Modelling and continuous stiffness control of robot with compliant wrist for misalignment shaft-hole assembly

Abstract

In this paper, a continuously variable stiffness control strategy for shaft-hole assembly with a compliant wrist is proposed. The compliant wrist adjusts stiffness by changing the cantilever length of a super-elastic Ni-Ti wire. Its core idea is that when the contact force of the robot exceeds a particular value, the wrist adjusts the stiffness and can deform in a specific direction that guarantees assembly, allows a relatively significant misalignment, and produces a small force. The advantage of the proposed strategy is that the shaft-hole assembly status is supervised by calculating the deformation of compliant wrist based on contact force information, this significantly decrease the requirements of shaft-hole alignment accuracy. On this basis, the kinetostatic coupling kinematic and static force model is built and the fuzzy PD stiffness control strategy is designed to realize the desired stiffness of the wrist in various directions. Finally, the shaft-hole assembly experiments under different misalignment error demonstrates the reliability of the wrist, indicating the efficacy of the control method.

Introduction

The manipulator has been widely applied in the field of assembly process in automatic factory. ¹ However, many confounding factors impinge upon the assembly process as robots interact with their environments. The robot itself cannot accurately identify the contact status.^2–3 Even minimal errors in the assembly process may cause assembly failure or damage to the assembly parts or robots. Additionally, with the acceleration of product development cycles, there are a greater number of assembly parts, especially in the assembly process of complex equipment. The stiffness and position information of the parts is usually incomplete and inaccurate, which brings potential risks to the robot in the process of rigid interactive operation. Therefore, the development of a compliant operation robot is of great interest.

In general, an interaction force will occur between the robot and its environment. Therefore, our research focuses on how to regulate the robot’s environmental interaction. The researcher addresses the main problem in the process of shaft-hole assembly and conducts a study on the robot’s compliance, which refers to the robot’s ability to show some adaptability to an unknown environment to assist in completing the assembly. There are two approaches to compliant assembly, passive and active compliance. Many studies have used active compliance control strategies to improve assembly performance. These active compliance control methods include stiffness control, ³ impedance control, ⁴ and hybrid F/P (force/position) control. ⁵ These studies mainly focused on two aspects: impedance control and hybrid F/P control. Impedance control methods include adaptive neural network, ⁶ adaptive, ⁷ adaptive learning, ⁸ variable-stiffness ⁹ and robust adaptive impedance control. ¹⁰ The force/position control includes hybrid, ¹¹ adaptive, ¹² robust, ¹³ neural network, ¹⁴ and learning F/P control. ¹⁵ Although these methods can achieve compliance control with the dynamic model of the manipulator, it is costly and complex, and the control algorithm relies on an accurate dynamic model of the manipulator, making it difficult to achieve the desired universal suppleness of function. Therefore, research has carried out on low-cost, more adaptable passive compliance devices.

In contrast to active suppleness, passive suppleness control is achieved by loading some additional auxiliary mechanism, such as a spring mechanism, at the end of the robot so that it can produce some compliance when it comes in contact with the environment. The compliant behavior is achieved through the interaction between the wrist and the contact object. Passive compliance effectively separates force control from the robot controller by introducing a compliant device independent of the robot. The interaction between the robot and the environment was realized by changing the stiffness of the contact point. It is feasible to design specific mechanical structures to achieve the robot’s weak or constant force control to contact the environment. For example, Seungjoon ¹⁶ designed the RCC (remote center compensation) structure to the passive compliance device and a robust industrial robotic assembly process is achieved. However, the compliant center of the RCC device is difficult to adjust, resulting in poor adaptability and versatility to different stiffness. SRCC (Spatial remote center compliance) introduces an additional axial rotation for accommodating the prismatic peg components. ¹⁷ An HRCC (horizontal remote center compliance) reduces the insertion force and avoids the jamming conditions in the dynamic insertion of chamfer-less peg-in-hole assembly along with horizontal and vertical directions.^18–19 VRCC (Variable remote center compliance) with stacks of elastomer shear pads and shear controller changes the compliance center concerning the insertion depth. ²⁰ Sangcheol and Liu^21,22 used the spring mechanism and the compliant linkage mechanism to realize passive precision assembly, respectively. Chen ²³ introduced an end pneumatic compliance mechanism to ensure compliance with the environment by controlling the input gas pressure. The above-mentioned passive compliant devices consider the assembly of a shaft-hole in a quasi-static situation, but the actual assembly process is dynamic. Therefore, several auxiliary sensors are used to assist with the dynamic assembly process, including vision and force sensors, to improve assembly precision. For example, Huang ²⁴ proposed a robust control method based on visual information. In paper, ²⁵ a robust passive control method based on visual information for the assembly task is proposed in the presence of attitude alignment errors. However, the stiffness of the contact tip cannot be dynamically adjusted.

From the discussion above, active compliance firstly requires precise identification of the contact environment, high requirements for force sensor accuracy, and complex control algorithms. Errors can produce an enormous impact, resulting in jamming, wedging, or surface damage. Secondly, passive compliance entirely relies on its fixed stiffness and lack of active adaptation to complex environments, which restricts its application in practical production. Therefore, if the advantage of two compliant methods can be combined, the robot can exhibit ideal compliant characteristics while reducing control difficulties and achieving high-precision assembly.

A continuous variable-stiffness wrist is used to achieve composite compliant operation, leveraging the advantages of both active and passive compliant approaches. The rigid robot can provide an accurate position with limited dependence, while a compliant wrist can effectively regulate and offset the forces exerted by the environment. The wrist is based on the Cosserat rod model and combines the characteristics of active and passive suppleness, which can continuously change the wrist stiffness according to the assembly interaction force. The wrist stiffness matches the stiffness of the acting object. Based on this, a stiffness controller with a fuzzy PD control method is designed to realize the desired stiffness of the end tip in various directions by adjusting the position of the slider. The wrist has the following features:

1) the tip stiffness is numerically obtained by solving the equations of the Cosserat rod theory.

2) the stiffness of the wrist can be adjusted continuously according to contact force.

3) the contact force is used as a means to determine the stiffness.

4) The proposed control strategy converts the stiffness control into the position control of the screw slider of the wrist.

Description of the compliant wrist

Assembly efficiency can be improved by adjusting the stiffness of parts during assembly to make sure the contact force satisfies the requirements. However, when the contact force exceeds a given value due to a relatively large misalignment, damage to the shaft and hole can be avoided by reducing the stiffness of the operation tool. These cases require the operation tool to have a variable stiffness during assembly. Here, a variable stiffness compliant wrist is used to realize variable stiffness performance to reduce contact force, as shown in Figure 1

Figure 1.

(a) Structure of complaint assemble system; (b) structure of wrist; (c) Variable-stiffness mechanism; (d) Structure diagram of wrist.

The compliant wrist consists of four hyper-elastic Ni-Ti alloy wires, two guide rods, one screw, one slider, and one servo actuator. The guide rods and the screw are rigid and the only connection with the robot. The four Ni-Ti alloy wires are flexible to connect the robot and the assembled shaft. The servo actuator drives the slider. The rigidity of the guide rods and screw cause the upper portion of the slider to be rigid and the lower portion to be flexible. This flexibility changes with the position of the slider.

Due to the use of Ni-Ti alloy wire, the wrist experiences elastic recoverable deformation, as shown in Figure 2. When the slider moves along the axial direction of the Ni-Ti wire, the length of the cantilever changes under the deformation principle diagram of the Ni-Ti wire. The screw-slider mechanism is used to adjust the extension length of the Ni-Ti alloy wire $l_i$ , to change the stiffness of the end of the Ni-Ti alloy wire $K=4k_i$ , here, $k_i=\left[\begin{array}{cc}{k}_{v,i}& k_{u,i}\end{array}\right]$ is the stiffness of single Ni-Ti wire, ²⁶ $k_{v,i}=\left[\begin{array}{ccc}{k}_{vx,i}& k_{vy,i}& k_{vz,i}\end{array}\right]$ and $k_{u,i}=\left[\begin{array}{ccc}{k}_{ux,i}& k_{uy,i}& k_{uz,i}\end{array}\right]$ is the translational and torsional stiffness along and around x,y,z-axis, as shown in Figure 1.

Figure 2.

(a) The deformation along x and y-axis under the force $F_x$ and $F_y$ ; (b) the deformation around x and y-axis under the torque $M_x$ and $M_y$ (in general, deformation around x and y-axis are neglected); (c) the deformation around z-axis under the torque $M_z$ ; (d) vertical view of the deformation around z-axis under the torque $M_z$ .

Once the contact force acts on the end of the wrist, the wrist will produce transverse deformation along x and y-axis, rotational deformation around z-axis to avoid force overshoots.

The force acting on the tip of the Ni-Ti wire is set as $f_i\left(i=\mathrm{1,2,3,4}\right)={\left[\begin{array}{cccccc}{f}_x^i& f_y^i& f_z^i& m_x^i& m_y^i& m_z^i\end{array}\right]}^T$ and the displacement is $u_i\left(i=\mathrm{1,2,3,4}\right)={\left[\begin{array}{cccccc}{u}_x^i& u_y^i& u_z^i& {\theta }_x^i& {\theta }_y^i& {\theta }_z^i\end{array}\right]}^T$ . The stiffness model of single Ni-Ti wire can be expressed as $f_i=k^{\prime }{k}_i{u}_i\left(i=\mathrm{1,2,3,4}\right)$ , here, $k^{\prime }=diag\left[\mathrm{1,1,0,0,0,1}\right]$ is the directional deformation coefficients of the stiffness matrix $k_i$ .

Thus, once the contact force $F={\left[\begin{array}{cccccc}{F}_x& F_y& F_z& M_x& M_y& M_z\end{array}\right]}^T$ acts on the end of the wrist, the relationship between contact force vector and the deformation force of Ni-Ti alloy wires is as $F_x=F_y={\displaystyle \sum _{i=1}^4{f}_i^x}={\displaystyle \sum _{i=1}^4{f}_i^y}$ , $M_z={\displaystyle \sum _{i=1}^4{k}_{vx,i}d{z}_i}$ and $d{z}_i=\sqrt{2}d\sin \left(\mathrm{\Delta }{\varphi }_z/2\right)$ is transverse deformation along x or y-axis of Ni-Ti alloy wires, $d$ is the length between two adjacent alloy wires, as shown in Figure 2.

The robot assembly system contains a 6-DOF manipulator, compliant wrist, and force sensor and control system, as shown in Figure 3. During insertion, the deformation and stiffness of the wrist are dependent on the horizontal offset error and contact force.

Figure 3.

Compliant robot system.

Kinematic and force mapping

Shown as in Figure 4, $\left\{W\right\}$ is the world coordinate system, $\left\{B\right\}$ , $\left\{S\right\}$ , $\left\{C\right\}$ and $\left\{\stackrel{⌢}{C}\right\}$ are attached on the wrist. $\left\{\stackrel{⌢}{C}\right\}$ and $\left\{C\right\}$ are contact tip coordinate system with and without the application of force $F\in R^6$ , which can be separated into force and torque applied on the contact tip.

Figure 4.

Structure diagram of the compliant robot system.

The transformation matrix between coordinate system $\left\{W\right\}$ and $\left\{B\right\}$ can be written as ${{\displaystyle {}^{B}T}}_W$ . It is also the Jacobian matrix of manipulator.

$${{\displaystyle {}^{B}T}}_W\left(q\right)=\left[\begin{array}{cc}{R}_{bw}\left(s\right)& P_{bw}\left(s\right)\\ 0& 1\end{array}\right]$$

(1)

here, $R_{bw}\left(s\right)\in SO\left(3\right)$ and $P_{bw}\left(s\right)\in SO\left(3\right)$ is the rotational matrix and the translation matrix of the manipulator. The equation can map the kinematic into the input joint variables $s\in Q$ and to the special Euclidean group $SE\left(3\right)$ .

Refers Figure 5, the transformation matrix between coordinate system $\left\{B\right\}$ and $\left\{C\right\}$ is ${{\displaystyle {}^{C}T}}_B$ .

$${{\displaystyle {}^{C}T}}_B\left(s,F\right)={{\displaystyle {}^{\stackrel{⌢}{C}}T}}_B\left(s\right)·{{\displaystyle {}^{C}T}}_{\stackrel{⌢}{C}}\left(s,F\right)$$

(2)

here, ${{\displaystyle {}^{\stackrel{⌢}{C}}T}}_B\left(q\right)$ is the transformation matrix between coordinate system $\left\{B\right\}$ and $\left\{\stackrel{⌢}{C}\right\}$ , which correspond to the unloaded state of the robot system, and ${{\displaystyle {}^{C}T}}_{\stackrel{⌢}{C}}\left(q,F\right)$ is the force-mapping transformation matrix between coordinate system $\left\{C\right\}$ and $\left\{\stackrel{⌢}{C}\right\}$ , which denotes the displacement associated with the deformation of the wrist due to the applied force.

Figure 5.

Deformation of a single Ni-Ti wire with force vector ${\varphi }_i$ and moment vector ${\eta }_i$ .

Therefore, the transformation matrix ${{\displaystyle {}^{C}T}}_B\left(s,F\right)$ is a mapping function for kinematic and external force.

The product ${{\displaystyle {}^{C}T}}_B\left(s,F\right)={{\displaystyle {}^{\stackrel{⌢}{C}}T}}_B\left(s\right)·{{\displaystyle {}^{C}T}}_{\stackrel{⌢}{C}}\left(s,F\right)$ depends on the Cosserat rod model.

Based on the virtual work principle, the force mapping can be obtained

$$\tau =J_{BC}^{T}F+\frac{\partial E\left(s,F\right)}{\partial \left(s\right)}$$

(3)

here, $\tau$ is the torque vector of the manipulator, $\partial E\left(s,F\right)$ is the elastic energy of the wrist, and $J_{BC}^T$ is the Jacobian matrix the of compliant robot system.

In general, when the wrist is in a rigid state, the equation can be rewritten as $\tau =J_{BC}^{T}F$ by excluding the elastic energy terms. The stiffness control of a rigid robot can map the desired tip force vector to the actuator by the force mapping equation. Therefore, we can realize the stiffness control by using the force mapping in equation (1).

Kinestostaic modelliing

To obtain the product ${{\displaystyle {}^{C}T}}_B\left(s,F\right)={{\displaystyle {}^{\stackrel{⌢}{C}}T}}_B\left(s\right)·{{\displaystyle {}^{C}T}}_{\stackrel{⌢}{C}}\left(s,F\right)$ , we approximate the wrist with a single elastic Ni-Ti wire with a single elastic Ni-Ti wire. Therefore, we choose a curvature to match curve $l\left(L,s\right)$ , and the calculation of stiffness relies on all elastic components.

When the tip force vector causes significant relative displacement between the elastic components of the wrist, several Cosserat rods can calculate the deviation deformation. ²⁷

Modelling of single Ni-Ti wire

A wrist consist of four Ni-Ti wires, the shape of the i-th Ni-Ti wire is defined by its position $P_i\left(s_i\right)\in R^3$ and the rotation matrix $R_i\left(s_i\right)\in SO\left(3\right)$ , forming a material-attached transformation matrix.

$${{\displaystyle {}^{S}T}}_B\left(s_i\right)=\left[\begin{array}{cc}{R}_i\left(s_i\right)& P_i\left(s_i\right)\\ 0& 1\end{array}\right]$$

(4)

here, ${{\displaystyle {}^{S}T}}_B\left(s_i\right)$ is the function of the arc length of the Ni-Ti wire, $s_i\in \left[\begin{array}{cc}0& L_i\end{array}\right]$ and $L_i$ is the length of Ni-Ti wire, which depends on $l\left(s\right)$ . The position and orientation along the length of the Ni-Ti wire relies on the linear strains $v_i\left(s\right)\in R^3$ and angular strains $u_i\left(s\right)\in R^3$ .

The differential of equation (4) is

$$\begin{array}{l}{\dot{P}}_i\left(s\right)=R_i\left(s\right)v_i\\ {\dot{R}}_i\left(s\right)=R_i\left(s\right)u_i\end{array}$$

(5)

In which, $u_i$ has the skew-symmetric matrix form

$$u_i=\left[\begin{array}{ccc}0& -u_3& u_2\\ u_3& 0& -u_1\\ -u_2& u_1& 0\end{array}\right]$$

(6)

Then, the deformation can be presented as

$$\xi =\left[\begin{array}{c}v\\ u\end{array}\right]=\left[\begin{array}{cccc}0& -u_3& u_2& v_1\\ u_3& 0& -u_1& v_2\\ -u_2& u_1& 0& v_3\\ 0& 0& 0& 0\end{array}\right]$$

(7)

Therefore, if the vectors $v_i\left(s\right)$ , $u_i\left(s\right)$ are known, we can calculate the initial frame ${{\displaystyle {}^{S}T}}_B\left(0\right)$ .

Thus, the differential of ${{\displaystyle {}^{S}T}}_B\left(s_i\right)$ is

$$\frac{d}{ds}{{\displaystyle {}^{S}T}}_B\left(s_i\right)={{\displaystyle {}^{S}T}}_B\left(s_i\right)\xi$$

(8)

The shape of wrist deflection with no load can be computed as follows

$$u\left(s\right)=R_i^{-1}\left(s_i\right)\frac{d{R}_i\left(s_i\right)}{ds}$$

(9)

Refers paper, ²⁸ the internal force vector $n_i$ and internal moment vector $m_i$ relate to the arc length $s_i$ of the Cosserat rod differential static equilibrium. The derivative to the arc length of the static equilibrium balance is shown in Figure 5.

The relationship between internal force vector and acting force vector is as follows.

$$\begin{array}{l}{\dot{n}}_i\left(s\right)=-f_i\\ {\dot{m}}_i\left(s\right)=-{\dot{P}}_i\times n_i\left(s\right)-T_i\end{array}$$

(10)

here, $f_i$ and $T_i$ are force and moment vectors that are applied on the end of Ni-Ti wire.

Kinetostatic modeling with contact loads

The constitutive model that relates strains u and v to the moment m and force f is used to compute rod deformations. The variables v and u are related to the material strain (shear, extension, bending, and torsion). Only the relationship between u and m is needed as the shear of Ni-Ti is neglected.

We defined the initial position of Ni-Ti wire without loads as $u^{\ast }\left(s\right)$ and $v^{\ast }\left(s\right)$ , and deformation position of Ni-Ti wire due to load as $u\left(s\right)$ and $v\left(s\right)$ . The force and moment along Ni-Ti wire is a linear constitutive relationship, so that:

$$\begin{array}{l}{n}_i\left(s\right)=R_i\left(s_i\right)K_u\left(u_i\left(s\right)-u_i^{\ast }\left(s\right)\right)\\ m_i\left(s\right)=R_i\left(s_i\right)K_v\left(v_i\left(s\right)-v_i^{\ast }\left(s\right)\right)\end{array}$$

(11)

here,

$$K=\left[\begin{array}{c}{K}_v\\ K_u\end{array}\right]$$

(12)

and $K_v$ and $K_u$ is the translational and torsional stiffness, respectively.

$$K_v=\left[\begin{array}{ccc}{k}_{vx}& 0& 0\\ 0& k_{vy}& 0\\ 0& 0& k_{vz}\end{array}\right]=\left[\begin{array}{ccc}{A}_i{G}_i& 0& 0\\ 0& A_i{G}_i& 0\\ 0& 0& A_i{E}_i\end{array}\right]$$

(13.a)

$$K_u=\left[\begin{array}{ccc}{k}_{ux}& 0& 0\\ 0& k_{uy}& 0\\ 0& 0& k_{uz}\end{array}\right]=\left[\begin{array}{ccc}{E}_i{G}_i& 0& 0\\ 0& E_i{G}_i& 0\\ 0& 0& J_i{E}_i\end{array}\right]$$

(13.b)

where, $A_i$ is the cross-sectional area, $G_i$ is shear modulus, $E_i$ is young’s module, $I_i$ is the moment around the x and y-axis, $J_i$ is the polar moment.

Combining equations (10)–(12), we get the kinestostaic model of the Ni-Ti wire.

Ignoring the type of mechanical connection of the Ni-Ti wire to the output flange, the static equilibrium about all forces and moments is as follows.

$$\begin{array}{l}{\displaystyle \sum _{i=1}^4{n}_i\left(s\right)}-F=0\\ {\displaystyle \sum _{i=1}^4\left[P_i\left(s\right)\times n_i\left(s\right)+m_i\left(s\right)\right]}-T=0\end{array}$$

(14)

here, $F$ and $T$ are the force and moment vector act on the end of wrist, respectively.

Based on the Cosserat rod model [30], the differential equation of moment m and force n about s is as follows.

$$\begin{array}{l}\frac{d{m}_i\left(s\right)}{ds}=\mathrm{\Delta }m\left(s\right)-u\left(s\right)m\left(s\right)-v\left(s\right)n\left(s\right)\\ \frac{d{n}_i\left(s\right)}{ds}=\mathrm{\Delta }f\left(s\right)-u\left(s\right)n\left(s\right)\end{array}$$

(15)

where $\mathrm{\Delta }m\left(s\right)$ and $\mathrm{\Delta }f\left(s\right)$ are the moment and force applied on the per unit length of a single Ni-Ti wire. When no contact force is applied on the end of Ni-Ti wire, $\mathrm{\Delta }m\left(s\right)=\mathrm{\Delta }f\left(s\right)=0,s\in \left[\begin{array}{cc}0& L\end{array}\right]$ .

Combing equations (10) and (14), the equation for m , n and u , v are obtained.

$$\begin{array}{l}\frac{d{u}_i\left(s\right)}{ds}=\frac{d{u}_i^{\ast }\left(s\right)}{ds}-K_u^{-1}(u_i\left(s\right)K_u\left(u_i\left(s\right)-u_i^{\ast }\left(s\right)\right)+v_i\left(s\right)n_i\left(s\right)+\frac{d{K}_u}{ds}\left(u_i\left(s\right)-u_i^{\ast }\left(s\right)\right))\\ \frac{d{n}_i\left(s\right)}{ds}=-u\left(s\right)n\left(s\right)\end{array}$$

(16)

$$\begin{array}{l}\frac{d{u}_i\left(s\right)}{ds}=\frac{d{u}_i^{\ast }\left(s\right)}{ds}-K_u^{-1}(u_i\left(s\right)K_u\left(u_i\left(s\right)-u_i^{\ast }\left(s\right)\right)+v_i\left(s\right)n_i\left(s\right)+\frac{d{K}_u}{ds}\left(u_i\left(s\right)-u_i^{\ast }\left(s\right)\right))\\ \frac{d{n}_i\left(s\right)}{ds}=-u\left(s\right)n\left(s\right)\end{array}$$

(17)

Therefore, the force vector $F_w$ mounted on the end of the wrist can be expressed as

$$\begin{array}{l}{F}_w={\left[\begin{array}{cc}{\displaystyle \sum _{i=1}^4{n}_i^T\left(s\right)}& {\displaystyle \sum _{i=1}^4{m}_i^T\left(s\right)}\end{array}\right]}^T\\ u_i\left(s\right)=K_u^{-1}{m}_i\left(s\right)+u_i^{\ast }\left(s\right)\end{array}$$

(18)

The desired end tip wrench $F_d$ of the wrist to be used in equation (18) is related to the force vector $F_w$ by

$$F_w=\left[\begin{array}{cc}{R}_{BC}^T& P_{BC}^T\\ 0& 1\end{array}\right]F^d$$

(19)

here, $R_{BC}^T$ and $P_{BC}^T$ are the rotational matrices of the translation matrix ${{\displaystyle {}^{C}T}}_B$ . Although the integral must be solved interactively now, it adjusts quickly once the contact force vector changes quickly, and the ideal position of the previous step is used to solve the above problem.

The inverse kinematic of the wrist corresponding to the position of the slider is $s$ .

$$s^d={\left[{{\displaystyle {}^{C}T}}_B\right]}^{-1}={\left[\begin{array}{cc}{{\displaystyle {}^{C}T}}_B& {{\displaystyle {}^{\stackrel{⌢}{C}}T}}_C^{-1}\left(s^d,F_W\right)\end{array}\right]}^{-1}$$

(20)

here, ${{\displaystyle {}^{\stackrel{⌢}{C}}T}}_C^{-1}\left(s,F_W\right)$ is calculated by Cosserat rod equations.

Therefore, we can get the end deformation position of the wrist caused by the force vector.

Design of fuzzy PD controller and stability analysis

Design of fuzzy PD controller

During the shaft-hole assembly process, the contact force changes rapidly, and the control of stiffness is exceedingly complicated. In this section, a stiffness controller is designed for the compliant robot system to adjust the contact tip stiffness, which aims to promote the compliance control performance during the assembly process. Simultaneously, the fuzzy PD (Proportional-Derivative) control strategy is used to improve the stiffness control performance in the contact force control loop.

The desired force acting on the tip of wrist is $F^d\in R^3$ , in which can be computed using a diagonal stiffness matrix $K^d\in R^{3\times 3}$ , which depends on the cantilever length of the Ni-Ti wire. Then, we have

$$F^d=K^d\mathrm{\Delta }P$$

(21)

here, $\mathrm{\Delta }P=P_{BC}^m-P_{BC}^r$ is the relative position error, $P_{BC}^m$ is the actual position and $P_{BC}^r$ is the desired position. Moreover, the desired orientation matrix of the end tip is defined as $R_{BC}^d\in SO\left(3\right)$ .

The desired force $F^d$ can be mapped to the position of the slider by $R_{BC}^d$ . Then, the desired force $F^d$ at the end tip of the robot will be generated. Therefore, the position and force vector of the slider can be calculated according to equation (1).

$$T_{BC}^{dm}=\left[\begin{array}{cc}{R}_{BC}^d& P_{BC}^m\\ 0& 1\end{array}\right]=T_{BC}\left(l^d,F_W^d\right)=T_{B\stackrel{⌢}{C}}\left(l^d\right)T_{C\stackrel{⌢}{C}}\left(l^d,F^d\right)$$

(22)

According to (19), ones have

$$F_W^d=\left[\begin{array}{c}{f}^d\\ {\left[\begin{array}{ccc}0& 0& 0\end{array}\right]}^T\end{array}\right]$$

(23)

To control the stiffness, we convert from the stiffness control into position control of the wrist by using the PD control algorithm. Then, the diagram of stiffness control is shown in Figure 6. The actual tip position and desired position are used to calculate stiffness and control the movement of the screw-slider, then, the cantilever length of the Ni-Ti wire varied with the change of stiffness. Therefore, the end stiffness of the robot system can be controlled by feed-forward feedback to the fuzzy PD controller.

Figure 6.

Diagram of stiffness control.

The traditional PD control method is

$$\eta =K_d\dot{e}+K_{p}e$$

(24)

here, $e=F^d-F$ , $K_d$ and $K_p$ is the proportional and integral control term, respectively.

The key point of fuzzy PD control is to optimize the two control parameters of the PD control system with fuzzy control theory. The two control parameters $(\mathrm{\Delta }{K}_d$ and $\mathrm{\Delta }{K}_p)$ of the fuzzy PD control system are real-timely calculated through the fuzzy controller to satisfy the control system. Then, the parameters of the fuzzy PD controller are given as

$$\begin{array}{l}{K}_p=K_{p0}+\mathrm{\Delta }{K}_p\\ K_d=K_{d0}+\mathrm{\Delta }{K}_d\end{array}$$

(25)

Based on the fuzzy control method, the fuzzy subset of $K_d$ , $c_e$ , $K_d$ and $K_p$ is defined as {NB, NM, NS, ZO, PS, PM, PB}. According to the experiment and research, the safe contact force maintained in the range of 60N, the fuzzy domain of $\mathrm{\Delta }{K}_d$ and $\mathrm{\Delta }{K}_p$ are defined in Table 1.

Table 1.

Fuzzy rule base of $\mathit{e},\dot{e},\mathrm{\Delta }{K}_p,\mathrm{\Delta }{K}_d$ .

$\dot{e}$	NB	NM	NS	ZO	PS	PM	PB
$\mathrm{\Delta }{K}_p,\mathrm{\Delta }{K}_d$
$\mathit{e}$
NB	PB	PB	PM	PM	PS	ZO	ZO
NM	PB	PB	PM	PS	PS	ZO	NS
NS	PM	PM	PM	PS	ZO	NS	NS
ZO	PM	PM	PS	ZO	NS	NM	NM
PS	PS	PS	ZO	NS	NS	NM	NM
PM	PS	ZO	NS	NM	NM	NM	NB
PB	ZO	ZO	NM	NM	NM	NB	NB

The range of $\dot{e}$ and $c_e$ is kept in the range of $-6<e<6$ , and $-9<c_e<9$ . When the range of $K_d$ and $K_p$ is set as $-0.2<K_p<0.6,0\text{<}{K}_d<0.8$ .

The parameters $K_{d0}$ and $K_{p0}$ of traditional PD control are chosen as

$$\begin{array}{l}{K}_{p0}=0.2\\ K_{d0}=0.4\end{array}$$

(26)

which has good force tracking performance according to the simulation and experiment, and the fuzzy domain of $\mathrm{\Delta }{K}_d$ and $\mathrm{\Delta }{K}_p$ are set as

$$\begin{array}{l}\mathrm{\Delta }{K}_p\left\{-0.3,-0.2,-0.1,\mathrm{0,0.1},0.2,0.3\right\}\\ \mathrm{\Delta }{K}_d\left\{-\mathrm{0.4.}-0.27,-\mathrm{0.14.0},0.14,0.27,0.4\right\}\end{array}$$

(27)

Then, the fuzzy base domain of $e$ and $ce$ are

$$e\left\{-6,-4,-\mathrm{2,0,2,4,6}\right\}{c}_e\left\{-9,-6,-\mathrm{3,0,3,6,9}\right\}$$

(28)

which covers most of the contact force situations, whereas the larger $\dot{e}$ and $c_e$ still exist because of stiffness uncertainty in control system.

A saturation sigmoid function is used to limit the range of output stiffness once the expected stiffness exceeds the allowed range. Then, the input of control is

$$U=sigmoid\left(\eta \right)K_{\max }$$

(29)

here, $K_{\max }$ is the maximum output stiffness of the wrist.

Stability analysis

To guarantee the stability and providing a robust stiffness control performance of the proposed controller, the stiffness control is reflected on the force control. The force control loop is simplified to complete the analysis of stability and calculation of steady-state error. The force control loop is illustrated as Figure 7, here $L_{fc}$ is the fuzzy control loop, $G_{cw}$ is the transfer function of compliant wrist, $G_f$ is the transfer function of measured contact force, $K^d$ is the desired contact stiffness.

Figure 7.

Simplified contact force with stiffness control loop.

Thus, $G_{cw}$ is expressed as

$$G_{cw}=\frac{1}{T_{c}s+1}\times \frac{1}{T_{w}s+1}$$

(30)

here, $T_c$ is the control cycle period that is short enough to be not considered. $T_w$ is the control response cycle of the compliant wrist which is about 20–50 ms.

And $G_f$ is expressed as

$$G_f=\frac{1}{T_{f}s+1}$$

(31)

here, $T_f$ is the contact force perception cycle time, refer Figure 8, the transfer function of overall contact force with stiffness control loop can be described as follows,

$$\begin{array}{c}G\left(s\right)=\frac{L_{fc}{K}^d{G}_{cw}}{L_{fc}{K}^d{G}_{cw}{G}_f+1-G_{cw}}\\ =\frac{L_{fc}{K}^d\left(T_f+1\right)}{L_{fc}{K}^d+T_w{T}_c{T}_f{s}^3+\left(T_w{T}_c+T_w{T}_f+T_c{T}_f\right)s^2+\left(T_w+T_c\right)s}\end{array}$$

(32)

Figure 8.

(a) Robot assemble system; (b) the compliant wrist.

According to the Routh criterion, the stability condition of the fuzzy PD control system is represented as

$$\left(T_w{T}_c+T_w{T}_f+T_c{T}_f\right)\left(T_w+T_c\right)-\left(L_{fc}{K}^d{T}_w{T}_c{T}_f\right)>0$$

(33)

Which is simplified as

$$L_{fc}<\frac{\left(T_w{T}_c+T_w{T}_f+T_c{T}_f\right)\left(T_w+T_c\right)}{\left(K^d{T}_w{T}_c{T}_f\right)}$$

(34)

Therefore, the stability of the control system is

$$\begin{array}{l}{L}_{fc}<\frac{\left(T_w{T}_c+T_w{T}_f+T_c{T}_f\right)\left(T_w+T_c\right)}{\left(K^d{T}_w{T}_c{T}_f\right)}\\ <\frac{2T_w{T}_c{T}_f}{K^d{T}_w{T}_c{T}_f}=\frac{2}{K^d}\end{array}$$

(35)

From above, the system control error is

$$E\left(s\right)=F^d\left(s\right)\left(1-G\left(s\right)G_f\left(s\right)\right)$$

(36)

The system stability error is

$$E_{ss}=\underset{s\to 0}{\lim }sE\left(s\right)=\underset{s\to 0}{\lim }s{F}^d\left(s\right)\left(1-G\left(s\right)G_f\left(s\right)\right)$$

(37)

In general, the control input signals of compliant wrist is pulse signal, it means $F^d\left(s\right)=F^d/s$ , then the steady-state error can be expressed as

$$\begin{array}{c}{E}_{ss}=\underset{s\to 0}{\lim }sE\left(s\right)=\underset{s\to 0}{\lim }s\frac{F^d}{s}\left(1-G\left(s\right)G_f\left(s\right)\right)\\ =\underset{s\to 0}{\lim }{F}^d\left(1-G\left(s\right)G_f\left(s\right)\right)\to 0\end{array}$$

(38)

Therefore, according to the Routh criterion, the contact force with stiffness control is stable.

Experiment verification and analysis

In this section, the compliant wrist is applied to connect the manipulator and the gripper (parts). A JAKA manipulator with the Advantech 610L PC and a force/torque sensor are used as the manipulator. The assembled system is shown in Figure 8(a), and the shape parameters of the compliant wrist are illustrated in Figure 8(b).

Simulation

Based on the above, a stiffness tracking control simulation is carried out with the input stiffness defined as:

$$k_i=1.5\ast \sin \left(t\right)$$

(39)

Simulation of stiffness tracking and tracking error shown as in Figure 9. Results shows that the fuzzy PD controller can effectively realize the tracking of the contact stiffness between parts.

Figure 9.

(a) Stiffness tracking control; (b) tracking error.

Experiment verification

To verify the effectiveness of the compliant robot system, shaft-hole assembly experiments are conducted as shown in Figure 10 under different conditions: (a) horizontal offset position $0mm$ , (b) horizontal offset position $3mm$ , (c) horizontal offset position $5mm$ , (d) horizontal offset position $8mm$ , respectively. And the position of the slider drives by servo motor under different horizontal deviations are measured as shown in Figure 11. The stiffness control performance based on measured deformation and force is as shown in Figure 12. From these figures, we can see that the stiffness adjusts quickly by controlling the slider once contact force overshoots.

Figure 10.

Shaft-hole assembly conditions: (a)–(d) horizontal deviation position error equal to $0mm$ , $3mm$ , $5mm$ , $8mm$ .

Figure 11.

(a)–(c) Control position of the slider under horizontal deviation error equal to $3mm$ , $5mm$ , $8mm$ .

Figure 12.

(a)–(c) Stiffness control performance under horizontal deviation error equal to $3mm$ , $5mm$ , $8mm$ .

The contact force of the shaft–hole assembly is measured and is shown in Figure 13, the contact force is small due to the compliant wrist even the horizontal deviations reaches to 8 mm. In addition, Figures 13(a)–(c) shows the contact force/torque comparison between the proposed method and the traditional method under different horizontal deviations equal to 3 mm, 5 mm and 8 mm. From these figures, it is observed that when the force and torque at the contact tip increases, the wrist can adjust the stiffness to decrease contact force (within a range of 15 N) and torque (within a range of 5 N-m) quickly without damaging the parts or the robot. In contrast, the contact force/torque of a rigid assembly without a compliant wrist under different horizontal deviations is extremely large (maximum contact force exceeding 200 N and maximum contact torque greater than 50 N-m). Because when the assembly contact force of shaft and hole exceeds the maximum allowable load of robot in all directions, the safety function of manipulator is activated and the joint holding brake is closed. At this time, the contact force will maintain a transient large contact force (more than 400 N or bigger), and the assembly process actually stops and failure. Thus, the assembly control strategy with compliant wrist in this study effectively reduced the contact force and impact between shaft and hole-parts even under large horizontal position error.

Figure 13.

Contact force/torque comparison between the proposed assembly and the traditional assembly under different horizontal deviations: (a) 3 mm; (b) 5 mm; (c) 8 mm.

Conclusions

In this paper, a robot with a continuously variable stiffness wrist is used to mitigate the impact of assembly errors and uncertainty according to the automatic assembly requirements. The wrist used a parallel mechanism based on Ni-Ti wire and a slider to achieve variable stiffness operation. As it can produce transverse and rotational deformation, the contact force during assembly is significantly reduced. Unlike the traditional robot assembly, it avoids a large contact force due to position deviations between a shaft and hole as the manipulator is fully rigid. The Cosserat rod theory is used to develop the kinetostatic model to calculate the tip stiffness of the wrist and to evaluate the stiffness of the robot system quantitatively. On this basis, a stiffness control strategy for the wrist is proposed that can be model as a loaded elastic rod and has a no-load kinematics model. The results of experimentation indicated that the proposed fuzzy PD controller possessed better stiffness control properties. Experiments are carried out for stiffness parameter calibration and stiffness control verification. The results of the numerical experiments demonstrated that stiffness variations could be achieved by changing the cantilever length of the Ni-Ti wires. The experiments of stiffness control for shaft-hole assembly verification show that the wrist can adjust the stiffness to decrease the contact force and respond to large deformations, which can effectively complete the shaft-hole assembly.