2011 IEEE International Conference on Robotics and Automation Shanghai International Conference Center May 9-13, 2011, Shanghai, China
Decoupled Kinematic Control of Terrestrial Locomotion for an ePaddle-Based Reconfigurable Amphibious Robot Yi Sun and Shugen Ma Abstract— In this paper we present a decoupled control method based on kinematic models of an amphibious reconfigurable robot called ePaddle-based quadruped robot (eQuad). The locomotion mechanism of eQuad is a novel eccentric paddle mechanism (ePaddle) that can perform wheeled, legged and paddling actions in both terrestrial and aquatic environments. We first introduce five terrestrial and aquatic gaits. A duty factor of up to 1.0 can be achieved for the legged walking. Therefore, the proposed robot eQuad can walk with a unique gait by eliminating the swing phase of the legs, and it has a large stable margin because all its legs are in contact with ground during walking. Kinematic models of this robot suggest that with this unique gait the reconfigurable ePaddle mechanism has the potential to achieve both legged and wheeled locomotion with the aid of a simple controller. A decoupled controller adapted from the wheeled robot is then built to evaluate this idea. Finally, simulations are performed to verify our proposed decoupled control and gait sequence planning methods.
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I. INTRODUCTION Animals are highly adaptive to their environments. The most common form of locomotion for terrestrial animals is legged walking [1]. For aquatic animals, hydrofoils are essential appendages for movement [2], [3]. For semi-aquatic animals, their ability to walk on land and swim in water might be their most important characteristic [4]. In the case of recent mobile robots, walking on legs [6], [7] and rolling on wheels [8], [9] are two major locomotion gaits used for performing terrestrial tasks, while paddling [10], [11] is widely used in aquatic environment. However, only certain robots can operate in amphibious environments. By manually assembled with semi-circular legs and paddles, AQUA can perform both walking and swimming locomotion [10], [12]. A robotic turtle that was inspired from sea turtles was developed by Low [7]. Santori et al. observed that all four legs of the thick-tailed opossum were used alternately, moving on a parasagittal plane when paddling, and propulsion was accomplished mainly by the hind limbs [5]. Based on this observation, we propose an eccentric paddle mechanism (ePaddle) as shown in Fig. 1 to achieve high mobility in amphibious tasks with the simple mechanism. A prototype quadruped robot (called eQuad) based on modularized ePaddle modules was designed. Several locomotion gaits of eQuad are introduced. The following kinematic models demonstrate the possibility of controlling wheeled and legged gaits with a simple controller adapted from wheeled vehicles [13], [14]. The design Y. Sun is with Department of Robotics, Ritsumeikan University, 525-8577 Shiga, Japan;
[email protected] S. Ma is with Department of Robotics, Ritsumeikan University, 525-8577 Shiga, Japan
[email protected]
978-1-61284-385-8/11/$26.00 ©2011 IEEE
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Fig. 2. Schematic sketch of an ePaddle module and its available configurations. (a). Major components in one ePaddle module. (b). Four actively actuated joints in ePaddle. (c).The wheel-like configuration is obtained by locating the paddle shaft to its topmost position. (d). The legged configuration is achieved by locating the paddle shaft to its bottom position.
of the decoupled control for posture and trajectory control is introduced and verified by simulations. II.
E Q UAD :
AN E PADDLE-BASED AMPHIBIOUS ROBOT
A. ePaddle Mechanism Fig 2 shows the mechanism schematic drawing of an ePaddle module. In this module, there are several major components: 1) a moveable paddle shaft that is driven by an linear slide screw with an revolving base by using two DC motors, 2) a set of four paddles that can rotate around
that paddle shaft freely, 3) an actively actuated shell that can rotate around the fixed shell axis, 4) four passive paddle hinges that are located radially at the edge of the shell and allow the paddle to slide through them, and 5) an actuated rotational hip joint for steering. Totally four actuators are used in one ePaddle module (one for the shell, one for the screw base, one for the slide screw, and one for the hip joint). Because the center of the paddle shaft can be made to be eccentric from the center of shell, we call this mechanism as the eccentric paddle mechanism. When the shell is driven to rotate around the shell shaft, hinges force the paddles to rotate around the paddle shaft accordingly. Because the paddle shaft is actively located inside of the shell with two actuators, the motion pattern of the tip point of the paddles can be alternated. The outer surface of each paddle protruding from the shell will serve as the control surface during swimming and as the leg during walking. The shell serves as the wheel during wheeled motion. This establishes the reconfigurability of the ePaddle mechanism’s gaits. B. Reconfigurable Gaits
contact areas, and wheels may fail to provide sufficient friction due to the sliding that occurs on contacting surfaces. Our robot can perform a unique wheel-leg-integrated gait (as shown in Fig. 3 (c)) to overcome these hazardous environmental conditions. Rotational paddling gait: In aquatic locomotion, thrust force is generated by pushing the surrounding water away from the protruding area of the rotating paddles. Two kinds of swimming gaits can be performed with the ePaddle module. The first swimming type is the rotational paddling mode, as shown in Fig. 3 (d). It is patterned after the paddling gaits used by the thick-tailed opossum [5]. In this gait, the paddle shaft in ePaddle is located eccentrically from the rotation center of the shell, with the a distance shown in Fig. 3 (d). The shell is actuated to rotate around its shaft in one direction. Oscillating paddling gait: The second type of swimming shown in Fig. 3 (e) is based on the oscillating motion of one of the paddles, and it is similar to the oscillating paddling motion applied by Georgiades et al. to AQUA [10]. III. KINEMATIC MODELS FOR E QUAD
Several gaits are available for eQuad, including wheeled, legged and wheel-leg-integrated gaits on a terrestrial terrain and paddling in an aquatic environment. Those gaits are listed in Figure 3. The designs of each gait were presented in [17], in this paper we give only a brief description of those gaits. Wheel-like gait: On solid, level terrain, a wheel-like motion such as that shown in Fig. 3 (a) can be performed. With this gait, the location of the paddle shaft will be kept at its uppermost position. The paddle near the ground is retracted into the shell. The outer surfaces of the shells are in contact with ground and serve as wheels to perform rolling motion. Legged gait: On rough terrain for which wheeled gaits cannot be used, the ePaddle-based robot can relocate the positions of the paddle shafts to extend the paddles so that they touch the ground and can operate as legs. By planning the trajectory of the paddle’s tip, a legged walking gait (such that shown in Fig. 3 (b)) can be achieved. It should be noted that the legged locomotion performed by ePaddle is different from conventional open-link-based legged gaits. It has two unique advantages: 1) Four paddles in one ePaddle module serve as supporting leg in turn. Hence, the time wasted in the recovery phase is eliminated, the robot can walk at a relatively high speed with a duty factor of 1.0; 2) there is always a paddle contacting with the ground in each ePaddle modules, the static stable margin of the robot during static walking can be kept at a high constant. Those two advantages imply that the static walking of this robot is high-speed and, even better, is stable at any instance. This lead us to design a simple locomotion controller by ignoring influence of leg transfer. The detailed design of this simple controller will be discussed in Sec.III. Wheel-leg-integrated gait: On sandy or muddy terrain, legs may fail to provide sufficient support due to the limited
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When design those gaits, we found that the wheel-like and legged gaits have similar wheel-rolling and paddle-shaftrelocating behaviors. Therefore, in this section we intend to seek the common portion in these behaviors by establishing the kinematic models for wheeled and legged configurations of eQuad. We define the fixed frame {0} , the robot body frame {B}, the contact frame for each leg {Ci }, and the wheel frame {Wi } (see Fig. 4 (a) and Fig. 4 (b)). We define the generalized velocity vector of the body motion vp as T T (1) vp = [p˙ TB , ωB ] where p˙ B = [x, ˙ y, ˙ z] ˙ T is the velocity of the body center with respect to the ground frame, ωB = [ωx , ωy , ωz ]T is the rotation rate vector of the body frame with respect to the ground frame. ωB can be computed by using the orientation angles φ = [α, β, γ]T of the body with respect to the ground (roll, pitch and yaw angles, respectively). The generalized leg posture parameter pAi is defined as pAi = [ATi , lAi ]T
(2)
where Ai = [xAi , yAi , zAi ]T is the position of the tip of the ith leg, and the lAi is the span length of the ith leg, as shown in Fig. 4. A. Forward Kinematic Model Fig 4 shows a schematic drawing of the ePaddle mechanism for kinematic modeling in both wheeled and legged configurations. In this figure, only one of the four paddles that are in contact with the ground is shown. The velocity of each contact point Ai with respect to the ground can be expressed by using the velocity composition principle: 0
vAi,Ai/0 =0 vAi,B/0 +0 vAi,Ai/B
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Fig. 3. Different gaits can be performed in different environments. Motion directions of joints are indicated by the arrows. (a). Wheel-like gait. (b). Legged gait. (c). Wheel-leg-integrated gait. (d). Rotational paddling gait. (e). Oscillating paddling gait.
The velocity component 0 vAi,Ai/B in (3) can be expressed
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where 0 JAi is the Jacobian matrix of the leg or wheel mechanism expressed in {0}. q˙ i = [θ˙Hi , θ˙W i , θ˙Si , r˙i ]T is the joint velocity vector. θ˙Hi , θ˙W i , and θ˙Si are the angular velocity of the hip joint, the wheel and the slide screw base, respectively. r˙i is the linear velocity of the slide screw. 0 JAi in (5) is different in the wheeled and legged configurations. For wheeled configuration as shown in Fig. 4 (a),
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where 0 vAi,Ai/0 is the velocity of the contact point Ai with respect to the ground, 0 vAi,B/0 is the velocity component of Ai due to the movement of the body frame {B} with respect to the ground, and 0 vAi,Ai/B is the velocity component of Ai due to the movement of the leg with respect to the body frame {B}. The superscript zero 0 means those variables are expressed in the fixed frame {0}. The velocity component 0 vAi,B/0 in (3) can be expressed as: 0 vAi,B/0 = 0 p˙ B + 0 ω × 0 ri where 0 ω is the angular velocity vector of the body expressed in {0}. 0 ri is the vector from the origin of the body frame to the contact point. This equation can be revised by using the body’s generalized velocity vector in (1) as 0 (4) vAi,B/0 = I3 S(0 ri ) 0 vp = Li 0 vp ⎡ ⎤ 0 −u3 u2 0 −u1 ⎦ is the skewwhere S(u) = ⎣ u3 −u2 u1 0 symmetric matrix of the cross-product operator.
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zP , zT , and zr are the z-axes of the paddle joint, the slide screw base and slide screw joint, respectively. Vector a, b and c are shown in Fig. 4 (b). In this paper, we focus only on the unique feature of the ePaddle gaits under the ideal condition. Hence,we assume that the wheeled and legged motion of the eQuad are supposed to satisfy the pure rolling without slipping condition. This means that the velocity of the contact point 0 vAi,Ai/0 is equal to zero. When project 0 vAi,Ai/0 to the contact frame {Ci }, we can get the expression of the contact condition as: T Ci vAi,Ai/0 = Ci vx,Ai/0 Ci vy,Ai/0 Ci vz,Ai/0 = 0 (8) where Ci vx,Ai/0 = 0 and Ci vy,Ai/0 = 0 denote the nonslippage conditions in the longitudinal and lateral directions. Ci vz,Ai/0 = 0 denotes the contact continuity condition. Then (3) becomes (9) Li 0 vp + 0 Ji q˙i = 0 The velocity model of the eQuad with all four ePaddle modules can be obtained: ⎛ ⎞ ⎛ ⎞⎛ ⎞ L1 JA1 0 0 0 q˙ 1 ⎜L 2 ⎟ 0 ⎜ 0 JA2 0 0 ⎟ ⎜q˙ 2 ⎟ ⎜ ⎟ vp + ⎜ ⎟⎜ ⎟ (10) ⎝L 3 ⎠ ⎝ 0 0 JA3 0 ⎠ ⎝q˙ 3 ⎠ = 0 0 0 0 JA4 L4 q˙ 4
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Fig. 5. General planar wheel model for both wheel-like and legged configuration of eQuad.
B. Steerability, Mobility and Maneuverability For the wheel-like configuration as shown in Fig. 4 (a), the contact point of each leg is the contact point between the wheel and the ground. In addition to the non-slippage condition described above, for the wheeled system, there is another implicit condition that the component velocity of the contact point orthogonal to the plane of the wheel must be equal to zero, namely 0 vAi,Ai/0 · 0 zW i = 0, which means 0 vAi,B/0 +0 vAi,W i/B +0 vAi,Ai/W i ·0 zW i = 0 (11) From Fig. 4 (a) we can see that the movement of the contact point occurs at the plane that is perpendicular to the z-axis of the wheel joint. Hereafter, this plane will be referred to as the wheel plane. Therefore, (11) can be simplified as 0
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vAi,B/0 · vW i + vAi,W i/B · zW i = 0
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For the legged configuration, as introduced in the preceding section, one of the four paddles in each leg is always in contact with the ground. It simplified the modeling for the legged gait because the issue of leg transferring can be avoided. Then the implicit velocity constraint for legged configuration can be expressed by the same equation used in (11). Additionally, the contact point Ai is the tip point of the paddle, and its movement occurs at the wheel plane also. This means that for the legged configuration the implicit constraint can also be simplified to (12). We build a general planar wheel model based on (12), as shown in Fig. 5, to analyze the maneuverability of both the wheel-like and legged configurations. By applying the parameters shown in Fig. 5, we can obtain the constraint equation for the ith leg as follows: [cos θ˜i , sin θ˜i , lAi + lB sin(θHi − τi )][x, ˙ y, ˙ γ] ˙ T =0
(13)
where θ˜i = θHi + γ. For all the four legs of the eQuad, the constraint equations can be obtained T
CS (θH1 , θH2 , θH3 , θH4 ) [x, ˙ y, ˙ γ] ˙ =0
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Based on the definitions in [15], we can get the degree of steerability and the degree of mobility of the eQuad as follows: δS = rank (CS ) , δm = 3−rank (CS ).
(15)
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The values of matrix CS in (14) and (15) depend on the steering angles. Hence, there are two cases similar to [8]: 1) If the four wheel axes intersect at the same point or are parallel with each other, three of the four constraint equations are linearly dependent. eQuad becomes a type(1,2) wheeled robot [15], which means δS = rank(Cβ ) = 2 and δm = 1. Therefore, the eQuad can turn around the point of the intersection of the four-wheel axis (for the parallel condition, the position of the intersection point can be treated as infinitely far away from the wheels and the robot moves along a straight line). The intersection point is called the instantaneous center of rotation (ICR). 2) If the wheel axes do not intersect at the same point or are not parallel with each other, all of the four constraint equations are linearly independent and δS = 3 and δm = 0. This means the body of the robot is not able to move at all. From the view point of control, the fact that CS depends only on the steering angles among the motion parameters of robot’s body means that for both the wheeled and legged configuration, the same trajectory control strategy can be used. C. Inverse Kinematic Model As the Jacobian J in (10) is not a square matrix, the locomotion mechanism of eQuad is a redundant system. By knowing the ICR position pr = [xr , yr ]T , we can easily obtain the steering angle θHi of the ith leg to be compatible with the body motion from (13): θHi = atan2(ySi − yr , xSi − xr )
(16)
where xSi = lB cos(τi + γ) and ySi = lB sin(τi + γ) are the absolute position of the steering joint center of the ith leg. After that, the number of joint parameters in (10) is reduced. For the wheeled configuration, the number of unknown joint parameters for each leg is reduced to 1 (namely, the angular velocity of the wheel θ˙W i ). For the legged configuration, the number of unknown joint parameters is reduced to 3 (the angular velocities of the wheel θ˙W i , the slide screw base θ˙Si , and the linear velocity of the slide screw joint r˙i ). We can then obtain the joint parameters by using the planar inverse kinematic model presented in [17]. IV. POSTURE AND TRAJECTORY CONTROL The posture of the robot will be measured by a tilt sensor mounted on robot’s body. The position of the robot is planed to be provided by localization method. Hence, we built a proportional feedback control based on these two parameters to verify the feature of the proposed ePaddle mechanism and its kinematic models. We first collect the posture-dependent velocity p˙ and the trajectory-dependent velocity v˙ t from the sensory information of the body’s movement. Because the steering angles in both wheeled and legged configurations are only relative to
the x, ˙ y, ˙ and γ˙ from the body’s generalized velocity vector (as demonstrated in (13)), we define the trajectory-dependent parameter u as: (17) u = [x, y, γ]T
However, in this paper, we only focus on the general control principle of the trajectory control for eQuad. Thus, a simple strategy based on proportional feedback [13] is used:
We then define the posture-dependent parameter p as:
where Kt is a positive diagonal gain matrix, Δu = ud − u is the tracking error. Compared to the posture-dependent gain matrices in (19), the trajectory-dependent gain matrix in (21) share the same values in both the wheeled and the legged configurations. This is because the trajectory parameters are only influenced by the steering angles of the robot.
p = [z, α, β, pTA1 , pTA2 , pTA3 , pTA4 ]T
(18)
A. Velocity Decoupling Matrix Similar to those decouple velocity models in [13], [14], [16], we introduce a decoupling matrix D to compute the platform velocity from the time-derivative of the postureand trajectory-dependent parameters:
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˙ vp = D(Cp p˙ + Ct u)
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pAi = CAi p˙
B. Posture Control The aim of posture control is to compute the desired joint velocity q˙ and apply it to each actuator to achieve the desired posture pd . One of the simplest posture-control strategies is achieved using a proportional feedback [13]: p˙ = Kp Δp
(21)
By applying both posture and trajectory control described earlier, the velocity of the robot body can be obtained and expressed as a combination of p˙ and u:
˙ γ] vp = D[x, ˙ y, ˙ z, ˙ α, ˙ β, ˙ T where
u˙ = Kt Δu
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where Kp is the positive gain matrix, Δp = pd − p and p˙ is the time-derivative of the posture parameters. It should be noticed that the gain matrices are different in the wheeled and legged configurations.
where Cp , Ct , and CAi are the corresponding component selection matrices. The joint velocities can be computed by using the inverse kinematic model described in preceding section. The systemic control architecture is shown in Fig. 6 9HORFLW\ 'HFRXSOLQJ
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C. Trajectory Control The goal of trajectory control is to control the body’s center point to follow a given path with minimal tracking error Δu. The current trajectory parameter of the robot body is u = [xp , yp , γp ]T . Assuming that the desired trajectory is specified in the horizontal plane by using a parametric function f (t) = [xd (t), yd (t)]T , the desired trajectory ud can be expressed as a function of time: ud = [xd , yd , γd ]T
(20)
where γd (s) = arctan(y˙ d /x˙ d ). As mentioned before, when all the wheel axes intersect at the same point, the eQuad robot in both the wheeled and legged configurations can be treated as a type(1,2) nonholonomic omnidirectional vehicle (which means the degree of steerability and the degree of mobility are equal to 2 and 1, respectively) described in [15]. The trajectory control of this kind of mobile robot has been well studied by using the ICR to describe the robot’s motion in [9]. We could apply this method to our robot in the future.
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V. SIMULATION RESULTS Because the prototype of eQuad has not been built yet, only simulations based on the prototype eQuad have been done to verify the ideas proposed in this paper. The simulations are aimed at evaluating the decoupled control method based on kinematic models by considering a desired posture and a given trajectory path. The desired trajectory is a straight line on irregular terrain as shown in Fig. 7. Because of the unique legged gait, the problems of the leg transferring can be avoided in the proposed robot eQuad. The static stability margin of the robot can be kept at a high constant value. Therefore, we simply set the posture that keeps the robot’s body horizontal as the desired posture in our simulations. Because the four legs are symmetric when applying this horizontal-aligned posture, only the simulation results of the right front module of eQuad are shown in Fig. 8. Fig. 8 (a) and Fig. 8 (b) present the values of the joint parameters. The traces of all the four paddles in the right front leg are shown in Fig. 8 (c). Those traces verified
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In future, we will focus on the optimal locomotion design for wheeled and legged gaits to avoid the rapid vertical vibration. Aquatic locomotion based on dynamics models will be studied as well. A locomotion controller that can integrates terrestrial and aquatic gaits will be designed. Gait experiments in terrestrial and aquatic environments will be carried out as well.
Fig. 8. Results of the straight line trajectory following simulation on irregular terrain. (a). Joint values for the slide screw. (b). Joint values for the wheel joint and the screw base joint. (c). The trace of four paddles and the trace of the gravity center of the ePaddle module.
that the duty factor of 1.0 is achievable by ePaddle-based legged walking. The trace of center gravity point of the module is denoted by ◦ in Fig. 8 (c). We observed that the vertical position of the module center vibrates with a amplitude of about 20mm. This vertical vibration motion always exists when we try to control the robot’s body center to follow a desired trajectory while keeping the robot’s body in horizontal plane with a 1.0-duty-factor gait. This will result the robot’s body to vibrate and a lose of energy for lifting up the body periodically. How to eliminate this vertical motion will be studied in our future research. VI. CONCLUSION In this paper, we have presented a novel locomotion mechanism called the eccentric paddle mechanism or ePaddle, for amphibious robots. An ePaddle-based prototype amphibious quadruped robot called eQuad has been designed. By using ePaddle modules, eQuad is able to walking with a static stable legged gait with a duty factor of up to 1.0. Kinematic models has been built for wheeled and legged motion. The kinematic equations show that the trajectory control of eQuad can be simplified by using a wheel-like control architecture, thanks to its high duty factor. As a result, a decoupled control architecture has been proposed based on the kinematic model with proportional feedback gains. Simulations on both legged gait planning and posture and trajectory control have been performed. Simulation results showed the effectiveness of the proposed method.
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[1] M. Hilderbrand, “Symmetrical gaits of primates,” American Journal of Physical Anthropology, vol. 26, 1967, pp. 119-130. [2] F. E. Fish, “Aquatic locomotion,” in Mammalian Energetics: Interdisciplinary Views of Metabolism and Reproduction, T. E. Tomasi and T. H. Horton, Ed Ithaca: Cornell University Press, 1992, pp. 34-63. [3] J. E. Colgate and K. M. Lynch, “Mechanics and control of swimming: a review,” IEEE Journal of Oceanic Engineering, vol. 29(3), 2004, pp. 660-673. [4] F. E. Fish, “Structure and mechanics of nonpiscine control surfaces,” IEEE Journal of Oceanic Engineering, vol. 29(3), 2004, pp. 605-621. [5] R. T. Santori, O. Rocha-Barbosa, M.V. Vieira, J.A. Magnan-Neto, and M.F.C. Loguercio, “Locomotion in Aquatic, Terrestrial, and Arboreal Habitat of Thick-Tailed Opossum, Lutreolina crassicaudata (Desmarest, 1804)” Journal of Mammalogy, vol. 86(5), 2005, pp. 902908. [6] A. German and M. Jenkin, “Gait Synthesis for Legged Underwater Vehicles,” in Proceedings of the 15th International Conference on Autonomic and Autonomous Systems,Valencia, Spain, 2009, pp. 189194. [7] K. H. Low, C.L. Zhou, T.W. Zhu, and J.Z. Yu, “Modular design and initial gait study of an amphibian robotic turtle,” in Proceedings of the 2007 IEEE International Conference on Robotics and Biomimetics, Sanya, China, 2007, pp. 535 - 540. [8] M. Lauria, I. Nadeau, P. Lepage, Y. Morin, P. Giguere, F. Gagnon,D. Letourneau, and F. Michaud, “Design and control of a four steered wheeled mobile robot.” in Proceedings of the 2006 IEEE 32nd Annual Conference on Industrial Electronics, Paris, France, 2006, pp. 40204025. [9] L. Clavien, M. Lauria, F. Michaud, “Instantaneous centre of rotation estimation of an omnidirectional mobile robot.” in Proceedings of the 2010 IEEE International Conference on Robotics and Automation, Alaska, USA, 2010, pp. 5435-5440. [10] C. Georgiades, M. Nahon and M. Buehler, “Simulation of an underwater hexapod robot,” Ocean Engineering, vol. 36(1) , 2009, pp. 39-47. [11] F. E. Fish, G.V. Lauder, R. Mittal, A.H. Techet, M.S. Triantafyllou, J.A. Walker and P.W. Webb, “Conceptual design for the construction of a biorobotic AUV based on biological hydrodynamics,” in Proceedings of the13th International Symposium on Unmanned Untethered Submersible Technology, Durham New Hampshire, 2003, pp. 207-209. [12] M. Buehler and C. Georgiades, “AQUA: An Amphibious Autonomous Robot,” Computer, vol. 40, 2007, pp. 46-53. [13] C. Grand, F. Benamar, F. Plumet, and P. Bidaud, “Decoupled control of posture and trajectory of the hybrid wheel-legged robot Hylos.” in Proceedings of the 2004 IEEE International Conference on Robotics and Automation, New Orieans, USA, 2004, pp. 5111-5116. [14] C. Grand, F. Benamar, F. Plumet, and P. Bidaud, “Stability and Traction Optimization of a Reconfigurable Wheel-Legged Robot.” The International Journal of Robotics Research, vol.23(10-11), 2004, pp. 1041-1058. [15] G. Campion, G. Bastin, and B. Dandrea-Novel, “Structural properties and classification of kinematic and dynamic models of wheeled mobile robots.” IEEE Transactions on Robotics and Automation, vol.12(1), 1996, pp. 47-62. [16] C. Grand, F. Benamar, and F. Plumet, “Motion kinematics analysis of wheeled-legged rover over 3D surface with posture adaptation.” Mechanism and Machine Theory, vol.45(3), 2010, pp. 477-495. [17] Y. Sun, S. Ma, and X. Luo, “Design of an amphibious robot based on eccentric paddle array mechanism.” in Proceedings of the 2010 IEEE International Conference on Robotics and Biomimetics, Tianjin, China, 2010, pp. 1098-1103.
