Torque sensor calibration using virtual load for a manipulator

INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 11, No. 2, pp. 219-225 APRIL 2010 / 219

DOI: 10.1007/s12541-010-0025-0

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 tasks.1,2 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

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

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 low-

quality 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

© KSPE and Springer 2010

220 / APRIL 2010 INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 11, No. 2

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.

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. 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

2

2

xF

t

w

γ δ= (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 =

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.

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.

p

C z τ⋅ = (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.

ˆT

J Fτ = ⋅ (3)

where τ 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:

2

1

ˆ( )minp

M

p p m mC m

C C z τ

=

= −∑ (4)

where M is the number of samples used for calibration, and zm and

ˆm

τ are the sensor output vector and estimated joint torque vector of

INTERNATIONAL JOURNAL OF PRECISION ENGINEERING AND MANUFACTURING Vol. 11, No. 2 APRIL 2010 / 221

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 ˆm

τ 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.

iτ ′

1τ ′

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:

1 1i i

Ef R F− −

′ ′= ⋅ (5)

1 1 1 1 1( )i i i i i

E E Eτ r R F r f

− − − − −

′ ′ ′= × ⋅ = × (6)

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-1rE 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

1 1 1 1 1 1[ ]

i i i i i i i T

x y z x y zL f f f τ τ τ− − − − − −

′ ′ ′ ′ ′ ′= (7)

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:

1 1 1 1 1[ ]

i i i i i i T

x y z x y iL f f f τ τ τ− − − − −

′ ′ ′ ′ ′ ′= (8)

The relationship between the virtual load iL and the desired

torque τi at torque sensor i can be described by

i i

iC L τ⋅ = (9)

where iC is the 1x6 secondary calibration matrix associated with the

virtual load at torque sensor i. The matrix iC can be obtained by

2

,

1

ˆ( )mini

M

i i i

m i m

C m

C C L τ

=

= ⋅ −∑ (10)

where M is the number of samples used for calibration, iLm is the

virtual load vector for sample m, and ,i m

τ is the torque of sample m.

Each sample is obtained by moving the manipulator for a given

external load F. iτ corresponds to the i-th element of the joint

torque vector ˆ,τ 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).

3.3 Error analysis of sensor calibration using virtual load

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

1 2 3 4 5 6

i i i i i i i i i i i i

i x y z x y zz C f C f C f C C Cτ τ τ= + + + + + (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

1 2 3 4 5

6 6

x z

i i i i i i i i i i

x y z x yi i

z i i

C f C f C f C Cz

C C

τ τ

τ

+ + + +

= − (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:

6

i i

z i

z

Cτ ′ = (13)

where iτ'z is the sensed torque which includes the crosstalk error.

The secondary calibration is given by

1 2 3 4 5

6 6

i i i i i i i i i i

x y z x yi i

z i i

C f C f C f C Cz

C C

τ τ

τ

′ ′ ′ ′ ′+ + + +′′ = − (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


1 6

6

i i i ii ix yi z z

p i i i

z z

C f Ce

C

ττ τ

τ τ

+ +′ −= =

�

(15)

1 6

6

( ) ( )i i i i i ii ix x y yi z z

s i i i

z z

C f f Ce

C

τ ττ τ

τ τ

′ ′− + + −′′ −= =

�

(16)

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 iτ′′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:

1

1 6

6

z

i i i

x

z

f h h

τ

τ

=

� � (17)

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:

1

1 6

6

z

i i i

x

z

f h h

τ

τ

′

′ = ′

� � (18)

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

expressed by

( ) ( )1 1 6 6

1 6

1 6

1 6

i ii i

z z z zx x

i i i

x z z

h hf f

f h h

τ τ τ τ

τ τ

′ ′− + + −′ −=

+ +

�

�

(19)

Substituting Eq. (15) into Eq. (19), the error associated with if′x

can be obtained by

1 1 6 6

1 6

1 6

1 6

max( )

i ii ip z p z ix x

pi i i

x z z

h e h ef fe

f h h

τ τ

τ τ

+ +′ −= ≤

+ +

�

�

(20)

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:

1 6

6

max( ) max( )

i i i i

x yi i i i

s p p pi i

z

C f Ce e e e

C

τ

τ

+ +

≤ ⋅ = ⋅

�

(21)

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.

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

transparent box represents the calibration process. Each of these

boxes was already detailed in the previous sections.

External load

Force/torque data

Encoder data

Primary calibration

Secondary calibration

Usual process

Calibration process

Virtual load

iL

Compensated torque τ

Encoder output θ

Estimation of external load

Primary calibration matrix Cp Secondary

calibration matrix Ci

Measured torque τ΄

FJτT

=ˆ

Torque sensor output

z

Fig. 5 Sensor calibration method using virtual load

4. Experiments

4.1 Experimental setup

To verify the validity of the proposed calibration method, a 3-

DOF 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.

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

(a) (b) (c)

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

connection frame, and (c) pitch-yaw connection frame


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.

Fig. 8 Experimental setup for calibration of load sensors

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

convention)

Table 1 Link parameters for wrist and hand

Link # a α d θ

1 0 -90° d1 θ 1

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,

i-1τy,

i-1τz)

of the virtual load vector iL in Eq. (8) can be obtained by

0 0

1 1 3

0 0

1 1 1 1

0

2

cos 0 sin

sin 0 cos

0 1 0 0

x y

y x

z

f

d f

τ θ θ τ

τ θ θ τ

τ τ

′ − − ′= + ′− −

(22)

1

3

1

1

1

2

x

y

z

τ τ

τ τ

τ τ

′ ′= ′−

(23)

4 1

1 4

4

4 1 3 4

4

5

cos

( )sin 0

0

x z

y

z

f

d l

τ τ θ

τ τ θ

τ τ

′ − ′= − + + ′−

(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 end-

effector 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

1 1

2 2

3 3

4 4

5 5

6 6

1.31 0 0 0 0 0

0 0.92 0.05 0 0 0.16

0 0.03 0.86 0 0 0.01

0 0 0 0.49 0 0

0 0 0 0 0.32 0

0 4.46 1.47 0 0 27.42

p

z

z

z

z

z

F z

τ

τ

τ

τ

τ

′ ′ − − ′

= ′

′

′ − C

��

(25)

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:

[ ] [ ]

�

1

1

0

0

0

1 0

0

1

0 0 0 0.03 0.06 1.04

x

y

z

x

y

F

F

Fτ

τ

τ

τ

= − − −

′

C

L

��

(26)

�

236

2

1

2 1

3

3

1

2

0 0 0 0.01 0.03 1.00

0 0 0 0.98 0.02 0.01

0.08 1.01 0.08 0.22 2.00 0.17

x

A

z

A

F

F

F

F

τ

τ

τ

τ

τ

′ − = ′ − − − − ′

′ − C

L

��

(27)

�

45

5

4

4

4

4

4

5

4

5

0 0 0.03 0.11 0.78 0.03

0 0.01 0 0.18 0.02 0.92

x

y

z

x

F

F

Fτ

τ τ

τ

τ

− −

= − − ′−

′ −

C

L

��

(28)

where 1L, 2L, and 5L are the virtual load vectors of each frame and

C1, C236, and C45 are the secondary calibration matrixes computed

by Eq. (10).


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 5.12 5.71 5.38 4.45

Error (%) 27.9 7.7 4.5 21.4 9.7 9.1

Secondary calibration 4.54 4.89 5.05 4.70 4.99 5.30

Error (%) 7.3 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.

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-

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).

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Torque sensor calibration using virtual load for a manipulator

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