a control law to accomplish this task. Finally, in order to demonstrate the validity of the derived model and effectiveness of the proposed control law, simulation ...
Proceedings of the 2006 IEEE International Conference on Robotics and Automation Orlando, Florida - May 2006
Cooperative Control of Two Snake Robots Motoyasu Tanaka, Fumitoshi Matsuno The University of Electro-Communications 1-5-1 Chofugaoka, Chofu, Tokyo 182-8585, Japan Email: {m-tanaka, matsuno}@hi.mce.uec.ac.jp Abstract— In this paper, we derive a dynamic model and a control law for cooperative task of two snake robots which have wheeled links and demonstrate the validity of proposed controller by simulation and experiment. First, we derive a dynamic model of single snake robot with redundancy by introducing the shape controllable points. Next, we extend it to the model for the cooperative task of two redundant snake robots and propose a control law to accomplish this task. Finally, in order to demonstrate the validity of the derived model and effectiveness of the proposed control law, simulation and experiment are carried out.
I. INTRODUCTION Unique and interesting gait of the snakes make them able to crawl, climb a hill, climb a tree by winding and move on very slippery floor [1]. Snake does not have hands and legs, however it has many function. It is useful to consider and understand the mechanism of the gait of the snakes for mechanical design and control law of snake robots. Snake robots have been developed by many researchers [2]-[10] There are some researches based on the control theoretic approach. Chirikijian and Burdick discuss the sidewinding locomotion of the snake robots based on the kinematic model [4]. Ostrowski and Burdick analyze the controllability of a class of nonholonomic systems that the snake robots are included on the basis of the gemotric approach [5]. The feedback control law for the snake head’s position using Lyapunov method has been developed by Prautesch et al. on the basis of the wheeled link model [6]. They point out the controller can stabilize the head position of the snake robot to its desired value, but the configuration of it converges to a singular configuration. Date et. al. proposed a new dynamic manipulability of the snake head which is realted to the constraint force of the passive wheels and a controller which minimizes the cost function related to the the constraint force and the tracking error of the snake-head [7]. As a results by applying the controller the sigularity avoidance would be accomplished by trade-off between the singulatity avoidance and the tracking errors. But this controller can not ensure the convergence of the trajectory tracking errors. We find that introduction of links without wheels and shape controllable points in the snake robot’s body makes the system redundancy controllable based on the kinematic model [8]. We proposed a trajectory tracking controller for the snake robot which can move in the 3D space based on the kinematic model [9] but not dynamic model. The proposed control law can ensure the exponentially convergence of the tracking errors.
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We also proposed a trajectory tracking controller for the snake robot based on the dynamic model [10]. Using the proposed control law and controlling shape controllable points, the robot can accomplish trajectory tracking of the head position without converging to the singular configuration. A mobile manipulator has two separate mechanisms which accomplish locomotion and manipulation tasks independently. By lifting up some links of a snake robot from the ground, the head part of it can act as a hand of a manipulator. A snake robot has two functions, locomotion and manipulation, and it can accomplish various task. A snake robot can change the functions for acheiving complicated tasks. However, in previous researches locomotion of snake robots has been mainly studied. There is few research which considers both the locomotion and the manipulation. In this paper, we consider cooperative transportation task by two snake robots, which is a typical application utilizing snake robot’s feature as a mobile manipulator. We consider a snake robot with active joints and wheeled links that can realize the friction mechanism of a real snake [2]. By introducing links without passive wheels, we can represent redundancy of the snake robot as shape controllable points [8]. The content of this paper is as follows. First, we derive dynamic model of a single snake robot and two cooperative snake robots that hold a common object. Second, we propose a controller which accomplishes the trajectory tracking of a holding object and the internal force control with avoiding the singular configuration. Finally, simulation and experiment have been carried out to demonstrate the validity of the control law. II. MODEL A. Notation We consider an n link passive wheeled snake robot with m unwheeled links as shown in Fig. 1. All wheels are passive and all joints are active. The passive wheel does not slide to the side direction. The length of the link is 2l and the wheel is attached at the middle point of the link. Removing passive wheels is modeled the action that the real snake lifts up some part of its body in winding motion. It changes contact points of its body to the envimonment. It means that the location of the passive wheels is changed and constraint conditions are switched in the model. As the hybrid swiching system is very difficult to treat, we assume that the location of the links without passive wheels is fixed in this paper. Following the paper [8] the joint angle related to the links without passive wheels is called as the shape controllable point. The joint angle
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corresponding to the shape controllable point is a sate variable to be controlled. T Let w = [xh , yh , θh ] be the position and the attitude of the snake head with respect to inertia coordinate frame O−xy. Let T φ = [φ1 , · · · , φn−1 ] be the set of relative joint angles. Let T τ = [τ1 , · · · , τn−1 ] be the set of joint torques. Let us define q = [wT , φT ]T ∈ Rn+2 . The joint angles corresponding to ˜ ∈ Rm−1 and the position and the shape controllable points φ the attitude of the snake head are regarded as the state variables to be controlled. Let us define the state vector to be controlled ˜T ]T ∈ Rm+2 and the generalized coordinate ¯ = [wT , φ as w T ¯ T , θ¯ ]T ∈ Rn+2 where θ¯ is the vector related to vector q¯ = [w the n−m joint angles that the corresponding links are wheeled. Other parameters of the i-th link are showed in Table I
Fig. 1.
An n-link snake robot
˙ A(q)w
= B(q)φ˙
PARAMETERS OF THE i- TH LINK length from center of mass to tip of link mass of the i-th link moment of inertia of the i-th link center position of the i-th link
Let us assume following conditions [8]: [Condition1]: All joints are active. [Condition2]: The 1st link is wheelless. [Condition3]: The tail link is wheeled. [Condition4]: The condition m < n − 3 is satisfied where m is the number of wheelless links. These assumption means following feature of the system [8]. As the active joint can be controlled directly, the first condition means maximization of the system redundancy. If the first link is wheeled, the position and attitude of the snake head are subjected to nonholonomic constraint. The second condition is needed to freely control the position and the attitude of the snake head. The third condition guarantees that all joints affect the movement of snake head. The fourth condition means that we introduce at least one unwheeled link and ensure redundancy of the system. These conditions imply that m + 2 is the dimension of the state vector to be controlled and n − 1 is the dimension of the input vector.
(1)
whereA(q) ∈ R(n−1)×3 , B(q) ∈ R(n−1)×(n−1) . The matrix ⎤ ⎡ l 0 ··· 0 ⎢ . ⎥ .. ⎢ ∗ . .. ⎥ l ⎥ (2) B=⎢ ⎥ ⎢ . . .. ... 0 ⎦ ⎣ .. ∗ ··· ∗ l ˙ is given, then is an invertible and lower triangular matrix. If w φ˙ can be determined uniquely. We find that this system does not have redundancy. Next we consider the n-link wheeled snake robot that the passive wheels of m links are removed. Let i1 , i2 , · · · , im be the numbers of unwheeled links where i1 = 1, im = n. As the links without passive wheels do not have the velocity constraint, the coresponding i1 , i2 , · · · , im -th row vectors are eliminated from the martices A and B in (2). Using the ˜ ∈ R(n−m)×n the new eliminated matrices A˜ ∈ R(n−m)×2 , B constraint equation is expressed as ˙ ˜ φ. ˙ =B A˜w
TABLE I
lg Mi J (xi , yi )
assume that the motion is restricted in the plane and neglect gravity force. As we introduce the assumption that the wheel does not slip to side direction, the system has the first order nonholonomic velocity constraint
(3)
In this system, attained joint velocity φ˙ corresponding to a ˙ of the snake head is not unique. In this sense given velocity w the system is redundant. A popular approach for control of redundant system is controlling internal dynamics related to the kinematical redundancy by using spanned vectors of the kernel space [11]. In this approach deriving the spanned vector is hard, especially for the complicated nonlinear system such as snake robots. So we consider that the shape contorollable point is treated as the representation of the redundancy of the redundant snake robot [10]. The joint angles corrsponding to the i1 , i2 , · · · , im -th unwheeled links, namely shape controllable points, are controlled directly. Now we set the joint variables corresponding to the i2 , · · · , ˜ ∈ Rm−1 and the im -th links as the shape controllable point φ ˜T ]T . The velocity ¯ = [wT φ state variable to be controlled as w constraint (1) is rewritten as ¯ θ¯˙ A¯w ¯˙ = B
(4)
where θ¯ is the state vector corresponding to the wheeled links. Let us introduce selection matrices S ∈ R(m+2)×(n+2) , S¯ ∈ ¯ = Sq ∈ R(m+2)×1 , θ¯ = R(n−m)×(n+2) which satisfy w (n−m)×1 ¯ . We find that the relations SS T = Im+2 , Sq ∈ R T ¯ ¯ S S = In−m , S S¯T = O are satisfied. It is easy to find that ¯ in a nonsingular matrix. B C. Dynamic equation of snake robot
B. Constraint conditon by passive wheel At first we consider the n-link wheeled snake robot that the first link is unwheeled and the other links are wheeled. We
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¯ to q. Let us set a transformation matrix T from q T =
¯ q =TT q S¯T S
(5)
TABLE II PARAMETERS OF THE OBJECT
where T ∈ R(n+2)×(n+2) is an orthogonal matrix. T −1 = T T
(xo , yo ) θo lo Mo Jo dxyo dθo
The velocity constraint (1) is expressed as Pfaffian system
w ¯˙ −F¯ In−m (6) = −F¯ In−m T T q˙ = 0 ˙θ ¯ ¯ By using the Lagrange multiplier method ¯ −1 A. where F¯ = B we obatin
−F¯ T ˙ M (θ)¨ q + C(θ, θ)q˙ + D(θ)q˙ + T λ = Eτ (7) In−m where λ∈R(n−m)×1 is the Lagrange multiplier, M (θ) ∈ ˙ θ) ∈ R(n+2)×(n+2) R(n+2)×(n+2) is an inertia matrix, C(θ, is Coriolis’and centrifugal forces, D(θ) ∈ R(n+2)×(n+2) is a damping martix, and E = [O(n−1)×3 , In−1 ]T ∈ R(n+2)×(n−1) . Considering n − m constraint conditions the n+2 dynamic equation is rewritten as m+2 dynamic equation ˜w ˜ ¨¯ + C˜ w M ¯˙ = Eτ where ˜ M C˜
˜ E
where subscript i (i = 1, 2) means that the variables and the matrices are corresponding to the snake robot i, Mo is an inertia matrix of the object, and Co is a damping matrix of the object motion. Assuming that two robots hold an object without changing contact points, there is the geometric constraint between the position of the head of the snake robot i; xih , yih and the position and the attitude of the object; xo , yo , θo
(8)
¯ > 0. The derived equation (8) In (8) it is easy to find that M with Conditions 1-4 is a dynamic model of the snake robot. D. Dynamic equation of cooprative task As shown in Fig. 2 we consider the model for the cooprative task of two snake robots. We assume that the tip of a head of each snake robot contacts with the object like a pin. Parameters of the object are shown in Table II.
= = = =
x1h y1h x2h y2h
Im+2 T T ¯ = [Im+2 F ]T M T ∈ R(m+2)×(m+2) F¯
Im+2 T T ¯ = [Im+2 F ]T (C + D)T F¯
Om+2 T T ¯ +[Im+2 F ]T M T ∈ R(m+2)×(m+2) F¯˙ = [Im+2 F¯ T ]T T E ∈ R(m+2)×(n−1) .
position of the center of mass orientation length from a contact point to center of mass mass moment of inertia coefficient of viscosity damping of translation coefficient of viscosity damping of rotation
xo + lo sin θo yo − lo cos θo xo − lo sin θo yo + lo cos θo .
(11)
By differentiating these equation and arranging, the velocity constraint of the cooprative two snake robots that hold a common object is expressed as ⎡ ⎤ w ¯˙ 1 ¯˙ 2 ⎦ = 0 (12) Ac ⎣ w ˙o w ˜T1 ]T , w¯2 = [wT2 , φ ˜T2 ]T , where w¯1 = [wT1 , φ Ac −1 6 0 4 0 0 2
= 0 −1 O4×m1 0 0
˛ ˛ 0 ˛ ˛ 0 ˛ −1 ˛ ˛ 0
0 0 O4×m2 0 −1
˛ ˛ ˛ ˛ ˛ ˛ ˛
1 0 1 0
0 1 0 1
3 lo cos θo lo sin θo 7 . −lo cos θo 5 −lo sin θo
Let mi be the number of the unwheeled links of the snake ¯ T2 , wTo ]T . By using the ˜ = [w ¯ T1 , w robot i. Let us set w Lagrange multiplier method we obtain
¯c τ 1 ¯ cw ¨ ˜ + C¯c w ˜˙ + Ac T λc = E (13) M τ2
Fig. 2.
Cooperative transportation task of 2 snake robots
Let us set wo = [xo , yo , θo ]T . Dynamic equation of each robots and the object are ˜ iw ¨¯ i + C˜i w M ¯˙ i ¨ o + Co w ˙o Mo w
˜i τ i = E = 0
(9) (10)
¯c M C¯c
=
¯c E
=
=
˜ 1, M ˜ 2 , Mo ] block diag[M block diag[C˜1 , C˜2 , Co ] ⎡ ⎤ ˜1 0 E ⎣ 0 E ˜2 ⎦ 0 0
where λc ∈ R4 is the Lagrange multiplier. Let us set wc = ˜1 T , θ2h , φ ˜2 T ]T . Eliminating xih , yih in (13) and [wo T , θ1h , φ using geometric constraint (11) yield
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˙c w ˜˙ = GT w
(14)
where
G
T
= [Gs |Go ] ⎡ 1 0 ⎢ 0 1 ⎢ ⎢ 0 0 ⎢ = ⎢ ⎢ 1 0 ⎢ 0 1 ⎢ ⎣ 0 0 I3
T
Gs T = Go T lo cos θo 0 0 lo sin θo 0 Im1 0 −lo cos θo 0 −lo sin θo 0 0 0
(15) 0 0 0 0 0 Im2 0
⎤ ⎥ ⎥ ⎥ ⎥ ⎥. ⎥ ⎥ ⎥ ⎦
(16) Fig. 3.
Contact force
The Lagrange multiplier λc can be eliminated by multiplying G from the left of (13). Substituting (14) into the obtained equation gives
˜1 0 E τ1 ˜ cw ¨ c + C˜c w ˙ c = Gs (17) M ˜2 τ2 0 E where ˜c M C˜c
˜ 1, M ˜ 2 , Mo ]GT = G block diag[M ˜ 1, M ˜ 2 , Mo ]G˙ T = G(block diag[M +block diag[C˜1 , C˜2 , Co ]GT ).
Fig. 4.
The derived equation (17) is a dynamic model of the cooper˜c > 0 ative two snake robots. In (17) it is easy to find that M ˜ because Mi > 0, Mo > 0 and G is row full rank. III. CONTROLLER DESIGN The control objective in the cooperative task is accomplishing the trajectory tracking of the holding object wo → wod and controlling of the internal force to keep contact between robots and the object with avoiding the singular configuration. Two snake robots can avoid the singular configuration by ˜→φ ˜d . The desired controlling the shape contrallable points φ ˜ value φd is decided by various indices, for examples, magnitude of constraint force of wheels and dynamic manipulability ˜d with considerating the constraint force [7]. In this paper, φ is set as a sinusoidal wave [10] for simplifying a controller and decreasing calculation amount.
Decomposition of contact force
˜ c in (21) is positive, we obtain the converAs the matrix M gence property wc → wcd . From (18) the input torque vector [τ T1 , τ T2 ]T is expressed as
τ1 ˜2 ])+ F + τ ker1 (22) ˜1 , E = (block diag[E τ2 τ ker2 where τ ker ˜i ). Ker(E
i
is an arbitrary vector which satisfies τ ker
i
∈
B. Internal force
(18)
The snake robots should push the object by controlling the internal force Fc , in order to hold the common object. The force that a robot exerts on the object is called the contact force. The internal force is a function of the contact force, and it doesn’t affect movement of the object [12]. The matrix Gs ∈ R(m1 +m2 +3)×(m1 +m2 +4) is row full rank and rank(Gs ) = m1 + m2 + 3. So rank([I − G+ s Gs ]) = 1 and the dimension of Fc is 1. From (5), the relation between w ¯˙ and q˙ is given by
Im+2 ˙ q˙ = T w. ¯ (23) F¯
˜ ¨ cd − Kv e˙ c − Kp ec ) + C˜c w ˙ c } + Fc F = G+ s {Mc (w
(19)
By applying the principle of virtual work to (23), we obtain
fi ˜i τ i (24) =E ∗
Fc = [I − G+ s Gs ]η
(20)
A. Trajectory tracking Let us define a new input F ∈ R2m+4 as follows
˜2 ] τ 1 . ˜1 , E F = block diag[E τ2 The new input F is set as
where η is an arbitrary vector, ec = wc − wcd , Kv , Kp ∈ R(2m+3) are positive symmetric matrices, and Fc ∈ Ker{Gs } which doesn’t affect movement of the object is a component related to internal force. Substituting (19) into (17) yields the closed-loop system ˜ c (¨ ec + Kv e˙ c + Kp ec ) = 0. M
(21)
where τ i is the input torque of the snake robot i and f i ∈ R2 is the contact force with respect to the inertia coordinate frame. ˜i τ i are the contact The first and second row elements of E forces f i of the snake robot i. So F is expressed as F = [f T1 , ∗, f T2 , ∗]T .
(25)
The internal force f ic ∈ R2 , that acts on the contact point of the object from the snake robot i, is determined by the
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TABLE III PARAMETERS FOR SIMULATION
corresponding contact force f i . Consequently, the internal force f ic in Fc and the contact forces f i in F are in the same row of each vector. So Fc is expressed as Fc = [f T1c , 0, · · · , 0, f T2c , 0, · · · , 0]T .
T Let f i = [fxi , fyi ] be the contact force with respect to the object coordinate frame O −x y (Fig. 3). Forces along x axis don’t contribute to the internal force because they give rise to the rotational or transrational motion of the object. Otherwise, forces along y contribute to the internal force. They cancel together and don’t give rise to the rotational motion of the is passing through the object, because the direction line of fyi can be center of mass of the object. As shown in Fig. 4 fyi expressed as = fm + fc fy1 fy2 = fm − fc
n Mi l lg J
(26)
(27)
8 0.5 0.05 0.02 0.001
[kg] [m] [m] [kgm2 ]
Mo lo α Kp Kv
1.0 0.1 10 2I 3I
[kg] [m]
TABLE IV I NITIAL CONFIGURATION AND DESIRED TRAJECTORY FOR SIMULATION wo (0) θ1h (0) θ2h (0) ffiT 1 (0) ffiT 2 (0) wcd fd
[−0.2, 0.3, −0.2]T −1.7 1.7 [−1.4, 0.035, 0.025, 0, −0.025, −0.035, −0.025] [1.4, 0.035, 0.025, 0, −0.025, −0.035, −0.025] [0.2t, 0, 0, − 34 π, − 14 π + 0.2 sin πt, 34 π, 14 π + 0.2 sin πt]T 2
Fig. 5.
Movement of 2 robots and object for the cooperative control
is a force that affects transrational movement of where fm the object and fc is a force that doesn’t affect movement of the object. The relation between the vectors f 1c , f 2c and the vector fc with respect to the inertia coordinate frame are expressed as
− sin θo f 1c = fc cos θo (28) f 2c = −f 1c
where fc = |fc | ∈ R is the magnitude of the internal force fc . By using (26) and (28), we obtain Fc = fc g ker
(29)
yo , θo , θ1h , φ11 , θ2h , φ21 . In Fig. 5 a solid line represents the trajectory of the center of mass of the object, a broken where line represents the desired trajectory and the arrows show the g ker = [− sin θo , cos θo , 0, · · · , 0, sin θo , − cos θo , 0, · · · , 0]T .internal forces. From Fig. 6 we find that all state variables (30) converge to the corresponding desired trajectories (- - - ). It is easy to find that Gs g ker = 0. If the condition fc > 0 is satisfied, snake robots can exert force whose direction is passing through the center of mass of the object. As a result, the internal force fc contributes to keep contact between the robots and the object. We set fc as (31) fc = fd + α (fd − f ) dt where α ∈ R, fd ∈ R is desired force of fc and f ∈ R is the real internal force measured by a force sensor. If α is positive, fc converges fd [13]. IV. SIMURATION In the simulation we consider two 8-link snake robots where the first and second link are unwheeled (m1 = m2 = 2) and wc = [xo , yo , θo , θ1h , φ11 , θ2h , φ21 ]T . Parameters of the simulation are shown in Table III and IV where I is an appropriate identity matrix. We set that the desired trajectory of shape contrallable points are sinusoids for avoiding the singular configuration and τ ker i = 0. Fig. 5 shows the movement of the two snake robots and the holding object. Fig. 6 shows the time responses for xo ,
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Fig. 6.
Time responces for cooperative control in simulation
V. EXPERIMENT Fig. 7 shows the experimental system of two 8-link snake robots where the first and second links are unwheeled and ball casters are installed at the links to compenzate gravitational force. Force sensor is mounted at the contact point of the head of snake robot to measure internal force. A metal plate is used as the object. Physical parameters are as follows : 2l = 0.1[m], Mi = 0.41[kg], lg = 0.021[m], J = 9.5 × 10−4 [kgm2 ], lo = 0.1[m], Mo = 0.2[kg], Jo = 2.0 × 10−3 [kgm2 ]. To satisfy the assumption that two robots pinned with the object. In this experiment, the input F in (19) is used as a cooperative controller where Kp = 5I, Kv = 0.2I, α = 2, fd = 0.8 [N] . In the case of t < 5 [s], the desired trajectory of the object is set so as to generate the constant linear acceleration motion whose acceleration is 0.002 [m/s2 ]. In the case of t ≥ 5 [s], the desired trajectory of the object is set so as to generate the uniform straight-line motion whose velocity is 0.02 [m/s2 ]. Fig. 8 shows the time responses for xo , yo , θo , θ1h , φ11 , θ2h , φ21 . The control objective in the cooperative control is accomplishing the trajectory tracking of the position and attitude of the object and controlling the internal force with avoiding the singular configuration. From Fig. 8 we find that two snake robots accomplish the desired task. But steady-state error occurs in the response of xo , vibration is found in the response of fc , and φ11 , φ21 don’t converge the desired trajectories. We can guess that these errors are affected by the modeling error caused by sliding of the passive wheels to the side direction and measurement errors of the stereo vision system that measures the position and the attitude of the holding object, and the attitude of the snake robots. Improvement of these problems is a future work. VI. CONCLUSION We have considered cooperative transportation task by two snake robots, which is a typical application utilizing snake robot’s feature as a mobile manipulator. We derived dynamic model of the snake robot and cooperative two snake robots, and proposed controller which accomplishes the trajectory tracking of the position and attitude of the object and control of the internal force with avoiding the singular configuration.
Fig. 7.
Fig. 8.
Time responses for cooperative control in experiment
Simulation and experiment results showed the validity of the proposed control law. For applications in real world, we should consider more realistic contact condition. We will discuss dynamic model and control law with considering the rolling contact between snake heads and the object. R EFERENCES [1] J. Gray, Animal Locomotion, pp. 166-193, Norton, 1968. [2] S. Hirose, Biologically Inspired Robots (Snake-like Locomotor and Manipulator), Oxford University Press, 1993. [3] T. Kamegawa, F. Matsuno and R. Chatterjee, “Proposition of Twisting Mode of Locomotion and GA based Motion Planning for Transition of Locomotion Modes of a 3-dimensional Snake-like Robot”, Proc. IEEE Int. Conf. on Robotics and Automation, pp. 1507-1512, 2002. [4] G. S. Chirikijian and J. W. Burdick, “The Kinematics of HyperRedundant Robotic Locomotion”, IEEE Trans. on Robotics and Automation, Vol. 11, No. 6, pp. 781-793, 1995. [5] J. Ostrowski and J. Burdick, “The Geometric Mechanics of Undulatory Robotic Locomotion”, Int. J. of Robotics Research, Vol. 17, No. 6, pp. 683-701, 1998. [6] P. Prautesch, T. Mita, H. Yamauchi, T. Iwasaki and G. Nishida, “Control and Analysis of the Gait of Snake Robots”, Proc. COE Super MechanoSystems Workshop’99, pp. 257-265, 1999. [7] H.Date, Y.Hoshi and M.Sampei,“Locomotion Control of a Snake-Like Robot based on Dynamic Manipulability”, Proc. IEEE Int. Conf. on Intelligent Robots and Systems, pp.2236-2241, 2001. [8] F. Matsuno and K. Mogi, “Redundancy Controllable System and Control of Snake Robot with Redundancy based on Kinematic Model”, Proc. IEEE Conf. on Decision and Control, pp. 4791-4796, 2000. [9] F. Matsuno and K. Suenaga, “Experimental Study on Control of Redundancy 3D Snake Robot based on Kinematic Model”, Proc. IEEE Conf. on Robotics and Automation, pp. 2061-2066, 2003. [10] F.Matsuno and H.Sato, “Trajectory tracking control of snake robot based on dynamic model”, Proc. IEEE Int. Conf. on Robotics and Automation, pp.3040-3046, 2005. [11] R.M.Murray, Z.Li and S.S.Sastry, “A Mathematical Introduction to Robotics Manipulation”, CRC Press, 1994. [12] Y.Nakamura, “Control of finger (Japanese)”, Journal of the Sosciety of Instrument and Control Engineers, Vol.30, No.5, p395-399, 1991. [13] T.Yoshikawa, “Foundations of Grasping and Manipulation 3.Control (Japanese)”, Journal of the Robotics Society of Japan, Vol.14, No.4, pp505-511, 1996.
Experimental system
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