Modified PSO Method for Robust Control of 3RPS Parallel Manipulators

Abstract

We propose an effective method to design a modified particle swarm optimization (MPSO) singularity control method for a fully parallel robot manipulator. By adopting MPSO to obtain simple and effective estimated damping values, the result automatically adjusts the damping value around a singular point and greatly improves the accuracy of system responses. This method works by damping accelerations of the end effector, so that accelerations along the degenerated directions are zero at a singular point. These velocities, however, may not be zero in some situations, in which case, fluctuations will be encountered around a singular point. To overcome this drawback, we propose a control scheme that uses both damped acceleration and damped velocity, called the hybrid damped resolved-acceleration control (HDRAC) scheme. The MPSO optimization method can immediately provide optimal damping factors when used in an online application. Our proposed approach offers such superior features as easy implementation, stable convergence characteristics, and good computational efficiency. The main advantage of the HDRAC with MPSO in the 3RPS parallel manipulator control system is that it is not necessary for the system to plan its path for avoiding the singular point; thus, the workspace can be improved. Illustrative examples are provided to show the effectiveness of this HDRAC in practical applications, and experimental results verifying the utility of the proposed control scheme are presented.

1. Introduction

Parallel robotics offers the advantages of heavy load capacity, high rigidity, and low inertia. The triaxial RPS parallel manipulator, which is equipped with revolute joints on the base platform and spherical joints on the moving platform, has been widely applied to electronic gaming machines, virtual reality movies, simulations of aircraft and vehicles, and so on. In their research on 3RPS parallel robotics, Lee and Shah [1] used loop closure equations to deduce kinematic equations for the relationship between link lengths and the position of the moving platform. In a separate study, they also deduced the dynamic equations that included the drive forces and reacting forces of 3RPS parallel robotics in the case of inverse dynamics [2, 3]. Following their work, Fang and Huang [4] simplified the mathematic models and established the relationship among three input-output variables. Li and Xu [5, 6] deal with the kinematic characteristics in terms of workspace and dexterity vary with differences in the arrangement of actuators are investigated in detail.

Generally, there are three types of singularity of parallel mechanisms, that is, inverse, direct, and combined singularities. In this study, we deal with the direct kinematic singularities. Therefore, the author divided the procedure of finding “direct kinematic singularities” of 3RPS platform mechanism. In horizontal position of the moving platform, to begin with, he assumed the direction of the moving platform and determined its horizontal position at this moment, that is, its projection on the fixed base. Then, since the direction of the moving platform has been assumed, also, a moving platform remaining in this direction can only translate vertically, it is possible to locate the limiting position while this direction has always been kept; this is the boundary of workspace, which makes the determinant of Jacobian matrix equal to zero and thus is the “direct kinematic singularities”.

There may exist singular configurations emerging from the motions of mechanisms, and resolved-rate control and resolved-acceleration control [7–9] are the two general approaches used to control robots. In the neighborhood of singularities, very high speeds are required to produce even small changes in the position or orientation of the end effector. In this paper, singular value decomposition (SVD) is used in order to show that fluctuations around the singular point are due to unnecessary nonzero end-effector velocities along the degenerated directions when the manipulator is at a singularity. This paper proposes a hybrid damped resolved-acceleration control (HDRAC) scheme, which considers both damped acceleration and velocity as a way to remove these unnecessary velocities.

Particle swarm optimization (PSO) is a stochastic population-based optimization approach and was first published by Kennedy and Eberhart in 1995 [10, 11]. PSO has been shown to be an efficient, roust, and simple optimization algorithm. Most studies of the PSO are empirical with only a few theoretical analyses that concentrate on understanding particle trajectories. A large amount of research has been proposed to improve the performance of PSO. From these studies, much effort has been invested to obtain a better understanding of its convergence properties. These studies concentrated mostly on a better understanding of the basic PSO control parameters, namely, the acceleration coefficients, inertia weight, velocity clamping, and swarm size [12–14]. In this paper, a modified PSO algorithm is proposed to improve the searching ability and prevent from being trapped in a local optimum. The main difference of the MPSO from the PSO is its fitness function which considers the “distance” to avoid converging to a local optimum. From these empirical studies, it can be concluded that the MPSO is sensitive to control parameter choices, specifically the inertia weight, acceleration coefficients, and velocity clamping [15]. However, wrong initialization of these parameters may lead to divergent or cyclic behavior.

In this study, singular value decomposition (SVD) is used to show that the abovementioned fluctuation is due to unnecessary nonzero end-effector velocities along the degenerated directions when the manipulator is at a singularity. The hybrid damped resolved-acceleration control scheme (HDRAC), which considers both damped acceleration and velocity, is proposed to remove these unnecessary velocities. Numerical simulations and the results of experiments are compared in order to demonstrate the applicability and effectiveness of the proposed algorithms at a singularity for a 3RPS parallel manipulator.

2. Singularity Analysis of the 3RPS Parallel Manipulator

As shown in Figure 1, one end of the 3RPS parallel manipulator is a moving platform and the other end is a fixed base. The moving platform is connected to the fixed base by three limbs via three spherical joints (𝐵₁, 𝐵₂, and 𝐵₃) and three revolute joints (𝐴₁, 𝐴₂, and 𝐴₃). Each limb consists of two links connected in a series by a prismatic joint. Figure 2 shows that the origin 𝑃 of the coordinate system [𝑥𝑦𝑧]^𝑇 is located at the mass center of the base moving platform. The Cartesian coordinate vector [𝑋𝑌𝑍𝛼𝛽𝛾]^𝑇  denotes the fixed coordinates, where [𝑋𝑌𝑍]^𝑇 and [𝛼𝛽𝛾]^𝑇  denote, respectively, the vectors of position and orientation. The fixed and moving platforms of the 3 RPS are circles with radius 𝑎 and 𝑏, respectively. The distance between the three spherical joints on the moving platform is 𝑤.

Figure 1 

The 3RPS parallel manipulator.

Figure 2 

Coordinate systems for the 3RPS manipulator.

2.1. Dynamics of the 3PRS Parallel Manipulator

Let us focus on the mass center 𝑂_𝑖 of the 𝑖th link, as shown in Figure 2. There are three components of the weight force of the moving platform that act on the 𝑖th link as ^𝑖𝐹_𝑝𝑖𝑥, ^𝑖𝐹_𝑝𝑖𝑦, and ^𝑖𝐹_𝑝𝑖𝑧. The weight force of the 𝑖th link is 𝑊_𝑖, the acting force of the separately excited DC servomotor on the revolution parts is ^𝑖𝐹_1motor, and the acting force on each vertical leg is ^𝑖𝐹_2motor. The 3RPS parallel manipulator dynamic motion of the links and moving platform is described by using Newton-Euler equations, which are formulated by separately considering the free-body diagrams of the links and the moving platform [16–19]. Through reduction and incorporation, we can calculate the acting force on each vertical leg by using the following force equation for the moving vertical leg [19]:^𝑖𝐹_2motor=^𝑖𝐅_𝑝𝑖𝑥+𝐑^𝑇_𝑖𝐖_𝑙𝑖𝑥, (2.1) where𝐑_𝑖=cos𝜙cos𝜓cos𝜃−sin𝜙sin𝜓−cos𝜙cos𝜃sin𝜓−sin𝜙cos𝜓cos𝜙sin𝜃sin𝜙cos𝜓cos𝜃+cos𝜙sin𝜓−sin𝜓cos𝜃sin𝜙+cos𝜓cos𝜙sin𝜙sin𝜃−sin𝜃cos𝜓sin𝜃sin𝜓cos𝜃, (2.2)𝐖_𝑙𝑖 is the weight of the 𝑖th link, ^𝑖𝐅_𝑝𝑖𝑥=𝐃⁻¹𝐐, and matrices 𝐃 and 𝐐 are given in Appendix A.

With regard to the revolution parts, we have the equation for the first motor as follows:𝑙_𝑎1𝑖̇_𝑎1+𝑟_𝑎1𝑖_𝑎1=𝑉_𝑇−𝑘_𝑏1𝜔,𝜏=𝐾_𝑡1𝑖_𝑎1, (2.3) where 𝐾_𝑡1 is the torque coefficient, 𝐾_𝑏1 is the back emf of the motor, 𝑙_𝑎1 is the armature inductance, 𝑟_𝑎1 is the armature resistance, 𝑉_𝑇 is the terminal voltage, 𝜔 is the rotor speed, and 𝑖_𝑎1 is the armature current.

2.2. Deduction of Direct Kinetic Singularities

In this section, we adopt a method [16–19] to formulate the direct singular position of the 3RPS parallel manipulator. The distance between any two points of 𝐵₁, 𝐵₂, and 𝐵₃ is always 𝑤, which gives us the following relationships: 𝑓₁=𝐵₁𝐵₂²−𝑤²𝑓=0,₂=𝐵₂𝐵₃²−𝑤²𝑓=0,₃=𝐵₃𝐵₁²−𝑤²=0. (2.4) By differentiating (2.4) with respect to time, and writing them in matrix form, we get𝑑𝑓₁𝑑𝑡𝑑𝑓₂𝑑𝑡𝑑𝑓₃=𝑎𝑑𝑡¹¹𝑎₁₂00𝑎₂₂𝑎₂₃𝑎₃₁0𝑎₃₃𝜃̇₁𝜃̇₂𝜃̇₃+𝑏₁₁𝑏₁₂00𝑏₂₂𝑏₂₃𝑏₃₁0𝑏₃₃𝑑̇₁𝑑̇₂𝑑̇₃𝜃̇=𝐀𝜃̇¹₂𝜃̇₃𝑑̇+𝐁𝑑̇¹₂𝑑̇₃=000, (2.5) where matrix 𝐀 and 𝐁 are given in Appendix B.

If the lengths of the three limbs are represented by 𝑑_𝑖(𝑖=1,2,3) and the link’s length is divided into two components, which are parallel and normal to the base and represented by 𝑑_𝑖𝑝 and 𝑑_𝑖𝑛, respectively, then we have 𝑑_1𝑝,𝑑_2𝑝,𝑑_3𝑝=𝑑₁cos𝜃₁,𝑑₂cos𝜃₂,𝑑₃cos𝜃₃,𝑑_1𝑛,𝑑_2𝑛,𝑑_3𝑛=𝑑₁sin𝜃₁,𝑑₂sin𝜃₂,𝑑₃sin𝜃₃, (2.6) where 𝜃₁, 𝜃₂, and 𝜃₃ are the rotating angles of the revolute joint on the first, second, and third limbs, respectively.

This study begins with projecting the moving platform onto the fixed base, where the projections of 𝐵₁, 𝐵₂, and 𝐵₃  are represented by 𝐵^′₁, 𝐵^′₂, and 𝐵^′₃, respectively. The length 𝐴₁𝐵^′₁ is 𝑑_1𝑝=𝑑₁cos(180−𝜃₁), and the lengths 𝑂𝐵^′₁, 𝑂𝐵^′₁, and 𝑂𝐵^′₃  are represented by 𝑒₁, 𝑒₂, and 𝑒₃, respectively, which gives us𝑒₁,𝑒₂,𝑒₃=𝑑_1𝑝+1,𝑑_2𝑝+1,𝑑_3𝑝+1. (2.7) For convenience of calculation, another coordinate system (ℎ₁,ℎ₂,ℎ₃) is defined as follows [19]:ℎ₁,ℎ₂,ℎ₃=𝑑_3𝑛−𝑑_2𝑛,𝑑_1𝑛−𝑑_3𝑛,𝑑_2𝑛−𝑑_1𝑛. (2.8) We must note that this manipulator is capable of translating upward or downward without changing the direction of the moving platform. The approach proposed here is to start by assuming the direction of the moving platform, which is 𝑑_2𝑛−𝑑_1𝑛, 𝑑_3𝑛−𝑑_2𝑛, 𝑑_1𝑛−𝑑_3𝑛, and equating (2.6) and (2.7) with (2.4) to solve the position of the three spherical joints, 𝑒₁, 𝑒₂, and 𝑒₃. The elevation of the moving platform can be represented by the levels of three spherical joints, as follows:𝑑_ave=𝑑_1𝑛+𝑑_2𝑛+𝑑_3𝑛3. (2.9) We then need to find a value for 𝑑_1𝑛+𝑑_2𝑛+𝑑_3𝑛 that makes the determinant of matrix 𝐀 equal to zero. We have det(𝐀)=𝑎₁₁𝑎₂₂𝑎₃₃+𝑎₁₂𝑎₂₃𝑎₃₁, while using𝑑_1𝑛,𝑑_2𝑛,𝑑_3𝑛=𝑑_ave+ℎ₂−ℎ₃,ℎ₃−ℎ₁,ℎ₁−ℎ₂3, (2.10) and this determinant is a cubic function of 𝑑_ave. Calling this function 𝐹(𝑑_ave), our purpose is then to find 𝑑_ave, such that [19] 𝐹𝑑_ave=det(𝐀)=Ω₁𝑑³_ave+Ω₂𝑑²_ave+Ω₃𝑑_ave+Ω₄=0, (2.11) where the coefficients are given in Appendix C. The process of finding a solution begins with assuming (ℎ₁,ℎ₂,ℎ₃)   values, and obtaining (𝑒₁,𝑒₂,𝑒₃) values and coefficients of the cubic polynomial equation, so that finally the roots of 𝑑_ave that satisfy det(𝐀)=0 can be found.

3. Optimization Method: Modifying Particle Swarm Optimization

3.1. Particle Swarm Optimization

Birds (particles) flocking optimizes a certain objective function in a PSO system. Each agent knows its best value so far (pbest) and its position. This information is an analogy of personal experiences of each agent. Moreover, each agent knows the best value so far in the group (gbest) among pbests. This information is an analogy of knowledge of how the other agents around them have performed. The PSO concept [10, 11] consists of changing the velocity of each particle toward its pbest and gbest locations. In the PSO, each particle moves to a new position according to new velocity and the previous positions of the particle. This is compared with the best position generated by previous particles in the fitness function, and the best one is kept; so each particle accelerates in the direction of not only the local best solution but also the global best position. If a particle discovers a new probable solution, other particles will move closer to it to explore the region more completely in the process.

In general, there are three attributes, current position 𝑥_𝑗, current velocity 𝑣_𝑗, and past best position 𝑝𝑏𝑒𝑠𝑡_𝑗, for particles in the search space to present their features. Each particle in the swarm is iteratively updated according to the aforementioned attributes. For example, [10–14], the 𝑗th particle is represented as 𝑥_𝑗=(𝑥_𝑗,1,𝑥_𝑗,2,…,𝑥_𝑗,𝑔) in the 𝑔-dimensional space. The best previous position of the𝑗th particle is recorded and represented as 𝑝𝑏𝑒𝑠𝑡_𝑗=(𝑝𝑏𝑒𝑠𝑡_𝑗,1,𝑝𝑏𝑒𝑠𝑡_𝑗,2,…,𝑝𝑏𝑒𝑠𝑡_𝑗,𝑔). The index of best particle among all particles in the group is represented by the 𝑔𝑏𝑒𝑠𝑡_𝑔. The rate of the position change (velocity) for particle 𝑗 is represented as 𝑣_𝑗=(𝑣_𝑗,1,𝑣_𝑗,2,…,𝑣_𝑗,𝑔). The modified velocity and position of each particle can be calculated using the current velocity and distance from 𝑝𝑏𝑒𝑠𝑡_𝑗,𝑔 to 𝑔𝑏𝑒𝑠𝑡_𝑗,𝑔 as shown in the following formulas [14]:𝑣^(𝑡+1)_𝑗,𝑔=𝑤⋅𝑣^(𝑡)_𝑗,𝑔+𝑐₁𝑅𝑎𝑛𝑑()⋅𝑝𝑏𝑒𝑠𝑡^(𝑡)_𝑗,𝑔−𝑥^(𝑡)_𝑗,𝑔+𝑐₂𝑅𝑎𝑛𝑑^∗()⋅𝑔𝑏𝑒𝑠𝑡^(𝑡)_𝑗,𝑔−𝑥^(𝑡)_𝑗,𝑔,𝑥^(𝑡+1)_𝑗,𝑔=𝑥^(𝑡)_𝑗,𝑔+𝑣^(𝑡+1)_𝑗,𝑔,𝑗=1,2,…,𝑛;𝑔=1,2,…,𝑚, (3.1) where 𝑛 is the number of particles in a group; 𝑚 is the number of members in a particle; 𝑡 is the pointer of iterations (generations); 𝑣^(𝑡)_𝑗,𝑔 is the velocity of the particle 𝑗 at iteration 𝑡, 𝑉_𝑔^min≤𝑣^(𝑡)_𝑗,𝑔≤𝑉_𝑔^max; 𝑤 is the inertia weighting factor; 𝑐₁, 𝑐₂ are the acceleration constants; Rand(), Rand^∗() are random numbers between 0 and 1; 𝑥^(𝑡)_𝑗,𝑔 is the current position of particle 𝑗 at iteration 𝑡; 𝑝𝑏𝑒𝑠𝑡_𝑗 is the pbest of particle 𝑗; 𝑔𝑏𝑒𝑠𝑡_𝑔 is the gbest of the group 𝑔.

In the above procedures, the parameter 𝑉_𝑔^max determines the resolution or fitness, with which regions are searched between the present position and the target position. If 𝑉_𝑔^max is too high, particles might fly past good solutions. If 𝑉_𝑔^max is too low, particles may not explore sufficiently beyond local solutions.

The constants 𝑐₁ and 𝑐₂ represent the weighting of the stochastic acceleration terms that pull each particle toward pbest and gbest positions. Low values allow particles to roam far from the target regions before being tugged back. On the other hand, high values result in an abrupt movement toward or past target regions.

Suitable selection of inertia weighting factor 𝑤 provides a balance between global and local explorations, thus requiring less iteration on average to find a sufficiently optimal solution. As originally developed, 𝑤 often decreases linearly from about 0.9 to 0.4 during a run. In general, the inertia weighting factor 𝑤 is set according to the following equation [11, 12]:𝑤=𝑤_max−𝑤_max−𝑤_miniter_max×iter, (3.2) where iter_max is the maximum number of iterations (generations), and iter is the current number of iterations.

3.2. Modified Particle Swarm Optimization

The main point of the MPSO differs from the PSO to consider the “distance” in its fitness function to avoid converging to a local optimum. Assign a rank (i.e., the number place 1,2,3,…,etc.) 𝑅𝐸_𝑘 to the calculated error of each new individual, 𝑣_𝑘, 𝑘=1,…,PS, PS is the population size. A combined population with 2×PS individuals is formed. Unlike previously developed statistic methods, the concept of “distance” is added to the fitness function to prevent from being trapped in a local minimum. The fitness score of the 𝑘th individual is modified by [20, 21]𝐹_𝑘=𝑅𝐸_𝑘+𝜌×𝑅𝐷_𝑘,𝑘=1,…,2×PS, (3.3) where 𝜌 is an adaptive decay scale, 𝜌_max is set as 0.7, and 𝜌_min is set as 0.005 in this paper. 𝑅𝐷_𝑘 is the rank of 𝐷_𝑘 assigned to the 𝑘th individual, where 𝐷_𝑘 is the distance from the individual to the current best solution vector and is given by𝐷_𝑘=𝑣_𝑘−𝑣_{𝑏𝑒𝑠𝑡}, (3.4) where 𝑣_𝑘 is the vector of the 𝑘th individual in the combined population, and 𝑣_{𝑏𝑒𝑠𝑡} is the current best solution vector.

An adaptive scheme is defined as [20]𝜌_Δ𝜌=𝑅×^max−𝜌_min𝑔_max,𝜌=𝜌𝑔𝑔+1−𝜌_Δ,𝐹_min𝑔=𝐹_min,𝜌𝑔𝑔−1,𝐹_min𝑔<𝐹_min,𝜌𝑔−1𝑔+1=𝜌_minif𝜌𝑔−𝜌_Δ<𝜌_min, (3.5) where 𝜌_Δ is the step size; 𝐹_min is the minimum value of fitness functions; 𝑅 is the regulating scale and is set as 1.25 in this paper, and 𝑔_max is the maximum allowable number of iterations.

Individuals will be ranked ascending according to their fitness scores by a sorting algorithm. The PS individuals are transcribed along with their fitness for the next generation. If the new population does not include the current best solution, the best solution must be replaced with the last individual in the new population. In addition, a gradually decreased decay scale can satisfy a successive statistic searching process by first using the diversification (bigger 𝜌) to explore more regions and then the intensification (smaller 𝜌) to exploit the neighborhood of an elite solution. The current best solution (point 𝐴) for a minimum fitness problem as shown in the Figure 3 may not reach the global optimum [20, 21], and there are three electable solutions. Generally, solutions with slightly better fitness (point 𝐶 or 𝐵) prevailed, so the solution was trapped into the valley prematurely. The more attractive solution (point 𝐺) is relatively far away from point 𝐴, but it nears the global optimal. To prevent prematurity, point 𝐺 with slightly worse fitness than 𝐶, it needs a higher rank to be selected. That is, a higher 𝑅𝐷_𝑘 is awarded to a longer 𝐷_𝑘.

Figure 3 

The concept of distances.

Stopping Criteria
Stopping criteria is given in the following order:(1)maximum allowable number of iterations reached.(2)number of iterations reached without improving the current best solution.Figure 4 shows the flow chart of the proposed algorithm.

Figure 4 

The flow chart of the MPSO.

4. Singularity Control Scheme

4.1. Resolved-Acceleration Control

The relationship between the velocities of the mass center 𝑃 of the moving platform and the joint velocity 𝐪̇ for parallel manipulators can be represented aṡ𝑟̇𝜔̇𝑋𝑌𝑍𝛼𝛽𝛾𝐱≡=𝐉𝐪,𝐪=,𝐪̇=𝑑𝐪=𝑣𝑑𝑡^𝑋𝑣_𝑌𝑣_𝑍𝜔_𝛼𝜔_𝛽𝜔_𝛾, (4.1) where 𝑟̇ and 𝜔 are the linear and angular velocities, respectively. The Cartesian coordinate vector 𝐪=(𝑋,𝑌,𝑍,𝛼,𝛽,𝛾)^𝑇 with six variables is chosen to describe the position and orientation of the moving base, that is, 𝐪̇ is the velocity and 𝐉 is the Jacobian matrix. Differentiating (4.1) gives𝑟̈𝛼𝐚≡=𝐉𝐪̈+𝐉̇𝐪̇, (4.2) where 𝑟̈ and 𝛼 are the linear and angular accelerations, respectively.

Now, by expanding (2.1)-(2.3), we can obtain the dynamic equation of the manipulator in the matrix form of𝝉=𝐌(𝐪)𝐪̈+𝐍(𝐪̇,𝐪)+𝐆(𝐪)=𝐌(𝐪)𝐪̈+𝐟(𝐪̇,𝐪), (4.3) where 𝐌(𝐪) is the inertia matrix of the manipulator, 𝐍(𝐪̇,𝐪) contains the Coriolis and centrifugal force/torque components, 𝐆(𝐪) contains the gravity force/torque components, and 𝝉 is the vector of actuator forces. The second-order nonlinear coupled dynamic equation (4.3) can be linearized by inputting the inverse dynamics as follows:𝝉=𝐌(𝐪)𝐪̈_𝑑+𝐟(𝐪̇,𝐪), (4.4) where 𝐪̈_𝑑 is the vector of the desired accelerations, so that 𝐪̈=𝐪̈_𝑑.

The resolved-acceleration control scheme [19] is𝐪̈_𝑑=𝐉⁻¹𝐚_𝑑−𝐉̇𝐪̇=𝐉⁻¹𝐚^∗_𝑑, (4.5a) with 𝐚_𝑑=𝑟̈_𝑑𝛼_𝑑+𝐊_𝐷𝑟̇_𝑑−𝑟̇𝜔_𝑑−𝜔+𝐊_𝑃𝜀_𝑟𝜀_𝑒,𝐚^∗_𝑑=𝐚_𝑑−𝐉̇𝐪̇, (4.5b)where 𝐊_𝐷 and 𝐊_𝑃 are gain matrices, subscript 𝑑 denotes the desired value, 𝜀_𝑟=𝜀_𝑑−𝑟 is the positional error, and 𝜀_𝑒 is the orientation error. Unfortunately, this control scheme (4.5a) breaks down when 𝐉⁻¹ does not exist, which occurs at a singular configuration.

4.2. Damped Least-Square Method

In this section, the damped least-square method for the inverse problem of (4.1) is applied to solve the following optimization problem [19]:min_̇𝐪𝜌𝐉𝐪̇_𝜌−𝑥̇²+𝜌²𝑞̇_𝜌², (4.6) where 𝜌 is the damping factor. The solution of (4.6) is𝐪̇_𝜌=𝐉^∗_𝜌𝑥̇, (4.7) where𝐉^∗_𝜌=𝐉^𝑇𝐉+𝜌²𝐈⁻¹𝐉^𝑇, (4.8) which always exists for 𝜌≠0. This solution is a compromise between the residual error, 𝐉𝐪̇−𝑥̇, and the velocity 𝐪̇ of the point 𝑃.

Singular value decomposition (SVD) can provide insight into the singularities of the inverse Jacobian [19, 22, 23]. This study deals with nonredundant manipulators, and there are two orthogonal matrices 𝐔=[𝑢₁⋯𝑢₆] and 𝐕=[𝑣₁⋯𝑣₆], such that𝐉=𝐔Σ𝐕^𝑇=⁶_𝑖=1𝜎_𝑖𝐮_𝑖𝐯^𝑇_𝑖, (4.9) where Σ=diag[𝜎₁,…,𝜎₆] and 𝜎_𝑖 are the singular values of 𝐉. The vectors 𝐮_𝑖 and 𝐯_𝑖 are the 𝑖th left and the 𝑖th right singular vectors, respectively. Substituting (4.9) into (4.8) gives𝐉^∗_𝜌=𝐕Σ^∗_𝜌𝐔^𝑇=⁶_𝑖=1𝜎_𝑖𝜎²_𝑖+𝜌²𝐯_𝑖𝐮^𝑇_𝑖, (4.10) where Σ^∗_𝜌=diag[𝜎₁/(𝜎²₁+𝜌²),…,𝜎₆/(𝜎²₆+𝜌²)]. When the manipulator is far away from a singular point (i.e., 𝜎_𝑖≫𝜌), we have 𝜎_𝑖/(𝜎²_𝑖+𝜌²)≈1/𝜎_𝑖, which implies that 𝐉^∗_𝜌≈𝐉⁻¹. When the manipulator is near a singular point, it can be seen from (4.10) that the solutions for the velocity 𝐪̇ have finite values.

4.3. The Hybrid Damped Resolved-Acceleration Control Scheme

In this paper, we propose a hybrid damped resolved-acceleration control scheme (HDRAC) that restricts velocity as well as acceleration. The HDRAC is used to solve the following optimization problem:min_{̈𝐪ℎ𝑑}𝐉𝐪̈_ℎ𝑑−𝐚^∗_𝑑²+𝜌²𝐪̈_ℎ𝑑+𝜌_ℎ𝑑𝐪̇², (4.11) where 𝐚^∗_𝑑 is defined in (4.2), 𝜌 is the damping factor, and 𝜌_ℎ𝑑 is an additional damping factor which is dependent on the smallest singular value of the Jacobian matrix. The joint acceleration commands 𝐪̈_ℎ𝑑 have to minimize (4.11). When the manipulator is far away from the singular point, there is no unnecessary joint velocity, so 𝜌_ℎ𝑑 will be zero. If the manipulator is at a singular point, the unnecessary joint velocity will have to be removed by joint acceleration commands, so 𝜌_ℎ𝑑 must be specified. The redundant components of the joint acceleration commands 𝐪̈_ℎ𝑑 will be equal to the unnecessary components of 𝐪̇ multiplied by 𝜌_ℎ𝑑 to minimize (4.11).

Equation (4.11) can be rewritten asmin_{̈𝐪ℎ𝑑}𝐉𝜌𝐼₆𝐪̈_ℎ𝑑−𝐚^∗_𝑑𝜌𝜌_ℎ𝑑𝐪̇², (4.12) where 𝐼₆ denotes a 6×6 identity matrix. In order to obtain the solutions for this optimization problem in a closed form, the discrete-time technique will be used. With discrete-time control, the back difference is always used to approximate the derivative, so that𝐪̈_ℎ𝑑(𝑘)=1𝐪̇Δ𝑡^ℎ𝑑(𝑘)−𝐪̇_ℎ𝑑(𝑘−1), (4.13) where Δ𝑡 is the sampling time. Through reduction and incorporation, the solution of the optimization problem is [19]𝐪̈_ℎ𝑑≡𝐪̈_𝑑𝑎+𝐪̈_ℎ𝑑𝑐=𝐉^∗_𝜌𝐚^∗_𝑑−𝜌_ℎ𝑑𝜌²𝐉^𝑇𝐉+𝜌²𝐼₆⁻¹𝐪̇. (4.14) Meanwhile, we have denoted the second term as 𝐪̈_ℎ𝑑𝑐.

The SVD theory can be reformulated as𝐪̈_ℎ𝑑𝑐=𝜌_ℎ𝑑⁶_𝑖=1−𝜌²𝜎²_𝑖+𝜌²𝐯^𝑇_𝑖𝐪̇𝐯_𝑖. (4.15) The size of the deceleration region of the HDRAC can be adjusted by a linear function 𝜌_ℎ𝑑, which is chosen as𝜌_ℎ𝑑=1𝜎Δ𝑡1−^min𝛿_ℎ𝑑,for0≤𝜎_min<𝛿_ℎ𝑑,0,for𝜎_min≥𝛿_ℎ𝑑, (4.16) where 𝜎_min is the smallest singular value or singular parameter [19] of the Jacobian matrix. 𝛿_ℎ𝑑 is a user-defined threshold value that represents the deceleration region. Unnecessary joint velocities will be partially removed when 𝜎_min is in this region and wholly removed at the next sampling interval when 𝜌_ℎ𝑑=1/Δ𝑡. Note that the maximum value of 𝜌_ℎ𝑑 is 1/Δ𝑡. If the manipulator is at a singular point of 𝜎_𝑖≫𝜌>0,  𝑖=1,…,𝑘, and 𝑗=𝑘+1,…,6, then (4.15) can be reduced to𝐪̈_ℎ𝑑𝑐≈−1Δ𝑡^6𝑖=𝑘+1𝐯^𝑇_𝑖𝐪̇𝐯_𝑖. (4.17)

We will next prove that this term removes unnecessary joint velocities at the next sampling interval. Multiplying both sides of (4.13) by 𝐯_𝑖𝐯^𝑇_𝑖 yields1𝐯Δ𝑡^𝑇𝑖𝐪̇_ℎ𝑑𝐯_𝑖=𝐯^𝑇_𝑖𝐪̈_ℎ𝑑𝐯_𝑖+1𝐯Δ𝑡^𝑇𝑖𝐪̇𝐯_𝑖. (4.18) By using an ideal computed-torque scheme, we force joint velocity 𝐯_𝑖 (i.e., 𝐯^𝑇_𝑖𝑞̇) along the degenerated direction to zero at the next sampling interval. The joint acceleration command along 𝐯_𝑖 (i.e., the product of 𝐯^𝑇_𝑖𝐪̈_ℎ𝑑) should then be (−1/Δ𝑡)(𝐯^𝑇_𝑖𝐪̇), which follows from (4.18). This implies that (4.17) will remove unnecessary joint velocities along all degenerated directions 𝐯_𝑖, 𝑖=𝑘+1,…,6. According to (4.18), the desired joint velocities along 𝐯_𝑖, 𝑖=1,…,𝑘 are also decreased by the amount of Δ𝑡𝐯^𝑇_𝑖𝐪̈_ℎ𝑑 (or [𝜌²/(𝜎²_𝑖+𝜌²)]𝐯^𝑇_𝑖𝐪̇). Note that the maximum coefficient of (𝐯^𝑇_𝑖𝐪̇)𝐯_𝑖 in (4.15) is −1/Δ𝑡 when 𝜎_𝑖=0.

5. Stability Analysis

In this section, we will show that the HDRAC is asymptotically stable and discuss its convergent property. Let the orientation error be 𝜀_𝑒=𝑓(𝜃_𝑒)𝑢_𝑒, where 𝑢_𝑒 and 𝜃_𝑒 are the rotational axis and the angle between the current orientation and the desired orientation, respectively, and 𝑓(𝜃_𝑒)=𝜃_𝑒 for very small values of 𝜃_𝑒.

Suppose that 𝜌>0 when the manipulator is at a singular point. A sufficient condition for the global asymptotic convergence of the HDRAC in the whole workspace of the manipulator in a situation, where 𝑟̈_𝑑=𝑟̇_𝑑=0 and 𝛼_𝑑=𝜔_𝑑=0, is that 𝐊_𝐷 is a positive-definite matrix and𝐊_𝑃=𝑘_𝑝𝑟𝐼₃00𝑘_𝑝𝑒𝐼₃, (5.1) where 𝑘_𝑝𝑟 and 𝑘_𝑝𝑒 are positive. The equilibrium is at the point where 𝐪̇=0. Let a Lyapunov function be [24]𝐿(𝐪̇)=1𝐪,2𝑘_𝑝𝑟𝜀^𝑇_𝑟𝜀_𝑟+𝑘_𝑝𝑒1𝐸+2𝐪̇^𝑇𝐉^𝑇𝐉+𝜌²𝐈₆𝐪̇, (5.2) where 𝜀_𝑟=𝑟_𝑑−𝑟 and 𝐸=∫^𝜃𝑒₀𝑓(𝜙)𝑑𝜙=(1/2)𝜃²_𝑒, for 𝑓(𝜙)=𝜙. Clearly, 𝐿(𝐪,𝐪̇)≥0. It is recognized that 𝜃̇_𝑒=𝑢_𝑒(𝜔_𝑑−𝜔). Differentiating 𝐸 with respect to time, we obtain𝐸̇=𝜕𝐸𝜕𝜃_𝑒𝜃̇_𝑒=𝜀_𝑒𝜔_𝑑−𝜔. (5.3) Evaluating 𝜕𝐿/𝜕𝑡 in terms of the solutions of (5.2) yields𝜕𝐿𝜕𝑡=𝑘_𝑝𝑟𝜀^𝑇_𝑟𝑟̇_𝑑−𝑟̇+𝑘_𝑝𝑒𝑓𝜃_𝑒𝑢^𝑇_𝑒𝜔_𝑑−𝜔+𝐪̇^𝑇𝐉^𝑇𝐉+𝜌²𝐈₆𝐪̈+𝐪̇^𝑇𝐉̇^𝑇𝐉𝐪̇. (5.4) If the ideal computed-torque control is used, 𝐪̈=𝐪̈_ℎ𝑑, (4.14) can be substituted into (5.4), which gives us [24]𝜕𝐿𝜕𝑡=𝑘_𝑝𝑟𝜀^𝑇_𝑟𝑟̇_𝑑+𝑘_𝑝𝑒𝜀^𝑇_𝑒𝜔_𝑑+𝐪̇^𝑇𝐉^𝑇𝑟̈_𝑑𝛼_𝑑+𝐪̇^𝑇𝐉^𝑇𝐊_𝐷𝑟̇_𝑑𝜔_𝑑−𝐉𝐪̇−𝜌_ℎ𝑑𝜌²𝐪̇^𝑇𝐪̇. (5.5) If 𝑟̈_𝑑=𝑟̇_𝑑=0 and 𝛼_𝑑=𝜔_𝑑=0, then𝜕𝐿𝐪̇𝜕𝑡=−^𝑇𝐉^𝑇𝐊_𝐷𝐉𝐪̇−𝜌_ℎ𝑑𝜌²𝐪̇^𝑇𝐪̇. (5.6) Since 𝐊_𝐷 is a positive-definite matrix, 𝜕𝐿/𝜕𝑡≤0 for all 𝐪̇, and 𝜕𝐿/𝜕𝑡=0 only for 𝐪̇=0. Combining the above results and 𝐿(𝐪,𝐪̇)≥0, it is shown by the Lyapunov theorem that 𝐪̇ is asymptotically stable, and the equilibrium point is 𝐪̇=0. However, for the proposed control scheme, 𝜕𝐿/𝜕𝑡 has an additional term (−𝜌_ℎ𝑑𝜌²𝐪̇^𝑇𝐪̇) more than that of the HDRAC, to ensure that 𝜕𝐿/𝜕𝑡 is zero only for 𝐪̇=0, that is, the equilibrium point is 𝐪̇=0.

6. Numerical Simulations and Experimental Results

The purpose of inverse dynamics analysis is to determine the required torque of the separately-excited DC servomotor for a given motion trajectory of the moving platform. The method [19] is adopted to analyze the 3RPS parallel manipulator dynamics of the links and moving platform by using Newton-Euler equations of motion, which are formulated by considering the free-body diagrams of the links and moving platform separately.

Let us focus on the mass center 𝑂_𝑖 of the 𝑖th link as shown in Figure 5, where^𝑖𝐹_𝑝𝑖𝑥, ^𝑖𝐹_𝑝𝑖𝑦, and ^𝑖𝐹_𝑝𝑖𝑧 are three components of the weight force of the moving platform acting on the 𝑖th link, 𝑊_𝑖 is the weight force of the 𝑖th link, ^𝑖𝐹_1motor is the acting force of the separately-excited DC servomotor on the revolution parts, and the ^𝑖𝐹_2motor is the acting force on each vertical leg. Each link is placed along the 𝑋_𝑖 axis; therefore, its inertia tensor about 𝑋_𝑖, 𝑌_𝑖 and 𝑍_𝑖 is a diagonal matrix with diagonal elements 𝐼_𝑖𝑥𝑥,𝐼_𝑖𝑦𝑦, and 𝐼_𝑖𝑧𝑧. The angular momentum [19] of the 𝑖th link is𝐇_𝑖=𝐼_𝑖𝑥𝑥𝜔_𝑖𝑥𝐢_𝑖+𝐼_𝑖𝑦𝑦𝜔_𝑖𝑦𝐣_𝑖+𝐼_𝑖𝑧𝑧𝜔_𝑖𝑧𝐤_𝑖. (6.1) The rate change of 𝐇_𝑖 is given by𝐇̇_𝑖=𝐇̇_{𝑖𝑥𝑦𝑧𝑖}+𝝎_𝑖×𝐇_𝑖. (6.2) It follows from Figures 2 and 5 that the external moment about the mass center 𝑂_𝑖 of the 𝑖th link is given by𝐌_𝑂𝑖=^𝑖𝐫_𝑝𝑖×^𝑖𝐅_𝑝𝑖, (6.3) where ^𝑖𝐫_𝑝𝑖 is the vector from 𝑂_𝑖 to 𝐵_𝑖 with respect to the link frame with unit vector [𝑖_𝑖𝑗_𝑖𝑘_𝑖].

Figure 5 

Free-body diagram of acting forces on the 𝑖th link.

The equations of rotational motion of the 𝑖th link can be obtained by equating the rate of change of 𝐇_𝑖 with the external moment. By equating (6.2) with (6.3), we obtain𝐼_𝑖𝑧𝑧𝜔_𝑖𝑧𝜔_𝑖𝑦−𝐼_𝑖𝑦𝑦𝜔_𝑖𝑦𝜔_𝑖𝑧+𝐼̇_𝑖𝑥𝑥𝜔_𝑖𝑥+𝐼_𝑖𝑥𝑥𝜔̇_𝑖𝑥=^𝑖𝑟^𝑖_𝑝𝑖𝑦𝐹_𝑝𝑖𝑧−^𝑖𝑟^𝑖_𝑝𝑖𝑧𝐹_𝑝𝑖𝑦,−𝐼_𝑖𝑧𝑧𝜔_𝑖𝑧𝜔_𝑖𝑥−𝐼_𝑖𝑥𝑥𝜔_𝑖𝑥𝜔_𝑖𝑧+𝐼̇_𝑖𝑦𝑦𝜔_𝑖𝑦+𝐼_𝑖𝑦𝑦𝜔̇_𝑖𝑦=−^𝑖𝑟^𝑖_𝑝𝑖𝑥𝐹_𝑝𝑖𝑧−^𝑖𝑟^𝑖_𝑝𝑖𝑧𝐹_𝑝𝑖𝑥,𝐼_𝑖𝑦𝑦𝜔_𝑖𝑦𝜔_𝑖𝑥−𝐼_𝑖𝑥𝑥𝜔_𝑖𝑥𝜔_𝑖𝑦+𝐼̇_𝑖𝑧𝑧𝜔_𝑖𝑧+𝐼_𝑖𝑧𝜔̇_𝑖𝑧=^𝑖𝑟^𝑖_𝑝𝑖𝑥𝐹_𝑝𝑖𝑦−^𝑖𝑟^𝑖_𝑝𝑖𝑦𝐹_𝑝𝑖𝑥. (6.4) Since 𝐼̇_𝑖𝑥𝑥=𝐼̇_𝑖𝑦𝑦=𝐼̇_𝑖𝑧𝑧=0, ^𝑖𝐹_𝑝𝑖𝑥 passing through the mass center 𝑂_𝑖 do not generate torques, and ^𝑖𝐫_𝑝𝑖 only has values in the 𝑋_𝑖-axis and couples with the separately-excited DC motor; the above equations (6.4) can be simplified as follows: ^𝑖𝑟^𝑖_𝑝𝑖𝑥𝐹_𝑝𝑖𝑧=𝐼_𝑖𝑧𝑧𝜔_𝑖𝑧𝜔_𝑖𝑥−𝐼_𝑖𝑥𝑥𝜔_𝑖𝑥𝜔_𝑖𝑧−𝐼_𝑖𝑦𝑦𝜔̇_𝑖𝑦+𝐽_𝑚1𝜔̇_𝑖𝑦+𝐵_𝑚1𝜔_𝑖𝑦−𝐾_𝑇1𝑖_𝑎1−𝐾_𝑇2𝑖_𝑎2,_𝑖𝑟^𝑖_𝑝𝑖𝑥𝐹_𝑝𝑖𝑦=𝐼_𝑖𝑦𝑦𝜔_𝑖𝑦𝜔_𝑖𝑥−𝐼_𝑖𝑥𝑥𝜔_𝑖𝑥𝜔_𝑖𝑦+𝐼_𝑖𝑧𝑧𝜔̇_𝑖𝑧+𝐽_𝑚1𝜔̇_𝑖𝑧+𝐵_𝑚1𝜔_𝑖𝑧−𝐾_𝑇1𝑖_𝑎1−𝐾_𝑇2𝑖_𝑎2, (6.5) where 𝐽_𝑚1 is the moment of inertia; 𝐵_𝑚1 is the damped coefficient; 𝐾_𝑇1 and 𝐾_𝑇2 are the torque constants; 𝑖_𝑎1 and 𝑖_𝑎2 are the armature currents.

We now process to find the total force acting on the mass center 𝑂_𝑖 as follows:^𝑖𝐹_𝑝𝑖𝑦=𝐼_𝑖𝑦𝑦𝜔_𝑖𝑦𝜔_𝑖𝑥−𝐼_𝑖𝑥𝑥𝜔_𝑖𝑥𝜔_𝑖𝑦+𝐼_𝑖𝑧𝑧𝜔̇_𝑖𝑧+𝐽_𝑚1𝜔̇_𝑖𝑧+𝐵_𝑚1𝜔_𝑖𝑧−𝐾_𝑇1𝑖_𝑎1−𝐾_𝑇2𝑖_𝑎2𝑖𝑟_{𝑝𝑖𝑥𝑖},𝐹_𝑝𝑖𝑧=𝐼_𝑖𝑧𝑧𝜔_𝑖𝑧𝜔_𝑖𝑥−𝐼_𝑖𝑥𝑥𝜔_𝑖𝑥𝜔_𝑖𝑧−𝐼_𝑖𝑦𝑦𝜔̇_𝑖𝑦+𝐽_𝑚1𝜔̇_𝑖𝑦+𝐵_𝑚1𝜔_𝑖𝑦−𝐾_𝑇1𝑖_𝑎1−𝐾_𝑇2𝑖_𝑎2𝑖𝑟_𝑝𝑖𝑥. (6.6) Once the force terms ^𝑖𝐹_𝑝𝑖𝑦 and ^𝑖𝐹_𝑝𝑖𝑧 have been obtained in (6.6) of link motion, one should take into consideration the force term ^𝑖𝐹_𝑝𝑖𝑥. The angular momentum [19] of the moving platform is described by𝐇_𝑝=𝐼_𝑝𝑥𝑥𝜔_𝑝𝑥𝐢_𝑝+𝐼_𝑝𝑦𝑦𝜔_𝑝𝑦𝐣_𝑝+𝐼_𝑝𝑧𝑧𝜔_𝑝𝑧𝐤_𝑝, (6.7) where 𝐼_𝑝𝑥𝑥, 𝐼_𝑝𝑦𝑦, and 𝐼_𝑝𝑧𝑧 are, respectively, the mass moments of inertia of the moving platform about the 𝑋, 𝑌, and 𝑍 axes. The rate change of angular momentum is𝐇̇_𝑝=𝐇̇_{𝑝𝑥𝑦𝑧𝑝}+𝝎_𝑝×𝐇_𝑝. (6.8) The total torque acting on mass center 𝑃 of the moving platform is given by𝐌_𝑃=^3𝑝_𝑖=1𝐏_𝑖×^𝑝𝐅_𝑝𝑖, (6.9) where ^𝑃𝐏_𝑖 is the vector from 𝑃 to 𝐵_𝑖, 𝑖=1,2,3.

6.1. Numerical Simulations

In this section, numerical simulations are offered to demonstrate the applicability and effectiveness of the proposed algorithms of the 3RPS manipulator platform. The physical model is shown in Figure 6. The moving platform of the manipulator is a circle with radius 𝑏=22.08cm, and the distance between any two of the three spherical joints on the moving platform is 𝑤=36.16cm. The fixed base is an inscribed triangle within a circle with radius 𝑎=41.72cm. In the physical model, the constraints of the lengths 𝑑₁, 𝑑₂, and 𝑑₃ are ±10 cm, and the revolution angles 𝜃₁, 𝜃₂, and 𝜃₃ are ±20°. A program has been written in order to calculate the singular point. It was found that when 𝑑₁=0.0717 cm, 𝑑₂=9.7488 cm, and 𝑑₃=0.0771 cm, and 𝜃₁=19.9834^∘, 𝜃₂=0.2438^∘, and 𝜃₃=19.9074^∘, then one singular point exists in the workspace. That is, the center coordinate of the moving platform with respect to the fixed base is (1.6144, −2.7933, 0.0308), and the unit is cm. In this study, point 𝑃 is required to move at a constant speed from the starting point (0, 0, 0), to then pass through the singular point (1.6144, −2.7933, 0.0308), and then it arrives at its final point (5, −5, 5).

Figure 6 

Physical model of the 3RPS parallel manipulator.

In this study, the HDRAC scheme is used to solve the damping factors 𝜌 by means of the MPSO optimization method. Damping factors 𝜌 are not fixed values and are used to damp the accelerations of point 𝑃 when the manipulator is in the neighborhood of singularities. The dependence of the damping factors on the accelerations can change. For on-line applications, the optimization method can immediately provide optimal damping factors.

Figures 7(a)–7(f) show numerical simulations of the trajectories and velocities along the 𝑋-axis, 𝑌-axis, and 𝑍-axis for the controller incorporating HDRAC with fixed damping values, and for the controller incorporating HDRAC with MPSO. For the controller incorporating HDRAC with MPSO, when the manipulator is near the singular point, the system functions are more smooth and there is no fluctuation.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 7 

Numerical simulations for the controller incorporating HDRAC with fixed damping values and the controller incorporating HDRAC with MPSO. (a) Translation responses along the 𝑋-axis. (b) Velocities along the 𝑋-axis. (c) Translation responses along the 𝑌-axis. (d) Velocities along the 𝑌-axis. (e) Translation responses along the 𝑍-axis. (f) Velocities along the 𝑍-axis.

6.2. Experimental Setup

In order to demonstrate the proposed control rules, PC-based experimental equipment was devised. The parallel manipulator was coupled with a DC motor. The translation position was measured by a sensor. A photo of the experimental equipment is shown in Figure 6. In general, such parallel robotic is controlled by hydraulic servo, that is, to control the extension or retraction of the limb through driving with hydraulic system. Limiter sensor or measuring tools are generally installed on the link of the limb to enable accurate changes in its length when controlling positions; the characteristic of position limiter sensor is shown in the Table 1. The controller is based on a PC with Pentium-586 CPU. In order to measure the angle, angular speed, position, and velocity of the manipulator, the interface device was implemented by the motion control card PCI-7344. The graphical software of Simulink and the control software LABVIEW were used to implement the proposed control rule. At the same time, this software also determines the linear converts between the physical scales and measured voltages from sensors.

Table 1 

The characteristic of position limiter sensor.

A block diagram of the overall system with the controller incorporating HDRAC with MPSO and acceleration feedback is illustrated in Figure 8. The experimental instrument of the 3RPS parallel manipulator was divided into three parts: an actuator, parallel manipulator, and controller. The actuator part consisted of a DC motor and a driver. The parameters of the DC motor were 𝐾_𝑡=0.21 Nm/A, 𝐽_𝑚=0.000037 Nms², 𝐵_𝑚=0.000013 Nms/rad, 𝑙_𝑎=2.4 mH, and 𝑟_𝑎=4.8 Ω.

Figure 8 

Block diagram of the controller incorporating HDRAC with MPSO and acceleration feedback.

6.3. Experimental Results

Figures 9(a)–9(f) show the experimental result of trajectory and velocity along the 𝑋-axis, 𝑌-axis, and 𝑍-axis with the DARAC controller that was obtained by using the 3RPS parallel manipulator, in which the end effector passed through a singular point. The DARAC controller functioned, such that joint acceleration along the degenerated direction was damped when the end effector was close to the singular point. It was found that when the manipulator approached the singular point, the system was unstable for the DARAC controller and displayed great fluctuations.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 9 

Experimental results for the controller incorporating DARAC with fixed damping values. (a) Translation responses along the 𝑋-axis. (b) Velocities along the 𝑋-axis. (c) Translation responses along the 𝑌-axis. (d) Velocities along the 𝑌-axis. (e) Translation responses along the 𝑍-axis. (f) Velocities along the 𝑍-axis.

Figures 10(a)–10(f) show the experimental result of trajectory and velocity along the 𝑋-axis, 𝑌-axis, and 𝑍-axis for the controller incorporating HDRAC with fixed damping values and for the controller incorporating HDRAC with MPSO. It was found that, by using the controller incorporating HDRAC with MPSO, a very good performance could be achieved under the same conditions and that the system was generally stable and had very slight fluctuations. These experimental results demonstrate that the HDRAC overcomes the problem of fluctuation when the manipulator arrives at the singular point. It also proves that the proposed controller incorporating HDRAC with MPSO has great robustness.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 10 

Experimental results for the controller incorporating HDRAC with fixed damping values and the controller incorporating HDRAC with MPSO. (a) Translation responses along the 𝑋-axis. (b) Velocities along the 𝑋-axis. (c) Translation responses along the 𝑌-axis. (d) Velocities along the 𝑌-axis. (e) Translation responses along the 𝑍-axis. (f) Velocities along the 𝑍-axis.

7. Conclusion

This paper proposes a simple HDRAC scheme with an MPSO method, which has the merits of offering a high-quality solution and better computation efficiency in the neighborhood of a singular point. The proposed method directly damps velocities and accelerations, such that unnecessary joint velocities are removed at a singular point. The proposed HDRAC scheme with MPSO control is asymptotically stable and remedies the problem of fluctuation.

In this study, HDRAC with MPSO is used to solve the damping factors 𝜌 by the optimization method. The damping factors 𝜌 are not fixed values for on-line applications, and the optimization method can immediately provide optimal damping factors. Because it shortens the detection processes, the MPSO method can satisfactorily predict optimal damping values.

Comparisons between our numerical simulations and experimental results showed that the proposed method offers greater stability and efficiency and solves the problems of instability and fluctuation more easily and quickly. The main advantage of using a controller incorporating HDRAC with MPSO is that the 3RPS parallel manipulator control system does not need to plan its path to avoid any singular point, because the controller will command the manipulator to move near the singular point with minimum trajectory errors. Moreover, in comparison with a controller incorporating HDRAC with fixed damping values, the proposed approach makes it easier and simpler to improve the performance of the motion in the neighborhood of singular points.

Appendices

A.

The items contained in matrix 𝐃 are as follows:𝐷_1𝑖=^𝑝𝑃_𝑖𝑥𝑅_𝑝𝑖21−^𝑝𝑃_𝑖𝑦𝑅_𝑝𝑖11,𝐷_2𝑖=−^𝑝𝑃_𝑖𝑥𝑅_𝑝𝑖31,𝐷_3𝑖=^𝑝𝑃_𝑖𝑦𝑅_𝑝𝑖31,𝐷_4𝑖=𝑅_𝑖11,𝐷_5𝑖=𝑅_𝑖21,𝐷_6𝑖=𝑅_𝑖31. (A.1) The items contained in matrix 𝐐 are as follows: 𝑄₁=𝐼_𝑝𝑧𝑧𝜔̇_𝑝𝑧+𝐼_𝑝𝑦𝑦−𝐼_𝑝𝑥𝑥𝜔_𝑝𝑦𝜔_𝑝𝑥−³_𝑖=1^𝑝𝑃_𝑖𝑥𝑅^𝑖_𝑝𝑖22𝐹_𝑝𝑖𝑦+𝑅^𝑖_𝑝𝑖23𝐹_𝑝𝑖𝑧−^𝑝𝑃_𝑖𝑦𝑅^𝑖_𝑝𝑖12𝐹_𝑝𝑖𝑦+𝑅^𝑖_𝑝𝑖13𝐹_𝑝𝑖𝑦+𝑅^𝑖_𝑝𝑖13𝐹_𝑝𝑖𝑧+𝐽_𝑚1𝜔̇_𝑝𝑧+𝐵_𝑚1𝜔_𝑝𝑧−𝐾_𝑇1𝑖_𝑎1−𝐾_𝑇2𝑖_𝑎2,𝑄₂=𝐼_𝑝𝑦𝑦𝜔̇_𝑝𝑦+𝐼_𝑝𝑥𝑥−𝐼_𝑝𝑧𝑧𝜔_𝑝𝑥𝜔_𝑝𝑧+^3𝑝_𝑖=1𝑃_𝑖𝑥𝑅^𝑖_𝑝𝑖32𝐹_𝑝𝑖𝑦+𝑅^𝑖_𝑝𝑖33𝐹_𝑝𝑖𝑧×𝐽_𝑚1𝜔̇_𝑝𝑦+𝐵_𝑚1𝜔_𝑝𝑦−𝐾_𝑇1𝑖_𝑎1−𝐾_𝑇2𝑖_𝑎2,𝑄₃=𝐼_𝑝𝑥𝑥𝜔̇_𝑝𝑥+𝐼_𝑝𝑧𝑧−𝐼_𝑝𝑦𝑦𝜔_𝑝𝑦𝜔_𝑝𝑧−^3𝑝_𝑖=1𝑃_𝑖𝑦𝑅^𝑖_𝑝𝑖32𝐹_𝑝𝑖𝑦+𝑅^𝑖_𝑝𝑖33𝐹_𝑝𝑖𝑧×𝐽_𝑚1𝜔̇_𝑝𝑥+𝐵_𝑚1𝜔_𝑝𝑥−𝐾_𝑇1𝑖_𝑎1−𝐾_𝑇2𝑖_𝑎2,𝑄₄=𝑚_𝑝𝑎_𝑝𝑥−³_𝑖=1𝑅^𝑖_𝑖12𝐹_𝑝𝑖𝑦+𝑅^𝑖_𝑖13𝐹_𝑝𝑖𝑧,𝑄₅=𝑚_𝑝𝑎_𝑝𝑦−³_𝑖=1𝑅^𝑖_𝑖22𝐹_𝑝𝑖𝑦+𝑅^𝑖_𝑖23𝐹_𝑝𝑖𝑧,𝑄₆=𝑚_𝑝𝑎_𝑝𝑧−³_𝑖=1𝑅^𝑖_𝑖32𝐹_𝑝𝑖𝑦+𝑅^𝑖_𝑖33𝐹_𝑝𝑖𝑧+𝑊_𝑝. (A.2)

B.

The items contained in matrix 𝐀 are as follows:𝑎₁₁=2cos𝛼₂−𝛼₁𝑑₁sin𝜃₁+𝑑₁𝑑₂sin𝜃₁cos𝜃₂−𝑑₁sin𝜃₁+𝑑₁𝑑₂sin𝜃₂cos𝜃₁,𝑎₁₂=2cos𝛼₂−𝛼₁𝑑₂sin𝜃₂+𝑑₁𝑑₂sin𝜃₂cos𝜃₁−𝑑₂sin𝜃₂+𝑑₁𝑑₂sin𝜃₁cos𝜃₂,𝑎₂₂=2cos𝛼₃−𝛼₂𝑑₂sin𝜃₂+𝑑₂𝑑₃sin𝜃₂cos𝜃₃−𝑑₂sin𝜃₂+𝑑₂𝑑₃sin𝜃₃cos𝜃₂,𝑎₂₃=2cos𝛼₃−𝛼₂𝑑₃sin𝜃₃+𝑑₂𝑑₃sin𝜃₃cos𝜃₂−𝑑₃sin𝜃₃+𝑑₂𝑑₃sin𝜃₂cos𝜃₃,𝑎₃₁=2cos𝛼₁−𝛼₃𝑑₁sin𝜃₁+𝑑₁𝑑₃sin𝜃₁cos𝜃₃−𝑑₁sin𝜃₁+𝑑₁𝑑₃sin𝜃₃cos𝜃₁,𝑎₃₃=2cos𝛼₁−𝛼₃𝑑₃sin𝜃₃+𝑑₁𝑑₃sin𝜃₃cos𝜃₁−𝑑₃sin𝜃₃+𝑑₁𝑑₃sin𝜃₁cos𝜃₃. (B.1) The items contained in matrix 𝐁 are as follows: 𝑏₁₁=cos𝜃₁𝑑₂cos𝜃₂+1+2𝑑₁+cos𝜃₁−𝑑₂sin𝜃₁sin𝜃₂,𝑏₁₂=cos𝜃₂𝑑₁cos𝜃₁+1+2𝑑₂+cos𝜃₂−𝑑₁sin𝜃₁sin𝜃₂,𝑏₂₂=cos𝜃₂𝑑₃cos𝜃₃+1+2𝑑₂+cos𝜃₂−𝑑₃sin𝜃₂sin𝜃₃,𝑏₂₃=cos𝜃₃𝑑₂cos𝜃₂+1+2𝑑₃+cos𝜃₃−𝑑₂sin𝜃₂sin𝜃₃,𝑏₃₁=cos𝜃₁𝑑₃cos𝜃₃+1+2𝑑₁+cos𝜃₁−𝑑₃sin𝜃₁sin𝜃₃,𝑏₃₃=cos𝜃₃𝑑₁cos𝜃₁+1+2𝑑₃+cos𝜃₃−𝑑₁sin𝜃₁sin𝜃₃. (B.2)

C.

The coefficients are as follows: Ω₁3=−43𝑒₁𝑒₂𝑒₃+𝑒₁𝑒²₂+𝑒²₃+𝑒₂𝑒²₁+𝑒²₃+𝑒₃𝑒²₁+𝑒²₂Ω₂,=34𝑒₁ℎ₁𝑒²₃+𝑒₃−𝑒²₂−𝑒₂+𝑒₂ℎ₂𝑒²₁+𝑒₁−𝑒²₃−𝑒₃+𝑒₃ℎ₃𝑒²₂+𝑒₂−𝑒²₁−𝑒₁,Ω₃=14−ℎ₁ℎ₂+ℎ₂ℎ₃+ℎ₃ℎ₁𝑒₁𝑒₂𝑒₃−𝑒₂−𝑒₃+4+𝑒₂𝑒₁𝑒₃−𝑒₁−𝑒₃+4+𝑒₃𝑒₁𝑒₂−𝑒₁−𝑒₂+4+ℎ₁ℎ₃−ℎ₂𝑒₃𝑒²₂+𝑒₂−2−𝑒₂𝑒²₃+𝑒₃−2+ℎ₂ℎ₁−ℎ₃𝑒₁𝑒²₃+𝑒₃−2−𝑒₃𝑒²₁+𝑒₁−2+ℎ₃ℎ₂−ℎ₁𝑒₂𝑒²₁+𝑒₁−2−𝑒₁𝑒²₂+𝑒₂,Ω−2₄=−169ℎ₁ℎ₂ℎ₂−ℎ₁+ℎ₂ℎ₃ℎ₃−ℎ₂+ℎ₃ℎ₁ℎ₁−ℎ₃×𝑒₁𝑒₂𝑒₃−𝑒₂−𝑒₃+4+𝑒₂𝑒₁𝑒₃−𝑒₁−𝑒₃+4+𝑒₃𝑒₁𝑒₂−𝑒₁−𝑒₂+4+3𝑒₂𝑒₃ℎ₂−ℎ₃²ℎ₃−ℎ₁𝑒₃−1+3𝑒₂𝑒₃ℎ₂−ℎ₃²ℎ₁−ℎ₂𝑒₂−1+3𝑒₁𝑒₃ℎ₃−ℎ₁²ℎ₂−ℎ₃𝑒₃−1+3𝑒₁𝑒₃ℎ₃−ℎ₁²ℎ₁−ℎ₂𝑒₁−1+3𝑒₁𝑒₂ℎ₁−ℎ₂²ℎ₂−ℎ₃𝑒₂−1+3𝑒₁𝑒₂ℎ₁−ℎ₂²ℎ₃−ℎ₁𝑒₁−1. (C.1)(1)