Joint control simulation method of pneumatic artificial muscle driven robot

Abstract

The application discloses a joint control simulation method of a robot driven by a pneumatic artificial muscle, and comprises the following steps: step 1, establishing a three-element model of a single pneumatic artificial muscle; step 2, calculating an adaptive backstepping sliding mode controller for driving the single pneumatic artificial muscle by using the three-element model; step 3, establishing a joint simulation model for driving the single pneumatic artificial muscle; and step 4, applying the pneumatic artificial muscle to the joint. The application has the characteristics of high control efficiency and high control precision of the pneumatic artificial muscle.

Description

TECHNICAL FIELD

[0001] The application relates to the technical field of flexible robots, in particular to a joint control simulation method of a robot driven by a pneumatic artificial muscle, which has high control precision and good control effect. BACKGROUND

[0002] With the continuous development of automation technology, the application field of flexible robots is more and more extensive. The pneumatic artificial muscle becomes a commonly used driver of the flexible robot due to the advantages of fast response speed, light weight, strong shape variability, high energy utilization efficiency and strong applicability. However, the pneumatic artificial muscle has the defects of slow control speed and high control difficulty. SUMMARY

[0003] The application aims at overcoming the defects of slow control speed and high control difficulty of the pneumatic artificial muscle in driving the joint to move in the prior art, and provides a joint control simulation method of a robot driven by a pneumatic artificial muscle, which has high control precision and good control effect.

[0004] In order to achieve the above-mentioned purpose, the application adopts the following technical scheme:

[0005] A joint control simulation method of a robot driven by a pneumatic artificial muscle comprises the following steps:

[0006] Step 1: establishing a three-element model of a single pneumatic artificial muscle;

[0007] Step 2: calculating an adaptive backstepping sliding mode controller for driving the single pneumatic artificial muscle by using the three-element model;

[0008] Step 3: building a joint simulation model for driving the single pneumatic artificial muscle;

[0009] Step 4: applying the pneumatic artificial muscle to the joint.

[0010] The three-element model is used as the mathematical model of the pneumatic artificial muscle in the application, which is composed of only three elements, is intuitive and easy to understand and use, can be combined with other models to build a more complex artificial muscle model, has strong scalability, and is relatively simple in mathematical equation, easy to calculate and implement; the three-element model is based on the physical control theory, and has better interpretability and practicability;

[0011] The pneumatic artificial muscle is used to simulate the extension and contraction of human muscles, and is relatively simple in structure compared with other models, and the difficulty of mathematical modeling of the model is reduced by using the knowledge of dynamics; the self-adaptive backstepping sliding mode controller has good control performance in controlling a single pneumatic artificial muscle and a joint model, improves the control of the pneumatic artificial muscle, improves the intelligent degree of the flexible robot, and effectively solves the problems of poor control efficiency and low control precision of the pneumatic artificial muscle.

[0012] As preferred,

[0013] The three-element model can be described as the following mathematical expression:

[0014]

[0015] When the pneumatic artificial muscle contracts: B(P) = B0 + B1P

[0016] When the pneumatic artificial muscle extends: B(P) = B2 + B3P

[0017] F(P) = F0 + F1P

[0018] Wherein, M is the mass of the load, g is the acceleration of gravity, P represents the input air pressure, F(P) is the effective force generated by the contraction unit, B(P) is the damping coefficient, K(P) is the spring coefficient, F(P), B(P) and K(P) are all first-order functions of pressure P; B0, B1, B2, B3, F0 and F1 are all constants; x is the displacement of the pneumatic artificial muscle, when x is in the initial state, x = 0; is the first derivative of x, which represents the speed of movement; is the second derivative of x, which represents the acceleration of movement.

[0019] As preferred, the step 2 comprises the following steps:

[0020] The three-element model is expressed as a state space expression:

[0021]

[0022]

[0023]

[0024] Wherein, x1 is the actual displacement of the pneumatic artificial muscle when moving, is the first derivative of x1, which represents the speed of the pneumatic artificial muscle when moving; x2 is the actual speed of the pneumatic artificial muscle when moving, x2 is the first derivative, representing the acceleration of the pneumatic artificial muscle during movement; u1 is the input air pressure P of the pneumatic artificial muscle; K0 and K1 are both positive constants, and d is the total disturbance during joint movement.

[0025] The joint position tracking error Z1 is set as follows:

[0026] Z1 = x1 - x 1d

[0027] Where, x 1d This represents the desired displacement during the movement of the pneumatic artificial muscle.

[0028] Then we have:

[0029]

[0030] in, The first derivative of Z1, For x 1d The first derivative of , i.e. the desired velocity during the movement of pneumatic artificial muscles;

[0031] Define the Lyapunov function V:

[0032]

[0033] Then we have:

[0034]

[0035] in, The first derivative of V;

[0036] Introduce a dummy variable α1, and set the state error between x2 and α1 to Z2, that is:

[0037] Z2=x2-α1

[0038] Let α1 be:

[0039]

[0040] At this point:

[0041]

[0042] Then we have:

[0043]

[0044] If Z2→0, that is, x2→α1, then we have:

[0045]

[0046] According to LaSalle invariance principle, when time t→∞, there is Z1→0, so x1→x 1d , the actual output of three-element model can track the predetermined trajectory, and the three-element model is stable; but there is not enough condition to get Z2→0, so it cannot be guaranteed that the three-element model is stable, at this time, the sliding mode function s is set as:

[0047]

[0048] wherein c is the sliding mode function coefficient, and is a positive constant;

[0049] Combined with the state space expression of three-element model, then there is:

[0050]

[0051] wherein, is the first order derivative of s, is the second order derivative of x 1d , that is, the expected acceleration when the pneumatic artificial muscle moves;

[0052] When t→∞, if s→0, there is Z1→0, and then Z2→0, so it can be concluded that the adaptive backstepping sliding mode controller is stable; at the same time, in order to avoid using the upper bound of total disturbance d, the parameter adaptive control is used to estimate it; it is assumed that the change of total uncertainty of adaptive backstepping sliding mode controller is very slow, that is, the first order derivative of d Therefore, a new Lyapunov function V1 is set as:

[0053]

[0054] wherein, is the estimation error of d, and has is the estimated value of d, and μ is a positive constant, so there is:

[0055]

[0056] wherein, is the first order derivative of V1, is the first order derivative of , is the first order derivative of d, is the first order derivative of , and is the first order derivative of ;

[0057] Since the change of total uncertainty is very slow, that is, so there is:

[0058]

[0059] Combined with Then we have:

[0060]

[0061] Further, the output u1 of the adaptive backstepping sliding mode controller is set as:

[0062]

[0063] wherein β and k2 are positive constants, sgn(s) is a sign function of s, -k2(s+βsgn(s)) is an exponential reaching law of the sliding mode controller, and the adaptive law is set as:

[0064]

[0065] The set controller output u1 and adaptive law are substituted into the function , and we have:

[0066]

[0067] wherein:

[0068] |s| is the absolute value of s;

[0069] If A is a positive definite matrix, then Further, we have:

[0070]

[0071] wherein |A| is the determinant of A; from the above formula, it can be seen that A can be a positive definite matrix by properly selecting the values of c, k2 and k1, and further, we have When t→∞, s→0, and thus Z1→0, and we have x1→x 1d Therefore, the set controller output u1 and adaptive law can guarantee the stability of the three-element model.

[0072] As a preferred, the step 3 comprises the following steps:

[0073] The joint simulation model is built by using the Simulink module of MATLAB:

[0074] Firstly, the desired trajectory x 1d of the joint simulation model is set by using the Signal module of Simulink, and the desired trajectory x 1d is differentiated by using the Derivative module of Simulink, and we have and The adaptive backstepping sliding mode controller and the three-element model are written by using a Function module of Simulink, parameters in the adaptive backstepping sliding mode controller and the three-element model are set by using a Constant module of Simulink, the output u1 of the adaptive backstepping sliding mode controller is limited by using a Saturation limiting module of Simulink, so as to prevent the output u1 from exceeding the working air pressure of the pneumatic artificial muscle, the total disturbance d of the controlled object is simulated by using a Uniform Random Number module of a random function generator, and the output of the adaptive backstepping sliding mode controller is limited by using a Saturation limiting module of Simulink After being integrated by an Integrator module of Simulink, x2 is obtained, the actual running track x1 of the pneumatic artificial muscle is obtained after being integrated again, and finally, the simulation results are recorded by using a Scope module of Simulink.

[0075] As preferred, the step 4 comprises the following steps:

[0076] Step 4-1, a joint mathematical model of a robot driven by the pneumatic artificial muscle is established.

[0077] Step 4-2, the adaptive backstepping sliding mode controller is improved according to different control characteristics of the joint object, and is applied to the joint mathematical model.

[0078] Therefore, the application has the beneficial effects that the control of the pneumatic artificial muscle is improved, the intelligent degree of the flexible robot is effectively improved, the control efficiency of the pneumatic artificial muscle is high, and the control precision is high. BRIEF DESCRIPTION OF DRAWINGS

[0079] Figure 1 is a flow chart of the application;

[0080] Figure 2 is a three-element model diagram of the application;

[0081] Figure 3 is a processor output waveform diagram under a step signal of the application;

[0082] Figure 4 is a processor output waveform diagram under a sine signal of the application;

[0083] Figure 5 is a robot joint model output waveform diagram under a step signal of the application;

[0084] Figure 6 is a robot joint model output waveform diagram under a half-sine signal of the application. DETAILED DESCRIPTION

[0085] The application will be further described in conjunction with the drawings and specific embodiments.

[0086] As Figure 1 shown in the embodiment is a joint control simulation method of a pneumatic artificial muscle driven robot, comprising the following steps:

[0087] Step 1, establishing a three-element model of a single pneumatic artificial muscle:

[0088] As Figure 2 shown, the three-element model can be described as the following mathematical expression:

[0089]

[0090] When the pneumatic artificial muscle contracts: B(P) = B0 + B1P

[0091] When the pneumatic artificial muscle extends: B(P) = B2 + B3P

[0092] F(P) = F0 + F1P

[0093] Wherein, M is the mass of the load, g is the gravity acceleration, P represents the input air pressure, F(P) is the effective force generated by the contraction unit, B(P) is the damping coefficient, K(P) is the spring coefficient, F(P), B(P) and K(P) are all linear functions of pressure P; B0, B1, B2, B3, F0 and F1 are all constants; x is the displacement of the pneumatic artificial muscle, when x is in the initial state, x = 0; is the first derivative of x, indicating the speed of movement; is the second derivative of x, indicating the acceleration of movement.

[0094] The specific values of the above parameters can be fitted out through experimental data. When identifying F(P) and K(P), the first and second derivatives of x can be both set to 0, then the relationship between the displacement and the load under the same air pressure is measured first, and the least square method can be used to fit a value of F(P) and K(P) under the air pressure, then the values of F(P) and K(P) under different air pressures are measured, so that the relationship between F(P) and K(P) and air pressure can be obtained, thereby identifying F(P) and K(P). For B(P), it needs to be measured respectively during the contraction and extension processes. First, a given air pressure is provided to make the pneumatic muscle in the contraction state, then the air pressure is suddenly reduced, the data relationship between the displacement and the pressure in this process is recorded, and the values of F(P) and K(P) obtained before are used, so that B(P) can be calculated. Similarly, the measurement method of B(P) during extension is similar. Taking the pneumatic artificial muscle produced by FESTO company with model DMSP-20-400N-RM-CM as an example, the final accurate model is as follows:

[0095]

[0096] B(P) = 132.51 - 17.34P (contraction)

[0097] B(P) = 129.924 - 10.433P (elongation)

[0098] F(P) = 8.9706 + 27.134P

[0099]

[0100] Step 2, using a three-element model to calculate the adaptive backstepping sliding mode controller for driving a single pneumatic artificial muscle:

[0101] Express the three-element model as a state space expression:

[0102]

[0103]

[0104]

[0105] where x1 is the actual displacement of the pneumatic artificial muscle when moving, is the first derivative of x1, indicating the speed of the pneumatic artificial muscle when moving; x2 is the actual speed of the pneumatic artificial muscle when moving, is the first derivative of x2, indicating the acceleration of the pneumatic artificial muscle when moving; u1 is the input air pressure P of the pneumatic artificial muscle; K0 and K1 are both positive constants, and d is the total disturbance when the joint is running;

[0106] Set the position tracking error Z1 of the joint as:

[0107] Z1 = x1 - x 1d

[0108] where x 1d is the desired displacement of the pneumatic artificial muscle when moving;

[0109] Then:

[0110]

[0111] where, is the first derivative of Z1, is the first derivative of x 1d , i.e., the desired speed of the pneumatic artificial muscle when moving;

[0112] Set the Lyapunov function V:

[0113]

[0114] Then we have:

[0115]

[0116] where, is the first derivative of V;

[0117] Introduce a virtual variable α1, and set the state error of x2 and α1 as Z2, i.e.,

[0118] Z2=x2-α1

[0119] Set α1 as:

[0120]

[0121] At this time we have:

[0122]

[0123] Then we have:

[0124]

[0125] If Z2→0, i.e., x2→α1, at this time we have:

[0126]

[0127] According to the LaSalle invariance principle, when time t→∞, we have Z1→0, so x1→x 1d , the actual output of the three-element model can track the predetermined trajectory, and the three-element model is stable; but there is not enough condition to get Z2→0, which cannot guarantee the stability of the three-element model, at this time, set the sliding mode function s as:

[0128]

[0129] where c is the sliding mode surface coefficient, which is a positive constant;

[0130] Combined with the state space expression of the three-element model, we have:

[0131]

[0132] where, is the first derivative of s, is the second derivative of x 1d , i.e., the expected acceleration when the pneumatic artificial muscle moves;

[0133] When t→∞, if s→0, we have Z1→0, Further, Z2→0, it can be concluded that the adaptive backstepping sliding mode controller is stable; at the same time, in order to avoid using the upper bound of the total disturbance d, the parameter adaptive control is used to estimate it; assuming that the change of the total uncertainty of the adaptive backstepping sliding mode controller is very slow, i.e. the first derivative of d Therefore, a new Lyapunov function V1 is set as:

[0134]

[0135] Wherein, is the estimation error of d, and has is the estimated value of d, and μ is a positive constant, then:

[0136]

[0137] Wherein, is the first derivative of V1, is the first derivative of , is the first derivative of d, is the first derivative of , is the first derivative of ;

[0138] Since the change of the total uncertainty is very slow, i.e. then:

[0139]

[0140] Combined with then:

[0141]

[0142] Further, the output u1 of the adaptive backstepping sliding mode controller is set as:

[0143]

[0144] Wherein, β and k2 are both positive constants, sgn(s) is the sign function of s, -k2(s+βsgn(s)) is the exponential approach law of the sliding mode controller, and the adaptive law is set as:

[0145]

[0146] The set controller output u1 and adaptive law are substituted into the function , then:

[0147]

[0148] Wherein:

[0149] |s| is the absolute value of s;

[0150] If A is a positive definite matrix, then Also:

[0151]

[0152] where |A| is the determinant of A; from the above formula, as long as the values of c, k2, k1 are appropriately selected, A can be a positive definite matrix, and thus When t→∞, s→0, so Z1→0, and thus x1→x 1d Therefore, the controller output u1 and the adaptive law set can guarantee the stability of the three-element model.

[0153] Step 3, build a joint simulation model driven by a single pneumatic artificial muscle;

[0154] The joint simulation model is built using the Simulink module of MATLAB:

[0155] First, use the Signal module of Simulink to set the desired trajectory x 1d of the joint simulation model, and use the Derivative module of Simulink to differentiate the desired trajectory x 1d , respectively, to obtain and The adaptive backstepping sliding mode controller and the three-element model are written using the Function module of Simulink, and the parameters in the adaptive backstepping sliding mode controller and the three-element model are set using the Constant module of Simulink. The output u1 of the adaptive backstepping sliding mode controller is limited using the Saturation limiting module of Simulink to prevent it from exceeding the working air pressure of the pneumatic artificial muscle. The total disturbance d of the controlled object is simulated by the Uniform Random Number module of the random function generator. The output of the adaptive backstepping sliding mode controller is integrated by the Integrator module of Simulink to obtain x2, and is integrated again to obtain the actual running trajectory x1 of the pneumatic artificial muscle. Finally, the simulation results are recorded using the Scope module of Simulink.

[0156] In this embodiment, the traditional sliding mode control, backstepping sliding mode controller, and the adaptive backstepping sliding mode controller of the present application are compared and analyzed as follows:

[0157] First, when the three controllers track the step signal x = 0.02m, the output waveform is as follows: Figure 3The performance indexes of the stability, the rapidity and the accuracy of the test model commonly used in the step response are adopted to embody the control effect. Table 1 is the specific parameter values of the performance indexes.

[0158] Table 1 Parameter values of the performance indexes of the tracking step signal

[0159]

[0160] As shown in Table 1, the three controllers all achieve the index of stability. In the aspect of the rise time, the rise time of the sliding mode control is the shortest, only 0.37s, because the sliding mode control has the least regulation parameters. The rise time of the backstepping sliding mode controller increases by 0.25s compared with the sliding mode control, because the backstepping control increases the regulation parameters. The rise time of the adaptive backstepping sliding mode controller is the longest, 0.75s, because the adaptive law design changes the structure parameters. In the aspect of the overshoot and the steady-state error, the sliding mode control has the largest overshoot, 25%, and the largest steady-state error, 0.13cm, because of the difficulty in parameter adjustment, chattering and other drawbacks. The overshoot of the backstepping sliding mode controller reduces by 9%, and the steady-state error reduces by 0.065cm, because the backstepping control improves the anti-interference ability. However, there is still a large error.

[0161] As for the adaptive backstepping sliding mode controller, there is almost no chattering, the overshoot is 0%, and the steady-state error is the smallest, only 0.02cm, because the adaptive control can estimate the total uncertainty of the three-element model, and eliminates the inherent drawbacks of the sliding mode control. The adaptive ability to different types of disturbances is enhanced, and the tracking performance is the best, the adjustment time is only 1.2s. Undoubtedly, the control effect of the adaptive backstepping sliding mode controller is the best.

[0162] Then, the output waveform of the controller tracking the sinusoidal signal x = 0.02sinx + 0.02 is as shown in Figure 4 The tracking performance is shown in the following table.

[0163] Table 2 Parameter values of the performance indexes of the tracking sinusoidal signal

[0164]

[0165] The rise time is the time when the actual output of each controller first tracks the sinusoidal signal, and the peak error represents the maximum error after the output tracks the sinusoidal signal.

[0166] From Table 2, when tracking the sine signal, the three controllers also achieve the performance index of model stability, the rise time of the sliding mode control is the shortest, only 0.52s, followed by the backstepping sliding mode controller, because the adjustment parameters are more, so the rise time increases by 0.22s compared with the sliding mode control, and the adaptive backstepping sliding mode controller has the longest rise time, which is 0.9s. In terms of peak error and steady-state error, the sliding mode control has the largest peak error and steady-state error due to the difficulty of parameter adjustment, chattering and other disadvantages, which are 0.11cm and 0.08cm respectively. The backstepping sliding mode controller has improved anti-interference ability, and the steady-state error is reduced by 0.054cm, and the peak error is reduced by 0.063cm.

[0167] The adaptive backstepping sliding mode controller can adapt to different types of disturbances, so the adjustment time is almost the same as the rise time, and the peak error and steady-state error are the smallest, which are 0.01cm and 0.002cm respectively, and it can also be concluded that the control performance of the adaptive backstepping sliding mode controller is the best.

[0168] Step 4, apply the pneumatic artificial muscle to the lower limb joint:

[0169] The lower limb joint simulation model mainly consists of a pneumatic artificial muscle, a spring and a movable pulley. The spring provides an extension force to make the rotation angle larger, and the movable pulley can convert the extension of the pneumatic artificial muscle in the radial direction into the size of the rotation angle, so as to simulate the process of human muscle stretching to drive the forward and backward movement of the lower limbs.

[0170] Step 4-1, establish a mathematical model of the lower limb joint of the robot driven by the pneumatic artificial muscle:

[0171]

[0172] Where m is the mass of the pulley, r is the radius of the pulley, K s is the spring coefficient of the spring, and a is the angular acceleration of the joint rotation.

[0173] Step 4-2, improve the adaptive backstepping sliding mode controller and apply it to the mathematical model of the lower limb joint:

[0174] The adaptive backstepping sliding mode controller driving the lower limb joint model has the following improvements compared with the adaptive backstepping sliding mode controller driving a single pneumatic artificial muscle in step 2:

[0175] Firstly, the output of the lower limb joint simulation model is the actual angle of the long rod rotated by the moving pulley, so the given predetermined trajectory needs to be improved to the predetermined angle of the long rod rotation; secondly, the mathematical model of the controlled object should be established by the mathematical relationship between the input air pressure P of the pneumatic artificial muscle and the actual rotation angle of the long rod, so the controller function of the controlled object needs to be re-established by using the rigid body fixed axis rotation law from the perspective of the dynamic model; finally, the given frequency and amplitude of the random signal used to simulate the external disturbance in the simulation model will increase because the external disturbance applied to the pneumatic artificial muscle applied to the joint model increases.

[0176] The output function u2 of the adaptive backstepping sliding mode controller for driving the lower limb joint model is:

[0177]

[0178] Wherein, the setting, derivation and stability verification process of the adaptive backstepping sliding mode controller are consistent with step 2, except that the original position signal is changed to the angle signal as the state variable.

[0179] The output waveform of the oscilloscope is analyzed under the input signal of the half-sine signal and the step signal, and the good control performance of the adaptive backstepping sliding mode controller in the lower limb joint mathematical model is verified.

[0180] When tracking the step signal of x=1 rad, the output waveform of the oscilloscope is as shown in Figure 5 It can be seen that when controlling the mathematical model of the lower limb joint, the controller also meets the stability index, and compared with the sliding mode control and the backstepping sliding mode controller, the chattering and overshoot of the adaptive backstepping sliding mode controller are almost eliminated, and there is almost no steady-state error, which also proves that the adaptive backstepping sliding mode control has better control performance.

[0181] When tracking the half-sine signal of x=2sint(rad), the output of the oscilloscope is as shown in Figure 6 It can also be seen that the adaptive backstepping sliding mode controller has good control effect in a certain range of rotation angle when controlling the mathematical model of the lower limb joint.

[0182] Therefore, it can be seen that in the present embodiment, the adaptive backstepping sliding mode controller is used to control the single pneumatic artificial muscle and the mathematical model of the lower limb joint driven by the pneumatic artificial muscle, and the best control effect is obtained.

[0183] The above only describes the preferred embodiments of the present application, and does not limit the present application, any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present application should be included in the protection scope of the present application.

Claims

1. A method for simulating the control of a joint of a pneumatic artificial muscle driven robot, characterized by, The method comprises the following steps: Step 1, establishing a three-element model of a single pneumatic artificial muscle: The three-element model can be described as the following mathematical expression: When the pneumatic artificial muscle contracts: B(P) = B0 + B1P When the pneumatic artificial muscle extends: B(P) = B2 + B3P F(P) = F0 + F1P Wherein, M is the mass of the load, g is the acceleration of gravity, P represents the input air pressure, F(P) is the effective force generated by the contraction unit, B(P) is the damping coefficient, K(P) is the spring coefficient, F(P), B(P) and K(P) are all first-order functions of pressure P; B0, B1, B2, B3, F0 and F1 are all constants; x is the displacement of the pneumatic artificial muscle, when x is in the initial state, x=0; is the first derivative of x, indicating the speed of movement; is the second derivative of x, indicating the acceleration of movement; Step 2, calculating an adaptive backstepping sliding mode controller for driving the single pneumatic artificial muscle by using the three-element model: The three-element model is expressed as a state space expression: wherein x1 is the actual displacement of the pneumatic artificial muscle during movement, is the first derivative of x1, indicating the speed of the pneumatic artificial muscle during movement; x2 is the actual speed of the pneumatic artificial muscle during movement, is the first derivative of x2, indicating the acceleration of the pneumatic artificial muscle during movement; u1 is the input air pressure P of the pneumatic artificial muscle; K0 and K1 are both positive constants, and d is the total disturbance during joint movement. The position tracking error Z1 of the joint is set as: Z1 = x1 - x 1d where x 1d is the desired displacement of the pneumatic artificial muscle during movement; Then, wherein is the first derivative of Z1, is the first derivative of x 1d , i.e. the desired velocity of the pneumatic artificial muscle during movement. The Lyapunov function V is set as: Then, wherein is the first derivative of V; A virtual variable alpha1 is introduced, and the state error Z2 of x2 and alpha1 is set as: Z2 = x2 - alpha1 Alpha1 is set as: At this time, Then, If Z2 tends to 0, i.e., x2 tends to alpha1, at this time, According to LaSalle invariance principle, when time t→∞, there is Z1→0, so x1→x 1d , the actual output of the three-element model can track the predetermined trajectory, and the three-element model is stable; but there is not enough condition to get Z2→0, which cannot guarantee the stability of the three-element model, at this time, the sliding mode function s is set as: Where c is a sliding mode surface coefficient, which is a positive constant; in combination with the state space expression of the three-element model, then, wherein is the first derivative of s, is the second derivative of x 1d is the second derivative of s, i.e. the desired acceleration during the movement of the pneumatic artificial muscle. When t→∞, if s→0, Z1→0, Further, Z2→0, it can be concluded that the adaptive backstepping sliding mode controller is stable; at the same time, in order to avoid using the upper bound of the total disturbance d, the parameter adaptive control is used to estimate it; assuming that the change of the total uncertainty of the adaptive backstepping sliding mode controller is very slow, i.e. the first derivative of d Therefore, a new Lyapunov function V1 is set as: wherein is the estimation error of d, and has is the estimation value of d, μ is a positive constant, then has: wherein is the first derivative of V1, is the first derivative of is the first derivative of is the first derivative of d, is the first derivative of is the first derivative of Since the total uncertainty changes very slowly, i.e. then we have: Combining then there is: Further, the output u1 of the adaptive backstepping sliding mode controller is set as: Where beta and k2 are both positive constants, sgn(s) is a sign function about s, -k2(s + beta sgn(s)) is an exponential type reaching law of the sliding mode controller, and the adaptive law is set as: The set controller output ui and the adaptive law Substitute the function Then, we have: Where: |s| is the absolute value of s; If A is a positive definite matrix, then Also, where |A| is the determinant of A; from the above equation, if the values of c, k2 and k1 are properly selected, A can be a positive definite matrix, and thus When t→∞, s→0, so that Z1→0, and thus x1→x 1d Therefore, the controller output u1 and the adaptive law can guarantee the stability of the three-element model. Step 3, establishing a joint simulation model for driving the single pneumatic artificial muscle: The joint simulation model is established by using the Simulink module of MATLAB: First, the desired trajectory x of the joint simulation model is set using Simulink's Signal module. 1d Using Simulink's Derivative module to determine the desired trajectory x 1d By performing differentiation, we obtain the following results: and The adaptive backstepping sliding mode controller and the three-element model were written using Simulink's Function module. The parameters in both the controller and the model were set using Simulink's Constant module. Simulink's Saturation limiting module was used to limit the output u1 of the adaptive backstepping sliding mode controller to prevent it from exceeding the working air pressure of the pneumatic artificial muscle. The total disturbance d of the controlled object was simulated using the Uniform Random Number module. The output of the adaptive backstepping sliding mode controller... After integration using Simulink's Integrator module, x2 is obtained. After further integration, the actual trajectory x1 of the pneumatic artificial muscle is obtained. Finally, the simulation results are recorded using Simulink's Scope module. Step 4, applying the pneumatic artificial muscle to the joint: Step 4-1, establishing a mathematical model of a robot driven by the pneumatic artificial muscle; Step 4-2, according to the different control characteristics of the joint object, the adaptive backstepping sliding mode controller is improved, and is applied to the mathematical model of the joint.

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