 Research
 Open Access
Realization of rolling locomotion by a wheelspiderinspired hexapod robot
 Takuma Nemoto^{1}Email author,
 Rajesh Elara Mohan^{2} and
 Masami Iwase^{1}
 Received: 15 July 2014
 Accepted: 11 May 2015
 Published: 7 July 2015
Abstract
This study aims to realize rolling locomotion by a multilegged robot inspired by “wheel spider” in nature. A novel robot mechanism and a strategy for rolling locomotion is proposed in this paper based on motion analysis of wheel spider species in nature. We also present details of a model of a wheelspiderinspired hexapod robot and design of its controller in realizing rolling locomotion. Results of numerical simulations validated the efficacy of the proposed approach in synthesizing rolling locomotion in a wheelspiderinspired hexapod robot.
Keywords
 Biologically inspired robot
 Multilegged robot
 Wheel spider
 Rolling locomotion
 Modeling
 Simulation
Background
Rolling locomotion which has high mobility has been applied many times to mobile robots. Spherical robots are an example of mobile robots performing rolling locomotion. Spherical robots operate in hostile industrial environment, other planets or a human place like office or home. Spherical robots have been expected to obtain environmental data with sensors, gather and convey an object with an arm installed inside a spherical body in those environments [1–7]. However, it is considered that these environments have obstacles or difference in level that interrupt rolling locomotion. In this case, robots should apply other locomotion.
This paper focuses on creatures found in nature that can perform rolling locomotion and other locomotion. Spiders called “wheel spider (carparachne aureoflava)” which can perform rolling locomotion is one of the creatures in nature. The wheel spiders can reconfigure its body structures as wheels by fixing its legs into constant positions and perform rolling locomotion on its side on a slope [8].
This paper aims to realize rolling locomotion by a wheelspiderinspired multilegged robot on the flat ground.
Biologically inspired robots have been studied in literatures, e.g., [9–11]. Especially, biologically inspiredrolling locomotion has been discussed in literatures, e.g., [12, 13]. Lin has focused on caterpillars that can escape rapidly from predators by reconfiguring their body structures like wheels. Caterpillarinspired soft robots have been developed and attempted rolling locomotion [12]. King has focused on rolling locomotion repeating somersault performed by a spider called “huntsman spider (cebrennus villosus)”. A quadruped robot which can somersault has been developed based on an image analysis of its rolling locomotion. The quadruped robot has performed rolling locomotion [13].
In those studies, although behavior of the creatures is analyzed by images, biologically inspired mathematical models are not developed. In this paper, a biologically inspired mathematical model of the wheel spiders is developed and rolling locomotion is analyzed through simulations. The model captures characteristics of rolling locomotion. In addition, motion analysis based on simulations applying the model allows physical parameters to be determined arbitrarily and it can be utilized to determine robot parameters.
To realize rolling locomotion by the wheelspiderinspired multilegged robot, it is necessary to develop the model considering the influence of the ground on the robot while rolling locomotion and design a controller. For such the issue, a model considering the influence of the ground on a system is developed by applying constraint force on the ground to a model without the influence of the ground and considering velocity transformation due to collision in a previous literature [14]. Applying the above method, a wheel spider rolling on a slope is modeled and analyzed through simulations at first. A mechanism of the multilegged robot and the strategy for rolling locomotion are proposed based on results of motion analysis of the wheel spider and the model of the multilegged robot is developed. The controller which can generate leg trajectories respectively in response to a robot state is designed. The effectiveness of the proposed controller is verified through numerical simulations of rolling locomotion by the multilegged robot on the flat ground and it is shown that the multilegged robot can achieve the rolling locomotion.
Method
2.1 Modeling of wheel spider rolling on the slope
The wheel spider can be found in the Namib Desert of Southern Africa and its size is 20 mm. The wheel spider reconfigures its body as a wheel by fixing its legs into constant positions after a short runup and goes down sand dunes when attacked by its nemesis. The wheel spider resumes walking with its legs straight as rotational speed reduces [8].
This section presents the model of the wheel spider rolling on the slope for motion analysis.

Assumption 1. The wheel spider does not fall down while rolling.

Assumption 2. The wheel spider rolls without slipping on the ground.

Assumption 3. Positional relationships between the cephalothorax, the abdomen and each leg of the wheel spider do not change while rolling.
According to assumptions 2 and 3, the wheel spider rolls on the slope with keeping a posture. Only some parts touching the ground will be changed. Constraint conditions between the parts of the wheel spider does not therefore change and only the constraint conditions between the ground and some touching parts will change.
The motion equation of the wheel spider model is derived by applying a projection method [15–17]. Assuming that a system is constituted of independent parts and the parts are connected by constraint conditions, the projection method involves a whole motion equation from motion equations and constraint conditions of each part.
In this paper, the motion equation of the rolling wheel spider is derived by applying constraint force on the ground to a motion equation of the wheel spider without the ground and considering velocity transformation due to collision [14, 18].
Simulation parameters of rolling wheel spider
Mass of cephalothorax and abdomen (kg)  5.00×10^{−4} 
Mass of legs (kg)  1.00×10^{−4} 
Inertia moment of cephalothorax and abdomen (kgm ^{2})  6.25×10^{−9} 
Inertia moment of legs (kgm ^{2})  5.00×10^{−11} 
Viscosity of cephalothorax and abdomen (Nms/rad)  5.00×10^{−12} 
Viscosity of legs (Nms/rad)  1.00×10^{−12} 
Length from cephalothorax to abdomen (m)  1.00×10^{−2} 
Length from cephalothorax to pair of first legs (m)  6.50×10^{−3} 
Length from cephalothorax to pair of second legs (m)  9.00×10^{−3} 
Length from cephalothorax to pair of third legs (m)  1.20×10^{−2} 
Length from cephalothorax to pair of last legs (m)  1.40×10^{−2} 
Radius of cephalothorax and abdomen (m)  5.00×10^{−3} 
Radius of legs (m)  1.00×10^{−3} 
Gravity acceleration (m/s ^{2})  9.81 
2.1.1 Model of wheel spider without the ground
An unconstrained motion equation is written to derive the motion equation of the wheel spider without the ground.
where M _{ s } is a generalized mass matrix and h _{ s } is a generalized force vector. The unconstrained motion equation (2) is constituted of motion equations of independent parts shown in Fig. 2a. The generalized mass matrix M _{ s } and the generalized force vector h _{ s } are given as
where I _{ n }, m _{ n } and c _{ n } are the inertia moment, mass and viscosity of each part, θ _{ slp } is the angle of the slope and g is the gravity acceleration shown in Fig. 1.
where l _{ a }, l _{ il } and l _{ ir } are the length from the cephalothorax to the abdomen, the left legs and the right legs shown in Fig. 1 and α _{ i } denote any constant angles shown in Fig. 2b. The constraint conditions (5) describe the linkages which connect the cephalothorax to other parts as shown in Fig. 2b.
Since (8) has redundant degrees of freedom, they are reduced.
2.1.2 Consideration of constraint force on the ground
The motion equation of the wheel spider rolling on the slope is derived by applying constraint force on the ground to the motion equation of the wheel spider without the ground [14].
and C _{ I } should satisfy \(\boldsymbol {C}_{I} \dot {\boldsymbol {x}}_{s} = \boldsymbol {0}\).
 1.
grounding parts roll without slipping,
 2.
height of grounding parts does not change.
where x _{ n0} and ψ _{ c0} are the xcoordinate of each part and the angle of the cephalothorax when the constraints occur, respectively.
where the constraint equation Φ _{ I }=0 is obtained from (15).
by projecting (13) on the space constrained by \(\boldsymbol {D}_{s}^{T}\) and transforming the coordinates of component vectors.
2.1.3 Velocity transformation
In the case of touching each part of the wheel spider to the ground, collisions occur. Thus velocities before the collision should be changed to velocities after the collision. Assuming that completely inelastic collision occurs when some parts touch the ground, the velocities after the collision is obtained from the velocities before the collision [14, 18].
where the dimensions of I _{30} should equal the dimensions of the generalized coordinates x _{ s }.
2.2 Simulation of rolling wheel spider
Behavior of the wheel spider rolling on the slope is simulated for motion analysis. The initial position and angle of the cephalothorax are set at (x _{ c },y _{ c })=(0.00,1.50×10^{−2}) m and ψ _{ c } deg, respectively. The rotational angles of the legs are set at α _{1}=60.0, α _{2}=95.0, α _{3}=1.30×10^{2} and α _{4}=1.55×10^{2} deg and the angle of the slop is set at θ _{ slp }=10.0 deg. The wheel spider rolls on the slope from the state of grounding the abdomen to the ground.
Figures 3, 4, 5 and 6 show that the angle of the cephalothorax and the x direction positions of the wheel spider increase over time. The wheel spider rotates about 11 times and moves about 0.7 m in 3 s. Here, it is found that the wheel spider rolls at a constant speed since displacement increase at a constant rate except when it starts rolling. Figures 7, 8 and 9 show that the z direction positions repeatedly increase and decrease without decreasing to below the height of the ground and at least one of the parts of the wheel spider constantly touch the ground. Furthermore, Figs. 10, 11 and 12 show that the wheel spider proceeds to the positive x direction with changing the height of each part.
From the above results, it is found that the wheel spider goes downhill at a constant speed with rolling. The wheel spider rolls only gravitationally since it is not provided with an initial velocity in the simulations. It is therefore effective in rolling locomotion by the multilegged robot on the flat ground to utilize gravity skillfully.
2.3 Rolling locomotion by wheelspiderinspired hexapod robot
The results of motion analysis of the rolling wheel spider show that the utilizing of gravity is effective in rolling locomotion by the multilegged robot. Rolling locomotion utilizing gravity is therefore proposed. It includes the raising of the center of gravity of the multilegged robot and the shifting of it forward on the overturned posture.
In this section, the mechanism of the multilegged robot is proposed and then the model of the multilegged robot with rolling locomotion is developed to realize the strategy for rolling locomotion.
The controller which can generate leg trajectories respectively in response to a multilegged robot state is designed.
The multilegged robot is not developed in this paper. The implementing and the testing of the multilegged robot are issues in the future.
2.3.1 Mechanism of wheelspiderinspired hexapod robot
The utilizing of gravity is adopted as the strategy for rolling locomotion on the flat ground. The rolling locomotion utilizing gravity includes the raising of the center of gravity of the multilegged robot and the shifting of it forward on the overturned posture. This paper describes the raising of the center of gravity of the multilegged robot as a raising movement and the shifting of it forward as a shifting movement.
The legs are composed of a supporting columnar linkage and two linkage connecting it with the body as shown in Fig. 13b. The legs move up and down and inside and outside with keeping the side of each leg parallel to the side of the body.
2.3.2 Modeling of wheelspiderinspired hexapod robot with rolling locomotion
The model of the wheelspiderinspired hexapod robot with rolling locomotion is developed in the manner similar to the modeling of the wheel spider.

Assumption 1. The robot does not fall down while rolling.

Assumption 2. The robot rolls without slipping on the ground.

Assumption 3. The robot can swing, elongate and contract its legs.
Variables of rolling wheelspiderinspired hexapod robot (j=1,⋯,6)
Center of gravity coordinates of body (m)  (x _{ b },z _{ b }) 
Center of gravity coordinates of first parts of legs (m)  (x _{ α j },z _{ α j }) 
Center of gravity coordinates of third parts of legs (m)  (x _{ γ j },z _{ γ j }) 
Translational distance of second parts of legs (m)  d _{ β j } 
Rotational angle of body (rad)  ψ _{ b } 
Rotational angle of first parts of legs (rad)  ψ _{ α j } 
Joint torque of first parts of legs (Nm)  τ _{ α j } 
Force of second parts of legs (N)  F _{ β j } 
According to assumptions 1 and 2, rolling locomotion by the robot is described on the vertical twodimensional surface as the rolling wheel spider. Xaxis describes the flat ground. The variables surrounded by the double line will be definitely independent.
According to assumptions 3, it is supposed that the legs of the robot have translational joints and rotational joints, respectively. Motor torque to power parallel linkages of legs are provided for translational joints as force.
Simulation parameters of wheelspiderinspired hexapod robot with rolling locomotion
Mass of body (kg)  1.70×10^{−1} 
Mass of first parts of legs (kg)  1.20×10^{−2} 
Mass of second parts of legs (kg)  1.30×10^{−2} 
Mass of third parts of legs (kg)  3.00×10^{−2} 
Inertia moment of body (kgm ^{2})  7.65×10^{−5} 
Inertia moment of first parts of legs (kgm ^{2})  2.00×10^{−6} 
Inertia moment of second parts of legs (kgm ^{2})  2.00×10^{−6} 
Inertia moment of third parts of legs (kgm ^{2})  2.10×10^{−6} 
Viscosity of body (Nms/rad)  1.70×10^{−9} 
Viscosity of first parts of legs (Nms/rad)  5.50×10^{−10} 
Damping coefficient of second parts of legs (Ns/m)  5.50×10^{−12} 
Spring constant of second parts of legs (N/m)  1.10×10^{−9} 
Length from body to center of gravity of first parts of legs (m)  1.00×10^{−2} 
Length from center of gravity of second parts to third parts of legs (m)  1.00×10^{−2} 
Radius of body (m)  3.00×10^{−2} 
Radius of third parts of legs (m)  1.50×10^{−2} 
Gravity acceleration (m/s ^{2})  9.81 
Although the motion equation of the robot is derived in the manner similar to the modeling of the rolling wheel spider, it is difficult to derive an unconstrained motion equation including relative distance and angle directly. The unconstrained motion equation including relative distance and angle is therefore derived by transforming coordinates after deriving an unconstrained motion equation letting center of gravity coordinates of the second parts of legs be absolute position and rotational angles of the legs be absolute angle.
where I _{ m } and m _{ m } are the inertia moment and mass of each part, c _{ b } and c _{ α j } are the viscosity of the body and the first parts of the legs, η _{ β j } is the damping coefficient of the second parts and k _{ β j } is the spring constant of the second parts shown in Fig. 16 The subscript m denotes the set of indexes m={b,α j,β j,γ j}.
Given that \({\boldsymbol {M}_{h}=\boldsymbol {A}_{h}^{T} \tilde {\boldsymbol {M}}_{h} \boldsymbol {A}_{h}}\) and \(\boldsymbol {h}_{h}=\boldsymbol {A}_{h}^{T} (\tilde {\boldsymbol {h}}_{h} \tilde {\boldsymbol {M}}_{h} \dot {\boldsymbol {A}}_{h} \dot {\boldsymbol {x}}_{h})\), (32) can be represented by \(\boldsymbol {M}_{h} \ddot {\boldsymbol {x}}_{h} = \boldsymbol {h}_{h}\).
where l _{ α j } is the length from the body to each center of gravity of first part of leg, l _{ β j } is the length from each center of gravity of second part to each third part and r _{ b } and r _{ γ j } are the radius of the body and the third parts shown in Fig. 16.
by projecting the constrained system (36) on the space constrained by \(\boldsymbol {D}_{h}^{T}\) and transforming the coordinates of component vectors.
where h _{ γ j }=z _{ γ j }−r _{ γ j } is the height of a grounding point and λ _{ γ j } is ground reaction force. \(x_{{\gamma j}_{0}}\phantom {\dot {i}\!}\), \(\psi _{b_{0}}\phantom {\dot {i}\!}\) and \(\psi _{{\alpha j}_{0}}\phantom {\dot {i}\!}\) are the xcoordinate of each third part of leg and the angle of the body and the third parts when constraints occur, respectively.
Utilizing the constraint conditions (40), the motion equation of the hexapod robot with rolling locomotion can be derived by applying constraint force on the ground to the motion equation of the hexapod robot without the ground and considering velocity transformation due to collision.
2.4 Design of controller
To realize rolling locomotion by the wheelspiderinspired hexapod robot on the flat ground, it is necessary to perform the raising movement and the shifting movement. In this case, the robot stands on two legs and the center of gravity is shifted by these legs. Legs getting off the ground return to the initial position after shifting the center of gravity. For that reason, each leg should behave differently. A controller which generates a target trajectory depending on each leg state and allows a joint to track it is therefore designed.
2.4.1 Target trajectory generation
when t=t _{ f }. A target state x _{ r }(t) is obtained by applying a _{0}, …, a _{5} to (41)  (43) within t _{0}≤t≤t _{ f }. When 0≠t _{ a }≤t≤t _{ b }, A target state is obtained from x _{ r }(t)=f(t−t _{ a }) and t _{ a }≤t≤t _{ b } after coefficients of trajectories were obtained for t _{0}=0 and t _{ f }=t _{ b }−t _{ a }.
2.4.2 PID controller
where K _{ p }, K _{ i } and K _{ d } denote the proportional, integral and differential gain, respectively. (46) calculates input and allows the joints to track the target trajectories.
Results and discussion
3.1 Simulation of rolling locomotion by wheelspiderinspired hexapod robot on the flat ground
The rolling locomotion utilizing gravity by the wheelspiderinspired hexapod robot on the flat ground is simulated to verify the effectiveness of the proposed controller. The initial rotational angles of the body and the legs are set at ψ _{ b }=30.0 and ψ _{ α j }=0.00 deg and initial translational distances of legs are set at d _{ β j }=1.50×10^{−2} m. The hexapod robot starts rolling locomotion from a state standing on two legs.
Final states of target trajectories and activating times
State  Target distance  Target angle  Activating 

(m)  (deg)  time (s)  
Grounded forward leg  3.50×10^{−2}  −25.0  5.00×10^{−1} 
Grounded backward leg  5.00×10^{−3}  0.00  5.00×10^{−1} 
Ungrounded leg  1.50×10^{−2}  0.00  1.00 
PID gains of translational and rotational joints
Joint  Proportional gain  Integral gain  Differential gain 

Translational joint  3.50×10^{3}  1.50×10^{3}  40.0 
Rotational joint  15.0  5.00  6.00×10^{−2} 
Figures 18 and 19 show that the angle of the body and the x direction positions of the robot increase over time. The robot rotates about 2.32 times and moves about 1.26 m in 10 s. Here, it is found that the robot rolls at a constant speed since displacement increase at a constant rate except when the robot starts rolling. Figure 20 shows that the z direction positions repeatedly increase and decrease without decreasing to below the height of the ground except when legs get off the ground. The exceptions when the legs get off the ground are due to push the ground with the legs. Figure 21 shows that the robot proceeds to a positive x direction with changing the height of each part.
Figures 21, 22 and 23 show that the raising movement and the shifting movement can be performed by tracking the trajectories which is generated in response to the leg states.
From the above results, the proposed controller is effective in realizing the proposed strategy for rolling locomotion and the wheelspiderinspired hexapod robot can achieve rolling locomotion.
Conclusions
This paper aimed to realize rolling locomotion by the wheelspiderinspired hexapod robot on the flat ground. To realize rolling locomotion by the robot, the wheel spider rolling on the slope was modeled and analyzed through simulations. The model of the rolling wheel spider is developed by applying constraint force on the ground to the model of the wheel spider without the ground and considering velocity transformation due to collision. The mechanism of the robot and the rolling locomotion utilizing gravity which includes the raising of the center of gravity of the robot and the shifting of it forward on the overturned posture were proposed based on the results of motion analysis of the wheel spider and the robot model was developed. The controller which can generate the leg trajectories respectively in response to a robot state was designed. Finally, the rolling locomotion by the robot on the flat ground was simulated applying the proposed controller. As a result, it was found that the proposed controller is effective in realizing the proposed strategy for rolling locomotion. In conclusion, it was verified through numerical simulations that the wheelspiderinspired hexapod robot can achieve the rolling locomotion utilizing gravity.
The proposed controller has the control issue that it does not consider the influence of touching the legs of the robot to the ground. Trajectory tracking errors hence occur when the legs are touching the ground. The designing of a controller considering the influence of the ground and the implementing and testing of the wheelspiderinspired hexapod robot are issues in the future.
Declarations
Acknowledgements
Mr. Shunsuke Nansai made enormous contribution to design simulation programs.
Authors’ Affiliations
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