CN113655712A - Vascular robot coupling modeling and robust self-adaptive control method - Google Patents

Vascular robot coupling modeling and robust self-adaptive control method Download PDF

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CN113655712A
CN113655712A CN202110604422.XA CN202110604422A CN113655712A CN 113655712 A CN113655712 A CN 113655712A CN 202110604422 A CN202110604422 A CN 202110604422A CN 113655712 A CN113655712 A CN 113655712A
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robot
vascular
blood
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CN113655712B (en
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周淼磊
陈文雪
张晨
高巍
于业伟
王一帆
聂琳琳
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Jilin University
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    • G05B13/00—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion
    • G05B13/02—Adaptive control systems, i.e. systems automatically adjusting themselves to have a performance which is optimum according to some preassigned criterion electric
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    • Y—GENERAL TAGGING OF NEW TECHNOLOGICAL DEVELOPMENTS; GENERAL TAGGING OF CROSS-SECTIONAL TECHNOLOGIES SPANNING OVER SEVERAL SECTIONS OF THE IPC; TECHNICAL SUBJECTS COVERED BY FORMER USPC CROSS-REFERENCE ART COLLECTIONS [XRACs] AND DIGESTS
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Abstract

A vascular robot coupling modeling and robust self-adaptive control method belongs to the technical field of control. The invention aims to establish a vascular robot attitude and orbit integrated kinematics and dynamics coupling model based on a spiral theory and dual quaternion, design a gravity-buoyancy compensation device, fully analyze the influence of blood pulsation flow field effect, and finally design a vascular robot coupling modeling and robust self-adaptive control method with a certain control precision robust self-adaptive control scheme. The invention establishes a model of the vascular robot for integrating the attitude and orbit kinematics and dynamics; building a gravity-buoyancy compensation device; the influence of the vessel wall motion on the blood flow velocity is combined to finally analyze the resistance borne by the vessel robot; and designing a robust adaptive controller. The invention can move spirally in blood without contacting with blood vessels, thereby realizing the effect that the vascular robot has no damage to vascular tissues, and having important significance for the application of the vascular robot in the medical field.

Description

Vascular robot coupling modeling and robust self-adaptive control method
Technical Field
The invention belongs to the technical field of control.
Background
Along with the development of society, people's income is higher and higher, and people also pay more and more attention to health. Cardiovascular and cerebrovascular diseases as three killers of human health are also increasingly paid attention. At present, the most perfect and advanced method for treating cardiovascular and cerebrovascular diseases is minimally invasive vascular interventional surgery. The minimally invasive vascular interventional operation needs medical care personnel to diagnose cardiovascular and cerebrovascular diseases under X-ray, and under a guiding device, a catheter guide wire needs to be inserted and the guide wire can move in a human body by controlling the tail end of the guide wire. Medical care personnel are required to be exposed to rays for working, so that the body of the medical care personnel is inevitably injured; the guide wire catheter needs to enter the body of a patient for a long time and a long distance, so that great trauma is caused to the patient; medical personnel carry out manual operation through the terminal wire that leads, highly rely on doctor's experience, therefore a little misoperation all can produce irreversible injury to the disease, and this makes vascular intervention operation more highly require medical personnel's operation accuracy and experience.
Disclosure of Invention
The invention aims to establish a vascular robot attitude and orbit integrated kinematics and dynamics coupling model based on a spiral theory and dual quaternion, design a gravity-buoyancy compensation device, fully analyze the influence of blood pulsation flow field effect, and finally design a vascular robot coupling modeling and robust self-adaptive control method with a certain control precision robust self-adaptive control scheme.
The method comprises the following steps:
s1, establishing a model of the vascular robot for integrating the attitude and orbit kinematics and dynamics;
the error-pair quaternion is defined as:
Figure RE-GDA0003281494100000011
wherein the content of the first and second substances,
Figure RE-GDA0003281494100000012
Figure RE-GDA00032814941000000110
represents a dual quaternion multiplication;
the vascular robot needs to firstly perform attitude conversion q by a body coordinate systemBDThen is translated again
Figure RE-GDA0003281494100000013
And when the position of the target coordinate system is reached, the vessel robot posture and track coupling kinematic equation of the spiral theory and the dual quaternion:
Figure RE-GDA0003281494100000014
wherein the content of the first and second substances,
Figure RE-GDA0003281494100000015
the angular velocity of the blood vessel robot body coordinate system relative to the inertial system is represented under the blood vessel robot body coordinate system;
the dual angular momentum can be expressed as:
Figure RE-GDA0003281494100000016
wherein the content of the first and second substances,
Figure RE-GDA0003281494100000017
expressed as a rigid dual inertia matrix;
according to the Euler equation:
Figure RE-GDA0003281494100000018
wherein the content of the first and second substances,
Figure RE-GDA0003281494100000019
the dual resultant force to which the vascular robot is subjected is expressed as follows:
Figure RE-GDA0003281494100000021
wherein the content of the first and second substances,
Figure RE-GDA0003281494100000022
is a resultant force acting on the center of mass of the vascular robot, an
Figure RE-GDA0003281494100000023
As a driving force, the driving force is,
Figure RE-GDA0003281494100000024
is the resultant force of gravity and buoyancy,
Figure RE-GDA0003281494100000025
in order to be a fluid resistance, it is,
Figure RE-GDA0003281494100000026
in order to be subjected to the disturbing force,
Figure RE-GDA0003281494100000027
the resultant moment acting on the centroid;
formula (1036) is simplified to yield:
Figure RE-GDA0003281494100000028
the vascular robot posture and track coupling kinetic equation of the spiral theory and the dual quaternion is as follows:
Figure RE-GDA0003281494100000029
the coupling terms are a second term and a fourth term, so that the coupling influence of the gesture motion and the track motion is analyzed;
the second term of the dynamic posture and track coupling model of the vascular robot is as follows:
Figure RE-GDA00032814941000000210
wherein the content of the first and second substances,
Figure RE-GDA00032814941000000211
is a cross-product factor of the velocity vector,
Figure RE-GDA00032814941000000212
and
Figure RE-GDA00032814941000000213
respectively are cross multiplication factors of an angular velocity vector and a linear velocity vector of the vascular robot; the concrete expression is substituted into formula (1040) to obtain:
Figure RE-GDA00032814941000000214
the fourth term of the kinetic model is simplified:
Figure RE-GDA00032814941000000215
s2, designing a gravity-buoyancy compensation device;
gravity force of blood vessel robot
Figure RE-GDA00032814941000000218
And buoyancy
Figure RE-GDA00032814941000000219
The resultant force is expressed as:
Figure RE-GDA00032814941000000216
the volume and magnetization of the robot, the magnetic force expression of the micro-robot is:
Figure RE-GDA00032814941000000217
wherein, gx,gy,gzThree components of the magnetic flux gradient, respectively; in order to move the robot in the horizontal direction, the magnetic force components in the x-axis and the y-axis should satisfy the following equation:
Figure RE-GDA0003281494100000031
wherein the values of the gradient of the magnetic induction along the x-axis and the y-axis should be equal, i.e. gx＝gyBy ghTo replace gx，gy. Magnetic force F in z-axis directionmzIs divided into two parts, wherein
Figure RE-GDA00032814941000000316
For counteracting
Figure RE-GDA00032814941000000315
Representing part of the vascular robot driving force, magnetic field gradient splitting at the z-axis
Figure RE-GDA00032814941000000317
And
Figure RE-GDA00032814941000000318
two parts;
the magnetic force and the magnetic field gradient along the z direction should satisfy the following relations:
Figure RE-GDA0003281494100000033
Figure RE-GDA00032814941000000319
s3, analyzing the relationship process of the pulsating flow field effect and the blood vessel wall motion and the blood flow velocity as follows:
resistance to the vascular robot
Figure RE-GDA0003281494100000034
Including bloodResistance to the spherical head of a tube robot
Figure RE-GDA0003281494100000035
And the resistance to which the helical tail is subjected
Figure RE-GDA0003281494100000036
The specific expression is as follows:
Figure RE-GDA0003281494100000037
where ρ isfExpressed as blood density, r represents the robot head radius,
Figure RE-GDA0003281494100000038
indicating the relative speed of the robot with respect to the blood,
Figure RE-GDA0003281494100000039
τ0is a dimensionless ratio related to local vessel occlusion;
will be provided with
Figure RE-GDA00032814941000000310
Is substituted into
Figure RE-GDA00032814941000000311
In (b), the following are obtained:
Figure RE-GDA00032814941000000312
wherein the content of the first and second substances,
Figure RE-GDA00032814941000000313
eta is blood viscosity;
pulsating fluid velocity of
Figure RE-GDA00032814941000000314
By a parabolic function vs(x) And time-varying periodic blood pulsatile flow function vt(t) composition, time-varying blood velocity vt(t)＝ζ1Expressed as an N-order truncated Fourier series;
Figure RE-GDA0003281494100000041
wherein, VrMean blood flow rate;
for the coupling effect between the vessel wall motion and the blood flow velocity, the following assumptions are made:
(1) the blood vessel is semi-infinitely long;
(2) the vessel wall is an isotropic Hooke elastomer;
(3) blood is a homogeneous newtonian viscous fluid and is incompressible and flows in an axisymmetric layer;
assuming that an original point is taken at the center of an inlet face, setting an x axis as an axial direction and an r axis as a longitudinal direction, and setting a cylindrical coordinate system x, r and theta;
establishing a mathematical model of blood flow:
Figure RE-GDA0003281494100000042
Figure RE-GDA0003281494100000043
Figure RE-GDA0003281494100000044
wherein the boundary conditions are:
Figure RE-GDA0003281494100000045
wherein, Vx＝Vx(r,x,t),Vr＝Vr(r,x,t),P＝P(x,t),υ0Is the characteristic velocity, P, of blood at the vascular opening0Is the mean pressure at the vascular inlet, ak,bk,gk,hkIs a constant number of times, and is,omega is the oscillation frequency;
suppose that
Figure RE-GDA0003281494100000046
Substitution into formula (1051) gives:
Figure RE-GDA0003281494100000047
obtaining the average blood inlet velocity Vr(ii) a Assumption (2) for the blood vessel wall that the deformation of the blood vessel wall following the blood is slight; the thickness of the vessel wall is h, the ratio of the thickness to the radius of the vessel is kept constant
Figure RE-GDA0003281494100000048
Are small;
the vascular wall can receive axial, radial effort, its expression respectively is:
Figure RE-GDA0003281494100000051
Figure RE-GDA0003281494100000052
where H is the actual vessel wall thickness, ρωH, R and E are respectively the density, thickness, radius and elastic modulus of the vessel wall, and mu is the poisson ratio of the vessel;
Figure RE-GDA00032814941000000517
ζ is the vessel axial and radial displacement, respectively. p, peRespectively the internal pressure and the external pressure of the vessel wall;
irrespective of the constraints of the surrounding tissue, H ═ H, ω 00 and for the blood flow rate, the radial velocity of the blood is much greater than the axial velocity, i.e. the velocity
Figure RE-GDA0003281494100000053
When neglecting the viscous term, inertia in the equation of motionAfter the term, the axial motion equation (1056) is:
Figure RE-GDA0003281494100000054
radial equation (1057) the left inertial term of the equation is much less affected than the right and therefore can be ignored, and the radial equation is written as:
Figure RE-GDA0003281494100000055
combining equations (1060) and (1061) yields the radial displacement equation:
Figure RE-GDA0003281494100000056
at the vascular wall R-R junction, the coupling condition is expressed as:
Figure RE-GDA0003281494100000057
s4, designing a robust adaptive controller according to the steps S1, S2 and S3; the robust adaptive control algorithm design process comprises the following steps:
modeling the robot kinematics and kinetic coupling established at S1 as:
Figure RE-GDA0003281494100000058
wherein the content of the first and second substances,
Figure RE-GDA0003281494100000059
in the form of a dual-moment matrix of inertia,
Figure RE-GDA00032814941000000510
in the form of a nominal matrix, the matrix,
Figure RE-GDA00032814941000000511
is an indeterminate portion;
firstly, set up
Figure RE-GDA00032814941000000512
The attitude and orbit coupling kinetic equation is the attitude angle of the motion of the vascular robot, and can be:
Figure RE-GDA00032814941000000513
wherein the content of the first and second substances,
Figure RE-GDA00032814941000000514
Θddesired attitude angle for robot movement, eΘ＝＝Θ-ΘdIn order to be an attitude angle tracking error,
Figure RE-GDA00032814941000000515
in order to control the total force for the system,
Figure RE-GDA00032814941000000516
external disturbance force to the system; and is
Figure RE-GDA00032814941000000518
The robust control law is designed as follows:
Figure RE-GDA0003281494100000061
note the book
Figure RE-GDA00032814941000000619
Combining the formulas (1065) and (1066) to obtain:
Figure RE-GDA0003281494100000062
defining the auxiliary signal:
Figure RE-GDA0003281494100000063
note the book
Figure RE-GDA0003281494100000064
Substituting (1068) into (1067) yields:
Figure RE-GDA0003281494100000065
designing a robust control item:
Figure RE-GDA0003281494100000066
wherein the content of the first and second substances,
Figure RE-GDA0003281494100000067
k1,k2,k3＞0；
Figure RE-GDA0003281494100000068
representing a robust term of the system for coping with external disturbances to which the vascular robot is bounded, i.e.
Figure RE-GDA0003281494100000069
And β (β > 0); definition of
Figure RE-GDA00032814941000000610
Wherein
Figure RE-GDA00032814941000000611
For a known nominal moment of inertia, the same applies
Figure RE-GDA00032814941000000612
Figure RE-GDA00032814941000000613
In order to deal with the uncertain part of the moment of inertia, an adaptive operator is designed:
Figure RE-GDA00032814941000000614
wherein a ═ a1,a2,a3]T；
Let I be the identity matrix, define
Figure RE-GDA00032814941000000615
The adaptive robust control law is designed as follows:
Figure RE-GDA00032814941000000616
wherein the content of the first and second substances,
Figure RE-GDA00032814941000000617
is thetaΔMAn estimated value of, and
Figure RE-GDA00032814941000000618
in the formula, gamma is a positive definite diagonal matrix.
The vascular robot researched by the invention consists of a magnetic spherical head and a spiral tail; the blood vessel robot can move spirally in the blood without contacting with the blood vessel, thereby realizing the effect that the blood vessel robot has no damage to the blood vessel tissue; in order to reduce the volume of the vascular robot and enable the vascular robot to flexibly move in a blood environment, a driving force needs to be provided for the vascular robot by virtue of an external three-dimensional magnetic field; in order to counteract the sinking motion under the action of gravity and buoyancy, ensure that the robot can perform expected motion in blood and reduce the contact to the vessel wall, an external three-dimensional magnetic field is needed to establish a gravity-buoyancy compensation device; in order to analyze the resistance of the vascular robot, the blood pulsation flow field effect and the influence of the vascular wall motion on the blood flow rate need to be analyzed; finally, a robust self-adaptive control scheme is determined, and trajectory tracking control simulation is carried out, so that the method has important significance for the application of the vascular robot in the medical field.
Drawings
Fig. 1 is a straight line view of PLUCKER;
FIG. 2 is a schematic representation of the rotational momentum of the rigid body velocity and the dual momentum;
FIG. 3 is a schematic diagram of a limited displacement screw principle;
FIG. 4 is a diagram of a quaternion representation coordinate transformation;
FIG. 5 is a schematic diagram of a transformation relationship between coordinate systems;
FIG. 6 is a diagram of a vascular robot gravity-buoyancy compensation analysis;
FIG. 7 is a schematic diagram of vascular robot gravity-buoyancy compensation;
FIG. 8 is a schematic diagram of the resistance experienced by the vascular robot;
FIG. 9 is a diagram showing the physical structure and parameters of each part of the vascular robot;
FIG. 10 is a schematic view of a linear irregular diameter vessel;
FIG. 11 is a graph of linear trajectory tracking of a vascular robot;
FIG. 12 is a graph showing the directional error of the linear trajectory of the vascular robot along each coordinate axis;
FIG. 13 is a graph of a linear trajectory tracking yaw angle tracking of a vascular robot;
FIG. 14 is a graph of the tracking error of the linear trajectory tracking yaw angle of the vascular robot;
FIG. 15 is a linear trajectory tracking pitch angle tracking graph of the vascular robot;
FIG. 16 is a graph of linear trajectory tracking pitch angle tracking error of a vascular robot;
FIG. 17 is a linear trajectory tracking roll rate tracking graph of a vascular robot;
FIG. 18 is a graph of the tracking error of the linear trajectory tracking roll angular velocity of the vascular robot;
FIG. 19 is a schematic view of a curvilinear vessel;
FIG. 20 is a graph of a curvilinear trajectory tracking of a vascular robot;
FIG. 21 is a graph showing the error in the direction of each coordinate axis of the curve-shaped trajectory of the vascular robot;
FIG. 22 is a graph of a vascular robot curve-type trajectory tracking yaw angle tracking;
FIG. 23 is a graph of a curve-type trajectory tracking yaw angle tracking error of a vascular robot;
FIG. 24 is a graph of a vascular robot curve-type trajectory tracking pitch angle tracking;
FIG. 25 is a graph of a curve-type trajectory tracking pitch angle tracking error of a vascular robot;
FIG. 26 is a graph of a vascular robot curve-type trajectory tracking roll rate tracking;
fig. 27 is a graph of the vascular robot curve-type trajectory tracking roll angular velocity tracking error.
Detailed Description
The invention aims at a vascular robot attitude and orbit integrated coupling model and a track tracking control method, and firstly establishes a vascular robot attitude and orbit integrated kinematics and dynamics coupling model based on a spiral theory and dual quaternion and analyzes the problem of the coupling effect between attitude motion and track motion. Aiming at the problem that the vascular robot generates sinking action under the action of gravity and buoyancy in the vertical direction in the blood movement process, the gravity-buoyancy compensation device is established to enable the vascular robot to move in the expected direction. Considering the influence of the pulsating flow field effect on the vascular robot, analyzing the coupling effect between the vascular wall motion and the blood flow velocity, and finally analyzing the resistance of the vascular robot in the blood complex environment. And finally, designing a robust adaptive controller, and performing linear and curve track tracking control simulation to prove the effectiveness of the control method.
The invention is described in detail below with reference to the attached drawing figures:
the invention comprises the following steps:
the method comprises the following steps: establishing a posture and orbit integrated kinematics and dynamics coupling model of the vascular robot based on a spiral theory and dual quaternion; step two: aiming at the sinking action of the vascular robot caused by the gravity and buoyancy in the vertical direction in the blood movement process, a gravity-buoyancy compensation device is established;
step three: analyzing the influence of the blood pulsation flow field effect according to the complexity of the blood environment, and finally analyzing the resistance borne by the vascular robot by combining the influence of the blood vessel wall motion on the blood flow rate;
step four: and designing a robust adaptive controller, and performing trajectory tracking control simulation.
Firstly, basic explanations are made on quaternion, even pair, dual quaternion and spiral theories:
quaternions, originally proposed by William Roman Hamilton, irish, in the nineteenth century, were applicable to rigid body kinematics, and are specifically defined as:
Figure RE-GDA0003281494100000081
wherein the content of the first and second substances,
Figure RE-GDA0003281494100000082
and q is0Is a scalar quantity of a quaternion,
Figure RE-GDA0003281494100000083
a vector that is a quaternion;
Figure RE-GDA0003281494100000084
the following conditions are satisfied for the unit vectors intersected two by two:
Figure RE-GDA0003281494100000085
when | | q | luminance2When 1, it is called a unit quaternion. When in use
Figure RE-GDA0003281494100000086
When q is called a scalar quaternion, when q is0When 0, it is called a vector quaternion.
For the even nineteenth century, Clifford invented the even number, which was defined as:
Figure RE-GDA0003281494100000087
wherein the content of the first and second substances,
Figure RE-GDA0003281494100000088
respectively a real part and a dual part of a dual number, and epsilon is a dual unit and satisfies epsilon 20, epsilon ≠ 0, which, as used herein,
Figure RE-GDA0003281494100000089
indicating belonging to an even pair.
The dual vector is developed on the basis of a dual number, the real part and the dual part of the dual vector are vectors, and the expression is as follows:
Figure RE-GDA00032814941000000810
wherein the content of the first and second substances,
Figure RE-GDA00032814941000000811
the invention introduces the concept of 'rotation amount' when a dual vector is applied to a space, when a straight line in the space is described by the dual vector, a real part of the dual vector is called a positioning vector, a dual part is called a free vector and is expressed as a moment between a unit direction vector of the straight line in the space and the straight line relative to a coordinate origin, and the dual vector is called the 'rotation amount'.
The straight line in the space coordinate system can be expressed as:
Figure RE-GDA00032814941000000812
where the dual vector may be referred to as the Plucker line, as shown in FIG. 1,
Figure RE-GDA00032814941000000813
is the real part of the signal,
Figure RE-GDA00032814941000000814
then the two-part is the dual part,
Figure RE-GDA00032814941000000815
Figure RE-GDA00032814941000000816
and
Figure RE-GDA00032814941000000817
are orthogonal to each other.
The dual vectors as described above may be referred to as "spin", as dual momentum and dual force vectors, and the like. The dual quaternion is a combination of even and quaternion as the name implies. The expression includes two types, one is that the expression is expressed as a quaternion with an even number of elements, and the expression is:
Figure RE-GDA0003281494100000091
the second is expressed as a pair of even numbers with elements being quaternions, and the expression is:
Figure RE-GDA0003281494100000092
wherein q and q' are quaternions.
Theory of helicity
The even number is used for describing the spiral amount and the spiral theory, such as:
Figure RE-GDA0003281494100000093
where ω represents angular velocity and v represents linear velocity. The following steps are repeated:
Figure RE-GDA0003281494100000094
wherein f represents the resultant force received by the rigid body, and T represents the resultant moment received by the rigid body.
Momentum:
Figure RE-GDA0003281494100000095
wherein, p represents the translational momentum of the rigid body, and h is the rotational momentum of the rigid body. From the above series of equations, it can be seen that the real part of the spiral quantity is a quantity independent of the selected point, while the dual part is dependent on the selected point.
The spiral amount has conversion property, and the expression is as follows:
Figure RE-GDA0003281494100000096
wherein the content of the first and second substances,
Figure RE-GDA0003281494100000097
expressed as the Hermitian matrix moving from point b to point a, the expression for which is:
Figure RE-GDA0003281494100000098
an expression of pose transformation is given by applying a finite spiral displacement theory:
Figure RE-GDA0003281494100000099
wherein the content of the first and second substances,
Figure RE-GDA00032814941000000910
the dual angle is expressed as the displacement of the rigid body, the real part is the angle of the rigid body rotating around the spiral shaft, and the dual part is the distance of the rigid body moving along the spiral shaft;
Figure RE-GDA00032814941000000911
the displacement is expressed as a rotation axis of the rigid body, the real part is a unit direction vector, and the dual part is a moment of the axis relative to the origin. A schematic diagram of which is shown in fig. 3.
A quaternion is utilized to establish the vascular robot attitude kinematics, a dynamic model is defined by the quaternion, and a unit quaternion can be obtained and is used for describing the attitude in rigid motion. The rotational motion can be represented by a unit quaternion as:
Figure RE-GDA00032814941000000912
wherein the vector part represents the position of the fixed shaft,
Figure RE-GDA00032814941000000913
the unit vector of Euler rotation axis direction is represented, and the scalar part represents the rotation angle theta
Figure RE-GDA0003281494100000101
The system O-X of the vascular robot represented by quaternion can be obtained through the analysisIYIZIRelative to the inertial system O-XBYBZBThe attitude motion equation of (a) is:
Figure RE-GDA0003281494100000102
the kinematic equation of the attitude of the vascular robot expectation system relative to the inertial system can be expressed as:
Figure RE-GDA0003281494100000103
wherein the content of the first and second substances,
Figure RE-GDA0003281494100000104
respectively representing angular velocity vectors of the spiral vascular robot body system relative to an inertial system under the body and inertial coordinate systems;
Figure RE-GDA0003281494100000105
for the angle of the vessel robot object system relative to the inertial systemAnd representing the velocity vector in the target and inertial coordinate system.
The error quaternion is the conversion from the body coordinate system of the vascular robot to the target coordinate system and is defined as:
Figure RE-GDA0003281494100000106
the error angular velocity is defined as:
Figure RE-GDA0003281494100000107
in the same way, the method for preparing the composite material,
Figure RE-GDA0003281494100000108
wherein the content of the first and second substances,
Figure RE-GDA0003281494100000109
the representation of the error angular velocity vector of the vascular robot target system relative to the system under the vascular robot system is shown.
Derivation of equation (1015) and substitution of equations (1013), (1014) into the equation of relative motion that yields the error quaternion representation:
Figure RE-GDA00032814941000001010
the vascular robot dynamics equation based on unit quaternion can be expressed as:
Figure RE-GDA00032814941000001011
wherein, F is the external resultant moment suffered by the vascular robot, and M is a positive definite matrix and is the rotational inertia of the vascular robot.
Substituting equation (1017) and the above equation into equation (1020) may result in the robot relative dynamics equation expressed in unit quaternion:
Figure RE-GDA00032814941000001012
based on the definition of the dual quaternion and the spiral theory, the pose and orbit integrated kinematics and dynamics model of the vascular robot based on the spiral theory can be expressed as the pose coupling kinematics equation of a coordinate system X relative to Y as follows:
Figure RE-GDA0003281494100000111
as can be known from the above formula, the vascular robot attitude motion equation based on dual quaternions is similar to the six-degree-of-freedom attitude motion equation represented by common quaternions, that is:
Figure RE-GDA0003281494100000112
wherein the dual angular velocities
Figure RE-GDA0003281494100000113
And
Figure RE-GDA0003281494100000114
the dual angular velocities of coordinate system X relative to coordinate system Y are represented under coordinate system X and coordinate system Y, respectively. Wherein
Figure RE-GDA0003281494100000115
And
Figure RE-GDA0003281494100000116
are respectively
Figure RE-GDA0003281494100000117
And
Figure RE-GDA0003281494100000118
the derivation of the time is carried out,
Figure RE-GDA0003281494100000119
and
Figure RE-GDA00032814941000001110
are respectively
Figure RE-GDA00032814941000001111
And
Figure RE-GDA00032814941000001112
associated rates of change in coordinate system X and coordinate system Y.
According to equation (1023), we can respectively express the kinematic equations of the vessel robot system and the vessel robot target system relative to the inertial system as:
Figure RE-GDA00032814941000001113
Figure RE-GDA00032814941000001114
the dual angular momentum is defined as:
Figure RE-GDA00032814941000001115
according to the Euler equation:
Figure RE-GDA00032814941000001116
wherein the content of the first and second substances,
Figure RE-GDA00032814941000001117
the dual resultant force to which the vascular robot is subjected is expressed as follows:
Figure RE-GDA00032814941000001118
wherein the content of the first and second substances,
Figure RE-GDA00032814941000001119
is the resultant force of the action points on the centroid of the vascular robot, an
Figure RE-GDA00032814941000001120
As a driving force, the driving force is,
Figure RE-GDA00032814941000001121
is the resultant force of gravity and buoyancy,
Figure RE-GDA00032814941000001122
in order to be a fluid resistance, it is,
Figure RE-GDA00032814941000001123
in order to be subjected to the disturbing force,
Figure RE-GDA00032814941000001124
is the resultant moment acting on the centroid.
Substituting equation (1007) and equation (1026) into equation (1027) yields:
Figure RE-GDA0003281494100000121
when the vascular robot mass is fixed with the defined moment of inertia, the above equation is simplified:
Figure RE-GDA0003281494100000122
then separating the real part and the dual part in the above equation into a kinetic equation of the trajectory motion and the attitude motion of the vascular robot:
Figure RE-GDA0003281494100000123
Figure RE-GDA0003281494100000124
vascular robot attitude and orbit integrated coupling mathematical model
The error-pair quaternion is defined as:
Figure RE-GDA0003281494100000125
wherein the content of the first and second substances,
Figure RE-GDA0003281494100000126
the vessel robot needs to transfer from the position of the body coordinate system to the target coordinate system, and then the vessel robot needs to perform attitude transformation q from the body coordinate systemBDThen is translated again
Figure RE-GDA0003281494100000127
When the position of the target coordinate system is reached, the correlation transformation between the coordinate systems is shown in fig. 5.
Derivation of equation (1032) in combination with equation (1022) yields the vessel robot kinematics equation represented by an error pair-even quaternion:
Figure RE-GDA0003281494100000128
wherein the content of the first and second substances,
Figure RE-GDA0003281494100000129
the dual angular momentum can also be expressed as:
Figure RE-GDA00032814941000001210
wherein the content of the first and second substances,
Figure RE-GDA00032814941000001211
representing a rigid dual inertia matrix.
Equation (1027) can be written as:
Figure RE-GDA00032814941000001212
simplified by equation (1023):
Figure RE-GDA0003281494100000131
this is obtained by the formula (1036):
Figure RE-GDA0003281494100000132
substituting equation (1038) into equation (1036) can obtain a vascular robot posture and trajectory coupling kinetic equation based on the spiral theory and dual quaternion:
Figure RE-GDA0003281494100000133
and the coupling terms are a second term and a fourth term, so that the coupling influence of the gesture motion and the track motion is analyzed.
Vascular robot mathematical model coupling effect analysis
The second term of the dynamic posture and track coupling model of the vascular robot is as follows:
Figure RE-GDA0003281494100000134
wherein the content of the first and second substances,
Figure RE-GDA0003281494100000135
is a cross-product factor of the velocity vector,
Figure RE-GDA0003281494100000136
and
Figure RE-GDA0003281494100000137
are respectively provided withIs the cross product factor of the angular velocity vector and the linear velocity vector of the vascular robot.
The concrete expression is substituted into formula (1040) to obtain:
Figure RE-GDA0003281494100000138
it can be seen that the real part appears
Figure RE-GDA0003281494100000139
The method shows that the track motion of the vascular robot has an influence on the posture motion, but the influence is small.
For the same reason, the fourth term of the kinetic model is simplified:
Figure RE-GDA00032814941000001310
as can be seen from the above formula, the dual part comprises
Figure RE-GDA00032814941000001311
It can be obtained that the relative angular velocity of the vascular robot affects the track motion of the vascular robot, i.e. the gesture motion affects the track motion of the vascular robot. The analysis can show that the coupling effect exists between the posture motion and the track motion of the vascular robot when the vascular robot moves in blood, and lays a foundation for designing the posture-orbit integrated robust self-adaptive control of the vascular robot.
Gravity-buoyancy compensation device for blood vessel designing robot
Gravity force of blood vessel robot
Figure RE-GDA0003281494100000141
And buoyancy
Figure RE-GDA0003281494100000142
The resultant force is expressed as:
Figure RE-GDA0003281494100000143
through the stress analysis of the vascular robot, it can be known that the vascular robot designs a gravity-buoyancy compensation device for counteracting the sinking motion of the vascular robot under the action of gravity and buoyancy in the vertical direction, and a compensation coordinate system P is established firstly, and the schematic diagram is shown in fig. 7.
Fig. 7 shows that when the component of the external three-dimensional magnetic field in the z-axis direction is equal to the resultant force of gravity and buoyancy, the sinking effect of the vascular robot can be completely cancelled, and when the volume and magnetization of the robot are known, the magnetic expression of the micro-robot is as follows:
Figure RE-GDA0003281494100000144
wherein, gx,gy,gzThree components of the magnetic flux gradient. In order to move the robot in the horizontal direction, the magnetic force components in the x-axis and the y-axis should satisfy the following equation:
Figure RE-GDA0003281494100000145
wherein the values of the gradient of the magnetic induction along the x-axis and the y-axis should be equal, i.e. gx＝gyBy ghTo replace gx，gyThe robot is subject to gravity
Figure RE-GDA0003281494100000146
Buoyancy force
Figure RE-GDA0003281494100000147
Acting to generate a magnetic force F in the z-axis directionmzIs divided into two parts, wherein
Figure RE-GDA0003281494100000148
For counteracting
Figure RE-GDA0003281494100000149
Fmz1Indicating vascular robot partial drivesDynamic, magnetic field gradient division at the z-axis
Figure RE-GDA00032814941000001410
And gz1Two parts.
The magnetic force and the magnetic field gradient along the z direction should satisfy the following relations:
Figure RE-GDA00032814941000001411
Figure RE-GDA00032814941000001412
therefore, the sinking effect of gravity and buoyancy in the vertical direction of the vascular robot can be compensated by applying the deflection angle of the external magnetic field.
Pulsating flow field effect analysis
Resistance to which the spiral vascular robot is subjected
Figure RE-GDA00032814941000001413
Including resistance to the spherical head of the vascular robot
Figure RE-GDA00032814941000001414
And the resistance to which the helical tail is subjected
Figure RE-GDA00032814941000001415
The specific expression is as follows:
Figure RE-GDA00032814941000001416
where ρ isfIs the blood density, r is the robot head radius,
Figure RE-GDA0003281494100000151
is the relative velocity of the robot with respect to the blood,
Figure RE-GDA0003281494100000152
τ0is a dimensionless ratio related to local vessel occlusion.
In FIG. 8
Figure RE-GDA0003281494100000153
Is substituted into
Figure RE-GDA0003281494100000154
In (b), the following are obtained:
Figure RE-GDA0003281494100000155
wherein the content of the first and second substances,
Figure RE-GDA0003281494100000156
eta is blood viscosity.
Considering the blood pulsation flow field effect, the vascular robot moves in the blood vessel to avoid the effect and takes a parabolic flow pattern, and the velocity of the pulsation fluid is
Figure RE-GDA0003281494100000157
By a parabolic function vs(x) And time-varying periodic blood pulsatile flow function vt(t) composition, time-varying blood velocity vt(t)＝ζ1Represented as an N-th truncated fourier series.
Figure RE-GDA0003281494100000158
Wherein, VrMean blood flow rate.
For the average flow rate of blood studied, the following assumptions were made:
(1) first assume that the vessel is semi-infinitely long;
(2) the vessel wall is an isotropic Hooke elastomer;
(3) blood is a homogeneous newtonian viscous fluid and is incompressible and flows in an axisymmetric layer.
Assuming that the origin is taken at the center of the inlet face and the x-axis is axial and the r-axis is longitudinal, a cylindrical coordinate system x, r, θ is set. Establishing a mathematical model of blood flow by using a Navier-Stokes equation:
Figure RE-GDA0003281494100000159
Figure RE-GDA00032814941000001510
Figure RE-GDA00032814941000001511
wherein the boundary conditions are:
Figure RE-GDA0003281494100000161
wherein, Vx＝Vx(r,x,t),Vr＝Vr(r,x,t),P＝P(x,t),υ0Is the characteristic velocity, P, of blood at the vascular opening0Is the mean pressure at the vascular inlet, ak,bk,gk,hkIs constant and ω is the oscillation frequency.
Suppose that
Figure RE-GDA0003281494100000162
Substitution into formula (1051) gives:
Figure RE-GDA0003281494100000163
solving the blood flow velocity motion equation based on the boundary conditions to obtain the blood inlet average velocity Vr. The derivation of the equation of motion for the arterial wall is also crucial, considering the pulsatile flow effect in the artery, and the blood motion is tightly coupled to the motion of the vessel wall.
Based on the assumed conditions for the vessel wall, it is additionally assumed that the vessel wall follows the bloodIs relatively minor; the thickness of the vessel wall is h, the ratio of the thickness to the radius of the vessel is kept constant
Figure RE-GDA0003281494100000164
Are small.
The vascular wall can receive axial, radial effort, its expression respectively is:
Figure RE-GDA0003281494100000165
Figure RE-GDA0003281494100000166
where H is the actual vessel wall thickness, ρωH, R and E are respectively the density, thickness, radius and elastic modulus of the vessel wall, and mu is the poisson's ratio of the vessel.
Figure RE-GDA00032814941000001610
ζ is the vessel axial and radial displacement, respectively. p, peThe internal and external pressure of the vessel wall, respectively (usually considered as constants, pe＝3.8kPa)。
Irrespective of the constraints of the surrounding tissue, H ═ H, ω 00 and for the blood flow rate, the radial velocity of the blood is much greater than the axial velocity, i.e. the velocity
Figure RE-GDA0003281494100000167
The axial and radial equations of motion can be:
Figure RE-GDA0003281494100000168
Figure RE-GDA0003281494100000169
when the viscous term and the inertia term in the motion equation are ignored, the axial motion equation (1058) is:
Figure RE-GDA0003281494100000171
radial equation (1059) the inertial term of the equation is much less influential than on the right and therefore negligible, and the radial equation is written as:
Figure RE-GDA0003281494100000172
combining equations (1060) and (1061) yields the radial displacement equation:
Figure RE-GDA0003281494100000173
the blood flow equation and the vessel wall motion equation can be solved through a boundary coupling condition, and at the juncture of the vessel wall R ═ R, the coupling condition is as follows:
Figure RE-GDA0003281494100000174
robust adaptive controller for designing vascular robot
Based on the established vascular robot kinematics and dynamics mathematical model, the attitude and orbit coupling dynamics equation is simplified
Figure RE-GDA0003281494100000175
Wherein the content of the first and second substances,
Figure RE-GDA0003281494100000176
in the form of a dual-moment matrix of inertia,
Figure RE-GDA0003281494100000177
in the form of a nominal matrix, the matrix,
Figure RE-GDA0003281494100000178
is an indeterminate portion. In accordance withAnd designing an adaptive robust controller according to the dynamics modeling of the spiral vascular robot.
Firstly, set up
Figure RE-GDA0003281494100000179
The attitude and orbit coupling kinetic equation is the attitude angle of the motion of the vascular robot, and can be:
Figure RE-GDA00032814941000001710
wherein the content of the first and second substances,
Figure RE-GDA00032814941000001711
Θddesired attitude angle for robot movement, eΘ＝＝Θ-ΘdIn order to be an attitude angle tracking error,
Figure RE-GDA00032814941000001712
in order to control the total force for the system,
Figure RE-GDA00032814941000001713
is the external disturbance force received by the system. And is
Figure RE-GDA00032814941000001714
The robust control law is designed as follows:
Figure RE-GDA00032814941000001715
note the book
Figure RE-GDA00032814941000001716
Combining the formulas (1065) and (1066) can obtain:
Figure RE-GDA00032814941000001717
defining the auxiliary signal:
Figure RE-GDA0003281494100000181
note the book
Figure RE-GDA0003281494100000182
Substituting (1068) into (1067) yields:
Figure RE-GDA0003281494100000183
designing a robust control item:
Figure RE-GDA0003281494100000184
wherein the content of the first and second substances,
Figure RE-GDA00032814941000001815
k1,k2,k3＞0。
Figure RE-GDA00032814941000001816
representing a robust term of the system for coping with external disturbances to which the vascular robot is bounded, i.e.
Figure RE-GDA0003281494100000185
And beta (beta > 0). Definition of
Figure DA00030936986163419736
Wherein
Figure RE-GDA0003281494100000187
For a known nominal moment of inertia, the same applies
Figure RE-GDA0003281494100000188
Figure RE-GDA0003281494100000189
In order to deal with the uncertain part of the moment of inertia, an adaptive operator is designed:
Figure RE-GDA00032814941000001810
wherein a ═ a1,a2,a3]T。
Let I be the identity matrix, define
Figure RE-GDA00032814941000001811
The adaptive robust control law is designed as follows:
Figure RE-GDA00032814941000001812
wherein the content of the first and second substances,
Figure RE-GDA00032814941000001813
is thetaΔMAn estimated value of, and
Figure RE-GDA00032814941000001814
in the formula, gamma is a positive definite diagonal matrix.
The robustness self-adaptive controller stability proves that:
in order to prove the stability of the controller, the method is divided into two steps, firstly, when the dynamic model of the vascular robot is considered to have bounded disturbance, the designed adaptive robust control is proved to be asymptotically stable; secondly, when the dynamic model of the vascular robot is considered to have bounded disturbance, the designed control method is proved to be applied to any eΘ(0) Are all final value bounded.
Before the controller stability certification is carried out, firstly, the assumed conditions are set:
(1) presence of a normal number λ1，λ2So that λ1I3×3≤M≤λ2I3×3；
(2)nΘBounded, i.e. nΘ||≤nΘm；
(3) Δ M is bounded and varies more smoothly with time, i.e.
Figure RE-GDA0003281494100000191
Assuming an inertia estimation error of
Figure RE-GDA0003281494100000192
(4)
Figure RE-GDA0003281494100000193
And
Figure RE-GDA0003281494100000194
are all bounded and there is a constant
Figure RE-GDA0003281494100000195
Make it
Figure RE-GDA0003281494100000196
Proving that:
under the above assumptions (1), (2), (3), a positive definite function is defined:
Figure RE-GDA0003281494100000197
to V1(t,nΘ) The derivation is carried out by the derivation,
Figure RE-GDA0003281494100000198
according to a designed control law (1074) and
Figure RE-GDA0003281494100000199
it is possible to obtain:
Figure RE-GDA00032814941000001910
from condition 2, it is known that a constant ρ exists so that
Figure RE-GDA00032814941000001911
Formula (1075) is thus arranged:
Figure RE-GDA00032814941000001912
wherein λ3Can be according to KcThe expression is derived, thus for ρ, β and the matrix K in the expressioncCan be obtained by limitation
Figure RE-GDA00032814941000001918
Thus proving the progressive stability of the closed loop system.
And (2) proving that:
under the assumptions (1), (2), (4), a positive definite function is defined:
Figure RE-GDA00032814941000001913
to V2(t,nΘ) The derivation yields:
Figure RE-GDA00032814941000001914
definition of
Figure RE-GDA00032814941000001915
From the condition 1, the dual inertia matrix M is bounded, so that a normal number λ exists1，λ2The following steps are performed:
Figure RE-GDA00032814941000001916
known from the above series of formulas
Figure RE-GDA00032814941000001917
This is true. From n toΘThe definition can be given as:
Figure RE-GDA0003281494100000201
let nΘ＝[n1,n2,n3]TFrom the euclidean norm relationship we can derive:
Figure RE-GDA0003281494100000202
Figure RE-GDA0003281494100000203
is provided with
Figure RE-GDA0003281494100000204
Then equation (1079) is:
Figure RE-GDA0003281494100000205
order to
Figure RE-GDA0003281494100000206
The following results were obtained:
Figure RE-GDA0003281494100000207
from the above equation, for an arbitrary initial tracking error eΘ(0) In other words, when t is sufficiently large, the tracking error eΘ(t)||≤C。
Thus, the certification is completed.
The following specific examples were employed to verify the control effect of the present invention.
Example one: and (4) performing track tracking control simulation on the linear motion track of the vascular robot by using the robust adaptive controller designed in the step four, and verifying the control effect of the robust adaptive control method.
The parameters of the vascular robot are set as follows:
TABLE 1 vascular robot simulation parameters
Figure RE-GDA0003281494100000208
Figure RE-GDA0003281494100000211
In the process of operation, because the vascular robot enters, vasospasm is easily caused, so that blood generates ineffectiveness to the robot, the following interference couple force is added in the track tracking control simulation, and the validity and the accuracy of a control algorithm are more powerfully verified. The interference couple force is as follows:
Figure RE-GDA0003281494100000212
assuming an initial velocity of 0mm/s and an average motion velocity of 1mm/s, the initial velocity
Figure RE-GDA0003281494100000215
Theta, psi are 0rad, 0rad/s, respectively.
First, linear trajectory tracking control simulation is performed on the spiral vascular robot, and assuming that the vascular robot performs approximate linear trajectory simulation, the robot starting point is located at (0,0,0), and the reference route shown in fig. 10 is followed.
Defining the linear trajectory equation as:
Figure RE-GDA0003281494100000213
and performing tracking control simulation on the linear track through an adaptive robust control algorithm, wherein the simulation result is shown in fig. 11-18. Fig. 11 is a robot trajectory tracking curve, and it can be known that the robust adaptive controller has better control accuracy. FIG. 12 is a graph showing the three different directional errors of the vascular robot in the coordinate system, indicating the linear motion of the vascular robotIn the process, the control precision and stability of the vascular robot in the blood can be met. FIGS. 13-18 show respectively
Figure RE-GDA0003281494100000216
The tracking curve and the error curve of theta and psi can be known from the figure, the spiral vascular robot can achieve the control precision specified by the attitude motion and the track motion in the track tracking simulation, and has certain stability.
Example two: and (4) performing track tracking control simulation on the curvilinear motion track of the vascular robot by using the robust adaptive controller designed in the step four, and verifying the control effect of the robust adaptive control method.
A schematic diagram of the reference path of the spiral vascular robot in a curved vessel is shown in fig. 19. In the simulation of the bending type track, assuming the origin (0,5,0) of the coordinate system of the starting point of the robot, the bending type track equation is defined as follows:
Figure RE-GDA0003281494100000214
the curve type track simulation diagram is shown in fig. 20-27. Fig. 20 is a curve trace tracing graph, and the adaptive robust trace controller can more accurately trace the trace of the robot in the curve trace simulation. FIG. 21 is a graph of the error in X, Y and Z directions during the movement of the robot, from which it can be seen that the tracking error of the controller is small and within the controllable range. FIGS. 22-27 show the movement of the vascular robot
Figure RE-GDA0003281494100000221
Theta, psi tracking curves and tracking errors. Fig. 27 shows that the error of the roll angular velocity ψ is 0.8953rad/s at the maximum, which occurs at the turning of the vascular robot, that is, the roll angular velocity error is large at the initial movement and the movement at the middle turning of the vascular robot, and fluctuates somewhat at the end stage. In conclusion, the adaptive robust control has stability for the curve track tracking of the vascular robot and certain control precision.

Claims (1)

1. A vascular robot coupling modeling and robust self-adaptive control method is characterized in that:
s1, establishing a model of the vascular robot for integrating the attitude and orbit kinematics and dynamics;
the error-pair quaternion is defined as:
Figure FDA0003093698600000011
wherein the content of the first and second substances,
Figure FDA0003093698600000012
Figure FDA00030936986000000119
represents a dual quaternion multiplication;
the vascular robot needs to firstly perform attitude conversion q by a body coordinate systemBDThen is translated again
Figure FDA0003093698600000013
And when the position of the target coordinate system is reached, the vessel robot posture and track coupling kinematic equation of the spiral theory and the dual quaternion:
Figure FDA0003093698600000014
wherein the content of the first and second substances,
Figure FDA0003093698600000015
the angular velocity of the blood vessel robot body coordinate system relative to the inertial system is represented under the blood vessel robot body coordinate system;
the dual angular momentum can be expressed as:
Figure FDA0003093698600000016
wherein the content of the first and second substances,
Figure FDA0003093698600000017
expressed as a rigid dual inertia matrix;
according to the Euler equation:
Figure FDA0003093698600000018
wherein the content of the first and second substances,
Figure FDA0003093698600000019
the dual resultant force to which the vascular robot is subjected is expressed as follows:
Figure FDA00030936986000000110
wherein the content of the first and second substances,
Figure FDA00030936986000000111
is a resultant force acting on the center of mass of the vascular robot, an
Figure FDA00030936986000000112
As a driving force, the driving force is,
Figure FDA00030936986000000113
is the resultant force of gravity and buoyancy,
Figure FDA00030936986000000114
in order to be a fluid resistance, it is,
Figure FDA00030936986000000115
in order to be subjected to the disturbing force,
Figure FDA00030936986000000116
the resultant moment acting on the centroid;
formula (1036) is simplified to yield:
Figure FDA00030936986000000117
the vascular robot posture and track coupling kinetic equation of the spiral theory and the dual quaternion is as follows:
Figure FDA00030936986000000118
the coupling terms are a second term and a fourth term, so that the coupling influence of the gesture motion and the track motion is analyzed;
the second term of the dynamic posture and track coupling model of the vascular robot is as follows:
Figure FDA0003093698600000021
wherein the content of the first and second substances,
Figure FDA0003093698600000022
is a cross-product factor of the velocity vector,
Figure FDA0003093698600000023
and
Figure FDA0003093698600000024
respectively are cross multiplication factors of an angular velocity vector and a linear velocity vector of the vascular robot; the concrete expression is substituted into formula (1040) to obtain:
Figure FDA0003093698600000025
the fourth term of the kinetic model is simplified:
Figure FDA0003093698600000026
s2, designing a gravity-buoyancy compensation device;
gravity force of blood vessel robot
Figure FDA0003093698600000027
And buoyancy
Figure FDA0003093698600000028
The resultant force is expressed as:
Figure FDA0003093698600000029
the volume and magnetization of the robot, the magnetic force expression of the micro-robot is:
Figure FDA00030936986000000210
wherein, gx,gy,gzThree components of the magnetic flux gradient, respectively; in order to move the robot in the horizontal direction, the magnetic force components in the x-axis and the y-axis should satisfy the following equation:
Figure FDA00030936986000000211
wherein the values of the gradient of the magnetic induction along the x-axis and the y-axis should be equal, i.e. gx＝gyBy ghTo replace gx，gy. Magnetic force F in z-axis directionmzIs divided into two parts, wherein
Figure FDA00030936986000000212
For counteracting
Figure FDA00030936986000000213
Figure FDA00030936986000000216
Representing part of the vascular robot driving force, magnetic field gradient splitting at the z-axis
Figure FDA00030936986000000217
And
Figure FDA00030936986000000218
two parts;
the magnetic force and the magnetic field gradient along the z direction should satisfy the following relations:
Figure FDA00030936986000000214
Figure FDA00030936986000000215
s3, analyzing the relationship process of the pulsating flow field effect and the blood vessel wall motion and the blood flow velocity as follows:
resistance to the vascular robot
Figure FDA0003093698600000031
Including resistance to the spherical head of the vascular robot
Figure FDA0003093698600000032
And the resistance to which the helical tail is subjected
Figure FDA0003093698600000033
The specific expression is as follows:
Figure FDA0003093698600000034
where ρ isfExpressed as blood density, r represents the robot head radius,
Figure FDA0003093698600000035
indicating the relative speed of the robot with respect to the blood,
Figure FDA0003093698600000036
τ0is a dimensionless ratio related to local vessel occlusion;
will be provided with
Figure FDA0003093698600000037
Is substituted into
Figure FDA0003093698600000038
In (b), the following are obtained:
Figure FDA0003093698600000039
wherein the content of the first and second substances,
Figure FDA00030936986000000310
eta is blood viscosity;
Figure FDA00030936986000000311
Figure FDA00030936986000000312
by a parabolic function vs(x) And time-varying periodic blood pulsatile flow function vt(t) composition, time-varying blood velocity vt(t)＝ζ1Expressed as an N-order truncated Fourier series;
Figure FDA00030936986000000313
wherein, VrMean blood flow rate;
for the coupling effect between the vessel wall motion and the blood flow velocity, the following assumptions are made:
(1) the blood vessel is semi-infinitely long;
(2) the vessel wall is an isotropic Hooke elastomer;
(3) blood is a homogeneous newtonian viscous fluid and is incompressible and flows in an axisymmetric layer;
assuming that an original point is taken at the center of an inlet face, setting an x axis as an axial direction and an r axis as a longitudinal direction, and setting a cylindrical coordinate system x, r and theta;
establishing a mathematical model of blood flow:
Figure FDA00030936986000000314
Figure FDA0003093698600000041
Figure FDA0003093698600000042
wherein the boundary conditions are:
Figure FDA0003093698600000043
wherein, Vx＝Vx(r,x,t),Vr＝Vr(r,x,t),P＝P(x,t),υ0Is the characteristic velocity, P, of blood at the vascular opening0Is the mean pressure at the vascular inlet, ak,bk,gk,hkIs constant, ω is oscillation frequency;
suppose that
Figure FDA0003093698600000044
Substitution into formula (1051) gives:
Figure FDA0003093698600000045
obtaining the average blood inlet velocity Vr(ii) a Assumption (2) for the blood vessel wall that the deformation of the blood vessel wall following the blood is slight; the thickness of the vessel wall is h, the ratio of the thickness to the radius of the vessel is kept constant
Figure FDA0003093698600000046
Are small;
the vascular wall can receive axial, radial effort, its expression respectively is:
Figure FDA0003093698600000047
Figure FDA0003093698600000048
where H is the actual vessel wall thickness, ρωH, R and E are respectively the density, thickness, radius and elastic modulus of the vessel wall, and mu is the poisson ratio of the vessel;
Figure FDA00030936986000000412
ζ is the vessel axial and radial displacement, respectively. p, peRespectively the internal pressure and the external pressure of the vessel wall;
irrespective of the constraints of the surrounding tissue, H ═ H, ω00 and for the blood flow rate, the radial velocity of the blood is much greater than the axial velocity, i.e. the velocity
Figure FDA0003093698600000049
When the viscous term and the inertia term in the motion equation are ignored, the axial motion equation (1056) is:
Figure FDA00030936986000000410
radial equation (1057) the left inertial term of the equation is much less affected than the right and therefore can be ignored, and the radial equation is written as:
Figure FDA00030936986000000411
combining equations (1060) and (1061) yields the radial displacement equation:
Figure FDA0003093698600000051
at the vascular wall R-R junction, the coupling condition is expressed as:
Figure FDA0003093698600000052
s4, designing a robust adaptive controller according to the steps S1, S2 and S3; the robust adaptive control algorithm design process comprises the following steps:
modeling the robot kinematics and kinetic coupling established at S1 as:
Figure FDA0003093698600000053
wherein the content of the first and second substances,
Figure FDA0003093698600000054
in the form of a dual-moment matrix of inertia,
Figure FDA0003093698600000055
in the form of a nominal matrix, the matrix,
Figure FDA0003093698600000056
is an indeterminate portion;
firstly, set up
Figure FDA0003093698600000057
The attitude and orbit coupling kinetic equation is the attitude angle of the motion of the vascular robot, and can be:
Figure FDA0003093698600000058
wherein the content of the first and second substances,
Figure FDA0003093698600000059
Θddesired attitude angle for robot movement, eΘ＝＝Θ-ΘdIn order to be an attitude angle tracking error,
Figure FDA00030936986000000510
in order to control the total force for the system,
Figure FDA00030936986000000511
external disturbance force to the system; and is
Figure FDA00030936986000000512
The robust control law is designed as follows:
Figure FDA00030936986000000513
note the book
Figure FDA00030936986000000514
Combining the formulas (1065) and (1066) to obtain:
Figure FDA00030936986000000515
defining the auxiliary signal:
Figure FDA00030936986000000516
note the book
Figure FDA00030936986000000517
Substituting (1068) into (1067) yields:
Figure FDA00030936986000000518
designing a robust control item:
Figure FDA00030936986000000519
wherein the content of the first and second substances,
Figure FDA00030936986000000520
representing a robust term of the system for coping with external disturbances to which the vascular robot is bounded, i.e.
Figure FDA0003093698600000061
And β (β > 0); definition of
Figure FDA0003093698600000062
Wherein
Figure FDA0003093698600000063
For a known nominal moment of inertia, the same applies
Figure FDA0003093698600000064
Figure FDA0003093698600000065
In order to deal with the uncertain part of the moment of inertia, an adaptive operator is designed:
Figure FDA0003093698600000066
wherein a ═ a1,a2,a3]T；
Let I be the identity matrix, define
Figure FDA0003093698600000067
The adaptive robust control law is designed as follows:
Figure FDA0003093698600000068
wherein the content of the first and second substances,
Figure FDA0003093698600000069
is thetaΔMAn estimated value of, and
Figure FDA00030936986000000610
in the formula, gamma is a positive definite diagonal matrix.
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