Wednesday, November 7, 2012

White paper - Chord half theory

Chord half theory of Four Bar Mechanism
Application of Surgical Instruments
Arun Soundararajan


In ground offset four bar mechanism, when horizontal movement of the right link and the left link in various swing circles are equal also within the limit of 180 deg angle of rotation. Hence, the upper link and ground link meets parallel to each other. When reverse the surgical instruments, ease of reverse the shape and critical dimensions. But functionally reverse the links are to be challenge through reverse engineering. Because, that links design decides the total output of the mechanism and links are operating by its hole positions. Hence I evaluated relation between the various swing circles of links and fix the angle rotation by equaling the chord length. Also, it’s proved by calculations as well as graphically by using 2D software. It has given a way to perfect design with low time consumption to attain the functional needs.

Key Words: Four Bar Mechanism, link design, surgical instruments, Offset ground holes, Crocodile action

PCD        Pitch Circle Diameter
R.L          Right Link
L.L          Left Link
G.L          Ground Link
RDS        Rotational Distance Method
CHT        Chord Half Theory


Crocodile is a type of surgical instrument used in laparoscopy surgery. It has to work through the small operating hole (<10mm). The working movement of the tool is restricted and small tip part has to open in the required angle. The above said are some of the key requirements for the instrument. It’s available in different sizes. When do the reverse engineering for scale up and scale down sizes, the design team found difficult to achieve the key requirements.

Crocodile – Surgical Instrument Fig.1
Instead of using a systematic mathematical solution, the design team was using iteration methods. The iteration method is very tedious and time taking process.
I approached the problem in a systematic manner. The link hole positions are the driving elements to achieve the key requirements. The four bar mechanism has been used to calculate the angle of rotation, cutting angle (q) and link hole positions. This documents talks about fixing the hole positions of the links with help of the mathematical model.

2. Related Work

In the iteration method, Pro-e /AutoCAD has been used to find the hole positions of links. The method is, fixing the ground holes constantly and moving the upper link holes along the length step by step to find relative position of holes. This method requires the extensive use of modeling software (Pro-e/AutoCAD) and highly experienced designers. It is increasing the design cost.

Exploded view and Part names Fig.2
My study on the design brings up me to solve the problem mathematically will take less time, less cost and importantly accurate.
In this instrument, the left and right link are working with various PCD (øM, øS) with offset ground holes.    (Refer the figure 3)
 I started with the assumption of the rotational distances of right link (a) and left link (b) are equal. And ended up with the relation between angle of rotations of the left and right links as below,  
Angle of Rotation (b), (From Figure.1)

This solution gave the required cutting angle but upper link and ground link is not touching them self all along. There was a gap between them  in end. In the closing/opening position, the ground link and upper link should be flushed together all along is one of the key requirement. So, the above mathematical model is not meeting all the requirements.
On further studies told me that, Horizontal movement of Link Right and Link Left should be same.
Summary of literature review/related work:

·         Iteration method is very tedious method, time taking process, high cost and experienced persons needed.
·         Equaling the rotational distance concept is mathematical solution not fulfilled the requirements. But better the iteration method.
·         Chord length method is a systematic mathematical approach meet the 100% design requirements.


Study Model of Four Bar Mechanism Fig.3

In reverse engineering of a crocodile instrument deals with outer profiles, cutting edges and the functionality. The outer profiles and critical dimensions can be measured through the various 3D scanning options. Even functionality can be obtained, if we do 1:1 Scale of reverse engineering. When it comes to scale up or scale down reverse engineering, it is difficult achieve the functionality for the different length and thickness.  The crocodile instrument is smaller in the size for child and bigger for adult. 
No issues on scaling the outer profile and cutting edges. Maintaining the cutting angle and upper link parallel motion are important requirements and difficult to achieve though reverse engineering. The requirements are driven by the following parameters,

1.        Ground link hole positions (O,P)
2.        Right link Swing Diameter (øM)
3.        Angle of rotation - Right link (a)

4.        Left link Swing Diameter (øS)
5.        Angle of rotation of the Left link (b)

The first four parameters are fixed by the width of the instrument. The width should not exceed 12mm at the hand end. The angle of rotation- Left link is driving the hole positions on the upper link. Hence the angle of rotation – Left Link decides the functionality of the mechanism.

Angle of Rotation – Left Link (b) =   ?


Find the angle of rotation (b) of the left link towards the pivot axis.

4. SYSTEM design
        The right link in rotating in larger pitch circle and left link rotating in smaller pitch circle and it’s pivoted in ground with offset holes. Upper link is connected to the ground link through the two links as shown in Figure 1.
Considering the larger pitch circle (øM) from fig.1
Right link design Fig.5
Initial Assumptions for design:-
·       Chord lengths are equal for RU and LL.
·         When right link start to rotate from R to U makes a chord in larger pitch circle diameter and forms a triangle PRU.
·         Pivot point P divides the side RU of triangle PRU symmetrically and forms two right angle triangles TRP and TUP. Hence, RT=TP.
·         Angle a is equal to both right angle triangles TRP and TUP.
Considering the smaller pitch circle (øS) from fig.1

Left link design Fig.6            
From the right angle triangle TRP,
Sin q = Opposite side / Hypotenuse
Sin a = RT/ PR
RT = Sin a * PR ---------------- Equ.1
Here, PR = Radius of  øM
Assume, Chord lengths (RU&LL) are equal.
Therefore,  RT=NK --------- Equ.2
Considering the smaller pitch circle (øP) from fig.1
Sin b = NK/ON -----------Equ.3
NK = Sin b * ON --------Equ.4
Here, ON = Radius of  øS
Substitute equ.2 in equ.3
Sin b = RT/ON
Hence, the angle of rotation (b) of øS,
b = Sin-1(RT/ON) in radians
b = Sin-1(RT/ON) X (180 ° / p ) in degrees.
By simplification,
Substitute Equ.1 & 4 in Equ.2
Radius of øM               Sin b                180
------------------- = ------------ X  -------- in deg.
Radius of øS               Sin a                  p

Angle of Rotation b in Degrees.

                           Radius of øM                        180
b   = Sin -1 {-------------------------- X Sin a } X -----------
                         Radius of øS                          p

6.       Evaluation

After find the angle of rotation of left link from the calculation, links of the Crocodile product is designed by this above method by equalling the chord length of swilling circle. Motion of the mechanism is tested by pro-mechanism by applying the limit for the angle of rotation by the calculated value. Hence, the upper link works perfectly.
Also, angle of rotation of left link can be calculated by graphical method. By using AutoCAD, When we draw the fig.5 for the requirements we can measure chord length from the diagram. Apply the same chord length in fig.6, directly measure the angle of rotation (b) from the diagram.
Instrument closed condition upper link flushed with ground link Fig.7

Instrument open condition upper link flushed with ground link with required cutting angle Fig.8
Figure 7 & 8 Shows the perfect working conditions of crocodile instrument. Equalling the chord length method is satisfying all the key requirements of the design.
Design validation of model four bar mechanism
 in Pro-E Figure.9
        Design validated for various angles and equalling the chord length of different pitch circles by using pro-e. Every design satisfied the motion of upper flushed with ground in Pro-E Mechanism in open & close condition.
Graphical method:-

Graphical method to find angle of rotation Fig.10
By using AutoCAD
From the fig10,
Ø Draw Swing circle for dia 30 and the inputs angle of rotation 20 deg as shown.
Ø Then measure the chord length = 10.26 mm
Ø Draw the dia 20 swing circle. Then offset the vertical axis by the half of the chord length at the both sides as shown
Ø Connect the intersection points. Then measure the angle of rotation 31.26 deg.
Performance matrix with time taken and cost for related concepts.  Time (hours) & Cost (INR)
Graph Plotted for time and cost for various methods
Ø Angle of rotation of left link should be with in 180 deg when calculated. Hence upper gets parallel motion.
Ø Initial design assumptions values are depend upon only the width of the instrument.
Ø Angle of rotation of right link is always decides the cutting angle of the tip part.



Link design is very significant in four bar mechanism. Relative motion of the links decides the output of the perfect mechanism. Every mechanical needs satisfied by exact relations in a mathematical approach.
Chord Half Theory model – Four bar mechanism
According to my Chord half theory, When axis of circle divide the chord symmetrically, half of the chord length of various swing circle of the right link and the left link are equal and also angle of rotation is less than 180 deg, the upper link and ground link flushed with each other parallel in the four bar mechanism.
                Where the crocodile action needed for operate a human body, hence we can use the chord half theory to design the link of the surgical instruments.

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