Robot manipulators trends and development 2010 Part 5 doc

Measurement Analysis and Diagnosis for Robot Manipulators using Advanced Nonlinear Control Techniques 153diagnose sensor biases in nonlinear systems, such as Vemuri, 2001; Wang et al., 1

diagnose sensor biases in nonlinear systems, such as (Vemuri, 2001); (Wang et al., 1997), is the

ability to diagnose piecewise constant bias with the same observer Moreover, the proposed

approach is not limited to sensor biases and can be used to diagnose measurement errors of

any harmonics

5 Measurement Error Identification for Low and High Frequencies

We now consider measurement errors of low frequencies determined by a cutoff frequency

ω l The SISO weighting ˆw l(s) = as+b s , (Zhou & Doyle, 1998), emphasizes this range with “b”

selected as ω l and “a” as an arbitrary small number for the magnitude of ˆw l(jω)as ω →∞

With a diagonal transfer matrix ˆW(s)that consists of these SISO weightings (and similar to

the approach adopted in section 4.1), the detection and identification objectives can be

com-bined in the unified framework represented by the weighted setup of Fig 5 In this case, the

augmented plant ¯G is given by:

where A θ=0p , B θ =I p , C θ =diag p(b)and D θ =diag p(a) This form also violates the assumptions

of Theorem 1 (note that(A, ¯B¯ 2) is not stabilizable) Similar to Section 4, we introduce the

modified weighting ˆw lmod(s)=as+b

s+λ ; with arbitrary small positive “λ” The augmented plant

¯

G is then the same as (41) except for A θ which is now given by the stable matrix diag p(− λ)

and C θ given by diag p(b − aλ) Similar to the narrow frequency band case, the assumptions

of Theorem 1 are now satisfied and the LMI approach in (Gahinet & Apkarian, 1994) can be

used to solve the H∞problem To this end, we define the H∞problem associated with the low

frequency range as follows:

Definition 7. (Low frequency H∞) Given λ > 0,  > 0, find S , the set of admissible controllers K

satisfying  ˆT ζ ¯τ ∞< γ for the setup in Fig 5 where ¯ G has the state space representation (41) with

A θ=diag p( − λ), B θ=I p , C θ=diag p( b − aλ)and D θ=diag p( a).

Based on all the above, we now present the main result of this section in the form of the

following definition for an optimal residual generator inL2sense:

Definition 8. (Optimal residual for low frequencies) An observer of the form (8)-(12) is an optimal

residual generator for the measurement error identification problem (with low frequency measurement

errors below the cutoff frequency ω l ) if the dynamic gain K ∈ S ∗ (the set of controllers solving the H∞

problem in Definition 7 for γ=1/α with the minimum possible λ).

Similar to the low frequency range, a proper weighting ˆw hmod(s) = s+(a×b) λs+b , (Zhou & Doyle,

1998), with an arbitrary small λ >0, could be selected to emphasize the high frequency range

[w h, ∞)with “b” selected as w h and “a” as an arbitrary small number for | ˆw h(jω)| as ω →0

With the help of ˆw hmod(s), a suitable weighting W that emphasizes the high frequency range

can be designed The augmented ¯G is also given from (41) (same as the low frequency case),

but with A θ , B θ , C θ and D θ given as diag p(− λ b), I p , diag p( a×b λ − λ b2)and diag p(1λ)

respec-tively It is straightforward that ¯G satisfies all of the assumptions of Theorem 1 and therefore,

similar to the low frequency range, an H∞problem related to the high frequency range can be

defined An optimal residual generator can be defined in the same way as Definition 8 for the

generalized low frequency case

6 Experimental Results

The experimental results presented in this section (Pertew, 2006) are intended to illustrate theapplicability of the theoretical results presented in this chapter for robotic systems

6.1 The ROTPEN: Models and Assumptions

The Quanser rotary inverted pendulum (ROTPEN) is shown schematically in Fig 6, Lynch (2004) The angle that the perfectly rigid link of length l1and inertia J1makes with the x-axis

of an inertial frame is denoted θ1(degrees) Also, the angle of the pendulum (of length l2and

mass m2) from the z-axis of the inertial frame is denoted θ2(degrees)

Fig 6 The Rotary Inverted Pendulum (ROTPEN)

The system has one input which is the scalar servomotor voltage input (Volt) Therefore, thesystem is a special case of the robot manipulator model discussed in Section 1: a planar robot

manipulator with two links (n=2), with only one torque applied at the first joint, while thesecond joint is subject to the gravitational force In fact, the ROTPEN has a state space model

of the form ˙x= f(x) +g(x)u, where x= [θ1θ2 ˙θ1 ˙θ2]T is the state vector, and u is the scalar

servomotor voltage input (Volt) More details about this model and its parameters can befound in Appendix 9.1

The system has an infinite number of equilibrium points, representing the following two librium points:

equi-1) Pendant position: x1=0 (rad), x2=π (rad), x3=x4=0 (rad/sec)

2) Inverted position: x1=x2=0 (rad), x3=x4=0 (rad/sec)

By separating the nonlinear terms, the model can be put in the form ˙x=Ax+Φ(x, u), where:

Measurement Analysis and Diagnosis for Robot Manipulators using Advanced Nonlinear Control Techniques 153

diagnose sensor biases in nonlinear systems, such as (Vemuri, 2001); (Wang et al., 1997), is the

ability to diagnose piecewise constant bias with the same observer Moreover, the proposed

approach is not limited to sensor biases and can be used to diagnose measurement errors of

any harmonics

5 Measurement Error Identification for Low and High Frequencies

We now consider measurement errors of low frequencies determined by a cutoff frequency

ω l The SISO weighting ˆw l(s) = as+b s , (Zhou & Doyle, 1998), emphasizes this range with “b”

selected as ω l and “a” as an arbitrary small number for the magnitude of ˆw l(jω)as ω →∞

With a diagonal transfer matrix ˆW(s)that consists of these SISO weightings (and similar to

the approach adopted in section 4.1), the detection and identification objectives can be

com-bined in the unified framework represented by the weighted setup of Fig 5 In this case, the

augmented plant ¯G is given by:

where A θ=0p , B θ =I p , C θ =diag p(b)and D θ =diag p(a) This form also violates the assumptions

of Theorem 1 (note that(A, ¯B¯ 2)is not stabilizable) Similar to Section 4, we introduce the

modified weighting ˆw lmod(s)=as+b

s+λ ; with arbitrary small positive “λ” The augmented plant

¯

G is then the same as (41) except for A θ which is now given by the stable matrix diag p( − λ)

and C θ given by diag p(b − aλ) Similar to the narrow frequency band case, the assumptions

of Theorem 1 are now satisfied and the LMI approach in (Gahinet & Apkarian, 1994) can be

used to solve the H∞problem To this end, we define the H∞problem associated with the low

frequency range as follows:

Definition 7. (Low frequency H∞) Given λ > 0,  > 0, find S , the set of admissible controllers K

satisfying  ˆT ζ ¯τ ∞< γ for the setup in Fig 5 where ¯ G has the state space representation (41) with

A θ=diag p( − λ), B θ=I p , C θ=diag p(b − aλ)and D θ=diag p( a).

Based on all the above, we now present the main result of this section in the form of the

following definition for an optimal residual generator inL2sense:

Definition 8. (Optimal residual for low frequencies) An observer of the form (8)-(12) is an optimal

residual generator for the measurement error identification problem (with low frequency measurement

errors below the cutoff frequency ω l ) if the dynamic gain K ∈ S ∗ (the set of controllers solving the H∞

problem in Definition 7 for γ=1/α with the minimum possible λ).

Similar to the low frequency range, a proper weighting ˆw hmod(s) = s+(a×b) λs+b , (Zhou & Doyle,

1998), with an arbitrary small λ >0, could be selected to emphasize the high frequency range

[w h, ∞)with “b” selected as w h and “a” as an arbitrary small number for | ˆw h(jω)| as ω →0

With the help of ˆw hmod(s), a suitable weighting W that emphasizes the high frequency range

can be designed The augmented ¯G is also given from (41) (same as the low frequency case),

but with A θ , B θ , C θ and D θ given as diag p( − b λ), I p , diag p( a×b λ − λ b2)and diag p( λ1)

respec-tively It is straightforward that ¯G satisfies all of the assumptions of Theorem 1 and therefore,

similar to the low frequency range, an H∞problem related to the high frequency range can be

defined An optimal residual generator can be defined in the same way as Definition 8 for the

generalized low frequency case

6 Experimental Results

The experimental results presented in this section (Pertew, 2006) are intended to illustrate theapplicability of the theoretical results presented in this chapter for robotic systems

6.1 The ROTPEN: Models and Assumptions

The Quanser rotary inverted pendulum (ROTPEN) is shown schematically in Fig 6, Lynch (2004) The angle that the perfectly rigid link of length l1and inertia J1makes with the x-axis

of an inertial frame is denoted θ1(degrees) Also, the angle of the pendulum (of length l2and

mass m2) from the z-axis of the inertial frame is denoted θ2(degrees)

Fig 6 The Rotary Inverted Pendulum (ROTPEN)

The system has one input which is the scalar servomotor voltage input (Volt) Therefore, thesystem is a special case of the robot manipulator model discussed in Section 1: a planar robot

manipulator with two links (n=2), with only one torque applied at the first joint, while thesecond joint is subject to the gravitational force In fact, the ROTPEN has a state space model

of the form ˙x= f(x) +g(x)u, where x= [θ1θ2 ˙θ1 ˙θ2]T is the state vector, and u is the scalar

servomotor voltage input (Volt) More details about this model and its parameters can befound in Appendix 9.1

The system has an infinite number of equilibrium points, representing the following two librium points:

equi-1) Pendant position: x1=0 (rad), x2=π (rad), x3=x4=0 (rad/sec)

2) Inverted position: x1=x2=0 (rad), x3=x4=0 (rad/sec)

By separating the nonlinear terms, the model can be put in the form ˙x=Ax+Φ(x, u), where:

mainly trigonometric terms, and using the symbolic MATLAB toolbox, an upper bound on

Φ(x, u) is found as 44.45, and hence the Lipschitz constant for the ROTPEN is α =44.45

This follows from the fact that if Φ : n × → mis continuously differentiable on a domain

D and the derivative of Φ with respect to the first argument satisfies  ∂Φ

∂x  ≤ α on D, then Φ

is Lipschitz continuous on D with constant α, i.e.:

Φ(x, u)−Φ(y, u) ≤ α  x − y , ∀ x, y ∈ D (42)

There are two encoders to measure the angle of the servomotor output shaft (θ1) and the angle

of the pendulum (θ2) An encoder is also available to measure the motor velocity ˙θ1, but

no one is available to measure the pendulum velocity ˙θ2 In the experiments, linear as well as

nonlinear control schemes are used to stabilize the pendulum at the inverted position (θ2=0),

while tracking a step input of 30 degrees for the motor angle

6.2 Case Study 1 - Lipschitz Observer Design

In this experiment, we focus on the nonlinear state estimation problem when no

measure-ment errors are affecting the system We consider situations in which the operating range of

the pendulum is either close or far from the equilibrium point, comparing the Luenberger

ob-server with the Lipschitz obob-server in these cases For the purpose of applying the Lipschitz

observer design, the nonlinear model discussed in section 6.1 is used We also compare the

dynamic Lipschitz observer of section 3 with the static design method in Reference (Raghavan

& Hedrick, 1994) In this case study the full-order linear and Lipschitz models are used for

observer design, where the output is assumed as y = [x1 x2]T (all the observer parameters

that are used in this experiment can be found in Appendix 9.2)

First, a linear state feedback controller is used to stabilize the system in a small operating

range around the inverted position, and three observers are compared:

1) Observer 1: A linear Luenberger observer where the observer gain is obtained by

plac-ing the poles of (A − LC) at{−24, −3.8, −4.8, −12.8} (see L 3−small in Appendix

9.2)

2) Observer 2: A high gain Luenberger observer, which has the same form of Observer

1 but with the poles placed at{−200, −70, −20+15i, −20− 15i } (see L 3−largein

Appendix 9.2)

3) Observer 3: A Lipschitz observer of the form (8)-(11), based on the full-order Lipschitz

model of the ROTPEN The dynamic gain is computed using the design procedure in

section 3.1, for α=44.45 (see K3in Appendix 9.2)

The three observers run successfully with stable estimation errors Table 1 shows the

maxi-mum estimation errors in this case It can be seen that both the Luenberger observer (large

poles) and the Lipschitz observer achieve comparable performance, which is much better than

the Luenberger observer with small poles The three observers are also tested in

observer-based control, and their tracking performance is compared in Table 2 We conclude that, due

to the small operating range considered in this case study, a high-gain Luenberger observer

achieves a good performance in terms of the state estimation errors and the tracking errors

We then consider a large operating range by using a nonlinear control scheme that stabilizes

the pendulum angle at the pendant position (see Appendix 9.2 for more details about the

controller used in this case study) Using this controller, a large operating range is obtained

as seen in Fig 7 The same observers (Observers 2 and 3) are used in parallel with this control

scheme, and the resulting estimation errors are compared in Fig 8 The two observers are also

Small-gain Luenberger High-gain Luenberger Lipschitz

Table 1 Case study 1 - Estimation errors “e1” and “e2” in degrees

pure state feedback High-gain Luenberger Lipschitz

−40

−20 0 20 40 60 80 100 120 140 160

time (sec)

(b)

Fig 7 Case Study 1 - (a) Motor Response, (b) Pendulum Response

Finally, we conduct a comparison between static and dynamic Lipschitz observers, namely theobserver (6)-(7) and the one in (8)-(11) The comparison is between the new design proposed

in Section 3 and the one in Reference (Raghavan & Hedrick, 1994) First, the design algorithm

in (Raghavan & Hedrick, 1994) is tested for different values of α and ε It fails for all values

of α > 1, and the maximum attainable value is α = 1 (see L5 in Appendix 9.2), while theLipschitz constant of the ROTPEN model is 44.45 as mentioned earlier This observer is then

compared to the dynamic Lipschitz observer having the dynamic gain K3, and the estimationerrors are shown in Fig 10 It is also important to note that the static Lipschitz observer fails

in stabilizing the system, when used in observer-based control, for both the small and largeoperating range experiments This shows the importance of the dynamic Lipschitz observerdesign in this case

Measurement Analysis and Diagnosis for Robot Manipulators using Advanced Nonlinear Control Techniques 155

mainly trigonometric terms, and using the symbolic MATLAB toolbox, an upper bound on

Φ(x, u) is found as 44.45, and hence the Lipschitz constant for the ROTPEN is α =44.45

This follows from the fact that if Φ : n × → mis continuously differentiable on a domain

D and the derivative of Φ with respect to the first argument satisfies  ∂Φ

∂x  ≤ α on D, then Φ

is Lipschitz continuous on D with constant α, i.e.:

Φ(x, u)−Φ(y, u) ≤ α  x − y , ∀ x, y ∈ D (42)

There are two encoders to measure the angle of the servomotor output shaft (θ1) and the angle

of the pendulum (θ2) An encoder is also available to measure the motor velocity ˙θ1, but

no one is available to measure the pendulum velocity ˙θ2 In the experiments, linear as well as

nonlinear control schemes are used to stabilize the pendulum at the inverted position (θ2=0),

while tracking a step input of 30 degrees for the motor angle

6.2 Case Study 1 - Lipschitz Observer Design

In this experiment, we focus on the nonlinear state estimation problem when no

measure-ment errors are affecting the system We consider situations in which the operating range of

the pendulum is either close or far from the equilibrium point, comparing the Luenberger

ob-server with the Lipschitz obob-server in these cases For the purpose of applying the Lipschitz

observer design, the nonlinear model discussed in section 6.1 is used We also compare the

dynamic Lipschitz observer of section 3 with the static design method in Reference (Raghavan

& Hedrick, 1994) In this case study the full-order linear and Lipschitz models are used for

observer design, where the output is assumed as y = [x1 x2]T (all the observer parameters

that are used in this experiment can be found in Appendix 9.2)

First, a linear state feedback controller is used to stabilize the system in a small operating

range around the inverted position, and three observers are compared:

1) Observer 1: A linear Luenberger observer where the observer gain is obtained by

plac-ing the poles of(A − LC)at{−24, −3.8, −4.8, −12.8} (see L 3−small in Appendix

9.2)

2) Observer 2: A high gain Luenberger observer, which has the same form of Observer

1 but with the poles placed at{−200, −70, −20+15i, −20− 15i } (see L 3−largein

Appendix 9.2)

3) Observer 3: A Lipschitz observer of the form (8)-(11), based on the full-order Lipschitz

model of the ROTPEN The dynamic gain is computed using the design procedure in

section 3.1, for α=44.45 (see K3in Appendix 9.2)

The three observers run successfully with stable estimation errors Table 1 shows the

maxi-mum estimation errors in this case It can be seen that both the Luenberger observer (large

poles) and the Lipschitz observer achieve comparable performance, which is much better than

the Luenberger observer with small poles The three observers are also tested in

observer-based control, and their tracking performance is compared in Table 2 We conclude that, due

to the small operating range considered in this case study, a high-gain Luenberger observer

achieves a good performance in terms of the state estimation errors and the tracking errors

We then consider a large operating range by using a nonlinear control scheme that stabilizes

the pendulum angle at the pendant position (see Appendix 9.2 for more details about the

controller used in this case study) Using this controller, a large operating range is obtained

as seen in Fig 7 The same observers (Observers 2 and 3) are used in parallel with this control

scheme, and the resulting estimation errors are compared in Fig 8 The two observers are also

Small-gain Luenberger High-gain Luenberger Lipschitz

Table 1 Case study 1 - Estimation errors “e1” and “e2” in degrees

pure state feedback High-gain Luenberger Lipschitz

−40

−20 0 20 40 60 80 100 120 140 160

time (sec)

(b)

Fig 7 Case Study 1 - (a) Motor Response, (b) Pendulum Response

Finally, we conduct a comparison between static and dynamic Lipschitz observers, namely theobserver (6)-(7) and the one in (8)-(11) The comparison is between the new design proposed

in Section 3 and the one in Reference (Raghavan & Hedrick, 1994) First, the design algorithm

in (Raghavan & Hedrick, 1994) is tested for different values of α and ε It fails for all values

of α > 1, and the maximum attainable value is α = 1 (see L5 in Appendix 9.2), while theLipschitz constant of the ROTPEN model is 44.45 as mentioned earlier This observer is then

compared to the dynamic Lipschitz observer having the dynamic gain K3, and the estimationerrors are shown in Fig 10 It is also important to note that the static Lipschitz observer fails

in stabilizing the system, when used in observer-based control, for both the small and largeoperating range experiments This shows the importance of the dynamic Lipschitz observerdesign in this case

−50 0 50 100 150 200

time (sec)

(b)

state feedback Lipschitz observer−based feedback

Fig 9 Case Study 1 - (a) Pendulum Angle, (b) Motor Angle

6.3 Case Study 2 - Lipschitz Measurement Error Diagnosis

In this experiment, the results of Sections 4 and 5 are assessed on the nonlinear Lipschitz

model A large operating range is considered by using a nonlinear, switching, LQR control

scheme (with integrator) that stabilizes the pendulum at the inverted position (starting from

the pendant position) while tracking a step input of 30 degrees for the motor angle as seen in

Fig 11 (the no-bias case) In the first part of this experiment, an important measurement error

that affects the ROTPEN in real-time is considered This is a sensor fault introduced by the

pendulum encoder The encoder returns the pendulum angle relative to the initial condition,

assuming this initial condition to be θ2 = 0 This constitutes a source of bias, as shown in

Fig 11(b), when the pendulum initial condition is unknown or is deviated from the inverted

position The effect of this measurement error on the tracking performance is also illustrated

in Fig 11(a) for two different bias situations The dynamic Lipschitz observer (discussed in

section 4) is applied to diagnose and tolerate this fault In addition to this bias fault, the

observer is also applied for a 2 rad/sec fault introduced in real-time, as well as for the case of

a low frequency fault in the range[0, 1 rad/sec]

−20 0 20 40 60 80 100

time (sec)

No bias Small bias Large bias

0 5 10 15 20 25 30 35 40 45 50

−30

−20

−10 0 10 20 30 40

time (sec)

No bias Small bias

bias= −8.965

bias= −13.623

Fig 11 Case Study 2 - (a) Tracking Performance, (b) Pendulum Angle

First, the design procedure in section 4 is used to accurately estimate and tolerate the bias

faults shown in Fig 11(b) This is the special case where ω o = 0 Using the reduced-order

Lipschitz model with α=44.45 (and using the LMI design procedure, the dynamic gain for

the observer (8)-(12) that achieves measurement error identification is obtained as K6(see pendix 9.3 for more details) Using this observer, the biases affecting the system in Fig 11 aresuccessfully estimated as shown in Fig 12 Moreover, by using this observer in an observer-based control scheme, the tracking performance in the large bias case is illustrated in Fig 13.The performance is much improved over the one with no fault tolerance as seen in Fig 13(b)

Ap-It also gives less overshoot than the no bias case, as seen in Fig 13(a) Similar results areobtained for the small bias case

The case of measurement error in the form of harmonics is now considered, with a sensorfault having a frequency of 2 rad/sec The dynamic gain for the observer (8)-(12) is computed

using the design approach discussed in section 5 This is the special case where ω o=2 The

gain is obtained at λ=10−12 as K7(see Appendix 9.3) Using this observer, Fig 14 shows thecorrect estimation of a measurement error of amplitude 20 degrees and frequency 2 rad/sec

Measurement Analysis and Diagnosis for Robot Manipulators using Advanced Nonlinear Control Techniques 157

−100

−50 0 50 100 150 200

time (sec)

(b)

state feedback Lipschitz observer−based feedback

Fig 9 Case Study 1 - (a) Pendulum Angle, (b) Motor Angle

6.3 Case Study 2 - Lipschitz Measurement Error Diagnosis

In this experiment, the results of Sections 4 and 5 are assessed on the nonlinear Lipschitz

model A large operating range is considered by using a nonlinear, switching, LQR control

scheme (with integrator) that stabilizes the pendulum at the inverted position (starting from

the pendant position) while tracking a step input of 30 degrees for the motor angle as seen in

Fig 11 (the no-bias case) In the first part of this experiment, an important measurement error

that affects the ROTPEN in real-time is considered This is a sensor fault introduced by the

pendulum encoder The encoder returns the pendulum angle relative to the initial condition,

assuming this initial condition to be θ2 = 0 This constitutes a source of bias, as shown in

Fig 11(b), when the pendulum initial condition is unknown or is deviated from the inverted

position The effect of this measurement error on the tracking performance is also illustrated

in Fig 11(a) for two different bias situations The dynamic Lipschitz observer (discussed in

section 4) is applied to diagnose and tolerate this fault In addition to this bias fault, the

observer is also applied for a 2 rad/sec fault introduced in real-time, as well as for the case of

a low frequency fault in the range[0, 1 rad/sec]

−20 0 20 40 60 80 100

time (sec)

No bias Small bias Large bias

0 5 10 15 20 25 30 35 40 45 50

−30

−20

−10 0 10 20 30 40

time (sec)

No bias Small bias

bias= −8.965

bias= −13.623

Fig 11 Case Study 2 - (a) Tracking Performance, (b) Pendulum Angle

First, the design procedure in section 4 is used to accurately estimate and tolerate the bias

faults shown in Fig 11(b) This is the special case where ω o = 0 Using the reduced-order

Lipschitz model with α=44.45 (and using the LMI design procedure, the dynamic gain for

the observer (8)-(12) that achieves measurement error identification is obtained as K6(see pendix 9.3 for more details) Using this observer, the biases affecting the system in Fig 11 aresuccessfully estimated as shown in Fig 12 Moreover, by using this observer in an observer-based control scheme, the tracking performance in the large bias case is illustrated in Fig 13.The performance is much improved over the one with no fault tolerance as seen in Fig 13(b)

Ap-It also gives less overshoot than the no bias case, as seen in Fig 13(a) Similar results areobtained for the small bias case

The case of measurement error in the form of harmonics is now considered, with a sensorfault having a frequency of 2 rad/sec The dynamic gain for the observer (8)-(12) is computed

using the design approach discussed in section 5 This is the special case where ω o=2 The

gain is obtained at λ=10−12 as K7(see Appendix 9.3) Using this observer, Fig 14 shows thecorrect estimation of a measurement error of amplitude 20 degrees and frequency 2 rad/sec

time (sec)

Residual Actual bias

Fig 12 Case Study 2 - (a) Estimation of the Small Bias, (b) Estimation of the Large Bias

0 5 10 15 20 25 30 35 40 45 50

−50 0 50 100 150 200 250

time (sec)

Large bias response Large bias with observer−based control

Fig 13 Case Study 2 - (a) No-bias versus based, (b) Large Bias versus

Observer-based

We then consider the case of low frequency sensor faults (in the range[0, 1 rad/sec]) Using

the design introduced in section 5 (and with a=0.1, b=1 and =0.1), the optimal observer

gain is obtained using the command hinflmi in MATLAB, with minimum λ as 10 −12 (see K8

in Appendix 9.3) Using this observer for measurement error diagnosis, a correct estimation

of a low frequency sensor fault (generated using the MATLAB command idinput) is shown in

Fig 15

7 Conclusion

The Lipschitz observer design approach provides an important framework for solving the

measurement error diagnosis problem in robot manipulators The classical observer structure

is not directly applicable to the detection and identification problems This is in part due to the

restrictive observer structure, and also due to the idealized assumptions inherent in this

struc-ture that do not take into account uncertain model parameters and disturbances The dynamic

observer structure offers two important advantages in that regard: (i) The observer stability

condition that ensures asymptotic convergence of the state estimates is satisfied by a family

time (sec)

actual fault residual

Fig 14 Case Study 2 - Frequency Band Estimation

time (sec)

Actual fault Residual

Fig 15 Case Study 2 - Diagnosis of Low Frequency Sensor Fault

of observers, adding extra degrees of freedom to the observer which lay the ground to the dition of the detection and identification objectives in the design, (ii) The observer design can

ad-be carried out using a systematic design procedure which is less restrictive than the existingdesign approaches and which is solvable using commercially available software The designdepends heavily on the nature of the objectives considered While an analytical solution can

be used for measurement error detection, the identification problem is more demanding andneeds a more general design framework This problem is shown to be equivalent to a standardconvex optimization problem which is solvable using Linear Matrix Inequalities (LMIs) Us-ing this generalized framework, different frequency patterns for the measurement errors thataffect the robot manipulator could be considered, and systematic design procedures could be

used to solve the problem A practical example, namely the Quanser rotary inverted

pendu-lum (ROTPEN) in the Control Systems Lab, Electrical and Computer Engineering department,University of Alberta, is used to illustrate these results The ROTPEN model falls in the cate-gory of planar robot manipulators, and the experimental results illustrate the applicability ofthe proposed techniques in the robotics field by showing the following:

Measurement Analysis and Diagnosis for Robot Manipulators using Advanced Nonlinear Control Techniques 159

time (sec)

Residual Actual bias

Fig 12 Case Study 2 - (a) Estimation of the Small Bias, (b) Estimation of the Large Bias

0 5 10 15 20 25 30 35 40 45 50

−50 0 50 100 150 200 250

time (sec)

Large bias response Large bias with observer−based control

Fig 13 Case Study 2 - (a) No-bias versus based, (b) Large Bias versus

Observer-based

We then consider the case of low frequency sensor faults (in the range[0, 1 rad/sec]) Using

the design introduced in section 5 (and with a=0.1, b=1 and =0.1), the optimal observer

gain is obtained using the command hinflmi in MATLAB, with minimum λ as 10 −12 (see K8

in Appendix 9.3) Using this observer for measurement error diagnosis, a correct estimation

of a low frequency sensor fault (generated using the MATLAB command idinput) is shown in

Fig 15

7 Conclusion

The Lipschitz observer design approach provides an important framework for solving the

measurement error diagnosis problem in robot manipulators The classical observer structure

is not directly applicable to the detection and identification problems This is in part due to the

restrictive observer structure, and also due to the idealized assumptions inherent in this

struc-ture that do not take into account uncertain model parameters and disturbances The dynamic

observer structure offers two important advantages in that regard: (i) The observer stability

condition that ensures asymptotic convergence of the state estimates is satisfied by a family

time (sec)

actual fault residual

Fig 14 Case Study 2 - Frequency Band Estimation

time (sec)

Actual fault Residual

Fig 15 Case Study 2 - Diagnosis of Low Frequency Sensor Fault

of observers, adding extra degrees of freedom to the observer which lay the ground to the dition of the detection and identification objectives in the design, (ii) The observer design can

ad-be carried out using a systematic design procedure which is less restrictive than the existingdesign approaches and which is solvable using commercially available software The designdepends heavily on the nature of the objectives considered While an analytical solution can

be used for measurement error detection, the identification problem is more demanding andneeds a more general design framework This problem is shown to be equivalent to a standardconvex optimization problem which is solvable using Linear Matrix Inequalities (LMIs) Us-ing this generalized framework, different frequency patterns for the measurement errors thataffect the robot manipulator could be considered, and systematic design procedures could be

used to solve the problem A practical example, namely the Quanser rotary inverted

pendu-lum (ROTPEN) in the Control Systems Lab, Electrical and Computer Engineering department,University of Alberta, is used to illustrate these results The ROTPEN model falls in the cate-gory of planar robot manipulators, and the experimental results illustrate the applicability ofthe proposed techniques in the robotics field by showing the following:

i) How to model a robot manipulator as a standard Lipschitz system.

ii) The importance of the dynamic Lipschitz observer in large operating regions where the

linear observer normally fails

iii) The accurate velocity estimations obtained using the dynamic observer, alleviating the

need to introduce velocity sensors in real-time

iv) How the static observer fails, compared to the dynamic observer, when applied to

Robotic Systems due to the large Lipschitz constant that these systems normally have

v) The efficiency of the dynamic observer in diagnosing and tolerating measurement

er-rors of different frequencies, including an important bias introduced by the error in the

initial conditions of the pendulum encoder

8 Acknowledgement

The author would like to thank the Advanced Control Systems Laboratory members at

Uni-versity of Alberta Special thanks to Dr Alan Lynch and to Dr Thomas Grochmal for

pro-viding the ROTPEN equations and the switching swingup control scheme used in the

experi-ments

9 Appendix

9.1 The ROTPEN Model

The system parameters are: l1 = 0.215 m, l2 =0.335 m, m2 =0.1246 Kg, β= 0.135Nm/s,

µ =0.2065Nm/V, b2 =0.0018Kg/s, g =9.81m/s2, and J1 = 0.0064 Kg.m2 With the state

defined as x= [x1x2x3x4]T = [θ1(rad)θ2(rad) ˙θ1(rad/s) ˙θ2(rad/s)]T, the state space model

has the form ˙x= f(x) +g(x)u as follows (This model was derived in Lynch (2004)):

9.2 Models and Parameters for Case Study 1

Luenberger observer with small gain:

Nonlinear “normal form” Controller:

By considering y=x2, and using the nonlinear model of the ROTPEN in Appendix 9.1, thefollowing coordinate transformation:

2c2

+x4l32

Static Lipschitz observer : (obtained for α=1, ε=0.5)

L5=

1.7108 −2.1247 1.9837 −5.40190.4338 −0.2089 1.1030 −2.8972

T

Measurement Analysis and Diagnosis for Robot Manipulators using Advanced Nonlinear Control Techniques 161

i) How to model a robot manipulator as a standard Lipschitz system

ii) The importance of the dynamic Lipschitz observer in large operating regions where the

linear observer normally fails

iii) The accurate velocity estimations obtained using the dynamic observer, alleviating the

need to introduce velocity sensors in real-time

iv) How the static observer fails, compared to the dynamic observer, when applied to

Robotic Systems due to the large Lipschitz constant that these systems normally have

v) The efficiency of the dynamic observer in diagnosing and tolerating measurement

er-rors of different frequencies, including an important bias introduced by the error in the

initial conditions of the pendulum encoder

8 Acknowledgement

The author would like to thank the Advanced Control Systems Laboratory members at

Uni-versity of Alberta Special thanks to Dr Alan Lynch and to Dr Thomas Grochmal for

pro-viding the ROTPEN equations and the switching swingup control scheme used in the

experi-ments

9 Appendix

9.1 The ROTPEN Model

The system parameters are: l1 =0.215 m, l2 = 0.335 m, m2 =0.1246 Kg, β =0.135Nm/s,

µ =0.2065Nm/V, b2 = 0.0018Kg/s, g= 9.81m/s2, and J1 = 0.0064 Kg.m2 With the state

defined as x= [x1x2x3x4]T= [θ1(rad)θ2(rad) ˙θ1(rad/s) ˙θ2(rad/s)]T, the state space model

has the form ˙x=f(x) +g(x)u as follows (This model was derived in Lynch (2004)):

9.2 Models and Parameters for Case Study 1

Luenberger observer with small gain:

Nonlinear “normal form” Controller:

By considering y=x2, and using the nonlinear model of the ROTPEN in Appendix 9.1, thefollowing coordinate transformation:

2c2

+x4l32

Static Lipschitz observer : (obtained for α=1, ε=0.5)

L5=

1.7108 −2.1247 1.9837 −5.40190.4338 −0.2089 1.1030 −2.8972

T

9.3 Models and Parameters for Case Study 2

Lipschitz reduced-order model for observer design ( ¯x= [θ2 ˙θ1 ˙θ2]T) :

˙¯x=



−25.140 −17.220 0.2210168.13 16.57 −0.599





10 References

Aboky, C., Sallet, G & Vivalda, J (2002) Observers for Lipschitz nonlinear systems, Int J of

Contr., vol 75, No 3, pp 204-212.

Adjallah, K., Maquin, D & Ragot, J (1994) Nonlinear observer based fault detection, IEEE

Trans on Automat Contr., pp 1115-1120.

Chen, R., Mingori, D & Speyer, J (2003) Optimal stochastic fault detection filter, Automatica,

vol 39, No 3, pp 377-390

Chen, J & Patton, R (1999) Robust model-based fault diagnosis for dynamic systems, Kluwer

Aca-demic Publishers

Doyle, J., Glover, K., Khargonekar P & Francis, B (1989) State spce solutions to standard H2

and H∞control problems, IEEE Trans Automat Contr., Vol 34, No 8, pp 831-847.

Frank, P (1990) Fault diagnosis in dynamic systems using analytical and knowledge-based

redundancy - A survey and some new results, Automatica, vol 26, No 3, pp 459-474 Gahinet, P & Apkarian, P (1994) A linear matrix inequality approach to H∞control, Int J of

Robust and Nonlinear Contr., vol 4, pp 421-448.

Garcia, E & Frank, P (1997) Deterministic nonlinear observer based approaches to fault

di-agnosis: A survey, Contr Eng Practice, vol 5, No 5, pp 663-670.

Hammouri, H., Kinnaert, M & El Yaagoubi, E (1999) Observer-based approach to fault

de-tection and isolation for nonlinear systems, IEEE Trans on Automat Contr., vol 44,

No 10

Hill, D & Moylan, P (1977) Stability Results for Nonlinear Feedback Systems, Automatica,

Vol 13, pp 377-382

Iwasaki, T & Skelton, R (1994) All controllers for the general H∞control problem: LMI

exis-tence conditions and state space formulas, Automatica, Vol 30, No 8, pp 1307-1317.

Kabore, P & Wang, H (2001) Design of fault diagnosis filters and fault-tolerant control for a

class of nonlinear systems, IEEE Trans on Automat Contr., vol 46, No 11.

Lynch, A (2004) Control Systems II (Lab Manual), University of Alberta.

Marino, R & Tomei, P (1995) Nonlinear Control Design - Geometric, Adaptive and Robust,

Pren-tice Hall Europe, 1995

Marquez, H (2003) Nonlinear Control Systems: Analysis and Design, Wiley, NY.

Pertew, A (2006) Nonlinear observer-based fault detection and diagnosis, Ph.D Thesis,

De-partment of Electrical and Computer Engineering, University of Alberta

Pertew, A., Marquez, H & Zhao, Q (2005) H∞ synthesis of unknown input observers for

nonlinear Lipschitz systems, International J Contr., vol 78, No 15, pp 1155-1165 Pertew, A., Marquez, H & Zhao, Q (2006) H∞observer design for Lipschitz nonlinear sys-

tems, IEEE Trans on Automat Contr., vol 51, No 7, pp 1211-1216.

Pertew, A., Marquez, H & Zhao, Q (2007) LMI-based sensor fault diagnosis for nonlinear

Lipschitz systems, IEEE Trans on Automat Contr., vol 43, pp 1464-1469.

Raghavan, S & Hedrick, J (1994) Observer design for a class of nonlinear systems, Int J of

Contr., vol 59, No 2, pp 5515-528.

Rajamani, R (1998) Observers for Lipschitz nonlinear systems, IEEE Trans on Automat Contr.,

vol 43, No 3, pp 397-401

Rajamani, R & Cho, Y (1998) Existence and design of observers for nonlinear systems:

rela-tion to distance of unobservability, Int J Contr., Vol 69, pp 717-731.

Scherer, C (1992) H∞optimization without assumptions on finite or infinite zeros, Int J Contr.

and Optim., Vol 30, No 1, pp 143-166.

Sciavicco, L & Sicliano, B (1989) Modeling and Control of Robot Manipulators, McGraw Hill

Stoorvogel, A (1996) The H∞control problem with zeros on the boundary of the stability

domain, Int J Contr., Vol 63, pp 1029-1053.

Vemuri, A (2001) Sensor bias fault diagnosis in a class of nonlinear systems, IEEE Trans on

Automat Contr., vol 46, No 6.

Wang, H., Huang, Z & Daley, S (1997) On the Use of Adaptive Updating Rules for Actuator

and Sensor Fault Diagnosis, Automatica, Vol 33, No 2, pp 217-225.

Measurement Analysis and Diagnosis for Robot Manipulators using Advanced Nonlinear Control Techniques 163

9.3 Models and Parameters for Case Study 2

Lipschitz reduced-order model for observer design ( ¯x= [θ2 ˙θ1 ˙θ2]T) :

˙¯x=



−25.140 −17.220 0.2210168.13 16.57 −0.599





10 References

Aboky, C., Sallet, G & Vivalda, J (2002) Observers for Lipschitz nonlinear systems, Int J of

Contr., vol 75, No 3, pp 204-212.

Adjallah, K., Maquin, D & Ragot, J (1994) Nonlinear observer based fault detection, IEEE

Trans on Automat Contr., pp 1115-1120.

Chen, R., Mingori, D & Speyer, J (2003) Optimal stochastic fault detection filter, Automatica,

vol 39, No 3, pp 377-390

Chen, J & Patton, R (1999) Robust model-based fault diagnosis for dynamic systems, Kluwer

Aca-demic Publishers

Doyle, J., Glover, K., Khargonekar P & Francis, B (1989) State spce solutions to standard H2

and H∞control problems, IEEE Trans Automat Contr., Vol 34, No 8, pp 831-847.

Frank, P (1990) Fault diagnosis in dynamic systems using analytical and knowledge-based

redundancy - A survey and some new results, Automatica, vol 26, No 3, pp 459-474 Gahinet, P & Apkarian, P (1994) A linear matrix inequality approach to H∞control, Int J of

Robust and Nonlinear Contr., vol 4, pp 421-448.

Garcia, E & Frank, P (1997) Deterministic nonlinear observer based approaches to fault

di-agnosis: A survey, Contr Eng Practice, vol 5, No 5, pp 663-670.

Hammouri, H., Kinnaert, M & El Yaagoubi, E (1999) Observer-based approach to fault

de-tection and isolation for nonlinear systems, IEEE Trans on Automat Contr., vol 44,

No 10

Hill, D & Moylan, P (1977) Stability Results for Nonlinear Feedback Systems, Automatica,

Vol 13, pp 377-382

Iwasaki, T & Skelton, R (1994) All controllers for the general H∞control problem: LMI

exis-tence conditions and state space formulas, Automatica, Vol 30, No 8, pp 1307-1317.

Kabore, P & Wang, H (2001) Design of fault diagnosis filters and fault-tolerant control for a

class of nonlinear systems, IEEE Trans on Automat Contr., vol 46, No 11.

Lynch, A (2004) Control Systems II (Lab Manual), University of Alberta.

Marino, R & Tomei, P (1995) Nonlinear Control Design - Geometric, Adaptive and Robust,

Pren-tice Hall Europe, 1995

Marquez, H (2003) Nonlinear Control Systems: Analysis and Design, Wiley, NY.

Pertew, A (2006) Nonlinear observer-based fault detection and diagnosis, Ph.D Thesis,

De-partment of Electrical and Computer Engineering, University of Alberta

Pertew, A., Marquez, H & Zhao, Q (2005) H∞ synthesis of unknown input observers for

nonlinear Lipschitz systems, International J Contr., vol 78, No 15, pp 1155-1165 Pertew, A., Marquez, H & Zhao, Q (2006) H∞observer design for Lipschitz nonlinear sys-

tems, IEEE Trans on Automat Contr., vol 51, No 7, pp 1211-1216.

Pertew, A., Marquez, H & Zhao, Q (2007) LMI-based sensor fault diagnosis for nonlinear

Lipschitz systems, IEEE Trans on Automat Contr., vol 43, pp 1464-1469.

Raghavan, S & Hedrick, J (1994) Observer design for a class of nonlinear systems, Int J of

Contr., vol 59, No 2, pp 5515-528.

Rajamani, R (1998) Observers for Lipschitz nonlinear systems, IEEE Trans on Automat Contr.,

vol 43, No 3, pp 397-401

Rajamani, R & Cho, Y (1998) Existence and design of observers for nonlinear systems:

rela-tion to distance of unobservability, Int J Contr., Vol 69, pp 717-731.

Scherer, C (1992) H∞optimization without assumptions on finite or infinite zeros, Int J Contr.

and Optim., Vol 30, No 1, pp 143-166.

Sciavicco, L & Sicliano, B (1989) Modeling and Control of Robot Manipulators, McGraw Hill

Stoorvogel, A (1996) The H∞control problem with zeros on the boundary of the stability

domain, Int J Contr., Vol 63, pp 1029-1053.

Vemuri, A (2001) Sensor bias fault diagnosis in a class of nonlinear systems, IEEE Trans on

Automat Contr., vol 46, No 6.

Wang, H., Huang, Z & Daley, S (1997) On the Use of Adaptive Updating Rules for Actuator

and Sensor Fault Diagnosis, Automatica, Vol 33, No 2, pp 217-225.

Willsky, A (1976) A survey of design methods for failure detection in dynamic systems,

Au-tomatica, vol 12, pp 601-611.

Yu, D & Shields, D (1996) A bilinear fault detection observer, Automatica, vol 32, No 11, pp.

1597-1602

Zhong, M., Ding, S., Lam, J & Wang, H (2003) An LMI approach to design robust fault

detection filter for uncertain LTI systems, Automatica, vol 39, No 3, pp 543-550 Zhou, K & Doyle, J (1998) Essentials of robust control, Prentice-Hall, NY.

Cartesian Control for Robot Manipulators 165

Cartesian Control for Robot Manipulators

Pablo Sánchez-Sánchez and Fernando Reyes-Cortés

0

Cartesian Control for Robot Manipulators

Pablo Sánchez-Sánchez and Fernando Reyes-Cortés

Benemérita Universidad Autónoma de Puebla (BUAP)

Facultad de Ciencias de la Electrónica

México

1 Introduction

A robot is a reprogrammable multi-functional manipulator designed to move materials, parts,

tools, or specialized devices through variable programmed motions, all this for a best

perfor-mance in a variety of tasks A useful robot is the one which is able to control its movements

and the forces it applies to its environment Typically, robot manipulators are studied in

con-sideration of their displacements on joint space, in other words, robot’s displacements inside

of its workspace usually are considered as joint displacements, for this reason the robot is

an-alyzed in a joint space reference These considerations generate an important and complex

theory of control in which many physical characteristics appear, this kind of control is known

as joint control.

The joint control theory expresses the relations of position, velocity and acceleration of the

robot in its native language, in other words, describes its movements using the torque and

an-gles necessary to complete the task; in majority of cases this language is difficult to understand

by the end user who interprets space movements in cartesian space easily The singularities in

the boundary workspace are those which occur when the manipulator is completely

streched-out or folded back on itself such as the end-effector is near or at the boundary workspace

It’s necessary to understand that singularity is a mathematical problem that undefined the

system, that is, indicates the absence of velocity control which specifies that the end-effector

never get the desired position at some specific point in the workspace, this doesn’t mean the

robot cannot reach the desired position structurally, whenever this position is defined inside

the workspace This problem was solved by S Arimoto and M Takegaki in 1981 when they

proposed a new control scheme based on the Jacobian Transposed matrix; eliminating the

possibility of singularities and giving origin to the cartesian control.

The joint control is used for determining the main characteristics of the cartesian control based

on the Jacobian Transposed matrix It is necessary to keep in mind that to consider the robot’s

workspace like a joint space, has some problems with interpretation because the user needs

having a joint dimensional knowledge, thus, when the user wants to move the robot’s

end-effector through a desired position he needs to understand the joint displacements the robot

needs to do, to get the desired position This interpretation problem is solved by using the

cartesian space, that is, to interpret the robot’s movements by using cartesian coordinates on

reference of cartesian space; the advantage is for the final user who has the cartesian

dimen-sional knowledge for understanding the robot’s movements Due this reason, learning the

mathematical tools for analysis by the robot’s movements on cartesian space is necessary, this

allows us to propose control structures, to use the dynamic model and to understand the

8

physical phenomenons on robot manipulators on cartesian space When we control the global

motion or position of general manipulators, we are confronted with the nonlinear dynamics

in a lot of degrees of freedom In literature focused with the dynamic control of manipulators,

the complexity of nonlinear dynamics is emphasized and some methods, compensating all

nonlinear terms in dynamics in real time, are developed in order to reduce the complexity in

system control However, these methods require a large amount of complicated calculation so

it is difficult to implement these methods with low level controllers such as microcomputers

In addition, the reliability of these methods may be lost when a small error in computation or a

small change in system parameters occurs, occurs because they are not considered in the

con-trol Most industrial robots, each joint of manipulator is independently controlled by a simple

linear feedback However, convergence for target position has not been enough investigated

for general nonlinear mechanical systems

This chapter is focused on the position control for robot manipulators by using control

struc-tures defined on the cartesian space because the robot move freely in its workspace, which

is understood by the final user like cartesian space Besides, the mathematical tools will be

detailed for propose, analyze and evaluating control structures in cartesian space

2 Preliminaries: forward kinematics and Jacobian matrix

A rigid multi-body system consists in a set of rigid objects, called links, joined together by

joints Simple kinds of joints include revolute (rotational) and prismatic (translational) joints

It is also possible to work with more general types of joints, and thereby simulate non-rigid

objects Well-known applications of rigid multi-bodies include robotic arms A robot

manip-ulator is modeled with a set of links connected by joints There are a variety of possible joint

types Perhaps the most common type is a rotational joint with its configuration described

by a single scalar angle value The key point is: ”the configuration of a joint is a continuous

function of one or more real scalars; for rotational joints“, the scalar is the angle of the joint

Complete configuration in robot manipulators is specified by vectors, for example the position

where q ∈Rn×1 We assume there are n joints and each q n value is called a joint position.

The robot manipulator will be controlled by specifying target positions by the end-effectors

The desired positions are also given by a vector

where q d i is the desired position for the ith end-effector We let ˜q i = q d i − q i, the desired

change in position of the ith end effector, also this vector is well-known as an error position.

The end-effector positions(x, y, z)are functions of the joint angles q; this fact can be expressed

as:

x i=f i(q) for i=1, 2, , k (3)

this equation is well-known as forward kinematics.

2.1 Case of study: Cartesian robot (forward kinematics

In order to understand application of cartesian control in robot manipulators a case of studywill be used, which all the concepts were evaluated In this section we will obtain the for-ward kinematics of a three degrees of freedom cartesian robot, Figure 1; and we will use thisinformation in the following sections

Fig 1 Three degrees of freedom cartesian robot

In order to obtain the forward kinematics of three degrees of freedom cartesian robot we need

to draw a system diagram, Figura 2,

where q1, q2, q3are join displacements; and m1, m2, m3represent the masses of each link As

it is observed, translation is the unique movement that realizes this kind of robots, then theforward kinematics are defined as:

We can observe, that in the first vector is contemplated only by the first displacement of value

q1, in the second one considers the movement of translation in q1and q2respecting the axis x and y, and finally the complete displacement in third axis described in the last vector, being

this representation the robot forward kinematics

Cartesian Control for Robot Manipulators 167

physical phenomenons on robot manipulators on cartesian space When we control the global

motion or position of general manipulators, we are confronted with the nonlinear dynamics

in a lot of degrees of freedom In literature focused with the dynamic control of manipulators,

the complexity of nonlinear dynamics is emphasized and some methods, compensating all

nonlinear terms in dynamics in real time, are developed in order to reduce the complexity in

system control However, these methods require a large amount of complicated calculation so

it is difficult to implement these methods with low level controllers such as microcomputers

In addition, the reliability of these methods may be lost when a small error in computation or a

small change in system parameters occurs, occurs because they are not considered in the

con-trol Most industrial robots, each joint of manipulator is independently controlled by a simple

linear feedback However, convergence for target position has not been enough investigated

for general nonlinear mechanical systems

This chapter is focused on the position control for robot manipulators by using control

struc-tures defined on the cartesian space because the robot move freely in its workspace, which

is understood by the final user like cartesian space Besides, the mathematical tools will be

detailed for propose, analyze and evaluating control structures in cartesian space

2 Preliminaries: forward kinematics and Jacobian matrix

A rigid multi-body system consists in a set of rigid objects, called links, joined together by

joints Simple kinds of joints include revolute (rotational) and prismatic (translational) joints

It is also possible to work with more general types of joints, and thereby simulate non-rigid

objects Well-known applications of rigid multi-bodies include robotic arms A robot

manip-ulator is modeled with a set of links connected by joints There are a variety of possible joint

types Perhaps the most common type is a rotational joint with its configuration described

by a single scalar angle value The key point is: ”the configuration of a joint is a continuous

function of one or more real scalars; for rotational joints“, the scalar is the angle of the joint

Complete configuration in robot manipulators is specified by vectors, for example the position

where q ∈Rn×1 We assume there are n joints and each q n value is called a joint position.

The robot manipulator will be controlled by specifying target positions by the end-effectors

The desired positions are also given by a vector

where q d i is the desired position for the ith end-effector We let ˜q i = q d i − q i, the desired

change in position of the ith end effector, also this vector is well-known as an error position.

The end-effector positions(x, y, z)are functions of the joint angles q; this fact can be expressed

as:

x i=f i(q) for i=1, 2, , k (3)

this equation is well-known as forward kinematics.

2.1 Case of study: Cartesian robot (forward kinematics

In order to understand application of cartesian control in robot manipulators a case of studywill be used, which all the concepts were evaluated In this section we will obtain the for-ward kinematics of a three degrees of freedom cartesian robot, Figure 1; and we will use thisinformation in the following sections

Fig 1 Three degrees of freedom cartesian robot

In order to obtain the forward kinematics of three degrees of freedom cartesian robot we need

to draw a system diagram, Figura 2,

where q1, q2, q3are join displacements; and m1, m2, m3represent the masses of each link As

it is observed, translation is the unique movement that realizes this kind of robots, then theforward kinematics are defined as:

We can observe, that in the first vector is contemplated only by the first displacement of value

q1, in the second one considers the movement of translation in q1and q2respecting the axis x and y, and finally the complete displacement in third axis described in the last vector, being

this representation the robot forward kinematics

2.2 Jacobian matrix

The Jacobian matrix J(q)is a multidimensional form of the derivative This matrix is used

to relate the joint velocity ˙q with the cartesian velocity ˙x, based on this reason we are able to

think about Jacobian matrix as mapping velocities in q to those in x:

where ˙x is the velocity on cartesian space; ˙q is the velocity in joint space; and J(q) is the

Jacobian matrix of the system

In many cases, we use modeling and simulation as a tool for analysis about the behavior

of a given system Even though at this stage, we have not formed the equations of motion

for a robotic manipulator, by inspecting the kinematic models, we are able to revel many

characteristics from the system One of the most important quantities (for the purpose of

analysis) in (5), is the Jacobian matrix J(q) It reveals many properties of a system and can

be used for the formulation of motion equations, analysis of special system configurations,

static analysis, motion planning, etc The robot manipulator’s Jacobian matrix J(q)is defined

where f(q)is the relationship of forward kinematics, equation (3); n is the dimension of q; and

m is the dimension of x We are interested about finding what joint velocities ˙q result in given

(desired) v Hence, we need to solve a system equations.

2.2.1 Case of study: Jacobian matrix of the cartesian robot

In order to obtain the Jacobian matrix of the three degrees of freedom cartesian robot it is

necessary to use the forward kinematics which is defined as:



 x y z

Cartesian Control for Robot Manipulators 169

2.2 Jacobian matrix

The Jacobian matrix J(q)is a multidimensional form of the derivative This matrix is used

to relate the joint velocity ˙q with the cartesian velocity ˙x, based on this reason we are able to

think about Jacobian matrix as mapping velocities in q to those in x:

where ˙x is the velocity on cartesian space; ˙q is the velocity in joint space; and J(q)is the

Jacobian matrix of the system

In many cases, we use modeling and simulation as a tool for analysis about the behavior

of a given system Even though at this stage, we have not formed the equations of motion

for a robotic manipulator, by inspecting the kinematic models, we are able to revel many

characteristics from the system One of the most important quantities (for the purpose of

analysis) in (5), is the Jacobian matrix J(q) It reveals many properties of a system and can

be used for the formulation of motion equations, analysis of special system configurations,

static analysis, motion planning, etc The robot manipulator’s Jacobian matrix J(q)is defined

where f(q)is the relationship of forward kinematics, equation (3); n is the dimension of q; and

m is the dimension of x We are interested about finding what joint velocities ˙q result in given

(desired) v Hence, we need to solve a system equations.

2.2.1 Case of study: Jacobian matrix of the cartesian robot

In order to obtain the Jacobian matrix of the three degrees of freedom cartesian robot it is

necessary to use the forward kinematics which is defined as:



 x y z

2.3 Jacobian transpose matrix

The transpose of a matrix J(q) is another matrix J(q)T created by anyone of the following

equivalent actions: write the J(q)T rows as the J(q)T columns; write the J(q)T columns as

the J(q)T rows; and reflect J(q)by its main diagonal (which starts from the top left) to obtain

J(q)T Formally, the transpose of an m × n matrix J(q)with elements J(q)ij is n × m matrix as

follow

J ji(q)T =J ij(q) for 1≤ i ≤ n, 1 ≤ j ≤ m. (13)The transposing of a scalar is the same scalar

2.3.1 Case of study: Jacobian transpose matrix of the cartesian robot

In order to obtain the Jacobian transpose matrix J(q)T we apply (13) leaving of the equation

(12) In particular case of cartesian robot the Jacobian matrix J(q)is equal to the identity matrix

I, thus its transposed matrix J(q)Tis the same, thus we have:

Singularities correspond certain configurations in robot manipulators which have to be

avoided because they lead to an abrupt loss of manipulator rigidity In the vicinity of these

configurations, manipulator can become uncontrollable and the joint forces could increase

considerably and may there would be risk to even damage the manipulator mechanisms The

singularities in a workspace can be identified mathematically when the determinant in the

Jacobian matrix is zero:

Mathematically this means that matrix J(q)is degenerated and there is, in the inverse

geomet-rical model, an infinity of solutions in the vicinity of these points

2.5 Singular configurations

Due to the tuning of derivative and proportional matrices from the control algorithms of

which objective is to maintain in every moment the error position nearest to zero, it exists

the possibility that in certain values of the determinant in Jacobian matrix the system is

singu-lar undefined It’s denominated singusingu-lar configurations of a robot those distributions in which

that determinant of the Jacobian matrix is zero, equation (15) Because of this circumstance,

in the singular configurations the inverse Jacobian matrix doesn’t exist For a undefine

Jaco-bian matrix, an infinitesimal increment in the cartesian coordinates would suppose an infinite

increment at joint coordinates, which is translated as movements from the articulations to

in-accessible velocities on some part of its links for reaching the desired position for a constant

velocity in the practice Therefore, in the vicinity of the singular configurations lost some

de-grees in the robot’s freedom, being impossible their end-effector moves in a certain cartesian

address

Different singular configurations on robot can be classified as:

• Singularities in the limits in the robot’s workspace These singularities are presented when

the robot’s boundary is in some point of the limit of interior or external workspace Inthis situation it is obvious the robot won’t be able to move in the addresses that weretaken away from this workspace

• Singularities inside the robot’s workspace They take place generally inside the work area

and for the alignment of two or more axes in the robot’s articulations

2.5.1 Case of study: determinant of the Jacobian matrix of the cartesian robot

In order to determine if there are singularities in the system, it is necessary to obtain the

determinant on the system det J(q), considering a general structure of the Jacobian matrix,thus we have:

deter-of freedom is described

2.6 Inverse Jacobian matrix

In mathematics, and especially in linear algebra, a matrix squared A with an order n × n it

is said is reversible, nonsingular, non-degenerate or regular if exists another squared matrix

with order n × n called inverse matrix A −1and represented matrix like

I is the identity matrix with order n × n and the used product is the usual product of matrices.

The mathematical definition in the inverse matrix is defined as follow:

J(q)−1= C T

where C is the co-factors matrix.

2.6.1 Case of study: co-factors matrix in the cartesian robot

In order to obtain the co-factor matrix it is necessary to apply the following procedure:

Con-sidering the matrix A defined like: