CN114939869A - Mechanical arm trajectory tracking method based on nonsingular rapid terminal sliding mode - Google Patents
Mechanical arm trajectory tracking method based on nonsingular rapid terminal sliding mode Download PDFInfo
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- CN114939869A CN114939869A CN202210588338.8A CN202210588338A CN114939869A CN 114939869 A CN114939869 A CN 114939869A CN 202210588338 A CN202210588338 A CN 202210588338A CN 114939869 A CN114939869 A CN 114939869A
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- B—PERFORMING OPERATIONS; TRANSPORTING
- B25—HAND TOOLS; PORTABLE POWER-DRIVEN TOOLS; MANIPULATORS
- B25J—MANIPULATORS; CHAMBERS PROVIDED WITH MANIPULATION DEVICES
- B25J9/00—Programme-controlled manipulators
- B25J9/16—Programme controls
- B25J9/1656—Programme controls characterised by programming, planning systems for manipulators
- B25J9/1664—Programme controls characterised by programming, planning systems for manipulators characterised by motion, path, trajectory planning
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- Y02P90/00—Enabling technologies with a potential contribution to greenhouse gas [GHG] emissions mitigation
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Abstract
The invention relates to a nonsingular rapid terminal sliding mode control method, wherein a mechanical arm track tracking method comprises a three-connecting-rod mechanical arm dynamic model and a dynamic controller based on the three-connecting-rod mechanical arm dynamic model, and the three-connecting-rod mechanical arm dynamic model comprises a dynamic equation. The invention relates to measurement information based on joint angles and angular velocities, which realizes three-degree-of-freedom mechanical arm trajectory tracking control, enables a system to be converged in limited time, has higher response speed, and reduces the buffeting phenomenon of control output by designing a saturated double-power approximation law.
Description
Technical Field
The invention belongs to the field of industrial robot control, and particularly relates to a mechanical arm track tracking method based on a nonsingular rapid terminal sliding mode.
Background
The tandem type multi-degree-of-freedom mechanical arm has the advantages of high flexibility, environmental adaptability and the like, is widely applied to various fields such as manufacturing industry and the like, and plays an increasingly important role. With the expansion of the application range, the improvement of the complexity of the mechanical structure and the requirement of task performance, higher requirements are put forward on the motion control of the mechanical arm, and the traditional position servo control is no longer applicable. The motion control system combined with dynamics can not only compensate the dynamic characteristic of the simple position servo control, but also improve the control precision and the system stability.
Because the mechanical arm is interfered by various factors such as internal disturbance, external disturbance, model uncertainty and the like in the motion process, the strong coupling of the mechanical structure of the mechanical arm brings about the control coupling of joint torque, and the factors can increase the difficulty of the design of the controller. At present, many control methods have the problems of unstable control output, system buffeting, incapability of completely converging system errors in limited time and the like, so that the tracking precision and the stability of a control system still have a great space for improvement.
The existing track tracking control methods include PID control, fuzzy control, self-adaptive control, neural network control, sliding mode control and the like. The method has the following characteristics in application: the PID control is difficult to ensure the dynamic performance of the mechanical arm, and the output torque is large when the mechanical arm is started, so that the mechanical arm is easy to damage; the fuzzy control has more parameters needing to be adjusted and is not applied to actual engineering; adaptive control requires on-line parameter discrimination, and the real-time performance is strict. The control of the sliding mode variable structure is not easily affected by external interference and model uncertainty, so that the control problem of the nonlinear system is solved.
The sliding mode control has the following problems: on one hand, when the traditional sliding mode control approaches to the sliding mode surface, the control output generates a buffeting phenomenon, and meanwhile, the linear sliding mode surface has a phenomenon that the system error and the convergence time are balanced, so that the control system cannot converge in limited time, and the dynamic response and the tracking precision of the system are influenced by the factors. On the other hand, if the terminal sliding mode control is adopted, the control torque tends to be infinite in certain specific areas, a singular point phenomenon is generated, the system stability is influenced, and even a task cannot be completed.
Disclosure of Invention
Aiming at the defects of the prior art, the invention provides a nonsingular rapid terminal sliding mode control method, which is used for realizing three-degree-of-freedom mechanical arm track tracking control based on the measurement information of joint angles and angular velocities, so that a system can be converged in limited time and has higher response speed.
The technical scheme of the invention is as follows: a mechanical arm track tracking method based on a nonsingular rapid terminal sliding mode comprises a three-connecting-rod mechanical arm dynamic model and a dynamic controller based on the three-connecting-rod mechanical arm dynamic model, wherein the dynamic controller realizes control over a system through output torque, the three-connecting-rod mechanical arm mechanical model comprises a dynamic equation, and a state space expression of the three-connecting-rod mechanical arm mechanical model system model is as follows:
x represents the state function vector with 6 components, the left side of the equation represents the derivative of x, f (x) represents the vector steering function with 3 components, g (x) represents the system saturation bipolarity approach law with two components.
The kinetic equation is:
wherein q is [ q ] 1 ,q 2 ,q 3 ] T As a joint angle vector, τ ═ τ 1 ,τ 2 ,τ 3 ] T For moment vectors, M (q) s R 3×3 Is a matrix of the inertia, and,
is a matrix of coriolis forces and centrifugal forces, and R is a real space.
Controller output torque u i Involving a steady-state control law τ ist And robust control law g i (x) Two parts, the steady state control law keeps the system state to move on the sliding mode surface, and the controller output torque is:
u i ＝δ i τ i ＝τ ist +g i (x)
the nonsingular fast terminal sliding mode has nonsingularity, limited time convergence and fast convergence.
Controller output torque u i Including 13 constant coefficients or parameters:
constant coefficient of delta 1 ＝δ 2 ＝δ 3 ＝1.5μ 1 ＝μ 2 ＝μ 3 ＝1、λ 1 ＝λ 2 ＝λ 3 0.5, 5 for index parameter m, 3 for n, 1 for α, and 1.7 for β.
The parameters of the saturated double power approximation law are as follows: gamma ray 1 ＝γ 2 ＝γ 3 ＝1.2，
k 1 ＝k 2 ＝k 3 ＝50，ξ 1 ＝ξ 2 ＝ξ 3 ＝0.8，η 1 ＝η 2 ＝η 3 ＝1.1。
Description of the drawings:
FIG. 1 is a schematic view of a three link robot arm of the present invention;
FIG. 2 is a diagram of a simulation result of tracking error of a three-joint trajectory according to the present invention;
FIG. 3 is a diagram showing the simulation result of the three-joint control torque variation according to the present invention.
The invention has the beneficial effects
1. The invention describes the dynamic process of moving the system from the initial state to the sliding surface. The quality of the method reflects the time of the system moving from an initial state to a sliding mode surface and the flutter amplitude after the system reaches a switching surface, and determines the sliding mode control effect to a certain extent, so that the method is based on a linear sliding mode surface, and reduces the buffeting phenomenon of control output by designing a saturated double-power approximation law.
2. Aiming at the problems of singular point and low convergence speed in the terminal sliding mode, the invention combines the nonsingular terminal sliding mode control method and the rapid terminal sliding mode control method, and provides a hybrid terminal sliding mode control method, so that the control system has the advantages of limited time convergence and faster response characteristic.
3. The invention applies the saturated double power approximation law to the method, provides the mixed terminal sliding mode control method based on the saturated double power approximation law, reduces the buffeting phenomenon of control output, and improves the robustness of the buffeting phenomenon.
The specific implementation mode is as follows:
the invention comprises a three-link mechanical arm dynamic model and a dynamic controller based on the three-link mechanical arm dynamic model, wherein the dynamic controller realizes the control of a system through output torque, and the three-link mechanical arm model is shown in figure 1, wherein L i Is the ith rod link length, K i The length from the centroid of the ith rod to the previous joint.
Because the three-link mechanical arm lacks the degree of freedom in the Z-axis direction, the kinetic equation of the mechanical arm can be simplified as follows:
wherein the joint angle vector is q ═ q 1 ,q 2 ,q 3 ] T The moment vector is τ ═ τ 1 ,τ 2 ,τ 3 ] T ，M(q)∈R 3×3 Is a matrix of the inertia, and,
is a matrix of coriolis forces and centrifugal forces. The matrix composition form of M (q) is:
each element in the inertial matrix m (q) is calculated as follows:
in the formula of gamma i For system configuration parameters, see table 1, the calculation is as follows:
TABLE 1 three-link mechanical arm structural parameters
Centrifugal force and Coriolis torque matrix
And angular velocity vector
The composition form of the composition is as follows:
the calculation of the elements in the matrix is as follows:
the kinetic model of the three-link arm system can be expressed in a matrix form as shown in equation (1):
(2) controller design based on dynamic model
Order to
The state space expression of the system model is formula (8):
the matrix expression form is shown as formula (9):
for convenience of description, the state space expression is simplified as formula (10):
wherein each function can be described as:
in the formula, g 1 (x) The method is a zero matrix of 3 multiplied by 3, and the angular acceleration in the formula (1) is obtained according to a mechanical arm dynamic model and is a formula (12):
accordingly, f (x) and g 2 (x) Respectively, formula (13):
let the expected joint angle trajectory of the robot arm be q d The expected angular velocity trajectory of the joint is
The actual joint angular velocity trajectory is q, and the actual joint angular velocity trajectory is q
The tracking error defining the ith joint angle and joint angular velocity is formula (14):
designing a nonsingular fast terminal sliding mode function of the ith joint as a formula (15):
in the formula, delta i ＞0、μ i > 0 and lambda i A constant coefficient > 0, an exponential parameter m > n and
beta > alpha and
and are all positive odd numbers. Derivation of the above formula yields formula (16):
according to equation (15), when the system state is in the slip form plane (i.e., S) i 0), let equation (15) be 0, we get:
order to
Then there is equation (18):
finishing to obtain formula (19):
substituting equation (19) into equation (17) yields:
the formula (20) is arranged to obtain:
when t is 0, y i (0) Integrated over both sides of the above formula:
when tracking error e i When equal to 0, y i 0. Assume final convergence time t s Then, there are:
finish the above formula to get t s Comprises the following steps:
because the index parameters m > n, beta > alpha are positive odd numbers, delta i > 0 and mu i A constant coefficient of > 0, knowing t s Is constant, i.e.: the convergence time exists and is constant.
According to formula (15), when S i When the value is equal to 0, the following results are obtained:
from equation (25), the following can be concluded:
when the system state is far away from the sliding mode (i.e. tracking error)When the difference is large), the convergence rate is controlled by
The items play a dominant role in the presentation,
the term plays an auxiliary role and rapidly converges in an exponential form;
when the system state approaches the sliding mode (i.e. the tracking error gradually converges), the convergence rate is determined by
The item plays a dominant role.
In conclusion, the designed nonsingular fast terminal sliding mode has the characteristics of nonsingularity, limited time convergence and fast convergence.
The inertia matrix M (q) is multiplied on both sides of the formula (16), and then the mechanical arm kinetic equation is combined, so that a formula (26) can be obtained:
according to the formulae (14) and (15) there are:
substituting the above formula into (16) yields:
for convenience in describing the above formula, let:
equation (28) can be further simplified as:
according to the designed nonsingular fast terminal sliding mode function, the available moment control law is shown as the formula (31):
u i ＝δ i τ i ＝τ ist +g i (x) (31)
master control law u i The method comprises two parts of a steady-state control law and a robust control law:
control law τ ist For a steady-state control law, the aim is to keep the system state moving on the sliding mode surface, and at the moment:
the steady state control law is obtained from equation (26), as shown in equation (32):
the term g (x) in equation (31) is a robust control law for attracting the system state from outside to top of the sliding mode surface, namely: reducing the system error and reaching the set system expected state. The robust control law can be defined as formula (33) based on the saturated double power approach law:
in the formula (I), the compound is shown in the specification,
k is a constant coefficient and controls gain for saturated double power approximation law, xi is more than 0 and less than 1, eta is more than 1 and is a constant, sat (S) i ) As a saturation function, expressed as:
wherein gamma is a constant and 1 < gamma < 2.
The overall control law (i.e., the controller output torque) can be obtained from equations (32) and (33), as shown in equation (35):
technical effects of the invention
In the method, MATLAB 2020a is used as a simulation environment, an expected track of a joint angle and an angular speed of a mechanical arm is given, the actual track and the deviation generated by the actual track are controlled, and the performance of the designed nonsingular fast terminal sliding mode control method is verified. The relevant parameters are set as follows:
the desired joint angle and angular velocity trajectory are: q. q.s d ＝[q 1d ,q 2d ,q 3d ]And
the actual joint angle and angular velocity trajectory sampled are: q ═ q 1 ,q 2 ,q 3 ]And
relevant parameters of the nonsingular fast terminal sliding mode function: constant coefficient delta 1 ＝δ 2 ＝δ 3 ＝1.5μ 1 ＝μ 2 ＝μ 3 ＝1、λ 1 ＝λ 2 ＝λ 3 0.5, 5 for index parameter m, 3 for n, 1 for α, and 1.7 for β.
Setting relevant parameters of a saturated double-power approximation law: gamma ray 1 ＝γ 2 ＝γ 3 ＝1.2，
The simulation results obtained from the above settings are shown in fig. 2 and 3. As can be seen from fig. 2 and 3: angular tracking error is [ -7,5 ]]*10 -3 The rad interval has smaller error; the maximum error convergence time is 10s, and the convergence speed is higher; the control torque shown in fig. 3 is output in a relatively stable form within 2n.m, so that the buffeting of the system is effectively reduced, and the robustness is relatively high. The conclusion shows that the nonsingular fast terminal sliding mode control method designed by the patent can improve the track control precision and the system stability and realize the limited time convergence.
Claims (6)
1. A mechanical arm track tracking method based on a nonsingular rapid terminal sliding mode is characterized by comprising the following steps: the mechanical arm track tracking method comprises a three-connecting-rod mechanical arm dynamic model and a dynamic controller based on the three-connecting-rod mechanical arm dynamic model, wherein the three-connecting-rod mechanical arm dynamic model comprises a dynamic equation, the dynamic controller realizes control over a system through output torque, and a state space expression of the three-connecting-rod mechanical arm dynamic model system model is as follows:
x represents the state function vector with 6 components, the left side of the equation represents the derivative of x, f (x) represents the vector steering function with 3 components, g (x) represents the system saturation bipolarity approach law with two components.
2. The mechanical arm trajectory tracking method based on the nonsingular fast terminal sliding mode according to claim 1, which is characterized in that: the kinetic equation is:
wherein q is [ q ] 1 ,q 2 ,q 3 ] T Is the angle direction of the jointQuantity, τ ═ τ [ τ ] 1 ,τ 2 ,τ 3 ] T For moment vectors, M (q) e R 3×3 Is a matrix of the inertia, and,
is a matrix of coriolis forces and centrifugal forces, and R is a real space.
3. The mechanical arm trajectory tracking method based on the nonsingular fast terminal sliding mode according to claim 1 or 2, which is characterized in that: output torque u of the dynamic controller i Involving a steady-state control law τ ist And robust control law g i (x) Two parts, a steady-state control law keeps the system state to move on a sliding mode surface, and the output torque of the controller is as follows:
u i ＝δ i τ i ＝τ ist +g i (x)
4. the mechanical arm trajectory tracking method based on the nonsingular fast terminal sliding mode according to claim 1, which is characterized in that: the nonsingular fast terminal sliding mode has nonsingularity, limited time convergence and fast convergence.
5. The mechanical arm trajectory tracking method based on the nonsingular fast terminal sliding mode as claimed in claim 3, wherein the method comprises the following steps: controller output torque u i Including 13 constant coefficients or parameters:
constant coefficient of delta 1 ＝δ 2 ＝δ 3 ＝1.5μ 1 ＝μ 2 ＝μ 3 ＝1、λ 1 ＝λ 2 ＝λ 3 0.5, 5 for index parameter m, 3 for n, 1 for α, 1.7 for β.
6. The mechanical arm trajectory tracking method based on the nonsingular fast terminal sliding mode according to claim 1, which is characterized in that: the parameters of the saturated double power approximation law are as follows: gamma ray 1 ＝γ 2 ＝γ 3 ＝1.2，
k 1 ＝k 2 ＝k 3 ＝50，ξ 1 ＝ξ 2 ＝ξ 3 ＝0.8，η 1 ＝η 2 ＝η 3 ＝1.1。
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Cited By (2)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN116277036A (en) * | 2023-05-16 | 2023-06-23 | 湖南工商大学 | Rapid fault-tolerant vibration suppression control method for flexible-base and flexible-arm space robot |
| CN117047782A (en) * | 2023-10-11 | 2023-11-14 | 中建四局安装工程有限公司 | Control method and device suitable for three-joint manipulator, terminal and medium |
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Cited By (4)
| Publication number | Priority date | Publication date | Assignee | Title |
|---|---|---|---|---|
| CN116277036A (en) * | 2023-05-16 | 2023-06-23 | 湖南工商大学 | Rapid fault-tolerant vibration suppression control method for flexible-base and flexible-arm space robot |
| CN116277036B (en) * | 2023-05-16 | 2023-08-22 | 湖南工商大学 | Rapid fault-tolerant vibration suppression control method for flexible-base and flexible-arm space robot |
| CN117047782A (en) * | 2023-10-11 | 2023-11-14 | 中建四局安装工程有限公司 | Control method and device suitable for three-joint manipulator, terminal and medium |
| CN117047782B (en) * | 2023-10-11 | 2023-12-08 | 中建四局安装工程有限公司 | Control method and device suitable for three-joint manipulator, terminal and medium |
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