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Ch85-I044963.fm Page 423 Monday, August 7, 2006 11:32 AM
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11:32 AM
Ch85-I044963.fm
Monday, August 7,2006
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Inverse Kinematics
The inverse kinematics are required to define the parameters of the actuators if the position and
orientation of the tool tip are given and the inverse kinematic model is used in position control of the
robot, the inverse kinematics model has been given in the reference [5]. According to the frames
defined in Fia.2 the models are

=E + R-EB :-UU, for cylinders L 4, L 5

l=UB,-UU,=UU' + R'-IT B i - UU, for cylinders L h L 2, L 3 (1)
Denoted
L,=fi(x,y,z, a,P) (i= 1,2,3,4,5) (2)

Where R, R' are rotational transformation matrixes
Forward Kinematic and Jacobian Matrix

The forward kinematics is required to find the position and orientation of the tool tip once the
parameters of the actuators are given. The forward kinematics can also be solved from Eqn.2.when the
lengths of the linear drivers are given. Since Eqn.2.ontains non-linear items, the forward kinematics is
difficult to solve directly. The numeric iterative method has been usually used to solve forward
kinematics problems. From Eqn.2, we can obtain the differential motion vector, thus

= j[SX,SY,SZ,5a,Sp] (3)

Where. J is the Jacobian matrix

J =

Then the inverse speed equation can be obtained

[Z 1,Z 2,Z 3,Z 4,Z, jf =j[x,Y,Z,a,p] (4)

When the Jacobian matrix J is singular, that is det|J|=0, the robot is then in a singular position and
cannot put out any Cartesian force.

Static Force

The force capacity of the robot should be investigated, which is for a certain payload inside the
workspace to calculate the static forces of cylinders. If we denote Eqn.2. as

Where 8Z.=(8/;, 8/ 2, hi3, 8U, Sis), h®=(SX, SY, SZ, Sa, SP)