Analytical Kinematic Model and RCM Control of a Parallel-Serial Steady-Hand Eye Robot

Abstract

Maintaining a precise remote center of motion is essential for safe and accurate tool manipulation in retinal microsurgery. However, existing numerical Jacobian identification methods for parallel manipulators often exhibit nonlinear, workspace-dependent inaccuracies and require frequent recalibration, limiting their reliability and clinical applicability. To address these challenges, this study presents an analytical kinematic framework for a hybrid parallel–serial robot designed for retinal surgery, known as the Steady-Hand Eye Robot (SHER 3.0). Closed-form solutions for forward and inverse kinematics, as well as analytical formulations of the direct and inverse Jacobians, are derived to ensure consistent motion estimation across the workspace. The kinematic performance of SHER 3.0 is analyzed to evaluate the manipulability and workspace efficiency. Building on these models, a model predictive control strategy is implemented on SHER 3.0, which maintains the remote center of motion constraint at the sclerotomy with sub-millimeter accuracy, with a root mean square error of 0.55 ± 0.12 mm, in a pilot study on a teleoperation experiment. Experimental results demonstrate the effectiveness of the proposed analytical models and control framework in enhancing robot motion stability and robustness of the controller in maintaining the remote center of motion constraints, enabling safe robot-assisted retinal microsurgery.

1. INTRODUCTION

Parallel manipulators (PMs) have attracted significant attention from both researchers and industry because of their distinct advantages over serial robot manipulators [1]. Given that their end-effector is supported by multiple kinematic chains, these systems exhibit superior structural rigidity and dynamic performance while maintaining a lower overall inertia [2,3,4]. PMs offer additional benefits such as simplified design and reduced manufacturing costs, while providing high precision and payload, even at high velocities [5]. They have found applications across a variety of fields, including pick-and-place operations [6], robotic legged landers [7], manufacturing [8, 9] and machine tools [10], and medical robotics for surgical [11,12] and rehabilitation purposes [13].

Focusing on medical robotics for ophthalmic surgeries, several parallel [14, 15, 16] or serial [17] robotic manipulators have been developed that enable manipulation of a surgical instrument inside the eye, provide micromillimeter scale positioning accuracy, and remove hand tremor while satisfying a remote center of motion (RCM) constraint at the sclerotomy to avoid damaging the eye [18, 19]. For example, Micron is a handheld micromanipulator that can filter out physiological hand tremor [20]. Its handheld version, however, lacked RCM enforcement capability because of insufficient degrees of freedom. To address this limitation, an upgraded Micron system incorporating a six-degree-of-freedom (6-DOF) piezoelectric Stewart-platform architecture was later developed [21]. Despite this advancement, Micron remains constrained to a fixed one-to-one force feedback ratio, a characteristic that may be suboptimal for the delicate force requirements of vitreoretinal surgical procedures.

The Steady-Hand Eye Robots (SHERs) [22] are 5-DOF manipulators, developed at Johns Hopkins University for retinal microsurgery applications. They come with several versions having different mechanical designs, including serial and parallel mechanisms [23]. SHER 1 employs a slider–crank mechanism to produce the instrument’s tilt motion [24]. Because this mechanism cannot mechanically enforce a remote center of motion (RCM), a virtual RCM (V-RCM) must be achieved through control and optimization strategies. Other versions of SHER, namely SHER 2.0 and SHER 2.1, are serial manipulators with a translation accuracy of $5\pm 2\mu \text{m}$ for tip position regulation independently along X, Y, and Z Cartesian coordinate axes in free space [25]. These versions employ a parallelogram six-bar mechanism, capable of enforcing RCM in the plane of tilt motion; however, general enforcement of the RCM constraint (restricting lateral out-of-plane motions around the trocar point) still requires V-RCM enforcement via optimization and a control algorithm. Previous studies developed a scleral force-based RCM enforcement using an adaptive force control algorithm and fiber Bragg grating (FBG)-based force-sensing instruments, for bimanual cooperative [26] and bimanual teleoperation [27] frameworks using SHERs 2.0 and 2.1. This method safely minimizes the scleral forces and dynamically enforces the RCM constraint, despite eye and/or head movements; however, it requires advanced FBG-based force-sensing needles, which could have limitations for clinical use. The latest version of this robot, SHER 3.0 (Fig. 1), is a hybrid robot comprising a linear parallel Delta platform with a serial roll–tilt mechanism [28]. SHER 3.0 adopts a four-bar linkage architecture that is optimized to reduce the linear travel required by the linear Delta stage when implementing a V-RCM during surgical tool tilting [29, 30].

Fig. 1:

The CAD model (left) and schematic diagram of the SHER 3.0 kinematic (right).

The linear Delta manipulator provides high translational stiffness, low moving inertia, and a compact structure, which are advantageous for precise and stable RCM-constrained motion in eye surgery. The springs in Fig. 1-a are utilized to preload the ball-joint bearings to minimize clearance. The two rotational degrees of freedom, roll and tilt (Fig. 1), reflect the natural manipulation of surgeons during retinal procedures. Axial rotation of the tool is generally not required and is thus omitted from the design. It is worth noting that SHER 3.0 does not have a mechanically enforced RCM; instead, the RCM is enforced virtually through a control algorithm. This is because SHER 3.0 was originally designed as a cooperative surgical robot, and a mechanically enforced RCM was not required. However, the development of a teleoperative control mode requires an RCM enforcement for patient safety. Therefore, this work presents a virtual RCM enforcement by developing analytical kinematic relations and a model predictive control (MPC) implementation, enabling safe tool–to–eye interaction.

Accurate position control of SHER 3.0 is important since retinal surgery operations typically require micromillimeter-scale manipulation accuracy. For example, microsurgical procedures such as subretinal injection require tool tip positioning accuracy of about $25-30\mu \text{m}$ [18]. Having an accurate kinematic model is an essential step to achieve repeatable and consistent position control accuracy; however, as opposed to serial manipulators, kinematic modeling and calibration of parallel mechanisms are challenging tasks [31]. A previous study employed a numerical method for kinematic calibration of SHER 3.0 using a Bernstein polynomial. The root mean square error (RMSE) of the robot position regulation about the X, Y, and Z axes in the tool tip coordinate $\left\{t\right\}$ (Fig. 1) in free space varied from $23\pm 1\mu \text{m}$ to $121\pm 80\mu \text{m}$ at different positions of the robot workspace (at $Z=-20,0,20\text{mm}$ in the frame $\left\{p\right\}$ of Fig. 1) [32]. This shows that the accuracy of numerical kinematic calibration methods may not remain consistent across different locations within the SHER 3.0 workspace, due to the nonlinear kinematic nature of its parallel mechanism. To reduce this variability in the robot positioning accuracy, the authors in [32] repeated this calibration procedure at multiple regions of interest (ROI), achieving an RMSE of less than $20\mu \text{m}$ with small variations at different locations of the robot workspace. This method requires repeated calibration at multiple ROIs, which is a challenging and time-consuming procedure.

Depending on the intended clinical application, a variety of surgical instruments can be mounted on the robot end-effector. For instance, microneedle-based tools have been utilized for subretinal injection [33, 34] and for retinal vein cannulation (RVC), enabling targeted cannulation of retinal veins and delivery of thrombolytic agents to dissolve intravascular clots and treat retinal vein occlusion (RVO) [16, 35]. In contrast, procedures such as epiretinal membrane (ERM) peeling require forceps-type instruments that allow surgeons to grasp and delaminate the membrane along a controlled helical trajectory. Achieving an optimal and controllable approach angle to the retinal surface using conventional straight gripper tools is challenging, as their limited dexterity restricts orientation control and hinders smooth membrane delamination, potentially increasing retinal distortion and collateral tissue damage. To address these limitations, several miniature continuum and snake-like robotic devices [36, 37] have been developed to provide wrist-like articulation at the tool tip, thereby enabling more dexterous manipulation and more efficient ERM peeling; however, accurate control of these robots is a challenging problem due to mechanical nonlinearities such as hysteresis [38, 39], coupling effects [40], challenges in shape reconstruction [41, 42], and several other limitations [43, 44].

To resolve these limitations and gaps in science, in this study, we developed analytical models for forward and inverse kinematics and the Jacobian of the parallel robot for retinal microsurgery, the SHER 3.0, and designed a model predictive control algorithm based on these models that accurately satisfies an RCM constraint at the eye sclerotomy. Moreover, we integrated the Improved Integrated Robotic Intraocular Snake (I²RIS) [45] to the end-effector of SHER 3.0 to achieve a higher level of dexterity and controlled the combined robotic systems to provide an optimum approach angle needed for the ERM peeling procedures. The target operational workspace is defined by the patient’s eyeball, which has an approximate diameter of 24 mm, and serves as the primary region of interest for evaluation.

The contributions of this work are listed as follows:

1. We developed analytical models for the forward and inverse kinematics as well as forward and inverse Jacobians, for a parallel-serial robot manipulator, SHER 3.0, for retinal microsurgery applications. We evaluated the kinematics performance of the SHER 3.0, using several kinematic-related metrics such as reachable and dexterous workspace analysis and manipulability index.

2. We implemented an MPC algorithm, using the proposed analytical kinematics and Jacobian models to enforce an RCM constraint that satisfies the clinical precision needed for retinal microsurgeries. We developed a teleoperation framework to evaluate the accuracy and performance of the kinematic models and the MPC controller for SHER 3.0 (integrated with a straight gripper tool attached to its end-effector) in maintaining the RCM constraint in an artificial eye phantom with a trocar constraint.

3. We increased the dexterity of SHER 3.0 by integrating the I²RIS robot at its end effector, allowing the surgical tool to achieve a more optimal approach angle to the retinal surface (eye phantom) compared to the straight gripper tool.

The remaining sections of this paper are as follows. Section 2 explains the methodologies in robot kinematics and control. Section 3 presents the experimental setup and procedure. Experimental results are presented in Section 4 and discussed in Section 5. Section 6 concludes the paper. For clarity, the main symbols used throughout this paper are summarized in Table 1.

Table 1:

Table of notations.

Symbol	Description
{b}	Base frame, fixed at the center of the base plate, with $xy$-plane parallel to the plate
{p}	Platform frame, centered at the Delta platform plane, defined by the three platform joints
{r}	Roll frame, with $x$-axis along the roll axis and origin located on the $z$-axis of $\left\{p\right\}$
{a}	Frame at joint A, with $x$-axis aligned with that of $\left\{r\right\}$ and $y$-axis along the joint axis
{t}	Tool frame, with $y$-axis aligned with that of $\left\{\text{a}\right\}$ and $z$-axis along the surgical tool shaft
$r_b$	Radius of the circumscribed circle of the three base joints in the $xy$-plane of $\left\{b\right\}$
$r_p$	Radius of the circumscribed circle of the three Delta platform joints in the $xy$-plane of $\left\{p\right\}$
${\theta }_{b_i}$	Angles in $\left\{b\right\}$ from $X_b$ to the projected base joint in the $X_b{Y}_b$ plane, where $i=\mathrm{1,2},3$
${\theta }_{p_i}$	Angles in $\left\{p\right\}$ from $X_p$ to the platform joint in the $X_p{Y}_p$ plane, where $i=\mathrm{1,2},3$
${\text{l}}_i$	Link vector of the $i$-th limb between universal joints, where $i=\mathrm{1,2},3$
${\mathbf{L}}_i$	Prismatic joint displacement vector of the $i$-th limb, along the $z$-axis in $\left\{b\right\}$, where $i=\mathrm{1,2},3$
r	Position vector of moving platform frame $\left\{p\right\}$ in base frame $\left\{b\right\}$
Q, R	Points at joints of the slider-crank mechanism, defined in the planar mechanism
A, B, C, D	Points at joints of the four-bar linkage, defined in the planar mechanism
P	Arbitrary point defined along the surgical tool shaft
I	Point defined as the intersection of link $AD$ and $BC$
J	Point defined as the intersection of link $AQ$ and the line through $\mathbf{R}$ along the $z$-axis of $\left\{r\right\}$
$s$	Stroke length of the linear actuator controlling the slider-crank mechanism
$\psi$	Angle between the $z$-axes of frames $\left\{p\right\}$ and $\left\{r\right\}$
$\theta$	Angle between the $x$-axes of frames $\left\{a\right\}$ and $\left\{t\right\}$
$\rho$	Angle between link $RQ$ and the $x$-axis of frame $\left\{r\right\}$
$\delta$	Angle between links $BD$ and $CD$
${\alpha }_1$	Angle between links $AB$ and $AD$
${\alpha }_3$	Angle between links $AR$ and $AQ$
${\alpha }_4$	Angle between link $AR$ and the negative $x$-axis of frame $\left\{a\right\}$
${\beta }_1$	Angle between links $AB$ and $BD$
${\beta }_2$	Angle between links $BC$ and $BD$
${\theta }_1$	Angle between link $AD$ and the $x$-axis of frame $\left\{a\right\}$
${\theta }_2$	Angle between link $CD$ and the $x$-axis of frame $\left\{a\right\}$
${\theta }_3$	Angle between link $BC$ and the $x$-axis of frame $\left\{a\right\}$
${\varphi }_1$	Angle between link $AB$ and the $x$-axis of frame $\left\{a\right\}$
${\varphi }_2$	Angle between links $AD$ and $AQ$
${\varphi }_3$	Angle between links $CD$ and $DP$
${\varphi }_4$	Angle between link $DP$ and the $z$-axis of frame $\left\{t\right\}$

2. MATERIALS AND METHODS

2.1. Forward Kinematics

The proposed eye surgical robot can be decomposed into two serially connected subsystems: a linear parallel delta mechanism and a roll–tilt mechanism mounted on the moving platform. This decomposition allows the forward kinematics to be derived for each subsystem separately, and the complete forward kinematics of the robot is then obtained by combining these two results.

2.1.1. Delta Platform Forward Kinematics

The derivation for the parallel delta mechanism kinematics follows the approach described in [46]. Fig. 2-a shows the linear Delta mechanism, which consists of a base platform, a moving platform, and three limbs, each formed by two parallel rods connected by spherical joints. This parallelogram configuration constrains the moving platform to pure translational motion by maintaining link parallelism and suppressing torsional deformation. For kinematic modeling, Fig. 2-b shows a simplified mechanism with universal joints, as described in [32]. The base and platform joint circle radii are denoted by $r_b$ and $r_p$. The fixed base frame $\left\{b\right\}$ is defined with its $Z_b$-axis aligned with the common direction of the three prismatic joints of the linear Delta platform. The projections of the three universal joint centers onto the $X_b{Y}_b$ plane form an equilateral triangle centered at the origin of the base frame. The configuration in Fig. 2-b is for illustrative purposes only. In the implemented design, the positive $X_b$-axis is aligned with the first projected joint centers, and the planar angles ${\theta }_{b_i}\in \left\{0,\frac{2\pi }{3},\frac{4\pi }{3}\right\}$, where $i=\mathrm{1,2},3$, are measured counterclockwise from the $X_b$-axis to each projected joint center. The moving platform frame $\left\{p\right\}$ is defined with axes parallel to $\left\{b\right\}$, and the joint centers are defined in the $X_p{Y}_p$ plane with angles ${\theta }_{p_i}\in \left\{0,\frac{2\pi }{3},\frac{4\pi }{3}\right\}$ relative to the $X_p$ axis. The link length between universal joints is $l$, with the corresponding link vector ${\mathbf{l}}_i={\left[x_{l_i},y_{l_i},z_{l_i}\right]}^T$, and the prismatic joint vector ${\mathbf{L}}_i={\left[\mathrm{0,0},q_i\right]}^T$. The moving platform position vector is denoted by $\mathbf{r}={\left[r_x,r_y,r_z\right]}^T$. With the above parameters, the $i$-th vector loop equation is:

$${\mathbf{l}}_i=-{\mathbf{L}}_i-{\mathbf{r}}_{bi}+\mathbf{r}+{\mathbf{r}}_{pi}$$ (1)

where

$${\mathbf{r}}_{bi}={\left[r_b\text{cos}{\theta }_{bi},r_b\text{sin}{\theta }_{bi},0\right]}^T$$

$${\mathbf{r}}_{pi}={\left[r_p\text{cos}{\theta }_{pi},r_p\text{sin}{\theta }_{pi},0\right]}^T$$

Fig. 2:

linear Delta platform: (a) 3D view of the mechanical structure; (b) simplified kinematic model

Let ${\mathbf{e}}_i={\mathbf{L}}_i+{\mathbf{r}}_{bi}-{\mathbf{r}}_{pi}$ and rewrite Eq. (1) as

$${\mathbf{l}}_i=\mathbf{r}-{\mathbf{e}}_i$$ (2)

Taking the dot product of Eq. (2) with itself on both sides:

$${\mathbf{l}}_i^T{\mathbf{l}}_i={\left(\mathbf{r}-{\mathbf{e}}_i\right)}^T\left(\mathbf{r}-{\mathbf{e}}_i\right)$$ (3)

which can be expanded into the scalar form:

$$l^2={\mathbf{r}}^T\mathbf{r}-2{\mathbf{r}}^T{\mathbf{e}}_i+{\mathbf{e}}_i^T{\mathbf{e}}_i$$ (4)

from which, three scalar equations can be written for $i=1,2,3$ as

$$\begin{gathered} l^2={\mathbf{r}}^T\mathbf{r}-2{\mathbf{r}}^T{\mathbf{e}}_1+{\mathbf{e}}_1^T{\mathbf{e}}_1 \\ {l}^2={\mathbf{r}}^T\mathbf{r}-2{\mathbf{r}}^T{\mathbf{e}}_2+{\mathbf{e}}_2^T{\mathbf{e}}_2 \\ {l}^2={\mathbf{r}}^T\mathbf{r}-2{\mathbf{r}}^T{\mathbf{e}}_3+{\mathbf{e}}_3^T{\mathbf{e}}_3 \end{gathered}$$ (5)

Pairwise subtraction of these equations and combining like terms:

$$\begin{gathered} 0=-2{\mathbf{r}}^T\left({\mathbf{e}}_2-{\mathbf{e}}_1\right)+\left({\mathbf{e}}_2^T{\mathbf{e}}_2-{\mathbf{e}}_1^T{\mathbf{e}}_1\right) \\ 0=-2{\mathbf{r}}^T\left({\mathbf{e}}_3-{\mathbf{e}}_1\right)+\left({\mathbf{e}}_3^T{\mathbf{e}}_3-{\mathbf{e}}_1^T{\mathbf{e}}_1\right) \end{gathered}$$ (6)

Let $h_1=\frac{{\mathbf{e}}_2^T{\mathbf{e}}_2-{\mathbf{e}}_1^T{\mathbf{e}}_1}{2},h_2=\frac{{\mathbf{e}}_3^T{\mathbf{e}}_3-{\mathbf{e}}_1^T{\mathbf{e}}_1}{2}$

Equation (6) can be written as:

$$\begin{gathered} {\mathbf{r}}^T\left({\mathbf{e}}_2-{\mathbf{e}}_1\right)-h_1=0 \\ {\mathbf{r}}^T\left({\mathbf{e}}_3-{\mathbf{e}}_1\right)-h_2=0 \end{gathered}$$ (7)

Since all of the elements in $\left({\mathbf{e}}_2-{\mathbf{e}}_1\right)$ and $\left({\mathbf{e}}_3-{\mathbf{e}}_1\right)$ are constants, let

$${\mathbf{e}}_2-{\mathbf{e}}_1={\left[x_{21},y_{21},z_{21}\right]}^T$$

$${\mathbf{e}}_3-{\mathbf{e}}_1={\left[x_{31},y_{31},z_{31}\right]}^T$$

Then Eq. (7) can be expressed as:

$$\begin{gathered} x_{21}{r}_x+y_{21}{r}_y+z_{21}{r}_z-h_1=0 \\ {x}_{31}{r}_x+y_{31}{r}_y+z_{31}{r}_z-h_2=0 \end{gathered}$$ (8)

Solving Eq. (8) for $r_x$ and $r_y$ :

$$\begin{gathered} r_x=\frac{\left(y_{31}{z}_{21}/y_{21}-z_{31}\right)}{\left(x_{31}-y_{31}{x}_{21}/y_{21}\right)}{r}_z+\frac{\left(h_2-y_{31}{h}_1/y_{21}\right)}{\left(x_{31}-y_{31}{x}_{21}/y_{21}\right)} \\ {r}_y=\frac{\left(x_{31}{z}_{21}/x_{21}-z_{31}\right)}{\left(y_{31}-x_{31}{y}_{21}/x_{21}\right)}{r}_z+\frac{\left(h_2-x_{31}{h}_1/x_{21}\right)}{\left(y_{31}-x_{31}{y}_{21}/x_{21}\right)} \end{gathered}$$ (9)

Let

$$k_1=\frac{\left(y_{31}{z}_{21}/y_{21}-z_{31}\right)}{\left(x_{31}-y_{31}{x}_{21}/y_{21}\right)},k_2=\frac{\left(h_2-y_{31}{h}_1/y_{21}\right)}{\left(x_{31}-y_{31}{x}_{21}/y_{21}\right)}$$

$$k_3=\frac{\left(x_{31}{z}_{21}/x_{21}-z_{31}\right)}{\left(y_{31}-x_{31}{y}_{21}/x_{21}\right)},k_4=\frac{\left(h_2-x_{31}{h}_1/x_{21}\right)}{\left(y_{31}-x_{31}{y}_{21}/x_{21}\right)}$$

Then, substituting Eq. (9) into the first row of Eq. (5), where ${\mathbf{e}}_1={\left[e_{1x},e_{1y},e_{1z}\right]}^T$ :

$$\begin{gathered} l^2=\left[k_1{r}_z+k_2,k_3{r}_z+k_4,r_z\right]\left[\begin{array}{c}{k}_{1}z+k_2\\ k_{3}z+k_4\\ r_z\end{array}\right] \\ -2\left[k_1{r}_z+k_2,k_3{r}_z+k_4,r_z\right]\left[\begin{array}{c}{e}_{1x}\\ e_{1y}\\ e_{1z}\end{array}\right]+{\mathbf{e}}_1^T{\mathbf{e}}_1 \end{gathered}$$ (10)

Let

$$T_1=k_1^2+k_3^2+1$$

$$T_2=k_1{k}_2+k_3{k}_4-e_{1x}{k}_1-e_{1y}{k}_3-e_{1z}$$

$$T_3=k_2^2+k_4^2-2e_{1x}{k}_2-2e_{1y}{k}_4-l^2+e_{1x}^2+e_{1y}^2+e_{1z}^2$$

Rearranging Eq. (10):

$$r_z=\frac{-T_2\pm \sqrt{T_2^2-T_1{T}_3}}{T_1}$$ (11)

Since the moving platform must be above the base plane, only a positive $r_z$ solution is valid.

Thus, Eqs. (9) and (11) form the forward kinematic solution $\mathbf{r}={\left[r_x,r_y,r_z\right]}^T$ of the linear delta platform. The homogeneous transformation matrix from base frame $\left\{b\right\}$ to platform frame $\left\{p\right\}$ is:

$$T_{bp}=\left[\begin{array}{cc}{\mathbf{I}}_{3\times 3}& {\mathbf{r}}_{3\times 1}\\ {\mathbf{0}}_{1\times 3}& 1\end{array}\right]$$ (12)

where ${\mathbf{I}}_{3\times 3}$ denotes a 3 × 3 identity matrix, ${\mathbf{r}}_{3\times 1}$ is the position vector of the moving platform, and ${\mathbf{0}}_{1\times 3}$ denotes a 1 × 3 zero matrix.

2.1.2. Roll–Tilt Mechanism Forward Kinematics

The roll–tilt mechanism is a two-DOF hybrid kinematic structure where roll motion is provided by a revolute joint and tilt motion is generated by a four-bar mechanism (Fig. 1). As shown in Fig. 3, the control inputs of this mechanism are the roll angle $\psi$ and the linear actuator stroke length $s$. Point $\mathbf{P}$ is a user-defined point that can be selected at any location along the tool shaft. Due to the nature of the four-bar linkage, the motion of the tool involves both rotation and translation. However, these two motions are kinematically coupled; thus, knowing one uniquely determines the other.

Fig. 3:

Side view of the roll–tilt mechanism showing the reference frames and geometric parameters.

The forward kinematics is relatively straightforward to compute since the lengths of all the links are known. The present derivation extends the approach introduced in [30]. Link $AB$ is fixed on the shaft of the roll motor, and point $P$ remains fixed relative to link $CD$. For simplicity, point $A$ is defined as the origin of the coordinate system. As the stroke length $s$ is known:

$$x_{AR}=x_{A{R}_{\mathit{\text{max}}}}-s$$ (13)

where $x_{A{R}_{\mathit{\text{max}}}}$ is the max horizontal distance from the origin to the slider, which is known based on the design parameters, and $z_{AR}$ is also a fixed value, representing the height from the linear guide rail to the frame $\left\{a\right\}$. Thus,

$$L_{AR}=\sqrt{x_{AR}^2+z_{AR}^2}$$ (14)

$${\alpha }_4=\mathit{\text{atan}}2\left(y_{AR},z_{AR}\right)$$ (15)

As the lengths of $AQ$ and $QR$ are known, from the law of cosines:

$${\alpha }_3=\text{arccos}\left(\frac{L_{AR}^2+L_{AQ}^2-L_{QR}^2}{2L_{AR}{L}_{AQ}}\right)$$ (16)

Since ${\varphi }_1$ and ${\varphi }_2$ are angles with fixed values, and ${\alpha }_3,{\alpha }_4$ are known:

$${\theta }_1={180}^{\circ }-\left({\varphi }_2+{\alpha }_3-{\alpha }_4\right)$$ (17)

$${\alpha }_1={\varphi }_1-{\theta }_1$$ (18)

From the law of cosines again:

$$L_{BD}=\sqrt{L_{AB}^2+L_{DA}^2-2L_{AB}{L}_{DA}\text{cos}\left({\alpha }_1\right)}$$ (19)

$${\beta }_1=\text{arccos}\left(\frac{L_{AB}^2+L_{BD}^2-L_{DA}^2}{2L_{AB}{L}_{BD}}\right)$$ (20)

$$\delta =\text{arccos}\left(\frac{L_{BD}^2+L_{CD}^2-L_{BC}^2}{2L_{BD}{L}_{CD}}\right)$$ (21)

$$\begin{gathered} {\theta }_2={180}^{\circ }-\left(\left({180}^{\circ }-{\beta }_1-{\varphi }_1\right)+\delta \right) \\ ={\beta }_1+{\varphi }_1-\delta \end{gathered}$$ (22)

Since ${\varphi }_3$ and ${\varphi }_4$ are also known and the lengths of $AD$ and $DP$ are constants, the $x$ and $y$ coordinates of point $\mathbf{P}$ in frame $\left\{a\right\}$ can be directly computed:

$$\begin{gathered} P_x=\text{cos}\left({\theta }_1\right)L_{DA}+\text{cos}\left({\theta }_2-{\varphi }_3\right)L_{DP} \\ {P}_z=\text{sin}\left({\theta }_1\right)L_{DA}+\text{sin}\left({\theta }_2-{\varphi }_3\right)L_{DP} \end{gathered}$$ (23)

Let $\theta ={90}^{\circ }-{\varphi }_4-\left({\varphi }_3-{\theta }_2\right)$ denote the angle between the x-axis of frame $\left\{a\right\}$ and the x-axis of frame $\left\{t\right\}$. Then, the transformation matrix can be expressed as:

$$T_{at}=\left[\begin{array}{cccc}\text{cos}(\theta )& 0& \text{sin}(\theta )& P_x\\ 0& 1& 0& 0\\ -\text{sin}(\theta )& 0& \text{cos}(\theta )& P_z\\ 0& 0& 0& 1\end{array}\right]$$ (24)

Thus, given the stroke length $s$, we can then determine the tool frame’s position and orientation relative to frame $\left\{a\right\}$. The next step is to determine $T_{pt}$, representing the forward kinematics of the roll-tilt mechanism from frame $\left\{p\right\}$ to frame $\left\{t\right\}$. As illustrated in Fig. 4, the roll angle $\psi$ is defined with respect to the $z$-axis of frame $\left\{p\right\}$. The roll axis is located a distance $d_1$ above the linear delta platform frame $\left\{p\right\}$, while frame $\left\{a\right\}$ is positioned a distance $d_3$ along the $z$-axis of frame $\left\{r\right\}$. In addition, the displacement between frames $\left\{r\right\}$ and $\left\{a\right\}$ along the $x$-axis is $d_2$.

Fig. 4:

Simplified front view of the roll-tilt mechanism

Accordingly, the transformation matrices $T_{pr}$ and $T_{ra}$ can be written as:

$$\begin{gathered} T_{pr}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& \text{cos}(\psi )& -\text{sin}(\psi )& 0\\ 0& \text{sin}(\psi )& \text{cos}(\psi )& d_1\\ 0& 0& 0& 1\end{array}\right] \\ {T}_{ra}=\left[\begin{array}{cccc}1& 0& 0& d_2\\ 0& 1& 0& 0\\ 0& 0& 1& d_3\\ 0& 0& 0& 1\end{array}\right] \end{gathered}$$ (25)

By composing the transformation matrices defined in Eqs. (25) and (24), we obtain the forward kinematics of the roll–tilt mechanism, expressed as the transformation matrix $T_{pt}$:

$$T_{pt}=T_{pr}{T}_{ra}{T}_{at}$$ (26)

2.1.3. System Forward Kinematics

The overall system forward kinematics can be obtained by composing Eqs. (12) and (26). Thus, the transformation from base frame $\left\{b\right\}$ to tool frame $\left\{t\right\}$ is:

$$\begin{gathered} T_{bt}=T_{bp}{T}_{pt} \\ =\left[\begin{array}{cccc}\text{cos}(\theta )& 0& \text{sin}(\theta )& x_{bt}\\ \text{sin}(\theta )\text{sin}(\psi )& \text{cos}(\psi )& -\text{cos}(\theta )\text{sin}(\psi )& y_{bt}\\ -\text{sin}(\theta )\text{cos}(\psi )& \text{sin}(\psi )& \text{cos}(\theta )\text{cos}(\psi )& z_{bt}\\ 0& 0& 0& 1\end{array}\right] \end{gathered}$$ (27)

where

$$x_{bt}=P_x+d_2+r_x$$

$$y_{bt}=-\text{sin}(\psi )P_z-\text{sin}(\psi )d_3+r_y$$

$$z_{bt}=\text{cos}(\psi )P_z+\text{cos}(\psi )d_3+d_1+r_z$$

Since $\mathbf{r}\left(q_1,q_2,q_3\right),\psi \left(q_4\right),\theta \left(q_5\right),P{\left(q_5\right)}_x,P{\left(q_5\right)}_z$ are all functions of the joint coordinates, this formulation establishes a direct mapping from the configuration space to the task space, thereby defining the forward kinematic model of the serially connected delta and rotation four-bar mechanism.

2.2. Inverse Kinematics

Similar to the forward kinematics derivation, the inverse kinematics of each subsystem is derived independently. The results are then combined to form the complete inverse kinematics of the robot.

2.2.1. Delta Platform Inverse Kinematics

Write Eq. (1) in matrix form:

$$\left[\begin{array}{l}{l}_{ix}\\ l_{iy}\\ l_{iz}\end{array}\right]=-\left[\begin{array}{c}0\\ 0\\ q_i\end{array}\right]-\left[\begin{array}{c}{r}_b\text{cos}\left({\theta }_{bi}\right)\\ r_b\text{sin}\left({\theta }_{bi}\right)\\ 0\end{array}\right]+\left[\begin{array}{l}{r}_x\\ r_y\\ r_z\end{array}\right]+\left[\begin{array}{c}{r}_p\text{cos}\left({\theta }_{pi}\right)\\ r_p\text{sin}\left({\theta }_{pi}\right)\\ 0\end{array}\right]$$ (28)

Since the length of each link is $l=\sqrt{l_{ix}^2+l_{iy}^2+l_{iz}^2}$, taking the square of the Eq. (28) results in:

$$\begin{gathered} l^2=l_{ix}^2+l_{iy}^2+l_{iz}^2 \\ ={\left(r_x+a_i\right)}^2+{\left(r_y+b_i\right)}^2+{\left(r_z-q_i\right)}^2 \end{gathered}$$ (29)

where

$$a_i=-r_b\text{cos}{\theta }_{bi}+r_p\text{cos}{\theta }_{pi}$$

$$b_i=-r_b\text{sin}{\theta }_{bi}+r_p\text{sin}{\theta }_{pi}$$

Reorganizing Eq. (29) yields the inverse kinematics of the linear Delta platform:

$$q_i=r_z-\sqrt{l^2-{\left(r_x+a_i\right)}^2-{\left(r_y+b_i\right)}^2}$$ (30)

This gives the prismatic actuator extension for each leg, where $i=\mathrm{1,2},3$.

2.2.2. Roll–Tilt Mechanism Inverse Kinematics

The inverse kinematics of this mechanism presents more complexity. Given the tool tilt angle $\theta$, the angle between link $CD$ and the horizontal axis can be easily obtained as ${\theta }_2=\theta +{\varphi }_3+{\varphi }_4-{90}^{\circ }$. However, substituting ${\theta }_2$ back into Eq. (22) to solve for ${\theta }_1$ and stroke length $s$ introduces difficulties, as both ${\beta }_1$ and $\delta$ remain unknown since the length of $BD$ varies. Solving for these variables would require handling a set of quadratic equations, which can be computationally expensive.

A more efficient approach is to derive the inverse kinematics directly from its geometry by translating edges $AB$ and $BC$ to construct a new parallelogram $\mathit{\text{ABCE}}$, as shown in Fig. 5.

Fig. 5:

Close up of the four-bar mechanism

It is obvious to get the following relations:

$$L_{DF}=\text{cos}\left({\theta }_2\right)L_{CD}-\text{cos}\left({\varphi }_1\right)L_{AB}$$

$$L_{EF}=\text{sin}\left({\varphi }_1\right)L_{AB}-\text{sin}\left({\theta }_2\right)L_{CD}$$

$$L_{DE}=\sqrt{L_{DF}^2+L_{EF}^2}$$

Given the lengths of AD, AE, and DE, using the law of cosines:

$$\angle ADH=\text{arccos}\left(\frac{L_{AD}^2+L_{DE}^2-L_{AE}^2}{2L_{AD}{L}_{DE}}\right)$$

$$\angle AHD=\angle FDH=\text{arctan}\left(L_{EF},L_{DF}\right)$$

$${\theta }_1={180}^{\circ }-\angle ADH-\angle AHD$$

Then, we can use ${\theta }_1$ to calculate the stroke length $s$.

Let $\alpha ={\alpha }_3+{\alpha }_4$ :

$$\alpha ={180}^{\circ }-{\theta }_1-{\varphi }_2$$

$$x_{AR}=\text{cos}(\alpha )L_{AQ}+\sqrt{L_{QR}^2-{\left(\text{sin}(\alpha )L_{AQ}-y_{AR}\right)}^2}$$

$$s=x_{{AR}_{\mathit{\text{max}}}}-x_{AR}$$

Since the roll angle and stroke length are directly controlled by the revolute and linear actuators, the roll and prismatic joint variables are given by $\psi =q_4,s=q_5$. This leads to the inverse kinematics of the roll-tilt mechanism:

$$\begin{gathered} q_4=\psi \\ {q}_5=x_{{AR}_{\text{max}}}-x_{AR}(\theta ) \end{gathered}$$ (31)

where $\psi$ and $\theta$ denote the prescribed orientation of the tool.

2.2.3. System Inverse Kinematics

Given point $\mathbf{P}$ expressed in frame $\left\{a\right\}$ as ${\mathbf{P}}_{\left\{a\right\}}$, its coordinates in the base frame $\left\{b\right\}$, denoted as ${\mathbf{P}}_{\left\{b\right\}}$, can be obtained from Eq. (27):

$${\mathbf{P}}_{\left\{b\right\}}=\left[\begin{array}{c}{P}_{x\left\{b\right\}}\\ P_{y\left\{b\right\}}\\ P_{z\left\{b\right\}}\end{array}\right]=\left[\begin{array}{c}{P}_{x\left\{a\right\}}+d_2+r_x\\ -\text{sin}(\psi )P_{z\left\{a\right\}}-\text{sin}(\psi )d_3+r_y\\ \text{cos}(\psi )P_{z\left\{a\right\}}+\text{cos}(\psi )d_3+d_1+r_z\end{array}\right]$$ (32)

The base-to-platform vector $\mathbf{r}={\left[r_x,r_y,r_z\right]}^T$ is then obtained as:

$$\left[\begin{array}{c}{r}_x\\ r_y\\ r_z\end{array}\right]={\mathbf{P}}_{\left\{b\right\}}-\left[\begin{array}{c}P{\left(q_5\right)}_{x\left\{a\right\}}+d_2\\ -\text{sin}\left(q_4\right)P{\left(q_5\right)}_{z\left\{a\right\}}-\text{sin}\left(q_4\right)d_3\\ \text{cos}\left(q_4\right)P{\left(q_5\right)}_{z\left\{a\right\}}+\text{cos}\left(q_4\right)d_3+d_1\end{array}\right]$$ (33)

Then plugging Eq. (33) into Eq. (30) gives the actuator displacements $q_1,q_2,q_3$.

Thus, Eqs. (30), (31), (33) form the inverse kinematics, proceed as follows: the inputs are the tool position ${\mathbf{P}}_{\left\{b\right\}}$ and orientation $(\psi ,\theta )$. From $\psi$ and $\theta$, the joint values $q_4$ and $q_5$ are obtained, which in turn determine $P_{x\left\{a\right\}}$ and $P_{z\left\{a\right\}}$. These allow the computation of $\mathbf{r}$, and finally lead to the actuator displacements $q_1,q_2,q_3$.

2.3. Jacobian

To relate the joint velocities to the tool-tip velocity, the system Jacobian is required. The tool-tip velocity and the joint velocities are defined as

$$\mathbf{V}={\left[\dot{x}\dot{y}\dot{z}{\dot{\theta }}_x{\dot{\theta }}_y{\dot{\theta }}_z\right]}_{\left\{b\right\}}^T,\dot{\mathbf{q}}={\left[{\dot{q}}_1{\dot{q}}_2{\dot{q}}_3{\dot{q}}_4{\dot{q}}_5\right]}^T$$

Throughout this section, the term Jacobian refers to the spatial Jacobian, where all quantities are expressed in the base frame $\left\{b\right\}$. The mapping between the joint velocities and the tool-tip velocity is given by

$$\mathbf{V}=\mathbf{J}\dot{\mathbf{q}}$$ (34)

where $\mathbf{J}\in {\mathbb{R}}^{6\times 5}$ is the Jacobian matrix.

2.3.1. Delta Platform Jacobian

Since $q_1,q_2,q_3$ only generate translational motions, the upper-left 3 × 3 block of $\mathbf{J}$ is the Jacobian of the linear delta platform, denoted ${\mathbf{J}}_D$. To find ${\mathbf{J}}_D$, let $\dot{\mathbf{r}}={\left[{\dot{r}}_x,{\dot{r}}_y,{\dot{r}}_z\right]}^T$ be the linear delta platform velocity, and ${\dot{\mathbf{L}}}_i={\left[\mathrm{0,0},{\dot{q}}_i\right]}^T$ the rate of change of each prismatic joint. Rearranging Eq. (1) and taking derivative of both sides:

$${\dot{\mathbf{L}}}_i=\dot{\mathbf{r}}-{\dot{\mathbf{l}}}_i-{\dot{\mathbf{r}}}_{bi}+{\dot{\mathbf{r}}}_{pi}$$ (35)

Since all of the elements in ${\mathbf{r}}_{bi}$ and ${\mathbf{r}}_{pi}$ are constants, ${\dot{\mathbf{r}}}_{bi}={\dot{\mathbf{r}}}_{pi}=0$. Thus,

$$\begin{gathered} {\dot{\mathbf{L}}}_i=\dot{\mathbf{r}}-{\mathbf{\text{l}}}_i \\ =\dot{\mathbf{r}}-\left({\omega }_i\times {\mathbf{l}}_i\right) \end{gathered}$$ (36)

Taking the dot product of both sides with ${\mathbf{\text{l}}}_i$:

$${\mathbf{l}}_i\cdot {\dot{\mathbf{L}}}_i={\mathbf{l}}_i\cdot \dot{\mathbf{r}}-{\mathbf{l}}_i\cdot \left({\omega }_i\times {\mathbf{l}}_i\right)$$ (37)

Since ${\mathbf{l}}_i$ is perpendicular to ${\omega }_i\times {\mathbf{l}}_i$ :

$${\mathbf{l}}_i^T{\dot{\mathbf{L}}}_i={\mathbf{l}}_i^T\dot{\mathbf{r}}$$ (38)

$$\left[l_{ix}{l}_{iy}{l}_{iz}\right]\left[\begin{array}{c}0\\ 0\\ {\dot{q}}_i\end{array}\right]=\left[l_{ix}{l}_{iy}{l}_{iz}\right]\left[\begin{array}{c}{\dot{r}}_x\\ {\dot{r}}_y\\ {\dot{r}}_z\end{array}\right]$$ (39)

$$l_{iz}{\dot{q}}_i=l_{ix}{\dot{r}}_x+l_{iy}{\dot{r}}_y+l_{iz}{\dot{r}}_z$$ (40)

Expand Eq. (40) with $i=\mathrm{1,2},3$:

$$\left[\begin{array}{ccc}{l}_{1z}& 0& 0\\ 0& l_{2z}& 0\\ 0& 0& l_{3z}\end{array}\right]\left[\begin{array}{c}{\dot{q}}_1\\ {\dot{q}}_2\\ {\dot{q}}_3\end{array}\right]=\left[\begin{array}{lll}{l}_{1x}& l_{1y}& l_{1z}\\ l_{2x}& l_{2y}& l_{2z}\\ l_{3x}& l_{3y}& l_{3z}\end{array}\right]\left[\begin{array}{c}{\dot{r}}_x\\ {\dot{r}}_y\\ {\dot{r}}_z\end{array}\right]$$ (41)

Thus we can obtain:

$$\dot{\mathbf{r}}={\mathbf{J}}_D\dot{\mathbf{q}}$$ (42)

where

$${\mathbf{J}}_D={\left[\begin{array}{lll}{l}_{1x}& l_{1y}& l_{1z}\\ l_{2x}& l_{2y}& l_{2z}\\ l_{3x}& l_{3y}& l_{3z}\end{array}\right]}^{-1}\left[\begin{array}{ccc}{l}_{1z}& 0& 0\\ 0& l_{2z}& 0\\ 0& 0& l_{3z}\end{array}\right]$$ (43)

is the 3×3 Jacobian matrix of the linear delta platform.

2.3.2. Roll–Tilt Mechanism Jacobian

The kinematic contributions of $q_4$ and $q_5$ require more careful consideration. As discussed previously, the four-bar linkage introduces coupled motion: actuation of $q_5$ generates both translation and rotation, while $q_4$ also contributes to the linear velocity due to the tool tip being offset from its axis of rotation.

Taking the derivative of Eq. (23) gives the linear velocity of the tool tip:

$$\begin{gathered} {\dot{P}}_x=-\text{sin}\left({\theta }_1\right)L_{DA}{\dot{\theta }}_1-\text{sin}\left({\theta }_2-{\varphi }_3\right)L_{DP}{\dot{\theta }}_2 \\ {\dot{P}}_z=\text{cos}\left({\theta }_1\right)L_{DA}{\dot{\theta }}_1+\text{cos}\left({\theta }_2-{\varphi }_3\right)L_{DP}{\dot{\theta }}_2 \end{gathered}$$ (44)

Although the velocity relations can be obtained by differentiating the loop-closure equations, the expressions for ${\dot{\theta }}_1$ and ${\dot{\theta }}_2$ involve derivatives of arctan and arccos functions, which lead to algebraically tedious forms. Thus, the instantaneous center of velocity (ICV) method is used because it yields a more compact and geometrically intuitive closed-form expression, which simplifies implementation and improves code readability. The principle is that, at any given instant, a rigid body in planar motion can be treated as a pure rotation about a unique point in space, referred to as its instantaneous center. For the four-bar mechanism, the center is denoted as point $\mathbf{I}$, and for the slider-crank loop, the center is denoted as point $\mathbf{J}$, as illustrated in Fig. 3-a.

For example, at any instant, link $\mathbf{Q}\mathbf{R}$ can be assumed to be in pure rotation with respect to the point $\mathbf{J}$, where $\mathbf{Q}\mathbf{J}$ is perpendicular to the velocity of point $\mathbf{Q}$ and $\mathbf{R}\mathbf{J}$ is perpendicular to the velocity of point $\mathbf{R}$.

Then, from the geometry above:

$$\begin{gathered} L_{JR}=\text{tan}(\alpha )\cdot x_{AR}-z_{AR} \\ {L}_{JQ}=\text{sec}(\alpha )\cdot x_{AR}-L_{AQ} \end{gathered}$$ (45)

$$\dot{\rho }=\frac{\dot{s}}{L_{JR}},L_{AQ}\cdot \dot{\alpha }=-L_{JQ}\cdot \dot{\rho }$$ (46)

$${\dot{\theta }}_1=\dot{\alpha }=\frac{L_{JQ}}{L_{AQ}{L}_{JR}}\dot{s}$$ (47)

Similarly, link $\mathbf{C}\mathbf{D}$ can be assumed to be in pure rotation with respect to the point $\mathbf{I}$, where $\mathbf{C}\mathbf{I}$ is perpendicular to the velocity of point $\mathbf{C}$ and $\mathbf{D}\mathbf{I}$ is perpendicular to the velocity of point $\mathbf{D}$.

$${\beta }_2=\text{arccos}\left(\frac{L_{BC}^2+L_{BD}^2-L_{CD}^2}{2L_{BC}{L}_{BD}}\right)$$ (48)

Let $\beta ={\beta }_1+{\beta }_2$:

$$\angle AIB={180}^{\circ }-{\alpha }_1-\beta$$ (49)

Then, apply the law of sines:

$$\frac{L_{AB}}{\text{sin}(\angle AIB)}=\frac{L_{IA}}{\text{sin}(\beta )}=\frac{L_{IB}}{\text{sin}\left({\alpha }_1\right)}$$ (50)

Since $\text{sin}\left({180}^{\circ }-{\alpha }_1-\beta \right)=\text{sin}\left({\alpha }_1+\beta \right)$ :

$$\begin{gathered} L_{IA}=\frac{L_{AB}\cdot \text{sin}(\beta )}{\text{sin}\left({\alpha }_1+\beta \right)} \\ {L}_{ID}=L_{IA}-L_{DA} \end{gathered}$$ (51)

Using the instantaneous center method:

$$\begin{gathered} {\dot{\theta }}_1{L}_{DA}=-{\dot{\theta }}_2{L}_{ID} \\ {\dot{\theta }}_2=-\frac{L_{DA}}{L_{ID}}{\dot{\theta }}_1 \end{gathered}$$ (52)

From Eq. (47), since ${\dot{\theta }}_1=\frac{L_{JQ}}{L_{AQ}{L}_{JR}}\dot{s}$ :

$${\dot{\theta }}_2=-\frac{L_{DA}{L}_{JQ}}{L_{ID}{L}_{AQ}{L}_{JR}}\dot{s}$$ (53)

Then, plugging Eqs. (47) and (53) into Eq. (44):

$$\begin{gathered} {\dot{P}}_{x\left\{a\right\}}=\left(-\text{sin}\left({\theta }_1\right)\frac{L_{DA}{L}_{JQ}}{L_{AQ}{L}_{JR}}+\text{sin}\left({\theta }_2-{\varphi }_3\right)\frac{L_{DP}{L}_{DA}{L}_{JQ}}{L_{ID}{L}_{AQ}{L}_{JR}}\right)\dot{s} \\ {\dot{P}}_{z\left\{a\right\}}=\left(\text{cos}\left({\theta }_1\right)\frac{L_{DA}{L}_{JQ}}{L_{AQ}{L}_{JR}}-\text{cos}\left({\theta }_2-{\varphi }_3\right)\frac{L_{DP}{L}_{DA}{L}_{JQ}}{L_{ID}{L}_{AQ}{L}_{JR}}\right)\dot{s} \end{gathered}$$ (54)

Let:

$$A=\left(-\text{sin}\left({\theta }_1\right)\frac{L_{DA}{L}_{JQ}}{L_{AQ}{L}_{JR}}+\text{sin}\left({\theta }_2-{\varphi }_3\right)\frac{L_{DP}{L}_{DA}{L}_{JQ}}{L_{ID}{L}_{AQ}{L}_{JR}}\right)$$

$$B=\left(\text{cos}\left({\theta }_1\right)\frac{L_{DA}{L}_{JQ}}{L_{AQ}{L}_{JR}}-\text{cos}\left({\theta }_2-{\varphi }_3\right)\frac{L_{DP}{L}_{DA}{L}_{JQ}}{L_{ID}{L}_{AQ}{L}_{JR}}\right)$$

This gives a clear and compact expression for the velocity of point $\mathbf{P}$ as a function of $\dot{s}$ :

$${\dot{P}}_{x\left\{a\right\}}=A\dot{s},{\dot{P}}_{z\left\{a\right\}}=B\dot{s}$$

All velocity components above are with respect to frame $\left\{a\right\}$. According to Fig. 3-b, the corresponding tooltip velocity in the linear Delta platform frame $\left\{p\right\}$ can be readily derived as:

$${\left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{z}\\ {\dot{\theta }}_x\\ {\dot{\theta }}_y\\ {\dot{\theta }}_z\end{array}\right]}_{\left\{p\right\}}=\left[\begin{array}{c}{\dot{P}}_{x\left\{a\right\}}\\ -\text{sin}(\psi ){\dot{P}}_{z\left\{a\right\}}-\text{cos}(\psi )\dot{\psi }\left(P_{z\left\{a\right\}}+d_3\right)\\ \text{cos}(\psi ){\dot{P}}_{z\left\{a\right\}}-\text{sin}(\psi )\dot{\psi }\left(P_{z\left\{a\right\}}+d_3\right)\\ \dot{\psi }\\ \text{cos}(\psi ){\dot{\theta }}_2\\ \text{sin}(\psi ){\dot{\theta }}_2\end{array}\right]$$ (55)

Since $\dot{\psi }={\dot{q}}_4$ and $\dot{s}={\dot{q}}_5$, Eq. (55) can be written as:

$${\left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{z}\\ {\dot{\theta }}_x\\ {\dot{\theta }}_y\\ {\dot{\theta }}_z\end{array}\right]}_{\left\{p\right\}}=\underset{{\mathbf{J}}_{RT}}{\underbrace{\left[\begin{array}{cc}0& A\\ -\text{cos}\left(q_4\right)\left(P_{z\left\{a\right\}}+d_3\right)& -\text{sin}\left(q_4\right)B\\ -\text{sin}\left(q_4\right)\left(P_{z\left\{a\right\}}+d_3\right)& \text{cos}\left(q_4\right)B\\ 1& 0\\ 0& \text{cos}\left(q_4\right)\frac{L_{DA}{L}_{JQ}}{L_{ID}{L}_{AQ}{L}_{JR}}\\ 0& \text{sin}\left(q_4\right)\frac{L_{DA}{L}_{JQ}}{L_{ID}{L}_{AQ}{L}_{JR}}\end{array}\right]}}\left[\begin{array}{l}{\dot{q}}_4\\ {\dot{q}}_5\end{array}\right]$$ (56)

2.3.3. System Jacobian

Because there is no rotation between the base frame $\left\{b\right\}$ and the linear delta platform frame $\left\{p\right\}$, the velocity of the tool tip expressed in $\left\{b\right\}$ is simply the sum of the linear delta platform’s velocity in $\left\{b\right\}$ and the tool tip’s velocity in fixed $\left\{p\right\}$ frame. This provides all the necessary components for assembling the spatial Jacobian:

$$\mathbf{J}=\left[\left.\begin{array}{c}{\mathbf{J}}_D\\ {\mathbf{0}}_{3\times 3}\end{array}\right|{\mathbf{J}}_{RT}\right]$$ (57)

where ${\mathbf{J}}_D\in {\mathbb{R}}^{3\times 3}$ is the Jacobian of the linear delta positioning platform, controlled by joints $q_1,q_2$, and $q_3.{\mathbf{J}}_{RT}\in {\mathbb{R}}^{6\times 2}$ is the Jacobian of the roll-tilt mechanism, controlled by joints $q_4$ and $q_5$.

2.4. Jacobian Pseudo-Inverse

Since SHER 3.0 is a 5-DOF robot, its Jacobian matrix is rectangular and a true matrix inverse does not exist. Therefore, a Jacobian pseudo-inverse is derived using an analytical formulation. The Jacobian Pseudo-Inverse is defined as:

$$\dot{\mathbf{q}}={\mathbf{J}}^{†}\mathbf{V}$$ (58)

where ${\mathbf{J}}^{†}\in {\mathbb{R}}^{5\times 6}$ is the Jacobian Pseudo-Inverse matrix.

2.4.1. Delta Platform Jacobian Inverse

From Eq. (40), the desired platform velocity can be directly related to the corresponding joint velocities as:

$${\dot{q}}_i=\frac{l_{ix}}{l_{iz}}{\dot{r}}_x+\frac{l_{iy}}{l_{iz}}{\dot{r}}_y+{\dot{r}}_z$$ (59)

Focusing on the linear delta platform, the joint velocities associated with a given linear platform velocity are:

$${\left[\begin{array}{l}{\dot{q}}_1\\ {\dot{q}}_2\\ {\dot{q}}_3\end{array}\right]}_{\mathit{\text{delta}}}=\left[\begin{array}{lll}\frac{l_{1x}}{l_{1z}}& \frac{l_{1y}}{l_{1z}}& 1\\ \frac{l_{2x}}{l_{2z}}& \frac{l_{2y}}{l_{2z}}& 1\\ \frac{l_{3x}}{l_{3z}}& \frac{l_{3y}}{l_{3z}}& 1\end{array}\right]\left[\begin{array}{c}{\dot{r}}_x\\ {\dot{r}}_y\\ {\dot{r}}_z\end{array}\right]$$ (60)

where the coefficient matrix depends on each limb’s position vector of the linear delta mechanism. Eq. (60) maps the platform velocities to joint-space velocities for the linear delta stage, defining the top-left block of the inverse Jacobian.

2.4.2. Roll–Tilt Mechanism Jacobian Inverse

For the roll and tilt mechanism, since the tool tip is not located along their axes of rotation, both roll and tilt introduce linear displacements in addition to their rotational effects. While angular velocities ${\dot{\theta }}_x,{\dot{\theta }}_y,{\dot{\theta }}_z$ will cause nonzero $\dot{x},\dot{y},\dot{z}$, these induced translations can be determined analytically, and these induced velocities can be compensated by subtracting them from the linear delta platform velocities. The translational velocities induced by the roll-tilt mechanism can be expressed as:

$${\left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{z}\end{array}\right]}_{\left\{p\right\}}=\left[\begin{array}{c}{\dot{P}}_{x\left\{a\right\}}\\ -\text{sin}(\psi ){\dot{P}}_{z\left\{a\right\}}-\text{cos}(\psi )\dot{\psi }\left(P_{z\left\{a\right\}}+d_3\right)\\ \text{cos}(\psi ){\dot{P}}_{z\left\{a\right\}}-\text{sin}(\psi )\dot{\psi }\left(P_{z\left\{a\right\}}+d_3\right)\end{array}\right]$$ (61)

By plugging Eq. (61) into Eq. (60), the joint velocities required to compensate the induced linear motion are:

$$\left[\begin{array}{l}{\dot{q}}_1\\ {\dot{q}}_2\\ {\dot{q}}_3\end{array}\right]=\left[\begin{array}{lll}\frac{l_{1x}}{l_{1z}}& \frac{l_{1y}}{l_{1z}}& 1\\ \frac{l_{2x}}{l_{2z}}& \frac{l_{2y}}{l_{2z}}& 1\\ \frac{l_{3x}}{l_{3z}}& \frac{l_{3y}}{l_{3z}}& 1\end{array}\right]{\left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{z}\end{array}\right]}_{\left\{p\right\}}$$ (62)

To analyze the contribution of the roll and tilt mechanisms to the induced velocity, Eq. (62) needs to be reformulated in terms of the roll and tilt rates $\dot{\psi }$ and ${\dot{\theta }}_2$. Plugging Eq. (52) into Eq. (44) gives the following expression:

$$\begin{gathered} {\dot{P}}_{x\left\{a\right\}}=\left(-\text{sin}\left({\theta }_1\right)L_{ID}+\text{sin}\left({\theta }_2-{\varphi }_3\right)L_{DP}\right){\dot{\theta }}_2 \\ {\dot{P}}_{y\left\{a\right\}}=\left(\text{cos}\left({\theta }_1\right)L_{ID}-\text{cos}\left({\theta }_2-{\varphi }_3\right)L_{DP}\right){\dot{\theta }}_2 \end{gathered}$$ (63)

Let:

$$\begin{gathered} C=-\text{sin}\left({\theta }_1\right)L_{ID}+\text{sin}\left({\theta }_2-{\varphi }_3\right)L_{DP} \\ D=\text{cos}\left({\theta }_1\right)L_{ID}-\text{cos}\left({\theta }_2-{\varphi }_3\right)L_{DP} \end{gathered}$$ (64)

Then

$$\begin{gathered} {\dot{P}}_{x\left\{a\right\}}=C{\dot{\theta }}_2 \\ {\dot{P}}_{z\left\{a\right\}}=D{\dot{\theta }}_2 \end{gathered}$$ (65)

Substituting Eq. (65) into Eq. (61), Eq. (62) becomes:

$$\begin{gathered} \left[\begin{array}{c}{\dot{q}}_1\\ {\dot{q}}_2\\ {\dot{q}}_3\end{array}\right]=\left[\begin{array}{lll}\frac{l_{1x}}{l_{1z}}& \frac{l_{1y}}{l_{1z}}& 1\\ \frac{l_{2x}}{l_{2z}}& \frac{l_{2y}}{l_{2z}}& 1\\ \frac{l_{3x}}{l_{3z}}& \frac{l_{3y}}{l_{3z}}& 1\end{array}\right]\left[\begin{array}{c}C{\dot{\theta }}_2\\ -\text{sin}(\psi )D{\dot{\theta }}_2-\text{cos}(\psi )\left(P_{z\left\{a\right\}}+d_3\right)\dot{\psi }\\ \text{cos}(\psi )D{\dot{\theta }}_2-\text{sin}(\psi )\left(P_{z\left\{a\right\}}+d_3\right)\dot{\psi }\end{array}\right] \\ =\underset{{\mathbf{\text{M}}}_{\text{D}}}{\underbrace{\left[\begin{array}{lll}\frac{l_{1x}}{l_{1z}}& \frac{l_{1y}}{l_{1z}}& 1\\ \frac{l_{2x}}{l_{2z}}& \frac{l_{2y}}{l_{2z}}& 1\\ \frac{l_{3x}}{l_{3z}}& \frac{l_{3y}}{l_{3z}}& 1\end{array}\right]\left[\begin{array}{cc}0& C\\ -\text{cos}(\psi )\left(P_{z\left\{a\right\}}+d_3\right)& -\text{sin}(\psi )D\\ -\text{sin}(\psi )\left(P_{z\left\{a\right\}}+d_3\right)& \text{cos}(\psi )D\end{array}\right]}}\left[\begin{array}{l}\dot{\psi }\hfill \\ {\dot{\theta }}_2\hfill \end{array}\right] \end{gathered}$$ (66)

From Eq. (55), the tool tip’s angular velocities in frame $\left\{p\right\}$ are:

$${\left[\begin{array}{l}{\dot{\theta }}_x\\ {\dot{\theta }}_y\\ {\dot{\theta }}_z\end{array}\right]}_{\left\{p\right\}}=\left[\begin{array}{c}\dot{\psi }\\ \text{cos}(\psi )\dot{{\theta }_2}\\ \text{sin}(\psi )\dot{{\theta }_2}\end{array}\right]$$ (67)

Since there is no rotational transformation between base frame $\left\{b\right\}$ and linear delta platform frame $\left\{p\right\},{\left[{\dot{\theta }}_x,{\dot{\theta }}_y,{\dot{\theta }}_z\right]}_{\left\{p\right\}}^T={\left[{\dot{\theta }}_x,{\dot{\theta }}_y,{\dot{\theta }}_z\right]}_{\left\{b\right\}}^T$. From Eq. (67), it is clear that ${\dot{\theta }}_y$ and ${\dot{\theta }}_z$ are coupled through $\psi$ and $\dot{{\theta }_2}$. In other words, specifying ${\dot{\theta }}_y$ determines a unique corresponding ${\dot{\theta }}_z$ and vice versa.

Since $\dot{\psi }$ represents the rotational velocity about the base frame’s $x$ axis, and the relationship between ${\dot{\theta }}_y$ and ${\dot{\theta }}_2$ has been shown in Eq. (67), substituting $\dot{\psi }={\dot{\theta }}_x$ and ${\dot{\theta }}_2=\frac{1}{\text{cos}(\psi )}{\dot{\theta }}_y$ into Eq. (66) gives:

$$\begin{gathered} \left[\begin{array}{l}{\dot{q}}_1\\ {\dot{q}}_2\\ {\dot{q}}_3\end{array}\right]={\mathbf{M}}_{\mathbf{D}}\left[\begin{array}{c}{\dot{\theta }}_x\\ \frac{1}{\text{cos}(\psi )}{\dot{\theta }}_y\end{array}\right] \\ \begin{array}{rr}& \\ & =\left[\begin{array}{ll}{\mathbf{M}}_{\mathbf{D}\left[\mathrm{1,1}\right]}& \frac{1}{\text{cos}(\psi )}{\mathbf{M}}_{\mathbf{D}\left[\mathrm{1,2}\right]}\\ {\mathbf{M}}_{\mathbf{D}\left[\mathrm{2,1}\right]}& \frac{1}{\text{cos}(\psi )}{\mathbf{M}}_{\mathbf{D}\left[\mathrm{2,2}\right]}\\ {\mathbf{M}}_{\mathbf{D}\left[\mathrm{3,1}\right]}& \frac{1}{\text{cos}(\psi )}{\mathbf{M}}_{\mathbf{D}\left[\mathrm{3,2}\right]}\end{array}\right]\left[\begin{array}{c}{\dot{\theta }}_x\\ {\dot{\theta }}_y\end{array}\right]\end{array} \end{gathered}$$ (68)

Let

$${\mathbf{M}}_{\mathbf{r}}=\left[\begin{array}{ll}{\mathbf{M}}_{\mathbf{D}\left[\mathrm{1,1}\right]}& \frac{1}{\text{cos}(\psi )}{\mathbf{M}}_{\mathbf{D}\left[\mathrm{1,2}\right]}\\ {\mathbf{M}}_{\mathbf{D}\left[\mathrm{2,1}\right]}& \frac{1}{\text{cos}(\psi )}{\mathbf{M}}_{\mathbf{D}\left[\mathrm{2,2}\right]}\\ {\mathbf{M}}_{\mathbf{D}\left[\mathrm{3,1}\right]}& \frac{1}{\text{cos}(\psi )}{\mathbf{M}}_{\mathbf{D}\left[\mathrm{3,2}\right]}\end{array}\right]$$ (69)

Then

$${\left[\begin{array}{l}{\dot{q}}_1\\ {\dot{q}}_2\\ {\dot{q}}_3\end{array}\right]}_{\mathit{\text{induced}}}={\mathbf{M}}_{\mathbf{r}}{\left[\begin{array}{l}{\dot{\theta }}_x\\ {\dot{\theta }}_y\end{array}\right]}_{\left\{b\right\}}$$ (70)

In this case, it is clear that ${\dot{\theta }}_z$ has no effect on ${\dot{q}}_1,{\dot{q}}_2,{\dot{q}}_3$. However, to maintain consistency with the full Jacobian inverse expression, ${\dot{\theta }}_z$ is included on the right-hand side of Eq. (70), yielding:

$${\left[\begin{array}{l}{\dot{q}}_1\\ {\dot{q}}_2\\ {\dot{q}}_3\end{array}\right]}_{\mathit{\text{induced}}}=\left[{\mathbf{M}}_{\mathbf{r}}\mathbf{0}\right]{\left[\begin{array}{c}{\dot{\theta }}_x\\ {\dot{\theta }}_y\\ {\dot{\theta }}_z\end{array}\right]}_{\left\{b\right\}}$$ (71)

Note that the joint velocities from Eq. (71) represent the linear velocity induced by roll and tilt. In the Jacobian inverse formulation, Eq. (71) is subtracted from Eq. (60) to eliminate the induced linear velocity.

$$\begin{gathered} \left[\begin{array}{l}{\dot{q}}_1\\ {\dot{q}}_2\\ {\dot{q}}_3\end{array}\right]={\left[\begin{array}{c}{\dot{q}}_1\\ {\dot{q}}_2\\ {\dot{q}}_3\end{array}\right]}_{\mathit{\text{delta}}}-{\left[\begin{array}{c}{\dot{q}}_1\\ {\dot{q}}_2\\ {\dot{q}}_3\end{array}\right]}_{\mathit{\text{induced}}} \\ =\left[\begin{array}{lll}\frac{l_{1x}}{l_{1z}}& \frac{l_{1y}}{l_{1z}}& 1\\ \frac{l_{2x}}{l_{2z}}& \frac{l_{2y}}{l_{2z}}& 1\\ \frac{l_{3x}}{l_{3z}}& \frac{l_{3y}}{l_{3z}}& 1\end{array}\right]\left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{z}\end{array}\right]-\left[{\mathbf{M}}_{\mathbf{r}}\mathbf{0}\right]\left[\begin{array}{c}{\dot{\theta }}_x\\ {\dot{\theta }}_y\\ {\dot{\theta }}_z\end{array}\right] \\ =\left[\begin{array}{ccc}\frac{l_{1x}}{l_{1z}}& \frac{l_{1y}}{l_{1z}}& 1-{\mathbf{M}}_{\mathbf{r}\left[\mathrm{1,1}\right]}-\begin{array}{cc}{\mathbf{M}}_{\mathbf{r}\left[\mathrm{1,2}\right]}& 0\end{array}\\ \frac{l_{2x}}{l_{2z}}& \frac{l_{2y}}{l_{2z}}& 1-{\mathbf{M}}_{\mathbf{r}\left[\mathrm{2,1}\right]}-\begin{array}{cc}{\mathbf{M}}_{\mathbf{r}\left[\mathrm{2,2}\right]}& 0\end{array}\\ \frac{l_{3x}}{l_{3z}}& \frac{l_{3y}}{l_{3z}}& 1-{\mathbf{M}}_{\mathbf{r}\left[\mathrm{3,1}\right]}-\begin{array}{cc}{\mathbf{M}}_{\mathbf{r}\left[\mathrm{3,2}\right]}& 0\end{array}\end{array}\right]{\left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{z}\\ {\dot{\theta }}_x\\ {\dot{\theta }}_y\\ {\dot{\theta }}_z\end{array}\right]}_{\left\{b\right\}} \end{gathered}$$ (72)

Since $\dot{\psi }={\dot{\theta }}_x={\dot{q}}_4,\dot{s}={\dot{q}}_5$ and ${\dot{\theta }}_2=\frac{1}{\text{cos}(\psi )}{\dot{\theta }}_y$, the relationship between ${\dot{q}}_5$ and ${\dot{\theta }}_y$ can be derived from Eq. (53):

$$\begin{gathered} {\dot{\theta }}_2=-\frac{L_{DA}{L}_{JQ}}{L_{ID}{L}_{AQ}{L}_{JR}}\dot{s} \\ \frac{1}{\text{cos}(\psi )}{\dot{\theta }}_y=-\frac{L_{DA}{L}_{JQ}}{L_{ID}{L}_{AQ}{L}_{JR}}{\dot{q}}_5 \\ {\dot{q}}_5=-\frac{L_{ID}{L}_{AQ}{L}_{JR}}{\text{cos}\left(q_4\right)L_{DA}{L}_{JQ}}{\dot{\theta }}_y \end{gathered}$$ (73)

Thus, all components needed to construct the system Jacobian inverse have been derived.

2.4.3. System Jacobian Pseudo-Inverse

By assembling Eq. (73) with Eq. (72), the joint velocities can be expressed as:

$$\left[\begin{array}{c}{\dot{q}}_1\\ {\dot{q}}_2\\ {\dot{q}}_3\\ {\dot{q}}_4\\ {\dot{q}}_5\end{array}\right]=\left[\begin{array}{cccccc}\frac{l_{1x}}{l_{1z}}& \frac{l_{1y}}{l_{1z}}& 1& -{\mathbf{\text{M}}}_{\mathbf{\text{r}}[1,1]}& -{\mathbf{\text{M}}}_{\mathbf{\text{r}}[1,2]}& 0\\ \frac{l_{2x}}{l_{2z}}& \frac{l_{2y}}{l_{2z}}& 1& -{\mathbf{\text{M}}}_{\mathbf{\text{r}}[2,1]}& -{\mathbf{\text{M}}}_{\mathbf{\text{r}}[2,2]}& 0\\ \frac{l_{3x}}{l_{3z}}& \frac{l_{3y}}{l_{3z}}& 1& -{\mathbf{\text{M}}}_{\mathbf{\text{r}}[3,1]}& -{\mathbf{\text{M}}}_{\mathbf{\text{r}}[3,2]}& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& -\frac{L_{ID}{L}_{AQ}{L}_{JR}}{\text{cos}(q_4)L_{DA}{L}_{JQ}}& 0\end{array}\right]\left[\begin{array}{c}\dot{x}\\ \dot{y}\\ \dot{z}\\ {\dot{\theta }}_x\\ {\dot{\theta }}_y\\ {\dot{\theta }}_z\end{array}\right]$$ (74)

The analytical expression of the spatial Jacobian

Pseudo-Inverse of the SHER 3.0 robot is:

$${\mathbf{J}}^{†}=\left[\begin{array}{cccccc}\frac{l_{1x}}{l_{1z}}& \frac{l_{1y}}{l_{1z}}& 1& -{\mathbf{\text{M}}}_{\mathbf{\text{r}}[1,1]}& -{\mathbf{\text{M}}}_{\mathbf{\text{r}}[1,2]}& 0\\ \frac{l_{2x}}{l_{2z}}& \frac{l_{2y}}{l_{2z}}& 1& -{\mathbf{\text{M}}}_{\mathbf{\text{r}}[2,1]}& -{\mathbf{\text{M}}}_{\mathbf{\text{r}}[2,2]}& 0\\ \frac{l_{3x}}{l_{3z}}& \frac{l_{3y}}{l_{3z}}& 1& -{\mathbf{\text{M}}}_{\mathbf{\text{r}}[3,1]}& -{\mathbf{\text{M}}}_{\mathbf{\text{r}}[3,2]}& 0\\ 0& 0& 0& 1& 0& 0\\ 0& 0& 0& 0& -\frac{L_{ID}{L}_{AQ}{L}_{JR}}{\text{cos}(q_4)L_{DA}{L}_{JQ}}& 0\end{array}\right]$$ (75)

2.5. Control Strategy

2.5.1. Definition of RCM point

In our system, the RCM is defined as a point on the surface of the eyeball. As shown in Fig. 1, the RCM frame is oriented parallel to the base frame, and its origin is located at the center of the insertion site on the eyeball.

2.5.2. Model Predictive Control with RCM Constraint

As discussed in the previous section, SHER 3.0 does not have a mechanically enforced RCM. Instead, a virtual RCM constraint is implemented to constrain the robot’s motion, and MPC is implemented as the primary control strategy. The system state at time $t$ is defined by the tool tip pose as:

$${\chi }_t={\left[x,y,z,{\theta }_x,{\theta }_y,{\theta }_z\right]}^T$$ (76)

and the system input $u_t$ at time $t$ is the joint velocities of the robot:

$$u_t={\left[{\dot{q}}_1,{\dot{q}}_2,{\dot{q}}_3,{\dot{q}}_4,{\dot{q}}_5\right]}^T$$ (77)

The cost function is defined as:

$$C=\mathit{\text{min}}\sum _{t=0}^{n-1} \left[\left({\chi }_t-{\chi }_{\text{des}}\right)Q{\left({\chi }_t-{\chi }_{\text{des}}\right)}^T+u_{t}R{u}_t^T+l_{t}D{l}_t^T\right]+{\chi }_n{Q}_f{\chi }_n^T$$ (78)

s.t.

$${\chi }_{t+1}={\chi }_t+\mathbf{J}\left(q_t\right)u_{t}dt,$$

$$q_{t+1}=q_t+u_{t}dt,$$

$$l_t=‖\left(p_{\text{rcm}}-p\right)\times {\hat{a}}_t‖,$$

$${\dot{\chi }}_{\text{min}}\le \mathbf{J}\left(q_t\right)u_t\le {\dot{\chi }}_{\text{max}}$$

$${\dot{q}}_{\text{min}}\le u_t\le {\dot{q}}_{\text{max},}$$

$$Q⪰0,R\succ 0,D⪰0,Q_f⪰0,$$

where the target end effector state is represented by ${\chi }_{\text{des}}$, and $\mathbf{J}\left(q_t\right)\in {\mathbb{R}}^{6\times 5}$ represents the Jacobian matrix of the SHER 3.0 system at joint configuration $q_t$. The scalar $l_t$ represents the distance between the gripper shaft and the RCM point, and $p_{\text{rcm}}\in {\mathbb{R}}^3$ is the position vector of the RCM point, $p$ refers to the position vector of the straight shaft tip, and ${\hat{a}}_t$ indicates the unit vector along the tool shaft direction.

2.5.3. Dual Robot Control Loop

In the SHER 3.0 robot control loop, the input device sends the target tool tip position $x_{\mathit{\text{des}}},y_{\mathit{\text{des}}}$ and $z_{\mathit{\text{des}}}$ to the MPC optimal controller. The desired values are sent to the optimal controller, shown in Eq. 78, which outputs the joint-velocity command ${\dot{q}}_{\mathit{\text{des}}}\in {\mathbb{R}}^5$. A motion-limiting block guarantees the safe execution of commanded motions. The limited joint velocity ${\dot{q}}_{\text{des}}^{\prime }$ is defined as

$${\dot{q}}_{\text{des}}^{\prime }=\left\{\begin{array}{ll}{E}^T{\dot{q}}_{\text{des}},& \alpha =1\\ 0,& \text{otherwise}\end{array}\right.$$ (79)

where $\alpha$ is a binary indicator. The mapping function $E\in {\mathbb{R}}^5(E_i=1\text{or}0)$ restricts motion along allowable directions determined by the translational inputs $x_{\text{des}},y_{\text{des}},z_{\text{des}}$ from the input device and the current ${\dot{\chi }}_{\text{real}}\in {\mathbb{R}}^6$ of the tool pose. The resulting target joint velocity ${\dot{q}}_{\text{real}}^{\prime }\in {\mathbb{R}}^5$ is then sent to the low-level velocity controller, which moves the five joints of SHER 3.0. The joint velocity ${\dot{q}}_{\text{real}}$ is used to update the Jacobian $\mathbf{J}$ in Eq. 57 for the next iteration and to compute the corresponding Cartesian tip velocity ${\dot{\chi }}_{\text{real}}$.

Algorithm 1.

RCM-Constrained Control Algorithm

Require: Initial joint state ${\mathbf{q}}_0$
Ensure: Joint velocity command $\dot{\mathbf{q}}$
1:	$\mathbf{q}\leftarrow {\mathbf{q}}_0$
2:	while system is enabled do
3:	${\mathbf{u}}_h\leftarrow$ read input device command
4:	${\mathbf{u}}_r\leftarrow 𝒯\left({\mathbf{u}}_h\right)$ {transform to robot frame}
5:	${\mathbf{u}}_r\leftarrow 𝒮\left({\mathbf{u}}_r\right)$ {scaling}
6:	${\mathbf{T}}_{bt}\leftarrow f(\mathbf{q})$ {forward kinematics, Eq. (27)}
7:	$\mathbf{J}\leftarrow J(\mathbf{q})$ {Jacobian, Eq. (57)}
8:	${\dot{\mathbf{q}}}_d\leftarrow \text{MPC}\left({\mathbf{u}}_r,\mathbf{q},\mathbf{J}\right)$ {Eq. (78)}
9:	apply ${\dot{\mathbf{q}}}_d$ to joint velocity controller
10:	$\mathbf{q}\leftarrow$ read joint feedback
11:	end while

Designed specifically for SHER 3.0, the ${\text{I}}^2$ RIS micromanipulator extends the system with two extra degrees of freedom [47]; however, in our experiment, it is operated as a straight surgical tool. The I²RIS control loop uses single PD control. The input device sends desired orientation ${\theta }_{x_{\mathit{\text{des}}}},{\theta }_{y_{\mathit{\text{des}}}}$, and ${\theta }_{z_{\mathit{\text{des}}}}$ to the micromanipulator’s motors. The target input drives the I²RIS to move in different directions.

Figure 6 shows the detailed control loop diagram for this MPC-based teleoperation controller. To avoid confusion, note that the notations used in the figure differ slightly from those in the main text: $BR$ denotes the Base Robot (SHER 3.0), and $SR$ denotes the snake robot (I²RIS). Algorithm 1 shows the steps of the SHER 3.0 control loop and the corresponding formulas.

Fig. 6:

Control loop diagram of the MPC-based teleoperation control for SHER 3.0

3. EXPERIMENTS

To evaluate the overall performance of the proposed kinematics and Jacobian models and the MPC-based RCM control algorithm, three distinct experiments were conducted, each targeting a specific goal:

1. Kinematic Model Validation: To quantitatively validate the analytical forward and inverse kinematics and Jacobian formulation.

2. Kinematic Analysis: To analyze the kinematic performance of SHER 3.0.

3. RCM Control Evaluation: To evaluate the performance of the implementation of the MPC control algorithm and analyze the accuracy of the robot kinematics and Jacobian in maintaining the RCM constraint in a teleoperation experiment using SHER 3.0 and a straight gripper tool.

4. Approach Angle Evaluation: To evaluate and compare the performance of the I²RIS snake robot and the straight gripper tool in providing the optimum approach angle to the retina surface while the RCM constraint is maintained by the SHER 3.0 and the MPC controller.

3.1. Experimental Setup

Figures 7 and 8 illustrate the experimental setup, which includes the SHER 3.0 with a motion controller (Galil Motion Control, Rocklin, CA, USA) for low-level joint velocity control. A 6-DOF force/torque sensor (Nano17, ATI Industrial Automation, Apex, NC, USA) is attached to SHER’s end-effector to measure the forces/torques applied by the user’s hand and to enable cooperative control in an admittance-based strategy [22]. The high-level MPC controller with RCM constraint, mid-level inverse kinematics solver, and low-level joint velocity controller were all implemented in C++ using the CISST-SAW libraries, running at an update rate of 200 Hz [48]. To evaluate the RCM control accuracy, a set of 3-millimeter ArUco markers were rigidly mounted on the robot’s end-effector, with their position and orientation tracked by a digital microscope (AD407, Andonstar, China) calibrated using a standard pinhole camera model. A 3D-printed eye phantom with a diameter of 24 mm (typical human eye diameter), incorporates a gimbal joint that functions as the RCM to simulate the sclerotomy. The robotic instrument shaft is inserted through the gimbal joint, simulating the RCM constraint during the eye surgery procedures. A Phantom Omni (Geomagic Touch) haptic interface is used for control of the actual SHER 3.0 with a straight gripper tool attached to its endeffector (Fig. 7). An Omega 7 haptic interface is used for teleoperation control of the Improved Integrated Robotic Intraocular Snake (I²RIS) attached to the SHER 3.0 end-effector (Fig. 8).

Fig. 7:

Experimental setup for the teleoperation system with the SHER 3.0 and a straight gripper tool attached to the end-effector. The setup includes a Phantom Omni haptic interface, a surgical microscope, and an eye phantom with a buzzer to confirm touching the target points at the bottom of the phantom with the tool tip.

Fig. 8:

Experiment setup for the teleoperation system with the SHER 3.0 and a snake-like robot with gripper (I²RIS) attached to the end-effector (left). The I²RIS robot provides a wrist-like motion with extra DOFs, pitch, and yaw (top right), and its shaft passes through a trocar (creating an RCM constraint) to enter the eye phantom (bottom right). The setup also includes an Omega 7 haptic interface, a surgical microscope, and a digital microscope to track the robot end-effector motion (left).

3.2. Experimental Procedures

3.2.1. Kinematic Model Validation

To quantitatively validate the analytical kinematic model, a numerical consistency study was performed using simulated joint trajectories within the feasible workspace. Random joint trajectories consisting of 150 configurations were generated and used as inputs to the analytical forward kinematics to produce a reference tooltip trajectory. The same joint trajectories were then evaluated using two independent methods: (i) velocity-based integration using the analytical Jacobian, and (ii) finite-difference approximation of the forward kinematics.

The trajectory error is defined as the position deviation of the Jacobian-based velocity integration from the analytical forward kinematics, $e_{\mathbf{x}}=‖{\mathbf{x}}_{FK}-{\mathbf{x}}_J‖$, where ${\mathbf{x}}_{FK}=f(\mathbf{q})$ denotes the end-effector position obtained from the analytical forward kinematics, and ${\mathbf{x}}_J$ is obtained by integrating the velocity predicted by the analytical Jacobian. The velocity error is defined as the discrepancy between the end-effector velocity predicted by the analytical Jacobian and that obtained via finite-difference approximation, $e_{\dot{\mathbf{x}}}=‖{\dot{\mathbf{x}}}_J-{\dot{\mathbf{x}}}_{FD}‖$, where ${\dot{\mathbf{x}}}_J=\mathbf{J}(\mathbf{q})\dot{\mathbf{q}}$ and ${\dot{\mathbf{x}}}_{FD}$ is computed from the analytical forward kinematics using the central finite-difference method:

$${\dot{\mathbf{x}}}_{FD}(k)=\frac{{\mathbf{x}}_{FK}(k+1)-{\mathbf{x}}_{FK}(k-1)}{2\Delta t},$$

where ${\mathbf{x}}_{FK}(k)=f(\mathbf{q}(k))$ is the end-effector position at the $k$-th sampled configuration and $\Delta t$ is the sampling interval.

For inverse kinematics validation, the reference tooltip poses obtained from the analytical forward kinematics, ${\mathbf{T}}_{bt}=f(\mathbf{q})$, were used as inputs to the analytical inverse kinematics, $\stackrel{\mathbf{ˆ}}{\mathbf{q}}=f^{-1}\left({\mathbf{T}}_{bt}\right)$. The consistency of the inverse kinematics was evaluated by comparing the reconstructed joint variables $\stackrel{\mathbf{ˆ}}{\mathbf{q}}$ with the original $\mathbf{q}$. In addition, a velocity-level validation was performed using the analytical Jacobian pseudo-inverse. The joint velocities recovered from the Jacobian pseudo-inverse, ${\dot{\mathbf{q}}}_{J^{†}}={\mathbf{J}}^{†}(\mathbf{q}){\dot{\mathbf{x}}}_{FD}$, were compared with the joint velocities obtained via the central finite-difference method:

$${\dot{\mathbf{q}}}_{FD}(k)=\frac{\mathbf{q}(k+1)-\mathbf{q}(k-1)}{2\Delta t}$$

3.2.2. Kinematic Analysis

The SHER 3.0 kinematic relations are modeled in Python and analyzed based on several kinematics-related metrics, such as reachable and dexterous workspaces, and manipulability index. The reachable and dexterous workspaces are evaluated and presented for the entire SHER 3.0 robot end-effector (linear Delta stage combined with the roll-tilt mechanism). The manipulability index is evaluated and presented for the linear parallel Delta stage and the roll-tilt mechanism separately.

In this work, the dexterous workspace of SHER 3.0 is defined as the subset of the reachable workspace that has more than one inverse kinematic solution. This concept aligns with prior microsurgical robot research [49].

The manipulability of the roll–tilt mechanism and the linear delta platform of SHER 3.0 are evaluated using Yoshikawa’s classical manipulability index [50], which quantifies the isotropy of velocity transmission between joint space and task space. The manipulability index is defined as:

$$w=\sqrt{\text{det}\left({\mathbf{J}}^T\mathbf{J}\right)}$$ (80)

where $\mathbf{J}$ is the Jacobian matrix of the robot, and $w$ represents the volume of the velocity manipulability ellipsoid, which indicates the ability of the end-effector to generate arbitrary motion directions given bounded joint inputs. A smaller $w$ corresponds to configurations where the mechanism loses directional controllability. It is worth noting that Yoshikawa’s manipulability index is meaningful only when the task-space variables are homogeneous, i.e., composed solely of translational or solely of rotational quantities. Since SHER 3.0 comprises decoupled translational and rotational subsystems, the manipulability analysis is performed separately for the linear Delta platform and the roll–tilt mechanism. Separate manipulability indices are defined using the corresponding Jacobians from Eq. (57), denoted as $w_D$ and $w_{RT}$, computed from ${\mathbf{J}}_{\mathbf{D}}$ and ${\mathbf{J}}_{\mathbf{R}\mathbf{T}}$ as shown below:

$$\begin{gathered} w_D=\sqrt{\text{det}({J_{\mathbf{\text{D}}}}^T{J}_{\mathbf{\text{D}}})} \\ {w}_{RT}=\sqrt{\text{det}({{\mathbf{\text{J}}}_{\mathbf{\text{RT}}}}^T{\mathbf{\text{J}}}_{\mathbf{\text{RT}}})} \end{gathered}$$ (81)

For the roll–tilt mechanism, although roll is actuated by a revolute joint and tilt is produced by a prismatic actuator through a four-bar linkage, these two motions are kinematically independent, and the roll joint does not influence the tilt velocity transmission, as shown in Fig. 16-a. Therefore, they are combined in the manipulability analysis. To facilitate interpretation of the manipulability distribution of the roll–tilt and the linear delta mechanism, a normalized manipulability index is introduced here:

$$w_{\mathit{\text{norm}}}=\frac{w}{w_{\mathit{\text{max}}}}$$ (82)

where $w_{\mathit{\text{max}}}$ is the maximum manipulability index within the considered joint and task space.

Fig. 16:

Normalized manipulability index of the roll–tilt mechanism: (a) overall distribution across the joint space; (b) near-singular configuration at $s=25\text{mm}$, with corresponding ${\theta }_{\mathit{\text{tool}}}={133.7}^{\circ }$; (c) near-singular configuration at $s=50\text{mm}$, with corresponding ${\theta }_{\mathit{\text{tool}}}={159.1}^{\circ }$. (d) Tool-to-stroke velocity ratio with stroke length ranging from 0 to 50 mm.

3.2.3. RCM Control Evaluation

To assess the performance of the robot kinematics and the MPC control algorithm in maintaining the RCM constraint safely and accurately, we conducted a pilot study experiment with five users in a teleoperation framework. Before the experiments began, the participants were asked to manipulate the digital twin of the robot using a haptic interface for 5 min, followed by another 5 min manipulation of the actual robot in free space (Figs. 7 and 9) to familiarize themselves with the setup and adjust the seat height, robot position, and microscope focus based on their preference (Fig. 10).

Fig. 9:

An example of the trajectory of the straight tool tip in a teleoperation experiment for the actual robot (a), and for that of its digital twin in AMBF (b).

Fig. 10:

Experimental procedure for testing the RCM control performance and the surgical tools’ approach angle (using both straight gripper tool and the I²RIS snake robot at the distal end of SHER 3.0) inside an eye phantom in a teleoperation framework.

This experiment includes inserting a surgical instrument – a straight gripper tool attached to the SHER 3.0 end-effector – through the trocar into the eye phantom, manipulating the instrument towards a set of target points (five points selected in a random order), and touching the points with the tool tip (this is considered as one trial). An electrical buzzer is used to confirm the touch. Each participant completed five trials, and the sequence of the target points was identical across all users for consistency. The robot end-effector motion was tracked using two external microscopes and ArUco markers for later analysis of RCM constraint accuracy.

3.2.4. Approach Angle Evaluation

It is challenging to approach the retinal surface with an optimum and controllable angle using the straight gripper tool, as it lacks a dexterous wrist-like motion capability, a problem that could be solved by integrating the I²RIS snake robot to the SHER 3.0 end-effector. In this experiment, we replaced the straight gripper tool with the I²RIS robot, which provides two additional DOFs, pitch and yaw bending angles at the distal end, enabling an optimum approach angle (for example, normal) to the retinal surface. The users were asked to manipulate the SHER 3.0 robot using the Omega 7 haptic interface and move it along the surface of a cylindrical eye phantom to reach 5 reference points along the surface, equally distributed between −20° and 20°. The users were supposed to bend the I²RIS robot to keep its tip as perpendicular as possible to the retina surface (see the top row of Fig. 18), while the MPC controller automatically enforces the RCM constraint for the shaft of the I²RIS robot passing through the trocar (bottom row of Fig. 18).

Fig. 18:

RCM constraint implementation on SHER 3.0: The shaft of the surgical instrument attached to the SHER 3.0 end-effector pivots through a trocar point (bottom row). The I²RIS snake robot, attached to the distal end of the surgical instrument, enables an optimum approach angle (perpendicular in this case) to the retinal surface (top row). Of note, the top row is the magnified view of the I²RIS distal end approaching the retina surface of the cylindrical eye phantom in the bottom row.

4. RESULTS

This section presents the results using three main approaches: a kinematic model validation that verifies the analytical formulation, a theoretical approach that performs a kinematic analysis of the SHER 3.0 robot, studying the kinematics performance of the robot mechanism design, followed by an experimental approach that validates the accuracy and performance of the RCM control implementation within a pilot teleoperation study experiment with five users.

4.1. Kinematic Model Validation

As shown in Fig. 11, the trajectories obtained from analytical forward kinematics, Jacobian-based, and finite-difference integration closely overlap. The corresponding errors in Fig. 12 remain small, on the order of 10⁻² mm in position, 10⁻⁷ mm/s in linear velocity, and 10⁻³ deg/s in angular velocity.

Fig. 11:

Tool-tip trajectories obtained from analytical forward kinematics, Jacobian-based integration, and finite-difference integration.

Fig. 12:

Forward kinematics validation: (a) Absolute position error between Jacobian-based integration and the analytical solution. (b) Linear velocity error between Jacobian and finite-difference methods. (c) Angular velocity error between Jacobian and finite-difference methods.

Similarly, Fig. 13 shows the joint position errors are on the order of 10⁻¹³ mm, while the roll joint error $q_4$ is effectively zero. The joint velocity validation shows small discrepancies on the order of 10⁻³ mm/s and 10⁻⁴ deg/s.

Fig. 13:

Inverse kinematics validation: (a) Joint position error between reconstructed and original values. (b) Velocity error between Jacobian pseudo-inverse and finite-difference methods.

4.2. Kinematic Analysis

4.2.1. Workspace Analysis

The reachable and dexterous workspaces of the tool tip of SHER 3.0 are shown in Fig. 14, expressed in the robot’s base frame, which includes the 3D workspaces (Fig. 14-a) and their corresponding 2D projections on the $X_b{Y}_b,X_b{Z}_b$, and $Y_b{Z}_b$ planes (Figs. 14-b to 14-d). The volumes of the reachable and dexterous workspaces are calculated as 8.878 × 10⁵ mm³ and 1.977 × 10⁵ mm³, respectively, with a dexterous-to-reachable ratio of 22.27% as shown in Table 2.

Fig. 14:

Reachable and dexterous workspaces of SHER 3.0: (a) 3D view and (b) $X_b{Y}_b$, (c) $X_b{Z}_b$, (d) $Y_b{Z}_b$ planar views.

Table 2:

Kinematic performance metrics of SHER 3.0. (Delta) and (RT) denote the linear Delta platform and the roll–tilt mechanism.

Metric	Value
Reachable workspace vol. (mm3)	8.878 × 105
Dexterous workspace vol. (mm3)	1.977 × 105
Dexterous-to-reachable ratio	22.27%
Max manipulability (Delta)	2.21
Normalized Min manipulability (Delta)	4.4 × 10−5
Normalized Mean manipulability (Delta)	0.426
Normalized Std. of manipulability (Delta)	0.275
Max manipulability (RT)	39.78
Normalized Min manipulability (RT)	0.046
Normalized Mean manipulability (RT)	0.286
Normalized Std. of manipulability (RT)	0.205

All experiments were conducted within a workspace limited by hardware limit switches. For the Delta platform, each limb prismatic actuator was constrained to a stroke of ± 30 mm, resulting in the translational workspace shown in Fig. 15-c. The slider crank linear actuator operated over a range of 0 to 50 mm, corresponding to a tool tilt angle of 109.6° to 159.1° between the $Y_t{Z}_t$ and $X_a{Y}_a$ planes. In addition, the tool roll angle was constrained to ± 90° between the $X_t{Z}_t$ and $X_p{Z}_p$ planes.

Fig. 15:

Normalized manipulability index of the delta platform: (a) distribution in joint space and (c) in task space. Plots (b) and (d) show corresponding 45° cross-sectional views of the joint space and task space, respectively.

4.2.2. Manipulability Analysis

For the linear delta platform, the normalized manipulability volume in joint space is plotted as a function of the three joint variables $q_1,q_2,q_3$, and in task space as a function of the Cartesian coordinates $x,y,z$ in the base frame (see Fig. 15). As shown in Figs. 15-a and 15-b, the linear delta mechanism has higher manipulability when the three joint values are similar. Correspondingly, Figs. 15-c and 15-d indicate that the region of highest manipulability forms an approximately cylindrical core near the center of the delta’s reachable task space.

The maximum, minimum, and mean values of the normalized manipulability index for the linear delta mechanism are 2.21, 4.4 × 10⁻⁵, and 0.426, respectively, with a standard deviation of 0.275, as shown in Table 2.

For the roll–tilt mechanism, the normalized manipulability surface is plotted as a function of the two joint variables $q_4=\psi$ and $q_5=s$, as shown in Fig. 16. It is observed that the roll-tilt mechanism has its minimum normalized manipulability index around the two near-singular configurations at joint variable $s\approx 25\text{mm}$ and $\approx 50\text{mm}$, with corresponding tool angles ${\theta }_{\mathit{\text{tool}}}={133.7}^{\circ }$ and ${\theta }_{\mathit{\text{tool}}}={159.1}^{\circ }$ (see Figs. 16-b and 16-c).

The maximum, minimum, and mean values of the normalized manipulability index for the roll-tilt mechanism are 39.78, 0.046, and 0.286, respectively, with a standard deviation of 0.205, as shown in Table 2.

4.3. RCM Control Results

In the implemented MPC framework, the system state is defined as the 6-dimensional tool-tip pose, $x=[x,y,z,\text{roll},\text{pitch},0{]}^T$, while the control input consists of the 5-dimensional joint-velocity command of the robotic manipulator, $u={\left[{\dot{q}}_1,{\dot{q}}_2,{\dot{q}}_3,{\dot{q}}_4,{\dot{q}}_5\right]}^T$, where the first three components correspond to the translational positioning joints and the remaining two correspond to the revolute tilt mechanism. Tool-tip motion is propagated over the prediction horizon using the robot Jacobian, with the prediction horizon set to 10 steps in the current implementation. The MPC cost function penalizes tooltip tracking error, joint-velocity magnitude, and RCM constraint violation using the weighting matrices $Q=\text{diag}(\mathrm{50,30,30,0},\mathrm{0,0}),R=\text{diag}(\mathrm{18,18,18,8},15)$, and $M=6000$, respectively. These weights prioritize accurate position tracking, particularly along the insertion direction, while strongly enforcing the RCM constraint for safe manipulation. Joint-velocity constraints are imposed according to the hardware limits, where ${\dot{q}}_1,{\dot{q}}_2$, and ${\dot{q}}_3$ are bounded within $[-\mathrm{5,5}],{\dot{q}}_4$ within [−1.0, 1.0], and ${\dot{q}}_5$ within [−2.8, 2.8]. The RCM constraint is formulated using a point-to-line distance metric between the trocar entry point and the tool shaft, computed from the tool-tip position and the roll/pitch-based tool-frame rotation matrix. The nonlinear MPC optimization problem is implemented in CasADi and solved using IPOPT with ipopt.max_iter=1000, ipopt.tol=10⁻¹, and ipopt.acceptable_obj_change_tol=10⁻¹. Warm-start initialization using the previous optimization solution is additionally employed to improve computational efficiency.

Five male users, with ages ranging from 24 to 40 (mean 29.8), participated in this pilot study teleoperation experiment. We employed the 10-item Edinburgh Handedness Inventory (EHI) score to evaluate the handedness of the users [51]. All users were right-hand dominant with a Laterality Quotient (LQ) of 100.00 ± 0.00. A total of 25 trials were collected from the users, and the results of the accuracy of the RCM control algorithm are summarized in Table 3. Fig. 17 illustrates an example of the trajectory (position and orientation) of the shaft of the straight gripper tool (attached to the SHER 3.0 end-effector), which passed through the trocar point on the eye phantom during the teleoperation experiments. The blue curve shows the tip trajectory of the surgical tool while touching the target points. The dashed lines represent the footprint of the position/orientation of the tool shaft during a single trial. The RCM error, defined as the distance between the closest point on the tool shaft axis and the center of the trocar entry point, was measured as 0.55 ± 0.12 mm, with a maximum observed error of 0.73 mm. These results demonstrate the feasibility of the proposed framework for maintaining the RCM constraint during robot-assisted eye manipulation in the phantom-based experimental setup.

Table 3:

Results for the RCM control experiment

Metric	Value
Max. RCM error (mm)	0.73
Min. RCM error (mm)	0.01
Mean RCM error (mm)	0.55
Std. of RCM error (mm)	0.12

Fig. 17:

Visualization of one trial from the RCM experiments: (a) tool tip path with the start and end tool positions highlighted, and (b) $X_b{Z}_b$ plane, (c) $Y_b{Z}_b$ plane projections.

4.4. Approach Angle Results

The approach angle of the straight tool gripper and I²RIS snake gripper were measured when the gripper was aligned with the reference points. It can be observed that it is challenging to maintain a consistent (close to normal) approach angle using the straight tool, as it varied from 105.6° to 74.9° (see the straight tool in Table 4 and Fig. 19). Conversely, with the use of the I²RIS snake robot, a relatively consistent approach angle of approximately 90° is observed for all reference points (see the I²RIS snake in Table 4 and Fig. 19). This limitation of the straight tool is because the orientation of the tool shaft is constrained at the trocar to satisfy the RCM requirement. The resulting lack of control over the distal approach angle reduces surgical dexterity during procedures such as epiretinal membrane (ERM) peeling, which requires smooth delamination of the membrane along a helical trajectory to minimize hemorrhage. In contrast, the I²RIS snake robot enables a wrist-like motion, providing pitch, yaw, and gripping motions that enhance surgeon dexterity and facilitate safer and more efficient ERM peeling.

Table 4:

Approach angle of I²RIS snake gripper vs. straight tool gripper (mean ± std) over five different reference points.

Reference Point	I2RIS snake	Straight tool
−20°	92.6° ± 1.8	105.6° ± 2.1
−10°	94.1° ± 3.3	96.6° ± 2.3
0°	92.8° ± 3.3	84.4° ± 1.4
10°	87.7° ± 1.8	79.4° ± 1.4
20°	88.7° ± 1.7	74.9° ± 1.0

Fig. 19:

Comparison of approach angles across five reference points at the bottom of the cylindrical eye phantom: using the I²RIS snake robot vs. the straight gripper tool.

5. DISCUSSION AND FUTURE WORK

The kinematic formulation of the SHER 3.0 robotic system, including the forward and inverse kinematics and their corresponding Jacobian and pseudo-inverse (Eq. (27), Eqs. (30)–(33), Eq. (57), and Eq. (75)), is derived in closed form. The kinematic model validation confirms the correctness of the derived forward and inverse kinematics, as well as the Jacobian formulation, demonstrating that the analytical model is mathematically consistent for control and planning applications. This closed-form formulation reduces computational load and facilitates real-time implementation. The workspace analysis demonstrates that the dexterous workspace of SHER 3.0 is sufficient to cover the target operational region, corresponding to the patient’s eyeball with an approximate diameter of 24 mm. This confirms that the robot can achieve full dexterity within the surgical workspace, enabling flexible tool positioning and orientation during microsurgical procedures. The manipulability analysis shows two notable singularities: at $s\approx 25\text{mm}$, the slider-crank linkage approaches a configuration in which the transmission of motion to the four-bar mechanism is reduced, which is consistent with the discovery in the prior work [30] for this type of tilt mechanism. In this configuration, the velocity of the joint connected to the four-bar linkage becomes significantly smaller relative to the actuator input, indicating a loss of velocity transmission efficiency. As a result, maintaining a desired tool-tip velocity would require a large actuator speed, corresponding to a near-singular configuration of the mechanism. At $s\approx 50\text{mm}$, the finite stroke of the linear actuator approaches its limit, restricting the feasible range of motion of the roll-tilt mechanism. These findings highlight the regions in which the robot operates with reduced manipulability and can inform safe operational boundaries during surgical tasks.

The RCM error was measured to be 0.55 ± 0.12 mm, with a maximum value of 0.73 mm. This level of error is comparable to the range of RCM errors reported by Alikhani et al. for a similar robotic application [52], supporting the feasibility of the proposed kinematics and control methods for phantom-based RCM constraint implementation. Assessment of clinical safety will require further experimental evaluation under clinically relevant conditions, including tissue compliance, trocar clearance, contact forces, and off-axis loading, which were beyond the scope of this study.

Regarding the gripper’s approach angle, the straight tool shows limited controllability over the approach angle at different locations of the retina, due to the RCM constraint enforcement at the trocar, which limits the tool shaft orientation. As such, it is difficult to approach the anatomy at optimal angles for microsurgical procedures such as ERM peeling, which requires smooth delamination of the ERM along a helical trajectory to minimize hemorrhage. The I²RIS snake robot allows for additional pitch, yaw, and gripping motions beyond the trocar, at the distal end of the instrument shaft. This wrist-like motion provides surgeons with the necessary dexterity to perform ERM peeling safely and efficiently.

One limitation of this work is the number of users (five). This is a pilot study that focuses on the validation of the proposed kinematics models and the feasibility of the SHER 3.0 and I²RIS robotic systems and the MPC control algorithm in enforcing the RCM constraint in an artificial eye phantom. Another limitation is that the eye phantom used in this experiment is a 3D-printed component that simulates the RCM constraint at the eye sclerotomy. Still, it has a relatively rigid trocar without sufficient deformability to represent that of human eye sclera tissue. To address these limitations in our future work and to improve experimental realism, we plan to extend the study to multiple users and use biological phantoms, such as ex vivo porcine eyes, to better mimic the structural and mechanical properties of real tissues, and to further analyze the performance of this robotic system in ERM peeling procedures.

6. CONCLUSION

This work presents, for the first time, a thorough analytical forward and inverse kinematics and Jacobian models for the SHER 3.0, featuring a Delta-based parallelserial mechanism. Based on these analytical models, an MPC algorithm is designed to maintain an RCM constraint at the eye sclerotomy during robot-assisted retinal microsurgical procedures. A kinematic analysis is performed to calculate the reachable and dexterous workspaces of the robot. Moreover, the manipulability performance of the SHER 3.0 is evaluated across the robot’s joint space and end-effector task space. The proposed robotic platform is also designed to accommodate the I²RIS snake robot, whose distal wrist-like articulation provides additional pitch, yaw, and gripping motions beyond the trocar. This capability addresses the limited approach angles of conventional straight instruments under RCM constraints and has the potential to improve dexterity during delicate retinal microsurgical tasks, such as epiretinal membrane (ERM) peeling. Finally, the accuracy of the MPC-based controller in enforcing the RCM constraint is evaluated in a pilot teleoperation experiment. Experimental results demonstrate that the proposed kinematics and Jacobian models, together with the MPC control algorithm, achieve accurate RCM constraint enforcement and support the phantom-based feasibility of the proposed robotic system for robot-assisted retinal surgery. Further experimental validation under clinically relevant conditions, including tissue compliance, trocar clearance, contact forces, and off-axis loading, will be required to establish clinical safety.

DATA AVAILABILITY STATEMENT

The authors attest that all data related to this study are included in the paper.