There are many elements that make up smooth riding, however, and understanding those elements - and putting numbers to them - can often help a rider improve in this area.
As an example, consider being asked to ride on the highway as smoothly as you possibly can. To do that, one requirement would be to keep your speed as constant as possible, avoiding any acceleration or deceleration. And if you did have to speed up or slow down, you would be as gentle as you possibly could on the throttle and brake, to avoid any sudden acceleration or deceleration.
Mathematically, acceleration is defined as the "rate of change" or derivative of speed over time, and we can look at the amount of acceleration to quantify that aspect of your smooth riding. Continuing with the same example, if you speed up on the highway from 60 km/h to 70 km/h in one second (quickly enough to stretch your arms a bit), you are accelerating at a rate of 10 km/h per second (km/h/s). A smoother rider may take two seconds to speed up the same amount, accelerating at a rate of 5 km/h/s. Ideally, the smoothest you could ride would be no acceleration or deceleration at all - a value of zero km/h/s.
Using data acquisition, we can look at instantaneous acceleration at any point in time; higher values indicate not-so-smooth riding in terms of holding a steady speed, while low values show smoother riding. Taking the concept one step further, we can apply this to practically any data stream and look at the derivative to get an indication of how smooth the rider is in that department. For example, the derivative of throttle position can show exactly how smoothly the rider opens the throttle exiting a corner, and put a number to that smoothness. The derivative of brake pressure or braking force can show precisely how smoothly the rider releases the brakes entering a turn. We can even apply the derivative channel to chassis data to see how smoothly the bike reacts to the rider's inputs; for example, the derivative of lean angle shows how smoothly the rider keeps to maximum lean in the middle of a long turn.
As an indication of how important smoothness is, consider this: In the late '80s and early '90s, Kenny Roberts' 500 cc Grand Prix team was at the forefront of data acquisition and applying new technology to road racing. New Zealander Mike Sinclair was largely responsible for the team's development in this area. In an interview with Sport Rider magazine several years ago, Sinclair noted that they were dealing with the second derivative - the rate of change of the rate of change - of some data as an important measure, taking the concept of smoothness to a whole different level.
What does all this mean for the average rider? Too often we see riders at the track trying to go faster by simply doing things quicker: opening the throttle more abruptly, flicking the bike from side to side faster, and so on. In some ways this works, but more importantly, smoothness - actually slowing things down in terms of the rate of change of those inputs - is the key to making time around the track, and doing it safely.