GAMformer: In-Context Learning for Generalized Additive Models

Generalized Additive Models (GAMs) are widely recognized for their ability to create fully interpretable machine learning models for tabular data. Traditionally, training GAMs involves iterative learning algorithms, such as splines, boosted trees, or neural networks, which refine the additive components through repeated error reduction. In this paper, we introduce GAMformer, the first method to leverage in-context learning to estimate shape functions of a GAM in a single forward pass, representing a significant departure from the conventional iterative approaches to GAM fitting. Building on previous research applying in-context learning to tabular data, we exclusively use complex, synthetic data to train GAMformer, yet find it extrapolates well to real-world data. Our experiments show that GAMformer performs on par with other leading GAMs across various classification benchmarks while generating highly interpretable shape functions.

The significance of the configuration space Lie group for the constraint satisfaction in numerical time integration of multibody systems

The dynamics simulation of multibody systems (MBS) using spatial velocities (non-holonomic velocities) requires time integration of the dynamics equations together with the kinematic reconstruction equations (relating time derivatives of configuration variables to rigid body velocities). The latter are specific to the geometry of the rigid body motion underlying a particular formulation, and thus to the used configuration space (c-space). The proper c-space of a rigid body is the Lie group SE(3), and the geometry is that of the screw motions. The rigid bodies within a MBS are further subjected to geometric constraints, often due to lower kinematic pairs that define SE(3) subgroups. Traditionally, however, in MBS dynamics the translations and rotations are parameterized independently, which implies the use of the direct product group $SO\left( 3\right) \times {\Bbb R}^{3}$ as rigid body c-space, although this does not account for rigid body motions. Hence, its appropriateness was recently put into perspective. In this paper the significance of the c-space for the constraint satisfaction in numerical time stepping schemes is analyzed for holonomicaly constrained MBS modeled with the 'absolute coordinate' approach, i.e. using the Newton-Euler equations for the individual bodies subjected to geometric constraints. It is shown that the geometric constraints a body is subjected to are exactly satisfied if they constrain the motion to a subgroup of its c-space. Since only the $SE\left( 3\right) $ subgroups have a practical significance it is regarded as the appropriate c-space for the constrained rigid body. Consequently the constraints imposed by lower pair joints are exactly satisfied if the joint connects a body to the ground. For a general MBS, where the motions are not constrained to a subgroup, the SE(3) and $SO\left( 3\right) \times {\Bbb R}^{3}$ yield the same order of accuracy.

Is there an optimal choice of configuration space for Lie group integration schemes applied to constrained MBS?

Recently various numerical integration schemes have been proposed for numerically simulating the dynamics of constrained multibody systems (MBS) operating. These integration schemes operate directly on the MBS configuration space considered as a Lie group. For discrete spatial mechanical systems there are two Lie group that can be used as configuration space: $SE\left( 3\right) $ and $SO\left( 3\right) \times \mathbb{R}^{3}$. Since the performance of the numerical integration scheme clearly depends on the underlying configuration space it is important to analyze the effect of using either variant. For constrained MBS a crucial aspect is the constraint satisfaction. In this paper the constraint violation observed for the two variants are investigated. It is concluded that the $SE\left( 3\right) $ formulation outperforms the $SO\left( 3\right) \times \mathbb{R}^{3}$ formulation if the absolute motions of the rigid bodies, as part of a constrained MBS, belong to a motion subgroup. In all other cases both formulations are equivalent. In the latter cases the $SO\left( 3\right) \times \mathbb{R}^{3}$ formulation should be used since the $SE\left( 3\right) $ formulation is numerically more complex, however.

Position: A Call to Action for a Human-Centered AutoML Paradigm

Automated machine learning (AutoML) was formed around the fundamental objectives of automatically and efficiently configuring machine learning (ML) workflows, aiding the research of new ML algorithms, and contributing to the democratization of ML by making it accessible to a broader audience. Over the past decade, commendable achievements in AutoML have primarily focused on optimizing predictive performance. This focused progress, while substantial, raises questions about how well AutoML has met its broader, original goals. In this position paper, we argue that a key to unlocking AutoML's full potential lies in addressing the currently underexplored aspect of user interaction with AutoML systems, including their diverse roles, expectations, and expertise. We envision a more human-centered approach in future AutoML research, promoting the collaborative design of ML systems that tightly integrates the complementary strengths of human expertise and AutoML methodologies.

An Overview of Formulae for the Higher-Order Kinematics of Lower-Pair Chains with Applications in Robotics and Mechanism Theory

The motions of mechanisms can be described in terms of screw coordinates by means of an exponential mapping. The product of exponentials (POE) describes the configuration of a chain of bodies connected by lower pair joints. The kinematics is thus given in terms of joint screws. The POE serves to express loop constraints for mechanisms as well as the forward kinematics of serial manipulators. Besides the compact formulations, the POE gives rise to purely algebraic relations for derivatives wrt. joint variables. It is known that the partial derivatives of the instantaneous joint screws (columns of the geometric Jacobian) are determined by Lie brackets the joint screws. Lesser-known is that derivative of arbitrary order can be compactly expressed by Lie brackets. This has significance for higher-order forward/inverse kinematics and dynamics of robots and multibody systems. Various relations were reported but are scattered in the literature and insufficiently recognized. This paper aims to provide a comprehensive overview of the relevant relations. Its original contributions are closed form and recursive relations for higher-order derivatives and Taylor expansions of various kinematic relations. Their application to kinematic control and dynamics of robotic manipulators and multibody systems is discussed.

Towards Continuous Time Finite Horizon LQR Control in SE(3)

The control of free-floating robots requires dealing with several challenges. The motion of such robots evolves on a continuous manifold described by the Special Euclidean Group of dimension 3, known as SE(3). Methods from finite horizon Linear Quadratic Regulators (LQR) control have gained recent traction in the robotics community. However, such approaches are inherently solving an unconstrained optimization problem and hence are unable to respect the manifold constraints imposed by the group structure of SE(3). This may lead to small errors, singularity problems and double cover issues depending on the choice of coordinates to model the floating base motion. In this paper, we propose the use of canonical exponential coordinates of SE(3) and the associated Exponential map along with its differentials to embed this structure in the theory of finite horizon LQR controllers.

Screw and Lie Group Theory in Multibody Dynamics -- Recursive Algorithms and Equations of Motion of Tree-Topology Systems

Screw and Lie group theory allows for user-friendly modeling of multibody systems (MBS) while at the same they give rise to computationally efficient recursive algorithms. The inherent frame invariance of such formulations allows for use of arbitrary reference frames within the kinematics modeling (rather than obeying modeling conventions such as the Denavit-Hartenberg convention) and to avoid introduction of joint frames. The computational efficiency is owed to a representation of twists, accelerations, and wrenches that minimizes the computational effort. This can be directly carried over to dynamics formulations. In this paper recursive $O\left( n\right) $ Newton-Euler algorithms are derived for the four most frequently used representations of twists, and their specific features are discussed. These formulations are related to the corresponding algorithms that were presented in the literature. The MBS motion equations are derived in closed form using the Lie group formulation. One are the so-called 'Euler-Jourdain' or 'projection' equations, of which Kane's equations are a special case, and the other are the Lagrange equations. The recursive kinematics formulations are readily extended to higher orders in order to compute derivatives of the motions equations. To this end, recursive formulations for the acceleration and jerk are derived. It is briefly discussed how this can be employed for derivation of the linearized motion equations and their time derivatives. The geometric modeling allows for direct application of Lie group integration methods, which is briefly discussed.

Screw and Lie Group Theory in Multibody Kinematics -- Motion Representation and Recursive Kinematics of Tree-Topology Systems

After three decades of computational multibody system (MBS) dynamics, current research is centered at the development of compact and user friendly yet computationally efficient formulations for the analysis of complex MBS. The key to this is a holistic geometric approach to the kinematics modeling observing that the general motion of rigid bodies as well as the relative motion due to technical joints are screw motions. Moreover, screw theory provides the geometric setting and Lie group theory the analytic foundation for an intuitive and compact MBS modeling. The inherent frame invariance of this modeling approach gives rise to very efficient recursive $O\left( n\right) $ algorithms, for which the so-called 'spatial operator algebra' is one example, and allows for use of readily available geometric data. In this paper three variants for describing the configuration of tree-topology MBS in terms of relative coordinates, i.e. joint variables, are presented: the standard formulation using body-fixed joint frames, a formulation without joint frames, and a formulation without either joint or body-fixed reference frames. This allows for describing the MBS kinematics without introducing joint reference frames and therewith rendering the use of restrictive modeling convention, such as Denavit-Hartenberg parameters, redundant. Four different definitions of twists are recalled and the corresponding recursive expressions are derived. The corresponding Jacobians and their factorization are derived. The aim of this paper is to motivate the use of Lie group modeling and to provide a review of the different formulations for the kinematics of tree-topology MBS in terms of relative (joint) coordinates from the unifying perspective of screw and Lie group theory.

Hamel's Equations and Geometric Mechanics of Constrained and Floating Multibody and Space Systems

Modern geometric approaches to analytical mechanics rest on a bundle structure of the configuration space. The connection on this bundle allows for an intrinsic splitting of the reduced Euler-Lagrange equations. Hamel's equations, on the other hand, provide a universal approach to non-holonomic mechanics in local coordinates. The link between Hamel's formulation and geometric approaches in local coordinates has not been discussed sufficiently. The reduced Euler-Lagrange equations as well as the curvature of the connection, are derived with Hamel's original formalism. Intrinsic splitting into Euler-Lagrange and Euler-Poincare equations, and inertial decoupling is achieved by means of the locked velocity. Various aspects of this method are discussed.

Review of the Exponential and Cayley Map on SE as relevant for Lie Group Integration of the Generalized Poisson Equation and Flexible Multibody Systems

The exponential and Cayley map on SE(3) are the prevailing coordinate maps used in Lie group integration schemes for rigid body and flexible body systems. Such geometric integrators are the Munthe-Kaas and generalized-alpha schemes, which involve the differential and its directional derivative of the respective coordinate map. Relevant closed form expressions, which were reported over the last two decades, are scattered in the literature, and some are reported without proof. This paper provides a reference summarizing all relevant closed form relations along with the relevant proofs. including the right-trivialized differential of the exponential and Cayley map and their directional derivatives (resembling the Hessian). The latter gives rise to an implicit generalized-alpha scheme for rigid/flexible multibody systems in terms of the Cayley map with improved computational efficiency.