Dynamics and stability of directional jumps in the desert locust

Introduction

Locusts are extremely capable jumpers. Whether escaping predators or merely getting from one location to another, they are able to aim their jumps at specific points in space (Collett & Paterson, 1991; Santer et al., 2005; Sobel, 1990), reaching distances of up to 20 times their own body length (Bennet-Clark, 1975). In adult locusts the jump also serves in flight take-off (Katz & Gosline, 1993), propelling the insect into the air to allow the initiation of flapping flight. Understanding the details of locust jumping behaviour, and the way by which specific aspects of the behaviour affect the properties of the resulting jump, is critical for revealing the underlying bio-mechanical principles of locust jumping and flight initiation. Such knowledge could also serve in designing much sought-after small jumping robots (e.g., Zaitsev et al., 2015a; Zaitsev et al., 2015b), as the similar size-scale is also resulted in similar challenges and difficulties in achieving high-performance and high-accuracy jumps.

The key to successful jumps lies in the production of sufficient power (Gabriel, 1984). The locust enhances its power by storing energy in the hind-legs’ cuticle and soft tissues prior to the jump and releasing it simultaneously with muscle action (Bennet-Clark, 1975). The motor program and mechanics responsible for producing and controlling the thrust of the jump have been widely studied (Burrows, 1995; Heitler & Burrows, 1977). Only a relatively few studies, however, have addressed the issue of trajectory control. For any jump, the initial trajectory is defined by the magnitude, azimuth and elevation of the take-off velocity vector. The locust controls its azimuth by rotating its body towards the desired direction through rapid movements of the fore- and meso-thoracic legs (Santer et al., 2005; Sutton & Burrows, 2008), whereas elevation is separately controlled by establishing the position of the hind legs through their rotation at the thoraco-coxal (TC) and coxo-trochanteral joints, accelerating the body along a line connecting the distal end of the tibia and the proximal end of the femur (Sutton & Burrows, 2008). These actions (hereafter referred to as the aiming manoeuvres) are very rapid and manifested shortly before and during jump initiation, thus enabling the locust a hasty escape in an appropriate direction when surprised by a predator. Although asynchrony between the hind legs might appear to offer an intuitive strategy for controlling the jump trajectory, it is not exploited by the locust (Santer et al., 2005; Sutton & Burrows, 2008). While the force applied by the hind legs accelerates the locust’s body, if the force vector of each leg does not pass through the centre of mass (COM) of the body, it will additionally produce a torque that causes rotation. Rotational velocity during the air-born phase can lead to difficulties in flight initiation or in safe landing. Cofer et al. (2010) suggested that the locust uses two mechanisms to minimize pitch rotations, also known as tumbling. In the first, setting the pitch according to the elevation angle, the COM is brought in-line with the force vector, thus minimizing the thrust force torque; in the second, a counter torque is produced by way of contraction of the dorso-longitudinal muscles during the jump. Those authors also observed that tumbling locusts were biased to rotate their body in a head-upwards direction, and hypothesized that the reason for this bias could be to enhance lift during flight initiation.

While mechanisms of elevation control have been well explained, the mechanics underlying azimuth control are much less understood. Cofer et al. (2010) focused on straight jumps, where due to the symmetry between the hind legs’ position, the locust rotates almost only about the pitch axis. In directional escape jumps, however, the symmetry of the hind legs position is lost, leading to the production of torque about all three principal axes (yaw, pitch and roll), and therefore to the development of rotational velocity about these axes. The nature of these additional rotations (about the yaw and roll axes) and the locust’s means to control them have not been studied to date.

In the current study we further argue that azimuth, elevation and stability control are coupled problems in the sense that they cannot be explained independently. To the best of our knowledge previous reports (e.g., cited above) have not addressed all three issues simultaneously. We aim to provide a dynamic model capturing the full spatial mechanics of the jump, and to determine the locust’s strategies for controlling the jump trajectory and rotational instability.

To accomplish this aim, we extended an existing two degrees of freedom (DOF), point mass dynamic model (Sutton & Burrows, 2008) into a six DOF, rigid body model. To complement the extended model we observed and monitored adult locust jumps through synchronous multi high-speed video cameras, enabling extraction of the full six DOF trajectory of the locust body during the jump. By comparing the real jump trajectories to trajectories predicted by computer simulations we validated the dynamic model, enabling us to explore further the locust jump through simulated experiments.

Materials and Methods

Simulations

The motion equations governing the dynamic model were derived using Maple (Maplesoft, Waterloo, ON, Canada), and solved with the MATLAB ODE45 solver. The main simplification for running the simulation was that aiming the jump is achieved solely by changing body posture prior to the jump, while there is no control factor during the jump itself. Each simulated jump was initialized using data obtained from the jump videos: contact points of the dorsal end of both hind-leg tarsi to the platform, distance between TC joints, initial body position and orientation angles, and jump duration. The mass was set according to the weight measurement. Two parameters could not be obtained from the video sequences: the position of the COM and the reaction forces exerted by the hind legs. The COM position with respect to O_l was set according to previous measurements (Taylor & Thomas, 2003). In locust jumps, the ground reaction force has a typical profile, starting at 0 at the beginning of the jump, peaking at approximately 75% of the jump duration and decreasing to 0 at take-off (Han et al., 2013). The force profile was approximated as a triangular, peaking at 75% of the jump force impulse length. As reported, there is no difference in the motor program of the left and right hind legs during side jumps (Santer et al., 2005), and measurements show that the reaction force of both hind legs is practically the same (Han et al., 2013). Hence we set the magnitude of the forces representing the hind legs to be equal. The maximum reaction force at the peak of the force profile was set manually so that at take-off the simulated and real locusts would propagate the same linear distance.

Results

Dynamic model validation

To validate the dynamic model and its underlying assumptions we examined how well the simulation was able to predict the outcome (trajectory) of recorded real jumps. The azimuth and elevation angles, compared between the real and the simulated jumps, demonstrated a strong linear correlation for both criteria (Figs. 3A and 3B), thus validating that the dynamic model indeed captures the governing principles of trajectory control. Real and simulated rotational velocities were also compared, but no correlations were found (see Fig. S3 and discussion for further details). Simulations were therefore not used as a tool to further investigate jump stability.

Development of rotational velocity during real jumps

As noted earlier, Cofer et al. (2010) reported that the locusts set their pitch prior to jumping according to jump elevation, moving their COM in line with the hind legs’ thrust force to minimize tumbling. This strategy will diminish tumbling caused by torques resulting from the thrust force. Tumbling, however, could also be a result of the aiming manoeuvres (see detailed explanation in the ‘Introduction’) prior to the thrust force initiation: once the locust detects a threat and decides to jump away, it begins the aiming manoeuvres, during which the jump is triggered. Only then do the hind legs start to extend and exert force on the body until take-off. To investigate whether rotational instability at take-off is a result of thrust exerted by the hind legs or of the earlier aiming manoeuvres, the rotational velocities at jump initiation and at take-off were compared (Figs. 5A–5C). We found that rotational velocities at the initial jump triggering moment were already of the same scale as the rotational velocities at take-off, reaching up to 500 deg/s in yaw and roll (Figs. 5A and 5C) and 400 deg/s pitching head upwards (Fig. 5B). The yaw velocities during triggering and during take-off were uncorrelated (Fig. 5A). A strong correlation between initial and take-off roll with a slope of near 1 indicates that velocities about this axis tend to remain practically constant throughout the jump (Fig. 5C). We found that while pitch velocities at jump triggering were almost always head-upwards, pitch velocity at take-off was either head-upwards or downwards with hardly any jumps free of tumbling (Fig. 5B). The most dramatic effect of the thrust force on tumbling could be seen in jumps in which the locust tumbled head upwards prior to jumping (at triggering), but had changed its tumbling direction to head-downward by take-off. These findings indicate that although there is a significant change in rotational velocity during the jump (mainly about the pitch axis), the aiming maneuvers prior to thrust force initiation also have an important role in the development of rotational velocity.

The effect of rotational velocity on the timing of flight initiation in real jumps

As noted, locusts are biased to tumble head-upwards when jumping (Cofer et al., 2010). To test the hypothesis that the purpose of the tumbling bias is to reduce risk of crashing during flight initiation, we tested correlations between the angular velocity at take-off and flight initiation timing. The angular velocity was measured just prior to take-off (2.5 ms), with take-off defined as the moment the hind-leg tarsi lost ground contact, and expressed in the locust-attached coordinate system. (In all the jumps analyzed in the current study loss of ground contact by the two legs was synchronous or within less than 2 ms.) Flight initiation timing was the difference between the time when initial hind leg extension was observed and the time that the wings started to spread. There was a linear correlation between pitch velocity and flight initiation timing such that head-downwards rotations led to earlier wing opening (Fig. 6B). No such correlation was found between flight initiation and either roll or yaw rotational velocities (Figs. 6A and 6C). Interestingly, the correlation between pitch velocity and flight initiation timing was even improved when testing pitch velocity at 5, 10 or even 15 ms prior to take-off (R² larger than 0.5 p < 0.01; see Fig. S4).

Discussion

In this study we proposed a single dynamic model for the locust jump trajectory control, explaining the control of azimuth, elevation and stability. The mechanisms and strategies that have been revealed are consistent with earlier reports regarding the mechanics of elevation control (Sutton & Burrows, 2008), and the locust’s behavior during escape jumps (Santer et al., 2005). The locust can potentially use all six DOF of its body to control the jump trajectory. Hence, the full six DOF trajectory of the locust body was both monitored during real jumps and simulated using our dynamic model. This allowed us for the first time to specifically and directly determine the relative importance of each of the initial (prior to the jump) locust body coordinates in the jump trajectory control. Our investigation of the full parameter-range and parameter-combination space suggests that it is the α and β angles that are instrumental in controlling the jumps’ azimuth and elevation angles, respectively, while the rest of the body coordinates (including the body’s orientation) have little effect on the jump trajectory parameters.

As noted, real and simulated rotational velocities were compared but no correlations were found. We believe that this is due to two main reasons: (1) rotational velocities are much more sensitive to subtle changes in the initial state of the locust body than azimuth and elevation, and inaccuracies in the data initializing the simulations were too large to enable accurate prediction of rotational velocities; and (2) the locust body in our model is rigid and does not allow the abdominal flexion that may contribute to jump stabilization (Cofer et al., 2010). Simulations were therefore not used as a tool to further investigate jump stability.

Rotational instability and angular velocities are byproducts of practically all locust jumps, with important and potentially undesirable effects on flight initiation. At the beginning of the jump, at the moment the hind legs start to extend, the locust body had already accumulated rotational velocity generated during the aiming manoeuvres (Fig. 2). As we observed, aiming manoeuvres resulted in pitch rotations that were usually head-upwards. This is in accordance with the report by Cofer et al. (2010) for tumbling at take-off, but it also shows that the head-upwards rotation bias exists even before the hind legs’ thrust is initiated. This is important because different sources of jump instability might require different means for controlling it. In addition, we noted that the timing of wing opening and flight initiation was strongly dependent on pitch angular velocity. Our study consequently supports the hypothesis correlating jump stability to successful flight initiation (Cofer et al., 2010). It also indicates that the locust is sensitive to pitch angular velocity throughout the jump (and can differentiate it from the yaw and roll rotations) and that this sensory input is coupled to activation of the flight motor pattern in a yet to be explored manner (see Camhi, 1969; Pond, 1972; Reichert, 1993; Taylor, 1981 and references within, for the role of sensory inputs, including those related to pitch, during flight).

As noted earlier, Santer et al. (2005) reported that no bilateral differences in the motor programs of the left and right hind legs correlated with jump trajectory. Hind-leg asynchronous action was also reported to have no effect on jump elevation (Sutton & Burrows, 2008). The two hind legs could also have a differential effect on jump trajectory as a simple result of asynchronous loss of ground contact: prior to and during sideways jumps the locust body translates and rotates to the side, resulting in a different distance between ground contact and the TC joint of the two legs. The leg furthest from the jump direction therefore loses ground contact earlier, resulting in a short time-period during which only one leg (that still in contact with the ground) exerts forces and torques on the body. In all the jumps analysed in the current study this time difference between the loss of ground contact of the two legs was shorter than 2 ms. Based on the force profile produced by each leg throughout the jump (Han et al., 2013), the magnitude of the thrust force during the final 2 ms period is very small and diminishing. This becomes more negligible still when comparing the time-period in which only one leg produces thrust to the much longer period in which both legs exert much larger thrust. We therefore conclude that hind-leg asynchronization has a negligible effect on trajectory control and jump stability.

The locust continues to serve as an important inspiration for the development of small jumping robots (Chen et al., 2011; Kai et al., 2012; Kovač et al., 2011; Nguyen & Park, 2012). In many robotics applications and tasks there is an advantage to designing multimodal robots, capable of multiple locomotion modes. However, these introduce new challenges related to the control mechanisms and integration between modes. A locust-inspired jumping-flying robot will encounter the need to perform an efficient transition from a ballistic trajectory (jumping) to flapping-flight. Our current study suggests that the locust utilizes control and stabilization mechanisms that are based on the timing of wing-spreading (in addition to the aerodynamics of the flapping wings). Development of a bio-inspired robot based on our findings is currently underway (Zaitsev et al., 2015a; Zaitsev et al., 2015b; G Kosa & A Ayali, 2016, unpublished data) and will provide further opportunities to evaluate the contribution and importance of the presented mechanisms to flight initiation.

Supplemental Information