Trajectory Optimization on Matrix Lie Groups with Differential Dynamic Programming and Nonlinear Constraints

Matrix Lie groups are an important class of manifolds commonly used in control and robotics, and the optimization of control policies on these manifolds is a fundamental problem. In this work, we propose a novel approach for trajectory optimization on matrix Lie groups using an augmented Lagrangian-based constrained discrete Differential Dynamic Programming. The method involves lifting the optimization problem to the Lie algebra in the backward pass and retracting back to the manifold in the forward pass. In contrast to previous approaches which only addressed constraint handling for specific classes of matrix Lie groups, the proposed method provides a general approach for nonlinear constraint handling for generic matrix Lie groups. We also demonstrate the effectiveness of the method in handling external disturbances through its application as a Lie-algebraic feedback control policy on SE(3). Experiments show that the approach is able to effectively handle configuration, velocity and input constraints and maintain stability in the presence of external disturbances.Comment: 10 pages, 7 figure

Learning Stable Robotic Skills on Riemannian Manifolds

In this paper, we propose an approach to learn stable dynamical systems evolving on Riemannian manifolds. The approach leverages a data-efficient procedure to learn a diffeomorphic transformation that maps simple stable dynamical systems onto complex robotic skills. By exploiting mathematical tools from differential geometry, the method ensures that the learned skills fulfill the geometric constraints imposed by the underlying manifolds, such as unit quaternion (UQ) for orientation and symmetric positive definite (SPD) matrices for impedance, while preserving the convergence to a given target. The proposed approach is firstly tested in simulation on a public benchmark, obtained by projecting Cartesian data into UQ and SPD manifolds, and compared with existing approaches. Apart from evaluating the approach on a public benchmark, several experiments were performed on a real robot performing bottle stacking in different conditions and a drilling task in cooperation with a human operator. The evaluation shows promising results in terms of learning accuracy and task adaptation capabilities.Comment: 16 pages, 10 figures, journa

Manipulation primitives: A paradigm for abstraction and execution of grasping and manipulation tasks

Sensor-based reactive and hybrid approaches have proven a promising line of study to address imperfect knowledge in grasping and manipulation. However the reactive approaches are usually tightly coupled to a particular embodiment making transfer of knowledge difficult. This paper proposes a paradigm for modeling and execution of reactive manipulation actions, which makes knowledge transfer to different embodiments possible while retaining the reactive capabilities of the embodiments. The proposed approach extends the idea of control primitives coordinated by a state machine by introducing an embodiment independent layer of abstraction. Abstract manipulation primitives constitute a vocabulary of atomic, embodiment independent actions, which can be coordinated using state machines to describe complex actions. To obtain embodiment specific models, the abstract state machines are automatically translated to embodiment specific models, such that full capabilities of each platform can be utilized. The strength of the manipulation primitives paradigm is demonstrated by developing a set of corresponding embodiment specific primitives for object transport, including a complex reactive grasping primitive. The robustness of the approach is experimentally studied in emptying of a box filled with several unknown objects. The embodiment independence is studied by performing a manipulation task on two different platforms using the same abstract description

Hybrid control trajectory optimization under uncertainty

Trajectory optimization is a fundamental problem in robotics. While optimization of continuous control trajectories is well developed, many applications require both discrete and continuous, i.e. hybrid controls. Finding an optimal sequence of hybrid controls is challenging due to the exponential explosion of discrete control combinations. Our method, based on Differential Dynamic Programming (DDP), circumvents this problem by incorporating discrete actions inside DDP: we first optimize continuous mixtures of discrete actions, and, subsequently force the mixtures into fully discrete actions. Moreover, we show how our approach can be extended to partially observable Markov decision processes (POMDPs) for trajectory planning under uncertainty. We validate the approach in a car driving problem where the robot has to switch discrete gears and in a box pushing application where the robot can switch the side of the box to push. The pose and the friction parameters of the pushed box are initially unknown and only indirectly observable

A probabilistic framework for learning geometry-based robot manipulation skills

Programming robots to perform complex manipulation tasks is difficult because many tasks require sophisticated controllers that may rely on data such as manipulability ellipsoids, stiffness/damping and inertia matrices. Such data are naturally represented as Symmetric Positive Definite (SPD) matrices to capture specific geometric characteristics of the data, which increases the complexity of hard-coding them. To alleviate this difficulty, the Learning from Demonstration (LfD) paradigm can be used in order to learn robot manipulation skills with specific geometric constraints encapsulated in SPD matrices. Learned skills often need to be adapted when they are applied to new situations. While existing techniques can adapt Cartesian and joint space trajectories described by various desired points, the adaptation of motion skills encapsulated in SPD matrices remains an open problem. In this paper, we introduce a new LfD framework that can learn robot manipulation skills encapsulated in SPD matrices from expert demonstrations and adapt them to new situations defined by new start-, via- and end-matrices. The proposed approach leverages Kernelized Movement Primitives (KMPs) to generate SPD-based robot manipulation skills that smoothly adapt the demonstrations to conform to new constraints. We validate the proposed framework using a couple of simulations in addition to a real experiment scenario

Multimodal Representation Learning for Place Recognition Using Deep Hebbian Predictive Coding

Recognising familiar places is a competence required in many engineering applications that interact with the real world such as robot navigation. Combining information from different sensory sources promotes robustness and accuracy of place recognition. However, mismatch in data registration, dimensionality, and timing between modalities remain challenging problems in multisensory place recognition. Spurious data generated by sensor drop-out in multisensory environments is particularly problematic and often resolved through adhoc and brittle solutions. An effective approach to these problems is demonstrated by animals as they gracefully move through the world. Therefore, we take a neuro-ethological approach by adopting self-supervised representation learning based on a neuroscientific model of visual cortex known as predictive coding. We demonstrate how this parsimonious network algorithm which is trained using a local learning rule can be extended to combine visual and tactile sensory cues from a biomimetic robot as it naturally explores a visually aliased environment. The place recognition performance obtained using joint latent representations generated by the network is significantly better than contemporary representation learning techniques. Further, we see evidence of improved robustness at place recognition in face of unimodal sensor drop-out. The proposed multimodal deep predictive coding algorithm presented is also linearly extensible to accommodate more than two sensory modalities, thereby providing an intriguing example of the value of neuro-biologically plausible representation learning for multimodal navigation

Increased gene sampling strengthens support for higher-level groups within leaf-mining moths and relatives (Lepidoptera: Gracillariidae)

Background: Researchers conducting molecular phylogenetic studies are frequently faced with the decision of what to do when weak branch support is obtained for key nodes of importance. As one solution, the researcher may choose to sequence additional orthologous genes of appropriate evolutionary rate for the taxa in the study. However, generating large, complete data matrices can become increasingly difficult as the number of characters increases. A few empirical studies have shown that augmenting genes even for a subset of taxa can improve branch support. However, because each study differs in the number of characters and taxa, there is still a need for additional studies that examine whether incomplete sampling designs are likely to aid at increasing deep node resolution. We target Gracillariidae, a Cretaceous-age (similar to 100 Ma) group of leaf-mining moths to test whether the strategy of adding genes for a subset of taxa can improve branch support for deep nodes. We initially sequenced ten genes (8,418 bp) for 57 taxa that represent the major lineages of Gracillariidae plus outgroups. After finding that many deep divergences remained weakly supported, we sequenced eleven additional genes (6,375 bp) for a 27-taxon subset. We then compared results from different data sets to assess whether one sampling design can be favored over another. The concatenated data set comprising all genes and all taxa and three other data sets of different taxon and gene sub-sampling design were analyzed with maximum likelihood. Each data set was subject to five different models and partitioning schemes of non-synonymous and synonymous changes. Statistical significance of non-monophyly was examined with the Approximately Unbiased (AU) test. Results: Partial augmentation of genes led to high support for deep divergences, especially when non-synonymous changes were analyzed alone. Increasing the number of taxa without an increase in number of characters led to lower bootstrap support; increasing the number of characters without increasing the number of taxa generally increased bootstrap support. More than three-quarters of nodes were supported with bootstrap values greater than 80% when all taxa and genes were combined. Gracillariidae, Lithocolletinae + Leucanthiza, and Acrocercops and Parectopa groups were strongly supported in nearly every analysis. Gracillaria group was well supported in some analyses, but less so in others. We find strong evidence for the exclusion of Douglasiidae from Gracillarioidea sensu Davis and Robinson (1998). Our results strongly support the monophyly of a G.B.R.Y. clade, a group comprised of Gracillariidae + Bucculatricidae + Roeslerstammiidae + Yponomeutidae, when analyzed with non-synonymous changes only, but this group was frequently split when synonymous and non-synonymous substitutions were analyzed together. Conclusions: 1) Partially or fully augmenting a data set with more characters increased bootstrap support for particular deep nodes, and this increase was dramatic when non-synonymous changes were analyzed alone. Thus, the addition of sites that have low levels of saturation and compositional heterogeneity can greatly improve results. 2) Gracillarioidea, as defined by Davis and Robinson (1998), clearly do not include Douglasiidae, and changes to current classification will be required. 3) Gracillariidae were monophyletic in all analyses conducted, and nearly all species can be placed into one of six strongly supported clades though relationships among these remain unclear. 4) The difficulty in determining the phylogenetic placement of Bucculatricidae is probably attributable to compositional heterogeneity at the third codon position. From our tests for compositional heterogeneity and strong bootstrap values obtained when synonymous changes are excluded, we tentatively conclude that Bucculatricidae is closely related to Gracillariidae + Roeslerstammiidae + Yponomeutidae