Calin Vaida 1 , Doina Pisla 1 , Josef Schadlbauer 2 , Manfred Husty 2 - - PowerPoint PPT Presentation

Calin Vaida1 , Doina Pisla1 , Josef Schadlbauer2 , Manfred Husty2 and Nicolae Plitea1

1Research Center for Industrial Robots Simulation and Testing, Technical University of Cluj-Napoca, Romania 2Unit for Geometry and CAD, University of Innsbruck, Austria

Nantes, July 9th , 2015

Introduction

Brachytherapy Robotics in brachytherapy

PARA-BRACHYROB - The medical parallel robot Singularities analysis using the Jacobi matrices The kinematics using Study parameters Singular poses of the manipulator Results interpretation and implementation The experimental model Conclusions

Brachytherapy (BT), also known as internal radiotherapy, sealed source radiotherapy, curietherapy or endocurietherapy, is an advanced cancer treatment technique, where radioactive seeds are delivered directly in the tumor area . It involves the placement of tiny radioactive miniaturized sources precisely in the tumor area, delivering high dosage of radiation in the cancerous cells. Its effectiveness is clearly demonstrated, its side effects are reduced to a minimum, but it involves an important condition: the catheters delivering the radioactive sources must be placed precisely as the radiation dose decreases abruptly from the base and incorrect positioning causes the necrosis of healthy tissue without affecting the tumor.

Why is BT undervalued? A report presented in the AAPM (American Association of Physicists in Medicine-

• prof. Podder and Fichtinger) meeting in 2010, shows that robotic BT is

underdeveloped as most of the solutions target only the prostate, without any device capable of performing the BT tasks on larger areas of the body. EUCLIDIAN MrBOT

A new approach

The development of a robotic system, able to perform general BT procedures, capable of targeting any organs in the thoracic and abdominal areas, like liver, lungs, paravertebral areas, breast, kidney, etc.

CT scan with needle placement (arrow) for transpulmonary biopsy CT-Sim scan with laser positioning system (Courtesy of the Oncology Institute Cluj-Napoca, Romania)

A new report presented in the AAPM (written by the same task force leaded by prof. Podder and Fichtinger) dating from august 2014, illustrates the progress made in the last four years, defining more clearly several aspects for robotic brachytherapy:

• Specific criteria for the robots;
• Emphasis on safety issues;
• Testing protocols for new robotic solutions validation;
• The need for further researches as still most robotic solutions cover only prostate

brachytherapy with one solution focused on lung treatment. Evolution of robotic brachytherapy

Kinematic scheme of the medical parallel robot

3 2 1

, , q q q

Patent pending, 2013

The medical task of the robot :

The robot should introduce, based on radiologic data, needles with diameters varying from 0.6 mm up to 2 mm and lengths from 100 mm up to 250 mm, on distances up to 200 mm, following a linear trajectory. For this task the robotic system receives a set of two points: point I ( 𝑌𝑗, 𝑍

𝑗, 𝑎𝑗 ) -

the insertion point and point T ( 𝑌𝑈, 𝑍

𝑈, 𝑎𝑈 ) - the target point. In addition, a third

point is introduced in the algorithm, the current robot position, point C ( 𝑌𝐷, 𝑍

𝐷,

𝑎𝐷, 𝜘𝐷, 𝜄𝐷). Based on this data, the final needle orientation is computed:  

                

T I 2 I T 2 I T IT I T I T IT

Z Z , ) X X ( ) Y Y ( 2 atan = X X , Y Y 2 atan =  

The medical task of the robot : This task will be achieved through a motion decomposed in two different parts:

• the approach stage (from point C to I), when the first five actuators are used, to

reach the point of entry inside the patient, with the final orientation, having the pose ( 𝑌𝐽𝑈, 𝑍

𝐽𝑈, 𝑎𝐽𝑈, 𝜘𝐽𝑈, 𝜄𝐽𝑈 );

• the needle insertion stage (from point I to T), when the sixth actuator is used to

push the needle, on a straight line, to reach the target point having the pose ( 𝑌𝑈, 𝑍

𝑈, 𝑎𝑈, 𝜘𝑈, 𝜄𝑈).

The system should also allow the control of the insertion force into the patient’s tissue.

Kinematic model was presented in [Plitea et al, 2014]

The system of implicit functions, which results using the geometrical relations between the coordinates of the active joints, the needle tip and the geometrical parameters of the structure, is:

 

                                                                               

2 4 5 2 2 2 2 4 5 2 2 2 2 E 2 O E 2 O E c 6 2 2 c 6 2 E 2 E 2 2 O 5 4 c 6 2 E 4 3 2 1 2 2 1 1 c 6 E 3 3 2 1 2 2 1 1 c 6 E 2 1 c 6 1 E 1

) q q ( d b 2 ) q q ( d b X X 2 ) ( sin Y ) ( cos ) X X ( ) ( sin ) l q ( 2 ) ( sin ) l q ( Y X X : f q ) ( cos ) l q ( l Z : f ) q ( sin ) q q ( d b ) ( sin ) ( sin ) l 2 q ( Y : f ) q ( cos ) q q ( d b ) ( cos ) ( sin ) l 2 q ( X : f q ) ( cos ) l 2 q ( l Z : f          

Where matrix A is: and matrix B is: and

Kinematic model

                                                      

5 5 5 5 4 3 3 2 2 1

1 1 1 1 f f Y f X f f f f f f f A

E E

                                        

5 5 4 5 3 3 2 3 1 3 3 2 2 2 1 2

1 1 q f q f q f q f q f q f q f q f B

 

 T

T E E E

q q q q q q Z Y X X

5 4 3 2 1

, , , , , , , , ,                

   T

T E E E

q q q q q q Z Y X X

5 4 3 2 1

, , , , , , , , ,    

 

 T

T E E E

q q q q q q Z Y X X

5 4 3 2 1

, , , , , , , , ,                            

Safety in use

The determinants of A Jacobi and B Jacobi matrices are:  

) ( cos Y ) ( sin ) X X ( ) ( sin l 2 = ) A ( det

E E 2 O 2 2 c

          

2 4 5 2 2 2 1 2 2 1 2 4 5 2 2 2 2 1 2 2 1 1 4 5 1 2

) q q ( d ) q q ( d ) q q ( d b ) q q ( d b ) q q ( ) q q ( = ) B ( det                            

= ) (A det

= lc

• this situation corresponds to the case when the distance between the two

Cardan joints is zero, meaning that the points and superpose; This situation cannot materialize in practice and can be discarded; 𝐷1 𝐷2

= ) ( sin2 

• this equation characterizes the vertical position of the needle module, when

the two Cardan joints are positioned on a vertical axis (parallel with the OZ axis). In this configuration the robot gains one degree of freedom, allowing the rotation of the needle positioning module around the axis defined by the points and . In the robot control algorithm this situation will be avoided by imposing that 𝐷1 𝐷2 𝜄𝐽𝑈 ≠ 𝜌 2

= ) ( cos Y ) ( sin ) X X (

E E 2 O

     

which is equivalent with

E 2 O E

X X Y = ) ( tan   

Considering the fact that the needle tip coordinates (point E) will be always located between the two robot arms, the ratio will be always positive, meaning that the angle will take values in the domain The configuration corresponds to the case when the coordinates of and E are located in the same plane. In the control algorithm this situation will be tested when a trajectory is generated, and in case a singular configuration an alternate trajectory will be computed.

E 2 O E

X X Y 



                    ,2 2 3 , 2

2 1 5 4

C , C , q , q

= ) B ( det

2 1

q = q

, respectively - these configurations define the case when one

• f the two pairs of translational active joints of the PARA-BRACHYROB

modules superpose. In a real scenario this case cannot appear, the experimental model of the robot having limit switches which avoid this situation;

5 4

q = q

1 2 1

q q = d 

, respectively - this configuration defines the maximum distance between the two pairs of translational active joints which corresponds to the case when the rod will be positioned vertically. This second case defines an extreme configuration which is avoided in the robot control system.

4 5 2

q q = d 

Basic structure of the first PRPRR chain Basic structure of the second PRPRR chain Equivalent serial structures for each chain

Direct kinematics:

2 , 1

3 5 2 4 3 2 1 1

         i M T M T T T T M L

i i i i

The transformation matrix L1 resp. L2 is representing a Euclidean displacement for all sets of parameters. These Euclidean displacements are mapped to the kinematic image space P7 using the kinematic mapping k introduced by Study:

 

                                                               

1 1 5 3 4 2 1 4 2 4 2 1 5 4 3 2 4 3 3 2 5 4 1 5 2 1 5 4 5 2 5 4 1 5 2 1 3 5 1 5 5 1 4 3 2 5 4 2 1 5 4 2 5 4 2 1 4 1 2 1 4 2 4 1 2 1 5 4 3 5 3 2 4 2 5 4 2 4 2 5 4 2 1

2 2 2) (2 ) ( 2 2 2 = t l l t t t t t t t l t t l t t t t t t t t t t t t t t t t l t t l t t l t t l t t t t l t l t t t t t t t t t t l t t t l t t t t t l t l t l t l t t t t t t t t t t t t t t t t

c c c c c c c c

L

The transformation matrix L1 resp. L2 is representing a Euclidean displacement for all sets of parameters. These Euclidean displacements are mapped to the kinematic image space P7 using the kinematic mapping k introduced by Study:

 

                                 

4 3 2 1 4 2 5 4 2 4 2 5 4 2 2

2 2 2) (2 ) ( 2 2 2 = p p p p s s s s s s s s s s L

1 1 5 3 5 12 4 2 1 4 2 4 2 1 5 4 3 2 5 4 2 12 4 4 3 3 2 4 12 2 12 5 4 1 5 2 1 5 4 5 2 5 4 1 5 2 1 3 3 12 5 1 5 5 1 4 3 2 4 2 12 5 4 2 1 5 4 2 5 4 2 1 2 4 1 2 1 4 2 4 1 2 1 5 4 3 5 3 2 5 4 12 5 2 12 1

= = = = s l l s s s d s s s s s l s s l s s s s s s s d p s s s s s d s d s s s s s s s s l s s l s s l s s l p s d s s s l s l s s s s s d s s s s s s s l s s s l p s s s s s l s l s l s l s s s s s s s s d s s d p

c c c c c c c c

                                     

 

2 .. 1 ,   i

i

L

• the 5-dimensional varieties of the Study quadric.

Their representation is computed using the LIA (Linear Implizitization Algorithm), in which process all the the passive joints (table 1) are eliminated. The degree of the resulting polynomials in Study parameters: has to be fixed.

3 2 1 3 2 1

, , , , , , , y y y y x x x x

For using LIA 4 linear and one quadric polynomial are found, where the first constraint polynomial is:

 

1

L  = ) 2 (2 4 ) 2 2 2 2 2 (2 ) 2 2 (

2 3 3 2 2 1 3 2 1 1 2 2 1 2 2 2 2 1 3 2 3 2 1 2 1 1 2 1 2 3 2 2 2 2 2 1 2 2 2 2 2 1 1 2 2 2 1

y t t t y t t y t l l t t t l t l x t t l t l l t t t t t l t t l t l

c c c c

                 

C

l l t t t , , , ,

1 3 2 1

• input parameters of the first chain

The ideal generated by the 5 polynomials is denoted with I1

For LIA returns no linear polynomial. Instead it returns 10 quadratic polynomials, 56 polynomials of degree 3 and 205 polynomials of degree 4. Further inspection shows that all polynomials of degrees 3 and 4 are included in the ideal generated by the quadratic polynomials. The ideal generated by these 10 quadratic polynomials is denoted by I2.

 

2

L 

One of the quadratic polynomial is:

= 2 2 ) ( ) ( ) ( ) (

3 2 1 1 3 1 1 2 1 1 3 1 1 1 2 1 1

y y y y y x s l l y x s l l y x s l l y x s l l

c c c c

              

Another quadratic polynomial is the Study quadric itself:

=

3 3 2 2 1 1

y x y x y x y x   

For the solution of the direct kinematics the zero set of the reunion of the two ideals have to be computed of every set of input parameters:

I= I 1 I 2

The vanishing set of an ideal are all common roots of the polynomial which generate the ideal. The first ideal I1 is generated by the 5 polynomials (4 linear and 1 quadratic ). Solving for all linearly, results:

3 .. ,  i yi . 1) 2( 2 = , 1) 2( 2 = , 1) 2( 2 = , 1) 2( 2 =

2 2 2 3 1 1 1 3 2 2 3 2 2 2 2 1 2 2 2 2 1 3 2 2 3 3 1 1 1 1 1 3 2 3 3 2 2 1 2 2 1 1 2 2 1 2 2 1 2 2 2 3 2 1 2 2 1 3 3 2 3 2 2 2 2 2 1 2 2 2 2 2 2 1 1 2 2 1 3 3 1 3 3 1 2 3 2 1 3 2 2 3 2 2 1 3 2 2 3 2 2 1

                                      t x t x t x l x l x t t x t t x t t x t l x t l y t x t x t x l x l x t t x t t x t t x t l x t l y t x t x t x l x l x t t x t t x t t x t l x t l y t x t x t x l x l x t t x t t x t t x t l x t l y

c c c c c c c c

Substituting the yi into the quadratic equation in I1 results:

=

3 2 1

x x x x 

equation which is contained also in I2. Substituting the yi into the polynomial equations of I2 will determine a new ideal, I2’ with the unknowns xi. The ideal I2’ is generated by 10 polynomials. Adding the normalizing condition:

= 1

2 3 2 2 2 1 2

    x x x x

Maple computed a basis with 5 other generating polynomials. As the ordering was chosen to be lexicographic, the basis contains a univariate polynomial which is of degree 8. It has to be noted that up to now all the computations could be done without specifying the design and input parameters.

The resulting ideal I2 ‘ is: A numerical example is presented with the following input parameters:

600. = 700, = , 2 1 = 100, = 300, = 65, = 100, = 500, =

3 3 2 1 1 1 12

s t t s t l l d

c

. > 827896 1598728406 4285 7273655324 97777920 1589702343 68000000 4616550215 480000000 ,340241940 36 3489996987 03 1571917965 81440 1019603323 00000 4485611520 00000 4430233600 , 72 7344062406 5 2186786038 45440 2029405276 00000 5980815360 00000 4430233600 245, 1115133370 60064 7154969951 69185280 1044084849 32000000 3062177464 200000001 2268279603 <

3 3 3 5 3 7 3 3 1 3 3 5 3 7 3 3 2 3 3 5 3 7 3 2 3 4 3 6 3 8 3

x x x x x x x x x x x x x x x x x x x                 

The complete solution is presented in the next table. A numerical example is presented with the following input parameters:

600. = 700, = , 2 1 = 100, = 300, = 65, = 100, = 500, =

3 3 2 1 1 1 12

s t t s t l l d

c

The complete solution is presented in the next table. A numerical example is presented with the following input parameters:

600. = 700, = , 2 1 = 100, = 300, = 65, = 100, = 500, =

3 3 2 1 1 1 12

s t t s t l l d

c

From the previous table result the 4 different solutions for the direct kinematics of this manipulator

The top views reveal that solutions 1 and 3 respectively 5 and 7 differ only in a 180° rotation around the z axis.

The study of singularities for the PARA-BRACHYROB consist in the analysis of the Jacobian of the ideal I =g1,.., g8.

       

j i j i

y g x g , = J

The determinant of J, det(J)=0 can be computed without specifying any design or input parameters. The resulting polynomial S, has as variable x3 and degree 7. It can be written as a product of 14 factors, meaning that:

14 .. 1 , ,    i f if S

i

The first two factors are:

1 1 1 1

2 2 : t s l l f

c

  

1 1 1 2 3 2

2 2 8 : t s l l x l f

c c

   

Introducing the design parameters (in mm):

65 = 100, = 500, =

1 12

l l d

c 1 1 1

330 : t s f   70. 3 800 :

1 1 2 2

   t s x f 330 =

1 1

s t 

the height of the first two prismatic joints is exactly the distance between A1 and A2, for example

100 = 430, =

1 1

s t

Introducing the input parameters:

700 = , 2 1 =

3 2

t t

The basis of the ideal I is:

) 24 (7 320000) 19531250( , 320000) 31250000( , 320000) 109375000( , 320000) 390625( =

3 4 2 3 1 3 2 3 2 3 2 3 4 2 3

x x s x s x s s        

I Note that the first polynomial is independent of x3. This means that its value can be chosen arbitrarily, so this is a one parametric motion, but value of s3 has to be s3 = 400 2 to make the first polynomial vanish. In this case the end effector can move without changing the input parameters, a so called self-

• motion. This motion is a pure rotation about the z-axis of the moving frame.

Self motion of the manipulator

For the second polynomial, f2 the following parameters are used: Singular pose of the manipulator

300 = , 39 336 3551 10 = 100, = , 100 = 65, = 500, =

1 3 1 1 12

t s s l l d

c

 15 20 3 = , 700 = , 2 1 =

3 3 2

x t t

The numerical values of the Study parameters are:

.580 = .402 = .403 = .578 =

3 2 1

x x x x    11. = 352. = 108. = 158. =

3 2 1

y y y y  

The two approaches used for the singularities analysis of PARA-BRACHYROB enable the implementation of a safe control system for the robot. The simple geometrical interpretation of the singular poses described using the Jacobi matrices enabled the definition of a workspace of the robot limited with position sensors. The complete description of the singular poses of the robot achieved using the Study parameters defined a polynomial (S) of 14 factors which are dependant only on the geometrical parameters of the structure. det(A), det(B), (S) Safe robot behavior

PowerPanel control unit Power supply unit PARA-BRACHYROB

CT Gantry PARA-BRACHYROB Brachytherapy robot Patient Mobile CT Table

MOVIE!!!

 An innovative parallel robot for brachytherapy, able of targeting any tumor in the thoraco-abdominal areas of the body with real-time CT monitoring has been presented  The kinematics of the robot have been achieved in two different ways using the Jacobi matrices and the Study parameters  The singularities were studied using the two approaches leading to a complete description of the singular robot configurations  Both approaches characterized the singularities using analytical expressions for the determinants which enabled the implementation

• f these equations in the control system of the robot which can thus

generate safe, singularity free trajectories for the needle.

This work was supported by the Project no. 173/2012, code PN-II- PCCA- 2011-3.2-0414, entitled ”Robotic assisted brachytherapy, an innovative approach of inoperable cancers - CHANCE” and the Bilateral Austria - Romania Project 745/2014, entitled ”Developing methods to evaluate the accuracy of potential parallel robots for medical applications” both financed by UEFISCDI.

Questions