Mili Shah

Solving the Robot-World/Hand-Eye Calibration Problem Using the Kronecker Product

This paper constructs a separable closed-form solution to the robot-world/hand-eye calibration problem AX1⁄4YB. Qualifications and properties that determine the uniqueness of X and Y as well as error metrics that measure the accuracy of a given X and Y are given. The formulation of the solution involves the Kronecker product and the singular value decomposition. The method is compared with existing solutions on simulated data and real data. It is shown that the Kronecker method that is presented in this paper is a reliable and accurate method for solving the robot-world/hand-eye calibration problem. [DOI: 10.1115/1.4024473]

An overview of robot-sensor calibration methods for evaluation of perception systems

In this paper, an overview of methods that solve the robotsensor calibration problem of the forms AX = XB and AX = YB is given. Each form will be split into three solutions: separable closed-form solutions, simultaneous closed-form solutions, and iterative solutions. The advantages and disadvantages of each of the solutions in the case of evaluation of perception systems will also be discussed.

A Symmetry Preserving Singular Value Decomposition

A reduced order representation of a large data set is often realized through a principal component analysis based upon a singular value decomposition (SVD) of the data. The left singular vectors of a truncated SVD provide the reduced basis. In several applications such as facial analysis and protein dynamics, structural symmetry is inherent in the data. Typically, reflective or rotational symmetry is expected to be present in these applications. In protein dynamics, determining this symmetry allows one to provide SVD major modes of motion that best describe the symmetric movements of the protein. In face detection, symmetry in the SVD allows for more efficient compression algorithms. Here we present a method to compute the plane of reflective symmetry or the axis of rotational symmetry of a large set of points. Moreover, we develop a symmetry preserving singular value decomposition (SPSVD) that best approximates the given set while respecting the symmetry. Interesting subproblems arise in the presence of noisy data or in situations where most, but not all, of the structure is symmetric. An important part of the determination of the axis of rotational symmetry or the plane of reflective symmetry is an iterative reweighting scheme. This scheme is rapidly convergent in practice and seems to be very effective in ignoring outliers (points that do not respect the symmetry).

Comparing two sets of corresponding six degree of freedom data

Abstract This paper is concerned with comparing two sets of corresponding six degree of freedom data that consist of both object position and object orientation. Specifically, the best rotation and translation that aligns the position and orientation of one data set to the other is constructed by solving an optimization problem. In addition, a statistical method that identifies outliers in the data sets is proposed.

Simulating nanoscale functional motions of biomolecules

We are describing efficient dynamics simulation methods for the characterization of functional motion of biomolecules on the nanometer scale. Multivariate statistical methods are widely used to extract and enhance functional collective motions from molecular dynamics (MD) simulations. A dimension reduction in MD is often realized through a principal component analysis (PCA) or a singular value decomposition (SVD) of the trajectory. Normal mode analysis (NMA) is a related collective coordinate space approach, which involves the decomposition of the motion into vibration modes based on an elastic model. Using the myosin motor protein as an example we describe a hybrid technique termed amplified collective motions (ACM) that enhances sampling of conformational space through a combination of normal modes with atomic level MD. Unfortunately, the forced orthogonalization of modes in collective coordinate space leads to complex dependencies that are not necessarily consistent with the symmetry of biological macromolecules and assemblies. In many biological molecules, such as HIV-1 protease, reflective or rotational symmetries are present that are broken using standard orthogonal basis functions. We present a method to compute the plane of reflective symmetry or the axis of rotational symmetry from the trajectory frames. Moreover, we develop an SVD that best approximates the given trajectory while respecting the symmetry. Finally, we describe a local feature analysis (LFA) to construct a topographic representation of functional dynamics in terms of local features. The LFA representations are low-dimensional, and provide a reduced basis set for collective motions, but unlike global collective modes they are sparsely distributed and spatially localized. This yields a more reliable assignment of essential dynamics modes across different MD time windows.

Principle Component Analysis and Model Reduction for Dynamical Systems with Symmetry Constraints

Dimension reduction in molecular dynamics simulation is often realized through a principle component analysis based upon a singular value decomposition (SVD) of the trajectory. The left singular vectors of a truncated SVD provide the reduced basis. In many biological molecules, such as HIV1 protease, reflective or rotational symmetry should be present in the molecular configuration. Determining this symmetry allows one to provide SVD major modes of motion that best describe the symmetric movements of the protein. We present a method to compute the plane of reflective symmetry or the axis of rotational symmetry of a large set of points. Moreover, we develop an SVD that best approximates the given set while respecting the symmetry. Interesting subproblems arise in the presence of noisy data or in situations where most, but not all of the structure is symmetric. An important part of the determination of the axis of rotational symmetry or the plane of reflection symmetry is an iterative re-weighting scheme. This scheme is rapidly convergent in practice and seems to be very effective in ignoring outliers (points that do not respect the symmetry).

Mathematical metrology for evaluating a 6DOF visual servoing system

In this paper we develop a homogeneous matrix transformation to fit two streams of dynamic six degree of freedom (6DOF) data for evaluating perception systems using ground truth. In particular, we compare object position and orientation results from a 6DOF laser tracker that we consider to be ground truth with results from a real-time visual servoing system from the Purdue Robot Vision Lab. A problem that arises when comparing these two data streams is that they are not necessarily in the same coordinate system. Therefore, a method to transform one coordinate system to the other is needed. We solve this problem by developing an optimization problem that minimizes the space between each coordinate system. In other words, we construct a rotation and translation which best transforms one coordinate space to the other.