An adaptive linearization and control strategy for mechanical manipulators

An adaptive linearization and control strategy for mechanical

manipulators

Citation for published version (APA):

Mazure, P. F. M. (1987). An adaptive linearization and control strategy for mechanical manipulators. (TH Eindhoven. Afd. Werktuigbouwkunde, Vakgroep Produktietechnologie : WPB; Vol. WPA0461). Technische Universiteit Eindhoven.

Document status and date: Published: 01/01/1987

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tAN ADAPTIVE LINEARIZATION AND CXJNTROL

STRATEGY FOR lfECHANICAL MANIPULATORS

~.S.A. AAZURE ~1vers1te de

~echnolog1e de ~omp1egne

Srance

.March - tuly 1987 'N~ppor t nr. ".14-0461

FOREWORD

-~cause several persons will work after me on the same subject and will continue my work. I had to develop the theory and put a lot of details in my report. especially all the formulae I have used to

deve-lop the software.

~ fact. the aim of this report is to render more convenient the task of the people who will follow me. ~or this reason. the thorough-ness of this report should not surprise my readers.

Note also that it was done with the agreement of my coach

~cknowledgements

I want to express my grati tude to my coach. Mr Henk Smi t. who with his competence and kindness. contributed to the progress of my training period.

Greatest thanks to P.C. Mulders. who welcomed me into his team. Leon Pijls. Arthur Udo. Willy Smits, Gerard Kreffer. Henk Van Rooij and also to the staff of the International Relations/external training projects for the perfect organization of the traineeships especially to IR. ~.,. ~rzilius.

AW.

~.~. den

aaan

and

AW.

~.A.J. van

sfme I svoor t .

I also want to thank L. Loron from the Compiegne

University of Technology who visited me and showed me his interest in the work I was doing.

Because of the work atmosphere and the different way of life in the Netherlands. I will never forget my traineeship in this nice country ...

Dank je weI en tot ziens

~ntents

§1.~ry . . . ~e ... 5

§ 2. ~e work environment. . . 7

§ 2.1. ~e engineering education in Netherlands... 7

§ 2.2. She 'indhoven ~iversity of ~echnology... 7

§ 2.3. She Aechanical engineering ~partment. .... ... 9

§ 2.4. She F.L.A. I.R. project... 10

§ 2.5.

Ay

training period project ... 10

§ 3. !Iobot arm dynamics. . . .. . . .. . .. . ... . .. . .. .... ... . . . 12

§ 3.1. Newton-4uler formulation of equations of motion ... 13

§ 3.1.1. ~sic dynamic equations ... 13

§ 3.1.2. ~losed-form dynamic equations ... 14

§ 3.2. ~rangian formulation of manipulator dynamics ... 16

§ 3.2.1. ~rangian dynamics ... 16

§ 3.2.2. She manipulator inertia tensor ... 17

§ 3.2.3. ~riving ~range's equations of motion ... 19

§ 3.3. Inverse dynamics.... . . .. 21

§ 3.3.1. !lecursive computation... 22

§ 3.3.2. Aoving coordinates... 25

§ 3.3.3. ~h-~ul-Walker·s algorithm ... 27

-§ 4. ~ptive control for robot manipulators ... 32

§ 4.1. ~e feed forward component ... 33

§ 4.2. ~e feedback component ... 36

§ 4.2.1. ~rivation of the state space representation ... 36

§ 4.2.1.1. ~erturbation equations of motion ... 36

§ 4.2.1.2. ~rivation of the system parameters A(t).B(t) 38 § 4.2.1.2.1. ~rivation of the t.acobian matrix M .... 39

§ 4.2.1.2.2. ~rivatlon of the 1acobian matrix C .... 41

§ 4.2.1.2.3. ~rivation of the f,acobian matrix K .... 45

§ 5. !!ynamic simulation... 51

§ 5.1 ~clusion of nonrigid body effects ... 51

§ 5.2 ~e simulation itself... 52

§ 5.2.1.'echnique for solving for the joint acceleration 54 § 5.2.2.~e different steps to test the simulation ... 56

§ 6. ne !:fof tware. . . 58

§ 7. dbout the work atmosphere ... 59

§ 8. !onclusion... 60

dppendix ~ontents

1. ~sume en franc;:ais 62

2. lIIbou t PC-MA TLAB 63

3. ~isting of the softwares 68

4. ~ibliography 105

-S.l. ~ry

~dustrial robots are mechanical manipulators whose dynamics

characteristics are highly nonlinear. d purpose of manipulator control is to maintain a prescribed motion for the manipulator along a desired time-based trajectory by applying corrective compensation torques to the actuators to adjust for any deviations of the manipulator from the trajectory.

~ere is a lot of possibilities to control a robot arm

manipulator. but most of the existing control schemes control the manipulator at the joint level or the hand level and emphasize nonlinear compensations of the interaction forces among the various

joints. ~ese control are inadequate because they neglect the changes of the load in the task cycle. ~ese changes are significant enough to render conventional feedback control strategies ineffective. ~

significant performance gain in robot arm control require the

consideration of adaptive control techniques. ~o control a manipulator which carries a variable or unknown load and moves along a planned path. it is required to compute the forces and the torques needed to drive all its joints accurately and frequently at an adequate sampling frequency (no less than 60 Hz since the resonant frequency of most of the mechanical manipulators are around 10 Hz). ~t necessity is in fact the bottleneck of a lot of control strategies because of the number of calculation. ~e aim is to find faster algorithms and also

to use all the possibilities of the existing computers (one tendency is the development of concurrent programming for instance).

~e chapter § 3.of this report presents a new approach of

computation based on the method of Xewton-Iuler formulation which is independent of the manipulator configuration. ~is method involves the successive transformation of velocities and accelerations from the

transformed back from the gripper to the base to obtain the joint torques. ~eoreticaly the mathematical model is "exact"and a further advantage of using this method is that the amount of computation

increases linearly with the number of links whereas the conventional method based on the ~rangian formulation increases as the quartic of the number of I inks.

~e chapter § 4. of this report deals with an adaptive control for a robot manipulator. ~e proposed adaptive control is based on the

linearization of the recursive formulation of ~'s Newton-'uler equations of motion. ~is idea from ~enk ~it allows us to obtain the state matrices A{k) and B{k). When the system parameters are known the regulator matrix is determined with a software like PC-MATLAB.

~e total torques applied to the joint actuators then consist of the nominal torques computed from the Newton-&uler equations and the variational torques computed from the feedback component.

-1.2. ~e work environment

~e ~echnische ~iversiteit lindhoven (T.U.E.lindhoven ~iversity

of ~echnology) has been founded in 1956. ~e former name of this school was : ~echnische ~ogeschool 'indhoven (T.H.E)

1.2.1. ~e engineering education in the Netherlands

'ngineering education at ~iversity level in the Netherlands is concentrated in separate institutes named '~iversiteit'; at this type of institute only academic engineering degrees are conferred. ~ere

are three institutes of that name :

~.~.!elft. founded in 1842, ~.~.8indhoven, founded in 1956,

~.~.~wente, founded in 1961.

~ey group about 20.000 students.

~e first degree which terminates a course of studies is the Jaster's degree. ~ou have three types of Jaster : ~enieur (Ir.) for

technical studies, ~eester (Mr.) for law studies and ~ctorandus

(Drs.) for the rest. ~ engineering subjects, the Jaster's degree entitles the graduate to use legally protected abreviation 'Ir.'

(~genieur in dutch) before his name.

~o obtain the ~ctor degree (~r.) no normal course of study is scheduled. ~e candidate must have an engineer's degree, and to carry out original research.

§.2.2. ~e lindhoven ~iversity of ~echnology

~e ~.~., offers nine courses of study in which students can qualify as graduate engineers specializing in the following fields

- ~echnology in its ~ocial ~plications

- ~dustrial Ingineering and Janagement ~cience

- ~echnical ~hysics - Aechanical gngineering - 'lectrical tngineering - ~emical 'ngineering

- ~rchitecture. ~tructural tngineering and ~rban ~lanning

' t includes about 6.000 students with only 7% of girls.

~ince the ~iversity opened. about 8.000 students have graduated from it. ~e operating costs was in 1985 250 million guilders.

Sive years ago, a new system of studies was introduced by the dutch government. ' t is not very well received by all students.

~e former system gave the students almost complete freedom to study or not study. ~ey could choose more or less freely the semester in which they wanted to take a particular examination.

~is system allowed students to manage their own time and to spent time in out of school activities like:

jobs with or without any educational interest, practicing sports, managing the students association. 'ach

department has its own students

association and their goal is to help students in obtaining extra facilities such as excursions to national

companies. annual excursions to foreign countries {lapan, ~orway, ~rmany. ~lgium. ~rance ... }. organizing international student meetings and many others.

~e nominal duration of the studies was five years. but averagely the students remained at university about seven years

~e new system decreases the freedom of the students and shortens the studies duration to four years, with a maximum of six years. ' t is said to be difficult and to require generally an extra year. ~ccording

to that. students under this system do not take so much time either for many external things. nor for extra training periods. ~eir elders are afraid about how the associative student life will decrease and about the ~iversity level. ~fter spending five months at the ~.~.6 ..

the students I know are in a very large scale proportion under the former system.

-§.2.3. ~e Aechanical Engineering ~partment

~sign and production are the two main groups into which the

highly varied tasks of mechanical engineers are divided. ~e nature of the tasks carried out by the mechanical engineers varies from

scientific research and development to industrial organisation. Next to their theoretical knowledge. mechanical engineers must have specific practical skills. ~o this end. the curriculum includes amongst other things. participation in the work done by the department in its four divisions

- Sundamentals of mechanical engineering - ~oduct design and development

- ~sign for industrial processing

- ~roduction engineering and production automation (W.~.d).

~e mechanical department has about two hundred employees (teachers and technical staff) and nine hundred students. I have worked and lived for five months among them. in the W.~.d. division.

r,....

.-.-§.2.4. ~e S.~.~.,.~. project (S~exible dutomation and 9ndustrial ~bots)

~e research project S.~.~.,.~. is financed and directed by the

~tch government. 'ts aim is to get some experience in flexible automation and industrial robot systems.

1n this project each ~iversity has its own task. ~t the ~iversity of 'indhoven. this project is called S.d.'.~.

~th the mechanical and electrical departments are involved in this project as well as several private companies.

~e project is divided in five sections : - ~e general aspect of automation - :7be handling of parts

- jinematics and dynamics of mechanical structures

- :7be drive systems. the control systems and applications of those systems

- :7be arc-welding and the sensory system

S.2.S.

Av

training period project

dfter the development of a one axis linear robot arm. a new

subject of research was defined in the laboratory; the aim of this new subject is to get more experience in robot control and to use parallel computers.

~e subject is defined as follows

~tudy adaptative control for robot manipulators and the possibility to use state space control techniques.

~sign a robot control based on state space techniques for a 2 or 3 dimensional robot arm (and consider possible parallallism in this control) .

-~imulate a 2 or 3 degrees of freedom robot-arm by implementing its dynamic behavior in a PC-Matlab program. test the robot control with

this simulation program with relation to model inaccuracies (e.g friction. changing payloads,dynamic behaviour of motor •... ).

~us. my work was included in this huge subject. SUring the first six weeks my coach gave me general books about robotics to read in order to obtain the basis which are necessary to perform my task

Sirst I studied :

~ial descriptions and transformations. Janipulators ~inematics.

Janipulators ~tatics.

Janipulators !ynamics.

~rajectory generation and control.

~sition control of manipulators. Sorce control of manipulators. '€Qmpliant motion control.

~fter this first step, I studied more specifics articles about

ddaptative control for robot manipulators

!ynamic computer simulation of robotic mechanisms On-line computational scheme ...

I decided in collaboration with my coach. to implemente a dynamic simulation software of a multi-dimensional robot.

§.3. ~obot arm dynamics

~ this chapter, we analyze the dynamic behavior of manipulator arms. 'he dynamic behavior is described in terms of the time rate of change of the arm configuration in relation to the joint torques exerted by the actuators. 'his relationship can be expressed by a set of differential equations, called equations of motion, that govern the dynamic response of the arm linkage to input joint torques.

'he dynamics of an n-joint manipulator can be derived either by ~grange-'uter (~.t), Mewton-'uter

(M.t),

~cursive ~grangian. or

~nerattzed d·~tembert formulattons. ~wo main approaches, the

~range-'uler and the Mewton-'uler formulations are described here to obtain two different forms of equations of motion suitable for

developing the proposed simulation of an adaptive control strategy. 'he Mewton-Buler formulation is derived by the direct

interpretation of Mewton's ~econd ~w of ~otion. which describes dynamic systems in terms of force and momentum. 'he equations

incorporate all the forces and moments acting on the individual arm links, including the coupling forces and moments between the links. 'he equations obtained from the Mewton-Iuler method include the constraint forces acting between adjacent links. 'hus. additional arithmetic operations are required to eliminate these terms and to obtain explicit relations between the joint torques and the resultant motion in terms of joint displacements.

~ the ~rangian formulation. on the other hand, the system's dynamic behavior is described in terms of work and energy using generalized coordinates. ~ll the workless forces are automatically eliminated in this method. 'he resultant equations are generally

compact and provide a closed-form expression in terms of joint torques and joint displacements. 'he derivation is Simpler and more systematic

than in the Mewton-'uler method.

'he manipulator's equations of motion are baSically a description of the relationship between the input joint torques and the output motion. i.e. the motion of the arm linkage. ~s in kinematics and in statics, we need to solve the inverse problem of finding the necessary input torques to obtain a desired output motion. Recently, efficient algotithms have been developed that allow the dynamic computations to be carried out on-line in real time.

We

will see one of them in the chapter § 3.3.3.

-§.3.1. Newton-euler formulation of equations of motion

§.3.1.1. Masic dynamic equations

'he dynamic equations of a rigid body can be represented by two equations : one describes the translational motion of the centroid. while the other describes the rotational motion about the centroid.~e

former is the Newton's equation of motion for a mass particle, and the latter is caled 'uler's equation of motion.

~or any given link i, and according to the figure 3.1, we derive the two equations

Jointi + 1

Yo

F i-I. i is the coupling force applied to link i by link i-I. Ni -1. i is the moment applied to link i by link i-I.

F 1. 1+1 is the coupling force applied to link i+l by link 1, Ni, 1+1 is the moment applied to link i+l by link 1,

Mi

is the mass of the link.

Ii is the centroidal inertia tensor of link i v

cl is the linear velocity of the centroid of link i.

wi is the angular velocity vector

g is the acceleration of gravity.

Note that all the vectors are defined with the reference to the base coordinate frame 0

-x

y

z .

o 0 0 0

And as the inertia tensor is

J

2 2 {(y-y )

+(z-z )

}pdV c c I _{i c c}

=

-J{X-X

)(y-y )pdV -J{Z-Z _c

)(x-x

_c )pdV

-J(

x-x

_c )(y-y ) pdV _c

J

{(z-z ) +(x-x )

2 2 }pdV c c -J(y-y _c

)(z-z

_c )pdV -J(Z-Z _c

){x-x

_c )pdV -J(y-y _c

)(z-z

_c )pdV

J

2 2

{{x-x)

+(y-Yc) pdV Where p is the mass density. x .y .z are the coordinates of the

c c c

centroid of the link. ' t is obvious that the inertia tensor varies with the orientation of the rigid body.

6quations (1)§(2) govern the dynamic behavior of an individual arm link. ~e complete set of equations for the whole manipulator arm is obtained by evaluating both equations for all the arm links. i=1 .... n.

§.3.1.2. ,nosed-form dynamic equations

~e main problem is that the Newton-6uler equations are not in an appropriate form for use in dynamic analysis and controller design.

~ey do not explicitly describe the input-output relationship. ~o

derive explicit input-output dynamic relations. we need to separate the input joints torques from the constraint forces and moments.

-9ndividual link motions are not independent. they must satisfy certain kinematic relationships to conform to the geometric

constraints. ~us individual centroid position variables are not appropriate for output variables since they are not independent.

~e appropriate form of the dynamic equations therefore consists of equations described in terms of all independent position variables and input forces. i.e. joint torques, that are explicitly involved in

the dynamic equations. !ynamic equations in such an explicit

input-output form are referred to as cLosed-fona dynaatc equattons 9n this context. "closed-form" means a solution method based on

analytic expressions or on the solution of a polynomial of degree 4 or less, such that noniterative calculations suffice to arrive at a

solution. ~s joint displacements q are a complete and independent set of generalized coordinates that locate the whole arm linkage, and as

joint torques T are a set of independent inputs that are separated from constraint forces and moment, dynamic equations in terms of joint displacements q and joint torques T are a closed-form dynamic

equations.

~fter arithmetic manipulations, the closed-form dynamic equations of a n-degree-of-freedom manipulator can be given in the form :

i=l •... ,n

or T{t}

=

H{q} q{t) + V{q,q} +G{q}

where. T is an nxl applied torque vector to joint actuators,

q(t) is the angular positions. q(t) is the angular velocities. q(t) is an nxl acceleration vector, C{q) is an nxl vector of gravity,

V{q.q} is an nxl vector of centrifugal and coriolis terms,

(3)

§.3.2. ¥agrangian formulation of manipulator dynamics

§ 3.2.1. ¥8grangian dynamics

~s we have seen in the last part of this report. arithmetic operations are needed to derive the closed-form dynamic equations. 'his represents a complex procedure which requires physical intuition.

dn

alternative is the ~rangian formulation. which describes the behavior of a dynamic system in terms of work and energy stored. 'hus.

the closed-form can be derived systematically in any coordinate system.

We define the ~rangian ~.by

T is the total kinetic energy stored in the dynamic system. U is the total potential energy stored in the dynamic system. ql •....• ~ is the generalized coordinates.

( 4)

~sing the ~rangian. equations of motion of the dynamic system are given by

i=1 •....• n (5)

where

Q.

is the generalized force corresponding to the generalized

1

coordinate q ..

1

-9.3.2.2 'he manipulator inertia tensor

~s shown in the figure 3.2. v . is the velocity vector of the Cl

centroid and wi the angular velocity vector, with reference to the base coordinate frame. which is an inertial referenc~ frame.

Yo

Figure 3.2 ~ntroidal velocity and angular velocity of link i

~us the kinetic energy of link i is given by

1 T I T

T. = _m.v . v . + -wi Iiw.

1 2 1 Cl Cl 2 1 (6)

~e total kinetic energy stored in the whole arm linkage is then given by

(7)

~e expression for the kinematic energy is written in terms of linear velocity and angular velocity of each link member. which are

(8)

i i

where

J

Lj and

J

Aj are the j-th column vectors of the 3 x n 1acobian matrices

J£

and

J!.

for Linear and Angular velocities of link i. respectively.

5'his yields to :

where H is the n x n matrix given by

n iT i iT .

H

=

~

(m.J

_L

J

_L

+

J

_{A I.}

J

_A

I )

1=1 1 1

(9)

(10)

5'he matrix H incorporates all the mass properties of the whole arm linkage. ~is matrix is named the Aanipulator 9nertia ~ensor.

~t is obvious that the Janipulator inertia tensor is

conFi.guration-dependent and represents the instantaneous composi te mass properties of the whole arm linkage at the current arm

configuration.

~f Hij are the components of these matrix. we can rewrite equation (9) in a scalar form so that.

n n T =_1 v v H · · .G j.G: 1 ij qi qj 2 1=1 (11) 18

-§.3.2.3. Seriving ¥8grange's equations of motion

~ order to derive ~range's equations of motion. we need to find the potential energy

U.

~e potential energy stored in the whole arm linkage is given by :

n

T

U = ~ m. g r i

i=1 1 o.e

(12)

.e can now derive ~range's equations of motion using the total kinetic energy

(II)

and the total potential energy

(12).

n d H ..

~{

a

~

}

=

~{ .~

Hij

qi

dt

a

q.

dt

J-I

1 } n

=

! j=1 •• IJ • Hi . q. + ! - - qJ' J J j=1 dt

8 T

a

{I

n n . . }

1

n

- - = -

-

.! ! Hjk q j qk

= -

.!

a

a

2 J=1 k=1 2 J=1 qi qi

au

n 8 r _{n T '} C i = ---1= ! T o.ej ! mjg r{i mjg =

a

qi j=1

a

qi j=1

'quations (13) through (I6) into (5) yields

..

(I3)

(14)

(15)

where (} Hij h ijk = -(} qk and (18) (19)

~e first term represents inertia torques. including interaction torques. while the second term accounts for the ~oriolis and

centrifugal effects. and the last term is the gravity torque.

'quation (3) is the same as equation (17) derived from Newton-€uler equations. ~us the ~grangian formulation provides the closed-form dynamic equations directly.

or we can also express equation (17) as

•• •

T(t}

=

H(q} q(t} + V(q,q) + C(q) (17' )

-T. I

9.3.3. ~verse dynamics

~e closed-form dynamic equations derived in the previous sections govern the dynamic responses of a manipulator arm to the input joint

torques generated by the actuators. ~is dynamic process can be

illustrated by the block diagram of figure 3.3 where the inputs are joint torques T

1(t) ...• Tn(t). and the outputs are generalized coordinates. joint displacements ql(t) •...• ~(t).

~verse problems are important to robot control and programming. since they allow one to find the appropriate inputs necessary for producing the desired outputs.

Time Arm linkage dynamics I nve~se dynamics q,/t) q/tJ • ... , q,/t) Ti

in this section, we discuss the inverse dynamics process, shown in the block diagramm at the bottom of figure 3.3. ~e inputs are the desired trajectories, described as time functions ql(t) through ~(t).

~e outputs are the joint torques to be applied at each instant by the actuators in order to follow the specified trajectories. and are

obtained by evaluating the right-hand side of the closed-form dynamic equations. ~sing the specified trajectory data.

t=l • . ..• n

~t each instant, we compute joint velocities

q

and joint

accelerations

q

from the given time functions.and then substitute them to the right-hand side of the above equation.

~e total amount of computation becomes extremely large because as. all the coefficients. H

ij, hijk, Gi are configuration-dependent. they need to be computed at each instant. ~us the extremely heavy computation load is a bottleneck for the inverse dynamics.

~t. the inverse dynamics approach is particularly important for control, since it allows us to compensate for the highly coupled and nonlinear arm dynamics. ~us we need to find fast computation

algorithms.

§.3.3.1 ~ecursive computation

~ efficient algorithm for inverse dynamics computation based on the Hewton-Iuler formulation have recently been developed. ' t reduce the computational complexity from O(n4) to

O(n).

She key concept of these method is to formulate dynamic equations in a recurst.ve form, so that the computation can be accomplished from

,

one link to another. ~igure 3.4 illustrates the outline of the

recursive computation algorithm. ~e algorithm can be applied to any manipulator arm with an open kinematic chain structure.

-~is formulation yields a set of forward and backward

recursive.equations which can be applied to the manipulator links sequentially. ~e forward recursion propagates kinematics information of each link from the base reference frame to the end-effector.

~e backward recursion transforms the forces that exert on each link from the end-effector to the base frame and the applied joint torques are computed from these forces.

Kinematics

Dynamics

-fn." .. 1

-Nn,n+ 1

Figure 3.4 ~cursive CORpUtation of kineaatic and dynamic equations

~e first phase of the recursive Newton-Suler formulation (forward computation) is to determine all the kinematic variables that are needed for evaluating the Newton-Suler equations. ~ese include the

linear and angular velocities and accelerations of each link member involved in the serial linkage.

~e algorithm starts with the first link.

- ~iven the joint displacement ql and the joint velocity and accelerations ql and ql' the linear and angular velocities and accelerations of the centroid C are determined.

- ~is procedure is repeated until all the centroidal velocities and accelerations. as well as the angular velocities and

accelerations. are determined for all the links involved.

~e second phase (backward computation) of the recursive formulation is to evaluate the Newton-'uler equations with the computed kinematic variables to determine the joint torques. We now proceed with the recursive computation starting from the last link back to the proximal links.

~o summarize. the backward recursive procedure can be formulated as

(20)

o 0.. 0 0 0 0 0 o·

N_{i - l •i}

="

_1.i+l

-

r _i~ci A F _i.i+l + ri-l.C1.A F_{i - l •1 +} I. ₁ w. ₁ +

(21)

~is procedure is repeated until the link number i reaches i

=

1.

Once the coupling and moment of each joint are determined. the joint torque can be computed with:

- for a prismatiC joint

~ To

D_{i -1 . Fi - l •i} (22)

- for a revolute joint

Ti

=

~i-lT·~i-l.i

(23)

1n

the first phase of the computation. we need to find the

velocity and acceleration of link i. given the motion of the previous link and the specified motion of link i relative to link i-I. ~o solve this problem. we need to analyze relative motions defined in a moving coordinate frame.

1n

the next section we derive basic results about moving coordinates. and then apply these results to the recursive computation algorithm.

-§ 3.3.2. Aoving coordinates

~ccording to the ~navit-~rtenberg convention, one coordinate system is attached to each link of the manipulator. ~us the

coordinates move together with links. ~onsider the following three coordinate systems as shown in figure 3.5 : the base coordinates

(xo'yo'zo) attached to link 0 with origin 0, the coordinates

*

(xi'Yi,zi) attached to link i with origin 0 • and the coordinates (xi+_{1 'Yi+l,zi+1) attached to link i+1 with origin 0'.}

!I+I

Figure 3.5 ~tationships among three coordinate systems

~us 0' is located by the position vector r relative to origin O. or the vector s relative to origin 0*. ~s shown in figure 3.5, these vectors are related by :

r = s + e (24)

o 0

~et v. and ~. be, respectively, the linear and angular velocity

1 1

of coordinate system (xi'Yi,zi) with reference to the base coordinates (x ,y ,z ). _o ~en the velocity of coordinate system (x. l'Yi l'z. I)

0 0 1+ + 1+

(25)

*

where,d /dt denotes the time derivative with respect to the moving coordinates (xi'Yi,zi)' ~us the acceleration is given by

(26)

in which the third and the fourth terms are respectively, the ~riolis

and centrifugal accelerations.

o i

~et w_i+₁ and wi+l be the angular velocity of coordinate system (xi +1'Yi+l,zi+l) with reference respectively the systems (xo'Yo,zo) and (xi'Yi,zi) ~en,

0 0

+ i

wi+l

=

w. 1 Wi+l (27) o· o· i·

wi+l

=

Wi + wi+l (29) ~t.

(29)

hence, (29) becomes

(3O)

tquations (25), (26), (27), (30) are basic relations for

velocities and accelerations among links and base of the manipulator.

-§ 3.3.3. ~-Walker-~ul's ~lgorithm

On the basis of the kinematic analysis on the moving coordinate frame. we now formulate the recursive computation algorithm of

Xewton-'uler dynamic equations. ~e algorithm was originally developed by ~. Walker and ~uL

~e first phase (Sorward computation) consist of kinematic

computations. We derive different recursive equations depending on the type of joint (prismatic or revolute).

Forward Equations with respect to the base frame: for i

=

1, 2 •.... n When the joint i+l is prismatic

0 0 (31) w1+1

=

Wi o· o· (32) w1+1

=

w. 1 0 0 _{• 0} 0 A 0 (33) v1+1

=

_{Vi + q1+1' b i + wi+l} r. "+1 _1,1

o· o· 0 o· 0

v1+1

=

_{Vi + q i+l' b i} + w1+1 A ri,i+l +

0

2

*

_{wi +1 A Q1+1'}

~i

+ 0 wi+l A _(Ow1+₁ A °r_{i •1+1]} (34)

When the joint i+l is revolute

(35)

~e Newton-Iuler equations are expressed in terms of centroidal accelerations. whereas the recursive formulation is expressed with

respect to the origin of the coordinate frame attached to each link.

~is is illustrated in figure 3.6. where v. and w. are.

1 1

respectively. the velocity at the origin of the coordinate frame attached to link i. and the angular velocity of the link.

VI

Figure 3.6 ~ntrotd velocity and Joint velocity

The centroidal velocity is then given by

0 0 0 0

v

ei

=

Vi + Wi A r i.ci o· o· o· 0 0

A [oWi A °ri.Ci] v _ei

₌

v. + w. A r. i + W.

1 1 I.e 1

With all these kinematic parameters. we can now evaluate the backward equations

(39)

(40)

Ba.cluoo.rd .Egua.ttons 'With respect to the base frame for i

=

n ... ,1

(41)

for a prismatic joint. (43)

for a revolute joint (44)

-~e equations derived in the preceding section are referenced to the base coordinate systems. ~fortunately the inertia tensor I. is

1

dependent of the orientation of the link i. which is changing. When we evaluate guIer's equation. the inertia tensor must be obtained for each arm configuration. ~is requires extra computation time. ~us.

the computation is more complicated and especially long.

~o solve this problem. it is better to rewrite the equations (31) through (44) in link coordinates (in spite of the base coordinate frame).

~et

_{i Ri+1} be a 3 x 3 matrix which transforms any vector with reference to (x1+1'Y1+1,z1+1) coordinate system to a new coordinate system whose coordinates are parallel to (xi'Yi'Zi)' i.e. a pure rotation. ~us the following equations are derived :

Forward Equations mith respect to the link coordinates: for i

=

l, ... n When the joint i+l is prismatic

i+l _[tR i+1]-1

*

i wi+l = w. ₁ (45) i+l-

C

_]-1 i · w1+1 = 1Ri+l

_*

wi (46) 1+1 [tR1+1)-1 * [iv1 + z.Qi+l] + 1+1 (" ]-1 1 v1+1

=

w1+1 A 1Ri+l * r 1. 1+1 (47) 1+1-

e

_)-1

[t.

..]

1+1-

e

_]-1 1

v1+1 = 1Ri+1 * vi + z·qi+l + _{w1+1 A 1Ri+l} * r. '+1 + _1.1

2* 1+1 _"'1+1 A [tR_i+₁]-1 _{* z·Qi+l +} •

When joint i+1 is revolute

1+1 "'1+1

i+l

vi+l

=

[i R_i+₁ ]-1 !If i vi + i+l wi+l A [iR i+l ]-1 M i r_{i • i}+₁

{

i+l {[. ]-1 A "'i+l A _{lRi +1} !If

'f'he centroidal velocity is then given by

i i vi + i A i v ei

=

"'i ri .el . i· i · i· A i i _{A [t"'i A i ri •Ci]} v ei

=

v. + 1

"'.

1. r. 1..ci + "'. 1. (49) (50) (51) (52) (53) (54)

Backward Equations with respect to the

link

coordinates: for i=n ....

l

i

[iRi+1] !If 1+1 + M. !If i· (55)

F i-I. i

=

F 1. i+l 1. vei i

[tRi +1] !If 1+1 i (i ] !If i+l

Ni-1. 1

=

Ni • i+l

-

r 1.ei A Ri +l F i • i +1 +

i i _{+ il} i- i

A [tli !If i"'i] (56)

r _{A F i - 1 • i} !If

"'i + "'i

i-l.ei i

-~inally we derive the joint torques as follow

{~-IRiJ-l

M

z}T

i for a prismatic joint (57)

Ti ₌

_*

_Fi-1.i

=

{[i-lRi]-1

*

z}T

i for a revolute joint (58) Ti

_*

_N

_{i -}

_1•

_t

§ 4. ~aptive control for robot manipulators

~ purpose of manipulator control is to maintain a prescribed motion for the manipulator along a desired time-based trajectory by applying corrective compensation torques to the actuators to adjust

for any deviations of the manipulator from the trajectory.

Aost of the existing control schemes control the manipulator at the joint level or the hand level and emphasize nonlinear

compensations of the interaction forces among the various joints. Yhese control are inadequate because they neglect the changes of the

load in the task cycle. Yhese changes are significant enough to render conventional feedback control strategies ineffective. dny significant performance gain in robot arm control require the consideration of adaptive control techniques.

Yhe proposed adaptive control is based on the linearization of the recursive formulation of ~h's Hewton-euler equations of motion. Yhis idea from ~enk ~it allow us to obtain directly the state matrices A(t) and B(t). With the determination of these two matrices, proper control laws can be designed to obtain the required correction torques

to reduce the position errors of the manipulator along the nominal trajectory.

Yhe controlled system is then characterized by feedforward and feedback components. ~sing the recursive Hewton-euler equations of motion seen in the previous chapter as an inverse dynamics of the manipulator. the feedforward component computes the nominal torques which compensate all the interaction forces among the the various

joints along the nominal trajectory. ~sing the linearization of the recursive formulation of ~'s Hewton-Iuler equations of motion. the feedback component computes the variational torques which reduce the position and the velocity errors to zero along the nominal trajectory.

Yhe total torques applied to the joint actuators then consist of the nominal torques computed from the Newton-£uler equations and the variational torques computed from feedback component.

Yhe figure 4.1 shows the different part of the proposed adaptive control.

-Robot link

Parameters Disturbances

A

₁

Trajectory

x (k) Newton-Euler _{un(k) L<7\u(k)} Robot

I EnviroomentI

planning n equations _.~ ¥ = _Manipulator System of motion 6 u(k) , - K ,. II Derivation of Measurements A(k) and B{k)

J

Linearization x (k) of Newton-Euler me equations

o

x(k .J( _~ ''CY x (k) n x me (k)

Figure 4.1 A control Block Diagram of The Adaptive Control System

§ 4.1. Yhe feedforward component

We

derive the nominal joint torques with the ~h-Walker-~ul's

algorithm seen in the chapter § 3.3.3 (equation 45 through 58). Yhis algorithm is the fastest of eXisting algorithms for dynamic

computation. It takes 4.5 milliseconds on average to compute the six joint torques on a PDP 11/45 minicomputer using floating point

assembly language.

Yhe computation procedure is summarized in Figure 4.2. Yhe left half of the figure shows the kinematics computation. while the right half shows the dynamics computation. Yhe kinematics computation

~e input data of joint motions are transmitted horizontally from the left to the right. ~e data in the last column are the output

joint torques computed through the operations shown by the blocks. ~e

equation numbers in each block indicate the computation to be performed at each stage.

~tarting from the top left corner, we first specify the velocity and the acceleration of the base link. Note that in this algorithm we can deal with the case when the base frame of the manipulator arm is

in motion. if the acceleration of the base frame is known.

Note

also that the acceleration of gravity is represented as part of

acceleration of the base frame. so that the effect of gravity can be included without extra computation.

~e first computation block give the velocities and acceleration of the first link according to the link coordinate. which are used in

the second step of the computation. ~lso. the data is transmitted to the right computation block. where the centroidal velocity and

acceleration are obtained. ~e results are further transmitted to the third column, where the Newton-guler equations are evaluated. and the coupling forces and moments are produced. ~e result is used to

compute the joint torque.

-• • • • • • • (45.46.47.48) or (49.50.51.52) (45.46.47.48) or (49.50.51.52) 53 54 1-V C1 'v •. ''''' •

'y •.

'':''If-:~=-11----t

• n-l n-l n-l· n-l· V_{n- 1 ·} "'n-l' V_{n- 1 ·} "'n-l 55 56

1IJ:

7 1 58 •

·

_• • • • • • • • • • • • • • • • • • • •

O.~.~~

~

~ ~

1n 1n 1n

1

n-l.n· n-l.n ~ n (45.46.47.48) or (49.50.51.52) n _v 55 56 n· cn' v cn

§ 4.2. flle feedback component

§ 4.2.1. ~rlvatlon of the state space representation

~fining a 2 x n-dimensional state vector for the system as

and a n-dimensional input vector as

(60)

~e closed-form dynamic equations of a n-degree-of-freedom manipulator can be expressed in state space representation as

x(t) = f(x(t). u(t»

where f(.) is a 2n x 1 nonlinear vector-valued function and continuously differentiable, and n is the number of D.O.F.

§ 4.2.1.1. 'erturbation equations of motion

(61)

~ppose that the nominal states xd(t) of the system (equation 61) are known from the planned trajectory, and the corresponding torques ud(t) are also known from the computations of the joint torques using

the Newton-8uler equations of motion. ~en both xd(t) and ud(t) satisfy equation (61).

(62)

.sing the taylor series expansion on equation (61) about the

nominal trajectory and substracting equation (62) from it and assuming that the higher order terms are negligible, the associated linearized perturbation model for this control system can be obtained :

-6 i(t)

=

A(t) 6

x(t) + B(t)

6

u(t)

(54)

where vxll_d and Vuf1d are the Jacobian matrices of l(x(t},u(t}}

evaluated at xd(t) and ud(t). respectively, 6 x(t)

=

x(t) - xd(t) and 6 u(t) = u(t) - ud(t).

'he system parameters. A(t) and B(t). of equation

(54)

depend on the instantaneous manipulator position and velocity along the nominal trajectory and are thus slowly varying in time. ~cause of the

complexity of the manipulator equations of motion. it is extremely difficult to find the elements of A(t) and B(t) explicitly.

~owever. the design of a feedback control law for the perturbation equations requires that the system parameters of equation

(54)

be known at all times. 'hus a special technique must be used to find the unknown elements in A(t} and B{t).

§ 4.2.1.2 ~rivation of the system parameters ACt) and Bet)

.

.

.

• s T is function of q. q. q. we can derive the differential as follow

a f a r

_•

a r·

_••

dT = _.dq + _ d q + _ d q (65)

aq

aq

aq

this Yields the following matrix form

•• •

AT=MAq+CAq+KAq (66)

..

•

AT1

_a

_~1

_a

_~1

Aq1

_a

_:1

_a

_{:1 Aql}

a

T1

a

T1 Aql

a

q1 _a~

a

ql _a~

a

q1 a~

=

+ +

aT

a T

a

T

aT

_•

a

T

aT AT --./l --./l

A~

-,Jl -,Jl _A~ - - l l - - l l n

_a

_{q1 a~}

_a

_{ql a~}

_a

_{ql a~} then - -1 • A q

=

M (A T - C A q - K A q)

When you use the equation (57) issued from equation (65) you can derive an equation of the same form as equation (64) i.e.:

6 x{t) = A(t) 6 x{t) + B(t) 6 u(t) • 6 Xl 0 0 1 _{6 Xl} 0 0 0 0 6 u l • 6 xn 0 0 I 6x 0 0 n •• =

---

---

• + -6 Xl 6 Xl _M- I K _M- I C +M-1 _{6 u} n •• • 6x _n 6x n 38 -A~ (57) (58)

'ormulated in this way, the derivation of the system parameters is obvious, we just need to evaluate the three matrices M, C and K to obtain A{t) and B{t).

When we have A{t) and B(t), the control problem is to find a feedback control law u{t)

=

g{x{t)} such that the closed loop control

•

system x{t}

=

f{x{t},g{x{t}» is asymptotically stable and tracks a desired trajectory as closely as possible over a wide range of payloads for all times.

§ 4,2.1.2.1. Slerivation of the Jacobian matrix M

~o derive the matrix M, we use the linearization about

q

of Xewton-tuler equations in the vicinity of a nominal trajectory, i.e. equations (45) through (58), this yields

Ltnearizat ton about

q

of the forv:nrd equat ions: When the joint 1+1 is prismatic

1+1 8 w1+1 _ _ _ _ =0 1+1 8 v1+1 _ _ _ _ =0 ••

8'\n

i-8 Wi

*

-(69) (70) (71)

When the joint i+1 is revolute 1+1

a

LJ_i+₁ _ _ _ _ =0 ₍₇₃₎

a

1+1~1+1

[1 ]-1

{a

i~1

8 qi+1} - - - - = Rl+1

*

+ Z. _ _ _ » " -a~ ·a~ 8~ (14) i+1

a

v_i+₁ _ _ _ _ =0 ₍₁₅₎

The linearizntion about

q

of the centroidal velocitu is given by

a

iv ci 0 = (17)

..

a~ i ·

_a

i · i ·

a

v ci Vi

a

LJi i + A T. i = ~.c ••

..

•• (78) 8~ 8~ a~ 40

-Lineartzat ton about

q

of the Backunrd Egua.t ions i i+l

i-o

F_{i -I.i i}

o

_{Fi .i+l} + lIi *

o

v ci

=

Ri+l*

..

••

..

0'\0 0'\0 0'\0 i i+l 1+1

o

_{Ni - I. i i}

o

Ni, i+l _{i i}

o

_{Fi •}i+l

=

R_{i+1 *}

-

r. A Ri+1 * •• •• l.ci •• o~ 0'\0 0'\0 i 0 i· i _{A 0 Fi -1.i + i l} wi r

_*

i-l.ci

..

i •• 0'\0 0'\0

Finally. we derive the coefficients of the matrix M by the linearization about

q

of the joint torques.

for a prismatic joint

for a revolute joint

(79)

+

(80)

(81)

(82)

Note. that it is necessary to compute all these terms for i

=

1 •..• n and for m

=

1 ... n to obtain the matrix M. which is:

o

:'1

o

:'1

o

q1 o~ M

=

o

T OT -Jl -Jl

o

q1 _o~

• Linearization of the forward equa.ttons about q

When the joint 1+1 is prismatic

i+l (" ]-1 i

o

(0)1+1

o

(0). IR1+1 i f I

=

•

-(83) o~ o~ 1+1- 1·

o

(0)1+1

[t

_{]-1 0 (0)1}

=

R1+1 if •

-o~ o~ (84)

o

i+l

[. r

t

iv.

a

qHI}

v i +1 _{IR if I + z.} + = i+l

-

-

_o~

o~ o~ 1+1

A {[iRi+1]-1

M

iri.i+l}

o

"'1+1 • o~ (85) 1+1 i -1 0 ql+1

{

.

}

2

M

"'1+1 A [RHI] .z.

a

q",

+

o

(0)1+1 1+1 1 -1 1 1+1 { { }}

a

ci",

A

"'HI A [R1+I]

M

r 1.1+1 +

1+I"'i+1 A

{a

:+::1+1 A

{[i

_R1+I]-1

M

irioi+I}}

(86)

-42-When the joint

1+1

Is revolute:

8

i+l~i+l

(i

)-1 {8

i~i

8 qi+l}

- - - - =

Rl+1

*

+ Z. _ _ _

8Q...

8Q...

8Q...

(87) (SS) (89)

1+1

f8

i+l~i+l

{[I

]-1

i }} "'1+1 A

a

ci,.

A Ri+l Of r i. 1+1 (90)

The linearization of the centroidal velocity about

9

is given by

1 1 8

VI

8 ~1 i _ _ + A r . .

=

1.C1

8Q...

8Q...

(91)

Linearization about

q

of the Ba.ck.unrd Equations i a HI i· a F. 1 . _Ft,i+l a v 1- ,1 iR

*

M

*

ci (93)

=

_i+l + i •

·

• a~ a~ a~ i i+l i+l aNi_I. i iR

*

a Nt,Hl i i a F1 • 1+1

=

-

r

_A

_{Ri +l}

_*

+ HI i.ci _• • • a~ a~ a~ i i· i a a i a F. 1 . + iI

*

wi w. A eli

*

1w1 ) + A 1- ,1 ₊ 1 r1-I.ci i

·

• • a~ a~ a'\n 1

A

{iIi

*

a>}

w. I a~

Finally. we derive the coefficients of the matrix C by the linearization about

q

of the joint torques :

a Ti {(i-l

]-1 }T

a iFi_l,i

- - =

Ri

*

Z

* ____ _

a

~

a~

for a prismatic joint

a Ti {[i-l

]-1 }T

a iNi_l.i

- - = Ri

*

Z

* ____ _

a~

a~

for a revolute joint

We

need to evaluate all these terms for i

=

1 •....• n and for m

=

1 •....• n to obtain the matrix C which is :

a :1 a :1 a ql a~ C= aT aT ~ ~ a ql a~ 44 -(94) (95) (96)

§ 4.2.1.2.3. Serivation of the Jacobian matrix K

~s in the two last section. to derive the matrix K. we use the linearization about q of equations

(45)

through

(58),

this yields Linea.rizat ion of the Forward equa.t tons about q

When the joint 1+1 is prismatic

8 1+1 v_i+₁ =

=

(97) (99) (OO)

8 i+lVi

+

1

=

8 [1R1

+

1]-1

*

[iV1 +

z

*

qi+l]

+ 8~ 8~ 1+1-

A {8

"i+l 1+1 8 "0+1 2

*

1 1+1 2

*

"'i+l A _{ 8 8 "'°+1 i i { 1+1 { -1 }} A (J

'1,.'

A

[Ri+l]

*

r1

,1+1

+ 46 -(tOO)

When the Joint i+l is revolute HI 8 [tRHl]-l

*

[twi + z·qHl] + [iRi+l]-l 8 i 8 w Hl w.

*

1

=

(lOl) 8'\0 8'\0 8'\0 i+l-

r

]-1

8 w HI 8 1RHl _{(i -}

-.

i A Z.qi+l] +

=

*

wi + z·QHl + wi 8'\0 8'\0

[i

r ..

t

i~i

8 1 wi } RHI + A z.QHl 8'\0 8'\0 (102) i+l _w

A {8

HI {l03}

8

i+l';'i+l

8

(iR

]-1

i+l t·

----=

- - - *

vi +

8

1+1~i+l

{(I

]-1 _ _ _ _ A RHI

i+1

{a

I+1~i+1

{(I ]-1 i }}

wl+1 A 8

~

A Rl+1

*

r1 ,1+1 +

(104)

The linearization about g of the centroidal velocity is given by

8 i v 8 i i i ci Vi 8 ~. i i 8 ri . + 1 A r i . + A .Cl ~.

=

.Cl 1 (105) 8~ 8~ 8~ 8~ i· i· i · i 8 v . 8 v. _{8 ~i} i i · 8 ri . Cl 1 + A r. i + A .Cl +

=

l,C Wi 8~ 8~ 8~ 8~ i { 1 } 8 Wi A

[t~i

A irl,Ci] + i 8 ~i i ~i _{A 8}

~

_A

ri,ci

₊ 8~ i A

{iWI

A 8

:r~Cl}

~i (106) 48

-Ltneartzat ton about q of the Backunrd Equat tons

-

1-a

v .

M

*

C1 t

=

a

[iR1+1] a~

a

1+1F 1+1 + lR

*

1.1+1

*

F1 •t+1 1+1 + a~

a

1+1N

*

1+1N + lR

*

1.1+1 1.1+1 1+1 (107) 1+1 i

a

i+al:i.i+l} +

*

F1 ,1+1 + Rl+1

*

---"111--1 1

a

F.

1 . r 1-1•ci A 1- ,1 a~

a l a

11 r. 1 1 1 1 . 1- ,C

A F

*

1- + - - - 1-1,1 + - - (,)1 (lOS)

ginally. we derive all the coefficients of the third matrix K by the linearization about q of the joint torques as follow

{C

t}

{ }

i 8 T _{8 1-1R.} T -1 T _{8 F. 1 .} 8

~ ~

a

'1,.'

.. '" ..

_{i Fi_l • t} + [i-lRt] .. Z .. 1- ,1 {l09} 8'\n 8 Ti

{B

[i-1Ri]-1

.. zr ..

{ -1

r

B

i t ₊ [t-1Rt] ,. Z ,. Nt - 1 •i _{llO}

=

Ni-t. i 8'\n 8'\n 8'\n

We

need here also to evaluate all these terms for i

=

1 •...• n and for M

=

1 •....• n to obtain K which is

~

8 T] 8 ql 8~ K = 8T 8 T - - 1 l - - 1 l 8 ql 8~

Now that we have the M. C. K matrices the derivation of A(t} and B(t) is easy. (see equation 68)

Note also that is it not necessary to compute all the terms which are known to be equal to zero. gor instance

w.

is only a function of

1 ~ when m is less or equal to i.

-§ 5. ~ic ~imulation

§ 5.1 . .1'nclusion of nonrigid body effects

' t is important to realize that the dynamic equations we have derived do not encompass all the effects acting on a manipulator. Yhey

include just those forces which arise from rigid body mechanics. Yhe most important source of forces that are not included is friction. ~ll

mechanisms are, of course affected by frictional forces . .1'n present day manipulators in which significant gearing is typical. the forces due to friction can actually be quite large (perhaps. 25% of the torque required to move the manipulator in typical situations.

~ order to make dynamics equations reflect the reality of the physical device. it is important to model (at least approximately)

these forces of friction. ~ very simple model for friction is viscious friction in which the torque due to the friction is proportional to the velocity of joint motion. Yhus we have:

•

T _{friction -}- vq _• (111 )

where v is a viscious friction constant. ~other possible simple model for friction, ~oulomb friction, is sometimes used. ~oulomb friction is constant except for a sign dependence on the joint velOCity:

•

Tfriction

=

c sgn(q}. (112)

where c is a ~oulomb friction constant. Yhe value of c is often taken

•

at one value when q

=

0, the static coefficient, and at a lower value. the dynamic coefficient when

q

¢ O. Whether a joint of a particular manipulator exhibits viscious or ~ulomb friction is a complicated

' t turns out that in many manipulator joints. friction also displays a dependence on the joint position. ~ major cause of this effect might be gears which are not perfectly round (their

eccentricity would cause friction to change according to joint position. ~o fairly complex friction model would have the form

•

Tfriction

=

f(g,g) (11~)

~se friction models are then added to the other dynamic terms derived from the rigid body model. yielding the more complete model

_ .

.

T(t)

=

H(q) q(t) + V(q,q) + C(q} + F(q.q). (ll5) ~ere are also other effects which are also neglected in this model. ~or example. the assumption of rigid body links means that we have failed to include bending effects (which give rise to resonances)

in our equations of motion.

§ 5.2. ~e simulation itself

~cause of the combinatorial complexity of the equations. researchers in the past have applied simplifying assumption to the model for the mechanism under consideration so that a solution could be obtained. ~owever. the results thus obtained may only be valid in a

limited range of operation. (for instance coriolis forces are non-negligible in high-speed motion).

~ecently. more general solutions have been obtained for the dynamic equations of motion in which most of the simplifying assumptions have been removed (Newton-Iuler equations).

~ block diagram showing the component part of a typical dynamic computer simulation for control law development is given in

Figure 5.1.

-Motion Design lqd Feedback ---+ <Alntrol

t

---+ Dynamic Equations

1

q

Motion Integration

1

q,q

Figure 5.1 Block diagram of a typical computer simulation

~s trained or commanded by human operator or as automatically generated through programmed algorithms. the complete trajectory is furnished by the Aotion !esign section. ~rough a control law. torques are applied by the actuators to the mechanism. ~rough solution of the dynamic equations. the joint accelerations may be obtained. and

through integration. the actual trajectory is determined.

~e basic problem as shown in Figure 5.1 is to solve for the relative joint accelerations given the input torques at a particular trajectory point (specified joint positions and velocities).

~s we use the Newton-Suler formulation to obtain the dynamic equations of the manipulator. these equations contain the forces and

the moments due to physical constraints at each joint. ~t. to

simulate the manipulator. these forces and moments are eliminated from the equations. ~ooker and Aargulies have shown how this can be done

!I't is

-

• •

T ·

H(q)q + V(q,q)q + C(q) + K(q) k + F(q,q)

=

T (116)

where

H(q) n X n symmetric, nonsingular moment of inertia matrix.

V(q.q) n x n matrix specifying centrifugal and coriolis effects. C(q) n x 1 vector specifying the effect due to the gravity.

K(q) 6 xn Jacobian matrix specifying the torques (forces) created at each joint due to external forces and moments exerted on link. the last link.

k 6x 1 vector of external moments and forces exerted on the last link

T nx 1 vector of torques (forces) at each joint actuator.

q n x 1 vector of joint variables.

.

F(q,q) nx 1 vector which represents the friction models

§ 5.2.1. ~echnigue for solving for the 'oint ~ccelerations

~rom equation (II6). note that the joint torques (forces) are linear functions of the joint accelerations. ~erefore if b is defined to be a bias vector which is equal to the torques (forces) due to the gravity, centrifugal and coriolis accelerations, and external forces and moments on the last link. We can also include the friction model

in the bias. ~en

• •

T ·

b

=

V(q,q)q + C(q) + K(q) k + F(q,q) (117)

~us the accelerations of the joint variables can be obtained by solving the linear equation

H{q)q

=

(T - b) (118)

-54-~is bias vector b can easily be computed by a function which compute the ~ewton-'uler equations with q.q and k to their current state. but letting

q

=

O. ~ fact. b is the torque computed

considering that

q

=

O. So take in account the friction we must first compute F(q.q) and after add that vector into the bias.

~e difficult part in solving equation (116) is in evaluating the elements of the matrix H. ~ere is different way to obtain it, the easiest is accomplished by setting q to its current state. but letting

..

_q

₌

_e

j • and use a function which calculate the ~-Walker and ~ul's algorithm where the velocity terms. the gravitational effects and the efffects due to external forces and moments are eliminated. Where e_j

is a n x 1 vector with the jth element equal to 1 and 0 everywhere else. ~e result is h

j which is the jth column of H. Shat is. hj is the torque (force) on the joint actuators when the joint velocities are zero; there are no external forces; there are no gravitational effects; and the joint accelerations.

q

are equal to e_j .

Once the elements of H are determined. then the joint

acceleration is obtained by solving equation 118. ~e exact procedure is: STEP 1: Compute the bias vector b, using function 1.

STEP 2: Compute the matrix H one column at a time. using function 2.

STEP 3: Solve the linear equation 118 for the joint accelerations

~is method to compute the joint acceleration is the easiest. but. more efficient methods exist. One of these takes into account the fact that the moment of inertia matrix H is symmetric. but also utilizes a recursive procedure for computing the mass. and the moment of inertia matrix of the composite system of links j through n. !ecause I did not have enough time I could not implemente that method. but you can find more information about it in the article "Efficient Dynamic Computer