Event-triggered double-power fractional-order sliding mode control method for lower limb exoskeletons

Abstract

The present invention is an event-triggered double-power fractional-order sliding mode control method for a lower limb exoskeleton. The control method includes the following steps: establishing a dynamic model of the lower limb exoskeleton using the Lagrangian equation; obtaining the lower limb movement data of a healthy human body using sensors, and obtaining the desired motion trajectories of the hip joint angle, hip joint angular velocity, knee joint angle, and knee joint angular velocity of the lower limb exoskeleton through function fitting; designing a double-power fractional-order sliding mode control law, and constructing an event-triggered mechanism based on the tracking error between the desired trajectory and the actual trajectory, and finally obtaining an event-triggered double-power fractional-order sliding mode controller, referred to as the controller for short; using the Lyapunov theory to prove that the event-triggered double-power fractional-order sliding mode controller is asymptotically stable. On the premise of ensuring the control accuracy, save communication resources as much as possible, reduce the wear of the actuator, and weaken the chattering existing in the traditional sliding mode control.

Description

Technical Field

[0001] The present invention relates to the field of motion control of lower limb exoskeleton robots, and specifically to a double-power fractional-order sliding mode control method for lower limb exoskeletons based on event triggering. Background Technique

[0002] Lower limb exoskeletons are typical non-linear and time-varying systems with limited joint space and activity space, and there are uncertain factors such as system modeling errors, high-frequency characteristics, joint friction, and signal detection errors. It is difficult to represent the dynamic performance of the system with an accurate mathematical model. These objective factors will cause the performance of the control system of the lower limb exoskeleton to deteriorate, resulting in the inability of conventional control techniques to meet the control requirements well. Therefore, the research on the control strategy of lower limb exoskeletons is very important.

[0003] Since sliding mode control can overcome the uncertainty of the system and has strong robustness to disturbances and unmodeled dynamics, especially has good control effects on the control of non-linear systems, sliding mode control has been widely used in the control field of lower limb exoskeletons. However, when the system state trajectory reaches the sliding mode surface, it is difficult to strictly slide along the sliding mode surface towards the equilibrium point, but approaches the equilibrium point by crossing back and forth on both sides of it. Therefore, chattering will occur in the actual application of sliding mode control. How to eliminate the chattering existing in sliding mode control has become a research hotspot in recent years.

[0004] In the research on the control of lower limb exoskeletons, most of them adopt time-triggered sampling methods, that is, the control signal is updated periodically at a given step size. This sampling method will cause waste of resources when the system tends to be stable. The event-triggered mechanism can reduce resource waste. Only when the system meets the preset conditions, the signal will be updated, which can save resources while ensuring the control effect. The mechanism of action of the event-triggered mechanism is: when the system remains stable, the control input is not updated; when the system tends to be unstable, the control input is updated. Since the control law does not need to be updated at all times, the amount of data transmitted and calculated inside the system is small, which can effectively save system resources and reduce the wear of the actuator. Summary of the Invention

[0005] In order to save communication resources as much as possible and reduce the wear of the actuator on the premise of ensuring control accuracy; at the same time, in order to weaken the chattering existing in traditional sliding mode control, the present invention proposes a double-power fractional-order sliding mode control method for lower limb exoskeletons based on event triggering.

[0006] The present invention is realized through the following technical solutions:

[0007] A double-power fractional-order sliding mode control method for lower limb exoskeletons based on event triggering, the control method includes the following steps:

[0008] Step 1: Establish a lower limb exoskeleton dynamics model using the Lagrange equation;

[0009] Step 2: Obtain the lower limb motion data of a healthy human body using sensors, and through function fitting, obtain the desired motion trajectories of the hip joint angle, hip joint angular velocity, knee joint angle, and knee joint angular velocity of the lower limb exoskeleton;

[0010] Step 3: Design a double-power fractional-order sliding mode control law, and based on the tracking error between the desired trajectory and the actual trajectory, construct an event-triggering mechanism, and finally obtain an event-triggered double-power fractional-order sliding mode controller, referred to as the controller;

[0011] The double-power fractional-order sliding mode control law is:

[0012]

[0013] where s is the sliding surface, c, k 1 , k 2 , β 1 , β 2 , α are all double-power fractional-order sliding mode control parameters, c is the sliding surface parameter, and c > 0, k 1 > 0, k 2 > 0, β 1 > 1, 0 < β 2 < 1; D α represents the fractional calculus operator, α represents the order of the fractional calculus; e is the tracking error; and are the first derivative and second derivative of the tracking error; M is the inertia matrix, G is the gravity-related term matrix, C is the Coriolis force and centrifugal force-related term matrix; sgn is the sign function; τ is the output torque of the controller, that is, the input torque of the controlled object (exoskeleton);

[0014] Define the tracking error e(t) of each joint:

[0015] e(t) = q d (t) - q(t) (5)

[0016] where, q d (t) is the desired angle of the joint, q(t) is the actual tracking angle, and t is the time; the desired trajectory is the input of the controller, and the actual trajectory is obtained by the action of the output torque of the controller on the controlled object;

[0017] The event-triggering mechanism is:

[0018]

[0019] where, t k , k ∈ Z +is the triggering moment of the k-th event, i.e., the update moment of the event-triggered double-power fractional-order sliding mode controller, Z + is a positive integer, t k-1 is the triggering moment of the (k - 1)-th event; τ(t k ) represents the update value of the controller at the triggering moment of the k-th event, τ et (t k ) stores this update value as an intermediate variable; e τ is the defined comparison error of the controller, representing the difference between the controller at the current moment and the controller at the previous triggering moment; d and r represent the parameters of the event-triggering mechanism, which are numbers greater than 0 and are set according to the actual situation;

[0020] So far, the event-triggered double-power fractional-order sliding mode controller is obtained;

[0021] Step 4: Use Lyapunov theory to prove that the event-triggered double-power fractional-order sliding mode controller is asymptotically stable, and the event-triggering mechanism can avoid the occurrence of Zeno phenomenon; and use the obtained event-triggered double-power fractional-order sliding mode controller to perform sliding mode control on the lower limb exoskeleton.

[0022] Build a lower limb exoskeleton simulation model in Matlab / Simulink, which is implemented using the FOMCON toolbox. The lower limb exoskeleton simulation model includes a controlled object module Plant, an input quantity module Input, a collection and control module Control, and two event-triggering mechanism modules Et,

[0023] The input quantity module Input is used to provide the desired angle and desired angular velocity to the system;

[0024] The collection and control module Control is used to execute the double-power fractional-order sliding mode control law, including a proportional module, a fractional-order module, and a derivative module;

[0025] The output of the collection and control module Control is connected to two parallel event-triggering mechanism modules Et. The two event-triggering mechanism modules Et are respectively used to perform event triggering on the hip joint and the knee joint. The event-triggering mechanism module Et uses rising edge triggering and sets a memory module, and the memory module is used to avoid generating algebraic loops;

[0026] The outputs of the two event-triggering mechanism modules Et act on the controlled object module Plant to obtain the actual trajectory, and then calculate the real-time tracking error between the actual trajectory and the desired trajectory.

[0027] The c = 10, α = 0.9, β 1 = 1.2, β 2 = 0.6, k 1= 5, k 2 = 0.5; d = 0.4, r = 0.4.

[0028] Compared with the existing exoskeleton control methods, the control method proposed by the present invention has the following significant improvements:

[0029] The present invention is a double-power fractional-order sliding mode control method for lower limb exoskeletons based on event triggering. Compared with time-triggered periodic sampling, event triggering can judge whether the system needs to sample and update the control law through triggering conditions. It can greatly reduce the update frequency of the control law while ensuring control accuracy, not only reducing the wear of the actuator, but also effectively reducing the computational amount and saving communication resources. At the same time, the fractional calculus theory is applied to the double-power reaching law of sliding mode control. After the state trajectory reaches the sliding surface in traditional sliding mode control, it will show a phenomenon of reciprocating penetration on both sides of the sliding surface, generating a chattering effect. Although the double-power sliding mode control can suppress the generation of chattering to a certain extent, there is still a certain amount of chattering in the control of lower limb exoskeletons. The double-power fractional-order sliding mode control law proposed by the present invention can eliminate the chattering generated by sliding mode control and achieve chattering-free sliding mode control of lower limb exoskeletons. Description of the Drawings

[0030] Figure 1 is a simulation model of a lower limb exoskeleton established in Matlab / Simulink, which includes an input quantity module, a sampling controller module, an event triggering mechanism module, and a controlled object module;

[0031] Figure 2(a) is a simulation result diagram of the hip joint angle tracking effect obtained by using the control method described in the present invention;

[0032] Figure 2(b) is a simulation result diagram of the knee joint angle tracking effect obtained by using the control method described in the present invention;

[0033] Figure 3(a) is a hip joint angle tracking error diagram obtained by using the control method described in the present invention;

[0034] Figure 3(b) is a knee joint angle tracking error diagram obtained by using the control method described in the present invention;

[0035] Figure 4(a) is a simulation result diagram of the hip joint angular velocity tracking effect obtained by using the control method described in the present invention;

[0036] Figure 4(b) is a simulation result diagram of the knee joint angular velocity tracking effect obtained by using the control method described in the present invention;

[0037] Figure 5(a) is a comparison diagram of the hip joint control input torque between the double-power fractional-order sliding mode control designed by the present invention and the traditional sliding mode control;

[0038] Figure 5(b) is a comparison diagram of the knee joint control input torque between the double-power fractional-order sliding mode control designed by the present invention and the traditional sliding mode control;

[0039] Figure 6(a) is a curve graph of the hip joint input torque for trajectory tracking using the event-triggered double-power fractional-order sliding mode control method described in the present invention;

[0040] Figure 6(b) is a curve graph of the knee joint input torque for trajectory tracking using the event-triggered double-power fractional-order sliding mode control method described in the present invention. Specific implementation manner

[0041] The following will introduce in detail the lower limb exoskeleton control method described in the present invention with reference to the accompanying drawings.

[0042] The event-triggered lower limb exoskeleton double-power fractional-order sliding mode control method of the present invention has the following specific operation steps:

[0043] Step 1: Establish a lower limb exoskeleton dynamics model using the Lagrangian equation.

[0044] Using the Lagrangian dynamics modeling method, the following lower limb exoskeleton dynamics equation can be obtained:

[0045]

[0046] In the formula, q ∈ R n is the joint angle vector, is the joint angular velocity vector, is the joint angular acceleration vector; M(q) ∈ R n × n is a real symmetric inertia matrix, is the matrix related to the Coriolis force and centrifugal force, G(q) ∈ R n is the matrix related to the gravity term, τ ∈ R n is the control torque of the exoskeleton, and n is the degree of freedom of the lower limb exoskeleton. In this embodiment, n = 2.

[0047] The expression of the inertia matrix M(q) is:

[0048]

[0049] Among them,

[0050]

[0051] The friction matrix has the following expression:

[0052]

[0053] Among them,

[0054]

[0055] The expression of the gravity matrix G(q) is as follows:

[0056]

[0057] Wherein,

[0058] In the formula, m h and m k are the masses of the thigh rod and the calf rod respectively, l h and l k are the lengths of the thigh rod and the calf rod respectively, q h is the hip joint angle, q k is the knee joint angle, is the hip joint angular velocity, is the knee joint angular velocity.

[0059] Step 2: Use the sensor to obtain the lower limb movement data of a healthy human body, and obtain the expected motion trajectories of the hip joint angle, hip joint angular velocity, knee joint angle, and knee joint angular velocity of the lower limb exoskeleton through function fitting. This part is the prior art;

[0060] Step 3: Design a double-power fractional-order sliding mode control law, and design an event-triggering mechanism according to the error between the expected trajectory and the actual trajectory, and finally obtain an event-triggered double-power fractional-order sliding mode controller (referred to as the controller or sliding mode controller);

[0061] According to the lower limb exoskeleton dynamics model, define the tracking error of each joint as follows:

[0062] e(t) = q d (t) - q(t) (5)

[0063] Where q d (t) is the expected angle of the joint, and q(t) is the actual tracking angle.

[0064] Take the first derivative of the tracking error and the second derivative

[0065]

[0066]

[0067] Take the sliding mode surface as:

[0068]

[0069] Taking the derivative of the sliding mode surface function and combining Eqs. (1) and (8), we get:

[0070]

[0071] Take the double power-law fractional order sliding mode reaching law:

[0072]

[0073] where k 1 > 0, k 2 > 0, β 1 > 1, 0 < β 2 < 1. D α represents the fractional order calculus operator, and α represents the order of the fractional order calculus.

[0074] According to the properties of fractional order calculus, taking the derivative of Eq. (10) gives:

[0075]

[0076] Combining and simplifying Eqs. (9) and (11), the control law is obtained as Eq. (12):

[0077]

[0078] The event-triggering mechanism and event-triggering conditions are designed as:

[0079]

[0080] where t k , k ∈ Z + is the event-triggering time, that is, the update time of the controller, τ(t k ) represents the updated value of the controller at the previous event-triggering time, and τ et (t k ) stores this updated value as an intermediate variable. T is the static threshold of the event-triggering condition. e τ is the defined comparison error of the sliding mode controller, representing the difference between the controller at the current time and the controller at the previous triggering time. As shown in Eq. (14):

[0081] e τ = τ(t) - τ et (t k ), t ∈ [t k , t k+1 ) (14)

[0082] According to the event-triggering condition (13), after a certain triggering time t k , when the controller comparison error e τ exceeds the static threshold T, the event-triggering time will change from tk Updated to t k+1 , t k+1 is the moment when the comparison error of the controller exceeds the static threshold. At the same time, the controller updates from τ(t k ) to τ(t k+1 ), and transmits the updated controller to the controlled object, so as to control the lower limb exoskeleton and extend the actuator life by reducing the update frequency of the controller. The controller remains unchanged within the time of [t k , t k+1 ), and the value of the controller is τ(t k ).

[0083] Step 4: Use Lyapunov theory to prove the stability of the involved event-triggered double-power fractional-order sliding mode controller and prove the feasibility of the event-triggered mechanism, which can avoid the occurrence of Zeno phenomenon.

[0084] (1) Stability analysis

[0085] Select the Lyapunov function as shown in Equation (15):

[0086]

[0087] Take the derivative of both sides of the selected Lyapunov function and substitute Equations (1) and (9) to get:

[0088]

[0089] Substitute Equation (9) into Equation (16) to get:

[0090]

[0091] According to the properties of fractional calculus, we can obtain According to the Lyapunov function stability theory, the designed event-triggered double-power fractional-order sliding mode controller is asymptotically stable.

[0092] (2) Feasibility analysis

[0093] Next, prove the feasibility of the proposed event-triggered mechanism by excluding the Zeno phenomenon.

[0094] Assume that there exists a positive constant t * , such that:

[0095] Since e τ = τ(t) - τ et (t), t ∈ [t k , t k+1 ), it can be known that:

[0096]

[0097] As can be seen from Equation (12), τ is differentiable, and there exists a positive constant a such that Since e(t k ) = 0 and Then Therefore, the Zeno phenomenon is eliminated.

[0098] Step Five: Build the lower limb exoskeleton simulation model as shown in Figure 1 In the Matlab / Simulink environment. Among them, the Input module is the input quantity module, which is used to provide the desired angle and desired angular velocity for the system. Here, q1_d represents the desired angle of the hip joint, q2_d represents the desired angle of the knee joint; dq1_d represents the desired angular velocity of the hip joint, dq2_d represents the desired angular velocity of the knee joint; ddq1_d represents the desired angular acceleration of the hip joint, and ddq2_d represents the desired angular acceleration of the knee joint; the Control module is the acquisition and control module, which executes the double power-law fractional-order sliding mode control law, including a proportional module, a fractional-order module, and a derivative module. In the figure, s represents the sliding mode surface, fo represents the variable of the fractional order, and de represents the first derivative of the error; the Et module is the event-triggering mechanism module, which uses the rising edge trigger and sets a memory module, and the memory module is used to avoid generating algebraic loops; the Plant is the controlled object module.

[0099] Verify the effectiveness of the event-triggered double power-law fractional-order sliding mode control method proposed in the present invention through this simulation model.

[0100] The dynamic equation of the lower limb exoskeleton system can be arranged in the following form:

[0101]

[0102] First, verify the influence of the double power-law fractional-order reaching law on the chattering of the sliding mode control. From Equations (9) and (14), the double power-law fractional-order sliding mode control law based on the double power-law fractional-order reaching law is

[0103]

[0104] The parameters of the double power-law fractional-order sliding mode control are selected as c = 10, α = 0.9, β 1 = 1.2, β 2 = 0.6, k 1 = 5, k 2 = 0.5.

[0105] The event-triggering mechanism is as follows:

[0106]

[0107] The parameters of the event-triggered mechanism are set as: d = 0.4, r = 0.4, and preferably the ranges of both are within 0 - 1.

[0108] The simulation results are shown in Figures 2(a) and 2(b), Figures 3(a) and 3(b), Figures 4(a) and 4(b), Figures 5(a) and 5(b), Figures 6(a) and 6(b).

[0109] Figures 2(a) and 2(b) show the tracking effects of the joint angle trajectories of the hip joint and the knee joint. The method proposed by the present invention can quickly and accurately track the desired trajectory.

[0110] Figures 3(a) and 3(b) show the tracking errors of the joint angles of the hip joint and the knee joint. The tracking error can approach 0 in a short time.

[0111] Figures 4(a) and 4(b) show the tracking of the joint angular velocities of the hip joint and the knee joint. It can be seen from the figures that the method proposed by the present invention can quickly and accurately track the joint angular velocities.

[0112] Figures 5(a) and 5(b) show the comparison of the control torques between the double power-law fractional-order sliding mode control and the traditional sliding mode control proposed by the present invention, where u 1 is the input torque of the double power-law fractional-order sliding mode control designed by the present invention, and u 2 is the input torque of the traditional sliding mode control. It can be seen from the figures that the method proposed by the present invention can effectively eliminate the chattering existing in the sliding mode control.

[0113] Figures 6(a) and 6(b) show the control torques of the hip joint and the knee joint based on the event-triggered mechanism. The present invention compares the control input torques before and after the event trigger. In the figure, u 1 is the control input torque curve after adding the event-triggered mechanism, and u 2 is the control input torque curve without the event-triggered mechanism. It can be seen from the figure that after adding the event-triggered mechanism, the output of the controller remains unchanged during the interval between two event triggers. The update times of the controller before and after adding the event-triggered mechanism are shown in the following table:

[0114] Table 1

[0115]

[0116] It can be analyzed from Table 1 that the event-triggered mechanism can effectively reduce the update times of the controller. The simulation experimental results show that: the control method proposed by the present invention can accurately and quickly track the reference trajectory, and eliminate the chattering phenomenon existing in the sliding mode control. Under the event-triggered mechanism, the control signal changes stepwise after being triggered, significantly reducing the update frequency of the signal, saving communication resources, and no Zeno phenomenon occurs in the lower limb exoskeleton system.

[0117] Matters not described in the present invention are applicable to the prior art.

Claims

1. An event-triggered double-power fractional-order sliding mode control method for lower limb exoskeletons, characterized in that, the control method includes the following steps: Step 1: Establish a dynamic model of the lower limb exoskeleton using the Lagrangian equation; Step 2: Obtain the lower limb movement data of a healthy human body using sensors, and obtain the desired motion trajectories of the hip joint angle, hip joint angular velocity, knee joint angle, and knee joint angular velocity of the lower limb exoskeleton through function fitting; Step 3: Design a double-power fractional-order sliding mode control law, and construct an event-triggered mechanism based on the tracking error between the desired trajectory and the actual trajectory, and finally obtain an event-triggered double-power fractional-order sliding mode controller, referred to as the controller; The double-power fractional-order sliding mode control law is: Among them, s is the sliding surface, c, k 1 , k 2 , β 1 , β 2、 , α are all double-power fractional-order sliding mode control parameters, c is the sliding surface parameter, and c > 0, k 1 > 0, k 2 > 0, β 1 > 1, 0 < β 2 < 1; D α represents the fractional calculus operator, and α represents the order of the fractional calculus; e is the tracking error; and are the first derivative and the second derivative of the tracking error; M is the inertia matrix, G is the gravity-related term matrix, and C is the Coriolis force and centrifugal force-related term matrix; sgn is the sign function; τ is the output torque of the controller; is the joint angular velocity vector; is the second derivative of the desired angle of the joint; Define the tracking error e(t) of each joint: e(t) = q d (t) - q(t) (5) where, q d (t) is the desired angle of the joint, q(t) is the actual tracking angle, and t is time; the desired trajectory is the input of the controller, and the actual trajectory is obtained by the output torque of the controller acting on the controlled object; The event-triggered mechanism is: where t k , k ∈ Z + is the k-th event triggering moment, i.e., the update moment of the controller, and Z + is a positive integer, t k-1 is the (k - 1)-th event triggering moment; τ(t k ) represents the update value of the controller at the k-th event triggering moment, and τ et (t k ) stores this update value as an intermediate variable; e τ is the comparison error of the defined sliding mode controller, representing the difference between the controller at the current moment and the controller at the previous triggering moment; d and r represent the event triggering mechanism parameters and are numbers greater than 0; Thus, an event-triggered double-power fractional-order sliding mode controller is obtained; Step 4: Use Lyapunov theory to prove that the event-triggered double-power fractional-order sliding mode controller is asymptotically stable, and the event-triggered mechanism can avoid the occurrence of Zeno phenomenon; and use the obtained event-triggered double-power fractional-order sliding mode controller to perform sliding mode control on the lower limb exoskeleton.

2. The event-triggered double-power fractional-order sliding mode control method for lower limb exoskeletons according to claim 1, characterized in that, in Step 1, the dynamic model of the lower limb exoskeleton is established using the Lagrangian equation, which is formula (1): where \(q\in\mathbb{R}\) n is the joint angle vector, \(n\) is the degree of freedom of the lower limb exoskeleton, is the joint angular velocity vector, is the joint angular acceleration vector; \(M(q)\in\mathbb{R}\) n×n is the real symmetric inertia matrix, is the matrix related to Coriolis force and centrifugal force, \(G(q)\in\mathbb{R}\) n is the matrix related to gravity, \(\tau\in\mathbb{R}\) n is the control torque of the exoskeleton; where the expression of the inertia matrix M(q) is: Among them, Friction matrix The expression of which is: Among them, The expression of the gravity matrix G(q) is: Among them, where m h and m k are the masses of the thigh rod and the calf rod respectively, l h and l k are the lengths of the thigh rod and the calf rod respectively, q h is the hip joint angle, q k is the knee joint angle, is the hip joint angular velocity, is the knee joint angular velocity.

3. The event-triggered double-power fractional-order sliding mode control method for lower limb exoskeletons according to claim 1, characterized in that, Build a lower limb exoskeleton simulation model in Matlab / Simulink, which is realized using the FOMCON toolbox. The lower limb exoskeleton simulation model includes a controlled object module Plant, an input quantity module Input, a collection control module Control, and two event-triggered mechanism modules Et. The input quantity module Input is used to provide the desired angle and desired angular velocity to the system; The collection control module Control is used to execute the double-power fractional-order sliding mode control law, including a proportional module, a fractional-order module, and a derivative module; The output of the collection control module Control is connected to two parallel event-triggered mechanism modules Et. The two event-triggered mechanism modules Et are respectively used to perform event triggering on the hip joint and the knee joint. The event-triggered mechanism module Et uses a rising edge trigger and sets a memory module, and the memory module is used to avoid generating algebraic loops; The outputs of the two event-triggered mechanism modules Et act on the controlled object module Plant to obtain the actual trajectory, and then calculate the real-time tracking error between the actual trajectory and the desired trajectory.

4. The event-triggered double-power fractional-order sliding mode control method for lower limb exoskeletons according to claim 1, characterized in that, where c = 10, α = 0.9, β 1 = 1.2, β 2 = 0.6, k 1 = 5, k 2 = 0.5; d = 0.4, r = 0.4.

Images