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Dynamic stabilization of rapid hexapedal locomotion

Devin L. Jindrich^* and Robert J. Full

Department of Integrative Biology, University of California at Berkeley, Berkeley, CA 94720-3140, USA

View larger version (24K): [in a new window]   Fig. 1. The rapid impulsive perturbation (RIP) apparatus. (A) Diagram of the RIP apparatus, which consisted of a plastic cylinder placed laterally on a balsawood base. The apparatus was mounted on the mesonotum of the animal using small bolts. The cylinder was loaded with flint, black powder and a steel ball bearing. (B) The triggering system for generating RIPs. Flint and black powder were ignited using a spark generated from the ignition module, which was triggered manually.

View larger version (15K): [in a new window]   Fig. 2. Calibration of rapid impulsive perturbations (RIPs). (A) The RIP apparatus was placed vertically on a miniature force platform and triggered. Following the explosion, the RIP apparatus and force platform oscillated at a frequency of approximately 100 Hz. The time to the first force peak (gray area) was assumed to be the time necessary to arrest the RIP apparatus, which had been accelerated by a very rapidly generated force impulse. (B) The force impulse generated by the RIP apparatus was determined by integrating force with respect to time during the period between the beginning of the explosion and the first force peak. Small negative deflections before the positive force generated by the RIP apparatus were due to electromagnetic interference from the spark used to ignite the RIP.

View larger version (24K): [in a new window]   Fig. 3. Coordinate frames used to express kinematic data. (A) Translational positions and rotations were expressed in a coordinate (X,Y,Z) frame based on the mean direction of movement before perturbations and the global horizontal plane. (B) Translational velocities were expressed in a coordinate (x,y,z) frame based on the orientation of the fore—aft axis of the animals. (C—E) Rotation was expressed using yaw, pitch and roll Euler angles. Reaction forces from perturbations were directed towards the positive lateral axis.

View larger version (77K): [in a new window]   Fig. 4. Sequence of video images from a perturbation trial. Arrows superimposed on the images indicate the relative magnitude and orientation of the velocity of the center of mass before, during and after the perturbation. (A) Movement direction 10 ms before perturbation. (B) Movement direction 2 ms following start of perturbation. The rapid impulsive perturbation apparatus generates force, but the movement direction has yet to deflect substantially. (C) Perturbation causes the movement direction to be deflected towards the positive lateral direction, shown 10 ms following the perturbation. (D) At 20 ms following the perturbation, the movement direction has returned to a direction closer to the fore—aft axis. However, return towards the mean reference direction is not sufficient to indicate recovery. Recovery also requires the velocity to be not significantly different from the mean reference trajectory for an appropriate time period. (E) Velocity 40 ms following the perturbation. If animals continued running at velocities that did not differ from reference velocities over a locomotory half-cycle, such as lateral velocity in this trial, recovery was considered to have occurred. (F) Animals were free to move in any direction following the perturbation.

View larger version (16K): [in a new window]   Fig. 5. Perturbation and recovery of lateral velocity from a representative trial. Lateral velocity was expressed in the fore—aft reference frame (see Fig. 3B). The thick solid line represents data from the perturbed trial. The thin solid line represents mean lateral velocity from reference trials scaled to the phase of the perturbed trial. Broken lines above and below the thin solid line represent reference mean ± 1 S.E.M. Vertical broken lines represent touchdown events for alternate tripods. `RF,LM,RR stance' indicates the beginning of the period when the right front, left middle and right hind legs were in stance. `LF,RM,LR stance' indicates the beginning of the period when the left front, right middle and left hind legs were in stance. Solid vertical lines indicate the time of perturbation and time at which maximum lateral velocity was reached. Horizontal lines below the lateral velocity represent the comparison of kinematics from steps of trial in which the animal was perturbed with reference kinematics. The perturbed trial is significantly different from the reference trial during the perturbed step, but not during subsequent steps. The horizontal line terminating near 100 ms indicates the time to recovery of the perturbed trial. Recovery was measured by comparing the kinematics of the trial in which the animal was perturbed with reference kinematics in a sliding window of length equal to the mean step period. The window began sliding at the time sample in which the perturbation occurred and moved forwards in time in 1 ms intervals. The perturbed trial ceased to be significantly different from reference trials 31 ms following perturbation.

View larger version (33K): [in a new window]   Fig. 6. Translational positions and velocities following perturbations relative to reference positions and velocities. Filled circles are values (mean ± 1 S.D.) for all perturbation trials. Velocities are `errors': the difference between perturbed velocities and mean reference velocities collected from unperturbed trials at equivalent phases of the step cycle. Fore—aft and lateral velocities are in the body orientation coordinate frame. Vertical positions and velocities are in the global coordinate frame. Data from perturbed trials are normalized so that perturbations occur 30 ms from the beginning of the data set (indicated by gray vertical lines). N=11 perturbed trials and N=12 unperturbed reference trials.

View larger version (35K): [in a new window]   Fig. 7. Rotational positions and velocities following perturbations relative to reference positions and velocities. Points represent mean rotational velocity `errors' normalized to reference velocities and to the time of perturbation (gray vertical lines) as in Fig. 6. Values are means ± 1 S.D. Yaw, pitch and roll Euler angles were calculated relative to a coordinate frame based on the initial movement direction of the animal (see Fig. 3A,C-E). N=11 perturbed trails and N=12 unperturbed reference trials.

View larger version (21K): [in a new window]   Fig. 8. Descriptive viscoelastic model fitted to the lateral recovery of cockroaches. (A) Voigt model for the mechanical behavior of cockroaches in the lateral direction. (B) Measured and calculated acceleration for a representative trial. The blue curve shows measured acceleration during recovery from a perturbation, and the red curve shows acceleration calculated from the Voigt model fitted to the cumulative trial data. The magenta curve shows the contribution of velocity-dependent acceleration (the damper) to calculated acceleration. The green curve shows the contribution of position-dependent acceleration (the spring) to calculated acceleration. The mean percentage error for the spring component alone for this trial was 40%, that for the damping component alone was 88% and that for the Voigt model was 17%.