Design and control of tensegrity morphing airfoils

Design and Control of Tensegrity Morphing Airfoils

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Design and Control of Tensegrity Morphing Airfoils Muhao Chen, Jiacheng Liu, Robert E. Skelton PII: DOI: Reference:

S0093-6413(20)30009-4 https://doi.org/10.1016/j.mechrescom.2020.103480 MRC 103480

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Mechanics Research Communications

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31 August 2019 13 January 2020 14 January 2020

Please cite this article as: Muhao Chen, Jiacheng Liu, Robert E. Skelton, Design and Control of Tensegrity Morphing Airfoils, Mechanics Research Communications (2020), doi: https://doi.org/10.1016/j.mechrescom.2020.103480

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Design and Control of Tensegrity Morphing Airfoils Muhao Chena,∗,1 , Jiacheng Liub,2 and Robert E. Skeltona,3 a Department b Department

of Aerospace Engineering, Texas A & M University, College Station, TX, USA of Ocean Engineering, Texas A & M University, College Station, TX, USA

ARTICLE INFO

ABSTRACT

Keywords: Tensegrity structures Morphing airfoil Nonlinear dynamics Nonlinear control Integrating structure and control design

We present a general approach of design, dynamics, and control for tensegrity morphing airfoils. Firstly, based on reduced order Class-𝑘 tensegrity dynamics, a shape control law for tensegrity systems is derived. Then, we develop a method for discretizing continuous airfoil curves based on shape accuracy. This method is compared with conventional methods (i.e. evenly spacing and cosine spacing methods). A tensegrity topology for shape controllable airfoil is proposed. A morphing tensegrity airfoil example is given to demonstrate successful shape control. This work paves a road towards integrating structure and control design, the principles developed here can also be used for 3D morphing airfoil design and control of various kinds of tensegrity structures.

integrates structure and control designs using a novel tensegrity morphing airfoil. The Wright Brothers made the first sustainable, controlTensegrity is a network of bars and strings, where bars lable, powered, heavier than air manned flight in 1903. The only take compression and strings take tension [34]. Perfundamental breakthrough was their invention of a threehaps biological systems provide the greatest evidence that axis control method for improving fly control stability of the tensegrity concepts yield the most efficient structures [34], wing box structure by morphing the shape of the wing during for example, the micromechanism of the spider fibers shows flight [21]. that the network structure has both tensile and compressive Morphing airfoil is gaining significant attention by remembers [33]. There are many advantages of tensegrity: searchers with the thriving prosperity of the aerospace indus1. All structural members are axially loaded, there is no try. Compared to a fixed-wing, a flexible profile is more admaterial bending. 2. All the structure members are unijustable to various flight conditions. The fundamental condirectionally loaded, so there is no reversal of load direction, cept of achieving certain aerodynamic performance by means and the uncertainties of the 1-dimensional material moveof changing the shape of a wing motivates us to solve the ment bring better stability margins. 3. The structural effollowing problems: find an efficient airfoil structure and ficiency in strength to mass is very high. 4. The tensegcontrol laws to adjust various flight regimes. The existing rity system is easy to integrate structure and control because morphing technologies (wing slats, flaps, spoiler, aileron, the dynamic models are more accurate. A structural memwinglet, and trims) can achieve some desired performance. ber can also serve as a sensor and actuator. The actuator And many researchers have pointed out the challenges of and sensor architecture can be easily optimized. One can this area. Sofla [36], Lachenal [13], Kuribayashi [12], and change shape/stiffness without changing stiffness/shape, and Liu [17] summarized shape morphing status and challenges, one can achieve minimal control energy (morphing from one mainly focusing on the direction of shape memory alloys equilibrium to another). The bar-string connection patterns (SMA), piezoelectric actuators (PZT), shape memory polyare like 3D fibers that allow engineers and artists to knit any mers (SMP), and stimulus-responsive polymers (SRP). Valasek structure that physics and their imagination allow. This new [39], Barbarino [1], and Reich [25] addressed important isdimension of engineering thought motivates engineers to resues on morphing aircraft, bio-inspiration, smart structures, think and study structures in a more fundamental way, for power requirements and smart actuators. Santer showed load- example, beam structures made of V-Expander cells [5, 6], , path-based topology optimization for adaptive wing structhe mechanical response of 3D tensegrity lattices [27], high tures [30, 31]. However, few of them start with a system performance robotics [2, 3, 23], minimum mass bridge [32], point of view to solve the structure and control problem in deployable antenna [38], landers [22, 24, 26, 37], tunable enan integrative manner. This study presents our system which ergy dissipation structure [40], and tensegrity spine[28, 29], ∗ Corresponding author. Tel.: +1 9799858285. etc. [email protected] (M. Chen); [email protected] (J. Liu); The conceptual design and physical model of the [email protected] (R.E. Skelton) rity wing are presented in the book Tensegrity System by ORCID (s): 0000-0003-1812-6835 (M. Chen); 0000-0002-6400-3759 (J. Skelton and Mauricio 2009, which was a DARPA sponsored Liu); 0000-0001-6503-9115 (R.E. Skelton) 1 Graduate Student, Department of Aerospace Engineering, Texas A “smart structures” research program. The design in their and & M University, College Station, TX, USA. book is a light weight fixed wing, composed of 2D airfoil 2 Graduate Student, Department of Ocean Engineering, Texas A& M solid pieces connected with tensegrity T-Bar topology [34]. University, College Station, TX, USA. Henrickson et al. [9] presents a 2D cross-bar topology air3 TEES Eminent Professor, Department of Aerospace Engineering, foil (a class-1 tensegrity structure) design and demonstrate Texas A&M University, College Station, TX, USA.

1. Introduction

M. Chen, J. Liu, and R.E. Skelton: Preprint submitted to Mechanics Research Communications

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the morphing ability. These design shows the tensegrity system brings more functionalities to wing design such as minimum mass, deployability, and shape control. In a similar way to a bird’s wing, the strings in the tensegrity wing are functioning as muscles to warp the whole wing to achieve various aerodynamic performance requirements. Some researchers have discussed their ideas in morphing wing design, for example, Moored studies the deflection [18] and shape optimization [19] of the wing in span-wise direction by tensegrity beams and plates, Jones shows his idea in fuzzy control strategy for morphing [11]. James presented a control theory to formulate computationally feasible procedures for aerodynamic design [10]. However, we argue that control design after wing design destroys the flying efficiency that is so carefully treated in the structure design in the first place [34]. Instead of using control systems to push the structure away from its equilibrium, we propose to simply modify the equilibrium of tensegrity structures to achieve the new desired shape with little control effort. As feedback, less control power also exerts less stress on structural components to accomplish the same objectives. Thus, the best performance cannot be achieved by separating structure and control designs. This paper starts from airfoil shape discretization methods, presents a practical airfoil topology design, and a nonlinear control law for any Class-𝑘 structures. This paper is structured as follows: Section 2 introduces tensegrity principles, nonlinear Class-𝑘 dynamics, reducedorder form, and nonlinear shape control laws. Section 3 discusses the error bound method for continuous airfoil shape discretization and the design of tensegrity airfoils. Section 4 gives a case study and results. Section 5 presents the conclusions.

2.1. Class-𝑘 Tensegrity Dynamics

The accurate quantitative knowledge of structural behavior should be given in a simple, compact form. Tensegrity dynamics were first analytically studied by Motro, Najari, and Jouanna in 1987 [20], since then many kinds of research followed. Skelton et al. (2001) introduced a non-minimal coordinates method without using the conventional angular velocities for rigid bodies [35] simplified the math a lot. Recent work by Goyal et al. give a complete description of tensegrity dynamics by including string mass, class-𝑘 barlength correction, and analytic solutions of Lagrange multiplier Ω [8]:

𝑠

𝑛𝑏 𝑏

𝑠

(1) (2) (3)

where 𝜆̂ is: ̇ − 1 𝑙̂−2 ⌊𝐵 𝑇 (𝑊 + Ω𝑃 𝑇 𝜆̂ = − 𝐽̂𝑙̂−2 ⌊𝐵̇ 𝑇 𝐵⌋ 2 𝑇 𝑇 − 𝑆 𝛾̂ 𝐶𝑠 )𝐶𝑛𝑏 𝐶𝑏 ⌋,

𝑚𝑏𝑖 𝑟2𝑏𝑖 𝑚𝑏𝑖 + , 12 4𝑙𝑖2 𝑖𝑡ℎ bar.

inertia matrix, which satisfies 𝐽𝑖 =

2. Tensegrity Dynamics and Control

̈ 𝑠 + 𝑁𝐾𝑠 = 𝑊 + Ω𝑃 𝑇 , 𝑁𝑀 [ 𝑇 𝑇 ] 𝑇𝑚 (𝐶𝑏 𝐽̂𝐶𝑏 + 𝐶𝑟𝑇 𝑚̂ 𝑏 𝐶𝑟 ) 𝐶𝑛𝑠 ̂𝑠 , 𝑀𝑠 = 𝐶𝑛𝑏 [ ] ̂ 𝑏 𝐶 𝑇 𝛾̂ 𝐶𝑠𝑠 , 𝐾𝑠 = 𝐶 𝑇 𝛾̂ 𝐶𝑠𝑏 − 𝐶 𝑇 𝐶 𝑇 𝜆𝐶

and the operator ⌊∙⌋ sets every off-diagonal element of the square matrix operand to zero, 𝑁 ∈ 𝑅3×𝑛 (𝑛 is the total number of nodes) is the nodal matrix with each column denotes 𝑥, 𝑦, and 𝑧 coordinates of each node, 𝑀𝑠 ∈ 𝑅𝑛×𝑛 is mass matrix of all the bars and strings, 𝐾𝑠 ∈ 𝑅𝑛×𝑛 is the stiffness matrix, 𝑊 ∈ 𝑅3×𝑛 contains the external force at each node, Ω ∈ 𝑅3×𝑐 (𝑐 is number of constraints 𝑁𝑃 = 𝐷), the Lagrange multipliers required to maintain these constraints can be thought of as contact forces at the Class-𝑘 nodes [4]. Ω is the matrix of Lagrange multipliers associated with the constraint 𝑁𝑃 = 𝐷, 𝑃 ∈ 𝑅𝑐×𝑛 is the constraint matrix, denoting which nodes are [ the Class-𝑘 nodes ] and which nodes are grounded. 𝐵 = 𝒃1 𝒃2 ⋯ 𝒃𝛽 ∈ 𝑅3×𝛽 and 𝑆 = [ ] 𝒔1 𝒔2 ⋯ 𝒔𝛼 ∈ 𝑅3×𝜎 (𝒃𝑖 and 𝒔𝑖 are bar and string vectors, 𝛽 and 𝜎 represent the number of bars and number of string) are bar and string matrices whose columns are bar or string vectors, 𝐶𝑏 and 𝐶𝑠 are the connectivity matrix of bars and strings (consists of a "−1" at the 𝑖𝑡ℎ column, a "+1" at the 𝑗 𝑡ℎ column, and zeros elsewhere to define a structure member connecting from 𝒏𝑖 to 𝒏𝑗 .), they satisfy 𝐵 = 𝑁𝐶𝑏𝑇 and 𝑆 = 𝑁𝐶𝑠𝑇 . The nodes have two types: bar nodes 𝑁𝑏 ∈ 𝑅3×2𝛽 , which are the endpoints of bars, and string nodes 𝑁𝑠 ∈ 𝑅3×𝜎 , which are the locations of stringto-string connections that [ ] have a point mass associated with them, 𝑁 = 𝑁𝑏 𝑁𝑠 . Then, the bar and string nodes can be extracted from the node matrix 𝑁 with the definition of two connectivity matrices, 𝐶𝑛𝑏 and 𝐶𝑛𝑠 . 𝐶𝑠 is divided into two parts: the first, 𝐶𝑠𝑏 , describing bar-to-string joints and the second, 𝐶𝑠𝑠 , describing string-to-string joints. ∙̂ is an operator that converts a vector into a diagonal matrix. 𝑚̂ 𝑏 , 𝑚̂ 𝑠 , 𝛾̂ , 𝜆̂ are bar mass, string mass, string force density, and bar force density matrices respectively. 𝐽̂ is the bar moment of

(4)

and 𝑟𝑏𝑖 and

𝑙𝑖 are the radius and length of the Adding the linear constraints to the dynamics will restrict the motion in certain dimensions. We separate the moving and stationary nodes by performing a Singular Value Decomposition of the matrix 𝑃 to eliminate unnecessary computations of stationary nodes, the order of dynamics equation can be reduced into [8]: ̃, 𝜂̈2 𝑀2 + 𝜂2 𝐾2 = 𝑊

(5)

where 𝜂 = [𝜂1 𝜂2 ] ≜ 𝑁𝑈 = [𝑁𝑈1 𝑁𝑈2 ], 𝑀2 = ̃ = 𝑊 𝑈2 − 𝜂 1 𝑈 𝑇 𝐾 𝑠 𝑈 2 , 𝜂 1 = 𝑈2𝑇 𝑀𝑠 𝑈2 , 𝐾2 = 𝑈2𝑇 𝐾𝑠 𝑈2 , 𝑊 [1 ] [ ] Σ1 −1 𝑇 𝐷𝑉 Σ1 , 𝑁𝑃 = 𝐷, 𝑃 = 𝑈 Σ𝑉 = 𝑈1 𝑈2 𝑉 𝑇. 0

2.2. Coordinate Transformation 2.2.1. X Matrix Definition

From the above, the reduced order dynamics can be written in a standard second order differential equation form. Let 𝑋 matrix be the product of the 𝜂2 and 𝑀2 matrices. Since 𝑀2 is the mass matrix for the tensegrity system and it is non-singular, an expression for 𝜂2 in terms of 𝑋 can then be written as: 𝑋 = 𝜂2 𝑀2 → 𝜂2 = 𝑋𝑀2−1

M. Chen, J. Liu, and R.E. Skelton: Preprint submitted to Mechanics Research Communications

(6)

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(7)

̇ −1 𝑋̇ = 𝜂̇2 𝑀2 → 𝜂̇2 = 𝑋𝑀 2 −1 ̈ ̈ 𝑋 = 𝜂̈2 𝑀2 → 𝜂̈2 = 𝑋𝑀 .

(8)

2

2.2.2. Conversion of Matrix Dynamics Substitute equations (6) - (8) into equation (1), one can get: (9)

̃. 𝑋̈ + 𝑋𝑀2−1 𝐾2 = 𝑊

Take the 𝑖𝑡ℎ element of the first and second terms of the ̂ = 𝒚𝒙, ̂ where 𝒙 and 𝒚 are equation (4) and use the fact 𝒙𝒚 vectors, equation (4) can be written as: − 𝜆𝑖 =

𝐽𝑖 ||𝒃̇ 𝑖 ||2 𝑙𝑖2

+

1 𝑇 𝑇 𝑇 𝒃𝑖 𝑊 𝐶𝑛𝑏 𝐶𝑏 𝒆𝑖 + 2𝑙𝑖2

1 𝑇 1 𝑇 𝑇 𝑇 𝑇 𝐶𝑇 𝒆 𝒃𝑖 Ω𝑃 𝑇 𝐶𝑛𝑏 𝐶𝑏 𝒆𝑖 − 𝒃𝑖 𝑆 𝐶𝑠 𝐶𝑛𝑏 𝛾, 𝑏 𝑖 2𝑙𝑖2 2𝑙𝑖2 ⋀

(10)

where 𝒃𝒊 is the vector of each bar, 𝒆𝑖 is a vector with 1 in the 𝑖𝑡ℎ elements and zeros else where. Stack all the elements of vector 𝜆 we get:

(13)

𝑌𝑐 = 𝐿𝑁𝑈2 𝑅 = 𝐿𝜂2 𝑅 = 𝐿𝑋𝑀2−1 𝑅.

𝑌̄ is a matrix containing the desired values for the node coordinates of interest. The error matrix 𝐸 between the current and desired node coordinates of interest can then be written: (14)

𝐸 = 𝑌𝑐 − 𝑌̄ .

2.3.2. Error Dynamics To achieve the desired shape control, the error matrix 𝐸 and its first and second-time derivatives should all go to zero. This goal is expressed as follows. Let Ψ and Φ be chosen matrices such that: (15)

𝐸̈ + Ψ𝐸̇ + Φ𝐸 = 0

is a stable equation about the value 𝐸 = 0. Using equations (9), (13), and (14), the equation (15) becomes: ̃ − 𝑋𝑀 −1 𝐾2 )𝑀 −1 𝑅 + Ψ𝐿𝑋𝑀 ̇ −1 𝑅+ 𝐿(𝑊 2 2 2 Φ𝐿𝑋𝑀 −1 𝑅 − Φ𝑌̄ = 0.

(16)

2

This can be expanded and rearranged: ⎡ 𝐽1 ||𝒃̇ 1 ||2 + 1 𝒃𝑇 𝑊 𝐶 𝖳 𝐶 𝑇 𝒆 + 1 𝒃𝑇 Ω𝑃 𝖳 𝐶 𝖳 𝐶 𝑇 𝒆 ⎤ ̃ 𝑀 −1 𝑅 𝑛𝑏 𝑏 1 𝑛𝑏 𝑏 1 ⎥ 𝐿𝑋𝑀2−1 𝐾2 𝑀2−1 𝑅 = 𝐿𝑊 ⎢ 𝑙12 2𝑙12 1 2𝑙12 1 2 ⎥ ⎢ ⋮ ̇ −1 𝑅 + Φ𝐿𝑋𝑀 −1 𝑅 − Φ𝑌̄ +Ψ𝐿𝑋𝑀 (17) ⎥ ⎢ 𝐽 ||𝒃̇ ||2 2 2 1 𝑇 1 𝑇 𝖳 𝑇 𝑖 𝑖 𝖳 𝖳 𝑇 −𝜆 = ⎢ 𝑙2 + 2𝑙2 𝒃𝑖 𝑊 𝐶𝑛𝑏 𝐶𝑏 𝒆𝑖 + 2𝑙2 𝒃𝑖 Ω𝑃 𝐶𝑛𝑏 𝐶𝑏 𝒆𝑖 ⎥ ⎥ Substitute 𝐾2 = 𝑈 𝑇 𝐾𝑠 𝑈2 and 𝐾𝑠 from equation (3) into ⎢ 𝑖 𝑖 𝑖 2 ⋮ ⎥ ⎢ the ⎥ left hand side of this equation: ⎢ 𝐽𝛽 ||𝒃̇ 𝛽 ||2 1 𝑇 1 𝑇 𝖳 𝑇 𝖳 𝑇 𝖳 + 2 𝒃𝛽 𝑊 𝐶𝑛𝑏 𝐶𝑏 𝒆𝛽 + 2 𝒃𝛽 Ω𝑃 𝐶𝑛𝑏 𝐶𝑏 𝒆𝛽 ⎥ ⎢ 𝑙2 2𝑙𝛽 2𝑙𝛽 ⎦ ⎣ 𝛽 𝐿𝑋𝑀2−1 𝐾2 𝑀2−1 𝑅 = 𝐿𝑋𝑀2−1 𝑈2𝑇 𝐾𝑠 𝑈2 𝑀2−1 𝑅 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ [ ] 𝖳 𝐶 𝖳 𝜆𝐶 ̂ 𝑏 𝐶 𝖳 𝛾̂ 𝐶𝑠𝑠 𝑈2 𝑀 −1 𝑅. = 𝐿𝑋𝑀2−1 𝑈2𝑇 𝐶𝑠𝖳 𝛾̂ 𝐶𝑠𝑏 − 𝐶𝑛𝑏 𝜏 𝑠 2 𝑏 (18) ⎡ ⎤ ⎢ 1 𝒃𝑇 𝑆 𝐶 𝐶 𝖳 𝐶 𝑇 𝒆 ⎥ 𝑠 𝑛𝑏 𝑏 1 ⎥ Let us take a look at the mass matrix again, ⎢ 2𝑙12 1 ⎢ ⎥ ⎢ ⎥ 𝑀2 = 𝑈2𝖳 𝑀𝑠 𝑈2 (19) ⋮ ⎢ ⎥ ⎢ 𝖳 𝐶𝑇 𝒆 ⎥ [ 𝖳 𝖳 ] − ⎢ 12 𝒃𝑇𝑖 𝑆 𝐶𝑠 𝐶𝑛𝑏 𝛾. (11) 𝖳𝑚 𝑏 𝑖⎥ 2𝑙 ̂𝑠 (𝐶𝑏 𝐽̂𝐶𝑏 + 𝐶𝑟𝖳 𝑚̂ 𝑏 𝐶𝑟 ) 𝐶𝑛𝑠 𝑀𝑠 = 𝐶𝑛𝑏 ⎢ 𝑖 ⎥ ⎢ ⎥ [ 𝑇 ] ⎡𝐽̂ 0 0 ⎤ ⎡𝐶𝑏 0 ⎤ ⋮ 𝐶𝑏 𝐶𝑟𝑇 0 ⎢ ⎢ ⎥ = 0 𝑚 ̂ 0 ⎥ ⎢ 𝐶𝑟 0 ⎥ 𝑏 ⎥ ⎢ 0 0 𝐼 ⎢ ⎥⎢ ⎥ 1 𝑇 𝖳 𝑇 ⎢ 2 𝒃𝛽 𝑆 𝐶 𝑠 𝐶 𝐶 𝒆 𝛽 ⎥ ⎣ 0 0 𝑚̂ 𝑠 ⎦ ⎣ 0 𝐼 ⎦ 𝑛𝑏 𝑏 ⎢ 2𝑙𝛽 ⎥ ⎣ ⎦ ]−1 [ [ ]𝖳 ⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟ Using 12 𝐶𝑏𝖳 2𝐶𝑟𝖳 = 𝐶𝑏𝖳 𝐶𝑟𝖳 , it can also be Λ shown that: We can express in a compact matrix form, ⋀

⋀

⋀

−𝜆 = Λ𝛾 + 𝜏.

2.3. Shape Control Law 2.3.1. Shape Objectives

(12)

𝐿 specifies the axes of interest for the system nodes, 𝑅 denotes which of those system nodes are nodes of interest. By multiplying 𝑌𝑐 = 𝐿𝜂2 𝑅, one can, therefore, extract the current values of the node coordinates of interest for the node configuration 𝑁:

[1 2

𝐶𝑏𝖳 0

2𝐶𝑟𝖳 0

0 𝐼

]−1

⎡𝐶𝑏 = ⎢𝐶𝑟 ⎢ ⎣0

0⎤ 0⎥ . ⎥ 𝐼⎦

(20)

Then, 𝑀𝑠−1 can be obtained, 𝑀𝑠−1

[1

𝐶𝖳 = 2 𝑏 0

2𝐶𝑟𝖳 0

M. Chen, J. Liu, and R.E. Skelton: Preprint submitted to Mechanics Research Communications

] ⎡𝐽̂ 0 ⎢ 0 𝐼 ⎢0 ⎣

0 𝑚̂ 𝑏 0

−1

0⎤ 0⎥ ⎥ 𝑚̂ 𝑠 ⎦

⎡ 12 𝐶𝑏 ⎢ 2𝐶 ⎢ 𝑟 ⎣ 0

0⎤ 0⎥ ⎥ 𝐼⎦

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[1

𝐶𝖳 = 2 𝑏 0

2𝐶𝑟𝖳 0

[1

] ⎡𝐽̂ 0 ⎢ 0 𝐼 ⎢0 ⎣

0 𝑚̂ 𝑏 0

0⎤ 0⎥ ⎥ 𝑚̂ 𝑠 ⎦

−1

⎡ 21 𝐶𝑏 𝐶𝑛𝑏 ⎤ ⎢ 2𝐶 𝐶 ⎥ ⎢ 𝑟 𝑛𝑏 ⎥ ⎣ 𝐶𝑛𝑠 ⎦

] 𝖳 𝐽̂ −1 𝐶 𝐶 + 4𝐶 𝖳 𝑚 −1 𝐶 𝐶 𝐶 ̂ 𝑏 𝑛𝑏 𝑟 𝑛𝑏 𝑟 𝑏 𝑏 = 4 . 𝑚̂ −1 𝑠 𝐶𝑛𝑠

(21)

𝐶𝑟 𝐶𝑛𝑏 and 𝑀𝑠2 = Let 𝑀𝑠1 = 14 𝐶𝑏𝖳 𝐽̂−1 𝐶𝑏 𝐶𝑛𝑏 + 4𝐶𝑟𝖳 𝑚̂ −1 𝑏 [ ] 𝑀𝑠1 −1 𝑚̂ 𝑠 𝐶𝑛𝑠 , 𝑀𝑠 can be simply written as 𝑀𝑠 = . Then, 𝑀𝑠2 [ ] 𝑀𝑠1 𝑀2−1 = 𝑈2𝑇 𝑈 . 𝑀𝑠2 2 Equation (18) can be written as: 𝐿𝑋𝑀2−1 𝐾2 𝑀2−1 𝑅 = 𝐿𝑋𝑀2−1 𝑈2𝑇 (𝐶𝑠𝖳 𝛾̂ 𝐶𝑠𝑏 𝑀𝑠1 𝖳 𝖳̂ −𝐶𝑛𝑏 𝐶𝑏 𝜆𝐶𝑏 𝑀𝑠1 + 𝐶𝑠𝖳 𝛾̂ 𝐶𝑠𝑠 𝑀𝑠2 )𝑈2 𝑅. (22)

Figure 1: Close loop system, where 𝑢 is control input and 𝑢𝑖 is the rest length of the 𝑖𝑡ℎ string, given by equation (30).

⋀

⋀

Γ𝑖 = 𝐿𝑋𝑀2−1 𝑈2𝑇 (𝐶𝑠𝑇 𝐶𝑠𝑏 𝑀𝑠1 𝑈2 𝑅𝒆𝑖 + 𝐶𝑠𝑇 𝐶𝑠𝑠 𝑀𝑠2 𝑈2 𝑅𝒆𝑖 ⋀

(28)

𝖳 𝖳 + 𝐶𝑛𝑏 𝐶𝑏 𝐶𝑏 𝑀𝑠1 𝑈2 𝑅𝒆𝑖 Λ).

Take the 𝑖𝑡ℎ column on both sides,

Since control variable 𝛾 is composed of force densities of 𝐿𝑋𝑀2−1 𝐾2 𝑀2−1 𝑅𝒆𝑖 = 𝐿𝑋𝑀2−1 𝑈2𝑇 (𝐶𝑠𝖳 𝛾̂ 𝐶𝑠𝑏 𝑀𝑠1 𝑈2 𝑅𝒆𝑖 the strings, it also satisfies 𝛾 ≥ 0 (strings are always in ten𝖳 𝖳̂ sion), the least square problem we choose to solve at each 𝐶𝑏 𝜆𝐶𝑏 𝑀𝑠1 𝑈2 𝑅𝒆𝑖 + 𝐶𝑠𝖳 𝛾̂ 𝐶𝑠𝑠 𝑀𝑠2 𝑈2 𝑅𝒆𝑖 ). (23) − 𝐶𝑛𝑏 increment of real time Δ𝑡, is min ||𝜇 − Γ𝛾||2 , 𝛾 ≥ 0. Let ̂ = 𝒚𝒙, ̂ we have: Using the fact 𝒙𝒚 the rest length of the 𝑖𝑡ℎ string be denoted by 𝑠𝑖0 , extensional stiffness by 𝑘𝑖 , damping constant by 𝑐𝑖 , and string vector by 𝒔𝑖 . Assuming that strings follow Hooke’s law and the viscous friction damping model, the tension in a string is: −1 −1 −1 𝑇 𝑇 𝐿𝑋𝑀2 𝐾2 𝑀2 𝑅𝒆𝑖 = 𝐿𝑋𝑀2 𝑈2 (𝐶𝑠 𝐶𝑠𝑏 𝑀𝑠1 𝑈2 𝑅𝒆𝑖 𝛾 ⋀

⋀

⋀

𝖳 𝖳 + 𝐶𝑠𝑇 𝐶𝑠𝑠 𝑀𝑠2 𝑈2 𝑅𝒆𝑖 𝛾 − 𝐶𝑛𝑏 𝐶𝑏 𝐶𝑏 𝑀𝑠1 𝑈2 𝑅𝒆𝑖 𝜆).

(24) Recalling that −𝜆 = Λ𝛾 + 𝜏,

if if

||𝒔𝑖 || ≥ 𝒔𝑖0

||𝒔𝑖 || < 𝒔𝑖0 (29)

Then, we have rest length 𝒔𝑖0 : ⋀

𝐿𝑋𝑀2−1 𝐾2 𝑀2−1 𝑅𝒆𝑖 = 𝐿𝑋𝑀2−1 𝑈2𝑇 [(𝐶𝑠𝑇 𝐶𝑠𝑏 𝑀𝑠1 𝑈2 𝑅𝒆𝑖 ⋀

⎧ 𝒔𝑇 𝒔̇ 𝑖 𝒔 ||𝒕𝑖 || ⎪ 𝑘𝑖 (1 − 𝑖0 ) + 𝑐𝑖 𝑖 ||𝒔𝑖 || 𝛾𝑖 = = ||𝒔𝑖 ||2 ||𝒔𝑖 || ⎨ ⎪0 ⎩

⋀

𝖳 𝖳 + 𝐶𝑠𝑇 𝐶𝑠𝑠 𝑀𝑠2 𝑈2 𝑅𝒆𝑖 + 𝐶𝑛𝑏 𝐶𝑏 𝐶𝑏 𝑀𝑠1 𝑈2 𝑅𝒆𝑖 Λ)𝛾 ⋀

𝖳 𝖳 + 𝐶𝑛𝑏 𝐶𝑏 𝐶𝑏 𝑀𝑠1 𝑈2 𝑅𝒆𝑖 𝜏].

𝒔𝑇 𝒔̇ 𝑖 1 (𝛾𝑖 − 𝑐𝑖 𝑖 )]. 𝑘𝑖 ||𝒔𝑖 ||2

(30)

We can also write the string tension (a product of string length and string force density) in a matrix form: 𝑇 ̇ 𝑇 = 𝑆 𝛾̂ = (𝑆 − 𝑆0 )𝑘̂ + 𝑆⌊𝑆 𝑇 𝑆⌋⌊𝑆 𝑆⌋−1 𝑐, ̂

(25)

2.3.3. Control Law From equations (17) and (25), one can simplify this into a compact matrix form 𝜇 = Γ𝛾 with definitions of 𝜇𝑖 and Γ𝑖 , in which 𝜇 is the stack of each 𝜇𝑖 matrix and Γ is similarly a stack of each Γ𝑖 matrix. (26)

𝜇 = Γ𝛾

𝒔𝑖0 = ||𝒔𝑖 ||[1 −

(31)

where 𝑆0 = 𝑆⌊𝑆 𝑇 𝑆⌋− 2 𝒔̂ 0 represents the matrix containing the rest length vectors. The overall control system is shown in Figure 1. 1

3. Tensegrity Airfoil Design

This section focuses on an error bound method for the discretization of continuous airfoils and a representation of the tensegrity airfoil topology based on matrix notations.

3.1. Error Bound Method

̃ 𝑀 −1 𝑅 + Ψ𝐿𝑋𝑀 ̇ −1 𝑅 + Φ𝐿𝑋𝑀 −1 𝑅 − Φ𝑌̄ )𝒆𝑖 Let us assume the cord length of an airfoil is 1, the num𝜇𝑖 = (𝐿𝑊 2 2 2 ber of discrete points is 𝑝. There are two widely used spacing methods (evenly and cosine spacing) for discretizing an 𝖳 𝖳 (27) − 𝐿𝑋𝑀2−1 𝑈2𝑇 𝐶𝑛𝑏 𝐶𝑏 𝐶𝑏 𝑀𝑠1 𝑈2 𝑅𝒆𝑖 𝜏 airfoil in the computational fluid dynamics (CFD) field [14, ⋀

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Figure 2: Illustration of error bound spacing method by NACA0012 with error bound 𝛿 = 0.008 m.

Algorithm 1: Error Bound Spacing Algorithm 1) Let 𝑦 = 𝑓 (𝑥), 𝑥 ∈ [0, 1] be the function of a continuous airfoil shape, an error bound value be 𝛿. 2) Let 𝑥0 ∈ 𝑥 be the start point of discretization, then 𝑥 ∈ [0, 𝑥0 ] is the continuous part (called the “D-Section”) and 𝑥 ∈ [𝑥0 , 1] is the discrete part. 3) Let (𝑥2 , 𝑦2 ) be the next discrete point, the line function of segment (𝑥0 , 𝑦0 ) and (𝑥2 , 𝑦2 ) is: (32)

𝐴𝑥 + 𝐵𝑦 + 𝐶 = 0, where 𝐴 = 𝑦2 − 𝑦0 , 𝐵 = −1, and 𝐶 = 𝑦0 − 𝐴𝑥0 . 4)The point (𝑥1 , 𝑦1 ), 𝑥1 ∈ (𝑥0 , 𝑥2 ) has the largest distance with the line segment: 𝑑=

|𝐴𝑥1 + 𝐵𝑦1 + 𝐶| . √ 𝐴2 + 𝐵 2

(33)

5) Obtain all the discrete points: while 𝑥 ≤ 1 do Figure 3: Comparison of cosine spacing, evenly spacing, and error bound spacing methods by NACA0012 with same amount of discrete points.

15]. The definition of evenly and cosine spacing is the xcoordinates of the discrete points on the airfoil satisfy a linear function 𝑥𝑖 = 𝑝𝑖 , (𝑖 = 1, 2, 3, . . . , 𝑝) or a cosine function

= 1, 2, 3, . . . , 𝑝). However, these 𝑥𝑖 = 0.5[1 − two methods could not quantify the shape accuracy of the discretized shape compared to a continuous shape. In other words, one cannot specify how big the shape error is merely by the control points one uses. It might not bother much when one is allowed to have a sufficient number of discrete points, but when it comes to describing an airfoil with limited points, it reveals the importance to obtain a better discrete shape. In consequence, this paper proposes an error bound spacing method, see Figure 2, which discretizes an airfoil and provides a quantitative representation of airfoil shape accuracy. This may improve the performance prediction of airfoil designs. Error Bound Method: Given the exact airfoil shape, approximate the shape with straight-line segments. Choose the location of the nodes of the straight-line segments such that the maximum error between the defined shape and each straight-line segment is less than a specified value 𝛿. Following the definition, an algorithm can be formulated to obtain the coordinates of the discrete points, shown in Algorithm 1. Let an example to illustrate the advantage of the error bound method comparing with evenly and cosine spacing ones, shown in Figure 3. It is clear that the error bound method has better accuracy for the same number of discrete points, and the error bound 𝛿 also provides a quantitative sense of the accuracy of the discrete airfoil.

⎧ ′ 𝑦2 −𝑦0 ⎪𝑓 (𝑥)|𝑥=𝑥1 = 𝑥2 −𝑥0 , solve for 𝑥2 ⎨ ||𝐴𝑥√1 +𝐵𝑦1 +𝐶|| =𝛿 ⎪ 𝐴2 +𝐵 2 ⎩ Store (𝑥2 , 𝑦2 ) and update 𝑥0 ← 𝑥2 . end while

𝑐𝑜𝑠( 𝜋𝑝 𝑖)], (𝑖

Figure 4: Tensegrity topology of airfoil NACA2412 with shape error bound 𝛿 = 0.001 m, along the chord 0−0.4 m (the shaded blue area) is ‘Rigid Structure’, and 0.4 − 1 m (the black and red lines) is ‘Flexible Structure’.

3.2. Topology of Tensegrity Airfoil

Having located the nodes of the straight-line approximation of the desired shape, we now must show the interior tensegrity structure of the airfoil. Inspired by vertebrae, we connect the discrete points in a similar pattern shown in Figure 4, where the black and lines represent rigid members (bars) and tendons (strings). The D-section, also called Dbox, is a structure in a letter D form in the front of the airfoil widely used in wing structure construction. This topology design is intuitive, compact, but effective for shape control ability. We define a tensegrity airfoil of complexity 𝑞 as the number of groups of vertical bars. Examples of an asymmetric and asymmetric airfoil with high complexity are shown in Figure 5.

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Figure 5: Tensegrity airfoil topology, error bound 𝛿 = 1.0 × 10−4 m, along the chord 0 − 0.2 m(the shaded blue area is the D-Section), NACA0012 (structure complexity 𝑞 = 21) and NACA2412 (structure complexity 𝑞 = 22).

3.3. General Modeling of Tensegrity Airfoil

To generate a tensegrity airfoil with any complexity 𝑞, the first thing is to define nodal, bar connectivity, and string connectivity matrices: 𝑁, 𝐶𝑏 , and 𝐶𝑠 . The nodal matrix 𝑁 contains the vectors of all the nodes and labeling bars and strings as shown in Figure 6, one can write 𝐶𝑏 and 𝐶𝑠 by starting with each unit coordinates. 𝐶𝑏𝑖𝑛 and 𝐶𝑠𝑖𝑛 whose two elements in each row denotes the start and end node of one bar or string: 𝐶𝑏𝑖𝑛 =

𝐶𝑠𝑖𝑛 =

⎧[𝑖, 𝑖 + 1], 1 ≤ 𝑖 ≤ 𝑞 ⎪ ⎨[𝑖 − 𝑞, 𝑖 + 1], 𝑞 + 1 ≤ 𝑖 ≤ 2𝑞 ⎪[𝑖 − 2𝑞, 𝑖 + 1], 2𝑞 + 1 ≤ 𝑖 ≤ 3𝑞 ⎩

,

(34)

⎧[𝑖 + 1 + 𝑞, 𝑖 + 2 + 𝑞], 1 ≤ 𝑖 ≤ 𝑞 − 1 ⎪[2𝑞 + 1, 𝑞 + 1] ⎪ ⎪[𝑖 − 𝑞, 𝑖 + 2], 𝑞 + 1 ≤ 𝑖 ≤ 2𝑞 − 1 (35) . ⎨ ⎪[𝑖 − 2𝑞 + 1, 𝑖 + 3], 2𝑞 ≤ 𝑖 ≤ 3𝑞 − 2 ⎪[𝑖 + 3 − 𝑞, 𝑖 + 4 − 𝑞], 3𝑞 − 1 ≤ 𝑖 ≤ 4𝑞 − 3 ⎪ ⎩[3𝑞 + 1, 𝑞 + 1]

Then a function can be written to convert 𝐶𝑏𝑖𝑛 and 𝐶𝑠𝑖𝑛 to 𝐶𝑏 and 𝐶𝑠 [7].

4. Numerical Study of a Morphing Airfoil

The primary goal of describing the kinematics of a flexible foil (i.e. fish swimming and wing flapping) is helping researchers study the fundamental motion mechanism and recreate the similar efficient flow types. In this study, we implement NACA0012 (D-Section 0 − 0.05 m, error bound 𝛿 = 0.002 m) to evaluate the performance of the controller. The control objective is to transform the airfoil from one shape to another, shown in the upper part of Figure 7. The morphing targets described in [16], which are: { 0, 0 ≤ 𝑥 ≤ 𝑐𝑟𝑔𝑑 𝑥−𝑐 𝑦(𝑥, 𝑡) = 𝐴0 ( 𝑐−𝑐 𝑟𝑔𝑑 )2 𝑠𝑖𝑛( 𝜋2 𝑡), 𝑐𝑟𝑔𝑑 ≤ 𝑥 ≤ 𝑐 , (36)

Figure 6: Node, bar, and string notations of a tensegrity airfoil with complexity 𝑞.

where 𝐴0 is the maximum amplitude achieved at the trailing edge of the airfoil, 𝑐 is the cord length, the rigid part is 𝑐𝑟𝑔𝑑 (𝑐𝑓 𝑙𝑒𝑥 = 𝑐−𝑐𝑟𝑔𝑑 ), 𝐴 is the amplitude envelop, 𝑥 is coordinate along the cord, and 𝑡 = [0, 1] s (tail bends maximum at one forth of a period). Then, based on equation (36), coordinates of control targets 𝑌̄ can be calculated. In the control simulation, amplitude 𝐴0 = −0.04 m, time step 𝑑𝑡 = 0.001𝑠, stability coefficients Ψ = 2.5 and Φ = 4, mass of the longest bar and string are 1 kg and 0.01 kg, and the masses of the shorter bars and strings are scaled accordingly. Nodes 𝑛1 , 𝑛𝑞+2 , and 𝑛2𝑞+2 in Figure 6 are fixed with the D-Section, in our case structure complexity 𝑞 = 5, 𝑛1 , 𝑛7 , and 𝑛12 are not controlled. Since we are controlling 𝑥 and 𝑦 coordinates, matrix 𝐿 = [1 0 0; 0 1 0; 0 0 0], matrix 𝑅16×13 (generated by 𝐼 16×16 with columns 1, 7 and 12 being deleted). The time-lapse in Figure 7 shows that the desired shape transformation is achieved in 20 seconds. For the above simulation, string tensions drive the desired shape transformation. To illustrate this, force densities (𝛾) time history is shown in Figure 8. Figure 9 shows the distance error between the current node positions to the target respect to time, which demonstrates the successful shape control at a final time.

5. Conclusion

This paper offers an approach to integrate structure and control design. Based on the non-linear reduced-order class𝑘 tensegrity dynamics, a non-linear control law is derived. The control variables (force densities in strings) appear linearly in the non-linear dynamics. An airfoil discretization method to bound the local error of each node is introduced and combined with the design of tensegrity airfoils. An ex-

𝑟𝑔𝑑

M. Chen, J. Liu, and R.E. Skelton: Preprint submitted to Mechanics Research Communications

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Figure 7: Tensegrity NACA0012, error bound 𝛿 = 0.002 m, along the chord 0 − 0.05 m, control objectives are big blue dots in the upper plot. There are three-time history plots in the lower figure, the magenta dotted lines are at 0s, the blue lines are at 7s, and the red lines are at the 20s.

Figure 9: Distance error between current node position and the target position with respect to time, node labels can be found in Figure 7.

Acknowledgments

The authors thank Mr. Xiaolong Bai for his many helpful discussions.

References

Figure 8: Force densities of all the strings, string labels can be found in Figure 7.

ample is given to demonstrate the feasibility of shape control of tensegrity airfoils. The approach can also be used for design and shape control of other tensegrity structures.

Conflicts of Interest Statement

The authors whose names are listed immediately below certify that they have NO affi liations with or involvement in any organization or entity with any fi nancial interest (such as honoraria; educational grants; participation in speakers’ bureaus; membership, employment, consultancies, stock ownership, or other equity interest; and expert testimony or patentlicensing arrangements), or non-fi nancial interest (such as personal or professional relationships, affi liations, knowledge or beliefs) in the subject matter or materials discussed in this manuscript.

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