The smoothness of hand trajectories, in general, and of cursive handwriting, in particular, has long been the subject of study and controversy. Is it embedded into the motor planning processes or is it a consequence of the dynamics of the neuromuscular system? Are the motor plans analogic/smooth or symbolic/discrete? Human movements are indeed smooth, in the sense that can be described by mathematical models that maximize some kind of smoothness criterion. On the other hand, the detailed analysis of the speed profile clearly shows that complex movements are segmented into sub-movements and it is quite tempting to assume that the global observed movement is just an emergent property, determined by the superposition of individually smooth primitives. In fact, an argument against a dynamic explanation is that it is difficult to reconcile it with the observed scale and shift invariance of handwriting patterns: the dynamic interaction forces in the arm are strongly non-linear and their effects are configuration-dependent, inducing patterns of anisotropy and deformation that are not invariant with scaling and shifting.

At the same time, a consistent body of knowledge is being accumulated that emphasizes the “surprising” effects of the non-linear dynamics of the “motor servo” (the muscular system and the associate segmental reflex mechanisms). In fact, most neuromotor models are based on an explicit/implicit linearity hypothesis that assigns a constant stiffness to the muscles, thus posing an unsolvable dilemma: if stiffness is “small” and compatible with physiologic values near equilibrium, then model-driven trajectories tend to be highly distorted; if stiffness is big enough to counteract the distorting effects of dynamics, then its value turns out to be beyond the normal physiologic range.

The muscle part of the model is a simplified, lumped-parameter model that does not reach the level of motor units but still attempts to be biologically realistic from the point of view of system dynamics. Its plausibility is based onto two different lines of evidence: structural/local and behavioral/global. From the structural point of view, the model attempts to incorporate, separately for each muscle, a number of non-linear effects characterized by experimentally evaluated parameters. From the behavioral point of view, we checked the consistency of a multiple-muscle system, constructed with such muscular components, with experimental data about end-point impedance, that depend on the global interaction of whole musculo-skeletal system. As mentioned above, this level of analysis has been demonstrated to be sufficient to explain the characteristic time-course of arm stiffness in reaching movements, thus motivating our study of the smoothness/invariance features of handwriting movements.

The main features of the model can be summarized as follows:

1. Muscle force is decomposed into a non-controllable passive component and a controllable active component:

(1)

f_m=f_p+f_a.

2. The passive component is modeled for simplicity as a linear spring.

3. The active component is modeled as the cascade of three mechanisms: (i) a controllable force generation system, which depends of the muscle activation A(t) and is compatible with the family of ?-models; (ii) a mechanism of graded force development, which takes into account the dynamics of fused tetanus formation; (iii) a non-linear force–velocity relationship which is related to the well known Hill’s law.

4. The force generation mechanism is modeled as follows:

(2)

f=?(e^cA?1),

where c is a universal parameter that characterizes the “muscle tissue” and is the same for all the muscles, whereas ? is specific for each muscle and is assumed to depend linearly on the physiological cross-sectional area. The exponential form of the function incorporates the size principle, which is known to characterize the recruitment of motor units during the graded build-up of muscle force. The muscle activation variable A(t) is a function of muscle length l, its rate of change dl/dt, and the controllable neural input ?:

(3)

A(t)=[l(t?d)??(t)+?dl(t?d)/dt]⁺,

where [ ]⁺ is a ramp function (it clips the output to 0 for negative inputs), d is the delay of the segmental tonic and phasic reflexes, and ? is the gain of the phasic segmental reflex;⁴ ? is the centrally specified rest-length of a muscle.

5. The mechanism of graded force development is approximated, for simplicity, by means of a low-pass filter, characterized by a time constant of 15 ms (it yields a “tetanic fusion“ of about 60 ms.).

6. The force–velocity relationship is approximated by a sigmoid, in agreement with the experimental data of Joyce and Rack (1969).

The dynamic model of the whole arm translates the muscle force vector f_m into the corresponding joint torque vector ?_m according to the moment-arm matrix J_m: ?=(J_m)^T f_m. This vector is balanced with the internal load (due to inertial torques) and the external load (due to friction of the of the pen with the writing surface, disturbances, etc.). The combined equation can then be written as follows:

(4)

I(q)d²q/dt²+C(q,dq/dt)dq/dt+(J_s)^Tf_load=(J_m)^Tf_m,

I is the inertial matrix, C the Coriolis matrix, q is the vector of joint angles; J_s is the “spatial” Jacobian matrix that maps end-effector forces into joint torques and is a function of q; J_m is the “muscle” Jacobian matrix and is constant in our case for the simplifying assumption about the moment arms of the muscles. The mechanical parameters of the arm, which determine the values of I, C, J_s, J_m, are compatible with typical human data. In the simulations we assumed that the pen friction can be neglected.