3.1. Assembly modeling for the nonconvex X–Y axisymmetric parts

In the previous discussion about the relationship between the high-dimensional attractive region and the low-dimensional one, Qiao et al. pointed out that the 3D constraint region (or attractive region) may contain multiple 2D attractive regions, which is helpful to make feasible and reliable manipulation strategies in the low-dimensional configuration space [Reference Qiao, Wang, Su, Jia and Li17]. Inspired by this conclusion, we can design a series of 2D attractive regions in the low-dimensional subspace of the 3D constraint region. By designing different state-independent inputs for each 2D attractive region, the stable point of the former attractive region falls within the latter one. Thus, the peg can move directionally step by step under a series of simple inputs and environmental constraints, and finally reach the global lowest point.

In another work about condition and strategy analysis for assembly based on ARIE, Li and Qiao further pointed out that the transversal-sectional shape (except the boundary state) of the constraint region is related to the shape of the hole. In particular, for round peg-in-hole assembly, the outline of the transversal section of the constraint region consists of three arcs (except the boundary state) [Reference Li and Qiao20]. Obviously, the conclusion is also applicable to the assembly where the transversal-sectional shape of the hole is X–Y axisymmetric convex polygon. As shown in Fig. 8, on the transversal section of the constraint region, there is a 2D attraction region along $\vec{e}_{tv}$ direction, where point $C_{2}$ ( $C^{\prime}_{2}$ ) is the strictly stable point corresponding to the two-point contact state. Therefore, when the noncritical peg-hole contact condition is satisfied, keep the peg fixed in posture and let it free in translation along the horizontal direction; then apply a transversal input ${\vec{u}_{tv}}$ , and the peg will finally keep stable in the two-point contact state. It should be noted that the convexity of the transversal section of the constraint region is only related to the shape of the hole, and the nonconvex structures on the peg do not affect geometric characteristics of the constraint region. In other words, the nonconvex structures on peg do not affect the existence of the transversal 2D attractive region.

Figure 8.

(a) Transversal section (except the boundary state) of the constraint region of polygon parts; (b) Transversal section (except the boundary state) of the constraint region of cylindrical parts.

From the continuity of motion, it is obvious that when the nonconvex area is ignored, the peg maintains two-point contact with the hole and slides down, and finally reaches the three-point contact state (global minimum point). So, we considered to search the attractive region on the longitudinal section of the constraint region including the two-point contact state and the three-point contact state. The mathematical model of the peg-hole two-point contact state is established as shown in Fig. 9.

Figure 9.

Mathematical model of two-point contact in the axisymmetric parts assembly.

The coordinates attached on the peg and the hole are $\{p\}$ and $\{h\}$ , respectively. $O_{p}$ is the geometric center of the lower surface of the peg, and $O_{h}$ is the geometric center of the upper surface of the hole. For the selection of the posture of the peg, it should be considered from the perspective of strategy design. The purpose of tilting the peg is to generate an appropriate constraint region conducive to assembly. According to the analysis of constraint regions in Section 2, an axisymmetric conical constraint region can be generated by rotating the peg about a single axis, which is conducive to modeling and analysis and is easy to operate in actual assembly. Therefore, it is advisable to take the rotation of the peg about ${\overrightarrow{{O_{p}Y_{p}}}}$ as an example to study the two-point contact state in the peg-in-hole assembly. Starting with the frame $O_{p}-X_{p}Y_{p}Z_{p}$ parallel to the frame $O_{h}-X_{h}Y_{h}Z_{h}$ , rotate the peg about ${\overrightarrow{{O_{p}Y_{p}}}}$ by an angle $\theta _{y}$ , and the orientation of the peg is described by the frame $O_{p}-X^{\prime}_{p}Y^{\prime}_{p}Z^{\prime}_{p}$ . The segment $B_{p}C_{p}$ and the segment $N_{p}M_{p}$ are two segments symmetrical about ${\overrightarrow{{O_{p}X^{\prime}_{p}}}}$ at the bottom edge of the peg. The straight line $B_{p}C_{p}$ and the straight line $N_{p}M_{p}$ intersect at the point $Q_{p}$ , and set the segment $O_{p}Q_{p}=r_{p}$ , and $\angle O_{p}Q_{p}C_{p}=\angle O_{p}Q_{p}M_{p}=\varphi _{p}$ . When the peg and the hole are in contact by two points, the segment $F_{h}E_{h}$ and the segment $J_{h}K_{h}$ are the two segments on the edge of the hole, which is related to the segment $B_{p}C_{p}$ and the segment $N_{p}M_{p}$ , respectively. The two contact points are $P_{c1}$ and $P_{c2}$ . The straight line $F_{h}E_{h}$ and the straight line $J_{h}K_{h}$ intersect at the point $T_{h}$ , and set the segment $O_{h}T_{h}=r_{h}$ , and $\angle O_{h}T_{h}E_{h}=\angle O_{h}T_{h}K_{h}=\varphi _{h}$ . The projections of the segment $B_{p}C_{p}$ and $N_{p}M_{p}$ , the point $Q_{p}$ and $O_{p}$ on $X_{h}O_{h}Y_{h}$ plane are the segment $B_{p}^{*}C_{p}^{*}$ and $N_{p}^{*}M_{p}^{*}$ , the point $Q_{p}^{*}$ and $O_{p}^{*}$ , respectively, where $\angle C_{p}^{*}Q_{p}^{*}O_{p}^{*}=\angle M_{p}^{*}Q_{p}^{*}O_{p}^{*}=\varphi _{p}^{*}$ .

The position of the geometric center of the lower surface of the peg in the frame $\{h\}$ is described by ${}^{h}{P}{_{Op}^{}}=(x_{{O_{p}}},y_{{O_{p}}},z_{{O_{p}}})$ . When the peg and the hole are in contact by two points, the equations are obtained to get ${}^{h}{P}{_{Op}^{}}$ as the following:

Equation of the line $F_{h}E_{h}$ :

(1) \begin{equation} y=\left(x+r_{h}\right)\tan \varphi _{h} \end{equation}

Equation of the line $B_{p}^{*}C_{p}^{*}$ :

(2) \begin{equation} y=-\left(x-x_{{O_{p}}}-r_{p}\cos \theta \right)\tan \varphi _{p}^{*} \end{equation}

where $\tan \varphi _{p}^{*}=\frac{\tan \varphi _{p}}{\cos \theta _{y}}, \varphi _{p},\varphi _{h},\theta _{y}\in \left(0,\frac{\pi }{2}\right)$ .

For the two-point contact state, the trigonometric relationship exists as following:

(3) \begin{equation} \tan \theta _{y}=\frac{\left| z_{{O_{p}}}\right| }{\left| x_{{O_{p}}}\right| -\left| x_{{P_{c1}}}\right| } \end{equation}

where $x_{{P_{c1}}}$ is the coordinate of the contact point $P_{c1}$ in the ${\overrightarrow{{O_{h}X_{h}}}}$ direction in the frame $\{h\}$ .

The result is obtained from symmetry and Eqs. (1)–(3):

(4) \begin{equation} {}^{h}{P}{_{Op}^{}}=\left(x_{{O_{p}}},0,-\frac{\sin \theta _{y}\tan \varphi _{h}}{\cos \theta _{y}\tan \varphi _{h}+\tan \varphi _{p}}x_{{O_{p}}}+\frac{r_{p}\sin \theta _{y}\tan \varphi _{p}-r_{h}\sin \theta _{y}\tan \varphi _{h}}{\cos \theta _{y}\tan \varphi _{h}+\tan \varphi _{p}}\right) \end{equation}

When the cross-sectional shape is X–Y axisymmetric, we can obtain $\varphi _{p}=\varphi _{h}$ . For the parts with approximately equal size, assume that $r_{p}=r_{h}$ . The further result is obtained from Eq. (4):

(5) \begin{equation} {}^{h}{P}{_{Op}^{}}=\left(x_{{O_{p}}},0,-x_{{O_{p}}}\tan \frac{\theta _{y}}{2}\right) \end{equation}

Particularly, for axisymmetric parts assemblies with circular cross-sections, the mathematical model of the peg-hole two-point contact state is established as shown in Fig. 10. The coordinates attached on the peg and the hole are $\{p\}$ and $\{h\}$ , respectively. $O_{pc}$ is the geometric center of the lower surface of the peg, and $O_{hc}$ is the geometric center of the upper surface of the hole. The radii of the peg and the hole are $r_{p}$ and $r_{h}$ , respectively. Starting with the frame $O_{pc}-X_{p}Y_{p}Z_{p}$ parallel to the frame $O_{hc}-X_{h}Y_{h}Z_{h}$ , rotate the peg about ${\overrightarrow{{O_{pc}Y_{p}}}}$ by an angle $\theta _{y}$ , and the orientation of the peg is described by the frame $O_{pc}-X^{\prime}_{p}Y^{\prime}_{p}Z^{\prime}_{p}$ . The ellipse $O_{pc}^{*}$ is the projection of the lower surface of the peg on the plane $X_{h}O_{hc}Y_{h}$ . The two contact points are $P_{c1}$ and $P_{c2}$ . The position of the geometric center of the lower surface of the peg in the frame $\{h\}$ is described by ${}^{h}{P}{_{Op_{c}}^{}}=(x_{{O_{pc}}},y_{{O_{pc}}},z_{{O_{pc}}})$ . For a pair of peg and hole with approximately equal radius in high-precision assembly, assume that $r_{p}=r_{h}=r$ . When the peg and the hole are in contact by two points, the equations are obtained to get ${}^{h}{P}{_{Op_{c}}^{}}$ as the following:

Figure 10.

Mathematical model of two-point contact in the cylindrical parts assembly.

Equation of the circle $O_{hc}$ :

(6) \begin{equation} x^{2}+y^{2}=r^{2} \end{equation}

Equation of the ellipse $O_{pc}^{*}$ :

(7) \begin{equation} \frac{\left(x-x_{{O_{pc}}}\right)^{2}}{\left(r\cos \theta _{y}\right)^{2}}+\frac{y^{2}}{r^{2}}=1 \end{equation}

where $\theta _{y}\in \left(0,\frac{\pi }{2}\right)$ .

The two contact points $P_{c1}$ and $P_{c2}$ are symmetrical about ${\overrightarrow{{O_{hc}X_{h}}}}$ , so we can obtain $x_{{P_{c1}}}=x_{{P_{c2}}}, y_{{P_{c1}}}=-y_{{P_{c2}}}$ , and $z_{{P_{c1}}}=z_{{P_{c2}}}=0$ . Then, from Eqs. (6) to (7), the coordinate of $P_{c1}$ can be obtained:

(8) \begin{equation} P_{c1}=\left(\frac{x_{{O_{pc}}}}{1+\cos \theta _{y}},\sqrt{r^{2}-\left(\frac{x_{{O_{pc}}}}{1+\cos \theta _{y}}\right)^{2}},0\right) \end{equation}

For the two-point contact state, the trigonometric relationship exists as following:

(9) \begin{equation} \tan \theta _{y}=\frac{\left| z_{{O_{pc}}}\right| }{\left| x_{{O_{pc}}}\right| -\left| x_{{P_{c1}}}\right| } \end{equation}

The result is obtained from symmetry and Eqs. (8) and (9):

(10) \begin{equation} {}^{h}{P}{_{Op_{c}}^{}}=\left(x_{{O_{pc}}},0,-x_{{O_{pc}}}\tan \frac{\theta _{y}}{2}\right) \end{equation}

See Appendix for specific calculation process.

Equations (5) and (10) show that, on the longitudinal section of the constraint region including the two-point contact state and the three-point contact state, the set of ${}^{h}{P}{_{Op}^{}}$ ( ${}^{h}{P}{_{Op_{c}}^{}}$ ) is a straight-line segment with a slope of $-\tan\!(\theta _{y}/2)$ . Based on this important geometric property, the input ${\overrightarrow{u}_{lt}}$ in the longitudinal section can be designed with the angle between the direction and the horizontal plane of $\theta _{y}/2$ , as shown in Fig. 11. Under the action of ${\overrightarrow{u}_{lt}}$ , the peg is able to stride over the nonconvex areas without considering their distribution, and then reaches the three-point contact state $C_{3}$ . It should be noted that although the direction of ${\overrightarrow{u}_{lt}}$ is not vertical downward, for a small $\theta _{y}$ , the peg will stop moving and remain stable under the resistance of the inside of the hole after reaching the three-point contact state.

Figure 11.

Longitudinal section of constraint region of the nonconvex axisymmetric polygon parts.

3.2. ARIE-based multi-step sliding strategy for nonconvex axisymmetric parts assembly

The classic ARIE-based assembly strategy contains two steps: (1) generate a constant force in the direction of $-{\overrightarrow{{O_{h}Z_{h}}}}$ to push the peg with a tilting posture angle into three-point contact state and (2) adjust the posture of the peg to finish the insertion.

Based on the previous analysis in Section 3.1, an ARIE-based multi-step sliding strategy for nonconvex X–Y axisymmetric peg-in-hole assembly is proposed. As shown in Fig. 12, the new strategy includes two main steps: (Step 1) peg sliding with a tilting posture and (Step 2) posture adjustment. In Step 1, there are four substeps as following:

Figure 12.

ARIE-based multi-step sliding strategy for nonconvex axisymmetric peg-in-hole assembly.

• Step (1a): Move the peg to the area above the hole and keep it in a tilting posture. Then, apply the input ${\overrightarrow{u}_{0}}$ in the direction of $-{\overrightarrow{{Z}}}$ to realize peg-hole one-point contact state;

• Step (1b): Keep the peg fixed in posture while free in transversal movement and design the input ${\overrightarrow{u}_{1}}$ with a proper value in the direction of $-{\overrightarrow{{X}}}$ . In the 2D attractive region parallel to the horizontal plane, the peg will be subjected to the combined effect of the input force and the contact force. As a result, the peg will move along the edge of the hole and reach the two-point contact state;

• Step (1c): Keep on the peg fixed in posture while free in transversal movement. Then, apply the obliquely downward input ${\overrightarrow{u}_{2}}$ whose direction is $-\theta _{y}/2$ from the horizontal. Ideally, the peg is not subjected to the contact force, so that it is able to pass over the nonconvex areas and reach the three-point contact state;

• Step (1d): To enable the peg to keep stable at the global lowest point, the vertically downward input ${\overrightarrow{u}_{3}}$ can be applied. Due to the friction between the two parts, ${\overrightarrow{u}_{3}}$ is unnecessary in many cases.

By executing Step 1, the position error is eliminated and the blocking problem is avoided. Then, the posture of the peg can be adjusted by executing Step 2 and the assembly will be finished successfully.

3.3. Impedance control

Considering the motion requirements in the strategy based on ARIE, impedance control is a good choice because it can keep the peg (robot end-effector) posture fixed while motion relaxed in Cartesian space. The behavior of robot is compliant with impedance controllers. The imposed external force can cause the trajectory error of end-effector of robot. The dynamics model of impedance control is based on virtual springs and dampers, which have force change due to the difference between the measured states and the desired states of the end-effector [Reference Albu-Schäffer, Ott and Hirzinger29, Reference Ott, Schäffer, Kugi and Hirzinger30]. The characteristics of the system are described by Eq. (11):

(11) \begin{equation} M_{d}\!\left(\ddot{x}_{d}-\ddot{x}_{c}\right)+D_{d}\!\left(\dot{x}_{d}-\dot{x}_{c}\right)+K_{d}\!\left(x_{d}-x_{c}\right)=F_{d}-F_{ext} \end{equation}

where $x_{c}$ and $x_{d}$ are the initial desired position and the new desired position, respectively; $M_{d}$ is the inertial matrix; $D_{d}$ is the damping coefficient; $K_{d}$ is the stiffness parameters of the virtual mass-spring damper system; $F_{d}$ and $F_{ext}$ are the desired contact force and the detected external force, respectively.

4.1. Comparison experiments design

The purpose of the experiments is to verify the effectiveness of the new assembly method in solving the blocking problem caused by the nonconvex structures on the peg. The algorithms based on ARIE classic strategy and the new proposed multi-step sliding strategy are, respectively, given as Algorithms 1 and 2. Based on the same initial conditions, the new strategy is compared with ARIE-based classic strategy. Assumed that the location of the hole needs to be positioned by using visual information. When the accuracy of the sensor is low or signal drift exists, there is a large error between the measured hole position and the actual hole position, so it is impossible to directly use position control for high-precision assembly. However, within the error range, the projection of the lowest point on the peg with a tilting posture in the vertical direction is still inside the hole, which satisfy the applying condition of the ARIE-based strategy. In order to keep the peg fixed in orientation while free in translation, the impedance parameters are set as $K_{d}=\text{diag}([200,200,200,300,300,300])$ . The first three parameters are the stiffness coefficient of translation $C_{T}$ (unit: N/m), and the last three parameters are the stiffness coefficient of rotation $C_{R}$ (unit: Nm/rad). It should be noted that, in the KUKA controller, the values of stiffness coefficients of translation along or rotation about three axes are required to be equal. $M_{d}$ and $D_{d}$ are set as the robot system default values. Here present an explanation of the effect of stiffness parameter setting. $C_{T}$ and $C_{R}$ meet the formula $\Delta l=F_{E}/C_{T}$ and $\Delta \theta =T_{E}/C_{R}$ , respectively, where $F_{E}(F_{Ex},F_{Ey},F_{Ez})$ (unit: N) and $T_{E}(T_{Ex},T_{Ey},T_{Ez})$ (unit: Nm) are the external force and the torque of the end-effector in Cartesian coordinate system; $\Delta l(\Delta l_{x},\Delta l_{y},\Delta l_{z})$ (unit: m) and $\Delta \theta (\Delta \theta _{x},\Delta \theta _{y},\Delta \theta _{z})$ (unit: rad) are the translation change and the rotation change of the end-effector under the action of $F_{E}$ and $T_{E}$ , respectively. Taking $C_{T}=200$ N/m and $C_{R}=300$ Nm/rad as an example, assume that $F_{E}=(1,0,0)$ , the maximum force arm is about 0.27 m (the distance from the lower surface of the peg to the geometric center of the joint 6 of the robot), so the corresponding $T_{E}=(0,-0.27,0)$ , and we can obtain that $\Delta l=(0.005,0,0)$ and $\Delta \theta =(0,-0.0009,0)$ (about $-0.05^{\circ}$ ). Therefore, in the contact process between the peg and the hole without blocking, under the action of a small external force (1N), the original desired trajectory of the peg can be changed relatively greatly (5 mm), so that the peg can slide smoothly along the edge of the hole. At the same time, under the action of the corresponding torque (0.27 Nm), the posture of the peg hardly changes ( $0.05^{\circ}$ ).

4.2. Experimental results

In the experiments, the position of the geometric center of the upper surface of the hole in the frame $\{r\}$ was set as ${{}^{r}}{P}_{{O}_{h}}=(453.3,-7.7,92.1)$ (unit:mm). The initial position error has been achieved by randomly changing the initial position of the peg within ± 40 mm along the radial direction of the hole.

A group of representative comparison experiments is selected here. The initial position and posture of the peg were set as ${{}^{r}}{P_{O_{p}0}}=(392.7,-4.3,122.5)$ and $\theta _{0}=(0,20^{\circ},0)$ , where there is a posture error $\Delta \theta _{z}\approx 10^{\circ}$ . The process of the assembly experiments with different strategies is shown, respectively, in Figs. 14 and 15.

Figure 14.

Assembly process with ARIE-based classic strategy: (a) position and posture initialization, (b) one-point contact (blocking), (c) posture adjustment, and (d) insertion (failed).

Figure 15.

Assembly process with ARIE-based multi-step sliding strategy: (a) position and posture initialization, (b) one-point contact, (c) two-point contact, (d) three-point contact, (e) posture adjustment, and (f) insertion.

In the assembly experiment with ARIE-based classic strategy, position error could not be eliminated because the peg was stuck by the groove, which ultimately led to assembly failure. In the assembly experiment with the new strategy, after the peg touched the hole by one point, the peg moved laterally to reach two points contact state. Then under the new input, it passed over the nonconvex area and reached the global lowest point. After adjusting the posture of the peg and insertion, the assembly was successfully complete. The trajectory and posture of the peg, and the information detected by the six-dimension force/torque sensor in the comparison experiments are shown in Figs. 16–19.

Figure 16.

Position and posture of the peg when using ARIE-based classic strategy. (a) position and posture initialization, (b) approaching, (c) one-point contact (blocking), (d) posture adjustment, and (e) insertion (failed).

Figure 17.

Position and posture of the peg when using ARIE-based multi-step sliding strategy. (a) position and posture initialization, (b) approaching, (c) sliding, (d) three-point contact, (e) posture adjustment, and (f) insertion.

Figure 18.

Six-dimension force/torque information when using ARIE-based classic strategy. (a) position and posture initialization, (b) approaching, (c) one-point contact (blocking), (d) posture adjustment, and (e) insertion (failed).

Figure 19.

Six-dimension force/torque information when using ARIE-based multi-step sliding strategy. (a) position and posture initialization, (b) approaching, (c) sliding, (d) three-point contact, (e) posture adjustment, and (f) insertion.

Figures 16 and 17 record the position and posture information of the peg in the experiment based on ARIE classic strategy and multi-step sliding strategy, respectively. In the experiment based on ARIE classic strategy, the period of $t=0\sim 2.8$ s is for the position and posture initialization of the peg. When the peg reaches the initial position and posture ( ${{}^{r}}{{P}_{O_{p}0}}=(392.4,-4.4,122.3)$ and ${{}^{r}}{\theta_{0}}=(0,20^{\circ},0)$ ), keep the peg fixed in orientation while free in translation, and let it gradually approach the hole. Then, the first contact between the peg and the hole occurs at $t=6.5$ s, and the position of the peg is ${{}^{r}}{P}{_{Op1}^{}}=(392.2,-4.4,116.1)$ . Because the groove on the peg is stuck at the edge of hole, the peg is prevented from further sliding, and thus the position of the peg does not change in the period of $t=6.5\sim 10.5$ s. The algorithm misjudges that the peg has reached the global stable state or the position error has been eliminated, thus entering the stage of posture adjustment ( $t=10.5\sim 15.5$ s). After adjusting the posture of the peg, the position and orientation of the peg are ${{}^{r}}{P}{_{Op2}^{}}=(442.9,-4.2,104.9)$ and ${{}^{r}}{\theta }{_{2}^{}}=(0,0,0)$ , respectively. The radial position deviation between the peg and the hole is about 11 mm (The coordinate of the hole ${{}^{r}}{P}_{{O}_{h}}=(453.3,-7.7,92.1), \Delta x=10.4$ mm and $\Delta y=3.5$ mm). The lower surface of the peg is in contact with the upper surface of hole, so the insertion cannot be completed and the assembly task fails.

In the experiment based on multi-step sliding strategy, similarly, the period of $t=0\sim 2.8$ s is for the position and posture initialization of the peg. When the peg reaches the initial position and posture ( ${{}^{r}}{P^{\prime}_{O_{p}0}}=(393.0,-4.3,122.7)$ and ${{}^{r}}{\theta^{\prime}_{0}}=(0,20^{\circ},0)$ ), keep the peg fixed in orientation while free in translation, and let it gradually approach the hole. When $t=7.5$ s, the peg touches the hole, and the position of the peg is ${{}^{r}}{P^{\prime}}{_{Op1}^{}}=(393.4,-4.3,116.3)$ . In the period of $t=7.5\sim 7.9$ s, the peg slides along the edge of the hole, moving 3.3 mm along $-{\overrightarrow{{O_{r}Y_{r}}}}$ and reached the two-point contact state. Then the peg continues to slide along the edge of the hole, moving 17.1 mm along ${\overrightarrow{{O_{r}X_{r}}}}$ and 7.6 mm along $-{\overrightarrow{{O_{r}Z_{r}}}}$ , respectively, and finally reaches the global lowest potential energy point ${{}^{r}}{P^{\prime}}{_{Op2}^{}}=(410.5,-7.5,108.0)$ at $t=8.9$ s. During $t=8.9\sim 13$ s, the position of the peg does not change significantly, thus entering the stage of posture adjustment ( $t=13\sim 17.5$ s). With the correction of the posture of the peg, the peg-in-hole insertion is completed at $t=17.5$ s. The final position and orientation of the peg are ${{}^{r}}{P^{\prime}}{_{Opf}^{}}=(453.5,-7.7,92.1)$ and ${{}^{r}}{\theta^{\prime}}{_{f}^{}}=(0,20^{\circ},0)$ respectively, and the assembly task is finished successfully.

Comparing the position and orientation information of the two experiments, the main difference is whether there is a complete sliding stage. In the experiment based on ARIE classic strategy, due to the blocking problem, the expected sliding stage is not completed, which leads to misjudgment and premature entry into the stage of posture adjustment (at $t=10.5$ s), resulting in assembly failure. In the experiment based on the new strategy, the peg successfully slides to the global lowest potential energy point and enters the stage of posture adjustment at $t=13$ s. With the correction of the posture of the peg, the peg is inserted into the hole, and the assembly task is successfully completed.

Figures 18 and 19, respectively, record the six-dimension force information in the experiment based on ARIE classic strategy and multi-step sliding strategy. It should be noted that the force sensor has been zeroed before the experiment. The data detected in the experiment are raw data, including the influence of the gravity of the pneumatic gripper and the peg.

In the experiment based on ARIE classic strategy, the period $t=0\sim 2.8$ s is for the stage of initialization. In the process of reaching the predetermined initial state, $F_{y}$ and $F_{z}$ are basically kept near 0. Due to the gravity of the gripper and the peg, $F_{x}$ increases from 0 to about 2.2 N. The corresponding torques $T_{x}$ and $T_{z}$ vibrate slightly near 0, and $T_{y}$ changes from 0 to $-0.09$ Nm. When $t=6.5$ s, the first contact between the peg and the hole occurs, and the detected forces/torques change significantly. Subsequently, due to blocking, the forces/torques basically remain unchanged ( $F_{x}\approx 1.2$ N, $F_{y}\approx 0$ N, $F_{z}\approx 2.8$ N, $T_{x}\approx -0.04$ Nm, $T_{y}\approx -0.007$ Nm, and $T_{z}\approx -0.028$ Nm) in the period of $t=7\sim 10.5$ s. Then, in the stage of posture adjustment ( $t=10.5\sim 15.5$ s), the forces/torques fluctuate greatly. The maximum amplitude of contact force in one direction is about 2 N, and the maximum amplitude of torque is about 0.1 Nm. In the period of $t=15.5\sim 30$ s, the lower surface of the peg is in contact with the upper surface of the hole, and the insertion cannot be completed. The robot further applies a pressure along $-{\overrightarrow{{O_{r}Z_{r}}}}, F_{z}$ increases from about 2.7 N to about 4.2 N, $F_{x}$ and $F_{y}$ basically remain near 0, $T_{x}\approx 0.03$ Nm, $T_{y}\approx 0.03$ Nm, and $T_{z}\approx 0.004$ Nm.

In the experiment based on multi-step sliding strategy, similarly, the period $t=0\sim 2.8$ s is for the stage of initialization. In this period, $F_{y}$ and $F_{z}$ are basically kept near 0 and $F_{x}$ increases from 0 to about 2.3 N. The corresponding torques $T_{x}$ and $T_{z}$ vibrate slightly near 0, and $T_{y}$ changes from 0 to $-0.09$ Nm. When $t=7.5$ s, the first contact between the peg and the hole occurs, and the detected forces/torques change significantly. When $t=7.9$ s, there is a pulse of $F_{y}\approx 8$ N and $T_{x}\approx -0.175$ Nm,

corresponding to the collision between the peg and the hole when the peg slides to the two-point contact state. When $t=8.9$ s, $F_{x}, F_{z}$ and $T_{y}$ jump to about $-7$ N, 9 N and 0.736 Nm respectively, which corresponds to the collision between the peg and the hole when the peg slides down to the global lowest potential energy point. In the period of $t=8.9\sim 13$ s, the peg is kept in the stable contact state, and the forces/torques basically remain unchanged ( $F_{x}\approx -2.9$ N, $F_{y}\approx 0.5$ N, $F_{z}\approx 2.6$ N, $T_{x}\approx 0.02$ Nm, $T_{y}\approx 0.35$ Nm, $T_{z}\approx 0.02$ Nm). The period of $t=13\sim 17.5$ s is for the posture adjustment, and the forces/torques fluctuate greatly. When $t=17.5$ s, the peg falls into the hole and $F_{z}$ declines from about 2.6 N to about 1.1 N. In the period of $t=17.5\sim 30$ s, the robot further applies a pressure along $-{\overrightarrow{{O_{r}Z_{r}}}}, F_{z}$ increases from about 1.1 N to about 2.7 N, $F_{x}\approx -2.2$ N, $F_{y}\approx -0.4$ N, $T_{x}\approx 0.01$ Nm, $T_{y}\approx 0.2$ Nm, and $T_{z}\approx -0.002$ Nm.

Comparing the six-dimension force information of the two experiments, it can be seen that in the experiment based on the classic strategy, due to the blocking problem, the expected sliding stage is not completed, and the force/torque change is relatively gentle in the whole process. In the experiment based on multi-step sliding strategy, the peg slides along the edge of the hole and collides with the hole, resulting in rigid impact. When it reaches the two-point contact state or the three-point contact state, large impact forces are generated.