Kinematic Analysis of a 3-Axis Parallel Manipulator: The P3

Abstract

This paper presents kinematic analysis of a 3-axis parallel manipulator, called P3. The P3 parallel manipulator is developed based on a 3-PRRU parallel mechanism, where P denotes prismatic pair, R revolute pair, and U universal joint. The P3 parallel manipulator can undergo two rotations and one translation without parasitic motion. First, a brief description of the P3 is presented. Then, forward and inverse kinematic solutions of the P3 are derived. Parasitic motion is discussed. The Jacobian matrix is obtained based on the velocity equations. Further, singular configurations of the P3 are identified. A special configuration where the actuation fails is also detected based on screw theory. The workspace of P3 is investigated.

1. Introduction

When a PM (parallel mechanism) has less than six DOF (degrees of freedom), it is called a lower-mobility parallel mechanism. The 3-DOFs PM with one translational DOF and two rotational DOFs is an important category of the lower-mobility PM. Such a kind of PM can be constructed by using three legs generating a planar-spherical kinematic bond [1]. Hence, such a kind of PM can be called 3-[P][S] PM. The 3-[P][S] family includes four typical types of architectures, namely, 3-RPS, 3-PRS, 3-RRS, and 3-PPS, where R denotes a revolute pair, P a prismatic pair, and S a spherical joint. Note that the 3-[P][S] PM is also called [PP]S PM [2].

Since the 3-RPS PM was proposed [3], the 3-[P][S] family has been used extensively because many tasks require one translational DOF and two rotational DOFs (1T2R). For example, Z3 head in machine tool [4], telescope application [5], motion simulator [6], micromanipulator [7], and coordinate measuring machine [8]. Much progress regarding kinematics analysis [9–12], dimensional synthesis [13–17], singularity [18, 19], dynamics [12, 20, 21], and parasitic motion [22] has been obtained. Particularly, a new 1T2R PM with motion bifurcation of the end tool proposed for machine tool application [23].

Parasitic motion is an important kinematic feature of the 3-[P][S] parallel mechanism. Carretero et al. [22] defined parasitic motion of the 3-PRS PM as the motion occurring in the constrained DOFs, including one rotation about the z-axis of the fixed frame and two translations along the x and y axes of the fixed frame. Though the amplitude of parasitic motion is often small, it may exert negative effects on manipulation accuracy and quality. For instance, when the 3-PRS PM is used in a telescope to perform tipping/tilting and focusing of the secondary mirror, the parasitic translations in x and y axes are unwanted displacements and cause image aberrations. Actually, the x and y parasitic translations cannot be compensated or corrected by the control system of a 3-PRS PM.

As it is pointed out in [24], the motion of a [P][S] PM without parasitic motion can be expressed as a product like {T(z)}{R(O, x)}{R(O, y)}. Such a motion belongs to the Cat. I submanifold [25]. Though the above notion for parasitic motion is coordinate free, the choice of parameterization used in practice for representing parasitic motion depends on the actual application and is often arbitrary. Bonev proposed a new parameterization of orientation angles, called Tilt-and-Torsion angles [26]. However, such a parameterization is not applicable to a [P][S] PM with nonsymmetrical limb arrangements.

The P3 parallel manipulator is developed based on a 3-PRRU PM belonging to the new family of the 3-[P][S] PM without parasitic motion proposed by Li and Hervé [24]. Moreover, the rotation center of the moving platform of the P3 is constrained to move in a line, and two rational axes can be specified. This fact means that the translation and rotations are decoupled and simple control can be achieved. Besides, the 3-PRRU parallel mechanism has no parasitic motion because of the constrained path of the rotation center. Thus, the P3 parallel manipulator is a simple-to-control PM without parasitic motion and has great potential in application.

2. Description of the P3 Parallel Manipulator

The prototype of the P3 parallel manipulator is shown in Figure 1. The P3 parallel manipulator is developed based on a 3-PRRU parallel mechanism as shown in Figure 2. In Figure 2, I is the base platform and Π is the moving platform. The moving platform is connected to the base through three identical PRRU limbs.

Figure 1:

Prototype of P3 parallel manipulator.

Figure 2:

3-PRRU PM.

The three prismatic pairs that are connected to the base are actuated and parallel to each other.

There are two revolute joints at the end of each link A _i B _i , and one is connected with the prismatic joint and the other is connected with link B _i C _i . At the end of B _i C _i , there is a universal joint, which is connected with the base platform. Counting from the moving platform, the first revolute axis of the universal joint is perpendicular to the moving platform.

The fixed base frame is denoted by O-xyz, and the moving frame is denoted by O′-uvw, as shown in Figure 1. Plane A _i B _i C _i is called the ith limb plane (LP). Note that the three planes, A₁B₁C₁, A₂B₂C₂, and O-xz, are coincident and the directions of the revolute joints are parallel to y axis. A₃B₃C₃ and O-yz are the same planes in which the directions of the revolute joints are parallel to x-axis. The first revolute axis of the universal joint is perpendicular to the base platform and the second revolute axis of the universal joint is parallel to the base platform in each limb. The length of the tool which is installed at the center of the moving platform is H. The parameters of P3 parallel manipulator are listed in Table 1.

Table 1:

Parameters of the P3 parallel manipulator.

Parameter	Value (mm)	Description
l 1	230	Length of A1B1 and A2B2
l 2	270	Length of O′B1 and O′B2
l 3	230	Length of A3B3
l 4	225	Length of O′B3
r	45	Distance from C1 or C2 to O′
D	270	Half distance between limb 1 and limb 2
E	250	Distance from O to limb 3
H	130	Tool length

3. Position Analysis

3.1. Forward Kinematics

The forward kinematics require finding the location and orientations of the moving platform given the input d _i . In this work, we use a y-x-z Euler angle about the moving frame to describe the orientation of the moving platform. The rotations are defined as follows: (i) rotation by an angle θ about y-axis, (ii) rotation by an angle ψ about the new x-axis, (iii) rotation by an angle φ about the new z-axis. The rotation matrix is given by

$\begin{array}{c}\begin{array}{c}\mathbf{T}={\mathbf{T}}_{yxz}={\mathbf{R}}_y\left(\theta \right){\mathbf{R}}_x\left(\psi \right){\mathbf{R}}_z\left(\phi \right)\end{array}\hspace{1em}=\left[\begin{array}{ccc}c\theta c\phi +s\psi s\theta s\phi & -c\theta s\phi +s\psi s\theta c\phi & c\psi s\theta \\ c\psi s\phi & c\psi c\phi & -s\psi \\ -s\theta c\phi +s\psi s\theta s\phi & s\theta s\phi +s\psi c\theta c\phi & c\psi c\theta \end{array}\right],\end{array}$

(1)

where c* and s* correspond to cos (*) and sin (*), respectively.

The fact that the 3-PRRU PM has no parasitic motion was explained by Lie's group theory in [25]. Here, we present a brief kinematic analysis for the reader's sake and show that such a property can simplify the following analysis. As shown in Figure 2, in the initial configuration, the axes of the base frame are parallel to the axes of the moving frame, respectively. Moreover, z and w are collinear. p = [x, y, z] ^T is the vector from O, the origin of the base frame, to O′, the origin of the moving frame, relative to O-xyz; q _i = [q _ix , q _iy , q _iz ] ^T is the vector from O to C _i relative to O-xyz; ${a^{\prime }}_i={\left[{a^{\prime }}_{ix},{a^{\prime }}_{iy},{a^{\prime }}_{iz}\right]}^T$ is the vector from O′ to C _i relative to O′-uvw; and r is the distance from C₁ or C₂ to O′.

Because the points C _i can only move in the limb plane, we have the following three constraint equations for each leg:

$q_{\mathrm{1}y}=\mathrm{0},\hspace{0.17em}{q}_{\mathrm{2}y}=\mathrm{0},\hspace{0.17em}{q}_{\mathrm{3}x}=\mathrm{0}.$

(2)

Apparently, we have ${\mathbf{a}}_{\mathrm{1}}^{\prime }={[-r\hspace{0.5em}0\hspace{0.5em}0]}^T$ , ${\mathbf{a}}_{\mathrm{2}}^{\prime }={[r\hspace{0.5em}0\hspace{0.5em}0]}^T$ , and ${\mathbf{a}}_{\mathrm{3}}^{\prime }={[0\hspace{0.5em}0\hspace{0.5em}0]}^T$ . As shown in Figure 2, we have ${\mathbf{q}}_i=\mathbf{T}{\mathbf{a}}_i^{\prime }+\mathbf{p}$ and the expressions of q_1y, q_2y, and q_3x can be obtained. Thus, (2) can be written as

$\begin{array}{c}{q}_{\mathrm{1}y}=-rc\psi s\phi +y=\mathrm{0},\\ q_{\mathrm{2}y}=rc\psi s\phi +y=\mathrm{0},\\ q_{\mathrm{3}x}=x=\mathrm{0}.\end{array}$

(3)

Solving the above equations yields

$x=y=\mathrm{0},\hspace{0.17em}c\psi s\phi =\mathrm{0}.$

(4)

Because ψ represents a continuous rotation of the moving platform, we have ψ ≠ 0 and φ = 0. Hence, the three parasitic motions are all equal to zero and the P3 parallel manipulator has no parasitic motion. Therefore, the rotation matrix can be simplified as follows:

$\mathbf{R}={\mathbf{R}}_v\left(\theta \right){\mathbf{R}}_u\left(\psi \right)=\left[\begin{array}{ccc}c\theta & s\theta s\psi & s\theta c\psi \\ \mathrm{0}& c\psi & -s\psi \\ -s\theta & c\theta s\psi & c\theta c\psi \end{array}\right].$

(5)

Then, let us focus on limb 1 and limb 2, as shown in Figure 3.

Figure 3:

Sketch of limb 1 and limb 2.

When d₁ and d₂ are given, all the parameters in Figure 3 can be obtained:

$l_{\mathrm{1}}c\alpha +l_{\mathrm{2}}c\theta =D,$

(6)

$\mathrm{2}{l}_{\mathrm{2}}s\theta =d_{\mathrm{2}}-d_{\mathrm{1}}⟹\theta =\arcsin \left(\frac{d_{\mathrm{2}}-d_{\mathrm{1}}}{\mathrm{2}{l}_{\mathrm{2}}}\right).$

(7)

Substituting (7) into (6) yields

$\alpha =\text{arccos}\left(\frac{D-\sqrt{l_{\mathrm{2}}^{\mathrm{2}}-{\left((d_{\mathrm{1}}-d_{\mathrm{2}})/\mathrm{2}\right)}^{\mathrm{2}}}}{l_{\mathrm{1}}}\right)$

(8)

$P_z=-d_{\mathrm{1}}-l_{\mathrm{1}}s\alpha +l_{\mathrm{2}}s\theta .$

(9)

Focusing on limb 3 as shown in Figure 4, we can use the x-y-z Euler angle to describe the orientation of the moving platform. The three rotations are defined as follows: (i) rotation by an angle γ about u-axis, (ii) rotation by an angle θ₃ about the new v-axis, (iii) rotation by an angle θ₄ about new w-axis. The rotation matrix is:

$\mathbf{R}={\mathbf{R}}_u\left(\theta \right){\mathbf{R}}_v\left(\psi \right){\mathbf{R}}_w\left(\phi \right)=\left[\begin{array}{ccc}c{\theta }_{\mathrm{3}}c{\theta }_{\mathrm{4}}& -c{\theta }_{\mathrm{3}}s{\theta }_{\mathrm{4}}& s{\theta }_{\mathrm{3}}\\ s\gamma s{\theta }_{\mathrm{3}}c{\theta }_{\mathrm{4}}+c\gamma s{\theta }_{\mathrm{4}}& -s\gamma s{\theta }_{\mathrm{3}}s{\theta }_{\mathrm{4}}+c\gamma c{\theta }_{\mathrm{4}}& -s\gamma c{\theta }_{\mathrm{3}}\\ -c\gamma s{\theta }_{\mathrm{3}}c{\theta }_{\mathrm{4}}+s\gamma s{\theta }_{\mathrm{4}}& c\gamma s{\theta }_{\mathrm{3}}s{\theta }_{\mathrm{4}}+s\gamma c{\theta }_{\mathrm{4}}& c\gamma c{\theta }_{\mathrm{3}}\end{array}\right].$

(10)

Figure 4:

Sketch of limb 3.

When d₃ is given, all the parameters in Figure 4 can be obtained. Thus, we have

$\begin{array}{c}{d}_{\mathrm{3}}-l_{\mathrm{3}}s{\alpha }_{\mathrm{3}}-l_{\mathrm{4}}s\gamma =P_z,\\ l_{\mathrm{3}}c{\alpha }_{\mathrm{3}}+l_{\mathrm{4}}c\gamma =E.\end{array}$

(11)

Then, sγ and cγ can be obtained from (11).

Although we use different Euler angles to describe the orientation of the moving platform, the rotation matrices are the same. Hence, we have (10) = (5) and

$\begin{array}{c}s\theta c\psi =s{\theta }_{\mathrm{3}},\\ -s\psi =-s\gamma c{\theta }_{\mathrm{3}}.\end{array}$

(12)

Then

$\psi =\text{arccos}\frac{c\gamma }{\sqrt{\mathrm{1}-{\left(s\theta s\gamma \right)}^{\mathrm{2}}}}.$

(13)

Hence, the forward kinematics are as follows:

$\begin{array}{c}\theta =\arcsin \left(\frac{d_{\mathrm{2}}-d_{\mathrm{1}}}{\mathrm{2}{l}_{\mathrm{2}}}\right),\\ P_z=-d_{\mathrm{1}}-l_{\mathrm{1}}s\alpha +l_{\mathrm{2}}s\theta ,\\ \psi =\text{arccos}\frac{c\gamma }{\sqrt{\mathrm{1}-{\left(s\theta s\gamma \right)}^{\mathrm{2}}}}.\end{array}$

(14)

Actually, there are two possibilities in the limbs 1 and 2 when the limb lengths d _i are given. As shown in Figure 3, the broken line is another possible configuration. And each possibility also has two consequences in the limb 3; see the broken line in Figure 4. Therefore, there are four solutions in the forward kinematics.

3.2. Inverse Kinematics

The inverse kinematics require finding the limb length d _i given the location and orientations of the moving platform. As shown in Figure 3, when P _z , θ, and ψ are given, we have

$\begin{array}{c}{d}_{\mathrm{1}}=P_z-l_{\mathrm{2}}s\theta \pm \sqrt{l_{\mathrm{1}}^{\mathrm{2}}-{\left(D-l_{\mathrm{2}}c\theta \right)}^{\mathrm{2}}},\\ d_{\mathrm{2}}=P_z+l_{\mathrm{2}}s\theta \pm \sqrt{l_{\mathrm{1}}^{\mathrm{2}}-{\left(D-l_{\mathrm{2}}c\theta \right)}^{\mathrm{2}}}.\end{array}$

(15)

Further, focusing on limb 3 in Figure 4, we have

$\begin{array}{c}E-l_{\mathrm{4}}c\gamma =l_{\mathrm{3}}c{\alpha }_{\mathrm{3}},\\ d_{\mathrm{3}}=P_z+l_{\mathrm{4}}s\gamma \pm l_{\mathrm{3}}s{\alpha }_{\mathrm{3}}.\end{array}$

(16)

Then,

$d_{\mathrm{3}}=P_z+l_{\mathrm{4}}s\gamma \pm \sqrt{l_{\mathrm{3}}^{\mathrm{2}}-{\left(E-l_{\mathrm{4}}c\gamma \right)}^{\mathrm{2}}},$

(17)

where γ can be calculated by (8):

$\gamma =\arcsin \left(\frac{s\psi }{\sqrt{\mathrm{1}-{\left(s\theta c\psi \right)}^{\mathrm{2}}}}\right).$

(18)

Apparently, every limb has two possibilities and the inverse kinematics have eight solutions. The “±” in (15) and (17) should be “+” in this text. So, we have

$\begin{array}{c}{d}_{\mathrm{1}}=P_z-l_{\mathrm{2}}s\theta +\sqrt{l_{\mathrm{1}}^{\mathrm{2}}-{\left(D-l_{\mathrm{2}}c\theta \right)}^{\mathrm{2}}},\\ d_{\mathrm{2}}=P_z+l_{\mathrm{2}}s\theta +\sqrt{l_{\mathrm{1}}^{\mathrm{2}}-{\left(D-l_{\mathrm{2}}c\theta \right)}^{\mathrm{2}}},\\ d_{\mathrm{3}}=P_z+l_{\mathrm{4}}s\gamma +\sqrt{l_{\mathrm{3}}^{\mathrm{2}}-{\left(E-l_{\mathrm{4}}c\gamma \right)}^{\mathrm{2}}}.\end{array}$

(19)

3.3. Tool Position

As shown in Figure 2, the length of the tool is H. The coordinate of G in the moving frame is G′ = (0,0, – H) ^T . So, the tool position is calculated as

$\mathbf{G}={\mathbf{P}}_z+\mathbf{R}{\mathbf{G}}^{\prime }\hspace{1em}=\left[\begin{array}{c}\mathrm{0}\\ \mathrm{0}\\ P_z\end{array}\right]+\left[\begin{array}{ccc}c\theta & s\theta s\psi & s\theta c\psi \\ \mathrm{0}& c\psi & -s\psi \\ -s\theta & c\theta s\psi & c\theta c\psi \end{array}\right]\left[\begin{array}{c}\mathrm{0}\\ \mathrm{0}\\ -H\end{array}\right]\hspace{1em}=\left[\begin{array}{c}-Hs\theta c\psi \\ Hs\psi \\ P_z-Hc\theta c\psi \end{array}\right].$

(20)

4. Forward and Inverse Velocity Analysis

Differentiating both sides of (14) with respect to time yields

$\mathbf{J}\left[\begin{array}{c}{\stackrel{.}{d}}_{\mathrm{1}}\\ {\stackrel{.}{d}}_{\mathrm{2}}\\ {\stackrel{.}{d}}_{\mathrm{3}}\end{array}\right]-\mathbf{G}\left[\begin{array}{c}\stackrel{.}{\theta }\\ \stackrel{.}{\psi }\\ {\stackrel{.}{v}}_z\end{array}\right]=\mathrm{0},$

(21)

where

$\begin{array}{c}\mathbf{J}=\left[\begin{array}{ccc}{J}_{\mathrm{11}}& & \\ & J_{\mathrm{22}}& \\ & & J_{\mathrm{33}}\end{array}\right],\hspace{0.17em}\mathbf{G}=\left[\begin{array}{c}{G}_{\mathrm{11}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{G}_{\mathrm{12}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{G}_{\mathrm{13}}\\ G_{\mathrm{21}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{G}_{\mathrm{22}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{G}_{\mathrm{23}}\\ G_{\mathrm{31}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{G}_{\mathrm{32}}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{G}_{\mathrm{33}}\hspace{0.17em}\end{array}\right],\\ J_{11}=d_1-P_z+l_{2}s\theta ,\\ J_{22}=d_2-P_z-l_{2}s\theta ,\\ J_{33}=d_3-P_z-l_{4}s\gamma \\ G_{11}=d_1-P_z+l_{2}s\theta ,\hspace{0.17em}{G}_{12}=0,\\ G_{13}=-l_2\left(\left(d_1-P_z\right)c\theta \right)+Ds\theta ,\\ G_{21}=d_2-P_z-l_{2}s\theta ,\hspace{0.17em}{G}_{22}=0,\\ G_{23}=-l_2\left(-\left(d_2-P_z\right)c\theta \right)+Ds\theta ,\\ G_{31}=d_3-P_z-l_{4}s\gamma ,\hspace{0.17em}{G}_{32}=A{R}_1+B{R}_2,\\ G_{\mathrm{33}}=A{Q}_1+B{Q}_2,\\ A=l_4\left(d_3-P_z\right)-l_4^{2}s\gamma ,\hspace{0.17em}B=l_4\left(E-l_{4}c\gamma \right),\\ R_1=-\frac{c\gamma \left(1-s^2\gamma s^2\theta \right)}{c\theta },\hspace{0.17em}{R}_2=s\gamma c\theta +\frac{s\gamma c^2\gamma s^2\theta }{c\theta },\\ Q_1=\frac{-s\gamma c^2\gamma s\theta }{c\theta },\hspace{0.17em}{Q}_2=\frac{s\theta c\gamma s^2\gamma }{c\theta }.\end{array}$

(22)

So, we can establish the velocity equation of the 3-PRRU PM as follows:

$\mathbf{J}\stackrel{.}{\mathbf{d}}\mathbf{-}\mathbf{G}\stackrel{.}{\mathbf{X}}\mathbf{=}\mathrm{0}.$

(23)

Hence, we have

$\stackrel{.}{\mathbf{X}}={\mathbf{G}}^{-\mathrm{1}}\mathbf{J}\stackrel{.}{\mathbf{d}}.$

(24)

Obviously, (24) describes both the forward and inverse velocity relations of the 3-PRRU PM.

5. Singularity Analysis

Singularity is an intrinsic property of a PM. When the PM is at its singular configuration, it will lose or gain one or more degrees of freedom and become incontrollable or lose its stiffness. There are three types of singularity based on the rank deficiency of the matrices J and G [27]. The first type of singularity is called inverse kinematic singularity which occurs when J is singular; the second type of singularity is called direct kinematic singularity which occurs when G is singular; the third type of singularity is called combined singularity which occurs when both J and G are singular.

5.1. Inverse Kinematic Singularities

An inverse kinematic singularity occurs when the determinant of J equals zero. Because J is a diagonal matrix as shown in (22), the determinant of J equals zero when J _ii = 0 (i = 1,2, 3). For the P3 manipulator with parameters in Table 1, J₁₁ and J₂₂ are not zero in the workspace. One can find that J₃₃ is equal to zero when γ = 84.9°. At this configuration, A₃B₃ is horizontal, as shown in Figure 5.

Figure 5:

Inverse kinematic singularity.

5.2. Direct Kinematic Singularities

A direct kinematic singularity occurs when the determinant of G is equal to zero. When γ = 56.67°, G₃₂ = G₂₂ = G₁₂ = 0 and the determinant of G is equal to zero. As shown in Figure 6, A₃B₃ and O′B₃ are collinear.

Figure 6:

Direct kinematic singularity.

5.3. Combined Singularities

A combined singularity occurs when the determinants of G and J are both zero. When θ = arccos ((D – l₁)/l₂), we have J₁₁ = J₂₂ = 0 and the determinant of J is equal to zero. When G₁₁ = G₂₁, G₁₂ = G₂₂, G₁₃ = G₂₃, and the determinant of G is equal to zero. As shown in Figure 7, A₁B₁ and A₂B₂ are horizontaly. Due to the length and constraint of the prismatic pair, the prototype cannot reach this configuration.

Figure 7:

Combined singularities.

6. Actuation Failure

When the moving platform is upright, as shown in Figure 8, the actuations in limb 1 and limb 2 fail. In other words, the mobility of the 3-PRRU PM is the same while the three motors cannot determine the position and orientation of the moving platform. Such a phenomenon is called actuation failure and can be investigated by screw theory.

Figure 8:

Actuation failure configuration.

The modified Grübler-Kutzbach criterion is applicable to mobility analysis of all kinds of lower-mobility PMs [28]. The modified Grübler-Kutzbach criterion is given by

$M=d\left(n-g-\mathrm{1}\right)+\sum _{i=\mathrm{1}}^g{f}_i+v-\zeta ,$

(25)

where M denotes the mobility of a mechanism, d is the order of a mechanism, n is the number of links including the frame, g is the number of kinematic joints, f _i is the number of freedoms of the ith joint, v is the number of redundant constraints except those forming common constraints, and ζ is the number of local freedoms.

The order of a mechanism d is given by

$d=\mathrm{6}-\lambda ,$

(26)

where λ is the common constraint of the mechanism.

A common constraint can be defined as a screw reciprocal to the unit twists associated with all kinematic pairs in a lower-mobility PM. Geometrically, a common constraint can be interpreted as a constraint formed by coaxial limb constraints of the same kind.

For convenience, ${\mathcal{S}}_{ij}$ is used to represent the unit twist associated with the jth kinematic pair in the ith limb; ${\mathcal{S}}_{ij}^r$ is used to represent the jth unit wrench exerted by the ith limb; and ${\mathcal{S}}_{mi}^r$ is used to represent the ith wrench in the mechanism constraint system.

As shown in Figure 9, we establish a limb reference coordinate system denoted by O₁-X₁Y₁Z₁ when the moving platform is upright. Then, the twist system of limb 1 is given as follows:

$\begin{array}{c}{\mathcal{S}}_{\mathrm{11}}=\left(\begin{array}{cccccc}0& 0& 0;& 0& 0& 1\end{array}\right),\\ {\mathcal{S}}_{\mathrm{12}}=\left(\begin{array}{cccccc}0& 1& 0;& a_{\mathrm{12}}& 0& c_{\mathrm{12}}\end{array}\right),\\ {\mathcal{S}}_{\mathrm{13}}=\left(\begin{array}{cccccc}0& 1& 0;& 0& 0& c_{13}\end{array}\right),\\ {\mathcal{S}}_{\mathrm{14}}=\left(\begin{array}{cccccc}1& 0& 0;& 0& 0& 0\end{array}\right),\\ {\mathcal{S}}_{\mathrm{15}}=\left(\begin{array}{cccccc}0& 1& 0;& 0& 0& 0\end{array}\right).\end{array}$

(27)

Figure 9:

First limb twist system.

Using reciprocal screw theory, we can obtain the limb constraint system as

$\begin{array}{c}{\mathcal{S}}_{\mathrm{11}}^r=\left(\begin{array}{cccccc}0& 1& 0;& 0& 0& 0\end{array}\right),\\ {\mathcal{S}}_{\mathrm{12}}^r=\left(\begin{array}{cccccc}0& 0& 0;& 0& 0& 1\end{array}\right).\end{array}$

(28)

Note that ${\mathcal{S}}_{\mathrm{11}}^r$ represents a constraint force being parallel to ${\mathcal{S}}_{\mathrm{12}}$ and ${\mathcal{S}}_{\mathrm{13}}$ and passing through point O₁. And ${\mathcal{S}}_{\mathrm{12}}^r$ represents a constraint couple being perpendicular to ${\mathcal{S}}_{\mathrm{13}}$ and ${\mathcal{S}}_{\mathrm{14}}$ . Apparently, limb 2 has the same constraint.

Note that in this configuration, the direction of the prismatic joints in limbs 1 and 2 is perpendicular to the three revolute joints ${\mathcal{S}}_{\mathrm{12}}$ , ${\mathcal{S}}_{\mathrm{13}}$ , and ${\mathcal{S}}_{\mathrm{15}}$ . So there exist two local freedom in limbs 1 and 2, that is, ζ = 2. This is the reason why the actuation failure happens in this configuration.

Similarly, we can establish another limb coordinate system O₀-X₀Y₀Z₀ for limb 3, as depicted in Figure 10. The twist system of limb 3 is given by

$\begin{array}{c}{\mathcal{S}}_{\mathrm{31}}=\left(\begin{array}{cccccc}0& 0& 0;& 0& 0& 1\end{array}\right),\\ {\mathcal{S}}_{\mathrm{32}}=\left(\begin{array}{cccccc}1& 0& 0;& 0& b_{32}& c_{32}\end{array}\right),\\ {\mathcal{S}}_{\mathrm{33}}=\left(\begin{array}{cccccc}1& 0& 0;& 0& b_{33}& 0\end{array}\right),\\ {\mathcal{S}}_{\mathrm{34}}=\left(\begin{array}{cccccc}0& 1& 0;& 0& 0& 0\end{array}\right),\\ {\mathcal{S}}_{\mathrm{35}}=\left(\begin{array}{cccccc}0& 0& 1;& 0& 0& 0\end{array}\right).\end{array}$

(29)

Figure 10:

Third limb twist system.

The limb constraint system of limb 3 can be obtained:

${\mathcal{S}}_{\mathrm{31}}^r=\left(\begin{array}{cccccc}1& 0& 0;& 0& 0& 0\end{array}\right).$

(30)

The ${\mathcal{S}}_{\mathrm{31}}^r$ represents a constraint force being parallel to Z₀ and passing through the center of the U joint center. Totally, there are five limb constraints acting on the moving platform as shown in Figure 11.

Figure 11:

Constraint system.

The five limb constraints form the mechanism constraint system. The combined effect of them can be represented by the standard base of the mechanism constraint system, which is given by

$\begin{array}{c}{\mathcal{S}}_{m1}^r=\left(\begin{array}{cccccc}1& 0& 0;& 0& 0& 0\end{array}\right),\\ {\mathcal{S}}_{m2}^r=\left(\begin{array}{cccccc}0& 1& 0;& 0& 0& 0\end{array}\right),\\ {\mathcal{S}}_{m\mathrm{3}}^r=\left(\begin{array}{cccccc}0& 0& 0;& 0& 0& 1\end{array}\right).\end{array}$

(31)

The three wrenchs in (31) constrain two translations of the moving platform along X₀ and Y₀ axes and one rotation about Z₀, respectively. There are no common constraints, and λ = 0. Two redundant constraints exist and v = 2.

Hence, the mobility of the 3-PRRU PM is

$M=\mathrm{6}\left(\mathrm{11}-\mathrm{12}-\mathrm{1}\right)+\mathrm{15}+\mathrm{2}-\mathrm{2}=\mathrm{3}.$

(32)

Although there are redundant constraints and local freedoms in this configuration, the number and pattern of the DOF are the same as the common configuration. Therefore, this configuration is not a singular configuration, but a configuration where the actuation fails. However, if l₃ < E, the moving platform will never be upright. Such a design is implemented in the prototype.

7. Workspace Analysis

The P3 PM has two rotational DOFs and one translational DOF. The moving platform can rotate about two axes intersecting at a point which is determined by the translation in Z. In other words, the two rotations are decoupled with the translation. Hence, we can investigate the workspace without considering the translation.

7.1. Tilt Ability

The tilt ability of the P3 PM is defined as the scope of the pose of the moving platform and is described by the maximal scope of θ and ψ. Note that the Jacobian matrix is not singular when the P3 PM is in the configurations shown in Figure 12. However, dexterity is very bad in these configurations, as shown in Section 8. Therefore, we choose the two configurations as the right and left limits.

Figure 12:

The left and right limit.

As shown in Figures 3 and 12, when the mechanism is between the two limits, the scope of θ is

$\left(-\text{arccos}\left(\frac{D}{l_1+l_2}\right),\text{arccos}\left(\frac{D}{l_1+l_2}\right)\right).$

(33)

Equation (13) shows the relation between θ and ψ. By using the conclusion of singularity analysis, we can also obtain the scope of the ψ. As shown in Figure 8, A₃B₃ and O′B₃ are collinear, and the mechanism is at its singular configuration. At this configuration, ψ reaches its upper limit value.

We have

$\gamma =\text{arccos}\frac{E}{l_{\mathrm{3}}+l_{\mathrm{4}}},$

(34)

$\psi =\arcsin \frac{s\gamma c\theta }{\sqrt{\mathrm{1}-{\left(s\theta s\gamma \right)}^{\mathrm{2}}}}.$

(35)

There are two possibilities when ψ reaches its lower limit value.

1. l₃ ≥ E: when the moving platform is upright, as shown in Figure 8, actuation failure occurs. At this configuration, γ = – 90°.

2. l₃ < E: the PM will never be upright. But when A₃B₃ is horizontal, as shown in Figure 5, the mechanism is at the inverse kinematic singularity configuration. We have

   $\gamma =-\text{arccos}\left(\frac{E-l_3}{l_4}\right).$

   (36)

Using (13), we can find the lower limit value of ψ.

In order to increase the tilt ability, we can maximize the scope of θ and ψ. From (33), the scope of θ can be enlarged by increasing l₁ + l₂ or decreasing D: From (34) and (35), the scope of ψ can be enlarged by increasing l₃ + l₄ or decreasing E: From (36), it is known that when l₃ < E, the scope of ψ can be enlarged by increasing l₃ and l₄ until l₃ = E.

The scope of θ can be calculated by (33). Because l₃ < E, the upper and the lower-limit values of ψ can be obtained by (34) and (36). The tilt ability of 3-PRRU PM is depicted in Figure 13.

Figure 13:

The tilt ability.

The area surrounded by curves is the poses where the 3-PRRU PM can reach. The lower curve means the inverse kinematic singularity when ψ is close to – 80°; see Figure 5. The upper curve means the direct kinematic singularity when ψ is close to 50°; see Figure 6. The two lines in the right and left of the picture mean the two configurations shown in Figure 12.

7.2. Tool End Workspace

When P _z is given, the tool end workspace is determined by the tilt ability which has already been described by (33)–(36). Using (20), we can find the tool end workspace. When P _z is given, say P _z = – 200, the tool end workspace is a part of spherical surface, as shown in Figure 14.

Figure 14:

Tool end workspace (P _z = – 200).

8. Conclusion

This paper presents the kinematic analysis of a 3-axis PM, called P3. The forward and inverse kinematics are presented. There are four solutions in the forward position analysis and eight solutions in the inverse position analysis. The inverse, direct, and combined kinematic singularities of the P3 PM are identified. It is also shown that the actuation failure can occur and can be avoided at the design stage by selecting appropriate link dimensions. The tilt ability and workspace of the P3 prototype are presented. The P3 has no parasitic motion and simple control can be achieved.