DE112019006884T5 - Method for making an automated seamless major corneal incision - Google Patents

Abstract

The present invention relates to a method for performing an automated, seamless corneal incision, comprising a machine operating arm and a scalpel installed on the machine operating arm. The three-dimensional coordinates of the scalpel in the eye and the position of the iris are calculated by the microscope camera system. The cutting path of the movement of the machine operating arm is set in such a way that the machine operating arm carries a scalpel along the cutting path in order to automatically complete the corneal main incision operation, whereby the method for performing a fully automatic corneal incision with a scalpel is realized and the injury to the wound enlargement caused by inaccurate manual operation as well other injuries to the corneal layer caused by the laser are avoided.

Description

Technical area

The present invention relates to the field of automated surgery and, more particularly, to a method for making an automated seamless major corneal incision.

State of the art

The first step in cataract surgery is to make a major corneal incision in the corneal layer of the patient's eye. The existing procedure is mainly performed by doctors manually with the help of instruments. But different doctors are different in every operation for different patients. The quality of the main cut depends on the technique and the condition of the chief doctor. However, when making a seamless main incision, it is easy to injure the incision due to the high precision required in manual surgery.

The revelation with the publication number CN102639088B discloses a laser system and method for suturing corneal cuts and corneal cuts that perform automated laser surgery on the cuts. However, the laser procedure can easily cause other damage to the corneal layer.

Content of the present invention

The present invention aims to solve the above technical problems, at least to some extent, and to provide a method for performing an automated, seamless corneal incision, thereby avoiding additional damage during corneal incision surgery.

• Schritt 1: Zwei Bilder werden mit einem Mikroskopkamerasystem mit a mm Vertikalverschiebung aufgenommen;
• Schritt 2: Jedes Bild wird vorverarbeitet, in ein Bild mit einem Grauwert umgewandelt, und das erste Bild wird einer Eckpunktdetektion unterzogen, um eine Vielzahl von Kandidateneckpunkten zu erhalten, und die Koordinaten der Eckpunkte werden erhalten;
• Schritt 3: Eine Vielzahl von Eckpunkten ist zentriert und R_j ist der Radius, um schwarze und weiße Punkte relativ zu den Eckpunkten zu ermitteln, und der schwarze Punkt befindet sich in dem weißen Pixelbereich des Eckpunkts und der weiße Punkt befindet sich in dem schwarzen Pixelbereich des Eckpunkts, und die Pixelgraustufe des schwarzen Punkts ist < 0,5 und die Pixelgraustufe des weißen Punkts ist > 0,5, dann ist der Eckpunkt der Skalpellpunkt P'_i, und die Berechnungsformel für die Position des schwarzen Punkts und des weißen Punkts ist wie folgt: $${\text{X}}_{\text{blacki}}=\frac{{\displaystyle {\sum }_{\text{j}\in {\text{R}}_{\text{j}}}\left(1-\text{I}\left({\text{X}}_{\text{j}}.{\text{ Y}}_{\text{j}}\right)\right){\text{X}}_{\text{j}}}}{{\displaystyle {\sum }_{\text{j}\in {\text{R}}_{\text{i}}}\left(1-\text{I}\left({\text{X}}_{\text{j}}.{\text{ Y}}_{\text{j}}\right)\right)}}$$
  
  $${\text{Y}}_{\text{blacki}}=\frac{{\displaystyle {\sum }_{\text{j}\in {\text{R}}_{\text{j}}}\left(1-\text{I}\left({\text{X}}_{\text{j}}.{\text{ Y}}_{\text{j}}\right)\right){\text{Y}}_{\text{j}}}}{{\displaystyle {\sum }_{\text{j}\in {\text{R}}_{\text{i}}}\left(1-\text{I}\left({\text{X}}_{\text{j}}.{\text{ Y}}_{\text{j}}\right)\right)}}$$
  
  $${\text{X}}_{\text{whitei}}=\frac{{\displaystyle {\sum }_{\text{j}\in {\text{R}}_{\text{j}}}\left(1-\text{I}\left({\text{X}}_{\text{j}}.{\text{ Y}}_{\text{j}}\right)\right){\text{X}}_{\text{j}}}}{{\displaystyle {\sum }_{\text{j}\in {\text{R}}_{\text{i}}}\left(1-\text{I}\left({\text{X}}_{\text{j}}.{\text{ Y}}_{\text{j}}\right)\right)}}$$
  
  $${\text{Y}}_{\text{whitei}}=\frac{{\displaystyle {\sum }_{\text{j}\in {\text{R}}_{\text{j}}}\left(1-\text{I}\left({\text{X}}_{\text{j}}.{\text{ Y}}_{\text{j}}\right)\right){\text{Y}}_{\text{j}}}}{{\displaystyle {\sum }_{\text{j}\in {\text{R}}_{\text{i}}}\left(1-\text{I}\left({\text{X}}_{\text{j}}.{\text{ Y}}_{\text{j}}\right)\right)}}$$
  
  $${\text{R}}_{\text{j}}\Rightarrow \left[{\left({\text{X}}_{\text{j}}-{\text{X}}_{\text{i}}\right)}^2+{\left({\text{Y}}_{\text{j}}-{\text{Y}}_{\text{i}}\right)}^2\le {\text{range}}^2\right]$$
  

In order to solve the above technical problems, the technical solution of the present invention is that a method for performing an automated seamless corneal main incision, comprising a machine operating arm and a scalpel installed on the machine operating arm, wherein the machine operating arm has a clamping device for clamping the scalpel and a drive arm for Driving the clamping device to vibrate, the drive arm being provided with a first linear motor and a second linear motor, the clamping device being provided with a third linear motor, comprising the following steps:

• Step 1: Two images are recorded with a microscope camera system with a mm vertical displacement;
• Step 2: Each image is preprocessed, converted into an image having a gray value, and the first image is subjected to corner point detection to obtain a plurality of candidate corner points, and the coordinates of the corner points are obtained;
• Step 3: A plurality of corner points are centered and R _j is the radius to find black and white points relative to the corner points, and the black point is in the white pixel area of the corner point and the white point is in the black pixel area the vertex and pixel gray level of the black dot is <0.5 and the pixel gray level of the white point is> 0.5, is the cornerstone of the scalpel point P _'i, and the formula for calculating the position of the black point and the white point is as follows: $${\text{X}}_{\text{blacki}}=\frac{{\displaystyle {\sum }_{\text{j}\in {\text{R.}}_{\text{j}}}\left(1-\text{I.}\left({\text{X}}_{\text{j}}.{\text{Y}}_{\text{j}}\right)\right){\text{X}}_{\text{j}}}}{{\displaystyle {\sum }_{\text{j}\in {\text{R.}}_{\text{i}}}\left(1-\text{I.}\left({\text{X}}_{\text{j}}.{\text{Y}}_{\text{j}}\right)\right)}}$$
  
  $${\text{Y}}_{\text{blacki}}=\frac{{\displaystyle {\sum }_{\text{j}\in {\text{R.}}_{\text{j}}}\left(1-\text{I.}\left({\text{X}}_{\text{j}}.{\text{Y}}_{\text{j}}\right)\right){\text{Y}}_{\text{j}}}}{{\displaystyle {\sum }_{\text{j}\in {\text{R.}}_{\text{i}}}\left(1-\text{I.}\left({\text{X}}_{\text{j}}.{\text{Y}}_{\text{j}}\right)\right)}}$$
  
  $${\text{X}}_{\text{whitei}}=\frac{{\displaystyle {\sum }_{\text{j}\in {\text{R.}}_{\text{j}}}\left(1-\text{I.}\left({\text{X}}_{\text{j}}.{\text{Y}}_{\text{j}}\right)\right){\text{X}}_{\text{j}}}}{{\displaystyle {\sum }_{\text{j}\in {\text{R.}}_{\text{i}}}\left(1-\text{I.}\left({\text{X}}_{\text{j}}.{\text{Y}}_{\text{j}}\right)\right)}}$$
  
  $${\text{Y}}_{\text{whitei}}=\frac{{\displaystyle {\sum }_{\text{j}\in {\text{R.}}_{\text{j}}}\left(1-\text{I.}\left({\text{X}}_{\text{j}}.{\text{Y}}_{\text{j}}\right)\right){\text{Y}}_{\text{j}}}}{{\displaystyle {\sum }_{\text{j}\in {\text{R.}}_{\text{i}}}\left(1-\text{I.}\left({\text{X}}_{\text{j}}.{\text{Y}}_{\text{j}}\right)\right)}}$$
  
  $${\text{R.}}_{\text{j}}\Rightarrow \left[{\left({\text{X}}_{\text{j}}-{\text{X}}_{\text{i}}\right)}^2+{\left({\text{Y}}_{\text{j}}-{\text{Y}}_{\text{i}}\right)}^2\le {\text{range}}^2\right]$$
  

• Schritt 4: Das P'_i des ersten Bildes wird an das zweite Bild angepasst, um einen Pixelabstand zwischen den Anpassungspunkten P''_i, P'_i und P''_i zu erhalten;
• Schritt 5: Die Formel zur Berechnung der Tiefe des Skalpellpunkts lautet wie folgt: $$\text{z}\text{'}=\left(\frac{\text{r}}{\mathrm{\Delta }\text{r}}+1\right)\text{a}$$
  

X _blacki and Y _{blacki are} the coordinate values of the black point, X _whitei and Y _whitei are the coordinate values of the white point, X _j and Y _j are the respective coordinates of all pixels in the area of the corner point as the center and R _j as the radius, I is the gray value of the point, range is the maximum value of the selected radius and X _i and Y _i are the coordinate values of the scalpel point P '_i;

• Step 4: The P ' _{i of} the first image is matched to the second image in order to obtain a pixel spacing between the matching points P'' _i , P' _i and P ''_i;
• Step 5: The formula for calculating the depth of the scalpel point is as follows: $$\text{z}\text{'}=\left(\frac{\text{r}}{\mathrm{\Delta }\text{r}}+1\right)\text{a}$$
  

• Schritt 6: Die Position und die relative Tiefe des Skalpells werden gemäß der Berechnung erhalten, und die Position der Iris wird ermittelt und die Schnittsposition wird an dem Punkt am Rand der Iris positioniert, der der Pupillenmitte am nächsten liegt;
• Schritt 7: Das standardmäßige nahtlose Hornhauthauptschnitt wird als lose Z-Form definiert, und der Maschinenbedienarm steuert die Bewegung des Skalpells entlang des eingestellten Schnittwegs.

Here r is the distance between the scalpel point P 'and the magnification center of the microscope, and Δr is the pixel distance between P' _i and P '' _i , which is referred to as parallax;

• Step 6: The position and the relative depth of the scalpel are obtained according to the calculation, and the position of the iris is found and the cutting position is positioned at the point on the edge of the iris which is closest to the pupil center;
• Step 7: The standard seamless corneal main cut is defined as a loose Z-shape and the machine operator controls the movement of the scalpel along the set cutting path.

$$\begin{array}{l}\text{d}\left(\text{P}{\text{'}}_{\text{ij}},\text{P}{\text{'}}_{\text{ij}}-\mathrm{\Delta }\text{r},{\text{I}}_{\text{z}},{\text{I}}_{\text{z}-\text{a}}\right)=\text{minP}{\text{'}}_{\text{ij}}-\mathrm{\Delta }\text{r}-0.5\\ \le \text{P}{\text{'}}_{\text{ij}}-\mathrm{\Delta }\text{r}+0.5\left\{\left|{\text{I}}_{\text{z}}\left(\text{P}{\text{'}}_{\text{ij}}\right)-{\text{I}}_{\text{z}-\text{a}}\left(\text{P}\text{'}{\text{'}}_{\text{ij}}\right)\right|\right\}\end{array}$$

It is preferably provided that there is in step 4, the adjustment method is a P _'i -centered base point block area in P' _i set up to define a point other than P'i as an adjacent point and the depth of each adjacent point relative to P _'i where the fitting points corresponding to the second image are calculated by fitting all the base points of the block area, the fitting point and the corresponding parallax being calculated by selecting the parallax that minimizes the value of the cost function, the formula being the cost function, $$\text{C.}\left(\text{P.}{\text{'}}_{\text{ij}},\mathrm{\Delta }\text{r}\right)=\text{min}\left\{\text{d}\left(\text{P.}{\text{'}}_{\text{ij}},\text{P.}{\text{'}}_{\text{ij}}-\mathrm{\Delta }\text{r},{\text{I.}}_{\text{Z}},{\text{I.}}_{\text{z}-\text{a}}\right),\text{d}(\text{P.}{\text{'}}_{\text{ij}}-\mathrm{\Delta }\text{r},\text{P.}{\text{'}}_{\text{ij}},{\text{I.}}_{\text{Z}-\text{a}},{\text{I.}}_{\text{z}}\right\}$$

$$\begin{array}{l}\text{d}\left(\text{P.}{\text{'}}_{\text{ij}},\text{P.}{\text{'}}_{\text{ij}}-\mathrm{\Delta }\text{r},{\text{I.}}_{\text{z}},{\text{I.}}_{\text{z}-\text{a}}\right)=\text{minP}{\text{'}}_{\text{ij}}-\mathrm{\Delta }\text{r}-0.5\\ \le \text{P.}{\text{'}}_{\text{ij}}-\mathrm{\Delta }\text{r}+0.5\left\{\left|{\text{I.}}_{\text{z}}\left(\text{P.}{\text{'}}_{\text{ij}}\right)-{\text{I.}}_{\text{z}-\text{a}}\left(\text{P.}\text{'}{\text{'}}_{\text{ij}}\right)\right|\right\}\end{array}$$

P ' _{ij is} the pixel position of the corresponding base point, P'' _ij is the pixel position of the corresponding adjustment point, I _z is the gray value of each pixel in the first image, I _za is the gray value of each pixel in the second image, I _z (P ' _ij ) is the gray value of P' _ij and I _za (P '' _ij ) is the gray value of P '' _ij .

It is preferably provided that a checking step is provided between step 4 and step 5, and if the position of the corresponding adjustment point P ″ _ij in a plurality of second images relative to P ″ _i with the position of the neighboring point relative to P ″ _i matches, the check is successful (where P ' _{ij is} the field point, P'' _ij is the adaptation point of the field point, P' _i is the center point and P ' _i is the matching point of the center point); A point is center P _{'i is} added, the total number of votes is 8 (8 field points), and 4 or more voices (including 4 parts) are considered to be successful adaptation; Otherwise, a base point block area is established around each neighboring point, the calculation formula in step 4 is executed again, the match point and corresponding parallax are calculated again, and the voting is carried out and the first person with more than 4 votes is considered the winner. More than half of the fitting points have the same relative position to the neighboring points, thereby avoiding the situation that P ' _{i is} a noise point in order to improve the correctness of the scalpel point.

Preferably, it is provided that the first successful adjacent check point as the new point P _'i is defined, and other adjacent points and unsuccessful verification points are discarded and then the remaining adjustment point P'_'ik in the second image as a base point for matching with the first image is used to get the verification point, and if the verification point coincides with the position of the base point, the adjustment is successful, otherwise the verification step is performed again. The correctness of the adjustment is checked again by reversing the adjustment direction in order to improve the accuracy of the scalpel point.

It is preferably provided that an interpolation optimization calculation is carried out on the depth calculation formula of the scalpel point in step 5. The calculation of the depth is made more accurate by interpolation optimization calculation. Each P '' _ik defines two interpolation points along the epipolar line L _ik , namely the left interpolation point P '' _ikl and the right interpolation point P '' _ikr , where the cost function is used to adapt the block area in order to calculate the costs C (P ' _i , Δr _ikr ), C (P ' _i , Δr _ikl ) and C (P' _i , Δr _ik ), remove the interpolation point or origin of minimum cost and remove the remaining two points to get the final depth formula like follows to get $${\text{Z}}_{\text{kj}}=\left(\frac{\text{r}}{\mathrm{\Delta }{\text{r}}_{\text{l}}}\Rightarrow {\text{min}}_{\text{il}=\text{ik},\text{ikr},\text{ikl}}\left\{\text{C.}\left(\text{P.}{\text{'}}_{\text{i}},\mathrm{\Delta }{\text{r}}_{\text{il}}\right)\right\}+1\right)\text{a}$$

P'_ij entspricht;L _{ik is} the epipolar line and Z _kj is the depth, the P _ik

P 'corresponds to _ij;

The formula to calculate the final depth: $$Z_{p_i^{\text{'}}}=Z_{p_i^{\text{'}\text{'}}}=\frac{{\displaystyle {\sum }_{kj}{Z}_{kj}}}{n_O}$$

Here, n _{o is} the number of Z _kj .

$$\text{P}\text{'}{\text{'}}_{\text{ikr}}=\left(\text{P}\text{'}{\text{'}}_{\text{ik}-\text{Li}}+\text{P}\text{'}{\text{'}}_{\text{ik}}\right)/2$$

It is preferably provided that the calculation formulas for the interpolation point are $$\text{P.}\text{'}{\text{'}}_{\text{ikl}}=\left(\text{P.}\text{'}{\text{'}}_{\text{ik}+\text{Li}}+\text{P.}\text{'}{\text{'}}_{\text{ik}}\right)/2$$

$$\text{P.}\text{'}{\text{'}}_{\text{ikr}}=\left(\text{P.}\text{'}{\text{'}}_{\text{ik}-\text{Li}}+\text{P.}\text{'}{\text{'}}_{\text{ik}}\right)/2$$

Where P '' _{ik + Li is} the value at which P '' _ik moves the distance of Li pixels forward along the epipolar line, and P '' _ik-Li is the value where P '' _{ik is} the distance of Li pixels moved backwards along the epipolar line.

• S1: Das Skalpell tritt in die Hornhautschicht ein;
• S2: Das Skalpell wird um 40-50 Grad angehoben, um eine RCM-Bewegung am ersten entfernten Mittelpunkt durchzuführen;
• S3: Das Skalpell bewegt sich vorwärts gemäß dem Winkel in S2, bis es den zweiten entfernten Mittelpunkt erreicht;
• S4: Nach der Durchführung der RCM-Bewegung wird das Skalpell um 40-50 Grad abgesenkt;
• S5: Das Skalpell bewegt sich vorwärts gemäß dem Winkel in S4 weiter, bis es die Hornhautschicht passiert.

It is preferably provided that the cutting path is defined as follows:

• S1: The scalpel enters the corneal layer;
• S2: The scalpel is raised 40-50 degrees to make an RCM move at the first distant center point;
• S3: The scalpel moves forward according to the angle in S2 until it reaches the second distant center;
• S4: After performing the RCM movement, the scalpel is lowered 40-50 degrees;
• S5: The scalpel continues to move forward according to the angle in S4 until it passes the corneal layer.

$${\text{y}}_{\text{R}}=\left({\text{L}}_{\text{tool}}+{\text{L}}_3\right)\text{cos}\left(\mathrm{\phi +\vartheta }-\frac{\mathrm{\pi }}{2}\right)-{\text{L}}_{\text{o}}\text{ cos}\left(\mathrm{\vartheta }\right)-{\text{d}}_{\text{m}}\text{sin}\left(\mathrm{\phi +\vartheta }-\frac{\mathrm{\pi }}{2}\right)$$

It is preferably provided that the scalpel performs an RCM movement in step 7, the position of the RCM point is determined, and the control strategy formula of the RCM point is as follows, $${\text{x}}_{\text{R.}}=\frac{{\text{L.}}_1+{\text{L.}}_2}{2}+\left({\text{L.}}_{\text{tool}}+{\text{L.}}_3\right)\text{sin}\left(\mathrm{\phi \; +\; \vartheta }-\frac{\mathrm{\pi }}{2}\right)+{\text{L.}}_{\text{O}}\text{cos}\left(\mathrm{\vartheta }\right)+{\text{d}}_{\text{m}}\text{cos}\left(\mathrm{\phi \; +\; \vartheta }-\frac{\mathrm{\pi }}{2}\right)$$

$${\text{y}}_{\text{R.}}=\left({\text{L.}}_{\text{tool}}+{\text{L.}}_3\right)\text{cos}\left(\mathrm{\phi \; +\; \vartheta }-\frac{\mathrm{\pi }}{2}\right)-{\text{L.}}_{\text{O}}\text{cos}\left(\mathrm{\vartheta }\right)-{\text{d}}_{\text{m}}\text{sin}\left(\mathrm{\phi \; +\; \vartheta }-\frac{\mathrm{\pi }}{2}\right)$$

Where x _{R is} the X-axis coordinate position of the RCM point; y _R is the Y-axis coordinate position of the RCM point; L ₁ is the linear displacement of the first linear motor; L ₂ is the linear displacement of the second linear motor; L _tool is the distance between the clamp and the tip of the scalpel; L ₃ is the linear displacement of the third linear motor; φ is the angle between the surface of the scalpel and the connecting end of the clamp; ϑ is the angle between the surface of the scalpel and the horizontal plane; L _o is the length of the surface of the scalpel; dm is the vertical distance from the third linear motor to the horizontal axis of the first linear motor or the second linear motor.

ϑ ϑ = f(t) ist, wobei die eingestellten Trajektorienformeln des ersten Linearmotors, des zweiten Linearmotors und des dritten Linearmotors wie folgt sind, $$\begin{array}{l}{\text{L}}_3=({\text{y}}_{\text{R}}+{\text{L}}_{\text{o}}\text{ sin}\left(\text{f}\left(\text{t}\right)\right)+{\text{d}}_{\text{m}}\text{sin}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)\\ -\left({\text{L}}_{\text{tool}}\right)\text{cos}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)/\text{cos}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)\end{array}$$

$$\begin{array}{l}{\text{L}}_2={\text{x}}_{\text{R}}-\left({\text{L}}_{\text{tool}}+{\text{L}}_3\right)\text{sin}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)-{\text{L}}_{\text{o}}\text{ cos}\left(\text{f}\left(\text{t}\right)\right)-{\text{d}}_{\text{m}}\text{ cos}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)\\ +\frac{\text{h}}{2\text{tan}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)}\end{array}$$

$$\begin{array}{l}{\text{L}}_1={\text{x}}_{\text{R}}-\left({\text{L}}_{\text{tool}}+{\text{L}}_3\right)\text{sin}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)-{\text{L}}_{\text{o}}\text{ cos}\left(\text{f}\left(\text{t}\right)\right)-{\text{d}}_{\text{m}}\text{ cos}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)\\ +\frac{\text{h}}{2\text{tan}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)}\end{array}$$

It is preferably provided that the scalpel rotates around the RCM point and the angle change trajectory of $\mathrm{\vartheta }=\text{arctan}\left(\frac{\text{H}}{{\text{L.}}_{\text{1}}-{\text{L.}}_2}\right)+\frac{\mathrm{\pi }}{2}-\mathrm{\phi },$

ϑ ϑ = f (t), where the set trajectory formulas of the first linear motor, the second linear motor and the third linear motor are as follows, $$\begin{array}{l}{\text{L.}}_3=({\text{y}}_{\text{R.}}+{\text{L.}}_{\text{O}}\text{sin}\left(\text{f}\left(\text{t}\right)\right)+{\text{d}}_{\text{m}}\text{sin}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)\\ -\left({\text{L.}}_{\text{tool}}\right)\text{cos}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)/\text{cos}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)\end{array}$$

$$\begin{array}{l}{\text{L.}}_2={\text{x}}_{\text{R.}}-\left({\text{L.}}_{\text{tool}}+{\text{L.}}_3\right)\text{sin}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)-{\text{L.}}_{\text{O}}\text{cos}\left(\text{f}\left(\text{t}\right)\right)-{\text{d}}_{\text{m}}\text{cos}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)\\ +\frac{\text{<figure-callout id="2" label="H" filenames="DE112019006884T5\_0058.png" state="{{state}}">H</figure-callout>}}{2\text{tan}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)}\end{array}$$

$$\begin{array}{l}{\text{L.}}_1={\text{x}}_{\text{R.}}-\left({\text{L.}}_{\text{tool}}+{\text{L.}}_3\right)\text{sin}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)-{\text{L.}}_{\text{O}}\text{cos}\left(\text{f}\left(\text{t}\right)\right)-{\text{d}}_{\text{m}}\text{cos}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)\\ +\frac{\text{<figure-callout id="2" label="H" filenames="DE112019006884T5\_0058.png" state="{{state}}">H</figure-callout>}}{2\text{tan}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)}\end{array}$$

Where h is the linear distance between the first linear motor and the second linear motor.

The scalpel calculates the movement trajectory of the first linear motor, the second linear motor, and the third linear motor according to the coordinate control strategy of the RCM point during the movement and rotation. At the same time, when moving, the movement trajectory of the first linear motor, the second linear motor and the third linear motor is also calculated according to the formula to achieve the effect of controlling the movement of the scalpel.

It is preferably provided that a control method for a machine operating arm with parallel series joints consists in that the first linear motor, the second linear motor and the third linear motor Define the position from the previous target position to the next target position by the linear interpolation method and set the step size between the two target positions.

It is preferably provided that the position of every millisecond for the target position of each step is obtained by calculating the cutting path.

It is preferably provided that the microscope system and the machine operating arm are initialized before step 1 is carried out, the coordinates of the X-axis and the Y-axis of the machine operating arm and the microscope camera system being consistent during the initialization process. The coordinates of the machine operator and the microscope system are consistent, and the coordinates for the execution movement of the machine operator are consistent with the coordinates for calculation, thereby ensuring the accuracy of the movement of the machine operator.

It is preferably provided that the width of the cut is not greater than the width of the surface of the scalpel. The machine operator arm controls the movement of the scalpel only along the cutting path and not shifting it in other directions, which ensures that the width of the cut is not greater than the width of the surface of the scalpel in order to prevent the cut from expanding.

In comparison to the prior art, the advantageous effect is that the method is based on a machine operating arm and a scalpel, whereby the method for performing a fully automatic corneal incision with a scalpel is implemented and the wound enlargement damage caused by inaccurate manual operation and others by the Laser-induced injuries to the corneal layer can be avoided.

Figure list

• 1 Figure 3 is a flow diagram of the method of the present invention;
• 2 Fig. 3 is a schematic diagram of the structure of the machine operating arm;
• 3 Figure 3 is a schematic diagram of the cutting path of the present invention.

1.

Skalpell;

2.

Klemmvorrichtung;

3.

Der erste Linearmotor;

4

Der zweite Linearmotor;

5.

Der dritte Linearmotor;

6.

Hornhautschicht;

7.

Der erste entfernte Mittelpunkt;

8.

Der zweite entfernte Mittelpunkt.

Reference number:

1.

Scalpel;

2.

Clamping device;

3.

The first linear motor;

4th

The second linear motor;

5.

The third linear motor;

6th

Corneal layer;

7th

The first distant center;

8th.

The second distant center.

Detailed embodiments

The drawings are only used for illustrative purposes and are not to be understood as limiting the present invention. In order to better illustrate the present embodiment, certain parts of the drawings may be omitted, enlarged or reduced that do not represent the size of the actual product. It will be understood by those skilled in the art that certain known structures and their descriptions can be omitted from the drawings. The terms describing the positional relationship in the drawings are used for illustrative purposes only and should not be construed as limiting the present invention.

Identical or similar reference symbols in the drawings of embodiments of the present invention correspond to identical or similar parts. In the description of the present invention it is to be understood that the azimuth or positional relationship indicated by the terms upper, lower, left, right, long, short is based on the azimuth or positional relationship illustrated in the drawings. Just to facilitate the description of the present invention and to simplify the description instead of To indicate or imply that said device or component must have a particular orientation and must be designed and operated in a particular orientation, the terms describing the positional relationship in the drawings are used for illustrative purposes only and should not be taken as a limitation on the present Invention to be understood. For those of ordinary skill in the art, the specific meaning of the above terms in the present invention can be understood on a case-by-case basis.

The technical solutions of the present invention are described in detail below through specific embodiments in conjunction with the drawings.

Embodiment 1

• Schritt 1: Zwei Bilder werden mit einem Mikroskopkamerasystem mit 6 mm Vertikalverschiebung aufgenommen;
• Schritt 2: Jedes Bild wird vorverarbeitet, in ein Bild mit einem Grauwert umgewandelt, und das erste Bild wird einer Eckpunktdetektion unterzogen, um eine Vielzahl von Kandidateneckpunkten zu erhalten, und die Koordinaten der Eckpunkte werden erhalten;
• Schritt 3: Eine Vielzahl von Eckpunkten ist zentriert und R_j ist der Radius, um schwarze und weiße Punkte relativ zu den Eckpunkten zu ermitteln, und der schwarze Punkt befindet sich in dem weißen Pixelbereich des Eckpunkts und der weiße Punkt befindet sich in dem schwarzen Pixelbereich des Eckpunkts, und die Pixelgraustufe des schwarzen Punkts ist < 0,5 und die Pixelgraustufe des weißen Punkts ist > 0,5, dann ist der Eckpunkt der Skalpellpunkt P'_i, und die Berechnungsformel für die Position des schwarzen Punkts und des weißen Punkts ist wie folgt: $${\text{X}}_{\text{blacki}}=\frac{{\displaystyle {\sum }_{\text{j}\in {\text{R}}_{\text{j}}}\left(1-\text{I}\left({\text{X}}_{\text{j}}.{\text{ Y}}_{\text{j}}\right)\right){\text{X}}_{\text{j}}}}{{\displaystyle {\sum }_{\text{j}\in {\text{R}}_{\text{i}}}\left(1-\text{I}\left({\text{X}}_{\text{j}}.{\text{ Y}}_{\text{j}}\right)\right)}}$$
  
  $${\text{Y}}_{\text{blacki}}=\frac{{\displaystyle {\sum }_{\text{j}\in {\text{R}}_{\text{j}}}\left(1-\text{I}\left({\text{X}}_{\text{j}}.{\text{ Y}}_{\text{j}}\right)\right){\text{Y}}_{\text{j}}}}{{\displaystyle {\sum }_{\text{j}\in {\text{R}}_{\text{i}}}\left(1-\text{I}\left({\text{X}}_{\text{j}}.{\text{ Y}}_{\text{j}}\right)\right)}}$$
  
  $${\text{X}}_{\text{whitei}}=\frac{{\displaystyle {\sum }_{\text{j}\in {\text{R}}_{\text{j}}}\left(1-\text{I}\left({\text{X}}_{\text{j}}.{\text{ Y}}_{\text{j}}\right)\right){\text{X}}_{\text{j}}}}{{\displaystyle {\sum }_{\text{j}\in {\text{R}}_{\text{i}}}\left(1-\text{I}\left({\text{X}}_{\text{j}}.{\text{ Y}}_{\text{j}}\right)\right)}}$$
  
  $${\text{Y}}_{\text{whitei}}=\frac{{\displaystyle {\sum }_{\text{j}\in {\text{R}}_{\text{j}}}\left(1-\text{I}\left({\text{X}}_{\text{j}}.{\text{ Y}}_{\text{j}}\right)\right){\text{Y}}_{\text{j}}}}{{\displaystyle {\sum }_{\text{j}\in {\text{R}}_{\text{i}}}\left(1-\text{I}\left({\text{X}}_{\text{j}}.{\text{ Y}}_{\text{j}}\right)\right)}}$$
  
  $${\text{R}}_{\text{j}}\Rightarrow \left[{\left({\text{X}}_{\text{j}}-{\text{X}}_{\text{i}}\right)}^2+{\left({\text{Y}}_{\text{j}}-{\text{Y}}_{\text{i}}\right)}^2\le {\text{range}}^2\right]$$
  

A method for performing an automated, seamless corneal incision, comprising a machine operator arm and a scalpel installed on the machine operator arm 1 , whereby a machine operating arm with parallel series joints has a clamping device 2 for clamping the scalpel 1 and a drive arm for driving the clamping device 2 comprises for vibrating, wherein the drive arm with a first linear motor 3 and a second linear motor 4th is provided, the clamping device 2 with a third linear motor 5 is provided, comprising the following steps:

• Step 1: Two images are recorded with a microscope camera system with 6 mm vertical displacement;
• Step 2: Each image is preprocessed, converted into an image having a gray value, and the first image is subjected to corner point detection to obtain a plurality of candidate corner points, and the coordinates of the corner points are obtained;
• Step 3: A plurality of corner points are centered and R _j is the radius to find black and white points relative to the corner points, and the black point is in the white pixel area of the corner point and the white point is in the black pixel area the vertex and pixel gray level of the black dot is <0.5 and the pixel gray level of the white point is> 0.5, is the cornerstone of the scalpel point P _'i, and the formula for calculating the position of the black point and the white point is as follows: $${\text{X}}_{\text{blacki}}=\frac{{\displaystyle {\sum }_{\text{j}\in {\text{R.}}_{\text{j}}}\left(1-\text{I.}\left({\text{X}}_{\text{j}}.{\text{Y}}_{\text{j}}\right)\right){\text{X}}_{\text{j}}}}{{\displaystyle {\sum }_{\text{j}\in {\text{R.}}_{\text{i}}}\left(1-\text{I.}\left({\text{X}}_{\text{j}}.{\text{Y}}_{\text{j}}\right)\right)}}$$
  
  $${\text{Y}}_{\text{blacki}}=\frac{{\displaystyle {\sum }_{\text{j}\in {\text{R.}}_{\text{j}}}\left(1-\text{I.}\left({\text{X}}_{\text{j}}.{\text{Y}}_{\text{j}}\right)\right){\text{Y}}_{\text{j}}}}{{\displaystyle {\sum }_{\text{j}\in {\text{R.}}_{\text{i}}}\left(1-\text{I.}\left({\text{X}}_{\text{j}}.{\text{Y}}_{\text{j}}\right)\right)}}$$
  
  $${\text{X}}_{\text{whitei}}=\frac{{\displaystyle {\sum }_{\text{j}\in {\text{R.}}_{\text{j}}}\left(1-\text{I.}\left({\text{X}}_{\text{j}}.{\text{Y}}_{\text{j}}\right)\right){\text{X}}_{\text{j}}}}{{\displaystyle {\sum }_{\text{j}\in {\text{R.}}_{\text{i}}}\left(1-\text{I.}\left({\text{X}}_{\text{j}}.{\text{Y}}_{\text{j}}\right)\right)}}$$
  
  $${\text{Y}}_{\text{whitei}}=\frac{{\displaystyle {\sum }_{\text{j}\in {\text{R.}}_{\text{j}}}\left(1-\text{I.}\left({\text{X}}_{\text{j}}.{\text{Y}}_{\text{j}}\right)\right){\text{Y}}_{\text{j}}}}{{\displaystyle {\sum }_{\text{j}\in {\text{R.}}_{\text{i}}}\left(1-\text{I.}\left({\text{X}}_{\text{j}}.{\text{Y}}_{\text{j}}\right)\right)}}$$
  
  $${\text{R.}}_{\text{j}}\Rightarrow \left[{\left({\text{X}}_{\text{j}}-{\text{X}}_{\text{i}}\right)}^2+{\left({\text{Y}}_{\text{j}}-{\text{Y}}_{\text{i}}\right)}^2\le {\text{range}}^2\right]$$
  

• Schritt 4: Das P'_i des ersten Bildes wird an das zweite Bild angepasst, um einen Pixelabstand zwischen den Anpassungspunkten P''_i, P'_i und P''_i zu erhalten;
• Schritt 5: Die Formel zur Berechnung der Tiefe des Skalpellpunkts lautet wie folgt: $$\text{z}\text{'}=\left(\frac{\text{r}}{\mathrm{\Delta }\text{r}}+1\right)\text{a}$$
  

X _blacki and Y _{blacki are} the coordinate values of the black point, X _whitei and Y _whitei are the coordinate values of the white point, X _j and Y _j are the respective coordinates of all pixels in the area of the corner point as the center and R _j as the radius, I is the gray value of the point, range is the maximum value of the selected radius and X _i and Y _i are the coordinate values of the scalpel point P '_i;

• Step 4: The P ' _{i of} the first image is matched to the second image in order to obtain a pixel spacing between the matching points P'' _i , P' _i and P ''_i;
• Step 5: The formula for calculating the depth of the scalpel point is as follows: $$\text{z}\text{'}=\left(\frac{\text{r}}{\mathrm{\Delta }\text{r}}+1\right)\text{a}$$
  

• Schritt 6: Die Position und die relative Tiefe des Skalpells werden gemäß der Berechnung erhalten, und die Position der Iris wird ermittelt und die Schnittsposition wird an dem Punkt am Rand der Iris positioniert, der der Pupillenmitte am nächsten liegt;
• Schritt 7: Das standardmäßige nahtlose Hornhauthauptschnitt wird als lose Z-Form definiert, und der Maschinenbedienarm steuert die Bewegung des Skalpells entlang des eingestellten Schnittwegs und wird wie folgt definiert:
  • S1: Das Skalpell 1 bewegt sich um 5 mm vorwärts und tritt in die Hornhautschicht 6 ein;
  • S2: Das Skalpell 1 wird um 45 Grad angehoben, um eine RCM-Bewegung am ersten entfernten Mittelpunkt 7 durchzuführen;
  • S3: Das Skalpell 1 bewegt sich um 1,5-2 mm vorwärts gemäß dem Winkel in S2, bis es den zweiten entfernten Mittelpunkt 8 erreicht;
  • S4: Nach der Durchführung der RCM-Bewegung wird das Skalpell 1 um 45 Grad abgesenkt;
  • S5: Das Skalpell 1 bewegt sich um 2 mm vorwärts gemäß dem Winkel in S4 weiter, bis es die Hornhautschicht 6 passiert.

Here r is the distance between the scalpel point P 'and the magnification center of the microscope, and Δr is the pixel distance between P' _i and P '' _i , which is referred to as parallax;

• Step 6: The position and the relative depth of the scalpel are obtained according to the calculation, and the position of the iris is found and the cutting position is positioned at the point on the edge of the iris which is closest to the pupil center;
• Step 7: The standard seamless corneal main cut is defined as a loose Z-shape and the machine operator controls the movement of the scalpel along the set cutting path and is defined as follows:
  • S1: The scalpel 1 moves forward 5 mm and enters the corneal layer 6th a;
  • S2: The scalpel 1 is raised 45 degrees to make an RCM move at the first distant center point 7th perform;
  • S3: The scalpel 1 moves forward 1.5-2 mm according to the angle in S2 until it is the second distant center point 8th achieved;
  • S4: After performing the RCM movement, the scalpel becomes 1 lowered by 45 degrees;
  • S5: The scalpel 1 moves forward 2 mm according to the angle in S4 until it reaches the corneal layer 6th happened.

How the present embodiment works: The scalpel point is the position of the scalpel. The three-dimensional coordinates of the scalpel are obtained by calculating the position of the scalpel and the depth relative to the microscope system. At the same time, the received position information such as the cornea and iris brings the scalpel into its starting position. The machine operating arm controls the movement of the scalpel along the set cutting path in order to automatically complete the cut.

The method is based on a machine operating arm and a scalpel, whereby the method for performing a fully automatic corneal incision with a scalpel is implemented and the injury to the wound enlargement caused by inaccurate manual operation and other injuries to the corneal layer caused by the laser are avoided.

Embodiment 2

• Das Anpassungsverfahren darin besteht, einen P'_i-zentrierten Basispunktblockbereich in P'_i einzurichten, einen anderen Punkt als P'_i als einen benachbarten Punkt zu definieren und die Tiefe jedes benachbarten Punkts relativ zu P'_i zu definieren, wobei die dem zweiten Bild entsprechenden Anpassungspunkte durch Anpassen aller Basispunkte des Blockbereichs berechnet werden, wobei der Anpassungspunkt und die entsprechende Parallaxe durch Auswählen der Parallaxe berechnet werden, die den Wert der Kostenfunktion minimiert, wobei die Formel der Kostenfunktion ist,

$$\text{C}\left(\text{P}{\text{'}}_{\text{ij}},\mathrm{\Delta }\text{r}\right)=\text{min}\left\{\text{d}\left(\text{P}{\text{'}}_{\text{ij}},\text{P}{\text{'}}_{\text{ij}}-\mathrm{\Delta }\text{r},{\text{I}}_{\text{Z}},{\text{I}}_{\text{z}-\text{a}}\right),\text{ d}(\text{P}{\text{'}}_{\text{ij}}-\mathrm{\Delta }\text{r},\text{P}{\text{'}}_{\text{ij}},{\text{I}}_{\text{Z}-\text{a}},{\text{I}}_{\text{z}}\right\}$$

$$\begin{array}{l}\text{d}\left(\text{P}{\text{'}}_{\text{ij}},\text{P}{\text{'}}_{\text{ij}}-\mathrm{\Delta }\text{r},{\text{I}}_{\text{z}},{\text{I}}_{\text{z}-\text{a}}\right)=\text{minP}{\text{'}}_{\text{ij}}-\mathrm{\Delta }\text{r}-0.5\\ \le \text{P}{\text{'}}_{\text{ij}}-\mathrm{\Delta }\text{r}+0.5\left\{\left|{\text{I}}_{\text{z}}\left(\text{P}{\text{'}}_{\text{ij}}\right)-{\text{I}}_{\text{z}-\text{a}}\left(\text{P}\text{'}{\text{'}}_{\text{ij}}\right)\right|\right\}\end{array}$$

In the present embodiment, a specific adjustment method, a checking method of the adjustment result, and a depth calculation optimization method based on Embodiment 1 are provided, in particular:

• The fitting method is to establish a P ' _i -centered base point block area in P' _i , define a point other than P ' _i as an adjacent point, and define the depth of each adjacent point relative to P' _i , being that of the second image corresponding fitting points are calculated by fitting all the base points of the block area, the fitting point and the corresponding parallax being calculated by selecting the parallax that minimizes the value of the cost function, where the formula of the cost function is

$$\text{C.}\left(\text{P.}{\text{'}}_{\text{ij}},\mathrm{\Delta }\text{r}\right)=\text{min}\left\{\text{d}\left(\text{P.}{\text{'}}_{\text{ij}},\text{P.}{\text{'}}_{\text{ij}}-\mathrm{\Delta }\text{r},{\text{I.}}_{\text{Z}},{\text{I.}}_{\text{z}-\text{a}}\right),\text{d}(\text{P.}{\text{'}}_{\text{ij}}-\mathrm{\Delta }\text{r},\text{P.}{\text{'}}_{\text{ij}},{\text{I.}}_{\text{Z}-\text{a}},{\text{I.}}_{\text{z}}\right\}$$

$$\begin{array}{l}\text{d}\left(\text{P.}{\text{'}}_{\text{ij}},\text{P.}{\text{'}}_{\text{ij}}-\mathrm{\Delta }\text{r},{\text{I.}}_{\text{z}},{\text{I.}}_{\text{z}-\text{a}}\right)=\text{minP}{\text{'}}_{\text{ij}}-\mathrm{\Delta }\text{r}-0.5\\ \le \text{P.}{\text{'}}_{\text{ij}}-\mathrm{\Delta }\text{r}+0.5\left\{\left|{\text{I.}}_{\text{z}}\left(\text{P.}{\text{'}}_{\text{ij}}\right)-{\text{I.}}_{\text{z}-\text{a}}\left(\text{P.}\text{'}{\text{'}}_{\text{ij}}\right)\right|\right\}\end{array}$$

P ' _{ij is} the pixel position of the corresponding base point, P'' _ij is the pixel position of the corresponding adjustment point, I _z is the gray value of each pixel in the first image, I _za is the gray value of each pixel in the second image, I _z (P ' _ij ) is the gray value of P' _ij and I _za (P '' _ij ) is the gray value of P '' _ij , and Δr is the pixel distance between P ' _i and P'' _i , which is called parallax.

Verification step: If the position of the corresponding adjustment point P '' _ij in a large number of second images relative to P '' _i coincides with the position of the neighboring point relative to P'i, the verification is successful (where P ' _{ij is} the field point, P '' _ij is the match point of the field point, P ' _i is the midpoint and P'i is the match point of the midpoint); A point is center P _{'i is} added, the total number of votes is 8 (8 field points), and 4 or more voices (including 4 parts) are considered to be successful adaptation; Otherwise, a base point block area is established around each neighboring point, the calculation formula in step 4 is executed again, the fitting point and the corresponding parallax are calculated again. More than half of the fitting points have the same relative position to the neighboring points, thereby avoiding the situation that P ' _{i is} a noise point in order to improve the correctness of the scalpel point.

The matching point and the corresponding parallax are recalculated and checked, that the vote is carried out, and the first person with more than 4 votes will be deemed the winner, the first successful neighboring observation point is _i defined as the new point P ', and other neighboring Points and unsuccessful checkpoints are discarded and then the remaining match point P '' _ik in the second image is used as the base point to match with the first image to get the check point, and if the check point coincides with the position of the base point it is the Adjustment successful, otherwise the verification step is carried out again. The correctness of the adjustment is checked again by reversing the adjustment direction in order to improve the accuracy of the scalpel point.

$$\text{P}\text{'}{\text{'}}_{\text{ikr}}=\left(\text{P}\text{'}{\text{'}}_{\text{ik}-\text{Li}}+\text{P}\text{'}{\text{'}}_{\text{ik}}\right)/2$$

$${\text{Z}}_{\text{kj}}=\left(\frac{\text{r}}{\mathrm{\Delta }{\text{r}}_{\text{l}}}\Rightarrow {\text{min}}_{\text{il}=\text{ik},\text{ikr},\text{ikl}}\left\{\text{C}\left(\text{P}{\text{'}}_{\text{i}},\mathrm{\Delta }{\text{r}}_{\text{il}}\right)\right\}+1\right)\text{a}$$

The depth calculation optimization is specifically the interpolation optimization of the depth calculation formula. The specific method is that a Interpolationsoptimierungsberechnung is performed on the depth calculation formula of the scalpel point and 'defines _ik two interpolation points along the epipolar line L _ik, namely the left interpolation point P' of each P '' _ikl and the right interpolation point P '' _ikr, wherein the cost function for adjusting the block area is used to calculate the costs C (P ' _i , Δr _ikr ), C (P' _i , Δr _ikl ) and C (P ' _i , Δr _ik ), the interpolation point or origin of the minimum Remove cost and remove the remaining two dots to get the final depth formula as follows, $$\text{P.}\text{'}{\text{'}}_{\text{ikl}}=\left(\text{P.}\text{'}{\text{'}}_{\text{ik}+\text{Li}}+\text{P.}\text{'}{\text{'}}_{\text{ik}}\right)/2$$

$$\text{P.}\text{'}{\text{'}}_{\text{ikr}}=\left(\text{P.}\text{'}{\text{'}}_{\text{ik}-\text{Li}}+\text{P.}\text{'}{\text{'}}_{\text{ik}}\right)/2$$

$${\text{Z}}_{\text{kj}}=\left(\frac{\text{r}}{\mathrm{\Delta }{\text{r}}_{\text{l}}}\Rightarrow {\text{min}}_{\text{il}=\text{ik},\text{ikr},\text{ikl}}\left\{\text{C.}\left(\text{P.}{\text{'}}_{\text{i}},\mathrm{\Delta }{\text{r}}_{\text{il}}\right)\right\}+1\right)\text{a}$$

P'_ij entspricht; P''_ik+Li ist der Wert, bei dem P''_ik die Entfernung von Li Pixeln entlang der Epipolarlinie vorwärts bewegt, und P''_ik-Li ist der Wert, bei dem P''_ik die Entfernung von Li Pixeln entlang der Epipolarlinie rückwärts bewegt.L _{ik is} the epipolar line and Z _kj is the depth, the P _ik

P 'corresponds to _ij; P '' _{ik + Li} is the value at which P '' _{ik advances} the distance of Li pixels along the epipolar line, and P '' _ik-Li is the value at which P '' _ik moves the distance of Li pixels along the epipolar line moves backwards.

Finally, the formula for calculating the depth is: $$Z_{p_i^{\text{'}}}=Z_{p_i^{\text{'}\text{'}}}=\frac{{\displaystyle {\sum }_{kj}{Z}_{kj}}}{n_O}$$

Here, n _{o is} the number of Z _kj .

The advantageous effects of the present invention are that by setting up the adjustment and checking the block area, the position of the scalpel can be found more precisely and the calculation of the depth becomes more precise through interpolation optimization.

Embodiment 3

• Das Skalpell 1 führt eine RCM-Bewegung durch, die Position des RCM-Punkts festgelegt ist, und die Steuerstrategieformel des RCM-Punkts wie folgt ist, $${\text{x}}_{\text{R}}=\frac{{\text{<figure-callout id="1" label="L" filenames="DE112019006884T5\_0058.png" state="{{state}}">L</figure-callout>}}_1+{\text{<figure-callout id="2" label="L" filenames="DE112019006884T5\_0058.png" state="{{state}}">L</figure-callout>}}_2}{2}+\left({\text{L}}_{\text{tool}}+{\text{L}}_3\right)\text{sin}\left(\mathrm{\phi +\vartheta }-\frac{\mathrm{\pi }}{2}\right)+{\text{L}}_{\text{o}}\text{ cos}\left(\mathrm{\vartheta }\right)+{\text{d}}_{\text{m}}\text{cos}\left(\mathrm{\phi +\vartheta }-\frac{\mathrm{\pi }}{2}\right)$$
  
  $${\text{y}}_{\text{R}}=\left({\text{L}}_{\text{tool}}+{\text{L}}_3\right)\text{cos}\left(\mathrm{\phi +\vartheta }-\frac{\mathrm{\pi }}{2}\right)-{\text{L}}_{\text{o}}\text{ cos}\left(\mathrm{\vartheta }\right)-{\text{d}}_{\text{m}}\text{sin}\left(\mathrm{\phi +\vartheta }-\frac{\mathrm{\pi }}{2}\right)$$
  

In the present embodiment, there are provided a movement trajectory control algorithm formula and a method for a machine operating arm having parallel series joints based on Embodiment 1 or Embodiment 2, specifically:

• The scalpel 1 performs an RCM move, the position of the RCM point is fixed, and the control strategy formula of the RCM point is as follows, $${\text{x}}_{\text{R.}}=\frac{{\text{L.}}_1+{\text{L.}}_2}{2}+\left({\text{L.}}_{\text{tool}}+{\text{L.}}_3\right)\text{sin}\left(\mathrm{\phi \; +\; \vartheta }-\frac{\mathrm{\pi }}{2}\right)+{\text{L.}}_{\text{O}}\text{cos}\left(\mathrm{\vartheta }\right)+{\text{d}}_{\text{m}}\text{cos}\left(\mathrm{\phi \; +\; \vartheta }-\frac{\mathrm{\pi }}{2}\right)$$
  
  $${\text{y}}_{\text{R.}}=\left({\text{L.}}_{\text{tool}}+{\text{L.}}_3\right)\text{cos}\left(\mathrm{\phi \; +\; \vartheta }-\frac{\mathrm{\pi }}{2}\right)-{\text{L.}}_{\text{O}}\text{cos}\left(\mathrm{\vartheta }\right)-{\text{d}}_{\text{m}}\text{sin}\left(\mathrm{\phi \; +\; \vartheta }-\frac{\mathrm{\pi }}{2}\right)$$
  

Where x _{R is} the X-axis coordinate position of the RCM point; y _R is the Y-axis coordinate position of the RCM point; L ₁ is the linear displacement of the first linear motor; L ₂ is the linear displacement of the second linear motor; L _tool is the distance between the clamp and the tip of the scalpel; L ₃ is the linear displacement of the third linear motor; φ is the angle between the surface of the scalpel and the connecting end of the clamp; ϑ is the angle between the surface of the scalpel and the horizontal plane; L _o is the length of the surface of the scalpel; dm is the vertical distance from the third linear motor to the horizontal axis of the first linear motor or the second linear motor.

ϑ ϑ = f(t) ist, sind die eingestellten Trajektorienformeln des ersten Linearmotors, des zweiten Linearmotors und des dritten Linearmotors wie folgt, $$\begin{array}{l}{\text{L}}_3=({\text{y}}_{\text{R}}+{\text{L}}_{\text{o}}\text{ sin}\left(\text{f}\left(\text{t}\right)\right)+{\text{d}}_{\text{m}}\text{sin}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)\\ -\left({\text{L}}_{\text{tool}}\right)\text{cos}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)/\text{cos}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)\end{array}$$

$$\begin{array}{l}{\text{L}}_2={\text{x}}_{\text{R}}-\left({\text{L}}_{\text{tool}}+{\text{L}}_3\right)\text{sin}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)-{\text{L}}_{\text{o}}\text{ cos}\left(\text{f}\left(\text{t}\right)\right)-{\text{d}}_{\text{m}}\text{ cos}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)\\ +\frac{\text{h}}{2\text{tan}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)}\end{array}$$

$$\begin{array}{l}{\text{L}}_1={\text{x}}_{\text{R}}-\left({\text{L}}_{\text{tool}}+{\text{L}}_3\right)\text{sin}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)-{\text{L}}_{\text{o}}\text{ cos}\left(\text{f}\left(\text{t}\right)\right)-{\text{d}}_{\text{m}}\text{ cos}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)\\ +\frac{\text{h}}{2\text{tan}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)}\end{array}$$

When the scalpel 1 rotates around the RCM point and the angle change trajectory of $\mathrm{\vartheta }=\text{arctan}\left(\frac{\text{H}}{{\text{L.}}_{\text{1}}-{\text{L.}}_2}\right)+\frac{\mathrm{\pi }}{2}-\mathrm{\phi },$

ϑ ϑ = f (t), the set trajectory formulas of the first linear motor, the second linear motor and the third linear motor are as follows, $$\begin{array}{l}{\text{L.}}_3=({\text{y}}_{\text{R.}}+{\text{L.}}_{\text{O}}\text{sin}\left(\text{f}\left(\text{t}\right)\right)+{\text{d}}_{\text{m}}\text{sin}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)\\ -\left({\text{L.}}_{\text{tool}}\right)\text{cos}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)/\text{cos}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)\end{array}$$

$$\begin{array}{l}{\text{L.}}_2={\text{x}}_{\text{R.}}-\left({\text{L.}}_{\text{tool}}+{\text{L.}}_3\right)\text{sin}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)-{\text{L.}}_{\text{O}}\text{cos}\left(\text{f}\left(\text{t}\right)\right)-{\text{d}}_{\text{m}}\text{cos}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)\\ +\frac{\text{<figure-callout id="2" label="H" filenames="DE112019006884T5\_0058.png" state="{{state}}">H</figure-callout>}}{2\text{tan}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)}\end{array}$$

$$\begin{array}{l}{\text{L.}}_1={\text{x}}_{\text{R.}}-\left({\text{L.}}_{\text{tool}}+{\text{L.}}_3\right)\text{sin}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)-{\text{L.}}_{\text{O}}\text{cos}\left(\text{f}\left(\text{t}\right)\right)-{\text{d}}_{\text{m}}\text{cos}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)\\ +\frac{\text{<figure-callout id="2" label="H" filenames="DE112019006884T5\_0058.png" state="{{state}}">H</figure-callout>}}{2\text{tan}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)}\end{array}$$

Where h is the linear distance between the first linear motor and the second linear motor.

The scalpel 1 calculates the movement trajectory of the first linear motor, the second linear motor, and the third linear motor according to the coordinate control strategy of the RCM point during the movement and rotation. At the same time, the movement trajectory of the first linear motor is also used when moving 3 , the second linear motor 4th and the third linear motor 5 calculated according to the formula to control the effect of the movement of the scalpel 1 to reach.

Additionally define the first linear motor 3 , the second linear motor 4th and the third linear motor 5 the position from the previous target position to the next target position by the linear interpolation method and set the step size between the two target positions.

In particular, the position of every millisecond for the target position of each step is obtained by calculating the cutting path.

In addition, the microscope system and the machine operator arm will be initialized before step 1 executed, the coordinates of the X-axis and the Y-axis of the machine operating arm and the microscope camera system being consistent during the initialization process. The coordinates of the machine operator and the microscope system are consistent, and the coordinates for the execution movement of the machine operator are consistent with the coordinates for calculation, thereby ensuring the accuracy of the movement of the machine operator.

In addition, the width of the cut is no greater than the width of the surface of the scalpel 1 . The machine operating arm controls the movement of the scalpel 1 takes place only along the cutting path and no displacement in other directions, which ensures that the width of the cut is not greater than the width of the surface of the scalpel, in order to avoid an expansion of the cut.

The advantageous effects of the present embodiment are that the step size of each step of the machine operator is adjusted until the cutting path is obtained and the corresponding movement trajectory of the machine operator along the cutting path and the corresponding movement trajectory of the linear motor are calculated so that the machine operator drives the scalpel, to move along the set cutting path.

Obviously, the above embodiment of the present invention is used only to clearly illustrate the examples of the present invention, rather than to limit the embodiments of the present invention. Other various forms of changes or modifications may be made on the basis of the above description to those of ordinary skill in the art. Not all embodiments can be listed here in detail. All changes, equivalent substitutions, and improvements made within the spirit and principles of the present invention fall within the scope of the claims of the present invention.

QUOTES INCLUDED IN THE DESCRIPTION

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Patent literature cited

• CN 102639088 B [0003]

Claims (10)

A method for performing an automated, seamless corneal incision, comprising a machine operating arm and a scalpel (1) installed on the machine operating arm, the machine operating arm comprising a clamping device (2) for clamping the scalpel (1) and a drive arm for driving the clamping device (2) to vibrate wherein the drive arm is provided with a first linear motor (3) and a second linear motor (4), the clamping device (2) being provided with a third linear motor (5), characterized in that it comprises the following steps: Step 1 : Two eye images are recorded with a microscope camera system with a mm vertical displacement; Step 2: Each image is preprocessed, converted into an image having a gray value, and the first image is subjected to corner point detection to obtain a plurality of candidate corner points, and the coordinates of the corner points are obtained; Step 3: A plurality of corner points are centered and R _j is the radius to find black and white points relative to the corner points, and the black point is in the white pixel area of the corner point and the white point is in the black pixel area the vertex and pixel gray level of the black dot is <0.5 and the pixel gray level of the white point is> 0.5, is the cornerstone of the scalpel point P _'i, and the formula for calculating the position of the black point and the white point is as follows: $${\text{X}}_{\text{blacki}}=\frac{{\displaystyle {\sum }_{\text{j}\in {\text{R.}}_{\text{j}}}\left(1-\text{I.}\left({\text{X}}_{\text{j}}.{\text{Y}}_{\text{j}}\right)\right){\text{X}}_{\text{j}}}}{{\displaystyle {\sum }_{\text{j}\in {\text{R.}}_{\text{i}}}\left(1-\text{I.}\left({\text{X}}_{\text{j}}.{\text{Y}}_{\text{j}}\right)\right)}}$$

$${\text{Y}}_{\text{blacki}}=\frac{{\displaystyle {\sum }_{\text{j}\in {\text{R.}}_{\text{j}}}\left(1-\text{I.}\left({\text{X}}_{\text{j}}.{\text{Y}}_{\text{j}}\right)\right){\text{Y}}_{\text{j}}}}{{\displaystyle {\sum }_{\text{j}\in {\text{R.}}_{\text{i}}}\left(1-\text{I.}\left({\text{X}}_{\text{j}}.{\text{Y}}_{\text{j}}\right)\right)}}$$

$${\text{X}}_{\text{whitei}}=\frac{{\displaystyle {\sum }_{\text{j}\in {\text{R.}}_{\text{j}}}\left(1-\text{I.}\left({\text{X}}_{\text{j}}.{\text{Y}}_{\text{j}}\right)\right){\text{X}}_{\text{j}}}}{{\displaystyle {\sum }_{\text{j}\in {\text{R.}}_{\text{i}}}\left(1-\text{I.}\left({\text{X}}_{\text{j}}.{\text{Y}}_{\text{j}}\right)\right)}}$$

$${\text{Y}}_{\text{whitei}}=\frac{{\displaystyle {\sum }_{\text{j}\in {\text{R.}}_{\text{j}}}\left(1-\text{I.}\left({\text{X}}_{\text{j}}.{\text{Y}}_{\text{j}}\right)\right){\text{Y}}_{\text{j}}}}{{\displaystyle {\sum }_{\text{j}\in {\text{R.}}_{\text{i}}}\left(1-\text{I.}\left({\text{X}}_{\text{j}}.{\text{Y}}_{\text{j}}\right)\right)}}$$

$${\text{R.}}_{\text{j}}\Rightarrow \left[{\left({\text{X}}_{\text{j}}-{\text{X}}_{\text{i}}\right)}^2+{\left({\text{Y}}_{\text{j}}-{\text{Y}}_{\text{i}}\right)}^2\le {\text{range}}^2\right]$$

X _blacki and Y _{blacki are} the coordinate values of the black point, X _whitei and Y _whitei are the coordinate values of the white point, X _j and Y _j are the respective coordinates of all pixels in the area of the corner point as the center and R _j as the radius, I is the gray value of the point, range is the maximum value of the selected radius and X _i and Y _i are the coordinate values of the scalpel point P '_i; Step 4: The P ' _{i of} the first image is matched to the second image in order to obtain a pixel spacing between the matching points P'' _i , P' _i and P ''_i; Step 5: The formula for calculating the depth of the scalpel point is as follows: $$\text{z}\text{'}=\left(\frac{\text{r}}{\mathrm{\Delta }\text{r}}+1\right)\text{a}$$

Here r is the distance between the scalpel point P ' _i and the magnification center of the microscope, and Δr is the pixel distance between P' _i and P '' _i , which is referred to as parallax; Step 6: The position and the relative depth of the scalpel are obtained according to the calculation, and the position of the iris is found and the cutting position is positioned at the point on the edge of the iris which is closest to the pupil center; Step 7: The machine operating arm controls the movement of the scalpel (1) along the set cutting path. Method for performing an automated, seamless corneal incision according to Claim 1 , characterized in that in step 4 the adaptation process is to establish a P ' _i -centered base point block area in P' _i , define a point other than P ' _i as an adjacent point, and the depth of each adjacent point relative to P' _i where the fitting points corresponding to the second image are calculated by fitting all the base points of the block area, the fitting point and the corresponding parallax being calculated by selecting the parallax that minimizes the value of the cost function, the formula being the cost function, $$\text{C.}\left(\text{P.}{\text{'}}_{\text{ij}},\mathrm{\Delta }\text{r}\right)=\text{min}\left\{\text{d}\left(\text{P.}{\text{'}}_{\text{ij}},\text{P.}{\text{'}}_{\text{ij}}-\mathrm{\Delta }\text{r},{\text{I.}}_{\text{Z}},{\text{I.}}_{\text{z}-\text{a}}\right),\text{d}(\text{P.}{\text{'}}_{\text{ij}}-\mathrm{\Delta }\text{r},\text{P.}{\text{'}}_{\text{ij}},{\text{I.}}_{\text{Z}-\text{a}},{\text{I.}}_{\text{z}}\right\}$$

$$\begin{array}{l}\text{d}\left(\text{P.}{\text{'}}_{\text{ij}},\text{P.}{\text{'}}_{\text{ij}}-\mathrm{\Delta }\text{r},{\text{I.}}_{\text{z}},{\text{I.}}_{\text{z}-\text{a}}\right)=\text{minP}{\text{'}}_{\text{ij}}-\mathrm{\Delta }\text{r}-0.5\\ \le \text{P.}{\text{'}}_{\text{ij}}-\mathrm{\Delta }\text{r}+0.5\left\{\left|{\text{I.}}_{\text{z}}\left(\text{P.}{\text{'}}_{\text{ij}}\right)-{\text{I.}}_{\text{z}-\text{a}}\left(\text{P.}\text{'}{\text{'}}_{\text{ij}}\right)\right|\right\}\end{array}$$

P ' _{ij is} the pixel position of the corresponding base point, P'' _ij is the pixel position of the corresponding adjustment point, I _z is the gray value of each pixel in the first image, I _za is the gray value of each pixel in the second image, I _z (P ' _ij ) is the gray value of P' _ij and I _za (P '' _ij ) is the gray value of P '' _ij . Method for performing an automated, seamless corneal incision according to Claim 2 , characterized in that a checking step is provided between step 4 and step 5, and if the position of the corresponding matching point P '' _ij in a plurality of second images relative to P '' _i with the position of the neighboring point relative to P ' _i matches, the check is successful; Otherwise, a base point block area is established around each neighboring point, and the fitting point and the corresponding parallax are recalculated according to the calculation formula in step 4. Method for performing an automated, seamless corneal incision according to Claim 3 Characterized in that at least half of the adjustment points corresponds relative to P '' _i with the position of neighboring points relative to P _'i. Method for performing an automated, seamless corneal incision according to Claim 3 , Characterized in that the first successful adjacent check point _'i is defined, and other adjacent points and unsuccessful verification points are discarded and then the remaining matching point P' as the new point P _'ik in the second image as a base point for matching with the first image is used to get the verification point, and if the verification point coincides with the position of the base point, the adjustment is successful, otherwise the verification step is performed again. Method for performing an automated, seamless corneal incision according to Claim 5 Characterized in that a Interpolationsoptimierungsberechnung at the depth calculation formula of the scalpel point in step 5 is performed, and 'defines _ik two interpolation points along the epipolar line L _ik, namely the left interpolation point P' of each P '' _ikl and the right interpolation point P '' _ikr, where the block area adjustment cost function is used to calculate costs C (P ' _i , Δr _ikr ), C (P' _i , Δr _ikl ) and C (P ' _i , Δr _ik ), the interpolation point or origin of the remove minimal cost and remove the remaining two dots to get the final depth formula as follows, $${\text{Z}}_{\text{kj}}=\left(\frac{\text{r}}{\mathrm{\Delta }{\text{r}}_{\text{l}}}\Rightarrow {\text{min}}_{\text{il}=\text{ik},\text{ikr},\text{ikl}}\left\{\text{C.}\left(\text{P.}{\text{'}}_{\text{i}},\mathrm{\Delta }{\text{r}}_{\text{il}}\right)\right\}+1\right)\text{a}$$

L _{ik is} the epipolar line and Z _kj is the depth that corresponds to _{P '' ik} and P '_ij; The formula to calculate the final depth: $$Z_{p_i^{\text{'}}}=Z_{p_i^{\text{'}\text{'}}}=\frac{{\displaystyle {\sum }_{kj}{Z}_{kj}}}{n_O}$$

Here, n _{o is} the number of Z _kj . Method for performing an automated, seamless corneal incision according to Claim 6 , characterized in that the formulas for calculating the interpolation point are $$\text{P.}\text{'}{\text{'}}_{\text{ikl}}=\left(\text{P.}\text{'}{\text{'}}_{\text{ik}+\text{Li}}+\text{P.}\text{'}{\text{'}}_{\text{ik}}\right)/2$$

$$\text{P.}\text{'}{\text{'}}_{\text{ikr}}=\left(\text{P.}\text{'}{\text{'}}_{\text{ik}-\text{Li}}+\text{P.}\text{'}{\text{'}}_{\text{ik}}\right)/2$$

Where P '' _{ik + Li is} the value at which P '' _ik moves the distance of Li pixels forward along the epipolar line, and P '' _ik-Li is the value where P '' _{ik is} the distance of Li pixels moved backwards along the epipolar line. Method for performing an automated, seamless corneal major incision according to one of Claims 1 until 7th , characterized in that the cutting path is defined as follows, S1: the scalpel (1) enters the corneal layer (6); S2: The scalpel (1) is raised 40-50 degrees to perform an RCM movement at the first distant center point (7); S3: The scalpel (1) moves forward according to the angle in S2 until it reaches the second distant center (8); S4: After performing the RCM movement, the scalpel is lowered 40-50 degrees; S5: The scalpel (1) continues to move forward according to the angle in S4 until it passes the corneal layer. Method for performing an automated, seamless corneal incision according to Claim 8 , characterized in that the scalpel (1) performs an RCM movement in step 7, the position of the RCM point is fixed, and the control strategy formula of the RCM point is as follows, $${\text{x}}_{\text{R.}}=\frac{{\text{L.}}_1+{\text{L.}}_2}{2}+\left({\text{L.}}_{\text{tool}}+{\text{L.}}_3\right)\text{sin}\left(\mathrm{\phi \; +\; \vartheta }-\frac{\mathrm{\pi }}{2}\right)+{\text{L.}}_{\text{O}}\text{cos}\left(\mathrm{\vartheta }\right)+{\text{d}}_{\text{m}}\text{cos}\left(\mathrm{\phi \; +\; \vartheta }-\frac{\mathrm{\pi }}{2}\right)$$

$${\text{y}}_{\text{R.}}=\left({\text{L.}}_{\text{tool}}+{\text{L.}}_3\right)\text{cos}\left(\mathrm{\phi \; +\; \vartheta }-\frac{\mathrm{\pi }}{2}\right)-{\text{L.}}_{\text{O}}\text{cos}\left(\mathrm{\vartheta }\right)-{\text{d}}_{\text{m}}\text{sin}\left(\mathrm{\phi \; +\; \vartheta }-\frac{\mathrm{\pi }}{2}\right)$$

Where x _{R is} the X-axis coordinate position of the RCM point; y _R is the Y-axis coordinate position of the RCM point; L ₁ is the linear displacement of the first linear motor; L ₂ is the linear displacement of the second linear motor; L _tool is the distance between the clamp and the tip of the scalpel; L ₃ is the linear displacement of the third linear motor; φ is the angle between the surface of the scalpel and the connecting end of the clamp; ϑ is the angle between the surface of the scalpel and the horizontal plane; L _o is the length of the surface of the scalpel; dm is the vertical distance from the third linear motor to the horizontal axis of the first linear motor or the second linear motor. Method for performing an automated, seamless corneal incision according to Claim 9 , characterized in that the scalpel (1) rotates around the RCM point and the angle change trajectory of $\mathrm{\vartheta }=\text{arctan}\left(\frac{\text{H}}{{\text{L.}}_{\text{1}}-{\text{L.}}_2}\right)+\frac{\mathrm{\pi }}{2}-\mathrm{\phi },$

ϑ ϑ = f (t), where the set trajectory formulas of the first linear motor (3), the second linear motor (4) and the third linear motor (5) are as follows, $$\begin{array}{l}{\text{L.}}_3=({\text{y}}_{\text{R.}}+{\text{L.}}_{\text{O}}\text{sin}\left(\text{f}\left(\text{t}\right)\right)+{\text{d}}_{\text{m}}\text{sin}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)\\ -\left({\text{L.}}_{\text{tool}}\right)\text{cos}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)/\text{cos}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)\end{array}$$

$$\begin{array}{l}{\text{L.}}_2={\text{x}}_{\text{R.}}-\left({\text{L.}}_{\text{tool}}+{\text{L.}}_3\right)\text{sin}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)-{\text{L.}}_{\text{O}}\text{cos}\left(\text{f}\left(\text{t}\right)\right)-{\text{d}}_{\text{m}}\text{cos}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)\\ +\frac{\text{H}}{2\text{tan}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)}\end{array}$$

$$\begin{array}{l}{\text{L.}}_1={\text{x}}_{\text{R.}}-\left({\text{L.}}_{\text{tool}}+{\text{L.}}_3\right)\text{sin}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)-{\text{L.}}_{\text{O}}\text{cos}\left(\text{f}\left(\text{t}\right)\right)-{\text{d}}_{\text{m}}\text{cos}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)\\ +\frac{\text{H}}{2\text{tan}\left(\mathrm{\phi }+\text{f}\left(\text{t}\right)-\frac{\mathrm{\pi }}{2}\right)}\end{array}$$

Where h is the linear distance between the first linear motor and the second linear motor.

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