve compliance during the interaction process, researchers have proposed introducing flexible structural elements between the actuator and the load, thereby developing flexible joint robots. These robots have strong impact resistance and better dynamic performance.

[0003] However, the introduction of flexible joints increases the complexity of robot design and the system order. Compared with rigid joint robots, the number of state variables in a flexible joint robot system doubles, placing higher demands on the design of the control system. Existing research has proposed a variety of control strategies to address this problem, including pushback control, sliding mode control, observer-based control, passive control, and intelligent control methods. Among them, pushback control has attracted much attention due to its systematic design framework and good robustness, and has achieved significant results in handling system uncertainties and external disturbances. However, pushback control faces two major challenges in practical applications:

[0004] First, it typically requires designing low-level control laws; second, it requires calculating higher-order derivatives of the robot's states. While low-level control laws can usually be obtained analytically, obtaining higher-order derivatives of states containing measurable noise remains a challenge.

[0005] To avoid introducing higher-order state derivatives in back-reasoning control, some scholars have introduced command filters to replace direct differentiation operations. However, these methods generally face a key challenge:

[0006] Filters inevitably introduce filtering errors, which means that the system stability can only reach a semi-global level, and cannot meet the requirements of global stability and control accuracy in practical industrial applications.

[0007] Therefore, how to achieve globally consistent bounded control of flexible joint robot systems while avoiding signal differentiation has become a key bottleneck in current technological development. Summary of the Invention

[0008] The purpose of this invention is to provide a global tracking control method for flexible joint robots based on obstacle functions. This method significantly improves tracking accuracy and system robustness while maintaining low computational complexity. It is suitable for robot systems with compliant actuation characteristics and has good engineering application value.

[0009] The specific technical solution adopted by this invention is as follows:

[0010] A global tracking control method for a flexible joint robot based on an obstacle function includes the following steps:

[0011] S1: Construct and reconstruct the dynamic model of the flexible joint robot to adapt it to the reverse control framework;

[0012] S2: Design a nonlinear filter that incorporates a barrier function, and use the barrier function to limit the filtering error within a preset range;

[0013] S3: Based on the nonlinear filter, a controller is recursively designed using the reverse method, an energy function is constructed, and stability analysis is performed to ensure that all signals in the closed-loop system are globally consistent and bounded, thus completing the global tracking control of the flexible joint robot.

[0014] The technical effects achieved by this invention are as follows:

[0015] This invention first transforms the dynamic model of a flexible joint robot to adapt it to a reverse control framework. Then, it designs a nonlinear filter that integrates obstacle functions to avoid direct differentiation of signals containing measurement noise. Simultaneously, it strictly constrains the filtering error within a preset range, effectively preventing error accumulation from damaging global stability. Based on the constructed filter and the reverse control framework, by constructing an appropriate energy function and combining it with Lyapunov stability theory, the globally uniform boundedness of all signals in the closed-loop system is proved. Simulation results show that this control method significantly improves tracking accuracy and system robustness while maintaining low computational complexity. It is suitable for robot systems with compliant actuation characteristics and has good engineering application value. It can solve the problem of semi-global stability limitations in existing command filtering reverse control methods applied to flexible joint robots. Attached Figure Description

[0016] Figure 1 This is a schematic diagram of the reference trajectory of the robot in this invention;

[0017] Figure 2 This is a schematic diagram of the position tracking error in this invention;

[0018] Figure 3 This is a schematic diagram of the controller containing white noise in this invention;

[0019] Figure 4This is a schematic diagram of the filtering error and corresponding constraint boundary in the first step of the reverse calculation method in this invention;

[0020] Figure 5 This is a schematic diagram of the filtering error and corresponding constraint boundary in the second step of the reverse calculation method in this invention;

[0021] Figure 6 This is the state of the reverse calculation method in this invention. A schematic diagram of the trajectory;

[0022] Figure 7 This is the state of the reverse calculation method in this invention. A schematic diagram of the trajectory;

[0023] Figure 8 This is the state of the reverse calculation method in this invention. A schematic diagram of the trajectory. Detailed Implementation

[0024] To make the objectives and advantages of this invention clearer, the invention will be specifically described below with reference to embodiments. It should be understood that the following text is merely used to describe one or more specific embodiments of the invention and does not strictly limit the scope of protection specifically claimed by the invention.

[0025] like Figures 1-8 As shown, a global tracking control method for flexible joint robots based on obstacle functions is proposed. This method aims to address the semi-global stability limitation of existing command-filter back-propagation control methods applied to flexible joint robots. First, the dynamic model of the flexible joint robot is transformed to fit the back-propagation control framework. Then, a nonlinear filter incorporating obstacle functions is designed to avoid direct differentiation of signals containing measurement noise. Simultaneously, the filtering error is strictly constrained within a preset range, effectively preventing error accumulation from damaging global stability. Based on the constructed filter and back-propagation control framework, the globally uniform boundedness of all signals in the closed-loop system is proved by constructing an appropriate energy function and combining it with Lyapunov stability theory. Simulation results show that this control method significantly improves tracking accuracy and system robustness while maintaining low computational complexity. It is suitable for robot systems with compliant actuation characteristics and has good engineering application value.

[0026] Includes the following steps:

[0027] S1: Construct and reconstruct the dynamic model of the flexible joint robot to adapt it to the reverse thrust control framework, and clarify the relevant preparatory knowledge, characteristics and technical principles;

[0028] The steps for constructing and reconstructing the dynamic model of a flexible joint robot are as follows:

[0029] The dynamic model of an n-link flexible joint robot is as follows:

[0030]

[0031] Equation (1) describes the dynamic characteristics of the linkage system, and This indicates the position of the connecting rod. It represents the gravity term. It is the joint torque. It is an external disturbance torque. It is the mass matrix. ， It is the Coriolis force centrifugal force matrix. ， External disturbance torque For transient disturbances, and for the sake of theoretical analysis in controller design, it can be assumed to be zero without loss of generality;

[0032] Equation (2) describes the dynamic characteristics of the compliant actuator, where It represents the joint stiffness matrix. As an auxiliary input for adjusting stiffness, under the quasi-static stiffness assumption, the dynamic influence of the actuator can be ignored. The impact on controller design is minimal; therefore, equation (2) can be simplified to:

[0033]

[0034] Equation (3) describes the dynamics of the motor, where the vector Indicates the motor position. Indicates the driving torque of the motor. This is the moment of inertia matrix of the motor.

[0035] To facilitate the design and analysis of the controller and adapt it to the reverse control framework, equations (1), (2), and (3) are reconstructed into a feedback form, as follows:

[0036]

[0037] S2: Design a nonlinear filter that incorporates a barrier function. The nonlinear filter does not require direct differentiation of the signal containing measurement noise, and the filtering error is limited to a preset range using the barrier function.

[0038] In the design of tracking control for flexible joint robots, a fundamental principle is to preserve the inherent dynamics of the mechanical system as much as possible. Advanced control techniques are only introduced when performance requirements exceed the system's inherent capabilities. This hierarchical strategy emphasizes the adaptability of the mechanical system to task demands, viewing control as an auxiliary mechanism to enhance functionality. Based on this principle, the following error transformation variables are defined, as shown in the equation:

[0039]

[0040] in, , , as well as These represent state errors, Represents reference error. The output of the filter represents the robot's state signal. To avoid directly differentiating the robot's state signal, a nonlinear filter based on the obstacle function is used, which is:

[0041]

[0042] Among them, the filter output and Used to track virtual input and The filtering error is defined as:

[0043]

[0044] And the initial conditions are met. and ,in, and , , where is a positive design parameter, and is a diagonal matrix. and Defined as:

[0045]

[0046] in, , Used to filter errors Limited to a predetermined range, Represents design parameters used to constrain filtering errors. .

[0047] S3: Based on the nonlinear filter, the controller is recursively designed using the reverse method. An energy function is constructed and stability analysis is performed using Lyapunov stability theory to ensure that all signals in the closed-loop system are globally consistent and bounded, thus completing the global tracking control of the flexible joint robot.

[0048] The back-reasoning recursive design of controllers includes:

[0049] S31: For the linkage dynamics equation (4), the first virtual input is constructed using a model-based PD control law. :

[0050]

[0051] in, Both are positive definite PD control gains, assuming torque If it can be accurately tracked, then the closed-loop error dynamics are:

[0052]

[0053] S32: Design a second virtual input for joint moment dynamics (5). :

[0054]

[0055] in, It is the control gain, and the corresponding error kinetics are:

[0056]

[0057] S33: Based on motor dynamics (18), actual control input The design is as follows:

[0058]

[0059] in, To control the gain, set Afterwards, the dynamics of the closed-loop system are:

[0060]

[0061] In the stability analysis of S3, the proposition is established as follows:

[0062] Define the vector as:

[0063]

[0064] And Opening Collection:

[0065]

[0066] For the closed-loop system equations (4), (5), and (6), combined with the control input equations (15), (17), and (19), there exists a unique maximum solution. Defined in time interval Inside, that is, satisfying , For all Established;

[0067] From equations (7), (8), (9), (10), (15), (17), and (19), it can be seen that the virtual control input and Actual control input and robot status , , Both can be represented as vectors The function is as follows:

[0068] ;

[0069] ;

[0070] ;

[0071] ;

[0072] ;

[0073] ;

[0074]

[0075] in, , as well as , Closed-loop system The dynamics are:

[0076] ;

[0077] ;

[0078] ;

[0079] ;

[0080]

[0081] The entire closed-loop system can be compactly represented as follows:

[0082]

[0083] Given initial conditions satisfy for and The initial state can be obtained as follows:

[0084]

[0085] In satisfying , , , It is bounded, and , , Under the premise that it is first-order continuous and differentiable, the function about Piecewise continuity, regarding Locally Lipschitz continuous, and according to the lemma that the initial value belongs to a set and the maximum solution exists, the system (34) is in the time interval The memory contains a unique maximum solution That is, for any All Established, and and .

[0086] The above theoretical results can be summarized into the theorem as follows:

[0087] For the flexible joint robot system equations (4), (5), and (6), using virtual control input equations (16) and (17), and actual motor control input equation (19), the following properties are ensured:

[0088] A1: All signals in a flexible joint robot system are globally bounded;

[0089] A2: Joint position Able to track reference trajectory .

[0090] This proof is divided into two parts. The first part establishes the forward completeness of the closed-loop system through proof by contradiction, that is, it proves the maximum existence time. This ensures that the system solution exists globally. The second part further demonstrates that the designed controller can achieve the control objective, thus completing the proof of this theorem.

[0091] For the proof of A1, the forward completeness of the closed-loop system is established by contradiction, that is, the maximum existence time is proven. This ensures that the solution of the flexible joint robot system exists in a bounded global domain.

[0092] Prove the solution of the controlled system by contradiction. It exists in any finite amount of time, that is Therefore, we assume the opposite is true, that is, the maximum existence time of the system solution is a finite value, i.e. The proof is as follows:

[0093] Constructing Lyapunov functions:

[0094]

[0095] right Taking the time derivative, we get:

[0096]

[0097] Substituting equations (16), (18), and (20) into equation (36), we get:

[0098]

[0099] Applying scaling principles and This is a property of skew-symmetric matrices, and the above expression (37) can be written as:

[0100]

[0101] in:

[0102]

[0103] From equation (38), it can be seen that for , belong This implies the existence of compactness. , making For all All are true;

[0104] Define the Lyapunov function as:

[0105]

[0106] right Taking the time derivative, we get:

[0107]

[0108] in:

[0109]

[0110] The above calculation formula is a continuous function, since and exist Bounded above, combined , , , It is bounded, we know and This interval is also bounded;

[0111] From equation (15), we can see that It is bounded, further combined with the proposition. The boundedness is known That is, in Bounded above, and finally, as can be seen from equation (4) It also has boundaries, and can be obtained. Bounded;

[0112] Rewrite equation (40) as:

[0113]

[0114] in, ， ， ， express The upper bound;

[0115] From equation (41), it can be seen that there exists a constant. ， ,satisfy ,when Sometimes, Furthermore, based on equation (41), we can derive... and ;

[0116] Therefore, the conclusion is:

[0117]

[0118] For all Established, thus obtaining That is, it is bounded throughout the entire time interval;

[0119] Subsequently, calculation Time derivative:

[0120]

[0121] in:

[0122]

[0123] The above calculation formula is a continuous function, combined with formula (11) and the relationship and to Based on previous analysis, it is inferred that , , and All are bounded, based on propositions, equation (10), and The facts lead to It is bounded, and then, as can be seen from equation (5), Bounded, therefore It also has boundaries;

[0124] Equation (42) can be restated as follows:

[0125]

[0126] in, , , , express The upper bound;

[0127] From equation (44), it can be seen that there exists a constant. , ,satisfy ,when Sometimes, ;

[0128] Furthermore, from equation (44), we can further deduce ,in ;

[0129] Therefore, for any ,get:

[0130]

[0131] Therefore, the control input is derived. In time interval The upper limit is bounded;

[0132] In summary, , , , , as well as exist All of the above belong to Thus, there exists a compact subset. ;

[0133] The above shows the maximum solution of equation (34). Existing interval Internally, combining the global boundedness theorem and the contradiction method, it is further proved that the system possesses forward completeness, that is... ,therefore, In time interval It can be defined.

[0134] See appendix Figure 5 Regarding the proof of A2, further demonstration is provided that the designed controller can achieve the control objective, thus completing the proof of this theorem. This is achieved by introducing recursive design into the back-reasoning control framework. , , , , , , , , , , , as well as In time interval Since all signals remain bounded, according to the global boundedness theorem, all signals are globally ultimately bounded.

[0135] S4: Based on the designed controller and the nonlinear filter, a control signal is generated to drive the flexible joint robot actuator to achieve global stable tracking control;

[0136] Experimental verification shows that:

[0137] To verify the effectiveness and superiority of the proposed obstacle function-based filtering control algorithm on a flexible joint robot, this paper conducts a simulation experiment on a seven-DOF FrankaPanda robot using the Simulink module in MATLAB R2023a. The simulation was run on a computer equipped with an NVIDIA RTX 4060 GPU, and the total task duration was 50 seconds. To more realistically evaluate the algorithm's performance in a real-world environment, zero-mean Gaussian white noise was added to the robot's position and velocity measurement data to simulate sensor measurement errors. The dynamic behavior of the flexible joint robot follows the formula... The model described is configured such that the joint stiffness matrix and the motor moment of inertia matrix are respectively set as follows: and ;

[0138] The control parameters are configured as follows: For the PD virtual control law, its proportional gain and derivative gain are set to... and For the virtual control law and the actual control law, the corresponding gains are set as follows: and In terms of filter design, the filter time constant is set to The filtering error constraint boundary is set as ,in , ;

[0139] The main objective of this simulation is to determine the position of the joint links. Capable of accurately tracking reference trajectories generated through curve fitting techniques Simultaneously, it ensures that all signals in the flexible joint robot system remain bounded globally. Simulation results are presented in... Figures 1-8 . Figure 1 The reference trajectory was shown; Figure 2 The tracking error responses of the seven joints relative to the reference trajectory are presented, demonstrating that the proposed control strategy can achieve high-precision trajectory tracking. Figure 3 The motor torque response is demonstrated in the presence of measurement noise; Figure 4 and Figure 5 The filtering error and its corresponding constraint boundary are presented respectively. It can be seen that the filtering error is always strictly limited within the set boundary throughout the process. Figures 6-8 The evolution of other state variables under sensor noise interference is shown, further verifying the global boundedness of the system state, where PD represents the proportional derivative.

[0140] This application innovatively designs a nonlinear filter that integrates a barrier function. This eliminates the need for direct differentiation of signals containing measurement noise, effectively limiting the filtering error to a preset range and preventing error accumulation from impacting system stability. By constructing an appropriate energy function and analyzing it based on Lyapunov stability theory, the globally uniform boundedness of all signals in the closed-loop system is proven. Simulation results demonstrate that this method significantly improves tracking control accuracy and system robustness while maintaining computational efficiency. It is suitable for the high-performance control requirements of flexible joint robots in industrial automation and precision manufacturing, and has significant engineering application value.

[0141] The above description is merely a preferred embodiment of the present invention. It should be noted that those skilled in the art can make various improvements and modifications without departing from the principles of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention. Structures, devices, and operating methods not specifically described or explained in this invention are implemented according to conventional methods in the art unless otherwise specified or limited.

Claims

1. A flexible joint robot global tracking control method based on barrier function, characterized by, The method comprises the following steps: S1: constructing and reconstructing a dynamics model of the flexible joint robot to adapt to a backstepping control framework; In the S1, the step of constructing and reconstructing the dynamics model of the flexible joint robot is: For an n-link flexible joint robot dynamics model, the formula (1), the formula (2) and the formula (3) are reconstructed into a feedback form, so that the flexible joint robot is adapted to the backstepping control framework, and the formula (1) is: ； wherein formula (1) describes the dynamics of the linkage system, and which represents the linkage position, which represents the gravity term, which is the joint torque, which is the external disturbance torque, which is the mass matrix ， which is the Coriolis force matrix ， external disturbance torque is the transient disturbance; The dynamic behavior of a compliant actuator is described by the equation of motion (2), where represents the joint stiffness matrix, is the auxiliary input for stiffness adjustment, the equation of motion (2) can be simplified to: ； The dynamics of the electric machine are described by the equation of motion (3), where the vector represents the electric machine position, represents the electric machine drive torque, is the electric machine moment of inertia matrix; S2: designing a nonlinear filter fused with an obstacle function, and limiting filter error in a preset range by using the obstacle function; ； S3: based on the nonlinear filter, a controller is recursively designed by using a backstepping method, an energy function is constructed, and stability analysis is performed, so that all signals in a closed loop system are globally uniformly bounded, and global tracking control of the flexible joint robot is completed; In the S3, the backstepping method recursively designing the controller comprises: In the tracking control design of the flexible joint robot in the S2, the following error transformation variables are defined, and the formula is: S31: For the link dynamics equation (4), a model-based PD control law is adopted to construct the first virtual input : ； where, are positive definite PD control gains, assuming that the torques can be accurately tracked, then the closed-loop error dynamics are: ； S32: Design a second virtual input for joint torque dynamics (5) : ； wherein, which is a control gain, the corresponding error dynamics are: ； S33: Actual control input based on motor dynamics (18) The design is as follows: ； wherein, is the control gain, set After that, the dynamics of the closed loop system are: 。 2. The global tracking control method for a flexible joint robot based on barrier function according to claim 1, characterized in that: In the stability analysis in the S3, the following proposition is established: ； where, , , and represent the state error, represent the reference error, represent the output of the filter, to avoid direct differentiation of the robot state signal, a nonlinear filter based on barrier functions is employed, which is given by: ； where the filter output and are used to track the virtual input and respectively, the filtering error is defined as: ； and satisfy the initial condition and wherein, and , is a positive design parameter, diagonal matrix and is defined as: ； wherein, , to limit the filter error to a predetermined range, the representing design parameters for constraining the filter error .

3. The global tracking control method for a flexible joint robot based on barrier function according to claim 1, characterized in that: The vector is defined as: And the open set is: ； The whole closed loop system is compactly represented as: ； For closed loop system equations (4), (5) and (6), combined with control input equations (15), (17) and (19), there exists a unique maximum solution defined over the time interval , i.e. satisfying , for all . From equations (7), (8), (9), (10), (15), (17), and (19), the virtual control input and the actual control input and the robot state , , may be expressed as functions of the vector , as follows: ； ； ； ； ； ； ； wherein , and , , the dynamics of the closed loop system with respect to are: ； ； ； ； ； ​ ； Given the initial conditions satisfy For and The initial state is: ； In satisfying , , , is bounded, and , , is continuously differentiable of the first order, the function is piecewise continuous with respect to and locally Lipschitz continuous with respect to , by the lemma that the initial value belongs to the set, the maximum solution exists, the system (34) exists in the time interval The unique maximum solution , that is, for any , is established, and and ; For flexible joint robot systems of the form (4), (5) and (6), the virtual control input of the form (16) and (17), and the actual motor control input of the form (19) are employed, such that "A1 : All signals in the flexible joint robot system are globally bounded" holds, while "A2: Joint position Ability to track reference trajectories " holds.

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