Kinematic and Dynamic Modelling for a Class of Hybrid Robots Composed of m Local Closed-Loop Linkages Appended to an n -Link Serial Manipulator

Featured Application: The systematic and generic method proposed in this study for the kinematic design and dynamic modelling for a class of complex hybrid robots is useful and applicable for the development of new hybrid robot products. Based on the proposed method, the mechanism of a new hybrid robot can be synthesized and analysed effectively; the dynamic model and control law of a complex robot can be formulated and simulated in a simplified and effective manner. Abstract: Recently, more and more hybrid robots have been designed to meet the increasing demand for a wide spectrum of applications. However, development of a general and systematic method for kinematic design and dynamic analysis for hybrid robots is rare. Most publications deal with the kinematic and dynamic issues for individual hybrid robots rather than any generalization. Hence, in this paper, we present a novel method for kinematic and dynamic modelling for a class of hybrid robots. First, a generic scheme for the kinematic design of a general hybrid robot mechanism is proposed. In this manner, the kinematic equation and the constraint equations for the robot class are derived in a generalized case. Second, in order to simplify the dynamic modelling and analysis of the complex hybrid robots, a Lemma about the analytical relationship among the generalized velocities of a hybrid robot system is proven in a generalized case as well. Last, examples of the kinematic and dynamic modelling of a newly designed hybrid robot are presented to demonstrate and validate the proposed method. Author Contributions: Conceptualization, A.M.C. and C.D.N.; methodology, A.M.C. and C.D.N.; software, M.H.V.; validation M.H.V., A.M.C. and X.B.D.; formal analysis A.M.C., C.H.L. and M.H.V.; investigation, A.M.C., C.H.L. and T.A.N.; writing—original draft preparation, A.M.C.; writing— review and editing, C.H.L. T.A.N.; administration, A.M.C.;


Introduction
Generally, the robotic mechanisms can be categorized into three main groups: the open-loop mechanism, the closed-loop mechanism and the hybrid mechanism. A robot of the first group is usually designed with an end-effector connected to a fixed base by means of a single serial kinematic chain. Meanwhile, each robot of the second group is a mechanism which is usually composed of a moving end-effector connected to a fixed base by at least two kinematic chains, such as the parallel robots. A hybrid manipulator is a mechanism which is usually composed of some open-loop kinematic chains combined with some closed-loop linkages .
The main advantages of the serial robots are the large workspace, the high dexterity of the endeffector and the large non-singular range of the joint variables. However, the robots of this group suffer from several drawbacks, such as relatively low stiffness and accuracy, low nominal load/weight ratio and heavy structure [21]. Therefore, in the last decades, several types of robots having closed-loop kinematic chains have been investigated in order to obtain better kinematic performances. The main advantages of these robots are low weight, compact structure, better accuracy and stiffness. Nevertheless, this robot group also have some disadvantages, such as a small workspace, a complex mechanical design and a complex procedure for dynamic modelling and control. For these reasons, in recent years, great attention has been paid to the development of hybrid robots which have combined advantages of the serial robots and the closed-loop architectures. More and more hybrid robots are designed and developed to meet the increasing demand for a wide spectrum of hybrid robotic applications .
In practice, the hybrid robots have been designed in a large variety of kinematic configurations and structures, and they can be classified into three main classes as follows: The first class (Class I) includes all the hybrid robots whose structure has several parallel modules connected in series [1][2][3][4][5][6][7][8][9][10] (Figure 1a).
Class III is a family of the hybrid robots which are composed of some local closed-loop chains appended to a main serial mechanism [25][26][27][28][29][30][31][32][33] (Figure 1c). It is clearly seen that the main architecture of a hybrid robot Class I or II is a parallel mechanism. Other parallel modules or serial links are connected to such the main structure to complete a hybrid serial-parallel robot.
Different from the robots Class I and II, each hybrid robot Class III is usually designed with a serial module as the main structure of a robot. To provide with desirable mechanical advantages for a hybrid robot, some local closed-loop mechanisms are added to the main serial module. For example, when designing manipulators which suffer a heavy payload and operate in a large workspace, parallelograms were added and hydraulic cylinders were used for driving some revolute joints of the manipulators [26,[28][29][30][31]. The addition of parallelograms to the main arm is to increase the rigidity of the robot, and also to restrict the orientation of the end-effector as desired. The use of the hydraulic cylinders is to increase the structural stiffness and to avoid the counterweights for the robot structure. The parallelograms and cylinders and other links of the serial arm naturally compose local closed loops of the entire robot mechanism. Other examples of using local closed-loop linkages to optimize the design and control for a robot can be found in [26][27][28][29][30][31][32][33][34].
In recent decades, a huge number of hybrid robot prototypes and products have been designed and developed . The dynamic modelling and control for some of these complex robots have been addressed as well. However, to develop a generic method for formulating the dynamic model of a generalized hybrid robot mechanism is challenging.
In the literature, there have been efforts that focused on the kinematic modelling of the hybrid robots Class I [1][2][3][4][5]. The kinematic modelling and analysis of two serially connected parallel mechanisms were investigated in [1]. In [2], the kinematic redundancy issue of a serial-parallel manipulator was addressed. The Screw theory and the Jacobian approach were applied for the kinematic modelling of a serial-parallel robot [3,4]. In particular, the dynamics and the structural stiffness models of some specific hybrid robots Class I were formulated and analysed [6][7][8][9].
There have also been attempts working on the modelling and analysis of the hybrid robots Class II [11][12][13][14][15][16][17][18][19][20][21]. A real-time implementation of path planning, trajectory generation, and servo control for a hybrid manipulation was presented in [11]. The kinematic modelling, the kinematic design, the workspace modelling and other aspects related to the kinematics of the hybrid robots Class II were studied [12][13][14][15][16][17]. The dynamic modelling and analysis of the robots Class II were also investigated [18,19]. In addition, the structural synthesis and the stiffness of the hybrid robots of this class were studied [20][21][22].
Apart from the aforementioned works, there have been investigations that emphasized on design and development of the hybrid robots Class III [25][26][27][28][29][30][31][32][33]. A valuable comparison of the conventional serial robot architectures and the hybrid robots Class III was presented in [34]. In the research [25], some theoretical issues related to the kinematic modelling of multi closed-loop mechanisms were addressed. In addition, some closed-form solutions to the kinematics problem of individual hybrid robots Class III were presented in [26][27][28]. These solutions played an important role in validating the design of the robots. The workspace and mobility analysis were also investigated in [28,29]. Other aspects related to the design analysis of heavy-duty robots were presented in [30,31]. The control issues were studied in [32,33]. It is noticeable that the dynamic modelling of a hybrid manipulator Class III (forestry robot) was investigated in [33]. However, the effects of the local closed-loops on the dynamic responses of this robotic system were overlooked.
Most of the efforts working on the hybrid robots Class III mainly focused on the design and control of some individual robots for the kinematical aspects. Though the dynamic modelling of the robots was addressed in a few researches, e.g., [33], however, the mass and inertia of the local closedloop linkages that have significant effects on the robot motion were neglected; the geometrical and kinematical constraints due to the presence of the local closed-loop linkages were usually ignored. Moreover, little attention has been paid to the development of a general method for the kinematic and dynamic modelling of this robot class. Therefore, in this paper, a new method for designing the kinematic chain and formulating the dynamic model of the hybrid robots is developed. First, a generic scheme for the kinematic design of a generalized hybrid robot mechanism is proposed. In this scheme, the generalized hybrid mechanism is synthesized with m local closed-loop linkages appended to a general n-link serial manipulator. Each locally closed linkage is regarded as a four bars mechanism which is made up of two successive links of the main serial arm, and other two added links. Second, the kinematic equation for the generalized hybrid mechanism and constraint equations for the closed loops are derived in a generalized case. Third, a Lemma about the relationship among the generalized velocities of the hybrid robots is proven, which is useful for transforming the dynamic equation and constraint equations into a minimal and compact form. Last, the kinematic and dynamic modelling of a real robot prototype are presented to demonstrate and validate the proposed method. It is shown clearly that, since the kinematic and dynamic modelling of a hybrid robot take into account the constraint equations and the dynamic effects of all the local closed loops, the kinematics and dynamics models of a hybrid robot are formulated in a better and more accurate manner. Therefore, the method proposed in this study is advantageous and plays a crucial role in the development of hybrid robot products.

Kinematics of a Generalized Mechanism for The Hybrid Robots Class III
Let us consider a manipulator which is designed with n serial links    Let us consider the main kinematic chain of the hybrid robot. In Figure 2,  can be calculated as follows: Note that the variables 1 k q and 2 k q in Equations (4)-(6) can be determined with respect to i q via the two constraint equations of a corresponding closed-loops k  , which will be presented later in the next section.

Constraint equations
Different from the formulation of the kinematics model for a serial robot, the formulation of the hybrid kinematics models of the hybrid robots Class III must take into account the constraints due to the presence of the local closed-loops.
As discussed before, each linkage k  includes two sub-kinematic chains that move relative to the base link. For example, as shown in Figure 4a, the first sub-chain of a linkage Type A is the link i l , and the second sub-chain includes the two link  On one hand, the point M, For a loop Type C and D the constraint equation can be written as follows: Note that, for a closed-loop Type A or Type C, the point M coincides the point In addition, for a closed-loop Type B or Type D, In essence, Equations (7) and (8) can be rewritten as three constraint equations corresponding to three axes of the coordinate system For every local linkage, only one joint variable among i q ,   ...
is the vector of applied torques/forces. By using the Lagrangian formulation, the equations of motion including constraint equations for the robot can be written as follows:

M s s C s s s g s
J λ τ f s 0    (10) Note that the size of the global matrices , t M s must be formulated with respect to all the links of the robot including n links of the serial chain and 2m links added to m local closed-loops.
The global mass matrix is calculated as follows: where j m and j I are the mass and inertia of a link j . The translational and rotational Jacobian matrices are calculated as where Cj r is the centre of mass, and j ω is the angular velocity of a link j , respectively.
The Coriolis and centrifugal matrix   , ,t C s s  is calculated by using Cristoffel notation.
The matrix   ,t g s is calculated as follows: where the total potential energy of the entire system is calculated as It is clear that the system of DAE equations seen in Equation (11) consists of   2 n m  differential equations of second order and 2m constraint equations   , t  f s 0 . Moreover, Equation (10) has not only n independent variables u , but also 2m dependent variables z and 2m unknown Lagrangian Multipliers λ . Therefore, if using this complex system of DAEs to analyse the dynamics behaviour and to design a control law for the robot, the time complexity of the formulation and computation will be increased dramatically. For this reason, it is necessary to transform Equation (10) into a more compact form in which the Lagrangian Multipliers λ should be cancelled, and the number of the differential equations is minimized.
By using the following Lemma,   As a consequence of this Lemma, the following important relationships, Equations (14) and (15), can be proved easily.
T T s  R J 0 (14) s = Ru + Ru     In Equation (14), the Jacobian matrix s J can be expressed as follows: Note that Equations (14) and (15) (14) and (15), multiplying T R with both sides of Equation (10) yields Different from the complex ADEs system in Equation (10), which has 2 n m  differential equations with the presence of 2m dependent variables z , and 2m unknown Lagrangian Multipliers λ , the minimal form of the dynamic equation, Equation (17), consists of only n differential equations which are expressed for n independent generalized coordinates u only.
Moreover, the matrix form of Equation (17) is similar to the form of any dynamic equation that is usually written for a conventional industrial robot. Hence, by using Equation (17), the dynamic analysis and control law computation for any complex hybrid robot Class III can be implemented in an effective and simplified manner.

Proof of the Lemma:
Taking a time derivative of the constraint equation   , t  f s 0 yields s  J s 0  Substituting Equation (16) into Equation (18) obtains Rewriting the relationship   T   s u z Ru     in the following form: Note that E is an identity matrix and T is an unknown matrix which needs to be proved.
With respect to Equation (20), the vector z  can be expressed as follows: Multiplying z J with both sides of Equation (21) yields Substituting Equation (22) into Equation (19) yields Finally, substituting Equation (24) into Equation (20) yields Equation (25) completes the proof.

Example: Kinematic and Dynamic Modelling of a Real Robot
In this section, the kinematic and dynamic modelling of a newly designed hybrid robot Class III are presented. The robot was designed and implemented with the purpose of handling material for a hot forging press shop floor. The 3D design of the robot and the robot prototype are shown in Figure  5a,b, respectively. The functional tests of the robot prototype and the numerical simulation of the kinematic and dynamic responses shows clearly the effectiveness and advantages of the method proposed in this study. The kinematic diagram of the robot is presented in Figure 6, which is designed with 4 serial links    Table 1 presents the Denavit-Hartenberg notations of the four loops. Table 2 presents the dynamics parameters of the robot.
In order to verify the formulation of the kinematics model, the following numerical experiments were carried out: 1. Calculate the trajectory of the end effector   , with the input data is given as Figure 8 shows the curves of    It is clearly shown in Figure 9 that the curves

Dynamic modelling and analysis of the robot
The dynamic equation of this robot can be derived directly by using the formulation Equation (17). In order to formulate  Table 2, and the inertias j I can be calculated based on the geometrical dimension of the links given in Table 2 as well. In Equation (11), the Jacobian matrices are calculated as , ,t C s s  is then calculated by using Cristoffel notation.
The component   ,t g s is calculated with Equation (13), where the total potential energy of the robot, , is calculated and presented in the Appendix also.
The matrix R is calculated with Equation (25), where the Jacobian matrices u J and z J are presented in the Appendix, respectively. Finally, all the terms of Equation (17) can be formulated properly that can be used for the dynamic analysis and the control law computation. In the following example, we demonstrate an inverse dynamic solution    Figure 10. It has shown that instead of the complex governing equations seen in Equation (10), the minimal form of the dynamic Equation (17) can be used directly for computation of the control (the inverse dynamic analysis) for the robot. In addition, analysing the dynamic model of a hybrid robot does not need to take into account the unknown Lagrangian Multipliers λ .

Conclusions
A new method for the kinematic and dynamic modelling for a class of hybrid robots was introduced in this study. The proposed method can be applied for the kinematic design of a robot mechanism composed of m local closed-loop linkages appended to a general n-link serial manipulator. In this manner, the kinematic equation for a generalized hybrid robot mechanism and the constraint equations due to the closed-loop linkages of a robot are derived effectively. Since a Lemma about the analytical relationship among the generalized velocities of a hybrid robot system was proven in a generalized case, the dynamic equation and constraint equations can be transformed into a minimal and compact form, so that the dynamics model of any hybrid robot can be formulated and analysed in an effective and simplified manner. Finally, to demonstrate and validate the proposed method, examples about the kinematic and dynamic modelling of a real robot are presented. It was shown clearly that, since the kinematic synthesis and the dynamic modelling of a hybrid robot take into account the constraint equations and the dynamic effects of all the local closed loops, the kinematics and dynamics model of a hybrid robot is formulated in a better and more accurate manner. In addition, since the method presented in this study was proposed and validated in a generalized case, it can be applied effectively for the kinematic and dynamic modelling of any individual hybrid robot Class III. In particular, when designing a new hybrid robot, the designers can follow all the steps of the kinematic design procedure presented in this paper to archive easily an optimal mechanism for the robot. The control law of the designing robot is then designed and simulated in an effective and useful manner by using the dynamic modelling procedure presented in this study as well. Therefore, the method proposed in this study is advantageous and plays an important role in the development of the hybrid robot products. All the experimental procedures for development of next versions of the robot prototype will be the future work of this study. The simulation of the kinematic and dynamic modelling of the hybrid robots having closed loops Type C and D will be the future study of the authors.

Conflicts of Interest:
The authors declare that there is no conflict of interest.