By Marcus Chin
Kinematics equations are often used to represent ideal situations that only exist within textbooks. While they provide an adequate basis to approach real-world problems, this doesn’t work in robotics. There are too many factors to consider, even when doing something as simple as driving forward one meter. Chances are, the robot will overshoot its target, slide a bit to the left or right, or maybe even tip over. Because of this, programmers and engineers need to apply control systems to robotics in order to provide more precise and smooth movement to robot systems.
The field of control theory incorporates advanced mathematics that can be difficult to understand, but Proportional Integral Derivative (PID), the most common control system applied to robotics, can be explained in basic terms. PID describes the different terms used to calculate output based on a specified setpoint and its current position, also known as the error (relative to the setpoint). For example, if a robot is at position 0 and wants to move to position 10, we could accomplish this by using a PID control system, we would specify the setpoint to be position 10, and the error that the control system identifies would be 10 units since it is 10 units away from the setpoint.
The main term that makes PID “work” is the proportional term. It is defined by the formula kP * ERROR, where kP is a constant. A proportional controller would use this formula to give the output of the system to reach the setpoint. The reason why the proportional controller works is that the error of a system will decrease as the controller runs. If the system runs for one second, the error decreases since the proportional term will specify an output that moves the system closer to its setpoint. Since the error decreases, the output also decreases, and this continues until the system slowly approaches its setpoint. Put plainly, the proportional controller makes a robot faster if it is further away from where it wants to go and vice versa.
However, a proportional controller will only be effective if it is tuned correctly, meaning a correct constant value for kP is selected. This value will differ based on the system. If kP is too high, the output will overshoot the setpoint and may result in an infinite oscillation. Consequently, if kP is too low, the output will undershoot and never reach the setpoint. When a correct value for kP is found, the system should stabilize at its setpoint without oscillations.
The integral and derivative terms of PID controllers are slightly more complex and use calculus, but they are still somewhat intuitive. Both these terms are added to the proportional term to generate the full equation, Output = Proportional + Integral + Derivative. Since I and D are mostly used to further refine the proportional term, they are relatively insignificant compared to the proportional term and are not always used. Standalone integral controllers are only used in niche situations, and standalone derivative controllers don’t work as they cannot identify a setpoint. Nevertheless, depending on the application, they can help to make PID controllers more smooth and more accurate.
The integral term is defined by kI * Integral(ERROR). All this term does is accentuate the output the further away the system is from the setpoint, reducing the steady-state error of the system, which is the difference between the ending state and setpoint (The PID Controller & Theory Explained, n.d.). It essentially “accumulates” the error, which you can visualize as the area under the error (position) graph, also known as the absement. Like the proportional term, the magnitude of the integral term can be accentuated by the kI constant. However, adding an integral term incorrectly can have adverse effects, including violent overshooting.
Lastly, the derivative term is defined by kD * dERROR/dt. The effect of the derivative term is that it scales with the slope of the error graph, essentially attempting to linearize it. This can remove a lot of overshooting and oscillation, but it is commonly not used due to instability when used in practical applications. It also can significantly reduce the response time of a controller due to the calculations for the derivative term being more complex (Libretexts, 2022).
When control systems come together, a robot can be programmed to be seamlessly smooth. Most robots in robotics competitions use some sort of PID control, whether that be for autonomous driving or arm control. Some even use more advanced control systems, such as Linear Quadratic Regulator (LQR) control which utilizes linear algebra to create a state-space control system that is more advanced and effective than PID for some use cases like auto-balancing. While the end product may seem to work seamlessly, understanding the complex theory behind robotics provides insight into how mathematical models are applied in the real world.
References
Libretexts. (2022, May 22). 9.2: P, I, D, PI, PD, and PID control. Engineering LibreTexts. https://eng.libretexts.org/Bookshelves/Industrial_and_Systems_Engineering/Book:_Chemical_Process_Dynamics_and_Controls_(Woolf)/09:_Proportional-Integral-Derivative_(PID)_Control/9.02:_P,_I,_D,_PI,_PD,_and_PID_control
The PID Controller & Theory Explained. (n.d.). NI. https://www.ni.com/en-sg/innovations/white-papers/06/pid-theory-explained.html
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