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A scheme for robust trajectory control of space robots
Simulation Modelling Practice and Theory 16 (2008) 1337–1349
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Simulation Modelling Practice and Theory journal homepage: www.elsevier.com/locate/simpat
A scheme for robust trajectory control of space robots Pushparaj Mani Pathak a,*, R. Prasanth Kumar b, Amalendu Mukherjee b, Anirvan Dasgupta b a b
Mechanical and Industrial Engineering Department, Indian Institute of Technology, Roorkee 247667, India Mechanical Engineering Department, Indian Institute of Technology, Kharagpur 721302, India
a r t i c l e
i n f o
Article history: Received 11 November 2006 Received in revised form 16 June 2008 Accepted 28 June 2008 Available online 8 July 2008
Keywords: Free-floating Space robotics Robust control Bond graph modeling Non-holonomic system
a b s t r a c t This paper presents a scheme for robust trajectory control of free-floating space robots. The idea is based on the overwhelming robust trajectory control of a ground robot on a flexible foundation and robust foundation disturbance compensation presented elsewhere. No external jets/thrusters are required or used in the scheme. An example of a three-link robot mounted on a free-floating space platform is considered for demonstrating the efficacy of the control scheme. Bond graph technique has been used for the purpose of modeling and simulation. Robustness of the control scheme is guaranteed since the controller does not require the knowledge of the manipulator parameters. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction As the number of space structures and satellites in orbit increases, the need for their maintenance increases. Extending the life of such space systems, and therefore reducing the associated costs, will require extensive inspection, assembly and maintenance capabilities in orbit. The cost of human life support facilities, the limited time available for astronaut’s extra vehicular activity (EVA), and high risks involved make space robotic devices indispensable for space operations. The typical tasks that can be performed by space robots are deploying or assembling space platforms and space stations, deploying large antennas or solar power stations, and service and maintenance of satellites. All these applications require precise manipulator motion control. In a free-floating space robotic system, the spacecraft’s position and the attitude are not actively controlled using external jets/thrusters during manipulator activity. This conserves attitude control fuel, and hence lengthens the life of the system. However, due to the conserved linear and the angular momenta, the spacecraft moves freely in response to the dynamical disturbances caused by the manipulator’s motion. Such disturbances may result in significant deviation of the end-effector from the desired trajectory, and are particularly significant when the manipulator is holding a heavy payload. Moreover, the angular momentum conservation constraints are non-integrable rendering the system non-holonomic. Hence, it is not possible to cancel this base disturbance using static state feedback. This complicates the motion planning and control issue further [1]. The concepts of free-flying and free-floating robots were introduced in the early 1980s. Vafa and Dubowsky [2] introduced the concept of virtual manipulator (VM) for the analytical modeling of space manipulators. Papadopoulos and Dubowsky [3] suggested that any control algorithm that can be used for fixed-based manipulators can also be used in the control of free-floating space manipulators systems, with the additional conditions of estimating or measuring a spacecraft’s
* Corresponding author. Tel.: +91 1332 285608; fax: +91 1332 285665. E-mail addresses: [email protected] (P.M. Pathak), rpkumar@scientific.net (R.P. Kumar), [email protected] (A. Mukherjee), [email protected] (A. Dasgupta). 1569-190X/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.simpat.2008.06.011
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orientation, and of avoiding dynamic singularities. Watanabe and Nakamura [4] proposed a free-flying space robot having a special kinematic structure and mass distribution such that the CM of the total system remains fixed to the base body regardless of the manipulators configuration. Liang et al. [5] mapped a free-floating space manipulator to a conventional fixed base manipulator. In the mapping, the dynamic and kinematic properties of the manipulator are preserved. Jain and Rodriguez [6] analyzed the relationship between base invariance symmetry (i.e. freedom in choosing any link as base body) and the dynamical equations for free-flying manipulators. Umetani and Yoshida [7] used the angular momentum conservation law to derive a new Jacobian matrix in generalized form for a space robotic arm. Using this new matrix, they developed a control method for the space manipulator based on the resolved motion rate control concept. Yoshida [8] demonstrated the concept of generalized Jacobian in ETS-VII. Yokokohji et al. [9] proposed an efficient computation algorithm for the trajectory control of multi arm free-flying space robot using the generalized Jacobian matrix technique. Huang et al. [10] presented a trajectory planning method of space manipulator that can track, approach and catch the uncontrolled spinning satellite (USS) in free-floating situation. They planed a spiral ascending trajectory for space manipulator to approach towards USS in Cartesian space. Nakamura and Mukherjee [1] proposed an approach for non-holonomic path planning to obtain a specified base attitude and an arm posture starting from an arbitrary initial state. Mukherjee and Kamon [11] studied the configuration control problem in free-flying planar space robots. Franch et al. [12] studied non-holonomic rate constraints for free-floating planar open chain robots to determine the design conditions under which the system exhibits differential flatness. Under these design conditions, they used the property of flatness for trajectory planning and feedback control under perturbations in the initial state. Chen [13] presented the optimal non-holonomic motion planning of free-floating space robot system with dual-arms. Based on the linear and angular momentum conservations of the system, the system state equations for control design are established. Alexander and Cannon [14] presented a control algorithm for control of the free-floating space robot without the strict attitude or position control of the robot body. The concept of disturbance map, and its use in non-holonomic motion planning of space robots with minimum attitude change was proposed by Dubowsky and Torres [15], and Legnani et al. [16]. Yoshida et al. [17] derived the conditions for manipulation with zero base disturbances, and determined the joint velocities from those conditions. In the coordinated control strategy [18], the deviation of the base due to the manipulator reaction is corrected by the reaction wheels mounted on the space vehicle. Taniwaki et al. [19] worked on coordinated control of a satellite-mounted manipulator handling a flexible payload. Papadopoulos and Dubowsky [20] studied the coordinated control of a space manipulator mounted on a spacecraft. Marchesi et al. [21] method on coordinated control, exploits both coupled dynamics of the spacecraft–manipulator system and non-holonomic constraint due to angular momentum. Yoshida et al. [22] presented the post-flight analysis with ETS-VII of Reaction Null-Space-based reaction-less manipulation or Zero Reaction Maneuver. To solve the communication time delay in the space robot teleoperation, Yoon et al. [23] introduced a new control method that modified model-based teleoperation system against the time delay, and this method was applied to the ETS-VII manipulator teleoperation in orbit. Zappa et al. [24] studied the case of planar polar manipulators, analyzed the peculiarities of their robot arm configurations. Menon et al. [25] investigated test-beds for free-flying robots with extended arms, emphasizing pros and cons of the different test-beds. Wang et al. [26] studied the dynamics modeling and control simulation of space robot with the dynamic effects due to orbital mechanics. Belousov et al. [27] presented motion planning algorithms for the large space robot manipulators with complicated dynamic behaviour. They proposed two ‘‘two-stage” iterative algorithms, which provide collision-free robot motion taking into account robot’s dynamics. Based on the insights developed from bond graph modeling, Ghosh [28] developed a robust overwhelming controller for the robotic manipulator, which does not require the knowledge of the robot parameters and the payload. Kumar and Mukherjee [29] further developed the overwhelming control strategy and applied it for robust trajectory control of a two-link planar manipulator on a flexible foundation. In this paper the basic ideas of the overwhelming controller [30] is first reviewed for the sake of completeness then a scheme for robust trajectory control of space robots is proposed. A numerical example of the three-link manipulator on a free-floating space platform is presented to demonstrate the efficacy of the proposed control strategy. The complete dynamics is modeled using bond graphs [31]. For the purpose of modeling and simulation, the bond graph package SYMBOLS Shakti [32] has been used. This software generates the equation of motion symbolically, which are then used for simulation.
2. Robust trajectory controller 2.1. The robust overwhelming controller The overwhelming control strategy [28–30] consists of a set of linear time invariant controllers coupled to the robotic manipulator through a set of high feed-forward gains and suitably defined feedbacks. The driving efforts to the manipulator actuator are decided by the linear controller’s parameters and the coupling between the robot and the controller ensures that the robotic manipulator is ‘‘dragged” along the trajectory of the controller model. Since the controller is linear time invariant, linear control theory can be used in the design of such systems. The high feed-forward gain renders the non-linear dynamics of the robot small perturbations on that of the linear controller and hence the dynamics of the robotic manipulator closely resembles that of the controller. The error in tracking can be reduced to any extent by the proper
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Fig. 1. Mechanical equivalent of overwhelming controller.
Fig. 2. Bond graph of realizable overwhelming controller.
choice of controller parameters and by increasing the feed forward gains within the bandwidth of the control system and the capacity of the actuators. Above discussed strategy can be illustrated by considering a system as shown in Fig. 1. Consider that a single DOF plant is to be driven by a similar single DOF controller through a mass less lever (amplifier). Plant parameters mp (mass), kp (stiffness), rp (resistance) may be undetermined functions of time or states whereas controller parameters Mc (mass), Kc (stiffness) and Rc (resistance) are constants. A force eref (t) drives the controller to generate the desired motion of the plant. The bond graph model of the system is shown in Fig. 2. It is to be noted that the I element in overwhelmer is differentially causalled. This can be removed by introducing an artificial compliance or pad. This consists of a spring and a damper and has been discussed further in detail later in this paper. The controllers are usually implemented by using amplifiers and the feedback system. Hence the physical entity of lever of Fig. 1 can be realized by resolving the transformer element (representing the lever) into two separate elements. One as modulated source of effort with modulus lH. The other as modulated source of flow with modulus as 1 as shown in Fig. 2. The modulated source of flow with modulus as 1 is the feedback path of the flow information with no power associated with it. The modulated source of effort with high gain (lH) is an effort amplifier. The actuation power at the output port of this amplifier comes from a tank system sustaining it. This way of splitting the transformer will put the controller in the digital domain, as the controller expressions can be in the computer. The effort signals can be passed from computer to amplifier whereas flow feedback signal can be received by the computer. The transfer function between robot (plant) output flow and force input can be evaluated from the bond graph shown in Fig. 2. From this figure it is seen that,
e5 ðtÞ ¼ e1 ðtÞ e2 ðtÞ e4 ðtÞ e3 ðtÞ or e5 ðtÞ ¼ eref ðtÞ M c
d fc ðtÞ Rc fc ðtÞ K c dt
Z
fc ðtÞdt:
ð1Þ
Here eref (t) is the reference force input, fc(t) is the common flow at 1c junction in Fig. 2. Mc, Rc and Kc are the inertia (differential gain), resistance (proportional gain) and stiffness (integral gain), respectively, of the overwhelming controller. If s is the Laplace variable, taking Laplace transform of both sides of Eq. (1) gives
E5 ðsÞ ¼ Eref ðsÞ M c sF c ðsÞ Rc F c ðsÞ K c F c ðsÞ=s or E5 ðsÞ ¼ Eref ðsÞ F c ðsÞðMc s2 þ Rc s þ K c Þ=s; or E5 ðsÞ ¼ Eref ðsÞ þ F c ðsÞCðsÞ;
ð2Þ
CðsÞ ¼ ðMc s2 þ Rc s þ K c Þ=s:
ð3Þ
where
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Similarly from Fig. 2,
e7 ðtÞ ¼ e15 ðtÞ e9 ðtÞ e8 ðtÞ; mp
d fp ðtÞ ¼ lH e5 ðtÞ r p fp ðtÞ kp dt
Z
fp ðtÞdt:
ð4Þ
Here mp, rp and kp are the inertia, resistance and stiffness, respectively, of the plant, lH is high feed-forward gain and fp is the flow at 1p junction in Fig. 2. Taking Laplace transform of both sides of Eq. (4) gives
mp sF p ðsÞ ¼ lH E5 ðsÞ r p F p ðsÞ kp F p ðsÞ=s; F p ðsÞðmp s2 þ r p s þ kp Þ=s ¼ lH E5 ðsÞ; F p ðsÞ=RðsÞ ¼ lH E5 ðsÞ;
ð5Þ
RðsÞ ¼ s=½mp s2 þ r p s þ kp :
ð6Þ
where
Substituting for E5(s) in Eq. (5) from Eq. (2), we get,
F p ðsÞ=RðsÞ ¼ lH ½Eref ðsÞ þ F c ðsÞCðsÞ:
ð7Þ
But as seen from Fig. 2,
F c ðsÞ ¼ F p ðsÞ:
ð8Þ
Substituting in Eq. (7) from Eq. (8) gives the transfer function between robot output flow Fp(s) and force input Eref(s) as,
F p ðsÞ=Eref ðsÞ ¼ lH RðsÞ=½1 lH RðsÞCðsÞ:
ð9Þ
When lH 1, one obtains from Eq. (9) the overall transfer function as,
F p ðsÞ=Eref ðsÞ ffi 1=CðsÞ ¼ s=ðMc s2 þ Rc s þ K c Þ:
ð10Þ
In order to control the plant the command eref(t) is taken of the form,
eref ðtÞ ¼ e1 ðtÞ ¼ M c €xref þ Rc x_ ref þ K c xref ;
ð11Þ
so that, it produces the desired motion xref(t) of the plant. Therefore,
Eref ðsÞ ¼ s2 Mc X ref ðsÞ þ sRc X ref ðsÞ þ K c X ref ðsÞ or X ref ðsÞ=Eref ðsÞ ¼ 1=ðM c s2 þ Rc s þ K c Þ:
ð12Þ
Since Xref(s) = Fref(s)/s, Eq. (12) can be written as,
F ref ðsÞ=Eref ðsÞ ¼ s=ðM c s2 þ Rc s þ K c Þ:
ð13Þ
From Eqs. (13) and (10) we get,
F p ðsÞ=F ref ðsÞ ¼ 1;
ð14Þ
i.e., the plant follows the command. The bond graph of Fig. 2 can be modified to give reference input as flow input. Here, the non-causal form of the controller is only artificial and is a consequence of differential causality in the overwhelmer. The controller can be brought to causal form by introducing a pad [30]. Pads are lumped flexibilities [31]. Choosing high values of damping and stiffness can reduce their dynamic effects. The effects of this are usually in the form of a high frequency vibration in the system at the onset of simulation. These frequencies however decay soon due to heavy damping. Thus an overwhelming controller can be designed which takes care of complex and highly non-linear dynamics of robot and provides robust trajectory control. The controller is fed with the error in trajectory and controller in turn provides the corrective torques to be applied in the joints of robot. By considering the controller in the world coordinate frame external disturbances and desired input can be directly conveyed in the world coordinate frame. Conventionally input forces from controller to joints are determined by the desired trajectory of the system and the inverse kinematics of the given robot. Hence the inverse kinematics transformations can be avoided by considering the controller in the workshop coordinate frame rather than joint coordinate frame. Under these conditions the velocities can be transformed from the joint coordinates to the world coordinate using the forward kinematics Jacobian. Next section will consider the behaviour of the overwhelming trajectory controller when the robot is placed on a flexible foundation. 2.2. Robust overwhelming controller for ground manipulator with foundation compensation The overwhelming control strategy for a ground manipulator with some modification has been applied to a ground robot on a flexible foundation by Kumar and Mukherjee [29]. The principle of operation can be explained through an example of a single DOF robot on a flexible foundation as shown in Fig. 3. To incorporate the foundation disturbance in the world
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Fig. 3. Single DOF robot on flexible foundation.
Fig. 4. Bond graph model for system shown in Fig. 3.
coordinates, the foundation velocity is sensed and feedback to the controller. The bond graph model of the system shown in Fig. 3 is presented in Fig. 4, where the flow signal from foundation is modulated by magnitude a and this signal is supplied as modulated source of flow through bond 13 to controller. Essentially a shows the feedback compensation. System shown in Fig. 4 is a closed loop system subjected to two inputs i.e., reference inputs Eref(s) and foundation disturbance Edis(s). When two inputs are present in a linear system each input can be treated independently of the other and the outputs corresponding to each input alone can be added to give the complete output. From bond graph shown in Fig. 4 we observe that
E15 ðsÞ ¼ Eref ðsÞ þ F c ðsÞCðsÞ;
ð15Þ
CðsÞ ¼ ðMc s2 þ Rc s þ K c Þ=s:
ð16Þ
where
Since E11(s) = lHE8(s) = lHE15(s), we get
E15 ðsÞ ¼ E11 ðsÞ=lH and since
F p ðsÞ=E11 ðsÞ ¼ PðsÞ ¼ 1=mp s; F p ðsÞ E15 ðsÞ ¼ ; lH PðsÞ
ð17Þ ð18Þ
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also,
F c ðsÞ ¼ F 15 ðsÞ ¼ ½F 20 ðsÞ þ F 13 ðsÞ; ¼ ½F 11 ðsÞ aF 19 ðsÞ; ¼ ½fF 1 ðsÞ F 18 ðsÞg aF f ðsÞ; ¼ ½F p ðsÞ þ F f ðsÞ aF f ðsÞ or F c ðsÞ ¼ ½F p ðsÞ F f ðsÞð1 aÞ:
ð19Þ
Substituting for E15(s) and Fc(s) from Eqs. (18) and (19) into Eq. (15) we get,
F p ðsÞ
lH PðsÞ
¼ Eref ðsÞ þ ½F p ðsÞ F f ðsÞð1 aÞCðsÞ:
ð20Þ
At 1f junction,
e3 ðtÞ ¼ e5 ðtÞ e10 ðtÞ e12 ðtÞ e18 ðtÞ; Z d Mf ff ¼ edis ðtÞ Rff ðtÞ K f ff ðtÞdt e18 ðtÞ: dt
ð21Þ
In Eq. (21) Mf (mass), Rf (damping), Kf (stiffness) are foundation parameters. Taking Laplace transform of both sides of Eq. (21) gives
sMf F f ðsÞ ¼ Edis ðsÞ Rf F f ðsÞ K f F f ðsÞ=s E18 ðsÞ; F f ðsÞ½s2 M f þ sRf þ K f =s þ E11 ðsÞ ¼ Edis ðsÞ; F f ðsÞ=FðsÞ þ F p ðsÞ=PðsÞ ¼ Edis ðsÞ;
ð22Þ
FðsÞ ¼ s=ðM f s2 þ Rf s þ K f Þ:
ð23Þ
where
If we take robot motion due to reference velocity command only i.e., Edis(s) to be zero, then finding Ff from Eq. (22) and substituting its value in Eq. (20) gives us the transfer function for the system between the force input to the robot velocity as,
F p ðsÞ lH PðsÞ ¼ : Eref ðsÞ ½1 lH PðsÞCðsÞ þ ð1 aÞlH CðsÞFðsÞ
ð24Þ
For the high feed-forward gains (i.e., lH 1) and a = 1, and on using Eqs. (16), (17) and (23) in Eq. (24) one obtains
F p ðsÞ=Eref ðsÞ ffi 1=CðsÞ ffi s=ðMc s2 þ Rc s þ K c Þ:
ð25Þ
To evaluate the effect of the foundation disturbance Edis(s), we assume that the system is at rest with Eref as zero. Then putting Eref as zero in Eq. (20) and substituting the value of Ff from Eq. (22) into Eq. (20), the transfer function between the robot end-effector velocity Fp(s), and the disturbance force Edis(s) for such a system can be derived as
F p ðsÞ ð1 aÞlH PðsÞCðsÞFðsÞ ¼ : Edis ðsÞ ½1 lH PðsÞCðsÞ þ ð1 aÞlH CðsÞFðsÞ
ð26Þ
The transfer functions in Eqs. (24) and (26) show that if a = 1, the foundation has no effect on the trajectory of the robot under the chosen feedback scheme. Thus the external disturbances can be completely compensated even in the absence of the knowledge of foundation parameters. A foundation disturbance sensor and a compensating feedback have been introduced to achieve this. In the above control scheme, the effort command to the controller has to be calculated based on the given trajectory and the controller parameters as evident from Eq. (25). By making a slight modification to the scheme the flow reference input can be directly used instead of the effort reference input. The scheme is extendable to multi-degree of freedom robotic systems and can be used effectively to track a desired endeffector trajectory. However there are practical implications of increasing the feed-forward gains as increasing it to high value also increases the noise. This noise can be reduced by low pass filtering. Low pass filtering may introduce sluggishness in the system, so there has to be a trade off between high gain and bandwidth of filtering. The scheme for robust control of ground manipulator with foundation compensation presented in this section can be extended for space robots. The only difference between a ground robot on flexible foundation and a space robot is that, in case of the space robot base is not anchored. So if we remove the foundation spring and damper from the robot on flexible foundation, we get a robot on floating base i.e. a space robot. In case of such robots trajectory control task is challenging due to floating base. Next section will present a control strategy for the robust trajectory control of space robot which will be based on base motion compensation by feeding the velocity of the space vehicle to the robust controller. 3. Robust controller for a space robot To illustrate the control strategy for the space robot, let us consider a three DOF free-floating space manipulator consisting of a serial robot mounted on a satellite base as shown in Fig. 5. Joint rotation (hi) of manipulator and Euler angle of the
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Fig. 5. A free-floating space manipulator.
base (/, h, w) in 3-2-1 convention are used as generalized coordinates. Links are numbered starting with zero from the base to the last link. All joints are assumed to be revolute. The joint between links i and i + 1 is numbered as i + 1. An absolute frame {A} is considered as shown in Fig. 5. The vehicle is denoted by frame {V} located at the CM of the vehicle. The manipulator base is mounted on the vehicle at a position (rx, ry, rz) from the frame {V}. A frame {0} is attached on the vehicle to represent the manipulator base location on the vehicle. Frame {1} is attached on the link 1 at its base. Frame {1} has relative motion with respect to frame {0} about y-axis. Frames {2} and {3} are attached at the base of second and third link. They represent the joint axes for the second and third joint. They have rotation about their z-axes. The moments of inertia are defined about the body-fixed principal coordinate system. Without loss of generality it is assumed that links of the manipulator are rigid and the specified task space trajectory is within the reachable workspace of the manipulator. The angular relative velocities of links are defined as 1
x1 ¼ ½ 0 h_ 1 0 T ;
2
x2 ¼ ½ 0 0 h_ 2 T ;
3
x3 ¼ ½ 0 0 h_ 3 T :
Here h_ 1 ; h_ 2 and h_ 3 are the joint velocities of the first, second and third joints, respectively. The bond graph sub-model for robust overwhelming controller (ROC) is shown in Fig. 6, which is obtained from Fig. 2, by converting it into a sub-model form. Reference flow input for plant is supplied to this sub-model from main bond graph shown in Fig. 7. This sub-model is also provided with robot tip velocity as input from main bond graph and it gives correction efforts as output to sub-model CTRL. In Fig. 6 pad sub-model is used to avoid differential causality. Three robust overwhelming controller sub-models (one for each X, Y and Z direction) are used to control the space robot as shown in Fig. 7. A signal structure (CTRL) can be created using kinematic relations to convert correction efforts from ROC into correction torques to be supplied to different joints. For creation of this signal structure (CTRL), the tip velocity with respect to robot base frame {0} expressed in absolute frame {A} can be evaluated as
ð V 4 Þ ¼ A0 R0 ð0 V 4 Þ:
A 0
ð27Þ
k j
Here, ( Vi) denotes the velocity of the origin of ith frame as observed from jth frame and expressed in kth frame, sents the orientation of a frame {A} with respect to a frame {B}. The term 0(0V4) in Eq. (27) can be derived as
Fig. 6. Robust overwhelming controller sub-model (ROC).
B AR
repre-
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Fig. 7. Multi bond graph model of a free-floating space robot.
2 6 0 0 ð V 4Þ ¼ 4
38 9 _ ðc1 c2 l2 þ c1 c23 l3 Þ c1 c23 l3 > = < h1 > 7 0 ðs2 l2 þ s23 l3 Þ s23 l3 5 h_ 2 ; > ; :_ > ðc1 s2 l2 þ c1 s23 l3 Þ ðs1 c2 l2 þ s1 c23 l3 Þ s1 c23 l3 h3 ðs1 s2 l2 þ s1 s23 l3 Þ
ð28Þ
where ci = coshi, si = sinhi, cij = cos(hi + hj), sij = sin(hi + hj). One can write Eq. (28) in a compact form as 0 0
ð V 4 Þ ¼ ½Q ½ h_ 1
h_ 2
h_ 3 T ;
ð29Þ
where Q represents the transformation matrix in Eq. (28). The transformation A0 R is computed as A0 R ¼ AV R V0 R where
0
A VR
1 c/ch c/shsw s/cw c/shcw þ s/sw B C ¼ @ s/ch s/shsw þ c/cw s/shcw c/sw A; sh chsw chcw
ð30Þ
and V0 R is identity matrix. Here sw = sinw, cw = cosw, sh = sinh, ch = cosh, s/ = sin/, c/ = cos/. Thus, Eq. (27) can be written as
ð V 4 Þ ¼ ½A0 R½Q h_ 1
A 0
h_ 2
h_ 3
T
ð31Þ
:
Thus, the relationship between joint correction torques and correction efforts from controller can be expressed as
½ 1 n1
2
n2
3
T T n3 ¼ ½Q ½0A R½ F x
Fy
F z T ;
ð32Þ
where ini is the torque exerted on link i by link (i 1), expressed in terms of frame {i}. Fx, Fy and Fz are corrective efforts in X, Y and Z directions, respectively. The created sub-model (CTRL) for signal structure for conversion of correction efforts to correction torques is shown in Fig. 7. In next section we present a three DOF space robot with the robust trajectory controller.
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4. A free-floating space manipulator with robust controller Modeling of a space manipulator consists of modeling of the linear and the angular dynamics of the manipulator, as well as the vehicle on which it is mounted. In this section various sub-systems required for drawing bond graph model of the space manipulator and vehicle will be discussed. 4.1. Kinematic relations A signal structure has been made for transforming the vehicle angular velocity to Euler angle rates using the relation
3 2 3 2 32 xx 1 thsw thcw w_ 7 6_7 6 76 cw sw 54 xy 5 or ½ w_ h_ /_ T ¼ ½EV R½ xx 4 h 5 ¼ 40 xz 0 sw=ch cw=ch /_
xy xz T ;
where th ¼ tan h; ½EV R is transformation matrix in Eq. (33). The absolute angular velocity sively using the following relation [33]
^ iþ1 ; ð xiþ1 Þ ¼ iiþ1 Ri ðA xi Þ þ h_ iþ1 iþ1 U
iþ1 A
ð33Þ i+1 A
( xi+1) can be computed recur-
ð34Þ
b iþ1 is a unit vector representing the direction of the joint axis. This computation is implemented through the subwhere U model AVP (angular velocity propagation) of link [34] in bond graph shown in Fig. 7. This sub-model shown in Fig. 7 takes the angular velocity of previous link (ith) and the joint velocity between the previous and current link (i + 1)th as input and the angular velocity of the current link is given out after calculation. The joint velocity ðh_ iþ1 Þ can be about any one of the axes of the joint coordinate frame. Based on the axis of the joint rotation, one of the transformer moduli (Mt[0], Mt[1] or Mt[2]) conb iþ1 . nected to the joint velocity junction will be unity, and the remaining will be zero. This is decided by the vector iþ1 U The translation velocity of the tip of the (i + 1)th link can be computed from the previous link’s translation and angular velocities as follows: iþ1
A A
ð V iþ1 Þ ¼ A ðA V i Þ þ Ai R½i ðA xi Þ i ði Piþ1 Þ:
ð35Þ
Writing in matrix form
½A ðA V iþ1 Þ ¼ ½A ðA V i Þ þ ½Ai R½i ði Piþ1 Þ½i ðA xi Þ;
ð36Þ
where
2 2 3 0 0 6 6 7 i i ð Piþ1 Þ ¼ 4 li 5 and i ði Piþ1 Þ ¼ 4 0 0 li
0
li
3
7 0 0 5: 0 0
The translation velocity of the CM of the (i + 1)th link can be obtained by using i ði PGi Þ ¼ ½ 0 lGi 0 T in place of i(iPi+1) in Eq. (36). The bond graph model of the created sub-system TVP (translation velocity propagation) of link is shown in Fig. 7. This sub-model takes the angular velocity of current link and tip velocity of previous link as input and gives velocity of CM of current link and tip velocity of current link as output. The CM velocity of the spacecraft, links are obtained from the linear inertia of the spacecraft and links, respectively. The 1h_ i junctions represent the joint relative velocities. The integrator sub-models attached to 1h_ i junctions are observers of joint angles. Here pad sub-models are used to avoid differential causality. A rigid body has translational and angular motion. Thus its dynamics has two parts. The translational dynamics is modeled using Euler’s first law [35], and the angular dynamics using the Euler’s second law [35]. In order to model the Euler’s equation the gyrator loop commonly known as an Euler junction structure (EJS) is used. However in multi bond graph representation a modulated gyrator [36–39] can be used. Hence, it can be used to represent the rotational dynamics of the links of the space robot as well as that of the space vehicle. The Euler equations given by Eqs. (37)–(39) are used in the creation of this sub-model
_ x þ ðIzz Iyy Þxy xz ; Nx ¼ Ixx x
ð37Þ
_ y þ ðIxx Izz Þxz xx ; Ny ¼ Iyy x _ Nz ¼ Izz xz þ ðIyy Ixx Þxx xy :
ð38Þ ð39Þ
Here Nx, Ny and Nz represents the moments acting on the vehicle expressed in vehicle frame in x, y and z directions, respectively. The created EJS bond graph for a link is shown in Fig. 7. Multi bond graph representation is adopted in order to represent it in compact form. It is to be noted that Eqs. (37)–(39) have been arrived with the assumption that the reference frame used for computing the inertia matrix coincide with the principal axes of the body. If the reference frame and principal axis of body do not coincide then Eqs. (37)–(39) will also have product of inertia terms Ixy, Iyz, Izx [40], and then EJS cannot be created using Euler’s equations. In that case a separate sub-model has to be created which will take care of Ixy, Iyz, Izx terms in Euler’s equation.
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The complete multi bond graph model of the free-floating space robot is shown in Fig. 7. The tip velocity, which is provided to the ROC sub-model (shown in Fig. 6), is compared with the reference flow command, and the error in the flow is sent to the controller. The controller returns the correction effort, which is then provided to the robot with a high feed-forward gain. This correction effort is transformed back to joint coordinates as correction joint torques using sub-model CTRL and applied at the joints. Here the signal structure CTRL from the joint velocities to the tip velocities is created based on the forward kinematics of the manipulator. In Fig. 7, integrators are used as detector of displacement [41]. From the bond graph of Fig. 7, it is seen that the joint torque at the first joint is given by 1
n1 ¼ 1 N1 þ 12 R2 n2 þ 1 ð1 PG1 Þ 1A RA F 1 þ 1 ð1 P2 Þ 1A RA f2 :
ð40Þ
A
1
Here f1 is the force exerted on link 1 by link 0 (i.e. vehicle), expressed in terms of absolute frame, n1 is the torque exerted on link 1 by link 0 (i.e. vehicle), expressed in terms of frame {1}, AF1 is the resultant of all the external forces acting on the link, and 1N1 are the resultant of all the external moments acting on the link. Similarly from the bond graph the force balance for link 1 can be given as A
f1 ¼ A F 1 þ A f2 :
ð41Þ
Here Af2 is the force exerted on link 2 by link 1, expressed in terms of absolute frame. Eqs. (40) and (41) are similar to the one which can be obtained by considering force balance and moment balance equations based on the free body diagram of a typical link [33]. Equations similar to Eqs. (40) and (41) can be written for other links also. In next section we present a numerical simulation of three DOF space robot with robust trajectory controller. 5. Numerical simulation The link parameters selected for the simulation of space robot are shown in Table 1. The overwhelming controller parameters are taken as Mc = 1.0 kg, Kc = 1000 N/m, Rc = 100 Ns/m and lH = 10.0. Joint resistance on each joint can be assumed as 0.1 Ns/m. The motor inertia for joint 1 is assumed as 0.026 kg m2. For joint 2 and 3, it is assumed as 0.001 kg m2. Reference velocity command in X, Y and Z direction in absolute frame, located at initial position of CM of space vehicle is taken as, X_ ref ¼ R x sinðxtÞ; Y_ ref ¼ A sinð2xtÞ and Z_ ref ¼ Rx cosðxtÞ, respectively. Here, R is the radius of the reference circle in X–Z plane; A is the amplitude of velocity in Y-direction. It is a circular trajectory in X–Z plane and parabolic trajectory in X–Y, and Y–Z plane. Fig. 8 shows the initial configuration of the space robot. At the beginning of the simulation, the overwhelmer initial position is initialized to robot tip position. In other words, the initial error between the controller and the robot is zero. The simulation is carried out for 25 s with R as 0.3 m and A as 0.01 m. Fig. 9a shows a three-dimensional plot of reference and actual tip movement. Fig. 9b shows the variation of Xtip trajectory along with Ztip trajectory. From these figures it is clear that tip is effectively dragged along the reference trajectory. Fig. 9c shows the root mean square (RMS) value of tip trajectory error. The error is taken as the difference between reference and
Table 1 Parameters used for modeling of space robot Parameters
Mass (kg)
Ixx (kg m2)
Iyy (kg m2)
Izz (kg m2)
Length (li) (m)
Length ðlGi Þ (m)
hi (rad)
Link 1 Link 2 Link 3 Vehicle
6.13 15.69 11.76 200
0.03 0.2153 0.0929 40.0
0.025 0.0126 0.0094 40.0
0.03 0.2153 0.0929 40.0
0.1 0.4 0.3 –
0.05 0.2 0.15
0
Fig. 8. Initial configuration of space robot.
p/2 p/2 –
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Fig. 9. (a): Xtip versus Ytip versus Ztip; (b): Xtip versus Ztip; (c): RMS value of tip trajectory errors; (d): XCM versus YCM versus ZCM; (e): XCM versus ZCM and (f): Euler angles w versus / versus h.
actual tip position in mm. Initial error is more than the later part of the error because the tip acquires a velocity from the rest position. The plots of the motion of the CM of spacecraft in three-dimensions are shown in Fig. 9d. Fig. 9e shows the variation of ZCM versus XCM. The plot shows a continuous drift in the vehicle CM location in X–Z plane due to conservation of the
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angular momentum of the free-floating space robot, i.e., due to the non-integrability of the angular momentum equation. Fig. 9f shows the three-dimensional plot for Euler angles / versus w versus h. It clearly shows that even after the tip reaches its initial position after one cycle, the Euler angles do not close. This is due to the non-holonomicity of the system. From the simulation of space manipulator model one can get an estimate of drift in the center-of-mass of the spacecraft. 6. Conclusions In this paper an overwhelming control strategy for free-floating space robots is presented. The strategy consists of a set of linear time invariant controllers coupled to the robotic manipulator through a set of high feed-forward gains, and suitably defined feedback paths. The driving efforts to the manipulator actuators are decided by the linear controller’s parameters and the coupling between the robot and the controller ensures that the robotic manipulator is ‘‘dragged” along the trajectory of the controller model. The controller is robust because it is independent of inertial parameters of the system. The three-link space manipulator is modeled, simulated and the robustness of the controller has been demonstrated. The simulation of the space manipulator model provides quantitative information on the drift of the spacecraft due to non-holonomic nature of the system. This will be useful for developing new path planning methods for minimizing the drift. References [1] Yoshihiko Nakamura, Ranjan Mukherjee, Non-holonomic path planning of space robots via a bidirectional approach, IEEE Transactions on Robotics and Automation 7 (4) (1991) 500–514. [2] Z. Vafa, S. Dubowsky, On the dynamics of manipulators in space using the virtual manipulator approach, in: Proceedings of 1987 IEEE International Conference on Robotics and Automation, 1987, pp. 579–585. [3] E. Papadopoulos, S. Dubowsky, On the nature of control algorithms for free-floating space manipulators, IEEE Transactions on Robotics and Automation 7 (6) (1991) 750–758. [4] Y. Watanabe, Y. Nakamura, A space robot of the center-of-mass invariant structure, in: Proceedings of the IEEE/RSJ International Conference on Intelligent Robots and Systems, Victoria, BC, Canada, 1998, pp. 1370–1375. [5] B. Liang, Y. Xu, M. Bergerman, Mapping a space manipulator to a dynamically equivalent manipulator, ASME Transactions of the Journal of Dynamic Systems, Measurement, and Control 120 (1998) 1–7. [6] A. Jain, G. Rodriguez, Base-invariant symmetric dynamics of free-flying manipulators, IEEE Transactions on Robotics and Automation 11 (4) (1995) 585–597. [7] Yoji Umetani, Kazuya Yoshida, Resolved motion rate control of space manipulators with generalized jacobian matrix, IEEE Transactions on Robotics and Automation 5 (3) (1989) 303–314. [8] K. Yoshida, Engineering test satellite VII flight experiments for space robot dynamics and control: theories on laboratory test beds ten years ago, now in orbit, International Journal of Robotic Research 22 (5) (2003) 321–335. [9] Y. Yokokohji, T. Toyoshima, T. Yoshikawa, Efficient computational algorithms for trajectory control of free-flying space robots with multiple arms, IEEE Transactions on Robotics and Automation 9 (5) (1993) 571–579. [10] Panfeng Huang, Yangsheng Xu, Bin Liang, Tracking trajectory planning of space manipulator for capturing operation, International Journal of Advanced Robotic Systems 3 (3) (2006) 211–218. [11] Ranjan Mukherjee, Masayuki Kamon, Almost smooth time-invariant control of planar space multibody systems, IEEE Transactions on Robotics and Automation 15 (2) (1999) 268–280. [12] J. Franch, S.K. Agrawal, A. Fattah, Design of differentially flat planar space robots: a step forward in their planning and control, in: Proceedings of the 2003 IEEE/RSJ International Conference on Intelligent Robots and Systems, Las Vegas, Nevada, 2003, pp. 3053–3058. [13] Li Chen, Xiaoteng Tang, Optimal motion planning of attitude control for space robot system with dual-arms, in: Proceedings of the Sixth World Congress on Intelligent Control and Automation, Dalian, China, 2006, pp. 8858–8861. [14] H.L. Alexander, R.H. Cannon Jr., An extended operational space control algorithm for satellite manipulators, The Journal of the Astronautical Sciences 38 (4) (1990) 473–486. [15] S. Dubowsky, M.A. Torres, Path planning for space manipulators to minimize spacecraft attitude disturbances, in: Proceedings of IEEE International Conference on Robotics and Automation, Sacramento, CA, vol. 3, 1991, pp. 2522–2528. [16] G. Legnani, B. Zappa, R. Adamini, F. Casolo, A contribution to the dynamics of free-flying space manipulators, Mechanism and Machine Theory 34 (1999) 359–372. [17] K. Yoshida, D.N. Nenchev, M. Uchiyama, Moving base robotics and reaction management control, in: G. Giralt, G. Hirzinger (Eds.), Robotics Research: The Seventh International Symposium, Springer Verlag, 1996, pp. 101–109. [18] M. Oda, Y. Ohkami, Coordinated control of spacecraft attitude and space manipulators, Control Engineering Practice 5 (1) (1997) 11–21. [19] S. Taniwaki, S. Matunaga, S. Tsurumi, Y. Ohkami, Coordinated control of a satellite-mounted manipulator with consideration of payload flexibility, Control Engineering Practice 9 (2001) 207–215. [20] E. Papadopoulos, S. Dubowsky, Coordinated manipulator/spacecraft motion control for space robotic systems, in: Proceedings of the IEEE International Conference on Robotics and Automation, Sacramento, CA, 1991, pp. 1696–1701. [21] M. Marchesi, F. Angrilli, R. Venezia, Coordinated control for free-flyer space robots, in: IEEE International Conference on Systems, Man, and Cybernetics, vol. 5, 2000, pp. 3550–3555. [22] K. Yoshida, K. Hashizume, S. Abiko, Zero reaction maneuver: validation with ETS-VII space robot and extension to kinematically redundant arm, in: Proceedings of the 2001 IEEE International Conference on Robotics and Automation, 2001, pp. 441–446. [23] Woo-Keun Yoon, Toshihiko Goshozono, Hiroshi Kawabe, Masahiro Kinami, Yuichi Tsumaki, Masaru Uchiyama, Mitsushige Oda, Toshitsugu Doi, Modelbased space robot teleoperation of ETS-VII manipulator, IEEE Transactions on Robotics and Automation 20 (3) (2004) 602–612. [24] Bruno Zappa, Giovanni Legnani, Riccardo Adamini, Path planning of free-flying space manipulators: an exact solution for polar robots, Mechanism and Machine Theory 40 (2005) 806–820. [25] Carlo Menon, S. Busolo, S. Cocuzza, A. Aboudan, A. Bulgarelli, C. Bettanini, M. Marchesi, F. Angrilli, Issues and solutions for testing free-flying robots, Acta Astronautica 60 (2007) 957–965. [26] Feng Wang, Fuchun Sun, Huaping Liu, Space robot modeling and control considering the effect of orbital mechanics, in: First International Symposium on Systems and Control in Aerospace and Astronautics, 2006, pp. 193–198. [27] Igor Belousov, Claudia Estevès, Jean-Paul Laumond, Etienne Ferré, Motion planning for the large space manipulators with complicated dynamics, in: IEEE/RSJ International Conference on Intelligent Robots and Systems, 2005, pp. 2160–2166. [28] A.K. Ghosh, Dynamics and robust control of robotic systems: a bond graph approach, Ph.D. thesis, Department of Mechanical Engineering, Indian Institute of Technology, Kharagpur, 1990.
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[29] C.S. Kumar, A. Mukherjee, Robust control of a robot manipulator on a flexible foundation, in: Proceedings of the Fourth International Conference on CAD/CAM, Robotics & Factories of Future, 1989, TATA McGraw-Hill, New Delhi, India, pp. 727–742. [30] Pushparaj Mani Pathak, Amalendu Mukherjee, Anirvan Dasgupta, Impedance control of space robots using passive degrees of freedom in controller domain, Transactions of ASME, Journal of Dynamic Systems, Measurement, and Control 127 (2005) 564–578. [31] A. Mukherjee, R. Karmakar, A.K. Samantaray, Bond Graph in Modeling, Simulation and Fault Identification. New Delhi: I.K. International Publishing House Pvt. Ltd., 2006, second ed., CRC press, 2006. [32] A.K. Samantaray, A. Mukherjee, Users Manual of SYMBOLS Shakti,
Contents lists available at ScienceDirect
Simulation Modelling Practice and Theory journal homepage: www.elsevier.com/locate/simpat
A scheme for robust trajectory control of space robots Pushparaj Mani Pathak a,*, R. Prasanth Kumar b, Amalendu Mukherjee b, Anirvan Dasgupta b a b
Mechanical and Industrial Engineering Department, Indian Institute of Technology, Roorkee 247667, India Mechanical Engineering Department, Indian Institute of Technology, Kharagpur 721302, India
a r t i c l e
i n f o
Article history: Received 11 November 2006 Received in revised form 16 June 2008 Accepted 28 June 2008 Available online 8 July 2008
Keywords: Free-floating Space robotics Robust control Bond graph modeling Non-holonomic system
a b s t r a c t This paper presents a scheme for robust trajectory control of free-floating space robots. The idea is based on the overwhelming robust trajectory control of a ground robot on a flexible foundation and robust foundation disturbance compensation presented elsewhere. No external jets/thrusters are required or used in the scheme. An example of a three-link robot mounted on a free-floating space platform is considered for demonstrating the efficacy of the control scheme. Bond graph technique has been used for the purpose of modeling and simulation. Robustness of the control scheme is guaranteed since the controller does not require the knowledge of the manipulator parameters. Ó 2008 Elsevier B.V. All rights reserved.
1. Introduction As the number of space structures and satellites in orbit increases, the need for their maintenance increases. Extending the life of such space systems, and therefore reducing the associated costs, will require extensive inspection, assembly and maintenance capabilities in orbit. The cost of human life support facilities, the limited time available for astronaut’s extra vehicular activity (EVA), and high risks involved make space robotic devices indispensable for space operations. The typical tasks that can be performed by space robots are deploying or assembling space platforms and space stations, deploying large antennas or solar power stations, and service and maintenance of satellites. All these applications require precise manipulator motion control. In a free-floating space robotic system, the spacecraft’s position and the attitude are not actively controlled using external jets/thrusters during manipulator activity. This conserves attitude control fuel, and hence lengthens the life of the system. However, due to the conserved linear and the angular momenta, the spacecraft moves freely in response to the dynamical disturbances caused by the manipulator’s motion. Such disturbances may result in significant deviation of the end-effector from the desired trajectory, and are particularly significant when the manipulator is holding a heavy payload. Moreover, the angular momentum conservation constraints are non-integrable rendering the system non-holonomic. Hence, it is not possible to cancel this base disturbance using static state feedback. This complicates the motion planning and control issue further [1]. The concepts of free-flying and free-floating robots were introduced in the early 1980s. Vafa and Dubowsky [2] introduced the concept of virtual manipulator (VM) for the analytical modeling of space manipulators. Papadopoulos and Dubowsky [3] suggested that any control algorithm that can be used for fixed-based manipulators can also be used in the control of free-floating space manipulators systems, with the additional conditions of estimating or measuring a spacecraft’s
* Corresponding author. Tel.: +91 1332 285608; fax: +91 1332 285665. E-mail addresses: [email protected] (P.M. Pathak), rpkumar@scientific.net (R.P. Kumar), [email protected] (A. Mukherjee), [email protected] (A. Dasgupta). 1569-190X/$ - see front matter Ó 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.simpat.2008.06.011
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orientation, and of avoiding dynamic singularities. Watanabe and Nakamura [4] proposed a free-flying space robot having a special kinematic structure and mass distribution such that the CM of the total system remains fixed to the base body regardless of the manipulators configuration. Liang et al. [5] mapped a free-floating space manipulator to a conventional fixed base manipulator. In the mapping, the dynamic and kinematic properties of the manipulator are preserved. Jain and Rodriguez [6] analyzed the relationship between base invariance symmetry (i.e. freedom in choosing any link as base body) and the dynamical equations for free-flying manipulators. Umetani and Yoshida [7] used the angular momentum conservation law to derive a new Jacobian matrix in generalized form for a space robotic arm. Using this new matrix, they developed a control method for the space manipulator based on the resolved motion rate control concept. Yoshida [8] demonstrated the concept of generalized Jacobian in ETS-VII. Yokokohji et al. [9] proposed an efficient computation algorithm for the trajectory control of multi arm free-flying space robot using the generalized Jacobian matrix technique. Huang et al. [10] presented a trajectory planning method of space manipulator that can track, approach and catch the uncontrolled spinning satellite (USS) in free-floating situation. They planed a spiral ascending trajectory for space manipulator to approach towards USS in Cartesian space. Nakamura and Mukherjee [1] proposed an approach for non-holonomic path planning to obtain a specified base attitude and an arm posture starting from an arbitrary initial state. Mukherjee and Kamon [11] studied the configuration control problem in free-flying planar space robots. Franch et al. [12] studied non-holonomic rate constraints for free-floating planar open chain robots to determine the design conditions under which the system exhibits differential flatness. Under these design conditions, they used the property of flatness for trajectory planning and feedback control under perturbations in the initial state. Chen [13] presented the optimal non-holonomic motion planning of free-floating space robot system with dual-arms. Based on the linear and angular momentum conservations of the system, the system state equations for control design are established. Alexander and Cannon [14] presented a control algorithm for control of the free-floating space robot without the strict attitude or position control of the robot body. The concept of disturbance map, and its use in non-holonomic motion planning of space robots with minimum attitude change was proposed by Dubowsky and Torres [15], and Legnani et al. [16]. Yoshida et al. [17] derived the conditions for manipulation with zero base disturbances, and determined the joint velocities from those conditions. In the coordinated control strategy [18], the deviation of the base due to the manipulator reaction is corrected by the reaction wheels mounted on the space vehicle. Taniwaki et al. [19] worked on coordinated control of a satellite-mounted manipulator handling a flexible payload. Papadopoulos and Dubowsky [20] studied the coordinated control of a space manipulator mounted on a spacecraft. Marchesi et al. [21] method on coordinated control, exploits both coupled dynamics of the spacecraft–manipulator system and non-holonomic constraint due to angular momentum. Yoshida et al. [22] presented the post-flight analysis with ETS-VII of Reaction Null-Space-based reaction-less manipulation or Zero Reaction Maneuver. To solve the communication time delay in the space robot teleoperation, Yoon et al. [23] introduced a new control method that modified model-based teleoperation system against the time delay, and this method was applied to the ETS-VII manipulator teleoperation in orbit. Zappa et al. [24] studied the case of planar polar manipulators, analyzed the peculiarities of their robot arm configurations. Menon et al. [25] investigated test-beds for free-flying robots with extended arms, emphasizing pros and cons of the different test-beds. Wang et al. [26] studied the dynamics modeling and control simulation of space robot with the dynamic effects due to orbital mechanics. Belousov et al. [27] presented motion planning algorithms for the large space robot manipulators with complicated dynamic behaviour. They proposed two ‘‘two-stage” iterative algorithms, which provide collision-free robot motion taking into account robot’s dynamics. Based on the insights developed from bond graph modeling, Ghosh [28] developed a robust overwhelming controller for the robotic manipulator, which does not require the knowledge of the robot parameters and the payload. Kumar and Mukherjee [29] further developed the overwhelming control strategy and applied it for robust trajectory control of a two-link planar manipulator on a flexible foundation. In this paper the basic ideas of the overwhelming controller [30] is first reviewed for the sake of completeness then a scheme for robust trajectory control of space robots is proposed. A numerical example of the three-link manipulator on a free-floating space platform is presented to demonstrate the efficacy of the proposed control strategy. The complete dynamics is modeled using bond graphs [31]. For the purpose of modeling and simulation, the bond graph package SYMBOLS Shakti [32] has been used. This software generates the equation of motion symbolically, which are then used for simulation.
2. Robust trajectory controller 2.1. The robust overwhelming controller The overwhelming control strategy [28–30] consists of a set of linear time invariant controllers coupled to the robotic manipulator through a set of high feed-forward gains and suitably defined feedbacks. The driving efforts to the manipulator actuator are decided by the linear controller’s parameters and the coupling between the robot and the controller ensures that the robotic manipulator is ‘‘dragged” along the trajectory of the controller model. Since the controller is linear time invariant, linear control theory can be used in the design of such systems. The high feed-forward gain renders the non-linear dynamics of the robot small perturbations on that of the linear controller and hence the dynamics of the robotic manipulator closely resembles that of the controller. The error in tracking can be reduced to any extent by the proper
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Fig. 1. Mechanical equivalent of overwhelming controller.
Fig. 2. Bond graph of realizable overwhelming controller.
choice of controller parameters and by increasing the feed forward gains within the bandwidth of the control system and the capacity of the actuators. Above discussed strategy can be illustrated by considering a system as shown in Fig. 1. Consider that a single DOF plant is to be driven by a similar single DOF controller through a mass less lever (amplifier). Plant parameters mp (mass), kp (stiffness), rp (resistance) may be undetermined functions of time or states whereas controller parameters Mc (mass), Kc (stiffness) and Rc (resistance) are constants. A force eref (t) drives the controller to generate the desired motion of the plant. The bond graph model of the system is shown in Fig. 2. It is to be noted that the I element in overwhelmer is differentially causalled. This can be removed by introducing an artificial compliance or pad. This consists of a spring and a damper and has been discussed further in detail later in this paper. The controllers are usually implemented by using amplifiers and the feedback system. Hence the physical entity of lever of Fig. 1 can be realized by resolving the transformer element (representing the lever) into two separate elements. One as modulated source of effort with modulus lH. The other as modulated source of flow with modulus as 1 as shown in Fig. 2. The modulated source of flow with modulus as 1 is the feedback path of the flow information with no power associated with it. The modulated source of effort with high gain (lH) is an effort amplifier. The actuation power at the output port of this amplifier comes from a tank system sustaining it. This way of splitting the transformer will put the controller in the digital domain, as the controller expressions can be in the computer. The effort signals can be passed from computer to amplifier whereas flow feedback signal can be received by the computer. The transfer function between robot (plant) output flow and force input can be evaluated from the bond graph shown in Fig. 2. From this figure it is seen that,
e5 ðtÞ ¼ e1 ðtÞ e2 ðtÞ e4 ðtÞ e3 ðtÞ or e5 ðtÞ ¼ eref ðtÞ M c
d fc ðtÞ Rc fc ðtÞ K c dt
Z
fc ðtÞdt:
ð1Þ
Here eref (t) is the reference force input, fc(t) is the common flow at 1c junction in Fig. 2. Mc, Rc and Kc are the inertia (differential gain), resistance (proportional gain) and stiffness (integral gain), respectively, of the overwhelming controller. If s is the Laplace variable, taking Laplace transform of both sides of Eq. (1) gives
E5 ðsÞ ¼ Eref ðsÞ M c sF c ðsÞ Rc F c ðsÞ K c F c ðsÞ=s or E5 ðsÞ ¼ Eref ðsÞ F c ðsÞðMc s2 þ Rc s þ K c Þ=s; or E5 ðsÞ ¼ Eref ðsÞ þ F c ðsÞCðsÞ;
ð2Þ
CðsÞ ¼ ðMc s2 þ Rc s þ K c Þ=s:
ð3Þ
where
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Similarly from Fig. 2,
e7 ðtÞ ¼ e15 ðtÞ e9 ðtÞ e8 ðtÞ; mp
d fp ðtÞ ¼ lH e5 ðtÞ r p fp ðtÞ kp dt
Z
fp ðtÞdt:
ð4Þ
Here mp, rp and kp are the inertia, resistance and stiffness, respectively, of the plant, lH is high feed-forward gain and fp is the flow at 1p junction in Fig. 2. Taking Laplace transform of both sides of Eq. (4) gives
mp sF p ðsÞ ¼ lH E5 ðsÞ r p F p ðsÞ kp F p ðsÞ=s; F p ðsÞðmp s2 þ r p s þ kp Þ=s ¼ lH E5 ðsÞ; F p ðsÞ=RðsÞ ¼ lH E5 ðsÞ;
ð5Þ
RðsÞ ¼ s=½mp s2 þ r p s þ kp :
ð6Þ
where
Substituting for E5(s) in Eq. (5) from Eq. (2), we get,
F p ðsÞ=RðsÞ ¼ lH ½Eref ðsÞ þ F c ðsÞCðsÞ:
ð7Þ
But as seen from Fig. 2,
F c ðsÞ ¼ F p ðsÞ:
ð8Þ
Substituting in Eq. (7) from Eq. (8) gives the transfer function between robot output flow Fp(s) and force input Eref(s) as,
F p ðsÞ=Eref ðsÞ ¼ lH RðsÞ=½1 lH RðsÞCðsÞ:
ð9Þ
When lH 1, one obtains from Eq. (9) the overall transfer function as,
F p ðsÞ=Eref ðsÞ ffi 1=CðsÞ ¼ s=ðMc s2 þ Rc s þ K c Þ:
ð10Þ
In order to control the plant the command eref(t) is taken of the form,
eref ðtÞ ¼ e1 ðtÞ ¼ M c €xref þ Rc x_ ref þ K c xref ;
ð11Þ
so that, it produces the desired motion xref(t) of the plant. Therefore,
Eref ðsÞ ¼ s2 Mc X ref ðsÞ þ sRc X ref ðsÞ þ K c X ref ðsÞ or X ref ðsÞ=Eref ðsÞ ¼ 1=ðM c s2 þ Rc s þ K c Þ:
ð12Þ
Since Xref(s) = Fref(s)/s, Eq. (12) can be written as,
F ref ðsÞ=Eref ðsÞ ¼ s=ðM c s2 þ Rc s þ K c Þ:
ð13Þ
From Eqs. (13) and (10) we get,
F p ðsÞ=F ref ðsÞ ¼ 1;
ð14Þ
i.e., the plant follows the command. The bond graph of Fig. 2 can be modified to give reference input as flow input. Here, the non-causal form of the controller is only artificial and is a consequence of differential causality in the overwhelmer. The controller can be brought to causal form by introducing a pad [30]. Pads are lumped flexibilities [31]. Choosing high values of damping and stiffness can reduce their dynamic effects. The effects of this are usually in the form of a high frequency vibration in the system at the onset of simulation. These frequencies however decay soon due to heavy damping. Thus an overwhelming controller can be designed which takes care of complex and highly non-linear dynamics of robot and provides robust trajectory control. The controller is fed with the error in trajectory and controller in turn provides the corrective torques to be applied in the joints of robot. By considering the controller in the world coordinate frame external disturbances and desired input can be directly conveyed in the world coordinate frame. Conventionally input forces from controller to joints are determined by the desired trajectory of the system and the inverse kinematics of the given robot. Hence the inverse kinematics transformations can be avoided by considering the controller in the workshop coordinate frame rather than joint coordinate frame. Under these conditions the velocities can be transformed from the joint coordinates to the world coordinate using the forward kinematics Jacobian. Next section will consider the behaviour of the overwhelming trajectory controller when the robot is placed on a flexible foundation. 2.2. Robust overwhelming controller for ground manipulator with foundation compensation The overwhelming control strategy for a ground manipulator with some modification has been applied to a ground robot on a flexible foundation by Kumar and Mukherjee [29]. The principle of operation can be explained through an example of a single DOF robot on a flexible foundation as shown in Fig. 3. To incorporate the foundation disturbance in the world
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Fig. 3. Single DOF robot on flexible foundation.
Fig. 4. Bond graph model for system shown in Fig. 3.
coordinates, the foundation velocity is sensed and feedback to the controller. The bond graph model of the system shown in Fig. 3 is presented in Fig. 4, where the flow signal from foundation is modulated by magnitude a and this signal is supplied as modulated source of flow through bond 13 to controller. Essentially a shows the feedback compensation. System shown in Fig. 4 is a closed loop system subjected to two inputs i.e., reference inputs Eref(s) and foundation disturbance Edis(s). When two inputs are present in a linear system each input can be treated independently of the other and the outputs corresponding to each input alone can be added to give the complete output. From bond graph shown in Fig. 4 we observe that
E15 ðsÞ ¼ Eref ðsÞ þ F c ðsÞCðsÞ;
ð15Þ
CðsÞ ¼ ðMc s2 þ Rc s þ K c Þ=s:
ð16Þ
where
Since E11(s) = lHE8(s) = lHE15(s), we get
E15 ðsÞ ¼ E11 ðsÞ=lH and since
F p ðsÞ=E11 ðsÞ ¼ PðsÞ ¼ 1=mp s; F p ðsÞ E15 ðsÞ ¼ ; lH PðsÞ
ð17Þ ð18Þ
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also,
F c ðsÞ ¼ F 15 ðsÞ ¼ ½F 20 ðsÞ þ F 13 ðsÞ; ¼ ½F 11 ðsÞ aF 19 ðsÞ; ¼ ½fF 1 ðsÞ F 18 ðsÞg aF f ðsÞ; ¼ ½F p ðsÞ þ F f ðsÞ aF f ðsÞ or F c ðsÞ ¼ ½F p ðsÞ F f ðsÞð1 aÞ:
ð19Þ
Substituting for E15(s) and Fc(s) from Eqs. (18) and (19) into Eq. (15) we get,
F p ðsÞ
lH PðsÞ
¼ Eref ðsÞ þ ½F p ðsÞ F f ðsÞð1 aÞCðsÞ:
ð20Þ
At 1f junction,
e3 ðtÞ ¼ e5 ðtÞ e10 ðtÞ e12 ðtÞ e18 ðtÞ; Z d Mf ff ¼ edis ðtÞ Rff ðtÞ K f ff ðtÞdt e18 ðtÞ: dt
ð21Þ
In Eq. (21) Mf (mass), Rf (damping), Kf (stiffness) are foundation parameters. Taking Laplace transform of both sides of Eq. (21) gives
sMf F f ðsÞ ¼ Edis ðsÞ Rf F f ðsÞ K f F f ðsÞ=s E18 ðsÞ; F f ðsÞ½s2 M f þ sRf þ K f =s þ E11 ðsÞ ¼ Edis ðsÞ; F f ðsÞ=FðsÞ þ F p ðsÞ=PðsÞ ¼ Edis ðsÞ;
ð22Þ
FðsÞ ¼ s=ðM f s2 þ Rf s þ K f Þ:
ð23Þ
where
If we take robot motion due to reference velocity command only i.e., Edis(s) to be zero, then finding Ff from Eq. (22) and substituting its value in Eq. (20) gives us the transfer function for the system between the force input to the robot velocity as,
F p ðsÞ lH PðsÞ ¼ : Eref ðsÞ ½1 lH PðsÞCðsÞ þ ð1 aÞlH CðsÞFðsÞ
ð24Þ
For the high feed-forward gains (i.e., lH 1) and a = 1, and on using Eqs. (16), (17) and (23) in Eq. (24) one obtains
F p ðsÞ=Eref ðsÞ ffi 1=CðsÞ ffi s=ðMc s2 þ Rc s þ K c Þ:
ð25Þ
To evaluate the effect of the foundation disturbance Edis(s), we assume that the system is at rest with Eref as zero. Then putting Eref as zero in Eq. (20) and substituting the value of Ff from Eq. (22) into Eq. (20), the transfer function between the robot end-effector velocity Fp(s), and the disturbance force Edis(s) for such a system can be derived as
F p ðsÞ ð1 aÞlH PðsÞCðsÞFðsÞ ¼ : Edis ðsÞ ½1 lH PðsÞCðsÞ þ ð1 aÞlH CðsÞFðsÞ
ð26Þ
The transfer functions in Eqs. (24) and (26) show that if a = 1, the foundation has no effect on the trajectory of the robot under the chosen feedback scheme. Thus the external disturbances can be completely compensated even in the absence of the knowledge of foundation parameters. A foundation disturbance sensor and a compensating feedback have been introduced to achieve this. In the above control scheme, the effort command to the controller has to be calculated based on the given trajectory and the controller parameters as evident from Eq. (25). By making a slight modification to the scheme the flow reference input can be directly used instead of the effort reference input. The scheme is extendable to multi-degree of freedom robotic systems and can be used effectively to track a desired endeffector trajectory. However there are practical implications of increasing the feed-forward gains as increasing it to high value also increases the noise. This noise can be reduced by low pass filtering. Low pass filtering may introduce sluggishness in the system, so there has to be a trade off between high gain and bandwidth of filtering. The scheme for robust control of ground manipulator with foundation compensation presented in this section can be extended for space robots. The only difference between a ground robot on flexible foundation and a space robot is that, in case of the space robot base is not anchored. So if we remove the foundation spring and damper from the robot on flexible foundation, we get a robot on floating base i.e. a space robot. In case of such robots trajectory control task is challenging due to floating base. Next section will present a control strategy for the robust trajectory control of space robot which will be based on base motion compensation by feeding the velocity of the space vehicle to the robust controller. 3. Robust controller for a space robot To illustrate the control strategy for the space robot, let us consider a three DOF free-floating space manipulator consisting of a serial robot mounted on a satellite base as shown in Fig. 5. Joint rotation (hi) of manipulator and Euler angle of the
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Fig. 5. A free-floating space manipulator.
base (/, h, w) in 3-2-1 convention are used as generalized coordinates. Links are numbered starting with zero from the base to the last link. All joints are assumed to be revolute. The joint between links i and i + 1 is numbered as i + 1. An absolute frame {A} is considered as shown in Fig. 5. The vehicle is denoted by frame {V} located at the CM of the vehicle. The manipulator base is mounted on the vehicle at a position (rx, ry, rz) from the frame {V}. A frame {0} is attached on the vehicle to represent the manipulator base location on the vehicle. Frame {1} is attached on the link 1 at its base. Frame {1} has relative motion with respect to frame {0} about y-axis. Frames {2} and {3} are attached at the base of second and third link. They represent the joint axes for the second and third joint. They have rotation about their z-axes. The moments of inertia are defined about the body-fixed principal coordinate system. Without loss of generality it is assumed that links of the manipulator are rigid and the specified task space trajectory is within the reachable workspace of the manipulator. The angular relative velocities of links are defined as 1
x1 ¼ ½ 0 h_ 1 0 T ;
2
x2 ¼ ½ 0 0 h_ 2 T ;
3
x3 ¼ ½ 0 0 h_ 3 T :
Here h_ 1 ; h_ 2 and h_ 3 are the joint velocities of the first, second and third joints, respectively. The bond graph sub-model for robust overwhelming controller (ROC) is shown in Fig. 6, which is obtained from Fig. 2, by converting it into a sub-model form. Reference flow input for plant is supplied to this sub-model from main bond graph shown in Fig. 7. This sub-model is also provided with robot tip velocity as input from main bond graph and it gives correction efforts as output to sub-model CTRL. In Fig. 6 pad sub-model is used to avoid differential causality. Three robust overwhelming controller sub-models (one for each X, Y and Z direction) are used to control the space robot as shown in Fig. 7. A signal structure (CTRL) can be created using kinematic relations to convert correction efforts from ROC into correction torques to be supplied to different joints. For creation of this signal structure (CTRL), the tip velocity with respect to robot base frame {0} expressed in absolute frame {A} can be evaluated as
ð V 4 Þ ¼ A0 R0 ð0 V 4 Þ:
A 0
ð27Þ
k j
Here, ( Vi) denotes the velocity of the origin of ith frame as observed from jth frame and expressed in kth frame, sents the orientation of a frame {A} with respect to a frame {B}. The term 0(0V4) in Eq. (27) can be derived as
Fig. 6. Robust overwhelming controller sub-model (ROC).
B AR
repre-
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Fig. 7. Multi bond graph model of a free-floating space robot.
2 6 0 0 ð V 4Þ ¼ 4
38 9 _ ðc1 c2 l2 þ c1 c23 l3 Þ c1 c23 l3 > = < h1 > 7 0 ðs2 l2 þ s23 l3 Þ s23 l3 5 h_ 2 ; > ; :_ > ðc1 s2 l2 þ c1 s23 l3 Þ ðs1 c2 l2 þ s1 c23 l3 Þ s1 c23 l3 h3 ðs1 s2 l2 þ s1 s23 l3 Þ
ð28Þ
where ci = coshi, si = sinhi, cij = cos(hi + hj), sij = sin(hi + hj). One can write Eq. (28) in a compact form as 0 0
ð V 4 Þ ¼ ½Q ½ h_ 1
h_ 2
h_ 3 T ;
ð29Þ
where Q represents the transformation matrix in Eq. (28). The transformation A0 R is computed as A0 R ¼ AV R V0 R where
0
A VR
1 c/ch c/shsw s/cw c/shcw þ s/sw B C ¼ @ s/ch s/shsw þ c/cw s/shcw c/sw A; sh chsw chcw
ð30Þ
and V0 R is identity matrix. Here sw = sinw, cw = cosw, sh = sinh, ch = cosh, s/ = sin/, c/ = cos/. Thus, Eq. (27) can be written as
ð V 4 Þ ¼ ½A0 R½Q h_ 1
A 0
h_ 2
h_ 3
T
ð31Þ
:
Thus, the relationship between joint correction torques and correction efforts from controller can be expressed as
½ 1 n1
2
n2
3
T T n3 ¼ ½Q ½0A R½ F x
Fy
F z T ;
ð32Þ
where ini is the torque exerted on link i by link (i 1), expressed in terms of frame {i}. Fx, Fy and Fz are corrective efforts in X, Y and Z directions, respectively. The created sub-model (CTRL) for signal structure for conversion of correction efforts to correction torques is shown in Fig. 7. In next section we present a three DOF space robot with the robust trajectory controller.
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4. A free-floating space manipulator with robust controller Modeling of a space manipulator consists of modeling of the linear and the angular dynamics of the manipulator, as well as the vehicle on which it is mounted. In this section various sub-systems required for drawing bond graph model of the space manipulator and vehicle will be discussed. 4.1. Kinematic relations A signal structure has been made for transforming the vehicle angular velocity to Euler angle rates using the relation
3 2 3 2 32 xx 1 thsw thcw w_ 7 6_7 6 76 cw sw 54 xy 5 or ½ w_ h_ /_ T ¼ ½EV R½ xx 4 h 5 ¼ 40 xz 0 sw=ch cw=ch /_
xy xz T ;
where th ¼ tan h; ½EV R is transformation matrix in Eq. (33). The absolute angular velocity sively using the following relation [33]
^ iþ1 ; ð xiþ1 Þ ¼ iiþ1 Ri ðA xi Þ þ h_ iþ1 iþ1 U
iþ1 A
ð33Þ i+1 A
( xi+1) can be computed recur-
ð34Þ
b iþ1 is a unit vector representing the direction of the joint axis. This computation is implemented through the subwhere U model AVP (angular velocity propagation) of link [34] in bond graph shown in Fig. 7. This sub-model shown in Fig. 7 takes the angular velocity of previous link (ith) and the joint velocity between the previous and current link (i + 1)th as input and the angular velocity of the current link is given out after calculation. The joint velocity ðh_ iþ1 Þ can be about any one of the axes of the joint coordinate frame. Based on the axis of the joint rotation, one of the transformer moduli (Mt[0], Mt[1] or Mt[2]) conb iþ1 . nected to the joint velocity junction will be unity, and the remaining will be zero. This is decided by the vector iþ1 U The translation velocity of the tip of the (i + 1)th link can be computed from the previous link’s translation and angular velocities as follows: iþ1
A A
ð V iþ1 Þ ¼ A ðA V i Þ þ Ai R½i ðA xi Þ i ði Piþ1 Þ:
ð35Þ
Writing in matrix form
½A ðA V iþ1 Þ ¼ ½A ðA V i Þ þ ½Ai R½i ði Piþ1 Þ½i ðA xi Þ;
ð36Þ
where
2 2 3 0 0 6 6 7 i i ð Piþ1 Þ ¼ 4 li 5 and i ði Piþ1 Þ ¼ 4 0 0 li
0
li
3
7 0 0 5: 0 0
The translation velocity of the CM of the (i + 1)th link can be obtained by using i ði PGi Þ ¼ ½ 0 lGi 0 T in place of i(iPi+1) in Eq. (36). The bond graph model of the created sub-system TVP (translation velocity propagation) of link is shown in Fig. 7. This sub-model takes the angular velocity of current link and tip velocity of previous link as input and gives velocity of CM of current link and tip velocity of current link as output. The CM velocity of the spacecraft, links are obtained from the linear inertia of the spacecraft and links, respectively. The 1h_ i junctions represent the joint relative velocities. The integrator sub-models attached to 1h_ i junctions are observers of joint angles. Here pad sub-models are used to avoid differential causality. A rigid body has translational and angular motion. Thus its dynamics has two parts. The translational dynamics is modeled using Euler’s first law [35], and the angular dynamics using the Euler’s second law [35]. In order to model the Euler’s equation the gyrator loop commonly known as an Euler junction structure (EJS) is used. However in multi bond graph representation a modulated gyrator [36–39] can be used. Hence, it can be used to represent the rotational dynamics of the links of the space robot as well as that of the space vehicle. The Euler equations given by Eqs. (37)–(39) are used in the creation of this sub-model
_ x þ ðIzz Iyy Þxy xz ; Nx ¼ Ixx x
ð37Þ
_ y þ ðIxx Izz Þxz xx ; Ny ¼ Iyy x _ Nz ¼ Izz xz þ ðIyy Ixx Þxx xy :
ð38Þ ð39Þ
Here Nx, Ny and Nz represents the moments acting on the vehicle expressed in vehicle frame in x, y and z directions, respectively. The created EJS bond graph for a link is shown in Fig. 7. Multi bond graph representation is adopted in order to represent it in compact form. It is to be noted that Eqs. (37)–(39) have been arrived with the assumption that the reference frame used for computing the inertia matrix coincide with the principal axes of the body. If the reference frame and principal axis of body do not coincide then Eqs. (37)–(39) will also have product of inertia terms Ixy, Iyz, Izx [40], and then EJS cannot be created using Euler’s equations. In that case a separate sub-model has to be created which will take care of Ixy, Iyz, Izx terms in Euler’s equation.
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The complete multi bond graph model of the free-floating space robot is shown in Fig. 7. The tip velocity, which is provided to the ROC sub-model (shown in Fig. 6), is compared with the reference flow command, and the error in the flow is sent to the controller. The controller returns the correction effort, which is then provided to the robot with a high feed-forward gain. This correction effort is transformed back to joint coordinates as correction joint torques using sub-model CTRL and applied at the joints. Here the signal structure CTRL from the joint velocities to the tip velocities is created based on the forward kinematics of the manipulator. In Fig. 7, integrators are used as detector of displacement [41]. From the bond graph of Fig. 7, it is seen that the joint torque at the first joint is given by 1
n1 ¼ 1 N1 þ 12 R2 n2 þ 1 ð1 PG1 Þ 1A RA F 1 þ 1 ð1 P2 Þ 1A RA f2 :
ð40Þ
A
1
Here f1 is the force exerted on link 1 by link 0 (i.e. vehicle), expressed in terms of absolute frame, n1 is the torque exerted on link 1 by link 0 (i.e. vehicle), expressed in terms of frame {1}, AF1 is the resultant of all the external forces acting on the link, and 1N1 are the resultant of all the external moments acting on the link. Similarly from the bond graph the force balance for link 1 can be given as A
f1 ¼ A F 1 þ A f2 :
ð41Þ
Here Af2 is the force exerted on link 2 by link 1, expressed in terms of absolute frame. Eqs. (40) and (41) are similar to the one which can be obtained by considering force balance and moment balance equations based on the free body diagram of a typical link [33]. Equations similar to Eqs. (40) and (41) can be written for other links also. In next section we present a numerical simulation of three DOF space robot with robust trajectory controller. 5. Numerical simulation The link parameters selected for the simulation of space robot are shown in Table 1. The overwhelming controller parameters are taken as Mc = 1.0 kg, Kc = 1000 N/m, Rc = 100 Ns/m and lH = 10.0. Joint resistance on each joint can be assumed as 0.1 Ns/m. The motor inertia for joint 1 is assumed as 0.026 kg m2. For joint 2 and 3, it is assumed as 0.001 kg m2. Reference velocity command in X, Y and Z direction in absolute frame, located at initial position of CM of space vehicle is taken as, X_ ref ¼ R x sinðxtÞ; Y_ ref ¼ A sinð2xtÞ and Z_ ref ¼ Rx cosðxtÞ, respectively. Here, R is the radius of the reference circle in X–Z plane; A is the amplitude of velocity in Y-direction. It is a circular trajectory in X–Z plane and parabolic trajectory in X–Y, and Y–Z plane. Fig. 8 shows the initial configuration of the space robot. At the beginning of the simulation, the overwhelmer initial position is initialized to robot tip position. In other words, the initial error between the controller and the robot is zero. The simulation is carried out for 25 s with R as 0.3 m and A as 0.01 m. Fig. 9a shows a three-dimensional plot of reference and actual tip movement. Fig. 9b shows the variation of Xtip trajectory along with Ztip trajectory. From these figures it is clear that tip is effectively dragged along the reference trajectory. Fig. 9c shows the root mean square (RMS) value of tip trajectory error. The error is taken as the difference between reference and
Table 1 Parameters used for modeling of space robot Parameters
Mass (kg)
Ixx (kg m2)
Iyy (kg m2)
Izz (kg m2)
Length (li) (m)
Length ðlGi Þ (m)
hi (rad)
Link 1 Link 2 Link 3 Vehicle
6.13 15.69 11.76 200
0.03 0.2153 0.0929 40.0
0.025 0.0126 0.0094 40.0
0.03 0.2153 0.0929 40.0
0.1 0.4 0.3 –
0.05 0.2 0.15
0
Fig. 8. Initial configuration of space robot.
p/2 p/2 –
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Fig. 9. (a): Xtip versus Ytip versus Ztip; (b): Xtip versus Ztip; (c): RMS value of tip trajectory errors; (d): XCM versus YCM versus ZCM; (e): XCM versus ZCM and (f): Euler angles w versus / versus h.
actual tip position in mm. Initial error is more than the later part of the error because the tip acquires a velocity from the rest position. The plots of the motion of the CM of spacecraft in three-dimensions are shown in Fig. 9d. Fig. 9e shows the variation of ZCM versus XCM. The plot shows a continuous drift in the vehicle CM location in X–Z plane due to conservation of the
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angular momentum of the free-floating space robot, i.e., due to the non-integrability of the angular momentum equation. Fig. 9f shows the three-dimensional plot for Euler angles / versus w versus h. It clearly shows that even after the tip reaches its initial position after one cycle, the Euler angles do not close. This is due to the non-holonomicity of the system. From the simulation of space manipulator model one can get an estimate of drift in the center-of-mass of the spacecraft. 6. Conclusions In this paper an overwhelming control strategy for free-floating space robots is presented. The strategy consists of a set of linear time invariant controllers coupled to the robotic manipulator through a set of high feed-forward gains, and suitably defined feedback paths. The driving efforts to the manipulator actuators are decided by the linear controller’s parameters and the coupling between the robot and the controller ensures that the robotic manipulator is ‘‘dragged” along the trajectory of the controller model. The controller is robust because it is independent of inertial parameters of the system. The three-link space manipulator is modeled, simulated and the robustness of the controller has been demonstrated. The simulation of the space manipulator model provides quantitative information on the drift of the spacecraft due to non-holonomic nature of the system. This will be useful for developing new path planning methods for minimizing the drift. References [1] Yoshihiko Nakamura, Ranjan Mukherjee, Non-holonomic path planning of space robots via a bidirectional approach, IEEE Transactions on Robotics and Automation 7 (4) (1991) 500–514. [2] Z. Vafa, S. 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Dubowsky, Coordinated manipulator/spacecraft motion control for space robotic systems, in: Proceedings of the IEEE International Conference on Robotics and Automation, Sacramento, CA, 1991, pp. 1696–1701. [21] M. Marchesi, F. Angrilli, R. Venezia, Coordinated control for free-flyer space robots, in: IEEE International Conference on Systems, Man, and Cybernetics, vol. 5, 2000, pp. 3550–3555. [22] K. Yoshida, K. Hashizume, S. Abiko, Zero reaction maneuver: validation with ETS-VII space robot and extension to kinematically redundant arm, in: Proceedings of the 2001 IEEE International Conference on Robotics and Automation, 2001, pp. 441–446. [23] Woo-Keun Yoon, Toshihiko Goshozono, Hiroshi Kawabe, Masahiro Kinami, Yuichi Tsumaki, Masaru Uchiyama, Mitsushige Oda, Toshitsugu Doi, Modelbased space robot teleoperation of ETS-VII manipulator, IEEE Transactions on Robotics and Automation 20 (3) (2004) 602–612. [24] Bruno Zappa, Giovanni Legnani, Riccardo Adamini, Path planning of free-flying space manipulators: an exact solution for polar robots, Mechanism and Machine Theory 40 (2005) 806–820. [25] Carlo Menon, S. Busolo, S. Cocuzza, A. Aboudan, A. Bulgarelli, C. Bettanini, M. Marchesi, F. Angrilli, Issues and solutions for testing free-flying robots, Acta Astronautica 60 (2007) 957–965. [26] Feng Wang, Fuchun Sun, Huaping Liu, Space robot modeling and control considering the effect of orbital mechanics, in: First International Symposium on Systems and Control in Aerospace and Astronautics, 2006, pp. 193–198. [27] Igor Belousov, Claudia Estevès, Jean-Paul Laumond, Etienne Ferré, Motion planning for the large space manipulators with complicated dynamics, in: IEEE/RSJ International Conference on Intelligent Robots and Systems, 2005, pp. 2160–2166. [28] A.K. 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