## Abstract

Robot-assisted femur repair has been of increased interest in recent literature due to the success of robot-assisted surgeries and current reoperation rates for femur fracture surgeries. The current limitation of robot-assisted femur fracture surgery is the lack of large force generation and sufficient workspace size in traditional mechanisms. To address these challenges, our group has created a 3-RRPS parallel mechanism, Robossis, which maintains the strength of parallel mechanisms while improving the translational and rotational workspace volume. In this paper, an optimal design methodology of parallel mechanisms for application to robot-assisted femur fracture surgery using a single-objective genetic algorithm is proposed. The genetic algorithm will use a single-objective function to evaluate the various configurations based on the clinical and mechanical design criteria for femur fracture surgery as well as the global conditioning index. The objective function is composed of the desired translational and rotational workspaces based on the design criteria, dynamic load-carrying capacity, and the homogeneous Jacobian global conditioning index. Lastly, experimental results of Robossis were obtained to validate the kinematic solution and the mechanism itself; Robossis had an average error of 0.31 mm during experimental force testing.

## 1 Introduction

The need for improved clinical outcomes, both intraoperative and postoperative, combined with the ever-advancing field of medical research, has brought surgical robotics to the forefront of surgical innovation [1]. Recently, robotic-assisted surgeries have been shown to decrease costs, improve recovery times, and improve patient outcomes [2,3]. These affirmative clinical outcomes coupled with the rise in fractures since the turn of the century have motivated research on robot-assisted fracture reduction and alignment surgeries. Femur fracture surgeries have been of particular interest, primarily due to undesirable clinical outcomes. The 1-year mortality rate for persons ≥50 is 25%, 21%, and 20% for proximal, shaft, and distal femur fractures, respectively. For males ≥50, the proximal femur fracture mortality rate is 32% [4]. A significant factor in this is the postoperative malalignment rate and the need for reoperation. The torsional malalignment rate for all femoral fractures has been reported up to 27.5%, where malalignment is defined as >5 deg of angulation in any coordinate plane [5]. These high rates of malalignment and 1-year mortality suggest that it is necessary to improve surgical procedure efficacy and patient outcomes for femur fractures.

During the surgical procedure for femur fracture repair, the reduction of fragments poses significant challenges. The difficulties associated with the reduction and alignment of these fractures are due to the complex anatomy of the femur and the significant passive muscle forces of the thigh (maximal recorded forces: 504.7 N and 38.3 Nm) [6].

Robot-assisted orthopedic trauma surgery is a developing field of research. This field aims to provide surgeons with techniques that allow for high precision and high force generation in the operating room. Despite the growing field, traditional and current mechanisms have not been shown to be capable of opposing the large forces observed during long-bone fracture reduction while maintaining a sufficiently large translational and rotational workspace. Table 1 illustrates the various parallel mechanisms that have been used for robot-assisted femur fracture surgery, their maximal load, and range of motion.

To address these load and workspace challenges, our group previously presented Robossis, a novel design of a 3-RRPS 6-degree-of-freedom (DoF) parallel mechanism, and has completed a comprehensive theoretical analysis, including the derivation of the inverse and forward kinematics, Jacobian via the theory of screws, and inverse dynamics. In addition, our group conducted an investigation into the kinematic and dynamic performance, focusing on singularity, force transmission, Jacobian, and performance index analysis [18–21] as well as preliminary benchtop and cadaver testing [22–25].

For the application of femur fractures, the size of the translational and rotational workspaces is an important indicator of the performance of a parallel mechanism. Based on the literature available, we used the reported femur fracture malalignment ranges to quantify certain translational and rotational ranges a mechanism must satisfy to be considered a viable solution for robot-assisted femur fracture surgery. We compared the workspace viability of Robossis alongside the traditional Gough–Stewart platform (GSP).

In this paper, we propose an optimal design methodology of parallel mechanisms for application to robot-assisted femur fracture surgery using a single-objective genetic algorithm. The genetic algorithm will use an objective function to evaluate the various configurations based on the clinical and mechanical design criteria as well as the homogenous Jacobian global conditioning index (GCI). Additionally, we demonstrate the performance of Robossis through force and position testing to validate the inverse kinematic solution.

## 2 Clinical and Mechanical Requirements

One of the significant and impactful complications in femur repair is the malrotation of the femur fragments. Regardless of the location of the fracture, proximal, shaft, or distal, there is no less than a 24% reported malrotation rate [5,26,27]. These malrotations have been shown to alter a person’s gait and quality of life [28].

In part, the reason a surgical robot has not yet seen widespread clinical use is due to the clinical and mechanical design requirements necessary for use in the operating room. A surgical robot designed for long-bone femur fracture surgeries is required to meet or exceed the required accuracy, rotational and translational movements, and maximal forces observed during the procedure. The required accuracy is derived from the Thoresen metric [29]. The highest score on the Thoresen metric, “Excellent,” is defined as alignment within ±1 cm and ±5 deg. A goal of this work is to show that Robossis can perform at least 50% better than the Thoresen “Excellent” metric of ±0.5 cm and ±2.5 deg.

To understand the maximal forces present during femur fracture repair, Zhu et al. measured the intraoperative passive muscle forces along the anatomical axes. The maximal anterior–posterior (AP), medial–lateral (ML), and femoral–shaft (FS) forces were 472.4, 203.0, and 504.7 N, respectively [6]. Zhu et al. reported maximal torques for the anterior–posterior and medial–lateral axes of 16.4 Nm and 38.3 Nm, respectively. Estimated forces from pressure readings were not included in Table 2.

In addition to the mechanical requirements of force and torque, there are clinical criteria that focus on the usability of the design. It is desired to have a surgical robot that can be used with all patient body types. McDowell et al. [30] reported the average male center thigh circumference as 53.8 cm ± 9.13 cm. Robossis, with a ring circumference greater than 81.1 cm (radius greater than 12.9 cm, Table 2), can be used with 99.7%, 3 standard deviations, of all patient body types.

Furthermore, the femur fracture malalignment range was determined to provide a preliminary approximation of the translations and rotations necessary for femur fractures. The required rotations were derived from the maximal reported malrotations for the AP, ML, and FS axes [27,31]. The required translations along the FS were derived from the maximal reported leg length discrepancy reported by Citak et al. [32]. Lowe et al. studied cadaveric femur fractures and found a medial and anterior translation of 17.1 mm ± 10.4 mm and 6.0 mm ± 4.3 mm, respectively, when reducing the fractures [33]. Using three standard deviations gives a maximal medial and anterior translation of 48.3 mm and 18.9 mm, respectively. These values are the femur fracture malalignment range listed in Table 2.

## 3 Robossis Architecture

The 6-DoF 3-RRPS parallel mechanism, Robossis, is shown in Fig. 1. The design is a three-arm parallel mechanism, with each arm affixed to both a moving and stationary ring using spherical and universal joints, respectively. The proposed application of Robossis is shown in Fig. 2. Robossis will be placed over the lower leg to allow clinicians access to the fracture surgical area. Surgical Schanz screws will be used to attach Robossis to the distal fracture (Fig. 2(2)). The robot will be attached after traction is pulled to eliminate large variances in the femoral–shaft axis (*z*-axis).

Robossis is actuated by two types of Autonics stepper motors. Affixed to the stationary ring are three Autonics-A8K (Busan, South Korea) stepper motors with a nominal holding torque of 0.814 Nm. The A8K motors are coupled with a 60:1 gearbox to increase the output to 48.6 Nm. Each arm consists of a HIWIN-KK40 (Hiwin, Taichung, Taiwan) linear actuator with a 1 mm pitch. The linear actuator is powered by an Autonics-A3K (Busan, South Korea) stepper motor, nominal holding torque = 0.235 Nm. Assuming a torque-to-force efficiency of 0.9, the maximal linear force generated is 1527 N for each arm, for a total theoretical load insertion of 4581 N.

## 4 Inverse Kinematics

*P*(

*x*,

*y*,

*z*,

*α*,

*β, γ*), the linear actuator length (

*d*

_{i}) and active joint angle (

*θ*

_{i}) are calculated. Referring to Fig. 3(a), signifying

*a*

_{i}and

*b*

_{i}(representing

*OA*

_{i}and

*PB*

_{i}) in reference frame {A}, it can be determined that

*d*

_{i}can be expressed using the Euclidean norm

*θ*

_{i}and

*ψ*

_{i}, respectively, are expressed as

## 5 Inverse Dynamics

*i*'s center of mass (CoM). $Fp$ is denoted as the sum of applied and inertial wrenches around the moving ring’s CoM. The wrench applied on the actuators is denoted by

**. $\delta xp,\delta xi,and\delta q$ are denoted as the virtual displacement and rotation about the center of the moving ring, center of link**

*τ**i*, and $q$, respectively, where $q\u02d9=[\theta \u02d91\theta \u02d92\theta \u02d93d\u02d91d\u02d92d\u02d93]T$ is the joint velocity vector, and $\delta qand\delta xp$ are related as

**denotes the Jacobian matrix. Designating**

*J*

*J*_{i}as link

*i*'s Jacobian

**provides the necessary motor torque to support a given position and speed of the mechanism**

*τ*## 6 Translational and Rotational Workspace Analysis

To understand the performance of Robossis, a comparison with the 3_3-SPS-GSP was made [35]. The 3_3 GSP has the best performance metrics of all the well-known GSP variations [21]. Robossis and the GSP were analyzed in terms of their theoretical translational and rotational workspaces. For both mechanisms, the fixed and moving rings g and h have a radius of 0.175 m (Figs. 3(b) and 3(c)). The radius is defined as the distance from the center of the ring to the center of the spherical joint. Both mechanisms have a linear actuator stroke length of 0.14 m but have different midrange positions (*d _{i}* in Figs. 3(b) and 3(c)). The linear actuator ranges,

*d*, for Robossis and the GSP are 0.214 m–0.354 m and 0.292 m–0.432 m, respectively. For the unconstrained comparison of Robossis and the GSP, the only limit on the inverse kinematic solutions of both mechanisms is the prismatic joint linear actuator range. This approach allows for a comparison of different parallel mechanisms based on their best possible theoretical workspace, not limited by any rotational joint angles.

_{i}### 6.1 Translational Workspace Comparison.

The translational workspace was determined by assuming an initial 3D space with a length and width equal to 0.8 m and a height from 0.1 to0.45 m. Each [*x*, *y*, *z*] point in the 3D space was input to the inverse kinematics solution (Eqs. (2)–(5)) to determine if it was within the linear actuator range, *d _{i}*. This analysis determines the translational workspace each mechanism can reach without exceeding the linear actuator range. The results are shown in Fig. 4. Robossis and the GSP have workspace volumes of 4.66E-2 m

^{3}and 8.53E-3 m

^{3}, respectively. Robossis has a larger workspace due to the nature of the mechanisms. The GSP workspace is the result of the intersection of six spheres, while Robossis is the result of the intersection of three spheres.

### 6.2 Rotational Workspace Comparison.

The rotational workspace was determined by first setting the point *P* (Fig. 3(a)) of the moving platform to be at the midrange position, [*x* = 0 m, *y* = 0 m, *z* = 0.284 m] and [*x* = 0 m, *y* = 0 m, *z* = 0.362 m], for Robossis and the GSP, respectively. The Euler angles *α*, *β*, and *γ*, based on the *XYZ* convention, were then incremented by 1 deg varying from [−180 deg, 180 deg]. At each step, the position, [*x*, *y*, *z*, *α*, *β*, *γ*] was input to the inverse kinematics solution (Eqs. (2)–(5)) to determine if the position was within the constraints of the linear actuators. The results are shown in Fig. 5 using cylindrical coordinates [*r*, Θ, *z*], where *r* = *α*, Θ = *β*, and *z* = *γ*. Due to the nature of cylindrical coordinates, only the positive values of *α* are shown. Based on Fig. 5, it can be observed that Robossis has a larger rotational workspace than the GSP and provides an advantage in applications requiring large rotational capabilities.

## 7 Mechanism Analysis Based on the Clinical and Mechanical Requirements

Based on the literature data in Table 2, it is desired to analyze the ability of the GSP and Robossis to meet the mechanical design requirements of femur fracture surgeries. Since 2004 (Table 1), the GSP has been investigated as a possible mechanism for robot-assistive femur fracture surgery, yet there is not currently a commercially available device for this application. This may be, in part, due to the GSP lacking the translational and rotational workspace necessary for clinical applications. To address this concern, we attempt to quantify specific minimum translations and rotations necessary for a device to be used in robot-assisted femur fracture surgery. From the literature, we were able to assign translation and rotation requirements based on the femur fracture malalignment range (Table 2). The maximal reported malalignments in Table 2 were linearly interpolated to create a 3D volumetric malalignment range.

In contrast with the unconstrained workspace in Sec. 6, both Robossis and the SPS-GSP are now also constrained by spherical joints ranging from ±25 deg in addition to the stroke length of the linear actuators. This provides a comparison of how both mechanisms would perform if manufactured with commercially available parts. The workspaces in Secs. 7.1 and 7.2 show all points the prototypes in Figs. 3(b) and 3(c) can reach. The mechanical interference limitations on the active and passive revolute joints do not affect the workspace volume and are not included as constraints.

### 7.1 Rotational Requirements: Malalignment Range.

We compare the constrained rotational workspaces of Robossis and the GSP and label the femoral–shaft axis rotational requirement from Table 2. Figure 6(a) provides the constrained rotational workspaces for Robossis and the GSP. Figure 6(b) provides a 2D slice at *β* = 0 deg. Robossis can reach the clinical rotation requirements around all axes while the GSP cannot.

### 7.2 Translational Requirements: Malalignment Range.

The comparison of the constrained translational workspaces of Robossis and the GSP with the malalignment range provides us with a malalignment range workspace in which we will focus on analyzing the performance of Robossis and the GSP.

### 7.3 Clinical Load Requirements.

The experimentally measured fracture reduction forces from Zhu et al. (Table 2) were used as a minimum required carrying capacity. We conducted a static load-carrying capacity (SLCC) simulation to determine the maximal loads Robossis can support within the malalignment workspace defined in Sec. 7.2. Based on the SLCC methodology in Fig. 8, the design of Robossis requires stepper motors with a nominal torque of at least 1.6 Nm to support the external loads in Table 2.

## 8 Optimization Problem Design and Formulation

Real-world optimization problems tend to be difficult due to the numerous variables as well as insufficient information about the relationship and trends of modifying variables. The genetic algorithm (GA) is population-based, where each population or generation has individuals with randomly assigned values for each parameter [36]. The individuals who are “fittest,” that is, perform the best based on the fitness test (objective function), have their parameter values passed onto the next generation [36]. The use of a genetic algorithm for this application is especially important as it robustly searches for a global solution. For optimizing the configuration of Robossis, these individuals have five parameters outlined in Table 3 and shown in Fig. 3(b).

### 8.1 Global Conditioning Index.

*m*is the dimension of matrix

**J**. This CN is not bound and can range from 1 to infinity. To bound the value between 0 and 1, the inverse is taken and is defined as the conditioning index (CI)

*W*is the parallel mechanism’s translational workspace and

*n*is the GCI. As the GCI approaches 1 from 0, the manipulator will have better performance [38]. The CI used to calculate the GCI has been called into question as it is based on the inhomogeneous Jacobian matrix [39]. This criticism stems from the argument that since the Jacobian has both dimensionless and dimensioned units and thus is dimensionally nonhomogeneous, there may not be a physical meaning for the condition number. Tandirci et al. [40] proposed the characteristic length (CL) to create a homogeneous Jacobian. The CL was further studied by Nawratil [41], and it was determined that the CL should be a predefined design constant

*R*is the radius of the operation sphere as defined by Nawratil. The CL is then used to create a dimensionally homogeneous Jacobian matrix

*L*is a 6 × 6 matrix,

*I*is a 3 × 3 identity matrix, and

*O*is a 3 × 3 zero matrix.

### 8.2 Objective Function Description.

*n*

_{in}is the number of points of the desired translational workspace the individual encompasses out of the total number of desired points

*N*

_{tot}, and a value of

*n*

_{in}/

*N*

_{tot}= 1 indicates the robot configuration can encompass the desired workspace. Similarly, for the rotational workspace,

*R*

_{i,in}is the number of points of the desired rotational workspace the individual encompasses out of the total number of desired points

*R*

_{tot}. Due to the nature of parallel mechanisms, the rotational workspace changes in size depending on where the end effector is in the translational workspace. Addressing this point, the points inside the desired workspace

*R*

_{i,in}/

*R*

_{tot}are evaluated at three positions (

*n*

_{R}is set to 3) in the translational workspace: at the (1) midrange position of Robossis and at the (2) maximum and (3) minimum

*x*-position of the medial–lateral malalignment range (Table 2). The

*x*-axis was chosen as this is the largest proportion of variance that is observed. After pulling traction, Robossis would then be attached to assist with the procedure. The action of pulling traction is anticipated to significantly reduce variance in the

*z*-direction (femoral–shaft axis), leaving the

*x*-axis (medial–lateral axis) as the largest axis of variation.

To ensure the individual can meet the forces observed during femur fracture surgery, the dynamics model (Eq. (9)) was used to determine the minimal force the individual can support. It is desired that the minimal force the individual can support is equal to or greater than the maximal force observed during femur fracture repair (Table 2). In Eq. (17), *F*_{tot} is the maximal force from Table 2 for each direction. *F*_{i,in} is the minimal amount of force the individual can support in each direction. The individual was evaluated on the ability to support loads in the *x*, *y*, and *z* directions (*n*_{F} is set to 3). For each direction, the SLCC simulation from Fig. 8 was used to determine the minimum guaranteed load within the femur fracture malalignment range. The smallest load recorded from the SLCC is *F*_{i,in}. *F*_{i,in}/*F*_{tot} = 1 indicates the individual can support all the maximal loads reported in Table 2. The motor torque used to determine the maximal load was 1.6 Nm, the minimum result from the SLCC analysis. Lastly, the GCI was calculated using the homogeneous Jacobian. Weights were used to apply significance to the different parameters (*w*_{1} − *w*_{4}). Figure 9 depicts the genetic algorithm pipeline used to obtain the optimal parameters for Robossis.

### 8.3 Single-Objective Genetic Algorithm.

Based on the results found by Yoon et al. [42], the GCI decreases as the rotational and translational workspaces increase. This inverse relationship provides a challenge when trying to optimize a mechanism for both parameters. To observe the effects of GCI and workspace size, the weights in the GA were set to only include GCI, and only include *R*_{i,in}. For maximizing GCI: (a) *w*_{1−4} = [0, 0, 0, 1]. For maximizing the rotational workspace: (b) *w*_{1−4} = [0, 1, 0, 0]. Figure 10 shows the result of the genetic algorithm for maximizing rotational workspace and GCI. The GCI and rotational volume for Robossis appear to have an inverse relationship.

Next, it is desired to have a robot configuration that meets the desired translation and rotation requirements as well as the force requirements (all ratios: *n*_{in}/*N*_{tot}, *R*_{i,in}/*R*_{tot}, *F*_{i,in}/*F*_{tot} = 1). The weights were all initially set to 1: *w*_{1−4} = [1, 1, 1, 1]. After the genetic algorithm was run, if a ratio did not equal 1, the weight for that ratio was altered, and the GA was run again. The final weights for the GA were [*w*_{1} = 1, *w*_{2} = 2.5, *w*_{3} = 1, *w*_{4} = 0.5]. Figure 11 compares the results of the three GA optimizations.

For parallel mechanisms, it is understood that the size of the rotational workspace decreases as the end effector moves from the midrange position. Figure 12 displays the rotational workspace at two end effector positions, the midrange position and the edge of the medial–lateral femur fracture malalignment range. The configurations from (a) and (c) can encompass all the requirements: *n*_{in}/*N*_{tot}, *R*_{i,in}/*R*_{tot}, *F*_{i,in}/*F*_{tot} = 1, as seen in the objective function parameter outputs in Table 4.

Table 4 includes the results of the GA for the three optimizations performed: (a) maximize rotations, (b) maximize GCI, and (c) meet all the OF requirements. In addition to the three GA optimizations, the current configuration of Robossis is shown for a comparison. Based on the information presented in Table 4, no configuration is the best in all areas of analysis. The GA result (c) meets all the objective function criteria and has a GCI of 0.407. Compared to the current Robossis design, the GA result (c) has a 29% decrease in translational workspace volume, a 38% increase in rotational volume, and a 7.3% decrease in GCI.

## 9 Experimental Testing

Experimental testing was conducted to study the accuracy, precision, and reliability under anatomically relevant loads. Three OptiTrack Flex-13 (NaturalPoint, Corvallis, OR) motion capture cameras were used alongside the motive software to track the 6-DoF motion of Robossis during the experimental testing using IR reflective markers (Fig. 13(a)). The testing was segmented into two parts: complex movements while under varying external forces (Fig. 13(b)) and repetitive precision testing under constant load (Fig. 13(c)). The error metric used was the Euclidean distance between the theoretical position of the robot and the actual position recorded by OptiTrack.

### 9.1 Repetitive Precision Testing.

To evaluate if there is an accruing error, repetitive testing was conducted. Robossis was commanded to move in isolated repetitive motions to gauge the accuracy and reliability of the mechanism. An isolated repetitive motion is defined as a 1D movement along each of the 6-DoFs (*x*, *y*, *z*, *α*, *β*, *γ*). The experimental results are displayed in Fig. 14. A one-sided *t* test was used to determine if the experimental results were less than the OptiTrack accuracy threshold of 0.088 mm. All isolated repetitive motion errors were significantly less than OptiTrack’s accuracy threshold (*p* < 0.05).

### 9.2 Complex Movements.

During the proposed use case, femur fracture repair, the movements of the robot will not be simple in nature, and the robot will move simultaneously along multiple axes. To evaluate the performance of Robossis, the robot was commanded to move along two paths: a 2D path and a 3D path. The first path was a circle with a varying external load, as seen in Fig. 15(a). The second path was along a 3D trajectory (Fig. 15(c)). Both experiments had an average trajectory error of less than 1 mm (Figs. 15(b)–15(d)). Additionally, both experiments were able to maintain the clinical criteria of at least 50% better than the Thoresen Metric “Excellent,” as shown by the green markers in Fig. 15.

## 10 Discussion

Based on our analysis and work, the need for improved clinical outcomes in femur fracture surgeries, the limitations with traditional parallel mechanisms for this application, an optimized solution of Robossis based on the femur fracture design criteria, and experimental force testing to evaluate the precision of Robossis has been demonstrated. The GSP is not able to meet the workspace limits outlined by our preliminary femur fracture malalignment range; this may be one of the reasons the GSP has not been used for femur fracture surgeries. Based on the static load-carrying capacity, Robossis needs motors capable of generating 1.6 Nm of torque; these motors can be adapted to the design without affecting the overall workspace. The genetic algorithm was able to find a solution to meet all the desired criteria of the objective function, increasing the rotational workspace volume by 38%. This result was only possible due to a 29% decrease in the translational workspace and a 7.3% decrease in GCI, as seen in Table 4. Further analysis and discussion may be necessary to determine if this tradeoff is overall beneficial.

The limitations of this study are the lack of literature-reported values on the necessary clinical and mechanical design criteria. The design criteria are based on the femur fracture malalignment range, which provides an estimate of the translational and rotational movements during femur fracture reduction surgery but does not give the exact movement ranges during surgery. In future work, it would be pertinent to design a musculoskeletal model to determine how the femur fragments can be manipulated before causing soft tissue damage and to record translational and rotational ranges during femur fracture repair surgeries. The goal would be to have an exact fracture reduction range similar to Lei and Wang’s work on pelvic fractures [43]. This would allow for a more accurate description of the workspace a robot must have to be used for femur fracture repair surgery and provide foresight on how much clinicians can manipulate the femur fragments before risking soft tissue damage.

## 11 Conclusion

Femur fracture surgery requires both large forces and rotations to reduce the fracture. Currently, these forces are generated manually by surgeons with the assistance of traction tables. We propose a parallel mechanism, Robossis, which is shown to be capable of generating the required forces, as well as meeting the clinical requirements to improve patient outcomes during femur fracture surgeries. In this study, we investigated the theoretical workspace and force generation limits and optimized the design to satisfy the mechanical and clinical design criteria. Through this analysis and optimization, it has been concluded that Robossis has the potential to improve clinical outcomes and reduce the chance of reoperations as a robot-assistive surgical device.

## Funding Data

The work presented in this paper was supported by the National Science Foundation (NSF) under Grant Nos. 2141099 and 2226489, and by the New Jersey Health Foundation (NJHF) under Grant No. PC 62-21.

## Conflict of Interest

There are no conflicts of interest.

## Data Availability Statement

The datasets generated and supporting the findings of this article are obtainable from the corresponding author upon reasonable request.