Shao Liu
email: shao.liu@monash.edu

Binbin Chen
email: binbin.chen@monash.edu

Stéphane Caro
email: stephane.caro@irccyn.ec-nantes.fr

Sébastien Briot
email: sebastien.briot@irccyn.ec-nantes.fr

Laurence Harewood
email: laurenceharewood@urologyvictoria.com

Chao Chen
email: chao.chen@monash.edu

A cable linkage with remote centre of motion

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Introduction

In a minimally-invasive surgery (MIS), the surgical instrument is constrained to have four degrees-of-freedom (DOF) through the incision port [START_REF] Kuo | Kinematic design considerations for minimally invasive surgical robots: an overview[END_REF]: pitch, yaw, translation along the longitudinal axis and roll. The first three DOF in combine function as a spherical coordinate system to define the position of end-effector inside the patient's body. Remote centre of motion (RCM) mechanisms provide the two rotational DOF while permitting the surgical instrument to pivot around the incision port, hence eliminate potential damage to the incision port and promote the safety of MIS procedures [START_REF] Taylor | Medical robotics in computer-integrated surgery[END_REF].The RCM function refers to the capability of a mechanism to rotate its link(s) around a remote point without having a physical revolute joint at the point [START_REF] Kuo | Robotics for minimally invasive surgery: a historical review from the perspective of kinematics[END_REF][START_REF] Zong | Classification and type synthesis of 1-DOF remote center of motion mechanisms[END_REF]. A remote centre (RC) can be constrained virtually or mechanically [START_REF] Chen | Novel linkage with remote center of motion[END_REF]. Mechanical RCM mechanisms are more reliable and considered suitable for clinical applications [START_REF] Kuo | Robotics for minimally invasive surgery: a historical review from the perspective of kinematics[END_REF]. Mechanical RCM mechanisms that generate single RC and are applied on robotic MIS systems include isocentres [START_REF] Ghodoussi | Robotic surgery -the transatlantic case[END_REF], circular tracking arcs [START_REF] Guerrouad | SMOS: stereotaxical microtelemanipulator for ocular surgery[END_REF][START_REF] Hempel | An MRI-compatible surgical robot for precise radiological interventions[END_REF], parallelograms [START_REF] Madhani | The black falcon: a teleoperated surgical instrument for minimally invasive surgery[END_REF][START_REF] Taylor | Remote Center-of-Motion Robot for Surgery[END_REF][START_REF] Blumenkranz | Manipulator for Positioning Linkage for Robotics Surgery[END_REF][START_REF] Feng | Development of a medical robot system for minimally invasive surgery[END_REF][START_REF] Kim | Design and evaluation of a teleoperated surgical manipulator with an additional degree of freedom for laparoscopic surgery[END_REF][START_REF] Zhu | Motion/Force/Image control of a diagnostic ultrasound robot[END_REF], synchronous transmissions [START_REF] Stoianovici | Remote Center of Motion Robotic System and Method[END_REF] and spherical linkages [START_REF] Lum | The REVAN: design and validation of a telesurgery system[END_REF]. In addition, there are RCM mechanisms that generate multiple RCs [START_REF] Bai | Kinematics and synthesis of a type of mechanisms with multiple remote centers of motion[END_REF].

A commonly used approach to synthetise a two-DOF RCM mechanism is to combine a planar RCM mechanism with a revolute joint [START_REF] Kuo | Robotics for minimally invasive surgery: a historical review from the perspective of kinematics[END_REF]. The axis of the revolute joint coincides with the one-DOF RC to add the second DOF. Such approach results in fully decoupled rotational DOF, whose benefits include reduced complexity in control, promoted level of confidence in safety as well as rapid and intuitive manual positioning of the entire mechanism or individual DOF [START_REF] Kuo | Kinematic design considerations for minimally invasive surgical robots: an overview[END_REF].

The translational DOF required in the MIS applications is often achieved by mounting an independent translational mechanism on the two-DOF RCM mechanism. A typical example of such three-DOF mechanism is the clinically-approved da Vinci series robotic surgical system [START_REF] Blumenkranz | Manipulator for Positioning Linkage for Robotics Surgery[END_REF][START_REF] Solomon | Multi-ply Strap Driver Trains for Robotic Arms[END_REF][START_REF] Haber | Novel robotic da Vinci instruments for laparoendoscopic single-site surgery[END_REF]. In other approaches, various types of RCM mechanisms that also provide translational DOF are explored [START_REF] Long | Type synthesis of 1R1T remote center of motion mechanisms based on pantograph mechanisms[END_REF][START_REF] Li | Kinematic design of a novel spatial remote center-of-motion mechanism for minimally invasive surgical robot[END_REF][START_REF] Hadavand | A novel remote center of motion mechanism for the force-reflective master robot of haptic tele-surgery systems[END_REF]. However, these mechanisms have coupled DOF [START_REF] Long | Type synthesis of 1R1T remote center of motion mechanisms based on pantograph mechanisms[END_REF], or are relatively bulky in terms of the transverse dimension [START_REF] Li | Kinematic design of a novel spatial remote center-of-motion mechanism for minimally invasive surgical robot[END_REF], or have large sweeping volume upon rotation of the planar RCM mechanism around the revolute joint, due to the large enclosed area by the outer boundary of the planar RCM mechanism [START_REF] Hadavand | A novel remote center of motion mechanism for the force-reflective master robot of haptic tele-surgery systems[END_REF].

The parallelogram-based structure is widely used as the planar RCM mechanism in the robotic MIS systems [START_REF] Kuo | Robotics for minimally invasive surgery: a historical review from the perspective of kinematics[END_REF]. However, there are footprint issues associated with the parallelogram-base linkages (PB-linkage), which the consequences being poor access for bedside assistance [START_REF] Haber | Novel robotic da Vinci instruments for laparoendoscopic single-site surgery[END_REF] and the compromise in optimal surgical functioning [START_REF] Taylor | Is smaller workspace a limitation for robot performance in laparoscopy?[END_REF]. The term "footprint" is mostly referred to as the sweeping volume of the RCM mechanism, which is generated by the rotation of the planar RCM mechanism around the revolute joint. The sweeping volume is thus related to the area enclosed by the outer boundary of the planar RCM mechanism.

When the output link of the parallelogram is short, the output joint of the PB-linkage is positioned closely to the incision ports. Given that the space around an incision port is often crowded with robotic or manual surgical tools, the collision-free workspace is reduced and the chance of interference is increased. The transmission for the translational mechanism mounted on the output link of the PB-linkage goes through the output joint, causing further expansion in size.

In opposite case where a longer output link is used to displace the output joint away from the mechanism, the size of the parallelogram and thus the enclosed area is increased. The consequence being the increase in the sweeping volume, which again leads to increase in chance of interference. Apart from the sweeping volume, longer links occupy more space even when the linkage is stationary. It also increases the weight and inertia of the system. Quantitative analysis on the footprint of the PB-linkage though three approaches is presented in Section 5.

This paper proposes a cable linkage with RCM, in the attempt to address the footprint issue associated with the PB-linkage. The entire RCM mechanism is kept relatively faraway from the RC, when the distance between the input joint and the RC is given. A cantilever is rigidly mounted on to the output link. It is the only part of the entire RCM mechanism that is operated near the RC. Therefore, the cable system can leave more collision-free workspace for the neighbouring robotic surgical arms or human surgeons to operate. In addition, the enclosed area of the planar RCM is relatively small, resulting in a smaller sweeping volume. Further, the links are relatively short, which reduces the space taken by the links when stationary, and potentially reduce the weight and inertia of the mechanism. A comparison between footprint of the proposed linkage and the PB-linkage is presented in Section 5.

Cable-pulley mechanism provides advantages such as structure simplicity, compactness, light weight, low friction and low backlash [START_REF] Hong | A method for representing the configuration and analyzing the motion of complex cable-pulley systems[END_REF]. Therefore it is widely applied on serial and parallel robotic manipulators, as summarised in [START_REF] Tsai | Design of tendon-driven manipulators[END_REF] and [START_REF] Tang | An overview of the development for cable-driven parallel manipulator[END_REF], respectively. In MIS applications, the evolution from linkage-based da Vinci system [START_REF] Blumenkranz | Manipulator for Positioning Linkage for Robotics Surgery[END_REF] to cable-based da Vinci systems [START_REF] Solomon | Multi-ply Strap Driver Trains for Robotic Arms[END_REF][START_REF] Haber | Novel robotic da Vinci instruments for laparoendoscopic single-site surgery[END_REF] shows significant deduction in the size of linkage. As such, the proposed planar RCM mechanism is developed based on cable-pulley mechanism.

Cable tension analysis is essential for proof of functioning of the cable linkage. Approaches for describing cable tension of cable-constrained open-chain linkage and multi-link parallel manipulator are available in [START_REF] Tsai | Kinematic analysis of tendon-driven robotic mechanisms using graph theory[END_REF] and [START_REF] Lau | Generalized modeling of multilink cable-driven manipulators with arbitrary routing using the cable-routing matrix[END_REF], respectively. However, a more generalised approach based on mechanical constraint [START_REF] Chen | Power analysis of epicyclic transmissions based on constraints[END_REF] is used in the analysis. The reason being that the proposed linkage is based on four-bar linkage and is affected by the singularity of an unconstrained four-bar linkage. The constraint approach provides indication on the constraint status of the mechanism, thus enable justification on the removal of singularity. The cable tension is solved as generalised force along mechanical constraints. Numerical solution of cable tension is obtained using QR decomposition and verified with static finite element simulation in ANSYS.

The rest of the paper is arranged as follows. Section 2 describes the design of the cable linkage and the proof of the RCM function. Section 3 applies the constraint-based analysis. Constraint equations are derived. Cable tension is solved and verified with finite element analysis. The functioning of the cable loops is hence proven. Section 4 calculates the minimum required cable stiffness to achieve a given overall stiffness of the linkage. Section 5 compares the footprints of the cable linkage and PB-linkage in a simplified surgical scenario. Section 6 introduces the prototype of the cable linkage.

The cable linkage with RCM

In this section, the design of cable linkage with RCM is presented, along with the proof of RCM under the condition that the cable is in tension.

Design of cable linkage

The design of the cable linkage with RCM is illustrated in Figs. 1 and2, where the schematic diagram of links (without cable loops) and the full schematic diagram are presented, respectively. The cable linkage consists of eight links and seven pulleys that are arranged in three cable loops. As illustrated in Fig. 1, the links are AF, AC, CE, EH, BG, DG, CI and IG, respectively. Joint I is a passive prismatic joint while all other joints (indicated by the small circles) are revolute joints. Note that joints B and D do not divide links AC and CE, respectively. For convenience, links CI, IG and passive prismatic joint I are grouped and termed "diagonal link CG" in the later descriptions. Different configurations of the cable linkage are presented in Figs. 3 and4. ROM in the figures stands for range of motion. The limits in ROM above and below the ground (AO), and the centre configuration where all links are overlapped with the ground are illustrated in Fig. 3. Two mid-point configurations in between the centre and two limits in ROM are illustrated in Fig. 4. Note that pulleys and cable loops are not drawn for clarity of figures. In Fig. 3, all labelled joints are for the centre configuration only, where all the links overlap with the ground while D, E and H are coincident with B, A and F, respectively.

Point O is the remote centre. Links AF, AC, CE and EH are the ground, input, connector and output links, respectively. Links AC and CE, with virtual links AO and EO, form virtual four-bar linkage ACEO. Links BG and DG constraint the motion of the linkage. Link CG is the diagonal link of the virtual four-bar linkage. It contains a passive prismatic joint, Joint I, to accommodate for the change in length of CG with respect to movement of linkage. The distance between Joints C and I is constant while the distance between Joints I and G changes. The shape and size of output Link EH can be optimised freely to suit specific MIS applications.

The three cable loops are called Loops PL, FB and PU, for the lower, middle and upper loops in Fig. 2, respectively. Loop PL connects Pulleys P1 and P2, which are rigidly attached to Links AF and BG, respectively. Loop FB connects Pulleys P3 to P5. Pulley P3 is rigidly attached to Link AC. Pulley P4 rotates freely at I. Pulley P5 is rigidly attached to Link CE. The cable connecting P4 and P5 has crossed configuration. Loop PU connects Pulleys P6 and P7, which are rigidly attached to Links DG and EH, respectively. Note that although there exist two forms of cable loop, which are the end-less tendon drive and the open-ended tendon drive, as classified in [START_REF] Tsai | Design of tendon-driven manipulators[END_REF], they are not distinguished in the 2D design of the cable linkage. The reason is that the synchronised rotation of pulleys, which is the essential mechanical constraint of the cable linkage, can be achieved equivalently using two forms of cable loop under the assumptions of zero cable slippage elongation.

Loops PL and PU ensure Links BG and EH are parallel to AF and DG, respectively. Slip of cable on pulleys in Loops PL and PU ruins the parallel constraint, thus cause failure in maintaining the RCM function. Loop FB permits the cable linkage to pass the configuration where all links overlap with the ground, which is the singular configuration of an unconstrained four-bar linkage. Slip of cable in Loop FB does not affect the RCM function. In non-overlapped configurations, Loop FB is redundant. In the overlap configuration, slip of cable in Loop FB can cause the cable linkage to turn into an undesired configuration, where the output link stays coincident with the ground regardless of the input angle. Thus the position of the remote centre is not affected, despite the lost of mobility. More details regarding the singularity are described in the later paragraphs of this section.

The outer shape of the cable linkage, which is virtual four-bar linkage ACEO, is symmetrical to simplify the geometry. Links AC and CE have equal link length. Virtual links AO and CO have another equal link length, which is longer than that of AC and CE. Such structure ensures that the link length of the RCM mechanism is always smaller than the distance between the input joint A and the remote centre O. Hence the RCM mechanism is considered to be relatively faraway from the remote centre.

The geometry of the linkage is fully defined by two parameters, v and r, expressed mathematically as

v = L AC L AO = L CE L EO (1)
and

r = L AC L AB = L CE L DE (2)
To ensure Links AC and CE are shorter than AO and CO, v is smaller than one. With a given length AO, the range of motion (ROM) of the RCM mechanism is fully defined by v. As shown in Fig. 3, the centre (diagonal) line CO is perpendicular to ACE at two ends of ROM. Therefore, the ROM is

ROM = 4 arcsin v ( 3
)
Parameter r is greater than one. It defines the geometry of four-bar linkage BCDG, where the length of Links BG and DG is

L BG = L DG = 1 - 1 r L AO (4)
Eq. ( 4) indicates that the larger the r, the longer Links BG and DG are. Four-bar linkage BCDG occupies additional space between Links AC, CE and the RC, thus a smaller r is desired to leave more clearance around the RC.

The configuration where all links overlap with ground is a bifurcation (singularity) configuration of an unconstrained four-bar linkage, where two possible configurations can be achieved upon crossing and

∠BCD = ∠BGD = 0 (5)
∠BCD = 2∠BCG (6)
where Eq. ( 6) can be equivalently expressed as

∠BGD = 2∠BGC (7)
The first configuration shown in Eq. ( 5) is an undesired configuration where Links BG, DG and EH stay overlapped with ground (AF/AO) regardless of the input angle. In this case the mobility of the linkage is lost. Therefore, Eq. ( 6) or [START_REF] Guerrouad | SMOS: stereotaxical microtelemanipulator for ocular surgery[END_REF] needs to be enforced to fully constrain the linkage. The cable linage achieves Eq. ( 6). In Loop FB, ∠BCG and ∠DCG are related to the relative rotations of Pulleys P3 and P5 with respect to diagonal link CG, respectively. Pulley P4 allows the two angles to be synchronised hence eliminates the chance of turning into Eq. [START_REF] Chen | Novel linkage with remote center of motion[END_REF]. In this design, P3 and P5 are placed at Joint C to maximise the clearance between Links AC, CE and the RC.

The cable on one side of a loop is in tension in one direction of motion of the linkage. The sides of loops that are in tension in the upward and downward motions of the linkage are shown in Figs. 5 and6, respectively.

Proof of RCM function

The proof of RCM function of the cable linkage is conducted based on the condition that the cable loops are functioning correctly hence the linkage is in the desired configuration. Such condition is proven in the Section 3. Given that an RCM mechanism requires a link to rotate around the remote centre, the proof is conducted through two steps:

1. Joint E rotates around O with a constant radius.

Point H rotates around O with a constant radius.

where E and H form output link EH. To simplify the expressions, the following link lengths are assigned.

L 1 = L AC = L CE = vL AO L 2 = L AB = L DE = L AC r = vL AO r L 3 = L AF = L BG = L DG = L EH = 1 - 1 r L AO (8)
Fig. 6. Cables in tension in downward motion.

Generalised coordinates

To conduct the proof, ten generalised coordinates are assigned to fully describe the linkage. The generalised coordinates q are listed in Table 1.

h 1 is the actuator input angle thus it is the independent generalised coordinate. All other generalised coordinates are dependent ones. The angles are measured with respect to the "previous" link, as shown in the "Reference", and in counter-clockwise (CCW) direction. There is no h 3 and h 5 since Pulleys P3 and P5 are rigidly attached to links and do not have their own independent rotation. The graphical representation of the generalised coordinates is shown in Fig. 7.

Step 1 -position of joint E

The distance between Joint E and O is

L EO = (xE -L AO) 2 + y 2 E ( 9
)
where x E and y E are the horizontal and vertical positions of Joint E with respect to Joint A, respectively

x E = L 1 cos h 1 + L 1 cos (h1 + h 8 + p + h 9) y E = L 1 sin h 1 + L 1 sin (h1 + h 8 + p + h 9) (10)
where the p term is added as h 9 is measured with respect to GC instead of CG. The objective of Step 1 is to prove that L EO is a constant. h 8 and h 9 need to be eliminated when Eq. ( 10)is substituted into Eq. ( 9). h 8 is calculated by subtracting the angle of Links AC with respect to ground from the angle of Link CG with respect to ground, i.e.

h 8 = ∠ CG -∠ AC (11)
Since Loop PL ensures that triangles BCG and ACO are similar triangles, the angle of Link CG with respect to ground is the same as the angle of diagonal line CO with respect to ground

∠ CG = ∠ CO = -arctan L 1 sin h 1 L AO -L 1 cos h 1 (12) Fig. 7. Generalised coordinates.
The angle of Link AC with respect to ground is simply h 1 ,thus

h 8 = ∠ CG -∠ AC = -arctan L 1 sin h 1 L AO -L 1 cos h 1 -h 1 (13)
Due to the symmetrical structure of four-bar linkage BCDG

h 9 = h 8 = -arctan L 1 sin h 1 L AO -L 1 cos h 1 -h 1 ( 14
)
Substituting Eqs. ( 10), ( 13) and ( 14) into Eq. ( 9) yields

L EO = L AO ( 15
)
Hence Joint E is proven to rotate around O.

Step 2 -position of point H

The position of Point H is determined through a similar approach as that of Joint E. The distance between Point H and O is

L HO = (xH -L AO) 2 + y 2 H ( 16
)
where x H and y H are

x H = L 1 cos h 1 + L 1 cos (h1 + h 8 + p + h 9) + L 3 cos (h1 + h 8 + p + h 9 + h 7) y H = L 1 sin h 1 + L 1 sin (h1 + h 8 + p + h 9) + L 3 sin (h1 + h 8 + p + h 9 + h 7) (17)
Given cable loop PU, the angles of Links EH and DG with respect to the ground are equal. Therefore, h 7 and h 6 are equal. From Eq. ( 7), the angle of Link DG with respect to ground is given by

∠ DG = 2∠ CG = -2 arctan L 1 sin h 1 L AO -L 1 cos h 1 ( 18
)
Hence

h 7 = h 6 = ∠ DG -∠ CE = -2 arctan L 1 sin h 1 L AO -L 1 cos h 1 -(h1 + h 8 + p + h 9) = h 1 -p (19)
Substituting Eqs. ( 12), ( 13), ( 17) and ( 19) into Eq. ( 16) yields

L HO = L AO r ( 20
)
Therefore L HO is a constant. Since both L EO and L HO are constant, Link EH is proven to rotate around the remote centre and the cable linkage is proven to be an RCM mechanism.

Cable tension analysis

The RCM function proven in Section 2 is conducted under the condition that the cable loops are functioning. To achieve such condition, the following criteria must be achieved:

1. The cable loops (especially Loop FB) must not obstruct the movement of the links. 2. The two sides of a cable loop must be in tension in each direction of motion of the linkage, respectively.

The functioning of cable linkage is proven through constraint approach analysis. Criterion 1 is proven based on the constraint equations derived from cable loops. Criteria 2 is proven based on the solutions of cable tension. In addition, the cable linkage is also proven to be fully-constrained (singularity-free) based on the number of constraint equations derived. The full proof is presented in this section.

The constraint approach

Such constraint-based analysis is based on a generalised constraint approach [START_REF] Chen | Power analysis of epicyclic transmissions based on constraints[END_REF], which is briefed below. The dynamics of a constrained physical system is described as [START_REF] Shabana | Computational Dynamics[END_REF]

Q i = Q c + Q e (21)
where Q i , Q c and Q e are the generalised inertia, constraint and external forces applied on the generalised coordinates q, respectively. The generalised constraint force Q c applied on q is related to the mechanical constraints of the systems through

Q c = -C T q k ( 22
)
where k is the Lagrange multiplier, which represents the generalised constraint force acting along the mechanical constraints. C q is the derivative of the constraint matrix c q with respect to the generalised coordinates q, which is

C q = ∂c q ∂q (23)
c q represents the mechanical constraints of the system. Assuming that c q and q have dimensions of m and n respectively, the dimensions of Q c , C q and k are n×1, m×n, and m×1, respectively. The cable linkage (one-DOF) is fully constrained when n equals (m + 1). In the cable linkage, the constraints from cable loops are contained in c q . Hence, c q can be used to prove Criterion 1.

C q and c q also satisfy ∂c ∂q

d qd + ∂c ∂q i qi = C qd qd + C qi qi = 0 (24)
where q i and q d are the independent and dependent generalised coordinates, respectively. C qi and C qd are the derivative matrices determined using q i and q d , respectively. In static analysis, the generalised inertia force Q i vanishes. Therefore, combining Eqs. ( 21) and [START_REF] Hadavand | A novel remote center of motion mechanism for the force-reflective master robot of haptic tele-surgery systems[END_REF] gives

Q e = C T q k (25)
or equivalently

Q ei = C T qi k Q ed = C T qd k ( 26
)
where Q ei and Q ed are the generalised external forces applied on q i and q d , respectively. Eq. ( 25) or ( 26) is used to determine the generalised constraint force k acting along the mechanical constraints, when external load Q e is given. Cable tension is contained in k and thus proves Criterion 2. One constraint equation for each pair of pulleys needs be derived to determine the cable tension, for each direction of motion of the linkage.

The direction of such constraint force is dependent on the way in which the constraint equation c q is written in. An example is given below. Consider an arbitrary constraint equation that is defined as

c q = p a -p b ( 27
)
where p a and p b are the physical quantities of Bodies A and B, respectively, which are used to construct the constraint equation.

In this case the k determined using such constraint equation is the generalised constraint force acting on Body B by Body A. In the opposite case, if the positions of p a and p b are swapped in Eq. ( 27), k will give the generalised constraint force acting on Body A by Body B.

Analysis on cable loops

In this analysis, it is assumed that the cable is inextensible and there is no slip between the cable and pulleys. All the bodies are assumed to be rigid bodies with zero mass.

Constraint equations from cable loops

Given constant length of cable section connecting two pulleys and constant distance between pulleys, the sum of lengths of cable sections wrapping on two pulleys is also a constant. This is expressed mathematically as

L PA + L PB = L const. ( 28
)
where L PA and L PB are the lengths of cable sections wrapping on arbitrary driving pulley PA and driven pulley PB. L const. is a constant that contains the overall length of cable section and the distance between two pulleys. Since the lengths of cable wrapping on pulleys are related to the rotation of pulleys and their adjacent links, generalised coordinates are embedded in Eq. ( 28). Rearranging Eq. ( 28) into the form of Eq. ( 27) yields

(Lconst. -L PB) -L PA = 0 (29)
thus according to the definition of Eq. ( 27), Eq. ( 29) gives the force applied by the cable section attached to Pulley PB on that attached to PA.

Since the cable can wrap on a pulley in either clockwise or counter-clockwise direction, the expressions for L PA and L PB need to take into account the direction. The cases corresponding to clockwise and counter-clockwise cable wrapping are illustrated in Figs. 8 and9, respectively.

In Figs. 8 and9, "Ref" represents an arbitrary reference where the angles are measured with respect to. h in , h q and h out are the angle where the cable starts to wrap on the pulley, generalised coordinate and angle where cable leaves the pulley, respectively.

In the clockwise case, h in is the largest angle, followed by h q and h out . The length of cable from h in to h q is

L CW = R (h in -h q) ( 30
)
and the length of cable from h q to h out is

L CW = R (hq -h out ) (31)
In the counter-clockwise case, h out is the largest angle, followed by h q and h in . The length of cable from h in to h q is

L CCW = R (hq -h in ) (32)
Fig. 9. Counter-clockwise cable wrapping. and the length of cable from h q to h out is

L CCW = R (hout -h q) ( 33
)
The h in , h q and h out for all the pulleys are summarised in Tables 2 and3. Tables 2 and3 correspond to the cable sections for upward and downward motions of the cable linkage, respectively, as illustrated in Figs. 5 and6, respectively. 0 is the additional constant angle introduced by crossed cable between P4 and P5. CW and CCW indicate clockwise and counter-clockwise directions, respectively. Substituting the h terms from Tables 2 and3 into Eq. ( 28) yields four constraint equations for each of the upward and downward motions of the cable linkage, respectively, which are between P1 and P2, P3 and P4, P4 and P5, and P6 and P7, respectively.

Constraint equation from joint positions

Apart from the cable loops, there are constraint equations from joint positions. Such constraint equations are derived based on shared joint position of two links. For example, position of Joint G can be derived from two paths: Links AB-BG and Links AC-CG, respectively. The position derived from the two paths must be identical

x G|BG -x G|CG = 0 y G|BG -y G|CG = 0 ( 34
)
where x G|BG and y G|BG are the positions derived from Links AB-BG, and x G|CG and y G|CG are the positions derived from Links AC-CG. Six constraint equations are obtained, the paths are summarised in Table 4. Note that for Point H, the positions derived from path AC-CE-EH are equated to the generalised coordinates x H and y H .

Proof of Criterion 1

Criterion 1 states that with given cable length, the rotation of pulleys must be consistent with the rotation of the links they are attached to, otherwise it will jam the linkage. It can be readily seen that Criterion 1 is met in Loops PL and PU. The proof for Loop FB based on constraint approach is presented below.

The aim is to prove that the output rotation of the cable loop (h 9 ) is consistent with the h 9 derived from the links, as shown in Eq. ( 13), when a constant cable length from P3 to P5 is given.

Substituting the h for P3 to P5 from Table 2 into Eq. ( 28) yields two constraint equations

R h 4 -- p 2 + R h 8 - p 2 -0 = L const.34 (35)
for P3 and P4, and

R p 2 + 0 -h 9 + R p 2 + 0 -h 4 = L const
R (p + h 8 -h 9 + 20) = L const.34 + L const.45 (37)
Eq. ( 37) indicates that when given the correct cable length between P3 and P5 to start with, h 9 equals h 8 within full ROM of the cable linkage. Such outcome is consistent with Eq. ( 13), which means that the rotation of P5 is consistent with that of Link CE, thus Loop FB is not obstructing the movement of the linkage.

Constraint matrix c q and derivative matrix C q

Assembling the constraint equations derived from cable loops and joint positions yields the constraint matrices c q . Individual constraint equations within the matrix are arranged in the following order 1. c qP12 for Pulleys P1 and P2 in Loop PL.

c q1 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ -R (h1 + h 2) R (h4 + h 8) R (p -h 4 -h 9 + 20) R (h6 -h 7) (L1 -L 2) cos h 1 -L 3 cos (h1 + h 2) + L CG cos (h1 + h 8) (L1 -L 2) sin h 1 -L 3 sin (h1 + h 2) + L CG sin (h1 + h 8) (L1 -L 2) (cos h 1 -cos (h1 + h 8 + h 9)) -L 3 (cos (h1 + h 2) + cos(h 1 + h 8 + h 9 + h 6 )) (L 1 -L 2 ) (sin h 1 -sin (h1 + h 8 + h 9)) -L 3 (sin (h1 + h 2) + sin (h1 + h 8 + h 9 + h 6)) x H -L 1 cos h 1 + L 1 cos (h1 + h 8 + h 0) + L 3 cos (h1 + h 8 + h 9 + h 7) y H -L 1 sin h 1 + L 1 sin (h1 + h 8 + h 0) + L 3 sin (h1 + h 8 + h 9 + h 7) ⎤ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ (38)
for the upward motion of cable linkage. Note that for the constraint equations derived from cable loops, the constants associated with cable lengths are not included, as they do not appear in the derivative matrix C q . For downward motion of the cable linkage, the joint constraint equations (last six) are the same. The cable constraint equations (first four) have negative signs to those in Eq. (38). The derivative matrix C q is given by where each column in the matrix contains the derivative of constraint matrix c q with respect to one of the generalised coordinates. The derivative matrices C qi and C qd for independent and dependent generalised coordinates, respectively, are which the two matrices are used for Eq. ( 26).

C q = ∂cq ∂h 1 ∂cq ∂h
C qi = ∂c q ∂h 1 (

Constraint status of cable linkage

With ten generalised coordinates and ten constraint equations, the cable linkage seems to be over-constrained. However, certain retain constraint equations are redundant or vanish, resulting in fully-constrained one-DOF RCM mechanism. The cases where the links in non-overlapped and overlapped configurations are analysed separately.

In non-overlapped configurations, Loop FB is redundant. Loop FB introduces one generalised coordinate h 4 and two constraint equations c qP34 and c qP45 . By excluding the generalised coordinate and constraint equations, the cable system is described by nine generalised coordinates with eight constraint equations. The cable linkage is fully constrained and has one DOF.

The overlapped configuration is the singular configuration of a four-bar linkage, thus c qGx|DG from four-bar linkage BCDG vanishes, resulting in fully-constrained cable linkage. According to the constraint approach, the elimination of c qGx|DG can be proven mathematically by observing zeros for its corresponding terms in C q .

Writing h 6 to h 9 , and L CG in terms of h 1 , and substitute into C q yields ∂ ∂q c qGx|DG = 0 0 0 0 0 0 0 0 0 (42)

where h 6 to h 9 are given by Eqs. ( 13), ( 14)and ( 19), respectively, and

L CG = 1 - 1 r (LAO -L 1 cos h 1) 2 + (L1 sin h 1) 2 (43)
Thus the cable linkage is described by ten generalised coordinates with night constraint equations. The cable linkage is fully constrained and has one DOF.

Calculation of cable tension

In the analysis, no external load is applied on Pulleys P2 to P6 and the passive prismatic joint I. Further, the upward and downward are simulated by applying vertical force F Hy at point H. The torque on Joint E (t 7 ) and horizontal force on Point H (F Hx ) are zero. The generalised external force Q e applied on the generalised coordinates is given by

Q e = t 1 t 2 • • • t 6 t 7 F CG F Hx F Hy = t 1 0 • • • 0 0 0 0 F Hy (44)
where t 1 is the actuator input torque. Since the actuator input torque t 1 is unknown, Eq. ( 26) is used instead of Eq. ( 25), such that the constraint force can be calculated based on given t 7 , F Hx and F Hy , and then determine t 1 using the constraint force k. The corresponding Q ei and Q ed are

Q ei = t 1 ( 45
)
and

Q ed = 0 • • • 0 0 0 0 ∓F Hy (46)
According to Eq. ( 26), the constraint force under given generalised external force is

k = C -T qd Q ed (47)
Following Section 3.5, Eq. ( 47) at the overlapped and non-overlapped configurations is solved separately, which yields analytical and numerical solutions, respectively.

Analytical solution of k at overlapped configuration

At the overlapped configuration, Eq. ( 47) which is solved with c qGx|DG is excluded from C qd . The dimension of C qd is 9 × 9 hence can be inversed directly. The analytical solution of cable tension, which are the first four elements in the constraint force k, is given by

k 1 = ⎡ ⎢ ⎢ ⎢ ⎢ ⎣ - F Hy L AO (-1+r+v+rv) rR(1+v) - F Hy vL AO rR - F Hy vL AO rR - F Hy L AO (-1+r) rR ⎤ ⎥ ⎥ ⎥ ⎥ ⎦ (48)
for the upward direction of motion, and

k 2 = -k 1 ( 49
)
for the downward direction of motion. In both k, the first to fourth rows correspond to cable tension between P1 and P2, P3 and P4, P4 and P5, and P6 and P7, respectively. In Eq. ( 48), since F Hy is negative, r is greater than 1, and R and v are positive, all the elements are positive. As such, the cable forces within Loops PL, FB and PU are all tension. Similarly, in Eq. ( 49), since F Hy is positive, the cable forces are all positive. Hence, the cable loops are in tension at the overlapped configuration in both directions of motion of the cable linkage.

Numerical solution of k at non-overlapped configuration

At the non-overlapped configurations, the redundant constraints cannot be removed, as cable tension needs to be solved. The dimension of C qd is 10 × 9 and cannot be inversed directly. QR decomposition is applied to obtain the numerical solution

C qd = Q 1 Q 2 R 1 0 ( 50
)
where h 6 to h 9 , and L CG in C qd are represented in terms of h 1 before QR decomposition is applied. Eq. ( 47) is written with Q 1 and R 1 as

k = Q 1 R -T 1 Q ed (51)
The numerical solution of constraint force is plotted in Fig. 10, for full ROM of a sample cable linkage whose dimensions are listed in Table 5. The numerical solutions are identical for the upward and downward motions of the cable linkage.

In Fig. 10, P12, P34, P45 and P67 represent the cable tension in between Pulleys P1 and P2, P3 and P4, P4 and P5, and P6 and P7, respectively. The maximum cable tension is observed at the overlapped configuration. The magnitudes in Loops PL (P12), FB (P34 and P45) and PU (P67) are 119.1 N, 52.48 N, and 38.4 N, respectively, which agree with the analytical solutions. The cable tension P34 and P45 are identical, which also agrees with the analytical solution. The minimum tension are observed at both ends of ROM. The numerical solution indicates that the cable tension in all loops is positive within the full ROM.

Verification of cable tension and proof of Criterion 2

The numerical solution obtained from the constraint approach is verified with static rigid body simulation conducted using ANSYS Rigid Dynamics package, with a 2D model of the same sample cable linkage. The model used is shown in Fig. 11. In the 2D model, links are used to mimic cable sections. The links are connected to pulleys with revolute joints, such that only tension and compression are allowed.

The cable tension in one side of the cable loop is obtained by summing the magnitudes of tensile and compressive of the two links which are connected to the same pulley. To justify the direction of cable force, the joint force at the link corresponding to the cable that is expected to be in tension must be tension as well. The external force F Hy is simulated by a remote force applied at the tip of Link EH.

The cable tension obtained through simulation is compared with the calculated results in Figs. 12 and13. The legends with "S" correspond to the simulation results.

The comparison indicates that the cable tension in Loops PL and PU, as well as the actuator input torque t 1 obtained through the constraint approach is accurate. In Loop FB, exact solution is obtained at the overlapped configuration. The errors are relatively large towards two ends of the ROM, but the trend in the change in cable tension with respect to input angle is well captured. Therefore, the cable tension calculated from the constraint approach is verified, and it can be concluded that Criterion 2 for functioning of cable loops is proven.

Calculation on cable stiffness

The stiffness of the cable linkage is defined as the deformation of Point H under external load

k = F Hy dy H = 8N 5mm = 1.6N/mm (52)
The stiffness of the cable must be sufficient to achieve the overall stiffness. Since cable tension and elongation change with respect to the movement of the linkage, the maximum value is determined as the minimum required cable stiffness

k = max k i dL i ( 53
)
where k is the minimum required cable stiffness in the cable loop. k i and dL i are the cable tension at different actuator input angle and its corresponding elongation, respectively. Due to the assumption of inextensible cable, conventional definition of elongation, which is defined as deformation of cable, cannot be applied in the analysis. Here, the elongation is defined as the infinitesimal travel distance of cable with respect to infinitesimal displacement at Point H dy H . The derivation of cable elongation is presented as follows.

The travel distance of cable is related length of cable wrapping on pulley and hence can be written in terms of generalised coordinates where the function is the suitable equation among Eqs. ( 30) to (33). h angles are listed in Table 2 or 3. The mathematical expressions are (use P1, P3 and P6 for upward motion as examples)

dL = function (R, h in , h q , h out ) (54)
dL PL = Rdh 1 dL FB = Rdh 8 dL PU = Rdh 6 (55)
where dL PL , dL FB and dL PU are the infinitesimal cable travel in Loops PL, FB and PU, respectively. dh 1 , dh 8 and dh 6 are the infinitesimal change in generalised coordinates. Eq. (55) needs to be further related to dy H using Eq. ( 24). Rearranging yields

dq d = -C -1 qd C qi dq i ( 56
)
where dq d and dq i are the infinitesimal change in dependent and independent generalised coordinates, respectively. dy H , dh 8 , and h 6 are contained in dq d , while dh 1 is dq i . Eq. ( 56) writes dh 1 in terms of dy H . All other dependent generalised coordinates are written in terms of dh 1 and hence related to dy H . Eq. ( 56) is solved numerically using QR decomposition as

dq d = -R -1 1 Q T 1 C qi dq i ( 57
)
where Q 1 and R 1 are obtained from Eq. ( 50). Substituting Eq. (57) into Eq. ( 55) yields the infinitesimal travel distance of cable in the cable loops. Further substituting such infinitesimal travel distance of cable into Eq. ( 53) gives the minimum required cable stiffness corresponding to the overall system stiffness. The results are illustrated in Fig. 14.

The minimum required cable stiffness are 240.5 N/mm, 137.7 N/mm and 77.51 N/mm for Loops PL, FB and PU, respectively. In all three cable loops, the largest k are observed at the overlapped configuration. The pattern of change in k with respect to input angle in Fig. 14 is very similar to the pattern of change in tension with respect to input angle in Fig. 10, which means the cable tension is the dominating factor in stiffness.

Comparison on device footprint

For a 2-DOF RCM mechanism based on the cable linkage, the footprint is described through three approaches 1. The sweeping volume of linkage. 2. The space behind the input joint to permit the rotation of input link.

The volume of links.

In this section, the footprint of cable-linkage is compared to that of a PB-linkage through the above-mentioned approaches. The comparison of footprint is conducted mathematically under a simplified surgical scenario, as shown in Figs. 15 and16, for the cable linkage and PB-linkage, respectively. Five configurations of both RCM mechanisms are shown, which are the two extremes in ROM, centre and two mid-point configurations as presented in Figs. 3 and4.

In the simplified surgical scenario, the incision port is Point O. The ROM to be achieved is symmetric around the ground axis. R C is a coefficient less than one. It gives the radius of the circular region in terms of distance L AO , where the links (except for the output link) of the RCM mechanisms are not allowed to enter such region. The region with R C is set, such that more clearance near the incision port is left for the output links of multiple surgical arms and human surgeons to occupy. L AO is the same for both RCM mechanism. The ROM is changed from 30 to 120 • . R C is changed from 0.5 to 0.825. The dimensions and hence the footprints of the RCM mechanisms are calculated based on the given set of ROM and R C .

For the cable linkage, the lengths of input and connector links (AC and CE) are determined from the ROM using Eq. ( 3). The lengths of BG and DG are calculated from Eq. ( 4), where r is assumed to be 1.22 and not changed in the simulation. For the PBlinkage, the length of input link AC is the same as the radius defined by R C and L AO , such that the output joint C is just on the edge of the circle. The length of connector link BC equals L AO .

The results of the comparison are presented as the percentage difference of the footprint of cable linkage, over that of the PB-linkage, expressed as

d = f CL -f PB f PB × 100% (58)
where f CL and f PB are the footprints of the cable linkage and PB-linkage, respectively.

Approach 1

In MIS applications, a planar RCM mechanism with smaller enclosed area is desired, such that the sweeping volume and the chance of collision between surgical arms are reduced.

In Approach 1, the footprint is described as the characteristic enclosed area. Such characteristic enclosed area is calculated by summing the four out of five configurations in Figs. 15 and16, such that the change in enclosed area with respect to movement of linkage is taken into account. The overlapped configuration is not considered, as the enclosed area is zero. The mathematical expression is

f 1 = f top + f mid-top + f mid-bottom + f bottom (59)
where f 1 is the characteristic enclosed area and the four remaining f terms are the enclosed area at four configurations, respectively. The enclosed area of the cable linkage and PB-linkage is defined by four-bar linkage ACEO and parallelogram ABCO in Figs. [START_REF] Stoianovici | Remote Center of Motion Robotic System and Method[END_REF] and 16, respectively. The results are illustrated as a contour plot in Fig. 17. The blank region at the top right corner of Fig. 17 is where the cable link fails to stay outside the radius with given R C . The edge of such region corresponds to the configuration of the cable linkage, where Joint G is on the edge of the circle, as illustrated in Fig. 15.

The result reveals that the enclosed area of the cable linkage is smaller than that of the PB-linkage in about half of the design points (sets of ROM and R C ). The cable linkage has smaller enclosed area at large R C (>0.5). The smallest enclosed area is achieved at the maximum R C of 0.81, and the corresponding percentage difference is -35 %. Cable linkage has a relatively smaller enclosed area because its geometry is fully defined by the ROM (when r is given). In the contrast, the length of Links AB and CO and hence the footprint of the PB-linkage increase with R C .

Approach 2

Both RCM linkage require additional clearance behind input joint A to allow the movement of linkage. As such, the region needs to be kept clear during the operation. A large region will increase the space needed to maneuver the linkage. In addition, a large region requires a larger base to hold the RCM mechanism, thus increases the overall dimension of the robotic manipulator.

The footprint in terms of the aforementioned region is represented by the sweeping L 4 in Figs. 18 and 19 around Joint A and behind Joint A. For cable linkage, L 4 equals L AC . Link AC is behind Joint A when actuator input angle is within p/2 to 3p/2. For PB-linkage, L 4 equals the length Link AB and thus the radius of circle. Link AB sweeps behind Joint A in the full ROM. The mathematical expressions are

f CL2 = 1 2 pL 2 4 ( 60
)
f PB2 = ROM 2p pL 2 4
The results are illustrated as a contour plot in Fig. 20.

The result shows that the footprint (Approach 2) of the cable linkage is smaller than that of the PB-linkage in more than half of the design points. The maximum reduction in footprint is 83% at ROM of 30 • and R C equals 0.8. The footprint of the cable linkage is smaller in the cases where ROM is small and/or R C is large. On the other hand, since the dimension of the cable linkage is defined by the ROM, the footprint is large at large ROM and/or small R C .

Approach 3

There are cases in MIS applications where a surgical manipulator stays stationary while the other robotic arms or human surgeons are conducting manipulations. Therefore, the footprint in terms of the volume taken by the stationary links themselves must be taken into account.

Since the volume of the links is proportional to the length of links, the sum of lengths of links is used as the measurement of footprint in Approach 3. For cable linkage, the links involved are Links AC, CE, BG and DG. Diagonal link CG is not considered as it is enclosed by BG and DG. For PB-linkage, the links involved are Links AB and BC. The mathematical expressions are

f CL3 = L AC + L BG + L DG + L CE (61) f PB3 = L AB + L BC
The results are illustrated as a contour plot in Fig. 21.

- The figure shows that the footprint (Approach 3) is smaller than that of the PB-linkage in all the achievable design points. The percentage reduction in footprint varies between -50% and -80 %, which is significant. Again, the cause of the difference is the fact that the footprints of two RCM mechanisms are dependent on ROM and R C , respectively, while irrelevant to the other.

The comparison in footprint through three approach shows that the cable linkage has a smaller sweeping volume and requires less space behind the input joint in more than half of the design points. In addition, the overall lengths of the links are shorter in all design points, resulting in smaller and lighter linkage. Therefore it can be concluded that the cable linkage has advantages over the PB-linkage in terms of footprint in MIS applications. The scenario that suits the cable link best is where the required ROM is relatively small while the RCM mechanism has to be kept faraway from the incision port.

Prototype of cable linkage

A passive concept-proving prototype of the cable linkage is built. The dimensions of the linkage are given in Table 5. The ROM is 69 • , which satisfies the requirement of 60 • ROM in most of the abdominal MIS applications [START_REF] Lum | The REVAN: design and validation of a telesurgery system[END_REF]. The additional 9 • are added such that v is 0.3 and L 1 is an integer. r is selected to be 1.22 to minimise rounding of L 2 and L 3 . The selection of v and r helps reducing machining error and thus the potential positioning error of the remote centre. Note that due to the limit on the available cable, the minimum required stiffness presented in Fig. 14 is not applied on the cable in Loop FB.

The mechanical constraint of a cable loop can be achieve identically through end-less tendon drive or open-ended tendon drive, as classified in [START_REF] Tsai | Design of tendon-driven manipulators[END_REF]. End-less tendon drive is literally closed-loop cable. It is structurally simpler and can theoretically be fully controlled by less actuators comparing to the open-ended tendon. On the other hand, open-ended tendon drive, which actively controls a pulley with two antagonistic tendons that are connected to their own actuators, respectively, yields zero or small backlash in the expanse of complexity. As the prototype is built for proof of concept, closed-loop cable is implemented in the prototype. To minimise the backlash, timing belt is used and the pre-tension is applied.

The CAD model of the cable linkage, showing two limits and the centre of the ROM, is illustrated in Fig. 22. The centre and lower limit configurations are combined in the figure by photoshop hence are semi-transparent. The sphere at the bottom right corner of the figure indicates the RC. The design of output link is arbitrary and yet to be optimised for specific MIS applications. The output ROM is 105 to 175 • from the positive x-direction. The belts and cable sections are not shown.

A side view of the prototype is illustrated in Fig. 23. In the prototype, the widths of Links AC and CE are 16 mm. The radius of pulleys in Loops PL and PU is reduced to 12 mm due to the available pulley size. It can be observed from the figure that the area enclosed by Loops PL and PU is not significantly larger than the sizes of Links AC and CE, despite the outstanding tensioners located in the middle of the links. In addition, the pulleys in Loop FB are mostly enclosed by Links BG and DG in the side view, thus do not cause significant increase in the size of mechanism.

Top views of the prototype are illustrated in Figs. 24 and 25. Fig. 24 annotates the links and Loops PL and PU. In both loops, timing belts are used to minimise backlash. It also eliminates the chance of slip of cable on pulleys in Loops PL and PU, where the consequence of such cable slip is the lost of RCM function. To increase the rigidity of the mechanism, a duplicated pair of Links AC and AF is introduced. Both links are located above Links CE, P6 and P7 in Fig. 24. Fig. 25 annotates Loop FB. In this prototype, Loop FB is achieved by using two cable loops, one between P3 and P4 and the other between P4 and P5. Two pulleys, P4 and P4 are rigidly connected to each other and mounted at Joint I. P4 is connected to P3, and P4 is connected to P5 with crossed cable. Pulleys P3 and P5 are rigidly connected to Links AC and CE, respectively, by 3D-printing the pulley and the link as one part. To enable the crossed configuration of cable between P4 and P5, a round belt is used.

The presence of tension in the correct sides of cable loops is vital for verifying the design concept. On the other hand, the exact magnitude of tension is of less importance, as it is primarily used as a prototyping guideline. The sign of cable tension can be readily observed through higher transverse cable stiffness on the driving side of a cable loop than that on the driven side. By manually applying force at the output link and comparing the cable stiffness, it is found that the presence of tension within the full range of motion is consistent with Figs. 5 and6, as well as the outcomes in Section 3 . Apart from that, the prototype is manipulated to the centre configuration, where loads are applied manually in the attempt to force the prototype into the undesired configuration. Jamming of linkage is observed, thus verifies that the linkage can only turn into the desired configuration.

Conclusion

This paper introduces a cable linkage with RCM. The design of the cable linkage is presented, and proof on RCM function is conducted mathematically.

The tension in all cable sections is determined by means of a constraint approach. The analytical solution at the overlapped configuration is derived. The numerical solution at non-overlapped configurations are obtained through QR decomposition. The tensions are positive within the full working range, validating the function of the linkage.

The cable tension calculated is verified through finite element simulation. The constraint approach yields exact solution of cable tension in Loops PL and PU within full ROM, and in Loop FB at overlapped configuration. The error in Loop FB at nonoverlapped configurations is small.

The constraint approach is further used to determine the minimum required cable stiffness, corresponding to a given overall system stiffness at the output link. Both the tension and infinitesimal displacement of cable are derived from the constraint approach and solved with QR decomposition.

Quantitative comparison between the footprints of the cable linkage and PB-linkage is conducted in a simplified surgical scenario. The footprint is described in three approaches. In two of the approaches, the cable linkage yields smaller footprint in half of the design points. In the last one the cable linkage achieves 50% to 80% reduction in footprint.

A prototype is constructed for proof of concept. Closed-loop cable is used to simplify the architecture. Timing belts are used in Loops PL and PU to prevent slip of cable on pulley. Loop FB is further divided into two loops, in between P3 and P4, and P4 and P5, respectively. Round belt is implemented between P4 and P5 to enable crossed cable configuration. The correct presence of tension within full range of motion, as calculated in Section 3, is validated. The prototype is also proved to be prevented from turning into the undesired configuration.

Fig. 1 .

 1 Fig. 1. Configuration of links.

Fig. 2 .

 2 Fig. 2. Configuration of links and cable loops.

Fig. 3 .

 3 Fig. 3. Configurations at two extremes and centre of ROM.

Fig. 4 .

 4 Fig. 4. Configurations at mid-points.

Fig. 5 .

 5 Fig. 5. Cables in tension in upward motion.

Fig. 8 .

 8 Fig. 8. Clockwise cable wrapping.

2 .

 2 c qP34 for Pulleys P3 and P4 in Loop FB. 3. c qP45 for Pulleys P4 and P5 in Loop FB. 4. c qP67 for Pulleys P6 and P7 in Loop PU. 5. c qGx|CG for x position of Joint G from paths AB-BG and AC-CG. 6. c qGy|CG for y position of Joint G from paths AB-BG and AC-CG. 7. c qGx|DG for x position of Joint G from paths AB-BG and AC-CD-DG. 8. c qGy|DG for y position of Joint G from paths AB-BG and AC-CD-DG. 9. c qHx|EH for x position of Point H from generalised coordinate x H and path AC-CE-EH. 10. c qHy|EH for y position of Point H from generalised coordinate y H and path AC-CE-EH. The constraint matrices are

Fig. 10 .

 10 Fig. 10. Numerical solution of k for sample cable linkage.

Fig. 11 .

 11 Fig. 11. 2D model in ANSYS.

Fig. 12 .

 12 Fig. 12. Comparison of cable tension in Loops PL and PU.

Fig. 13 .

 13 Fig. 13. Comparison of cable tension in Loop FB.

Fig. 14 .

 14 Fig. 14. Cable stiffness.

Fig. 15 .

 15 Fig. 15. Cable linkage in surgical scenario.

Fig. 16 .

 16 Fig.16. PB-linkage in surgical scenario.

Fig. 17 .

 17 Fig. 17. Percentage difference in footprint -Approach 1.

Fig. 18 .

 18 Fig.18. Cable linkage in surgical scenario.

Fig. 19 .Fig. 20 .

 1920 Fig. 19. PB-linkage in surgical scenario.

Fig. 21 .

 21 Fig. 21. Percentage difference in footprint -Approach 3.

Fig. 22 .

 22 Fig. 22. CAD model of cable linkage.

Fig. 23 .

 23 Fig. 23. Right side view of prototype.

Fig. 24 .

 24 Fig. 24. Top view of prototype 1.

Fig. 25 .

 25 Fig. 25. Top view of prototype 2.

Table 1

 1 List of generalised coordinates.

	q	Definition	Reference
	h 1	Angle of Link AC	AF
	h 2	Angle of Link BG and P2	AC
	h 4	Angle of P4	CG
	h 6	Angle of Link DG and P6	CE
	h 7	Angle of Link EH and P7	CE
	h 8	Angle of Link CG	AC
	h 9	Angle of Link CE	GC
	L CG	Distance between CG	n/a
	x H	Horizontal position of H	A
	y H	Vertical position of H	A

Table 2

 2 Cable and pulley angles for upward motion.

	Pulley	h in (rad)	hq (rad)	hout (rad)	Direction	Reference
	P1	n/a	0	h 1 + p/2	C W	AF
	P2	p/2	h 2	n/a	CW	AC
	P3	n/a	0	h 8 -p/2	CCW	AC
	P4	-p/2	h 4	p/2 + 0	CCW	CG
	P5	p/2 + 0	h 9	n/a	CW	GC
	P6	n/a	h 6	p/2	C W	CE
	P7	p/2	h 7	n/a	CW	CE

Table 3

 3 Cable and pulley angles for downward motion.

	Pulley	h in (rad)	hq (rad)	hout (rad)	Direction	Reference
	P1	n/a	0	h 1 -p/2	CCW	AF
	P2	-p/2	h 2	n/a	CCW	AC
	P3	n/a	0	h 8 + p/2	C W	AC
	P4	p/2	h 4	-p/2 -0	CW	CG
	P5	-p/2 -0	h 9	n/a	CCW	GC
	P6	n/a	h 6	-p/2	CCW	CE
	P7	-p/2	h 7	n/a	CCW	CE

Table 4

 4 Joints for constraint equations.

	Joint	Path 1	Path 2
	G	AB-BG	AC-CG
	G	AB-BG	AC-CD-DG
	H	x H and y H	AC-CE-EH

Table 5

 5 Dimensions of sample cable linkage.

	Parameters	Magnitude
	L AO (mm)	400
	v	0.3
	r	1.220
	L 1 (mm)	120
	L 2 (mm)	98.4
	L 3 (mm)	72
	R (mm)	15
	F Hy (N)	∓8


Christel Bidet-Ildei
email: christel.ildei@upmf-grenoble.fr

David Méary

Jean-Pierre Orliaguet

David Meary

Visual Perception of Elliptic movements in 7-to-11-year-old children : Influence of Motor Rules

 : dès 7 ans les ajustements perceptifs sont conformes au principe d'isochronie. Ces résultats permettent de discuter les liens motricité-perception.

INTRODUCTION

The human visual system is extremely sensitive to human motion. In the past, several studies showed that, even when movement is represented by a simplified point-light display, observers can discriminate human body movements from moving objects [START_REF] Bingham | Dynamics and the orientation of kinematic forms in visual event recognition[END_REF]. They identify actions such as walking or dancing [START_REF] Johansson | Visual perception of biological motion and a model for its analysis[END_REF][START_REF] Johansson | Visual motion perception[END_REF], the gender and the identity of a person (Cutting & Kozlowski, 1977;Kozlowski & Cutting, 1977) and even the properties of handled objects such as the weight of a lifted object [START_REF] Runeson | Visual perception of lifted weight[END_REF], 1983). Other results showed that when subjects have to visually evaluate the velocity of human movements (i.e., pointing movement, handwriting, drawing an ellipse) they prefer those that conformed to motor laws [START_REF] Meary | Visual perception of writing and pointing movements[END_REF][START_REF] Viviani | The effect of movement velocity on form perception: geometric illusions in dynamic displays[END_REF]. For example when a subject is asked to adjust the velocity of a reaching movement, a writing movement or an elliptic movement, he/she tends to choose durations of movement which are respectively in line with the Fitts' Law, the isochrony principle and the two-third power law. To explain this high sensibility to human movement, it has been suggested that visual identification of human movement could be not only based on visual experience but also on motor experience (i.e., [START_REF] Jeannerod | Neural simulation of action: a unifying mechanism for motor cognition[END_REF][START_REF] Jeannerod | Mental imaging of motor activity in humans[END_REF]. In others words the recognition of human movements would be the result of motor-perceptual interactions. Several results are in accord with this view. Observers better recognise point-light displays representing their own movement than movements of their friends [START_REF] Beardsworth | The ability to recognize oneself from a video recording of one's movements without seeing one's body[END_REF][START_REF] Loula | Recognizing people from their movement[END_REF]. Patients with motor deficits often have difficulties to recognise human movements as this is the case for dysgraphic [START_REF] Chary | Influence of motor disorders on the visual perception of human movements in a case of peripheral dysgraphia[END_REF] and apraxic [START_REF] Heilman | Two forms of ideomotor apraxia[END_REF] patients. Finally, neuroimaging studies show that both observation and execution of movements activate common brain regions [START_REF] Hari | Activation of human primary motor cortex during action observation: a neuromagnetic study[END_REF][START_REF] Peuskens | Specificity of regions processing biological motion[END_REF][START_REF] Saygin | Point-light biological motion perception activates human premotor cortex[END_REF].

Some developmental studies also indicate that the motor competence of the observer could be involved, at least in part, in the visual perception of human movements. For example, children with articulatory disorders (i.e., d/b confusion) have more difficulties in lips reading than children without such motor difficulties [START_REF] Desjardins | An exploration of why preschoolers perform differently than do adults in audiovisual speech perception tasks[END_REF]. In the same way, reading deficits is often associated with motor disorders [START_REF] Felmingham | Visual and visuomotor performance in dyslexic children[END_REF]. Finally, an experiment carried out by Louis-Dam, [START_REF] Louis-Dam | Anticipation motrice et anticipation perceptive[END_REF] showed that in children the ability to visually anticipate the forthcoming movement in a motor sequence is directly influenced by their level of motor competence.

In this theoretical context, the aim of the present research is to bring additional evidence that, in children, visual perception of human movement is influenced by motor rules. To this end, 7 to 11 year-old children were asked to evaluate and to adjust the velocity of a dot depicting an elliptic movement. In motor production, [START_REF] Viviani | A developmental study of the relationship between geometry and kinematics in drawing movements[END_REF] showed that from 7 years, the duration of elliptic movements conforms to the isochrony principle: the duration of the movement tended to be constant irrespective of the perimeter of the ellipse. If, as showed by former studies that visual perception of human movement tends to conform to motor rules, the visual evaluation of time movement will also tend to conform to the isochrony principle from 7 years of age.

METHOD Participants

Forty-five children participated in the experiment. They were divided into 3 age groups of 15 children each: 7 years (mean age 6 years 11 months), 9 years (mean age 8 years 9 months) and 11 years (mean age 10 years 9 months). A group of 15 adults, students at the university, was used as control. All subjects were right-handed and had normal or corrected to normal vision.

Stimuli

The visual stimuli consisted of 6 elliptic movements. The perimeter of the ellipse was respectively 2. 94, 5.25, 9.36, 16.69, 29.74 or 53 cm long. The semi axis ratio b/a (0.425) and the eccentricity of the ellipse ∑ (0.9) were constant whatever the perimeter of the ellipse (see figure 1). The major semi axis of the ellipse a was rotated by 45 degrees counter-clockwise. A software permitted to modify the trajectory perimeter (Pe) or the movement period (P). For all period values, the velocity profile of the movement respected the twothird power law (covariation velocity-curvature) observed in the production of movement [START_REF] Lacquaniti | The law relating the kinematic and figural aspects of drawing movements[END_REF].

Procedure

Participants seated at a distance of 50 cm from a computer screen (17', resolution 1024*768 pixels, sampling rate 85 Hz) in a dimly illuminated room (see figure 2). Each trial consisted in the presentation of a black spot (Ø = 0.4 cm) depicting an elliptic movement on a white background area (22*22 cm).

Participants were asked to adjust the period of movement (that is to find their "preferred velocity") by pressing the arrows (← or →) of the keyboard. The adjustments allowed increasing or decreasing of the duration of the period by step of 25 ms.

The experiment was run in a single session including 6 blocks of trials. Each block comprised 6 trials corresponding to each perimeter of the ellipse. Therefore each subject performed 36 trials. The order of presentation of blocks and trials, and the initial period value of each trial (Pi) were randomized. Short periods of rest separated each block of trial.

Data analysis

The results were analysed by using the same formalisation and the same procedure than those used in Viviani & Schneider's experiment (1991). The relation between the perimeter of the ellipse (Pe) and the final period (Pf) chosen by the participants was formalised by the power function Pf = P0 * Pe γ where γ represents the exponent of the function and P0 a baseline period which depends of each subject (see figure 3). Then, we evaluated for each subject the coherence of the results, the degree of isochrony and the movement speed by measuring the correlation coefficient r), the exponent γ) and the baseline period (P0) according to age.

RESULTS

The mean of the correlation coefficient, of the exponent, and of the baseline period were calculated and statistically evaluated with an ANOVA with age as between factor.

Because the correlation coefficients and the baseline period (P0) did not follow a normal distribution, data has been transformed. For the coefficients of correlation we used the Fisher hyperbolic tangent transform [START_REF] Kendall | The advanced theory of statistics[END_REF]) and the baseline period was analysed by using the logarithm of P0. Results obtained in children and in adults were analysed separately. Correlation coefficient (r): the statistical analysis revealed a significant effect of age [F (2,42) = 6.09, p < 0.01]. We observed an increase of r value between 7 (0.71) and 9 years (0.88), a stabilisation between 9 and 11 years (0.85) and again an increase between 11 year-old children and adults (0.95) [t (28) = 3.67, p < 0.01]. This result indicates that the within variability is higher at 7 years and tends to decrease with age. However, it should be noted that whatever the age, the values of the coefficient correlations are very high. This result shows that the link between the final period and the perimeter of the ellipse is well approximated by the power function.

Exponent γ): there was no effect of age [F (2,42) = 1.54, p = 0.22]. It is noteworthy that the mean value of γ (0.41 ± 0.14) is not different from the value (0.44) obtained by [START_REF] Viviani | A developmental study of the relationship between geometry and kinematics in drawing movements[END_REF] in the motor task [t (44) = 1.61, p = 0.11]. In addition, we did not observe any significant difference between children (0.41 ± 0.14) and adults (0.46 ± 0.19) [t (58) = 0.98, p = 0.33].

Baseline period (P0): age has no effect on the performances [F (2,42) = 0.40, p = 0.67].

Whatever the age, the mean baseline period was 0.43 s and no significant difference was observed between children (0.42 ± 0.29) and adults (0.37 ± 0.20) [t (58) = 0.73, p = 0.46].

DISCUSSION

The purpose of this experiment was to know if the visual perception of human movements was influenced by the rules of motor production. Our results showed a very high similarity between the performances observed in the visual perception of elliptic movements and those obtained by Viviani & Schneider in the motor production of ellipses (1991). Indeed the exponents of the power function which defines the relation between perimeter and time movement (the final period chosen by the subject) are not different in the perceptive (0.41) and motor tasks (0.44). Therefore these findings indicate that in children, from 7 year-old, the isochrony principle is observable both in production and in perception. Though the perimeter was multiplied by 18 (2.53 to 53 cm), the final period chosen by the participant was only multiplied by 3 (930 to 2606 ms). Therefore, these findings demonstrate that a similar principle determines motor and perceptual performances and suggest that the perceptual abilities of children may depend on their level of motor development.

Such similarities between perceptual and motor behaviours may be interpreted within the motor simulation theory [START_REF] Jeannerod | Neural simulation of action: a unifying mechanism for motor cognition[END_REF]. According to this theory, the motor system is considered at a covert stage as a simulation system that is activated in self-intended action and also in the recognition of others' action. At this covert stage, action is not executed but the way to reach the goal and the consequences of the action on both the organism and the external world are simulated. Therefore it can be hypothesised that when perceiving a movement, children based their decision on an internalized simulation of the ellipse movement that leads to prefer stimuli that share common kinematic properties with their own motor productions, that is those which conform to the isochrony principle. This assumption is in accordance with clinical observations showing the role of motor competence in the visual perception of human movement (i.e., [START_REF] Chary | Influence of motor disorders on the visual perception of human movements in a case of peripheral dysgraphia[END_REF] and with neuroimaging studies showing the activation of neuronal motor structures during the visual perception of human movements [START_REF] Chaminade | Is perceptual anticipation a motor simulation? A PET study[END_REF][START_REF] Decety | Brain activity during observation of actions. Influence of action content and subject's strategy[END_REF][START_REF] Hari | Activation of human primary motor cortex during action observation: a neuromagnetic study[END_REF][START_REF] Nishitani | Temporal dynamics of cortical representation for action[END_REF][START_REF] Rizzolatti | Localization of grasp representations in humans by PET: 1. Observation versus execution[END_REF].

Though, the numerous evidences for a connection between perception and action, few authors consider that the visual perception of dynamic events could be determined by general invariants which have emerged from evolution of species and which would be present from the beginning of the development, that is long before the emergence of motor competences [START_REF] Shepard | Path-guided apparent motion[END_REF][START_REF] Vallortigara | Visually inexperienced chicks exhibit spontaneous preference for biological motion patterns[END_REF]. This hypothesis is supported by some results obtained in infants and in patients with impaired motor functions. [START_REF] Fox | The perception of biological motion by human infants[END_REF] demonstrated that 8 week-old infants exhibit a preference for a point-light walker over the same configuration inverted 180 degrees. Moreover it has been showed that by 3-5 months of age, infants discriminate a point-light walker from displays with perturbed local rigidity [START_REF] Bertenthal | Perception of biomechanical motions by infants: implementation of various processing constraints[END_REF] or with scrambled spatial relations between the dots [START_REF] Bertenthal | Infant sensitivity to figural coherence in biomechanical motions[END_REF]. In addition Pavlova and her colleagues (2003) have shown that adolescents with congenitally impaired locomotion can exhibit high sensitivity to a point-light walker. Therefore these results show that motor competences are not necessary to perceive and recognize human movements.

Animal studies point to a similar conclusion by suggesting that the perception of biological movement could be an intrinsic capacity of the vertebrate visual system. For example, [START_REF] Vallortigara | Visually inexperienced chicks exhibit spontaneous preference for biological motion patterns[END_REF] reported that chicks, hatched and reared in darkness, exhibit a preference to biological movements from the first presentation of the stimuli after birth. They tend to prefer the biological movement of a hen than a rigid or random motion. It is noteworthy that a similar behaviour is observed when the stimuli represent the movement of a cat. This latter result suggests that the preference for biological motion in newborn chick is not species-specific and therefore does not depend on motor ability.

Taken together these results suggest that the isochrony principle observed in visual perception could be explained either by an activation of the motor system or by a genetic predisposition. Our experiment carried out in 7-to-11-year-old children does not permit to conclude, though it seems hardly plausible that all the motor rules are available at birth. It however demonstrates that in children visual perception of an elliptic movement is directly influenced by some intrinsic properties of the motor system, i.e., the isochrony principle.

Figure 2 :

 2 Figure 2: Kinematic characteristics of the stimulus. A) Form of the elliptic trajectory. B) Velocity profile for one period (one cycle) with Vt corresponding to the tangential velocity and t to the movement time.

Figure 2 :

 2 Figure 2: Experimental set up. Subject was in front of a screen. A dot depicted an elliptic trajectory. The task consisted in adjusting the velocity by using the arrows of the keyboard.

Figure 3 :

 3 Figure 3: Example of the power approximation Pf = P0 * Pe γ for one participant chosen randomly in the adult group.

Figure 4 :

 4 Figure 4: Mean of the correlation coefficients (r), of the exponents γ) and the baseline periods (P0) as a function of age.