Kinematic and Dynamic Analysis of Spatial Six Degree of Freedom ...

Chapter 5
DYNAMIC ANALYSIS
Dynamics deals with the forces and/or torques required to cause the motion of a
system of bodies. The study includes inertia forces as one of the principal concerns. The
dynamics of a robot manipulator is a very complicated subject. Typically, the end effector is
to be guided through a given path with certain prescribed motion characteristics. A set of
torque and/or force functions must be applied at the actuated joints in order to produce that
motion. These actuating force and/or torque functions depend not only on the spatial and
temporal attributes of the given path but also on the mass properties of the links, the payload,
the externally applied forces and so on.
Dynamical analysis deals with the derivation of the equations of motion of a given
manipulator. There are two types of dynamical analysis problems: direct (or forward) and
inverse dynamics. Direct dynamics can be defined as follows: Given a set of actuated joint
torque and/or force functions, calculate the resulting motion of the end effector as a function
of time; and inverse dynamics as: Given a trajectory of the end effector as a function of time,
find a set of actuated joint torque and/or force functions which will produce that motion.
Direct dynamics is mainly used for computer simulations of robot manipulators. On the other
hand inverse dynamics is primarily used for real-time model-based control of a manipulator.
This chapter is devoted to the inverse dynamics of the six DOF spatial parallel manipulator
given in Chapter 4.
5.1 Determination of the Reduced Moments and Forces
An external force f acting to the moving platform of the manipulator in figure 1 can be
represented as a moment equation as:
m = ρ × f
where ρ( x,
y,
z)
is the acting point of forcef . The equation can be rewritten as follows:
M x =
Fz
y − Fy
z M y = Fx
z − Fz
x M z = Fy
x − M x y
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