2.1. Concept

The parallel robot presented herein consists of four actuated chains that connect the moving platform to the base of the robot, that is, the bone. The connection of the robot to the bone is realized by directly attaching the robot’s legs to the bone. The legs’ position relative to the bone could be fixed based on a noninvasive concept, for example, suction cups [Reference Eugster, Oliveira Barros, Cattin and Rauter54] or balloon catheters. Details on the attachment of the legs to the bone are a research topic per se, and will therefore not be further discussed in this article. The moving platform of the parallel robot is the endoscope’s tip segment, which houses the laser optics. Each actuated chain is of the RRP type, that is, two passive rotary joints followed by an actuated prismatic joint. In total, there are four prismatic actuators; thus, this robot structure is abbreviated as 4-RRP (Fig. 3, visualization of the robot topology and motion available as supplementary video material). This structure allows movement of the laser optics (see (5) in Fig. 3) in two translational DoFs $\left\lbrace \vec{e}^B_x, \vec{e}^B_y \right\rbrace$ , and one rotational DoF $\varphi_e$ (see (3) in Fig. 3). The origin B of the robot’s base frame is defined to be located at the center point of the connecting line between the two attachment points of the robot on the bone. The x-axis of the frame is directed toward the right attachment point $\vec{p}_r$ , and the z-axis is defined to be parallel to the robot’s legs ( $\vec{n}_1$ and $\vec{n}_2$ , Fig. 3).

Figure 3. The parallel robot has two legs (1) that are firmly attached to the bone at $\left\lbrace \vec{p}_l,\vec{p}_r \right\rbrace$ and directed along $\left\lbrace \vec{n}_1,\vec{n}_2 \right\rbrace$ . The right leg (attached at $\vec{p}_r$ ) consists of two revolute joints $\left\lbrace R_1,R_3 \right\rbrace$ with rotation axis $\vec{n}_1$ . Revolute joint $R_1$ is connected to a link $L_1$ (2) with a fixed length of $d_a$ . The other end of $L_1$ is connected to a further revolute joint $R_2$ , which can slide along the supporting link $A_1B_1$ . The motion of $R_2$ along $A_1B_1$ is controlled by prismatic actuator $P_1$ , which adjusts the distance between $A_1$ and $R_2$ . The second link $L_2$ is connected to $R_3$ and $R_4$ , with $R_4$ sliding along the supporting link $A_2B_2$ . Prismatic actuator $P_2$ controls the motion of $R_4$ along $A_2B_2$ . The same arrangement is established for the other leg, which is anchored on the bone at $\vec{p}_l$ with actuators $ \left\lbrace P_3,P_4 \right\rbrace$ . The positions of the prismatic joints are denoted by $\rho_i, i \in \left\lbrace 1,2,3,4 \right\rbrace$ . The two legs anchored on the bone define the base (3) of the parallel robot. The endoscope tip (4), which houses the laser optics (5), is the moving platform of the mechanism. The robot enables the movement of the laser optics, that is, the end-effector frame E, in three DoFs.

When the robot is attached to a surface, the number of controlled DoFs (three) of the end effector is smaller than the number of actuators (four); thus, the mechanism is a so-called redundant mechanism. If one of the two legs is not attached to a surface, that is, is moving in free space, the robot is not redundant. Since two actuators control the movement of each parallel robot leg, the legs can be positioned in two DoFs when they are not attached to the bone. This allows the leg and arm structure to be brought close to the robot’s main body for reducing the robot’s cross section during insertion into and retraction from the patient. Additionally, the actuation redundancy enables the robot to translate along the bone by successively repositioning its legs and therefore expanding the workspace [Reference Eugster, Oliveira Barros, Cattin and Rauter54]. A schematic visualization of this insertion, retraction, and leg repositioning is provided as Supplementary video material. Although these two features are not the focus of this work, it is essential for future work that the mechanical design of the miniature robot, the actuation concept, and the position sensing will allow the independent movement of each leg of the parallel robot.

2.2. Actuation

For minimally invasive surgery, extrinsic actuation, that is, actuators that are placed outside the patient, can be beneficial as it facilitates the design of simpler, smaller, and lighter components of the instrument parts that enter the patient. Furthermore, possible safety risks can be avoided, for example, electrical current or high-pressure hydraulics do not have to be transmitted inside the patient. However, placing the actuator outside the patient means that motion transmission elements are required, which have the drawbacks of energy losses (e.g., due to friction), backlash, and compliance. Despite existing drawbacks of extrinsic actuation, high-energy densities can be introduced in small structures from outside, while the possibility for further miniaturization remains.

The upscaled prototype of the parallel structure presented in previous work [Reference Eugster, Cattin, Zam and Rauter51] was actuated based on tendons, Bowden tubes, and pulleys. The tendons were attached to sliders that moved on linear rails. However, this prototype was built with the usage of low friction and high accuracy linear rails with built-in encoders. While tendon-driven mechanisms allow miniaturization, friction between the tendons and Bowden tubes, tendon wear, and tendon elongation under load decrease the accuracy and dynamic response of the actuator. These effects do not decrease with decreased size of the device. Furthermore, the usage of pulleys reduces the miniaturization potential since the smallest realizable pulley size depends on the limited bending radius of the tendons. Also, the used encoders for slider position feedback were not available in the required dimensions. Thus, we decided to design a novel actuation concept for the miniaturized version of the end effector.

In the design for the miniaturized parallel robot, the prismatic actuators are realized as follows: Motors outside the patient are connected to flexible shafts. These flexible shafts are guided through the robotic endoscope and transfer the motor rotation to leadscrews inside the endoscope tip. These leadscrews then transfer the rotary motion of the flexible shafts into linear movements of nuts, that is, the prismatic actuators (Fig. 4). This extrinsic actuation facilitates the device miniaturization and allows using off-the-shelf electrical motors. Furthermore, by realizing the prismatic actuation as a leadscrew–nut pair, a transmission gear is directly implemented at the last moving element of the actuator. Due to the gear ratio, it is possible to measure the position of the nut by measuring the rotation of the flexible shaft using the motor encoders outside the patient body, thus eliminating the need for integration of very accurate sensors in the miniature device. To the best of our knowledge, there are only a few other devices for MIS where the benefits of the combination of flexible shafts and screws were exploited, that is, refs. [Reference Ibrahim, Ramadan, Fanni, Kobayashi, Abo-Ismail and Fujie55] and [Reference Sekiguchi, Kobayashi, Tomono, Watanabe, Toyoda, Konishi, Tomikawa, Ieiri, Tanoue, Hashizume and Fujie56].

Figure 4. The actuation concept for the miniature parallel robot: Flexible shafts are transmitting the rotation $\alpha_{i}$ from the externally placed electrical motors to the rotation of the leadscrews inside the end effector. These leadscrews then transfer the rotary motion of the flexible shafts into linear movements of nuts $\rho_{i}$ , that is, the prismatic actuators. The miniature robot houses four such prismatic actuators.

2.3. Position sensing

For closed-loop control, direct absolute measurements of the positions of the actuated prismatic joints of the parallel robot and, thus, the nut positions on the leadscrews would be desirable. However, no commercial sensor is available with dimensions that would allow both direct integration into the endoscope tip and measurement of the nut positions in the submillimeter range. Therefore, for the setup presented herein, the nut positions are calculated based on the motor angles $\alpha_{i}$ , which are measured by encoders integrated into the motors located outside the patient. The motor rotation is transmitted to the leadscrews inside the endoscope tip by flexible shafts, which may introduce a rotation error $\beta_{i}$ due to torsional elasticity. The leadscrews transform the rotary motion into linear movements of the nuts. This transformation may introduce an error due to backlash $\overline{\delta}_i$ between the leadscrew and the nut. With the leadscrew thread pitch p and under the assumption of a perfect initial calibration, the nut positions can be calculated as follows:

(1) \begin{equation}\begin{aligned}\overline{\rho}_i = \frac{p}{{360}^{\circ}} \left( \overline{\alpha}_{i}+ \overline{\beta}_{i} \right) + \overline{\delta}_i \quad \forall i \in \left\lbrace 1,2,3,4 \right\rbrace\!,\end{aligned}\end{equation}

where $\overline{\rho}_i,\overline{\alpha}_i$ , $\overline{\beta}_i$ , and $\overline{\delta}_i$ are the true values of the nut position $\rho_i$ , the motor angle $\alpha_{i}$ , the rotation error $\beta_{i}$ , and the backlash $\delta_i$ . The measurement result $\rho_i$ is only an estimate of the true value of the nut position $\overline{\rho}_i$ and thus is complete only with a statement of the expected measurement error $\Delta_{\rho_i}\!$ , which is assumed to be smaller than the expanded measurement uncertainty $U_{\rho_i}$ :

(2) \begin{equation}\begin{aligned}\rho_i &= \overline{\rho}_i + \Delta_{\rho_i}, \quad \text{with} \quad \lvert\Delta_{\rho_i}\rvert \leq U_{\rho_i} \quad \forall i \in \left\lbrace 1,2,3,4 \right\rbrace.\end{aligned}\end{equation}

Since we use these nut position measurements $\rho_i$ for closed-loop control of the end effector, we have to ensure that the uncertainty in the end-effector position due to the uncertainty of the nut positions leads to an error that is smaller than the positioning accuracy we want to achieve ( $0.25\,\text{mm}$ ). To provide an interval that is expected to encompass a large fraction of the distribution of values around the true nut positions, we calculate the expanded measurement uncertainty $U_{\rho_i}$ with a coverage factor of $k=2$ , according to ref. [57]:

(3) \begin{equation}\begin{aligned}U_{\rho_i} &= k \cdot u_{\rho_i} \quad \forall i \in \left\lbrace 1,2,3,4 \right\rbrace.\end{aligned}\end{equation}

Three contributors to the measurement uncertainty $u_{\rho_i}$ are considered; the limited resolution of the motor encoders, the torsional elasticity of the flexible shafts, and the axial play of the leadscrew–nut pair. The limited resolution of the motor encoders leads to a measurement uncertainty $u_{\alpha_{i}}$ on the motor angles. In addition, the flexible shafts have a torsional elasticity that allows deformations. This leads to an additional uncertainty $u_{\beta_{i}}$ . The axial play of the leadscrew–nut pair introduces backlash that leads to an uncertainty of the nut position $u_{\delta_i}$ . Due to the linearity of (1), these three uncertainties can be combined to obtain the uncertainty of the measured nut position $\rho_i$ as follows:

(4) \begin{equation}\begin{aligned}u_{\rho_i} &= \sqrt{ \left( \frac{p}{{360}^{\circ}} \right) ^2 \left( u_{\alpha_{i}}^2+u_{\beta_{i}}^2 \right) +u_{\delta_i}^2}.\end{aligned}\end{equation}

The uncertainties $u_{\alpha_{i}}$ , $u_{\beta_{i}}$ , and $u_{{\delta}_i}$ are taken to follow an approximately uniform distribution on a corresponding interval $\left[a_{\gamma}^-, a_{\gamma}^+ \right]$ :

(5) \begin{equation}\begin{aligned}u_{\gamma} &= \frac{a_\gamma^+ - a_\gamma^-}{2\sqrt{3}}, \quad \gamma \in \left\lbrace \alpha_i,\beta_i,\delta_i \right\rbrace.\end{aligned}\end{equation}

The resulting measurement error $\vec{\Delta}_{\rho}$ (with $\lvert\Delta_{\rho_i}\rvert \leq U_{\rho_i }$ ) on the positions of the prismatic actuators will influence the pose error $\vec{\Delta}_{\chi_e}$ of the end effector. A linear relationship exists between these two quantities:

(6) \begin{equation}\begin{aligned}\vec{\Delta}_{{\chi}_e} &= \textbf{J} (\vec{\rho}) \vec{\Delta}_{\rho},\end{aligned}\end{equation}

where $\textbf{J}$ is the pose-dependent Jacobian matrix of the parallel robot. The Jacobian matrix can be directly determined by differentiating the direct kinematics equations (see Section 2.4.1). All entries of the Jacobian matrix are converted into positive values to calculate the maximal expected uncertainty of the end-effector pose.

These calculations allow us to estimate the maximal expected uncertainty of the end-effector position based on the modeled uncertainty of the nut position measurement for a given joint configuration. This value is a lower bound for the achievable positioning accuracy. Since not all errors can be known in advance, the lower bound of achievable positioning accuracy has to be smaller than the required positioning accuracy. To verify that this criterion is met, the theoretical minimal end-effector position errors will be calculated for the conducted path-following experiment. We are aware that additional error sources such as mechanical play or manufacturing inaccuracies will be present. However, this analysis is a first necessary step for choosing appropriate hardware components. The determination of other uncertainties first requires an experimental setup.

2.4. Kinematics

The end-effector pose is characterized by the pose of the body-fixed end-effector coordinate frame E in relation to the bone-fixed inertial coordinate frame I. Since the end effector moves in a planar way, the motion can be described by three variables:

(7) \begin{equation}\vec{\chi}_e = \left( \begin{array}{c c c}x_e & y_e & \varphi_e\end{array} \right)^T \in \mathbb{R}^3.\end{equation}

Here, the end-effector position $\left\lbrace x_e, y_e \right\rbrace$ corresponds to the point $\vec{p}_e$ on the moving platform, which is the location at which the laser exits. The orientation $\varphi_e$ corresponds to the angle of the moving platform with respect to the base, as shown in Fig. 5. The actuator positions are the positions of the nuts on the leadscrews:

(8) \begin{equation}\vec{\rho} = \left( \begin{array}{c c c c }\rho_1 & \rho_2 & \rho_3 & \rho_4\end{array} \right)^T.\end{equation}

Figure 5. Top view of the parallel mechanism and variable notation: The origin $\vec{p}_e$ of the end-effector frame E is located at the point, where the laser exits the robot. The y-axis of the end-effector coordinate frame E is parallel to the line of symmetry of the moving platform, and the x-axis is oriented parallel to the plane spanned by $\vec{e}_x^I$ and $\vec{e}_y^I$ and points toward the right leg ( $\vec{p}_r$ ). The joint coordinates $\rho_i, i \in \left\lbrace 1,2,3,4 \right\rbrace$ , are defined as the y components of the nut positions in the frame of the moving platform, M. The geometrical parameters of the mechanism are the arm length, $d_a$ ; the distance between the connecting lines $\left\lbrace L_l,L_r \right\rbrace$ between the rotary joints on the left and right sides of the platform, $d_s$ ; the position coordinates of the end effector on the moving platform, $\left\lbrace d_{e_x},d_{e_y} \right\rbrace$ ; the length of the leadscrew, $d_r$ ; and the distance between the attachment points $\left\lbrace \vec{p}_l, \vec{p}_r \right\rbrace$ of the left and right legs, $d_{lr}$ . The lines $L_{r,l}$ represent the supporting links $\left\lbrace A_1B_1, A_2B_2\right\rbrace$ and $ \left\lbrace A_3B_3, A_4B_4 \right\rbrace$ , as shown in Fig. 3.

The definitions of the notation used in the following are illustrated in Fig. 5. The parameters $h_l$ and $h_r$ denote the distances between the left and right legs, respectively, and the connecting line between the two rotational joints around which the arms connected to the corresponding leg rotate. These parameters can be calculated based on the joint positions and the arm length $d_a$ , as follows:

(9) \begin{equation}\begin{aligned}h_l &= \sqrt{d_a^2-\frac{1}{4}(\rho_3-\rho_4)^2}, \\[3pt] h_r &= \sqrt{d_a^2-\frac{1}{4}(\rho_2-\rho_1)^2}.\end{aligned}\end{equation}

These parameters are used to calculate the direct and inverse kinematics of the parallel robot in the following sections. The direct kinematics are presented in previous work [Reference Eugster, Cattin, Zam and Rauter51]. For the inverse kinematics, a new derivation is formulated that facilitates the workspace calculation.

2.4.1. Direct kinematics

The position and orientation of the end effector, $ \left\lbrace x_e, y_e , \varphi_e \right\rbrace$ , given a specific configuration $\vec{\rho}$ of the robot, can be calculated based on the distance between the right anchor point and the end-effector position in the end-effector frame as follows:

(10) \begin{equation}\begin{aligned}x_e &= \frac{d_{lr}}{2}- \left( h_r +\frac{d_s}{2}-d_{e_x} \right) c\varphi_e- \left(d_{e_y}-\frac{\rho_1+\rho_2}{2} \right) s\varphi_e, \\[3pt] y_e &= \left( d_{e_y}-\frac{\rho_1+\rho_2}{2} \right) c\varphi_e - \left( h_r +\frac{d_s}{2}-d_{e_x} \right) s\varphi_e,\\[3pt] \varphi_e &= \arctan\left(\frac{\rho_3 + \rho_4 - (\rho_2 + \rho_1)}{2(h_l+d_s+h_r)}\right).\end{aligned}\end{equation}

Alternatively, the direct kinematics can also be calculated based on the distance between the left anchor point and the end-effector position, see Appendix B.

2.4.2. Inverse kinematics

To calculate the joint coordinates $\vec{\rho}$ based on a known end-effector pose $\vec{\chi}_e$ , we can rearrange the direct kinematic equations to obtain

(11) \begin{equation}\begin{aligned}\rho_{1,2} &= a_1 \mp \sqrt{a_2},\\[3pt] \rho_{3,4} &= a_3 \pm \sqrt{a_4},\end{aligned}\end{equation}

where

(12) \begin{align}a_1 &= x_e s\varphi_e -y_e c\varphi_e - b_1 s\varphi_e + b_2 c\varphi_e, \nonumber\\[2pt] a_2 & = d_a^2-\left(\!-x_e c\varphi_e - y_e s\varphi_e + b_1 c\varphi_e + b_2 s\varphi_e\right)^2,\nonumber\\[2pt] a_3 &= x_e s\varphi_e -y_e c\varphi_e - b_3 s\varphi_e + b_4 c\varphi_e,\nonumber\\[2pt] a_4 & = d_a^2-\left(x_e c\varphi_e+y_e s\varphi_e-b_3 c\varphi_e+b_4 s\varphi_e\right)^2,\nonumber\\[2pt] b_1 &= \frac{d_{lr}}{2}-d_{e_y} s\varphi_e-\left(\frac{d_{s}}{2}-d_{e_x}\right) c\varphi_e, \\[2pt] b_2 &= d_{e_y} c\varphi_e-\left(\frac{d_{s}}{2}-d_{e_x}\right) s\varphi_e, \nonumber\\[2pt] b_3 &= -\frac{d_{lr}}{2}-d_{e_y} s\varphi_e+\left(\frac{d_{s}}{2}+d_{e_x}\right) c\varphi_e, \nonumber\\[2pt] b_4 &= d_{e_y} c\varphi_e+\left(\frac{d_{s}}{2}+d_{e_x}\right) s\varphi_e.\nonumber\end{align}

We define $\rho_1$ and $\rho_4$ as the position of the nuts that are closer to the origin of the frame M on the moving platform (see Fig. 5). More details on the derivation of the inverse kinematics is provided in Appendix B.

2.5. Workspace

The parallel robot has three DoFs; two correspond to translational motion, $ \left\lbrace \vec{e}^I_x, \vec{e}^I_y \right\rbrace$ , and one corresponds to rotational motion, $\varphi_e$ . For each point $\left\lbrace x_e,y_e \right\rbrace$ that is part of the robot’s workspace, there is at least one nonempty interval for the angle $\varphi_e$ . The workspace of the robot is restricted by various factors, such as the mechanical limits of the passive joints, self-collision between the elements of the robot, and limitations due to actuators and singularities [Reference Merlet58]. Based on the constraints that limit the robot’s movement, the robot’s workspace can be calculated. For a specified robot position $\left\lbrace x_e,y_e \right\rbrace$ , we can obtain exactly the rotational workspace as a set of intervals. The constraints that limit the robot’s workspace and the procedure to calculate the rotational workspace for a known robot position $\left\lbrace x_e,y_e \right\rbrace$ are described in the following.

The first constraint is based on the condition that the inverse kinematic equations (11) must yield a real-valued solution for the square roots. Therefore, the values under the square roots must be larger than or equal to zero:

(13) \begin{equation}\begin{aligned}a_2 \geq 0 \quad \text{and} \quad a_4 \geq 0.\end{aligned}\end{equation}

Furthermore, there is a mechanical constraint that requires that the anchor points must be beside the moving platform and cannot be below it, since this would lead to collision between the arms and the leadscrews. To formulate this constraint, we introduce additional variables as follows. The lines $L_r$ and $L_l$ represent the supporting links $\left\lbrace A_1 B_1, A_2 B_2 \right\rbrace$ and $\left\lbrace A_3 B_3, A_4 B_4 \right\rbrace$ :

(14) \begin{equation}\begin{aligned}L_{l,r} \left( x,y \right) |y &= m x +n_{l,r}.\end{aligned}\end{equation}

The slope m of these lines can be calculated as

(15) \begin{equation}\begin{aligned}m = - \tan\left(\frac{\pi}{2} - \varphi_e\right).\end{aligned}\end{equation}

The y-intercepts of the lines $L_{l,r}$ in the B frame can be calculated by inserting the points $\{ _{B} \vec{p}_{L_l}, _{B}\vec{p}_{L_r}\}$ into the line equations defined in (14), which leads to

(16) \begin{equation}\begin{aligned}n_l = \left(y_e - \left(\frac{d_{s}}{2}+d_{e_x}\right) s\varphi_e \right) -m\left(x_e - \left(\frac{d_{s}}{2}+d_{e_x}\right) c\varphi_e\right),\\[3pt] n_r = \left(y_e + \left(\frac{d_{s}}{2}-d_{e_x}\right) s\varphi_e \right) -m\left(x_e + \left(\frac{d_{s}}{2}-d_{e_x}\right) c\varphi_e\right).\end{aligned}\end{equation}

The constraint that the positions of the legs must be beside the moving platform and cannot be below it can be formulated by limiting the x-intercept of the line $L_l$ in the B frame to be located on the right side of the left anchor point and the x-intercept of the line $L_r$ on the right side of the robot to be located on the left side of the right anchor point:

(17) \begin{equation}\begin{aligned}x_{L_l(y = 0)} &\geq -\frac{d_{lr}}{2} &\Rightarrow & \qquad\frac{d_{lr}}{2}-\frac{n_l}{m} \geq 0, \\[3pt] x_{L_r(y = 0)} &\leq \frac{d_{lr}}{2} &\Rightarrow & \qquad \frac{d_{lr}}{2}+\frac{n_r}{m} \geq 0.\end{aligned}\end{equation}

The robot has two relevant singular configurations, which correspond to the cases in which the arms are parallel on one side of the robot. The first configuration occurs when the nuts are separated by the maximal distance and the second occurs when the nuts are in the same position and the arms are coincident. The closeness to such a singularity can be limited by specifying limits on the variables $h_r$ and $h_l$ . These variables denote the distances between the right and left legs, respectively, and the connecting line between the two rotational joints around which the arms connected to the corresponding leg rotate. This constraint results in four inequalities:

(18) \begin{equation}\begin{aligned}h_{l_{\min}} &\leq h_l \leq h_{l_{\max}}, \\[3pt] h_{r_{\min}} &\leq h_r \leq h_{r_{\max}}.\end{aligned}\end{equation}

In addition, the actuator strokes are limited, meaning that the nut positions $\rho_i$ are bounded:

(19) \begin{equation}\begin{aligned}\rho_i \in \left[\rho_{\min},\rho_{\max}\right] \quad \forall i \in \left\lbrace 1,2,3,4 \right\rbrace.\end{aligned}\end{equation}

Note that based on the inverse kinematics Eqs. (11), the actuators are further constrained such that $\rho_1\ \leq\ \rho_2$ and $\rho_4\ \leq\ \rho_3$ , respectively.

Based on these constraints, the robot’s rotational workspace can be calculated using interval arithmetic. More exactly, the end effector is at the workspace border if any of the inequality variables (13), (17), (18), or (19) reaches its limit. By investigating the edge case of each inequality equation, we can find the border of the workspace. Each of the edge case equalities can be transformed into second order polynomials in T using the Weierstrass substitution

(20) \begin{equation}\begin{aligned}\sin(\varphi_e) = \frac{2T}{1+T^2}, \quad \cos(\varphi_e) = \frac{1-T^2}{1+T^2}. \end{aligned}\end{equation}

The real-valued roots $T_{{r}_j}$ of all 16 edge case polynomials are calculated and stored in a value sorted list. A value smaller than the first entry and a value bigger than the last entry are appended at the beginning and end of the list, respectively. This sorted root list forms the intervals that must be checked to be part of the reachable workspace. For each interval $\left[ T_{{r}_j}, T_{{r}_{j+1}} \right]$ , the midpoint is calculated as $T_{mid_j} = (T_{{r}_j}+T_{{r}_{j+1}})/2$ , and at this midpoint, the inequality variables (13), (17), (18), and (19) are tested. If none of the inequalities is violated, then the interval $\left[ T_{{r}_j}, T_{{r}_{j+1}} \right]$ is part of the reachable rotational workspace; otherwise, the interval is discarded. The resulting boundaries of the reachable rotational workspace are translated back into bounds on the angle $\varphi_e$ by using the transformation $\left[ \varphi_{e_{j}}, \varphi_{e_{j+1}} \right]= 2\arctan(\left[ T_{{r}_j}, T_{{r}_{j+1}} \right])$ . This procedure yields the set of valid intervals of the rotation angle $\varphi_e$ for each end-effector point $\left\lbrace x_e,y_e \right\rbrace$ . Each point $\left\lbrace x_e,y_e \right\rbrace$ for which there exists at least one nonempty interval for $\varphi_e$ is part of the translational workspace.

The geometrical parameters of the robot directly influence the shape and size of the workspace. For the prototype that is described in this work, we selected these parameters based on a compromise between minimizing the dimensions of the prototype, maximizing the resulting translational workspace of the robot, and simplifying the manufacturing and assembly process while reducing production costs. For that purpose, the translational workspace was calculated for all feasible parameter sets, and the set that resulted in the largest translational workspace was selected. The resulting workspace of the miniature robot for the geometrical parameters $d_s = {7.8}\,\text{mm}, d_a = {3}\,\text{mm}, d_{lr} = {11.5} \mathrm{mm}, d_{e_x} = {0}\mathrm{mm}$ and $d_{e_y} = {7}\,\mathrm{mm}$ and constraints of $\rho_{\min} = {0}\,\mathrm{mm},\rho_{\max} = {13}\,\mathrm{mm}, h_{{l,r}_{\min}} = {0}\,\mathrm{mm}, $ and $h_{{l,r}_{\max}} = {3}\,\mathrm{mm}$ is shown in Fig. 6.

Figure 6. The robot’s translational and rotational workspace: The rotational workspace range is shown by the marked section of the circle at each corresponding position in the translational workspace. The gray values of the circle sections encode the range of the feasible angle. The gray value of the circle border encodes the total size of the feasible angle range at that position. The filled green circles mark locations where the valid interval for $\varphi_e$ consists of two disjoint intervals. The arrows indicate the maximal straight cut lines in the workspace: vertical, horizontal, and overall.

3.1. Hardware design

The hardware of the miniature parallel robot prototype consisted of a housing, which held the optical components for the laser, four drivetrains, which enabled the linear movement of the nuts along the leadscrews, and two legs, which connected the two arms on each side of the robot to the ground (Fig. 7). The movement of the nuts was enabled by the leadscrews (standard M1.2 thread), which transformed rotary motion into linear motion. The leadscrews were mounted in the housing via ball bearings (SD 1425XZRY, MPS Microsystems, Biel–Bienne, Switzerland). The rotation of the nuts was constrained by the leg structure and a guiding rod. The leg structure aligned the orientation of the two nuts on each side of the mechanism, while the rod constrained the orientation of the upper two nuts. The legs were constructed based on ejector pins with a diameter of 1 mm, which could be mounted in drill holes on a ground plate. The distance to the ground was defined by an aluminum socket. The rotary motion was generated by extrinsic electrical motors and was transmitted to the endoscope tip segment by flexible shafts (Schmid & Wezel GmbH, Sinsheim, Germany) with an outer diameter of 1.4 mm and a length of 325 mm. These flexible shafts were constructed of steel wire ( $\phi$ 0.2 mm), wound into coils, alternately twisted in the right or left direction. This construction provides torsional stiffness while maintaining bending flexibility. The minimal bending radius of the flexible shafts was approximately 20 mm, and their dynamic torque capacity in both directions of rotation was estimated to be 5Ncm. Due to the manufacturing technique, each flexible shaft had a hollow core ( $\phi$ 0.6 mm), which was used to mount it on the leadscrew. Since there is not sufficient space to implement interlocking mechanisms such as keys or screws, each leadscrew was glued inside the corresponding flexible shaft. The laser optics inside the prototype consisted of a ferrule (SFLC127-10, Thorlabs Inc., Newton, NJ, USA) holding the laser fiber (FG105LCA, Thorlabs Inc., Newton, NJ, USA), which guided the laser beam from the source to the miniature parallel robot. Shortly after the laser beam exited the fiber, the laser was redirected toward the surface below by a small mirror mounted on the robot’s distal endplate. The mirror consisted of an aluminum cylinder with a reflective surface tilted at 45 ${^\circ}$ . The laser source was a diode laser (CPS532, Thorlabs Inc., Newton, NJ, USA) producing 4.5 mW output at 532 nm for visualization purposes.

Figure 7. The manufactured prototype of the parallel robot for laser positioning and stabilization: The housing of the device consists of two endplates (1) and a connection plate (2) made of aluminum. A ferrule (3), which holds the optical fiber (4), is mounted on the proximal endplate. The distal endplate holds a small mirror (5), which redirects the laser (6) toward the surface below the robot. The four drivetrains each consist of a leadscrew (7) and a nut (8), which can travel along the leadscrew as it rotates. The leadscrews are mounted on both endplates via ball bearings (9). The upper nuts are guided by a rod (10) mounted between the two endplates. Each nut contains a rotary joint (11), which holds one of the robots arms (12). The two arms on each side of the robot are mounted on the corresponding leg (13) by means of rotational slide bearings. The distance to the ground plate is defined by aluminum sockets (14). The leadscrews are connected to flexible shafts (15) using glue. A small aluminum coupling (16) houses the flexible shaft and prevents the glue from entering the bearing.

The robot was mounted on a ground plate, which consisted of two separate plates connected by two pins. This setting allowed the distance between the two plates to be continuously varied while they remained parallel, thus allowing the mounting distance between the robot’s legs, $d_{lr}$ , to be adjusted (Fig. 8). The movement of the robot was actuated by four flexible shafts connected to DC motors ( $\phi$ 25 mm, 263,349, Maxon Motor AG, Sachseln, Switzerland) using aluminum couplings. On the robot side, the flexible shafts were connected to leadscrews, which transformed the rotary motion of the motors into linear movements of the nuts and, thus, of the parallel robot’s joints. The system is controlled by TwinCAT 3 automation software running on a real-time CPU module (CX2020, Beckhoff Automation GmbH & Co. KG, Verl, Germany) (Fig. 8). The motors were driven by positioning controllers (MAXPOS 50/5, Maxon Motor AG). Using the Simulink Coder software suite (MathWorks, MA, USA), Simulink models could be compiled and integrated into the TwinCAT 3 environment. A graphical user interface was implemented (App designer from MATLAB R2018b, MathWorks), which communicated with the CPU module and enabled the user to define a reference path in the operational space to be followed by the parallel robot. The CPU module controlled the motor drives via EtherCAT. The App Designer from MATLAB has been used to implement a graphical user interface on the developer computer that enabled high-level control via a device- and fieldbus- independent interface (Automation Device Specification, ADS).

Figure 8. Hardware and software framework: The developer computer runs TwinCAT 3 and enables programming of the real-time CPU module (CX2020) which controls the motor drives via EtherCAT. A graphical user interface on the developer computer enables high-level control via a device- and fieldbus-independent interface (Automation Device Specification, ADS). The robot is mounted on a ground plate (1) that allows adjustment (2) of the mounting distance between the robot’s legs, $d_{lr}$ . The flexible shafts (3) driving the robot are connected to four Maxon DC motors (4) mounted on a tilted plate. A rubber tube (5), which represents the endoscope shaft, constrains the flexible shafts to pass through a cylinder before reaching the parallel robot. A diode laser (6) is coupled into a fiber (7) and guided to the robot.

3.2. Control

The cutting process will be based on pointwise ablation by a pulsed laser. Therefore, it is not necessary to realize continuous accurate movement of the laser; it is merely necessary to ensure that each ablation point along the desired cut can be reached with the desired accuracy. Each laser pulse is triggered only when the parallel robot has positioned the laser at the correct location, that is, has reduced the error in the operational space, $\Delta_{{xy}_{control}}$ , below a specified threshold, $\delta_{xy_{\max}}$ . The desired cut was specified as a path, which we defined offline as a series of end-effector poses with a defined order but without a time variable.

Based on the desired path in the operational space defined by the user, the corresponding sequence of joint positions was calculated using inverse kinematics. This joint position sequence was sent to the real-time system, where the control scheme was executed in real time with an update rate of 1 kHz. The control scheme consisted of two subcontrollers: first, a path controller decided whether to advance along the path and second, a proportional gain controller set the velocity for each motor based on the differences between the desired joint positions and the actual joint positions (Fig. 9). This controller used the sensor feedback from the motor encoders that actuated the flexible shafts.

Figure 9. The control structure of the parallel robot: A desired path is defined offline. The control system then regulates the movement of the robot based on the position feedback from the motor encoders. For experimental evaluation, the actual position of the robot is measured using a tracking system. $\textbf{P}_{des}$ : desired joint position sequence, $\delta_{xy_{\max}}$ : position error threshold, $\delta_{\rho_{\max}}$ : joint error threshold, $\vec{v}_{m}$ : desired motor velocity, $\vec{\alpha}$ : angular motor positions.

The path controller consisted of two different instances that decided when the next point along the path would be set as the desired position: a parallel structure supervisor and a checkpoint supervisor. The parallel structure supervisor had higher priority, was implemented in the joint space, and ensured that advancement along the path was allowed only when the absolute error between the desired and actual joint positions was below a certain threshold $\delta_{\rho_{\max}}$ for all four joints:

(21) \begin{equation}\begin{aligned}|\rho_{i_{act}}-\rho_{i_{des}}| < \delta_{\rho_{\max}} \quad \forall i \in \left\lbrace 1,2,3,4 \right\rbrace.\end{aligned}\end{equation}

This supervision was necessary since the parallel robot is overactuated and the four prismatic joints (the nuts on the leadscrews) must move synchronously to avoid introducing strain on the robot. The checkpoint supervisor operated at a lower priority, was implemented in the operational space, and permitted the user to define checkpoints along the path at which the error between the desired and actual positions in the operational space, $\Delta_{{xy}_{control}}$ , must satisfy a specified threshold, $\delta_{xy_{\max}}$ , before advancement along the path was allowed:

(22) \begin{equation}\begin{aligned}\Delta_{{xy}_{control}} &= \sqrt{(x_{e_{des}}-x_{e_{act}})^2+(y_{e_{des}}-y_{e_{act}})^2} \\&< \delta_{xy_{\max}}.\end{aligned}\end{equation}

The actual and desired positions in the operational space were calculated based on the measured and desired joint positions using direct kinematics (Section 2.4). The direct kinematics was calculated starting from both the left and the right anchor points and the mean value was used as the calculated end-effector pose to reduce the influence of play, manufacturing inaccuracies, and measurement errors.

The resulting desired joint positions $\vec{\rho}_{des}$ were compared against the actual joint positions $\vec{\rho}_{act}$ , and the proportional gain controller sent the corresponding velocity commands $\vec{v}_{m}$ to the motor drives. Based on the motor positions $\vec{\alpha}$ measured by the encoders, the actual joint positions $\vec{\rho}_{act}$ were calculated in accordance with the thread pitch of the leadscrews. Safety limitations were imposed on the maximal velocity and torque values to avoid hardware damage. For performance evaluation of the device, the visual tracking system measured the actual position of the robot. The corresponding data were logged synchronously in real-time and postprocessed offline.

This control scheme enables the mechanism to follow a path and reach certain checkpoints on that path with the desired accuracy in operational space based on the encoder feedback of the motors that rotate the flexible shafts.

4.1. Evaluation path

To enable the execution of a continuous cut employing a pointwise laser ablation scheme with a beam diameter of 0.5 mm, the robot must achieve a positioning accuracy below 0.25 mm in the xy-plane. A path-following experiment was performed in which the robot was controlled to move along a predefined path in the operational space. A subset of points along the path were selected for which the robot was controlled to reach an error below a threshold of $\delta_{xy_{\max}}\ \leq\ {0.01}\,\text{mm}$ for (22). For the performed experiment, the threshold $\delta_{\rho_{\max}}$ of the parallel structure supervisor was set to 0.1 mm and the gain for the proportional controller was set to 20/s. These values were selected to establish a trade-off between limiting the duration of the path-following experiment and not impeding the performance of the parallel robot.

The reference path that was followed in this experiment was designed based on the shape of a subspace of the robot’s maximal reachable workspace that was defined to avoid working at the limits of the theoretically possible workspace (Fig. 10). The limits on the minimal and maximal joint positions were set to $\rho_{\min} = {1.5}\,\text{mm}$ and $\rho_{\max} = {12.5}\,\text{mm}$ , respectively. The minimal and maximal distances from the leadscrews to the legs were limited to $h_{{l,r}_{\min}} = {0.5}\,\text{mm}$ and $h_{{l,r}_{\max}} = {2.5}\,\text{mm}$ . In addition, the workspace was limited to the subspace in which the rotational workspace has a size of at least 6 ${^\circ}$ to avoid the workspace boundaries. The distance between adjacent points along the path in the xy-plane was set to 0.05 mm. Subsequently, the corresponding feasible rotational workspace was calculated for each point on the path as described in Section 2.5. The angle $\varphi_{e_{des}}$ is not relevant to the laser cutting process since the laser is rotationally symmetric. However, the resulting path for $\varphi_{e_{des}}$ must lie within the feasible rotational workspace, and the transition from one point on the path to the next must be possible with a continuous change in $\varphi_{e_{des}}$ . Thus, the angle was selected to lie in the middle of the corresponding feasible range $\left[ \varphi_{e_{\min}}, \varphi_{e_{\max}}\right]$ . In addition, a moving average filter was applied to this path to smooth the resulting rotational reference path. The path started in the middle of the workspace, at $x_e = 0$ and $y_e = 0$ . Checkpoints were placed at all edge points of the path. Further checkpoints were placed in between the path edge points (Fig. 10).

Figure 10. The path-following experiment: The reference path is located in the safe workspace and avoids the biorientation workspace (areas in which the rotational workspace consists of two disjoint subspaces). On the left, the orientation for checkpoints 6–12 and the orientation workspace borders are shown. The checkpoints are visualized as circles, and some are labeled in accordance with their order along the path.

4.2. Joint position measurement

The expected error on the calculated end-effector position based on the uncertainty of the joint position measurement was estimated for the prototype, as described in Section 2.3 (Eq. (3)). We estimated the backlash due to axial play of the leadscrew–nut pair to approximately ${0.01}\,\text{mm}$ . This estimate is based on measurements of axial movement of the leadscrew using a dial gauge, while the nut was fixed. With a leadscrew pitch of $p = {0.25}\,\text{mm}$ , a maximal deformation of the flexible shafts of ${20}{^\circ}$ , and the limited resolution of the motors with 500 counts per revolution, we calculated $U_{\rho_i} = {0.02}\,\text{mm}$ . To calculate the corresponding position error of the end effector, the end-effector pose error (Eq. (6)) was evaluated at each checkpoint on the path. The resulting expected error of the measured end-effector position $\Delta_{{xy}_{est}}$ due to the measurement uncertainty of the nut positions was calculated as follows:

(23) \begin{equation}\Delta_{{xy}_{est}} = \sqrt{\Delta{x_e}^2+\Delta{y_e}^2}.\end{equation}

4.3. Visual tracking

An external tracking system (Vicon Motion Systems, Oxford, United Kingdom) was used to evaluate the robot’s accuracy in the operational space. Reflective markers mounted on the robot were tracked by a camera system consisting of six Vicon Vantage V5 cameras (Vicon Motion Systems, Oxford, United Kingdom) mounted on tripods around the system setup. Reflective hemispherical markers were placed on the ground plate and on the parallel robot (Fig. 11). The tracked positions of the markers were acquired by a dedicated workstation running the tracking software (Nexus 2.8, Vicon Motion Systems). The tracked marker positions were forwarded to the real-time CPU module to synchronize the measured data. Since the robot moved with a relatively slow velocity, the delay of the tracking system was not relevant. The accuracy of the tracking system was assessed by analyzing the distances between the five stationary markers on the ground plate, the positions of which were fixed relative to each other. Over all performed measurements, the maximal deviation of a distance measurement from the corresponding mean value was in the range of $\pm17\mu {\text{m}}$ . Since our aim was to track the parallel robot in the submillimeter range, this measurement setup was considered sufficiently accurate.

4.4. Experimental procedure and data processing

Before the experiment started, the robot was calibrated by manually rotating the motor shafts such that the nuts reached the first endplate (Fig. 7). Then, the encoder values were calibrated accordingly.

Figure 11. Experimental setup: For performance evaluation using the tracking system, reflective hemispherical markers were mounted on the robot as well as on the ground plate. An aluminum plate (2) was mounted on top of the robot, which was used to host two markers ( $\phi$ 3 mm) (3) to enable tracking of the end effector ( $\vec{p}_e$ ) and the orientation of the mobile platform ( $\varphi_e$ ). Two further markers were mounted on the tops of the legs to track their movement during the experiment. Five hemispherical markers ( $\phi$ 4 mm) were placed on the ground plate to establish a reference coordinate frame I (4). The tracking information obtained by the motion capture cameras and corresponding software was forwarded to the real-time system via User Data Protocol (UDP).

The robot was controlled to follow the reference path (Fig. 10) 10 times in alternating directions. For the reversed direction, the path was flipped around the y-axis. At the start of each run, the tracked end-effector position was zeroed.

The real-time system logged the tracked marker positions as well as the path controller information during the experiment in a synchronized manner for later evaluation. The data were postprocessed in MATLAB R2018b. The distances between the desired and tracked positions in the xy-plane in the operational space were calculated to evaluate the performance of the parallel robot:

(24) \begin{equation}\Delta_{{xy}_{tracked}} = \sqrt{(x_{e_{des}}-x_{e_{tracked}})^2+(y_{e_{des}}-y_{e_{tracked}})^2}.\end{equation}

As soon as the position error dropped below the threshold in accordance with (22), the advancement along the path was delayed for 2 s before the next checkpoint was set as the target position by the controller to assess the influence of noise on the measurement. The mean and maximal values of the tracked error over the static 2 s were calculated for each of the $N = 33$ checkpoints, and the 10 experiments. The mean value of the tracked error $ \Delta_{{xy}_{mean}}$ allows to cancel out noise originating from the tracking system and the neglectable residual movement of the robot for a static reference. The maximal value of the tracked error $\Delta_{{xy}_{\max}}$ is provided for completeness and to determine potential systematic errors of the tracking system. The average expected accuracy of the miniature parallel robot in the considered workspace is calculated as the grand average over all checkpoints and experimental runs $\left( k = 1,2,... 10 \cdot N \right)$ of the mean errors over the static 2 s at the checkpoints $avg_{k}(\Delta_{{xy}_{mean,k}})$ . The expected worst-case accuracy in the considered workspace is calculated as the maximum of the mean errors over the static 2 s at the checkpoints $\max_{k}(\Delta_{{xy}_{mean,k}})$ . To allow stating a guaranteed accuracy, also the maximum of the maximal errors is provided $\max_{k}(\Delta_{{xy}_{\max,k}})$ . The mean travel speed of the robot along the path between the checkpoints $v_{e_{mean}}$ was calculated by dividing the distance of the desired path by the time of travel for each path segment between two checkpoints. The maximal value of the mean travel speed is calculated $\max_{k}(v_{e_{mean,k}})$ to provide an estimate of the robot’s maximal mean travel speed that was validated with the performed experiment.