Dynamics of the Shell Class of Tensegrity Structures

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18.1 Introduction 18.2 Tensegrity Deﬁnitions A Typical Element • Rules of Closure for the Shell Class 18.3 Dynamics of a Two-Rod Element 18.4 Choice of Independent Variables and Coordinate Transformations Robert E. Skelton 18.5 Tendon Forces University of California, San Diego 18.6 Conclusion Appendix 18.A Proof of Theorem 18.1 Jean-Paul Pinaud Appendix 18.B Algebraic Inversion of the Q Matrix University of California, San Diego Appendix 18.C General Case for (n, m) = (i, 1) D. L. Mingori Appendix 18.D Example Case (n,m) = (3,1) University of California, Los Angeles Appendix 18.E Nodal Forces

Abstract

A tensegrity structure is a special truss structure in a stable equilibrium with selected members designated for only tension loading, and the members in tension forming a continuous network of cables separated by a set of compressive members. This chapter develops an explicit analytical model of the nonlinear dynamics of a large class of tensegrity structures constructed of rigid rods connected by a continuous network of elastic cables. The kinematics are described by positions and velocities of the ends of the rigid rods; hence, the use of angular velocities of each rod is avoided. The model yields an analytical expression for accelerations of all rods, making the model efﬁcient for simulation, because the update and inversion of a nonlinear mass matrix are not required. The model is intended for shape control and design of deployable structures. Indeed, the explicit analytical expressions are provided herein for the study of stable equilibria and controllability, but control issues are not treated.

18.1 Introduction

The history of structural design can be divided into four eras classiﬁed by design objectives. In the prehistoric era, which produced such structures as Stonehenge, the objective was simply to oppose gravity, to take static loads. The classical era, considered the dynamic response and placed design constraints on the eigenvectors as well as eigenvalues. In the modern era, design constraints could be so demanding that the dynamic response objectives require feedback control. In this era, the

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control discipline followed the classical structure design, where the structure and control disciplines were ingredients in a multidisciplinary system design, but no interdisciplinary tools were developed to integrate the design of the structure and the control. Hence, in this modern era, the dynamics of the structure and control were not cooperating to the fullest extent possible. The post-modern era of structural systems is identified by attempts to unify the structure and control design for a common objective. The ultimate performance capability of many new products and systems cannot be achieved until mathematical tools exist that can extract the full measure of cooperation possible between the dynamics of all components (structural components, controls, sensors, actuators, etc.). This requires new research. Control theory describes how the design of one component (the controller) should be influenced by the (given) dynamics of all other components. However, in systems design, where more than one component remains to be designed, there is inadequate theory to suggest how the dynamics of two or more components should influence each other at the design stage. In the future, controlled structures will not be conceived merely as multidisciplinary design steps, where a plate, beam, or shell is first designed, followed by the addition of control actuation. Rather, controlled structures will be conceived as an interdisciplinary process in which both material architecture and feedback information architecture will be jointly determined. New paradigms for material and structure design might be found to help unify the disciplines. Such a search motivates this work. Preliminary work on the integration of structure and control design appears in Skelton1,2 and Grigoriadis et al.3 Bendsoe and others4-7 optimize structures by beginning with a solid brick and deleting finite elements until minimal mass or other objective functions are extremized. But, a very important factor in determining performance is the paradigm used for structure design. This chapter describes the dynamics of a structural system composed of axially loaded compression members and tendon members that easily allow the unification of structure and control functions. Sensing and actuating functions can sense or control the tension or the length of tension members. Under the assumption that the axial loads are much smaller than the buckling loads, we treat the rods as rigid bodies. Because all members experience only axial loads, the mathematical model is more accurate than models of systems with members in bending. This unidirectional loading of members is a distinct advantage of our paradigm, since it eliminates many nonlinearities that plague other controlled structural concepts: hysteresis, friction, deadzones, and backlash. It has been known since the middle of the 20th century that continua cannot explain the strength of materials. While science can now observe at the nanoscale to witness the architecture of materials preferred by nature, we cannot yet design or manufacture manmade materials that duplicate the incredible structural efficiencies of natural systems. Nature’s strongest fiber, the spider fiber, arranges simple nontoxic materials (amino acids) into a microstructure that contains a continuous network of members in tension (amorphous strains) and a discontinuous set of members in compression (the β-pleated sheets in Figure 18.1).8,9 This class of structure, with a continuous network of tension members and a discontinuous network of compression members, will be called a Class 1 tensegrity structure. The important lessons learned from the tensegrity structure of the spider fiber are that 1. Structural members never reverse their role. The compressive members never take tension and, of course, tension members never take compression. 2. Compressive members do not touch (there are no joints in the structure). 3. Tensile strength is largely determined by the local topology of tension and compressive members. Another example from nature, with important lessons for our new paradigms is the carbon nanotube often called the Fullerene (or Buckytube), which is a derivative of the Buckyball. Imagine

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amorphous chain

β-pleated sheet

entanglement

hydrogen bond

y z

x 6nm

FIGURE 18.1 Nature’s strongest ﬁber: the Spider Fiber. (From Termonia, Y., Macromolecules, 27, 7378–7381, 1994. Reprinted with permission from the American Chemical Society.)

FIGURE 18.2 Buckytubes.

a 1-atom thick sheet of a graphene, which has hexagonal holes due to the arrangements of material at the atomic level (see Figure 18.2). Now imagine that the flat sheet is closed into a tube by choosing an axis about which the sheet is closed to form a tube. A specific set of rules must define this closure which takes the sheet to a tube, and the electrical and mechanical properties of the resulting tube depend on the rules of closure (axis of wrap, relative to the local hexagonal topology).10 Smalley won the Nobel Prize in 1996 for these insights into the Fullerenes. The spider fiber and the Fullerene provide the motivation to construct manmade materials whose overall mechanical, thermal, and electrical properties can be predetermined by choosing the local topology and the rules of closure which generate the three-dimensional structure from a given local topology. By combining these motivations from Fullerenes with the tensegrity architecture of the spider fiber, this chapter derives the static and dynamic models of a shell class of tensegrity structures. Future papers will exploit the control advantages of such structures. The existing literature on tensegrity deals mainly11-23 with some elementary work on dynamics in Skelton and Sultan,24 Skelton and He,25 and Murakami et al.26

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FIGURE 18.3 Needle Tower of Kenneth Snelson, Class 1 tensegrity. Kröller Müller Museum, The Netherlands. (From Connelly, R. and Beck, A., American Scientist, 86(2), 143, 1998. With permissions.)

18.2 Tensegrity Deﬁnitions

Kenneth Snelson built the first tensegrity structure in 1948 (Figure 18.3) and Buckminster Fuller coined the word “tensegrity.” For 50 years tensegrity has existed as an art form with some architectural appeal, but engineering use has been hampered by the lack of models for the dynamics. In fact, engineering use of tensegrity was doubted by the inventor himself. Kenneth Snelson in a letter to R. Motro said, “As I see it, this type of structure, at least in its purest form, is not likely to prove highly efficient or utilitarian.” This statement might partially explain why no one bothered to develop math models to convert the art form into engineering practice. We seek to use science to prove the artist wrong, that his invention is indeed more valuable than the artistic scope that he ascribed to it. Mathematical models are essential design tools to make engineered products. This chapter provides a dynamical model of a class of tensegrity structures that is appropriate for space structures. We derive the nonlinear equations of motion for space structures that can be deployed or held to a precise shape by feedback control, although control is beyond the scope of this chapter. For engineering purposes, more precise definitions of tensegrity are needed. One can imagine a truss as a structure whose compressive members are all connected with ball joints so that no torques can be transmitted. Of course, tension members connected to compressive members do not transmit torques, so that our truss is composed of members experiencing no moments. The following definitions are useful. Definition 18.1 A given configuration of a structure is in a stable equilibrium if, in the absence of external forces, an arbitrarily small initial deformation returns to the given configuration. Definition 18.2 A tensegrity structure is a stable system of axially loaded members. Definition 18.3 A stable structure is said to be a “Class 1” tensegrity structure if the members in tension form a continuous network, and the members in compression form a discontinuous set of members.

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1 2

FIGURE 18.4 Class 1 and Class 2 tense grity structures.

FIGURE 18.5 Topology of an (8,4) Class 1 tense grity shell.

Deﬁnition 18.4 A stable structure is said to be a “Class 2” tense grity structure if the members in tension form a continuous set of members, and there are at most tw o members in compression connected to each node. Figure 18.4 illustrates Class 1 and Class 2 tensegrity structures. Consider the topology of structural members given in Figure 18.5, where thick lines indicate rigid rods which tak e compressi ve loads and the thin lines represent tendons. This is a Class 1 tense grity structure. Deﬁnition 18.5 Let the topology of Figure 18.5 describe a three-dimensional structure by con- necting points A to A, B to B, C to… C,, I to I. This constitutes a“ Class 1 tense grity shell” if there α → β → γ → exists a set of tensions in all tendons tαβγ , = 1 ( 10, = 1 n, = 1 m) such that the structure is in a stable equilibrium.

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FIGURE 18.6 A typical ij element.

18.2.1 A Typical Element The axial members in Figure 18.5 illustrate only the pattern of member connections and not the actual loaded configuration. The purpose of this section is two-fold: (i) to define a typical “element” which can be repeated to generate all elements, and (ii) to define rules of closure that will generate a “shell” type of structure. Consider the members that make the typical ij element where i = 1, 2, …, n indexes the element to the left, and j = 1, 2, …, m indexes the element up the page in Figure 18.5. We describe the

axial elements by vectors. That is, the vectors describing the ij element, are t1ij, t2ij, … t10ij and r1ij, r2ij, where, within the ij element, tαij is a vector whose tail is fixed at the specified end of tendon number α, and the head of the vector is fixed at the other end of tendon number α as shown in Figure 18.6 where α = 1, 2, …, 10. The ij element has two compressive members we call “rods,”

shaded in Figure 18.6. Within the ij element the vector r1ij lies along the rod r1ij and the vector r2ij lies along the rod r2ij. The ﬁrst goal of this chapter is to derive the equations of motion for the dynamics of the two rods in the ij element. The second goal is to write the dynamics for the entire system composed of nm elements. Figures 18.5 and 18.7 illustrate these closure rules for the case (n, m) = (8,4) and (n, m) = (3,1). Lemma 18.1 Consider the structure of Figure 18.5 with elements deﬁned by Figure 18.6. A Class 2 tensegrity shell is formed by adding constraints such that for all i = 1, 2, …, n, and for m > j > 1,

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FIGURE 18.7 Class 1 shell: (n,m) = (3,1).

−+ = tt14ij ij 0, += tt23ij ij 0, += (18.1) tt5ij 6ij 0, += tt78ij ij 0.

This closes nodes n2ij and n1(i+1)(j+1) to a single node, and closes nodes n3(i–1)j and n4i(j–1) to a single node (with ball joints). The nodes are closed outside the rod, so that all tension elements are on the exterior of the tensegrity structure and the rods are in the interior. The point here is that a Class 2 shell can be obtained as a special case of the Class 1 shell, by imposing constraints (18.1). To create a tensegrity structure not all tendons in Figure 18.5 are → → necessary. The following deﬁnition eliminates tendons t9ij and t10ij, (i = 1 n, j = 1 m). Deﬁnition 18.6 Consider the shell of Figures 18.4. and 18.5, which may be Class 1 or Class 2

depending on whether constraints (18.1) are applied. In the absence of dotted tendons (labeled t9 and t10), this is called a primal tensegrity shell. When all tendons t9, t10 are present in Figure 18.5, it is called simply a Class 1 or Class 2 tensegrity shell. The remainder of this chapter focuses on the general Class 1 shell of Figures 18.5 and 18.6.

18.2.2 Rules of Closure for the Shell Class

Each tendon exerts a positive force away from a node and fαβγ is the force exerted by tendon tαβγ ˆ and fαij denotes the force vector acting on the node nαij. All fαij forces are postive in the direction α of the arrows in Figure 18.6, where wαij is the external applied force at node nαij, = 1, 2, 3, 4. At the base, the rules of closure, from Figures 18.5 and 18.6, are

t9i1 = – t1i1, i = 1, 2, …, n (18.2)

t6i0 = 0 (18.3)

t600 = – t2n1 (18.4)

t901 = t9n1 = –t1n1 (18.5)

0 = t10(i–1)0 = t5i0 = t7i0 = t7(i–1)0, i = 1, 2, …, n. (18.6)

At the top, the closure rules are

t10im = –t7im (18.7)

t100m = –t70m = –t7nm (18.8)

t2i(m+1) = 0 (18.9)

0 = t1i(m+1) = t9i(m+1) = t3(i+1)(m+1)

= t1(i+1)(m+1) = t2(i+1)(m+1). (18.10)

At the closure of the circumference (where i = 1):

t90j = t9nj, t60(j–1) = t6n(j–1), t70(j–1) = t7n(j–1) (18.11)

t80j = t8nj, t70j = t7nj, t100(j–1) = t10n(j–1). (18.12)

From Figures 18.5 and 18.6, when j = 1, then

0 = f7i(j–1) = f7(i–1)(j–1) = f5i(j–1) = f10(i–1)(j–1), (18.13)

and for j = m where,

0 = f1i(m+1) = f9i(m+1) = f3(i+1)(m+1) = f1(i+1)(m+1). (18.14)

Nodes n11j, n3nj, n41j for j = 1, 2, …, m are involved in the longitudinal “zipper” that closes the structure in circumference. The forces at these nodes are written explicitly to illustrate the closure rules. In 18.4, rod dynamics will be expressed in terms of sums and differences of the nodal forces, so the forces acting on each node are presented in the following form, convenient for later use.

The deﬁnitions of the matrices Bi are found in Appendix 18.E. The forces acting on the nodes can be written in vector form:

f = Bd f d + Bo f o + Wo w (18.15)

where

f d    1  o    f1   f11w   f d      f = M ,,,f d = 2 f o = MMw = ,    M       o  f    f  w  m  d  m m fm 

o L L W = BlockDiag [, W1, W1, ],

L L BB3400 BB1200  OO M  OO M BB5 6   0 B7  Bdo=  00B OO , B=  MOOO0   5    MOO MOO  BB6 4   B2   K   LL   00BB5 8   00B7 

and

 f   2   f3   f  w   4   1  f  f w o 5 d  6   2  f =   , f = , w = (18.16) ij f  ij  f  ij w  1 ij  7   3   f  w 8  4 ij  f   9  f  10 ij Now that we have an expression for the forces, let us write the dynamics.

18.3 Dynamics of a Two-Rod Element

Any discussion of rigid body dynamics should properly begin with some decision on how the motion of each body is to be described. A common way to describe rigid body orientation is to use three successive angular rotations to define the orientation of three mutually orthogonal axes fixed in the body. The measure numbers of the angular velocity of the body may then be expressed in terms of these angles and their time derivatives. This approach must be reconsidered when the body of interest is idealized as a rod. The reason is that the concept of “body fixed axes” becomes ambiguous. Two different sets of axes with a common axis along the rod can be considered equally “body fixed” in the sense that all mass particles of the rod have zero velocity in both sets. This remains true even if relative rotation is allowed along the common axis. The angular velocity of the rod is also ill defined because the component of angular velocity along the rod axis is arbitrary. For these reasons, we are motivated to seek a kinematical description which avoids introducing “body-fixed” reference frames and angular velocity. This objective may be accomplished by describing the configuration of the system in terms of vectors located only the end points of the rods. In this case, no angles are used. We will use the following notational conventions. Lower case, bold-faced symbols with an underline indicate vector quantities with magnitude and direction in three-dimensional space. These are the usual vector quantities we are familiar with from elementary dynamics. The same bold- faced symbols without an underline indicate a matrix whose elements are scalars. Sometimes we also need to introduce matrices whose elements are vectors. These quantities are indicated with an upper case symbol that is both bold faced and underlined. As an example of this notation, a position vector can be expressed as

 p   i1  pe= []. e e p = Ep i 123 i2  i   pi3 

In this expression, pi is a column matrix whose elements are the measure numbers of p for the mutually i ˆ orthogonal inertial unit vectors e1, e2, and e3. Similarly, we may represent a force vector fi as

ˆˆ= fEfi i.

Matrix notation will be used in most of the development to follow.

FIGURE 18.8 A single rigid rod.

ˆ ˆ We now consider a single rod as shown in Figure 18.8 with nodal forces f1 and f 2 applied to the ends of the rod. The following theorem will be fundamental to our development. Theorem 18.1 Given a rigid rod of constant mass m and constant length L, the governing equations may be described as:

qKqHf˙˙ +=˜ (18.17)

where

q  pp+  = 1 = 12 q    −  q2  pp21

ˆˆ+  I3 0  00 ~ ff12 2 f =  ,, H =   K =  − . ˆˆ−  m  0 3 q~ 2  0qqIL 2 ˙˙T  ff12  L2 2  223

The notation r~ denotes the skew symmetric matrix formed from the elements of r:

 0 −rr r   32  1  ~ = − r rr310 , r = r2  −     rr210  r3 

and the square of this matrix is

−−r2 r2 rr rr   2 3 12 13  ~2 = −−2 2 r  rr21 r 1 r3 rr23 .  −−2 2   rr31 rr 32 r 1 r2 

The matrix elements r1, r2, r3, q1, q2, q3, etc. are to be interpreted as the measure numbers of the corresponding vectors for an orthogonal set of inertially ﬁxed unit vectors e1, e2, and e3. Thus, using the convention introduced earlier,

r = Er, q = Eq, etc.

The proof of Theorem 18.1 is given in Appendix 18.A. This theorem provides the basis of our dynamic model for the shell class of tensegrity structures. Now consider the dynamics of the two-rod element of the Class 1 tensegrity shell in Figure 18.5. Here, we assume the lengths of the rods are constant. From Theorem 18.1 and Appendix 18.A, the motion equations for the ij unit can be described as

 m ^^ 1ij .. =+  2 qff1ij 12ij ij   m 1ij ×=×−.. ^ ^  ()()qq2ij 2ij q2ij ff21ij ij  6 , (18.18)  q. .. q. += q q.. 0  22ij ij 2ij 2ij  = 2  q22ij . q ijL 1 ij

 m2ij qff˙˙ =+ˆˆ  2 334ij ij ij  m2ij ×=×−˙˙ ˆˆ  6 (qq44ij ij)( q 443 ij ff ij ij )  , (18.19) q˙ . q˙ += q . q˙˙ 0  44ij ij 44 ij ij   q . q = L2  44ij ij2 ij α where the mass of the rod ij is mαij and rαij = Lαij. As before, we refer everything to a common inertial reference frame (E). Hence,

q  q  q  q   11ij   21ij   31ij   41ij  ∆∆∆∆ q1ij q12ij ,,,, q2ij q22ij  q3ij q32ij  q4ij q42ij          q13ij  q23ij  q33ij  q43ij 

T ∆ T T T T qq11ij[] ij,, q 2ij q 3ij ,, q 4ij

and the force vectors appear in the form

I 0  I 0  223 3 H =  2  H =  2  1ij 3 ˜ , 2ij 3 ˜ , 0 2 q2ij 0 2 q4ij mm1ij  L1ij 2ij  L2 ij 

ffˆˆ+   12ij ij    ˆˆ−  H 0  ff12ij ij H =  1ij , f ∆  . ij 0 H ij    2ij  ˆˆ+ ff34ij ij    ˆˆ−  ff34ij ij 

Using Theorem 18.1, the dynamics for the ij unit can be expressed as follows:

˙˙ +=ΩΩ qqHfij ij ij ij ij,

where

00 00 ΩΩΩ=  −−, Ω =  , 1ij 2 ˙˙T 2ij 2 ˙˙T 0 LL1ijqqI223 ij ij  0 2ijqqI443 ij ij 

ΩΩ 0  ΩΩ =  1ij  ij ΩΩ ,  0 2ij

= T TT T T T T qq[]11,..., qqn 1 , 12 ,..., qn 2 ,..., q 1m ,..., qnm .

The shell system dynamics are given by

˙˙ += qKqHfr , (18.20)

where f is deﬁned in (18.15) and

= T TT T T T T qq[]11,..., qqn 1 , 12 ,..., qn 2 ,..., q 1m ,..., qnm ,

= ΩΩΩΩΩΩΩΩΩΩΩΩ KrnnmnmBlockDiag[]11,..., 1 , 12 ,..., 2 ,..., 1 ,..., ,

H = BlockDiag[]HHHH11,...,nnmnm 1 , 12 ,..., 2 ,..., H 1 ,..., H.

18.4 Choice of Independent Variables and Coordinate Transformations

Tendon vectors tαβγ are needed to express the forces. Hence, the dynamical model will be completed by expressing the tendon forces, f, in terms of variables q. From Figures 18.6 and 18.9, it follows ˆ that vectors ij and ij can be described by

i i−1 j j−1 ρρρ=+ρ − + + − ij 11 ∑∑rt11k 11k ∑ t1ik ∑ tr5ik 1ij (18.21) k==1 k 1 k=2 k=1

ρρρˆ ρ − ij = ij + r1 ij + t5ij r2ij . (18.22)

To describe the geometry, we choose the independent vectors {r1ij, r2ij, t5ij, for i = 1, 2, …, n, j = ρρ 1, 2, …, m} and { 11, t1ij, for i = 1, 2, …, n, j = 1, 2, …, m, and i < n when j = 1}. This section discusses the relationship between the q variables and the string and rod vectors

tαβγ and rβij. From Figures 18.5 and 18.6, the position vectors from the origin of the reference frame, E, to the nodal points, p1ij, p2ij, p3ij, and p4ij, can be described as follows:

FIGURE 18.9 Choice of independent variables.

p = ρρ  1ij ij pr=+ρρ  21ij ij ij  (18.23) = ρρˆ p ij ij  3  =+ρρˆ pr42ij ij ij We deﬁne

 =+=∆∆ ρρ + qp12ij ij p 1 ij2 ijr 1 ij   ∆∆ qpp=−=r  2211ij ij ij ij  (18.24)  ∆∆ qpp=+=2ρρˆ +r  34ij ij 3 ij 12ij ij   =−=∆∆ qpp44ij ij 32 ijr ij Then, q   II 00 p   1   33   1  ∆∆ q −II 00p q = 2  =  33   2  ij q   00II p   3   33  3  q 00−IIp  4 ij 33  4  ij (18.25) 2II 00 ρρ  33    000I r =  3   1   002II ρρˆ   33   000I r  3 ij  2 ij

In shape control, we will later be interested in the p vector to describe all nodal points of the structure. This relation is

p = Pq P = BlockDiag […,P1, …,P1,…] (18.26)

−1 −  II3300 II3300 −    ∆∆ II 00 1 II 00 P = 33  =  33  1  00II 2  00II−   33  33 −  00 II33  00II33

The equations of motion will be written in the q coordinates. Substitution of (18.21) and (18.22)

into (18.24) yields the relationship between q and the independent variables t5, t1, r1, r2 as follows:

 i i−1 j j−1  qrtttr=+2ρρ ∑∑∑ −∑ + +  − 111ij  11k 11kik 1 5ik  1ij  k===1 k=1 k 2 k 1  = qr21ij ij (18.27)  i i−1 j j  qrttt=+2ρρ ∑∑∑ −∑ + +  − r 311ij  11k 11kik 1 5ik  2ij  k===1 k= k 2 k 11  = qr42ij ij

To put (18.27) in a matrix form, deﬁne the matrices:

r   1ij  r2ij  l = for jm= 23 , ,..., , ij t   5ij  t1ij 

ρρ  t   11   111(-)i  r r l =  111 , l =  11i  for in= 2 ,... , , 11 r  i1  r   211   21i  t511  t51i 

and

T l= [] lTT l KKKK lTT l lT lT lT 11,,,,,,,,,, 21n 1 12n 2 1m nm ,

2II 00 −2II 00  33   33  000I 000I A =  3 , B =  3  22IIII− 2 −−22IIII 2  3333  3333  00I3 0  00I3 0

−  II33002  22II00    33  I 000 0000 C =  3 , D =    0 −III22 22II00  333  33   000I3   0000

−22II00 22II0 2 I  33   33 3 0000 0000 E =  , F =   −22II00 22II0 2 I  33   33 3  0000  0000

0022II −22II0 2 I  33  33 3 00 0 0 0000 J =  , G=   0022II  22II0 2 I  33  33 3 00 0 0  0000

Then (18.27) can be written simply q = Ql, (18.28) where the 12nm × 12nm matrix Q is composed of the 12 × 12 matrices A–H as follows:

LL Q11 00  OM QQ21 22  QQQ OM  Q =  21 32 22 , (18.29) QQQQ21 32 32 22 0  QQQQQ   21 32 32 32 22   MMMMOO

A 00LLL    OM DB  DEBOM = × × Q11   n n blocks of 12 12 matrices, DEEBOM  MMMOO0    L  DEE EB

F 00LLL    OM DG  DEGOM = × Q21  12n 12n matrix, DEEOO M  MMMOO0    DEEEEG … … Q22 = BlockDiag [ ,C, ,C], … … Q32 = BlockDiag [ ,J, ,J],

× where each Qij is 12n 12n and there are m row blocks and m column blocks in Q. Appendix 18.B provides an explicit expression for the inverse matrix Q, which will be needed later to express the tendon forces in terms of q. Equation (18.28) provides the relationship between the selected generalized coordinates and an independent set of the tendon and rod vectors forming l. All remaining tendon vectors may be written as a linear combination of l. This relation will now be established. The following equations are written by inspection of Figures 18.5, 18.6, and 18.7 where

=+−ρρρρ tr11nnn 1 11 11 (18.30)

and for i = 1, 2, …, n, j = 1, 2, …,m we have

ρρρ−>ρˆ t2121ij = ij( i() j−− + r i () j ), (j 1 )

t =−ρρρˆ ρ 3ij ()i−1 j ij =−ρρρ + −ρˆ tt431ij ij + r ij = ij r 1 ij() i− 1 j =−+<ρρρρˆ t6ij ()()i++11 j( ij r 2 ij ),( j mm) (18.31) =−ρρρˆ ρ < t711ij ij()() i++ j ,( jm ) =+−=−−+ρρρρˆ tr81ij ij1 ij ij rtr ij 5ij 2ij =−+ρρρρ tr91ij() i+ j() ij 1 ij

tr=+ρρρˆˆ −ρ . 10ijij()++ 1 2 () ij 1 ij

For j = 1 we replace t2ij with =−ρρρρ t21ii 1() i+ 11.

For j = m we replace t6ij and t7ij with

=+ρρρˆ −+ρˆ trr6im ()im++121 () im( imim 2) =−ρρρˆ ρˆ + tr7121im im( () i++ m () i m ).

ρρρ∆∆∆ρρρρˆˆ∆ρ where 0 jnjojnj, , and i + n = i. Equation (18.31) has the matrix form,

−−  t2  00 II 0000    33   t  3  00 0 0 00I3 0  t  00 0 0 ρρ 00−I 0 ρρ  4       3    ∆∆  t6  00 0 0 r1 0000 r1 td = =   +   ij  t  00 0 0 ρρˆ  0000 ρρˆ   7           t8  00 0 0 rr2  −−0000 r2    ij()1 () i1 j t 00 0 0 0000  9      t  00 0 0 0000  10 ij    

 I3 000  000 0 −     I3 000  000 0  II 00  ρρ  000 0  ρρ  33        00−−IIr 000 0r +  33  1  +    1   00I 0  ρρˆ   0 000 0  ρρˆ   3         II− I0  r  000 0 r 33 3  2 ij  2  ()ij+1 −−II00 I 00 0  33   3   −     00I3 0  00II33

 0 000    0 000  0 000  ρρ      I 000 r1 + 3   , −I 000  ρρˆ   3     0 000 r2  ++   ()()ij11  0 000  0 000

 t2  00 0 0  I3 000       t −I 000  3  00I3 0  3   t  00−I 0  ρρ  II 00  ρρ  4   3   3 3    −−   t6  00 0 0 r1  00 II33 r1 td ∆∆ =   +   i1  t  00 0 0  ρρˆ   00I 0  ρρˆ   7       3    −  t8  00 0 0 r22  −  II33 I 30  r2    ()i 11   i1 t 00 0 0 −−II00  9     33  t  00 0 0  00−I 0  10 i1  3  (18.32) −  I3 00 0  0 000      0000  0 000  0000  ρρ  0 000  ρρ          0000 r1  I 000 r1 +   + 3   ,  0000  ρρˆ  −I 000  ρρˆ       3     0000 r2  +  0 000 r2  +   ()i 11 ()i 12 I 00 0  0 000  3     00II  0 000  33

−−  t2   00 I II   0000    3 3    t  3   00 0 0  00I3 0  t   00 0 0 ρρ  00−I 0 ρρ  4       3     t6   00 0 0 r1  0000 r1 td ∆∆ =   +   im  t   00 0 0 ρρˆ   0000 ρρˆ   7           t8   00 0 0 r2  −  0000 r2  −   im()1 ()im1 t  00 0 0  0000  9      t   00 0 0  00 0 00   10 im    

 I3 000  00 0 0 −     I3 000  00 0 0  II00 ρρ  00 0 0 ρρ  3 3         00−−II r  00 II r + 33 1  + 33 1  .  00I 0 ρρˆˆ  00−−II ρρ  3     33    II− I0  r  00 0 0 r 333  2 im  2 ()im+1 −−II00 I 00 0  33   3   00−I 0  00 II  3   33

Equation (18.25) yields

ρρ 1 − 1    2 II2 00    3 3  r1   000I3  = q (18.33) ρρˆ   001 II− 1  ij    2 3 2 3  r 000I  2 ij  3 

Hence, (18.32) and (18.33) yield

 t2  00–– I I 00 0 0    33   t  3  00 0 0 00 I33– I  t  00 0 0 00– I I  4     33 ∆∆  t6  1 00 0 0 1 00 0 0 td = = q + q ij  t  2 00 0 0 ij( −−−11)()2 00 0 0 ij  7       t8  00 0 0 00 0 0   t 00 0 0 00 0 0  9      t  00 0 0 00 0 0  10 ij    

 II00–   0000  33    –II33 0 0  0000  II 00  0000  33    1  00II–– 1  0000 + 33q + q 2  00I– I ij 2  0000 ()ij+1  33    I333III– 33  0000     ––II00 II00–  33   33 

 00II– 33  00II33

 0000    0000  0000   1  II00–  + 33 q 2 –II00 ()(),ij++11  33   0000    0000  0000

−  t2  00 0 0  II0033        t − −II 0 0  3  00 I33 I  33   t  00− I I  II 00  4   33  33  −−  t6  1 00 0 0 1  00II33 td ∆∆ = q + q i1  t  2 00 0 0 (()ii−112  00I− I 1  7     33 −  t8  00 0 0  II33 II 33     t 00 0 0 −−II00  9     33  t  00 0 0  00II−   10 i1  33 −  II33 00  0000      0000  0000  0000  0000     1  0 0000 1  II00−  + q + 33 q 2  0000 ()i+11 2 −II00 (),i+12    33   0000  0000   II00−  0000  33       00II33  0000 (18.34)

−−  t2  00 I I 00 0 0    33   t −  3  00 0 0 00 I33 I  t  00 0 0 00− I I  4     33  t6  1 00 0 0 1 00 0 0 td ∆∆ = qq+ im  t  2 00 0 0 im()−−112 00 0 0 () i m  7       t8  00 0 0 00 0 0   t 00 0 0 00 0 0  9      t  00 0 0 00 0 0  10 im     −  II0033   0000 −     II33 0 0  0000  II 00  0000  33    1  000II−− 1  00I I + 33q + 33q . 2  00I− I im 2  00−− I I ()im+1  33  33 −  II33 II 33  0000 −−II00 II00−   33   33  −  00II33  00I33 I

Also, from (18.30) and (18.32)

 ρρ  ρρ     r r tI=−[]000,,  1  + []II 00,  1  11n 3,,, ρρˆ  33  ρρˆ      r r  2 11  2 n1

=−1 1 + 1 1 tIIqIIq11nn[]2 3,,2 311, 00[]2 3 ,,,,2 31 00 = ++ Eq6 11 Eq71n , (18.35) q   11  q = []E0,,L ,, 0E 21  ,EE∈∈312×× ,R312 , 6 7  M  6 7   qn1 

tRqRqRR==0 ,.∈ 312× n 11n 0 1[] 0, 0 × With the obvious deﬁnitions of the 24 12 matrices E1, E2, E3, E4, Ê4, E4, E5, equations in (18.34) are written in the form, where q01 = qn1, q(n+1)j = qij,

d =+++ˆ tEqEqEqEqil 21131411()iiii−++ ()5 ()12,

d =++++ tEqEqEqEqEqij 11ij(−− ) 21 () i j 3 ij 41 () i + j5 ()() i ++11 j , (18.36)

d =+++ tEqEqim 11im()−− 21 () i m EEq341im Eq() i+ m .

Now from (18.34) and (18.35), deﬁne

T d = dT dT dT KKKdT dT dT dT lttt[]1n 1,,,, 11 21 ttn 1 12 ,, tn 2 , tnm

= []dT dT dT K dT T ttt11n ,,,, 1 2 tn , to get lldnmnmnmdnm=∈∈∈Rq,,,, R R()24+× 3 12 q R 12 R(+) 24 3 (18.37)

LLL  R0 00  ˆ OM RR11 12  RRR OM  21 11 12  = OM ∈∈24nn×× 12 312 n R  0 RRR21 11 12 ,,,RRij RR0  MORR O 0   21 11  M OOO  R12   LL   00RR21 11  LL LL EE340 E 2  00E5 0  O MM   OM EEE234   00E5   0 EEEOM  MOOOOM R =  234 , R =   11 MO 12 MOOO  EEE2340   0   0 OOOE   0 OOE   4   5  L LL E04230 EE E5 000 => => Ri(()ik+ 00ifkk11, R(iki+ ) if

= LL∈=→24× 12 REE21BlockDiag[],,, 1 1 ERi i 15 1 RE= [],,00L ,, EE ,=−[] II , ,, 00 , 0 6 7 6 2 33 1 EI= [],,,. I 00 7332 ˆ ˆ R11 and R11 have the same structure as R11 except E4 is replaced by E4 , and E4 , respectively. Equation (18.37) will be needed to express the tendon forces in terms of q. Equations (18.28) and

(18.37) yield the dependent vectors (t1n1, t2, t3, t4, t6, t7, t9, t10) in terms of the independent vectors (t5, t1, r1, r2). Therefore,

d = l RQl. (18.38) 18.5 Tendon Forces

Let the tendon forces be described by

t = αij fααijF ij (18.39) tαij

For tensegrity structures with some slack strings, the magnitude of the force Fαij can be zero, for taut strings Fαij > 0. Because tendons cannnot compress, Fαij cannot be negative. Hence, the magnitude of the force is =− Fkααααij ij()t ij L ij (18.40) where   0 , if Lααα> tα k ∆∆ ij ij ααij >≤ kifLαααij0 , αα ijtα ij

−+o ≥ LuLααij ij α ij 0 (18.41)

o > where Lαij 0 is the rest length of tendon tαij before any control is applied, and the control is uαij, the change in the rest length. The control shortens or lengthens the tendon, so uαij can be positive o > or negative, but Lαij 0 . So uαij must obey the constraint (18.41), and

≤>o uLααij ij 0. (18.42)

α d Note that for t1n1 and for = 2, 3, 4, 6, 7, 8, 9, 10 the vectors tαij appear in the vector l related d α to q from (4.7) by l = Rq, and for = 5, 1 the vectors tαij appear in the vector l related to q from –1 α (18.28), by l = Q q. Let Pαij denote the selected row of R associated with tαij for ij = 1n1 and α –1 α for = 2, 3, 4, 6, 7, 8, 9, 10. Let Pαij also denote the selected row of Q when = 5, 1. Then,

312× nm tqααα=∈RRαα, α (18.43) ij ij ij

2 T T tqααα= RRααα q (18.44) ij ij ij From (18.39) and (18.40),

fαij = – Kαij (q)q + bαij (q)uαij where

∆∆  − 1  Kq() = kLoT() qT q2 −1 , K∈R312× nm αααijαα ij αα ij RRααij ααij Rαα ijα ij (18.45)

∆∆ − 1 × bq() =∈ k ( qT RRT q )2 R qb , R31 (18.46) αijααα ij ααij ααijαα ijα ij Hence,

 f   K   2ij   2ij   f3ij   K3ij   f  K   4ij   4ij   f   K  f d = 6ij =− 6ij q ij  f   K   7ij   7ij   f8ij   K8ij   f   K   9ij   9ij      f10ij  K10ij 

b  u   2ij   2ij   b3ij   u3ij   b  u   4ij   4ij   b  u  + 6ij 6ij  b  u   7ij   7ij   b8ij   u8ij   b  u   9ij   9ij       b10ij  u10ij  or

d =−d + d d fKqPuij ij ij ij , (18.47) and

f  K  b 0 u  o = 5ij =− 5ij + 5ij 5ij fij    q    f1ij  K1ij   0 b1ij u1ij  or

o =−o + o o fKqPuij ij ij ij . (18.48)

Now substitute (18.47) and (18.48) into

f11n  K11n  P11n u11n   d   d   d  d   f11   K11   P11 u11  d d d d d ddd f1 =  f  =−K q +  P u  =−Kq11 + Pu  21   21   21  21  1  M   MOM     d   d   d  d   fn1  Kn1   Pn1un1 

f d  Kd  Pd ud   12   12   12  12  f d  Kd   Pd ud  f d = 22 =− 22 q + 22 22 =−Kqd + Pddu . 2  MM    O M 22 222        d d d d fn2  Kn2   Pn2 un2 

Hence, in general,

d =−d + d d fKqPuj j j j

or by deﬁning

 d   d  K1 P1     Kd  Pd  Kd = 2 , Pd = 2 (18.49)  MM        d   d  Km  Pm 

fd = –Kdq + Pdud.

o Likewise, for f11 forces (18.48),

f o  Ko  Po  uo   11   11   11   11  f o  Ko   Po  uo  f o = 21 =− 21 q + 21 21 1  MM    O  M          o   o   o   o  fn1 Kn1  Pn1 un1

f o  Ko  Po  uo   1j   1j   1j   1j  f o  Ko   Po  uo  f o = 2 j =− 2 j q + 2 j 2 j j  MM    O  M         f o  Ko   Po  uo   nj   nj   nj   nj 

o =−o + o o fKqPuj j j j (18.50)

fo = – Koq + Pouo.

Substituting (18.49) and (18.50) into (18.E.21) yields

f = –(BdKd + BoKo)q + BdPdud + BoPouo + Wow, (18.51)

which is written simply as

fKqBuWw=−˜˜ + + o , (18.52)

by deﬁning,

∆ K˜ =+ BKdd B oo K ,

∆ BBP˜ =[]dd,, BP oo

 d d  BP1 BP 00LL  3 42  d ddOM BP5 1 BP6 243 BP   dd  0 BP5 BP OO M BPdd=  2 6 3 ,  MOBPd OO 0   5 3  MOOOd  BP4 m   LL d d   00BP5 m−18 BPm 

BPoo BP 00L   11 2 2  ooOM  0 BP72 BP 23  BPoo=  MOBPo O0   73  MOOo  BP2 m   LL o   00BP7 m 

 BKdo++ BK BK d + BK o   31 11 42 22  dddoo++++  BK5 1 BK6 2437223 BK BK BK   BKdddoo++++ BK BK BK BK   5 2 6 3447324 ˜ =+=dd oo dddo+++++ o KBKBK  BK5 3 BK5 44 BK5 BK74 BK2 5  (18.53)  M     d ++++d d o o  BK5 m−−2 BK6 m 14 BKm BK 712m − BKm  d ++d o   BK5 m−1 BK6 m BK7 m 

 d   ˆ d  u1 u1  d   d  u2  u2  ud  ud   3   3  d d u4  u4   M   M      ud  uˆ d  ˜ = m = m u  o , u  o  (18.54) u1  u1   o   o  u2 u2  o   o  u3  u3  uo  uo   4   4   M   M   o   o  um  um 

d In vector u˜ in (18.54), u1n1 appears twice (for notational convenience u1n1 appears in u1 and in o … u1 ). From the rules of closure, t9i1 = – t1i1 and t7im = – t10im, i = 1, 2, , n, but t1i1, t7im, t9i1, t10im all appear in (18.54). Hence, the rules of closure leave only n(10m – 2) tendons in the structure, but (18.54) contains 10nm + 1 tendons. To eliminate the redundant variables in (18.54) deﬁne u˜ = Tu, where u is the independent set u ∈nm()10− 2 , and u˜ ∈R10nm+ 1 is given by (18.54). We choose

to keep t7im in u and delete t10im by setting t10im = – t7im. We choose to keep t1i1 and delete t9i1 by … setting t9i1 = – t1i1, i = 1, 2, , n. This requires new deﬁnitions of certain subvectors as follows in (18.57) and (18.58). The vector u˜ is now deﬁned in (18.54). We have reduced the u˜ vector by 2n + 1 scalars to u. The T matrix is formed by the following blocks,

 000L 001  TS  1   OO   TS  1    I8  O     I   8  T =  2  T  O    (18.55)  T2   I   2   O     I2   I   2   O      I2

()10nm+ 1×−() n() 10 m 2 ∈R

where   I6610 ×   87× T = 0 × 0 ∈R 1  16    016× 1

 I  T =  7  ∈R87× 2  0000 − 100 

  0  62×  S =  01−  ∈R82× . (18.56)

 0 0   

≤ There are n blocks labeled T1, n(m – 2) blocks labeled I8 (for m 2 no I8 blocks needed, see appendix 18.D), n blocks labeled T2, nm blocks labeled I2 blocks, and n blocks labeled S.

d The u1 block becomes

u   21n  uˆ d  u  11   31n  ˆ d u  u21  41n ∆∆   uˆ d = uˆ d , uˆ d = u ∈=R71× , in123 , , , ... ,, j = 1 (18.57) 1  31  i1  61n  M u     71n  ˆ d  u un1   81n    u10n 1 

d The um block becomes

u   2nm  uˆ d  u  1m   3nm  ˆ d u  u2m  4nm ∆∆   uˆ d = uˆ d , uˆ d = u ∈==R71× , injm123 , , , ... , . (18.58) m  3m  im  6nm  M u     7nm   ˆ d  u unm   8nm    u9nm 

d d o The u1 block is the u1 block with the ﬁrst element u1n1 removed, because it is included in un1. From (18.17) and (18.52),

˙˙ ++˙ =+ qKqKqqBquDqw((rp) ()) () () , (18.59)

where,

Kp = H(q)(K˜ q),

B = H(q)(B˜ q)T,

D = H(q)Wo.

The nodal points of the structure are located by the vector p. Suppose that a selected set of nodal points are chosen as outputs of interest. Then

yp = Cp = CPq (18.60) where P is deﬁned by (18.26). The length of tendon vector tαij = ααij q is given from (18.44). Therefore, the output vector yl describing all tendon lengths, is

 M  1   TT 2 yy= αα , yqααα= ()α q. l  ij  ij ααij ij  M   

Another ouput of interest might be tension, so from (18.40) and (18.44)

 M  =   =− yf FFkαααij,() αα ijαα ijyαα ij Lα ij .  M   

The static equilibria can be studied from the equations

Kp(q)q = B(q)u + D(q)w, yp = CPq. (18.61)

Initial Conditions Perspective View 10 Initial Conditions Top View 20 8 18 6 16 4 14 12 2 10 0 8 2 6 4 4 2 6 0 8 10 10 5 0 0 5 10 5 10 10 5 10 8 6 4 2 0 2 4 6 8 10

FIGURE 18.10 Initial conditions with nodal points on cylinder surface.

Steady State Equilibrium Perspective View Steady State Equilibrium Top View 10 20 8 18 6 16 4 14 12 2 10 0 8 -2 6 -4 4 2 -6 0 -8 10 10 5 0 0 5 -10 5 10 10 5 10 8 6 4 2 0 2 4 6 8 10

FIGURE 18.11 Steady-state equilibrium.

Tendon Dynamics 10 t1, t7, t9, t10, (Diagonal Strings) 9

7 t4, t8, (Saddle Strings) 6

5 t5, t3, (Vertical Strings) Tendon Length

4 t2,t6, (Base, Top Strings) 3 0 5 10 15 20 25 30 35 40 45 50 time

FIGURE 18.12 Tendon dynamics.

Of course, one way to generate equilibria is by simulation from arbitrary initial conditions and record the steady-state value of q. The exhaustive definitive study of the stable equilibria is in a separate paper.27 Damping strategies for controlled tensegrity structures are a subject of further research. The example case given in Appendix 18.D was coded in Matlab and simulated. Artificial critical damping was included in the simulation below. The simulation does not include external disturbances or control inputs. All nodes of the structure were placed symmetrically around the surface of a cylinder, as seen in Figure 18.10. Spring constants and natural rest lengths were specified equally for all tendons in the structure. One would expect the structure to collapse in on itself with this given initial condition. A plot of steady-state equilibrium is given in Figure 18.11 and string lengths in Figure 18.12.

18.6 Conclusion

This chapter developed the exact nonlinear equations for a Class 1 tensegrity shell, having nm rigid rods and n(10m – 2) tendons, subject to the assumption that the tendons are linear-elastic, and the rods are rigid rods of constant length. The equations are described in terms of 6nm degrees of freedom, and the accelerations are given explicitly. Hence, no inversion of the mass matrix is required. For large systems this greatly improves the accuracy of simulations. Tensegrity systems of four classes are characterized by these models. Class 2 includes rods that are in contact at nodal points, with a ball joint, transmitting no torques. In Class 1 the rods do not touch and a stable equilibrium must be achieved by pretension in the tendons. The primal shell class contains the minimum number of tendons (8nm) for which stability is possible. Tensegrity structures offer some potential advantages over classical structural systems composed of continua (such as columns, beams, plates, and shells). The overall structure can bend but all elements of the structure experience only axial loads, so no member bending. The absence of bending in the members promises more precise models (and hopefully more precise control). Prestress allows members to be uni-directionally loaded, meaning that no member experiences reversal in the direction of the load carried by the member. This eliminates a host of nonlinear problems known to create difﬁculties in control (hysteresis, dead zones, and friction).

Acknowledgment

The authors recognize the valuable efforts of T. Yamashita in the ﬁrst draft of this chapter.

Appendix 18.A Proof of Theorem 18.1

Refer to Figure 18.8 and deﬁne

qppqpp=+,, =− 121221

using the vectors p and p which locate the end points of the rod. The rod mass center is located 1 2 by the vector,

1 pq= . (18.A.1) c 2 1

Hence, the translation equation of motion for the mass center of the rod is

m mpqff˙˙==+ ˙˙ (ˆˆ), (18.A.2) c 2 1 12

where a dot over a vector is a time derivative with respect to the inertial reference frame. A vector p locating a mass element, dm, along the centerline of the rod is

1 1 x pp=+ρρρρ (pp) − = q +− (), q01 ≤≤ , = , (18.A.3) 12112 2 2 L

and the velocity of the mass dm, v, is

1 1 vp==˙˙ q +()ρ − q˙ . (18.A.4) 2 122

The angular momentum for the rod about the mass center, hc, is

hppp=−−×()˙ dm (18.A.5) c ∫ c m

where the mass dm can be described using dx = L dρ as

 m dm = ()Ld ρρ= md . (18.A.6)  L 

Hence, (18.A.2) can be rewritten as follows:

1 hppp(=−×()˙ md ρ) (18.A.7) c ∫ c 0

where (18.A.1) and (18.A.3) yield

1 pp−=−().ρ q (18.A.8) c 2 2

(18.A.4)–(18.A.8) yield

1 1 1 1  hqqq=−×+−md()ρρρ ˙ ()˙  c ∫ 21 2 0 2 2 2 

 1 1 1 1 1  =×mddqq˙ ()ρρ − + q˙ () ρ −2 ρ 21 2∫∫  0 2 2 0 2  (18.A.9)  1 1 1 1 1 1 1  =×mqqρρ2 − ˙ +−() ρ33 q˙ 2    1   2   2 2 2 0 3 2 0 

m =×qq˙ 12 22 ττ The applied torque about the mass center, c, is

1 ττ =×−qff(ˆˆ) c 2 2 21 ττ Then, substituting hc and c from (18.A.9) into Euler’s equations, we obtain

˙ = h cc or m hqqqq˙ =(˙˙ ×+× ˙˙) c 12 2222 (18.A.10) m 1 =×=×−qq˙˙ q( ffˆˆ) 12 222 2 21

Hence, (18.A.2) and (18.A.10) yield the motion equations for the rod:

 m q˙˙ =+ ffˆˆ  2 1 12 (18.A.11)  m (qq×=×−˙˙ )( q ffˆˆ)  6 22 2 21

We have assumed that the rod length L is constant. Hence, the following constraints for q hold: 2

qq⋅=L2 22 d ()qq⋅ =⋅+⋅= qq˙˙˙ qq20 qq ⋅= dt 22 22 22 22 d (qq⋅=⋅+⋅=˙ ) qq˙˙ qq ˙˙ 0 dt 22 22 22

Collecting (18.A.11) and the constraint equations we have

 m qff˙˙ =+ˆˆ  2 1 12  m (qq×=×−˙˙ )( q ffˆˆ)  6 22 2 21 (18.A.12) qq˙˙⋅+⋅ qq ˙˙ =0  22 22 qq⋅=L2  22

We now develop the matrix version of (18.A.12). Recall that

q== E ; fˆˆ Ef i qiii

Also note that ET⋅E = 3 × 3 identity. After some manipulation, (18.A.12) can be written as:

m ˙˙ =+ˆˆ 2 qff112

m ˜ ˙˙ =−˜ ˆˆ 6 qq22 q 2( f 2 f 1) (18.A.13) TT˙˙=− ˙ ˙ qq22 qq 22 T = 2 qq22 L .

Introduce scaled force vectors by dividing the applied forces by m and mL

∆∆26 gff=+(ˆˆ), gff =− (ˆˆ). 112mmL 212 2

Then, (18.A.13) can be rewritten as

˙˙ = qg11 qq˜ ˙˙ =− q˜ () gL2 22 2 2 (18.A.14) TT˙˙=− ˙ ˙ qq22 qq 22 T = 2 qq22 L .

˙˙ Solving for q2 requires,

q˜   0  q˜   2 q˙˙ =   −  2 g L2 (18.A.15) T 2 − ˙˙T 2 q2   qq22  0 

Lemma For any vector q, such that qTq = L2,

T  q˜˜  q      = L I qT  qT  23

Proof:

0 −qq 0 qqq−   32  321 qq0 − −qqq0  31 = L2I ∆.  312 −qq 0  3  −  21 qq210 q 3   qqq123

˙˙ Since the coefﬁcient of q2 in (18.A.15) has linearly independent columns by virtue of the Lemma, ˙˙ the unique solution for q2 is

+ q˜   0  q˜   q˙˙ =  2    −  2 g L2 , (18.A.16) 2 T  − ˙˙T 2  q2   qq22  0  

where the pseudo inverse is uniquely given by

− + T 1 T T q˜˜˜˜ q  q  q  q ˜ 2 = 2 2 2 = −2 2  TT    TT   L  T  q2  q2  q2  q2  q2 

˙˙ It is easily veriﬁed that the existence condition for q2 in (18.A.15) is satisﬁed since

+  q˜˜q   0  q˜   I −  2  2     −  2 g L2 = 0, TT − ˙˙T 2   q2 q2   qq22  0  

Hence, (18.A.16) yields

qq˙˙T q˙˙ =−22qqg +˜ 2 (18.A.17) 2 L2 222

Bringing the ﬁrst equation of (18.A.14) together with (18.A.17) leads to

˙˙        q1 00q1 I3 0 g1   +  qq˙˙T   =    (18.A.18) ˙˙ 0 22I ˜ 2 q2   L2 3 q2   0 q2 g2 

Recalling the deﬁnition of g1 and g2 we obtain

˙˙        q1 00q1 2 I3 0 f1   +  qq˙˙T   =    (18.A.19) ˙˙ 0 22I 0 3 q˜ 2 q2   LL223 q2  m  2 f2 

where we clarify

f  ffˆˆ+   1  =  12. f ˆˆ−  2  ff12

Equation (18.A.19) is identical to (18.17), so this completes the proof of Theorem 18.1. Example 1

 pp+  q  qpp=+=11 21 = 11 , 112 +    pp12 22  q12 

 pp−  q  qpp=−=21 11 = 21 , 221 −    pp22 12  q22 

The generalized forces are now deﬁned as

FIGURE 18.A.1 A rigid bar of the length L and mass m.

22 ff+  6 6 ff−  gff=+=(ˆˆ),(11 21 g=−= ffˆˆ) 21 11 112 +  2 2 21 2  −  mm ff12 22  mL mL  ff22 12 

From (18.A.16),

q˙˙  00 0 0q  10 0 0  11    11    q˙˙ 00 0 0 q 01 0 0 g   12    12    1 = ˙˙2 + 2 +  − qq21 22  2   q˙˙  002 0 q  00 q −qqq g 21  L  21 22 21 22  2    ˙˙2 + 2     q˙˙ 00 0 − qq21 22  q 00−qq q2  22   L2  22   21 22 21 

Example 2 Using the formulation developed in 18.A.3, 18.A.4, and 18.A.5 we derive the dynamics of a planar tensegrity. The rules of closure become: =− tt5 4 = tt81 =− tt72 =− tt6 3 We deﬁne the independent vectors lo and ld: ρ   t  r  1  lod=  1 , l = t r   2  2     t3  t5  The nodal forces are

ffˆˆ+   ()()ffw−+ + ffw −+   12  32 1 5 12 ffˆˆ−  ()()ffw−+ − ffw −+ = 12=  32 1 5 12 f   ffˆˆ+ ()()ffw++ +−−+ ffw  34  12 3 35 4  ˆˆ−  (ffw++ )(−−ffw − + ) ff34  1233 3 5 4 

We can write

 I  −−III II 00  2   222  22  −I III− II− 00 f =  2  f od+  222 f +  22  w −I   II− I  00II  2   22 2  22 −  I2   III222  00II22

where

w  f   1   1  w fffod= [],,= f w =  2  . 5  2  w    3 f3    w4 

o d Or, with the obvious deﬁnitions for B , B , and W1, in matrix notation:

= o o d d f B f + B f + W1w. (18.A.20)

The nodal vectors are deﬁned as follows:

p = ρρ  1 pr=+ρρ  21  , = ρρˆ p3   =+ρρˆ pr42 and ρρρˆ =++ρ − rt1 5 r2.

We deﬁne  =+=∆∆ ρρ + qp12 p 12 r 1   =−=∆∆ qppr2211   =+=∆∆ ρρρˆ +=ρ ++− qpp34 322 r 2() rt 15 r2   =−=∆∆ qpp44 32 r

The relation between q and p can be written as follows:

q   II 00p   1   22  1  q −II 00p q =  2  =  22  2  , q   00IIp   3   22 3  − q4   00 II22p4 

FIGURE 18.A.2 A planar tensegrity.

and

q  2II 00ρρ  1   22   q 000I r q =  2  =  2  1  = Qlo. (18.A.21) q  22IIII− 2r   3   2222 2  q4   00I2 0t5 

We can now write the dependent variables ld in terms of independent variables lo. From (18.30) and (18.31):

ρρ ρρ t1 = + r1 –( + r1 + t5 – r2), ρρ ρρ t2 = – ( + r1 + t5 – r2).

By inspection of Figure 18.A.2, (18.30) and (18.31) reduce to:

t1 = r2 – t5

t2 = –r1 + r2 – t5

t3 = r1 + t5, (18.A.22)

or,

ρρ t  00II−    1   22 r Id = t = 0 −−II I 1   2   22 2r      2 t3  00II22  t5 

Equation (18.A.21) yields

ρρ  II− 00 q     22   1  r 1 0002I q − lo =  1  =  2  =  2  = Qq1 . r  2  0002I  q   2   2   3  t −−IIIIq  5   2222  4  Hence,

 −  II22 II 22 d 1   l =  IIIIqRq−−  = 2 2222 −   II22 II 22

We can now write out the tendon forces as follows:

f  K  b u   1   1   1  1  d = − f f2  = K2 q + b2 u2 , f  K   b u   3   3   3  3  or

fd = –Kdq + Pdud

and

o f = [f5] = –[K5] q + [b5] [u5],

fo = – Koq + Pouo,

using the same deﬁnitions for K and b as found in (18.45) and (18.46), simply by removing the ij element indices. Substitution into (18.A.20) yields:

fB=−oo()() KqPuB +oo +−dd KqPu + dd

=−oo +dd +ooo + ddd ()().BK BK q BPu BPu

With the matrices derived in this section, we can express the dynamics in the form of (18.20):

qK˙˙ +() + KqB =u + Dw , rp

= K r 1,

KHK= ˜ , p

BHB = ˜ ,

DHWHW == o , 11 where

I000  2  0q003 ˜ 2 2  L2 2  HH== 1 , 1 m  00I 0  2  3 ˜ 2  0002 q2   L2 

0000  −  0qqI00L 2 ˙˙T K =  1 222 , r 1 0000   −2 ˙˙T 000qqIL2 444

KBKBK˜ ∆∆ dd+ oo,

˜ dd oo BBPBP∆∆[,],

and

 −qqq2   −qqq2  qq˜ 2 =  22 21 22 , ˜ 2 =  42 41 42 , 2 − 2 4 − 2 qq21 22 q 21  qq41 42 q 41 

˙˙TT=+2 2 ˙˙ =+2 2 qq22qq˙˙ 21 22, qq44 qq˙˙ 41 42.

Appendix 18.B Algebraic Inversion of the Q Matrix

This appendix will algebraically invert a 5 × 5 block Q matrix. Given Q in the form:

Q 0000  11  QQ21 22 000 Q = QQQ 00 (18.B.1)  21 32 22  QQQQ21 32 32 22 0  QQQQQ  21 32 32 32 22  we deﬁne x and y matrices so that

Qx = y (18.B.2)

where

x  y   1   1  x2  y2  x = x , y = y  (18.B.3)  3   3  x4  y4  x  y   5   5  Solving (18.B.2) for x we obtain

x = Q–1y (18.B.4)

Substituting (18.B.1) and (18.B.3) into (18.B.2) and carrying out the matrix operations we obtain

Qx= y  11 1 1 Qx+= Qx y  21 1 22 2 2 ++ = Qx21 1 Qx 32 2 Qx 22 3 y 3 (18.B.5)  Qx++ Qx Qx+ Qx = y  21 1 32 2 32 3 22 4 4 Qx++ Qx Qx+ Qx + Qx = y  21 1 32 2 32 3 32 4 22 55 Solving this system of equations for x will give us the desired Q–1 matrix. Solving each equation for x we have

 = −1 xQy1111  −1 xQQxy =−( + )  22221 1 2 =−−1 − xQQxQxy3222113223 ( + ) (18.B.6)  −1 x =− Q( Qx − Qx − Qx+ y)  4 22 21 1 32 2 32 3 4  =−−1 − − − x5 Q22( Qx 21 1 Qx 32 2 Qx 32 3 Qx 32 + y5 ) 4 Elimination of x on the right side of (18.B.6) by substitution yields

xy = ΛΛ  1111 xyy =+ΛΛΛΛ  2 21 1 22 2 =+ΛΛΛΛΛ +Λ xyyy3311322333 (18.B.7)  xyyyy =+++ΛΛΛΛΛΛΛΛ  4 41 1 42 2 43 3 44 4 xyyyyy =++++ΛΛΛΛΛΛΛΛΛΛ  5511 52 2 53 3 54 4 55 5 Or, in matrix form,

ΛΛ  11 0000 ΛΛΛΛ   21 22 000 Q−1 =ΛΛΛΛΛΛ 00 (18.B.8)  31 32 33  ΛΛΛΛΛΛΛΛ  41 42 43 44 0  ΛΛΛΛΛΛΛΛΛΛ   51 52 53 54 55  Λ where we have used the notation row, column with the following deﬁnitions

ΛΛ = −1 11Q 11 ΛΛΛ=−−−1 1 =− −1 ΛΛ =−ΛΛΛ 21QQQ 22 21 11 QQ22 21 11 22 Q 21 11 ΛΛ =−−−−−−1 1 1 1 1 31QQQQQ 22 32 22 21 11 QQQ22 21 11 ΛΛ =−−−−1 1 1 –1 + −−−1 1 1 − −−1 1 41QQQQQQQ 22 32 22 32 22 21 11 2 QQQQQ22 32 22 21 11 Q2222QQ21 11 ΛΛ =−−−−−−−−1 1 1 1 + 1 1 1 51 33QQQQQQQ22 32 22 32 22 21 11 QQQQQ22 32 22 21 11 +−−−−−−−−1 1 1 1 1 1 1 QQQQQQQQQ22 32 22 32 22 32 22 21 11 QQQ22 21 11 ΛΛΛ====ΛΛΛΛΛ −1 22 33 44 55 Q22 ΛΛΛ= Λ ==−ΛΛ −−1 1 32 4343 54 QQQ22 32 22

ΛΛΛ==Λ −−−−−1 1 1 −1 1 42 53 QQQQQ22 32 22 32 22 QQQ22 32 22 ΛΛ =−−−−−−−−−−1 1 1 1 +1 1 1 − 1 1 52 QQQQQQQ22 32 22 32 22 32 22 2 QQQQQ22 32 22 32 22 QQQ22 32 22

Using repeated patterns, the inverse may be computed by

ΛΛ  11  ΛΛΛΛ   21 22  ΘΘΛΛΛΛΛΛ   21 32 22  −1 = ΘΘΛ2 ΛΘΘΛΛΛΛΛΛ Q  21 32 32 22  (18.B.9) ΘΘΛ3ΛΘΘΛ2 ΛΘΘΛΛΛΛΛΛ   21 31 32 32 22  ΘΘΛ4 ΛΘΘΛ3ΛΘΘΛ2 ΛΘΘΛΛΛΛΛΛ  21 31 31 32 32 22    M M O O OOO

using

ΘΘΛ− Λ =()IQ 22 32 ΛΛ −1 11= Q 11 ΛΛΛ−ΛΛΛ 21= 22Q 21 11 ΛΛ −1 22= Q 22 ΛΛΛ= − ΛΛQ Λ . 32 22 32 22 ΛΛΛΛΛΛΛΛ Θ −1 Only 11,,,, 22 21 32 and powers of need to be calculated to obtain Q for any (n, m). −1 −1 −1 The only matrix inversion that needs to be computed to obtain Q is Q11 andQ22 , substantially reducing computer processing time for computer simulations.

Appendix 18.C General Case for (n, m) = (i, 1)

In Appendix E the forces acting on each node are presented, making special exceptions for the case when (j = m = 1). The exceptions arise for (j = m = 1) because one stage now contains both closure rules for the base and the top of the structure. In the following synthesis we usefˆ and ˆ ˆ ˆ 111 f211 from (18.E.1) and f311 and f411 from (18.E.4). At the right, where (i = 1, j = 1):

ffˆˆ+   111 211  ffˆˆ−  = 111 211 f11   ffˆˆ+  311 411 ˆˆ−  ff311 411

()()−+ffffw +++ + ffffw −− − +   211 311 1nn 1 2 1 111 511 111 411 811 211  ()(−+ffffw ++ + − fff −− −−+fw) =  211 311 1nn 1 2 1 111 511 111 411 811 211 . ()()fff−− + fw + + ffffw −−++   811 711 321 421 311 611 511 61n 7n 1 411  ()()fff−− + fw + − ffffw −−++  811 711 321 421 311 611 511 61n 7n 1 411  At the center, where (1 < i < n, j = 1):

ffˆˆ+   11ii 21 ffˆˆ−  = 11ii 21 fi1   ffˆˆ+  31ii 41 ˆˆ−  ff31ii 41

 +−+++−−+−+  ()(ffffwffff111211()ii−− () 21ii 31 11 i 11 ii 41 51i 8iii121w )  +−++−−−+−+  ()()ffffwffffw111211()ii−− () 21ii 31 11 i 11 ii 41 51i 81ii 21  = . ()(fff−− + f + w +−+ ff−−+ff+ w)  81ii 71 311411(+)ii (+) 31i 51i 61i 61()ii−− 711 () 41i  −− + + −−+− + ()()fff81ii 71 311411(+)ii f (+) w31i fff51i 61i 61 ()ii−− f 711 ()+ w41i 

At the left end of the base in Figure 18.5, where (i = n, j = 1):

ffˆˆ+   11nn 21 ffˆˆ−  = 11nn 21 fn1   ffˆˆ+  31nn 41 ˆˆ−  ff31nn 41 (18.C.1) ++−++−−+−+ ()(ffffwffff111211()nn−− () 31nn 21 11 n 11 nn 41 51n 8nnn121w )   ++−+−−−+−+  ()()ffffwffffw111211()nn−− () 31nn 21 11 n 11 nn 41 51n 81nn 21  = . ()(−+−+ffffw + +− f − fff + + + w )  7nn 1 8 1 311 411 3 n 1 61151()nn− 61n 7711()n− 41n  −+−+ + −− − + + + ()()ffffw7nn 1 8 1 311 411 3 n 1 f61151()nn−− fff61n 711 ()n w41n 

Using,

 f   2   f3   f  w   4   1  f  f w o 5 d  6   2  f =   ,,f = w = , ij f  ij  f  ij w  1 ij  7   3   f  w 8  4 ij  f   9  f   10  ij

I 000 0 000 −−II I0 0 − I00  3   33 3 3  I 000 0 000 −III 0 0 I 00 f =  3  f d +  333 3  f d 110 00− I I 000 n 1  000 I− II 00 11  33   333  − −− 0 00I33 I 000  000 I333 II 00

− 0 0 0 00000  II33 I       3  0 0 0 00000 −II I +  f do+  33 f +  3  f 0− I I 00000 21 −I0 11  0  11n  33   3    − 0 I33 I 00000  I03   0 

II 00  33  III00− + 33  w .  00II 11  33 −  00I33 I

Or, in matrix notation, with the implied matrix deﬁnitions,

=++++ˆ d dddddoo + fBfBfBfBfBfWw11n 1n 1 11 11 21 21 11 11 1nn 1 1 1 11 . (18.C.2)

I00 0 0 000 −−II I 00 − I 00  3   33 3 3  I00 0 0 000 −I I I 00 I 00 f =  3  f d +  33 3 3  f d i1 0 00−I I 000 ()i−11  00 0 I− I I 00 i1  33   333  − −− 0 00 II33 000  00 0000II333 I 

0 0 0 00000 0 I     3  0 0 0 00000 0 I +   f d +  3  f o 0−II 00000 ()i+1100 (i− 1))1  33  −   0II33 00000 00

 II−  I I00   33  33  −I I II− 00  33 o  33  + f + w . −I 0  i1100 I I i  3   33 −  I3 0  00 II33

Or, in matrix form,

fBf=++++ˆ d dd BfBfdddo Bfoo BfWwo + . (18.C.3) i101()i−−11 11i 1 21 (i+11) 01 (i 11) 11i 1i 1

I 00 0 0 000 −−II I 0 0 − I00  3   33 3 3  I 00 0 0 000 −II I 0 0 I 00 f =  3  f d +  33 3 3  f d n1 0 00− I I 000 ()n−11  000 I− II 00 n1  33   333  − −− 0 00 I33 I 000  0 000 I333 II00 

0 0 0 00000 0I  I03     3    0 0 0 00000 0I −I0 +   f d +  3  f oo +  3  f o 0− I I 00000 11 00 ()n−11 −I0 nl  33     3  − 0 I33 I 00000 00  I03  −  I  II0033   3    I II00− +  3  f +  33  w .  0  11nn00II  1    33 −  0  00I 33 I

Or, in matrix form,

=++++++ˆ d dddddoo o o fBfBfBfBfBfBfWwn101()n−11 11n 1 21 11 01 ()n−11 n1n 1 1111nn n 1. (18.C.4)

Now assemble (18.C.2)–(18.C.4) into the form

=++do fBfBfWw1311111, (18.C.5)

where

f  f o  w   1j   1j   1j    o w f2 j  f11n f2 j   2 j  f = ,,,f d =   f o = w = j  M  1  f d  j  M  j  M    1     f  f o  w   nj   nj   nj 

BBB0dd L 0Bˆ d   1n 1 11 21n 1  B0o LL 0  0Bˆ d OOO 0  11  01 Bo OO M  MOOOOOM  01  B =  , B =  0 OO O M, 3  00OOOO  1    MOOB0o   OOOOd  11  0B21   o o   000BBL   dddL ˆ  01n 1 BB1n 1 21 0 0BB 01 11 

= []KK WWW 1 BlockDiag ,,,.

Or, simply, (18.C.5) has the form (18.E.21), where

ffffffwwWW = ddoo== = o = 1111,,, , 1 ,

do== BBBB31,.

The next set of necessary exceptions that apply to the model (i,1) arises in the form of the R d = matrix that relates the dependent tendons set to the generalized coordinates ( l Rq) . For any (i,1) case R takes the form following the same procedure as in (18.32) and (18.33).

R0  R=  ˜ , (18.C.6) R11

where

EE0˜ LL E  34 2 ˜ O EEE234 0   0 EEE˜ OM R˜ =  234  11 MO ˜  EEE0234  OOO ˜  0E4   ˜ L  E0423 0 EE

−I I 00   33   0000   0000    1  00 I I E˜ = 33. 4 2  00 −− II  33  0000   II− 00   33     00 I33 I

The transformation matrix T that is applied to the control inputs takes the following form. The

only exception to (18.55) is that there are no I8 blocks due to the fact that there are no stages between the boundary conditions at the base and the top of the structure. The second set of I2 blocks is also not needed since m = 1. Hence, the appropriate T matrix for u˜ = Tu is

 001L   ′  T1 S   OO   ′ ∈ ()10nn+ 1× 8 T= T1 S  R ,  I   2   O     I2 

where S is deﬁned by (18.56), and

1000 0 0   01 00 0 0 0010 0 0   0001 0 0 8× T′ = ∈ R 6. 1 0000 1 0   0000 01   0000 0 0 0000− 10

′ There are n T1 blocks, n S blocks, and n I2 blocks. The control inputs are now deﬁned as

ud  u =  1 . uo  1 

d The u1 block becomes

u21n  u′d     11  u31n  u′d  21 u  d   d 41n 61× u′ = u′d ,,,,,,u′ =   ∈===R injm123L 1 1  31  i1 u61n   M      u71n u′d    n1    u81n 

Appendix 18.D will explicitly show all matrix forms for the speciﬁc example (n,m) = (3,1).

Appendix 18.D Example Case (n,m) = (3,1)

Given the equation for the dynamics of the shell class of tensegrity structures:

˙˙ ++˙ =+ qKqKqqBquDqw((rp) ()) () () .

We explicitly write out the matrices that deﬁne the problem:

q  ρρ   111   11  q211  r111  q  r   311   211  q411  r511  q  r  q   121  A00 l  A00  111   11  q   11  r  == = 221 ==  =   121 qq1 q21    Ql11 1 DB0 l21  DB0   , q321 r221 q    DEB l  DEB   31 q  31 r   421   521  q131  r121  q  r   231   131  q331  r321      q431  r531 

× where Q11 is (36 36). Furthermore,

ΩΩ 00   11  ˙ ΩΩ Kqr ( ) =  0021   ΩΩ   00 31

which is also a (36 × 36) matrix.

==˜ dd + oo = d + o0 Kqp () HKHBK()() BK HBK131 BK 11

K   ddd 131 H 00  B BBBˆ    11  1n 1 11 21n 1  Kd   ddd 11  K = 00H  0 BBBˆ  p  21   01 11 21  d   ddd K21  00 H  B BBBˆ    31   1n 1 21 01 11  d  K31 

Bo 000  Ko   11   11  + oo o BB01 11 0  K21   o o   o   0 BB01n 1  K31 

(36 × 36) = (36 × 36) [(36 × 75) ∗ (75 × 36) + (36 × 18) ∗ (18 × 36)]

d1 o –1 In order to form K1 , R is needed. In order to form K1 , Q is needed. Therefore, we obtain R as follows:

td  R  lldd== 131  = Rq =  0  [],q 1 d R˜ 1  t1   11 

td  EE0   131  6 7 q  d  ˜  11  t  EEE342  11 =   q ,  d   ˜   21  t EEE234  21    q   td  EEE˜  31  31  423 where the matrix is dimension (75 × 36).

Bq()== HBT˜ HBP[,]dd BP oo T

Pd 000   ddˆ d  11n B1n 1 BBB 11 21n 1      000Pd  []BPdd= [] BP d=  0 BBBˆ ddd 11 31 01 11 21  00 Pd 0  dddˆ  21 B BBB    1n 1 21 01 11   000 Pd   31  where the dimensions are (36 × 25) = (36 × 75) ∗ (75 × 25).

 o   o  B11 00 P11 00 oo o  oo   o  [][]BP= BP11 = BB01 11 0   00P21   0 BBo o   00 Po   01n 1   31  and the dimensions are (36 × 6) = (36 × 18) ∗ (18 × 6).

 0001LL   ′  TS1 00 00 H 00   0000TS′   11   1  do ′ B= 00H21  []BP31, BP 11  00TS1 00   00 H  000I 00  31   2   0000I2 0    00000I2 

and the dimensions are (36 × 24) = (36 × 36) ∗ (36 × 31) ∗ (31 × 24).

The control inputs uαij are deﬁned as

u′d   11  ′d u21  ud  u′d  u′d  = =  1  =  31  × u  o  o o ()24 1 u   u1  u11    uo  21   o  u31 

where

u   21n  u31n  u  ′d = 41n ∈==61 × ui1   R ,,,.ij123 1 u61n  u   71n  u81n 

Bu∈R()36× 1 .

The external forces applied to the nodes arise in the product Dw, where

H 00  W 00   11  o = =   D()= q HW H []W1  00H21   00W   00 H  00 W  31  with dimensions: (36 × 36) = (36 × 36) ∗ (36 × 36).

w   11  = × ww= 1 w21  ()36 1   w31 

Dw ∈R (36 × 1).

Appendix 18.E Nodal Forces

At the base, right end of Figure 18.5, where (i = 1, j = 1):

 ffˆˆ+   111 211   ffˆˆ−  = 111 211 f11   f+ˆˆ f  311 411  ˆˆ−  ff311 411 

()()−+ffffw ++ + + ffffw −− − +   211 311 1nn 1 2 1 111 511 111 411 811 211  ()(−+ffffw ++ + − fff −− −+fw) =  211 311 1nn 1 2 1 111 511 111 411411 811 211   ()()fff−+ −++ ffw + ffffw −+++   811 711 1011 321 42 1 311 611 511 212 112 411  −+ −++ − −+++ ()()fff811 711 1011 ffw 321 42 1 311 ffffw611 511 212 112 411 

At the center of the base, where (1 < i < n, j = 1):

 ffˆˆ+   11ii 21  ffˆˆ−  = 11ii 21 fi1   f+ˆˆ f  31ii 41  ˆˆ−  ff31ii 41

 ()(ffffwffff−−+−+++−−+−+ w )   1()iiiiiii 11 2 () 11 21 31 11 11 41 51i 881ii 21  ()()ffffwffffw+−++−−−−−+ =  1()iiiiiii−− 11 2 () 11 21 31 11 11 41 51i 81ii 21  ()(fff−+ − f + f + w +− f+−ff ++ ff+ w)  8ii 1 7 1 10 i 1 3() i++ 1 1 4 () i 1 1 3 i 1 5i11 61i 10()iiii− 1 1 2 2 2 4 1  −+ − + + −−+− ++ ()()fff8ii 1 7 1 10 i 1 f 3() i++ 1 1 f 4 () i 1 1 w 3 i 1 fff51i 61i 10 ()iiii − 1 ff+ 1 2 2 2 w 4 1 

At the left end of the base in Figure 18.5, where (i = n, j = 1):

 ffˆˆ+   11nn 21  ffˆˆ−  = 11nn 21 fn1   f+ˆˆ f  31nn 41  ˆˆ−  ff31nn 41

()(ffffwffff−−++−++−−+−+ w )  (18.E.1)  1()nnnnnnn 11 2 () 11 31 21 11 11 41 51n 81 8nn 21   ++−+−−−+−+  ()()ffffwffffw1()nnnnnnn−− 11 2 () 11 31 21 11 11 41 51n 81nn 21 =    −+ + − + + +− − + +++  ()(fff7nn 1 8 1 10 n 1 ffw 311 411 3 n 1 f 10() n− 1 1 f51n ffffw61n 12nn 22 41 n)   −+ + − + + −− − + + + + ()()fff7nn 1 8 1 10 n 1 ffw 311 411 3 n 1 f 10() n− 1 1 ffffw51n 61n 12nn 22 41 n

At the second stage, where (1 ≤ i ≤ n, j = 2):

ffˆˆ+   112 212  ffˆˆ−  = 112 212 f12   ffˆˆ+  312 412  ˆˆ−  ff312 412 

()−+fffffw − − + + +−() fffffw − + − + +   61n 7nn 1 9 2 212 312 112 112 412 512 812 912 212  ()−+fffff − − + + wfffffw−−() − + − + +  = 61n 7nn 1 9 2 212 312312 112 112 412 512 812 912 212 ()−+fff + − ffw + + +−() fff + − + ffw + +   712 812 1012 322 422 312 512 612 10n 2 113 213 412  ()−+ + − + + −−(() + − + + +  fff712 812 1012 ffw 322 422 312 fff512 612 10n 2 ffw 113 213 412 

ffˆˆ+   12ii 22 ffˆˆ−  = 12ii 22 fi2   ffˆˆ+  32ii 42 ffˆˆ−  32ii 42 (18.E.2) ()−+fffff − −++ wfffffw+−() − + − + +   611()i− 711912()iiii−− () 22 3 22121242iii52i 82ii 92 22 i

()−+fffffwfffffw− −− − −++−−() − + − + +  =  611()i 7()iiiiiii 11 9 () 12 22 32 12 12 42 52i 82ii 92 2 i22  −+ + − + + +− + − + + + ()fff7ii 2 8 2 10 i 2 f 3() i++ 1 2 f 4 () i 1 2 w 3 i 2 () fff52i 62i 10 ()iiii − 1 2 ffw 1 3 2 3 4 2  ()−+ + − + + −−() + − + + +   fff7ii 2 8 2 10 i 2 f 3() i++ 1 2 f 4 () i 1 2 w332i fff52i 62i 10()iiii− 1 2 ffw 1 3 2 3 4 2 

ffˆˆ+   12nn 22 ffˆˆ−  = 12nn 22 fn2   ffˆˆ+  32nn 42 ˆˆ−  ff32nn 42

()−+fffffwfffffw − −+++−() − + − + +   611()n− 7()nnnnnnn−− 11 9 () 12 22 32 12 12 42 52n 82nn 92 22 n

()−+ffff− −− − −+ffw+ −−() fffffw − + − + +  =  611()n 71191222()nnn () 32nn 12 12 nn 42 52n 82nn 92 22 n ()−+ + −++ +− + − + + +  f7nn 2 f 8 2 f 10 n 2 f 312 f 412 w 3 n 2 () f52n f62n f10()nnnn− 1 2 f 1 3 f 2 3 w 4 2  ()−+ +−++ −−() + − + + +   f72n ff8nn 2 f 10 2 f 312 f 412 w 3 n 2 f52n f62n f10()nnnn− 1 2 f 1 3 f 2 3 w 4 2 

At the typical stage (1 ≤ j < m, 1 ≤ i ≤ n):

ffˆˆ+   11jj 21  ffˆˆ−  f = 11jj 21 1j ˆˆ+  ff31jj 41  ˆˆ−  ff31jj 41 

 ()−+f f −−++ fffw+−(() fffffw − + − + +   61nj()− 71nj()− 9 nj 213111 j j j 1141 j j51 81 j 91 j 21 j    ()−+f− f− −−++ f f f w−−() f − f + f − f + f + w =  61nj() 71n() j 9 nj 213111 j j j 1141 j j 51j 81jj 91 21 j   ()−+fff + − ffw + + +−() fff + − + f+ +++fw+   71jj 81 101 jjj 32 42 31 j 51j 61j 10nj 11() j 1 21()jj 1 41  ()−+ + − + + −−() + − + + +  fff71jj 81 101 jjj ffw 32 42 31 j fff51j 61j 10nj f 11() j++ 1 f 21 () j 1 w 41 j 

ffˆˆ+   12ij ij  ffˆˆ−  f = 12ij ij ij ˆˆ+  ff34ij ij  ˆˆ−  ff34ij ij 

()(−+fffffwfff−− −− −−+++−−+− − fffw++ )   61()()ij 1 71()()i j 1 91 () i j 2 ij 3 ij 1 ij 1 ij 4 ij 5ij 89ij ij 2 ij  ()()−+fffffwfffffw −−++−−−+−++ =  61()()ij−− 1 71()()i−− j 1 91 () i − j 2 ij 3 ij 1 ij 1 ij 4 ij 5ij 89ij ij 2 ij  (−+fff + − f + f ++−+−+wfffffw)( + + )  78103141ij ij ij() i++ j () i )()()()jij3 5ij 6ij 10i−++ 1 j 1 ij 1 2 ij 1 4 ij  −+ + − + + −−+ − + + + ()(fff7ij 8 ij 10 ij f 3() i++ 1 j f 4 () i 1 j w 3 ij fff5ij 6ij 10 ()ij − 1 f 1 ij ( + 1 ) f221ij()+ w 4 ij) (18.E.3)

ffˆˆ+   1nj 2nj  ffˆˆ−  f = 1nj 2nj nj ˆˆ+  ff34nj nj  ˆˆ−  ff34nj nj 

(−+fff−− −− −− − fffw++)( +−−+−++ fffffw )  61()()nj 1 71()()nj 1 91 () nj 23nj nj 1 nj 14 nj nj 5nj 89nj nj 2 nj  ()()−+fffffwfffffw −−++−−−+−++ =  61()()nj−− 1 71()()n−− j 1 91 () n − j 2 nj 3 nj 1 nj 1 nj 4 nj 5nj 89nj nj 2 nj   (−ffff++ −++ ffw)( +−+− fff + f + f + w )   7nj 8 nj 10 nj 31 j 41 j 3 nj 5nj 6nj 10()n−++ 1 j 1 nj () 1 2 nj () 1 4 nj  −++ − + + −−+ − + ++  ()(fff7nj 8 nj 10 nj ffw 31 j 41 j 3 nj fff5nj 6nj 10(nj− 1) f1n(()jnjnj++121fw () 4)  (1 ≤ i ≤ n, j = m)

ffˆˆ+   11mm 21  ffˆˆ−  = 11mm 21 f1m   ffˆˆ+  31mm 41  ˆˆ−  ff31mm 41 

()(−f− + f− −+−+ fffw +−+−−++ fffffw)  61nm() 7n() m 1 21 m 31 m 9 nm 1 nm 41 m 51m 81mmm 11 91 2nmnm  ()()−f + f −+−+ fffw −−+−−++ fffffw =  61nm()− 7n() m− 1 21 m 31 m 9 nm 1 nm 41 m 51m 81mmm 11 91 2 nm ()()−+−++f f f f w +−+−++ f f f f w   71mmmm 81 32 42 31 m 51m 61mnm 6 741nm m  − +−++ −−+−++ ( ff71mmmm f 81 f 32 f 42 w 31 m)( f51m f61mnm f 6 f741nm w m ) 

ffˆˆ+   12im im  ffˆˆ−  = 12im im (18.E.4) fim   ffˆˆ+  34im im  ˆˆ−  ff34im im 

(−f −−+−−+++−−+−++ffffwfffffw−− − )( )  661()()im 1 71()()i m 1 91 () i m 2 im 3 im 1 im 1 im 4 im 5im 89im im 2 im  (−+fffffw −−++)(−−fffffw − + − + + ) =  61()()im−− 1 71()()i−− m 1 91 () i − m 2 im 3 im 111im im 45imim89im im 2 im  ()()−+f f − f + f + w +−− f f + f + f + w   7831413im im() i++ m () i m im 5im 61()im− 6 im 71()im− 4 im  −+ − ++−−−+++ ( fff7831im im() i+ mmimimf41()+ w 3)( f5im f61()im− f 6 im f71()im− w 4 im ) 

ffˆˆ+   12nm nm  ffˆˆ−  = 12nm nm fnm   ffˆˆ+  34nm nm  ˆˆ−  ff34nm nm 

()()−+fffffwfffffw−− −− −−+++−−+−++ −   61()()nm 1 71()()n m 1 91 () n m 2 nm 3 nm 1 nm 1 nm 4 nm 5nm 89nm nm 2 nm  (−+ffff −−+++fw)( −−−+−++ fffffw ) =  61()()nm−− 1 71()()nm−− 1 91 () nm − 2 nm 31nm nm 14 nm nm 5nm 89nm nm 2 nm  ()()−+f f − f + f + w +−+ f f − f + f + w   7831413nm nm m m nm 5nm 661nm() n− m 71()nm− 4 nm  −+ − + + −− + − + + ( ffffw7831413nm nm m m nm ))(f5nm f661nm f() n− m f71()nm− w 4 nm ) 

 f   2   f3     f  w1 4   f   f  w f o = 5 ,,f d =  6  w =  2  ij   ij ij   f1   f7  w3 ij     f w  8   4 ij  f9  f   10 ij

−− − I 0000000  II33 I 3 0 000I3   3    I 0000000 −II I 0 000 I ff=  3  d +  33 3 3  f d 11  0 0000000 n1  00 0 I−I I0 I 11    3 33 3 − −  0 0000000  00 0 I3 I33 I00 I3 

 0 0000000 0 0 000000     0 0000000 0 0 000000 +   f d +   f d  I 0000000 12 0II0− 0000 21  3   33  − −  I3 0000000 0II033 0000 −  I3 I  0 0  I  I33 I00   3     3    −I I 0 0 I II− 00 +  3 3 foo+  f +  3  f +  33  w . −I 0  11 0 I  12  0  11n  00I I I 11  3   3     33 − −  I3 0  0 I3   0   00 II33

Or, in matrix notation, with the implied matrix deﬁnitions,

=++++++d dddddddoooo + fBfBfBfBfBfBfBfWw11n 1n 1 11 11 12 12 21 21 11 11 12 12 1nn 1 1 1 11. (18.E.5)

I 000000 0 −−II I 0 000−I   3   33 3 3  I 000000 0 −II I 0 000 I f =  3  f d +  33 3 3  f d ii1  0 000000I  ()−11  000 I −I I0 I i1  3   3 33 3 − −  0 000000I3   000 I3 I33 I0 I 3

 0 0000000 00 000000     0 0000000 00 000000 +   f d +   f d  I 0000000 i2 00−II 0000 ()i+11  3   33  − −  I3 0000000 00II33 0000 − 0 I   I3 I  0 0  II3300  3   3      0 I −I I 0 0 II− 00 +  3  f o +  3 3  fo +   f o +  33  w 00 0  ()i−11 −I 0  i1 0 I  i2  00II i1    3   3   33 0 0 I 0 0 −I 000I− I    3   3   33

Or, in matrix form,

=++++d dddddddoo fBfBfBfBfBfi1 0111111122()i−+−i i 21110111 ()i ()i (18.E.6) ++o ooo + Bf11i 1 Bf 12i 2 Wwi 1.

I 000000 0 −−II I 0 000−I   3   33 3 3  I 000000 0 −II I 0 000I f =  3  f d +  33 3 3  f d nn1  0 000000I  ()−11  000 I −II0 I n1  3   3 33 3 0 000000I 000−I −II0 I  3   3 33 3  0 0000000  00 000000     0 0000000 00 000000 +   ffdd+    I 0000000 n2  0II0− 0000 11  3   33  − −  I3 0000000  0II033 0000 −  0 I   I3 0   0 0   I   3       3  0 I −I 0 0 0 I +  33  f o +  3  f o +   f o +  3  f  0 0  ()n−11 −I 0  n1  0 I  n2  0  11n    3   3    −  0 0   I3 0   0 I3   0 

 II00  33  III00− +  33  w . 00II n1  33 − 00I33 I

Or, in matrix form,

=++++d dddddddoo fBfBfBfBfBfn1 011111112221110111()n−−n n ()n (18.E.7) +++o ooo + Bfn11n Bf 122n B 1111nn f Ww n 1.

−−II I 0 00−II  00 000000  33 3 33    −II I 0 00II− 00 000000 f =  33 3 33 f d +   f d 12  000 I −II0 I 12 0II0− 0000 22  3 33 3  33  − − −  000 I3 II330 I 3 0II033 0000

 0 0000000 000− II 000    3 3  0 0000000 000− II 000 +   f d +  3 3  f d  I 0000000 13 000 0 0 000 n1  3    −  I3 0000000 000 0 0 000

 000000− I 0  I −I  0 0   3   3 3    000000− I 0 −I I 0 0 +  3  f do+  3 3  f+   f o  000000 0− I n2 −I 0  12 0 I  1313  3   3   3  −  000000 0 I3   I3 0  0 I3 

II 00  33  II− 00 +  33  w .  00II 12  33 −  00II33

Or, in matrix form,

=++++dd dd dd d d d d fBfBfBfBfBf12 12 12 21 22 12 13n 1n 1n 2n 2 (18.E.8) +++oo oo B11 f 12 B 12 f 13 Ww 12.

− −− − 000 II 000  II33 I 30 00II33   3 3    000− II 000 −II I 0 00II− ff=  3 3  d +  33 3 33 f d ii2 000 0 0 000 ()−11  000 I −II0 I i2    3 33 3 − − 000 0 0 000  000 I3 I3 III330 

 0 0000000 000000− I 0    3  0 0000000 000000− I 0 +   f d +  3  f d  I 0000000 i3 000000 0− I ()i−12  3   3  −  I3 0000000 000000 0 I3 

00 000000  I −I     3 3  00 000000 −I I +   f d +  3 3  f o 0II0− 0000 ()i+12 −I 0  i2  33   3  − 0II033 0000  I3 0 

00 0  II 00    33  0 0 II− 00 +   f o +  33  w . 0 I  i3  00II i2  3   33 − − 0 I3   00II33

Or, in matrix form,

=++++d ddddd d ddd fBfBfBfBfBfin2 1()i−−+ 1 1 12i 2 12i 3n 2 ()i 1 2 21 ()i 1 2 (18.E.9) +++o ooo B11 fi 2 B 12 fi 3 Wwi 2.

00 000000 000− II 000    3 3  00 000000 000− II 000 ff=   d +  3 3  fd n2 0II0− 0000 12 000 00 000 ()n−11  33    0II0− 0000 000 00 000  33   

000000− I 0 −−II I 0 00−II   3   33 3 33  000000− I 0 −II I 0 00II− + 3  f d +  33 3 33 f d 000000 0− I ()n−12  000 I −II0 I n2  3   3 33 3 − − 000000 0 I3   000 I3 II3300 I3 

 0 0000000 0 0   I −I       3 3  0 0000000 0 0 −I I +  fd +   f o +  3 33  f o  I 0000000 n3 0 I  n3 −I 0  n2  3   3   3  − −  I3 0000000 0 I3   I3 0 

II 00  33  II− 00 + 33  w .  00II n2  33 −  00II33

Or, in matrix form,

=+dd d d +d dd +++dddoo fBfBfBfBfBfBfn22112111212122123123n ()n−−n ()n n n n (18.E.10) ++o o B11 fn 2 Wwn 2.

− −− − 000 II 000  II33 I 30 00II33   3 3    000− II 000 −II I 0 00II− ff=  3 3  d +  33 3 33 f d 1jnj000 0 0 000 ()−1  000 I −II0 I 1j    3 33 3 − − 000 0 0 000  000 I3 II3 3330 I 

000000− I 0 00 000000  3    000000− I 0 00 000000 +  3  f d +   f d 000000 0− I nj 0II0− 0000 2 j  3   33  − 000000 0 I3  0II033 0000

 0 0000000  I −I     3 3  0 0000000 −I I +   f d +  3 3  f o  I 0000000 11()j+ −I 0  1j  3   3  −  I3 0000000  I3 0 

0 0  II 00    33  0 0 II− 00 +   f o +  33  w . 0 II  11()j+  00II 1j  3   33 − − 0 I3   00II33

Or, in matrix form,

=++++d ddd d ddddd fBfBfBfBfBf1jn 1nj()−+ 1 121j n 2nj 212j 121 ()j 1 (18.E.11) ++o ooo + B11 f 1j B 12 f 1()j+ 1 Ww 1j .

− −− − 000 II 000  II33 I 30 00II33   3 3    000− II 000 −II I 0 00II− ff=  3 3  d +  33 3 33 f d ij000 0 0 000 ()() i−−11 j  000 I −II0 I ij    3 33 3 − − 000 0 0 000  000 I3 III330 I 3

000000− I 0 00 000000  3    000000− I 0 00 000000 + 3  ffd +   d 000000 0− I ()ij−+1 0II0− 0000 (i 1)) j  3   33  − 000000 0 I3  0II033 0000

 0 0000000  I −I     3 3  0 0000000 −I I +  f d +  3 3  f o  I 0000000 ij()+1 −I 0  ij  3   3  −  I3 0000000  I3 0 

0 0  II 00    33  0 0 II− 00 +  f o +  33  w . 0 I  ij()+1  00II ij  3   33 0 −I 00II−  3   33 Or, in matrix form,

=++++d ddd d ddddd fBfij n1()()ij−− 1 1 BfBf 12ij n 2 ()ij − 1 Bf 21 ()ij + 1 Bf 12ij ( + 1 ) (18.E.12) ++o ooo + B11 fij B 12 fij()+ 1 Wwij .

− −− − 000 II 000  II33 I 30 00II33   3 3    000− II 000 −II I 0 00II− ff=  3 3  d +  33 3 33 f d nj000 0 0 000 ()() n−−11 j  000 I −II0 I nj    3 33 3 − − 000 0 0 000  000 I3 III330 I 3

000000− I 0 00 000000  3    000000− I 0 00 000000 +  3  f d +   f d 000000 0− I ()nj−1 0II0− 00000 1j  3   33  − 000000 0 I3  0II033 0000

 0 0000000  I −I  0 0     3 3    0 0000000 −I I 0 0 +   f d +  3 3 f o +   f o  I 0000000 nj()+1 −I 0  nj 00 I  nj()+1  3   3   3  − −  I3 0000000  I3 0  0 I3 

II 00  33  II− 00 +  33  w .  00II nj  33 00II−  33

Or, in matrix form,

=++++d ddd d ddddd fBfnj n1()()nj−− 1 1 BfBf 12nj n 2 ()nj − 1 BfBf 21 1j 12nj () + 1 (18.E.13) ++o ooo + B11 fnj B 12 fnj()+ 1 Wwnj .

−−II I 0 00−II  00 000000  33 3 33   −II I 0 00II− 00 000000 f =  33 3 33 f d +   f d 1m  000 I −II 00 1m 0II0− 0000 2m  3 33   33  − − −  000 I3 II3300 0II033 0000

000 0 0 000 000− II 000    3 3  000 0 0 000 000− II 000 +  ffd +  3 3  d 000− I I 000 nm 000 0 0 000 nm()−1  3 3    − 000 I3 I3 000 000 0 0 000

 I −I  II 00  3 3   33  −II I II− 00 + 3 3  f o +  33  W . −I 0  1m  00II 1m  3   33 −  I3 0   00II33

Or, in matrix form,

=+++d ddd d d d do ++o (18.E.14) fBfBfBfBfBfWw111212mmm m nm nm n 1nm()−1 11 1m 1m.

− − 000 II 000 000 0 00 I03   3 3    000− II 000 000 0 00− I0 ff=  3 3  d +  3  f d im 000 0 0 000 ()im−11()− 000− I I000 ()im−1    3 3  − 000 0 0 000 000 I3 I0003 

−−II I 0 00−II  00 000000  33 3 33   −II I 0 00II− 00 000000 +  33 3 33 f d +   f d  000 I −II 00 1m 0II0− 00000 ()im+1  3 33   33  − − −  000 I3 II3300 0II033 0000

 I −I  II 00  3 3   33  −I I II− 00 +  3 3  f o +  33  w . −I 0  im  00II im  3   33 −  I3 0   00II33

Or, in matrix notation,

=+++d d d d d ddd fBfim n1 ()im−11()− BfBfBfnm ()im− 1 121m im ()im+1 (18.E.15) o o + + B11 fim Wwim.

− − 000 II 000 000 0 00 I03   3 3    000− II 000 000 0 00− I0 ff=  3 3  d +  3  f d nm000 0 0 000 ()() n−−11 m 000− I I000 ()nm−1    3 3  − 000 0 0 000 000 I3 I0003 

−−II I 0 00−II  00 000000  33 3 33   −II I 0 00II− 00 000000 +  33 3 33 f d +   f d  000 I −II 00 nm 0II0− 00000 1m  3 33   33  − − −  000 I3 II3300 0II033 0000

 I −I  II 00  3 3   33  −I I II− 00 +  3 3  f o +  33  w . −I 0  nm  00II nm  3   33 −  I3 0   00II33

Or, in matrix notation,

d d +++d d d ddd fBfBfBnm = n1()()nm−− 1 1nm 1 ()nm − 1 1m fnm Bf 21 1m (18.E.16) ++o o Bf11 nm Wwnm.

Now assemble (18.E.5)–(18.E.16) into the form

ddoo++++ fBfBfBfBfWw13142112211 = (18.E.17)

where

f d  f o  w  f   1j   1j   1j   1j    d o w f11n f2 j  f2 j   2 j  f2 j  f d =  ,,,f d = f o = w = ,f = 1  f d  j  M  j  M  j  M  j  M  1         f d  f o  w  ff   nj   nj   nj   nj 

BBB0dd L 0Bd   1n 1 11 21n 1 d OOO  0B01 0  MOOOOOM B =   3  00OOOO    0BOOOOd  21  dddL  BB1n 1 21 0 0BB 01 11 

=……dd BBB41212BlockDiag[,,,], =……od BBB21212BlockDiag[,,,],

Bo 00LL   11  o OO M B01  B =  0 OO O M , 1   MOO o  B11 0   L o o   00BB01n 1

W1 = BlockDiag […, W, W, …].

Also, from (18.E.5)–(18.E.16)

dddoo+++++ f=Bf2 5 2 Bf6 243722312 Bf Bf Bf Ww, (18.E.18) where B0,BB= []= BlockDiag[]LL BBoo 55,71111 ,,,,

BBdd00L Bd   00L 0Bd   12 21n 2   n1  Bd OOO 0  Bd OO 0  n2 n1  0 OOOO M B =  0 OO O M, B =  . 5   6  MOOOO0  MOOO  0   d    0 OO O B 00L Bd 0 21   n1   d L dd BBB2100n 2 12 

Also from (18.E.5)–(18.E.16)

=+++++dddoo f3 BfBfBfBfBfWw5 2 6 344732413, (18.E.19)

d +++d o f=Bfmm 5 ()−18 BfBfWw,m 7m 1m (18.E.20)

BB0dd L 0Bd   121m nm  OOO B0nm   0 OOOO M B =   8  MOOOO0    0BOO O d  21   d d d  B021 0BBnm 1m 

Or, simply, the vector form of (18.E.17)–(18.E.19) is

dd++ oo o f=Bf Bf Ww, (18.E.21)

where f d   f   1  f o   w   1  f d   11   f = M ,,,f d = 2 f o = MMw = ,    M       o  f    f  w  m f d  m m  m 

o = LL W BlockDiag [,,,], W11 W

L L  BB34 0 0 BB12 0 0  OO M  OO M BB5 6   0B7  Bdo=  0BOO 0,.B =  MOOO0   5    MOO MOO  BB6 4   B2   L   LL   00BB5 8   00B7 

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