Controlling a system with redundant degrees of freedom: II. Solution of the force distribution problem without a body model

Abstract

What strategies may insects use when controlling redundant degrees of freedom? We investigate this question in standing stick insects. Specifically, the question is addressed how the changes of the torques are coordinated that are produced by the 18 leg joints in a still standing animal. Using a generalization of the principal component analysis, three coordination rules have been identified. These rules are sufficient to describe more than half of the variation observed in the data. To move from a descriptive approach to hypotheses on how the neuronal system may be structured, two simulation approaches are proposed. In both cases, torques are decreased by randomly selected values. In the first simulation, the coordination rules derived from the principal components are used to produce changes in torques. In the second simulation, the individual joint torques are modified using a simple local approach. In both approaches, the resulting torques are re-adjusted by Integral controllers applied in each joint. The results show that the torque distribution problem can be solved by a local approach without requiring a body model.

Appendix

Modelling the body and position control

For the simulation using the Matlab interface Simulink, the animal was represented using measurements for masses and lengths taken from Ekeberg et al. (2004). The animal is represented by a rigid body with a mass concentrated at the center of gravity between middle leg coxae. It has three translational (body fixed coordinates) and three rotational (yaw, pitch and roll angles) degrees of freedom. Each one of the 18 leg joints is represented by a joint with one rotational degree of freedom, i.e., is considered as hinge joint. α-joints are connected to the body after being rotated using the values of the ψ and φ-angles. Coxa, femur and tibia are defined as rigid body segments with a mass concentrated at their center of gravity. Each leg is connected to the ground via a joint with three rotational degrees of freedom. These joints ensure that leg endpoints remain at fixed positions, but do not affect the workspace of the insect. Gravity force was set to act on the body.

α- β-, and γ-joints are supplied with a position controller that controls the size of the angle values. The P- (Proportional), I- (Integral), and D- (Derivative) components of the negative feedback controller were set manually. The P- and the D- parts are small. The I-part is large. The reason for this setting is to guarantee that the size of the individual joint angles and therefore the position of the insect remains constant during the complete simulation as was observed in the experiments.

Minimization of torques

Before starting the simulation run, the body model was given the position the insect has chosen in the experiment to be simulated. These 18 angle values are given as reference values to the PID joint controllers. At the beginning, the joint motors are provided with the torque values measured at the beginning of this experiment. Due to errors in the measurements these torque values do not exactly produce the position introduced. Therefore, torques were allowed to relax to a torque distribution that fits to the body position given. This torque distribution is determined by the PID controllers and is similar to the original torques values. After a stable situation is reached, i.e., a situation where torques match the given position, the actual simulation is started.

The simulation consists of a repetition of the following iteration process. First the torques are changed following one of the two given procedures as explained below. In general, the changed torques do not match the position constraint. Therefore, governed by the PID controller, the torque values are allowed to relax for 1,300 iterations until a new torque distribution is found that matches the angular reference values, i.e., the given body position. This is illustrated by Fig. 11, which shows two consecutive iteration procedures for one selected joint. As mentioned earlier, at the beginning of the first iteration process, torques are given a value (left upper panel, circle, about −45.68 mNmm). As the torques (in combination with the other 17 torques values and the rigid connection via the body and leg segments) do not match the desired body position which is maintained via the PID position controllers, the torque relaxes to adopt a new value (also the other 17 torques), in this example relaxes to −45.87 mNmm. After relaxation, in a next time step, the torque is changed again according one of the two procedures explained below (11, right upper panel, circle, new torque value is about −45.55 mNmm). Again the torques do not fit the position, and are allowed to relax during the iteration process. The two lower panels of Fig. 11 show the time courses of the deviation from the angular reference value, i.e., the error signal received by the PID controller, which approximate zero during the relaxation. Note that the deviations are very small in absolute terms.

Fig. 11

Two consecutive iteration processes of a simulation (from left to right). The upper panel shows the time course of the torque. The lower panel shows the corresponding error signals used by the negative feedback controller. The starting values are marked by circles

In the following, the two procedures are explained that are used to simulate the minimization of torques.

Procedure 1: application of PCs

The first procedure that is applied to change the torques is based on the PARAFAC and on the correlation structure observed during the torque minimization process. Correctly weighted and added, the PCs correspond to a decrement of torques between the two time steps t _i and t _i+1. In the simulation, weights are taken at random from normal distributions whose parameters $({\mu}_{a_{q}} ,{\sigma}_{a_{q}}^{2} ,{\mu}_{c_{q}} ,{\sigma}_{c_{q}}^{2})$ were estimated from the experimental data. Using the weighted PCs, the set of torques obtained at the end of an iteration process is changed and the new set of torques is used for the initiation of the new iteration. The minimization process between the two time steps t _i and t _i+1 is illustrated as follows:

$$ \begin{aligned} x_{t_{i+1} j} &=x_{t_{i} j} -\sum_{q=1}^{3} a_{t_{i} q} c_{q} b_{jq}\\ X&=\left[ \begin{array}{lll} x_{t_{i} 1}&\cdots&x_{t_{i} 18}\\ x_{t_{i+1} 1}&\cdots&x_{t_{i+1} 18}\\ \end{array} \right], B=\left[ \begin{array}{lll} b_{1,1}&b_{1,2}&b_{1,3}\\ \vdots&\vdots&\vdots\\ b_{18,1}&b_{18,2}&b_{18,3}\\ \end{array} \right],\\ A&=\left[ a_{t_{i} 1} a_{t_{i} 2} a_{t_{i} 3} \right], a_{t_{i} q} \sim N(\mu _{a_{q}} ,\sigma _{a_{q}}^{2}),\\ C&=\left[ c_{1} c_{2} c_{3}\right], c_{q} \sim N(\mu _{c_{q}} ,\sigma _{c_{q}}^{2}) \end{aligned} $$

‘X’ is the matrix containing the set of torques. ‘B’ is the PC-matrix, containing the three PCs calculated from the PARAFAC analysis and is the same for each simulation. ‘A’ is a score matrix whose entries are taken at random from a normal distribution for each time step t _i . ‘C’ is another score matrix whose entries are taken at random from a normal distribution at the beginning of each simulation and stays the same for each time step.

Procedure 2: application of a local rule

The second procedure represents a local rule, which is not based on the PCs. In this simulation, each torque is reduced between two iteration processes proportionally to its actual size. These reduction factors $({\hat{p}_{t_{i} j}})$ are not fixed, but, for each time step t _i taken at random from normal distributions whose parameters $(\hat{\mu}{{_{p}}}_{\tau},{\hat{\sigma}{{_{p}}}_{\tau}}^{2})$ depend on the absolute sum of all torques, denoted by τ in Table 5. The minimization process between the two time steps t _i and t _i+1 is illustrated as follows:

$$ x_{t_{i+1} j} =x_{t_{i} j}{\hat{p}_{t_{i} j}} , {\hat{p}_{t_{i} j}} \sim N(\hat{\mu}{{_{p}}}_{\tau},{\hat{\sigma}{{_{p}}}_{\tau}}^{2}) $$

The parameters $\hat{\mu}{{_{p}}}_{\tau}$ and ${\hat{\sigma}{{_{p}}}_{\tau}}^{2}$ were estimated in a way that the amount of torque decrease and variation was large for a high sum of absolute torques but became smaller when the sum of absolute torques decreases.