1466 Part G Human-Centered and Life-Like Robotics perform the computations of a forward model,an inverse obtaining insight into cerebellar function on cellular model or a responsibility predictor,but all receiving the level.The first steps in this direction were taken by same input.A single internal model i is considered to the Schweighofer-Arbib model. be a controller which generates a motor command ti Functional models:From the computer-science and a predictor which predicts the current acceleration. point of view,the most interesting models are based Each predictor is a forward model of the controlled sys- on functional understanding of the cells.In this case, tem,while each controller contains an inverse model of we obtain only a basic insight of the functions of the the system in a region of specialization.The responsibil- parts and apply it as a crude approximation.This ity signal weights the contribution that this model will kind of approach is very promising and MPFIM, make to the overall output of the microzone.Indeed, with its emphasis on the use of responsibility sig- MPFIM further assumes that each microzone contains nals to combine models appropriately,provides an n internal models of situations occurring in the control interesting example of this approach task.Model i generates motor command ti,and esti- mates its own responsibility ri.The feedforward motor 62.3.3 Cerebellar Models and Robotics command rf consists only of the output of the single models adjusted by the sum of responsibility signals: From the previous discussions,it is clear that a popu- 难=∑ii/∑ri. lar view is that the function of the cerebellum within The PCs are considered to be roughly linear.The MF the motor control loop is to represent a forward model inputs carry all necessary information including state in- of the skeletomuscular system.As such it predicts the formation,efference copies of the last motor commands movements of the body,or rather the perceptually coded as well as desired states.Granule cells,and eventually (e.g.,through muscle spindles,skin-based positional in- the inhibitory interneurons as well,nonlinearly trans- formation,and visual feedback)representation of the form the state information to provide a rich set of basis movements of the body.With this prediction a fast con- functions through the PFs.A climbing fiber carries trol loop between motor cortex and cerebellum can be a scalar error signal while each Purkinje cell encodes realized,and motor programs are played before being a scalar output-responsibilities,predictions,and con- sent to the spinal cord(Fig.62.4).Proprioceptive feed- troller outputs are all one-dimensional values.MPFIM back is used for adaptation of the motor programs as has been introduced with different learning methods:its well as for updating the forward model stored in the first implementations were done using gradient descent cerebellum.However.the Schweighofer-Arbib model methods;subsequently,expectation maximization (EM) is based on the view that the cerebellum offers not so batch-learning and hidden Markov chain EM learning much a total forward model of the skeletomuscular sys- have been applied. tem as a forward model of the difference between the crude model of the skeletomuscular system available to Comparison of the Models the motor planning circuits of the cerebral cortex,and Summing up,we can categorize the cerebellar models the more intricately parameterized forward model of the CMAC,APG,Schweighofer-Arbib,and MPFIM as skeletomuscular system needed to support fast,graceful follows. movements with minimal use of feedback.This hypoth- esis is reinforced by the fact that cerebellar lesions do State-encoder-driven models:This kind of model Part not prohibit motion but substantially reduce its quality, assumes that the granule cells are on-off types of since the forward model of the skeletomuscular system 0 entities which split up the state space.This kind of is of lesser quality. 23 model is best suited for,e.g.,simple function ap- As robotic systems move towards their biological proximation,and suffers strongly from the curse of counterparts,the control approaches can or must do the dimensionality. same.There are many lines of research investigating Cellular-level models:Obviously,the most realistic the former part;cf.Chap.13 Robots with Flexible Arms simulations would be at the cellular level.Unfortu-and Chap.60 Biologically Inspired Robots.It should be nately,modeling only a few Purkinje cells at realistic noted that the drive principle that is used to move the conditions is an immense computational challenge,joints does not necessarily have a major impact on the and other relevant neurons are even less well under-outer control loop.Whether McKibben muscles,which stood.Still,from the biological point of view this are intrinsically flexible but bulky (see van der Smagt kind of model is the most important since it allows et al.[62.90]),low-dynamics polymer linear actuators,1466 Part G Human-Centered and Life-Like Robotics perform the computations of a forward model, an inverse model or a responsibility predictor, but all receiving the same input. A single internal model i is considered to be a controller which generates a motor command τi and a predictor which predicts the current acceleration. Each predictor is a forward model of the controlled sys￾tem, while each controller contains an inverse model of the system in a region of specialization. The responsibil￾ity signal weights the contribution that this model will make to the overall output of the microzone. Indeed, MPFIM further assumes that each microzone contains n internal models of situations occurring in the control task. Model i generates motor command τi, and esti￾mates its own responsibility ri . The feedforward motor command τff consists only of the output of the single models adjusted by the sum of responsibility signals: τff = riτi/ ri . The PCs are considered to be roughly linear. The MF inputs carry all necessary information including state in￾formation, efference copies of the last motor commands as well as desired states. Granule cells, and eventually the inhibitory interneurons as well, nonlinearly trans￾form the state information to provide a rich set of basis functions through the PFs. A climbing fiber carries a scalar error signal while each Purkinje cell encodes a scalar output – responsibilities, predictions, and con￾troller outputs are all one-dimensional values. MPFIM has been introduced with different learning methods: its first implementations were done using gradient descent methods; subsequently, expectation maximization (EM) batch-learning and hidden Markov chain EM learning have been applied. Comparison of the Models Summing up, we can categorize the cerebellar models CMAC, APG, Schweighofer–Arbib, and MPFIM as follows. • State-encoder-driven models: This kind of model assumes that the granule cells are on–off types of entities which split up the state space. This kind of model is best suited for, e.g., simple function ap￾proximation, and suffers strongly from the curse of dimensionality. • Cellular-level models: Obviously, the most realistic simulations would be at the cellular level. Unfortu￾nately, modeling only a few Purkinje cells at realistic conditions is an immense computational challenge, and other relevant neurons are even less well under￾stood. Still, from the biological point of view this kind of model is the most important since it allows obtaining insight into cerebellar function on cellular level. The first steps in this direction were taken by the Schweighofer–Arbib model. • Functional models: From the computer-science point of view, the most interesting models are based on functional understanding of the cells. In this case, we obtain only a basic insight of the functions of the parts and apply it as a crude approximation. This kind of approach is very promising and MPFIM, with its emphasis on the use of responsibility sig￾nals to combine models appropriately, provides an interesting example of this approach. 62.3.3 Cerebellar Models and Robotics From the previous discussions, it is clear that a popu￾lar view is that the function of the cerebellum within the motor control loop is to represent a forward model of the skeletomuscular system. As such it predicts the movements of the body, or rather the perceptually coded (e.g., through muscle spindles, skin-based positional in￾formation, and visual feedback) representation of the movements of the body. With this prediction a fast con￾trol loop between motor cortex and cerebellum can be realized, and motor programs are played before being sent to the spinal cord (Fig. 62.4). Proprioceptive feed￾back is used for adaptation of the motor programs as well as for updating the forward model stored in the cerebellum. However, the Schweighofer–Arbib model is based on the view that the cerebellum offers not so much a total forward model of the skeletomuscular sys￾tem as a forward model of the difference between the crude model of the skeletomuscular system available to the motor planning circuits of the cerebral cortex, and the more intricately parameterized forward model of the skeletomuscular system needed to support fast, graceful movements with minimal use of feedback. This hypoth￾esis is reinforced by the fact that cerebellar lesions do not prohibit motion but substantially reduce its quality, since the forward model of the skeletomuscular system is of lesser quality. As robotic systems move towards their biological counterparts, the control approaches can or must do the same. There are many lines of research investigating the former part; cf. Chap. 13 Robots with Flexible Arms and Chap. 60 Biologically Inspired Robots. It should be noted that the drive principle that is used to move the joints does not necessarily have a major impact on the outer control loop. Whether McKibben muscles, which are intrinsically flexible but bulky (see van der Smagt et al. [62.90]), low-dynamics polymer linear actuators, Part G 62.3