A dental implant robot path generation method

Abstract

This invention provides a method for generating a path for a dental implant robot, specifically including the following steps: constructing a spatial pose vector of the tooth; combining the spatial pose features of the contralateral corresponding tooth and the upper and lower occlusal teeth to construct a regression model predicting the spatial features of the target missing tooth, predicting the spatial position and orientation angle of the target missing tooth; performing TCP calibration on the dental implant robot; performing TCF calibration on the dental implant robot; constructing the spatial transformation relationship between the robot arm's end effector and the external visual positioning system; using point cloud registration technology to construct the spatial transformation matrix between the digital model and the physical printed model; constructing the initial obstacle avoidance path of the robot arm based on the fast extended random tree sampling algorithm, while combining the improved algorithm RRT*, and smoothing and controlling the joint trajectory and speed through polynomial interpolation and B-spline curve interpolation. The technical solution of this invention overcomes the problems of large implantation errors and low fit in the prior art.

Description

Technical Field

[0001] This invention relates to the field of path generation technology, and more specifically to a method for generating paths for a dental implant robot. Background Technology

[0002] Because dental implants function similarly to natural teeth, offer significant post-treatment results with no adverse reactions, they are the optimal treatment option for conditions such as missing teeth and edentulous teeth currently seen in clinical practice. However, the precision and effectiveness of manual implantation are highly dependent on the dentist's experience and the operating environment, such as visual obstruction that may cause implantation errors or differences in treatment outcomes due to the patient's unique physiological structure.

[0003] Therefore, there is a need for a method for generating the path of a dental implant robot with small implantation error and high fit. Summary of the Invention

[0004] The main objective of this invention is to provide a method for generating a path for a dental implant robot, in order to solve the problems of large implantation errors and poor fit in the prior art.

[0005] To achieve the above objectives, the present invention provides a method for generating a path for a dental implant robot, specifically including the following steps:

[0006] S1, based on CBCT images, extracts the central axis of the tooth space, expresses the spatial relationship between the central axis of the tooth and the standard Cartesian coordinate system, and constructs the tooth space pose vector.

[0007] S2, combining the spatial pose features of the contralateral tooth and the upper and lower occlusal teeth, constructs a regression model to predict the spatial features of the target missing tooth, and predicts the spatial position and pose angle of the target missing tooth.

[0008] S3, TCP calibration of the dental implant robot.

[0009] S4, perform TCF calibration on the dental implant robot.

[0010] S5, construct the spatial transformation relationship between the robot arm end effector and the external vision positioning system.

[0011] S6 uses point cloud registration technology to construct the spatial transformation matrix between the digital model and the physical printed model.

[0012] S7 constructs the initial obstacle avoidance path of the robotic arm based on the fast extended random tree sampling algorithm, and combines the improved algorithms RRT-Connect and RRT*, and smooths the joint trajectory and controls the speed through polynomial interpolation and B-spline curve interpolation.

[0013] Furthermore, step S1 specifically includes the following steps:

[0014] S1.1 Import CBCT scan data into a medical image processing platform for three-dimensional reconstruction to obtain a tooth voxel model.

[0015] S1.2, Select three layers in the tooth voxel model, corresponding to the upper, middle, and apical regions of the tooth root, respectively, denoted as layers 1, 2, and 3. On each layer, obtain the set of boundary points of the tooth's outer contour. Where k = 1, 2, 3 represents the layer number, and n k This represents the number of boundary points extracted in this layer, and the two-dimensional geometric center C of each layer. (k) The calculation is as follows:

[0016]

[0017] The two-dimensional geometric center of each layer is determined according to the corresponding layer height z. k Upgrading to three-dimensional coordinates yields three sets of spatial points P1, P2, and P3:

[0018] P1=(x1,y1,z1), P2=(x2,y2,z2), P3=(x3,y3,z3).

[0019] S1.3, using the least squares method, P1, P2, and P3 are fitted to the tooth midline, and the direction vector of the tooth midline is... Represented as:

[0020]

[0021] Where (a,b,c) is a unit vector, representing the components of the tooth's central axis in the X, Y, and Z directions.

[0022] S1.4, the spatial relationship between the tooth's central axis and the standard Cartesian coordinate system is expressed as follows:

[0023]

[0024] Where α, β, and γ represent the direction cosine angles of the X-axis, Y-axis, and Z-axis, respectively.

[0025] S1.5, the tooth spatial pose vector is represented as:

[0026] ξ=[x,y,z,α,β,γ] T ;

[0027] Where (x,y,z) are the three-dimensional spatial coordinates of the tooth.

[0028] Furthermore, step S2 specifically includes the following steps:

[0029] S2.1, Let the spatial pose of the target missing tooth be ξ. tar :

[0030] ξ tar =[x tar ,y tar ,z tar ,α tar ,β tar ,γ tar ] T .

[0031] S2.2, Define the spatial pose vector ξ of the contralateral tooth with the same name. contra ∈R 6 Spatial pose vector ξ of the upper and lower occlusal teeth occlu ∈R 6 S2.3, Construct the spatial features ξ of the predicted target missing tooth. tar Regression model:

[0032] ξ tar =W1·ξ contra +W2·ξ occlu +b+ε;

[0033] Where W1, W2 ∈ R 6×6 , represent the coefficient matrices of the predictive contributions of the contralateral teeth and occlusal teeth to the target tooth, respectively; b∈R 6 For the intercept term; ε∈R 6 This is the residual term.

[0034] The regression model expands to the following form:

[0035]

[0036] Where, ξ tar,m This represents the m-th feature component of the target missing tooth location.

[0037] Furthermore, step S3 specifically includes the following steps:

[0038] S3.1, Select a fixed spatial reference point. The robotic arm of the dental implant robot holds the tool through the clamping mechanism and repeatedly positions the tool tip at the spatial reference point.

[0039] S3.2, control the end flange of the robotic arm to contact a spatial reference point in at least four significantly different postures.

[0040] S3.3 Record the corresponding coordinates of the end flange of the robotic arm.

[0041] S3.4, construct the calibration equation system and fit it using the least squares method to obtain the TCP position of the tool center point.

[0042]

[0043] in, and These represent the rotation matrices of the end flange coordinate system relative to the dental implant robot's base coordinate system under four different postures of the robotic arm.

[0044] Furthermore, step S4 specifically includes the following steps:

[0045] S4.1, mark the center point of the end flange under the initial pose of the robotic arm movement as point o; drag the robotic arm from point o along the X-axis by a distance Δx; drag the robotic arm from point o along the Z-axis by a distance Δz; establish the following coordinate relationship:

[0046]

[0047] in, The coordinates of the center point of the end flange in the base coordinate system after moving a distance Δx; The coordinates of the center point of the end flange in the base coordinate system under the initial pose; The coordinates of the tool's end point in the base coordinate system after moving a distance Δx; The coordinates of the tool end point in the base coordinate system under the initial pose; This represents the coordinates of the center point of the end flange in the base coordinate system after moving a distance Δz. This represents the coordinates of the tool's end point in the base coordinate system after moving a distance Δz.

[0048] S4.2, construct the rotation matrix of the tool coordinate system based on the three calibration points, and obtain the orthogonal basis through vector difference and cross product operations:

[0049]

[0050] o = a × n;

[0051] in, This is the inverse of the rotation matrix of the end flange center point in the base coordinate system under the initial pose. Let be the rotation matrix of the center point of the end flange in the base coordinate system. The coordinates of the tool end point in the end flange coordinate system under the initial pose; n is the rotation matrix relative to the x-axis, a is the rotation matrix relative to the z-axis, and o is the rotation matrix relative to the y-axis.

[0052] Furthermore, step S5 specifically includes the following steps:

[0053] S5.1, Since the marker C is fixed to the end flange E, then at time t and at time t+1 we have:

[0054]

[0055] in, This is the rotation matrix of the end flange coordinate system relative to the end positioning marker coordinate system.

[0056] S5.2, the transformation matrix between the coordinate system of the end flange and the coordinate system of the end positioning marker is given by the homogeneous transformation relationship:

[0057]

[0058] in, Let be the transformation matrix between the robot base coordinate system and the end flange coordinate system. Let be the transformation matrix of the optical positioning and tracking device coordinate system relative to the robot base coordinate system. It is the inverse of the transformation matrix of the optical positioning and tracking instrument coordinate system with respect to the end position marker.

[0059] Further conclusions can be drawn:

[0060]

[0061] After equivalent transformation, the basic expression for hand-eye calibration is obtained as AX = XB:

[0062]

[0063] Where A and B are the transformation matrices of the robotic arm system and the vision system between two adjacent poses, respectively, and X is the fixed transformation matrix to be determined; R A R X and R B Let b be a rotation submatrix of the matrix. A b X and b B These are the translation submatrices of the matrix.

[0064] S5.3, after decomposing the homogeneous matrix into rotation and translation components, it is further decomposed into:

[0065]

[0066] Among them, t X It is a translation vector.

[0067] S5.4, the Tasi-Lenz method is used to model the rotation matrix using axis-angle representation, based on R. A R X =R X R B ,have to right Taking the logarithm, we have:

[0068]

[0069] Using the similarity transformation property of matrix logarithms:

[0070]

[0071] Since a three-dimensional rotation matrix is ​​obtained by performing matrix exponentiation operations on the antisymmetric matrix form of the rotation vector:

[0072]

[0073] Among them, [α] × and [β] × It is an antisymmetric matrix generated by rotation vectors α and β.

[0074] S5.5, this problem is a classic absolute orientation problem, solved using Singular Value Decomposition (SVD) to find R. X :

[0075] S5.6, After solving the rotation part, the translation vector t is solved using the following formula. X :

[0076] (R Ai -I)t X =R X t Bi -t Ai .

[0077] S5.7 introduces the LM algorithm for nonlinear optimization. The iterative increment of the LM algorithm satisfies:

[0078] (J T J+λI)Δp=-J T r;

[0079] in, ρ is the parameter vector, r is the rotation part of the rotation vector parameterization, and r is the residual vector composed of rotation and translation residuals; J is the Jacobian matrix of residuals with respect to parameters; I is the identity matrix; Δp is the parameter update amount.

[0080] S5.8, the objective function E(p) is defined as the sum of squared rotation and translation errors:

[0081]

[0082] Where R(ρ)=exp([ρ] × ) represents the rotation matrix corresponding to the rotation vector.

[0083] vee(·) converts an antisymmetric matrix into a three-dimensional vector; They are The corresponding rotation matrix; They are The corresponding translation vector.

[0084] S5.9, through iterative solution of equation (J) T J+λI)Δp=-J T r, update parameter p until the objective function E(p) converges to the preset threshold, and obtain the optimal estimate of the rotation matrix and displacement vector.

[0085] Furthermore, step S6 specifically includes the following steps:

[0086] S6.1, downsampling the original point cloud, the calculation expression is:

[0087]

[0088] in, p is the center point within the voxel. i is the original point within the voxel, and n is the number of voxel points;

[0089] S6.2 uses the FPFH algorithm to construct a feature histogram by statistically analyzing the relative angular relationships between the point normal vectors.

[0090] S6.3, based on the initial transformation matrix obtained from coarse registration, the ICP algorithm is used for fine registration. By iteratively minimizing the Euclidean distance between two sets of corresponding points, accurate alignment of the point clouds is achieved. The objective function of ICP is:

[0091]

[0092] Where, p i Let q be the i-th point in the source set. i Let R be the point corresponding to the target point set, t be the rotation matrix, t be the translation vector, and min be the minimum value.

[0093] Furthermore, step S7 specifically includes the following steps:

[0094] S7.1 uses axis-aligned bounding boxes (AABBs) for structured modeling of objects. AABBs enclose the object Ω within a minimal cuboid parallel to the coordinate axes, defined as:

[0095] AABB(Ω)=[x min ,x max ]×[y min ,y max ]×[z min ,z max ];

[0096] Where, x min With x max Let x and y represent the minimum and maximum boundaries of object Ω on the x-axis, respectively.min ,y max Let z represent the minimum and maximum boundaries of object Ω on the y-axis, respectively. min ,z max These represent the minimum and maximum boundaries of object Ω on the z-axis, respectively.

[0097] Collision determination is based on whether the AABB coordinates of two objects A and B overlap in all three axes.

[0098]

[0099] S7.2, Path search is based on collision detection results, and the RRT algorithm is first used; RRT randomly samples points q in the continuous configuration space C. rand And find the node q that is closest to the random sampling point in the existing path tree. near :

[0100]

[0101] Where, ‖·‖ represents the Euclidean distance, Q is the set of all nodes in the current tree, and arg min is the position of the minimum value.

[0102] Subsequently, a new node q is generated using the step-size constraint function Steer. new :

[0103]

[0104] Where η is the maximum step size.

[0105] S7.3 introduces the RRT-Connect algorithm, simultaneously starting from the starting point q. start and the endpoint q goal Each grows a random tree and The two trees are expanded alternately and attempts to connect; the connection condition is the number of nodes in the two trees. and Distance not exceeding threshold ∈ conn :

[0106]

[0107] S7.4 introduces the RRT* algorithm, which is used to expand new nodes q. new Then, the path is optimized through a local rewiring mechanism.

[0108] S7.5 uses fifth-order polynomial interpolation and B-spline curve interpolation to smooth the path points and transform them into continuous, dynamically controllable joint angle trajectories.

[0109] Furthermore, step S7.4 specifically includes the following steps:

[0110] S7.4.1, at radius r n Search for the set of neighboring nodes Q near :

[0111] Q near ={q∈Q∣‖qq new ||≤r n};

[0112] Where, r n The number of samples will be dynamically adjusted.

[0113] S7.4.2, Select the nearest neighbor node q that minimizes the total cost. min ∈Q near As q new Parent node:

[0114]

[0115] Where c(q) start Cost(q, q) represents the path cost from the starting point to node q. new ) represents from q to q new The cost is usually expressed as path length.

[0116] S7.4.3, perform rewiring operation on neighboring nodes, if via q new If a lower path cost can be obtained, then update the parent node and the path cost.

[0117] The present invention has the following beneficial effects:

[0118] (1) By combining statistical regression and geometric fitting to construct a personalized initial path, the systematic error caused by unreasonable direction setting is reduced;

[0119] (2) Using sampling algorithms such as RRT-Connect / RRT*, obstacle avoidance is completed in Cartesian space to ensure the safety of the path in an anatomical sense;

[0120] (3) Apply seventh-order polynomial interpolation in joint space to optimize trajectory smoothing, suppress end jitter caused by sudden changes in velocity and acceleration, and transform the "feasible path" into a "highly executable path".

[0121] Therefore, this invention achieves comprehensive optimization of existing methods in terms of path rationality, execution stability, and result consistency. Attached Figure Description

[0122] To more clearly illustrate the specific embodiments of the present invention or the technical solutions in the prior art, the drawings used in the description of the specific embodiments or the prior art will be briefly introduced below. Obviously, the drawings described below are some embodiments of the present invention. For those skilled in the art, other drawings can be obtained based on these drawings without creative effort. In the drawings:

[0123] Figure 1 A flowchart of a method for generating a path for a dental implant robot according to the present invention is shown.

[0124] Figure 2 A spatial mapping diagram is shown.

[0125] Figure 3 A diagram showing the tooth distribution in this embodiment is provided.

[0126] Figure 4 The diagram shows the rotation angle changes of each axis of the upper teeth.

[0127] Figure 5 The diagram shows the rotation angle changes of each axis of the lower teeth.

[0128] Figure 6 The graph shows the rotation angle data for each axis after removing the mean.

[0129] Figure 7 The diagram shows the two-dimensional obstacle avoidance and path planning of the RRT algorithm.

[0130] Figure 8 The diagram shows the 3D obstacle avoidance and path planning of the RRT algorithm.

[0131] Figure 9 The initial obstacle avoidance path diagram is shown. Detailed Implementation

[0132] The technical solution of the present invention will now be clearly and completely described with reference to the accompanying drawings. Obviously, the described embodiments are only some, not all, of the embodiments of the present invention. Based on the embodiments of the present invention, all other embodiments obtained by those skilled in the art without creative effort are within the scope of protection of the present invention.

[0133] like Figure 1 The method for generating a path for a dental implant robot, as shown, specifically includes the following steps:

[0134] S1, extract the tooth space midline based on CBCT images, express the spatial relationship between the tooth space midline and the standard Cartesian coordinate system, and construct the tooth space pose vector;

[0135] S2, combining the spatial pose features of the contralateral tooth and the upper and lower occlusal teeth, constructs a regression model to predict the spatial features of the target missing tooth, and predicts the spatial position and pose angle of the target missing tooth.

[0136] S3, TCP calibration of the dental implant robot;

[0137] S4, TCF calibration of the dental implant robot;

[0138] S5, Construct the spatial transformation relationship between the robot arm end effector and the external vision positioning system;

[0139] S6 uses point cloud registration technology to construct the spatial transformation matrix between the digital model and the physical printed model;

[0140] S7 constructs the initial obstacle avoidance path of the robotic arm based on the fast extended random tree sampling algorithm. It also combines the improved algorithms RRT-Connect and RRT*, and smooths the joint trajectory and regulates the speed through polynomial interpolation and B-spline curve interpolation. The aim is to achieve safe, high-precision and continuous motion control of the robotic arm of the dental implantation robot in complex spaces.

[0141] Specifically, step S1 includes the following steps:

[0142] S1.1 Import CBCT scan data into a medical image processing platform for 3D reconstruction to obtain a tooth voxel model. Import the CBCT scan data into medical image processing platforms such as Mimics and 3D Slicer in DICOM format. Based on the difference in grayscale (HU) values ​​between the tooth and bone tissue, multiple threshold intervals are set for segmentation. Given the drastic density changes in the root apex and crown-root junction, a single threshold is insufficient to effectively distinguish boundaries; therefore, a multi-layer cross-sectional segmentation strategy is adopted to semi-automatically identify boundaries and extract point sets for each tooth layer. S1.2 Select three layers in the tooth voxel model, corresponding to the upper segment (near the crown), middle segment, and root apex region, denoted as layers 1, 2, and 3. On each layer, obtain the point set of the tooth's outer contour boundary. Where k = 1, 2, 3 represents the layer number, and n k This represents the number of boundary points extracted in this layer, and the two-dimensional geometric center C of each layer. (k) The centroid is calculated as follows:

[0143]

[0144] The two-dimensional geometric center of each layer is determined according to the corresponding layer height z. k Upgrading to three-dimensional coordinates yields three sets of spatial points P1, P2, and P3:

[0145] P1=(x1,y1,z1), P2=(x2,y2,z2), P3=(x3,y3,z3);

[0146] S1.3, using the least squares method, P1, P2, and P3 are fitted to the tooth midline, and the direction vector of the tooth midline is... Represented as:

[0147]

[0148] Where (a,b,c) is a unit vector, representing the components of the tooth's central axis in the X, Y, and Z directions;

[0149] S1.4, the spatial relationship between the tooth's central axis and the standard Cartesian coordinate system is expressed as follows:

[0150]

[0151] Where α, β, and γ represent the direction cosine angles of the X-axis, Y-axis, and Z-axis, respectively;

[0152] S1.5, the three angle parameters (α, β, γ) mentioned above together constitute a complete pose vector, used to describe the orientation characteristics of the tooth in Cartesian space. This vector, combined with the tooth's position coordinates (x, y, z) in the CBCT image, constructs a spatial pose vector. The tooth's spatial pose vector is expressed as:

[0153] ξ=[x,y,z,α,β,γ] T ;

[0154] Where (x,y,z) are the three-dimensional spatial coordinates of the tooth.

[0155] These six-DOF pose parameters not only form the basic input for subsequent implant path generation and guide direction control, but also serve as key indicators for target pose setting in robot trajectory planning. To facilitate regression modeling and parameter training, this invention standardizes all tooth pose parameters, unifying numerical scales and effectively eliminating interference from individual differences. Furthermore, statistical analysis of Euler angle parameters for different tooth positions reveals the spatial distribution regularity of jawbone anatomy, providing solid data support for an anatomically driven adaptive implant path generation method. Given the high correlation between left-right symmetry and maxillary-mandibular occlusal coupling in human oral structures, this invention proposes a multiple linear regression model combining the spatial pose features of the contralateral corresponding tooth and the upper and lower occlusal teeth to predict the spatial position and pose angle of missing teeth, thereby achieving preliminary deduction and automated generation of personalized implant paths.

[0156] Specifically, step S2 includes the following steps:

[0157] Variable definition and eigenvector construction:

[0158] S2.1, Let the spatial pose of the target missing tooth be ξ. tar :

[0159] ξ tar =[x tar ,y tar ,z tar ,α tar ,β tar ,γ tar ] T ;

[0160] S2.2, Define the spatial pose vector ξ of the contralateral tooth with the same name. contra ∈R 6 Spatial pose vector ξ of the upper and lower occlusal teeth occlu ∈R 6 ;

[0161] Regression model construction:

[0162] S2.3, Construct the spatial features ξ of the predicted target missing tooth. tar Regression model:

[0163] ξ tar =W1·ξ contra +W2·ξ occlu +b+ε;

[0164] Where W1, W2 ∈ R 6×6 , represent the coefficient matrices of the predictive contributions of the contralateral teeth and occlusal teeth to the target tooth, respectively; b∈R 6 For the intercept term; ε∈R 6 For the residual term; assume it follows a zero-mean normal distribution.

[0165] The regression model can be expanded into the following form (taking the m-th feature as an example):

[0166]

[0167] Where, ξ tar,m This represents the m-th feature component of the target missing tooth location.

[0168] To enhance the robustness and generalization ability of the regression model, this invention employs a 5-fold cross-validation strategy for model training and performance evaluation. The coefficient of determination R0 is used as the benchmark. 2Using Mean Squared Error (MSE) as the primary evaluation metric, the system compares the impact of different model structures and feature combinations on predictive performance, ultimately determining the regression model that performs best in both fitting accuracy and interpretability. This method can effectively predict the six-degree-of-freedom spatial pose of any missing tooth, providing quantitative parameter support for implant initial path fitting and robot trajectory planning, and realizing intelligent modeling of personalized implant paths based on geometric correlation. To uniformly describe the position and orientation relationships of each component of the system in three-dimensional space, multiple Cartesian coordinate systems are introduced for geometric modeling: robot base coordinate system {B}, patient model coordinate system {Q}, model marker coordinate system {D}, optical tracking system coordinate system {M}, end marker coordinate system {C}, end flange coordinate system {E}, and drill tool coordinate system {T}. Figure 2 The mapping structure and calibration process between the various coordinate systems are shown.

[0169] In the coordinate architecture of this invention, the spatial transformation relationship between each joint of the robotic arm and the end effector flange is first established using the robot's forward kinematics equations, thus establishing its basic motion model. Subsequently, the tool center point TCP and tool coordinate system TCF calibration method are used to accurately obtain the position and orientation of the tool carried by the end effector in the flange coordinate system. Next, the rigid transformation matrix between the optical tracking system and the robot coordinate system is analyzed using the classic hand-eye calibration algorithm to achieve a unified mapping between the two spatial systems. Finally, combined with 3D point cloud registration technology, the spatial alignment between the model marker and the patient's anatomical structure is achieved, thereby constructing a complete mapping link from the virtual path to the physical execution space, providing a high-precision spatial foundation for implantation path interpolation, trajectory planning, and obstacle avoidance control.

[0170] To ensure sufficient spatial accuracy and directional control for the robot when executing the implantation path, the actual spatial position and orientation of the surgical tool (such as an implantation handpiece) mounted on the end effector must be calibrated. Since clamping and assembly may introduce orientation deviations, the end effector needs to be modeled and error-compensated using TCP and TCF calibration methods to accurately obtain its pose parameters in the end flange coordinate system.

[0171] This invention employs a four-point method for geometric modeling of the tool center point (TCP). This method requires that, with the tool tip fixed at a stationary point in space, the end effector of the robotic arm is controlled to contact that point in multiple different postures, and the spatial coordinates of the end flange under each posture are recorded. Based on the geometric constraint equations constructed from these discrete postures, the least squares method is used to solve for the three-dimensional displacement vector of the TCP in the flange coordinate system, thereby obtaining the precise spatial position.

[0172] Specifically, step S3 includes the following steps:

[0173] S3.1 Select a fixed spatial reference point (such as a tip reference object), and the robotic arm of the dental implant robot holds the tool (such as a dental drill) through the gripping mechanism, repeatedly positioning the tool tip at the spatial reference point;

[0174] S3.2, Control the end flange of the robotic arm to contact a spatial reference point in at least four significantly different postures;

[0175] S3.3, record the corresponding coordinates of the end flange of the robotic arm;

[0176] S3.4, construct the calibration equation system and fit it using the least squares method to obtain the TCP position of the tool center point.

[0177] During calibration, the tool is rigidly mounted to the end flange of the robotic arm via a clamping mechanism, and its relative pose to the flange remains unchanged. Therefore, this tool system can be modeled as the "seventh joint" of the robotic arm. To describe the spatial transformation relationship between the tool and the end flange, this invention introduces a homogeneous transformation matrix representation, whose transformation relationship can be formally expressed as:

[0178]

[0179] Expanding this transformation relationship into a block matrix yields the linear relationship between the position vectors:

[0180]

[0181] Since all calibration points are the same point in space (i.e., the calibration tip remains constant), we can conclude that:

[0182]

[0183] By combining the above two equations and performing difference processing on the data under different attitudes, the following system of linear equations can be constructed:

[0184]

[0185] Ideally, the system of equations should satisfy the condition that both the augmented matrix and the coefficient matrix have a rank of 3. However, due to measurement errors and repetitive positioning errors of the robotic arm, incompatible solutions may arise. Therefore, the least squares method is used for generalized inverse solving to obtain the optimal approximate solution:

[0186]

[0187] Since the coefficient matrix is ​​not a square matrix and has full column rank, this invention uses the normal equation method for calculation, which has the following form: A + =(A T A) -1 A T ;

[0188] Substituting A into the equation, we get:

[0189]

[0190] in, and These represent the rotation matrices of the end flange coordinate system relative to the base coordinate system of the dental implant robot under four different postures of the robotic arm;

[0191] To improve calibration accuracy, it is necessary to properly filter the collected end-effector attitude data. Specifically, data sets with small attitude changes should be removed to reduce the impact of TCP offset error on the calibration results. The attitude information of the robotic arm's end effector is represented using Euler angles (RPY) and further converted into a rotation matrix form for attitude calculation and pose unification modeling within the homogeneous transformation matrix.

[0192]

[0193] This is the rotation matrix of the end flange coordinate system relative to the robot base coordinate system.

[0194] By substituting the acquired end-effector posture data into the aforementioned geometric model, the three-dimensional coordinates of the tool center point (TCP) in the end-effector coordinate system can be calculated, thereby completing the consistency calibration between the end-effector tool and the robotic arm coordinate system. This calibration process not only provides accurate initial parameters for subsequent spatial trajectory interpolation and pose control of the planting path, but also significantly reduces path deviations caused by tool installation errors, ensuring spatial and accuracy consistency between the robot's execution path and the planned path.

[0195] After completing the TCP positioning of the end-effector center point, the TCF coordinate system needs to be calibrated to further achieve precise control of the end-effector attitude. TCF calibration aims to establish a complete orientation reference system for the end-effector in three-dimensional space, providing accurate coordinate basis for the attitude interpolation of the planting path and the end-effector control of the robotic arm.

[0196] This invention employs the Three-Point Method along ZX axes for TCF calibration. This method guides the end effector of the robotic arm to move along the Z-axis and X-axis directions in which the tool is actually oriented, and records three representative spatial points. Based on this, the three orthogonal axial vectors of the tool coordinate system are derived, thereby establishing a right-handed coordinate frame.

[0197] Specifically, step S4 includes the following steps:

[0198] S4.1, mark the center point of the end flange under the initial pose of the robotic arm movement as point o; drag the robotic arm from point o along the X-axis by a distance Δx; drag the robotic arm from point o along the Z-axis by a distance Δz;

[0199] Since the tool is rigidly connected to the end flange, its posture remains unchanged during dragging. Therefore, the following coordinate relationship is established:

[0200]

[0201] in, The coordinates of the center point of the end flange in the base coordinate system after moving a distance Δx; The coordinates of the center point of the end flange in the base coordinate system under the initial pose; The coordinates of the tool's end point in the base coordinate system after moving a distance Δx; The coordinates of the tool end point in the base coordinate system under the initial pose; This represents the coordinates of the center point of the end flange in the base coordinate system after moving a distance Δz. The coordinates of the tool's end point in the base coordinate system after moving a distance Δz;

[0202] S4.2, construct the rotation matrix of the tool coordinate system based on the three calibration points, and obtain the orthogonal basis through vector difference and cross product operations:

[0203]

[0204] o = a × n;

[0205] in, This is the inverse of the rotation matrix of the end flange center point in the base coordinate system under the initial pose. Let be the rotation matrix of the center point of the end flange in the base coordinate system. The coordinates of the tool end point in the end flange coordinate system under the initial pose; n is the rotation matrix relative to the x-axis, a is the rotation matrix relative to the z-axis, and o is the rotation matrix relative to the y-axis.

[0206] Considering the existence of experimental errors, the three vectors initially calculated may not fully satisfy the orthogonality condition. Therefore, the z-axis vector needs to be corrected to ensure the orthogonality of the basis matrices. Finally, the rotation matrix R of the tool coordinate system is constructed through normalization, thereby completing the attitude calibration of the TCF.

[0207] This method achieves a good balance between accuracy and practicality, and is particularly suitable for dental implant robot tasks that require high precision in end-effector posture control. Combined with TCP calibration results, TCF calibration effectively ensures the spatial consistency and motion stability of the robot's end effector during path planning and posture control.

[0208] To achieve accurate mapping of the target planting path within the robotic arm's space, it is essential to establish the spatial transformation relationship between the robotic arm's end effector and the external vision positioning system. Hand-eye calibration, a commonly used spatial registration technique, aims to solve for the rigid transformation matrix between the robot's end effector and the optical tracking system. Depending on the installation method of the vision sensor, hand-eye calibration can be divided into two modes: "eye on the hand" and "eye outside the hand." This invention adopts the "eye outside the hand" configuration, where a spatial marker is installed at the end effector of the robotic arm, and its pose information is collected in real time by a fixed optical positioning system.

[0209] Specifically, step S5 includes the following steps:

[0210] S5.1, the movement of the robotic arm causes a change in the pose of the end effector. A mapping relationship can be established between the robot coordinate system and the vision coordinate system through a fixed rigid transformation X. Since the end effector C is fixed to the end flange E, the following relationships exist between time t and time t+1:

[0211]

[0212] in, This is the rotation matrix of the end flange coordinate system relative to the end positioning marker coordinate system;

[0213] S5.2, the transformation matrix between the coordinate system of the end flange and the coordinate system of the end positioning marker is given by the homogeneous transformation relationship:

[0214]

[0215] in, Let be the transformation matrix between the robot base coordinate system and the end flange coordinate system. Let be the transformation matrix of the optical positioning and tracking device coordinate system relative to the robot base coordinate system. This is the inverse of the transformation matrix of the optical positioning and tracking instrument's coordinate system with respect to the end-positioning marker;

[0216] Further conclusions can be drawn:

[0217]

[0218] After equivalent transformation, the basic expression for hand-eye calibration is obtained as AX = XB:

[0219]

[0220] Where A and B are the transformation matrices of the robotic arm system and the vision system between two adjacent poses, respectively, and X is the fixed transformation matrix to be determined; R A R X and R B Let b be a rotation submatrix of the matrix. A b X and b B These are the translation submatrices of the matrix;

[0221] S5.3, after decomposing the homogeneous matrix into rotation and translation components, it is further decomposed into:

[0222]

[0223] Among them, t X It is a translation vector;

[0224] S5.4, To reduce solution redundancy, this invention employs the Tasi-Lenz method to model the rotation matrix using axis-angle representation, thus avoiding the parameter redundancy problem present in traditional rotation matrices. The axis-angle form uses a three-parameter representation, significantly reducing computational dimensionality and improving robustness. Efficient conversion between the axis-angle representation and the rotation matrix can be achieved using the Rodriguez formula. (From R...) A R X =R X R B ,have to right Taking the logarithm, we have:

[0225]

[0226] Using the similarity transformation property of matrix logarithms:

[0227]

[0228] Since a three-dimensional rotation matrix is ​​obtained by performing matrix exponentiation operations on the antisymmetric matrix form of the rotation vector:

[0229]

[0230] Among them, [α] × and [β] × It is an antisymmetric matrix generated by rotation vectors α and β;

[0231] S5.5, Solve for R X This is a classic absolute orientation problem, solved using Singular Value Decomposition (SVD) to find R. X :

[0232]

[0233] S5.6, After solving the rotation part, the translation vector t is solved using the following formula. X :

[0234] (R Ai -I)t X =R X t Bi -t Ai ;

[0235] The linear equations in steps S5.7 and S5.6 can be estimated using the least squares method. However, considering the influence of noise in actual measurement data, to improve the accuracy and robustness of the solution, the LM algorithm is introduced for nonlinear optimization. The LM algorithm combines the Gauss-Newton method and the gradient descent method, and dynamically adjusts the iteration step size and direction by adjusting the damping factor λ. The iteration increment of the LM algorithm satisfies:

[0236] (J T J+λI)Δp=-J T r;

[0237] in, ρ is the parameter vector, r is the rotation part of the rotation vector parameterization, and r is the residual vector composed of rotation and translation residuals; J is the Jacobian matrix of residuals with respect to parameters; I is the identity matrix; Δp is the parameter update amount.

[0238] S5.8, the objective function E(p) is defined as the sum of squared rotation and translation errors:

[0239]

[0240] Where R(ρ)=exp([ρ] × ) represents the rotation matrix corresponding to the rotation vector.

[0241] vee(·) converts an antisymmetric matrix into a three-dimensional vector; They are The corresponding rotation matrix (an orthogonal matrix with a determinant of 1); They are The corresponding translation vector.

[0242] Jacobian matrix J i The expression form is:

[0243]

[0244] By iteratively solving the equation (J) T J+λI)Δp=-J T r, update parameter p until the objective function E(p) converges to the preset threshold, and obtain the optimal estimate of the rotation matrix and displacement vector.

[0245] To achieve accurate mapping of the virtual implantation path in the real surgical space, this invention employs point cloud registration technology to construct a spatial rigid transformation matrix between the digital model and the physical printed model. This method discretizes the 3D implantation model into a point cloud and combines this with the pose information of the physical sample obtained by an optical positioning system to achieve high-precision pose alignment between the virtual simulation space and the physical space, ensuring the accurate placement of the regression prediction path in the real environment.

[0246] To address the large-scale errors and local noise present in the initial point cloud data, a rigid registration process combining coarse and fine registration was designed, including downsampling, feature extraction and coarse registration, and fine registration.

[0247] Specifically, step S6 includes the following steps:

[0248] S6.1, Voxel Raster Downsampling

[0249] To reduce the computational complexity of point clouds while preserving geometric integrity, the original point cloud is first downsampled. Specifically, the point cloud is divided into a three-dimensional voxel lattice with fixed side lengths, and each voxel is replaced with the geometric center of all points, forming a sparse point set. The computational expression is as follows:

[0250]

[0251] in, p is the center point within the voxel. i Let n be the original point within the voxel, and n be the number of voxel points. This process reduces the number of points while preserving local geometric information.

[0252] S6.2, FPFH Feature Extraction and RANSAC Coarse Registration

[0253] To achieve initial coarse registration of the point cloud, it is necessary to describe the local geometric features of each point. The FastPoint Feature Histogram (FPFH) algorithm is used to construct a feature histogram by statistically analyzing the relative angular relationships between the point normal vectors. The specific steps are as follows:

[0254] ① Calculate the simplified point feature histogram (SPFH) of the query point and its neighboring points, and encode the angle features α, φ, θ, which represent the relative directions between normal vectors;

[0255] ② Use weighted neighborhood information to aggregate SPFH features to form the final FPFH features:

[0256]

[0257] Where, ω i To query point p to its neighbor p iThe weights are the inverse of the Euclidean distance between them, where k is the number of neighborhood points.

[0258] Based on the FPFH descriptor, coarse registration is performed using RANSAC (Random Sample Consensus). This method constructs a candidate registration model through multiple random samplings, eliminates outliers, selects a set of interior points, and then calculates the rigid transformation matrix. This process enhances the stability and robustness of the initial registration.

[0259] S6.3, ICP Fine Registration: Based on the initial transformation matrix obtained from coarse registration, the ICP (Iterative Closest Point) algorithm is used for fine registration. By iteratively minimizing the Euclidean distance between two sets of corresponding points, accurate alignment of the point clouds is achieved. The objective function of ICP is:

[0260]

[0261] Where, p i Let q be the i-th point in the source set. i Let R be the point corresponding to the target point set, R be the rotation matrix, and t be the translation vector.

[0262] The solution steps are as follows:

[0263] ① Calculate the weighted centroid of the source and target point sets, denoted as . and Let the partial derivative of the translation vector be zero, that is:

[0264]

[0265] The translation vector is derived as follows:

[0266]

[0267] ② Construct the covariance matrix:

[0268]

[0269] ③ Perform singular value decomposition (SVD) on the covariance matrix H:

[0270] H=UΣV T ;

[0271] ④ The optimal rotation matrix is ​​calculated as follows:

[0272] R = VU T ;

[0273] ⑤ Substitute the rotation matrix into the translation calculation formula Obtain the complete rigid transformation matrix to achieve the optimal rigid matching between the two point clouds.

[0274] This registration process combines voxel downsampling, FPFH-based feature matching and RANSAC coarse registration, and SVD-based ICP fine registration, effectively solving the problems of point cloud data errors and noise. It achieves sub-millimeter-level spatial alignment between the virtual digital model and the physical printed model, providing a reliable geometric basis for subsequent robotic arm trajectory import and high-precision surgical execution.

[0275] Within the limited space of the oral cavity, path planning for dental implant robots must overcome numerous spatial constraints and complex obstacles to ensure that the robotic arm's motion path avoids critical anatomical structures while also meeting real-time response and path executability requirements. To address this, a collision detection mechanism based on geometric boundaries was designed, and combined with RRT and its improved algorithms RRT-Connect / RRT*, efficient path search and generation were achieved.

[0276] Specifically, step S7 includes the following steps:

[0277] S7.1, Environment Modeling and Collision Detection: The surgical environment includes various obstacles such as the patient's oral cavity model, the robotic arm body, the guide plate device, and the operating table.

[0278] Axis-aligned bounding boxes (AABBs) are used for structured modeling of each object. AABBs enclose the object Ω within a minimal cuboid parallel to the coordinate axes, defined as:

[0279] AABB(Ω)=[x min ,x max ]×[y min ,y max ]×[z min ,z max ];

[0280] Where, x min With x max Let x and y represent the minimum and maximum boundaries of object Ω on the x-axis, respectively. min ,y max Let z represent the minimum and maximum boundaries of object Ω on the y-axis, respectively. min ,z max These represent the minimum and maximum boundaries of object Ω on the z-axis, respectively.

[0281] Collision determination is based on whether the AABB coordinates of two objects A and B overlap in all three axes.

[0282]

[0283] If the above conditions are met, a collision is determined to be possible, and the system will activate a more refined collision detection algorithm to reduce false positives. This method has low computational complexity and is suitable for dynamic detection in real-time environments.

[0284] S7.2, Pathfinding Algorithms: RRT, RRT-Connect, and RRT*:

[0285] Path search is based on collision detection results, and the RRT algorithm is first used; RRT randomly samples points q in the continuous configuration space C. rand And find the node q that is closest to the random sampling point in the existing path tree. near :

[0286]

[0287] Where, ‖·‖ represents the Euclidean distance. Let be the set of all nodes in the current tree, and arg min be the position of the minimum value;

[0288] Subsequently, a new node q is generated using the step-size constraint function Steer. new :

[0289]

[0290] Where η is the maximum step size, ensuring a smooth path and reasonable increments. After a new node is generated, the collision detection function CollisionFree(q) is called. near ,q new Ensure that the path segments are free of obstacles. The path tree gradually approaches the target node by continuously expanding.

[0291] In S7.3, to improve search efficiency and path quality, the RRT-Connect algorithm is introduced, while starting from the starting point q... start and the endpoint q goal Each grows a random tree and The two trees are expanded alternately and attempts to connect; the connection condition is the number of nodes in the two trees. and Distance not exceeding threshold ∈ conn :

[0292]

[0293] Once the connection is successful, a collision-free path from the starting point to the ending point is obtained. RRT-Connect significantly reduces search time and path redundancy, adapting to the path planning needs of surgical robots with high degrees of freedom and multiple constraints, but its generated path is still not globally optimal.

[0294] S7.4, to achieve asymptotic optimality of the path, the RRT* algorithm is introduced. RRT* expands the new node q... new Then, the path is optimized through a local rewiring mechanism.

[0295] S7.5 employs fifth-order polynomial interpolation and B-spline curve interpolation to smooth the path points and transform them into continuous, dynamically controllable joint angle trajectories. The path generated by RRT* is a discrete node sequence, exhibiting trajectory discontinuities and abrupt velocity changes. Direct execution would lead to robotic arm vibration and positioning errors. Therefore, it is necessary to smooth the path points and transform them into continuous, dynamically controllable joint angle trajectories. This invention uses the following two interpolation methods:

[0296] (1) Fifth-order polynomial interpolation

[0297] Fifth-order polynomial interpolation is widely used in robot trajectory planning due to its good continuity and computational efficiency. Its trajectory expression is:

[0298] p(t) = a0 + a1t + a2t 2 +a3t 3 +a4t 4 +a5t 5 ,t∈[t0,t f ];

[0299] Where p(t) represents position, and the coefficient a i (i = 0, ..., 5) is determined by satisfying the boundary conditions for position, velocity, and acceleration:

[0300]

[0301] The velocity and acceleration are respectively:

[0302]

[0303] Substituting the boundary conditions into the system of equations forms a linear system:

[0304] Ma = b;

[0305] Where a = [a0, a1, a2, a3, a4, a5] T b = [p0, v0, a0, p f ,v f ,a f ] T ;

[0306] The coefficient matrix M is based on t0 and t f After constructing and solving, the coefficient 'a' is obtained, enabling the generation of a continuous and smooth trajectory. This significantly reduces the impact and vibration during the movement of the robotic arm, meeting the requirements for fine oral manipulation.

[0307] (2) B-spline curve interpolation

[0308] B-spline curves utilize multi-segment polynomial basis functions and control points to achieve local adjustment and overall smoothing of the trajectory. The B-spline basis function is recursively defined as:

[0309]

[0310] Where, N i,p (t) is a basis function of order p, {t i} represents the node vector. The trajectory of a cubic B-spline curve (p=3) is expressed as:

[0311]

[0312] Among them, P i The coordinates of the control points are given. The trajectory velocity and acceleration are as follows:

[0313]

[0314] B-spline curves guarantee high-order continuity (at least second order), effectively avoiding the boundary oscillation problem common in polynomial interpolation, and are suitable for scenarios with dense path points and high requirements for motion continuity. Their disadvantages lie in their higher computational complexity and more demanding requirements for node and weight configuration.

[0315] Specifically, step S7.4 includes the following steps:

[0316] S7.4.1, at radius r n Search for the set of neighboring nodes Q near :

[0317] Q near ={q∈Q∣‖qq new ||≤r n};

[0318] Where, r n The sampling quantity is dynamically adjusted to meet the asymptotic optimal condition.

[0319] S7.4.2, Select the nearest neighbor node q that minimizes the total cost. min ∈Q near As q new Parent node:

[0320]

[0321] Where c(q) start Cost(q, q) represents the path cost from the starting point to node q. new ) represents from q to q new The cost is typically expressed as path length;

[0322] S7.4.3, perform rewiring operation on neighboring nodes, if via q new If a lower path cost can be obtained, the parent node and path cost are updated to achieve local path optimization.

[0323] Although RRT* cannot guarantee absolute optimality within a finite time, the path quality gradually approaches global optimality as the number of samples increases, making it suitable for offline planning tasks with high requirements for trajectory quality.

[0324] In summary, the RRT algorithm, with its simple structure and speed, is suitable for initial path exploration. RRT-Connect accelerates the planning process through bidirectional search, while RRT* achieves incremental path optimization through path cost functions and rewiring mechanisms. Depending on the specific task requirements, the dental implant robot path planning can flexibly select the above algorithms to achieve a balance between efficiency and path quality.

[0325] To achieve accurate initial estimation and spatial orientation reference for implantation pathways, this invention utilizes CBCT voxel data to extract the central axis and perform three-dimensional geometric fitting on the entire dentition of the patient. In this process, firstly, multi-layer cross-sectional images are extracted from each tooth in the CBCT images, and four boundary points (mesiodistal, labial, and lingual) are marked within each cross-section. These boundary points constitute the cross-sectional area of ​​the tooth, and then the position of the center point of each cross-section is calculated based on the coordinates of these points. Using this method, we obtain the sequence of center points for each tooth in three-dimensional space, and further use these center points for central axis fitting in subsequent analyses, as detailed in Tables 1 and 2.

[0326] Table 1. Information on Upper Tooth Form

[0327]

[0328] Table 2. Information on lower tooth cartilage

[0329]

[0330] After extracting the center points of all tooth cross-sections, a three-dimensional linear model of the center point sequence for each tooth was created using the least squares fitting method to obtain the midline equation for each tooth. This method ensures minimal error in fitting the midline and accurately represents the geometry and spatial position of the teeth. The fitting results for all teeth are finally summarized in a set of spatial vectors (see Table 3). These vectors represent the spatial position and orientation information of each tooth, providing a reference for the subsequent implantation path direction.

[0331] Table 3. Tooth Fitting Axis Table

[0332]

[0333] A three-dimensional spatial distribution map of the tooth midline was generated using the Mimics and 3-matic 3D visualization platforms, such as... Figure 3 As shown, this illustrates the tilt direction and spatial distribution of the patient's upper and lower jaw dentition. Figure 3 The location of the missing areas is clearly revealed, especially the specific positions of LR6 and LR7 in the lower right region, which is of great reference value for subsequent path planning and implant placement selection. Visualizing the three-dimensional distribution of the patient's teeth allows for a direct observation of the continuity of the dentition and its spatial distribution patterns. Specifically, the figure shows the tilt direction of the patient's dentition, and through intuitive spatial distribution analysis, it helps confirm whether the arrangement relationship between the teeth conforms to physiological expectations.

[0334] Figure 3 The analysis also revealed the occlusal relationship between the patient's upper and lower dentitions, clearly reflecting the spatial location of the missing teeth. This analysis provides essential data support for subsequent path planning, including direction setting, implant depth determination, and obstacle avoidance space modeling. For example, by displaying the missing LR6 and LR7 areas, it can be verified whether these areas provide sufficient distance for obstacle avoidance, ensuring the feasibility of the implant path planning. In three-dimensional space, accurate positioning of the missing teeth helps ensure that the implant placement is completely consistent with the predetermined path, avoiding any potential errors or path conflicts.

[0335] To ensure the accuracy of implant pathway planning and its spatial adaptability during the planning process, a detailed statistical analysis of the spatial pose of the patient's teeth was conducted. While 3D display data alone can show the spatial distribution of teeth and the location of missing teeth, this method fails to deeply analyze the specific spatial characteristics of the teeth, potentially leading to flaws in the planning results. Therefore, a more comprehensive analysis of spatial pose and position data was employed to reveal the distribution characteristics and interrelationships of teeth in three-dimensional space.

[0336] When analyzing the position and pose of the upper and lower teeth, the changes in the rotation angles of each axis were first observed. Figure 4 and Figure 5 The figure shows the rotation angle changes of the upper and lower teeth along the X, Y, and Z axes. As can be seen from the figure, there is a strong negative correlation between the rotation angle changes along the X-axis, meaning that the rotation trends of the upper and lower teeth on this axis are opposite. Specifically, the rotation angle of the lower teeth is smaller, a phenomenon that can be attributed to the occlusal relationship between the upper and lower teeth. During occlusion, the mandible, as the primary force-generating tissue, has teeth that are more concentrated on the inner side of the oral cavity, resulting in a smaller X-axis rotation angle for the lower teeth.

[0337] A negative correlation was also observed in the changes of the Y-axis rotation angle, indicating that the rotation angles of the upper and lower teeth show similar trends along the Y-axis. However, a significant positive correlation was found in the changes of the Z-axis rotation angle, meaning that the rotation angles of the upper and lower teeth tend to be consistent, with significant numerical differences. Overall, the upper and lower teeth show a strong correlation in tilt angles, although the rotation angle changes are not significant in some specific tooth positions (such as L6 and R1), which may be related to occlusal habits or bone density conditions.

[0338] To make these data features more intuitive, the rotation angle data underwent mean removal. Figure 6 This study demonstrates the changes in the rotation angles of the upper and lower teeth after removing the mean. The data after removing the mean shows a negative correlation between the rotation angles of the upper and lower teeth along the X and Y axes, and a positive correlation along the Z axis, further validating the correlation between the poses of the upper and lower teeth. The distribution of the upper and lower teeth after removing the mean shows a significant positive correlation, especially in the positional data along the X and Y axes. Comparing the X and Y positional distribution maps after removing the mean, a mirror-symmetry feature was observed between the contralateral teeth, indicating that the upper and lower teeth have regularity and symmetry in spatial position. This finding provides important evidence for subsequent implant path regression prediction.

[0339] To accurately predict the location of missing teeth, this invention selects the contralateral teeth and occlusal teeth for each tooth position as key elements for regression analysis. The spatial location and orientation of the missing teeth are then determined through regression analysis. Based on this method, regression analysis is performed on all tooth positions using the spatial location and orientation data of the upper and lower teeth, ultimately yielding a set of regression equations. The calculation results of these regression equations accurately predict the spatial location of each tooth position, especially providing precise localization of the missing teeth (LR6 and LR7).

[0340] Specifically, the regression equations consider multiple factors, such as the symmetry between the upper and lower teeth, the position of the occlusal surface, and the distribution pattern of the surrounding teeth. Each regression equation is based on the spatial relationship of adjacent teeth and statistical data to estimate the accurate location of the missing tooth.

[0341]

[0342] The regression analysis results, calculated using five regression equations, determined the position and orientation of the dental implants. Table 4 shows the implant path data predicted by the regression equations, which serves as a crucial basis for customizing implant plans for missing teeth LR6 and LR7. Specific calculation results are as follows:

[0343] Table 4. Planting pathways obtained based on regression equations.

[0344]

[0345] Table 4 lists the spatial coordinates and orientation angles of LR6 and LR7, where x, y, and z represent spatial coordinates, and α, β, and γ represent the rotational orientation of the tooth. These calculations allow for the accurate determination of the implant path's direction and depth, ensuring precise implant positioning and occlusal engagement.

[0346] These regression equations provide crucial data support for subsequent implantation path planning. Especially in multi-implant bridging, selecting the average rotation angle from the regression equations ensures the parallelism of the implants, thereby guaranteeing precise implant alignment and functionality. Furthermore, all predicted implantation path data will ultimately be converted into image spatial data for use in subsequent surgical procedures and implantation simulations.

[0347] Through this series of regression analyses and data verifications, this invention successfully provides reliable prediction results for the implantation paths of LR6 and LR7, and provides a mathematical basis for the precise implementation of implantation surgery.

[0348] To ensure the safe and stable arrival of the dental implant robot at the target implantation site in narrow and anatomically complex surgical areas, the overall trajectory planning is divided into two stages: first, a collision-free path that meets anatomical constraints is generated in Cartesian space; second, the path is smoothed and its dynamic executability is optimized in joint space. The first stage focuses on path accessibility and obstacle avoidance reliability, while the second stage emphasizes the continuity and controllability of dynamic parameters such as velocity and acceleration.

[0349] Obstacle avoidance analysis and path generation results:

[0350] During the Cartesian space path search, a geometric collision detection strategy is employed: axis-aligned bounding boxes (AABBs) are constructed for each link of the robotic arm, the support platform, and the patient's oral tissue. In the path expansion phase, these bounding boxes are used for rapid filtering, and fine-grained collision detection is triggered only when the bounding boxes intersect, thus significantly reducing computational load while maintaining detection accuracy.

[0351] Regarding path search algorithms, two-dimensional and three-dimensional simulations were conducted to compare three typical sampling-based planning methods: RRT, RRT-Connect, and RRT*, as shown in the figure. Figure 7 and Figure 8See Table 5. The results show that RRT-Connect, with its bidirectional tree expansion strategy, exhibits extremely short planning times (0.002s and 0.003s) in both 2D and 3D environments, making it suitable for rapid online response during surgery. While RRT* significantly increases planning time (11.67s in 2D and 6.66s in 3D), it boasts the shortest path length (12.67mm in 2D and 16.62mm in 3D), demonstrating asymptotic optimality and making it more suitable for offline generation of high-quality baseline trajectories. RRT's performance falls between the two, achieving a 100% success rate, but it is not superior in terms of both time consumption and path optimization. Considering both real-time performance and security requirements, the workflow prioritizes using RRT-Connect to quickly obtain feasible paths, and then uses RRT* for offline path quality verification and optimization when time permits, balancing efficiency and optimality. Figure 9 The initial obstacle avoidance path's morphology is further illustrated: the first collision-free path obtained using a sampling planner (such as RRT*) in Cartesian space consists of a series of discrete nodes and their connections. The starting and target points are highlighted with different colors or markers, and intermediate nodes are connected segment by segment in a polygonal line, clearly reflecting the algorithm's search process under obstacle distribution constraints. The main obstacles in the robotic arm's working environment and their geometric boundaries are also depicted, with the path remaining outside the envelope without penetration. It can be seen that the trajectory exhibits an overall "zigzag" pattern with numerous turns and large curvature variations. This is a typical "feasible but not smooth" initial solution generated by random sampling, laying the foundation for subsequent smoothing optimization.

[0352] Table 5 Comparison of three algorithms in two-dimensional and three-dimensional environments

[0353]

[0354] The original sampling data from the three planting experiments are listed in Table 6; the corresponding top / root end point position deviations and attitude angle deviations are listed in Table 7.

[0355] Table 6. Locations obtained through optical positioning

[0356]

[0357] Table 7. Deviation Analysis Between Experimental Results and Planned Path

[0358]

[0359] This invention, based on CBCT data, constructs a closed-loop dental implant robot system encompassing "image reconstruction, path planning, multi-coordinate system registration, robot execution, and effect verification." Compared to traditional free-hand operation and semi-automatic guide systems, this system exhibits significant advantages in spatial positioning and posture control during cavity preparation. Experimental results show that the linear deviations of the top and root endpoints of the implant path are (1.03±0.34) mm and (1.06±0.23) mm, respectively, and the posture deviation is (3.47°±1.99°). Compared to the reported errors of free-hand operation in existing literature—positional deviations of (1.43±0.49) mm and (2.20±0.79) mm, and angular deviations of 6.78°±3.31°—this invention improves positional accuracy by approximately 28%–52% and nearly halves the posture deviation. Compared to the >1.5 mm positional deviation and >5° posture error commonly found in conventional semi-automatic guide systems, this invention also demonstrates superior precision control capabilities.

[0360] Of course, the above description is not intended to limit the present invention, and the present invention is not limited to the examples given above. Any changes, modifications, additions or substitutions made by those skilled in the art within the scope of the present invention should also fall within the protection scope of the present invention.

Claims

1. A dental implant robot path generation method characterized by, Specifically, the steps include the following: S1, extract the tooth space midline based on CBCT images, express the spatial relationship between the tooth space midline and the standard Cartesian coordinate system, and construct the tooth space pose vector; S2, combining the spatial pose features of the contralateral tooth and the upper and lower occlusal teeth, constructs a regression model to predict the spatial features of the target missing tooth, and predicts the spatial position and pose angle of the target missing tooth. S3, TCP calibration of the dental implant robot; S4, TCF calibration of the dental implant robot; S5, Construct the spatial transformation relationship between the robot arm end effector and the external vision positioning system; S6 uses point cloud registration technology to construct the spatial transformation matrix between the digital model and the physical printed model; S7 constructs the initial obstacle avoidance path of the robotic arm based on the fast extended random tree sampling algorithm, and combines the improved algorithms RRT-Connect and RRT*, and smooths the joint trajectory and controls the speed through polynomial interpolation and B-spline curve interpolation. Step S7 specifically includes the following steps: S7.1 uses axis-aligned bounding boxes (AABB) for structured modeling of each object. AABB binds the objects... Enclosed in the smallest cuboid parallel to the coordinate axes, it is defined as: ； in, and Representing objects exist Minimum and maximum boundaries on the axis, Representing objects exist Minimum and maximum boundaries on the axis, Representing objects exist Minimum and maximum boundaries on the axis; The collision determination is based on whether the AABBs of two objects A and B exist in interval overlap in three-axis directions : ； S7.2, Path search is based on collision detection results, and the RRT algorithm is first used; RRT is performed in a contiguous configuration space. Random sampling points And find the node that is closest to the random sampling point in the existing path tree. : ； wherein, denotes the Euclidean distance, is the set of all nodes in the current tree, is the position of the minimum value; Subsequently, a new node is generated using a step limit function Steer : ； wherein is the maximum step size; S7.3, introduce RRT-Connect algorithm, while growing from start and end each a random tree and , alternate expanding and trying to connect the two trees; connection condition is that nodes and in the two trees are within a threshold distance : ； S7.4, introduce RRT* algorithm, RRT* optimizes the path after extending a new node by a local re-wiring mechanism; S7.5 uses fifth-order polynomial interpolation and B-spline curve interpolation to smooth the path points and transform them into continuous, dynamically controllable joint angle trajectories. Step S7.4 specifically includes the following steps: S7.4.1, in radius Internal search for neighboring node set : ； wherein, Dynamic adjustment with sampling number; S7.4.2, select the neighboring node that minimizes the total cost as parent node: ； wherein, denotes the cost of a path from the start node to the node denotes the cost from to , typically the path length; S7.4.

3. Perform a re-wiring operation to a neighboring node if a lower path cost is obtained, updating the parent node and path cost. If a lower path cost is obtained, update the parent node and path cost.

2. The dental implant robot path generation method of claim 1, wherein, Step S1 specifically includes the following steps: S1.1 Import CBCT scan data into a medical image processing platform for three-dimensional reconstruction to obtain a tooth voxel model; S1.2, select three layers in the dental voxel model, respectively corresponding to the upper segment of the root, the middle segment and the apical region, recorded as the first, second and third layers, on each layer, obtain the dental outer contour boundary point set wherein, denotes the layer number, denotes the number of boundary points extracted in the layer, the two-dimensional geometric center of each layer is calculated as follows: ； The two-dimensional geometric center of each layer is lifted to three-dimensional coordinates according to the corresponding layer height to obtain three groups of spatial points 、 and : ； S1.3, using least squares, fit , and to the dental axis, the dental axis direction vector is represented as: ； wherein is a unit vector, representing the components of the dental axis in the X, Y, Z directions; S1.4, the spatial relationship between the tooth's central axis and the standard Cartesian coordinate system is expressed as follows: ； wherein respectively represent the direction cosine angles of the X-axis, Y-axis, and Z-axis. S1.5, the tooth spatial pose vector is represented as: ； wherein, is a three-dimensional spatial coordinate of the tooth.

3. The dental implant robot path generation method of claim 1, wherein, Step S2 specifically includes the following steps: S2.1, set the spatial position and pose of the target missing tooth as : ； S2.2, defining a spatial pose vector of the contralateral homonymous tooth ; a spatial pose vector of the upper and lower occlusal teeth ; S2.3, constructing a predictive target edentulous space feature regression model: ； in, , represent the coefficient matrices of the contribution of the contralateral teeth and occlusal teeth to the prediction of the target tooth, respectively; For the intercept term; For residual terms; The regression model expands to the following form: ； in, Indicates the position of the target missing tooth. Each feature component.

4. The method for generating a path for a dental implant robot according to claim 1, characterized in that, Step S3 specifically includes the following steps: S3.1, Select a fixed spatial reference point, and the robotic arm of the dental implant robot clamps the tool through the clamping mechanism, repeatedly positioning the tool tip at the spatial reference point; S3.2, Control the end flange of the robotic arm to contact a spatial reference point in at least four significantly different postures; S3.3, record the corresponding coordinates of the end flange of the robotic arm; S3.4, construct the calibration equation system and fit it using the least squares method to obtain the TCP position of the tool center point. : ； in, , , and These represent the rotation matrices of the end flange coordinate system relative to the dental implant robot's base coordinate system under four different postures of the robotic arm.

5. The method for generating a path for a dental implant robot according to claim 1, characterized in that, Step S4 specifically includes the following steps: S4.1, mark the center point of the end flange in the initial pose of the robotic arm movement as point O; drag the robotic arm a certain distance along the X-axis from point O. Drag the robotic arm a certain distance along the z-axis from point O. Establish the following coordinate relationship: ； in, For the distance traveled Then, the coordinates of the center point of the end flange in the base coordinate system; The coordinates of the center point of the end flange in the base coordinate system under the initial pose; For the distance traveled Then, the coordinates of the tool's end point in the base coordinate system; The coordinates of the tool end point in the base coordinate system under the initial pose; For the distance traveled Then, the coordinates of the center point of the end flange in the base coordinate system; For the distance traveled Then, the coordinates of the tool's end point in the base coordinate system; S4.2, construct the rotation matrix of the tool coordinate system based on the three calibration points, and obtain the orthogonal basis through vector difference and cross product operations: ； ； ； in, Let be the inverse of the rotation matrix of the end flange center point in the base coordinate system under the initial pose. Let be the rotation matrix of the center point of the end flange in the base coordinate system. The coordinates of the tool end point in the end flange coordinate system under the initial pose; ; , Let x be the rotation matrix relative to the x-axis. Let Z be the rotation matrix relative to the z-axis. is the rotation matrix relative to the y-axis.

6. The method for generating a path for a dental implant robot according to claim 1, characterized in that, Step S5 specifically includes the following steps: S5.1, since the marker C is fixed to the end flange E, then Time and Always: ； in, This is the transformation matrix between the end flange coordinate system and the end positioning marker coordinate system; S5.2, The transformation matrix between the coordinate system of the end flange and the coordinate system of the end positioning marker is given by the homogeneous transformation relationship: ； in, Let be the transformation matrix between the robot base coordinate system and the end flange coordinate system. Let be the transformation matrix of the optical positioning and tracking device coordinate system relative to the robot base coordinate system. This is the inverse of the transformation matrix of the optical positioning and tracking instrument's coordinate system with respect to the end-positioning marker; Further conclusions can be drawn: ； The basic expression for hand-eye calibration is obtained after equivalent transformation. : ； ； in, and These are the transformation matrices of the robotic arm system and the vision system between two adjacent poses, respectively. Let be the fixed transformation matrix to be found; , and For a rotation submatrix of a matrix, , and These are the translation submatrices of the matrix; S5.3, after decomposing the homogeneous matrix into rotation and translation components, it is further decomposed into: ； in, It is a translation vector; S5.4, the Tasi–Lenz method is used to model the rotation matrix using axis-angle representation, by ,have to ,right Taking the logarithm, we have: ； Using the similarity transformation property of matrix logarithms: ； Since a three-dimensional rotation matrix is ​​obtained by performing matrix exponentiation operations on the antisymmetric matrix form of the rotation vector: ； in, and It is composed of rotation vectors and The generated antisymmetric matrix; S5.5 This problem is a classic absolute orientation problem, solved using Singular Value Decomposition (SVD). : S5.6, After solving for the rotation part, the translation vector is solved using the following formula. : ； S5.7 introduces the LM algorithm for nonlinear optimization. The iterative increment of the LM algorithm satisfies: ； ； in, It is a parameter vector. For the rotation part parameterized by the rotation vector, It is a residual vector composed of rotation and translation residuals; It is the Jacobian matrix of the residuals with respect to the parameters; It is the identity matrix; It is the parameter update amount; S5.8, Objective Function Defined as the sum of squares of rotation and translation errors: ; in, This represents the rotation matrix corresponding to the rotation vector. ; ; To convert an antisymmetric matrix into a three-dimensional vector; They are , The corresponding rotation matrix; They are , The corresponding translation vector; S5.9, solving the equation through repeated iterations Update parameters until the objective function The system converges to a preset threshold, yielding the optimal estimates of the rotation matrix and displacement vector.

7. The method for generating a path for a dental implant robot according to claim 1, characterized in that, Step S6 specifically includes the following steps: S6.1, downsampling the original point cloud, the calculation expression is: ； in, The center point within the voxel. The origin point within the voxel. Voxel count; S6.2 uses the FPFH algorithm to construct a feature histogram by statistically analyzing the relative angular relationships between the point normal vectors; S6.3, based on the initial transformation matrix obtained from coarse registration, the ICP algorithm is used for fine registration. Accurate alignment of the point clouds is achieved by iteratively minimizing the Euclidean distance between two sets of corresponding points. The objective function of ICP is: ； in, The first point of the source set One point, For the target point set corresponding to the point, For rotation matrix, It is a translation vector. To take the minimum.

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