Synchronization of Robot Manipulators Actuated By Induction Motors with Velocity Estimator

A complete modeling (including the actuator dynamics) of a robot manipulator that uses three-phase induction motors is presented in this paper. A control scheme is designed to synchronize robot manipulators actuated by induction motors under a masterslave scheme in the case where the joint velocity of the slave robots is estimated. All of the research on the synchronization of robot manipulators assumes the use of ideal actuators to drive the joints; for that reason, in this work, a three-phase induction motor is considered to be a direct-drive actuator for each joint. An entire model of the mated system is obtained by a combination of the dynamics of the induction motor and robot manipulator. Thus, the synchronization control algorithm for a master-slave scheme in both the joint space and workspace of robot manipulators driven by induction motors is developed. An observer based on the entire model is proposed to estimate the joint velocity of the slave robot manipulators. Through the Lyapunov criterion, a stability analysis of the synchronization control with a velocity estimator is detailed. The analytical results show the synchronization and estimation errors are globally, uniformly, and ultimately bounded. Simulations with multiple robots demonstrate the performance of the proposed control algorithm.


Introduction
The controlled synchronization of robot manipulators has become a subject of interest within the scientific community during the last decade, mainly because of the efficiency and quality requirements in the production processes, in addition to the flexibility and maneuverability in the execution of tasks that a single robot cannot perform. This has led to the demand that two or more robots complete the same activity in a synchronized manner. This synchronization consists of matching two or more dynamic systems developing singly on a common trajectory from a certain instant onward, one of them called the master and the others the slaves. In the literature, diverse algorithms of synchronization control have been stated, and all of these works assume ideal actuators. Thus, Nijmeijer et al. [Nijmeijer and Rodriguez-Angeles (2003)] applied a feedback control with access only to the robot's position. Chung et al. [Chung and Slotine (2009)], from contraction analysis, tracked a common trajectory while maintaining a formation through a directed graph of interconnection. Chopra et al. [Chopra and Spong (2008)] solved the previous features combined with constant delays of communications by using a passivity analysis. Abdessameud et al. [Abdessameud, Polushin and Tayebi (2014)] and Cicek et al. [Cicek and Dasdemir (2018)] used a passivity theorem to consider varying delays. Nuno et al. [Nuño, Ortega, Jayawardhana et al. (2013)] considered the interconnection in the synchronization scheme as a non-directed graph. Regarding the synchronization problems, different control algorithms have been applied; for instance, Zhang et al. [Zhang, Wang and Guo (2014)] required backstepping; Bouaziz et al. [Bouaziz, Bouteraa and Medhaffar (2013)] made use of cross-coupling; Mei et al. [Mei, Ren and Ma (2011)] used sliding mode controllers; and Chen et al. [Chen and Lewis (2011)] and Cui et al. [Cui and Yan (2012)] achieved synchronization through neural networks (NNs). Moreover, synchronization approaches in the workspace have been reported. In the study by Kyrkjebo et al. [Kyrkjebo and Pettersen (2007)], in which a virtual manipulator to synchronize robot manipulators in a leader-follower scheme was used, the leader velocity was unknown. Liu et al. [Liu and Chopra (2012)] obtained the synchronization of heterogeneous robot manipulators with varying delays using the passivity property. Wang [Wang (2013)] synchronized robot manipulators without a leader, but rather used directed graphs strongly connected to develop an adaptable control law with attention to parametric uncertainties. Aldana et al. [Aldana, Nuño, Basañez et al. (2014)] synchronized the position and orientation of the robot manipulators: The position synchronization was designed with the use of the Jacobian, while the orientation was synchronized by means of the quaternions. Cicek et al. [Cicek, Dasdemir and Zergeroglu (2015)] synchronized the end effectors of robot manipulators in both the joint space and workspace, considering the parametric uncertainties. In practice, a great number of industrial robot manipulators use electric motors as actuators, where the DC-brushless permanent magnet (PMBLDC) servomotors are the most common. The advantage of PMBLDC motors is given by their relative ease control of the position and desired trajectory; however, their main disadvantage is cost, which is due to the use of rare earths (neodymium-iron-boron or samarium-cobalt) in the manufacturing process of the permanent magnet and, furthermore, by the necessity of periodic mechanical maintenance. For this reason, an alternative to the use of servomotors is set by employing induction motors (IMs) as actuators. These devices generate a high-output torque and have a low manufacturing cost; furthermore, they are robust and can operate in any environmental conditions, while their disadvantage lies in their control difficulty, which is caused by their high non-linearity. Owing to this fact, Guerrero et al. [Guerrero-Ramírez and Tang (2001)] and Diniz et al. [de Diniz, Júnior, Honório et al. (2012)] researched robot manipulators driven by IMs only to track the desired trajectory. A synchronization approach that considers ideal conditions in the joint space was addressed in Torres et al. [Torres, Guerrero, Garcia et al. (2016)], where robot manipulators are driven by IMs, and, consequently, the IMs were synchronized in the task execution as well. In this sense, Sun et al. [Sun, Gong, Yang et al. (2019)] synchronized multiple induction motors for a tracking system. In this work, the definition of the tracking error in relation to the angular velocity was considered. Therefore, it was necessary to avoid the chattering phenomena that might be depicted by noise in the velocity and acceleration measurements, leading to other problems in synchronization. The objective of this work is to develop a master-slave synchronization control scheme for both the joint space and workspace of robot manipulators directly driven by IMs (designated as IM-Robot), considering the load torque for the IMs as the torque derived from the synchronization controller. Hence, the angular positions q i (t) ∈ R n , as well as the position and orientation χ i (t) ∈ R m of the ith slave IM-Robot are synchronized with respect to q j (t) ∈ R n and χ j (t) ∈ R m of the master IM-Robot through the synchronization control approaches based on velocity observer, for the case in which only the angular position measurement of the joints of the slave IM-Robot is available. The rest of this paper is organized as follows. Dynamic models of the robot manipulator and induction motor are presented in Section 2. In Section 3, the combination of IM dynamics and robot manipulator dynamics is depicted. In Section 4, the synchronization control designs with a velocity estimator in both the joint space and workspace are developed, and in Section 5, the results of the simulations are shown. Finally, conclusions are given in Section 6.
2 Dynamic models

Dynamic model of a robot manipulator
Consider p robot manipulators fully actuated in which the friction losses are neglected, with k = 1, 2, · · · , n joints. The vector of the angular position of the robot joints is q i (t) ∈ R n , i = 1, 2, · · · , p. Using the Euler-Lagrange formalism, the dynamic model of the ith robot is given by Behal et al. [Behal, Dixon, Dawson et al. (2009)]: where M (q) ∈ R n×n is the inertia matrix, C (q,q) ∈ R n×n the Coriolis and centrifugal forces matrix, g (q) ∈ R n the vector of gravitational forces, and τ (t) ∈ R n the vector of input torques. This model comprises the following properties: 1. The inertia matrix M (q) ∈ R n×n is symmetric and positive definite for all q i ∈ R n .

The matrix
The position and orientation in the workspace of the end-effector of the ith robot manipulator are denoted by χ i (t) ∈ R m through Cartesian coordinates: where f (·) ∈ R m is a nonlinear function of the direct kinematics. The relationships between the joint velocitiesq i ∈ R n and the time derivative of the endeffector coordinates of the robotχ i ∈ R m in the workspace are given bẏ where J ac ∈ R m×n represents the Jacobian of the manipulator, defined as The pseudo-inverse of J ac ∈ R m×n , described by J ac + ∈ R n×m , is defined as

Dynamic model of IM
The vectors of the currents i α − i β and flows λ α − λ β of the stationary reference frame fixed to the stator α − β of the IM are used to express the equations in a field-oriented frame d − q. In this sense, the model of the IM mechanical and electrical dynamics without consideration of the effects of viscous friction is given by [Marino, Tomei and Verrelli (2010) where ρ = arctan λβ λα , µ = 3 2 n p Lm JLr , α = Rr . ω m is the rotor speed, i d and i q are the currents on the d − q axes, and λ d is the rotor flux linkage on the d axis. T L and n p are the load torque and number of pole pairs. J is the moment of inertia, which is defined as constant. L m is the mutual inductance, and L s and L r are the self-inductances of the stator and rotor, respectively. f is the nominal frequency in (Hz). R s and R r are the resistances of the stator and rotor, respectively, in Ω. Finally, u d and u q are the non-linear state feedback control inputs, described by Substitution of Eq. (7) into Eq. (6) results in the closed-loop system v d and v q are the new control inputs obtained by applying the next PI loops: where λ dref , T ref and ω ref are the references for rotor flux linkage, torque, and angular speed, respectively. K d1 , K d2 , K q1 , K q2 , K τ 1 and K τ 2 are the positive constant gains. T em is the electromagnetic torque, defined as T em = µJλ d i q .

Mated system: robot manipulator and induction motor
Consider that the kth joint, k = 1, 2, · · · , n, of each ith robot manipulator, i = 1, 2, · · · , p, being directly driven by an IM. It is also assumed that the amplitude of the flux linkage λ d,ik is regulated to the reference constant value λ dref,ik through the control loop Eq. (9); hence, the closed-loop system for each IM is reduced to Let The closed-loop reduced model representing all the k induction motors for the ith robot manipulator is expressed as whereΩ i = dΩi dt ∈ R n . The vector of the angular position θ i ∈ R n from all the k induction motors for the ith robot manipulator is defined as To mate the robot manipulator dynamics and induction motor dynamics, we consider the following assumptions.
Assumption 1. There exists a direct mechanical coupling between the rotor of the IM and the joint of the robot manipulator. This might reduce the periodical mechanical maintenance with respect to the PMBLDC motors; therefore, Assumption 2. The synchronization control law causes the input torque or required torque for each joint of the robot manipulator, which is considered as the load torque applied to each induction motor; thus, This gives This model has the same properties as listed in Section 2, becauseḊ 4 Synchronization control and velocity observer

Synchronization control in joint space
Synchronization errors s i ,ṡ i ∈ R n are defined as in [Rodriguez-Angeles and Nijmeijer (2004)]: The reference signals q ri andq ri ,q ri are established to consider the interactions between robots and ensure the synchronous behavior, as where q d (t) is the common desired trajectory that each IM-Robot will be forced to track; K cp_i,j ,K cv_i,j , K ca_i,j ∈ R n×n are diagonal positive semi-definite matrices, and, for simplicity, it is assumed that they satisfy K cp_i,j = K cv_i,j = K ca_i,j = K i,j . Thus, from Eq. (14) and Eq. (18), the reference signals for the mated IM-Robot system are obtained: Assumption 3. The torque necessary to achieve synchronization for each joint τ i shall be the reference torque T ref for the corresponding IM. This avoids the control loop PI Eq. (11) in the control scheme of the IM. Taken from Torres et al. [Torres, Guerrero, Garcia et al. (2016)], the control law to achieve a master-slave synchronization of IM-Robot systems in the joint space is given by where K di and K pi are positive-definite gain matrices.

Synchronization control design in workspace
To attain synchronization of robot manipulators driven by IMs in the workspace, a controller design is given below, more details of which appear in Appendix A. The position error in the workspace e i (t) ∈ R m is defined as The error dynamic equation is expressed aṡ where αe i has been added and subtracted to facilitate the formulation of the control, I n ∈ R n×n is the identity matrix, α ∈ R m×m denotes a positive-definite gain matrix, and ϑ i (t) ∈ R n is a signal constructed in accordance with the required control objective. Based on the structure of Eq. (22), the filtered tracking error r i (χ d , e i ,q i ) ∈ R n is used to reduce the order of the error dynamic equation, which is defined as Thus, the position error of the IM-Robot system in the workspace could be written using The dynamics of the filtered tracking error is obtained by applying the time derivative of Eq. (23): Thus, the dynamic equation of the IM-Robot system in an open loop is written as where the regression matrix/parameters vector Y i Φ i is defined by where Y i χ d ,χ d , χ i , q i ,q i , ϑ i ,θ i ∈ R n×r is the regression matrix, and Φ i ∈ R r denotes the constant parameters of the system. A candidate Lyapunov function is proposed to design the synchronization control in the workspace of IM-Robot systems: The function V (r i , e i ) is positive-definite for all r i , e i and V (r i , e i ) = 0 if and only if r i = 0, e i = 0.
The time derivative of the proposed Lyapunov function V (r i , e i ) is given bẏ By the skew-symmetric property of the is obtained. An equality is established to achieveV (r i , e i ) < 0: where K ∈ R n×n is a constant positive-definite gain matrix.
In accordance with Behal et al. [Behal, Dixon, Dawson et al. (2009)], it is said the synchronization error e i is global asymptotically stable. A controller leading to master-slave synchronization of IM-Robot systems in the workspace is developed via Eq. (31) as

Joint variables estimator
Owing to the problems caused by the velocity measurements, an observer based on the model is used to estimate q i andq i in the slave IM-Robot systems. Then, the stability analysis is developed. For more details, see Appendix B. Thus, the proposed observer is where ξ 1 and ξ 2 are diagonal positive-definite matrices. The position estimation errorq i ∈ R n and velocity estimation errorq i ∈ R n are given bỹ The estimation errors' dynamics is expressed as The candidate Lyapunov function for the stability analysis is proposed as where P 1 and P 2 are diagonal positive-definite matrices. The time derivative of the proposed Lyapunov function is given separately bẏ The closed-loop error dynamic equation of the IM-Robot system is expressed as The Lyapunov function derivativeV s s i ,ṡ i ,q i ,q i manifests an inequality given bẏ where

As of Eq. (42), the analysis is centered on the expression
This equation is ≤ 0 if and only if Therefore, the Lyapunov function derivative is rewritten aṡ Another condition for the negative definition of the Lyapunov function derivative is for the positive-definiteness of the matrix ∆ to remain, which is ensured in accordance with Sylvesters criterion as if and only if where the sub-indexes m and M indicate the minimum and maximum matrix eigenvalues, respectively.
Definition 1. Letting w i = s iṡiqiqi T and the matrix In accordance with Khalil [Khalil (1996) Thus, w i ∈ Ω ε and the solutions are GUUB (globally uniformly ultimately bounded) by the bound

Synchronization control with velocity estimator
The synchronization control law of IM-Robot systems in the joint space with a velocity estimator of the slave IM-Robot is given by the expression In contrast, the synchronization control law of IM-Robot systems in the workspace with a velocity estimator of the slave IM-Robot is expressed as wherê

Simulation results
With the intention of proving the proposed synchronization approaches, robot manipulators of type SCARA with k = 4 joints driven by IMs were simulated in a scheme shown in Fig. (1) with one master IM-Robot system and three slave IM-Robot systems. The robot parameters are shown in Tab. 1, taken from the BOSH R SR-8 datasheet. The parameters for each of the IMs appear in Tab. 2. Simulations were carried out on the Simulink R platform through the S-function level-2 with ode-45 solver, variable-step, and a simulation time of 10 s. The voltage sources can supply the required levels in the simulation settings.

IM-Robot synchronization in joint space
The desired trajectory in the joint space is described by the expression  where t f is the final time, set as 4 s. q d in denotes the initial desired position for each joint k = 1, 2, 3, 4. q d f is the final desired position. The values of these references are The initial conditions for the joint position of the master IM-Robot j and the three slave IM-Robots i = 1, 2, 3 with k = 1, 2, 3, 4 joints are established as The gain matrices for the synchronization controller in the joint space are set as It should be noted that the process for gain tuning in relation to the proposed controller was guided based on our prior experience with these types of synchronization controllers, attending to the positive-definite gain matrices. The simulation results of the IM-Robot synchronization in the joint space for each slave IM-Robot are shown in Figs. 2-4. A solid line denotes the master IM-Robot trajectory, and a dashed line the slave IM-Robot system. In Fig. 5, all the trajectories are shown for each joint, including the desired trajectory. Synchronization errors and tracking error for each joint are displayed in Fig. 6, where ζ j,k is the tracking error of the master IM-Robot with respect to the desired trajectory. Remark 1. In the joint space, all the trajectories of the slave IM-Robot systems synchronize with the trajectory of the master IM-Robot system for each joint after an initial transient of 1.5 s, depending on the initial conditions. Therefore, the synchronization errors and the tracking error converge to zero in the case in which the velocity measurement is estimated.

IM-Robot synchronization in workspace
The desired trajectory in the workspace is described by The initial position of the master IM-Robot end-effector is set as For each slave IM-Robot, i = 1, 2, 3, and the initial position for the end-effector is established as The gain matrices used in the synchronization controller in the workspace are The simulation results for IM-Robot synchronization in the workspace are presented as follows: In Fig. 7, the synchronization is seen in the x − y plane; in Fig. 8, it is seen in the x − z plane; finally, the workspace synchronization errors on the x and y axes are displayed in Fig. 9. while tracking a common desired trajectory. Furthermore, the synchronization errors converge to zero without velocity measurements. The gain matrices for the velocity observer in both the joint space and workspace are established as Note that this choice of the gains satisfies the conditions given in Eq. (47) for each slave IM-Robot system in the scheme. The position estimation error is shown in Fig. 10, and the velocity estimation position is presented in Fig. 11. Remark 3. It is clear that the estimation errors converge to zero in both the joint space and workspace. The dynamics of the robot manipulator and IM have been combined to obtain the entire dynamic model of the mated IM-Robot system. Thus, a novel synchronization scheme including the actuator dynamics within the synchronization control with a velocity observer based on the entire IM-Robot model has been satisfactorily achieved, in both the joint space and workspace. Considering a direct mechanical coupling between the rotor of the IM and the joint of the robot manipulator, the proposed approach realizes synchronization of both the IM and robot manipulator. The IM-Robot systems synchronize before tracking the desired trajectory, which is similar to the case in prior works in the literature in which ideal actuators were considered, so the use of an actuator with high nonlinear dynamics, like the IM, might allow reliable performance while minimizing costs. Based on the results of Lyapunov analysis, the proposed velocity observer exhibited GUUB estimation closed-loop errors. The simulation results, assuming knowledge of the parameters and availability of the partial state and ideal sources in the simulations, show the feasibility of the proposed synchronization approach. Future works will be directed at obtaining experimental results to compare the effects of chattering caused by the noise in the velocity measurements against the performance of the proposed observer. Moreover, future approaches will be undertaken that include time delays and parametric uncertainties. Therefore, Considering a matrix representation, where Z = P 1 ξ 1 −P 1 P 2 ξ 2 P 2 D −1 i (q i ) C i q i ,q i . The expressionq T i εP 2 ξ 2qi is added and subtracted to Eq. (69), with 0 < ε < 1, where the time derivative of the Lyapunov functionV s s i ,ṡ i ,q i ,q i manifests an inequality given by Eq. (42).