International Journal of Advanced Robotic Systems an Analytic Method for the Kinematics and Dynamics of a Multiple-backbone Continuum Robot Regular Paper

which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract Continuum robots have been the subject of extensive research due to their potential use in a wide range of applications. In this paper, we propose a new continuum robot with three backbones, and provide a unified analytic method for the kinematics and dynamics of a multiple‐backbone continuum robot. The robot achieves actuation by independently pulling three backbones to carry out a bending motion of two‐degrees‐ of‐freedom (DoF). A three‐dimensional CAD model of the robot is built and the kinematical equation is established on the basis of the Euler‐Bernoulli beam. The dynamical model of the continuum robot is constructed by using the Lagrange method. The simulation and the experiment's validation results show the continuum robot can exactly bend into pre‐set angles in the two‐dimensional space (the maximum error is less than 5% of the robot length) and can make a circular motion in three‐dimensional space. The results demonstrate that the proposed analytic method for the kinematics and dynamics of a continuum robot is feasible.


Introduction
A continuum robot is a new type of biologically inspired robot. Compared to traditional rigid link robots, continuum robots feature a continuous backbone without joints [1], redundant degrees of freedom and exhibit significant compliance that provides exceptional operational capacities during environmental interaction and object manipulation. Due to inherent flexibility, a continuum robot has great potential in applications that include operation inside complex, unstructured environments [2][3], such as collapsed buildings in search and rescue operations [4][5][6] or minimally invasive surgery (MIS) in medical applications [7][8][9][10].
The outstanding compliance and adaptability of invertebrate limbs such as elephant trunks and octopus arms have motivated a recent surge of research activity in continuum robots [11]. Walker et al. designed a continuum robot inspired by the elephant trunk, which is actuated by a single flexible backbone with wires [10][11][12][13]. Simaan et al. presented a continuum robot with multiple flexible push-pull rods and derived a novel control method based on force sensing [7][8][9]15]. Webster III et al. proposed a novel continuum robot using the geometry and elastic interaction of precured concentric tubes to achieve a wide variety of shaft curves [16]. They put forward a framework for the kinematics of multi-link active cannulas and derived a set of differential equations that capture both bending and torsional effects continuously along the active cannulas backbone [17][18][19]. Trivedi et al. proposed a continuum robot with pneumatic actuation and derived an approach for modelling soft robotic manipulators that incorporates the effect of material nonlinearities, and distributed weight and payload [20].
There also has been much research into dynamical models of biologically inspired robot manipulators. Chirikjian developed an efficient formulation of approximate hyper-redundant manipulator dynamics, based on the principle of continuum mechanics [21]. Matsuno et al. presented the dynamic modelling of an underwater snake-like robot, based on the Newton-Euler equations [22]. The two methods considered hyperredundant serial rigid-link systems, but cannot model the continuous nature of continuum robots. Gravagne et al. proposed a large-deflection dynamic model [23][24]. But this kind of dynamic modelling cannot be used in the case of general three dimensions. In [25] and [26], Mochiyama et al. developed a three-dimensional dynamic model for an inextensible (constant length) continuum manipulator based on a slice of the arm, which is regarded as a rigid link with an infinitesimal width of a serial rigid chain manipulator. However, the dynamic model does not include extensible manipulators. Tatlicioglu put forward a new dynamic model for continuum robot manipulators by using the geometric model of extensible continuum robot manipulators with no torsional effects [27][28].
In this paper, a novel continuum robot with three backbones was proposed and provided a new analytic method based on the Euler-Bernoulli beam for kinematics and dynamics of a robot. In Section 2, a new wire-driven continuum robot using super-elastic NiTi alloy is presented. Then the kinematics of the continuum robot is analysed on the basis of the Euler-Bernoulli beam in Section 3. By using the Lagrange method, the dynamical model of the robot is obtained in Section 4. Simulation results are presented in Section 5. Experiments on the performance of the robot are described in Section 6. Finally, conclusions and future work are reported in Section 7.
2. Design of the robot and modelling assumptions Fig. 1 shows the structure of the continuum robot, which is composed of a base disk, an end disk, several spacer disks and four super elastic NiTi wires as its backbones.
The central wire is the primary backbone and the remaining three wires are secondary backbones. The secondary backbones are equidistant from the primary backbone and from each other. The central backbone is glued to all the disks, while the secondary backbones are only attached to the end disk and are free to slide through appropriately toleranced holes in the base and spacer disks. The secondary backbones are used for actuating the continuum robot. The spacer disks are distributed uniformly to prevent buckling of the central and secondary backbones and to maintain an equal distance between the secondary backbones and the central backbone. Through independent actuation of the three secondary backbones, a 2-DoF bending motion of the continuum robot can be achieved. The multi-DoF continuum robot can be realized by using series 2-DoF unit. The modelling assumptions of the continuum robot are summarized as follows.
1. The super elastic NiTi wire is assumed to have linear and isotropic relations between stain and stress [29] in the presented robot. The only external force acting on the continuum robot is driving force and the backbones behave like Euler-Bernoulli beams [9]. 2. The robot disks are thin and rigid and friction between the backbones and the disks is neglected [9]. 3. Compared to the elastic deformation energy, gravitational potential energy is much smaller. Gravity can be ignored in the analysis. 4. The shape of continuum robot is assumed to be a smooth and continuous curve and each backbone has constant curvature. 5. The robot is under static equilibrium [15]. 6. The primary and the secondary backbones are always perpendicular to the base, the spacer and the end disks.

Flexible beam with end-moment
where, EI is the stiffness of flexible beam.
According to assumption (4), we can get the follow equation:

Kinematic nomenclature
According to the structural characteristics of a continuum robot, it can be considered as a flexible beam. Fig. 3 shows the continuum robot with only the primary and two secondary backbones illustrated. There are three coordinate systems defined [8]: (1) the base disk coordinate system {X, Y, Z}, (2) the bending plane coordinate system {X1, Y1, Z1}, (3) the end disk coordinate system {Xe, Ye, Ze}. The base disk coordinate system is attached to the base disk. The X axis points from the centre to the first secondary backbone, the Z axis is perpendicular to the disk, and the Y axis is established according to the "right-hand rule". The bending plane coordinate system can be obtained by rotating the base disk coordinate around the Z axis for angle  . The end disk coordinate system is obtained from the base disk coordinate system by four steps: (1) translating the centre O to Oe, (2) rotating the local coordinate around its Z axis for angle  , (3) rotating the local coordinate around its Y axis for angle  , (4) rotating the local coordinate around its Z axis for angle   . The following symbols are used: s -arc-length parameter of the segment OP (s=0 at the base disk and s=l at the end disk). r -the distance from the primary backbone to each secondary backbone on the disk. ro -curvature radius of the primary backbone, defined in the bending plane.
p  -the bending angle of the primary backbone tangent in the X1Z1 plane at the point P.  is the bending angle at the end disk.  -the rotation angle of the bending plane.
, is the number of secondary backbones). m1 -the total mass of the primary backbone. m2 -the total mass of the secondary backbone. h -distance between adjacent disks.
The kinematics analysis consists of two transformation parts: (1) transformation between joint space and operational space; (2) transportation between driving space and joint space.

Transformation between joint space and operational space
According to Eq. (2), we can get the equation as follows: Based on the flexible beam model, as mentioned above, the position of point P can be obtained as follows: The coordinates of point P in the bending plane coordinate system can be translated into the base disk coordinate system as follows: where, Rot(Z,  ) is given by: The position of any point P on the primary backbone can be expressed as: (1 cos ) cos (1 cos ) sin sin Substituting Eq. (3) for Eq. (7), the coordinates of point P in the base disk coordinate system can be given by: The rotation matrix can be written as: where, sin s , the interval of the bending The transformation matrix from the end disk coordinate system to the base coordinate system can consist of Eqs. 8 and 9 and be expressed as:

Transformation between joint space and driving space
In the motion of the continuum robot, the position and shape can be controlled by changing the lengths of the three driving NiTi wires, which are equally spaced by 120 degrees around the primary backbone, as shown in Fig. 4. ri {i=1, 2, 3} represents the curvature radius of each second backbone. The change in length of each wire is realized by a stepping motor and the motion of each motor is directly proportional to the displacement of the wires. The transformation between driving space and joint space can be expressed as: where, qi {i=1, 2, 3} is the length of each driving wire and r is the distance from each secondary backbone to the primary backbone (the secondary backbones are equidistant from the primary backbone), and 2 / 3    .

Kinetic energy of the continuum robot
The structure of the continuum robot consists of a primary backbone、a secondary backbone and disks. So the kinetic energy of the continuum robot should consist of these three parts.
Firstly, the kinetic energy of the primary backbone can be obtained. Calculating the derivative of the Eq. (8) The kinetic energy of the primary backbone can be obtained as: where  is the density of the primary backbone and A is the cross-section area of the primary backbone.
Substituting Eq. (12) for Eq. (13), we can obtain the following: where m1 is the mass of the primary backbone and K1 and K2 are the kinetic energy equivalent factor.
From Eq. (15) and (16), we can see that K1 and K2 can be expressed as a function of bending angle  . The two equations can be simplified by using least square fit, as shown in the following:  5 and 6 show the comparison and error analysis of the kinetic energy equivalent factor. From Fig. 5(a) and 6(a), we can see that the fitting curves are closely consistent with the actual curve. Fig. 5(b) shows that the calculated relative error of the kinetic energy equivalent coefficient is less than 0.8%, which indicates that the simplified expression is very accurate. The continuum robot has a small deflection near the base disk, which makes the relative error slightly larger, as shown in Fig.  6(b). But when the rotation angle  >0.5rad, the relative error becomes smaller. According to the above analysis, the simplifications of the kinetic energy equivalent factors are available.  Secondly, the kinetic energy of the secondary backbone, which consists of two parts, can be obtained. One part of the kinetic energy of the secondary backbone is the same as with the primary backbone. Another part of the kinetic energy comes from driven kinetic energy.
From Eq. (12), the first part of the kinetic energy can be expressed as: where  is the density of the secondary backbone and A is the cross-section area of the secondary backbone.
By combining Eq. (12) and (19) we can get the following equation: where m1 is the mass of the secondary backbone, K1 and K2 are kinetic energy equivalent factors (as mentioned above).
Calculating the derivative of Eq. (11), the driven velocity can be given as: Then the driven kinetic energy of the secondary backbone can be obtained as follows: Substituting Eq. (21) for Eq. (22), we can get the following equations: [ sin 2 sin 2( ) sin 2( )] where m1 is the mass of the secondary backbone and K3, K4 and K5 are kinetic energy equivalent factors.
Combining Eq. (20) and (23), the total kinetic energy of the secondary backbone can be written as: where k=1, 2, 3 …, n is the total number of spacer and end disks (n=l/h).
The total kinetic energy of all the spacer and end disks can be expressed as: By combining Eq. (28) and (29), we have the following equation:   Fig. 7(a) and 8(a), we can see that the fitting curves are very consistent with the actual curve. Fig. 7(b) shows that the calculated relative error of the kinetic energy equivalent coefficient is close to zero. It indicates that the simplified expression is very accurate. From Fig. 8(b) we can see that when the rotation angle  is smaller, the relative error is slightly larger. But when the rotation angle  >0.5rad, the relative error is less than 0.01%. The analysis from the above proves that the simplifications of the kinetic energy equivalent factors are feasible.  By combining Eq. (14), (27) and (30), the total kinetic energy of continuum robot can be obtained as follows:

Potential energy of the continuum robot
For a continuum robot, the potential energy consists of two parts: elastic potential energy and gravitational potential energy. Relative to the elastic potential energy, its gravitational potential energy can be ignored. The reason will be shown in the following section.
Each disk of the continuum robot in Fig. 1 weighs 0.848g. The density of NiTi is 6.2 grams/cm 3 . It is assumed that the shape of the continuum robot is circular. The ratio of gravitational energy to elastic energy can be calculated and is a function of the bending angle  , which is plotted in Fig. 9. The maximum value of gravitational potential energy is less than 1.4% of the elastic deformation energy for the continuum robot. And it decreases with the increase of the bending angle  . So compared to the elastic deformation energy, the gravitational potential energy can be ignored. The conclusion proves the feasibility of assumption (3). The elastic energy of the continuum robot is given by [31]: where EI is the stiffness of NiTi alloy wire, / p s l    as mentioned above.

Generalized force of the continuum robot
Generalized force is a force or torque acting on the generalized coordinates. As the robot has two degrees of freedom, the pose of the continuum robot can be determined by two driving forces. The structure of continuum robot is centrosymmetric, as shown in Fig. 10. So the driving force F1, F2 and F3 have the characteristic of space rotation symmetric. The robot achieves actuation by independently pulling two backbones at the same time.
The following formulas are derived based on the assumption that F1 and F2 take effect. Given the displacement of driving force F1 and F2 are q1 and q2, respectively. qi can be defined as l-li, where l is the length of the primary backbone and li is the length of secondary backbone. The generalized force of the continuum robot can be expressed as: From Eq. (11) and (37), we can get the following equations: where r is the offset from the primary backbone to the secondary backbone.

Dynamical model of the continuum robot
The Lagrange equation can be expressed as: where Qj is the generalized force of system, 1 p   and 2 p   (as mentioned in Fig. 3).

Simulation examples
To underline the validity of the proposed kinematic and dynamical models, we present some simulation results obtained for the continuum robot using Matlab7.0. The parameters of the simulated continuum robot are given in Table 1. These values correspond to the robot in Fig. 3 and for the rest of this paper. The pre-set end trajectory of the continuum robot in a base disk coordinate system is given as follows: where R=71.62mm is the radius of the circle, / 6    (rad/s) is the sports angular velocity and hz=124.05mm is the height of the circle plane in the Z direction.
For the given end trajectory of the continuum robot in a base disk coordinate system, the movement of all the driven components can be obtained through the kinematics analysis and the driven force can be obtained through dynamics analysis. The initial position of the robot is perpendicular to the base disk (the central axis of the robot coincides with the Z-axis direction). According to the special structure of the continuum robot, the trajectory of the continuum robot should be simulated in two steps: (1) bending simulation and (2) circle simulation.
(1) Bending simulation Figure 11. Joint angles, driving displacement and driving force for the planar simulation.
In this step, the end point of the continuum robot will be pulled to the given circle path position by only one secondary backbone. Fig. 11 shows the joint angles、driving displacement and the driving force for the two-dimensional simulation. The results show that when the continuum robot is bending at uniform speed, its driving force is linear to the time. The bending angular velocity v  is / 12  rad/s and the rotation angular velocity v  is 0.

(2) Circle simulation
In this step the secondary backbones of the continuum robot should act to drive the end disk to a certain pose. Fig. 12 shows joint angles, driving displacement and driving force for the three dimensional space simulations. The results show that only one or two driven forces are needed to drive the robot to a certain position and the result corresponds to the DOF of the continuum robot. The bending angular velocity v  is 0 and the rotation angular velocity v  is / 6  rad/s.

Experiments
The prototype of the continuum robot is shown in Fig. 13. The length of the continuum robot is 150mm and its diameter is 10mm. The robot can bend ±180°in any direction. The primary and secondary backbones are made of super elastic NiTi wires with 1.2mm diameters. The secondary backbones are connected with three stepping motors which provide the driving force for the continuum robot. The tension sensor is used to measure the driving force provided by the motors. The motion ability of the continuum robot was tested to prove the feasibility of the analytic method for kinematics and dynamics.

Bending ability experiment of the robot
The continuum robot has two DoF, which are bending angle  and rotation angle  . The experiment is carried out in two-dimensional space (rotation angle 0   ) with one secondary backbone applying the driving force.   Fig. 15 shows the error curve between the theoretical trajectory and the actual trajectory of the end of the robot. The desired trajectory was drawn on the coordinate paper, which was divided into 15 sections equally by 13 points. The robot was driven to bend from 0°to 180°. There was a pen on the end of the robot, so the actual trajectory can be drawn on the paper during the process. We can estimate the errors by measuring the distance between the desired trajectory and actual trajectory at every pre-set point. The result shows the maximum error is no larger than 5mm, which means the relative error is less than 4% of the robot length. Reasons for the error include: (1) manufacture and assembly error, (2) measurement error and (3) the deformation of the NiTi wires.

Driving force experiment of the robot
The pre-set end trajectory of the continuum robot is the circle defined by Eq. 41 and mentioned in Section 5. The driving forces are measured by the tension sensor and its signal is recorded by the computer. The experiment is also divided into two steps: (1) bending experiment and (2) circle experiment.

(1) Bending experiment
In this step, the end point of the continuum robot will be pulled to the given circle path position by one NiTi wire with the driving force of F1. Fig. 16 shows the theoretical value, actual value of F1 and the errors between them. The result shows that the theoretical value of the driving force is less than the actual value, which indicates that the secondary backbone has an initial pretension force in the free state. In addition, because of the friction between the NiTi wires and disks, the error will increase with the increasing driving force. (2) Circle experiment In this step, the secondary backbones of the continuum robot will drive the end disk to track the pre-set trajectory. The process of the experiment is shown in Fig.  17. The bending angle  remains at 30° and the velocity of rotation angle v  is / 6  rad/s.

Conclusion
In this paper an analytic method for the kinematics and dynamics of a new type of continuum robot with multiple flexible backbones has been derived. With consideration of the large deformation character of the flexible backbone, the deformation curve of the flexible beam was obtained on the basis of the Euler-Bernoulli beam equation. Then the kinematic model was formulated on the basis of the deformation. The dynamical model is derived by using the Lagrange method. The simulation and experimental results demonstrated that the kinematics analysis approach can accurately predict the shape of the continuum robot. The results of the bending ability experiment showed that the maximum error was less than 5mm, or less than 4% of the total robot length. The driving force results also showed that the actual driving forces were approximately consistent with the theoretical driving forces and validated the dynamical model.
The methods developed in this paper can be a useful design tool and also provide the theoretical foundations for control algorithms for continuum robots with multiple-backbones. However, much work remains in the modelling, design and control of continuum robots. Future work includes the study of the deformation characteristic and the stiffness of the robot with an external load, which is an important step forward for future practical applications. The extension of our model should take the friction and pretension force of the backbone into consideration. In addition, the feedback control of the robot should be addressed. On the basis of the two-DoF unit, the kinematics and dynamical model of multiple-DoF continuum robot will be studied.