We present a method for solving two minimal problems for relative camera pose estimation from three views, which are based on three view correspondences of (i) three points and one line and the novel case of (ii) three points and two lines through two of the points. These problems are too difficult to be efficiently solved by the state of the art Gröbner basis methods. Our method is based on a new efficient homotopy continuation (HC) solver framework MINUS, which dramatically speeds up previous HC solving by specializing hc methods to generic cases of our problems. We characterize their number of solutions and show with simulated experiments that our solvers are numerically robust and stable under image noise, a key contribution given the borderline intractable degree of nonlinearity of trinocular constraints. We show in real experiments that (i) sift feature location and orientation provide good enough point-and-line correspondences for three-view reconstruction and (ii) that we can solve difficult cases with too few or too noisy tentative matches, where the state of the art structure from motion initialization fails.

We present a method for solving two minimal problems for relative camera pose estimation from three views, which are based on three view correspondences of (i) three points and one line and (ii) three points and two lines through two of the points. These problems are too difficult to be efficiently solved by the state of the art Grobner basis methods. Our method is based on a new efficient homotopy continuation (HC) solver, which dramatically speeds up previous HC solving by specializing HC methods to generic cases of our problems. We show in simulated experiments that our solvers are numerically robust and stable under image noise. We show in real experiment that (i) SIFT features provide good enough point-and-line correspondences for three-view reconstruction and (ii) that we can solve difficult cases with too few or too noisy tentative matches where the state of the art structure from motion initialization fails.

Bertini_real is a compiled command line program for numerically decomposing the real portion of a positive-dimensional complex component of an algebraic set. The software uses homotopy continuation to solve a series of systems via regeneration from a witness set to compute a cell decomposition. The implemented decomposition algorithms are similar to the well-known cylindrical algebraic decomposition (CAD) first established by Collins in that they produce a set of connected cells. In contrast to the CAD, Bertini_real produces cells with midpoints connected to boundary points by homotopies, which can easily be numerically tracked. Furthermore, the implemented decomposition for surfaces naturally yields a triangulation. This CAD-like decomposition captures the topological information and permits further computation on the real sets, such as sampling, visualization, and three-dimensional printing.

Impedance control is well-suited to robot manipulation applications because it gives the designer a measure of control over how the manipulator to conforms to the environment. However, in the context of end-effector impedance control when the robot manipulator is redundant with respect to end-effector configuration, the question arises regarding how to control the impedance of the redundant joints. This paper considers multi-priority impedance control where a second-priority joint space impedance operates in the null space of a first-priority Cartesian impedance at the end-effector. A control law is proposed that realizes both impedances while observing the priority constraint such that a weighted quadratic error function is optimized. This control law is shown to be a generalization of several motion and impedance control laws found in the literature. The paper makes explicit two forms of the control law. In the first, parametrization by passive inertia values allows the control law to be implemented without requiring end-effector force measurements. In the second, a class of parametrizations is introduced that makes the null space impedance independent of end-effector forces. The theoretical results are illustrated in simulation.

A unified formulation for the calibration of both serial-link robots and robotic mechanisms having kinematic closed-loops is presented and applied experimentally to two 6-degree-of-freedom devices: the RSI 6-DOF hand controller and the MEL "modified Stewart platform". The unification is based on an equivalence between end-effector measurements and constraints imposed by the closure of kinematic loops. Errors are allocated to the joints such that the loop equations are satisfied exactly, which eliminates the issue of equation scaling and simplifies the treatment of multi-loop mechanisms. For the experiments reported here, no external measuring devices are used; instead we rely on measurements of displacements in some of the passive joints of the devices. Using a priori estimates of the statistics of the measurement errors and the parameter errors, the method estimates the parameters and their accuracy, and tests for unmodeled factors.>

Conventionally, the kinematics for serial-link manipulators are formed in terms of the joint angles. The author presents an alternative formulation which uses unit vectors along the joint axes as the variables. For the task of updating the manipulator pose for an incremental change in position and orientation, it is shown that the new formulation is approximately equal in computational cost to the traditional method. Compared to the conventional approach, the joint vector formulation has a simple, but larger, sparse Jacobian. Efficiency is achieved by using a sparse matrix solver.>

A class of industrial tasks requires two-degree-of-freedom wrists to point an axisymmetric tool mounted at the end of a robot arm in a specified direction. Due to inherent deficiencies of wrists, such as joint limits and singular positions, the workspace of the tool on a pointing sphere is limited. A three-degree-of-freedom wrist is a redundant mechanism for point tasks. Using inverse functions, it is shown that with a redundant mechanism the workspace on the pointing sphere can be significantly improved by positioning degenerate regions at suitable areas in the workspace. The authors review arguments toward the design of an ideal redundant wrist for pointing tasks and propose an optimal modification of the wrist of an existing PUMA-560 arm. The main considerations in the modified design are existing joint limits and collisions of the total structure with the forearm. Plots of the joint angles and the joint rates in the workspace are presented. The experimental results confirm that the predicted workspace improvements are attained.>

By upgrading to a manipulator with a three-axis wrist, it is possible to expand the workspace and largely eliminate the effects of singularities. The author achieves these goals using the inverse kinematic function approach to redundancy. He shows that an ideal wrist should be mounted with its first axis perpendicular to the major axis of the manipulator's forearm. With regard to commercially available manipulators, he shows that the pointing workspace of a PUMA-560 can be expanded by approximately 20% by using all three wrist axes as compared to using only the first two wrist axes. The results can also be applied to kinematic control of a four-axis spherical wrist applied to three-degree-of-freedom orientation tasks.>

Many conventional nonredundant manipulators have singular configurations, near which some small motions of the end-effector require excessive and physically unrealizable joint speeds. Consequently, the usable workspace of the manipulator is effectively reduced. It has been proposed that high joint speeds could be avoided by introducing redundant joints and using an appropriate kinematic inversion algorithm. For a very general class of kinematic inversion algorithms, the theorems of this paper state some fundamental relations between the properties of the algorithm and its ability to resolve these problems. These results have practical implications in the design of controllers for redundant manipulators, especially when real-time sensory input is used to modify the manipulator's trajectory.

A general approach to the kinematic control of redundant manipulators using inverse kinematic functions is presented. The principal objective is to find an inverse kinematic function and a workspace such that the ratio of joint speed to workspace speed is bounded. The definition of such feasible workspaces leads naturally to optimization problems aimed at reducing this bound. The application of these results to the inverse kinematics of a redundant wrist illustrates both the theoretical importance of the approach and its immediate practicality. However, for general redundant manipulators, the development of algorithms for constructing feasible workspaces and the refinement of the optimization problem and its solution remain open questions ripe for further research.

Inverse kinematic solutions are used in manipulator controllers to determine corrective joint motions for errors in end-effector position and orientation. Previous formulations of these solutions, based on the Jacobian matrix, are inefficient and fail near kinematic singularities. Vector formulations of inverse kinematic problems are developed that lead to efficient computer algorithms. To overcome the difficulties encountered near kinematic singularities, the exact inverse problem is reformulated as a damped least-squares problem, which balances the error in the solution against the size of the solution. This yields useful results for all manipulator configurations.