Evaluating Morphological Computation in Muscle and DC-Motor Driven Models of Hopping Movements



In the context of embodied artificial intelligence, morphological computation refers to processes, which are conducted by the body (and environment) that otherwise would have to be performed by the brain. Exploiting environmental and morphological properties are an important feature of embodied systems. The main reason is that it allows to significantly reduce the controller complexity. An important aspect of morphological computation is that it cannot be assigned to an embodied system per se, but that it is, as we show, behavior and state-dependent. In this work, we evaluate two different measures of morphological computation that can be applied in robotic systems and in computer simulations of biological movement. As an example, these measures were evaluated on muscle and DC-motor driven hopping models. We show that a state-dependent analysis of the hopping behaviors provides additional insights that cannot be gained from the averaged measures alone. This work includes algorithms and computer code for the measures.



  • [PDF] K. Ghazi-Zahedi, D. F. B. Haeufle, G. F. Montufar, S. Schmitt, and N. Ay, “Evaluating morphological computation in muscle and dc-motor driven models of hopping movements,” Frontiers in robotics and ai, vol. 3, iss. 42, 2016.
    Author = {Ghazi-Zahedi, Keyan and Haeufle, Daniel F.B. and Montufar, Guido Francisco and Schmitt, Syn and Ay, Nihat},
    Issn = {2296-9144},
    Journal = {Frontiers in Robotics and AI},
    Number = {42},
    Pdf = {http://www.frontiersin.org/computational_intelligence/10.3389/frobt.2016.00042/abstract},
    Title = {Evaluating Morphological Computation in Muscle and DC-motor Driven Models of Hopping Movements},
    Volume = {3},
    Year = {2016}}

In a nutshell

In this publication, we quantified the contribution of two different muscle models and a DC-motor model with respect to a hopping movement. In a previous publication (Haeufle et al., 2014), the authors were able to show that the non-linear, biologically realistic muscle model requires significantly less control effort compared to a linear muscle model and a dc-motor model. The question of this paper is, can we quantify how much the different hopping models contribute to the observed behaviour. For this purpose, we recorded the behaviour of the three hopping models and quantified morphological computation with two different measures.

Hopping Models

The force that a muscle can exceed depends on two factors, the length of the muscle y and the velocity of the contraction or expansion. This is captured in the following equation:

\fn_phv F(t) = A(t)\,Fl(y)\,Fv(\dot{y})\,F_\mathrm{max}

where \fn_phv F_\mathrm{max} is the maximal force, that the muscle can exert, is a length-dependent component of the muscle force, is the velocity-dependent component, and \fn_phv A(t) are the neural signals that the muscle receives. The resulting force is denoted by \fn_phv F(t).

For the non-linear muscle model (MusFib), both functions \fn_phv Fl(y) and \fn_phv Fv(\dot{y}) are non-linear functions. The linear muscle model has a constant force-length dependency and a linear force-velocity dependency. The DC-motor model uses standard equations. Please see the paper for a full discussion of the models.

The control commands \fn_phv A(t) are learned, such that the three models show approximately, the same hopping behaviour. All systems are controlled with an open-loop controller.

Quantifying Morphological Computation

We evaluated two different measures on the data, namely \fn_phv \mathrm{MC}_\mathrm{W} and \fn_phv \mathrm{MC}_\mathrm{MI}. To understand what they are measuring and how they differ, we need a causal model of the sensorimotor loop, which is presented for reactive systems next:



Sensorimotor Loop for Reactive Systems

This sensorimotor loop and the first quantification \fn_phv \mathrm{MC}_\mathrm{W} are explained [here]. The second measure is explained along the following variation of the sensorimotor loop.





Causal Graph for MI_MI
where W and W’ are the current and next world state (environment and body state), S is the current sensor signal and A is the current actuator signal (see schematics above).



The mutual information I(W’;W) captures the complexity of the behaviour. Yet, it also captures the information that passes through the brain or controller, i.e., the sensors S and the actuators A. Hence, to isolate the information that is contained in W about W’ that does not pass through the controller (light blue), we subtract the mutual information I(S;A) (orange). This leads to the following formalisation:

\fn_phv \mathrm{MI}_\mathrm{MI} = I(W';W) - I(A;S)

Both measures can be evaluated point-wise, which means that the contribution of the different hopping models can be evaluated at every point in time. This is shown in the next section.


We first present the results as they were published in the paper, based on estimating the measure on the discretised data:


The measure confirm previous results, which showed that the non-linear, biologically realistic model requires less control effort, by showing that this muscle model contributes most on average (not shown here, but in the paper). Futhermore, we could also analyse the contribution of the muscles models over time (see videos above).

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