Neville Hogan (1984a) noted that smoothness can be quantified as a function of jerk, which is the time derivative of acceleration. Hence, jerk is the third time derivative of location (i.e., position). If the location of a system is specified by variable , then the jerk of that system is:
For your CNS to move your hand or some other end effector smoothly from one point to another, it should minimize the sum of the squared jerk along its trajectory. For a particularly trajectory that starts at time and ends at time , you can measure smoothness by calculating a jerk cost:
Note that the jerk cost is a scalar; the expression above assigns a number to the function . Hogan wondered what function most smoothly connects a starting point to a target in a given amount of time. This function , among all possible functions, has the minimum jerk cost. Some people find this fact interesting, others find it simply tedious. If you find yourself in the latter category, you might consider skipping over the remainder of this chapter.
To make the issue concrete, imagine that you wish to move something 10cm in a 0.5sec period. The object will be at rest at start time and at the end of the movement. You can write this as:
What trajectory has the smoothest path? To find , you need to assign a jerk cost to each possible trajectory, and then find the trajectory with the least cost. Mathematically, this calculation corresponds to minimizing the functional:
(The ½ factor in the front makes the calculations come out a little prettier; otherwise it has no special significance). To find the minimum of this functional, Hogan used a technique called the calculus of variations. The idea resembles finding the minimum of a function: you find the derivative of the function with respect to a small perturbation and when that derivative is zero, you have found a minimum. The variation is a function that you can name .
Figure 1 shows an example of . The variation has the special property that it smoothly goes away at the boundary conditions, i.e., at the beginning and end of movement:
To minimize , you can replace by a variation and you can find the derivative of the new functional with respect to the variation.
Using integration by parts, you can rewrite this integral as:
where means 4th derivative of function . Continuing the integration by parts,
This final integral is the derivative of your functional, and you have:
The above property must hold true for any function , and therefore you have the fact that
which means that some function that happens to have its sixth derivative equal to zero will minimize your jerk function. The differential equation has the general solution of:
To find the constants, you can plug in what you know about at the boundaries:
Thus, you arrive at the function that most smoothly travels 10 cm in 0.5 s:
Figure 2 plots this function along with its first three derivatives: velocity, acceleration, and jerk. The function represents the minimum jerk trajectory in one dimension. Hogan noted that, in general, if something you wanted to move something from location to in seconds, the minimum jerk trajectory would be:
Flash and Hogan (1985) found that, for end-effector locations specified as a vector of two or more dimensions, Eq. (1) described the minimum jerk trajectory for each dimension. For example, for a movement in two dimensions, the functional to minimize is:
Eq. (2) implies that a minimum jerk trajectory in two or three dimensions always corresponds to a straight line.
Figure 3 exemplifies this relationship, for a two-joint arm moving from an initial- to a final location in 0.5 s. Note how each component of location in cartesian coordinate moves smoothly to its final value and end-effector location moves along a straight line.
The third derivative of location with respect to time is called jerk. The fourth, fifth, and sixth derivatives are called snap, crackle, and pop, respectively. How can you know that a minimum-jerk description provides the best description of your reaching movements: Why not minimum snap?
It turns out that as the order of the derivative n increases, the solution to the functional approaches a step function.
Figure 4A shows the minimum jerk, snap, and crackle trajectories. Note how the first derivative (speed) of each trajectory becomes narrower and taller as you minimize jerk, snap and crackle. Therefore, if you wish to minimize snap, the fourth derivative of location, you get a movement with a higher peak speed than a trajectory that minimizes jerk. This means that as you increase n in Eq. (3), the solution yields a trajectory with a larger peak speed to average speed.
If you call this ratio of peak speed to average speed r, then a minimum-acceleration trajectory [i.e., where n = 2 in Eq. (3)], has a ratio of r = 1.5. For a minimum-jerk trajectory, n = 3 and r = 1.875; for a minimum-snap trajectory, n = 4 and r = 2.186. Psychophysical experiments reveal that your reaching movements have a ratio r = 1.75, and thus most resemble minimum-jerk trajectories.