Reduced-Order Model With Foot Tipping Allowance for Legged Balancing

2021 ◽  
Author(s):  
William Z. Peng ◽  
Hyunjong Song ◽  
Joo H. Kim
Keyword(s):  
Trajectory Optimization ◽  
Inverted Pendulum ◽  
A Priori ◽  
Contact Forces ◽  
Phase Contact ◽  
Order Model ◽  
Reduced Order ◽  
Optimization Framework ◽  
Direct Collocation ◽  
Multi Phase

Abstract Tipping is an instrumental aspect of multi-phase contact situations that arise during common tasks such as the locomotion of legged systems. Despite its importance in balance recovery, tipping is often ignored in trajectory optimization due to the lack of existing methods that are able to actively plan and optimize for unspecified contacts. Trajectory Optimization based on nonlinear programming requires a priori knowledge about anticipated contact changes, such as their order and timing, in order to generate physically feasible motions. In this paper, an optimization framework with conditional constraints is established for direct collocation in trajectory optimization for legged balancing with foot tipping allowance. The proposed approach can evaluate the timing of contact phases without preplanned contact forces or sequences of events, which is not possible with conventional methods. This optimization framework is demonstrated by computing the balanced regions of two reduced-order models of a legged system, namely, inverted-pendulum-based models without and with a flywheel, and is verified with control simulations. The contribution of tipping to balance stability is quantified and compared to prior results obtained without tipping allowance. The framework presented can also be generalized to other multi-phase contact scenarios, such as rolling and sliding, where unspecified discontinuous changes in contact occur with important consequences in the performance of legged systems.

scholarly journals An Evolve-Then-Correct Reduced Order Model for Hidden Fluid Dynamics

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 570 ◽  
Author(s):  
Suraj Pawar ◽  
Shady E. Ahmed ◽  
Omer San ◽  
Adil Rasheed
Keyword(s):  
Process Model ◽  
Correction Term ◽  
A Priori ◽  
Basis Functions ◽  
Least Squares Regression ◽  
Physical Processes ◽  
Order Model ◽  
Reduced Order ◽  
Real Time Simulations ◽  
Proper Orthogonal

In this paper, we put forth an evolve-then-correct reduced order modeling approach that combines intrusive and nonintrusive models to take hidden physical processes into account. Specifically, we split the underlying dynamics into known and unknown components. In the known part, we first utilize an intrusive Galerkin method projected on a set of basis functions obtained by proper orthogonal decomposition. We then present two variants of correction formula based on the assumption that the observed data are a manifestation of all relevant processes. The first method uses a standard least-squares regression with a quadratic approximation and requires solving a rank-deficient linear system, while the second approach employs a recurrent neural network emulator to account for the correction term. We further enhance our approach by using an orthonormality conforming basis interpolation approach on a Grassmannian manifold to address off-design conditions. The proposed framework is illustrated here with the application of two-dimensional co-rotating vortex simulations under modeling uncertainty. The results demonstrate highly accurate predictions underlining the effectiveness of the evolve-then-correct approach toward real-time simulations, where the full process model is not known a priori.

scholarly journals Model Order Reduction for convection dominated problems

2018 ◽  
Author(s):  
Harshit Bansal ◽  
Laura Iapichino ◽  
Stephan Rave ◽  
Wil Schilders ◽  
Nathan van de Wouw
Keyword(s):  
Order Reduction ◽  
Benchmark Problems ◽  
Reduced Basis ◽  
Order Model ◽  
Model Order ◽  
Reduced Order ◽  
Non Linear ◽  
Order Representation ◽  
Multi Phase ◽  
Convection Dominated

Model Order Reduction (MOR) of systems of non-linear(parameterized) Hyperbolic Partial Differential Equations (PDEs) is still an uncharted territory in the scientific community. Moving discontinuities are representative features of this class of problems and pose a major hindrance to obtain effective reduced-order model representations, since typically bases with high spatial frequency are needed to accurately capture these moving discontinuities. We will discuss a MOR framework to efficiently capture the travelling dynamics of such systems. The motivation of this work is to enable the usage of multi-phase hydraulic models, such as the Drift Flux Model (DFM) [2] in developing drilling automation strategies for real-time down-hole pressure management. The DFM is a system of multiscale non-linear PDEs, whose convective subset is conditionally hyperbolic. Convection dominated problems, such as the DFM, admit solutions, which possess a diagonal structure in space-time diagram and high solution variability. As a first step, we apply standard MOR approaches [4] to obtain a reduced-order representation of the DFM for a representative multi-phase shock tube test case. We capture the dynamics in an essentially non-oscillatory manner but we obtain a small dimensionality reduction. Since the dimension of the reduced model is still too large, we develop new techniques for deriving more efficient alternative reduced-order models for this class of problems. We invoke the idea of the method of freezing [1] and combine it with non-linear reduced basis approximations [3] to develop an efficient reduced-order model representation, which we demonstrate for several benchmark problems. These benchmark problems embody the challenges faced in the reduced-order representation of the DFM. However, the existing MOR framework [3] lacks consideration of boundary conditions and multiple fronts. The main novelty of this work is in investigating the performance of combined approach of Method of Freezing and reduced basis approximations in dealing with merging (discontinuous) wave-fronts. Finally, we present numerical experiments and discuss the efficacy of above mentioned approach in terms of computational speed up and computational accuracy compared with standard numerical techniques.

scholarly journals Trajectory Optimization for Hybrid Wheeled-Legged Robots in Challenging Terrain

2020 ◽  
Author(s):  
Vivian Suzano Medeiros ◽  
Marco Antonio Meggiolaro
Keyword(s):  
Trajectory Optimization ◽  
Experimental Tests ◽  
Contact Forces ◽  
Legged Robots ◽  
Whole Body ◽  
Planning Problem ◽  
Average Speed ◽  
Optimization Framework ◽  
Promising Solution ◽  
Reference Trajectories

Wheeled-legged robots are a promising solution for agile locomotion in challenging terrain, combining the speed of the wheels with the ability of the legs to cope with unstructured environments. This paper presents a trajectory optimization framework that allows wheeled-legged robots to navigate in challenging terrain, e.g., steps, slopes, gaps, while negotiating these obstacles with dynamic motions. The framework generates the robot’s base motion as well as the wheels’ positions and contact forces along the trajectory, accounting for the terrain map and the dynamics of the robot. The knowledge of the terrain map allows the optimizer to generate feasible motions for obstacle negotiation in a dynamic manner, at higher speeds. To take full advantage of the hybrid nature of wheeled-legged robots, driving and stepping motions are both considered in a single planning problem that can generate trajectories with purely driving motions or hybrid driving-stepping motions. The optimization is formulated as a Nonlinear Programming Problem (NLP) employing a phase-based parametrization to optimize over the wheels’ motion and contact forces. The reference trajectories are tracked by a hierarchical whole-body controller that computes the torque actuation commands for the robot. The motion plans are verified on the quadrupedal robot ANYmal equipped with non-steerable torque-controlled wheels in simulations and experimental tests. Agile hybrid motions are demonstrated in simulations with discontinuous obstacles, such as floating steps and gaps, at an average speed of 0.75 m/s.

scholarly journals Time Stable Reduced Order Modeling by an Enhanced Reduced Order Basis of the Turbulent and Incompressible 3D Navier–Stokes Equations

2019 ◽  
Vol 24 (2) ◽  
pp. 45 ◽  
Author(s):  
Nissrine Akkari ◽  
Fabien Casenave ◽  
Vincent Moureau
Keyword(s):  
Stokes Equations ◽  
A Priori ◽  
Order Reduction ◽  
Reduced Order Modeling ◽  
Reduced Order Model ◽  
Navier Stokes ◽  
Reduced Basis ◽  
Order Model ◽  
Navier Stokes Equations ◽  
Reduced Order

In the following paper, we consider the problem of constructing a time stable reduced order model of the 3D turbulent and incompressible Navier–Stokes equations. The lack of stability associated with the order reduction methods of the Navier–Stokes equations is a well-known problem and, in general, it is very difficult to account for different scales of a turbulent flow in the same reduced space. To remedy this problem, we propose a new stabilization technique based on an a priori enrichment of the classical proper orthogonal decomposition (POD) modes with dissipative modes associated with the gradient of the velocity fields. The main idea is to be able to do an a priori analysis of different modes in order to arrange a POD basis in a different way, which is defined by the enforcement of the energetic dissipative modes within the first orders of the reduced order basis. This enables us to model the production and the dissipation of the turbulent kinetic energy (TKE) in a separate fashion within the high ranked new velocity modes, hence to ensure good stability of the reduced order model. We show the importance of this a priori enrichment of the reduced basis, on a typical aeronautical injector with Reynolds number of 45,000. We demonstrate the capacity of this order reduction technique to recover large scale features for very long integration times (25 ms in our case). Moreover, the reduced order modeling (ROM) exhibits periodic fluctuations with a period of 2 . 2 ms corresponding to the time scale of the precessing vortex core (PVC) associated with this test case. We will end this paper by giving some prospects on the use of this stable reduced model in order to perform time extrapolation, that could be a strategy to study the limit cycle of the PVC.

Reduced Order Model and Decoupled Control Architecture for Two Manipulators Holding a Rigid Object

1991 ◽  
Vol 113 (4) ◽  
pp. 646-654 ◽  
Author(s):  
A. J. Koivo ◽  
M. A. Unseren
Keyword(s):  
Inverse Dynamics ◽  
Equations Of Motion ◽  
Functional Relation ◽  
Contact Forces ◽  
Reduced Order Model ◽  
Dynamical Model ◽  
Control Architecture ◽  
Order Model ◽  
Rigid Object ◽  
Reduced Order

A dynamical model and a control architecture are developed for the closed chain motion of two N-joint manipulators holding a rigid object in a three-dimensional workspace. Dynamic and kinematic constraints are determined and combined with the equations of motion of the manipulators to obtain a dynamical model of the entire system in the joint space. Reduced order equations of motion and a functional relation for the generalized contact forces are derived. The problem of solving the reduced order model for the forward and inverse dynamics is discussed. Control laws are determined for the reduced order model so as to decouple the force and position (velocity) controlled degrees of freedom (DOF). A simulation example is presented to illustrate the approach.

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