2.1. Cubic Bézier basis functions with three shape parameters

Definition 1. The Bernstein basis functions of $t,\:t\in\left[0,1\right]$ with degree three and three shape parameters are defined as:

(1) \begin{equation}\begin{cases}b_{0}(t)= & (1-t)^{3}\left(1-\lambda_{1}t\right)\\[8pt]b_{1}(t)= & 3(1-t)^{2}t\left[1+\dfrac{\lambda_{1}}{3}(1-t)-\dfrac{\lambda_{2}}{2}t\right]\\[18pt]b_{2}(t)= & 3(1-t)t^{2}\left[1+\dfrac{\lambda_{2}}{2}(1-t)-\dfrac{\lambda_{3}}{3}t\right]\\[12pt]b_{3}(t)= & t^{3}\left[1+\lambda_{3}(1-t)\right]\end{cases}\end{equation}

where $-2<\lambda_{1}<1$ , $-1<\lambda_{2}<2$ , and $-1<\lambda_{3}<3$ . Figure 1 shows the curves of the Bernstein basis functions with degree three for different values of $\lambda_{1}$ , $\lambda_{2}$ , and $\lambda_{3}$ .

Figure 1. The Bernstein basis functions with degree three for $\lambda_{1}=-1.5$ , $\lambda_{2}=1$ , $\lambda_{3}=-0.8$ (solid lines), for $\lambda_{1}=-1$ , $\lambda_{2}=-0.5$ , $\lambda_{3}=1.5$ (dashed lines) and for $\lambda_{1}=0.7$ , $\lambda_{2}=1.5$ , $\lambda_{3}=2.5$ (dotted lines).

Proof. For $t\in\left[0,1\right]$ , and $-2<\lambda_{1}<1$ , $-1<\lambda_{2}<2$ and $-1<\lambda_{3}<3$ , it is obvious from the Eq. (1) that the proofs of the Theorems 1.1, 1.3, and 1.4 can be seen in ref. [Reference Yang and Zeng26]. For the Theorem 1.2:

$\sum\limits_{i=0}^{3} b_{i}(t)=1+ \sum\limits_{i=1}^{3}\lambda_{i}\left(\frac{1}{3-i+1}(1-t)B_{i,3}(t)-\frac{1}{i}tB_{i-1,3}(t)\right)\equiv1$ , in which $B_{i,3}(t)$ and $B_{i-1,3}(t)$ are the original Bernstein basis functions.

2.2. Properties of cubic Bézier curve functions with three shape parameters

Definition 2. The cubic Bézier curve with three shape parameters is defined by:

(2) \begin{equation}\boldsymbol{r}(t)= \sum\limits_{i=0}^{3}\boldsymbol{P}_{i}b_{i}(t),\:t\in\left[0,1\right],\,\lambda_{1}\in\left(-2,1\right),\,\lambda_{2}\in\left(-1,2\right),\,\lambda_{3}\in\left(-1,3\right)\end{equation}

where $\boldsymbol{P}_{i}$ is control points in $R^{2}$ or $R^{3}$ .

Proof. The proofs of all the above properties can be easily obtained in ref. [Reference Yang and Zeng26].

The shape parameters $\lambda_{1}$ , $\lambda_{2}$ , and $\lambda_{3}$ provide the local control on the cubic Bézier curve according to Proposition 3 in ref. [Reference Yang and Zeng26] as shown in Fig. 2.

Figure 2. Effect of the altered shape parameters on the shape of the cubic Bézier curve.

Proposition 2. A regular curve $\boldsymbol{r}(t)$ is given in $R^{3}$ . The curvature of this curve is defined by:

(3) \begin{equation}\kappa=\frac{\left\Vert \boldsymbol{r}^{\prime\prime}\times\boldsymbol{r}^{\prime}\right\Vert }{\left\Vert \boldsymbol{r}^{\prime}\right\Vert ^{3}}\end{equation}

where the $\times$ and ^′ denote the cross product and $\dfrac{d}{dt}$ , respectively.

Proof. The proof of this proposition can be obtained from ref. [Reference Pressley29].

Proposition 3. A regular curve $\boldsymbol{r}(t)$ is given with nowhere-vanishing in $R^{3}$ . The torsion of this curve is given by:

(4) \begin{equation}\tau=\frac{\left(\boldsymbol{r}^{\prime}\times\boldsymbol{r}^{\prime\prime}\right).\boldsymbol{r}^{\prime\prime\prime}}{\left\Vert \boldsymbol{r}^{\prime}\times\boldsymbol{r}^{\prime\prime}\right\Vert ^{2}}.\end{equation}

Proof. The proof of this proposition can be found from ref. [Reference Pressley29].

Definition 2. The arc length of a curve $\boldsymbol{r}(t)$ starting at the point $\boldsymbol{r}\left(t_{0}\right)$ is given by the function $L(t)$ as [Reference Pressley29]:

(5) \begin{equation}L(t)= \int\limits_{t_{0}}^{t} \left\Vert \boldsymbol{r}^{\prime}\left(u\right)\right\Vert du.\end{equation}

The curvature and the torsion of the cubic Bézier curve with three shape parameters can be calculated using the Eqs. (3) and (4). Since the Bézier curve has shape parameters, the curvature and the torsion of this curve will be affected. Figure 3 shows the influence of the three shape parameters on curvature and torsion of cubic Bézier curve.

Figure 3. The curvature and the torsion curves of the cubic Bézier curve for different values of shape parameters.

3.1. Proposed method

Hierarchical clustering is a clustering method that finds successive clusters using a predetermined ordering from top to bottom. These successive clusters can be presented as a tree called a dendrogram. The divisive and the agglomerative hierarchical clustering are the clustering types, where the former constructs the dendrogram top-down and the latter constructs the dendrogram bottom-up. Agglomerative hierarchical clustering generates the dendrogram by taking each element as a separate cluster and then merging them iteratively. On the other hand, the divisive hierarchical clustering generates the dendrogram by taking the data as a single cluster and then separating this cluster into successively smaller clusters iteratively [Reference Ranka, Singh and Alsabti30].

The optimal path planning method can be carried out using a cubic Bézier curve with three shape parameters and hierarchical clustering. Different curve pairs with the same control points are generated with different shape parameters, since the main goal is to control the path.

Suppose that the robotic arm’s starting and goal points and obstacles placed in the workspace are known. Note that this study focuses on general cases, that is, extreme cases are not included, such as all obstacles placed through a line between starting and goal points. Next, several cubic Bézier curves are formed according to the obstacles in the workspace and altered by changing the shape parameters randomly within their range given as in the Eq. (2). The feature vector F, including the curvature, torsion, and path length of each curve, is obtained for each curve, using Eqs. (3), (4), and (5):

(6) \begin{equation} F = \left[\kappa, \tau, L \right] \end{equation}

Moreover, we need to check whether any obstacle collides to the optimal curve pair candidate and remove them if they collide with obstacles. To cluster the optimal path candidates, the hierarchical clustering method is used. The multi-objective function value $OP_{sum} = OP_{1} + OP_{2}$ for each curve pair is determined using the following equation for $i=1, 2$ :

(7) \begin{align} OP_{i} & = \alpha_{1}O_{Ji} + \alpha_{2}O_{\kappa i} + \alpha_{3}O_{di}+ \alpha_{4}O_{Li}\nonumber\\[3pt] & = \alpha_{1}\left(\int_{u_{s}}^{u_{f}}\left\Vert j\left(u\right)\right\Vert^{2} du \right) + \alpha_{2}\left( \int_{u_{s}}^{u_{f}}\left\Vert \kappa\left(u\right)\right\Vert^{2} du\right)\nonumber \\[3pt] & \quad + \alpha_{3} \left( mean\left(\sum_{i=1}^{4} d_{i} \right)\right) + \alpha_{4}\left( \int_{a}^{b}\left\Vert \boldsymbol{r}^{\prime}\left(u\right)\right\Vert du\right),\quad \left(i=1,2 \right) \end{align}

inwhich the coefficient values are denoted by $\alpha_{i}$ , $\left( i=1,2,3,4\right)$ and provide $\sum_{i=1}^{4}\alpha_{i} = 1 $ . Also, $O_{Ji}$ , $O_{\kappa i}$ , $O_{di}$ , and $O_{Li}$ , $\left(i=1,2 \right) $ are the objective functions according to the jerk, curvature, Euclidean distance to the original line path passes through the obstacles, and path length for the optimal path candidate pairs, respectively. The optimum curve is the one with the minimum OP value between two path candidate pairs as in Eq. (8):

(8) \begin{equation} OP_{\min} = \min \left\lbrace OP_{1},OP_{2}\right\rbrace \end{equation}

where OP1 and OP2 belongs to $OP_{sum}$ that has the minimum value. The steps of the proposed method are given in Algorithms 1, 2, and 3. Algorithm 1 computes the multi-objective function for the optimal path candidates.

If the optimal path can not be determined after the selected iteration number, the proposed Optimal Path Algorithm with Hierarchical Cluster (OPA-H) should be executed again with different shape parameters. The proposed OPA-H is shown in Algorithm 2:

To check whether any obstacle collides to the optimal curve pair candidate, the Algorithm 3 is utilized.

The original line path, which is the convex hull of the cubic Bézier path, passes through the waypoints, which are obstacles, and therefore this path collides with the obstacles. Hence, the cubic Bézier curve with three shape parameters is used to obtain the optimal path because this curve does not collide with other obstacles, since this curve does not pass through the control points except the starting and end points.

We need to obtain the optimal path candidate pairs among different path candidates. Therefore, this study utilizes hierarchical clustering, since it is the most appropriate method for using the bottom-up approach. The optimal path is selected during experiments regarding the multi-objective function among the path candidates. The optimal path provides efficiency and flexibility because it provides more options by altering shape parameters and controlling the optimal path besides minimizing the multi-objective function. Users can adjust path length, jerk, acceleration, which is related to the curvature of the path, and smoothness, which is related to the curvature and torsion, based on the shape parameters. The reason is that altering the shape parameters without changing the obstacles offers different options to the user to control the path, such as if the user desires to control the jerk increasing the $\alpha_{1}$ coefficient value, the user may determine the optimal path by altering the shape parameters as it is done in Tables II, 4, and 6.

4.1. Example 1

Let the starting and the goal points be $\boldsymbol{P}_{0}=\left(27.25,\,9,\,14\right)$ and $\boldsymbol{P}_{19}=\left(37.25,\,29,\,29\right)$ , respectively. Also, the obstacles in the workspace are given in the matrix form by the following Eq. (9) as:

(9) \begin{equation}O=\left[\begin{array}{ccc}\boldsymbol{P}_{1}=\left(31,19,18\right); & & \boldsymbol{P}_{2}=\left(17,24,14\right)\\[3pt]\boldsymbol{P}_{3}=\left(17,24,21\right); & & \boldsymbol{P}_{4}=\left(27,20,18\right)\\[3pt]\boldsymbol{P}_{5}=\left(27,12,23\right); & & \boldsymbol{P}_{6}=\left(18,22,17\right)\\[3pt]\boldsymbol{P}_{7}=\left(22,15,12\right); & & \boldsymbol{P}_{8}=\left(22,14,22\right)\\[3pt]\boldsymbol{P}_{9}=\left(23,19,18\right); & & \boldsymbol{P}_{10}=\left(23,16,16\right)\\[3pt]\boldsymbol{P}_{9}=\left(15,18,25\right); & & \boldsymbol{P}_{10}=\left(9,13,12\right)\\[3pt]\boldsymbol{P}_{11}=\left(21,21,28\right); & & \boldsymbol{P}_{12}=\left(11,19,7\right)\\[3pt]\boldsymbol{P}_{13}=\left(20,24,11\right); & & \boldsymbol{P}_{14}=\left(14,14,26\right)\\[3pt]\boldsymbol{P}_{15}=\left(13,19,9\right); & & \boldsymbol{P}_{16}=\left(24,28,15\right)\\[3pt]\boldsymbol{P}_{17}=\left(10,13,4\right); & & \boldsymbol{P}_{18}=\left(13,29,24\right)\end{array}\right].\end{equation}

In this example, we generated 10 cubic Bézier path candidates where each path takes the points $\boldsymbol{P}_{0}$ and $\boldsymbol{P}_{19}$ as the starting and the goal points, and each row of the matrix in the Eq. (9), which corresponds to obstacles, like other control points of the path of the robotic arm. Next, we carried out the proposed method (OPA-H) given in the Algorithm 2 by changing the shape parameters randomly within their range. The results can be seen from the following Table I. Here, the curves in each row represent the curves that have the same control points with different shape parameters.

Table I. Test results.

The best values are shown in bold face.

As seen from Table I, the curve pair $\left\{r_{4},r_{14}\right\} $ in row 9 has the minimum multi-objective value, which is $1.2976$ . Note that this value is the sum of $OP_1$ and $OP_2$ . From this pair $r_{14}$ is selected as the optimal curve, since it has lower OP value than $r_{4}$ .

On the other hand, the optimal path between $r_{4}$ and $r_{14}$ can be obtained for the Kinova Gen3 robotic arm using its joints velocities and accelerations from the Fig. 6.

Figure 6. Trapezoidal position, velocity, and acceleration profiles of seven joints of the Kinova Gen3 robotic arm for the paths $r_{4}$ (straight) and $r_{14}$ (dashed).

Figure 7. The optimal path $r_{14}$ for different values of the shape parameters in Table I.

As seen from Fig. 6 and Table I, the path $r_{14}$ , which is obtained according to the multi-objective using the proposed algorithm, is the optimum one for the robotic arm. The optimal path $r_{14}$ can be controlled via three shape parameters as given in Table I and can be seen from different perspectives in Fig. 7.

If all shape parameters are accepted as zero, the curve transforms to the classical Bézier curve CBC. The proposed method provides flexibility to fine-tune the optimal path regarding user requirements, which are given by the multi-objective function in the Eq. (7), for the robotic arm in the workspace. This study compares the optimal curve for different shape parameters and the CBC in path length. Therefore, Fig. 8 presents the superiority of the optimal curve to the CBC due to its flexibility which means we can fine-tune the optimal curve according to the priorities of the user, such as velocity, acceleration, jerk, or path length. Because shape parameters are real numbers, they can have an infinite number of values in their range.

Figure 8. The path lengths of the optimal path $r_{14}$ for different values of the shape parameters and the CBC (black).

Also, the path lengths of the optimal path and CBC are given in Table II.

Table II. The comparision of optimal curve with different shape parameters and CBC.

As seen from Table II, we have different options to alter the path besides providing being more productive in terms of the multi-objective function in (7) than the CBC.

4.2. Example 2

In this example, let the initial and the end points $\boldsymbol{P}_{0}=\left(5,\,0.5,\,2\right)$ and $\boldsymbol{P}_{19}=\left(16,\,10,\,10\right)$ be given, respectively. Also, the obstacles placed in the workspace are given in the below Eq. (10):

(10) \begin{equation}O = \left[\begin{array}{ccc}\boldsymbol{P}_{1}=\left(12,5,5\right); & & \boldsymbol{P}_{2}=\left(3,8,8\right)\\[5pt]\boldsymbol{P}_{3}=\left(10,8,8\right); & & \boldsymbol{P}_{4}=\left(6,12,12\right)\\[5pt]\boldsymbol{P}_{5}=\left(-2,4,8\right); & & \boldsymbol{P}_{6}=\left(-4,5,12\right)\\[5pt]\boldsymbol{P}_{7}=\left(2,4,6\right); & & \boldsymbol{P}_{8}=\left(4,5,7\right)\\[5pt]\boldsymbol{P}_{9}=\left(13,4,6\right); & & \boldsymbol{P}_{10}=\left(14,5,7\right)\\[5pt]\boldsymbol{P}_{9}=\left(15,6,8\right); & & \boldsymbol{P}_{10}=\left(17,15,7\right)\\[5pt]\boldsymbol{P}_{11}=\left(16,3,3\right); & & \boldsymbol{P}_{12}=\left(18,10,5\right)\\[5pt]\boldsymbol{P}_{13}=\left(2,3,3\right); & & \boldsymbol{P}_{14}=\left(5,10,5\right)\\[5pt]\boldsymbol{P}_{15}=\left(26,26,6\right); & & \boldsymbol{P}_{16}=\left(28,28,8\right)\\[5pt]\boldsymbol{P}_{17}=\left(-6,-6,6\right); & & \boldsymbol{P}_{18}=\left(-8,-8,8\right)\end{array}\right].\end{equation}

We again generated ten cubic Bézier path candidates where each path takes the points $\boldsymbol{P}_{0}$ and $\boldsymbol{P}_{19}$ as the starting and the goal points, and each row in (10) as other control points for the robotic arm. Next, the proposed method OPA-H in the Algorithm 2 was carried out to determine optimal path curve pair in terms of feature vector by changing the shape parameters randomly within their range. The results are given in Table III.

Table III. Test Results.

The best values are shown in bold face.

The curve pair $\left\{ \boldsymbol{r}_{5},\boldsymbol{r}_{15}\right\} $ will be the optimal curve as seen from Table III. Moreover, even the optimal path can be determined based on the multi-objective function between $r_{5}$ and $r_{15}$ , it can be supported using the robotic arm’s joints velocities and accelerations in Fig. 9.

Figure 9. Trapezoidal position, velocity, and acceleration profiles of seven joints of the Kinova Gen3 robotic arm for the paths $r_{5}$ (straight) and $r_{15}$ (dashed).

The path $r_{5}$ is the optimum path for the given robotic arm. The optimal path $r_{5}$ can be controlled via three shape parameters as given in Table III and can be seen from different perspectives in Fig. 10.

Figure 10. The optimal path $r_{5}$ for different values of the shape parameters in Table III.

Figure 11 presents the superiority of the optimal curve to the CBC based on flexibility.

Figure 11. The path lengths of the optimal path $r_{5}$ for different values of the shape parameters and the CBC (black).

Additionally, the path lengths of the optimal path and CBC are given in Table IV.

As seen from the Table IV, the shape parameters give opportunity to control the optimal path besides providing being more productive in terms of the multi-objective function in (7) than the CBC.

Table IV. The comparison of optimal curve with different shape parameters and CBC.

4.3. Example 3

Assume that the starting and the goal points are given as $\boldsymbol{P}_{0}=\left(10,\,0,\,0\right)$ and $\boldsymbol{P}_{19}=\left(40,\,40,\,0\right)$ , respectively. Also, the positions of obstacles in the environment are placed by the following equation:

(11) \begin{equation} O=\left[\begin{array}{ccc} \boldsymbol{P}_{1}=\left(40,0,15\right); & & \boldsymbol{P}_{2}=\left(0,40,8\right)\\[5pt] \boldsymbol{P}_{3}=\left(10,8,8\right); & & \boldsymbol{P}_{4}=\left(10,30,12\right)\\[5pt] \boldsymbol{P}_{5}=\left(-2,4,8\right); & & \boldsymbol{P}_{6}=\left(-4,5,12\right)\\[5pt] \boldsymbol{P}_{7}=\left(2,4,6\right); & & \boldsymbol{P}_{8}=\left(4,5,7\right)\\[5pt] \boldsymbol{P}_{9}=\left(13,4,6\right); & & \boldsymbol{P}_{10}=\left(14,5,7\right)\\[5pt] \boldsymbol{P}_{9}=\left(15,6,8\right); & & \boldsymbol{P}_{10}=\left(17,15,7\right)\\[5pt] \boldsymbol{P}_{11}=\left(16,3,3\right); & & \boldsymbol{P}_{12}=\left(18,10,5\right)\\[5pt] \boldsymbol{P}_{13}=\left(2,3,3\right); & & \boldsymbol{P}_{14}=\left(5,10,5\right)\\[5pt] \boldsymbol{P}_{15}=\left(26,26,6\right); & & \boldsymbol{P}_{16}=\left(28,28,8\right)\\[5pt] \boldsymbol{P}_{17}=\left(-6,-6,6\right); & & \boldsymbol{P}_{18}=\left(-8,-8,8\right) \end{array}\right]. \end{equation}

Some obstacles have the same positions as in Example 2. Our goal with this example is to present the case in which the starting and goal points, besides some obstacles, are farther away from each other than the other two examples. Similarly, 10 cubic Bézier path candidates are formed where each path takes the points $\boldsymbol{P}_{0}$ and $\boldsymbol{P}_{19}$ as the starting and the goal points, and each row in (11) indicates other control points for the robotic arm. Next, the proposed method OPA-H in the Algorithm 2 is applied to determine optimal path curve pair in terms of feature vector by changing the shape parameters randomly within their range. The results are given in Table V.

Table V. Test results.

The best values are shown in bold face.

The curve pair $\left\{ \boldsymbol{r}_{5},\boldsymbol{r}_{15}\right\} $ will be the optimal curve as seen from Table V regarding to $OP_{sum}$ .

Additionally, the optimal path between $r_{5}$ and $r_{15}$ can be enforced by using the robotic arm’s joints velocities and accelerations in Fig. 12, although it can be reliazed using the multi-objective function.

Figure 12. Trapezoidal position, velocity, and acceleration profiles of seven joints of the Kinova Gen3 robotic arm for the paths $r_{5}$ (straight) and $r_{15}$ (dashed).

The path $r_{15}$ is the optimum path for the given robotic arm. The optimal path $r_{15}$ can be controlled via three shape parameters as given in Table V and can be seen from different perspectives inFig. 13.

Figure 13. The optimal path $r_{15}$ for different values of the shape parameters in Table V.

Figure 14 presents the superiority of the optimal curve to the CBC based on flexibility.

Figure 14. The path lengths of the optimal path $r_{15}$ for different values of the shape parameters and the CBC (black).

Moreover, the path lengths of the optimal path and CBC are given in Table VI.

Table VI. The comparision of optimal curve with different shape parameters and CBC.

As seen from Table VI, the shape parameters give opportunity to control the optimal path besides providing being more productive in terms of the multi-objective function in (7) than the CBC.

5.1. Ruled and developable ruled surfaces

Definition 3. A union of straight lines is called a ruled surface. Straight lines are called rulings of the ruled surface [Reference Pressley29]. A ruled surface is defined by:

(12) \begin{equation}\boldsymbol{R}(v,t)=\boldsymbol{r}(t)+v\boldsymbol{d}(t)\end{equation}

in which $\boldsymbol{r}(t)$ is the directrix or base curve and $\boldsymbol{d}(t)$ is the direction vector of the ruling at each point on the directrix. Alternatively, the ruled surface can be represented as:

(13) \begin{equation}\boldsymbol{R}(v,t)=(1-t)\boldsymbol{r}_\boldsymbol{A}(t)+v\boldsymbol{r}_\boldsymbol{B}(t),\:v,t\in\left[0,1\right]\end{equation}

where $\boldsymbol{r}_\boldsymbol{A}$ and $\boldsymbol{r}_\boldsymbol{B}$ are directrices, and $\boldsymbol{r}(t)=\boldsymbol{r}_\boldsymbol{A}(t)$ and $\boldsymbol{d}(t)=\boldsymbol{r}_\boldsymbol{B}(t)-\boldsymbol{r}_\boldsymbol{A}(t)$ .

Definition 4. Developable surfaces are surfaces that unfolded onto a plane without stretching or tearing. For developable ruled surfaces, the tangent plane is constant along with each ruling. Cylinders, cones, and planes are the most known developable surfaces. A ruled surface is called developable ruled surface if and only if the vectors $\boldsymbol{r}^{\prime}(t)$ , $\boldsymbol{d}(t)$ , and $\boldsymbol{d}^{\prime}(t)$ are linearly independent, that is [Reference Chalfant31]

(14) \begin{equation}\left|\boldsymbol{r}^{\prime}(t)\:\boldsymbol{d}(t)\:\boldsymbol{d}^{\prime}(t)\right|=0\end{equation}

5.2. The optimal Bézier ruled and developable ruled surface path of robotic arm

The optimal curves in Examples 1 and 2 are determined using the proposed method OPA-H. Next, the optimal shortest robot pose ruled surface can be obtained using the Eq. in (13). If the only one shape parameter of the directrix curve $\boldsymbol{r}_\boldsymbol{A}(t)$ is altered to find the other directrix curve $\boldsymbol{r}_\boldsymbol{B}(t)$ , the condition in the Eq. (14) will be satisfied. Consequently, the optimal developable robot pose ruled surface will be obtained.

Figure 15 shows the robot pose ruled surface and developability degree for the curve pair $\left\{ \boldsymbol{r}_{4},\boldsymbol{r}_{14}\right\} $ in Example 1. Since the developability degree is different than zero (see Fig. 15(b)), the robot pose ruled surface (see Fig. 15(a)) is not a developable robot pose ruled surface.

Figure 15. The robot pose ruled surface belong to the curve pair $\left\{ \boldsymbol{r}_{4},\boldsymbol{r}_{14}\right\} $ with the shape parameters $\lambda_{1}=-1.2258$ , $\lambda_{2}=0.2262$ , $\lambda_{3}=0.3796$ and $\lambda_{1}^{*}=-1.2134$ , $\lambda_{2}^{*}=0.8085$ , $\lambda_{3}^{*}=0.8449$ .

If the shape parameters are changed such as $\lambda_{1}=-1.2258$ , $\lambda_{2}=0.2262$ , $\lambda_{3}=0.3796$ and $\lambda_{1}^{*}=-1.2258$ , $\lambda_{2}^{*}=0.2262$ , $\lambda_{3}^{*}=0.8449$ , then the formed robot pose ruled surface will be developable robot pose ruled surface as seen from Fig. 16.

Figure 16. The developable robot pose ruled surface belong to the curve pair $\left\{ \boldsymbol{r}_{4},\boldsymbol{r}_{14}\right\} $ .

Figure 17 shows the ruled surface and developability degree for the curve pair $\left\{ \boldsymbol{r}_{5},\boldsymbol{r}_{15}\right\} $ in Example 2. Since the developability degree is different than zero (see Fig. 17(b)), the robot pose ruled surface (see Fig. 17(a)) is not a developable robot pose ruled surface.

Figure 17. The robot pose ruled surface belong to the curve pair $\left\{ \boldsymbol{r}_{5},\boldsymbol{r}_{15}\right\} $ with the shape parameters $\lambda_{1}=-1.8709$ , $\lambda_{2}=-0.4930$ , $\lambda_{3}=0.5965$ and $\lambda_{1}^{*}=0.1952$ , $\lambda_{2}^{*}=0.9432$ , $\lambda_{3}^{*}=-0.1963$ .

If the shape parameters are changed such as $\lambda_{1}=-1.8709$ , $\lambda_{2}=-0.4930$ , $\lambda_{3}=0.5965$ and $\lambda_{1}^{*}=0.1952$ , $\lambda_{2}^{*}=-0.4930$ , $\lambda_{3}^{*}=0.5965$ , then the formed robot pose ruled surface will be developable robot pose ruled surface as seen from Fig. 18.

Figure 18. The developable robot pose ruled surface belong to the curve pair $\left\{ \boldsymbol{r}_{5},\boldsymbol{r}_{15}\right\} $ .

The ruled surface and developability degree for the curve pair $\left\{ \boldsymbol{r}_{5},\boldsymbol{r}_{15}\right\} $ in Example 3 are presented in Fig. 19. Since the developability degree is different than zero (see Fig. 19(b)), the robot pose surface (see Fig. 19(a)) is the robot pose ruled surface.

Figure 19. The robot pose ruled surface belong to the curve pair $\left\{ \boldsymbol{r}_{5},\boldsymbol{r}_{15}\right\} $ with the shape parameters $\lambda_{1}=-0.8945$ , $\lambda_{2}=0.8769$ , $\lambda_{3}=1.1209$ and $\lambda_{1}^{*}=-1.7566$ , $\lambda_{2}^{*}=1.7882$ , $\lambda_{3}^{*}=1.1029$ .

If the shape parameters is changed such as $\lambda_{1}=-0.8945$ , $\lambda_{2}=1.7882$ , $\lambda_{3}=1.1029$ and $\lambda_{1}^{*}=-1.7566$ , $\lambda_{2}^{*}=1.7882$ , $\lambda_{3}^{*}=1.1029$ , then the formed robot pose ruled surface will be developable robot pose ruled surface as seen from Fig. 20.

Figure 20. The developable robot pose ruled surface belong to the curve pair $\left\{ \boldsymbol{r}_{5},\boldsymbol{r}_{15}\right\} $ .