Welcome to kdFlex’s documentation!

Contents:

• Chatbot
• Quick Start
  • Installing
• Diving In
  • 1. Dipping toes
  • 2. Free swim
  • 3. Deep dive
• Reference Guide
  • Basics
  • Frames Layer
  • Multibody Layer
  • Simulation Layer
  • Geometry Layer
  • Simulation Core
• Examples
  • Python Tutorial Notebooks
• Program skeletons
  • Serial-chain/Tree multibody system dynamics simulation
  • Closed-chain multibody dynamics simulation
  • Contact/Collision dynamics simulation
  • Closed-loop simulations
  • Robotics system analysis
  • Embedded model manager in control stack
  • Training simulator
  • Flexible body dynamics simulation
• Recipes
  • Frames layer
  • Rigid multibody models
  • Using Bilateral Closure Constraints
  • Working with sub-trees
  • Working with sub-graphs
  • Flexible body dynamics
  • Kinematics
  • Inverse kinematics
  • State Propagation
  • Data logging
  • Using events
  • Plotting
  • Graphics visualization
• API
  • kdFlex Python API
  • kdFlex C++ API

Indices and tables

• Index

• Module Index

• Search Page

Concepts

kdFlex is a fast, hi-fidelity, rigid and flexible body multibody dynamics modeling and simulation toolkit. kdFlex is architected to serve as a comprehensive solution for diverse modeling needs—spanning multiple engineering disciplines, domains, projects, and applications. kdFlex is built on the Spatial Operator Algebra (SOA) cutting-edge, high-performance, computational dynamics techniques. SOA is mature, and has been used across multiple mission critical applications.

kdFlex key themes

Core Methodology: Minimal Coordinates and Spatial Operator Algebra (SOA)

• kdFlex primarily employs a minimal coordinates approach, aiming to avoid the complexities associated with explicit constraints and DAE integration methods found in non-minimal coordinate approaches. The software is fundamentally based on the Spatial Operator Algebra (SOA) methodology. SOA leverages mathematical operator techniques to exploit the inherent structure of minimal coordinate models, leading to the development of “low-cost recursive computational algorithms.”

• SOA enables the derivation of “analytical operator expressions for the factorization of the mass matrix and its inverse for arbitrary tree multibody systems.” These expressions are the foundation for efficient O(N) (linear scaling with the number of bodies) recursive methods for solving equations of motion. More information on the SOA methodology can be found in the SOA book, the SOA lecture series and SOA publications.

Handling Complex System Topologies

• Serial/Tree Rigid/Flexible Systems: kdFlex implements the O(N) articulated-body inertia (ABI) recursive method for solving the equations of motion for tree-topology systems. A key advantage highlighted is that “SOA shows that these structure-based recursive methods for rigid multibody systems directly extend to flexible bodies as well.” kdFlex supports both rigid and flexible body dynamics using assumed mode, small deformation models for flexible bodies.

• Closed-Chain Topology Systems: Recursive methods are not directly applicable to systems with loops. kdFlex offers two approaches:

• Tree-Augmented (TA) Method: This non-minimal coordinate approach introduces loop-cuts to create a spanning tree, applies O(N) methods to the spanning tree, and then corrects the solution for the loop-cuts using a SOA-based low-cost method for computing operational space inertias.

• Constraint Embedding (CE) Technique: This is a SOA-based minimal coordinate method that “transforms the graph-topology systems into a minimal coordinate tree-topology system using compound, variable-geometry bodies.” This transformation is structure-preserving, allowing the use of O(N) ABI recursive methods and ODE integrators even for closed-loop systems. Both CE and TA methods are available in kdFlex. (coming soon)

Dynamics Features and Capabilities

• Run-Time Configuration Changes: A significant feature of kdFlex is its ability to handle “run-time structural changes in the system topology such as from the attachment/detachment and addition/deletion of bodies.” This is crucial for applications involving deployment, manipulation, and model refinement. The structure-based SOA algorithms seamlessly adapt to these changes by simply following the new system topology.

• Contact/Collision Dynamics: kdFlex supports non-smooth dynamics. It includes “penalty-based methods” and “impulsive complementarity based solution methods that exploit minimal coordinate to reduce the size of the complementarity problem.” Both linear and non-linear complementarity solution algorithms are available.

• Broad Family of Computational Algorithms: Beyond standard forward dynamics, kdFlex offers:

• Jacobians: Methods to compute the Jacobian matrix mapping generalized velocities to spatial velocities of any set of frames.

• Hybrid Dynamics: An O(N) recursive algorithm where input can be a mix of generalized forces and prescribed generalized accelerations, allowing computation of unknown forces or accelerations. This algorithm can reduce to standard forward dynamics or inverse dynamics depending on the inputs.

• Operational Space Compliance Matrix (OSCM): A low-cost recursive algorithm (based on SOA) for computing the mass matrix reflected to a specific node, valuable for robotics control.

• Composite Body Properties: Methods for computing configuration-dependent spatial inertia, momentum, and kinetic energy for subsets of bodies.

• CM Frame: Automatically creates and tracks the center of mass (CM) location for any subgraph.

• Interbody Forces: Provides an “inexpensive expression for computing this value for any pair of connected bodies” using the SOA algorithm, even though the standard ABI forward dynamics doesn’t require it.

• Pruning Bodies: Allows users to reduce computational cost by freezing and coalescing rigidly connected bodies. (Coming soon)

• DCA Algorithm: Includes an implementation of the divide-and-conquer algorithm for forward dynamics, suitable for parallelization with large models (e.g., molecular dynamics). (Coming soon)

• Linearization: Provides methods for automatically computing and exporting linearized dynamics models, useful for control system design.

• Statistical Dynamics: Supports statistical properties for molecular dynamics simulations, including a generalized equipartition principle (using SOA) and the use of a Fixman potential to correct statistical biases (also using SOA).

Simulation Architecture and Design

• Frames Layer: A general-purpose layer supporting “on demand computation of relative pose, velocity and accelerations for any pair of frames.” Values are computed only when requested and are cached for efficiency. This layer supports run-time creation, deletion, and reattachment of frames.

• Multibody Elements: Bodies, connector hinges, and nodes are represented as distinct C++ objects. Body and node classes are derived from a base frame class. Custom hinge types can be created, and bilateral loop constraints are defined via specific hinges.

• Subgraphs: While computations are typically for the full system, SOA algorithms can be applied to “connected subgraphs of bodies.” The full system is considered a subgraph. This distinction is important, particularly for the CE method for closed-loop systems.

• Python Interface: A comprehensive Python interface is auto-generated. This allows users to “conveniently set up simulations using Python scripts without losing the speed benefits from the compiled C++ layer.” It provides flexibility for model definition, configuration, and run-time interaction.

• Dynamics Solvers and Integrators: kdFlex includes solver C++ classes to ensure compatibility between selected solution methods (e.g., O(N) ABI, TA) and appropriate numerical integrators (ODE integrators for minimal coordinate methods, DAE solvers or constraint error management for non-minimal coordinate methods).

• Visualization: Includes 3D graphics visualization. A built-in capability generates a stick figure model for debugging, and users can attach primitive geometry or CAD parts for richer representation. Different graphics engines can be used.

• Other Features: Includes nonlinear solvers for loop constraints and state initialization, and GUIs for model browsing, adjustment, and interactive animation.

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Key Questions

What is distinctive and unique about kdFlex?

• Short answer: kdFlex uses advanced algorithms, which are known to be the lowest cost and most robust techniques available for multi-body dynamics. These algorithms known as minimal coordinate techniques, are notoriously complex, and the few implementations that exist until now have narrow and very limited capabilities. kdFlex employs novel mathematical techniques from the Spatial Operator Algebra (SOA) (add link) to drive for improved efficiency as model complexity increases.

• Long version: kdFlex employs minimal coordinate techniques, which offer several key advantages.

  • Firstly, by using minimal coordinates, (in contrast to redundant absolute coordinates), these methods avoid the need for managing constraints and constraint errors. They also reduce the size of the equations of motion that need to be solved.

  • Additionally, the use of minimal coordinates also allows the use of standard and significantly simpler ordinary differential equations for propagating system state. This is in contrast with a need for complex Differential Algebraic Equation solvers called DAE methods which propagate the state under algebraic constraints. The DAE methods require special purpose complex solvers, and the use of much smaller time steps for robust propagation. The minimal coordinate techniques on the other hand completely avoid the need for constraint error management.

  • Going beyond this, the minimal coordinate techniques have been shown to support what are referred to as order(N) recursive algorithms. So, while absolute coordinate techniques require methods whose cost goes up cubically with the number of bodies , which can grow expensive very quickly, minimal coordinate technique’s exact algorithms cost only goes up linearly with the number of bodies. This is an extraordinary benefit in cost for even moderate to large size systems.

  • The combination of these benefits makes the use of minimal coordinates a far more powerful approach. There have also been studies showing that the minimal coordinate techniques are also numerically more robust and accurate.

Why are kdFlex’s recursive methods not used more widely?

• Great question! Dynamics models are complex, and minimal coordinated models ever more so. Conventional methods for minimal coordinate dynamics struggle with even modest size and are limited to simple topologies and narrow (e.g., rigid body) capabilities. kdFlex overcomes the complexity of minimal coordinate dynamics by using the powerful Spatial Operator Algebra (SOA) methodology. SOA provides a mathematical framework and rich vocabulary for exploiting the rich underlying structure of minimal coordinate dynamics for analysis and the development of fast and recursive low cost, general purpose computational algorithms for rigid and flexible multibody dynamics The structure-based SOA algorithms also make it easy to easily accommodate run-time changes to system configuration from the addition/deletion of bodies or their reattachment.

kdFlex seems new – how can I trust it?

• kdFlex is a new and modern software product, but one based upon the proven and mature SOA methodology used over multiple decades for a variety of NASA and other mission critical applications including

  • NASA’s Cassini mission to Saturn, the Mars Pathfinder mission, and more recently for the Mars lander missions such as Mars Science Lab, Mars 2020, Mars Phoenix and Insight, the development to flight of the Mars Ingenuity helicopter,

  • the NASA Artemis I day of launch flight simulations ,and for ground vehicle simulations including platforms for DOD and planetary rovers for NASA. SOA has been used for several robotics research and technology development projects. These have included robotics manipulators, multi-limb robotic systems, and autonomous sampling systems. The SOA methods have also been used to develop advanced novel methods for protein folding simulation of large bio-molecular proteins under NIH research grants.

What is the ROI with kdFlex?

• Dynamics modeling is a complicated and expensive process requiring specialized expertise and tools used in engineering design process, for early concept development, to engineering design, to software development, V&V and operation of platforms in the field. Organizations often end up with expensive and fragmented solutions and and a plethora of tools and processes to meet such a variety of dynamics modeling needs. kdFlex is designed to provide a one-stop solution to enable the consolidation and streamlining of dynamics modeling across such life-cycles. kdFlex supports an extensive API for high fidelity flexible body models needs for loads and structures analysis as well as for guidance and control design and stability performance analysis, the speed and fidelity for closed-loop simulations including hardware in the loop, the versatility for robotics applications to handle ever-changing system constraints and topologies, an API to support the tailoring of system level models for multiple domain needs, and support for functionality well beyond solving the equations.

How usable is kdFlex?

• The kdFlex design is based on our belief (to borrow a phrase) that an informed customer is our best customer. Towards this, kdFlex includes a well structured, and feature complete API meant to allow a user to adapt and tailor its use as an SDK for the modeling and simulation application at hand. The APIs are available at the C++ level (for computational heavy lifting), and mirrored at the Python level (for prototyping and scripting). Undoubtedly there will be a familiarization phase, but one that will empower a simulation engineer to use the toolkit far more effectively than GUIs - which tend to obfuscate rather than illuminate the powerful capabilities. Having said that, web-based 3D graphics support is built in to allow visualization and inspection of the model.

• Furthermore, the kdFlex libraries are object oriented, with a family of objects for frames, bodies, hinges, subhinges, constraints etc, each with a well defined functionality and design. The objects serve a convenient way to interface with external user software, while also providing the benefits of information hiding and encapsulation that avoid fragile and error prone bookkeeping of the large amount of data within the simulation. For instance, adding or removing a body, or changing a hinge type and the number of degrees of freedom is trivial with no significantly impact on interface code such as is common in many applications.

• The built-in general-purpose frames layer allows users to seamlessly make arbitrary relative pose, velocity and acceleration queries across the system. This does away with the burden on users to develop custom code - something that should be straight forward, but is notoriously error prone.

• There is built in support for tagging input deck parameters with units and quantity types, to allow the use of such data from multiple sources as is a common. Avoiding the need for the user to carry out conversions avoids errors and maintain traceability.

• There are built in important simulation services for 3D graphics visualization (web based), model importers, data logging, plotting, controlling message verbosity, exporting input decks in multiple formats etc.

• There are architectural features that identify not ready parameters, improper clean up during configuration changes etc - that catch errors sooner than later - essential for developing and maintaining large scale, industrial strength complex simulations!

Why is flexible body dynamics important?

• Rigid bodies do not exist in reality, but the notion of idealized rigid bodies is extremely useful. However, there are situations where the deformation of bodies is very important, such as for loads and stress analysis, and for handling control/structure interaction and developing controllers that handle servo-elastic, aero-elastic effects. The data for such deformation properties typically originates from finite-element modeling using structural analysis tools. Accurate modeling the nonlinear dynamics of multibody systems with deformable bodies requires the integration of FEM data, component mode synthesis, model reduction and correctly modeling of the rigid/elastic dynamics coupling. There are only a small handful of tools, that do the this rigorously, but none with the with the fast computational speed kdFlex’s fast SOA recursive methods and minimal coordinate formulation. kdFlex’s unique and unparalleled ability to provide an integrated solution that combines fast performance with accurate flexible multibody dynamics modeling provides a critical bridge between the structures and control design disciplines. This bridge addresses gaps in tool-chains, modeling solutions and workflows between these disciplines that have for long plagued these communities and allows the streamlining of design iterations and simplify workflow pipelines with enormous savings in time and cost.

Conventions

• no absolute positions, velocities, etc. - everything is relative

• Unit quaternions (see the Pose representations section)

• Spatial vectors (see the Spatial notation section)

• Inertia positive/negative integral convention (see the Spatial notation section)

• Q, U, Udot, T for subhinges, hinges etc (see the Subhinges section)

• oframe, pframe (see the Creating a frame container section)

• onode/pnode (see the Connecting bodies via hinges section)

• quantities/units

• subtree virtual root does not belong to the subtree

• use create/discard methods, not constructors/destructors

• use object level getQ() methods and not take apart the CoordData etc level methods (see the The CoordData container section)

• call ensureHealthy

• call allReady

• hop definition

• Ktime for time