In this article we present a unified geometric treatment of robot dynamics. Using standard ideas from Lie groups and Riemannian 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 expresses the dynamics entirely in terms of coordinate-free Lie algebraic operations. The Lagrangian formulation also expresses 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 explicitly 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 effective and easily differentiated and factored with respect to any of the robot parameters. This latter feature makes the geometric formulation attractive for applications such as robot design and calibration, motion optimization, and optimal control, where analytic gradients involving the dynamics are required.
We present an algorithm for generating a twice-differentiable curve on the rotation group SO(3) that interpolates 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 metric, 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.
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.
This paper presents a coordinate-invariant differential geometric analysis of manipulability for closed kinematic chains containing active and passive joints. The formulation treats both redundant and nonredundant mechanisms, as well as over-actuated and exactly actuated ones, in a uniform manner. Dynamic characteristics of the mechanism and manipulated object can also be naturally included by an appropriate choice of Riemannian metric. We illustrate the methodology with several closed chain examples, and provide a practical algorithm for manipulability analysis of general chains.
This paper presents an optimization‐based framework for emulating the low‐level capabilities of human motor coordination and learning. Our approach rests on the observation that in most biological motor learning scenarios some form of optimization with respect to a physical criterion is taking place. By appealing to techniques from the theory of Lie groups, we are able to formulate the equations of motion of complex multibody systems in such a way that the resulting optimization problems can be solved reliably and efficiently—the key lies in the ability to compute exact analytic gradients of the objective function without resorting to numerical approximations. The methodology is illustrated via a wide range of optimized, “natural” motions for robots performing various human‐like tasks—for example, power lifting, diving, and gymnastics. © 2001 John Wiley & Sons, Inc.
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