Locomotion Optimization and Manipulation Planning of a TetrahedronBased Mobile Mechanism with Binary Control
 Ran Liu^{1},
 YanAn Yao^{1}Email author,
 Wan Ding^{2} and
 XiaoPing Liu^{1, 3}
https://doi.org/10.1186/s1003301802158
© The Author(s) 2018
Received: 4 April 2016
Accepted: 15 January 2018
Published: 27 February 2018
Abstract
Locomotion and manipulation optimization is essential for the performance of tetrahedronbased mobile mechanism. Most of current optimization methods are constrained to the continuous actuated system with limited degree of freedom (DOF), which is infeasible to the optimization of binary control multiDOF system. A novel optimization method using for the locomotion and manipulation of an 18 DOFs tetrahedronbased mechanism called 5TET is proposed. The optimization objective is to realize the required locomotion by executing the least number of struts. Binary control strategy is adopted, and forward kinematic and tipping dynamic analyses are performed, respectively. Based on a developed genetic algorithm (GA), the optimal number of alternative struts between two adjacent steps is obtained as 5. Finally, a potential manipulation function is proposed, and the energy consumption comparison between optimal 5TET and the traditional wheeled robot is carried out. The presented locomotion optimization and manipulation planning enrich the research of tetrahedronbased mechanisms and provide the instruction to the successive locomotion and operation planning of multiDOF mechanisms.
Keywords
1 Introduction
Terrestrial mobile robots mainly include wheeled, tracked, legged, hybrid, snakelike, and spherical robots [1, 2], which have different adaptability and locomotion modes. Different from conventional robots that actuated by motors inside to hold a constant shape, some mechanisms realize locomotion and manipulation functions by shifting shapes with the change of the linkages [3]. In recent years, some linkage structures constructed by basic or special geometry configurations, such as Euclidean polyhedron, have been proposed. Conformable tetrahedrons are the simplest spacefilling form in the same way triangles are the simplest planefilling facets [4]. The tetrahedronbased mobile mechanism has been commonly explored.
Tetrahedronbased mobile mechanism is a class of hyperredundant robots, which has multiple DOFs and kinematic redundancy [5]. Different from wheeled, tracked, and legged robots, hyperredundant mechanisms arise from internally induced deforming. Since the advantage of fault tolerance [6], it is superior for operation in highly constrained environments, for instance, in uneven terrain with some simple operation tasks, such as nuclear reactor cores, toxic abandoned factories, and many other conceivable environments [7].
Several tetrahedronbased robots have been designed, and the emphasis mainly on building novel structures to produce feasible gait patterns. Based on the mathematical models of 4TET, 8TET, 12TET Walkers and tetrahedron worm built by Abrahantes et al. [4, 8], the choreographed gaits of these robots were designed according to the geometric relationships of the struts. For a novel tensegrity duct robot constructed with two linked tetrahedrons [9, 10], the main focus is also on the design of climbing gaits. Through the structure design and kinematic analysis of a steering crawling tetrahedron robot which links a pushing element on one of its four nodes [11], the slope crawling gait was presented. However, the research on successive path planning of such linkagebased mobile mechanism is relatively rare. And its successive motion much depends on complex control system. A light source tracking 1TET designed by Yu and Nagpal [12] realize its rolling motion by control the selfadapting system contains amounts of sensors. And a spine tensegrity robot simulated its successive rolling by using central pattern generators (CPGs) [13].
Since the large redundancy of multiDOF robots, the key issue of the realization of successive locomotion lies on the inverse kinematics, which is a nonlinear problem that has multiple solutions [14]. A twolinked tetrahedron robot adopted force density method to solve the inverse kinematics [9] for the minimum of elastic potential energy contained. In Ref. [15], a GA was used to solve the inverse kinematics of a redundant robot, which mainly focuses on finding the best solution among multiple solutions with the objective of minimum joint displacements.
Tetrahedronbased robots are always overconstrained, which requires to embed large extension ratio actuators to acquire large deformation and high mobility [16]. Since there are few actuators meet this special requirement, researchers have explored using ordinary extension ratio actuators to design modular reconfigurable robotic systems with high environmental adaptability [17, 18]. One of the modular reconfigurable robots called Tetrobot [5] that can be reconstructed as tetrahedron module, octahedral module, sixlegged walker, and tetrahedronbased manipulator. Each of these four constructions can meet the respectively given tasks. Another modular reconfigurable robot [12] has three reconfigurable constructions, including an adaptive gripper, a modular tetrahedron robot, and a modular hexahedron robot. However, the three constructions can only meet its corresponding task. If there were only one construction that could meet more than one task, it would be very desirable [19].
In previous work, we reported a closedloop mechanism consisted of two tetrahedron units with two DOFs that could realize rolling locomotion on the ground [20]. And a pneumatic driving tetrahedron mechanism with multiDOF was proposed [21]. In this paper, we focused mainly on the locomotion optimization and manipulation planning of the tetrahedronbased mobile mechanism constructed as 5TET. A binary control strategy [22, 23] using binary actuators is adopted to simplify the control system. A GA based on binary codes is developed to optimize the locomotion of 5TET by altering minimum actuators, which further simplifies the control of the mechanism. Double control simplification provides a reference for multiDOF inputs combination strategy.
This paper is organized as follows. The structure and locomotion of 5TET are described in Section 2. Section 3 illustrates the optimal locomotion based on kinematic and dynamic analyses. Section 4 presents a potential operation function of 5TET. Energy consumption is analyzed Section 5. And the optimal locomotion paths and manipulation gaits are simulated Section 6.
2 Structure and Locomotion Description
2.1 Structure Description
2.2 Binary Control and System Implementation
Binary control is one of the control concept applied to multiDOF mechanisms [24]. In this concept, tens or hundreds of binary actuators are embedded in a structure, which is analogous to the digital computer replacing the analog computer. These digital mechanisms can perform precise, discrete motions without the need of sensing, complex electronics or feedbacks, so the control elements of such devices are simple [25]. Since the binary control only has signals of 0 and 1 that correspond to off and on, the control operation of such devices is also simple [23].
Taking the advantages of quick response, large strengthweight ratio, and lower control complexity, pneumatic cylinders are selected as the binary actuators for 5TET. Pneumatic cylinder has only ‘min/max’ length, which correspond to the binary codes of ‘0/1’. However, since the binary behavior, the feasible configurations of binary mechanisms would decrease. To remedy this limitation, it is preferred to assemble several modules in the architecture to realize a quasicontinuous mobility [26].
For mobile robot, the integration design concept is a good choice. Since the embedding of pneumatic elements, the implementation of 5TET calls for the pneumatic power supply, which is a dominant bottleneck in practical of pneumatic mobile applications. Many researchers have been studied on this problem, and new advances in portable pneumatic power source are presented [27–29]. For example, a portable pneumatic supply called Dry Ice Power Cell, which could provide sustaining 0.42 MPa gas for about 218 NL with portable size [29]. The achievements from these researches can be used directly to equip 5TET to realize mobile function, so do other elements.
Structure parameters of 5TET mechanism
Structure items  Parameters 

Length range of pneumatic cylinder l/mm  360–460 
Mass of a pneumatic cylinder m_{L}/kg  0.5 
Pressure range of pneumatic cylinder P/MPa  0.1–0.9 
Radius of a spherical joint r_{N}/mm  30 
Mass of a spherical joint m_{N}/kg  0.3 
The implemental system of integration design concept mainly contains the 5TET mechanism, the pneumatic system (including a pneumatic power supply, 18 electric solenoid valves and some pneumatic tube), a control unit with battery, a wireless unit and a remote computer. 5TET is an executive mechanism that activated by electronic solenoid valves when the control unit receives the optimal binary codes from the computer by the wireless unit.
2.3 Locomotion Feasibility of 5TET
For 5TET, in any static stable state, the support triangle surface is composed of three of the eight nodes. And one of the three nodes is chosen from four nodes of the basic tetrahedron ABCD, while the other two are chosen from the four external nodes E, F, G and H. But during the moving, the support area of 5TET is always changing, which deeply varies the kinematic analysis. Fortunately, 5TET is a fully symmetric configuration that has isotropic feature [30], so that it has identical structures with different support nodes. Using this significant feature, the complexity in kinematic analysis would be reduced.
3 Motion Gait Optimization and Optimal Path Planning
For mobile mechanism, it is meaningful to find the optimal input of the required locomotion. To improve the efficiency and accuracy of the optimization, an analytical kinematic model based on geometric relationship rather than coordinate transformation was built. Then based on the tipping dynamic analysis, the motion gaits of a single step and successive locomotion were optimized.
3.1 GeometricBased Forward Kinematics
For tetrahedronbased mechanism, it is complicated and timeconsuming to analyze the forward kinematics with coordinate transformation, which directly affects the efficiency of the locomotion optimization. However, based on a fixed coordinate system, the analytical solution could be quickly obtained by the proper use of the geometrical relationship of the mechanism.
The same goes for the rest nodes E, F and G, which could also obtain its position coordinates from the corresponding ternary quadratic groups according to nodes A, C, D, nodes A, B, C, and nodes B, C, D, respectively.
3.2 Dynamics of the Tipping Motion
The first phase is to manipulate the controlled nodes to generate the tipping motion. The actuating forces from DA and DB are exerted on nodes A and B during this phase while the base nodes are placed on the ground. The second phase comprises a tipping motion when the base node D leaves the ground and the new node C falls down to the ground. And 5TET turns into the third phase after node C hits the ground. Assume that all the actuators are locked in this process, in other words, 5TET can be regarded as a rigid body during the tipping motion.
3.2.1 Before Tipping Phase
Dynamically, the critical condition of the tipping is that the ground support force equals to zero, which corresponds to that the CM goes on the tipping axis in kinematics.
3.2.2 Tipping Phase
3.2.3 Contact with the Ground Phase
According to dynamic analysis, 5TET stays in unstable state in the first two phases, while it is stable in contact with the ground phase.
3.3 Motion Gait Optimization for the First Step Rolling
Based on the isotropic feature of 5TET, no matter which support area is the initial one, the same result will be gotten. In Figure 3, the initial support area is given as ΔDEG. The rolling feasibility analysis infers that 5TET could statically tip towards all of the three directions over edge DE, DG and EG, respectively. Thus, 5TET is capable of rolling successively by binary codes.
LimitedDOF [31] mechanism is a popular concept in parallel mechanism which has great potential in practical applications with advantages of simple and compact constructions, easy control and low cost. Taking these advantages into the analysis of multiDOF 5TET, every step tipping is regarded as the rolling of a limitedDOF mechanism. The objective of the optimization is defined as executing the minimum number of struts. The motion optimization model of 5TET was set up.
Design vector
X = [x_{1}, x_{2}, …, x_{18}]^{Τ} = [AD, AC, AB, DC, BC, BD, ED, EC, EA, GD, GC, GB, HD, HA, HB, FA, FC, FB]^{Τ};
Objective function
min f(X) = ∑ _{i=1} ^{18} x_{ i } – x_{i0};
Subject to
g_{ i } (X) = x_{ i } (x_{ i } − 1) = 0, (i = 1, 2, …, 18);
g_{19}(X) < 0;
g_{20}(X) < 0.
Where, design variables x_{ i } (i = 1, 2, …, 18) is the present state of actuators that could make 5TET roll to the specific direction, while x_{i0} (i = 1, 2, …, 18) is the former state. The constraint g_{19}(X) < 0 is the condition that the CM goes beyond the current support area, while g_{20}(X) < 0 is the constraint that the CM falls in the new support area.
According to the requirement of the optimization model, and considering binary thinking is also applied in GA [15], a GA combined with binary codes is developed to optimize the motion of 5TET. Where, the crossover and mutation rate are determined by the traditional selection method trialanderror based on the principle of crossover is expected to be higher while mutation is expected to be lower [32], which respectively equal to 0.2 and 0.02. According to the genome length of 18, assumed that the population number is 500 and the maximum generation equals to 50. In GA, the implementation of the objective function and constraints are realized within the fitness function,which is a raw measure of the solution value [33]. For fitness function has no requirement for continuity in the derivatives, virtually any cost function can be selected [14]. Here, to get rapid calculation, the fitness function is defined as Fit = M – f (X), M is the maximum possible value of objective function that equals to 18.
Considering that axis DE and DG are in mirror position, which means these two tipping directions have isotropic feature, so that tipping in Direction I and III are analyzed.
3.3.1 Tipping in Direction I Over Edge DE
 Code 1:

[0,0,1,0,0,0,0,0,0,0,1,1,0,0,0,0,0,0];
 Code 2:

[0,0,0,0,0,0,0,1,0,0,1,1,0,0,0,0,0,0];
 Code 3:

[0,0,0,0,0,0,0,0,1,0,1,1,0,0,0,0,0,0];
 Code 4:

[0,0,0,0,0,0,0,0,0,0,1,1,0,0,0,0,1,0].
3.3.2 Relationship between Direction I and II
Correspondence of telescopic struts between two mirrored tipping axes
Tipping axis  Corresponding edges  

DE  AD  AC  AB  DC  BC  BD  ED  EC  EA  GD  GC  GB  HD  HA  HB  FA  FC  FB 
DG  BD  BC  AB  DC  AC  AD  GD  GC  GB  ED  EC  EA  HD  HB  HA  FB  FC  FA 
 Code 5:

[0,0,1,0,0,0,0,1,1,0,0,0,0,0,0,0,0,0];
 Code 6:

[0,0,0,0,0,0,0,1,1,0,1,0,0,0,0,0,0,0];
 Code 7:

[0,0,0,0,0,0,0,1,1,0,0,1,0,0,0,0,0,0];
 Code 8:

[0,0,0,0,0,0,0,1,1,0,0,0,0,0,0,0,1,0].
3.3.3 Tipping in Direction III over Edge EG
 Code 9:

[1,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0];
 Code 10:

[1,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0];
 Code 11:

[0,0,0,0,0,1,0,1,0,0,0,0,0,0,0,0,0,0];
 Code 12:

[0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,1,0,0].
3.4 Motion Optimization for Successive Rolling
Possible combinations of triangle support area
Support nodes of A, B, C, D  A  A  A  B  B  B  C  C  C  D  D  D 

Support nodes of E, F, G, H  EF  EH  FH  FG  FH  GH  EF  EG  FG  EG  EH  GH 
Triangle support area  ΔAEF  ΔAEH  ΔAFH  ΔBFG  ΔBFH  ΔBGH  ΔCEF  ΔCEG  ΔCFG  ΔDEG  ΔDEH  ΔDGH 
Based on the optimization of single step rolling of 5TET, a successive moving optimization algorithm is developed. The objective of the algorithm is to reach the target by executing the least pneumatic cylinders between every two adjacent steps, which requires every step moves with limitedDOF [31]. The tipping axis of each step depends on the intersection of the line segment between target and CM projection on the ground.
5TET can automatically judge the tipping direction by the tipping rule, and the new touchdown node is also shown automatically according to Table 3.
For the special rolling mode, the mechanism not only has the motion of straight going, but also turning motion.
3.4.1 Straight Path
 Quadrant I:

(1500 mm, 2000 mm);
 Quadrant II:

(− 1500 mm, 2000 mm);
 Quadrant III:

(− 1500 mm, − 2000 mm);
 Quadrant IV:

(1500 mm, − 2000 mm).
3.4.2 Turning Path
According to nodes combination principle of triangle support area, the turning motion can be divided into two cases. Case I turns around nodes A, B, C or D; Case II around nodes E, F, G or H.
 Target 1:

(0 mm, − 100 mm);
 Target 2:

(− 100 mm, 0 mm);
 Target 3:

(0 mm, 100 mm).
 Target 1:

(200 mm, 0 mm);
 Target 2:

(0 mm, 200 mm);
 Target 3:

(− 200 mm, 0 mm);
 Target 4:

(0 mm, − 200 mm).
From the detailed elaboration of straight paths and turning paths, as Figures 17, 19, and 21 shown, 5TET can move to the target by executing at most 5 pneumatic cylinders between every two adjacent steps, which indicates that the 5TET is a 18 DOFs mechanism in total while it is a 5 DOFs mechanism in every step of movement. This feather simplifies the inputs control and program of the rolling, which makes the mechanism have effective foundations for inputs combination strategy.
4 OptimizationBased Manipulation Planning
For gripper manipulators, the gripping range of the endeffector is a major parameter, which has effects on the capacity of gripper manipulation [34]. For 5TET mechanism, the gripping range manifests as the variable range of the dihedral angle of HBAF. Here, in order to guarantee a stable gripper, nodes F and H should be fully symmetric with strut AB. That makes the variable range of distance between F and H equivalent to the variable range of the dihedral angle HBAF. In the initial state of 5TET, the gripper F and H are at a certain distance of 600 mm. However, to realize the gripper action, the dihedral angle HBAF should open and close based on the initial state, which would be represented by the distance between nodes F and H. The optimization model is set up to get the inputs of the gripper manipulation of 5TET. Note that, in the optimization model, the number of state change struts is determined as even numbers to maintain symmetry. The optimal model of the gripper is described as follows.
Design vector
X = [x_{1}, x_{2}, …, x_{18}]^{T} = [AD, AC, AB, DC, BC, BD, ED, EC, EA, GD, GC, GB, HD, HA, HB, FA, FC, FB]^{T};
Objective function
min f (X);
Subject to
g_{ i }(X) = x_{ i } (x_{ i } −1) = 0, (i=1, 2, …, 18);
Where g_{20}(X) is the constraint of the stability. And similar with the optimal parameters of motion planning, the crossover and mutation rates respectively equal to 0.2 and 0.02, population number is 500, and the maximum generation is 50. The fitness function is Fit = M – f (X), M is the maximum possible value of the objective function.
4.1 Manipulation Planning of Opening Wide
4.2 Manipulation Planning of Turning Down
For the optimization manipulation model of turning down, \(f({\mathbf{X}}) = \sqrt {({}_{H}^{O} x  {}_{F}^{O} x )^{2} { + }({}_{H}^{O} y  {}_{F}^{O} y )^{2} + ({}_{H}^{O} z  {}_{F}^{O} z )^{2} } .\)Here, assume that nodes A, B, F and H are coplanar, which means the dihedral angle HBAF is at the maximum. And the longest distance between F and H can be easily obtained, l_{FHmax} = 846.64 mm, rounded to M = 850.
4.3 Implementation of Gripper Manipulation
In gripper manipulation mode, the number of the executed pneumatic cylinders between every two adjacent steps is no more than 4, so that 5TET mechanism is an 18 DOFs mechanism in total while it is a 4 DOFs mechanism in every step manipulation of gripper. And taking rolling motion into consideration, 5TET is a 5 DOFs mechanism with rolling locomotion mode and gripper manipulation mode.
5 Energy Consumption Analysis
The 5TET realizes locomotion and manipulation functions by deformation, and the CM of the mechanism frequently changed by driving actuators, so that the energy and locomotion efficiency may be lower than that of traditional mobile mechanisms, such as wheeled and tracked robots.
Taking wheeled robot as example to be compared with 5TET on energy consumption and locomotion efficiency. The total weight of the robot is set to be the same with 5TET as 12 kg. And the moving path is set as the same trajectory of going straight from the origin (0, 0) to the target (1500, 2000) in Quadrant I corresponding to Figure 17(b), a total of 2.5 m.
5.1 The Wheeled Robot
5.2 The 5TET Mechanism
From the results of energy consumption analyses, the traditional wheeled robot is the most energyefficient. And 5TET supplied by Dry Ice Power Cell consumes less energy than that supplied by compressed air, where the energysaving rate reaches to approximately 48%.
6 Simulations of Successive Gait and Gripper Function
To verify the feasibility of 5TET, the dynamic simulation of locomotion and manipulation were carried out. 5TET has a total of 18 DOFs, 18 pneumatic cylinders were used to drive the 18 prismatic joints, so that the spherical joints are passive joints.
7 Conclusions
 (1)
An optimization method based on a developed GA is proposed to optimize the motion of binary control 5TET mechanism. The 5TET currently realizes its locomotion by executing at most 5 pneumatic cylinders between each two steps, rather than using multiple actuators as before.
 (2)
A potential manipulation function of the mechanism that operated with two nonsupport TETs synergistically is presented based on the proposed optimization method.
 (3)
The energy consumptions of the 5TET and the traditional wheeled robot are compared. Though the wheeled robot is more energysaving, the 5TET using portable supply of Dry Ice Power Cell saves approximately 48% energy than that supplied by general air compressor.
 (4)
Dynamic simulations are carried out to validate the proposed algorithm. As a result, the dynamic simulation trajectories consistently match with that of the mathematic analyses, which indicates that the optimization is effective both on the locomotion and manipulation planning. Through the multiterrain optimization and the whole system design in the next step, it will prospectively have better performance in more complex environments.
Declarations
Authors' information
Ran Liu born in 1991, is currently a PhD candidate at School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, China. She received her bachelor degree from Hebei University of Engineering, China, in 2013. Her research interests include mechanisms and mobile robotics.
YanAn Yao born in 1972, is currently a professor at School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, China. He received his PhD degree from Tianjin University, China, in 1999. His main research interests include mechanisms and mobile robotics.
Wan Ding born in 1987, is currently a postdoctoral fellow at Department of Mechanism Theory and Dynamics of Machines, RWTH Aachen University, Germany. He received his PhD degree in 2015 from Beijing Jiaotong University, China. His research interests include mechanisms and mobile robotics.
XiaoPing Liu born in 1970, is currently a professor at School of Mechanical, Electronic and Control Engineering, Beijing Jiaotong University, China. He is a Canada Research Chair Professor at Department of Systems and Computer Engineering, Carleton University, Canada. He received his PhD degree from the University of Alberta, Canada. His research interests include intelligent systems and robotics.
Authors’ contributions
RL carried out gait optimization studies, participated in the dynamic analysis and drafted the manuscript. YY completed the mechanism design and modeling, participated in the gripper function planning. WD presented the kinematic analysis, participated in the dynamic simulation. XL participated in the improved genetic algorithm. All authors read and approved the final manuscript.
Acknowledgements
Supported by National ScienceTechnology Support Plan Projects of China (Grant No. 2015BAK04B00), and 2015 SinoGerman Postdoc Scholarship Program (Grant No. 57165010).
Competing interests
The authors declare that they have no competing interests.
Ethics approval and consent to participate
Not applicable.
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Authors’ Affiliations
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