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Research Article
Dynamic stabilization of rapid hexapedal locomotion
Devin L. Jindrich, Robert J. Full
Journal of Experimental Biology 2002 205: 2803-2823;
Devin L. Jindrich
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Robert J. Full
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Figures

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  • Fig. 1.
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    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.

  • Fig. 2.
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    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.

  • Fig. 3.
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    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.

  • Fig. 4.
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    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.

  • Fig. 5.
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    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.

  • Fig. 6.
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    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.

  • Fig. 7.
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    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.

  • Fig. 8.
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    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%.

Tables

  • Figures
  • Table 1.

    Responses to rapid impulsive perturbations during running

    All trials (N=11)Perturbation occurs during stance phase of tripod LF,RM,LR (N=6)Perturbation occurs during stance phase of tripod RF,LM,RR (N=5)
    Phase of perturbation in stride (%)37±2835±2640±32
    Maximum lateral velocity (cm s-1)21.0±6.920.5±7.321.5±7.1
    Time to lateral velocity decrease (ms)*13±512±613±4
    Minimum yaw velocity (degrees s-1)-451±283-438±293-465±315
    Time to yaw velocity decrease (ms)14±912±616±12
    Minimum pitch velocity (degrees s-1)-498±364-530±338-467±435
    Time to pitch velocity decrease (ms)10±126±114±16
    Minimum roll velocity (degrees s-1)-596±837-806±911-336±746
    Time to roll velocity decrease (ms)14±2621±356±1
    • Values are means ± S.D.

      Reported velocities are increases or decreases in velocity relative to scaled mean reference velocities from unperturbed trials.

      LF, left front leg; RM, right middle leg; LR, left hind leg; RF, right front leg; LM, left middle leg; RR, right hind leg.

    • ↵ * Indicates the time from the beginning of the perturbation until the magnitude of relative velocity begins to decrease from its maximal value.

  • Table 2.

    Moments of inertia about the principal axes of inertia

    Fore—aft, x, axis (roll)Medio—lateral, y, axis (pitch)Vertical, z, axis (yaw)
    RIP apparatus absent (kg m2)0.4×10-7±0.2×10-73.0×10-7±0.9×10-73.7×10-7±1.7×10-7
    RIP apparatus present (kg m2)2.5×10-7±0.7×10-73.9×10-7±0.73×10-76.0×10-7±1.3×10-7
    Percentage change740±400135±33209±153
    • RIP, rapid impulsive perturbation.

      Values are means ± S.D. (N=9).

  • Table 3.

    Stride, stance and swing periods for individual legs during unperturbed and perturbed running

    LeftRight
    FrontMiddleHindFrontMiddleHind
    Stride period (ms)
    Unperturbed101±12 (8)103±3 (8)99±10 (8)103±4 (8)100±10 (8)101±13 (8)
    During perturbation112±20 (9)106±18 (7)111±19 (7)108±18 (7)111±19 (9)107±16 (6)
    P 0.240.670.180.490.170.51
    After perturbation106±12 (9)113±17 (8)109±14 (7)112±16 (6)115±24 (7)109±17 (7)
    P 0.460.120.170.190.150.37
    Stance duration (ms)
    Unperturbed46±5 (8)55±11 (8)54±12 (8)49±9 (8)57±5 (8)52±9 (8)
    During perturbation55±22 (5)66±4 (6)60±19 (4)46±7 (5)75±22 (3)60±5 (3)
    P 0.350.050.550.560.090.19
    After perturbation50±7 (8)67±13 (8)51±5 (9)55±17 (6)60±8 (9)60±15 (7)
    P 0.230.080.610.430.340.24
    Swing duration (ms)
    Unperturbed56±7 (8)44±5 (8)45±6 (8)53±8 (8)45±3 (8)47±6 (8)
    During perturbation54±9 (4)42±3 (4)49±10 (5)56±12 (5)44±6 (5)52±14 (3)
    P 0.720.530.450.660.880.49
    After perturbation58±5 (8)41±6 (8)49±11 (9)56±11 (6)43±8 (9)45±2 (7)
    P 0.670.320.370.630.650.43
    • `After perturbation' refers to all steps occurring after the stride during which the perturbation occurred.

      Numbers in parentheses indicate the number of samples used in the comparison.

      Numbers below values indicate P-values for comparisons with unperturbed running.

      Values are means ± S.D.

  • Table 4.

    Phase relationships among legs during unperturbed and perturbed running

    UnperturbedDuring perturbationPAfter perturbationP
    Ipsilateral legs
    Left front in left middle52±5 (8)52±7 (5)0.8753±5 (8)0.89
    Left middle in left hind48±4 (8)45±3 (5)0.1947±3 (7)0.73
    Right front in right middle53±6 (8)56±4 (5)0.3350±8 (7)0.51
    Right middle in right hind45±3 (8)45±3 (3)0.8446±8 (7)0.67
    Contralateral legs
    Left front in right front50±5 (8)48±6 (5)0.5651±4 (6)0.72
    Left middle in right middle51±3 (8)50±3 (6)0.5448±6 (7)0.27
    Left hind in right hind48±2 (8)50±7 (4)0.5452±5 (7)0.09
    • `Left front in left middle' indicates that the value corresponds to the phase during the step cycle of the left middle leg (touchdown-to-touchdown) at which the left front leg touches down and begins stance. For this comparison, the left middle leg is the `reference leg'. The leg listed second is the reference leg for each comparison.

      Values are percentages of the stride period of the reference leg.

      Numbers in parentheses indicate the number of samples used in the comparison.

  • Table 5.

    Recovery from lateral perturbations

    Significantly different, perturbed step (number of trials)*Significantly different, step after perturbation (number of trials)Mean number of steps to recoveryNumber of trials recovering within first stepPhase of perturbation in step when animal recovered within first step (%)Phase of perturbation in step when animal did not recover within first step (%)Time to recovery (ms)
    Fore—aft velocity682.0±1.0 (4)234±4838±2599±45 (5)
    Lateral velocity1161.6±1.0 (9)516±1554±2327±12 (9)
    Vertical velocity1081.9±1.0 (7)363±2628±2348±24 (8)
    Yaw velocity681.3±0.5 (5)12938±3037±15 (5)
    Pitch velocity992.3±1.1 (7)260±3632±2541±40 (7)
    Roll velocity8113 (1)0NA37±30101±17 (3)
    • Values are means ± S.D.

      Numbers in parentheses indicate the number of samples used in the comparison.

      NA, not applicable.

    • ↵ * 11 trials were used in the analysis.

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Research Article
Dynamic stabilization of rapid hexapedal locomotion
Devin L. Jindrich, Robert J. Full
Journal of Experimental Biology 2002 205: 2803-2823;
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Research Article
Dynamic stabilization of rapid hexapedal locomotion
Devin L. Jindrich, Robert J. Full
Journal of Experimental Biology 2002 205: 2803-2823;

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