An intrinsic homotopy for intersecting algebraic varieties

Recently we developed a diagonal homotopy method to compute a numerical representation of all positive dimensional components in the intersection of two irreducible algebraic sets. In this paper, we rewrite this diagonal homotopy in intrinsic coordinates, which reduces the number of variables, typically in half. This has the potential to save a significant amount of computation, especially in the iterative solving portion of the homotopy path tracker. There numerical experiments all show a speedup of about a factor two

Numerical algebraic intersection using regeneration

In numerical algebraic geometry, algebraic sets are represented by witness sets. This paper presents an algorithm, based on the regeneration technique, that solves the following problem: given a witness set for a pure-dimensional algebraic set Z, along with a system of polynomial equations f: Z�C n, compute a numerical irreducible decomposition of V�Z�V f. An important special case is when Z�A B for irreducible sets A and B and f x,y�x y for x A, y B, in which case V is isomorphic to A�B. In this way, the current contribution is a generalization of existing diagonal intersection techniques. Another important special case is when Z�A C k, so that the projection of V dropping the last k coordinates consists of the points x A where there exists some y in a new set of variables such that f x,y�0. This arises in many contexts, such as finding the singularities of A, in which case f x,y can be a set of singularity conditions that involve new variables associated to the tangent space of A. The combining of multiple intersection scenarios into one common scheme brings new capabilities and organizational simplification to numerical algebraic geometry

A fast branch-and-prune algorithm for the position analysis of spherical mechanisms

The final publication is available at link.springer.comDifferent branch-and-prune schemes can be found in the literature for numerically solving the position analysis of spherical mechanisms. For the prune operation, they all rely on the propagation of motion intervals. They differ in the way the problem is algebraically formulated. This paper exploits the fact that spherical kinematic loop equations can be formulated as sets of 3 multi-affine polynomials. Multi-affinity has an important impact on how the propagation of motion intervals can be performed because a multi-affine polynomial is uniquely determined by its values at the vertices of a closed hyperbox defined in its domain.Peer ReviewedPostprint (author's final draft

Method and apparatus for calibrating multi-axis load cells in a dexterous robot

A robotic system includes a dexterous robot having robotic joints, angle sensors adapted for measuring joint angles at a corresponding one of the joints, load cells for measuring a set of strain values imparted to a corresponding one of the load cells during a predetermined pose of the robot, and a host machine. The host machine is electrically connected to the load cells and angle sensors, and receives the joint angle values and strain values during the predetermined pose. The robot presses together mating pairs of load cells to form the poses. The host machine executes an algorithm to process the joint angles and strain values, and from the set of all calibration matrices that minimize error in force balance equations, selects the set of calibration matrices that is closest in a value to a pre-specified value. A method for calibrating the load cells via the algorithm is also provided

Tension Distribution in a Tendon-Driven Robotic Finger

A method is provided for distributing tension among tendons of a tendon-driven finger in a robotic system, wherein the finger characterized by n degrees of freedom and n+1 tendons. The method includes determining a maximum functional tension and a minimum functional tension of each tendon of the finger, and then using a controller to distribute tension among the tendons, such that each tendon is assigned a tension value less than the maximum functional tension and greater than or equal to the minimum functional tension. The method satisfies the minimum functional tension while minimizing the internal tension in the robotic system, and satisfies the maximum functional tension without introducing a coupled disturbance to the joint torques. A robotic system includes a robot having at least one tendon-driven finger characterized by n degrees of freedom and n+1 tendons, and a controller having an algorithm for controlling the tendons as set forth above

Hierarchical Robot Control System and Method for Controlling Select Degrees of Freedom of an Object Using Multiple Manipulators

A robotic system includes a robot having manipulators for grasping an object using one of a plurality of grasp types during a primary task, and a controller. The controller controls the manipulators during the primary task using a multiple-task control hierarchy, and automatically parameterizes the internal forces of the system for each grasp type in response to an input signal. The primary task is defined at an object-level of control, e.g., using a closed-chain transformation, such that only select degrees of freedom are commanded for the object. A control system for the robotic system has a host machine and algorithm for controlling the manipulators using the above hierarchy. A method for controlling the system includes receiving and processing the input signal using the host machine, including defining the primary task at the object-level of control, e.g., using a closed-chain definition, and parameterizing the internal forces for each of grasp type

Applied Joint-Space Torque and Stiffness Control of Tendon-Driven Fingers

Existing tendon-driven fingers have applied force control through independent tension controllers on each tendon, i.e. in the tendon-space. The coupled kinematics of the tendons, however, cause such controllers to exhibit a transient coupling in their response. This problem can be resolved by alternatively framing the controllers in the joint-space of the manipulator. This work presents a joint-space torque control law that demonstrates both a decoupled and significantly faster response than an equivalent tendon-space formulation. The law also demonstrates greater speed and robustness than comparable PI controllers. In addition, a tension distribution algorithm is presented here to allocate forces from the joints to the tendons. It allocates the tensions so that they satisfy both an upper and lower bound, and it does so without requiring linear programming or open-ended iterations. The control law and tension distribution algorithm are implemented on the robotic hand of Robonaut-2