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Exponential Stabilization of Driftless Nonlinear Control Systems" Robert Thomas M'closkey I11

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Exponential Stabilization of Driftless Nonlinear Control Systems TECHNICAL MEMORANDUM NO. CIT-CDS 95-003 January 1995 "Exponential Stabilization of Driftless Nonlinear Control Systems" Robert Thomas M'Closkey I11 Control and Dynamical Systems California Institute of Technology Pasadena, CA 91125 Exponential Stabilization of Driftless Nonlinear Control Systems Robert Thomas M7Closkey I11 Department of Mechanical Engineering California Institute of Technology Abstract This dissertation lays the foundation for practical exponential stabilization of driftless control systems. Driftless systems have the form, Such systems arise when modeling mechanical systems with nonholonomic con- straints. In engineering applications it is often required to maintain the mechan- ical system around a desired configuration. This task is treated as a stabilization problem where the desired configuration is made an asymptotically stable equi- librium point. The control design is carried out on an approximate system. The approximation process yields a nilpotent set of input vector fields which, in a special coordinate system, are homogeneous with respect to a non-standard dilation. Even though the approximation can be given a coordinate-free in- terpretation, the homogeneous structure is useful to exploit: the feedbacks are required to be homogeneous functions and thus preserve the homogeneous struc- ture in the closed-loop system. The stability achieved is called p-exponential stability. The closed-loop system is stable and the equilibrium point is exponen- tially attractive. This extended notion of exponential stability is required since the feedback, and hence the closed-loop system, is not Lipschitz. However, it is shown that the convergence rate of a Lipschitz closed-loop driftless system cannot be bounded by an exponential envelope. The synthesis methods generate feedbacks which are smooth on Rn \ (0). The solutions of the closed-loop system are proven to be unique in this case. In addition, the control inputs for many driftless systems are velocities. For this class of systems it is more appropriate for the control law to specify actuator forces instead of velocities. We have extended the kinematic velocity controllers to controllers which command forces and still p-exponentially stabilize the sys- tem. Perhaps the ultimate justification of the methods proposed in this thesis are the experimental results. The experiments demonstrate the superior con- vergence performance of the p-exponential stabilizers versus traditional smooth feedbacks. The experiments also highlight the importance of transformation conditioning in the feedbacks. Other design issues, such as scaling the mea- sured states to eliminate hunting, are discussed. The methods in this thesis bring the practical control of strongly nonlinear systems one step closer. Contents 1 Introduction 1 2 Homogeneous Systems 6 2.1 Definitions and Properties ...................... 7 2.1.1 Definitions .......................... 7 2.1.2 Properties of homogeneous functions ............ 8 2.1.3 Stability definitions ..................... 10 2.1.4 Properties of homogeneous degree zero vector fields .... 12 2.2 Approximations of Vector Fields .................. 15 2.3 Lyapunov Functions ......................... 19 3 Analysis Results 26 3.1 Uniqueness of Solutions ....................... 26 3.2 Averaging Results .......................... 31 4 Driftless Control Systems 4 1 4.1 Lipschitz Feedback .......................... 42 4.2 Approximations ............................ 47 4.3 Synthesis Methods .......................... 49 4.3.1 Extension of Pomet's algorithm ............... 51 4.3.2 Modification of smooth controllers into p-exponential sta- bilizers ............................. 58 4.4 Practical Considerations ....................... 65 4.4.1 Filtering of measurements .................. 68 4.4.2 Torque inputs and dynamic extension ........... 72 5 Experimental Validation 83 5.1 Description of Experiment ...................... 83 5.2 Control Laws ............................. 87 5.3 Experimental Results ......................... 91 5.3.1 Open loop inputs ....................... 92 5.3.2 Stabilization of the car .................... 92 5.3.3 Stabilization of the car and one trailer ........... 94 5.3.4 Discussion ........................... 94 6 Conclusion 106 6.1 Future Directions ........................... 107 A Controllability and Stabilizabilization 110 A.1 Controllability ............................110 A.2 Stabilization ..............................112 B Coordinates Adapter to a Filtration 114 B.l Background and Motivation ..................... 114 B.2 Results ................................. 115 B.3 Improvements to the Algorithm ...................119 C Three Dimensional System 122 List of Figures . 2.1 2~-periodiccoefficient in ..................... 3.1 Nonunique solutions .......................... 3.2 Homogeneous balls used in proof: use the dilation to map 22 to XI thus extending the trajectory starting at xo ........... 3.3 Lyapunov function derivative on Si ................. 3.4 Simulation of averaged system (3.12) and original system (3.11) forc=0.1. .............................. 3.5 System (3.11) is unstable for 6 = 1.................. 4.1 Extension of Pomet's algorithm ................... 4.2 Log of the homogeneous norm of the state............. 4.3 Comparison of modified feedback and smooth feedback ...... 4.4 Comparison of energy in control signal............... 4.5 Norm of states for the systems .................... 4.6 Norm of control command ...................... 4.7 Kinematic state response ....................... 4.8 Extended state response ........................ 4.9 Controller output ........................... 4.10 Norm of control output ........................ The nonholomobile mobile robot ................... Experimental apparatus........................ Coordinate systems .......................... Open-loop control ........................... Experimental comparison of homogeneous and smooth feedbacks for the car with no trailers ...................... Experimental controller effort and convergence rates for the car with no trailers ............................. Pomet control law response and actuator effort for the experi- mental system with no trailers .................... Numerical simulation of the homogeneous and smooth Pomet controllers for the car with no trailers ................ Experimental comparison of homogeneous and smooth feedbacks for the car with one trailer ...................... vii 5.10 Experimental controller effort and convergence rates for the car with one trailer .............................100 5.11 Hunting inside a homogeneous ball ..................103 5.12 Affect of ill-conditioned diffeomorphism on system performance . 104 6.1 Unstable equilibrium .........................108 C.l Level sets of V(t.x) = 0.85 are transverse to XE.......... 124 C.2 Closed-loop system is asymptotically stable............. 125 List of Tables 5.1 Kinematic parameters ......................... 86 5.2 Control law parameters for the car with no trailers ......... 94 5.3 Control law parameters for the car and one trailer .........101 List of Symbols Symbol : Description and page when applicable : Recursive Lie bracket on X : Convex hull : Dilation with parameter A, 8 : Upper right Dini derivative : Euler vector field, 12 : Flow of differential equation, 8 : Generalized Jacobian of map F, 53 : Homogeneous sphere, 12 : Lie bracket of vector fields : Lie derivative of function f with respect to the vector field X : Euclidean norm : Tangent space map of function : Homogeneous norm, 9 Chapter 1 Introduction This thesis studies the problem of locally exponentially stabilizing analytic drift- less control systems. Driftless systems have the form where the control inputs ui are real valued and the Xiare analytic "input" vector fields. A diverse set of mechanical systems may be modeled as driftless control systems. The special form of the model is often the result of nonholo- nomic constraints that the kinematic variables of the system must satisfy. A mobile robot with wheels that roll without slipping is an example of a system with nonholonomic constraints 1241, [32]. Dextrous manipulation with multi- fingered robotic hands is another application where driftless control systems arise from nonholonomic constraints [27], [26], [35]. Reorientation of rigid bod- ies with zero angular momentum through internal motion may be studied as a driftless control system. In this case, the angular momentum constraint enforces a nonholonomic-like constraint on the system model [25], [21], [12]. Finally, non- holonomic actuators are studied in [5]. A problem of practical interest is how to transfer the system to some desired final state. The change in state may be affected by two different approaches. The first approach is an open-loop strategy. This involves defining the control inputs as functions of time so that the initial state of the model (1.1) is transferred to the desired final state. Initial efforts to control driftless systems were directed at open-loop path planning. The literature in this area is large. A recent paper containing a comprehensive reference list is [32]. However, the limitations of an open-loop methodology restrict its use in physical systems: without feedback the system performance is degraded by modeling errors and external distur- bances. In other words, small errors in the initial state measurement or the model resulted in poor performance (the performance being measured by the deviation from the desired final
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