Advanced Engineering Dynamics 2010 Part 11 pps

8.3 Kinematics of robot arms In this section we shall first revise and extend the study of the kinematics of a rigid body with particular reference to rotation about a point and change

8

8.1 Introduction

In this chapter we examine the way in which three-dimensional dynamics is applied to a sys- tem of rigid bodies connected by various types of joints Initially we shall describe some typical arrangements of robot arms together with their end effectors We shall only be con- cerned with the overall dynamics and not with the detail This is a vast subject area of which dynamics is a substantial and vital part

8.2 Typical arrangements

8.2.1 CARTESIAN CO-ORDINATES

Figure 8.1 shows the arrangement of a rectangular robot arm where the position of the end effector is located by specifying the x, y, z co-ordinates Each joint responds to one co-ordi- nate, and all joints in this arrangement are sliding joints An end effector is usually a grip- per or hand-like mechanism; these will be briefly described later

Fig 8.1

Typical arrangements 195

8.2.2 CYLINDRICAL CO-ORDINATES

A typical cylindrical co-ordinate arm is shown in Fig 8.2 In this case the joints respond to

r, 8 and z co-ordinates with the joints being sliding, revolute and sliding respectively,

Fig 8.2

8.2.3 SPHERICAL CO-ORDINATES

As can be seen from Fig 8.3 this arm is controlled by specifying r, 0 and 0 with the joints being sliding and two revolute

Fig 8.3

8.2.4 REVOLUTE ARM

A very common layout is shown in Fig 8.4(a) in which all joints are revolute; this is a ver- satile system and more akin to the human arm

8.2.5 END EFFECTOR

A simple end effector in the form of a gripper is shown in Fig 8.5 This example has three degrees of freedom plus a gripping action The movements at the wrist are often referred to

Fig 8.5

Kinematics of robot arms 197

as roll, pitch and yaw It is quite common to find that for some end effectors only roll and pitch are provided

8.3 Kinematics of robot arms

In this section we shall first revise and extend the study of the kinematics of a rigid body with particular reference to rotation about a point and change of reference axes The con- cept of homogeneous transformation matrices will then be introduced so that a systematic description of arm position and displacement can be made

The most common task to be performed is: given the path of the end effector, determine the magnitudes of the joint displacements as functions of time This is referred to as the inverse kinematic problem and is usually more difficult than the forward problem of calcu- lating the path of the end effector given the joint positions The obvious exception is the case

of the Cartesian system

For one position of a cylindrical system

r = \ (x’ + y’)

z = z

e = arctanb/x)

r = , (x’ + y’ + z’)

e = arctan [ z / , (x’ +yz) 3

0 = arctan(y/x)

and for a spherical system

For the revolute arm of Fig 8.4(b)

8, = arctan b/x)

Y = \ (x’ + y 2 )

c = , (r‘ + 2’)

A = arccos [ (L: + c’ - L : ) / (~L, c) 1

B = arcsin [ ( L , / L , ) sinA 3

8, = arctan (ZIT) - B

~ , = A + B

A position vectorp (shown in Fig 8.6) has scalar componentsp,,p, andp, when referred to

the xyz frame This is written

P = ip, + ip? + kp,

which, in matrix form, becomes

198 Robot a m dynamics

Fig 8.6

If we let

(PI = @XP."Pd'

and

(e) = ( i j k l T

then

If the same vector is viewed from the set of primed axes as shown in Fig 8.7

Fig 8.7

8.3.2 CO-ORDINATE TRANSFORMATION

Since

Kinematics of a robot arm 199 let us premultiply both sides by (e’), it being understood that the products of the unit vec-

tors shall be the scalar products

Thus

(e’)(e’)T(p’) = (ef>(e)T@> (8.8)

Now

3 3 il.,*f 3 kl

(e‘)(e’)T - ( ’:) ( 3 jl kl ) = [ j f il j f j l j l kl ]

k’ k’ i’ k1.j’ k’ k’

so that equation (8.8) reads

and

3 i 3 j 3 k

(e’)(e)T = ( ; : ) ( i j k ) = [ j ’ i j ’ j j ’ k ]

k‘ k’ i k’ j k’ k

1: rn: n,

where I,, m,, n, are the direction cosines between the x’ axis and the x, y z axes, as shown

in Fig 8.8

Now let [ E ] = (e’)(e)T so that equation ( M a ) becomes ( p ‘ ) = [ Q ] ( p ) ; therefore [ E ] is a

transformation matrix From the definition of the inverse of a matrix ( p ) = [Q]-’(p’) but by

Fig 8.8

200 Robot arm dynamics

premultiplying equation (8.7) by (e) and noting that (e)(e)’ = [Z], the identity matrix, we

obtain ( p ) = (e)(e’)T(p’) By inspection it is seen that (e)(e’)’ is the transpose of (e’ e

This is also seen from the rule for transposing the product of matrices, that is [(e’)(e) 3 -

From this argument it is apparent that [a]-’ = [a]’, so by definition [a] is an orthogonal

F T ’ 1

(e)(e’IT

IlWtriX

8.3.3 FINITE ROTATION

We shall now consider a closely related problem, that of rotating a vector

Consider a vector pI relative to fixed axes X; I: 2 A further set of axes, U , K W, moves withp, and may be regarded as rigidly fixed top, If the U W a x e s are rotated about the ori- gin then relative to the fixed axes p, moves topz as shown in Fig 8.9

Fig 8.9

Using the prime to indicate components seen from the UVWaxes we have that initially

( p ; ) = ( p , ) We now look at p2 from the rotated axes U W so that its components ( p i ) =

[11c](p2), but because the vector is fixed relative to the UWaxes, ( p i ) = ( p , ’ ) = ( p l ) and thus

( P A = [‘Ql-I(PI 1

If we define the rotation matrix [R] by ( p J = [ R ] ( p l ) then

[RI = [a]-’ = [a]’ = (e)(e’)’

ROTATION ABOUT X, Y AND 2 A X E S

(8.1 1 )

8.3.4

In general the rotation matrix is given by

( 1) [ i i’ i j ’ i k ’ ]

[ R ] = j ( T j ’ k ’ ) = ji’ j j ‘ j k ‘ (8.12)

k i’ k j’ k k‘

So for rotation of the U W a x e s by an angle a about the Xaxis, refemng to Fig 8.10, and

noting that i = i’ and that j * j ’ = cos(ang1e between the Y axis and the V axis) = cos a, etc., the rotation matrix is

Kinematics of a robot arm 201

Fig 8.10

[ 1 s i n a cos a ]

This result should be verified by simple trigonometry

Similarly for a rotation of p about the Y axis

cos p 0 sin p

[ -:np : c:sp]

and for a rotation of y about the Z axis

cosy -sin y 0

Note that by inspection

That is, the transpose is the same as the inverse which is also the same as rotation by a neg- ative angle

8.3.5

In this section we shall adopt a simpler notation for rotation matrices, replacing [ R ] , , by

[X,a] to mean a rotation of a about the fixed X axis

If a vector with components ( p l ) as seen from the fixed axes is rotated about the Xaxis

by an angle a, then the new components are

SUCCESSIVE ROTATIONS ABOUT FIXED AXES

202 Robot arm dynamics

If now this vector is rotated about the Y axis by an angle j3 then the components will be

It follows that any firther rotations result in successive premultiplications by the appropri- ate rotation matrix

In the above case the new composite rotation matrix is

P I = [Y,Pl[X,al = [ 0 1 0 ] [ i coo; -Si.si;]

-

- [ -sin c: P SPSa Ca 0 cos j3 -Sa SPCa ]

where the usual abbreviations are made by writing C for cosine and S for sine

because [X,al[Y,Pl * [Y,Pl[Xal

It must be emphasized that reversing the order of the rotations produces a different result

8.3.6

If we wish to form the rotation matrix for a rotation of 0 about an axis defined by the unit vector n as shown in Fig 8.1 1 , one method is given in the following steps:

1 Rotate the axis of rotation so that it coincides with one of the fixed axes

2 Rotate the body by 0 about that axis

3 Rotate the axis back to its original position

ROTATION ABOUT AN ARBITRARY AXIS

Fig 8.1 1

Kinematics of a robot arm 203

Refemng again to Fig 8.1 1,

Step 1: Rotate the axis about the Y axis by p followed by a rotation of y about the Z axis;

tan j3 = n/l and sin y = m, where 1, m and n are the components of the unit vector n Note that in this example y would be numerically negative

Step 2: Rotate by 0 about the Xaxis

Step 3: Rotate back

In matrix form

ERI,, = {[Y,-PI [Z,YI) {[x,0I) {[Z,-Yl[Y,Pl)

(Remember that [Y,P]-' = [Y,-p].)

Alternative method

A vectorial relationship can be achieved as is shown in Fig 8.12 Here n is the unit vector

in the direction of the rotation and 0 is the finite angle of rotation Owing to the rotation the vector r becomes r' The vector r generates the surface of a right circular cone; the head of the vector moves on a circular arc PQ N is the centre of the circular arc so

n - r = I r I c o s a = O N

and

I n x r l = I r I s i n a = NP = NQ

Note also that the direction of n X r is that of VQ +

Now

r ' = O N + NV + VQ

= n(n * r) + [ r - n(n r)] cos 0 + (n x r) sin 0

Fig 8.12

204 Robot ann dynamics

= r cos 0 + n(n - r)(l - cos 0 ) + (n x r) sin 0 (8.19)

If we use the same basis for all vectors then the above vector equation may be written in

matrix form (see Appendix 1 on vector-trix algebra) as

(8.20)

(r)' = (r)cos 0 + (n)(n)T(r)(l - cos 0) + [nix (r)sin 0

where

(n) = (/mnlT (4 = ( v Y z ) T

0 -n m

[nIX = [ -; ; ; ]

(l, m and n are the components of n referred to the chosen set of axes)

8.3.7 ROTATION ABOUT BODY A X E S

It is very common for rotation to take place about axes which are fixed to the body and not

to axes which are fixed in space For example, with the end effector, or hand, the axes of pitch, roll and yaw are fixed with respect to the hand

Let us first consider a simple case of just two successive rotations In Fig 8.13 a body with body axes UVW is initially lined up with the fixed XYZ axes The body is first rotated

by a about the Xaxis and then by y about the Z axis Exactly the same result can be obtained

by a rotation of y about the W axis followed by a rotation of Q about the U axis This can

best be demonstrated by using a marked box as shown in Fig 8.13(a)

The rotation matrix for the first case is

[RI = [Z,rl[x,aI

The form of a transformation matrix for rotation about the Xaxis is identical to that for rota- tion by the same angle about the U axis, similarly for the Y and V axes and also the Z and

Waxes So [Z,y][X,a] must be equivalent to [ W,y][U,a] Note that the matrix for the second rotation now -multiplies the matrix for the first rotation rather than =multiplying as it

Fig 8.13 (a)

Kinematics of a robot arm 205

Fig 8.13 (b)

did in the case of rotation about fixed axes Because the first two rotations were completely abitrary it follows that the rule is general However, further justification will now be given After the two rotations just made a further rotation p is now made about the V axis This could be treated as a rotation about an arbitrary axis by rotating the body back to the initial position, rotating about the Y (or V ) axis and then rotating the body back again That is,

{rotate back} {[Y,p]} {return to base) {first two rotations)

[RI = [Z,rI[Xal [Y,PI [x,-al[Z,-rI [Z,rI[Xal

Note that [Z,-y][Z,y] = [X,-a][X,a] = [fl, the identity matrix This process can clearly be repeated for any further rotations about body axes

In summary, for rotation about a fixed axis the new rotation matrix premultiplies the exist- ing rotation matrix and for rotation about a body axis the new rotation matrix postmultiplies the existing rotation matrix

8.3.8 HOMOGENEOUS CO-ORDINATES

The objective of this section is to find a way of producing transformation matrices which

will allow for translation of a body as well as rotation

For a pure translation u of a body, a point defined by a vectorp, from some origin will be transformed to a vector pz where pz = p I + u, or in terms of their components

Pzr = Plr + 4

P2, = PI, + 4

P2 = PI; 4- u:

or

206 Robot arm dynamics

For a combined rotation followed by a translation

If we now introduce an equation

(where (0) = (0 0 O)T, a null vector), we may now combine equations (8.23) and (8.24) to give

(8.25)

or, in abbreviated form,

Here ( 3 ) is the 4 X 1 homogeneous vector and [TI is the 4 X 4 homogeneous transfor- mation matrix In projective geometry the null vector and unity are replaced by variables so that the transformation can also accomplish scaling and perspective, but these features are not required in this application

For pure rotation (u) = (0) and for a pure translation [R] = [ I ] (the identity matrix) There-

fore if we carry out the translation first (which is simply the vector addition ofp, and u ) and then perform the rotation the combined transformation matrix will be

so the transformed vector is

(P2) = [ R l ( P l ) + [Rl(u) = [ R l ( ( P l ) + (0

as would be expected Note that rotation followed by translation produces a different result This is because the rotation is about the origin and not a point fixed on the body

8.3.9

Figure 8.14 shows a Cartesian co-ordinate robot arm It is required to express co-ordinates

in UVWaxes in terms of the XYZ axes This can be achieved by starting with the UVWaxes

coincident with the XYZ axes and then moving the axes by a displacement L parallel to the

2 axis, M parallel to the Xaxis and then by N parallel to the Y axis (The order of events in

this case is not important.) Writing this out in full we obtain the overall transformation matrix

TRANSFORMATION MATRIX FOR SIMPLE ROBOT ARM

O l O N 0 1 0 0 0 1 0 0

[: : : :I[: : : :I[ ," ," : :I_ [ 8 : : ,I

This result is equivalent to a single displacement of (M N L)T

We now consider a spherical co-ordinate arm as shown in Fig 8.15 again starting with the two sets of axes in coincidence First we could translate by d, along the Xaxis, then rotate

Kinematics of a robot arm 207

Fig 8.15

by 0 about the Y axis followed by 8 about the Z axis The overall transformation matrix is

[Z,~l[Y,0l[d,l

ce -se o o c0 o s0 o l O O d ,

208 Robot arm dynamics

c e c a -se ces0 dxcecO

-

The above sequence could be interpreted as a rotation of 0 about the Waxis, a rotation of 0 about the V axis and finally a translation along the U axis, as shown in Fig 8.16

K W

Fig 8.16

8.3.10 THE DENAVIT-HARTENBERG REPRESENTATION

For more complicated arrangements it is preferable to use a standardized notation describ- ing the geometry of a robot arm Such a scheme was devised in 1955 by Denavit and Harten- berg and is now almost universally adopted

Figure 8.17 shows an arbitrary rigid link with a joint at each end The joint axis is desig- nated the z axis and the joint may either slide parallel to the axis or rotate about the axis To make the scheme general the joint axes at each end are taken to be two skew lines Now it

is a fact of geometry that a pair of skew lines lie in a unique pair of parallel planes; a clear visualization of this fact is very helpful in following the definitions of the notation The ith link is defined to have joints which are labelled (i - 1) at one end and (i) at the other It is another geometric fact that there is a unique line which is the shortest distance between the

two z axes, shown as Nf-l to 0, on Fig 8.17, and is normal to both axes (and both planes)

If the joint axes are parallel then there is not a unique pair of planes, so choose the pair which are normal to N,-, 0, The origin of the (i - 1) set of axes by definition lies on the

z , - ~ axis but the location along this axis and the orientation of the x , - ~ axis have been deter- mined by the previous links

The ith set of axes have their origin at N, and the x, axis is the continuation of the line Nf-l

to 0, If the joint axes are in the same plane it follows that x, is normal to that plane This can be seen if the two planes are almost coincident