On-line Mobile Robotic Dynamic Modeling Using Integrated .
On-line Mobile Robotic Dynamic Modeling usingIntegrated Perturbative DynamicsForrest Rogers-MarcovitzCMU-RI-TR-10-15Submitted in partial fulfillment of therequirements for the degree ofMaster of Science in Robotics.The Robotics InstituteCarnegie Mellon UniversityPittsburgh, Pennsylvania 15213April 2010Thesis CommitteeAlonzo Kelly, ChairThomas Howard, Jet Propulsion LaboratoryMihail PivtoraikoCopyright c 2010 by Forrest Rogers-Marcovitz. All rights reserved.
AbstractMobile robotic dynamics modeling is necessary for reliable planning and control of unmannedground vehicles on rough terrain. Autonomous vehicle research has continuously demonstrated thata platform’s precise understanding of its own mobility is a key ingredient of competent machineswith high performance. I will investigate the feasibility and mechanism of enabling a platform tobetter predict its own mobility by learning from its own experience. The autonomy system willcalibrate, in real-time, vehicle dynamics models, based on residual differences between the motionoriginally predicted by the platform and the motion ultimately experienced by the platform.This thesis develops an integrated perturbative dynamics method for real-time identification ofwheel-terrain interaction models for enhanced autonomous vehicle mobility. I develop perturbativedynamics model which predict vehicle slip rates. The slip rates are first learned for steady stateconditions and interpolated to slip rate surfaces. An Extended Kalman Filter uses the residualpose differences for on-line identification of the perturbative parameters on a six wheel, skid steeredvehicle. An order of magnitude change in relative pose prediction was observed on loose and muddygravel.
AcknowledgmentsMy advisor Al Kelly helped me develop much of the math and vehicle dynamics in this paper.His encyclopedic understanding of vehicle dynamics was critical in pushing me to understand everyaspect of the algorithms while never simplifying the details. My thesis committee members, ThomasHoward and Mihail Pivtoraiko, were diligent and gave constructive criticism.The engineers and researchers at NREC have been a great supportive team. I’d like to singleout the USDA positioning team - Michael George, Jean-Philippe Tardif, and Michel Laverne - forhelping me on prior projects. The SACR team - Jason Ziglar, Nicholas Chan, and Rob Meyers helped collect the vehicle data and were always willing to help, even though I had a bad tendencyto break their code.The teachers and students of the Robotics Institute challenged me and pushed me. It is a greatenvironment to tackle the many difficult and complex problems involved with robotics as they havefor the past thirty years.Long before I started at Carnegie Mellon, my family has support my educational curiosities be they robotics, spacecrafts, or dance. For that I am very grateful.i
Contents1 Introduction11.1VGMI Project. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .11.2Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .22 Related Work32.1Autonomous Vehicles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .32.2Dynamic Modeling in Other Robotic Applications . . . . . . . . . . . . . . . . . . .32.3Skid-Steered Vehicle Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .42.4Predictive Models for Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .63 Perturbative Dynamics Modeling73.1Systematic Error Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .73.2Linearized Systematic Perturbation Dynamics . . . . . . . . . . . . . . . . . . . . . .83.3Non-linear Systematic Perturbation Dynamics . . . . . . . . . . . . . . . . . . . . . .94 Steady State104.1Path Segmentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.2Slip Rate Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104.3Uncertainty Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124.4Experimental Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.54.4.1Experimental Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.4.2Vehicle Platform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 Slip Rate Surfaces5.1Least Square Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205.1.15.2Least Squares Regression Results . . . . . . . . . . . . . . . . . . . . . . . . . 21Gaussian Process Regression5.2.15.320. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Gaussian Process Regression Results . . . . . . . . . . . . . . . . . . . . . . . 22Bayes Linear Regression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 235.3.1Bayes Linear Regression Results . . . . . . . . . . . . . . . . . . . . . . . . . 24ii
6 Transient Dynamics266.1Vehicle Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.2Gradient Descent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266.2.16.3Extended Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.3.16.4Gradient Descent Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Extended Kalman Filter Results . . . . . . . . . . . . . . . . . . . . . . . . . 31Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327 Conclusions348 Future Work358.1Model Improvements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358.1.1Pose Extended Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . 358.2Data Gathering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368.3Incorporation of Perception and Terrain Prediction . . . . . . . . . . . . . . . . . . . 368.4Motion Planning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37iii
List of Figures2.1Bandit & Akoya . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .43.1Vehicle Dynamic Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .74.1Path Segments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114.2Optimized Slip Curves from Path Segments . . . . . . . . . . . . . . . . . . . . . . . 114.3Optimized Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154.4Generated Path Segments via Learned Uncertainty . . . . . . . . . . . . . . . . . . . 154.5Terminal spread due to initial heading errors . . . . . . . . . . . . . . . . . . . . . . 164.6Terminal spread due to process noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 164.7Experiment Commands . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174.8Land Tamer Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184.9Optimized Angular Slip Rates for Multiple Curvatures . . . . . . . . . . . . . . . . . 184.10 Optimized Translational Slip Rates for Multiple Curvatures . . . . . . . . . . . . . . 195.1Slip Surfaces learned via Least Square Regression . . . . . . . . . . . . . . . . . . . . 225.2Slip Surfaces learned via Gaussian Process Regression . . . . . . . . . . . . . . . . . 245.3Slip Surfaces learned via Bayes Linear Regression . . . . . . . . . . . . . . . . . . . . 256.1Simulated Vehicle Path Compared to Ideal Path . . . . . . . . . . . . . . . . . . . . 276.2Simulated Vehicle Commands and Perturbations . . . . . . . . . . . . . . . . . . . . 276.3Relative Pose Residuals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 296.4Parameter Convergence during Gradient Descent . . . . . . . . . . . . . . . . . . . . 306.5EKF Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316.6Uncertainty Matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326.7Slip Surfaces learned via EKF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338.1Available Platforms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36iv
List of Tables5.1Gaussian Process Regression Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 236.1Slip Parameters Extended Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . 316.2Transient Relative Pose Error Comparison . . . . . . . . . . . . . . . . . . . . . . . . 32v
Chapter 1IntroductionMobile robotic dynamic modeling is necessary for reliable planning and control of unmanned groundvehicles (UGVs) on rough terrain. Autonomous vehicle research has continuously demonstratedthat a platforms precise understanding of its own mobility is a key ingredient of competent machineswith high performance. Since ground vehicles are propelled over the earth by the two forcesof gravity and traction, agile autonomous mobility relies fundamentally on understanding andexploiting the interactions between the terrain and tractive devices like wheels and tracks.I will investigate the feasibility and mechanism of enabling a platform to better predict its ownmobility by learning from its own experience. It will calibrate vehicle models, in real-time, basedon residual differences between the motion originally predicted by the platform and the motionultimately experienced by the platform. Although the model does not consider the direct forces,it provides an accurate model of the underlying dynamics. This thesis develops an integratedperturbative dynamics model for real-time identification of wheel-terrain interaction that enhanceautonomous vehicle mobility.1.1VGMI ProjectThe work presented in this thesis is part of the larger Vehicle Ground Model Identification (VGMI)project to improve the accuracy of predictive models which are the key enabler for high performanceunmanned ground vehicles. The proposed work will: Gather data sets specifically targeted to on-line calibration of unmanned ground vehicle motion prediction algorithms Investigate numerous approaches for representing the terrain properties and the vehicle dynamics Elaborate the merits of each approach Fit systematic and stochastic models to the data off-line1
CHAPTER 1. INTRODUCTION2 Develop methods for real-time calibration of the models while a vehicle operates Evaluate the merits of the identification system in full-up UGV experimentsThe main objective of the work is to find a way to calibrate the specialized faster-than-realtime models which are ubiquitous in unmanned ground vehicles. Such models are at one extremeof the spectrum where fidelity is relatively low and speed is relatively high. Adequate fidelity isdefined as the capacity to adequately predict motion for the purposes of model predictive controland sequential motion planning.On-line calibration (identification) techniques are necessary because one of the most influentialaspects of wheeled vehicle motion prediction is wheel-terrain interaction and such interactions varycontinuously on time scales far shorter than typical missions. Finally, the work will pursue theobjective of investigating the use of terrain perception information in order to impart the modelingsystem with both associative memory and perceptive prediction of the properties of terrain in view.This technique is intended to better enable systems to adapt to terrain transitions before drivingover them.The project will use simulation, analysis, controlled experimentation and evaluative field testingto accomplish these goals. Simulation will be used to generate ground truth data during algorithmdevelopment and will be used for Monte Carlo studies. Analysis will be conducted on large datasets to assess the performance and robustness of candidate approaches. Controlled experimentswill be conducted on several available robotic platforms to gather data under specific conditions.Field testing will demonstrate the value of the calibrated predictive models in improving UnmannedGround Vehicle performance.1.2Thesis OverviewThe following chapters develop the vehicle integrated perturbative dynamics model and show resultsfrom vehicle testing. Chapter 2 of this paper discusses previous work in mobile robotics relatedto off road vehicle modeling and estimation of slip parameters. The vehicle perturbative dynamicsmodel is developed in Chapter 3 which describes how the slip rates affect the vehicles motion onrough terrain. This model is used to learn steady state perturbative parameters, along with theuncertainty, of a skid-steered vehicle in Chapter 4. Next, Chapter 5 expands these the optimizedperturbative parameters to general surfaces interpolated for new vehicle states; three regressiontechniques are applied. In Chapter 6, I develop and test on-line filters to learn the slip rate surfaceparameters during general driving conditions with transient dynamics. The paper will concludewith an assessment of the lessons and observations that can be drawn from this work in Chapter 7and a discussion of further areas of investigation for the future of the VGMI project in Chapter 8.
Chapter 2Related WorkAutonomous vehicles operate in complex environments including field robotics, exploration robotics,agricultural and mining systems, autonomous automobiles, and mobile manipulators. All of theseapplications have vehicle models used for motion planning and navigation, yet very few of themexplicitly handle perturbations such as slip. Of the slip based modeling, none are based off residualpose error based on integrated perturbative dynamics.2.1Autonomous VehiclesAutonomous vehicles in city settings have recently become a reality as show by Boss, the autonomous car that on the 2007 DARPA Urban Challenge [28]. Recent advances in robotic groundvehicles have led to the autonomous traversal of increasingly complex terrain at higher rates ofspeed. With these advances, the motion of a vehicle becomes progressively more difficult to predictas wheel slip slip increases due to larger momentum. Solving this problem is essential for selectingcommands to avoid collisions with obstacles.Examples of ground vehicles using vehicle models include agricultural tractors [26] and theCrusher platform [4]. Unfortunately, these models do not normally take into account realisticterrain interactions that can adapt to terrain changes. These models are characterized by thecoefficient of rolling resistance, the coefficient of friction, and the shear deformation modulus, whichhave terrain-dependent values. Information is not well known or measurable such as coefficient offriction in braking or maximum shear strength in aggressive turns. This causes the vehicle to skidand slip over the terrain. Tire to terrain interaction must be taken into account during motionprediction to know the set of feasible motions and reachable states [15].2.2Dynamic Modeling in Other Robotic ApplicationsDynamic modeling and system identification has been applied to a number of robotic systems thatare not ground vehicles. An acceleration based parameterization has been used for learning helicopter dynamics [1]. An kinematic implicit loop method was used to calibrate robotic mechanisms3
CHAPTER 2. RELATED WORK4using displacement measurements [32]; at heart, this is simply a least-square fit weighted to thevariances of the sources of uncertainty. Gaussian Process Regression has been combined with anUnscented Kalman Filter for control of an autonomous blimp [17] [18].In space, on-line dynamic modeling has been developed forproximity operations. Very small spacecraft are well-suited forproximity activities and are of growing interest in the aerospacecommunity. However, due to size and power constraints, smallvehicles cannot carry traditional precision navigation systems andgenerally have noisy sensor and actuator options. Developed atWashington University, the Bandit inspector vehicle docks withthe parent Akoya spacecraft, Figure 2.1. Bayes Linear Regressionmodels the time-varying thruster dynamics for the inspector spacecraft [23]. The BLR has replaced the noisy accelerometer as inputto the Unscented Kalman Filter, improving performance [3].2.3Figure 2.1: Bandit & AkoyaSkid-Steered Vehicle ModelingThe basic kinematic and dynamic equations of motion have been developed for skid-steered vehiclesfor flat surfaces with little or no friction or slip [19]. Using the rotational transformation, we canderive the world-frame kinematic equation from the body-frame longitudinal speed, vx , lateralspeed, vy , and angular speed, ω. Ẋcos θ sin θ 0 vx q̇ Ẏ sin θ cos θ 0 vy θ̇001ω(2.1)The left and right wheel speeds, ωL and ωR , control the longitudinal and angular speeds.vx rωL ωR ωL ωR, ω r,22c(2.2)while r is the effective wheel radius, and 2c is the spacing between wheel tracks. The lateral speeddepends on the coordinate of instantaneous center of rotation (IRC), xIRC . These equation describethe nonholonomic constraints of the system.vy xICR ω 0(2.3)In order to simplify the mathematical model of the skid-steered vehicle, the following assumptions are made in the previously mentioned paper: plane motion is considered only, achievable linear and angular velocities of the robot are relatively small,
CHAPTER 2. RELATED WORK5 wheel contacts with surface at geometrical point (tire deformation is neglected), vertical forces acting on wheels are statically dependent on weight of the vehicle, viscous friction phenomenon is assumed to be negligible,Other algorithms have been developed to learn soil parameters given wheel-terrain dynamicmodels [24]. The field of terra-mechanics identifies five primary material properties (soil cohesion,internal friction angle, sinkage coefficient, shear deformation modulus and the maximum shearbefore soil failure) which influence the forces generated by the terrain and in turn the resultingvehicle motion. Analytical models exist for steering maneuvers of planetary rovers on loose soil[12].Related work takes into account the shear stress-shear displacement relationship on the trackground interface assuming firm ground with minimal sinkage [33]. With track sinkage and bulldozing effect being neglected, the resultant shear stress, τ , on the track-ground interface is assumedto be an exponential function of shear displacement, j,τ τmax (1 e j/K )(2.4)where τmax is the maximum shear stress between the track and the ground, and K is the sheardeformation modulus.Another method is to replace all the unknown soil parameters with slip ratios, iL and iR , anda slip angle, α. An Extended Kalman Filter (EKF) or a Sliding Mode Observer (SMO) can bedeveloped to learn these parameters [35] [20] [25]. Sliding Mode Observer controllers have beenproven to be robust enough to ignore the knowledge of the forces within the wheel-soil interaction,in the presence of benign sliding phenomena and ground level fluctuations [20] [21].Ẋ vx (cos θ sin θ tan α) vx (sin θ sin θ tan α)vxωL (1 iL )vxωR (1 iR )θ̇ r rccccẎ(2.5)(2.6)(2.7)Another technique models resistive wheel torques as functions of terrain properties and vehiclestate. The terrain properties can be learned by placing current monitors on the wheels to calculatewheel torques instead of measuring forces [34]. This model can be extended to generate a powermodel of the vehicle [7].Model-based approaches have been applied to estimate longitudinal wheel slip and detect immobilized conditions of mobile robots [31]. In this work, a explicitly differentiable traction/breakingmodel is used - expressed as a function of the wheel’s relative velocity, rather than slip which mayhave singularities. The simplified model includes traction and rolling resistive forces.Ftraction N (sign(
Mobile robotic dynamic modeling is necessary for reliable planning and control of unmanned ground vehicles (UGVs) on rough terrain. Autonomous vehicle research has continuously demonstrated that a platforms precise understanding of its own mobility is a key ingredient of competent machines with high performance.
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