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First published online April 18, 2006
Journal of Experimental Biology 209, 1617-1629 (2006)
Published by The Company of Biologists 2006
doi: 10.1242/jeb.02166

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Task-level control of rapid wall following in the American cockroach

N. J. Cowan^1,*, J. Lee¹ and R. J. Full²

¹ Department of Mechanical Engineering, Johns Hopkins University, Baltimore, MD 21218, USA
² Department of Integrative Biology, University of California at Berkeley, Berkeley, CA 94720, USA

View larger version (11K): [in this window] [in a new window]   Fig. 1. (A) Depiction of a cockroach following a straight wall. L is the farthest point ahead of the cockroach's point of rotation (POR), as measured along the body axis, that the antenna contacts the wall. The bold arrow at the bottom indicates the leading point on the antenna that is in contact with the wall. (B) Unicycle model of the running cockroach. The model parameters are l, the preview distance; d, the antenna measurement; v, the forward running speed; , the angle of the cockroach body relative to the wall [positive is measured counter clockwise (CCW) for all angles, angular velocities and moments; note that <0 in this figure]; , the angular velocity of the body; u, the moment applied by the legs about the POR. The preview distance l may be less than L due to neural and muscle activation delays. In the model, the angle of the POR velocity, , is the same as the body angle, , so is not shown for clarity.

View larger version (9K): [in this window] [in a new window]   Fig. 2. Block diagram of simplified control model. The `mechanics' box represents the torsional dynamics, and relates the body moment, u, to the body angle, . The antenna box is a simplified model of the antenna sensing kinematics, and it dynamically relates the cockroach angle, , to the antenna sensor measurement, d. We fit a simplified neural controller (in the broken box), in which the error between a nominal `desired' wall-following distance, d, and the measured distance, d, is fed back through a PD-controller. This control model enabled us to test PD-control (KD=0) against P-control (KD=0).

View larger version (19K): [in this window] [in a new window]   Fig. 3. Wall-following arena. (A) Two high-speed cameras were positioned above an enclosed arena. The field-of-view of each camera was centered on an observation wall. Half-silvered mirrors in front of each camera reflected light from a fiber-optic illuminator onto the retroreflective running substrate, providing a stark silhouette of the cockroach despite very low ambient light (see Fig. 4). (B) The arena viewed from above showing the two cameras' overlapping fields of view.

View larger version (46K): [in this window] [in a new window]   Fig. 4. Multiple exposures of a cockroach running along an angled wall from a single trial. Superimposed on the images are plots of the corresponding POR position (magenta, left axis) and body angle (blue, right axis). The markers (*) indicate the location of the POR (x,y) and the body angle at the beginning of each stride (as measured by the PEP of the leg that is contralateral to the wall). The cockroach is shown every two strides.

View larger version (10K): [in this window] [in a new window]   Fig. 5. Instantaneous motion of the unicycle model. The two empty circles correspond to the two retroreflective markers that are used to locate the position of the POR (denoted as two concentric circles). v is the forward velocity; is the rotational velocity; v is the component of the velocity of the rear marker perpendicular to the body's fore–aft axis; is the distance between the rear marker and the POR.

View larger version (21K): [in this window] [in a new window]   Fig. 6. Relationship between stride-averaged and . Each data point consists of the averaged and values during a stride. We analyzed 1079 strides observed in 59 trials from 11 individuals. The best fit line (solid line) and the model, = (broken line), are both shown.

View larger version (21K): [in this window] [in a new window]   Fig. 7. Average cockroach distance to the wall y (A,B) and body angle (C,D) as a function of distance traveled along the 45° angled wall for two different speed groups: Slow (A,C) and Fast (B,D). The actual cockroach data (black) are compared to predictions from the PD-control model (red) using the parameters from Table 1. To show the importance of the derivative gain, KD, we tested the controller with the KD=0 (P-control, blue); note that for P-control, performance degrades with increasing speed as expected. The derivative gain significantly improved the fit for the speeds tested.

View larger version (15K): [in this window] [in a new window]   Fig. 8. Root locus plots (see Franklin et al., 1994) of the transfer function of G(s) given by Eqn 7 for five characteristic values of the dimensionless constant, . Each plot depicts the locus of poles (roots of the denominator) of the closed-loop system (Fig. 2) under P-control. The three open-loop poles (roots of the denominator of Eqn 7) are indicated by crosses, and therefore there are three branches of the root locus (magenta, green, blue). There is an open-loop zero (root of the numerator of Eqn 7) at –1, indicated by a circle. The small inset plot (d vs t) for each root locus depicts a typical response of the hypothetical closed-loop system. For stability, all of the poles of the closed-loop system must be in the open left-half-plane, that is, they must have negative real parts. (A) For <1/9, all of the poles are in the left-half-plane; the inset shows an over-damped response of dvst. (B) For =1/9, the system would be critically damped with KP=3. (C) For 1/9<<1, the system would be underdamped under P-control. (D) For =1, the system would be oscillatory for all choices in gain, KP. (E) For >1, the system would be unstable. Since approaches or exceeds 1 for behaviorally relevant running speeds (Eqn 22), these graphs preclude the possibility of P-control. Stability can be greatly improved by adding a derivative feedback term, as in Eqn 11, enabling larger values of . Imag., imaginary.

View larger version (7K): [in this window] [in a new window]   Fig. 9. Three Nyquist plots of the system in Eqn 20 are shown for three characteristic values of the dimensionless neural delay, T, assuming that =0. Delay cannot be handled using the root locus method; thus, we resort to Nyquist's stability criterion (see Franklin et al., 1994). (A) T<1. (B) T=1. (C) T>1. Each plot is constructed by evaluating the transfer function in Eqn 20 along the imaginary axis. Because the open-loop system has no open right-half-plane poles, the closed-loop system is stable if the Nyquist plot does not encircle –1 on the complex plane. As can be seen, this is only possible for the case that (A) T<1, whereas for (B,C) T1, there will always be at least two encirclements of –1, and thus at least two right-half-plane poles. Stability can be greatly improved by adding a derivative feedback term, as in Eqn 11, enabling larger values of T. Imag., imaginary.