On the Dynamics and Computed Torque Control of an Asymmetric Four-Limb Parallel Schoenflies Motion Generator

This paper deals with the dynamic analysis and control of a four-limb parallel Schoenflies-motion generator dedicating to fast pick-and-place operations, with an asymmetric topological architecture for saving mounting space. The concise equation of motion is built with principle of virtual work to characterize robot dynamics. Computed torque control is adopted to design the control scheme, of which the control variables are optimized by minimizing the errors of joint variables. Simulation tests are carried out along a trajectory of pick-and-place testing cycle to evaluate the control design and dynamic performances, which is verified experientially by observing the joint dynamics. The results show the effectiveness of the developed dynamic model and controller for the parallel robot under study.


Introduction
Parallel robots with three independent translations and one rotation around an axis of a fixed direction, a.k.a. SCARA a robots or Schoenflies motion generators (SMGs), dedicate to material handling in terms of pick-and-place (PnP) operations, thanks to their advantages of high speed/stiffness and lightweight architecture. For instance, the well-known Delta robot [1] that was commercialized by ABB Flexible Automation in the late 1990s, serving in three industry sectors, i.e., the food, pharmaceutical, and electronics industries. Amongst all types of the parallel SMGs, the robot counterparts with four identical RRПR b limbs in-parallel received the most attentions from both industry and academia. The research group "Laboratoire d'Informatique, de Robotique et de Microelectronique de Montpellier (LIRMM)" firstly developed the H4 [2] robot in two versions, with four identical limbs and an articulated traveling plate [3]. Later on, the similar versions of H4 robot, i.e., the I4 robot [4] and the symmetrical Par4 [5] were developed. The main difference among those robots lies in the different structures of articulated platforms [6]. The commercial version of Par4, namely, the Quattro robot by Adept Technologies Inc., hit the market in 2007, which is the fastest industrial PnP robot available. Besides, the LIRMM group proposed the Heli4 robot with a compact articulated traveling plate that is connected by a screw pair featuring full-circle end-effector rotation [7], of which the commercial version "Veloce. Robot" was developed by the Penta Robotics. Other four-limb parallel SMGs should be noticeable for their high performance [8][9][10][11][12][13][14].
ones with the elimination of the nonlinear factors such as the modeling error and the nonlinear friction [36]. If both structured and unstructured modeling errors exist, neural network CTC provides a better solution for robot control [37,38]. On the other hand, the controller enhancement may lead to the increased complexity and cost of design and implementation of the control schemes. In this work, CTC is applied to the dynamic control of presented robot under study, wherein the control variables are obtained by solving an optimization problem, aiming to achieve a tradeoff between the control precision and complexity.
In this paper, the dynamics of a four-limb fully parallel robot adapted for the PnP operations, towards the efficient utilization of shop-floor space, is investigated. A concise dynamic model for the robot under study is built with principle of virtual work. Computed torque controller is developed with the optimized control variables, which is numerically illustrated and evaluated experimentally. The comparisons show the effectiveness of the developed dynamic control strategy for the parallel robot. Figure 2 depicts the kinematic stricture of the robot with a screw-pair based mobile platform. Similar to its existing robot vantages of the SMG counterparts, an asymmetric robot with Ragnar architecture and screw-pair based mobile platform (MP) was developed [16] as depicted in Figure 1, for efficient use of space with high rotational capability.

Manipulator Architecture
In general, the parallel PnP robots work in the domain of high frequencies with complex and coupled nonlinear dynamic models, which is a challenging task in their dynamic control. The major difficulty lies in finding a solution that not only can sufficiently describe the real robotic system, but also can possibly be calculated in real time for implementation into control algorithms [17]. Since the robot links usually do not have regular geometries, some assumptions will be made for simplification and computational efficiency [18][19][20][21], leading to simplified dynamic models but introducing modeling errors in turn [22,23], which calls for a highly advanced control algorithm to achieve a satisfactory dynamic performance. Computed-torque control (CTC) is well suited for trajectory tracking control when the dynamic model of a robot is accurately known [24]. Some illustrative examples of the application of CTC to parallel robots can be found in the literature [25][26][27][28][29]. If there are unmodeled errors unignorable, some adaptive CTC methods are able to capture the parameter variations [30][31][32][33][34][35]. For instance, the linear proportional-derivative (PD) parameters can be replaced with nonlinear

Kinematic Geometry and Jacobian Matrices
The robot's kinematic problem of this family has been extensively studied [39], which will be revisited by skipping the details. Let i, j and k stand for the unit vectors of x-, yand z-axis, respectively, the axis of rotation of the i th actuated joint is denoted by [ ] b = cos cos sin -sin cos where [ ] By making use of the geometric constraint equation be-counterparts, each limb is composed of an actuated proximal link arm and a distal link to connect the base and mobile platforms. The mobile platform consists of two sub-platforms that are connected by a screw pair to transform the relative translation along vertical axis of two sub-platforms into rotational movement about the axis. For the lightweight design of the robot prototyping, as displayed in Figure 1, the proximal and distal link are fabricated with carbon fibre tubes, and most of the connecting components between the links and platforms are made of composite material by 3D printing. The previous work with a footprint ratio comparison [15] has revealed that the topological robot architecture under study determines a long and narrow workspace envelope well adapted to the PnP operations, with respect to (w.r.t.) the probability distribution of end-effector locations.
The reference coordinate frame F b is built with the origin located at the geometric center of the base frame. The z-axis points upward, and x-axis is parallel to segment A 1 A 2 between actuated joints 1 and 2. The moving coordinate frame F p is located at geometric center of the lower platform, and the Zand X-axis are parallel to zand x-axis, respectively, in the neutral configuration. Point A i represents the center of actuated joint, and B i , C i represent the center of short connecting bar at both ends of the parallelogram. The i th limb consists of an inner and an outer arms with link lengths b and l, respectively. The rotational capability can be enhanced by selecting an appropriate screw lead that is denoted by pitch h.  henceforce, the Jacobian matrices mapping the velocities of the upper and lower sub-platforms and actuated joint rates are written as

Dynamic Modeling by Principle of Virtual Work
The main approaches for robot dynamic modeling usually include the Lagrangian formulation [40,41], the Newton-Euler equations [42] and the principle of virtual work [43]. Since the kinematic mapping of the movements between the input and output components of the robot linkage in the reference frame has been built in previous section and the links are simply connected, the relationship between actuation torques and inertia forces can be directly determined. Henceforth, the principle of virtual work is adopted to model robot dynamics for computational efficiency, without the operations of the differentiation on the system energy in Lagrangian formulation or recursive matrix propagation by Newton-Euler equations. Prior to modeling, it is supposed that the centers of mass of the links are located at their geometric centers, and the torques caused by the rotation of the outer arms are ignored due to the lightweight structure. The mass of each outer arm is simplified as two lumped masses located at the two ends of the distal link, as shown in Figure 3.
the solutions to the inverse geometry problem of the robot can be solved as For a given pose of the robot end-effector, each limb can have two postures that are characterized by the sign "-/+" in Eq. (3), which means that the robot can have up to 16 working modes (WMs). To avoid the mechanical collision, the working mode "--+ +" is selected for the robot.

Kinematic Jacobian matrix of the robot
Differentiating Eq. (2) w.r.t. time yields where J x and J θ are the forward and inverse Jacobians, respectively, and As long as the forward kinematic Jacobian is nonsingular, the kinematic Jacobian matrix is obtained as

Jacobian matrices of the sub-platforms
Due to that the mobile platform consists of two sub-platforms, the upper sub-platform can translate in vertical direction with respect to the lower one, which introduces different velocities between the end-effector and sub-platforms. Besides, the Jacobian matrix between the upper platform and the actuated joints is also different from that of the lower one.
The velocities of the upper and lower sub-platforms, i.e.,   T  T  T  T  T  T  T  b pu up pl low b pu up pl low τ δθ For any virtual displacement θ δ Eq. (19) should meet After rearranging the Eq. (20) in connection with Eq. (17), the simplified dynamic model is derived as below: Dynamic model (21) has been evaluated [44], applicable for the characterization of the robot dynamics, calculation of torques for selection of actuators and gearboxes, robot control, etc.

Control Scheme Design
Dynamic control plays an important role in the robot applications, thus, dynamic model (21) will be integrated in the controller for the robot. It is noteworthy that Eq. (21) does not take the frictions and other disturbances into account, namely, unmodeled errors, which may decrease the control precision. In light of this, the dynamic model is reformed as below where f (θ, t) characterizes the unmodeled errors in terms of the frictions and time-varying stochastic disturbances, and here we let ( ) ( ) Amongst various control approaches for the robotic manipulators, computed torque control that is easily understood and of good performances, is an effective motion control strategy for robotic systems [45], which will be adopted to design the control scheme. According, the proportional-de-   The geometric parameters and masses of the links and sub-platforms.

Link mass [kg]
[a  Table 1, together with the geometric parameters.

Simulation tests
A testbed, as depicted in Figure 6, was built in the Simscape module of Matlab/simulink to analyze the dynamic behavior of the robot and to validate the controller parameters.   Simulation tests and experimental measurements are carried out along a trajectory of a picking testing cycle, both of which shows that the pose errors of the mobile platform are acceptable for industrial application. Experiments are implemented by observing the joint dynamics to evaluate the developed controller in comparison with the simulation results. The comparison shows a good agreement between the simulation and experimental results and the measurements validates the effectiveness of the developed dynamic control strategy. Future work will focus on the integrated advanced control design for improvement of control precision and buffeting suppression. It should be noted that the solved control variables may not be the global optimum, while, they can provide acceptable accuracy of trajectory tracking without high computational cost.
With the control variables in Eq. (27), dynamic simulation results along the trajectory defined in Figure 5 are shown in Figure 7, Figure 8 and Figure 9. Figure 7a depicts the tracked errors of the actuated joint variables, from which it is seen that the magnitudes of the joint errors increase from the starting point to the end, without the convergence. The corresponding time-varying joint torques are displayed in Figure  7b, in which the maximum torques occur in the period of acceleration/deceleration, equalling to 48 Nm approximately for joints 2 and 4. Similar to the joint errors θ-θ d , the errors of the joint rates The tracked MP pose errors are displayed in Figure 9, wherein the maximum positioning errors appears at the z-coordinate with δ z = -0.491 mm, and the maximum orientation error is equal to 0.8˚. This means that the design control scheme can meet the precision requirements in most PnP operations for industrial applications.

Experimental evaluation
Tracking the prescribed PnP testing trajectory, the actual joint errors and torques are read from the motor encoders, as depicted in Figure 10. Compared to the simulation results, the experimental measurements have significant buffetings, particularly, in the period of acceleration/deceleration. The possible reasons may include the irregular friction and backlash of the kinematic pairs.
Referring to the MP pose errors in Figure 9, the predicted actual position and orientation errors generated by the joint errors in Figure 10a may reach to 1 mm and 1.5˚, which are still acceptable for most industrial sectors in terms of PnP operations that do not require quite high accuracies. Comparing Figure 10b to Figure 7b, the changes of the measured joint torques have similar varying trends with simulated ones, which reaches to a good agreement, despite the presence of fluctuations that are ignorable when reduction gearbox is installed between the motors and active links. The comparison between the simulation tests and experimental measurements implies that the developed dynamic model and controller can work effectively for the robot under study, with a reasonable tradeoff between control accuracy and implementation complexity.

Conclusion
This paper presents the robot dynamics of a four-limb parallel Schoenflies-motion generator, which is characterized along with a physical prototype of this kinematic chain. The concise robot dynamics is modeled with principle of virtual work for the deployment in control design. The computed torque controller is designed by optimizing the control vari-