Parallel Manipulators New Developments Part 14 docx

The resulting consistency condition that the generalized accelerations must satisfy is obtained from equation 15 as Hence the time trajectory st of the deployment motion should be select

1 1 3 13 1 1 3 13

2 2 4 24 2 2 4 24 T

The drive singularities are found from A =0 as sin(θ1+θ3−θ2−θ4) 0= , i.e as the

positions when points A, B and D become collinear Hence, drive singularities occur inside

the workspace and avoiding them limits the motion in the workspace Defining a path for

the operational point P which does not involve a singular position would restrict the motion

to a portion of the workspace where point D remains on one side of the line joining A and D

In fact, in order to reach the rest of the workspace (corresponding to the other closure of the

closed chain system) the manipulator has to pass through a singular position

When the end point comes to s L= d=0.80m, θ1 θ3 becomes equal to π θ+ 2+θ4, hence a

drive singularity occurs At this position the third row of A becomes T r r3/4 times the

fourth row Then, for consistency of equation (8), the third row of the right hand side of

equation (8) should also be r r3/4 times the fourth row The resulting consistency condition

that the generalized accelerations must satisfy is obtained from equation (15) as

Hence the time trajectory s(t) of the deployment motion should be selected such that at the

drive singularity the generalized accelerations satisfy equation (36)

An arbitrary trajectory that does not satisfy the consistency condition is not realizable This

is illustrated by considering an arbitrary third order polynomial for ( )s t having zero initial

and final velocities, i.e ( ) 3L t22 2L t33

s t = − The singularity position is reached when 0.48s

t = The actuator torques are shown in Figure 2 The torques grow without bounds as

the singularity is approached and become infinitely large at the singular position (In Figure

2 the torques are out of range around the singular position.)

For the time function s(t), a polynomial is chosen which satisfies the consistency condition at

the drive singularity in addition to having zero initial and final velocities The time T when d

the singular position is reached and the velocity of the end point P at T d, v T P( )d can be

arbitrarily chosen The loop closure relations, the specified angle of the acceleration of P and

the consistency condition constitute four independent equations for a unique solution of

T and v T are chosen by trial and error to prevent any overshoot in s or s The values P( )d

used are T = d 0.55 s and v T = P( ) 3.0 m/sd , yielding a T = P( ) 18.2 m/sd 2 s(t) so obtained is

given by equation (37) and shown in Figure 3

2 3 4 5 6( ) 30.496 154.909 311.148 265.753 81.318

Figure 2 Motor torques for the trajectory not satisfying the consistency condition: 1 T1, 2 T2

Furthermore, even when the consistency condition is satisfied, AT is ill-conditioned in the

vicinity of the singular position, hence τ cannot be found correctly from equation (8)

Deletion of a linearly dependent equation in that neighborhood would cause task violations

due to the removal of a task For this reason the modified equation (17) is used to replace the

dependent equation in the neighborhood of the singular position The modified equation,

which relates the actuator forces to the system jerks, takes the following form

which in general do not vanish at the singular position if the system is in motion

Once the trajectory is chosen as above such that it renders the dynamic equations to be

consistent at the singular position, the corresponding θi, θ and i θ are obtained from i

inverse kinematics, and when there is no actuation singularity, the actuator torques T1 and

T (along with the constraint forces λ1and λ2) are obtained from equation (8) However in the neighborhood of the singular position, equation (22) is used in which the third row of equation (8) is replaced by the modified equation (38) The neighborhood of the singularity where equation (22) is utilized is taken as θ1+θ3−θ2−θ4−180o < =ε 1o The motor torques necessary to realize the task are shown in Figure 4 At the singular position the motor torques are found as T = −1 138.07 Nm and T = −2 30.66Nm To test the validity of the modified equations, when the simulations are repeated with ε=0.5o and ε=1.5o, no significant changes occur and the task violations remain less than 10− 4m

Figure 3 Time function satisfying the consistency condition

4.2 Three degree of freedom 2-RPR planar parallel manipulator

The 2-RPR manipulator shown in Figure 5 has 3 degrees of freedom (n=3) Choosing the

revolute joint at D for disconnection (among the passive joints) the joint variable vector of

the open chain system is [ ]T

1 1 1 3

P

c c

Figure 4 Motor torques for the trajectory satisfying the consistency condition: 1 T1, 2 T2

Figure 5 2-RPR planar parallel manipulator

Let the joints whose variables are θ ζ1, 1 and ζ2 be the actuated joints The actuator force

vector can be written as [ ]T

G

5 G

1

3

x y

θ , and F1 and F2 are the translational actuator forces corresponding to ζ1 and ζ2,

respectively Consider a deployment motion where the platform moves with a constant

( ) ( ) sin( ) ( ) cos

The link dimensions and mass properties are arbitrarily chosen as follows The link lengths

are AC a= =1.0m, BD b= =0.4m, BP c= =0.2m, 0∠PBD= = The masses and the α

centroidal moments of inertia are m =1 2kg, m =2 1.5kg, m =3 2kg, m =4 1.5kg, m =5 1.0kg,

kg m2

1 0.05

I = , I =2 0.03kg m2, I =3 0.05kg m2, I =4 0.03kg m2 and I =5 0.02kg m2 The mass

center locations are given by AG1=g1=0.15m, BG2=g2=0.15m, CG3=g3=0.15m,

m

4 4 0.15

DG =g = , BG5=g5=0.2m and ∠G BD5 = = β 0

The generalized mass matrix M and the generalized inertia forces involving the second

order velocity terms R are

33 44

R R R R R

where M and ij R are given in the Appendix i

For the set of actuators considered, the actuator direction matrix Z is

Since A =bζ2sin(θ2−θ3), drive singularities occur when ζ2= or 0 sin(θ2 θ3) 0= Noting that ζ2 does not become zero in practice, the singular positions are those positions where

points B, D and C become collinear

Hence, drive singularities occur inside the workspace and avoiding them limits the motion

in the workspace Avoiding singular positions where θ2−θ3= ±nπ (n =0,1,2, ) would

restrict the motion to a portion of the workspace where point D is always on the same side

of the line BC This means that in order to reach the rest of the workspace (corresponding to

the other closure of the closed chain system) the manipulator has to pass through a singular position

When point P comes to s L= d=0.662m, a drive singularity occurs since θ2 becomes equal

to θ3+ π At this position the third and fifth rows of A become linearly dependent as T 2

s t = − The singularity occurs when t =0.46s The actuator forces are shown in Figures 6 and 7 (In the figures the forces are out of range around the singular position.)

Figure 6 Motor torque for the trajectory not satisfying the consistency condition

Figure 7 Actuator forces for the trajectory not satisfying the consistency cond.: 1 F1, 2 F2

For the time function s(t) a polynomial is chosen that renders the dynamic equations to be

consistent at the singular position in addition to having zero initial and final velocities The

time T d when singularity occurs and the velocity of the end point when t T= d, v T P( )d can

be arbitrarily chosen The acceleration level loop closure relations, the specified angle of the

acceleration of P (γ =200o), the specified angular acceleration of the platform (θ3=0) and

the consistency condition constitute five independent equations for a unique solution of

Bad choices for T d and v T P( )d would cause local peaks in s(t) implying back and forth

motion of point P during deployment along its straight line path

However, even when the equations are consistent, in the neighborhood of the singular

positions AT is ill-conditioned, hence τ cannot be found correctly from equation (8) This

problem is eliminated by utilizing the modified equation valid in the neighborhood of the

singular position The modified equation (17) takes the following form

Figure 8 A time function that satisfies the consistency condition

Once the trajectory is specified, the corresponding η , η and η are obtained from inverse

kinematics, and when there is no actuation singularity, the actuator forces T1, F1 and F2

(and the constraint forces λ1 and λ2) are obtained from equation (8) However in the

neighborhood of the singularity, A is ill-conditioned So the unknown forces are obtained

from equation (22) which is obtained by replacing the third row of equation (8) by the

modified equation (48) The neighborhood of the singular position where equation (22) is

θ −θ + < =ε The motor torques and the translational actuator forces necessary to realize the task are shown in Figures 9 and 10, respectively At the

singular position the actuator forces areT =1 30.31Nm,F =1 26.3NandF =2 1.61N The joint

displacements under the effects of the actuator forces are given in Figures 11 and 12 To test

the validity of the modified equations in a larger neighborhood, when the simulations are

repeated with ε=1o, no significant changes are observed, the task violations remaining less

than 10− 5m

5 Conclusions

A general method for the inverse dynamic solution of parallel manipulators in the presence

of drive singularities is developed It is shown that at the drive singularities, the actuator

forces cannot influence the end-effector accelerations instantaneously in certain directions

Hence the end-effector trajectory should be chosen to satisfy the consistency of the dynamic

equations when the coefficient matrix of the drive and constraint forces, A becomes singular

The satisfaction of the consistency conditions makes the trajectory to be realizable by the actuators of the manipulator, hence avoids the divergence of the actuator forces

Figure 9 Motor torque for the trajectory satisfying the consistency condition

Figure 10 Actuator forces for the trajectory satisfying the consistency condition: 1 F1, 2 F2

To avoid the problems related to the ill-condition of the force coefficient matrix, A in the

neighborhood of the drive singularities, a modification of the dynamic equations is made using higher order derivative information Deletion of the linearly dependent equation in that neighborhood would cause task violations due to the removal of a task For this reason the modified equation is used to replace the dependent equation yielding a full rank force coefficient matrix

Figure 11 Rotational joint displacements: 1.θ1, 2.θ2

Figure 12 Translational joint displacements: 1.ζ1, 2.ζ2

6 References

Alıcı, G (2000) Determination of singularity contours for five-bar planar parallel

manipulators, Robotica, Vol 18, No 5, (September 2000) 569-575

Daniali, H.R.M.; Zsombor-Murray, P.J & Angeles, J (1995) Singularity analysis of planar

parallel manipulators, Mechanism and Machine Theory, Vol 30, No 5, (July 1995)

665-678

Di Gregorio, R (2001) Analytic formulation of the 6-3 fully-parallel manipulator’s

singularity determination, Robotica, Vol 19, No 6, (September 2001) 663-667

Gao, F.; Li, W.; Zhao, X.; Jin, Z & Zhao, H (2002) New kinematic structures for 2-, 3-, 4-,

and 5-DOF parallel manipulator designs, Mechanism and Machine Theory, Vol 37,

No 11, (November 2002) 1395-1411

Gunawardana, R & Ghorbel, F (1997) PD control of closed-chain mechanical systems: an

experimental study, Proceedings of the Fifth IFAC Symposium on Robot Control, Vol 1,

79-84, Nantes, France, September 1997, Cambridge University Press, New York

Ider, S.K (2004) Singularity robust inverse dynamics of planar 2-RPR parallel manipulators,

Proceedings of the Institution of Mechanical Engineers, Part C: Journal of Mechanical

Engineering Science, Vol 218, No 7, (July 2004) 721-730

Ider, S.K (2005) Inverse dynamics of parallel manipulators in the presence of drive

singularities, Mechanism and Machine Theory, Vol 40, No 1, (January 2005) 33-44

Ji, Z (2003) Study of planar three-degree-of-freedom 2-RRR parallel manipulators,

Mechanism and Machine Theory, Vol 38, No 5, (May 2003) 409-416

Kong, X & Gosselin, C.M (2001) Forward displacement analysis of third-class analytic

3-RPR planar parallel manipulators, Mechanism and Machine Theory, Vol 36, No 9,

(September 2001) 1009-1018

Merlet, J.-P (1999) Parallel Robotics: Open Problems, Proceedings of Ninth International

Symposium of Robotics Research, 27-32, Snowbird, Utah, October 1999,

Springer-Verlag, London

Sefrioui, J & Gosselin, C.M (1995) On the quadratic nature of the singularity curves of

planar three-degree-of-freedom parallel manipulators, Mechanism and Machine

Theory, Vol 30, No 4, (May 1995) 533-551

Appendix

The elements of M and R of the 2-RPR parallel manipulator shown in equation (41) are

given below, where m i, i =1, ,5 are the masses of the links, I i, i =1, ,5 are the centroidal

moments of inertia of the links and the locations of the mass centers G , 1, ,5 i i = are

Control of a Flexible Manipulator with Noncollocated Feedback: Time Domain

Passivity Approach

Jee-Hwan Ryu1, Dong-Soo Kwon2 and Blake Hannaford3

Korea University of Technology and Education1, Korea Advanced Institute of Science and Technology2,

University of Washington3

R of Korea1,2 USA3

1 Introduction

Flexible manipulators are finding their way in industrial and space robotics applications due

to their lighter weight and faster response time compared to rigid manipulators Control of flexible manipulators has been studied extensively for more than a decade by several researchers (Book 1993, Cannon and Schmitz 1984, De Luca and Siciliano 1989, Siciliano and Book 1988, Vidyasagar and Anderson 1989, and Wang and etc 1989) Despite their applications, control of flexible manipulators has proven to be rather complicated

It is well known that stabilization of a flexible manipulator can be greatly simplified by collocating the sensors and the actuator, in which the input-output mapping is passive (Wang and Vidyasagar 1990), and a stable controller can be easily devised independent of the structure details However, the performance of this collocated feedback turns out to be not satisfactory due to a week control of the vibrations of the link (Chudavarapu and Spong 1996) This initiated finding other noncollocated output measurements like the position of the end-point of the link to increase the control performance (Cannon and Schmitz 1984) However, if the end-point is chosen as the output and the joint torque is chosen as the input, the system becomes a nonminimum phase one, hence possibly behave actively As a result, the small increment of the output feedback controller gains can easily make the closed-loop system unstable This had led many researchers to seek other outputs for which the passivity property is enjoyed

Wang and Vidyasagar (1990) proposed the so-called reflected tip position as such an output This corresponds to the rigid body deflection minus the deflection at the tip of the flexible manipulator Pota and Vidyasagar (1991) used the same output to show that in the limit, for

a non uniform link, the transfer function from the input torque to the derivative of the reflected tip position is passive whenever the ratio of the link inertia to the hub inertia is sufficiently small Chodavarapu and Spong (1996) considered the virtual angle of rotation, which consists of the hub angle of rotation augmented with a weighted value of the slope of the link at its tip They showed that the transfer function with this output is minimum phase and that the zero dynamics are stable

Despite the fact that these previous efforts have succeeded in numerous kinds of applications, the critical drawback was that these are model-based approaches requiring the system parameters or the dynamic structure information at the least However, interesting systems are uncertain and it is usually hard to obtain the exact dynamic parameters and structure information

In this paper, we introduce a different way of treating noncollocated control systems without any model information Recently developed stability guaranteed control method based on time-domain passivity control (Hannaford and Ryu 2002, Ryu, Kwon, and

by the addition of a virtual feedback of the conjugate variable For a motion control system, the trajectory generator output would be a desired velocity ( )v d , and the virtual feedback would be equal to the controller output ( ) τ (Fig 1)

To show that this consideration is generally possible for motion control systems, we physically interpret these energy flows We consider a general tracking control system with

a position PID and feed forward controller for moving a mass ( )M on the floor with a desired velocity ( )v d The control system can be described by a physical analogy with Fig

2 The position PD controller is physically equivalent to a virtual spring and damper whose reference position is moving with a desired velocity ( )v d In addition Integral Controller

( )u I and the feed forward controller ( )u FF can be regarded as internal force sources Since the mass and the reference position are connected with the virtual spring and damper, we

can obtain the desired motion of the mass by moving the reference position with the desired velocity The important point is that if we want to move the reference position with the desired velocity ( )v d , force is required This force is determined by the impedance of the controller and the plant Physically this force is equivalent to the controller (PID and feed forward) output ( ) τ As a result, the conjugate pair (v dandτ ) simulates the flow of virtual input energy from the trajectory generator, and the conjugate pair (vandτ ) simulates the flow of real output energy to the plant Through the above physical interpretation, we can construct a network model for general tracking control systems (Fig 1), and this network model is equivalently described with Fig 3 whose trajectory generator is a current (or velocity) source with electrical-mechanical analogy Note that electrical-mechanical analog networks enforce equivalent relationships between effort and flow For the mechanical systems, forces replace voltages in representing effort, while velocities representing currents

d

Fig 3 Network view of general motion control systems