Forward Dynamic Algorithm Research Articles

A recursive, numerically stable, and efficient simulation algorithm for serial robots with flexible links

A methodology for the formulation of dynamic equations of motion of a serial flexible-link manipulator using the decoupled natural orthogonal complement (DeNOC) matrices, introduced elsewhere for rigid bodies, is presented in this paper. First, the Euler Lagrange (EL) equations of motion of the system are written. Then using the equivalence of EL and Newton–Euler (NE) equations, and the DeNOC matrices associated with the velocity constraints of the connecting bodies, the analytical and recursive expressions for the matrices and vectors appearing in the independent dynamic equations of motion are obtained. The analytical expressions allow one to obtain a recursive forward dynamics algorithm not only for rigid body manipulators, as reported earlier, but also for the flexible body manipulators. The proposed simulation algorithm for the flexible link robots is shown to be computationally more efficient and numerically more stable than other algorithms present in the literature. Simulations, using the proposed algorithm, for a two link arm with each link flexible and a Space Shuttle Remote Manipulator System (SSRMS) are presented. Numerical stability aspects of the algorithms are investigated using various criteria, namely, the zero eigenvalue phenomenon, energy drift method, etc. Numerical example of a SSRMS is taken up to show the efficiency and stability of the proposed algorithm. Physical interpretations of many terms associated with dynamic equations of flexible links, namely, the mass matrix of a composite flexible body, inertia wrench of a flexible link, etc. are also presented.

Cluster computing of mechanisms dynamics using recursive formulation

This paper presents the O(n) recursive algorithm for forward dynamics of closed loop kinematic chains adapted to parallel computations on a cluster of workstations. The Newton–Euler equations of motion are formulated in terms of relative coordinates. Closed loop kinematic chains are transformed into open loop chains by cut joint technique. Cut joint constraint and Lagrange multipliers are introduced to complete the equations of motion. Constraint stabilization is performed using the Baumgarte stabilization technique with application to multibody systems with large number of degrees of freedom. Numerical simulations are carried out to study the influence of the degrees of freedom of the multibody system on computational efficiency of the algorithm using the Message Passing Interface (MPI). We also consider the ways of minimization of communication overhead which has significant impact on efficiency in case of cluster computing.

Special Coordinates Associated With Recursive Forward Dynamics Algorithm for Open Loop Rigid Multibody Systems

The recursive forward dynamics algorithm (RFDA) for a tree structured rigid multibody system has two stages. In the first stage, while going down the tree, certain equations are associated with each node. These equations are decoupled from the equations related to the node’s descendants. We refer them as the equations of RFDA of the node and the current paper derives them in a new way. In the new derivation, associated with each node, we recursively obtain the coordinates, which describe the system consisting of the node and all its descendants. The special property of these coordinates is that a portion of the equations of motion with respect to these coordinates is actually the equations of RFDA associated with the node. We first show the derivation for a two noded system and then extend to a general tree structure. Two examples are used to illustrate the derivation. While the derivation conclusively shows that equations of RFDA are part of equations of motion, it most importantly gives the associated coordinates and the left out portion of the equations of motion. These are significant insights into the RFDA.

Control of Robot Manipulators in Terms of Quasi-Velocities

In this paper we present new control algorithms for robots with dynamics described in terms of quasi-velocities (Kozlowski, Identification of articulated body inertias and decoupled control of robots in terms of quasi-coordinates. In: Proc. of the 1996 IEEE International Conference on Robotics and Automation, pp. 317---322. IEEE, Piscataway, 1996a; Zeitschrift fur Angewandte Mathematik und Mechanik 76(S3):479---480, 1996c; Robot control algorithms in terms of quasi-coordinates. In: Proc. of the 34 Conference on Decision and Control, pp. 3020---3025, Kobe, 11---13 December 1996, 1996d). The equations of motion are written using spatial quantities such as spatial velocities, accelerations, forces, and articulated body inertia matrices (Kozlowski, Standard and diagonalized Lagrangian dynamics: a comparison. In: Proc. of the 1995 IEEE Int. Conf. on Robotics and Automation, pp. 2823---2828. IEEE, Piscataway, 1995b; Rodriguez and Kreutz, Recursive Mass Matrix Factorization and Inversion, An Operator Approach to Open- and Closed-Chain Multibody Dynamics, pp. 88---11. JPL, Dartmouth, 1998). The forward dynamics algorithms incorporate new control laws in terms of normalized quasi-velocities. Two cases are considered: end point trajectory tracking and trajectory tracking algorithm, in general. It is shown that by properly choosing the Lyapunov function candidate a dynamic system with appropriate feedback can be made asymptotically stable and follows the desired trajectory in the task space. All of the control laws have a new architecture in the sense that they are derived, in the so-called quasi-velocity and quasi-force space, and at any instant of time generalized positions and forces can be recovered from order $O({\cal N})$ recursions, where ${\cal N}$ denotes the number of degrees of freedom of the manipulator. This paper also contains the proposition of a sliding mode control, originally introduced by Slotine and Li (Int J Rob Res 6(3):49---59, 1987), which has been extended to the sliding mode control in the quasi-velocity and quasi-force space. Experimental results illustrate behavior of the new control schemes and show the potential of the approach in the quasi-velocity and quasi-force space.

A recursive, numerically stable, and efficient simulation algorithm for serial robots

Traditionally, the dynamic model, i.e., the equations of motion, of a robotic system is derived from Euler–Lagrange (EL) or Newton–Euler (NE) equations. The EL equations begin with a set of generally independent generalized coordinates, whereas the NE equations are based on the Cartesian coordinates. The NE equations consider various forces and moments on the free body diagram of each link of the robotic system at hand, and, hence, require the calculation of the constrained forces and moments that eventually do not participate in the motion of the coupled system. Hence, the principle of elimination of constraint forces has been proposed in the literature. One such methodology is based on the Decoupled Natural Orthogonal Complement (DeNOC) matrices, reported elsewhere. It is shown in this paper that one can also begin with the EL equations of motion based on the kinetic and potential energies of the system, and use the DeNOC matrices to obtain the independent equations of motion. The advantage of the proposed approach is that a computationally more efficient forward dynamics algorithm for the serial robots having slender rods is obtained, which is numerically stable. The typical six-degree-of-freedom PUMA robot is considered here to illustrate the advantages of the proposed algorithm.

A survey of equations of motion in terms of inertial quasi-velocities for serial manipulators

In this work we compare equations of motion using the so-called inertial quasi-velocities. As a result of these velocities we obtain two first-order decoupled equations of motion instead of one second-order differential equation of motion. The methods presented here, solve in a way, the problem of nonlinear dynamic decoupling. The first and the second method result from diagonalized Lagrangian robot dynamics (Jain and Rodriguez, IEEE Trans Robot Autom 11:571–584, 1995) and are known as normalized and unnormalized quasi-velocities. The third method described by Junkins and Schaub (J Astronaut Sci 45:279–295, 1997) offers eigenfactor quasi-coordinate velocities formulation for multibody dynamics. As a consequence of using transformation given by Loduha and Ravani (Trans ASME J Appl Mech 62:216–222, 1995) we obtain decoupled equations of motion in terms of modified generalized velocity components. Here we limit all these methods to serial manipulators. The novelty of this paper consists in physical interpretation of the quasi-velocities and discussion concerning equations of motion, the kinetic energy shaping, relationship between each of them and properties useful for simulation and control purposes. Also forward dynamics algorithms and their computational complexity in terms of new velocities are given. Simulation results illustrate the theoretical investigations. We conclude that all methods offer interesting possibilities for dynamic simulation and future control investigations.

Lot-sizing problem with several production centers

In this paper, a case study is carried out concerning the lot-sizing problem involving a single item production planning in several production centers that do not present capacity constraints. Demand can be met with backlogging or not. This problem results from simplifying practical problems, such as the material requirement planning (MRP) system and also lot-sizing problems with multiple items and limited production capacity. First we propose an efficient implementation of a forward dynamic programming algorithm for problems with one single production center. Although this does not reduce its complexity, it has shown to be rather effective, according to computational tests. Next, we studied the problem with a production environment composed of several production centers. For this problem two algorithms are implemented, the first one is an extension of the dynamic programming algorithm for one production center and the second one is an efficient implementation of the first algorithm. Their efficiency are shown by computational testing of the algorithms and proposals for future research are presented.

Hybrid backward and forward dynamic programming based Lagrangian relaxation for single machine scheduling

In this paper we consider the single machine scheduling problem with precedence constraints to minimize the total weighted tardiness of jobs. This problem is known to be strongly NP-hard. A solution methodology based on Lagrangian relaxation is developed to solve it. In this approach, a hybrid backward and forward dynamic programming algorithm is designed for the Lagrangian relaxed problem, which can deal with jobs with multiple immediate predecessors or successors as long as the underlying precedence graph representing the precedence relations between jobs does not contain cycles. All precedence constraints are considered in this dynamic programming algorithm, unlike the previous work using forward dynamic programming where some precedence relations were ignored. Computational experiments are carried out to analyze the performance of our algorithm with respect to different problem sizes and to compare the algorithm with a forward dynamic programming based Lagrangian relaxation method. The results show that the proposed algorithm leads to faster convergence and better Lagrangian lower bounds.

The single-item lot-sizing problem with immediate lost sales

We introduce a profit maximization version of the well-known Wagner–Whitin model for the deterministic uncapacitated single-item lot-sizing problem with lost sales. Demand cannot be backlogged, but it does not have to be satisfied, either. Costs and selling prices are assumed to be time-variant, differentiating our model from previous models with lost sales. Production quantities and levels of lost sales in different periods represent a twofold decision problem. We first transform the total profit function into a special total cost function. We then prove several properties of an optimal solution. A forward recursive dynamic programming algorithm is developed to solve the problem optimally in O( T 2) time, where T denotes the number of periods in the problem horizon. The proposed algorithm can solve problems of sizes up to 400 periods in less than a second on a 500 MHz Pentium ® III processor.

DETERMINISTIC TIME‐VARYING DEMAND LOT‐SIZING MODELS WITH LEARNING AND FORGETTING IN SETUPS AND PRODUCTION

We study the deterministic time‐varying demand lot‐sizing problem in which learning and forgetting in setups and production are considered simultaneously. It is an extension of Chiu's work. We propose a near‐optimal forward dynamic programming algorithm and suggest the use of a good heuristic method in a situation in which the computational effort is extremely intolerable. Several important observations obtained from a two‐phase experiment verify the goodness of the proposed algorithm and the chosen heuristic method.

Simulation of Industrial Manipulators Based on the UDU T Decomposition of Inertia Matrix

The UDUT – U and D are respectively the upper triangular and diagonal matrices – decomposition of the generalized inertia matrix of an n-link serial manipulator, introduced elsewhere, is used here for the simulation of industrial manipulators which are mainly of serial type. The decomposition is based on the application of the Gaussian elimination rules to the recursive expressions of the elements of the inertia matrix that are obtained using the Decoupled Natural Orthogonal Complement matrices. The decomposition resulted in an efficient order n, i.e., O(n), recursive forward dynamics algorithm that calculates the joint accelerations. These accelerations are then integrated numerically to perform simulation. Using this methodology, a computer algorithm for the simulation of any n degrees of freedom (DOF) industrial manipulator comprising of revolute and/or prismatic joints is developed. As illustrations, simulation results of three manipulators, namely, a three-DOF planar manipulator, and the six-DOF Stanford arm and PUMA robot, are reported in this paper.

2P2-L09 O(N) 順動力学計算法と陰積分による衝突・接触の高速シミュレーション

2P2-L09 O(N) 順動力学計算法と陰積分による衝突・接触の高速シミュレーション

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 n3 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 system’s 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.

Coordinate-invariant algorithms for robot dynamics

We present, using methods from the theory of Lie groups and Lie algebras, a coordinate-invariant formulation of the dynamics of open kinematic chains. We first re-formulate the recursive dynamics algorithm originally given by Park et al. (1995) for open chains in terms of standard linear operators on the Lie algebra of the special Euclidean group. Using straight forward algebraic manipulations, we then recast the resulting algorithm into a set of closed-form dynamic equations. We then reformulate Featherstone's (1987) articulated body inertia algorithm using this same geometric framework, and re-derive Rodriguez et al.'s (1991, 1992) square factorization of the mass matrix and its inverse. An efficient O(n) recursive algorithm for forward dynamics is also extracted from the inverse factorization. The resulting equations lead to a succinct high-level description of robot dynamics in both joint and operational space coordinates that minimizes symbolic complexity without sacrificing computational efficiency, and provides the basis for a dynamics formulation that does not require link reference frames in the description of the forward kinematics.

Efficient dynamic constraints for animating articulated figures.

This paper presents an efficient dynamics-based computer animation system for simulating and controlling the motion of articulated figures. A non-trivial extension of Featherstone's O(n) recursive forward dynamics algorithm is derived which allows enforcing one or more constraints on the animated figures. We demonstrate how the constraint force evaluation algorithm we have developed makes it possible to simulate collisions between articulated figures, to compute the results of impulsive forces, to enforce joint limits, to model closed kinematic loops, and to robustly control motion at interactive rates. Particular care has been taken to make the algorithm not only fast, but also easy to implement and use. To better illustrate how the constraint force evaluation algorithm works, we provide pseudocode for its major components. Additionally, we analyze its computational complexity and finally we present examples demonstrating how our system has been used to generate interactive, physically correct complex motion with small user effort.

Forward Dynamics, Elimination Methods, and Formulation Stiffness in Robot Simulation

The numerical simulation problem of tree-structured multi body systems, such as robot manipulators, is usually treated as two separate problems: 1) the forward dynamics problem for computing system accelerations, and 2) the numerical integra tion problem for advancing the state in time. The interaction of these two problems can be important, and has led to new conclusions about the overall efficiency of multibody simula tion algorithms (Cloutier, Pai, and Ascher 1995). In particular, the fastest forward dynamics methods are not necessarily the most numerically stable, and in ill-conditioned cases may slow down popular adaptive step-size integration methods. This phenomenon is called formulation stiffness. In this article, we first unify the derivation of both the com posite rigid-body method (Walker and Orin 1982) and the articulated-body method (Featherstone 1983, 1987) as two elimination methods for solving the same linear system, with the articulated-body method taking advantage of sparsity. Then the numerical instability phenomenon for the composite rigid- body method is explained as a cancellation error that can be avoided, or at least minimized, when using an appropriate version of the articulated-body method. Specifically, we show that a variant of the articulated-body method is better suited to deal with certain types of ill-conditioning than the composite rigid-body method. The unified derivation also clarifies the un derlying linear algebra of forward dynamics algorithms, and is therefore of interest in its own right.

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.

Multiobjective Analysis of Service-Water-Transmission Systems

A multiobjective analysis technique is presented to determine the optimum operation of pumps and reservoirs in service-water-transmission systems. Three major objectives were identified as (1) economics in terms of energy cost minimization; (2) stability in pump operation; and (3) reliability in meeting stochastically varying demands. These objectives were evaluated based on the required number of times the pumps were turned on or off, the required total energy cost, and the minimum required storage during the operating horizon. A set of nondominated solutions to show the explicit trade-off relationships between the considered objectives was generated using a combination of the ε-constraint method and the weighting method. A forward dynamic programming algorithm was applied to find the optimal solution to the discretized multiobjective problem. Decision makers can evaluate the developed trade-off relationship to identify potential operational alternatives for more detailed evaluation. The methodology was illustrated by application to the Kumi service-water-transmission system in the Republic of Korea.

Recursive dynamics algorithm for multibody systems with prescribed motion

This paper uses spatial operator techniques to develop a new algorithm for the dynamics of multibody systems with hinges undergoing prescribed motion. This algorithm is spatially recursive, and its computational complexity grows only linearly with the number of degrees of freedom in the system. Its structure is a hybrid of known recursive forward and inverse dynamics algorithms for regular multibody systems. Changes to the prescribed/nonprescribed nature of hinges can be implemented during run time since they are handled with very low overhead in the algorithm.

A single-machine replacement model with learning

The replacement or upgrade of productive resources over time is an important decision for a manufacturing organization. The type of technology used in the productive resources determines how effectively the manufacturing operations can support the product and marketing strategy of the organization. Increasing operating costs (cost of maintenance, labor, and depreciation) over time force manufacturing organizations to periodically consider replacement or upgrade of their existing productive resources. We assume that there is a setup cost associated with the replacement of a machine, and that the setup cost is a nonincreasing function of the number of replacements made so far due to learning in setups. The operating cost of a newer machine is assumed to be lower than the operating cost of an older machine in any given period, except perhaps in the first period of operation of the new machine when the cost could be unusually high due to higher initial depreciation. A forward dynamic programming algorithm is developed which can be used to solve finite-horizon problems. We develop procedures to find decision and forecast horizons such that choices made during the decision horizon based only on information over the forecast horizon are also optimal for any longer horizon problem. Thus, we are able to obtain optimal results for what is effectively an infinite-horizon problem while only requiring data over a finite period of time. We present a numerical example to illustrate the decision/forecast horizon procedure, as well as a study of the effects of considering learning in making a series of machine replacement decisions. © 1993 John Wiley & Sons. Inc.