# Trajectory Tracking Control Study of a New Parallel Mechanism with Redundant Actuation

Authors: Haiqiang Zhang; Hairong Fang; Dan Zhang; Qi Zou; Xueling Luo
Journal: International Journal of Aerospace Engineering
 (2020)

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## Abstract

Parallel mechanisms with redundant actuation are attracting numerous scholars’ research interest due to their inherent advantages. In this paper, an efficient trajectory tracking control scheme for the new redundantly actuated parallel mechanism by integrating force/position hybrid control with the combination of inertia feed-forward control and back propagation (BP) neural network PID control is proposed. The dynamic models including the joint space and task space are formulated explicitly in efficient and compact form by means of the principle of virtual work and d’Alembert formulations. The force/position hybrid control is implemented to perform trajectory tracking and optimize the driving force configuration in MATLAB/Simulink environment, before being applied to an actual parallel mechanism. The illustrative simulation results demonstrate that the force/position hybrid control scheme is available to provide good trajectory tracking performance. Simultaneously, the feasibility of the proposed control scheme is verified by comparison analysis with the aforementioned conventional control method.


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## Body

## 1. Introduction

Compared to counterpart serial mechanisms, parallel mechanisms exhibit much greater advantages such as high stiffness, lower inertia, high loading capability, high precision, large acceleration, and high dexterity, which are desirable properties for use in high speed machine, high precision assisted surgery, high speed pick and place of delta robots, and some other applications [1–3]. However, small workspace and many singularities within the workspace limit their wide applications. Under such circumstance, parallel mechanisms with redundant actuation are expected to tackle their shortcomings, as redundant actuation can eliminate or decrease singularities, increase the dexterous workspace, optimize the configuration of driving force, and so on. At present, the research works on redundant actuation are mainly focused on the singularities and the workspace. However, the intelligent control strategies in the existing literature are seldom studied [4–6], which has just gotten more attention to make the most use of their advantages; in particular, for high speed, high accuracy machine tools, trajectory tracking control, for instance, adaptive control and robust control, is very essential for more prosperous applications [7–11].Redundant actuation can be achieved in a different method; one is to add the kinematic chain including nonconstrained subchains or constrained subchains without increasing the degree of freedom, and the other is to change the passive joint into active joint [12]. Gosselin and Schreiber [13] increased a kinematically redundant leg in order to preserve the force transmission properties of the well-known Gough-Stewart parallel mechanism, which can avoid singularities and extend the rotational workspace. Liu et al. [14] studied exhaustively the force/motion transmissibility of redundantly actuated parallel mechanisms, and six simulation examples illustrate that the actuation redundancy can significantly improve the kinematic performance and enlarge the singularity-free workspace. Yao et al. [15] presented a redundant actuated 5UPS-PRPU parallel mechanism by adding an actuator for the middle PRPU passive constraint branch; the simulations illustrated that redundant actuation can greatly improve the performance and ability of the parallel mechanism. Boudreau et al. [16, 17] presented the dynamic model of a cinematically redundant planar parallel mechanism and an optimization to minimize the torques when the end-effector performs the trajectory tracking. Wang et al. [18] derived the inverse dynamic model of a spatial 3-DOF parallel mechanism with redundant actuation by adding actuators to passive rotational joints, and the driving force was optimized in terms of the Moore-Penrose inverse matrix. The motion control is very complicated for the coupled dynamics of the parallel mechanism with redundant actuation. Wen et al. [19] investigated a new hybrid control algorithm for the redundantly actuated 6PUS-UPU parallel robot, i.e., the inner model control in the position loop and the fractional order fuzzy proportional-integral derivative control in the force loop. In addition, in [20], two model predictive controllers including proportional, integral, and differential algorithm and fuzzy algorithm were proposed for a five-degree-of-freedom redundantly actuated parallel mechanism. And the simulation results showed that the fuzzy model predictive control can effectively improve real-time performance of the control system compared with the traditional PID model predictive control algorithm. Liu et al. [21] proposed a model-based driving force synchronous control method with neural network, which can effectively reduce the driving force errors of the parallel mechanism. Similarly, Luces et al. [22] presented an in-depth discussion of past research on parallel mechanism kinematic redundancy and indicated that primary advantages of the parallel mechanism with redundant actuation contain workspace enlargement, singularity elimination/avoidance, and joint-torque optimum distribution, yet redundant actuation is facing a valuable opportunity as well as great challenges, such as motion planning and control and calibration. Flohic et al. [23] adopted a hybrid force/position control algorithm to conduct parallel kinematic machine for mechanical testing, and the control algorithm is illustrated on a Nooru-Mohammed test performed on a Gough-Stewart platform and validated experimentally on a five-bar parallel mechanism. Cazalilla et al. [24, 25] developed an adaptive controller for the 3-PRS parallel mechanism with consideration of some uncertainties; the simulation results show that the adaptive controller can greatly improve the trajectory tracking precision compared with the passivity-based controllers. Urrea and Kern [26] presented a summary of the controllers considering for the evaluation of the redundant robot model with their actuators and corresponding performance. Callegari et al. [27] presented the inverse dynamic model by means of the principle of virtual work; three control strategies were employed and the results show that hybrid position/force schemes should be able to provide better performance than the other position control or force control. Achievements that are both important and probably of a very scientific perspective were achieved by Siciliano in [28]; in these recent works, they took profit from a general solution for the inverse dynamic of parallel mechanism with redundant actuation to construct a computationally efficient control scheme compared with a conventional control one, which makes it available in control applications with singularity avoidance. In this paper, the conventional PID control algorithm is unavailable to be implemented on the proposed parallel mechanism, so the force/position hybrid control algorithm is carried out on the redundantly actuated parallel mechanism to perform trajectory tracking.The remainder of this paper is organized as follows: Section2 describes the structure of the redundantly actuated 2RPU-2SPR parallel mechanism and the coordinate frames. The kinematic relations of the parallel mechanism are developed in detail in Section 3. In Section 4, the explicit dynamic model is derived in terms of the principle of virtual work and d’Alembert formulation. Next, in Section 5, the force/position hybrid control algorithm is presented. In Section 6, a force/position hybrid control simulation case is given. Finally, some brief contributions and future works are presented.


## 2. System Description

The serial-parallel hybrid kinematic machine (HKM) structure was selected as a candidate architecture for complicated surface milling in aerospace domain [29] (as shown in Figure 1), whose parallel module is a 2RPU-2SPR parallel mechanism with redundant actuation (R, P, U, and S standing for rotational, prismatic, universal, and spherical kinematic joint, respectively). The underline format (P) represents the actuated joint. The structural diagram is depicted in Figure 2. The new parallel mechanism mainly consists of the fixed platform, the moving platform, and four limbs with two of the same identical kinematic configuration. The prismatic joints described by the linear joint variables di connects the lower platform to the upper platform by a rotational joint followed by a universal joint or a spherical joint followed by a rotational joint. It is worth noting that the used spherical joint is designed by means of three mutually perpendicular rotational joints so as to enlarge the rotation range.Figure 1
A five-axis hybrid kinematic machine tool.Figure 2
Schematic diagram of the 2RPU-2SPR parallel mechanism.For modeling purposes, as shown in Figure2, we assume a fixed coordinate system B‐xyz at the centered point B of the fixed platform and a moving coordinate system A‐uvw on the square moving platform at the centered point A, along with the z-axis and w-axis perpendicular to the platform, and the x-axis and y-axis parallel to the u-axis and v-axis, respectively. Both A1A2A3A4 and B1B2B3B4 are designed to be squares so as to yield a symmetric architecture of the parallel mechanism. With these geometric conditions satisfied, the moving platform of parallel mechanism will pose with three independent degrees of freedom in the Cartesian space by controlling the movement of four servo actuators, that is, two rotational degrees of freedom around the x-axis and v-axis and one translational degree of freedom along the z-axis; in this case, the spatial rotations are coupled with the change of four nonidentical limbs [30]. What is more, as the number of actuated joints is greater than that of the degree of freedom, it belongs to a redundantly actuated parallel mechanism. Nevertheless, if the fourth limb, i.e., the SPR limb, is removed, the parallel mechanism will turn into a general nonredundant parallel mechanism.


## 3. Kinematic Analysis

The moving platform can move along thez direction and rotate around the x-axis and v-axis with the consideration of mobility characteristics. The homogeneous transformation matrix T of the reference coordinate point A can be expressed as follows:
(1)T=Rp01=Rotx,αTransz,dRotv,β=cosβ0sinβxsinαsinβcosα−sinαcosβy−sinβcosαsinαcosαcosβz0001,where p=x,y,zT is the position coordinates of point A in the fixed coordinate system, α is the rotation angle around the x-axis, and β is the rotation angle around the v-axis.Additionally, once the rotation matrixR is obtained, then the rotational Euler angles will be obtained by the following (2), which are important for achieving the orientation angles in the feedback control proposed in this paper:
(2)α=arctan2R3,2,R2,2,β=arctan2R1,3,R1,1,where R⋅,⋅represents the row and column elements of the rotation matrix R.The inverse Jacobian matrix relationship between input and output can be represented as follows:(3)q̇=J0Ẋ=JqJpẊ,where q̇=ḋ1,ḋ2,ḋ3,ḋ4T, J0is the velocity Jacobian matrix, and
(4)Jq=s1Ta1×s1Ts2Ta2×s2Ts3Ta3×s3Ts4Ta4×s4T,Jp=JvpJwp=000−tanα−z⋅sec2α010001000cosα00sinα,where si denotes the direction vector of i-th limb, ai=R⋅aAi, aAi is the description of the position vector in the moving coordinate system, and X=zαβT is the independent motion degree of freedom parameters.By differentiating both sides of (3), the acceleration between the actuated joints and corresponding end-effector can be expressed as
(5)q¨=J̇0Ẋ+J0X¨where
(6)J̇0=w1×s1Twp×a1×s1+a1×w1×s1Tw2×s2Twp×a2×s2+a2×w2×s2Tw3×s3Twp×a3×s3+a3×w3×s3Tw4×s4Twp×a4×s4+a4×w4×s4T⋅Jp+Jq⋅000−1−tan2α⋅α̇−ż⋅sec2α+2z⋅sinαcos3α⋅α̇000000000−sinα⋅α̇00cosα⋅α̇,where wi is the angular velocity of the i-th limb defined in the fixed coordinate system.The determination ofJ̇0 is actually very important for the next section about dynamic analysis.It is analogous to (3); the forward Jacobian matrix Jf which characterized the relationship between the end-effector and corresponding actuating joints can be expressed as
(7)Ẋ=Jfq̇,Jf=J0+,where ⋅+ denotes the Moore-Penrose inverse matrix. The Moore-Penrose inverse of the matrix J0 can be denoted as J0+=J0TJ0J0T−1 and the condition J0J0+=I should be satisfied.Hence, the following relationship should be satisfied:(8)Jf⋅J0=J0⋅Jf.Taking the derivative of (8) with respect to time, yield
(9)ddtJf⋅J0=0⇒J̇f⋅J0+Jf⋅J̇0=0⇒J̇f=−J0−1⋅J̇0⋅J0−1.The structural diagram of thei-th limb is depicted in Figure 2; each of that consists of an upper sublimb and a lower sublimb. Let e1 be the distance between the centroid Bi and the center of mass of the lower sublimb, while e2 be the distance of the centroid Ai and the center of mass of the upper sublimb. The position vectors of the centers of mass of the i-th limb are given with respect to the fixed coordinate system:
(10)r1i=bi+e1si,(11)r2i=bi+di−e2si.The inverse kinematics should be differentiated with respect to time, and the velocity of kinematic jointsAi connected to the moving platform can be obtained in the fixed coordinate system:
(12)vai=vp+ωp×ai=Jai⋅Jp⋅Ẋ,where
(13)Jai=1000aiz−aiy010−aiz0aix001aiy−aix0.The angular velocity of thei-th limb
(14)ωi=si×vaidi=JwiẊ,where Jwi=s~iJaiJp/di, and s~i is antisymmetric operator corresponding to the vector si.Taking the derivative of (10) and (11) with respect to time, the velocity of the center of mass of the lower sublimb and upper sublimb can be obtained in the fixed coordinate system.
(15)v1i=e1ωi×si=Jv1iẊ,where Jv1i=−e1⋅s~iJwi
(16)v2i=di−e2ωi×si+ḋi⋅si=Jv2iẊ(17)Jv2i=e2−di⋅s~iJwi+siJdiJpTurning to accelerations, the acceleration of the pointAi can be carried out by differentiating (12) with respect to time:
(18)v̇ai=v̇p+ω̇p×ai+ωp×ωp×ai.The angular acceleration velocity of thei-th limb can be expressed as
(19)ω̇i=si×v̇ai−2ḋiωidi=JwiX¨+J̇wiẊ,where
(20)J̇wi=s~iJ̇vp−a~iJ̇wp−JwpẊ×a~iJwp−2JdiJ0ẊJwidi.Finally, the acceleration of the centers of mass of thei-th limb can be obtained by differentiating once again (15) and (16) with respect to time:
(21)v̇1i=e1ω̇i×si+e1ωi×ωi×si=Jv1iX¨+J̇v1iẊ,where
(22)J̇v1i=−e1s~iJ̇wi−e1si⋅JwiẊTJwi,v̇2i=d¨i⋅si+2ḋiωi×si+di−e2ω̇i×si+di−e2ωi×ωi×si=Jv2iX¨+J̇v2iẊ,J̇v2i=si⋅J̇0i−2J0i⋅Ẋ⋅s~i⋅Jwi+e2−dis~i⋅J̇wi+JwiẊ×s~⋅J̇wi.The detailed kinematic analysis has been developed in-depth in our previous work [31], so herein, we do not repeat it.


## 4. Dynamic Analysis

The inverse dynamic problem of the parallel mechanism is to determine the required driving force under the requirement to generate a prescribed motion trajectory. For the 2RPU-2SPR parallel mechanism, the dynamic model in the task space can be deduced by applying the principle of virtual work and d’Alembert’s formulation stated as follows [32, 33]:
(23)FX=MXX¨+CẊ,XẊ+NX,where
(24)MX=JpTmaJvpIBpJwp+∑i4Jvω1iTm1iJv1iIB1iJwi+∑i4Jvω2iTm2iJv2iIB2iJwi,CẊ,X=JpTmaJ̇vpIBpJ̇wp+JwpẊ×IBpJwp+∑iJvω1iTm1iJ̇v1iIB1iJ̇wi+JwiẊ×IB1iJwi+∑iJvω2iTm2iJ̇v2iIB2iJ̇wi+JwiẊ×IB2iJwi,NX=JpTmag0+∑iJvw1Tm1ig0+∑iJvw2Tm2ig0−JpTfene,where Mq is the positive definite inertia matrix, Cq is the nonlinear matrix including the centrifugal and the Coriolis terms, and NX is the sum of gravitational f and external generalized force terms.To facilitate force control, it is usually necessary to transform the dynamic formulation in the task operational space to that in the joint space.In the static equilibrium state, the linear mapping relationship between the actuated joint force and corresponding task operation space force can be expressed as(25)FX=J0Tτ.Combining (23) and (25) leads to
(26)J0Tτ=MXJ0+q¨+J̇0+q̇+CẊ,XJ0+q̇+NX.Then, the dynamic formulation in the joint space can be expressed with the deduced kinematic relations.(27)τ=Mqq¨+Cq̇,qq̇+Nq,where
(28)Mq=J0T+MXJ0+,Cq̇,q=J0T+CX−MXJ0+J̇0J0+,Nq=J0T+NX.In addition, the dynamic model has several properties that can be exploited to facilitate dynamic controller designs; for example:Property 1.
The inertia matrixMq is symmetric and positive definite for any q∈Rn.Proof.
(29)MqT=J0+TMXJ0+T=J0+TMXJ0+=Mq.Property 2.
Ṁq−2Cq̇,q is skew symmetric for any X, and the condition should be satisfied
(30)XTṀq−2Cq̇,qX=0.Proof.
(31)Ṁq−2Cq̇,q=J̇0+TMXJ0++J0+TṀXJ0++J0+TMXJ̇0+−2J0+TCX−MXJ0+J̇0J0+=J0+TṀX−2CXJ0++J̇0+TMXJ0+−J0+TMXJ̇0+,due to
(32)J̇0+TMXJ0+−J0+TMXJ̇0++J̇0+TMXJ0+−J0+TMXJ̇0+T=0,and ṀX−2CX is skew symmetric matrix.
Hence,Ṁq−2Cq̇,q can be proven that it is a skew symmetric matrix.


## 5. Control Design

With due consideration of the fact that the dynamic formulations of the parallel mechanisms possess the same abovementioned properties as those of serial robots, we can apply the control scheme utilized for serial robots to parallel mechanisms with redundant actuation [34–36]. In order to achieve a good performance of parallel mechanism in trajectory tracking problem, an effective control scheme based on the nonlinear control theory is required. In this paper, we proposed a force/position hybrid control scheme to improve the trajectory tracking performance. Details are as follows:


### 5.1. Task Space PID (Proportional-Integral-Derivative) Position Control

Nonredundant limb (the 1st, 2nd, and 3rd limbs) adopts position control law combining inertia feed-forward control and back propagation (BP) neural network (NN) PID control to guarantee the transient tracking precision performance. The control law for the proposed scheme can be expressed as follows:
(33)up=MXX¨d+CẊ,XẊd+NX+Kpe+Ki∫0tedt+Kdė,in which Xd is the desired trajectory in the task (operational) workspace; ẊdandX¨d are the desired velocity and acceleration, respectively; and Kp, Ki, and Kd are positive definite suitable gain matrices.Combining (31) and (41), and simplifying, we can get
(34)MXX¨d−X¨+CẊ,XẊd−Ẋ+Kpe+Ki∫0tedt+Kdė=0,namely,
(35)MXe¨+CẊ,Xė+Kpe+Ki∫0tedt+Kdė=0,where e=Xd−X and ė=Ẋd−Ẋ; if time is infinity, and the error e will evolve towards zero, then the derivative of the error ė will evolve towards zero as well.Position control can guarantee the trajectory movement precision. The nonredundant limbs are controlled by the BP neural network PID controller. Figure3 shows the block diagram of the BP neural network model which is employed in the intelligent gain tuning by online learning [37–41]. In this model, the NN has four input layers, eight hidden layers, and three output layers, and wo and wi are the weight factors of input layers and output layers that can be continuously updated by machine learning. The BP neural network PID controller is implemented by resorting to the “S-Function.”Figure 3
The diagram of the BP neural network PID controller.Here, the input of the control function is considered as the output of the NN, which can be formulated as follows:(36)uk=uk−1+KKpek−ek−1+Kiek+Kdek−2ek−1+ek−2,ek=rk−yk,where uk is the output of the controller in time k and ek is the control error between input rk and output yk. It is worth noting that the variables uk, rk, and yk are all in the range between zero and one, so we do not have to normalize them. However, we should define an amplification gain factor K that can modify the magnitude of PID, and Kp, Ki, and Kd are the gain matrices of proportion, integration, and differentiation, respectively.


### 5.2. Joint Space P (Proportional) Force Control

Utilizing the force control as input function of the redundantly actuated limb (the 4th limb), the proportion control law can be defined as follows:
(37)uf=τd+Kvτd−τ,where τd is the desired driving force, τ is the actual joint force, and KV is the positive gain matrices.Force control can guarantee the distribution of the driving forces for preventing the damage of unexpected dash. Because the forces in joint space can be more easily measured than in the task space, so we select the dynamic control in joint space.Up to now, the whole hybrid control law applied to the parallel mechanism with redundant actuation can hence be described as(38)Γ=Srm×J0+Tup+Srf×uf=Srm×J0+TMXX¨d+CẊ,XẊd+NX+Kpe+Ki∫0tedt+Kdė+Srf×τd+Kvτd−τ,where
(39)Srm=100010001000,Srf=0001.Therefore,Srm determines the nonredundantly actuated joint to be controlled with position control, while Srf determines the redundant actuated joint to be controlled with force/torque control. The integrated control scheme can guarantee the stability at high speed and high acceleration engineering application.In order to validate the stability of the control law, the following candidate Lyapunov function is proposed:(40)V=12ėTMXė+12eTKpe.Taking into consideration positive defined matrices ofMX and Kp, leads to
(41)V≥0.DifferentiatingV with regard to time, meanwhile, noting (31) and (38), yields
(42)V̇=ėTMXe¨+12ėTṀXė+ėTKpe=ėTMXe¨+CẊ,Xė+Kpe=−ėTKdė−ėTKi∫0tedt≤−ėTKdė≤0.The parallel mechanism system will be stable, ande→0 and ė→0, as the time is infinity.In order to demonstrate that the force/position hybrid control scheme combining the BP neural network PID control can effectively improve the trajectory tracking accuracy performance, the traditional PID also has been formulated for comparison analysis, as shown in Figure4. The parallel manipulator is modeled by SolidWorks software, then is imported to SimMechanics through the SimMechanics Link plug-in; finally, Simulink module is established.Figure 4
Force/position hybrid control scheme diagram of parallel mechanism with redundant actuation.The desired trajectory parameterXd can be obtained once the trajectory function was given. Herein, we present two methods to require the actual operational task space parameter X. One is forward kinematics with the black line depicted in Figure 4. In the forward kinematics, the position of the actuated joints is known, then the position and orientations of the moving platform can be obtained by adopting the back propagation (BP) neural network optimization method or Newton-Raphson iteration, which converges rather quickly when the initial values are close to the desired solution [42, 43]. However, the solving procedure for forward kinematics is very complicated and the computational efficiency is very low. Another efficient method to require the position and orientations of the moving platform is exactly what we want to introduce in this paper. We know that the position z. The rotation matrix R defining the orientation of the moving platform with respect to the fixed coordinate system can be obtained by position sensors. We can obtain the rotational Euler angles by interfacing with MATLAB in terms of the relation equation (2), just as shown in Figure 4, the dotted line within magenta. It is computationally simple and it does not require measurement of the end-effector velocity and acceleration as well. In fact, during the real prototype test, we can measure the plane change of the detected moving platform through the binocular version sensors; that is to say, we can obtain the rotation matrix. By resorting to the relationship equation (2), we can directly obtain the orientation angles α and β in the workspace. The advantage of the proposed simplified method is that it can efficiently avoid the complex forward kinematics.


## 5.1. Task Space PID (Proportional-Integral-Derivative) Position Control

Nonredundant limb (the 1st, 2nd, and 3rd limbs) adopts position control law combining inertia feed-forward control and back propagation (BP) neural network (NN) PID control to guarantee the transient tracking precision performance. The control law for the proposed scheme can be expressed as follows:
(33)up=MXX¨d+CẊ,XẊd+NX+Kpe+Ki∫0tedt+Kdė,in which Xd is the desired trajectory in the task (operational) workspace; ẊdandX¨d are the desired velocity and acceleration, respectively; and Kp, Ki, and Kd are positive definite suitable gain matrices.Combining (31) and (41), and simplifying, we can get
(34)MXX¨d−X¨+CẊ,XẊd−Ẋ+Kpe+Ki∫0tedt+Kdė=0,namely,
(35)MXe¨+CẊ,Xė+Kpe+Ki∫0tedt+Kdė=0,where e=Xd−X and ė=Ẋd−Ẋ; if time is infinity, and the error e will evolve towards zero, then the derivative of the error ė will evolve towards zero as well.Position control can guarantee the trajectory movement precision. The nonredundant limbs are controlled by the BP neural network PID controller. Figure3 shows the block diagram of the BP neural network model which is employed in the intelligent gain tuning by online learning [37–41]. In this model, the NN has four input layers, eight hidden layers, and three output layers, and wo and wi are the weight factors of input layers and output layers that can be continuously updated by machine learning. The BP neural network PID controller is implemented by resorting to the “S-Function.”Figure 3
The diagram of the BP neural network PID controller.Here, the input of the control function is considered as the output of the NN, which can be formulated as follows:(36)uk=uk−1+KKpek−ek−1+Kiek+Kdek−2ek−1+ek−2,ek=rk−yk,where uk is the output of the controller in time k and ek is the control error between input rk and output yk. It is worth noting that the variables uk, rk, and yk are all in the range between zero and one, so we do not have to normalize them. However, we should define an amplification gain factor K that can modify the magnitude of PID, and Kp, Ki, and Kd are the gain matrices of proportion, integration, and differentiation, respectively.


## 5.2. Joint Space P (Proportional) Force Control

Utilizing the force control as input function of the redundantly actuated limb (the 4th limb), the proportion control law can be defined as follows:
(37)uf=τd+Kvτd−τ,where τd is the desired driving force, τ is the actual joint force, and KV is the positive gain matrices.Force control can guarantee the distribution of the driving forces for preventing the damage of unexpected dash. Because the forces in joint space can be more easily measured than in the task space, so we select the dynamic control in joint space.Up to now, the whole hybrid control law applied to the parallel mechanism with redundant actuation can hence be described as(38)Γ=Srm×J0+Tup+Srf×uf=Srm×J0+TMXX¨d+CẊ,XẊd+NX+Kpe+Ki∫0tedt+Kdė+Srf×τd+Kvτd−τ,where
(39)Srm=100010001000,Srf=0001.Therefore,Srm determines the nonredundantly actuated joint to be controlled with position control, while Srf determines the redundant actuated joint to be controlled with force/torque control. The integrated control scheme can guarantee the stability at high speed and high acceleration engineering application.In order to validate the stability of the control law, the following candidate Lyapunov function is proposed:(40)V=12ėTMXė+12eTKpe.Taking into consideration positive defined matrices ofMX and Kp, leads to
(41)V≥0.DifferentiatingV with regard to time, meanwhile, noting (31) and (38), yields
(42)V̇=ėTMXe¨+12ėTṀXė+ėTKpe=ėTMXe¨+CẊ,Xė+Kpe=−ėTKdė−ėTKi∫0tedt≤−ėTKdė≤0.The parallel mechanism system will be stable, ande→0 and ė→0, as the time is infinity.In order to demonstrate that the force/position hybrid control scheme combining the BP neural network PID control can effectively improve the trajectory tracking accuracy performance, the traditional PID also has been formulated for comparison analysis, as shown in Figure4. The parallel manipulator is modeled by SolidWorks software, then is imported to SimMechanics through the SimMechanics Link plug-in; finally, Simulink module is established.Figure 4
Force/position hybrid control scheme diagram of parallel mechanism with redundant actuation.The desired trajectory parameterXd can be obtained once the trajectory function was given. Herein, we present two methods to require the actual operational task space parameter X. One is forward kinematics with the black line depicted in Figure 4. In the forward kinematics, the position of the actuated joints is known, then the position and orientations of the moving platform can be obtained by adopting the back propagation (BP) neural network optimization method or Newton-Raphson iteration, which converges rather quickly when the initial values are close to the desired solution [42, 43]. However, the solving procedure for forward kinematics is very complicated and the computational efficiency is very low. Another efficient method to require the position and orientations of the moving platform is exactly what we want to introduce in this paper. We know that the position z. The rotation matrix R defining the orientation of the moving platform with respect to the fixed coordinate system can be obtained by position sensors. We can obtain the rotational Euler angles by interfacing with MATLAB in terms of the relation equation (2), just as shown in Figure 4, the dotted line within magenta. It is computationally simple and it does not require measurement of the end-effector velocity and acceleration as well. In fact, during the real prototype test, we can measure the plane change of the detected moving platform through the binocular version sensors; that is to say, we can obtain the rotation matrix. By resorting to the relationship equation (2), we can directly obtain the orientation angles α and β in the workspace. The advantage of the proposed simplified method is that it can efficiently avoid the complex forward kinematics.


## 6. Simulation Results

In order to validate the performance of the proposed control method, first of all, the control scheme for the parallel mechanism simulation examples has been actually implemented within a modular MATLAB/Simulink programming environment [44]. The geometric parameters used for the parallel mechanism are listed as follows: the radius of the fixed platform is ra=0.2 m, the radius of the moving platform is rb=0.339 m, the mass of the moving platform is ma=12.75 kg, the mass of the lower sublimb is m1i=0.95 kg, the mass of the upper sublimb is m2i=1.1 kg, the distance between the centroid of the lower sublimb and the kinematic joint Bi is e1=0.23 m, the distance between the centroid of the upper sublimb and the joint Ai is e2=0.22 m, the gravity acceleration is g=00−9.807T, and the inertia matrix about the center of mass of the i-th limb can be determined by the SolidWorks software as follows (unit: kg × m2):
(43)IBp=0.770001.070001.83×10−1,Ii1i=2.880002.890000.05×10−2,Ii2i=2.650002.670000.03×10−2A prescribed trajectory in spatial space is tracked using the proposed control scheme. Specifically, the desired trajectory function of the moving platform in the workspace is defined as follows:(44)αt=π6sin2t,βt=π6cos2t,z=0.7+0.02sin2t.The following values can be determined with tuning in the present traditional PID position control and P force control simulations:(45)Kp=diag1100,1100,1100,1100,Ki=20,20,20,20,Kd=20,20,20,20,Kv=diag120,120,120,120.In BP neural network PID simulations, the weight factors are initialized from zero, while BP neural network PID (Kp, Ki, and Kd) control gain matrices are set to one, the amplification gain matrix K is set as diag[8700,300,70], and Kv is the same as the abovementioned. In the simulation process, the fixed simulation step (fundamental sample time) is set to 1 ms, the model was simulated using the Bogacki-Shampine method, and the simulation time is set to 5 seconds.The Simulink simulation model and desired trajectory are depicted in Figure5.Figure 5
Parallel mechanism model in Simulink and desired trajectory.First of all, the proposed control scheme is applied to the parallel mechanism for the prescribed trajectory tracking control, and the position errors of the actuated joints are drawn in Figures6–9. When the control system reaches a steady state, the errors will decrease gradually to zero, although the errors oscillated at the beginning or within some certain time.Figure 6
The position error of the first actuated jointd1.Figure 7
The position error of the second actuated jointd2.Figure 8
The position error of the third actuated jointd3.Figure 9
The position error of the fourth actuated jointd4.From the figures above, we can see that the errors of the actuated joints are very small, and the variation mainly appears within one second. In the next four seconds, the errors almost approach zero, especially in Figures7 and 9, while the errors in Figures 6 and 8 have small fluctuations that are approximately close to zero, which demonstrate that the designed controller has the advantage of quick response speed and excellent trajectory tracking results.Similarly, the errors of driving forces are very important to validate the controller performance, and the simulation results including three nonredundant forces in the actuated joints are illustrated in Figures10–12.Figure 10
The driving force error of the first actuated joint.Figure 11
The driving force error of the second actuated joint.Figure 12
The driving force error of the third actuated joint.From the figures above, we can see that the errors are getting very fast, and they are close to zero after one second; however, the errors have a large fluctuation within one second. The unexpected dash may destroy the structural components, so we should improve significantly and pay attention to this problem in practical application.The errors including actuated positions and actuated forces in the above figures tend to zero as the control system reaches a steady state. During the transient response of the control system, the values of the errors produced will be oscillating.Figures13 and 14 show the required movement position and driving force for each actuated joint of the parallel mechanism to follow the specified trajectory; herein, the force/position hybrid control scheme (the 1st limb, 2nd limb, and 3rd limb utilized position control, while the 4th limb employed force control) is adopted.Figure 13
The desired position of the actuated jointd.Figure 14
The desired driving force of the actuated joint.In Figure13, the changes of the actuated joint distances are stable and the amplitudes are approximately consistent. However, in Figure 14, the changes of the driving force are very apparent; what is more, the forces have a big difference between the 1st driving force and the 4th one. The comparison analysis demonstrates that the redundantly actuated parallel mechanism has the advantage of optimizing the driving force configuration over the movement process. But most importantly, we should consider the force control in trajectory tracking process, and it is very significant for the parallel mechanism with redundant actuation. In addition, Figure 14 illustrates that the driving forces τ1, τ2, τ3, and τ4 are the function about time. The values of actuator forces can be applied to choose the ideal drive motor. Apparently, the parallel mechanism with redundant actuation can effectively decrease the magnitude of driving force and optimize the force configuration, which is a feature of the proposed new parallel mechanism.In order to demonstrate the effectiveness of the proposed force/position hybrid control scheme for the execution of trajectory tracking, the two controllers, the BP neural network PID controller, and the traditional PID controller have been simulated simultaneously. As can readily be appreciated in Figures15–17, the simulation results presented with both controllers are very similar. However, the traditional PID controller response speed is much worse, and we can see that the designed control scheme has a much better trajectory tracking performance than the traditional PID control method. The variation of Euler angle α is not apparent in the tracking process, as shown in Figure 16, but the designed control scheme has a quick response through the comparison analysis of Figures 15 and 17. The simulation results verify the trajectory tracking performance of the proposed controller, which is available to control the parallel mechanism with redundant actuation for the most important application.Figure 15
The trajectory tracking comparison in thez direction.Figure 16
The trajectory tracking comparison about Euler angleα.Figure 17
The trajectory tracking comparison about Euler angleβ.The kinematic and dynamic analyses established the theoretical foundation for intelligent control of the parallel mechanism. In the short run, we will concentrate on calibrating the errors by experiments. The experimental prototype is shown in Figure18; the experimental system mainly consists of the parallel mechanism, four driver controllers, a multiaxis controller, and a master CPU controller. The real position and velocity of the four actuators will be transmitted to the computer for data monitoring and implementing control in real time.Figure 18
The experimental prototype of the parallel mechanism.


## 7. Conclusions

In this paper, the trajectory tracking control scheme for a new 2RPU-2SPR parallel mechanism with redundant actuation is systematically investigated, and the following contributions can be obtained:
(1)
The inverse kinematic analysis of the proposed parallel mechanism has been formulated, and the mapping relationship of velocity and acceleration between the joint space and task space are straightforwardly deduced including the inverse Jacobian matrix and the forward Jacobian matrix
(2)
In terms of the principle of virtual work and the d’Alembert formulation, the inverse and forward dynamic standard formulation of the proposed parallel mechanism with redundant actuation is established and the optimal driving force value is obtained by utilizing the generalized Moore-Penrose inverse matrix
(3)
On the basis of the kinematic and dynamic formulations, the force/position hybrid control by means of inertia feed-forward control and BP neural network PID control was implemented in MATLAB/Simulink. As an illustrative case, the simulation results verified the effectiveness of the force/position hybrid control law which can possess good trajectory tracking performance as compared with the traditional PID controller. Our future works will concentrate upon the effective intelligent controllers for versatile applications of high speed and high precision serial-parallel hybrid kinematic machine tool



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*Source: 7178103-2020-01-11.xml*