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Predictive force model for haptic feedback in bone sawing
Medical Engineering & Physics 35 (2013) 1638–1644
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Predictive force model for haptic feedback in bone sawing Thomas P. James ∗ , John J. Pearlman, Anil Saigal Laboratory for Biomechanical Studies, Department of Mechanical Engineering, Tufts University, 200 College Avenue, Medford, MA 02155, USA
a r t i c l e
i n f o
Article history: Received 14 December 2012 Received in revised form 13 May 2013 Accepted 26 May 2013 Keywords: Orthopedics Haptic feedback Surgical simulator Bone sawing force
a b s t r a c t Bone sawing simulators with force feedback represent a cost effective means of training orthopedic surgeons in various surgical procedures, such as total knee arthroplasty. To develop a machine with accurate haptic feedback, giving a sensation of both cutting force and rate of material removal, algorithms are required to forecast bone sawing forces based on user input. Presently, studies on forces generated while machining bone are not representative of the high cutting speeds and low depths of cut common to the bone sawing process. The objective of this research was to quantify sawing forces in cortical bone as a function of blade speed and depth of cut. A fixture was developed to simulate linear bone sawing over a range of speeds comparable to surgical reciprocating and oscillating (sagittal) bone saws. A single saw blade tooth was isolated and used to create a slotted cut in bovine cortical bone. Over a range in linear sawing speed from 1700 to 7000 mm/s, a t-test (˛ = 0.05) revealed there was no statistically significant effect of blade speed on either cutting or thrust force. However, an increase in depth of cut from 2 to 10 m resulted in a 30% increase in thrust force, while cutting force remained constant. The increase in thrust force with depth of cut was relatively linear, R2 = 0.80. Using a two factor, two level design of experiments approach, regression equations were developed to relate sawing forces to changes in blade speed and depth of cut. These equations can be used to predict forces in a haptic feedback model. © 2013 IPEM. Published by Elsevier Ltd. All rights reserved.
1. Introduction Haptic devices assist in the training of orthopedic surgeons through virtual simulations of procedures involving bone removal processes, such as sawing and drilling [1–3]. In haptic devices, the visual experience is reproduced, as well as the physical sensation of forces, through the use of mechanical actuators attached to surrogate tools. Hsieh and colleagues recently developed a surgical simulator for bone amputation by sawing [3]. With their device, feedback of sawing forces is generated by the application of a mechanistic bandsawing model developed by Ko and Kim [4]. However, the bandsawing model was developed to predict forces when cutting metals at relatively low speeds. Bone is known to be a viscoelastic material so forces must be determined at higher speeds appropriate to surgical saws [5]. Bone saws create high linear cutting speeds and low depths of cut per tooth, which distinguishes sawing from conventional machining. Sagittal bone saws oscillate the blade through an angle of approximately 5◦ at rates up to 20,000 oscillations per minute. Depending on blade length and oscillation rate, the resulting average linear velocity of saw blade teeth is between 4000 and 7000 mm/s. Under these conditions, the depth of cut per tooth in
∗ Corresponding author at: Tufts University, Mechanical Engineering, Anderson Hall, 200 College Avenue, Medford, MA 01921, United States. Tel.: +1 978 771 8945. E-mail address: [email protected] (T.P. James).
cortical bone is typically less than 5 m. This depth of cut is generally less than the cutting edge radius of saw blade teeth, which creates a cutting condition analogous to a cylinder ploughing across the surface of the bone, which generally crates discontinuous bone fragments under highly compressive conditions. Previous research into the machining of bone has primarily focused on lower cutting velocities (<1000 mm/s) and larger depths of cut (>10 m) than are common to power sawing. Krause was one of the earliest investigators of orthogonal machining of cortical bone [6,7], but the maximum cutting speed investigated was 409 mm/s and the depth of cut was 68 m. These conditions resulted in Krause proposing a dependence of force on cutting velocity. A relationship between velocity and force was also postulated by Jacobs et al. where it was determined that chips were formed primarily via fracture while machining bone at 7.7 mm/s [8,9]. Cutting performed later by Wiggins and Malkin at nearly an identical speed of 8.4 mm/s, and relatively high depths of cut, also resulted in fracture based material removal [10,11]. Recognizing the limitations of previous work on achieving high cutting speeds and low depths of cut, Plaskos constructed a pendulum mechanism to investigate orthogonal bone cutting at speeds up to 3471 mm/s and depths of cut down to 2 m. These conditions are similar to those encountered during sawing with surgical power tools. Here, in a contradiction to prior work, Plaskos concluded there was no dependence between velocity and cutting force [12]. In recent work by Yeager et al. over a velocity range of 310 to 1130 mm/s, there was a weak dependence of force on
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velocity, with a much greater dependence of force on depth of cut and on anisotropy of the bone [13]. In each of the previous investigations, orthogonal cutting was simulated. In the orthogonal arrangement, the cutting edge is much wider than a saw blade tooth and rather than cutting a slot, the cutting edge extends over the edges of the workpiece. In sawing, the cutting edge width is less than the workpiece so shear along the slot edges of the saw blade tooth can contribute to the cutting force. In addition, previous work has generally regarded the cutting edge as perfectly sharp when compared to the depth of cut. In sawing, the cutting edge radius is on the same order of magnitude as the depth of cut and therefore particular care must be taken to measure and report the cutting edge radius and to be mindful of the effect of cutting edge wear on measured forces. The primary objective of this research is to quantify the relationship between forces and velocity at sub-radius depths of cut and speeds representative of a bone sawing process. The results from previous research on bone machining provide conflicting results on the dependency of force on velocity. In addition, sub-radius cutting conditions at speeds greater than 4000 mm/s have not been investigated. Finally, the ratio of cutting force to thrust force in a sawing process has not been studied. Sub-radius cutting is expected to yield high thrust forces [14].
2. Methods and materials 2.1. Experimental fixture Limitations of previous research to achieve cutting speeds common to power sawing appear to be a result of using standardized machining equipment as a means to accelerate the workpiece into the cutting tool. For example, while a milling machine or a shaper may create the necessary linear cutting conditions and accuracy for small depths of cut, the bed of the machines moves too slowly to represent sawing conditions. Alternatively, lathe turning equipment can be used to achieve the proper cutting speeds, but in this case the cutting is not linear, which exposes the cutting tool to different material properties along a single cutting path. This is due to a combination of a curved cutting path and a highly anisotropic workpiece. To generate linear cutting conditions representative of the high speeds encountered in bone sawing, a new fixture was designed to meet the following specifications: (1) Linear cutting velocity of at least 7000 mm/s, (2) Minimum incremental tool feed of 2.5 m, and (3) Linear cutting distance of approximately 50 mm. The first two specifications for cutting velocity and depth of cut were determined from measurements of cutting rate and blade speed during preliminary investigations with a surgical bone saw. The final specification for sample length was based upon general experience with using dynamometers in high speed cutting applications, where it takes a sufficient length of cut to both reach steady state cutting conditions and to obtain sufficient data for the determination of average forces. The new fixture was designed with consideration of the high speeds that could be obtained when rotating a workpiece in a lathe arrangement. However, compensation for the curved cutting path was necessary to prevent the effect of material anisotropy on measured cutting forces. From a design standpoint, the problem of a curved cutting path was overcome by observing that it is possible to adjust the timing of two counter-rotating disks such that they produce a linear cutting path when viewed from a fixed point. When operating the high speed linear sawing machine, both the rotor and workpiece holder were rotated while the cutting tool (saw blade tooth) remained fixed to ground. A schematic of this high speed linear cutting system is shown in Fig. 1(A). The depth of cut
Fig. 1. (A) Conceptual model of the test apparatus used to achieve high linear speeds and low depths of cut. Rotation of the bone sample holder in a direction counter clockwise to the main rotor, and at a speed equal to one half of the rotor speed, ω, creates a linear cutting path at the point where the saw blade tooth contacts the bone. The depth of cut control creates precise micro-movement of the saw blade tooth by using a mechanical screw to advance a shallow wedge. Cutting and thrust forces are measured with the dynamometer. (B) The workpiece holder is shown in two positions to demonstrate how a linear path is created by using rotary motion (not to scale). In this depiction, rotation of the rotor by a positive 20 degrees results in rotation of the workpiece holder by a negative 10 degrees. This relative and apposing rotary motion creates a constant depth of cut along a straight line relative to a fixed point (cutting tooth) as shown. The size of the workpiece and depth of cut are highly exaggerated for illustrative purposes.
remained constant so long as the workpiece holder rotated in the opposite direction of the rotor at one half the speed, Fig. 1(B). This was accomplished with a 2:1 gear set, where a 1/3 HP permanent magnet motor (Model LM90T033, Dart Controls Inc., Zionsville, IN) was used to drive the rotor under variable speed control (Dart 250G, Dart Controls Inc., Zionsville, IN). Rotational inertia of the system was sufficient to avoid changes in speed during the cutting process. This permitted speed to be investigated independently of cutting forces. In order to minimize the effect of vibration on force measurement, the system was dynamically balanced with an adjustable counterweight at a fixed radial distance. 2.2. Feed rate control Unlike traditional one-pass methods of collecting force data, such as with a milling machine, shaper, or pendulum device, the new fixture used an incremental tool feed mechanism to facilitate data collection over several passes so that a greater statistical power could be achieved. In order to build a robust tool feed system, which would prevent the tool from inadvertently crashing into the
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Fig. 2. 3D CAD model of the depth of cut control mechanism. The power screw drives a shallow wedge to deflect a stiff cantilevered beam. This arrangement results in precise feed rate control with very little spring back as the saw blade tooth exits the kerf in the bone. The tool holder, which is attached directly to the dynamometer, contains a single saw blade tooth.
workpiece, a purely mechanical system was developed. This was accomplished by using a power screw to drive a self locking wedge to deflect a cantilever beam. At the end of the beam, the force gage and cutting tool were attached as shown conceptually in Fig. 1(A) and as a 3D solid model in Fig. 2. When a screw is used to drive a wedge, which has a very low wedge angle, movement perpendicular to the wedge is precise and there is virtually no backlash. By using a wedge to bend a sufficiently stiff beam, there is essentially no spring-back during or after the cutting event. A combination of the power screw pitch and sliding wedge angle provided micrometer level resolution in tool advancement as verified in calibration experiments. Ultimately, the system provided rigid mechanical feed of the cutting tool between successive workpiece passes so that force data could be collected and averaged for several cuts. The complete tool feed system was mounted alongside the rotor mechanism on top of a granite surface plate, Fig. 3. A full description of the high speed linear sawing fixture and design methodology was previously published [14]. 2.3. Cutting tool (saw blade tooth) Cutting tools were fashioned by isolating single teeth from surgical saw blades (KM-458, Brasseler USA, Savannah, GA). The blade was constructed from 0.64 mm thick stainless steel and incorporated an alternating tooth set pattern (Left–Right–Left) with a pitch of 18 teeth per inch. Rake and clearance angles were approximately −10 and 60◦ , respectively, with measured edge radii ranging
Fig. 3. High speed linear cutting machine mounted to a granite surface plate. The drive motor, speed control, and related gearing lie below the surface plate. The counterweight is adjustable to account for changes in mass of the workpiece during dynamic calibration of the dynamometer.
Fig. 4. Workpiece holder attached to the dynamometer. The arrow is pointing at a single saw blade tooth, which is appropriately mounted in the tool holder to preserve the proper rake angle.
between 10 m and 17 m. Initial cutting edge radii were measured by mounting a single tooth in epoxy and then sanding and polishing the surface to various depths to determine changes in cutting edge radius along the face of the tooth by optical microscope. In addition, a three dimensional view of edge radii was evaluated by scanning electron microscopy (SEM). The saw blade teeth were held in a custom tool holder attached to the force gage, where the −10◦ rake face angle on the cutting face was maintained, Fig. 4. Initial cutting tests were performed to determine the effect of multiple cutting passes on edge wear. For the number of cuts performed with each saw blade tooth, which was typically less than 100 passes over a 50 mm workpiece, there was no evidence of edge wear upon periodic inspection with an optical microscope and SEM. 2.4. Force measurement From prior research, the magnitude of cutting forces in bone sawing was expected to be less than 100 N for single tooth cutting conditions. In addition, the cutting event was expected to last <50 ms for a single pass over a 50 mm length. Under these conditions, a force gage capable of quickly capturing dynamic cutting events was required. A 3-axis piezoelectric dynamometer (9047C dynamometer, Kistler Instrument Corp., Amherst, NY) was used to capture forces during cutting. The dynamometer had a dynamic range of 30 kN in the thrust force direction, Fz , and 15 kN in the cutting force direction, Fx , with a measurement sensitivity of 8.1 pC/N in Fx , and 3.7 pC/N in Fz . Data from the dynamometer was transmitted to a charge amplifier (ICAM 5073, Kistler Instrument Corp., Amherst, NY), which had a sensitivity of 1 ± 0.5% pC/mV. Collection of sufficient data to calculate representative average cutting forces required a high speed data collection system. As the maximum speed of 7000 mm/s was approached, the time to acquire data over the 50 mm cut was approximately 7 ms. High speed data acquisition was achieved with a multifunction data acquisition board (PCI 6132, National Instruments, Austin, TX), which had a simultaneous sampling rate of 2.5 × 106 samples per second. A LabVIEW (National Instruments, Austin, TX) virtual instrument was written to stream the sampled data to a series of TDMS binary files. These were then converted to LVMS files through a separate virtual instrument, at which point they were imported into MATLAB (MathWorks, Natick, MA) for data smoothing (4th order, low pass, Butterworth filter) and determination of the average cutting and thrust forces.
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HBSS, while being stored in a refrigerator during the 24 h adhesive curing time. Once cured, the sample assemblies were secured in a mill and the bone surfaces were machined to normalize thickness and to expose subperiosteal cortical bone. To prevent surface degradation due to heat from machining and subsequent drying, the bone samples were fully submerged in HBSS during the milling operation. 2.6. Cutting procedure
Fig. 5. Segments of bovine cortical bone after removal from the mid-diaphysis of the femur. The numbers and line markings signify the sample identification used in the Design of Experiments.
2.5. Bone sample preparation Prior to conducting single tooth cutting experiments, suitable bone samples were selected, rough sawn, secured to a mounting plate, and finish machined. During the steps involved in sample preparation, special care was taken to keep the bone moist through either submersion or misting with Hanks’ balanced salt solution (HBSS). For consistency, two fresh femurs from a 20 to 30 month adult bovine were procured from a local abattoir. Two 50 mm sections were removed by band saw from the mid-diaphysis of each femur. Once removed, the cortex of each section was sawn radially into six separate samples, Fig. 5. Through visual inspection, the bone samples were representative of homogeneous cortical bone with no apparent defects. Sample orientation was chosen such that cutting experiments would produce a slot parallel to the primary osteon direction, which falls along the axis of a long bone. After processing of the bulk femur, rectangular bone samples were attached to metal backing plates. After flattening the interior face (marrow side) of the bone by hand with a block plane, the surface was slightly roughened with a rasp file. Adhesive (Fast Cure 5200, 3M, St. Paul, MN) was applied to the roughened surface to secure the bone to an aluminum backing plate, Fig. 6. The completed samples were covered with towels, which were soaked with
For each experiment, a wedge angle and appropriate power screw rocker arm were selected for the desired depth of cut (feed) per cutting pass. Upon clear formation of a kerf along the full 50 mm of the bone surface, cutting and thrust forces were recorded with the dynamometer for a minimum of 10 passes. In post processing of force measurements for each cut, the data was clipped to eliminate the erroneous response of the dynamometer to the impact event as the tooth first entered the bone sample and again to the disrupted surface morphology near the exit of the kerf. The data eliminated from each cutting pass represented approximately the first 5 mm and the last 2 mm of the 50 mm cutting distance. The remaining force data was considered representative of steady state cutting conditions. 2.7. Screening studies Screening experiments were initially conducted to determine the appropriate factor levels. From the perspective of designing an experiment, cutting velocity and depth of cut represent the experimental factors, while cutting and thrust force represent the response variables. In order to minimize the number of experimental trials, while capturing both the effect of the individual factors and the effect of factor interactions on the response variables, a full factorial Design of Experiments (DOE) approach and related statistical methods was pursued [15]. A screening study on depth of cut was performed while holding velocity constant at 3980 mm/s. A range in depth of cut from a low of 2.5 m to a high of 10.2 m was investigated in increments of 2.5 m. A second screening study was performed on velocity while holding depth of cut constant at 7.6 m. Levels for cutting velocity were chosen to represent the clinical application of a sagittal saw, which operates between 12,000 and 20,000 cycles per minute. Actual tip speed of a saw blade tooth depends on both oscillation rate and blade length, where a typical sagittal saw blade varies in length from between 60 and 100 mm. Considering the variation in blade length and oscillation rate, the screening study investigated cutting velocity from a low of 1764 mm/s to a high of 7050 mm/s in increments of 880 mm/s. 2.8. Full factorial experimental design
Fig. 6. Cortical bone sample mounted to an aluminum backing plate. The sample shows five cuts, constituting an entire experimental replicate.
Upon completion of the screening studies, a two factor, two level, full factorial experiment was designed with four unique treatments. The lower and upper levels for depth of cut were set at 2.5 m and 10 m, respectively. For velocity, the lower level was set at 2645 mm/s and the upper level at 6172 mm/s, representing the tooth tip speeds for common reciprocating and oscillating surgical saws and falling within the range of the screening study. Per the experimental design, two sets of treatment orders were then defined and each was applied to a single bone sample, Table 1. There were sufficient bone samples to replicate the full experiment five times, adding to the statistical power of the regression analysis. While implementing the experiment, randomization was employed wherever possible to negate procedural bias and physical differences in the mechanical properties between bone samples. From the original twelve bone samples, seven were suitable for use
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Table 1 Four trials for each treatment of the two factor, two level, Design of Experiments. Two treatment orders were specified to aid in randomization. Treatment order 1
Treatment order 2
Velocity (mm/s)
Depth of cut (m)
Velocity (mm/s)
Depth of cut (m)
6172 2645 2645 6172
2.5 2.5 10 10
6172 2645 2645 6172
10 10 2.5 2.5
after mounting to the aluminum backing plates and finish machining. From these seven, five bone samples, one for each experiment replication, were chosen at random. One of the two defined treatment orders was then assigned to each sample with a coin toss. The randomized treatment order and sample assignments are recorded in Table 2. Lastly, a new saw blade tooth was randomly chosen for each experimental run from a large supply of previously prepared teeth. Before collecting data, each new saw blade tooth was “broken in” by making a single cut within the assigned bone sample used in the particular replication. 2.9. Statistical methods Following the Design of Experiments methodology, the higher level of each factor was coded as +1 and the lower level of each factor was coded as −1. Given the predominantly linear relationship between the factors and the response variables, a two factor, two level, full factorial DOE was pursued. To avoid confounding, the full factorial design required four unique treatments (+1+1, −1+1, +1−1, −1−1). With this scheme, the significance of each factor and their interaction on the response variable can be uniquely determined. Dynamometer data for both cutting force and thrust force was captured and analyzed for 10 passes of the cutting tool over the workpiece for each replication. This resulted in a total of 200 individual cuts (10 cuts per replication, multiplied by 5 replications of each unique treatment, multiplied by 4 unique treatments) from which to perform the statistical analysis and to develop the cutting and thrust force predictive models. Following the DOE methodology, the main factor effects were determined by taking the difference between the average response of the high and low levels of a factor. The standard deviation of the total experiment, Se , was determined by taking the square root of the average of the variances for each unique treatment. Once the standard deviation for the experiment was known, the standard deviation of the main factor effects, Seff , for the two level exper-
iment were determined by Seff = Se 4/N, where N = 20, which is the total number of replications. Finally, the decision limits for statistical significance of the factor effects were determined by multiplying the t-test value (˛ = 0.05) by the standard deviation of the factor effects. Table 2 Randomization of treatment orders and sample assignment for each of the five replicates. Replicate number
Sample assignment (See Note)
Treatment order (see Table 1)
1 2 3 4 5
S2-4 S1-1 S1-6 S2-6 S2-1
2 1 1 2 1
S2-4 refers to bone sample 2, Section 4, as identified in Fig. 5.
Fig. 7. Results from the screening study on depth of cut. Cutting force (solid diamond markers) and thrust force (solid square markers) represent the average of 10 cutting passes at a constant velocity of 6172 mm/s. The error bars represent + and −1 standard deviation from the mean.
Following an analysis of variance, linear equations were developed to predict cutting and thrust forces, FC and FT , respectively, according to Eq. (1) [15]. FC,T = FAVE +
E(v) E(d) E(vd) v+ vd d+ 2 2 2
(1)
Regarding the effect of the factors on force, E(v) is the effect of velocity, E(d) is the effect of depth of cut, and E(vd) is the effect of the interaction. The first term in Eq. (1), FAVE , represents the average of the measured cutting or thrust force components for all DOE treatments. The regression coefficients for the equation are determined by taking one half of the factor effects. Finally, v and d are assigned the coded values between +1 and −1 to represent a linear mapping of the actual high and low factor levels, respectively. 3. Results Results from the screening study on depth of cut demonstrated a linear trend of lower thrust force at lower depths of cut, R2 = 0.80. Meanwhile, cutting force appeared less sensitive to changes in depth of cut, Fig. 7. For this study, depth of cut was varied from 1.3 m to 10.2 m while velocity was held constant at 6172 mm/s. The average cutting forces measured were in the range of 4 N/mm to 10 N/mm, while thrust forces were in the range of 8 N/mm to 16 N/mm. For all depths of cut, thrust force exceeded cutting force, with a relatively constant thrust to cutting force ratio of between 1.5 and 2.0. During the screening study on velocity, cutting and thrust force remained relatively constant, Fig. 8. For this screening study, the depth of cut was held constant at 7.6 m, while the velocity was increased from 2625 mm/s to 10,583 mm/s. Cutting force was relatively constant at 8 N/mm over the full range of cutting velocities. The response of thrust force to cutting velocity showed a slight increase from 10 N/mm to 13 N/mm at higher velocities. As a result of the screening studies, there was reasonable evidence of a linear response in both cutting and thrust force to changes in the factor levels, which supported the choice for a two level, two factor, experimental design. When considering the average forces for each treatment combination, forces in the thrust direction were greater than those in the cutting direction by a factor of approximately 1.5 and the ratio was slightly greater at the higher depth of cut level, Fig. 9.
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Fig. 8. Results from the screening study on velocity. Cutting force (solid diamond markers) and thrust force (solid square markers) represent the average of 10 cutting passes at a constant 7.6 m depth of cut. The error bars represent + and −1 standard deviation from the mean.
The main factor and interaction effects demonstrate that depth of cut has the greatest impact on thrust force while velocity has the greatest impact on cutting force, Fig. 10. The interaction of velocity with depth of cut has only a negligible effect. The decision limit for statistical significance (˛ = 0.05) related to the effect of velocity on cutting and thrust force was determined to be 0.80 N, while the decision limit for the effect of depth of cut on force was determined to be 0.71 N. For factor effect values below the decision limit, the results may be related to the factor or they may be simply due to variation inherent within the experiment. In strict application of the decision limits to the data, there is no statistically significant effect of velocity on either cutting or thrust force, Fig. 10. However, an increase in depth of cut did have a sta-
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Fig. 10. Absolute value of the individual factor effects and interaction effect (velocity with depth of cut) as determined from the Design of Experiments. Due to the inherent variation in the force measurements, the t-test limit for statistical significance (˛ = 0.05) is 0.80 N for cutting force and 0.71 N for thrust force. Based on these criteria, only the effect of depth of cut on thrust force is statistically significant.
tistically significant effect on thrust force. An increase in depth of cut from 2.5 m to 10.2 m resulted in a 30% increase in average thrust force, while the effect on cutting force fell below the threshold limit. The interaction of velocity and depth of cut fell below the decision limit for both cutting and thrust force. The application of Eq. (1), following the ANOVA to determine the regression coefficients, resulted in the following equations for predicting cutting and thrust force when sawing bovine cortical bone with a single saw blade tooth: FCUTTING = 4.54 + 0.356v + 0.289d − 0.178vd
(2)
FTHRUST = 7.07 + 0.200v + 0.912d − 0.178vd
(3)
It should be noted that the magnitude of each effect in Eqs. (2) and (3) is dependent on the units of force. Here, the units of force are Newton for a cutting edge width of 0.64 mm. The regression coefficients in Eqs. (2) and (3) are equal to one half of the factor effects. While it is traditional practice to show the absolute values of the factor effects, Fig. 10, the regression equations include the appropriate sign. Following the DOE and statistical analysis, three additional experiments were conducted for independent verification of the regression equations. Table 3 contains a summary of the predicted and measured forces, along with the experimental conditions and corresponding parameters for v, d, and vd used in Eqs. (2) and (3). The predicted forces were within approximately 10% of the measured forces in all but one case. This percent difference is within the standard deviation of results from the screening experiments. 4. Discussion
Fig. 9. Results from the two factor, two level, full factorial Design of Experiments, which consisted of 8 treatment combinations. Each bar represents the average thrust force or cutting force for various combinations of cutting velocity and depth of cut.
The average cutting force in Eq. (2), 4.54 N, and the average thrust force in Eq. (3), 7.07 N, is an order of magnitude greater than the influence of either of the factor effects. In addition, the average thrust force in Eq. (3) is approximately 1.5 times larger than the
Table 3 Verification experiments and corresponding predictions from regression model, Eqs. (2) and (3). Factor levels
Parameters used in Eqs. (2) and (3)
Cutting force (N)
Thrust force (N)
Depth of cut, d (m)
Velocity, v (mm/s)
d
v
vd
Empirical
Eq. (2)
Empirical
Eq. (3)
5.1 7.6 7.6
6172 3509 4406
−0.33 0.33 0.33
1.00 −0.51 0.00
−0.33 −0.17 0.00
5.9 5.0 4.2
4.9 4.5 4.6
7.5 7.1 7.5
7.0 7.3 7.4
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average cutting force in Eq. (2). With these two conditions present, the regression equations predict that thrust force will always be greater than cutting force for all admissible factor levels, resulting in a thrust to cutting force ratio that is always greater than one. A thrust to cutting force ratio greater than one is expected under conditions where depth of cut is less than the cutting edge radius and therefore ploughing dominates chip creation [16,17]. From inspection of chips created during the sawing experiments, there was no evidence of continuous chips, but rather an accumulation of very fine particles. This too is an indication of ploughing and abrasive chip creation, rather than the existence of a well defined shear plane that could facilitate continuous chips. A predominantly linear relationship was found between tool forces and depth of cut, which was also reflected in prior research on machining of bone [8,11,12]. The interaction between factors, depth of cut and cutting velocity, had a similar impact on the cutting and thrust force response. The negative signs in Eqs. (2) and (3) i.e. (−0.178), can be interpreted to mean that higher values of both depth of cut and cutting velocity tend to reduce the average cutting and thrust force. Force measurements from the linear sawing fixture were in relatively close agreement with previous results reported by Plaskos et al. [12]. Cutting force measurements correlated most closely, falling within the range of 4 N/mm to 10 N/mm as compared to the 6 N/mm to 8 N/mm for Plaskos. The same could not be said for thrust forces, however, as they were 8 N/mm to 16 N/mm in this study as compared to 3 N/mm to 12 N/mm for Plaskos. The Plaskos results were produced with a neutral rake tool of unspecified edge radius and their tool width was greater than workpiece thickness. The higher thrust forces determined here may be attributed to the tool having a larger edge radius than that used by Plaskos. The regression equations developed as part of this research represent the forces encountered when cutting parallel to the primary osteon direction. It is well known that the bulk material properties of cortical bone, as well as the forces measured during machining, are dependent on the cutting direction, parallel, transverse or radial with respect to the osteons [5,12]. The length of cut was not sufficient to develop steady state cutting conditions for the radial or transverse directions, where thickness of the bovine cortical bone is typically less than 10 mm. When developing algorithms for force feedback, the forces predicted by the regression equations must be extrapolated from single teeth to a multi-toothed saw blade. When doing so, tooth set must be considered. Teeth are set out of plane on a saw blade to reduce the friction between the backing of the blade and the sides of the kerf and to provide clearance for chips. The oblique cutting face of a set tooth may create out-of-plane cutting and thrust force components. 5. Conclusion Mechanistic models are required to predict bone sawing forces when developing haptic feedback systems for surgical simulators. To fully investigate the effect of cutting velocity and depth of cut on bone sawing forces a full factorial Design of Experiments approach was utilized, resulting in a few observations:
Abiding strictly by the decision limits for statistical significance, a t-test (˛ = 0.05) indicated that depth of cut had a significant impact on thrust force, but not on cutting force; Cutting velocities between 2645 mm/s and 6172 mm/s, representative of sagittal bone saws, did not have a statistically significant effect on either cutting or thrust force; The two factor interaction of cutting velocity and depth of cut did not have a significant effect on either cutting or thrust force; When investigating depths of cut from 2.5 m to 10.2 m, the thrust force increased 30%, but cutting force was relatively unaffected; The regression equations revealed that thrust force will always be greater than cutting force for the range of velocities and depths of cut investigated. Author declarations Competing interests: None declared. Funding: No external funding was received. Internal funding was received from the School of Engineering at Tufts University, Medford, MA, USA. Ethical approval: Not required. Acknowledgement This research was supported financially by Tufts University, School of Engineering, Medford, Massachusetts, USA. References [1] Vankipuram M, Kahol K, McLaren A, Panchanathan S. A virtual reality simulator for orthopedic basic skills: A design and validation. J Biomed Inform 2010;43(5):661–8. [2] Tsai MD, Hsieh MS, Tsai CH. Bone drilling haptic interaction for orthopaedic simulator. Comput Biol Med 2007;37:1709–18. [3] Hsieh MS, Tsai MD, Yeh YD. An amputation simulator with bone sawing haptic interaction. Biomed Eng Appl Basis Commun 2006;18(5):229–36. [4] Ko TJ, Kim HS. Mechanistic cutting force model in band sawing. Int J Mach Tools 1999;39(8):1185–97. [5] Reilly DT, Burstein AH. The mechanical properties of cortical bone. J Bone Joint Surg 1974;56:1001–22. [6] Krause WR. Orthogonal bone cutting: Saw design and operating characteristics. J Biomech Eng 1987;109:263–71. [7] Krause WR. Mechanical effects of orthogonal bone cutting. Clemson, South Carolina, USA.: Clemson University; 1996. PhD Thesis. [8] Jacobs CH, Pope MH, Berry JT, Hoaglund F. A study of the bone machining process–orthogonal cutting. J Biomech 1976;7:131–6. [9] Jacobs CH, Pope MH, Berry JT, Hoaglund F. A study of the bone machining process–drilling. J Biomech 1976;9(5):343–9. [10] Wiggins KL, Malkin S. Drilling of bone. J Biomech 1976;9(9):553–9. [11] Wiggins KL, Malkin S. Orthogonal machining of bone. J Biomech Eng 1978;100:122–30. [12] Plaskos C, Hodgson AJ, Cinquin P. Modeling and optimization of bone-cutting forces in orthopaedic surgery. Lect Notes Comput Sci 2003;2878:254–61. [13] Yeager C, Nazari A, Arola D. Machining of cortical bone: surface texture, surface integrity and cutting forces. Mach Sci Technol 2008;12:100–18. [14] Pearlman JJ. Cutting Velocity Effects in Bone Sawing. Medford. MA, USA: Tufts University; 2011 [MS Thesis]. [15] Antony J. Design of Experiments for Engineers and Scientists. New York: Butterworth Heinemann; 2003. [16] Albrecht P. New development in the theory of metal-cutting process, part I. The ploughing process in metal cutting. J Eng Ind – Trans ASME 1960;82: 348–57. [17] Basuray PK, Misra BK, Lal GK. Transition from ploughing to cutting during machining with blunt tools. Wear 1977;43(3):341–9.
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Medical Engineering & Physics journal homepage: www.elsevier.com/locate/medengphy
Predictive force model for haptic feedback in bone sawing Thomas P. James ∗ , John J. Pearlman, Anil Saigal Laboratory for Biomechanical Studies, Department of Mechanical Engineering, Tufts University, 200 College Avenue, Medford, MA 02155, USA
a r t i c l e
i n f o
Article history: Received 14 December 2012 Received in revised form 13 May 2013 Accepted 26 May 2013 Keywords: Orthopedics Haptic feedback Surgical simulator Bone sawing force
a b s t r a c t Bone sawing simulators with force feedback represent a cost effective means of training orthopedic surgeons in various surgical procedures, such as total knee arthroplasty. To develop a machine with accurate haptic feedback, giving a sensation of both cutting force and rate of material removal, algorithms are required to forecast bone sawing forces based on user input. Presently, studies on forces generated while machining bone are not representative of the high cutting speeds and low depths of cut common to the bone sawing process. The objective of this research was to quantify sawing forces in cortical bone as a function of blade speed and depth of cut. A fixture was developed to simulate linear bone sawing over a range of speeds comparable to surgical reciprocating and oscillating (sagittal) bone saws. A single saw blade tooth was isolated and used to create a slotted cut in bovine cortical bone. Over a range in linear sawing speed from 1700 to 7000 mm/s, a t-test (˛ = 0.05) revealed there was no statistically significant effect of blade speed on either cutting or thrust force. However, an increase in depth of cut from 2 to 10 m resulted in a 30% increase in thrust force, while cutting force remained constant. The increase in thrust force with depth of cut was relatively linear, R2 = 0.80. Using a two factor, two level design of experiments approach, regression equations were developed to relate sawing forces to changes in blade speed and depth of cut. These equations can be used to predict forces in a haptic feedback model. © 2013 IPEM. Published by Elsevier Ltd. All rights reserved.
1. Introduction Haptic devices assist in the training of orthopedic surgeons through virtual simulations of procedures involving bone removal processes, such as sawing and drilling [1–3]. In haptic devices, the visual experience is reproduced, as well as the physical sensation of forces, through the use of mechanical actuators attached to surrogate tools. Hsieh and colleagues recently developed a surgical simulator for bone amputation by sawing [3]. With their device, feedback of sawing forces is generated by the application of a mechanistic bandsawing model developed by Ko and Kim [4]. However, the bandsawing model was developed to predict forces when cutting metals at relatively low speeds. Bone is known to be a viscoelastic material so forces must be determined at higher speeds appropriate to surgical saws [5]. Bone saws create high linear cutting speeds and low depths of cut per tooth, which distinguishes sawing from conventional machining. Sagittal bone saws oscillate the blade through an angle of approximately 5◦ at rates up to 20,000 oscillations per minute. Depending on blade length and oscillation rate, the resulting average linear velocity of saw blade teeth is between 4000 and 7000 mm/s. Under these conditions, the depth of cut per tooth in
∗ Corresponding author at: Tufts University, Mechanical Engineering, Anderson Hall, 200 College Avenue, Medford, MA 01921, United States. Tel.: +1 978 771 8945. E-mail address: [email protected] (T.P. James).
cortical bone is typically less than 5 m. This depth of cut is generally less than the cutting edge radius of saw blade teeth, which creates a cutting condition analogous to a cylinder ploughing across the surface of the bone, which generally crates discontinuous bone fragments under highly compressive conditions. Previous research into the machining of bone has primarily focused on lower cutting velocities (<1000 mm/s) and larger depths of cut (>10 m) than are common to power sawing. Krause was one of the earliest investigators of orthogonal machining of cortical bone [6,7], but the maximum cutting speed investigated was 409 mm/s and the depth of cut was 68 m. These conditions resulted in Krause proposing a dependence of force on cutting velocity. A relationship between velocity and force was also postulated by Jacobs et al. where it was determined that chips were formed primarily via fracture while machining bone at 7.7 mm/s [8,9]. Cutting performed later by Wiggins and Malkin at nearly an identical speed of 8.4 mm/s, and relatively high depths of cut, also resulted in fracture based material removal [10,11]. Recognizing the limitations of previous work on achieving high cutting speeds and low depths of cut, Plaskos constructed a pendulum mechanism to investigate orthogonal bone cutting at speeds up to 3471 mm/s and depths of cut down to 2 m. These conditions are similar to those encountered during sawing with surgical power tools. Here, in a contradiction to prior work, Plaskos concluded there was no dependence between velocity and cutting force [12]. In recent work by Yeager et al. over a velocity range of 310 to 1130 mm/s, there was a weak dependence of force on
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velocity, with a much greater dependence of force on depth of cut and on anisotropy of the bone [13]. In each of the previous investigations, orthogonal cutting was simulated. In the orthogonal arrangement, the cutting edge is much wider than a saw blade tooth and rather than cutting a slot, the cutting edge extends over the edges of the workpiece. In sawing, the cutting edge width is less than the workpiece so shear along the slot edges of the saw blade tooth can contribute to the cutting force. In addition, previous work has generally regarded the cutting edge as perfectly sharp when compared to the depth of cut. In sawing, the cutting edge radius is on the same order of magnitude as the depth of cut and therefore particular care must be taken to measure and report the cutting edge radius and to be mindful of the effect of cutting edge wear on measured forces. The primary objective of this research is to quantify the relationship between forces and velocity at sub-radius depths of cut and speeds representative of a bone sawing process. The results from previous research on bone machining provide conflicting results on the dependency of force on velocity. In addition, sub-radius cutting conditions at speeds greater than 4000 mm/s have not been investigated. Finally, the ratio of cutting force to thrust force in a sawing process has not been studied. Sub-radius cutting is expected to yield high thrust forces [14].
2. Methods and materials 2.1. Experimental fixture Limitations of previous research to achieve cutting speeds common to power sawing appear to be a result of using standardized machining equipment as a means to accelerate the workpiece into the cutting tool. For example, while a milling machine or a shaper may create the necessary linear cutting conditions and accuracy for small depths of cut, the bed of the machines moves too slowly to represent sawing conditions. Alternatively, lathe turning equipment can be used to achieve the proper cutting speeds, but in this case the cutting is not linear, which exposes the cutting tool to different material properties along a single cutting path. This is due to a combination of a curved cutting path and a highly anisotropic workpiece. To generate linear cutting conditions representative of the high speeds encountered in bone sawing, a new fixture was designed to meet the following specifications: (1) Linear cutting velocity of at least 7000 mm/s, (2) Minimum incremental tool feed of 2.5 m, and (3) Linear cutting distance of approximately 50 mm. The first two specifications for cutting velocity and depth of cut were determined from measurements of cutting rate and blade speed during preliminary investigations with a surgical bone saw. The final specification for sample length was based upon general experience with using dynamometers in high speed cutting applications, where it takes a sufficient length of cut to both reach steady state cutting conditions and to obtain sufficient data for the determination of average forces. The new fixture was designed with consideration of the high speeds that could be obtained when rotating a workpiece in a lathe arrangement. However, compensation for the curved cutting path was necessary to prevent the effect of material anisotropy on measured cutting forces. From a design standpoint, the problem of a curved cutting path was overcome by observing that it is possible to adjust the timing of two counter-rotating disks such that they produce a linear cutting path when viewed from a fixed point. When operating the high speed linear sawing machine, both the rotor and workpiece holder were rotated while the cutting tool (saw blade tooth) remained fixed to ground. A schematic of this high speed linear cutting system is shown in Fig. 1(A). The depth of cut
Fig. 1. (A) Conceptual model of the test apparatus used to achieve high linear speeds and low depths of cut. Rotation of the bone sample holder in a direction counter clockwise to the main rotor, and at a speed equal to one half of the rotor speed, ω, creates a linear cutting path at the point where the saw blade tooth contacts the bone. The depth of cut control creates precise micro-movement of the saw blade tooth by using a mechanical screw to advance a shallow wedge. Cutting and thrust forces are measured with the dynamometer. (B) The workpiece holder is shown in two positions to demonstrate how a linear path is created by using rotary motion (not to scale). In this depiction, rotation of the rotor by a positive 20 degrees results in rotation of the workpiece holder by a negative 10 degrees. This relative and apposing rotary motion creates a constant depth of cut along a straight line relative to a fixed point (cutting tooth) as shown. The size of the workpiece and depth of cut are highly exaggerated for illustrative purposes.
remained constant so long as the workpiece holder rotated in the opposite direction of the rotor at one half the speed, Fig. 1(B). This was accomplished with a 2:1 gear set, where a 1/3 HP permanent magnet motor (Model LM90T033, Dart Controls Inc., Zionsville, IN) was used to drive the rotor under variable speed control (Dart 250G, Dart Controls Inc., Zionsville, IN). Rotational inertia of the system was sufficient to avoid changes in speed during the cutting process. This permitted speed to be investigated independently of cutting forces. In order to minimize the effect of vibration on force measurement, the system was dynamically balanced with an adjustable counterweight at a fixed radial distance. 2.2. Feed rate control Unlike traditional one-pass methods of collecting force data, such as with a milling machine, shaper, or pendulum device, the new fixture used an incremental tool feed mechanism to facilitate data collection over several passes so that a greater statistical power could be achieved. In order to build a robust tool feed system, which would prevent the tool from inadvertently crashing into the
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Fig. 2. 3D CAD model of the depth of cut control mechanism. The power screw drives a shallow wedge to deflect a stiff cantilevered beam. This arrangement results in precise feed rate control with very little spring back as the saw blade tooth exits the kerf in the bone. The tool holder, which is attached directly to the dynamometer, contains a single saw blade tooth.
workpiece, a purely mechanical system was developed. This was accomplished by using a power screw to drive a self locking wedge to deflect a cantilever beam. At the end of the beam, the force gage and cutting tool were attached as shown conceptually in Fig. 1(A) and as a 3D solid model in Fig. 2. When a screw is used to drive a wedge, which has a very low wedge angle, movement perpendicular to the wedge is precise and there is virtually no backlash. By using a wedge to bend a sufficiently stiff beam, there is essentially no spring-back during or after the cutting event. A combination of the power screw pitch and sliding wedge angle provided micrometer level resolution in tool advancement as verified in calibration experiments. Ultimately, the system provided rigid mechanical feed of the cutting tool between successive workpiece passes so that force data could be collected and averaged for several cuts. The complete tool feed system was mounted alongside the rotor mechanism on top of a granite surface plate, Fig. 3. A full description of the high speed linear sawing fixture and design methodology was previously published [14]. 2.3. Cutting tool (saw blade tooth) Cutting tools were fashioned by isolating single teeth from surgical saw blades (KM-458, Brasseler USA, Savannah, GA). The blade was constructed from 0.64 mm thick stainless steel and incorporated an alternating tooth set pattern (Left–Right–Left) with a pitch of 18 teeth per inch. Rake and clearance angles were approximately −10 and 60◦ , respectively, with measured edge radii ranging
Fig. 3. High speed linear cutting machine mounted to a granite surface plate. The drive motor, speed control, and related gearing lie below the surface plate. The counterweight is adjustable to account for changes in mass of the workpiece during dynamic calibration of the dynamometer.
Fig. 4. Workpiece holder attached to the dynamometer. The arrow is pointing at a single saw blade tooth, which is appropriately mounted in the tool holder to preserve the proper rake angle.
between 10 m and 17 m. Initial cutting edge radii were measured by mounting a single tooth in epoxy and then sanding and polishing the surface to various depths to determine changes in cutting edge radius along the face of the tooth by optical microscope. In addition, a three dimensional view of edge radii was evaluated by scanning electron microscopy (SEM). The saw blade teeth were held in a custom tool holder attached to the force gage, where the −10◦ rake face angle on the cutting face was maintained, Fig. 4. Initial cutting tests were performed to determine the effect of multiple cutting passes on edge wear. For the number of cuts performed with each saw blade tooth, which was typically less than 100 passes over a 50 mm workpiece, there was no evidence of edge wear upon periodic inspection with an optical microscope and SEM. 2.4. Force measurement From prior research, the magnitude of cutting forces in bone sawing was expected to be less than 100 N for single tooth cutting conditions. In addition, the cutting event was expected to last <50 ms for a single pass over a 50 mm length. Under these conditions, a force gage capable of quickly capturing dynamic cutting events was required. A 3-axis piezoelectric dynamometer (9047C dynamometer, Kistler Instrument Corp., Amherst, NY) was used to capture forces during cutting. The dynamometer had a dynamic range of 30 kN in the thrust force direction, Fz , and 15 kN in the cutting force direction, Fx , with a measurement sensitivity of 8.1 pC/N in Fx , and 3.7 pC/N in Fz . Data from the dynamometer was transmitted to a charge amplifier (ICAM 5073, Kistler Instrument Corp., Amherst, NY), which had a sensitivity of 1 ± 0.5% pC/mV. Collection of sufficient data to calculate representative average cutting forces required a high speed data collection system. As the maximum speed of 7000 mm/s was approached, the time to acquire data over the 50 mm cut was approximately 7 ms. High speed data acquisition was achieved with a multifunction data acquisition board (PCI 6132, National Instruments, Austin, TX), which had a simultaneous sampling rate of 2.5 × 106 samples per second. A LabVIEW (National Instruments, Austin, TX) virtual instrument was written to stream the sampled data to a series of TDMS binary files. These were then converted to LVMS files through a separate virtual instrument, at which point they were imported into MATLAB (MathWorks, Natick, MA) for data smoothing (4th order, low pass, Butterworth filter) and determination of the average cutting and thrust forces.
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HBSS, while being stored in a refrigerator during the 24 h adhesive curing time. Once cured, the sample assemblies were secured in a mill and the bone surfaces were machined to normalize thickness and to expose subperiosteal cortical bone. To prevent surface degradation due to heat from machining and subsequent drying, the bone samples were fully submerged in HBSS during the milling operation. 2.6. Cutting procedure
Fig. 5. Segments of bovine cortical bone after removal from the mid-diaphysis of the femur. The numbers and line markings signify the sample identification used in the Design of Experiments.
2.5. Bone sample preparation Prior to conducting single tooth cutting experiments, suitable bone samples were selected, rough sawn, secured to a mounting plate, and finish machined. During the steps involved in sample preparation, special care was taken to keep the bone moist through either submersion or misting with Hanks’ balanced salt solution (HBSS). For consistency, two fresh femurs from a 20 to 30 month adult bovine were procured from a local abattoir. Two 50 mm sections were removed by band saw from the mid-diaphysis of each femur. Once removed, the cortex of each section was sawn radially into six separate samples, Fig. 5. Through visual inspection, the bone samples were representative of homogeneous cortical bone with no apparent defects. Sample orientation was chosen such that cutting experiments would produce a slot parallel to the primary osteon direction, which falls along the axis of a long bone. After processing of the bulk femur, rectangular bone samples were attached to metal backing plates. After flattening the interior face (marrow side) of the bone by hand with a block plane, the surface was slightly roughened with a rasp file. Adhesive (Fast Cure 5200, 3M, St. Paul, MN) was applied to the roughened surface to secure the bone to an aluminum backing plate, Fig. 6. The completed samples were covered with towels, which were soaked with
For each experiment, a wedge angle and appropriate power screw rocker arm were selected for the desired depth of cut (feed) per cutting pass. Upon clear formation of a kerf along the full 50 mm of the bone surface, cutting and thrust forces were recorded with the dynamometer for a minimum of 10 passes. In post processing of force measurements for each cut, the data was clipped to eliminate the erroneous response of the dynamometer to the impact event as the tooth first entered the bone sample and again to the disrupted surface morphology near the exit of the kerf. The data eliminated from each cutting pass represented approximately the first 5 mm and the last 2 mm of the 50 mm cutting distance. The remaining force data was considered representative of steady state cutting conditions. 2.7. Screening studies Screening experiments were initially conducted to determine the appropriate factor levels. From the perspective of designing an experiment, cutting velocity and depth of cut represent the experimental factors, while cutting and thrust force represent the response variables. In order to minimize the number of experimental trials, while capturing both the effect of the individual factors and the effect of factor interactions on the response variables, a full factorial Design of Experiments (DOE) approach and related statistical methods was pursued [15]. A screening study on depth of cut was performed while holding velocity constant at 3980 mm/s. A range in depth of cut from a low of 2.5 m to a high of 10.2 m was investigated in increments of 2.5 m. A second screening study was performed on velocity while holding depth of cut constant at 7.6 m. Levels for cutting velocity were chosen to represent the clinical application of a sagittal saw, which operates between 12,000 and 20,000 cycles per minute. Actual tip speed of a saw blade tooth depends on both oscillation rate and blade length, where a typical sagittal saw blade varies in length from between 60 and 100 mm. Considering the variation in blade length and oscillation rate, the screening study investigated cutting velocity from a low of 1764 mm/s to a high of 7050 mm/s in increments of 880 mm/s. 2.8. Full factorial experimental design
Fig. 6. Cortical bone sample mounted to an aluminum backing plate. The sample shows five cuts, constituting an entire experimental replicate.
Upon completion of the screening studies, a two factor, two level, full factorial experiment was designed with four unique treatments. The lower and upper levels for depth of cut were set at 2.5 m and 10 m, respectively. For velocity, the lower level was set at 2645 mm/s and the upper level at 6172 mm/s, representing the tooth tip speeds for common reciprocating and oscillating surgical saws and falling within the range of the screening study. Per the experimental design, two sets of treatment orders were then defined and each was applied to a single bone sample, Table 1. There were sufficient bone samples to replicate the full experiment five times, adding to the statistical power of the regression analysis. While implementing the experiment, randomization was employed wherever possible to negate procedural bias and physical differences in the mechanical properties between bone samples. From the original twelve bone samples, seven were suitable for use
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Table 1 Four trials for each treatment of the two factor, two level, Design of Experiments. Two treatment orders were specified to aid in randomization. Treatment order 1
Treatment order 2
Velocity (mm/s)
Depth of cut (m)
Velocity (mm/s)
Depth of cut (m)
6172 2645 2645 6172
2.5 2.5 10 10
6172 2645 2645 6172
10 10 2.5 2.5
after mounting to the aluminum backing plates and finish machining. From these seven, five bone samples, one for each experiment replication, were chosen at random. One of the two defined treatment orders was then assigned to each sample with a coin toss. The randomized treatment order and sample assignments are recorded in Table 2. Lastly, a new saw blade tooth was randomly chosen for each experimental run from a large supply of previously prepared teeth. Before collecting data, each new saw blade tooth was “broken in” by making a single cut within the assigned bone sample used in the particular replication. 2.9. Statistical methods Following the Design of Experiments methodology, the higher level of each factor was coded as +1 and the lower level of each factor was coded as −1. Given the predominantly linear relationship between the factors and the response variables, a two factor, two level, full factorial DOE was pursued. To avoid confounding, the full factorial design required four unique treatments (+1+1, −1+1, +1−1, −1−1). With this scheme, the significance of each factor and their interaction on the response variable can be uniquely determined. Dynamometer data for both cutting force and thrust force was captured and analyzed for 10 passes of the cutting tool over the workpiece for each replication. This resulted in a total of 200 individual cuts (10 cuts per replication, multiplied by 5 replications of each unique treatment, multiplied by 4 unique treatments) from which to perform the statistical analysis and to develop the cutting and thrust force predictive models. Following the DOE methodology, the main factor effects were determined by taking the difference between the average response of the high and low levels of a factor. The standard deviation of the total experiment, Se , was determined by taking the square root of the average of the variances for each unique treatment. Once the standard deviation for the experiment was known, the standard deviation of the main factor effects, Seff , for the two level exper-
iment were determined by Seff = Se 4/N, where N = 20, which is the total number of replications. Finally, the decision limits for statistical significance of the factor effects were determined by multiplying the t-test value (˛ = 0.05) by the standard deviation of the factor effects. Table 2 Randomization of treatment orders and sample assignment for each of the five replicates. Replicate number
Sample assignment (See Note)
Treatment order (see Table 1)
1 2 3 4 5
S2-4 S1-1 S1-6 S2-6 S2-1
2 1 1 2 1
S2-4 refers to bone sample 2, Section 4, as identified in Fig. 5.
Fig. 7. Results from the screening study on depth of cut. Cutting force (solid diamond markers) and thrust force (solid square markers) represent the average of 10 cutting passes at a constant velocity of 6172 mm/s. The error bars represent + and −1 standard deviation from the mean.
Following an analysis of variance, linear equations were developed to predict cutting and thrust forces, FC and FT , respectively, according to Eq. (1) [15]. FC,T = FAVE +
E(v) E(d) E(vd) v+ vd d+ 2 2 2
(1)
Regarding the effect of the factors on force, E(v) is the effect of velocity, E(d) is the effect of depth of cut, and E(vd) is the effect of the interaction. The first term in Eq. (1), FAVE , represents the average of the measured cutting or thrust force components for all DOE treatments. The regression coefficients for the equation are determined by taking one half of the factor effects. Finally, v and d are assigned the coded values between +1 and −1 to represent a linear mapping of the actual high and low factor levels, respectively. 3. Results Results from the screening study on depth of cut demonstrated a linear trend of lower thrust force at lower depths of cut, R2 = 0.80. Meanwhile, cutting force appeared less sensitive to changes in depth of cut, Fig. 7. For this study, depth of cut was varied from 1.3 m to 10.2 m while velocity was held constant at 6172 mm/s. The average cutting forces measured were in the range of 4 N/mm to 10 N/mm, while thrust forces were in the range of 8 N/mm to 16 N/mm. For all depths of cut, thrust force exceeded cutting force, with a relatively constant thrust to cutting force ratio of between 1.5 and 2.0. During the screening study on velocity, cutting and thrust force remained relatively constant, Fig. 8. For this screening study, the depth of cut was held constant at 7.6 m, while the velocity was increased from 2625 mm/s to 10,583 mm/s. Cutting force was relatively constant at 8 N/mm over the full range of cutting velocities. The response of thrust force to cutting velocity showed a slight increase from 10 N/mm to 13 N/mm at higher velocities. As a result of the screening studies, there was reasonable evidence of a linear response in both cutting and thrust force to changes in the factor levels, which supported the choice for a two level, two factor, experimental design. When considering the average forces for each treatment combination, forces in the thrust direction were greater than those in the cutting direction by a factor of approximately 1.5 and the ratio was slightly greater at the higher depth of cut level, Fig. 9.
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Fig. 8. Results from the screening study on velocity. Cutting force (solid diamond markers) and thrust force (solid square markers) represent the average of 10 cutting passes at a constant 7.6 m depth of cut. The error bars represent + and −1 standard deviation from the mean.
The main factor and interaction effects demonstrate that depth of cut has the greatest impact on thrust force while velocity has the greatest impact on cutting force, Fig. 10. The interaction of velocity with depth of cut has only a negligible effect. The decision limit for statistical significance (˛ = 0.05) related to the effect of velocity on cutting and thrust force was determined to be 0.80 N, while the decision limit for the effect of depth of cut on force was determined to be 0.71 N. For factor effect values below the decision limit, the results may be related to the factor or they may be simply due to variation inherent within the experiment. In strict application of the decision limits to the data, there is no statistically significant effect of velocity on either cutting or thrust force, Fig. 10. However, an increase in depth of cut did have a sta-
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Fig. 10. Absolute value of the individual factor effects and interaction effect (velocity with depth of cut) as determined from the Design of Experiments. Due to the inherent variation in the force measurements, the t-test limit for statistical significance (˛ = 0.05) is 0.80 N for cutting force and 0.71 N for thrust force. Based on these criteria, only the effect of depth of cut on thrust force is statistically significant.
tistically significant effect on thrust force. An increase in depth of cut from 2.5 m to 10.2 m resulted in a 30% increase in average thrust force, while the effect on cutting force fell below the threshold limit. The interaction of velocity and depth of cut fell below the decision limit for both cutting and thrust force. The application of Eq. (1), following the ANOVA to determine the regression coefficients, resulted in the following equations for predicting cutting and thrust force when sawing bovine cortical bone with a single saw blade tooth: FCUTTING = 4.54 + 0.356v + 0.289d − 0.178vd
(2)
FTHRUST = 7.07 + 0.200v + 0.912d − 0.178vd
(3)
It should be noted that the magnitude of each effect in Eqs. (2) and (3) is dependent on the units of force. Here, the units of force are Newton for a cutting edge width of 0.64 mm. The regression coefficients in Eqs. (2) and (3) are equal to one half of the factor effects. While it is traditional practice to show the absolute values of the factor effects, Fig. 10, the regression equations include the appropriate sign. Following the DOE and statistical analysis, three additional experiments were conducted for independent verification of the regression equations. Table 3 contains a summary of the predicted and measured forces, along with the experimental conditions and corresponding parameters for v, d, and vd used in Eqs. (2) and (3). The predicted forces were within approximately 10% of the measured forces in all but one case. This percent difference is within the standard deviation of results from the screening experiments. 4. Discussion
Fig. 9. Results from the two factor, two level, full factorial Design of Experiments, which consisted of 8 treatment combinations. Each bar represents the average thrust force or cutting force for various combinations of cutting velocity and depth of cut.
The average cutting force in Eq. (2), 4.54 N, and the average thrust force in Eq. (3), 7.07 N, is an order of magnitude greater than the influence of either of the factor effects. In addition, the average thrust force in Eq. (3) is approximately 1.5 times larger than the
Table 3 Verification experiments and corresponding predictions from regression model, Eqs. (2) and (3). Factor levels
Parameters used in Eqs. (2) and (3)
Cutting force (N)
Thrust force (N)
Depth of cut, d (m)
Velocity, v (mm/s)
d
v
vd
Empirical
Eq. (2)
Empirical
Eq. (3)
5.1 7.6 7.6
6172 3509 4406
−0.33 0.33 0.33
1.00 −0.51 0.00
−0.33 −0.17 0.00
5.9 5.0 4.2
4.9 4.5 4.6
7.5 7.1 7.5
7.0 7.3 7.4
1644
T.P. James et al. / Medical Engineering & Physics 35 (2013) 1638–1644
average cutting force in Eq. (2). With these two conditions present, the regression equations predict that thrust force will always be greater than cutting force for all admissible factor levels, resulting in a thrust to cutting force ratio that is always greater than one. A thrust to cutting force ratio greater than one is expected under conditions where depth of cut is less than the cutting edge radius and therefore ploughing dominates chip creation [16,17]. From inspection of chips created during the sawing experiments, there was no evidence of continuous chips, but rather an accumulation of very fine particles. This too is an indication of ploughing and abrasive chip creation, rather than the existence of a well defined shear plane that could facilitate continuous chips. A predominantly linear relationship was found between tool forces and depth of cut, which was also reflected in prior research on machining of bone [8,11,12]. The interaction between factors, depth of cut and cutting velocity, had a similar impact on the cutting and thrust force response. The negative signs in Eqs. (2) and (3) i.e. (−0.178), can be interpreted to mean that higher values of both depth of cut and cutting velocity tend to reduce the average cutting and thrust force. Force measurements from the linear sawing fixture were in relatively close agreement with previous results reported by Plaskos et al. [12]. Cutting force measurements correlated most closely, falling within the range of 4 N/mm to 10 N/mm as compared to the 6 N/mm to 8 N/mm for Plaskos. The same could not be said for thrust forces, however, as they were 8 N/mm to 16 N/mm in this study as compared to 3 N/mm to 12 N/mm for Plaskos. The Plaskos results were produced with a neutral rake tool of unspecified edge radius and their tool width was greater than workpiece thickness. The higher thrust forces determined here may be attributed to the tool having a larger edge radius than that used by Plaskos. The regression equations developed as part of this research represent the forces encountered when cutting parallel to the primary osteon direction. It is well known that the bulk material properties of cortical bone, as well as the forces measured during machining, are dependent on the cutting direction, parallel, transverse or radial with respect to the osteons [5,12]. The length of cut was not sufficient to develop steady state cutting conditions for the radial or transverse directions, where thickness of the bovine cortical bone is typically less than 10 mm. When developing algorithms for force feedback, the forces predicted by the regression equations must be extrapolated from single teeth to a multi-toothed saw blade. When doing so, tooth set must be considered. Teeth are set out of plane on a saw blade to reduce the friction between the backing of the blade and the sides of the kerf and to provide clearance for chips. The oblique cutting face of a set tooth may create out-of-plane cutting and thrust force components. 5. Conclusion Mechanistic models are required to predict bone sawing forces when developing haptic feedback systems for surgical simulators. To fully investigate the effect of cutting velocity and depth of cut on bone sawing forces a full factorial Design of Experiments approach was utilized, resulting in a few observations:
Abiding strictly by the decision limits for statistical significance, a t-test (˛ = 0.05) indicated that depth of cut had a significant impact on thrust force, but not on cutting force; Cutting velocities between 2645 mm/s and 6172 mm/s, representative of sagittal bone saws, did not have a statistically significant effect on either cutting or thrust force; The two factor interaction of cutting velocity and depth of cut did not have a significant effect on either cutting or thrust force; When investigating depths of cut from 2.5 m to 10.2 m, the thrust force increased 30%, but cutting force was relatively unaffected; The regression equations revealed that thrust force will always be greater than cutting force for the range of velocities and depths of cut investigated. Author declarations Competing interests: None declared. Funding: No external funding was received. Internal funding was received from the School of Engineering at Tufts University, Medford, MA, USA. Ethical approval: Not required. Acknowledgement This research was supported financially by Tufts University, School of Engineering, Medford, Massachusetts, USA. References [1] Vankipuram M, Kahol K, McLaren A, Panchanathan S. A virtual reality simulator for orthopedic basic skills: A design and validation. J Biomed Inform 2010;43(5):661–8. [2] Tsai MD, Hsieh MS, Tsai CH. Bone drilling haptic interaction for orthopaedic simulator. Comput Biol Med 2007;37:1709–18. [3] Hsieh MS, Tsai MD, Yeh YD. An amputation simulator with bone sawing haptic interaction. Biomed Eng Appl Basis Commun 2006;18(5):229–36. [4] Ko TJ, Kim HS. Mechanistic cutting force model in band sawing. Int J Mach Tools 1999;39(8):1185–97. [5] Reilly DT, Burstein AH. The mechanical properties of cortical bone. J Bone Joint Surg 1974;56:1001–22. [6] Krause WR. Orthogonal bone cutting: Saw design and operating characteristics. J Biomech Eng 1987;109:263–71. [7] Krause WR. Mechanical effects of orthogonal bone cutting. Clemson, South Carolina, USA.: Clemson University; 1996. PhD Thesis. [8] Jacobs CH, Pope MH, Berry JT, Hoaglund F. A study of the bone machining process–orthogonal cutting. J Biomech 1976;7:131–6. [9] Jacobs CH, Pope MH, Berry JT, Hoaglund F. A study of the bone machining process–drilling. J Biomech 1976;9(5):343–9. [10] Wiggins KL, Malkin S. Drilling of bone. J Biomech 1976;9(9):553–9. [11] Wiggins KL, Malkin S. Orthogonal machining of bone. J Biomech Eng 1978;100:122–30. [12] Plaskos C, Hodgson AJ, Cinquin P. Modeling and optimization of bone-cutting forces in orthopaedic surgery. Lect Notes Comput Sci 2003;2878:254–61. [13] Yeager C, Nazari A, Arola D. Machining of cortical bone: surface texture, surface integrity and cutting forces. Mach Sci Technol 2008;12:100–18. [14] Pearlman JJ. Cutting Velocity Effects in Bone Sawing. Medford. MA, USA: Tufts University; 2011 [MS Thesis]. [15] Antony J. Design of Experiments for Engineers and Scientists. New York: Butterworth Heinemann; 2003. [16] Albrecht P. New development in the theory of metal-cutting process, part I. The ploughing process in metal cutting. J Eng Ind – Trans ASME 1960;82: 348–57. [17] Basuray PK, Misra BK, Lal GK. Transition from ploughing to cutting during machining with blunt tools. Wear 1977;43(3):341–9.