Servo - February 2009

Time vs Theta

Implementing Two Motors

0.12

0.1

Theta (rad)

0.08

0.06

0.04

0.02

0

1

2

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4

567

Time (s)

8

9

10 11 12

Motors with propellers have a lot of inertia. If a motor is spinning in one direction and it's suddenly commanded to turn in the opposite direction, it will take a lot of time to stop and start spinning in reverse. The implementation with one motor relies on gravity to push the pendulum in the opposite direction when needed. By adding a second motor, we avoid the inertia problem and we avoid relying on gravity to stabilize the system.

■ FIGURE 8. Kp=0.028, Ki=0.7, Kd=0, Tetharef =0.08 radians

Theta Theta ref

Controlling The Robot Using Two Motors

well. You should also have in mind how the proportional, integral, and derivative controllers affect the behavior of the system. From a theoretical point of view, one can never achieve a given set point with just a proportional controller. Even if it were possible, just a proportional controller by itself would not suffice.

Any system will wear out with time, requiring the controller to apply different control efforts, depending on those changes. In order to avoid this problem, one must use an integral controller. Integral controllers scale the integrated (accumulated) error in time. This means that even if the error is zero, the control effort will not necessarily be zero. This behavior could be thought of as the system building up memory as time passes by. It has a drawback, though. If friction is huge or the system gets stuck for any reason, the control effort will keep building up until it saturates. This could possibly damage the controller or the actuators.

The last control element is the derivative which slows down the response of the system. This controller scales the rate of change of the error, decreasing the control effort if the system reacts too fast. This is useful to avoid overshoot. However, it may slow the system's response too much.

■ FIGURE 9. System stability.

Stability

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1

0.8

0.6

Stability

0.4

0.2

In this case, we used the same PID we designed previously. The only difference is that the control effort is split to activate each motor individually. The second motor takes a negative control effort such that if one motor needs to speed up to push the pendulum in one direction, the other slows down. Figure 10 shows the block diagram for the new setup.

Software Architecture For Two Motors

The architecture for two motors is pretty much the same as the one for a single motor. The main difference is the addition of a calibration routine. We did this to avoid determining the bias for each motor individually. The bias is approximately 50% of the maximum control effort each motor can take. This allows the controller to speed up or slow down the motors within their operation region while keeping the system in equilibrium. Figure 11 shows the new flow diagram.