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Design of a 1R1T planar mechanism with remote center of motion
Mechanism and Machine Theory 149 (2020) 103845
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Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmachtheory
Research paper
Design of a 1R1T planar mechanism with remote center of motion Wei Ye, Bo Zhang, Qinchuan Li∗ Faculty of Mechanical Engineering and Automation, Zhejiang Sci-Tech University, Hangzhou, Zhejiang Province 310018, PR China
a r t i c l e
i n f o
Article history: Received 12 December 2019 Revised 12 January 2020 Accepted 12 February 2020
Keywords: Remote-center-of-motion mechanism Kinematics analysis Singularity Optimization design
a b s t r a c t Compared with traditional open surgery, minimally invasive surgery (MIS) is safer and more beneficial to patients’ recovery. However, this type of surgery is an arduous task for doctors. Using robots to carry out MIS is a good choice. In this paper, a new remote-centerof-motion (RCM) mechanism is presented. First, the direct kinematics and inverse kinematics are analyzed. Second, singularity analysis is conducted based on the Jacobian matrix, and a three-dimensional workspace is sketched. Next, the motion/force transmission index is established using screw theory, and optimization design is conducted to improve the performance of the mechanism. Finally, workspace validation on a prototype is conducted. The proposed RCM mechanism has several advantages: a good actuation scheme, analytical direct kinematic solutions, and no singular configurations in the prescribed workspace. The proposed RCM mechanism is suitable for single minimal incision surgery, such as biopsy and cryosurgery. © 2020 Elsevier Ltd. All rights reserved.
1. Introduction Many surgical procedures, such as minimally invasive biopsies, brachytherapy, and cryotherapy, involve the insertion of a slender surgical instrument into a patient’s body through an incision point, and the instrument is controlled by a doctor from outside the body [1,2]. Surgical robots can help doctors to perform surgery in a safe and efficient manner because of the well-established kinematics modeling and control system [3]. Because this type of minimally invasive surgery (MIS) has to be performed through a small incision, the surgical robotic mechanism should have four degrees of freedom (DOFs): one translational DOF that passes through the incision point and three rotational DOFs around the incision point. It should be noted that the incision point should be far away from the mechanism to avoid mechanical interference. Mechanisms that can perform this type of motion are called remote-center-of-motion (RCM) mechanisms [4]. A variety of RCM mechanisms have been proposed based on some special mechanisms, such as parallelograms, spherical linkages, and gear chains [5–8]. Generally, there are two approaches to obtain RCM output. One approach is to use serial mechanisms with high DOFs, and ensure that the end effector of the mechanism always passes through the incision point using an advanced control strategy [9,10]. However, a serious security problem occurs when the control strategy fails. The other approach is to design RCM mechanisms whose RCM output is mechanically guaranteed. The da Vinci robot [11,12] is a typical example that has been extensively applied. Several other mechanically constrained RCM mechanisms exist, such as the gear chain surgical robot proposed by Lehman et al. [13] and the generalized double parallelogram mechanism proposed by Kong et al. [14]. ∗
Corresponding author. E-mail addresses: [email protected] (W. Ye), [email protected] (B. Zhang), [email protected] (Q. Li).
https://doi.org/10.1016/j.mechmachtheory.2020.103845 0094-114X/© 2020 Elsevier Ltd. All rights reserved.
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Fig. 1. Structure of the 1R1T RCM mechanism.
Huang et al. [15] presented type synthesis of 1R1T remote center of motion mechanisms based on pantograph mechanisms. Chen et al. [16] proposed a new type of planar two degree-of-freedom remote center of motion mechanism inspired by the Peaucellier–Lipkin straight-line linkage. It is widely acknowledged that parallel mechanisms can provide high rigidity because of their multiple closed-loop structures, which is very welcome in surgical applications. Therefore, researchers have focused on the structural design and analysis of RCM parallel mechanisms. Li et al. [17] introduced a type synthesis method of RCM based on intersecting motion planes. Vischer et al. [18] proposed a 3-DOF RCM parallel mechanism using parallelograms. Di Gregorio [19] proposed a simple 3RRS parallel manipulator that has RCM. Navarro et al. [20] proposed a 3UPS-1S parallel mechanism that is suitable for laparoscopic surgery. It is worth noting that, generally, most parallel mechanisms have small workspaces, complex direct kinematic solutions, and singularity configurations inside the workspace. Another major concern about RCM mechanisms is their performance. Some widely used kinematics performance indices, such as the condition number [21] and manipulability [22], have been used for RCM mechanisms [23]. However, indices generated from the Jacobian matrix have obscure physical significance when the mechanisms have mixed rotational DOFs and translational DOFs [24], which means that they cannot evaluate performance credibly. Liu and coworkers [25–27] introduced an index that can describe power transfer efficiency from actuators to the end effector in a parallel mechanism, that is, the motion/force transmission index. It is a type of dimensionless index that is not affected by the location of the reference coordinate system. In this paper, a planar 1R1T (R: rotation, T: translation) RCM mechanism is proposed. Its kinematics and dimensional optimization are conducted. A good actuation scheme, simple direction solutions, and large singularity-free workspace are the main features of the mechanism. This paper is organized as follows: In Section 2, the design process is presented. Position analysis of the mechanism is conducted in Section 3. In Section 4, the Jacobian matrix, singularity configurations, and workspace are analyzed. In Section 5, the performance analysis of the mechanism based on the motion/force transmission index is presented. In Section 6, optimization design is conducted. In Section 7, the workspace validation is conducted. Finally, in Section 8, the paper is summarized. 2. Structural design MIS requires three rotational DOFs around the incision point and one translational DOF through the incision point (3R1T RCM motion). Since the 3R1T RCM motion can be generated by connecting two serial rotations to a planar 1R1T RCM mechanism, the design objective can be identified as designing a planar 1R1T RCM mechanism. Because the parallelogram mechanism is widely used in RCM mechanisms to guarantee remote center rotation, a parallelogram and a linear motion mechanism that is used as a translation generator are used to construct a planar 1R1T RCM mechanism in this paper. Fig. 1(a) shows the basic structure of the parallelogram mechanism. It contains two sets of parallel links connected by revolute joints. It is obvious that the virtual link denoted by EI always passes through point O. Hence, the virtual link EI can output an RCM rotation. Fig. 1(b) shows a 1-DOF four bar linkage. Link FK is parallel to link EO, and link IK is parallel to link EG. Because the IO FK special length condition EO EG = IK = F G is satisfied, points G, K, and O are collinear [16,28]. Therefore, we can add a block and link (denoted by the virtual lines in Fig. 1(b)) to the four bar linkage without affecting its DOFs. This mechanism is called a linear motion mechanism because there is a relative translation between the added link and block. By adding the linear motion mechanism to the parallelogram shown in Fig. 1(a), a 2-DOF mechanism can be constructed, as shown in Fig. 1(c). It should be noted that some special conditions for link lengths should be satisfied. Link GM in this
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Fig. 2. Schematic diagram of the initial mechanism.
Fig. 3. Modified mechanism and final mechanism.
mechanism can perform a rotation around virtual point O and a translation passing through point O. The mechanism is a planar 1R1T RCM mechanism. However, to control the translation of link GM, an actuator needs to be added in the linear motion mechanism, which increases the moving mass. To enhance the stiffness and optimize the actuation scheme, the mechanism in Fig. 1(c) is duplicated symmetrically. A novel planar mechanism is obtained, as shown in Fig. 2. It has the same DOFs as the mechanism in Fig. 1(c) because the newly added parts introduce only virtual constraints. Links E1 G and E2 G remain parallel to lines OE2 and OE1 , respectively, during motion. The motion of output link GM depends on the shape change of the parallelogram OE1 GE2 , and consists of a rotation around point O and a translation passing through point O. When joints C1 and C2 are locked, points E1 and E2 are fixed, and all the links cannot move because E1 E2 O becomes a fixed triangle. Therefore, joints C1 and C2 can be used as inputs. Actuators for this mechanism are mounted near the fixed base, which results in good dynamic performance. This mechanism will be referred to as the initial mechanism hereafter. Note that in the initial mechanism, many links introduce only virtual constraints. The topological structure of the mechanism is complex. To simplify it, some of the links that do not affect the mechanism’s DOFs are removed. A modified mechanism is obtained, as shown in Fig. 3(a). This mechanism is the same with that presented by Chen et al. [16]. It has the same motion characteristic as the initial mechanism, and is also fixed when the two revolute joints C1 and C2 are locked. We detected that, for the modified mechanism shown in Fig. 3(a), the translational range of output link GM depends on the shape-changing ability of quadrangle OE1 GE2 . Provided that the translational range is about 250 mm, which is usually adopted in mechanism design for MIS [29], the longest link in quadrangle OE1 GE2 can be approximately 600 mm if some
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Fig. 4. A 3R1T RCM mechanism.
Fig. 5. Diagram of the 1R1T RCM mechanism.
angle limitations are set between links to avoid mechanical interference. Clearly, such a length is too large and will result in problems for manufacturing and accuracy. Therefore, the modified mechanism is further improved, and a novel mechanism is obtained, as shown in Fig. 3(b), which is referred to as the final mechanism. Of note is that the final mechanism also belongs to the mechanisms family synthesized in Ref. [16]. However, it has different topological structure, which will result in different kinematic modeling and performance. Furthermore, the translational range of link GM in the final mechanism is the summation of the diagonal length changes in quadrangles OE1 KE2 and KF1 GF2 . The link length can be decreased while still satisfying the output link’s translational range. It is worth noting that all three mechanisms in Figs. 2 and 3 can perform 1R1T RCM motion. The motion of output link GM in these mechanisms can be fully controlled by actuators mounted on joints C1 and C2 because a fixed triangle E1 E2 O forms once the input joints are locked. Of note is that line OE1 is not necessarily parallel to link GE2 in Fig. 3(a), and line OE1 is not necessarily parallel to link F1 E2 in Fig. 3(b). Taking into consideration the structural complexity and linkage dimensions, the final mechanism shown in Fig. 3(c) is selected, and investigated in the following sections. By connecting the final mechanism to a fixed base using a revolute joint, and adding a rotational DOF to the surgical tool around its axis, a 3R1T RCM mechanism can be constructed as shown in Fig. 4. Constructing spatial 3R1T RCM mechanisms from 1R1T RCM mechanism is a common practice [15]. Such an implement has two sides. On one hand, because of the added DOFs are serially connected, the rotational ranges are relatively large, and kinematics of the resulted mechanism is simple. On the other hand, large power input is required for the base-connected rotational DOF because the whole mechanism rotates around that axis. 3. Position analysis In this section, both direct position analysis and inverse position analysis of the proposed 1R1T RCM mechanism are conducted. Attach a reference coordinate system Oxy to the fixed base with its origin coincident with point O, as shown in Fig. 5. Axis x is coincident with OA2 , and axis y is vertical. Let Mx and My denote the coordinates of point M in the reference coordinate system, and θi (i = 1, 2) denote the angle between link Ci Di and axis x. The symbols used to denote the lengths of links are listed in Table 1.
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Table 1 Link length symbols. Links
Ai Bi
Ei K
Fi K
Fi G
GM
Length
l1
l2
l3
l4
l
3.1. Direct position analysis The direct position problem is to identify coordinates Mx and My given θi (i = 1, 2). The distance from point Ei (i = 1, 2) to link GM is denoted by p:
p = l1 sin α .
(1)
θ1 −θ2
where α = 2 is the angle between GO and E1 O. The coordinates of point K can be obtained as
Kx = l1 cos θ1 + l2 cos β , Ky = l1 sin θ1 + l2 sin β
(2)
where β is the angle between link E1 K and axis x, which can be expressed as
β = arccos
p + l2
θ1 + θ2 2
−
π 2
.
(3)
The distance from point K to point G is denoted by lKG :
lKG = l3 cos α1 + l4 cos α2 ,
(4)
where α 1 is the angle between E1 K and KM, and α 2 is the angle between GM and GF1 , as shown in Fig. 5. We have
α1 = arcsin lp2 α1 . α2 = arcsin l3 sin l4
(5)
Because points K, M, and O are collinear, the coordinates of point M can be obtained as
Mx = Kx − (l − lKG ) · Kx / Kx2 + Ky2 . My = Ky − (l − lKG ) · Ky / Kx2 + Ky2
(6)
Eq. (6) means that analytical expressions for forward position solutions are obtained, which makes the real-time control of the mechanism easier. 3.2. Inverse position analysis The inverse position problem is to identify the input angles θi (i = 1, 2) given the coordinates of point M. The coordinates of point G in the coordinate system can be expressed as
Gx = Mx − l Mx / Mx2 + My2 Gy = My − l My / Mx2 + My2
.
(7)
Considering the distance between points G and O, we have the following equation:
l1 cos α + (l2 + l3 ) cos α1 + l4 cos α2 =
G2x + G2y .
(8)
Substituting Eqs. (1) and (5) into Eq. (8), α can be derived. Then, the inverse position solutions can be obtained as
y θ1 = arc tan M +α Mx . My θ2 = arc tan Mx − α
(9)
4. Velocity, singularity, and workspace analysis 4.1. Velocity analysis Velocity analysis establishes a mapping between the input joint velocity and end-effector velocity, which can be derived from the position model. To simplify the analysis, the angle between link GM and axis x denoted by ϕ , and the length lKG are regarded as the output parameters. From the geometrical relationship of the mechanism, we have
ϕ = θ2 +
θ1 − θ2 2
=
θ1 + θ2 2
,
(10)
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Fig. 6. Singular configurations.
2 l32 + lKG − l42 2l3 lKG
2
+
l1 sin l2
θ1 − θ2
2 = 1.
2
(11)
Taking derivation on both sides of Eqs. (10) and (11) with respect to time, velocity equations can be obtained, and rearranged in matrix form as
ϕ˙
JA ˙ lKG where
θ˙ = JB 1 , θ˙ 2
J JA = A11 JA21
(12)
JA12 J , JB = B11 JA22 JB21
JB12 JB22
JA11 = 1, JA12 = 0, JA21 = 0, JA22 = JB11 =
1 ,J 2 B12
=
1 ,J 2 B21
=−
l12
cos
2 2 −l42 ) (l42 +lKG −l32 ) (l32 +lKG ,
θ1 −θ2
3 2l32 lKG
sin
2
l22
θ1 −θ2 2
, JB22 =
l12 cos
θ1 −θ2
sin
2
θ1 −θ2
.
2
l22
4.2. Singularity analysis Singularity analysis is very important for a mechanism. When a mechanism is in singular configurations, it gains or loses some DOFs, which makes the mechanism unable to perform the task. Singularities of a mechanism can be identified based on its Jacobian matrix. For the mechanism analyzed in this paper, two cases of singularities are obtained based on Eq. (12). (1) Case I: det(JB ) = 0. We can obtain the condition that satisfies this singularity as θ 1 − θ 2 = 0. This means that point E1 is coincident with point E2 , and link F2 E1 is coincident with link F1 E2 , as shown in Fig. 6(a). 2 − l 2 = 0 and l 2 + l 2 − l 2 = (2) Case II: det(JA ) = 0. Two conditions that satisfy this singularity can be obtained as l32 + lKG 4 4 KG 3 0. This means that link GK is perpendicular to link E1 K and link F1 G, respectively. The two corresponding configurations are shown in Fig. 6(b) and (c), respectively. 4.3. Workspace Workspace analysis verifies whether the moving range of the end effector satisfies the requirement of the MIS application. The link dimensions are given as l1 =197 mm, l2 =124 mm, l3 =129 mm, l4 =255 mm, and l = 690 mm. Considering the moving ranges and physical interference of the joints in this mechanism, several constraints of the workspace are given as follows: 1. θ 1 ∈ [30, 170◦ ], θ 2 ∈ [10◦ , 150◦ ], and θ1 − θ2 > 20◦ . This constraint is introduced to keep the mechanism far away from singularity since it is singular when θ 1 =θ =2 as shown in Fig. 6(a). 2. Any angle between two links should be larger than 10° to avoid link interference. 3. No singularities in the workspace.
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Fig. 7. Workspace.
Fig. 8. Equivalent model of the 1R1T RCM mechanism.
To better show the appropriateness of the mechanism for MIS application, the additional serial rotational DOF denoted by θ 3 in Fig. 4 is considered. The moving range of θ 3 can be very large because it corresponds to a serial DOF, so we only consider the region with θ3 ∈ [−40◦ , 40◦ ] for simplicity. Search the potential region for points that satisfy the aforementioned constraints. The workspace of the mechanism is numerically determined as shown in Fig. 7. Clearly, the workspace is much larger than the required workspace in MIS, which is generally a cone with a coning angle equal to 60° [30]. 5. Motion/force transmission analysis The workspace of the mechanism is larger than that required for MIS. It is desired that the mechanism has good performance in the desired workspace. Because in 90% of MIS operations the surgical tool works in a cone region with a coning angle of 60° and a length of 250 mm [29,30], a circular sector whose central angle is 60° of the workspace will be the focus of this section, within which performance is analyzed and optimized. 5.1. Local transmission index Motion/force transmission analysis is an effective method when evaluating the performance of a mechanism, and calculates the input transmission index and output transmission index, respectively. The smaller one in the two indices is regarded as the local transmission index (LTI). In this study, the structure of the mechanism is complex, which makes motion/force transmission analysis difficult. To reduce the complexity, three simplifications are used. First, the slider at point K that has the same motion as link GM is regarded as the output link because it is closer to the actuators compared with link GM. Second, links OE1 and OE2 are added, which are treated as input links. Finally, a virtual chain composed of a revolute joint and prismatic joint are added between the base and slider K. The role of this chain is to constrain the motion of the output link so that the parts above slider K can be removed. After these simplifications, an equivalent model in Fig. 8 is
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obtained, in which links OE1 and OE2 are two input links, and slider K is the output link. It has a parallel structure, whose motion/force transmission analysis is much easier. In motion/force transmission analysis, the input transmission index represents the efficiency of power transmitted from the actuated joints to the limbs, whereas the output transmission index represents the efficiency of power transmitted from the limbs to the moving platform. For the equivalent model shown in Fig. 8, power is transmitted from the actuators to the moving platform through two limbs. Limb 1 is composed of links OE1 and E1 K, and three associated revolute joints. Limb 2 is composed of links OE2 and E2 K, and three associated revolute joints. Motion/force transmission analysis is conducted for these two limbs using the method in [25]. For limb 1, the twists of the revolute joints are expressed as
⎧ ⎨$11 = (0 0 1; 0 0 0)
$12 = 0 0 1; − E1y E1x 0 ,
(13)
⎩
$13 = (0 0 1; − Ky Kx 0 )
where $11 is the twist associated with the input revolute joint at point O, $12 is the twist related to the revolute joint at point E1 , and $13 is the twist related to the revolute joint at point K. E1 x , E1 y , Kx , and Ky are the coordinates of points E1 and K. The constraint wrench system that is reciprocal to the twist system in Eq. (13) is obtained as
$r11 = (0 0 1; 0 0 0) $r12 = (0 0 0; 1 0 0) . $r13 = (0 0 0; 0 1 0)
(14)
In the constraint wrench system, $r11 denotes a constraint force along axis z. $r12 and $r13 denote constraint couples around axes x and y, respectively. For limb 1, its transmission wrench $T1 should not be included in the wrench system in Eq. (14), and is reciprocal to the twist screw system of the passive joints (the last two twists in Eq. (13)), which is expressed as
$T 1 ◦ $1i = 0 i = 2 , 3 .
(15)
Therefore, transmission wrench $T1 is obtained as
$T1 = (a1 b1 0; 0 0 f1 ), where a1 =
Kx −E1x D1 , b1
=
Ky −E1y D1 , f 1
=
Kx E1y −E1x Ky , D1 D1
=
(16)
(E1y − Ky )2 + (E1x − Kx )2 .
According to screw theory, the wrench in Eq. (16) represents a force along link E1 K. Similarly, transmission wrench $T2 in limb 2, which represents a force wrench along linkage E2 K, is obtained as
$T2 = (a2 b2 0; 0 0 f2 ), where a2 =
Kx −E2x D2 , b2
=
Ky −E2y D2 , f 2
=
Kx E2y −E2x Ky , D2 D2
=
(17)
(E2y − Ky )2 + (E2x − Kx )2 .
By only keeping the actuator in limb 1 active and locking the actuator in limb 2, the transmission wrench $T2 in limb 2 becomes a constraint wrench. Under this condition, the slider has only one remaining DOF, that is, a rotational DOF around point O1 (the intersecting point of line KE2 and line OO1, which is perpendicular to link KO). The rotational center O1 can be obtained using instantaneous centers. The remaining DOF is expressed using a twist screw as
$O1 = (0 0 1; d3 e3 0),
(18)
where d3 = Kx T1 , e3 = Ky T1 , T1 = (E2x Ky − Kx E2y )/(Kx2 + Ky2 − E2x Kx − E2y Ky ). Similarly, output twist screw $O2 for limb 2, which represents a rotation around point O2 , is expressed as
$O2 = (0 0 1; d4 e4 0),
(19)
(Kx E1y − E1x Ky )/(Kx2
where d4 = Kx T2 , e4 = Ky T2 , T2 = + − E1x Kx − E1y Ky ). It should be noted that, in a real mechanism, the directly driven links are C1 D1 and C2 D2 rather than links OE1 and OE2 . Therefore, the transmission efficiency from linkage Ai Bi to linkage OEi (i = 1, 2) should not be neglected. This efficiency can be expressed as μi = sin θi according to the concept of the transmission angle. Therefore, the input transmission performance for limb i (i = 1, 2) is expressed as
λi =
Ky2
|$1i ◦ $Ti | μ. |$1i ◦ $Ti |max i
(20)
The output transmission performance is expressed as
ηi =
|$Ti ◦ $Oi | , |$Ti ◦ $Oi |max
(21)
where $1i denotes the input twist of limb i, $Ti denotes the transmission wrench of limb i, and $Oi denotes the output twist of limb i (i = 1, 2).
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Fig. 9. LTI distribution over the workspace. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
The total transmission index γ is then defined as
γ = min{λi , ηi }, (i = 1, 2 ).
(22)
Because γ only reflects the performance of the mechanism in a certain configuration, it is a local index, which is called the LTI. Clearly, γ is dimensionless, and varies from 0 to 1. When γ tends to 1, the mechanism has good motion/force transmissibility, whereas when γ tends to 0, the mechanism has poor motion/force transmissibility and is close to a singular configuration. For the same set of parameters given in Section 4, the LTI performance of the mechanism over the target workspace is shown in Fig. 9. The result shows that the performance is symmetric about axis y, which is in accordance with the mechanism’s structure. 5.2. Global transmission index Because the LTI can only denote the motion/force transmission performance in a single configuration, the global performance of the mechanism remains unclear. To address this issue, a global index that can evaluate the performance of the mechanism over the prescribed workspace is considered. When γ ≥ 0.7 [31], the motion/force transmissibility of the mechanism is regarded as sufficiently good and all configurations that satisfy this condition form a good transmission workspace. Then, a global index σ can be defined as
σ=
GTW , PW
(23)
where GTW represents the area of the good transmission workspace, and PW represents the area of the entire prescribed workspace. Index σ ranges from 0 to 1. The closer σ is to 1, the better the global performance of the mechanism. 6. Optimization of the design parameters The GTW index σ for Fig. 9 is calculated as 0.256, which means that the GTW only occupies 25.6% of the prescribed workspace. Such global performance is not satisfactory, and should be improved through parameter optimization. In this paper, l1 , l2 , and l3 are chosen as design parameters, and the other parameters are dependent on l1 , l2 , and l3 to ensure that the workspace of the mechanism still satisfies the requirements of MIS. Using the parameter-finiteness normalization method [32,33], l1 , l2 , and l3 are normalized, and non-dimensional parameters r1 , r2 , and r3 are obtained as
r1 =
l1 l2 l3 , r2 = , r3 = , D D D
(24) l +l +l
where D is a normalized factor that equals 1 32 3 . Considering that the size of the mechanism is too large when the length of linkage l1 is smaller than that of linkages l2 and l3 , the following conditions should be satisfied:
r2 , r3 ≤ r1 . 0 < r1 , r2 , r3 < 3
(25)
The parameter design space (PDS) can be obtained as shown in Fig. 10(a), in which the shaded region represents the possible solutions. For visualization reasons, the three-dimensional (3D) PDS is transformed into a plan view, as shown in
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Fig. 10. PDS.
Fig. 11. Distribution of the GTW index. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table 2 GTW in the PDS. Group
s
t
r1
r2
r3
l1 /mm
l2 /mm
l3 /mm
σ
1 2 3 4 5 6 7 8 9
1.275 1.315 1.315 1.335 1.415 1.465 1.515 1.515 1.385
−0.04 −0.02 −0.08 −0.29 −0.15 −0.05 0.01 −0.29 −0.44
1.275 1.315 1.315 1.335 1.415 1.465 1.515 1.515 1.385
0.827 0.825 0.773 0.581 0.662 0.724 0.751 0.491 0.426
0.897 0.859 0.911 1.083 0.922 0.810 0.733 0.993 1.188
191.25 197.25 197.25 200.25 212.25 219.75 227.25 227.25 207.75
124.17 123.77 115.98 87.20 99.38 108.62 110.07 73.70 63.96
134.57 128.97 136.76 162.54 138.36 121.62 112.67 149.04 178.28
0.231 0.268 0.353 0.658 0.586 0.538 0.550 0.866 0.897
Fig. 10(b). The relationship between the parameters in 3D space (r1 , r2 , r3 ) and those in two-dimensional (2D) space (s, t) is as follows:
⎧ r =s ⎪ ⎨1 r2 =
3 2
−
r3 =
3 2
+
⎪ ⎩
√
3 t 2 √ 3 t 2
−
s 2
−
s 2
or
s = r1
t=
√ 3(r3 −r2 ) 3
.
(26)
The procedures for the optimal design are summarized as follows: 1. Obtain the distribution of the GTW index σ of the mechanism in 2D design space as shown in Fig. 11. The colored region represents design region, from which we can choose design parameters with different global performance. 2. Table 2 shows nine groups of parameters chosen from the design region. For each group, obtain the normalized parameters (r1 , r2 , r3 ) using Eq. (26).
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Fig. 12. LTI distribution of the mechanism with parameters in group 9. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 13. Prototype of the final mechanism.
3. Determine the normalized factor D. Considering the practical conditions in surgical applications, D is determined as 150 mm in this paper. Then obtain the link parameters l1 , l2 , and l3 using Eq. (24) and determine the other parameters accordingly considering the workspace requirement. 4. Check whether the linkage parameters obtained in step 3 are suitable for actual use. The process is complete if the necessary assembly conditions are satisfied. Otherwise, return to step 2 and choose another group of data from the design region, and repeat steps 3 and 4. Compared with the performance shown in Fig. 9, whose link parameters are close to those in group 2, a mechanism with parameters in group 9 has much better performance, which is shown in Fig. 12. Its GTW index σ is 0.897, which means that the GTW occupies 89.7% of the prescribed workspace. We select parameters in group 9 as the optimized results. 7. Prototype and workspace validations To validate the design, a prototype of the final mechanism was built, as shown in Fig. 13. Note that the rotational DOF around a base-fixed axis was included. Therefore, the prototype had three DOFs in total. The prototype had three motors, which were used to control its three DOFs. A belt transmission was used to make the motor’s installation easy and reduce the size of the prototype. Workspace validation was conducted on the prototype. Fig. 14 shows 12 configurations. The configurations in the first and third rows demonstrate the rotational capability of the mechanism where the inserted distance of the needle is large, whereas those in the second and fourth rows demonstrate the rotational capability of the mechanism where the inserted distance of the needle is small. The results demonstrated that the mechanism has RCM and the rotational region can reach −40°–40°, which satisfies the workspace requirement of MIS. Future experiments will focus on the calibration and accuracy of the prototype.
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Fig. 14. Workspace validation.
8. Conclusion In this paper, a new type of 1R1T RCM mechanism was proposed. The actuators of the mechanism were arranged near to the base, which resulted in a quick dynamic response. Analytical expressions for forward position solutions were obtained, which benefited the real-time control of the mechanism. The singularity-free workspace of the mechanism was much larger than that demanded in MIS. A performance evaluation of the mechanism over the prescribed workspace was conducted using the method of motion/force transmission. Optimization design was conducted and parameters with good performance were identified. A prototype was built and preliminary experiments showed that the mechanism satisfied the workspace requirement of MIS. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments The authors would like to acknowledge the financial support of the National Natural Science Foundation of China (NSFC) under Grant Nos. 51525504, and 51705465. References [1] G. Strassmann, P. Olbert, A. Hegele, et al., Advantage of robotic needle placement on a prostate model in hdr brachytherapy, Strahlenther. Onkol. 187 (6) (2011) 367–372.
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[2] E.M. Boctor, M.A. Choti, E.C. Burdette, et al., Three-dimensional ultrasound-guided robotic needle placement: an experimental evaluation, Int. J. Med. Robot. Comput. Assist. Surg. 4 (2) (2008) 180–191. [3] C.H. Kuo, J.S. Dai, P. Dasgupta, Kinematic design considerations for minimally invasive surgical robots: an overview, Int. J. Med. Robot. Comput. Assist. Surg. 8 (2) (2012) 127–145. [4] C.H. Kuo, J.S. Dai, Kinematics of a fully-decoupled remote center-of-motion parallel manipulator for minimally invasive surgery, J. Med. Device 6 (2) (2012) 021008. [5] Y. Kobayashi, S. Chiyoda, K. Watabe, et al., Small occupancy robotic mechanisms for endoscopic surgery, in: Proceedings of International Conference on Medical Image Computing and Computer-Assisted Intervention, Berlin, Heidelberg, Springer, 2002, pp. 75–82. [6] A.J. Madhani, G. Niemeyer, J.K. Salisbury, The black falcon: a teleoperated surgical instrument for minimally invasive surgery, in: Proceedings of the1998 IEEE/RSJ International Conference on Intelligent Robots and Systems. Innovations in Theory, Practice and Applications, 2, IEEE, 1998, pp. 936–944. Cat. No. 98CH36190. [7] P. Berkelman, P. Cinquin, J. Troccaz, et al., A compact, compliant laparoscopic endoscope manipulator, in: Proceedings of the 2002 IEEE International Conference on Robotics and Automation, 2, IEEE, 2002, pp. 1870–1875. Cat. No. 02CH37292. [8] H. Kang, J.T. Wen, Robotic assistants aid surgeons during minimally invasive procedures, IEEE Eng. Med. Biol. Mag. 20 (1) (2001) 94–104. [9] O. Schneider, J. Troccaz, A six-degree-of-freedom passive arm with dynamic constraints (PADyC) for cardiac surgery application: preliminary experiments, Comput. Aided Surg. 6 (6) (2001) 340–351. [10] E. Dombre, M. Michelin, F. Pierrot, et al., MARGE project: design, modeling and control of assistive devices for minimally invasive surgery, in: Proceedings of International Conference on Medical Image Computing and Computer-Assisted Intervention, Berlin, Heidelberg, Springer, 2004, pp. 1–8. [11] G.P. Haber, M.A. White, R. Autorino, et al., Novel robotic da vinci instruments for laparoendoscopic single-site surgery, Urology 76 (6) (2010) 1279–1282. [12] E.J. Hanly, M.A. Talamini, Robotic abdominal surgery, Am. J. Surg. 188 (4) (2004) 19–26. [13] A.C. Lehman, M.M. Tiwari, B.C. Shah, et al., Recent advances in the cobrasurge robotic manipulator and dexterous miniature in vivo robotics for minimally invasive surgery, Proc. Inst. Mech. Eng. Part C J. Mech. Eng. Sci. 224 (7) (2010) 1487–1494. [14] K. Kong, J. Li, H. Zhang, et al., Kinematic design of a generalized double parallelogram based remote center-of-motion mechanism for minimally invasive surgical robot, J. Med. Device 10 (4) (2016) 041006. [15] L. Huang, Y. Yang, J. Xiao, et al., Type synthesis of 1R1T remote center of motion mechanisms based on pantograph mechanisms, J. Mech. Des. 138 (1) (2016) 014501. [16] G. Chen, J. Wang, H. Wang, A new type of planar two degree-of-freedom remote center-of-motion mechanism inspired by the peaucellier–lipkin straight-line linkage, J. Mech. Des. 141 (1) (2019) 015001. [17] J. Li, G. Zhang, A. Müller, et al., A family of remote center of motion mechanisms based on intersecting motion planes, J. Mech. Des. 135 (9) (2013) 091009. [18] P. Vischer, R. Clavel, Argos: a novel 3-DoF parallel wrist mechanism, Int. J. Robot. Res. 19 (1) (20 0 0) 5–11. [19] R. Di Gregorio, The 3-RRS wrist: a new, simple and non-overconstrained spherical parallel manipulator, J. Mech. Des. 126 (5) (2004) 850–855. [20] J.S. Navarro, N. Garcia, C. Perez, et al., Kinematics of a robotic 3UPS1S spherical wrist designed for laparoscopic applications, Int. J. Med. Robot. Comput. Assist. Surg. 6 (3) (2010) 291–300. [21] J. Angeles, C.S. López-Cajún, Kinematic isotropy and the conditioning index of serial robotic manipulators, Int. J. Robot. Res. 11 (6) (1992) 560–571. [22] R.S. Stoughton, T. Arai, A modified stewart platform manipulator with improved dexterity, IEEE Trans. Robot. Autom. 9 (2) (1993) 166–173. [23] J. Li, Y. Xing, K. Liang, et al., Kinematic design of a novel spatial remote center-of-motion mechanism for minimally invasive surgical robot, J. Med. Device 9 (1) (2015) 011003. [24] J.P. Merlet, Jacobian, manipulability, condition number, and accuracy of parallel robots, J. Mech. Des. 128 (1) (2006) 199–206. [25] J. Wang, C. Wu, X.J. Liu, Performance evaluation of parallel manipulators: motion/force transmissibility and its index, Mech. Mach. Theory 45 (10) (2010) 1462–1476. [26] X. Chen, F. Xie, X. Liu, Evaluation of the maximum value of motion/force transmission power in parallel manipulators, J. Mech. Eng. 50 (3) (2014) 1–9. [27] X.J. Liu, J. Wang, A new methodology for optimal kinematic design of parallel mechanisms, Mech. Mach. Theory 42 (9) (2007) 1210–1224. [28] G. Zong, X. Pei, J. Yu, et al., Classification and type synthesis of 1-DOF remote center of motion mechanisms, Mech. Mach. Theory 43 (12) (2008) 1585–1595. [29] N. Zhang, P. Huang, Q. Li, Modeling, design and experiment of a remote-center-of-motion parallel manipulator for needle insertion, Robot. Comput. Integr. Manuf. 50 (2018) 193–202. [30] M.J.H. Lum, J. Rosen, M.N. Sinanan, et al., Kinematic optimization of a spherical mechanism for a minimally invasive surgical robot, in: Proceedings of IEEE International Conference on Robotics and Automation, ICRA’04, 1, IEEE, 2004, pp. 829–834. [31] C. Wu, X.J. Liu, L. Wang, et al., Optimal design of spherical 5R parallel manipulators considering the motion/force transmissibility, J. Mech. Des. 132 (3) (2010) 031002. [32] X.J. Liu, L.P. Wang, F. 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Mechanism and Machine Theory journal homepage: www.elsevier.com/locate/mechmachtheory
Research paper
Design of a 1R1T planar mechanism with remote center of motion Wei Ye, Bo Zhang, Qinchuan Li∗ Faculty of Mechanical Engineering and Automation, Zhejiang Sci-Tech University, Hangzhou, Zhejiang Province 310018, PR China
a r t i c l e
i n f o
Article history: Received 12 December 2019 Revised 12 January 2020 Accepted 12 February 2020
Keywords: Remote-center-of-motion mechanism Kinematics analysis Singularity Optimization design
a b s t r a c t Compared with traditional open surgery, minimally invasive surgery (MIS) is safer and more beneficial to patients’ recovery. However, this type of surgery is an arduous task for doctors. Using robots to carry out MIS is a good choice. In this paper, a new remote-centerof-motion (RCM) mechanism is presented. First, the direct kinematics and inverse kinematics are analyzed. Second, singularity analysis is conducted based on the Jacobian matrix, and a three-dimensional workspace is sketched. Next, the motion/force transmission index is established using screw theory, and optimization design is conducted to improve the performance of the mechanism. Finally, workspace validation on a prototype is conducted. The proposed RCM mechanism has several advantages: a good actuation scheme, analytical direct kinematic solutions, and no singular configurations in the prescribed workspace. The proposed RCM mechanism is suitable for single minimal incision surgery, such as biopsy and cryosurgery. © 2020 Elsevier Ltd. All rights reserved.
1. Introduction Many surgical procedures, such as minimally invasive biopsies, brachytherapy, and cryotherapy, involve the insertion of a slender surgical instrument into a patient’s body through an incision point, and the instrument is controlled by a doctor from outside the body [1,2]. Surgical robots can help doctors to perform surgery in a safe and efficient manner because of the well-established kinematics modeling and control system [3]. Because this type of minimally invasive surgery (MIS) has to be performed through a small incision, the surgical robotic mechanism should have four degrees of freedom (DOFs): one translational DOF that passes through the incision point and three rotational DOFs around the incision point. It should be noted that the incision point should be far away from the mechanism to avoid mechanical interference. Mechanisms that can perform this type of motion are called remote-center-of-motion (RCM) mechanisms [4]. A variety of RCM mechanisms have been proposed based on some special mechanisms, such as parallelograms, spherical linkages, and gear chains [5–8]. Generally, there are two approaches to obtain RCM output. One approach is to use serial mechanisms with high DOFs, and ensure that the end effector of the mechanism always passes through the incision point using an advanced control strategy [9,10]. However, a serious security problem occurs when the control strategy fails. The other approach is to design RCM mechanisms whose RCM output is mechanically guaranteed. The da Vinci robot [11,12] is a typical example that has been extensively applied. Several other mechanically constrained RCM mechanisms exist, such as the gear chain surgical robot proposed by Lehman et al. [13] and the generalized double parallelogram mechanism proposed by Kong et al. [14]. ∗
Corresponding author. E-mail addresses: [email protected] (W. Ye), [email protected] (B. Zhang), [email protected] (Q. Li).
https://doi.org/10.1016/j.mechmachtheory.2020.103845 0094-114X/© 2020 Elsevier Ltd. All rights reserved.
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W. Ye, B. Zhang and Q. Li / Mechanism and Machine Theory 149 (2020) 103845
Fig. 1. Structure of the 1R1T RCM mechanism.
Huang et al. [15] presented type synthesis of 1R1T remote center of motion mechanisms based on pantograph mechanisms. Chen et al. [16] proposed a new type of planar two degree-of-freedom remote center of motion mechanism inspired by the Peaucellier–Lipkin straight-line linkage. It is widely acknowledged that parallel mechanisms can provide high rigidity because of their multiple closed-loop structures, which is very welcome in surgical applications. Therefore, researchers have focused on the structural design and analysis of RCM parallel mechanisms. Li et al. [17] introduced a type synthesis method of RCM based on intersecting motion planes. Vischer et al. [18] proposed a 3-DOF RCM parallel mechanism using parallelograms. Di Gregorio [19] proposed a simple 3RRS parallel manipulator that has RCM. Navarro et al. [20] proposed a 3UPS-1S parallel mechanism that is suitable for laparoscopic surgery. It is worth noting that, generally, most parallel mechanisms have small workspaces, complex direct kinematic solutions, and singularity configurations inside the workspace. Another major concern about RCM mechanisms is their performance. Some widely used kinematics performance indices, such as the condition number [21] and manipulability [22], have been used for RCM mechanisms [23]. However, indices generated from the Jacobian matrix have obscure physical significance when the mechanisms have mixed rotational DOFs and translational DOFs [24], which means that they cannot evaluate performance credibly. Liu and coworkers [25–27] introduced an index that can describe power transfer efficiency from actuators to the end effector in a parallel mechanism, that is, the motion/force transmission index. It is a type of dimensionless index that is not affected by the location of the reference coordinate system. In this paper, a planar 1R1T (R: rotation, T: translation) RCM mechanism is proposed. Its kinematics and dimensional optimization are conducted. A good actuation scheme, simple direction solutions, and large singularity-free workspace are the main features of the mechanism. This paper is organized as follows: In Section 2, the design process is presented. Position analysis of the mechanism is conducted in Section 3. In Section 4, the Jacobian matrix, singularity configurations, and workspace are analyzed. In Section 5, the performance analysis of the mechanism based on the motion/force transmission index is presented. In Section 6, optimization design is conducted. In Section 7, the workspace validation is conducted. Finally, in Section 8, the paper is summarized. 2. Structural design MIS requires three rotational DOFs around the incision point and one translational DOF through the incision point (3R1T RCM motion). Since the 3R1T RCM motion can be generated by connecting two serial rotations to a planar 1R1T RCM mechanism, the design objective can be identified as designing a planar 1R1T RCM mechanism. Because the parallelogram mechanism is widely used in RCM mechanisms to guarantee remote center rotation, a parallelogram and a linear motion mechanism that is used as a translation generator are used to construct a planar 1R1T RCM mechanism in this paper. Fig. 1(a) shows the basic structure of the parallelogram mechanism. It contains two sets of parallel links connected by revolute joints. It is obvious that the virtual link denoted by EI always passes through point O. Hence, the virtual link EI can output an RCM rotation. Fig. 1(b) shows a 1-DOF four bar linkage. Link FK is parallel to link EO, and link IK is parallel to link EG. Because the IO FK special length condition EO EG = IK = F G is satisfied, points G, K, and O are collinear [16,28]. Therefore, we can add a block and link (denoted by the virtual lines in Fig. 1(b)) to the four bar linkage without affecting its DOFs. This mechanism is called a linear motion mechanism because there is a relative translation between the added link and block. By adding the linear motion mechanism to the parallelogram shown in Fig. 1(a), a 2-DOF mechanism can be constructed, as shown in Fig. 1(c). It should be noted that some special conditions for link lengths should be satisfied. Link GM in this
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Fig. 2. Schematic diagram of the initial mechanism.
Fig. 3. Modified mechanism and final mechanism.
mechanism can perform a rotation around virtual point O and a translation passing through point O. The mechanism is a planar 1R1T RCM mechanism. However, to control the translation of link GM, an actuator needs to be added in the linear motion mechanism, which increases the moving mass. To enhance the stiffness and optimize the actuation scheme, the mechanism in Fig. 1(c) is duplicated symmetrically. A novel planar mechanism is obtained, as shown in Fig. 2. It has the same DOFs as the mechanism in Fig. 1(c) because the newly added parts introduce only virtual constraints. Links E1 G and E2 G remain parallel to lines OE2 and OE1 , respectively, during motion. The motion of output link GM depends on the shape change of the parallelogram OE1 GE2 , and consists of a rotation around point O and a translation passing through point O. When joints C1 and C2 are locked, points E1 and E2 are fixed, and all the links cannot move because E1 E2 O becomes a fixed triangle. Therefore, joints C1 and C2 can be used as inputs. Actuators for this mechanism are mounted near the fixed base, which results in good dynamic performance. This mechanism will be referred to as the initial mechanism hereafter. Note that in the initial mechanism, many links introduce only virtual constraints. The topological structure of the mechanism is complex. To simplify it, some of the links that do not affect the mechanism’s DOFs are removed. A modified mechanism is obtained, as shown in Fig. 3(a). This mechanism is the same with that presented by Chen et al. [16]. It has the same motion characteristic as the initial mechanism, and is also fixed when the two revolute joints C1 and C2 are locked. We detected that, for the modified mechanism shown in Fig. 3(a), the translational range of output link GM depends on the shape-changing ability of quadrangle OE1 GE2 . Provided that the translational range is about 250 mm, which is usually adopted in mechanism design for MIS [29], the longest link in quadrangle OE1 GE2 can be approximately 600 mm if some
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Fig. 4. A 3R1T RCM mechanism.
Fig. 5. Diagram of the 1R1T RCM mechanism.
angle limitations are set between links to avoid mechanical interference. Clearly, such a length is too large and will result in problems for manufacturing and accuracy. Therefore, the modified mechanism is further improved, and a novel mechanism is obtained, as shown in Fig. 3(b), which is referred to as the final mechanism. Of note is that the final mechanism also belongs to the mechanisms family synthesized in Ref. [16]. However, it has different topological structure, which will result in different kinematic modeling and performance. Furthermore, the translational range of link GM in the final mechanism is the summation of the diagonal length changes in quadrangles OE1 KE2 and KF1 GF2 . The link length can be decreased while still satisfying the output link’s translational range. It is worth noting that all three mechanisms in Figs. 2 and 3 can perform 1R1T RCM motion. The motion of output link GM in these mechanisms can be fully controlled by actuators mounted on joints C1 and C2 because a fixed triangle E1 E2 O forms once the input joints are locked. Of note is that line OE1 is not necessarily parallel to link GE2 in Fig. 3(a), and line OE1 is not necessarily parallel to link F1 E2 in Fig. 3(b). Taking into consideration the structural complexity and linkage dimensions, the final mechanism shown in Fig. 3(c) is selected, and investigated in the following sections. By connecting the final mechanism to a fixed base using a revolute joint, and adding a rotational DOF to the surgical tool around its axis, a 3R1T RCM mechanism can be constructed as shown in Fig. 4. Constructing spatial 3R1T RCM mechanisms from 1R1T RCM mechanism is a common practice [15]. Such an implement has two sides. On one hand, because of the added DOFs are serially connected, the rotational ranges are relatively large, and kinematics of the resulted mechanism is simple. On the other hand, large power input is required for the base-connected rotational DOF because the whole mechanism rotates around that axis. 3. Position analysis In this section, both direct position analysis and inverse position analysis of the proposed 1R1T RCM mechanism are conducted. Attach a reference coordinate system Oxy to the fixed base with its origin coincident with point O, as shown in Fig. 5. Axis x is coincident with OA2 , and axis y is vertical. Let Mx and My denote the coordinates of point M in the reference coordinate system, and θi (i = 1, 2) denote the angle between link Ci Di and axis x. The symbols used to denote the lengths of links are listed in Table 1.
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Table 1 Link length symbols. Links
Ai Bi
Ei K
Fi K
Fi G
GM
Length
l1
l2
l3
l4
l
3.1. Direct position analysis The direct position problem is to identify coordinates Mx and My given θi (i = 1, 2). The distance from point Ei (i = 1, 2) to link GM is denoted by p:
p = l1 sin α .
(1)
θ1 −θ2
where α = 2 is the angle between GO and E1 O. The coordinates of point K can be obtained as
Kx = l1 cos θ1 + l2 cos β , Ky = l1 sin θ1 + l2 sin β
(2)
where β is the angle between link E1 K and axis x, which can be expressed as
β = arccos
p + l2
θ1 + θ2 2
−
π 2
.
(3)
The distance from point K to point G is denoted by lKG :
lKG = l3 cos α1 + l4 cos α2 ,
(4)
where α 1 is the angle between E1 K and KM, and α 2 is the angle between GM and GF1 , as shown in Fig. 5. We have
α1 = arcsin lp2 α1 . α2 = arcsin l3 sin l4
(5)
Because points K, M, and O are collinear, the coordinates of point M can be obtained as
Mx = Kx − (l − lKG ) · Kx / Kx2 + Ky2 . My = Ky − (l − lKG ) · Ky / Kx2 + Ky2
(6)
Eq. (6) means that analytical expressions for forward position solutions are obtained, which makes the real-time control of the mechanism easier. 3.2. Inverse position analysis The inverse position problem is to identify the input angles θi (i = 1, 2) given the coordinates of point M. The coordinates of point G in the coordinate system can be expressed as
Gx = Mx − l Mx / Mx2 + My2 Gy = My − l My / Mx2 + My2
.
(7)
Considering the distance between points G and O, we have the following equation:
l1 cos α + (l2 + l3 ) cos α1 + l4 cos α2 =
G2x + G2y .
(8)
Substituting Eqs. (1) and (5) into Eq. (8), α can be derived. Then, the inverse position solutions can be obtained as
y θ1 = arc tan M +α Mx . My θ2 = arc tan Mx − α
(9)
4. Velocity, singularity, and workspace analysis 4.1. Velocity analysis Velocity analysis establishes a mapping between the input joint velocity and end-effector velocity, which can be derived from the position model. To simplify the analysis, the angle between link GM and axis x denoted by ϕ , and the length lKG are regarded as the output parameters. From the geometrical relationship of the mechanism, we have
ϕ = θ2 +
θ1 − θ2 2
=
θ1 + θ2 2
,
(10)
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Fig. 6. Singular configurations.
2 l32 + lKG − l42 2l3 lKG
2
+
l1 sin l2
θ1 − θ2
2 = 1.
2
(11)
Taking derivation on both sides of Eqs. (10) and (11) with respect to time, velocity equations can be obtained, and rearranged in matrix form as
ϕ˙
JA ˙ lKG where
θ˙ = JB 1 , θ˙ 2
J JA = A11 JA21
(12)
JA12 J , JB = B11 JA22 JB21
JB12 JB22
JA11 = 1, JA12 = 0, JA21 = 0, JA22 = JB11 =
1 ,J 2 B12
=
1 ,J 2 B21
=−
l12
cos
2 2 −l42 ) (l42 +lKG −l32 ) (l32 +lKG ,
θ1 −θ2
3 2l32 lKG
sin
2
l22
θ1 −θ2 2
, JB22 =
l12 cos
θ1 −θ2
sin
2
θ1 −θ2
.
2
l22
4.2. Singularity analysis Singularity analysis is very important for a mechanism. When a mechanism is in singular configurations, it gains or loses some DOFs, which makes the mechanism unable to perform the task. Singularities of a mechanism can be identified based on its Jacobian matrix. For the mechanism analyzed in this paper, two cases of singularities are obtained based on Eq. (12). (1) Case I: det(JB ) = 0. We can obtain the condition that satisfies this singularity as θ 1 − θ 2 = 0. This means that point E1 is coincident with point E2 , and link F2 E1 is coincident with link F1 E2 , as shown in Fig. 6(a). 2 − l 2 = 0 and l 2 + l 2 − l 2 = (2) Case II: det(JA ) = 0. Two conditions that satisfy this singularity can be obtained as l32 + lKG 4 4 KG 3 0. This means that link GK is perpendicular to link E1 K and link F1 G, respectively. The two corresponding configurations are shown in Fig. 6(b) and (c), respectively. 4.3. Workspace Workspace analysis verifies whether the moving range of the end effector satisfies the requirement of the MIS application. The link dimensions are given as l1 =197 mm, l2 =124 mm, l3 =129 mm, l4 =255 mm, and l = 690 mm. Considering the moving ranges and physical interference of the joints in this mechanism, several constraints of the workspace are given as follows: 1. θ 1 ∈ [30, 170◦ ], θ 2 ∈ [10◦ , 150◦ ], and θ1 − θ2 > 20◦ . This constraint is introduced to keep the mechanism far away from singularity since it is singular when θ 1 =θ =2 as shown in Fig. 6(a). 2. Any angle between two links should be larger than 10° to avoid link interference. 3. No singularities in the workspace.
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Fig. 7. Workspace.
Fig. 8. Equivalent model of the 1R1T RCM mechanism.
To better show the appropriateness of the mechanism for MIS application, the additional serial rotational DOF denoted by θ 3 in Fig. 4 is considered. The moving range of θ 3 can be very large because it corresponds to a serial DOF, so we only consider the region with θ3 ∈ [−40◦ , 40◦ ] for simplicity. Search the potential region for points that satisfy the aforementioned constraints. The workspace of the mechanism is numerically determined as shown in Fig. 7. Clearly, the workspace is much larger than the required workspace in MIS, which is generally a cone with a coning angle equal to 60° [30]. 5. Motion/force transmission analysis The workspace of the mechanism is larger than that required for MIS. It is desired that the mechanism has good performance in the desired workspace. Because in 90% of MIS operations the surgical tool works in a cone region with a coning angle of 60° and a length of 250 mm [29,30], a circular sector whose central angle is 60° of the workspace will be the focus of this section, within which performance is analyzed and optimized. 5.1. Local transmission index Motion/force transmission analysis is an effective method when evaluating the performance of a mechanism, and calculates the input transmission index and output transmission index, respectively. The smaller one in the two indices is regarded as the local transmission index (LTI). In this study, the structure of the mechanism is complex, which makes motion/force transmission analysis difficult. To reduce the complexity, three simplifications are used. First, the slider at point K that has the same motion as link GM is regarded as the output link because it is closer to the actuators compared with link GM. Second, links OE1 and OE2 are added, which are treated as input links. Finally, a virtual chain composed of a revolute joint and prismatic joint are added between the base and slider K. The role of this chain is to constrain the motion of the output link so that the parts above slider K can be removed. After these simplifications, an equivalent model in Fig. 8 is
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obtained, in which links OE1 and OE2 are two input links, and slider K is the output link. It has a parallel structure, whose motion/force transmission analysis is much easier. In motion/force transmission analysis, the input transmission index represents the efficiency of power transmitted from the actuated joints to the limbs, whereas the output transmission index represents the efficiency of power transmitted from the limbs to the moving platform. For the equivalent model shown in Fig. 8, power is transmitted from the actuators to the moving platform through two limbs. Limb 1 is composed of links OE1 and E1 K, and three associated revolute joints. Limb 2 is composed of links OE2 and E2 K, and three associated revolute joints. Motion/force transmission analysis is conducted for these two limbs using the method in [25]. For limb 1, the twists of the revolute joints are expressed as
⎧ ⎨$11 = (0 0 1; 0 0 0)
$12 = 0 0 1; − E1y E1x 0 ,
(13)
⎩
$13 = (0 0 1; − Ky Kx 0 )
where $11 is the twist associated with the input revolute joint at point O, $12 is the twist related to the revolute joint at point E1 , and $13 is the twist related to the revolute joint at point K. E1 x , E1 y , Kx , and Ky are the coordinates of points E1 and K. The constraint wrench system that is reciprocal to the twist system in Eq. (13) is obtained as
$r11 = (0 0 1; 0 0 0) $r12 = (0 0 0; 1 0 0) . $r13 = (0 0 0; 0 1 0)
(14)
In the constraint wrench system, $r11 denotes a constraint force along axis z. $r12 and $r13 denote constraint couples around axes x and y, respectively. For limb 1, its transmission wrench $T1 should not be included in the wrench system in Eq. (14), and is reciprocal to the twist screw system of the passive joints (the last two twists in Eq. (13)), which is expressed as
$T 1 ◦ $1i = 0 i = 2 , 3 .
(15)
Therefore, transmission wrench $T1 is obtained as
$T1 = (a1 b1 0; 0 0 f1 ), where a1 =
Kx −E1x D1 , b1
=
Ky −E1y D1 , f 1
=
Kx E1y −E1x Ky , D1 D1
=
(16)
(E1y − Ky )2 + (E1x − Kx )2 .
According to screw theory, the wrench in Eq. (16) represents a force along link E1 K. Similarly, transmission wrench $T2 in limb 2, which represents a force wrench along linkage E2 K, is obtained as
$T2 = (a2 b2 0; 0 0 f2 ), where a2 =
Kx −E2x D2 , b2
=
Ky −E2y D2 , f 2
=
Kx E2y −E2x Ky , D2 D2
=
(17)
(E2y − Ky )2 + (E2x − Kx )2 .
By only keeping the actuator in limb 1 active and locking the actuator in limb 2, the transmission wrench $T2 in limb 2 becomes a constraint wrench. Under this condition, the slider has only one remaining DOF, that is, a rotational DOF around point O1 (the intersecting point of line KE2 and line OO1, which is perpendicular to link KO). The rotational center O1 can be obtained using instantaneous centers. The remaining DOF is expressed using a twist screw as
$O1 = (0 0 1; d3 e3 0),
(18)
where d3 = Kx T1 , e3 = Ky T1 , T1 = (E2x Ky − Kx E2y )/(Kx2 + Ky2 − E2x Kx − E2y Ky ). Similarly, output twist screw $O2 for limb 2, which represents a rotation around point O2 , is expressed as
$O2 = (0 0 1; d4 e4 0),
(19)
(Kx E1y − E1x Ky )/(Kx2
where d4 = Kx T2 , e4 = Ky T2 , T2 = + − E1x Kx − E1y Ky ). It should be noted that, in a real mechanism, the directly driven links are C1 D1 and C2 D2 rather than links OE1 and OE2 . Therefore, the transmission efficiency from linkage Ai Bi to linkage OEi (i = 1, 2) should not be neglected. This efficiency can be expressed as μi = sin θi according to the concept of the transmission angle. Therefore, the input transmission performance for limb i (i = 1, 2) is expressed as
λi =
Ky2
|$1i ◦ $Ti | μ. |$1i ◦ $Ti |max i
(20)
The output transmission performance is expressed as
ηi =
|$Ti ◦ $Oi | , |$Ti ◦ $Oi |max
(21)
where $1i denotes the input twist of limb i, $Ti denotes the transmission wrench of limb i, and $Oi denotes the output twist of limb i (i = 1, 2).
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Fig. 9. LTI distribution over the workspace. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
The total transmission index γ is then defined as
γ = min{λi , ηi }, (i = 1, 2 ).
(22)
Because γ only reflects the performance of the mechanism in a certain configuration, it is a local index, which is called the LTI. Clearly, γ is dimensionless, and varies from 0 to 1. When γ tends to 1, the mechanism has good motion/force transmissibility, whereas when γ tends to 0, the mechanism has poor motion/force transmissibility and is close to a singular configuration. For the same set of parameters given in Section 4, the LTI performance of the mechanism over the target workspace is shown in Fig. 9. The result shows that the performance is symmetric about axis y, which is in accordance with the mechanism’s structure. 5.2. Global transmission index Because the LTI can only denote the motion/force transmission performance in a single configuration, the global performance of the mechanism remains unclear. To address this issue, a global index that can evaluate the performance of the mechanism over the prescribed workspace is considered. When γ ≥ 0.7 [31], the motion/force transmissibility of the mechanism is regarded as sufficiently good and all configurations that satisfy this condition form a good transmission workspace. Then, a global index σ can be defined as
σ=
GTW , PW
(23)
where GTW represents the area of the good transmission workspace, and PW represents the area of the entire prescribed workspace. Index σ ranges from 0 to 1. The closer σ is to 1, the better the global performance of the mechanism. 6. Optimization of the design parameters The GTW index σ for Fig. 9 is calculated as 0.256, which means that the GTW only occupies 25.6% of the prescribed workspace. Such global performance is not satisfactory, and should be improved through parameter optimization. In this paper, l1 , l2 , and l3 are chosen as design parameters, and the other parameters are dependent on l1 , l2 , and l3 to ensure that the workspace of the mechanism still satisfies the requirements of MIS. Using the parameter-finiteness normalization method [32,33], l1 , l2 , and l3 are normalized, and non-dimensional parameters r1 , r2 , and r3 are obtained as
r1 =
l1 l2 l3 , r2 = , r3 = , D D D
(24) l +l +l
where D is a normalized factor that equals 1 32 3 . Considering that the size of the mechanism is too large when the length of linkage l1 is smaller than that of linkages l2 and l3 , the following conditions should be satisfied:
r2 , r3 ≤ r1 . 0 < r1 , r2 , r3 < 3
(25)
The parameter design space (PDS) can be obtained as shown in Fig. 10(a), in which the shaded region represents the possible solutions. For visualization reasons, the three-dimensional (3D) PDS is transformed into a plan view, as shown in
10
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Fig. 10. PDS.
Fig. 11. Distribution of the GTW index. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.) Table 2 GTW in the PDS. Group
s
t
r1
r2
r3
l1 /mm
l2 /mm
l3 /mm
σ
1 2 3 4 5 6 7 8 9
1.275 1.315 1.315 1.335 1.415 1.465 1.515 1.515 1.385
−0.04 −0.02 −0.08 −0.29 −0.15 −0.05 0.01 −0.29 −0.44
1.275 1.315 1.315 1.335 1.415 1.465 1.515 1.515 1.385
0.827 0.825 0.773 0.581 0.662 0.724 0.751 0.491 0.426
0.897 0.859 0.911 1.083 0.922 0.810 0.733 0.993 1.188
191.25 197.25 197.25 200.25 212.25 219.75 227.25 227.25 207.75
124.17 123.77 115.98 87.20 99.38 108.62 110.07 73.70 63.96
134.57 128.97 136.76 162.54 138.36 121.62 112.67 149.04 178.28
0.231 0.268 0.353 0.658 0.586 0.538 0.550 0.866 0.897
Fig. 10(b). The relationship between the parameters in 3D space (r1 , r2 , r3 ) and those in two-dimensional (2D) space (s, t) is as follows:
⎧ r =s ⎪ ⎨1 r2 =
3 2
−
r3 =
3 2
+
⎪ ⎩
√
3 t 2 √ 3 t 2
−
s 2
−
s 2
or
s = r1
t=
√ 3(r3 −r2 ) 3
.
(26)
The procedures for the optimal design are summarized as follows: 1. Obtain the distribution of the GTW index σ of the mechanism in 2D design space as shown in Fig. 11. The colored region represents design region, from which we can choose design parameters with different global performance. 2. Table 2 shows nine groups of parameters chosen from the design region. For each group, obtain the normalized parameters (r1 , r2 , r3 ) using Eq. (26).
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Fig. 12. LTI distribution of the mechanism with parameters in group 9. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 13. Prototype of the final mechanism.
3. Determine the normalized factor D. Considering the practical conditions in surgical applications, D is determined as 150 mm in this paper. Then obtain the link parameters l1 , l2 , and l3 using Eq. (24) and determine the other parameters accordingly considering the workspace requirement. 4. Check whether the linkage parameters obtained in step 3 are suitable for actual use. The process is complete if the necessary assembly conditions are satisfied. Otherwise, return to step 2 and choose another group of data from the design region, and repeat steps 3 and 4. Compared with the performance shown in Fig. 9, whose link parameters are close to those in group 2, a mechanism with parameters in group 9 has much better performance, which is shown in Fig. 12. Its GTW index σ is 0.897, which means that the GTW occupies 89.7% of the prescribed workspace. We select parameters in group 9 as the optimized results. 7. Prototype and workspace validations To validate the design, a prototype of the final mechanism was built, as shown in Fig. 13. Note that the rotational DOF around a base-fixed axis was included. Therefore, the prototype had three DOFs in total. The prototype had three motors, which were used to control its three DOFs. A belt transmission was used to make the motor’s installation easy and reduce the size of the prototype. Workspace validation was conducted on the prototype. Fig. 14 shows 12 configurations. The configurations in the first and third rows demonstrate the rotational capability of the mechanism where the inserted distance of the needle is large, whereas those in the second and fourth rows demonstrate the rotational capability of the mechanism where the inserted distance of the needle is small. The results demonstrated that the mechanism has RCM and the rotational region can reach −40°–40°, which satisfies the workspace requirement of MIS. Future experiments will focus on the calibration and accuracy of the prototype.
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Fig. 14. Workspace validation.
8. Conclusion In this paper, a new type of 1R1T RCM mechanism was proposed. The actuators of the mechanism were arranged near to the base, which resulted in a quick dynamic response. Analytical expressions for forward position solutions were obtained, which benefited the real-time control of the mechanism. The singularity-free workspace of the mechanism was much larger than that demanded in MIS. A performance evaluation of the mechanism over the prescribed workspace was conducted using the method of motion/force transmission. Optimization design was conducted and parameters with good performance were identified. A prototype was built and preliminary experiments showed that the mechanism satisfied the workspace requirement of MIS. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgments The authors would like to acknowledge the financial support of the National Natural Science Foundation of China (NSFC) under Grant Nos. 51525504, and 51705465. References [1] G. Strassmann, P. Olbert, A. Hegele, et al., Advantage of robotic needle placement on a prostate model in hdr brachytherapy, Strahlenther. Onkol. 187 (6) (2011) 367–372.
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