Tool Position Estimation of a Flexible Industrial Robot ...

Tool Position Estimation of a Flexible IndustrialRobot using Recursive Bayesian Methods

Patrik Axelsson, Rickard Karlsson, andMikael Norrlöf

Division of Automatic ControlDepartment of Electrical EngineeringLinköping University, Sweden

Patrik Axelsson, Rickard Karlsson, and Mikael Norrlöf

Tool Position Estimation of a Flexible Industrial Robot using Recursive Bayesian Methods

AUTOMATIC CONTROLREGLERTEKNIK

LINKÖPINGS UNIVERSITET

Outline 2(12)

1. Introduction• Problem Formulation• Bayesian Estimation

2. Modelling• Robot Model• Accelerometer Model• Estimation Model

3. Experimental Results

4. Conclusions and Future Work





Problem Formulation 3(12)

Controller Robot

Observer

qrefm u

d

X

qmXref

Can control

Want to measure

Want to control the TCP. Can only measure the motor angles qm.

Feedback of qm does not work sufficiently due to flexible joints.

If the TCP can be measured it is natural to feedback it.

What can we do instead of measuring the TCP?






Controller Robot

Observer

qrefm u

d

X

qmXref

Want to control

Can measure










Controller Robot

Observer

Xref u

d

X

qm










Controller Robot

Observer

Xref u

d

X

qm

Let the acceleration of the tool be a measurement.

Use an observer to estimate the TCP.

How can we estimate the TCP?






Controller Robot

Observer

Xref u

d

X

qmX









Controller Robot

Observer

Xref u

d

X

qmX

X








Bayesian State-space Estimation 5(12)

Model:

xt+1 = f (xt, ut, wt),

yt = h(xt) + et.

Bayesian inference:

p(xt+1|Yt) =∫

Rnp(xt+1|xt)p(xt|Yt)dxt,

p(xt|Yt) =p(yt|xt)p(xt|Yt−1)

p(yt|Yt−1).

The Kalman filter is the optimal choice for linear models.Approximative filters have to be used for nonlinear models.

In this work

• Extended Kalman filter (EKF)– Approximate the system with a linearisation of the nonlinear

equations.– Assume additive Gaussian noise.

• Particle filter (PF)– Approximate the posterior distribution with a large number of

particles.– The optimal proposal distribution approximated using an EKF.






Model:


yt = h(xt) + et.

Bayesian inference:

p(xt+1|Yt) =∫



p(yt|Yt−1).

The Kalman filter is the optimal choice for linear models.Approximative filters have to be used for nonlinear models.In this work• Extended Kalman filter (EKF)

– Approximate the system with a linearisation of the nonlinearequations.

– Assume additive Gaussian noise.

• Particle filter (PF)– Approximate the posterior distribution with a large number of

particles.– The optimal proposal distribution approximated using an EKF.






Model:


yt = h(xt) + et.

Bayesian inference:

p(xt+1|Yt) =∫



p(yt|Yt−1).

The Kalman filter is the optimal choice for linear models.Approximative filters have to be used for nonlinear models.In this work• Extended Kalman filter (EKF)

– Approximate the system with a linearisation of the nonlinearequations.

– Assume additive Gaussian noise.• Particle filter (PF)

– Approximate the posterior distribution with a large number ofparticles.

– The optimal proposal distribution approximated using an EKF.





Robot Model 6(12)

Serial robot with 2 DOF.

Linear stiffness.

No friction.

qa =(qa1 qa2

)T ,

qam =

(qm1/η1 qm2/η2

)T ,

τam =

(τm1η1 τm2η2

)T ,

bx

bz

1aq

2aq

SxSz

bρ

Ma(qa)qa + C(qa, qa) + G(qa) + T(qa − qam) + D(qa − qa

m) = 0,Mmqa

m − T(qa − qam)−D(qa − qa

m) = τam.





Accelerometer Model 7(12)

Acceleration from the motion.

Acceleration due to thegravity.

Rotation matrix from Oxbzb toOxszs.

Bias parameter. bx

bz

1aq

2aq

SxSz

bρ

ρs(qa) = Rb/s (qa) (ρb(qa) + Gb) + bACC





Estimation Model 8(12)

State space vector:

x =(xT

1 xT2 xT

3)T

=(qT

a qTa qT

a)T

Linear dynamic model (Double integrator in discrete time):

xk+1 = Fxk + Guuk + Gvvk

The input signal is the jerk of the arm angle reference.

Measurement equation:

yk =

(x1,k + K−1 (Ma(x1,k)x3,k + C(x1,k, x2,k) + G(x1,k))

Rb/s(x1,k)(

JACC(x1,k)x3,k +(

ddt JACC(x1,k)

)x2,k + Gb

))+ ek





Experimental Setup 9(12)

Experiments performed on an ABB IRB4600.

Only joints 2 and 3 are used.

The true path is measured by a laser system from Leica.

Synchronization errors and kinematic errors are present.

EKF and PF are compared to the forward kinematic usingmeasured motor angles, TTCP(qm).

1.1 1.2 1.3 1.40.75

0.85

0.95

1.05

x [m]

z[m

]





Experimental Results 10(12)

Both filters follow the true path.

The EKF passes in the corners.

TTCP(qm) out of phase.

Bias states for both motor andaccelerometer measurements arerequired to get good results.Root Mean Square Error

EKF PF TTCP(qm)

RMSE 0.124 0.083 0.167

The current MATLAB implementationnot in real time for the EKF and PF.

• True path • EKF • PF • TTCP(qm)

0.998

1.002

1.006

z[m

]

1.15 1.2 1.25 1.3 1.350.795

0.8

0.805

x [m]

z[m

]

0 0.2 0.4 0.6 0.8 1.0 1.2 1.40

1

2

3

4

5

6

Time [s]

RM

SE

[mm

]





Conclusions and Future Work 11(12)

Conclusions

Estimation of the robot’s TCP using Bayesian methods and anaccelerometer.

Experimental evaluation of an EKF and PF on an ABB IRB4600.

The Bayesian methods show significant improvement.

Future Work

Sensitivity analysis of model parameters.

Time varying covariance matrices for the EKF to handle sharpturns.

Extend to 6 DOF. More sensors? Where to place the sensors?

Control using the estimated position.





Tool Position Estimation of a Flexible IndustrialRobot using Recursive Bayesian Methods

Patrik Axelsson, Rickard Karlsson, andMikael Norrlöf

Division of Automatic ControlDepartment of Electrical EngineeringLinköping University, Sweden





Tool Position Estimation of a Flexible Industrial Robot ...

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