5 B I O I N S P I R E D G E N E R A T I O N O F O P T I M A L D R I V I N G W A V E F O R M S abstract
Walking is the most common way of locomotion in land living animals and poses a problem whose solution has been fine-tuned by nature during millions of years of evolution. Multi-legged animals like insects gracefully coordinate their legs on different substrates and maintain stability even under strong disturbances. Their superb walking performance is a natural inspiration for a novel drive strategy in the walking motor. This chapter discusses different possibilities of an architectural mapping between a bioinspired model of hexapod walking and the piezomotor. Specifically, a novel drive strategy is proposed in which all piezoelectric legs are allowed to be driven independently and not in pairs as in the classical drive strategy. Based on the physical model of motor dynamics, it is shown that the bioinspired strategy significantly improves the performance of the motor in terms of its force generating capabilities as well as maximal drive velocity. Furthermore, the novel approach is described by a moderate number of intuitive parameters and produces a variety of velocity-dependent gaits as known from the research on animals.
The choice of driving waveforms has a significant influence on the performance of the walking motor (cf. sect.3.2.2). Simu et al. [186] proposed two rudimentary drive strategies similar to the stick-slip and impact drive mechanism (called inertial stepping in their work) from sect.2.3.3which result in a relatively high motor speed on the cost of high velocity fluctuations and a presumably poor load capacity.
As an alternative, in the same work, they proposed a trapezoidal and the well-known sinusoidal waveforms which are practically used in the commercial product. Building on their results, Merry et al. [145] parametrized the sinusoidal waveform and derived an asymmetrical waveform which improves motor velocity constancy on the cost of lower drive velocity. In a later work, Merry et al. followed this strategy and proposed an optimization strategy based on 4th order Fourier series description of the waveforms [146]. Each of the 4waveforms was described by 8 different parameters, resulting in altogether 32 parameters. As a result, motor velocity constancy could be improved on the cost of further motor velocity decrease. However, their strategy is based on an extensive optimization process with a dedicated solver.
The high-dimensional solution is highly susceptible to manufacturing differences in individual motors, thus questioning the actual perfor-mance improvement in practical applications. So far, the waveform optimization efforts in the literature were concentrated on improving motor performance in load-free nanopositioning tasks and assumed the legs to move pairwise to ensure static stability. This contrasts with the high-load capacity, high-speed objectives of this thesis. But even more importantly, it seems that the engineering efforts fostered with computationally-intensive, high-dimensional optimization processes are not able to deliver significant improvements in motor performance.
An analogous observation comes from robotics, where the immense differences in comparing the walk of an animal and a robot are clearly visible. Following this analogy and considering the fact that the drive principle of the motor is based on “walking”, it is useful to look for a biologically inspired solution [149].
The abundance of biological literature on walking is not surprising if one considers that one of the defining characteristics of animals is their movement [24]. The scientific analysis of walking began in 1870s (promptly before the discovery of piezoelectricity) as the result of a dispute, lingering from ancient times, about whether or not all four feet of a galloping horse are ever off the ground at the same time [158]. The answer, given in1872by the photographer Eadweard Muybridge, triggered a broad interest in walking and, particularly, in how animals are able to generate the rhythmic walking movement. The scientific efforts led to the conclusion that the mechanism by which the nervous system generates the rhythmic movements of the leg during walking is basically the same in animals as diverse as the cat and the cockroach [158]. Experiments have been performed with cat [85],
crayfish [42], locust [171], ants [233], flies [225], cockroach [159, 55] and stick insects [47,49]. Generally speaking, walking can be seen as a coordinated action of (at least two) legs which has to account for the stability of the whole body while at the same time maximizing propulsion and minimizing strain across the body [61]. While walking, animals have to control the number of degrees of freedom which is usually larger than that necessary to perform the task and be able to adapt to unpredictably changing environment or even a loss of a leg [123]. Considering the difficulty of the task from the engineering point of view, it is stunning with what ease even simple animals like insects can “solve” the problem of walking. In this context, insects are an especially interesting group of interest not only for the biologists due to the simplicity of their nervous system but also for this work because of their multitude of legs which they need to coordinate during walking.
The research on insects carried out at Bielefeld University since1981 by Holk Cruse and his coworkers has brought about a repertoire of findings and a kinematic model of the walking behavior of a stick insect Carausius morosus.1 A part of this model is used in this chapter (see next section) to realize a novel coordination mechanism between the legs of the walking motor. However, before proceeding to the description of the model, it is worthwhile to focus on the differences and similarities between the leg of an insect and a piezoelectric leg.
Fig.44 illustrates the legs schematically during one walking cycle.
Perhaps the most noticeable difference in this depiction is the reversed concept of a “ground”. Neglecting the climbing or upside down walking situations, an insect stands on a ground which is under its body and toward which it is pulled by the gravitational force.
Furthermore, normally it is the insect that moves on a static (immobile) ground. In case of the motor and neglecting its specific mountings, the drive rod plays the role of a movable ground which is placed on top of the fixed drive unit with legs and which is pushed against them by means of a preload (leaf springs in Fig.20). The thick black arrows in Fig.44indicate the direction of forward motion either of the insect’s body or the drive rod of the motor. While the insect moves preferably forward and takes a turn in order to change the direction of motion, the forward and backward motion are fully symmetrical cases in the motor. In the following discussion only forward motion is considered (for a comment on backward motion refer to the discussion in sect.5.5).
Further differences are exhibited in the legs themselves. An insect leg is functionally a3DOF serial manipulator consisting of multiple segments connecting rotary joints. A piezoelectric leg is a 2 DOF
1This research has its root in much earlier works and dates back to1921and the publication on the stick insectDyxippusby Buddenbrock [36] and later works of Wendler [222], B¨assler [15] and Cruse [45,46].
femur
tibia
tarsus trochanter
ground coxa
AEP
PEP
bimorph leg
drive unit drive rod
body segment (thorax)
(a) (b)
x
AEP
PEP friction pad
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Figure 44: Schematic depiction of an insect leg and a piezoelectric leg per-forming one walking cycle. (a) the insect leg consists of several segments connected with joints resulting in three functional DOF.
(b) the piezoelectric leg is a multilayer bimorph bender which can perform an in-plane movement. Each walking cycle consists of two main stages, the stance stage (dashed line) in which the leg contacts the ground (or the drive rod) and the swing stage (dot-ted line) in which the leg is freely repositioned. The trajectories in (b) are highly exaggerated. The contact spots tarsus/ground and friciton pad/drive rod are marked with an x-symbol. The resulting directions of motion are indicated with a thick arrow for both insect’s body and the drive rod of the motor.
piezoelectric bender. For an in-plane motion, the former is redundant while the latter is not. The walking cycle of an insect leg consists functionally of two stages – the power stroke (also called the stance or support phase) and the return stroke (also swing or recovery phase).
During the power stroke, the leg contacts the ground where it can support and propel the body. During the return stroke, the leg is lifted off the ground and swung freely to the starting position for the next power stroke [47]. This starting position is called theanterior extreme position (AEP) (also touch-down position).2 While performing the power stroke, which is a retraction/backward movement in forward motion, the leg moves toward and ends at theposterior extreme position (PEP) (also lift-up position) from which the next return stroke is started.
Adapting the biological terms, the power stroke can be defined as the in-contact motion from AEP to PEP and the return stroke as the contactless motion from PEP to AEP. These are indicated in Fig.44 with dashed and dotted lines, respectively. Note that due to the difference in the static vs. movable “ground” concept, the directions of power and return strokes are swapped for the insect and piezoelectric legs given the indicated direction of motion. The current leg position in the walking cycle is marked with an x-marker by which the position of a leg tip (tarsus base or friction pad center) is meant. Thus both legs in the schematic depiction are in the middle of their power strokes.
2Anterior(Latinante; before) is an anatomical term of location and refers to the front (head) end of an animal if it has a distinct head. The opposite (rear) end is called theposterior(Latinpost; after) end.
Also note the difference in the shape of the power stroke trajectory.
While it resembles a straight line compared to the arched return stroke in case of the insect, it has an arched shape in case of both the power and return strokes in the motor. These shapes are due to the differences in the relative compliance of the legs and the “grounds”
which they touch. An insect leg is relatively compliant compared to the stiff ground it walks on while a piezoelectric leg is relatively stiff compared to the stiffness of the preload (cf. sect.3.4.1). The last difference to be mentioned here concerns the movement velocity of the legs. The velocity of a piezoelectric leg can vary from a theoretically infinitely small (cf. next chapter on drive electronics) to as large as several thousands of cycles (steps) per second. This is in a strong contrast to the relatively narrow velocity bandwidth of insect leg movement [81,123,86].3 Fortunately, none of the above differences seems to be a fundamental issue in terms of an adaptation of the bioinspired strategy in the motor.
Having the one-leg considerations in mind, the following discussion turns to the problem of coordination of multiple legs. The subsequent sections are organized as follows. The next section introduces the kinematic model of leg coordination in hexapod walking and argues how this bioinspired model can be transformed to suit the architecture of the walking motor. Next, sect.5.3describes the implementation of the bioinspired trajectory generator together with leg coordination rules which can be optimized to improve specific aspects of motor performance. In sect.5.4, an actual optimization in terms of maximal stall force and maximal motor velocity is performed with the physical motor model from chapt.3. The results show a significant improve-ment in motor performance as compared to the classical drive strategy utilizing theforceorsinewaveforms. Finally, a critical discussion of the proposed drive strategy is presented in sect.5.5.