Dynamic modeling and configuration adaptive control for modular reconfigurable robot

Abstract

A unit module of the mobile modular reconfigurable robot (M²rBot) with the 12 connection ways is first introduced. For expressing the spatial connection relations and the traversing paths of the M²rBot, a novel method of the automatic generation of the spatial connection matrix is presented based on our proposed double-module space pose transformation. The motion equations are also automatically generated to solve the reconstruction problems of mechanical configuration, kinematics, and dynamics. This method enables to analyze the kinematics and dynamics for both serial and multi-branch modular robots with different configurations. As for the control strategies, a fuzzy robust adaptive controller is also developed to adapt to the different configurations without adjusting any control parameters. The results show that the novel method is effective, and the proposed controller can satisfy tracking performances and quickly adapt to the change of the configurations, the frictions, and the external disturbances. The proposed method and controller are easy to extend to other modular robots.

Introduction

Modular reconfigurable robot (MRR), composed of multiple modular components, is a variable topology system that can manually rearrange the modules’ connectedness and form a large variety of robot configurations. Due to the versatility and the replace ability, the MRR would have a great potential application to meet varieties of task requirements in unstructured and dynamic environments.^1–3 In a plain field, it can change its configuration into a loop to move fast. On the bumpy road, the robot can reconfigure itself into the four-leg or multi-leg shape to easily crawl over obstacles. The robot can also perform tasks that one single module or a fixed-structure robot cannot accomplish, and it can also append specialized units to the original configuration. The specialized units are grippers, welding guns, or sensor units such as cameras.

The MRR can be traced back to the CEBOT ⁴ introduced by Toshio Fukuda in the late 1980s and is divided into three categories: mobile architecture, lattice architecture, and chain architecture. ⁵ Utilizing the mobility, it can form the chain or the lattice architecture through the locking mechanism, and it can also implement more kinds of configurations. Several literatures of the mobile architecture were published, such as Trimobot, ⁶ UBot, ⁷ ModRED, ⁸ Scout robot, ⁹ AMOEBA, ¹⁰ Sambot, ¹¹ 360bot, ¹² M²sbot, ¹³ and OMNIMO. ¹⁴ Previous researchers mainly focused on the structure design^12–14 and the locomotion and control.^{6,7,9,15–17} However, the research of automatic generation of motion equations is not enough, and especially the configuration adaptive control is less.

The common kinematics methods are the Denavit–Hartenberg (D-H) method ¹⁸ and the product of exponential (POE) method. ¹⁹ Compared with the traditional D-H method, the POE method has more explicit geometric meaning and is more simple. Fei and Zhao ¹⁹ proposed the automatic generation of the forward kinematics for cuboid and cubic modules with POE and only analyzed the theoretical configuration of the reconfigurable mechanical system. Pan et al. ²⁰ derived the forward kinematics of the multi-chain MRR with different joints, links, bases, and end effectors by the assembly incidence matrix (AIM) and the path matrix. The described methodologies in the above works, however, are useful only for the special configurations, and the assembly representation is also very complex. When the assembly sequences are changed, the automatic generation becomes difficult due to the large number of intermediate variables. Bi et al. ²¹ applied the specific finite element method (FEM) to the robotic configuration with serial, parallel, or hybrid structures. Each module was regarded as a conceptual element which could be predefined. The method simplified the number of variables, but the spatial expression was more complex than the above two.

For a fixed-configuration robot, the dynamics equations are generally derived by manual. However, it can be challenging and time-consuming to obtain the dynamics equation of each configuration on the MRR. It is impossible and impractical to derive and compute the dynamics equations of each configuration because the configuration cannot be known in advance. Yang ²² proposed one unified modeling method of the underwater self-reconfigurable robot based on graph theory and path matrix using Kane’s method. Djuric and ElMaraghy ²³ provided any automatic generation of the dynamics equation matrices elements based on Newton–Euler method. The calculated parameters were all computed containing the joint position, velocity, acceleration, and input torque or forces. Fei et al. ²⁴ presented a compensative and recursive method for generating the dynamics equations automatically. The dynamics parameters could not be accurately computed with the compensative method. The dynamics equations based on Newton–Euler method are simple, fast, and precise, but the forward and inverse recursive equations destroy the whole structure of the dynamics model, so it is difficult to design the compensation controller.

The generated dynamics model is a nonlinear model. There are different possibilities to deal with nonlinearity either using Jacobi linearization or compensating the nonlinear terms through feedback techniques. One of the most used methods is to linearize the model at a certain operation point by use of Jacobi method. In recent years, the compensatory control method has attracted lots of researches and probably becomes the first choice.^25,26 Compared with the Jacobi method, the compensatory control method is to more precisely approximate the model uncertainty terms. For the MRR, the compensatory controller is designed not only for compensating uncertainties of one fixed-configuration model but also for adapting to the dynamics models with different configurations. The research on compensating uncertainties of different dynamics models using one single controller is a huge challenge, especially without adjusting any control parameters.

In this article, the main researches based on the above problems include the following. To solve the kinematics generation problem of multiple connection ways, a unique double-module space pose transformation (DMSPT) based on the screw form is first proposed. Second, inspired by the work from Chen and Yang, ²⁷ the kinematic and dynamics equations for multiple branches are derived by the recursive method with a more simple expression of spatial connection relationships among the modules. In order to avoid destroying the whole structure of the dynamics models, the closed-form dynamics equations can be derived according to different configurations. And then, a fuzzy robust adaptive controller is presented, which contains a feedback plus a robust term for the model uncertainties, frictions, and external disturbances. Finally, the simulation results are implemented to verify the effectiveness of the proposed controller for the motion equations of different configurations without adjusting any control parameters and to compensate simultaneously the frictions and external disturbances of different configurations.

M²rbot unit module description

For the M²rbot, the following characteristics are considered: the simple module structure which can move independently, the rich connecting surfaces and connecting ports, and the reliable locking mechanism. The basic design of the M²rbot is identical to the former 360bot ¹² and M²sbot ¹³ as shown in Figure 1. The M²rbot has similar sizes and similar physical properties to the 360bot and M²sbot, but it can be more firmly locked and be less weight. The robot has a slight mobile ability to adapt to the ground, and it can realize a reliable physical connection between modules. Table 1 summarizes the major specification parameters between the 360bot and the M²rbot.

Figure 1.

Developed modular robot: (a) 360bot, (b) M²sbot, and (c) M²rbot.

Table 1.

Major specification parameters between 360bot and M²rbot.

Parameters	360bot	M2rbot
Size and weight	100 mm × 100 mm × 128 mm, 1.41 kg	120 mm × 120 mm × 120 mm, 0.66 kg
Material	Duralumin	Acrylonitrile butadiene styrene
Motor	Link: proMOTION CDS5516 robotic steering engine	Link: 35YF22GN120R-TF0 step motor (power 12 V, 400 mA) Locking: 24BYJ-48 step motor (power 12 V, 600 mA)
Controller	Multi FLEX 2-AVR controller	Main: ARM Cortex™-A8 core Sub: STM32F103 (32 bit)
Maximum torque	16 kgf·cm	15 kgf·cm
Battery	Lithium 7.4 V850 mA h	Lithium 12 V1800 mA h
Docking mechanism	Manual	Pin-hole connecting mechanism

The M²rbot unit module is shown in Figure 2. It mainly comprises two U-type blocks and a rotating shaft. The inner U_b block can independently rotate around the shaft by ±90°. The U_b is used as the active connection surface (C1) to connect with the adjacent module. The outer U_a block has three passive connection surfaces (C2, C3, and C4) which to be connected with the other adjacent modules. Viewed from the top, the three passive surfaces are sequentially defined in a counter-clockwise direction. Although the three passive surfaces are not identical, they can be connected symmetrically in four possible connection ports and it can achieve more topological configurations. The typical topology configurations consisting of some modules are shown in Figure 3. With the increase in the modules, the number of configurations will increase exponentially.

Figure 2.

M²rbot unit module.

Figure 3.

Different configurations of M²rbot: (a) double-branch configuration, (b) quadruped configuration, (c) ring configuration, and (d) snake configuration.

All blocks are connected by dyads (a dyad is defined as two adjacent modules connected through a passive block and an active one). The dyad structure of the M²rbot unit module has the following properties. (1) Each block is always a neighbor of opposite block as a dyad. (2) The dyad has four connection ports, which can provide more rich connection configurations.

Module assemblies and kinematic generation

The unit modular robots can bring some possible assembly sequences. The kinematics varies according to the different configurations and assembly sequences between the modules. For the M²rbot, the kinematics starts with a two-module assembly model and a spatial connection matrix (SCM) representation.

Two-module assembly model

A unit module of the M²rbot can be simply abstracted as the model with two rigid bodies and a rotating shaft. The U_a of the ith module is noted as ia and the U_b as ib. As shown in Figure 2, the center point of the rotation shaft is the origin of the $\mathrm{xyz}$ coordinate system belonging to the ia, and the $x\prime y\prime z\prime$ belongs to the ib. The homogeneous representation of the ib with respect to the ia is given as follows

$$T_{ia,ib}=T_{ia,ib}(0)e^{{\widehat{s}}_i{\alpha }_i}=e^{{\widehat{s}}_i{\alpha }_i}$$

(1)

where ${\alpha }_i$ is the ib rotation angle with respect to ia, ${\hat{s}}_i\in \mathrm{se}(3)$ is the motion screw, and $T_{\mathrm{ia},\mathrm{ib}}(0)\in \mathrm{SE}(3)$ denotes the initial position.

There are 12 different connecting dyads for the DMSPT as listed in Table 2. The connected dyad between the connector $C_1$ of the ith module and the connector $C_k$ of the jth module with N corresponds to an edge and can be expressed as $e_{\mathrm{ij}}=〈C_1,N,C_k〉$ from the point of view of the module i. The connection dyad ja with respect to the ib can be directly obtained as follows

$$T_{\mathrm{ib},\mathrm{ja}}=\mathrm{Transy}(l)·I_k·(O_k{)}^N$$

(2)

where k denotes the passive connector of module j, $k=1,2,3$ , and N is the connecting port of the connector C_k, $N=1,\dots ,4$ . $\mathrm{Transy}(l)$ expresses the unit translation matrix, and l is the distance between the two modules’ coordinate frame. When $k$ and $N$ can be obtained, the relative pose of the connecting dyads can be obtained from equation (2). The representation of different passive connectors and different connecting ports can be easier to be described than other methods.^20,22,23,27

Table 2.

Double-modules space pose transformation (DMSPT).

k	Initial matrix	Port matrix	$N=0$	$N=1$	$N=2$	$N=3$
$k=1$	$I_1=\left[\begin{array}{cccc}0\hfill & -1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{1}& 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right]$	$O_1=\left[\begin{array}{cccc}1\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & -1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right]$
			$e_{\mathrm{ij}}=〈C_1,0,C_2〉$	$e_{\mathrm{ij}}=〈C_1,1,C_2〉$	$e_{\mathrm{ij}}=〈C_1,2,C_2〉$	$e_{\mathrm{ij}}=〈C_1,3,C_2〉$
$k=2$	$I_2=\left[\begin{array}{cccc}1\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right]$	$O_2=\left[\begin{array}{cccc}0\hfill & 0\hfill & 1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{-1}& 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right]$
			$e_{\mathrm{ij}}=〈C_1,0,C_3〉$	$e_{\mathrm{ij}}=〈C_1,1,C_3〉$	$e_{\mathrm{ij}}=〈C_1,2,C_3〉$	$e_{\mathrm{ij}}=〈C_1,3,C_3〉$
$k=3$	$I_3=\left[\begin{array}{cccc}0\hfill & 1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{-1}& 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right]$	$O_3=\left[\begin{array}{cccc}1\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& -1\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right]$
			$e_{\mathrm{ij}}=〈C_1,0,C_4〉$	$e_{\mathrm{ij}}=〈C_1,1,C_4〉$	$e_{\mathrm{ij}}=〈C_1,2,C_4〉$	$e_{\mathrm{ij}}=〈C_1,3,C_4〉$

Note that the initial matrix and the port matrix can be expressed by the screw form.

SCM representation with a branching structure

The assembly of multiple modules can represent not only a simple chain structure but also a complicated branching structure. When a base module is determined at a chain structure, the transformation order of the module starts from the base module. For a branching one, the shortest traversing paths can be obtained using the breadth-first search (BFS) or the depth-first search (DFS) algorithms. Note that the kinematics of a closed-loop configuration requires additional constraints and is not considered here.

The SCM, similar to a kinematic graph as a kinematic chain, is proposed for representing the spatial assemblies between the modules and their neighbor modules. A huge advantage of the SCM representation method is that one smaller matrix is only needed to store the spatial assemblies, and there is a one-to-one relationship between the spatial assemblies and the SCM. Compared with the AIM, ²⁷ the kinematics calculation can be done simultaneously for all branches by the SCM containing the traversing paths. The double-branch configuration in Figure 3(a) can be depicted as shown in Figure 4. The graph differs from an ordinary graph since it can express the connector directions. The directions can be expressed from the active connectors to the passive ones. The SCM can also be represented by a vertex-edge matrix. Furthermore, in order to adapt to the recursive kinematics, an additional row and additional column are inserted to the matrix for the joint angles and the connecting ports. The six-module configuration with two branches in Figure 4 can be obtained as follows.

$$\begin{array}{c}\mathrm{SCM}=\begin{array}{c}\begin{array}{c}{v}_1\hfill \\ \multicolumn{1}{c}{v_2}\\ \multicolumn{1}{c}{v_3}\\ \multicolumn{1}{c}{v_4}\\ \multicolumn{1}{c}{v_5}\\ \multicolumn{1}{c}{v_6}\\ \multicolumn{1}{c}{N}\end{array}\hfill \end{array}\left[\begin{array}{c}\begin{array}{cccccc}{e}_1\hfill & e_2\hfill & e_3\hfill & e_4\hfill & e_5\hfill & \mathrm{joints}\hfill \\ 1\hfill & 0\hfill & 0\hfill & 0\hfill & 0\hfill & {\alpha }_1\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 4\hfill & 0\hfill & 0\hfill & {\alpha }_2\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill & 0\hfill & 0\hfill & {\alpha }_3\hfill \\ \multicolumn{1}{c}{3}& 4\hfill & 0\hfill & 2\hfill & 0\hfill & {\alpha }_4\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill & 2\hfill & {\alpha }_5\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 0\hfill & 1\hfill & {\alpha }_6\hfill \\ 0\hfill & 0\hfill & 3\hfill & 0\hfill & 1\hfill & \hfill \end{array}\hfill \end{array}\right]\hfill \end{array}$$

(3)

Figure 4.

Kinematic graph representation.

This SCM can provide a flexible way to calculate the sequences of all possible transformations along the traversing paths during run time. For a branching configuration, the forward kinematics can be obtained from a chosen base module to each end link in all branches. The forward kinematic transformations for the path k with m branches, based on the dyad kinematics, can be formulated as follows

$$G=\left[\begin{array}{c}{G}_1\hfill \\ \multicolumn{1}{c}{G_2}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{G_k}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{G_m}\end{array}\right]=\left[\begin{array}{c}\cdots \hfill \\ \multicolumn{1}{c}{\cdots }\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{\underset{i=1}{\overset{n}{\Pi }}{T}_{ia,\mathrm{ib}}^k·T_{\mathrm{ib},(i+1)a}^k}\\ \multicolumn{1}{c}{⋮}\\ \multicolumn{1}{c}{\cdots }\end{array}\right]$$

(4)

where $G_k$ represents the 4 × 4 homogeneous matrix of the poses of the branch.

Robot dynamics modeling

Dynamics for the links assembly

The rotation shaft of the unit module is considered as “joint.” The U_a locked with one U_b or more is considered as “links.” Let the links ia and ib be the two links connected by joint i. If the two or more adjacent modules are rigidly connected, we consider the connection of joint i and link i as the links assembly i. Taken the six-module robot configuration with two branches as an example, the link assembly of the robot is shown in Figure 5.

Figure 5.

Link assembly of six-module configuration.

Suppose frame i is the body (or local) coordinate frame. The mass frame i* can be located on the mass center of the links assembly. In general, the two coordinate frames do not coincide with each other. The two frames belong to the same rigid link assembly, and then, for the ith unit module, we can get following

$$T_{\mathrm{ia},i{a}^{*}}=\left(\begin{array}{cccc}1\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0\hfill & l/4\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right),T_{\mathrm{ib},i{b}^{*}}=\left(\begin{array}{cccc}1\hfill & 0\hfill & 0\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 1\hfill & 0\hfill & -l/4\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 1\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0\hfill & 0\hfill & 1\hfill \end{array}\right)$$

(5)

Suppose links assembly i is composed of one joint i and more links, for a branched structure, $p_{i^{*},i}$ can be obtained as follows

$$p_{i,i^{*}}=\frac{(p_{i,{i_1}^{*}}+p_{i,{i_2}^{*}}+\cdots +p_{i,{i_n}^{*}})}{n}$$

(6)

where n denotes the number of the links, and there are $p_{i,i^{*}}=-p_{i^{*},i}$ and $R_{i,i^{*}}=I_{3\times 3}$ . The Newton–Euler equation of a rigid body with respect to its body coordinate at the mass center is given by the following ²⁷

$$\begin{array}{l}{F}_{i^{\ast }}=\left[\begin{array}{cc}{m}_{i}I& 0\\ 0& J_{i^{\ast }}\end{array}\right]\left[\begin{array}{c}{\stackrel{.}{v}}_{i^{\ast }}\\ {\stackrel{.}{\omega }}_{i^{\ast }}\end{array}\right]\\ -\left[\begin{array}{cc}-{\widehat{\omega }}_{i^{\ast }}& 0\\ -{\widehat{v}}_{i^{\ast }}& -{\widehat{\omega }}_{i^{\ast }}\end{array}\right]\left[\begin{array}{cc}{m}_{i}I& 0\\ 0& J_{i^{\ast }}\end{array}\right]\left[\begin{array}{c}{v}_{i^{\ast }}\\ {\omega }_{i^{\ast }}\end{array}\right]\end{array}$$

(7)

$$J_{i^{\ast }}=R_{i^{\ast },i_1^{\ast }}·J_1·{R^T}_{i^{\ast },i_1^{\ast }}+R_{i^{\ast },i_2^{\ast }}·J_2·{R^T}_{i^{\ast },i_2^{\ast }}+\mathrm{\dots }+R_{i^{\ast },i_n^{\ast }}·J_n·{R^T}_{i^{\ast },i_n^{\ast }}$$

(8)

where $F_{i^{*}}$ is the generalized force of the links assembly i. $f_{i^{*}}$ and ${\tau }_{i^{*}}$ are the forces and moments, respectively. $M_{i^{*}}$ is the generalized mass matrix. $m_i$ is the total mass of the links assembly i, which contains the mass of the joint i, the link ib and the link (i + 1)a. $J_{i^{*}}$ is the inertia tensor. The above equations can be transformed into the adjoint representation

$$F_{i^{*}}=M_{i^{*}}{\stackrel{·}{V}}_{i^{*}}-{\mathrm{ad}}_{V_{i^{*}}}^{*}\left(M_{i^{*}}{V}_{i^{*}}\right)$$

(9)

where $V_{i^{*}}$ is the generalized body velocity, ${\stackrel{·}{V}}_{i^{*}}$ is the generalized body acceleration, and ${\mathrm{ad}}_{V_{i^{*}}}^{*}$ is the transpose of the adjoint matrix $a{d}_{V_{i^{*}}}$ as follows

$${\mathrm{ad}}_{V_{i^{*}}}^{*}={\left(a{d}_{V_{i^{*}}}\right)}^T={\left[\begin{array}{cc}{\hat{\omega }}_{i^{*}}\hfill & {\hat{v}}_{i^{*}}\hfill \\ \multicolumn{1}{c}{0}& {\hat{\omega }}_{i^{*}}\hfill \end{array}\right]}^T=\left[\begin{array}{cc}-{\hat{\omega }}_{i^{*}}\hfill & 0\hfill \\ \multicolumn{1}{c}{-{\hat{v}}_{i^{*}}}& -{\hat{\omega }}_{i^{*}}\hfill \end{array}\right]$$

(10)

Equation (9) is not relative to the above kinematic equation of the body frame, so it must be transformed to the body frame i. So, we use the following identities

$$F_i={\mathrm{Ad}}_{T_{i^{*},i}}^{*}{F}_{i^{*}}={\left(A{d}_{T_{i^{*},i}}\right)}^T{F}_{i^{*}}$$

(11)

$$V_{i^{*}}=A{d}_{T_{i^{*},i}}{V}_i,{\stackrel{·}{V}}_{i^{*}}=A{d}_{T_{i^{*},i}}{\stackrel{·}{V}}_i$$

(12)

$${\stackrel{·}{V}}_{i^{*}}=A{d}_{T_{i^{*},i}}{\stackrel{·}{V}}_i$$

(13)

$$A{d}_{T_{i^{*},i}}(a{d}_{V_i})=a{d}_{A{d}_{T_{i^{*},i}}}(V_i)A{d}_{T_{i^{*},i}}$$

(14)

Substituting equations (11)–(13) into equation (9), and using equation (14), Newton–Euler equation of generalized link assembly i with respect to the body frame i can be rewritten as follows

$$F_i=M_i{\stackrel{·}{V}}_i-{\mathrm{ad}}_{V_i}^T(M_i{V}_i)$$

(15)

where $M_i={\mathrm{Ad}}_{T_{i^{*},i}}^{*}{M}_{i^{*}}A{d}_{T_{i^{*},i}}=\left[\begin{array}{cc}{m}_{i}I\hfill & m_i{R}_{i^{*},i}^T{\hat{p}}_{i^{*},i}{R}_{i^{*},i}\hfill \\ \multicolumn{1}{c}{-m_i{R}_{i^{*},i}^T{\hat{p}}_{i^{*},i}{R}_{i^{*},i}}& R_{i^{*},i}^T(J_{i^{*}}-m_i{\hat{p}}_{i^{*},i}^2)R_{i^{*},i}\hfill \end{array}\right]$ .

Newton–Euler recursive equation

The first considering point of the MRR is that the dynamics model of the robot assembly cannot be known a priori. Therefore, the robot should be able to generate its own equations autonomously without human intervention. The original idea for recursive equations and its closed forms were introduced by Chen and Yang. ²⁷ In this article, starting with the SCM that contains all the information how to be assembled, the equations of motion can be done by following steps. Starting from the base module to end actuators, the SCM is gradually filled based on path search algorithms such as BFS or DFS. After the SCM is built and all paths are determined, the recursive equations can be started to be calculated.

The forward recursive velocity and acceleration can be written as follows

$$V_i=A{d}_{T_{i-1,i}^{-1}}(V_{i-1})+s_i{\stackrel{·}{q}}_i$$

(16)

$${\stackrel{·}{V}}_i=s_i{\stackrel{··}{q}}_i+A{d}_{T_{i-1,i}^{-1}}({\stackrel{·}{V}}_{i-1})-a{d}_{s_i{\stackrel{·}{q}}_i}(V_i)$$

(17)

$A{d}_{T_{i-1,i}^{-1}}$ is considered as the adjoint matrix of $T_{i-1,i}^{-1}$ used to transform from $V_{i-1}$ to $V_i$ , and then, the relationship can be expressed as $A{d}_{T_{i-1,i}^{-1}}=(A{d}_{T_{i-1,i}}{)}^{-1}$ ; $a{d}_{s_i{\stackrel{·}{q}}_i}(V_i)$ indicates $V_i$ linear mapping via Lie bracket. Suppose $s_i{\stackrel{·}{q}}_i=x=(v_1,{\omega }_1)$ and $V_i=y=(v_2,{\omega }_2)$ , then we can obtain the following $a{d}_x(y)=\left[\begin{array}{cc}{\hat{\omega }}_1\hfill & {\hat{v}}_1\hfill \\ \multicolumn{1}{c}{0}& {\hat{\omega }}_1\hfill \end{array}\right]\left[\begin{array}{c}{v}_2\hfill \\ \multicolumn{1}{c}{{\omega }_2}\end{array}\right]$ .

As for the backward iteration, equation (15) can be written in the following form

$$F_i={\mathrm{Ad}}_{T_{i,i+1}^{-1}}^T(F_{i+1})+M_i{\stackrel{·}{V}}_i-{\mathrm{ad}}_{V_i}^T(M_i{V}_i)$$

(18)

$${\tau }_i=s_i^T{F}_i$$

(19)

where ${\tau }_i$ is the applied torque/force to the link assembly i by actuators.

Closed-form equations of dynamics

By expanding Newton–Euler equations (16)–(19) in the body coordinates, the generalized velocity, generalized acceleration, and generalized force equations can be obtained from the following global matrix representations

$$V=\mathrm{Gs}\stackrel{·}{q}$$

(20)

$$\stackrel{·}{V}=\mathrm{Gs}\stackrel{··}{q}+\mathrm{Ga}{d}_{s\stackrel{·}{q}}(\Gamma V)+G{P}_0{\stackrel{·}{V}}_0$$

(21)

$$F=G^{T}M\stackrel{·}{V}+G^T{\mathrm{ad}}_V^{*}(\mathrm{MV})+G^T{P}_t^T{F}_{n+1}$$

(22)

$$\tau =s^{T}F$$

(23)

Substituting equations (20)–(22) into equation (23), we can obtain the closed-form equations of dynamics for a branching structure with (n + 1) modules

$$M(q)\stackrel{··}{q}+C(q,\stackrel{·}{q})\stackrel{·}{q}+N(q)=\tau$$

(24)

where

$$M(q)=s^T{G}^T\mathrm{MGs}$$

(25)

$$C(q,\stackrel{·}{q})=s^T{G}^T(\mathrm{MGa}{d}_{s\stackrel{·}{q}}\Gamma +{\mathrm{ad}}_V^{*}M)\mathrm{Gs}$$

(26)

$$N(q)=s^T{G}^T\mathrm{MG}{P}_0{\stackrel{·}{V}}_0+s^T{G}^T{P}_t^T{F}_{n+1}$$

(27)

where $M(q)$ is the mass matrix. $C(q,\stackrel{·}{q})$ is the centrifugal and Coriolis accelerations. $N(q)$ represents the gravitational force and external forces.

Configuration adaptive control

Previous researchers have focused on the dynamics modeling and the compensatory control method for a fixed configuration, and the researches of the configuration adaptive control for the MRR have been less in the past years. It is difficult to compensate the different model uncertainties without adjusting any control parameters. To deal with approximation errors, external disturbances, and frictions on different models, the adaptive fuzzy controller is appended by a robust compensator. For practical purposes, the controller may be applicable for the MRR due to dealing with the complex systems and implementing the controller without any precise knowledge of the exact dynamics model.

According to equation (24), the dynamics of the MRR incorporating the external disturbance and the friction can be represented by the following form

$$M(q)\stackrel{··}{q}+C(q,\stackrel{·}{q})\stackrel{·}{q}+G(q)+F(\stackrel{·}{q})=\tau -{\tau }_e$$

(28)

where $G(q)$ is the gravity vector, $F(\stackrel{·}{q})$ denotes the friction term, and ${\tau }_e$ represents the external disturbances. The control objective is to design adaptive fuzzy control laws based on equation (28), and the MRR can show a desired trajectory, guarantee boundedness of all variables of the closed-loop system, and achieve the tracking performances.

Rewriting equation (28), the equation can be shown as follows

$$\stackrel{··}{q}=M^{-1}(q)\tau -M^{-1}(q)[C(q,\stackrel{·}{q})\stackrel{·}{q}+G(q)+F(\stackrel{·}{q})-{\tau }_e]$$

(29)

Let us suppose

$$H(q)=-M^{-1}(q)[C(q,\stackrel{·}{q})\stackrel{·}{q}+G(q)+F(\stackrel{·}{q})-{\tau }_e]$$

(30)

Substituting equation (30) into equation (29), we can get the following

$$\stackrel{··}{q}=H(q)+M^{-1}(q)\tau$$

(31)

Define $e_i(t)=q_{\mathrm{di}}-q_i$ as the trajectory tracking error vector and $s_i(t)=(d/\mathrm{dt}+{\lambda }_i)e_i(t)$ as the filtering tracking error vector where ${\lambda }_i>0$ and $q_{\mathrm{di}}$ are the desired trajectory vectors of the respective joint position. It follows that $e_i(t)\to 0$ asymptotically as $s_i(t)\to 0$ for $i=1,\dots ,n$ . The control objective is to force $s_i(t)\to 0$ . Using the Newton binomial theorem, the time derivatives of the filtered errors can be written with the compact form

$$\stackrel{·}{s}=v-H(q)-M^{-1}(q)\tau$$

(32)

where $v_i(t)={\stackrel{··}{q}}_{\mathrm{di}}+{\lambda }_i{\stackrel{·}{e}}_i$ .

If the nonlinear function $H(q)$ is known, we consider the following nonlinear control law for achieving the control objectives

$$\tau =M(q)(-H(q)+v+K_{0}s)$$

(33)

where $K_0=\mathrm{diag}[k_{01},\dots ,k_{0p}]$ with $k_{0i}>0(i=1,\dots ,p)$ , and it is a positive-definite constant diagonal matrix. However, the uncertainty term $H(q)$ cannot be obtained accurately. Therefore, the control law (33) cannot be applied directly. In this article, the fuzzy system is proposed to approximate the unknown nonlinear function $H(q)$ . For one unfixed configuration model, the uncertainty term $H(q)$ is drastically changed. It is difficult to approximate $H(q)$ only by the fuzzy systems. So, we append a robust term to adapt to the unfixed model error for different configurations.

Let us suppose $\hat{H}(q,{\theta }_h)$ is the fuzzy approximation of the unknown nonlinear function $H(q)$ . The control law (33) for compensating torque using the adaptive fuzzy approximations $\hat{H}(q,{\theta }_h)$ instead of the actual function $H(q)$ can be rewritten as follows

$${\tau }_{\mathrm{c}}=M(q)(-\hat{H}(q,{\theta }_{\mathrm{h}})+v+K_{0}s)$$

(34)

where ${\hat{h}}_i(q,{\theta }_{h_i})={\xi }_{h_i}^T(q){\theta }_{h_i}$ , ${\xi }_{h_i}(q)$ is the fuzzy radial basis function vector, and ${\theta }_{h_i}$ is the adaptive adjustable parameter vector. The optimal approximation parameter ${\theta }_{h_i}^{*}$ , the parameter estimation error ${\tilde{\theta }}_{h_i}$ , and the minimum fuzzy approximation error ${\epsilon }_{h_i}(q)$ of the ${\theta }_{h_i}$ can be defined as follows

$${\theta }_{h_i}^{*}=\arg \underset{{\theta }_{h_i}}{min}\{\underset{q\in D_q}{sup|}{h}_i(q)-{\hat{h}}_i(q,{\theta }_{h_i})|\}$$

(35)

$${\tilde{\theta }}_{h_i}={\theta }_{h_i}^{*}-{\theta }_{h_i}$$

(36)

$${\epsilon }_{h_i}(q)=h_i(q)-{\hat{h}}_i(q,{\theta }_{h_i}^{*})$$

(37)

where ${\theta }_{h_i}^{*}$ is an artificial constant quantity introduced only for analytical purpose and not needed for the implementation. Let us suppose $q\in D_q$ , $D_q$ is the compact set, ${\epsilon }_{h_i}(q)$ is bounded with $|{\epsilon }_{h_i}(q)|\le {\overline{\epsilon }}_{h_i}$ , and ${\overline{\epsilon }}_{h_i}$ is a given constant. From the above analysis, we have the following

$$H(q)-\hat{H}(q,{\theta }_h)=\hat{H}(q,{\theta }_h^{*})-\hat{H}(q,{\theta }_h)+{\epsilon }_h(q)$$

(38)

Up to now, even though the control law (34) is always well defined, it cannot guarantee the stability of the robot’s closed-loop system alone. The reason is partly due to the unavoidable unfixed errors of the unknown function $H(q)$ and partly due to the unmeasured and unpredictable external disturbance ${\tau }_e$ in $H(q)$ . To reduce these approximations errors, we append a robust term ${\tau }_{\mathrm{r}}$ to the controller (34)

$$\tau ={\tau }_{\mathrm{c}}+{\tau }_{\mathrm{r}}$$

(39)

$${\tau }_{\mathrm{r}}=\frac{s\left|s^T\right|({\overline{\epsilon }}_h(q)+\left|{\tau }_0\right|)}{{\sigma }_0{\Vert s\Vert }^2+\delta }$$

(40)

where $\delta$ is a time-varying design parameters, ${\sigma }_0$ is a specific real number which satisfied $M^{-1}(q)\ge {\sigma }_0{I}_n$ , and ${\tau }_0$ is defined as follows

$${\tau }_0={\epsilon }_0(-\hat{H}(q,{\theta }_h)+v+K_{0}s)$$

(41)

To meet the control objectives, the adaptive parameter ${\theta }_{h_i}$ and the design parameter $\delta$ are updated by the following adaptive laws

$${\stackrel{·}{\theta }}_{h_i}=-{\eta }_{h_i}{\xi }_{h_i}^T(q)s_i$$

(42)

$$\stackrel{·}{\delta }=-{\eta }_0\frac{1}{{\sigma }_0{\Vert s\Vert }^2+\delta }\left|s^T\right|({\overline{\epsilon }}_h(q)+\left|{\tau }_0\right|)$$

(43)

Substituting equations (39)–(43) into equation (32), we get the following

$$\begin{array}{c}\multicolumn{1}{c}{s^T\stackrel{·}{s}=-\sum _{i=1}^p({\xi }_{h_i}^T(q)({\theta }_{h_i}^{*}-{\theta }_{h_i})s_i)-s^T{K}_{0}s}\\ \multicolumn{1}{c}{-s^T{M}^{-1}(q){\tau }_{\mathrm{r}}-s^T{\epsilon }_{\mathrm{h}}(q)}\end{array}$$

(44)

The following Lyapunov function can be defined by

$$V=\frac{1}{2}{s}^{T}s+\frac{1}{2}\sum _{i=1}^p\frac{1}{{\eta }_{h_i}}{\tilde{\theta }}_{h_i}^T{\tilde{\theta }}_{h_i}+\frac{1}{2{\eta }_0}{\delta }^2$$

(45)

Differentiating equation (45), we can get the following

$$\stackrel{·}{V}=s^T\stackrel{·}{s}-\sum _{i=1}^p\frac{1}{{\eta }_{h_i}}{\tilde{\theta }}_{h_i}^T{\stackrel{·}{\theta }}_{h_i}+\frac{1}{{\eta }_0}\delta \stackrel{·}{\delta }$$

(46)

Substituting equation (44) into equation (46), the equation can be obtained as follows

$$\stackrel{·}{V}=-s^T{K}_{0}s+{\stackrel{·}{V}}_1+{\stackrel{·}{V}}_2$$

(47)

where

$${\stackrel{·}{V}}_1=-\sum _{i=1}^p({\xi }_{h_i}^T(q)({\theta }_{h_i}^{*}-{\theta }_{h_i})s_i)-\sum _{i=1}^p\frac{1}{{\eta }_{h_i}}{\tilde{\theta }}_{h_i}^T{\stackrel{·}{\theta }}_{h_i}$$

(48)

$${\stackrel{·}{V}}_2=-s^T{M}^{-1}(q){\tau }_{\mathrm{r}}-s^T{\epsilon }_{\mathrm{h}}(q)+\frac{1}{{\eta }_0}\delta \stackrel{·}{\delta }$$

(49)

Substituting equation (42) into equation (48), we can obtain ${\stackrel{·}{V}}_1=0$ . Substituting equation (40) into equation (49), it is easy to get ${\stackrel{·}{V}}_2\le -\left|s^T\right|\left|{\tau }_0\right|\le 0$ . From the above results, the derivation of Lyapunov function must be satisfied as follows

$$\stackrel{·}{V}\le -s^T{K}_{0}s-\left|s^T\right|\left|{\tau }_0\right|\le -s^T{K}_{0}s=-\sum _{i=1}^p{k}_{0i}{s}_i^2$$

(50)

It is clear that $\stackrel{·}{V}$ is the negative semi-definite outside a compact set $D_q$ . So, the variables $s_i(t)$ , ${\tilde{\theta }}_{h_i}(t)$ , ${\theta }_{h_i}(t)$ , $q$ , $\tau$ , and ${\stackrel{·}{s}}_i(t)$ are bounded.

By integrating equation (50) from $t=0$ to $t=\infty$ , we have

$$\underset{0}{\overset{\infty }{\int }}\sum _{i=1}^p{k}_i{s}_i^2(t)\mathrm{d}t\le V(0)-V(t)<\infty$$

(51)

where $s_i(t)\in L^2$ .

Using Barbalat’s Lemma, ²⁸ we can have a conclusion that $\underset{t\to \infty }{lim}{s}_i(t)\to 0$ . Therefore, the tracking errors $\underset{t\to \infty }{lim}{e}_i(t)\to 0$ and $\underset{t\to \infty }{lim}{\stackrel{·}{e}}_i(t)\to 0$ can be obtained.

Simulation results

To evaluate the automatic generation of motion equations, an example of the six-module configuration with two branches has been implemented. ²⁹ Another example with one branch of the three different configurations can also be automatically generated. The proposed adaptive controller is applied to the second example as shown in Figure 6. The end module of the branch can be supposed as the terminal actuator without rotating. Therefore, the configurations (a) and (b) shown in Figure 6(a) and (b) have the same two degrees of freedom (DOFs), but its connection ports are different. The configuration (c) shown in Figure 6(c) has three DOFs. Once the SCMs and the parameters of the module ( $m=1\mathrm{kg}$ , $l=0.1\mathrm{m}$ , and $g=9.8\mathrm{m}/{\mathrm{s}}^2$ ) are input into the system, the dynamics models with the above three configurations are generated by

• Configuration (a)

$$\begin{array}{c}M(q)=\left[\begin{array}{cc}m{L}^2(c_2+1.83)\hfill & m{L}^2(0.5c_2+0.42)\hfill \\ \multicolumn{1}{c}{m{L}^2(0.5c_2+0.42)}& 0.42m{L}^2\hfill \end{array}\right]\hfill \\ C(q,\stackrel{·}{q})=\left[\begin{array}{cc}-m{L}^2{\stackrel{·}{q}}_2{s}_2\hfill & -0.5m{L}^2{\stackrel{·}{q}}_2{s}_2\hfill \\ \multicolumn{1}{c}{0.5m{L}^2({\stackrel{·}{q}}_1-{\stackrel{·}{q}}_2)s_2}& 0.5m{L}^2{\stackrel{·}{q}}_1{s}_2\hfill \end{array}\right]\hfill \\ G(q)=\left[\begin{array}{c}\mathrm{mgL}(0.5s_{12}+1.5s_1)\hfill \\ \multicolumn{1}{c}{0.5\mathrm{mgL}{s}_{12}}\end{array}\right]\hfill \end{array}$$

(52)

• Configuration (b)

$$\begin{array}{c}M(q)=\left[\begin{array}{cc}m{L}^2(c_2+0.25c_2^2+1.58)\hfill & 0\hfill \\ \multicolumn{1}{c}{0}& 0.42m{L}^2\hfill \end{array}\right]\hfill \\ C(q,\stackrel{·}{q})=\left[\begin{array}{cc}-0.5m{L}^2{\stackrel{·}{q}}_2(c_2+2)s_2\hfill & 0\hfill \\ \multicolumn{1}{c}{0.25m{L}^2{\stackrel{·}{q}}_1(c_2+2)s_2}& 0\hfill \end{array}\right]\hfill \\ G(q)=\left[\begin{array}{c}\mathrm{mgL}(0.5c_2+1.5)s_1\hfill \\ \multicolumn{1}{c}{0.5\mathrm{mgL}{c}_1{s}_2}\end{array}\right]\hfill \end{array}$$

(53)

• Configuration (c)

$$\begin{array}{c}M(q)=\left[\begin{array}{ccc}m{L}^2(s_2{s}_3+0.25s_3^2-c_3-0.25s_2^2{s}_3^2-4.25)\hfill & -0.25m{L}^2{c}_2{s}_3(c_3+2)\hfill & 0.08m{L}^2(5s_2-6s_3+6c_3{s}_2)\hfill \\ \multicolumn{1}{c}{-0.25m{L}^2{c}_2{s}_3(c_3+2)}& m{L}^2(0.25s_3^2+0.33)\hfill & 0\hfill \\ \multicolumn{1}{c}{0.08m{L}^2(5s_2-6s_3+6c_3{s}_2)}& 0\hfill & 0.42m{L}^2\hfill \end{array}\right]\hfill \\ C(q,\stackrel{·}{q})=\left[\begin{array}{ccc}\begin{array}{c}-0.5m{L}^2(2{\stackrel{·}{q}}_3{s}_3-{\stackrel{·}{q}}_2{c}_2{s}_2\hfill \\ \multicolumn{1}{c}{+2{\stackrel{·}{q}}_2{c}_2{s}_3+2{\stackrel{·}{q}}_3{c}_3{s}_2}\\ \multicolumn{1}{c}{+{\stackrel{·}{q}}_2{c}_2{c}_3^2{s}_2+{\stackrel{·}{q}}_3{c}_2^2{c}_3{s}_3)}\end{array}\hfill & \begin{array}{c}0.25m{L}^2({\stackrel{·}{q}}_3{c}_2-2{\stackrel{·}{q}}_3{c}_2{c}_3\hfill \\ \multicolumn{1}{c}{+2{\stackrel{·}{q}}_2{s}_2{s}_3-2{\stackrel{·}{q}}_3{c}_2{c}_3^2}\\ \multicolumn{1}{c}{+{\stackrel{·}{q}}_2{c}_3{s}_2{s}_3)}\end{array}\hfill & \begin{array}{c}-0.08m{L}^2(5{\stackrel{·}{q}}_2{c}_2-6{\stackrel{·}{q}}_3{c}_3\hfill \\ \multicolumn{1}{c}{+6{\stackrel{·}{q}}_2{c}_2{c}_3-6{\stackrel{·}{q}}_3{s}_2{s}_3)}\end{array}\hfill \\ \multicolumn{1}{c}{\begin{array}{c}m{L}^2(0.25{\stackrel{·}{q}}_3{c}_2-0.5{\stackrel{·}{q}}_3{c}_2{c}_3\hfill \\ \multicolumn{1}{c}{-0.25{\stackrel{·}{q}}_1{c}_2{s}_2+0.5{\stackrel{·}{q}}_1{c}_2{s}_3}\\ \multicolumn{1}{c}{+0.5{\stackrel{·}{q}}_2{s}_2{s}_3-0.5{\stackrel{·}{q}}_3{c}_2{c}_3^2}\\ \multicolumn{1}{c}{+0.25{\stackrel{·}{q}}_2{c}_3{s}_2{s}_3+0.25{\stackrel{·}{q}}_1{c}_2{c}_3^2{s}_2)}\end{array}}& \begin{array}{c}0.25m{L}^2(2{\stackrel{·}{q}}_1{s}_2-2{\stackrel{·}{q}}_3{c}_3\hfill \\ \multicolumn{1}{c}{+{\stackrel{·}{q}}_1{c}_3{s}_2)s_3}\end{array}\hfill & 0.08m{L}^2{\stackrel{·}{q}}_1{c}_2(6c_3+5)\hfill \\ \multicolumn{1}{c}{\begin{array}{c}0.25m{L}^2(2{\stackrel{·}{q}}_3{c}_3-{\stackrel{·}{q}}_2{c}_2\hfill \\ \multicolumn{1}{c}{+2{\stackrel{·}{q}}_1{s}_3+2{\stackrel{·}{q}}_1{c}_3{s}_2+2{\stackrel{·}{q}}_3{s}_2{s}_3}\\ \multicolumn{1}{c}{+{\stackrel{·}{q}}_2{c}_2{c}_3^2+{\stackrel{·}{q}}_1{c}_2^2{c}_3{s}_3)}\end{array}}& \begin{array}{c}-0.08m{L}^2(3{\stackrel{·}{q}}_2{s}_3{c}_3+5{\stackrel{·}{q}}_1{c}_2\hfill \\ \multicolumn{1}{c}{-3{\stackrel{·}{q}}_1{c}_2{c}_3^2)}\end{array}\hfill & -0.5m{L}^2{\stackrel{·}{q}}_1(c_3+s_2{s}_3)\hfill \end{array}\right]\hfill \\ G(q)=\left[\begin{array}{c}-\mathrm{mgL}(1.5c_1-2.5s_1+0.5c_1{c}_3+0.5s_1{s}_2{s}_3)\hfill \\ \multicolumn{1}{c}{0.5\mathrm{mgL}{c}_1{c}_2{s}_3}\\ \multicolumn{1}{c}{0.5\mathrm{mgL}(s_1{s}_3+c_1{c}_3{s}_2)}\end{array}\right]\hfill \end{array}(54)$$

(54)

where $c_i=\cos q_i$ , $s_i=\sin q_i$ , $c_{\mathrm{ij}}=\cos (q_i+q_j)$ , and $s_{\mathrm{ij}}=\sin (q_i+q_j)$ .

Figure 6.

Three different configurations with one branch: (a) 2 DOF configuration with the same connection way, (b) 2 DOF configuration with the different connection way, and (c) 3 DOF configuration with the different connection way.

The control objective is to force the joint outputs $q_1$ , $q_2$ , and $q_3$ to track the desired trajectories $q_{d1}=(\pi /2)\cos (\pi t/4)$ , $q_{d2}=(\pi /2)\sin (\pi t/4)$ , and $q_{d3}=(\pi /2)\sin (\pi t/4+\pi /4)$ , respectively. The friction is $f_i=0.5\mathrm{Sgn}({\stackrel{·}{q}}_i)+2u_t$ , and the disturbance is assumed as ${\tau }_{\mathrm{ei}}=0.5\sin t+2u_t$ , where $u_t$ is a unit step signal and $t\in [3.8,4.2]$ . The fuzzy adaptive controller does not require the system’s dynamics model. Suppose the components of $H(q)$ are unknown, the fuzzy systems are used to approximate the components of $H(q)$ . For each state variable $x$ , we can define the following three Gaussian membership functions

$$\begin{array}{c}{u}_i(1)=\exp (-\frac{1}{2}{\left(\frac{x_i+1.25}{0.6}\right)}^2)\hfill \\ u_i(2)=\exp (-\frac{1}{2}{\left(\frac{x_i}{0.6}\right)}^2)\hfill \\ u_i(3)=\exp (-\frac{1}{2}{\left(\frac{x_i-1.25}{0.6}\right)}^2)\hfill \\ \{\begin{array}{c}i=1,\dots ,4(n=2)\hfill \\ \multicolumn{1}{c}{i=1,\dots ,6(n=3)}\end{array}\hfill \end{array}$$

(55)

The robot’s initial conditions are $x(0)=[0.500.250{]}^T$ and $x(0)=[0.500.2500.250{]}^T$ and the initial values of the estimated parameters are ${\theta }_f(0)={\theta }_g(0)=0$ . The control parameters are chosen as ${\sigma }_0=0.2$ , $K_0=0.2I$ , ${\epsilon }_0=0.2$ , ${\overline{\epsilon }}_h=0.1$ , ${\eta }_{\mathrm{hi}}=0.5$ , ${\eta }_0=0.001$ , and ${\lambda }_i=\mathrm{diag}(25,25)$ .

The simulation results of configuration (a) without the friction and disturbance are shown in Figure 7, and those with the friction and disturbance are also shown in Figure 8. The results for configurations (b) and (c) with the same friction and disturbance are shown in Figures 9 and 10, respectively.

Figure 7.

Configuration (a) without $F(\stackrel{·}{q})$ and ${\tau }_e$ : (a) joint position, (b) tracking error, (c) joint velocity, and (d) control input torque.

Figure 8.

Configuration (a) with $F(\stackrel{·}{q})$ and ${\tau }_e$ : (a) joint position, (b) tracking error, (c) joint velocity, and (d) control input torque.

Figure 9.

Configuration (b) with $F(\stackrel{·}{q})$ and ${\tau }_e$ : (a) joint position, (b) tracking error, (c) joint velocity, and (d) control input torque.

Figure 10.

Configuration (c) with $F(\stackrel{·}{q})$ and ${\tau }_e$ : (a) joint position, (b) tracking error, (c) joint velocity, and (d) control input torque.

For configuration (a), we can see that the desired and actual trajectories shown in Figures 7 and 8 almost overlap with each other regardless of the frictions and disturbances. When the sudden friction and disturbance appears, the tracking trajectory has almost no change and the proposed control method can rapidly adapt to the sudden changes. But the tracking error, the joint velocity, and the input torque are affected by the sudden disturbance. When the sudden disturbance passed, the above results can quickly restore. Comparing Figure 7(b) with Figure 8(b), we can see that the tracking errors are quickly converging into a small limit area near zero.

From Figures 9 and 10, we can see that the actual outputs can similarly track the desired trajectories with two different configurations. Therefore, the control method can approximate the dynamics equations with the unknown terms. Because the tracking errors are quickly converging into a small limit area near zero, the algorithm can adapt to the different configurations. From Figures 8 to 10, the joint tracking errors, based on the periodic disturbances $0.5\sin t$ , have a rapid following response and enter quickly into a steady state when the controller is appended by the sudden disturbance.

From these local amplification diagrams, the response curves are clearer to be seen. It can be shown that the control method can effectively compensate the whole unknown terms. The tracking velocities will almost overlap the desired velocity except for the time area $t\in [3.8,4.2]$ . The input torques is also relatively smooth except for the time area $t\in [3.8,4.2]$ and always within a bounded state.

In the actual experiment, more mechanistic interferences are needed to be considered such as the increment in the torque caused by the friction, the mechanical and communicational delay, the vibration caused by the two adjacent modules to be unstably locked and the decrease in the control performance caused by the system modeling errors.

Conclusion

In this article, a novel method for automatic generation of motion equations is proposed for the multi-branch MRR, and the fuzzy adaptive robust controller with adapting the different configurations is also presented. The method enables to analyze the kinematics and dynamics of modular robots, and the adaptive controllers are also implemented for both the serial and the multi-branch configuration with different connectors and different connecting ports by the DMSPT. The dynamics equations of both the serial and the multi-branch configuration are derived by the Newton–Euler recursive method. Therefore, when the SCM is input, the dynamics equations can be automatically generated. Different examples are used to evaluate the serial and the multi-branch configurations. The proposed fuzzy adaptive robust controller can compensate the dynamics model uncertainties of different configurations. The simulation results can not only illustrate the effectiveness on tracking performance, but also it quickly adapt to the change in the configurations, the frictions, and the external disturbances. The method will be used to the reconfigurable fixture system in the sheet metal assembly line for the automotive industry, which can satisfy the ever-increasing demand for automotive product variety as well as shortens the product changeover time. We will also replace the end effectors of the robot for adapting the different tasks, that is, wielding or grinding.