Lecture 12 : Trajectory planning I ( point to point and continuous trajectories)
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The global (X, Y) coordinates of the end-effector are:
(12.1.1)
(12.1.2)
Given and i.e., the joint positions, finding the (X, Y) coordinates of the end-effector involves solution of simple Linear Algebraic Equations (LAE). However, if we are given the desired end-effector positions , finding and will require solution of trigonometric (hence nonlinear) equations.
Let us now differentiate the two position equations with respect to time to get the velocities:
(12.1.3)
(12.1.4)
i.e
(12.1.5)
i.e, Or
At a given position (, , ,) for known joint velocity and finding or vice-versa involves solution of simple LAE. Of course there could be some configurations (i.e ,) where may not exist (singularities) and may pose difficulties. Similarly we can relate the joint-space and Cartesian-space accelerations.
Given joint position, velocity and acceleration, finding the Cartesian space position, velocity and acceleration of the end-effector is referred to as the FORWARD KINEMATICS problem. Given end effector Cartesian position, velocity, acceleration, finding the joint coordinate position, velocity and acceleration is termed the INVERSE KINEMATICS problem. Formal description of the kinematics of a robot manipulator in 3-D space and intricacies of FORWARD/INVERSE kinematics would be discussed at a later stage. For the purpose of present discussion, we simply state that the motion of a robot manipulator when specified as desired end effector motion can be converted into joint space motion and vice-versa. Thus, in order to plan the motion of the end-effector we could simply plan the motion of each joint. This is known as Joint Interpolated Motion.
and 
Or 
,
) for known joint velocity 
and 
finding 
or vice-versa involves solution of simple LAE. Of course there could be some configurations (i.e 
,
) where 
may not exist (singularities) and may pose difficulties. Similarly we can relate the joint-space and Cartesian-space accelerations.
