Springs and Muscles, Part 4

[From Bruce Abbott (2015.02.13.1445 EST)]

Part 3 of this series presented the Level 1 muscle control system developed by Bill Powers and described in Chapter 7 of his book, Behavior: The Control of Perception, published in 1973. At this point no attempt had been made to develop a computer simulation that could demonstrate whether the model actually behaves realistically. However, with the availability of relatively inexpensive personal computers, Powers eventually was able to work toward implementing control-system models capable of “behaving” realistically. These included two versions of what came to be known as the “Little Man” demos, program names “Arm1” and “Arm2.”

Little Man

image00175.jpgThe Little Man simulation was designed to demonstrate visually and kinesthetically guided reaching. As shown in the figure at right, the simulation presents the image of a “little man,” represented by a triangular chest, a rectangular head with two small circles for eyes, and a right arm consisting of an upper arm and forearm. This figure is set within a box that can be rotated by the viewer to provide different perspective views of the scene within. Also visible within the box is a triangular target. Controls were provided that allowed the user to move the target around in all three spatial directions: left-right, up-down, and in-out (relative to the Little Man figure). The Little Man’s job was to keep his “finger” (really just the end of the forearm) touching the center of the target as the user moved the target around.

Models vary in the level of detail they include, depending on the purpose to be served by the model. For example, a static scale model airplane may be designed to look exactly like the full-size airplane it models, yet be incapable of flying. In contrast, a radio-controlled model airplane may be designed to fly well, but to do so it will have to sacrifice some ability to match the appearance of the full-scale airplane. This is because the aerodynamic forces acting on the full-scale airplane do not scale down linearly to those acting on the smaller model, requiring that some adjustments be made. (In particular, the tail surfaces will have to be made proportionately larger.) In the original Little Man simulation (Arm1), interest was not so much in modeling all the way down to the level of muscle-control as in demonstrating that systems controlling joint angle are capable of producing adequate guided reaching. These joint-angle control systems acted directly on joint angle (based on joint-angle sensors) rather than calling on a lower-level system to contract muscles and thereby bring about the requested joint-angle change.

The model included a simple mechanism that would allow Little Man to “visually” locate the target’s position in three dimensions. This system set references to turn and nod the head so as to center the target in the visual field, represented by the circle in the Little Man diagram. The circle shows the position of the target within the visual field as seen by each eye. The head was rotated both horizontally and vertically, and the eyes moved horizontally, as needed to center the target within the visual field and to converge the double-image of the target (one for each eye) to the same position, as shown in the figure. This convergence was used to determine distance to target, effectively giving Little Man stereoscopic depth perception.

With the parameters of each control system properly adjusted, Little Man demonstrated the ability to accurately track the target position both visually and with its finger tip while the target’s position is being manipulated by the user. This demonstrates that control systems like these are capable of realistically producing the required behavior.

In 1999 Powers published a new version of the Little Man demo (Arm 2) that for the first time implemented Level 1 control over muscle length and force. The paper’s abstract nicely summarizes the intent behind creating the model:

This paper describes a preliminary simulation of kinesthetic control systems that operate a humanoid arm having three degrees of freedom. The design is in part a literal

interpretation of the stretch and tendon reflexes considered as control systems. A second level of control converts independent control of three joint angles into a trio of systems

controlling the tip of the arm in pitch, yaw and distance coordinates centered on the shoulder. The basic properties of muscles are included, and the arm movements are

calculated using equations describing the physical dynamics of the arm. A “visual servo’’ level of control is included in preliminary form. The model exhibits realistic behavior,

producing stable and fast control without computing either inverse kinematics or inverse dynamics.

[Note: The relevance of the reference to inverse kinematics and inverse dynamics found at the end of this abstract will become apparent when I discuss, in a later part, a competing approach to motor control in which it is assumed that the brain produces movements by computing the forces that the muscles will need to exert in order to produce those movements and then determines the neural “commands” that will cause the muscles to generate those forces.]

In the real system operating the joints, opposing sets of muscles (flexors and extensors) pull on the bones to either flex or extend the joint. The reason for this arrangement is that muscles can act only in one direction: they can pull but they cannot push. As contraction of one set of muscles closes the joint, the opposing muscles are stretched, and vice versa. In the simulation, however, Power chose to replace the opposing pairs of muscle with a single actuator capable of both pulling and pushing. This simplifies the model without materially changing its behavior. The model implemented is identical to the one illustrated in Powers (1973), save for representing opposing muscles as a single push-pull muscle. According to Powers,

When the equations for two one-way muscle systems are combined, all the resting lengths drop out and we are left with a single zero-centered linear system valid in the regions where both muscles are active; the effect of muscle tone is absorbed into the effective linear spring constant term. Also, this way of combining opposed systems automatically takes care of reciprocal connections between the reflex signals from opposing muscles, and crossovers of the driving signals as well.

The model with the higher levels of control switched off and the alpha and gamma reference signals set to hold the arm out straight and horizontal exhibited a rather odd property when the effect of gravity was switched on: The arm slowly sagged. I am not clear as to the reason for this behavior, but Powers suggested that it might in fact duplicate the initial behavior sometimes observed after spinal transaction, which cuts off communication with higher-level control systems. Adding systems controlling the perception of joint angle corrected this problem. These higher-level systems act by setting the alpha and gamma references of the bottom-level muscle control systems. Any “sagging” produced by control at the muscle level will result in joint-angle error, causing the joint-angle control system to set the alpha-gamma references to higher values, increasing muscle contraction to eliminate the sag.

In 1998, Powers and mathematician Richard Kennaway produced a paper describing a revised muscle model that was able to account for certain muscle-associated phenomena better than the standard model most commonly in use at that time. The model was of the muscle only; it did not include the neural circuits that control muscle length and tension. I have not investigated whether the substitution of this model for the muscle model found in the Little Man demonstration would materially affect its operation. Although the new muscle model paper apparently was never formally published, it is available as a free download at http://www.livingcontrolsystems.com/intro_papers/muscle_model.pdf .

A Windows version of the Arm1 demo is available at the Living Control Systems Publishing website at http://www.livingcontrolsystems.com/demos/tutor_pct.html . Unfortunately, Arm2 currently exists only as a DOS program for the PC.

Up Next: The top-down “computational” approach to motor control.

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