Visually Controlled Locomotion: 40 years Later.

Transcription

ECOLOGICAL PSYCHOLOGY, 10(3-4), 177-219
Copyright © 1998, Lawrence Erlbaum Associates, Inc.
Visually Controlled Locomotion:
40 Years Later
William H. Warren, Jr.
Department of Cognitive and Linguistic Sciences
Brown University
Gibson's article, "Visually Controlled Locomotion and Visual Orientation in Animals" (1958/this issue), is the leading statement of a nonrepresentational, information-based approach to visual control. The core ideas he introduced 40 years ago resurface, explicitly or implicitly, in much contemporary work on perception and action
in humans, insects, robots, and autonomous agents. The purpose of this special issue is
to assess the continuing pertinence of these insights and illustrate current directions
in research on visually controlled locomotion. In this article, I locate the 1958 article
in the context of Gibson's emerging theory of perception, contrast information-based
control with standard model-based and cybernetic control architectures, evaluate the
current status of Gibson's visual control formulae, and situate visual control within an
informational-dynamical approach to agent--environment systems.
Locomotion is a biologically basic function, and if that can be accounted for then the
problem of human space perception may appear in a new light. The question, then, is
how an animal gets about by vision.
-Gibson (1958, p. 183/this issue)
How do animals get about by vision? The question is of fundamental importance to
our understanding of both perception and behavior. The capacity for locomotion is
a defining characteristic distinguishing the kingdoms Animalia and Plantae, and the
imperative of controlling locomotion has clearly shaped the evolution of the visual
system. The function of the most primitive visual organs such as invertebrate ocelli
is to regulate locomotion and orientation, including orienting to the vertical,
phototaxis, and shadow escape responses. Insects have specialized visual-motor
systems that support locomotor behavior such as the optomotor response and sophisticated flight control. We are only beginning to learn about cortical motion arRequests for reprints should be sent to William H. Warren, Jr., Department of Cognitive and Linguistic Sciences, Brown University, Box 1978, Providence, R1 02912. E-mail: [email protected]
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eas in mammals that appear to be specialized for the detection of optic flow patterns.
The significance of vision for locomotor guidance has also become apparent with recent attempts to build autonomous robots and intelligent vehicles, although computer vision systems remain unequal to this most basic perceptual task. However,
when Gibson posed the question in 1958, it was quite novel to the field of perception, and in its pedestrian familiarity, locomotion is something most psychologists
and cognitive scientists continue to take for granted. It is typically assumed that the
perception of space will provide a general basis for behavior, but seldom considered
that the latter might condition the former. In reversing this formulation, Gibson argued that visual perception might better be understood in the context of its biological function of guiding action, with locomotion as a paradigmatic example.
In many ways, the field is only now catching up with Gibson circa 1958. The
ideas he proposed 40 years ago are resurfacing, explicitly or implicitly, in much of
the recent work on perception and action in humans, insects, and machines. The
current interest in this area is leading to a new appreciation of the deep relations
between behavior and vision (Aloimonos, 1993; Milner & Goodale, 1995). The
purpose of this special anniversary issue is to draw attention to this lineage, assess
the continuing pertinence of Gibson's insights, and illustrate current directions in
research on visually controlled locomotion. In this article, I begin by locating the
1958 article in the context of Gibson's emerging theory of perception and highlighting what I view as its core contributions. Gibson's account of information-based control is contrasted with standard model-based control and cybernetic
control architectures. My own research program can be viewed as an attempt to
work through Gibson's "formulae" for the visual control oflocomotion experimentally, and the heart of this article is an effort to evaluate and update his hypotheses
in light of recent findings. Finally, I discuss how our understanding of visual control
might be advanced by situating it within an informational-dynamical approach to
agent--environment systems.
THE 1958 ARTICLE AND ITS ANTECEDENTS
The article "Visually Controlled Locomotion and Visual Orientation in Animals"
was drafted during 1955-1956 while Gibson was on a Fulbright fellowship at Oxford,
following a year at Berkeley in the company of Edward Tolman, Egon Brunswik, and
Gordon Walls. The fmal version appeared in the BritishloumalofPsychology in 1958,
when Gibson was 54. It represents a milestone in the gradual development of Gibson's pragmatic, information-based theory of perception. The "animal" in the title is
indicative of his ambitions-to propose control principles generic to any mobile
agent, from aerial bird to aquatic fish to terrestrial human to, potentially, Martian robot. Although the article itself may not have had a high "impact factor" by the somewhat cynical standards of today, his biographer Ed Reed (1988b) cited it as among his
most important. Core concepts such as the optic array, affordances, visual kinaesthesis, and visual control had their first full expression therein and were subsequently
VISUALLY CONTROLLED LOCOMOTION: 40 YEARS LAlER
179
elaborated in Gibson's last two influential books (1966; 1979/1986). In each of these
threads of thought, his engagement with the problem of locomotion played a catalyzing role and reilluminated the problem of spatial perception.
Gibson's early interest in locomotion was manifested in an analysis of automobile
driving (Gibson & Crooks, 1938). Arguing even then that "locomotion is guided
chiefly by vision" (p. 454), Gibson and Crooks, an automotive engineer, offered a
functional description of the task of driving based on the concept of the field of safe
travel, defmed as the possible paths a car can currently take given the destination of
the driver, the array ofstationary and moving obstacles, and the speed of the vehicle.
Following Lewin (1936), they asserted that the environment possesses positive and
negative valences that attract one toward a destination and repel one from obstacles,
and they suggested that the sum ofthese two force fields yields the path oflocomotion
and might even account for detour behavior-anticipating current potential field
models of robot navigation (Khatib, 1986). However, whereas Lewin conceived of
valences as subjective values attributed to objects, Gibson and Crooks stated that
the field of safe travel existed objectively, independent of the driver's awareness, as a
field of "behavioral opportunities." This clearly contains the germ of the later notion
of affordances. Further, they claimed that "When they [fields of possible behavior!
are perceived as such by the driver, the seeing and the doing are merged in the same
experience" (p. 460), an early assertion of the mutuality of perception and action.
Here Gibson began the transmutation of the motor theory ofperception he inherited
from his professor, the behaviorist Edwin Holt (see Reed, 1988b, pp. 67-73), according to which spatial experience is determined by patterns of motor responses to the
environment, into a theory in which perception and action are complementary aspects of adaptive behavior. As he stated in 1950, "Spatial behavior and spatial perception are coordinate with one another" (Gibson, 1950, p. 225). Notably absent,
however, was any conception of the visual information for the field of safe travel, but
by 1958 Gibson was prepared to argue that the optic array "specifies" goals, obstacles,
and paths of locomotion.
During Wodd War II, Gibson turned his attention to the practical problems of
pilot testing and training, an experience that greatly influenced his thinking about
perception (Gibson, 1947,1950). As he recounted it, tests of depth cues for points
in empty space, such as the 3-point test of stereo acuity, failed to predict an individual pilot's ability to land an airplane. This realization led Gibson to reject the
long-standing "air theory" of spatial perception and propose a reconceived "ground
theory," in which perception depends on a continuous ground surface and associated optical gradients. His work on flying also led to the discovery of the information that had been missing from his analysis of driving-the optic flow field (see also
Langewiesche, 1944, which Gibson cited). Optic flow was the offspring of a meeting between Helmholtz' idea of motion parallax and Gibson's ground theory: the
generalization of local parallax to a gradient of optical velocities surrounding the
moving observer, which he termed motion perspective and represented as a vector
field (Figure 1). First, the gradient of vector magnitudes specifies the structure of
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the environment, including continuous distance along the ground and the orientation of surfaces. Second, the pattern of vector directions specifies one's own path of
locomotion, for they radiate outward from a focus of expansion that corresponds to
the current direction of self-motion or heading. Optic flow thus revealed that we
not only perceive a stable environment but co-perceive self-motion through that
environment, which Gibson termed visual kinaesthesis. Third, Gibson realized that
optic flow might reciprocally be used to control locomotion. This was only hinted
at in 1950, when he suggested that steering is "a matter of keeping the focus of expansion in the direction one must go" (Gibson, 1950, p. 128). This circular causality of
self.produced stimulation being used to guide action contributed to Gibson's rejection of the causal theory of perception and stimulus-response (S-R) theories of behavior, in which perception and behavior are responses to stimuli, in favor of active
perception that seeks out information and purposive action that is visually guided
but not causally determined. In 1955, Paul Olum and Frank Rosenblatt (later of
perceptron fame) helped Gibson perform a mathematical analysis of optic flow and
characterize the information it contains for relative distance, direction of locomotion, and time-to-contact (Gibson, Olum, & Rosenblatt, 1955).
osnsss
aX
--
....-
-
--..
/'
,/"
as
--..
~
/
"'"
FIGURE 1 Gibson's representation of optic flow as a two-dimensional
velocity field, for observer rranslation. Each vector represents the optical
motion of the corresponding environmental element. All vectors radiate
outward from a focus of expansion in the direction of heading.
Note. From The Ecological Approach to Visual Perception (p. 124), by J. J.
Gibson, 1986, Hillsdale, NJ: Lawrence Erlbaum Associates, Inc. (Original work published 1979). Copyright 1986 by Lawrence Erlbaum Associates, Inc. Reprinted by permission.
VISUALLY CONTROLLED LOCOMOTION: 40 YEARS LATER
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The 1958 article represents the culmination of Gibson's hard-won insights about
perception and locomotion. Foremost among them is the first exposition of the optic
array, his crucial move away from the retinal image as the foundation for understanding perception. During the 1940s and 1950s, Gibson wrestled with the fundamental
problem of perceptual constancy-how a moving observer can see a rigid world with
objects of constant size and shape despite continually varying stimulation and a kaleidoscopic flux ofvisual sensations. The traditional explanation was that mental operations are required to reconstruct constant objects from inadequate sense-data.
Yet if the perceived world is an artifact of the mind, how is it that we are able to get
about in the real one as accurately and precisely as we do? And how is it that different
individuals perceive the same world well enough to interact and communicate? By
1950 Gibson arrived at the opposite conclusion-that perception is not a mental
construction and constancy must be based on invariant properties of stimulation.
Although Gibson relied on the retinal image in 1950, shortly thereafter he realized
that it suffered from the same problem as sensations-it was highly variable, fluctuating with movement of the eyes, head, and body, and historically had not yielded up
correlates for the constant properties of objects. In the 1958 article, he took the step
to the optic array, the pattern of "different intensities (and frequency compositions)
in different directions" in the light available at a point ofobservation, independent of
the optics or physiology of any particular eye. The higher order variables he sought,
confounded in the retinal image, would be more readily revealed in the optic array-specifically, the family ofcontinuous transformations that corresponded to the
observer's path oflocomotion, and the mathematical invariants that corresponded
to the constant structure of the environment. Subsequent formal work on structure
from motion has shown that information for three-dimensional structure and the observer's path is indeed available in the pattern of optical motion.
The second notable contribution of the 1958 article is the first articulated account of affordances (although only the verb form afford is used). Gibson carefully
distinguished between visual control reactions based on optic flow, such as steering
toward an object, and identifying reactions based on object perception, such as perceiving whether the object is a bump-into-able obstacle or edible food, a
mate-with-able conspecific or a dangerous predator. The important new claim was
that these higher order properties of objects are specified by higher order variables
in the optic array. To the extent that their textures, colors, and motions discriminate behaviorally significant classes of objects, discriminative reactions are possible. However, betraying a lingering behaviorist influence, each object was still
conceived as belonging to only one behavioral class rather than possessing multiple
affordances, temporarily salvaging an "S-R theory of identifying reactions," with
stimulus-response links between objects and appropriate actions. With regard to
locomotion, features of the ground surface were described in functional terms as
paths, barriers, jumping-off places, and falling-off places, all of which "depend on the
locomotor capacities of the animal" (p. 192). To the extent that the arrangement
of surfaces was specified in the optic array, so were these functional features, al-
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though their scaling with respect to the actor was not yet incorporated into the account (see Gibson, 1979/1986; Warren, 1984; Warren & Whang, 1987). The
claim that an animal "perceives the possibilities of locomotion surrounding him"
(Gibson, 1958, p. 192) clearly echoed the definition of the field of safe travel 20
years before, but these new details anticipated the full theory of affordances to
come 20 years later. When Gibson went on to say, "And this is probably what should
be meant by asserting that an animal has visual space perception" (p. 192), the implication was that perceiving the layout of surfaces and what it affords for locomotor behavior is the evolutionarily primary function of spatial perception.
Third, Gibson developed his 1950 observations about visual kinaesthesis. The
notion that vision, in addition to vestibular, joint, and muscle proprioception,
could detect bodily movement subsequently led Gibson (1966) to question the traditional anatomical classification of the senses and argue that all perceptual systems serve both proprioceptive and exteroceptive functions. His classic examples
of the fish swimming upstream and the bird flying upwind first appeared in 1958 to
make the case that visual kinaesthesis is essential for sensing movement relative to
the environment, although Gibson also allowed that terrestrial animals who push
against the ground may also exploit touch and muscle-joint proprioception as well.
The "sea of energy" around us, he said, contains "redundant information" that can
be picked up in multiple ways and in various combinations. Far from being inadequate, information provides an embarrassment of riches, allowing for behavior to
be robust over a wide range of conditions.
However, the central hypothesis of the 1958 article was that the optic array and
optic flow provide information for the control of locomotion relative to the visible
environment. Gibson argued that visual kinaesthesis is "an invariable accompaniment of locomotor behavior and therefore provides 'feedback' stimulation for the
control and guidance of locomotor behavior" (1958, p. 185/this issue)-a clear
statement of optic flow as self-produced stimulation and the circular causality of visual control. It also revealed a cybernetic turn in his thinking, with its concepts of
control loops, feedback, and error-correction, which I believe colored his framing
of the formulae. I I Most important for present purposes, Gibson offered a fully conceived set of five descriptive "formulae" for the visual control oflocomotion, which
govern starting and stopping, steering toward a goal, braking, obstacle avoidance,
and pursuit of and flight from moving objects. He later called them "rules" while
cautioning that, "The rules are not commands from a brain; they emerge from the
animal-environment system" (Gibson, 1979/1986, p. 232); I subsequently called
them "laws" (Warren, 1988) to emphasize their principled basis in the systematic
relations between information and behavior. It is to the problem of visual control
that I now turn.
I He later noted that visual kinaesthesis does not, strictly speaking, constitute feedback about
purposive movement because it is produced by passive as well as active self-motion (Gibson,
1979/1986).
VISUALLY CONlROLLED LOCOMOTION: 40 YEARS LATER
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VISUAL CONTROL ARCHITECIURES
Locomotion and manipulation are controlled not by the brain, but by information.
Control lies in the animal-environment system. Behavior is regular without being regulated. The question is how this can be.
--Gibson (1979/1986, p. 225)
One approach to the guidance oflocomotion common in mobile robotics (Moravec,
1981) and human navigation (Loomis & Beall, this issue) is model-based control (see
Figure 2a). On this view, input from vision and other sensors is used to construct a
metric three-dimensional model of the world, which is in turn used to explicitly plan
a path of movement, which is finally executed. This approach is consonant with the
prevailing cognitivist paradigm in which all behavior is based on internal representations such as models and plans, and consequently also confronts that paradigm's inherent foundational problems (see Clark & Toribio, 1994; Epstein, 1993; Putnam,
1994; Searle, 1980; Shaw, Turvey, & Mace, 1981; vanGelder, 1995). The approach
also assimilates the standard view that the function of perception is to create and update a general-purpose three-dimensional scene description, independent of the behavior it serves--ironically, a view that owes much to Marr's (1982) interpretation
of Gibson (1950). However, considered in the biological context of the survival of
the organism, the function of perception is better stated as creating contact between
the animal and its environment that supports adaptive behavior. The reification of
the environment as an internal representation is not necessary for such contact, and
if analogous knowledge is required for other purposes, it is likely to be derived from
this primary contact.
Gibson's article was the leading statement of an alternative nonrepresentational
approach, which I call information-based control. Rather than being governed by an
internal model and a precomputed plan, behavior is controlled "online," if you will,
by occurrent information 2 about the relation between the observer and the environment. Steering through a field of obstacles, for instance, could be controlled on the
basis of optic flow by perceiving heading and time-to-contact with respect to them,
without computing a model of their three-dimensional layout. The realized path of
locomotion is thus emergent rather than planned in advance, resulting from the dynamic interaction between the structure of the environment and the animal's strategies for steering and obstacle avoidance, as suggested in the quote from Gibson
(1979/1986) at the beginning of this section.
There was, however, a tension lurking in Gibson's (1958/this issue) discussion of
information-based control, specifically, whether it is based on general structural information for the objects and three-dimensional layout of the environment, or specific control information found in optic flow patterns (see also Loomis & Beall, this
issue). On the one hand, the central thrust of his emerging theory of perception was
21nformation that is currently available to the observer.
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(a)
~~~~I~
Info:
3D World
Model
H Pa~
Planmng
f::Ac~onl
Action 2
(b)
(c)
Info I ~ Controllaw1 - - . Action I
Info 2 ~ Controllaw2 - - . Action 2
FIGURE 2 Possible architectures for visual control. (a) Model-based control. (b) Information-based control, predicated on generalsttuctural information. (c) Information-based control,
predicated on task-specific control information.
to account for the human ability to perceive the layout of the environment, by means
of a rich array of static and dynamic information. Relating this account of perception
to the problem of visual control, he emphasized at several points that behavior is oriented to the environment, based on the perception of constant objects. Furthermore, Gibson explicitly criticizes the notion that proximal stimulus patterns serve as
'''cues' for behavior" (p. 191), reflecting his struggle with an increasingly creaky behaviorism. This view echoes Holt's insistence on the "recession of the distal stimulus," the idea that mature behavior is no longer a reflexive response to proximal
stimuli but an integrated response to distal objects. I have schematized this in Figure
lb, where a variety of information, such as texture gradients, disparity gradients, motion perspective, dynamic occlusion, and eye-height information may contribute to
the perception of the three-dimensional environment surrounding the observer,
which guides adaptive action. The proposal that behavior is oriented to the environmentratherthan composed ofstereotyped responses to stimuli allows for action to be
flexible and context-conditioned, so that it is functionally specific to the situation at
hand (Reed, 1988a). However, although based on occurrent information rather
than an internal representation, it is also reminiscent of the general-purpose view
that vision recovers the three-dimensional structure of the world, incidentally providing a common basis for any action.
On the other hand, when it comes to detailing control principles for locomotion, Gibson's formulae read very much as though behavior is controlled by proximal stimulation-in the form of optic flow patterns-rather than by perceived
environmental structure. Particular actions, such as steering to a goal or making
contact without collision, are each governed by particular informational variables (or combinations of variables) according to corresponding control laws, as
schematized in Figure lc (Warren, 1988). This view seems to sacrifice a general
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basis for functional specificity and lead to a proliferation of control laws for every
conceivable situation. In sum, there appears to be a rift between the role of information in specifying the environment and its use in controlling behavior.
I think this tension between specification and control is dissolved by the understanding that optic flow patterns constitute not merely proximal stimulation, but information about the ecological state of affairs encompassing both the environment
and the observer. In his deSCription of visual kinaesthesis, Gibson emphasized that
optic flow specifies the relative motion between the observer and surfaces in the
world and conversely can be used to control that relative motion. Thus, locomotion
is neither a response to proximal stimulus cues (as in both S-R and cybernetic theory), nor governed by perception of the environment per se (exteroception), but is
controlled by information about the motion of the observer relative to that environment (which David Lee called ex-proprioception) and motion of the environmentrelative to the observer (which Bob Shaw has christened pro-exteroception). AB I have
argued previously, laws of control are reciprocals of laws of specification (Warren,
1988). If information adequately specifies the state of the econiche, corresponding
control laws are avaUable that wUl allow behavior to be functionally specific to the
situation at hand.
An account of visual control in the spirit of Gibson (1958/this issue) thus implies three specific claims: (a) control laws are nonrepresentational, in that visual
control does not employ an internal model of the environment; (b) control laws are
predicated on special-purpose control information, rather than general-purpose
structural information about the three-dimensional layout of the environment;
and (c) control laws are task-specific, such that different actions may be regulated
by different sets of informational variables (which may overlap). Control laws for
obstacle avoidance, braking, and pursuing a moving object, for example, might involve different relations with different optic flow variables.
The adequacy of the control architectures sketched in Figure 2 is ultimately an
empirical matter. The first divide, between model-based and information-based
control (Figure 2a vs. 2b,c) , poses the question of whether locomotion is based on an
internal representation or occurrent information. Although this is largely a
metatheoretical issue that is difficult to submit to direct experimental test, certain
implications may be drawn from experiments that force participants to rely on memory for the environment or a preplanned action sequence. To the extent that performance deteriorates under such conditions, it would imply that occurrent
information is ordinarily used for visual control and would undermine reliance on a
memory representation or advanced path planning; however, it might be argued that
this does not rule out some sort of transient world model that depends on occurrent
input. Conversely, maintaining accurate performance under such conditions would
imply the persistence of spatial knowledge or an action plan sufficient to gUide behavior; however, it could be of a partial and task-specific variety as opposed to a rich
world model, and it would not follow that online visual control is normally based on
such knowledge. Evidence for a more complete memory representation might in-
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volve showing that, after blindfolding, action can be accurately redirected to unexpected target locations, even after partial execution to expected locations. For
example, research on blind walking has shown that people can walk to targets or
point to a target during walking after brief visual exposure to the layout (Fukusima,
Loomis, & Da Silva, 1997; Loomis, Da Silva, Fujita, & Fukusima, 1992), suggesting
that they can preserve and update some knowledge of spatial position. On the other
hand, recent work on change blindness and saccadic integration, which shows that
observers fail to notice large changes in a scene during a blank interval or a saccade,
suggests that people do not have a very complete memory representation for unattended parts of the world (Rensink, O'Regan, & Clark, 1997).
The second divide for visual control is between general structural infonnation
about the three-dimensional layout of the scene and specific control infonnation
for particular tasks (Figure 2a,b vs. 2c). One way to approach this is to manipulate
three-dimensional information and control infonnation selectively, to detennine
which variables are tracked by perfonnance. To the extent that (a) task-specific
variables are sufficient for behavior in the absence of three-dimensional infonnation and (b) they predict behavior even when three-dimensional infonnation is
available, this would constitute evidence in favor of task-specific control infonnation and vice versa. However, given that optic flow provides infonnation both for
control and for three-dimensional layout, the experimental dissociation of the two
may be difficult; for example, motion perspective specifies both self-motion and
surface orientation, and relative time-to-contact with an array of objects in the
scene is fonnally equivalent to a relative depth map of the scene. A second challenge is that, even in the presence of general three-dimensional infonnation, systematic distortions in the perception of metric structure are frequently reported,
such as compression along the depth axis (Todd, Tittle, &Nonnan, 1995), so one
must consider whether these perceptual effects could also account for the observed
behavior. Finally, the task-specificity of control infonnation can be tested by determining whether different tasks, such as obstacle avoidance, braking, and pursuit,
actually depend on different infonnational variables.
We recently probed the first divide by manipulating the visibility ofobjects during
a joystick steering task (Duchon & Warren, 1997). Participants were instructed to
steer through a slalom course of"gates" (pairs of vertical posts) while we manipulated
(a) the number of gates that were visible and (b) the number of gates immediately in
front of the observer that disappeared from view before the observer reached them.
For example, of the four upcoming gates, the farthest two might be visible while the
closest two are invisible. Steering accuracy was as high when participants could see
only one visible gate as when they could see three, suggesting that they do not take
advantage of preview infonnation in this demanding task. J However, when two or
more gates became invisible directly in front of them, forcing them to plan ahead, ac3A stronger test of this question is planned by varying the position of far gates to see whether they
influence current behavior.
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curacy plummeted so that, on average, they missed every one. This implies that participants control steering with respect to the upcoming gate and cannot accurately
plan their actions more than one gate ahead, either because they cannot remember
and update the metric position ofmore than one gate (impoverished world model) or
because they cannot plan an action sequence longer than one steering maneuver
(impoverished path planning). They do, however, usually tum in the right general
direction even with two invisible gates, suggesting the persistence ofsome crude spatial knowledge or left/right tum sequence. The results appear contrary to modelbased control and consistent with online control, at least for the normal guidance of
steering with respect to visible objects.
Even if an information-based view is granted for online visual control with respect
to the visible environment, it is often argued that when it comes to long-range navigation with destinations beyond the range of vision, mental representations such as
cognitive maps become necessary. However, in 1958 Gibson began to extend his
approach to what he called way-finding. He suggested that we remain oriented to remote parts ofour environment because we are linked to them via sequences of transformations of the optic array, which specify both the invariant structure of the
environment and the paths we have traveled. The orderly opening up and closing of
vistas over time offers an informational basis for apprehending the three-dimensional
environmental layout, which Gibson (1979/1986, p. 198-199) distinguished from
both an S-R chain oflocations and actions and a plan-view cognitive map, but is more
like "being everywhere at once." Most controversially, he stretched the traditional
definition of perception to cover the awareness of existing environmental surfaces
that are both visible and currently occluded, which thereby "includes within perception a part of memory, expectation, knowledge, and meaning," (1979/1986, p. 255) .
In some instances purported evidence for metric cognitive maps has turned out to
depend on the presence of occurrent information for orientation, such as
long-distance landmarks or vistas (Dyer, 1991; Gould, 1986). In collaboration with
Michael Tarr and Leslie Kaelbling, we are currently setting up a visual navigation
laboratory at Brown University that will use virtual reality techniques to investigate
this form of spatial knowledge and its dependence on exploratory behavior, in both
humans and robots.
The past decade has seen a reaction against the model-based approach in both
computer vision and robotics. Work in animate vision (Aloimonos, 1993; Bajcsy,
1988; Ballard & Brown, 1992) has rejected the aim of constructing a general-purpose world model, and uses active vision systems to extract information for
particular behavioral goals. Work in behavior-based robotics (Braitenberg, 1984;
Brooks, 1991) seeks to build adaptive robots using a repertoire of simple behaviors
based on direct sensory-motor mappings, without reference to a central model.
Complex behavior emerges from the interaction between the agent and the environment rather than being prescribed by a plan. I view these projects as contemporary avatars of Gibson (1958/this issue), a vindication (often unacknowledged) of
his earlier insights. The empirical adequacy and range of applicability of informa-
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tion-based control, however, remains to be determined. Let us consider his formulae for the visual control of locomotion in detail.
LAWS OF CONTROL FOR LOCOMOTION
Gibson (1958/this issue) articulated five control formulae for guiding locomotion
with respect to stationary and moving objects, which remained substantively similar
in 1979. There are 12 generic situations: the global array may be flowing outward,
static, or flowing inward, specifying whether the observer is moving forward, stationary, or backing up with respect to the environment; and an object's contour
may expand, remain constant, or contract, specifying whether the object is approaching, remaining a constant distance, or retreating with respect to the observer. If the local expansion is consistent with the global flow, then one is approaching a stationary object; otherwise, one is dealing with a moving object. The
12 combinations correspond to various forms of goal seeking (approaching, keeping
a constant distance from, or avoiding a stationary object), pursuit (gaining on, shadowing, or falling behind a moving object), flight (being gained on, shadowed by, or
outdistancing a moving object), and vigilance (a stationary observer with an approaching, stationary, or retreating object).
Although they represent an important new way of thinking about visual control, the specific formulae were based on an intuitive analysis of flow patterns
and should be treated as empirical hypotheses open to experimental test. Alternative hypotheses abound, some of them not based on optic flow at all. Despite
his early exploitation of film and shadow-caster displays, Gibson lamented the
lack of adequate techniques for studying optic flow and active perception, which
have recently become available in the form of real-time computer graphics and
virtual realiry technology. Over the past 10 years, our Perception and Action
Laboratory at Brown has carried out a number of passive perception experiments
on optic flow, demonstrating that humans can judge their direction of heading
from global radial flow patterns with an accuracy on the order of 1 of visual angle (see Warren, 1998, for a review). This is theoretically sufficient for guiding a
variety of locomotor behaviors. The ultimate agenda of this research is to determine how optic flow is actually used to control posture and locomotion, and we
have recently begun to tackle this problem directly using interactive displays and
virtual environments. In what follows, I try to assess the current status of Gibson's five formulae. In each case, the "structural" alternative is to perceive the
metric structure of the scene from spatial information and choose a path based
on the three-dimensional locations and motions of goals and obstacles.
0
Starting and Stopping
To begin locomotion, Gibson proposed, one should move so as to make the optic array flow outward; to stop, make the flow cease; and to back up, make the array flow in-
189
ward. The best evidence for such effects is Lee's classic work on postural sway in the
"moving room" (Lee & Aronson, 1974; Lee & Lishman, 1975), where the task of
standing still is analogous to stopping by making the flow cease. When the visual surround was moved toward the observer along the anterior-posterior axis, creating a
radial outflow pattern in the optic array, it induced compensatory backward sway,
whereas radial inflow induced compensatory forward sway. Note that in the moving
room, other spatial information about the distance of the walls and the participant's
position in the room is also available, but similar effects have been observed with
large-screen computer displays that isolate monocular optic flow, albeit with a somewhat lower gain. In addition, experiments with stroboscopic illumination indicate
that the visual system is responding to optical motion per se rather than the position
of features in the scene (Amblard, Cremieux, Marchand, & Carblanc, 1985; van
Asten, Gielen, & van der Gon, 1988b).
There is considerable evidence that such postural responses are functionally specific to the ecological state ofaffairs specified by optic flow. Their directional specificity indicates that the system responds appropriately to both outflow and inflow,
consistent with the hypotheses about starting and backing up (Delorme, Frigon, &
Lagace, 1989; van Asten, Gielen, & vanderGon, 1988a); directional specificity has
also been observed along lateral and diagonal axes for postural sway during walking
(Warren, Kay, & Yilmaz, 1996). The type of postural response (horizontal movement or body tilt) also depends on the class of optic flow (translation or pitch/roll
stimulation; van Asten et al., 1988b; Warren et al., 1996). Consistent with motion
perspective, optic flow generated by a three-dimensional environment is more effective than that generated by a flat frontal plane, indicating that differential motion is
used to stabilize posture (Bardy, Warren, & Kay, 1996). Further, behavior is not simply driven by the current flow stimulation, for with oscillatory displays postural sway
is adapted to match the display frequency (Dijkstra, Schaner, Giese, & Gielen, 1994;
Schaner, Dijkstra, &Jeka, this issue). There is also evidence that for terrestrial animals such as ourselves, postural sway is not determined solely by vision but is also influenced by somatosensory information from the feet and ankles (Diener, Dichgans,
Bruzek, &Selinka, 1982; Diener, Dichgans, Guschlbauer, & Mau, 1984; Howard,
1986) , indicating that we take advantage of typically redundant information (but see
Stoffregen & Riccio, 1988, for a different interpretation).
Gibson further hypothesized that to speed up one should act so as to increase
the flow rate, and to slow down, decrease it. This is appropriate if environmental
structure is constant, but a change in the distance of surrounding surfaces such as
walls or objects can also affect the flow rate. There is indeed some evidence that
the overall flow rate affects the speed of walking. When a "moving hallway" translates at a constant velOCity opposite the direction of walking, Konczak (1994) reported that participants reduced their speed, but they did not systematically speed
up with hall motion in the same direction as walking. Analogous results were found
with an apparatus that projected moving spots of light on a stationary floor
(Pailhous, Ferrandez, Fliickiger, & Baumberger, 1990).
190
WARREN
Steering Toward a Goal
The basic function of locomotion is to get to a destination, often some goal object
with positive affordances. Thus, the most essential of Gibson's formulae deals with
steering toward a stationary goal. He assumed that the object projects a "form" in
the optic array, an optical contour with internal texture. Of course, given that background surfaces are textured, too, Gibson (197911986) later rephrased this in terms
of dynamic occlusion. As the observer moves, an object is specified by the gain or
loss of texture outside a closed contour, whereas an aperture is specified by gain or
loss of texture inside a closed contour.
The focus of expansion hypothesis. To aim locomotion at a goal, move so
as to keep the focus of expansion close to the object's form in the optic array (Figure
3a). This deceptively simple formulation is one of Gibson's most elegant proposals, a
solution that will bring the observer to the goal regardless of environmental conditions. Self-motion through the world generates a global flow pattern with a focus of
expansion in the direction of travel, so keeping this focus ofexpansion on a stationary
goal will take one to it. He also pointed out that when making a turn, the focus of expansion shifts in the scene through a visual angle that is equivalent to the angle of the
turn. This implies that to steer toward a goal, the required turning angle is specified
by the current visual angle between the focus ofexpansion and the goal; thus, turning
might be controlled either continuously or discretely. The cybernetic interpretation
of this hypothesis is straightforward, for the visual angle between the focus of expansion and the object can be treated as an error signal in a closed-loop system that is reduced to achieve a desired state (0°). Because the focus of expansion is a fixed point
in the flow field, the goal will remain in a roughly constant position in the optic array.
However, there are a number of alternative control laws that might also govern
steering toward a goal. Those I have identified are sketched in Figure 3 with optic
flow strategies4 in the top row and positional strategies in the bottom row, although
surely there are others, such as strategies based on the tau function that Lee analyzes in this issue.
The heading hypothesis. This is essentially the same as the preceding
hypothesis, except that one's perceived heading, rather than the focus of expansion per se, is kept near the goal. In recent experiments, we found that perceived
heading can be biased away from the focus of expansion by a few degrees when
an independently moving object crosses the path of travel (Royden & Hildreth,
1996; Warren & Saunders, 1995a, 1995b). The observer's locomotion generates a focus of expansion for stationary background surfaces (circle in Figure
4b), but a moving object can create a secondary focus of expansion (square in
41 use this term as shorthand for a possible law of control, and do not mean to imply an explicit
cognitive strategy.
goal
goal
"'~I~.,.,
-;,e
goal
~
FOE
-+- ,-,,---t§~
--
~
VL
VR
.......
~
!
!
:gent
:gent
.......
! ~gent
(a) FOE/heading
(b) Magnification
goal
goal
goal
D
D
D
,
~
191
agent
:~
"b
,
~
T
""
agent
(c) Equalization
,
,,,
,,
,,
,
.6,
~
T",,,,
+F
(d) Thrust
(e) Constant bearing
(f) Centering
FIGURE 3 Diagram of visual control hypotheses for steering toward a goal or aperture. FOE =
focus of expansion.
Figure 4b) such that the perceived heading determined by the total flow field lies
in between them (X in Figure 4b). Interestingly, this suggests that the visual system does not resolve the structure and motion of all objects in the scene to perceive heading, but rather determines heading directly from the global flow pattern. Thus, this control law states that to steer toward a goal, keep the perceived
heading close to the goal.
The magnification hypothesis. To steer toward a goal, move so that its
form is magnified or expands symmetrically (Figure 3b). This strategy can be distinguished from the focus of expansion hypothesis because it depends on the flow of
the object itself, rather than the global flow pattern, and does not rely on the radial
directions of flow vectors, but on the pattern of their speeds or magnitudes. There
192
WARREN
/
(a) Stationary
scene
(b) Obstacle
texture motion
(c) Sidewall
texture motion
o
Actual heading
X
Perceived heading
•
Obstacle FOE
~
Texture motion
Resultant flow
FIGURE 4 Schematic of corridor displays to test the heading and equalization strategies for goal seeking and obstacle avoidance. (a) Optic flow
due to self-motion with no added motion in scene. (b) Optic flow resulting
from added obstacle texture motion to the right: perceived heading shifts
to the left, flow rate increases on the right. (c) Optic flow resulting from
added texture motion on side wall: no shift in perceived heading, flow rate
increases on the left. The symmetrical conditions were also tested.
are cases in which the two strategies diverge, for example, when heading toward a
flat surface at an oblique angle, there is radial flow from the heading point but the
expansion is not symmetrical. In most cases, however, keeping the focus of expansion on the goal is accompanied by approximately symmetrical expansion, and a visual system tuned to such flow patterns would respond similarly to radial flow or
symmetrical expansion.
193
The thrust hypothesis. On the other hand, one need not use optic flow at
all to steer to a goal-one could use positional information instead. The next three
strategies are all based on the egocentric direction of the goal, its position with respect to the observer's body. Most simply, one might detect the position of the goal
and locomote toward it; specifically, one could perceive its egocentric direction and
apply thrust force in the opposite direction (Figure 3d). This hypothesis assumes
that the visual direction of a target can be related to the direction of effector movement, for which there is ample evidence; one can point accurately to a light in the
dark, for instance. It also relies on the fact of terrestrial locomotion that there is typically a unique relation between a push against the ground and the resulting displacement of the body, so a known thrust force will move the observer in the visually
specified direction. The optical result is that the goal remains in a roughly constant
position in the optic array.
The constant bearing hypothesis. Similarly, one could steer toward a goal
by moving so that it expands at a constant bearing in the optic array (Le., a fixed
egocentric direction, assuming no body rotation; Figure 3e). This differs from the
previous hypothesis because it does not rely on a known mapping from visual direction to direction of thrust. Instead, it capitalizes on a local feature of optic flow: If
the goal is at the focus of expansion, it will remain at a fixed egocentric position,
othetwise it will drifr. A simple way to do this is to move so as to match the optical
velocity of the goal, such that drift goes to zero. Once again, the optical result is that
the goal remains in a roughly constant position in the optic array.
The centering hypothesis. A closely related hypothesis is to move so as to
keep the goal in the center of the visual field or at the midline of the body (Figure
3f). This is essentially the same as the previous strategy but with a privileged egocentric direction of "straight ahead." Its effect is to line up body orientation and forward locomotor progression with the direction of the goal. This strategy would
move the goal to a particular egocentric position.
Research on strategies for steering to a goal is only now beginning to appear. A
major obstacle is that most of these hypotheses are redundant. Whether relying on
the focus of expansion, perceived heading, symmetrical expansion, egocentric direction, or constant bearing, they generally predict that the goal will end up at the
focus of expansion in a fixed egocentric position. We have recently obtained evidence consistent with the heading strategy in an active joystick task, in which participants steered toward a target in the presence of a moving object (J urn, Duchon,
& Warren, 1998). The moving object induces steering biases that were consistent
with those observed in passive heading judgments, implying that joystick control
194
WARREN
has a basis in perceived heading, rather than the focus of expansion per se or positional infonnation. This provides evidence for a task-specific optic flow strategy.
To examine steering control in legged locomotion, we have recently begun teasing apart some of these hypotheses using a "virtual treadmill" that allows us to manipulate the optic flow of a virtual environment during actual treadmill walking. A
computer display is projected on a large screen in front of the treadmill, and an
electromagnetic tracker on the participant's forehead is used to update the center
of projection in real time. The results are quite preliminary and what follows should
be regarded as a tentative progress report. In our initial experiments CWarren &
Kay, 1997a, 1997b), we simulated motion toward a doorway surrounded by a textured frontal wall, ground plane, and ceiling. The door could appear directly in
front of the participant or 5° on either side, and the task on each trial was simply to
walk through the center of the doorway. To dissociate the thrust strategy from the
optic flow strategies, we rotated the tracker's coordinate system so that the focus of
expansion was offset by 5° from the actual direction of walking. Thus, when the
participant walked toward the doorway, the scene did not expand from the doorway, but rather from a point 5° to the left or right. If, as predicted by the thrust
strategy (Figure Sa), the participant walked in the egocentric direction of the door,
the heading error (~) between the walking direction and the doorway would tend
to zero. (The door would subsequently drift, but the participant would chase it
across the screen, keeping ~ near zero.) If, on the other hand, the participant
walked so as to keep the focus of expansion within the doorway (or make the doorway expand symmetrically), then they would have to walk 5° away from the door,
so beta would go to 5° (Figure 5b). The mean-time series of heading error (Figure
5c) indicated that participants walked in the egocentric direction of the door for
the first second (~ = 0°), but then switched to an optic flow strategy (~ = 5°).
However, a similar pattern of behavior could also be produced by either the
constant bearing or centering strategies, for keeping the doorway in a fixed egocentric direction would have also yielded walking at a 5° angle to the door. We thus
tested a control condition in which a vertical target line was present at the same location as the doorway, but all other optic flow was eliminated. The mean time series of heading error was quite similar to the doorway condition, such that
participants initially walked in the egocentric direction of the target (~ = 0°) but
then, as it began to drift, heading at an angle to it (~ = 5°). Further analysis indicated that they may have been tracking the horizontal motion of the target in front
of them on the screen, for their lateral position was within 2 cm of the horizontal
position of the target, consistent with the centering strategy. However, this behavior could be an artifact of the screen and treadmill apparatus, so we plan to repeat
the study with a head-mounted display. At this point in our work, it seems that
steering toward an expanding doorway might be guided by either an optic flow
strategy or the centering strategy.
Rushton, Harris, Lloyd, and Wann (1998) recently reported steering behavior
that is consistent with the centering strategy. They asked participants to walk to a
195
-t,t: ~=5·
iT
(b) Optic flow
hypothesis
(a) Thrust force
hypothesis
10
(c)
Offset
5
0;
CD
~
0
~
al
-5
... - - -
-..
;"
- ~ - - ... - ------- --........... - -Control
N=10
-10
0
2
3
4
t (s)
FIGURE 5 Steering toward a doorway in the virtual treadmill. (a) Prediction of thrust hypothesis. (b) Prediction of optic
flow hypotheses. (c) Mean-time series of heading error (Ii)
when focus of expansion is offset from direction of walking by 50
(Offset) and with no offset (Control). N = 10.
target-a small ball 10 to 15 m away-while wearing prism glasses that displaced the
entire visual scene by 16°. The prisms acted to displace both the egocentric direction
of the target and the flow pattern together. so that if observers kept the focus of expansion near the target they would walk on a straight path to its actual location. On
the other hand. ifthey kept the target in a fixed egocentric direction. they would walk
with a constant heading error of 16°. yielding a curved path to the target (with an increasing curvature). Videotapes revealed curved rather than straight paths. consistent with any of the positional strategies; assuming that participants walked with
forward progression. the results were indicative of centering. However. a small target
held in the air may not have generated significant flow until the last seconds before
contact. and even if it did. the prism deflection was large enough to create a noticeable discrepancy between the optic flow and somatosensory information for walking
direction. which could have led participants to discount the flow.
The results on steering to a goal are thus suggestive but not definitive. At this
early stage. it remains possible that both egocentric initial position and optic flow
playa role. Similar studies of obstacle aYOidance in the virtual treadmill. to be discussed next. clearly indicate a dependence on optic flow.
196
WARREN
Obstacle Avoidance
Steering away from an obstacle is the converse of steering toward a goal, and Gibson
(1958/this issue) proposed a reciprocal formula. Another way of framing the task is
to steer toward an opening or bounded apenure, in which case the control laws for
obstacle avoidance are identical to those for goal seeking. Gibson described both.
The focus of expansion/heading hypothesis. To steer away from an obstacle, keep the focus of expansion (or more likely, one's perceived heading) outside
its optical contour and within a region of the optic anay that specifies an opening.
This strategy thus relies on the focus of expansion (or perceived heading) by placing
it in openings between obstacles.
The magnification hypothesis. In the same breath, Gibson described the
magnification hypothesis. To steer away from an obstacle, move so that it undergoes
a skewed magnification (i.e., an asymmetrical expansion). The alternative formulation is to move so that the optical contour of an aperture expands symmetrically.
This strategy is based on the local expansion of a bounded contour rather than the
global flow pattern.
The equalization hypothesis. A different approach is suggested by the observation that once a goal is specified, all other objects become obstacles. Consequently, the observer need not steer with respect to them individually, or even segment them from the background, but could simply lump the flow from all surfaces
together. The equalization hypothesis thus proposes that one move so as to equate
or balance the global rate of flow on either side of the goal, the focus of expansion, or
the midline of the body (Figure 3c). This strategy leads the observer to steer away
from regions of high flow and toward regions oflow flow. It is surprisingly effective
for obstacle avoidance because when one nears an object or surface, its rate of flow
increases, leading one to tum away from it. If the observer is aimlessly wandering,
the equalization strategy would prevent collisions with objects, but if the agent is
seeking a goal it would dovetail nicely with the focus of expansion/heading strategy.
While the focus of expansion (or heading) is kept near the goal, obstacles would be
avoided without having to explicitly reset the focus of expansion for each opening.
The equalization strategy was originally discovered in honeybees, based on the
observation that they tend to fly through the middle of apenures (as in Figure 3c)
and down the center of corridors (Srinivasan, this issue; Srinivasan, Lehrer,
Kirchner, & Zhang, 1991). To test the hypothesis, Srinivasan and his colleagues
added a longitudinal motion to one side wall of the corridor, and found that bees
flew down the hallway on a trajectory that precisely balanced the speed of flow in
the two eyes (Srinivasan, this issue, Figure 2). Specifically, when the side wall
moved against the bee's direction of travel, this increased the flow rate on that side
and the bee took up a position fanher from the moving wall, but when it moved in
VISUALLY CONTROLLED LOCOMOTION: 40 YEARS LA1ER
197
the same direction as the bee this decreased the rate of flow on that side and the
bee flew nearer to the moving wall. This implies that the bee steers by means of a
task-specific control law that equalizes the rate of flow, rather than using other
structural information about the three-dimensional environment and its position
in the corridor.
The positional hypotheses. Each of the three positional hypotheses also
has a straightforward reciprocal that could apply to avoiding obstacles. The thrust
hypothesis can be rephrased as, "apply force opposite the direction of an opening";
the constant bearing hypothesis as, "move so as to keep an opening in a fixed egocentric position"; and the centering hypothesis as, "move so as to keep an opening
in the center of the visual field (or at the midline of the body) and progress forward."
In each case, these strategies assume segmentation of objects and openings.
There is even less experimental evidence regarding human control laws for obstacle avoidance than there is for goal seeking, let alone the usual combination of the
two. The most systematic work on this problem has been carried out by Andrew
Duchon at Brown. Duchon initially evaluated the adequacy of the equalization strategy by implementing it in a mobile robot and testing model agent simulations
(Duchon & Warren, 1994; Duchon, Warren, & Kaelbling, 1998); similar work was
done at about the same time by other robotics groups (Coombs, Herman, Hong, &
Nashman, 1998; Sandini, Santos-Victor, Curotto, & Girabaldi, 1993; Weber,
Venkatesh, &Srinivasan, 1997). The robot had a fixed camera with an optical axis in
the direction of translation, so the focus of expansion was generally at the center of
the visual field. The control law simply determined the robot's turning rate as a function of the difference between the summed magnitudes of optic flow on the left and
right sides of the visual field (normalized by the total sum). This simple strategy was
surprisingly robust, allowing the robot to wander for extended periods while successfully avoiding walls, chairs, people, and potted plants without collision, as long as
there was sufficient lighting and surface texture for optic flow extraction. This confirmed that equalization could function as an effective obstacle-avoidance strategy
without performing object segmentation. But do people use it?
A recent perceptual experiment by Dyre and Anderson (1996) bears on this
question, for they reported that heading judgments indeed depend on the rate of
optic flow, not just vector directions. They found that greater flow magnitudes on
one side of the focus of expansion bias perceived heading toward the opposite side.
However, this predicts that compensatory steering should be biased back toward
the side with greater flow, contrary to the equalization hypothesis. A different interpretation of this fmding, consistent with participants' ratings, is that the asymmetrical flow pattern was perceived as movement on a curved path.
In his dissertation work, Duchon (1998; Duchon & Warren, 1998) first replicated the honeybee experiment in humans, using both steering with a joystick and
walking in the virtual treadmill. Somewhat to our surprise, he found that people be-
198
WARREN
have just like bees in both set-ups. They steer to the predicted position in the corridor
when added side wall motion is 0.5 times observer speed (in the opposite direction),
although they undershoot slightly to about 80% of the predicted position when wall
motion is 1.0 times observer speed. This indicates that humans rely on the equalization strategy rather than geometric position information such as texture gradients,
texture scale, or perspective, at least until extreme positions are specified, consistent
with task-specific control information. The result is also contrary to what one would
expect for active steering based on Dyre and Anderson's (1996) perceptual data.
Duchon then turned to the fundamental question of integrating obstacle avoidance
with goal seeking. An intriguing possibility suggested earlier is that one may simultaneously use the heading strategy to steer toward a goal and the equalization strategy to
avoid obstacles. To examine this question, a target line was added at the far end of the
corridor and an obstacle was placed in the path, a frontal surface that ran from the right
wall almost to the center of the corridor (symmetrical conditions were also tested). Figure 4a is a schematic observer's-eye view of the scene, with the optic flow due to
self-motion represented by black vectors (the target is behind the obstacle). Presumably,
parricipants would try to head toward the target but detour around the obstacle and
through the opening. To do so, they might use (a) the equalization strategy, equating the
global rate of flow on either side of the opening, (b) the heading strategy, placing the perceived heading in the middle of the opening, or (c) any of the positional strategies, keeping the middle of the opening in a fixed egocentric direction. To dissociate these
strategies, two conditions were tested in a crossed design. First, we added a horizontal
motion to the texture on the obstacle, say to the right (gray vectors in Figure 4b) ; when
summed with the optic flow due to self-motion, the resultant flow vectors (dashed lines)
created a secondary focus of expansion (square). Consider the consequences for the
heading strategy. AI, mentioned earlier, we previously showed that such independent
motion biases perceived heading by a couple of degrees in the opposite direction (to the
left; X in Figure 4b), 5 which thus predicts a compensatory steering bias in the direction of
texture motion (to the right). Now consider the equalization strategy. The added texture
motion increases the rate offlowon the right side of the visual field, which should lead the
observer to steer away from the obstacle, thus predicting a steering bias opposite the direction of texture motion (to the left). The two flow strategies thus make opposite predictions. Finally, the positional strategies predict no bias from texture motion, because the
geometric structure of the obstacle and opening remains constant. The second manipulation tested the equalization strategy by adding longitudinal motion of the texture on
the left wall, opposite the direction of travel (gray vectors in Figure 4c). Motion that is
generally parallel to the observer's path does not shift the focus of expansion or perceived
heading, for it only affects the magnitudes of the resultant flow vectors. This increase in
the rate offlow on the left predicts a steering bias to the right ifthe equalization strategy is
in play, but not if the heading strategy alone governs steering.
5We used speeds of lateral texture motion that, when added to the optic flow produced by
self-motion, yielded this bias (technically, by creating a path angle of 8°).
199
7
6
SideWall
Texture
6
p < .001
p < .001
5
5
4
E
obstacle
4
3
3
2
2
1
Obstacle
Texture
~ stationary
--.- moving
O+--,---r--.-~~~~
-0.25 -0.2 -0.15 -0.1
obstacle
i-left
--e- stationary
__ right
O+--'---r--.--'--~~
-0.25 -0.2 -0.15 -0.1 -0.05
0
0.05
lateral posttion on treadmill (m)
FIGURE 6 Mean time series of lateral position on the treadmill with corridor displays
(Figure 4). Left panel: Effect of side wall texture motion against the direction of travel.
12.
Right panel: Effect of obstacle texture motion to the right and left. N
=
All panicipants detoured around the obstacle even though a straight path to
the target was possible, and the mean time series oflateral position reveal a consistent steering pattern. For active walking on the treadmill, added motion of the texture on the obstacle significantly biased steering in the same direction as the
texture motion (Figure 6, right panel), consistent with the heading strategy. In addition, added side wall motion also induced a significant steering bias away from
the side wall (Figure 6, left panel), consistent with the equalization strategy. The
same pattern of results was obtained in the joystick task. It thus appears that both
strategies were running simultaneously and may have partially canceled each other
out; the positional strategies, on the other hand, cannot account for these effects of
texture motion. Duchon modeled these results with a simulated agent-environment system using both strategies simultaneously and linearly summing their outputs (turn angle). To reproduce the data, it was necessary for the heading strategy
to explicitly set the goal point in the middle of the opening until the agent passed
the obstacle, and then reset it to the target. These results are consistent with the
idea that steering toward a goal is controlled by the heading strategy, whereas obstacle avoidance involves both the heading and equalization strategies, and behavior is determined by the linear combination of the two. It remains to be seen
whether this interpretation will generalize to more complex situations.
Braking
Gibson described braking in terms of approaching an object so as to make contact
without collision. When approaching at a constant velocity, the visual angle of the
object will expand at an accelerating rate, at first indicating a "looming" object until
200
WARREN
the final explosive expansion specifies an "imminent collision." The moment of the
eye's contact with the surface occurs when the visual angle reaches 180 Thus, Gibson proposed, to make contact without collision, slow down so as to cancel this expansion at the moment of contact. The intuition here is that one can use the rate of
optical expansion to control one's deceleration, but there are a number of ways in
which this might be done, and several versions have been proposed.
0
•
The tau-dot hypothesis. Lee (1976) first formalized this idea in terms of
the tau-dot variable (refer to Figure 7 top). Tau denotes the current visual angle of
an object (9) divided by its rate of expansion (9), which corresponds to the
time-to-contact with an object if observer velocity is constant. Tau-dot is the
time-derivative of tau, which can be thought of as expressing the rate of change in
time-to-contact.A tau-dot of -1.0 corresponds to a constant velocity approach
with no deceleration. Lee showed that if i < -0.5, one's current deceleration is too
low and will result in a collision, so one should increase braking; if i > -0.5, the current deceleration is too high and one will stop short of the object, so one should decrease braking; and if i = -0.5, holding the current deceleration constant will bring
one to a stop at the moment of contact. Thus, one could control braking by adjust-
v, d
Q . :~~--J-_~_~~-_-_-_-_ _-_-_-_]
t =
aI e
z
x
~(])----.
brake: d =ex
point
tau-dot
attractor
i = (zdlv 2 )-1
Displacement
Z
= zu-.5t
2
v = vo-td
d
Braking action
x = xo+b(tm-i)+E
'"
brake
d=ex
t..
FIGURE 7 Hypothesized dynamics of braking. = margin
value of i; £ = noise tenn; x = brake position; d = observer
deceleration; \I = observer velocity; z = distance from observer to object.
VISUALLY CONTROLLED LOCOMOTION: 40 YEARS LA'fER
201
ing deceleration so that tau-dot is near the margin value of -0.5, either by holding
tau-dot constant at that value, by treating it as a set-point in a control loop, or by using the current value of tau-dot to control the direction and magnitude ofbraking.
The rate of expansion hypothesis. Alternatively, one might keep the simple rate of expansion theta-dot near a constant positive value (Yilmaz & Warren,
1995). Any such margin value will lead to a smooth decrease in velocity that approaches zero at contact, with higher values of expansion rate requiring larger initial decelerations. This strategy would produce positive values of tau-dot near + 1.0
for most of the approach, so it is empirically distinguishable from the preceding hypothesis. Flach, Stanard, and Smith (in press) made a persuasive case that the perception of time-to-contact may be based on the rate of expansion rather than tau,
thereby accounting for data showing that time-to-contact judgments are influenced by object size and velocity, and they favor the idea that braking is governed by
optical expansion as well. They propose that braking could be initiated at a ctitical
boundary value of optical expansion, but do not show how deceleration would actually be regulated by this variable.
The deceleration hypothesis. Finally, there are a number of ways that the
deceleration (d) required to stop at an object could be computed from spatial information about object distance (z), object size (r), and observer velocity (v), together
with optical variables such as tau or the visual angle of the object (Yilmaz & War2
ren, 1995); forexample,d = v /2z (refer to Figure 7 top). Of course , these strategies
assume that the corresponding spatial properties can be accurately perceived. The
computed deceleration is identical to that produced by holding tau-dot constant at
-0.5, given the same initial conditions, and conversely the unique correct deceleration will yield a tau-dot of -0.5. However, the deceleration hypothesis can be tested
by selectively removing spatial information about distance, size, and velOcity.
The data on braking strategies are hotly contested. Lee and his colleagues have
reported several naturalistic studies that are consistent with the tau-dot strategy
(Lee, Davies, Green, &vanderWeel, 1993; Lee, Reddish, & Rand, 1991). For example, Lee et al. (1991) found that, during the last 100 msec of approach to a
feeder, hummingbirds decelerate with a mean tau-dot of -0. 71 (computed for the
tip of the bird's bill), a slightly low value consistent with the fact that the bill does
not stop at "contact" but enters the feeding tube. However, it is questionable
whether active visual regulation is occurring over such a short 100 msec interval.
In tasks such as slowing down to grasp a doorknob, Wann, Edgar, and Blair (1993)
found that adults have a mean tau-dot of -0.45 to -0.5 for most of the approach
(computed for the eye), but switch to a rapid deceleration and a high tau-dot in the
fmal350 msec. This is understandable, because the hand (not the eye) must make
contact with the door, and if the eye stops short, tau-dot necessarily escalates to infmity. Such results are consistent with a tau-dot strategy. However, it is a general
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WARREN
weakness of observational studies that participants could be using a spatial strategy
that yields the appropriate deceleration and coincidentally generates mean tau-dot
values near -0.5. To distinguish competing hypotheses the information available
for the control of braking must be manipulated experimentally and the time-series
of tau-dot (rather than just mean values) must be analyzed.
Tau-dot information was experimentally isolated by Kim, Turvey, and Carello
(1993), who presented computer displays of an approaching surface at various constant tau-dot values and had observers judge whether they would result in a "hard" or
"soft" collision. They found a category boundary centered at a tau-dot of-0.5, consistent with the successful perception ofcollision states. Kaiser and Phatak (1993) criticized this finding because holding tau-dot constant at any value greater than -1.0
will, in the limit, result in a smooth stop at the object, and thus -0.5 does not demarcate a critical point between hard and soft collisions. However, if tau-dot is viewed as
information about the appropriateness of one's current deceleration, the value of
-0.5 does indeed specify a critical point dividing adequate from inadequate deceleration, which distinguishes different adaptive actions. Thus, I believe that Kim et al.'s
(1993) results show that observers can reliably judge a pragmatic boundary between
crash states and safe states, providing evidence for reliance on tau-dot.
In an attempt to study the active control of braking, we manipulated the visual
information available in interactive displays that simulated an approach to a set of
three diamond-shaped "road signs" (Yilmaz & Warren, 1995). Participants tried to
stop at the object using a hand-held spring-loaded brake in which deceleration was
a linear function of brake position (with a maximum of 0.7 g), modeled on the linear range of an automobile brake. On half the trials, a checkerboard ground surface
provided information about object distance, size, and observer velocity; when it
was absent, only optical expansion and tau-dot information were available. The
data reveal that mean tau-dot during the approach was -0.51, as predicted by the
tau-dot strategy. Further, the presence of the ground surface made little difference,6 suggesting that a tau-dot strategy was used whether or not spatial information was available. A detailed analysis of the time-series data showed that, in both
conditions, each brake adjustment acted, on average, to bring tau-dot to a critical
value of -0.52 (Figure 8), indicative of discrete adjustments to a tau-dot margin
value. These data also clearly contradict a constant expansion strategy, for the observed tau-dot values are nowhere near the predicted value of + 1.0.
In sum, our results are consistent with the use of tau-dot to control the direction
and magnitude of braking, whether or not spatial information is available, at least
for the simplified case of a linear brake with no plant dynamics. The tau-dot strategy may be particularly appropriate when the controller regulates deceleration directly, as in an automobile or on foot. Kaiser and Phatak (1993) argued that a
6'fhe only exception was a slightly lower mean tau-dot (...{).43) with the checkerboard ground
surface on the shottest high-velocity trials (initial time-to-contact 3 sec), suggesting that panicipants
used the fast ground motion as a cue to decelerate faster early in the trial.
VISUALLY CONTROLLED LOCOMOTION: 40 YEARS LAlER
203
0.8
0.6
0.4
0.2
FIGURE 8 Mean change in tau-dot
on each brake adjustment as a function
of tau-dot at the onset of the adjustment. The regression line has a slope of
-1.04 and a zero-crossing at i = 4l.52,
with r = 0.98. Shaded area represents
± 1 SE (the mean within-subject SE).
Nate. From "Visual Control of Braking:
A Test of the i Hypothesis" by E. H.
Yilmaz and W. H. Warren, 1995,JouT-
nal of Experimental Ps,chology: Human
PerceptionandPe!j'armance, 21, p.l01O.
Copyright e 1995 by the American
Psychological Association. Reprinted
by permission.
~~
,,.
·c S
"'O~
0
.,...~
<l "
-0.2
-0.4
-0.6
-0.8
+-~---.-~~-r-~---.-~~.,..-~-l
-1.0
-0.8
-0.6
-0.4
-0.2
o
-1 at onset of
adjustment
simple tau-dot strategy cannot account for deceleration profiles observed in tasks
such as helicopter landing, raising the possibility that there may be task-specific
braking strategies that depend on the nature of the controller and the dynamics of
the vehicle. As I emphasize later, ecosystem dynamics provide an important constraint that may influence visual control strategies.
Pursuit and Flight
Gibson cast pursuit and flight in terms of predator-prey relations, but this formula is
relevant to the general problem of steering with respect to a moving object. In principle, given that optic flow is determined by the relative motion between the observer and any surface, steering with respect to that surface could be controlled on
the basis of its flow alone, whether it is stationary or moving. Thus, Gibson's solution was to rely on the local flow of the object itself.
The magnificationlminification hypothesis. To pursue a moving object,
move so as to magnify its contour in the field of view; reciprocally, to flee, move so as
to minify its contour. As before, this hypothesis relies on the symmetrical expansion
or contraction of the object, and hence on the speeds of its flow vectors. There are,
of course, a number of alternatives.
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WARREN
The object focus of expansion hypothesis. The local focus of expansion,
defined by the flow vector directions for the object alone, specifies whether one is
currently on a collision course. Thus, to pursue a moving object, keep the local focus
of expansion within the object's form in the optic array. Conversely, to avoid a moving object, shift the local focus of expansion away from it, and to flee, keep the local
focus of contraction near it. Gaining on, shadowing, or falling behind the object
would be specified by whether it is expanding, constant, or contracting.
The background focus of expansion hypothesis. Given that the focus of
expansion defined by optic flow from background surfaces specifies one's current direction of heading in the environment, one could use this information to steer with
respect to moving objects. Specifically, to pursue a moving object, keep the background focus of expansion near its form in the optic arrayj to flee, keep the focus of
contraction from the background near it. The limitation of this hypothesis is that it
does not take the object's motion into account and is thus not predictive about
whether one is on a collision course. The resulting closed-loop behavior over time
would usually yield successful pursuit, but would not guarantee effective avoidance.
The perceived heading hypothesis. The preceding strategies assume that
the flow of a moving object can be segmented from the flow of the surrounding
background surfaces. However, the finding that perceived heading is biased by a
moving object indicates that this might not be the case
arren & Saunders,
1995b). In addition, when pursuing a moving object that is embedded in a global radial flow field, detection latencies for expansion are affected by the surrounding
flow (Probst, Krafczyk, Brandt, & Wist, 1984). Thus, a more likely hypothesis is: To
pursue a moving object, keep one's perceived heading near its form in the optic arraYj to flee, keep one's perceived point of origin near it.
rw
The constant bearing hypothesis. There is, however, a strategy familiar
from sailing that is based on predictive information about a collision course.
To intercept a moving object, move so that its form expands at a constant bearing in the optic array, or equivalently, at a constant visual angle to one's heading direction (the object-heading angle). This can be achieved by translating
so as to match its horizontal optical velocity. Reciprocally, to flee, move so that
its form contracts at a constant bearing, or constant object-heading angle. To
avoid a moving object, move so that its bearing or object-heading angle is not
constantj if the object-heading angle is increasing, one will pass in front of the
object, and if it is decreasing, one will pass behind it (Cutting, Vishton, &
Braren, 1995). This strategy is predicated on two assumptions: that both the
object and the observer are traveling on linear paths and moving at constant
velocities.
To test the perceptual basis for this strategy, Cutting et al. (1995) presented
displays of an approaching pedestrian on a ground surface and asked participants
VISUALLY CON1ROLLED LOCOMOTION: 40 YEARS LATER
205
to judge whether they would pass in front, behind, or collide with him. The displays contained a component of rotation to simulate observer fixation of the pedestrian, such that he remained at the center of the screen. The pedestrian thus
always had a constant bearing with respect to the observer, forcing participants to
use the object-heading angle. Judgments of one's path with respect to the pedestrian were quite accurate, suggesting that observers can use the object-heading
angle to perceive when they are on a collision course. Paradoxically, however,
judgments of heading with respect to the scene were highly inaccurate, raising
the question of whether collision judgments were actually based on some other
information. In a post hoc analysis, the authors found that differential motion
parallax of the ground surface about the pedestrian correlated with passing in
front, behind, or colliding, but this hypothesis was not tested experimentally.
Nevertheless, these results indicate that humans can perceive future hits and
misses with a moving object with sufficient accuracy to control steering, although
the informational basis for this ability remains unclear.
The thrust and centering hypotheses. Finally, one could pursue a moving
object by applying force opposite its current egocentric direction, or moving so as to
keep it centered in the visual field. The centering strategy has been observed in insects, where it is referred to as fixation or tracking (Land & Collett, 1974; Reichardt
& Poggio, 1976). When presented with a moving bar or a flying conspecific, a fly will
turn to fixate the object as a function of both its retinal position and velocity. This is
equivalent to the constant bearing strategy with a privileged egocentric position in
the center of the visual field. When implemented in a mobile robot, the centering
strategy yields quite robust pursuit behavior (Duchon et al., 1998).
Lane Following
There is a special case of steering that should also be considered. which I call lane following. Here the task is to follow a preexisting path defined by lane markings or
edges, as when driving down a road, landing on a runway, or walking on a sidewalk,
down a hallway, or along a wall. One could think of this as a form of goal seeking,
where the far end of the lane is the goal, or a form of obstacle avoidance, where the
edges of the lane are barriers, but the geometrical features of edges or markings also
allow for specialized strategies.
The heading hypothesis. To travel in a lane, keep the perceived heading
centered on the far end of the lane, such that the radial flow vectors line up with the
edges of the lane. A change in position in the lane is specified by a heading error.
This strategy requires radial optic flow from texture or other structure on the
ground surface.
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WARREN
The splay angle hypothesis. The splay angle of an edge line is defined as its
optical angle with respect to the vertical in the field of view. Lateral translation of
the observer, such as a change of position in the lane, is accompanied by a change in
splay angle (optical rotation of the edge lines), whereas a pure rotation of the observer is accompanied by a simple displacement of the edge lines in the field of view
(Calvert, 1954; Riemersma, 1981). Hence, the splay angle can be used to control
one's lateral deviation in a lane. Specifically, to travel in the center of a lane, keep
the splay angle of its left and right edges symmetrical about the vertical; or to maintain a constant distance from one edge, as when driving on the right side of an unmarked road, keep its splay angle at a fixed margin value. This strategy is available
when only edge lines are visible and there is no other optic flow, as when driving at
night.
There is empirical evidence for both strategies. In Mclean and Hoffmann's
(1973) study of driving a straight lane, cross-cortelations with the steering wheel
angle indicated that heading angle and heading rate were continuously controlled,
whereas lateral deviation in the lane, specified by splay angle, was adjusted only intermittently. Similarly, in a driving simulator study with lateral cross-wind disturbances, Weir and Wojcik (1971) found that the best model of the data included a
short high-frequency control loop that reduced heading ertors and a long
low-frequency loop that occasionally corrected for accumulated lateral deviation.
These results indicate that heading is more important than splay information, but
other results suggest that splay angle is dominant. Beall and Loomis (1996) found
that performance in simulated driving with cross-winds was the same with edge
lines alone, providing only splay information, as it was with both edge lines and
random dots on the ground surface, which added heading information. Beusmans
(1995) added optic flow to the ground texture independent of the edge lines,
which should have shifted the perceived heading, but found no cross-correlation
between the texture motion and steering adjustments. However, he only tested
two participants, who directly controlled lateral position in the lane (i.e., splay angle itself) rather than actual steering (which has a rotational component), which
may have biased them toward the splay angle strategy.
In Duchon's (1998) corridor studies, he also included a condition in which the
side walls ended at an implicit black floor, so that the base of the walls provided
lane edges and, hence, splay angles. In each of Duchon's experiments, the presence of splay information attenuated but did not eliminate the previously described
effects of optic flow. Notably, in the corridor with the highest side wall motion (1.0
times observer speed), participants responded in the first 6 sec just as they had
without the floor, consistent with the equalization strategy, but then they stopped
their lateral movement and at 12 sec swerved back to the center of the corridor,
consistent with the splay strategy. This is reminiscent of a short loop based on optic
flow and a long loop based on splay angle. Finally, other studies that have examined curve-taking when following a lane indicate that, in addition to this
VISUALLY CONTROLLED LOCOMOTION: 40 YEARS LA1ER
207
closed-loop control, there is a long-range preview that anticipates steering adjustments based on the upcoming curvature of the road (Land & Horwood, 1995;
Land & Lee, 1994; McRuer, Allen, Weir, & Klein, 1977). In sum, existing results
suggest that both optic flow and splay angle, when it is available, are exploited in
lane following.
Finally, Gibson (1958/this issue) also noted the special complexities of terrestrial locomotion, including the first reference I know of to the "rough terrain"
problem of finding footing (Patla, Robinson, Samways, & Armstrong, 1989;
Warren & Yaffe, 1989; Warren, Young, & Lee, 1986). Patla and his colleagues
continue to be one of the few groups studying how vision actually modulates human gait with respect to the contingencies of the terrain, such as stepping over
obstacles, changing direction, or choosing foot placement (Patla, 1991; Patla,
this issue; Patla, Martin, Rietdyk, & Prentice, 1996). New work by Adolph and
her colleagues (Adolph, 1997; Adolph & Eppler, this issue) examines the development of visually guided locomotion in varying terrain such as slopes, illuminating the task-specificity of the coupling of perception to action (see also Gibson,
Riccio, Schmuckler, & Stoffregen, 1987).
THE DYNAMICS OF VISUAL CONTROL
Up to this point my focus, like Gibson's, has been on the informational basis for visual control. However, the broader point with which I wish to close is that a theory
of visually controlled behavior is going to require both informational and dynamical
concepts. An agent perceiving and acting in an environment together form a nonlinear dynamical system, and the next advance in our understanding of adaptive behavior must take this as a starting point. Behavior is not simply determined by a control formula or action plan in a straightforward manner, but corresponds to stable
attractors in the dynamics of the ecosystem as a whole. The control problem should
thus be reformulated in terms of using available information to tweak the ecosystem
dynamics so as to yield an attractor for the desired behavior. Rather than a centralized command-and-control architecture, a more apt metaphor is surfing. One reads
the waves and makes adjustments so as to get in the groove and ride the dynamics as
far as possible. But in this metaphor the actions of the surfer can reciprocally shape
the perfect wave, creating a recurrent system with its own interaction dynamics.
We have observed this kind of behavioral complexity already in the discussion of visual control laws. Even though the heading strategy and the equalization strategy are based on simple informational variables and are conceptually
quite straightforward, to predict the result of simply summing them together linearly, simulations of the agent~nvironment interaction were needed (Duchon,
1998; Duchon et al., 1998). The situation is further complicated by control laws
that are nonlinear. The tau-dot strategy, for instance, is described by a simple set
of second-order differential equations that characterize the relations between
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WARREN
tau-dot and the state variables of position, velocity, and deceleration, yet the resulting behavior can be wildly unintuitive. These observations exemplify the
tenet that combinations of simple rules can generate surprisingly complex behavior. This argues for simulation analysis of agent-environment dynamics and the
collection of active control data when ttying to evaluate competing visual control strategies.
In my current version of such a framework (see Figure 9), the agent and the environment are treated as a pair of coupled dynamical systems. This is a truism for
the environment. Changes in environmental state variables (e) are a function of
their initial state (e) and any external forces if) applied by the agent (or other actors), according to the laws of physics. For purposes of analysis, the state variables
are defined as those relevant to a given task, and may include geometric, kinematic, and kinetic properties of the world's furniture, such as positions, masses,
spring properties, and so on, as well as the position of the observer. But it is equally
the case that the agent can be considered a dynamical system, parameterized by
task-relevant state variables of the action system (a), with changes in these variables (a) being a function of their initial state and occurrent information (i). In the
absence of occurrent information, the dynamics of the action system are referred to
as its intrinsic dynamics, and the biomechanical properties of the body contribute to
their determination. Kelso, Turvey, and their colleagues devoted the better part of
the past two decades to showing that the action system exhibits stable movement
patterns that can be identified with attractors of a dynamical system and transition
behavior that can be identified with bifurcations in the dynamics (Kelso, 1995;
Kugler & Turvey, 1987; Schmidt, Shaw & Turvey, 1993). The high dimensionality
of the action state space is reduced to a low-dimensional, task-specific, dissipative
dynamic, with stable attractors and only a few free parameters. Some of these parameters act to tune the attractor states, and hence the preferred behavior (its amplitude, frequency, etc.), whereas others may act as control parameters that yield a
qualitative change in the layout of attractors, and hence transitions in behavior
(e.g. from one coordination pattern to another). The role of information is to
modulate the free parameters of an action system that is organized for a particular
task, thereby shaping these dynamics.
The agent and environment are coupled in two ways. A mechanical coupling
expresses the fact that the agent's movements translate into forces that affect the
state of the environment (including the position of the agent), by coupling action variables into the environmental force term. An informational coupling captures the laws of ecolOgical optics, acoustics, haptics, olfaction, and so forth,
expressing Gibson's essential point that the environment structures energy arrays
for a stationary or moving observer in a lawful manner; hence, environmental
variables are coupled into the information term. Thus, following the perception-action cycle, forces applied by the agent change the state of the environment (including the position of the agent), making new information available,
which acts to modulate the free parameters of the action system and affect the
209
behavioral
attractor
Environment
e=<l>(e, f)
..
, , . - - - - - - it
Agent
='P(a, i)
mechanics
f=9(a)
FIGURE 9 A framework for the dynamics of the agent-environment system. a = state variables of action system; e = state
variables of environment; i = informational coupling; f = mechanical coupling.
movements of the agent. The resulting behavior is emergent, determined by the
dynamics of the coupled system and corresponding to its attractor states. Thus,
to govern its behavior, the agent cannot simply dictate a path or trajectory, but
must use the levers at its disposal, using available information to adjust the parameters of the action system in a way that creates an attractor in the ecosystem
dynamics for the intended behavior.
Similar general characterizations of the dynamics of agent-environment systems have been offered independently in the literature on autonomous agents
(Beer, in press; Smithers, 1994). The most systematic and rigorous approach has
been developed by Schoner and his colleagues, applied both to modeling biological
behavior (Schoner, 1991, 1994; Schoner et aI., this issue) and to path planning and
control in mobile robots (Schoner & Dose, 1992; Schoner, Dose, & Engels, 1995).
Although the visual information is not elaborated, those of us seeking to analyze
and model the dynamics of visual control can learn from his programmatic formal
strategy.
Although this is merely a sketch, the present framework does make several
conceptual commitments. First, the dynamics are defined over both the agent
and the environment, thereby attributing the structure of behavior to the ecosystem rather than the cognitive or neural system alone. Second, the dynamics are
embodied, such that the physics of the environment and the biomechanics of the
body contribute to constrain stable solutions. This reflects the belief that biological systems exploit physical and informational regularities to order their behavior. For example, the moment of inertia of a limb constrains the eigenfrequency
of rhythmic behavior, which in tum constrains phase transitions in coordination.
This also suggests that, when it comes to vehicular control, the dynamics of the
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WARREN
vehicle and the controller may condition the observed control strategies. Third,
information is also taken to be a fundamental constituent of the ecosystem dynamics. The central problem for visual control is how information couples into
the dynamics so as to affect behavior (Schaner et al., this issue; Warren, 1988).
Part of the solution lies in characterizing occurrent informational variables-optic flow patterns, chemical gradients, and so on-in the same units and coordinates as behavioral variables, so they can modulate the action system directly.
Another part lies in understanding exactly what influence informational variables have on the action system, such as driving the state of the system, or tuning a parameter on the system, or allowing the action system to adapt its own
parameters to attain a certain informational state, or perhaps all of the above.
Fourth, behavior involves both performatory and exploratory activity, in Gibson's sense. Movements in the service of performing a task will reveal information, and movements may also be performed solely to obtain information, which
in both cases can be used to guide further behavior.
As an example, consider Gibson's basic task of steering toward a goal (Figure
10). The dynamics of the environment are relatively trivial in this case, involving
the effect of locomotor forces on the displacement of the observer. The observer's
motion (described by vector f) with a heading error (P) with respect to the goal
creates a focus of expansion at a visual angle (a) from the goal (a = p, normally).
Following the perception-action cycle, this visual angle governs the agent's turning rate according to a law of control, determining the change in the direction of
thrust force applied in the environment. This in turn alters the direction of observer motion and the rate of change in heading error (13), which determines a new
value of visual angle (a). The dynamics of this simple system can be parameterized
in intrinsic coordinates by beta, the current direction of heading with respect to the
goaL The goal functions as an asymptotically stable fixed point (Strogatz, 1994)
that attracts the agent's heading direction:
~ = a sin(P-1t)
(1)
where the fixed point is at P = 0 and the slope (a) determines the strength of the attractor.7 The behavior of this system can be represented in a phase portrait, which
plots the rate of change in beta as a function of beta (see Figure 10 inset).
Zero-crossings with negative slopes correspond to stable point attractors, those with
positive slopes to unstable repellors. This simple idea makes testable predictions, for
example, that turning rate should increase monotonically with heading error from
the goal and heading should be stable under perturbation. Further, the observed behavior will depend on the underlying control strategy and the available information. In the virtual treadmill, for example, we created two competing attractors, one
7Schoner, Dose, and Engels (1995) proposed a sinusoidal form for this function, because angle is a
circular variable.
211
o
goal
"t:
FOE .....,-...
vJl,~\
,
:z
,
,
,
, a :
\f:
T~
agent
"
;,.,
.......
:y~
.
F
~
FOE angle
a=(3
Displacement
Turn rate
y= -bsin(a+Jt)
T = g(-F)
.
-'"
(3 = T/z
point
attractor
thrust angle
A
F==y
FIGURE 10 Hypothesized dynamiCS of steering toward a goal.\i = heading error; <X = visual
angle between goal and focus of expansion (FOE); 'Y = angle between goal and thrust vector; F =
thrust force, t
observer translation.
=
fixed point at ~ = 0° for the thrust strategy (Figure 5a), and another at ~ = 5° for the
optic flow strategies (Figure 5b).
The cybernetic approach was the first to analyze such visual-motor behavior by
describing it in terms of closed-loop control, borrowing heavily from classical control theory. It was this potential for characterizing the circular causality of perception and action, lacking in S-R formulations, that appealed to Gibson in 1958 and
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WARREN
lent a cybernetic flavor to his control formulae. Perhaps its greatest achievement is
the empirical analysis and modeling of insect flight as a control system (Heisenberg
& Wolf, 1993; Reichardt & Poggio, 1976), incorporating efference copies,
reafferent loops, and error correction. Such linear models are, in fact, a subclass of
dynamical systems, formalized in similar terms as sets of differential equations. Although there is much to be learned in these analyses, from the present point of view
the cybernetic approach has two limitations.
First, on the cybernetic view, consistent with its engineering roots, behavior is
governed by uninterpreted sensory and control signals (e.g., error signals), rather
than by meaningful information. Consequently, the system functions adaptively by
virtue of its design, including its components, their relations in control loops, the
values of reference levels, and so on, rather than by virtue of its basis in laws of
specification and control that relate the agent to ecological states of affairs.
Consider, as another example, visual control of braking. On the one hand, the
margin value of i == -0.5 can be treated as a set-point in a control loop. But one can
go beyond this description by grounding the margin value in an analysis of the
agent-environment interaction, which reveals it to be a critical point demarcating
adequate from inadequate levels of deceleration. A simple model of the dynamics
of braking appears in Figure 7 (bottom). Following the perception-action cycle, the
change in brake position (x) is governed by the difference between the current
value of tau-dot and the margin value of -0.5. The resulting deceleration (d) is determined by the characteristics of the linear brake, and in tum alters the observer's
velocity (v) and distance (z) from the obstacle, generating a new value of tau-dot.
The dynamics of this system are again characterized by a simple attractor with a
fixed point at i = -0.5 (inset in Figure 7). The data in Figure 8, which plot the
change in tau-dot as a function of tau-dot, have the characteristics of a point attractor: a phase portrait with a negative zero-crossing at -0.5. This does not imply
that viewing the margin value as a set-point in a control loop is incorrect, only that
this description fails to capture the richness of the situation, particularly the physical regularity that determines the critical value.
The second limitation is that the cybernetic approach emphasizes linear control
systems and their conditions for stability, and has difficulty with discontinuous or
nonlinear phenomena. The advantage of this approach is that linear sensor-driven
control achieves a given behavior reliably and stably. On the other hand, it limits
the agent's flexibility and adaptiveness, particularly when facing what are referred
to as the problems of "action selection" or adopting a new action mode, and
so-called "reflex interaction" or interference between action modes. For example,
the control model for a behavior such as target tracking in insects is based on a continuous mapping from sensory input to motor output. Although this results in stable orienting to the target, it is not clear how to handle discrete switching between
continuous modes, whether it be switching to a different goal object (e.g., another
conspectlic), a different control strategy (e.g., from the centering to the object focus of expansion strategy), a different behavior (e.g., foraging), or integrating two
213
or more behaviors to perform several tasks at once (e.g., tracking and the
optomotor response).
One approach is to extend linear control principles. Consider the problem of
competition between tracking and the optomotor response in insects. If a target is
moving against a stationary background, the target will induce a tracking response.
However, this yields global motion of the background, which will in tum induce an
opposite optomotor response. Various researchers have suggested that the two
control modes are simultaneously active and sum linearly (Collett & Land, 1975),
as we observed for the heading and equalization strategies in steering and obstacle
avoidance, or that they operate with different gains or temporal frequencies
(Egelhaaf, 1987; Wagner, 1986), or that an efference copy of the tracking command is used to correct the optomotor response (Collett, 1980). However, there
are clearly limits to how far this kind of analysis can be extended to more complex
interactions and nonlinear switching.
A second approach is to add a discrete lOgic atop an otherwise continuous control system, such as a behavioral hierarchy (Tyrrell, 1993), a subsumption architecture (Brooks, 1986), a set of state transition rules (Duchon et al., 1998), or some
other process that selects appropriate action modes. Gibson's (1958/this issue) solution was of this sort, implied by his distinction between "identifying reactions" for
affordances and "control reactions" for visual guidance, and I adopted a similar
view in Warren (1988). Given an intentional constraint that defines a task ("get
coffee"), the affordances of objects specify immediate goals (the door) and obstacles (the chairs) and allow one to select an appropriate action (goal seeking and obstacle avoidance) and corresponding control laws (heading and equalization
strategies). The problem of discontinuous switching is thus regressed onto intentions and affordances, which effectively function as switches between continuous
control modes. Although this may prove to be adequate, it is conceptually unsatisfying because it conflates the languages of discrete logic and continuous dynamics.
One would like to cast both linear and nonlinear phenomena within a common
framework.
A nonlinear dynamics approach offers a natural set of concepts that encompass
not only stability but the instability that underlies flexible behavior, transitions,
and the formation of new action patterns. Although this does not solve the problem of intentionality, it does provide a context in which the selection of tasks,
goals, action modes, and control strategies can be framed in terms of competition
between attractors, within intentional and informational constraints. Assuming
that a task and an immediate goal object have been specified, for example, consider
the phenomenon of switching between control laws. In the case of steering, one interpretation consistent with our data on walking through doorways is that steering
is initially governed by the thrust strategy, but then switches to the heading strategy. As just noted, this can be modeled dynamically in terms of competition between two attractors, one at ~ = 0° and one at ~ = 5°, whose relative strength is
influenced by a variable such as the rate of optic flow. As the observer approaches
214
WARREN
the doorway, its flow rate increases, raising the competitive advantage of the heading strategy, until a bifurcation occurs corresponding to a transition between strategies. An occurrent variable (flow rate) thus acts as a control parameter so that
switching can be accounted for within the dynamics of the agent-environment system. Schoner and Dose (1992) proposed a nonlinear form for such competitive dynamics in a different context (competition between goals and obstacles in robot
navigation), and with Tjeerd Dijkstra's help we successfully adapted it to model
our data on strategy switching. One can envision similar analyses of switching between goals or action modes that incorporate information about the affordances of
objects, specifying the behavior they afford for a given task. The problem of scaling
up such dynamical descriptions to more complicated tasks and a larger repertoire of
behaviors while avoiding local minima was recently addressed by Large,
Christensen, and Bajcsy (in press).
In principle, both the continuous control modes that Gibson (1958/this issue)
proposed in his formulae for locomotion and the discontinuous switching between
them required for flexible behavior can be accommodated by a nonlinear dynamical framework for visual control. I believe that such an informational-dynamical
approach will ultimately allow us to understand how it can be, in the deepest sense,
that control lies not in the brain, but in the animal-environment system.
ACKNOWLEDGMENT
This research was supported by grants from the National Eye Institute (EYI0923)
and the National Institute of Mental Health (MH01353).
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Visually Controlled Locomotion: 40 years Later.

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