Movement Primitives as a Robotic Tool to Interpret Trajectories Through Learning-by-doing
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Articulated movements are fundamental in many human and robotic tasks. While humans can learn and generalise arbitrarily long sequences of movements, and particularly can optimise them to fit the constraints and features of their body, robots are often programmed to execute point-to-point precise but fixed patterns. This study proposes a new approach to interpreting and reproducing articulated and complex trajectories as a set of known robot-based primitives. Instead of achieving accurate reproductions, the proposed approach aims at interpreting data in an agent-centred fashion, according to an agent’s primitive movements. The method improves the accuracy of a reproduction with an incremental process that seeks first a rough approximation by capturing the most essential features of a demonstrated trajectory. Observing the discrepancy between the demonstrated and reproduced trajectories, the process then proceeds with incremental decompositions and new searches in sub-optimal parts of the trajectory. The aim is to achieve an agent-centred interpretation and progressive learning that fits in the first place the robots' capability, as opposed to a data-centred decomposition analysis. Tests on both geometric and human generated trajectories reveal that the use of own primitives results in remarkable robustness and generalisation properties of the method. In particular, because trajectories are understood and abstracted by means of agent-optimised primitives, the method has two main features: 1) Reproduced trajectories are general and represent an abstraction of the data. 2) The algorithm is capable of reconstructing highly noisy or corrupted data without pre-processing thanks to an implicit and emergent noise suppression and feature detection. This study suggests a novel bio-inspired approach to interpreting, learning and reproducing articulated movements and trajectories. Possible applications include drawing, writing, movement generation, object manipulation, and other tasks where the performance requires human-like interpretation and generalisation capabilities.
KeywordsMovement primitives learning pattern matching trajectory decomposition perception
Humans and animals are capable of learning, perfecting and reproducing complex trajectories that allow them to perform a variety of tasks, from coordinated body movements to catching, and particularly in humans, object manipulation, writing and drawing. The mechanisms underlying motor skills, from the learning of basic primitives to their organisation in higher-level cognitive structures, are fundamental in understanding how humans accomplish advanced motor skills. Object manipulation, skilful movements and generalised trajectories are considered fundamental in the evolution of intelligence and modern technology. The autonomous learning of robotic movements and their organisation are an increasingly important research focus.
Object manipulation and precise movements are implemented in industrial robots for manufacturing and production processes. However, a considerable limitation in such movements is that trajectories are often pre-programmed and executed point-to-point, therefore lacking a general and symbolic representation of the movement, as well as the capability of adapting and improving.
One solution to point-to-point representations are movement primitives, short movement or strokes that represent elementary building blocks for more complex movements. Motor primitives represent a biological hypothesis on how complex movements are formed in human and animals[3, 4, 5]. One intrinsic feature of motor primitives is that they generate basic and general movements that can then be combined to compose arbitrary long and convoluted trajectories. While biological studies continue to reveal the neurological bases of motor primitives[6, 7], computational models provide examples of computer generated primitives. Among those, dynamic movement primitives (DMP)[8,9], Gaussian mixture models (GMM)[10,11], feedforward neural networks, and recurrent neural networks (RNN)[13,14] have gained considerable attention as mathematical tools to generate simple primitive-like movements.
Fundamental questions that arise with the use of primitives are: What are the features of a set of primitives? How are primitives composed to perform articulated movements? And what role do they play in interpreting, coding, and learning complex movements? Some approaches start by analysing the demonstrated trajectory employing polynomial decomposition, hidden markov models, non-negative matrix factorisation, detection of critical points, and Guassian observation model. Other methods employ reinforcement learning to refine an approximation over time by means of reward signals[20, 21, 22, 23, 24]. Finally, methods have been proposed to join segments to achieve natural-looking trajectories by blending[25,26] and co-articulation[27,28]. Algorithms that combine primitive or shape-identification, trajectory segmentation and on-line learning have also been proposed[18,24, 29] to integrate various subproblems in more capable learning algorithms.
Most algorithms attempt to learn both primitives and their use in the decomposition of long trajectories simultaneously. In contrast, the present method focuses entirely on the decomposition, assuming that primitives are previously learnt and already available to an agent. This separation between learning of primitives and decomposition of demonstrations implies that existing and established methods for learning and executing primitives (e.g., DMP[8’9], stable estimator of dynamical systems (SEDS), extreme learning machine (ELM)[12’30]) can be employed in combination with the current algorithm. In other words, any set of primitives can be chosen and used in combination with the proposed decomposition method. This ensures that existing robotic platforms and primitives can be enhanced with the current algorithm to decompose and reproduce articulated trajectories.
The agent-centred approach implies that, as opposed to other approaches[16, 17, 18], the algorithm does not analyse directly the demonstrated trajectory to find features, e.g. , inflection points, points of discontinuous derivative, critical points, etc. The demonstrated trajectory is approximated instead by means of a process of learning-by-doing in which the performance, or the accuracy of a reproduction, is improved over time with increasingly refined decompositions. Avoiding the analysis of demonstrated data results in two main features of the algorithm. The first is that a reproduction of a demonstration is biased by the agent’s set of primitives. In this respect, the reproduction represents an interpretation of a demonstration. In other words, any demonstration, which was generated by an unknown process, is being fitted with the agent’s fixed primitives. While this may appear as a limitation, it also means that no assumptions on the demonstrated trajectory are required. The agent attempts to achieve a best approximation with its current primitives, which are used as tools to interpret input data. A second feature of the algorithm is that it copes well with highly noisy and corrupted data, and is capable of noise suppression and feature detection.
One fundamental aspect of the proposed method is that, similarly to , the decomposition starts as a rough approximation based on one single movement primitive. Interestingly, a complex trajectory with many convoluted parts is not likely to be adequately represented by one single stroke. Yet, by adopting this counter-intuitive approach, a fundamental step in an iterative process can be achieved towards further and more precise decompositions. Points of decomposition are progressively discovered during the iterative process. At each iteration, the part of the reproduction with the maximum discrepancy with the demonstrated trajectory is considered for improvement. Thus, segmentation points are introduced with the simple but effective heuristic of observing the point of maximum error. Decomposition points can also be later suppressed if more general primitives are discovered to fit a part of the demonstration. The deletion of segmentation points is a bio-inspired search that, once some main features of a demonstration are captured, it relaxes constraints to find better solutions and overcome local optima. Such an approach is inspired by the early/later practice phases in motor learning. Finally, combining primitives as symbolic entities supports biological theories on the construction of motor skills as mental representations of existing building blocks.
It is important to note that the present algorithm only focuses on geometrical properties of the trajectories, while it is agnostic to the velocity profiles. This apparent limitation in reality allows for a more flexible interpretation of trajectories, which may not be necessarily determined by the velocity profile used during generation. Once more, no assumption on the demonstration implies that any demonstration can be observed, decomposed and reproduced with the proposed algorithm.
The algorithm is an extension of that proposed in . In this paper, as opposed to , the algorithm uses one additional set of primitives learnt from human data. This test is essential to confirm the claim that the algorithm can perform well with very diverse sets of primitives. Various criteria to compare trajectories are proposed to underline that trajectory matching may vary according to specific domains and requirements. Additionally, tests are extended particularly to assess the capability of reconstructing noisy data, and the performance is verified on an extended set of exemplary trajectories.
The proposed decomposition algorithm lends itself to promising extensions including learning trajectories from multiple examples, hand-writing recognition, decomposition of complex movement patterns for manipulation and combination of skills. The method is tested only in simulation. Tests on robotic platforms, e.g. the iCub hu-manoid robot or the KUKA lightweight robot arm, are promising extensions.
The decomposition algorithm is explained in detail in the next section. Decomposition tests from simple to complex trajectories are shown in Section 3. The implications and possible extensions are then discussed in Section 4 before the conclusions in Section 5.
2 Search, decomposition and interpretation of trajectories
This section explains the algorithm in all its parts, motivates the bio-inspired approach and illustrates all the steps to reproduce the method.
2.1 Sets of primitives as decomposition tools
The decomposition algorithm requires a set of pre-learnt primitives that can be freely chosen and generated by means of a variety of methods. This feature is particularly important to integrate the proposed algorithm with the well established methods for primitive generations cited above. The performance of the algorithm in the decomposition varies in accuracy and approximation according to the set of primitives, as later tests show. Nevertheless, the method can decompose even with very poor sets of primitives.
2.2 Trajectory matching and iterative decomposition
Equalities (1) – (4) can be used independently or linearly combined to assess how similar two trajectories are. Equalities (1) – (3) are null for perfectly matching trajectories, while (4) is equal to 1 for matching trajectories. Visual observation over many examples revealed that deriving a measure of similarities between two different trajectories is not immediate. In effect, evaluating similarities between trajectories may be domain-dependent or even subjective. The focus of the study is neither to compare the performance of (1) – (4), nor to propose a best criterion. Different matching criteria are proposed here as alternatives which can be chosen to work with the present algorithm. The tests in the current study use by default (2) because it produced predictable segmentations on a large variety of demonstrations. Equalities (1) and (3) are also employed in tests to show the robustness of the method.
The heuristic that identifies candidate segmentation points is not based on an optimality measure, which is difficult to infer in an iterative process. Instead, the algorithm identifies potentially appropriate points to improve further a reproduction. The underlying idea is that the furthest point on the demonstration from the current reproduction lays potentially in a part of the demonstration that is not correctly represented by the reproduction, and thus requires further segmentation. This simple heuristic proves effective as demonstrated later in simulations.
2.3 From sequences of points to sequences of primitives
When decomposed finely, any trajectory can be represented as a sequence of close points united by straight lines. A decomposition that reproduced exactly the demonstration in such a way minimises the reproduction error. However, such a decomposition merely copies a demonstrated trajectory without generalising the overall shape of a movement. The problem is effectively for both a classification problem (i.e., finding the best matching primitive) and an optimisation problem (i.e., reducing the number of segmentation points). A trade-off between generality, with few decomposition points, and precision, with many segmentation points, is desired and sought. As a rule, generality of one solution is accompanied by a residual error with one particular demonstration. The implication is that decomposing a trajectory to minimise the error may lead to a high number of segmentation points. Most algorithms use an error threshold below which the segmentation is considered satisfactory. This problem derives also from the arguable assumption that trajectories have a length but dimensionless thickness. In robotic and real world scenarios, trajectories are both executed and perceived with a certain tolerance. Accounting for such an aspect is a key aspect to avoid over-fitting, unnecessary computation and excessing segmentation.
The method in this study mimics a trajectory with given primitives, which guarantee generality. Primitives may be devised to guarantee also efficiency, optimality, or to conform to particular robotic requirements without necessary minimising an error measure. For example, the minimum-jerk model used in the current experiments guarantees energy minimisation and is biologically plausible, while the ELM-set uses a neural learning paradigm that reproduces human drawn trajectories.
2.3.1 Precision of primitives and intersections
Instead of considering the error between demonstration and reproduction as stopping criterion, the current algorithm looks at whether the demonstration and the reproduction have intersections. If they have at least one intersection, the demonstration is assumed to have further features that need decomposing. If there are no intersections, the current primitive is assumed to be the best approximation: Further decompositions may reduce the error but also reduce generality.
Intersections are intended as two trajectories crossing each other: However, two noisy and overlapping trajectories have many local intersections that would not be considered such by a human observer. Thus, to detect significant intersections, the algorithm associates a precision value to the primitives. Such a precision is an index of how thin a trajectory may be with respect to its length. In effect, this parameter may encode the precision of a mechanical arm, or may be adjusted to account for the variance of many samples, if those are executed by imprecise human movements. In short, the precision parameter is a necessary element in the interpretation of an observed trajectory. It answers the questions: What are the agent’s perception and execution capabilities? What is a realistic precision to be implemented when reproducing a demonstration?
In the experiments of this paper, the smaller sets (MJM and ELM) have a precision value p = 20 = 1/0.05, where 0.05 is the thickness of the primitive normalised to the shortest side of the drawing area. The more accurate 51-primitive set has a precision p = 100 = 1/0.01, i.e., the thickness of a primitive is 1 % of the drawing area. A thickness of 0 corresponds to infinite precision, a concept that does not describe real data from a demonstration and clearly underlines the importance of considering thickness values higher than 0. Higher precision values can be adopted when the demonstration is known to be very accurate. Intersections are detected analytically by computing the cross products between the direction of the primitive and the error vectors of all points: If the cross products have different signs and the error vectors are greater than the line thickness, then an intersection is detected.
The intersection criterion attempts to capture features of the demonstration that are observable with set of primitives used. It is nevertheless possible to use a more traditional stopping criterion, e.g., requiring that the maximum error is decreased below a certain threshold. Such an approach may be used when more emphasis on minimising the error is necessary and an approximation that respects an error constraint is desired. This variation was experimented in the current algorithm and is easy implementable by letting the algorithm continue the segmentation until the maximum error falls under a threshold. A similar variation may also include a measure of parallelism between two trajectories. The algorithm may be required to continue segmenting until a certain threshold is reached. These variations of the algorithm require more human supervision in setting such an error threshold and understanding what matching criteria are needed in a particular scenario. In some cases, introducing additional matching measurements and stopping criteria may lead to reproductions that are perceived visually as better approximations.
2.3.2 Deleting segmentation points
The iterative process implies that the interpretation of the demonstration (i.e., the solution) varies and improves at each further decomposition. One question is whether decomposition points that were found initially during the process are still good segmentation points later as the accuracy improves. Inspired by theories of motor learning in humans, the proposed method introduces a type of search that releases early constraints when a new segment is added. At the insertion of a new decomposition point, primitives are searched to the left and to the right of the candidate point. The search may go beyond the immediate left and right segments. It is possible to search further left and further right, thereby attempting larger generalisations. Neighbouring decomposition points are eliminated if more general primitives without intersections are discovered.
This check guides the search to avoid local optima, and at the same time it helps reduce the number of overall segmentation points, thereby achieving more general solutions. Releasing constraints implies more computation while searching larger primitives that may skip segmentation points. This type of search is nevertheless far from exhaustive: The further exploration relies on the current segmentation. It represents an attempt to reorganise parts of the trajectory according to new knowledge that was gathered during the iterative segmentation process.
2.4 The iterative algorithmic procedure
The algorithm starts selecting one primitive that best matches the demonstrated trajectory (Fig. 4, blocks 1 and 2). The best match is obtained comparing all primitives with the demonstration and choosing the primitive that minimises a measure of discrepancy (1)–(3) or maximises a measure of similarity (4). In the next step (block 3), the algorithm finds the point of maximum error between the demonstration and the reproduction (5). This is a candidate segmentation point and is located in a part of the demonstration that is poorly approximated. Initially there is only one segment. As the iterations proceed, more segments are created. When created, each segment is labelled as non-finalised, meaning that further decompositions are possible. The point of maximum error is sought on a non-finalised segment (blocks 3 and 4). The algorithm now checks whether the primitive intersects the demonstration or not (block 5). As illustrated in Figs.3(a) and (b), an intersection suggests the presence of a relevant feature that can be captured with further decompositions. If the best matching primitive does not intersect the demonstration (block 5), the demonstration may be laying outside the reachable area of the primitives (block 6). This case, or the case in which there is an intersection, mean that there are additional features in the demonstration that need to be captured. Therefore, the algorithm proceeds with the segmentation (block 7). Otherwise, the current segment is finalised (block 10). The search in block 7 is carried out by exploring primitives that approximate the left and right parts of the demonstration from the candidate segmentation point. Such a search involves also the elimination of older segmentation points when better approximations are found. The search for better primitives in block 7 may or may not result in an improvement of (1)–(4). If no improvements can be achieved, the segmentation point is rejected and that segment is labelled as finalised (blocks 8 and 10). If an improvement is found, the segmentation point and the left and right primitives are promoted as part of the current segmentation (block 9).
Throughout the process, the representation of a demonstration is updated. Table 1 shows how the primitive-based symbolic trajectory is described. At the first iteration, only one row is present. Further, segmentations add more rows describing primitives, start point, scaling and angle, and whether the segment is finalised.
Representation of a trajectory as a sequence of primitives. Segments (i.e., rows in the table) are added and occasionally removed during the iterative process. For each segment, it is necessary to specify which primitive is used (2nd column), the starting point (3rd column), the scaling and angle (4th and 5th column) and whether the segment can be further decomposed (6th column)
3 Simulation results
The current section reports the simulation results of the algorithm applied to a variety of demonstrated trajectories, from simple to complex.
3.1 Reconstructing short demonstrations
From this first test, it emerges one important and bio-inspired feature of the algorithm. The method appears to reconstruct in a way to recognise those trajectories that are similar to the known primitives. The reconstruction by the MJM 51-set in the third row is more abstract and less similar to the original than the reconstruction by the ELM-set, despite the considerably larger library of primitives in the 51-set. However, while the ELM-set performed well in this particular test, the 51-set is more generic and is likely to perform better on other trajectories with arbitrary geometry.
The primitives are executed sequentially without transformations (e.g., smoothing) at the points of junction. Therefore, points of discontinuous derivative are noticeable where primitives join. Smoothing a trajectory requires the understanding of whether a point is a cuspid, i.e., the trajectory has a discontinuous derivative, or it can be rounded with a co-articulation algorithm[26, 28]. As this particular problem was not the focus of the present algorithm, all primitives are joined without blending or coarticulation.
3.2 Decomposition of hand writing
The decomposition of human-generated writing trajectories is a task in which the symbolic aspect is more important than the exact geometry. In other words, global features in a trajectory are fundamental in distinguishing different letters more than the precise geometry of the trajectory. The proposed algorithm was shown in the previous section to be suited to extract high level representations from noisy data. It is natural to ask whether this feature may be of use as a step towards abstracting human hand writing. Note that the experiment in this section decomposes and represents hand writing as a set of primitives, but it does not interpret or map trajectories to letters.
The original idea in the proposed algorithm is to decompose an arbitrarily complex trajectory using the agent’s pre-learnt primitives during an iterative process of learning-by-doing. The process starts with a rough approximation of the demonstrated trajectory and learns step by step the features of the input data by a progressive decomposition. Segmentation points are discovered simply by a criterion of maximum error between demonstration and reproduction. Such a trivial criterion that ignores features of both demonstration and reproduction proved nevertheless surprisingly effective and robust. The final result is a sequence of primitives that is in effect an intelligent reading of a demonstrated trajectory represented as a general and abstract concept. The strength of the algorithm lies in the primitive-centred and progressive search, which uses existing skills and implicitly solves data-induced problems like noise and discontinuous derivatives.
Finding segmentation points and fitting sub-trajectories is potentially an intractable problem if considered exhaustively. The proposed method suggests candidate segmentation points taking advantage of progressive approximations. The computation required to generate a reproduction increases with the number of iterations and the number of available primitives. The removal of constraints, i.e., the search of primitives that bypass segmentation points, is done at a the computational cost of matching the locally segmented demonstration with primitives. However, removing segmentation points results in more general solutions, which justify the additional computation. The removal of constraints is effectively a search procedure to avoid local minima in a high dimensional search landscape.
For simplicity, the current study considers finite sets of primitives in which each primitive has a fixed geometry. An alternative approach is to use primitives with variable geometry: In such a case, a parameter can be used to change certain features of a primitive as the curvature for example. The use of infinite-set primitives requires a different representation, but does not increase the computational complexity of the search. In fact, a larger variety of geometries can be implemented with fewer tuneable primitives. The extension of the algorithm to infinite-set primitives is promising particularly in the cases where high precision and compact representations are required.
The algorithm appears to have generalisation capabilities even if it decomposes trajectories from one single demonstration. The generalisation capability, noticeable particularly in Fig. 5 (rows 1 and 3), derives from the interpretation of the demonstration according to the agent’s set of primitives. The reconstruction from noisy data in particular shows the generalisation capability in reconstructing straight lines, identify correct curvatures, as well as maintaining cuspids, as clearly shown in Fig. 11.
The criteria upon which the algorithm is constructed (Section 2) represent the intelligence of the decomposition, which is intended to mimic loosely human processes of understanding, acquiring and reproducing articulated movement or trajectories. For this reason, the proposed algorithm focuses less on the input data itself and more on the quality of the procedure applied to interpret it. The use of primitives implies inevitably the classification of imprecise and noise-affected demonstration into well defined trajectories. Therefore, such a process causes the loss of accuracy from the demonstrated data. However, such an accuracy may not be descriptive of features of the demonstration. Abstract representations are more compatible with hypotheses on how humans and animals represent and execute movements.
The precision parameter, encoding the second dimension or thickness of a primitive, determines effectively to which level small details in the demonstration need to be reproduced. As a consequence, high precision means that a noisy demonstration is reproduced accurately down to small details, while low precision means that the trajectory is more heavily interpreted according to the agent’s primitives. It is important to note that a low precision parameter is not equivalent to high noise filtering. In fact, cuspids and prominent features of the demonstrations are nevertheless captured as shown in Fig. 11.
The method is tested here using one demonstration only for each trajectory. A promising extension is to use multiple demonstrations of the same trajectory to increase the generalisation properties of the algorithm. In particular, more observations of one demonstration are likely to have variations but retain relevant features. One extension is to increase its capability of generalising trajectories by finding one decomposition of a set of similar demonstrations.
The algorithm uses primitives and demonstration in a two dimensional space. The method can be extended and applied to a 3-dimensional (3D) scenario because primitives and matching functions can be equally generated and computed in 3D space. The increased dimensionality implies also a larger search space, extended sets of primitives and more computation required. It is conceivable that primitives in a 3D space may nevertheless lay on a two-dimensional plane, and that truly 3D trajectories like a helix are relatively rare. The extension presents challenges but is a promising venue for reproducing fully-fledged robotic movements in space.
The trajectories considered in this study were only determined by the geometry without velocity profiles. In effect, releasing the constraints on velocity allows agents to reproduce complex demonstrations by freely choosing from their own primitives with given velocities representing their own capabilities. Extensions of the algorithm could include velocity profiles. The addition of kinematics may imply that velocity cannot drop to zero at segmentation points, introducing strict constraints to the search. In effect, whereas kinematics are essential in dynamics movements such as walking, they become less stringent in object manipulation and marginal in drawing and writing. As the respect of kinematics constraints depends heavily on the precise field of application, tailored algorithms may be required.
The proposed method focuses on the decomposition of trajectories and does not consider the learning of new primitives. The results in this paper showed that the set of primitives is important to achieve particular required performance, and in particular is crucial in interpreting noisy or corrupted data. It is natural to ask how the algorithm can be adapted to extend the available set of primitives while decomposing. A promising research direction is that of integrating the current method in a more powerful algorithm that learns additional primitives with experience. Additionally, certain sequences of primitives that repeat themselves frequently could be assimilated as a new longer primitive, thereby accelerating the search in future occurrences of the given sequence.
The variety of tasks in which simple movements are combined to achieve complex movements extends to numerous scenarios. The proposed method can be applied to those scenarios in which imprecisely perceived movements need to be decomposed, learnt and reproduced. In particular, robots with different dimensions, joint structures and degree of freedom can attempt to perform complex movements according to their own features and capabilities. The implication is that the current method, by adopting an agent-centred and iterative approach to decomposition, is suited to a large variety of robotic platforms, particularly animallike and humanoid robots that are required to perform a large variety of tasks, not all of them perfectly fitting their anatomical features.
A new approach to decompose and reconstruct complex trajectories is proposed. The method starts decomposing a complex trajectory with one initial single primitive and progressively increases the accuracy of the approximation through an iterative process. This approach allows for an initial reduction of the search space with the identification of prominent features of a demonstrated trajectory. Subsequently, the iterative search makes use of newly found segmentation points to search locally better solutions and escape local optima. The agent-centred process offers a new way of interpreting data as function of the agent’s skills, which may represent various optimal primitives generated with established methods. The algorithm proves robust and displays remarkable generalisation and feature extraction capabilities. In particular, the algorithm is suited to reconstructing trajectories from corrupted and noisy data. Diverse robotic platforms with different degrees of accuracy and motor patterns could benefit from this method while learning progressively and autonomously the decomposition of complex trajectories. Promising extensions of the algorithm include the applications to a variety of tasks such as imitation learning, learning of complex motor patterns, gestures, object manipulation, software-based and robotic hand-writing.
Generating the primitives with the minimum-jerk model (MJM)
The minimum-jerk model (MJM) is used in the current study to generate two primitives sets (compare Figs. 1 (a) and (b)) and sample trajectories for the testing in Section 3.1. This model plans a trajectory starting from a given starting point to a given end-point through a via-point. The constraint for planning the trajectory is to be as smooth as possible (minimum jerk). In the generated data-sets, the via-point is located at the maximum of each shape, which is reached at t = 0.5 of the movement duration.
Generating trajectories with ELM
A point-to-point motion is driven by a vector field (i.e., a mapping from position x to vector v) represented by a data driven learning method called extreme learning machine (ELM). The ELM is a feedforward neural network that comprises three different layers of neurons: The input layer x ∈ R I , the hidden layer h ∈ R R , and the output neurons v ∈ R I . The input is connected to the hidden layer through the input matrix W inp ∈ R R×I that is unchanged after random initialisation. A supervised learning schema was adopted to compute the output weight matrix to generate stable movements. Each new primitive is learnt from at least three human demonstrated trajectories. The sequence of motion can be computed by discretisation of \(\dot x = \hat v\)(x), where x (0) ∈ R d denotes the starting point. Different movements can be generated depending on the starting point relative to the target. The mean starting point x ms of all demonstration was used to learn each primitive. Using such a starting point, the learnt primitive is likely to be similar to the average demonstration. The movement is then normalised such that the start point is at x ms → x S = (–1,0) and the end point at x T = (0,0). The normalised primitives are used in the algorithm with correct rotations and scaling accordingly to the initial (I) and final (F) points as described in the paper.
The authors thank William Land for discussions and comments on earlier version of the manuscript.
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