Failure control and energy optimization of multi-axes space manipulator through genetic algorithm approach

Abstract

This paper presents intelligent approaches for swinging failure control and fuel saving of space robot by switching over the actuators. The space manipulators mainly work in extremely harsh and remote locations; hence, there may be a possibility of catastrophic failure in the manipulator. The primary concern of this work is to identify the faulty joints and subsequently switching over the actuators intelligently to control these joints and then finally optimize the trajectory based on the energy conservation criterion implementing a genetic algorithm approach. The purpose is to develop a capable robot for tackling joint failure problem in the multi-links robotic manipulators, by using redundant motorized joints. By anticipating joint failure in the form of free swinging, a power-saving six-axis space robot has been modeled. To increase the flexibility of space robots, redundancy is provided. The bond graph technique and MATLAB are used in tandem to validate and verify the proposed systems. The simulation results show that the proposed redundancy by switching over the actuator works well for failure control and optimizes 37.85% onboard fuel, which has a significant impact on satellite life. As a result, the proposed controlling method can achieve the operation of the floating body in an unknown environment with failure in the manipulator's joints. Furthermore, due to redundancy, flexibility will be economically rendered by the proposed control strategy.

Appendix

1.1 Bond graph model to generate dynamic equations

The bond graph technique adds flexibility to system modeling. It generates system dynamic equations automatically based on kinematic input and provides an exclusive method of modeling systems from various engineering domains, such as the proposed work, which includes mechanical, electrical and electronics. Because of its simplicity and higher information content, this method is more appealing for a complex system.

The proposed robot system's bond graph model (BGM) has been developed (Fig. 11), and the kinematic relations (7)–(10) are fed into the expression section as inputs. After compiling the model, the dynamic equations are derived. To draw the bond graph model, the transformer moduli (µ) can be developed from the kinematic relations (7)–(10).

For the first time user, understanding the bond graph model (BGM) may be difficult. As shown in Fig. 

Fig. 10

Word bond graph (WBG) of the multi-links space robot

10, before constructing the bond graph model, the word bond graph (WBG) model is developed for better understanding. The WBG includes words as standard symbol substitutes for the main components and bonds for power and signal exchange. It is divided into two major sections. Part I depicts the dynamics of a space robot, while Part II depicts the design of controllers. As shown in the WBG, the main components of the robot system are the robot base, actuators, links, PD, PID controllers and Jacobian. The modulated source of efforts is represented by the terms MSe. This WBG depicts each component's role in completing the desired task. PD controllers are used to activate or fix joints as needed. The transformers $tf_{1} , \ldots ,tf_{6}$ and $tf_{1}^{\prime} , \ldots ,tf_{6}^{\prime}$ act as torque switches for PD and PID controllers, respectively, and are discussed in A2 and A3.

Let us now replace words with standard elements containing precise functional or mathematical relationships, as shown in Fig. 

Fig. 11

Bond graph model of the multi-links space robot

11, by developing a bond graph model (BGM). For clarity, the Jacobian, PD and PID controllers are depicted as block diagrams in this figure, with details provided in the following sub-sections.

At $1\dot{\phi }$ junction Element I depicts rotational inertia (I_V) and at $1\dot{x}_{CM}$ and $1\dot{y}_{{_{CM} }}$ junctions it represents translational inertias (M_V) of robot base. Element I at 0-junctions represents translational inertias (M) in the X and Y directions of the first link. Element I at junction $1\dot{\theta }_{\phi 1}$ represents rotational inertia of the first link, where $\dot{\theta }_{\phi 1} = \dot{\phi } + \dot{\theta }_{1}$. Similarly, inertias of other links are appended in bond graph.

The transfer moduli (µ) are represented as modulated transformers (MTF). Here, flows are translational $(\dot{X}_{tip} ,\dot{Y}_{tip} )$ and rotational velocities $(\dot{\phi },\dot{\theta }_{1} , \ldots \ldots ,\dot{\theta }_{6} )$. Hence, for clarity, only the base and first link are represented; μ₇ to μ₂₆ are not shown in Fig. 3. Elements R at junctions $1\dot{\theta }_{1}$…$1\dot{\theta }_{6}$ represent the damping present at joints. The transformer moduli $tf_{1} \ldots ..tf_{6}$ and $tf_{1}^{\prime} \ldots ..tf_{6}^{\prime}$ are torque switches for PD and PID controllers, which are discussed in the coming sub-sections.

The angular and translational positions can be known by incorporating integrators and position sensor at 1-junctions, respectively.

While deriving dynamic equations, the problem generally encountered is the differential causality and stuck. To eradicate this problem, artificial compliances (pads) are used in the model. The pad has C element as compliance and R element as resistance. The dynamic consequences of the pads are canceled out by giving high input values to the C and R elements.

Let X_v and Y_v (shown in Fig. 11) denote the body-fixed coordinates and space robot modeling that has been done in a non-inertial frame. The base of robot rotates about the Z-axis. For a planar case, where, ω_x = 0, ω_y = 0 and v_z = 0, the net force acting on the space robot base can be given as

$$ F_{x} = M_{V} \ddot{x}_{CM} - M_{V} \dot{\phi }\dot{y}_{CM} , $$

(A.1)

$$ F_{y} = M_{V} \ddot{y}_{CM} + M_{V} \dot{\phi }\dot{x}_{CM} , $$

(A.2)

$$ F_{z} = 0. $$

(A.3)

Here, M_v is the mass of the vehicle (robot base) and ϕ is the attitude or orientation of the space robot base. A gyrator relates effort to flow and flow to efforts. A gyrator element can be used to establish Eqs. (A.1) and (A.2) in the bond graph as shown in Fig. 11.

Jacobian, PD and PID controllers are shown in Fig. 

Fig. 12

Architecture of controllers of redundant space robot

12, and the details are presented in Sects. Appendix A1, Appendix A2 and Appendix A3, respectively. The bond graph models are built-in Bond pad module of SYMBOL SHAKTI® software (Mukherjee et al., 2006) and are compiled using Borland compiler to create simulation codes.

1.2 Appendix A1: Derivation of Jacobian

Jacobian maps the joint velocity into tip velocity, which is represented in the signal form in Fig. 12. Equations (4) and (5) can be written in compact forms that lead to Jacobian as

$$ \left[ {\begin{array}{*{20}c} {\dot{X}_{{{\text{tip}}}} } \\ {\dot{Y}_{{{\text{tip}}}} } \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\dot{X}_{CM} } \\ {\dot{Y}_{C1M} } \\ \end{array} } \right] + \left[ {\begin{array}{*{20}l} {K_{1x} } \hfill & {K_{2x} } \hfill & {K_{3x} } \hfill & {K_{4x} } \hfill & {K_{5x} } \hfill & {K_{6x} } \hfill \\ {K_{1y} } \hfill & {K_{2y} } \hfill & {K_{3y} } \hfill & {K_{4y} } \hfill & {K_{5y} } \hfill & {K_{6y} } \hfill \\ \end{array} } \right]\left[ {\dot{\theta }} \right]^{T} + \left[ {\begin{array}{*{20}c} {K_{7x} } \\ {K_{7y} } \\ \end{array} } \right]\left[ {\dot{\phi }} \right] $$

(A.4)

where

$\left[ {\begin{array}{*{20}c} {K_{1x} } & {K_{2x} } & {K_{3x} } & {K_{4x} } & {K_{5x} } & {K_{6x} } \\ {K_{1y} } & {K_{2y} } & {K_{3y} } & {K_{4y} } & {K_{5y} } & {K_{6y} } \\ \end{array} } \right] = \left[ J \right]$, [J] denotes Jacobian of the manipulator which contains the gains required to model Jacobian as shown in Fig. 12. The expressions of gains are not given due to space constraint. Also,

$$ \left[ {\dot{\theta }} \right]^{T} = \left[ {\begin{array}{*{20}c} {\dot{\theta }_{1} } & . & . & . & . & {\dot{\theta }_{6} } \\ \end{array} } \right]^{T} $$

1.3 Appendix A2: PD controllers

The PD controllers are exercised to control joint motions by appending each joint to make the joints active/fixed (Fig. 12) in the signal block diagrams. The control law for the actuator at the joint is given in the general form as

$$ \tau_{n} = K_{pn} (\theta_{dn} - \theta_{an} ) - K_{dn} \dot{\theta }_{an} $$

(A.5)

where τ_n is the actuator torque for n^th joint, here, n = 1,…,6. The terms θ_dn andθ_an are the desired and the actual joint position of nth joint, and $\dot{\theta }_{an}$ is the actual joint angular velocity of nth joint. The terms $K_{pn}$ and K_dn are the proportional and the derivative gain parameters of nth joint, respectively.

The torques $\tau_{1} , \ldots ..,\tau_{6}$ come from the PD controller to the joints. The moduli $tf_{1} , \ldots ..,tf_{6}$ act as switch on/off of the PD controller. Switch on means moduli have 1 value, i.e.; $tf_{1} = tf_{2} = \cdots \cdot = tf_{6} = 1,$ subsequent to which, the joints become fixed. Similarly, switch off means moduli have 0 value, i.e. $tf_{1} = tf_{2} = \cdots \cdot = tf_{6} = 0,$ it means no PD controller is appended to the joints. Moreover, the joints become active for the desired motion planning of the manipulator.

1.4 Appendix A3: PID controllers

PID controllers are used to control end-effector motion, as shown in Fig. 12, by comparing the actual velocity signal to the reference velocity signal and correcting it if there is an error. The corrected signals are then transmitted to each actuated joint via the Jacobian. As a result, close trajectory tracking is achieved during the task. The PID controllers are represented in the form of signal block diagrams in Fig. 12. In these block diagrams, K_p, K_i and K_d represent proportional gain, integral gain and derivative gain, respectively. $\dot{X}_{tip}$ and $\dot{Y}_{tip}$ are the actual velocity signals and $\dot{X}_{ref}$ and $\dot{Y}_{ref}$ are the reference velocity signals in the X and Y directions, respectively. The terms $\varepsilon_{x}$ and $\varepsilon_{y}$ are the trajectory errors in X and Y directions, respectively.

The torques, $\tau_{1}^{\prime} , \ldots ..,\tau_{6}^{\prime}$ come from the PID controller to the joints through Jacobian. The moduli $tf_{1}^{\prime} , \ldots ..,tf_{6}^{\prime}$ act as switch on/off of the PID controller. Switch on means the moduli have 1 value, i.e. $tf_{1}^{\prime} = tf_{2}^{\prime} = \cdots . = tf_{6}^{\prime} = 1,$ and emphasize that the joints become controllable and responsible for the desired motion planning. Similarly, switch off means the moduli have 0 value, i.e., $tf_{1}^{\prime} = tf_{2}^{\prime} = \cdots . = tf_{6}^{\prime} = 0,$ and emphasize on the uncontrollable nature of joints; hence, the desired motion of robot tip cannot be realized.