Ola Dahl

Nonlinear and Adaptive Observers for Perspective Dynamic Systems

Estimation of 3D structure and motion from 2D images in computer vision systems can be performed using a dynamic system, often referred to as a perspective dynamic system. This paper presents a novel parametrization of the nonlinear perspective dynamic system, from which different estimators for rigid body structure as well as motion can be derived in a straightforward manner. The parametrization allows a structure estimator to be formulated as a nonlinear observer which estimates 3D position, assuming knowledge of angular and linear velocities. The observer performance is demonstrated using simulation examples, where it is also shown how a time scaling parameter can be used to tune the transient response. The parametrization also allows a motion estimator to be formulated as an adaptive observer, estimating angular velocity and 3D position assuming knowledge of the linear velocity. This is demonstrated by deriving an estimator and illustrating its performance in a simulation example. The presented investigations and simulations indicate that the parametrization has a potential for future development of estimators for structure as well as motion in perspective dynamic systems, and for the investigation of similarities and differences in comparison to discrete, projective geometry based, methods.

Linear Design of a Nonlinear Observer for Perspective Systems

Estimation of three-dimensional information from two-dimensional images is an important requirement in many computer vision applications. The estimation task can often be formulated as a problem of estimating states and/or parameters in nonlinear dynamic systems. This paper presents an algorithm for recursive state estimation in nonlinear dynamic systems, where the estimated states correspond to three-dimensional positions of feature points on an observed object. The algorithm is designed as a nonlinear observer, with a gain matrix that can be determined using methods from linear control theory. A stability criterion for the resulting nonlinear system is derived, and simulations are presented in order to illustrate the estimation performance.

Brief paper: Observer forms for perspective systems

The estimation of three-dimensional position information from two-dimensional images in computer vision systems can be formulated as a state estimation problem for a nonlinear perspective dynamic system. The multi-output state estimation problem has been treated by several authors using methods for nonlinear observer design. This paper shows that a perspective system can be transformed to two observer forms, and provides constructive methods for arriving at the transformations. These observer forms lead to straightforward observer designs. First, it is shown that, using an output transformation, the system admits an observer form which leads to an observer with linear error dynamics. A second observer design is based on a time-scaled block triangular form. Both designs assume a commonly used observability condition. The designs are demonstrated in simulation.

Dynamic structure from motion based on nonlinear adaptive observers

Structure and motion estimation from long image sequences is a an important and difficult problem in computer vision. We propose a novel approach based on nonlinear and adaptive observers based on a dynamic model of the motion. The estimation of the three-dimensional position and velocity of the camera as well as the three-dimensional structure of the scene is done by observing states and parameters of a nonlinear dynamic system, containing a perspective transformation in the output equation, often referred to as a perspective dynamic system. An advantage of the proposed method is that it is filter-based, i.e. it provides an estimate of structure and motion at each time instance, which is then updated based on a novel image in the sequence. The observer demonstrates a trade-off compared to a more computer vision oriented approach, where no specific assumptions regarding the motion dynamics are required, but instead additional feature points are needed. Finally, the performance of the proposed method is shown in simulated experiments.

Path Following for a Flexible Joint Robot

Abstract A path following algorithm for flexible joint robots is presented. The algorithm is an on-line modification of the reference trajectory used by the robot’s control system. The path is represented by a vector function f ( s ), and the algorithm is based on on-line modification of the fourth order time derivative of the scalar path parameter s(t). The purpose of the modification is to achieve a reference trajectory f ( s ( t )), that leads to admissible torques, which may not be the case if the nominal, unmodified reference trajectory is used. A simulation of a flexible joint robot, where a nominally unfollowable reference trajectory is converted to a trajectory which gives good path following, is presented. We also present an analysis where a nonlinear dynamic system, used for trajectory generation is rewritten as a linear system by interpreting the path parameter s as a transformed time variable.

An Interactive Environment for Real Time Implementation of Control Systems

Abstract An approach to efficient implementation of real time control systems is presented. A compiler for translation of control algorithms is used in combination with a general program for real time control. The compiler translates control algorithms, written in a design language, to an implementation language, and generates code for connecting the control algorithms to the user interface. The translated algorithms are then automatically incorporated in the real time control program. The resulting executable program have a number of interactive facilities such as interconnection of controllers, plotting and textual display of all variables, and data logging. The design language is chosen as Simnon, a language for simulation of nonlinear systems, and the implementation language is chosen as Modula-2. The system has been used in research and education, and has reduced the implementation time considerably, e.g. when developing new laboratory exercises, or when a control algorithm is tested in a laboratory experiment

Provably convergent structure and motion estimation for perspective systems

Estimation of structure and motion in computer vision systems can be performed using a dynamic systems approach, where states and parameters in a perspective system are observed. This paper presents a new approach to the structure estimation problem, where the estimation of the 3D-positions of feature points on a moving object is reformulated as a parameter estimation problem. For each feature point, a constant parameter is estimated, from which it is possible to calculate the time-varying 3D-position. The estimation method is extended to the estimation of motion, in the form of angular velocity estimation. The combined structure and angular velocity observer is shown stable using Lyapunov theory and persistency of excitation based arguments. The estimation method is illustrated with simulation examples, demonstrating the estimation convergence.

Observer Forms for Perspective Systems

Abstract Estimation of 3D position information from 2D images in computer vision systems can be formulated as a state estimation problem for a nonlinear perspective dynamic system. The multi-output state estimation problem has been treated by several authors using methods for nonlinear observer design. This paper shows that a perspective system can be transformed to two observer forms, and provides constructive methods for arriving at the transformations. These observer forms lead to straightforward observer designs. First, it is shown that using an output transformation, the system admits an observer form which leads to an observer with linear error dynamics. A second observer design is based on a time scaled block triangular form. Both designs assume a commonly used observability condition. The designs are demonstrated in simulation.

On Observer Error Linearization for Perspective Dynamic Systems

State estimation in perspective dynamic systems can be performed using different kinds of nonlinear observers. In this paper we investigate observer error linearization for perspective dynamic systems, where the goal is to find a coordinate transformation that results in a system for which a linear observer can be constructed. We present preliminary results in this direction, showing that such linearizing coordinate transformations indeed exist, subject to certain constraints on the angular and linear velocities. Specifically, it is shown that in many situations a linearizing coordinate transformation can be found only when additional states are added to the original system. This results in what is referred to as dynamic observer error linearization. Our investigations further provide additional insight into observability issues for perspective dynamic systems by showing how an observability condition can be interpreted in terms of the focus of expansion, and how the condition is related to the derivation of appropriate coordinate transformations.

Recursive structure and motion estimation based on hybrid matching constraints

Motion estimation has traditionally been approached either from a pure discrete point of view, using multi-view tensors, or from a pure continuous point of view, using optical flow. This paper builds upon a novel framework of hybrid matching constraints for motion estimation, combining the advantages of both discrete and continuous methods. We will derive both bifocal and trifocal hybrid constraints and use them together with a structure estimate based on filtering techniques. A feedback from the structure estimate will be used to further refine the motion estimate. This gives a complete iterative structure and motion estimation scheme. Its performance will be demonstrated in simulated experiments.