Dynamics and Momentum Equalization Control of ... - AIAA ARC

Dynamics and Momentum Equalization Control of Redundant Space Robot with Control Moment Gyroscopes for Joint Actuation Xiao Feng,∗ Yinghong Jia, † and Shijie Xu

‡

Beihang University, Beijing, 100191, People’s Republic of China

This paper focuses on the dynamic modeling and momentum equalization control algorithm for a space robot comprised of a platform and a redundant manipulator actuated by control moment gyroscopes(CMGs). A computer-oriented dynamic model is developed for simulation and controller design, and the effect of CMGs is introduced through torque equation neglecting the varying part of CMGs’ inertia. The translational degrees of freedom of the platform are decoupled from the system dynamic equation, since platform orientation and manipulator motion are the major considerations in many cases. A model-based controller with acceleration-level redundancy resolution of the manipulator is developed for coordinate control of platform orientation and manipulator motion. The platform can maintain a fixed orientation while the manipulator tracks an operational space trajectory. To deal with the saturation problem of the CMGs, manipulator redundancy is used to equalize the CMGs momentum usage among manipulator links by decreasing a momentum equalization index. Simulation is used to verify the control technique.

Nomenclature n Manipulator Degrees of Freedom R Platform Position Fi Body Frame of i-th Body Fref(i) Reference Frame of i-th Joint FEE End Effector Frame Θ Platform Rotation Angle ψ Yaw θ Pitch φ Roll ω0 Platform Angular Velocity G Kinematics Matrix Γi Axis Vector of i-th Joint VEE End Effector Velocities vEE End Effector Linear Velocities ωEE End Effector Angular Velocities q Joint Angle Jg Geometric Jacobian Jt Task Jacobian XEE End Effector Pose L Kinematics Matrix relating different Jacobians p v Partial Linear Velocity p ω Partial Angular Velocity ∗ Ph.D

candidate, School of Astronautics, No.37 Xueyuan Rd, Haidian Dist, Beijing, 100191. Professor, School of Astronautics, No.37 Xueyuan Rd, Haidian Dist, Beijing, 100191 . ‡ Professor, School of Astronautics, No.37 Xueyuan Rd, Haidian Dist, Beijing, 100191. † Associate

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u Generalized Speed M Mass Matrix F It Remainder Terms of Generalized Inertial Force F A Generalized Active Force Mi Contribution From i-th Body to Mass Matrix FiIt Contribution From i-th Body to Remainder Terms of Generalized Inertial Force FiA Contribution From i-th Body to Generalized Active Force mi Mass of i-th Body Si Static Moment of i-th Body Ji Inertia Tensor of i-th Body ωi Aangular Velocity of i-th p T vi Partial Linear Velocity p T ωi Partial Angular Velocity Remainder Term of Acceleration ati αti Remainder Term of Angular Acceleration Aref(j),i Transformation Matrix from Frame Fi to Frame Fjref lref(j) Location of Frame Fjref in Frame Fi Aj,ref(j) Transformation Matrix from Frame Fjref to Frame Fj Rhj Location of Frame Fj in Frame Fjref vhj Linear Velocity of Frame Fj in Frame Fjref ωhj Angular Velocity of Frame Fj in Frame Fjref p vhj Relative Partial Linear Velocity. p ωhj Relative Partial Angular Velocity. athj Relative Acceleration Remainder Term. αthj Relative Angular Acceleration Remainder Term. T Torque on Platform Ti Torque on i-th Link hi CMGs Momentum of i-th Link h CMGs Momentum H CMGs Torque Remainder Terms B CMGs Torque Gain Matrix ¯ M Decoupled Mass Matrix F¯ It Decoupled Remainder Terms of Generalized Inertial Force ¯ C Input Gain Matrix ¯ B Decoupled CMGs Torque Remainder Terms ¯ H Decoupled CMGs Torque Gain Matrix ¯˙ c u Decoupled Generalized Speed d Task Space Reference XEE ¨ c Task Space Control Input X EE e Task Space Error KdX Task Space Derivative Gain KpX Task Space Proportional Gain q¨c Joint Acceleration Command θ Arbitrary Vector ¨ c Platform Angular Acceleration Command Θ KdΘ Platform Orientation Derivative Gain KpΘ Platform Orientation Proportional Gain ¯˙ c u Decoupled Generalized Acceleration Command η Momentum Equalization Index hmax Momentum Capacity of the i-th Body i pi Normalized Momentum α Algorithm Parameter Subscript i Body number

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I.

Introduction

Spase robot, with its great potential in on-orbit servicing, has drawn wide attention since the 1980s. Several successful demonstration missions (e.g. ETS-VII, Orbital Express) have shown its capability of onorbit inspection, capture and maintenance. A detailed review of the modeling, planning, control, and ground verification techniques for space robot can be found in Ref. 1 and 2 Space robots differ from ground-based ones in that a manipulator is not fixed to the ground but is mounted on a satellite platform and is working in micro gravity environment.1 No fixed base raise the problems of kinematic and dynamic coupling, which are major concerns in the research of space robot. The manipulator of a space robot may lose its tracking accuracy due to platform motion, while the movement of the manipulator can cause large platform orientation error, which will result in power, communication and fuel problems. To reduce the dynamic coupling, researchers have made effort from different respects. Vafa3 and Torres4 introduce Disturbance Map(DM), and Enhanced Disturbance Map(EDM), respectively, to guide trajectory planning in joint space that reduces platform disturbance. Yoshida5 and Nenchev6 utilize Reaction NullSpace(RNS) in planning reaction-less manipulator trajectory. Huang7 introduces the concept of degree of controllability to the self-correcting method presented by Vafa3 to plan efficient manipulator cyclic motion that corrects platform orientation error. Recently, Peck8, 9 suggests using control moment gyroscopes(CMGs) instead of joint motors for manipulator actuation to reduce disturbance torque exerted on the platform. Carpenter10 shows that CMGs actuator can reduce disturbance on the platform, and studies the energy-optimal problem of such system in Ref. 11 and 12. Brown studies the design and energetics of CMGs actuated space robots in Ref. 13 and 14. However, These works have not considered the saturation problem of CMGs, which will degrade tracking accuracy of CMGs actuated manipulator. A frequently encountered case is that CMGs on only one or two links go into saturation while the others still have momentum capacity. For a redundant manipulator, this kind of saturation can be dealt with by introducing null motion that tries to equalize momentum usage among links. This method is similar to the singularity avoidance algorithm for redundant manipulator. The rest of the paper is organized as follow. Section II gives a description of the CMGs actuated space robot system, and define several kinematic quantities. Section III introduce a computer-oriented dynamic model for simulation and controller design. Section IV develop in detail the momentum equalization controller. In section V a planar space robot with a 3-degree-of-freedom manipulator is used to test the algorithm. In section VI we draw conclusions.

II.

System Description and Kinematics

We consider a space robot system composed of a satellite platform and a serial n-link manipulator, as shown in figure 1. Manipulator links are connected through n single-degree-of-freedom revolute joints, thus the space robot has 6+n degrees of freedom. Each joint is driven by a single scissored pair of control moment gyroscopes(CMGs). A scissored pair means two identical single-gimbal CMGs with gimbal axes in the same direction working in such a mode that the gimbal angles are equal in magnitude but with opposite sign. Figure 2 is a sketch of a scissored pair of CMGs. II.A.

Kinematics

The platform is denoted B0 , and the i-th link of the manipulator is denoted Bi . A reference frame F0 is attached to B0 arbitrarily as its body frame. F0 can move with 6 degrees of freedom in inertial frame. The position of F0 ’s origin in inertial space R and the yaw-pitch-roll rotation angles Θ = [ ψ θ φ ]T are chosen as the translational and rotational coordinates, respectively. The well known kinematics equation ˙ to the angular velocity ω0 of F0 relates the derivatives of rotation angle Θ ˙ ω0 = G(Θ)Θ

(1)

where G is a matrix function of Θ formed by the three unit vectors of rotation. A pair of reference frames, Fi on the i-th link and Firef on the (i − 1)-th link, are introduced for the i-th revolute joint, with their origins 3 of 13 American Institute of Aeronautics and Astronautics

F3ref

F3

F4ref

F2

F4

FEE ref 2

F

Fn

Fnref

F1ref F1

F0

Figure 1. Space robot system with a satellite platform and a serial manipulator driven by scissored pairs of CMGs

coincide and related by a single rotation of angle qi along joint axis Γi . A final frame FEE is fixed to the n-th link that represents the position and orientation of the end effector. The 6-degree-of-freedom pose of the end effector XEE = [ xEE yEE zEE ψEE θEE φEE ]T in F1ref usually defines a task space. The manipulator is redundant when the dimension of task space is less than n. II.B.

Manipulator Jacobian

The geometric Jacobian Jg of a manipulator represents a linear mapping between joint velocities q˙ and end T ]T T effector velocities VEE = [ vEE ωEE VEE = Jg (q)q˙ (2) where vEE and ωEE are the linear and angular velocity of FEE in F1ref expressed in F1ref and FEE , respectively. The task Jacobian Jt (also called analytical Jacobian), on the other hand, represents the linear mapping

Net momentum (along joint axis)

Gimbal angle 1

Gimbal angle 2

CMG 1

CMG 2

Joint axis

Figure 2. Sketch of a scissored pair of control moment gyroscopes

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between joint velocities q˙ and velocities of task space variables. X˙ EE = Jt (q, XEE )q˙

(3)

The relation between Jt and Jg varies with the choice of task space variables, but it can usually be found to have the following form. Jt (q, XEE ) = L(XEE )Jg (q) (4) where L(XEE ) can be found to include inverse of matrix similar to G in Eq. (1). Another set of kinematics quantities, partial velocity and partial angular velocity introduced by Kane,15 is closely related to the geometric Jacobian. In a holonomic system, as is the case for manipulator, the linear and angular velocity of a rigid body can expressed as a linear combination of the generalized speeds of the system # # " " p v v = p u (5) ω ω where p v and p ω are called the partial velocity and partial angular velocity of the rigid body, respectively, and u is a vector of generalized speeds, which is typically chosen as q˙ in the case of manipulator. Comparison between Eq. (5) and Eq. (2) with u = q˙ shows that geometric Jacobian is a special case of partial velocity and partial angular velocity. A recursive formula for computing partial velocity and partial angular velocity of each manipulator link is developed in section III as a part of dynamic model, thus the computation fo geometric Jacobian can be done in the same framework.

III.

Computer Oriented Dynamics

It is necessary to derive a dynamic model for the space robot system for both simulation and model-based controller design. Ref. 16 has offered a computer oriented model for general flexible multi-body system with motion constraints, which can be simplified to model the rigid multi-body space robot in the present work. As a preliminary approximation, the effect of CMGs can be introduced through torque equation neglecting the varying part of CMGs’ inertia. In many situation, the translational degrees of freedom of the platform are not of interest, thus it is preferable to decouple these degrees of freedom from the system dynamics equations as in Ref. 17. In this section, a computer oriented model with the above features is briefly derived for the space robot considered. III.A.

Dynamic Equations

In Ref. 16, the dynamic equations for general flexible multi-body system with motion constraints is derived from Kane’s equation.15 These equations are written in a manner that facilitate computer implementation. By applying these equations to a special case where all the bodies in the system are rigid and no motion constraints is included, we can have a simplified algorithm suitable for this research. The dynamic equations for a the space robot system has the following structure16 M u˙ + F It = F A

(6)

where u is a vector of generalized speeds, M is a square matrix called mass matrix, F It is called the remainder or nonlinear term of generalized inertial force, and F A is called generalized active force. Both F It and F A are vectors and have compatible dimensions. Specifically, vector u is u=

h

R˙ T

ω0T

q˙ T

iT

(7)

The mass matrix M and nonlinear term of generalized inertial force F It can be obtain by summing up contributions over all bodies Pn Pn (8) M = i=0 Mi F It = i=0 FiIt where subscript i indicates contribution from the i-th body. The expression for F A depends on the characteristic of the actuators used and will be derived in section III.B for our CMGs actuated space robot.

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The contributions Mi and FiIt for a generic rigid body in the system can be calculated with five kinematic quantities and the inertia parameters of the body, specifically Mi = p viT (mi p vi − Si× p ωi ) + p ωiT (Si× p vi + J p ωi )

(9)

FiIt = p viT (mi ati − Si× αti + ωi× ωi× Si ) + p ωiT (Si× ati + Ji αti + ωi× Ji ωi )

(10)

where mi , Si , and Ji are the mass, static moment, and inertia tensor of the i-th body, ωi is the angular velocity, and p viT , p ωiT , ati , and αti are called the partial linear velocity, partial angular velocity, remainder(nonlinear) term of acceleration and angular acceleration. The superscript × means the cross product matrix of a vector. The five kinematics quantities ωi , p viT , p ωiT , ati , and αti can be easily obtain through a recursive procedure. Consider a generic body Bj and its inner body Bi , connected by joint j. The recursive relations about rigid body Bi are ωref(j) = Aref(j),i ωi (11) p

× p vref(j) = Aref(j),i (p vi − lref(j) ωi ) p

atref(j)

=

p

ωref(j) = Aref(j),i ωi

Aref(j),i (ati

−

× lref(j) αti

+

ωi× ωi× lref(j) )

αtref(j) = Aref(j),i αti ;

(12) (13) (14) (15)

where Aref(j),i is the transformation matrix from frame Fi to frame Fjref , lref(j) is the location of frame Fjref in frame Fi and subscript ref(j) denote quantities related to frame Fjref . The recursive relations about joint j are ωj = Aj,ref(j) ωref(j) + ωhj (16) p

×p ωref(j) ) + p vhj vj = Aj,ref(j) (p vref(j) − Rhj p

ωj = Aj,ref(j) p ωref(j) + p ωhj

(17) (18)

× t × × × atj = Aj,ref(j) (atref(j) − Rhj αref(j) + ωref(j) ωref(j) Rhj ) − 2vhj Aj,ref(j) ωref(j) + athj

(19)

αtj = Aj,ref(j) αtref(j) + αthj ;

(20)

where Aj,ref(j) is the transformation matrix from frame Fjref to frame Fj , Rhj is the location of frame Fj in frame Fjref , vhj and ωhj are the linear and angular velocities of frame Fj in frame Fjref , and p vhj , p ωhj , athj and αthj denote partial linear and angular velocities and acceleration remainder terms defined using relative velocities. The specific expressions for the terms above depend on the type of joint(revolute, fictitious, etc), thus will not be shown here for simplicity. III.B.

Generalized Active Force

The generalized active force in Eq. (6) can be obtain by summing up contributions over all bodies as in the case of Mi and FiIt n X A F = FiA (21) i=0

The actuator on the platform can be modeled as a force F and a torque T around it. From Eq. (17) and Eq. (18), the partial linear and angular velocities for the platform are p p

v0 = p vh0

(22)

ω0 = p ωh0

(23)

With R˙ and ω0 chosen as generalized speeds, the expressions for the above quantities are h i p v0 = A0,ref(0) 03×(n+3)

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(24)

p

ω0 =

h

03×3

E3

03×n

i

(25)

Thus the contribution of the platform to the generalized active force is h iT F0A = (Aref(0),0 F )T T T 0T n×1

(26)

Neglecting the varying part of CMGs’ inertia, the net angular momentum of the scissored pair of CMGs actuating the i-th joint is along joint axis Γi and can be written as hi = Γi hi

(27)

where hj is the projection of hi on Γ. The torque exert by the scissored pair on the Body Bi can be obtain by differentiating Eq. (27) and add a minus sign Ti = −(Γi h˙i + ωi× Γi hi )

(28)

Thus the contribution of a CMGs actuated body to the generalized active force is FiA = −p ωiT Γi h˙i − p ωiT ωi× Γi hi

(29)

Then the contribution from the CMGs actuated manipulator is n X

˙ ˙ − B(Θ, q, )h FiA = −H(Θ, q, ω0 , q)

(30)

i=1

where ˙ = H(Θ, q, ω0 , q)

n X

p

ωiT ωi× Γi hi

i=1

B(Θ, q) =

h

h=

p

ω1T Γ1

h

h1

p

h2

ω2T Γ2 ...

. . . p ωnT Γn iT

i

(31)

hn

The generalized active force for the system can be found using Eq. (21)(26) and (30). III.C.

Decouple Translational DOFs

The translational degrees of freedom of the platform can be decoupled from the system dynamic equation. The dynamic equation of the CMGs actuated space robot can be express using Eq. (6),(26)and(30)as   (Aref(0),0 F )   ˙ M u˙ + F It =  (32) T  − H − Bh 0n×1 Note that H and B do not depend on the translational variables. A careful review of the construction of M and F It could show that they are also independent of translational variables. Thus the only translational ¨ in the derivative of the generalized speed u. The dynamic variable in the system dynamic equation is R ¨ equation without R can be derived by simple matrix manipulations. The matrices in Eq. (32) can be partitioned as " # " # " # M11 M12 R˙ F1It It M= u= F = ¯ M21 M22 u F2It " # " # (33) H1 B1 H= B= H2 B2 ¯ consists of the rest of generalized where the upper part has 3 rows and is the linear momentum equation, u speeds. The dynamic equation with translational degrees of freedom decoupled is obtain by subtracting −1 M21 M11 times the upper part from the lower part, the result is " # T ¯u ¯ ¯ − M21 M −1 Aref(0),0 F ¯˙ + F¯ It = C M −H (34) 11 ˙ h 7 of 13 American Institute of Aeronautics and Astronautics

where " ¯ = M

−1 −M21 M11 M12

¯= B

−1 −M21 M11 B1

+ M22 + B2

F¯ It =

−1 It −M21 M11 F1

¯ = H

−1 −M21 M11 H1

+ F2It

¯= C

E3 0n×3

# ¯ −B

(35)

+ H2

When the control actuator on the platform can be modeled as a torque, the third term in Eq. (34) should be dropped.

IV.

Momentum Equalization Controller

The task for the manipulator of a space robot is usually defined as tracking a trajectory in task space. During operation a fixed platform orientation is desirable in many cases for power and communication reasons. Thus it is desirable to coordinately control platform orientation and manipulator motion in task space. For a CMGs actuated manipulator, it is possible that a few pairs of CMGs will saturate during operation while the rest still have momentum capacity. This kind of saturation can be avoided through equalization of momentum usage among links for a redundant manipulator. In this section, a model-based controller is proposed for coordinate control of platform orientation and end effector motion. A accelerationlevel redundancy resolution is used to equalize momenta usage among links. The controller is a two-level hierarchy with the higher works in task space and the lower in joint space. IV.A.

Task Space Controller

The model used in the design of task space controller is ¨ EE = X ¨c X EE

(36)

¨ c such that the pose of end effector XEE ¨ c is the control input. The control problem is to design X where X EE EE d d d ˙ ˙ as t → ∞). For this double integrator track a desired trajectory XEE (i.e.XEE → XEE and XEE → XEE system, the control problem can be solved using the control law ¨c = X ¨ d + K X e˙ + K X e X EE EE d p

(37)

d − XEE and KdX and KpX are constant gain matrices. Substituting Eq. (37) into Eq. (36) where e = XEE leads to the error dynamics e¨ + KdX e˙ + KpX e = 0 (38)

If the gain KdX and KpX are chosen as positive definite symmetric matrices, then the tracking error will go to zero exponentially, thus solve the task space control problem. The task space variable can be obtained through direct measurement or forward kinematics with manipulator joint angles. IV.B.

Joint Space Controller

The joint space controller is designed for three goals . 1. establish task space control input in Eq. (37) 2. stabilize platform orientation 3. equalize CMGs momentum usage among manipulator links For the first goal, it is necessary to recall the relation between task space and joint space in Eq. (3). Differentiating Eq. (3) gives the acceleration level relation ¨ EE = Jt q¨ + J˙t q˙ X

(39)

An acceleration level redundancy resolution establish the task space control input with joint space control input ¨ c − J˙t q) ˙ + (En − Jt+ Jt )θ q¨c = Jt+ (X (40) EE 8 of 13 American Institute of Aeronautics and Astronautics

To stabilize the platform orientation, the following PD control law is used ¨ c = −K Θ Θ ˙ − K ΘΘ Θ d p

(41)

The acceleration commands in Eq. (40) and (40) are convert to torque and CMGs momentum commands using a model-based control law(i.e. inverse dynamics controller) # " T ¯ −1 (M ¯u ¯ ¯˙ c + F¯ It + H) =C (42) ˙ h ¯˙ c is the desired acceleration from Eq. (40) and (40). Note that in designing the model-based control where u law, the control force on the platform is assumed to be zero. The only remaining task is to introduce momentum equalization mechanism in the controller so as to avoid CMGs saturation. A equalization index similar to that in Ref. 18 can be define to measure the performance of momentum equalization η= where

1 (ps − p¯ms )T (ps − p¯ms ) 2

(43)

iT h ps = p21 p22 . . . p2n h iT p¯ms = pms pms . . . pms pms =

1 2 (p + p22 + · · · + p2n ) n 1 hi pi = max hi

(44)

where hmax is the momentum capacity of the CMGs on the i-th body and pi is called normalized momentum. i The magnitude of pi represents the proportion of momentum used and the sign of it indicates the direction of momentum. It can be easily seen that η ≥ 0, and η = 0 if and only if p21 = p22 = · · · = p2n . The smaller η is, the better momentum equalization the CMGs holds. Differentiating Eq. (43) gives the relation between the derivative of η and that of h η˙ =

∂η ˙ (d [hmax ])−1 h ∂h

where d [] construct a diagonal matrix from a vector,

∂η ∂h

(45)

and hmax have the following expression

∂η 1 = 2(ps − p¯ms )T (d [p] − 1n×1 pT ) ∂h n h iT max max max h = h1 h2 . . . hmax n

(46)

where 1n×1 is a n × 1 matrix with all elements equal to 1. By substituting Eq. (39)(42) into Eq. (45), we have h i ∂η ¯ −1 M ¯ 21 Θ ¨c +M ¯ 22 J + (X ¨ c − J˙t q) ¯ 2 − ∂η (d [hmax ])−1 B ¯ −1 M ¯ 22 (En −J + Jt )θ ˙ + F¯2It + H η˙ = − (d [hmax ])−1 B t t EE 2 2 ∂h ∂h (47) ¯2 M ¯ 21 M ¯ 22 F¯ It H ¯ 2 are sub-matrix of B ¯ M ¯ F¯ It H ¯ with compatible dimension from the lower part where B 2 of Eq. (42). The first term of Eq. (47) is associated with achieving the first and second goal of control and can not be modified, while the second part term (specifically, θ) can be designed to minimize the index η thus achieving the third goal. It is shown in Ref. 19 that (En − Jt+ Jt ) is idempotent, so θ can be chosen as ¯ T (B ¯ −1 )T (d [hmax ])−1 θ = αM 22 2

∂η T ∂h

where α is a positive algorithm parameter and should be carefully selected. 9 of 13 American Institute of Aeronautics and Astronautics

(48)

V.

Simulation

In this section the momentum equalization control algorithm is tested with a planar space robot, as shown in figure 3. The manipulator of the space robot has 3 links with length of 1m, 1m, and 1.15m. Each link is actuated by a single scissored pair of CMGs with momentum capacity of 5Nms. The task space

Figure 3. Planar simulation model

trajectory to be followed is a 2D end effector position trajectory(end effector orientation not controlled), thus the 3-degree-of-freedom manipulator is redundant for the task. The trajectory is a straight line with zero velocity at both the start and the end with a duration of 15s. To demonstrate the functionality of momentum equalization control algorithm, two sets of simulation results are presented with algorithm parameter α = 0 and α = 0.01. With α = 0, the controller does not have the functionality of equalizing CMGs’ momenta. From the sketch in figure 4 it can be seen that the platform orientation can be stabilized while the manipulator failed to track the trajectory accurately, which can be confirmed with figure 5, 6, and 7. The cause of the problem can be found from figure 8, which indicates saturation of the CMGs on the first and the second links. It can also be seen that only a small portion of the CMGs momentum of the third link is used. This is case where the momentum equalization control algorithm is designed to work. On the other hand, with the momentum equalization functionality turned on with α = 0.01, the controller gives satisfactory results, as can be seen from figure 4, 5, 6, and 7. Figure 8 shows that in the case with α = 0.01, the CMGs on the first two links are kept away from saturation, while the CMGs on the third link is more actively involved in the control process. In figure 9, a comparison is drawn about the equalization index. In the case with α = 0.01, the index is smaller, which indicates better equalization. This verified the idea that by equalizing momentum usage among links saturation can be avoided to a certain degree.

VI.

Conclusion

This paper considers the dynamic modeling and control problem of a redundant space robot actuated by control moment gyroscopes. A computer-oriented model is briefly derived that facilitates simulation and controller design. A model-based controller is developed to coordinately control platform orientation and manipulator motion. Special attention is paid to the problem of CMGs saturation, which results in manipulator tracking error. A momentum equalization control algorithm based on acceleration-level redundancy resolution is introduced to equalize CMGs momentum usage among links, thus avoids unsuccessful tracking caused by saturation of only one or two pairs of CMGs.

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XdEE XEE

XdEE XEE

(a) α = 0

(b) α = 0.01.

3

platform attitude, deg

platform attitude, deg

Figure 4. Task space tracking with platform orientation stabilized(center of platform fixed)

2 1 0 -1

3 2 1 0 -1

0

5

10

15

0

5

time, s

10

15

time, s

(a) α = 0

(b) α = 0.01.

Figure 5. Platform orientation motion with coordinate control

0.15 x direction y direction

0.1

tracking error, m

tracking error, m

0.15

0.05 0 -0.05 -0.1

x direction y direction

0.1 0.05 0 -0.05

0

5

10

15

0

time, s

5

10

time, s

(a) α = 0

(b) α = 0.01. Figure 6. Task space position tracking error

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15

0

-0.05

x direction y direction

-0.1 0

5

10

15

tracking velocity error, m/s

tracking velocity error, m/s

0.05

0.02 0 x direction y direction

-0.02 -0.04 -0.06 -0.08 0

5

time, s

10

15

time, s

(a) α = 0

(b) α = 0.01.

2

normalized momentum

normalized momentum

Figure 7. Task space velocity tracking error

p1 p2 p3

1 0 -1 -2 0

5

10

1.5 p1 p2 p3

1 0.5 0 -0.5 -1

15

0

5

time, s

10

15

10

15

time, s

(a) α = 0

(b) α = 0.01. Figure 8. Momentum usage during operation

0.5

equalization index

equalization index

0.5 0.4 0.3 0.2 0.1 0

0.4 0.3 0.2 0.1 0

0

5

10

15

0

5

time, s

time, s

(a) α = 0

(b) α = 0.01. Figure 9. Momentum equalization index

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Acknowledgments This paper reports the work carried out in the project supported by the National Natural Science Foundation of China (11272027).

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