Twenty-five years of natural coordinates

In the early eighties, the author and co-workers created and further developed the natural coordinates to describe the motion of 2-D and 3-D multibody systems. Natural coordinates do not need angles or angular parameters to define orientation, leading to constant inertia matrices and to the simplest form of the constraint equations. Natural coordinates are composed by the Cartesian coordinates of some points and the Cartesian components of some unit vectors distributed on the different bodies of the system. The points and vectors can be located in the joints, being shared by contiguous bodies, decreasing or even eliminating the need to set joint constraints and reducing the total number of variables. However, other authors prefer not to share variables in order to get even simpler equations and to keep a bigger decoupling of equations, which is preferable in some cases. In this paper the history of natural coordinates is reviewed, as well as the main contributions coming from other research groups. In the second part of the paper some application areas in which natural coordinates can be particularly advantageous are examined. Commemorative Contribution.     

Driving simulator with double-wishbone suspension using efficient block-triangularized kinematic equations

When modeled with ideal joints, many vehicle suspensions contain closed kinematic chains, or kinematic loops, and are most conveniently modeled using a set of generalized coordinates of cardinality exceeding the degrees-of-freedom of the system. Dependent generalized coordinates add nonlinear algebraic constraint equations to the ordinary differential equations of motion, thereby producing a set of differential-algebraic equations that may be difficult to solve in an efficient yet precise manner. Several methods have been proposed for simulating such systems in real time, including index reduction, model simplification, and constraint stabilization techniques. In this work, the equations of motion for a double-wishbone suspension are formulated symbolically using linear graph theory. The embedding technique is applied to eliminate the Lagrange multipliers from the dynamic equations and obtain one ordinary differential equation for each independent acceleration. Symbolic computation is then used to triangularize a subset of the kinematic constraint equations, thereby producing a recursively solvable system for calculating a subset of the dependent generalized coordinates. Thus, the kinematic equations are reduced to a block-triangular form, which results in a more computationally efficient solution strategy than that obtained by iterating over the original constraint equations. The efficiency of this block-triangular kinematic solution is exploited in the real-time simulation of a vehicle with double-wishbone suspensions on both axles, which is implemented in a hardware- and operator-in-the-loop driving simulator.     

Singular value decomposition for dynamic system design sensitivity analysis

A computer-based method for automatic generation and efficient numerical solution of mixed differential-algebraic equations for dynamic and design sensitivity analysis of dynamic systems is developed. The equations are written in terms of a maximal set of Cartesian coordinates to facilitate general formulation of kinematic and design constraints and forcing functions. Singular value decomposition of the system Jacobian matrix generates a set of composite generalized coordinates that are best suited to represent the system. The coordinates naturally partition into optimal independent and dependent sets, and integration of only the independent coordinates generates all of the system information. An adjoint variable method is used to compute design sensitivities of dynamic performance measures of the system. A general-purpose computer program incorporating these capabilities has been developed. A numerical example is presented to illustrate accuracy and properties of the method.     

运动滑道含摩擦多体系统的建模和数值方法

研究了运动约束面含摩擦多体系统动力学方程的建立和算法问题．首先利用第一类Lagrange方程给出了系统的动力学方程,并以矩阵形式给出了这类系统摩擦力的广义力的一般表达式．为便于摩擦力和铰链约束力的分析与计算,采用笛卡尔坐标和约束方程的局部方法,使得系统的约束力与Lagrange乘子一一对应．应用增广法将微分一代数方程组转化为常微分方程组并用分块矩阵的形式给出,以便于方程的编程与计算,提高计算效率．最后用一个算例验证了该方法的有效性．     

Triangularizing kinematic constraint equations using Gröbner bases for real-time dynamic simulation

Real-time simulation is an essential component of hardware- and operator-in-the-loop applications, such as driving simulators, and can greatly facilitate the design, implementation, and testing of dynamic controllers. Such applications may involve multibody systems containing closed kinematic chains, which are most readily modeled using a set of redundant generalized coordinates. The governing dynamic equations for such systems are differential-algebraic in nature—that is, they consist of a set of ordinary differential equations coupled with a set of nonlinear algebraic constraint equations—and can be difficult to solve in real time. In this work, the equations of motion are formulated symbolically using linear graph theory. The embedding technique is applied to eliminate the Lagrange multipliers from the dynamic equations and obtain one ordinary differential equation for each independent acceleration. The theory of Gröbner bases is then used to triangularize the kinematic constraint equations, thereby producing a recursively solvable system for calculating the dependent generalized coordinates given values of the independent coordinates. The proposed approach can be used to generate computationally efficient simulation code that avoids the use of iteration, which makes it particularly suitable for real-time applications.     

Initial condition correction in multibody dynamics

A simple procedure is presented to correct initial conditions for the coordinates and velocities prior to performing a kinematic or forward dynamic analysis of multibody systems. Such corrections are crucial since slight amount of constraint violations at the start of any numerical integration of equations of motion can lead to erroneous results. The correction process is based on the well-known method of minimizing the sum-of-squares of adjustments in the coordinates or velocities. The process provides a solution that is closest to the estimated values. It should be a simple task to implement this methodology as a preprocessing step for any kinematic or forward dynamic analysis program regardless of the formulation. Commemorative Contribution.     

Stabilization Methods for the Integration of DAE in the Presence of Redundant Constraints

The use of multibody formulations based on Cartesian or naturalcoordinates lead to sets of differential-algebraic equations that haveto be solved. The difficulty in providing compatible initial positionsand velocities for a general spatial multibody model and the finiteprecision of such data result in initial errors that must be correctedduring the forward dynamic solution of the system equations of motion.As the position and velocity constraint equations are not explicitlyinvolved in the solution procedure, any integration error leads to theviolation of these equations in the long run. Another problem that isvery often impossible to avoid is the presence of redundant constraints.Even with no initial redundancy it is possible for some systems toachieve singular configurations in which kinematic constraints becometemporarily redundant. In this work several procedures to stabilize thesolution of the equations of motion and to handle redundant constraintsare revisited. The Baumgarte stabilization, augmented Lagrangian andcoordinate partitioning methods are discussed in terms of theirefficiency and computational costs. The LU factorization with fullpivoting of the Jacobian matrix directs the choice of the set ofindependent coordinates, required by the coordinate partitioning method.Even when no particular stabilization method is used, a Newton–Raphsoniterative procedure is still required in the initial time step tocorrect the initial positions and velocities, thus requiring theselection of the independent coordinates. However, this initialselection does not guarantee that during the motion of the system otherconstraints do not become redundant. Two procedures based on the singlevalue decomposition and Gram–Schmidt orthogonalization are revisited forthe purpose. The advantages and drawbacks of the different procedures,used separately or in conjunction with each other and theircomputational costs are finally discussed.     

Modeling and simulation of structural components in recursive closed-loop multibody systems

Passenger cars, transit buses, railroad vehicles, off-highway trucks, earth moving equipment and construction machinery contain structural and light-fabrications (SALF) components that are prone to excessive vibration due to rough terrains and work-cycle loads’ excitations. SALF components are typically modeled as flexible components in the multibody system allowing the analysts to predict elastic deformation and hence the stress levels under different loading conditions. Including SALF component in the multibody system typically generates closed-kinematic loops. This paper presents an approach for integrating SALF modeling capabilities as a flexible body in a general-purpose multibody dynamics solver that is based on joint-coordinates formulation with the ability to handle closed-kinematic loops. The spatial algebra notation is employed in deriving the spatial multibody dynamics equations of motion. The system kinematic topology matrix is used to project the Cartesian quantities into the joint subspace, leading to a condensed set of nonlinear equations with minimum number of generalized coordinates. The proposed flexible body formulation utilizes the component mode synthesis approach to reduce the large number of finite element degrees of freedom to a small set of generalized modal coordinates. The resulting reduced flexible body model has two main characteristics: the stiffness matrix is constant while the mass matrix depends on the elastic modal coordinates. A consistent set of pre-computed inertia shape integrals are identified and used to update the modal mass matrix at each time step. The implementation of the component mode synthesis approach in a closed-loop recursive multibody formulation is presented. The kinematic equations are modified to include the effect of the flexible body modal elastic coordinates. Also, modified constraint equations that include the effect of flexibility at the joint connections and the necessary details of the Jacobian matrix are presented. Baumgarte stabilization approach is used to stabilize the constraint equations without using iterative schemes. A sample results for flexible body impeded in a closed system will be presented to demonstrate the above mentioned approach.     

Application of Fuzzy Logic for the Solution of Inverse Kinematics and Hierarchical Controls of Robotic Manipulators

In this paper, hierarchical control techniques is used for controlling a robotic manipulator. The proposed method is based on the establishment of a non-linear mapping between Cartesian and joint coordinates using fuzzy logic in order to direct each individual joint. The hierarchical control will be implemented with fuzzy logic to improve the robustness and reduce the run time computational requirements. Hierarchical control consists of solving the inverse kinematic equations using fuzzy logic to direct each individual joint. A commercial Microbot with three degrees of freedom is utilized to evaluate this methodology. A decentralized fuzzy controller is used for each joint, with a Fuzzy Associative Memories (FAM) performing the inverse kinematic mapping in a supervisory mode. The FAM determines the inverse kinematic mapping which maps the desired Cartesian coordinates to the individual joint angles. The individual fuzzy controller for each joint generates the required control signal to a DC motor to move the associated link to the new position. The proposed hierarchical fuzzy controller is compared to a conventional controller. The simulation experiments indeed demonstate the effectiveness of the proposed method.     

An algorithmic approach to coordinate transformation for robotic manipulators

Coordinate transformation is one of the most important issues in robotic manipulator control. Robot tasks are naturally specified in work space coordinates, usually a Cartesian frame, while control actions are developed on joint coordinates. Effective inverse kinematic solutions are analytical in nature; they exist only for special manipulator geometries and geometric intuition is usually required. Computational inverse kinematic algorithms have recently been proposed; they are based on general closed-loop schemes which perform the mapping of the desired Cartesian trajectory into the corresponding joint trajectory. The aim of this paper is to propose an effective computational scheme to the inverse kinematic problem for manipulators with spherical wrists. First an insight into the formulation of kinematics is given in order to detail the general scheme for this specific class of manipulators. Algorithm convergence is then ensured by means of the Lyapunov direct method. The resulting algorithm is based on the hand position and orientation vectors usually adopted to describe motion in the task space. The analysis of the computational burden is performed by taking the Stanford arm as a reference. Finally a case study is developed via numerical simulations.     

Flatness Conservation in the n-trailer System Equipped with a Sliding Kingpin Mechanism

Nonlinear systems, which are differentially flat, have several properties that can be useful on designing effective controllers. In this paper we show that the n-trailer system equipped with a sliding kingpin mechanism is a differentially flat system, like its non-sliding kingpin counter part. The sliding kingpin technique is used to eliminate the undesired deviation of the path of each intermediate vehicle from that of the leading one (off-tracking phenomenon). The linearizing outputs of the flat system are the Cartesian coordinates of the middle of the last semi-trailer's axle. The state space and the kinematic equations of the new modified system are derived and the conditions for flatness are examined. The flatness conservation is also checked relatively to several kinds of dynamic sliding feedback control.     

大范围运动矩形板动力刚化分析

针对大范围运动下弹性矩形板，采用有限元技术和Lagrange方程建立了系统刚柔耦合一次近似动力学方程组．不同于传统动力学建模方法，本文采用两个弧长变量和一个笛卡尔坐标变量来描述板的变形，利用有限元方法离散，在动力学方程中得到了动力刚度项．数值仿真表明，在大范围运动下，传统的动力学模型不能正确的预示系统动力学行为；而本文动力学模型能够较好的预测系统的动力学行为，且比采用假设模态离散板变形的方法更为精确．     

A computer-oriented method for reducing linearized multibody system equations by incorporating constraints

Consider a spatial multibody system with rigid and elastic bodies. The bodies are linked by rigid interconnections (e.g. revolute joints) causing constraints, as well as by flexible interconnections (e.g. springs) causing applied forces. Small motions of the system with respect to a given nominal configuration can be described by linearized dynamic equations and kinematic constraint equations. We present a computer-oriented procedure which allows to develop a minimum number of these equations. There are three problems. First: algorithmic selection of position coordinates; second: condensation of the dynamic equations; third: evaluation of the constraint forces. To demonstrate the procedure, a closed loop multibody system is used as an example.     

Pulse control of flexible multibody systems

An active pulse control method is developed to reduce the vibrations of multibody systems resulting from impact loadings. The pulse, which is a function of system generalized coordinates and velocities, is determined analytically using energy and momentum balance equations of the impacting bodies. Elastic components in the multibody system are discretized using the finite element method. The system equations of motions and nonlinear algebraic constraint equations describing mechanical joints between different components are written in the Lagrangian formulation using a finite set of coupled reference position and local elastic generalized coordinates. A set of independent differential equations are identified by the generalized coordinate partitioning of the constraint Jacobian matrix. These equations are written in the state space formulation and integrated forward in time using a direct numerical integration method. Dependent coordinates are then determined using the constraint kinematic relations. Points in time at which impact occurs are monitored by an impact predictor function, which controls the integration algorithms and forces for the solution of the momentum relation, to define the jump discontinuities in the composite velocity vector as well as the system reaction forces. The effectiveness of the active pulse control in reducing the vibration of flexible multibody aircraft during the touchdown impact is investigated and numerical results are presented.     

On the constraints violation in forward dynamics of multibody systems

It is known that the dynamic equations of motion for constrained mechanical multibody systems are frequently formulated using the Newton–Euler’s approach, which is augmented with the acceleration constraint equations. This formulation results in the establishment of a mixed set of partial differential and algebraic equations, which are solved in order to predict the dynamic behavior of general multibody systems. The classical solution of the equations of motion is highly prone to constraints violation because the position and velocity constraint equations are not fulfilled. In this work, a general and comprehensive methodology to eliminate the constraints violation at the position and velocity levels is offered. The basic idea of the described approach is to add corrective terms to the position and velocity vectors with the intent to satisfy the corresponding kinematic constraint equations. These corrective terms are evaluated as a function of the Moore–Penrose generalized inverse of the Jacobian matrix and of the kinematic constraint equations. The described methodology is embedded in the standard method to solve the equations of motion based on the technique of Lagrange multipliers. Finally, the effectiveness of the described methodology is demonstrated through the dynamic modeling and simulation of different planar and spatial multibody systems. The outcomes in terms of constraints violation at the position and velocity levels, conservation of the total energy and computational efficiency are analyzed and compared with those obtained with the standard Lagrange multipliers method, the Baumgarte stabilization method, the augmented Lagrangian formulation, the index-1 augmented Lagrangian, and the coordinate partitioning method.     

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.     

Translational Parallel Manipulators: New Proposals

Translational parallel manipulators are parallel manipulators wherein the end‐effector performs only spatial translations. This paper presents a new family of translational parallel manipulators. The manipulators of this family are independent constraint manipulators. They have three limbs that are topologically identical and have no rotation singularity. The limbs of these manipulators feature five one‐degree‐of‐freedom kinematic pairs in series. Four joints are revolute pairs and the remaining one, called T‐pair, is a kinematic pair that can be manufactured in different ways. In each limb, three adjacent revolute pairs have parallel axes and the remaining revolute pair has an axis that is not parallel to the axes of the other revolute pairs. The mobility analysis of the manipulators of this new family is addressed by taking into account two different choices for the actuated pairs. One of the results of this analysis is that the geometry of a translational parallel manipulator free from singularities can be defined for a particular choice of the actuated pairs. © 2002 Wiley Periodicals, Inc.     

三维几何约束求解的变分算法

研究了运动学变分原理在三维几何约束求解中的应用,提出了变分求解算法．该算法采用相对坐标,将求解域从笛卡儿空间转换到相对坐标空间．对于约束开环,依次选取相对坐标即可获得满足几何约束的刚体位姿;对于约束闭环,通过切断闭环铰,将约束闭环转化为等价的广义开环与切断铰约束代数方程,其切断铰约束代数方程的相对坐标雅克比矩阵解析式可通过变分关系显式获得．最后通过实例验证,说明该算法具有较高的求解效率与稳定性．     

基于动力学约束的机械臂最优关节轨迹规划

本文提出了基于机械臂关节驱动力矩约束方程规划其关节最优运动轨迹的一种有效方法.该方法运用矩阵范数理论简化机械臂的动力学约束方程;在机械臂的关节空间内采用归一化的无因次量运用非线性规划法优化其运动轨迹.将所规划的无因次量轨迹方程作为机械臂产生实际运动轨迹的发生器,通过给定机械臂各运动段的起始和终止关节坐标,由系统的动力学约束方程计算出整个运动段所允许的最短运行时间,即生成所期望的运动轨迹.本文的轨迹规划方法计算效率高,可用于在线轨迹规划,文中通过算例证实了该方法的实用性.     

Recursive and Residual Algorithms for the Efficient Numerical Integration of Multi-Body Systems

In this paper a family of methods for multi-body dynamic simulation is introduced. Equations of motion are obtained using a set of Cartesian coordinates and projected onto a set of independent relative coordinates using the concept of velocity transformation. Open-chain systems are solved directly following either a fully recursive or a semi-recursive procedure. Closed-chain systems are solved in two steps; kinematic loops are opened by removing either some kinematic joints or a rigid body, and the resulting open-chain system is solved; closure-of-the-loop conditions are imposed by means of a second velocity transformation. The dynamic formalisms have been developed so as to handle both non-stiff and stiff systems. Non-stiff systems are solved by means of an Adams–Bashforth–Moulton numerical integration scheme, which requires the computation of the function derivatives. Stiff problems are integrated by using either BDF or NDF methods, which require the computation of the residual of the equations of motion and, optionally, the evaluation of the Jacobian matrix. The proposed algorithms have been implemented using an Object-Oriented Programming approach that makes it possible to re-use the source code, keeping programs smaller, cleaner and easier to maintain. Practical examples that illustrate the performance of these implementations are included. These examples have also been solved using a commercial multi-body simulation package and comparative results are included. In most cases, the algorithms here presented outperform those implemented in the commercial package, leading to important savings in terms of total computation times.