1 Introduction

Typical civil-engineering structures are designed mainly to be passive and static for pre-determined loads or indirect actions. Therefore, active adaption is a more challenging matter of current research for civil-engineering structures. Active control of civil-engineering structures is a rising research trend, which was first introduced by Yao (1972). There are two concepts that are related to active structural control: first, active structures (AS) are structures consisting of both passive (static) members and active (dynamic) members (Soong and Manolis, 1987); second, adaptive structures are structures that can change their geometric configuration and physical properties to meet requirements in response to external stimulation (Miura and Furuya, 1988; Wada, 1990).

Although most studies of structural control focus on enhancing safety during vibration caused by earthquakes or high winds, maintaining serviceability is another important goal in this field (Zuk, 1968; Yao, 1972). In practical applications, there is often considerable uncertainty about in-service loading or conditions, which can significantly influence the serviceability of some flexible structures. Some of these structures governed by serviceability criteria are targeted as the primary focus of active structural control. Shape control of AS has been a research topic since the end of the 20th century (Kawaguchi et al., 1996; Skelton and Sultan, 1997; Sultan and Skelton, 1997; Djouadi et al., 1998; Kanchanasaratool and Williamson, 2002). Equipped with sensors and actuators, structures are able to control their shapes to adapt themselves to changing loads or environments. Furthermore, advanced computing methods can be used to enable AS to learn, so that they can improve their performance based on previous experience.

AS usually perform actuations by adjusting member lengths. Adjustable members such as struts and cables can function through elongation or contraction of embedded actuators. To satisfy the serviceability objective, the actuations (length changes of the active members) have to be determined in shape control of AS, so that the structure can be driven to the target configuration. Approaches based mainly on stochastic search methods have been proposed to find feasible actuations (Shea et al., 2002; Domer et al., 2003b; Domer and Smith, 2005; Xu and Luo, 2008; 2009). Fest et al. (2003; 2004) designed AS prototypes with telescopic struts and proposed control strategies based on stochastic search to maintain a top-surface slope.

Being in a position to respond to external actions, AS are essentially a kind of bio-inspired system of civil structures. They have properties that are similar to those of living organisms and thus have been referred to as biomimetic structures (Veuve et al., 2014). This term does not include those civil structures whose structural shape simply mimics a natural shape. The word ‘biomimetic’ means that AS can interact with their environment and reproduce behavior inspired by nature. If AS are designed and equipped appropriately, bio-inspired ideas can be introduced into the control of AS (Smith, 2009a; 2009b). Thus, AS are capable of self-diagnosis, self-repair, and learning. Adam and Smith (2007a; 2007b; 2008) used a multi-objective approach for AS control to satisfy serviceability and additional robustness objectives. They proposed a control framework involving multi-objective search, reinforcement-learning, and self-diagnosis. This can be applied to situations of partially defined loading events. Korkmaz et al. (2011; 2012) studied self-repair of an active tensegrity bridge to meet safety and serviceability requirements and proposed a multi-objective damage tolerance strategy using stochastic search. However, these existing control approaches are based on numerical models of AS. To some extent, the reliance on AS models limits the performance of shape control of AS because of the large amount of calculation, the long search time of control solutions, and deviations between numerical models and real physical structures.

Compared with traditional model-based control methods, model-free bio-inspired control is more distinct, self-organized, and real-time. Instead of being manipulated only by high-level control, bio-inspired control strategies imitate movement patterns of animals and can be driven directly by sensing. Since the 1990s, studies on bio-inspired control have focused mainly on central pattern generator (CPG)-based control methods (Ijspeert, 2008; Wu et al., 2009; Yu et al., 2014) that are suitable for locomotion control of bio-inspired robots with multiple degrees of freedom (DOFs). Some researchers have used this type of method to control the dynamic properties of AS (Bliss et al., 2008). Studies on shape control of AS using bio-inspired methods are rare, mainly due to a lack of appropriate tools.

Nowadays advances in wireless technology and smart sensor technology are creating a favorable environment for the development of civil-engineering structures. Wireless sensor network (WSN) has shown potential for structural health monitoring (SHM) (Straser and Kiremidjian, 1998; Spencer et al., 2004; Lynch, 2007), because its installation and maintenance are easier and less expensive than a wired network. In many structural applications of WSNs, however, it is not enough to observe only the structural state. The networks are expected also to respond to the sensed information by performing actions on the structure. There have been some studies of structural vibration control using WSN. These designs are mainly implemented through hydraulic dampers wired to wireless sensor nodes (Wang et al., 2007; Swartz and Lynch, 2009; Wang and Law, 2011). However, WSNs usually work only as passive devices for most of civil-engineering structures.

Actuators that are capable of wireless communication can be incorporated into WSN to form a wireless sensor and actuator network (WSAN) (Akyildiz and Kasimoglu, 2004; Stankovic, 2008; Atzori et al., 2010). WSANs consist of groups of sensors and actuators that are connected to a wireless medium. Sensors serve to detect information about the physical world, while actuators can perform appropriate actions to affect the physical world according to the sensory information. WSANs are closed-loop systems that can interact with the physical world automatically. In recent years, researchers started to use WSANs for centralized or decentralized control systems (Chen et al., 2008; 2010; Cao et al., 2010).

Being able not only to observe but also to respond, AS is a conceptual extension of SHM, and in the same way, WSAN is an extension of WSN. Moreover, WSANs can be the best choice for active structural control, because the entire control system can perform via easily deployed devices. They can also be potentially applied to spatially distributed control units embedded in the structure, especially for AS whose members unite spatially. However, few studies of AS using WSANs have been published.

This paper presents the concept of a compound system named WAS (short for WSAN-AS) which consists of an AS and embedded WSAN. The concept treats AS and WSAN equally and considers them as a whole, and its core principle is that the structural arrangement should harmonize with the topology of wireless hardware devices. Moreover, to overcome the limitation of previous work on shape control of AS, a two-hierarchy bio-inspired control framework is proposed for WAS to give immediate response to unknown loads.

In this paper, the concept and the modeling of WAS are described first. Then the control framework is introduced. At last, a simple example of WAS is given to illustrate the concept and validate the framework.

2 Concept and basic model

2.1 Motivation and difficulty

As stated above, existing studies on AS have obtained shape control solutions mainly by stochastic searches. In each search process, the computer has to generate adequate samples of actuation results for a single solution. In each calculation process, the computer has to solve nonlinear finite-element-method (FEM) problems repetitively to evaluate (or to predict) the state of AS and get an actuation result through a numerical method, such as dynamic relaxation (Fest et al., 2003). In short, to generate effective solutions, substantial search work has to be done. That can place several limitations on the practical applications of AS:

  1. 1.

    It is sometimes difficult to implement real-time control. Without case-base or any learning methods, it takes a relatively long time to give an effective solution in response to a new loading event (Shea et al., 2002; Smith, 2003; Fest et al., 2004). It is long even for quasi-static control, not to mention for real-time control. With an ordinary low-cost computer, there can be a heavy computational burden, especially in the case of a relatively complex structure.

  2. 2.

    The system may not have sufficient reliability. On the whole, the computer has to manage all the computation, communication, and coordination works to keep the system running. If, by any chance, the computer happens to crash (actually not uncommon), the entire system will break down.

  3. 3.

    Adaptability to loads can sometimes be limited. In those model-based stochastic search procedures for the numerical FEM calculation, the load conditions have to be known or prescribed to the computer or the human controllers (the users) (Shea et al., 2002; Fest et al., 2003; 2004), or at least the load has to be partially defined so that the load identification work can be done first (Adam and Smith, 2007a; 2008). In situations where the only piece of information is the structural deformation (from totally unknown loads), the control task will be difficult.

  4. 4.

    There may be the influence of discrepancy between the calculation and the physical world. The computational process relies on the numerical FEM model and other assumptions. In fact, construction of real structures cannot be 100% the same as the blueprint, and structural damage or loss of pre-stress also makes a physical structure deviate from the numerical design model. Researchers have considered this issue and offered some solutions. For example, calibration of a numerical model with a correction based on measurements (Shea et al., 2002; Fest et al., 2003) or an artificial neural network (ANN) used to modify numerical models (Domer et al., 2003a) may help reduce the discrepancy, locate some simple structural damage (Adam and Smith, 2006), and identify self-stress to some extent (Angellier et al., 2012). Nevertheless, the effect and generality of such measures are limited. The difference between the numerical model and the real structure can often affect the accuracy of control in experimental tests or the practical application of AS, not to mention the limitation of the accuracy of actuators.

By contrast, bio-inspired methods for locomotion control in the field of robotics are rather attractive for the shape control of AS (Table 1). However, applying bio-inspired methods to shape control of AS is not that easy. Unlike multi-degree-of-freedom (multi-DOF) robots, such as multi-legged robots, AS are geometrically stable, and they are always statically determinate or indeterminate during the in-service period. All members of AS unite as a whole and interact with each other. A slight move in one active member can cause redistribution of the internal forces of all the members, and it definitely affects the shape of the entire structure. This means the correlation between each DOF of AS is more significant, and AS cannot change the shape (or configuration) as casually as robots do. Meanwhile, all the actuators and sensors are fixed-wired to the central computer, and the centralized one-to-many relation is not suitable for distributed control of bio-inspired methods. This precludes AS from behaving as flexibly as robots.

Table 1 Comparison between model-based and bio-inspired approaches
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2.2 Definition and formulas

A WSAN is used to enhance AS. WSAN is not a simple superposition of WSN and wireless actuators. Instead, it is heterogeneous and closed-loop, while WSN is homogeneous and open-loop. The feature of WSAN is cooperative communication and control.

As the medium and functional component, WSAN is embedded in AS. This integrated system that consists of AS and WSAN is abbreviated to WAS. It is intended to be a bio-inspired compound system. In WAS, the role of WSAN is the ‘nervous system’ that coordinates the activity of the ‘muscles’ (actuators) and processes the input from the sensors. With the wireless ‘nerve fibers’, the ‘nervous system’ of WAS can be self-organized, and it can change its topology to adapt to different situations.

As the ‘skeletons’ and ‘tendons’ of WAS, the arrangement of structural members should be compatible with the ‘nervous system’ and ‘muscles’. In a word, all the components of WAS should be designed to harmonize with each other. Taking cable-strut structures as example, the physical process of WAS can be described as the following four parts:

$$M\ddot{D} = {F_{{\rm i}}} + {F_{{\rm e}}},$$
(1)
$$S = {\Lambda _{\rm{s}}}P + {\Gamma _{\rm{s}}},$$
(2)
$$\Delta A = {\Omega _{\rm{a}}}C{\Omega _{\rm{s}}}S + {\Gamma _{\rm{a}}},$$
(3)
$${L_{\rm{r}}} = {L_{{\rm{r}},{\rm{o}}}} + {\Lambda _{\rm{a}}}A,$$
(4)

where all structural members are uniaxial elements with three DOFs at each node, struts are tension-compression members, and cables are tension-only members. Here are some details of expressions (1)–(4).

The mechanical process (1) is expressed as Newton’s second law of motion, where M is nn×nn mass matrix of structural nodes, and nn is the number of all the structural nodes. M can be expressed as

$$M = {\rm{diag}}(\ldots, \;{m_k},\; \ldots),$$
(5)

where m k is the equivalent mass of the kth structural node (k∈{1, 2, …, nn}). D is nn×3 displacement matrix of structural nodes, and

$$D = [ \ldots, {d_k}, \ldots ]^{\rm{T}},$$
(6)

where dk is displacement vector of the kth structural node (k∈{1, 2, …, nn}), and

$${d_k} = [{d_{x,k}},\;{d_{y,k}},\;{d_{z,k}}]^{\rm{T}}.$$
(7)

\(\ddot D\) is nn×3 acceleration matrix of structural nodes. Fi is nn×3 internal force matrix of structural nodes and can be determined by

$${F_{\rm{i}}} = {\Lambda _{\rm{t}}}{\hat F_{\rm{a}}}{D_{\rm{c}}},$$
(8)

where Λt is nn×nm layout matrix of structural topology and nm is the number of all the structural members. Λt can be expressed as

$${\Lambda _{\rm{t}}} = \{ {\lambda _{{\rm{t}},i,j}}\},$$
(9)

where i∈{1, 2, …, nn}, and j∈{1, 2, …, nm}. If the ith node is the starting point of the jth member, λt,i,j=1; or if the ith node is the ending point of the jth member, λt,i,j=−1; otherwise, λt,i,j=0. Fa is nm×1 axial force vector of the structural members, and

$${F_{\rm{a}}} = {[ \ldots, \;{f_{{\rm{a}},k}},\; \ldots ]^{\rm{T}}},$$
(10)

where fa,k is the axial force of the kth structural member (k∈{1, 2, …, nm}). \({\hat F_{\rm{a}}}\) is the nm×nm diagonal form of Fa, i.e.,

$${{\hat F}_{\rm{a}}} = {\rm{diag}}(\ldots, \;{f_{{\rm{a}},k}},\; \ldots),$$
(11)

where k∈{1, 2, …, nm}. Dc is nm×3 direction cosine matrix of structural members, and

$${D_{\rm{c}}} = {[ \ldots, \;{d_{{\rm{c}},}}_k,\; \ldots ]^{\rm{T}}},$$
(12)

where dc,k is the direction cosine of the kth structural member (k∈{1, 2, …, nm}), and

$${d_{{\rm{c}},k}} = {[{d_{{\rm{c}},x,k}},\;{d_{{\rm{c}},y,k}},\;{d_{{\rm{c}},z,k}}]^{\rm{T}}}{.}$$
(13)

Fe is nn×3 external force matrix of structural nodes, and

$${F_{\rm{e}}} = {F_{\rm{l}}} + {F_{\rm{d}}} + {F_{\rm{r}}}{.}$$
(14)

Fl is nn×3 load matrix of structural nodes, and

$${F_{\rm{l}}} = {[ \ldots, \;{f_{\rm{l},k}},\; \ldots ]^{\rm{T}}}, $$
(15)

where fl,k is the load vector of the kth structural node (k∈{1, 2, …, nn}), and

$${f_{{\rm{l}},k}} = {[{f_{{\rm{l}},x,k}},\;{f_{{\rm{l}},y,k}},\;{f_{{\rm{l}},z,k}}]^{\rm{T}}}.$$
(16)

Fd and Fr are nn×3 damping force matrix and nn×3 support reaction force matrix of structural nodes, respectively.

In the sampling process (2), the sensors measure the structural physical quantities, such as deformation or internal forces. S is ns×1 vector of sampled data of sensors, and ns is the number of all the sensors. S can be expressed as

$$S = {[ \ldots, \;{s_k},\; \ldots ]^{\rm{T}}}, $$
(17)

where sk is the effective feedback of the kth sensor (k∈{1, 2, …, ns}). P is np×1 vector of measurable physical quantities, and np is the number of all the measurable physical quantities. Usually,

$$P = {[{\bar D^{\rm{T}}},\;F_{\rm{a}}^{\rm{T}}]^{\rm{T}}}, $$
(18)

\(\bar D\) where is the 3nn×1 single-column form of D, and

$$\bar D = {[ \ldots, \;d_k^{\rm{T}},\; \ldots ]^{\rm{T}}}, $$
(19)
$${n_{\rm{p}}} = 3{n_{\rm{n}}} + {n_{\rm{m}}}{.} $$
(20)

Λs is ns×np layout matrix of sensors, and

$${\Lambda _{\rm{s}}} = \{ {\lambda _{{\rm{s}},i,j}}\},$$
(21)

where i∈{1, 2, …, ns}, and j∈{1, 2, …, np}. If the ith sensor is deployed on the jth physical quantity, λs,i,j=1; otherwise, λs,i,j=0. S can be influenced by the noise Γs, which is ns×1 measurement noise vector of sensors.

In the control process (3), with the feedback received from S, the controllers calculate and transmit the output to the actuators. A is na×1 vector of actual actuations of actuators, and na is the number of all the actuators. A can be expressed as

$$A = {[ \ldots, \;{a_k},\; \ldots ]^{\rm{T}}}, $$
(22)

where a k is the length adjustment of the kth actuator (k∈{1, 2, …, na}). ΔA is the increment of A. C is na×ns transfer matrix of controllers. Ωa and Ωs are na×na wireless transmission matrix of actuations and ns×ns wireless transmission matrix of sensor feedback, respectively. They can be expressed as

$${\Omega _{\rm{a}}} = {\rm{diag}} (\ldots, \;{\omega _{{\rm{a}},}}_k,\; \ldots),$$
(23)
$${\Omega _{\rm{s}}} = {\rm{diag}}(\ldots, \;{\omega _{{\rm{s}},}}_k,\; \ldots){.}$$
(24)

If the kth (k∈{1, 2, …, na}) actuator receives wireless instruction, ωa,k=1; otherwise, ωa,k=0. If the data packet of the kth (k∈{1, 2, …, ns}) sensor is not lost, ωs,k =1; otherwise, ωs,k=0. Γa is na×1 movement error vector of actuators.

In the performing process (4), the rest lengths of the active members are determined by actuations. Lr is nm×1 rest length vector of structural members, and

$${L_{\rm{r}}} = {[ \ldots, {l_{{\rm{r}},k}}, \ldots ]^{\rm{T}}}, $$
(25)

where k∈{1, 2, …, nm}, and lr,k is the rest length of the kth structural member. Lr,o is the original Lr at the initial state. Λa is na×nm layout matrix of actuators, and

$${\Lambda _{\rm{a}}} = \{ {\lambda _{{\rm{a}},i,j}}\},$$
(26)

where i∈{1, 2, …, na}, and j∈{1, 2, …, nm}. If the ith actuator is embedded on the jth member, λa,i,j=1; otherwise, λa,i,j=0. Thus, the structural internal forces, i.e., the axial forces,

$${F_{\rm{a}}} = {A_{\rm{c}}}E{E_{\rm{a}}}, $$
(27)

change with the actuations, where Ac is nm×nm cross-section area matrix of structural members, E is nm×nm elastic modulus matrix of structural members, and Ea is nm×1 axial strain vector of structural members. They can be expressed as

$${A_{\rm{c}}} = {\rm{diag}}(\ldots, \;{a_{{\rm{c}},k}},\; \ldots), $$
(28)
$$E = {\rm{diag}}(\ldots, \;{e_k},\; \ldots), $$
(29)
$$E_{\rm{a}} = {[ \ldots, \;{\varepsilon _{{\rm{a}},k}},\; \ldots ]^{\rm{T}}}, $$
(30)

where k∈{1, 2, …, nm}, and ac,k, ek, and εa,k are the cross-section area, the elastic modulus, and the axial strain of the kth structural member, respectively. Ea can be influenced by Lr via

$${E_{\rm{a}}} = {\Phi _{\rm{c}}}(\ln \;L - \ln \;{L_{\rm{r}}}),$$
(31)

where L is nm×1 length vector of structural members, and

$$L = {[ \ldots, \;{l_k},\; \ldots ]^{\rm{T}}}, $$
(32)

where l k is the current length of the kth structural member (k∈{1, 2, …, nm}). Φc is nm×nm amendment matrix of cable members, and

$${\Phi _{\rm{c}}} = {\rm{diag}}(\ldots, \;{\phi _{{\rm{c}},k}},\; \ldots), $$
(33)

where k∈{1, 2, …, nm}. If the kth member is a cable and lk<lr,k, Φc,k=0; otherwise, Φc,k=1.

2.3 Discretization in the time domain

From the perspective of control theory, WAS can drive the structure to reach requirements step by step by analyzing the measurements reported by the sensors, and each step can be interpreted as a step of control. Meanwhile, according to Section 2.2, WAS is a non-linear time variant system. If it is assumed that there is no time delay in control, the state-space expression of WAS can be described as a discrete form:

$$P(t) = f({P^{k - 1}},\;\Delta {A^{k - 1}},\;t), $$
(34)
$${P^k} = P({t^k}), $$
(35)
$${S^k} = {\Lambda _{\rm{s}}}{P^k} + {\Gamma _{\rm{s}}}({t^k}), $$
(36)

where k means the kth control step, and t means time. Eq. (36) comes from Eq. (2). Function f is a general description related to Eqs. (1) and (4) which are continuous mechanical processes in the time domain. Thus, the time of the mechanical process is subdivided into equal time intervals (or time steps), Δt, and Eq. (34) can be rewritten as

$$P(t) = g(P(t - \Delta t),\;\ldots, \;P({t^{k - 1}}),\;\Delta {A^{k - 1}},\;\Delta t){.} $$
(37)

The control output is determined by Eq. (3), and for the kth control step,

$$\Delta {A^k} = {\Omega _{\rm{a}}}({t^k}){C^k}{\Omega _{\rm{s}}}({t^k}){S^k} + {\Gamma _{\rm{a}}}({t^k}), $$
(38)

and

$${A^k} = {A^{k - 1}} + \Delta {A^k}{.} $$
(39)

Fig. 1 is an illustration of one ordinary control step. Some parameters of time are considered, such as time cost of wireless transmission tw, time duration of sensor sampling ts, time duration of actuator performing ta, and response time of controller tc. In addition, tp is the total time of the control step, fs is the sampling frequency of sensors, and va is the performing velocity of actuators (i.e., the velocity of length changing of an active member).

Fig. 1
figure 1

Discretization of the k th control step (TX means transmission, and RX means reception)

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In fact, function g in Eq. (37) is not required to contain those arguments. From the point of computation, the equation of motion (1) can be solved by a simple explicit time integration, such as the central difference method (Bathe and Wilson, 1976; Luo and Yang, 2014; Yang et al., 2014). The variation of nodal displacement is assumed to consist of a series of straight-line segments, and the nodal velocity between two time points is assumed to be constant (Fig. 1). The abrupt change of the velocity at each time point is caused by an acceleration pulse. According to the central difference method, the nodal displacement can be calculated through the recursion:

$$D(t + \Delta t) = 2D(t) - D(t - \Delta t) + \ddot D(t)\Delta {t^2}, $$
(40)

where the time step Δt should be below the critical value for integration stability (Belytschko et al., 2000).

3 Two-hierarchy control

3.1 A bio-inspired control framework: reflex

As stated in Sections 1 and 2.1, for bio-inspired control in robotics, CPG-based control methods are commonly applicable to the locomotion (or gait) of multi-legged robots, but they are not suitable for shape control of AS. As mentioned in Section 2.2, WAS is a new concept that means the AS is embedded with a WSAN. It can model itself after animals by developing a bio-inspired self-control mechanism (architecture) which takes WSAN as its ‘nervous system’ (Fig. 2).

Fig. 2
figure 2

Bio-inspired architecture of WSAN materialized with IEEE Standard 802.15.4

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Here are some details about how the architecture imitates animals:

  1. 1.

    The nervous system of animals is commonly divided into a central nervous system (CNS) and a peripheral nervous system (PNS). CNS is divided into brain and spinal cord. CNS and PNS are connected to each other through nerve fibers. Sensory receptors and muscles are connected with the nervous system (Morris and Fillenz, 2003). Similarly, as Fig. 2 shows, WSAN consists of one single central controller (CC) and groups of distributed actuators and sensors (DAs and DSs). Usually, the CC and the DAs are resource-rich devices with more energy and better processing capabilities than DSs (IEEE Computer Society, 2011). Thus, the CC is assigned to be the higher level of ‘CNS’, i.e., the ‘brain’. The DAs are the ‘muscles’, and their processing units play the role of spinal cord. The DSs represent the ‘sensory receptors’ and the ending of ‘PNS’. Meanwhile, all these components communicate with each other via wireless pathways, the ‘nerve fibers’.

  2. 2.

    The motor system of animals is a distributed organization. Sensory information relating to movement is processed in different circuits that work in parallel. The circuits that process sensory inputs and commands to muscles are distributed in the spinal cord. Each circuit can organize specific motor responses through its input and output connections (Kandel, 2013). Likewise, the WSAN contains multiple parallel personal area networks (PANs) as the ‘response circuits’ (Fig. 2). The PANs are connected to the CC directly or indirectly, and they can also communicate with each other. As the PAN coordinators, the DAs are in charge of the responses. The connection and input-output relation of DAs and DSs in each PAN can be different and relatively independent from the other PANs.

  3. 3.

    Locomotion of animals is hierarchically controlled by the CNS from the brain to the spinal cord level (Delcomyn, 1980). The spinal cord deals with some basic and simple movements autonomously, while the brain can directly influence the spinal cord for coordinating and planning complex sequences of movement. Imitating this feature, WAS establishes the architecture of WSAN between centralized control and decentralized control (Fig. 2). As the lower level of ‘CNS’, the DAs of different PANs generate their respective responses. At the same time, the CC, the higher level, can give direct instructions to the DAs.

Components of the architecture presented above are organized in a control framework based on the term ‘reflex’ (Fig. 3). A reflex, often known as a spinal reflex, is a sudden, involuntary contraction of muscles in response to a stimulus, such as knee-jerk reflex and withdrawal reflex (Starr et al., 2008). As the basic units for movement (Sherrington, 1906), spinal reflexes are intrinsic and rapid, because they usually involve the spinal cord instead of the brain. As Fig. 3 shows, DSs, DAs, and related pathways constitute reflex arcs. In the reflex arcs, after being triggered by DAs (the ‘spinal cord’), DSs (the ‘receptors’) take samples from the structure (the ‘body’) and pass the samples back to DAs (the ‘spinal cord’) through wireless pathways (the ‘sensory neuron’). Then DAs (the ‘spinal cord’) generate outputs (the ‘motor neuron’) according to the intrinsic methods written in their flash memories and perform the output actuations (the ‘effectors’) on the structure.

Fig. 3
figure 3

Two-hierarchy control framework of WAS

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The reflex that results from the lower part of the CNS is known as an inborn reflex or unconditioned reflex (UR). Some complex URs are generally known as instincts. By linking up the action of a new stimulus with a UR, a conditioned reflex (CR) is obtained. For instance, in the famous example of classical conditioning, the dog’s salivary reflex to food was eventually elicited by the bell alone (Pavlov, 1927). The proposed control framework has a similar mechanism, and it is split into two hierarchies (Fig. 3). In the lower hierarchy, namely the UR, the DAs respectively respond to the DS feedback according to their own intrinsic methods, and the CC is rarely involved. On the other hand, the higher hierarchy, namely the CR, gets a higher priority. The CC can directly command the DAs to perform specified actuations and ask the DAs to submit data or report their status. A key feature of the CR hierarchy lies in the ‘conditioning’. To be specific, the CC can reserve the outputs of actuations in response to certain inputs of sensations. If the DAs have responded to an input of sampled data of the DSs in the UR mode, the CC memorizes the sampling values of the DSs and the actuations performed by the DAs. Thus, when the same or similar DS data appears next time, the CC will recall the stored case and provide the control solution to the DAs directly.

3.2 Rhythmic movement in UR and establishment of CR

Locomotion of animals relies mainly on the sequential and rhythmic movements generated properly by the spinal cord, and the timing of muscle activations that constitute any rhythmic movement is regulated by the intrinsic properties of the spinal cord (Delcomyn, 1980; Duysens and van de Crommert, 1998; Goulding, 2009). In rhythmic movements, all or part of the body moves in a cyclic and repetitive way, as in walking, swimming, scratching, etc. The proposed control framework borrows this idea for the UR implementation. In the UR hierarchy, the DAs, i.e., the lower level of the ‘CNS’ (Fig. 3), can perform their intrinsic responses according to a specific rhythm, such as alternation or something more complex. The DAs can communicate with each other and create an intrinsic mechanism of the rhythm, and the CC can also exert additional influence on the rhythm. Therefore, on one hand, the higher level of the ‘CNS’ can help to regulate and stabilize the rhythmic pattern of the UR activity and, on the other hand, even if the CC may malfunction (i.e., the ‘brain’ does not work), the lower level of the ‘CNS’ is still capable of organizing the UR autonomously (Brown, 1914). Apparently this can make the proposed control framework more robust in some extreme situations.

In the control framework, the CR can be ‘conditioned’ through an association between the stimulus (sensations of DSs) and the response (DA actuations) of the UR, which is termed ‘associative learning’ (Raven and Johnson, 2002). The learning algorithm can be very intelligent with the help of some advanced techniques, such as data mining, expert systems, and reinforcement learning. It all depends on the energy and the processing capability of the CC. In fact, for this model-free control framework, the learning does not have to be complex, and the computational burden can be light or none. In the simplest scenario, the CC just needs to store each association for case-matching next time, which requires little processing capability. In another example, the CC can be trained with an ANN, which can be simple enough to run on an ordinary single-chip microprocessor.

The execution flow of the proposed control framework can be simple. It is outlined in Fig. 4 in the form of the pseudo codes for microprocessors. The interrupt function _service() will wake up the device if any applied load has been detected. After awaking, the DSs take samples from the structure, and then functions CR() and UR() will be called successively with the sampled S. In CR(), the decision of the actuations, i.e., A, is made according to previous experience. Then the operative mode is switched to the UR mode. In UR(), the system keeps doing intrinsic rhythmic reflexes, function _rhythm(), until the timer overflows (time to sleep). At last, after executing function _learn(S, A), the system will enter the sleep state to save power.

Fig. 4
figure 4

Basic two-hierarchy control algorithm

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4 Example using the control framework: a numerical prototype

4.1 A simple WAS designed to maintain height under unknown loads: postural reflex

As proof of the concept WAS proposed in Section 2.2, a triangular tensegrity prism (TTP) with embedded WSAN was designed. The TTP is composed of three struts, nine cables, one central controller (CC), three distributed actuators (DAs), and three distributed sensors (DSs) (Fig. 5). It covers a circular area of radius 0.5 m, has a height of 1 m, and rests on three supports. The struts and the cables are uniaxial, as stated in Section 2.2. The CC is fixed inside the structural skeleton, the three DAs are embedded in the three struts to adjust their lengths, and the three DSs are deployed on the top surface of the TTP to acquire vertical displacement data. As the root of the entire network, the CC rules the three PANs that are coordinated by the DAs. Some further details of the TTP can be found in Tables 2 and 3.

Fig. 5
figure 5

A tensegrity prism with embedded WSAN: isometric view (a), top view (b), and side view (c)

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Table 2 Specification of tensegrity
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Table 3 Specification of WSAN
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The TTP is intended to have a capability of ‘postural reflex’ (Squire, 2013) so that it can adapt itself to maintain its height under different unknown loads (unknown to the TTP). When subjected to an external load, the TTP is always trying to keep the vertical displacements of its top surface to zero, i.e., has a ‘virtual infinite’ vertical stiffness.

Some values of physical parameters and initial conditions in this example are listed in Tables 2 and 3. For instance, it is assumed that the external environment is ideal. In particular, all wireless transmission is successful (no packet loss), samples of sensors are always noise free, actuator adjustments are undisturbed by the interference of the environment, and the time taken by analog and digital devices to process signals is too short to be noticeable. Meanwhile, limitations are set up to conform to objective reality, such as sensor resolution, actuator resolution, actuator adjusting velocity, actuator stroke range, and time cost of wireless transmission. In addition, initial conditions are given to reduce unnecessary outside factors. Specifically, the system self-weight is not considered, the structure is in the state of null self-stress equilibrium geometry (Motro, 2003) before time t=0 (i.e., the initial internal forces are zero), and the damping force of the structural nodes is simplified to

$${F_{\rm{d}}} = {\alpha _{\rm{d}}}M\dot D,$$
(41)

\(\dot D\) where is nn×3 velocity matrix of structural nodes, and αd is the global damping coefficient (Table 2). The value of the time step is set according to the stability condition of the central difference method, as stated in Section 2.3.

4.2 Settings

Diverse load cases including different types (single point or multi-point), distributions (uniform or not), magnitudes, and directions (vertical or oblique) are carried out to test the TTP (Table 4). In case C1 and its subcases C1-1–C1-9, the TTP responds just in UR mode, while in cases C2–C8, in CR mode. The TTP is allowed to learn in cases C1–C8.

Table 4 Load cases for the test of the TTP
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In each load case, the TTP is in the initial state of null self-stress equilibrium geometry at first. Then a set of unexpected loads is suddenly applied to the structural nodes. At that point, the TTP wakes and starts the timer (t=0). The stabilization time variables in Fig. 4, namely tds, tdc, and tdr, are set to 1.5 s, 0.5 s, and 2 s, respectively.

Some functions presented in Fig. 4 are defined specifically for the TTP (Fig. 6). In function A= _rhythm(), once the DA of the kth PAN is excited (activated), it will keep responding intrinsically according to the DS feedback, which provides a linear output feedback control:

$$\Delta {a_k} = {c_{\rm{t}}}{s_k},$$
(42)

where ∀k∈{1, 2, …, na} (na=3), and ct=−0.5. As stated in the macro definition, a critical displacement dc (=0.05 mm) is defined as the criterion whether the DAs should enter the state of inhibition or not. Once the extent of the feedback sk is less than dc, the kth DA inhibits its excitation and excites the coordinator of the next PAN, i.e., the (k+1)th DA, and then the (k+2)th, …, the nath DA, so that excitatory reflex happens in one PAN after another. The system repeats these sequential excitatory reflexes until the timer overflows, which takes on a regular rhythm. The CC simply uses a 10-layer feed-forward neural network (FNN) Nf for case learning and decision making. Function _learn(S, A) adds a new group of input S and output A into the case libraries (Li and Lo) and trains Nf. Function A=_decision(S) directly gives the output of Nf according to an input S.

Fig. 6
figure 6

Intrinsic action and learning mode of the TTP

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It can be seen that dc actually determines the control accuracy. The final value of sk will always be in the interval of (−dc, dc). Thus, the lower dc is, the more accurate the control effect will be, and the more time the control process will take. The coefficient ct determines the feedback extent of each single intrinsic response. The lower ct is, the finer the actuations will be. Actually, there can be some other more complex feedback forms than Eq. (42). It all depends on the specific layout of WAS and the requirement of the designers.

4.3 The first UR

Because there is no knowledge in the ‘brain’ (the CC) at the beginning of the simulation, case C1 is the first pure UR test. Fig. 7 shows two contour plots (isometric view) of vertical displacement in case C1. The upper plot shows the status when the load has been applied and UR has not yet started (t=3.99 s), and the lower one is the result at t=39.99 s when all PANs have been in a state of inhibition. Apparently, at last UR has made all vertical displacements of the top surface to be within the criterion dc (0.05 mm). That is to say, the vertical deformation of the top surface has reverted. Some more details about the first UR are stated as follows.

Fig. 7
figure 7

Contour plots of vertical displacement before (a) and after (b) UR in case C1 (unit: mm)

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Fig. 8 shows the wireless transmission record in case C1 (from t=0 s to t=40.0 s), containing the record of the CC and the DAs. This can be viewed as the representation of signal transmission among the ‘CNS’ of the TTP. On the ordinate, 1 stands for transmitting, and 0 stands for not. At t=1.5 s, the CC asks each DA to update and submit the DS sample. From t=4.0 s, the DAs start to act one by one. While one DA is excited (activated), the other two are not. In the rhythmic movement, one excited DA keeps asking its corresponding DS for performing feedback until the DA is inhibited. Then the DA passes its excitement to the next, and in the meantime, it submits a report to the CC for confirmation. As time goes by, the alternation among the DAs becomes more and more frequent, which means the rhythmic movement gets closer to a stable state.

Fig. 8
figure 8

Time-history data in case C1: wireless transmission of the CC (a) and DAs (b)–(d)

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The rhythmic movement can be surveyed more clearly in Fig. 9. The DAs perform alternately, and the DSs also sample in turn. In each PAN, the DA actuation and the DS sample change synchronously with each other. One difference is that the displacements do not change alternately one node after another. No matter which DA performs, it can induce that all the three nodal displacements fluctuate synchronously. In fact, this phenomenon has been stated in Section 2.1, which is also the reason why UR motion should be rhythmic (orderly). If each DA reflexes casually on its own without any rhythm, the actions of the DAs will interact with each other, and the structure shape may vary disorderedly.

Fig. 9
figure 9

Time-history data in case C1: sampled values of DSs (a), actuations of DAs (b), and nodal vertical displacements (c)

‘LV’ is short for last updated value of sampling displacement; ‘RP’ is short for relative position in actuator stroke (i.e., adjustment amount); ‘UZ’ stands for displacement on z-axis direction; ‘C1’ means case C1

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As time goes by, the actuation extents of the DAs and the sampled values of the DSs both start to decrease, and the fluctuation of the displacements reduces gradually (Fig. 9). Approximately at t=35.0 s, the DAs all stop performing (enter a state of inhibition), and the displacements do not vary any more.

The UR process can be seen more visually in Fig. 10. An actuation path of the rhythmic movement starts from the initial state (before any actuation is performed), wriggles around, narrows the encirclement, and then stabilizes to an inhibition state (Fig. 10a). The inhibition state is the state of t≥ti when all the DAs have entered the state of inhibition, where ti is called the inhibition time. From a macroscopic view (Fig. 10b), the top surface slants, swings, and finally reverts back to the original height (within the margin of error).

Fig. 10
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Process of the UR in case C1: (a) convergence of the actuations; (b) reversion of the top surface

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Fig. 11 shows some internal force results of structural members including M1, M4, and M7. The stress values of the three members fluctuate synchronously with the curves of time-displacement in Fig. 9. A negative correlation can be found between strut stress and cable stress. Moreover, compared with the initial state, the stress magnitudes in struts and cables have significantly increased after entering the stable state. In other words, to maintain the top surface height, the TTP has to sacrifice some internal force margin to neutralize the structural vertical deformation.

Fig. 11
figure 11

Time-history data in case C1: axial stress (SAXL) of structural members

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4.4 Performance

In case C1 and its subcases C1-1–C1-9, the TTP responds only in the UR mode. The actuation amount and inhibition time are related to the magnitudes of loads in these cases (Fig. 12). Basically, both the extent of actuation and the required time of inhibition are positively correlated with the load magnitudes. The maximum and minimum values of the actuations determine the maximum stroke of the actuators. Both maximum stroke and inhibition time indicate the control efficiency. Without any previous experience, the larger the effect that is caused by external loads, the more energy and time it takes to control.

Fig. 12
figure 12

Actuations and inhibition times of single vertical load cases

‘VL’ is short for vertical load; ‘MAX’ means maximum; ‘MIN’ means minimum; ‘EVT’ means eventual results of the inhibition state; ‘IT’ is short for inhibition time

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For different types of loads, the TTP successfully returns its top surface back to the original height (Fig. 13). In fact, the accuracy of the reversion depends on the displacement criterion dc. The smaller dc is, the more accurate the reversion will be.

Fig. 13
figure 13

Vertical displacement plot of the inhibition state: cases C2–C8 (unit: mm)

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Unlike pure UR in case C1, the CR solution A(t=2.0 s) can be generated in cases C2–C8 according to the input S (t=1.5 s) (Table 5). With the previous experience, ti in cases C2–C8 reduces significantly compared with case C1. It means that the DAs enter the inhibition state sooner. Apparently, because of the FNN in the CC, the full process with CR is more efficient than with pure UR.

Table 5 Performance summary of cases C1–C8
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With the experience of case C1, the rhythmic movement inhibits more quickly in case C2 (Fig. 14a). For similar types of loads, the control performance improves with more cases carried out in sequence (Figs. 14b and 14c). In case C5 particularly, the DAs do not need to act any more after the CR performs, because the CR solution given by the FNN is exactly the same as the eventual actuation of case C2.

Fig. 14
figure 14

Time-history data of different load cases

(a) The wireless transmission log, the actuation amounts, and the vertical displacements in case C2; (b) The actuation amounts in cases C3–C5; (c) The vertical displacements in cases C3–C5; (d) The actuation amounts in cases C6–C8; (e) The vertical displacements in cases C6–C8

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For those load types that have not been experienced before, the TTP seems somewhat inexperienced (Figs. 14d and 14e). The more dissimilar the load types are from the previous experience, the less effective the CR solution is, and then the longer time it takes to inhibit. Nevertheless, the CR mode is much faster than the pure UR model. In a word, the more the TTP practices, the more sophisticated it will be.

5 Conclusions

Inspired by the locomotion of animals and the control strategies of multi-legged robots, this paper presents a method to enhance the shape control of AS in civil engineering. A concept of compound system, WAS, has been proposed and modeled. In WAS, AS is integrated with a WSAN. A two-hierarchy bio-inspired control framework is designed for WAS by imitating reflex activity and rhythmic movements of animals. A typical example of a triangular tensegrity prism that aims to maintain its original height has been carried out for verification. The results demonstrate that the deformed structure successfully reverts to its original height after being subjected to different types, distributions, magnitudes, or directions of unknown loads. The performance of the proposed control framework is also discussed. In summary, the WAS with the proposed bio-inspired control framework possesses the following attractive features compared with recent related work of AS:

  1. 1.

    Instant response and little computational burden. The proposed control framework is model-free and does not need stochastic search methods or a large number of FEM solving processes. The hardware can provide the control solution directly according to built-in parameters.

  2. 2.

    Less dependence on hardware performance and a more robust system. Because of avoidance of heavy computational burden, most low-cost microprocessors can meet the requirement for hardware performance. Meanwhile, multiple distributed controllers greatly reduce the risk that the whole system might break down.

  3. 3.

    Greater adaptability to different types of loads. For the proposed framework, the load conditions do not have to be known or prescribed to the computer or the human controllers, because there is no need for the numerical FEM calculation during the control process. In the framework, the types and extents of loads are allowed to be totally unknown.

  4. 4.

    Greater flexibility for design of control strategy. Compared with wired control systems, the proposed WAS with WSAN can change its communication topology without plugging or unplugging wires when the layout or the control strategy of WAS system is modified.

  5. 5.

    Fewer sources of error (model-free and closed-loop control). The proposed model-free control framework eliminates the error caused by the difference between the numerical model and the real structure. The closed-control mechanism in the framework guarantees that the control effect only depends on the parameter setting and sensor accuracy.

  6. 6.

    Easier installation or deployment, easier maintenance, lower cost, etc. Using wireless devices avoids a vast wiring workload, and can save time and cost for installation or replacement.

Moreover, if applicable, the proposed control framework can act as a supplementary means for improving the efficiency and accuracy of control frameworks based on a common stochastic search. Although the proposed concept has been verified in this study, there are still some limitations for shape control of WAS:

  1. 1.

    To achieve the stable bio-inspired feature, the structural form and device parameters of WAS require delicate design, which limits the universality of the proposed control framework.

  2. 2.

    Similar to a common stochastic search, the control solution provided by the proposed control framework also has the limitation that although it is certainly feasible, it is not guaranteed to be the optimum solution.

  3. 3.

    The stability of the proposed control framework has not been proved yet.

Overcoming the limitations and implementing an experimental demonstration will be the main orientation of further research. Besides, considering the potential impacts of WAS, there is room for improvement in the future. For example:

  1. 1.

    The situations that WAS adapts to can be extended from loading to rapid temperature changes, pre-stress loss, or structural damage.

  2. 2.

    Shape control of WAS is expected to extend from single-object control to multi-object control.

  3. 3.

    The proposed control framework can be enhanced and enriched by more advanced learning algorithms, more complex design of rhythmic patterns or self-evolution algorithms.

  4. 4.

    WAS provides a possibility for creating large-scale active grid-structures.