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Trajectory generation for autonomous unmanned aircraft using inverse dynamics

By R. G. Drury


The problem addressed in this research is the in-flight generation of trajectories for autonomous unmanned aircraft, which requires a method of generating pseudo-optimal trajectories in near-real-time, on-board the aircraft, and without external intervention. The focus of this research is the enhancement of a particular inverse dynamics direct method that is a candidate solution to the problem. This research introduces the following contributions to the method. A quaternion-based inverse dynamics model is introduced that represents all orientations without singularities, permits smooth interpolation of orientations, and generates more accurate controls than the previous Euler-angle model. Algorithmic modifications are introduced that: overcome singularities arising from parameterization and discretization; combine analytic and finite difference expressions to improve the accuracy of controls and constraints; remove roll ill-conditioning when the normal load factor is near zero, and extend the method to handle negative-g orientations. It is also shown in this research that quadratic interpolation improves the accuracy and speed of constraint evaluation. The method is known to lead to a multimodal constrained nonlinear optimization problem. The performance of the method with four nonlinear programming algorithms was investigated: a differential evolution algorithm was found to be capable of over 99% successful convergence, to generate solutions with better optimality than the quasi- Newton and derivative-free algorithms against which it was tested, but to be up to an order of magnitude slower than those algorithms. The effects of the degree and form of polynomial airspeed parameterization on optimization performance were investigated, and results were obtained that quantify the achievable optimality as a function of the parameterization degree. Overall, it was found that the method is a potentially viable method of on-board near- real-time trajectory generation for unmanned aircraft but for this potential to be realized in practice further improvements in computational speed are desirable. Candidate optimization strategies are identified for future research

Topics: Autonomy, differential evolution, direct methods, inverse dynamics, near-real-time, negative-g, nonlinear programming, numerical optimization, optimal control, quaternions, trajectory generation, UAV, unmanned aircraft
Publisher: Cranfield University
Year: 2010
OAI identifier:
Provided by: Cranfield CERES

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  1. 3D Dubins Sets Based Coordinated Path Planning for Swarm of UAVs", doi
  2. (1996). A Compact Differential Formula for the First Derivative of a Unit Quaternion Curve", doi
  3. (2005). A Comprehensive Study on Flight Path Selection Algorithms", doi
  4. (2002). A Constraint Handling Approach for the Differential Evolution Algorithm", doi
  5. (2002). A Convergent Variant of the NelderMead Algorithm",
  6. (1996). A Practical Guide to Pseudospectral Methods, doi
  7. (1994). A Practical Guide to the Solution of Real-Life Optimal Control Problems",
  8. (2009). A Quaternion-Based Inverse Dynamics Model for Real-Time Trajectory Generation", doi
  9. (1965). A Simplex Method for Function Minimization", doi
  10. (1993). A Survey of Attitude Representations",
  11. (2007). A View of Algorithms for Optimization Without Derivatives",
  12. (2007). Advancement and Analysis of a Gauss Pseudospectral Transcription for Optimal Control Problems",
  13. (2008). Advances in Pseudospectral Methods for Optimal Control", doi
  14. (2003). Aim Point Guidance: an Extension of Proportional Navigation to the Control of Terminal Guidance", doi
  15. (1988). Algorithms for Polynomials in Bernstein Form", doi
  16. (2004). An Extension of Matlab to Continuous Functions and Operators", doi
  17. (1986). An Interpretation of Airplane General Motion and Control as Inverse Problem", doi
  18. (1999). Analytical Dynamics,
  19. (1985). Animating Rotation with Quaternion Curves", doi
  20. Applied Optimal Control, Optimization, Estimation, and Control, Revised Printing 1988, Taylor and Francis,
  21. (2004). Barycentric Lagrange Interpolation", doi
  22. (1984). Basic Optimisation Methods, doi
  23. (2002). Benchmarking Optimization Software with Performance Profiles", doi
  24. (1989). BNDSCO - a Program for the Numerical Solution of Optimal Control Problems",
  25. (2001). Chebyshev and Fourier Spectral Methods, 2nd ed., doi
  26. (1998). Collocation vs Differential Inclusion in Direct Optimization", doi
  27. (1993). Comment on Generalized Technique for Inverse Simulation Applied to Aircraft Maneuvers", doi
  28. (1972). Comparison of Pseudospectral and Spectral Approximations",
  29. (1999). Computation of Optimal Control Trajectories Using Chebyshev Polynomials, Parameterization, and Quadratic Programming", doi
  30. (1999). Computational Method Based on State Parameterization for Solving Constrained Nonlinear Optimal Control Problems", doi
  31. (2009). Computational Optimal Control: Tools and Practice, doi
  32. (2006). Computing Real Roots of a Polynomial in Chebyshev Series Form Through Subdivision with Linear Testing and Cubic Solves", doi
  33. (2008). Computing Short-Time Aircraft Maneuvers Using Direct Methods", doi
  34. (2006). Computing the Zeros, Maxima, and Inflection Points of Chebyshev, Legendre and Fourier Series: Solving Transcendental Equations by Spectral Interpolation and Polynomial Rootfinding", doi
  35. (2008). Connections Between the Covector Mapping Theorem and Convergence of Pseudospectral Methods for Optimal Control", doi
  36. Convergence Properties of the Nelder-Mead Simplex Algorithm in Low Dimensions", doi
  37. (1997). Conversion of Optimal Control Problems into Parameter Optimization Problems", doi
  38. (2004). Cooperative Control of Small UAVs for Naval Applications", doi
  39. (2006). Coordinated Control of Multiple UAVs for Time-Critical Applications", doi
  40. (2001). Costate Estimation by a Legendre Pseudospectral Method", doi
  41. (1970). Derivation of the Quaternion Scheme via the Euler Axis and Angle", doi
  42. (2005). Differential Evolution: a Practical Approach to Global Optimization, doi
  43. (1965). Differential Games and Optimal Pursuit Evasion Strategies", doi
  44. (2005). Differential Geometric Guidance Based on the Involute of the Target's Trajectory", doi
  45. (1992). Direct and Indirect Methods for Trajectory Optimization", doi
  46. (2005). Direct Intercept Guidance using Differential Geometry Concepts", doi
  47. (2000). Direct Method for Rapid Prototyping of Near-Optimal Aircraft Trajectories", doi
  48. (1986). Direct Method of Calculus of Variations in Boundary Problems of Flight Dynamics,
  49. (1961). Direct Search Solution of Numerical and Statistical Problems", doi
  50. (2006). Direct Trajectory Optimization and Costate Estimation via an Orthogonal Collocation Method", doi
  51. (2002). Direct Trajectory Optimization by a Chebyshev Pseudospectral Method", doi
  52. (1987). Direct Trajectory Optimization using Nonlinear Programming and Collocation", doi
  53. (1992). Discrete Approximations to Optimal Trajectories Using Direct Transcription and Nonlinear Programming", doi
  54. Dynamic Programming, doi
  55. (1968). Experience of Employment of Ritz's, Poincaré's and Lyapunov's Methods for Solving the Problems of Flight Dynamics", Soviet Air Force Engineering Academy,
  56. (2008). Feedback Systems : an Introduction for Scientists and Engineers,
  57. (2003). Flat Systems, Equivalence and Trajectory Generation",
  58. (1995). Flatness and Defect of Nonlinear Systems: Introductory Theory and Examples", doi
  59. (1988). Flight Control Design Using Nonlinear Inverse Dynamics", doi
  60. (2004). Flight Dynamics, doi
  61. (1991). Generalized Technique for Inverse Simulation Applied to Aircraft Maneuvers", doi
  62. (1996). Improved Methodology for Inverse Simulation",
  63. (1995). Interpolating Solid Orientations with Circular Blending Quaternion Curves", doi
  64. (2002). Introduction to Numerical Analysis, doi
  65. (1992). Inverse and Optimal Control for Desired Outputs", doi
  66. (1996). Inverse and Optimal Control for Precision Aerobatic Maneuvers", doi
  67. (1993). Inverse Dynamics Approach to Trajectory Optimization for an Aerospace Plane", doi
  68. (1999). Iterative Methods for Optimization, doi
  69. (2007). Low-Thrust, High Accuracy Trajectory Optimization", doi
  70. (2005). MILP and its Application in Flight Planning", doi
  71. (1973). Minimization Without Derivatives, doi
  72. (2010). MQ-1 Predator and MQ-9 Reaper Operations",
  73. (1990). Navigation Path Planning for Autonomous Aircraft: Voronoi Diagram Approach", doi
  74. (1999). Near-optimal trajectory generation for autonomous aircraft landing", doi
  75. (2011). Negative-g Trajectory Generation Using Quaternion-Based Inverse Dynamics", doi
  76. (2007). Nonlinearly-Constrained Optimization Using Asynchronous Parallel Generating Set Search", doi
  77. (1993). Nonsmooth Trajectory Optimization: an Approach Using Continuous Simulated Annealing", doi
  78. (2007). Numerical Recipes: The Art of Scientific Computing, 3rd ed., doi
  79. (1957). On Curves of Minimal Length With a Constraint on Average Curvature and With Prescribed Initial and Terminal Positions and Tangent", doi
  80. (1992). On Differentially Flat Nonlinear Systems", doi
  81. (1997). On the Convergence of Pattern Search Algorithms", doi
  82. (1996). On the Optimal Stability of the Bernstein Basis", doi
  83. (2006). Optimality Measures for Performance Profiles", doi
  84. (1998). Optimization - Algorithms and Consistent Approximations, doi
  85. (2003). Optimization by Direct Search: New Perspectives on Some Classical and Modern Methods", doi
  86. (1983). Optimization by Simulated Annealing", doi
  87. (2006). Path Generation, Path Following and Coordinated Control for Time Critical Missions of Multiple UAVs", doi
  88. (2000). Path Planning for UAVs", doi
  89. Path Planning of Multiple UAVs Using Dubins Sets", doi
  90. (2007). Path-Planning with Avoidance using Nonlinear Branch-and-Bound Optimization", doi
  91. (2001). Practical Methods for Optimal Control Using Nonlinear Programming, doi
  92. (1981). Practical Optimization, doi
  93. (2006). Preliminary Modeling, Control, and Trajectory Design for Minature Autonomous Tailsitters", doi
  94. (1972). Problems Related to Unconstrained Optimization", in Numerical Methods for Unconstrained Optimization, doi
  95. (1998). Pseudospectral Chebyshev Optimal Control of Constrained Nonlinear Dynamical Systems",
  96. (2005). Pseudospectral Methods for Infinite Horizon Nonlinear Optimal Control Problems", doi
  97. (2004). Pseudospectral Methods for Optimal Motion Planning of Differentially Flat Systems", doi
  98. (1978). Quaternion from Rotation Matrix",
  99. (1998). Quaternion Parameterization and a Simple Algorithm for Global Attitude Estimation", doi
  100. (1979). Quaternion Singularity Revisited", doi
  101. (2009). Quaternion-Based Inverse Dynamics Model for Evaluating Aerobatic Aircraft Trajectories", doi
  102. (2005). Real-Time Dynamic Trajectory Smoothing for Unmanned Air Vehicles", doi
  103. (2002). Real-Time Motion Planning for Agile Autonomous Vehicles", doi
  104. (2003). Real-Time Optimal Trajectory Generation for Constrained Dynamical Systems",
  105. (2010). Real-Time Trajectory Generation: Improving the Optimality and Speed of an Inverse Dynamics Method", doi
  106. (2002). Receding Horizon Control of Autonomous Aerial Vehicles", doi
  107. (2001). Review of Attitude Representations Used for Aircraft Kinematics", doi
  108. (2005). Revisiting Asynchronous Parallel Pattern Search for Nonlinear Optimization", doi
  109. (2001). Second Look at Approximating Differential Inclusions", doi
  110. (1989). Short-Range Nonlinear Feedback Strategies for Aircraft Pursuit-Evasion", doi
  111. (1996). Should Controls be Eliminated While Solving Optimal Control Problems via Direct Methods?", doi
  112. (1993). Simulated Annealing: Practice vs Theory", doi
  113. (2002). SNOPT: an SQP Algorithm for Large-Scale Constrained Optimization ", doi
  114. (2000). Spectral Methods in Matlab, doi
  115. (2006). Spectral Methods: Fundamentals in Single Domains, doi
  116. (1998). Survey of Numerical Methods for Trajectory Optimization", doi
  117. (1968). Survey of Penalty, Exact-Penalty and Multiplier Methods from doi
  118. (1995). The Pseudospectral Legendre Method for Discretizing Optimal Control Problems", doi
  119. Time-Optimal Reorientation of a Spacecraft Using a Direct Optimization Method Based on Inverse Dynamics", doi
  120. (1994). Trajectory Optimization Based on Differential Inclusions", doi
  121. (2000). Trajectory Optimization by Indirect Spectral Collocation Methods", doi
  122. (2004). Trajectory Planning for Autonomous Aerospace Vehicles amid Known Obstacles and Conflicts", doi
  123. (1938). Trigonometric Interpolation of Empirical and Analytical Functions",
  124. (2007). User's Guide for SNOPT Version 7: Software for Large-Scale Nonlinear Programming",
  125. (2010). Watchkeeper in the UK - Update",

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