We will use a linear control system in our design, which is an approximation of the nonlinear nature of the dynamics equations of the system, which are more properly represented by nonlinear differential equations. This is a reasonable approximation, and it is used in current industrial practice.
We will assume that there are sensors at each joint to measure the joint angle and velocity, and there is an actuator at each joint to apply a torque on the neighboring link. Our goal is to cause the manipulator joints to follow a desired trajectory. The readings from the sensors will constitute the feedback of the control system. By choosing appropriate gains we can control the behavior of the output function representing the actual trajectory generated. Minimizing the error between the desired and actual trajectories is our main concern. Figure 2 shows a high level block diagram of a robot control system.
Figure 2: High-level block diagram of a robot control system.
When we talk about control systems, we should consider several issues
related to that field, such as: stability,
controllability, and observability. For any control system to
be stable, its poles should be negative, since the output equation
contains terms of the form 
; if 
is positive,
the system is said to be unstable. We can guarantee the stability of
the system by choosing certain values for the feedback gains.
We will assume a second order control system of the form:
Another desired property of the control system is that it be
critically damped, which means that the output will reach the desired
position in minimum time without overshooting. This can be
accomplished by making 
. Figure 3 shows
the three types of damping: underdamped, critically
damped, and overdamped.
Figure 3: Different types of damping in a second order control system.
Figure 4 shows a block diagram for the controller, and the role of each of the robot modules in the system.
Figure 4: Block diagram of the controller of a robot manipulator.
More about robot control can be found in [3, 33, 45].