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Development of a new 4-DOF endoscopic parallel manipulator based on screw theory for laparoscopic surgery
Mechatronics xxx (2015) xxx–xxx
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Mechatronics journal homepage: www.elsevier.com/locate/mechatronics
Development of a new 4-DOF endoscopic parallel manipulator based on screw theory for laparoscopic surgery Khalil Ibrahim a,⇑, Ahmed Ramadan b, Mohamed Fanni c, Yo Kobayashi d, Ahmed Abo-Ismail e, Masakatus G. Fujie d a
Mechatronics Engineering, Faculty of Engineering, Assiut University, Assiut, Egypt Computers and Automatic Control Engineering Dep., Tanta University, Tanta, Egypt Mechanical Department, Mansoura University, Mansoura, Egypt d The Faculty of Science and Engineering, Waseda University, Tokyo, Japan e Mechatronics and Robotics Engineering, Egypt-Japan University of Science and Technology, Alexandria, Egypt b c
a r t i c l e
i n f o
Article history: Received 30 January 2014 Accepted 26 February 2015 Available online xxxx Keywords: Parallel manipulators Medical robotics Trajectory control Virtual chain type synthesis method
a b s t r a c t Development of rigid manipulators for Minimally Invasive Surgery (MIS) becomes essential to replace wire driven manipulators which are problematic due to possible cutting of the wire during the surgery. In this paper, a 4-DOF dexterous endoscopic parallel manipulator for MIS is designed and implemented. The manipulator is developed based on the concept of virtual chain and screw theory. The manipulator consists of four links; two links are 2-PUU (each leg consists of one active prismatic joint and two passive hook joints); the other two links are 2-PUS (each leg consists of one active prismatic join, one passive hook joint and one passive spherical joint). The inverse and forward kinematics solutions are derived analytically and numerically, respectively. The singularity analysis is investigated using screws algebra. The manipulator workspace is obtained using MATLAB software. The known problem of limited bending angles found in previous existing surgical manipulators was solved as the proposed manipulator can reach ±90° in any direction. The proposed surgical manipulator is designed, manufactured and tested successfully. The system model utilizing PID and PI controllers has been built using MATLAB software. Co-simulation using ADAMS/MATLAB software is implemented to validate the achieved bending angles and the proposed tracking control. The results show that the performance of the tracking control is satisfactory since the tracking error is about 2.5%. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Recently, Minimally Invasive Robotic Surgery (MIRS) has been increasingly introduced into therapy to reduce trauma and recovery time in hospitals. The current state of the art surgical robots are using wire driven mechanisms which are problematic. Development of endoscopic manipulator suitable for such kind of MIRS, as shown in Fig. 1, becomes essential. This allows the surgeon to carry out the surgery with high accuracy and more comfortable. In contrast to flexible endoscopes, laparoscopic instruments are required to resist much higher forces [1–9]. In the case of a laparoscopic liver surgery, 5 N are needed at the
⇑ Corresponding author. E-mail addresses: [email protected] (K. Ibrahim), ahmed.ramadan@ ejust.edu.eg (A. Ramadan), [email protected] (M. Fanni), [email protected] (Y. Kobayashi), [email protected] (A. Abo-Ismail), [email protected] (M.G. Fujie).
instrument tip. The existing endoscopic manipulators can be generally classified into two broad categories namely: flexible mechanisms, usually driven by wires, and rigid mechanism manipulators. Most of the conventional endoscopic systems in the market for MIS are of the wire-actuated type [10]. However, the rigidity and the durability of wires are poor. For flexible mechanisms, there are two commercial surgical systems which are currently in clinical use and using the wire-driven mechanisms, namely, da Vinci system (Intuitive Surgical, Inc.) [11] and ZEUS system (Computer Motion, Inc.) [12]. For rigid mechanism manipulators, they use rigid-links mechanism to transmit the motion to the gripper and perform the surgery. This approach is challenging because of the difficulty of manufacturing the mechanism and controlling the motion of the gripper with sufficient dexterity. Robotic forceps with link mechanisms have been developed, which can bend in two dimensions and grasp [13]. However, the bending angle is limited to ±70°. Ishii et al. [14] developed a robotic forceps manipulator using a new screw-drive mechanism
http://dx.doi.org/10.1016/j.mechatronics.2015.02.006 0957-4158/Ó 2015 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Ibrahim K et al. Development of a new 4-DOF endoscopic parallel manipulator based on screw theory for laparoscopic surgery. Mechatronics (2015), http://dx.doi.org/10.1016/j.mechatronics.2015.02.006
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dynamic simulation of the controlled robot is implemented using ADAMS software. This paper is organized as follows. The parallel manipulator based on virtual chain approach is introduced in Section 2. In Section 3, the kinematics analysis is presented. The singularity analysis is investigated in Section 4. The mechanical design and manufacturing of the parallel manipulator is described in Section 5. In Section 6, the optimization and simulation of the control scheme is investigated by MATLAB/ADAMS to test the motion capability of the proposed design. The results and discussion of the motion simulation and control of the proposed parallel manipulator are presented in Section 7. Finally, conclusions and future work are given in Section 8.
Fig. 1. Concept of MIRS.
2. Parallel manipulator synthesis based on virtual chain approach 2.1. Virtual chain approach
(DSD forceps). However, the fabrication accuracy of the manipulator needs improvement. Röse et al. [15] developed an endoscopic manipulator using parallel mechanism which produces different bending angles in different directions. However, the maximum bending angles is unequal in all directions. It achieves ±80° in one direction and ±40° in the perpendicular direction. Also, the kinematics analysis of this manipulator is very difficult and not reported. Yamashita et al. [16] have built an endoscopic forceps manipulator with 3-DOF, where 2-DOF are used for bending via complicated multi-slider linkage mechanism and 1-DOF for the wire driven gripper. Accuracy problems and insufficient power were reported. Kobayashi et al. [17] developed one of the Omnidirectional driven-type forceps manipulators, an active forceps manipulator in the form of a platform. Although it has high rigidity, its bending range is 40–50°, and it is difficult to expand the bending range due to inherent constraints in the mechanism. A snake-like manipulator with a diameter of 4.2 mm was developed in [18]. It can reach bending angles between ±90° in any direction by push and pull modes of three super-elastic tubes. However, rigidity and large bending force are not expected. Ibrahim et al. [19] proposed a 4-DOF parallel manipulator where computer simulations showed the feasibility of the new design and proved that the workspace of the bending mechanism was increased significantly and reached the limits of ±90°. However, maximum bending angles are not equal in all directions. Snake-like endoscopic tool, of 5 mm diameter, with 2-DOF and wire actuation is developed. It uses a wire actuated NiTi tube with laser-machined flexure joints. However, there is a hysteresis in the relation between the wire force and the bending angle [20]. The use of parallel mechanisms can be beneficial for the design of robotic surgical instruments due to their characteristics such as their high accuracy, high stiffness, and high load capacity. But most of the parallel mechanisms do not have a closed-form solution for either the forward or inverse kinematics problems mainly due to complex constraints imposed on the closed-chains [21,22]. Conventional Jacobian for an n-DOF parallel manipulator (n 6) is not a square matrix (6 n), which adds difficulty to kinematic modeling [25]. Fang et al. [26] presented the inverse velocity analysis for lower-mobility serial mechanism with the reciprocal screw. Tsai et al. [27] employed the reciprocal screw in Jacobian analysis where the Jacobian of constraints and actuations are used to build a square (6 6) overall Jacobian. In this work, the design of a novel parallel endoscopic manipulator that would facilitate performing MIRS is presented. To improve the bending motion and trajectory- tracking accuracy of the robot, a dynamic control scheme is developed and the simplified dynamic model of the proposed structure is used. Finally, the
For a known motion pattern, a virtual chain is proposed by a comprehensive wrench system analysis of a large number of serial chains and parallel chains and by considering the possible changes in the wrench system with a change of the configuration of the serial chain or the parallel chain. Also, as a general rule, the simplest possible virtual chain will be selected [28]. Wrenches of 0-pitch, 1-pitch and h-pitch are represented by f0 ; f1 and fh respectively, where h is any finite number larger than zero. For type synthesis of a Parallel Manipulator (PM) that mimics the motion pattern of a proposed Parallel Virtual Chain (PVC), a general procedure is proposed in 0 which consists of four main steps, namely: 1. 2. 3. 4.
The The The The
decomposition of the number of legs. type synthesis of sub-Parallel Kinematic Chains (PKCs). combination of sub-PKCs to generate PKCs. selection of the actuated joints.
2.2. Screw theory The normalized screw coordinates 0 may be given by:
$¼
S
ð1Þ
S0 S þ hs
where S ¼ ½ sx
sy
T
sz denotes a unit vector along the direction of
T the screw axis, and the vector S0 ¼ ½ s0x s0y s0z denotes the position vector of a point lying on the screw axis as shown in Fig. 2. Generally, the movement between a pair of bodies is determined by either a rotation or a prismatic joint. For a rotation joint, the pitch of the twist is null h = 0. In this case, the normalized screw of the ith joint is expressed by:
Fig. 2. Screw representation.
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$¼
Si S0 Si
ð2Þ
For a prismatic joint, the pitch of the twist goes to infinity and the normalized screw of the ith joint is reduced to:
$¼
0 Si
ð3Þ
2.3. Parallel Virtual Chain (PVC) In this section the manipulator parallel chain will be designed systematically using screw theory and virtual chain approach. According to Grubler’s formula, the DOF; F, can be computed as follows [30]:
F ¼ 6ð n g 1Þ þ
g X fi
ð4Þ
i¼1
where n is the number of link of PVC, g is the number of joints in PVC and f i is the number of connectivity. From screw theory approach, we can calculate the wrench system of PVC. The total wrench system of the PVC is 2 fh . 2.4. Type synthesis of the proposed PM The synthesis is restricted to 4 legs-PM with active prismatic joints located at the bottom of the four legs to be actuated by four linear motors. This restriction is dictated by the requirement of laparoscopic surgery. To get the total wrench system of PM (which is the summation of the wrench system of each leg; ci ), it can be calculated from the following Eq. (28):
F ¼6
w X ci þ D
3. Kinematics analysis 3.1. The inverse kinematics solution The main objective of this section is to derive an inverse kinematics closed-form solution for that class of parallel manipulators. Fig. 5 shows the model for the proposed PM. Points Ai and Bi (i = 1, . . . , 4) are symmetrically arranged on the circumference of the circle with radius R as shown in Fig. 6b. Reference frame O-XYZ is attached to the center of the base platform with the X-axis in the direction OP 1 , the Z-axis is perpendicular to the plane of the base and Y-axis satisfies the right hand rule. A mobile platform frame C-UVW, which has the same pose as the frame O-XYZ in the stretched configuration, is attached to the center of the mobile platform. Link Pi Ai is connected to the base platform through active Pjoint. Links Pi Ai and Ai Bi are connected through U-joints as shown in Fig. 6a. Links Ai Bi are of fixed length Li , while links P i Ai are of variable lengths di in the Z-direction due to the linear displacements of the actuators. The coordinates of the points Pi with respect to the fixed base reference frame O-XYZ are given by:
2 3 R 6 7 P 1 ¼ 4 0 5; 0
where w is the number of legs of PM and D is the number of overconstraints. Let D ¼ 0 to avoid over-constrain, w = 4 and F = 4. Substituting all parameters into (5), we get:
ð6Þ
i¼1
There are two possible solutions of Eq. (6): c1 ¼ 2 and c2 ¼ c3 ¼ c4 ¼ 0 or c1 ¼ c2 ¼ 1 and c3 ¼ c4 ¼ 0 By applying the procedure described in Section 2.1 to the second solution. The first proposed parallel manipulator 2-PRRRR_2PRRRRR can be obtained as shown in Fig. 3. The bending angles of this parallel manipulator are not the same in all directions. The maximum bending angle reaches ±90° but in only two directions. Moreover, the kinematics analysis and design are difficult [19]. For the requirements of symmetrical bending angles, the second solution is selected. In order to get equal bending angles in different directions, PVC is designed of two identical serial chains. Each serial chain consists of two hook joints (U) and one prismatic joint (P) in the bottom (PUU) as shown in Fig. 4. Applying the procedure described in section II-A with the constraints described above, one can obtain 2-PUU_2-PUS PM which consists of two PUU-legs and two PUS-legs where (S) is spherical joint as shown in Fig. 4. For manufacturing considerations, the spherical joint can be replaced by one hook joint followed by one revolute joint whose axis is perpendicular to the axes of the hook joint [31]. The limiting angles of hook joints in small scale will be taken into consideration during kinematics analysis. This new proposed parallel manipulator simplifies the design and kinematics analysis compared to the previous manipulators and has the merits of wide
2 3 0 6 7 P 2 ¼ 4 R 5; 0
2
R
3
2
0
3
6 7 6 7 P3 ¼ 4 0 5 and P4 ¼ 4 R 5 0 0
ð7Þ
Similarly, the coordinates of the points Bim referred to the mobile frame C-UVW are given by
ð5Þ
i¼1
m X ci ¼ c1 þ c2 þ c3 þ c4 ¼ 2
and symmetrical bending angles. Besides, the size of the PM will be more compact.
B1m
2 3 r 6 7 ¼ 4 0 5; 0
B2m
2 3 0 6 7 ¼ 4 r 5; 0
2
B3m
3 r 6 7 ¼ 4 0 5; 0
2
B4m
3 0 6 7 ¼ 4 r 5 0
ð8Þ
The coordinates of points Ai referred to the location of the first hook joint (U) for each chain are:
Ai ¼
P i þ di Z
ð9Þ
where Z is a unit vector in Z-direction. Substituting (7) into (9), gives:
2
R
3
6 7 A1 ¼ 4 0 5; d1
2
0
3
6 7 A2 ¼ 4 R 5 ; d2
2
R
3
2
0
3
6 7 6 7 A3 ¼ 4 0 5 and A4 ¼ 4 R 5 d4 d3 ð10Þ
The transformation from the mobile frame to the fixed base frame is expressed by a position vector Pc equal to OC and a rotation matrix b Rm described by the direction cosines of unit vectors U, V, and W, as specified in (11).
2
3 P cx 6 7 Pc ¼ 4 Pcy 5; P cz
2
Ux 6 b Rm ¼ 4 U y Uz
b
Rm can be expressed by:
b
Rm ¼ RðX; ;ÞRðY; hÞRðZ; wÞ
Vx
Wx
3
Vy
7 Wy 5
Vz
Wz
ð11Þ
ð12Þ
where RðX; ;Þ is the rotational matrix around X-axis, RðY; hÞ is the rotational matrix around Y-axis and RðZ; wÞ is the rotational matrix around Z-axis. The coordinates of points Bi referred to the base fixed frame O-XYZ are:
Please cite this article in press as: Ibrahim K et al. Development of a new 4-DOF endoscopic parallel manipulator based on screw theory for laparoscopic surgery. Mechatronics (2015), http://dx.doi.org/10.1016/j.mechatronics.2015.02.006
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(a) First Parallel Virtual Chain 2-PRRRR
(b) first proposed 2-PRRRR_2-PRRRRR PM
Fig. 3. Parallel manipulator based on PVC [19].
Bi ¼ b Rm Bim þ Pc
ð13Þ
Substituting (8) and (11) into (13), and after simplifying we get:
2
Pcx þ RU x
3
2
Pcx þ RV x
3
2
Pcx RU x
3
6 7 6 7 6 7 B1 ¼ 4 Pcy þ RU y 5; B2 ¼ 4 Pcy þ RV y 5; B3 ¼ 4 Pcy RU y 5 and 2
Pcz þ RU z Pcx RV x
Pcz þ RV z
3
Pcz RU z
6 7 B4 ¼ 4 Pcy RV y 5 Pcz RV z
ð14Þ
The kinematic closure of each elementary chain can be written as:
kAi Bi k2 ¼ L2i
ð15Þ
Substituting (10) and (14), we obtain: Fig. 4. Second parallel virtual chain.
Ai Bi ¼ b Rm Bim þ Pc Pi di Z
ð16Þ
Once, we get the orientation matrix b Rm and have the center position P c vector, using Fig. 6a, we can proceed as follows:
qi ¼ b Rm Bim þ Pc Pi
ð17Þ
where qi is a vector of the prismatic joint P to the second hook joint U in one serial chain as shown in Fig. 6a. Substituting (17) into (16), we can get:
jAi Bi j ¼ qi di Z
ð18Þ
Substituting (18) into (15), we obtain: 2
kAi Bi k2 ¼ qTi qi þ di 2di qi Z ¼ L2i 2
di 2di qi Z þ qTi qi L2i ¼ 0
ð19Þ ð20Þ
The two solutions of (20) can be expressed in simple form as
dij1;2 ¼ qi Z
Fig. 5. The proposed 2-PUU_2-PUS PM based on PVC.
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðqi Z Þ2 qTi qi þ L2i
ð21Þ
Eq. (21) indicates that there are two solutions for the actuator displacements representing the intersection of the displacement vector di Z with the surface of a sphere of radius Li and centered
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(a) First chain PUU
(b) Base platform
Fig. 6. Schematic representation of 2-PUU_2-PUS PM.
Fig. 8. Dexterous workspace of 2-PUU_2-PUS PM.
Fig. 7. Serial chain with two hook joints.
at Bi , as shown in Fig. 6a. For minimum displacement of linear actuators, we choose the negative sign for the square root, which corresponds to the lower intersection point so,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðqi Z Þ2 qTi qi þ L2i
motion of the actuated joints. Forward kinematics has a practical use in the control of the end-effector motion. Supposing d1 ; d2 ; d3 and d4 are known, we can solve for the position vector and the orientation angles P cx P cy Pcz ; h w . The configuration of two hook joints in serial chain is described in Fig. 7. The kinematic closure of the second and fourth chain (PUS) can be written as (15). Consequently, each chain provides us with the following equation:
ðA2 B2 Þ2x þ ðA2 B2 Þ2y þ ðA2 B2 Þ2z ¼ L22
ð23Þ
Eq. (22) represents the closed-form solution for the inverse kinematic problem of the 2-PUU_2-PUS PM [32].
ðA4 B4 Þ2x þ ðA4 B4 Þ2y þ ðA4 B4 Þ2z ¼ L24
ð24Þ
3.2. The forward kinematics solution
B1 ¼ P 1 þ ½ 0 0 d1 þ ðA1 B1 Þ Z
ð25Þ
ðA1 B1 Þ Z ¼ RX ða1 Þ RY ðb1 Þ L1
ð26Þ
R ¼ RX ða1 Þ RY ðb1 Þ RY ðc1 Þ RX ðd1 Þ
ð27Þ
di ¼ qi Z
ð22Þ
Direct kinematics solution addresses the problem of determining the position and orientation of the movable platform due to the
The kinematic closure of first chain (PUU) can be written: T
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Fig. 9. Flow chart for singularity program.
where RX ða1 Þ is rotation matrix around X axis related to the first hook joint angle a1 ; RX ðd1 Þ is rotation matrix around X axis related to the second hook joint angle d1 , where: RY ðb1 Þ is rotation matrix around Y axis related to the first hook joint angle b1 and RY ðc1 Þ is rotation matrix around Y axis related to the second hook joint angle c1 . Similarly, for the third chain PUU: T
To obtain the angles of all passive hook joints in the first and third chains, we can do the following: Eqs. (26) and (29) are transformable into the following:
RX ða1 Þ RY ðb1 Þ ¼ RX ða3 Þ RY ðb3 Þ RY ðc3 Þ RX ðd3 Þ RX ðd1 Þ RY ðc1 Þ ð31Þ
B3 ¼ P3 þ ½ 0 0 d3 þ ðA3 B3 Þ Z ðA3 B3 Þ Z ¼ RX ða3 Þ RY ðb3 Þ L3
ð28Þ ð29Þ
The first row of the second column of Eq. (30) is given by:
R ¼ RX ða3 Þ RY ðb3 Þ RY ðc3 Þ RX ðd3 Þ
ð30Þ
sin ðb3 þ c3 Þ sin ðd1 d3 Þ ¼ 0
where RX ða3 Þ is rotation matrix around X-axis related to the first hook joint angle a3 ; RX ðd3 Þ is rotation matrix around X-axis related to the second hook joint angle d3 . Also, RY ðb3 Þ is rotation matrix around Y-axis related to the first hook joint angle b3 and RY ðc3 Þ is rotation matrix around Y-axis related to the second hook joint angle c3 .
ð32Þ
B3 from Eq. (14) with (30) are substituted into (27).
Pc ¼ A3 þ ðA3 B3 Þ Z RX ða3 Þ RY ðb3 Þ RY ðc3 Þ RX ðd3 Þ B3m
ð33Þ
B1 from Eq. (14) with (30), (31), and (33) are substituted into Eq. (25).
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ðP cx ; Pcy ; P cz ; ;; h and wÞ. MATLAB is used to solve (25) and (28) to obtain the solution of forward kinematics [32]. 3.3. Workspace determination The reachable workspace of the proposed PM is defined here as all the positions that can be reached by the central point of the mobile platform with arbitrary orientation (up to ±90°) in a plane formed by the mobile platform center and the Z-axis of the fixed frame without violation of the angle limitation of the small scale hook joint. As stated by the manufacturer of small scale hook joint (U2–5.5 mm in diameter) and of the (A-B MSS) type, the angle limitation is ±90° for one angle and ±30° for the other angle [33]. Linear interpolation is used to get the angle limitation beyond the 90° limit. The workspace of the proposed PM can be determined numerically according to the inverse and forward kinematics analysis. The resulted total workspace is shown in Fig. 8, where we can notice that the bending angles increased significantly and reached more than ±90° in all directions. 4. Singularity analysis Fig. 10. Singular position of a 2-PUU_2-PUS PM.
A1 þ RX ða3 Þ RY ðb3 Þ RY ðc3 Þ RX ðd3 Þ RX ðd1 Þ RY ðc1 ÞL1 ¼ A3 þ ðA3 B3 Þ Z RX ða3 Þ RY ðb3 Þ RY ðc3 ÞRX ðd3 Þ B3m þ RX ða3 Þ RY ðb3 Þ RY ðc3 Þ RX ðd3 Þ B1m
ð34Þ
B2 and B4 from Eqs. (14), (30) and (33) are substituted into Eqs. (23) and (24).
½A3 þ ðA3 B3 Þ Z RX ða3 Þ RY ðb3 Þ RY ðc3 Þ RX ðd3 ÞB3m 2
þRX ða3 Þ RY ðb3 Þ RY ðc3 Þ RX ðd3 Þ B2m A2 ¼
L22
Sp ¼ Si ¼ d_ i $1;i þ h_ 2;i $2;i þ h_ 3;i $3;i þ h_ 4;i $4;i h_ 5;i $5;i ð35Þ
½A3 þ ðA3 B3 Þ Z RX ða3 Þ RY ðb3 Þ RY ðc3 Þ RX ðd3 ÞB3m þRX ða3 Þ RY ðb3 Þ RY ðc3 Þ RX ðd3 Þ B4m A4 2 ¼ L24
The theory of reciprocal screws is an effective way to derive the Jacobian matrix of a parallel manipulator [34]. The twist of the T movable platform can be expressed as Sp ¼ xT mT where x is the angular velocity of the movable platform and m is the linear velocity of a point in the movable platform which is instantaneously coincident with the origin of the reference frame in which the screws are defined. Concerning a 2-PUU_2-PUS PM, the connectivity of two limbs PUU is equal to 5, hence the instantaneous twist T of these limbs can be expressed as a linear combination of the five instantaneous twists:
ð37Þ
for i = 1, 3, where h_ j;i is the intensity and $j;i represents a unit screw associated with the jth joint of the ith limb.
ð36Þ
The previous nonlinear Eqs. (32), (34), (35), and (36) are four equations with six unknown variables (a3 ; b3 ; c3 ; d3 ; c1 and d1 ). MATLAB is used to solve the resulted equations numerically. From (30) the other two passive joints a1 and b1 of the first hook joint in the first serial chain can be determined. Finally, forward kinematics is reduced to six nonlinear equations represented by (25) and (28), each one has three equations with six unknown variables
(a) CAD Model of S joint
(b) CAD Model of U joint
Fig. 11. Different joints used in 2-PUU_2_PUS PM.
Fig. 12. SolidWorks model of the proposed endoscopic PM based on PVC.
Please cite this article in press as: Ibrahim K et al. Development of a new 4-DOF endoscopic parallel manipulator based on screw theory for laparoscopic surgery. Mechatronics (2015), http://dx.doi.org/10.1016/j.mechatronics.2015.02.006
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(a) CAD Model of S joint
(b) Real model of S joint
(c) Real model of MAAS-1.0 U joint
(d) CAD Model of P joint
(e) Real model of P joint
Fig. 13. Different manufactured joints used in 2-PUU_2-PUS PM.
(a) CAD model of Transmission joint
(b) Real model of Transmission joint
Fig. 14. Transmission joint.
$1;i ¼ $4;i ¼
s1;i
;
$2;i ¼
0 Bi s4;i s4;i
Ai s2;i s2;i
and $5;i ¼
;
$3;i ¼
Bi s5;i
Ai s3;i s3;i
;
Sp ¼ Sk ¼ d_ k $1;k þ h_ 2;k $2;k þ h_ 3;k $3;k þ h_ 4;k $4;k þ h_ 5;k $5;k þ h_ 6;k $6;k ð39Þ where k = 2, 4, and h_ j;k is the intensity and $j;k denotes a unit screw associated with the jth joint of the kth limb.
s5;i
sj;i is a unit vector along the jth joint of the ith limb. From Fig. 7a, we can get the following:
Ai ¼ Bi RX ðai Þ RY ðbi Þ Li
$1;k ¼ ð38Þ
Similarly, the connectivity of the other two limbs PUS is equal to 6. So we can describe these limbs as a linear combination of the six instantaneous twists as follow:
s1;k
;
$2;k ¼
0 Bk s4;k ; ¼ s4;k
Ak s2;k
$4;k
$5;k
;
$3;k ¼
Ak s3;k
;
s2;k s3;k Bk s5;k Bk s6;k and $6;k ¼ ¼ s5;k s6;k
Ak ¼ Bk RX ðak Þ RY ðbk Þ Lk
ð40Þ
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Universal joint
(Set screw) for S joint
Fig. 15. The proposed endoscopic manipulator based on PVC ‘‘EndoEJUST’’ [35].
Fig. 16. Load simulation of 5 N vertical forces on the movable platform.
Fig. 17. Control block diagram for tracking motion of PM.
Please cite this article in press as: Ibrahim K et al. Development of a new 4-DOF endoscopic parallel manipulator based on screw theory for laparoscopic surgery. Mechatronics (2015), http://dx.doi.org/10.1016/j.mechatronics.2015.02.006
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Fig. 18. Signal constraint graphically shaping the desired response.
Fig. 19. Sine signal position for each prismatic joints.
4.1. Jacobian of Constraint Those screws are reciprocal to all the joint screws of the ith limb form a ð6 ci Þ reciprocal screw system. Hence, ð6 ci Þ reciprocal basis screws can be identified.
$Trc;i Si ¼ 0 c ¼ 1; 2; . . . ; 6 ci
ð41Þ
where $rc;i denotes the cth reciprocal basis screw of the ith limb. The reciprocal systems for the first and third limbs are [34]:
$rc;i ¼
0
ð42Þ
ri
where r i is a unit vector along the direction defined by ðs2;i s3;i Þ or ðs4;i s5;i Þ. The Jacobian of constraints can be expressed as:
J c Si ¼ 0 " 013 Jc ¼ 013
r T1 r T3
#
ð43Þ ð44Þ
26
4.2. Jacobian of actuations
Fig. 20. Joints position control response for PM.
Now, lock the actuated joint in each limb. With the actuator locked, the dimension of the reciprocal screw system increases by 1 [27]. Then, when making the actuators of PM locked, the reciprocal screws of the first and third limbs form a 2-system which includes the screw $rc;i . One additional basis screw $ra;i being reciprocal to all the passive joint screws of the ith limb can be identified as a zero pitch screw along the direction passing through the two U joints:
Please cite this article in press as: Ibrahim K et al. Development of a new 4-DOF endoscopic parallel manipulator based on screw theory for laparoscopic surgery. Mechatronics (2015), http://dx.doi.org/10.1016/j.mechatronics.2015.02.006
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R ¼ r ¼ 3:5 mm, L1;2;3;4 ¼ 20 mm, m = 5 mm and all hook joints’ angles are zero. Then, we calculated the determinant of the overall Jacobian J which is found to be equal to zero as this is one of the singular positions. This is shown in Fig. 10. The MATLAB program can be applied to any position of the mobile platform to check the singularity.
5. Development of parallel mechanism 5.1. SolidWorks model The proposed conceptual design of the parallel manipulator is implemented using SolidWorks software as shown in Figs. 11 and 12. As a first trial, a scaled-up prototype of the proposed PM with a scale (2:1) was developed using plastic as the working material and machined using 3D prototyping machine [31]. This first trial prototype is served to understand and verify the mechanism motion and to check the feasibility of the proposed PM. Fig. 21. Error in each joint and position of movable platform.
$ra;i ¼
ðAi Bi Þ Z
5.2. Mechanical design and manufacturing
ðAi Bi Þ Z Bi
Similarly, for the second and fourth limb (PUS) with actuators locked, the reciprocal screws are as follows:
$ra;k ¼
ðAk Bk Þ Z ðAk Bk Þ Z Bk
J a Sp ¼ q_
ð45Þ h
From (39), q_ ¼ d_ 1
2
ððAi Bi ÞZ ÞT ððAi Bi ÞZ ÞT s1;1
6 6 ððA B ÞZÞT k k 6 6 ððAk Bk ÞZÞT s1;2 Ja ¼ 6 6 ððAi Bi ÞZÞT 6 6 ððAi Bi ÞZÞT s1;3 4 T ððAk Bk ÞZ Þ ððAk Bk ÞZ ÞT s1;4
d_ 2
d_ 3
ðAi Bi ÞZBi ððAi Bi ÞZ ÞT s1;1 ðAk Bk ÞZBk ððAk Bk ÞZ ÞT s1;2 ðAi Bi ÞZBi ððAi Bi ÞZ ÞT s1;3 ðAk Bk ÞZBk ððAk Bk ÞZ ÞT s1;4
d_ 4
iT
denotes the actuated joint.
All parts of the proposed 2-PUU_2-PUS were designed and then fabricated using high precision machines as shown in Fig. 13b, c, and e [35]. Transforming the rotary motion of the motor to translational motion of the prismatic joint is illustrated in Fig. 14. A real model of the proposed parallel mechanism, in actual size, is manufactured from AISI 347 annealed stainless steel as shown in Fig. 15. This prototype is served to understand and verify the mechanism motion also and to prove the feasibility of the design. The real prototype illustrates the capability of making large bending angles.
3 7 7 7 7 7 7 7 7 5
Point (0,2 and 4)
ð46Þ Point (3)
46
JSp ¼ q_ o
Point (1)
ð47Þ
where J is the overall Jacobian that we can get by the following Eq. (26):
J¼
Ja Jc
ð48Þ 66
h where q_ o ¼ d_ 1
d_ 2
d_ 3
d_ 4
0
0
iT
.
The first four rows of J represent four forces of actuation acting along the four limbs. Clearly, the manipulator will be under singularity if these four forces lie on a common plane, such as the case when moving the mobile platform up and down vertically when all links are equally stretched, or are parallel to each other as shown in Fig. 8. On the other hand, the last two rows of J represent two moments of constraint, each being perpendicular to the universal joint of a limb [27]. Constraint singularity occurs when J c loses its rank (rank = 2) whereas architecture singularity occurs when det(J) = 0 but J c has a full rank of 2 [32]. 4.3. Numerical result MATLAB program is build using the flowchart shown in Fig. 9 to determine the singular positions of the PM. The following numerical values are used for the PM parameters:
Fig. 22. Joints position control response (Sine signal) for PM.
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5.3. Load simulation For the load simulation, the mechanism is in straight configuration with all prismatic joints are locked and the movable platform is loaded with forces in the Z- direction as shown in Fig. 16. The mechanism maintains a force of 5 N in the Z-direction which deforms it slightly [36]. The AISI 347 annealed stainless steel has been selected as the material for manufacturing the endoscopic parallel manipulator. The maximum stress in this case is about 30 MPa which is below the yield point of the material (275 MPa). Also, the maximum deflection observed is within the acceptable limits.
6. Control simulation Simulation of the virtual prototype (4-DOF PM) has been accomplished by combining MATLAB/Simulink with ADAMS. During the simulation, the control algorithm is executed under MATLAB to generate the command forces, which are then exported to ADAMS environment and applied to the model as actuation forces at each sample time. The resulted outputs of the virtual prototype (velocity of the prismatic joints are measured by sensors in ADAMS) are then feedback to the controller in MATLAB/Simulink for calculation of the next command signal. The schematic diagram in Fig. 17 illustrates the inputs and outputs’ variables for the virtual prototype. The Simulink Design Optimization (SDO) is used to automatically tune the controller parameters to meet time-domain specifications. We can optimize any design criterion by expressing it as a Simulink signal and connecting this signal to the Signal Constraint block provided by SDO. The Signal Constraint block can constrain the signal either by graphically shaping the desired response or by specifying a reference signal trajectory as shown in Fig. 18. It then adjusts the values of chosen controller parameters to satisfy the constraints [37]. The signal constraint block GUI is updated during optimization so that we can visually monitor the optimization progress. To follow smoothly and accurately the commands of the surgeon, it is necessary to control both, the position and the velocity of the linear motors. Each DOF is controlled by a PID-position/PI-velocity loop as shown in Fig. 17. A PI-velocity loop is enclosed within a proportional position loop. The velocity is measured by ADAMS™ software. In laparoscopic surgery, the surgeon needs to make trajectory planning
Fig. 23. Error in each joint and position of movable platform.
during the surgery. Some signals that mimic the surgeon motion during the surgery are chosen. One of these signals is Sine signal as shown in Fig. 19. Several signal constraint blocks are used simultaneously to optimize multiple design criteria such as the error in the position of each prismatic joint.
7. Results and discussion 7.1. Result of bending motion To test the capability of the simulation control, a step input is applied to each prismatic joint in the PM as follows: Each DOF is controlled by a PID-position/PI-velocity loop as shown in Fig. 17. PI-velocity loop is enclosed within a proportional position loop. The velocity is estimated from the position signal by differentiation. PID controller is used to control the positions of each prismatic joint in the PM. Signal constraint blocks in Simulink/MATLAB are used to tune the parameters of the controllers. Overshoot is not desired especially in position control systems. It can be seen in Fig. 20 that signal constraint block adjusted the controllers parameters for each prismatic joint to avoid overshoots in most of the position responses. The simulation results of the steady state error as shown in Fig. 21 are almost 2.5% in each joint position with very small overshoot occurs. This interface between ADAMS and Simulink can be very useful to facilitate the approach known as design for control. Complex mechanisms can be designed and tested very easily using this interface. 7.2. Results of motion tracking The sinusoidal signal inputs that have been used for each prismatic joint are given as: The simulation results are illustrated in Fig. 22. The error of tracking in the joint position is almost 5% for each joint as shown in Fig. 23. The results show that the position of the movable platform center started and ended almost at the same location. Hence the movable platform is not affected by the tracking error in each joint.
8. Conclusion and future work Laparoscopic instrument was designed based on parallel mechanisms to serve as a rigid surgical manipulator. The design of the parallel manipulator that is based on the screw theory and parallel virtual chain approach is presented. The proposed manipulator consists of four links; two links are 2-PUU (each leg consists of one active prismatic joint and two passive hook joints); the other two links are 2-PUS (each leg consists of one active prismatic join, one passive hook joint and one passive spherical joint). The direct and inverse kinematics analysis of the parallel mechanism is developed. A closed-form solution for the inverse kinematics problem is obtained. Computer simulations (MATLAB/ADAMS) and control system design confirmed the feasibility of the new design. The results proved that the workspace of the bending mechanism is increased significantly and reached ±90° uniformly in all directions. The tracking control simulation results are satisfactory with an error of maximum 5% for each joint. An annealed stainless steel prototype is manufactured and tested successfully. In future work, a master/slave configuration will be prepared to test the performance of the whole system. Also, a hybrid intelligent control strategy could be implemented to improve the positioning accuracy and response of the surgical manipulator.
Please cite this article in press as: Ibrahim K et al. Development of a new 4-DOF endoscopic parallel manipulator based on screw theory for laparoscopic surgery. Mechatronics (2015), http://dx.doi.org/10.1016/j.mechatronics.2015.02.006
K. Ibrahim et al. / Mechatronics xxx (2015) xxx–xxx
Acknowledgments This research work is funded from the Science and Technology Development Fund (STDF – Project: ID 4100) and from the Missions Department, Ministry of Higher Education (MOHE), Egypt. The research team appreciates very much the support of Prof. Fujie through his invitation to the first author (Exchange Researcher) for a long visit to Waseda University, Japan.
References [1] Sackier JM, Wang Y. Robotically assisted laparoscopic surgery. Surg Endosc 1994;8:63–6. [2] Ghodoussi M, Butner SE, Wang Y. Robotic surgery-the transatlantic case. In: Proceedings of the IEEE International Conference on Robotics and Automation, vol. 2; 2002. p. 1882–8. [3] Guthart GS, Salisbury JK. The intuitive telesurgery system: overview and application. In: Proceedings of the IEEE International Conference of Robotics and Automation, vol. 1. San Francisco (CA), USA; 2000. p. 618–21. [4] Kim SK, Shin WH, Ko SY, Kim J, Kwon DS. Design of acompact 5-DOF surgical robot of a spherical mechanism: cures. In: Proceedings of the IEEE/ASME International Conference on Advances in Intelligent Mechatronics, July. Xian (China); 2008. p. 990–5. [5] Berkelman P, Ma J. A compact modular teleoperated robotic system for laparoscopic surgery. Int J Robot Res 2009;28:1198–215. [6] Lum MJH, Friedman DCW, Sankaranarayanan G, King H, Fodero II K, Leuschke R, et al. The RAVEN: design and validation of a telesurgery system. Int J Robot Res 2009;28:1183–97. [7] Takahashi H, Warisawa S, Mitsuishi M, Arata J, Hashizume M. Development of high dexterity minimally invasive surgical system with augmented force feedback capability. In: Proceedings of the IEEE/RAS-EMBS international conference on biomedical robotics and biomechatronics; 2006. p. 284–9. [8] Choi J, Park JW, Kim DJ, Shin J, Park CY, Lee JC, et al. A compact telesurgcal robot system for minimally invasive surgery, Part I: system description. Minimally Invas Ther Allied Technol 2011:1–7. [9] Feng M, Fu Y, Pan B, Liu C. Development of a medical robot system for minimally invasive surgery. Int J Med Robot Compute Assist Surg 2012;8:85–96. [10] Ikuta K, Yamamoto K, Sasaki K. Development of remote microsurgery robot and new surgical procedure for deep and narrow space. In: Proceedings of the IEEE International Conference on Robotics and Automation. Taipei (Taiwan); 2003. p. 1103–8. [11] Guthart G, Salisbury J. The intuitive telesurgery system: overview and application. In: Proceedings of the IEEE International Conference on Robotics and Automation. San Francisco (CA); 2000. p. 618–21. [12] Berkelman P, Ma J. A compact modular teleoperated robotic system for laparoscopic surgery. Int J Robot Res 2009;1198-215. [13] Arata J, Mitsuishi M, Warisawa S, Hashizume M. Development of a dexterous minimally-invasive surgical system with augmented force feedback capability. In: IEEE/RSJ international conference on intelligent robotics and systems; 2005. p. 3738-43. [14] Chiharu I, Kosuke K, Yusuke K, Yosuke N. Robotic forceps manipulator with a novel bending mechanism. IEEE/ASME Trans Mechatron 2010;15(5). [15] Röse A, Wohlleber C, Kassner S, Schlaak HF, Werthschützky R. A novel piezoelectric driven laparoscopic instrument with multiple degree of freedom parallel kinematic structure. In: 4th International conference on intelligent robots and systems, October 11–15. St. Louis (USA); 2009. [16] Yamashita H, Iimura A, Aoki E, Suzuki T, Nakazawa T, Kobayashi E, et al. Development of endoscopic forceps manipulator using multi-slider linkage mechanisms. In: Proceeding of the 1st Asian symposium on computer aided surgery – robotic and image guided surgery, April 28. Ibaraki, Japan; 2005. [17] Kobayashi Y, Chiyoda S, Watabe K, Okada M, Nakamura Y. Small occupancy robotic mechanisms for endoscopic surgery. In: Proceedings of the international conference on medical computer-assistance intervention; 2002. p. 75–82. [18] Simaan N, Taylor R, Flint P. A dexterous system for laryngeal surgery: multibackbone bending snake-like slaves for teleoperated dexterous surgical tool manipulation. In: Proceedings of the IEEE International Conference on Robotics and Automation. New Orleans (LA); 2004. p. 351–7. [19] Ibrahim K, Ramadan A, Abdellatif M, Fath El-Bab A, Abo-Ismail A. Development of a dexterous endoscopic manipulator based on parallel mechanism ‘‘Parallel Mechanism’’. In: IEEE international conference on intelligent computing and intelligent systems, ICIS2011, November 18–20. Guangzhou (China); 2011. [20] Peirs J, Reynaerts D, Brussel HV, Genem GD, Tang H-W. Design of an advanced tool guiding system for robotic surgery. In: Proceedings of the IEEE international conference on robotics and automation. Taipei (Taiwan); 2003. p. 2651–6. [21] Kim SM, Kim W, Chung J, Yi B-Ju. Kinematic analyses of a 1T2R and a 1T3R parallel mechanisms with closed-form position solutions. In: IEEE international conference on control and automation, December 9–11. Christchurch (New Zealand); 2009.
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[22] Ramadan A, Takubo T, Mae Y, Oohara K, Arai T. Development process of a chopstick-like hybrid structure two-fingered micromanipulator hand for 3D manipulation of microscopic objects. IEEE Trans Indust Electron 2009;56(4). [25] Si Zhu J, Huang Zhen, Ding Hua F. Forward/reverse velocity and acceleration analysis for a class of lower-mobility parallel mechanisms. Trans ASME 2007;129. [26] Fang YF, Tsai LW. Inverse velocity and singularity analysis of low-dof serial mechanisms. J Rob Syst 2003;204:177–88. [27] Joshi SA, Tsai LW. Jacobian analysis of limited-DOF parallel manipulators. ASME J Mech Des 2002:254–8. [28] Kong X, Gosselin C. Type synthesis of parallel mechanisms. Berlin, Heidelberg: Springer-Verlag; 2007. [31] Ibrahim K, Ramadan A, Fanni M, Kobayashi Yo, Abo-Ismail A, Fujie Masakatus G. Design and workspace analysis of a new endoscopic parallel manipulator. In: 12th International conference on control, automation and system ICCAS 2012, October 17–21. ICC, Jeju Island (Korea); 2012. [32] Ibrahim K, Ramadan A, Fanni M, Kobayashi Yo, Abo-Ismail A, Fujie Masakatus G. Kinematic analysis and control of limited 4-DOF parallel manipulator based on screw theory. In: IEEE international conference on systems, man, and cybernetics IEEE SMC 2012, COEX, October 14–17. Seoul (Korea); 2012. [33]
Further reading [23] Yang Chifu, Qu Zhiyong, Han Junwei. Decoupled-space control and experimental evaluation of spatial electro-hydraulic robotic manipulators using singular value decomposition algorithms. IEEE Trans Indust Electron 2014;61(7):3427–38. http://dx.doi.org/10.1109/TIE.2013.2278958. [24] Allais AA, McInroy JE, O’Brien JF. Locally decoupled micromanipulator using an even number of parallel force actuators. IEEE Trans Robot 2012;28(6):1323–34. http://dx.doi.org/10.1109/TRO.2012.2209229. [29] Davidson J, Hunt K. Robots and screw theory – applications of kinematics and statics to robotics. Great Britain: Oxford University Press; 2004. [30] Tsai LW. Robot analysis the mechanics of serial and parallel manipulators. New York: Wiley-Science; 1999.
Khalil Ibrahim received the B.E. and M.Sc. degrees in mechanical engineering from the Mechanical Engineering Department, Faculty of Engineering, University, Assiut, Egypt, in 2001 and 2006, respectively. Then he worked toward the Ph.D. degree from late 2010 till September, 2013 at Mechatronics and Robotics Engineering, Graduate School of Innovation, Egypt-Japan University of Science and Technology EJUST, Egypt. He worked as exchange researcher in Waseda University, Japan around one year. During this period he got experiences in surgery robot design. Starting from September, 2013 he is a Lecturer/Assistant professor in the mechanical department, Faculty of Engineering, and Assiut University. His major research interests include robotics engineering, automatic control, and artificial intelligence techniques. His current research focuses on Tele surgical manipulators, endoscopic manipulator, and control system design.
Ahmed A. Ramadan received his B.E. and M.Sc. degrees from Electrical Engineering department in Computer Engineering and Automatic Control from Tanta University, Egypt, in 1996 and 2002 respectively. Then he worked toward the Ph.D. degree from late 2005 till March, 2009 at Systems Innovation Department, Graduate School of Engineering Science, OSAKA University, Japan. Starting from May, 2009 he is a Lecturer/Assistant Professor in the Computer and Control Engineering department in Tanta University. As of the beginning of April 2010, he granted a research fellowship to work as an Assistant Professor in Mechatronics and Robotics Engineering department, Egypt-Japan University and science and technology (EJUST), Egypt. He is a
Please cite this article in press as: Ibrahim K et al. Development of a new 4-DOF endoscopic parallel manipulator based on screw theory for laparoscopic surgery. Mechatronics (2015), http://dx.doi.org/10.1016/j.mechatronics.2015.02.006
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K. Ibrahim et al. / Mechatronics xxx (2015) xxx–xxx
member of the Institute of Electrical and Electronics Engineers (IEEE). His research interests are in the fields of Robotics Engineering, Automatic Control, and Artificial Intelligence Techniques. His current research interests focuses on the design and control of Surgical Manipulators, Master/ Slave teleoperation systems, Aerial manipulation and control of Flying Robots.
Mohamed Fanni received the B.E. and M.Sc. degrees in mechanical engineering from Faculty of Engineering of both Cairo University and Mansoura University, Egypt, in 1981 and 1986, respectively and the Ph.D. degree in engineering from Karlsruhe University, Germany, 1993. He is an Associate Professor with Innovation, Graduate School of Engineering Science, Egypt-Japan University of Science and Technology E-JUST, Alexandria, on leave from Production Engineering & Mechanical Design Department, Faculty of Engineering, Mansoura University, Egypt. His major research interests include robotics engineering, automatic control, and Mechanical Design. His current research focuses on Design & Control of Mechatronic Systems, Surgical Manipulators, Teleoperation systems and Flying/ Walking Robots.
Yo Kobayashi (M’08) received the B.S. and M.S. degrees from the Graduate School of Science and Engineering, Waseda University, Tokyo, Japan, in 2004 and 2005, and the Ph.D. degree in engineering from Waseda University in 2008. He was a Visiting Research Associate with the Graduate School of Science and Engineering, Waseda University, from 2005 to 2006. He was a recipient of the Japan Society for the Promotion of Science Re-search Fellowship for Young Scientists in 2007. In 2008, he was a Research Associate with the Institute for Biomedical Engineering, Waseda University, whereas in 2009, he was a Research Associate with the Faculty of Science and Engineering, Waseda University. He has been a Lecturer (Junior Researcher) with the Faculty of Science and Engineering, Waseda University, since 2010. His current research interests include medical robots and rehabilitation robots. Dr. Kobayashi is a member of the Japan Society of Mechanical Engineers, the Society of Instrument and Control Engineers, the Robotics Society of Japan, and the Japan Society of Computer-Aided Surgery.
Ahmed Abo-Ismail received his Ph.D. degree from Tokyo Institute of Technology, TIT, Japan, 1979.He is a full professor of Automatic Control and Mechatronics, EJUST, Egypt. He is an Honorary Professor of Budapest Tech., Hungary. He is granted a Fulbright award fellowship as a Visiting Professor to PSU, USA, 1987. He is an IFAC Chair (International Federation of Automatic Control) for African, Asian, South American Countries, 2003–2006 and he is a Member of the technical Editorial Board of the ACTA Polytechnica Hungaraica Journal of Applied Sciences, Budapest, Hungary. He is also the Provost, Vice President for Education and Academic Affairs, EJUST, Egypt and former dean of faculty of Engineering, Assuit University, Assuit, Egypt. His major research interests are in the fields of Intelligent and Robust Control, Smart Grippers design, Intelligent Mechatronics System, VRHepatic Surgical Simulators, Surgical Robots, Assistive Devises for elderly people design and control, Aerial manipulation and control, and Flying/Walking Robots.
Masakatsu G. Fujie (M’90–SM’08) received the M.S. degree from the Graduate School of Science and Engineering, Waseda University, Tokyo, Japan, in 1971, and the Ph.D. degree in engineering from Waseda University in 1999. From 1971 to 2000, he was with the Mechanical Engineering Research Laboratory, Hitachi Ltd., where he was a Senior Researcher during 1984 and a Principal Researcher and Project Leader for the Medical and Welfare Apparatus Development Project during 1995. He was also the Head of Researchers in the Mechanical Engineering Research Laboratory, Hitachi Ltd., and the Director o f the Medical and Welfare Apparatus Development Research Laboratory during 1999. Since 2001, he has been a Professor with the Faculty of Science and Engineering, Waseda University, where he is the Director of The Grobal Robot Academia for MEXT Grobal COE Program. His current research interests include surgical robots, image guided surgery, endoscopic surgery, and assistive and rehabilitation robots. Prof. Fujie is a Fellow o f the Japan Society of Mechanical Engineering (JSME) and a member of the International Society of Computer-Aided Surgery. He received the Atomic Energy Society of Japan’s Technology Development. Award (1991), the Notable Invention Award from the Science and Technology Agency Director General (1994), the Technical Innovations Awards from the Robotics Society of Japan (1999), the JSME Medal for New Technology from the Japan Society of Mechanical Engineers (2000), the Nikkei BP Technology Award, Medical/Biotechnology Division (2000), the Japan Society of Mechanical Engineers, Robotics and Mechatronics.
Please cite this article in press as: Ibrahim K et al. Development of a new 4-DOF endoscopic parallel manipulator based on screw theory for laparoscopic surgery. Mechatronics (2015), http://dx.doi.org/10.1016/j.mechatronics.2015.02.006
Contents lists available at ScienceDirect
Mechatronics journal homepage: www.elsevier.com/locate/mechatronics
Development of a new 4-DOF endoscopic parallel manipulator based on screw theory for laparoscopic surgery Khalil Ibrahim a,⇑, Ahmed Ramadan b, Mohamed Fanni c, Yo Kobayashi d, Ahmed Abo-Ismail e, Masakatus G. Fujie d a
Mechatronics Engineering, Faculty of Engineering, Assiut University, Assiut, Egypt Computers and Automatic Control Engineering Dep., Tanta University, Tanta, Egypt Mechanical Department, Mansoura University, Mansoura, Egypt d The Faculty of Science and Engineering, Waseda University, Tokyo, Japan e Mechatronics and Robotics Engineering, Egypt-Japan University of Science and Technology, Alexandria, Egypt b c
a r t i c l e
i n f o
Article history: Received 30 January 2014 Accepted 26 February 2015 Available online xxxx Keywords: Parallel manipulators Medical robotics Trajectory control Virtual chain type synthesis method
a b s t r a c t Development of rigid manipulators for Minimally Invasive Surgery (MIS) becomes essential to replace wire driven manipulators which are problematic due to possible cutting of the wire during the surgery. In this paper, a 4-DOF dexterous endoscopic parallel manipulator for MIS is designed and implemented. The manipulator is developed based on the concept of virtual chain and screw theory. The manipulator consists of four links; two links are 2-PUU (each leg consists of one active prismatic joint and two passive hook joints); the other two links are 2-PUS (each leg consists of one active prismatic join, one passive hook joint and one passive spherical joint). The inverse and forward kinematics solutions are derived analytically and numerically, respectively. The singularity analysis is investigated using screws algebra. The manipulator workspace is obtained using MATLAB software. The known problem of limited bending angles found in previous existing surgical manipulators was solved as the proposed manipulator can reach ±90° in any direction. The proposed surgical manipulator is designed, manufactured and tested successfully. The system model utilizing PID and PI controllers has been built using MATLAB software. Co-simulation using ADAMS/MATLAB software is implemented to validate the achieved bending angles and the proposed tracking control. The results show that the performance of the tracking control is satisfactory since the tracking error is about 2.5%. Ó 2015 Elsevier Ltd. All rights reserved.
1. Introduction Recently, Minimally Invasive Robotic Surgery (MIRS) has been increasingly introduced into therapy to reduce trauma and recovery time in hospitals. The current state of the art surgical robots are using wire driven mechanisms which are problematic. Development of endoscopic manipulator suitable for such kind of MIRS, as shown in Fig. 1, becomes essential. This allows the surgeon to carry out the surgery with high accuracy and more comfortable. In contrast to flexible endoscopes, laparoscopic instruments are required to resist much higher forces [1–9]. In the case of a laparoscopic liver surgery, 5 N are needed at the
⇑ Corresponding author. E-mail addresses: [email protected] (K. Ibrahim), ahmed.ramadan@ ejust.edu.eg (A. Ramadan), [email protected] (M. Fanni), [email protected] (Y. Kobayashi), [email protected] (A. Abo-Ismail), [email protected] (M.G. Fujie).
instrument tip. The existing endoscopic manipulators can be generally classified into two broad categories namely: flexible mechanisms, usually driven by wires, and rigid mechanism manipulators. Most of the conventional endoscopic systems in the market for MIS are of the wire-actuated type [10]. However, the rigidity and the durability of wires are poor. For flexible mechanisms, there are two commercial surgical systems which are currently in clinical use and using the wire-driven mechanisms, namely, da Vinci system (Intuitive Surgical, Inc.) [11] and ZEUS system (Computer Motion, Inc.) [12]. For rigid mechanism manipulators, they use rigid-links mechanism to transmit the motion to the gripper and perform the surgery. This approach is challenging because of the difficulty of manufacturing the mechanism and controlling the motion of the gripper with sufficient dexterity. Robotic forceps with link mechanisms have been developed, which can bend in two dimensions and grasp [13]. However, the bending angle is limited to ±70°. Ishii et al. [14] developed a robotic forceps manipulator using a new screw-drive mechanism
http://dx.doi.org/10.1016/j.mechatronics.2015.02.006 0957-4158/Ó 2015 Elsevier Ltd. All rights reserved.
Please cite this article in press as: Ibrahim K et al. Development of a new 4-DOF endoscopic parallel manipulator based on screw theory for laparoscopic surgery. Mechatronics (2015), http://dx.doi.org/10.1016/j.mechatronics.2015.02.006
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dynamic simulation of the controlled robot is implemented using ADAMS software. This paper is organized as follows. The parallel manipulator based on virtual chain approach is introduced in Section 2. In Section 3, the kinematics analysis is presented. The singularity analysis is investigated in Section 4. The mechanical design and manufacturing of the parallel manipulator is described in Section 5. In Section 6, the optimization and simulation of the control scheme is investigated by MATLAB/ADAMS to test the motion capability of the proposed design. The results and discussion of the motion simulation and control of the proposed parallel manipulator are presented in Section 7. Finally, conclusions and future work are given in Section 8.
Fig. 1. Concept of MIRS.
2. Parallel manipulator synthesis based on virtual chain approach 2.1. Virtual chain approach
(DSD forceps). However, the fabrication accuracy of the manipulator needs improvement. Röse et al. [15] developed an endoscopic manipulator using parallel mechanism which produces different bending angles in different directions. However, the maximum bending angles is unequal in all directions. It achieves ±80° in one direction and ±40° in the perpendicular direction. Also, the kinematics analysis of this manipulator is very difficult and not reported. Yamashita et al. [16] have built an endoscopic forceps manipulator with 3-DOF, where 2-DOF are used for bending via complicated multi-slider linkage mechanism and 1-DOF for the wire driven gripper. Accuracy problems and insufficient power were reported. Kobayashi et al. [17] developed one of the Omnidirectional driven-type forceps manipulators, an active forceps manipulator in the form of a platform. Although it has high rigidity, its bending range is 40–50°, and it is difficult to expand the bending range due to inherent constraints in the mechanism. A snake-like manipulator with a diameter of 4.2 mm was developed in [18]. It can reach bending angles between ±90° in any direction by push and pull modes of three super-elastic tubes. However, rigidity and large bending force are not expected. Ibrahim et al. [19] proposed a 4-DOF parallel manipulator where computer simulations showed the feasibility of the new design and proved that the workspace of the bending mechanism was increased significantly and reached the limits of ±90°. However, maximum bending angles are not equal in all directions. Snake-like endoscopic tool, of 5 mm diameter, with 2-DOF and wire actuation is developed. It uses a wire actuated NiTi tube with laser-machined flexure joints. However, there is a hysteresis in the relation between the wire force and the bending angle [20]. The use of parallel mechanisms can be beneficial for the design of robotic surgical instruments due to their characteristics such as their high accuracy, high stiffness, and high load capacity. But most of the parallel mechanisms do not have a closed-form solution for either the forward or inverse kinematics problems mainly due to complex constraints imposed on the closed-chains [21,22]. Conventional Jacobian for an n-DOF parallel manipulator (n 6) is not a square matrix (6 n), which adds difficulty to kinematic modeling [25]. Fang et al. [26] presented the inverse velocity analysis for lower-mobility serial mechanism with the reciprocal screw. Tsai et al. [27] employed the reciprocal screw in Jacobian analysis where the Jacobian of constraints and actuations are used to build a square (6 6) overall Jacobian. In this work, the design of a novel parallel endoscopic manipulator that would facilitate performing MIRS is presented. To improve the bending motion and trajectory- tracking accuracy of the robot, a dynamic control scheme is developed and the simplified dynamic model of the proposed structure is used. Finally, the
For a known motion pattern, a virtual chain is proposed by a comprehensive wrench system analysis of a large number of serial chains and parallel chains and by considering the possible changes in the wrench system with a change of the configuration of the serial chain or the parallel chain. Also, as a general rule, the simplest possible virtual chain will be selected [28]. Wrenches of 0-pitch, 1-pitch and h-pitch are represented by f0 ; f1 and fh respectively, where h is any finite number larger than zero. For type synthesis of a Parallel Manipulator (PM) that mimics the motion pattern of a proposed Parallel Virtual Chain (PVC), a general procedure is proposed in 0 which consists of four main steps, namely: 1. 2. 3. 4.
The The The The
decomposition of the number of legs. type synthesis of sub-Parallel Kinematic Chains (PKCs). combination of sub-PKCs to generate PKCs. selection of the actuated joints.
2.2. Screw theory The normalized screw coordinates 0 may be given by:
$¼
S
ð1Þ
S0 S þ hs
where S ¼ ½ sx
sy
T
sz denotes a unit vector along the direction of
T the screw axis, and the vector S0 ¼ ½ s0x s0y s0z denotes the position vector of a point lying on the screw axis as shown in Fig. 2. Generally, the movement between a pair of bodies is determined by either a rotation or a prismatic joint. For a rotation joint, the pitch of the twist is null h = 0. In this case, the normalized screw of the ith joint is expressed by:
Fig. 2. Screw representation.
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$¼
Si S0 Si
ð2Þ
For a prismatic joint, the pitch of the twist goes to infinity and the normalized screw of the ith joint is reduced to:
$¼
0 Si
ð3Þ
2.3. Parallel Virtual Chain (PVC) In this section the manipulator parallel chain will be designed systematically using screw theory and virtual chain approach. According to Grubler’s formula, the DOF; F, can be computed as follows [30]:
F ¼ 6ð n g 1Þ þ
g X fi
ð4Þ
i¼1
where n is the number of link of PVC, g is the number of joints in PVC and f i is the number of connectivity. From screw theory approach, we can calculate the wrench system of PVC. The total wrench system of the PVC is 2 fh . 2.4. Type synthesis of the proposed PM The synthesis is restricted to 4 legs-PM with active prismatic joints located at the bottom of the four legs to be actuated by four linear motors. This restriction is dictated by the requirement of laparoscopic surgery. To get the total wrench system of PM (which is the summation of the wrench system of each leg; ci ), it can be calculated from the following Eq. (28):
F ¼6
w X ci þ D
3. Kinematics analysis 3.1. The inverse kinematics solution The main objective of this section is to derive an inverse kinematics closed-form solution for that class of parallel manipulators. Fig. 5 shows the model for the proposed PM. Points Ai and Bi (i = 1, . . . , 4) are symmetrically arranged on the circumference of the circle with radius R as shown in Fig. 6b. Reference frame O-XYZ is attached to the center of the base platform with the X-axis in the direction OP 1 , the Z-axis is perpendicular to the plane of the base and Y-axis satisfies the right hand rule. A mobile platform frame C-UVW, which has the same pose as the frame O-XYZ in the stretched configuration, is attached to the center of the mobile platform. Link Pi Ai is connected to the base platform through active Pjoint. Links Pi Ai and Ai Bi are connected through U-joints as shown in Fig. 6a. Links Ai Bi are of fixed length Li , while links P i Ai are of variable lengths di in the Z-direction due to the linear displacements of the actuators. The coordinates of the points Pi with respect to the fixed base reference frame O-XYZ are given by:
2 3 R 6 7 P 1 ¼ 4 0 5; 0
where w is the number of legs of PM and D is the number of overconstraints. Let D ¼ 0 to avoid over-constrain, w = 4 and F = 4. Substituting all parameters into (5), we get:
ð6Þ
i¼1
There are two possible solutions of Eq. (6): c1 ¼ 2 and c2 ¼ c3 ¼ c4 ¼ 0 or c1 ¼ c2 ¼ 1 and c3 ¼ c4 ¼ 0 By applying the procedure described in Section 2.1 to the second solution. The first proposed parallel manipulator 2-PRRRR_2PRRRRR can be obtained as shown in Fig. 3. The bending angles of this parallel manipulator are not the same in all directions. The maximum bending angle reaches ±90° but in only two directions. Moreover, the kinematics analysis and design are difficult [19]. For the requirements of symmetrical bending angles, the second solution is selected. In order to get equal bending angles in different directions, PVC is designed of two identical serial chains. Each serial chain consists of two hook joints (U) and one prismatic joint (P) in the bottom (PUU) as shown in Fig. 4. Applying the procedure described in section II-A with the constraints described above, one can obtain 2-PUU_2-PUS PM which consists of two PUU-legs and two PUS-legs where (S) is spherical joint as shown in Fig. 4. For manufacturing considerations, the spherical joint can be replaced by one hook joint followed by one revolute joint whose axis is perpendicular to the axes of the hook joint [31]. The limiting angles of hook joints in small scale will be taken into consideration during kinematics analysis. This new proposed parallel manipulator simplifies the design and kinematics analysis compared to the previous manipulators and has the merits of wide
2 3 0 6 7 P 2 ¼ 4 R 5; 0
2
R
3
2
0
3
6 7 6 7 P3 ¼ 4 0 5 and P4 ¼ 4 R 5 0 0
ð7Þ
Similarly, the coordinates of the points Bim referred to the mobile frame C-UVW are given by
ð5Þ
i¼1
m X ci ¼ c1 þ c2 þ c3 þ c4 ¼ 2
and symmetrical bending angles. Besides, the size of the PM will be more compact.
B1m
2 3 r 6 7 ¼ 4 0 5; 0
B2m
2 3 0 6 7 ¼ 4 r 5; 0
2
B3m
3 r 6 7 ¼ 4 0 5; 0
2
B4m
3 0 6 7 ¼ 4 r 5 0
ð8Þ
The coordinates of points Ai referred to the location of the first hook joint (U) for each chain are:
Ai ¼
P i þ di Z
ð9Þ
where Z is a unit vector in Z-direction. Substituting (7) into (9), gives:
2
R
3
6 7 A1 ¼ 4 0 5; d1
2
0
3
6 7 A2 ¼ 4 R 5 ; d2
2
R
3
2
0
3
6 7 6 7 A3 ¼ 4 0 5 and A4 ¼ 4 R 5 d4 d3 ð10Þ
The transformation from the mobile frame to the fixed base frame is expressed by a position vector Pc equal to OC and a rotation matrix b Rm described by the direction cosines of unit vectors U, V, and W, as specified in (11).
2
3 P cx 6 7 Pc ¼ 4 Pcy 5; P cz
2
Ux 6 b Rm ¼ 4 U y Uz
b
Rm can be expressed by:
b
Rm ¼ RðX; ;ÞRðY; hÞRðZ; wÞ
Vx
Wx
3
Vy
7 Wy 5
Vz
Wz
ð11Þ
ð12Þ
where RðX; ;Þ is the rotational matrix around X-axis, RðY; hÞ is the rotational matrix around Y-axis and RðZ; wÞ is the rotational matrix around Z-axis. The coordinates of points Bi referred to the base fixed frame O-XYZ are:
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(a) First Parallel Virtual Chain 2-PRRRR
(b) first proposed 2-PRRRR_2-PRRRRR PM
Fig. 3. Parallel manipulator based on PVC [19].
Bi ¼ b Rm Bim þ Pc
ð13Þ
Substituting (8) and (11) into (13), and after simplifying we get:
2
Pcx þ RU x
3
2
Pcx þ RV x
3
2
Pcx RU x
3
6 7 6 7 6 7 B1 ¼ 4 Pcy þ RU y 5; B2 ¼ 4 Pcy þ RV y 5; B3 ¼ 4 Pcy RU y 5 and 2
Pcz þ RU z Pcx RV x
Pcz þ RV z
3
Pcz RU z
6 7 B4 ¼ 4 Pcy RV y 5 Pcz RV z
ð14Þ
The kinematic closure of each elementary chain can be written as:
kAi Bi k2 ¼ L2i
ð15Þ
Substituting (10) and (14), we obtain: Fig. 4. Second parallel virtual chain.
Ai Bi ¼ b Rm Bim þ Pc Pi di Z
ð16Þ
Once, we get the orientation matrix b Rm and have the center position P c vector, using Fig. 6a, we can proceed as follows:
qi ¼ b Rm Bim þ Pc Pi
ð17Þ
where qi is a vector of the prismatic joint P to the second hook joint U in one serial chain as shown in Fig. 6a. Substituting (17) into (16), we can get:
jAi Bi j ¼ qi di Z
ð18Þ
Substituting (18) into (15), we obtain: 2
kAi Bi k2 ¼ qTi qi þ di 2di qi Z ¼ L2i 2
di 2di qi Z þ qTi qi L2i ¼ 0
ð19Þ ð20Þ
The two solutions of (20) can be expressed in simple form as
dij1;2 ¼ qi Z
Fig. 5. The proposed 2-PUU_2-PUS PM based on PVC.
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðqi Z Þ2 qTi qi þ L2i
ð21Þ
Eq. (21) indicates that there are two solutions for the actuator displacements representing the intersection of the displacement vector di Z with the surface of a sphere of radius Li and centered
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(a) First chain PUU
(b) Base platform
Fig. 6. Schematic representation of 2-PUU_2-PUS PM.
Fig. 8. Dexterous workspace of 2-PUU_2-PUS PM.
Fig. 7. Serial chain with two hook joints.
at Bi , as shown in Fig. 6a. For minimum displacement of linear actuators, we choose the negative sign for the square root, which corresponds to the lower intersection point so,
qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðqi Z Þ2 qTi qi þ L2i
motion of the actuated joints. Forward kinematics has a practical use in the control of the end-effector motion. Supposing d1 ; d2 ; d3 and d4 are known, we can solve for the position vector and the orientation angles P cx P cy Pcz ; h w . The configuration of two hook joints in serial chain is described in Fig. 7. The kinematic closure of the second and fourth chain (PUS) can be written as (15). Consequently, each chain provides us with the following equation:
ðA2 B2 Þ2x þ ðA2 B2 Þ2y þ ðA2 B2 Þ2z ¼ L22
ð23Þ
Eq. (22) represents the closed-form solution for the inverse kinematic problem of the 2-PUU_2-PUS PM [32].
ðA4 B4 Þ2x þ ðA4 B4 Þ2y þ ðA4 B4 Þ2z ¼ L24
ð24Þ
3.2. The forward kinematics solution
B1 ¼ P 1 þ ½ 0 0 d1 þ ðA1 B1 Þ Z
ð25Þ
ðA1 B1 Þ Z ¼ RX ða1 Þ RY ðb1 Þ L1
ð26Þ
R ¼ RX ða1 Þ RY ðb1 Þ RY ðc1 Þ RX ðd1 Þ
ð27Þ
di ¼ qi Z
ð22Þ
Direct kinematics solution addresses the problem of determining the position and orientation of the movable platform due to the
The kinematic closure of first chain (PUU) can be written: T
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Fig. 9. Flow chart for singularity program.
where RX ða1 Þ is rotation matrix around X axis related to the first hook joint angle a1 ; RX ðd1 Þ is rotation matrix around X axis related to the second hook joint angle d1 , where: RY ðb1 Þ is rotation matrix around Y axis related to the first hook joint angle b1 and RY ðc1 Þ is rotation matrix around Y axis related to the second hook joint angle c1 . Similarly, for the third chain PUU: T
To obtain the angles of all passive hook joints in the first and third chains, we can do the following: Eqs. (26) and (29) are transformable into the following:
RX ða1 Þ RY ðb1 Þ ¼ RX ða3 Þ RY ðb3 Þ RY ðc3 Þ RX ðd3 Þ RX ðd1 Þ RY ðc1 Þ ð31Þ
B3 ¼ P3 þ ½ 0 0 d3 þ ðA3 B3 Þ Z ðA3 B3 Þ Z ¼ RX ða3 Þ RY ðb3 Þ L3
ð28Þ ð29Þ
The first row of the second column of Eq. (30) is given by:
R ¼ RX ða3 Þ RY ðb3 Þ RY ðc3 Þ RX ðd3 Þ
ð30Þ
sin ðb3 þ c3 Þ sin ðd1 d3 Þ ¼ 0
where RX ða3 Þ is rotation matrix around X-axis related to the first hook joint angle a3 ; RX ðd3 Þ is rotation matrix around X-axis related to the second hook joint angle d3 . Also, RY ðb3 Þ is rotation matrix around Y-axis related to the first hook joint angle b3 and RY ðc3 Þ is rotation matrix around Y-axis related to the second hook joint angle c3 .
ð32Þ
B3 from Eq. (14) with (30) are substituted into (27).
Pc ¼ A3 þ ðA3 B3 Þ Z RX ða3 Þ RY ðb3 Þ RY ðc3 Þ RX ðd3 Þ B3m
ð33Þ
B1 from Eq. (14) with (30), (31), and (33) are substituted into Eq. (25).
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ðP cx ; Pcy ; P cz ; ;; h and wÞ. MATLAB is used to solve (25) and (28) to obtain the solution of forward kinematics [32]. 3.3. Workspace determination The reachable workspace of the proposed PM is defined here as all the positions that can be reached by the central point of the mobile platform with arbitrary orientation (up to ±90°) in a plane formed by the mobile platform center and the Z-axis of the fixed frame without violation of the angle limitation of the small scale hook joint. As stated by the manufacturer of small scale hook joint (U2–5.5 mm in diameter) and of the (A-B MSS) type, the angle limitation is ±90° for one angle and ±30° for the other angle [33]. Linear interpolation is used to get the angle limitation beyond the 90° limit. The workspace of the proposed PM can be determined numerically according to the inverse and forward kinematics analysis. The resulted total workspace is shown in Fig. 8, where we can notice that the bending angles increased significantly and reached more than ±90° in all directions. 4. Singularity analysis Fig. 10. Singular position of a 2-PUU_2-PUS PM.
A1 þ RX ða3 Þ RY ðb3 Þ RY ðc3 Þ RX ðd3 Þ RX ðd1 Þ RY ðc1 ÞL1 ¼ A3 þ ðA3 B3 Þ Z RX ða3 Þ RY ðb3 Þ RY ðc3 ÞRX ðd3 Þ B3m þ RX ða3 Þ RY ðb3 Þ RY ðc3 Þ RX ðd3 Þ B1m
ð34Þ
B2 and B4 from Eqs. (14), (30) and (33) are substituted into Eqs. (23) and (24).
½A3 þ ðA3 B3 Þ Z RX ða3 Þ RY ðb3 Þ RY ðc3 Þ RX ðd3 ÞB3m 2
þRX ða3 Þ RY ðb3 Þ RY ðc3 Þ RX ðd3 Þ B2m A2 ¼
L22
Sp ¼ Si ¼ d_ i $1;i þ h_ 2;i $2;i þ h_ 3;i $3;i þ h_ 4;i $4;i h_ 5;i $5;i ð35Þ
½A3 þ ðA3 B3 Þ Z RX ða3 Þ RY ðb3 Þ RY ðc3 Þ RX ðd3 ÞB3m þRX ða3 Þ RY ðb3 Þ RY ðc3 Þ RX ðd3 Þ B4m A4 2 ¼ L24
The theory of reciprocal screws is an effective way to derive the Jacobian matrix of a parallel manipulator [34]. The twist of the T movable platform can be expressed as Sp ¼ xT mT where x is the angular velocity of the movable platform and m is the linear velocity of a point in the movable platform which is instantaneously coincident with the origin of the reference frame in which the screws are defined. Concerning a 2-PUU_2-PUS PM, the connectivity of two limbs PUU is equal to 5, hence the instantaneous twist T of these limbs can be expressed as a linear combination of the five instantaneous twists:
ð37Þ
for i = 1, 3, where h_ j;i is the intensity and $j;i represents a unit screw associated with the jth joint of the ith limb.
ð36Þ
The previous nonlinear Eqs. (32), (34), (35), and (36) are four equations with six unknown variables (a3 ; b3 ; c3 ; d3 ; c1 and d1 ). MATLAB is used to solve the resulted equations numerically. From (30) the other two passive joints a1 and b1 of the first hook joint in the first serial chain can be determined. Finally, forward kinematics is reduced to six nonlinear equations represented by (25) and (28), each one has three equations with six unknown variables
(a) CAD Model of S joint
(b) CAD Model of U joint
Fig. 11. Different joints used in 2-PUU_2_PUS PM.
Fig. 12. SolidWorks model of the proposed endoscopic PM based on PVC.
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(a) CAD Model of S joint
(b) Real model of S joint
(c) Real model of MAAS-1.0 U joint
(d) CAD Model of P joint
(e) Real model of P joint
Fig. 13. Different manufactured joints used in 2-PUU_2-PUS PM.
(a) CAD model of Transmission joint
(b) Real model of Transmission joint
Fig. 14. Transmission joint.
$1;i ¼ $4;i ¼
s1;i
;
$2;i ¼
0 Bi s4;i s4;i
Ai s2;i s2;i
and $5;i ¼
;
$3;i ¼
Bi s5;i
Ai s3;i s3;i
;
Sp ¼ Sk ¼ d_ k $1;k þ h_ 2;k $2;k þ h_ 3;k $3;k þ h_ 4;k $4;k þ h_ 5;k $5;k þ h_ 6;k $6;k ð39Þ where k = 2, 4, and h_ j;k is the intensity and $j;k denotes a unit screw associated with the jth joint of the kth limb.
s5;i
sj;i is a unit vector along the jth joint of the ith limb. From Fig. 7a, we can get the following:
Ai ¼ Bi RX ðai Þ RY ðbi Þ Li
$1;k ¼ ð38Þ
Similarly, the connectivity of the other two limbs PUS is equal to 6. So we can describe these limbs as a linear combination of the six instantaneous twists as follow:
s1;k
;
$2;k ¼
0 Bk s4;k ; ¼ s4;k
Ak s2;k
$4;k
$5;k
;
$3;k ¼
Ak s3;k
;
s2;k s3;k Bk s5;k Bk s6;k and $6;k ¼ ¼ s5;k s6;k
Ak ¼ Bk RX ðak Þ RY ðbk Þ Lk
ð40Þ
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9
Universal joint
(Set screw) for S joint
Fig. 15. The proposed endoscopic manipulator based on PVC ‘‘EndoEJUST’’ [35].
Fig. 16. Load simulation of 5 N vertical forces on the movable platform.
Fig. 17. Control block diagram for tracking motion of PM.
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Fig. 18. Signal constraint graphically shaping the desired response.
Fig. 19. Sine signal position for each prismatic joints.
4.1. Jacobian of Constraint Those screws are reciprocal to all the joint screws of the ith limb form a ð6 ci Þ reciprocal screw system. Hence, ð6 ci Þ reciprocal basis screws can be identified.
$Trc;i Si ¼ 0 c ¼ 1; 2; . . . ; 6 ci
ð41Þ
where $rc;i denotes the cth reciprocal basis screw of the ith limb. The reciprocal systems for the first and third limbs are [34]:
$rc;i ¼
0
ð42Þ
ri
where r i is a unit vector along the direction defined by ðs2;i s3;i Þ or ðs4;i s5;i Þ. The Jacobian of constraints can be expressed as:
J c Si ¼ 0 " 013 Jc ¼ 013
r T1 r T3
#
ð43Þ ð44Þ
26
4.2. Jacobian of actuations
Fig. 20. Joints position control response for PM.
Now, lock the actuated joint in each limb. With the actuator locked, the dimension of the reciprocal screw system increases by 1 [27]. Then, when making the actuators of PM locked, the reciprocal screws of the first and third limbs form a 2-system which includes the screw $rc;i . One additional basis screw $ra;i being reciprocal to all the passive joint screws of the ith limb can be identified as a zero pitch screw along the direction passing through the two U joints:
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R ¼ r ¼ 3:5 mm, L1;2;3;4 ¼ 20 mm, m = 5 mm and all hook joints’ angles are zero. Then, we calculated the determinant of the overall Jacobian J which is found to be equal to zero as this is one of the singular positions. This is shown in Fig. 10. The MATLAB program can be applied to any position of the mobile platform to check the singularity.
5. Development of parallel mechanism 5.1. SolidWorks model The proposed conceptual design of the parallel manipulator is implemented using SolidWorks software as shown in Figs. 11 and 12. As a first trial, a scaled-up prototype of the proposed PM with a scale (2:1) was developed using plastic as the working material and machined using 3D prototyping machine [31]. This first trial prototype is served to understand and verify the mechanism motion and to check the feasibility of the proposed PM. Fig. 21. Error in each joint and position of movable platform.
$ra;i ¼
ðAi Bi Þ Z
5.2. Mechanical design and manufacturing
ðAi Bi Þ Z Bi
Similarly, for the second and fourth limb (PUS) with actuators locked, the reciprocal screws are as follows:
$ra;k ¼
ðAk Bk Þ Z ðAk Bk Þ Z Bk
J a Sp ¼ q_
ð45Þ h
From (39), q_ ¼ d_ 1
2
ððAi Bi ÞZ ÞT ððAi Bi ÞZ ÞT s1;1
6 6 ððA B ÞZÞT k k 6 6 ððAk Bk ÞZÞT s1;2 Ja ¼ 6 6 ððAi Bi ÞZÞT 6 6 ððAi Bi ÞZÞT s1;3 4 T ððAk Bk ÞZ Þ ððAk Bk ÞZ ÞT s1;4
d_ 2
d_ 3
ðAi Bi ÞZBi ððAi Bi ÞZ ÞT s1;1 ðAk Bk ÞZBk ððAk Bk ÞZ ÞT s1;2 ðAi Bi ÞZBi ððAi Bi ÞZ ÞT s1;3 ðAk Bk ÞZBk ððAk Bk ÞZ ÞT s1;4
d_ 4
iT
denotes the actuated joint.
All parts of the proposed 2-PUU_2-PUS were designed and then fabricated using high precision machines as shown in Fig. 13b, c, and e [35]. Transforming the rotary motion of the motor to translational motion of the prismatic joint is illustrated in Fig. 14. A real model of the proposed parallel mechanism, in actual size, is manufactured from AISI 347 annealed stainless steel as shown in Fig. 15. This prototype is served to understand and verify the mechanism motion also and to prove the feasibility of the design. The real prototype illustrates the capability of making large bending angles.
3 7 7 7 7 7 7 7 7 5
Point (0,2 and 4)
ð46Þ Point (3)
46
JSp ¼ q_ o
Point (1)
ð47Þ
where J is the overall Jacobian that we can get by the following Eq. (26):
J¼
Ja Jc
ð48Þ 66
h where q_ o ¼ d_ 1
d_ 2
d_ 3
d_ 4
0
0
iT
.
The first four rows of J represent four forces of actuation acting along the four limbs. Clearly, the manipulator will be under singularity if these four forces lie on a common plane, such as the case when moving the mobile platform up and down vertically when all links are equally stretched, or are parallel to each other as shown in Fig. 8. On the other hand, the last two rows of J represent two moments of constraint, each being perpendicular to the universal joint of a limb [27]. Constraint singularity occurs when J c loses its rank (rank = 2) whereas architecture singularity occurs when det(J) = 0 but J c has a full rank of 2 [32]. 4.3. Numerical result MATLAB program is build using the flowchart shown in Fig. 9 to determine the singular positions of the PM. The following numerical values are used for the PM parameters:
Fig. 22. Joints position control response (Sine signal) for PM.
Please cite this article in press as: Ibrahim K et al. Development of a new 4-DOF endoscopic parallel manipulator based on screw theory for laparoscopic surgery. Mechatronics (2015), http://dx.doi.org/10.1016/j.mechatronics.2015.02.006
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5.3. Load simulation For the load simulation, the mechanism is in straight configuration with all prismatic joints are locked and the movable platform is loaded with forces in the Z- direction as shown in Fig. 16. The mechanism maintains a force of 5 N in the Z-direction which deforms it slightly [36]. The AISI 347 annealed stainless steel has been selected as the material for manufacturing the endoscopic parallel manipulator. The maximum stress in this case is about 30 MPa which is below the yield point of the material (275 MPa). Also, the maximum deflection observed is within the acceptable limits.
6. Control simulation Simulation of the virtual prototype (4-DOF PM) has been accomplished by combining MATLAB/Simulink with ADAMS. During the simulation, the control algorithm is executed under MATLAB to generate the command forces, which are then exported to ADAMS environment and applied to the model as actuation forces at each sample time. The resulted outputs of the virtual prototype (velocity of the prismatic joints are measured by sensors in ADAMS) are then feedback to the controller in MATLAB/Simulink for calculation of the next command signal. The schematic diagram in Fig. 17 illustrates the inputs and outputs’ variables for the virtual prototype. The Simulink Design Optimization (SDO) is used to automatically tune the controller parameters to meet time-domain specifications. We can optimize any design criterion by expressing it as a Simulink signal and connecting this signal to the Signal Constraint block provided by SDO. The Signal Constraint block can constrain the signal either by graphically shaping the desired response or by specifying a reference signal trajectory as shown in Fig. 18. It then adjusts the values of chosen controller parameters to satisfy the constraints [37]. The signal constraint block GUI is updated during optimization so that we can visually monitor the optimization progress. To follow smoothly and accurately the commands of the surgeon, it is necessary to control both, the position and the velocity of the linear motors. Each DOF is controlled by a PID-position/PI-velocity loop as shown in Fig. 17. A PI-velocity loop is enclosed within a proportional position loop. The velocity is measured by ADAMS™ software. In laparoscopic surgery, the surgeon needs to make trajectory planning
Fig. 23. Error in each joint and position of movable platform.
during the surgery. Some signals that mimic the surgeon motion during the surgery are chosen. One of these signals is Sine signal as shown in Fig. 19. Several signal constraint blocks are used simultaneously to optimize multiple design criteria such as the error in the position of each prismatic joint.
7. Results and discussion 7.1. Result of bending motion To test the capability of the simulation control, a step input is applied to each prismatic joint in the PM as follows: Each DOF is controlled by a PID-position/PI-velocity loop as shown in Fig. 17. PI-velocity loop is enclosed within a proportional position loop. The velocity is estimated from the position signal by differentiation. PID controller is used to control the positions of each prismatic joint in the PM. Signal constraint blocks in Simulink/MATLAB are used to tune the parameters of the controllers. Overshoot is not desired especially in position control systems. It can be seen in Fig. 20 that signal constraint block adjusted the controllers parameters for each prismatic joint to avoid overshoots in most of the position responses. The simulation results of the steady state error as shown in Fig. 21 are almost 2.5% in each joint position with very small overshoot occurs. This interface between ADAMS and Simulink can be very useful to facilitate the approach known as design for control. Complex mechanisms can be designed and tested very easily using this interface. 7.2. Results of motion tracking The sinusoidal signal inputs that have been used for each prismatic joint are given as: The simulation results are illustrated in Fig. 22. The error of tracking in the joint position is almost 5% for each joint as shown in Fig. 23. The results show that the position of the movable platform center started and ended almost at the same location. Hence the movable platform is not affected by the tracking error in each joint.
8. Conclusion and future work Laparoscopic instrument was designed based on parallel mechanisms to serve as a rigid surgical manipulator. The design of the parallel manipulator that is based on the screw theory and parallel virtual chain approach is presented. The proposed manipulator consists of four links; two links are 2-PUU (each leg consists of one active prismatic joint and two passive hook joints); the other two links are 2-PUS (each leg consists of one active prismatic join, one passive hook joint and one passive spherical joint). The direct and inverse kinematics analysis of the parallel mechanism is developed. A closed-form solution for the inverse kinematics problem is obtained. Computer simulations (MATLAB/ADAMS) and control system design confirmed the feasibility of the new design. The results proved that the workspace of the bending mechanism is increased significantly and reached ±90° uniformly in all directions. The tracking control simulation results are satisfactory with an error of maximum 5% for each joint. An annealed stainless steel prototype is manufactured and tested successfully. In future work, a master/slave configuration will be prepared to test the performance of the whole system. Also, a hybrid intelligent control strategy could be implemented to improve the positioning accuracy and response of the surgical manipulator.
Please cite this article in press as: Ibrahim K et al. Development of a new 4-DOF endoscopic parallel manipulator based on screw theory for laparoscopic surgery. Mechatronics (2015), http://dx.doi.org/10.1016/j.mechatronics.2015.02.006
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Acknowledgments This research work is funded from the Science and Technology Development Fund (STDF – Project: ID 4100) and from the Missions Department, Ministry of Higher Education (MOHE), Egypt. The research team appreciates very much the support of Prof. Fujie through his invitation to the first author (Exchange Researcher) for a long visit to Waseda University, Japan.
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[22] Ramadan A, Takubo T, Mae Y, Oohara K, Arai T. Development process of a chopstick-like hybrid structure two-fingered micromanipulator hand for 3D manipulation of microscopic objects. IEEE Trans Indust Electron 2009;56(4). [25] Si Zhu J, Huang Zhen, Ding Hua F. Forward/reverse velocity and acceleration analysis for a class of lower-mobility parallel mechanisms. Trans ASME 2007;129. [26] Fang YF, Tsai LW. Inverse velocity and singularity analysis of low-dof serial mechanisms. J Rob Syst 2003;204:177–88. [27] Joshi SA, Tsai LW. Jacobian analysis of limited-DOF parallel manipulators. ASME J Mech Des 2002:254–8. [28] Kong X, Gosselin C. Type synthesis of parallel mechanisms. Berlin, Heidelberg: Springer-Verlag; 2007. [31] Ibrahim K, Ramadan A, Fanni M, Kobayashi Yo, Abo-Ismail A, Fujie Masakatus G. Design and workspace analysis of a new endoscopic parallel manipulator. In: 12th International conference on control, automation and system ICCAS 2012, October 17–21. ICC, Jeju Island (Korea); 2012. [32] Ibrahim K, Ramadan A, Fanni M, Kobayashi Yo, Abo-Ismail A, Fujie Masakatus G. Kinematic analysis and control of limited 4-DOF parallel manipulator based on screw theory. In: IEEE international conference on systems, man, and cybernetics IEEE SMC 2012, COEX, October 14–17. Seoul (Korea); 2012. [33]
Further reading [23] Yang Chifu, Qu Zhiyong, Han Junwei. Decoupled-space control and experimental evaluation of spatial electro-hydraulic robotic manipulators using singular value decomposition algorithms. IEEE Trans Indust Electron 2014;61(7):3427–38. http://dx.doi.org/10.1109/TIE.2013.2278958. [24] Allais AA, McInroy JE, O’Brien JF. Locally decoupled micromanipulator using an even number of parallel force actuators. IEEE Trans Robot 2012;28(6):1323–34. http://dx.doi.org/10.1109/TRO.2012.2209229. [29] Davidson J, Hunt K. Robots and screw theory – applications of kinematics and statics to robotics. Great Britain: Oxford University Press; 2004. [30] Tsai LW. Robot analysis the mechanics of serial and parallel manipulators. New York: Wiley-Science; 1999.
Khalil Ibrahim received the B.E. and M.Sc. degrees in mechanical engineering from the Mechanical Engineering Department, Faculty of Engineering, University, Assiut, Egypt, in 2001 and 2006, respectively. Then he worked toward the Ph.D. degree from late 2010 till September, 2013 at Mechatronics and Robotics Engineering, Graduate School of Innovation, Egypt-Japan University of Science and Technology EJUST, Egypt. He worked as exchange researcher in Waseda University, Japan around one year. During this period he got experiences in surgery robot design. Starting from September, 2013 he is a Lecturer/Assistant professor in the mechanical department, Faculty of Engineering, and Assiut University. His major research interests include robotics engineering, automatic control, and artificial intelligence techniques. His current research focuses on Tele surgical manipulators, endoscopic manipulator, and control system design.
Ahmed A. Ramadan received his B.E. and M.Sc. degrees from Electrical Engineering department in Computer Engineering and Automatic Control from Tanta University, Egypt, in 1996 and 2002 respectively. Then he worked toward the Ph.D. degree from late 2005 till March, 2009 at Systems Innovation Department, Graduate School of Engineering Science, OSAKA University, Japan. Starting from May, 2009 he is a Lecturer/Assistant Professor in the Computer and Control Engineering department in Tanta University. As of the beginning of April 2010, he granted a research fellowship to work as an Assistant Professor in Mechatronics and Robotics Engineering department, Egypt-Japan University and science and technology (EJUST), Egypt. He is a
Please cite this article in press as: Ibrahim K et al. Development of a new 4-DOF endoscopic parallel manipulator based on screw theory for laparoscopic surgery. Mechatronics (2015), http://dx.doi.org/10.1016/j.mechatronics.2015.02.006
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member of the Institute of Electrical and Electronics Engineers (IEEE). His research interests are in the fields of Robotics Engineering, Automatic Control, and Artificial Intelligence Techniques. His current research interests focuses on the design and control of Surgical Manipulators, Master/ Slave teleoperation systems, Aerial manipulation and control of Flying Robots.
Mohamed Fanni received the B.E. and M.Sc. degrees in mechanical engineering from Faculty of Engineering of both Cairo University and Mansoura University, Egypt, in 1981 and 1986, respectively and the Ph.D. degree in engineering from Karlsruhe University, Germany, 1993. He is an Associate Professor with Innovation, Graduate School of Engineering Science, Egypt-Japan University of Science and Technology E-JUST, Alexandria, on leave from Production Engineering & Mechanical Design Department, Faculty of Engineering, Mansoura University, Egypt. His major research interests include robotics engineering, automatic control, and Mechanical Design. His current research focuses on Design & Control of Mechatronic Systems, Surgical Manipulators, Teleoperation systems and Flying/ Walking Robots.
Yo Kobayashi (M’08) received the B.S. and M.S. degrees from the Graduate School of Science and Engineering, Waseda University, Tokyo, Japan, in 2004 and 2005, and the Ph.D. degree in engineering from Waseda University in 2008. He was a Visiting Research Associate with the Graduate School of Science and Engineering, Waseda University, from 2005 to 2006. He was a recipient of the Japan Society for the Promotion of Science Re-search Fellowship for Young Scientists in 2007. In 2008, he was a Research Associate with the Institute for Biomedical Engineering, Waseda University, whereas in 2009, he was a Research Associate with the Faculty of Science and Engineering, Waseda University. He has been a Lecturer (Junior Researcher) with the Faculty of Science and Engineering, Waseda University, since 2010. His current research interests include medical robots and rehabilitation robots. Dr. Kobayashi is a member of the Japan Society of Mechanical Engineers, the Society of Instrument and Control Engineers, the Robotics Society of Japan, and the Japan Society of Computer-Aided Surgery.
Ahmed Abo-Ismail received his Ph.D. degree from Tokyo Institute of Technology, TIT, Japan, 1979.He is a full professor of Automatic Control and Mechatronics, EJUST, Egypt. He is an Honorary Professor of Budapest Tech., Hungary. He is granted a Fulbright award fellowship as a Visiting Professor to PSU, USA, 1987. He is an IFAC Chair (International Federation of Automatic Control) for African, Asian, South American Countries, 2003–2006 and he is a Member of the technical Editorial Board of the ACTA Polytechnica Hungaraica Journal of Applied Sciences, Budapest, Hungary. He is also the Provost, Vice President for Education and Academic Affairs, EJUST, Egypt and former dean of faculty of Engineering, Assuit University, Assuit, Egypt. His major research interests are in the fields of Intelligent and Robust Control, Smart Grippers design, Intelligent Mechatronics System, VRHepatic Surgical Simulators, Surgical Robots, Assistive Devises for elderly people design and control, Aerial manipulation and control, and Flying/Walking Robots.
Masakatsu G. Fujie (M’90–SM’08) received the M.S. degree from the Graduate School of Science and Engineering, Waseda University, Tokyo, Japan, in 1971, and the Ph.D. degree in engineering from Waseda University in 1999. From 1971 to 2000, he was with the Mechanical Engineering Research Laboratory, Hitachi Ltd., where he was a Senior Researcher during 1984 and a Principal Researcher and Project Leader for the Medical and Welfare Apparatus Development Project during 1995. He was also the Head of Researchers in the Mechanical Engineering Research Laboratory, Hitachi Ltd., and the Director o f the Medical and Welfare Apparatus Development Research Laboratory during 1999. Since 2001, he has been a Professor with the Faculty of Science and Engineering, Waseda University, where he is the Director of The Grobal Robot Academia for MEXT Grobal COE Program. His current research interests include surgical robots, image guided surgery, endoscopic surgery, and assistive and rehabilitation robots. Prof. Fujie is a Fellow o f the Japan Society of Mechanical Engineering (JSME) and a member of the International Society of Computer-Aided Surgery. He received the Atomic Energy Society of Japan’s Technology Development. Award (1991), the Notable Invention Award from the Science and Technology Agency Director General (1994), the Technical Innovations Awards from the Robotics Society of Japan (1999), the JSME Medal for New Technology from the Japan Society of Mechanical Engineers (2000), the Nikkei BP Technology Award, Medical/Biotechnology Division (2000), the Japan Society of Mechanical Engineers, Robotics and Mechatronics.
Please cite this article in press as: Ibrahim K et al. Development of a new 4-DOF endoscopic parallel manipulator based on screw theory for laparoscopic surgery. Mechatronics (2015), http://dx.doi.org/10.1016/j.mechatronics.2015.02.006