Although our idea of sequential input being a key to cerebellar function was taken seriously by most commentators, there were also objections, based in part on experimental evidence that seems to contradict our intuitions and in part on commentators' preferences for different schemes. Several were suspicious of experiments (performed on slices of cerebellar tissue) that may have severed some of the synaptic connections, particularly the inhibitory ones. It is our feeling that a modi- fication of our theory that could satisfy most critics would not have to be very radical.

This target article presents a critical survey of the scientific literature dealing with the speed/accuracy trade-offs in rapid-aimed movements. It highlights the numerous mathematical and theoretical interpretations that have been proposed in recent decades. Although the variety of points of view reflects the richness of the field and the high degree of interest that such basic phenomena attract in the understanding of human movements, it calls into question the ability of many models to explain the basic observations consistently reported in the field. This target article summarizes the kinematic theory of rapid human movements, proposed recently by R. Plamondon (1993b; 1993c; 1995a; 1995b), and analyzes its predictions in the context of speed/accuracy trade-offs. Data from human movement literature are reanalyzed and reinterpreted in the context of the new theory. It is shown that the various aspects of speed/ accuracy trade-offs can be taken into account by considering the asymptotic behavior of a large number of coupled linear systems, from which a delta-lognormal law can be derived to describe the velocity profile of an end-effector driven by a neuromuscular synergy. This law not only describes velocity profiles almost perfectly, it also predicts the kinematic properties of simple rapid movements and provides a consistent framework for the analysis of different types of speed/accuracy trade-offs using a quadratic (or power) law that emerges from the model.

Recent accounts of the speed/accuracy relation for motor tasks have focused on the concept of motor output variability. We outline the advantages of this approach and the limitation of Plamondon's model in explaining movement error. We also examine and present complimentary data for rapid timing tasks. While these tasks do not meet the presented assumptions, the data still fit the model predictions.

Though capable of reproducing experimentally observed velocity profiles, the model proposed by Plamondon & Alimi (P&A) does not provide a viable theoretical framework for the understanding of trajectory formation and speed/accuracy trade-offs. The issues of variation and stability can be better understood by considering movement as resulting from an underlying dynamic rather than from an impulse-response of the system.

Visual feedback regulation during movement is not fully captured in Plamondon's kinematic theory. However, numerous studies indicate that visual response-produced feedback is a powerful determinant of performance and kinematic characteristics of target-directed movement.

Notwithstanding its overwhelming descriptive power for existing data, it is not clear whether the kinematic theory of Plamondon & Alimi could generate new insights into biomechanical constraints and psychological processes underlying the way organisms trade off speed for accuracy. The kinematic model should elaborate on the role of neuromotor noise and on biomechanical strategies for reducing endpoint variability related to such noise.

Plamondon's kinematic theory is very powerful from a descriptive point of view. Unfortunately, the fact that it neglects some fundamental features of the motor system, such as nonlinear inertial torque interactions or joint redundancies, limits its explanatory power and biological validity. As a consequence, the data presented by Plamondon & Alimi should be analyzed and interpreted with caution. There appears to be a gap between the observations reported by the authors and some of the conclusions they draw.

Plamondon & Alimi's model will gain substantially in credibility when it is able to come up with predictions for new (rather than old) experimental results that discriminate between various models. Moreover, the present model is nothing more than a descriptive “black box,” not an explanation for motor performance. A link to the contribution of various neuronal mechanisms involved in motor control and of muscle properties to the performance of the model is crucial.

Plamondon & Alimi (P&A) have unified much data on speed/accuracy trade-offs during reaching movements using a delta-lognormal form factor that describes “the asymptotic behavior of a large number of dependent linear systems,” notably neuromuscular systems. Their approach raises questions about whether a large number of systems is needed, whether they are linear, and whether the results disclose the neural design principles that control reaching behaviors. The authors admit that “it is difficult . . . to provide a direct biological interpretation for the system parameters” (sect. 6, para. 4).

In Schmidt's experiments, only properties of actually produced movements systems are measured; in Fitts' experiments, external task parameters are measured too (target size and distance). Thus, the laws contain variables of different natures and cannot be reduced to each other even formally. These difference especially reveals itself in modeling: a model of variability can be simpler if it deals with the performance variables only. On the other hand, modeling Fitts' law, one should take into account not only the human effector system but features of the experimental apparatus as well. These differences have not been sufficiently reflected in Plamondon & Alimi's target article.

Even though Plamondon's kinematic model fits the data well, we do not share the view that it explains movements other than ballistic ones. The model does not account for closed-loop control, which is the more common type of movement in everyday life, nor does it account for recent data indicating interference with ongoing processing.

This commentary cites several findings of neuromuscular research that are consistent with aspects of Plamondon's kinematic theory. In addition, we point out certain biomechanical properties of the limb that influence the requirements for the production of accurate movement, and might thus compromise the global applicability of any law governing speed/accuracy trade-offs.

Movement time and accuracy, as defined by Plamondon & Alimi, do not conform to empirical definitions. When definitions are used that conform better to empirical ones, the original predictions of the kinematic theory are no longer valid – as is demonstrated by simulations. Thus the theoretically derived quadratic law and the successful empirical quadratic law seem to be independent of each other.

It is important to study movement data from individual subjects rather than by averaging data across subjects or trials, because averaged data may follow different laws than those followed by the individual data. This fact can be shown mathematically. In addition, clear assumptions and a thorough understanding of their consequences are a necessary component of any realistic model.

Plamondon & Alimi's derivation of the kinematic model is mathematically flawed. By simply naming a particular parameter combination “variability,” the model fails to explicate the sources of variability. As a result, the model cannot distinguish between various sources of error, such as those resulting from task demands and those resulting from movement execution.

A purely kinematic theory of movement runs the risk of having no explanatory power because it neglects the internal generative structures of the central nervous system. Distributed interaction between the agonist and antagonist systems would better simulate physiological mechanisms of oscillation, lateral inhibition, and synchronization, all of which have important roles in motor control.

We address two main issues: the distinction between time-constrained and spatially constrained tasks, and the separable A and W effects on movement time (MT) in spatially- constrained tasks. We consider MT and 3-D kinematic data from human adults pointing to targets in human-computer interaction. These are better fit by Welford's (1968) two- part model, than Fitts' (1954; Fitts & Peterson 1964) ID model. We identify theoretical and practical implications.

Fitts' law and the ΔΛ model are “weak” theories of motor control because they are limited to the kinematic aspects of movement and do not capture its essential dynamic nature. The internal source of “noise” that determines the speed/ accuracy trade-off can be associated with the partial compensation of movement-generated “parasitic” forces.

In Plamondon & Alimi's target article, a bell-shaped velocity profile typically observed in fast movements is used as a basis for the “kinematic theory” of motor control. In our opinion, kinematics is a necessary but insufficient ground for a theory of motor control. Relationships between different kinematic characteristics are an emergent property of the system dynamics controlled by the brain in a specific way. In particular, bell-shaped velocity profiles with or without additional waves are a trivial consequence of shifts in the equilibrium state of the system as suggested, for example, in the λ-model of motor control.

While explaining a large proportion of any variance, accounts of the speed and accuracy of targetting movements use techniques (e.g., log transforms) that typically reduce variability before “explaining” the data. Therefore the predictive power of such accounts are important. We consider whether Plamondon's model can account for kinematics of targetting movements of clinical populations.