Neuromuscular function is the interaction between the nervous system and the physical world. Limbs and fingers are, therefore, the ultimate mechanical filters between the motor commands that the nervous system issues and the physical actions that result. In this chapter we present a mathematical approach to understanding how their anatomy (i.e., physical structure) defines their mechanical capabilities. We call them "mechanical filters" because they attenuate, amplify, and transform neural signals into mechanical output. We explicitly distinguish between limbs and fingers because their subtle anatomical differences have profound effects on their mechanical properties. Our main message is that many aspects of neuromuscular function such as co-contraction, posture selection, muscle redundancy, optimality of motor command, are fundamentally affected (if not defined) by the physical structure of limbs and fingers. We attempt to present the fundamental filtering properties of limbs and fingers in a unified manner to allow for a direct and useful application of powerful mathematical concepts to the study of neuromuscular function. Every researcher of motor control is well advised to consider these filtering properties to properly understand the co-evolution and synergistic interactions between brain and body. At the end of the day, every inquiry in neuromuscular function can be reduced to the fundamental question whether and how the nervous system can perform the necessary sensorimotor functions to exploit and reach the mechanical capabilities of limbs and fingers.
Limbs and Fingers Transform the Output of the Nervous System into Mechanical ActionsThe anatomical structure of limbs and fingers takes as inputs muscular actions and produces as outputs motions and forces. We will use the generic term "limb" for both limbs and fingers, and will distinguish between them where necessary. We begin by applying the mathematical tools of robotics where the limb is a serial kinematic chain of rigid links articulated by rotational joints, and muscles are idealized as linear actuators that produce joint torques (for a more detailed discussion see (Yoshikawa, 1990;Valero-Cuevas et al., 1998)). The propagation of information and/or energy through the system can be described as a forward model or, alternatively, as a serial cascade of filters ( Fig. 1). In this chapter we call the neuromuscular system a series of "mechanical filters" because they attenuate, amplify, and transform neural signals into mechanical output.To analyze the idealized 3-joint limb shown in Fig. 2, the first step in defining the mapping from muscular actions to mechanical outputs is to establish the mapping from joint angles (the vector of three generalized rotational coordinates q⃗ = {q 1 , q 2 , q 3 } T ) to limb endpoint position and orientation (the vector of three endpoint coordinates x⃗ = {x, y, α} T ) for a given set of link lengths:
NIH Public Access( 1) where G(q⃗ ) is the geometric model defined by the trigonometric equations for the end point...