Estimate a Flexible Link’s Shape by the Use of Strain Gauge Sensors

is paper presents a method for estimating the �exible link’s shape by �nite number of sensors. e position and orientation of �exible link are expressed as a function of curvature of the link. An interpolation technique gives this continuous curvature function from a �nite set of the �heatstone bridge made with strain gauges. For interpolation we can use different functions to �nd better way for estimation of link’s shape. Comparison between different types of function can show us best corresponding with nature of the link. �ur case study is a single �exible link robot. A high-precision data logger is used as data acquisition instrument.


Introduction
Research into �exible manipulator �elds has become more important for more than a decade on account of new robotic applications in different areas such as industrial tasks in which the trend is to use lightweight materials (e.g., carbon �ber and composites� in the structures of manipulators in order to improve the performance of the industrial robots which are large and heavy.In aerospace applications the lightness of the materials is also an important requirement; the control of large structures, such as boom cranes and �re rescue turntable ladders, which are treated as �exible link robots is discussed by [1,2]; minimally invasive surgery is performed with thin �exible instruments in which precise automatic manipulators are necessary [3].ese �exible manipulators exhibit important advantages in comparison to rigid ones, such as reduce energy consumption, smaller actuators, safety for humans, and more agility.Moreover, the effects of the collision with obstacles or humans are decreased because of the �exibility of the links.In spite of these advantages, the control problem of the �exible link manipulators becomes more complicated because the structural �exibility has to be controlled in such a way that the vibrations are cancelled if a good tip position and/or force control is to be gained.
Most of the work in the �eld of elastic robotic manipulator is con�ned to theoretical investigations.For validation the theoretical results by experimental testbed, it can be pointed out to the works of Cannon and Schmitz [4] as a pioneer in experimental study of single-link elastic robotic manipulator.Naganathan and Soni [5] also prepared an experimental setup composed of a single-link mounted on a stepper motor to verify their FEM results.Bellezza et al. [6] conducted experiments to study quasi-clamped and quasi-pinned end conditioned in slewing links.Luo [7] used a strain gauge sensor to measure the �exible robot arm de�ections.�ang [8] presented a modal data-based method to estimate the dynamic response of an elastic robotic manipulator.Martins et al. [9] conducted experiments for a single-link elastic robotic manipulator to validate their suggested FEM based analytical model.More experimental testbed for vibration control of elastic robotic arm can be found in the investigation of Ladkany [10] and Peza Solis et al. [11].
So study and research on de�ections and vibration of �exible link robots is an important problem.Increase of accuracy of measurements with �nite number of sensors can help us.Flexible link's shapes can be approximated with different functions each one has speci�c nature.e usage of the best function that correspond nature of �exible link helps us to reduce sensor numbers and increases accuracy.

Kinematics of the Single-Link
Elastic Robotic Manipulator e single-link �exible manipulator system considered in this work is shown in Figure 1, where  and  represent the stationary and moving coordinate's frames, respectively.e position vector of arbitrary differential element  with respect to the local reference system  is shown by ⃗   .e approach of modal analysis is used to incorporate the de�ection of the link.So, where   =     is the position vector of differential element  with respect to , when the elastic link is unreformed and , , and  are small displacements in , , and  directions, respectively.To express these small displacements, the approach of assumed mode method is used as Here    =         is the Eigen function vector whose components   ,   , and   are th longitudinal and transverse mode shapes of the link;   is the -th time-dependent modal generalized coordinate of the link, and  is the number of modes used to de�ne the de�ection of the link.e total transverse displacement of the centerline of differential element  is due to bending and shear.So the total slopes of the de�ected centerline about  and  directions due to bending and shear deformation can be represented as where   and   are the slope of the de�ected centerline due to shear and   ,   are the slope of the de�ected centerline due to bending.Shear has no in�uence on rotating the differential element , and this differential element undertakes rotations only due to bending and torsion.So the rotation of this element in , , and  directions can be considered as   ,   , and   , respectively.ese small angles can be represented by truncated modal expansion as where    =         is the Eigen function vector whose components   ,   , and   are -th rotational mode shapes of the link in , , and  directions, respectively.

Tangent to centerline
Parallel with centerline F 1: Single-link elastic robotic manipulator.

The System Gibbs Function and Its Derivatives
In this section the system's acceleration energy and its derivatives with respect to quasi-accelerations are developed for construction the G-A formulation.With the assumption of TBT, the acceleration energy of a differential element  can be represented as In the above expressions,   and   are angular velocity and angular acceleration of the link, respectively.Also ̇⃗   and ̈⃗   are velocity and acceleration of differential element  with respect to the origin of the local reference system, respectively.Inserting (6) into (5) and integrating over the link from  to , the total acceleration energy of the link will be obtained as where   is the skew-symmetric tensor associated with   vector.Also there is a term named as "irrelevant terms." Motion equation with G-A formulation will be constructed by taking the partial derivatives of Gibbs' function with respect to quasi-accelerations.So, the term in Gibbs' function that does not contain q and δ can be ignored.e variables that appeared in ( 7) can be calculated as, where In the above equations,   and    are skew-symmetric tensors associated with   and    vectors.As mentioned above, one part of dynamic equations of the system using (G-A) formulation will be obtained by differentiating of Gibbs' function with respect to quasi-accelerations. ese two terms can be represented as But for the second assumption, the �rst term in the above integral will be omitted.In (11),  and  are modulus of elasticity and shear modulus, respectively;   is the polar area moment of inertia about  axis;   and   are area moment of inertia about  and  axes, respectively;  is the cross section area of the link and  is shear correction factor.As noted in the previous section the small angles   ,   ,   and small displacements , ,  can be expressed with a truncated modal approximation.By inserting these expressions in (11), the strain potential energy for the link will be obtained as where Generalized forces due to internal and external damping are obtained by taking partial derivatives of Rayleigh's dissipation function with respect to generalized velocities.ese two terms may be presented as

Dynamic Equations of Flexible Link Manipulator
Motion equation of viscoelastic robotic manipulators will be completed by considering the generalized forces which are caused by the remaining external force terms.Let us assume that there is no external load on the links of the considered robotic manipulator.So, the generalized forces in the de�ection equations will be zero.e generalized force in the joint equations is the torque  that applies to the joint.With this assumption, the dynamic equations of motion by (G-A) formulation will be completed as follows (i) e joint equations of motion: (ii) e de�ection equations of motion: 6.1.Strain and Curvature.If the strain in different sections of an elastic beam is known, the shape of the beam can be estimated by interpolation.e differential element in yellow color in Figures 2 and 3 shows the beam's neutral axis before and aer bending, respectively.e strain of this differential element can be presented as: Based on (21), the strain on the surface of the beam will be obtained as where  is the thickness of the beam.On the other hand from elementary calculus the curvature of a plane curve can be expressed as: us, the strain of the beam in terms of the function () can be represented as: But, in the case of the elastic curve of a beam, the slope  � is very small, and its square is negligible compared to unity.So, erefor, the strain is a function of curvature and thickness.e magnitude of strain in  position will be obtained with a strain gauge on that point.As we cannot use in�nite number of sensors, the strain is de�nite only in �nite points.We can guess () as a suitable function that can approximate link's shape.e nature of the function must be the same with the nature of the �exible link.If one uses suitable function for link's shape, fewer constants and fewer sensors will be needed.
In this paper, the following functions will be used: In experimental setup, three full bridge strain gauges attached in three different positions.By the use of (25) and considering the three strain gauge bridges, we can organize three equations.In addition, from boundary condition, slope and de�ection at the beginning of the link are zero.So,  and  � in this point will be zero.With these three strains and two boundary conditions, �ve equations with �ve unknowns can be solved.But if we consider the boundary conditions at the end point of the link, two more equations will be added.In fact at the end point of the link shear and moment are zero.So,  �� and  ��� will be zero in this point.In this condition, we have seven equations with seven unknowns and we can guess a function with more constants and more accuracy.

Experimental Setup
e experimental testbed is the �exible manipulator that is prepared at the Department of Mechanical Engineering of the Iran University of Science and Technology Figure 4. e experimental setup consists of a single-link �exible manipulator driven by a servo-tech AC servo motor.is 400 W AC servomotor can be drived in position, speed, and toque mode.In order to remove additional effect of bearing friction, the �exible manipulator is directly attached to the motor sha.In Table 1 the physical properties of the �exible manipulator are summari�ed.Also, the Schematic diagram of the experimental setup is shown in Figure 5.
e �exible manipulator has a planar motion thus the effect of gravity can be ignored.An incremental encoder with the resolution of 2500 pulse per revolution is used to calculate the rotation angle of the motor.Motor angular velocity is determined by time derivation of the motor angular position data.e tip de�ection and vibration of the link are measured by three strain gauge bridges mounted on the link.e analogue signals of the strain gauges amplify through a signal conditioner circuit and sent to a data acquisition card.
A 64-bit personal computer was used for control motor output via an Advantech PCI-1710HG data acquisition card.PCI-1710HG is a multifunction card with 12 bits resolution and 100 kS/s sampling rate.
Strain gauges can be attached in many ways for different purposes.For recording vibrations without temperature compensation, we have three-type layouts.Quarter bridge, half bridge and full bridge layout can be used for this purpose.Quarter bridge (Figure 6) has less sensitivity than two other types.So, it can be economic.But the weakness of signals makes the necessity of expensive signal conditioners.Half bridge (Figure 7) and full bridge layouts have more sensitivity but need more number of sensors, respectively.Half bridge layout uses two strain gauges, and if it is required, two other strain gauges in normal directions can be added in order to compensate the temperature effects.
Full bridge layout (Figure 8) uses four strain gauges and has best sensitivity and simplest equation.But as the bridge is full, it is not possible to use temperature compensator in this type.
e following equations can be used for calculation the exact magnitude of strain: ese three equations show relations between   and strain in quarter bridge, half bridge, and full bridge, respectively.It is clear that quarter bridge equation has nonlinear form, but the other two types have linear and simple relations.In the above equations  Ex is excitation voltage exerted on the Wheatstone bridge.  in these equations exhibit output voltage.Excitation voltage affects   .On the other hand, increasing in excitation voltage causes increasing in output voltage and provides more resolution.But increasing in exoitation voltage causes increasing in ampere.It can generates heat in strain gauges.Generated heat in strain gauges must be compensated with temperature compensator strain gauges.Otherwise, error increases in output voltage.

Results and Discussion
In this section, simulation and experimental results of the response of a �exible manipulator and the effect of functions are presented.e torque input implemented to the manipulator is a bang-bang torque with amplitude of ±0.13 N ⋅ m and duty cycle of 0.3 s. is torque has a positive (acceleration) and a negative (deceleration) period allowing the manipulator to initially accelerates and then decelerates and eventually stops at a target location.ere are 400 data from each strain gauge bridges that can help to have a comparison between experimental results and theoretical results.Sampling frequency is 200 Hz.
At �rst, we compare a sixth-order polynomial with a fourth-order polynomial.It's clear in Figure 9, 6th-order polynomial exhibits better results from sensors data.If we have free conditions at the end of the link, it's better to use this function.
Figure 10 shows that the interpolation with 5th-order trigonometric function leading to very bad results.e function has no enough agility for tracking vibrations.So the behavior of trigonometric functions is far from link vibrations.
At the next stage, the behavior of hyperbolic functions will be studied.As shown in Figure 11, comparison between hyperbolic functions with different constants shows similar results.Nevertheless, the use of hyperbolic function with seven constants yields better results.
In the next step comparison between trigonometric, hyperbolic, and polynomial functions with seven constants will be shown.As seen in Figure 12 vibration behavior of these functions is the same.Although the nature of these three functions is not similar, very good �tting will be observed.
Continuously experimental results will be compared with the Gibbs-Apple theoretical results.For experimental estimation, hyperbolic function with seven contents is used.
In Figure 13 experimental results follow theoretical results closely.So hyperbolic and polynomials are suitable for estimating �exible links vibrations.

Conclusions
In this paper the equation of motion of a single-like �exible manipulator is derived by the Gibbs-Appell formulation.Also an experimental test bed was prepared to verify the simulation results.e obtained results from strain gauges converted to end point de�ection by di�erent functions.�ese functions can be used for monitoring �exible link vibrations.Among these functions, the obtained result from hyperbolic function follows theoretical results much better than the other functions.

F 5 :F 6 :
Schematic view of the experimental setup.Quarter bridge layout for strain gauges.

F 7 :F 8 :
Half bridge layout for strain gauges.a i n g a u g e S t r a i n g a u g e S t r a i n g a u g e S t r a i n g a u g e Output voltage Amplifier Full bridge layout for strain gauges.

5 F 9 : 4 F 10 :
Estimation of end point de�ection with 6th-order and 4th-order polynomials.Estimation of end point de�ection with 5th and 7th order trigonometric functions.

5 F 12 :F 13 :
Comparison between polynomial, trigonometric, and hyperbolic functions with seven constants.Experiment and theoretical result comparison.
To express the strain potential energy stored in elastic link, let us assume two theories hold that: TBT and EBBT.�or the �rst assumption the strain potential energy will be expressed in terms of de�ections and rotations as e potential energy of the system arises from two sources, �rst potential energy due to gravity and second potential energy due to the elastic deformations.e corresponding energy due to the gravity can be considered simply by inserting ̈⃗   =  , where   is the acceleration of gravity.