A Discrete Geometric Optimal Control Framework for Systems with Symmetries

This paper studies the optimal motion control of mechanical systems through a discrete geometric approach. At the core of our formulation is a discrete Lagrange-d’Alembert- Pontryagin variational principle, from which are derived discrete equations of motion that serve as constraints in our optimization framework. We apply this discrete mechanical approach to holonomic systems with symmetries and, as a result, geometric structure and motion invariants are preserved. We illustrate our method by computing optimal trajectories for a simple model of an air vehicle flying through a digital terrain elevation map, and point out some of the numerical benefits that ensue

End-effector vibrations reduction in trajectory tracking for mobile manipulator

A method of motion planning for a mobile manipulator taking into account damping the end-effector vibrations is presented. The primary task of the robot is to trace a given end-effector trajectory. The redundant degrees of freedom are used to fulfil secondary objectives such as minimisation of platform kinetic energy and maximisation of holonomic manipulability measure, which leads to reduction of the end-effector vibrations. The method is based on Jacobian pseudo inverse at the acceleration level. Nonholonomic constraints in a Pfaffian form are explicitly incorporated to the control algorithm. A computer example involving a mobile manipulator consisting of a nonholonomic platform (2, 0) class and SCARA-type holonomic manipulator operating in two-dimensional task space is also presented

Control of Mechanical Systems With Rolling Constraints: Application to Dynamic Control of Mobile Robots

There are many examples of mechanical systems which require rolling contacts between two or more rigid bodies. Rolling contacts engender nonholonomic constraints in an otherwise holonomic system. In this paper, we develop a unified approach to the control of mechanical systems subject to both holonomic and nonholonomic constraints. We first present a state space realization of a constrained system and show that it is not input-state linearizable. We then discuss the input-output linearization and zero dynamics of the system. This approach is applied to the dynamic control of mobile robots. Two types of control algorithms for mobile robots are investigated: (a) trajectory tracking, and (b) path following. In each case, a smooth nonlinear feedback is obtained to achieve asymptotical input-output stability, and Lagrange stability of the overall system. Simulation results are presented to demonstrate the effectiveness of the control algorithms and to compare the performance of trajectory tracking and path following algorithms

Mobile manipulators collision-free trajectory planning with regard to end-effector vibrations elimination

A sub-optimal point-to-point trajectory planning method for mobile manipulators operating in the workspace including obstacles taking into account the damping of the end-effector vibrations is presented. The proposed solution is based on extended Jacobian approach and redundancy resolution at the acceleration level. Fulfilment of the condition stopping the mobile manipulator at the destination point is guaranteed, which leads to elimination of the end-effector vibrations and significantly increases positioning accuracy. The effectiveness of the presented method is shown and compared to the classical Jacobian pseudo inverse approach. A computer example involving a mobile manipulator consisting of a nonholonomic platform (2, 0) class and SCARA-type holonomic manipulator operating in two-dimensional task space including obstacle is also presented

The Hamiltonian and Lagrangian Approaches to the Dynamics of Nonholonomic Systems

This paper compares the Hamiltonian approach to systems with nonholonomic constraints (see Weber [1982], Arnold [1988], and Bates and Sniatycki [1993], van der Schaft and Maschke [1994] and references therein) with the Lagrangian approach (see Koiller [1992], Ostrowski [1996] and Bloch, Krishnaprasad, Marsden and Murray [1996]). There are many differences in the approaches and each has its own advantages; some structures have been discovered on one side and their analogues on the other side are interesting to clarify. For example, the momentum equation and the reconstruction equation were first found on the Lagrangian side and are useful for the control theory of these systems, while the failure of the reduced two form to be closed (i.e., the failure of the Poisson bracket to satisfy the Jacobi identity) was first noticed on the Hamiltonian side. Clarifying the relation between these approaches is important for the future development of the control theory and stability and bifurcation theory for such systems. In addition to this work, we treat, in this unified framework, a simplified model of the bicycle (see Getz [1994] and Getz and Marsden [1995]), which is an important underactuated (nonminimum phase) control system

Stabilizability and Motion Tracking Conditions for Mechanical Nonholonomic Control Systems

This paper addresses formulation of stabilizability and motion tracking conditions for mechanical systems from the point of view of constraints put on them. We present a new classification of constraints, which includes nonholonomic constraints that arise in both mechanics and control. Based on our classification we develop kinematic and dynamic control models of systems subjected to these constraints. We demonstrate that a property of being a “hard-to-control” nonholonomic system may not be related to the nature of the constraints. It may result from the formulation of control objectives for a system. We examine two control objectives which are stabilization to the target equilibrium by a continuous static state feedback control and motion tracking. Theory is illustrated with examples of control objective formulations for systems with constraints of various types

Forces Associated with Nonlinear Nonholonomic Constraint Equations

A concise method has been formulated for identifying a set of forces needed to constrain the behavior of a mechanical system, modeled as a set of particles and rigid bodies, when it is subject to motion constraints described by nonholonomic equations that are inherently nonlinear in velocity. An expression in vector form is obtained for each force; a direction is determined, together with the point of application. This result is a consequence of expressing constraint equations in terms of dot products of vectors rather than in the usual way, which is entirely in terms of scalars and matrices. The constraint forces in vector form are used together with two new analytical approaches for deriving equations governing motion of a system subject to such constraints. If constraint forces are of interest they can be brought into evidence in explicit dynamical equations by employing the well-known nonholonomic partial velocities associated with Kane's method; if they are not of interest, equations can be formed instead with the aid of vectors introduced here as nonholonomic partial accelerations. When the analyst requires only the latter, smaller set of equations, they can be formed directly; it is not necessary to expend the labor to form the former, larger set first and subsequently perform matrix multiplications

Geometric path planning without maneuvers for nonholonomic parallel orienting robots

Current geometric path planners for nonholonomic parallel orienting robots generate maneuvers consisting of a sequence of moves connected by zero-velocity points. The need for these maneuvers restrains the use of this kind of parallel robots to few applications. Based on a rather old result on linear time-varying systems, this letter shows that there are infinitely differentiable paths connecting two arbitrary points in SO(3) such that the instantaneous axis of rotation along the path rest on a fixed plane. This theoretical result leads to a practical path planner for nonholonomic parallel orienting robots that generates single-move maneuvers. To present this result, we start with a path planner based on three-move maneuvers, and then we proceed by progressively reducing the number of moves to one, thus providing a unified treatment with respect to previous geometric path planners.Peer ReviewedPostprint (author's final draft