Instability suppression method and system for a flexible robotic arm based on SMA actuation

Abstract

The application provides a flexible machine arm instability suppression method and system based on SMA driving, comprising the following steps: step 1: analyzing the constitutive property of SMA and the influence of the discontinuous output force characteristic introduced by the SMA on system dynamics; step 2: combining the physical configuration of the flexible machine arm to construct a universal state equation; step 3: introducing trajectory constraint conditions to plan a reference trajectory and output a desired trajectory; step 4: combining the obtained state equation and the desired trajectory to reconstruct an error dynamics equation, designing a sliding mode control surface and a sliding mode reaching law, and giving a sliding mode control amount of a multiple-input multiple-output (MIMO) system, so as to output a MIMO time sequence control amount while meeting the trajectory tracking accuracy requirement. Through the trajectory constraint method, the application can effectively avoid the problems of control algorithm mismatch or model instability caused by system parameter uncertainty, and ensure that the flexible machine arm works in a model-known and controllable region.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of flexible robot arm control, in particular to a flexible robot arm instability suppression method and system based on SMA driving. BACKGROUND

[0002] The development of aerospace, automation control and advanced manufacturing fields puts forward higher standard requirements for the compliance, flexibility and safety of robots. Traditional rigid robot arms often have high structural stiffness, are relatively bulky, are easy to control, but are difficult to overcome some inherent defects of the structure, such as large actuator volume, high power consumption and low load ratio. Therefore, some new flexible robot arms have gradually attracted attention. The lightweight structure has a high power-to-weight ratio, and its precise positioning and precision control play an important role in space engineering applications, can achieve motion forms that rigid robot arms cannot achieve, and has higher degrees of freedom.

[0003] Patent document CN113858187A (application number: CN202111227151.7) discloses a flexible robot arm switching control method and system based on SMA spring driving, relating to the technical field of flexible robot arm control. The method comprises the following steps: step S1: obtaining the SMA spring vector expression and calculating the numerical value; step S2: identifying the subsystem, boundary, switching set and mapping according to the calculated numerical value; step S3: introducing physical constraints, judging the number of subsystems and the current subsystem; step S4: real-time monitoring of each platform pose vector, outputting the current subsystem when system switching occurs, providing a reference for the control method.

[0004] SMA is a kind of intelligent material with shape memory effect, has high power density and strong driving force, and can drive the robot arm to move more flexibly. However, due to its special constitutive characteristics, it presents high nonlinearity, theoretically infinite degrees of freedom, and force discontinuity, so the control method of SMA-driven flexible robot arm faces great challenges.

[0005] 1. Discontinuity of the system

[0006] Due to its special constitutive characteristics, SMA spring has different mechanical properties in compression and stretching states. At the critical moment of switching between the two, the output force has discontinuity, which causes the phenomenon of system switching during the movement of the flexible robot arm. In the compression process, the SMA spring is prone to bending, which introduces bending moment and uncertainty of model parameters, resulting in a decline in the effectiveness of the control method, or even failure. Therefore, during the movement of the robot arm, certain trajectory constraints need to be set to limit the occurrence of compression.

[0007] 2. High nonlinearity and multiple degrees of freedom of the system

[0008] SMA controlled flexible robot arm can be bent with high curvature, theoretically has infinite degrees of freedom, and the relationship between the deformation amount and the offset angle of the SMA spring and the input presents a high nonlinear characteristic, and the forward and inverse kinematics and dynamics modeling of the flexible robot arm is more difficult, and the multiple groups of SMA are coupled with each other, in this case with complex motion characteristics, it is necessary to set up a control method with strong anti-interference ability and in line with multiple input multiple output, and the MIMO control method based on sliding mode can meet the demand. SUMMARY

[0009] In view of the defects in the prior art, the purpose of the present application is to provide a SMA driven flexible robot arm instability suppression method and system.

[0010] The SMA driven flexible robot arm instability suppression method provided by the present application comprises:

[0011] Step 1: analyze the constitutive property of SMA and the influence of the introduced output force discontinuity characteristic on system dynamics;

[0012] Step 2: combine the physical configuration of the flexible robot arm to build a universal state equation;

[0013] Step 3: introduce trajectory constraint conditions, plan the reference trajectory, and output the expected trajectory;

[0014] Step 4: combine the obtained state equation and expected trajectory to reconstruct the error dynamics equation, design the sliding mode control surface and sliding mode reaching law, give the SMA sliding mode control amount of the MIMO system, and output the MIMO time sequence control amount when the trajectory tracking accuracy requirement is met.

[0015] Preferably, the step 1 comprises:

[0016] The SMA driven flexible robot arm comprises M platforms, each platform comprises N groups of SMA springs below, the spatial distribution of the SMA springs in each platform is reasonably configured to drive each platform to move in any direction, considering that the SMA is prone to bending when compressed, the output force direction is not the axial direction at this time, and a non-negligible bending moment may be introduced, and the constitutive model of the SMA spring is combined to obtain:

[0017]

[0018] wherein F ij represents the output force of the jth group of SMA springs of the ith platform; ΔY ij represents the deformation amount of the jth group of SMA springs of the ith platform; ξ ijf i,j (T i,j, f i,j ) represents the martensitic volume fraction of the jth group of SMA springs in the ith platform; f 0 represents the initial martensitic volume fraction; T represents the temperature; T 0 represents the ambient temperature; Ω represents the phase transition coefficient; Θ represents the thermoelastic coefficient; σ ij f i,j (T i,j, f i,j ) represents the martensitic volume fraction of the jth group of SMA springs in the ith platform; f 0 represents the initial martensitic volume fraction; T represents the temperature; T 0 represents the ambient temperature; Ω represents the phase transition coefficient; Θ represents the thermoelastic coefficient; σ e f i,j (T i,j, f i,j ) represents the martensitic volume fraction of the jth group of SMA springs in the ith platform; f 0 represents the initial martensitic volume fraction; T represents the temperature; T 0 represents the ambient temperature; Ω represents the phase transition coefficient; Θ represents the thermoelastic coefficient; σ e and Y e are proportional parameters determined by the SMA spring structure; a ij represents the equivalent parameter of the output force of the jth group of SMA springs in the ith platform when it is compressed in the axial direction, which is 1 when the axial compression has no bending, and is between 0 and 1 when there is bending;

[0019] In formula (1), F e and Y e are determined by the SMA spring structure, and the temperature and the martensitic volume fraction can be controlled by an external voltage or current signal, then the controllable input u of the jth group of SMA springs in the ith platform is defined as ij

[0020]

[0021] Preferably, the step 2 comprises:

[0022] For the ith platform, the Euler angles thereof relative to the fixed coordinate system are defined as The rotation matrix R of the ith platform to the fixed coordinate system is (i)

[0023]

[0024] where s. = sin(·), c. = cos(·);

[0025] For the jth group of SMA springs in the ith platform, the vector is represented as:

[0026]

[0027] where, represents the vector of the upper fixed end of the jth group of SMA springs in the ith platform in the i+1th reference system; represents the vector of the lower fixed end of the jth group of SMA springs in the ith platform in the ith reference system; L (i) represents the vector of the origin of the i+1th platform in the ith reference system;

[0028] Based on formula (4), the deformation variable ΔY of the jth group of SMA springs in the ith platform ij is written as:

[0029] ​​

[0030] where ||·||2 represents the Euclidean norm; Y0 represents the memory length of each SMA spring;

[0031] The dynamic modeling is performed by the Lagrange method or the Newton-Euler method, and is converted into the form of a state equation. For any platform, the state equation has a form similar to equation (6):

[0032]

[0033] where f1, f2, f3 describe the influence of system parameters of the dynamic model; k1, k2, k3 represent the influence of control quantities on the dynamic model; ξ1, ξ2, ξ3 represent external disturbances and errors of the system model; and sign(·) represents a sign function.

[0034] Preferably, the step 3 comprises:

[0035] The reference trajectory is given as The trajectory constraint is performed, and the desired trajectory is obtained as The trajectory constraint is introduced as:

[0036]

[0037] When the jth group of SMA springs of the ith platform is in a stretched state, ΔY ij is positive, the control target is trajectory tracking, and the reference trajectory is the desired trajectory at this time; when the jth group of SMA springs of the ith platform is in a compressed state, ΔY ij is negative, the state equation parameters have uncertainties, the reference trajectory should be constrained, and the control target is a stabilization problem, and the desired trajectory is the intersection of the reference trajectory and the boundary at this time.

[0038] Preferably, the step 4 comprises:

[0039] The error of the dynamic system is defined as:

[0040]

[0041] The state equation of equation (6) can be rewritten as the following error dynamic equation:

[0042]

[0043] According to the extended Lyapunov method, the sliding mode control surface and are respectively:

[0044] The sliding mode reaching law is taken as an exponential reaching law:

[0045]

[0046] Then the equation group is:

[0047]

[0048] The control amount of the i-th platform and the j-th group of SMA springs in the equation group in equation (12) is:

[0049]

[0050] Wherein, z is a serial number, less than or equal to N; alpha is a gain coefficient, and beta is a reaching law;

[0051] The loop is ended after the trajectory tracking is ended and the desired accuracy is met, and the MIMO time sequence control amount in the whole trajectory tracking process is output.

[0052] The flexible robot arm instability suppression system based on SMA driving provided by the application comprises:

[0053] Module M1: analyzing the constitutive property of SMA and the influence of the introduced output force discontinuous property on system dynamics;

[0054] Module M2: combining the physical configuration of the flexible robot arm, constructing a universal state equation;

[0055] Module M3: introducing a trajectory constraint condition, planning a reference trajectory, and outputting a desired trajectory;

[0056] Module M4: combining the obtained state equation and the desired trajectory, reconstructing an error dynamics equation, designing a sliding mode control surface and a sliding mode reaching law, giving a SMA sliding mode control amount of a multiple input multiple output system MIMO, and outputting a MIMO time sequence control amount when the trajectory tracking accuracy requirement is met.

[0057] Preferably, the module M1 comprises:

[0058] The SMA driven flexible robot arm comprises M platforms, each platform comprises N groups of SMA springs below, the spatial distribution of the SMA springs in each platform is reasonably configured, each platform is driven to move in an arbitrary direction, considering that the SMA is prone to bending when being compressed, at this time, the direction of the output force is not the axial direction, and a non-negligible bending moment can be introduced, the constitutive model of the SMA spring is combined to obtain:

[0059]

[0060] Wherein, F ij represents the output force of the i-th platform and the j-th group of SMA springs; delta Y ijrepresents the deformation of the jth group of SMA springs of the ith platform; ξ ij represents the martensite volume fraction of the jth group of SMA springs of the ith platform; ξ0 represents the initial martensite volume fraction; T ij represents the temperature of the jth group of SMA springs of the ith platform; T0 is the ambient temperature; Ω represents the phase transition coefficient; Θ represents the thermoelastic coefficient; σ e is the SMA elastic stress limit; F e and Y e are proportional parameters determined by the SMA spring structure; a ij represents the equivalent parameter of the output force of the jth group of SMA springs of the ith platform in the axial direction when it is compressed, which is equal to 1 when the axial compression has no bending, and is between 0 and 1 when there is bending;

[0061] In formula (1), F e and Y e are determined by the SMA spring structure, and the temperature and the martensite volume fraction can be controlled by an external voltage or current signal, then the controllable input u of the jth group of SMA springs of the ith platform is defined as: ij

[0062]

[0063] Preferably, the module M2 comprises:

[0064] For the ith platform, the Euler angles thereof relative to the fixed coordinate system are defined as The rotation matrix R of the ith platform to the fixed coordinate system is (i)

[0065]

[0066] where s. = sin(·), c. = cos(·);

[0067] For the jth group of SMA springs of the ith platform, the vector is represented as:

[0068]

[0069] where, represents the vector of the upper fixed end of the jth group of SMA springs of the ith platform in the ith+1 reference system; represents the vector of the lower fixed end of the jth group of SMA springs of the ith platform in the ith reference system; L (i) represents the vector of the origin of the ith+1 platform in the ith reference system;

[0070] Based on formula (4), the deformation ΔY of the jth group of SMA springs of the ith platform is ij ​​is written as:

[0071]

[0072] where ||·||2 represents the Euclidean norm; Y0 represents the memory length of each SMA spring;

[0073] The dynamic modeling is performed by the Lagrange method or the Newton-Euler method, and is converted into the form of a state equation. For any platform, the state equation has a form similar to equation (6):

[0074]

[0075] where f1, f2, f3 describe the influence of system parameters of the dynamic model; k1, k2, k3 represent the influence of control quantities on the dynamic model; ξ1, ξ2, ξ3 represent external disturbances and errors of the system model; sign(·) represents a sign function.

[0076] Preferably, the module M3 comprises:

[0077] The reference trajectory is given as The trajectory constraint is performed, and the desired trajectory is obtained as The trajectory constraint is introduced as:

[0078]

[0079] When the jth group of SMA springs of the ith platform is in a stretched state, ΔY ij is positive, the control target is trajectory tracking, and the reference trajectory is the desired trajectory at this time; when the jth group of SMA springs of the ith platform is in a compressed state, ΔY ij is negative, the state equation parameters have uncertainties, the reference trajectory should be constrained, and the control target is a stabilization problem, and the desired trajectory is the intersection of the reference trajectory and the boundary at this time.

[0080] Preferably, the module M4 comprises:

[0081] The error of the dynamic system is defined as is:

[0082]

[0083] The state equation of equation (6) can be rewritten as the following error dynamic equation:

[0084]

[0085] According to the extended Lyapunov method, the sliding mode control surface and are respectively:

[0086] The sliding mode approach law is exponential approach:

[0087]

[0088] Then the equation group is:

[0089]

[0090] The control amount of the i-th platform j-th group of SMA spring in the equation group in formula (12) is:

[0091]

[0092] Wherein, z is a serial number, less than or equal to N; alpha is a gain coefficient, and beta is an approach law;

[0093] The cycle is ended after the trajectory tracking is ended and the desired accuracy is met, and the MIMO time sequence control amount in the whole trajectory tracking process is output.

[0094] Compared with the prior art, the present application has the beneficial effects as follows:

[0095] (1) The present application avoids the situation that the model is unstable or the controller is adapted due to the parameter uncertainty caused by the compression of the SMA by introducing trajectory constraints;

[0096] (2) The present application overcomes the high nonlinearity of the SMA spring and external disturbance by introducing sliding mode control, the sliding mode controller is not sensitive to the internal error of the system modeling and the external disturbance caused by the environmental disturbance, and the influence of the SMA spring nonlinearity and disturbance can be reduced by proper design of the sliding surface and given sliding approach law, so that the flexible robot arm can be driven to track the trajectory;

[0097] (3) The sliding surface designed by the reference extended Lyapunov method can give the control amount of the MIMO system in real time, and the system stability is met;

[0098] (4) The present application combines the trajectory constraints, sliding mode control, sliding surface of the reference extended Lyapunov method, and can provide a theoretical reference for the control method of other flexible robot arms with switching system, MIMO, intelligent material based driving, such as pneumatic artificial muscle, etc. BRIEF DESCRIPTION OF DRAWINGS

[0099] Other features, objects and advantages of the present application will become more apparent from the following detailed description of non-limiting embodiments, made with reference to the accompanying drawings:

[0100] Figure 1 The present application is a flow chart of the instability suppression method for the SMA driven flexible robot arm. DETAILED DESCRIPTION

[0101] The application will be described in detail below with specific examples. The following examples will help those skilled in the art to further understand the application, but do not limit the application in any form. It should be noted that those skilled in the art can make several changes and improvements without departing from the concept of the application. These are within the scope of protection of the application.

[0102] Example 1:

[0103] As Figure 1 , the application proposes a SMA-driven flexible robot arm instability suppression method, including: step 1: analyzing the constitutive property of SMA, and the influence of the introduced output force discontinuity on system dynamics; step 2: combining the physical configuration of the flexible robot arm, constructing a universal state equation; step 3: introducing trajectory constraint conditions, planning the reference trajectory, and outputting the desired trajectory; step 4: combining the obtained state equation and desired trajectory, reconstructing the error dynamics equation, and designing the sliding mode control surface and sliding mode reaching law, giving the SMA sliding mode control quantity of the multi-input multi-output system MIMO, and outputting the MIMO time sequence control quantity when the trajectory tracking accuracy requirement is met.

[0104] The step 1 includes: the SMA-driven flexible robot arm contains M platforms, each platform contains N groups of SMA springs below, by reasonably configuring the spatial distribution of SMA springs in each platform, driving each platform to move in any direction, considering that SMA is easy to bend when compressed, at this time the direction of output force is not the axial direction, and a non-negligible bending moment may be introduced, combining the constitutive model of SMA spring, obtaining:

[0105]

[0106] Where, F ij represents the output force of the jth group of SMA springs of the ith platform; ΔY ij represents the deformation of the jth group of SMA springs of the ith platform; ξ ij represents the martensite volume fraction of the jth group of SMA springs of the ith platform; ξ0 represents the initial martensite volume fraction; T ij represents the temperature of the jth group of SMA springs of the ith platform; T0 is the environmental temperature; Ω represents the phase change coefficient; Θ represents the thermal elastic coefficient; σ e is the SMA elastic stress limit; F e and Y e are proportional parameters determined by the SMA spring structure; α ijrepresents the equivalent parameter of the output force of the jth group of SMA springs of the ith platform in the axial direction when the SMA springs are compressed, and is equal to 1 when there is no bending in the axial compression, and is between 0 and 1 when there is bending;

[0107] In formula (1), F e and Y e The controllable input u of the jth group of SMA springs of the ith platform is defined as: ij

[0108]

[0109] The step 2 comprises: for the ith platform, defining the Euler angles of the ith platform relative to the fixed coordinate system as The rotation matrix R of the ith platform to the fixed coordinate system is: (i)

[0110]

[0111] Where s. = sin(·), c. = cos(·);

[0112] For the jth group of SMA springs of the ith platform, the vector is expressed as:

[0113]

[0114] Where, represents the vector of the upper fixed end of the jth group of SMA springs of the ith platform in the ith+1 reference system; represents the vector of the lower fixed end of the jth group of SMA springs of the ith platform in the ith reference system; L (i) represents the vector of the origin of the ith+1 platform in the ith reference system;

[0115] Based on formula (4), the deformation variable ΔY of the jth group of SMA springs of the ith platform is written as: ij

[0116]

[0117] Where ||·||2 represents the Euclidean norm; Y0 represents the memory length of each SMA spring;

[0118] The dynamics modeling is performed by the Lagrange method or the Newton Euler method, and is converted into the form of a state equation, and for any platform, the state equation has a form similar to formula (6):

[0119]

[0120] ​​​where f1, f2, f3 describe the influence of the system parameters of the dynamic model; k1, k2, k3 represent the influence of the control variables on the dynamic model; ξ1, ξ2, ξ3 represent the external disturbance and the error of the system model; sign(·) represents the sign function.

[0121] The step 3 comprises: performing trajectory constraint on the given reference trajectory to obtain the desired trajectory The trajectory constraint is introduced as:

[0122]

[0123] When the jth group of SMA springs of the ith platform is in a tensile state, ΔY ij is positive, the control target is trajectory tracking, and the reference trajectory is the desired trajectory at this time; when the jth group of SMA springs of the ith platform is in a compression state, ΔY ij is negative, the state equation parameters have uncertainty, the reference trajectory should be constrained, and the control target is a stabilization problem, and the desired trajectory is the intersection of the reference trajectory and the boundary at this time.

[0124] The step 4 comprises: defining the error of the dynamic system as:

[0125]

[0126] The state equation of formula (6) can be rewritten as the following error dynamic equation:

[0127]

[0128] According to the extended Lyapunov method, the sliding mode control surface and are respectively:

[0129]

[0130] The sliding mode reaching law is taken as an exponential reaching:

[0131]

[0132] Then the equation group is:

[0133]

[0134] The control variable of the jth group of SMA springs of the ith platform is:

[0135]

[0136] where z is a serial number, less than or equal to N; α is a gain coefficient, and β is a reaching law.

[0137] After the trajectory tracking ends and the desired accuracy is met, the loop ends and the MIMO timing control quantity in the whole trajectory tracking process is output.

[0138] Embodiment 2

[0139] The application also provides a flexible robot arm instability suppression system based on SMA driving, which can be realized by executing the process steps of the flexible robot arm instability suppression method based on SMA driving, that is, the flexible robot arm instability suppression method based on SMA driving can be understood by those skilled in the art as the preferred embodiment of the flexible robot arm instability suppression system based on SMA driving.

[0140] The flexible robot arm is connected in sequence by a plurality of joints, each joint contains a plurality of groups of shape memory alloys (SMA), spherical hinge straight rods, disc platforms and the like. The SMA spring has residual deformation at the initial moment, and can contract back to the memory length after being heated by power supply, and at the same time, does work to the outside, drives the flexible robot arm to move, and through the single-chip microcomputer adjusting the amplitude and frequency of power supply, the expected driving force output by the SMA spring to the outside can be controlled. The flexible robot arm has a plurality of attitude angle inputs and a plurality of SMA control quantity outputs, and is a multiple input multiple output (MIMO) system. Combined with forward and inverse kinematics and trajectory planning, the flexible robot arm can complete specific tasks such as target grabbing and space obstacle avoidance.

[0141] The application is divided into four modules, in the first module, the constitutive property of SMA is mainly analyzed, and the influence of the discontinuous output force introduced by the constitutive property on the system dynamics is analyzed; in the second module, the universal state equation is constructed in combination with the physical configuration of the flexible robot arm; in the third module, the reference trajectory is planned by introducing the trajectory constraint method, and the expected trajectory is output; in the fourth module, the error dynamics equation is reconstructed in combination with the obtained state equation and the expected trajectory, the sliding mode control surface and the sliding mode reaching law are designed, the SMA sliding mode control quantity of MIMO is given, and the MIMO timing control quantity is output when the trajectory tracking accuracy requirement is met.

[0142] SMA spring constitutive property module

[0143] Consider this SMA-driven flexible robot arm contains M platforms, each platform contains N groups of SMA springs. By reasonably configuring the spatial distribution of SMA springs in each section of the platform, each section of the platform can be driven to move in any direction. Generally, SMA springs output tensile force, and the direction of the force is the axial direction of the SMA spring. But when the platform is subjected to a larger external force impact or has a larger initial angular velocity, the SMA spring may be compressed during movement. At this critical position, the output force of the SMA has a large jump, which has a discontinuous phenomenon. A small deformation will produce a huge axial output force. The influence of this axial output force on the flexible robot arm system is almost a jump, which triggers the switching of the system. At the same time, considering that when the SMA is compressed, it is easy to bend, at this time the direction of the output force is not the axial direction, and a non-negligible bending moment may be introduced. Combined with the constitutive model of SMA spring, we can get:

[0144]

[0145] Where F ij is the output force of the jth group of SMA springs in the ith platform; ΔY ij is the deformation of the jth group of SMA springs in the ith platform; ξ ij is the martensite volume fraction of the jth group of SMA springs in the ith platform; ξ0 is the initial martensite volume fraction; T ij is the temperature of the jth group of SMA springs in the ith platform; T0 is the ambient temperature; Ω is the phase transition coefficient; Θ is the thermal elastic coefficient; σ e is the SMA elastic stress limit; F e and Y e are proportional parameters determined by the SMA spring structure; α ij is the equivalent parameter of the output force of the jth group of SMA springs in the ith platform in the axial direction when it is compressed. When there is no bending in the axial compression, the value is 1. When bending, the value is between 0 and 1. However, the direction and angle of SMA spring bending have high randomness, uncertainty and time-varying nature. It is difficult to measure and estimate α ij when the SMA spring is actually bent.

[0146] In equation (1), F e and Y e are determined by the SMA spring structure, and the temperature and martensite volume fraction can be controlled by the external voltage or current signal. Then define the controllable input u ij of the jth group of SMA springs in the ith platform as:

[0147]

[0148] Flexible robot arm state equation module

[0149] For the i-th platform, define its Euler angles relative to the fixed coordinate system as The rotation matrix R of the i-th platform to the fixed coordinate system (i) is:

[0150]

[0151] where s. = sin(·), c. = cos(·).

[0152] For the j-th group of SMA springs of the i-th platform, its vector can be expressed as:

[0153]

[0154] where, represents the vector of the upper fixed end of the j-th group of SMA springs of the i-th platform in the i+1-th reference system; represents the vector of the lower fixed end of the j-th group of SMA springs of the i-th platform in the i-th reference system; L (i) represents the vector of the origin of the i+1-th platform in the i-th reference system.

[0155] Based on equation (4), the deformation variable ΔY of the j-th group of SMA springs of the i-th platform ij can be written as:

[0156]

[0157] where ||·||2 represents the Euclidean norm; Y0 represents the memory length of each SMA spring.

[0158] Through the Lagrange method or the Newton-Euler method for dynamic modeling, it is converted into the form of state equation. For any platform, the state equation has a similar form to equation (6):

[0159]

[0160] where f1, f2, f3 describe the influence of system parameters of the dynamic model; k1, k2, k3 represent the influence of control variables on the dynamic model; ξ1, ξ2, ξ3 represent external disturbances and errors of the system model; sign(·) represents the sign function; N is the number of SMA spring groups.

[0161] Trajectory constraint module

[0162] Considering the discontinuous output force characteristics of SMA at the compression and stretching positions, and the bending phenomenon easily occurring when the SMA spring is compressed, which causes the model parameter uncertainty phenomenon, i.e., the equivalent coefficient α in equation (1)ij Its high degree of randomness, uncertainty, and time-varying nature has a significant negative impact on control effectiveness, and may even lead to system uncontrollability or instability. Therefore, given a reference trajectory... Next, trajectory constraints need to be applied to obtain the desired trajectory. To avoid SMA compression, trajectory constraints are introduced as follows:

[0163]

[0164] When the j-th group of SMA springs on the i-th platform is in a stretched state, ΔY ij When the value is positive, the control objective is trajectory tracking, and the reference trajectory is the desired trajectory; when the j-th group of SMA springs on the i-th platform is in a compressed state, ΔY ij If the value is negative, the parameters of the state equation have uncertainty, the reference trajectory should be constrained, the control objective is a stabilization problem, and the desired trajectory is the intersection of the reference trajectory and the boundary.

[0165] MIMO sliding mode control module

[0166] Define the error of a dynamic system for:

[0167]

[0168] The state equation of equation (6) can then be rewritten as the following error dynamics equation:

[0169]

[0170] Referring to the extended Lyapunov method, sliding mode control surfaces and They are respectively:

[0171]

[0172] The sliding mode reaching law is taken as exponential reaching:

[0173]

[0174] Then we have a system of equations:

[0175]

[0176] The control quantity of the SMA spring in the j-th group of equations in equation (12) is:

[0177]

[0178] Where α is the gain coefficient and β is the reaching law;

[0179] The stability of the flexible robot arm can be proved by Lyapunov method, and the form of the selected extended Lyapunov function V(s) is given, Based on this, Lyapunov stability can be proved. After the end of trajectory tracking and the satisfaction of the desired accuracy, the loop is ended, and the MIMO time sequence control quantity in the whole trajectory tracking process is output.

[0180] Those skilled in the art know that, in addition to implementing the system, device and each module thereof provided by the present application in the form of pure computer readable program code, the same program can also be implemented in the form of logic gates, switches, application specific integrated circuits, programmable logic controllers and embedded microcontrollers, etc. by logically programming the method steps. Therefore, the system, device and each module thereof provided by the present application can be considered as a hardware component, and the modules included therein for implementing various programs can also be considered as structures within the hardware component; the modules for implementing various functions can also be considered as both software programs for implementing methods and structures within the hardware component.

[0181] The specific embodiments of the present application are described above. It needs to be understood that the present application is not limited to the specific embodiments described above, and various changes or modifications can be made by those skilled in the art within the scope of the claims, which does not affect the essential content of the present application. The embodiments of the present application and the features in the embodiments can be arbitrarily combined with each other without conflict.

Claims

1. A method for instability suppression of a flexible robot arm based on SMA actuation, characterized in that, The application relates to a method for controlling a flexible robot arm driven by SMA (Shape Memory Alloy) springs. The method comprises the following steps: Step 1: analyzing the constitutive property of SMA and the influence of the discontinuous output force property introduced by SMA on system dynamics; Step 2: combining the physical configuration of the flexible robot arm to build a universal state equation; Step 3: introducing trajectory constraint conditions to plan a reference trajectory and output a desired trajectory; Step 4: combining the obtained state equation and desired trajectory to reconstruct error dynamics equations, designing a sliding mode control surface and a sliding mode reaching law, giving a MIMO (Multiple Input Multiple Output) SMA sliding mode control quantity, and outputting a MIMO time sequence control quantity when the trajectory tracking accuracy requirement is met. The step 1 comprises the following steps: …………(1) in, Indicates the first i The first platform j Output force of the SMA spring group; Indicates the first i The first platform j Deformation of the SMA spring assembly; Represented as the first i The first platform j Martensite volume fraction of SMA springs; This is expressed as the initial martensite volume fraction; Represented as the first i The first platform j Temperature of the SMA spring assembly; The ambient temperature; Represented as the phase transition coefficient; Expressed as the thermoelastic coefficient; This represents the elastic stress limit of the SMA. and The proportional parameters are determined by the SMA spring structure; Indicates the first i The first platform j When an SMA spring is compressed, its output force in the axial direction is equivalent to a parameter. When there is no bending during axial compression, the value is 1. When there is bending, the value is between 0 and 1. In formula (1), and The temperature and martensite volume fraction can be controlled by an external voltage or current signal, determined by the SMA spring structure, then the controllable input of the jth group of SMA springs in the ith platform is defined as: ​ …………(2)； The SMA-driven flexible robot arm comprises M platforms, each platform comprising N groups of SMA springs. For the i-th platform, define its Euler angles relative to the fixed coordinate system as The rotation matrix of the i-th platform to the fixed coordinate system is …………(3) wherein , ; For the ith platform, jth group of SMA springs, the vector is represented as: …………(4) wherein, represents the vector of the upper fixed end of the jth group of SMA springs of the ith platform in the ith+1 reference frame; represents the vector of the lower fixed end of the jth group of SMA springs of the ith platform in the ith reference frame; represents the vector of the origin of the ith+1 platform in the ith reference frame; Based on equation (4), the deformation of the jth group of SMA springs of the ith platform is is written as: …………(5) wherein, denotes the Euclidean norm; denotes the memory length of each SMA spring; Considering that SMA is prone to bending when compressed, the output force direction is not the axial direction, and a non-negligible bending moment is introduced, the constitutive model of the SMA spring is combined to obtain the following equation: …………(6) wherein , , describing the influence of the system parameters of the kinetic model; , , denoting the influence of the control quantities on the kinetic model; , , denoting external disturbances and errors of the system model; denotes a sign function; The step 2 comprises the following steps: The reference trajectory is given The trajectory constraint is performed after the reference trajectory, and the desired trajectory is obtained The trajectory constraint is introduced as …………(7) when the jth group of SMA springs of the ith platform is in the stretched state, is positive, the control objective is trajectory tracking, and the reference trajectory is the desired trajectory at this time; when the jth group of SMA springs of the ith platform is in the compressed state, is negative, the state equation parameters have uncertainties, the reference trajectory should be constrained, and the control objective is the stabilization problem, and the desired trajectory is the intersection of the reference trajectory and the boundary at this time; The dynamics modeling is carried out through the Lagrange method or the Newton-Euler method, and is converted into the form of a state equation. Error in defining a dynamic system , , is: …………(8) For any platform, the state equation has the form of equation (6): …………(9) Referring to the extended Lyapunov method, the sliding mode control surface , and are respectively: …………(10) The step 3 comprises the following steps: …………(11) The step 4 comprises the following steps: …………(12) The state equation of equation (6) can be rewritten as the following error dynamics equation: …………(13) wherein z is a sequence number, less than or equal to N; is a gain coefficient, is a reaching law; The sliding mode reaching law is exponential reaching: Then the equation group is: The control quantity of the i-th platform and the j-th group of SMA springs is: After the trajectory tracking is completed and the desired accuracy is met, the cycle is ended, and the MIMO time sequence control quantity in the whole trajectory tracking process is output.

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