2.1. Problem statement

Various shapes of workpieces need to be grinded in industrial scenes. How to keep the constant contact force along the normal vector of the surface at the contact point is very important for the polishing accuracy. However, the location of the contact point is hard to determine because the geometry of the environment may be unknown. In addition, most industrial robots do not have torque sensors in each joint. Thus, we cannot get the exact force at the contact point based on the joint torques and Jacobian. Some papers try to obtain the contact point according to analyzing contact forces by simplifying the contact characteristics [Reference Jiang, Xu and Xie5–Reference Yip and Camarillo8] or learning Jacobian matrix [Reference Namvar and Aghili9]. However, some complex contacts are often difficult to be simplified. To solve the problem of the normal vectors estimation, we try to use learning method to obtain the normal vectors. Model-based method and learning based method for getting the normal vectors are shown in Fig. 1. In order to control the contact force along the normal direction of unknown surfaces, we should solve the following problems. The first is the estimation of the normal vector of the surface at the contact point. The second is accuracy in the case of unknown surface geometry. Finally, the effect of friction on direction control should be considered.

Figure 1. Model-based method and learning-based method for normal vectors learning. (a) Spherical tool in contact with environment where P is the contact point and $\boldsymbol{n}$ is the normal vector of environment. The problem is to estimate $\boldsymbol{n}$ . (b) Model-based method to obtain the relationship between angle $\theta$ and force where $F_N$ is the normal force, $\mu$ is the friction coefficient, and f is the friction. (c) Learning-based method to obtain the relationship between angle $\theta$ and force F.

To illustrate our method, we build the model of robot performing tasks as shown in Fig. 2 and define the relevant parameters as follows.

Figure 2. Robot performing constrained tasks on the unknown surfaces. The key idea is to learn the normal vectors on a known plane $d({x,y,z}) = 0$ and using the learning results on an unknown surface $g({x,y,z}) = 0$ .

The known plane equations and its normal vectors: $d\!\left ({x,y,z} \right ) = 0$ , $\boldsymbol{n}_h$ .

The unknown surface equations and its normal vectors: $f\!\left ({x,y,z} \right ) = 0$ , $\boldsymbol{n}_f$ .

Tangent plane equation and tangent vector on the reference plane of the unknown surface: $t\!\left ({x,y,z} \right ) = 0$ , $\boldsymbol{t}_f$ .

Robot reference plane: ${r_\varphi }\!\left ({x,y,z} \right ) = 0$ . The reference plane is formed by $Y_w$ and $Z_w$ of the coordinate system ${W}$ . When the ${}^w{F_x} = 0$ , the reference plane is called the initial plane with ${r_0}\!\left ({x,y,z} \right ) = 0$ . The force sensor data in initial plane are $\boldsymbol{F}_{\textbf{0}}$ .

The normal vector of the reference plane: $\boldsymbol{n}_r$ .

Tool orientation: $\boldsymbol{z}_t$ .

$\left \{{{X_C}\;{Y_C}\;{Z_C}} \right \}$ : Constrained coordinate system; $\left \{{{X_{\rm{0}}}\;{Y_0}\;{Z_0}} \right \}$ : 0 coordinate system of the robot arm;

$\left \{{{X_w}\;{Y_w}\;{Z_w}} \right \}$ : Wrist coordinate system of robot arm;

The angle between tool direction and surface normal vector in the reference plane: $\theta$ ;

The angle between reference plane and initial plane: $\varphi$ .

The force sensor data in reference plane: ${\boldsymbol{F}}_\varphi$ .

The problem is to use learning method to obtain the relationship between force sensor data and normal vectors as

(1) \begin{equation} f\,:\,{{\boldsymbol{F}}_\varphi } \to{{\boldsymbol{n}}_h} \end{equation}

2.2. The overall control framework

To achieve end-effector force and direction control, we develop the normal vectors learning and generalizing method with adaptive force control and iterative learning control. Figure 3 shows the normal vectors learning and generalizing framework of the robot. The framework consists of two parts, learning the relation on the demonstrated plane and using the relation on the unknown surface. The panda robot control library provides Cartesian space controller and joint space controller. The joint space controller uses PD with gravity compensation. The gravity calculation requires the desired positions of joints which are obtained through inverse kinematics by official open source library provided by Franka. Considering that most of the current industrial arms use joint space control, the joint space controller is chosen. It helps to apply the method to current arms.

Figure 3. Normal vectors learning and generalizing framework of the robot. $\boldsymbol{X}_d$ is the desired pose of the end effector.

3.1. Learning from a known plane

For different end tools, the force characteristics may be completely different. Force analysis based on specific end tools to determine contact points, surface information, and motion strategies has poor adaptability. Therefore, we learn the relation from the force sensor information to the normal vectors of the surface on the known plane. First, we get the training data. $\theta$ is traversed with a certain interval in a reference plane ${r_\varphi }\!\left ({u,v} \right ) = 0$ . Recording the force sensor data ${\boldsymbol{F}}_\varphi$ and the normal vector ${\boldsymbol{n}}_h$ , we can get p group of raw data $\left [{\begin{array}{*{20}{c}}{{{\boldsymbol{F}}_\varphi }}&{{{\boldsymbol{n}}_h}} \end{array}} \right ]$ . Through normalization, we can get p group of training data $\left [{\begin{array}{*{20}{c}}{{{{\bar{\boldsymbol{F}}}}_\varphi }}&{{{{\bar{\boldsymbol{n}}}}_h}} \end{array}} \right ]$ . Using locally weighted regression, we can get:

(2) \begin{equation}{{\bar{\boldsymbol{n}}}_h} ={{\bar{\boldsymbol{F}}}^T_\varphi }{\boldsymbol{\omega }} \end{equation}

Compared with linear regression, locally weighted regression increases weight parameters and can smoothly fit nonlinear functions. For p groups of samples ${{\bar{\boldsymbol{F}}}_\varphi } = {\left[{\begin{array}{*{20}{c}}{{\bar{\boldsymbol{F}}}_\varphi ^1}&{{\bar{\boldsymbol{F}}}_\varphi ^j}&{\ldots }&{{\bar{\boldsymbol{F}}}_\varphi ^p} \end{array}}\right]^T}$ and ${{\bar{\boldsymbol{n}}}_h} = {\left [{\begin{array}{*{20}{c}}{{\bar{\boldsymbol{n}}}_h^1}&{{\bar{\boldsymbol{n}}}_h^j}&{\ldots }&{{\bar{\boldsymbol{n}}}_h^p} \end{array}} \right ]^T}$ , the cost function is established as

(3) \begin{equation} J ={\sum \nolimits _{j = 1}^p{{\Phi _j}\!\left ({{\bar{\boldsymbol{n}}}_h^j -{{\left ({{\bar{\boldsymbol{F}}}_0^j} \right )}^T}\omega } \right )} ^2} \end{equation}

where $\Phi$ is the kernel function. The most commonly used is Gaussian kernel function, and the corresponding weight is shown as the following equation:

(4) \begin{equation}{{\Phi _j} = {\rm{exp}}\left ({\frac{{\left |{{\bar{\boldsymbol{F}}}_\varphi ^c -{\bar{\boldsymbol{F}}}_\varphi ^j} \right |}}{{ - 2{\sigma ^2}}}} \right ).} \end{equation}

where ${\bar{\boldsymbol{F}}}_\varphi ^c$ is the current normalized force data. It follows that,

(5) \begin{equation}{\boldsymbol{\omega }} = \frac{\bar{\boldsymbol{F}}^T_\varphi {\boldsymbol{\Gamma }}{\bar{\boldsymbol{n}}}_h}{\bar{\boldsymbol{F}}^T_\varphi {\boldsymbol{\Gamma}}{\bar{\boldsymbol{F}}_\varphi}} \end{equation}

where ${\boldsymbol{\Gamma }} = \left [{\begin{array}{*{20}{c}}{{\Phi _1}}&{}&{}\\{}&{\ldots }&{}\\{}&{}&{{\Phi _p}} \end{array}} \right ]$ .

3.2. Reference pose estimation

The normal vector of the unknown surface at the contact point can be obtained based on the learned relation on the demonstrated plane. The normal vector can be calculated by

(6) \begin{equation}{{\boldsymbol{n}}_f} ={{\bar{\boldsymbol{F}}}^T_\varphi }{\boldsymbol{\omega }}. \end{equation}

The tangent vector on the reference plane $\boldsymbol{t}_f$ can be calculated by

(7) \begin{equation}{{\boldsymbol{t}}_f} ={{\boldsymbol{n}}_r} \times{{\boldsymbol{n}}_f}. \end{equation}

The desired pose of the next point of the robotic arm can be estimated as the Eq. (8):

(8) \begin{equation}{ \left \{ \begin{array}{l}{{\boldsymbol{x}}_d}\!\left ({t + 1} \right ) ={{\boldsymbol{x}}_d}\!\left ( t \right ) + v{t_s}{{\bf{t}}_f}\\[5pt]{{\boldsymbol{n}}_{fd}}\!\left ({t + 1} \right ) ={{{\bar{\boldsymbol{F}}}}^T_\varphi }\!\left ( t \right ){\boldsymbol{\omega }} \end{array} \right .} \end{equation}

where $v$ is the speed of the end effector, xd is the desired position of the end effector, and $t_s$ is the sampling time. The desired transformation matrix can be calculated as follows:

(9) \begin{equation}{\boldsymbol{A}}_{0d}^7\!\left ({t + 1} \right ) ={\boldsymbol{A}}_0^7{{\boldsymbol{n}}_{fd}}\!\left ({t + 1} \right ), \end{equation}

(10) \begin{equation}{\boldsymbol{T}}_{0d}^7\!\left ({t + 1} \right ) = \left [{\begin{array}{*{20}{c}}{{\boldsymbol{A}}_{0d}^7}&\quad {{{\boldsymbol{x}}_d}}\\[3pt] 0&\quad 1 \end{array}} \right ]. \end{equation}

where ${\boldsymbol{A}}_0^7$ is the current rotation matrix, ${\boldsymbol{A}}_{0d}^7$ is the desired rotation matrix, and ${\boldsymbol{T}}_{0d}^7$ is the desired transformation matrix.

4.1. Adaptive force control

There is a certain deviation between the expected pose obtained from Eq. (8) and the actual pose of the curved surface because of the sampling time, straight line approximation, and the tangent vector estimated error. In order to control the contact force along the surface normal vector, a PID-type adaptive force control method is developed. The adaptive force control law is as follows.

(11) \begin{equation} \ddot e + \dot e = \left ({{f_e} -{f_d}} \right ) +{K_i}\int{\left ({{f_e} -{f_d}} \right )dt} +{K_d}\!\left ({{{\dot f}_e} -{{\dot f}_d}} \right ) \end{equation}

where $f_e$ and $f_d$ are the actual normal force and desired normal force, respectively. $\dot f_e$ and $\dot f_d$ are their differentiations. $\dot e$ and $\ddot e$ are the velocity error and acceleration error respectively. $K_i$ is the integral coefficient. $K_d$ is the differential coefficient.

The accumulation of force errors is used as the input of the admittance control in refs. [Reference Duan, Gan, Chen and Dai18, Reference Jung, Hsia and Bonitz19], which can adaptively compensate the environment errors. Results show that the method can achieve high force control accuracy on unknown surfaces. Compared with the method in refs. [Reference Duan, Gan, Chen and Dai18, Reference Jung, Hsia and Bonitz19], the differential term is added and the mass and damping parameters are deleted. The proposed method in this paper can achieve high force control accuracy on geometric unknown surface and reduce the dynamic force error by the differential term. The parameter $e$ will be added to the desired position as the following equation:

(12) \begin{equation}{{\boldsymbol{x}}_r} ={{\boldsymbol{x}}_d} +{\boldsymbol{A}}_0^7{\left [{\begin{array}{*{20}{c}} 0&\quad 0&\quad e \end{array}} \right ]^{\rm{T}}} \end{equation}

where ${\boldsymbol{x}}_r$ is the reference position. ${\boldsymbol{A}}_0^7$ is the rotation matrix. The reference transformation matrix can be calculated as follows:

(13) \begin{equation}{\boldsymbol{A}}_{0r}^7\!\left ({t + 1} \right ) ={\boldsymbol{A}}_{0d}^7\!\left ({t + 1} \right ), \end{equation}

(14) \begin{equation}{\boldsymbol{T}}_{0r}^7\!\left ({t + 1} \right ) = \left [{\begin{array}{*{20}{c}}{{\boldsymbol{A}}_{0r}^7} & \quad {{{\boldsymbol{x}}_r}}\\[4pt] 0&\quad 1 \end{array}} \right ]. \end{equation}

4.2. Iterative learning for direction control

The direction control accuracy will be poor when the friction characteristics are different between the demonstrated plane and the operating surface. In this paper, we developed the iterative learning control for direction control considering the symmetrical repetitive movement. We suppose that the frictions are opposite at the same point during the symmetrical movement of the robot. The deviation angle of the direction control due to the friction force is reversed. Therefore, we design the control law for direction control as the following equation:

(15) \begin{equation}{ \begin{aligned} {\boldsymbol{n}}_f^{k + 1}\!\left ( t \right ) &= \gamma{\left ({{\bar{\boldsymbol{F}}}_e^{k + 1}\!\left ( t \right ) +{\bar{\boldsymbol{F}}}_e^k\!\left ( t \right )} \right )^T}{\boldsymbol{\omega }}\\ & = 2\gamma \!\left ({{\boldsymbol{n}}_f^k\!\left ( t \right ) + 0.5{{\left ({{\bar{\boldsymbol{F}}}_e^{k + 1}\!\left ( t \right ) -{\bar{\boldsymbol{F}}}_e^k\!\left ( t \right )} \right )}^T}{\boldsymbol{\omega }}} \right ) \end{aligned} } \end{equation}

where the superscript $k$ represents the number of iterations. ${\bar{\boldsymbol{F}}}_e^k\!\left ( t \right )$ is the normalized end force. $\gamma$ is the learning rate. Using Eq. (15), the tool of the robotic arm is gradually aligned with the surface normal vector.

4.3. Stability and convergence analysis of adaptive force control

Assuming the environment model is a pure stiffness model, then the contact force can be obtained by

(16) \begin{equation}{{f_e}} ={K_e}\!\left ({x -{x_e}} \right ) ={K_e}e\!\left ( t \right ) \end{equation}

where $K_e$ is the environmental stiffness, $x_e$ is the environmental position, and $x$ is the end position of the robot. We can get,

(17) \begin{equation}{ e\!\left ( t \right ) = \frac{{{f_e}}}{{{K_e}}}\quad \dot e\!\left ( t \right ) = \frac{{{{\dot f}_e}}}{{{K_e}}}\quad \ddot e\!\left ( t \right ) = \frac{{{{\ddot f}_e}}}{{{K_e}}} } \end{equation}

Considering the uncertainty of the environmental location, we use as the estimation of the environmental location. Then,

(18) \begin{equation} \delta{x_e} ={\hat x_e} -{x_e} \end{equation}

(19) \begin{equation} \hat e = e + \delta{x_e} \end{equation}

We use the following equation for adaptive force control:

(20) \begin{equation} \ddot{\hat{e}} + \dot{\hat{e}} = \left ({{f_e} -{f_d}} \right ) +{K_i}\int{\left ({{f_e} -{f_d}} \right )dt} +{K_d}\!\left ({{{\dot f}_e} -{{\dot f}_d}} \right ) \end{equation}

Substituting Eqs. (12) (13) into Eq. (14), we can get

(21) \begin{equation} \begin{aligned} & \left ({{{\ddot f}_d} -{{\ddot f}_e}} \right ) + \left ({{{\dot f}_d} -{{\dot f}_e}} \right ) + {K_e}\!\left ({{f_d} -{f_e}} \right )\\ & \quad +{K_i}{K_e}\int{\left ({{f_d} -{f_e}} \right ){\rm{dt}}} +{K_d}{K_e}\!\left ({{{\dot f}_d} -{{\dot f}_e}} \right )\\ & \quad = {{\ddot f}_d} +{{\dot f}_d} -{K_e}\delta{{\ddot x}_e} -{K_e}\delta{{\dot x}_e}\\ & \quad = \left ({{{\ddot f}_d} -{{\ddot{\bar{f}}}_e}} \right ) + \left ({{{\dot f}_d} -{{\dot{\bar{f}}}_e}} \right ) \end{aligned} \end{equation}

Let $\varepsilon = {f_d} -{f_e}$ , $\xi = {f_d} -{\bar f_e}$ and take Laplace transformation of Eq. (15), we can get

(22) \begin{equation} \begin{aligned} & \varepsilon \!\left ( s \right ){s^2} + \varepsilon \!\left ( s \right )s + {K_e}\varepsilon \!\left ( s \right ) +{K_i}{K_e}\varepsilon \!\left ( s \right )/s\\ & \quad +{K_d}{K_e}\varepsilon \!\left ( s \right )s = \xi \!\left ( s \right ){s^2} + \xi \!\left ( s \right )s \end{aligned} \end{equation}

(23) \begin{equation} \frac{{\varepsilon \!\left ( s \right )}}{{\xi \!\left ( s \right )}} = \frac{{{s^3} + {s^2}}}{{{s^3} +{K_d}{K_e}{s^2} + {K_e}s +{K_i}{K_e}}} \end{equation}

According to Routh stability criterion, we can get,

(24) \begin{equation} \begin{array}{*{20}{c}}{{s^3}}&1&{{K_e}}\\{{s^2}}&{{K_d}{K_e}}&{{K_i}{K_e}}\\{{s^1}}&{{K_e} -{K_i}/{K_d}}&0\\ 1&{{K_i}{K_e}}&0 \end{array} \end{equation}

The stability condition is as the following equation.

(25) \begin{equation} 0 \lt{K_i} \lt{K_d}{K_e} \end{equation}

The convergence can be further analyzed by the final-value theorem.

(26) \begin{equation} \begin{aligned}{e_s} & = \mathop{\lim }\limits _{t \to \infty } e\!\left ( t \right ) = \mathop{\lim }\limits _{s \to 0} se\!\left ( s \right ) = \mathop{\lim }\limits _{s \to 0} s\!\left ({\varepsilon \!\left ( s \right ) - \xi \!\left ( s \right )} \right )\\ & = \mathop{\lim }\limits _{s \to 0} s\!\left ({\frac{{{s^3} + {s^2}}}{{{s^3} +{K_d}{K_e}{s^2} + {K_e}s +{K_i}{K_e}}}\xi \!\left ( s \right ) - \xi \!\left ( s \right )} \right ) \end{aligned} \end{equation}

When the input is a step signal, that is, $\xi \!\left ( s \right ) = \dfrac{1}{s}$ ,

(27) \begin{equation}{e_s} = \mathop{\lim }\limits _{s \to 0} \left ({\frac{{{s^3} + {s^2}}}{{{s^3} +{K_d}{K_e}{s^2} + {K_e}s +{K_i}{K_e}}} - 1} \right ) = - 1 \end{equation}

(28) \begin{equation} \mathop{\lim }\limits _{s \to 0} s\varepsilon \!\left ( s \right ) = 0 \Rightarrow \mathop{\lim }\limits _{t \to \infty } \varepsilon \!\left ( t \right ) = 0 \end{equation}

Therefore, ${F_e}$ tends to ${F_d}$ as the time ${t}$ is infinite.

4.4. Stability and convergence analysis of iterative learning

The linear time-invariant system can be expressed by

(29) \begin{equation} \left \{ \begin{array}{l} y\!\left ( t \right ) = pu\!\left ( t \right ) + d\!\left ( t \right )\\[3pt]{u^{k + 1}}\left ( t \right ) = Q\!\left ({{u^k}\!\left ( t \right ) + L{e^k}\!\left ( t \right )} \right ) \end{array} \right. \end{equation}

where ${t}$ is the time variable, ${k}$ is the number of iterations, $u\!\left ( t \right )$ is the input variable, $y\!\left ( t \right )$ is the output variable, and $d\!\left ( t \right )$ is an external iteration signal. The progressive stability condition of the system is that the initial state of each iteration of the system is identical, and

(30) \begin{equation} \left |{Q\!\left ({1 - Lp} \right )} \right | \lt 1 \end{equation}

If the system is asymptotically stable and $Q = 1$ , then the system error converges gradually to zero [Reference Bristow, Tharayil and Alleyne24]. For iterative learning control algorithm for direction control, it follows that

(31) \begin{equation}{ \left \{ \begin{array}{l}{\boldsymbol{n}}_f^{k + 1}\!\left ( t \right ) = 2\gamma \!\left ({{\boldsymbol{n}}_f^k\!\left ( t \right ) + 0.5{{\left ({{\bar{\boldsymbol{F}}}_e^{k + 1}\!\left ( t \right ) -{\bar{\boldsymbol{F}}}_e^k\!\left ( t \right )} \right )}^T}\omega } \right )\\{\boldsymbol{F}}_e^k\!\left ( t \right ) = {{\boldsymbol{K}}_e}{{\boldsymbol{x}}^k} -{{\boldsymbol{K}}_e}{\boldsymbol{x}}_e^k \end{array} \right. } \end{equation}

If the environmental location $x_e^k = x_e^0$ , progressive stability conditions is

(32) \begin{equation} \left |{I - \gamma{{\boldsymbol{\omega }}^T}{K_e}} \right | \lt 1 \end{equation}

If $Q = 1$ , the system error converges gradually to 0. It follows that $\gamma = 0.5$ . Then, we can get

(33) \begin{equation}{ \left \{ \begin{array}{l}{\bf{n}}_f^{k + 1}\!\left ( t \right ) ={\bf{n}}_f^k\!\left ( t \right ) + 0.5{\left ({{\bar{\boldsymbol{F}}}_e^{k + 1}\!\left ( t \right ) -{\bar{\boldsymbol{F}}}_e^k\!\left ( t \right )} \right )^T}{\bf{\omega }}\\{\bf{F}}_e^k\!\left ( t \right ) = {{\bf{K}}_e}{{\bf{x}}^k} -{{\bf{K}}_e}{\bf{x}}_e^k \end{array} \right. } \end{equation}

Assuming that the force deviation due to friction at the same point in the symmetrical movement of the robotic arm is opposite, that is, ${\tilde{\boldsymbol{F}}}_e^{k + 1}\!\left ( t \right ) = -{\tilde{\boldsymbol{F}}}_e^k\!\left ( t \right )$ . It follows that

(34) \begin{equation}{ \begin{aligned}{\hat{\boldsymbol{n}}}_f^{k + 1}\!\left ( t \right ) & = 0.5{\left ({{\bar{\boldsymbol{F}}}_e^{k + 1}\!\left ( t \right ) +{\bar{\boldsymbol{F}}}_e^k\!\left ( t \right )} \right )^T}{\boldsymbol{\omega }}\\ &={\left ({0.5\left ({{\bar{\boldsymbol{F}}}_d^{k + 1}\!\left ( t \right ) -{\tilde{\boldsymbol{F}}}_e^{k + 1}\!\left ( t \right ) + {\bar{\boldsymbol{F}}}_d^k\!\left ( t \right ) -{\tilde{\boldsymbol{F}}}_e^k\!\left ( t \right )} \right )} \right )^T}{\boldsymbol{\omega }}\\ &={\left ({0.5\left ({{\bar{\boldsymbol{F}}}_d^{k + 1}\!\left ( t \right ) + {\bar{\boldsymbol{F}}}_d^k\!\left ( t \right )} \right )} \right )^T}{\boldsymbol{\omega }} ={\boldsymbol{n}}_f^{k + 1}\!\left ( t \right ) \end{aligned} } \end{equation}

When $\gamma = 0.5$ , ${\hat{\boldsymbol{n}}}_f^{k + 1}\!\left ( t \right )$ converges gradually to ${\boldsymbol{n}}_f^{k + 1}\!\left ( t \right )$ .

4.5. Normal vectors learning and generalizing process

Algorithm 1 gives the process of normal vectors learning and generalization. We first learn the relation between the normal vector and the force sensor information on the demonstrated plane through trajectory planning of the robotic arm. We record the force sensor information and use the locally weighted regression algorithm to obtain the relationship between the normal vector and the force sensor information. All variables are represented in the wrist coordinate system.

The generalization process on the unknown surface uses the strategy learned on the known plane. Based on the force sensor information, the normal vector of the surface is estimated to obtain the expected pose of the robot arm at the next moment. Adaptive force control is used to compensate for surface estimation errors. Iterative learning control is used to eliminate the influence of friction during the symmetrical repetitive motion and further improves the direction control accuracy. After obtaining the reference position of the robotic arm, the desired angle of the joint is obtained through inverse kinematics. The robotic arm can be driven by position control algorithm in the joint space. With these algorithms, the robotic arm can achieve high precision trajectory planning, direction and force control on unknown surfaces.

Algorithm 1. Normal vectors learning and generalizing process

5.1. Simulations of normal vectors learning

In order to verify the normal vectors learning method, we performed simulations of normal vectors learning on the plane. We built the model system with the 7-DOF arm called panda and a plane in ADAMS and implemented the control algorithms in MATLAB. The spherical tool is used in the simulation. We considered the simple case that the range of the angle $\theta$ is [–60 $^{\circ }$ , 60 $^{\circ }$ ] in the initial plane where $\varphi$ = 0. The simulation time is 10 s. The length of the steps is 0.01 s. In the learning process on the known plane, the friction is ignored. The force sensor information are measured to obtain 1000 sets of data. The relationship between ${\bar{\boldsymbol{F}}}_0$ and ${\bar{\boldsymbol{n}}}_h$ is obtained by locally weighted regression. The normal vectors learning process is shown in Fig. 4.

Figure 4. Normal vectors learning process on a plane. The raw data include force and angle are measured. LWR is used to train the relation between force and angle.

5.2. Simulation of adaptive force control

In order to verify the adaptive force control algorithm, we performed simulations of trajectory planning and force control on the curved surface. The curved surface is part of the ellipsoid. The simulation time is 5 s. The sampling time is 0.01 s. The velocity of the end-effector is 0.015 m/s. In the generalization process on the unknown surface, the friction is ignored first. The direction error caused by friction will be discussed in next section.

We carried out the comparative simulations using traditional admittance control in ref. [Reference Qin, De, Zhang and Li16], adaptive admittance control in refs. [Reference Duan, Gan, Chen and Dai18, Reference Jung, Hsia and Bonitz19], and adaptive force control in this paper, respectively. The stiffness is set as $K_e=10^6$ N/m. The parameters of traditional admittance control are set as $M$ = 1, $B$ = 100, $K=10^6$ . The parameters of adaptive admittance control are set as $M$ = 1, $B$ = 20, $\eta$ = 0.01. The adaptive force control parameters are set as $K_i$ = 1, $K_d$ = 0.03. These parameters are chosen by manual adjustment according to the curves. The simulation process of adaptive force control is shown in Fig. 5.

The attitude error and force error are defined as

(35) \begin{equation}{e_\theta } = \frac{{{{\boldsymbol{n}}_f} \cdot{{\boldsymbol{z}}_t}}}{{\left \|{{{\boldsymbol{n}}_f}} \right \|\left \|{{{\boldsymbol{z}}_t}} \right \|}} \end{equation}

(36) \begin{equation}{e_F} = \left ({{F_d} -{F_e}} \right )\cos \!\left ({{e_\theta }} \right ) \end{equation}

The force error curves are shown in Fig. 6. Through adaptive force control in this paper and adaptive admittance control in refs. [Reference Jung, Hsia and Bonitz19, Reference Yang, Peng, Li, Cui, Cheng and Li20], the force control accuracy can be guaranteed well. The force error is better than 0.1 N. The traditional admittance control has poor force control performance. Compared with the adaptive admittance control in refs. [Reference Duan, Gan, Chen and Dai18, Reference Jung, Hsia and Bonitz19], the adaptive force control does not care mass and damping parameters but $K_d$ and $K_e$ instead.

Figure 5. The simulation process of adaptive force control on the bulb surface. Three force control methods are used for comparing force tracking performance.

Figure 6. Force error curves comparison using different force control methods. The admittance control in this paper achieves lower force error and initial mutation.

5.3. Effects of frictions on direction

In the simulation of adaptive force control, we ignore the friction. In this part, we carried out the comparative simulations under different friction coefficients.

Under different frictions, the configurations of the robotic arm at the same point are shown in Fig. 7. Direction control error curves under different friction coefficients are shown in Fig. 8. From the curves, we can find that the direction control error is not more than 0.4 $^{\circ }$ without the friction. The higher friction coefficient, the higher force control error. However, the direction deviation caused by friction does not have a great influence on the accuracy of force control. Assuming that the direction error is 16 $^{\circ }$ and the force control error along the tool direction is 1 N, we can calculate that the force error is 0.96 N according to Eq. (27). This force error deviation caused by frictions is only 0.04 N.

Figure 7. Effects of frictions on direction. As the friction increases, the direction gradually deviates from the normal vector.

Figure 8. Direction control error curves under different friction coefficient. As the friction increases, the direction error gradually becomes larger.

Therefore, although the adaptive force control can achieve high precision force control, it cannot guarantee the direction control performance. In the next part, we use the iterative control method to improve the direction control accuracy in the symmetric repetitive motion.

5.4. Simulations of iterative learning

In order to verify the iterative learning for direction control, we performed simulations of trajectory planning and force control on the surface. Different friction coefficients are selected in simulations. The simulation time is 20s containing four symmetric repetitive movements. The simulation process when $\mu = 0.25$ is shown in Fig. 9.

Figure 9. The simulation process when $\mu = 0.25$ . The first iteration is between 0 and 5 s; The second iteration is between 5 and 10 s; The third iteration is between 10 and 15 s; The fourth iteration is between 15 and 20 s.

The attitude error curves under different friction coefficients using adaptive force control and iterative learning control are shown in Fig. 10 (left). The force error curves are shown in Fig. 10 (right). With the increase of the number of iterations, the accuracy of the direction control is gradually improved. After three iterations, the direction control accuracy is better than 0.5 $^{\circ }$ . The force control accuracy is better than 0.1 N except for the time when the direction of movement changes. Results show that the force and direction control accuracy are high using the adaptive force control and iterative learning control in the symmetric repetitive movements.

Figure 10. Direction control error and force error curves under different friction coefficients using adaptive force control and iterative learning control.

6.1. Prototype and experiment setup

In order to verify the normal vectors learning method, we conducted the experiments of learning the relation on the plane and force control on the curved surface. We established the experimental system as shown in Fig. 11. The system includes the franka robot and its control system, a demonstrated plane, an object with curved surface, a power supply, a 6-axis force sensor, and its data acquisition system. The control algorithms are implemented by C++ programming language in robot operating system (ROS). The control frequency is 1000 Hz. The communication bus is EtherCAT. The video can be seen in ref. [Reference Han34].

Figure 11. The prototype of the system.

6.2. Surface learning process

In surface learning process, the most difficult job is to get good data. The force data need to be mean filtered. In actual robot tasks, the limitations of the joint need to be considered. It is difficult to reach the angle of $\pm$ 60 $^{\circ }$ . So, we take $\pm$ 45 $^{\circ }$ in the experiment. The robotic arm rotates around the contact point in the initial plane from –45 $^{\circ }$ to 45 $^{\circ }$ through fifth polynomial planning. The planning time is 10 s. The force sensor data are sampled every 0.001 s. Due to packet loss, we obtain 9976 samples. Because the robotic arm works in the initial plane, the force along Y axis is almost zero. Each sample is denoted as $[F_{ex}, 0, F_{ez}]$ . We process the data as $[F_{ex}/F_{ez}, 0, 1]$ and normalize the training data so that $\left \|{{{\boldsymbol{F}}_e}} \right \|=1$ . Using locally weighted regression, we can get the relation from the force sensor data to the normal vectors. The learning process is shown in Fig. 12. To better observe the results of the training, we give the fitted curves between the force ratio and angle in Fig. 13. From Fig. 13, the relationship between force and normal vector can be accurately fitted by locally weighted regression. We can use this relationship in the force control test.

Figure 12. The learning process on the plane. The experiment process is same as it in the simulation.

Figure 13. The relationship between force and normal vectors. For simplicity, we used angles instead of normal vectors. The angle and the normal vector are in one-to-one correspondence.

6.3. Fixed-point force and direction control

To verify the proposed method for pose and force control, we first carry out the fixed-point attitude and force control test. According to the relationship between force and normal vector learned on the plane, the desired pose of the arm can be obtained based on the force sensor information. The force control along the tool direction is achieved by the adaptive admittance control proposed in this paper. The desired force along the tool direction is set to 5 N. The adaptive force control parameters are set as $K_i=1, K_d=0.03$ . The experimental process is shown in Fig. 14. The force and direction tracking curves are shown in Fig. 15.

Figure 14. Fixed-point force and direction control. We first carry out the single-point force control and directional control to validate the algorithm. Comparison experiments will be performed on inclined surface and bulb surface.

Figure 15. Force and direction control curves. The red curves are desired values and the blue curves are actual values.

From Fig. 15, we can find that using adaptive force control and the learned relationship between force and normal vector, the force and direction of the end effector can converge to given values. There are oscillations during 0–3 s, which could be caused by the sliding of the robot against the surface and parameters selection. In the fixed-point attitude and force control experiment, the force control accuracy is better than $\pm$ 0.5 N, and the direction control accuracy is less than $\pm$ 0.5 $^{\circ }$ .

6.4. Force and direction control on the bulb surface

In order to verify the proposed method for force and direction control on the curved surface, we carry out the comparative experiments between the traditional adaptive force and direction control in refs. [Reference Oba and Kakinuma10, Reference Armesto, Moura, Ivan, Erden, Sala and Vijayakumar11, Reference Duan, Gan, Chen and Dai18, Reference Jung, Hsia and Bonitz19] and the proposed method in this paper. References [Reference Oba and Kakinuma10, Reference Armesto, Moura, Ivan, Erden, Sala and Vijayakumar11] achieve the direction control by controlling the torque to be zero. References [Reference Duan, Gan, Chen and Dai18, Reference Jung, Hsia and Bonitz19] achieve the adaptive force control by the accumulation of force errors. In the force and direction control test, the arm moves repeatedly on the curved surface. According to the relationship between force and normal vector learned on the plane, the desired pose of the arm can be obtained based on the force sensor information. The desired force along the tool direction is set to 5 N. The parameters of traditional admittance control are set as $M=1, B=100, K=10^6$ . The adaptive force control parameters are set as $K_i=1, K_d=0.03$ . The experimental process is shown in Fig. 16. The force error curve and direction error curves are shown in Fig. 17. Using adaptive admittance control in refs. [Reference Duan, Gan, Chen and Dai18, Reference Jung, Hsia and Bonitz19], the force error is about $\pm$ 1 N. Using force-based direction control in refs. [Reference Oba and Kakinuma10, Reference Armesto, Moura, Ivan, Erden, Sala and Vijayakumar11], the direction control error is about $\pm$ 5 $^{\circ }$ . Using adaptive force control is this paper, the force control accuracy is about $\pm$ 0.6 N. In the first two iterations, the direction angle is about $\pm$ 5 $^{\circ }$ , which is consistent with the results of the method in refs. [Reference Oba and Kakinuma10, Reference Armesto, Moura, Ivan, Erden, Sala and Vijayakumar11]. After the fourth iteration, the angle error is reduced to $\pm$ 0.5 $^{\circ }$ . The reason for the big initial error is due to friction. This is the main limitation of the proposed method in this paper.

Figure 16. Force and direction control on the bulb surface. The six images above are obtained using the method in [Reference Oba and Kakinuma10, Reference Armesto, Moura, Ivan, Erden, Sala and Vijayakumar11, Reference Duan, Gan, Chen and Dai18, Reference Jung, Hsia and Bonitz19]. The six images below are obtained using the proposed method in this paper.

Figure 17. Force and direction tracking curves. The red curves are desired values, the black curves are actual values produced by the method in refs. [Reference Oba and Kakinuma10, Reference Armesto, Moura, Ivan, Erden, Sala and Vijayakumar11, Reference Duan, Gan, Chen and Dai18, Reference Jung, Hsia and Bonitz19], and the blue curves are actual values produced by the proposed method in this paper.

Figure 18. Trajectory planning and force control on the inclined surface. The six images above are obtained using the method in refs. [Reference Oba and Kakinuma10, Reference Armesto, Moura, Ivan, Erden, Sala and Vijayakumar11, Reference Duan, Gan, Chen and Dai18, Reference Jung, Hsia and Bonitz19]. The six images below are obtained using the proposed method in this paper.

6.5. Force and direction control on the inclined surface

In order to fully verify the proposed method for trajectory planning and force control on the inclined surface, we carry out the comparative experiments between the traditional adaptive force and direction control in refs. [Reference Oba and Kakinuma10, Reference Armesto, Moura, Ivan, Erden, Sala and Vijayakumar11, Reference Duan, Gan, Chen and Dai18, Reference Jung, Hsia and Bonitz19] and the proposed method in this paper. In the force and direction control test, the arm moves repeatedly on the inclined surface. According to the relationship between force and normal vector learned on the plane, the desired pose of the arm can be obtained based on the force sensor information. The desired force along the tool direction is set to 1 N. The parameters are set as the same as them in the trajectory planning and force control experiment on the bulb surface. The experimental process on inclined surface is shown in Fig. 18. The force error curve and direction error curves on inclined surface are shown in Fig. 19. From Fig. 18, we can find that when using adaptive admittance control in refs. [Reference Duan, Gan, Chen and Dai18, Reference Jung, Hsia and Bonitz19], the force error is about $\pm$ 1N. Using force-based direction control in refs. [Reference Oba and Kakinuma10, Reference Armesto, Moura, Ivan, Erden, Sala and Vijayakumar11], the direction control error is about $\pm$ 0.2 rad. Using adaptive force control is this paper, the force control accuracy is about $\pm$ 1 N. The direction control error is about $\pm$ 0.1 rad. From Fig. 19, we can find that when using adaptive admittance control in refs. [Reference Duan, Gan, Chen and Dai18, Reference Jung, Hsia and Bonitz19], the force error is about $\pm$ 1 N. Using force-based direction control in refs. [Reference Oba and Kakinuma10, Reference Armesto, Moura, Ivan, Erden, Sala and Vijayakumar11], the direction control error is about $\pm$ 0.1 rad. The settling time is about 35s. Using adaptive force control is this paper, the force control accuracy is about $\pm$ 2 N. The direction control error is about $\pm$ 0.1 rad. The settling time is about 5s.

Figure 19. Force error and direction control error curves on the inclined surface. The red curves are desired values, the black curves are actual values produced by the method in refs. [Reference Oba and Kakinuma10, Reference Armesto, Moura, Ivan, Erden, Sala and Vijayakumar11, Reference Duan, Gan, Chen and Dai18, Reference Jung, Hsia and Bonitz19], and the blue curves are actual values produced by the proposed method in this paper.

From Figs. 18 and 19, the method in refs. [Reference Oba and Kakinuma10, Reference Armesto, Moura, Ivan, Erden, Sala and Vijayakumar11] has a long settling time, which lead to a large delay on the bulb surface experiment. The proposed has short settling time and perform better. In five iterations, the angle error is decreasing, but it is not obvious. The reason for the big initial error is due to friction. The reason why the iteration effect is not obvious is the noise of the force sensor, which lead to the low accuracy of surface normal vector measurement. This is the main limitation of the proposed method in this paper.