A robot system tracking control method based on a bounded-jerk type output constraint

Abstract

The application discloses a robot system tracking control method based on a bounded mutation type output constraint, and comprises the following steps: establishing an uncertain robot system dynamics model; formulating and describing a corresponding mutation time range based on a bounded mutation type constraint boundary function; constructing a transfer function and building a conversion mechanism based on the constructed transfer function, so as to convert any multiple bounded mutation type constraint boundary functions into globally continuous functions; designing an adaptive control scheme based on the converted constraint boundary functions, relaxing the restriction condition that the constraint boundary function is always a positive function or a negative function in a traditional control method, ensuring that the output of the system does not violate the bounded mutation type constraint condition, and ensuring that the output of the system closely tracks the expected trajectory; and applying the designed control scheme to a continuous constraint scene by changing the discontinuous time. The application solves the problem that the output of the robot system is limited by the bounded mutation type constraint condition during operation.

Description

Technical Field

[0001] This invention belongs to the field of constraint control technology for robot systems, and specifically relates to a tracking control method for robot systems based on bounded mutation type output constraints. Background Technology

[0002] Robotic systems have wide applications in industry. However, when these systems operate in complex dynamic environments, their outputs are often limited by various constraints, which can seriously threaten their safe and stable operation. It is particularly important to note that most existing control strategies are only applicable to continuous output constraints. However, in real-world industrial scenarios, constraints are often discontinuous. Taking palletizing robot operations as an example, due to significant differences in the location, volume, and weight of transported goods, the robot system often needs to cope with abrupt dynamic constraints during actual operation. This condition can easily lead to sudden changes in system load, resulting in decreased positioning accuracy of the end effector or even system instability. Clearly, traditional control methods are ineffective in handling such discontinuous constraints. In recent years, some researchers have conducted studies on control problems under discontinuous constraints. For example, by designing transfer functions and other methods, the control problem of switching between constrained and unconstrained states of system output has been successfully solved. However, existing research still has limitations: on the one hand, these results mainly focus on the transition of system output between constrained and unconstrained states; on the other hand, related studies usually require the constraint boundary function to maintain a constant positive or negative property, and have not addressed the problem that the alternating positive and negative constraints on the system output at a certain moment are abrupt but bounded. Therefore, in-depth research on a robot system tracking control method based on bounded abrupt output constraints has significant research significance and potential application value. Summary of the Invention

[0003] To address the problem of bounded mutational output constraint control for uncertain robot systems, the present invention aims to provide a robot system tracking control method based on bounded mutational output constraints, thereby solving the problem that the robot system output is limited by bounded mutational constraints during operation.

[0004] To achieve the above objectives, the present invention adopts the following technical solution:

[0005] A tracking control method for a robot system based on bounded mutation-type output constraints includes the following steps:

[0006] Step 1: Establish a dynamic model of an uncertain robot system with n joints;

[0007] Step 2: Based on the bounded mutation constraint boundary function, formulate a description of the corresponding mutation time range;

[0008] Step 3: Construct a transfer function for smoothing the boundary of the mutation constraint, and build a transformation mechanism based on the constructed transfer function to convert any number of bounded mutation constraint boundary functions into globally continuous functions.

[0009] Step 4: Based on the transformed constraint boundary function, design an adaptive control scheme, relax the restriction that the constraint boundary function is always a positive or negative function in the traditional control method, and ensure that the system output does not violate the bounded mutation constraint condition, and that the system output closely tracks the desired trajectory;

[0010] Step 5: By changing the discontinuous moments, the designed control scheme is applied to continuous constraint scenarios.

[0011] Furthermore, in step 1, the dynamic model of the uncertain robot system with n joints is established as follows:

[0012]

[0013] Where, q∈R n , R represents the angular displacement, angular velocity, and angular acceleration of the joint, respectively. n Represent an n-dimensional column vector; p∈R n M(q,p)∈R is an unknown constant vector. n×n It is a positive definite symmetric inertia matrix. It is a Coriolis matrix, G(q,p)∈R n×n Let D(p,t) ∈ R represent the gravity vector. n Represents unknown external disturbances, u∈R n It is the torque input vector; the vector dimension n is the same as the number of joints n.

[0014] Let x1 = [x 11 ,...,x 1n ] T =qsum Equation (1) is then transformed into the following system:

[0015]

[0016] Furthermore, step 2 is detailed below:

[0017] Known constraint boundary functions:

[0018] k dfi <x 1i <k ufi (3)

[0019]

[0020] Where i = 1,...,n, Indicates the number of joints. Let v represent a positive integer, v = j + 1, k ufi and k dfi Then, these represent the upper and lower constraint boundary functions of the system output at t∈[0,+∞); T0=0, T w These are known discontinuous moments, and adjacent constraint boundary functions are discontinuous, i.e., k uiw (T w )≠k uir (T w ), k diw (T w )≠k dir (T w ), w = 1, ..., j, r = w + 1;

[0021] Next, define the boundary function k. uiw k uir k diw and k dir Two fixed times T wL and T wR It meets the following conditions:

[0022] T w-1 <T wL <T w <T wR <T r (6)

[0023]

[0024] Where, when w = j, T j+1 =+∞; Therefore, the discontinuity problem of the constraint boundary function is described by equation (7).

[0025] Furthermore, step 3 is detailed below:

[0026] Based on the upper constraint boundary function of mutation, two types of transfer functions F are constructed. Rw (t) and F Lw (t);

[0027] Transfer function F Rw (t) is:

[0028]

[0029] Where e is the natural constant, w = 1,...,j and Δ w =(tT) w ) / (T wR -t);

[0030] Transfer function F Rw (t) has the following properties:

[0031] ①F Rw (t) in t∈[0,T) w The value of ] is always 1, in t∈[T wR The value of (+∞) is always 0;

[0032] ②F Rw (t) in t∈(T) w ,T wR Strictly monotonically decreasing;

[0033] ③F Rw (t), and The condition is continuous and bounded in t∈[0,+∞);

[0034] Transfer function F Lw (t) is:

[0035]

[0036] Where e is the natural constant, w = 1,...,j and Λ w =(T w -t) / (tT wL );

[0037] Transfer function F Lw (t) has the following properties:

[0038] ①F Lw (t) in t∈[0,T) wL The value of ] is always 0, in t∈[T w The value of (+∞) is always 1;

[0039] ②F Lw (t) in t∈(T) wL ,T w Strictly monotonically increasing;

[0040] ③F Lw (t), and The condition is continuous and bounded in t∈[0,+∞);

[0041] Based on the lower constraint boundary function of mutation, two types of transfer functions W are constructed. Rw (t) and W Lw (t); transfer function W Rw (t) is as follows:

[0042]

[0043] Where e is the natural constant, w = 1,...,j and β w =(T wR -t) / (tT w );

[0044] Transfer function W Rw (t) has the following properties:

[0045] ①W Rw (t) in t∈[0,T) w The value of ] is always 0, in t∈[T wR The value of (+∞) is always 1;

[0046] ②W Rw (t) in t∈(T) w ,T wR Strictly monotonically increasing;

[0047] ③W Rw (t), and The condition is continuous and bounded in t∈[0,+∞);

[0048] Transfer function W Lw (t) is as follows:

[0049]

[0050] Where e is the natural constant, w = 1,...,j and τ w =(tT) wL ) / (T w -t);

[0051] Transfer function W Lw (t) has the following properties:

[0052] ①W Lw (t) in t∈[0,T) wL The value of ] is always 1, in t∈[T w The value of (+∞) is always 0;

[0053] ②W Lw (t) in t∈(T) wL ,T w Strictly monotonically decreasing;

[0054] ③W Lw (t), and The condition is continuous and bounded in t∈[0,+∞);

[0055] Next, based on the transfer function, transformation mechanisms are constructed for the upper constraint boundary function and the lower constraint boundary function, as follows:

[0056] ① If in t∈(T) wL ,T w )inner k uiw >k uir ,but

[0057]

[0058] ②If in t∈(T) w ,T wR )inner k uiw <k uir ,but

[0059]

[0060] ③ If in t∈(T) wL ,T w )inner k diw <k dir ,but

[0061]

[0062] ④ If in t∈(T) w ,T wR )inner k diw >k dir ,but

[0063]

[0064] Where w = 1, ..., j, r = w + 1, T j+1 =+∞, k ui Let k represent the transformed upper constraint boundary function. di This represents the transformed lower constraint boundary function.

[0065] Furthermore, in step 3, the transformed constraint boundary function k ui and k di It has the following properties:

[0066] ①k ui >k di This holds true for all t∈[0,+∞).

[0067] ②k ui ,k di Their first derivatives and their second derivatives The condition is continuous and bounded in t∈[0,+∞).

[0068] Furthermore, step 4 specifically includes the following sub-steps:

[0069] Step 4.1, consider the alternating positive and negative constraint boundary functions, and select a nonlinear function of the following form:

[0070]

[0071] Where i = 1,...,n, therefore, we obtain

[0072]

[0073] Among them, a 1i =(k ui -k di )l 1i , as well as Next, we get

[0074]

[0075] Where, A1 = diag(a 1i ) and B1 = [b 11 ,b 12 ,...,b 1n ] T ;

[0076] When k di (0) <x 1i (0) <k ui (0) if and only if x 1i →k di Or x 1i →k ui , ζ 1i →∞ holds true; therefore, if ζ is guaranteed 1i It is bounded, which ensures x 1i ∈(k di ,k ui );

[0077] Because of k ui >k di It is easy to obtain a 1i The value >0 always holds true, meaning the constraint boundary function alternates between positive and negative values;

[0078] Step 4.2, to facilitate controller design, define the generalized tracking error z1 and the virtual error z2, as follows:

[0079]

[0080] Where, ζ1=[ζ 11 ,....,ζ 1n ] T , z1 = [z 11 ,....,z 1n] T z2 = [z 21 ,....,z 2n ] T χ is the output of the first-order filter, ζ d =[ζ d1 ,....,ζ dn ] T ,and

[0081]

[0082] Among them, y di Represents the known desired signal y d The i-th component; the first-order filter is represented as:

[0083]

[0084] in, These are design parameters. For virtual controllers;

[0085] Next, define the filtering error:

[0086]

[0087] From equations (19) and (22):

[0088]

[0089] From equations (21) and (22):

[0090]

[0091] Step 4.3: Design the controller using backstepping technology.

[0092] Furthermore, step 4.3 specifically includes the following sub-steps:

[0093] Step 4.3.1, select the Lyapunov function Then V was launched 10 The derivative with respect to time t:

[0094]

[0095] According to Young's inequality, it is easy to obtain:

[0096]

[0097] Next, the virtual controller Designed as follows:

[0098]

[0099] Where k1>0 represents the design parameters;

[0100] Choose the Lyapunov function V1 as:

[0101]

[0102] The derivative of V1 with respect to time t is:

[0103]

[0104] in, It is a constant.

[0105] Step 4.3.2, select the Lyapunov function as:

[0106]

[0107] V 20 The derivative with respect to time t is:

[0108]

[0109] From equation (2):

[0110]

[0111] make roll out:

[0112]

[0113] in, Indicates an unknown parameter. Let denote a computable function, c1>0, g1>0, d1>0, λ m >0 are all constants;

[0114] Next, using Young's inequality, we derive:

[0115]

[0116] The design of the real controller and adaptive law is as follows:

[0117]

[0118] Where k2>0, μ1>0, and δ>0 are all design parameters. This represents an estimate of the unknown parameter θ. Represents unknown parameters, and It is a computable function;

[0119] Next, we select the Lyapunov function V2 as:

[0120]

[0121] in, This represents the estimation error; the derivative of V2 with respect to time t is:

[0122]

[0123] in, It is an unknown constant term;

[0124] Consider an uncertain robotic system (1), if the initial conditions satisfy k di (0) <x 1i (0) <k ui (0), and there exists a constant E>0 such that V n (0)≤E, then under the action of the controller (37) and the parameter adaptive law (38), by selecting appropriate design parameters, it is possible to achieve that all signals of the closed-loop system are bounded, the output of the robot system closely tracks the desired trajectory, and the system output does not violate the constraint conditions of bounded mutation.

[0125] because

[0126]

[0127] Both are compact sets, therefore in There exist constants K>0 such that ||H||≤K holds, where E0>0, E1>0, E2>0, and E3>0 all represent constants;

[0128] Next, we can derive from equation (40):

[0129]

[0130] Where M2 = M1 + K 2 , Therefore, the launch Right now The initial value of the system satisfies k di (0) <x 1i (0) <k ui When (0), we get as well as Depend on roll out but Next, launch but Through similar analysis, it is proven that all signals within the closed-loop system are bounded;

[0131] Derived from equation (45) That is, by selecting appropriate design parameters, the generalized tracking error z1 can be made sufficiently small;

[0132] Next, from equation (19), the generalized tracking error z is derived. 1i Compared with the actual tracking error e 1i =x 1i -y di The relationship is:

[0133]

[0134] Where i = 1,...,n; since y di -k di >0 and k ui -y di >0, launch Where D>0 is a constant; by The conclusion is Therefore, by selecting the design parameters, the actual tracking error e can also be reduced. 1i Small enough;

[0135] Depend on To obtain k di <x 1i <k ui According to equations (4) and (12), when t∈[T] w-1 ,T wL When k ui =k ufi =k uiw When t∈[T] w ,T r When k ui =k ufi =k uir Therefore, x 1i <k ufi In t∈[T] w-1 ,T wL ] and t∈[T w ,T r It holds true for all t ∈ (T); wL ,T w When ), from equation (12), we can deduce:

[0136]

[0137] According to equations (4) and (47), we obtain that when t∈(T) wL ,T w When x 1i <k ui ≤k ufi This holds true for all; therefore, when k uiw >kuir At that time, x 1i <k ui ≤k ufi In any t∈[T] w-1 ,T r All of these hold true, i.e., x 1i <k ui ≤k ufi It holds true for all t ∈ [0, +∞); through similar analysis, it is proved that when k uiw <k uir At that time, x 1i <k ui ≤k ufi This still holds true for t∈[0,+∞), meaning the system output does not violate the bounded catastrophe constraint boundary function.

[0138] Next, from equations (5) and (14), we can deduce that when t∈[T] w-1 ,T wL ], k di =k dfi =k diw It always holds true when t∈[T] w ,T r When k di =k dfi =k dir It always holds true; when t∈(T) wL ,T w When ), it is easy to obtain (k) dir -k diw (1-W) Lw If )≥0, then (1-W) Lw )k dir +W Lw k diw ≥k diw That is, when t∈(T) wL ,T w When k di ≥k dfi It is true; therefore, x 1i >k di ≥k dfi In any t∈[T] w-1 ,T r All of these hold true, i.e., x 1i >k di ≥k dfi It holds true on t∈[0,+∞); by similar analysis, it is proved that when k diw >k dir At that time, x 1i >k di ≥k dfi This still holds true for t∈[0,+∞), meaning the system output does not violate the bounded boundary function under bounded mutation.

[0139] In summary, the robot system output does not violate the bounded mutation constraint, that is, when t∈[0,+∞), k dfi ≤k di <x 1i <k ui ≤k ufi It always holds true.

[0140] Furthermore, step 5 is detailed below:

[0141] When T w When j = +∞, w = 1, k ufi =k ui1 k dfi =k di1 k ui =k ui1 , and k di =k di1 At this point, the system output constraint condition is k. di1 <x 1i <k ui1 .

[0142] Beneficial effects: Compared with the prior art, the present invention has the following advantages:

[0143] 1. The present invention proposes a robot system tracking control method based on bounded mutation output constraints, which can effectively solve the problem that the constraints on the system output are bounded but discontinuous at some time. It can ensure that the system output closely tracks the desired trajectory and does not violate the constraints at each stage, thereby improving the practicality of the control strategy.

[0144] 2. Unlike most existing methods that require the upper constraint boundary function to always be a positive function and the lower constraint boundary function to always be a negative function, this invention only requires the upper constraint boundary function to be greater than the lower constraint boundary function, allowing the constraint boundary function to alternate between positive and negative, thereby greatly relaxing the restrictions on the constraint conditions and making the controller applicable to a wider range of scenarios.

[0145] 3. The control strategy proposed in this invention has strong flexibility. Under the premise of keeping the controller structure unchanged, the control strategy can also be applied to traditional continuous constraint scenarios, further enhancing the practical applicability of the control strategy and significantly improving the adaptability and feasibility of the algorithm under different working conditions. Attached Figure Description

[0146] Figure 1 This invention presents a control block diagram for tracking control of a robot system based on bounded mutation-type output constraints. Detailed Implementation

[0147] The invention will now be further explained with reference to the accompanying drawings.

[0148] like Figure 1 The diagram shows the control block diagram of the robot system tracking control based on bounded mutational output constraints according to the present invention. In practical engineering, the state of the robot system is measured by sensors. The difference between the measured joint angular displacement and the desired signal, along with the transformed constraint boundary function and transformation variables, are all transmitted to the adaptive controller. The controller processes the received signals according to the control law and then uses actuators to control the robot system.

[0149] The present invention provides a robot system tracking control method based on bounded mutation type output constraints, the specific steps of which are as follows:

[0150] Step 1: Establish the following dynamic model of an uncertain robot system with n joints:

[0151]

[0152] Where, q∈R n , R represents the angular displacement, angular velocity, and angular acceleration of the joint, respectively. n Represent an n-dimensional column vector; p∈R n M(q,p)∈R is an unknown constant vector. n×n It is a positive definite symmetric inertia matrix. It is a Coriolis matrix, G(q,p)∈R n×n Let D(p,t) ∈ R represent the gravity vector. n Represents unknown external disturbances, u∈R n It is the torque input vector; the vector dimension n is the same as the number of joints n.

[0153] Let x1 = [x 11 ,...,x 1n ] T =qsum Equation (1) is then transformed into the following system:

[0154]

[0155] Step 2: Based on the bounded mutation constraint boundary function, formulate a description of the corresponding mutation time range;

[0156] Known constraint boundary functions:

[0157] k dfi <x 1i <k ufi (3)

[0158]

[0159] Where i = 1,...,n, Indicates the number of joints. Let v represent a positive integer, v = j + 1, k ufi and k dfi Then, these represent the upper and lower constraint boundary functions of the system output at t∈[0,+∞); T0=0, T w These are known discontinuous moments, and adjacent constraint boundary functions are discontinuous, i.e., k uiw (T w )≠k uir (T w ), k diw (T w )≠k dir (T w ), w = 1, ..., j, r = w + 1;

[0160] Next, define the boundary function k. uiw k uir k diw and k dir Two fixed times T wL and T wR It meets the following conditions:

[0161] T w-1 <T wL <T w <T wR <T r (6)

[0162]

[0163] Where, when w = j, T j+1 =+∞; Therefore, the discontinuity problem of the constraint boundary function is described by equation (7).

[0164] Step 3: Construct a transfer function for smoothing the boundary of the mutation constraint, and build a transformation mechanism based on the constructed transfer function to convert any number of bounded mutation constraint boundary functions into globally continuous functions.

[0165] Based on the upper constraint boundary function of mutation, two types of transfer functions F are constructed. Rw (t) and F Lw (t);

[0166] Transfer function F Rw (t) is:

[0167]

[0168] Where e is the natural constant, w = 1,...,j and Δ w =(tT) w ) / (T wR -t);

[0169] Transfer function F Rw (t) has the following properties:

[0170] ①F Rw (t) in t∈[0,T) w The value of ] is always 1, in t∈[T wR The value of (+∞) is always 0;

[0171] ②F Rw (t) in t∈(T) w ,T wR Strictly monotonically decreasing;

[0172] ③F Rw (t), and The condition is continuous and bounded in t∈[0,+∞);

[0173] Transfer function F Lw (t) is:

[0174]

[0175] Where e is the natural constant, w = 1,...,j and Λ w =(T w -t) / (tT wL );

[0176] Transfer function F Lw (t) has the following properties:

[0177] ①F Lw (t) in t∈[0,T) wL The value of ] is always 0, in t∈[T w The value of (+∞) is always 1;

[0178] ②F Lw (t) in t∈(T) wL ,T w Strictly monotonically increasing;

[0179] ③F Lw (t), and The condition is continuous and bounded in t∈[0,+∞);

[0180] Based on the lower constraint boundary function of mutation, two types of transfer functions W are constructed. Rw (t) and W Lw (t);

[0181] Transfer function W Rw (t) is as follows:

[0182]

[0183] Where e is the natural constant, w = 1,...,j and β w =(T wR -t) / (tT w );

[0184] Transfer function W Rw (t) has the following properties:

[0185] ①W Rw (t) in t∈[0,T) w The value of ] is always 0, in t∈[T wR The value of (+∞) is always 1;

[0186] ②W Rw (t) in t∈(T) w ,T wR Strictly monotonically increasing;

[0187] ③W Rw (t), and The condition is continuous and bounded in t∈[0,+∞);

[0188] Transfer function W Lw (t) is as follows:

[0189]

[0190] Where e is the natural constant, w = 1,...,j and τ w =(tT) wL ) / (T w -t);

[0191] Transfer function W Lw (t) has the following properties:

[0192] ①W Lw (t) in t∈[0,T) wL The value of ] is always 1, in t∈[T w The value of (+∞) is always 0;

[0193] ②W Lw (t) in t∈(T) wL ,T w Strictly monotonically decreasing;

[0194] ③W Lw (t), and The condition is continuous and bounded in t∈[0,+∞);

[0195] Next, based on the transfer function, transformation mechanisms are constructed for the upper constraint boundary function and the lower constraint boundary function, as follows:

[0196] ① If in t∈(T) wL ,T w )inner k uiw >k uir ,but

[0197]

[0198] ②If in t∈(T) w ,T wR )inner k uiw <k uir ,but

[0199]

[0200] ③ If in t∈(T) wL ,T w )inner k diw <k dir ,but

[0201]

[0202] ④ If in t∈(T) w ,T wR )inner k diw >k dir ,but

[0203]

[0204] Where w = 1, ..., j, r = w + 1, T j+1 =+∞, k ui Let k represent the transformed upper constraint boundary function. di This represents the transformed lower constraint boundary function.

[0205] Wherein, the transformed constraint boundary function k ui and k di It has the following properties:

[0206] ①k ui >k di This holds true for all t∈[0,+∞).

[0207] ②k ui ,k di Their first derivatives and their second derivatives The condition is continuous and bounded in t∈[0,+∞).

[0208] Step 4: Based on the transformed constraint boundary function, design an adaptive control scheme, relax the restriction that the constraint boundary function is always a positive or negative function in the traditional control method, and ensure that the system output does not violate the bounded mutation constraint condition, and that the system output closely tracks the desired trajectory;

[0209] Step 4.1, consider the alternating positive and negative constraint boundary functions, and select a nonlinear function of the following form:

[0210]

[0211] Where i = 1,...,n, therefore, we obtain

[0212]

[0213] Among them, a 1i =(k ui -k di )l 1i , as well as Next, we get

[0214]

[0215] Where, A1 = diag(a 1i ) and B1 = [b 11 ,b 12 ,...,b 1n ] T ;

[0216] When k di (0) <x 1i (0) <k ui (0) if and only if x 1i →k di Or x 1i →k ui , ζ 1i →∞ holds true; therefore, if ζ is guaranteed 1i It is bounded, which ensures x 1i ∈(k di ,k ui );

[0217] Because of k ui >k di It is easy to obtain a 1i The value >0 always holds true, meaning the constraint boundary function alternates between positive and negative values;

[0218] Step 4.2, to facilitate controller design, define the generalized tracking error z1 and the virtual error z2, as follows:

[0219]

[0220] Where, ζ1=[ζ 11 ,....,ζ 1n ] T , z1 = [z 11 ,....,z 1n ] T z2 = [z 21 ,....,z 2n ] T χ is the output of the first-order filter, ζ d =[ζ d1 ,....,ζ dn ] T ,and

[0221]

[0222] Among them, y di Represents the known desired signal y d The i-th component; the first-order filter is represented as:

[0223]

[0224] in, These are design parameters. For virtual controllers;

[0225] Next, define the filtering error:

[0226]

[0227] From equations (19) and (22):

[0228]

[0229] From equations (21) and (22):

[0230]

[0231] Step 4.3: Design the controller using backstepping technology;

[0232] Step 4.3.1, select the Lyapunov function Then V was launched 10 The derivative with respect to time t:

[0233]

[0234] According to Young's inequality, it is easy to obtain:

[0235]

[0236] Next, the virtual controller Designed as follows:

[0237]

[0238] Where k1>0 represents the design parameters;

[0239] Choose the Lyapunov function V1 as:

[0240]

[0241] The derivative of V1 with respect to time t is:

[0242]

[0243] in, It is a constant.

[0244] Step 4.3.2, select the Lyapunov function as:

[0245]

[0246] V 20 The derivative with respect to time t is:

[0247]

[0248] From equation (2):

[0249]

[0250] make roll out:

[0251]

[0252] in, Indicates an unknown parameter. Let denote a computable function, c1>0, g1>0, d1>0, λ m >0 are all constants;

[0253] Next, using Young's inequality, we derive:

[0254]

[0255] The design of the real controller and adaptive law is as follows:

[0256]

[0257] Where k2>0, μ1>0, and δ>0 are all design parameters. This represents an estimate of the unknown parameter θ. Represents unknown parameters, and It is a computable function;

[0258] Next, we select the Lyapunov function V2 as:

[0259]

[0260] in, This represents the estimation error; the derivative of V2 with respect to time t is:

[0261]

[0262] in, It is an unknown constant term;

[0263] Consider an uncertain robotic system (1), if the initial conditions satisfy k di (0) <x 1i (0) <k ui (0), and there exists a constant E>0 such that V n (0)≤E, then under the action of the controller (37) and the parameter adaptive law (38), by selecting appropriate design parameters, it is possible to achieve that all signals of the closed-loop system are bounded, the output of the robot system closely tracks the desired trajectory, and the system output does not violate the constraint conditions of bounded mutation.

[0264] because

[0265]

[0266] Both are compact sets, therefore in There exist constants K>0 such that ||H||≤K holds, where E0>0, E1>0, E2>0, and E3>0 all represent constants;

[0267] Next, we can derive from equation (40):

[0268]

[0269] Where M2 = M1 + K 2 , Therefore, the launch Right now The initial value of the system satisfies k di (0) <x 1i (0) <k ui When (0), we get as well as Depend on roll out but Next, launch but Through similar analysis, it is proven that all signals within the closed-loop system are bounded;

[0270] Derived from equation (45) That is, by selecting appropriate design parameters, the generalized tracking error z1 can be made sufficiently small;

[0271] Next, from equation (19), the generalized tracking error z is derived. 1i Compared with the actual tracking error e 1i =x 1i -y di The relationship is:

[0272]

[0273] Where i = 1,...,n; since y di -k di >0 and k ui -y di >0, launch Where D>0 is a constant; by The conclusion is Therefore, by selecting the design parameters, the actual tracking error e can also be reduced. 1i Small enough;

[0274] Depend on To obtain k di <x 1i <k ui According to equations (4) and (12), when t∈[T] w-1 ,T wL When k ui =k ufi =k uiw When t∈[T] w ,T r When k ui =k ufi =k uir Therefore, x 1i <k ufi In t∈[T] w-1 ,T wL ] and t∈[T w ,T r It holds true for all t ∈ (T); wL ,T w When ), from equation (12), we can deduce:

[0275]

[0276] According to equations (4) and (47), we obtain that when t∈(T) wL ,T w When x 1i <k ui ≤k ufi This holds true for all; therefore, when k uiw >k uir At that time, x 1i <k ui ≤k ufi In any t∈[T] w-1 ,T r All of these hold true, i.e., x 1i <k ui ≤k ufi It holds true for all t ∈ [0, +∞); through similar analysis, it is proved that when k uiw <k uir At that time, x 1i <k ui ≤k ufi This still holds true for t∈[0,+∞), meaning the system output does not violate the bounded catastrophe constraint boundary function.

[0277] Next, from equations (5) and (14), we can deduce that when t∈[T] w-1 ,T wL ], k di =k dfi =k diw It always holds true when t∈[T] w ,T r When k di =k dfi =k dir It always holds true; when t∈(T) wL ,T w When ), it is easy to obtain (k) dir -k diw (1-W) Lw If )≥0, then (1-W) Lw )k dir +W Lw k diw ≥k diw That is, when t∈(T) wL ,T w When k di ≥k dfi It is true; therefore, x 1i >k di ≥k dfi In any t∈[T] w-1 ,T r All of these hold true, i.e., x 1i >k di≥k dfi It holds true on t∈[0,+∞); by similar analysis, it is proved that when k diw >k dir At that time, x 1i >k di ≥k dfi This still holds true for t∈[0,+∞), meaning the system output does not violate the bounded boundary function under bounded mutation.

[0278] In summary, the robot system output does not violate the bounded mutation constraint, that is, when t∈[0,+∞), k dfi ≤k di <x 1i <k ui ≤k ufi It always holds true.

[0279] Step 5: By changing the discontinuous time points, apply the designed control scheme to continuous constraint scenarios:

[0280] When T w When j = +∞, w = 1, k ufi =k ui1 k dfi =k di1 k ui =k ui1 , and k di =k di1 At this point, the system output constraint condition is k. di1 <x 1i <k ui1 .

[0281] Similar to the design and analysis process in step 4, the feasibility and effectiveness of the designed control strategy can be demonstrated.

[0282] The above description is only a preferred embodiment of the present invention. It should be noted that for those skilled in the art, several improvements and modifications can be made without departing from the principle of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention.

Claims

1. A tracking control method for a robot system based on bounded mutation-type output constraints, characterized in that: Includes the following steps: Step 1: Establish a dynamic model of an uncertain robot system with n joints; Step 2: Based on the bounded mutation constraint boundary function, formulate a description of the corresponding mutation time range; Step 3: Construct a transfer function for smoothing the boundary of the mutation constraint, and build a transformation mechanism based on the constructed transfer function to convert any number of bounded mutation constraint boundary functions into globally continuous functions. Step 4: Based on the transformed constraint boundary function, design an adaptive control scheme, relax the restriction that the constraint boundary function is always a positive or negative function in the traditional control method, and ensure that the system output does not violate the bounded mutation constraint condition, and that the system output closely tracks the desired trajectory; In step 4, considering the alternating positive and negative constraint boundary functions, a nonlinear function of the following form is selected: in, ; , , q Indicates the angular displacement of the joint. This represents the transformed upper constraint boundary function. This represents the transformed lower constraint boundary function; To facilitate controller design, a generalized tracking error is defined. and virtual error The details are as follows: in, , , , It is the output of a first-order filter. ,and in, Represents the known desired signal The One component; Finally, the controller was designed using backstepping technology. Step 5: By changing the discontinuous moments, the designed control scheme is applied to continuous constraint scenarios.

2. The robot system tracking control method based on bounded mutation type output constraints according to claim 1, characterized in that: In step 1, the dynamic model of the uncertain robot system with n joints is established as follows: in, These represent the angular displacement, angular velocity, and angular acceleration of the joint, respectively. Represent an n-dimensional column vector; It is an unknown constant vector. It is a positive definite symmetric inertia matrix. It is a Coriolis matrix. Represents the gravity vector. This indicates unknown external interference. It is the torque input vector; the vector dimension n is the same as the number of joints n. make and Then equation (1) is transformed into the following system: （2）。 3. The robot system tracking control method based on bounded mutation type output constraints according to claim 2, characterized in that: Step 2 is as follows: Known constraint boundary functions: in, , n represents the number of joints. Represents positive integers. , and These respectively represent the system output at The upper and lower constraint boundary functions; , These are known discontinuous moments, and adjacent constraint boundary functions are discontinuous, i.e. , , , ; Next, define the boundary function. , , and Two definite moments and It meets the following conditions: Among them, when hour, Therefore, the discontinuity problem of the constraint boundary function is described by equation (7).

4. The robot system tracking control method based on bounded mutation type output constraints according to claim 3, characterized in that: Step 3 is as follows: Based on the upper constraint boundary function of mutation, two types of transfer functions are constructed. and ; Transfer function for: Where e is the natural constant, as well as ; Transfer function It has the following properties: ① exist The value is always 1, in The value of is always 0; ② exist Strictly monotonically decreasing; ③ , and exist All are continuous and bounded; Transfer function for: Where e is the natural constant, as well as ; Transfer function It has the following properties: ① exist The value is always 0, in The value of is always 1; ② exist Strictly monotonically increasing; ③ , and exist All are continuous and bounded; Based on the lower constraint boundary function of mutation, two types of transfer functions are constructed. and ; Transfer function as follows: Where e is the natural constant, as well as ; Transfer function It has the following properties: ① exist The value is always 0, in The value of is always 1; ② exist Strictly monotonically increasing; ③ , and exist All are continuous and bounded; Transfer function as follows: Where e is the natural constant, as well as ; Transfer function It has the following properties: ① exist The value is always 1, in The value of is always 0; ② exist Strictly monotonically decreasing; ③ , and exist All are continuous and bounded; Next, based on the transfer function, transformation mechanisms are constructed for the upper constraint boundary function and the lower constraint boundary function, as follows: ①If in Inside ,but ②If in Inside ,but ③If in Inside ,but ④ If in Inside ,but in, , , .

5. The robot system tracking control method based on bounded mutation type output constraints according to claim 4, characterized in that: In step 3, the transformed constraint boundary function and It has the following properties: ① exist Hengcheng was established; ② , Their first derivatives , and their second derivatives , exist All are continuous and bounded.

6. The robot system tracking control method based on bounded mutation type output constraints according to claim 5, characterized in that: Step 4 specifically includes the following sub-steps: Step 4.1, obtained from equation (16) in, , ,as well as , Next, we get in, as well as ; when When, if and only if or , It is valid; therefore, if it is guaranteed It is bounded, which ensures ; because Easy to obtain This always holds true, meaning the constraint boundary functions alternate between positive and negative values; Step 4.2, the first-order filter is expressed as: in, These are design parameters. For virtual controllers; Next, define the filtering error: From equations (19) and (22): From equations (21) and (22): Step 4.3: Design the controller using backstepping technology.

7. The robot system tracking control method based on bounded mutation type output constraints according to claim 6, characterized in that: Step 4.3 specifically includes the following sub-steps: Step 4.3.1, select the Lyapunov function And then to introduce Regarding time The derivative: According to Young's inequality, it is easy to obtain: Next, the virtual controller Designed as follows: in, Indicate design parameters; Choosing Lyapunov functions for: roll out Regarding time The derivative is: in, , It is a constant. ; Step 4.3.2, select the Lyapunov function as: Regarding time The derivative is: From equation (2): make ,roll out: in, Indicates an unknown parameter. Represents a computable function. , , , All are constants; Next, using Young's inequality, we derive: The design of the real controller and adaptive law is as follows: in, , , These are all design parameters. Indicates the unknown parameter The estimate, Represents unknown parameters, and It is a computable function; Next, we select the Lyapunov function. for: in, Indicates the estimation error; Regarding time The derivative is: in, It is an unknown constant term; Consider an uncertain robotic system (1), if the initial conditions are satisfied And there exists a constant. , making Then, under the action of the controller (37) and the parameter adaptive law (38), by selecting appropriate design parameters, it is possible to achieve that all signals of the closed-loop system are bounded, the output of the robot system closely tracks the desired trajectory, and the system output does not violate the constraint conditions of bounded mutation. because Both are compact sets, therefore in There exists a constant in , making Established, among which, Both represent constants; Next, we can derive from equation (40): in, , ; Therefore, the launch ,Right now ; when the initial values ​​of the system satisfy At that time, , , , as well as ;Depend on , roll out ,but Next, we will launch , ,but Through similar analysis, it is proven that all signals within a closed-loop system are bounded. Derived from equation (45) That is, by selecting appropriate design parameters, the generalized tracking error can be reduced. Small enough; Next, from equation (19), the generalized tracking error is derived. Compared with the actual tracking error The relationship is: in, ;because as well as ,roll out ,in >0 is a constant; by , and thus Therefore, by selecting design parameters, the actual tracking error can also be reduced. Small enough; Depend on , and thus Based on equations (4) and (12), we can derive the result when... hour, ;when hour, ;therefore, exist and Internal constancy is established; when Then, from equation (12), we can deduce: According to equations (4) and (47), we obtain that when hour, It is always true; therefore, when hour, In any Both are true, that is exist The upper limit is established; through similar analysis, it is proven that when hour, exist This still holds true, meaning the system output does not violate the bounded boundary function of the bounded mutation type. Next, from equations (5) and (14), we can deduce when... , Heng is established; when hour, Heng is established; when Time, easy to obtain ,roll out That is, when hour, Established; therefore, In any Both are true, that is exist The above holds true; through similar analysis, it is proven that when hour, exist This still holds true, meaning the system output does not violate the bounded boundary function under bounded mutation. In summary, the robot system output does not violate the bounded mutation constraint, that is, when hour, It always holds true.

8. The robot system tracking control method based on bounded mutation type output constraints according to claim 7, characterized in that: Step 5 is as follows: when , hour, , , , ,as well as At this point, the system output constraints are: .

Images