Christopher J. Damaren

Simulation of Flexible-Link Manipulators With Inertial and Geometric Nonlinearities

Several important issues relevant to modeling of flexible-link robotic manipulators are addressed in this paper. First, we examine the question of which inertial nonlinearities should be included in the equations of motion for purposes of simulation. A complete model incorporating all inertial terms that couple rigid-body and elastic motions is presented along with a rational scheme for classifying them. Second, the issue of geometric nonlinearities is discussed. These are terms whose origin is the geometrically nonlinear theory of elasticity, as well as the terms arising from the interbody coupling due to the elastic deformation at the link tip. Accordingly, a general way of incorporating the well-known geometric stiffening effect is presented along with several schemes for treating the elastic kinematics at the joint interconnections. In addition, the question of basis function selection for spatial discretization of the elastic displacements is also addressed. The finite element method and an eigenfunction expansion techniques are presented and compared. All issues are examined numerically in the context of a simple beam example and the Space Shuttle Remote Manipulator System. Unlike a single-link system, the results for the latter show that all terms are required for accurate simulation of faster maneuvers. Hence, the conclusions of the paper are contrary to some of the previous findings on the validity of various models for dynamics simulation of flexible-body systems

On the design of strictly positive real transfer functions

The synthesis of strictly positive real transfer functions is considered. For a given Hurwitz polynomial of degree n comprising the denominator polynomial, necessary and sufficient conditions on the numerator which render a rational function strictly positive real are given. In the case where the function is strictly proper, a parameterization of the polynomial numerator by n real numbers satisfying a simple constraint is provided. The approach taken employs factorization of a polynomial into its even and odd parts. The relationship of the results to those provided by the Kalman-Yakubovich lemma is given and the present method shown to have certain advantages. >

Passivity analysis for flexible multilink space manipulators

The important input-output property of passivity is explored for a general flexible space manipulator with chain topology. The manipulator is assumed to consist of rigid and/or flexible links interconnected via revolute joints, and a free rigid spacecraft and cantilevered payload are modeled at the base and tip, respectively. Actuation on the spacecraft and torques at the joints serve as control inputs and a suitably modified input variable is constructed. The notion of reflected tip position introduced by Wang and Vidyasagar for a single flexible link is extended to the multilink case and used to define a corresponding modified output variable. The dynamics governing the system are developed using a Lagrangian approach and both linearized and nonlinear forms of the mapping relating modified inputs to modified outputs are examined. Our major result shows that the transfer function in the linear case is positive real when the spacecraft and payload are much more massive than the manipulator links. The corresponding nonlinear analysis shows that the mapping is, in fact, passive and uncovers an approximate static relationship between the elastic coordinates and applied torques. A numerical example employing the Space Shuttle, remote manipulator system, and payload is used to demonstrate the validity of the theoretical results. Applications to control system design are indicated.

Modal Properties and Control System Design for Two-Link Flexible Manipulators

The vibration modes of a generic two-link fiexible manipulator are studied as a function of the link, rotor, and tip (statorlpayload) mass distribution. Necessary and sufficient conditions are devel oped for all vibration modes to exhibit a node at the manipulator endpoint. A rigorous treatment of the relevant kinematics and dy namics shows that this property can be closely achieved for large tip/link mass ratio and sufficiently small rotor inertia. The major impacts of this result on feedforward/feedback controller design are uncovered. First, the nonlinear joint torque to end-effector motion dynamics become essentially equivalent to those of the rigid case. Second, an output involving the endpoint rates and elastic motions is shown to possess the passivity property for suitably defined inputs. This permits the design of simple controllers that furnish endpoint stabilization with simultaneous vibration suppression. A numeri cal example is used to illustrate the results and demonstrate the achievable performance using the controller design concepts.

Optimal control of large space structures using distributed gyricity

An optimal formulation is developed for the shape control of large flexible spacecraft possessing a distribution of control moment gyros. The structure is modeled as a continuum in mass, stiffness, and gyricity (stored angular momentum). A small, linear viscous damping term completes the dynamical description. The equation of motion is formulated in continuum form, and a brief eigenanalysis is presented that permits the modal equations of motion to be derived. The optimal control problem is treated using distributed-parameter concepts, and a modal expansion for the resulting Riccati operator reduces the problem to the solution of a matrix Riccati equation. Such an approach permits pointwise control moment gyros as well as the distributed analog to be handled with the same theory. By means of an example, the use of distributed gyricity is demonstrated to be very effective for shape control of large space structures. Moreover, the notion of a continuous distribution of gyricity is shown to be beneficial in modeling the dynamics and control of flexible spacecraft employing many control moment gyros.

Approximate inverse dynamics and passive feedback for flexible manipulators with large payloads

A derivation is presented of an approximate form of the dynamics governing a structurally flexible manipulator carrying a massive payload at its end-effector. An output called the /spl mu/-tip rate which incorporates end-effector and elastic motions is introduced. The input-output mapping relating a transformed version of the joint torques to the /spl mu/-tip rates is shown to be passive for large payloads. A feedforward torque strategy is developed which preserves the passivity property in the error dynamics and a suitable Lyapunov function is used to demonstrate global asymptotic stability of the tracking provided by a PD law. Implementation of the controllers without measurements of the elastic coordinates and rates is shown to be possible. Simulation studies of a six DOF manipulator with flexible links, modeled after the Shuttle Remote Manipulator System, demonstrate excellent tracking in all six Cartesian end-effector coordinates, even for payloads with modest mass properties. A major conclusion is that some of the problems normally associated with lack of collocation in flexible manipulators can be surmounted when large (massive) payloads are involved.

On the Dynamics and Control of Flexible Multibody Systems with Closed Loops

The motion control problem for cooperating flexible robot arms manipulating a large rigid payload is considered. An output that depends on the payload position and contributions from the joint motion of each arm is constructed whose rate yields the passivity property with respect to a special input. The input is a combination of the torques from each arm and contains a free load-sharing parameter. The passivity property is shown to depend on the payload mass properties, and in cases where the payload is large, a passivity-based controller combining feedforward and feedback as elements is devised, which yields tracking. An experimental facility consisting of two planar 3-DoF arms is used to implement the strategies. Good tracking is observed and compared with simulation predictions for closed-loop flexible multibody systems.

Correspondence: Comments on Fully magnetic attitude control for spacecraft subject to gravity gradient

The Lyapunov argument used in Wisniewski and Blanke, (Automatica 35 (1999) 1201) to establish asymptotic stability of a magnetic control law for an earth-orbiting spacecraft is incorrect. It is shown here that a small change to the assumed form of the control law can remedy the situation.

Adaptive control of flexible manipulators carrying large uncertain payloads

An adaptive controller is presented for a manipulator with revolute joints and structurally flexible links which carries a rigid payload with unknown mass properties. Under the assumption that the payload mass is much greater than that of the manipulator, globally stable tracking of the Cartesian end-effector coordinates is established. Key ideas underlying the controller development are the passivity of a mapping involving the end-effector rates as part of the output and a fixed parameter feedforward which preserves this property. The concept of filtered error is borrowed from previous work on rigid arms and suitably modified in developing the adaptive law. Although measurements of the tip positions and rates are needed, there is no requirement for sensing of the elastic coordinates. A numerical example involving a six DOF manipulator with flexible links demonstrates excellent tracking with respect to a simulation based on the exact motion equations.

Controllability and observability of gyroelastic vehicles

Stored angular momentum devices such as reaction wheels and control moment gyros have been used extensively for space vehicle attitude control. They represent a potential source of actuation for vibration and shape control of large space structures where they can potentially be distributed in large numbers. The vibration characteristics of these gyroelastic vehicles are affected by the presence of stored angular momentum and, hence, so are the conditions for controllability and observability. In this paper, these conditions are derived for systems modeled as gyroelastic continua, i.e., vehicles with continuous distributions of mass, stiffness, and gyricity (stored angular momentum). The conditions are expressed in terms of the gyroelastic modes and cover the case of pointwise actuators and those modeled in a continuum fashion. A numerical example is used to show that the degree of controllability in the continuum case can be interpreted as that corresponding to the limit of a sequence of pointwise control problems. The observability conditions are developed for a general class of measurements. The concept of a gyroelastic node is introduced and related to the problem of locating sensors.