tieu luanrobot ppt

2.3 Parameterization of manipulators via twists
Using the product of exponentials formula the kinematics of a manipulator
is completely characterized by the twist coordinates for each of the joints.
We now consider some issues related to parameterizing robot motion using
twist
Choice of base frame and reference configuration
In the examples above we chose the base frame for the manipulator to be
at the base of the robot other choice of the base frame are possible and
can sometimes lead to simplified calculations. One natural choice is to
place the base frame coincident with the tool frame in the reference
configuration. That is we choose a base frame which is fixed relative to the
base of the robot and which lines up with the tool frame when θ=0. This
simplifies calculations since g
st
(0)=I with this choice of base frame and
hence

A further degree of freedom in specifying the manipulator kinematics is the
choice of the reference configuration for the manipulator. Recall that the
reference configuration was the configuration corresponding to setting all of
the joint variable to 0. By adding or subtracting a fixed offset from each joint
variable. We can define any configuration of the manipulator as the
reference configuration. The twist coordinates for the individual joint of a
manipulator depend on the choice of reference configuration(as well as
base frame) and so the reference configuration is usually chosen such that
the kinematic analysis is as simple as possible.
For example a common choice is to define the reference configuration

such that points on the twist axes for the joint have a simple form as in all
of the example above.
Consider the scara manipulator with base frame coincident with the tool

frame at θ=0 as shown in figure3.5. the twist are now calculated with
respect to the new base frame:


And similar calculation yield

Expanding the product of exponentials formula gives

Note that g
st
(0)

= I which is consistent with the fact that the base and tool
frame are coincident at θ =0. Compare this formula with the kinematics
map derived in example3.1
Relationship with denavit-hartenberg parameters
Given a base frame S and tool frame T the coordinates of the twist
corresponding to each joint of a robot manipulator provide a complete
parameterization of the kinematics of the manipulator. An alternative
parameterization which is the de facto standard in robotics is the use of
denavit-hartenberg parameters [25 ]. In the section we discuss the
relationships between these two parameterization and their relative merits
Denavit-hartenberg parameters are obtained by applying a set of rules
which specify the position and orientation of frames L
i
attached to each link
of the robot and then constructing the homogeneous transformations
between frames denoted . By convention we identify the base frame
s with L
0

. The kinematics of the mechanism can be written as
just as in equation(3.1). Each of the transformations gl
i-1,
l
i

has the form
Where the flour scalars α
i
,ai,di and ɸ
i
are the parameters for the i
th
link.
For revolute joints, ɸ
i
corresponds to the joint variable θ
i


. while for
prismatic joint, d
i
corresponds to the joint variable θ
i
. Denavit-hartenberg
parameters are available for standard industrial robots and are used by
most commercial robot simulation and programming systems
It may seem somewhat surprising that only four parameters are needed to
specify the relative link displacements since the twist for each joint have

six independent parameters. This is achieved by cleverly choosing
The link frames so that certain cancellations occur. In fact it is possible to
give physical interpretations to the various parameters based on
relationships between adjacent link frames. An excellent discussion can be
found in spong and vidyasagar [111]
There is not a simple one to one mapping between the twist coordinates
for the joint of a robot manipulation and the denavithartenberg parameters.
This is because the twist coordinates for each joint are specified with
respect to a single base frame and hence do not directly represent the
relative motion of each link with respect to the previous link. To see this, let
be the twist for the i
th
link relative to the previous link frame. Then gl
i-
1l
i
is
given by
And the forward kinematics map becomes
This is evidently not the same as the product of exponential formula though
it bears some resemblance to it
The relationship between the twist and the pairs and can be
determined using the adjoint mapping. We first rewrite equation(3.8) as
We can simplify this equation by making use of the relationship
to obtain
It follows immediately that
ξ
i
= Ad
gl 0l i− 1

( 0) ξ
i− 1 ,i
. (3.10)
This formula verifies that the twist ᶓ
i
is the joint twist for the i
th
joint in its
reference configuration and written relative to the base coordinate frame
Given the Denavit-hartenberg parameters for a manipulator the
corresponding twists ξ
i

can be determined by first parameterizing g
i-1,i
using
exponential coordinates as in equation(3.7), and then applying
equation(3.10). However in almost all instances it is substantially easier to
construct the joint twists ᶓ
i
directly by writing down the direction of the joint
axes and in the case of revolute joints choosing a convenient point on each
axis. Indeed one of the most attractive features of the product of
exponential formula is its usage of only two coordinate frames the base
frame S and the tool frame T. tis property combined with the geometric
significance of the twists ξ
i
,make the product of exponentials representation
a superior alternative to the use of denavithartenberg parameters
2.4 Manipulator workspace

The workspace of a manipulator is defined as the set of all end effector
configurations which can be reach by some choice of joint angles. If q is the
configuration space of a manipulator and g
st
:Q->SE(3) is the forward
kinematics map then the workspace W is defined as the set
W={ g
st
(θ): θ ϵ Q =(p,R) } ⊂ SE(3) (3.11)
The workspace is used when planning a task for the manipulation to
execute all desired motion of the manipulation must remain within the
workspace. We refer to this notion of workspace as the complete
workspace of a manipulator
Characterizing the workspace as a subset of SE(3) is often somewhat
difficult to interpret. Instead one can consider the set of positions(in R
3
)
which can be reached by some choice of joint angles. This set is called the
reachable workspace and is defined as
W
R
= {p(θ) : θ ∈Q}⊂R3 , (3.12)
Where p(θ):Q -> R
3
is the position component of the forward kinematics
map g
st
. The reachable workspace is the volume of R
3
which can be

reached at some orientation
Since the reachable workspace does not consider ability to arbitrarily orient
the end effector for some task it is not a useful measure of the range of a
manipulator. The dexterous workspace of a manipulator is the volume of
space which can be reached by the manipulator with arbitrary orientation:
WD = {p ∈R3 : ∀R ∈SO(3), ∃θ with g
st
(θ) = (p,R) }⊂R3 . (3.13)
The dexterous workspace is useful in the context of task planning since it
allows the orientation of the end effector to be ignored when positioning
objects in the dextrous workspace
For a general robot manipulator the dextrous workspace can be very
difficult to calculate. A common feature of industrial manipulator is to
Figure 3.6 workspace calculation for a planar three link robot (a). The
construction of the workspace is illustrated in (b). the reachable workspace
is shown in (c) and the dextrous workspace is shown in (d)
Place a spherical wrist at the end of the manipulator as in the elbow
manipulator given in example 3.2. Recall that a spherical wrist consists of
three orthogonal revolute axes which intersect at a point. If the end effector
frame is placed at the origin of the wrist exes then the spherical wrist can
be used to achieve any orientation at a given end effector position. Hence
for a manipulator with a spherical wrist the dextrous workspace is equal to
the reachable workspace,W
D
=W
R
. Furthermore the complete workspace for
the end effector satisfies W =W
R
×SO(3) . this analysis only holds when the

end effector frame is placed at the center of the spherical wrist if an offset
is present the analysis becomes more complex
Example 3.4 workspace for a planar three link robot
Consider the planar manipulator shown in figure 3.6a. let g=(x,y, )ɸ
Represent the position and orientation of the end effector. The forward
kinematics of the mechanism can be derived using the product of
exponentials formula, but are more easily derived using plane geometry:
x = l1 cos θ1 + l2 cos ( θ1 + θ2 ) + l3 cos ( θ1 + θ2 + θ3 )
y = l1 s in θ1 + l2 s in(θ1 + θ2 ) + l3 s in(θ1 + θ2 + θ3 ) (3.14)
φ = θ1 + θ2 + θ3 .
We take l
1
>l
2
>l
3
, and assume l
1
>l
2
+l
3
The reachable workspace is calculated by ignoring the orientation of the
end effector. To generate it, we first take and as fixed. The set of reachable
points becomes a circle of radius l3 formed by sweeping angles to get an
annulus with inner radius l2-l3 and outer radius l2+l3 centered at the end of
the first link. Finally, we generate the reachable workspace by sweeping the
annulus through all values of , to give the reachable workspace. The final
construction is shown in figure3.6c. wr s an annulus with inner radius l
1

-l
2
-
l
3
and outer radius l
1
+l
2
+l
3
.
The dextrous workspace for this manipulator is somewhat subtle. Although
the manipulator has the planar equivalent of a spherical wrist, the end
effector frame is not aligned with the center of the wrist. This reduces the
size of the dextrous workspace by 2l
3
on the inner and outer edges, as
shown in figure 3.6d.