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# Robotics Toolbox for Matlab

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16
3
Tutorial
3
Manipulator kinematics
Kinematics is the study of motion without regard to the forces which cause it. Within kine-
matics one studies the position, velocity and acceleration, and all higher order derivatives of
the position variables. The kinematics of manipulators involves the study of the geometric
and time based properties of the motion, and in particular how the various links move with
respect to one another and with time.
Typical robots are serial-link manipulators comprising a set of bodies, called links, in a
chain, connected by joints
1
. Each joint has one degree of freedom, either translational or
rotational. For a manipulator with n joints numbered from 1 to n, there are n
numbered from 0 to n. Link 0 is the base of the manipulator, generally fixed, and link n
carries the end-effector. Joint i connects links i and i
- 1.
A link may be considered as a rigid body defining the relationship between two neighbour-
ing joint axes. A link can be specified by two numbers, the link length and link twist, which
define the relative location of the two axes in space. The link parameters for the first and
last links are meaningless, but are arbitrarily chosen to be 0. Joints may be described by
two parameters. The link offset is the distance from one link to the next along the axis of the
joint. The joint angle is the rotation of one link with respect to the next about the joint axis.
To facilitate describing the location of each link we affix a coordinate frame to it -- frame i
is attached to link i. Denavit and Hartenberg[6] proposed a matrix method of systematically
assigning coordinate systems to each link of an articulated chain. The axis of revolute joint
i is aligned with z
i
-1
. The x
i
-1
axis is directed along the normal from z
i
-1
to z
i
and for
intersecting axes is parallel to z
i
-1
× z
i
. The link and joint parameters may be summarized
as:
a
i
the offset distance between the z
i
-1
and z
i
axes along the
x
i
axis;
i
the angle from the z
i
-1
axis to the z
i
i
axis;
d
i
the distance from the origin of frame i
- 1 to the x
i
axis
along the z
i
-1
axis;
joint angle
i
the angle between the x
i
-1
and x
i
i
-1
axis.
For a revolute axis
i
is the joint variable and d
i
is constant, while for a prismatic joint d
i
is variable, and
i
is constant. In many of the formulations that follow we use generalized
coordinates, q
i
, where
q
i
=
i
for a revolute joint
d
i
for a prismatic joint
1
Parallel link and serial/parallel hybrid structures are possible, though much less common in industrial manip-
ulators.

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Summary :