Subir Kumar Saha

Dynamics of Nonholonomic Mechanical Systems Using a Natural Orthogonal Complement

The dynamics equations governing the motion of mechanical systems composed of rigid bodies coupled by holonomic and nonholonomic constraints are derived. The underlying method is based on a natural orthogonal complement of the matrix associated with the velocity constraint equations written in linear homogeneous form. The method is applied to the classical example of a rolling disk and an application to a 2-dof Automatic Guided Vehicle is outlined.

Dynamics of Serial Multibody Systems Using the Decoupled Natural Orthogonal Complement Matrices

Constrained dynamic equations of motion of serial multibody systems consisting of rigid bodies in a serial kinematic chain are derived in this paper. First, the Newton-Euler equations of motion of the decoupled rigid bodies of the system at hand are written. Then, with the aid of the decoupled natural orthogonal complement (DeNOC) matrices associated with the velocity constraints of the connecting bodies, the Euler-Lagrange independent equations of motion are derived. The De NOC is essentially the decoupled form of the natural orthogonal complement (NOC) matrix, introduced elsewhere. Whereas the use of the latter provides recursive order n-n being the degrees-of-freedom of the system at hand-inverse dynamics and order n 3 forward dynamics algorithms, respectively, the former leads to recursive order n algorithms for both the cases. The order n algorithms are desirable not only for their computational efficiency but also for their numerical stability, particularly, in forward dynamics and simulation, where the systems accelerations are solved from the dynamic equations of motion and subsequently integrated numerically. The algorithms are illustrated with a three-link three-degrees-of-freedom planar manipulator and a six-degrees-of-freedom Stanford arm.

RECURSIVE KINEMATICS AND DYNAMICS FOR PARALLEL STRUCTURED CLOSED-LOOP MULTIBODY SYSTEMS*

A kinematic formulation for the parallel structured closed-loop multibody mechanical systems, such as Stewart Platform and Hexapod machine tools, is presented in this paper, which is recursive in nature. This also leads to the minimum order representation of the dynamic equations of motion. The recursive algorithms are known for their efficiency when a system is large. They also provide many physical interpretations. On the other hand, the minimum order dynamic equations of motion are desired in control and simulation. For the latter a minimum set of dynamic equations of motion leads to a numerically stable integration algorithm that does not violate the kinematic constraints. Two recursive algorithms, one for the inverse and another for the forward dynamics, are proposed. The overall complexity of either problem is O(n) + n O(m), where n and m are the number of legs and the total number of rigid bodies in each leg, respectively. Hence the proposed formulation exploits the advantages of both minimum order representation and recursive algorithms, which earlier were available only for the open-loop systems such as serial manipulators. The method is illustrated with three examples: a one-degree-of-freedom (DOF) slider-crank mechanism, four-bar linkage, and a two-DOF five-bar planar parallel manipulator. *Communicated by J. McPhee.

A decomposition of the manipulator inertia matrix

A decomposition of the manipulator inertia matrix is essential, for example, in forward dynamics, where the joint accelerations are solved from the dynamical equations of motion. To do this, unlike a numerical algorithm, an analytical approach is suggested in this paper. The approach is based on the symbolic Gaussian elimination of the inertia matrix that reveal recursive relations among the elements of the resulting matrices. As a result, the decomposition can be done with the complexity of order n, O(n), where n being the degrees of freedom of the manipulator, as opposed to an O(n/sup 3/) scheme, required in the numerical approach. In turn, O(n) inverse and forward dynamics algorithms can be developed. As an illustration, an O(n) forward dynamics algorithm is presented.

Kinematics and dynamics of a three-wheeled 2-DOF AGV

A systematic method for the kinematic and dynamic modeling of a two-degree-of-freedom (DOF) automatic guided vehicle (AGV) is presented. This type of methodology can be used to analyze, design, simulate, and control any kind of rolling robots. The concept of orthogonal complement is used to develop the dynamical equations of motion. The vehicle is analyzed for simulation purposes. Simulation results are reported. It is shown that, using the natural orthogonal complement of the matrix of velocity constraint equations, it is possible to derive systematically the Euler-Lagrange equations of motion of nonholonomic robotic mechanical systems. Moreover, the introduction of the orthogonal complement leads naturally to an efficient computational algorithm.<<ETX>>

The design of kinematically isotropic rolling robots with omnidirectional wheels

Rolling robots with omnidirectional wheels (ODW), and hence, three degrees of freedom, are the subject of this paper. We focus on one kind of ODW, namely, those consisting of a hub with rollers on its periphery, but the main ideas discussed here are readily portable to other kinds. The full mobility of these robots is due to the free rollers. The choice of the orientation of the roller axes with respect to the hub axis, along with the hub orientations with respect to the platform, the number of wheels etc., playing an important role in the operation of the robot, are design issues considered here. A design criterion based on the isotropy of the underlying Jacobian matrices is reported in this paper. Various designs are produced that meet this criterion.

Recursive Kinematics and Inverse Dynamics for a Planar 3R Parallel Manipulator

We focus on the development of modular and recursive formulations for the inverse dynamics of parallel architecture manipulators in this paper. The modular formulation of mathematical models is attractive especially when existing sub-models may be assembled to create different topologies, e.g., cooperative robotic systems. Recursive algorithms are desirable from the viewpoint of simplicity and uniformity of computation. However, the prominent features of parallel architecture manipulators-the multiple closed kinematic loops, varying locations of actuation together with mixtures of active and passive joints-have traditionally hindered the formulation of modular and recursive algorithms. In this paper, the concept of the decoupled natural orthogonal complement (DeNOC) is combined with the spatial parallelism of the robots of interest to develop an inverse dynamics algorithm which is both recursive and modular. The various formulation stages in this process are highlighted using the illustrative example of a 3R Planar Parallel Manipulator.

Analytical Expression for the Inverted Inertia Matrix of Serial Robots

This paper presents the analytical derivation of the inertia matrix and its inverse for an open-loop, serial-chain mbot. The derivation allows one to write a recursiveforward-dynamics algorithmforsimulation purposes whose computational complexity is of order a, i.e., O(n) n being the degrees offreedom of the robot under study The proposed methodology is based on the Gaussian elimination of the inertia matrix, in contrast to, say, Kalmanftltering; which is proposed elsewhere. The derivation is illustrated with a three-degrees-of-freedom planar robot.

A unified approach to Space robot kinematics

An essential step in deriving kinematic models of free-flying space robots, consisting of a free-base and a manipulator mounted on it, is to write the total momenta of the system at hand. The momenta are, usually, expressed as the functions of the velocities of a preselected body that belongs to the robot, e.g., the free-base. In this paper, no preselection is recommended. On the contrary, the total momenta are expressed as the functions of the velocities of an arbitrary body of the space robot, namely, the primary body (PB). The identity of the PB, unlike the conventional approaches, need not be known at this stage. Therefore, the generalized expressions for the total momenta are obtained. The resulting expressions can explain the existing kinematic models and how they affect the efficiencies of the associated control algorithms. Based on the proposed approach, it is shown that if the end-effector motion is the only concern, as desired in kinematic control, it should be selected as the PB. This leads to the most efficient algorithms.

Constraint Wrench Formulation for Closed-Loop Systems Using Two-Level Recursions

In order to compute the constraint moments and forces, together referred here as wrenches, in closed-loop mechanical systems, it is necessary to formulate a dynamics problem in a suitable manner so that the wrenches can be computed efficiently. A new constraint wrench formulation for closed-loop systems is presented in this paper using two-level recursions, namely, subsystem level and body level. A subsystem is referred here as the serial- or tree-type branches of a spanning tree obtained by cutting the appropriate joints of the closed loops of the system at hand. For each subsystem, unconstrained Newton-Euler equations of motion are systematically reduced to a minimal set in terms of the Lagrange multipliers representing the constraint wrenches at the cut joints and the driving torques/forces provided by the actuators. The set of unknown Lagrange multipliers and the driving torques/forces associated to all subsystems are solved in a recursive fashion using the concepts of a determinate subsystem. Next, the constraint forces and moments at the uncut joints of each subsystem are calculated recursively from one body to another. Effectiveness of the proposed algorithm is illustrated using a multiloop planar carpet scraping machine and the spatial RSSR (where R and S stand for revolute and spherical, respectively) mechanism.