CN112549010B - Design method of multi-joint snake-shaped robot self-adaptive trajectory tracking controller based on improved Serpenoid curve - Google Patents
Design method of multi-joint snake-shaped robot self-adaptive trajectory tracking controller based on improved Serpenoid curve Download PDFInfo
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- B—PERFORMING OPERATIONS; TRANSPORTING
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- B25J9/00—Programme-controlled manipulators
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Abstract
The invention discloses a design method of a self-adaptive trajectory tracking controller of a multi-joint snake-shaped robot based on an improved Serpenoid curve, and belongs to the field of motion control of bionic robots. The invention comprises the following steps: establishing a kinematic model of the robot through coordinate transformation according to the self structure and joint characteristics of the multi-joint snake-shaped robot; formulating a control target of the trajectory tracking controller; providing a control equation of the improved Serpenoid curve of the multi-joint snake-shaped robot; designing a track tracking controller to realize the control function of the robot on the angle of the connecting rod, the angle of the joint and the movement position; searching a Lyapunov function to verify the gradual stability of the controller; through experiments, the motion mode of the multi-joint snake-shaped robot is observed, the motion track of the robot is analyzed, and the effectiveness of the proposed controller is verified. The invention aims to solve the technical problem of designing an adaptive trajectory tracking controller of the multi-joint snake-shaped robot based on an improved Serpenoid curve, and laying a theoretical foundation for path tracking control of the snake-shaped robot.
Description
Technical Field
The invention belongs to the field of motion control of bionic robots, and particularly relates to a design method of an adaptive trajectory tracking controller of a multi-joint snake-shaped robot based on an improved Serpenoid curve.
Background
As organisms continue to evolve and evolve, many organisms exhibit unique advantages in locomotor patterns and environmental adaptation. The special movement mode enables the biological snakes to have extremely strong environment adaptability and movement stability. This has attracted the attention of scientists, and a multi-joint bionic snake robot combining bionics and robots has come into force. The motion characteristic of the snake-shaped robot is similar to that of a snake, the motion gait of the snake-shaped robot in snaking and crawling enables the snake-shaped robot to work on rugged and complicated terrain or areas where human beings cannot enter, and a plurality of dangerous operations can be completed by replacing human beings, so that the articulated snake-shaped robot has a very wide application prospect.
In the construction of an articulated snake robot, the controller is an important component of whether the robot can achieve the desired effect. The controller can realize the control of the motion gait and the motion direction of the snake-shaped robot, so that the snake-shaped robot moves in a gait of winding crawling and tracks the expected path to move ahead. On the basis, the snake-shaped robot can be optimized or expanded to have more functions, so that the design of the multi-joint snake-shaped robot trajectory tracking controller can establish a good foundation for researching the two-dimensional motion of the snake-shaped robot and the expansion of subsequent functions.
Disclosure of Invention
Aiming at the problem that the existing articulated snake-shaped robot tracks the expected path in the motion process, the invention aims to solve the technical problems that: the design method of the multi-joint snake-shaped robot self-adaptive trajectory tracking controller based on the improved Serpenoid curve is provided, and the robot can track an expected path in the motion process.
The purpose of the invention is realized by the following technical scheme:
the invention discloses a design method of a self-adaptive trajectory tracking controller of a multi-joint snake-shaped robot based on an improved Serpenoid curve, which comprises the following steps:
the method comprises the following steps: and obtaining a robot kinematic model based on the connecting rod structure through coordinate transformation according to the body structure and joint characteristics of the multi-joint snake-shaped robot.
Step two: four control targets of the self-adaptive trajectory tracking controller of the multi-joint snake-shaped robot are formulated, and the four control targets are respectively: the aim is to realize the angle error of the connecting rod of the robot
Converge to 0 and asymptotically stabilize; second, the aim is to realize the joint angle error of the robot
Converge to 0 and asymptotically stabilize; the third aim is to realize the normal motion position error e of the robot y Convergence and normal movement velocity
Consistently bounded; the aim is to realize the tangential movement of the robotError of dynamic position
And tangential motion velocity error
Converge to 0 and asymptotically stabilize.
Step three: the Serpenoid gait curve control equation of the articulated snake-shaped robot is improved, the fixed swing amplitude in the traditional Serpenoid curve is replaced by the time-varying swing amplitude related to the state, and the axial bending moment constraint of the robot with the self-adaptive time-varying swing amplitude is obtained.
Step four: and the LOS guidance rate is utilized to realize that the motion direction of the articulated snake-shaped robot points to a desired path.
Step five: setting the expected link angle of the articulated snake-shaped robot to be
The actual link angle is θ. The link angle error target is to control the actual link angle theta of the robot to track the desired link angle theta
Finally realizing the angle error of the connecting rod
Converge to 0 and asymptotically stabilize.
Step six: setting the expected joint angle of the multi-joint snake-shaped robot into
The actual joint angle is phi. The joint angle error target is to control the actual joint angle phi to track the expected joint angle
Finally reach the joint angle error
Converge to 0 and asymptotically stabilize.
Step seven: setting the actual motion track of the robot in the y-axis direction according to the third target formulated in the second step
And the ideal path
Has a normal motion position error of
The normal movement speed of the robot is
The aim is to realize the normal motion position error e of the robot y Convergence and normal movement velocity
Is consistently bounded.
Step eight: setting the actual motion track of the robot in the x-axis direction according to the target IV formulated in the step two
And the ideal path
Has a tangential motion position error of
The tangential motion speed of the robot along the ideal path is
At the same time satisfy
The aim is to achieve a tangential movement position error of the robot
And tangential motion velocity error
Converge to 0 and asymptotically stabilize.
Step nine: and constructing a Lyapunov function L, and verifying that the link angle error in the step five, the joint angle error in the step six, the normal motion position error and the normal motion speed in the step seven are consistent and bounded, and the tangential motion position error and the tangential motion speed error in the step eight are stable.
Step ten: and (3) verifying the effectiveness of the multi-joint snake-shaped robot self-adaptive track tracking controller based on the improved Serpenoid curve through an MATLAB simulation experiment.
Has the advantages that:
1. the invention discloses a design method of an adaptive trajectory tracking controller of a multi-joint snake-shaped robot based on an improved Serpenoid curve, which formulates a control target of the adaptive trajectory tracking controller of the multi-joint snake-shaped robot, and the robot can track not only a set linear path but also a set curve path.
2. The invention discloses a design method of a self-adaptive trajectory tracking controller of a multi-joint snake-shaped robot based on an improved Serpenoid curve, which improves a Serpenoid gait curve control equation of the multi-joint snake-shaped robot, considers the self-adaptability of the moving gait of the robot, obtains the axial bending moment constraint of the robot with a self-adaptive time-varying swing amplitude value, and improves the moving efficiency of the robot.
3. The design method of the self-adaptive trajectory tracking controller of the articulated snake-shaped robot based on the improved Serpenoid curve eliminates the negative influence caused by uncertain environmental friction coefficient and time variation, realizes the estimation of the robot on time variation errors, improves the convergence speed of the trajectory tracking errors of the robot, and increases the stability of the controller.
Drawings
FIG. 1 is a coordinate transformation of a multi-joint serpentine robot;
FIG. 2 is a kinematic model of a multi-joint serpentine robot;
FIG. 3 is an adaptive trajectory tracking controller architecture;
FIG. 4 is a Line-of-Sight boot law;
FIG. 5 is a normal motion trajectory error contrast curve of the articulated snake robot;
FIG. 6 is a tangential motion trajectory error contrast curve of the articulated snake robot;
FIG. 7 is a plot of link angle versus ideal link angle for an articulated serpentine robot;
FIG. 8 is a link angle error contrast curve for a multi-joint serpentine robot;
FIG. 9 is a curve of joint angles of an articulated serpentine robot under the control of a modified Serpenoid method;
fig. 10 is a curve of joint angles of the articulated serpentine robot under the control of the serpoioid method.
Detailed Description
The invention will be further explained with reference to the drawings.
The embodiment starts from the requirements of the articulated snake-shaped robot, combines the motion characteristics of the articulated snake-shaped robot, and discloses a design method of an adaptive trajectory tracking controller of the articulated snake-shaped robot based on an improved Serpenoid curve, which comprises the following steps:
the method comprises the following steps: the articulated snake-shaped robot consists of N rigid connecting rods with the length of 2h, N-1 rotating joints are arranged between the connecting rods, and a driven wheel is arranged in the middle of each connecting rod. Through coordinate transformation (1), as shown in fig. 1. Get the new centroid position of the robot as
A kinematic model (2) of the robot is established, as shown in fig. 2.
Wherein the articulated snake robotThe joint is composed of N rigid connecting rods with the length of 2h, and the position coordinate of the robot moves along the tangential direction by the distance
The link angle of the robot is set as
The center of mass of the joint of the robot is p = [ p ] x ，p y ] T The angular velocity of the connecting rod of the robot is
The normal vector velocity of the robot is v n ∈R N The normal vector velocity after coordinate transformation is
Wherein the content of the first and second substances,
and
the set of joint angles of the multi-joint snake-shaped robot is
The joint angular velocity of the robot is set as
Tangential velocity v of the robot t ∈R N . The tangential and normal friction coefficients in the multi-joint snake-shaped robot connecting rod model are respectively lambda 1 > 0 and lambda 2 > 0, the magnitude of the friction coefficient is influenced by the geographical environment. Setting the mapping proportion of the rotation speed of the robot joint to the rotation acceleration to be mu 1 ＞0. Setting the mapping proportion of the mean value of the angle of the joint of the robot and the tangential velocity to the rotation acceleration as mu 2 Is greater than 0. The joint mass of the robot is m, and the auxiliary matrix is
The control input of the system is
Step two: four control targets of the multi-joint snake-shaped robot trajectory tracking controller are formulated, the self-adaptive trajectory tracking controller of the multi-joint snake-shaped robot is shown in figure 3, and the formulated four targets are respectively: the goal is to set the desired link angle of the articulated serpentine robot to
The actual link angle is θ. The link angle error target is to control the actual link angle theta of the robot to track the desired link angle theta
Finally realize the angle error of the connecting rod
Converge to 0 and asymptotically stabilize; the second objective is to set the expected joint angle of the multi-joint snake-shaped robot as
The actual joint angle is phi. The joint angle error target is to control the actual joint angle phi to track the expected joint angle
Finally reach the joint angle error
Converge to 0 and asymptotically stabilize; the third goal is the actual motion track of the robot in the y-axis direction
And the ideal path
Has a normal motion position error of
The normal movement speed of the robot is
The aim is to realize the normal motion position error e of the robot y Convergence and normal movement velocity
Consistently bounded; the fourth target is the actual motion track of the robot in the x-axis direction
And the ideal path
Has a tangential motion position error of
The tangential motion speed of the robot along the ideal path is
At the same time satisfy
Achieving tangential motion position error of a robot
And tangential motion velocity error
Converge to 0 and asymptotically stabilize. When the target is achieved, the link angle of the articulated serpentine robot reaches the desired target. When the second objective is realizedWhen the joint angle of the articulated snake robot reaches the desired target. At this time, the robot is directed in a prescribed direction, and the links of the robot make periodic snaking oscillations, which urge the forward motion of the robot. When the third target is realized, the convergence of the normal motion position error of the articulated snake-shaped robot is realized, and the actual normal motion position of the robot fluctuates up and down near the ideal path, and the fluctuation is slightly bounded. When the target four is finished, the actual tangential motion position and the actual tangential motion speed of the multi-joint snake-shaped robot realize convergence, and the robot realizes track tracking motion in the complete sense. The control target of the multi-joint snake-shaped robot self-adaptive trajectory tracking controller based on the improved Serpenoid curve is realized, and all state errors of the robot can be converged to 0.
Step three: and (3) providing a control equation of the improved Serpenoid curve of the articulated snake-shaped robot. The improved Serpenoid curve control equation establishes the relation between the swing amplitude of the robot and the motion position and the motion speed of the robot, and is defined as formula (3) as the ith joint angle of the robot.
Wherein, the amplitude of the time-varying gain of the vibration of the joints of the multi-joint snake-shaped robot is
The robot has a joint offset of
The phase shift of the robot joint is delta, the motion frequency of the robot joint is omega, and the solution of two compensators for controlling the advancing speed and the offset direction of the robot is delta
And phi 0 。
In particular, desired joint angle of articulated snake robot
Equation (4) is obtained in association with the governing equation for improving the Serpenoid curve.
Wherein the content of the first and second substances,
step four: the Line-of-Sight (LOS) guidance law is adopted to enable the motion direction of the articulated snake-shaped robot to point to a desired path. The LOS method is a common method of guiding a robot to track a desired path. In the LOS method, the ideal link angle of the robot is (5), as shown in fig. 4.
Wherein the overall desired link angle of the articulated snake robot is
The angle is the desired value for the actual direction of motion of the robot. Tracking distance error of the robot is e y The forward distance is Δ. The articulated snake-shaped robot is at t 1 The closest point of the time and the expected path is F, and the motion path guide point (x) of the robot is obtained by taking the point F as a tangent with the length delta LOS ,y LOS ). The robot generates a link angle theta to track the guidance point. In this process, it is desirable for the actual link angle θ of the robot to track the desired link angle
When the multi-joint snake-shaped robot moves to t 2 At the moment, the guiding point of the robot changes, which results in the desired link angle
Change occursAnd (4) transforming. Therefore, the actual link angle θ of the robot also needs to be adjusted. The above steps are repeated in a circulating way to push the motion track of the multi-joint snake-shaped robot to continuously approach to the expected path, so that the track tracking purpose is achieved. Therefore, although the desired path is set manually, the set desired path can be tracked regardless of the position of the robot. Except for the desired link angle that the robot tracks at different positions and at different times
The time required for the robot to track the path varies. Influencing the tracking of an articulated serpentine robot by an expected link angle
Includes a normal position error e y And a forward distance delta. Wherein the tracking distance error e y Related to the motion state of the robot, the forward distance Δ affects the convergence speed and the convergence accuracy of the error, and is a parameter designed manually. Δ > 0 indicates that the moving direction of the robot is forward, and Δ < 0 indicates that the moving direction of the robot is backward. Also, the forward distance Δ also affects the link angular velocity of the robot. The larger the Δ, the larger the
The smaller, the link angular velocity v θ The slower. Since the link angular velocity is linked to the joint rotation, the joint angular rotation velocity v of the robot is higher when Δ is larger φ It will also slow down. Similarly, when Δ is smaller, then
The greater the link angular velocity v θ And joint angular rotational velocity v φ It will be accelerated. Since the rate of convergence of the motion trail of the articulated snake robot to the desired path is greatly influenced by delta. Therefore, the robot can obtain an efficient and smooth track tracking process by selecting a proper delta value.
Step five: setting the link angle error variable of the multi-joint snake-shaped robot according to the first target established in the step twoIs e θ . Constructing a link angle feedback auxiliary function of the robot, and designing a joint angle compensator phi by using a Backstepping method 0 And the error convergence of the connecting rod angle of the robot to 0 is realized.
The link angle error and the link angular velocity error of the articulated snake-shaped robot are respectively (6) and (7).
And differentiating the connecting rod angle error and the connecting rod angular speed error to obtain (8) and (9).
The secondary function of the link angle error is set to (10). Another form from which the differential of the link angle error can be derived is (11).
Designing a connecting rod direction angle compensator of the robot to be phi by using Backstepping control method 0 。
Step six: setting the joint angle error variable of the multi-joint snake-shaped robot as e according to the second target established in the step two φ . And designing a friction coefficient estimation value function and a joint angle input control function of the robot by using a self-adaptive control method, and realizing that the joint angle error is converged to 0.
The joint angle error of the robot is e φ 。
The differential of the joint angle error of the articulated serpentine robot is (14).
In order to achieve convergence of the joint angle error, an assist function with an estimated value of the joint angle error coefficient is set to (15).
Wherein
Is a parameter k φ Time variable estimate > 0.
Has the effects of inhibiting
Unknown bounded function of (1)
Another form of deriving the derivative of the joint angle error is (16).
The joint angular velocity error and the differential thereof of the articulated snake-like robot are (17) and (18), respectively.
The input to the system is set to u using an adaptive control method. At the same time, a feedback input-output controller is designed
Wherein the content of the first and second substances,
and
are each lambda 1 And λ 2 The estimated value of the time-variable of (c),
k v,i and > 0 is a diagonal array of normal numbers.
Step seven: and setting the normal movement speed and the normal movement position error of the multi-joint snake-shaped robot according to the third target established in the second step. And the controller is designed through robust control, so that the controlled condition of the error is found, and the normal motion position error of the robot is quickly converged.
The expected normal motion path tracked by the multi-joint snake-shaped robot is
The actual normal motion track of the robot is
Wherein the content of the first and second substances,
as a function of time. The normal movement speed of the robot is
From the mean inequality (21) can be obtained.
The error between the actual normal motion position and the expected normal motion path of the articulated snake robot is (22).
Since the link angle error e has already been realized in step five θ Convergence is 0. Moreover, according to the LOS method, the trigonometric function relationship can be obtained as
And
therefore (23) can be simplified to (24). At the same time, have v t ∈[V min ,V max ]＞0。
Step eight: and C, according to the object IV formulated in the step II, using the improved Serpenoid gait curve control equation as an additional control item to control the tangential motion position and the tangential motion speed of the robot. An estimated value of the multi-joint snake-shaped robot swing amplitude compensation is designed by using a self-adaptive control method, and the estimated value is used for replacing an actual swing amplitude to adjust the controller in real time, so that the tangential motion position error and the tangential motion speed error of the robot are converged to 0.
Defining the tangential velocity of the robot along the ideal path as
The tangential position of the ideal path of the robot is
At the same time satisfy
The actual tangential movement position of the robot is
Actual speed of movement v t . The tangential motion position error of the robot is e x Error of tangential motion velocity of e s 。
The differential of (25) is substituted into (2) to obtain (26).
Setting the auxiliary function σ 1 。
And designing a tracking function of the robot swing amplitude compensation to be beta by using an adaptive control method.
To make it possible to
Setting the auxiliary function to σ 2 。
The time-varying swing amplitude of the articulated snake robot is defined as (32).
Step nine: and verifying the stability of the link angle error in the fifth step, the joint angle error in the sixth step, the normal motion position error and the normal speed in the seventh step and the tangential motion position error and the tangential speed error in the eighth step by using a Lyapunov method.
Setting Lyapunov candidate equation L 1 Is (33).
To obtain
Is (35).
Inequality relations (36) and (37) exist.
To obtain
Has the scaled form of (39).
Setting Lyapunov candidate equation V 21 Is (40).
Setting Lyapunov candidate equation V 22 Is (42).
Wherein eta is 1 > 0 is the normal gain.
Setting Lyapunov candidate equation L 2 Is (44).
Wherein eta is 2 > 0 and η 3 And the gains are normal gains when the gain is more than 0.
Setting time variable estimation values
And
are respectively differentiated by
And
at the same time, will
(20) And (19) into (45) to give (46).
We always keep
Wherein, χ φ And > 0 is a normal number gain. To obtain
Is (47).
And the consistency and the boundedness of the normal movement speed are proved. The Lyapunov candidate equation for the normal motion velocity is set to (48).
Can obtain
Another form of (50).
Setting boundary conditions as
And
since Y > 0, provided that
Small enough to make-p 1 Is negative. For is to
Scaling was performed to obtain (51).
Solving for (51) yields (52).
Therefore, the temperature of the molten metal is controlled,
is bounded and converges to
Due to the fact that
V in (1) θ Is bounded, so v n Is also bounded and does not set v n Is bound by n I.e. | | v n ||≤ε n 。
The Lyapunov equation for the normal motion position error is designed to be (53).
According toThe mean inequality can be obtained
And
at the same time, the user can select the desired position,
is bounded, i.e.
Can obtain
Is (55).
Without being provided with
And
at the same time should satisfy
To obtain
Is (56).
Solving for (56) yields (57).
Thus, the normal motion position error of the robot is bounded and converges to
To increase the convergence rate of the error, there may be an increase in V min Decreasing the forward distance delta, increasing the friction of the robot in the normal direction, decreasing the normal velocity v n Boundary value of n And decreasing the value of | X |.
Setting Lyapunov candidate equation V 41 Is (58).
Setting Lyapunov candidate equation V 42 Is (61).
To obtain
Is (63).
Setting Lyapunov function L 4 Is (64).
To obtain
Is (66).
The Lyapunov function of the system is L = L 1 +L 2 +L 3 +L 4 。
Therefore, the adaptive trajectory tracking controller for articulated serpentine robots based on modified Serpenoid curves is consistently bounded and stable. The error can be controlled within a small acceptable range by adjusting the value of the parameter.
Step ten: a simulation experiment is carried out on the articulated snake-shaped robot based on the improved Serpenoid curve by MATLAB, the normal motion track error contrast curve of the articulated snake-shaped robot is shown in figure 5, the tangential motion track error contrast curve of the articulated snake-shaped robot is shown in figure 6, the link angle and ideal link angle curve of the articulated snake-shaped robot is shown in figure 7, the link angle error contrast curve of the articulated snake-shaped robot is shown in figure 8, the joint angle curve of the articulated snake-shaped robot under the control of the improved Serpenoid method is shown in figure 9, and the joint angle curve of the articulated snake-shaped robot under the control of the Serpenoid method is shown in figure 10. According to simulation results, the self-adaptive trajectory tracking controller of the multi-joint snake-shaped robot based on the improved Serpenoid curve can enable the robot to have higher trajectory tracking convergence speed, higher connecting rod angle error convergence speed, more stable joint angle error curve, higher connecting rod angle speed error stability and better joint angle speed error stability compared with the original Serpenoid control method. This fully embodies the superiority of the proposed adaptive trajectory tracking controller that improves serpoioid curves.
The above is only a preferred embodiment of the present invention, and it should be noted that: it will be apparent to those skilled in the art that various modifications and adaptations can be made without departing from the principles of the invention and these are intended to be within the scope of the invention.
Claims (3)
1. A design method of a self-adaptive trajectory tracking controller of a multi-joint snake-shaped robot based on an improved Serpenoid curve is characterized by comprising the following steps: the method comprises the following steps:
the method comprises the following steps: analyzing the motion mechanism of the articulated snake-shaped robot, and establishing a kinematic model of the articulated snake-shaped robot through coordinate transformation;
step two: formulating a control target of the multi-joint snake-shaped robot trajectory tracking controller, which comprises the following steps: the objective is to control the actual link angle theta of the robot to track the desired link angle theta
Finally realizing the angle error of the connecting rod
Converge to 0 and asymptotically stabilize; second, the target is to control the actual joint angle phi to track the expected joint angle
Finally reach the joint angle error
Converge to 0 and asymptotically stabilize; the third goal is to set the reality of the robot in the y-axis directionMotion trail
And the ideal path
A normal movement position error of
Realizing the normal motion position error e of the robot y Convergence and normal movement velocity
Consistently bounded; setting the actual motion track of the articulated snake-shaped robot in the x-axis direction
And the ideal path
Has a tangential motion position error of
The tangential motion speed of the robot along the ideal path is
At the same time satisfy
Achieving tangential motion position error of a robot
And tangential motion velocity error
Converge to 0 and asymptotically stabilize;
step three: providing a control equation of an improved Serpenoid curve of the multi-joint snake-shaped robot;
step four: enabling the motion direction of the articulated snake-shaped robot to point to a desired path by using the LOS guidance rate;
step five: setting an ideal angle function of the connecting rod of the multi-joint snake-shaped robot according to the target I set in the step II
Adding an auxiliary function, constructing a feedback controller, and designing a connecting rod direction angle compensator phi 0 The progressive stability of the angle error of the connecting rod is realized;
step six: according to the second target made in the second step, according to the control equation of the improved Serpoid curve of the multi-joint snake-shaped robot, a feedback controller is designed by setting a feedback input-output controller to control the input u of the system, so as to realize gradual stabilization of the joint angle error;
step seven: setting the normal motion speed and the normal motion position error of the multi-joint snake-shaped robot according to the third target established in the second step, and finding out the controlled condition of the error through a robust control design controller to ensure that the normal motion position error of the robot is quickly converged;
step eight: according to the fourth target formulated in the second step, an improved Serpenoid gait curve control equation is used as an additional control item to control the tangential motion position and the tangential motion speed of the robot, an estimated value of the multi-joint snake-shaped robot swing amplitude compensation is designed by using an adaptive control method, the estimated value replaces the actual swing amplitude to adjust the controller in real time, and the tangential motion position error and the tangential motion speed error of the robot are converged to 0 and asymptotically stabilized;
step nine: constructing a Lyapunov function L, and verifying the stability of the robot connecting rod angle error in the fifth step, the robot joint angle error in the sixth step, the robot normal motion speed and normal motion position error in the seventh step and the robot tangential motion position and tangential motion speed error in the eighth step;
the multi-joint snake-shaped robot kinematic model established in the step one obtains a new center of mass position of the robot through coordinate transformation (1)
Establishing a kinematic model (2) of the robot;
wherein, the joints of the multi-joint snake-shaped robot are composed of N rigid connecting rods with the length of 2h, and the distance of the movement of the position coordinate of the robot along the tangential direction is
The link angle of the robot is set as
The center of mass of the joint of the robot is p = [ p ] x ，p y ] T Angular velocity of the link of the robot is
The normal vector velocity of the robot is v n ∈R N The normal vector velocity after coordinate transformation is
Wherein, the first and the second end of the pipe are connected with each other,
and
the joint angles of the multi-joint snake-shaped robot are combined into a set
The angular velocity of the joints of the robot is set as
Tangential velocity of the robot is v t ∈R N (ii) a The tangential and normal friction coefficients in the multi-joint snake-shaped robot connecting rod model are respectively lambda 1 > 0 and lambda 2 The friction coefficient is more than 0, and the size of the friction coefficient is influenced by the geographical environment; setting the mapping ratio of the rotation speed of the robot joint to the rotation acceleration to be mu 1 Is greater than 0; setting the mapping proportion of the mean value of the angle of the joint of the robot and the tangential velocity to the rotation acceleration as mu 2 Is greater than 0; the joint mass of the robot is m, and the auxiliary matrix is
The control input of the system is
The concrete implementation method of the third step is that,
providing a control equation of an improved Serpenoid curve of the multi-joint snake-shaped robot, wherein the improved Serpenoid curve control equation establishes the relation between the swing amplitude of the robot and the motion position and the motion speed of the robot, and the relation is defined as a formula (3) and is used as the ith joint included angle of the robot;
wherein the time-varying gain amplitude of the joint vibration of the multi-joint snake-shaped robot is
The robot has a joint offset of
The phase shift of the robot joint is delta, the frequency of the robot joint motion is omega, and the solution of two compensators for controlling the advancing speed and the offset direction of the robot is
And phi 0 ；
Desired joint angle of articulated snake-like robot
Correlating the control equation of the improved Serpenoid curve to obtain a formula (4);
wherein the content of the first and second substances,
the concrete implementation method of the step five is that,
the link angle error and the link angular velocity error of the articulated snake-shaped robot are respectively (6) and (7);
differentiating the connecting rod angle error and the connecting rod angular velocity error to obtain (8) and (9);
setting the auxiliary function of the connecting rod angle error to be (10), and obtaining another form of connecting rod angle error differential to be (11);
designing a connecting rod direction angle compensator of the robot to be phi by using Backstepping control method 0 ；
The concrete realization method of the step six is that,
the joint angle error of the robot is e φ ；
The differential of the joint angle error of the multi-joint snake-shaped robot is (14);
in order to realize convergence of joint angle errors, an auxiliary function with joint angle error coefficient estimated values is set as (15);
wherein
Is a parameter k φ A time variable estimate value of > 0;
another form of deriving the derivative of the joint angle error is (16);
the joint angular velocity error and the differential thereof of the multi-joint snake-shaped robot are respectively (17) and (18);
setting the input of the system as u by using an adaptive control method; at the same time, a feedback input-output controller is designed
Wherein the content of the first and second substances,
and
are each lambda 1 And λ 2 The estimated value of the time-variable of (c),
is a normal number diagonal matrix;
the concrete realization method of the step seven is that,
the expected normal motion path tracked by the multi-joint snake-shaped robot is
The actual normal motion track of the robot is
Wherein the content of the first and second substances,
is a function of time; the normal movement speed of the robot is
Obtaining (21) according to the mean inequality;
the error between the actual normal motion position and the expected normal motion path of the articulated snake-shaped robot is (22);
since the link angle error e has already been realized in step five θ Convergence is 0; moreover, according to the LOS method, the trigonometric function relation of
And
therefore (23) can be simplified to (24); at the same time, there is v t ∈[V min ,V max ]＞0；
The concrete implementation method of the step eight is that,
defining the tangential velocity of the robot along the ideal path as
The tangential position of the ideal path of the robot is
At the same time satisfy
The actual tangential movement position of the robot is
Actual speed of movement v t (ii) a The tangential motion position error of the robot is e x Error of tangential motion velocity of e s ；
Differentiating (25) and substituting (2) to obtain (26);
setting an auxiliary function sigma 1 ；
Designing a tracking function of the swing amplitude compensation of the robot to be beta by using an adaptive control method;
to make it possible to
Setting the auxiliary function to σ 2 ；
The time-varying swing amplitude of the articulated snake robot is defined as (32);
the specific implementation method of the step nine is that,
setting Lyapunov candidate equation L 1 Is (33);
to obtain
Is (35);
inequality relations (36) and (37) exist;
to obtain
Is (39);
setting Lyapunov candidate equation V 21 Is (40))；
Setting Lyapunov candidate equation V 22 Is (42);
wherein eta is 1 The gain is a normal number when the value is more than 0;
setting Lyapunov candidate equation L 2 Is (44);
wherein eta is 2 > 0 and η 3 The gain is normal number when the value is more than 0;
setting time variable estimation values
And
are respectively differentiated by
And
at the same time, will
(20) And (19) into (45) to obtain (46);
always keep at
Wherein, χ φ The gain is a normal number when the value is more than 0; to obtain
Is (47);
the consistency and the boundedness of the normal movement speed are proved; setting the Lyapunov candidate equation of the normal movement speed to be (48);
can obtain
Is (50);
setting boundary conditions as
And
since Y > 0, provided that
Small enough to make-p 1 Is negative; for is to
Scaling to obtain (51);
solving the (51) to obtain (52);
therefore, the temperature of the molten metal is controlled,
is bounded and converges to
Due to the fact that
V in (1) θ Is bounded, so v n Is also bounded, without v being set n Is bound by n I.e. | | v n ||≤ε n ；
Designing a Lyapunov equation of the normal motion position error as (53);
can be obtained according to the mean inequality
And
at the same time, the user can select the desired position,
is bounded, i.e.
Can obtain
Is (55);
without being provided with
And
at the same time should satisfy
To obtain
Is (56);
solving (57) for (56);
thus, the normal motion position error of the robot is bounded and converges to
To increase the convergence speed of the error, there may be an increase in V min Decreasing the forward distance delta, increasing the friction of the robot in the normal direction, decreasing the normal velocity v n Boundary value of n And decreasing the value of | X |;
setting Lyapunov candidate equation V 41 Is (58);
setting Lyapunov candidate equation V 42 Is (61);
to obtain
Is (63);
setting Lyapunov function L 4 Is (64);
to obtain
Is (66);
the Lyapunov function of the system is L = L 1 +L 2 +L 3 +L 4 ；
Therefore, the adaptive trajectory tracking controller of the articulated snake robot based on the improved Serpenoid curve is consistently bounded and stable; the error can be controlled within a small acceptable range by adjusting the value of the parameter.
2. The design method of the adaptive trajectory tracking controller of the articulated snake robot based on the improved Serpenoid curve as claimed in claim 1, wherein the method comprises the following steps: and step ten, carrying out simulation experiments through MATLAB, and verifying the effectiveness of the multi-joint snake-shaped robot self-adaptive trajectory tracking controller based on the improved Serpenoid curve.
3. The design method of the adaptive trajectory tracking controller of the articulated snake robot based on the improved Serpenoid curve as claimed in claim 1 or 2, wherein: the concrete implementation method of the step four is that,
adopting LOS guide law to enable the motion direction of the articulated snake-shaped robot to point to a desired path; in the LOS method, the ideal link angle of the robot is (5);
wherein the desired link angle of the articulated snake robot is
The angle is an expected value of the actual motion direction of the robot, and the tracking distance error of the robot is e y The forward distance is Δ.
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