A parallel robot with remote centre‐of‐motion for eye surgery: design, kinematics, prototype, and experiments

Background: Millions of patients suffering from eye disease cannot receive proper treatment due to the lack of qualified surgeons. Medical robots have the potential to solve this problem and have attracted significant attention in the research community. Method: This paper proposes a novel parallel robot with a remote centre of motion for minimally invasive eye surgery. Ki-nematics models, singularity and workspace analyses, and dimension optimisation are conducted. A prototype was developed, and experiments were conducted to test its mobility, accuracy, precision and stiffness. Results: The prototype robot can successfully perform the required motions, and has a precision ranging from 7 � 2 μm to 30 � 8 μm, accuracy from 21 � 10 μm to 568 � 374 μm, and stiffness ranging from 1.22 � 0.39 N/mm to 10.53 � 5.18 N/mm. Conclusion: The prototype robot has a great potential for performing the minimally invasive surgery. Its stiffness meets the design requirement, but its accuracy and precision need to be further improved.


| Introduction
Globally, more than 2.2 billion individuals grapple with visual impairments, with at least 200 million potentially benefiting from eye surgery [1,2].However, a substantial number of patients face barriers to receiving adequate treatment due to a shortage of qualified surgeons and the prohibitive costs involved [3,4].The deficiency of training eye surgeons is witnessed worldwide and is particularly pronounced in developing nations.The intricate nature of eye surgery demands highly refined motor and sensory skills, which may not be consistently attainable through training and practice.The training duration for eye surgeons varies worldwide but tends to be universally prolonged [5].Typically commencing their careers in their mid-30s, most eye surgeons retire by their late 50s due to declining operational capabilities with age [6].The integration of robots in eye surgery holds the promise of assisting surgeons, providing enhanced capabilities, and improving the affordability, accessibility, and efficacy of eye surgery.Robotic assistance can potentially extend surgeons' careers and shorten training periods, thereby enabling more patients to benefit from improved quality of life.
Current eye surgery predominantly employs a secure and efficient technique known as minimally invasive surgery (MIS).MIS employs slender surgical instruments to perform procedures through small incisions.Compared to open surgery, MIS offers advantages such as minimal scarring, reduced blood loss and pain, lower infection risk, shorter hospital stays, and faster recovery, making it a widely favoured procedure [7][8][9].However, during MIS, eye surgeons face the challenge of precisely controlling the surgical instruments within these small incisions without enlarging them, adding an extra layer of complexity to the procedure.A robotic system designed to assist in MIS is referred to as an MIS robot, aiding surgeons in holding and manoeuvring surgical instruments while limiting their movement at the incision points.In terms of robot design, the incision point is designated as the robot's remote centre of motion (RCM) point.
Figure 1 illustrates MIS being performed on an eye with incision points typically located at the cornea or sclera.The movement of each instrument is constrained to four degrees of freedom (DOFs), including three rotational (3R) DOFs (θ Y : instrument yaw motion, θ P : instrument pitch/tilt motion, θ R : instrument rolling motion) pivoting at the incision point, and one translational (1T) DOF (l T : instrument translation motion) moving through the incision point.
When designing MIS robots, a pivotal challenge is restricting the motion of surgical instruments in 3R1T DOFs at an RCM point for performing MIS.Two methods address this challenge.One approach involves controlling the surgical instrument's motion about the RCM point through specialised robot control, with a larger workspace than that constrained by the RCM point [10][11][12].The other method involves mechanically constraining the surgical instrument's motion using an RCM mechanism, limiting its movement to rotation or translation through a virtual fixed point (i.e., the RCM point) distal from the mechanical structure, without any physical joint over the point.Compared to control-based motion limitation, mechanical constraining is considered safer and more reliable, as it can still limit motion when powered down or in the presence of control errors.During MIS, the RCM point is positioned at the incision point, mechanically constraining the surgical instrument's motion in a 3R1T DOF pattern.
In existing 3R1T MIS robots, the instrument rolling and translation motions (1R1T DOF motion) are typically achieved by a revolute joint and prismatic joint on the instrument holder.Meanwhile, the instrument yaw and pitch motions (2R DOF motion) are usually facilitated by one of four types of RCM mechanisms as shown in Figure 2, that is, circular rail, serial spherical, parallelogram, and parallel spherical RCM mechanisms [13][14][15].
The circular rail RCM mechanism features a circular track that defines the RCM point at the centre of the circular track [14], as depicted in Figure 2a.The first robot-assisted MIS was performed by a 'Probot' [16,17] in 1991, utilising a circular rail RCM mechanism.Subsequently, circular rail RCM mechanisms have been employed in developing eye surgery robots at the University of Western Australia [18], University of Tokyo [19,20], and University of California, Los Angeles [21,22], conducting surgical experiments on animal eyes.While the circular rail RCM mechanism boasts a simple structure and an intuitive principle for defining the RCM point, its relatively large circular track poses challenges for placement in the operating room [23].Moreover, manufacturing a circular track to ensure the accuracy of instrument pitch motion can be a difficult task.
The serial spherical RCM mechanism comprises spherical linkages forming a serial chain, with all joint axes intersecting at one point, that is, the RCM point [15], as shown in Figure 2b.Serial spherical RCM mechanisms have been incorporated into MIS robots such as 'Raven' [24], 'CURES' [23], and the MIS robot from Harbin Institution of Technology [25,26].Although the serial spherical RCM mechanism features a simple structure for defining the RCM point, its serial nature introduces coupled rotational DOFs, leading to accumulated and amplified errors from linkage to end-effector [27].These issues adversely impact its dynamic characteristics, including accuracy and precision.
The parallelogram RCM mechanism involves at least two parallelograms in one plane, rotating parallel linkages at the same angle to achieve one rotational DOF at the RCM point [14], as illustrated in Figure 2c.This type of RCM mechanism is the most prevalent in MIS robots [14] and can be modified to gain an additional translational DOF [28][29][30] or be driven by cables [31,32].It is notably employed in the successful da Vinci Surgical System [33].Parallelogram RCM mechanisms have also been used to develop many eye surgery robots, involving 'Steady-Hand' [34,35], 'Preceyes' [15,36,37], and robotic systems from KU Leuven [38], Beihang University [39], and King's College London [40].While the parallelogram RCM mechanism offers advantages such as a relatively large rotational movement range, simple structure, and an easy gravity balance method using counterweights, its planar structure results in characteristics such as accuracy, precision, and stiffness being excellent in the parallelogram plane but suboptimal in the direction orthogonal to the plane.In addition, a revolute joint had to be deployed to rotate the whole parallelogram structure for instrument yaw motion, which inevitably increases the moment of inertia of the moving structure, negatively affecting the accuracy and precision of the mechanism.To address these limitations, many research groups have recently begun developing spatial parallel RCM mechanisms based on parallelogram structures [41][42][43][44].
The parallel spherical RCM mechanism comprises spherical linkages forming a closed-loop chain, where all joint axes intersect at one point, that is, the RCM point, as illustrated in Figure 2d.This mechanism has found applications in various fields, including a robotic arm for endoscopic surgery [45], 'ARAS-Diamond' [46,47], a craniotomy surgery robot [48], a neurosurgery robot [49], a neuro-endoscopic robot [50,51], and a laparoscopic robot [52].With its spatial parallel structure, the parallel spherical RCM mechanism offers advantages over serial and planar parallel structures, including relatively high accuracy, precision, stiffness, and substantial payload-carrying capability in three dimensions (3D) [27].Nevertheless, this spatial parallel structure presents challenges such as limited workspace, singularities within its workspace, non-linear dynamic characteristics, difficulties in balancing gravity, and intricate kinematics and control due to coupled non-linear motions [27].If these challenges are addressed, the parallel spherical RCM mechanism holds promise as a suitable option for MIS robots.
Implementing an MIS operation requires a 'master-slave' system in which 'master' devices (such as joysticks) are typically driven manually by a surgeon to remotely control the slave devices, that is, robots, for performing the surgical operations on patients.Various sensors such as visual, audio, and haptic devices would be utilised to provide real-time feedback to the master end.The master-slave technique is widely utilised in surgical robots, facilitating telesurgery where surgeons can operate surgery from a distance, particularly impactful in remote or hazardous areas such as battlefields.
To date, there is no commercial robotic system applying an RCM mechanism specially designed for eye surgery.Preceyes [15,36,37] stands as the first and only eye surgery robot applying the RCM mechanism to receive CE (European Conformity) marking approval, currently in the process of FDA approval.While some other surgical robots [14] for eye surgery have been proposed, they are yet to reach the CE-marked or FDA-approved stage.The research of eye surgical robots is still in its infancy.It is worthy to explore the possibility of different mechanisms/structures and to find the potential best options for developing ophthalmic robots.Further research is needed to develop eye surgery robots incorporating various mechanism structures, considering specific requirements, such as small workspace, high accuracy and precision (at the 10-μm level [53,54]), operating room layout, back-drivability, gravity balance, biocompatibility, and sterilisability.
Compared with serial structures, parallel structures have the advantages of higher stiffness (larger payload), better dynamic characteristics (smaller inertia, higher speed and reduced power consumption), easier control (easier solving inverse kinematics), and disadvantages of smaller workspace.However, intraocular surgery only requires a small working area inside the eye, and the small workspace could be an additional safety guarantee for limiting the instrument from moving out of the target area and damaging eye tissues.As for accuracy, parallel structures' higher stiffness and better dynamic characteristics could contribute to higher accuracy than serial structures, but the manufacturing errors and difficulty in calibration may affect the accuracy of parallel structures.Under the condition of proper design, modelling, manufacturing and calibration, parallel RCM mechanism has the potential for eye surgery robots.
With these considerations, this paper introduces a new 7-DOF parallel robot providing 3R1T DOFs at an RCM point for MIS procedures and additional 3T DOFs for positioning the RCM point at the incision.This robot can function as a slave robot in a master-slave robotic system for eye surgery to achieve high repeatability, high stiffness, good dexterity, small size, and low cost.Taking cataract surgery as an example, the design requirements are determined through interviews with surgeons and analysis of existing manual cataract surgical procedures.Considering these requirements, the conceptual design, kinematic design, and detailed design of the proposed robot are conducted.Models of kinematics, singularity, and workspace are formulated and discussed.Considering backdrivability, gravity balance, biocompatibility, and sterilisability, a prototype is built and a control system is developed.Experiments are carried out to assess the robot's mobility, accuracy, precision, and stiffness.
The subsequent sections of this paper are organised as follows: Section 2.1 determines the design requirements of an eye surgery robot, Section 2.2 focuses on architecture generation and kinematic design of the new eye surgery robot mechanism, Section 2.3 conducts detailed design, prototype construction, and control system development, Section 2.4 presents experiments evaluating the robot's mobility, accuracy, precision, and stiffness, Section 3 shows the experimental results and discussion on them, and finally, conclusions are drawn in Section 4. Cataract surgery is taken as an example to illustrate the design requirements.It is a procedure aimed at treating cataracts by removing the diseased lens in the eye and replacing it with an artificial lens through two incisions in MIS incisions, as illustrated in Figure 3.The procedure for robot-assisted cataract surgery can be categorised into five phases from a robot control perspective, as depicted in Figure 4. Phase I 'approaching' involves the robot approaching the patient and positioning its RCM point at the incision point on the patient's eye.Phase II 'incision making/insertion' entails moving the surgical instrument to create an incision on the patient's eye or inserting it into the eye through the incision.Phase III 'manipulation' comprises the robot manipulating the surgical instrument in an MIS manner to accomplish tasks such as breaking the lens, removing the diseased lens, and injecting the artificial lens.Phase IV 'extraction' refers to removing the surgical instrument from the patient's eye.Phase V 'homing' involves the robot moving away from the patient to return to its home position.

| Materials and Methods
An 'instrument change' phase is necessary following Phase IV to switch the surgical instrument for performing various tasks such as making incisions, dissecting the lens, removing the lens, and implanting the artificial lens.The phases of 'insertion', 'manipulation', 'extraction', and 'instrument change' can be repeated in sequence to complete the cataract surgery tasks.Phases II, III, and IV involve motions of the instrument tip inside the patient's eye, necessitating robot manipulation for greater accuracy and precision compared to the human hand, which can have positioning errors exceeding 100 μm due to hand tremors [55].Conversely, the instrument tip motions during Phases I, V, and the 'instrument change' phase occur outside the patient's eye, requiring less stringent accuracy and precision.Two methods can be employed to complete these phases: one is automation or remote control by a human for robotic completion, and the other is manual assistance from a human to move the robot and change instruments.Robots offer superior accuracy and precision, whereas human hands are more adept, and humans possess greater judgement and decision-making abilities.
Taking the right eye as an example, the task frame of robotassisted eye surgery {O Task } is established, as shown in Figure 5. Here, the task frame origin O Task is at the incision point on the eye near the right ear.In the patient's supine position, the X-axis of {O Task } is horizontal towards the inside of the eye and towards the direction of the nose.The Y-axis of {O Task } is horizontal towards the forehead direction.The Z-axis of {O Task } is vertically pointing upwards, as illustrated in Figure 5c.

| Design Specification
In this section, the design specifications for eye surgery robots are established through interviews with surgeons and references to literature on eye surgery [53,54,[56][57][58][59] and the dimensions of the human head [60] and human eye [61].Previous studies have indicated that an accuracy/precision of 500 μm in locating an RCM point on the surface of the incision is required for robot-assisted cataract surgery [57].Considering the potential applications in vitreoretinal surgery, the robot proposed herein aims to achieve an operational accuracy/precision of 10 μm [53,54], a speed less than 0.5 mm/s during Phase II, III and IV [58], and a speed less than 700 mm/s during Phase I and V [59].
According to [60], the workspace requirement of Phases I and V is at least 136.5 mm � 138 mm � 25 mm, allowing the robot's RCM point to move from the far side of the head to the eye in X direction, from the top of the head to the eye in Y direction, and from the nose to the eye in Z direction.
The required workspace of Phase II, III and IV is obtained by inserting a surgical instrument through an incision and then moving the instrument to ensure that the instrument tip can cover the lens while recording the yaw angle movement, the pitch angle movement and the insertion depth.This workspace is highlighted in yellow in Figure 6, where the yaw angle movement is 80°(range of instrument yaw motion), the pitch angle movement is 63°(range of instrument pitch motion), and the insertion depth is 10 mm (range of instrument translation motion).The lens-eye model depicted in Figure 6 is constructed based on average human ocular dimensions in [61].Since most forces applied during in vitro retinal manipulation are below  7.5 mN [56], the robot's stiffness is set to at least 0.75 N/mm for achieving 10 μm accuracy/precision.In summary, the design specifications for the eye surgery robot are summarised in Table 1.Note that although this robot is focused on assisting cataract surgery, the potential application of other eye surgeries such as vitreoretinal surgery is also applicable.Compared with cataract surgery, vitreoretinal surgery requires higher accuracy/ precision to suit the small size of vitreoretinal tissues (e.g., retinal vessel at 120 μm average diameter), a larger workspace to cover the vitreoretinal area, and a larger workspace to cover the vitreoretinal area [15].

| General Safety Requirements
When designing a medical device, it is imperative to adhere to the requirements outlined in relevant regulations and standards.As an eye surgery robot is classified as a medical device, it must comply with regulations such as the FDA regulations in the US, the EU Medical Device Regulation (MDR) (CE mark included), and the UK Medical Device Regulation with UK Conformity Accessed (UKCA) mark, as well as the standards such as EN ISO 14971, 13485, 60601-1-6 and 62366, and legislations like 90/ 385/EEC, 98/79/EC and 93/42/EEC.For the robot discussed in this paper, seven fundamental safety requirements related to its design have been identified based on these regulations and standards, and are summarised below: 1. Eye surgery robots must not collide with the patient's body with each other or block the microscope vision or other devices in the operating room.A typical operating room layout for manual cataract surgery is shown in Figure 7a.A surgeon sits towards the headrest, with monitoring equipment on the right side of the headrest and an assistant on the right side to clean the patient's eye and pass instruments to the surgeon when necessary.An instrument table stands next to the assistant to place instruments.An ophthalmic microscope is placed above the patient's eye to allow the surgeon to view the patient's eye and visualise the instrument tip position.Robots must avoid collisions with the patient or other equipment and should not impede the assistant's work.For robot-assisted cataract surgery, it is preferable to maintain the operating room layout in manual cataract surgery, as shown in Figure 7b.In this layout, two robots are placed on either side of the headrest to assist in operating surgery.When using robots applying master-slave technique, the surgeon is not required to be present in the operating room; instead, they can control the robots remotely using a master device.An assistant can be seated beside the headrest to facilitate tasks such as cleaning the patient's eye, manually changing instruments, and, if necessary, manually adjusting the position of the robots.The instrument table is positioned to the left of the headrest for easy access by the assistant.To prevent collisions with the ophthalmic microscope, the headrest, monitoring equipment, instrument table, and robots should be designed to be as compact as possible.
2. The robot must have high accuracy/precision (10 μm).One of the primary advantages of robot-assisted surgery over manual procedures is its high level of accuracy and precision.The robot's accuracy and precision are primarily attributed to its precisely manufactured components, meticulous mechanical designs and assemblies, accurate
Excessive drift can occur during eye surgery due to unexpected forces, such as unintentional interactions with the robot.To mitigate this issue, the robot must possess sufficient stiffness to ensure both safety and accuracy/precision.The required stiffness for the robot is set at a minimum of 0.75 N/mm, as detailed in Section 2.1.2.
4. The robot must be backdrivable.Good backdrivability is essential to ensure safety during robotic surgery, enabling the robot to be manually moved in the event of power loss.Additionally, Perret and Vercruysse [62] have highlighted several other advantages of backdrivable medical applications, including improved force control, reduced system complexity, and increased dependability.Achieving good backdrivability in the instrument manipulator requires careful consideration of both the motor and transmission systems.For eye surgery robots, DC motors are commonly chosen due to their favourable backdrivability.Regarding transmission systems, options such as gearing, leadscrew/ballscrew mechanisms, and cable/wire/tendon driving systems are commonly employed for their backdrivable characteristics [63][64][65][66][67].
5. The robot must be gravity balanced.
In backdrivable robotic systems, gravitational forces can cause the robot to deviate from its intended position.To prevent this, the gravity of the eye surgery robot must be effectively balanced.Three common methods are employed to achieve this balance, including one active and two passive approaches [68][69][70][71].The active method involves using auxiliary actuators to apply force or torque to counteract gravity.One passive method entails adding a counterweight to align the mass centre with the pivot point.The other passive method involves utilising the energy stored in springs to counteract gravitational effects.However, the active method is ineffective in the event of power loss.Adding a counterweight increases the overall weight and inertia of the system.Employing springs for gravity balance necessitates the use of zero-free length springs [70,71].The term 'zero-free length spring' refers to a coil spring designed to exert zero force if it were compressed to zero length, but a coil spring cannot physically contract to zero length [70].The modification of springs to achieve zero-free length and their placement to effectively balance the robot's gravity pose significant challenges.
6.The robot must use biocompatible materials.The use of biocompatible materials is necessary for all medical devices intended for insertion into the human body.Commonly used metals include stainless steel AISI 420, AISI 316, aluminium alloy, and titanium alloy, while plastics such as PPSU, POM, and UHMWPE are suitable.
The selection of the most appropriate material depends on specific requirements such as density, elastic modulus, and tensile strength.

The robot must use sterilisable components.
For an eye surgery robot, all the components exposed to the operating room air should be sterilised.For components that cannot be sterilised, sterile drapes should be used to cover them.joints C i , D i and E i intersecting at one point, that is, the RCM point O RCM , as shown in Figure 8a.Joints R E1 and R E2 are coincident with the surgical instrument axis.As shown in Figure 8b, the three actuated joints on the base (P A1 , P A2 and R A1 ) provide 3T motions to position the RCM point (according to the kinematic analysis shown in Section 2.2.2), and the four actuated joints (R C1 , R C2 , R E1 and P E2 ) manipulate the instrument to execute 3R1T DOF motions at O RCM in MIS fashion.
Figure 9 shows the procedure for robot-assisted cataract surgery employing the new 7-DOF parallel RCM mechanism (see Video S1).In Phase I 'approaching' or Phase V 'homing', the robot provides 3T DOFs to approach the patient's eye and locate the RCM point at the incision point or move away from the patient to the home position, as shown in Figure 9a,b.In Phase II 'insertion' or Phase IV 'extraction', the robot has 1R1T DOFs for rolling the instrument to adjust its posture and insert the instrument into the patient's eye through the located incision, or extract the instrument from the patient's eye, as shown in Figure 9d.In Phase III 'manipulation', the robot top section provides 3R1T DOF motions (2R DOFs for orientating the instrument þ 1R1T DOFs for instrument rolling and translation motions, as shown in Figure 9c,d) to manipulate the instrument in MIS fashion, completing the surgical tasks.In the 'instrument change' phase, the instruments held by the mechanism can be manually changed.

| Kinematic Modelling
The frames are assigned as depicted in Figure 10b, where the zcoordinate axis of each frame at the joint points outward along the corresponding joint.For the frames {O A1 }, {O A2 }, {O B1 } and {O B2 } in the bottom section, the y-coordinate axis of each frame is along the linkage that links the next joint.In the top section, the y-coordinate axis of the frame {O C1 } is in the plane O RCM C The position of the RCM point P ORCM (x ORCM , y ORCM , z ORCM ) can be calculated by Equations ( 3) and ( 4), By solving Equations ( 1)-( 4), the forward kinematic model is obtained in Equation ( 5) (refer to Appendix A for more details).
The results clearly indicate that the position of the RCM point is determined by the three actuator displacements on the base, that is, x ORCM solely controlled by l 2 , z ORCM solely controlled by l 1 , and y ORCM controlled by combined l 1 , l 2 and θ A1 .As a result, these three translational motions are partially decoupled.
For the 3R1T motions at the RCM point to position and orient the instrument, the x, y and z coordinates of the instrument tip (x Tip , y Tip , z Tip ) are controlled by l 1 , l 2 , θ A1 , θ C1 , θ C2 and l T .The instrument rolling posture θ R is solely controlled by θ E2 from R E2 .When P A1 , P A2 and R A1 are fixed, thereby fixing the position of O RCM , the x, y and z coordinates of the instrument tip are controlled by θ C1 from R C1 , θ C2 from R C2 and l T from P E2 .Thus, the 3R1T motions in the top section are partially decoupled from the 3T motions in the bottom section.
Moreover, the forward kinematics reveal two solutions as shown in Figure 11 (refer to Appendix A for more details).One solution, with ψ larger than 180°, is termed the Upconfiguration, while the other, with ψ smaller than 180°, is termed the Down-configuration.Here, ψ is the angle from linkage E 2 D 2 to E 1 D 1 anticlockwise.Actuation joints It's worth noting that when the seven input parameters are specified, the bottom section exhibits two possible postures, as depicted in Figure 12.For the two postures of the linkages A 2 B 2 and B 2 C 2 , one towards the inner side of the mechanism with θ B2 smaller than 180°, while the other towards the outer side with θ B2 larger than 180°.When θ B2 = 180°, the mechanism loses the DOF to control y ORCM , leading to singularity.Hence, in this study, to avoid the singularity and possible collisions with other linkages, θ B2 should be kept larger than 180°, favouring the posture towards the outer side of the mechanism.
There are eight solutions in inverse kinematics, with two solutions each for θ A1 , θ C1 and θ C2 , as depicted in Figure 13.For each of these angles, one solution towards the outer side of the mechanism, while the other towards the inner side of the mechanism.Considering that the linkages towards the inner side of the mechanism may cause collisions with other linkages, the solutions towards the outer side are selected in this study by consistently selecting the smaller solution of θ A1 , the larger solution of θ C1 and the smaller solution of θ C2 .

| Singularity and Workspace Analyses
The distinction between the two solutions of forward kinematics, namely the Up-configuration and Down-configuration (illustrated in Figure 11), lies in the top section of the new 7-DOF parallel RCM mechanism.In order to find which configuration is more suitable for robot-assisted eye surgery, the  singularity analysis was conducted and six types of singularities of the 2-DOF 5-bar spherical parallel mechanism were found [73], and used to draw the boundaries of the workspace.Consequently, the Down-configuration is selected for the proposed mechanism.

| Dimension Optimisation
This section focuses on determining the linkage dimensions of the proposed robot through task-oriented dimension optimisation, aligning with the workspace requirement outlined in Section 2.1.2.The optimisation process involves fine-tuning parameters such as α 0 , α 1 and α 2 .Additionally, it entails designing and analysing the layout of two proposed robots, ensuring they coordinate effectively to perform eye surgery operations without colliding with each other or the patient's head.Moreover, measures are taken to prevent obstruction of the ophthalmic microscope's vision.Key parameters such as r, d 1 and d 2 are carefully determined within this framework.
2.2.4.1 | Task-Oriented Dimension Optimisation (α 0 , α 1 and α 2 ).Dimension optimisation aims to refine the physical geometry parameters of the linkages, joints, base, and moving platform to enhance workspace performance.This section focuses on optimising three angle parameters, namely α 0 , α 1 and α 2 , while adhering to a specified constraint and objective.This constraint dictates that the optimised robot must align with the prescribed workspace requirements outlined in Section 2.1.2.
Meanwhile, the objective is to maximise the robot's dexterity within the workspace.
Taking into account the determined workspace outlined in Figure 6, a total of 1453 points are uniformly distributed within a lens model to represent the prescribed workspace, as depicted in Figure 16.This lens model is constructed based on the average dimensions of the human eye, as elucidated by Charles and Brown [61], and positioned within the task frame {O Task } defined in Figure 5.To facilitate analysis, the lens model is segmented into layers along the Z-direction, with each layer spaced at 0.5 mm intervals, as illustrated in Figure 16c,d.Within each layer, points are uniformly distributed based on X and Y coordinates, with a step size of 0.5 mm in both directions, as demonstrated in Figure 16b.While decreasing the distances between adjacent layers and points could yield a denser dataset, it would entail a longer traversal time across these points.
The robot must have the capability to access all 1453 designated points.The Condition Index C I (J) is employed to ascertain the reachability of each point.Furthermore, the Generalised Condition Index (GCI), as introduced by Gosselin and Angeles [74], serves to assess the robot's dexterity within its operational domain, as delineated in Equation (A20) provided in the Appendix A.
Constraint: C I (J) > C I (J) min for each of the 1435 points (the prescribed workspace).
The process for task-oriented dimension optimisation is delineated in Figure 17.Initially, for each designated point (X(i), Y(i), Z(i)) within the specified workspace, the Jacobian matrix is computed through inverse kinematics in the robot frame {O RCM }, as illustrated in Figure 18.Herein, the joint axes C 1 and C 2 are oriented horizontally, the robot's RCM point coincides with the incision point, and the instrument tip resides at the designated workspace point (x RCM = 0, y RCM = 0, z RCM = 0, x Tip = sin(α 0 /2)Y(i) − cos(α 0 /2)X(i), y Tip = Z(i), z Tip = −sin(α 0 /2) Y(i) − cos(α 0 /2)X(i)).To satisfy the constraint, the Jacobian matrix at each point must be solvable with a condition index greater than C I (J) min .If, at any point, the Jacobian matrix proves unsolvable or the condition index falls below C I (J) min , the current α 0 , α 1 and α 2 values are deemed unsatisfactory.Once the Jacobian matrix is solvable, and the condition index exceeds C I (J) min for all 1453 points, the current set is deemed viable.
Given the proximity of α 0 in the optimisation findings to 90°, and recognising that an α 0 of 90°aligns the robot base axes OA 1 and OA 2 orthogonally, simplifying the alignment and installation of the robot base, the parameter set of α 0 = 90°, α 1 = 63°a nd α 2 = 69°undergoes testing using the process outlined in Figure 17.The results affirm the qualification of these parameter values yielding a GCI of 0.445272, merely 0.35% smaller than the optimised results at 0.446847.Consequently, the dimensions of α 0 = 90°, α 1 = 63°and α 2 = 69°are employed in this study, as delineated in Table 4.
2.2.4.2 | Dimension Optimisation (r, d 1 and d 2 ).In this section, three linkage dimensions, r, d 1 and d 2 , are refined through the design and analysis of the layout of the two proposed robots.These robots are coordinated to perform eye surgical operations while mitigating the risk of three potential collision types: � Collision between the robot and the patient's head.
� Collision between the two robots.

� Collision between the robot and the vision pathway of an ophthalmic microscope.
The top sections of the two proposed robots are depicted in Figure 19, with their respective RCM points positioned on an eye model based on average human ocular dimensions [61].
In the configuration for the robot positioned near the ear, joint axes C 1 and C 2 are horizontally aligned, aligning the robot's base axes OA 1 and OA 2 horizontally as well.This arrangement facilitates control over both horizontal and vertical movements of the RCM point.Conversely, the robot positioned near the nose is set at a 30°angle from the horizontal plane to circumvent the nose.A minimum clearance of 5 mm between the instrument and the nose is maintained.
Various values of r are evaluated.Compared to the ear-side robot, the nose-side robot is situated closer to the patient's head.To prevent collisions with the patient's head, r is initially set at 70 mm, ensuring a 5 mm distance between joint C 1 and the nose.
In the bottom sections of the robots, as illustrated in Figure 19, the distance between the base and the eye measures 110 mm, while the maximum vertical distance between the eye and the nose is 25 mm.Thus, the combined lengths of d 1 and d 2 must exceed 135 mm.Therefore, both d 1 and d 2 are set to 70 mm in this scenario.Through simulations, it has been observed that the linkages of the nose side robot may obstruct the microscope's field of vision, the two robots may encounter collisions with each other, and the joints B 1 /B 2 may come into contact with the patient's head, as highlighted by the red circle areas in Figure 19.To mitigate the obstruction of the microscope's vision, the angles α 1 and α 2 for the nose side robot are optimised to 59°and 63°, respectively.Utilising the methodology depicted in Figure 19, the parameter values α 0 = 90°, α 1 = 59°and α 2 = 63°are determined, ensuring compliance with the prescribed workspace requisite for robotassisted cataract surgery, with a calculated Generalised Condition Index (GCI) of 0.442.In order to prevent collisions between the two robots, the value of r for the nose side robot is increased to 90 mm, thereby ensuring that the minimum distance between the two robots exceeds 7 mm.To avert collision incidents between the robot and the patient's head, adjustments are made to linkages B 1 C 1 and B 2 C 2 , introducing a 50 mm offset between the two joints, as illustrated in Figure 19.This design modification effectively positions the joints B 1 and B 2 at a distance of 50 mm from the patient, effectively eliminating the risk of collisions.
The optimised dimensions are shown in Table 5.The optimised results of the CAD models of a human head, an ophthalmic microscope, and two surgical robots with optimised dimensions are shown in Figure 19. Figure 20 shows the robot CAD model with detailed design and an exploded view of the robot top section.The robot top section comprises a spherical linkage assembly, two orientational motor assemblies and an instrument assembly with 1R1T motion.Additionally, a gravity balance assembly is incorporated to counterbalance the weight of the robot top section.
Within the spherical linkage assembly, four spherical linkages are interconnected using bolts, caps, bearings, washers, nuts, and a bush.Bearings within the joint connecting linkage D 1 E 1 and linkage D 2 E 2 are housed on a bush, leaving space for surgical instruments and tool shafts to go through.To mitigate friction, the bush features an inner diameter of 4 mm, matching the 3 mm diameter of the tool shaft.Interference fits are employed between the bush and bearings, between bolts and bearings, as well as between caps and linkages, providing axial constraints to ensure stability.Similar interference fits are adopted in the orientational motor assemblies and the instrument assembly, enhancing the axial constraints.Within the 1R1T box, a guide rail (45 mm length) is installed to ensure smooth translation, while two buttons positioned on either side of the guide serve to generate stop signals upon activation, preventing the instrument from exceeding its designated range of motion.The bottom of the 1R1T motor box features a slot designed to accommodate the subsequent gravity balance mechanism.The motor responsible for instrument translation motion integrates an incremental encoder (1024 CPT) and is connected to an NSK ballscrew measuring 45 mm in length and 6 mm in diameter and featuring a 2 mm lead via a coupling.The Maxon DCX 14L motor, a brushed DC motor, is characterised by its back-drivability, compact size (14 mm diameter), and lightweight construction (26 g).To meet sterilisation requirements in the operating room, sterile drapes can be utilised to cover the motor.
In the instrument rolling motion part, the nut link is affixed to the NSH ballscrew nut using M2 screws and secured to the guide block via two hex set screws.A shaft coupling establishes the connection between the motor shaft and the tool shaft.One end of the tool shaft features M8 threads, accommodating an RS PRO one-piece mini chuck to grasp the instrument securely.
The motor responsible for instrument rolling motion integrates a planetary gearbox (ratio 103:1) and an incremental encoder (1024 CPT).
Regarding the gravity balance assembly, a spring is employed to counterbalance the robot's weight, following a geometric approach outlined by [75].To achieve balance in a parallel spherical mechanism, each linkage requires a zero-free-length spring connected to a point directly above its revolute joint axis.In the context of a parallel spherical RCM mechanism, a point above the RCM point serves as a suitable connection point for all the springs, as the RCM point coincides with the intersection of all revolute joint axes.In this approach, one end of the zero-free length spring should align with the mass centre of the linkage, while the other end should be positioned directly above the linkage's revolute joint.The stiffness k of the spring is determined through Equation ( 7), where m, g and h represent the mass of the corresponding linkage, the gravitational acceleration, and the height from the corresponding revolute joint to the fixed spring end, respectively.
The connection point for the four springs is established 200 mm above the RCM point.Utilising Equation (7) and estimating the weight of each component, a zero-free-length spring with a stiffness of 23.6 N/m is selected to link the 1R1T motor box, aiming to balance an anticipated weight of 482.3 g.However, commercially available extension springs with a stiffness of approximately 23.6 N/m typically cannot support a maximum load of 4.73 N (equivalent to the gravity of a 482.3 g weight).
Consequently, an RS PRO stainless steel extension spring with a stiffness of 48 N/m and a maximum load capacity of 6.838 N is chosen.Though the stiffness exceeds 23.6 N/m, proper adjustment of the spring tension can ensure balanced support for the robot's weight.
In the gravity balance assembly, four brackets and a crossbar are affixed to establish a point 200 mm above the RCM point, as illustrated in Figure 20.The crossbar features a curved design to prevent obstruction of the ophthalmic microscope's field of view.Cable ties are employed to secure the spring to the hole on the crossbar and the slot on the 1R1T motor box.These cable ties facilitate adjustment of the spring tension as needed.
Figure 21 displays the assembled prototype alongside a dummy human head model.The weight of the robot prototype's top section totals 968.1 g, inclusive of the four Maxon motors.All bespoke parts are fabricated from biocompatible materials.Stainless steel grade 316 is chosen for the prototype's linkages and shafts to ensure high stiffness.Conversely, the 1R1T box and motor supports are crafted from 6082T6 aluminium alloy to prioritise lightweight construction and cost-effectiveness in manufacturing.

| Control Strategy and Control System
Development 2.3.2.1 | Closed-Loop Control. Figure 22 illustrates the closed-loop control mechanism governing the 3R1T motion of the robot.Initially, the desired position and orientation of the instrument, manipulated by the robot, are determined through trajectory planning by seven parameters (x Tip , y Tip , z Tip , x ORCM , y ORCM , z ORCM , θ R ).Once the RCM point is precisely positioned at the incision, the robot's bottom section is immobilised.Subsequently, employing inverse kinematics, the displacements of the four motors, namely, θ C1 , θ C2 , θ E2 and l T , are calculated.These input values are then processed in accordance with the transmission system.Following this, the four motors situated in the top section are individually actuated by four Maxon controllers, employing position PID control (provided by the Maxon controllers), thereby enabling the realisation of 3R1T motions at the RCM point.To enhance precision and mitigate backlash in the gearboxes, position feedback control is implemented for the two motors governing instrument yaw and pitch motions, utilising two absolute encoders to measure θ C1 and θ C2 .The hardware configuration of the prototype control system is available in Appendix B. the selected set of four motors can supply adequate torque throughout the robot's operation.PID control is implemented in the four Maxon DC motors to mitigate jerk.In this investigation, trajectory planning entails the uniform allocation of path points along linear and circular trajectories traversed by the instrument tip.
Figure 23 illustrates the uniformly allocated path points in trajectory planning within the eye frame for surgical operations.The distances between adjacent points are consistent and denoted by l Traj , set at 1 mm.The time interval between consecutive points, denoted as t Traj , is fixed at 3 s to ensure a speed not exceeding 0.5 mm/s, which aligns with the maximum permissible speed during retinal surgery [58].

| Robot Instrument Tip Measurement
There are three main challenges in accurately measuring the position of the robot instrument tip: (1) Ensuring measurement without direct contact, as any contact could perturb the instrument's position; (2) Meeting stringent accuracy and precision requirements, that is, at least 10 μm level; (3) Acquiring 3D coordinates of the instrument tip simultaneously.
To address these challenges, a camera was employed to measure the instrument tip position without physical contact, offering resolution at the 10-μm level.A grid paper sheet, featuring 1 mm � 1 mm grids, was positioned near the instrument tip, approximately 1 mm away, aiding in aligning the camera with the robot and facilitating focus on the instrument tip.Utilising an instrument with a straight tip, videos of the instrument's motion were captured by a camera with a resolution of 1920 � 1080 pixels.By adjusting the camera's view to encompass a 19 mm � 10 mm area on the grid paper, each pixel in the videos corresponded to 10 μm � 10 μm.However, due to equipment limitations, the resolution achieved (10 μm � 10 μm) fell short of one-tenth of the desired accuracy/precision.Moreover, the camera employed in our experiment captures only 2D data, so the instrument tip was controlled to move within the corresponding 2D plane.
After recording, images of the instrument tip were extracted from the video footage.As depicted in Figure 24, postprocessing involved contrast adjustments and enlargement, facilitating the facile identification and measurement of the instrument tip position in each image.

| Accuracy and Precision Test
In surgical robotics, accuracy refers to the degree of conformity between the actual test position and the intended target, while precision pertains to the consistency among all actual test results.This study evaluates the accuracy and precision of the instrument's 3R1T motion driven by the newly proposed robot mechanism.Note that the instrument's rolling motion is not carried out due to a malfunction encoder in the corresponding motor.
The schematic of the accuracy and precision testing procedure is delineated in Figure 25.Two Maxon ECXSP 16L brushless DC   approximately 1 mm below the instrument tip to ensure the well focus and coverage by the camera on top of the sheet, as shown in Figure 26a-c.For instrument pitch motion, a vertical grid paper sheet was positioned approximately 1 mm beside the instrument tip using the same approach, as shown in Figure 26d,e.The robot bottom section, which is used for positioning the RCM point, was locked during the experiment.
Figure 27 depicts the predetermined points representing the target positions of the instrument tip to be attained during the accuracy and precision tests.The predefined points for the instrument yaw motion test are illustrated in Figure 27a.In accordance with the trajectory planning, five points sharing the same Z coordinate were predetermined in the CAD models, lying on the trajectory of instrument yaw motion.The angle between any two adjacent points relative to the RCM point was set at 8°.To facilitate measurement, the distance between point 1 and the RCM point was set at 9 mm (l T = 29 mm), matching the lens size and inducing a noticeable position change.Employing inverse kinematics, the required inputs θ C1 and θ C2 for the two motors were calculated, followed by controlling the motors using the closed-loop control.Each run involved the instrument tip following the sequence of 1-2-3-2-1-4-5-4-1 to execute the instrument yaw motion.
The predefined points for the instrument pitch motion test are presented in Figure 27b.Similarly, four points within the X-Z plane were predefined.The angles between two adjacent points relative to the RCM point were set at 4°.The angle between the instrument axis at point 1 and its initial operation configuration was 10°.Following inverse kinematic calculations, the required inputs θ C1 and θ C2 were determined, and the two motors were controlled using the closed-loop control.Each run involved the instrument tip following the sequence of 1-6-7-8-7-6-1 to execute the instrument pitch motion.
The predefined points for the instrument translation motion test are depicted in Figure 27c.Similarly, six points along the X-direction were predetermined, with the distances between two adjacent points set at 1 mm.Employing inverse kinematics, the required motor input l T was computed, and the motor was then controlled using the closed-loop control.In each run, the instrument tip followed the sequence of 9-10-11-12-13-14-13-12-11-10-9 to execute the instrument translation motion.
During these experiments, upon reaching the required inputs for each predefined point, all relevant motors stopped for 1 s to facilitate measurement before proceeding to the next point.The prototype was programmed to execute the instrument yaw/ pitch/translation motion sequences 10 times each.Consequently, a total of 80 points were measured for the instrument yaw motion, 60 points for the instrument pitch motion, and 100 points for the instrument translation motion.
For analysis purposes, images of the instrument tip at each target point were extracted from the videos.At each target point, the distances between the position of the instrument tip and the target point were measured to determine tracking errors.The average of the absolute tracking errors served as an accuracy measure at each target point, while the standard deviation of all tracking errors was utilised as a precision measure.Subsequently, for each motion, the average and standard deviation of the accuracy and precision measures across all target points were computed.The results are shown in Section 3.

| Stiffness Test
To validate whether the robot meets the stiffness requirement, the stiffness of the robot was assessed when forces were applied in the X-or Y-or Z-directions, respectively.All motors were immobilised during the stiffness test, and the robot was set to its initial operational configuration.Similar to the accuracy and precision test, the camera was utilised to capture videos of the instrument tip's motion.
Considering that the robot may employ various types of instruments and that exerting forces directly on the instrument tip could obstruct the camera's vision, forces were applied to joint E 1 .This joint facilitates the attachment of strings for applying weights in different directions.The experimental setup is illustrated in Figure 28.Throughout the experiment, the distance between the instrument tip and joint E1 remained constant at 72 mm.A weight of 100 g (equivalent to a force of 0.98 N) was exerted in the X-or Y-or Z-directions, respectively, as depicted in Figure 28a-c.The displacements of the tool tip were measured from the recorded videos.For each trial, the deformation was determined as the difference in the position of joint E 1 with and without the applied force.This deformation was then divided by 0.98 N to calculate the stiffness.

FIGURE 27 |
The predefined points in the accuracy and precision test.
Table 6 shows the experimental results of the accuracy and precision.Table 7 shows the experimental results of the stiffness.Results in Table 7 indicate that all three stiffness measures meet the 0.75 N/mm stiffness requirement, and the results in Table 6 show that the precision of instrument pitch motion complies with the 10 μm precision standard.However, the accuracy measures fall short with the best value at 21 � 10 μm, and the poorest performance recorded at 568 � 374 μm.Several factors may contribute to this low accuracy: � Calibration was not conducted on the prototype, which is essential for enhancing robot accuracy.Proper calibration procedures will be implemented to address this in the future.
� Resolution of the used camera did not meet one-tenth of the targeted accuracy/precision, potentially leading to inaccuracies in position measurement.Utilising cameras with higher resolutions or alternative equipment such as optical coherence tomography could yield more reliable experimental data.
� Joint clearances within the prototype due to internal forces pose a challenge despite the application of interference fits, bolts, and self-lock nuts.A redesign of joints will address this problem considering alternative constraint methods such as manufacturing linkages with shoulders for axial constraints or employing circlips.
� Manufacturing errors also contribute to inaccuracies.Errors may arise during the transfer and setting of part profiles as well as from material stress during fabrication.These errors can be mitigated by enhancing the fabrication and assembly processes.
� Imperfections in the gravity balance design introduce errors.Further design improvement is favoured to dislocate the heavy motor box perfectly from the end-effector to the base.
� Inaccuracies are influenced by sensors and control strategies.Alternative sensors capable of directly measuring the instrument tip position for comprehensive closed-loop feedback control and determining the robot's initial configuration, such as 3D cameras or tracking systems, should be considered.

| Conclusion
In this paper, a novel 7-degree-of-freedom (DOF)    The forward kinematics are calculated first in 3T motions phase, then in 2R motions phase, and finally in 1R1T motions phase.In 3T motions phase, the three actuated joints are P A1 , P A2 and R A1 , while the other four actuated joints are locked.This phase is to position the RCM point, aiming to calculate the position of the RCM point P ORCM (x ORCM , y ORCM , z ORCM ), using the inputs l 1 , l 2 and θ A1 .
First, by solving Equations ( 3) and ( 4), parameters θ B1 , θ A2 and θ B2 are eliminated, P ORCM is calculated by Equation (A1), In 2R phase, the two actuated joints are R C1 and R C2 , while the other actuated joints are locked.This phase changes the orientation of the instrument towards the RCM point represented by the vector E 2 O RCM , aiming to calculate the position of joint E 2 , P E2 (x E2 , y E2 , z E2 ), using the inputs l 1 , l 2 , θ A1 , θ C1 and θ C2 , referring to the kinematic analysis of 2-DOF 5-bar spherical parallel mechanism from Ouerfelli and Kumar [77].
P E1 and P E2 can be calculated by Equations (A2) and (A3), Since the distance of E 2 O RCM is r, Equation (A4) is obtained as follows: By solving Equations (A2)-(A4), and P E1 = P E2 , parameters θ D1 , θ D2 , x E2 and y E2 are eliminated, then an equation with z E2 , Equation (A5) is obtained, and z E1 has two solutions, x E2 is calculated by Equation (A8).
x E2 = x ORCM + r × In 1R1T phase, the two actuated joints are R E1 and P E2 , while the other actuated joints are locked.In the final phase, the aim is to calculate the positions and orientations of the instrument tip (x Tip , y Tip , z Tip , θ R ) using l 1 , l 2 , θ A1 , θ C1 , θ C2 , θ E2 and l T .For the instrument orientation, θ R = θ E2 .The instrument position P Tip (x Tip , y Tip , z Tip ) is calculated by Equation (A10), The instrument yaw angle θ Y and pitch angle θ P are defined by Equation (A11), In conclusion, according to Equations (A1), (A5), (A8)-(A10), the forward kinematics are summarised in Equation (5).

FIGURE 6 |
FIGURE 6 | Required workspace shown in a lens-eye model.
and IV ≤0.5 mm/s [58] Workspace I and V ≥136.5 mm � 138 mm � 25 mm [60] Workspace II, III and IV ≥80°� 63°� 10 mm Force All ≤7.5 mN [56] Stiffness All ≥0.75 N/mm actuators, high-resolution sensors, and appropriate control methods.This heightened level of accuracy enhances the safety and quality of eye surgical operations.For the robot discussed in this study, the accuracy and precision are specified to be 10 μm during Phases II, III, and IV of the robot-assisted cataract surgery procedure [53, 54].

2. 2 | 2 . 2 . 1 |
Figure8shows the schematic diagram of the proposed 7-DOF parallel RCM mechanism, which is derived from[72].It has two legs, and each leg has a PRRRRR configuration with the first three R joints A i , B i and C i in parallel, and the last three R

FIGURE 7 |
FIGURE 7 | Layout of the operating room.

Figure
Figure 10a illustrates the kinematic parameters with all angles illustrated in positive direction.The instrument features a straight tip.Regarding the robot's base frame {O-x O y O z O }, z O axis aligns with the robot base axis OA 1 , while y O axis is orthogonal to both the robot base axes OA 1 and OA 2 .Table2shows all the kinematic parameters.The input kinematic parameters include (l 1 , l 2 , θ A1 , θ C1 , θ C2 , θ E2 , l T ) for the seven actuation joints P A1 , P A2 , R A1 , R C1 , R C2 , P E2 and R E2 .Here, l 1 represents the distance

2. 2 . 2 . 2 |
Inverse Kinematics.Inverse kinematics is used to calculate the displacements of the actuators l 1 , l 2 , θ A1 , θ C1 , θ C2 , θ E2 and l T , given the output positions and orientations of the instrument tip represented by (x Tip , y Tip , z Tip , x ORCM , y ORCM , z ORCM , θ R ).The inverse kinematics is summarised in Equation (6) (refer to Appendix A for more details),

FIGURE 11 |FIGURE 12 |
FIGURE 11 | Two solutions in forward kinematics for the new parallel RCM mechanism.

FIGURE 13 |
FIGURE 13 | Solutions in inverse kinematics for the new 7-DOF parallel RCM mechanism.

Figure 14
Figure 14 illustrates the two regions of the reachable workspace for E 2 as verified by the CAD model.The dark grey area represents the Up-configuration workspace, while the light grey area represents the Down-configuration workspace.Considering the six types of singularities alongside the two workspace

Figure 15
Figure15illustrates the singularities and workspace of the 2-DOF 5-bar spherical parallel mechanism as α 0 varies.In Conditions 1, 2 and 3, the Down-configuration workspace diminishes, while the Up-configuration workspace expands when α 0 increases.In Condition 4, with α 0 reduced to 0°, the Downconfiguration workspace vanishes, yielding the largest Upconfiguration workspace without a singularity area inside.

2. 3 | 2 . 3 . 1 |
Detailed Design, Prototype and Control Detailed Design and Prototype Each of the two identical orientational motor assemblies integrates two Maxon ECXSP 16L motors, furnishing 2R DOF motion for orienting the instrument.These motors are coupled with planetary gearboxes (ratio 111:1) and incremental encoders (1024 CPT).The Maxon ECXSP 16L motor, a sterilisable brushless DC motor, boasts compact dimensions (16 mm diameter) and lightweight construction (73 g), offering backdrivability.The planetary gearbox amplifies output torque while reducing rotational displacement.A magnetic sensor and magnetic ring (Renishaw AksIM-2 absolute encoder) are positioned separately on the motor support and linkage C 1 D 1 (C 2 D 2 ) to compensate for gearbox backlash and furnish feedback on the orientation of linkage C 1 D 1 (C 2 D 2 ).Furthermore, an accelerometer affixed to linkage C 1 D 1 (C 2 D 2 ) facilitates determining the robot's initial operational configuration by aligning linkages C 1 D 1 and C 2 D 2 with the gravity direction sensed by the accelerometers.In the instrument assembly with 1R1T motion, two Maxon DCX 14L motors provide 1R1T DOF motion for translating and rolling the instrument.The instrument translation motion part is affixed to linkage C 2 D 2 via screws on the 1R1T motor box.

FIGURE 20 |
FIGURE 20 | CAD model of the robot with the detailed design.

FIGURE 22 |
FIGURE 22 | Closed-loop control of the four motors for 3R1T motion.

Figure 26 FIGURE 24 |FIGURE 25 |FIGURE 26 |
Figure26shows the experimental setup for capturing and measuring the instrument motions including yaw, translation, and pitch motions.A horizontal grid paper sheet was positioned

FIGURE 28 |
FIGURE 28 | Experimental setup for the stiffness test.

TABLE 1 |
Design specification of the eye surgery robot.

Table 2
OA 1 , l 2 represents the distance of OA 2 , and l T represents the distance of E 2 O Tip .Additionally, d 1 denotes the link length A 1 B 1 (A 2 B 2 ); and d 2 denotes the link length B 1 C 1 (B 2 C 2 ).r signifies the radius of the robot top section with O RCM at the centre.α0representstheangle of ∠A 1 OA 2 (the same as∠C 1 O RCM C 2 ), α 1 the angle of ∠D 1 O RCM C 1 (the same as ∠D 2 O RCM C 2 ), α 2 the angle of ∠E 1 O RCM D 1 (the same as ∠E 2 O RCM D 2 ), θ A1 is the angle from the y O -axis to A 1 B 1 about A 1 joint, θ Ci the angle of ∠B i C i D i about C i joint (i = 1, 2), with directions as shown in Figure10a.θE2signifies the angle from the y axis of frame {O E2 } to the y axis of frame {O Tip }, as shown in Figure10b.The output positions and orientations of the instrument tip are represented by (x Tip , y Tip , z Tip , x ORCM , y ORCM , z ORCM , θ R ), in which (x Tip , y Tip , z Tip ) are the coordinates of the instrument tip, (x ORCM , y ORCM , z ORCM ) are the coordinates of the RCM point, and θ R is the rolling angle of the instrument, which equals to θ E2 shows all the kinematic parameters.The input kinematic parameters include (l 1 , l 2 , θ A1 , θ C1 , θ C2 , θ E2 , l T ) for the seven actuation joints P A1 , P A2 , R A1 , R C1 , R C2 , P E2 and R E2 .Here, l 1 represents the distance FIGURE 8 | Schematics of the new 7-DOF parallel RCM mechanism.FIGURE 9 | Motions of the new 7-DOF parallel RCM mechanism.FIGURE 10 | Schematics for the 7-DOF parallel RCM mechanism.of Forward kinematics is used to calculate the output positions and orientations of the instrument tip represented by (x Tip , y Tip , z Tip , x ORCM , y ORCM , z ORCM , θ R ) using the input displacements of the actuators l 1 , l 2 , θ A1 , θ C1 , θ C2 , θ E2 and l T .The forward kinematic transformations from the base O to the instrument tip O Tip of the two legs are expressed in Equations (1) and (2), 1 D 1 , the y-coordinate axis of the frame {O C2 } is in the plane O RCM C 2 D 2 , the y-coordinate axis of the frame {O D1 } is in the plane O RCM D 1 E 1 , the y-coordinate axis of the frame {O D2 } is in the plane O RCM D 2 E 2 , and the y-coordinate axis of the frame {O D2 } is also in the plane O RCM D 2 E 2 .The frame {O'-x O' y O' z O' } is derived by rotating the frame {O} about y O axis in α 0 , where z O' axis coincides with the robot base axis OA 2 .The frame {O Tip } is obtained by rotating the frame {O E2 } at z-coordinate axis by an angle θ E2 , and translating along z-coordinate axis by a distance l T . 2.2.2.1 | Forward Kinematics.

TABLE 2 |
Actuation joints and corresponding input and output parameters.
However, in Condition 5, when α 0 becomes excessively large (120°), the Down-configuration workspace shrinks, leading to a reduced workspace yaw angle (60°), which falls short of the workspace requirement outlined in Section 2.1.2(80°yawangle).Therefore, among the five conditions, Condition 4 is deemed the most favourable for the Up-configuration, boasting the largest Up-configuration workspace devoid of any singularity area.Conversely, for the Down-configuration, Condition 1 emerges as the optimal choice, offering the largest Down-configuration workspace.Both Condition 4 for the Up-configuration workspace and Condition 1 for the Down-configuration workspace satisfy the workspace requirement detailed in Section 2.1.2.However, Condition 4 is unsuitable for the proposed 7-DOF parallel RCM mechanism.With α 0 set to 0°, joints C 1 and C 2 become coaxial, as do joints A 1 and A 2 , thereby reducing one DOF of the proposed mechanism and rendering the parallel support from chain A 1 B 1 C 1 and chain A 2 B 2 C 2 meaningless.
FIGURE 19|Layout of the two robots' top sections, human head, and an ophthalmic microscope.

TABLE 7 |
Results of the stiffness test.

TABLE 6 |
Results of the accuracy and precision tests.