Structure-Based Computational Modeling Architecture For .

Structure-Based Computational Modeling Architecture for RoboticsAbhinandan Jain1Abstract— We describe a computational architecture formeeting a diverse range of robot modeling needs encompassinganalysis, simulation and embedded modeling for robotic systems. The architecture builds upon the spatial operator algebratheoretical framework for computational dynamics. It allowsapplications to meet the broad range of computational modelingneeds coherently and with fast, structure-based computationalalgorithms. The paper describes the SOA computational architecture, the DARTS computational dynamics software, andapplication modeling layers.I. INTRODUCTIONComputational models pervade all aspects of robotics,from the design and analysis of systems, to use withinonboard planning and control architectures for system development and test, and during their operation. RecentDARPA programs such as Autonomous Robotics Manipulation (ARM) and the DARPA Robotics Challenge (DRC)are focused on increasing autonomous robot operation inunstructured environments. Autonomous manipulation andmobility requires increased sophistication and robustness,and this in turn increases modeling needs for manipulationand control, grasp design and analysis, task planning, leggedlocomotion, online calibration etc [1].Such robotics demands have stimulated the developmentof physics-based simulators to support development andtesting [2–5]. However significant challenges remain and theDRC program is investing in the development of simulatorsfor use by the robotics community. Challenges in computational modeling include the large variety of modelingrequirements, the complexity of the numerical algorithms,designing models that adequately capture the physics, obtaining correct parameters for seeding the models, and obtainingfast computational performance.The challenges are even greater for models used withinautonomy software. Such embedded models support task andmotion planning, state estimation in the presence of noise anduncertainty and real-time closed-loop control. Computationalspeed requirements are significantly higher and the types ofinformation needed span a much broader range. Thus whilea simulator is required to do one function well, i.e., simulatethe system state time history, embedded models have toperform several functions well.Due to their inherent complexity, embedded computationalmodels are often platform specifid, over-simplified or pointsolutions to meet a narrow a range of functions or tailoredto the needs of specific autonomy modules. Low-fidelity and1 A. Jain is with the Jet Propulsion Laboratory, California Instituteof Technology, 4800 Oak Grove Drive, Pasadena, CA 91109, USAAbhi.Jain@jpl.nasa.govfragmented models degrade the overall system performance.Fragmentation requires extra measures to ensure the consistency of modeling assumptions and expectations as well asto keep changing model parameters and data in sync acrossthe system modules. In addition, there is the added cost ofimplementing, testing and validating these special purposemodels.In this paper we describe a computational modeling architecture for robotic systems that has been designed to overcome these challenges for robotics analysis, simulations andembedded modeling. Our approach is to develop a highly capable modeling layer to serve as a foundation for the diverseand demanding needs of application layers. The architectureis illustrated in Figure 1. Its lowest layer is the mbedded modelsDARTS computationaldynamicsSOA theoreticalframeworkFig. 1. Overall robotics computational mechanics architecture built uponthe SOA theory, the DARTS software library for analysis, simulation andembedded modeling applications.operator algebra (SOA) theoretical framework for computational mechanics [6]. SOA provides mathematical tools forexpressing, analyzing and computing a very broad range ofrobot mechanics quantities. It’s expressiveness allows it tocoherently meet the large variety of mechanics modelingneeds for robot models. It uses spatial operators for theconcise mathematical description and analysis of dynamicsquantities, as well as the generation of fast, structure-basedcomputational algorithms. Section II provides an overviewof the SOA framework, and Section II-A describes theSOA processes for developing low-order, structure-basedcomputational algorithms.The next layer of the architecture is the DARTS computational dynamics C library whose design is based upon theSOA framework. DARTS provides methods for the computation of a broad range of robot modeling quantities and isdescribed in Section III. The DARTS library also includes anumber of dynamics solvers needed for time simulation ofrobot dynamics. These solvers utilize the underlying DARTSmethods and differ in the type of dynamics they handle, as

well as the algorithms they use. These solvers are describedin Section III-A.The robot modeling applications build upon the SOAand DARTS layers. In Section IV we discuss models forsupporting robotics analysis. In particular we detail thePyCraft toolkit for the development, testing and maturationof advanced computational dynamics techniques. Section Vturns to the important area of dynamics simulation. DARTSand its solvers form the heart of the Dshell simulation framework for the development of system level physics-basedsimulations. Dshell provides ways to organize and managethe large number of component models in these simulationsand to reuse these component and sub-system models acrosssimulations. Dshell has been adapted to develop rover andflight dynamics system simulations. Finally, in Section VIwe describe the RoboDarts embedded modeling layer thatis built upon DARTS. RoboDarts is designed for use byautonomy software to meet the diverse modeling needs forplanning, estimation and real-time control. Specific attentionis paid to performance speed, time variability of the tasksand environment and the robot platform’s interactions withthem.II. SOA THEORETICAL FRAMEWORKThe spatial operator algebra (SOA) theory and mathematical framework for multibody dynamics [6] has beendeveloped over two decades of research. SOA provides amathematical language for succinctly characterizing and analyzing complex dynamics quantities for articulated roboticssystems. It also provides a natural avenue for developing fast,structure-based computational dynamics algorithms. Thesefeatures make possible its use as a powerful architecture forthe computational mechanics of robotic systems.SOA makes use of minimal coordinate system representations. Thus inter-link hinge motion is parameterized by minimal set of coordinates instead of with redundant coordinatessubject to constraints. This reduces the size of the equationsof motion and avoids the need to manage constraint violationerrors required with redundant coordinates. A side-effect ofusing minimal coordinates is that the mass matrix, and othersystem matrices, are dense and configuration dependent andthus, more complex. However, these quantities have richstructure that the SOA provides the mathematics tools tohandle.The mapping between joint and link velocities serves as asimple example to introduce the notion of spatial operators.The 6-dimensional spatial velocity V(k) of the kth link depends on that of its parent link’s spatial velocity V(k 1) viathe relationship V(k) φ (k 1, k)V(k 1) H (k) θ̇ (k),where φ (k 1, k) denotes the rigid body propagation matrixfor the pair of links, H (k) characterizes the permissiblehinge motion across the connecting hinge and θ(k) the kthhinge coordinates. This component level relationship canbe converted into a system level relationship of the formV φ H θ̇ , where V and θ̇ are system level vectorsobtained by stacking up the vector contributions from eachlink. φ and H are block matrices whose elements are definedby the component φ (k 1, k) and H (k) link-level matrices.φ and H are examples of spatial operators.The concise V φ H θ̇ relationship reflects the richstructure of the φ operator. Indeed, for tree/serial topologysystems, φ (I Eφ ) 1 , where Eφ is another spatialoperator whose structure is closely related to the adjacencymatrix for the graph associated with the connection topologyof the linkages [7]. Eφ is always nilpotent for serial/treetopology systems.The mathematical structure of the spatial operators makepossible several transformations and simplifications of dynamics quantities. A seminal example of this is the systemmass matrix, M(θ). It has been shown to have the factoredform M(θ) HφMφ H , where M is a block-diagonalspatial operator with link spatial inertias. While this factorization involves non-square factors, the following sequenceof analytical spatial operator expressions with alternativefactorizations involving square factors. and an expression forits inverse, can be derived [8]:M HφMφ H [I HφK]D[I HφK] [I HφK] 1 I HψK(1)M 1 [I HψK] D 1 [I HψK]These expressions involve the additional ψ, D, and K spatialoperators described in [6]. The analytical expressions in Eq. 1are remarkable in that they hold generally for arbitrary sizeserial/tree topology systems. They are direct consequences ofthe intrinsic mathematical structure of the spatial operators.Indeed, the Eφ and φ operators are special instances of abroader family of spatial kernel operator (SKO) and spatialpropagation operator (SPO) operators [6]. The ψ operatoris also an instance of an SPO operator.It has been shown using graph theory ideas that spatialoperator expressions, such as in Eq. 1, generalize to thecase where the component bodies and hinges are flexible[6]. Remarkably, they hold even when the system graphis transformed into simpler graphs by aggregating groupsof bodies into new variable geometry compound bodies.This observation forms the basis of constraint embeddingtechniques that transform non-tree topology graphs into treegraphs, and thus extend the applicability of Eq. 1 to evennon-tree topology systems [6].Spatial operator analysis applies to a number of robotdynamics problems such as joint space dynamics, operationalspace dynamics, under-actuated systems, sensitivity analysis,diagonalized dynamics etc. The SOA provides a unified wayto tie together results and analysis obtained from disparateapproaches, and to understand the relationships among them.Also, the SOA has been used for several novel analyses andresults not possible by other means [6]. The rich structureof spatial operators arises, perhaps not coincidentally, fromthe close mathematical parallels with concepts and analysistechniques developed in the optimal estimation arena.