Modelling and Control of Inverse Dynamics for a 5-DOF Parallel Kinematic Polishing Machine

Abstract

Currently, about 60% of the finishing works for freeform parts are done manually, in this paper a parallel kinematic polishing machine (PKPM) with five degrees of freedom (DOFs) and with structural configurations of the type T3R2 (three translations and two rotations) is presented. The PKPM is a variation of the existing METROM Pentapod. The PKPM has a large yaw angle space and, with the aid of an NC rotation table, it can access any given point of the freeform part. A kinematic analysis is developed, an inverse dynamic model based on Kane's equation and the affine projection method is set up in detail, and a real-time and highly efficient inverse dynamic algorithm is developed; the simulation results show that the presented PKPM could achieve an acceleration which is twice that of gravity. A mixed ℋ₂/ℋ _∞ control method is presented and investigated in order to track the error control of the inverse dynamic model; the simulation results from different conditions show that the mixed ℋ₂/ℋ _∞ control method could achieve an optimal and robust control performance. This work shows that the presented PKPM has a higher dynamic performance than conventional machine tools.

1. Introduction

Parts with freeform surfaces, such as blades, impellers, moulds etc., have been popularly applied in modern industries. According to the statistics, 60% of the finishing works for parts with freeform surfaces are done by hand globally. The manual polishing process is not satisfactory for the rapid development of modern industry due to its instability, unreliability, inefficiency and high cost. High precision and flexibility are the most important avenues to explore for making the whole polishing industry more efficient, stable and reliable. High precision is the basis for the efficient implementation of the finishing process and flexibility ensures the finishing processes are optimized and their stability and reliability are controlled. Therefore, the development of an automatic polishing machine is very important to the manufacturing industry.

Recently, lots of research has focused on the automatic polishing process. An XYZ Desktop NC machine tool was designed by Fusaomi Nagata to polish the PET bottle mould [1]. A 5-DOF polishing machine, consisting of an XYZ desktop NC machine table and a two-axis polishing robot, was developed to carry out the polishing task for freeform surfaces [2,3]. So far, many automatic polishing systems have been developed based on open architectural industrial robots [4,5]; both the dynamic performance and the precision of the industrial robot are quite poor due to its open-chain architecture.

The parallel manipulators are well known for their high speed, high structural rigidity and high precision etc. Parallel kinematic machines based on parallel manipulators begun to be used increasingly as innovative machining equipment over the last ten years, particularly for freeform parts. Much research focusing on PKMs, such as Delta, Tricept and METROM Pentapod, can be found in the literature [6–11]. Y. Hu and B. Li used a robust design method to propose a hybrid kinematic machine with five DOFs based on the structural configuration of a 4PUS-1RPU parallel mechanism; both the kinematic and dynamic performance indices of the proposed mechanism were carried out [12]. H. Huang, B. Li et al. proposed a novel adaptive parallel manipulator with six DOFs and with a large tilting capacity [13]. The manipulator consists of four identical peripheral limbs and one doubly actuated centre limb. Due to the special architecture, the doubly actuated centre limb could have infinite inverse solutions. In every configuration of the end-effector, the manipulator can adapt its centre limb to the position and orientation with the best dexterity. B. Li, X. Yang et al. analysed the inverse kinematics, dexterity and developed a prototype for a proposed 5-DOF hybrid parallel kinematic machine [14]. Y. Li and Q. Xu introduced the passivity-based robust control method into the tracking control of the PKM with parametric uncertainties [15].

The PKM-based polishing system has the characteristics of high flexibility, lower inertia, high stiffness performance and ease of control in the polishing process. The developed polishing system will be one of the most effective ways to achieve the goals with high precision and flexibility. D. Zhang designed a 3-DOF parallel manipulator with a small tilting angle and used an XY table to finish the freeform surface [16]. J. Zhao and M. Yu developed a hybrid polishing kinematics machine tool for freeform surface finishing [17, 18]. Liao Liang et al. developed a dual-purpose compliant tool-head for PKM for polishing and deburring [19].

In robotics technology, inverse dynamics modelling is one of the most important aspects for controlling a robot with high position precision [20, 21]. The classical computed torque is widely used in industrial robots for its easy realization and real-time control, the computed torque method with static gain is applied to the parallel manipulator to obtain the real-time tracking controller [22]. B. Achili et al. used the coupling of sliding modes and multi-layer perception neural networks to control the trajectory tracking of a C5 parallel robot without an inverse dynamic model [23]. H. Abdellatif et al. used a sliding mode to design the six DOFs parallel robot controller [24]. A robust ℋ _∞ method for tracking error control with disturbance is discussed in comparison to common robotic dynamic control [25, 26].

In this paper a parallel kinematic polishing machine with the motion type of T3R2 and a redundant linear motion actuator in the moving platform is proposed, the modelling, and an analysis of the kinematics and the inverse dynamics of the PKPM are investigated in detail.

This paper is organized as follows: the structural scheme of the parallel kinematic polishing machine (PKPM) is presented in section 2; then the kinematic modelling is developed and a dexterous workspace is obtained in section 3; in section 4 the inverse dynamics of the PKPM based on Kane's equation is developed, and the efficient inverse dynamics algorithm based on Kane's method is presented; in section 5 the mixed ℋ₂/ℋ _∞ control method is introduced to the PKPM dynamic tracking error control and the performance of the mixed ℋ₂/ℋ _∞ control method is discussed using simulations; the last section contains the conclusions.

2. Mechanism scheme

In this paper a mechanism scheme for the PKPM with a structural configuration of 4URHU-1URHR and a redundant translation is presented. The PKPM is shown in Fig. 1(a), from the figure we can see that the presented PKPM is composed of a moving platform, a fixed platform and five kinematic chains which connect the moving platform and the fixed platform. In order to control the polishing force during the polishing process, the linear motion actuator in the moving platform of the PKPM is introduced to carry a spindle with the polishing tool attached. In addition, a numerical control (NC) rotary table is located in the middle of the fixed platform. The spindle with the polishing tool attached is installed on the moving platform.

Figure 1.

PKPM and the illustrative diagram

The illustrative diagram of the proposed PKPM is shown in Fig. 1(b). The PKM is composed of five kinematic chains, the central kinematic chain L1 consists of a universal joint U, a revolute joint R, a helical joint H and a revolute joint R, the chain can be denoted as URHR and it has five DOFs. The chains of L2, L3, L4 and L5 are four identical 6-DOF kinematic chains denoted by URHU. According to screw theory, the DOF of the moving platform is the same as the central kinematic chain L1, which means that the PKM can provide five DOFs with three translations and two rotations. The actuator installed on the moving platform of the PKM provides the redundant linear motion.

A 3D model of the PKPM can be found in Fig. 2(a). The detailed design of the moving platform is shown in Fig. 2(b), the force sensor and the linear motor are mounted between the moving platform and the spindle, which are used to control the polishing force in the polishing process in order to guarantee that a high surface finish will be obtained. This paper will focus on the modelling and analysis of the dynamic performance of the PKPM in the polishing process.

A famous PKM, the METROM Pentapod, was proposed in 2002 [9]. The PKM structure of the PKPM in this paper is a variation of the METROM Pentapod. The differences between the METROM Pentapod and the PKPM can be described as follows. For the METROM Pentapod, the revolute joints of the kinematic chains connected to the moving platform are in a special configuration, the axes of the revolute joints of the four 6-DOF kinematic chains are collinear with the axis of the spindle, and the axis of revolute joint of the central 5-DOF kinematic chain is perpendicular to the spindle. This special arrangement means that the cutting moment along the axis of the spindle acting on the moving platform will only be carried out by the central kinematic chain, but the other four kinematic chains are not devoted to bearing the cutting moment, so the METROM Pentapod has a relatively low moment-load capability, so the METROM Pentapod is not very suitable for cases where a high cutting moment is carried.

Figure 2.

3D model of PKPM and local geometry of the moving platform

The presented PKPM contains two parts: the 5-DOF parallel kinematic mechanism and the linear motion actuator, so the PKPM is the redundant parallel kinematic machine with six driving motors, the structural configuration of the 5-DOF end-effector is T3R2, and the redundant linear translation motion is used to control the polishing force. In order to enlarge the moment-load capability and the stiffness, a variation of the METROM

Pentapod, the PKPM, is developed. The axes of the revolute joints of all the kinematic chains are distributed in three parallel lines, which are not coplanar and are perpendicular to the axis of the spindle, as shown in Fig. 2(b) and Fig. 3. The axis ${\overrightarrow{e}}_{26}$ and ${\overrightarrow{e}}_{36}$ of the kinematic chains L₂ and L₃ are collinear; the axis ${\overrightarrow{e}}_{46}$ and ${\overrightarrow{e}}_{56}$ of the kinematic chains L4 and L5 are collinear; and the axis ${\overrightarrow{e}}_{15}$ of the central kinematic chain L1 is in the third line. No matter what the position and orientation of the moving platform is, this configuration can ensure that all the kinematic chains bear the cutting force and moment, so the PKPM has a higher stiffness and a higher load capability than those of the METROM Pentapod.

Figure 3.

General notations of the kinematic chain Li

3. Kinematic analysis of the PKPM

3.1 Parametric definitions and coordinate systems

As shown in Fig.3, let P denote a point of a moving platform, it is also the centre point of the revolute joint R at the end of kinematic chain L₁, the coordinates of P are x_p, y_p, z_p; ϑ denotes the rotation angle of the moving platform along the axis ${\overrightarrow{e}}_{14}$ of the kinematic chain L₁; φ denotes the rotation angle of the moving platform along the axis ${\overrightarrow{e}}_{15}$ . To unify the formula for all the kinematic chains, the virtual axis ${\overrightarrow{e}}_{16}$ is introduced to the moving platform with the rotation angle θ₁₆ = 0. In the task space the variables x_p, y_p, z_p, ϑ,φ are regarded as generalized variables of the PKPM.

According to the definitions, the generalized variables ϑ and φ are not defined in the Cartesian coordinate system, the orientation of the moving platform depends not only on ϑ and φ, but also depends on the position (x_p,y_p,z_p) of the point P in other words, the orientation of the moving platform is coupled with x_p, y_p, z_p, ϑ, φ.

Due to the dynamic equation being set up in the Cartesian coordinate system, it is necessary to transform the velocity and acceleration of the generalized variables into linear/angular velocity and linear/angular acceleration. For convenience of notation, B _ij represents the j-th body of the i-th kinematic chain, where B_i1 is the first rotating joint of the universal joint; B_i2 is the stator of the hollow-shaft motor (fixed to the second rotating joint of the universal joint); B_i3 is the rotor of the hollow-shaft motor; B_i4 is the ball-screw; B_i5 is the Cardan-member (moving platform when i=1).

To obtain the velocity and acceleration of the moving platform described in the Cartesian coordinate system by the generalized variables, the intermediate variables θ₁₁, θ₁₂, θ₁₃, θ₁₄, θ₁₅, which denote the joint angles of the central kinematic chain, are introduced to simplify the relationship between the velocity, the acceleration of the moving platform and the generalized variables.

3.2 Kinematic analysis of the central kinematic chain

Note that A₁ (x_A1, y_A1, z_A1) denotes the central point of the universal joint of the central kinematic chain; it is a fixed point in relation to the global coordinate system. Let δ_x=x_A1 – x_p, δ_y = y_A1 – y_p and δ_z=z_A1 – z_p′ with the supplementary equations of l = K · (θ₁₃ – ϑ) and $l=\sqrt{{\delta }_x^2+{\delta }_y^2+{\delta }_z^2}$ , we can obtain the following equation,

$\{\begin{array}{c}{\theta }_{11}=\text{arctan}(-{\delta }_y,{\delta }_z)\\ {\theta }_{12}=\arcsin (\frac{{\delta }_x}{\sqrt{{\delta }_x^2+{\delta }_y^2+{\delta }_z^2}})\\ {\theta }_{13}=\frac{\sqrt{{\delta }_x^2+{\delta }_y^2+{\delta }_z^2}}{K}+\vartheta \\ {\theta }_{14}=\vartheta \\ {\theta }_{15}=\phi \end{array}$

(1)

where K= P/2π, P is the screw pitch, l·cosθ₁₂ > 0 ⇒ θ₁₂ ∈ (–90°, 90°) and θ₁₁ ∈ (–90°, 90°) Eq. (1) shows that x_p, y_p, z_p only affect θ₁₁, θ₁₂ and θ₁₃, and θ ₁₁ and θ₁₂ rather than θ₁₃ will decide the orientation of the ball-screw The transformation matrix of the moving platform relative to x_p,y_p,z_p,ϑ and φ can be written as,

$R_p(x_p,y_p,z_p,\vartheta ,\phi )=\left[\begin{array}{cc}{R}_s& P\\ 0& 1\end{array}\right]$

(2)

where R_s = R_x(θ₁₁) · Ry (θ₁₂) · R_z(ϑ) · R_x(φ); R_x(θ₁₁), R_y(θ₁₂), R_z(ϑ) and R_x(φ) denote the rotation matrices of their corresponding rotation axis, and $\overrightarrow{P}$ denotes the vector of point D₁ in the moving platform.

For a given vector $\overrightarrow{v}={\left[v_x,v_y,v_z\right]}^{\mathrm{T}}$ , an operator v̂ is introduced to simplify the calculation

$\widehat{v}=\left[\begin{array}{ccc}0& -v_z& v_y\\ v_z& 0& -v_x\\ -v_y& v_x& 0\end{array}\right]$

So the angular velocity of the moving platform can be expressed as,

${\widehat{\mathit{\omega }}}_p={\stackrel{.}{R}}_s{R}_s^T$

(3)

where

$\begin{array}{l}{\stackrel{.}{R}}_s={\stackrel{.}{R}}_x(\frac{\pi }{2})\cdot R_x({\theta }_{11})\cdot R_y({\theta }_{12})\cdot R_z(\vartheta )\cdot R_x(\phi )\cdot {\stackrel{.}{\theta }}_{11}+R_x({\theta }_{11})\cdot {\stackrel{.}{R}}_y(\frac{\pi }{2})\cdot R_y({\theta }_{12})\cdot R_z(\vartheta )\cdot R_x(\phi )\cdot {\stackrel{.}{\theta }}_{12}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+R_x({\theta }_{11})\cdot R_y({\theta }_{12})\cdot {\stackrel{.}{R}}_z(\frac{\pi }{2})\cdot R_z(\vartheta )\cdot R_x(\phi )\cdot \stackrel{.}{\vartheta }+R_x({\theta }_{11})\cdot R_y({\theta }_{12})\cdot R_z(\vartheta )\cdot {\stackrel{.}{R}}_x(\frac{\pi }{2})\cdot R_x(\phi )\cdot \stackrel{.}{\phi }\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}={\stackrel{.}{R}}_x(\frac{\pi }{2})\cdot R_s\cdot {\stackrel{.}{\theta }}_{11}+R_x({\theta }_{11})\cdot {\stackrel{.}{R}}_y(\frac{\pi }{2})\cdot R_x^{\mathrm{T}}({\theta }_{11})\cdot R_s\cdot {\stackrel{.}{\theta }}_{12}+R_s\cdot R_x^{\mathrm{T}}(\phi )\cdot {\stackrel{.}{R}}_z(\frac{\pi }{2})\cdot R_x(\phi )\cdot \stackrel{.}{\vartheta }+R_s\cdot {\stackrel{.}{R}}_x(\frac{\pi }{2})\cdot \stackrel{.}{\phi }\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=C_1\cdot {\stackrel{.}{\theta }}_{11}+C_2\cdot {\stackrel{.}{\theta }}_{12}+C_3\cdot \stackrel{.}{\vartheta }+C_4\cdot \stackrel{.}{\phi }\end{array}$

The angular acceleration of the moving platform can be written as,

${\widehat{\mathit{\epsilon }}}_p={\ddot{R}}_s{R}_s^T+{\stackrel{.}{R}}_s{\stackrel{.}{R}}_s^T$

(4)

where

$\begin{array}{l}{\ddot{R}}_s={\stackrel{.}{R}}_x(\frac{\pi }{2})\cdot R_s\cdot {\ddot{\theta }}_{11}+R_x({\theta }_{11})\cdot {\stackrel{.}{R}}_y(\frac{\pi }{2})\cdot R_x^{\mathrm{T}}({\theta }_{11})\cdot R_s\cdot {\ddot{\theta }}_{12}+R_s\cdot R_x^{\mathrm{T}}(\phi )\cdot {\stackrel{.}{R}}_z(\frac{\pi }{2})\cdot R_x(\phi )\cdot \ddot{\vartheta }+R_s\cdot {\stackrel{.}{R}}_x(\frac{\pi }{2})\cdot \ddot{\phi }\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\left(R_x({\theta }_{11})\cdot \left({\stackrel{.}{R}}_x(\frac{\pi }{2}){\stackrel{.}{R}}_y(\frac{\pi }{2})+{\stackrel{.}{R}}_y(\frac{\pi }{2}){\stackrel{.}{R}}_x^{\mathrm{T}}(\frac{\pi }{2})\right)\cdot R_x^{\mathrm{T}}({\theta }_{11})\cdot R_s\cdot {\stackrel{.}{\theta }}_{11}+R_s+R_x({\theta }_{11})\cdot {\stackrel{.}{R}}_y(\frac{\pi }{2})\cdot R_x^{\mathrm{T}}({\theta }_{11})\cdot {\stackrel{.}{R}}_s\right)\cdot {\stackrel{.}{\theta }}_2\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\left(R_s\cdot R_x(\phi )\cdot \left({\stackrel{.}{R}}_x(\frac{\pi }{2}){\stackrel{.}{R}}_z(\frac{\pi }{2})+{\stackrel{.}{R}}_z(\frac{\pi }{2}){\stackrel{.}{R}}_x^{\mathrm{T}}(\frac{\pi }{2})\right)\cdot R_x^{\mathrm{T}}(\phi )\cdot \stackrel{.}{\phi }+R_s+{\stackrel{.}{R}}_s\cdot R_x(\phi )\cdot {\stackrel{.}{R}}_z(\frac{\pi }{2})\cdot R_x^{\mathrm{T}}(\phi )\right)\cdot \stackrel{.}{\vartheta }\hspace{0.17em}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\left(R_s+{\stackrel{.}{R}}_x(\frac{\pi }{2})\cdot {\stackrel{.}{R}}_s\right)\cdot {\stackrel{.}{\theta }}_{11}+\left({\stackrel{.}{R}}_s\cdot {\stackrel{.}{R}}_x(\frac{\pi }{2})+R_s\right)\cdot \stackrel{.}{\phi }\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=C_1\cdot {\ddot{\theta }}_{11}+C_2\cdot {\ddot{\theta }}_{12}+C_3\cdot \ddot{\vartheta }+C_4\cdot \ddot{\phi }+C_{11}\cdot {\stackrel{.}{\theta }}_{11}\cdot {\stackrel{.}{\theta }}_{11}+C_{12}\cdot {\stackrel{.}{\theta }}_{11}\cdot {\stackrel{.}{\theta }}_{12}+C_{13}\cdot {\stackrel{.}{\theta }}_{11}\cdot \stackrel{.}{\vartheta }+C_{14}\cdot {\stackrel{.}{\theta }}_{11}\cdot \stackrel{.}{\phi }\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+C_{22}\cdot {\stackrel{.}{\theta }}_{12}\cdot {\stackrel{.}{\theta }}_{12}+C_{23}\cdot {\stackrel{.}{\theta }}_{12}\cdot \stackrel{.}{\vartheta }+C_{24}\cdot {\stackrel{.}{\theta }}_{12}\cdot \stackrel{.}{\phi }+C_{33}\cdot \stackrel{.}{\vartheta }\cdot \stackrel{.}{\vartheta }+C_{34}\cdot \stackrel{.}{\vartheta }\cdot \stackrel{.}{\phi }+C_{44}\cdot \stackrel{.}{\phi }\cdot \stackrel{.}{\phi }\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+R_s\cdot {\stackrel{.}{\theta }}_{11}+R_s\cdot {\stackrel{.}{\theta }}_{12}+R_s\cdot \stackrel{.}{\vartheta }+R_s\cdot \stackrel{.}{\phi }\\ {\stackrel{.}{R}}_s{\stackrel{.}{R}}_s^T=\left(C_1\cdot {\stackrel{.}{\theta }}_{11}+C_2\cdot {\stackrel{.}{\theta }}_{12}+C_3\cdot \stackrel{.}{\vartheta }+C_4\cdot \stackrel{.}{\phi }\right)\cdot {\left(C_1\cdot {\stackrel{.}{\theta }}_{11}+C_2\cdot {\stackrel{.}{\theta }}_{12}+C_3\cdot \stackrel{.}{\vartheta }+C_4\cdot \stackrel{.}{\phi }\right)}^{\mathrm{T}}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}=C_1{C}_1^{\mathrm{T}}\cdot {\stackrel{.}{\theta }}_{11}\cdot {\stackrel{.}{\theta }}_{11}+C_1{C}_2^{\mathrm{T}}\cdot {\stackrel{.}{\theta }}_{11}\cdot {\stackrel{.}{\theta }}_{12}+C_1{C}_3^{\mathrm{T}}\cdot {\stackrel{.}{\theta }}_{11}\cdot \stackrel{.}{\vartheta }+C_1{C}_4^{\mathrm{T}}\cdot {\stackrel{.}{\theta }}_{11}\cdot \stackrel{.}{\phi }\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\cdots +\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\hspace{0.17em}{C}_4{C}_1^{\mathrm{T}}\cdot \stackrel{.}{\phi }\cdot {\stackrel{.}{\theta }}_{11}+C_4{C}_2^{\mathrm{T}}\cdot \stackrel{.}{\phi }\cdot {\stackrel{.}{\theta }}_{12}+C_4{C}_3^{\mathrm{T}}\cdot \stackrel{.}{\phi }\cdot \stackrel{.}{\vartheta }+C_4{C}_4^{\mathrm{T}}\cdot \stackrel{.}{\phi }\cdot \stackrel{.}{\phi }\end{array}$

According to Eq. (1) the velocity and the acceleration of θ₁₁ and θ₁₂ can be obtained,

${\stackrel{.}{\theta }}_{11}=\frac{{\delta }_z{\stackrel{.}{y}}_p-{\delta }_y{\stackrel{.}{z}}_p}{{\delta }_y^2+{\delta }_z^2}$

(5)

${\ddot{\theta }}_{11}=\frac{\left({\delta }_z^3+{\delta }_z{\delta }_y^2\right){\ddot{y}}_p-\left({\delta }_z^2{\delta }_y+{\delta }_y^3\right){\ddot{z}}_p+2\left({\delta }_z^2-{\delta }_y^2\right){\stackrel{.}{y}}_p{\stackrel{.}{z}}_p+2{\delta }_z{\delta }_y\left({\stackrel{.}{y}}_p^2-{\stackrel{.}{z}}_p^2\right)}{{\left({\delta }_y^2+{\delta }_z^2\right)}^2}$

(6)

${\stackrel{.}{\theta }}_{12}=\frac{-\left({\delta }_y^2+{\delta }_z^2\right){\stackrel{.}{x}}_p+{\delta }_x{\delta }_y{\stackrel{.}{y}}_p+{\delta }_x{\delta }_z{\stackrel{.}{z}}_p}{\left({\delta }_x^2+{\delta }_y^2+{\delta }_z^2\right)\sqrt{{\delta }_y^2+{\delta }_z^2}}$

(7)

${\ddot{\theta }}_{12}={\mathrm{C}}_{{\ddot{x}}_p}{\ddot{x}}_p+{\mathrm{C}}_{{\ddot{y}}_p}{\ddot{y}}_p+{\mathrm{C}}_{{\ddot{z}}_p}{\ddot{z}}_p+{\mathrm{C}}_{{\stackrel{.}{x}}_p^2}{\stackrel{.}{x}}_p^2+{\mathrm{C}}_{{\stackrel{.}{y}}_p^2}{\stackrel{.}{y}}_p^2+{\mathrm{C}}_{{\stackrel{.}{z}}_p^2}{\stackrel{.}{z}}_p^2+{\mathrm{C}}_{{\stackrel{.}{x}}_p{\stackrel{.}{y}}_p}{\stackrel{.}{x}}_p{\stackrel{.}{y}}_p+{\mathrm{C}}_{{\stackrel{.}{x}}_p{\stackrel{.}{z}}_p}{\stackrel{.}{x}}_p{\stackrel{.}{z}}_p+{\mathrm{C}}_{{\stackrel{.}{y}}_p{\stackrel{.}{z}}_p}{\stackrel{.}{y}}_p{\stackrel{.}{z}}_p$

(8)

where $C_{den}={\left({\delta }_x^2+{\delta }_y^2+{\delta }_z^2\right)}^2{\left({\delta }_y^2+{\delta }_z^2\right)}^{3/2}$

${\mathrm{C}}_{{\ddot{x}}_p}=-\left(3\hspace{0.17em}{\delta }_y^4{\delta }_z^2+{\delta }_z^4{\delta }_x^2+3\hspace{0.17em}{\delta }_y^2{\delta }_z^4+{\delta }_y^4{\delta }_x^2+2\hspace{0.17em}{\delta }_y^2{\delta }_x^2{\delta }_z^2+{\delta }_y^6+{\delta }_z^6\right)/{\mathrm{C}}_{den}$

$\begin{array}{l}{\mathrm{C}}_{{\ddot{y}}_p}=\left(2\hspace{0.17em}{\delta }_x{\delta }_y^3{\delta }_z^2+{\delta }_x^3{\delta }_y^3+{\delta }_x{\delta }_y^5+{\delta }_x^3{\delta }_y{\delta }_z^2+{\delta }_x{\delta }_y{\delta }_z^4\right)/{\mathrm{C}}_{den}\\ {\mathrm{C}}_{{\ddot{z}}_p}=\left({\delta }_x{\delta }_z{\delta }_y^4+2\hspace{0.17em}{\delta }_x{\delta }_z^3{\delta }_y^2+{\delta }_x^3{\delta }_z{\delta }_y^2+{\delta }_x{\delta }_z^5+{\delta }_x^3{\delta }_z^3\right)/{\mathrm{C}}_{den}\\ {\mathrm{C}}_{{\stackrel{.}{x}}_p^2}=-\left(2\hspace{0.17em}{\delta }_x{\delta }_y^4+2\hspace{0.17em}{\delta }_x{\delta }_z^4+4\hspace{0.17em}{\delta }_x{\delta }_y^2{\delta }_z^2\right)/{\mathrm{C}}_{den}\\ {\mathrm{C}}_{{\stackrel{.}{y}}_p^2}=\left(2\hspace{0.17em}{\delta }_x{\delta }_y^4+{\delta }_x{\delta }_y^2{\delta }_z^2-{\delta }_x{\delta }_z^4-{\delta }_x^3{\delta }_z^2\right)/{\mathrm{C}}_{den}\\ {\mathrm{C}}_{{\stackrel{.}{z}}_p^2}=\left({\delta }_x{\delta }_y^2{\delta }_z^2+2\hspace{0.17em}{\delta }_x{\delta }_z^4-{\delta }_x{\delta }_y^4-{\delta }_x^3{\delta }_y^2\right)/{\mathrm{C}}_{den}\\ {\mathrm{C}}_{{\stackrel{.}{x}}_p{\stackrel{.}{y}}_p}=\left(2\hspace{0.17em}{\delta }_y{\delta }_z^2{\delta }_x^2+2\hspace{0.17em}{\delta }_y^3{\delta }_x^2-2\hspace{0.17em}{\delta }_y^5-2\hspace{0.17em}{\delta }_y{\delta }_z^4-4\hspace{0.17em}{\delta }_y^3{\delta }_z^2\right)/{\mathrm{C}}_{den}\\ {\mathrm{C}}_{{\stackrel{.}{x}}_p{\stackrel{.}{z}}_p}=\left(2\hspace{0.17em}{\delta }_z{\delta }_y^2{\delta }_x^2+2\hspace{0.17em}{\delta }_z^3{\delta }_x^2-4\hspace{0.17em}{\delta }_z^3{\delta }_y^2-2\hspace{0.17em}{\delta }_z^5-2\hspace{0.17em}{\delta }_z{\delta }_y^4\right)/{\mathrm{C}}_{den}\\ {\mathrm{C}}_{{\stackrel{.}{y}}_p{\stackrel{.}{z}}_p}=2\hspace{0.17em}{\delta }_x^3{\delta }_y{\delta }_z+6\hspace{0.17em}{\delta }_x{\delta }_y^3{\delta }_z+6\hspace{0.17em}{\delta }_x{\delta }_y{\delta }_z^3/{\mathrm{C}}_{den}\end{array}$

Combining Eq. (3), Eq. (5) and Eq. (7), the angular velocity of the moving platform can be expressed as,

${\overrightarrow{\mathit{\omega }}}_p=W_p\cdot \stackrel{.}{\overrightarrow{q}}$

(9)

The linear and angular velocity of the moving platform can be written as,

$\left[\begin{array}{l}{\overrightarrow{v}}_p\\ {\overrightarrow{\mathit{\omega }}}_p\end{array}\right]=N\cdot {\left[\begin{array}{ccccc}{\stackrel{.}{x}}_p& {\stackrel{.}{y}}_p& {\stackrel{.}{z}}_p& \stackrel{.}{\vartheta }& \stackrel{.}{\phi }\end{array}\right]}^{\mathrm{T}}=N\cdot \stackrel{.}{\overrightarrow{q}}$

(10)

where ${[N]}_{6\times 5}=\left[\begin{array}{l}{I}_{3\times 3}\hspace{0.17em}\hspace{0.17em}{0}_{3\times 3}\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{W}_p\end{array}\right]$ , $\stackrel{.}{\overrightarrow{q}}={\left[\begin{array}{ccc}{\overrightarrow{v}}_p^{\mathrm{T}}& \stackrel{.}{\vartheta }& \stackrel{.}{\phi }\end{array}\right]}^{\mathrm{T}}$ .

The angular acceleration ${\overrightarrow{\mathit{\epsilon }}}_p$ of the moving platform can be obtained by Eqs. (4)–(8).

${\overrightarrow{\mathit{\epsilon }}}_p=M_p\ddot{\overrightarrow{q}}+{\stackrel{.}{\overrightarrow{q}}}^{\mathrm{T}}{W}_p\stackrel{.}{\overrightarrow{q}}$

(11)

3.3 Velocity analysis of the kinematic mode

Fig. 4 is a diagram of the kinematic chain L _i (i=2, 3, 4 or 5) for the closure vector modelling with the centre kinematic chain L₁. The closure equation of the i-th kinematic chain is

$\overrightarrow{{\text{OA}}_i}+\overrightarrow{{\mathrm{A}}_i{\mathrm{D}}_i}=\overrightarrow{\text{OP}}+\overrightarrow{{\text{PD}}_i}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(i=2,3,4,5)$

(12)

Figure 4.

Geometric model of the i-th kinematic chain

Let $\overrightarrow{{\mathrm{A}}_i{\mathrm{D}}_i}=-l_i{\overrightarrow{e}}_{i4},\overrightarrow{{\text{PD}}_i}={\overrightarrow{r}}_{bi}$ , where ${\overrightarrow{e}}_{i4}$ is the unit vector along $\overrightarrow{{\mathrm{D}}_i{\mathrm{A}}_i}$ .

The first-order derivative of Eq. (12) with respect to t can be written as,

$-{\overrightarrow{\mathit{\omega }}}_{i4}\times \left(l_i{\overrightarrow{e}}_{i4}\right)-{\stackrel{.}{l}}_i{\overrightarrow{e}}_{i4}={\overrightarrow{v}}_p+{\overrightarrow{\mathit{\omega }}}_p\times {\overrightarrow{r}}_{bi}$

(13)

In Eq. (13) ${\overrightarrow{\mathit{\omega }}}_{i4}$ is the angular velocity of component B_i4 relative to the global coordinate system; l_i is the extension speed of the i-th kinematic chain; ${\overrightarrow{v}}_p$ is the velocity of point P fixed on the moving platform relative to the global coordinate system; ${\overrightarrow{\mathit{\omega }}}_p$ is the angular velocity of the moving platform relative to the global coordinate system. The angular velocity ${\overrightarrow{\mathit{\omega }}}_{i4}$ can be expressed as,

${\overrightarrow{\mathit{\omega }}}_{i4}={\overrightarrow{\mathit{\omega }}}_p+{\overrightarrow{\mathit{\omega }}}_{i4}^p$

(14)

where ${\overrightarrow{\mathit{\omega }}}_{i4}^p$ is the angular velocity of component B_i4 in the i-th chain relative to the moving platform, the ball-screw of the i-th chain and the moving platform are connected by the universal joint with its rotation axis being ${\overrightarrow{e}}_{i5}$ and ${\overrightarrow{e}}_{i6}$ respectively, the direction of the angular velocity ${\overrightarrow{\mathit{\omega }}}_{i4}^p$ relative to the moving platform lies on the plane spanned by ${\overrightarrow{e}}_{i5}$ and ${\overrightarrow{e}}_{i6}$ , substituting Eq. (14) to Eq. (13), we can obtain,

$-{\overrightarrow{\mathit{\omega }}}_{i4}^p\times \left(l_i{\overrightarrow{e}}_{i4}\right)-{\stackrel{.}{l}}_i{\overrightarrow{e}}_{i4}={\overrightarrow{v}}_p+{\overrightarrow{\mathit{\omega }}}_p\times \left({\overrightarrow{r}}_{bi}+l_i{\overrightarrow{e}}_{i4}\right)$

(15)

Take the inner product with ${\overrightarrow{e}}_{i4}$ for both sides of Eq. (15),

${\stackrel{.}{l}}_i=-{\overrightarrow{v}}_p\cdot {\overrightarrow{e}}_{i4}-{\overrightarrow{\mathit{\omega }}}_p\times {\overrightarrow{r}}_{bi}\cdot {\overrightarrow{e}}_{i4}$

(16)

Let ${\overrightarrow{\lambda }}_i=-{\stackrel{.}{l}}_i{\overrightarrow{e}}_{i4}-{\overrightarrow{v}}_p-{\overrightarrow{\mathit{\omega }}}_p\times \left({\overrightarrow{r}}_{bi}+l_i{\overrightarrow{e}}_{i4}\right)$ , we have

${\overrightarrow{\mathit{\omega }}}_{i4}^p\times \left(l_i{\overrightarrow{e}}_{i4}\right)={\overrightarrow{\lambda }}_i$

(17)

where ${\overrightarrow{\mathit{\omega }}}_{i4}^p$ is perpendicular to the vector ${\overrightarrow{\lambda }}_i$ , and ${\overrightarrow{\mathit{\omega }}}_{i4}^p$ lies on the plane spanned by ${\overrightarrow{e}}_{i5}$ and ${\overrightarrow{e}}_{i6}$ , so that ${\overrightarrow{\mathit{\omega }}}_{i4}^p$ is perpendicular to vector ${\overrightarrow{e}}_{i56}={\overrightarrow{e}}_{i5}\times {\overrightarrow{e}}_{i6}$ , so the unit direction vector ${\overrightarrow{n}}_{i4}^p$ of ${\overrightarrow{\mathit{\omega }}}_{i4}^p$ is

${\overrightarrow{\mathit{\omega }}}_{i4}^p=K\cdot {\overrightarrow{n}}_{i4}^p=K\cdot \left({\overrightarrow{\lambda }}_i\times {\overrightarrow{e}}_{i56}\right)$

(18)

where K is a scalar, substituting Eq. (18) to Eq. (17)

$K\cdot \left({\overrightarrow{\lambda }}_i\times {\overrightarrow{e}}_{i56}\right)\times \left(l_i{\overrightarrow{e}}_{i4}\right)={\overrightarrow{\lambda }}_i$

(19)

Eq. (17) implies ${\overrightarrow{e}}_{i4}\cdot {\overrightarrow{\lambda }}_i=0$ . Simplifying the numerator items of Eq. (19), we have,

${\overrightarrow{\lambda }}_i\times {\overrightarrow{e}}_{i56}\times {\overrightarrow{e}}_{i4}=\left({\overrightarrow{e}}_{i4}\cdot {\overrightarrow{\lambda }}_i\right){\overrightarrow{e}}_{i56}-\left({\overrightarrow{e}}_{i4}\cdot {\overrightarrow{e}}_{i56}\right){\overrightarrow{\lambda }}_i=-\left({\overrightarrow{e}}_{i4}\cdot {\overrightarrow{e}}_{i56}\right){\overrightarrow{\lambda }}_i$

(20)

Substituting Eq. (20) to Eq. (19), the angular velocity ${\overrightarrow{\mathit{\omega }}}_{i4}^p$ can be written as,

$K=\frac{-\left|{\overrightarrow{\lambda }}_i\times {\overrightarrow{e}}_{i56}\right|}{l_i\left({\overrightarrow{e}}_{i4}\cdot {\overrightarrow{e}}_{i56}\right)}$

(21)

Combining Eq. (18) and Eq. (21), the vector ${\overrightarrow{\mathit{\omega }}}_{i4}^p$ is obtained,

${\overrightarrow{\mathit{\omega }}}_{i4}^p=K\cdot {\overrightarrow{n}}_{i4}^p=\frac{{\overrightarrow{e}}_{i56}\times {\overrightarrow{\lambda }}_i}{l_i\left({\overrightarrow{e}}_{i4}\cdot {\overrightarrow{e}}_{i56}\right)}=\frac{{\overrightarrow{e}}_{i5}\times {\overrightarrow{e}}_{i6}\times {\overrightarrow{\lambda }}_i}{l_i\left[{\overrightarrow{e}}_{i4},{\overrightarrow{e}}_{i5},{\overrightarrow{e}}_{i6}\right]}$

(22)

[·,·,·] denotes a mixed product of three vectors, the angular velocity of the ball-screw in the global coordinate system can be written as,

${\overrightarrow{\mathit{\omega }}}_{i4}={\overrightarrow{\mathit{\omega }}}_p+{\overrightarrow{\mathit{\omega }}}_{i4}^p={\overrightarrow{\mathit{\omega }}}_p+\frac{{\overrightarrow{e}}_{i5}\times {\overrightarrow{e}}_{i6}\times {\overrightarrow{\lambda }}_i}{l_i\left[{\overrightarrow{e}}_{i4},{\overrightarrow{e}}_{i5},{\overrightarrow{e}}_{i6}\right]}$

(23)

In order to calculate the angular velocities, the affine projection method is introduced to solve the affine coordinates. ${\overrightarrow{\mathit{\omega }}}_{i4}$ can be expressed as a vector {α_i1 α_i2 α_i3} with the affine basis $\left\{\begin{array}{ccc}{\overrightarrow{e}}_{i1}& {\overrightarrow{e}}_{i2}& {\overrightarrow{e}}_{i4}\end{array}\right\}$ , which satisfies ${\overrightarrow{e}}_{i1}\perp {\overrightarrow{e}}_{i2}$ and ${\overrightarrow{e}}_{i2}\perp {\overrightarrow{e}}_{i4}$ , then ${\overrightarrow{\mathit{\epsilon }}}_{i4}={\beta }_{i1}{\overrightarrow{e}}_{i1}+{\beta }_{i2}{\overrightarrow{e}}_{i2}+{\beta }_{i3}{\overrightarrow{e}}_{i4}$ .

Combining ${\overrightarrow{e}}_{i1}\perp {\overrightarrow{e}}_{i2}$ and ${\overrightarrow{e}}_{i2}\perp {\overrightarrow{e}}_{i4}$ , the affine coordinates can be obtained

$\{\begin{array}{l}{\alpha }_{i1}=\frac{e_{i1}^{\mathrm{T}}-e_{i1}^{\mathrm{T}}{e}_{i4}{e}_{i4}^{\mathrm{T}}}{1-{\left(e_{i1}^{\mathrm{T}}{e}_{i4}\right)}^2}{\mathit{\omega }}_{i4}\\ {\alpha }_{i3}=\frac{e_{i4}^{\mathrm{T}}-e_{i4}^{\mathrm{T}}{e}_{i1}{e}_{i1}^{\mathrm{T}}}{1-{\left(e_{i1}^{\mathrm{T}}{e}_{i4}\right)}^2}{\mathit{\omega }}_{i4}\end{array}$

(24)

The angular velocity ${\overrightarrow{\mathit{\omega }}}_{i1},{\overrightarrow{\mathit{\omega }}}_{i2},{\overrightarrow{\mathit{\omega }}}_{i3}$ , and ${\overrightarrow{\mathit{\omega }}}_{i5}$ can be expressed as follows,

$\{\begin{array}{l}{\overrightarrow{\mathit{\omega }}}_{i1}={\alpha }_{i1}{\overrightarrow{e}}_{i1}\\ {\overrightarrow{\mathit{\omega }}}_{i2}={\overrightarrow{\mathit{\omega }}}_{i4}-{\alpha }_{i3}{\overrightarrow{e}}_{i4}\\ {\overrightarrow{\mathit{\omega }}}_{i3}={\overrightarrow{\mathit{\omega }}}_{i4}+\left({\stackrel{.}{l}}_i\cdot {\overrightarrow{e}}_{i3}\right)/k\\ {\overrightarrow{\mathit{\omega }}}_{i5}={\overrightarrow{\mathit{\omega }}}_p+\left({\overrightarrow{\mathit{\omega }}}_{i4}^p\cdot {\overrightarrow{e}}_{i6}\right){\overrightarrow{e}}_{i6}\end{array}$

(25)

3.4 Acceleration analysis of the kinematic model

Take the first-order derivative of Eq. (13) with respect to t, we obtain,

$\begin{array}{l}-{\overrightarrow{\mathit{\epsilon }}}_{i4}\times \left(l_i{\overrightarrow{e}}_{i4}\right)-{\overrightarrow{\mathit{\omega }}}_{i4}\times \left({\overrightarrow{\mathit{\omega }}}_{i4}\times l_i{\overrightarrow{e}}_{i4}+2{\stackrel{.}{l}}_i{\overrightarrow{e}}_{i4}\right)-{\ddot{l}}_i{\overrightarrow{e}}_{i4}=\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}={\overrightarrow{a}}_p+{\overrightarrow{\mathit{\epsilon }}}_p\times {\overrightarrow{r}}_{bi}+{\overrightarrow{\mathit{\omega }}}_p\times \left({\overrightarrow{\mathit{\omega }}}_p\times {\overrightarrow{r}}_{bi}\right)\end{array}$

(26)

where ${\overrightarrow{\mathit{\epsilon }}}_{i4}$ is the angular acceleration of the component B_i4 in a global coordinate system, angular acceleration ${\overrightarrow{\mathit{\epsilon }}}_{i4}$ can be written as,

${\overrightarrow{\mathit{\epsilon }}}_{i4}={\overrightarrow{\mathit{\epsilon }}}_p+{\overrightarrow{\mathit{\epsilon }}}_{i4}^p$

(27)

where ${\overrightarrow{\mathit{\epsilon }}}_{i4}^p$ is the angular acceleration of the component B_i4 relative to the moving platform, ${\overrightarrow{\mathit{\epsilon }}}_{i4}^p$ also lies on the plane spanned by ${\overrightarrow{e}}_{i5}$ and ${\overrightarrow{e}}_{i6}$ , substituting Eq. (27) into Eq. (26) results in

$\begin{array}{l}-{\overrightarrow{\mathit{\epsilon }}}_{i4}^p\times \left(l_i{\overrightarrow{e}}_{i4}\right)-{\overrightarrow{\mathit{\omega }}}_{i4}\times \left({\overrightarrow{\mathit{\omega }}}_{i4}\times l_i{\overrightarrow{e}}_{i4}+2{\stackrel{.}{l}}_i{\overrightarrow{e}}_{i4}\right)-{\ddot{l}}_i{\overrightarrow{e}}_{i4}=\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}={\overrightarrow{a}}_p+{\overrightarrow{\mathit{\epsilon }}}_p\times \left({\overrightarrow{r}}_{bi}+l_i{\overrightarrow{e}}_{i4}\right)+{\overrightarrow{\mathit{\omega }}}_p\times \left({\overrightarrow{\mathit{\omega }}}_p\times {\overrightarrow{r}}_{bi}\right)\end{array}$

(28)

Take the inner product with ${\overrightarrow{e}}_{i4}$ for both sides of Eq. (28),

$\begin{array}{l}{\ddot{l}}_i=-{\overrightarrow{\mathit{\omega }}}_{i4}\times \left({\overrightarrow{\mathit{\omega }}}_{i4}\times l_i{\overrightarrow{e}}_{i4}\right)\cdot {\overrightarrow{e}}_{i4}-{\overrightarrow{a}}_p\cdot {\overrightarrow{e}}_{i4}-\\ \hspace{0.17em}-{\overrightarrow{\mathit{\epsilon }}}_p\times {\overrightarrow{r}}_{bi}\cdot {\overrightarrow{e}}_{i4}-{\overrightarrow{\mathit{\omega }}}_p\times \left({\overrightarrow{\mathit{\omega }}}_p\times {\overrightarrow{r}}_{bi}\right)\cdot {\overrightarrow{e}}_{i4}\end{array}$

(29)

Let

$\begin{array}{l}{\overrightarrow{\zeta }}_i=-{\overrightarrow{\mathit{\omega }}}_{i4}\times \left({\overrightarrow{\mathit{\omega }}}_{i4}\times l_i{\overrightarrow{e}}_{i4}+2{\stackrel{.}{l}}_i{\overrightarrow{e}}_{i4}\right)-{\ddot{l}}_i{\overrightarrow{e}}_{i4}-\\ -{\overrightarrow{a}}_p-{\overrightarrow{\mathit{\epsilon }}}_p\times \left({\overrightarrow{r}}_{bi}+l_i{\overrightarrow{e}}_{i4}\right)-{\overrightarrow{\mathit{\omega }}}_p\times \left({\overrightarrow{\mathit{\omega }}}_p\times {\overrightarrow{r}}_{bi}\right),\end{array}$

Eq. (29) can be written as,

${\overrightarrow{\mathit{\epsilon }}}_{i4}^p\times \left(l_i{\overrightarrow{e}}_{i4}\right)={\overrightarrow{\zeta }}_i$

(30)

where ${\overrightarrow{\mathit{\epsilon }}}_{i4}^p$ is perpendicular to vector ${\overrightarrow{\zeta }}_i$ , and ${\overrightarrow{\mathit{\epsilon }}}_{i4}^p$ is on the plane spanned by ${\overrightarrow{e}}_{i5}$ and ${\overrightarrow{e}}_{i6}$ , then ${\overrightarrow{\mathit{\epsilon }}}_{i4}^p$ is perpendicular to vector ${\overrightarrow{e}}_{i56}={\overrightarrow{e}}_{i5}\times {\overrightarrow{e}}_{i6}$ , so the unit direction vector ${\overrightarrow{\mu }}_{i4}^p$ of ${\overrightarrow{\mathit{\epsilon }}}_{i4}^p$ is

${\overrightarrow{\mathit{\epsilon }}}_{i4}^p=K^{\prime }\cdot {\overrightarrow{\mu }}_{i4}^p=K^{\prime }\cdot \left({\overrightarrow{\zeta }}_i\times {\overrightarrow{e}}_{i56}\right)$

(31)

Substituting Eq. (31) to Eq. (30)

$K^{\prime }\cdot \left({\overrightarrow{\zeta }}_i\times {\overrightarrow{e}}_{i56}\right)\times \left(l_i{\overrightarrow{e}}_{i4}\right)={\overrightarrow{\zeta }}_i$

(32)

Simplifying Eq. (32) we have

$K^{\prime }=\frac{-\left|{\overrightarrow{\zeta }}_i\times {\overrightarrow{e}}_{i56}\right|}{l_i\left({\overrightarrow{e}}_{i4}\cdot {\overrightarrow{e}}_{i56}\right)}$

(33)

The relative angular acceleration ${\overrightarrow{\mathit{\epsilon }}}_{i4}^p$ of the ball-screw B_i4 is

${\overrightarrow{\mathit{\epsilon }}}_{i4}^p=K^{\prime }\cdot {\overrightarrow{\mu }}_{i4}^p=\frac{{\overrightarrow{e}}_{i5}\times {\overrightarrow{e}}_{i6}\times {\overrightarrow{\zeta }}_i}{l_i\left[{\overrightarrow{e}}_{i4},{\overrightarrow{e}}_{i5},{\overrightarrow{e}}_{i6}\right]}$

(34)

We have the angular acceleration of the ball-screw,

${\overrightarrow{\mathit{\epsilon }}}_{i4}={\overrightarrow{\mathit{\epsilon }}}_p+{\overrightarrow{\mathit{\epsilon }}}_{i4}^p={\overrightarrow{\mathit{\epsilon }}}_p+\frac{{\overrightarrow{e}}_{i5}\times {\overrightarrow{e}}_{i6}\times {\overrightarrow{\zeta }}_i}{l_i\left[{\overrightarrow{e}}_{i4},{\overrightarrow{e}}_{i5},{\overrightarrow{e}}_{i6}\right]}$

(35)

In order to calculate the angular velocities, the affine projection method is used to solve the affine coordinates. ${\overrightarrow{\mathit{\epsilon }}}_{i4}$ can be expressed as a vector (β_i1 β_i2 β_i3} with the affine basis $\left\{\begin{array}{ccc}{\overrightarrow{e}}_{i1}& {\overrightarrow{e}}_{i2}& {\overrightarrow{e}}_{i4}\end{array}\right\}$ , which satisfies ${\overrightarrow{e}}_{i1}\perp {\overrightarrow{e}}_{i2}$ and ${\overrightarrow{e}}_{i2}\perp {\overrightarrow{e}}_{i4}$ , then ${\overrightarrow{\mathit{\epsilon }}}_{i4}={\beta }_{i1}{\overrightarrow{e}}_{i1}+{\beta }_{i2}{\overrightarrow{e}}_{i2}+{\beta }_{i3}{\overrightarrow{e}}_{i4}$ .

Combining with ${\overrightarrow{e}}_{i1}\perp {\overrightarrow{e}}_{i2}$ and ${\overrightarrow{e}}_{i2}\perp {\overrightarrow{e}}_{i4}$ , the affine coordinates can be obtained

$\{\begin{array}{l}{\beta }_{i1}=\frac{e_{i1}^{\mathrm{T}}-e_{i1}^{\mathrm{T}}{e}_{i4}{e}_{i4}^{\mathrm{T}}}{1-{\left(e_{i1}^{\mathrm{T}}{e}_{i4}\right)}^2}{\mathit{\epsilon }}_{i4}\\ {\beta }_{i3}=\frac{e_{i4}^{\mathrm{T}}-e_{i4}^{\mathrm{T}}{e}_{i1}{e}_{i1}^{\mathrm{T}}}{1-{\left(e_{i1}^{\mathrm{T}}{e}_{i4}\right)}^2}{\mathit{\epsilon }}_{i4}\end{array}$

(36)

The angular acceleration ${\overrightarrow{\mathit{\epsilon }}}_{i1}$ , ${\overrightarrow{\mathit{\epsilon }}}_{i2}$ , ${\overrightarrow{\mathit{\epsilon }}}_{i3}$ and ${\overrightarrow{\mathit{\epsilon }}}_{i5}$ are

$\{\begin{array}{l}{\overrightarrow{\mathit{\epsilon }}}_{i1}={\beta }_{i1}{\overrightarrow{e}}_{i1}\\ {\overrightarrow{\mathit{\epsilon }}}_{i2}={\overrightarrow{\mathit{\epsilon }}}_{i4}-{\beta }_{i3}{\overrightarrow{e}}_{i4}\\ {\overrightarrow{\mathit{\epsilon }}}_{i3}={\overrightarrow{\mathit{\epsilon }}}_{i4}+{\ddot{l}}_i\cdot {\overrightarrow{e}}_{i3}/k\\ {\overrightarrow{\mathit{\epsilon }}}_{i5}={\overrightarrow{\mathit{\epsilon }}}_p+\left({\overrightarrow{\mathit{\epsilon }}}_{i4}^p\cdot {\overrightarrow{e}}_{i6}\right){\overrightarrow{e}}_{i6}\end{array}$

(37)

3.5 Jacobian matrix and the workspace analysis

${\overrightarrow{\mathit{\omega }}}_{i3}$ is the driving angular velocity of the i-th hollow-shaft motor. From Eq. (25), Eq. (23) and Eq. (10), ${\text{ω}}_{i3}^s$ , the scale of ${\overrightarrow{\mathit{\omega }}}_{i3}$ , can be expressed by the generalized velocities in matrix form,

${\text{ω}}_{i3}^s={\overrightarrow{e}}_{i3}\cdot {\overrightarrow{\mathit{\omega }}}_{i3}=J_i^{\mathrm{T}}\cdot \stackrel{.}{\overrightarrow{q}}$

(38)

with

$\begin{array}{l}{J}_i=e_{i3}^{\mathrm{T}}[\begin{array}{cc}\frac{{\widehat{e}}_{i56}\left(e_{i4}{e}_{i4}^{\mathrm{T}}-I_{3\times 3}\right)}{l_i\left[{\overrightarrow{e}}_{i4},{\overrightarrow{e}}_{i5},{\overrightarrow{e}}_{i6}\right]}-k{e}_{i4}{e}_{i4}^{\mathrm{T}}& ,\end{array}\\ \frac{{\widehat{e}}_{i56}\left({\widehat{r}}_{bi}+l_i{\widehat{e}}_{i4}-e_{i4}{e}_{i4}^{\mathrm{T}}{\widehat{r}}_{bi}\right)}{l_i\left[{\overrightarrow{e}}_{i4},{\overrightarrow{e}}_{i5},{\overrightarrow{e}}_{i6}\right]}+I_{3\times 3}+k{e}_{i4}{e}_{i4}^{\mathrm{T}}{\widehat{r}}_{bi}]\left[N_{6\times 5}\right]\end{array}$

(39)

then the kinematic Jacobian matrix can be obtained

$J={\left[J_1,J_2,J_3,J_4,J_5\right]}^{\mathrm{T}}$

(40)

Table 1 contains the basic geometric parameters of the PKPM, D′ _i and E′_i6 are the local coordinates in the moving platform.

Table 1.

Basic geometric parameters

A1 =(0.0,0.687,0.223)	D′1 =(0,0,0)	E11=(1.0,0,0)	E′16=(1.0,0,0)
A2=(0.334,0.434,0.223)	D′2=(0.088,-0.16,-0.02)	E21=(0.8192,-0.5736,0)	E′26=(0.8192,-0.5736,0)
A3=(-0.334,0.434,0.223)	D′3=(-0.088,-0.16,-0.02)	E31=(0.8192,0.5736,0)	E′36=(0.8192,0.5736,0)
A4=(-0.486,-0.306,0.223)	D′4=(-0.163,-0.06,-0.105)	E41=(-0.7071,0.7071,0)	E′46=(-0.7071,0.7071,0)
A5=(0.486,-0.30,0.223)	D′5=(0.163,-0.06,-0.105)	E51=(-0.7071,-0.7071,0)	E′56=(-0.7071,-0.7071,0)

Let μ(J) denote the condition number of the Jacobian matrix, the dexterous workspace can be searched with the limitation μ(J) <20 by the bound workspace search method, the dexterous workspace of the yaw angle [0°, 90°] and roll angle [-25°, 25°] is shown in Fig. 5, where Fig. 5(a) is the top view of the dexterous workspace and Fig. 5(b) is the oblique view.

Figure 5.

Dexterous workspace of the PKPM

The volume of the dexterous workspace of the PKPM is 0.7 × 0.4 × 0.3 m³, in the dexterous workspace the yaw angle of the moving platform can rotate from 0° to 90°, which means the PKPM is suitable for a large workspace five-axis machine tool.

Fig. 6(a) is the top view of the workspace of the PKPM obtained by fixing the yaw and roll angle of the moving platform to zero, and Fig. 6(b) is the oblique view. As Fig. 6(a)-(b) shows, the volume of the workspace of the PKPM is 1.0 × 1.2 × 0.65 m³, it is larger than that (0.8×0.8×0.5 m³) of the METROM Pentapod.

Figure 6.

Workspace of the PKPM with zero yaw and roll angles

4. Inverse dynamics based on Kane's equation

4.1 The Kane's equation method

The Lagrange's equation and the Newton-Euler method are the most common and popular dynamic modelling methods in robotics. They are very efficient and convenient for dynamic systems with low degrees of freedom and few components. The Newton-Euler method requires an analysis of each component of the robot and needs to take the internal forces into account. Finally, it combines the dynamic equations of all the components to obtain the final dynamic equations. So it needs much more work in order to deal with lots of internal forces to obtain the final dynamic equations. The Lagrange's equation based on the energy function does not need to take the internal forces between the components of the robot into account, but it requires that we calculate the second-order derivatives of the Lagrange function. Therefore, the Lagrange's equation is low in efficiency for a system with high DOFs and lots of components.

The Kane's equation method retains the advantages of the vector analysis method and the energy method, and also avoids the drawbacks of Lagrange's equation and the Newton-Euler method. The efficiency of the algorithm based on Kane's method reflects the high efficiency of the modelling and the simple expression of the final dynamic formula. The Kane's equation method does not care about the internal force, so its dynamic modelling is more efficient than the Newton-Euler method. Compared to the Lagrange's equation method, the equations of the dynamic formula of the PKPM based on Kane's equation method are expressed directly in the linear simple matrix form, it can be calculated highly efficiently and can easily be programmed for control.

4.1 Velocity and acceleration of the PKPM components

The mass centre of the body is represented by “C”; each body mass centre of the parallel mechanism is essentially coincident with the geometric centre, so the geometric centre is regarded as the mass centre of the body. As the motion of the component B_i1 is a fixed-point rotation motion through their centres of mass, the linear velocity ${\overrightarrow{v}}_{Ci1}$ and the acceleration ${\overrightarrow{a}}_{Ci1}$ are both zero.

Let the distance of component B_i2 deviated from the centre of the universal joint be ρ_Ci2, then the eccentric vector of the mass centre is ${\overrightarrow{r}}_{Ci2}={\rho }_{Ci2}{\overrightarrow{e}}_{i4}$ , take the first-order derivatives of ${\overrightarrow{r}}_{Ci2}$ with respect to time t, then the mass centre velocity can be expressed as follows,

${\overrightarrow{v}}_{Ci2}={\stackrel{.}{\overrightarrow{r}}}_{Ci2}={\rho }_{Ci2}\left({\overrightarrow{\mathit{\omega }}}_{i4}\times {\overrightarrow{e}}_{i4}\right)$

(41)

The mass centre acceleration of the component B_i3 is

${\overrightarrow{a}}_{Ci2}={\stackrel{.}{\overrightarrow{v}}}_{Ci2}={\rho }_{Ci2}\left({\overrightarrow{\mathit{\epsilon }}}_{i4}\times {\overrightarrow{e}}_{i4}+{\overrightarrow{\mathit{\omega }}}_{i4}\times \left({\overrightarrow{\mathit{\omega }}}_{i4}\times {\overrightarrow{e}}_{i4}\right)\right)$

(42)

The mass centre of component B_i3 is on the axis of the ball-screw, suppose its eccentric vector is ${\overrightarrow{r}}_{Ci3}={\rho }_{Ci3}{\overrightarrow{e}}_{i4}$ , then the mass centre velocity of component B_i3 is

${\overrightarrow{v}}_{Ci3}={\stackrel{.}{\overrightarrow{r}}}_{Ci3}={\rho }_{Ci3}\left({\overrightarrow{\mathit{\omega }}}_{i4}\times {\overrightarrow{e}}_{i4}\right)$

(43)

The mass centre acceleration of the component B_i3 is

${\overrightarrow{a}}_{Ci3}={\stackrel{.}{\overrightarrow{v}}}_{Ci3}={\rho }_{Ci3}\left({\overrightarrow{\mathit{\epsilon }}}_{i4}\times {\overrightarrow{e}}_{i4}+{\overrightarrow{\mathit{\omega }}}_{i4}\times \left({\overrightarrow{\mathit{\omega }}}_{i4}\times {\overrightarrow{e}}_{i4}\right)\right)$

(44)

Supposing the total length of the ball-screw is L, the mass centre vector of the component B_i4 is ${\overrightarrow{r}}_{Ci4}=\left(L/2-l_i\right){\overrightarrow{e}}_{i4}$ , the mass centre velocity of the component B_i4 is

${\overrightarrow{v}}_{Ci4}=-{\stackrel{.}{l}}_i{\overrightarrow{e}}_{i4}+\left(L/2-l_i\right){\overrightarrow{\mathit{\omega }}}_{i4}\times {\overrightarrow{e}}_{i4}$

(45)

By taking the first-order derivative of Eq. (45) with respect to time t, the mass centre acceleration is

$\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\overrightarrow{a}}_{Ci4}=-{\ddot{l}}_i{\overrightarrow{e}}_{i4}-2{\stackrel{.}{l}}_i{\overrightarrow{\mathit{\omega }}}_{i4}\times {\overrightarrow{e}}_{i4}+\\ +\left(L/2-l_i\right)\left[{\overrightarrow{\mathit{\epsilon }}}_{i4}\times {\overrightarrow{e}}_{i4}+{\overrightarrow{\mathit{\omega }}}_{i4}\times \left({\overrightarrow{\mathit{\omega }}}_{i4}\times {\overrightarrow{e}}_{i4}\right)\right]\end{array}$

(46)

Similarly, the mass centre velocity and acceleration of the component t B_i5 are

${\overrightarrow{v}}_{Ci5}={\rho }_{Ci5}{\overrightarrow{\mathit{\omega }}}_{i5}\times {\overrightarrow{e}}_{i5}-{\stackrel{.}{l}}_i{\overrightarrow{e}}_{i4}-l_i{\overrightarrow{\mathit{\omega }}}_{i4}\times {\overrightarrow{e}}_{i4}$

(47)

$\begin{array}{l}{\overrightarrow{a}}_{Ci5}={\rho }_{Ci5}\left({\overrightarrow{\mathit{\epsilon }}}_{i5}\times {\overrightarrow{e}}_{i5}+{\overrightarrow{\mathit{\omega }}}_{i5}\times \left({\overrightarrow{\mathit{\omega }}}_{i5}\times {\overrightarrow{e}}_{i5}\right)\right)-{\ddot{l}}_i{\overrightarrow{e}}_{i4}-\\ -2{\stackrel{.}{l}}_i{\overrightarrow{\mathit{\omega }}}_{i4}\times {\overrightarrow{e}}_{i4}-l_i\left[{\overrightarrow{\mathit{\epsilon }}}_{i4}\times {\overrightarrow{e}}_{i4}+{\overrightarrow{\mathit{\omega }}}_{i4}\times \left({\overrightarrow{\mathit{\omega }}}_{i4}\times {\overrightarrow{e}}_{i4}\right)\right]\end{array}$

(48)

In order to obtain the partial velocity of Kane's equation, the angular velocity of each component and the linear velocity of the mass centre relative to the generalized velocity $\stackrel{.}{\overrightarrow{q}}={\left[\begin{array}{ccccc}{\stackrel{.}{x}}_p& {\stackrel{.}{y}}_p& {\stackrel{.}{z}}_p& \stackrel{.}{\vartheta }& \stackrel{.}{\phi }\end{array}\right]}^{\mathrm{T}}$ can be obtained.

$\{\begin{array}{l}{\overrightarrow{\mathit{\omega }}}_{i1}=\frac{e_{i1}{e}_{i1}^{\mathrm{T}}-e_{i1}{e}_{i1}^{\mathrm{T}}{e}_{i4}{e}_{i4}^{\mathrm{T}}}{1-{\left(e_{i1}^{\mathrm{T}}{e}_{i4}\right)}^2}\left[\begin{array}{cc}\frac{{\widehat{e}}_{i56}\left(e_{i4}{e}_{i4}^{\mathrm{T}}-I_{3\times 3}\right)}{l_i\left[{\overrightarrow{e}}_{i4},{\overrightarrow{e}}_{i5},{\overrightarrow{e}}_{i6}\right]}& ,\frac{{\widehat{e}}_{i56}\left({\widehat{r}}_{bi}+l_i{\widehat{e}}_{i4}-e_{i4}{e}_{i4}^{\mathrm{T}}{\widehat{r}}_{bi}\right)}{l_i\left[{\overrightarrow{e}}_{i4},{\overrightarrow{e}}_{i5},{\overrightarrow{e}}_{i6}\right]}+I_{3\times 3}\end{array}\right]\left[N_{6\times 5}\right]\cdot \stackrel{.}{\overrightarrow{q}}\\ {\overrightarrow{\mathit{\omega }}}_{i2}=\left(I_{3\times 3}-\frac{e_{i4}{e}_{i4}^{\mathrm{T}}-e_{i4}{e}_{i4}^{\mathrm{T}}{e}_{i1}{e}_{i1}^{\mathrm{T}}}{1-{\left(e_{i1}^{\mathrm{T}}{e}_{i4}\right)}^2}\right)\left[\begin{array}{cc}\frac{{\widehat{e}}_{i56}\left(e_{i4}{e}_{i4}^{\mathrm{T}}-I_{3\times 3}\right)}{l_i\left[{\overrightarrow{e}}_{i4},{\overrightarrow{e}}_{i5},{\overrightarrow{e}}_{i6}\right]}& ,\frac{{\widehat{e}}_{i56}\left({\widehat{r}}_{bi}+l_i{\widehat{e}}_{i4}-e_{i4}{e}_{i4}^{\mathrm{T}}{\widehat{r}}_{bi}\right)}{l_i\left[{\overrightarrow{e}}_{i4},{\overrightarrow{e}}_{i5},{\overrightarrow{e}}_{i6}\right]}+I_{3\times 3}\end{array}\right]\left[N_{6\times 5}\right]\cdot \stackrel{.}{\overrightarrow{q}}\\ {\overrightarrow{\mathit{\omega }}}_{i3}=\left[\begin{array}{cc}\frac{{\widehat{e}}_{i56}\left(e_{i4}{e}_{i4}^{\mathrm{T}}-I_{3\times 3}\right)}{l_i\left[{\overrightarrow{e}}_{i4},{\overrightarrow{e}}_{i5},{\overrightarrow{e}}_{i6}\right]}-k{e}_{i4}{e}_{i4}^{\mathrm{T}}& ,\frac{{\widehat{e}}_{i56}\left({\widehat{r}}_{bi}+l_i{\widehat{e}}_{i4}-e_{i4}{e}_{i4}^{\mathrm{T}}{\widehat{r}}_{bi}\right)}{l_i\left[{\overrightarrow{e}}_{i4},{\overrightarrow{e}}_{i5},{\overrightarrow{e}}_{i6}\right]}+I_{3\times 3}+k{e}_{i4}{e}_{i4}^{\mathrm{T}}{\widehat{r}}_{bi}\end{array}\right]\left[N_{6\times 5}\right]\cdot \stackrel{.}{\overrightarrow{q}}\\ {\overrightarrow{\mathit{\omega }}}_{i4}=\left[\begin{array}{cc}\frac{{\widehat{e}}_{i56}\left(e_{i4}{e}_{i4}^{\mathrm{T}}-I_{3\times 3}\right)}{l_i\left[{\overrightarrow{e}}_{i4},{\overrightarrow{e}}_{i5},{\overrightarrow{e}}_{i6}\right]}& ,\frac{{\widehat{e}}_{i56}\left({\widehat{r}}_{bi}+l_i{\widehat{e}}_{i4}-e_{i4}{e}_{i4}^{\mathrm{T}}{\widehat{r}}_{bi}\right)}{l_i\left[{\overrightarrow{e}}_{i4},{\overrightarrow{e}}_{i5},{\overrightarrow{e}}_{i6}\right]}+I_{3\times 3}\end{array}\right]\left[N_{6\times 5}\right]\cdot \stackrel{.}{\overrightarrow{q}}\\ {\overrightarrow{\mathit{\omega }}}_{i5}=\left[\begin{array}{cc}\frac{e_{i6}{e}_{i6}^{\mathrm{T}}{\widehat{e}}_{i56}\left(e_{i4}{e}_{i4}^{\mathrm{T}}-I_{3\times 3}\right)}{l_i\left[{\overrightarrow{e}}_{4i},{\overrightarrow{e}}_{5i},{\overrightarrow{e}}_{6i}\right]}& ,\frac{e_{i6}{e}_{i6}^{\mathrm{T}}{\widehat{e}}_{i56}\left({\widehat{r}}_{bi}+l_i{\widehat{e}}_{i4}-e_{i4}{e}_{i4}^T{\widehat{r}}_{bi}\right)}{l_i\left[{\overrightarrow{e}}_{i4},{\overrightarrow{e}}_{i5},{\overrightarrow{e}}_{i6}\right]}+I_{3\times 3}\end{array}\right]\left[N_{6\times 5}\right]\cdot \stackrel{.}{\overrightarrow{q}}\end{array}$

(49)

$\{\begin{array}{l}{\overrightarrow{v}}_{Ci2}=\overrightarrow{0}\\ {\overrightarrow{v}}_{Ci2}=\left[\begin{array}{cc}-{\rho }_{Ci2}\frac{{\widehat{e}}_{i4}{\widehat{e}}_{i56}\left(e_{i4}{e}_{i4}^{\mathrm{T}}-I_{3\times 3}\right)}{l_i\left[{\overrightarrow{e}}_{i4},{\overrightarrow{e}}_{i5},{\overrightarrow{e}}_{i6}\right]}& ,-{\rho }_{Ci2}{\widehat{e}}_{i4}\left(\frac{{\widehat{e}}_{i56}\left({\widehat{r}}_{bi}+l_i{\widehat{e}}_{i4}-e_{i4}{e}_{i4}^{\mathrm{T}}{\widehat{r}}_{bi}\right)}{l_i\left[{\overrightarrow{e}}_{i4},{\overrightarrow{e}}_{i5},{\overrightarrow{e}}_{i6}\right]}+I_{3\times 3}\right)\end{array}\right]\left[N_{6\times 5}\right]\cdot \stackrel{.}{\overrightarrow{q}}\\ {\overrightarrow{v}}_{Ci3}=\left[\begin{array}{cc}-{\rho }_{Ci3}\frac{{\widehat{e}}_{i4}{\widehat{e}}_{i56}\left(e_{i4}{e}_{i4}^{\mathrm{T}}-I_{3\times 3}\right)}{l_i\left[{\overrightarrow{e}}_{i4},{\overrightarrow{e}}_{i5},{\overrightarrow{e}}_{i6}\right]}& ,-{\rho }_{Ci3}{\widehat{e}}_{i4}\left(\frac{{\widehat{e}}_{i56}\left({\widehat{r}}_{bi}+l_i{\widehat{e}}_{i4}-e_{i4}{e}_{i4}^{\mathrm{T}}{\widehat{r}}_{bi}\right)}{l_i\left[{\overrightarrow{e}}_{i4},{\overrightarrow{e}}_{i5},{\overrightarrow{e}}_{i6}\right]}+I_{3\times 3}\right)\end{array}\right]\left[N_{6\times 5}\right]\cdot \stackrel{.}{\overrightarrow{q}}\\ {\overrightarrow{v}}_{Ci4}=\left[\begin{array}{cc}{e}_{i4}{e}_{i4}^{\mathrm{T}}-\left(L/2-l_i\right)\frac{{\widehat{e}}_{i4}{\widehat{e}}_{i56}\left(e_{i4}{e}_{i4}^{\mathrm{T}}-I_{3\times 3}\right)}{l_i\left[{\overrightarrow{e}}_{i4},{\overrightarrow{e}}_{i5},{\overrightarrow{e}}_{i6}\right]}& ,-e_{i4}{e}_{i4}^{\mathrm{T}}{\widehat{r}}_{bi}-\left(L/2-l_i\right){\widehat{e}}_{i4}\left(\frac{{\widehat{e}}_{i56}\left({\widehat{r}}_{bi}+l_i{\widehat{e}}_{i4}-e_{i4}{e}_{i4}^{\mathrm{T}}{\widehat{r}}_{bi}\right)}{l_i\left[{\overrightarrow{e}}_{i4},{\overrightarrow{e}}_{i5},{\overrightarrow{e}}_{i6}\right]}+I_{3\times 3}\right)\end{array}\right]\left[N_{6\times 5}\right]\cdot \stackrel{.}{\overrightarrow{q}}\\ {\overrightarrow{v}}_{Ci5}=\left[\begin{array}{cc}{e}_{i4}{e}_{i4}^{\mathrm{T}}+\frac{\left(l_i{\widehat{e}}_{i4}-{\rho }_{Ci5}{\widehat{e}}_{i5}{e}_{i6}{e}_{i6}^{\mathrm{T}}\right){\widehat{e}}_{i56}\left(e_{i4}{e}_{i4}^{\mathrm{T}}-I_{3\times 3}\right)}{l_i\left[{\overrightarrow{e}}_{i4},{\overrightarrow{e}}_{i5},{\overrightarrow{e}}_{i6}\right]}& ,-e_{i4}{e}_{i4}^{\mathrm{T}}{\widehat{r}}_{bi}+\frac{\left(l_i{\widehat{e}}_{i4}{\widehat{e}}_{i56}-{\rho }_{Ci5}{\widehat{e}}_{i5}{e}_{i6}{e}_{i6}^{\mathrm{T}}\right)\left({\widehat{r}}_{bi}+l_i{\widehat{e}}_{i4}-e_{i4}{e}_{i4}^{\mathrm{T}}{\widehat{r}}_{bi}\right)}{l_i\left[{\overrightarrow{e}}_{i4},{\overrightarrow{e}}_{i5},{\overrightarrow{e}}_{i6}\right]}+l_i{\widehat{e}}_{i4}-{\rho }_{Ci5}{\widehat{e}}_{i5}\end{array}\right]\left[N_{6\times 5}\right]\cdot \stackrel{.}{\overrightarrow{q}}\end{array}$

(50)

To simplify the formula expression, we rewrite Eq. (49) and Eq. (50) as Eq. (51):

$\{\begin{array}{l}{\overrightarrow{\mathit{\omega }}}_{ij}=W_{ij}\left(\overrightarrow{q}\right)\cdot \stackrel{.}{\overrightarrow{q}}\\ {\overrightarrow{v}}_{Cij}=V_{Cij}\left(\overrightarrow{q}\right)\cdot \stackrel{.}{\overrightarrow{q}}\end{array}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i,j=1,\mathrm{\dots },5$

(51)

Similarly, we can combine Eq. (35), Eq. (37), Eq. (42), Eq. (44), Eq. (46) and Eq. (48) into the matrix form:

$\{\begin{array}{l}{\overrightarrow{\mathit{\epsilon }}}_{ij}=M_{\mathit{\epsilon }ij}\left(\overrightarrow{q}\right)\cdot \ddot{\overrightarrow{q}}+C_{\mathit{\epsilon }ij}\left(\overrightarrow{q},\stackrel{.}{\overrightarrow{q}}\right)\cdot \stackrel{.}{\overrightarrow{q}}\\ {\overrightarrow{a}}_{Cij}=M_{aij}\left(\overrightarrow{q}\right)\cdot \ddot{\overrightarrow{q}}+C_{aij}\left(\overrightarrow{q},\stackrel{.}{\overrightarrow{q}}\right)\cdot \stackrel{.}{\overrightarrow{q}}\end{array}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i,j=1,\mathrm{\dots },5$

(52)

The coefficient matrices W_ij, V_Cij, M_εij, M_aij depend on the generalized variables $\overrightarrow{q}$ , and C_εij, C_aij depend on both $\overrightarrow{q}$ and $\stackrel{.}{\overrightarrow{q}}$ .

4.2 Force analysis

(1) Active force and inertia force of the moving platform

The cutting force and moment and gravity act on the moving platform in the polishing process, so the principal force of the moving platform is,

${\overrightarrow{F}}_p={\overrightarrow{F}}_p^t+m_p\overrightarrow{g}$

(53)

where ${\overrightarrow{F}}_p^t$ denotes the cutting force simplified to the moving platform, m_p refers to the mass of the moving platform, and $\overrightarrow{g}$ stands for the gravity acceleration vector.

The principal moment acting upon the moving platform can be written as follows,

${\overrightarrow{M}}_p={\overrightarrow{L}}_p^t$

(54)

where ${\overrightarrow{L}}_p^t$ is the principal moment of the cutting force simplified to the mass centre of the moving platform.

The generalized inertia force ${\overrightarrow{F}}_p^{*}$ and the generalized inertia moment ${\overrightarrow{M}}_p^{*}$ of the moving platform are written as

${\overrightarrow{F}}_p^{*}=-m_p{\overrightarrow{a}}_p,\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\overrightarrow{M}}_p^{*}=-[I_p{\overrightarrow{\mathit{\epsilon }}}_p+{\overrightarrow{\mathit{\omega }}}_p\times (I_p{\overrightarrow{\mathit{\omega }}}_p)]$

(55)

where I_p is the inertia matrix of the moving platform in the global coordinate system; ${\overrightarrow{a}}_p,{\overrightarrow{\mathit{\omega }}}_p,{\overrightarrow{\mathit{\epsilon }}}_p$ are the mass centre acceleration, angular velocity and the angular acceleration of the moving platform.

2) Active force and inertia force of each component

The principal forces and moments acting upon component B _ij can be expressed as,

${\overrightarrow{F}}_{ij}=\{\begin{array}{l}{\overrightarrow{F}}_p+{\overrightarrow{F}}_{ij}^f\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=1\hspace{0.17em}and\hspace{0.17em}j=5\\ m_{ij}\overrightarrow{g}+{\overrightarrow{F}}_{ij}^f\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}otherwise\end{array}$

(56)

${\overrightarrow{M}}_{ij}=\{\begin{array}{l}{\overrightarrow{M}}_p+{\overrightarrow{M}}_{ij}^f\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}i=1\hspace{0.17em}and\hspace{0.17em}j=5\\ M_i{\overrightarrow{e}}_{ij}+{\overrightarrow{M}}_{ij}^f\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}j=3\\ {\overrightarrow{M}}_{ij}^f\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}otherwise\end{array}$

(57)

where m_ij is the mass of B _ij , M _i is a driving moment of the hollow-shaft motor in the i-th kinematic chain, ${\overrightarrow{F}}_{ij}^f$ and ${\overrightarrow{M}}_{ij}^f$ , the unknown variables, denote the friction force and friction torque of the body B_ij. The PKPM is driven by the hollow-shaft motors.

The principal inertia force ${\overrightarrow{F}}_{ij}^{*}$ and inertia moment ${\overrightarrow{M}}_{ij}^{*}$ acting upon component B _ij can be expressed as

${\overrightarrow{F}}_{ij}^{*}=-m_{ij}{\overrightarrow{a}}_{Cij},\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{\overrightarrow{M}}_{ij}^{*}=-[I_{ij}{\overrightarrow{\mathit{\epsilon }}}_{ij}+{\overrightarrow{\mathit{\omega }}}_{ij}\times (I_{ij}{\overrightarrow{\mathit{\omega }}}_{ij})]$

(58)

where I_ij is the inertia matrix of the j-th component in the i-th kinematic chain in the global coordinate system; ${\overrightarrow{a}}_{Cij},{\overrightarrow{\mathit{\omega }}}_{ij},{\overrightarrow{\mathit{\epsilon }}}_{ij}$ are mass centre acceleration, angular velocity and angular acceleration of component B_ij.

Suppose that $I_{ij}^0$ is the moment of inertia about component B _ij in its local coordinate system and its orientational transformation matrix from local coordinate system to global coordinate system is R_ij, then the inertia matrix of component I_ij in the global coordinate system is $I_{ij}=R_{ij}{I}_{ij}^0{R}_{ij}^{\mathrm{T}}$ .

4.3 Inverse dynamics of Kane's equation

Variables ẋ_p, ẏ_p, ż_p, ${\stackrel{.}{x}}_p,{\stackrel{.}{y}}_p,{\stackrel{.}{z}}_p,\stackrel{.}{\vartheta },\stackrel{.}{\phi }$ are selected as the generalized velocities of the PKPM, so the generalized driving force and the generalized inertia force can be expressed based on the partial velocities ${\overrightarrow{v}}_{ij}^{(s)}$ and ${\overrightarrow{\mathit{\omega }}}_{ij}^{(s)}$

${\overrightarrow{F}}_s={\displaystyle \sum _{i=1,j=1}^{5,5}{\overrightarrow{F}}_{ij}\cdot {\overrightarrow{v}}_{ij}^{(s)}}+{\displaystyle \sum _{i=1,j=1}^{5,5}{\overrightarrow{M}}_{ij}\cdot {\overrightarrow{\mathit{\omega }}}_{ij}^{(s)}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(s=1,2,3,4,5)$

(59)

${\overrightarrow{F}}_s^{*}={\displaystyle \sum _{i=1,j=1}^{5,5}{\overrightarrow{F}}_{ij}^{*}\cdot {\overrightarrow{v}}_{ij}^{(s)}}+{\displaystyle \sum _{i=1,j=1}^{5,5}{\overrightarrow{M}}_{ij}^{*}\cdot {\overrightarrow{\mathit{\omega }}}_{ij}^{(s)}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(s=1,2,3,4,5)$

(60)

W_ij and V_Cij denote the partial velocity matrix of component B _ij 's linear velocity and angular velocity relative to the generalized variables ${\stackrel{.}{x}}_p,{\stackrel{.}{y}}_p,{\stackrel{.}{z}}_p,\stackrel{.}{\vartheta },\stackrel{.}{\phi }$ which is shown in Eq. (51). The partial velocities ${\overrightarrow{v}}_{ij}^{(s)}$ and ${\overrightarrow{\mathit{\omega }}}_{ij}^{(s)}$ denote the s-th column vector of W_ij and V_Cij respectively.

Kane's equation shows that all the algebra sums of the generalized driving force ${\overrightarrow{F}}_s$ and the generalized inertia force ${\overrightarrow{F}}_s^{*}$ in the same partial velocity should be zero.

${\overrightarrow{F}}_s+{\overrightarrow{F}}_s^{*}=0\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}(s=1,2,3,4,5)$

(61)

With the partial velocities ${\overrightarrow{v}}_{ij}^{(s)}$ and ${\overrightarrow{\mathit{\omega }}}_{ij}^{(s)}$ , we can derive the following dynamic equations about the PKPM

$J_{drv}{\overrightarrow{T}}_{drv}=M\left(\overrightarrow{q}\right)\cdot \ddot{\overrightarrow{q}}+C\left(\overrightarrow{q},\stackrel{.}{\overrightarrow{q}}\right)\cdot \stackrel{.}{\overrightarrow{q}}+F\left(\stackrel{.}{\overrightarrow{q}}\right)+G\left(\overrightarrow{q}\right)$

(62)

where $\overrightarrow{q}={\left[\begin{array}{ccccc}{x}_p& y_p& z_p& \vartheta & \phi \end{array}\right]}^{\mathrm{T}}$ denotes the generalized variables, ${\overrightarrow{T}}_{drv}={\left[\begin{array}{ccccc}{T}_1& T_2& T_3& T_4& T_5\end{array}\right]}^{\mathrm{T}}$ refers to the driving moment of the hollow-shaft motor, J_drv the driving force Jacobian matrix, $M\left(\overrightarrow{q}\right)$ the generalized positive definite inertia matrix, $C\left(\overrightarrow{q},\stackrel{.}{\overrightarrow{q}}\right)$ the Coriolis and centripetal generalized force matrix, $F\left(\stackrel{.}{\overrightarrow{q}}\right)$ the velocity-dependent frictional generalized force vector, and $G\left(\overrightarrow{q}\right)$ the generalized gravitational force. The friction force $F\left(\stackrel{.}{\overrightarrow{q}}\right)$ in Eq. (62) is unknown.

4.4 Driving torque simulation of inverse dynamics

Given the geometric parameters of the PKPM as shown in Table 1, it is easy to calculate the mass and inertia of each component; the geometric and inertia parameters of the ball-screw and the hollow-shaft motor are known, and the inertia of the hollow-shaft motor is 3.8 Kg^*m² and the limit torque is 15 N-m. The property parameters of every component are listed in Table 2.

Table 2.

Property parameters of each component

Body	Mass(Kg)	Inertia (Kg*m2)	Eccentricity (m)
Bi1 (i = 1…5)	6.72	diag(3.89, 5.03, 8.53)*0.01	0
Bi2(i = 1…5)	8.07	diag(2.77, 2.21, 2.42)*0.01	0
Bi3(i=1…5)	3.84	diag(1.03,1.03,0.38)*0.01	0.08
Bi4(i = 1…5)	19.42	diag(3.44,3.44.00561)	0
Bi5 (i = 2…5)	1.34	diag([1.05, 1.37, 2.09)*0.001	0
B15	37.22	diag(1.55, 2.0, 0.628) *0.1	0.135

Suppose the load is just the gravity of the PKPM in the following inverse dynamic torque simulation. Let the moving platform make a circular motion on the plane parallel to the XY plane, and keep both ϑ and φ as zero, the centre of the moving platform is moving in a circular trajectory, $\left[x_p,y_p,z_p\right]=\left[\begin{array}{ccc}{r}_{sim}\cos \left(t\right)& r_{sim}\sin \left(t\right)& -0.6\end{array}\right],\left[{\stackrel{.}{x}}_p,{\stackrel{.}{y}}_p,{\stackrel{.}{z}}_p\right]=r_{sim}{\mathit{\omega }}_{sim}\left[\begin{array}{ccc}\text{-sin}\left({\mathit{\omega }}_{sim}t\right)& \cos \left({\mathit{\omega }}_{sim}t\right)& 0\end{array}\right],\left[{\ddot{x}}_p,{\ddot{y}}_p,{\ddot{z}}_p\right]=r_{sim}{\mathit{\omega }}_{sim}^2\left[\begin{array}{ccc}\text{-cos}\left({\mathit{\omega }}_{sim}t\right)& -\sin \left({\mathit{\omega }}_{sim}t\right)& 0\end{array}\right],{\mathit{\omega }}_{sim}$ stands for the angular velocity and r_sim refers to the radius.

The simulation results of the angular velocities and the driving moments of the motors are shown in Fig. 7

Figure 7.

Simulation results (v_p =18 m/min, a_p = 0.3 m/s²)

The radius and angular velocity of the circular motion is set at r_sim = 03m and ω_sm = 1 rad/s, so that the linear velocity is r_simω_sim = 0.3m/s and the linear acceleration is $r_{sim}{\mathit{\omega }}_{sim}^2=0.3{\text{m/s}}^2$ . The maximum angular velocities of the motors have reached 200 rad/s, but the derived driving moments are less than the maximum moment of the motor of 6.5 N-m. It shows that the polishing feedrate of the PKPM could reach 18m/min. We can also see that the driving characteristic of the PKPM is quite good as it only needs relatively small driving moments to drive.

Let r_sim = 0.0045 m, ω_sim = 66.7 rad/s, the polishing feedrate is 18m/min, the acceleration could reach $r_{sim}{\mathit{\omega }}_{sim}^2=20{\text{m/s}}^2$ , which is twice the gravity acceleration. The simulation results are shown in Fig. 8. From Fig.8 (b) we can see that the driving moments are less than 6.5 N·m. The result also demonstrates that the PKPM can bear much higher acceleration, so that the presented PKPM has a high dynamic performance.

Figure 8.

Simulation results (v_p =18 m/min, a_p =20m/s²)

4. Tracking error control based on the inverse dynamics

4.1 Methods for tracking error control

Currently, there exist many control techniques, such as PID control, computed torque control, slide control, fuzzy control, robust control and so on. The parallel manipulator has a complex structure. From Eq. (62) we know that the dynamic parameters of the parallel mechanism are time-varying.

In the polishing process the trajectory of the polishing tool requires it to be smooth, as the sharp motion and vibration of the polishing tool will deeply affect the polishing precision. As is well known, the slide mode control keeps the robotic motion state around the sliding mode surface, but the switch function of the sliding mode control could make the robot, at a high-frequency, chatter around the sliding mode surface; the chattering would result in a low control accuracy and joint wear occurring quickly.

The friction component $F\left(\stackrel{.}{\overrightarrow{q}}\right)$ in the dynamic model of Eq. (62) is the unknown force acting on the robot, on the other hand, the system parameters, determined by theoretical calculation, are not exactly the same as the real parameters, the difference between the nominal and the real parameters can be regarded as a perturbation to the parallel mechanism and there also exist various uncertain external disturbances in the polishing process. Taking into account all kinds of disturbances, the control method for the PKPM should be robust and adaptive to friction and perturbations; the control law should adapt the dynamical parameters of the PKPM in different configurations and velocities in the whole polishing process.

The control gain of the classical PID control method is a constant; hence the classical PID control gain does not satisfy the control of the dynamic parameters in the trajectory motion, so the adaptive control techniques should be introduced to compute the real-time dynamic control gain in order to obtain the optimal tracking error control. The computed torque control is a basic and effective control method in robotic control; with the measurement of the position and velocity of the robot it removes all the nonlinear components of the dynamic equations, so that the nonlinear dynamic control problem is changed into a linear control problem. The control laws of advanced robotic control methods are approximately based on the computed torque control model. In this paper the adapted robust control method with computed torque control is introduced to design the PKPM's trajectory controller.

4.2 Tracking error model and the mixed H₂/H_∞ control method

In order to simplify the discussion of designing a motion controller, the inverse dynamic Eq. (62) can be rewritten as,

$\mathit{\tau }=M\left(q\right)\cdot \ddot{q}+C\left(q,\stackrel{.}{q}\right)\cdot \stackrel{.}{q}+F\left(\stackrel{.}{q}\right)+G\left(q\right)$

(63)

where τ is used to denote $J_{drv}{\overrightarrow{T}}_{drv}$ and q is used to denote $\overrightarrow{q}$ for the purposes of simplification. The parameters M(q), C(q,q̇) and G(q) are divided into the nominal part and the unknown part,

$\begin{array}{c}M\left(q\right)=\widehat{M}\left(q\right)+\mathrm{\Delta }M\left(q\right)\\ C\left(q,\stackrel{.}{q}\right)=\widehat{C}\left(q,\stackrel{.}{q}\right)+\mathrm{\Delta }C\left(q,\stackrel{.}{q}\right)\\ G\left(q\right)=\widehat{G}\left(q\right)+\mathrm{\Delta }G\left(q\right)\end{array}$

(64)

where M̂ (q), Ĉ (q,q) and Ĝ (q) are the nominal parametric matrices, and ΔM(q), ΔC(q,q̇) and ΔG(q) are the uncertainties. Let τ_d denote a finite energy exogenous disturbance force, τ_d is divided into a constant disturbance component τ_d and a variable disturbance ${\mathit{\tau }}_d^v$ , i.e. ${\mathit{\tau }}_d={\mathit{\tau }}_d^c+{\mathit{\tau }}_d^v$ , they both are unknown in the inverse dynamic equation, so Eq. (63) with a disturbance force can be rewritten as,

$\mathit{\tau }+{\mathit{\tau }}_d^c+{\mathit{\tau }}_d^v=\widehat{M}\left(q\right)\cdot \ddot{q}+\widehat{C}\left(q,\stackrel{.}{q}\right)\cdot \stackrel{.}{q}+\widehat{G}\left(q\right)$

(65)

with

$\{\begin{array}{l}{\mathit{\tau }}_d^c=-\mathrm{\Delta }G\left(q\right)\\ {\mathit{\tau }}_d^v=-\left(\mathrm{\Delta }M\left(q\right)\cdot \ddot{q}+\mathrm{\Delta }C\left(q,\stackrel{.}{q}\right)\cdot \stackrel{.}{q}+F\left(\stackrel{.}{q}\right)\right)\end{array}$

(66)

The tracking error of the motion controller decides the motion precision of a robot or machine. For a given motion control object, the best way is to build up its dynamic model with the exact parameters and all the disturbance information, but in the real world there will be some information that cannot be exactly measured or obtained so we need to attenuate the effects of the unknown parameters and the perturbation on the tracking error. For a given trajectory of the moving platform, q_d, with its first and second order derivatives, q̇_d and ${\ddot{q}}_d$ , the tracking error vector is defined as,

$\stackrel{..}{x}=\left[\begin{array}{l}q-q_d\\ \stackrel{.}{q}-{\stackrel{.}{q}}_d\end{array}\right]=\left[\begin{array}{l}\stackrel{..}{q}\\ \stackrel{.}{\stackrel{..}{q}}\end{array}\right]$

(67)

The minimized control action, which includes the minimal applied torque and energy consumption on the motion control, is introduced by R. Johansson [27], considering a more general control action, a state transformation is given by

$\stackrel{..}{z}=\left[\begin{array}{l}{\stackrel{..}{z}}_1\\ {\stackrel{..}{z}}_2\end{array}\right]=T_0\stackrel{..}{x}=\left[\begin{array}{cc}I& 0\\ T_{11}& T_{12}\end{array}\right]\stackrel{..}{x}$

(68)

The applied torque with a variable u for the assignment of the control law is introduced into the robust control design by a minimized control action [26, 27].

$\begin{array}{l}\mathit{\tau }=\widehat{M}\left(q\right)\left({\ddot{q}}_d-T_{12}^{-1}{T}_{11}\stackrel{.}{\stackrel{..}{q}}-T_{12}^{-1}{\widehat{M}}^{-1}\left(q\right)\left(\widehat{C}\left(q,\stackrel{.}{q}\right)B^T{T}_0\stackrel{..}{x}-u\right)\right)+\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}+\widehat{C}\left(q,\stackrel{.}{q}\right){\stackrel{.}{q}}_d+\widehat{F}\left(\stackrel{.}{q}\right)+\widehat{G}\left(q\right)-\delta \left(q,\stackrel{.}{q},\ddot{q}\right)\end{array}$

(69)

Substituting Eq. (69) and Eq. (67) into Eq. (65), the tracking error model based on unknown parameter matrices and external disturbances can be obtained,

$\stackrel{.}{\stackrel{..}{x}}=A_T\left(\stackrel{..}{x},t\right)\stackrel{..}{x}+B_T\left(\stackrel{..}{x},t\right)u+B_T\left(\stackrel{..}{x},t\right)w$

(70)

where

$\begin{array}{l}\hspace{0.17em}\hspace{0.17em}{A}_T\left(\stackrel{..}{x},t\right)=T_0^{-1}\left[\begin{array}{cc}-T_{12}^{-1}{T}_{11}& T_{12}^{-1}\\ 0& -{\widehat{M}}^{-1}\left(q\right)\widehat{C}\left(q,\stackrel{.}{q}\right)\end{array}\right]T_0\\ \hspace{0.17em}\hspace{0.17em}{B}_T\left(\stackrel{..}{x},t\right)=T_0^{-1}\left[\begin{array}{c}0\\ I\end{array}\right]{\widehat{M}}^{-1}\left(q\right)\\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}w=\widehat{M}\left(q\right)T_{12}{\widehat{M}}^{-1}\left(q\right){\mathit{\tau }}_d\end{array}$

(71)

Let disturbance τ_d denote the sum of the constant disturbance ${\mathit{\tau }}_d^c$ and the variable disturbance ${\mathit{\tau }}_d^v$ , the ℋ _∞ control method attenuates the effect of the disturbance τ_d, and makes the system into a global asymptotic stability. The ℋ _∞ controller only considers the attenuation of the perturbation performance but not the tracking error performance. In contrast, the optimal control with ℋ₂ quadratic optimization only considers the tracking error performance but not the disturbance; the effect of the disturbance is always enlarged to the tracking error of the end-effector in the optimal control, because the optimal controller is used in the high gain control system in order to obtain the high responses. A mixed ℋ₂/ℋ _∞ performance tracking error control method is proposed to obtained the attenuation of the disturbance performance and high response performance.

For the given positive definite weight matrices Q₁, Q₂, R₁, R₂ and an attenuation level γ, the ℋ _∞ tracking error performance for the positive definite matrix Q_1f can be expressed as,

$\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{J}_1\left(u,w\right)={\stackrel{..}{x}}^T\left(t\right)Q_{1f}\stackrel{..}{x}\left(t\right)+\\ +{\displaystyle {\int }_0^{t_f}\left({\stackrel{..}{x}}^T\left(t\right)Q_1\stackrel{..}{x}\left(t\right)+u^T\left(t\right)R_{1}u\left(t\right)-{\gamma }^2{w}^T\left(t\right)w\left(t\right)\right)\mathrm{d}t}\end{array}$

(72)

The ℋ₂ optimal tracking error performance for the positive definite matrix Q_2f is written as,

$\begin{array}{l}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{J}_2\left(u,w\right)={\stackrel{..}{x}}^T\left(t\right)Q_{2f}\stackrel{..}{x}\left(t\right)+\\ +{\displaystyle {\int }_0^{t_f}\left({\stackrel{..}{x}}^T\left(t\right)Q_2\stackrel{..}{x}\left(t\right)+u^T\left(t\right)R_{2}u\left(t\right)\right)\mathrm{d}t}\end{array}$

(73)

Hence, the mixed ℋ₂/ℋ _∞ performance tracking control problem is to find the a pair (u^*, w^*) which satisfies the minimax problem,

$J_1\left(u^{\ast },w\right)\le J_1\left(u^{\ast },w^{\ast }\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\forall w\left(t\right)\in L_2[0,t_f]$

(74)

$J_2\left(u^{\ast },w^{\ast }\right)\le J_2\left(u,w^{\ast }\right)\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\forall u\left(t\right)\in L_2[0,t_f]$

(75)

In the mixed ℋ₂/ℋ _∞ tracking error control problem, we take R₁ = R₂ = R to define the same effects of the control variable u(t) in both the ℋ₂ and ℋ _∞ performance indices.

The solution of the minimax problem described by Eq. (74) and Eq. (75) turns out to solve a coupled nonlinear time-varying Riccati-like equation, but the nonlinear Riccati-like equations are too difficult to use in engineering. The simple solution for the ℋ₂/ℋ _∞ performance tracking error problem is given by introducing some restrictions [26]. For a desired disturbance attenuation level γ, with 0<α<1 and the positive definite weighting matrix of ℋ _∞ , the performance index can be written as,

$Q_1=\left[\begin{array}{cc}{Q}_{11}^{\mathrm{T}}{Q}_{11}& Q_{12}\\ Q_{21}& Q_{22}^T{Q}_{22}\end{array}\right]$

(76)

where Q₂₂ = q₂₂I and $Q_{11}^{\mathrm{T}}{Q}_{22}+Q_{22}^{\mathrm{T}}{Q}_{11}>Q_{12}^{\mathrm{T}}+Q_{12}$ , then the weighting matrix R and the state translation matrix T can be obtained,

$R=\left(\frac{1-\alpha }{2-\alpha }\right){\gamma }^{2}I$

(77)

$T_0=\left[\begin{array}{cc}I& 0\\ {(1-\alpha )}^{1/2}\gamma Q_{11}& {(1-\alpha )}^{1/2}\gamma Q_{22}\end{array}\right]$

(78)

Hence, the mixed ℋ₂/ℋ _∞ optimal applied torque is

$\begin{array}{l}{\mathit{\tau }}^{\ast }=\widehat{M}\left(q\right)\left({\ddot{q}}_d-T_{12}^{-1}{T}_{11}\stackrel{.}{\stackrel{..}{q}}\right)+\widehat{C}\left(q,\stackrel{.}{q}\right)\\ \left(\stackrel{.}{q}-T_{12}^{-1}{T}_{11}\stackrel{..}{q}\right)+\widehat{F}\left(\stackrel{.}{q}\right)+\widehat{G}\left(q\right)+T_{12}^{-1}{u}^{\ast }\end{array}$

(79)

with

$u^{\ast }=-R^{-1}{B}^{\mathrm{T}}{T}_0\stackrel{..}{x}$

(80)

Eq. (79) and Eq. (80) can be rewritten in the form of a computed torque,

$\begin{array}{l}{\mathit{\tau }}^{\ast }=\widehat{M}\left(q\right)\left({\ddot{q}}_d-K_v\stackrel{.}{\stackrel{..}{q}}-K_p\stackrel{..}{q}\right)+\\ \hspace{0.17em}+\widehat{C}\left(q,\stackrel{.}{q}\right){\stackrel{.}{q}}_d+\widehat{F}\left(\stackrel{.}{q}\right)+\widehat{G}\left(q\right)\end{array}$

(81)

with mixed ℋ₂/ℋ _∞ performance control gain matrices,

$\{\begin{array}{l}{K}_v=T_{12}^{-1}{T}_{11}+{\widehat{M}}^{-1}\left(q\right)T_{12}^{-1}{R}^{-1}{T}_{12}\\ K_p={\widehat{M}}^{-1}\left(q\right)\widehat{C}\left(q,\stackrel{.}{q}\right)T_{12}^{-1}{T}_{11}+{\widehat{M}}^{-1}\left(q\right)T_{12}^{-1}{R}^{-1}{T}_{11}\end{array}$

(82)

Compared to the classical static gain computed torque control, Eq. (82) shows that the mixed ℋ₂/ℋ _∞ performance control has dynamic gain. Hence, the mixed ℋ₂/ℋ _∞ performance control is more adaptive than the computed torque control. When the attenuation level γ=∞ is reached, the mixed ℋ₂/ℋ _∞ control is degenerated into the ℋ₂ optimal control.

4.3 Simulations with four control methods

In this section the computed torque method, the optimal control method, the ℋ _∞ control method and the mixed ℋ₂/ℋ _∞ control method for tracking the error control of the PKPM are compared. The classical computed torque control method does not tell us how to obtain the optimal control gain k_p and k_v, so that we select the control gain k_p =200 and k_v = 25 for the computed torque control method by experience, and select the attenuation level γ = 0.5 for the ℋ _∞ and the mixed ℋ₂/ℋ _∞ method.

The tracking error control focuses on control precision, the attenuation of disturbance, stability and computed driving torque. Four control methods are discussed in this paper. The following simulation is concerned with the disturbance condition. The disturbance is divided into two components, the constant component ${\mathit{\tau }}_d^c$ and the variable component ${\mathit{\tau }}_d^v$ . Fig. 9 shows the disturbance inputs in the simulation.

Figure 9.

Disturbances for simulation

Case 1: Continuous trajectory command input with disturbance ${\mathit{\tau }}_d^c$ and ${\mathit{\tau }}_d^v$

The continuous trajectory command input is shown in Fig. 10. Each trajectory component of the generalized variables of the moving platform is a sinusoidal function relative to the home configuration [0, 0, –0.8, 0, 0]^T.

Figure 10.

Continuous trajectory command relative to the home position

The simulation results of the continuous trajectory command are shown in Fig. 11(a)-(d). We can see that the optimal control obtains the highest tracking error precision and the lowest one is ℋ _∞ control. This is because the optimal controller is only designed to obtain the high tracking performance and the ℋ _∞ controller is only designed to attenuate the effect of disturbance and unknown parameters. The mixed ℋ₂/ℋ _∞ control has a higher tracking error precision than those obtained from computed torque control and ℋ _∞ control.

Figure 11.

Simulation results of continuous trajectory command with disturbances

Case 2: Step trajectory command with disturbances ${\mathit{\tau }}_d^c,{\mathit{\tau }}_d^v$ and torque limit

A step function command is used as the simulation's command input,

$q_d=\{\begin{array}{l}{\left[0,\hspace{0.17em}0,\hspace{0.17em}-0.8,\hspace{0.17em}0,\hspace{0.17em}0\right]}^{\mathrm{T}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}t<0\\ {\left[0,\hspace{0.17em}0,\hspace{0.17em}-0.7,\hspace{0.17em}0,\hspace{0.17em}0\right]}^{\mathrm{T}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}t\ge 0\end{array}$

(83)

The position [0, 0, –0.8, 0, 0]^T is the origin of the PKPM, the step command trajectory moves the robot to [0, 0, –0.7, 0, 0]^T at initial time t₀. The step command is shown in Fig. 12.

Figure 12.

Step function command

Actually, the driving torque of the hollow-shaft motor is limited by the current passing through the motor, the limited torque of the hollow shaft motor is 15 N·m, hence, we add this torque limit into the dynamic simulation, the simulation results of the step command input are shown in Fig. 13(a)-(d) with the constant disturbance ${\mathit{\tau }}_d^c$ and variable disturbance ${\mathit{\tau }}_d^v$ .

Figure 13.

Simulation results of step trajectory command with disturbances and torque limit

The computed torque controller can track the step trajectory command well, but when the disturbance ${\mathit{\tau }}_d^c$ and ${\mathit{\tau }}_d^v$ appear at time t from 1.25s and 2.5s, the controller cannot attenuate the effect of the disturbance.

The simulation results of the optimal control with driving torque limit show that the tracking error is enlarged to two times that of the step trajectory command input by the optimal control and the dynamic tracking is unstable, so that the optimal control method is not suitable for the step trajectory command.

Because of the disturbance attenuation, the tracking error with the ℋ _∞ control method is insensitive to the external disturbances, but the responsibility of the ℋ _∞ control method is too low to use in tracking the step trajectory command.

From the figure we can see that the tracking error is still smooth when the disturbances ${\mathit{\tau }}_d^c$ and ${\mathit{\tau }}_d^v$ appear at time t from 1.25s and 2.5s. This implies that the mixed ℋ₂/ℋ _∞ control method has a high attenuation of external disturbances. Hence, the mixed ℋ₂/ℋ _∞ control method has a high combination performance of tracking error control and attenuation of external disturbances.

Considering the worst conditions of the step trajectory command with disturbances and driving torque limit, the mixed ℋ₂/ℋ _∞ control exhibits both optimal and robust control performance, which means that the mixed ℋ₂/ℋ _∞ control method is capable of the attenuation of disturbance with a high tracking error performance.

5. Conclusions

A 5-DOF PKPM with a motion type of T3R2 and a redundant linear motion actuator is developed in this paper, there are two advantages to the structural scheme: one is the large yaw angle workspace which is suitable for polishing parts with freeform surfaces; the second is that the PKPM could perform constant polishing-force control with the 3D force sensor and linear motion actuator mounted on the moving platform. This paper focuses on the inverse dynamic modelling and tracking error control.

The kinematic analysis was developed and a closure vector method was introduced to express the geometric relations of the PKPM linkages, which could avoid a lot of tedious derivations in formulating the linear velocities and angular velocities. The affine projection method was presented as a means to deal with the angular velocities and the angular accelerations projection problem efficiently, so that the linear velocities and the linear accelerations at the mass centre are obtained. Based on these variables an inverse dynamic algorithm based on Kane's equation is developed: it is highly efficient and works in real-time to obtain the driving control torque of the motors. The inverse dynamic driving torque simulation results show that the parallel kinematic machine can work at high speeds and high acceleration. The presented PKPM can achieve an acceleration two times that of gravity, while the conventional machine tool can only achieve an acceleration of one time greater than gravity.

The mixed ℋ₂/ℋ _∞ control approach is introduced for PKPM tracking error control. The simulation results from comparing the computed torque method, the optimal control and the ℋ _∞ control method, show that the mixed ℋ₂/ℋ _∞ control method with dynamic control gain has a high tracking precision performance and attenuation of disturbances, and a high control performance can also be obtained even under the worst conditions.