Milos Zefran

On the generation of smooth three-dimensional rigid body motions

This paper addresses the problem of generating smooth trajectories between an initial and a final position and orientation in space. The main idea is to define a functional depending on velocity or its derivatives that measures smoothness of trajectories and find a trajectory that minimizes this functional. In order to ensure that the computed trajectories are independent of the parametrization of positions and orientations, we use the notions of Riemannian metric and covariant derivative from differential geometry and formulate the problem as a variational problem on the Lie group of spatial rigid body displacements. We show that by choosing an appropriate measure of smoothness, the trajectories can be made to satisfy boundary conditions on the velocities or higher order derivatives. Dynamically smooth trajectories can be obtained by incorporating the inertia of the system into the definition of the Riemannian metric. We state the necessary conditions for the shortest distance, minimum acceleration and minimum jerk trajectories.

On the 6 × 6 cartesian stiffness matrix for three-dimensional motions

Abstract It has been recently observed by Pigoski et al. [1, 2], and by Ciblak and Lipkin [3] that the Cartesian stiffness matrix associated with a linear elastic coupling between two rigid bodies is, in general, asymmetric if the resulting forces and moments do not sum to zero. In this paper, methods of differential geometry and properties of Lie groups are used to show that in any conservative system subjected to a non-zero external load, if motions in SE (3), the special Euclidean group of rigid body motions in three dimensions, are considered, the resulting 6 × 6 Cartesian stiffness matrix is asymmetric in any inertial (fixed) or body-fixed reference frame. We prove the general result that in any subgroup of SE (3), if the system is subject to external forces and moments, the Cartesian stiffness matrix is symmetric if finite displacements along the basis twists are used to generate the stiffness matrix commute. Further, we derive several useful properties of stiffness matrices using ideas from Lie theory. In particular, we offer a simple proof to show that the stiffness matrix in the body-fixed reference frame is the transpose of the stiffness matrix in the inertial reference frame, a result also derived in [3]. Finally, we outline a method to construct a symmetric stiffness matrix by choosing an appropriate moving reference frame that is not fixed to any rigid body.

Design of switching controllers for systems with changing dynamics

We present a framework for designing stable control schemes for systems with changing dynamics (SCD). Such systems form a subset of hybrid systems; their stabilization is therefore a problem in hybrid control. It is often difficult or even impossible to design a single controller that would stabilize a SCD. An appealing alternative are switching control schemes, where a different controller is employed in each dynamic regime and the stability of the overall system is ensured through an appropriate switching scheme. We formulate a set of sufficient conditions for the stability of a switching control scheme. We show that by imposing a hierarchy among the controllers, sufficient conditions can be formulated in a form suitable for the controller design. The hierarchy is formally defined through a partial order. Our methodology is applied to stabilization of a two-wheel mobile robot of the Hilare type, where the wheels are allowed to slip.

A Geometrical Approach to the Study of the Cartesian Stiffness Matrix

The stiffness of a rigid body subject to conservative forces and moments is described by a tensor, whose components are best described by a 6X6 Cartesian stiffness matrix. We derive an expression that is independent of the parameterization of the motion of the rigid body using methods of differential geometry. The components of the tensor with respect to a basis of twists are given by evaluating the tensor on a pair of basis twists. We show that this tensor depends on the choice of an affine connection on the Lie group, SE (3). In addition, we show that the definition of the Cartesian stiffness matrix used in the literature [1,2] implicitly assumes an asymmetric connection and this results in an asymmetric stiffness matrix in a general loaded configuration. We prove that by choosing a symmetric connection we always obtain a symmetric Cartesian stiffness matrix. Finally, we derive stiffness matrices for different connections and illustrate the calculations using numerical examples.

Affine connections for the Cartesian stiffness matrix

We study the 6/spl times/6 Cartesian stiffness matrix. We show that the stiffness of a rigid body subjected to conservative forces and moments is described by a (0,2) tensor which is the Hessian of the potential function. The key observation of the paper is that since the Hessian depends on the choice of an affine connection in the task space, so will the Cartesian stiffness matrix. Further, the symmetry of the Hessian and thus of the stiffness matrix depends on the symmetry of the connection. The connection that is implicit in the definition of the Cartesian stiffness matrix through the joint stiffness matrix (Salisbury, 1980) is made explicit and shown to be symmetric. In contrast, the direct definition of the Cartesian stiffness matrix in Griffis (1993), Ciblak and Lipkin (1994) and Howard et al. (1996) is shown to be derived from an asymmetric connection. A numerical example is provided to illustrate the main ideas of the paper.

Planning of smooth motions on SE(3)

This paper addresses the general problem of generating smooth trajectories between an initial and a final position and orientation. A functional depending an velocity and its higher derivatives involving a left invariant Riemannian metric on SE(3) is used to measure the smoothness of a trajectory. The problem of determining a smooth trajectory between two points is formulated as a variational problem on SE(3). The authors derive necessary conditions for the shortest distance and minimum jerk trajectories and solve the resulting two-point boundary value problem.

Modeling and controllability for a class of hybrid mechanical systems

This paper studies a class of hybrid mechanical systems that locomote by switching between constraints defining different dynamic regimes. We develop a geometric framework for modeling smooth phenomena such as inertial forces, holonomic and nonholonomic constraints, as well as discrete features such as transitions between smooth dynamic regimes through plastic and elastic impacts. We focus on devices that are able to switch between constraints at an arbitrary point in the configuration space. This class of hybrid mechanical control systems can be described in terms of affine connections and jump transition maps that are linear in the velocity. We investigate two notions of local controllability, the equilibrium and kinematic controllability, and provide sufficient conditions for each of them. The tests rely on the assumption of zero velocity switches. We illustrate the modeling framework and the controllability tests on a planar sliding, clamped, and rolling device. In particular, we show how the analysis can be used for motion planning.

Optimal control of systems with unilateral constraints

Problems in robotics and biomechanics such as trajectory planning or resolution of redundancy can be effectively solved using optimal control. Such systems are often subject to unilateral constraints. Examples include tasks involving contacts (e.g., walking, running, multifingered or multiarm manipulation), and other tasks that may not involve contacts but in which the system state or the inputs must satisfy inequality conditions (e.g., limits on actuator forces). This paper shows how problems of optimal control in robotics that involve unilateral constraints can be efficiently solved by first formulating the constrained optimal control problem as an unconstrained problem of the calculus of variations and then solving it using an integral formulation. This method has several advantages over the Pontryagin minimum principle which is traditionally employed to solve such problems. An example of two-arm manipulation with inequality constraints due to Coulomb friction is used to demonstrate the formulation of the problem and the algorithms.

Optimal trajectories and force distribution for cooperating arms

The optimization of trajectories and actuator torques for a dual arm manipulation system is considered. Given the starting and final configurations, we find the trajectories that minimize: (a) the integral of the norm of the vector of derivatives of the actuator forces; and (b) the integral of the norm of the actuator forces. In this way both kinematic and actuator redundancy are resolved. The optimization problem reduces to solving a two-point boundary valve problem for coupled, nonlinear differential equations. The effect of different parameters such as preload and inertia are investigated and the results are compared with those obtained using other well-known cost functions.<<ETX>>

Stabilization of systems with changing dynamics by means of switching

We present a framework for designing stable control schemes for systems whose dynamics change. The idea is to develop a controller for each of the regions defined by different dynamic characteristics and design a switching scheme that guarantees the stability of the overall system. We derive sufficient conditions for the stability of the switching scheme for systems evolving on a sequence of embedded manifolds. An important feature of the proposed framework is that if the conditions are satisfied by pairs of controllers adjacent in the hierarchy, the overall system will be stable. This makes the application of our results particularly straight forward. The methodology is applied to stabilization of a shimmying wheel, where changes in the dynamic behaviour are due to switches between sliding and rolling.