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ScienceDirect - Computer Methods in Applied Mechanics and Engineering : Inverse control of a three-link manipulator

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Computer Methods in Applied Mechanics and Engineering

Volume 190, Issues 40-41, 20 July 2001, Pages 5311-5324Font Size:


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Inverse control of a three-link manipulator

C. Frangosa and Y. Yavin , , b

a Department of Statistics, The Rand Afrikaans University, P.O. Box 524, Auckland Park 2006, Johannesburg, South Africa



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b Department of Electrical and Electronic Engineering, University of Pretoria, Pretoria 0002, South Africa

Received 14 June 2000. Available online 5 July 2001.



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Abstract

This work deals with the control of a three-link planar manipulator. The problem that is addressed is to develop control laws for the manipulator such that various end-

effector maneuvers can be performed. A particular maneuver that is considered here is for the end-effector to follow a given trajectory in the vertical plane. In addition, the




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problem of moving the end-effector during a given time interval from an initial state to a final state is also considered. A procedure for determining control laws for solving the

above-mentioned problems is proposed.

Author Keywords: Three-link planar manipulator; Inverse control; Trajectory following; Point-to-point maneuver; Path controllability




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p pp g g p

Article Outline

1. Introduction

2. Dynamical model


Control of a three-link manipulator with an inequality ...


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3. Inverse control law

4. Trajectory following

5. Point-to-point maneuver


Computer Methods in Applied Mechanics and Engineering

Control of a three-link manipulator with a constraint o...


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6. Computational results

6.1. Example 1: trajectory following

6.2. Example 2: point-to-point maneuver


Computers & Mathematics with Applications

Control of a three-link manipulator with inequality con...


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7. Conclusion

References

1. Introduction


Computers & Mathematics with Applications


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This work deals with the control of a three-link planar manipulator. It is assumed here that the motion of all parts of the manipulator is confined to a vertical ( X ,Z )-plane. It is

further assumed that the motion of the manipulator is driven by three motors. The first motor is located at the base of the first link, the second motor is located at the joint r 1

between the first and second links, and the third motor is located at the joint r 2 between the second and third links (see Fig. 1).



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Fig. 1. The configuration of the three-link manipulator.



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The problem that is addressed is to develop a control law for the manipulator such that various end-effector maneuvers can be performed. A particular maneuver that is

considered here is for the end-effector to follow a given trajectory in the ( X ,Z )-plane. In addition, the problem of moving the end-effector during a given time interval from an

initial state to a final state, is also considered.




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A procedure is proposed for deriving contro l laws to solve the above-mentioned problems using a kind of inverse con trol.

In principle, optimal control theory [2] can be applied to solve the above-mentioned problems. However, the proposed procedure is straightforward, direct and does not involve

an iterative process as is usually needed in solving optimal control problems.




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2. Dynamical model

In this work, we consider the control of the motion of a three-link planar manipulator. Let I ,J and K be unit vectors along an inertial ( X ,Y ,Z )-co-ordinate system. Denote by i k




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i k =cos(θ

k )I +sin(θ

k )K ,

a unit vector along the k th link, k =1,2,3 (see Fig. 1), and let

(1)

(2)



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j k =−sin(θ

k )I +cos(θ

k )K ,

be a unit vector perpendicular to i k , k =1,2,3, respectively. Note that

(3)




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The motion of the manipulator is driven by three motors. The first motor is located at the origin (see Fig. 1), the second motor is located at the point r 1, r 1=l 1i 1, and the third motor

is located at r 2, r 2=l 1i 1+l 2i 2.

Here l k denotes the length of link k , k =1,2,3. Also, r

Ck , the location of the center of mass of link k , k =1,2,3, is given by r

C 1=l

C 1i 1, r C 2

=l 1i 1+l C 2i 2 and r C 3

=l 1i 1+l 2i 2+l C 3i 3,




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0


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where

a11=m1l C 12+I 1+(m2+mR 2

+m3+mR 3)l 1

2,

(4)




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a22=m2l C 22+I 2+(m3+mR 3

)l 22, a33=m3l C 3

2+I 3,




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a12=m2l 1l C 2+(m3+mR 3

)l 1l 2, a13=m3l 1l C 3,




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a23=m3l 2l C 3, V

O1=m1l C 1

+(m2+mR 2+m3+mR 3

)l 1,




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V O2

=m2l C 2+(m3+mR 3

)l 2, V O3=m3l C 3

.




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In the expressions above , I k denotes the moment of inertia of link k about a vector in the direction of J located at r

Ck , k =1,2,3. Note that the motors are considered here as

point masses, as their rotational kinetic energies are assumed to be negligible with respect to the corresponding rotational kinetic energies of the links.

Denote




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The vector r EF

is given by r EF

= x EF

I +z EF

K where x EF

=∑i =1

3l i cos(θ

i ) and z

EF =∑

i =13l

i sin(θ

i ). Hence, v

EF =dr

EF /dt is given by,

(5)




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and

The equations of motion of the system are determined by the Lagrange equations ([ 1]),

(6)

(7)




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where T j is the torque applied by motor j on link j , j =1,2,3.

Thus, using the Lagrangian function ( 4) and the Lagrange equations, (7), the following equations are obtained:




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where, using the notations T =[T 1,T 2,T 3]T ,M =[m

ij ],i , j =1,2,3, and h =[h1,h2,h3]

T , and where,

(8)

(9)




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m11=a11, m12=a12cos(θ2−θ1), m13=a13cos(θ3−θ1),

(10)




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m21=m12, m22=a22, m23=a23cos(θ3−θ2),

(11)




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m31=m13, m32=m23, m33=a33,

(12)




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(13)

(14)




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It can be shown, for a proper set of parameters, that det(M (q ))>0, for all θk ,k =1,2,3. (8), (9), (10), (11), (12), (13) and (14) constitute the dynamical model for the system dealt

with here.




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3. Inverse control law

In this Section, a procedure is proposed for determining the manipulator control law. First, we begin by introducing the inverse dynamics control law,




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T =M (q )v +h (q ,p ),

where v =[v 1,v 2,v 3]T , and where the v

i are auxiliary control functions, i =1,2,3. For the case where det(M (q ))≠0 for all q , (8) and (15) imply

(15)

(16)




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As is evident from Eq. (16), the inverse control separates the kinematics from the nonlinear manipulator dynamics.

Second, the auxiliary control functions v i are taken here as




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where ωni

>0,0


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are Hurwitz. Eq. (17) can be written in matrix form as follows:

where Z =diag(ζ1,ζ2,ζ3), Ω=diag(ωn1,ωn2,ωn3), Ωs=diag(ω2n1

,ω2n2

,ω2n3

), and θcom=[θcom,1,θcom,2,θcom,3]T .

(19)




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Third, compute θcom so as to satisfy the required maneuvering goals. Two examples are considered below, namely, trajectory following and a point-to-point maneuver.

4. Trajectory following

In this Section a procedure is described for the computation of θcom using the kinematic manipulator model (19), (5) and (6), such that the end-effector follows a given




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smooth reference trajectory in the ( X ,Z )-plane with representation {( x r (t ),z

r (t )), t 0}. Let s

r (t )=[ x

r (t ),z

r (t ), d x

r (t )/dt , dz

r (t )/dt ]

T ,t 0, and define the end-effector state vector s :

s =[ x EF

,z EF

, d x EF

/dt , dz EF

/dt ]T .




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The objective is to find an expression for θcom such that the end-effector moves according to(20)

(21)

(22)




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during a finite time interval [0, t f ], and where s (0) and s

r (0) are given, and ω

di >0,0


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g i (s)=s2+2ζ

di ω

di s+ω

di 2, i =1,2

are Hurwitz. In this case it follows from (21) and (22) that

(23)

(24)




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and

(25)




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Combining Eqs. ((6), (19) and (21)), and (22) we obtain

where a =[a1,a2]T is given by the right-hand side of (21) and (22). By re-arranging the terms of Eq. (26) we obtain,

(26)

(27)




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In Eq. (27) A0 R 2×3. If rank(A0) = 2, then A0A0

T R 2×2 has an inverse and a solution to Eq. (27) is given by

(28)




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where

Thus, the following set of equations has to be solved simultaneously:

(29)

(30)




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(31)




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(32)

(33)




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(34)




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in any region of q such that rank(A0)=2. The values of θcom are computed from Eq. (34), and v is computed from Eq. (35). Thus, once v is obtained, the manipulator torque vector

(35)




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T can be computed using Eq. (15).

5. Point-to-point maneuver

In this Section a procedure is described for the computation of θcom using the kinematic manipulator model ( 19), ( 5), ( 6), such that the end-effector moves during a finite




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time interval [0,t f ] from an initial state s (0)=Q 1 R

4 to a final state s (t f )=Q 2 R

4. If a link angle reference command θcom can be found so as to perform this maneuver, then

we say that the end-effector is path controllable .

Using (19) and (6), we assign the right-hand side of Eq. (6) to w ,




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Then, using Eq. (6), it follows that

(36)

(37)




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In addition we define,

where w =[w 1,w 2]T and u =[u1,u2]

T is an auxiliary variable. The additional state variables w 1 and w 2 are introduced in order to obtain a smoother trajectory for the end-

(38)

(39)




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effector acceleration d2r EF

/dt 2. If we define a new state vector σ=[s T ,w 1,w 2]

T , then (37) and (38) can be written is state-space form as follows:

The linear system given by Eq. (39) is controllable [3]. This means that for any two points R 6 and R 6 and any finite

(40)




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time interval [0,t f ], one can find a control function u (·) such that σ(·) will move from to . One such control is given by [3]

u (t )=(eE

1(t f −t )E 2)

T S −1(σ(t

f )−e

E 1t f σ(0)),t [0,t f

],




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where

S =∫0t f eE

1(t f −t )E 2(e

E 1(

t f −t )E 2)

T dt .

(41)




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Thus, the following set of equations has to be solved simultaneously:

u (t )=(eE

1(t f −t )E 2)

T S −1(σ(t

f )−e

E 1t f σ(0)), t [0,t f

],

(42)




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(43)

(44)




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w 1=σ5, w 2=σ6,

(45)




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(46)




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in any region of q such that rank(A0)=2. The values of θcom are computed from Eq. (45).

Therefore, if the manipulator's variables are in any region where rank( A0) = 2, then the end-effector is path controllable .

6. Computational results




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In the examples dealt with here, the following set of parameters has been used: l k =1 m and l

Ck =0.5 m, k =1,2,3; m1=10 kg, m2=8 kg, m3=6 kg; mR 1

=mR 2

=2 kg, mR 3

=1.5 kg, I k =

(1/12)mk l k 2, k =1,2,3. It can be shown, by calculating the values of det(M (q )), that, for the above-mentioned values of parameters, det(M (q ))>87.111 for all −π


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A fourth-order Runge –Kutta algorithm was applied to solve the following two cases: the case of a trajectory tracking maneuver ( (30), (31), (32), (33), (34) and (35)), and the case

of a point-to-point maneuver ( (42), (43), (44), (45) and (46)). A time step of Δt =0.005 s was used throughout.

6.1. Example 1: trajectory following




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In this example the end-effector is required to move around a circle with radius r 0=1 m and centered at r e=1.9I . The circular trajectory is specified by (see (21) and ( 22)): d2 x

r (t )/

dt 2=−ω20r 0 cos(ω0t ),t 0, d2z r (t )/dt 2=−ω20r 0 sin(ω0t ),t 0, x r (0)=2.9 m, z r (0)=0, d x r (0)/dt =0, dz r (0)/dt =ω0r 0,ω0=1 rad/s. The time horizon is t f =6.3 s.

The initial angles and angular velocities of the three-link manipulator are as follows: θ1(0)=π/6, θ2(0)=0.0, θ3(0)=−π/6, dθi (0)/dt =0.0, i =1,2,3. Using the definition of r EF , and Eq.




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(5), we obtain that x EF

(0)=2.73025 m, z EF

(0)=0.0 m, d x EF

(0)/dt =dz EF

(0)/dt =0.0.

The tracking of the given circular trajectory has been achieved by the end-effector, and already at t =1 s,




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m. Some of the results obtained are presented in Fig. 2, Fig. 3, Fig. 4, Fig. 5, Fig. 6 and Fig. 7.



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Fig. 2. Example 1: plots of the values of θcom,1(t ) (line 1), θcom,2(t ) (line 2), θcom,3(t ) (line 3), t [0,t f ].



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http://www.sciencedirect.com/science?_ob=MiamiCaptionURL&_method=retrieve&_udi=B6V29-43F3SN3-8&_image=fig3&_ba=3&_user=686465&_rdoc=1&_fmt=full&_orig=search&_cdi=5697&view=c&_acct=C000037638&_version=1&_urlVersion=0&_userid=686465&md5=b5a76fcdfd6af9fc11a0ec67cae90055http://www.sciencedirect.com/science?_ob=MiamiCaptionURL&_method=retrieve&_udi=B6V29-43F3SN3-8&_image=fig3&_ba=3&_user=686465&_rdoc=1&_fmt=full&_orig=search&_cdi=5697&view=c&_acct=C000037638&_version=1&_urlVersion=0&_userid=686465&md5=b5a76fcdfd6af9fc11a0ec67cae90055http://www.sciencedirect.com/science?_ob=MiamiCaptionURL&_method=retrieve&_udi=B6V29-43F3SN3-8&_image=fig3&_ba=3&_user=686465&_rdoc=1&_fmt=full&_orig=search&_cdi=5697&view=c&_acct=C000037638&_version=1&_urlVersion=0&_userid=686465&md5=b5a76fcdfd6af9fc11a0ec67cae90055


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Fig. 3. Example 1: plots of the values of θ1(t ) (line 1), θ2(t ) (line 2), θ3(t ) (line 3), t [0,t f ].




http://www.sciencedirect.com/science?_ob=MiamiCaptionURL&_method=retrieve&_udi=B6V29-43F3SN3-8&_image=fig4&_ba=4&_user=686465&_rdoc=1&_fmt=full&_orig=search&_cdi=5697&view=c&_acct=C000037638&_version=1&_urlVersion=0&_userid=686465&md5=f583ed64eb4f569c9aef6b6a51c8f846http://www.sciencedirect.com/science?_ob=MiamiCaptionURL&_method=retrieve&_udi=B6V29-43F3SN3-8&_image=fig4&_ba=4&_user=686465&_rdoc=1&_fmt=full&_orig=search&_cdi=5697&view=c&_acct=C000037638&_version=1&_urlVersion=0&_userid=686465&md5=f583ed64eb4f569c9aef6b6a51c8f846http://www.sciencedirect.com/science?_ob=MiamiCaptionURL&_method=retrieve&_udi=B6V29-43F3SN3-8&_image=fig4&_ba=4&_user=686465&_rdoc=1&_fmt=full&_orig=search&_cdi=5697&view=c&_acct=C000037638&_version=1&_urlVersion=0&_userid=686465&md5=f583ed64eb4f569c9aef6b6a51c8f846


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Fig. 4. Example 1: plot of the values of v EF

(t ) (line 1), t [0,t f ].



http://www.sciencedirect.com/science?_ob=MiamiCaptionURL&_method=retrieve&_udi=B6V29-43F3SN3-8&_image=fig5&_ba=5&_user=686465&_rdoc=1&_fmt=full&_orig=search&_cdi=5697&view=c&_acct=C000037638&_version=1&_urlVersion=0&_userid=686465&md5=cf2acefbd34bb642850f8a73db8f008f


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Fig. 5. Example 1: plots of the values of T 1(t ) (line 1), T 2(t ) (line 2), T 3(t ) (line 3), t [0,t f ].



http://www.sciencedirect.com/science?_ob=MiamiCaptionURL&_method=retrieve&_udi=B6V29-43F3SN3-8&_image=fig5&_ba=5&_user=686465&_rdoc=1&_fmt=full&_orig=search&_cdi=5697&view=c&_acct=C000037638&_version=1&_urlVersion=0&_userid=686465&md5=cf2acefbd34bb642850f8a73db8f008fhttp://www.sciencedirect.com/science?_ob=MiamiCaptionURL&_method=retrieve&_udi=B6V29-43F3SN3-8&_image=fig5&_ba=5&_user=686465&_rdoc=1&_fmt=full&_orig=search&_cdi=5697&view=c&_acct=C000037638&_version=1&_urlVersion=0&_userid=686465&md5=cf2acefbd34bb642850f8a73db8f008fhttp://www.sciencedirect.com/science?_ob=MiamiCaptionURL&_method=retrieve&_udi=B6V29-43F3SN3-8&_image=fig5&_ba=5&_user=686465&_rdoc=1&_fmt=full&_orig=search&_cdi=5697&view=c&_acct=C000037638&_version=1&_urlVersion=0&_userid=686465&md5=cf2acefbd34bb642850f8a73db8f008f


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http://www.sciencedirect.com/science?_ob=MiamiCaptionURL&_method=retrieve&_udi=B6V29-43F3SN3-8&_image=fig6&_ba=6&_user=686465&_rdoc=1&_fmt=full&_orig=search&_cdi=5697&view=c&_acct=C000037638&_version=1&_urlVersion=0&_userid=686465&md5=efd569a777023a4a21c8642729639d05http://www.sciencedirect.com/science?_ob=MiamiCaptionURL&_method=retrieve&_udi=B6V29-43F3SN3-8&_image=fig6&_ba=6&_user=686465&_rdoc=1&_fmt=full&_orig=search&_cdi=5697&view=c&_acct=C000037638&_version=1&_urlVersion=0&_userid=686465&md5=efd569a777023a4a21c8642729639d05http://www.sciencedirect.com/science?_ob=MiamiCaptionURL&_method=retrieve&_udi=B6V29-43F3SN3-8&_image=fig6&_ba=6&_user=686465&_rdoc=1&_fmt=full&_orig=search&_cdi=5697&view=c&_acct=C000037638&_version=1&_urlVersion=0&_userid=686465&md5=efd569a777023a4a21c8642729639d05


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Fig. 6. Example 1: plots of the values of z EF

( x EF

) (line 1), z r ( x

r ) (line 2).




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Fig. 7. Example 1: plots of the manipulator links at time instants t =k Δd ,Δ

d =0.3 s, k =0,1,…,21. Note that the scale along the horizontal axis differs from the scale along the

vertical axis. This leads to a distortion in the length of the links in the figure.




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6.2. Example 2: point-to-point maneuver

For the point-to-point maneuver, the initial angles and angular velocities of the three-link manipulator are as follows: θ1(0)=−1.5 rad, θ2(0)=0.5 rad, θ3(0)=0.9 rad, dθi (0)/




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dt =0.0, i =1,2,3. Using the definition of r EF

, and Eq. (5), we obtain that x EF

(0)=1.56993 m, z EF

(0)=0.26526 m, d x EF

(0)/dt =dz EF

(0)/dt =0.0.

In this example, the end-effector is required to move during the time interval [0, t f ],t

f =4.5 s, from s (0)=Q 1=[1.56993,0.26526,0,0]

T , to s (t

f )=Q 2=[−2.0,1.0,0,0]

T .

The specified point-to-point maneuver has been achieved by the end-effector, and some of the results obtained are presented in Fig. 8, Fig. 9, Fig. 10, Fig. 11, Fig. 12 and Fig. 13.




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Fig. 8. Example 2: plots of the values of θcom,1(t ) (line 1), θcom,2(t ) (line 2), θcom,3(t ) (line 3), t [0,t f ].



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Fig. 9. Example 2: plots of the values of θ1(t ) (line 1), θ2(t ) (line 2), θ3(t ) (line 3), t [0,t f ].



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Fig. 10. Example 2: plot of the values of v EF

(t ) (line 1), t [0,t f ].




http://www.sciencedirect.com/science?_ob=MiamiCaptionURL&_method=retrieve&_udi=B6V29-43F3SN3-8&_image=fig11&_ba=11&_user=686465&_rdoc=1&_fmt=full&_orig=search&_cdi=5697&view=c&_acct=C000037638&_version=1&_urlVersion=0&_userid=686465&md5=beec0cc94bf827b5c879b1d88894e626http://www.sciencedirect.com/science?_ob=MiamiCaptionURL&_method=retrieve&_udi=B6V29-43F3SN3-8&_image=fig11&_ba=11&_user=686465&_rdoc=1&_fmt=full&_orig=search&_cdi=5697&view=c&_acct=C000037638&_version=1&_urlVersion=0&_userid=686465&md5=beec0cc94bf827b5c879b1d88894e626http://www.sciencedirect.com/science?_ob=MiamiCaptionURL&_method=retrieve&_udi=B6V29-43F3SN3-8&_image=fig11&_ba=11&_user=686465&_rdoc=1&_fmt=full&_orig=search&_cdi=5697&view=c&_acct=C000037638&_version=1&_urlVersion=0&_userid=686465&md5=beec0cc94bf827b5c879b1d88894e626


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Fig. 11. Example 2: plots of the values of T 1(t ) (line 1), T 2(t ) (line 2), T 3(t ) (line

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