INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 11, No. 2, pp. 219-225

APRIL 2010 / 219

DOI: 10.1007/s12541-010-0025-0

Torque Sensor Calibration Using Virtual Load for a Manipulator Sang-Hyuk Lee1, Young-Loul Kim1 and Jae-Bok Song1,# 1 Division of Mechanical Engineering, Korea University, 5Ga-1 Anam-dong, Sungbuk-gu, Seoul, South Korea, 136-713 # Corresponding Author / E-mail: [email protected], TEL: +82-2-3290-3363, FAX: +82-2-3290-3757 KEYWORDS: Torque Sensor, Calibration, Crosstalk, Manipulator

Accurate load sensing of a manipulator becomes increasingly important in performing various tasks involving contact with an environment. Most of the research has been focused on improving the hardware of a force/torque sensor. The torque sensors for a manipulator suffer from crosstalk, which is difficult to compensate for even with sophisticated calibration. This research proposed a novel calibration method composed of two steps. Through the primary calibration, the torque sensor output can be related to the joint torques. The secondary calibration, which is based on a virtual load, is conducted to compensate for the crosstalk of a torque sensor. The virtual load is obtained from the sensed joint torques and manipulator configuration. Using the proposed calibration method, the external load acting on the end-effector of a manipulator can be accurately measured even with relatively lowquality torque sensors. The experimental results showed that the error in the load sensing was significantly reduced by the proposed calibration method. Manuscript received: October 21, 2008 / Accepted: November 23, 2009

1. Introduction Accurate sensing of the external load at the end-effector of a manipulator has become increasingly important as robots are required to perform the tasks involving contact with the environment. Service robots, which have drawn a lot of attention in recent years, especially need such load sensing capability in order to assist humans in various tasks1,2. Since expensive force/torque sensors are not suitable in service robots for economic reasons, load sensing should be conducted by means of relatively cheap load sensors. Various techniques have been used to measure the external load applied to the robot arm. Six-axis force/torque sensors3-5 or joint torque sensors are popular among these techniques. The six-axis force/torque sensor, which is frequently mounted at the wrist of a manipulator, can accurately sense the external load acting on the robot hand. However, its structure is very complicated for accurate sensing, and thus it is too expensive to be used in service robots. Therefore, joint torque sensors are usually adopted in the practical manipulators. The joint torque sensors require calibration for accurate sensing, where typical calibration methods include the compliance matrix computed by the structural analysis,6 the least-squares method5 and © KSPE and Springer 2010

the approach based on the motion of a robot arm,7,8 etc. However, the crosstalk error caused by the load components other than the target load (either force or torque) of the sensor cannot be compensated for by these calibration methods. Although the calibration method based on a neural network can cope with this problem,9 it cannot be applied to joint torque sensors. Therefore, most torque sensors have been designed so that the crosstalk is minimized by the structural consideration, 10-13 as shown in Fig. 1. In most cases, however, such crosstalk problems cannot be avoided due to both the manufacturing tolerances and the misalignment in strain gauge bonding.

Fig. 1 Hollow hexaform joint torque sensor13 To cope with this problem, a new calibration method consisting of the primary and secondary calibration schemes is proposed in this research. The joint torques can be obtained through the primary calibration from the torque sensor outputs, but these joint torques

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contain the crosstalk interference. Then, a virtual load acting on each joint torque sensor is estimated from the torque sensor output and manipulator configuration. This virtual load contains the components which cause the crosstalk interference. Then, the desired joint torque, which is free of crosstalk, can be obtained by the secondary calibration scheme. Once the primary and secondary calibration matrices are obtained off-line, an estimation of the desired joint torque, and thus the external load under consideration, can be conducted in real-time. The main contribution of this proposed calibration method is that the relatively cheap load sensors or the custom designed sensors, which have a simple structure and relatively low accuracy, can be used for accurate sensing of the external load. The rest of this paper is organized as follows. The concept of the crosstalk error related to the joint torque sensors is presented in section II. The proposed calibration method using the virtual load is detailed in section III and the experimental verification of the proposed scheme is discussed in section IV.

0.1mm, γFx becomes 0.011 from Eq. (1), which results in a crosstalk error of 0.011 Fx.

3. Sensor calibration using virtual load The crosstalk error of a load sensor discussed in the previous section cannot be compensated by means of general calibration methods. The novel calibration method minimizing this crosstalk is proposed in this section.

3.1 Primary calibration The purpose of the load sensor calibration is to accurately estimate the applied load (i.e., force and/or torque) from the sensor outputs. Consider a six-link serial manipulator with 6 revolute joints, as shown in Fig. 3. Assume that each link frame is assigned at the distal joint of each link and torque sensor i (i = 1, …, 6) is installed at each joint i. Suppose the external load F, which is accurately known in both magnitude and direction, is applied to the end-point E.

2. Crosstalk interference When a specific load (either force or torque) component applied to the object is to be measured using a strain gauge load sensor, the other load components acting on the object also affect the sensor output to some extent. The sensor response to the unwanted load is called crosstalk. The crosstalk error of a load sensor is primarily caused by the error involved in the installment of strain gauges and the manufacturing tolerances of the structure. An example is given below in order to gain insight into the crosstalk interference.

Fig. 3 Coordinate systems and torque sensors for a 6-DOF serial manipulator As a result of the external load F, each joint will be subject to joint torque τi (i =1, .., 6). Without considering the crosstalk, the following relation holds. Cp ⋅ z = τ

Fig. 2 Error caused by position error Figure 2 represents a simple force sensor composed of a cantilever beam and a strain gauge. Suppose the strain gauge is bonded to the top of the beam with a position error δ. If the strain gauge was mounted on the centerline of the beam, then the lateral force Fx would not have affected the strain induced by the vertical force Fy. The position error δ, however, causes the crosstalk given by γF

x

2t = 2δ w

(1)

where γFx represents the signal-to-crosstalk ratio due to Fx, while t and w are the thickness and width of the cantilever beam, respectively. The force measured by the strain gauge is Fy + γFx Fx instead of Fy. That is, if δ = 0 in the ideal case, then γFx = 0, but if δ ≠ 0 in the practical case, then the non-zero Fx causes the measurement error. For example, with w = 6 mm, t = 2mm, and δ =

(2)

where z is the 6x1 sensor output vector, τ is the 6x1 joint torque vector, and Cp is the primary calibration matrix. The primary calibration problem is to find Cp that satisfies Eq. (2). Since the torque sensors are not yet calibrated, the joint torque vector τ should be computed from the known external load F by the following Jacobian relation.

τˆ = J T ⋅ F

(3)

where τˆ denotes the joint torque estimate computed from Eq. (3). The Jacobian matrix is computed from the geometric parameters of a manipulator and the manipulator configuration. The primary calibration matrix Cp is computed by the least-squares method as follows: M

C p = min ∑ (C p zm − τˆm ) 2 C p m =1

(4)

where M is the number of samples used for calibration, and zm and τˆ m are the sensor output vector and estimated joint torque vector of

INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 11, No. 2

sample number m, respectively. Each sample is obtained by changing the manipulator configuration and/or the magnitude of the external load. For each sample, the corresponding joint torque vector τˆ m is computed by Eq. (3).8

3.2 Secondary calibration As discussed before, a load sensor suffers from crosstalk, and this crosstalk can be compensated for to some extent once the load causing it can be found. In this research, the load causing the crosstalk interference will be referred to as a virtual load because it cannot be physically measured by the sensors, but can only be estimated from the other sensor outputs. The method for finding a virtual load is presented below.

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The relationship between the virtual load iL and the desired torque τi at torque sensor i can be described by i

C ⋅ L =τ i

(9)

i

where iC is the 1x6 secondary calibration matrix associated with the virtual load at torque sensor i. The matrix iC can be obtained by i

M

C = min ∑ ( iC ⋅ i Lm − τˆi m ) i

2

(10)

,

C m =1

where M is the number of samples used for calibration, iLm is the virtual load vector for sample m, and τˆi m is the torque of sample m. ,

Each sample is obtained by moving the manipulator for a given external load F. τˆ corresponds to the i-th element of the joint i

torque vector τˆ, which is computed by Eq. (3). Once the secondary calibration matrix iC is obtained by Eq. (10), the crosstalk compensated joint torque τi is found from Eq. (9).

τ i′

3.3 Error analysis of sensor calibration using virtual load

τ 1′ Fig. 4 Free-body diagrams for a 6-axis serial manipulator Consider again a 6 DOF serial manipulator with 6 revolute joints in Fig. 4. For the external load F applied to the end-point, the torque sensors produce the torque values of τ′1, ..., τ′6 using Eq. (2) because the primary calibration matrix Cp was obtained during the primary calibration process. However, since these torque outputs are contaminated by crosstalk errors, they are different from the desired torque values, τ1, .., τ6, which would be measured for no crosstalk. Suppose the external load is computed from τ′1, …, τ′6 by the Jacobian relation. Then the estimate F′ of the external load will slightly differ from the original load F mainly because of the crosstalk errors contained in τ′1, …, τ′6. Now consider the free body diagram shown in Fig. 4. From the force and moment balance, the force (i-1f′ ) and torque (i-1τ′) acting on torque sensor i by the estimated load F′ can be obtained as follows: i −1 i −1

τ′ =

i −1

f′=

i −1

RE ⋅ F ′

i −1

rE × ( RE ⋅ F ′) =

L = [ i −1 f x′

i −1

f y′

i −1

f z′

L = [ i −1 f x′

i −1

f y′

i −1

f z′

i −1

τ x′

i −1

i

i

i

i

i

i

x

i

i

y

rE ×

i −1

f′

i −1

τ ′y

τ ′z ]T

(7)

τ ′x

i −1

τ ′y τ i′]T

(8)

i

i

i

x

i

y

z

(11)

where iC1, …, iC6 are the coefficients associated with the load components (i.e., ifx, …) acting on torque sensor i. Suppose the torque iτz is to be measured by the torque sensor. Then the other components cause the crosstalk interference. From Eq. (11), the desired torque component is given by C1 f + C2 f + C3 f + C4 τ + C5 τ z − C6 C6 i

τ =

i

z

i

i

i

i

x

x

i

i

i

y

i

i

i

z

z

i

x

i

y

(12)

However, the desired torque iτz cannot be obtained by Eq. (12) since the load components causing the crosstalk interference cannot be measured. Therefore, the primary calibration can be given by neglecting the second term of Eq. (12) as follows: τ ′z =

(6)

i −1

i

z

i

i −1

This virtual load vector can become more accurate with the replacement of the component i-1τ′z by the actual output of torque sensor i, since the round-off error is compensated as follows: i

z = C1 f + C2 f + C3 f + C4 τ + C5 τ + C6 τ

(5)

where i-1f′ and i-1τ′ are the internal force and torque vectors applied to the torque sensor i, respectively, i-1RE is the rotation matrix describing the end-effector frame E relative to link frame i-1 and i-1 rE is the position vector from the origin of link frame i-1 to the end-point E. The virtual load vector associated with joint torque sensor i can be described by i

The previous section explained the process of sensor calibration using a virtual load. This section shows how the crosstalk error can be reduced through the secondary calibration process. All the internal load (i.e., force and torque) components applied to torque sensor i have an effect on the torque sensor output. Therefore, the torque sensor output zi can be expressed by

i

zi C

(13)

6

where iτ'z is the sensed torque which includes the crosstalk error. The secondary calibration is given by C1 f ′ + C2 f ′ + C3 f ′ + C4 τ ′ + C5 τ ′ z − C6 C6 i

τ ′′ =

i

z

i

i

i

i

x

i

i

i

y

i

z

i

i

i

x

i

y

(14)

where iτ′′z is the crosstalk-compensated torque, and if′x, …, iτ′y are the virtual load components computed from iτ′z to compensate for crosstalk. We conjecture that Eq. (14) provides a more accurate torque value than Eq. (13) because the former includes the load components (i.e., if′x, …) to compensate for the crosstalk although these load components are not the same as the actual ones. This conjecture will be proved below. A comparison of the torques from Eq. (13) and (14) with the theoretically correct sensed torque iτz yields

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C1 f +
+ C6 τ τ′ − τ = τ C6 τ

i i

e =

i

i

z

p

i

i

i

y

(15)

i

z

z

C1 ( f − f ′) +
+ C6 ( τ − τ ′ ) τ ′′ − τ = τ C6 τ

i i

i

x

z

i

e =

i

i

z

s

i

i

x

z

i

i

i

x

y

i

i

y

i

z

i

f =  h1
i

i

x

 1τ  h6     6τ 

z

z

    

(17)

f ′ =  h1
i

i

x

 1τ ′    h6      6τ ′   

(18)

i

From Eqs. (17) and (18), the error associated with f′x can be expressed by

h1 ( 1τ ′ − 1τ ) +
+ h6 ( 6τ ′ − 6τ f′− f = f h1 1τ +
+ h6 6τ i

i

z

x

z

i

z

i

z

)

(19)

i

x

z

z

Substituting Eq. (15) into Eq. (19), the error associated with if′x can be obtained by i

h1 1e 1τ +
+ h6 6 e 6τ f′− f = ≤ max( e ) f h1 1τ +
+ h6 6τ i

i

x

i

p

x

i

z

p

i

z

i

(20)

p

i

x

z

Primary calibration matrix C p

Secondary calibration matrix C i

Torque sensor output z

Virtual load L

Measured torque τ΄

i

Compensated torque τ Force/torque data Estimation of external load

Primary calibration Secondary calibration

External load

Fig. 5 Sensor calibration method using virtual load

4. Experiments 4.1 Experimental setup

z

i

Encoder output θ

Encoder data

z

i

τˆ = J T F

Calibration process

where ih1, …, ih6 are the coefficients relating the joint torques to the internal load (in this case ifx). Since these coefficients depend on the manipulator configuration, the identical coefficients can be used to describe the virtual load, if′x , using τ′1, .., τ′6 as follows:

x

Usual process

(16)

z

where iep and ies are the normalized errors of the primary and secondary calibration, respectively. Note that iep represents the crosstalk error associated with iτ′z and ies is the error associated with i τ′′z after the crosstalk compensation. In order to compare iep and ies, the relationship between the internal load components applied to the torque sensors must be analyzed. One of the internal load components, say ifx, can be represented by a linear combination of joint torques as follows:

i

transparent box represents the calibration process. Each of these boxes was already detailed in the previous sections.

z

To verify the validity of the proposed calibration method, a 3DOF wrist composed of the roll, pitch, and yaw joints were constructed, as shown in Fig. 6. Each sensing frame was designed so that the strain gauges were properly bonded, as shown in Fig. 7. The torque τ1 is measured by the roll frame sensor, the torques τ2, τ 3 and the force F6 by the sensors installed at the roll-pitch connection frame and finally the torques τ 4, τ 5 by the sensors at the pitch-yaw connection frame. For the calibration based on the static force, the weight (500g) is installed at the end of the wrist, as shown in Fig. 8.

where max(iep) is the maximum value of iep for i = 1, .., 6. The crosstalk error associated with the other virtual load components can be described in the same manner. Substituting Eq. (20) into Eq. (16) yields the relationship between |iep| and |ies| as follows: i i

e ≤ s

C1 f +
+ C6 τ i

i

x

i

C6 τ

i

y

⋅ max( e ) = i

p

i

i

e ⋅ max( e ) i

p

p

(21)

z

Since both |iep| and |ies| are less than 1, we conclude that |ies| is less than or much less than |iep|. Therefore, the crosstalk error is reduced through the secondary calibration process. Furthermore, the overall error is affected by the max(iep) which is the error of the torque sensor at which the maximum crosstalk occurs.

Fig. 6 3DOF wrist composed of roll, pitch and yaw joints

3.4 Overall calibration process In this research, the new calibration method consisting of the primary and secondary calibration is proposed to cope with the crosstalk problem. The overall process of the proposed calibration scheme is illustrated in Fig. 5. The shaded box (usual process) is the real-time process for sensing of the external load, and the

(a)

(b)

(c)

Fig. 7 Wrist frames for strain gauges (a) Roll frame, (b) roll-pitch connection frame, and (c) pitch-yaw connection frame

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 4τ x   −τ1′ cosθ 4  1 fz  4       τ y  =  −τ 4′  + ( d1 + l3 )sin θ 4  0   4τ z   −τ 5′  0    

For analysis, link frames are assigned to the wrist and hand according to the D-H convention,14 as shown in Fig. 9 and Table 1.

(24)

Note that although the virtual load vector can be calculated for each torque sensor, the identical virtual load vector can be used for the torque sensors which are installed in the same rigid frame.

4.2 Experimental results In this experiment, a weight of 500 g is attached to the endeffector to obtain sample data for calibration. A total of 200 sample data (the sensor outputs and joint angles) were collected for various configurations of the manipulator. Substituting these data into Eq. (4) and (10) yields the primary and secondary calibration matrices. The torques and force (i.e., τ΄1, τ΄2, τ΄3, τ΄4, τ΄5, F΄6) which include the crosstalk error can be given by Fig. 8 Experimental setup for calibration of load sensors

0 0 0 0 0   z1   τ1′  1.31 τ ′    z  − − 0 0.92 0.05 0 0 0.16  2    2 τ 3′   0 0.03 0.86 0 0 0.01   z3   =    0 0 0.49 0 0   z4  τ 4′   0 τ ′   0 0 0 0 0.32 0   z5   5     0 0 27.42   z6   0 −4.46 1.47  F6′
  

(25)

Cp

where z1 is the sensor output of the roll frame, z2, z3, and z6 are the sensor outputs of the roll-pitch connection frame, and z4, and z5 are the sensor outputs of the pitch-yaw connection frame. The results of the secondary calibration can be described as follows:

0 0 0 −0.03 −0.06 −1.04] [τ1 ] = [
 

Fig. 9 Coordinate systems for force and torque sensors (D-H convention)

C1

 0 Fx  0   Fy   0 Fz  0   τx   0τ   y τ1′  
1

Table 1 Link parameters for wrist and hand Link #

a

α

d

θ

1

0

-90°

d1

θ1

2

0

-90°

0

90°

3

lm

-90°

0

180°

4

lt

90°

0

θ4

5

lh

0

0

θ5

In this experiment, the torque components (i.e., i-1τx, of the virtual load vector iL in Eq. (8) can be obtained by

 0τ x  cosθ1 0 − sin θ1  0     τ y  =  sin θ1 0 cosθ1   0τ z   0 0  −1  

i-1

− 0 f y   τ 3′   τ′  + d  0 f   1  1 x   0   −τ 2′   

τy,

0 0 0.01 0.03 −1.00  τ 2   0 τ  =  0 0 0 0.98 0.02 0.01   3   FA   −0.08 −1.01 0.08 −0.22 2.00 −0.17 
   C236

L

 1Fx     FA′   1Fz    (27)  τ 3′   τ′   1   −τ 2′ 
2

i-1

τz)

(22)

1

(23)

L

 4 Fx  4   Fy  τ − − 0 0 0.03 0.11 0.78 0.03  4    4 Fz    τ  = 0 0.01 0 0.18 −0.02 −0.92   4τ x   5  
    −τ ′  C45  4  −τ 5′ 
5

1

 τ x   τ 3′  1     τ y  =  τ 1′   1τ z   −τ 2′   

(26)

2

5

(28)

L

where L, L, and L are the virtual load vectors of each frame and C1, C236, and C45 are the secondary calibration matrixes computed by Eq. (10).

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In the secondary calibration matrix, the elements associated with the load causing the crosstalk have relatively large values. For example, if torque τ4 sensed in the pitch-yaw connection frame did not contain the crosstalk error, the first row of the secondary calibration matrix of (28) would be [0 0 0 0 1 0]. However, since crosstalk occurs due to 4τx , 4Fz, and so on, this row is calculated as [0 0 -0.03 0.11 -0.78 0.03]. In other words, 11% of 4τx affects τ4 as a crosstalk error. Table 2 Comparison of forces estimated by primary and secondary calibrations Roll axis angle

0°

60°

120°

180°

240°

300°

Primary calibration

3.75

4.52

Error (%)

27.9

7.7

5.12

5.71

5.38

4.45

4.5

21.4

9.7

9.1

Secondary calibration

4.54

Error (%)

7.3

4.89

5.05

4.70

4.99

5.30

0.2

3.1

4.1

1.8

8.1

A comparison of the external force after the primary calibration with that after the secondary calibration is shown in Table 2 and Fig. 10. For this comparison, a weight of 4.9N was applied to the origin of link frame 5 pointing downward, meaning the gripper grasped the weight. When only the primary calibration was conducted without the crosstalk compensation, the maximum sensor error reached 28%. However, after the secondary calibration, the maximum sensor error was significantly reduced to 8% since the crosstalk errors were minimized.

quality load sensors can be reliably used to accurately measure the external load applied at the end-effector of a manipulator.

ACKNOWLEDGEMENT This work was supported by the Center for Autonomous Intelligent Manipulation under Human Resources Development Program for Robot Specialists (Ministry of Knowledge Economy).

REFERENCES 1. Kim, Y. H., “Laboratory-level Telesurgery with industrial robots and haptics devices communicating via the internet,” Int. J. Precis. Eng. Manuf., Vol. 10, No. 2, pp. 25- 29, 2009. 2. Kim, B. H., “Modeling and Analysis of Robotic Dual Softfingered Writing,” Int. J. Precis. Eng. Manuf., Vol. 10, No. 2, pp. 17- 23, 2009. 3. Kang, C. G., “Performance Improvement of a 6-Axis Forcetorque Sensor via Novel Electronics and Cross-shaped Doublehole Structure,” Int. Journal of Control, Automation, and Systems, Vol. 3, No. 3, pp. 469- 476, 2005. 4. Kim, G. S., “The design of a six-component force/moment sensor and evaluation of its uncertainty,” Measurement Science and Technology, Vol. 12, No. 9, pp. 1445-1455, 2001. 5. Shimano, B. E., “Kinematic design and force control of computer controlled manipulators,” Artificial Intelligence Lab., Stanford Univ., AI Memo No. 313, 1978. 6. Bicchi, A., “A Criterion for Optimal Design of Multiaxis Force Sensors,” Journal of Robotics and Autonomous Systems, Vol. 10, No. 4, pp. 269-286, 1992.

Fig. 10 Experimental results showing external force estimated by primary and secondary calibrations

5. Conclusion In this research, a new calibration scheme composed of both primary and secondary calibration processes was proposed to cope with the problem of crosstalk. Using the primary calibration matrix, the virtual load acting on each joint torque sensor can be computed. This virtual load contains the components which cause the crosstalk error. Then the desired joint torque, which is free of the crosstalk error, can be obtained by the secondary calibration, which is computed from the virtual load. Using the proposed calibration method, the error in the load sensing was significantly reduced, which means that relatively low-

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