International Journal of Aerospace Engineering

Volume 2016 (2016), Article ID 7819540, 14 pages

http://dx.doi.org/10.1155/2016/7819540

## Trajectory Optimization for Velocity Jumps Reduction considering the Unexpectedness Characteristics of Space Manipulator Joint-Locked Failure

School of Automation, Beijing University of Posts and Telecommunications, Beijing 100876, China

Received 24 October 2015; Revised 25 December 2015; Accepted 5 January 2016

Academic Editor: Paul Williams

Copyright © 2016 Qingxuan Jia et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Aiming at reducing joint velocity jumps caused by an unexpected joint-locked failure during space manipulator on-orbit operations without shutting down manipulator, trajectory optimization strategy considering the unexpectedness characteristics of joint-locked failure is proposed in the paper, which can achieve velocity jumps reduction in both operation space and joint space simultaneously. In the strategy, velocity in operation space concerning task completion directly is treated as equality constraints, and velocity in joint space concerning motion performance is treated as objective function. Global compensation vector which consists of coefficient, gradient of manipulability, and orthogonal matrix of null space is constructed to minimize the objective function. For each particular failure time, unique optimal coefficient can be obtained when the objective function is minimal. As a basis, a method for optimal coefficient function fitting is proposed based on a priori failure information (possible failure time and the corresponding optimal coefficient) to guarantee the unexpectedness characteristics of joint-locked failure. Simulations are implemented to validate the efficiency of trajectory optimization strategy in reducing velocity jumps in both joint space and operation space. And the feasibility of coefficient function is also verified in reducing velocity jump no matter when joint-locked failure occurs.

#### 1. Introduction

Space manipulator [1, 2] is a kind of complex electromechanical system, which plays an important role in space station maintenance, rendezvous and docking, and space exploration. During its long-time service on orbit, affected by radiation, energetic particle, and high-low temperature alternating of space environment, many kinds of malfunctions are inevitable, such as link deformation, performance degradation, and joint failure [3, 4]. Joint as a precision unit, integrating with machinery, communications, and automatic control technology, is the core component of space manipulator to achieve safe and smooth operation. Joint failure may bring about performance degradation of space manipulator and lead to bad effect on task completion.

For general manipulator on the ground, when joint fails during manipulator motion, the manipulator can be shut down to maintain or replace the failure joint. However, if shutting down space manipulator once joint failure occurs, the inertia force acting on the manipulator will cause serious impact force on the end-effector and joints, as well as cause instability of the base. When joint fails especially during load operations such as space cabin docking [5] and astronauts assistant [6], the bigger mass and inertia will cause huger impact, which may exceed the rated torque and cause serious hurt on the manipulator parts and operating objects. On the other hand, when trying to execute task continuously after joint failure, control strategy should be adjusted in real time. As a result, velocity jumps will be caused in joint space and operation space, which will also generate impact on manipulator structure. Thus, it is of great significance to smooth motion parameters when joint failure occurs during on-orbit operations no matter whether space manipulator will be shut down or execute task continuously.

Joint failure can be divided into joint-locked [7] and free swing [8] according to whether joint can be operated or not after failure occurs. In free swing state, failure joint moves freely, making the manipulator out of control. In joint-locked state, failure joint is locked at the angle of failure time. Although the degree of freedom (DOF) of space manipulator is reduced, the healthy joints can move normally, and the task can be executed continuously when joint-locked failure occurs. Thus in order to avoid the impact force caused by shutting down space manipulator after joint-locked failure, the paper is devoted into trajectory optimization research to achieve task completion and motion parameters smoothness (velocity jumps reduction especially) without shutting down manipulator.

In order to achieve task completion after manipulator failure, scholars concentrate on the following researches. First, in order to obtain the failure information such as failure time and failure joint, fault detection, fault isolation, and fault identification [9–12] are researched. Second, abilities of manipulator are analyzed in failure condition such as fault-tolerant workspace [13, 14] and dexterity [15, 16]. Based on these researches above, pseudoinverse method [17], self-motion manifold method [18], quadratic programming algorithm [19], and configuration optimization [20, 21] are applied to achieve task completion after failure occurs. Meanwhile, constraints are analyzed during fault tolerance to guarantee the practical application. Ralph and Pai [22] analyzed the minimal constraints necessary in fault tolerance, Jamisola Jr. et al. [23] took environment obstacle constraints into account, and Xie et al. [24] introduced the singularity constraints into the fault-tolerant method.

In the mentioned fault-tolerant researches, some of them shut down manipulator after failures; then fault-tolerant method is introduced to achieve task completion. However, it is not suitable for space manipulator fault-tolerance. The others execute task via fault-tolerant method in real time. Unfortunately, when guaranteeing velocity and position in operation space for task completion, velocity jumps are brought about into joint space. Scholars try to reduce the velocity jumps; Abdi et al. [25] concentrated on the minimum velocity jump in operation space and proposed an optimization strategy for joint velocity redistribution. Meanwhile, the velocity jump in joint space is reduced with the application of least square method and matrix perturbation method [26]. Jing and Cheng [27] achieved velocity jump reduction in joint space by optimizing the initial configuration of a dual-arm manipulator. In the authors’ former work [28], velocity jumps in joint space were reduced by constructing compensation vectors based on the gradient of manipulability. Further, the coefficient form of compensation vector was discussed to achieve better reduction performance.

However, simultaneous reduction for velocity jumps in operation space and joint space is not achieved. What is more, velocity jumps reduction in the operation space may even increase the velocity jumps in joint space. Since velocity jumps in operation space are concerned with task completion and velocity jumps in joint space are concerned with joint structure safety, it is of equal importance to achieve velocity jumps reduction in the two spaces to guarantee task completion and motion safety. According to the self-motion characteristics of space manipulator, a set of velocities in operation space corresponds to multiple sets of joint velocities. So when the velocity in operation space is fixed, joint velocity can be adjusted to find an optimal value minimizing the jumps. On the basis, the trajectory optimization strategy can be formed by treating velocity in operation space as constraints and treating velocity in joint space as optimization objective.

Besides, the mentioned fault-tolerant methods depend on failure information; only after the failure condition is confirmed can they begin to work. However, in practical application, joint-locked failure occurs unexpectedly. It needs time to obtain failure information with fault detection method, so time delay is inventible between failure discovery and fault-tolerant method application, which makes fault-tolerant performance worse than the ideal condition. Although controllers are designed based on neural networks to achieve real-time fault tolerance [29–31], the adaptation of controllers for rapid response and stable operation is not enough for on-orbit application. Therefore, in order to avoid time delay problem of fault detection method and to guarantee the unexpectedness characteristics of joint-locked failure, global compensation for joint velocity during the entire path planning is proposed in the paper, based on which the trajectory optimization is not relevant to the failure information and can handle an unexpected joint-locked failure.

In conclusion, the trajectory optimization strategy proposed in the paper aims at dealing with three problems. First, the task completion on-orbit is guaranteed without shutting down space manipulator, and the parameters of space manipulator particularly the velocity jumps should be smoothed after joint-locked failure occurs. Second, the reduction for velocity jumps in both operation space and joint space should be achieved simultaneously. It can be achieved by considering velocity in operation space as equality constraints and considering velocity in joint space as optimization objective in the strategy. Third, the avoidance of time delay problem and the unexpectedness characteristics of joint-locked failure should be handled. For this purpose, a global compensation vector is constructed for velocity compensation over the entire path planning task, which makes the optimization strategy irrelevant with the real-time failure information acquisition. In addition, with global compensation for joint velocity, a unique coefficient which makes objective function minimal can be obtained related to a particular failure time. Taking it as a priori failure information, the law of optimal coefficient related to arbitrary failure time is concluded. A coefficient function is obtained to deal with velocity jump reduction caused by unexpected joint-locked failure.

The presented paper is organized as follows: in Section 2, velocity jumps in operation space and joint space caused by joint-locked failure are analyzed. The mathematical expression of trajectory optimization strategy is derived, during which velocity jumps in operation space are expressed as equality constraint and velocity jumps in joint space are expressed as objective function. In Section 3, global compensation vector for joint velocity is constructed, and the solving method is proposed to obtain the optimal coefficient of global compensation when joint-locked failure occurs at a particular time. Then considering the unexpectedness characteristics of joint-locked failure, coefficient of global compensation vector is fitted in function form based on a priori failure information. In Section 4, simulations are implemented to validate the effectiveness of trajectory optimization strategy in reducing the velocity jump in both joint space and operation space when joint-locked failure occurs at arbitrary time. The last part is conclusion.

#### 2. Mathematical Expression of Trajectory Optimization Strategy

Joints are the key parts of space manipulator operations. When joint-locked failure occurs, it will bring about velocity jumps in operation space and joint space, which would tremendously reduce the operability of space manipulator, especially the ability of pose-reaching and task-completing. In this section, trajectory optimization is proposed considering simultaneous velocity jumps reduction in two spaces. Based on the analysis of velocity jumps, the constraints and objective of trajectory optimization are derived.

Trajectory optimization is discussed on a typical space manipulator path planning task, which starts at time and ends at time . The control interval is represented as , and the execution time of path planning can be represented as , , .

##### 2.1. Analysis on Velocity Jumps in Operation Space and Joint Space

For a degree space manipulator, when joints are healthy during path planning task, the Jacobian matrix and joint velocity vector at time can be expressed as

The Jacobian matrix is the function of joint angle vector at current time . And represents the dimension of operation space. represents the th column of Jacobian matrix.

When joint-locked failure occurs in joint at time , the dimension of joint space degenerates to . Correspondingly, the Jacobian matrix degenerates to , which is named as reduced Jacobian matrix [32]. The reduced Jacobian matrix and joint velocity vector at time can be expressed as

Correspondingly, represents the th column of reduced Jacobian matrix. And represents the reduced joint angle vector. After joint fails at time , the velocity of failure joints turns to 0, and velocity of healthy joints keeps constant. Thus, at time , it has

Then the velocity jump vector in joint space at time can be expressed as

represents the nominal velocity of th joint when the joint is healthy. On the basis, the velocity jump in operation space caused by joint-locked failure is expressed as

According to (5), it will lead to accumulative pose (position and orientation) deviation in operation space:

When joint failure occurs unexpectedly during path planning task, an average velocity deviation in the operation space can be obtained:

Equation (7) can be used to evaluate the influence of velocity deviation caused by an unexpected joint-locked failure, while, for a general path planning task, the pose accuracy should fulfill accuracy threshold at each time to guarantee task completion. Set the threshold value as ; when , the task cannot be completed. Therefore, the velocity jump in operation space should be eliminated preferentially to guarantee task completion.

Generally, velocity jump in operation space can be reduced by compensation for healthy joint. When compensation vector is introduced, (5) is turned into

When the velocity deviation in operation space is completely eliminated, namely, , the compensation vector for joint velocity can be expressed as

represents the pseudoinverse of the reduced Jacobian matrix. By given the compensation as (9), the velocity jump in operation space can be completely eliminated. The contribution of failure joint is redistributed to healthy joints. However, additional velocity jumps are introduced into healthy joints. The velocity compensation may be too large for safety threshold of joint velocity and the impact caused by compensation may damage the joint structure, leading to terribly concatenate failure of joints. Thus, it is of equal importance to decrease the velocity jumps in operation space and joint space.

##### 2.2. Constraints and Objective of Trajectory Optimization

In order to achieve simultaneous velocity jumps reduction in both operation space and joint space, the velocity in operation space is considered as equality constraint, which is preferentially guaranteed for task completion. And joint velocity is used to construct the objective function, which is adjusted to achieve joint velocity jumps minimal. The constraints and objective function of trajectory optimization can be expressed as follows.

###### 2.2.1. Constraints

Assume only position accuracy in operation space is considered during path planning task. The actual value of linear velocity in operation space should be in accordance with the nominal value; namely,

represent the actual and nominal value of the linear velocity in the operation space. Correspondingly, the position in operation space should be tracked as the nominal value:

represent the actual and nominal position in operation space, respectively. Equations (10) and (11) are named as task constraints. When task constraints are guaranteed, it has (; ).

Besides, the parameters of space manipulator should fulfill the thresholds in path planning. Constraints of joint velocity and joint angular acceleration are taken into account:wherein represents the number of joints and and represent the threshold value of joint velocity and angular acceleration of the th joint, respectively. When (12) is fulfilled, it has .

###### 2.2.2. Objective Function

The objective function is constructed as the joint velocity deviation at failure time . Since the dimension of joint velocity is reduced after joint failure, in order to guarantee the dimension, is defined based on :

Then the objective function can be expressed as

Based on the analysis above, the mathematical expression of trajectory optimization strategy is derived as

#### 3. Solution for Trajectory Optimization considering Unexpectedness Characteristics of Joint-Locked Failure

In this section, trajectory optimization when joint fails at a particular time is solved firstly. A global compensation vector for joint velocity is constructed based on the orthogonal basis of Jacobian matrix null space. Then the optimal coefficient of compensation vector is obtained when the objective function is minimal. Since global compensation vector compensates joint velocity in each control interval ignoring the failure time, time delay problem can be efficiently avoided. Taking the failure time and corresponding optimal coefficient as a priori failure information, the law of optimal coefficient related to arbitrary failure time is concluded, and a method for coefficient function fitting is proposed based on a priori failure information to guarantee the unexpectedness characteristics of joint-locked failure.

##### 3.1. Global Compensation Vector for Joint Velocity

In order to construct the global compensation vector, the orthogonal basis of Jacobian matrix null space is used to represent the self-motion characteristics of space manipulator. Manipulability [33] is introduced into the vector, whose gradient can be used to represent the velocity compensation weight of each joint. When joint-locked failure occurs, it is reduced as . Besides, a constant coefficient is used to adjust the velocity compensation rate for each joint. Then for each time, it has

In (16a) and (16b), and are used to compensate joint velocity at each time in path planning. Before failure occurs, precompensation is implemented with the vector of (16a). After failure occurs, joint velocity is compensated according to (16b). Namely, joint velocity at each time corresponds to a vector or . Then the global compensation vector for joint velocity over the entire task is constructed:

With the compensation of , joint velocity can be expressed as , which is detailed in

Joint angle at time can be obtained as

By substituting and into (16a) and (16b), and can be obtained. The recurrence relation between , , and is generated, which is shown in Figure 1.