Frank C. Park

A Lie group formulation of robot dynamics

In this article we present a unified geometric treatment of robot dynamics. Using standard ideas from Lie groups and Rieman nian geometry, we formulate the equations of motion for an open chain manipulator both recursively and in closed form. The recursive formulation leads to an O(n) algorithm that ex presses the dynamics entirely in terms of coordinate-free Lie algebraic operations. The Lagrangian formulation also ex presses the dynamics in terms of these Lie algebraic operations and leads to a particularly simple set of closed-form equations, in which the kinematic and inertial parameters appear explic itly and independently of each other. The geometric approach permits a high-level, coordinate-free view of robot dynamics that shows explicitly some of the connections with the larger body of work in mathematics and physics. At the same time the resulting equations are shown to be computationally ef fective and easily differentiated and factored with respect to any of the robot parameters. This latter feature makes the ge ometric formulation attractive for applications such as robot design and calibration, motion optimization, and optimal control, where analytic gradients involving the dynamics are required.

Robot sensor calibration: solving AX=XB on the Euclidean group

The equation AX=XB on the Euclidean group arises in the problem of calibrating wrist-mounted robotic sensors. In this article the authors derive, using methods of Lie theory, a closed-form exact solution that can be visualized geometrically, and a closed-form least squares solution when A and B are measured in the presence of noise. >

Singularity Analysis of Closed Kinematic Chains

This paper presents a coordinate-invariant differential geometric analysis of kinematic singularities for closed kinematic chains containing both active and passive joints. Using the geometric framework developed in Park and Kim (1996) for closed chain manipulability analysis, we classify closed chain singularities into three basic types: (i) those corresponding to singular points of the joint configuration space (configuration space singularities), (ii) those induced by the choice of actuated joints (actuator singularities), and (iii) those configurations in which the end-effector loses one or more degrees of freedom of available motion (end-effector singularities). The proposed geometric classification provides a high-level taxonomy for mechanism singularities that is independent of the choice of local coordinates used to describe the kinematics, and includes mechanisms that have more actuators than kinematic degrees of freedom.

Kinematic Dexterity of Robotic Mechanisms

In this article we develop a mathematical theory for optimizing the kinematic dexterity of robotic mechanisms and obtain a collection of analytical tools for robot design. The performance criteria we consider are workspace volume and dexterity; by the latter we mean the ability to move and apply forces in arbitrary directions as easily as possible. Clearly, dexterity and workspace volume are intrinsic to a mechanism, so that any mathematical formulation of these properties must necessarily be independent of the particular coordinate representation of the kinematics. By regarding the forward kinematics of a mechanism as defining a mapping between Riemannian manifolds, we apply the coordinate-free language of differential geometry to define natural measures of kinematic dexterity and workspace volume. This approach takes into account the geometric and topolog ical structures of the joint and workspaces. We show that the functional associated with harmonic mapping theory provides a natural measure of the kinematic dexterity of a mechan ism. Optimal designs among the basic classes of mechanisms are determined as extrema of this measure. We also examine the qualitative connections between kinematic dexterity and workspace volume.

Smooth invariant interpolation of rotations

We present an algorithm for generating a twice-differentiable curve on the rotation group SO(3) that interpolated a given ordered set of rotation matrices at their specified knot times. In our approach we regard SO(3) as a Lie group with a bi-invariant Riemannian metriac, and apply the coordinate-invariant methods of Riemannian geometry. The resulting rotation curve is easy to compute, invariant with respect to fixed and moving reference frames, and also approximately minimizes angular acceleration.

Design and analysis of a redundantly actuated parallel mechanism for rapid machining

This paper describes the design, construction, and performance analysis of the Eclipse, a redundantly actuated six-degree-of-freedom parallel mechanism intended for rapid machining. The Eclipse is a compact mechanism capable of performing five-face machining in a single setup while retaining the advantages of high stiffness and high accuracy characteristic of parallel mechanisms. We compare numerical and algebraic algorithms for the forward and inverse kinematics of a class of the Eclipse and formalize the notion of machine tool workspace. We also develop a simple method for the first-order elasto-kinematic analysis of parallel mechanisms that is amenable to design iterations. A complete characterization of the singularities of the Eclipse is given, and redundant actuation is proposed as a solution. The Eclipse case study demonstrates how diverse analytical tools originally developed in a robotics context can be synthesized into a practical design methodology for parallel mechanisms.

Bézier Curves on Riemannian Manifolds and Lie Groups with Kinematics Applications

In this article we generalize the concept of Bezier curves to curved spaces, and illustrate this generalization with an application in kinematics. We show how De Casteljaus algorithm for constructing Bezier curves can be extended in a natural way to Riemannian manifolds. We then consider a special class of Riemannian manifold, the Lie groups. Because of their group structure Lie groups admit an elegant, efficient recursive algorithm for constructing Bezier curves. Spatial displacements of a rigid body also form a Lie group, and can therefore be interpolated (in the Bezier sense) using this recursive algorithm. We apply this algorithm to the kinematic problem of trajectory generation or motion interpolation for a moving rigid body. The orientation trajectory of motions generated in this way have the important property of being invariant with respect to choices of inertial and body-fixed reference frames

Kinematic sensitivity analysis of the 3-UPU parallel mechanism

This paper addresses the kinematic sensitivity of the three degree-of-freedom 3-UPU parallel mechanism, a mechanism consisting of a fixed base and a moving platform connected by three serial UPU chains. Although a mathematical mobility analysis confirms that the mechanism has three degrees of freedom, hardware prototypes exhibit unexpected large motions of the platform even when the three prismatic joints are locked at arbitrary configurations. Existing mathematical classifications of kinematic singularities also fail to explain the gross motions of the 3-UPU. This paper resolves this apparent paradox. We show that the 3-UPU is highly sensitive to certain minute clearances in the universal joint, and that a careful kinematic sensitivity analysis of the 3-UPU augmented with virtual joints satisfactorily explains the gross motions. Observations with a hardware experimental prototype confirm the results of our sensitivity analysis.

Computational aspects of the product-of-exponentials formula for robot kinematics

In this article we investigate the modeling and computational aspects of the product-of-exponentials (POE) formula for robot kinematics. While its connections with Lie groups and Lie algebras give the POE equations mathematical appeal, little is known regarding its usefulness for control and other applications. We show that the POE formula admits a simple global interpretation of an open kinematic chain and possesses several useful device-independent features absent in the Denavit-Hartenberg (DH) representations. Methods for efficiently computing the forward kinematics and Jacobian using these equations are presented. In particular, the computational requirements for evaluating the Jacobian from the POE formula are compared to those of the recursive methods surveyed in Orin and Schrader (1984). >

Eclipse II: a new parallel mechanism enabling continuous 360-degree spinning plus three-axis translational motions

This paper presents the Eclipse II, a new six-degrees-of-freedom parallel mechanism, which can be used as a basis for general motion simulators. The Eclipse II is capable of x, y, and z axes translations, and a, b, and c axes rotations. In particular, it has the advantage of enabling continuous 360/spl deg/ spinning of the platform. We first describe the computational procedures for the forward and inverse kinematics of the Eclipse II. Next, the complete singularity analysis is presented for the two cases of end-effector singularity and actuator singularity. Finally, two additional actuators are added to the original mechanism to eliminate both types of singularity within the workspace.