2.1. Architecture of the parallel manipulators

The 3-DOF translational parallel manipulator DELTA was proposed by Clavel [Reference Clavel3]. As shown in Fig. 1(a), the considered manipulator is composed of a platform connected to the base by three kinematic legs. Each i-th leg is composed of a link (2) connected to the base by an active revolute pair (A_i) with axis Δ_i. Two bars (3) and (3^′) are connected to the link (2) by two spherical joints centered at C_3i and C_4i. The line C_3iC_4i is taken parallel to Δ_i. These two bars (3) and (3^′) are also connected to the moving platform (4) by two spherical joints centered at C_1i and C_2i. The points C_1iC_2iC_3iC_4i form a parallelogram structure. The points G_i and F_i correspond to the middles of C_1iC_2i and C_3iC_4i, respectively.

Figure 1.

Architecture of two 3-DOF translational parallel manipulators.

The 3-Translational Parallel Manipulator (RAF) was proposed by Romdhane et al. [Reference Romdhane, Affi and Fayet1] (Fig. 1(b)). The platform (4) is connected to the base (1) by three active legs of type SPS (S and P stand for spherical and active prismatic pairs, respectively) and two passive kinematic legs (PKLs), which eliminate all possible rotations of the platform with respect to the base. Each i-th PKL is composed of a link (2) connected to the base (1) by a revolute pair (A_i) with axis Δ_i. Two bars (3) and (3 $^{\prime}$ ) are connected to the link (2) by two spherical joints centered at C_3i and C_4i, respectively, and to the platform (4) by two revolute joints (D_1i and D_2i) with axes orthogonal to Δ_i. Thus, a parallelogram form is obtained by the points D_1iD_2iC_3iC_4i. The body (5^′), which is connected to the platform by a revolute joint (G $^{\prime}$ _i) with orthogonal axis is used to solve the over constrained problem. The two points G_i and F_i are, respectively, the midpoints of segment D_1iD_2i and segment C_3iC_4i.

2.2. Inverse kinematic model of the manipulators

In this section, the inverse kinematic model for both parallel manipulators is demonstrated. The following model will be used in order to show the orientation error of the manipulators due to the passive joint clearances.

For the three legs of the DELTA manipulator and the two passive kinematic legs (PKLs) of the RAF manipulator (Fig. 2), the position of the platform with respect to the reference system fixed to the base is given by the following equation:

(1) \begin{equation}\mathbf{O}_{\mathrm{b}}\mathbf{O}_{\mathrm{p}}=\mathbf{O}_{\mathrm{b}}\mathbf{A}_{\mathrm{i}}+\mathbf{A}_{\mathrm{i}}\mathbf{F}_{\mathrm{i}}+\mathbf{F}_{\mathrm{i}}\mathbf{G}_{\mathrm{i}}+\mathbf{G}_{\mathrm{i}}\mathbf{O}_{\mathrm{p}}\end{equation}

By Projection of the Eq. (1) on the x _b, y _b and z _b axes, the expression of the coordinates of the reference point O_p is given as follows:

(2) \begin{equation} \begin{cases} x=\cos \unicode{x03B1} _{\mathrm{i}}.\left(\mathrm{r}+\mathrm{l}_{2}.\sin \unicode{x03C6} _{1\mathrm{i}}+\mathrm{l}_{3}.\cos \unicode{x03C6} _{2\mathrm{i}}.\sin \unicode{x03C6} _{3\mathrm{i}}\right)-\mathrm{l}_{3}.\sin \unicode{x03B1} _{\mathrm{i}}.\sin \unicode{x03C6} _{2\mathrm{i}}\\[3pt] y=\sin \unicode{x03B1} _{\mathrm{i}}.\left(\mathrm{r}+\mathrm{l}_{2}.\sin \unicode{x03C6} _{1\mathrm{i}}+\mathrm{l}_{3}.\cos \unicode{x03C6} _{2\mathrm{i}}.\sin \unicode{x03C6} _{3\mathrm{i}}\right)+\mathrm{l}_{3}.\cos \unicode{x03B1} _{\mathrm{i}}.\sin \unicode{x03C6} _{2\mathrm{i}}\\[3pt] z=\mathrm{l}_{2}.\cos \unicode{x03C6} _{1\mathrm{i}}+\mathrm{l}_{3}.\cos \unicode{x03C6} _{3\mathrm{i}}.\cos \unicode{x03C6} _{2\mathrm{i}} \end{cases} \end{equation}

Figure 2.

Geometric model of translational parallel manipulators

(i = 1, 2, 3 for DELTA manipulator, i = 1, 2 for RAF manipulator)

Where

(3) \begin{equation}\mathrm{r}=\mathrm{r}_{\mathrm{b}}-\mathrm{r}_{\mathrm{p}}\end{equation}

By solving the system of equation given before, two possible solutions are obtained for the angle $\unicode{x03C6}$ _2i:

(4) \begin{equation}\unicode{x03C6} _{2\mathrm{i}}=\left\{\arcsin \left(\frac{\mathrm{y}.\cos \unicode{x03B1} _{\mathrm{i}}-\mathrm{x}.\sin \unicode{x03B1} _{\mathrm{i}}}{\mathrm{l}_{3}}\right),\unicode{x03C0} -\arcsin \left(\frac{\mathrm{y}.\cos \unicode{x03B1} _{\mathrm{i}}-\mathrm{x}.\sin \unicode{x03B1} _{\mathrm{i}}}{\mathrm{l}_{3}}\right)\right\}\end{equation}

According to the limits of the parallelogram, the solution that satisfied the condition–(π/2) < $\unicode{x03C6}$ _2i < (π/2) is considered.

For the angle $\unicode{x03C6}$ _i3, two possible solutions can be obtained:

(5) \begin{align}&\unicode{x03C6} _{3\mathrm{i}}=\left\{\arccos \left(\frac{\unicode{x03C7} _{\mathrm{i}}}{\sqrt{\left(\mathrm{x}.\cos \unicode{x03B1} _{\mathrm{i}}+\mathrm{y}.\sin \unicode{x03B1} _{\mathrm{i}}-\mathrm{r}\right)^{2}+\mathrm{z}^{2}}}\right)+\unicode{x03B2} _{\mathrm{i}},\right.\nonumber\\[3pt] &\left.-\arccos \left(\frac{\unicode{x03C7} _{\mathrm{i}}}{\sqrt{\left(\mathrm{x}.\cos \unicode{x03B1} _{\mathrm{i}}+\mathrm{y}.\sin \unicode{x03B1} _{\mathrm{i}}-\mathrm{r}\right)^{2}+\mathrm{z}^{2}}}\right)+\unicode{x03B2} _{\mathrm{i}}\right\}\end{align}

where

(6) \begin{equation}\unicode{x03C7} _{\mathrm{i}}=\frac{\left(\mathrm{x}.\cos \unicode{x03B1} _{\mathrm{i}}+\mathrm{y}.\sin \unicode{x03B1} _{\mathrm{i}}-\mathrm{r}\right)^{2}+\mathrm{z}^{2}+\left(\mathrm{l}_{3}.\cos \unicode{x03C6} _{2\mathrm{i}}\right)^{2}-{\mathrm{l}}_{2}^{2}}{2.\mathrm{l}_{2}.\cos \unicode{x03C6} _{2\mathrm{i}}}\end{equation}

(7) \begin{equation}\unicode{x03B2} _{\mathrm{i}}=\arctan \left(\frac{\mathrm{x}.\cos \unicode{x03B1} _{\mathrm{i}}+\mathrm{y}.\sin \unicode{x03B1} _{\mathrm{i}}-\mathrm{r}}{\mathrm{z}}\right)\end{equation}

For a given $\unicode{x03C6}$ _i3, one possible solution is obtained for $\unicode{x03C6}$ _i1:

(8) \begin{equation}\unicode{x03C6} _{1\mathrm{i}}=2.\arctan \left(\frac{2.\mathrm{l}_{2}.\mathrm{z}-\sqrt{{\Gamma }_{\mathrm{i}}^{2}+4\left(\mathrm{l}_{2}.\mathrm{z}\right)^{2}-{\mathrm{E }}_{\mathrm{i}}^{2}}}{\mathrm{E }_{\mathrm{i}}-\Gamma _{\mathrm{i}}}\right)\end{equation}

where

(9) \begin{equation}\Gamma _{\mathrm{i}}=2.\mathrm{l}_{2}.\left(\mathrm{r}-\mathrm{x}.\cos \unicode{x03B1} _{\mathrm{i}}-\mathrm{y}.\sin \unicode{x03B1} _{\mathrm{i}}\right)\end{equation}

(10) \begin{equation}\mathrm{E }_{\mathrm{i}}=\mathrm{r}^{2}+\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{z}^{2}+{\mathrm{l}}_{2}^{2}-{\mathrm{l}}_{3}^{2}-2.\mathrm{r}.\left(\mathrm{x}.\cos \unicode{x03B1} _{\mathrm{i}}+\mathrm{y}.\sin \unicode{x03B1} _{\mathrm{i}}\right)\end{equation}

According to Eq. (8), the inverse kinematic model of the manipulator has a solution when the following relationship is fulfilled:

(11) \begin{equation}{\Gamma }_{\mathrm{i}}^{2}+4\!\left(\mathrm{l}_{2}.\mathrm{z}\right)^{2}-{\mathrm{E }}_{\mathrm{i}}^{2}\geq 0\end{equation}

3.1. Static model of DELTA manipulator

For the DELTA manipulator, the goal is to determine the forces and moments transmitted by the joints of the three legs. This model will be used for computing the orientation error of the manipulator caused by the clearance in the passive joints.

Let $\unicode{x1D6D5}$ _ext = [F M]^T be the external wrenches applied to the reference point of the platform O_p (Fig. 3). The forces transmitted by the spherical joints centered at C_1i and C_2i in the i-th leg can be written, respectively, as

(12) \begin{equation}{\unicode{x1D6D5} }_{1\mathrm{i}}=\left[\begin{array}{c} \mathrm{f}_{1\mathrm{i}}\,{.}\,\mathbf{u}_{\mathrm{i}}\\ \\[-9pt] \mathbf{0} \end{array}\right];\quad {\unicode{x1D6D5} }_{2\mathrm{i}}=\left[\begin{array}{c} \mathrm{f}_{2\mathrm{i}}\,{.}\,\mathbf{u}_{\mathrm{i}}\\ \\[-9pt] \mathbf{0} \end{array}\right]\end{equation}

where u _i is the unit vector of the direction of the bars (3). The analytic expression of vector u _i is given as

(13) \begin{equation}\mathbf{u}_{\mathrm{i}}=\left[\begin{array}{c} \cos \unicode{x03B1} _{\mathrm{i}}.\cos \unicode{x03C6} _{2\mathrm{i}}.\sin \unicode{x03C6} _{3\mathrm{i}}+\sin \unicode{x03B1} _{\mathrm{i}}.\sin \unicode{x03C6} _{2\mathrm{i}}\\ \\[-9pt] \sin \unicode{x03B1} _{\mathrm{i}}.\sin \unicode{x03C6} _{3\mathrm{i}}.\cos \unicode{x03C6} _{2\mathrm{i}}+\cos \unicode{x03B1} _{\mathrm{i}}.\sin \unicode{x03C6} _{2\mathrm{i}}\\ \\[-9pt] \cos \unicode{x03C6} _{3\mathrm{i}}.\cos \unicode{x03C6} _{2\mathrm{i}} \end{array}\right]\end{equation}

Figure 3.

Forces and moments applied on the moving platform.

According to the equilibrium of the platform (4), one can obtain

(14) \begin{align}\begin{cases} {\sum }_{\mathrm{i}=1}^{3}\!\left(\mathrm{f}_{1\mathrm{i}}+\mathrm{f}_{2\mathrm{i}}\right).\ \mathbf{u}_{\mathrm{i}}-\mathbf{F}=\mathbf{0}\\ \\[-8pt] {\sum }_{\mathrm{i}=1}^{3}\!\left(\mathrm{f}_{1\mathrm{i}}.\mathbf{v}_{1\mathrm{i}}\times \mathbf{u}_{\mathrm{i}}+\mathrm{f}_{2\mathrm{i}}.\mathbf{v}_{2\mathrm{i}}\times \mathbf{u}_{\mathrm{i}}\right)-\mathbf{M}=\mathbf{0} \end{cases}\end{align}

Where the analytic expression of the vectors v _1i and v _2i are given as

(15) \begin{equation}\mathbf{v}_{1\mathrm{i}}=\mathbf{O}_{\mathbf{p}}\mathbf{C}_{1\mathrm{i}}=\left[\begin{array}{c} \mathrm{r}_{\mathrm{p}}.\cos \unicode{x03B1} _{\mathrm{i}}-\dfrac{\mathrm{a}}{2}\sin \unicode{x03B1} _{\mathrm{i}}\\ \\[-7pt] \mathrm{r}_{\mathrm{p}}.\sin \unicode{x03B1} _{\mathrm{i}}+\dfrac{\mathrm{a}}{2}\cos \unicode{x03B1} _{\mathrm{i}}\\ \\[-7pt] 0 \end{array}\right];\quad \mathbf{v}_{2\mathrm{i}}=\mathbf{O}_{\mathbf{p}}\mathbf{C}_{2\mathrm{i}}=\left[\begin{array}{c} \mathrm{r}_{\mathrm{p}}.\cos \unicode{x03B1} _{\mathrm{i}}+\dfrac{\mathrm{a}}{2}\sin \unicode{x03B1} _{\mathrm{i}}\\ \\[-7pt] \mathrm{r}_{\mathrm{p}}.\sin \unicode{x03B1} _{\mathrm{i}}-\dfrac{\mathrm{a}}{2}\cos \unicode{x03B1} _{\mathrm{i}}\\ \\[-7pt] 0 \end{array}\right]\end{equation}

The system of Eq. (14) can be written in matrix form:

(16) \begin{equation}\left[\begin{array}{c@{\quad}c} \mathbf{A} & \mathbf{A}\\ \\[-9pt] \mathbf{B} & \mathbf{C} \end{array}\right]\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} \mathrm{f}_{11} & \mathrm{f}_{12} & \mathrm{f}_{13} & \mathrm{f}_{21} & \mathrm{f}_{22} & \mathrm{f}_{23} \end{array}\right]^{\mathrm{T}}=\left[\begin{array}{c} \mathbf{F}\\ \\[-9pt] \mathbf{M} \end{array}\right]\end{equation}

Where the expression of the matrices A, B and C are given as follows:

(17) \begin{equation}\mathbf{A}=\left[\begin{array}{c@{\quad}c@{\quad}c} \mathbf{u}_{1} & \mathbf{u}_{2} & \mathbf{u}_{3} \end{array}\right]\end{equation}

(18) \begin{equation}\mathbf{B}=\left[\begin{array}{c@{\quad}c@{\quad}c} \mathbf{v}_{\mathbf{11}}\times \mathbf{u}_{\mathbf{1}} & \mathbf{v}_{\mathbf{12}}\times \mathbf{u}_{\mathbf{2}} & \mathbf{v}_{\mathbf{13}}\times \mathbf{u}_{\mathbf{3}} \end{array}\right]\end{equation}

(19) \begin{equation}\mathbf{C}=\left[\begin{array}{c@{\quad}c@{\quad}c} \mathbf{v}_{\mathbf{21}}\times \mathbf{u}_{\mathbf{1}} & \mathbf{v}_{\mathbf{22}}\times \mathbf{u}_{\mathbf{2}} & \mathbf{v}_{\mathbf{23}}\times \mathbf{u}_{\mathbf{3}} \end{array}\right]\end{equation}

Thus, the forces transmitted by the spherical joint connected to the platform are computed (the matrices A and C-B must be non-singular):

(20) \begin{equation}\left[\begin{array}{c} \mathbf{f}_{\mathbf{11}}\\ \\[-9pt]\mathbf{f}_{\mathbf{12}}\\ \\[-9pt]\mathbf{f}_{\mathbf{13}}\\ \\[-9pt]\mathbf{f}_{\mathbf{21}}\\ \\[-9pt]\mathbf{f}_{\mathbf{22}}\\ \\[-9pt]\mathbf{f}_{\mathbf{23}} \end{array}\right]=\left[\begin{array}{c@{\quad}c} \mathbf{A}^{-\mathbf{1}}+\left(\mathbf{C}-\mathbf{B}\right)^{-\mathbf{1}}\mathbf{B}\mathbf{A}^{-\mathbf{1}} & -\left(\mathbf{C}-\mathbf{B}\right)^{-\mathbf{1}}\\ \\[-9pt]-\left(\mathbf{C}-\mathbf{B}\right)^{-\mathbf{1}}\mathbf{B}\mathbf{A}^{-\mathbf{1}} & \left(\mathbf{C}-\mathbf{B}\right)^{-\mathbf{1}} \end{array}\right]\left[\begin{array}{c} \mathbf{F}\\ \\[-9pt] \mathbf{M} \end{array}\right]\end{equation}

The expressions of the forces transmitted by the i-th leg are

(21) \begin{equation}\mathbf{f}_{\mathbf{1}\mathbf{i}}=\left[\mathbf{L}_{\mathbf{i}}\right]\left[\begin{array}{c} \mathbf{F}\\ \\[-9pt] \mathbf{M} \end{array}\right];\quad \mathbf{f}_{\mathbf{2}\mathbf{i}}=\left[\mathbf{L}_{\mathbf{i}+\mathbf{3}}\right]\left[\begin{array}{c} \mathbf{F}\\ \\[-9pt] \mathbf{M} \end{array}\right]\end{equation}

Where L_i and L_{i + 3} are, respectively, the i-th and (i + 3)-th line vectors (1 $\,{\times}\,$ 6) from the matrix (6 $\,{\times}\,$ 6) defined in Eq. (20).

According to the Eqs. (12, 21), the forces transmitted by the spherical joint centered at C_1i and C_2i are given by

(22) \begin{equation}{\unicode{x1D6D5} }_{1\mathrm{i}}=\mathbf{H}_{1\mathrm{i}}\left[\begin{array}{c} \mathbf{F}\\ \\[-9pt] \mathbf{M} \end{array}\right];\quad {\unicode{x1D6D5} }_{2\mathrm{i}}=\mathbf{H}_{2\mathrm{i}}\left[\begin{array}{c} \mathbf{F}\\ \\[-9pt] \mathbf{M} \end{array}\right]\end{equation}

Where

(23) \begin{equation}\mathbf{H}_{1\mathrm{i}}=\left[\begin{array}{c@{\quad}c} \mathbf{u}_{\mathrm{i}} & \mathbf{0}\\ \\[-9pt] \mathbf{0} & \mathbf{0} \end{array}\right]\left[\begin{array}{c} \mathbf{L}_{\mathrm{i}}\\ \\[-9pt] \mathbf{0} \end{array}\right];\quad \mathbf{H}_{2\mathrm{i}}=\left[\begin{array}{c@{\quad}c} \mathbf{u}_{\mathrm{i}} & \mathbf{0}\\ \\[-9pt] \mathbf{0} & \mathbf{0} \end{array}\right]\left[\begin{array}{c} \mathbf{L}_{\mathrm{i}+3}\\ \\[-9pt] \mathbf{0} \end{array}\right]\end{equation}

According to the equilibrium of bars (3) and (3 $^{\prime}$ ), the forces transmitted by the spherical joint centered at C_3i and C_4i are given, respectively, as

(24) \begin{equation}{\unicode{x1D6D5} }_{3\mathrm{i}}=-{\unicode{x1D6D5} }_{1\mathrm{i}};\quad {\unicode{x1D6D5} }_{4\mathrm{i}}=-{\unicode{x1D6D5} }_{2\mathrm{i}}\end{equation}

3.2. Static model of RAF manipulator

For the RAF manipulator, the static model is already developed ([Reference Chouaibi, Chebbi, Affi and Romdhane11, Reference Affi and Romdhane29] present more details).

If a $\unicode{x1D6D5}$ _ext = [F M]^T is applied to O_p, the forces will be transmitted by the SPS legs and the moments will be supported by the PKL legs (Fig. 4).

Figure 4.

Forces and moments transmitted by the joints of the PKL legs.

The expression of the moments applied to the platform of the manipulator in the reference system fixed to the base can be written as

(25) \begin{equation}\mathbf{M}={\left[\begin{array}{c@{\quad}c@{\quad}c} \mathrm{L}_{\mathrm{p}} & \mathrm{M}_{\mathrm{p}} & \mathrm{N}_{\mathrm{p}} \end{array}\right]}_{\mathrm{S}_{\mathrm{b}}}^{\mathrm{T}}\end{equation}

According to the geometric model of both PKLs, the reaction forces and moments transmitted by the revolute joints D_1i and D_2i with axes $\mathbf{x}_{\mathrm{i}3}$ expressed in the reference system $\mathrm{S}_{3\mathrm{i}}$ can be written, respectively, by

(26) \begin{equation} {\unicode{x1D6D5} }_{\mathbf{1}\mathbf{i}}=\left[\begin{array}{c} \mathbf{f}_{\mathbf{1}\mathbf{i}}\\ \mathbf{m}_{\mathbf{1}\mathbf{i}} \end{array}\right];\quad {\unicode{x1D6D5} }_{\mathbf{2}\mathbf{i}}=\left[\begin{array}{c} \mathbf{f}_{\mathbf{2}\mathbf{i}}\\ \\[-8pt] \mathbf{m}_{\mathbf{2}\mathbf{i}} \end{array}\right]\end{equation}

The reaction forces transmitted by the spherical joints centered at points C_3i and C_4i expressed in the reference system $\mathrm{S}_{3\mathrm{i}}$ can be written, respectively, as

(27) \begin{equation}{\unicode{x1D6D5} }_{\mathbf{3}\mathbf{i}}=\left[\begin{array}{c} \mathbf{f}_{\mathbf{3}\mathbf{i}}\\ \mathbf{0} \end{array}\right];\quad {\unicode{x1D6D5} }_{\mathbf{4}\mathbf{i}}=\left[\begin{array}{c} \mathbf{f}_{\mathbf{4}\mathbf{i}}\\ \mathbf{0} \end{array}\right]\end{equation}

The equilibrium of the body (5), bars (3) and (3^′) are, respectively, given by the following set of equations:

(28) \begin{align}\begin{cases} {\sum }_{\mathbf{j}=\mathbf{1}}^{\mathbf{2}}\,\mathbf{f}_{\mathbf{ji}}=\mathbf{0}\\ \\[-9pt]\mathbf{f}_{\mathbf{1}\mathbf{i}}-\mathbf{f}_{\mathbf{3}\mathbf{i}}=\mathbf{0}\\ \\[-9pt]\begin{array}{l} \mathbf{f}_{\mathbf{2}\mathbf{i}}-\mathbf{f}_{\mathbf{4}\mathbf{i}}=\mathbf{0}\\ \\[-9pt]{\sum }_{\mathbf{j}=\mathbf{1}}^{\mathbf{2}}\mathbf{m}_{\mathbf{ji}}+\mathbf{f}_{\mathbf{1}\mathbf{i}}\times \mathbf{D}_{\mathbf{1}\mathbf{i}} \mathbf{D}_{\mathbf{2}\mathbf{i}}=\left[\begin{array}{c} \mathrm{L}_{\mathrm{i}}\quad 0\quad \mathrm{N}_{\mathrm{i}} \end{array}\right]^{\mathbf{T}}\\ \\[-9pt]\begin{array}{l} \mathbf{m}_{\mathbf{1}\mathbf{i}}+\mathbf{f}_{\mathbf{1}\mathbf{i}}\times \mathbf{D}_{\mathbf{1}\mathbf{i}}\mathbf{C}_{\mathbf{3}\mathbf{i}}=\mathbf{0}\\ \\[-9pt]\mathbf{m}_{\mathbf{2}\mathbf{i}}+\mathbf{f}_{\mathbf{2}\mathbf{i}}\times \mathbf{D}_{\mathbf{2}\mathbf{i}}\mathbf{C}_{\mathbf{4}\mathbf{i}}=\mathbf{0} \end{array} \end{array} \end{cases}\end{align}

By solving the system of Eqs. (28), the forces and the moments transmitted by the joints expressed in the reference system S_3i are given as

(29) \begin{equation}\mathbf{f}_{\mathbf{1}\mathbf{i}}=\mathbf{f}_{\mathbf{3}\mathbf{i}}=-\mathbf{f}_{\mathbf{2}\mathbf{i}}=-\mathbf{f}_{\mathbf{4}\mathbf{i}}={\left[\begin{array}{c@{\quad}c@{\quad}c} \dfrac{\mathrm{N}_{\mathrm{i}}}{\mathrm{a}} & -\dfrac{\mathrm{L}_{\mathrm{i}}}{\mathrm{a}}\tan \unicode{x03C6} _{2\mathrm{i}} & -\dfrac{\mathrm{L}_{\mathrm{i}}}{\mathrm{a}} \end{array}\right]}_{\mathbf{S}_{\mathbf{3}\mathbf{i}}}^{\mathbf{T}}\end{equation}

(30) \begin{equation}\mathbf{m}_{\mathbf{1}\mathbf{i}}=-\mathbf{m}_{\mathbf{2}\mathbf{i}}={\left[\begin{array}{c@{\quad}c@{\quad}c} 0 & -\dfrac{\mathrm{N}_{\mathrm{i}}}{\mathrm{a}}.\mathrm{l}_{3}.\mathrm{c}\unicode{x03C6} _{2\mathrm{i}} & \dfrac{\mathrm{N}_{\mathrm{i}}}{\mathrm{a}}.\mathrm{l}_{3}.\mathrm{s}\unicode{x03C6} _{2\mathrm{i}} \end{array}\right]}_{\mathbf{S}_{\mathbf{31}}}^{\mathbf{T}}\end{equation}

According to the equilibrium of the link (2), the reaction forces and moments transmitted by this revolute joint expressed in the reference system S_3i can be determined:

(31) \begin{align}\begin{cases} \mathbf{f}_{{\mathbf{A}_{\mathbf{i}}}}=\mathbf{0}\\ \\[-9pt] \mathbf{m}_{{\mathbf{A}_{\mathbf{i}}}}=\left[\begin{array}{c@{\quad}c@{\quad}c} \mathrm{L}_{\mathrm{i}} & 0 & \mathrm{N}_{\mathrm{i}} \end{array}\right]^{\mathbf{T}} \end{cases} \end{align}

Li and Ni moments of the first PKL are given by

(32) \begin{align}\begin{cases} \mathrm{L}_{\mathrm{i}}=\left(\mathrm{L}_{\mathrm{p}}\mathrm{c}\unicode{x03B1} _{\mathrm{i}}+\mathrm{M}_{\mathrm{p}}\mathrm{s}\unicode{x03B1} _{\mathrm{i}}\right).\mathrm{c}\unicode{x03C6} _{3\mathrm{i}}\\ \\[-9pt] \mathrm{N}_{\mathrm{i}}=\left(\mathrm{L}_{\mathrm{p}}\mathrm{c}\unicode{x03B1} _{\mathrm{i}}+\mathrm{M}_{\mathrm{p}}\mathrm{s}\unicode{x03B1} _{\mathrm{i}}\right).\mathrm{s}\unicode{x03C6} _{3\mathrm{i}} \end{cases}\end{align}

Li and Ni moments of the second PKL are given by

(33) \begin{align}\begin{cases} \mathrm{L}_{\mathrm{i}}=\left(\mathrm{L}_{\mathrm{p}}\mathrm{c}\unicode{x03B1} _{\mathrm{i}}+\mathrm{M}_{\mathrm{p}}\mathrm{s}\unicode{x03B1} _{\mathrm{i}}\right).\mathrm{c}\unicode{x03C6} _{3\mathrm{i}}-\mathrm{N}_{\mathrm{p}}\mathrm{s}\unicode{x03C6} _{3\mathrm{i}}\\ \\[-9pt] \mathrm{N}_{\mathrm{i}}=\left(\mathrm{L}_{\mathrm{p}}\mathrm{c}\unicode{x03B1} _{\mathrm{i}}+\mathrm{M}_{\mathrm{p}}\mathrm{s}\unicode{x03B1} _{\mathrm{i}}\right).\mathrm{s}\unicode{x03C6} _{3\mathrm{i}}+\mathrm{N}_{\mathrm{p}}\mathrm{c}\unicode{x03C6} _{3\mathrm{i}} \end{cases}\end{align}

5.1. Maximum orientation error

According to Eq. (39), the analytic expression of orientation error caused by the clearance in passive joints for a given external wrench applied to the reference point Op of the platform can be written as

(40) \begin{equation}\boldsymbol\Gamma_{\mathrm{k}}=\mathrm{h}(\mathrm{x}) \end{equation}

where

$\boldsymbol{\Gamma}$ _k ( $\mathrm{k}=4,5,6$ ) represents the k-th component of the vector $\boldsymbol{\Delta\Gamma}$ .

x is the vector (6 ×1) of variables which represents the external wrench applied on the reference point O_p of the platform $\unicode{x1D6D5}$ _ext.

The index intended is to find the maximum orientation error of the manipulator in each position of the workspace without depending on the direction of $\unicode{x1D6D5}$ _ext.

The maximum orientation error corresponds to the optimization of the objective function defined by

(41) \begin{align}\begin{cases} \max \left(\mathrm{h}\!\left(\mathbf{x}\right)\right)\\ \\[-9pt] \mathbf{l}_{\mathrm{b}}\leq \mathbf{x}\leq \mathbf{u}_{\mathrm{b}} \end{cases}\end{align}

Where l _b and u _b correspond to the vectors of the lower and the upper bounds of the variables, respectively. Since the orientation error of the platform depends on the direction of the external wrench $\unicode{x1D6D5}$ _ext and does not depend on its magnitude [Reference Chebbi, Affi and Romdhane2, Reference Chouaibi, Chebbi, Affi and Romdhane11]. The magnitude of the external wrench does not affect the orientation of the platform when considering the clearance in the joints. However, the direction of the external wrench can affect this orientation. Therefore, by selecting all possible values for the six components, from {–1, 0, 1} of the external wrench one can cover all its possible directions (3⁶ = 729). Then, the vectors l _b and u _b are defined as follows:

(42) \begin{equation}\mathbf{u}_{\mathrm{b}}=-\mathbf{l}_{\mathrm{b}}=\left[\begin{array}{c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c} 1 & 1 & 1 & 1 & 1 & 1 \end{array}\right]^{\mathrm{T}}\end{equation}

Using the value of the vectors u _b and l _b and taking an arbitrary initial guess, x ₀, of the vector of variable x, the problem defined by Eq. (41) can be solved. But the obtained result represents a local maximum orientation error of the objective function h(x). One of the best alternatives to solve the problem is to consider a combination of external wrench applied to the platform $\unicode{x1D6D5}$ _ext, as an initial guess. Thus, the maximum orientation error chosen corresponds to the maximum of all the local maxima of h(x). The maximum orientation error of the platform due to the clearance in passive joints which will be taken as index in order to compare TPMs. A flow chart of the calculation process is illustrated by Fig. 5.

Figure 5.

Flow chart of the maximum orientation error.

The MATLAB Optimization Toolbox can be used to solve the problem defined by Eq. (41).

5.2. Condition index

In order to analyze the sensitivity of the orientation error of the manipulator to the geometric parameters variations and the clearance in the passive joints, a condition index is used [Reference Merlet31, Reference Gosselin and Angeles32]. The relationships between the variation of the orientation error and the variations of the geometric parameters and the joint clearance are given by

(43) \begin{equation}{\unicode{x1D6C5}{\unicode{x03B8}} }=\mathbf{J}_{1\mathrm{M}}.{\unicode{x1D6C5} }\mathbf{q}_{1}\end{equation}

(44) \begin{equation}{\unicode{x1D6C5}{ \unicode{x03B8}} }=\mathbf{J}_{\mathbf{2}\mathbf{M}}.{\unicode{x1D6C5} }\mathbf{q}_{\mathbf{2}}\end{equation}

Where

$\unicode{x1D6C5}{\unicode{x03B8}}$ = [δ $\unicode{x03B8}$ ₁ δ $\unicode{x03B8}$ ₂ δ $\unicode{x03B8}$ ₃]^T corresponds to the vector of the orientation errors variation of the manipulator.

J _1M and J _2M are a (3 ×n_p) matrices. Where n_p is the number of geometric parameters and joint clearances.

δq ₁ and δq ₂ stand for the vectors of the variations in geometric parameters and joint clearances, respectively.

The condition number of the geometric parameters and joint clearances are defined as follows:

(45) \begin{equation}\mathbf{k}_{\mathbf{p}}\!\left(\mathbf{J}_{\mathbf{1}\mathbf{M}}\right)=\left|\left|\mathbf{J}_{\mathbf{1}\mathbf{M}}\right|\right|.\left|\left|{\mathbf{J}}_{\mathbf{1}\mathbf{M}}^{-\mathbf{1}}\right|\right|\end{equation}

(46) \begin{equation}\mathbf{k}_{\mathbf{j}}\!\left(\mathbf{J}_{\mathbf{2}\mathbf{M}}\right)=\left|\left|\mathbf{J}_{\mathbf{2}\mathbf{M}}\right|\right|.\left|\left|{\mathbf{J}}_{\mathbf{2}\mathbf{M}}^{-\mathbf{1}}\right|\right|\end{equation}

k_p and kj characterize the sensitivity of the orientation error of the manipulator to variations in the geometric parameters and the joint clearances.

The condition number that expresses how a relative variation in q (geometric parameters and joint clearances) gets multiplied and leads to a relative variation in $\unicode{x03B8}$ (orientation error) will be used as a performance index.

The condition number ranges from 1 to infinity. For more practical representation, the 1/k (0 ≤ (1/k) ≤ 1), referred to the inverse condition index, is used. When the value of (1/k) is close to 1 it indicates that the matrix is less sensitive, if it is small that means the sensitivity of the matrix to the parameters is important. The value of (1/k) is equal to zero, corresponds to a singular Jacobian matrix J. To give a physical meaning to the condition number, we have analyzed the limit values. It is apparent that the robot which has limit values close to 1 is the least sensitive.

For the RAF, the expression of the q ₁(1 ×12) and q ₂(1 ×22) vectors is, respectively, given by

(47) \begin{equation}{\unicode{x1D6C5} }\mathbf{q}_{\mathbf{1}}=\left[\begin{array}{c@{\quad}c} {{\unicode{x1D6C5} }\mathbf{q}_{\mathbf{1}}}_{\mathbf{1}} & {{\unicode{x1D6C5} }\mathbf{q}_{\mathbf{1}}}_{\mathbf{2}} \end{array}\right]^{\mathrm{T}}\end{equation}

(48) \begin{equation}{{\unicode{x1D6C5} }\mathbf{q}_{\mathbf{1}}}_{i}=\left[\begin{array}{c@{\quad}c} \begin{array}{c@{\quad}c@{\quad}c@{}} \unicode{x03B4} \mathrm{l}_{2\mathrm{i}} & \unicode{x03B4} \mathrm{l}_{3\mathrm{i}} & \unicode{x03B4} \mathrm{a}_{\mathrm{i}} \end{array} & \begin{array}{c@{\quad}c@{\quad}c} \unicode{x03B4} \unicode{x03BB} _{\mathrm{i}} & \unicode{x03B4} {\mathrm{r}_{\mathrm{b}}}_{\mathrm{i}} & \unicode{x03B4} {\mathrm{r}_{\mathrm{p}}}_{\mathrm{i}} \end{array} \end{array}\right]^{\mathrm{T}}\end{equation}

Where $\unicode{x03B4} \mathrm{l}_{2\mathrm{i}}$ , $\unicode{x03B4} \mathrm{l}_{3\mathrm{i}}$ , $\unicode{x03B4} \mathrm{a}_{\mathrm{i}}$ , $\unicode{x03B4} \unicode{x03BB} _{\mathrm{i}}$ , $\unicode{x03B4} {\mathrm{r}_{\mathrm{b}}}_{\mathrm{i}}$ and $\unicode{x03B4} {\mathrm{r}_{\mathrm{p}}}_{\mathrm{i}}$ describe the variations in the geometric parameters $\mathrm{l}_{2\mathrm{i}}$ , $\mathrm{l}_{3\mathrm{i}}$ , $\mathrm{a}_{\mathrm{i}}$ , $\unicode{x03BB} _{\mathrm{i}}$ , ${\mathrm{r}_{\mathrm{b}}}_{\mathrm{i}}$ and ${\mathrm{r}_{\mathrm{p}}}_{\mathrm{i}}$ , respectively, of the i-th leg.

(49) \begin{equation}{\unicode{x1D6C5} }\mathbf{q}_{\mathbf{2}}=\left[\begin{array}{@{}c@{\quad}c@{}} {{\unicode{x1D6C5} }\mathbf{q}_{\mathbf{2}}}_{\mathbf{1}} & {{\unicode{x1D6C5} }\mathbf{q}_{\mathbf{2}}}_{\mathbf{2}} \end{array}\right]^{\mathrm{T}}\end{equation}

(50) \begin{equation}{\unicode{x1D6C5} }{\mathbf{q}_{\mathbf{2}}}_{\mathbf{1}}=\left[\begin{array}{@{}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{\quad}c@{}} \unicode{x03B4} \mathbf{E}_{\mathrm{G}\mathrm{'}11} & \unicode{x03B4} \mathbf{E}_{\mathrm{G}11} & \unicode{x03B4} \mathbf{E}_{\mathrm{D}11} & \unicode{x03B4} \mathbf{E}_{\mathrm{D}21} & \unicode{x03B4} \mathrm{E}_{\mathrm{C}31} & \unicode{x03B4} \mathrm{E}_{\mathrm{C}41} & \unicode{x03B4} \mathbf{E}_{\mathrm{A}11} \end{array}\right]^{\mathrm{T}}\end{equation}

(51) \begin{equation}{\unicode{x1D6C5} }{\mathbf{q}_{\mathbf{2}}}_{\mathbf{2}}=\left[\begin{array}{@{}c@{\quad}c} \begin{array}{c@{\quad}c@{}} \unicode{x03B4} \mathbf{E}_{\mathrm{G}12} & \begin{array}{c@{\quad}c} \unicode{x03B4} \mathbf{E}_{\mathrm{D}12} & \unicode{x03B4} \mathbf{E}_{\mathrm{D}22} \end{array} \end{array} & \begin{array}{c@{\quad}c@{\quad}c@{}} \unicode{x03B4} \mathrm{E}_{\mathrm{C}32} & \unicode{x03B4} \mathrm{E}_{\mathrm{C}42} & \unicode{x03B4} \mathbf{E}_{\mathrm{A}12} \end{array} \end{array}\right]^{\mathrm{T}}\end{equation}

Where $\unicode{x03B4} \mathbf{E}_{\mathrm{G}\mathrm{'}11}=\unicode{x03B4} \mathbf{E}_{\mathrm{G}1\mathrm{i}}=\unicode{x03B4} \mathbf{E}_{\mathrm{A}1\mathrm{i}}=\unicode{x03B4} \mathbf{E}_{\mathrm{D}1\mathrm{i}}=\unicode{x03B4} \mathbf{E}_{\mathrm{D}2\mathrm{i}}=\left[\begin{array}{@{}c@{\quad}c@{}} \unicode{x03B4} \unicode{x025B} _{\mathrm{dji}} & \unicode{x03B4} \unicode{x025B} _{\mathrm{aji}} \end{array}\right]^{\mathrm{T}}$ describes the variations in the radial and axial clearances, respectively, in each j-th revolute joint of the i-th leg.

$\unicode{x03B4} \mathrm{E}_{\mathrm{Cji}}=\unicode{x03B4} \unicode{x025B} _{\mathrm{ji}}$ describe the variations in the clearance of j-th spherical joint of the i-th leg.

The expression of the J _1M (3 ×12) and J _2M (3 ×22) matrices becomes

(52) \begin{equation}\mathbf{J}_{\mathbf{M}\mathbf{1}}=\left(\begin{array}{c@{\quad}c@{\quad}c} \begin{array}{c@{\quad}c@{\quad}c} \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{1}}}{\partial \mathbf{l}_{\mathbf{21}}} & \ldots & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{1}}}{\partial {\mathbf{r}_{\mathbf{p}}}_{\mathbf{1}}}\\ \\[-7pt] \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{2}}}{\partial \mathbf{l}_{\mathbf{21}}} & \ldots & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{2}}}{\partial {\mathbf{r}_{\mathbf{p}}}_{\mathbf{1}}}\\ \\[-7pt] \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{3}}}{\partial \mathbf{l}_{\mathbf{21}}} & \ldots & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{3}}}{\partial {\mathbf{r}_{\mathbf{p}}}_{\mathbf{1}}} \end{array} & & \begin{array}{c@{\quad}c@{\quad}c} \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{1}}}{\partial \mathbf{l}_{\mathbf{22}}} & \ldots & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{1}}}{\partial {\mathbf{r}_{\mathbf{p}}}_{\mathbf{2}}}\\ \\[-7pt] \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{2}}}{\partial \mathbf{l}_{\mathbf{22}}} & \ldots & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{2}}}{\partial {\mathbf{r}_{\mathbf{p}}}_{\mathbf{2}}}\\ \\[-7pt] \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{3}}}{\partial \mathbf{l}_{\mathbf{22}}} & \ldots & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{3}}}{\partial {\mathbf{r}_{\mathbf{p}}}_{\mathbf{2}}} \end{array} \end{array}\right)\end{equation}

(53) \begin{equation}\mathbf{J}_{\mathbf{M}\mathbf{2}}=\left(\begin{array}{c@{\quad}c@{\quad}c} \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{1}}}{\partial {{\unicode{x1D6C6}}_{\mathbf{d}}}_{\mathbf{G'}\mathbf{11}}} & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{1}}}{\partial {{\unicode{x1D6C6} }_{\mathbf{a}}}_{\mathbf{G'}\mathbf{11}}} & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{1}}}{\partial {{\unicode{x1D6C6} }_{\mathbf{d}}}_{\mathbf{G}\mathbf{11}}}\\ \\[-7pt] \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{2}}}{\partial {{\unicode{x1D6C6} }_{\mathbf{d}}}_{\mathbf{G'}\mathbf{11}}} & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{2}}}{\partial {{\unicode{x1D6C6} }_{\mathbf{a}}}_{\mathbf{G'}\mathbf{11}}} & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{2}}}{\partial {{\unicode{x1D6C6} }_{\mathbf{d}}}_{\mathbf{G}\mathbf{11}}}\\ \\[-7pt] \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{3}}}{\partial {{\unicode{x1D6C6} }_{\mathbf{d}}}_{\mathbf{G'}\mathbf{11}}} & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{3}}}{\partial {{\unicode{x1D6C6} }_{\mathbf{a}}}_{\mathbf{G'}\mathbf{11}}} & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{3}}}{\partial {{\unicode{x1D6C6} }_{\mathbf{d}}}_{\mathbf{G}\mathbf{11}}} \end{array}\begin{array}{c@{\quad}c} \begin{array}{c} \begin{array}{c} \ldots \\ \\[-7pt] \end{array}\\ \\[-7pt] \begin{array}{c} \ldots \\ \\[-7pt] \end{array}\\ \\[-7pt] \ldots \end{array} & \begin{array}{c@{\quad}c@{\quad}c} \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{1}}}{\partial {\unicode{x1D6C6} }_{\mathbf{42}}} & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{1}}}{\partial {{\unicode{x1D6C6} }_{\mathbf{d}}}_{\mathbf{A}\mathbf{12}}} & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{1}}}{\partial {{\unicode{x1D6C6} }_{\mathbf{a}}}_{\mathbf{A}\mathbf{12}}}\\ \\[-7pt] \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{2}}}{\partial {\unicode{x1D6C6} }_{\mathbf{42}}} & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{2}}}{\partial {{\unicode{x1D6C6} }_{\mathbf{d}}}_{\mathbf{A}\mathbf{12}}} & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{2}}}{\partial {{\unicode{x1D6C6} }_{\mathbf{a}}}_{\mathbf{A}\mathbf{12}}}\\ \\[-7pt] \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{3}}}{\partial {\unicode{x1D6C6} }_{\mathbf{42}}} & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{3}}}{\partial {{\unicode{x1D6C6} }_{\mathbf{d}}}_{\mathbf{A}\mathbf{12}}} & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{3}}}{\partial {{\unicode{x1D6C6} }_{\mathbf{a}}}_{\mathbf{A}\mathbf{12}}} \end{array} \end{array}\right)\end{equation}

For the DELTA manipulator, the components of the geometric parameters vector q ₁ and the joint clearances vector q ₂ are, respectively, given by

(54) \begin{equation}{\unicode{x1D6C5} }\mathbf{q}_{\mathbf{1}}=\left[\begin{array}{c@{\quad}c@{\quad}c} {{\unicode{x1D6C5} }\mathbf{q}_{\mathbf{1}}}_{\mathbf{1}} & {{\unicode{x1D6C5} }\mathbf{q}_{\mathbf{1}}}_{\mathbf{2}} & {{\unicode{x1D6C5} }\mathbf{q}_{\mathbf{1}}}_{\mathbf{3}} \end{array}\right]^{\mathrm{T}}\end{equation}

(55) \begin{equation}{{\unicode{x1D6C5} }\mathbf{q}_{\mathbf{1}}}_{i}=\left[\begin{array}{@{}c@{\quad}c} \begin{array}{c@{\quad}c@{\quad}c@{}} \unicode{x03B4} \mathrm{l}_{2\mathrm{i}} & \unicode{x03B4} \mathrm{l}_{3\mathrm{i}} & \unicode{x03B4} \mathrm{a}_{\mathrm{i}} \end{array} & \begin{array}{c@{\quad}c@{}} \unicode{x03B4} {\mathrm{r}_{\mathrm{b}}}_{\mathrm{i}} & {\unicode{x03B4} \mathrm{r}_{\mathrm{p}}}_{\mathrm{i}} \end{array} \end{array}\right]\end{equation}

Where $\unicode{x03B4} \mathrm{l}_{2\mathrm{i}}$ , $\unicode{x03B4} \mathrm{l}_{3\mathrm{i}}$ , $\unicode{x03B4} \mathrm{a}_{\mathrm{i}}$ , $\unicode{x03B4} {\mathrm{r}_{\mathrm{b}}}_{\mathrm{i}}$ and $\unicode{x03B4} {\mathrm{r}_{\mathrm{p}}}_{\mathrm{i}}$ depicts the variations in the geometric parameters $\mathrm{l}_{2\mathrm{i}}$ , $\mathrm{l}_{3\mathrm{i}}$ , $\mathrm{a}_{\mathrm{i}}$ , ${\mathrm{r}_{\mathrm{b}}}_{\mathrm{i}}$ and ${\mathrm{r}_{\mathrm{p}}}_{\mathrm{i}}$ , respectively, of the i-th leg.

(56) \begin{equation}{\unicode{x1D6C5} }\mathbf{q}_{\mathbf{2}}=\left[\begin{array}{c@{\quad}c@{\quad}c} {{\unicode{x1D6C5} }\mathbf{q}_{\mathbf{2}}}_{\mathbf{1}} & {{\unicode{x1D6C5} }\mathbf{q}_{\mathbf{2}}}_{\mathbf{2}} & {{\unicode{x1D6C5} }\mathbf{q}_{\mathbf{2}}}_{\mathbf{3}} \end{array}\right]^{\mathrm{T}}\end{equation}

(57) \begin{equation}\boldsymbol{\unicode{x03B4} }{\mathbf{q}_{\mathbf{2}}}_{i}=\left[\begin{array}{c@{\quad}c} \begin{array}{c@{\quad}c} \unicode{x03B4} \unicode{x025B} _{1\mathrm{i}} & \unicode{x03B4} \unicode{x025B} _{2\mathrm{i}} \end{array} & \begin{array}{c@{\quad}c@{}} \unicode{x03B4} \unicode{x025B} _{3\mathrm{i}} & \unicode{x03B4} \unicode{x025B} _{4\mathrm{i}} \end{array} \end{array}\right]^{\mathrm{T}}\end{equation}

The expression of the J _1M (3 ×15) and J _2M (3 ×12) matrices becomes

(58) \begin{equation}\mathbf{J}_{\mathbf{M}\mathbf{1}}=\left(\begin{array}{c@{\quad}c@{\quad}c} \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{1}}}{\partial \mathbf{l}_{\mathbf{21}}} & \ldots & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{1}}}{\partial {\mathbf{r}_{\mathbf{p}}}_{\mathbf{1}}}\\ \\[-9pt] \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{2}}}{\partial \mathbf{l}_{\mathbf{21}}} & \ldots & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{2}}}{\partial {\mathbf{r}_{\mathbf{p}}}_{\mathbf{1}}}\\ \\[-9pt] \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{3}}}{\partial \mathbf{l}_{\mathbf{21}}} & \ldots & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{3}}}{\partial {\mathbf{r}_{\mathbf{p}}}_{\mathbf{1}}} \end{array}\begin{array}{c@{\quad}c@{\quad}c} \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{1}}}{\partial \mathbf{l}_{\mathbf{22}}} & \ldots & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{1}}}{\partial {\mathbf{r}_{\mathbf{p}}}_{\mathbf{2}}}\\ \\[-9pt] \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{2}}}{\partial \mathbf{l}_{\mathbf{22}}} & \ldots & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{2}}}{\partial {\mathbf{r}_{\mathbf{p}}}_{\mathbf{2}}}\\ \\[-9pt] \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{3}}}{\partial \mathbf{l}_{\mathbf{22}}} & \ldots & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{3}}}{\partial {\mathbf{r}_{\mathbf{p}}}_{\mathbf{2}}} \end{array}\begin{array}{c@{\quad}c@{\quad}c} \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{1}}}{\partial \mathbf{l}_{\mathbf{23}}} & \ldots & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{1}}}{\partial {\mathbf{r}_{\mathbf{p}}}_{\mathbf{3}}}\\ \\[-9pt] \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{2}}}{\partial \mathbf{l}_{\mathbf{23}}} & \ldots & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{2}}}{\partial {\mathbf{r}_{\mathbf{p}}}_{\mathbf{3}}}\\ \\[-9pt] \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{3}}}{\partial \mathbf{l}_{\mathbf{23}}} & \ldots & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{3}}}{\partial {\mathbf{r}_{\mathbf{p}}}_{\mathbf{3}}} \end{array}\right)\end{equation}

(59) \begin{align} \nonumber\\[-25pt] \mathbf{J}_{\mathbf{M}\mathbf{2}}=\left(\begin{array}{c@{\quad}c@{\quad}c} \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{1}}}{\partial {\unicode{x1D6C6} }_{\mathbf{11}}} & \ldots & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{1}}}{\partial {\unicode{x1D6C6} }_{\mathbf{41}}}\\ \\[-7pt] \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{2}}}{\partial {\unicode{x1D6C6} }_{\mathbf{11}}} & \ldots & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{2}}}{\partial {\unicode{x1D6C6} }_{\mathbf{41}}}\\ \\[-7pt] \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{3}}}{\partial {\unicode{x1D6C6} }_{\mathbf{11}}} & \ldots & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{3}}}{\partial {\unicode{x1D6C6} }_{\mathbf{41}}} \end{array}\begin{array}{c@{\quad}c@{\quad}c} \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{1}}}{\partial {\unicode{x1D6C6} }_{\mathbf{12}}} & \ldots & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{1}}}{\partial {\unicode{x1D6C6} }_{\mathbf{42}}}\\ \\[-7pt] \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{2}}}{\partial {\unicode{x1D6C6} }_{\mathbf{12}}} & \ldots & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{2}}}{\partial {\unicode{x1D6C6} }_{\mathbf{42}}}\\ \\[-7pt] \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{3}}}{\partial {\unicode{x1D6C6} }_{\mathbf{12}}} & \ldots & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{3}}}{\partial {\unicode{x1D6C6} }_{\mathbf{42}}} \end{array}\begin{array}{c@{\quad}c@{\quad}c} \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{1}}}{\partial {\unicode{x1D6C6} }_{\mathbf{13}}} & \ldots & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{1}}}{\partial {\unicode{x1D6C6} }_{\mathbf{43}}}\\ \\[-7pt] \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{2}}}{\partial {\unicode{x1D6C6} }_{\mathbf{13}}} & \ldots & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{2}}}{\partial {\unicode{x1D6C6} }_{\mathbf{43}}}\\ \\[-7pt] \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{3}}}{\partial {\unicode{x1D6C6} }_{\mathbf{13}}} & \ldots & \dfrac{\partial {\unicode{x1D6C9} }_{\mathbf{3}}}{\partial {\unicode{x1D6C6} }_{\mathbf{43}}} \end{array}\right)\end{align}

6.1. Comparison of maximum orientation error

In this section, two manipulators are compared with regard to maximum orientation errors.

Figure 6 represents a comparative of the ranges of orientation errors ( $\unicode{x03B8}$ ₁, $\unicode{x03B8}$ ₂ and $\unicode{x03B8}$ ₃) for the two manipulators evaluated throughout the workspace.

Figure 6.

Maximum orientation error of manipulators.

Based on the simulated results, it can be noted that for the DELTA manipulator, the orientation error $\unicode{x03B8}$ ₃ around z-axis varying between 4 10^–2 and 13 10^–2 deg is the more dominant than the errors $\unicode{x03B8}$ ₁ around x-axis and $\unicode{x03B8}$ ₂ around y-axis were variations in the ranges [2.3 10^–2, 6.9 10^–2] deg and [2.7 10^–2, 8.5 10^–2] deg, respectively.

For the RAF manipulator, the orientation error $\unicode{x03B8}$ ₃ ranging between 6.1 10^–2 and 17.6 10^–2 deg is smaller than the errors $\unicode{x03B8}$ ₁ and $\unicode{x03B8}$ ₂ were ranging between 17.4 10^–2 and 23.3 10^–2 deg and 15.5 10^–2 and 21.4 10^–2 deg, respectively.

In addition, it can be noted that the minimum orientation errors of $\unicode{x03B8}$ ₁ and $\unicode{x03B8}$ ₂ of RAF manipulator are higher than the maximum orientation error of the DELTA robot. Similarly, we can notice that the maximum orientation errors $\unicode{x03B8}$ ₁ and $\unicode{x03B8}$ ₂ of the DELTA does not exceed 40% of the maximum orientation errors $\unicode{x03B8}$ ₁ and $\unicode{x03B8}$ ₂ of the RAF manipulator. This result is due to the use of the spherical joints only in the parallelogram structure of the DELTA manipulator. As shown, the revolute joint clearance is an important factor that affects the orientation errors of the RAF manipulator and should be treated with caution.

For the same workspace, the orientation errors of RAF manipulator have a larger range than the DELTA manipulator. However, there is no significant difference in the orientation error $\unicode{x03B8}$ ₃ for both manipulators. Therefore, if we consider the maximum orientation errors, the DELTA structure will be a better choice for the tasks.

6.2. Sensitivity of the orientation error

In this section, the sensitivity of the orientation errors of the manipulators DELTA and RAF to the geometric parameters and joint clearances is presented. This study will be divided in two parts. The first part is related to the sensitivity to the geometric parameters variations and the second part is related to the sensitivity to the joint clearances.

The upper and lower bounds of the sensitivity of the orientation error to the variations in the geometric parameters of both manipulators are illustrated in Fig. 7. According to the obtained results, it can be seen that the orientation error of the DELTA robot is less sensitive than that the RAF manipulator. Moreover, the variation of the sensitivity of the orientation error for the RAF manipulator is more important than the DELTA robot. Also, the sensitivity of RAF is relatively uniform throughout the workspace by comparing it with the DELTA.

Figure 7.

Limits of the sensitivity of the orientation error to the geometric parameters.

Figure 8 sheds light on the distribution of the upper and lower bounds of the sensitivity of the orientation error of manipulators to the joint clearances. First, it can be observed that the orientation error of the RAF manipulator is more sensitive to the variations in the joint clearances than the variations in the geometric parameters. Then, it can be noticed that the sensitivity of the orientation error to the geometric parameters of the DELTA manipulator is the same as the joint clearances. This result is due to the only use of the spherical joints in the manipulator. Finally, it can be concluded that the manipulator DELTA is less sensitive to the variations of the joint clearances by comparing it with the RAF. However, this advantage is limited in a reduced workspace free from all the singular configurations. Therefore, the manipulator RAF represents a high sensitivity to the joint clearances which has as origin the existence of the revolute joints. Moreover, this manipulator maintains the same level of sensitivity throughout the workspace.

Figure 8.

Limits of the sensitivity of the orientation error to the joint clearances.