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Abstract

A redundant manipulator usually has multiple kinematic solutions at the velocity level and can be used to optimize other criteria. Although the pseudo inverse of the Jacobian matrix generates a solution with the least 2 norm (LN), it does not consider other constraints imposed by the system or the environment. The general weighted least norm (GWLN) method handles these general constraints via the concept of virtual joints, but this is not always feasible when the number of constraints exceeds the degrees of freedom. This paper proposes a unified weighted least norm method (UWLN) to unify the LN and the GWLN methods. The UWLN method merges the constraint tasks into a quadratic criterion that poses no limitations on the number of constraint tasks. It also generates a least 2 norm solution when all constraints are deactivated, thereby unifying the LN and GWLN methods. A comparative simulation on a 7-DoF planar manipulator demonstrates the validity of the algorithm.

Keywords

Constrained Manipulator Control Redundancy Resolution Unified Weighted Least Norm Method

1. Introduction

Manipulators are today widely used in both industry manufacture and services in daily life, e.g., welding, spray-painting, polishing, surgery, etc. A well-tuned manipulator system can execute various manipulation tasks with good precision. In addition to manipulation tasks, manipulators are also subject to various types of constraints. Some constraints are imposed by joint actuators, e.g., the configuration coordinates should remain in a feasible range and not violate physical limits, the joint velocity should not overflow the limitations of the harmonic drive, the available joint torque is limited by the capability of the actuator, etc. Some constraints are enforced by the operation environment: the link of the manipulator should not collide with the objects in the environment if doing so is not on purpose, the legged robot should keep its centre of mass in the supported polygon, the target being monitored should stay in the field of vision during a visual servo process, etc. These constraints lead to a constrained control problem, which complicates the control of the manipulator.

When the manipulator has extra degrees of freedom with respect to the dimensions of the task, it is redundant and has more dexterity. This dexterity can be used to optimize other criteria or to accomplish the constraints without interfering with the main task. Significant research aiming to provide a solution to the constrained control problem has been carried out. Generally, the algorithms that solve the constrained control problem can be classified into three groups, according to the relative priority of the manipulation task and the constraint tasks.

The first class assigns the manipulation task a higher priority, while the constraint tasks are solved via optimization of some criteria in the feasible set, defined by the manipulation task. For example, Liegeois proposes the gradient projected method (GPM) for solving the joint limit problem by minimizing the quadratic potential constructed in the joint space, by decreasing it along the projection of its gradient into the null space of the Jacobian matrix [1]. Hollerbach used joint acceleration in the null space of the Jacobian matrix to minimize torque deviation from the centre point in the joint torque space [2]. However, the gradient projected method may fail to carry out the constraint task if the constraint task conflicts with the manipulation task and sometimes this method will cause oscillations in the control variable [3].

When the constraint tasks have the same priority as the manipulation task, they coexist or compete with each other and are solved together. Khatib combined the collision avoidance task and the manipulation task in the joint acceleration space via an artificial potential field method [4]. However, the potential field of manipulation task may be interfered with by the field of constraints, even when the system can use its redundancy to fulfil both the manipulation task and constraint task. Baillieul introduced the extended Jacobian method by treating the constraint tasks as an additional dimension of manipulation task [5]; Maciejewski used this method to solve the collision avoidance problem in dynamic varying environments [6]. However, when the constraint task conflicts with the manipulation task, only a least square solution is derived by the extended Jacobian method, which may result in deviations in both the constraint task and the manipulation task. Cheng and Cheol formulated the manipulation task and the constraints as feasible when set within a quadratic programming (QP) problem and in this way solved the joint angle limits, joint velocity limits and joint torque limits [7,8]. Zhang solved the quadratic programming problem through an evolving dynamic system based on neural networks and discovered the joint torque constraint and obstacle avoidance [9, 10, 11]. However, these QP-based methods do not address the case when the constraint task conflicts with the manipulation task, which leads to an empty feasible set.

In some instances, the constraint task deserves higher priority, e.g., the self balance task of a humanoid robot. The violation of these constraint tasks may cause a safety problem or result in deviation from the desired task trajectory. Thus, the manipulation task must be accomplished strictly in the feasible domain defined by the constraint task. Flacco resolved the limitation on joint velocity by actively saturating the corresponding joint velocity on its capability boundary [12], which can be viewed as an special case of the priority-based hierarchy [13,14]. Kanoun generalized the task-priority framework to inequality constraints and maintained the pose of a humanoid robot while performing a reaching task [15]. However, this priority-based hierarchy is difficult to maintain: when the priorities are reversed or the constraint tasks are activated/deactivated, the switching law is complicated and discontinuities in the control solution may occur [16].

In [17], we propose the general weighted least norm method and solved both the joint limit constraint and the collision avoidance problem. The GWLN method has a constant structure and ensures the constraint tasks higher priority. By optimizing the weighted norm of the virtual joint variable, the manipulation task is not affected by the constraint tasks if they are compatible and the constraint tasks can take priority over the manipulation task via damping if they conflict. It is a pity that the GWLN method is not good at handling multiple constraints; here, the number of constraint tasks is limited by the size of the transform matrix and any two combined constraint tasks should not simultaneously near their thresholds. This limitation restricts the performance of the GWLN method. Furthermore, the transform matrix suffers from singularity and may result in numerical instability if the subtasks are linearly dependent. Furthermore, the solution derived via the GWLN method does not have the property of least 2-norm, even when all the constraints are deactivated. This property will raise the amplitude of motion and the energy consumed during the manipulation process.

To improve the performance of GWLN method, in this paper, we solved the constrained control problem via a unified weighted least norm (UWLN) method. Different from the GWLN method, the UWLN method combines constraints into a quadratic criterion that is able to contain arbitrary constraints. The criterion matrix always has full rank and is free from the singularity in the transform matrix. Moreover, the criterion matrix coincides with the criterion of the LN method when all the constraint tasks are not activated, which establishes a smooth transition between the LN solution and the constrained solution. This approach proves that the unified weighted least norm method can accomplish the constraint tasks as well as the GWLN method.

This paper is organized as follows: in section II, we recall the GWLN method and illustrate its problems. In section III, we propose the UWLN method, which overcomes the drawbacks of the GWLN method and unifies the constrained solution and the LN solution. In section IV, simulation of a constrained control process of a 7-DoF planar manipulator is carried out using the Robotics Tool Box in Matlab, demonstrating the validity of the algorithm and its superiority over the GWLN method. Finally, some conclusions are drawn.

2. The GWLN method

Describe the kinematic mapping of a manipulator on the velocity level as:

\dot{x} = J (q) \dot{q}

(1)

where $q \in R^{n}$ is the joint coordinate of the manipulator and $\dot{q} \in R^{n}$ is the joint velocity. $x \in R^{m}$ is the coordinate of the manipulation task and $\dot{x} \in R^{m}$ is the rate in the task space. $J (q) = \partial x / \partial q \in R^{m \times n}$ is the Jacobian matrix of the manipulation task with respect to joint coordinates. Throughout this paper, we consider the redundant manipulator, i.e., m<n. In this case, the solution set of (1) for a command manipulation task rate ${\dot{x}}_{c}$ is:

\dot{q} = J^{†} {\dot{x}}_{c} + (I_{n} - J^{†} J) z

(2)

where $J^{†} \in R^{n \times m}$ is the Moore-Penrose inverse of J and $z \in R^{n \times 1}$ is an arbitrary vector. To deal with uncertainties and disturbances, ${\dot{x}}_{c}$ can function as

{\dot{x}}_{c} = {\dot{x}}_{d} - k_{p} e

(3)

to form a close loop error dynamic. Here, x_d and ${\dot{x}}_{d}$ are the desired task coordinate and rate and $e = x - x_{d}$ is the error of the manipulation task. k_p is the proportional feedback gains.

When z is chosen as the zero vector, the resulting solution $\dot{q}$ is:

\dot{q} = J^{†} {\dot{x}}_{c}

(4)

which minimizes the scope of the action by minimizing the pseudo kinetic energy ${‖ \dot{q} ‖}_{2}^{2} = {\dot{q}}^{T} I_{n} \dot{q}$ and is sometimes referred to as the least norm (LN) solution [3].

In addition to the manipulation task, a manipulator also has to work within the constraints that the system and the environment impose. These constraints restrict the feasible joint coordinate of a manipulator. Assume that these constraints can be summarized in unilateral form:

H_{i} (q) \geq 0

(5)

where $H_{i} : R^{n} \mapsto R^{1}$ is the $i -$ th constraint function of joint coordinate q. For the constraint in operational space, $H_{i} (q)$ may be a highly nonlinear function that varies with joint coordinate q. Therefore, it is difficult to define the feasibility region directly in joint space. The LN method does not consider the constraint tasks and therefore will not guarantee said constraint tasks.

If we think of the state variable q as drifting in the feasible domain with velocity $\dot{q}$ , (5) can be realized by forbidding the evolution of q along the normal direction $\partial H_{i} / \partial q$ when it tends to decrease and across the boundary $H_{i} (q) = 0$ . Then, (5) can be reformulated as:

\lim_{H_{i} (q) \to 0} {\dot{H}}_{i} (q) \to 0

(6)

For a nonlinear boundary constraint, the normal direction $\partial H_{i} / \partial q$ may vary according to the configuration q. Note that the constraint matrix $C_{i} (q) = \partial H_{i} / \partial q \in R^{1 \times n}$ , (6) can be written as:

\lim_{H_{i} (t) \to 0} C_{i} (q) \dot{q} \to 0

(7)

As long as $C_{i} (q) \dot{q} = 0$ is maintained inside region $H_{i} (q) \geq 0$ , the manipulator complies with the boundary constraints, since ${\dot{H}}_{i} = 0$ , so that $H_{i} (q)$ will not cross boundary $H_{i} (q) = 0$ . Thus, (5) can be preserved by setting ${\dot{H}}_{i} (q) = C_{i} (q) \dot{q} = 0$ , which is a local linear constraint on the joint rate.

Note $C (q) \in R^{p \times n}$ as the stack of all $C_{i} (q)$ s. Combining the constraint task with the manipulation task, the control problem on the boundary of the constraints can be stated as:

\begin{matrix} ​ & \min {‖ J \dot{q} - {\dot{x}}_{c} ‖}_{2} \\ s . t . & C \dot{q} = 0 \end{matrix}

(8)

Here, the manipulation task is formulated as a minimization of the gap between the command task rate ${\dot{x}}_{c}$ and the actual task rate $\dot{x} = J (q) \dot{q}$ , since there may not exist an exact solution in the feasible set $C \dot{q} = 0$ .

The GWLN method solves the constrained manipulation problem (8) by introducing the virtual joint variable, which is derived via the coordinate transform matrix T:

{\dot{q}}_{v} = T (q) \dot{q}

(9)

where the transform matrix T is constructed as:

T = [\begin{matrix} C \\ C^{⊥} ​^{T} \end{matrix}]

(10)

here, $C^{⊥} \in R^{n \times (n - p)}$ is the complementary of C.

The manipulation task can be rewritten in the virtual joint variable as:

J_{v} {\dot{q}}_{v} = \dot{x}

(11)

where $J_{v} = J T^{- 1}$ . Solve (11) for ${\dot{q}}_{v}$ through the weighted least norm method with the weight matrix W constructed as:

W = [\begin{matrix} W_{1} & 0 \\ 0 & I \end{matrix}] .

(12)

The virtual joint variable solution minimizing the weighted norm ${‖ {\dot{q}}_{v} ‖}_{W} = \sqrt{{\dot{q}}_{v}^{T} W^{2} {\dot{q}}_{v}}$ can be derived by the Moore-Penrose inverse of $J_{v} W^{- 1}$ :

{\dot{q}}_{v} = W^{- 2} J_{v}^{T} {(J_{v} W^{- 2} J_{v}^{T})}^{- 1} \dot{x} .

(13)

Here $W_{1} \in R^{p \times p}$ is a diagonal matrix with its diagonal elements $w_{i} \in [1, \infty)$ , which is the weight factor for the i-th constraint task. The original joint velocity $\dot{q}$ can be derived by inverse transformation from ${\dot{q}}_{v}$ :

\begin{array}{r} \dot{q} & = & T^{- 1} W^{- 2} J_{v}^{T} {(J_{v} W^{- 2} J_{v}^{T})}^{- 1} \dot{x} \\ ​ & = & W_{T} J^{T} {(J W_{T} J^{T})}^{- 1} \dot{x} \end{array}

(14)

where $W_{T} = T^{- 1} W^{- 2} T^{- T}$ . In fact, (14) solves:

\begin{matrix} ​ & \min {\dot{q}}^{T} Q \dot{q} = {‖ W T \dot{q} ‖}_{2}^{2} \\ s . t . & J \dot{q} = \dot{x} \end{matrix}

(15)

where $Q = {(W T)}^{T} (W T) = W_{T}^{- 1} \in R^{n \times n}$ is a symmetric positive defined matrix that weights joint velocity $\dot{q}$ .

When w_i s tends toward ∞, the solution (14) satisfies that:

C_{i} \dot{q} \to 0

(16)

Thus, (7) and subsequently, (5) can be ensured through the GWLN method by setting $w_{i} \to \infty$ as $H_{i} (q) \to 0$ .

However, the GWLN method is not good at handling multiple constraints. Due to the finite degrees of freedom of the manipulator, multiple constraint tasks must be combined. These constraint tasks to be combined are required to not simultaneously be near their thresholds, which limits the application of the algorithm. Additionally, if constraint tasks C_i s are linearly dependent, the transform matrix T may not be fully ranked, which will result in numerical instability when inverted in W_T.

When the system leaves the boundary of the constraints, all weight factors w_i are equal to 1. In this case, the GWLN method still generates a solution that minimizes $\dot{q} T^{T} T \dot{q}$ , which may differ from the least 2-norm solution (4), that is, of the minimal velocity norm. The solution (14) may have a projection in the null space of the Jacobian matrix $N (J)$ and will increase the amplitude of motion. This motivated us to improve the performance of the GWLN method.

3. The unified weighted least norm method

In the previous section, we discussed the defects of the GWLN method, which mainly result from the construction of the criterion matrix. In this section, we address these problems and propose the unified weighted least norm method, which has a stable criterion matrix and the capability of merging multiple constraint tasks.

The criterion for the constrained control problem (15) is:

\begin{array}{r} Q = C^{T} W_{1}^{2} C + C^{⊥} C^{⊥} ​^{T} \end{array}

(17)

which is dominated by $C^{T} W_{1}^{2} C$ as $W_{1} \to \infty$ . When the constraint tasks C_i s are linearly dependent, the criterion Q is singular and may results in numerical instability in (14).

On the other hand, the least 2-norm solution (4) solves the problem:

\begin{array}{r} ​ & \min {\dot{q}}^{T} \dot{q} \\ s . t . & J \dot{q} = {\dot{x}}_{c} \end{array}

(18)

which is equivalent to (15) with $Q = I_{n}$ . Although the LN method does not ensure the constraint tasks, the criterion of (18), (4) is numerically stable. In order to acquire a stable constrained solution, we need to construct a full rank criterion matrix, which is limited by $C^{T} W_{1}^{2} C$ as $W_{1} \to \infty$ . One natural choice is:

\bar{Q} = I_{n} + \sum_{i = 1}^{p} C_{i}^{T} {\bar{w}}_{i}^{2} C_{i} = I_{n} + C^{T} {\bar{W}}_{1} C

(19)

where ${\bar{W}}_{1} \in R^{p \times p}$ is a diagonal matrix with its diagonal elements ${\bar{w}}_{1}, {\bar{w}}_{2}, \dots, {\bar{w}}_{p}$ and the weight factor ${\bar{w}}_{i}$ varies from 0 to ∞. It is obvious that (19) is a symmetric, positively defined matrix with full rank and $\bar{Q}$ is dominated by $C^{T} {\bar{W}}_{1}^{2} C$ as ${\bar{W}}_{1} \to \infty$ and coincides with I_n when: ${\bar{W}}_{1} \to 0$ . Now, the optimal problem becomes:

\begin{matrix} ​ & \min {\dot{q}}^{T} \bar{Q} \dot{q} \\ s . t . & J \dot{q} = {\dot{x}}_{c} \end{matrix} .

(20)

Since (20) makes a smooth transition between the least norm problem and the GWLN problem, we call (20) the unified weighted least norm (UWLN) problem.

According to the weighted least norm method, the least norm solution of (20) is:

\begin{array}{r} \dot{q} & = & {\bar{Q}}^{- 1} J^{T} {(J {\bar{Q}}^{- 1} J^{T})}^{- 1} {\dot{x}}_{c} \end{array} .

(21)

In case of singularities in the Jacobian matrix, a damped solution is used for practical purposes:

\begin{array}{r} \dot{q} & = & {\bar{Q}}^{- 1} J^{T} {(J {\bar{Q}}^{- 1} J^{T} + λ^{2} I_{m})}^{- 1} {\dot{x}}_{c} \end{array} .

(22)

The validity of the solution (22) is shown by the following theorem:

Theorem 3.1 (Validity of the UWLN method) When the weight factor ${\bar{w}}_{i}$ tends to ∞, the solution $\dot{q}$ in (22) satisfies $C_{i} \dot{q} \to 0$ .

Proof 1 Without loss of generality, we assume that $i = 1$ . According to (21), the constraint task $C_{1} \dot{q}$ is:

\begin{array}{r} C_{1} \dot{q} & = C_{1} {\bar{Q}}^{- 1} J^{T} {(J {\bar{Q}}^{- 1} J^{T} + λ^{2} I_{m})}^{- 1} {\dot{x}}_{c} \\ ​ & = C_{1} {(I_{n} + C^{T} {\bar{W}}_{1}^{2} C)}^{- 1} J^{T} \\ \times & {(J {(I_{n} + C^{T} {\bar{W}}_{1}^{2} C)}^{- 1} J^{T} + λ^{2} I_{m})}^{- 1} {\dot{x}}_{c} \end{array}

(23)

According to [18], when P and R make up a positive, definite matrix, the following identity holds:

\begin{matrix} P B^{T} {(B P B^{T} + R)}^{- 1} \\ = {(P^{- 1} + B^{T} R^{- 1} B)}^{- 1} B^{T} R^{- 1} \end{matrix}

(24)

Applying (24), we get:

\begin{array}{l} ​ & C_{1} {(I_{n} + C^{T} {\bar{W}}_{1}^{2} C)}^{- 1} \\ = & C_{1} {(I_{n} + C_{1}^{T} {\bar{w}}_{1}^{2} C_{1} + \sum_{j = 2}^{p} C_{j}^{T} {\bar{w}}_{j}^{2} C_{j})}^{- 1} \\ = & {\bar{w}}_{1}^{- 2} {[{\bar{w}}_{1}^{- 2} + C_{1}^{T} {(I_{n} + \sum_{j = 2}^{p} C_{j}^{T} {\bar{w}}_{j}^{2} C_{j})}^{- 1} C_{1}]}^{- 1} \\ \times & C_{1} {(I_{n} + \sum_{j = 2}^{p} C_{j}^{T} {\bar{w}}_{j}^{2} C_{j})}^{- 1} \end{array}

(25)

When ${\bar{w}}_{1} \to \infty$ , (25) is approximate to:

\begin{array}{l} ​ & C_{1} {(I_{n} + C^{T} {\bar{W}}_{1}^{2} C)}^{- 1} \\ \approx & {\bar{w}}_{1}^{- 2} {[C_{1}^{T} {(I_{n} + \sum_{j = 2}^{p} C_{j}^{T} {\bar{w}}_{j}^{2} C_{j})}^{- 1} C_{1}]}^{- 1} \\ \times & C_{1} {(I_{n} + \sum_{j = 2}^{p} C_{j}^{T} {\bar{w}}_{j}^{2} C_{j})}^{- 1} \\ \approx & 0 \end{array}

(26)

Since

\begin{array}{l} {‖ {(J {(I_{n} + C^{T} {\bar{W}}_{1}^{2} C)}^{- 1} J^{T} + λ^{2} I_{m})}^{- 1} ‖}_{2} \leq \frac{1}{λ^{2}} \end{array}

(27)

Then, we have $C_{1} \dot{q} \approx 0$ , provided J and ${\dot{x}}_{c}$ is bounded. The case when multiple ${\bar{w}}_{i}$ s tend to infinity can be analysed in a similar way. Thus, we have proven that solution (22) satisfies $C \dot{q} \to 0$ as ${\bar{w}}_{i} \to \infty$ .

Note that (19) is a summation of the weighted quadratic criteria $C_{i}^{T} {\bar{w}}_{i}^{2} C_{i}$ and is able to include an arbitrary number of constraint tasks. By making ${\bar{w}}_{i}$ vary in $[0, \infty)$ , we gain a compromise between a normal manipulation task and the constrained manipulation task (8). To achieve a smooth command curve, weight factors should also gradually vary. Any continuous setting of ${\bar{w}}_{i}$ from 0 to infinity as $H_{i} (q)$ approaches its threshold is acceptable and one option is:

{\bar{w}}_{i} = {\begin{matrix} α \cos (\frac{π H_{i} (q)}{2 ζ_{i}}) \frac{1}{H_{i} (q)} & if {\dot{H}}_{i} (q) < 0 \\ 0 & else \end{matrix}

(28)

where α is a scaler for adjusting the intensity of the algorithm and $ζ_{i}$ is the value of $H_{i} (q)$ at the instant that the constraint task $C_{i} (q)$ is activated. This value can be selected as a fixed threshold for the activation of the constraint task or sampled when ${\dot{H}}_{i} (q)$ becomes negative. Note that the depression on ${\dot{H}}_{i}$ is only necessary when $H_{i} (q)$ decreases; thus, ${\bar{w}}_{i}$ is set to zero when ${\dot{H}}_{i} (q) > 0$ . It is easy to verify that ${\bar{w}}_{i} = 0$ when $H_{i} (q) = ζ_{i}$ , which ensures a smooth increase in $\bar{Q}$ and the solution $\dot{q}$ , and ${\bar{w}}_{i} \to \infty$ as $H_{i} (q) \to 0$ , which ensures the efficiency of the unified weighted least norm method. Thus, the weight setting (28) is competent for complying with the constraint (7) in the unified weighted least norm problem (20).

Compared with the GWLN method, the UWLN method has the following advantages:

The UWLN method contains the constraint tasks in the quadratic criterion $\bar{Q}$ via weighted summary; thus, it does not have any restriction on the number of constraint tasks. Contrastingly, the GWLN method cannot deal with constraint tasks that exceed the manipulator's degree of freedom, since the constraint matrix always occupies the transform matrix and the combined constraint tasks are required not to be simultaneously near the thresholds.

The UWLN method constructs a full rank criterion matrix $\bar{Q}$ , in contrast to the criterion matrix Q in the GWLN method, which may be singular when the constraint matrix C_i s are linear dependent and result in numerical instability.

The computational burden of GWLN and UWLN methods are almost the same, since the formula (14) and (21) requires the same flops; therefore, the time consumed will not differ too much. However, the UWLN method directly constructs the quadratic criterion, thus avoiding the construction of the transform matrix in the GWLN method, which requires $O (n^{3})$ flops to compute $C^{⊥}$ .

The UWLN method unifies the least 2-norm solution and the constrained control solution. It will generate an LN solution if all constraint tasks are deactivated. However, the GWLN method will generate a solution with a much larger amplitude, since the criterion matrix Q differs form the criterion of the LN method I_n. This means that the GWLN method will consume more energy during the manipulation process. Later, we will observe this phenomenon in the simulation.

4. Simulation result

We simulated the manipulation process of a 7-DoF planar manipulator with collision avoidance control. The D-H parameter of the manipulator is shown in Table 1. The simulation step length we chose was $T_{s} = 0.005$ seconds, which can be satisfied by the majority of manipulators; we used the Euler method to simulate joint movement. To reduce the error resulting from discretization, we added position error feedback to the task velocity command with $k_{p} = 400$ .

Table 1.

DH parameters of the 7 DOF manipulator

No.	θ_i(rad)	a_i(Unit)
1	θ ₁	1
2	θ ₂	1
3	θ ₃	1
4	θ ₄	1
5	θ ₅	1
6	θ ₆	1
7	θ ₇	1
E.E	θ_ee	0

We placed the manipulator at an initial configuration as follows: $q (0) = [π, - \frac{π}{2}, - \frac{π}{4}, - \frac{π}{4}, - \frac{π}{4}, - \frac{π}{4}, - \frac{π}{4}]$ and required the end-effector to draw a quadrant in π seconds. The equation of the required end effector trajectory is:

\begin{array}{r} x (t) & = & \frac{\sqrt{2}}{2} + 3 \sin (ω) \\ y (t) & = & - \frac{\sqrt{2}}{2} + 3 \cos (ω) \end{array}

(29)

where ω is the centre angle of the quadrant trajectory:

ω (t) = \frac{3}{π^{4}} t^{5} - \frac{15}{2 π^{3}} t^{4} + \frac{5}{π^{2}} t^{3}

(30)

here, $t \in [0, π]$ is the time coordinate. This was an easy task, since the manipulator was redundant. However, an obstacle bar limited the workspace of the manipulator from the positive Y-direction. The bar is modeled as a line alongside its equation $y_{o b s} = 3$ . For safety reasons, the links of the manipulator should not cross this line.

To simplify the geometric model for the collision avoidance task, the radius of the links were ignored. Note that all the links had the same length of $a_{i} = 1$ ; we could therefore require that all mid-points of the links should have y-coordinates of less than $y_{t h r e s} = 2.5$ to avoid the calculation of critical points. When this requirement is satisfied, the collision avoidance task is continuous in the constraint matrix $C (q)$ .

For the LN method, the link of the manipulator will hit the obstacle at $t = 1.8$ s due to a lack of collision avoidance control. The trajectories of the mid-points of the links are shown in Figure 1. Each coloured line segment represents a link of the 7-Dof planar manipulator and the blue line segment represents link 1. It is obvious that the mid-points of the final three links crossed the safety line $y_{t h r e s} = 2.5$ , which indicates a possible danger. The trajectories of the joints confirm this; see Figure 2. From Figure 2, we can see that the end point of link 5 crossed over the obstacle boundary $y_{o b s} = 3$ .

Figure 1.

Trajectories of mid points of the links without collision avoidance control. The mid points of link 5–7 crossed the safe line y_thres=2.5. Each coloured line segment represents a link of the 7-Dof planar manipulator and the blue line segment represents link 1.

Figure 2.

Trajectories of joints without collision avoidance control. Due to a lack of collision avoidance control, joint 5 crossed the obstacle line y_obs=3, which indicates a collision with the obstacle.

To prevent the mid-point of each link from crossing the safety line $y_{t h r e s} = 2.5$ , we used the proposed UWLN method to solve the constrained manipulation problem. Position $x_{p i}$ of the mid-point of each link i can be calculated via forward kinematic equations, i.e., distance d_i between $x_{p i}$ and the safety line $y_{t h r e s} = 2.5$ , the gradient $J_{p i} = \partial d_{i} / \partial q$ and relative velocity ${\dot{d}}_{i}$ . Then, the collision avoidance task for link i can be formulated as:

\lim_{d_{i} \to 0} {\dot{d}}_{i} = J_{p i} \dot{q} = C_{i} \dot{q} \to 0

(31)

Without losing generality, we chose $C_{i} = J_{p i} / {‖ J_{p i} ‖}_{2}$ to exclude the scaling effect by ${‖ J_{p i} ‖}_{2}$ . Stacking all the activated collision avoidance tasks together, we achieved the constraint matrix $C (q)$ . We could then use (21) to solve the constrained manipulation task (8). We selected the weight factors ${\bar{w}}_{i}$ s as (28) with $α = 0.2$ . To smooth the deactivation process, the weight factors ${\bar{w}}_{i}$ s were diminished to zero in 10 steps.

The paths of the mid-points of the links are shown in Figure 3. It is clear that none of the mid-points crossed the safety line $y_{t h r e s} = 2.5$ , which indicates that the solution given by (21) for solving the constrained manipulation task (8) succeeded in carrying out the collision avoidance task. The trajectories of the joints confirmed this success; see Figure 4. The distances d_i s are shown in Figure 4. All d_i s had positive values, supporting the success of the collision avoidance task. The relative velocities ${\dot{d}}_{i}$ between the mid-points and the safety line are shown in Figure 5. It can be seen that ${\dot{d}}_{i}$ is depressed to 0 when d_i tends to 0 especially for links 5–7. The weight factors given by (28) is shown in Figure 7. All weight factors ${\bar{w}}_{i}$ grew from 0 and increased gradually, ensuring the smoothness of the joint velocity command, as shown in Figure 8. The activation states s_i s of the collision avoidance task for each link are shown in Figure 9. For the i -th link, s_i take values from ${0.1 * i,1 + 0.1 * i,2 + 0.1 * i,3 + 0.1 * i}$ , which indicates the deactivated, switching out, switching in and activated state of constraint task, respectively. The figure shows that there six collision avoidance tasks are activated simultaneously during the period $t = [1.8,2.1]$ seconds, which is even higher than the redundancy number. However, since the weight factors in (28) do not increase to infinity, the manipulator is still able to accomplish the manipulation task. Although the constraint task is activated and deactivated 12 times during the manipulation process, the joint velocity command generated by the UWLN method is continuous, which ensures the smooth operation of the system. The error of the manipulation task is shown in Figure 10, which mainly results from discretization, because the precision of the computation error is less than $10^{- 14}$ . To achieve a better error dynamic, the feedback control laws can be designed as in [21] and [22]. Since the manipulation task is accomplished with good precision and the collision avoidance task is also carried out successfully, the validity of the UWLN method is indisputable.

Figure 3.

Trajectories of the mid-points of the links with collision avoidance control effected by the UWLN method. None of the mid-points crossed safe line y_thres=2.5, which indicates success in the collision avoidance control.

Figure 4.

The trajectory of each joint with collision avoidance control carried out by the UWLN method. None of the joints crossed the obstacle line y_obs=3, confirming the success of the collision avoidance control.

Figure 5.

Distance between obstacle and each mid-point of the links. All of the distances took on a positive value, which confirmed the success of collision avoidance.

Figure 6.

Approaching velocity between obstacle and each mid-point of the links. It is obvious that the velocity of the mid- points of links 5–7 are depressed, as the distance between them and the obstacle tends to be zero.

Figure 7.

The weight factor of the collision avoidance task of each link. As the distance between obstacle and each mid-point of the links decreases, the weight factor increases.

Figure 8.

Joint velocity command generated by the UWLN method. The command curves are continuous, which results in a smooth running of system.

Figure 9.

Activation state of collision avoidance task. The states take values from ${0.1 * i,1$ $+ 0.1 *$ $i,2$ $+ 0.1 *$ $i,3 +$ $0.1 * i}$ for link i, which indicates a deactivated, deactivating, activating and activated state of constraint task, respectively. Note that six constraint tasks are activated during the time period [1.8, 2.1] seconds.

Figure 10.

Manipulation task errors during collision avoidance control effected by the UWLN method

We also executed the GWLN algorithm to solve this constrained manipulation problem. Since the initial configuration results in a dependency between the constraint tasks, the original GWLN method has a singular transform matrix that leads to numerical instability. We therefore only carried out the GWLN algorithm with a combined constraint task. The trajectories of the centres of each link and the final configuration is shown in Figure 11. Although none of the mid points crossed the safe line, the movement generated by the GWLN method was rather big compared to the LN method (Figure 1) and the UWLN method (Figure 3). Since all the constraint tasks are combined by algebraic addition, the resulting movement appears somewhat odd.

Figure 11.

Trajectories of each mid-point with collision avoidance control carried out by the GWLN method. None of the mid-points crossed the safety line $y_{t h r e s} = 2.5$ , but the amplitude of the motion was rather large.

The 2 norm of the joint velocity command of the LN, UWLN and GWLN method, which is equal to the kinetic energy of the manipulator if all the links have a unit mass, is shown in Figure 12. It is obvious that the UWLN algorithm generates a command curve that consumes less energy than the GWLN method. During the time period $[0,0.8]$ seconds, the norm of the UWLN method almost coincides with that of the LN method, since the weight factors are small and the criterion matrix $\bar{Q}$ approximates the identity matrix. This confirms the conclusion that UWLN method consumes less energy than GWLN method.

Figure 12.

The 2 norm of joint velocity commands generated by least norm, the GWLN method and the UWLN method; it is obvious that the UWLN method consumes less energy than the GWLN method and its command curve is closer to that generated by the LN method

5. Conclusion

In this paper, we used the unified weighted least norm method to solve the constrained manipulation problem in redundant manipulator control. The criterion matrix in the UWLN method is capable of combining arbitrary constraint tasks and always has full rank, which guarantees the numerical stability. Furthermore, the proposed UWLN method establishes a smooth transition between the LN solution and the constrained control solution. This unification eliminates motion in the null space of the Jacobian matrix when the constraint tasks are not activated and therefore generates motion with a smaller amplitude compared to the GWLN method. The validity of the UWLN method is confirmed in this paper by both providing statistical proof and the simulation of a constrained manipulation process of a 7-DoF planar manipulator.

Footnotes

6. Acknowledgements

This work is supported by a grant from the National Natural Science Foundation of China (61374174), the Zhejiang Province Natural Science Fund (LY13F030001) and Hangzhou City major scientific and technological innovation projects of industrial chain (20132111A04).

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A Unified Weighted Least Norm Method for Redundant Manipulator Control

Abstract

Keywords

1. Introduction

2. The GWLN method

3. The unified weighted least norm method

4. Simulation result

5. Conclusion

Footnotes

6. Acknowledgements

References