General-Weighted Least-Norm Control for Redundant Manipulators under Time-Dependent Constraint

Abstract

Previous research dealing with constrained kinematic redundancy problems focuses on manipulators with time-independent constraints. This paper extends the general-weighted least-norm (GWLN) method to manipulators with time-dependent constraints by introducing time-dependent virtual joints. In the virtual joint space, corresponding task space velocity is revised to encapsulate the effects of time-dependent parameters of constraints. This is done so that an intermediate kinematic control problem with only joint limit is obtained. Then, the inverse kinematic problem is solved in the virtual joint space. A new inverse-weighted matrix setting criterion is proposed to replace the one-step prediction that originally complicated implementation of the GWLN method. To demonstrate the efficacy of the method, simulations and experiments are carried out on a five-link welding manipulator to track the welding trajectory. Time-dependent orientation constraint and preventing joint limits is guaranteed using this method.

1. Introduction

In comparison to non-redundant manipulators, redundant manipulators have greater flexibility and dexterity in motion [1, 2]. For kinematically redundant manipulators, there are infinite solutions in the joint space for tracking a desired end-effector trajectory, which is referred to as the main task. In order not to affect the motions of the main task, redundancy in the main task can be used to obtain a solution simultaneously while performing other subtasks. These are usually formulated as constraints, such as joint limits or obstacle avoidance, and are called constrained kinematic redundancy problems [3].

A large body of research has appeared in the literature regarding control strategies, which effectively exploits the potential advantages of kinematically redundant mechanisms to achieve subtasks. The gradient-projection method (GPM) is a widely used method. A quadratic cost function to relate subtasks is presented. Joint motions are obtained without deteriorating the main task. This is achieved by projecting the gradient of the cost function onto the null space of the Jacobian matrix, which is the quickest decreasing orientation for the cost function [4, 5]. Mansard and Chaumette presented a directional GPM used to cast the gradient vector of the subtask into a space, which accelerated the convergence of the main task [6].

The task-priority framework was addressed based on GPM. The main task and constraints were typically classified into two distinct levels. The lower level task was realized in the null space of the higher task [7]. Then, the method was extended to any number of priority levels [8]. A unified method for unilateral constraints was proposed in [9], and a varied weights method was proposed for the kinematic control of redundant manipulators with multiple constraints [10]. In particular, a quadratic program solver was used to deal with a task-priority problem with inequalities in [11, 12].

Alternatively, constrained kinematic redundancy problems can be considered as a quadratic programming (QP) problem. Compact-Inverse QP method and duality theory are introduced to simplify the QP problem [3, 13]. At the same time, a primal-dual neural network based on linear variational inequalities (LVI) has been used to solve the QP problem [14–16]. A minimum-velocity-norm (MVN) scheme based on a primal-dual neural network has been presented to solve the QP problem with a minimumvelocity solution satisfying constraints. The MVN scheme has been used to successfully resolve the robotic redundancy problem with inequality-based obstacle avoiding tasks [17, 18]. However, solutions generally are computationally intensive.

Chan and Dubey proposed the weighted least-norm method (WLN) for joint limits avoidance [19]. A larger efficiency was demonstrated towards the utilization of redundant degrees of freedom (DOFs) in comparison to GPM. This is because the method preplans no motion for the subtask but does not allow any motion to violate the subtask. To deal with constraints other than joint limits, the general-weighted least-norm (GWLN) method was proposed in [20]. In GWLN, a homogeneous transformation is used to map the actual joint space to a virtual joint space, and the virtual joints with joint limits are used to present the constraints. Using the weight factor setting rule with one-step prediction, the constraint was satisfied by applying the WLN method on virtual joints.

The research focus has been limited to controlling redundant manipulators under time-independent constraints, such as joint limits and obstacle avoidance. However, in practice there are cases with time-dependent constraint, for example, end-effector of the manipulator tracks a trajectory, while avoiding a moving obstacle.

Another example of time-dependent constraint is welding torch orientation constraint in welding process control. In welding process control, the manipulator is required to simultaneously get the end-effector to track the welding seam and keep track of the orientation of the welding torch to satisfy constraints, which guarantee welding quality.

The usual strategy is to take welding process control as a five DOFs task, which includes three DOFs for position task and two DOFs for orientation [21, 22]. In this paper, welding process control is realized by simultaneously performing a three-dimensional position task with a welding torch orientation constraint. The orientation is defined by the intersection angle α between the preplanned desired welding torch direction vector O_d and actual welding torch direction vector O_a, which is smaller than the threshold angle β. Both O_d and O_a are three-dimensional unit vectors. As shown in Fig. 1, O_a is in the inner of the right circular cone, whose axis is O_d and apex angle is 2β. Thus, additional redundant degrees of freedom are obtained to do other tasks, such as joint limit avoidance. Actual orientation O_a is determined by joint position, while the desired direction O_d is usually decided by the desired position of the end-effector. Since it is difficult to describe the complicated welding trajectory using an analytic expression, which is a function of position, the preplanned end-effector trajectory is usually represented as a function of time. Thus, the desired orientation O_d is time-dependent. Welding orientation constraint can be formulated as

Figure 1.

Relation between actual and desired welding torch direction. Here, α is the intersection angle between preplanned desired welding torch direction O_d and actual welding torch direction O_a and β is the maximum acceptable deviation angle.

$O_a^T\cdot O_d\left(t\right)\ge \cos \beta ,$

(1)

Eq. (1) is a time-dependent constraint.

A majority of the methods discussed previously for problems with time-independent constraint probably cannot deal with time-dependent constraints. A small proportion of the methods consider the constrained kinematic redundancy problem as a dynamic-system-based QP problem, which are available for time-dependent constraints. The MVN scheme falls into this category of methods. However, as stated before, these are computationally intensive.

Here, the GWLN method, which inherits the advantages of the WLN method, is extended to make it applicable for the constrained kinematic redundancy problem with time-dependent constraint. The transformation from virtual joint space and actual joint space is non-homogeneous for a case with time-dependent constraint. Task space velocity is modified to guarantee homogenous transformation. The WLN method can then be applied on the virtual joint space to guarantee time-dependent constraints.

In addition, improvements to the GWLN method are proposed. This is achieved by setting the inverse-weighted matrix to guarantee joint velocity in a continuous manner instead of using the 0–1 switching law [20]. This relaxes the one-step prediction requirement to set the inverse-weighted matrix factors.

The paper is organized as follows: The GWLN method for the inverse kinematic problem with time-independent constraints is reviewed in Section 2. In Section 3, the problem with time-dependent constraint is formulated and the inverse kinematic problem with a single time-dependent constraint is solved using GWLN. The result is then extended to multiple time-dependent constraints in Section 4. The performances of the GWLN method and MVN scheme on welding process control with the time-dependent orientation constraint are compared in Section 5. In addition, experiments on a five-link welding manipulator using the proposed GWLN method are also presented in Section 5.

2. Kinematics control with time-independent constraint and the GWLN method

2.1 Constrained kinematic redundancy problem with time-independent constraint

For a manipulator, relationship between end-effector velocity ẋ _d ∈ ℝ and joint velocity q̇ ∈ ℝ ⁿ can be represented as:

${\stackrel{.}{x}}_d=J\stackrel{.}{q}$

(2)

where J is the m×n Jacobian matrix. In this paper, a situation where n>m is considered, which means redundant DOFs are available to deal with subtasks in the application. A general subtask is represented in the form of an inequality:

$H_g\left(q\right)\ge h_{gl}$

(3)

where H_g (q) is the time-independent constraint function, h_gl is the lower bound of the constraint.

2.2 Review of the GWLN method

GWLN method was proposed to solve Eq. (2) under Eq. (3) in [20]. GWLN takes H_g (q) as a virtual joint variable. To map the real joint velocities into virtual joint velocities, a coordinate transformation matrix T is used and given as:

$T\left(q\right)=\left[\begin{array}{c}\frac{\partial H_g}{\partial q}\\ N_q\left(q\right)\end{array}\right]$

(4)

where $\frac{\partial H_g}{\partial q}\in {ℝ}^{1\times n},N_q\in {ℝ}^{\left(n-1\right)\times n}$ is composed of unit orthogonal vectors, which are bases of orthogonal complementary space of $\frac{\partial H_g}{\partial q}$ , i.e., $N_q{\left(\frac{\partial H_g}{\partial q}\right)}^T=0$ . Virtual joint velocity q̇ _v ∈ ℝ ⁿ could be represented as:

${\stackrel{.}{q}}_v=T\stackrel{.}{q}\mathrm{.}$

(5)

Kinematic control equation in a virtual joint space is:

$\stackrel{.}{x}=J_v{\stackrel{.}{q}}_v$

(6)

where J_v=JT⁻¹ is the virtual Jacobian matrix. Virtual joint velocity is determined by:

${\stackrel{.}{q}}_v=W_v^{-1}{J}_v^T{\left(J_v{W}_v^{-1}{J}_v^T\right)}^{-1}{\stackrel{.}{x}}_d$

(7)

where W_v ⁻¹ is inverse-weighted matrix, defined as:

$W_v^{-1}=\left[\begin{array}{cccc}{\bar{w}}_{v1}& 0& \cdots & 0\\ 0& {\bar{w}}_{v2}& \cdots & 0\\ \cdots & \cdots & \ddots & \cdots \\ 0& 0& \cdots & {\bar{w}}_{vn}\end{array}\right]\mathrm{.}$

(8)

By setting the corresponding weighted factor, virtual joint motion will be suppressed as it reaches a predetermined limit. In the presence of only one constraint defined in Eq. 3, all diagonal elements are set as 1 except for w̄_v1, which is defined as follows:

${\bar{w}}_{v1}=\{\begin{array}{ccc}1& \mathrm{if}\hspace{1em}{q}_{v1}-h_{gl}>\epsilon \hspace{1em}\mathrm{or}\hspace{1em}{\stackrel{.}{q}}_{v1}>0& \hspace{0.17em}\\ 0& if\hspace{1em}{q}_{v1}-h_{gl}\le \epsilon \hspace{1em}\mathrm{and}\hspace{1em}{\stackrel{.}{q}}_{v1}\le 0& \hspace{0.17em}\end{array}$

(9)

where ε is a positive scalar defining a safety region length.

Closed loop solution to (2) under constraint defined by Eq. (3) is:

$\stackrel{.}{q}=T^{-1}{W}_v^{-1}{J}_v^T{\left(J_v{W}_v^{-1}{J}_v^T\right)}^{-1}\left({\stackrel{.}{x}}_d-k_{e}e\right)$

(10)

where k_e is the feedback gain, and e is the tracking error. The tracking error is defined as:

$e\left(t\right)=x\left(t\right)-x_d\left(t\right)\mathrm{.}$

(11)

In practice, there are two problems with the weights setting law in Eq. (9): First, a sign of virtual velocity q̇_v1 should be known beforehand to set the value of w̄_v1, but the latter decides the former. Second, switch of w̄_v1 leads to discontinuity in q̇_v1. To solve these problems, a one-step prediction along with a low-pass filter on w̄_v1, was proposed for setting w̄_v1 in [20].

3. GWLN method for time-dependent constraint

3.1 Time-dependent constraint

Here, we assume that a general time-dependent constraint on the manipulator can be formulated in the form of bilateral inequality with time-variant parameters, that is:

$h_{gu}\ge H_g\left(q,k\right)\ge h_{gl}$

(12)

where H_g(q,k):ℝ ⁿ × ℝ ^l ↦ ℝ is the time-dependent constraint function, k ∈ ℝ ^l is a time-variant parameter, l is the number of time-variant parameters, h_gl is the lower bound of the constraint, and h_gu is the upper bound of the constraint. One simple case is when k = t. The problem is to find the inverse kinematic solution to Eq. (2), which is subjected to constraints defined in Eq. (12).

3.2 GWLN method solution for time-dependent constraint

For time-dependent constraints, besides the transformation matrix T, an extra transformation matrix M is added as:

$T\left(q,k\right)=\left[\begin{array}{c}\frac{\partial H_g}{\partial q}\\ N_q\end{array}\right],\hspace{0.17em}M\left(q,k\right)=\left[\begin{array}{c}\frac{\partial H_g}{\partial k}\\ Z_k\end{array}\right]$

(13)

where N_q∈ℝ^(n-1)×n is the same as defined before.

$\frac{\partial H_g}{\partial k}\in {ℝ}^{1\times l}$ and Z_k ∈ ℝ^{(n −1)× l} is a zero matrix (in theory, it can be any bounded matrix).

The virtual joint velocity q̇ _v and virtual joint kinematic equation are described as:

${\stackrel{.}{q}}_v=T\stackrel{.}{q}+M\stackrel{.}{k},$

(14)

$\stackrel{.}{x}=J{T}^{-1}{\stackrel{.}{q}}_v-J{T}^{-1}M\stackrel{.}{k}\mathrm{.}$

(15)

Both transformations from the virtual joint space to the task space and from the joint space to the virtual joint space are non-homogeneous. The weighted technique defined in Eq. (9) is not able to guarantee that the virtual joints satisfy their limits. Due to this, task velocity needs to revised to keep the transformation from the virtual joint space to the task space homogenous.

Kinematic equation on virtual joint velocity can be rewritten as:

$J_v{\stackrel{.}{q}}_v=\stackrel{.}{\tilde{x}}$

(16)

where J_v=JT ⁻¹ is the virtual Jacobian matrix with respect to the virtual velocity, and $\stackrel{.}{\tilde{x}}=\stackrel{.}{x}+J{T}^{-1}M\stackrel{.}{k}$ is the revised task space velocity. It can be seen that if k̇=0, the revised task space equals to the original task space. This can be considered as an influence arising from time-dependent parameters encapsulated into the revised task space. As a result, GWLN method can be applied here. It should be pointed out that the time-dependent transformation from the virtual joint space to real joint space guarantees that the manipulator joints would perform the original task even though the virtual joints perform a revised task.

Virtual velocity can be obtained by referring to the GWLN method for time-independent constraint, as follows:

${\stackrel{.}{q}}_v=W_v^{-1}{J}_v^T{\left(J_v{W}_v^{-1}{J}_v^T\right)}^{-1}\stackrel{.}{\tilde{x}}$

(17)

where W_v⁻¹ is the inverse-weighted matrix defined in Eq. (8) and Eq. (9). Note that since solution in Eq. (17) only relates to the inverse matrix of the weighted matrix, W_v⁻¹ can be directly defined to avoid calculating the inverse of W_v as in the GWLN method. Since, one-step prediction to set weight factors [20] is complicated for calculations, a new method is proposed for setting diagonal elements of W_v ⁻¹.

A performance function for avoiding virtual joint limits was proposed by Zghal et al. [5] and is given as

$G\left(q_{vi}\right)=\frac{1}{4}\frac{{\left(h_{gu}-h_{gl}\right)}^2}{\left(h_{gu}-q_{vi}\right)\left(q_{vi}-h_{gl}\right)}\mathrm{.}$

(18)

Since w̄_v1∈(0,1], the weighted law proposed in [19] is invalid. The following weighted law for w̄_v1 is deduced

${\bar{w}}_{v1}=\{\begin{array}{cc}1-\arctan \left(\left|\frac{\partial G\left(q_{v1}\right)}{\partial q_{v1}}\right|\right)\cdot \frac{2}{\pi }& \Delta \left|\frac{\partial G\left(q_{v1}\right)}{\partial q_{v1}}\right|\ge 0\\ 1& \Delta \left|\frac{\partial G\left(q_{v1}\right)}{\partial q_{v1}}\right|<0\end{array}$

(19)

where $\Delta \left|\frac{\partial G\left(q_{v1}\right)}{\partial q_{v1}}\right|$ is the change rate of $\left|\frac{\partial G\left(q_{v1}\right)}{\partial q_{v1}}\right|$ . Other diagonal elements of the matrix W_v¹ are set to 1.

When the virtual joint moves towards its limits, w̄_v1 will decrease. This gradually suppresses virtual joint velocity q̇ _v1 and eventually the first virtual joint will stop before it reaches the limit (proved in Section 3.3). As the main task moves on, the component of end-effect velocity cast into the first virtual joint direction may change the sign, causing it to move away from the limit. As a result $\Delta \left|\frac{\partial G\left(q_{v1}\right)}{\partial q_{v1}}\right|<0$ . Thereby, w̄ _v1 switches to 1, removing suppression over the virtual joint, which could accelerate the virtual joint moving away from the joint limit. It is worthwhile to note that Eq. (19) is still a switching criterion. Different to Eq. (9), when $\Delta \left|\frac{\partial G\left(q_{v1}\right)}{\partial q_{v1}}\right|$ changes sign, the virtual joint is either in the middle of the joint range where $\left|\frac{\partial G\left(q_{v1}\right)}{\partial q_{v1}}\right|$ is zero, or the virtual joint velocity direction changes, and for both situations there is no chatter in the value of q̇ _v .

Eq. (18) is the performance function for a bilateral time-dependent constraint. For a unilateral constraint represented by H_g(q,k)≥h_gl, the criterion for deducing weighted factor w̄_v1 is defined as:

$G\left(q_{vi}\right)=\frac{q_{vi}-h_{gl}+\delta }{q_{vi}-h_{gl}}$

(20)

where δ is a small positive scalar. Then, Eq. (19) can be applied to set the inverse-weighted matrix W_v⁻¹.

Based on the previous result, the closed-loop control law for Eq. (2) with constraint defined in Eq. (12) is given as:

$\stackrel{.}{q}=T^{-1}\left(W_v^{-1}{J}_v^T{\left(J_v{W}_v^{-1}{J}_v^T\right)}^{-1}\left({\stackrel{.}{x}}_d+J_{v}M\stackrel{.}{k}-k_{e}e\right)-M\stackrel{.}{k}\right)\mathrm{.}$

(21)

Remark 1 Eq. (21) can be written as:

$\begin{array}{l}\stackrel{.}{q}={\left(J^{vw}\right)}^{+}\left({\stackrel{.}{x}}_d-k_{e}e\right)+PM\stackrel{.}{k}\hfill \end{array}$

(22)

where (J ^vw)+=T⁻¹(W_v⁻¹J_v^T (J_vW_v⁻¹J_v^T)⁻¹), and P=T⁻¹W_v⁻¹J_v^T (J_vW_v⁻¹J_v^T)⁻¹J_v−T⁻¹, satisfies JP = 0, is the projection operator onto the null space of the matrix J. GWLN solution to the problem with time-dependent constraint consists of two parts. The first part of Eq. (22) is the same with the GWLN method solution used to a problem with time-independent constraint, provided that the parameter is not time varying. The second part is the projection of the main task onto null space. This is done to overcome the effect of the time varying parameters on the subtask, which does not deteriorate the main task. Eq. (10), obtained from [20], can be regarded as a special case of Eq. (21), provided that k̇ = 0.

3.3 Performance analysis

Definition of the control law given in Eq. (21) and the associated weight setting criterion in Eq. (19) lead to the following theorem.

Theorem 1 Provided matrix J_vW⁻¹J_v^T is full rank, control law in Eq. (21) with the associated weight setting criterion in Eq. (19) makes the tracking error from Eq. (11) asymptotically converge to zero.

Proof. Substitute Eq. (21) into Eq. (2), we get

$\stackrel{.}{e}=-k_{e}e\mathrm{.}$

(23)

This guarantees that the main task tracking error asymptotically converges to zero. □

On the other hand, more attention is drawn on whether the constraint will be broken, when the weight setting criterion Eq. (19) is applied. In [20], the weight setting law defined in Eq. (9) sets w̄₁=0, which directly stops the virtual joint motion approaching the limit to guarantee the constraint. Here the weight setting law in Eq. (19) defines w̄₁ close to but not equal to 0. Before to show the compliance of constraints, the following lemma is presented firstly.

Lemma 1u(t)is a differentiable function, if u̇ ≥ − ku and u(t₀) > 0, k is a positive constant scalar, then u(t)>0 whent ≥ t₀.

Proof. Construct function F(t)=e^ktu. There is

$\stackrel{.}{F}=k{e}^{kt}u+e^{kt}\stackrel{.}{u}\ge 0.$

(24)

This demonstrates F(t) is a monotonously increasing function. Referring to u(t₀)>0, there is F(t₀)>0. Thus, u(t)>0, when t≥t₀. □

Theorem 2 Assume that matrix J_vW_v⁻¹J_v^T holds full rank, compliance of constraint defined in Eq. (12) is guaranteed for the manipulator in Eq. 2 under the control law in Eq. (21) with weight setting criterion defined in Eq. (19).

Proof. For the time-dependent constraint function there is:

$\begin{array}{ll}{\stackrel{.}{H}}_g=\hfill & \frac{\partial H_g}{\partial q}{T}^{-1}{W}_v^{-1}{J}_v^T{\left(J_v{W}_v^{-1}{J}_v^T\right)}^{-1}\left({\stackrel{.}{x}}_d+J_{v}M\stackrel{.}{k}-k_{e}e\right)\hfill \\ \hspace{0.17em}\hfill & +\left(\frac{\partial H_g}{\partial k}\stackrel{.}{k}-\frac{\partial H_g}{\partial q}{T}^{-1}M\stackrel{.}{k}\right)\mathrm{.}\hfill \end{array}$

(25)

According to the definition of matrix T, ${\left[{\frac{\partial H_g}{\partial q}}^T,N_q^T\right]}^T{T}^{-1}=I$ , which means $\frac{\partial H_g}{\partial q}{T}^{-1}=\left[\mathrm{1,0},\dots ,0\right]$ .

Therefore, the second term on the right side of Eq. (25) is always equal to zero and the first term on the right side of Eq. (25) can be simplified as:

$\begin{array}{ll}\hspace{0.17em}\hfill & \frac{\partial H_g}{\partial q}{T}^{-1}{W}_v^{-1}{J}_v^T{\left(J_v{W}_v^{-1}{J}_v^T\right)}^{-1}\left({\stackrel{.}{x}}_d+J_{v}M\stackrel{.}{k}-k_{e}e\right)\hfill \\ =\hfill & \left[{\bar{w}}_{v1},0,\dots ,0\right]J_v^T{\left(J_v{W}_v^{-1}{J}_v^T\right)}^{-1}\left({\stackrel{.}{x}}_d+J_{v}M\stackrel{.}{k}-k_{e}e\right)\mathrm{.}\hfill \end{array}$

(26)

Below the method of reduction to absurdity is adopted. Without loss of generality, Hg is assumed to fall below the lower threshold, which breaks the constraint, that is, H=h_gl at t = t₀. Then, a small time interval [t₀-ε,t₀] can always be found, where H_g monotonically decreases from H_g| _t =_t− ε∈(h_gl,(h_gl+h_gu)/2) to h_gl at t = t₀, and ε is small enough to guarantee 0<(H_g-H_gl) | _t0-ε≤1. There is:

$\begin{array}{ll}{\stackrel{.}{H}}_g=\hfill & {\bar{w}}_{v1}{J}_{e1}=\left(1-\arctan \left(\left|\frac{\partial G\left(q_{v1}\right)}{\partial q_{v1}}\right|\right)\cdot \frac{2}{\pi }\right)J_{e1}\hfill \\ =\hfill & \left(1-\frac{2}{\pi }\arctan \frac{{\left(h_{gl}-h_{gu}\right)}^2\left|h_{gl}+h_{gu}-2q_v\right|}{4{\left(h_{gl}-q_v\right)}^2{\left(h_{gu}-q_v\right)}^2}\right)J_{e1}\hfill \end{array}$

(27)

where J_e1 is the first element of J_v^T(J_vW_v⁻¹ J_v^T)⁻¹(ẋ_d + J_vMk̇-k_ee). To simplify Eq. (27), middle variables r = q_v-h_gl and L =h_gu−h_gl were defined. There is:

${\stackrel{.}{H}}_g=\stackrel{.}{r}=\left(1-\frac{2}{\pi }\arctan \frac{L^2\left|L-2r\right|}{4r^2{\left(L-r\right)}^2}\right)J_{e1}\mathrm{.}$

Referring to the property of inverse trigonometric function,

$\stackrel{.}{r}=\frac{2}{\pi }\arctan \frac{4r^2{\left(L-r\right)}^2}{L^2\left|L-2r\right|}{J}_{e1}\mathrm{.}$

For the matrix (J_vW_v ⁻¹J_v^T) is full rank, J_v^T (J_vW_v ⁻¹J_v^T)-¹ is bounded, J_v^T(J_vW_v ⁻¹ J_v^T)⁻¹(ẋ_d+J_vMk−k̇_ee) is also bounded. Thus, there exists a constant scalar φ such that ‖J_e1‖ ≤ φ, when t ∈ [t₀-ε,t₀]. According to r is monotonically decreasing, when t ∈[t₀-ε,t₀], there is 0≥J_e1 ≥−φ,

$\stackrel{.}{r}\ge -\frac{2}{\pi }\arctan \frac{4r^2{\left(L-r\right)}^2}{L^2\left|L-2r\right|}\phi \mathrm{.}$

Since, H_g|_t=i0−ε<(h_gl+h_gu)/2 and H_g is monotonically decreasing, L−2r>0 holds, t ∈[t₀−ε,t₀]. Thus:

$\stackrel{.}{r}\ge -\frac{2}{\pi }\arctan \frac{4r^2{\left(L-r\right)}^2}{L^2\left(L-2r\right)}\phi \ge -\frac{2}{\pi }\arctan \frac{4r^2}{\left(L-2r\right)}\phi \mathrm{.}$

Referring to arctanx≤x, if x≥0 and r≤1, there is,

$\stackrel{.}{r}\ge -\frac{2}{\pi }\frac{4r^2}{\left(L-2r\right)}\phi \ge -\frac{8}{\pi }\frac{\phi }{\left(L-2r{|}_{t_0-\epsilon }\right)}r\mathrm{.}$

Since, 8φ/(π(L−2r |_t0−ε))>0 and r (t₀−ε) >0. Lemma 1 leads to the conclusion r(t)>0, as t ∈[t₀−ε,t₀]. This contradicts the assumption that r(t₀)=0. This implies that the time interval, at which the performance function H_g falls to h_gl, does not exist. This shows that H_g>h_gl under control law defined in Eq. (21). Similarly, it can be shown that H_g<h_gu. □

The aforementioned proof clearly shows that the control law in Eq. (21) and the associated weight setting criterion in Eq. (19) guarantees time constraint and allows a more strict result such that H(q,k) satisfies h_gl<H (q,k)<h_gu. If the constraint is unilateral, then the proof is perfectly analogous.

Compared with the weight setting criterion defined in Eq. (19), Eq. (9) sets w̄_v1=0 at the virtual joint limit. This directly stops any motion of the related virtual joint. However, this criterion also locks the virtual joint as the virtual joint reaches the joint limit. Allowing the virtual joint to quit the limit is another reason to introduce one-step prediction. On the other hand, the weight setting criterion in Eq. (19) keeps the virtual joint within a feasible range, while the joints have not reached the virtual joint limit. This guarantees that w̄_v1>0 always holds. Thus, the situation where the virtual joint can get locked is completely avoided.

Remark 2Full rank of J_vW_v⁻¹J_v^T guarantees that the solution is non-singular. When both matrix T and J are full rank, matrix J_v is also full rank, but w̄_v1→0 as H_g→h_gl (or H_g→h_gu), J_vW_v⁻¹J_v^T may become singular. To avoid the singularity situation, damped least-square technique is applied at the price of a low precision [23], as the method proposed in [20]. The solution with damping factor λ is given as:

$\begin{array}{rr}\hfill \stackrel{.}{q}=& \hfill T^{-1}(W_v^{-1}{J}_v^T{\left(J_v{W}_v^{-1}{J}_v^T+{\lambda }^{2}I\right)}^{-1}\\ \hfill \hspace{0.17em}& \hfill \left(\stackrel{.}{x}+J{T}^{-1}M\stackrel{.}{k}-k_{e}e\right)-M\stackrel{.}{k})\mathrm{.}\end{array}$

(28)

The damping factor λ is determined according to [23].

${\lambda }^2=\{\begin{array}{ll}0\hfill & if\hspace{0.17em}{\sigma }_m\ge \rho \hfill \\ \left(1-{\left(\frac{{\sigma }_m}{\rho }\right)}^2\right){\lambda }_{max}^2\hfill & if\hspace{0.17em}{\sigma }_m<\rho \hfill \end{array}$

(29)

where σ_m is the minimal singular value, ρ is a positive constant, which defines the size of the singular region. λ_max is the maximum value of the damping factor. It should be pointed out that the damping factor λ≠0 and tracking error does not asymptotically converge to zero. This will deteriorate main task precision. Although J_vW_v⁻¹J_v^T is not full rank, J_v^T (J_vW_v ⁻¹ J_v^T + λ ² I)⁻¹ is still bounded. For control law defined in Eq. (28), theorem 2 still holds.

4. Inverse kinematics under multiple constraints

In this section the results from the previous section are extended to the case of multiple subtasks. This includes both time-dependent and independent constraints. Without loss of generality, it can be assumed that there are r time-dependent constraints. A time-independent constraint can be considered as a special case of time-dependent constraint as stated in Remark 1. Multiple time constraints are formulated as follows:

$h_{gu}^i\ge H_g^i\left(q,k\right)\ge h_{gl}^i\hspace{0.17em},\hspace{1em}i\in 1,\dots ,r$

(30)

k ∈ ℝ ^l is a time-dependent parameter, l is the number of time-dependent parameters. H_gⁱ is the i th constraint function, h_gⁱ_u and h_gⁱ _l upper threshold and lower threshold of the i th constraint, there must be r≤n; or else constraints can be combined into one constraint. This method to combine multiple constraints into a single constraint has been discussed later in this section. Virtual joint velocity can be defined as follows:

${\stackrel{.}{q}}_v=T\stackrel{.}{q}+M\stackrel{.}{k},$

(31)

T and M matrices are defined as:

$T\left(q,k\right)=\left[\begin{array}{c}\frac{\partial H_g^1}{\partial q}\\ \frac{\partial H_g^2}{\partial q}\\ ⋮\\ \frac{\partial H_g^r}{\partial q}\\ N_q\end{array}\right]\hspace{0.17em},\hspace{0.17em}M\left(q,k\right)=\left[\begin{array}{c}\frac{\partial H_g^1}{\partial k}\\ \frac{\partial H_g^2}{\partial k}\\ ⋮\\ \frac{\partial H_g^r}{\partial k}\\ Z_k\end{array}\right]$

(32)

where N_q ∈ ℝ^{(n − r} )× ⁿ is composed of unit orthogonal vectors, which satisfies $\frac{\partial H_g^i}{\partial q}{N}_q^T=0$ , for all i = 1,…,r. Z k ∈ ℝ⁽ⁿ − ^{r)x l} is a zero matrix.

Correspondingly, the inverse-weighted matrix is defined as:

$\begin{array}{ll}{\bar{w}}_{vi}\hfill & =\{\begin{array}{cc}1-\arctan \left(\left|\frac{\partial G\left(q_{vi}\right)}{\partial q_{vi}}\right|\right)\cdot \frac{2}{\pi }& \text{if}\hspace{0.17em}\Delta \left|\frac{\partial G\left(q_{vi}\right)}{\partial q_{vi}}\right|\ge 0\\ 1& \text{if}\hspace{0.17em}\Delta \left|\frac{\partial G\left(q_{vi}\right)}{\partial q_{vi}}\right|<0\end{array},\hfill \\ \hspace{0.17em}\hfill & \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}i=1,\dots ,r\hfill \\ {\bar{w}}_{vi}=1,\hfill & \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}i=r+1,\dots ,n,\hfill \end{array}$

(33)

where G(q_vi) is the performance function defined in Eq. (18). If the constraints are unilateral, performance functions follow the definition in Eq. (20) and the solution is the same as Eq. 28. Corresponding to proofs for the case with a single time-dependent constraint system, Theorem 1 and Theorem 2 can be easily extended to the case with multiple constraints.

For (n−m) diagonal elements of W_v⁻¹ are near zero and Jacobian matrix J is full rank. Matrix J_vW_v ¹ J_v^T can still be full rank. If J_vW_v⁻¹ J_v^T is singular, the damped least-square technique can be used.

For the case where the number of the time-dependent constraints is higher than the number of DOFs of the manipulator system, the strategy proposed in [20] is used here. For this case, several constraints that cannot simultaneously converge to their limits are merged into a new compact constraint H_c (q, k) as follows:

$\mu >H_c\left(q,k\right)={\displaystyle \sum _{i\in \Phi }}G\left(H_g^i\left(q,k\right)\right)$

(34)

where H_gⁱ (q, k) is the i th original constraint function. G is the performance function defined in Eq. (18) or Eq. (20), and Φ denotes the index set composed of constraints, which are not triggered simultaneously. μ is the threshold of compact constraint. The compact constraint can be treated as a normal constraint within the GWLN framework.

Remark 3 There may be some special configurations, where constraints are not compatible to each other at some time instant. $\frac{\partial H_g^1}{\partial q},\frac{\partial H_g^2}{\partial q},\dots ,\frac{\partial H_g^r}{\partial q}$ will be linearly dependent to each other, as result the transformation matrix T is not full rank. The relationship between the actual joint and the virtual joint is no longer one-to-one mapping. In such configurations, both error convergence and constraints do not hold and the GWLN method is invalid.

5. Application to welding control

Here, the GWLN method is applied to industrial welding control to verify efficacy. In this paper, the welding task is realized by performing a three-dimensional position task with a welding torch orientation constraint, shown in Eq. (1). In Eq. (1), O_d is time-dependent welding torch direction, pipe-to-pipe intersection seam welding task is selected as an example to illustrate the selection method for O_d. The entire welding seam is composed of segments $\overline{CD},\overline{DE}$ and $\overline{EF}$ , shown in Fig. 2. O_d is set as follows:

Figure 2.

Pipe-to-pipe intersection seam welding task with orientation constraints. Red line represents intersection line of the pipes to be welded. The green arrow represents the desired torch direction O_d at an instantaneous moment, and the blue arrow represents the actual torch direction O_a.

$O_d=\{\begin{array}{cc}\left(\overrightarrow{OA}+\overrightarrow{OB}\right)/2& \mathrm{in}\hspace{0.17em}\text{segment}\hspace{0.17em}\overline{CD}\hspace{0.17em}\mathrm{and}\hspace{0.17em}\overline{EF}\\ \overrightarrow{OA}& \mathrm{in}\hspace{0.17em}\text{segment}\hspace{0.17em}\overline{DE}\end{array}$

(35)

where point O is actual position of end-effector at any selected moment. $\overrightarrow{OA}$ is the unit normal vector to the surface of pipe 1. $\overrightarrow{OB}$ is the unit normal vector to surface of pipe 2 as shown in Fig. 2. Variables in Eq. (35) can be decided by the end-effector. Since the end-effector is determined by joint position q, O_d can be expressed as a function of q. Resulting from tracking errors in the welding process, point O may not be on the surface of the pipes, making Eq. (35) invalid. To calculate O_d, end-effector tracking errors can be ignored and the actual position of the end-effector O can be replaced with the preplanned one, which is a function of time. Thus, O_d decided by Eq. (35) is a function of time. In Fig. 3, green arrows demonstrate the discrete time sequence of desired orientations. Thus, welding torch direction constraint is time dependent, which is formulated as follows:

Figure 3.

End-effector trajectory under GWLN method, green arrows present the desired welding torch directions, red arrows present the actual welding torch directions under the GWLN method

$H_g\left(q,t\right)=O_d{\left(t\right)}^T\cdot O_a\left(q\right)=\cos \alpha \left(t\right)\ge \cos \beta \mathrm{.}$

(36)

A welding control problem can be treated as a kinematic problem with time-dependent constraints.

Simulations were carried out by applying the GWLN method to ZT-010 welding manipulator. The manipulator has five DOFs and was designed by Zhejiang Zhengte Group Co., Ltd and Zhejiang University for welding pipe-to-pipe intersection seam of fences. The welding manipulator has five joints. There are three prismatic joints near the base and two revolute joints at the end, as shown in Fig. 4. Danavit-Hartenberg (DH) parameters and limits for the joints are listed in Table 1. Pose and position of the end-effector at the last joint coordinate are P_E ⁵=[87,0,147] ^T and O_E ⁵ = [−sin(π/6),0,cos(π/6] ^T .

Table 1.

DH parameters

Joints	DH-α	DH-a(mm)	DH-d(mm)	DH-θ	d regions	θ regions
1	π/2	0	d 1	π	[0,4000]	–
2	π/2	0	d 2	π/2	[0,560]	–
3	π/2	0	d 3	0	[200,1000]	–
4	0	0	0	θ 4	–	[244, 4.18]
5	−π/2	0	0	θ 5	–	[−2.97, 0.17]

Figure 4.

Five-link welding manipulator

5.1 Simulation result

5.1.1 GWLN method

In this section, simulations based on the five-link welding manipulator for welding control are carried out to verify efficacy of the proposed GWLN method.

The GWLN method was simulated on the five-link welding manipulator for welding control The end-point of the welding torch should track the. pipe intersection line shown in Fig 2 In addition welding torch orientation should satisfy the time-dependent constraint in eq (36) Constraint threshold for torch orientation is set as cosβ=0 85 At the same time the manipulator should avoid joint limits shown in Table 1 To verify efficacy of the GWLN method start joint position was set as q(0) = [1928.91,525.67,402.22,3.04,−2.23] ^T at which the second joint is near the joint limit Since. orientation constraint in eq (36) is a unilateral time-dependent inequality eq (20) is used to set it as first subtask Second task is the compacted constraint H which is combined by joint limit avoidance constraints referring to Eq. (34). μ is set to 20. Then, Eq. (28) can be applied to solve this problem.

Results are shown in Fig. 3 and Fig. 5–7. Considering the manipulator has both prismatic and revolute joints, normalized joint velocity q̇_N is defined as q̇ⁱ_N=q̇_i / ‖q̇ⁱ_max‖, where q̇_i and q̇_imax is velocity of i th joint and maximum allowable velocity of i th joint, respectively. At the initial instant, desired torch direction and actual direction are the same. As the manipulator moves, torch direction successfully follows the desired orientation. Joint limit constraint is also guaranteed. Fig. 7 illustrates additional details related to the simulation. Time-dependent orientation constraint function H_g(q,t) value is close to the threshold at t =10.3s. Adjusting weight value w̄_v1 stops further deterioration of H_g. Note that at about 14s, the value of H_g increases, and weight value w̄_v1 switches sharply from near zero value to one based on Eq. (33). This accelerates H_g to move away from the lower threshold. It can be seen from Fig. 7(d) that the related virtual velocity q̇_v1 is approximately zero at that moment. Chatter in the weight value does not induce a discontinuity in both virtual and actual joint velocities and is verified by Fig. 5. The same situation occurs for w̄₂ at t =9.8 s. These examples verify the conclusion that discontinuity in weight setting criterion in Eq. (19) does not induce a discontinuity in the joint velocity. During the time interval between 8 to 10s, the second joint is near the joint limit, value of w̄₂ has a sharp valley, falling to 0.15, at the same time, H is near the lower threshold and the related inverse weighted factor value is small. However, the system has two degrees of redundant freedom. No singularity occurs and both constraints are guaranteed. As seen from Fig. 7(f), tracking error is less than 5 × 10-² mm.

Figure 5.

Trajectories of normalized joint velocities under the GWLN method.

Figure 6.

Joint trajectories under the GWLN method, with dash lines denoting joint limits

Figure 7.

End-effector tracks the intersection line of pipes under GWLN method. (a) Orientation constraint function value H_g. (b) Joint limits compact constraint function value H_c. (c) Inverse weighted factor w̄_v1 and w̄_v2. (d) Virtual joint 1 velocity profile. (e) Virtual joint 2 velocity profile. (f) Norm of tracking error.

Figure 8.

Normalized joint velocity profiles using MVN scheme

5.1.2 MVN scheme

For comparison, the MVN scheme proposed in [17] is applied to the five-link welding manipulator. MVN scheme treats constraints as dynamic changing equalities and converts the inverse kinematic problem to a dynamic-system-based QP problem Then a LVI-based numerical algorithm is exploited to solve the QP problem Thus it can be applied to the case with time-dependent constraint The welding control problem is reformulated as a QP problem as follows:

$\begin{array}{ccc}\mathrm{minimize}& \hspace{0.17em}& {\Vert \stackrel{.}{q}\Vert }_2^2\\ \mathrm{subject}& \mathrm{to}& J\stackrel{.}{q}=\stackrel{.}{x}\\ \hspace{0.17em}& \hspace{0.17em}& J_G\left(q\right)\stackrel{.}{q}\le b_G\\ \hspace{0.17em}& \hspace{0.17em}& {\xi }^{-}\le \stackrel{.}{q}\le {\xi }^{+}\end{array}$

(37)

where $J_G\left(q\right)=-\partial H_g/\partial q$ and $b_G:=s\left(H_g,d_1,d_2\right)max\left\{{\left(J_G\stackrel{.}{q}-\partial H_g/\partial t\right)|}_{H_g=d_1},0\right\}+\frac{\partial H_g}{\partial t},d_1$ and d₂ are the outer and inner safety thresholds s(H_g) is the distance-based smoothing function [17][18] ξ⁺=κ (q⁺−q) ξ⁻=κ(q⁻−q) q⁺ and q⁻ are the upper and lower joint limits with κ=0.25 For simulations the safety threshold was set as d₁ = 0.87 and d₂ = 0.85 Then the LVI based numerical algorithm proposed in [17] was applied to get the q in each control period.

Fig 9 and Fig 10 show that the MVN scheme guarantees orientation constraint and keeps joints away from joint limits Position errors were less than 1×10⁻¹ mm which is larger in comparison to the result obtained using the GWLN method However this can be improved by setting a stricter stop iteration condition for the linear variational inequality-based numerical algorithm at the cost of more computations According to simulation results MVN scheme is comparatively still computationally intensive Table 2 compared simulation time for MVN scheme with GWLN method Simulations were carried out using MATLAB software with simulation control period Δt=0 01s and the task duration t_total =20s Based on the results it can be stated that GWLN method has an advantage over the MVN scheme The MVN scheme reformulates the constrained kinematic problem into a QP problem for which obtaining an analytical solution is difficult, Although the LVI-based iteration algorithm for the QP problem has a global exponential convergence property to some extent it still complicates the process to obtain the solution [18] The average computation time of the MVN scheme is even longer than the control period shown in Table 2 Meanwhile the GWLN method directly allows the determination of the analytic solution and is comparatively faster than the iteration algorithm Considering the calculations required the MVN scheme is more suitable for off-line calculations in industrial applications.

Table 2.

Simulation Time of Tracking Intersection Line of Two Pipes by MVN Scheme [17] and GWLN Method (28)

	total time ttotal (s)	average time taverage (s)
MVN scheme	40.23	2.011×10−2
GWLN method	6.433	3.216×10−3

Figure 9.

Joint position trajectories using MVN scheme, with dash lines denoting joint limits

Figure 10.

End-effector tracks intersection line of pipes using MVN scheme (a) Orientation constraint function value H_g. (b). Norm of tracking error.

5.2 Experimental results

The GWLN method was applied to the ZT-010 welding manipulator with five DOFs, as shown in Fig. 4. The entire system is composed of a workstation and a manipulator. The workstation is a personal digital computer with a Pentium E5300 2.6 GHz CPU, 2 GB memory, which sends instructions to the manipulator's motion controller, Turbo PMAC2-Eth-Lite stacked with PMAC2A-PC/104 (Delta Tau Data Systems Inc.). All joints of the manipulator are driven by AC servo motors (models: GYS101D5-RC2, GYS101D5-RC2, GYS401D5-RC2, GYS751D5-RC2-B and GYS102D5-RC2) manufactured by FUJI company. Servo amplifiers are matching models from the FUJI company. Servo amplifiers operate in velocity mode. Encoders attached to the motors were used to obtain the joint position.

The objective of the experiments is similar to the simulation discussed in the previous section. The start joint position is set as q(0)= 1612.68,227.95,356.91,2.64,−1.80] ^T , which is different from simulation. This is because the start configuration needs to set the end-effector at the start point of the pipe-to-pipe intersection seam in actual manufacturing environments and is decided by the pipe fasteners. Experimental results are shown in Fig. 11, 12 and 13. The joints successfully keep away from the joint limits. Orientation constraint function H_g nearly breaks the threshold at 19s.

Figure 11.

Normalized joint velocity trajectories of five-link welding manipulator in welding process

Figure 12.

Actual joint position trajectories of five-link welding manipulator in welding process, with dash lines denoting joint limits

Figure 13.

Trajectories in the experiment. (a) Orientation constraint function value H_g. (b) Joint limits compact constraint function value H_c. (c) Inverse weighted factor w̄_v1 and w̄_v2. (d) Norm of tracking error.

By setting a weight factor w̄_v1 less than 9×10⁻⁵, the GWLN method keeps the manipulator from breaking the orientation constraint at the end of welding process. Tracking error is less than 1.5×10⁻¹ mm, which is 190% higher than errors in simulation part. The difference is due to the disturbance on the joint velocities, shown in Fig. 11. Disturbances or iginate from the fluctuation of the mechanical payload and tracking errors of the motor servo loop.

6. Conclusion

In this paper, the GWLN method was extended to an inverse kinematic problem with a time-dependent problem, which has not been discussed in the literature and is an important area of research in recent years. By introducing time-dependent virtual joints to represent time-dependent constraints, an intermediate kinematic control problem from the virtual joint space to the revised task space is obtained. The GWLN method was then applied to obtain a uniform control law to deal with both time-dependent and time-independent constraints. In addition, a new inverse-weighted matrix setting criterion was proposed here, which allowed the simplification of the implementation of the GWLN method in comparison to previous works reported in the literature. Simulation results, including a comparison to the MVN scheme, demonstrate the efficacy and advantages of extending the GWLN method. Experiments implemented on a five-link welding manipulator verified that it was possible to realize the proposed method and to demonstrate the efficacy of the method.

However, there are two drawbacks of the GWLN method. First, the method is unable to avoid the singularity configuration. This is because the transformation of the joint space will change the criterion for singularity-avoidance joints, which is part of the transformation. Secondly, GWLN is invalid if the matrix T is not full rank, since the relationship between the virtual joint and actual joint is not mapped one-to-one. One of the major future goals of this work is related to overcoming the drawbacks of this method.