Abstract
Robot learning from demonstration (LfD) emerges as a promising solution to transfer human motion to the robot. However, because of the open-loop between the learner and task constraints, the precision of the reproduction at the desired task constraints cannot always be guaranteed and the model is not robust to changes of the training data. This paper proposes a closed-loop framework of LfD based on the bagging method of Gaussian Mixture Model and Gaussian Mixture Regression (GMM/GMR) to obtain a robust learner of LfD with high precision reproduction. The original demonstration data are divided into several sub-training data, from which multiple Gaussian mixture models are developed and combined through weighted average to provide predictions. A closed-loop is built between the reproduction of the combined learner and task constraints, and the weights that satisfy task constraints are estimated in the closed-loop. The prediction uncertainty of the models is automatically eliminated by the closed-loop, therefore, the low robustness of the LfD model to the training date is overcome. In experiments, tasks of the position and velocity are both constrained in dual closed-loop. It is shown that the proposed method can significantly meet the task constraints without increasing the complexity of the algorithm.
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Abbreviations
- t :
-
Time step
- \(\varvec{\xi }^o\), \(\varvec{\xi }\) :
-
Training data before/after DTW
- \(\varvec{\xi }^s\) :
-
Spatial component of the training data
- \(\varvec{\Theta }\) :
-
Parameters of GMM
- \(\pi _k\) :
-
Prior probability of a Gaussian distribution in a GMM
- \(\varvec{\beta }_k\) :
-
The probability of the component k to be responsible for t
- \(\varvec{\mu }_k\) :
-
Mean of a Gaussian component
- \(\varvec{\Sigma }_k\) :
-
Covariance matrix of a Gaussian distribution
- K :
-
Number of Gaussian components
- \({\mathcal {N}}(\varvec{\mu _k},\varvec{\Sigma _k})\) :
-
Gaussian distribution described by mean \(\varvec{\mu }_k\) and covariance matrix \(\varvec{\Sigma }_k\)
- \({\mathcal {N}}(\varvec{\xi }_i;\varvec{\mu }_k,\varvec{\Sigma }_k) \) :
-
Probability of \(\varvec{\xi }_i\) where the density function is a Gaussian distribution
- \(\hat{\varvec{\xi }^s}\) :
-
Expected mean of the reproduction (spatial component)
- \(\hat{\varvec{\Sigma }}^{ss}\) :
-
Expected covariance of the reproduction (spatial component)
- Q :
-
The number of the base learners
- D :
-
Spatial dimensionality
- N :
-
Number of datapoints
- M :
-
Demonstration number
- L :
-
Key-points number
- \(G,G^{'}\) :
-
Combined learner
- \(G_i\), \(G^{'}_i\) :
-
The i-th base learner
- \(y_i\), y :
-
The output of \(G_i\)/G
- \(\omega _i\) :
-
The generalized weight of base learner \(G_i\)
- \({\mathbb {C}}\) :
-
Task constraints
- \({\mathbf {A}}^d\) :
-
The d-th dimension of \({\mathbb {C}}\)
- \({\mathbf {B}}^d\) :
-
\(L \times Q\) matrix consisted of the output of each base learners
- \({\mathbf {C}}_q\) :
-
\(D \times D\) diagonal matrix
- \(p_i\) :
-
The i-th key-position point
- \(\varvec{d}_i\),\(\varvec{d}^{'}_i\) :
-
The distance between the reproduction of GMR/Bagging-GMR and the key-points \(p_i\)
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Acknowledgements
This research is supported by the Key Research and Development Plan under Grant No. 2018YFB1308700, the National Natural Science Foundation of China (NSFC) under Grant No. 51535004, and the Fundamental Research Funds for the Central Universities under Grant No. 2020kfyXJJS064.
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Ye, C., Yang, J. & Ding, H. Bagging for Gaussian mixture regression in robot learning from demonstration. J Intell Manuf 33, 867–879 (2022). https://doi.org/10.1007/s10845-020-01686-8
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DOI: https://doi.org/10.1007/s10845-020-01686-8