Gonzalez-Fallas 2021a - Revision history

Rimni at 13:36, 31 March 2022

2022-03-31T13:36:46Z

Rimni at 13:36, 31 March 2022

2022-03-31T13:36:19Z

Rimni at 13:29, 31 March 2022

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Rimni: /* 3. Results */

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‎3. Results

Rimni at 13:24, 31 March 2022

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Rimni at 13:21, 31 March 2022

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Rimni: /* 2.1 Equilibrium formulations */

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‎2.1 Equilibrium formulations

Rimni: /* 2.1 Equilibrium formulations */

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‎2.1 Equilibrium formulations

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 |}
-==4 Conclusions==
+==4. Conclusions==
 In this paper, a methodology for removing rigid body motions from a tensegrity structure has been presented. The formulation of the equilibrium equations for a tensegrity structure is described, and the significance of the vector spaces of the equilibrium matrix is discussed. The left null space of the equilibrium matrix contains the affine motions of the tensegrity structure, including rigid body motions and infinitesimal mechanisms. It was described that it is possible to restrict some nodes of a tensegrity to remove rigid body motions by deleting from the equilibrium matrix the rows corresponding to these nodes. Although not indispensable, it was explained that it is desirable to remove the columns corresponding to the elements connecting the fixed nodes as well, to bring the matrix to its reduced form. The left null space of this reduced form of the equilibrium matrix will contain the infinitesimal mechanisms of the structure.

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	The rows <math>\{1, 2, 3, 7, 8, 9, 13, 14, 15\}</math>, corresponding to the components <math>\{ n_{1x}, \, n_{2x},\, n_{3x}, \,n_{1y}, \, n_{2y},</math> <math>\,n_{3y},\, n_{1z}, \,n_{2z}, \, n_{3z}\}</math> are then deleted from matrix A; in the same way, columns 1, 2 and 3, corresponding to <math>m_1</math>, <math>m_2</math> and <math>m_3</math> are also deleted, resulting in the matrix described in [[#tab-2\|Table 2]]. The nullspace of the reduced equilibrium matrix <math>{\bf A}'</math> can be calculated, resulting in <math>\nu =1</math>; this value is expected, as the structure is still self-standing and the fixed nodes only remove motions, so a prestress set is yet required. Because <math>{\bf A}'</math> is square, its left nullspace is also 1, indicating the presence of <math>\rho = 1</math>. Since the structure is fixed, this motion can only correspond to an infinitesimal mechanism. [[#img-6\|Figure 6]]a shows the resulting structure, where it is seen how the bottom nodes are fixed and the cables connecting them have been removed; the geometry in equilibrium, and the excited mode are also presented, where the latter is described as the shadow geometry. The rotation of the top nodes produced by the infinitesimal mechanism can be observed more clearly in [[#img-6\|Figure 6]]b.		The rows <math>\{1, 2, 3, 7, 8, 9, 13, 14, 15\}</math>, corresponding to the components <math>\{ n_{1x}, \, n_{2x},\, n_{3x}, \,n_{1y}, \, n_{2y},</math> <math>\,n_{3y},\, n_{1z}, \,n_{2z}, \, n_{3z}\}</math> are then deleted from matrix A; in the same way, columns 1, 2 and 3, corresponding to <math>m_1</math>, <math>m_2</math> and <math>m_3</math> are also deleted, resulting in the matrix described in [[#tab-2\|Table 2]]. The nullspace of the reduced equilibrium matrix <math>{\bf A}'</math> can be calculated, resulting in <math>\nu =1</math>; this value is expected, as the structure is still self-standing and the fixed nodes only remove motions, so a prestress set is yet required. Because <math>{\bf A}'</math> is square, its left nullspace is also 1, indicating the presence of <math>\rho = 1</math>. Since the structure is fixed, this motion can only correspond to an infinitesimal mechanism. [[#img-6\|Figure 6]]a shows the resulting structure, where it is seen how the bottom nodes are fixed and the cables connecting them have been removed; the geometry in equilibrium, and the excited mode are also presented, where the latter is described as the shadow geometry. The rotation of the top nodes produced by the infinitesimal mechanism can be observed more clearly in [[#img-6\|Figure 6]]b.
		+

	<div class="center" style="font-size: 75%;">'''Table 2'''. Equilibrium matrix for a constrained three-bar single unit</div>		<div class="center" style="font-size: 75%;">'''Table 2'''. Equilibrium matrix for a constrained three-bar single unit</div>

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 | colspan="1" style="padding:10px;"| '''Figure 6'''. Three-bar tensegrity structure with fixed nodes
 |}
 ===3.2 Tensegrity column===

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−	As mentioned before, for robotic applications such as positioning or active shape control, it is desirable to fix the bottom nodes of the tensegrity structure to avoid rigid body motions. In the case of the three-bar structure, nodes <math>n_1</math>, <math>n_2</math> and <math>n_3</math> are restricted, allowing the motion of the system without displacement. Because the distance between these nodes will not change, the elements <math>m_1</math>, <math>m_2</math> and <math>~~m_2~~</math> can be removed.	+	As mentioned before, for robotic applications such as positioning or active shape control, it is desirable to fix the bottom nodes of the tensegrity structure to avoid rigid body motions. In the case of the three-bar structure, nodes <math>n_1</math>, <math>n_2</math> and <math>n_3</math> are restricted, allowing the motion of the system without displacement. Because the distance between these nodes will not change, the elements <math>m_1</math>, <math>m_2</math> and <math>m_3</math> can be removed.

	The rows <math>\{1, 2, 3, 7, 8, 9, 13, 14, 15\}</math>, corresponding to the components <math>\{ n_{1x}, \, n_{2x},\, n_{3x}, \,n_{1y}, \, n_{2y},</math> <math>\,n_{3y},\, n_{1z}, \,n_{2z}, \, n_{3z}\}</math> are then deleted from matrix A; in the same way, columns 1, 2 and 3, corresponding to <math>m_1</math>, <math>m_2</math> and <math>m_3</math> are also deleted, resulting in the matrix described in [[#tab-2\|Table 2]]. The nullspace of the reduced equilibrium matrix <math>{\bf A}'</math> can be calculated, resulting in <math>\nu =1</math>; this value is expected, as the structure is still self-standing and the fixed nodes only remove motions, so a prestress set is yet required. Because <math>{\bf A}'</math> is square, its left nullspace is also 1, indicating the presence of <math>\rho = 1</math>. Since the structure is fixed, this motion can only correspond to an infinitesimal mechanism. [[#img-6\|Figure 6]]a shows the resulting structure, where it is seen how the bottom nodes are fixed and the cables connecting them have been removed; the geometry in equilibrium, and the excited mode are also presented, where the latter is described as the shadow geometry. The rotation of the top nodes produced by the infinitesimal mechanism can be observed more clearly in [[#img-6\|Figure 6]]b.		The rows <math>\{1, 2, 3, 7, 8, 9, 13, 14, 15\}</math>, corresponding to the components <math>\{ n_{1x}, \, n_{2x},\, n_{3x}, \,n_{1y}, \, n_{2y},</math> <math>\,n_{3y},\, n_{1z}, \,n_{2z}, \, n_{3z}\}</math> are then deleted from matrix A; in the same way, columns 1, 2 and 3, corresponding to <math>m_1</math>, <math>m_2</math> and <math>m_3</math> are also deleted, resulting in the matrix described in [[#tab-2\|Table 2]]. The nullspace of the reduced equilibrium matrix <math>{\bf A}'</math> can be calculated, resulting in <math>\nu =1</math>; this value is expected, as the structure is still self-standing and the fixed nodes only remove motions, so a prestress set is yet required. Because <math>{\bf A}'</math> is square, its left nullspace is also 1, indicating the presence of <math>\rho = 1</math>. Since the structure is fixed, this motion can only correspond to an infinitesimal mechanism. [[#img-6\|Figure 6]]a shows the resulting structure, where it is seen how the bottom nodes are fixed and the cables connecting them have been removed; the geometry in equilibrium, and the excited mode are also presented, where the latter is described as the shadow geometry. The rotation of the top nodes produced by the infinitesimal mechanism can be observed more clearly in [[#img-6\|Figure 6]]b.

← Older revision		Revision as of 13:24, 31 March 2022
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−	As mentioned before, for robotic applications such as positioning or active shape control, it is desirable to fix the bottom nodes of the tensegrity structure to avoid rigid body motions. In the case of the three-bar structure, nodes n1, n2 and n3 are restricted, allowing the motion of the system without displacement. Because the distance between these nodes will not change, the elements m1, m2 and m3 can be removed.	+	As mentioned before, for robotic applications such as positioning or active shape control, it is desirable to fix the bottom nodes of the tensegrity structure to avoid rigid body motions. In the case of the three-bar structure, nodes <math>n_1</math>, <math>n_2</math> and <math>n_3</math> are restricted, allowing the motion of the system without displacement. Because the distance between these nodes will not change, the elements <math>m_1</math>, <math>m_2</math> and <math>m_2</math> can be removed.

	The rows <math>\{1, 2, 3, 7, 8, 9, 13, 14, 15\}</math>, corresponding to the components <math>\{ n_{1x}, \, n_{2x},\, n_{3x}, \,n_{1y}, \, n_{2y},</math> <math>\,n_{3y},\, n_{1z}, \,n_{2z}, \, n_{3z}\}</math> are then deleted from matrix A; in the same way, columns 1, 2 and 3, corresponding to <math>m_1</math>, <math>m_2</math> and <math>m_3</math> are also deleted, resulting in the matrix described in [[#tab-2\|Table 2]]. The nullspace of the reduced equilibrium matrix <math>{\bf A}'</math> can be calculated, resulting in <math>\nu =1</math>; this value is expected, as the structure is still self-standing and the fixed nodes only remove motions, so a prestress set is yet required. Because <math>{\bf A}'</math> is square, its left nullspace is also 1, indicating the presence of <math>\rho = 1</math>. Since the structure is fixed, this motion can only correspond to an infinitesimal mechanism. [[#img-6\|Figure 6]]a shows the resulting structure, where it is seen how the bottom nodes are fixed and the cables connecting them have been removed; the geometry in equilibrium, and the excited mode are also presented, where the latter is described as the shadow geometry. The rotation of the top nodes produced by the infinitesimal mechanism can be observed more clearly in [[#img-6\|Figure 6]]b.		The rows <math>\{1, 2, 3, 7, 8, 9, 13, 14, 15\}</math>, corresponding to the components <math>\{ n_{1x}, \, n_{2x},\, n_{3x}, \,n_{1y}, \, n_{2y},</math> <math>\,n_{3y},\, n_{1z}, \,n_{2z}, \, n_{3z}\}</math> are then deleted from matrix A; in the same way, columns 1, 2 and 3, corresponding to <math>m_1</math>, <math>m_2</math> and <math>m_3</math> are also deleted, resulting in the matrix described in [[#tab-2\|Table 2]]. The nullspace of the reduced equilibrium matrix <math>{\bf A}'</math> can be calculated, resulting in <math>\nu =1</math>; this value is expected, as the structure is still self-standing and the fixed nodes only remove motions, so a prestress set is yet required. Because <math>{\bf A}'</math> is square, its left nullspace is also 1, indicating the presence of <math>\rho = 1</math>. Since the structure is fixed, this motion can only correspond to an infinitesimal mechanism. [[#img-6\|Figure 6]]a shows the resulting structure, where it is seen how the bottom nodes are fixed and the cables connecting them have been removed; the geometry in equilibrium, and the excited mode are also presented, where the latter is described as the shadow geometry. The rotation of the top nodes produced by the infinitesimal mechanism can be observed more clearly in [[#img-6\|Figure 6]]b.

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 |}
-Although the diagram in Figure 1 represents the force <math>{\bf F}_4  </math> in compression, its force is stated as positive in Equations (2) and (4) to maintain uniformity in the formulation of these expressions. However, the force density corresponding to this element is expected to have a negative value as a necessary condition for the existence of equilibrium in the node. The necessary and sufficient stability conditions for tensegrity structures can be found in Zhang and Ohsaki [20]. Equation (4) can also be expressed in a matrix form as
+Although the diagram in [[#img-1|Figure 1]] represents the force <math>{\bf F}_4  </math> in compression, its force is stated as positive in Equations (2) and (4) to maintain uniformity in the formulation of these expressions. However, the force density corresponding to this element is expected to have a negative value as a necessary condition for the existence of equilibrium in the node. The necessary and sufficient stability conditions for tensegrity structures can be found in Zhang and Ohsaki [20]. Equation (4) can also be expressed in a matrix form as
 {| class="formulaSCP" style="width: 100%; text-align: center;"

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 | style="width: 5px;text-align: right;white-space: nowrap;" | (2)
 |}
 <div id='img-1'></div>

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 | style="width: 5px;text-align: right;white-space: nowrap;" | (1)
 |}
 For the static analysis of tensegrity structures, three assumptions are made: the elements are only connected at the nodes forming friction-less joints, external forces are applied at the nodes only, and yielding and buckling are not considered. For simplification purposes, [[#img-1|Figure 1]] presents the forces acting on one node of an arbitrary tensegrity structure. If these forces <math>{\bf F}</math> are represented as a three-dimensional system of equations, and assuming a self-equilibrated structure without external forces, the <math>{\bf F}_i</math> forces acting on the <math display="inline"> i </math> elements can be expressed as
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 | style="width: 5px;text-align: right;white-space: nowrap;" | (2)
 |}
 According to the force density method, the forces on each element can be replaced by a force-to-length ratio, called force density, as in

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 Tensegrity structures belong to the class of pin-jointed structures and differ from other pin-jointed structures in that they are prestressed, and tensegrities are composed of both tensile and compressive elements. Following the definition by Connelly and Back [7], tensegrity structures can be seen as a set of points in space, connected by either compressive or tensile elements; the location of the nodes in space, and how are they connected, is known as topology. Generally speaking, the connectivity scheme can be arbitrarily defined by the designer, according to a preliminary desired shape; however, the nodal coordinates must be determined through a form-finding method, since not any configuration results in a stable structure. The technique used in this paper is called the force density method; it linearizes the equilibrium system of equations that describe a tensegrity system by introducing a force to length ratio. In this way, the forces in a tensegrity are dependent on the length of the elements and not on the nodal coordinates.
-For a tensegrity structure composed of <math display="inline"> m </math> elements and <math display="inline"> n </math> nodes, the connection between elements can be described by the connectivity matrix <math display="inline"> \bf C </math> <math display="inline">\(\in {R}^{m\times n}\)</math>, whose only non-zero entries correspond to the initial and final nodes of the subsequent member. Zhang and Ohsaki [6] express that, if an element <math display="inline">m_k</math> connects nodes <math display="inline"> i</math> and <math display="inline"> j </math> <math display="inline">\left(j<k\right)</math>, the entries <math display="inline"> i</math> and <math display="inline"> j </math> of the <math display="inline">k</math> -th row of the matrix <math display="inline"> \bf C </math> will be -1 or 1, according to the condition
+For a tensegrity structure composed of <math display="inline"> m </math> elements and <math display="inline"> n </math> nodes, the connection between elements can be described by the connectivity matrix <math display="inline"> \bf C </math> <math display="inline">\left(\in {R}^{m\times n}\right)</math>, whose only non-zero entries correspond to the initial and final nodes of the subsequent member. Zhang and Ohsaki [6] express that, if an element <math display="inline">m_k</math> connects nodes <math display="inline"> i</math> and <math display="inline"> j </math> <math display="inline">\left(j<k\right)</math>, the entries <math display="inline"> i</math> and <math display="inline"> j </math> of the <math display="inline">k</math> -th row of the matrix <math display="inline"> \bf C </math> will be -1 or 1, according to the condition
 {| class="formulaSCP" style="width: 100%; text-align: center;"