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Fig. 8: The workspace and singularity of the SDelta with the orientation e = [0,0, 1]’ and @ = 15° (a) Design I (b) Design II 3.4 Case Study: the Fixed-Orientation Singularity Locus

Figure 8 The workspace and singularity of the SDelta with the orientation e = [0,0, 1]’ and @ = 15° (a) Design I (b) Design II 3.4 Case Study: the Fixed-Orientation Singularity Locus

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Fig. 1: Topology of the SDelta robot
Fig. 1: Topology of the SDelta robot
Fig. 3: Architecture of the SDelta Robot
Fig. 3: Architecture of the SDelta Robot
Fig. 5: Notation for the kinematic chain of the jth limb of the SDelta Robot
Fig. 5: Notation for the kinematic chain of the jth limb of the SDelta Robot
Under this notation, g; and e; represent the unit vectors parallel to lines Kj, and 4j2, respectively, for j = 1,2,3. We will henceforth denote Aj; and Kj2 as G; and £;, respectively, for simplicity, as shown in Fig. 5, their Pliicker arrays being T T kj = [g),(g;xpy)’] andkjz = [e7,(e; x pj)’] . Once K is available, D is straightforward to derive as a diagonal matrix, whose ith component is the reciprocal product of the line corresponding to the ith row of K and that associated with the joint of the ith component of @. D is found to bear the form Next, consider line 4j2, associated with the wrench corresponding to the translational degree of freedom of the jth C- Irive. Since {j2 passes through Sj, it is coplanar with J;;, and normal to J;3; we can then conclude that Aj2 passes through 5; and is parallel to the axis of the jth C-drive, as shown, again, in Fig. 5. Based on the above analysis, we can obtain the K natrix as
Under this notation, g; and e; represent the unit vectors parallel to lines Kj, and 4j2, respectively, for j = 1,2,3. We will henceforth denote Aj; and Kj2 as G; and £;, respectively, for simplicity, as shown in Fig. 5, their Pliicker arrays being T T kj = [g),(g;xpy)’] andkjz = [e7,(e; x pj)’] . Once K is available, D is straightforward to derive as a diagonal matrix, whose ith component is the reciprocal product of the line corresponding to the ith row of K and that associated with the joint of the ith component of @. D is found to bear the form Next, consider line 4j2, associated with the wrench corresponding to the translational degree of freedom of the jth C- Irive. Since {j2 passes through Sj, it is coplanar with J;;, and normal to J;3; we can then conclude that Aj2 passes through 5; and is parallel to the axis of the jth C-drive, as shown, again, in Fig. 5. Based on the above analysis, we can obtain the K natrix as
Fig. 6: The reference posture of the MP We have obtained K as shown in Eqn. (7), whose six rows can be regarded as the Pliticker coordinates of six actuated wrenches, intersecting pairwise at the center of the three spherical joints. Since these six wrench lines of K intersect in three pairs in our case, it has been found that K becomes singular when the four planes—three planes composed by the three pairs of intersecting wrench lines together with the plane of the MP triangle—share at least one common point [28—31,50,51]. We denote the plane defined by the intersecting lines £;, G; as II;, its normal as nj, for j = 1,2,3. It is noteworthy that n; = f; for the SDelta Robot, but we use n; for generality. Moreover, we denote the MP plane as II4, its normal as ny. Next we conduct the singularity analysis based on this geometrical interpretation. First, denote the intersecting line between IT; and Ty as £;, for j = 1,2,3; it is noteworthy that £; passes through S;, and lies in the common plane IT4; the foregoing geometrical condition is then equivalent to requiring that the three lines £;, for 7 = 1,2,3, share common points. In this way, we can reduce the analysis to the plane II4. Furthermore, let us denote the intersecting point of £; with £, and with £ as R2 and R3, respectively; then, the condition leads to requiring that the position vectors of R2 and R3 be identical. + a = ‘ome aa
Fig. 6: The reference posture of the MP We have obtained K as shown in Eqn. (7), whose six rows can be regarded as the Pliticker coordinates of six actuated wrenches, intersecting pairwise at the center of the three spherical joints. Since these six wrench lines of K intersect in three pairs in our case, it has been found that K becomes singular when the four planes—three planes composed by the three pairs of intersecting wrench lines together with the plane of the MP triangle—share at least one common point [28—31,50,51]. We denote the plane defined by the intersecting lines £;, G; as II;, its normal as nj, for j = 1,2,3. It is noteworthy that n; = f; for the SDelta Robot, but we use n; for generality. Moreover, we denote the MP plane as II4, its normal as ny. Next we conduct the singularity analysis based on this geometrical interpretation. First, denote the intersecting line between IT; and Ty as £;, for j = 1,2,3; it is noteworthy that £; passes through S;, and lies in the common plane IT4; the foregoing geometrical condition is then equivalent to requiring that the three lines £;, for 7 = 1,2,3, share common points. In this way, we can reduce the analysis to the plane II4. Furthermore, let us denote the intersecting point of £; with £, and with £ as R2 and R3, respectively; then, the condition leads to requiring that the position vectors of R2 and R3 be identical. + a = ‘ome aa
Fig. 7: The workspace and singularity of the SDelta under the reference orientation (a) Design I (b) Design I
Fig. 7: The workspace and singularity of the SDelta under the reference orientation (a) Design I (b) Design I
Fig. 9: The workspace and singularity of the SDelta with the orientation e = [0,1,0]’ and = 15° (a) Design I (b) Design |
Fig. 9: The workspace and singularity of the SDelta with the orientation e = [0,1,0]’ and = 15° (a) Design I (b) Design |
Fig. 10: Illustration of the workspace formulation based on the geometric method
Fig. 10: Illustration of the workspace formulation based on the geometric method
Fig. 11: The workspace of the Sdelta under the reference orientation (a) Design I (b) Design II
Fig. 11: The workspace of the Sdelta under the reference orientation (a) Design I (b) Design II
Fig. 12: The workspace of the SDelta with the orientation e = [0,0, 1]’ and @ = 15° (a) Design I (b) Design II
Fig. 12: The workspace of the SDelta with the orientation e = [0,0, 1]’ and @ = 15° (a) Design I (b) Design II
Fig. 13: The workspace of the SDelta with the orientation e = [0,1,0]’ and @ = 15° (a) Design I (b) Design II
Fig. 13: The workspace of the SDelta with the orientation e = [0,1,0]’ and @ = 15° (a) Design I (b) Design II
Fig. 14: The workspace volume w.r.t. the ratio of a/b, under different orientations The above result indicates that, when the range of the passive limb length is fixed, the workspace volume generally decreases as the ratio a/b increases; however, for a larger MP, (e.g., a = b), the robot allows for a larger range of the passive limb length!!, which can end up with a bigger workspace volume. and e = [0,1,0]’ with @ = 15°, respectively, as shown in Fig. 14. Apparently, the workspace volume V under the reference orientation remains largely unchanged for different values of a/b; however, V decreases significantly when the MP is rotated about the Z-axis; when the MP is rotated about other directions, the workspace volume can either decrease or increase. It is noteworthy that the above workspace volume is different from those of Designs I and II, indicating that the range of the passive limb length affects the workspace as well.
Fig. 14: The workspace volume w.r.t. the ratio of a/b, under different orientations The above result indicates that, when the range of the passive limb length is fixed, the workspace volume generally decreases as the ratio a/b increases; however, for a larger MP, (e.g., a = b), the robot allows for a larger range of the passive limb length!!, which can end up with a bigger workspace volume. and e = [0,1,0]’ with @ = 15°, respectively, as shown in Fig. 14. Apparently, the workspace volume V under the reference orientation remains largely unchanged for different values of a/b; however, V decreases significantly when the MP is rotated about the Z-axis; when the MP is rotated about other directions, the workspace volume can either decrease or increase. It is noteworthy that the above workspace volume is different from those of Designs I and II, indicating that the range of the passive limb length affects the workspace as well.
The characteristic length is defined as the value L that minimizes the condition number of K;,; as reported in a forthcoming paper, the characteristic length of the SDelta robot can be obtained as the radius of the circle circumscribing the equilateral- triangular MP, ie., as L = a/\/3. Then, the dexterity is analyzed via a numerical algorithm based on discretization. Four different Z-values are chosen, while the dimensions in the other directions are discretized with an interval of b/100; the dexterity distribution is plotted in Figs. 15 and 16 for different orientations!”. The value log,(1/«-(K;)), denoted A for brevity, is plotted vs. X-Y, those surfaces with higher altitudes representing a layer with smaller Z, i.e., the condition Given that the entries of the Jacobian matrices bear different units, we introduce a pertinent characteristic length L [54] to resolve this dimensional inhomogeneity. To this end, we redefine the twist and K in their dimensionally homogeneous form:
The characteristic length is defined as the value L that minimizes the condition number of K;,; as reported in a forthcoming paper, the characteristic length of the SDelta robot can be obtained as the radius of the circle circumscribing the equilateral- triangular MP, ie., as L = a/\/3. Then, the dexterity is analyzed via a numerical algorithm based on discretization. Four different Z-values are chosen, while the dimensions in the other directions are discretized with an interval of b/100; the dexterity distribution is plotted in Figs. 15 and 16 for different orientations!”. The value log,(1/«-(K;)), denoted A for brevity, is plotted vs. X-Y, those surfaces with higher altitudes representing a layer with smaller Z, i.e., the condition Given that the entries of the Jacobian matrices bear different units, we introduce a pertinent characteristic length L [54] to resolve this dimensional inhomogeneity. To this end, we redefine the twist and K in their dimensionally homogeneous form:
Fig. 15: The dexterity of the SDelta on the layers of z = 0.3,0.6,0.9, 1.2, with the orientation e = (0,0, 1]’ and @ = 15° (a Design I (b) Design II
Fig. 15: The dexterity of the SDelta on the layers of z = 0.3,0.6,0.9, 1.2, with the orientation e = (0,0, 1]’ and @ = 15° (a Design I (b) Design II
Fig. 16: The dexterity of the SDelta on the layers of z = 0.3,0.6,0.9, 1.2, with the orientation e = [0, 1,0)? and @ = 15° (a Design I (b) Design II
Fig. 16: The dexterity of the SDelta on the layers of z = 0.3,0.6,0.9, 1.2, with the orientation e = [0, 1,0)? and @ = 15° (a Design I (b) Design II
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