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Proportional
The proportional term produces an output value that is proportional to the current error value. The proportional response can be adjusted by multiplying the error by a constant Kp, called the proportional gain constant.
The proportional term is given by: P(out) = Kp * e(t)
Making a change that is too large when the error is small is equivalent to a high gain controller and will lead to overshoot. If the controller were to repeatedly make changes that were too large and repeatedly overshoot the target, the output would oscillate around the setpoint in either a constant, growing, or decaying sinusoid.
If the oscillations increase with time then the system is unstable, whereas if they decrease the system is stable. If the oscillations remain at a constant magnitude the system is marginally stable.
Derivational
The Derivative Controller is designed to avoid the overshoot. This allows the robot to gracefully approach its target trajectory, assuming appropriate settings of differential gain.
The derivative of the process error is calculated by determining the slope of the error over time and multiplying this rate of change by the derivative gain Kd. The magnitude of the contribution of the derivative term to the overall control action is termed the derivative gain, Kd.
The derivative term is given by:
D(out) = Kd * d/dt e(t)

In PD-control the torque is not just related to the crosstrack error by virtue of the gain parameter (Kp), but also to the time derivative of the crosstrack error. This means that when the robot has moved enough to reduce the crosstrack error, the error becomes smaller over time and it will not just go on trying to reach the the desired point but will notice that it has already reduced the