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Edited by: Claudius Gros, Goethe University Frankfurt, Germany

Reviewed by: Dimitrije Marković, Dresden University of Technology, Germany; Sam Neymotin, State University of New York, USA; Gennaro Esposito, Univertitat Politecnica de Catalunya, Spain

Specialty section: This article was submitted to Computational Intelligence, a section of the journal Frontiers in Robotics and AI

This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) or licensor are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

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

Morphological computation (MC), in the context of embodied (artificial) intelligence, refers to processes, which are conducted by the body (and environment) that otherwise would have to be performed by the brain (Pfeifer and Bongard,

MC is relevant in the study of biological and robotic systems. To illustrate the utility of MC in robotics, we will first briefly discuss the Passive Dynamic Walker (McGeer,

This said, one should not mistake MC to be synonymous for energy efficiency. One example in which MC does not relate to energy efficiency is human running, in particular, running in an outdoor environment, such as a downhill off-road path. The runner is not able to detect every change of slope or see every stone, tree branch, etc., on the ground. This means that most irregularities are sensed at the moment when the foot touches the ground (Müller et al.,

For biological systems, energy efficiency and adaptivity are important and evolutionary advantages. A strong indication for the importance of energy efficiency is given by the fact that the human brain accounts for only 2% of the body mass but is responsible for 20% of the entire energy consumption (Clark and Sokoloff,

In biological systems, movements are typically generated by muscles. Several simulation studies have shown that the muscles’ typical non-linear contraction dynamics can be exploited in movement generation with very simple control strategies (Schmitt and Haeufle,

In view of these results, we expect that MC plays an important role in the control of muscle driven movement. To study this quantitatively, a suitable measure for MC is required. There are several approaches to formalize MC. Hauser et al. (_{MI} (see Section

In our previous work, we have focused on a direct quantification of the embodiment, whereas most other approaches quantify MC indirectly through the controller complexity. In particular, in our first publication (Zahedi and Ay,

The main contribution of this publication is to evaluate two measures of MC on biologically realistic hopping models. With this, we want to demonstrate their applicability in non-trivial, realistic scenarios. Based on our previous findings (see above), we hypothesize that MC is higher in hopping movements driven by a non-linear muscle compared to those driven by a simplified linear muscle or a DC-motor. Furthermore, our experiments show that a state-dependent analysis of MC for the different models leads to insights, which cannot be gained from the averaged measures alone. Finally, we provide detailed instructions on how to apply these measures to robotic systems and to computer simulations, including MATLAB^{®},

The quantifications of MC require a formal representation of the sensorimotor loop (see Figure

The conceptual idea of the sensorimotor loop is similar to the basic control loop systematics, which is the basis of robotics and also of computer simulations of human movement. In our understanding, a cognitive system consists of a brain or controller, which sends signals to the system’s actuators, thereby affecting the system’s environment. We prefer to capture the body and environment in a single random variable named

For simplicity, we only discuss the sensorimotor loop for reactive systems in this work (for a detailed discussion, please see Ay and Zahedi (

Starting with the initial distribution over world states, denoted by

To understand the function of the world dynamics kernel

Based on this notation, we can now formulate quantifications of MC in the next section.

In the introduction, we stated that MC relates to the computation that the body (and environment) performs that otherwise would have to be conducted by the controller (or brain). This means that we want to measure the extent to which the system’s behavior is the result of the world dynamics (i.e., the body’s internal dynamics and it’s interaction with its world) and how much of the behavior is determined by the policy

The first quantification, which is used in this work, was introduced in Zahedi and Ay (_{KL}(

The second quantification follows concept one of Zahedi and Ay (_{A} ∝–

The new measure compares the complexity of the behavior with the complexity of the controller. The complexity of the behavior can be measured by the mutual information of consecutive world states, _{x} _{2} _{x,y} _{2}

Equation

For deterministic systems, as those studied in this work, the two proposed measures, MC_{W} and MC_{MI} [see equations (_{W} − MC_{MI} = _{W} ≥ MC_{MI} may not be satisfied always, because discretization can introduce stochasticity.

Note that in the case of a passive observer, i.e., a system that observes the world but in which there is no causal dependency between the action and the next world state (i.e., missing connection between _{MI}, although the actuator state does not influence the world dynamics. This might be perceived as a potential shortcoming. In the context discussed in this paper, e.g., data recorded from biological or robotic systems, we think that this will not be an issue.

The next section introduces the hopping models on which the two measures are evaluated.

This section presents the algorithms to calculate both measures in pseudo-code. Implementations of the algorithms in MATLAB^{®} and

Note that we use a compressed notion in Algorithms

The first algorithm (see Algorithm

_{x} |

1: _{y} |

2: repeat previous step analogously for |

3: |

4: The previous step must be applied to sensors and actuators, if they result from more than one time series |

5: |

6: |

7: |

8: |

Once the discretised uni-variate random variables _{W} (see Algorithm _{MI} (see Algorithm

_{W}

1: _{|W| ×|W| ×|A|} {Matrix with | |

2: _{t+1}_{t}_{t} |

3: _{t+1}_{t}, a_{t}_{t+1}_{t}, a_{t} |

4: |

5: |

6: Estimate |

7: |

8: |

9: |

The state-dependent calculations of the two measures on require minimal changes to the original algorithms. Instead of averaging over all states, which leads to a single number as a result, the measures are evaluated _{W}, the logarithm is evaluated for every triple _{t+1}_{t}, a_{t}_{MI} are evaluated for every 4-tupel _{t+1}_{t}, a_{t}, s_{t}_{t}_{W}(_{MI}(

_{W}(

1: Perform steps 1–8 from Algorithm |

2: |

3: |

4: |

_{MI}

1: Estimate |

2: Estimate |

3: |

4: |

5: |

6: |

7: MC_{MI} = |

_{MI}(

1: Perform step 1–2 from Algorithm |

2: |

3: |

4: |

This concludes the discussion of the measures and their implementation. The next section presents the muscle and DC-motor model against which these measure were evaluated.

In a reduced model, hopping motions can be described by a one-dimensional differential equation (Haeufle et al., ^{2}) in negative _{L}_{0} = 1 m). Hopping motions are then characterized by alternating flight and stance phases. For this manuscript, we investigated three different models for the leg force (Figure ^{®} Simulink™ (Ver2014b) and solved with ode45 Dormand-Prince variable time step solver with absolute and relative tolerances of 10^{−12}. To evaluate and compare the results of the models, a time-discrete output with constant sampling frequency is required (see Section

_{M}_{DC}

A biological muscle generates its active force in muscle fibers whose contraction dynamics are well studied. It was found that the contraction dynamics are qualitatively and quantitatively (with some normalizations) very similar across muscles of all sizes and across many species. In the MusFib model, the leg force is modeled to incorporate the active muscle fibers’ contraction dynamics. The model has been motivated and described in detail elsewhere (Haeufle et al.,

The first term _{fib} considers the force-length and force-velocity relation of biological muscle fibers. It is a function of the system state, i.e., the muscle length _{M}_{0} and constant _{M}_{0} _{0}:

Here, we use a maximum isometric muscle force _{max} = 2.5 kN, an optimal muscle length _{opt} = 0.9 m, force-length parameters

In this model, periodic hopping is generated with a controller representing a mono-synaptic force-feedback. The neural muscle stimulation
_{L,MusFib}. Please note that this delay corresponds to the biophysical time delay due to the signal propagation velocity of neurons. The feedback gain is _{max}, and the stimulation at touch down _{0} = 0.027.

This model neither considers leg geometry nor tendon elasticity and is therefore the simplest hopping model with muscle-fiber-like contraction dynamics. The model output was the world state _{L,MusFib}(

This model differs from the model MusFib only in the representation of the force–length–velocity relation, i.e., _{max} and stimulation at touch down _{0} = 0.19 were chosen to achieve the same hopping height as the MusFib model.

An approach to mimic biological movement in a technical system (robot) is to track recorded kinematic trajectories with electric motors and a PD-control approach. The DCMot model implements this approach [slightly modified from Haeufle et al. (_{T}_{DC}_{DC}_{DC}_{nominal} = 0.212 Nm). As this relatively small motor would not be able to lift the same mass, the body mass was adapted to guarantee comparable accelerations

The controller implemented in this technical model is a standard PD-controller. The controller tries to minimize the error between a desired kinematic trajectory (_{des}(

Here, the feedback gains are _{P}_{D}_{des}(_{MusFib}(

This model is the simplest implementation of negative feedback control that allows to enforce a desired hopping trajectory on a technical system. The model output was the world state _{DC}

This section discusses the experiments that were conducted with the hopping models and the preprocessing of the data. Algorithms for the calculations are provided in the previous section (Section

The possible range of actuator values is different for the motor and muscle models. For the muscle models, the values are in the unit interval, i.e.,

The hopping models are deterministic, which means that only a few hopping cycles are necessary to estimate the required probability distributions. To ensure comparability of the results, we parameterized the hopping models to achieve the same hopping height.

The following paragraphs discuss the findings based on the averaged results presented in Table

_{W} [Morphological Computation as Conditional Mutual Information, see equation (14)] and MC_{MI} [Morphological Computation as the comparison between behavior and controller complexity, see equation (

Muscle fiber model MusFib | Lineraized muscle model MusLin | DC motor model DCMot | |
---|---|---|---|

MC_{W} |
7.219 bits | 4.975 bits | 4.960 bits |

MC_{MI} |
7.310 bits | 5.153 bits | 4.990 bits |

_{W} on the three hopping models (upper plot)

All three models generated a similar movement, i.e., periodic hopping with a hopping height of 1.07 m. However, the control signals and the trajectories of the center of mass vary between models (Figure

To analyze the differences between the models in more detail, we plotted the state-dependent MC (see Algorithm _{W} for each state of the models during two hopping cycles. We chose to discuss MC_{W} only, because the corresponding values of MC_{MI} are very similar to those of MC_{W}, and hence, a discussion of the state-dependent MC_{MI} will not provide any additional insights. The plots for all models and the entire data are shown in Figure

The orange line shows the state-dependent MC for the linear muscle model (MusLin) and the blue line for the non-linear muscle model (MusFib). The green line shows the state-dependent MC for the motor model (DCMot). In the figure, the lower lines show the position

The first observation is that MC is very similar for all models during most of the flight phase (position above the red line) and that it is proportional to the velocity of the systems. By that we mean that MC decreases when the velocity during flight decreases and increases when the system’s speed (toward the ground) increases. During flight, the behavior of the system is governed only by the interaction of the body (mass, velocity) and the environment (gravity) and not by the actuator models. Also, all actuator control signals are constant during flight. This explains why the values coincide for the three models.

For all models, MC drops as soon as the systems touch the ground. DCMot and MusLin reach their highest values only during the flight phase, which can be expected at least from a motor model that is not designed to exploit MC. The graphs also reveal that the MusLin model shows slightly higher MC around mid-stance phase, compared to the DCMot model. For the non-linear muscle model, the behavior is different. Shortly after touching the ground, the system shows a strong decline of MC, which is followed by a strong incline during the deceleration with the muscle. Contrary to the other two models, the non-linear muscle model MusFib shows the highest values when the muscle is contracted the most (until mid-stance). This is an interesting result, as it shows that the non-linear muscle is capable of showing more MC while the muscle is operating, compared to the flight phase, in which the behavior is only determined by the interaction of the body and environment.

This work presented two different quantifications of morphological computation including algorithms, MATLAB^{®}, and _{W}, measures MC as the conditional mutual information of the world and actuator states. Morphological computation is the additional information that the previous world state _{MI}, compares the behavior and controller complexity to determine the amount of MC.

The numerical results of the two quantifications MC_{W} and MC_{MI} confirm our hypothesis that the MusFib model should show significantly higher MC, compared to the two other models (MusLin, DCMot). These results complement previous findings showing that the minimum information required to generate hopping is reduced by the material properties of the non-linear muscle fibers compared to the DC-motor driven model (Haeufle et al.,

We also showed that a state-dependent analysis of MC leads to additional insights. Here, we see that the non-linear muscle model is capable of showing significantly more morphological computation in the stance phase, compared to the flight phase, during which the behavior is only determined by the interaction of the body and environment. This shows that morphological computation is not only behavior dependent but also state-dependent. Future work will include the analysis of additional behaviors, such as walking and running, for which we expect, based on the findings of this work, to see more morphological computation of the non-linear muscle model MusFib.

To summarize the previous paragraphs, in this work we have showed that the results obtained from the two measures MC_{W} and MC_{MI} correspond to the intuitive understanding of morphological computation in muscle and DC-motor models. Furthermore, the results are in accordance with previous work on the control complexity of these models (Haeufle et al.,

In our first publication (Zahedi and Ay, _{A} = _{MI}. In this work, we showed that MC_{MI} and MC_{W} are almost equivalent in the context of deterministic systems. In our second work into account, we applied an information decomposition to the sensorimotor loop (Ghazi-Zahedi and Rauh, _{W}, can be decomposed into the unique information that the current world state _{W} than MC_{MI} for the following reasons. (1) The decomposition of MC_{W} leads to quantities, which nicely reflect intuitive understanding of morphological computation (Ghazi-Zahedi and Rauh, _{W} and MC_{MI} lead to almost identical results for deterministic systems (see _{W} can be computed more easily. It requires the joint distribution of only three random variables (_{MI} requires the joint distributions of _{W} is the final answer to the question on how to quantify morphological computation. Yet, these are strong indications, that the second concept proposed in Zahedi and Ay (_{W} can already be used to quantify morphological computation in realistic systems. Hence, the next step is to apply MC_{W} on high-dimensional data collected from natural systems. The first issue that needs to be solved here is the formulation of MC_{W} for the continuous domain, which is our next step.

We believe that quantifying morphological computation is also useful in the context of robotics, e.g., in the optimization of morphology and control. Hence, in currently ongoing work, we are applying MC_{W} (and the provided software, which we hope proves to be beneficial also to others) to study the effect of morphological design and controller variations on morphological computation in the context of soft robotics.

The paper was mainly written by KG-Z and DH but with strong contributions by the other three authors. The experiments were conducted by KG-Z and DH with strong contributions by GM. All authors contributed significantly to the discussion of the results and the final manuscript.

The authors declare that the research was conducted in the absence of any commercial or financial relationships that could be construed as a potential conflict of interest.

The DC-motor model is based on a Simulink model provided by Roger Aarenstrup (

This work was partly funded by the DFG Priority Program Autonomous Learning (DFG-SPP 1527). DH and SS would like to thank the German Research Foundation (DFG) for financial support of the project within the Cluster of Excellence in Simulation Technology (EXC 310/1) at the University of Stuttgart.

In the case where

From the following equality

For deterministic systems, the conditional entropies _{MI} – MC_{W} =