Ashitava Ghosal

Comparison of the assumed modes and finite element models for flexible multilink manipulators

The objective of this paper is to compare two discretization models—namely, the assumed modes and finite element mod els—to efficiently represent the link flexibility of robot manip ulators. We present a systematic modeling procedure based on homogeneous transformation matrices for spatial multilink flexible manipulators with both revolute and prismatic joints. The Lagrangian formulation of dynamics and computer algebra are employed to derive closed-form equations of motion. We show that fewer mathematical operations are required for in ertia matrix computation in the finite element model compared with the assumed modes formulation; however, because the number of state space equations are more, the numerical sim ulation time may be greater for finite element models. Use of the finite element model to approximate flexibility usually gives rise to overestimated stiffness matrix. We analytically show that overestimation of structure stiffness may lead to unstable closed-loop response of the original manipulator system, using a model-based control law. We illustrate the complexity owing to the time-dependent frequency equation of the assumed modes model arising in a prismatic jointed flexible link with payload and in manipulators with more than one link with revolute joints. We describe a novel method based on the differential form of the frequency equation to simulate such systems. A model-based decoupling control law is used to compare the dynamic responses of the manipulator system. The results are illustrated by numerical simulation of a flexible spatial RRP configuration robot.

Modeling of slip for wheeled mobile robots

Wheeled mobile robots (WMRs) are known to be non-holonomic systems, and most dynamic models of WMRs assume that the wheels undergo rolling without slipping. This paper deals with the problem of modeling and simulation of motion of a WMR when the conditions for rolling are not satisfied at the wheels. The authors use a traction model where the adhesion coefficient between the wheels of a WMR and a hard flat surface is a function of the wheel slip. This traction model is used in conjunction with the dynamic equations of motion to simulate the motion of the WMR. The simulations show that controllers which do not take into account wheel slip give poor tracking performance for the WMR and path deviation is small only for large adhesion coefficients. This work shows the importance of wheel slip and suggests use of accurate traction models for improving tracking performance of a WMR. >

Singularity analysis of platform-type multi-loop spatial mechanisms

In parallel manipulators and multi-loop mechanisms, singularity is associated with either loss or gain of a degree of freedom. This paper deals with the singularity analysis associated with gain of degree of freedom in a class of spatial mechanisms. We present a geometric condition for platform-type, multi-loop, mechanisms and parallel manipulators, containing spherical joints on the platform, whose existence ensures singularity in such mechanisms. The geometric condition is based on the concept of a common tangent. We show that this condition also implies that the determinant of certain matrices, formed by the differentiation of the loop-closure equations, are zero. We illustrate our results with the help of several multi-loop mechanisms. In particular, we describe the singularity surface for the three-degree-of- freedom RPSSPR-SPR ‘wrist’ mechanism.

Singularity and controllability analysis of parallel manipulators and closed-loop mechanisms

This paper presents a study of kinematic and force singularities in parallel manipulators and closed-loop mechanisms and their relationship to accessibility and controllability of such manipulators and closed-loop mechanisms, Parallel manipulators and closed-loop mechanisms are classified according to their degrees of freedom, number of output Cartesian variables used to describe their motion and the number of actuated joint inputs. The singularities in the workspace are obtained by considering the force transformation matrix which maps the forces and torques in joint space to output forces and torques ill Cartesian space. The regions in the workspace which violate the small time local controllability (STLC) and small time local accessibility (STLA) condition are obtained by deriving the equations of motion in terms of Cartesian variables and by using techniques from Lie algebra.We show that for fully actuated manipulators when the number ofactuated joint inputs is equal to the number of output Cartesian variables, and the force transformation matrix loses rank, the parallel manipulator does not meet the STLC requirement. For the case where the number of joint inputs is less than the number of output Cartesian variables, if the constraint forces and torques (represented by the Lagrange multipliers) become infinite, the force transformation matrix loses rank. Finally, we show that the singular and non-STLC regions in the workspace of a parallel manipulator and closed-loop mechanism can be reduced by adding redundant joint actuators and links. The results are illustrated with the help of numerical examples where we plot the singular and non-STLC/non-STLA regions of parallel manipulators and closed-loop mechanisms belonging to the above mentioned classes

Robust control of multilink flexible manipulators

This paper deals with some robustness aspects of a model based controller used for trajectory tracking in multi-link flexible manipulators. It is known in literature that the finite element formulation over-estimates the natural frequencies of the original system. We show that over-estimation of natural frequencies may lead to unstable closed-loop response for flexible manipulators using a model based inversion control algorithm. A robust controller design based on the second method of Lyapunov using simple quantitative bounds on the model uncertainties is illustrated for use during the trajectory tracking phase in multi-link manipulator control. In order to actively suppress the link vibrations excited during the trajectory tracking phase, a second controller based on end-point sensing and the rigid Jacobian of the manipulator is used. The performance of the two-stage controller is illustrated with the help of numerical simulations of a flexible elbow manipulator.

Modeling of flexible-link manipulators with prismatic joints

The axially translating flexible link in flexible manipulators with a prismatic joint can be modeled using the Euler-Bernoulli beam equation together with the convective terms. In general, the method of separation of variables cannot be applied to solve this partial differential equation. In this paper, we present a nondimensional form of the Euler-Bernoulli beam equation using the concept of group velocity and present conditions under which separation of variables and assumed modes method can be used. The use of clamped-mass boundary conditions lead to a time-dependent frequency equation for the translating flexible beam. We present a novel method to solve this time-dependent frequency equation by using a differential form of the frequency equation. We then present a systematic modeling procedure for spatial multi-link flexible manipulators having both revolute and prismatic joints. The assumed mode/Lagrangian formulation of dynamics is employed to derive closed form equations of motion. We show, using a model-based control law, that the closed-loop dynamic response of modal variables become unstable during retraction of a flexible link, compared to the stable dynamic response during extension of the link. Numerical simulation results are presented for a flexible spatial RRP configuration robot arm. We show that the numerical results compare favorably with those obtained by using a finite element-based model.

A Differential-Geometric Analysis of Singularities of Point Trajectories of Serial and Parallel Manipulators

In this paper, we present a differential-geometric approach to analyze the singularities of task space point trajectories of two and three-degree-of-freedom serial and parallel manipulators. At non-singular configurations, the first-order, local properties are characterized by metric coefficients, and, geometrically, by the shape and size of a velocity ellipse or an ellipsoid. At singular configurations, the determinant of the matrix of metric coefficients is zero and the velocity ellipsoid degenerates to an ellipse, a line or a point, and the area or the volume of the velocity ellipse or ellipsoid becomes zero. The degeneracies of the velocity ellipsoid or ellipse gives a simple geometric picture of the possible task space velocities at a singular configuration. To study the second-order properties at a singularity, we use the derivatives of the metric coefficients and the rate of change of area or volume. The derivatives are shown to be related to the possible task space accelerations at a singular configuration. In the case of parallel manipulators, singularities may lead to either loss or gain of one or more degrees-of-freedom. For loss of one or more degrees-of-freedom, ther possible velocities and accelerations are again obtained from a modified metric and derivatives of the metric coefficients. In the case of a gain of one or more degrees-of-freedom, the possible task space velocities can be pictured as growth to lines, ellipses, and ellipsoids. The theoretical results are illustrated with the help of a general spatial 2R manipulator and a three-degree-of-freedom RPSSPR-SPR parallel manipulator.

Dynamic Modeling and Simulation of a Wheeled Mobile Robot for Traversing Uneven Terrain Without Slip

It is known in literature that a wheeled mobile robot (WMR), with fixed length axle, will undergo slip when it negotiates an uneven terrain. However, motion without slip is desired in WMRs, since slip at the wheel-ground contact may result in significant wastage of energy and may lead to a larger burden on sensor based navigation algorithms. To avoid slip, the use of a variable length axle (VLA) has been proposed in the literature and the kinematics of the vehicle has been solved depicting no-slip motion. However, the dynamic issues have not been addressed adequately and it is not clear if the VLA concept will work when gravity and inertial loads are taken into account. To achieve slip-free motion on uneven terrain, we have proposed a three-wheeled WMR architecture with torus shaped wheels, and the two rear wheels having lateral tilt capability. The direct and inverse kinematics problem of this WMR has been solved earlier and it was shown by simulation that such a WMR can travel on uneven terrain without slip. In this paper, we derive a set of 27 ordinary differential equations (ODEs) which form the dynamic model of the three-wheeled WMR. The dynamic equations of motion have been derived symbolically using a Lagrangian approach and computer algebra. The holonomic and nonholonomic constraints of constant length and no-slip, respectively, are taken into account in the model. Simulation results clearly show that the three-wheeled WMR can achieve no-slip motion even when dynamic effects are taken into consideration.

Chaos in Robot Control Equations

The motion of a feedback controlled robot can be described by a set of nonlinear ordinary differential equations. In this paper, we examine the system of two second-order, nonlinear ordinary differential equations which model a simple two-degree-of-freedom planar robot, undergoing repetitive motion in a plane in the absence of gravity, and under two well-known robot controllers, namely a proportional and derivative controller and a model-based controller. We show that these differential equations exhibit chaotic behavior for certain ranges of the proportional and derivative gains of the controller and for certain values of a parameter which quantifies the mismatch between the model and the actual robot. The system of nonlinear equations are non-autonomous and the phase space is four-dimensional. hence, it is difficult to obtain significant analytical results. In this paper, we use the Lyapunov exponent to test for chaos and present numerically obtained chaos maps giving ranges of gains and mismatch parameters which result in chaotic motions. We also present plots of the chaotic attractor and bifurcation diagrams for certain values of the gains and mismatch parameters. From the bifurcation diagrams, it appears that the route to chaos is through period doubling

Nonlinear Dynamics and Chaotic Motions in Feedback-Controlled Two-and Three-Degree-of-Freedom Robots

The dynamics of a feedback-controlled rigid robot is most commonly described by a set of nonlinear ordinary differential equations. In this paper we analyze these equations, representing the feedback-controlled motion of twoand three-degrees-of-freedom rigid robots with rnvolute (R) and prismatic (P) joints in the absence of compliance, friction, and potential energy for the possibility of chaotic motions. We first study the unforced or inertial motions of the robots, and show that when the Gaussian or Riemannian curvature of the configuration space of a robot is negative, the robot equations can exhibit chaos. If the curvature is zero or positive, then the robot equations cannot exhibit chaos. We show that among the two-degrees-of-freedom robots, the PP and the PR robot have zer Gaussian curvature while the RP and RR rohots have negative Gaussian curvatures. For the three-degrees-offireedom robots, we analyze the two well-known RRP and RRR configurations of the Stanford arm and the PUMA manipulator, respectively, and derive the conditions for negative curvature and possible chaotic motions. The criteria of negative curvature cannot be usedfor the forced or feedback-controlled motions. For the foired motion, we resort to the well-known numerical techniques and compute chaos maps, Poincard maps, and bifurcation diagrams. Numerical results are presentedfor the two-degreesf-ffreedom RP and RR robots, and we show that these robot equations can exhibit chaos for low controller gains and for large underestimated models. From the bifurcation diagrams, the mute to chaos appears to he through period doubling.