Mechanical arm control system based on robust adaptive control

Abstract

According to the robot self-adaptive controller adopting robust control, control methods of robust control and self-adaptive control are selected, a mechanical arm is adopted as a research object, modeling and control research results of a two-degree-of-freedom mechanical arm can be expanded to an n-degree-of-freedom mechanical arm, and the robot self-adaptive controller adopting robust control is provided. Therefore, the method can be applied to the field of industrial robot control development. Different from other nonlinear control methods such as traditional PID (Proportion Integration Differentiation) control and the like which have the characteristic that different constraints in an actual system are not considered, the mechanical arm is controlled by adopting a model robust adaptive algorithm, and the robust adaptive control method can realize trajectory tracking control without establishing an accurate system model.

Description

Technical Field

[0001] The present invention belongs to the field of robot control technology, and specifically relates to a robot arm control system based on robust adaptive control. This product solves the problem that most industrial robots currently use traditional PID and cannot ensure accurate and smooth movement of the robot arm. The robust adaptive control strategy is used to achieve the function of accurate tracking of the robot arm trajectory. Background Art

[0002] As an automated mechanical device widely used in the field of robotics, the robot arm can carry goods and sort items to replace heavy and monotonous physical labor, complete mechanized and automated industrial production processes due to its unique operational flexibility; it can also protect personal safety in situations where it is not suitable for direct human contact, such as high temperature, corrosion, and toxic gases. It is widely used in industrial production, medical diagnosis, military science, semiconductor manufacturing, and space exploration. From the perspective of control theory, the robot arm is a complex system with multiple inputs and outputs, high nonlinearity, and strong coupling, and there are uncertain factors such as parameter vibration, unknown external interference, and unknown modeling quantities.

[0003] Many industrial robots perform simple and repetitive tasks. This kind of work has relatively high requirements for position accuracy, but has relatively low requirements for trajectory control during the movement of the robot arm. Therefore, in the past, most of our traditional industrial robots only focused on the position accuracy of the robot arm and ignored the trajectory stability and trajectory accuracy control during its operation. However, for tasks with strict requirements on the path, simply controlling the position of the robot arm will produce a certain deviation between the theoretical trajectory and the actual trajectory. Under high-speed or heavy-load operation, the deviation on the motion path will be very large, and even strong shaking will occur. The present invention mainly focuses on the modeling and control research results of the two-degree-of-freedom robot arm, which can be extended to the n-degree-of-freedom robot arm. Therefore, it can be applied in the control development field of industrial robots, and is committed to improving the trajectory tracking accuracy of the robot arm during movement and its ability to resist external interference and modeling inaccuracy, and the effectiveness of the proposed algorithm is verified through simulation experiments. Summary of the invention

[0004] In order to solve the problem of strict path requirements in practical applications, a simple position control robot arm will produce a certain deviation between the theoretical trajectory and the actual trajectory. In the case of high-speed or heavy-load operation, the deviation on the motion path will be very large, and even strong jitter will occur. The purpose of the present invention is to provide a robot arm control system based on robust adaptive control, which has the function of solving the uncertainty in the system and realizing the precise tracking of the robot arm trajectory. And it is achieved through the following technical solutions:

[0005] A robotic arm control system based on robust adaptive control mainly includes an n-degree-of-freedom robotic arm, characterized in that mathematical modeling is first performed on the robotic arm, mainly taking a two-degree-of-freedom robotic arm as an example, and the robotic arm is modeled according to the Lagrangian method based on the kinematic characteristics of the robotic arm. Then, the stability of the system is judged by the Lyapunov theory, the reference trajectory of the robotic arm is set, and simulation verification is performed to verify the effectiveness of the control method.

[0006] Consider the target workspace as the plane formed by the x-axis and the y-axis. The two-degree-of-freedom robot is in the xy plane, and gravity is along the -y direction. Convert the rectangular coordinates (x, y) of the joint end node in the workspace to the joint angle position q 1 ,q 2 problem in order to establish the dynamic model of the robotic arm.

[0007] like Figure 1 As shown, (x 1 ,y 1 ) is the end coordinate of connecting rod 1, then the linear velocity of connecting rod 1 is:

[0008]

[0009]

[0010] like Figure 1 As shown, (x 2 ,y 2 ) is the end coordinate of connecting rod 2, and the linear velocity of connecting rod 2 is:

[0011]

[0012]

[0013] After calculation, the kinetic energy of each connecting rod is obtained from formula (3)-(3):

[0014]

[0015]

[0016] The potential energy of each connecting rod is:

[0017] u 1 =m 1 gy 1 =m 1 gl 1 sinq 1 (7)

[0018] u 2 =m 2 gy 2 =m2 gl 1 sinq 1 +m 2 l 2 sin(q 1 +q 2 ) (8)

[0019] u 1 is the potential energy of connecting rod 1, u 2 is the potential energy of connecting rod 2

[0020] The Lagrange equation is:

[0021]

[0022] Using kinematics, we can find the torque as:

[0023]

[0024]

[0025] The state space equation is:

[0026]

[0027]

[0028]

[0029] According to the above analysis and calculation, the dynamic equation of the two-degree-of-freedom manipulator is:

[0030]

[0031] Among them, q is the joint angular displacement, τ is the joint torque, D(q) is the n×n mass matrix of the robot, are the centrifugal force and Coriolis force terms, and G(q) is the gravity term.

[0032] The present invention uses a robust adaptive control robot control system, which has the function of solving the uncertainty in the system and realizing the precise tracking of the robot trajectory. As shown in the figure, the robot is modeled by the Lagrangian method, and then the stability of the system is judged by the Lyapunov theory, and then the controller design with bounded interference and unbounded interference is set based on the robust adaptive theory, and has a good trajectory tracking effect.

[0033] In summary, due to the adoption of the above technical solution, the beneficial effects of the present invention are:

[0034] 1. The present invention adopts a robust adaptive control robot control system to achieve trajectory tracking of the robot under parameter uncertainty and effectively improve the accuracy of controller motion control.

[0035] 2. Robust controllers can compensate for structural uncertainty and unstructured uncertainty in the system, such as suppressing external interference and compensating for unmodeled dynamic uncertainty, and are easy to implement. Adaptive controllers can modify their own characteristic parameters to adapt to changes in object dynamic characteristics and disturbances. The control system is simple and has excellent dynamic performance. BRIEF DESCRIPTION OF THE DRAWINGS

[0036] Figure 1 Schematic diagram of the robotic arm used in the present invention.

[0037] Figure 2 This is a simulation diagram of the robotic arm used in the present invention.

[0038] Figure 3 This is a simulation diagram of the joint trajectory tracking of the upper-bounded robotic arm of the present invention.

[0039] Figure 4 This is a graph of the upper bound parameter approximation result of the present invention.

[0040] Figure 5 This is a simulation diagram of the upper-bound robot arm joint trajectory tracking after changing parameters of the present invention.

[0041] Figure 6 This is a diagram of parameter approximation results after changing parameters in the present invention.

[0042] Figure 7 This is a simulation diagram of the unbounded robotic arm joint trajectory tracking of the present invention.

[0043] Figure 8 This is the result diagram of the upper bound parameter approximation of the present invention.

[0044] Fig. 9 This is a simulation diagram of the joint trajectory tracking of the unbounded robotic arm after changing parameters of the present invention.

[0045] Fig.10 This is a graph of parameter approximation results after changing parameters without an upper bound in the present invention. DETAILED DESCRIPTION

[0046] In order to make the purpose, technical solution and advantages of the present invention more clearly understood, the present invention is further described in detail below in conjunction with the accompanying drawings and embodiments. It should be understood that the specific embodiments described herein are only used to explain the present invention and are not intended to limit the present invention.

[0047] A robotic arm control system based on robust adaptive control, characterized in that it comprises the following steps:

[0048] S1. Establish the dynamic model of the robotic arm;

[0049] S2. Design of robust adaptive controller with bounded disturbance;

[0050] S3. Design a robust adaptive controller with unbounded disturbances;

[0051] S4. Prove stability through Lyapunov theory;

[0052] S5. Adjust parameters according to the designed controller and perform simulation verification.

[0053] Consider the target workspace as the plane formed by the x-axis and the y-axis. The two-degree-of-freedom robot is in the xy plane, and gravity is along the -y direction. Convert the rectangular coordinates (x, y) of the joint end node in the workspace to the joint angle position q 1 ,q 2 problem in order to establish the dynamic model of the robotic arm.

[0054] Connecting rod center of mass linear velocity

[0055] like Figure 1 As shown, (x 1 ,y 1 ) is the end coordinate of connecting rod 1, then the linear velocity of connecting rod 1 is:

[0056]

[0057]

[0058] like Figure 1 As shown, (x 2 ,y 2 ) is the end coordinate of connecting rod 2, and the linear velocity of connecting rod 2 is:

[0059]

[0060]

[0061] After calculation, the kinetic energy of each connecting rod is obtained from formulas (1)-(4):

[0062]

[0063]

[0064] The potential energy of each connecting rod is:

[0065] u 1 =m 1 gy 1 =m i gl 1 sinq1 (7)

[0066] u 2 =m 2 gy 2 =m 2 gl 1 sinq 1 +m 2 l 2 sin(q 1 +q 2 ) (8)

[0067] u 1 is the potential energy of connecting rod 1, u 2 is the potential energy of connecting rod 2

[0068] The Lagrange equation is:

[0069]

[0070] The state space equation is:

[0071]

[0072]

[0073]

[0074] According to the above analysis and calculation, the dynamic equation of the two-degree-of-freedom manipulator is:

[0075]

[0076] Among them, q is the joint angular displacement, τ is the joint torque, D(q) is the n×n mass matrix of the robot, are the centrifugal force and Coriolis force terms, and G(q) is the gravity term.

[0077] We consider that the dynamic performance of an n-joint robot can be described by the following second-order nonlinear differential equation:

[0078]

[0079] Where q is the joint angular displacement, τ is the joint control torque, D(q) is the n×n mass matrix of the manipulator, are the centrifugal force and Coriolis force terms, G(q) is the gravity term, and w is various errors and disturbances.

[0080] Make the following two assumptions:

[0081] Assumption 1: If we represent q d The first and second derivatives of the desired joint angular displacement exist.

[0082] Assumption 2: The norms of the error and disturbance w satisfy:

[0083]

[0084] Among them, d 1 , d 2 , d 3 are positive constants, the error e is the difference between the actual joint angle and the ideal joint angle, is the difference between the derivative of the actual joint angle and the derivative of the ideal joint angle.

[0085] Design of robust adaptive controllers with bounded disturbances:

[0086] For the robot system shown in (12), when the supremum of the error disturbance signal is known, the sufficient condition for the global asymptotic stability of the system is to use the following controller and adaptive law:

[0087]

[0088]

[0089] but The parameter estimation law of is taken as:

[0090]

[0091] in:

[0092]

[0093] K p1 =diag(k p11 , k p12 , …, k p1n ), K p2 =diag(k p21 , k p22 , …, k p2n )

[0094] K v1 =diag(k v11 , k v12 , …, k v1n ), K v2 =diag(k v21 , k v22 , …, k v2n )

[0095]

[0096]

[0097] Among them, kp1i , k p2i , k v2i , k v2i , δ i , σ i (i=1, 2, ..., n) are all numbers greater than zero, and Γ is a positive definite symmetric matrix.

[0098] Design of robust adaptive controller with unbounded disturbances:

[0099]

[0100]

[0101]

[0102]

[0103] Where: K p , K v The value is the same as Theorem 1, the estimated value of P By formula (13), we can get: d = d 1 +d 2 +d 3 , are all arbitrary normal numbers.

[0104] Proven stability:

[0105] set up: Then define the Lyapunov function as:

[0106]

[0107] Since the symmetric positive definite matrix is:

[0108]

[0109]

[0110] Then from equations (14)-(17) we get:

[0111]

[0112] According to formula (10), (11), we can get:

[0113]

[0114]

[0115] Depend on The parameter estimation law is:

[0116]

[0117]

[0118] Substituting (19) and (20) into (18), we get:

[0119]

[0120] By Features is a skew-symmetric matrix:

[0121]

[0122] but:

[0123]

[0124] From formula (5), we can get:

[0125]

[0126] but:

[0127]

[0128] but:

[0129]

[0130] because: Head From formula (2-18), we get:

[0131]

[0132] but:

[0133]

[0134] in:

[0135]

[0136]

[0137] Similarly, we can get:

[0138]

[0139]

[0140] but:

[0141]

[0142] because but:

[0143]

[0144]

[0145] like but:

[0146]

[0147]

[0148] Then we have:

[0149]

[0150] Let k p1i , k p1i , δ i , σ i The value of satisfies:

[0151]

[0152]

[0153] but:

[0154]

[0155] because:

[0156]

[0157] Where:

[0158]

[0159] but:

[0160]

[0161] Right now:

[0162]

[0163] Then, through the definition of Lyapunov function, we can get lim t→∞ e=0,lim t→∞ The controller is globally asymptotically stable.

[0164] The following describes the embodiments of the present invention by specific examples, and those skilled in the art can easily understand other advantages and effects of the present invention from the contents disclosed in this specification. The present invention can also be implemented or applied through other different specific embodiments, and the details in this specification can also be modified or changed in various ways based on different viewpoints and applications without departing from the spirit of the present invention. It should be noted that the following embodiments and features in the embodiments can be combined with each other without conflict.

[0165] Example 1

[0166] S1. Establish the dynamic model of the robotic arm;

[0167] S2. Design of robust adaptive controller with bounded disturbance;

[0168] S3. Design a robust adaptive controller with unbounded disturbances;

[0169] S4. Prove stability through Lyapunov theory;

[0170] S4. Adjust parameters according to the designed controller and perform simulation verification.

[0171] Comparative Example 1

[0172] In order to verify the accuracy of the robot arm motion trajectory tracking using robust adaptive control in the present invention, the parameters of the two connecting rods of the robot arm are adjusted under the condition that the disturbance has an upper bound to conduct simulation comparison results. Figure 3 and Figure 4 shown.

[0173] Assume the simulation data of the robot is: when m 1 =0.5, m 2 =0.5; l 1 =1,l 2 =0.8;

[0174] Disturbance error, desired position control, the initial state of the system is:

[0175] d 1 =2,d 2 =3,d 3 =6,

[0176]

[0177]

[0178] The following conclusions can be drawn by analyzing the experimental results. By observing the control strategy in the figure in Example 1, it can be found that under the initial conditions of the robotic arm, the error of joint 1 controlled by the controller is the largest at t=0, and then the error decreases. After a period of fluctuation, the error approaches 0 after t=2.5s; the error of joint 2 controlled by the controller is the largest at t=0.043s, and then the error decreases. After t=3.18s, the error approaches 0, achieving a good tracking effect.

[0179] like Figure 4 is the parameter p 1 , p 2 , p 3 And its approximation result, the parameter starts from the initial value, first increases and then decreases, and after a period of fluctuation, it stabilizes near a certain value, where p 1 Stable at value 1, p 2 Stable at a value of 0.32, p 3 Stable at a value of 0.4.

[0180] Example 2

[0181] S1. Establish the dynamic model of the robotic arm;

[0182] S2. Design of robust adaptive controller with bounded disturbance;

[0183] S3. Design a robust adaptive controller with unbounded disturbances;

[0184] S4. Prove stability through Lyapunov theory;

[0185] S4. Adjust parameters according to the designed controller and perform simulation verification.

[0186] Comparative Example 2: In order to verify the accuracy of the present invention in tracking the motion trajectory of the robot arm using robust adaptive control, the parameters of the two connecting rods of the robot arm are adjusted under the condition that the disturbance has an upper bound to conduct simulation comparison results. Figure 5 and Figure 6 shown.

[0187] Assume the simulation data of the robot is: when m 1 =2,m 2 =3; l 1 =1,l 2 =0.8;

[0188] Disturbance error, desired position control, the initial state of the system is:

[0189] d 1 =2,d 2 =3,d 3 =6,

[0190]

[0191]

[0192] By observation Figure 6 The experimental results can draw the following conclusions: In Example 2, by comparing with Example 1, it can be found that when the interference is bounded, adjusting the robot arm m 1 , m 2 , the present invention can be seen in different m 1 , m 2 Under the value of , the robust adaptive controller of the present invention can achieve a better control effect on the tracking robotic arm.

[0193] Example 3

[0194] S1. Establish the dynamic model of the robotic arm;

[0195] S2. Design of robust adaptive controller with bounded disturbance;

[0196] S3. Design a robust adaptive controller with unbounded disturbances;

[0197] S4. Prove stability through Lyapunov theory;

[0198] S4. Adjust parameters according to the designed controller and perform simulation verification.

[0199] Comparative Example 3

[0200] In order to verify the accuracy of the robot arm motion trajectory tracking using robust adaptive control in the present invention, the parameters of the two connecting rods of the robot arm are adjusted under the condition of no upper bound on the disturbance to conduct simulation comparison results. Figure 7 and Figure 8 shown.

[0201] Assume the simulation data of the robot is: when m 1 =0.5, m 2 =0.5; l 1 =1,l 2 =0.8;

[0202] Disturbance error, desired position control, the initial state of the system is:

[0203] d 1 =2,d 2 =3,d 3 =6,

[0204]

[0205]

[0206] By observation Figure 7 and Figure 8 The following conclusions can be drawn from the experimental results: under the initial conditions of the robot arm, the error of joint 1 controlled by the controller is the largest at t=0, and then the error decreases. After a period of fluctuation, the error approaches 0 after t=2.5s; the error of joint 2 controlled by the controller is the largest at t=0.043s, and then the error decreases. After a period of fluctuation, the error approaches 0 after t=3.18s, achieving good tracking effect.

[0207] like Figure 8 is the parameter p 1 , p 2 , p 3 And its approximation result, the parameter starts from the initial value, first increases and then decreases, and after a period of fluctuation, it stabilizes near a certain value, where p 1 Stable at value 1, p 2 Stable at a value of 0.32, p 3 Stable at a value of 0.4.

[0208] Example 4

[0209] S1. Establish the dynamic model of the robotic arm;

[0210] S2. Design of robust adaptive controller with bounded disturbance;

[0211] S3. Design a robust adaptive controller with unbounded disturbances;

[0212] S4. Prove stability through Lyapunov theory;

[0213] S4. Adjust parameters according to the designed controller and perform simulation verification.

[0214] Comparative Example 4

[0215] In order to verify the accuracy of the robot arm motion trajectory tracking using robust adaptive control in the present invention, the parameters of the two connecting rods of the robot arm are adjusted under the condition of no upper bound on the disturbance to conduct simulation comparison results. Fig. 9 and Fig.10 shown.

[0216] Assume the simulation data of the robot is: when m 1 =2,m 2 =3; l 1 =1,l 2 =0.8;

[0217] Disturbance error, desired position control, the initial state of the system is:

[0218] d1 =2,d 2 =3,d 3 =6,

[0219]

[0220]

[0221] By observation Fig.10 The experimental results can draw the following conclusions: In Example 4, by comparing with Example 3, it can be found that when the interference has no upper bound, adjusting the robot arm m 1 , m 2 , the present invention can be seen in different m 1 , m 2 Under the value of , the robust adaptive controller of the present invention can achieve a better control effect on the tracking robotic arm.

[0222] The above-mentioned embodiments only express the specific implementation methods of the present invention, and the descriptions thereof are relatively specific and detailed, but they cannot be understood as limiting the protection scope of the present invention. It should be pointed out that, for those skilled in the art, without departing from the concept of the present invention, any other changes, modifications, substitutions, combinations, simplifications made without departing from the spirit and principle of the present invention should be equivalent replacement methods and are included in the protection scope of the present invention.

Claims

1. A robotic arm control system based on robust adaptive control, mainly comprising an n-degree-of-freedom robotic arm, characterized in that: First, mathematical modeling is performed on the robotic arm, mainly taking a two-degree-of-freedom robotic arm as an example. According to the kinematic characteristics of the robotic arm, the robotic arm is modeled using the Lagrangian method. Then, the stability of the system is judged using the Lyapunov theory, the reference trajectory of the robotic arm is set, and simulation verification is performed to verify the effectiveness of the control method.

2. A robotic arm control system based on robust adaptive control according to claim 1, characterized in that: A mathematical model of the robotic arm is established. The robotic arm is mainly modeled for a two-degree-of-freedom robotic arm. The Lagrangian method is used to model the robotic arm, and the kinetic energy of the robotic arm connecting rod needs to be solved. The method for solving the kinetic energy is as follows: Considering that the target workspace is a plane formed by the x-axis and the y-axis, the two-degree-of-freedom robot is in the xy plane, and the gravity is along the -y direction. The problem of converting the rectangular coordinates (x, y) of the joint end nodes in the workspace into the joint angle positions q1, q2 is to establish the dynamic model of the robot. As shown in Figure 1, (x1, y1) is the end coordinate of connecting rod 1, and the linear velocity of connecting rod 1 is: As shown in Figure 1, (x2, y2) is the end coordinate of connecting rod 2, and the linear velocity of connecting rod 2 is: After calculation, the kinetic energy of each connecting rod is obtained from formulas (1)-(4): 。 3. The robotic arm control system based on robust adaptive control according to claim 1, characterized in that: A mathematical model of the robotic arm is established. The robotic arm is mainly modeled for a two-degree-of-freedom robotic arm. The Lagrangian method is used to model the robotic arm, and the potential energy of the robotic arm connecting rod needs to be solved. The method for solving the potential energy is as follows: u1=m1gy1=m1gl1sinq1 (7) u2=m2gy2=m2gl1sinq1+m2l2sin(q1+q2) (8) u1 is the potential energy of connecting rod 1, and u2 is the potential energy of connecting rod 2.

4. The robotic arm control system based on robust adaptive control according to claim 1, characterized in that: The Lagrange equation is established for the given robot arm, and the torque is obtained using kinematics to obtain the dynamic equation of the two-degree-of-freedom robot arm, which is as follows: The Lagrange equation is: Using kinematics, we can find the torque as: The state space equation is: According to the above analysis and calculation, the dynamic equation of the two-degree-of-freedom manipulator is: Among them, q is the joint angular displacement, τ is the joint torque, D(q) is the n×n mass matrix of the robot, are the centrifugal force and Coriolis force terms, and G(q) is the gravity term.

5. The robotic arm control system based on robust adaptive control according to claim 1, characterized in that: A robust adaptive control strategy is adopted for the two-degree-of-freedom manipulator to design a controller, mainly processing the dynamic equations described in claim 4, and setting relevant parameters in the simulation for verification.

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