Rolling contact type continuum mechanical arm with symmetrical structure and statics modeling method of rolling contact type continuum mechanical arm

Abstract

The invention relates to the technical field of robots, in particular to a rolling contact type continuum mechanical arm with a symmetrical structure and a statics modeling method thereof, the rolling contact type continuum mechanical arm comprises a plurality of rolling joints which are connected in sequence, and the rolling joints comprise one-degree-of-freedom joints and two-degree-of-freedom joints; the one-degree-of-freedom joints and the two-degree-of-freedom joints are alternately stacked, and every two adjacent rolling joints are connected in a rolling contact mode; the one-degree-of-freedom joint and the two-degree-of-freedom joint are each provided with a wire passing hole, and a driving cable is arranged in each wire passing hole in a penetrating mode. The continuum mechanical arm is of a symmetrical structure, the tail end of the continuum mechanical arm is provided with the most complete working space, and the problem that the working space at the tail end of an existing continuum mechanical arm has defects is solved; according to the static modeling method for the rolling contact type continuum mechanical arm, the problem that static analysis of rolling joints of different structures is inconsistent is solved, and the problem that calculation is complex due to the fact that integral terms exist in a traditional beam theoretical model is solved.

Description

Technical Field

[0001] The present invention relates to the field of robotics technology, and in particular to a rolling contact type continuum robotic arm with a symmetrical structure and a statics modeling method thereof. Background Art

[0002] In fields such as space exploration and medical surgery, the demand for robots to perform various tasks is increasing. However, traditional rigid robotic arms, due to their limited degrees of freedom, are unsuitable for operating in unstructured, complex environments. Continuum robotic arms, primarily inspired by continuous structures found in nature, such as elephant trunks, octopus tentacles, and snakes, offer high flexibility and good environmental adaptability. They have been increasingly applied in various fields in recent years. For example, in the medical field, continuum robotic arms, leveraging their superior compliance and flexibility, are used for endoscopic diagnosis of human cavities and therapeutic procedures in minimally invasive surgery. In industrial fields such as space exploration and nuclear reactor maintenance, continuum robotic arms can access confined spaces such as deep cavities or pipelines for monitoring, welding, cutting, and cleaning. Wire-driven continuum robotic arms, featuring continuous structures or super-redundant rigid joints, offer low mass, fast response time, and flexible wire layout. Furthermore, their drive motors are typically located at the base, making them easy to miniaturize and control, resulting in enhanced dexterity, safety, and maneuverability in confined environments. These advantages offer a broad range of applications in confined, unstructured, and complex environments.

[0003] Continuum manipulators, especially those based on rolling contact, are not continuously deformed as a whole due to their structure consisting of multiple rigid joints connected in series. This can lead to inconsistencies in the overall bending angle and attitude angle of the rolling contact continuum manipulator, resulting in defects in the workspace at the end of the continuum manipulator. In addition, in common continuum manipulator modeling methods, it is generally assumed that each part is bent into an arc, and the virtual work principle and Cosserat rod theory are generally used. However, rolling contact continuum manipulators are composed of multiple rigid joints, and the method of assuming an arc is not applicable to this type of manipulator. Analysis methods based on the Euler beam model are often used to establish the static equations of continuum manipulators. However, the equations established by the Euler beam model contain complex integral terms, making the equations more complex to solve. In addition, rolling contact continuum manipulators have rolling joints with different structures, and each joint is subjected to different forces. The above commonly used continuum manipulator modeling methods are no longer applicable. Summary of the Invention

[0004] The purpose of the present invention is to provide a rolling contact continuum manipulator with a symmetrical structure and a static modeling method thereof, which can solve the problems of inconsistent static analysis of rolling joints with different structures and defects in the end workspace.

[0005] Based on the above purpose, the present invention adopts the following technical solutions:

[0006] A rolling contact continuum robotic arm with a symmetrical structure includes multiple rolling joints connected in sequence, and the rolling joints include one-degree-of-freedom joints and two-degree-of-freedom joints; the one-degree-of-freedom joints and the two-degree-of-freedom joints are alternately stacked, and every two adjacent rolling joints are connected by rolling contact; the one-degree-of-freedom joints and the two-degree-of-freedom joints are both provided with wire holes, and drive cables are passed through the wire holes.

[0007] Preferably, the one-degree-of-freedom joint and the two-degree-of-freedom joint are both hollow structures, and four wire holes are symmetrically arranged along the center holes of the one-degree-of-freedom joint and the two-degree-of-freedom joint; the continuum robotic arm consists of two segments connected in sequence, and each segment is provided with 5 one-degree-of-freedom joints and 4 two-degree-of-freedom joints.

[0008] Preferably, the rolling contact angle 2α between every two adjacent rolling joints needs to satisfy:

[0009]

[0010] Where R represents the radius of the rolling contact surface, D represents the outer diameter of the rolling joint, and H represents the height of a single rolling joint.

[0011] A static modeling method for a continuum robotic arm comprises the following steps:

[0012] S1. Establishing a torque balance equation: Establishing a torque balance equation for each rolling joint to obtain the bending angle of each rolling joint when the continuum manipulator is subjected to a predetermined cable tension;

[0013] S2. Establishing a transformation matrix: Establishing a transformation matrix between each rolling joint, and obtaining the overall posture of the continuum manipulator based on the bending angle of each rolling joint and the transfer relationship between the transformation matrices of each rolling joint;

[0014] S3. Establish torque expressions: Based on the overall posture of the continuum manipulator, establish the cable tension and the torque expressions of the adjacent rolling joints on each rolling joint;

[0015] S4. Obtain the static equilibrium equation: Substitute the cable tension and the torque expression of the adjacent rolling joints on each rolling joint into the torque equilibrium equation of each rolling joint to obtain the static equilibrium equation of the entire continuum robotic arm.

[0016] Preferably, the specific process of establishing the moment balance equation in step S1 includes:

[0017] The moment balance equation for each rolling joint is:

[0018]

[0019] Where i represents the joint number, j represents the cable number, is the cable tension torque, represents the contact force torque, Indicates torque.

[0020] Preferably, the specific process of establishing the transformation matrix in step S2 includes:

[0021] The coordinate system is established in the geometric center of each joint, and the transformation matrix between the joints is established as follows:

[0022] 0 T n = 0 T1 1 T2… i-1 T i ... n-1 T n

[0023] Where n represents the number of joints, 0 T1 represents the transformation matrix from the base to the center of joint 1, 1 T2 represents the transformation matrix from joint 1 to joint 2, i-1 T i represents the transformation matrix from joint i-1 to joint i, n-1 T n Represents the transformation matrix from joint n-1 to joint n;

[0024] in, i-1 T i Expressed as:

[0025] i-1 T i =Rot(Z i-1 ±90°)Rot(Y i-1 ,θ i-1 )Trans(0,0,a)

[0026] in, "+" means clockwise rotation, "-" means counterclockwise rotation; θ i-1 is the bending angle of the i-th joint, and a is the distance the coordinate system translates along the Z axis.

[0027] Preferably, the specific process of establishing the torque expression in step S3 includes:

[0028] Establish the torque expression of the cable tension:

[0029]

[0030] Among them, F i,j is the tension of the jth cable in the i-th joint, r i,j is the displacement vector corresponding to the point of force application;

[0031] Establish the contact torque expression of the rolling joint:

[0032]

[0033] in, is the contact force on the i-th joint, is the displacement vector corresponding to the point of force application.

[0034] Preferably, the tension of the j-th driving cable at the top of the i-th rolling joint is:

[0035]

[0036] The displacement vector of the force point is:

[0037]

[0038] The tension of the j lowest driving cables at the bottom of the i-th rolling joint is:

[0039]

[0040] The displacement vector of the force point is:

[0041]

[0042] Where β1 is the angle of the joint top contact line relative to the Y axis, β2 is the angle of the joint bottom contact line relative to the Y axis; θ i is the bending angle of the i-th joint, is the phase angle of the jth cable of the i-th joint relative to the X-axis.

[0043] Preferably, the tension of the driving cable at the top of the i-th rolling joint is:

[0044]

[0045] The displacement vector expression is:

[0046]

[0047] The tension of the driving cable at the bottom of the i-th rolling joint is:

[0048]

[0049] The displacement vector expression is:

[0050]

[0051] in, It represents the component of the top cable tension on the X axis, It represents the component of the top cable tension on the Y axis, Indicates the component of the top cable tension on the Z axis; represents the component of the bottom cable tension on the X-axis, represents the component of the bottom cable tension on the Y axis, Indicates the component of the bottom cable tension on the Z axis.

[0052] Preferably, the specific process of obtaining the static equilibrium equation in step S4 includes:

[0053] Substitute the driving cable tension obtained in step S3 and the torque expression of the adjacent rolling joint on each rolling joint into the torque balance equation of each rolling joint obtained in step S2 to obtain the static equilibrium equation of the entire continuum manipulator:

[0054]

[0055] The beneficial effects of the present invention are:

[0056] The present invention adopts a rolling contact continuum robotic arm with a symmetrical stacked structure, in which the bending angle and the posture angle always remain consistent during the bending movement, and solves the problem of the end workspace defect of the existing rolling contact continuum robotic arm; and proposes a static modeling method for the rolling contact continuum robotic arm, which solves the problem of inconsistent static analysis of rolling joints with different structures in the robotic arm, and the static equilibrium equation finally obtained does not contain an integral term, which solves the problem of complex calculation caused by the presence of integral terms in the traditional beam theory model.

[0057] The present invention analyzes the symmetrical structure of a rolling contact continuum robotic arm and its end workspace. Based on the static equilibrium equation, a static equation is established that simultaneously considers the friction of the linear-driven continuum robotic arm. After experimental verification, the results calculated by this method have a small error compared with the experimental results, which effectively improves the analysis accuracy. BRIEF DESCRIPTION OF THE DRAWINGS

[0058] Figure 1 This is a schematic diagram of the overall structure of Example 1 of the present invention;

[0059] Figure 2 Schematic diagram of the structure of the rolling joint in Example 1 of the present invention;

[0060] Figure 3 Schematic diagram of the dimensions of the rolling joint in Example 1 of the present invention;

[0061] Figure 4 This is an analysis diagram of the bending angle and posture angle of the robotic arm in Example 1 of the present invention;

[0062] Figure 5 Schematic diagram of coordinate matrix transformation between rolling joints of a robotic arm in Example 2 of the present invention;

[0063] Figure 6 Schematic diagram of angles related to the rolling joint in Example 2 of the present invention;

[0064] Figure 7 Schematic diagram of force analysis of a single rolling joint in Example 2 of the present invention;

[0065] Figure 8 Schematic diagram of the friction model between the drive cable and the cable hole in Example 2 of the present invention;

[0066] Figure 9 The working space and maximum trajectory diagram of the end motion of the rolling contact continuum manipulator in Example 3 of the present invention;

[0067] Figure 10 Schematic diagram comparing the statics modeling method of the robotic arm in Example 3 of the present invention and the horizontal plane motion of the robotic arm obtained from actual experiments;

[0068] Figure 11 Schematic diagram comparing the statics modeling method of the robotic arm in Example 3 of the present invention and the vertical motion of the robotic arm obtained from actual experiments;

[0069] Figure 12 Schematic diagram comparing the statics modeling method of the robotic arm in Example 3 of the present invention and the spatial motion of the robotic arm obtained from actual experiments.

[0070] In the figure: one-degree-of-freedom joint 1; two-degree-of-freedom joint 2; drive cable 3; cable hole 4. DETAILED DESCRIPTION

[0071] Example 1

[0072] The following is a further explanation of the present invention with reference to specific embodiments. Figure 1 As shown, this embodiment is a rolling contact continuum robotic arm with a symmetrical structure, including two major segments (Seg1 and Seg2); each segment is composed of 9 rolling joints, and the rolling joints are divided into two types, namely one-degree-of-freedom joint 1 and two-degree-of-freedom joint 2.

[0073] like Figure 2As shown, the one-degree-of-freedom joint 1 and the two-degree-of-freedom joint 2 are both hollow structures, with a center hole at the center; four wire holes 4 are evenly distributed around the center hole; the axis of the wire holes 4 is set parallel to the axis of the center hole, and a drive cable 3 is passed through the wire holes 4. By applying tension through the drive cable 3, the joints in the robotic arm can be driven to roll and perform continuous bending deformation along the length direction, thereby driving the continuum robotic arm to perform overall bending movement.

[0074] like Figure 1 As shown, the two rolling joints in each segment are rollingly connected to each other in an interval arrangement and are stacked in series through drive cables 3 to form a robotic arm segment with a symmetrical structure; the two sets of rolling contact surfaces in the one-degree-of-freedom joint 1 are symmetrically arranged, while the two sets of rolling contact surfaces in the two-degree-of-freedom joint are arranged with a ninety-degree twist along the central axis of the center hole; in one robotic arm segment, there are a total of 5 one-degree-of-freedom joints 1 and 4 two-degree-of-freedom joints 2, among which the one-degree-of-freedom joint 1 at the center position is the symmetry axis of the segment; except for the one-degree-of-freedom joints 1 at both ends, the remaining internal one-degree-of-freedom joints 1 are rollingly connected on both sides with two two-degree-of-freedom joints 2, and the two two-degree-of-freedom joints 2 are symmetrically arranged; the end faces of the two one-degree-of-freedom joints 1 at both ends are set as planes, and the rolling joints in the two robotic arm segments are driven by two different sets of drive cables 3 respectively.

[0075] like Figure 3 As shown, in this embodiment, the rolling contact angle 2α needs to satisfy the following formula (1) to ensure that the minimum thickness of the side of the one-degree-of-freedom joint 1 is not 0:

[0076]

[0077] The outer diameter of the rolling joint is D=18 mm, the radius of the rolling contact surface is R=13 mm, the spacing of the wire holes 4 is d=10.75 mm, and the height of a single rolling joint is H=8 mm.

[0078] For a single-segment continuum robot, such as Figure 4 As shown in the figure, when the rolling joints of the continuum manipulator bend at the same angle θ, the total bending angle of the continuum manipulator is θ w , define its curvature radius as ρ, the center of the curvature radius as O0, the coordinates of the end joint center as (x, y); define the posture angle of the continuum manipulator as θ z The intersection of the extended lines of the proximal and distal planes of the continuum manipulator is O1, and their curvature radii are l1 and l2 respectively.

[0079] The bending angle θ of the robot arm can be obtained through the relevant geometric dimensions w and attitude angle θ z The expression is:

[0080] x 2 +(ρ-y) 2 =ρ 2 (2)

[0081]

[0082] θ w =4θ (4)

[0083] like Figure 4 As shown in (a), due to the symmetry of the structure of the continuum manipulator, as the rolling joint bends, the continuum manipulator is always symmetrical about the symmetry axis l, and the bisector of the bending angle is the symmetry axis of the manipulator, so the lengths of l1 and l2 are the same, which means that the intersection O1 is the center of the curvature radius of the manipulator O0; Substituting x and y into equations (2) and (3) we can get θ z =4θ, that is, θ z =θ w , indicating that the bending angle and the attitude angle always remain equal during the bending movement.

[0084] Example 2

[0085] This embodiment is a method for static modeling of the rolling contact continuum manipulator with a symmetrical structure in Example 1, comprising the following steps:

[0086] S1. Establishing a torque balance equation: Establishing a torque balance equation for each rolling joint to obtain the bending angle of each rolling joint when the continuum manipulator is subjected to a predetermined cable tension.

[0087] The moment balance equation for each rolling joint is:

[0088]

[0089] Where i represents the joint number, j represents the cable number, is the cable tension torque, represents the contact force torque, Indicates torque.

[0090] By solving equation (5), we can obtain the bending angles of each rolling joint when a constant tension is applied to the cable of the continuum robot arm. Then, based on these bending angles, the overall posture of the continuum robot arm can be obtained through the matrix transformation relationship between the joints.

[0091] S2. Establish a transformation matrix: Establish a transformation matrix between each rolling joint, and obtain the overall posture of the continuum robot arm based on the bending angle of each rolling joint and the transfer relationship between the transformation matrices of each rolling joint.

[0092] like Figure 5 As shown, in order to better describe the continuum manipulator, the coordinate matrix transformation relationship between the rolling joints of the continuum manipulator is established. The coordinate system is established at the internal geometric center point of the joint. The X axis is parallel to the contact line at the bottom of the joint, the Y axis is parallel to the contact line at the top of the joint, and the Z axis is parallel to the center of the joint and points from the bottom to the top. Then, the transformation matrix between each joint is established as follows:

[0093] 0 T n = 0 T1 1 T2… i-1 T i … n-1 T n (6)

[0094] Where n represents the number of joints, 0 T1 represents the transformation matrix from the base to the center of joint 1, 1 T2 represents the transformation matrix from joint 1 to joint 2, i-1 T i represents the transformation matrix from joint i-1 to joint i, n-1 T n Represents the transformation matrix from joint n-1 to joint n.

[0095] in, i-1 T i Expressed as:

[0096] i-1 T i =Rot(Z i-1 ±90°)Rot(Y i-1 ,θ i-1 )Trans(0,0,a) (7)

[0097] in, "+" means clockwise rotation, "-" means counterclockwise rotation; θ i-1 is the bending angle of the i-th joint, and a is the distance the coordinate system translates along the Z axis.

[0098] The kinematic relationship of the robotic arm described above clearly defines the positional relationship of each action point. The points of action of the cable tension are located at the top and bottom of the hole, and the point of action of the contact force is equivalent to the center of the contact line. On this basis, when analyzing the effect of cable tension on the robotic arm, it is necessary to simultaneously consider the influence of friction between the cable and the hole.

[0099] S3. Establish torque expressions: Based on the overall posture of the continuum robot arm, establish the cable tension and the torque expressions of the adjacent rolling joints on each rolling joint.

[0100] First, establish the torque expression of the cable tension:

[0101]

[0102] Among them, F i,j is the tension of the jth cable in the i-th joint, r i,j is the displacement vector corresponding to the point of force application.

[0103] Then, the contact torque expression of the rolling joint is established:

[0104]

[0105] in, is the contact force on the i-th joint, is the displacement vector corresponding to the point of force application.

[0106] like Figure 6 As shown in the figure, before analyzing the force on the rolling joint, the relevant angles involved are first defined. Since the structures of the one-degree-of-freedom rolling joint and the two-degree-of-freedom rolling joint are different, the expressions of the joint forces are different. In order to conveniently unify the relevant expressions of the two rolling joints, a two-degree-of-freedom rolling joint is taken as an example, where β1 is defined as the angle of the joint top contact line relative to the Y axis, β1 is 0° or 90°; β2 is the angle of the joint bottom contact line relative to the Y axis, β2 is 0° or 90°, and the phase angle of the j-th cable of the i-th joint relative to the X axis is

[0107] like Figure 7 As shown, the overall force analysis of a single rolling joint is performed, and the top of the joint is subjected to contact force and cable tension The bottom of the joint will also be subject to contact forces and cable tension

[0108] The expression of the tension of the j-th cable at the top of the i-th joint is:

[0109]

[0110] The expression of the tension of the jth cable at the bottom of the i-th joint is:

[0111]

[0112] Among them, the expression of the contact force on the top of the i-th joint is:

[0113]

[0114] Among them, the expression of the contact force at the bottom of the i-th joint is:

[0115]

[0116] in, It represents the component of the top cable tension on the X axis, It represents the component of the top cable tension on the Y axis, Indicates the component of the top cable tension on the Z axis; represents the component of the bottom cable tension on the X-axis, represents the component of the bottom cable tension on the Y axis, Indicates the component of the bottom cable tension on the Z axis.

[0117] The position of each force application point can be calculated from the relevant geometric dimensions of the joint:

[0118] The expression for the position of the tension point of the jth cable at the top of the i-th joint is:

[0119]

[0120] Among them, the expression for the position of the action point of the tension of the jth cable at the bottom of the i-th joint is:

[0121]

[0122] Among them, the expression of the point of action of the contact force at the top of the i-th joint is:

[0123]

[0124] Among them, the expression of the contact force acting point at the bottom of the i-th joint is:

[0125]

[0126] like Figure 8 As shown in the figure, due to the friction between the drive cable 3 and the cable hole 4, the cable tension at the top of the joint is not equal to the cable tension at the bottom. Therefore, the friction model expression for the cable passing through the hole is established as follows:

[0127]

[0128] in, μ represents the friction coefficient.

[0129] S4. Obtain the static equilibrium equation: Substitute the cable tension and the torque expression of the adjacent rolling joints on each rolling joint into the torque equilibrium equation of each rolling joint to obtain the static equilibrium equation of the entire continuum robotic arm.

[0130] Substituting the expression of the action force and the expression of the displacement vector into equation (5) can obtain the total force balance and torque balance equations of the continuum manipulator:

[0131]

[0132] The contact torque vector acting on the bottom of the joint can be obtained from the contact torque vector at the top of the previous joint, and the relationship expression is:

[0133]

[0134] The static model of the continuum manipulator can be realized by using Equations (19) and (20). The static equilibrium equation does not contain integral or differential terms, so it is easy to solve. In addition, by numerically solving Equations (19) and (20), the bending angles of each joint of the manipulator under a given cable tension can be obtained. Then, the overall posture of the manipulator can be calculated through the transformation matrix between the joints, i.e., Equation (6).

[0135] Example 3

[0136] This embodiment uses actual experimental data to test the accuracy of the static modeling method of the rolling contact continuum manipulator involved in Example 2; Figure 9 As shown, first, the working space and the maximum motion trajectory range of the continuum robot arm in Example 1 are analyzed in the Matlab environment. It can be seen that the working space of the rolling contact continuum robot arm is symmetrical, and the maximum motion trajectory of the end is circular and very complete. Therefore, there is no defect in the working space of the rolling contact continuum robot arm involved in Example 1.

[0137] like Figure 10-12 As shown, in order to verify the effectiveness of the static modeling method of the rolling contact continuum manipulator proposed in Example 2 of the present invention, three experiments were carried out respectively. Under the condition of a constant tension in the given driving cable 3, the theoretical posture of the manipulator obtained by the static model and the posture obtained by the actual experiment were compared and analyzed; the three experiments respectively included the movement of the continuum manipulator in two directions in the plane, as well as the experiment of spatial movement. After experimental comparison, it can be obtained that the error between the terminal result calculated by the static modeling method of Example 2 of the present invention and the terminal result obtained by the actual experimental measurement is about 4 mm, which proves the accuracy of the static modeling method of the continuum manipulator proposed in the present invention.

[0138] To sum up, the rolling contact continuum robotic arm involved in the present invention has a symmetrical structure, the bending angle and posture angle of the continuum robotic arm always remain consistent during the bending movement, and its end also has a complete working space; the static equilibrium equation of the continuum robotic arm established by the present invention can effectively solve the static problems of the rolling contact continuum robotic arm, and the result obtained is a set of equations about trigonometric functions, which does not contain integral terms, so its calculation is simple; for the complex calculation of multiple combinations of continuum robotic arms, the present invention can uniformly express the forces of the two rolling joints, has more efficient processing capabilities, and can maintain accuracy to meet actual use requirements.

[0139] The above description is merely a further explanation of the present invention in conjunction with specific embodiments. All descriptions do not limit the scope of protection of the present invention. Any changes or replacements that can be easily thought of by any technician in this field within the technical scope disclosed by the present invention should be included in the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be based on the scope of protection of the claims.

Claims

1. A rolling contact continuum robot arm with a symmetrical structure, characterized in that: It comprises a plurality of rolling joints connected in sequence, wherein the rolling joints comprise a one-degree-of-freedom joint and a two-degree-of-freedom joint; the one-degree-of-freedom joints and the two-degree-of-freedom joints are alternately stacked, and every two adjacent rolling joints are connected by rolling contact; the one-degree-of-freedom joints and the two-degree-of-freedom joints are both provided with wire holes, and drive cables are passed through the wire holes.

2. The rolling contact continuum robot arm with a symmetrical structure according to claim 1, characterized in that: The one-degree-of-freedom joint and the two-degree-of-freedom joint are both hollow structures, and four wire holes are symmetrically arranged along the center holes of the one-degree-of-freedom joint and the two-degree-of-freedom joint; the continuum robotic arm consists of two segments connected in sequence, and each segment is provided with 5 one-degree-of-freedom joints and 4 two-degree-of-freedom joints.

3. The rolling contact continuum robot arm with a symmetrical structure according to claim 2, characterized in that: The rolling contact angle 2α of each two adjacent rolling joints needs to satisfy: Where R represents the radius of the rolling contact surface, D represents the outer diameter of the rolling joint, and H represents the height of a single rolling joint.

4. The static modeling method of a continuum manipulator according to claims 1 to 3, comprising the following steps: S1. Establishing a torque balance equation: establishing a torque balance equation for each rolling joint to obtain a bending angle of each rolling joint when the continuum manipulator is subjected to a predetermined cable tension; S2. Establishing a transformation matrix: establishing a transformation matrix between each of the rolling joints, and obtaining the overall posture of the continuum manipulator according to the bending angle of each rolling joint and the transfer relationship between the transformation matrices of each rolling joint; S3. Establishing a torque expression: establishing a torque expression for the cable tension and the action of the adjacent rolling joints on each rolling joint according to the overall posture of the continuum manipulator; S4. Obtaining the static equilibrium equation: Substitute the cable tension and the torque expression of the adjacent rolling joints on each rolling joint into the torque equilibrium equation of each rolling joint to obtain the static equilibrium equation of the entire continuum robotic arm.

5. The static modeling method of a continuum manipulator according to claim 4, characterized in that: The specific process of establishing the moment balance equation in step S1 includes: The moment balance equation of each rolling joint is: Where i represents the joint number, j represents the cable number, is the cable tension torque, represents the contact force torque, Indicates torque.

6. The static modeling method of a continuum manipulator according to claim 5, characterized in that: The specific process of establishing the transformation matrix in step S2 includes: The coordinate system is established in the geometric center of each joint, and the transformation matrix between the joints is established as follows: 0 T n = 0 T1 1 T2… i-1 T i… n-1 T n Where n represents the number of joints, 0 T1 represents the transformation matrix from the base to the center of joint 1, 1 T2 represents the transformation matrix from joint 1 to joint 2, i-1 T i represents the transformation matrix from joint i-1 to joint i, n-1 T n Represents the transformation matrix from joint n-1 to joint n; in, i-1 T i Expressed as: i-1 T i =Rot(Z i-1 ±90°)Rot(Y i-1 ,θ i-1 )Trans(0,0,a) in, "+" means clockwise rotation, "-" means counterclockwise rotation; θ i-1 is the bending angle of the i-th joint, and a is the distance the coordinate system translates along the Z axis.

7. The static modeling method of a continuum manipulator according to claim 6, characterized in that: The specific process of establishing the torque expression in step S3 includes: Establish the torque expression of the cable tension: Among them, F i,j is the tension of the jth cable in the i-th joint, r i,j is the displacement vector corresponding to the point of force application; Establish the contact torque expression of the rolling joint: in, is the contact force on the i-th joint, is the displacement vector corresponding to the point of force application.

8. The static modeling method of a continuum manipulator according to claim 7, characterized in that: The tension of the j-th driving cable at the top of the i-th rolling joint is: The displacement vector of the force point is: The tension of the j lowest driving cables at the bottom of the i-th rolling joint is: The displacement vector of the force point is: Where β1 is the angle of the contact line at the top of the joint relative to the Y axis, and β2 is the angle of the contact line at the bottom of the joint relative to the Y axis; θ i is the bending angle of the i-th joint, is the phase angle of the jth cable of the i-th joint relative to the X-axis.

9. The static modeling method of a continuum manipulator according to claim 8, characterized in that: The tension of the driving cable at the top of the i-th rolling joint is: The displacement vector expression is: The tension of the driving cable at the bottom of the i-th rolling joint is: The displacement vector expression is: in, It represents the component of the top cable tension on the X axis, It represents the component of the top cable tension on the Y axis, Indicates the component of the top cable tension on the Z axis; represents the component of the bottom cable tension on the X-axis, represents the component of the bottom cable tension on the Y axis, Indicates the component of the bottom cable tension on the Z axis.

10. The static modeling method of a continuum manipulator according to claim 9, characterized in that: The specific process of obtaining the static equilibrium equation in step S4 includes: Substitute the driving cable tension obtained in step S3 and the torque expression of the adjacent rolling joint on each rolling joint into the torque balance equation of each rolling joint obtained in step S2 to obtain the static equilibrium equation of the entire continuum manipulator:

Images