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Figure 17: Persistent S42 systems.

Figure 17 Persistent S42 systems.

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Figure 1: (a) Rigid-body transformation g,» of body frame b with respect to reference frame a; (b) Conjugation of a rigid motion g due to change of reference frame and body frame by go.
Figure 1: (a) Rigid-body transformation g,» of body frame b with respect to reference frame a; (b) Conjugation of a rigid motion g due to change of reference frame and body frame by go.
and it is often called a twist (see Fig. 2(a)). The velocity q(t) of a point q(t) on the moving body is given by:
and it is often called a twist (see Fig. 2(a)). The velocity q(t) of a point q(t) on the moving body is given by:
Figure 3: (a) A wrench ¢ acting on a body moving with twist €; (b) geometry of the cylindroid.
Figure 3: (a) A wrench ¢ acting on a body moving with twist €; (b) geometry of the cylindroid.
Figure 4: Illustration of 1-D Lie subgroup generators: (a) the R joint; (b) the P joint; (c) the H joint. Lie’s third theorem [4] states that there is a one-to-one correspondence, via the exponential map, between Lie subalgebras g of se(3) (up to an Adjoint transformation) and connected Lie subgroups G of SE(3) (up to conjugation). Moreover, each Lie subgroup G shares the same Lie group and Lie algebra properties as those of SE(3). The systematic classification of a total of ten classes of connected Lie subgroups of SE(3) was (probably) first studied in [9] based on a classification of Lie subalgebras of se(3); see also [1]. The complete classification of Lie subgroups of SE(3) is now presented for quick reference. 1_-T) Tie ewhoeroune of SE(2) are of the form eyn(l6l_) for come twict £ C «e(2) After
Figure 4: Illustration of 1-D Lie subgroup generators: (a) the R joint; (b) the P joint; (c) the H joint. Lie’s third theorem [4] states that there is a one-to-one correspondence, via the exponential map, between Lie subalgebras g of se(3) (up to an Adjoint transformation) and connected Lie subgroups G of SE(3) (up to conjugation). Moreover, each Lie subgroup G shares the same Lie group and Lie algebra properties as those of SE(3). The systematic classification of a total of ten classes of connected Lie subgroups of SE(3) was (probably) first studied in [9] based on a classification of Lie subalgebras of se(3); see also [1]. The complete classification of Lie subgroups of SE(3) is now presented for quick reference. 1_-T) Tie ewhoeroune of SE(2) are of the form eyn(l6l_) for come twict £ C «e(2) After
Table 2: A complete list of connected Lie subgroups of SE(3).
Table 2: A complete list of connected Lie subgroups of SE(3).
Figure 5: Traces of a rigid body undergoing motions of the: (a) cylindrical group C(z) = exp {t;, Tz}sp; (b) planar translation group J2 = exp({tx, ty}sp). (c) spatial translation group J3 = exp {tx, ty, &}sp; (d) planar Euclidean group £(z) = exp({tx, &, Tz }sp);(e) planar helical group Y = exp({tx, ty, kp}sp); (f) spatial rotation group S = exp({Tx, Ty, Tz hsp).
Figure 5: Traces of a rigid body undergoing motions of the: (a) cylindrical group C(z) = exp {t;, Tz}sp; (b) planar translation group J2 = exp({tx, ty}sp). (c) spatial translation group J3 = exp {tx, ty, &}sp; (d) planar Euclidean group £(z) = exp({tx, &, Tz }sp);(e) planar helical group Y = exp({tx, ty, kp}sp); (f) spatial rotation group S = exp({Tx, Ty, Tz hsp).
Figure 6: Several instances of serial mechanical generators for X. (a) An HH HH chain; (b) An RRRP chain; (c) A PRR chain.
Figure 6: Several instances of serial mechanical generators for X. (a) An HH HH chain; (b) An RRRP chain; (c) A PRR chain.
Figure 7: (a) A redundant HH HHH chain generating the Schénflies group X; (b) A 1-DOF linkage resulting from closing the loop of the HAH HH H chain.
Figure 7: (a) A redundant HH HHH chain generating the Schénflies group X; (b) A 1-DOF linkage resulting from closing the loop of the HAH HH H chain.
Figure 8: Illustration of a 73 parallel mechanism with | X(v;) chains (v1, v2 and v3 must span R?, namely they must not be perpendicular to the same direction).
Figure 8: Illustration of a 73 parallel mechanism with | X(v;) chains (v1, v2 and v3 must span R?, namely they must not be perpendicular to the same direction).
Figure 9: Serial mechanism of a binary product £(u)£(v). (a) A redundant serial generator; (b) Redundant subgroup ?(u x v); (c) A non-redundant serial generator.
Figure 9: Serial mechanism of a binary product £(u)£(v). (a) A redundant serial generator; (b) Redundant subgroup ?(u x v); (c) A non-redundant serial generator.
Figure 10: (a) A 1-DOF linkage resulting from closing the loop of a 6-DOF redundant £(u)£(v) chain; (b) A purely translational parallel mechanism comprising 3 or more ££ chains.
Figure 10: (a) A 1-DOF linkage resulting from closing the loop of a 6-DOF redundant £(u)£(v) chain; (b) A purely translational parallel mechanism comprising 3 or more ££ chains.
Figure 11: Persistent S31 system (H RH type).
Figure 11: Persistent S31 system (H RH type).
Figure 15: Persistent S44 systems.
Figure 15: Persistent S44 systems.
Figure 19: Persistent S4,5 systems. Table 3: A complete list of 3-D persistent POE manifolds of SE(3).
Figure 19: Persistent S4,5 systems. Table 3: A complete list of 3-D persistent POE manifolds of SE(3).
Table 4: A complete list of 4-D persistent POE manifolds of SE(3).
Table 4: A complete list of 4-D persistent POE manifolds of SE(3).
Figure 20: (a) Example of a 5-R, homokinetic chain mirror-symmetric about the homokinetic plane TU; (b) and (c) Two examples of symmetric space: (b) the unit 2-D sphere; (c) the Euclidean space R™ (m = 2 here). and an infinite-pitch screw perpendicular to H; the homokinetic chain is inherently redundant and maintain the homokinetic (mirror-symmetry) condition only within a closed-loop (e.g.. parallel) mechanism. As we shall shortly see, the congruence-invariance of the corresponding 3-D motion manifold arises from yet another class of persistent manifolds, called symmetric ee
Figure 20: (a) Example of a 5-R, homokinetic chain mirror-symmetric about the homokinetic plane TU; (b) and (c) Two examples of symmetric space: (b) the unit 2-D sphere; (c) the Euclidean space R™ (m = 2 here). and an infinite-pitch screw perpendicular to H; the homokinetic chain is inherently redundant and maintain the homokinetic (mirror-symmetry) condition only within a closed-loop (e.g.. parallel) mechanism. As we shall shortly see, the congruence-invariance of the corresponding 3-D motion manifold arises from yet another class of persistent manifolds, called symmetric ee
Figure 21: Physical interpretation of Msp (a) and Mz, (b) motions. of their corresponding LTSs in se(3) [28], leading to the discovery of a total of seven classes ot non-trivial symmetric subspaces of SE(3), which are listed in Tab. 5 for quick reference. We shall follow [3] and denote a m-D LTS with finite-pitch screws parallel to a single direction by m4, and a m-D LTS with finite-pitch screws parallel to a single plane by m,,z; the corresponding symmetric subspace, derived algebra, completion algebra and completion group shall be indicated by the same subscripts. When there is only one LTS for the specified dimension, the subscript ()ma or ()me will be simply written as ()m.- There are three classes of 2-D symmetric subspaces of SE(3):
Figure 21: Physical interpretation of Msp (a) and Mz, (b) motions. of their corresponding LTSs in se(3) [28], leading to the discovery of a total of seven classes ot non-trivial symmetric subspaces of SE(3), which are listed in Tab. 5 for quick reference. We shall follow [3] and denote a m-D LTS with finite-pitch screws parallel to a single direction by m4, and a m-D LTS with finite-pitch screws parallel to a single plane by m,,z; the corresponding symmetric subspace, derived algebra, completion algebra and completion group shall be indicated by the same subscripts. When there is only one LTS for the specified dimension, the subscript ()ma or ()me will be simply written as ()m.- There are three classes of 2-D symmetric subspaces of SE(3):
Figure 22: A motion in Mgp corresponds to the tilt of the symmetry-axis (the z-axis) via a rotation about an axis in the xy-plane that is perpendicular to the initial and final location of the z-axis.
Figure 22: A motion in Mgp corresponds to the tilt of the symmetry-axis (the z-axis) via a rotation about an axis in the xy-plane that is perpendicular to the initial and final location of the z-axis.
Finally, M, characterizes the motion of the z-axis (called the characteristic line) to a tar- get location by twisting along the common perpendicular of the two lines without incurring redundant self-spin and sliding, which is represented by Hy = C(z) (see Fig. 23). M34 may be considered a submanifold of M, that maintains the perpendicularity of the line to the x-axis.
Finally, M, characterizes the motion of the z-axis (called the characteristic line) to a tar- get location by twisting along the common perpendicular of the two lines without incurring redundant self-spin and sliding, which is represented by Hy = C(z) (see Fig. 23). M34 may be considered a submanifold of M, that maintains the perpendicularity of the line to the x-axis.
Figure 24: Two examples of symmetric chains for M3p.
Figure 24: Two examples of symmetric chains for M3p.
Figure 25: Example of a Msp parallel mechanism: (a) chain geometry; (b) constraint wrenches. 6. Conclusion
Figure 25: Example of a Msp parallel mechanism: (a) chain geometry; (b) constraint wrenches. 6. Conclusion
Related topics:
Lie AlgebraMechanism DesignLie GroupsKinematics of MachinesParallel manipulatorsScrew TheoryMechanism Design: Analysis and Synthesis
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