A direct approach for control is possible when the external load in- formation is available directly by measurement of all joint torques via joint torque sensors.
A Joint Torque Sensor for Robots F. Aghili and M. Buehler
J. M. Hollerbach
Dept. of Mechanical Engineering McGill University Montreal, QC H3A 2A7, Canada
Dept. of Computer Science University of Utah Salt Lake City, UT 84112, USA
Abstract This paper describes the design of the elastic body of a new torque sensor to measure torsion moment faithfully in the presence of strong overhang and thrust forces as well as bending moments. This feature, in conjunction with high sti ness makes this sensor an ideal choice for robotic applications. The structure inherently exhibits minimum strain sensitivities and compliances to all non-torsional moments and forces such that their e ect can easily be cancelled in the Wheatstone bridge circuit. On the other hand, maximum possible local strain sensitivity near the material limit, and maximum torsional sti ness are achieved through the structure's strain concentration characteristic. Several other important design criteria are also discussed, as well. The sensor geometry is analysed and further optimized using the nite element method. The sensor was constructed, and experimental data validated the nite element model and documented the sensor's performance.
1 Introduction As direct drive robots become increasingly popular, there is a need for high performance motion controllers which achieve the dynamic accuracy in practice. A direct approach for control is possible when the external load information is available directly by measurement of all joint torques via joint torque sensors. Direct positive feedback of the torque signal can then compensate the dynamics of the robot manipulator [1, 10, 16]. Although this approach o ers a far simpler control solution than model based control, accurate joint torque measurements encounter several
challenges. In the design of robot manipulators, it is desirable to contribute much of the torque/force reaction of the link load on the joints in the form of non-torsional components as less actuation e orts are demanded (e.g. the SCARA design [12] prevents gravity torques acting on the joint motors). Moreover, in direct-drive robots the torque sensor is directly attached to the motor's distal link. Therefore, they must bear the large non-torsional components of the generalized force/torque vector exerted at the joint called them exogenous components hereafter. The rst challenge is how to measure the torsion torque faithfully without being in uenced by any of the exogenous components. Second, torque sensors rely on measuring the local strain or displacement of their elastic bodies. In order to increase the signal-to-noise ratio and obtain a high resolution, it is desirable to design a structure which generate a large strain for a given load torque and therefore has a large sensitivity. However, the resulting compliance introduces a joint angle error which should be minimized. Thus there are two con icting requirements; high mechanical sti ness and high torque sensitivity, and both quantities should be maximized simultaneously. The solution to both of these main challenges will be described in this paper and is new compared to existing designs. There is a large literature on the systematic design of six degree-of-freedom force/torque sensors [7, 17, 19] and on various techniques to instrument geared motors for torque sensing [5, 11, 14, 20], while little attention has been paid to pure joint torque sensing [3, 9]. It is important to note that the design criteria for the two types of sensors are very different. For instance, isotropy (i.e. uniform sensitivity) is a desirable property of a six degree-of-freedom force/torque
sensor, hence its elastic structure tends to be fragile and compliant in all directions. In contrast, the elastic sensitivity of a torque sensor has to be maximized only around its torsional axis. Insensitivity to the other ve non-torsional generalized force components provides high corresponding sti nesses and devotes a higher maximum strain to torsion resulting in high torsional sensitivity in addition to natural decoupling. In commercial torque [2, 6] sensors exogenous components are not permitted or are highly restricted. Furthermore, they usually come in bulky packages, and thus they are not suitable for integration in a robot joint. This paper is arranged as follows. In Sec. 2 the conditions on the mechanical design and the location of the strain gauges so that the Wheatstone bridge decouples the joint torque output signal from the exogenous components is proposed. How mechanical design is further constrained by additional design considerations, like sti ness, sensitivity, and size is discussed in Sec. 3. The constructed torque sensor prototype, and how the nite element method was used to exploit fully the advantages of the structure, to analyze its elastic behaviour, and to nalize the dimensions are described in Sec. 4. Experimental results and the sensor's characteristics are shown in Sec. 5.
2 Design for Decoupling In this section the general conditions on the elastic structure of the sensor such that the output signal from the Wheatstone bridge depends only upon the applied torque and not on the exogenous components are formulated. In general, the torque is derived via strain measurements at several locations on the elastic body. Assuming linear elastic material, there is a linear relationship between the applied forces and torques and the resultant strains described by = C fs ; (1) T where = [1 ; 2 ; ; n ] is the vector of n measured strains, fs = [fx; fy ; fz ; mx ; my ; mz ]T is the generalized force/moment vector acting at the center of the sensor body where z-axis and joint axis are identical, and C 2 IRn6 denotes the sensitivity matrix. Unlike in six-degree-offreedom force/torque sensors, it is desired to reconstruct only the torsion moment mz , from the measured strains . The natural question arises concerning the condition under which there exists such a mapping. In the most general case, the torque mz must be the summation of weighted measured strains, mz = T v ; (2) T where v = [v1 ; v2 ; ; vn ] is the gain vector. Substituting from (1) into (2) we have ? (3) mz = fs T CT v :
It is evident from (3) that the output signal is proportional to torsion and independent of the exogenous force/torque components provided
CT v = e;
(4)
where e = [0; 0; 0; 0; 0; 1]T . Equation (4) clearly says there exists a gain vector v if the vector e belongs to the range of matrix CT or equivalently in the support space of matrix C [4]. The number of solutions then depends on the rank of matrix C. If rank C < n, then there are in nitely many solutions, and the Moore-Penrose inverse, denoted by C+ , can calculate the minimum norm gain vector,
v = C+e: A
C
B
fy
1
2 Y
Y
mz
X
4
fx
fz
Z mx
3
Figure 1: Basic torque sensor structure. A: solid discs; B: elastic element; C: strain gauge. For the geometry in Fig. 1 and with four strain gauges located on the numbered locations, the parametric form of the sensitivity matrix can be found merely based on the symmetric property of the structure with respect to the external forces and torques as 2
c1 6 c2 C = 64 c2 c1
c3 c3 c4 c4
3
c5 c6 c7 ?c8 c5 c6 ?c7 c8 77 : c5 ?c6 ?c7 ?c8 5 c5 ?c6 c7 c8
This matrix permits the reconstruction of the torsion moment from the output signal with the gain vector v = 1 T c8 [?1; 1; ?1; 1] . In this case, one can use the additive properties of the Wheatstone bridge to implement the gain vector. In general, provided the gains have identical absolute values, the Wheatstone bridge can achieve decoupling without the need for any subsequent arithmetic. The resulting advantage is a reduction of instrumentation and the number of wires by completing the bridge wiring locally in the sensor, and simpli cation of the tedious calibration. There are two main reasons in practice that violate the above assumption of exact symmetry among the measured
strains. First, strain gauges exhibit variations in their Gauge Factor, GF = R=R = ;
which is the ratio of the fractional change in resistance R, to the fractional change in strain . Second, the strain gauges will be placed on areas with high strain gradients which makes the gauge outputs sensitive to placement errors. This can also be modelled as a gauge gain error. As a consequence, exact cancellation of the exogenous components may not be achieved with the theoretical gain vector. By virtue of the linear mapping (1), the exogenous components produce no output, if all elements of the sensitivity matrix except of the last column is suciently small. This implies that the strain sensitivity to the exogenous components has to be held minimum by mechanical design. This condition in conjunction with the decoupling property of the sensitivity matrix actually determines the capability of the sensor to reject the e ect of exogenous force/torque to the output and providing high delity output signal.
3 Further Design Considerations In this section additional design issues, like torsional strain sensitivity, torsional sti ness, the number of strain gauges, the sensor shape and material will be discussed. Maximum torsional strain sensitivity: To increase the S/N ratio and the resolution of the sensor, it is desirable to design the elastic component to provide large output signals. Therefore one of the design criteria is to increase the torsional sensitivity, subject to not exceeding the allowable strain. In the absence of exogenous components, the maximum attainable strain sensitivity depends solely on material properties as the strain due to the maximum load should be close to the maximum allowable material strain or stress. However, exogenous components produce strains which add to the strain caused by torsion. To ensure that the allowable maximum material strain is not exceeded, we consider the worst-case scenario where the generalized torque/force vector has its maximum force, fmax, and torque , mmax components. Then by virtue of (1), the following condition should be satis ed, (jc1 j + jc3 j + jc5 j) fmax + (jc6 j + jc7 j + jc8 j) mmax ja j: (5) Apparently, to exploit maximum torsion sensitivity, jc8 j, the other sensitivities components must be minimized by proper geometry design. This design requirement is consistent with a decoupling property of the sensor. It is interesting to note that cylinders are mainly used in the design of commercial torque sensors. By elementary strength of
material analysis, one can inspect that bending moments produce two times stress, or equivalently strain, than the same magnitude torsion moment. That is why shear and trust forces and bending moment must be kept small in these sensors. High sti ness: Torsional de ection degrades the position accuracy of the joint angle controller. Therefore, one of the critical design challenges is to maximize the sti ness while maintaining high sensitivity. These contradictory requirements can be captured by de ning a performance index: as the ratio of the (local, maximum) strain to the (overall) torsional de ection caused by the same torque,
(6)
def = : The dimensionless index is a decisive factor in the sen-
sor design and should be maximised. It is independent of material properties, and demonstrates the ratio of the local and global strains. The index is maximized in elastic structures which produce high strain concentration in torsion. In theory, there is no limit on the strain concentration in an elastic body. However, the high strain concentration takes place in a very small area, which might be smaller than the physical size of available strain gauges. Moreover, since strain gauges average the strain eld over their area, the detected strain can be signi cantly lower than the calculated maximum. Therefore, it is important to generate high strain over a suciently large area. This objective is dicult to formulate and satisfy analytically, but can be inspected by nite element methods. Introducing a torque sensor in a robot joint adds exibility. While torsional exibility can be compensated via sophisticated controllers, de ection in the other axes is more problematic. Consequently, another design criterion dictates high sti ness in non-torsional directions. Fortunately, the requirements for low de ection and low strain sensitivity for exogenous components are consistent. Practical shape considerations: The addition of a torque sensor to a robot joint must not require the redesign of the joint and should result in a minimum change in the manipulator kinematics, in particular the link o set. Hence, in general a small width shape is desirable. Minimizing the e ects of thermal stresses is another design factor which cannot be ignored. Motors are a source of heat which ows from the motor to the attached link through the sensor body. Therefore, an axis-symmetric con guration is desirable which constrains the heat ow in the axial direction where usually no position constraint exists. It is worth noting that hub-sprocket structures are potentially prone to thermal stresses because of the temperature di erence between the hub and wheel. Since the sensor is speci cally designed for a direct drive motor with hollow shaft, ange mounting is preferred. Finally, the body should be designed
for ease of manufacture. It should be machined from a single piece of material to avoid the need for welding, brazing, or mechanical fastening which can cause position backlash or torque response hysteresis. Material properties: So far only geometric properties of the elastic body were considered. Nevertheless, the sti ness and sensitivity characteristics of the torque sensor are determined by the material properties. The maximum allowable strain for foil strain gauges is typically 3% which is at least one order of magnitude higher than that of industrial metals a , making the materials the limiting factor for sensitivity. Furthermore, the sti ness depends linearly on Young's modulus E , of the material. By virtue of Hook's law, a = E a one can conclude that high sensitivity and sti ness are achievable simultaneously only by high strength material. Since a linear response of the sensor is desired, a linear strain-stress relationship from the selected sensor material is required. Steel is the best available industrial material which has good linearity properties within a large stress range. Moreover, due to the oscillatory nature of the loading, steel can work with in nite fatigue life [15] as the allowable strains are determined based on the endurance limit.1
its X-axis compared to that around its Z-axis (as a comparison, it is worth to note that a simple cylinder is 0:77 times less sti in bending than in torsion). Fig. 1 shows an elastic structure resulting from the aforementioned considerations. It consists of two exible elements attached to two rigid discs which carry the external loads. The exible elements are subjected to combination of torsion and bending when a torque is applied. The produced strain is the superposition of both torsion and bending strain contributions. The strain contribution of bending depends upon the beam's length, the distance between the two disks, which adversely in uences the compliance. But, thin section rectangular bars experience high stress/strain concentrations under torsion loads, which yield high sensitivity without sacri cing sti ness. This fact suggest that an appropriate structure should be primarily stressed by torsion. A drawback of the generic structure is its poor sti ness around the Y-axis ( see Fig. 1 ). This problem can be solved simply by incorporating additional wing pairs as shown in Fig. 2.
4 Design Procedure 4.1 Basic Sensor Geometry Since the sensor design must optimize and trade o between several con icting design criteria, an analytical procedure yielding a unique and optimum design is not available. In addition, many design iterations were required to arrive at a nal design. Despite this complexity, it is still possible to provide the main reasoning for the resulting novel basic sensor design. At this stage it is important to de ne general properties of an elastic body satisfying the design requirements. First, it should be noted that insensitivity to exogenous components automatically maximize the corresponding sti nesses. Secondly, as discussed earlier, strain concentration is the design key to achieve high torsional sensitivity and high sti ness. Consequently, from the elastic property point of view, the optimal body must have high lateral sti ness as well as high torsional strain concentration properties. Similar to a thin wall beam, one can observe that the structure should have a large second moment of area around 1 The endurance limit or fatigue limit is the maximum stress under which mechanical failure will not occur, independent of the number of load cycles. Only ferrous metals and alloys have an endurance limit.
Figure 2: Elastic structure of the torque sensor.
4.2 Design via FEM Analysis Given the basic sensor shape, the FEM analysis package IDEAS (Structural Dynamics Research Corporation) was used to nalize the dimensions such that performance is optimized. FEM was also used to select the location and proper size of the strain gauges, as to obtain maximum sensitivity, strain gauges should be located where maximum induced strain due to the torsion load occurs. Since the strain eld is averaged over the area covered by the strain gauges, it is very important to rst determine the locus of the peak strain, and second, ensure the creation of a suciently large strain eld. FEM is ideally suited to solve this problem. The body is modeled by tetrahedral solid elements as shown in Fig. 3 (A). Since it is symmetric in geometry and boundary conditions, it suces to analyze one half of the body, provided adequate position constraints are imposed on the nodes of the cutting plane.
Figure 5: The static test setup.
ers built into the sensor boost the signal level of Wheatstone bridge output before A/D conversion. Fig. 4 shows also that we took advantage of our hollow motor shaft, which is common in direct drive motors, to locate the electronic board beside the sensor. The local signal conditioning provides a stronger output signal and improves the S/N ratio. Moreover, since the electronic circuit is totally enclosed by the motor's hollow shaft, it is well shielded from the powerful magnetic noise created by the motor.
5 Calibration and Experiments 5.1 Static Test In order to calibrate the sensor, static torsional moments are applied. Fig. 5 illustrates the experimental setup for these static tests. The torque sensor is xed rigidly in one side to a bracket while an aluminium bar is attached to the other side. The end of the bar is connected to a mechanical level via a rope in which a load cell (MLP-50 from Transducer Techniques [18]) is installed. The level increases and decreases the tension in the cord gradually from zero to maximum and back to zero. During the loading and unloading, the load cell and the torque sensor outputs are being sampled by a two channel data acquisition system. The force transducer signal is scaled to torque (having arm length) and then is plotted versus the torque sensor output voltage in Fig. 6 for 2; 000 sample points. The slope of the line indicates the sensor calibration coecient
6 5
Sensor output (Volt)
Figure 4: The torque sensor prototype.
V . All deviations from linearity are less than 0:2% of 0:03 Nm full scale. This is close to the accuracy of the reference force transducer used for calibration, 0:1% full scale. In addition, cross sensitivity measurements were also possible with this setup. The bending moment is applied via an axial bar rmly connected to the sensor. Again, a measurable overhang force is applied at the end of the bar by a rope, resulting in a bending torque on the sensor. The direction of the force is set by a pulley, as shown in Fig. 5. The torsion torque is also applied by the radial arm. The worst case ratio of the two load sensitivities, the cross sensitivity, is only 0:6%. This con rms that the sensor e ectively decouples the e ect of the exogenous components on the measured torque signal.
4 3 2 1 0
0
50
100 Torque (Nm)
150
200
Figure 6: Sensor output versus applied torque.
5.2 Dynamic Test The dynamic test serves mainly to validate the FEM on which the analysis is based. The experiment is arranged to extract the general sti ness of the sensor prototype. Again the sensor is held rigidly by the bracket while a steel disk is anged to the other side. The disk is very massive with a polar inertia of 0:24 kgm2 and the whole system behaves like a second order system. In order to
detect all of the vibration modes due to all compliance directions, the cross sensitivity is deliberately increased by electrically by-passing the strain of all strain gauge pairs, except one. Therefore, the torque sensor no longer has the decoupling property, and its output is the summation of all torque/force components weighted by their corresponding gains. Since the value of the torque/force and subsequently their strain e ect is proportion to their relative de ections, the output signal is proportional to the combination of the torsion and bending vibrations. The system is excited impulsively by a hammer and the subsequent vibration is recorder by a data acquisition system at a frequency rate of 3:2 kHz . Fig. 7 shows the impulse response of the system in the time domain which closely resembles that of a second order system. 6
Sensor output (Volt)
4 2 0 −2 −4 −6
0
0.1
0.2 0.3 Time (sec)
0.4
0.5
Figure 7: Impulse response of the integrated torque sensor with an 0:24 kgm4 inertia (solid). The exponential envelope decays according to = 0:01 (dashed).
Magnitude (dB)
30
20
10
0
−10
0
500 1000 Frequency (Hz)
1500
Figure 8: Frequency response of the sensor system. Since the torsion gain is highest, the sensor signal is dominated by torsion vibration which decays due to the structural damping of the sensor material. Nevertheless, the sensor's modes are revealed clearly in the frequency domain. To this end, a short interval of the signal is taken via a Hamming window and then its spectrum is found by FFT. Fig. 8 reveals two distinct modal frequencies occur at 150 Hz and 980 Hz which belong to torsion and bending modes, respectively. Due to the low damping, the modal
frequencies are almost the same as the natural frequencies. Hence the torsion sti ness can be simply calculated by r
!z = kI z ; z
which results in the high torsional sti ness of kz = 2:4 105 Nm rad . A comparison with the FEM predictions reveals a 20% error which con rms its validity. The bending sti ness can be found in the same fashion, however, it should be noticed that the relative inertia is half of the polar inertia for disks, i.e. Ix = 21 Iz . The bending sti ness is calculated to be twenty times higher than the torsion sti ness, kx = 4:8 106 Nm rad . The internal structural damping is modelled as viscous friction . Suppose n is the amplitude ratio of two nth consecutive oscillations. Then the damping ratio is 1 ln = 0:01: = 4n n Fig. 7 validates the damping model where the corresponding exponential envelope matches the oscillation decay. The e ect of internal viscous friction suggest that the mechanical model of the sensor can be a spring in parallel with a damper. The strain gauges deliver a voltage signal as soon as they are exposed to strain. The external torque mz on the sensor is then equal to mz = kz z + b_z ; where z is a small angular de ection and b is the viscous coecient which can be obtained by the damping ratio . Since the sensor is calibrated when a static load is applied, the sensor voltage is proportion to kz z . Therefore, the transfer function G(s) from the applied torque to the output voltage is rst order: G(s) = 1 b 1 + kz s The time constant, kbz , for the sensor based on the experimental results is 21 s which is reasonably small.
6 Conclusion Motivated by the need for a joint torque sensor for robots, we described the systematic design of an elastic body for torque measurements in the presence of strong exogenous force/torque components. A new sensor is proposed whose key feature is its extremely high sti ness as well as insensitivity to all these exogenous components. It was shown that the maximum strain sensitivity to torsion can be maintained by not sacri cing torsional sti ness if the
elastic body exhibits strain concentration to torsion loads. The sensor has further been designed such that the e ect of non-torsional moments and forces on the local strains is minimal, which has been achieved by a combination of mechanical design and electrical summation of strain gauge signals. The sensor geometry is analysed and further optimized using the nite element method. The sensor was constructed and tested extensively. The tests con rm that the sensor met all its design goals and is ideally suited as a torque sensing device in robots or other industrial applications like machine tools.
Acknowledgements This project was supported by the PRECARN TDS Project, through MPB Technologies of Montreal, Quebec. The authors also thank Dr. R. Patterson and Mr. S. Boelen of MPB Technologies for their technical support.
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