Wittenburg J (1977) Dynamics of Systems of Rigid Bodies. B.G. Teubner, Stuttgart  Home/Search   Document Not in Database   Summary   Related Articles   Check

....error that causes the constraint to be violated will be damped out automatically. Although larger values of ff can cause greater stabilization, the stabilizing term must not become the dominant term in the differential equation, as that 11 would introduce numerical stiffness in the equation [40]. In our implementation, we have found ff =0:5worked well in all cases. Based on (11) 12) becomes Cq q ffC =0: 13) Putting it all together: Using (10) 13) becomes Cq (f q ) ffC =0# (14) which we can solve for and obtain = CqC (ffC Cq (f d q) 15) However, we do not ....

Jens Wittenburg. Dynamics of systems of rigid bodies. Teubner, Stuttgart, 1977.

....are presented in section 5. 2 Background 2.1 Previous work The field of articulated solids have been thoroughly investigated by mechanical engineering and robotics research. Paul studied the kinematics of manipulators[14] Numerous algorithms for direct or inverse dynamics have been proposed[21, 9, 3, 13] and a variety of simulation systems are available[16] In the field of animation synthesis, Wilhelms and Barsky[19] presented a general method to compute the motion of an articulated structure subject to external forces. Armstrong and Green[1] proposed a fast algorithm for acyclic structures ....

....an iterative solution. Since no cubic term is involved in the time complexity, this approach is well suited for structures including a large number of closed loops. 2. 2 Problem formulation Numerous ways of formulating the dynamics of articulated bodies can be found in standard texts, e.g. [21, 9, 16]. The formulation we use, which has been extensively presented by Baraff[5] is comparatively simple and involves only sparse matrices, which is an important feature when dealing with large structures. In the remaining of the paper, we use bold letters to denote vector and matrices related to the ....

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J. Wittenburg. Dynamics of Systems of Rigid Bodies. B.G. Teubner, Stuttgart, Germany, 1977. 20

....2.1 Review on the Previous Work Most virtual camera models have seven degrees of freedom: i.e. one for the focal length, and three for each position and orientation. In representing the orientations, the unit quaternions are quite useful since they are free of singularities such as gimbal lock [8, 11, 14]. Each unit quaternion consists of four parameters (q w ; q x ; q y ; q z ) with the constraint: q 2 w q 2 x q 2 y q 2 z = 1. When quaternions are used, a total of eight parameters instead of seven parameters are required to represent the status of a virtual camera. Given m points p 1 ....

....) be the rotated point of p i by the 3D rotation of the unit quaternion q(t) Then we have p 0 i (t) t) Theta p i (t) See [9] for more details on the derivation. When is interpreted as the angular velocity, this equation is exactly the same as the formula given in classical dynamics [14]. Consequently, for the transformation T i : S 3 Gamma R 3 q 7 Gamma R q (p i ) p i the differential d(T i ) q is given by d(T i ) q : T q (S 3 ) Gamma R 3 q 0 7 Gamma Theta p i Since the isomorphism F : T q (S 3 ) Gamma R 3 q 0 = 1 2 Delta q 7 Gamma ....

Wittenburg, J., Dynamics of Systems of Rigid Bodies, B.G. Teubner, Stuttgart, 1977. 8

....will be used in subsequent steps to replace the across variables for these cotree elements with expressions involving only the branch coordinates fqg. If a joint tree is used, then these transformation equations for cotree bodies will be identical to the velocity transformation equations [18, 19] employed by other researchers in multibody dynamics. The chord transformation equations that are used in conventional graph theoretic formulations [1, 4] are not required in this virtual work approach. 4.2 Step 2: Kinematic Equations Depending on the topology of the physical system and the ....

J. Wittenburg, Dynamics of Systems of Rigid Bodies, B.G. Teubner, Stuttgart, 1977.

....dynamic deformable models raise interesting challenges related to the application of constraints to construct composite models and control animation. We describe a method for computing generalized constraint forces between our models which is based on Baumgarte s constraint stabilization technique [4, 18]. As in [17, 3] our algorithm may be used to assemble complex objects satisfying constraints from initially mispositioned and misshaped parts, and it enables us to construct and animate articulated objects composed of rigid or nonrigid components. The remainder of this paper describes our ....

....satisfied (i.e. C(q; 0) 6= 0) Second, even if the constraints may be satisfied at a given time step of the animation (i.e. C(q; t) 0) they may not be satisfied at the next time step (i.e. C(q; t t) 6= 0) because of numerical errors, etc. The constraint stabilization method of Baumgarte [4, 18] remedies these problems. The constraint equation C = 0 is replaced in (14) by the damped second order equation C 2 C 2 C = 0, where and are stabilization factors. This replaces the lower entry of the vector on the rhs of (14) to 2 C 2 C. Fast stabilization means ....

Wittenburg, J., (1977) Dynamics of systems of rigid bodies, Tubner, Stuttgart. (a) (b) (c) (d)

.... Gossard 1988) STRUCTURAL DESIGN EVALUATOR (Fisher and Nguyen 1989) GALILEO3 (Bowen and Bahler 1992) STAURN (Fohn, et al. 1994) XCODOMAS (Burke et al. 1994) FDL (Imamura 1994) and FRODO (Kolb and Bailey 1993) The concept of graph theory applied to kinematic and dynamic analysis was used by Wittenburg (1977) to simplify the representation of mechanisms using a computer. A mechanism is modeled into a spanning tree where a body is defined as a node and a kinematic joint is defined as an edge. If there are no closed loops in the system graph, the system is said to have a tree structure. If a graph is ....

....environments to increase the automation of mechanical design. It can also be used to study different scenarios of a particular design. The ultimate goal is to extend this work to include the propagation of dynamics in mechanisms and machines. 2 Cut Joint Constraint Formulation Cut joint methods (Wittenburg 1977 and Haug 1989) are used to handle closed loop systems to form a spanning tree that has no closed loops. Joints that are cut in the topology analysis process are replaced by a set of constraint equations. Partial derivatives of basic constraints with respect to design variables are derived in ....

Wittenburg, J., 1977, Dynamics of systems of rigid bodies, (B G Teubner), Stuttgart, 1977.

....and spherical mechanisms under mobility constraints [12] A similar analysis was applied to the mechanism s path generators [13] Sheth and Uicker [14] implemented graph theory to analyze the topology of multibody systems in terms of relative coordinates. Graph theory was also used by Wittenburg [15] to handle closed loop systems by cutting joints to form a spanning tree. Wittenburg and Wolz [16] also presented a cut body method in a computer program for articulated multibody dynamics. The recursive formulation used in this paper was adapted from the work of Bae and Haug [17, 18] who used ....

J. Wittenburg, Dynamics of Systems of Rigid Bodies, (B G Teubner), Stuttgart, 1977.

....at both initial and final keyframes. This is done by blending two dynamic motion curves, each satisfying one of the two boundary conditions; the resulting blended motion curve satisfies all the boundary conditions. Many efficient methods have been developed for forward dynamic simulation [2, 3, 4, 6, 12, 13]; based on simulating the law of physics, they produce quite natural dynamic motions. The blended motion curve generates slightly less natural dynamic motion; however, this is inevitable when the motion curve must satisfy all the boundary conditions. The SC method tries to minimize awkwardness by ....

....Original Blended Energy [t1, t2] Figure 9: New Energy Peaks 9 3.2 Open Chain Multi Linked Body System We consider how the motion blending technique can be applied to the case of open chain multilinked body system. There are many ways to formulate the multi link body kinematics and dynamics [2, 6, 9, 12, 13]. For the implementation here, we use the explicit formulas developed in Reference [9] for the Lagrangian and the kinetic and potential energy terms. The formulas are very compact using Lie group and Lie algebra notations. Although they are somewhat difficult to read, the compact formulations ....

Wittenburg, J., Dynamics of Systems of Rigid Bodies , B.G. Teubner, Stuttgart, 1977.

....system. It consists of (n k) equations with n generalized coordinates q i ; i = 1; n and k unknown Lagrangian multipliers j ; j = 1; k. 2 Reduction of the dynamic model: the Appel s form The Lagrange multipliers may be computed using the classical method (see, e.g. [4]) Namely, by means of differentiating the equations of the nonholonomic constraints (1.1) along (1.3) and taking into account the nonsingularuty of the matrices M and Omega Gamma one can solve the resulting equation for as a function of (q; q; u) Then, substituting in (1.3) results in a ....

J. Wittenburg, Dynamics of Systems of Rigid Bodies. Stuttgart: B. G. Teubner, 1977.

....equations for the calculation of the adjoint equations which are commonly used with numerical nonlinear control methods. I. Introduction The investigation into the dynamics of multibody systems has been an active topic of research for many years [11] 13] 21] 22] 35] 36] 39] 42] [48]. As we will elaborate below, significant breakthroughs have been made in the last two decades in the attempt by researchers to try to unravel the complex articulated dynamics of a rigid, multibody system such as those found in robotics [10] 29] 37] 5] 31] The complexity and nonlinearity ....

....= l = 0 B B 0 Gammaz y z 0 Gammax Gammay x 0 1 C C A and I is the identity operator. Then the relations in (1) may be expressed as V y = OE y;x V x ; f x = OE T y;x f y : The dynamics of a rigid body at its center of mass are well known and may be found in such texts as Wittenburg [48]. In order to define an operator expression for the dynamics we first define the spatial inertia and the spatial momentum. Assume that C represents the center of mass of the rigid body, m denotes its mass, and JC is its inertia tensor at the center of mass. Then the spatial inertia MC and the ....

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J. Wittenburg, Dynamics of Systems of Rigid Bodies, B.G. Teubner, Stuttgart, 1977.

....eliminate terms to achieve simpler equations. However, little is said about dealing with complex systems with numerous rigid bodies subject to constraints. In contrast, multibody formalisms have been developed and published which offer a systematic analysis method based on matrix representations [25, 26, 85, 87, 97, 110, 134, 137]. In these methods, the analysis consists of setting up matrices which are subsequently manipulated to yield the equations of motion. Because all of the details of the analysis are handled as matrix manipulations, it is more difficult for the analyst to apply simplifications. An analysis method ....

Wittenburg, J. Dynamics of Systems of Rigid Bodies. 1977, B.G. Teubner. Stuttgart.

.... R is the focal length, u x ; u y ; u z ) 2 R 3 is the camera position, and (q w ; q x ; q y ; q z ) 2 S 3 is the unit quaternion for the camera rotation [16, 23] In representing the orientations, unit quaternions are quite useful since they are free of singularities such as gimbal lock (see [16, 23, 28] and Section 8.1) Thus, instead of three parameters, the unit quaternion (i.e. four parameters with one constraint) is used to represent the camera rotation. Given m points p 1 ; p m 2 R 3 , the perspective viewing transformation for the m points, V P : R 4 Theta S 3 R 2m , is ....

.... q(t) Delta p Delta q(t) v(t) Delta q(t) Delta p Delta q(t) 2v(t) Delta q(t) Delta p Delta q(t) 2v(t) Delta p(t) 2v(t) Theta p(t) 15 When we interpret (t) 2v(t) 2 R 3 as the angular velocity, the above is exactly the same as the formula given in classical dynamics [19, 28]: p 0 (t) t) Theta p(t) 16) 3.3 The Jacobian Matrix of the Transformation U Given fixed 3D points p i 2 R 3 (for i = 1; m) let p i (t) be the rotated point of p i by the 3D rotation R q(t) of the unit quaternion q(t) Then, we have p 0 i (t) t) Theta p i (t) ....

Wittenburg, J., Dynamics of Systems of Rigid Bodies, B.G. Teubner, Stuttgart, 1977. 45

....equations which are deduced from the application of the Newton Euler formulation. The assumptions are: ffl the bodies system composed by (1 axle, 2 wheels) can be considered as a gyrostat since its inertia properties (mass center, inertia momentum) are constant in this system of bodies [11] ffl the links between axles and plate form (fig.1) or between modules (fig.7) are considered massless. By substitution of forces exerted at internal links and by including contact forces (function of the relatives velocities and displacements) the motion equations system can be written as ....

J. Wittenburg. Dynamic of Systems of Rigid Bodies, B. G. Teubner Stuttgart, 1977.

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Wittenburg J (1977) Dynamics of Systems of Rigid Bodies. B.G. Teubner, Stuttgart

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Wittenburg, J., Dynamics of Systems of Rigid Bodies, B. G. Teubner, Stuttgart, 1977.

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Wittenburg, J., Dynamics of Systems of Rigid Bodies, B.G. Teubner, Stuttgart, 1977.

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J. Wittenburg, Dynamics of systems of rigid bodies, Teubner,

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Wittenburg, J. Dynamics of Systems of Rigid Bodies. 1977, B.G. Teubner. Stuttgart.

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J. Wittenburg, Dynamic of systems of rigid bodies, Teubner Verlag, Stuttgart, 1997

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J. Wittenburg, Dynamics of Systems of Rigid Bodies, B.G. Teubner, Stuttgart,

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J. Wittenburg, Dynamics of Systems of Rigid Bodies, B.G. Teubner, Stuttgart,

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J. Wittenburg, The Dynamics of Systems of Rigid Bodies, B. G. Teubner, Stuttgart, 1977.

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