Research on the motion characteristics of a flexible joint-flexible link space manipulator

. Research on motion characteristics of the space flexible-joint flexible-link manipulator is investigated. The motion equations of the N degree of freedom (DOF) flexible manipulator are established by means of the Newton-Euler method and finite segment method considering both computational efficiency and accuracy. The genetic algorithm is used to identify the parameters of the torsional stiffness and damping related to the link flexibility. The derived model involves the dynamic factors, such as joint and link flexibility, joint friction and the end lumped mass of the link. Then the joint critical stiffness is adopted to decide whether to consider joint flexibility. After that, spatial motions of a two flexible-joint flexible-link manipulator are performed with different joint stiffness and friction. The results show that the spatial vibration of the flexible manipulator can be reduced by increasing both the joint stiffness and friction, and the vibration can be effectively suppressed when the joint stiffness is greater than the critical value. Meanwhile, the validity of the presented model is verified. It lays the foundation for the reliability analysis and controller design of the flexible-joint flexible-link space manipulator.


Introduction
Space manipulators have been widely employed in space on-orbit missions due to the increasing of space exploration missions.To reduce energy consumption and improve operational speed, the space manipulator is designed slender and lighter.The harmonic reducer with no backlash and high payload-to-weight ratio is introduced in joints.These lead to the structural flexibility of joints and links, which affect the system reliability and the control accuracy of the end-effector.Therefore, it is necessary to investigate the highperformance controller.Meanwhile, the accurate modelling of the flexible manipulator and identification of model parameters are also significantly important ( [1][2][3]).
Many scholars have studied the dynamic models of flexible manipulators to analyse multi-body systems and design efficient controllers.By means of the dual mass system ( [4]), Zhang et al. ( [5]) established a flexible joint model with a harmonic reducer, in which the joint friction adopts the first degree polynomial of Coulomb viscous friction.Albu-Schaffer et al. ( [6]) pointed out that the joint friction of the space manipulator may increase by 20%-50% in space compared with the ground.They used the cubic polynomial of Coulomb viscous friction for joints and carried out the parameter identification, and the results were satisfactory compared with the first degree polynomial.Zhang et al. ([7]) proposed an improved dynamic method based on Hamilton's principle for a single flexiblejoint flexible-link manipulator.The flexible joint was transformed into a flexible link with a rigid joint and elastic constraint boundaries, which reduced the number of DOFs of the system and facilitated controller design.However, compared with the dual mass system model, this method made no obvious difference in calculation accuracy.Meng et al. ([8]) established the dynamics of a single flexible-joint flexible-link manipulator by ignoring the joint friction, and analysed the end vibration characteristics with different elastic modulus and damping ratio of flexible links, joint stiffness and motion planning.The results showed that the elastic modulus and joint stiffness are very important to suppress the vibration.Wu et al. ( [9]) adopted the finite segment method proposed by Yoshikawa et al. ( [10]) to model a planar single flexible-link manipulator.The flexible link was discretized into several virtual rigid links, which were connected by virtual passive joints.However, the research ignored the dynamics of active joints and the lumped mass at the end of the link.Bai et al. ( [11]) established the dynamic model of a planar two flexible-link manipulator.The reliability optimization for multiple failure modes of the system was investigated.A master planning required minimized component mass in the condition of satisfying component reliability.Yang et al. ([12]) established the dynamic equation of the two-link flexible manipulator based on the Lagrange assumed mode method.The decoupling method of nonlinear feedback control was used to decouple the joint angle and elastic deformation partially.An extended state observer was introduced to estimate the nonlinear term of the system.The nonlinear controller of the flexible manipulator system was designed based on the backstepping method.Hami et al. ( [13]) presented a dynamic modelling behaviour for a two-DOF rigid-flexible manipulator by means of the principle of Hamilton and Euler-Lagrange equations.A set of simulation results were given to show flexibility behaviour with and without damping effect.
In this paper, a general dynamic model that could describe the spatial motion of multi-DOF flexible space manipulator was established based on the finite segment method and Newton Euler equations.The dynamic factors, such as the flexibility of joints and links, joint friction, the end lumped mass of links, are comprehensively considered.The finite segment method is improved to involve the spatial motion of the link under the action of the end lumped mass.The critical stiffness method whether to consider the joint flexibility is adopted.Joint frictions are identified by using the nonlinear cubic Coulomb viscous model.According to the main feature of flexible slender links of the 7-DOF space manipulator, it is simplified into a 2-DOF system to investigate the spatial vibration characteristics with different joint stiffness and friction.

Kinematic description of manipulator 2.1 Coordinate system and symbol
Consider the case of the N-DOF flexible manipulator.The flexible link is discretized into a finite number of rigid links by using the finite segment method, and the rigid links are connected through virtual passive joints containing a torsion spring and damper.The discrete model and coordinate systems are shown in Fig. 1 and Fig. 2.
Three types of Cartesian coordinate systems are employed, i.e. the inertial reference frame O 0 -x 0 y 0 z 0 , the link-fixed reference frame O i -x i y i z i (i=1, 2, ..., N), and the rigid links reference frame O ij -x ij y ij z ij (j=1, 2, ..., n), n is the number of rigid links of each flexible arm.The virtual passive joint of the virtual rigid link (i,j) is indexed as (i,j). s and ij I ji represent the mass of the virtual link (i,j), the position vector from the origin O ij to O ij+1 expressed in {ij}, the vector from the origin O ij to the center of gravity of the link (i,j) expressed in {ij}, and the inertia tensor expressed at the center of mass of the link (i,j) in {ij}, respectively.The left superscript indicates the reference frame.
The manipulator incorporates harmonic drives for the speed reduction introducing torsional elasticity into joints ( [4]).To model the flexible behaviour in the active joint i, the link i is considered to be rigid and connect to the rotor through a torsional spring of stiffness K gi .The link i is driven by a servo motor of the active joint i, as shown in Fig. 3.The quantities related to the motor i include the rotation angle θ mi , moment of inertia J mi , output torque τ mi and friction torque τ mfi .And θ i , m i , C J i and COG i represent the rotation angle, mass, inertia tensor and center of mass of the link i, respectively.τ i , τ fi and τ ei are the driving torque, bearing friction torque and environmental loads acting on the link i, respectively.The system has two sets of generalized coordinates due to the joint flexibility.Let the column vector of the generalized coordinates be is the joint variable which represents the rotation angles of links, is the rotation angle of the motor shaft after the reducer, and η i is the gear reduction ratio.

Kinematics of N-DOF flexible manipulator
Kinematic transformations are established by using the Denavit-Hartenberg method.The homogeneous transformation matrix 1 a a − T of reference frame {a} expressed in {a-1} is: where a = (1,1), (1,2), …(i,j), …, (N,n), 1 a a − R is rotation transformation matrix, 1 a aO − P is position vector of the origin of {a} expressed in {a-1}.They can be written as: ( ) where otherwise [ ] where l a-1 , α a-1 and d a are the length, torsion angle and offset of link (a-1), respectively.The rotation transformation matrix ( ) where The rigid link (i,1) is fixed with the active joint i when a = (i,1) When a ≠ (i, 1), there are ϕ ijx = ϕ ijy = 0 for the link (i,j).

Dynamic model of flexible manipulator
The flexible dynamic model of N-DOF manipulator based on Newton-Euler equation is derived as follows.Angular velocity a ω a of link a with respect to {a} is: where ( ,1) where otherwise Angular acceleration can be expressed as:  Ca  v at the center of gravity satisfy: ( ) Let a f a and a n a be the force and moment of the joint a acting on the virtual link a, respectively.Then the equilibrium equations for the virtual link a can be written as: ( ) where a+1 f a+1 and a+1 n a+1 represent the external force and moment acting on the {Nn} at the end of the manipulator when a = (N,n), respectively.a F a and a N a are inertia force and moment expressed in {a}, respectively.According to Newton-Euler equation, we have: where the origin of reference frame {C a } is located at the center of mass of rigid link a, the initial orientation coincides with the coordinate {i} of the active joint i.The end lumped mass of the flexible link i satisfy m a = m ei when a = (N,n).The motion equation of the active joint i around the axis T [0, 0,1] i z =  can be written as: T 1 where ( ,1) where a n a is the torque of active joint i when a = (i,1).When a ≠ (i,1), a n a is written as:

Mathematical model of flexible joint
According to equation ( 15), the dynamic model of active joint i can be expressed as: where is the moment of inertia of the joint i, T fi is friction torque, τ gwi is the transmission torque of the motor i coupled with the link i through the harmonic reducer.
Driving torque τ gwi of flexible joint i can be expressed as: ( ) In order to decide whether the joint flexibility needs to be considered, the critical ratio r ci of the first-order natural frequency of the joint-link coupling system and the cantilever beam is utilized.r ci is a nonlinear function of the link flexibility parameter n i and the critical joint stiffness K gci .When K gi is greater than K gci , the joint is considered to be rigid.
There is a linear relationship between K gci and n i ( [7]): where n i = E i I i /l i reflects the link flexibility, E i , I i and l i are the elastic modulus, crosssectional moment of and length of the link i, respectively.

Numerical simulation and analysis
In this section, the spatial dynamic characteristics of the 2-DOF flexible manipulator equivalent from a 7-DOF system are analysed.Each flexible link is discretized into rigid links with n = 3, which can satisfy the accuracy requirements ( [9]) and expend the least computational cost.The length of rigid links and the stiffness of passive joints are optimized by genetic algorithm ( [9]).The effect of gravity and the torsional deformation around the axis x ij are ignored, and only the bending deformation of the passive joint along the axes y ij and z ij is considered.The end of the manipulator is subjected to a constant load of 24 f 24 = [0,0,2]N along the z 23 direction.The initial angles of active joints are θ 1 = 30º, θ 2 = -25º, the time step is 1 ms.Joint driving torque τ 11 = 6sin(4πt) when t ≤ 0.5 s, and τ 12 = 0.
The physical parameters of the flexible links are shown in Table 1 (i = 1,2), and the parameters of the rigid links are shown in Table 2 (j = 1,2,3).Fig. 7 and Fig. 8 show that the vibration of the end in y 0 decreases significantly with the increase of K gi .When K gi =200 Nm/rad, the end vibration in y 0 direction is obvious.Moreover, there is a phase advance in z 0 and φ y0 directions, which indicates that the flexible deformations of the links are excited rapidly when the joint stiffness is low.When K gi > K gci , the vibrations in the directions of the y 0 and φ z0 decrease significantly in 8-12s.

Conclusion
In this paper, the dynamics of a flexible-joint flexible-link space manipulator is investigated, and the effects of different active joint stiffness and friction on the motion characteristics of the flexible manipulator are discussed.
In order to analyse the dynamic characteristics of the flexible manipulator, a space dynamic model of N-DOF flexible manipulator taking both computational efficiency and accuracy into account is established based on the finite segment method and Newton-Euler equation.Then the effects of different joint stiffness on the spatial motion characteristics of 2-DOF flexible manipulator are analysed.The results show that the vibration amplitudes decrease with the increase of the stiffness, especially greater than critical values.The joint angles, flexible deformations of the links and the end vibrations will be suppressed as the increase of joint stiffness.The validity of the established model has been verified through the above analysis.In the future, the models of an actual manipulator will be established.

Fig. 1 .
Fig. 1.The finite segment model of N-link flexible manipulator.

Fig. 2 .
Fig. 2. The spatial coordinate system of virtual rigid links of Link i with an end mass.The angular deformations of the passive joint (i,j) are ϕ ijx 、 ϕ ijy and ϕ ijz around the axes x ij , y ij and z ij .The spring stiffness are k ijx , k ijy and k ijz .The damping coefficients are d ijx , d ijy and d ijz .Let the deformation vector be ϕij = [ϕ ijx , ϕ ijy , ϕ ijz ] T , the spring stiffness kij = [k ijx , k ijy , k ijz ] T and the damping coefficient dij = [d ijx , d ijy , d ijz ] T .The lumped mass at the end of the Link i is m ei .The parameters m ij , ij ij  p , ij ij

Fig. 3 .
Fig. 3. Integrated model of the i-th flexible revolute joint and rigid link.
is angular acceleration of rigid link a.The linear velocity a a v , the linear acceleration a a  v at the origin O a , and the linear acceleration a

Fig. 6 .
Fig. 6.Time history curves of the passive joint angles in y orientation.

Fig. 7 .
Fig. 7. Position time history of the end vibrations of the flexible manipulator.

Fig. 8 .
Fig. 8. Orientation time history of the end vibrations of the flexible manipulator.

Table 1 .
Physical parameters of the two flexible links.