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A miniaturised force-torque sensor with six degrees of freedom for dental measurements
This content has been downloaded from IOPscience. Please scroll down to see the full text. 1992 Clin. Phys. Physiol. Meas. 13 241 (http://iopscience.iop.org/0143-0815/13/3/003) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 155.69.4.4 This content was downloaded on 23/08/2015 at 03:34
Please note that terms and conditions apply.
Clin. Phys. Physiol. Meas., 1992, Vol. 13, No. 3, 241-248. Printed in the UK
A miniaturised force-torque sensor with six degrees of freedom for dental measurements J Plane*,
H Modlea, K Liideckeg and M Egerg
'IDental Clinic of t h e University of Gdttingen, Department ofProsrherics, Roben- KochSu.40,3400 GBttingen, Germany i Institute for Medical Physics and Biophysics, Univenity of Gattinpen, Gosslersu. lOf, 3400 Gottingen, Germany Weender Landstrase, 3400 Getringen, Germany
5 Sanorius GmbH,
Received 30 September 1991, in final form 29 January 1992 Abstract. Forcetorque sensor Systems capable of simultaneously measuring all six spatial degrees of freedom on solid bodies are not very widespread in &e medical field, panicularly because of the hitherto unacceptably large external dimensions. A sensor based on the strain gauge technique has been developedwith a diameter of only 32.5 mm and height of 29 mm. T h e miniaturised supporting framework for the strain gauges has been cast in a single piece from a wax-plasric model by the one-way mould principle. A sensitivity of 10 mN with an upper limit of 50 N is attained.
1. Introduction
Force-torque sensor systems capable of simultaneously measuring all six spatial degrees of freedom on solid bodies are not very widespread. Systems like those employed for kinematic closed-loop control of automated handling devices typically have dimensions in the range of several dm3 (Dillmann et ai 1982, 1986). For this reason, the use of one or more such sensor systems is seldom possible in the medical field (Bourauel et a1 1990). Besides the hitherto unacceptably large extemal dimensions, further shortcomings have resulted in a virtual lack of force-torque sensors in medical research. Among other factors, the absence of vector components in the case of sensors with less than six degrees of freedom frequently precludes any useikl application. Moreover, those force components which are located outside the limited measuring range of such sensors with less than six degrees of freedom must leave the measurable components unaffected. For this purpose, however, the degrees of freedom for a sensor of this kind must be completely uncoupled, that is, with a factor of 100% in the ideal case. This value cannot be attained in mechanical structures; hence, a corresponding measuring error must be tolerated from the very start. A major problem associated with the manufacture of very small sensors is the construction of the framework which supports the strain gauge; the elastic mechanical deformation of this structure as a function of the force is a prerequisite for the generation of a signal for measurement. The flexural beam and other components of the framework can be machined by milling, grinding, etc. with a level of precision which satisfies all requirements. However, this technology presents problems which result mainly from the joining of the individual components. Non-linear deformations occur at the joints under load 0143-08151921030241 + 08 802.50 0 1992 lnsrirute ofPhysical Sciences in Medicine
24 1
242
J Planen et al
and falsify the results of measurements. Plastic deformations must be expected if very small sensors are subjected to a torque exceeding 10 N cm, for instance. Besides these mechanical problems, additional difficulties are associated with the generation of sufficiently strong electrical signals. Since the geomemcal expansion of the applicable strain gauge is small, the supply voltage must be kept low because of the resulting low maximal permissible power dissipation. The amplitude of the useful electrical signals is correspondingly small. With due consideration of the aspects just mentioned, various sensors have been developed at the Dental Clinic of the University of Gottingen on the basis of the theoretical wrist model for automated manufacturing devices at the Institute of Robotics in Karlsruhe (Dillmann et a2 1982). For these sensors, the supporting framework for the strain gauges has been cast in a single piece from a wax-plastic model by the one-way mould principle. A nickelbased alloy (Ni,oCrl,Mo,MnAl) has been employed for this purpose; the technical data are as follows: Modulus of elasticity (Young’s modulus) Yield point (0.2 %) Elongation at fracture Average coefficient of thermal expansion Vicker’s hardness (HV 5 / HV 10) Density
220 000 N mm-* 790 N mm-2 3 Yo 14.3 wm K-l 14.3 8.2 g ~ m - ~
The mechanical parameters of the first prototype are as follows: Dimensions of sensor framework Diameter 53 mm Height 26 mm Mass 75 g Mechanical load-bearing capability Rotational force Translational forces
Dimensions of housing 71 mm 28 mm Volume 100 cm3 25 N cm 80 N
Figure 1. V ~ e winside the pr~totypesensor wirh housing removed The M8 nut and the four frames can be clearly seen.
Force-torque sensor for dental measurements
243
2. Mechanics The shaping of the sensor allows resolution of the force vector and the resultant torque into the respective three components. T h e body of the sensor consists of a base plate and a threaded sleeve, which are joined by four frames (figure 1). T h e frames are small rectangles constructed of thin metal rods; one comer of each frame is fastened to the base plate, and the diagonally opposite, comer is fastened to the threaded sleeve. The threaded sleeve is a cylindrical M8 nut in whose thread an appropriate component can be inserted for introducing the force. The frames are fastened to the lower edge of the cylinder in such a way that the frames are in tangential contact with the cylinders; the planes in which the frames lie are oriented in parallel with the axis of the cylinder or thread and are mutually perpendicular. The frames, and thus the thread axis, are perpendicular to the base plate. An M8 threaded pin projects from the base plate in the direction of the thread axis; by means of this pin, the entire device can be fastened onto an appropriate support. If a rectangular coordinate system is oriented such that the z axis coincides with the thread axis, the plane spanned by the x and y axes is parallel with the base plate. By rotation of the system ofaxes about the z axis, two frames each can be brought into the direction of the x a n d y axes. Such a position of the coordinate system is useful for illustration, but is not necessary. In practice, the coordinate system is specified by the calibration measurements and is independent of the mechanical structure of the sensor. Each frame is cemented at two adjacent spring beams with two strain gauges each; these are situated opposite one another on the inside and outside of the rod. T h e strain gauges are connected as conventional half-bridge circuits (Schanz 1986); hence, two signals are received from each frame. Thus, a total of eight signals are available for determining an introduced forcetorque vector. T h e electrical load capacity ofthe eight strain gauges employed is extremely low. As a result of sensor miniaturisation, damage to one strain gauge due to electrical overload can be repaired only by tedious disassembly, re-cementing with all strain gauges, rewiring, etc. For this reason, a n expressly designed and constructed electronic system has been employed, instead of a commercially available amplifier module. T h e selected amplifier concept requires no switching of amplification or matching circuit, has the necessary eight channels, and is individually adapted to the sensor. The necessary amplification of the electrical signals has been determined experimentally and specified with a factor of 1000. Satisfactory long-term stability has been achieved with the use of highly accurate voltage references, special wire-wound resistors, chopper amplifier, etc. Because of the four-conductor circuit, the length of the cable between the sensor and amplifier exerts no effect up to a length of about 10 m. T h e measuring system constructed for practical application of the sensor comprises a load cell, which includes the sensor body with the strain gauge, the current supply and amplifier unit, an analogue-digital converter circuit board in the personal computer, and the computer itself. 3. Mathematics and calibration
In order to determine the force or torque values, the load cell must be calibrated for establishing a mathematical relationship between an acting force and the output signals. Because of an overdetermination of the flexural beams associated with the design, more than the necessary number of strain gauges, and thus eight output
J Planen et al
244
channels, are available. The signals, Si,with 1 2 i 5 8, are present at the outputs if a force and the associated torque act on the load cell. The relationship between a signal, S , and the six components of force and torque, with 1 5 j 5 6, is given by
5,
(1) Si= C,,F, + CizFz+ C,F, + C,F4 + C,F5 + C,F, Thus, a system of eight equations (1) is available for determining the six unknowns, F , if the 48 coefficients, Cp are known. These coefficients are determined by J . calibration measurements. Before the measurements, a consensus must be reached on the orientation of the coordinate system for measurement and on the indexing rules which are to be observed for all subsequent measurements. A left-handed Cartesian coordinate system has been selected with the origin on the surface of the load cell at the centre of the M 8 thread. The x axis is thereby parallel with the longitudinal axis of the support plate of the load cell. Because of the selected indexing, F,, F2, and F3 are the x , y , and z components of the force; correspondingly, F4, F,, and F6 are the x, y. and z components of the torque. For determining the C?, at least six measurements with different loads on the load cell are required. The loads must be selected in such a way that equations (1) are linearly independent (Dillmann et a2 1982) and can be generated simply by the weight of a standard mass. A threaded rod with notches at intervals of 1 cm is inserted into the load cell for use as a lever. On a frame specially constructed for the purpose, the support plate with the load cell is then tilted in such a way that the direction of the
Figure 2. Calihrarirm of ~ l i cproc~xypcw i w r .
coordinate system in which the load is to act faces downward. After the zero measurement for determining the signals in the absence of load, the weight is suspended on a thin filament in one of the notches (figure 2). The six measurements for a calibration are performed with the same weight, while the direction of loading and torque arm of the lever are varied. Six times eight equations of type (1) are then available as a result of one calibration, and the following matrix is thus obtained:
S = C.F
(2)
S is the signal matrix, in which the eight output signals from each of the six measurements are comprised. F is the force matrix, whose elements represent the force or torque components of the calibration loads. By calculation of these components from the applied weights and lever arms employed, the units of force and torque are also specified for all later measurements. C denotes the coupling matrix
Force-torque sensor for dental measurements
245
and contains the coefficients, Cg,being sought. C is obtained from
C = sF-1
(3)
The decoupling matrix, C', must be calculated from C, since it is required for transforming the output signals to forces and torques by means of the following: F = C'.S
(4)
Since C is not a square mauix, it cannot be directly inverted; instead, C' is computed from C as a pseudoinverse as follows:
c'= (CT.C)-l.CT
(5)
C is saved in the measuring program for further use as a constant. In fact, the decoupling matrix, C', ultimately employed, is obtained by averaging the results from several calibrations. Calibrations have been performed with a total of seven different masses from 20 to 2OOOg. On the one hand, these series of measurements confirm the expected good linearity of the signals from the strain gauges as a function of the load. On the other hand, they offer the possibility of obtaining a decoupling matrix in which the effects of statistical fluctuations in the measured values are minimised. For this purpose, the output signals are plotted as a function of the weight loads for each channel, and the regression line is determined for the measured points. For calculating the coupling mamx (3), the signal matrix, S, is then constructed with values derived from the regression lines for an arbitrary weight. Subsequently, (5) yields the desired decoupling mamx. The mechanical stiffness of the support system, the supply voltage for the strain gauges, and the background noise of the amplifiers are three among several parameters which are decisive for the resolution of the measuring system. In the case of the sensor just described and the associated amplifier, the sensitivity of the overall system was 0.1 N as a maximum, which corresponds to about 2.4 mV, for all degrees of freedom. In figure 3, an example is presented for application of the measuring system just described. The forces and torques which arise during the use of various dental instruments for removing fixed bridgework (tooth crowns and dental bridges) were thereby compared. The experimental results are presented in figure 4.
Fi3. Experimental set-up for measuring the force acting on a tooth during loosening of a tooth crown with different instruments. In the foreground lie the pliers used.
J Planert er a1
246
Figure 4. Time development of absolute force and torque acting on a tooth during loosening of a tooth crown either with a lever or with pliers. The curves are the mean of ten measurements, the measuring times of about 10 to 25 s are normalised to ten.All the measurements were done by the same person to minimise fluctuations. (-) Mean of measuremenrs. (- - - - -) Region of standard dwiatian.
Figure 5 . A palatinal arc acting un a twtli ufsn uppcr li(w modci. 'l'hc liiw and all its teeth are fixed except the tooth in the middle above the sensor which is mounted to the MX nut of the sensor without touching the jaw. The arc was bowed so that no forces act on the teeth.
Components of force
Components of torque
Figure 6. Components of force and torque measured with three types of palatinal arcs with the experimental set-up of figure 5. The mean of ten measurements for each type are shown. The arcs have been bowed so tbat the forces are zero. Dotted line, standard deviation.
Force-torque sensorjor dental measurements
Figure 7 . A senior o f IIIC new turfhcr ~
247
I ~ I I . I L Lgeneration. ~ W ~
A measurement series performed in the range of considerably weaker forces is illustrated in figure 5 . The object is to determine whether the procedure hitherto applied for determining the expected forces by means of a simple geometrical-optical estimate is really useful with the use of auxiliary components (palatinal arcs) for operative dentistry. If the values of the forces actually measured differ appreciably, the former procedure is questionable. The measured values are compiled in figure 6. (Hitherto, the values were theoretically assumed to be zero!) For the in vitro simulation of dental problems, the spacing of several objects under investigation, for instance, teeth, must be kept within realistic limits, in order IO avoid parameters which are difficult to take into account. Consequently, a further, considerably miniaturised sensor has been developed from the experience gained during the construction of the initial prototype. This effort constitutes a further step toward intra-oral in vivo measurements with all degrees of keedom (figure 7). A limited series of sensors with the following properties has been constructed from the design principle of the prototype: Extemal dimensions of Diameter Height Volume
housing 32.5" 29 mm 23 cm3
Mechanical load-bearing capability Translational force 50 N Rotational force 15 Ncm
In order further to improve the resolution and minimise the hazard of electrical overload (power dissipation limit), special 1000 0 strain gauges have been employed with these sensors. With a grid load of 0.7Wcm-*, the accuracy is very high for dynamic measurements, and average for static measurements. 4. Measured data
With a gain factor of 1000, a sensitivity of about 5 mV per 10 mN is attained. In correspondence with the maximal load-bearing capability of the sensor and an
248
J Planen et al
average level of elaboration for the electronics, the resolution of 4096 steps for a twelve-bit analogue-digital convener can be fully utilised with about 2.5 mN per step. T h e characteristic features worthy of note include the reaction of the sensor to loads in the maximal force range. For example, with a torque of 2 N per 5 cm and a duty cycle ratio of 20 min / 20 min for loading, the maximal hysteresis is better than 8 pV o n all channels; at 1 min / 25 s, it is better than 1.5 pV. T h e temperature variation of the system is about 3 pVper"C.
5. Prospects A miniaturised sensor can be employed, for instance, for in vitro examination of socalled prosthetic auxiliary components, that is, connecting elements such as articulated joints and attachments, comparative measurements on the evolution of forces on dental prostheses and instruments, spring elements or arcs for operative dentistry, or intraoral in vivo measurements of periodontal behavior (intrinsic physiological mobility) of teeth under the action of forces. Miniaturisation by a factor of about 2.5 (by volume) is the object of current endeavours and appears feasible without excessively elaborate measures. References Bourauel Ch, Drescher D and Thier M 1990 Krafr-Mamenren-Aufnehmer fiir die Kieferorthopzdie Feinwerktech. Merrtech. 98 419-421 Dillmann R and Faller B 1982 Kraft-Momenten-Sensorsystem fiir IndustrieroboterElrkmnik 8 89-95 Dillmann R, Hugel Th and Meier W 1986 Ein sensarinregrierter Greifer als modulares Teilsystem fur Monrngeroborer Robotmystma 2 247-252 Schanz G W 1986 Sensoren (Heidelberg: H"rdg Vedag).
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My IOPscience
A miniaturised force-torque sensor with six degrees of freedom for dental measurements
This content has been downloaded from IOPscience. Please scroll down to see the full text. 1992 Clin. Phys. Physiol. Meas. 13 241 (http://iopscience.iop.org/0143-0815/13/3/003) View the table of contents for this issue, or go to the journal homepage for more
Download details: IP Address: 155.69.4.4 This content was downloaded on 23/08/2015 at 03:34
Please note that terms and conditions apply.
Clin. Phys. Physiol. Meas., 1992, Vol. 13, No. 3, 241-248. Printed in the UK
A miniaturised force-torque sensor with six degrees of freedom for dental measurements J Plane*,
H Modlea, K Liideckeg and M Egerg
'IDental Clinic of t h e University of Gdttingen, Department ofProsrherics, Roben- KochSu.40,3400 GBttingen, Germany i Institute for Medical Physics and Biophysics, Univenity of Gattinpen, Gosslersu. lOf, 3400 Gottingen, Germany Weender Landstrase, 3400 Getringen, Germany
5 Sanorius GmbH,
Received 30 September 1991, in final form 29 January 1992 Abstract. Forcetorque sensor Systems capable of simultaneously measuring all six spatial degrees of freedom on solid bodies are not very widespread in &e medical field, panicularly because of the hitherto unacceptably large external dimensions. A sensor based on the strain gauge technique has been developedwith a diameter of only 32.5 mm and height of 29 mm. T h e miniaturised supporting framework for the strain gauges has been cast in a single piece from a wax-plasric model by the one-way mould principle. A sensitivity of 10 mN with an upper limit of 50 N is attained.
1. Introduction
Force-torque sensor systems capable of simultaneously measuring all six spatial degrees of freedom on solid bodies are not very widespread. Systems like those employed for kinematic closed-loop control of automated handling devices typically have dimensions in the range of several dm3 (Dillmann et ai 1982, 1986). For this reason, the use of one or more such sensor systems is seldom possible in the medical field (Bourauel et a1 1990). Besides the hitherto unacceptably large extemal dimensions, further shortcomings have resulted in a virtual lack of force-torque sensors in medical research. Among other factors, the absence of vector components in the case of sensors with less than six degrees of freedom frequently precludes any useikl application. Moreover, those force components which are located outside the limited measuring range of such sensors with less than six degrees of freedom must leave the measurable components unaffected. For this purpose, however, the degrees of freedom for a sensor of this kind must be completely uncoupled, that is, with a factor of 100% in the ideal case. This value cannot be attained in mechanical structures; hence, a corresponding measuring error must be tolerated from the very start. A major problem associated with the manufacture of very small sensors is the construction of the framework which supports the strain gauge; the elastic mechanical deformation of this structure as a function of the force is a prerequisite for the generation of a signal for measurement. The flexural beam and other components of the framework can be machined by milling, grinding, etc. with a level of precision which satisfies all requirements. However, this technology presents problems which result mainly from the joining of the individual components. Non-linear deformations occur at the joints under load 0143-08151921030241 + 08 802.50 0 1992 lnsrirute ofPhysical Sciences in Medicine
24 1
242
J Planen et al
and falsify the results of measurements. Plastic deformations must be expected if very small sensors are subjected to a torque exceeding 10 N cm, for instance. Besides these mechanical problems, additional difficulties are associated with the generation of sufficiently strong electrical signals. Since the geomemcal expansion of the applicable strain gauge is small, the supply voltage must be kept low because of the resulting low maximal permissible power dissipation. The amplitude of the useful electrical signals is correspondingly small. With due consideration of the aspects just mentioned, various sensors have been developed at the Dental Clinic of the University of Gottingen on the basis of the theoretical wrist model for automated manufacturing devices at the Institute of Robotics in Karlsruhe (Dillmann et a2 1982). For these sensors, the supporting framework for the strain gauges has been cast in a single piece from a wax-plastic model by the one-way mould principle. A nickelbased alloy (Ni,oCrl,Mo,MnAl) has been employed for this purpose; the technical data are as follows: Modulus of elasticity (Young’s modulus) Yield point (0.2 %) Elongation at fracture Average coefficient of thermal expansion Vicker’s hardness (HV 5 / HV 10) Density
220 000 N mm-* 790 N mm-2 3 Yo 14.3 wm K-l 14.3 8.2 g ~ m - ~
The mechanical parameters of the first prototype are as follows: Dimensions of sensor framework Diameter 53 mm Height 26 mm Mass 75 g Mechanical load-bearing capability Rotational force Translational forces
Dimensions of housing 71 mm 28 mm Volume 100 cm3 25 N cm 80 N
Figure 1. V ~ e winside the pr~totypesensor wirh housing removed The M8 nut and the four frames can be clearly seen.
Force-torque sensor for dental measurements
243
2. Mechanics The shaping of the sensor allows resolution of the force vector and the resultant torque into the respective three components. T h e body of the sensor consists of a base plate and a threaded sleeve, which are joined by four frames (figure 1). T h e frames are small rectangles constructed of thin metal rods; one comer of each frame is fastened to the base plate, and the diagonally opposite, comer is fastened to the threaded sleeve. The threaded sleeve is a cylindrical M8 nut in whose thread an appropriate component can be inserted for introducing the force. The frames are fastened to the lower edge of the cylinder in such a way that the frames are in tangential contact with the cylinders; the planes in which the frames lie are oriented in parallel with the axis of the cylinder or thread and are mutually perpendicular. The frames, and thus the thread axis, are perpendicular to the base plate. An M8 threaded pin projects from the base plate in the direction of the thread axis; by means of this pin, the entire device can be fastened onto an appropriate support. If a rectangular coordinate system is oriented such that the z axis coincides with the thread axis, the plane spanned by the x and y axes is parallel with the base plate. By rotation of the system ofaxes about the z axis, two frames each can be brought into the direction of the x a n d y axes. Such a position of the coordinate system is useful for illustration, but is not necessary. In practice, the coordinate system is specified by the calibration measurements and is independent of the mechanical structure of the sensor. Each frame is cemented at two adjacent spring beams with two strain gauges each; these are situated opposite one another on the inside and outside of the rod. T h e strain gauges are connected as conventional half-bridge circuits (Schanz 1986); hence, two signals are received from each frame. Thus, a total of eight signals are available for determining an introduced forcetorque vector. T h e electrical load capacity ofthe eight strain gauges employed is extremely low. As a result of sensor miniaturisation, damage to one strain gauge due to electrical overload can be repaired only by tedious disassembly, re-cementing with all strain gauges, rewiring, etc. For this reason, a n expressly designed and constructed electronic system has been employed, instead of a commercially available amplifier module. T h e selected amplifier concept requires no switching of amplification or matching circuit, has the necessary eight channels, and is individually adapted to the sensor. The necessary amplification of the electrical signals has been determined experimentally and specified with a factor of 1000. Satisfactory long-term stability has been achieved with the use of highly accurate voltage references, special wire-wound resistors, chopper amplifier, etc. Because of the four-conductor circuit, the length of the cable between the sensor and amplifier exerts no effect up to a length of about 10 m. T h e measuring system constructed for practical application of the sensor comprises a load cell, which includes the sensor body with the strain gauge, the current supply and amplifier unit, an analogue-digital converter circuit board in the personal computer, and the computer itself. 3. Mathematics and calibration
In order to determine the force or torque values, the load cell must be calibrated for establishing a mathematical relationship between an acting force and the output signals. Because of an overdetermination of the flexural beams associated with the design, more than the necessary number of strain gauges, and thus eight output
J Planen et al
244
channels, are available. The signals, Si,with 1 2 i 5 8, are present at the outputs if a force and the associated torque act on the load cell. The relationship between a signal, S , and the six components of force and torque, with 1 5 j 5 6, is given by
5,
(1) Si= C,,F, + CizFz+ C,F, + C,F4 + C,F5 + C,F, Thus, a system of eight equations (1) is available for determining the six unknowns, F , if the 48 coefficients, Cp are known. These coefficients are determined by J . calibration measurements. Before the measurements, a consensus must be reached on the orientation of the coordinate system for measurement and on the indexing rules which are to be observed for all subsequent measurements. A left-handed Cartesian coordinate system has been selected with the origin on the surface of the load cell at the centre of the M 8 thread. The x axis is thereby parallel with the longitudinal axis of the support plate of the load cell. Because of the selected indexing, F,, F2, and F3 are the x , y , and z components of the force; correspondingly, F4, F,, and F6 are the x, y. and z components of the torque. For determining the C?, at least six measurements with different loads on the load cell are required. The loads must be selected in such a way that equations (1) are linearly independent (Dillmann et a2 1982) and can be generated simply by the weight of a standard mass. A threaded rod with notches at intervals of 1 cm is inserted into the load cell for use as a lever. On a frame specially constructed for the purpose, the support plate with the load cell is then tilted in such a way that the direction of the
Figure 2. Calihrarirm of ~ l i cproc~xypcw i w r .
coordinate system in which the load is to act faces downward. After the zero measurement for determining the signals in the absence of load, the weight is suspended on a thin filament in one of the notches (figure 2). The six measurements for a calibration are performed with the same weight, while the direction of loading and torque arm of the lever are varied. Six times eight equations of type (1) are then available as a result of one calibration, and the following matrix is thus obtained:
S = C.F
(2)
S is the signal matrix, in which the eight output signals from each of the six measurements are comprised. F is the force matrix, whose elements represent the force or torque components of the calibration loads. By calculation of these components from the applied weights and lever arms employed, the units of force and torque are also specified for all later measurements. C denotes the coupling matrix
Force-torque sensor for dental measurements
245
and contains the coefficients, Cg,being sought. C is obtained from
C = sF-1
(3)
The decoupling matrix, C', must be calculated from C, since it is required for transforming the output signals to forces and torques by means of the following: F = C'.S
(4)
Since C is not a square mauix, it cannot be directly inverted; instead, C' is computed from C as a pseudoinverse as follows:
c'= (CT.C)-l.CT
(5)
C is saved in the measuring program for further use as a constant. In fact, the decoupling matrix, C', ultimately employed, is obtained by averaging the results from several calibrations. Calibrations have been performed with a total of seven different masses from 20 to 2OOOg. On the one hand, these series of measurements confirm the expected good linearity of the signals from the strain gauges as a function of the load. On the other hand, they offer the possibility of obtaining a decoupling matrix in which the effects of statistical fluctuations in the measured values are minimised. For this purpose, the output signals are plotted as a function of the weight loads for each channel, and the regression line is determined for the measured points. For calculating the coupling mamx (3), the signal matrix, S, is then constructed with values derived from the regression lines for an arbitrary weight. Subsequently, (5) yields the desired decoupling mamx. The mechanical stiffness of the support system, the supply voltage for the strain gauges, and the background noise of the amplifiers are three among several parameters which are decisive for the resolution of the measuring system. In the case of the sensor just described and the associated amplifier, the sensitivity of the overall system was 0.1 N as a maximum, which corresponds to about 2.4 mV, for all degrees of freedom. In figure 3, an example is presented for application of the measuring system just described. The forces and torques which arise during the use of various dental instruments for removing fixed bridgework (tooth crowns and dental bridges) were thereby compared. The experimental results are presented in figure 4.
Fi3. Experimental set-up for measuring the force acting on a tooth during loosening of a tooth crown with different instruments. In the foreground lie the pliers used.
J Planert er a1
246
Figure 4. Time development of absolute force and torque acting on a tooth during loosening of a tooth crown either with a lever or with pliers. The curves are the mean of ten measurements, the measuring times of about 10 to 25 s are normalised to ten.All the measurements were done by the same person to minimise fluctuations. (-) Mean of measuremenrs. (- - - - -) Region of standard dwiatian.
Figure 5 . A palatinal arc acting un a twtli ufsn uppcr li(w modci. 'l'hc liiw and all its teeth are fixed except the tooth in the middle above the sensor which is mounted to the MX nut of the sensor without touching the jaw. The arc was bowed so that no forces act on the teeth.
Components of force
Components of torque
Figure 6. Components of force and torque measured with three types of palatinal arcs with the experimental set-up of figure 5. The mean of ten measurements for each type are shown. The arcs have been bowed so tbat the forces are zero. Dotted line, standard deviation.
Force-torque sensorjor dental measurements
Figure 7 . A senior o f IIIC new turfhcr ~
247
I ~ I I . I L Lgeneration. ~ W ~
A measurement series performed in the range of considerably weaker forces is illustrated in figure 5 . The object is to determine whether the procedure hitherto applied for determining the expected forces by means of a simple geometrical-optical estimate is really useful with the use of auxiliary components (palatinal arcs) for operative dentistry. If the values of the forces actually measured differ appreciably, the former procedure is questionable. The measured values are compiled in figure 6. (Hitherto, the values were theoretically assumed to be zero!) For the in vitro simulation of dental problems, the spacing of several objects under investigation, for instance, teeth, must be kept within realistic limits, in order IO avoid parameters which are difficult to take into account. Consequently, a further, considerably miniaturised sensor has been developed from the experience gained during the construction of the initial prototype. This effort constitutes a further step toward intra-oral in vivo measurements with all degrees of keedom (figure 7). A limited series of sensors with the following properties has been constructed from the design principle of the prototype: Extemal dimensions of Diameter Height Volume
housing 32.5" 29 mm 23 cm3
Mechanical load-bearing capability Translational force 50 N Rotational force 15 Ncm
In order further to improve the resolution and minimise the hazard of electrical overload (power dissipation limit), special 1000 0 strain gauges have been employed with these sensors. With a grid load of 0.7Wcm-*, the accuracy is very high for dynamic measurements, and average for static measurements. 4. Measured data
With a gain factor of 1000, a sensitivity of about 5 mV per 10 mN is attained. In correspondence with the maximal load-bearing capability of the sensor and an
248
J Planen et al
average level of elaboration for the electronics, the resolution of 4096 steps for a twelve-bit analogue-digital convener can be fully utilised with about 2.5 mN per step. T h e characteristic features worthy of note include the reaction of the sensor to loads in the maximal force range. For example, with a torque of 2 N per 5 cm and a duty cycle ratio of 20 min / 20 min for loading, the maximal hysteresis is better than 8 pV o n all channels; at 1 min / 25 s, it is better than 1.5 pV. T h e temperature variation of the system is about 3 pVper"C.
5. Prospects A miniaturised sensor can be employed, for instance, for in vitro examination of socalled prosthetic auxiliary components, that is, connecting elements such as articulated joints and attachments, comparative measurements on the evolution of forces on dental prostheses and instruments, spring elements or arcs for operative dentistry, or intraoral in vivo measurements of periodontal behavior (intrinsic physiological mobility) of teeth under the action of forces. Miniaturisation by a factor of about 2.5 (by volume) is the object of current endeavours and appears feasible without excessively elaborate measures. References Bourauel Ch, Drescher D and Thier M 1990 Krafr-Mamenren-Aufnehmer fiir die Kieferorthopzdie Feinwerktech. Merrtech. 98 419-421 Dillmann R and Faller B 1982 Kraft-Momenten-Sensorsystem fiir IndustrieroboterElrkmnik 8 89-95 Dillmann R, Hugel Th and Meier W 1986 Ein sensarinregrierter Greifer als modulares Teilsystem fur Monrngeroborer Robotmystma 2 247-252 Schanz G W 1986 Sensoren (Heidelberg: H"rdg Vedag).
