CN115981144B - Global nonlinear sliding mode finite time control method for gear inspection robot - Google Patents
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Abstract
A global nonlinear sliding mode finite time control method of a gear inspection robot constructs an error equation of a tracking track and a target track based on a mathematical model of an N-joint mechanical arm of the gear inspection robot system; on the basis, a nonlinear function and an inertial matrix related to a model are combined, a nonlinear function initial item of tracking error is output by a gear inspection robot system, a global finite time nonlinear sliding mode surface is established, the purposes of small-gain amplification and large-gain suppression of error are achieved, and the dynamic initial item is ensured to be on the sliding mode surface; based on the accessibility condition of the sliding mode surface, the system model is designed to be related to a general constant-speed approach law, so that the robustness and the self-adaptability under disturbance are improved.
Description
Technical Field
The invention relates to the technical field of industrial robot control systems, in particular to a global nonlinear sliding mode limited time control method of a gear inspection robot.
Background
The gear inspection robot is used as an industrial robot in an automatic assembly line, can be used for detecting gear products in the assembly line and guaranteeing the yield of finished products, and is composed of a multi-joint mechanical arm, wherein the multi-joint mechanical arm is a degree-of-freedom nonlinear system, the system has the problems of uncertainty, high nonlinearity, strong coupling and the like, and the conventional linear control method at present, such as a cooperative robot control method based on limited time tracking control disclosed in patent CN112223275A, can not meet the requirement of high-precision control although the control of the joint movement of the robot is realized. As a highly nonlinear system, how to effectively control the motion trail is a difficult problem, and there are problems of poor repeated positioning accuracy and the like, which still need to be solved.
Disclosure of Invention
Aiming at solving the technical problems of low track precision, repeated positioning offset and the like of a gear inspection robot system, the invention provides the following technical scheme: constructing an error equation of a tracking track and a target track based on a mathematical model of an N-joint mechanical arm of the gear inspection robot system; utilizing the gear inspection robot system to output tracking error, nonlinear function, the gear inspection robot system N-joint mechanical arm inertia matrix and the gear inspection robot system to output a tracking error nonlinear function initial term to construct a global finite-time nonlinear sliding mode surface; based on the accessibility condition of the sliding mode surface, designing a system model related general constant-speed approach law and gear inspection robot system N joint mechanical arm controller tau (t), and verifying stability.
A gear inspection robot global nonlinear sliding mode limited time control method comprises the following steps:
step 1, constructing an error equation of a tracking track and a target track based on a mathematical model of an N-joint mechanical arm of a gear inspection robot system;
step 2, utilizing a gear inspection robot system to output tracking errors, a nonlinear function, an N-joint mechanical arm inertia matrix of the gear inspection robot system and a gear inspection robot system to output a tracking error nonlinear function initial term to construct a global finite-time nonlinear sliding mode surface;
and step 3, designing a system model related general constant-speed approach law and gear inspection robot system N joint mechanical arm controller tau (t) based on the accessibility condition of the sliding die surface, and verifying stability.
In step 1, the gear inspection robot system includes an image operation center, a high-power camera, and an N-joint mechanical arm, and the high-power camera collects the gear surface quality image.
Further, in step 1, the mathematical model of the N-joint mechanical arm of the gear inspection robot system is as follows:
wherein q (t) represents the actual track of the N joint mechanical arm of the gear inspection robot system, the first order and the second order respectively represent the angular velocity and the angular acceleration, and the left side of the equation is respectively: n-joint mechanical arm inertia force item of gear inspection robot systemCentrifugal force and coriolis force item of N-joint mechanical arm of gear inspection robot system>Gravity item G (q (t)) n×n Friction force item of N-joint mechanical arm of gear inspection robot system>External disturbance term τ d (t)∈R n×n ；M(q(t))∈R n×n For gear inspection robot system N joint arm inertial matrix, +.> The centrifugal force matrix and the coriolis force matrix of the N-joint mechanical arm of the gear inspection robot system are adopted; the right side of the equation is the control force term tau (t) epsilon R of the N-joint mechanical arm of the gear inspection robot system n×n ；
Defining the tracking error of the N joint mechanical arm of the gear inspection robot system as follows:
e(t)＝q * (t)-q(t)
wherein q * (t) is a target track of the N-joint mechanical arm of the gear inspection robot system, and e (t) is a tracking error of the N-joint mechanical arm of the gear inspection robot system;
taking first-order differentiation of tracking error of the N-joint mechanical arm of the gear inspection robot system to obtain:
wherein,is a first order derivative of e (t); />Is q * (t) first order differentiation; />Is the first derivative of q (t);
taking a second-order differential from the tracking error of the N-joint mechanical arm of the gear inspection robot system to obtain:
wherein,is the second order derivative of e (t); />Is q * Second order differentiation of (t); />Is the second order derivative of q (t).
Further, in step 2, in combination with the output tracking error of the gear inspection robot system, the nonlinear function, the inertia matrix M (q (t)) of the N-joint mechanical arm of the gear inspection robot system, the gear inspection robot system outputs an initial item of the nonlinear function of the tracking error, and a global finite-time nonlinear sliding mode surface is constructed:
s(t)＝[s 1 (t)，s 2 (t)...s n (t)] T
k gnp ＝diag[k 1gnp ，j 2gnp ...j ngnp ] T
k gni ＝diag[k 1gni ，k 2gni ...k ngni ] T
e(t)＝[e 1 (t)，e 2 (t)...e n (t)] T
e(0)＝[e 1 (0)，e 2 (0)...e n (0)] T
k＝diag[k 1 ，k 2 ...k n ] T
fal[e(t)，γ，η]
＝[fal[e 1 (t)，γ，η] fal[e 2 (t)，γ，η] … fal[e n (t)，γ，η]] T
fal[e(0)，γ，η]
＝[fal[e 1 (0)，γ，η] fal[e 2 (0)，γ，η] … fal[e n (0)，γ，η]] T
wherein s is 1 (t)，s 2 (t)…s n (t) is a sub-slip plane of the globally finite time nonlinear slip plane s (t); k (k) gnp And k gni Parameter adjusting gains of a nonlinear proportional term and a nonlinear integral term of the tracking error output by the gear inspection robot system are respectively k is a direct feedback gain, e (0) is an initial state of the tracking error output by the gear inspection robot system, fal [ e (0), gamma, eta]Is the initial item of the nonlinear function of the output tracking error of the gear inspection robot system, 0<γ<1，η>0，
Taking first-order differentiation of the global limited time nonlinear sliding mode surface to obtain:
wherein,is a first order derivative of s (t);
when |e (t) | > η and e (t) > 0:
when |e (t) | > η and e (t) < 0:
when |e (t) | < η:
wherein C is an arbitrary constant, t s Is a finite time.
Further, in step 2, the slip plane reachability condition is defined:
obtaining a system model related general isokinetic approach law:
wherein α=diag [ α ] 1 ，α 2 …α n ] T For regulating the gain of the ginseng and determining tau md (t)∈R n×1 >Max(τ d (t)) is the upper bound of the disturbance,
further, in step 3, the first-order differentiation of the global finite-time nonlinear sliding mode surface and the system model-related general isovelocity approach law are combined to obtain:
for a pair ofTaking the first-order differential to obtain:
when |e (t) | > η:
when |e (t) | < η:
according to the first-order differential and the second-order differential of the tracking error of the N-joint mechanical arm of the gear inspection robot system, the method comprises the following steps of:
when |e (t) | > η:
when |e (t) | < η:
further, in step 3, the mathematical model of the N-joint mechanical arm of the gear inspection robot system is combined, and the N-joint mechanical arm controller τ (t) of the gear inspection robot system is designed:
when |e (t) | > η:
when |e (t) | < η:
the invention has the beneficial effects that:
(1) The method achieves the purposes of small-gain amplification and large-gain suppression of errors, and ensures that a dynamic initial item is on a sliding mode surface;
(2) Based on the accessibility condition of the sliding mode surface, designing a system model related general constant-speed approach law, and improving the robustness and the self-adaptability under disturbance;
(3) Experiments prove that the method is superior to the iterative learning control method ILC in three angles of robustness, rapidness and vibration suppression;
(4) High robustness and small errors of dynamic control of the target track are achieved.
Drawings
Fig. 1 is a schematic diagram of a control principle frame of a global nonlinear sliding mode finite time control method of a gear inspection robot according to an embodiment of the invention.
Fig. 2 is a schematic diagram of track tracking of a target track (1) of a dual-knuckle mechanical arm and a track tracking schematic diagram of a global nonlinear sliding mode finite time control method (GNSMFTC) of a gear inspection robot according to an embodiment of the present invention.
Fig. 3 is a schematic diagram of track tracking of a target track (2) of a dual-knuckle mechanical arm and a track tracking of a global nonlinear sliding mode finite time control method (GNSMFTC) of the gear inspection robot according to the embodiment of the present invention.
Fig. 4 is a schematic diagram of track tracking of a target track (1) of a double-joint mechanical arm and an iterative learning control method (ILC) of a global nonlinear sliding mode finite time control method of a gear inspection robot according to an embodiment of the present invention.
Fig. 5 is a schematic diagram of track tracking of a target track (2) of a double-joint mechanical arm and an iterative learning control method (ILC) of a global nonlinear sliding mode finite time control method of a gear inspection robot according to an embodiment of the present invention.
Detailed Description
The technical scheme of the invention is further described in detail below with reference to the attached drawings.
Referring to fig. 1, for an embodiment of the present invention, a method for controlling a global nonlinear sliding mode finite time of a gear inspection robot is provided, where the method organically unifies a mathematical model of an N-joint mechanical arm of the gear inspection robot system, an output tracking error of the gear inspection robot system, a global finite time nonlinear sliding mode surface, and a system model related general constant velocity approach law based on a accessibility condition of the sliding mode surface into a design of an N-joint mechanical arm controller u (t) of the gear inspection robot system, referring to fig. 1, and specifically includes:
s1: and constructing an error equation of a tracking track and a target track based on the mathematical model of the N-joint mechanical arm of the gear inspection robot system.
The gear inspection robot system comprises an image operation center, a high-power camera and an N-joint mechanical arm.
The gear inspection robot system N joint mechanical arm mathematical model:
wherein, the left side of the equation is respectively: n-joint mechanical arm inertia force item of gear inspection robot systemCentrifugal force and coriolis force item of N-joint mechanical arm of gear inspection robot system>Gravity item G (q (t)) n×n N-joint mechanical arm friction force item of gear inspection robot systemExternal disturbance term τ d (t)∈R n×n ；M(q(t))∈R n×n Robot for gear inspectionInertia matrix of N-joint mechanical arm of system +.>The centrifugal force matrix and the coriolis force matrix of the N-joint mechanical arm of the gear inspection robot system are adopted; the right side of the equation is the control force term tau (t) epsilon R of the N-joint mechanical arm of the gear inspection robot system n×n 。
Defining the tracking error of the N joint mechanical arm of the gear inspection robot system as follows:
e(t)＝q * (t)-q(t)
wherein q * (t) is a target track of the N-joint mechanical arm of the gear inspection robot system, and e (t) is a tracking error of the N-joint mechanical arm of the gear inspection robot system.
Taking first-order differentiation of tracking error of the N-joint mechanical arm of the gear inspection robot system to obtain:
wherein,is a first order derivative of e (t); />Is q * (t) first order differentiation; />Is the first derivative of q (t).
Taking a second-order differential from the tracking error of the N-joint mechanical arm of the gear inspection robot system to obtain:
wherein,is the second order derivative of e (t); />Is q * Second order differentiation of (t); />Is the second order derivative of q (t).
S2: and constructing a global finite-time nonlinear sliding mode surface by utilizing the output tracking error of the gear inspection robot system, the nonlinear function, the inertia matrix of the N-joint mechanical arm of the gear inspection robot system and the output tracking error nonlinear function initial term of the gear inspection robot system.
Combining the output tracking error of the gear inspection robot system, the nonlinear function, the inertia matrix M (q (t)) of the N joint mechanical arm of the gear inspection robot system, the gear inspection robot system outputting the initial item of the tracking error nonlinear function, and constructing a global finite-time nonlinear sliding mode surface:
s(t)＝[s 1 (t)，s 2 (t)...s n (t)] T
k gnp ＝diag[k 1gnp ，k 2gnp ...k ngnp ] T
k gni ＝diag[k 1gni ，k 2gni ...k ngni ] T
e(t)＝[e 1 (t)，e 2 (t)...e n (t)] T
e(0)＝[e 1 (0)，e 2 (0)...e n (0)] T
k＝diag[k 1 ，k 2 ...k n ] T
fal[e(t)，γ，η]
＝[fal[e 1 (t)，γ，η] fal[e 2 (t)，γ，η] … fal[e n (t)，γ，η]] T
fal[e(0)，γ，η]
＝[fal[e 1 (0)，γ，η] fal[e 2 (0)，γ，η] … fal[e n (0)，γ，η]] T
wherein s is 1 (t)，s 2 (t)…s n (t) is a sub-slip plane of the globally finite time nonlinear slip plane s (t); k (k) gnp And k gni Parameter adjusting gains of a nonlinear proportional term and a nonlinear integral term of the tracking error output by the gear inspection robot system are respectively k is a direct feedback gain, e (0) is an initial state of the tracking error output by the gear inspection robot system, fal [ e (0), gamma, eta]Is the initial item of the nonlinear function of the output tracking error of the gear inspection robot system, 0<γ<1，η>0，
Taking first-order differentiation of the global limited time nonlinear sliding mode surface to obtain:
wherein,is a first order derivative of s (t).
When |e (t) | > η and e (t) > 0:
when |e (t) | > η and e (t) < 0:
when |e (t) | < η:
/>
wherein C is an arbitrary constant, t s Is a finite time.
S3: based on the accessibility condition of the sliding mode surface, designing a system model related general constant-speed approach law and gear inspection robot system N joint mechanical arm controller tau (t), and verifying stability.
Defining the slip plane accessibility condition:
obtaining a system model related general isokinetic approach law:
wherein α=diag [ α ] 1 ，α 2 …α n ] T For regulating the gain of the ginseng and determining tau md (t)∈R n×1 >Max(τ d (t)) is the upper bound of the disturbance,
the first-order differentiation of the global finite-time nonlinear sliding mode surface and the system model related general isokinetic approach law are combined to obtain:
for a pair ofTaking the first-order differential to obtain:
when |e (t) | > η:
when |e (t) | < η:
according to the first-order differential and the second-order differential of the tracking error of the N-joint mechanical arm of the gear inspection robot system, the method comprises the following steps of:
when |e (t) | > η:
when |e (t) | < η:
the mathematical model of the N-joint mechanical arm of the gear inspection robot system is established simultaneously, and a N-joint mechanical arm controller tau (t) of the gear inspection robot system is designed:
when |e (t) | > η:
when |e (t) | < η:
to demonstrate the stability of the controller, the Lyapunov function was increased as:
where V is the Lyapunov function.
/>
Referring to fig. 1, which is a schematic diagram of a control principle frame of a global nonlinear sliding mode finite time control method of a gear inspection robot, the method of the invention is further described, and a main control diagram process is as follows: firstly, importing target tracks (1) and (2) of a double-joint mechanical arm; secondly, outputting a tracking error, a nonlinear function and an initial item of the nonlinear function by a gear inspection robot system, and establishing a global finite time nonlinear sliding mode surface by the gear inspection robot system; finally, a gear inspection robot system N-joint mechanical arm controller tau (t) is designed through a system model related general constant-speed approach law of the accessibility condition of the sliding die surface.
Preferably, the embodiment also needs to explain that, compared with the prior art, the invention discloses a method for controlling the global nonlinear sliding mode finite time of the gear inspection robot, which aims to track the target tracks (1) and (2) of the double-joint mechanical arm by adopting the global nonlinear sliding mode finite time, and then achieve the dynamic response of the approaching stage by the system model related general constant-speed approaching law. The dynamic control performance of the target track is realized: high robustness and small error.
Referring to fig. 2 to 5, in another embodiment of the present invention, unlike the first embodiment, a test verification of a global nonlinear sliding mode limited time control method of a gear inspection robot is provided, including:
in order to verify and explain the technical effects adopted in the method, in the embodiment, an iterative learning control method (ILC) is selected to be compared with the method of the invention, and the test results are compared by a scientific demonstration means to verify the true effects of the method.
In order to verify that the method has higher precision compared with the traditional method, the method adopts a global nonlinear sliding mode finite time control method (GNSMFTC), and the real-time measurement comparison is carried out on the tracking track of the N-joint mechanical arm of the gear inspection robot system under the condition of the target tracks (1) and (2) of the double-joint mechanical arm and the iterative learning control method (ILC) respectively.
Test environment: referring to fig. 1, a gear inspection robot system is operated on a simulation platform to simulate and track target tracks (1) and (2) of a double-joint mechanical arm, and is tested by a global nonlinear sliding mode finite time control method (GNSMFTC) and an iterative learning control method (ILC) respectively, so as to obtain test result data. All tests are performed by starting automatic test equipment and realizing simulation test of a comparison method by using MATLAB software programming, and simulation data are obtained according to experimental results; each method tests 3 groups of data, each group of data is sampled for 10s, and each group of data tracking tracks is obtained through calculation.
Referring to fig. 2 to 5, a track tracking schematic diagram is shown for comparing a global nonlinear sliding mode finite time control method (GNSMFTC) and an iterative learning control method (ILC) under the target tracks (1) and (2) of the double-joint mechanical arm.
Iterative learning control method (ILC):
wherein u is k+1 (t) is the input of K+1 times, u k (t) is the input of K times, K d Is the feedback gain.
Double-joint mechanical arm parameters: connecting rod 1 mass m 1 Length l of connecting rod 1 =1 kg 1 Distance l of centroid to joint 1 =1m c1 1/2m, connecting rod 1 moment of inertia I 1 =1/12 kg·m, connecting rod 2 mass m e Distance l of link 2 to joint 2 =3 kg ce =1m, connecting rod 2 moment of inertia I e =2/5 kg·m, centroid and joint 2 angle δ e Coefficient of friction e =0 1 = -7/12, gravitational acceleration e 2 ＝9.81。
q(t)＝[q 1 (t) q 2 (t)] T
τ(t)＝[τ 1 (t) τ 2 (t)] T
Wherein,ε＝m e l 1 l ce cos(δ e )，η＝m e l 1 l ce sin(δ e )。
e(t)＝q * (t)-q(t),
α=6.73, β=3.4, ε=3, η=0 is calculated.
/>
Referring to fig. 2 to 5, k 1gnp ＝1，k 1 ＝100，k gni ＝1，γ＝0.5，η＝0.5，k 2gnp ＝1，k 2 ＝2，k 2gni ＝0.02，/> The iterative learning control method (ILC) is of an M2_PD iterative type; as can be seen from fig. 2, when tracking the target track (1) of the double-joint mechanical arm, the global nonlinear sliding mode finite time control method (GNSMFTC) can completely track and can effectively inhibit switching vibration; as can be seen from fig. 3, when tracking the target track (2) of the double-joint mechanical arm, the target track period is enlarged by 2 times, and the global nonlinear sliding mode finite time control method (GNSMFTC) can also completely track with slight vibration phenomenon; as can be seen from fig. 4, when the target track (1) of the dual-joint mechanical arm is tracked, the iterative learning control method (ILC) can be completely tracked in the early stage, but when the time reaches 7s, an error accumulation phenomenon occurs and becomes larger and larger; as can be seen from fig. 5, when the target track (2) of the dual-joint mechanical arm is tracked, the iterative learning control method (ILC) cannot effectively track and has a larger tracking error in the early stage, and the target track (2) can be gradually tracked in the later stage along with the lapse of time, which illustrates that the rapidity of the iterative learning control method (ILC) is inferior to that of the global nonlinear sliding mode finite time control method (GNSMFTC).
In summary, the method for controlling the global nonlinear sliding mode finite time of the gear inspection robot is superior to the iterative learning control method (ILC) in three angles of robustness, rapidness and vibration suppression, is attributed to a nonlinear function, an initial term and a global finite time nonlinear sliding mode surface formed by an N-joint mechanical arm inertia matrix of the gear inspection robot system, and can suppress vibration phenomena by a designed system model related general constant-speed approach law based on the accessibility condition of the sliding mode surface.
It should be appreciated that embodiments of the invention may be implemented or realized by computer hardware, a combination of hardware and software, or by computer instructions stored in a non-transitory computer readable memory. The methods may be implemented in a computer program using standard programming techniques, including a non-transitory computer readable storage medium configured with a computer program, where the storage medium so configured causes a computer to operate in a specific and predefined manner, in accordance with the methods and drawings described in the specific embodiments. Each program may be implemented in a high level procedural or object oriented programming language to communicate with a computer system. However, the program(s) can be implemented in assembly or machine language, if desired. In any case, the language may be a compiled or interpreted language. Furthermore, the program can be run on a programmed application specific integrated circuit for this purpose.
Furthermore, the operations of the processes described herein may be performed in any suitable order unless otherwise indicated herein or otherwise clearly contradicted by context. The processes (or variations and/or combinations thereof) described herein may be performed under control of one or more computer systems configured with executable instructions, and may be implemented as code (e.g., executable instructions, one or more computer programs, or one or more applications), by hardware, or combinations thereof, collectively executing on one or more processors. The computer program includes a plurality of instructions executable by one or more processors.
Further, the method may be implemented in any type of computing platform operatively connected to a suitable computing platform, including, but not limited to, a personal computer, mini-computer, mainframe, workstation, network or distributed computing environment, separate or integrated computer platform, or in communication with a charged particle tool or other imaging device, and so forth. Aspects of the invention may be implemented in machine-readable code stored on a non-transitory storage medium or device, whether removable or integrated into a computing platform, such as a hard disk, optical read and/or write storage medium, RAM, ROM, etc., such that it is readable by a programmable computer, which when read by a computer, is operable to configure and operate the computer to perform the processes described herein. Further, the machine readable code, or portions thereof, may be transmitted over a wired or wireless network. When such media includes instructions or programs that, in conjunction with a microprocessor or other data processor, implement the steps described above, the invention includes these and other different types of non-transitory computer-readable storage media. The invention also includes the computer itself when programmed according to the methods and techniques of the present invention. The computer program can be applied to the input data to perform the functions described herein, thereby converting the input data to generate output data that is stored to the non-volatile memory. The output information may also be applied to one or more output devices such as a display. In a preferred embodiment of the invention, the transformed data represents physical and tangible objects, including specific visual depictions of physical and tangible objects produced on a display.
The above description is merely of preferred embodiments of the present invention, and the scope of the present invention is not limited to the above embodiments, but all equivalent modifications or variations according to the present disclosure will be within the scope of the claims.
Claims (4)
1. A gear inspection robot global nonlinear sliding mode finite time control method is characterized in that: the method comprises the following steps:
step 1, constructing an error equation of a tracking track and a target track based on a mathematical model of an N-joint mechanical arm of a gear inspection robot system;
step 2, utilizing a gear inspection robot system to output tracking errors, a nonlinear function, an N-joint mechanical arm inertia matrix of the gear inspection robot system and a gear inspection robot system to output a tracking error nonlinear function initial term to construct a global finite-time nonlinear sliding mode surface;
in step 2, the gear inspection robot system outputs a tracking error, the nonlinear function is combined, the gear inspection robot system N-joint mechanical arm inertia matrix M (q (t)), and the gear inspection robot system outputs an initial item of the tracking error nonlinear function to construct a global finite-time nonlinear sliding mode surface:
s(t)＝＝[s 1 (t)，s 2 (t)...s n (t)] T
k gnp ＝diag[k 1gnp ，k 2gnp ...k ngnp ] T
k gni ＝diag[k 1gni ，k 2gni ...k ngni ] T
e(t)＝[e 1 (t)，e 2 (t)...e n (t)] T
e(0)＝[e 1 (0)，e 2 (0)...e n (0)] T
k＝diag[k 1 ，k 2 ...k n ] T
fal[e(t)，γ，η]＝[fal[e 1 (t)，γ，η] fal[e 2 (t)，γ，η]…fal[e n (t)，γ，η]] T
fal[e(0)，γ，η]＝[fal[e 1 (0)，γ，η] fal[e 2 (0)，γ，η]…fal[e n (0)，γ，η]] T
wherein s is 1 (t)，s 2 (t)...s n (t) is a sub-slip plane of the globally finite time nonlinear slip plane s (t); k (k) gnp And k gni Parameter adjusting gains of a nonlinear proportional term and a nonlinear integral term of the tracking error output by the gear inspection robot system are respectively k is a direct feedback gain, e (0) is an initial state of the tracking error output by the gear inspection robot system, fal [ e (0), gamma, eta]Is the output heel of the gear inspection robot systemThe initial term of the nonlinear function of the trace error is 0 < gamma < 1, eta >0,
taking first-order differentiation of the global limited time nonlinear sliding mode surface to obtain:
wherein,is a first order derivative of s (t);
when |e (t) | > η and e (t) > 0:
when |e (t) | > η and e (t) < 0:
when |e (t) | < η:
wherein C is an arbitrary constant, t s Is a finite time;
step 3, designing a system model related general constant-speed approach law and gear inspection robot system N joint mechanical arm controller tau (t) based on the accessibility condition of the sliding die surface, and verifying stability;
in step 3, the first-order differentiation of the global finite-time nonlinear sliding mode surface and the system model related general constant velocity approach law are combined to obtain:
for a pair ofTaking first-order differentiation to obtain:
when |e (t) | > η:
when |e (t) | < η:
according to the first-order differential and the second-order differential of the tracking error of the N-joint mechanical arm of the gear inspection robot system, the method comprises the following steps of:
when |e (t) | > η:
when |e (t) | < η:
in the step 3, the mathematical model of the N-joint mechanical arm of the gear inspection robot system is combined, and a controller tau (t) of the N-joint mechanical arm of the gear inspection robot system is designed:
when |e (t) | > η:
when |e (t) | < η:
2. the method for controlling the global nonlinear sliding mode finite time of the gear inspection robot according to claim 1, wherein the method is characterized by comprising the following steps of: in step 1, the gear inspection robot system comprises an image operation center, a high-power camera and an N-joint mechanical arm, wherein the high-power camera collects quality images of the surface of the gear.
3. The method for controlling the global nonlinear sliding mode finite time of the gear inspection robot according to any one of claims 1-2, which is characterized by comprising the following steps: in step 1, the mathematical model of the N-joint mechanical arm of the gear inspection robot system is as follows:
wherein q (t) represents the actual track of the N joint mechanical arm of the gear inspection robot system, the first order and the second order respectively represent the angular velocity and the angular acceleration, and the left side of the equation is respectively: n-joint mechanical arm inertia force item of gear inspection robot systemCentrifugal force and coriolis force item of N-joint mechanical arm of gear inspection robot system>Gravity item G (q (t)) n×n Friction force item of N-joint mechanical arm of gear inspection robot system>External disturbance term τ d (t)∈R n×n ；M(q(t))∈R n×n For gear inspection robot system N joint arm inertial matrix, +.> The centrifugal force matrix and the coriolis force matrix of the N-joint mechanical arm of the gear inspection robot system are adopted; the right side of the equation is the control force term tau (t) epsilon R of the N-joint mechanical arm of the gear inspection robot system n×n ；
Defining the tracking error of the N joint mechanical arm of the gear inspection robot system as follows:
e(t)＝q * (t)-q(t)
wherein q * (t) is a target track of the N-joint mechanical arm of the gear inspection robot system, and e (t) is a tracking error of the N-joint mechanical arm of the gear inspection robot system;
taking first-order differentiation of tracking error of the N-joint mechanical arm of the gear inspection robot system to obtain:
wherein,is a first order derivative of e (t); />Is q * (t) first order differentiation; />Is the first derivative of q (t);
taking a second-order differential from the tracking error of the N-joint mechanical arm of the gear inspection robot system to obtain:
wherein,is the second order derivative of e (t); />Is q * Second order differentiation of (t); />Is the second order derivative of q (t).
4. The method for controlling the global nonlinear sliding mode finite time of the gear inspection robot according to claim 3, wherein the method comprises the following steps of: in step 2, a slip plane reachability condition is defined:
obtaining a system model related general isokinetic approach law:
wherein α=diag [ α ] 1 ，α 2 ...α n ] T For regulating the gain of the ginseng and determining tau md (t)∈R n×1 ＞Max(τ d (t)) is the upper bound of the disturbance,
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