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Formulation and Calibration of Fast, Accurate Vehicle Motion Models

Formulation and Calibration of Fast, Accurate Vehicle Motion Models. Thesis proposal Neal Seegmiller November 30, 2012. Outline. Terms: motion model WMR (wheeled mobile robot) model predictive control (MPC) model predictive planning (MPP). Problem Definition

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Formulation and Calibration of Fast, Accurate Vehicle Motion Models

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  1. Formulation and Calibration of Fast, Accurate Vehicle Motion Models Thesis proposal Neal Seegmiller November 30, 2012

  2. Outline • Terms: • motion model • WMR (wheeled mobile robot) • model predictive control (MPC) • model predictive planning (MPP) • Problem Definition • How should we produce motion models for MPP? • Progress to Date • Research Plan • Contributions to Robotics

  3. Prior work on Model Predictive Planning and Control

  4. Prior work on MPC & MPP cont. *where does the model come from?

  5. WMR motion models must be accurate planned actual • MPP requires accurate models to plan safe, feasible paths • Sometimes relying on feedback isn’t good enough!

  6. Fukushima nuclear power plant (models must be 3D)

  7. WMR motion models must be fast Receding horizon motion planning [Howard 2009] Lattice motion planning [Pivtoraiko 2009]

  8. Challenges to model generation Tradeoff between fidelity and speed in model formulation Difficulty of Calibration

  9. Challenge 1. Fidelity/speed tradeoff CarSim Adams/Car ROAMS [Jain 2003] High fidelity/slow Low fidelity/fast 2D Dubinscar http://planning.cs.uiuc.edu/node788.html

  10. General approaches to formulating models Related work on planning with moderate to high-fidelity models: • [Ishigami 2011] complete dynamic simulation of planetary rovers • [Iagnemma 2001] A* planning for planetary rovers • [Yu 2010] Skid-steered vehicles • [Howard 2009] multiple platforms A general approach by [Tarokh & McDermott 2005]. Extends the 2D approach by [Muir & Neuman1986]

  11. Challenge 2. Difficulty of calibration 1. Analyze subsystems in isolation Single-wheel testbed [Ishigami 2007] 2. Execute preprogrammed maneuvers UMBmark [Borenstein 1996] 3. Self-calibrate during normal operation Fast and easy odometery calibration [Kelly 2004]. See also [Antonelli 2005], [Martinelli 2007]

  12. What is desired? • Formulation requirements: • Account for 3D articulation (in a modular way) • Account for wheel slip • Account for powertrain dynamics • Predict the onset of extreme conditions • Be capable of simulation 100x real time (an order of magnitude faster than SOA) My hypothesis: Can be met by enhanced 3D velocity kinematic models. • Calibration requirements: • Run online during normal operation • Require only intermittent observations of pose • Be computationally tractable • Learn a model of non-systematic error (uncertainty) • Adapt quickly without overfitting. Can be met by IEE calibration method

  13. Outline • Problem Definition • Progress to Date • Formulation • Calibration • How I plan to address limitations • Research Plan • Contributions to Robotics

  14. A Vector algebra-based approach to velocity kinematics Applies to position and its derivatives (linear velocity, etc.) The Transport Theorem: (o)bject Notation (m)oving (f)ixed wrt frame f r: position v: velocity ω: angular velocity of frame m See [Kelly 2012] FSR paper Based on [Luh 1980] Recursive Newton-Euler Algorithm for manipulators

  15. The wheel equation (offset steering example) s c Velocity of wheel/terrain contact point Linear vehicle velocity Dimensions x v y Angular vehicle velocity Steering rate w Frames: (w)orld (v)ehicle (s)teer (c)ontact

  16. Inverse and Forward Kinematics • Inverse Kinematics • Use wheel equations directly! • Steer angle should align forward axis of wheel frame with velocity vector (not aligned) c • Forward Kinematics • Rearrange wheel equations into a linear system • Solve for vehicle velocity using pseudoinverse Use skew-symmetric matrices to represent cross products Vehicle frame velocity (4 wheel offset steering example)

  17. Video: 4 wheel offset steering example

  18. Formulating 3D motion prediction as the solution of a DAE Semi-explicit DAE consists of ODE + constraints: (constraints are provided by the wheel equations!) The ODE. x: state, u: inputs Holonomic constraints, e.g. terrain following Non-holonomic constraints, e.g. no wheel slip Unconstrained integration + non linear optimization Solve directly for constrained motion using DAE vs.

  19. 3D example, the Zoë rover 2 passive DOF for each axle. 4 independently driven wheels

  20. Zoë ramp experiment Similar accuracy to dynamics simulation but computationally cheaper! Physical Experiment Dynamics Simulation (2nd order) Kinematics Simulation (1st order) 2.5°

  21. Formulation Limitations Not modular How best to solve DAE? Only no-slip constraints supported Speed comparison not possible yet

  22. Integrated Equation Error (IEE) approach to Model Identification System Differential Equation System Integral vs. state inputs parameters IEE pro’s • No numerical differentiation • Compounded errors (a good thing!) • Optimize for chosen horizon directly IEE con’s • More computation (but tractable) • Must linearizean integral to compute Jacobian • Tricky to account for measurement uncertainty

  23. Linearized systematic & stochastic error dynamics [Stengel 1994] observations parameters systematic stochastic t t0

  24. IEE applied to vehicle model identification Suspension deflections are ignored Motion is restricted to a tangent plane The differential equation Body frame velocity consists of (n)ominal and (s)lip components Slip is parameterized over nominal velocity, centripetal acceleration, gravity (in C) y x See [Seegmiller 2011] ISRR paper

  25. Crusher at Camp Roberts Crusher 6 wheel skid-steer, active suspension Image captured by one of Crusher’s cameras Crusher traversed steep grassy slopes and a dirt road Path at Camp Roberts Roll: -28 to 29° Pitch: -22 to 17° Top speed: 6 m/s, 4 rad/s (commanded)

  26. Crusher at Camp Roberts Predicted path with slip calibration Prediction uncertainty (1σ, .683) Actual path (GPS) Predicted path using basic kinematic model VR VL x y assumes no slip! W (track width)

  27. Crusher at Camp Roberts, Systematic Calibration

  28. Crusher at Camp Roberts, Stochastic Calibration

  29. Calibrating powertrain dynamics using IEE angular acceleration (commanded) angular velocity See [Seegmiller 2012] ICRA paper time constant time delay LandTamer

  30. Calibration Method Limitations • Currently the method only supports: • Simplified models • Slip parameterization about body frame inputs • Insufficient experimental validation

  31. Related work on modeling wheel slip Rigid wheel on rigid terrain Use Coulomb friction model 2. Deformable wheel on rigid terrain Use an empirical model [Salaani 2007] 3. Rigid Wheel on deformable terrain Use a terramechanics-based model [Ishigami 2007]

  32. How to compute wheel reaction forces? 6 x 3n 6 x 1 [Iagnemma 2001] Use the force balance equation underdetermined for > 2 wheels! [Hung 1999] suggests choosing an objective function and formulating as a Linear or Quadratic Programming problem

  33. Related work on stability margins Predict rollover [Diaz-Calderon 2005] based on [Papadopoulos & Rey 1996] Predict loss of traction [Brach 2009]

  34. Related work on Rigid Body Dynamics Multilegged vehicles [McMillan & Orin 1998] [Featherstone 1987] The most computationally efficient rigid body dynamics algorithms were developed by roboticists Inverse vs. forward dynamics Maximal vs. generalized coordinates Various ways to handle contact between the wheels & ground

  35. Outline • Problem Definition • Progress to Date • Research Plan • Theoretical objectives • Experimental objectives • Schedule • Contributions to Robotics

  36. Research Plan, Theoretical objectives Formulate motion prediction as the solution of a DAE in a modular way Formulate and enforce slip constraints Incorporate a powertrain dynamics model Incorporate an extreme conditions predictor Calibrate enhanced 3D kinematic models using the IEE approach Deliverables to make my approach accessible: Software library Documentation of step-by-step approach

  37. Research Plan, Experimental objectives Quantitative comparison my proposed models with alternatives (accuracy and speed) Quantitative comparison of slip models Experimental verification of adaptability

  38. Research Plan, Resources Software resources for physics-based simulation: Open Dynamics Engine CarSim Available platforms: Zoë rover MiniCrusher

  39. Proposed Schedule

  40. Contributions to Robotics What is novel? An automated, modular approach to simulating WMR velocity kinematics (compare to [Tarokh & McDermott 2005]) Enhanced models: wheel slip, powertrain dynamics, extreme conditions Convenient self-calibration method Analysis of the trade offs between kinematic vs. dynamic models Why it’s useful: Broadly applicable, works for any WMR design Readily accessible (software library, documentation) A foundation for future research in planning, mobile manipulation, etc.

  41. References Antonelli, G., et al.: A calibration method for odometry of mobile robots based on the least-squares technique. IEEE Trans. Robot. 21(5), 994-1004 (2005) Borenstein, J., Feng, L.: Measurement and correction of systematic odometry errors in mobile robots. IEEE Trans. Robot. Autom. 12(6), 869-880 (1996) Brach, R.M., Brach, R.M.: Tire models for vehicle dynamic simulation and accident reconstruction. SAE Technical Paper 2009-01-0102 (2009) Diaz-Calderon, A., Kelly, A.: On-line stability margin and attitude estimation for dynamic articulating mobile robots. IJRR 24(10) (2005) Featherstone, R.: Robot dynamics algorithms. Kluwer, Boston/Dordrecht/Lancaster (1987) Howard, T.: Adaptive model-predictive motion planning for navigation in complex environments. Tech. Report, CMU-RI-TR-09-32 (2009) Iagnemma, K.: Rough-terrain mobile robot planning and control with application to planetary exploration. MIT Ph.D. Thesis (2001) Ishigami, G., et al.: Terramechanics-based model for steering maneuver of planetary exploration rovers on loose soil. JFR 24(3), 233-250 (2007) Ishigami, G., et al.: Path planning and evaluation for planetary rovers based on dynamic mobility index. ICRA (2011) Jain, A., et al.: ROAMS: Planetary surface rover simulation environment. i-SAIRAS (2003) Kelly, A.: Fast and easy systematic and stochastic odometry calibration. ICRA (2004) Kelly, A., Seegmiller, N.: A vector algebra formulation of mobile robot velocity kinematics. FSR (2012) Luh, J., Walker, M., Paul, R.: On-line computational scheme for mechanical manipulators. J. Dyn. Sys., Meas., Control 102(2), 69-76 (1980) Martinelli, A., et al.: Simultaneous localization and odometry self calibration for mobile robot. Autonomous Robots 22(1), 75-85 (2007) McMillan, S., Orin, D.: Forward dynamics of multilegged vehicles using the composite rigid body method. ICRA (1998) Muir, P., Neuman, C.: Kinematic modeling of wheeled mobile robots. Tech. Report, CMU-RI-TR-86-12 (1986) Pivtoraiko, M., et al.: Differentially constrained mobile robot motion planning in state lattices. JFR 26(1), 308-333 (2009) Salaani, M.K.: Analytical tire forces and moments model with validated data. SAE World Congress 2007-01-0816 (2007) Seegmiller, N., et al.: A unified perturbative dynamics approach to vehicle model identification. ISRR (2011) Seegmiller, N., et al.: Online calibration of vehicle powertrain and pose estimation parameters using integrated dynamics. ICRA (2012) Stengel, R.: Optimal Control and Estimation, Dover, New York (1994) Tarokh, M., McDermott, G.: Kinematics modeling and analyses of articulated rovers. IEEE Trans. Robot. 21(4), 539-553 (2005) Yu, W., et al.: Analysis and experimental verification for dynamic modeling of a skid-steered wheeled vehicle. IEEE Trans. Robot. 26(2), 340-353 (2010)

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