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126
Nonholonomic motion planning: Steering using sinusoids
 IEEE fins. Auto. Control
, 1993
"... AbstractIn this paper, we investigate methods for steering systems with nonholonomic constraints between arbitrary configurations. Early work by Brockett derives the optimal controls for a set of canonical systems in which the tangent space to the configuration manifold is spanned by the input vec ..."
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Cited by 251 (15 self)
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AbstractIn this paper, we investigate methods for steering systems with nonholonomic constraints between arbitrary configurations. Early work by Brockett derives the optimal controls for a set of canonical systems in which the tangent space to the configuration manifold is spanned by the input vector fields and their first order Lie brackets. Using Brockett’s result as motivation, we derive suboptimal trajectories for systems which are not in canonical form and consider systems in which it takes more than one level of bracketing to achieve controllability. These trajectories use sinusoids at integrally related frequencies to achieve motion at a given bracketing level. We define a class of systems which can be steered using sinusoids (chained systems) and give conditions under which a class of twoinput systems can be converted into this form. I.
Randomized Kinodynamic Motion Planning with Moving Obstacles
, 2000
"... We present a randomized motion planner for robots that must avoid moving obstacles and achieve a specified goal under kinematic and dynamic constraints. The planner samples the robot's statetime space by picking control inputs at random and integrating the equations of motion. The result is a roa ..."
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Cited by 190 (12 self)
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We present a randomized motion planner for robots that must avoid moving obstacles and achieve a specified goal under kinematic and dynamic constraints. The planner samples the robot's statetime space by picking control inputs at random and integrating the equations of motion. The result is a roadmap of sampled statetime points, called milestones, connected by short admissible trajectories. The planner does not precompute the roadmap as traditional probabilistic roadmap planners do; instead, for each planning query, it generates a new roadmap to find a trajectory between an initial and a goal statetime point. We prove in this paper that the probability that the planner fails to find such a trajectory when one exists quickly goes to 0, as the number of milestones grows. The planner has been implemented and tested successfully in both simulated and real environments. In the latter case, a vision module estimates obstacle motions just before planning starts; the planner is then allocated a small, fixed amount of time to compute a trajectory. If a change in the obstacle motion is detected while the robot executes the planned trajectory, the planner recomputes a trajectory on the fly. 1
Nonholonomic Mechanics and Locomotion: The Snakeboard Example
 In Proc. IEEE Int. Conf. Robotics and Automation
, 1994
"... Analysis and simulations are performed for a simplified model of a commercially available variant of the skateboard, known as the Snakeboard 1 . Although the model exhibits basic gait patterns seen in a large number of locomotion problems, the analysis tools currently available do not apply to this ..."
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Cited by 59 (25 self)
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Analysis and simulations are performed for a simplified model of a commercially available variant of the skateboard, known as the Snakeboard 1 . Although the model exhibits basic gait patterns seen in a large number of locomotion problems, the analysis tools currently available do not apply to this problem. The difficulty lies primarily in the way in which the nonholonomic constraints enter into the system. As a first step towards understanding systems represented by our model we present the equations of motion and perform some controllability analysis for the snakeboard. We also perform numerical simulations of possible gait patterns which are characteristic of snakeboard locomotion. Introduction This paper investigates a simplified model of a commercially available derivative of a skateboard known as the Snakeboard . The Snakeboard (Figure 1) allows the rider to propel him/herself forward without having to make contact with the ground. This motion is roughly accomplished by coupl...
Pricing of American PathDependent Contingent Claims
, 1994
"... We consider the problem of pricing pathdependent contingent claims. Classically, this problem can be cast into the BlackScholes valuation framework through inclusion of the pathdependent variables into the state space. This leads to solving a degenerate advectiondiffusion Partial Differential Eq ..."
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Cited by 39 (1 self)
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We consider the problem of pricing pathdependent contingent claims. Classically, this problem can be cast into the BlackScholes valuation framework through inclusion of the pathdependent variables into the state space. This leads to solving a degenerate advectiondiffusion Partial Differential Equation (PDE). Standard Finite Difference (FD) methods are known to be inadequate for solving such degenerate PDE. Hence, pathdependent European claims are typically priced through MonteCarlo simulation. To date, there is no numerical method for pricing pathdependent American claims. We first establish necessary and sufficient conditions amenable to a Lie algebraic characterization, under which degenerate diffusions can be reduced to lowerdimensional nondegenerate diffusions on a submanifold of the underlying asset space. We apply these results to pathdependent options. Then, we describe a new numerical technique, called Forward Shooting Grid (FSG) method, that efficiently copes with de...
A Kalman Filterbased Algorithm for IMUCamera Calibration
, 2007
"... Visionaided Inertial Navigation Systems (VINS) can provide precise state estimates for the 3D motion of a vehicle when no external references (e.g., GPS) are available. This is achieved by combining inertial measurements from an IMU with visual observations from a camera under the assumption that ..."
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Cited by 37 (8 self)
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Visionaided Inertial Navigation Systems (VINS) can provide precise state estimates for the 3D motion of a vehicle when no external references (e.g., GPS) are available. This is achieved by combining inertial measurements from an IMU with visual observations from a camera under the assumption that the rigid transformation between the two sensors is known. Errors in the IMUcamera calibration process causes biases that reduce the accuracy of the estimation process and can even lead to divergence. In this paper, we present a Kalman filterbased algorithm for precisely determining the unknown transformation between a camera and an IMU. Contrary to previous approaches, we explicitly account for the time correlations of the IMU measurements and provide a figure of merit (covariance) for the estimated transformation. The proposed method does not require any special hardware (such as spin table or 3D laser scanner) except a calibration target. Simulation and experimental results are presented that validate the proposed method and quantify its accuracy.
A LieBäcklund approach to equivalence and flatness of nonlinear systems
 922–937. IN INDIRECT OPTIMAL CONTROL OF MULTIVARIABLE SYSTEMS 317
, 1999
"... Abstract — In this paper, a new system equivalence relation, using the framework of differential geometry of jets and prolongations of infinite order, is studied. In this setting, two systems are said to be equivalent if any variable of one system may be expressed as a function of the variables of t ..."
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Cited by 33 (4 self)
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Abstract — In this paper, a new system equivalence relation, using the framework of differential geometry of jets and prolongations of infinite order, is studied. In this setting, two systems are said to be equivalent if any variable of one system may be expressed as a function of the variables of the other system and of a finite number of their time derivatives. This is a Lie–Bäcklund isomorphism. This quite natural, though unusual, equivalence is presented in an elementary way by the inverted pendulum and the vertical takeoff and landing (VTOL) aircraft. The authors prove that, although the state dimension is not preserved, the number of input channels is kept fixed. They also prove that a Lie–Bäcklund isomorphism can be realized by an endogenous feedback, i.e., a special type of dynamic feedback. Differentially flat nonlinear systems, which were introduced by the authors in 1992 via differential algebraic techniques, are generalized here and the new notion of orbitally flat systems is defined. They correspond to systems which are equivalent to a trivial one, with time preservation or not. Trivial systems are, in turn, equivalent to any linear controllable system with the same number of inputs, and consequently flat systems are linearizable by endogenous feedback. The endogenous linearizing feedback is explicitly computed in the case of the VTOL aircraft to track given reference trajectories with stability; simulations are presented. Index Terms — Dynamic feedback, flatness, infiniteorder prolongations, Lie–Bäcklund equivalence, nonlinear systems.
2001): A geometric approach to nonlinear fault detection and isolation
 IEEE Trans. Automat. Contr
"... We present in this article a differentialgeometric approach to the problem of fault detection and isolation for nonlinear systems. A necessary condition for the problem to be solvable is derived in terms of an unobservability distribution, which is computable by means of suitable algorithms. The ex ..."
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Cited by 33 (4 self)
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We present in this article a differentialgeometric approach to the problem of fault detection and isolation for nonlinear systems. A necessary condition for the problem to be solvable is derived in terms of an unobservability distribution, which is computable by means of suitable algorithms. The existence and regularity of such a distribution implies the existence of changes of coordinates in the state and in the output space which induce an “observable ” quotient subsystem unaffected by all fault signals but one. For this subsystem a fault detection filter is designed.
Dynamic Nonprehensile Manipulation: Controllability, Planning, and Experiments
 International Journal of Robotics Research
, 1998
"... We are interested in using low degreeoffreedom robots to perform complex tasks by nonprehensile manipulation (manipulation without a form or forceclosure grasp). By not grasping, the robot can use gravitational, centrifugal, and Coriolis forces as virtual motors to control more degreesof freedo ..."
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Cited by 31 (14 self)
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We are interested in using low degreeoffreedom robots to perform complex tasks by nonprehensile manipulation (manipulation without a form or forceclosure grasp). By not grasping, the robot can use gravitational, centrifugal, and Coriolis forces as virtual motors to control more degreesof freedom of the part. The extra motion freedoms of the part are exhibited as rolling, slipping, and free flight.
Steering nonholonomic systems using sinusoids
 in: Proc. 29th IEEE Conf. Decis. Contr
, 1990
"... In this paper we investigate methods for steering systems with nonholonomic constraints between arbitrary configurations. Early work by Brockett derives the optimal controls for a set of canonical systems in which the tangent space to the configuration manifold is spanned by the input vector fields ..."
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Cited by 31 (2 self)
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In this paper we investigate methods for steering systems with nonholonomic constraints between arbitrary configurations. Early work by Brockett derives the optimal controls for a set of canonical systems in which the tangent space to the configuration manifold is spanned by the input vector fields and their (first order) Lie brackets. Using Brockett's result as motivation, we derive suboptimal trajectories for systems which are not in canonical form and consider systems in which it takes more than one level of bracketing to achieve controllability. These trajectories use sinusoids at integrally related frequencies to achieve motion at a given bracketing level. Examples and simulation results are presented. 1
Steering a ThreeInput Nonholonomic System using Multirate Controls
 The International Journal of Robotics Research
, 1993
"... this paper are in finding paths for nonholonomic systems. These systems are characterized by having nonintegrable constraints on the state velocities which do not restrict the reachable configuration space. There is a classical result from nonlinear control theory [6] which can be used to prove that ..."
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Cited by 28 (5 self)
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this paper are in finding paths for nonholonomic systems. These systems are characterized by having nonintegrable constraints on the state velocities which do not restrict the reachable configuration space. There is a classical result from nonlinear control theory [6] which can be used to prove that such systems are completely controllable, implying that a feasible path exists between any two points in free space. This result, however, does not have a constructive proof. The problem we consider in this paper is the construction of a feasible path between the start and goal states. We transform the pathplanning problem with constraints on the velocities into the dual control problem with fewer control inputs than degrees of freedom. Some early work by Murray and Sastry (see [5]) used sinusoidal inputs at varying frequencies to plan paths for systems in a special chained canonical form. More recently, it has been shown by Monaco and NormandCyrot [4] that piecewise constant controls may be used for nonholonomic path planning. The system that we use as our example was originally examined in [1]. It has three control inputs and can be shown to be completely controllable. In addition, the system can be transformed into a specific class of nilpotent systems called chained form systems. Using these chained form coordinates, We apply the multirate control scheme to the example system. We also present some simulation results, comparing the multirate and sinusoidal algorithms on three different trajectories. Our findings are that the paths generated by the multirate control scheme are shorter and more direct than the paths generated by the sinusoidal pathplanning algorithm.