Optimal Trajectory Generation in Reaching and Tracking: Minimum Acceleration versus Minimum Jerk

Matthew B. Gibson &Peter D. Neilson

Neuroengineering Laboratory,
School of Electrical Engineering and Telecommunications,
The University of New South Wales, Sydney, Australia


This project concerns optimal movement trajectories computed according to both minimum acceleration and minimum jerk criteria. Simulations were developed for a discrete reaching movement and a continuous tracking task. For tracking, the minimum jerk simulation proved "jerky" compared with the minimum acceleration simulation. For reaching, the minimum jerk criterion produced a larger overshoot than did minimum acceleration. In real-time movements, actual values of the initial and final states for the next trajectory are not known at the time of planning. Thus predictions of initial and final states are required. The above results depended on estimates of instantaneous velocity and acceleration derived from differences in the predicted position at sample time intervals of 50 ms. Subsequent interpolation of data enabled the time between position predictions to be reduced to 2 ms, allowing more accurate estimation of initial conditions. The minimum acceleration model now produced an excellent fit for reaching with minimal overshoot. The minimum jerk model still produced overshoot and the simulation easily became unstable. The minimum acceleration model requires predictions of position and velocity only, whereas the minimum jerk model requires predictions of position, velocity and acceleration.

Predictions of acceleration may compromise the latter model. To date, few researchers have applied the minimum jerk criterion to tracking tasks. In reaching, both velocity and acceleration at the movement endpoint are constrained to zero, possibly masking any difficulty in predicting non-zero acceleration values. We suggest that the "jerkiness" observed in the minimum jerk tracking simulation is attributable to error in prediction of target acceleration. By utilizing the concept of intermittency in movement planning, we show that both simulations produce velocity and acceleration profiles to match experimentally observed features of reaching movements, such as single and double peaked velocity profiles. In summary, we suggest that the minimum acceleration model may provide a better representation of experimentally observed data as well as being more economical.