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Handling task prioritization as a product of Gaussians
Sylvain CalinonSenior ResearcherIdiap Research Institute, Martigny, Switzerland
LecturerEPFL, Lausanne, Switzerland
Research Groups:• Speech & Audio Processing• Natural Language Understanding• Perception & Activity Understanding• Machine Learning• Social Computing• Biometrics Security and Privacy• Biosignal Processing• Computational Bioimaging• Energy Informatics • Uncertainty Quantification and Optimal Design• Robot Learning & Interaction
Artificial Intelligence for Society
MARTIGNY
ResearchEducation
Technology transfer
Learning from demonstration Observational learning
Kinesthetic teachingCorrespondence problems
Superposition Fusion
t=0 t=1 t=2t=0 t=1 t=2
Motivating example:
A probabilistic view on segment crossing!
Many algorithms in robotics can be re-interpreted as products of Gaussians, including task prioritization!
(superposition)
(fusion)
center of the Gaussian
covariance matrix
precision matrix
Combination of primitives as a fusion problem
Scalar superposition as a product of Gaussians with :
Combination of primitives as a fusion problem
The full weight matrices approach covers both scalar weights
(with isotropic diagonal matrix) and null space projection
operations!
Null space projection(hierarchy constraints)
Scalar superposition
Tasks prioritization as PoGPrincipal task:track horizontal reference
Secondary task:track desired posture
Principal task
Secondary task
Principal task
Secondary task
Learning tasks prioritization
Parallel organization of motion/skill primitives
Task-parameterized Gaussian mixture model (TP-GMM)
[Canal, Pignat, Alenya, Calinon and Torras, ICRA’2018]
Calinon, S. (2016). A Tutorial on Task-Parameterized Movement Learning and Retrieval. Intelligent Service Robotics (Springer), 9:1, 1-29.
Task-parameterized Gaussian mixture model (TP-GMM)
[Calinon, Alizadeh and Caldwell, IROS’2013]
Coordinatesystem as task
parameter
Task-parameterized Gaussian mixture model (TP-GMM)
Candidate hierarchy
Candidate hierarchy
Demonstration
Reproduction
Demonstration
Reproduction
[Silvério, Calinon, Rozo and Caldwell, IEEE T-RO 2019]
Priority on left hand
Learning tasks prioritization
Dr João Silvério
Candidate hierarchy
Candidate hierarchy
Demonstration
Reproduction
Demonstration
Reproduction
Learning tasks prioritization
[Silvério, Calinon, Rozo and Caldwell, IEEE T-RO 2019]
Priority on right hand
Dr João Silvério
Candidate hierarchy
Candidate hierarchy
Demonstration
Reproduction
Demonstration
Reproduction
Learning tasks prioritization
[Silvério, Calinon, Rozo and Caldwell, IEEE T-RO 2019]
Equal priority
Dr João Silvério
Learning tasks prioritization
[Silvério, Calinon, Rozo and Caldwell, IEEE T-RO 2019]
Centaurorobot
Dr João Silvériobase position > end-effector positions > end-effector orientations
Learning tasks prioritization
[Silvério, Calinon, Rozo and Caldwell, IEEE T-RO 2019]
Centaurorobot
Dr João Silvérioend-effector positions > base position > end-effector orientations
Ongoing work
Learning tasks prioritization
Expert A>B>CExpert A>C>BExpert B>A>CExpert B>C>AExpert C>A>BExpert C>B>A
Expert A>BExpert B>AExpert B>CExpert C>B
Geodesic interpolation between nullspace structures
Representing task prioritization as a product of Gaussians provides a way to geometrically interpolate task hierarchies (geodesics on symmetric positive semidefinite manifolds)
Task 1 Task 2
Task 1 > Task 2 Task 2 > Task 1
Task 1 Task 1 Task 2Task 2
Exploiting redundancy in planning tasks
[Girgin and Calinon, arXiv:1905.09679, 2019]Hakan Girgin
Products of Uni-Gauss experts
Uni-Gauss experts are distributions combining the task constraint in the form of a Gaussianwith a uniform distribution
Intl Conf. on Artificial Neural Networks (ICANN’99)
Product of Gaussians
Product of Uni-Gauss experts
with the same πi
(no hierarchy; simply go from OR to AND)
Product of Uni-Gauss experts
with different πi
(task 1 is set as being obligatory)
Products of Uni-Gauss experts
Instead of providing a solution as an exact compromise between objectives (which might fulfill none of them), Uni-Gauss experts allow to represent tasks in which we might prefer to fulfill one or another, but not both.
Talos robot
Products of Uni-Gauss expertsVariational inference can be used to create distributions of good and diversifiedconfigurations.
Combined with Uni-Gauss experts, the system can sample robot configurations satisfying constraints.
[Pignat, Lembono and Calinon, arXiv:1905.09597, 2019]
Emmanuel Pignat
Conclusion
A product of Gaussians can be used to represent nullspace projection operations and weighted superposition of tasks (fusion problem)
Learning task prioritization can be achieved by a statistical analysis of the data transformed with different candidate projections (TP-GMM)
Demo
Repro
Variational inference with Uni-Gauss distributions is a promising approach to learn task constraints individually, and then combine them with prioritization
Nullspace structures are present in a wide range of robotics problems, including planning problems computed with model predictive control (MPC)
Robot Learning & Interaction Group at Idiap:
Contact:
[email protected]://calinon.ch
Source codes (Matlab/Octave, C++ and Python):
http://www.idiap.ch/software/pbdlib/
Photo: Basilio Noris
Thibaut Kulak
Emmanuel Pignat
NoémieJaquier
Dr Antonio Paolillo
HakanGirgin
MartinTroussard
TeguhLembono
?
