4.2 Making Use of Redundancies for the Integration of Internal Models
4.2.1 Redundancy in Sensorimotor Tasks
In mathematical terms, redundancy can be interpreted in terms of a many-to-many mapping: For a given task specification, such as a target position for the hand in an inverse kinematics model, there can be many possible values for the output variable (postural configurations in this case) that are a solution for the task. Thus, the mapping is not restricted to be a function (i.e., amany-to-one mapping). As a simple example1, consider the case of a planar robot with two rotational joints and the task to control the vertical position of its end-effector, disregarding its horizontal position. For each target vertical position (apart from the upper and lower extremes), the robot has infinitely many configurations to solve its task, see Figure 4.2.
Formally, the forward kinematics of the robot is given by the function (i.e.,
many-1example adapted from (Rolf et al., 2010)
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Figure 4.3: A second task with redundant solutions for the robot shown in Figure 4.2.
(a) The robot should obtain a certain orientation of its hand, that is, all configurations shown correspond to solutions to the task of obtaining the orientation α =−0.25π. (b) The robot’s joint space, showing sets of redundant solutions corresponding to directions indicated by the lines in (a).
to-one mapping)
f : Θ⊂R2 −→ Y ⊂R, (4.1)
where the input domain Θ is the set of possible (two-dimensional) joint configurations of the robot and is mapped onto the output domainY, which is the interval of vertical positions that the robot can reach. In contrast, the inverse kinematics of the robot is given by a many-to-many mapping
g : Y −→ P(Θ), (4.2)
whereP(Θ) denotes the power set of Θ, such that
∀ θi,θj ∈ g(y), y∈Y : f(θi) = f(θj), (4.3) meaning that g(y) is the subset of Θ of angular configurations θ that will bring the robot to the same vertical positon y. In Figure 4.2(b), each colored line represents the set of solutions g(y) for one particular y ∈Y, that is, any point along one line is mapped by the forward kinematics functionf(·) onto the same value y. Thus, to reach its task to bring its end-effector to a certain vertical position, the robot can choose any one solution that lies on the corresponding line, representing the set of redundant solutions for that task.
As a second task that the simple robot in Figure 4.2 might have, assume that it should control the orientation of its end-effector. That is, we introduce another inverse
4.2 Making Use of Redundancies for the Integration of Internal Models
schema system
y α θ
motivation system
sensory system
motor system
g h
Figure 4.4: Layout of a system with two internal models, both transforming sensory information and target values into arm configurations θ. When the robot is given task specifications for bothy andα, the system needs to select a single solutions from multiple candidate solutions.
mapping, which captures the relation between end-effector orientations α∈[−π, π] of the robot arm and angular configurationsθ,
h : [−π, π] −→ P(Θ). (4.4)
The robot has infinitely many configurations for all target end-effector orientations, which also makes this second inverse mapping a many-to-many mapping, see Figure 4.3.
Now assume a system with the layout shown in Figure 4.4, that has acquired two internal models, corresponding to the two inverse mappings g and h, respectively. It could be given two tasks, on the one hand to bring its hand to a certain vertical position y, and on the other hand to obtain a certain orientation α of the hand. If the two internal models of the system would both independently produce a single solution,θg andθh respectively, these two solutions would most likely differ from each other. None of the methods for the integration of internal models that were described above in Section 4.1 would be suited to produce a solution that satisfies both tasks simultaneously: Using a serialized integration, the robot could only perform one task after the other by switching between the two arm configurationsθg andθh; using some form of linear combinationθ∗=a·θg+b·θh would not yield a valid solution to either task, as the two internal models describe entirely different sensorimotor mappings, see Figure 4.5. Only if both internal models can restore the knowledge about redundancies in the task can the system decide on a solution that solves both tasks simultaneously.
The approaches for the integration of internal models using fuzzy command integration
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Figure 4.5: (a) The two arm configurations that solve the two tasks of bringing the hand to the vertical positiony=−0.9 and obtaining a hand orientation ofα=−0.25π simulta-neously. (b) The robot’s joint space, showing the sets of redundant solutions corresponding to the respective tasks. If the system employed its two internal models independently from each other, thus producing two uninformed candidate solutions θg and θh, shown as a blue and a red cross respectively, neither of them would satisfy both tasks. Also a linear combination of the two would not yield a solution for both tasks, but in most cases would produce an arm configuration that would satisfy neither task instead. Only the two con-figurations that lie in the intersections of both sets, shown as black crosses, are solutions to both tasks.
(see Section 4.1.3) pursue a similar idea, as in their cases each internal model produces a fuzzy set as response, instead of a single vector. However, it is not easily possible to represent non-trivial sets of solutions, such as the set of solutions g(−0.9) shown in blue in Figure 4.5(b), using fuzzy Sets.
Therefore, it is here proposed to integrate internal models in a generic way by relying on a learning method that can maintain the information about redundancy in the training data, and a form of representation that allows to retrieve not only single solutions, butarbitrary sets of solutions upon a query. We then want to apply a prioritization scheme for the integration of internal models, which favors those solutions that lie in the intersection of as many solution sets as possible. In Figure 4.5(b), exactly two solutions exist that lie in the intersection of both solution sets for the two tasks.
These solutions correspond to the arm configurations that can be seen in Figure 4.5(a).
Furthermore, in most sensorimotor tasks, gradual changes in the target value also correspond to gradual changes in the associated variables. Thus, when for example the target position or orientation for the robot’s hand is changed gradually, also the set of solutions moves continuously, not abruptly. Therefore, the selection of new solutions as the task changes over time should be dependent on the current solution, and the system should not abruptly select entirely different solutions, which would result in an
4.2 Making Use of Redundancies for the Integration of Internal Models
unstable behavior of the robot. Finally, one of the tasks could have a higher priority for the robot than others. For example, actually brining the hand to the target position might be crucial, while obtaining a certain orientation of the hand might be optional.
Thus, the former task should be given a higher priority than the latter.
4.2.2 Dynamic Selection of Solutions Using Dynamic Neural Fields