Behaviour-based modelling of hexapod locomotion: linking biology and technical application

Walking in insects and most six-legged robots requires simultaneous control of up to 18 joints. Moreover, the number of joints that are mechanically coupled via body and ground varies from one moment to the next, and external conditions such as friction, compliance and slope of the substrate are often unpredictable. Thus, walking behaviour requires adaptive, context-dependent control of many degrees of freedom. As a consequence, modelling legged locomotion addresses many aspects of any motor behaviour in general. Based on results from behavioural experiments on arthropods, we describe a kinematic model of hexapod walking: the distributed artificial neural network controller walknet. Conceptually, the model addresses three basic problems in legged locomotion. (I) First, coordination of several legs requires coupling between the step cycles of adjacent legs, optimising synergistic propulsion, but ensuring stability through flexible adjustment to external disturbances. A set of behaviourally derived leg coordination rules can account for decentralised generation of different gaits, and allows stable walking of the insect model as well as of a number of legged robots. (II) Second, a wide range of different leg movements must be possible, e.g. to search for foothold, grasp for objects or groom the body surface. We present a simple neural network controller that can simulate targeted swing trajectories, obstacle avoidance reflexes and cyclic searching-movements. (III) Third, control of mechanically coupled joints of the legs in stance is achieved by exploiting the physical interactions between body, legs and substrate. A local positive displacement feedback, acting on individual leg joints, transforms passive displacement of a joint into active movement, generating synergistic assistance reflexes in all mechanically coupled joints.     

Modelling memory functions with recurrent neural networks consisting of input compensation units: I. Static situations

Humans are able to form internal representations of the information they process—a capability which enables them to perform many different memory tasks. Therefore, the neural system has to learn somehow to represent aspects of the environmental situation; this process is assumed to be based on synaptic changes. The situations to be represented are various as for example different types of static patterns but also dynamic scenes. How are neural networks consisting of mutually connected neurons capable of performing such tasks? Here we propose a new neuronal structure for artificial neurons. This structure allows one to disentangle the dynamics of the recurrent connectivity from the dynamics induced by synaptic changes due to the learning processes. The error signal is computed locally within the individual neuron. Thus, online learning is possible without any additional structures. Recurrent neural networks equipped with these computational units cope with different memory tasks. Examples illustrate how information is extracted from environmental situations comprising fixed patterns to produce sustained activity and to deal with simple algebraic relations.     

Regulation of an embryogenic carrot gene (DC 2.15) and identification of its active promoter sites

According to previous studies the expression of the geneDC 2.15 is induced in cultured carrot cells after a transfer to an auxin-free medium, where somatic embryo development occurs. This embryogenic gene encodes a prolinerich protein, which resembles proteins involved in auxin-controlled developmental processes. To understand the mechanism underlying the regulation ofDC 2.15, an experimental approach has been employed which allows the direct identification of theDC 2.15 promoter structure by applying PCR techniques. We demonstrate the presence of five distinct promoter sequences highly similar in structure, but slightly different in a common region of about 15 nucleotides, which contain the binding site for the GATA factor originally found in the human HOX gene. Activity of each promoter structure was assessed in developing somatic embryos containing the specific sequence fused to the -glucuronidase (GUS) reporter gene. For two of the five promoter structures a drastic increase in activity was registered during the torpedo stage while the remaining three were inactive throughout the stages of somatic embryogenesis.     

A cell wall protein down-regulated by auxin suppresses cell expansion in Daucus carota (L.)

We investigated the function of the auxin-regulated cell wall gene DC 2.15, a member of a small gene family, present in Daucus carota (L.) and other plants. Cultured cells derived from carrot hypocotyls transformed by the DC 2.15 cDNA in antisense direction were ten-fold longer than wild-type cells, indicating a function of the corresponding protein in suppression of cell expansion. The analysis of carrot plants expressing the DC 2.15 gene in antisense direction showed that the corresponding protein and/or related proteins probably are involved in leaf and vascular bundle development. The antisense plants generally displayed a retarded growth phenotype and delayed greening in comparison to wild-type plants. The asymmetric architecture of the wild-type leaves was degenerated in the DC 2.15 antisense plants and the leaves showed a torsion within and along their major vein. The vascular bundles showed a lowered ratio of the phloem/xylem area in cross sections of the leaf middle vein whereas the bundle sheath and the cambium showed no obvious phenotype. Expression of a promoter-GUS construct was found primarily in vascular bundles of stems, leaves and in the nectar-producing flower discs. The observed pleiotropic antisense phenotype indicates, by loss of function, that one or several related cell wall proteins of this gene family are necessary to realize several complex developmental processes.     

Stick insects walking along inclined surfaces

In the experiments stick insects walk on an inclined substratesuch that the legs of one side of the body point uphill andthe legs of the other side point downhill. In this situationthe vertical axis of the body is rotated against the inclinationof the substrate as if to compensate for the effect of substrateinclination. A very small effect has been found when the experimentwas performed with animals standing on a tilted platform whichshows that the effect depends on the behavioral context. When,however, animals first walked along the inclined surface andthen, before measurement, stopped walking spontaneously, a rotationof the body has been observed similar to that in walking animals.In a second experiment it was tested whether the observed bodyrotation is caused by the change of direction of gravity vectoror by the fact that on an inclined surface gravity necessarilyhas a component pulling the body sideways. Experiments withanimals standing on horizontal ground and additional weightsapplied pulling the body to the side showed similar body rotationssupporting the latter idea. In a simulation study it could beshown that the combined activity of proportional feedback controllersin the leg joints is sufficient to explain the observed behavior.This is however only possible if the gain factors of coxa-trochanterjoint controller and of femur-tibia joint controller show aratio in the order of 1 : 0.05 to 1 : 1.8. In order to describethe behavior of animals standing on a tilted platform, a ratioof 1 : 1.7 is necessary. In walking animals, this body rotationrequires to change the trajectories of stance and swing movements.The latter have been studied in more detail. During swing, thefemur-tibia joint is more extended in the uphill legs. Conversely,the coxa-trochanter joint appears to be more elevated in thedownhill legs which compensates the smaller lift in the femur-tibiajoint. The results are discussed in the context of differenthypotheses.     

Winching up heavy loads with a compliant arm: a new local joint controller

A closed kinematic chain, like an arm that operates a crank, has a constrained movement space. A meaningful movement of the chain’s endpoint is only possible along the free movement directions which are given implicitly by the contour of the object that confines the movement of the chain. Many technical solutions for such a movement task, in particular those used in robotics, use central controllers and force–torque sensors in the arm’s wrist or a leg’s ankle to construct a coordinate system (task frame formalism) at the local point of contact the axes of which coincide with the free and constrained movement directions. Motivated by examples from biology, we introduce a new control system that solves a constrained movement task. The control system is inspired by the control architecture that is found in stick insects like Carausius morosus. It consists of decentral joint controllers that work on elastic joints (compliant manipulator). The decentral controllers are based on local positive velocity feedback (LPVF). It has been shown earlier that LPVF enables contour following of a limb in a compliant motion task without a central controller. In this paper we extend LPVF in such a way that it is even able to follow a contour if a considerable counter force drags the limb away along the contour in a direction opposite to the desired. The controller extension is based on the measurement of the local mechanical power generated in the elastic joint and is called power-controlled relaxation LPVF. The new control approach has the following advantages. First, it still uses local joint controllers without knowledge of the kinematics. Second, it does not need a force or torque measurement at the end of the limb. In this paper we test power-controlled relaxation LPVF on a crank turning task in which a weight has to be winched up by a two-joint compliant manipulator.     

Controlling a system with redundant degrees of freedom. I. Torque distribution in still standing stick insects

The question is investigated as to how a stick insect solves the task of distributing its body weight onto its six legs, i.e., how are the torques coordinated that are produced by the 18 joints (3 per leg). Three-dimensional force measurements of ground reaction forces have been used to calculate the torques developed by each of the 18 joints. Torques were found to change considerably although the body and the legs of the animal did not move. This result implies a tight cooperation between the 18 joint controllers. Indeed, in each individual experiment, strong correlations could be observed between specific pairs of joints. However, in spite of thorough analysis, no general correlation rules between torques could be detected. The only common attribute found for all experiments was that high absolute torques observed at the beginning of the experiment tend to converge to some minimum over time. Thus, the insects tend to decrease the torques while standing still, but do not use fixed rules. Rather they appear to exploit their extra degrees of freedom and produce time courses that can strongly vary between experiments. Possible mechanisms underlying this behaviour are discussed in a companion paper [Lévy and Cruse (2008) Controlling a system with redundant degrees of freedom: ii. solution of the force distribution problem without a body model, submitted].