Behind every robotic movement
An industrial robot is a general-purpose manipulator
that consists of several rigid bodies, called links, con-
nected in series by revolute or prismatic joints (Figure 1).
One end of the chain is attached to a supporting base,
while the other end is free and equipped with a tool to
manipulate objects or perform assembly tasks. The mo-
tion of the joints results in relative motion of the links.
Mechanically, a robot is composed of an arm (or primary
frame) and a wrist subassembly plus a tool and is designed
to reach a workpiece located within its work volume. The
work volume is a sphere of influence for a robot whose
arm can deliver the wrist subassembly unit to any point
within the sphere. The arm subassembly typically consists
of three degree-of-freedom movements, which together
place or position the wrist unit at the workpiece. The wrist
subassembly unit usually consists ofthree rotary motions,
often called pitch, yaw, and roll, and their combination
orients the tool according to the configuration of the ob-
ject to ease pickup. Hence for a six-joint robot, the arm
subassembly is the positioning mechanism, while the wrist
subassembly is the orientation mechanism.
Many commercially available industrial robots are
widely used in simple material handling, spot/arc weld-
ing, and parts assembly, including the Unimate 2000B
and PUMA 260/550/560 series robots by Unimation
Inc.; the T3 by Cincinnati Milacron; the Versatran by
Prab; the Asea robot; and the Sigma by Olivetti of Italy.
These robots, which exhibit their characteristics in mo-
tion and geometry, fall into one of four basic motion-
defining categories (Figure 1):
* Cartesian coordinate (three linear axes),
* cylindrical coordinate (two linear and one rotary
axes),
* spherical or polar coordinate (one linear and two
rotary axes), and
is a series of complex geometric
evaluations and equations that
describe motion dynamics.
Conformance to system goals
a must also be considered.
Robot Arm
Kinematics,
Dynamics,
and Control
C. S. George Lee, University of Michigan, Ann Arbor
* revolute or articulated coordinate (three rotary
axes).
Most automated manufacturing tasks are done by
special-purpose machines that are designed to perform
prespecified functions in a manufacturing process. The
inflexibility of these machines makes the computer-con-
trolled manipulators more attractive and cost-effective in
various manufacturing and assembly tasks. Today's in-
dustrial robots, though controlled by mini-/micro-
computer, are basically simple positional machines. They
execute a given task by playing back prerecorded or
preprogrammed sequences of motions that have been
previously guided or taught by a user with a hand-held
control/teach box. Moreover, because the robots are
equipped with few or no external sensors (both contact
and noncontact), they cannot obtain vital information
about their working environment. More research needs to
be directed towards improving the overall performance of
the manipulator systems, and one way is through the
study of robot arm kinematics, dynamics, and control.
Robot arm kinematics deals with the geometry ofrobot
arm motion with respect to a fixed-reference coordinate
system as a function of time without regard to the
forces/moments that cause the motion. Thus it deals with
the spatial configuration of the robot as a function of
time, in particular the relations between the joint-variable
space and the position and orientation of a robot arm.
The kinematics problem usually consists oftwo subprob-
lems-the direct and inverse kinematics problems.
The direct kinematics problem is to find the position
and orientation of the end effector of a manipulator with
respect to a reference coordinate system, given the joint-
angle vector , = (D1, t92, t93, t94, tq5, 96)t of the robot arm.
The inverse kinematics problem (or arm solution) is to
calculate the joint-angle vector e given the position and
001 8-9162/82/1200-406200.75 ( 1982 IEEE62 COMPUTER
orientation of the end effector with respect to the refer-
ence coordinate system. Since the independent variables
in a robot arm are the joint angles, and a task is generally
stated in terms of the base or world coordinate system, the
inverse kinematics solution is used more frequently in
computer applications. The direct kinematics results in a
4 x 4 homogeneous transformation matrix that relates the
spatial configuration between neighboring links. These
homogeneous transformation matrices are useful in
deriving the dynamic equations of robot arm motion.
Robot arm dynamics deals with the mathematical for-
mulations of the equations of robot arm motion. The
dynamic equations of manipulator motion are a set of
equations describing the dynamic behavior of the manip-
ulator. Such equations of motion are useful for computer
simulation of robot arm motion, the design of suitable
control equations for a robot arm, and the evaluation of
the kinematic design and structure of a robot arm.
The purpose of robot arm control is to maintain the
dynamic response of a computer-based manipulator in
accordance with some prespecified system performance
and goals. In general, the control problem consists of ob-
taining suitable dynamic models ofthe physical robot arm
for designing the controller and specifying corresponding
control laws or strategies to achieve the desired system
response and performance. This article details the com-
puted torque technique in the joint-variable space and an
adaptive control strategy.
Robot arm kinematics
Vector and matrix algebra* are used to develop a syste-
matic and generalized approach to describing and repre-
senting the location of robot arm links with respect to a
fixed reference frame. Since the robot arm links can
rotate and/or translate with respect to a reference coor-
dinate frame, a body-attached coordinate frame is estab-
lished at the joint for each link. The direct kinematics
problem is then reduced to finding a transformation
matrix that relates the body-attached coordinate frame to
the reference coordinate frame. A 3 x 3 rotation matrix
'Vectors are represented in lowercase bold letters, and matrices are in up-
percase bold.
Figure 1. Various robot arm categories.
December 1982
,-ff
CARTESIAN OR x-y-z
SPHERICAL
CYLINDRICAL
REVOLUTE
63
can be used to describe the rotational operations of the
body-attached frame with respect to the reference frame.
The homogeneous coordinates are then used to represent
position vectors in a 3-D space, and the rotation matrices
are expanded to 4 x 4 homogeneous transformation ma-
trices to include the translational operations of the body-
attached coordinate frames. This matrix representation
of a rigid mechanical link to describe the spatial geometry
of a robot arm was first used by Denavit and Hertenberg. I
The advantage of using the Denavit-Hartenberg represen-
tation of linkages is its algorithmic universality in deriving
the kinematic equation of a robot arm.
Rotation matrices. A 3 x 3 rotation matrix can be de-
fined as a transformation matrix that operates on a posi-
tion vector in a 3-D Euclidean space and maps its coor-
dinates expressed in a rotated coordinate system OUVW
(body-attached frame) to a reference coordinate system
OXYZ. Figure 2 shows two righthand rectangular coor-
dinate systems, namely the OXYZ coordinate system with
OX, OY, and OZ as its coordinate axes, and the OUVW
coordinate system with OU, OV, and OW as its coor-
dinate axes. The origins of both coordinate systems coin-
cide at point 0. The OXYZ coordinate system is fixed in
the 3-D space and is considered the reference frame. The
OUVW coordinate frame is rotating with respect to the
reference frame OXYZ. Physically we can consider the
OUVWcoordinate system as a body-attached coordinate
frame. That is, it is permanently and conveniently at-
tached to the rigid body (e.g., an aircraft or a link of a
robot arm) and moves together with it. Let (ixs jy, kz) and
(iu, iv, kw) be the unit vectors along the coordinate axes of
the OXYZ and OUVWsystems, respectively. A point p in
the space can be represented by its coordinates with
respect to both coordinate systems. For ease of discus-
sion, assume that p is at rest and fixed with respect to the
OUVW coordinate frame. Then point p can be repre-
sented by its coordinates with respect to the OUVWand
OXYZ coordinate systems, respectively, as
puvw =- (Pup p, pw) t and Pxyz = (PxPY PZ)f (1)
pxyz and puvw represent the same point p in the space with
reference to different coordinate systems.
We want a 3 x 3 transformation matrix A that will
transform the coordinates of pusw to the coordinates
expressed with respect to the OXYZ coordinate system,
after the OUVW coordinate system has been rotated.
That is,
PXYZ= A puvw (2)
Note that the point PUVW has been rotated together with
the OUVWcoordinate system.
Recalling the definition of the component of a vector,
we have
Puvw = Puil + Pvjv + pwkw (3)
andp, py, andpz represent the components of p along the
OX, OY, and OZ axes, respectively, or the projections of
p onto the respective axes. Thus using the definition of a
scalar product and equation 3, we have
Px= ix. P = ixiupu +ix jvPv+ ix kwpw
py=jy-p=jy-iupu+jy-jvpv+jy-kwpw
PZ=kz-p=kziupu+kzjpv+kzkwpw
or expressed in matrix form
[Px] xiIxju ix kw Pu
Py = Jy'iu Jyiju jy*kw Pv
Pz kziu kzjv kz.kw Pw
and
(4)
(5)
'x, u 'x jv ix kw
A= jyiu jy iv jy kw (6)
Lkz iu kz jv kz kw
Similarly, we can obtain the coordinates of puvw from the
coordinates of pxyz
Pvw =B Pxyz
Pu Fu lx uAJy i ukkz Px
Pv = Jv*ix JvuJy Jv* kz py (7)
LPw kw*ix kw jy kw*kzj LPZJ
Since dot products are commutative, we can see that from
equations 6 and 7,
B=A-= At (8)
and
BA=AtA=A -A=13 (9)
The transformation is called an orthogonal transforma-
tion, and since the vectors in the dot products are all unit
vectors, it is also called an orthonormal transformation.
The prime interest of developing the above transforma-
tion matrix is to find the rotation matrices that represent
rotations of the OUVWcoordinate system at each of the
three principal axes in the reference coordinate system
OXYZ.
If the OUVWcoordinate system is rotated at an a angle
about the OXaxis to arrive at a new location in the space,
then point puUw having coordinates (pu. pv Pw)I with
respect to the OUVW system will have different coor-
dinates (Px py, pz) t with respect to the reference system
OXYZ. The necessary transformation matrix Rxa, is
called the rotation matrix about the OXaxis with a angle.
Rx a can be derived from the above transformation matrix
concept, that is:
Figure 2. Coordinate systems for a rigid body. PxyZ = Rx, * Puuw
COM PUTER
z
P
y
(10)
64
and
Ix.iu ix-jv ix kw [ 0 0 1
Rx,a jy iu jy jv jy-kw = 0 cosa -sinla (11)
kz*iu kzj*v kz kwj L0 sina cosaj
(Note that ix= iu. ) Similarly the 3 x 3 rotation matrices
for rotation about the O Yaxis with 'p angle and about the
OZ axis with tQ angle are, respectively (Figure 3),
cos'p 0 sinmp cos - sint O
Ry"< 0 1 0 R;Rz = sind cost 0 (12)
sin'p 0 cosj 0 0 1
Rx, Ry, and Rz,,o are called the basic rotation
matrices. Other finite rotation matrices can be found
from these matrices.
Example. Find the resultant rotation matrix that
represents a rotation of 'p angle about the OY axis fol-
lowed by a rotation of i5 angle about the O Waxis follow-
ed by a rotation of a angle about the OU axis.
Solution
R=Ry,s0Rw,6,Ru,a
C(p 0 S(1 co -SWO 1 0 0
= 0 10IO SW C9 O 0 Cc, -Sa
0SpO Cfj E 0] L0 Sa cac
CPC SpSa-C'pSOCa CsOSdSa+SpCa1
= fsoc0ca - C jSo
-SPCtg S9Osdca + C0 (17)
0 0 OOs s
The physical Cartesian coordinates of the vector are:
=x =y z 5PX=S , Py=Ys Pz= , W=5-= 1 (18)
Therefore the fourth diagonal element in the homo-
geneous transformation matrix has the effect of globally
reducing the coordinates if s> 1 and enlarging the coor-
dinates if 0
本文档为【机器人手臂的运动解析】,请使用软件OFFICE或WPS软件打开。作品中的文字与图均可以修改和编辑,
图片更改请在作品中右键图片并更换,文字修改请直接点击文字进行修改,也可以新增和删除文档中的内容。
该文档来自用户分享,如有侵权行为请发邮件ishare@vip.sina.com联系网站客服,我们会及时删除。
[版权声明] 本站所有资料为用户分享产生,若发现您的权利被侵害,请联系客服邮件isharekefu@iask.cn,我们尽快处理。
本作品所展示的图片、画像、字体、音乐的版权可能需版权方额外授权,请谨慎使用。
网站提供的党政主题相关内容(国旗、国徽、党徽..)目的在于配合国家政策宣传,仅限个人学习分享使用,禁止用于任何广告和商用目的。