Motion space

Fig. 5.10. Noise-induced front motion Space-time plots of the electron density for (a) D=0 (no noise), (b) D = 0.5 s /m, (c) D = jm . Light and dark shading...

Tools and methods that are used in VE and that permit information to be obtained about the product in the early development phases offer attractive optimization potentials (time, costs, quality) in engineering. Among other things, in the foreground here are product aspects and features that essentially permit qualitative information to be obtained (e.g., formal aspects) and can be primarily assessed by evaluation (e.g., motion spaces in connection with capability of assembly). Human interaction with the virtual prototypes and human perception and immersion in the VE is decisive for handling the tasks on hand (Figure 2). The verification of qualitative information has normally been possible so far only on the basis of real, physical prototypes. 

The matrix, is of special significance. It is of dimension 6 x m, resolved in the coordinate firame of body i, and it has full column rank, nj. It represents the free modes of joint t, and its columns make up a basis for this free vector space. This will also be referred to as the motion space of joint i. Because 4>i is resolved in the tth fiame, it is constant It is always possible to construct a 6x6 matrix as shown [3] ... 

Since the motion of a revolute joint is confined to rotation about the ti unit vector, the motion space is ... 

The motion space which corresponds to this degree of freedom is [36] ... 

The columns of the Jacobian matrix are nothing m e than basis vectors for the motion space of the manipulator. A dot product of these columns with both sides of Equation 3.S will pxiject each term onto the manipulator joint (X motion space. Thus, for the one-link manipulator, we may write ... 

By definition, the projection of fi onto the motion space of the single joint is t. The unknown constraint forces and moments at joint 1 have been eliminated. It is also true that ai = Ji iji, since the joint velocities are all assumed to be zefo. Thus, Equation 3.6 becomes ... 

The term f2 represents the projection of onto the basis vectors of the motion space, which are expressed with respect to the cotndinate system of link 2. By Equation 3.4, we may write the two-link Jacobian as ... 

The t-link Jacobian matrix is used to project the force terms onto the motion space of the augmented manipulator... 

A number of important recurrence relations in the articulated-body fwmula-tion may be derived by analyzing a simple two-link articulated body as shown in Figure 4.3. This articulated body is made up of links 1 and 2 with spatial link inertias of Ii and I2, respectively. The two links are connected by a general joint, joint 2, which is characterized by its motion space, < 2> Recall from Chapter 2 that 2 represents the free modes of this joint In this analysis, it is assumed that the two bodies are initially at rest and experience no applied joint forces or torques on or about 2 (t2 = 0). The tip force, f, is zero b ause the chain is unconstrained. The only spatial force vector exerted on the articulated body is fi, which is applied to link 1, the handle of the articulated body. For the moment, we will assume that the physical parameters describing the links and joints, including the spatial link inertias, are given in absolute coordinates. 

The component of a spatial vector which lies on or along a specific axis of a joint may be determined by performing a simple dot product between that spatial vector and the vector which represents the joint axis. This may be refen to as projecting the spatial vector onto the joint axis . Thus, we may project the terms of Equation 4.32 onto the motion space of joint 2 via the following spatial dot product ... 

See Anstey, Philosophy of Robert Boyle, pp. 153-154. Anstey argues, I believe correctly, that Acre is no prospect of unearthing a coherent physics in Boyle s works... Boyle was first and foremost a philosopher of the qualities, not motion, space or time. ... 

Nespor, J. (1994). Knowledge in motion Space, time, and curriculum in undergraduate physics and management. London Fahner Press. 

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