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In order to control the execution of limb movements, the central nervous system must solve complex problems of mechanics. A substantial body of evidence supports the view that, in solving these problems the nervous system develops internal representations of the mechanics of the body coupled with its environment. Thus, through the process of motor learning the brain becomes implicitly an ‘expert’ in classical mechanics. After discussing the problems of kinematics and dynamics associated with the control of movement, this research paper introduces the formal deﬁnitions and the empirical evidence that constitute the underpinnings of the theory of internal representations in motor control.

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## 1. Dynamics

According to the laws of Newtonian physics, if one wants to impress a motion upon an object with mass m, one must apply a force, F, that is directly proportional to the desired acceleration, a. This is Newton’s equation:

A desired motion may be expressed as a sequence of positions, x, that one wishes the object to occupy at subsequent instants of time, t. Such a sequence is called a trajectory and is mathematically represented as a function, x=x(t). To use Newton’s equation for deriving the needed time-sequence of forces, one must calculate the ﬁrst temporal derivative of the trajectory, the velocity, and then the second temporal derivative, the acceleration. Finally, one obtains the desired force from this acceleration. The above calculation is an example of an inverse dynamic problem. The direct dynamic problem is that of computing the trajectory resulting from the application of a force, F(t). Direct problems are a common challenge for physicists who are concerned, for example, with predicting the motion of a comet from the known pattern of gravitational forces. Unlike physicists, the brain deals most often with inverse problems: we routinely recognize objects and people from their visual images—an ‘inverse optical problem’—and we ﬁnd out eﬀortlessly how to distribute the forces exerted by several muscles to move our limb in the desired way: an inverse dynamics problem.

One of the central questions in motor control is how the central nervous system may solve the inverse dynamics problem and generate the motor commands that guide our limbs (Hollerbach and Flash 1982). In the biological context, however, the inverse dynamic problem assumes a somewhat more complex form than the one described above. A system of second-order nonlinear diﬀerential equations is generally considered to be an adequate representation for the passive dynamics of a limb. A compact expression for such a system is:

where q, q, and q represent the limb conﬁguration vector—for example the vector of joint angles—and its ﬁrst and second time derivatives. D is a non-linear vector, valued mapping from the current state and its rate of change to the vector of joint torques, τ. In practice, the expression for D may have a few terms for a two-joint planar arm, or it may take several pages for more realistic models of the arm’s multi-joint geometry. The inverse dynamics approach to the control of multi-joint limbs consists in solving explicitly for a torque trajectory, τ(t), given a desired trajectory of the limb, q_{D}(t). This is done by replacing q_{D}(t) for the variable q on the left side of Eqn. (1):

## 2. Kinematics, Statics, And Coordinate Systems

A signiﬁcant computational challenge comes from the need to perform changes of representation—or, more technically, coordinate transformations—between the description of a task and the speciﬁcation of the body motions. Tasks, such as ‘hitting a ball with a racket,’ are described most eﬃciently and parsimoniously with respect to some ﬁxed reference points in the environment. In this example, the racket is the site at which one interacts with the environment. Borrowing some terminology from robotics, such a site is called an ‘endpoint.’ The position of the racket is fully determined by six coordinates. These coordinates may be measured with respect to three orthogonal axes originating, for example, from the shoulder. Then, a position in endpoint coordinates may be speciﬁed as a point p = (x, y, z, θ_{X}, θ_{Y}, θ_{Z}). The coordinates, x, y, and z determine a translation with respect to the orthogonal axes. The angular coordinates, θ_{X}, θ_{Y} , and θ_{Z} determine an orientation with respect to the same axes. Consistent with this notation, a force in endpoint coordinates is a vector with three linear and three angular components, F = (F_{X}, F_{Y}, F_{Z}, τ_{X}, τ_{Y}, τ_{Z}).

A diﬀerent way of describing the position of an arm is to provide the set of joint angles that deﬁne the orientation of each skeletal segment either with respect to ﬁxed axes in space or with respect to the neighboring segments. Joint angles are a particular instance of generalized coordinates. According to the standard deﬁnitions of analytical mechanics, generalized coordinates are independent variables, which are suitable for describing the dynamics of a system (see for example Goldstein 1980).

Once we have deﬁned a set of generalized coordinates we may also deﬁne a set of corresponding generalized forces. For example, if we use joint angles as generalized coordinates, the corresponding generalized forces are the torques measured at each joint. The dynamics of any mechanical system with N generalized coordinates are described by N second order diﬀerential equations relating the generalized coordinate to their ﬁrst and second time derivatives and to the generalized forces. The dynamics Eqn. (1) is an example of formulation in generalized coordinates.

Movements are executed by the central nervous system activating of a multitude of muscles. Muscle coordinates aﬀord the most direct representation for the motor output of the central nervous system. A position in this coordinate system is described by a collection of muscle lengths, l =(l_{1}, l_{2}, …, l_{M}). Accordingly, a force is a collection of muscle tensions, f=( f_{1}, f_{2}, …, f_{M}).

Both the transformations from generalized coordinates to endpoint coordinates, and from generalized coordinates to actuator coordinates are nonlinear mappings. In the case of the arm, the transformation from joint to endpoint coordinates is a nonlinear function:

The transformation from joint to muscle coordinates is another nonlinear mapping:

Some experimental studies (see Flash and Hogan 1985, Morasso 1981) have suggested that actions are planned by the brain in terms of endpoint coordinates. For example, one can mentally formulate (and execute) commands such as ‘move the hand 10 cm to the right’ without being concerned with the set of muscle commands that are involved with this action. However, once one has decided a plan of action one must somehow choose which muscles to activate and in what temporal order. In carrying out this task the brain must faces the challenges associated with kinematic redundancy: the imbalance between the number of joints that may participate in a movement, the number of degrees of freedom of the hand, and the number of independently controlled muscles acting upon the joints. There are typically fewer hand coordinates than joint angles and fewer joint angles than muscles. Such imbalance renders both transformations (3) and (4) non-invertible.

## 3. Internal Models Of Limb Dynamics

The ability to generate a variety of complex behaviors cannot be attained by just storing somewhere the control signals for each action and recalling these signals when subsequently needed. Simple considerations about the geometrical space of meaningful behaviors are suﬃcient to establish that this approach would be inadequate (see Bizzi and Mussa-Ivaldi 1998). To achieve its typical competence, the motor system must take advantage of experience for going beyond experience itself, by constructing internal representations of the controlled dynamics. These representations allow us to generate new behaviors and to handle situations that have not yet been encountered. A vivid illustration of how explicit representations of dynamics, also called internal models, may facilitate motor learning is oﬀered by work of Atkeson and Schaal (1997, Schaal 1999) who studied the task of balancing an inverted pendulum on the hand of a robotic arm. They found that robots learn to carry out this task successfully when they can build an internal model of the dynamics associated with the balancing act. Such a model may be constructed using data derived from the observation of humans engaging competently in the same task.

The term ‘internal model’ refers to two distinct mathematical transformations: (a) the transformation from a motor command to the consequent behavior; and, (b) the transformation from a desired behavior to the corresponding motor command (see Kawato and Wolpert 1998). A model of the ﬁrst kind is called a ‘forward model.’ Forward models provide the control system with the means not only to predict the outcome of a command, but also to estimate the current state in the presence of feedback delay. A representation of the mapping from planned actions to motor commands is called an ‘inverse model.’ Strong experimental evidence for the biological and behavioral relevance of internal models has been oﬀered by experiments that involved the adaptation of arm movements to a perturbing force ﬁeld generated by an instrumented manipulandum (Sabes et al. 1998, Shadmehr and Mussa-Ivaldi 1994). The major ﬁndings of these studies are as follows: (a) when exposed to a complex but deterministic ﬁeld of velocity-dependent forces, arm movements are ﬁrst distorted and, after repeated practice, the initial kinematics are recovered; (b) if, after adaptation, the ﬁeld is suddenly removed, after eﬀects are clearly visible as mirror images of the initial perturbations; (c) adaptation is achieved by the motor system through the formation of a local map that associates the states (positions and velocities) visited during the training period with the corresponding forces; and, (d) after adaptation this map—that is the internal model of the ﬁeld—undergoes a process of consolidation (see Brashers-Krug et al. 1996).

## 4. The Neurobiological ‘Building Blocks’ Of Internal Models

Once it has been established that the motor system creates internal representations of complex multi-joint dynamics, it remains to determine how these representations may come about. As pointed out by David Marr (1982), any mathematical transformation may be carried out in diﬀerent ways depending upon which building blocks or ‘primitives’ are employed. Electrophysiological studies involving the stimulation of muscles and of the spinal cord in frogs (Bizzi et al. 1991) indicated: (a) that the stimulation of a site in the lumbar spinal cord results in the activation of multiple muscles acting on the leg on the same side of the stimulation; (b) that concomitant or ‘synergistic’ muscle recruitment generates a ﬁeld of viscoelastic forces over a broad region of the leg workspace; and, (c) that the simultaneous activation of multiple spinal sites leads to the vectorial summation of the corresponding force ﬁelds. These and similar ﬁndings suggest that motor commands reaching the spinal cord from higher brain centers are not directed at controlling the forces of individual muscles or single joint torques. Instead, the descending motor commands modulate the viscoelastic force ﬁelds produced by speciﬁc sets of muscles. These force ﬁelds have inﬂuence over broad regions of the limb state space as each active muscle within a synergy contribute a signiﬁcant force over a large range of positions and velocities.

From a mathematical standpoint, the force ﬁelds generated by neural modules in the spinal cord are nonlinear functions of limb position and velocity and of time: φ_{i}(q, q, t). Consistent with the ﬁnding of vector summation, the net force ﬁeld induced by a pattern of K motor commands may be represented as a linear combination:

In this expression, each spinal ﬁeld is a force that depends upon the state of motion of the limb, (q, q) and upon time, t, in a ﬁxed stereotyped way. The descending commands, (u_{1}, u_{2}, …, u_{K}), act as coeﬃcients that modulate the degree with which each spinal ﬁeld participates in the combination. These commands can just select the modules by determining how much each one contributes to the net control policy. The linear combination (5) generates the torque that drives the limb inertia. Substituting it for τ(t) in Eqn. (1) one obtains:

Least squares approximation can eﬃciently determine an optimal set of tuning coeﬃcients given a desired trajectory, q_{D}(t):

with

The symbol ● indicates the ordinary inner product.

While spinal force ﬁelds oﬀer a practical way to generate movement, they also provide the central nervous system with a movement’s representation. This representation is geometrically similar to the representation of space by a set of Cartesian coordinates. In the latter case, we may take three directions—represented by three independent vectors—and then project any point in space along these directions. As a result, an arbitrary point in space is represented by three numbers, the coordinates x, y, and z. The movements of a limb can be considered as ‘points’ in an abstract geometrical space. In this abstract geometrical space, the force ﬁelds produced by a set of modules play a role equivalent to that of the Cartesian axes and the selection parameters that generate a particular movement may be regarded as generalized projections of this movement along the module’s ﬁelds.

### 4.1 A Computational Approach To Motor Adaptation

If the dynamics change while the modules remain unchanged, then the representation of the movement must change accordingly. This is shown by the following argument. Suppose that a trajectory, q(t), is represented by a selection vector c =(c_{1}, c_{2}, …, c_{K}) for a limb with the dynamics of Eqn. (6). Now, suppose that the limb dynamics are suddenly modiﬁed by an additional load, E(q, q, q). Leaving the representation and the ﬁelds unchanged we have now a new diﬀerential equation

whose solution is a trajectory q(t), generally diﬀerent from the original q(t). The original set of coeﬃcients, c_{m}, now generates the trajectory q(t) and can, accordingly, be considered as its representation within the modiﬁed environment, D+E. The old trajectory is recovered by changing the selection coeﬃcients to a new set, c´=c+e with

With these new coeﬃcients, the new dynamics become equivalent to the old dynamics along the original trajectory.

The modiﬁed coeﬃcients c´ oﬀer a new representation of the old movement q(t) in the altered dynamics. This procedure for forming a new representation and for recovering the original movement is consistent with the empirical observation of aftereﬀects in load adaptation (see Shadmehr and Mussa-Ivaldi 1994). If the load is removed after the new representation is formed, the dynamics become

that can be rewritten as

Therefore, removing the load with the new representation corresponds approximately to applying the opposite load with the old representation.

Is it necessary for the motor system to modify a movement’s representation each time the limb dynamics changes? Or is there a way for restoring the previously existing representations? From a computational point of view, whenever a dynamical change becomes permanent—as when we undergo growth or damage—it would seem convenient for the central nervous system to have the ability to restore the previously learned motor skills (that is, the previously learned movement representations) without need to relearn them all. It is possible for the adaptive system to restore, at least partially, the motor representations that preexist a change in dynamics by modifying the modules and their force ﬁelds. A speciﬁc modiﬁcation is obtained when we may express the coeﬃcients e=(e_{1}, e_{2}, …, e_{K}) as a linear transformation of the original coeﬃcients c =(c_{1}, c_{2}, …, c_{K}):

This transformation is a coordinate transformation of the selection vector and may be implemented by a linear associative network. With a minimum of algebra one sees that

where the old ﬁelds φ_{m} have been replaced by the new ﬁelds

This is, again, a coordinate transformation of the original ﬁelds that may be implemented by a neural network intervening between the descending commands and the original ﬁelds. By means of such coordinate transformations, one obtains the import- ant result that the movement representation—that is the selection vector c—can be maintained invariant after a change in limb dynamics.

In conclusion, the system of motor primitives induced by independent modules within the spinal cord—as well as within higher structures of the nervous system—provides us with an alphabet for representing the mechanics of the body and for modifying this representation as required by changes in limb and environmental dynamics.

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