2.1 Mechanical Architecture Concept

External ring fixators are widely used in orthopedics for fracture fixation, bone lengthening, and deformation correction purposes. The clinician typically brings the bone fragments to an anatomically desired position by changing the length of the rods connecting the fixator rings. This task is accomplished by the clinician based on the experience and expertise. As an alternative, the commercial system exists where the same task is automated with the help of an accompanying software that is implemented with the mathematical model of the fixator. The Stewart platform was selected considering its analogy with widely used Ilizarov's external fixation device. The Stewart platform has several advantageous characteristics to act as a robotic fracture reducer, including high stiffness and precision, acceptable repeatability, 6 DoF (3 angular and 3 linear) and high load to weight ratio. But the workspace of this architecture appears to be limited. Indeed, the range of angular motion rapidly decreases as soon as the end effector moves away from the workspace center. Also, the possibility of rotating the bone around its longitudinal axis is very limited, while it is an important feature for bone reduction in order to guaranty the surface correspondence of bone fragments. It is proposed to use a double triangle planar parallel manipulator with an equilateral triangular fixed and moving platforms respectively fixed to the proximal and distal fragments of the fracture bone, using pins or wires. This double triangle structure is composed of one 3 DoF parallel manipulator using 3 Revolute–Prismatic–Spherical (RPS) introduced by Hunt (1983) and one Triangular planar parallel robot using 3 PRP arms suggested by Daniali et al. (1993). The concept of this novel robot is shown in Fig. 4. The arms of the 3-RPS mechanism consist in connecting the base by the first joint (R) to the moving platform by the last joint (S-joint). The planar manipulator is a special symmetrical closed-loop mechanism that is composed of a pair of triangles: one base and one platform. The platform triangle is placed on the top of the base triangle and is moved by 3 PRP arms fixed to the edges. The first prismatic joint of each arm allows them to slide along each edge of the base triangle. A planar parallel mechanism has the specific performance, whereas the links are moving in the planar motions. The RPS manipulator has 2^∘ of orientation freedom and one degree of translation freedom. One advantage of the present architecture compared to the Stewart mechanism is that the double triangular structure can provide a much larger range of longitudinal rotation of the bone fragment for most angular position of the end effector given by the tripod mechanism.

Figure 3Schematic of the basic concept and kinematic representation of the mechanical architecture.

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Figure 4Schematic of the basic concept and kinematic representation of the mechanical architecture.

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2.2 Kinematic Analysis of the Mechanism

In order to determine the kinematic model of the present mechanism, several reference frames are attached with different items of the mechanism. The reference frame $\mathit{\{}O,\mathit{x},\mathit{y},\mathit{z}\mathit{\}}$ is attached with the fixed base A with O, the center of the equilateral triangle A₁A₂A₃ that orthogonal to the z-axis. The frames $\mathit{\{}{O}_{\mathrm{1}},{\mathit{x}}_{\mathrm{1}},{\mathit{y}}_{\mathrm{1}},{\mathit{z}}_{\mathrm{1}}\mathit{\}}$ and $\mathit{\{}{O}_{\mathrm{2}},{\mathit{x}}_{\mathrm{2}},{\mathit{y}}_{\mathrm{2}},{\mathit{z}}_{\mathrm{2}}\mathit{\}}$ are respectively attached with the moving platforms B and C. They are represented by the equilateral triangles B₁B₂B₃ and C₁C₂C₃ and they are respectively orthogonal to axis z₁ and z₂. The moving platform B is connected with the spatial 3-RPS parallel mechanism by three passive spherical joints located at the triangle corners B₁, B₂, and B₃. The fixed base has three passive revolute joints at the corners of the equilateral triangle A₁A₂A₃. The moving platform B is put in motion in the reference frame $\mathit{\{}O,\mathit{x},\mathit{y},\mathit{z}\mathit{\}}$ by the mean of three active prismatic joints, each located between the revolute and spherical joints of each RPS linkage.

It is assumed that the mechanism end-effector is located by the point O₂ of the moving platform C. Its linear position in the coordinate system $\mathit{\{}O,\mathit{x},\mathit{y},\mathit{z}\mathit{\}}$ is described by x, y and z variables. The orientation of the end-effector is given by the three angles α, β and γ. The general coordinates of the ith joints are given by the length q_i ($i=\mathrm{1},\mathrm{\dots },\mathrm{6}$).

The position of the revolute joints A₁, A₂ and A₃ in $\mathit{\{}O,\mathit{x},\mathit{y},\mathit{z}\mathit{\}}$ are given by the vectors below:

With r_a, the radius of the circumcircle of center O passing by A₁, A₂, and A₃. The position of the spherical joints in the coordinate system $\mathit{\{}{O}_{\mathrm{1}},{\mathit{x}}_{\mathrm{1}},{\mathit{y}}_{\mathrm{1}},{\mathit{z}}_{\mathrm{1}}\mathit{\}}$ of the moving platform B are written as below:

With r_b, the radius of the circumcircle of center O₁ passing by B₁, B₂, and B₃.

And the corners C₁, C₂, and C₃ of the moving platform C in $\mathit{\{}{O}_{\mathrm{2}},{\mathit{x}}_{\mathrm{2}},{\mathit{y}}_{\mathrm{2}},{\mathit{z}}_{\mathrm{2}}\mathit{\}}$ are located as followed:

The position of spherical joints B_k ($k=\mathrm{1},\mathrm{2},\mathrm{3}$) given in $\mathit{\{}{O}_{\mathrm{1}},{\mathit{x}}_{\mathrm{1}},{\mathit{y}}_{\mathrm{1}},{\mathit{z}}_{\mathrm{1}}\mathit{\}}$ in Eq. (2) can be expressed in $\mathit{\{}O,\mathit{x},\mathit{y},\mathit{z}\mathit{\}}$ as followed:

Where q_k (with $k=\mathrm{1},\mathrm{2},\mathrm{3}$) is the prismatic joint variable directed by the vector s_k3 as seen in Fig. 3. And where OO₁ gives the coordinates [x₁  y₁  z₁] of the center of the moving platform B, expressed in $\mathit{\{}O,\mathit{x},\mathit{y},\mathit{z}\mathit{\}}$ and R is the orientation matrix of frame $\mathit{\{}{O}_{\mathrm{1}},{\mathit{x}}_{\mathrm{1}},{\mathit{y}}_{\mathrm{1}},{\mathit{z}}_{\mathrm{1}}\mathit{\}}$ with respect to $\mathit{\{}O,\mathit{x},\mathit{y},\mathit{z}\mathit{\}}$ with α, β, and φ being the Roll-Pitch-Yaw and written as:

Where c^∗ and s^∗ correspond to cos (∗) and sin (∗), respectively.

Similarly, the original position of the center of the moving platform C in the frame $\mathit{\{}{O}_{\mathrm{1}},{\mathit{x}}_{\mathrm{1}},{\mathit{y}}_{\mathrm{1}},{\mathit{z}}_{\mathrm{1}}\mathit{\}}$ is given by:

With d, the distance between the moving platform B and the moving platform C.

The 3-RPS parallel mechanism is one of the lower-mobility parallel mechanism which has an important kinematic feature is parasitic motion (Carretero et al., 2000; Li et al., 2011). The 3-RPS parallel mechanism generates three parasitic motions: two translations along axis x, y and one rotation about the z-axis with respect to the fixed frame. In other words, the moving platform B has a small parasitic motion of the fixed base (x₁, y₁, φ). These unwanted parasitic translations (x₁, y₁) are compensated by the linear motion of the 2nd moving platform.

Hence, the position vector of the end effector O₂ expressed in the fixed frame $\mathit{\{}O,\mathit{x},\mathit{y},\mathit{z}\mathit{\}}$ is given by:

Considering the constraints applied by the revolute joints at points A₁, A₂ and A₃, the parasitic motions of the moving platform B are calculated.

Where s_k2 (with k=1, 2, 3) gives the direction of the axis of rotation for the angle θ_k as seen in Fig. 3. From Eq. (9), it can be found that:

The actuating lengths of the links for a prescribed position and orientation of the moving platform B are obtained by taking the Euclidean norm of the Eq. (4) as bellow:

By solving the Eqs. (4) and (5), the rotation angle θ₁, θ₂ and θ₃ of the kth legs can be written as:

The position vector of the active prismatic joints R₁, R₂, and R₃ expressed in frame $\mathit{\{}{O}_{\mathrm{1}},{\mathit{x}}_{\mathrm{1}},{\mathit{y}}_{\mathrm{1}},{\mathit{z}}_{\mathrm{1}}\mathit{\}}$.

And assuming $l_k(k=\mathrm{1},\mathrm{2},\mathrm{3})$, the distance between the corner C_k moving platform C and the closest passive prismatic joint $R_k^{\prime }$, namely $C_{\mathrm{1}}{R}_{\mathrm{1}}^{\prime }$, $C_{\mathrm{2}}{R}_{\mathrm{2}}^{\prime }$ and $C_{\mathrm{3}}{R}_{\mathrm{3}}^{\prime }$ respectively, the position of $R_k^{\prime }$ expressed relative to O₂:

In the frame $\mathit{\{}{O}_{\mathrm{1}},{\mathit{x}}_{\mathrm{1}},{\mathit{y}}_{\mathrm{1}},{\mathit{z}}_{\mathrm{1}}\mathit{\}}$, the position the passive prismatic joints $R_i^{\prime }$ are calculated as followed:

Where γ is the rotation angle of the moving platform C around z₁-axis and R_z(γ) is the rotation matrix around z₁-axis of angle γ.

The coordinate constraints of the revolute joints between two prismatic joints R_k and $R_k^{\prime }$ can be defined as followed:

By solving Eq. (20), the length of the active prismatic joints q₄, q₅, q₆ are determined as followed:

2.3 Velocity Model and Singular Configurations of the Mechanism

In order to identify the mechanism singular configurations that could result trajectory problems, its velocity model is studied to isolate the Jacobian matrices. The velocity model of the mechanical architecture can be obtained by differentiating Eq. (8) with respect to time. In order to simplify the calculation, it is possible to separate this mechanism architecture into two parallel mechanisms: one is the velocity model of the 3-RPS parallel mechanism and another is the velocity model of the 3-PRP parallel mechanism. The velocity model of the 3-RPS mechanical architecture is calculated by reformulating Eq. (4) as followed:

Where s_k3 is the unit vector directing the three legs q_k (Fig. 3). Equation (5) is then substituted and differentiated to obtain:

Where ϑ_k is the angular velocity of leg kth and ω is the angular velocity of the moving platform B expressed in the fixed platform. This yields:

The parasitic motion given in Eqs. (10) to (12) is differentiated with respect to time to obtain:

Where $K=\frac{s\mathit{\alpha }s\mathit{\beta }c\mathit{\phi }-c\mathit{\alpha }s\mathit{\phi }+c\mathit{\beta }s\mathit{\phi }}{\mathrm{1}+c\mathit{\alpha }c\mathit{\beta }}$.

By setting up Eqs. (26) to (29):

Where

The velocity model of the double triangular mechanical architecture is obtained by directly differentiating Eqs. (21) to (23) and arranging as followed:

Where

For the RPS tripod mechanism, the forward singular configuration is determined by using the Jacobean matric J_B to solve the following equation:

In this specific application, the singularity loci happened in the range of workspace will be considered. Figure 5 shows the singularity distribution of RPS mechanism in range of workspace −20^∘ < α < 20^∘, −20^∘ < β < 20^∘. According to the present simulations, there is a risk for the tripod mechanism to reach a singularity when the z coordinate goes from 0 to 111.8 mm. Several points in Fig. 5 are selected as examples to present the singularity configurations in Fig. 6. These configurations are similar to RO-type singularity that classified in Zlatanov et al. (2002) and Li et al. (2015).

Figure 5Singularity distribution of RPS mechanism in range of workspace −20^∘ < α < 20^∘, −20^∘ < β < 20^∘.

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Figure 6Three examples of singular configurations for the 3-RPS mechanism.

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For the double triangular mechanism, the inverse singularity is identified by using the Jacobian matric J_C to solve the following equation:

By solving this equation, the first singularity condition will be:

In this configuration, the moving platform B of the double triangular mechanism will be at the same position as moving platform A. The forward singularity is found by using the same method with the Jacobian matrix J_D. This yields the following equation:

By substituting Eqs. (21) to (23) into Eq. (51):

Assuming $s\mathit{\gamma }+\sqrt{\mathrm{3}}c\mathit{\gamma }\ne \mathrm{0}$, the solution is written as:

The inverse singularity conditions given by Eqs. (48) and (52) lead to the same configuration illustrated in Fig. 7.

Figure 7Singular configuration associated with double triangular mechanism.

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4.1 Pre-operative Simulation Method

The task of the mechanism studied above is to perform the required motions to manipulate bone fragments in bone reduction surgeries. While the base of the mechanism is attached to one fragment, the moving platform 2 is attached to the other one. But the operating this mechanism is only the last phase of a whole surgery that also includes pre-operative planning. Indeed, it is planned that the mechanism will automatically generate the bone reduction, i.e. the repositioning of the bone. The motions that the mechanism will have to perform is determined during the pre-operative phase by the use of a specific software, named PhysiGuide. It has been developed by the Bio-images and Clinical Assistant Laboratory of the National Central University (NCU) to carry out simulation and to provide surgical planning for bone reduction (Lee et al., 2014). In the present study, this software is used to perform the bone reduction simulation and to determine the corresponding motion. The entire procedure of bone reduction surgery is illustrated in Fig. 11. The clinical imaging phase consists in collecting the patient's CT-scan images. Based on these successive section views, a 3-D model of the patient's broken bones is reconstructed. The software PhysiGuide is then used to perform the segmentation of the different bone fragments. After this operation, these fragments can be interpreted as independent solids that can be manipulated separated. The second phase is also performed by PhysiGuide and provides the simulated bone reduction to reposition and re-assemble the bone fragments. At the same time, the software measures the relative linear and angular motions of one bone fragment from the other. These motions correspond to the mechanism moving platform. The last step is the actual intra-operative surgery where the mechanism is attached with the patient bone fragments. Based on the bone fragment motions measured by PhysiGuide, the implemented inverse kinematic model can determine the mechanism input variable necessary to perform the bone reduction. Indeed, the kinematic parameters of the bone fragments are defined to be matched with those used by the mechanism, i.e. x, y, z, α, β and γ.

Figure 11Robotic-assisted bone reduction procedure. From pre-operative to intra-operative surgery.

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4.2 Results and discussion on the Bone Reduction Surgery Simulation

To verify the feasibility of performing the fractured bone reduction using the mechanism introduced in Sect. 2, a simulation has been conducted based on real CT-scan images from a patient showing a fracture of the femur. The images have been provided by the Orthopaedic Department of the Show Chwan Memorial Hospital. After being fully reconstructed in a 3-D environment, the PhysiGuide software has been used to perform a simulated bone reduction. As the femur fragment has been repositioned, the parameters x, y, z, α, β and γ have been numerically measured directly using the software functionalities and they are shown in Table 3.

Table 2Motion variables measured for the repositioning of femur fragments.

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Figure 12Adams simulation of a bone reduction case using the hybrid mechanism.

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The graphic models of the two bone fragments have been imported in the Adams model of the hybrid mechanism. Their respective positions have been set exactly as measured by PhysiGuide software. One fragment has been attached to the base of the mechanism and the other to its end effector. During the pre-operative planning of a classical bone reduction, the surgeon will estimate the position of the bone fragments. A trajectory will be then roughly defined to move one bone fragment to the other while avoiding collisions. So the robot trajectory in the present case between the initial and final positions illustrated in Fig. 12 cannot be simply defined as a straight line. Several steps must be taken between these initial and final positions. There is no specific method to establish a trajectory. The main concern is to avoid collision between the structures of the bone fragments before the final fracture surface matching, which can be visually performed based on the medical images. Although the surgeon has no concern about numerical coordinates, the use of the kinematic data measured from PhysiGuide allows to verify the suitability of the proposed mechanism for bone reduction surgery.

In the present case, the trajectory has been defined so the initial and final positions of the bone correspond to the position measured by PhyisiGuide before and after the simulated bone reduction. In order to avoid collisions, the different trajectory steps have been set as followed: first, the top moving platform is rotated −22.2^∘ about the z axis. The moving platform is then translated along x, y and z axes of with $x=-\mathrm{8.8}$ mm, $y=-\mathrm{50}$ mm, z=43.8 mm. It is then rotated about x axis of α=2.7^∘ and about y axis of $\mathit{\beta }=-\mathrm{10.9}$^∘. After that, a linear translation of the top moving platform in z=30 mm. Finally, rotating the moving platform 1 about x axis with α=3^∘ and about y axis with $\mathit{\beta }=-\mathrm{12}$^∘. These motions are represented in Fig. 13. On the other hand, the mechanism input variables that correspond to this trajectory have been calculated and they are displayed in Fig. 14.

Figure 13Evolution of the end effector coordinated during the simulated bone reduction trajectory.

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Figure 14Evolution of the mechanism input variables during the simulated bone reduction trajectory.

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The different mechanism configurations corresponding to the present case of bone reduction trajectory are illustrated in Fig. 15a to e in xz plane view.

Figure 15Adams view of the mechanism configuration for each step of the bone reduction trajectory in xz plane view (a–e).

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In a real intra-operative robotic assisted surgery, the bone fragments would be attached by the means of orthopaedic pins or wires. The mechanism prismatic joints would be actuated until the final configuration is reached and then locked. A prototype can be designed to allow motors to be removed from the robot. The architecture would be secured by the use of irreversible prismatic joints and the remaining frames would act as an external fixation device until the fracture heals. The actual robotic operation can be performed automatically under the supervision of the orthopaedic surgeon who would be able to adjust the manipulation speed or to interrupt the process. Although the robot motion will be performed automatically, the supervising surgeon's control over the system could be offered by a Graphic User Interface that would provide manual minor adjustment on the bone reduction. Such interventions can be enforced by the mean of intra-operative imagery of the patient. As demonstrated above, the proposed hybrid mechanism is capable of providing a complete reduction trajectory as the ranges of motion of all its input variables are respected. On the other hand, it is also noted for this same clinical case, the Stewart manipulator would be in difficulties to perform the bone reduction. Indeed, at the final linear coordinates ($x=-\mathrm{8.8}$, $y=-\mathrm{50}$, z=283.7), its angular workspace is insufficient to reach the required angles β and γ and to reposition the bone fragment in the appropriate configuration. This problem could be solved by determining an accurate initial position of the manipulator which would ensure that complete trajectory remain within its workspace. However, there is no guaranty such a position actually exists since its workspace is limited. Also, in the case it would exist, this would require an accurate and delicate registration of the manipulator with the bone fragments. This explains the importance of proposing a mechanism with a workspace and a more adapted kinematic. On the other hand, it can be anticipated that the hybrid mechanism may have smaller stiffness than the Stewart robot. Also, dynamic issues can be predicted due to the presence of floating motors (on the triangular device). But the registration aspects will be far less of a concern since its workspace shows significant improvement.